TPTP Problem File: SLH0914^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% 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 : Khovanskii_Theorem/0008_Khovanskii/prob_00750_027506__13549940_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1429 ( 514 unt; 157 typ; 0 def)
% Number of atoms : 3615 (1244 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10965 ( 437 ~; 78 |; 281 &;8530 @)
% ( 0 <=>;1639 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 7 avg)
% Number of types : 14 ( 13 usr)
% Number of type conns : 612 ( 612 >; 0 *; 0 +; 0 <<)
% Number of symbols : 147 ( 144 usr; 25 con; 0-3 aty)
% Number of variables : 3723 ( 254 ^;3230 !; 239 ?;3723 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:14:59.527
%------------------------------------------------------------------------------
% Could-be-implicit typings (13)
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
list_list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
set_list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
set_set_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__List__Olist_It__Extended____Nat__Oenat_J,type,
list_Extended_enat: $tType ).
thf(ty_n_t__Set__Oset_It__Extended____Nat__Oenat_J,type,
set_Extended_enat: $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__Extended____Nat__Oenat,type,
extended_enat: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (144)
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_Finite__Set_Ofinite_001t__Extended____Nat__Oenat,type,
finite4001608067531595151d_enat: set_Extended_enat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
finite8170528100393595399st_nat: set_list_list_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Nat__Onat_J,type,
finite8100373058378681591st_nat: set_list_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
finite7047420756378620717st_nat: set_set_list_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
minus_1139252259498527702_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > list_nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Extended____Nat__Oenat,type,
minus_3235023915231533773d_enat: extended_enat > extended_enat > extended_enat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__List__Olist_It__Extended____Nat__Oenat_J,type,
minus_388785356899630291d_enat: list_Extended_enat > list_Extended_enat > list_Extended_enat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
minus_3911745200923244873st_nat: list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__List__Olist_It__Nat__Onat_J,type,
minus_minus_list_nat: list_nat > list_nat > list_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
minus_7954133019191499631st_nat: set_list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Nat__Oenat,type,
plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__List__Olist_It__Extended____Nat__Oenat_J,type,
plus_p2879455426997772835d_enat: list_Extended_enat > list_Extended_enat > list_Extended_enat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
plus_p2116291331692525561st_nat: list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__List__Olist_It__Nat__Onat_J,type,
plus_plus_list_nat: list_nat > list_nat > list_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__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
plus_p884110394369815071st_nat: set_list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Nat__Onat_J,type,
plus_plus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Extended____Nat__Oenat,type,
zero_z5237406670263579293d_enat: extended_enat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Extended____Nat__Oenat,type,
groups5145338220374282879d_enat: list_Extended_enat > extended_enat ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Nat__Onat,type,
groups4561878855575611511st_nat: list_nat > 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_Khovanskii_OKhovanskii_Olength__sum__set,type,
length_sum_set: nat > nat > set_list_nat ).
thf(sy_c_Khovanskii_OKhovanskii_Olist__incr,type,
list_incr: nat > list_nat > list_nat ).
thf(sy_c_Khovanskii_OKhovanskii_Ominimal__elements,type,
minimal_elements: set_list_nat > set_list_nat ).
thf(sy_c_Khovanskii_OKhovanskii_Ominimal__elementsp,type,
minimal_elementsp: ( list_nat > $o ) > list_nat > $o ).
thf(sy_c_Khovanskii_OKhovanskii__axioms_001t__List__Olist_It__Nat__Onat_J,type,
khovan1553326461689229922st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Khovanskii_OKhovanskii__axioms_001t__Nat__Onat,type,
khovan4585363760863428690ms_nat: set_nat > set_nat > $o ).
thf(sy_c_Khovanskii_Omax__pointwise,type,
max_pointwise: nat > set_list_nat > list_nat ).
thf(sy_c_Khovanskii_Omin__pointwise,type,
min_pointwise: nat > set_list_nat > list_nat ).
thf(sy_c_Khovanskii_Opointwise__le,type,
pointwise_le: list_nat > list_nat > $o ).
thf(sy_c_Khovanskii_Opointwise__less,type,
pointwise_less: list_nat > list_nat > $o ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__List__Olist_It__Nat__Onat_J_001t__Extended____Nat__Oenat,type,
lattic5912464335902825515d_enat: ( list_nat > extended_enat ) > set_list_nat > list_nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
lattic5785867957632790475at_nat: ( list_nat > nat ) > set_list_nat > list_nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Extended____Nat__Oenat,type,
lattic8926238025367240251d_enat: ( nat > extended_enat ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Nat__Onat,type,
lattic7446932960582359483at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_List_Oappend_001t__Extended____Nat__Oenat,type,
append_Extended_enat: list_Extended_enat > list_Extended_enat > list_Extended_enat ).
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__Nat__Onat_001t__Nat__Onat,type,
bind_nat_nat: list_nat > ( nat > list_nat ) > list_nat ).
thf(sy_c_List_Odrop_001t__List__Olist_It__Nat__Onat_J,type,
drop_list_nat: nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Odrop_001t__Nat__Onat,type,
drop_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Extended____Nat__Oenat,type,
cons_Extended_enat: extended_enat > list_Extended_enat > list_Extended_enat ).
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__Extended____Nat__Oenat,type,
nil_Extended_enat: list_Extended_enat ).
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_Ohd_001t__List__Olist_It__Nat__Onat_J,type,
hd_list_nat: list_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__Nat__Onat_J_001t__Extended____Nat__Oenat,type,
map_li1708695312048858560d_enat: ( list_nat > extended_enat ) > list_list_nat > list_Extended_enat ).
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__Extended____Nat__Oenat,type,
map_na5247803048364578384d_enat: ( nat > extended_enat ) > list_nat > list_Extended_enat ).
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__Extended____Nat__Oenat,type,
set_Extended_enat2: list_Extended_enat > set_Extended_enat ).
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_Osize__list_001t__List__Olist_It__Nat__Onat_J,type,
size_list_list_nat: ( list_nat > nat ) > list_list_nat > nat ).
thf(sy_c_List_Olist_Osize__list_001t__Nat__Onat,type,
size_list_nat: ( nat > nat ) > list_nat > nat ).
thf(sy_c_List_Olist__update_001t__List__Olist_It__Nat__Onat_J,type,
list_update_list_nat: list_list_nat > nat > list_nat > list_list_nat ).
thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
list_update_nat: list_nat > nat > 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__Extended____Nat__Oenat,type,
nth_Extended_enat: list_Extended_enat > nat > extended_enat ).
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__Nat__Onat,type,
product_lists_nat: list_list_nat > 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_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__Extended____Nat__Oenat_J,type,
size_s3941691890525107288d_enat: list_Extended_enat > 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__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
numera1916890842035813515d_enat: num > extended_enat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
bot_bot_list_nat_o: list_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,type,
bot_bo4199563552545308370d_enat: extended_enat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__List__Olist_It__Nat__Onat_J,type,
bot_bot_list_nat: list_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
bot_bo7653980558646680370d_enat: set_Extended_enat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
bot_bot_set_list_nat: set_list_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
bot_bo3886227569956363488st_nat: set_set_list_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__Extended____Nat__Oenat,type,
ord_le72135733267957522d_enat: extended_enat > extended_enat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__List__Olist_It__Extended____Nat__Oenat_J,type,
ord_le4445849522347344536d_enat: list_Extended_enat > list_Extended_enat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__List__Olist_It__Nat__Onat_J,type,
ord_less_list_nat: list_nat > list_nat > $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_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J_J,type,
ord_le6558929396352911974_nat_o: ( list_nat > list_nat > $o ) > ( list_nat > list_nat > $o ) > $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 ).
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ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__List__Olist_It__Extended____Nat__Oenat_J,type,
ord_le769749158434378124d_enat: list_Extended_enat > list_Extended_enat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__List__Olist_It__Nat__Onat_J,type,
ord_less_eq_list_nat: list_nat > list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__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_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_OCollect_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
collect_set_list_nat: ( set_list_nat > $o ) > set_set_list_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Wellfounded_OwfP_001t__Extended____Nat__Oenat,type,
wfP_Extended_enat: ( extended_enat > extended_enat > $o ) > $o ).
thf(sy_c_Wellfounded_OwfP_001t__List__Olist_It__Nat__Onat_J,type,
wfP_list_nat: ( list_nat > list_nat > $o ) > $o ).
thf(sy_c_Wellfounded_OwfP_001t__Nat__Onat,type,
wfP_nat: ( nat > nat > $o ) > $o ).
thf(sy_c_member_001t__Extended____Nat__Oenat,type,
member_Extended_enat: extended_enat > set_Extended_enat > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
member_list_list_nat: list_list_nat > set_list_list_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
member_set_list_nat: set_list_nat > set_set_list_nat > $o ).
thf(sy_v_U,type,
u: set_list_nat ).
thf(sy_v_Ua____,type,
ua: set_list_nat ).
thf(sy_v_VF____,type,
vf: nat > nat > set_list_nat ).
thf(sy_v_V____,type,
v: set_list_nat ).
thf(sy_v_delete____,type,
delete: list_nat > list_nat ).
thf(sy_v_i____,type,
i: nat ).
thf(sy_v_r,type,
r: nat ).
thf(sy_v_ra____,type,
ra: nat ).
thf(sy_v_t____,type,
t: nat ).
thf(sy_v_u____,type,
u2: list_nat ).
thf(sy_v_ua____,type,
ua2: list_nat ).
thf(sy_v_v____,type,
v2: list_nat ).
% Relevant facts (1266)
thf(fact_0__092_060open_062u_A_092_060unlhd_062_Av_092_060close_062,axiom,
pointwise_le @ ua2 @ v2 ).
% \<open>u \<unlhd> v\<close>
thf(fact_1_pointwise__le__refl,axiom,
! [X: list_nat] : ( pointwise_le @ X @ X ) ).
% pointwise_le_refl
thf(fact_2_pointwise__le__trans,axiom,
! [X: list_nat,Y: list_nat,Z: list_nat] :
( ( pointwise_le @ X @ Y )
=> ( ( pointwise_le @ Y @ Z )
=> ( pointwise_le @ X @ Z ) ) ) ).
% pointwise_le_trans
thf(fact_3_pointwise__le__antisym,axiom,
! [X: list_nat,Y: list_nat] :
( ( pointwise_le @ X @ Y )
=> ( ( pointwise_le @ Y @ X )
=> ( X = Y ) ) ) ).
% pointwise_le_antisym
thf(fact_4_that_I2_J,axiom,
member_list_nat @ v2 @ ( vf @ i @ t ) ).
% that(2)
thf(fact_5_that_I1_J,axiom,
member_list_nat @ ua2 @ ( vf @ i @ t ) ).
% that(1)
thf(fact_6_pointwise__le__Nil,axiom,
! [X: list_nat] :
( ( pointwise_le @ nil_nat @ X )
= ( X = nil_nat ) ) ).
% pointwise_le_Nil
thf(fact_7_pointwise__le__Nil2,axiom,
! [X: list_nat] :
( ( pointwise_le @ X @ nil_nat )
= ( X = nil_nat ) ) ).
% pointwise_le_Nil2
thf(fact_8_pointwise__less__def,axiom,
( pointwise_less
= ( ^ [X2: list_nat,Y2: list_nat] :
( ( pointwise_le @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% pointwise_less_def
thf(fact_9_pointwise__le__iff__less__equal,axiom,
( pointwise_le
= ( ^ [X2: list_nat,Y2: list_nat] :
( ( pointwise_less @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% pointwise_le_iff_less_equal
thf(fact_10_pointwise__append__le__iff,axiom,
! [U: list_nat,X: list_nat,Y: list_nat] :
( ( pointwise_le @ ( append_nat @ U @ X ) @ ( append_nat @ U @ Y ) )
= ( pointwise_le @ X @ Y ) ) ).
% pointwise_append_le_iff
thf(fact_11_pairwise__minus__cancel,axiom,
! [Z: list_nat,X: list_nat,Y: list_nat] :
( ( pointwise_le @ Z @ X )
=> ( ( pointwise_le @ Z @ Y )
=> ( ( ( minus_minus_list_nat @ X @ Z )
= ( minus_minus_list_nat @ Y @ Z ) )
=> ( X = Y ) ) ) ) ).
% pairwise_minus_cancel
thf(fact_12_u,axiom,
member_list_nat @ u2 @ ua ).
% u
thf(fact_13_len__delete,axiom,
! [U: list_nat] :
( ( member_list_nat @ U @ ( vf @ i @ t ) )
=> ( ( size_size_list_nat @ ( delete @ U ) )
= ra ) ) ).
% len_delete
thf(fact_14_delete__def,axiom,
( delete
= ( ^ [V: list_nat] : ( append_nat @ ( take_nat @ i @ V ) @ ( drop_nat @ ( suc @ i ) @ V ) ) ) ) ).
% delete_def
thf(fact_15_pointwise__le__imp___092_060sigma_062,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( pointwise_le @ Xs @ Ys )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ Xs ) @ ( groups4561878855575611511st_nat @ Ys ) ) ) ).
% pointwise_le_imp_\<sigma>
thf(fact_16__092_060open_062i_A_092_060le_062_Ar_092_060close_062,axiom,
ord_less_eq_nat @ i @ ra ).
% \<open>i \<le> r\<close>
thf(fact_17__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062u_O_A_092_060lbrakk_062u_A_092_060in_062_AU_059_A_092_060And_062y_O_Ay_A_092_060lhd_062_Au_A_092_060Longrightarrow_062_Ay_A_092_060notin_062_AU_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [U2: list_nat] :
( ( member_list_nat @ U2 @ ua )
=> ~ ! [Y3: list_nat] :
( ( pointwise_less @ Y3 @ U2 )
=> ~ ( member_list_nat @ Y3 @ ua ) ) ) ).
% \<open>\<And>thesis. (\<And>u. \<lbrakk>u \<in> U; \<And>y. y \<lhd> u \<Longrightarrow> y \<notin> U\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_18_zmin,axiom,
! [Y: list_nat] :
( ( pointwise_less @ Y @ u2 )
=> ~ ( member_list_nat @ Y @ ua ) ) ).
% zmin
thf(fact_19_False,axiom,
ua != bot_bot_set_list_nat ).
% False
thf(fact_20_Suc_Oprems,axiom,
! [X: list_nat] :
( ( member_list_nat @ X @ ua )
=> ( ( size_size_list_nat @ X )
= ( suc @ ra ) ) ) ).
% Suc.prems
thf(fact_21_assms,axiom,
! [X: list_nat] :
( ( member_list_nat @ X @ u )
=> ( ( size_size_list_nat @ X )
= r ) ) ).
% assms
thf(fact_22_minus__Nil,axiom,
! [Xs: list_nat] :
( ( minus_minus_list_nat @ nil_nat @ Xs )
= nil_nat ) ).
% minus_Nil
thf(fact_23__092_060open_062_092_060And_062v_At_Ai_O_Av_A_092_060in_062_AVF_Ai_At_A_092_060Longrightarrow_062_Alength_Av_A_061_ASuc_Ar_092_060close_062,axiom,
! [V2: list_nat,I: nat,T: nat] :
( ( member_list_nat @ V2 @ ( vf @ I @ T ) )
=> ( ( size_size_list_nat @ V2 )
= ( suc @ ra ) ) ) ).
% \<open>\<And>v t i. v \<in> VF i t \<Longrightarrow> length v = Suc r\<close>
thf(fact_24_pointwise__less__Nil,axiom,
! [X: list_nat] :
~ ( pointwise_less @ nil_nat @ X ) ).
% pointwise_less_Nil
thf(fact_25_pointwise__less__Nil2,axiom,
! [X: list_nat] :
~ ( pointwise_less @ X @ nil_nat ) ).
% pointwise_less_Nil2
thf(fact_26_minimal__elementsp_Ocases,axiom,
! [U3: list_nat > $o,A: list_nat] :
( ( minimal_elementsp @ U3 @ A )
=> ~ ( ( U3 @ A )
=> ~ ! [Y3: list_nat] :
( ( U3 @ Y3 )
=> ~ ( pointwise_less @ Y3 @ A ) ) ) ) ).
% minimal_elementsp.cases
thf(fact_27_minimal__elementsp_Ointros,axiom,
! [U3: list_nat > $o,X: list_nat] :
( ( U3 @ X )
=> ( ! [Y4: list_nat] :
( ( U3 @ Y4 )
=> ~ ( pointwise_less @ Y4 @ X ) )
=> ( minimal_elementsp @ U3 @ X ) ) ) ).
% minimal_elementsp.intros
thf(fact_28_minimal__elementsp_Osimps,axiom,
( minimal_elementsp
= ( ^ [U4: list_nat > $o,A2: list_nat] :
? [X2: list_nat] :
( ( A2 = X2 )
& ( U4 @ X2 )
& ! [Y2: list_nat] :
( ( U4 @ Y2 )
=> ~ ( pointwise_less @ Y2 @ X2 ) ) ) ) ) ).
% minimal_elementsp.simps
thf(fact_29_drop__all,axiom,
! [Xs: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
=> ( ( drop_nat @ N @ Xs )
= nil_nat ) ) ).
% drop_all
thf(fact_30_drop__eq__Nil,axiom,
! [N: nat,Xs: list_nat] :
( ( ( drop_nat @ N @ Xs )
= nil_nat )
= ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_31_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_nat] :
( ( nil_nat
= ( drop_nat @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_32_append__take__drop__id,axiom,
! [N: nat,Xs: list_nat] :
( ( append_nat @ ( take_nat @ N @ Xs ) @ ( drop_nat @ N @ Xs ) )
= Xs ) ).
% append_take_drop_id
thf(fact_33_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_34_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_35_append__eq__append__conv__if,axiom,
! [Xs_1: list_nat,Xs_2: list_nat,Ys_1: list_nat,Ys_2: list_nat] :
( ( ( append_nat @ Xs_1 @ Xs_2 )
= ( append_nat @ Ys_1 @ Ys_2 ) )
= ( ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs_1 ) @ ( size_size_list_nat @ Ys_1 ) )
=> ( ( Xs_1
= ( take_nat @ ( size_size_list_nat @ Xs_1 ) @ Ys_1 ) )
& ( Xs_2
= ( append_nat @ ( drop_nat @ ( size_size_list_nat @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ ( size_size_list_nat @ Xs_1 ) @ ( size_size_list_nat @ Ys_1 ) )
=> ( ( ( take_nat @ ( size_size_list_nat @ Ys_1 ) @ Xs_1 )
= Ys_1 )
& ( ( append_nat @ ( drop_nat @ ( size_size_list_nat @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
= Ys_2 ) ) ) ) ) ).
% append_eq_append_conv_if
thf(fact_36_minimal__elements_Ocases,axiom,
! [A: list_nat,U3: set_list_nat] :
( ( member_list_nat @ A @ ( minimal_elements @ U3 ) )
=> ~ ( ( member_list_nat @ A @ U3 )
=> ~ ! [Y3: list_nat] :
( ( member_list_nat @ Y3 @ U3 )
=> ~ ( pointwise_less @ Y3 @ A ) ) ) ) ).
% minimal_elements.cases
thf(fact_37_minimal__elements_Osimps,axiom,
! [A: list_nat,U3: set_list_nat] :
( ( member_list_nat @ A @ ( minimal_elements @ U3 ) )
= ( ? [X2: list_nat] :
( ( A = X2 )
& ( member_list_nat @ X2 @ U3 )
& ! [Y2: list_nat] :
( ( member_list_nat @ Y2 @ U3 )
=> ~ ( pointwise_less @ Y2 @ X2 ) ) ) ) ) ).
% minimal_elements.simps
thf(fact_38_minimal__elements_Ointros,axiom,
! [X: list_nat,U3: set_list_nat] :
( ( member_list_nat @ X @ U3 )
=> ( ! [Y4: list_nat] :
( ( member_list_nat @ Y4 @ U3 )
=> ~ ( pointwise_less @ Y4 @ X ) )
=> ( member_list_nat @ X @ ( minimal_elements @ U3 ) ) ) ) ).
% minimal_elements.intros
thf(fact_39_same__append__eq,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_40_append__same__eq,axiom,
! [Ys: list_nat,Xs: list_nat,Zs: list_nat] :
( ( ( append_nat @ Ys @ Xs )
= ( append_nat @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_41_append__assoc,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( append_nat @ ( append_nat @ Xs @ Ys ) @ Zs )
= ( append_nat @ Xs @ ( append_nat @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_42_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_43_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_44_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A3: set_list_nat] :
( ( collect_list_nat
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A3: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ! [X3: list_nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_list_nat @ P )
= ( collect_list_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_48_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_49_append__is__Nil__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= nil_nat )
= ( ( Xs = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_50_Nil__is__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( nil_nat
= ( append_nat @ Xs @ Ys ) )
= ( ( Xs = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_51_self__append__conv2,axiom,
! [Y: list_nat,Xs: list_nat] :
( ( Y
= ( append_nat @ Xs @ Y ) )
= ( Xs = nil_nat ) ) ).
% self_append_conv2
thf(fact_52_append__self__conv2,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_nat ) ) ).
% append_self_conv2
thf(fact_53_self__append__conv,axiom,
! [Y: list_nat,Ys: list_nat] :
( ( Y
= ( append_nat @ Y @ Ys ) )
= ( Ys = nil_nat ) ) ).
% self_append_conv
thf(fact_54_append__self__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_nat ) ) ).
% append_self_conv
thf(fact_55_append__Nil2,axiom,
! [Xs: list_nat] :
( ( append_nat @ Xs @ nil_nat )
= Xs ) ).
% append_Nil2
thf(fact_56_append_Oright__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ A @ nil_nat )
= A ) ).
% append.right_neutral
thf(fact_57_append__eq__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat,Us: list_nat,Vs: list_nat] :
( ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
| ( ( size_size_list_nat @ Us )
= ( size_size_list_nat @ Vs ) ) )
=> ( ( ( append_nat @ Xs @ Us )
= ( append_nat @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_58_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_59_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_nat] :
( ( size_size_list_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_60_append__eq__append__conv2,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ts: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Zs @ Ts ) )
= ( ? [Us2: list_nat] :
( ( ( Xs
= ( append_nat @ Zs @ Us2 ) )
& ( ( append_nat @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_nat @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_nat @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_61_append__eq__appendI,axiom,
! [Xs: list_nat,Xs1: list_nat,Zs: list_nat,Ys: list_nat,Us: list_nat] :
( ( ( append_nat @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_nat @ Xs1 @ Us ) )
=> ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_62_take__equalityI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ! [I2: nat] :
( ( take_nat @ I2 @ Xs )
= ( take_nat @ I2 @ Ys ) )
=> ( Xs = Ys ) ) ).
% take_equalityI
thf(fact_63_eq__Nil__appendI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append_nat @ nil_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_64_append_Oleft__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ nil_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_65_append__Nil,axiom,
! [Ys: list_nat] :
( ( append_nat @ nil_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_66_take__Nil,axiom,
! [N: nat] :
( ( take_nat @ N @ nil_nat )
= nil_nat ) ).
% take_Nil
thf(fact_67_drop__Nil,axiom,
! [N: nat] :
( ( drop_nat @ N @ nil_nat )
= nil_nat ) ).
% drop_Nil
thf(fact_68_append__eq__conv__conj,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Zs )
= ( ( Xs
= ( take_nat @ ( size_size_list_nat @ Xs ) @ Zs ) )
& ( Ys
= ( drop_nat @ ( size_size_list_nat @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_69_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_70_Suc_OIH,axiom,
! [U3: set_list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ U3 )
=> ( ( size_size_list_nat @ X3 )
= ra ) )
=> ( finite8100373058378681591st_nat @ ( minimal_elements @ U3 ) ) ) ).
% Suc.IH
thf(fact_71_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_72_lift__Suc__antimono__le,axiom,
! [F: nat > set_list_nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_le6045566169113846134st_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_le6045566169113846134st_nat @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_73_lift__Suc__antimono__le,axiom,
! [F: nat > extended_enat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_le2932123472753598470d_enat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_le2932123472753598470d_enat @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_74_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_75_lift__Suc__mono__le,axiom,
! [F: nat > set_list_nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_le6045566169113846134st_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_le6045566169113846134st_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_76_lift__Suc__mono__le,axiom,
! [F: nat > extended_enat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_le2932123472753598470d_enat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_le2932123472753598470d_enat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_77_diff__shunt__var,axiom,
! [X: set_list_nat,Y: set_list_nat] :
( ( ( minus_7954133019191499631st_nat @ X @ Y )
= bot_bot_set_list_nat )
= ( ord_le6045566169113846134st_nat @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_78_nth__delete,axiom,
! [U: list_nat,K: nat] :
( ( member_list_nat @ U @ ( vf @ i @ t ) )
=> ( ( ord_less_nat @ K @ ra )
=> ( ( ( ord_less_nat @ K @ i )
=> ( ( nth_nat @ ( delete @ U ) @ K )
= ( nth_nat @ U @ K ) ) )
& ( ~ ( ord_less_nat @ K @ i )
=> ( ( nth_nat @ ( delete @ U ) @ K )
= ( nth_nat @ U @ ( suc @ K ) ) ) ) ) ) ) ).
% nth_delete
thf(fact_79_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_80_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_81_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_82_empty__Collect__eq,axiom,
! [P: list_nat > $o] :
( ( bot_bot_set_list_nat
= ( collect_list_nat @ P ) )
= ( ! [X2: list_nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_83_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_84_Collect__empty__eq,axiom,
! [P: list_nat > $o] :
( ( ( collect_list_nat @ P )
= bot_bot_set_list_nat )
= ( ! [X2: list_nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_85_pointwise__less__imp___092_060sigma_062,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( pointwise_less @ Xs @ Ys )
=> ( ord_less_nat @ ( groups4561878855575611511st_nat @ Xs ) @ ( groups4561878855575611511st_nat @ Ys ) ) ) ).
% pointwise_less_imp_\<sigma>
thf(fact_86_Diff__eq__empty__iff,axiom,
! [A3: set_list_nat,B2: set_list_nat] :
( ( ( minus_7954133019191499631st_nat @ A3 @ B2 )
= bot_bot_set_list_nat )
= ( ord_le6045566169113846134st_nat @ A3 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_87_Diff__cancel,axiom,
! [A3: set_list_nat] :
( ( minus_7954133019191499631st_nat @ A3 @ A3 )
= bot_bot_set_list_nat ) ).
% Diff_cancel
thf(fact_88_empty__Diff,axiom,
! [A3: set_list_nat] :
( ( minus_7954133019191499631st_nat @ bot_bot_set_list_nat @ A3 )
= bot_bot_set_list_nat ) ).
% empty_Diff
thf(fact_89_Diff__empty,axiom,
! [A3: set_list_nat] :
( ( minus_7954133019191499631st_nat @ A3 @ bot_bot_set_list_nat )
= A3 ) ).
% Diff_empty
thf(fact_90_empty__subsetI,axiom,
! [A3: set_list_nat] : ( ord_le6045566169113846134st_nat @ bot_bot_set_list_nat @ A3 ) ).
% empty_subsetI
thf(fact_91_subset__empty,axiom,
! [A3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ bot_bot_set_list_nat )
= ( A3 = bot_bot_set_list_nat ) ) ).
% subset_empty
thf(fact_92_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_93_empty__iff,axiom,
! [C: list_nat] :
~ ( member_list_nat @ C @ bot_bot_set_list_nat ) ).
% empty_iff
thf(fact_94_all__not__in__conv,axiom,
! [A3: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat @ X2 @ A3 ) )
= ( A3 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_95_all__not__in__conv,axiom,
! [A3: set_list_nat] :
( ( ! [X2: list_nat] :
~ ( member_list_nat @ X2 @ A3 ) )
= ( A3 = bot_bot_set_list_nat ) ) ).
% all_not_in_conv
thf(fact_96__092_060open_062t_A_060_Au_____A_B_Ai_092_060close_062,axiom,
ord_less_nat @ t @ ( nth_nat @ u2 @ i ) ).
% \<open>t < u__ ! i\<close>
thf(fact_97_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_98_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_99_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_100_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_101_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_102_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_103_length__drop,axiom,
! [N: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( drop_nat @ N @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% length_drop
thf(fact_104_take__append,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat] :
( ( take_nat @ N @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( take_nat @ N @ Xs ) @ ( take_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys ) ) ) ).
% take_append
thf(fact_105_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_106_drop__append,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat] :
( ( drop_nat @ N @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( drop_nat @ N @ Xs ) @ ( drop_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_107_nth__minus__list,axiom,
! [I: nat,Xs: list_list_nat,Ys: list_list_nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( nth_list_nat @ ( minus_3911745200923244873st_nat @ Xs @ Ys ) @ I )
= ( minus_minus_list_nat @ ( nth_list_nat @ Xs @ I ) @ ( nth_list_nat @ Ys @ I ) ) ) ) ) ).
% nth_minus_list
thf(fact_108_nth__minus__list,axiom,
! [I: nat,Xs: list_Extended_enat,Ys: list_Extended_enat] :
( ( ord_less_nat @ I @ ( size_s3941691890525107288d_enat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s3941691890525107288d_enat @ Ys ) )
=> ( ( nth_Extended_enat @ ( minus_388785356899630291d_enat @ Xs @ Ys ) @ I )
= ( minus_3235023915231533773d_enat @ ( nth_Extended_enat @ Xs @ I ) @ ( nth_Extended_enat @ Ys @ I ) ) ) ) ) ).
% nth_minus_list
thf(fact_109_nth__minus__list,axiom,
! [I: nat,Xs: list_nat,Ys: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
=> ( ( nth_nat @ ( minus_minus_list_nat @ Xs @ Ys ) @ I )
= ( minus_minus_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).
% nth_minus_list
thf(fact_110__C_K_C,axiom,
! [V2: list_nat] :
( ( member_list_nat @ V2 @ v )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ ra )
& ( ord_less_nat @ ( nth_nat @ V2 @ I2 ) @ ( nth_nat @ u2 @ I2 ) ) ) ) ).
% "*"
thf(fact_111_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_112_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_113_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_114_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_115_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_116_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_117_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_118_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_119_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_120_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_121_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_122_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_123_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_124_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_125_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_126_lift__Suc__mono__less__iff,axiom,
! [F: nat > extended_enat,N: nat,M: nat] :
( ! [N3: nat] : ( ord_le72135733267957522d_enat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_le72135733267957522d_enat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_127_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_128_lift__Suc__mono__less,axiom,
! [F: nat > extended_enat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_le72135733267957522d_enat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ord_le72135733267957522d_enat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_129_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_nat,Z2: list_nat] : ( Y5 = Z2 ) )
= ( ^ [Xs3: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs3 ) )
=> ( ( nth_nat @ Xs3 @ I3 )
= ( nth_nat @ Ys2 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_130_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X4: nat] : ( P @ I3 @ X4 ) ) )
= ( ? [Xs3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_nat @ Xs3 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_131_nth__equalityI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I2 )
= ( nth_nat @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_132_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_133_bot__set__def,axiom,
( bot_bot_set_list_nat
= ( collect_list_nat @ bot_bot_list_nat_o ) ) ).
% bot_set_def
thf(fact_134_finite__maxlen,axiom,
! [M3: set_list_nat] :
( ( finite8100373058378681591st_nat @ M3 )
=> ? [N3: nat] :
! [X5: list_nat] :
( ( member_list_nat @ X5 @ M3 )
=> ( ord_less_nat @ ( size_size_list_nat @ X5 ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_135_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_136_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_137_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_138_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_139_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_140_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_141_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_142_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_143_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_144_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_145_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_146_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_147_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_148_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_149_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ N )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_150_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_151_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_152_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_153_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M4: nat] :
( ( M
= ( suc @ M4 ) )
& ( ord_less_nat @ N @ M4 ) ) ) ) ).
% Suc_less_eq2
thf(fact_154_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_155_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_156_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_157_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_158_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_159_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_160_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
& ( M5 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_161_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_162_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_nat @ M5 @ N4 )
| ( M5 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_163_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_164_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_165_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_166_nth__append,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat] :
( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
= ( nth_nat @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
= ( nth_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_167_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs2: list_nat] :
( ! [Ys3: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys3 ) @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_168_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_169_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_170_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_171_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_172_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_173_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_174_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_175_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_176_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_177_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_178_nth__take__lemma,axiom,
! [K: nat,Xs: list_nat,Ys: list_nat] :
( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( nth_nat @ Xs @ I2 )
= ( nth_nat @ Ys @ I2 ) ) )
=> ( ( take_nat @ K @ Xs )
= ( take_nat @ K @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_179_pointwise__le__iff__nth,axiom,
( pointwise_le
= ( ^ [X2: list_nat,Y2: list_nat] :
( ( ( size_size_list_nat @ X2 )
= ( size_size_list_nat @ Y2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ X2 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ X2 @ I3 ) @ ( nth_nat @ Y2 @ I3 ) ) ) ) ) ) ).
% pointwise_le_iff_nth
thf(fact_180_pointwise__less__iff2,axiom,
( pointwise_less
= ( ^ [X2: list_nat,Y2: list_nat] :
( ( pointwise_le @ X2 @ Y2 )
& ? [K3: nat] :
( ( ord_less_nat @ K3 @ ( size_size_list_nat @ X2 ) )
& ( ord_less_nat @ ( nth_nat @ X2 @ K3 ) @ ( nth_nat @ Y2 @ K3 ) ) ) ) ) ) ).
% pointwise_less_iff2
thf(fact_181_drop__take,axiom,
! [N: nat,M: nat,Xs: list_nat] :
( ( drop_nat @ N @ ( take_nat @ M @ Xs ) )
= ( take_nat @ ( minus_minus_nat @ M @ N ) @ ( drop_nat @ N @ Xs ) ) ) ).
% drop_take
thf(fact_182_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_183_emptyE,axiom,
! [A: list_nat] :
~ ( member_list_nat @ A @ bot_bot_set_list_nat ) ).
% emptyE
thf(fact_184_equals0D,axiom,
! [A3: set_nat,A: nat] :
( ( A3 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A3 ) ) ).
% equals0D
thf(fact_185_equals0D,axiom,
! [A3: set_list_nat,A: list_nat] :
( ( A3 = bot_bot_set_list_nat )
=> ~ ( member_list_nat @ A @ A3 ) ) ).
% equals0D
thf(fact_186_equals0I,axiom,
! [A3: set_nat] :
( ! [Y4: nat] :
~ ( member_nat @ Y4 @ A3 )
=> ( A3 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_187_equals0I,axiom,
! [A3: set_list_nat] :
( ! [Y4: list_nat] :
~ ( member_list_nat @ Y4 @ A3 )
=> ( A3 = bot_bot_set_list_nat ) ) ).
% equals0I
thf(fact_188_ex__in__conv,axiom,
! [A3: set_nat] :
( ( ? [X2: nat] : ( member_nat @ X2 @ A3 ) )
= ( A3 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_189_ex__in__conv,axiom,
! [A3: set_list_nat] :
( ( ? [X2: list_nat] : ( member_list_nat @ X2 @ A3 ) )
= ( A3 != bot_bot_set_list_nat ) ) ).
% ex_in_conv
thf(fact_190_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_191_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_192_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_193_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_194_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_195_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_196_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_197_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_198_size__neq__size__imp__neq,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( size_size_list_nat @ X )
!= ( size_size_list_nat @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_199_sum__list__minus,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( pointwise_le @ Xs @ Ys )
=> ( ( groups4561878855575611511st_nat @ ( minus_minus_list_nat @ Ys @ Xs ) )
= ( minus_minus_nat @ ( groups4561878855575611511st_nat @ Ys ) @ ( groups4561878855575611511st_nat @ Xs ) ) ) ) ).
% sum_list_minus
thf(fact_200_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_201_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_202_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_203_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M7: nat] :
( M6
= ( suc @ M7 ) ) ) ).
% Suc_le_D
thf(fact_204_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_205_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_206_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_207_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_208_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_209_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y4: nat,Z3: nat] :
( ( R @ X3 @ Y4 )
=> ( ( R @ Y4 @ Z3 )
=> ( R @ X3 @ Z3 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_210_elem__le__sum__list,axiom,
! [K: nat,Ns: list_Extended_enat] :
( ( ord_less_nat @ K @ ( size_s3941691890525107288d_enat @ Ns ) )
=> ( ord_le2932123472753598470d_enat @ ( nth_Extended_enat @ Ns @ K ) @ ( groups5145338220374282879d_enat @ Ns ) ) ) ).
% elem_le_sum_list
thf(fact_211_elem__le__sum__list,axiom,
! [K: nat,Ns: list_nat] :
( ( ord_less_nat @ K @ ( size_size_list_nat @ Ns ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Ns @ K ) @ ( groups4561878855575611511st_nat @ Ns ) ) ) ).
% elem_le_sum_list
thf(fact_212_sum__list__mono2,axiom,
! [Xs: list_Extended_enat,Ys: list_Extended_enat] :
( ( ( size_s3941691890525107288d_enat @ Xs )
= ( size_s3941691890525107288d_enat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3941691890525107288d_enat @ Xs ) )
=> ( ord_le2932123472753598470d_enat @ ( nth_Extended_enat @ Xs @ I2 ) @ ( nth_Extended_enat @ Ys @ I2 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( groups5145338220374282879d_enat @ Xs ) @ ( groups5145338220374282879d_enat @ Ys ) ) ) ) ).
% sum_list_mono2
thf(fact_213_sum__list__mono2,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ Xs ) @ ( groups4561878855575611511st_nat @ Ys ) ) ) ) ).
% sum_list_mono2
thf(fact_214_min__pointwise__ge__iff,axiom,
! [U3: set_list_nat,R2: nat,X: list_nat] :
( ( finite8100373058378681591st_nat @ U3 )
=> ( ( U3 != bot_bot_set_list_nat )
=> ( ! [U2: list_nat] :
( ( member_list_nat @ U2 @ U3 )
=> ( ( size_size_list_nat @ U2 )
= R2 ) )
=> ( ( ( size_size_list_nat @ X )
= R2 )
=> ( ( pointwise_le @ X @ ( min_pointwise @ R2 @ U3 ) )
= ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ U3 )
=> ( pointwise_le @ X @ X2 ) ) ) ) ) ) ) ) ).
% min_pointwise_ge_iff
thf(fact_215_max__pointwise__le__iff,axiom,
! [U3: set_list_nat,R2: nat,X: list_nat] :
( ( finite8100373058378681591st_nat @ U3 )
=> ( ( U3 != bot_bot_set_list_nat )
=> ( ! [U2: list_nat] :
( ( member_list_nat @ U2 @ U3 )
=> ( ( size_size_list_nat @ U2 )
= R2 ) )
=> ( ( ( size_size_list_nat @ X )
= R2 )
=> ( ( pointwise_le @ ( max_pointwise @ R2 @ U3 ) @ X )
= ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ U3 )
=> ( pointwise_le @ X2 @ X ) ) ) ) ) ) ) ) ).
% max_pointwise_le_iff
thf(fact_216_min__pointwise__le,axiom,
! [U: list_nat,U3: set_list_nat] :
( ( member_list_nat @ U @ U3 )
=> ( ( finite8100373058378681591st_nat @ U3 )
=> ( pointwise_le @ ( min_pointwise @ ( size_size_list_nat @ U ) @ U3 ) @ U ) ) ) ).
% min_pointwise_le
thf(fact_217_ex__min__if__finite,axiom,
! [S2: set_list_nat] :
( ( finite8100373058378681591st_nat @ S2 )
=> ( ( S2 != bot_bot_set_list_nat )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ S2 )
& ~ ? [Xa: list_nat] :
( ( member_list_nat @ Xa @ S2 )
& ( ord_less_list_nat @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_218_ex__min__if__finite,axiom,
! [S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ S2 )
& ~ ? [Xa: nat] :
( ( member_nat @ Xa @ S2 )
& ( ord_less_nat @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_219_ex__min__if__finite,axiom,
! [S2: set_Extended_enat] :
( ( finite4001608067531595151d_enat @ S2 )
=> ( ( S2 != bot_bo7653980558646680370d_enat )
=> ? [X3: extended_enat] :
( ( member_Extended_enat @ X3 @ S2 )
& ~ ? [Xa: extended_enat] :
( ( member_Extended_enat @ Xa @ S2 )
& ( ord_le72135733267957522d_enat @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_220_infinite__growing,axiom,
! [X6: set_list_nat] :
( ( X6 != bot_bot_set_list_nat )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ X6 )
=> ? [Xa: list_nat] :
( ( member_list_nat @ Xa @ X6 )
& ( ord_less_list_nat @ X3 @ Xa ) ) )
=> ~ ( finite8100373058378681591st_nat @ X6 ) ) ) ).
% infinite_growing
thf(fact_221_infinite__growing,axiom,
! [X6: set_nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X6 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X6 )
& ( ord_less_nat @ X3 @ Xa ) ) )
=> ~ ( finite_finite_nat @ X6 ) ) ) ).
% infinite_growing
thf(fact_222_infinite__growing,axiom,
! [X6: set_Extended_enat] :
( ( X6 != bot_bo7653980558646680370d_enat )
=> ( ! [X3: extended_enat] :
( ( member_Extended_enat @ X3 @ X6 )
=> ? [Xa: extended_enat] :
( ( member_Extended_enat @ Xa @ X6 )
& ( ord_le72135733267957522d_enat @ X3 @ Xa ) ) )
=> ~ ( finite4001608067531595151d_enat @ X6 ) ) ) ).
% infinite_growing
thf(fact_223_finite__has__minimal,axiom,
! [A3: set_list_nat] :
( ( finite8100373058378681591st_nat @ A3 )
=> ( ( A3 != bot_bot_set_list_nat )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ A3 )
& ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ A3 )
=> ( ( ord_less_eq_list_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_224_finite__has__minimal,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A3 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_225_finite__has__minimal,axiom,
! [A3: set_set_list_nat] :
( ( finite7047420756378620717st_nat @ A3 )
=> ( ( A3 != bot_bo3886227569956363488st_nat )
=> ? [X3: set_list_nat] :
( ( member_set_list_nat @ X3 @ A3 )
& ! [Xa: set_list_nat] :
( ( member_set_list_nat @ Xa @ A3 )
=> ( ( ord_le6045566169113846134st_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_226_finite__has__minimal,axiom,
! [A3: set_Extended_enat] :
( ( finite4001608067531595151d_enat @ A3 )
=> ( ( A3 != bot_bo7653980558646680370d_enat )
=> ? [X3: extended_enat] :
( ( member_Extended_enat @ X3 @ A3 )
& ! [Xa: extended_enat] :
( ( member_Extended_enat @ Xa @ A3 )
=> ( ( ord_le2932123472753598470d_enat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_227_finite__has__maximal,axiom,
! [A3: set_list_nat] :
( ( finite8100373058378681591st_nat @ A3 )
=> ( ( A3 != bot_bot_set_list_nat )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ A3 )
& ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ A3 )
=> ( ( ord_less_eq_list_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_228_finite__has__maximal,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A3 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_229_finite__has__maximal,axiom,
! [A3: set_set_list_nat] :
( ( finite7047420756378620717st_nat @ A3 )
=> ( ( A3 != bot_bo3886227569956363488st_nat )
=> ? [X3: set_list_nat] :
( ( member_set_list_nat @ X3 @ A3 )
& ! [Xa: set_list_nat] :
( ( member_set_list_nat @ Xa @ A3 )
=> ( ( ord_le6045566169113846134st_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_230_finite__has__maximal,axiom,
! [A3: set_Extended_enat] :
( ( finite4001608067531595151d_enat @ A3 )
=> ( ( A3 != bot_bo7653980558646680370d_enat )
=> ? [X3: extended_enat] :
( ( member_Extended_enat @ X3 @ A3 )
& ! [Xa: extended_enat] :
( ( member_Extended_enat @ Xa @ A3 )
=> ( ( ord_le2932123472753598470d_enat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_231_max__pointwise__mono,axiom,
! [X7: set_list_nat,X6: set_list_nat,R2: nat] :
( ( ord_le6045566169113846134st_nat @ X7 @ X6 )
=> ( ( finite8100373058378681591st_nat @ X6 )
=> ( ( X7 != bot_bot_set_list_nat )
=> ( pointwise_le @ ( max_pointwise @ R2 @ X7 ) @ ( max_pointwise @ R2 @ X6 ) ) ) ) ) ).
% max_pointwise_mono
thf(fact_232_subset__antisym,axiom,
! [A3: set_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ B2 )
=> ( ( ord_le6045566169113846134st_nat @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ).
% subset_antisym
thf(fact_233_psubsetI,axiom,
! [A3: set_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ B2 )
=> ( ( A3 != B2 )
=> ( ord_le1190675801316882794st_nat @ A3 @ B2 ) ) ) ).
% psubsetI
thf(fact_234_subsetI,axiom,
! [A3: set_nat,B2: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( member_nat @ X3 @ B2 ) )
=> ( ord_less_eq_set_nat @ A3 @ B2 ) ) ).
% subsetI
thf(fact_235_subsetI,axiom,
! [A3: set_list_nat,B2: set_list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A3 )
=> ( member_list_nat @ X3 @ B2 ) )
=> ( ord_le6045566169113846134st_nat @ A3 @ B2 ) ) ).
% subsetI
thf(fact_236_DiffI,axiom,
! [C: list_nat,A3: set_list_nat,B2: set_list_nat] :
( ( member_list_nat @ C @ A3 )
=> ( ~ ( member_list_nat @ C @ B2 )
=> ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A3 @ B2 ) ) ) ) ).
% DiffI
thf(fact_237_DiffI,axiom,
! [C: nat,A3: set_nat,B2: set_nat] :
( ( member_nat @ C @ A3 )
=> ( ~ ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B2 ) ) ) ) ).
% DiffI
thf(fact_238_Diff__iff,axiom,
! [C: list_nat,A3: set_list_nat,B2: set_list_nat] :
( ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A3 @ B2 ) )
= ( ( member_list_nat @ C @ A3 )
& ~ ( member_list_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_239_Diff__iff,axiom,
! [C: nat,A3: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B2 ) )
= ( ( member_nat @ C @ A3 )
& ~ ( member_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_240_finite__Diff2,axiom,
! [B2: set_list_nat,A3: set_list_nat] :
( ( finite8100373058378681591st_nat @ B2 )
=> ( ( finite8100373058378681591st_nat @ ( minus_7954133019191499631st_nat @ A3 @ B2 ) )
= ( finite8100373058378681591st_nat @ A3 ) ) ) ).
% finite_Diff2
thf(fact_241_finite__Diff2,axiom,
! [B2: set_nat,A3: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ B2 ) )
= ( finite_finite_nat @ A3 ) ) ) ).
% finite_Diff2
thf(fact_242_finite__Diff,axiom,
! [A3: set_list_nat,B2: set_list_nat] :
( ( finite8100373058378681591st_nat @ A3 )
=> ( finite8100373058378681591st_nat @ ( minus_7954133019191499631st_nat @ A3 @ B2 ) ) ) ).
% finite_Diff
thf(fact_243_finite__Diff,axiom,
! [A3: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ B2 ) ) ) ).
% finite_Diff
thf(fact_244_DiffE,axiom,
! [C: list_nat,A3: set_list_nat,B2: set_list_nat] :
( ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A3 @ B2 ) )
=> ~ ( ( member_list_nat @ C @ A3 )
=> ( member_list_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_245_DiffE,axiom,
! [C: nat,A3: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B2 ) )
=> ~ ( ( member_nat @ C @ A3 )
=> ( member_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_246_DiffD1,axiom,
! [C: list_nat,A3: set_list_nat,B2: set_list_nat] :
( ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A3 @ B2 ) )
=> ( member_list_nat @ C @ A3 ) ) ).
% DiffD1
thf(fact_247_DiffD1,axiom,
! [C: nat,A3: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B2 ) )
=> ( member_nat @ C @ A3 ) ) ).
% DiffD1
thf(fact_248_DiffD2,axiom,
! [C: list_nat,A3: set_list_nat,B2: set_list_nat] :
( ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A3 @ B2 ) )
=> ~ ( member_list_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_249_DiffD2,axiom,
! [C: nat,A3: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B2 ) )
=> ~ ( member_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_250_Diff__mono,axiom,
! [A3: set_list_nat,C2: set_list_nat,D: set_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ C2 )
=> ( ( ord_le6045566169113846134st_nat @ D @ B2 )
=> ( ord_le6045566169113846134st_nat @ ( minus_7954133019191499631st_nat @ A3 @ B2 ) @ ( minus_7954133019191499631st_nat @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_251_Diff__subset,axiom,
! [A3: set_list_nat,B2: set_list_nat] : ( ord_le6045566169113846134st_nat @ ( minus_7954133019191499631st_nat @ A3 @ B2 ) @ A3 ) ).
% Diff_subset
thf(fact_252_double__diff,axiom,
! [A3: set_list_nat,B2: set_list_nat,C2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ B2 )
=> ( ( ord_le6045566169113846134st_nat @ B2 @ C2 )
=> ( ( minus_7954133019191499631st_nat @ B2 @ ( minus_7954133019191499631st_nat @ C2 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_253_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_254_psubset__imp__ex__mem,axiom,
! [A3: set_list_nat,B2: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A3 @ B2 )
=> ? [B3: list_nat] : ( member_list_nat @ B3 @ ( minus_7954133019191499631st_nat @ B2 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_255_psubset__imp__ex__mem,axiom,
! [A3: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A3 @ B2 )
=> ? [B3: nat] : ( member_nat @ B3 @ ( minus_minus_set_nat @ B2 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_256_subset__iff__psubset__eq,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A4: set_list_nat,B4: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_257_subset__psubset__trans,axiom,
! [A3: set_list_nat,B2: set_list_nat,C2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ B2 )
=> ( ( ord_le1190675801316882794st_nat @ B2 @ C2 )
=> ( ord_le1190675801316882794st_nat @ A3 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_258_subset__not__subset__eq,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [A4: set_list_nat,B4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A4 @ B4 )
& ~ ( ord_le6045566169113846134st_nat @ B4 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_259_psubset__subset__trans,axiom,
! [A3: set_list_nat,B2: set_list_nat,C2: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A3 @ B2 )
=> ( ( ord_le6045566169113846134st_nat @ B2 @ C2 )
=> ( ord_le1190675801316882794st_nat @ A3 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_260_psubset__imp__subset,axiom,
! [A3: set_list_nat,B2: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A3 @ B2 )
=> ( ord_le6045566169113846134st_nat @ A3 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_261_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_262_Collect__mono__iff,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) )
= ( ! [X2: list_nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_263_set__eq__subset,axiom,
( ( ^ [Y5: set_list_nat,Z2: set_list_nat] : ( Y5 = Z2 ) )
= ( ^ [A4: set_list_nat,B4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A4 @ B4 )
& ( ord_le6045566169113846134st_nat @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_264_subset__trans,axiom,
! [A3: set_list_nat,B2: set_list_nat,C2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ B2 )
=> ( ( ord_le6045566169113846134st_nat @ B2 @ C2 )
=> ( ord_le6045566169113846134st_nat @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_265_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_266_Collect__mono,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ! [X3: list_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_267_subset__refl,axiom,
! [A3: set_list_nat] : ( ord_le6045566169113846134st_nat @ A3 @ A3 ) ).
% subset_refl
thf(fact_268_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A4 )
=> ( member_nat @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_269_subset__iff,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A4: set_list_nat,B4: set_list_nat] :
! [T2: list_nat] :
( ( member_list_nat @ T2 @ A4 )
=> ( member_list_nat @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_270_psubset__eq,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [A4: set_list_nat,B4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% psubset_eq
thf(fact_271_equalityD2,axiom,
! [A3: set_list_nat,B2: set_list_nat] :
( ( A3 = B2 )
=> ( ord_le6045566169113846134st_nat @ B2 @ A3 ) ) ).
% equalityD2
thf(fact_272_equalityD1,axiom,
! [A3: set_list_nat,B2: set_list_nat] :
( ( A3 = B2 )
=> ( ord_le6045566169113846134st_nat @ A3 @ B2 ) ) ).
% equalityD1
thf(fact_273_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A4 )
=> ( member_nat @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_274_subset__eq,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A4: set_list_nat,B4: set_list_nat] :
! [X2: list_nat] :
( ( member_list_nat @ X2 @ A4 )
=> ( member_list_nat @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_275_equalityE,axiom,
! [A3: set_list_nat,B2: set_list_nat] :
( ( A3 = B2 )
=> ~ ( ( ord_le6045566169113846134st_nat @ A3 @ B2 )
=> ~ ( ord_le6045566169113846134st_nat @ B2 @ A3 ) ) ) ).
% equalityE
thf(fact_276_psubsetE,axiom,
! [A3: set_list_nat,B2: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A3 @ B2 )
=> ~ ( ( ord_le6045566169113846134st_nat @ A3 @ B2 )
=> ( ord_le6045566169113846134st_nat @ B2 @ A3 ) ) ) ).
% psubsetE
thf(fact_277_subsetD,axiom,
! [A3: set_nat,B2: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A3 @ B2 )
=> ( ( member_nat @ C @ A3 )
=> ( member_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_278_subsetD,axiom,
! [A3: set_list_nat,B2: set_list_nat,C: list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ B2 )
=> ( ( member_list_nat @ C @ A3 )
=> ( member_list_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_279_in__mono,axiom,
! [A3: set_nat,B2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A3 @ B2 )
=> ( ( member_nat @ X @ A3 )
=> ( member_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_280_in__mono,axiom,
! [A3: set_list_nat,B2: set_list_nat,X: list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ B2 )
=> ( ( member_list_nat @ X @ A3 )
=> ( member_list_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_281_finite__psubset__induct,axiom,
! [A3: set_list_nat,P: set_list_nat > $o] :
( ( finite8100373058378681591st_nat @ A3 )
=> ( ! [A5: set_list_nat] :
( ( finite8100373058378681591st_nat @ A5 )
=> ( ! [B5: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ B5 @ A5 )
=> ( P @ B5 ) )
=> ( P @ A5 ) ) )
=> ( P @ A3 ) ) ) ).
% finite_psubset_induct
thf(fact_282_finite__psubset__induct,axiom,
! [A3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A3 )
=> ( ! [A5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ! [B5: set_nat] :
( ( ord_less_set_nat @ B5 @ A5 )
=> ( P @ B5 ) )
=> ( P @ A5 ) ) )
=> ( P @ A3 ) ) ) ).
% finite_psubset_induct
thf(fact_283_not__psubset__empty,axiom,
! [A3: set_list_nat] :
~ ( ord_le1190675801316882794st_nat @ A3 @ bot_bot_set_list_nat ) ).
% not_psubset_empty
thf(fact_284_finite__subset,axiom,
! [A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A3 ) ) ) ).
% finite_subset
thf(fact_285_finite__subset,axiom,
! [A3: set_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ B2 )
=> ( ( finite8100373058378681591st_nat @ B2 )
=> ( finite8100373058378681591st_nat @ A3 ) ) ) ).
% finite_subset
thf(fact_286_infinite__super,axiom,
! [S2: set_nat,T3: set_nat] :
( ( ord_less_eq_set_nat @ S2 @ T3 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ T3 ) ) ) ).
% infinite_super
thf(fact_287_infinite__super,axiom,
! [S2: set_list_nat,T3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ S2 @ T3 )
=> ( ~ ( finite8100373058378681591st_nat @ S2 )
=> ~ ( finite8100373058378681591st_nat @ T3 ) ) ) ).
% infinite_super
thf(fact_288_rev__finite__subset,axiom,
! [B2: set_nat,A3: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A3 @ B2 )
=> ( finite_finite_nat @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_289_rev__finite__subset,axiom,
! [B2: set_list_nat,A3: set_list_nat] :
( ( finite8100373058378681591st_nat @ B2 )
=> ( ( ord_le6045566169113846134st_nat @ A3 @ B2 )
=> ( finite8100373058378681591st_nat @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_290_Diff__infinite__finite,axiom,
! [T3: set_list_nat,S2: set_list_nat] :
( ( finite8100373058378681591st_nat @ T3 )
=> ( ~ ( finite8100373058378681591st_nat @ S2 )
=> ~ ( finite8100373058378681591st_nat @ ( minus_7954133019191499631st_nat @ S2 @ T3 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_291_Diff__infinite__finite,axiom,
! [T3: set_nat,S2: set_nat] :
( ( finite_finite_nat @ T3 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ T3 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_292_finite__has__minimal2,axiom,
! [A3: set_list_nat,A: list_nat] :
( ( finite8100373058378681591st_nat @ A3 )
=> ( ( member_list_nat @ A @ A3 )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ A3 )
& ( ord_less_eq_list_nat @ X3 @ A )
& ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ A3 )
=> ( ( ord_less_eq_list_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_293_finite__has__minimal2,axiom,
! [A3: set_nat,A: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ A @ A3 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A3 )
& ( ord_less_eq_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A3 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_294_finite__has__minimal2,axiom,
! [A3: set_set_list_nat,A: set_list_nat] :
( ( finite7047420756378620717st_nat @ A3 )
=> ( ( member_set_list_nat @ A @ A3 )
=> ? [X3: set_list_nat] :
( ( member_set_list_nat @ X3 @ A3 )
& ( ord_le6045566169113846134st_nat @ X3 @ A )
& ! [Xa: set_list_nat] :
( ( member_set_list_nat @ Xa @ A3 )
=> ( ( ord_le6045566169113846134st_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_295_finite__has__minimal2,axiom,
! [A3: set_Extended_enat,A: extended_enat] :
( ( finite4001608067531595151d_enat @ A3 )
=> ( ( member_Extended_enat @ A @ A3 )
=> ? [X3: extended_enat] :
( ( member_Extended_enat @ X3 @ A3 )
& ( ord_le2932123472753598470d_enat @ X3 @ A )
& ! [Xa: extended_enat] :
( ( member_Extended_enat @ Xa @ A3 )
=> ( ( ord_le2932123472753598470d_enat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_296_finite__has__maximal2,axiom,
! [A3: set_list_nat,A: list_nat] :
( ( finite8100373058378681591st_nat @ A3 )
=> ( ( member_list_nat @ A @ A3 )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ A3 )
& ( ord_less_eq_list_nat @ A @ X3 )
& ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ A3 )
=> ( ( ord_less_eq_list_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_297_finite__has__maximal2,axiom,
! [A3: set_nat,A: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ A @ A3 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A3 )
& ( ord_less_eq_nat @ A @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A3 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_298_finite__has__maximal2,axiom,
! [A3: set_set_list_nat,A: set_list_nat] :
( ( finite7047420756378620717st_nat @ A3 )
=> ( ( member_set_list_nat @ A @ A3 )
=> ? [X3: set_list_nat] :
( ( member_set_list_nat @ X3 @ A3 )
& ( ord_le6045566169113846134st_nat @ A @ X3 )
& ! [Xa: set_list_nat] :
( ( member_set_list_nat @ Xa @ A3 )
=> ( ( ord_le6045566169113846134st_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_299_finite__has__maximal2,axiom,
! [A3: set_Extended_enat,A: extended_enat] :
( ( finite4001608067531595151d_enat @ A3 )
=> ( ( member_Extended_enat @ A @ A3 )
=> ? [X3: extended_enat] :
( ( member_Extended_enat @ X3 @ A3 )
& ( ord_le2932123472753598470d_enat @ A @ X3 )
& ! [Xa: extended_enat] :
( ( member_Extended_enat @ Xa @ A3 )
=> ( ( ord_le2932123472753598470d_enat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_300_infinite__imp__nonempty,axiom,
! [S2: set_nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( S2 != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_301_infinite__imp__nonempty,axiom,
! [S2: set_list_nat] :
( ~ ( finite8100373058378681591st_nat @ S2 )
=> ( S2 != bot_bot_set_list_nat ) ) ).
% infinite_imp_nonempty
thf(fact_302_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_303_finite_OemptyI,axiom,
finite8100373058378681591st_nat @ bot_bot_set_list_nat ).
% finite.emptyI
thf(fact_304_max__pointwise__ge,axiom,
! [U: list_nat,U3: set_list_nat] :
( ( member_list_nat @ U @ U3 )
=> ( ( finite8100373058378681591st_nat @ U3 )
=> ( pointwise_le @ U @ ( max_pointwise @ ( size_size_list_nat @ U ) @ U3 ) ) ) ) ).
% max_pointwise_ge
thf(fact_305_le__Nil,axiom,
! [X: list_nat] :
( ( ord_less_eq_list_nat @ X @ nil_nat )
= ( X = nil_nat ) ) ).
% le_Nil
thf(fact_306_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_307_Collect__empty__eq__bot,axiom,
! [P: list_nat > $o] :
( ( ( collect_list_nat @ P )
= bot_bot_set_list_nat )
= ( P = bot_bot_list_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_308_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X2: nat] : ( member_nat @ X2 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_309_bot__empty__eq,axiom,
( bot_bot_list_nat_o
= ( ^ [X2: list_nat] : ( member_list_nat @ X2 @ bot_bot_set_list_nat ) ) ) ).
% bot_empty_eq
thf(fact_310_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_311_dual__order_Orefl,axiom,
! [A: set_list_nat] : ( ord_le6045566169113846134st_nat @ A @ A ) ).
% dual_order.refl
thf(fact_312_dual__order_Orefl,axiom,
! [A: extended_enat] : ( ord_le2932123472753598470d_enat @ A @ A ) ).
% dual_order.refl
thf(fact_313_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_314_order__refl,axiom,
! [X: set_list_nat] : ( ord_le6045566169113846134st_nat @ X @ X ) ).
% order_refl
thf(fact_315_order__refl,axiom,
! [X: extended_enat] : ( ord_le2932123472753598470d_enat @ X @ X ) ).
% order_refl
thf(fact_316_psubsetD,axiom,
! [A3: set_list_nat,B2: set_list_nat,C: list_nat] :
( ( ord_le1190675801316882794st_nat @ A3 @ B2 )
=> ( ( member_list_nat @ C @ A3 )
=> ( member_list_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_317_psubsetD,axiom,
! [A3: set_nat,B2: set_nat,C: nat] :
( ( ord_less_set_nat @ A3 @ B2 )
=> ( ( member_nat @ C @ A3 )
=> ( member_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_318_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_319_nle__le,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ~ ( ord_le2932123472753598470d_enat @ A @ B ) )
= ( ( ord_le2932123472753598470d_enat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_320_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_321_le__cases3,axiom,
! [X: extended_enat,Y: extended_enat,Z: extended_enat] :
( ( ( ord_le2932123472753598470d_enat @ X @ Y )
=> ~ ( ord_le2932123472753598470d_enat @ Y @ Z ) )
=> ( ( ( ord_le2932123472753598470d_enat @ Y @ X )
=> ~ ( ord_le2932123472753598470d_enat @ X @ Z ) )
=> ( ( ( ord_le2932123472753598470d_enat @ X @ Z )
=> ~ ( ord_le2932123472753598470d_enat @ Z @ Y ) )
=> ( ( ( ord_le2932123472753598470d_enat @ Z @ Y )
=> ~ ( ord_le2932123472753598470d_enat @ Y @ X ) )
=> ( ( ( ord_le2932123472753598470d_enat @ Y @ Z )
=> ~ ( ord_le2932123472753598470d_enat @ Z @ X ) )
=> ~ ( ( ord_le2932123472753598470d_enat @ Z @ X )
=> ~ ( ord_le2932123472753598470d_enat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_322_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_323_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_list_nat,Z2: set_list_nat] : ( Y5 = Z2 ) )
= ( ^ [X2: set_list_nat,Y2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X2 @ Y2 )
& ( ord_le6045566169113846134st_nat @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_324_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: extended_enat,Z2: extended_enat] : ( Y5 = Z2 ) )
= ( ^ [X2: extended_enat,Y2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X2 @ Y2 )
& ( ord_le2932123472753598470d_enat @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_325_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_326_ord__eq__le__trans,axiom,
! [A: set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( A = B )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ord_le6045566169113846134st_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_327_ord__eq__le__trans,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( A = B )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ord_le2932123472753598470d_enat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_328_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_329_ord__le__eq__trans,axiom,
! [A: set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( B = C )
=> ( ord_le6045566169113846134st_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_330_ord__le__eq__trans,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( B = C )
=> ( ord_le2932123472753598470d_enat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_331_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_332_order__antisym,axiom,
! [X: set_list_nat,Y: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X @ Y )
=> ( ( ord_le6045566169113846134st_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_333_order__antisym,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ Y )
=> ( ( ord_le2932123472753598470d_enat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_334_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_335_order_Otrans,axiom,
! [A: set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ord_le6045566169113846134st_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_336_order_Otrans,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ord_le2932123472753598470d_enat @ A @ C ) ) ) ).
% order.trans
thf(fact_337_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_338_order__trans,axiom,
! [X: set_list_nat,Y: set_list_nat,Z: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X @ Y )
=> ( ( ord_le6045566169113846134st_nat @ Y @ Z )
=> ( ord_le6045566169113846134st_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_339_order__trans,axiom,
! [X: extended_enat,Y: extended_enat,Z: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ Y )
=> ( ( ord_le2932123472753598470d_enat @ Y @ Z )
=> ( ord_le2932123472753598470d_enat @ X @ Z ) ) ) ).
% order_trans
thf(fact_340_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A6: nat,B3: nat] :
( ( ord_less_eq_nat @ A6 @ B3 )
=> ( P @ A6 @ B3 ) )
=> ( ! [A6: nat,B3: nat] :
( ( P @ B3 @ A6 )
=> ( P @ A6 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_341_linorder__wlog,axiom,
! [P: extended_enat > extended_enat > $o,A: extended_enat,B: extended_enat] :
( ! [A6: extended_enat,B3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A6 @ B3 )
=> ( P @ A6 @ B3 ) )
=> ( ! [A6: extended_enat,B3: extended_enat] :
( ( P @ B3 @ A6 )
=> ( P @ A6 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_342_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A2: nat,B6: nat] :
( ( ord_less_eq_nat @ B6 @ A2 )
& ( ord_less_eq_nat @ A2 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_343_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_list_nat,Z2: set_list_nat] : ( Y5 = Z2 ) )
= ( ^ [A2: set_list_nat,B6: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B6 @ A2 )
& ( ord_le6045566169113846134st_nat @ A2 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_344_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: extended_enat,Z2: extended_enat] : ( Y5 = Z2 ) )
= ( ^ [A2: extended_enat,B6: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B6 @ A2 )
& ( ord_le2932123472753598470d_enat @ A2 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_345_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_346_dual__order_Oantisym,axiom,
! [B: set_list_nat,A: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B @ A )
=> ( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_347_dual__order_Oantisym,axiom,
! [B: extended_enat,A: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B @ A )
=> ( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_348_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_349_dual__order_Otrans,axiom,
! [B: set_list_nat,A: set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B @ A )
=> ( ( ord_le6045566169113846134st_nat @ C @ B )
=> ( ord_le6045566169113846134st_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_350_dual__order_Otrans,axiom,
! [B: extended_enat,A: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B @ A )
=> ( ( ord_le2932123472753598470d_enat @ C @ B )
=> ( ord_le2932123472753598470d_enat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_351_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_352_antisym,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ord_le6045566169113846134st_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_353_antisym,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_354_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A2: nat,B6: nat] :
( ( ord_less_eq_nat @ A2 @ B6 )
& ( ord_less_eq_nat @ B6 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_355_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_list_nat,Z2: set_list_nat] : ( Y5 = Z2 ) )
= ( ^ [A2: set_list_nat,B6: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B6 )
& ( ord_le6045566169113846134st_nat @ B6 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_356_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: extended_enat,Z2: extended_enat] : ( Y5 = Z2 ) )
= ( ^ [A2: extended_enat,B6: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A2 @ B6 )
& ( ord_le2932123472753598470d_enat @ B6 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_357_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,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_358_order__subst1,axiom,
! [A: nat,F: set_list_nat > nat,B: set_list_nat,C: set_list_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ! [X3: set_list_nat,Y4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_359_order__subst1,axiom,
! [A: nat,F: extended_enat > nat,B: extended_enat,C: extended_enat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_360_order__subst1,axiom,
! [A: set_list_nat,F: nat > set_list_nat,B: nat,C: nat] :
( ( ord_le6045566169113846134st_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le6045566169113846134st_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_361_order__subst1,axiom,
! [A: set_list_nat,F: set_list_nat > set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ ( F @ B ) )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ! [X3: set_list_nat,Y4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y4 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le6045566169113846134st_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_362_order__subst1,axiom,
! [A: set_list_nat,F: extended_enat > set_list_nat,B: extended_enat,C: extended_enat] :
( ( ord_le6045566169113846134st_nat @ A @ ( F @ B ) )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le6045566169113846134st_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_363_order__subst1,axiom,
! [A: extended_enat,F: nat > extended_enat,B: nat,C: nat] :
( ( ord_le2932123472753598470d_enat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le2932123472753598470d_enat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_364_order__subst1,axiom,
! [A: extended_enat,F: set_list_nat > extended_enat,B: set_list_nat,C: set_list_nat] :
( ( ord_le2932123472753598470d_enat @ A @ ( F @ B ) )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ! [X3: set_list_nat,Y4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le2932123472753598470d_enat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_365_order__subst1,axiom,
! [A: extended_enat,F: extended_enat > extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ ( F @ B ) )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le2932123472753598470d_enat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_366_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,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_367_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_list_nat,C: set_list_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le6045566169113846134st_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le6045566169113846134st_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_368_order__subst2,axiom,
! [A: nat,B: nat,F: nat > extended_enat,C: extended_enat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_369_order__subst2,axiom,
! [A: set_list_nat,B: set_list_nat,F: set_list_nat > nat,C: nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_list_nat,Y4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_370_order__subst2,axiom,
! [A: set_list_nat,B: set_list_nat,F: set_list_nat > set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ord_le6045566169113846134st_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_list_nat,Y4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y4 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le6045566169113846134st_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_371_order__subst2,axiom,
! [A: set_list_nat,B: set_list_nat,F: set_list_nat > extended_enat,C: extended_enat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ ( F @ B ) @ C )
=> ( ! [X3: set_list_nat,Y4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_372_order__subst2,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > nat,C: nat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_373_order__subst2,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > set_list_nat,C: set_list_nat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_le6045566169113846134st_nat @ ( F @ B ) @ C )
=> ( ! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le6045566169113846134st_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_374_order__subst2,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ ( F @ B ) @ C )
=> ( ! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_375_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_376_order__eq__refl,axiom,
! [X: set_list_nat,Y: set_list_nat] :
( ( X = Y )
=> ( ord_le6045566169113846134st_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_377_order__eq__refl,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( X = Y )
=> ( ord_le2932123472753598470d_enat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_378_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_379_linorder__linear,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ Y )
| ( ord_le2932123472753598470d_enat @ Y @ X ) ) ).
% linorder_linear
thf(fact_380_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,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_381_ord__eq__le__subst,axiom,
! [A: set_list_nat,F: nat > set_list_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le6045566169113846134st_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_382_ord__eq__le__subst,axiom,
! [A: extended_enat,F: nat > extended_enat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le2932123472753598470d_enat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_383_ord__eq__le__subst,axiom,
! [A: nat,F: set_list_nat > nat,B: set_list_nat,C: set_list_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ! [X3: set_list_nat,Y4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_384_ord__eq__le__subst,axiom,
! [A: set_list_nat,F: set_list_nat > set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ! [X3: set_list_nat,Y4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y4 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le6045566169113846134st_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_385_ord__eq__le__subst,axiom,
! [A: extended_enat,F: set_list_nat > extended_enat,B: set_list_nat,C: set_list_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ! [X3: set_list_nat,Y4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le2932123472753598470d_enat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_386_ord__eq__le__subst,axiom,
! [A: nat,F: extended_enat > nat,B: extended_enat,C: extended_enat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_387_ord__eq__le__subst,axiom,
! [A: set_list_nat,F: extended_enat > set_list_nat,B: extended_enat,C: extended_enat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le6045566169113846134st_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_388_ord__eq__le__subst,axiom,
! [A: extended_enat,F: extended_enat > extended_enat,B: extended_enat,C: extended_enat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le2932123472753598470d_enat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_389_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,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_390_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_list_nat,C: set_list_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le6045566169113846134st_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_391_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > extended_enat,C: extended_enat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_392_ord__le__eq__subst,axiom,
! [A: set_list_nat,B: set_list_nat,F: set_list_nat > nat,C: nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_list_nat,Y4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_393_ord__le__eq__subst,axiom,
! [A: set_list_nat,B: set_list_nat,F: set_list_nat > set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_list_nat,Y4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y4 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le6045566169113846134st_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_394_ord__le__eq__subst,axiom,
! [A: set_list_nat,B: set_list_nat,F: set_list_nat > extended_enat,C: extended_enat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_list_nat,Y4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_395_ord__le__eq__subst,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > nat,C: nat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_396_ord__le__eq__subst,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > set_list_nat,C: set_list_nat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le6045566169113846134st_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_397_ord__le__eq__subst,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_398_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_399_linorder__le__cases,axiom,
! [X: extended_enat,Y: extended_enat] :
( ~ ( ord_le2932123472753598470d_enat @ X @ Y )
=> ( ord_le2932123472753598470d_enat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_400_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_401_order__antisym__conv,axiom,
! [Y: set_list_nat,X: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ Y @ X )
=> ( ( ord_le6045566169113846134st_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_402_order__antisym__conv,axiom,
! [Y: extended_enat,X: extended_enat] :
( ( ord_le2932123472753598470d_enat @ Y @ X )
=> ( ( ord_le2932123472753598470d_enat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_403_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_404_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_405_less__imp__neq,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le72135733267957522d_enat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_406_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_407_order_Oasym,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ~ ( ord_le72135733267957522d_enat @ B @ A ) ) ).
% order.asym
thf(fact_408_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_409_ord__eq__less__trans,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( A = B )
=> ( ( ord_le72135733267957522d_enat @ B @ C )
=> ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_410_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_411_ord__less__eq__trans,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( B = C )
=> ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_412_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y3: nat] :
( ( ord_less_nat @ Y3 @ X3 )
=> ( P @ Y3 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_413_less__induct,axiom,
! [P: extended_enat > $o,A: extended_enat] :
( ! [X3: extended_enat] :
( ! [Y3: extended_enat] :
( ( ord_le72135733267957522d_enat @ Y3 @ X3 )
=> ( P @ Y3 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_414_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_415_antisym__conv3,axiom,
! [Y: extended_enat,X: extended_enat] :
( ~ ( ord_le72135733267957522d_enat @ Y @ X )
=> ( ( ~ ( ord_le72135733267957522d_enat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_416_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_417_linorder__cases,axiom,
! [X: extended_enat,Y: extended_enat] :
( ~ ( ord_le72135733267957522d_enat @ X @ Y )
=> ( ( X != Y )
=> ( ord_le72135733267957522d_enat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_418_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_419_dual__order_Oasym,axiom,
! [B: extended_enat,A: extended_enat] :
( ( ord_le72135733267957522d_enat @ B @ A )
=> ~ ( ord_le72135733267957522d_enat @ A @ B ) ) ).
% dual_order.asym
thf(fact_420_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_421_dual__order_Oirrefl,axiom,
! [A: extended_enat] :
~ ( ord_le72135733267957522d_enat @ A @ A ) ).
% dual_order.irrefl
thf(fact_422_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X8: nat] : ( P2 @ X8 ) )
= ( ^ [P3: nat > $o] :
? [N4: nat] :
( ( P3 @ N4 )
& ! [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
=> ~ ( P3 @ M5 ) ) ) ) ) ).
% exists_least_iff
thf(fact_423_exists__least__iff,axiom,
( ( ^ [P2: extended_enat > $o] :
? [X8: extended_enat] : ( P2 @ X8 ) )
= ( ^ [P3: extended_enat > $o] :
? [N4: extended_enat] :
( ( P3 @ N4 )
& ! [M5: extended_enat] :
( ( ord_le72135733267957522d_enat @ M5 @ N4 )
=> ~ ( P3 @ M5 ) ) ) ) ) ).
% exists_least_iff
thf(fact_424_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A6: nat,B3: nat] :
( ( ord_less_nat @ A6 @ B3 )
=> ( P @ A6 @ B3 ) )
=> ( ! [A6: nat] : ( P @ A6 @ A6 )
=> ( ! [A6: nat,B3: nat] :
( ( P @ B3 @ A6 )
=> ( P @ A6 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_425_linorder__less__wlog,axiom,
! [P: extended_enat > extended_enat > $o,A: extended_enat,B: extended_enat] :
( ! [A6: extended_enat,B3: extended_enat] :
( ( ord_le72135733267957522d_enat @ A6 @ B3 )
=> ( P @ A6 @ B3 ) )
=> ( ! [A6: extended_enat] : ( P @ A6 @ A6 )
=> ( ! [A6: extended_enat,B3: extended_enat] :
( ( P @ B3 @ A6 )
=> ( P @ A6 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_426_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_427_order_Ostrict__trans,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( ord_le72135733267957522d_enat @ B @ C )
=> ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_428_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_429_not__less__iff__gr__or__eq,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ~ ( ord_le72135733267957522d_enat @ X @ Y ) )
= ( ( ord_le72135733267957522d_enat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_430_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_431_dual__order_Ostrict__trans,axiom,
! [B: extended_enat,A: extended_enat,C: extended_enat] :
( ( ord_le72135733267957522d_enat @ B @ A )
=> ( ( ord_le72135733267957522d_enat @ C @ B )
=> ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_432_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_433_order_Ostrict__implies__not__eq,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_434_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_435_dual__order_Ostrict__implies__not__eq,axiom,
! [B: extended_enat,A: extended_enat] :
( ( ord_le72135733267957522d_enat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_436_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_437_linorder__neqE,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( X != Y )
=> ( ~ ( ord_le72135733267957522d_enat @ X @ Y )
=> ( ord_le72135733267957522d_enat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_438_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_439_order__less__asym,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le72135733267957522d_enat @ X @ Y )
=> ~ ( ord_le72135733267957522d_enat @ Y @ X ) ) ).
% order_less_asym
thf(fact_440_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_441_linorder__neq__iff,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( X != Y )
= ( ( ord_le72135733267957522d_enat @ X @ Y )
| ( ord_le72135733267957522d_enat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_442_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_443_order__less__asym_H,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ~ ( ord_le72135733267957522d_enat @ B @ A ) ) ).
% order_less_asym'
thf(fact_444_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_445_order__less__trans,axiom,
! [X: extended_enat,Y: extended_enat,Z: extended_enat] :
( ( ord_le72135733267957522d_enat @ X @ Y )
=> ( ( ord_le72135733267957522d_enat @ Y @ Z )
=> ( ord_le72135733267957522d_enat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_446_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_447_ord__eq__less__subst,axiom,
! [A: extended_enat,F: nat > extended_enat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_448_ord__eq__less__subst,axiom,
! [A: nat,F: extended_enat > nat,B: extended_enat,C: extended_enat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le72135733267957522d_enat @ B @ C )
=> ( ! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_449_ord__eq__less__subst,axiom,
! [A: extended_enat,F: extended_enat > extended_enat,B: extended_enat,C: extended_enat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le72135733267957522d_enat @ B @ C )
=> ( ! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_450_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_451_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > extended_enat,C: extended_enat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_452_ord__less__eq__subst,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > nat,C: nat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_453_ord__less__eq__subst,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_454_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_455_order__less__irrefl,axiom,
! [X: extended_enat] :
~ ( ord_le72135733267957522d_enat @ X @ X ) ).
% order_less_irrefl
thf(fact_456_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,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_457_order__less__subst1,axiom,
! [A: nat,F: extended_enat > nat,B: extended_enat,C: extended_enat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le72135733267957522d_enat @ B @ C )
=> ( ! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_458_order__less__subst1,axiom,
! [A: extended_enat,F: nat > extended_enat,B: nat,C: nat] :
( ( ord_le72135733267957522d_enat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_459_order__less__subst1,axiom,
! [A: extended_enat,F: extended_enat > extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ ( F @ B ) )
=> ( ( ord_le72135733267957522d_enat @ B @ C )
=> ( ! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_460_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,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_461_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > extended_enat,C: extended_enat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le72135733267957522d_enat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_462_order__less__subst2,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > nat,C: nat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_463_order__less__subst2,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( ord_le72135733267957522d_enat @ ( F @ B ) @ C )
=> ( ! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_464_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_465_order__less__not__sym,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le72135733267957522d_enat @ X @ Y )
=> ~ ( ord_le72135733267957522d_enat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_466_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_467_order__less__imp__triv,axiom,
! [X: extended_enat,Y: extended_enat,P: $o] :
( ( ord_le72135733267957522d_enat @ X @ Y )
=> ( ( ord_le72135733267957522d_enat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_468_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_469_linorder__less__linear,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le72135733267957522d_enat @ X @ Y )
| ( X = Y )
| ( ord_le72135733267957522d_enat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_470_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_471_order__less__imp__not__eq,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le72135733267957522d_enat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_472_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_473_order__less__imp__not__eq2,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le72135733267957522d_enat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_474_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_475_order__less__imp__not__less,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le72135733267957522d_enat @ X @ Y )
=> ~ ( ord_le72135733267957522d_enat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_476_bot__list__def,axiom,
bot_bot_list_nat = nil_nat ).
% bot_list_def
thf(fact_477_Nil__le__Cons,axiom,
! [X: list_nat] : ( ord_less_eq_list_nat @ nil_nat @ X ) ).
% Nil_le_Cons
thf(fact_478_not__less__Nil,axiom,
! [X: list_nat] :
~ ( ord_less_list_nat @ X @ nil_nat ) ).
% not_less_Nil
thf(fact_479_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_480_leD,axiom,
! [Y: set_list_nat,X: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ Y @ X )
=> ~ ( ord_le1190675801316882794st_nat @ X @ Y ) ) ).
% leD
thf(fact_481_leD,axiom,
! [Y: extended_enat,X: extended_enat] :
( ( ord_le2932123472753598470d_enat @ Y @ X )
=> ~ ( ord_le72135733267957522d_enat @ X @ Y ) ) ).
% leD
thf(fact_482_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_483_leI,axiom,
! [X: extended_enat,Y: extended_enat] :
( ~ ( ord_le72135733267957522d_enat @ X @ Y )
=> ( ord_le2932123472753598470d_enat @ Y @ X ) ) ).
% leI
thf(fact_484_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_485_nless__le,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( ~ ( ord_le1190675801316882794st_nat @ A @ B ) )
= ( ~ ( ord_le6045566169113846134st_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_486_nless__le,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ~ ( ord_le72135733267957522d_enat @ A @ B ) )
= ( ~ ( ord_le2932123472753598470d_enat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_487_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_488_antisym__conv1,axiom,
! [X: set_list_nat,Y: set_list_nat] :
( ~ ( ord_le1190675801316882794st_nat @ X @ Y )
=> ( ( ord_le6045566169113846134st_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_489_antisym__conv1,axiom,
! [X: extended_enat,Y: extended_enat] :
( ~ ( ord_le72135733267957522d_enat @ X @ Y )
=> ( ( ord_le2932123472753598470d_enat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_490_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_491_antisym__conv2,axiom,
! [X: set_list_nat,Y: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X @ Y )
=> ( ( ~ ( ord_le1190675801316882794st_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_492_antisym__conv2,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ Y )
=> ( ( ~ ( ord_le72135733267957522d_enat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_493_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ~ ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_494_less__le__not__le,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [X2: set_list_nat,Y2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X2 @ Y2 )
& ~ ( ord_le6045566169113846134st_nat @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_495_less__le__not__le,axiom,
( ord_le72135733267957522d_enat
= ( ^ [X2: extended_enat,Y2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X2 @ Y2 )
& ~ ( ord_le2932123472753598470d_enat @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_496_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_497_not__le__imp__less,axiom,
! [Y: extended_enat,X: extended_enat] :
( ~ ( ord_le2932123472753598470d_enat @ Y @ X )
=> ( ord_le72135733267957522d_enat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_498_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B6: nat] :
( ( ord_less_nat @ A2 @ B6 )
| ( A2 = B6 ) ) ) ) ).
% order.order_iff_strict
thf(fact_499_order_Oorder__iff__strict,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A2: set_list_nat,B6: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A2 @ B6 )
| ( A2 = B6 ) ) ) ) ).
% order.order_iff_strict
thf(fact_500_order_Oorder__iff__strict,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [A2: extended_enat,B6: extended_enat] :
( ( ord_le72135733267957522d_enat @ A2 @ B6 )
| ( A2 = B6 ) ) ) ) ).
% order.order_iff_strict
thf(fact_501_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A2: nat,B6: nat] :
( ( ord_less_eq_nat @ A2 @ B6 )
& ( A2 != B6 ) ) ) ) ).
% order.strict_iff_order
thf(fact_502_order_Ostrict__iff__order,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [A2: set_list_nat,B6: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B6 )
& ( A2 != B6 ) ) ) ) ).
% order.strict_iff_order
thf(fact_503_order_Ostrict__iff__order,axiom,
( ord_le72135733267957522d_enat
= ( ^ [A2: extended_enat,B6: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A2 @ B6 )
& ( A2 != B6 ) ) ) ) ).
% order.strict_iff_order
thf(fact_504_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_505_order_Ostrict__trans1,axiom,
! [A: set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ord_le1190675801316882794st_nat @ B @ C )
=> ( ord_le1190675801316882794st_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_506_order_Ostrict__trans1,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_le72135733267957522d_enat @ B @ C )
=> ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_507_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_508_order_Ostrict__trans2,axiom,
! [A: set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A @ B )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ord_le1190675801316882794st_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_509_order_Ostrict__trans2,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ord_le72135733267957522d_enat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_510_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A2: nat,B6: nat] :
( ( ord_less_eq_nat @ A2 @ B6 )
& ~ ( ord_less_eq_nat @ B6 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_511_order_Ostrict__iff__not,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [A2: set_list_nat,B6: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B6 )
& ~ ( ord_le6045566169113846134st_nat @ B6 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_512_order_Ostrict__iff__not,axiom,
( ord_le72135733267957522d_enat
= ( ^ [A2: extended_enat,B6: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A2 @ B6 )
& ~ ( ord_le2932123472753598470d_enat @ B6 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_513_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B6: nat,A2: nat] :
( ( ord_less_nat @ B6 @ A2 )
| ( A2 = B6 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_514_dual__order_Oorder__iff__strict,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [B6: set_list_nat,A2: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ B6 @ A2 )
| ( A2 = B6 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_515_dual__order_Oorder__iff__strict,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [B6: extended_enat,A2: extended_enat] :
( ( ord_le72135733267957522d_enat @ B6 @ A2 )
| ( A2 = B6 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_516_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B6: nat,A2: nat] :
( ( ord_less_eq_nat @ B6 @ A2 )
& ( A2 != B6 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_517_dual__order_Ostrict__iff__order,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [B6: set_list_nat,A2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B6 @ A2 )
& ( A2 != B6 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_518_dual__order_Ostrict__iff__order,axiom,
( ord_le72135733267957522d_enat
= ( ^ [B6: extended_enat,A2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B6 @ A2 )
& ( A2 != B6 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_519_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_520_dual__order_Ostrict__trans1,axiom,
! [B: set_list_nat,A: set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B @ A )
=> ( ( ord_le1190675801316882794st_nat @ C @ B )
=> ( ord_le1190675801316882794st_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_521_dual__order_Ostrict__trans1,axiom,
! [B: extended_enat,A: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B @ A )
=> ( ( ord_le72135733267957522d_enat @ C @ B )
=> ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_522_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_523_dual__order_Ostrict__trans2,axiom,
! [B: set_list_nat,A: set_list_nat,C: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ B @ A )
=> ( ( ord_le6045566169113846134st_nat @ C @ B )
=> ( ord_le1190675801316882794st_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_524_dual__order_Ostrict__trans2,axiom,
! [B: extended_enat,A: extended_enat,C: extended_enat] :
( ( ord_le72135733267957522d_enat @ B @ A )
=> ( ( ord_le2932123472753598470d_enat @ C @ B )
=> ( ord_le72135733267957522d_enat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_525_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B6: nat,A2: nat] :
( ( ord_less_eq_nat @ B6 @ A2 )
& ~ ( ord_less_eq_nat @ A2 @ B6 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_526_dual__order_Ostrict__iff__not,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [B6: set_list_nat,A2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B6 @ A2 )
& ~ ( ord_le6045566169113846134st_nat @ A2 @ B6 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_527_dual__order_Ostrict__iff__not,axiom,
( ord_le72135733267957522d_enat
= ( ^ [B6: extended_enat,A2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B6 @ A2 )
& ~ ( ord_le2932123472753598470d_enat @ A2 @ B6 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_528_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_529_order_Ostrict__implies__order,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A @ B )
=> ( ord_le6045566169113846134st_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_530_order_Ostrict__implies__order,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ord_le2932123472753598470d_enat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_531_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_532_dual__order_Ostrict__implies__order,axiom,
! [B: set_list_nat,A: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ B @ A )
=> ( ord_le6045566169113846134st_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_533_dual__order_Ostrict__implies__order,axiom,
! [B: extended_enat,A: extended_enat] :
( ( ord_le72135733267957522d_enat @ B @ A )
=> ( ord_le2932123472753598470d_enat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_534_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_535_order__le__less,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [X2: set_list_nat,Y2: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_536_order__le__less,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [X2: extended_enat,Y2: extended_enat] :
( ( ord_le72135733267957522d_enat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_537_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_538_order__less__le,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [X2: set_list_nat,Y2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_539_order__less__le,axiom,
( ord_le72135733267957522d_enat
= ( ^ [X2: extended_enat,Y2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_540_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_541_linorder__not__le,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ~ ( ord_le2932123472753598470d_enat @ X @ Y ) )
= ( ord_le72135733267957522d_enat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_542_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_543_linorder__not__less,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ~ ( ord_le72135733267957522d_enat @ X @ Y ) )
= ( ord_le2932123472753598470d_enat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_544_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_545_order__less__imp__le,axiom,
! [X: set_list_nat,Y: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ X @ Y )
=> ( ord_le6045566169113846134st_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_546_order__less__imp__le,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le72135733267957522d_enat @ X @ Y )
=> ( ord_le2932123472753598470d_enat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_547_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_548_order__le__neq__trans,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( A != B )
=> ( ord_le1190675801316882794st_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_549_order__le__neq__trans,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( A != B )
=> ( ord_le72135733267957522d_enat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_550_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_551_order__neq__le__trans,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( A != B )
=> ( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ord_le1190675801316882794st_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_552_order__neq__le__trans,axiom,
! [A: extended_enat,B: extended_enat] :
( ( A != B )
=> ( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ord_le72135733267957522d_enat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_553_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_554_order__le__less__trans,axiom,
! [X: set_list_nat,Y: set_list_nat,Z: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X @ Y )
=> ( ( ord_le1190675801316882794st_nat @ Y @ Z )
=> ( ord_le1190675801316882794st_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_555_order__le__less__trans,axiom,
! [X: extended_enat,Y: extended_enat,Z: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ Y )
=> ( ( ord_le72135733267957522d_enat @ Y @ Z )
=> ( ord_le72135733267957522d_enat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_556_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_557_order__less__le__trans,axiom,
! [X: set_list_nat,Y: set_list_nat,Z: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ X @ Y )
=> ( ( ord_le6045566169113846134st_nat @ Y @ Z )
=> ( ord_le1190675801316882794st_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_558_order__less__le__trans,axiom,
! [X: extended_enat,Y: extended_enat,Z: extended_enat] :
( ( ord_le72135733267957522d_enat @ X @ Y )
=> ( ( ord_le2932123472753598470d_enat @ Y @ Z )
=> ( ord_le72135733267957522d_enat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_559_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,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_560_order__le__less__subst1,axiom,
! [A: nat,F: extended_enat > nat,B: extended_enat,C: extended_enat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le72135733267957522d_enat @ B @ C )
=> ( ! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_561_order__le__less__subst1,axiom,
! [A: set_list_nat,F: nat > set_list_nat,B: nat,C: nat] :
( ( ord_le6045566169113846134st_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_le1190675801316882794st_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le1190675801316882794st_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_562_order__le__less__subst1,axiom,
! [A: set_list_nat,F: extended_enat > set_list_nat,B: extended_enat,C: extended_enat] :
( ( ord_le6045566169113846134st_nat @ A @ ( F @ B ) )
=> ( ( ord_le72135733267957522d_enat @ B @ C )
=> ( ! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ( ord_le1190675801316882794st_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le1190675801316882794st_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_563_order__le__less__subst1,axiom,
! [A: extended_enat,F: nat > extended_enat,B: nat,C: nat] :
( ( ord_le2932123472753598470d_enat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_564_order__le__less__subst1,axiom,
! [A: extended_enat,F: extended_enat > extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ ( F @ B ) )
=> ( ( ord_le72135733267957522d_enat @ B @ C )
=> ( ! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_565_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,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_566_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_list_nat,C: set_list_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le1190675801316882794st_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le1190675801316882794st_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_567_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > extended_enat,C: extended_enat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le72135733267957522d_enat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_568_order__le__less__subst2,axiom,
! [A: set_list_nat,B: set_list_nat,F: set_list_nat > nat,C: nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_list_nat,Y4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_569_order__le__less__subst2,axiom,
! [A: set_list_nat,B: set_list_nat,F: set_list_nat > set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ord_le1190675801316882794st_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_list_nat,Y4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y4 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le1190675801316882794st_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_570_order__le__less__subst2,axiom,
! [A: set_list_nat,B: set_list_nat,F: set_list_nat > extended_enat,C: extended_enat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ord_le72135733267957522d_enat @ ( F @ B ) @ C )
=> ( ! [X3: set_list_nat,Y4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_571_order__le__less__subst2,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > nat,C: nat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_572_order__le__less__subst2,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > set_list_nat,C: set_list_nat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_le1190675801316882794st_nat @ ( F @ B ) @ C )
=> ( ! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le1190675801316882794st_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_573_order__le__less__subst2,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_le72135733267957522d_enat @ ( F @ B ) @ C )
=> ( ! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_574_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,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_575_order__less__le__subst1,axiom,
! [A: set_list_nat,F: nat > set_list_nat,B: nat,C: nat] :
( ( ord_le1190675801316882794st_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le1190675801316882794st_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_576_order__less__le__subst1,axiom,
! [A: extended_enat,F: nat > extended_enat,B: nat,C: nat] :
( ( ord_le72135733267957522d_enat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_577_order__less__le__subst1,axiom,
! [A: nat,F: set_list_nat > nat,B: set_list_nat,C: set_list_nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ! [X3: set_list_nat,Y4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_578_order__less__le__subst1,axiom,
! [A: set_list_nat,F: set_list_nat > set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A @ ( F @ B ) )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ! [X3: set_list_nat,Y4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y4 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le1190675801316882794st_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_579_order__less__le__subst1,axiom,
! [A: extended_enat,F: set_list_nat > extended_enat,B: set_list_nat,C: set_list_nat] :
( ( ord_le72135733267957522d_enat @ A @ ( F @ B ) )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ! [X3: set_list_nat,Y4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_580_order__less__le__subst1,axiom,
! [A: nat,F: extended_enat > nat,B: extended_enat,C: extended_enat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_581_order__less__le__subst1,axiom,
! [A: set_list_nat,F: extended_enat > set_list_nat,B: extended_enat,C: extended_enat] :
( ( ord_le1190675801316882794st_nat @ A @ ( F @ B ) )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le1190675801316882794st_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_582_order__less__le__subst1,axiom,
! [A: extended_enat,F: extended_enat > extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ ( F @ B ) )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y4 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_583_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,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_584_order__less__le__subst2,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > nat,C: nat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_585_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_list_nat,C: set_list_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le6045566169113846134st_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_le1190675801316882794st_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le1190675801316882794st_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_586_order__less__le__subst2,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > set_list_nat,C: set_list_nat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( ord_le6045566169113846134st_nat @ ( F @ B ) @ C )
=> ( ! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ( ord_le1190675801316882794st_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le1190675801316882794st_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_587_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > extended_enat,C: extended_enat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
=> ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_588_order__less__le__subst2,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > extended_enat,C: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ ( F @ B ) @ C )
=> ( ! [X3: extended_enat,Y4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Y4 )
=> ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_589_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_590_linorder__le__less__linear,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ Y )
| ( ord_le72135733267957522d_enat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_591_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_592_order__le__imp__less__or__eq,axiom,
! [X: set_list_nat,Y: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X @ Y )
=> ( ( ord_le1190675801316882794st_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_593_order__le__imp__less__or__eq,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ Y )
=> ( ( ord_le72135733267957522d_enat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_594_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_595_bot_Oextremum,axiom,
! [A: set_list_nat] : ( ord_le6045566169113846134st_nat @ bot_bot_set_list_nat @ A ) ).
% bot.extremum
thf(fact_596_bot_Oextremum,axiom,
! [A: extended_enat] : ( ord_le2932123472753598470d_enat @ bot_bo4199563552545308370d_enat @ A ) ).
% bot.extremum
thf(fact_597_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_598_bot_Oextremum__unique,axiom,
! [A: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ bot_bot_set_list_nat )
= ( A = bot_bot_set_list_nat ) ) ).
% bot.extremum_unique
thf(fact_599_bot_Oextremum__unique,axiom,
! [A: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ bot_bo4199563552545308370d_enat )
= ( A = bot_bo4199563552545308370d_enat ) ) ).
% bot.extremum_unique
thf(fact_600_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_601_bot_Oextremum__uniqueI,axiom,
! [A: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ bot_bot_set_list_nat )
=> ( A = bot_bot_set_list_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_602_bot_Oextremum__uniqueI,axiom,
! [A: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ bot_bo4199563552545308370d_enat )
=> ( A = bot_bo4199563552545308370d_enat ) ) ).
% bot.extremum_uniqueI
thf(fact_603_bot_Oextremum__strict,axiom,
! [A: set_list_nat] :
~ ( ord_le1190675801316882794st_nat @ A @ bot_bot_set_list_nat ) ).
% bot.extremum_strict
thf(fact_604_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_605_bot_Oextremum__strict,axiom,
! [A: extended_enat] :
~ ( ord_le72135733267957522d_enat @ A @ bot_bo4199563552545308370d_enat ) ).
% bot.extremum_strict
thf(fact_606_bot_Onot__eq__extremum,axiom,
! [A: set_list_nat] :
( ( A != bot_bot_set_list_nat )
= ( ord_le1190675801316882794st_nat @ bot_bot_set_list_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_607_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_608_bot_Onot__eq__extremum,axiom,
! [A: extended_enat] :
( ( A != bot_bo4199563552545308370d_enat )
= ( ord_le72135733267957522d_enat @ bot_bo4199563552545308370d_enat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_609_sum__list__incr,axiom,
! [I: nat,X: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ X ) )
=> ( ( groups4561878855575611511st_nat @ ( list_incr @ I @ X ) )
= ( suc @ ( groups4561878855575611511st_nat @ X ) ) ) ) ).
% sum_list_incr
thf(fact_610_finite__length__sum__set,axiom,
! [R2: nat,N: nat] : ( finite8100373058378681591st_nat @ ( length_sum_set @ R2 @ N ) ) ).
% finite_length_sum_set
thf(fact_611_id__take__nth__drop,axiom,
! [I: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( Xs
= ( append_nat @ ( take_nat @ I @ Xs ) @ ( cons_nat @ ( nth_nat @ Xs @ I ) @ ( drop_nat @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_612_Khovanskii__axioms_Ointro,axiom,
! [A3: set_nat,G: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ G )
=> ( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( khovan4585363760863428690ms_nat @ G @ A3 ) ) ) ) ).
% Khovanskii_axioms.intro
thf(fact_613_Khovanskii__axioms_Ointro,axiom,
! [A3: set_list_nat,G: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ G )
=> ( ( finite8100373058378681591st_nat @ A3 )
=> ( ( A3 != bot_bot_set_list_nat )
=> ( khovan1553326461689229922st_nat @ G @ A3 ) ) ) ) ).
% Khovanskii_axioms.intro
thf(fact_614_Khovanskii__axioms__def,axiom,
( khovan4585363760863428690ms_nat
= ( ^ [G2: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ G2 )
& ( finite_finite_nat @ A4 )
& ( A4 != bot_bot_set_nat ) ) ) ) ).
% Khovanskii_axioms_def
thf(fact_615_Khovanskii__axioms__def,axiom,
( khovan1553326461689229922st_nat
= ( ^ [G2: set_list_nat,A4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A4 @ G2 )
& ( finite8100373058378681591st_nat @ A4 )
& ( A4 != bot_bot_set_list_nat ) ) ) ) ).
% Khovanskii_axioms_def
thf(fact_616_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_617_arg__min__if__finite_I2_J,axiom,
! [S2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ~ ? [X5: nat] :
( ( member_nat @ X5 @ S2 )
& ( ord_less_nat @ ( F @ X5 ) @ ( F @ ( lattic7446932960582359483at_nat @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_618_arg__min__if__finite_I2_J,axiom,
! [S2: set_list_nat,F: list_nat > nat] :
( ( finite8100373058378681591st_nat @ S2 )
=> ( ( S2 != bot_bot_set_list_nat )
=> ~ ? [X5: list_nat] :
( ( member_list_nat @ X5 @ S2 )
& ( ord_less_nat @ ( F @ X5 ) @ ( F @ ( lattic5785867957632790475at_nat @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_619_arg__min__if__finite_I2_J,axiom,
! [S2: set_nat,F: nat > extended_enat] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ~ ? [X5: nat] :
( ( member_nat @ X5 @ S2 )
& ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ ( lattic8926238025367240251d_enat @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_620_arg__min__if__finite_I2_J,axiom,
! [S2: set_list_nat,F: list_nat > extended_enat] :
( ( finite8100373058378681591st_nat @ S2 )
=> ( ( S2 != bot_bot_set_list_nat )
=> ~ ? [X5: list_nat] :
( ( member_list_nat @ X5 @ S2 )
& ( ord_le72135733267957522d_enat @ ( F @ X5 ) @ ( F @ ( lattic5912464335902825515d_enat @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_621_list_Oinject,axiom,
! [X21: nat,X222: list_nat,Y21: nat,Y222: list_nat] :
( ( ( cons_nat @ X21 @ X222 )
= ( cons_nat @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_622_list__incr__Cons,axiom,
! [I: nat,K: nat,Ks: list_nat] :
( ( list_incr @ ( suc @ I ) @ ( cons_nat @ K @ Ks ) )
= ( cons_nat @ K @ ( list_incr @ I @ Ks ) ) ) ).
% list_incr_Cons
thf(fact_623_length__list__incr,axiom,
! [I: nat,X: list_nat] :
( ( size_size_list_nat @ ( list_incr @ I @ X ) )
= ( size_size_list_nat @ X ) ) ).
% length_list_incr
thf(fact_624_list__incr__Nil,axiom,
! [I: nat] :
( ( list_incr @ I @ nil_nat )
= nil_nat ) ).
% list_incr_Nil
thf(fact_625_append1__eq__conv,axiom,
! [Xs: list_nat,X: nat,Ys: list_nat,Y: nat] :
( ( ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) )
= ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_626_nth__Cons__Suc,axiom,
! [X: nat,Xs: list_nat,N: nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ ( suc @ N ) )
= ( nth_nat @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_627_take__Suc__Cons,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( take_nat @ ( suc @ N ) @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ ( take_nat @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_628_drop__Suc__Cons,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( drop_nat @ ( suc @ N ) @ ( cons_nat @ X @ Xs ) )
= ( drop_nat @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_629_nth__append__length,axiom,
! [Xs: list_nat,X: nat,Ys: list_nat] :
( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ Ys ) ) @ ( size_size_list_nat @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_630_transpose_Ocases,axiom,
! [X: list_list_nat] :
( ( X != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X3: nat,Xs2: list_nat,Xss: list_list_nat] :
( X
!= ( cons_list_nat @ ( cons_nat @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_631_not__Cons__self2,axiom,
! [X: nat,Xs: list_nat] :
( ( cons_nat @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_632_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_633_list__induct2_H,axiom,
! [P: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X3 @ Xs2 ) @ nil_nat )
=> ( ! [Y4: nat,Ys4: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y4 @ Ys4 ) )
=> ( ! [X3: nat,Xs2: list_nat,Y4: nat,Ys4: list_nat] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_634_neq__Nil__conv,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
= ( ? [Y2: nat,Ys2: list_nat] :
( Xs
= ( cons_nat @ Y2 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_635_remdups__adj_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ( ! [X3: nat] :
( X
!= ( cons_nat @ X3 @ nil_nat ) )
=> ~ ! [X3: nat,Y4: nat,Xs2: list_nat] :
( X
!= ( cons_nat @ X3 @ ( cons_nat @ Y4 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_636_min__list_Ocases,axiom,
! [X: list_nat] :
( ! [X3: nat,Xs2: list_nat] :
( X
!= ( cons_nat @ X3 @ Xs2 ) )
=> ( X = nil_nat ) ) ).
% min_list.cases
thf(fact_637_list_Oexhaust,axiom,
! [Y: list_nat] :
( ( Y != nil_nat )
=> ~ ! [X212: nat,X223: list_nat] :
( Y
!= ( cons_nat @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_638_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X222: list_nat] :
( ( List
= ( cons_nat @ X21 @ X222 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_639_list_Odistinct_I1_J,axiom,
! [X21: nat,X222: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_640_Nil__less__Cons,axiom,
! [A: nat,X: list_nat] : ( ord_less_list_nat @ nil_nat @ ( cons_nat @ A @ X ) ) ).
% Nil_less_Cons
thf(fact_641_Cons__eq__appendI,axiom,
! [X: nat,Xs1: list_nat,Ys: list_nat,Xs: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_nat @ Xs1 @ Zs ) )
=> ( ( cons_nat @ X @ Xs )
= ( append_nat @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_642_append__Cons,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat] :
( ( append_nat @ ( cons_nat @ X @ Xs ) @ Ys )
= ( cons_nat @ X @ ( append_nat @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_643_list__encode_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ~ ! [X3: nat,Xs2: list_nat] :
( X
!= ( cons_nat @ X3 @ Xs2 ) ) ) ).
% list_encode.cases
thf(fact_644_length__Suc__conv,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y2: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ Y2 @ Ys2 ) )
& ( ( size_size_list_nat @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_645_Suc__length__conv,axiom,
! [N: nat,Xs: list_nat] :
( ( ( suc @ N )
= ( size_size_list_nat @ Xs ) )
= ( ? [Y2: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ Y2 @ Ys2 ) )
& ( ( size_size_list_nat @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_646_impossible__Cons,axiom,
! [Xs: list_nat,Ys: list_nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) )
=> ( Xs
!= ( cons_nat @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_647_list__induct4,axiom,
! [Xs: list_nat,Ys: 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 @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( 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,Y4: nat,Ys4: list_nat,Z3: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys4 ) @ ( cons_nat @ Z3 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_648_list__induct3,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat,P: list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y4: nat,Ys4: list_nat,Z3: nat,Zs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys4 ) @ ( cons_nat @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_649_list__induct2,axiom,
! [Xs: list_nat,Ys: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y4: nat,Ys4: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_650_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_651_append__eq__Cons__conv,axiom,
! [Ys: list_nat,Zs: list_nat,X: nat,Xs: list_nat] :
( ( ( append_nat @ Ys @ Zs )
= ( cons_nat @ X @ Xs ) )
= ( ( ( Ys = nil_nat )
& ( Zs
= ( cons_nat @ X @ Xs ) ) )
| ? [Ys5: list_nat] :
( ( Ys
= ( cons_nat @ X @ Ys5 ) )
& ( ( append_nat @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_652_Cons__eq__append__conv,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( append_nat @ Ys @ Zs ) )
= ( ( ( Ys = nil_nat )
& ( ( cons_nat @ X @ Xs )
= Zs ) )
| ? [Ys5: list_nat] :
( ( ( cons_nat @ X @ Ys5 )
= Ys )
& ( Xs
= ( append_nat @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_653_rev__exhaust,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ~ ! [Ys4: list_nat,Y4: nat] :
( Xs
!= ( append_nat @ Ys4 @ ( cons_nat @ Y4 @ nil_nat ) ) ) ) ).
% rev_exhaust
thf(fact_654_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_655_nth__via__drop,axiom,
! [N: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ( drop_nat @ N @ Xs )
= ( cons_nat @ Y @ Ys ) )
=> ( ( nth_nat @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_656_minus__Cons,axiom,
! [Y: list_nat,Ys: list_list_nat,X: list_nat,Xs: list_list_nat] :
( ( minus_3911745200923244873st_nat @ ( cons_list_nat @ Y @ Ys ) @ ( cons_list_nat @ X @ Xs ) )
= ( cons_list_nat @ ( minus_minus_list_nat @ Y @ X ) @ ( minus_3911745200923244873st_nat @ Ys @ Xs ) ) ) ).
% minus_Cons
thf(fact_657_minus__Cons,axiom,
! [Y: extended_enat,Ys: list_Extended_enat,X: extended_enat,Xs: list_Extended_enat] :
( ( minus_388785356899630291d_enat @ ( cons_Extended_enat @ Y @ Ys ) @ ( cons_Extended_enat @ X @ Xs ) )
= ( cons_Extended_enat @ ( minus_3235023915231533773d_enat @ Y @ X ) @ ( minus_388785356899630291d_enat @ Ys @ Xs ) ) ) ).
% minus_Cons
thf(fact_658_minus__Cons,axiom,
! [Y: nat,Ys: list_nat,X: nat,Xs: list_nat] :
( ( minus_minus_list_nat @ ( cons_nat @ Y @ Ys ) @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ ( minus_minus_nat @ Y @ X ) @ ( minus_minus_list_nat @ Ys @ Xs ) ) ) ).
% minus_Cons
thf(fact_659_Cons__less__Cons,axiom,
! [A: nat,X: list_nat,B: nat,Y: list_nat] :
( ( ord_less_list_nat @ ( cons_nat @ A @ X ) @ ( cons_nat @ B @ Y ) )
= ( ( ord_less_nat @ ( size_size_list_nat @ X ) @ ( size_size_list_nat @ Y ) )
| ( ( ( size_size_list_nat @ X )
= ( size_size_list_nat @ Y ) )
& ( ( ord_less_nat @ A @ B )
| ( ( A = B )
& ( ord_less_list_nat @ X @ Y ) ) ) ) ) ) ).
% Cons_less_Cons
thf(fact_660_Cons__less__Cons,axiom,
! [A: extended_enat,X: list_Extended_enat,B: extended_enat,Y: list_Extended_enat] :
( ( ord_le4445849522347344536d_enat @ ( cons_Extended_enat @ A @ X ) @ ( cons_Extended_enat @ B @ Y ) )
= ( ( ord_less_nat @ ( size_s3941691890525107288d_enat @ X ) @ ( size_s3941691890525107288d_enat @ Y ) )
| ( ( ( size_s3941691890525107288d_enat @ X )
= ( size_s3941691890525107288d_enat @ Y ) )
& ( ( ord_le72135733267957522d_enat @ A @ B )
| ( ( A = B )
& ( ord_le4445849522347344536d_enat @ X @ Y ) ) ) ) ) ) ).
% Cons_less_Cons
thf(fact_661_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
= ( ? [X2: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ X2 @ Ys2 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_662_same__length__different,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs != Ys )
=> ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ? [Pre: list_nat,X3: nat,Xs4: list_nat,Y4: nat,Ys6: list_nat] :
( ( X3 != Y4 )
& ( Xs
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X3 @ nil_nat ) @ Xs4 ) ) )
& ( Ys
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y4 @ nil_nat ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_663_length__Suc__conv__rev,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y2: nat,Ys2: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ Y2 @ nil_nat ) ) )
& ( ( size_size_list_nat @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_664_Cons__le__Cons,axiom,
! [A: nat,X: list_nat,B: nat,Y: list_nat] :
( ( ord_less_eq_list_nat @ ( cons_nat @ A @ X ) @ ( cons_nat @ B @ Y ) )
= ( ( ord_less_nat @ ( size_size_list_nat @ X ) @ ( size_size_list_nat @ Y ) )
| ( ( ( size_size_list_nat @ X )
= ( size_size_list_nat @ Y ) )
& ( ( ord_less_nat @ A @ B )
| ( ( A = B )
& ( ord_less_eq_list_nat @ X @ Y ) ) ) ) ) ) ).
% Cons_le_Cons
thf(fact_665_Cons__le__Cons,axiom,
! [A: extended_enat,X: list_Extended_enat,B: extended_enat,Y: list_Extended_enat] :
( ( ord_le769749158434378124d_enat @ ( cons_Extended_enat @ A @ X ) @ ( cons_Extended_enat @ B @ Y ) )
= ( ( ord_less_nat @ ( size_s3941691890525107288d_enat @ X ) @ ( size_s3941691890525107288d_enat @ Y ) )
| ( ( ( size_s3941691890525107288d_enat @ X )
= ( size_s3941691890525107288d_enat @ Y ) )
& ( ( ord_le72135733267957522d_enat @ A @ B )
| ( ( A = B )
& ( ord_le769749158434378124d_enat @ X @ Y ) ) ) ) ) ) ).
% Cons_le_Cons
thf(fact_666_Cons__nth__drop__Suc,axiom,
! [I: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( cons_nat @ ( nth_nat @ Xs @ I ) @ ( drop_nat @ ( suc @ I ) @ Xs ) )
= ( drop_nat @ I @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_667_arg__min__least,axiom,
! [S2: set_nat,Y: nat,F: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S2 )
=> ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_668_arg__min__least,axiom,
! [S2: set_list_nat,Y: list_nat,F: list_nat > nat] :
( ( finite8100373058378681591st_nat @ S2 )
=> ( ( S2 != bot_bot_set_list_nat )
=> ( ( member_list_nat @ Y @ S2 )
=> ( ord_less_eq_nat @ ( F @ ( lattic5785867957632790475at_nat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_669_arg__min__least,axiom,
! [S2: set_nat,Y: nat,F: nat > extended_enat] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S2 )
=> ( ord_le2932123472753598470d_enat @ ( F @ ( lattic8926238025367240251d_enat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_670_arg__min__least,axiom,
! [S2: set_list_nat,Y: list_nat,F: list_nat > extended_enat] :
( ( finite8100373058378681591st_nat @ S2 )
=> ( ( S2 != bot_bot_set_list_nat )
=> ( ( member_list_nat @ Y @ S2 )
=> ( ord_le2932123472753598470d_enat @ ( F @ ( lattic5912464335902825515d_enat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_671_length__append__singleton,axiom,
! [Xs: list_nat,X: nat] :
( ( size_size_list_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_append_singleton
thf(fact_672_length__Cons,axiom,
! [X: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X @ Xs ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_Cons
thf(fact_673_list__incr__def,axiom,
( list_incr
= ( ^ [I3: nat,X2: list_nat] : ( list_update_nat @ X2 @ I3 @ ( suc @ ( nth_nat @ X2 @ I3 ) ) ) ) ) ).
% list_incr_def
thf(fact_674_take__hd__drop,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( append_nat @ ( take_nat @ N @ Xs ) @ ( cons_nat @ ( hd_nat @ ( drop_nat @ N @ Xs ) ) @ nil_nat ) )
= ( take_nat @ ( suc @ N ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_675_upd__conv__take__nth__drop,axiom,
! [I: nat,Xs: list_nat,A: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ Xs @ I @ A )
= ( append_nat @ ( take_nat @ I @ Xs ) @ ( cons_nat @ A @ ( drop_nat @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_676_list__update__overwrite,axiom,
! [Xs: list_nat,I: nat,X: nat,Y: nat] :
( ( list_update_nat @ ( list_update_nat @ Xs @ I @ X ) @ I @ Y )
= ( list_update_nat @ Xs @ I @ Y ) ) ).
% list_update_overwrite
thf(fact_677_list__update__nonempty,axiom,
! [Xs: list_nat,K: nat,X: nat] :
( ( ( list_update_nat @ Xs @ K @ X )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% list_update_nonempty
thf(fact_678_length__list__update,axiom,
! [Xs: list_nat,I: nat,X: nat] :
( ( size_size_list_nat @ ( list_update_nat @ Xs @ I @ X ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_list_update
thf(fact_679_nth__list__update__neq,axiom,
! [I: nat,J: nat,Xs: list_nat,X: nat] :
( ( I != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
= ( nth_nat @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_680_list__update__id,axiom,
! [Xs: list_nat,I: nat] :
( ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ I ) )
= Xs ) ).
% list_update_id
thf(fact_681_list__update__beyond,axiom,
! [Xs: list_nat,I: nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I )
=> ( ( list_update_nat @ Xs @ I @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_682_hd__append2,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs != nil_nat )
=> ( ( hd_nat @ ( append_nat @ Xs @ Ys ) )
= ( hd_nat @ Xs ) ) ) ).
% hd_append2
thf(fact_683_take__update__cancel,axiom,
! [N: nat,M: nat,Xs: list_nat,Y: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( take_nat @ N @ ( list_update_nat @ Xs @ M @ Y ) )
= ( take_nat @ N @ Xs ) ) ) ).
% take_update_cancel
thf(fact_684_drop__update__cancel,axiom,
! [N: nat,M: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( drop_nat @ M @ ( list_update_nat @ Xs @ N @ X ) )
= ( drop_nat @ M @ Xs ) ) ) ).
% drop_update_cancel
thf(fact_685_list__update__length,axiom,
! [Xs: list_nat,X: nat,Ys: list_nat,Y: nat] :
( ( list_update_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ Ys ) ) @ ( size_size_list_nat @ Xs ) @ Y )
= ( append_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_686_nth__list__update__eq,axiom,
! [I: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ I )
= X ) ) ).
% nth_list_update_eq
thf(fact_687_list__update__swap,axiom,
! [I: nat,I4: nat,Xs: list_nat,X: nat,X9: nat] :
( ( I != I4 )
=> ( ( list_update_nat @ ( list_update_nat @ Xs @ I @ X ) @ I4 @ X9 )
= ( list_update_nat @ ( list_update_nat @ Xs @ I4 @ X9 ) @ I @ X ) ) ) ).
% list_update_swap
thf(fact_688_list__update__code_I1_J,axiom,
! [I: nat,Y: nat] :
( ( list_update_nat @ nil_nat @ I @ Y )
= nil_nat ) ).
% list_update_code(1)
thf(fact_689_list__update_Osimps_I1_J,axiom,
! [I: nat,V2: nat] :
( ( list_update_nat @ nil_nat @ I @ V2 )
= nil_nat ) ).
% list_update.simps(1)
thf(fact_690_list_Osel_I1_J,axiom,
! [X21: nat,X222: list_nat] :
( ( hd_nat @ ( cons_nat @ X21 @ X222 ) )
= X21 ) ).
% list.sel(1)
thf(fact_691_take__update__swap,axiom,
! [M: nat,Xs: list_nat,N: nat,X: nat] :
( ( take_nat @ M @ ( list_update_nat @ Xs @ N @ X ) )
= ( list_update_nat @ ( take_nat @ M @ Xs ) @ N @ X ) ) ).
% take_update_swap
thf(fact_692_list__update__code_I3_J,axiom,
! [X: nat,Xs: list_nat,I: nat,Y: nat] :
( ( list_update_nat @ ( cons_nat @ X @ Xs ) @ ( suc @ I ) @ Y )
= ( cons_nat @ X @ ( list_update_nat @ Xs @ I @ Y ) ) ) ).
% list_update_code(3)
thf(fact_693_hd__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( Xs = nil_nat )
=> ( ( hd_nat @ ( append_nat @ Xs @ Ys ) )
= ( hd_nat @ Ys ) ) )
& ( ( Xs != nil_nat )
=> ( ( hd_nat @ ( append_nat @ Xs @ Ys ) )
= ( hd_nat @ Xs ) ) ) ) ).
% hd_append
thf(fact_694_longest__common__prefix,axiom,
! [Xs: list_nat,Ys: list_nat] :
? [Ps: list_nat,Xs4: list_nat,Ys6: list_nat] :
( ( Xs
= ( append_nat @ Ps @ Xs4 ) )
& ( Ys
= ( append_nat @ Ps @ Ys6 ) )
& ( ( Xs4 = nil_nat )
| ( Ys6 = nil_nat )
| ( ( hd_nat @ Xs4 )
!= ( hd_nat @ Ys6 ) ) ) ) ).
% longest_common_prefix
thf(fact_695_list__update__append1,axiom,
! [I: nat,Xs: list_nat,Ys: list_nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ I @ X )
= ( append_nat @ ( list_update_nat @ Xs @ I @ X ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_696_nth__list__update,axiom,
! [I: nat,Xs: list_nat,J: nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
= X ) )
& ( ( I != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
= ( nth_nat @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_697_list__update__same__conv,axiom,
! [I: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ( list_update_nat @ Xs @ I @ X )
= Xs )
= ( ( nth_nat @ Xs @ I )
= X ) ) ) ).
% list_update_same_conv
thf(fact_698_drop__update__swap,axiom,
! [M: nat,N: nat,Xs: list_nat,X: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( drop_nat @ M @ ( list_update_nat @ Xs @ N @ X ) )
= ( list_update_nat @ ( drop_nat @ M @ Xs ) @ ( minus_minus_nat @ N @ M ) @ X ) ) ) ).
% drop_update_swap
thf(fact_699_list__update__append,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat,X: nat] :
( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ N @ X )
= ( append_nat @ ( list_update_nat @ Xs @ N @ X ) @ Ys ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ N @ X )
= ( append_nat @ Xs @ ( list_update_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ X ) ) ) ) ) ).
% list_update_append
thf(fact_700_hd__drop__conv__nth,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( hd_nat @ ( drop_nat @ N @ Xs ) )
= ( nth_nat @ Xs @ N ) ) ) ).
% hd_drop_conv_nth
thf(fact_701_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_702_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_703_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_704_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_705_WFP,axiom,
wfP_list_nat @ pointwise_less ).
% WFP
thf(fact_706_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I5: nat] :
( ( ord_less_nat @ K2 @ I5 )
=> ( P @ I5 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_707_finite__transitivity__chain,axiom,
! [A3: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ A3 )
=> ( ! [X3: nat] :
~ ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y4: nat,Z3: nat] :
( ( R @ X3 @ Y4 )
=> ( ( R @ Y4 @ Z3 )
=> ( R @ X3 @ Z3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ? [Y3: nat] :
( ( member_nat @ Y3 @ A3 )
& ( R @ X3 @ Y3 ) ) )
=> ( A3 = bot_bot_set_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_708_finite__transitivity__chain,axiom,
! [A3: set_list_nat,R: list_nat > list_nat > $o] :
( ( finite8100373058378681591st_nat @ A3 )
=> ( ! [X3: list_nat] :
~ ( R @ X3 @ X3 )
=> ( ! [X3: list_nat,Y4: list_nat,Z3: list_nat] :
( ( R @ X3 @ Y4 )
=> ( ( R @ Y4 @ Z3 )
=> ( R @ X3 @ Z3 ) ) )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A3 )
=> ? [Y3: list_nat] :
( ( member_list_nat @ Y3 @ A3 )
& ( R @ X3 @ Y3 ) ) )
=> ( A3 = bot_bot_set_list_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_709_infinite__nat__iff__unbounded__le,axiom,
! [S2: set_nat] :
( ( ~ ( finite_finite_nat @ S2 ) )
= ( ! [M5: nat] :
? [N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
& ( member_nat @ N4 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_710_infinite__nat__iff__unbounded,axiom,
! [S2: set_nat] :
( ( ~ ( finite_finite_nat @ S2 ) )
= ( ! [M5: nat] :
? [N4: nat] :
( ( ord_less_nat @ M5 @ N4 )
& ( member_nat @ N4 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_711_unbounded__k__infinite,axiom,
! [K: nat,S2: set_nat] :
( ! [M7: nat] :
( ( ord_less_nat @ K @ M7 )
=> ? [N5: nat] :
( ( ord_less_nat @ M7 @ N5 )
& ( member_nat @ N5 @ S2 ) ) )
=> ~ ( finite_finite_nat @ S2 ) ) ).
% unbounded_k_infinite
thf(fact_712_wfP__if__convertible__to__nat,axiom,
! [R: list_nat > list_nat > $o,F: list_nat > nat] :
( ! [X3: list_nat,Y4: list_nat] :
( ( R @ X3 @ Y4 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( wfP_list_nat @ R ) ) ).
% wfP_if_convertible_to_nat
thf(fact_713_wfP__less,axiom,
wfP_list_nat @ ord_less_list_nat ).
% wfP_less
thf(fact_714_wfP__less,axiom,
wfP_nat @ ord_less_nat ).
% wfP_less
thf(fact_715_wfP__less,axiom,
wfP_Extended_enat @ ord_le72135733267957522d_enat ).
% wfP_less
thf(fact_716_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_717_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_718_subset__emptyI,axiom,
! [A3: set_nat] :
( ! [X3: nat] :
~ ( member_nat @ X3 @ A3 )
=> ( ord_less_eq_set_nat @ A3 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_719_subset__emptyI,axiom,
! [A3: set_list_nat] :
( ! [X3: list_nat] :
~ ( member_list_nat @ X3 @ A3 )
=> ( ord_le6045566169113846134st_nat @ A3 @ bot_bot_set_list_nat ) ) ).
% subset_emptyI
thf(fact_720_wfP__subset,axiom,
! [R2: list_nat > list_nat > $o,P4: list_nat > list_nat > $o] :
( ( wfP_list_nat @ R2 )
=> ( ( ord_le6558929396352911974_nat_o @ P4 @ R2 )
=> ( wfP_list_nat @ P4 ) ) ) ).
% wfP_subset
thf(fact_721_bounded__nat__set__is__finite,axiom,
! [N6: set_nat,N: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ N6 )
=> ( ord_less_nat @ X3 @ N ) )
=> ( finite_finite_nat @ N6 ) ) ).
% bounded_nat_set_is_finite
thf(fact_722_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N7: set_nat] :
? [M5: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N7 )
=> ( ord_less_nat @ X2 @ M5 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_723_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N7: set_nat] :
? [M5: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N7 )
=> ( ord_less_eq_nat @ X2 @ M5 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_724_finite__indexed__bound,axiom,
! [A3: set_list_nat,P: list_nat > nat > $o] :
( ( finite8100373058378681591st_nat @ A3 )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A3 )
=> ? [X_12: nat] : ( P @ X3 @ X_12 ) )
=> ? [M7: nat] :
! [X5: list_nat] :
( ( member_list_nat @ X5 @ A3 )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ M7 )
& ( P @ X5 @ K2 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_725_finite__indexed__bound,axiom,
! [A3: set_nat,P: nat > nat > $o] :
( ( finite_finite_nat @ A3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ? [X_12: nat] : ( P @ X3 @ X_12 ) )
=> ? [M7: nat] :
! [X5: nat] :
( ( member_nat @ X5 @ A3 )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ M7 )
& ( P @ X5 @ K2 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_726_finite__indexed__bound,axiom,
! [A3: set_list_nat,P: list_nat > extended_enat > $o] :
( ( finite8100373058378681591st_nat @ A3 )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A3 )
=> ? [X_12: extended_enat] : ( P @ X3 @ X_12 ) )
=> ? [M7: extended_enat] :
! [X5: list_nat] :
( ( member_list_nat @ X5 @ A3 )
=> ? [K2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ K2 @ M7 )
& ( P @ X5 @ K2 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_727_finite__indexed__bound,axiom,
! [A3: set_nat,P: nat > extended_enat > $o] :
( ( finite_finite_nat @ A3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ? [X_12: extended_enat] : ( P @ X3 @ X_12 ) )
=> ? [M7: extended_enat] :
! [X5: nat] :
( ( member_nat @ X5 @ A3 )
=> ? [K2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ K2 @ M7 )
& ( P @ X5 @ K2 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_728_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M3: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M3 ) )
=> ~ ! [M7: nat] :
( ( P @ M7 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M7 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_729_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_730_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% minf(8)
thf(fact_731_minf_I8_J,axiom,
! [T: extended_enat] :
? [Z3: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Z3 )
=> ~ ( ord_le2932123472753598470d_enat @ T @ X5 ) ) ).
% minf(8)
thf(fact_732_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ord_less_eq_nat @ X5 @ T ) ) ).
% minf(6)
thf(fact_733_minf_I6_J,axiom,
! [T: extended_enat] :
? [Z3: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Z3 )
=> ( ord_le2932123472753598470d_enat @ X5 @ T ) ) ).
% minf(6)
thf(fact_734_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ord_less_eq_nat @ T @ X5 ) ) ).
% pinf(8)
thf(fact_735_pinf_I8_J,axiom,
! [T: extended_enat] :
? [Z3: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z3 @ X5 )
=> ( ord_le2932123472753598470d_enat @ T @ X5 ) ) ).
% pinf(8)
thf(fact_736_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% pinf(6)
thf(fact_737_pinf_I6_J,axiom,
! [T: extended_enat] :
? [Z3: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z3 @ X5 )
=> ~ ( ord_le2932123472753598470d_enat @ X5 @ T ) ) ).
% pinf(6)
thf(fact_738_pinf_I1_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P5 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_739_pinf_I1_J,axiom,
! [P: extended_enat > $o,P5: extended_enat > $o,Q: extended_enat > $o,Q2: extended_enat > $o] :
( ? [Z4: extended_enat] :
! [X3: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: extended_enat] :
! [X3: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z3 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P5 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_740_pinf_I2_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P5 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_741_pinf_I2_J,axiom,
! [P: extended_enat > $o,P5: extended_enat > $o,Q: extended_enat > $o,Q2: extended_enat > $o] :
( ? [Z4: extended_enat] :
! [X3: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: extended_enat] :
! [X3: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z3 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P5 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_742_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_743_pinf_I3_J,axiom,
! [T: extended_enat] :
? [Z3: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_744_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_745_pinf_I4_J,axiom,
! [T: extended_enat] :
? [Z3: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_746_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ~ ( ord_less_nat @ X5 @ T ) ) ).
% pinf(5)
thf(fact_747_pinf_I5_J,axiom,
! [T: extended_enat] :
? [Z3: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z3 @ X5 )
=> ~ ( ord_le72135733267957522d_enat @ X5 @ T ) ) ).
% pinf(5)
thf(fact_748_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ord_less_nat @ T @ X5 ) ) ).
% pinf(7)
thf(fact_749_pinf_I7_J,axiom,
! [T: extended_enat] :
? [Z3: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z3 @ X5 )
=> ( ord_le72135733267957522d_enat @ T @ X5 ) ) ).
% pinf(7)
thf(fact_750_minf_I1_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P5 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_751_minf_I1_J,axiom,
! [P: extended_enat > $o,P5: extended_enat > $o,Q: extended_enat > $o,Q2: extended_enat > $o] :
( ? [Z4: extended_enat] :
! [X3: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: extended_enat] :
! [X3: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Z3 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P5 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_752_minf_I2_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P5 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_753_minf_I2_J,axiom,
! [P: extended_enat > $o,P5: extended_enat > $o,Q: extended_enat > $o,Q2: extended_enat > $o] :
( ? [Z4: extended_enat] :
! [X3: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: extended_enat] :
! [X3: extended_enat] :
( ( ord_le72135733267957522d_enat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Z3 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P5 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_754_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_755_minf_I3_J,axiom,
! [T: extended_enat] :
? [Z3: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_756_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_757_minf_I4_J,axiom,
! [T: extended_enat] :
? [Z3: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_758_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ord_less_nat @ X5 @ T ) ) ).
% minf(5)
thf(fact_759_minf_I5_J,axiom,
! [T: extended_enat] :
? [Z3: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Z3 )
=> ( ord_le72135733267957522d_enat @ X5 @ T ) ) ).
% minf(5)
thf(fact_760_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ~ ( ord_less_nat @ T @ X5 ) ) ).
% minf(7)
thf(fact_761_minf_I7_J,axiom,
! [T: extended_enat] :
? [Z3: extended_enat] :
! [X5: extended_enat] :
( ( ord_le72135733267957522d_enat @ X5 @ Z3 )
=> ~ ( ord_le72135733267957522d_enat @ T @ X5 ) ) ).
% minf(7)
thf(fact_762_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 )
& ! [X5: nat] :
( ( ( ord_less_eq_nat @ A @ X5 )
& ( ord_less_nat @ X5 @ C3 ) )
=> ( P @ X5 ) )
& ! [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_763_complete__interval,axiom,
! [A: extended_enat,B: extended_enat,P: extended_enat > $o] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ C3 )
& ( ord_le2932123472753598470d_enat @ C3 @ B )
& ! [X5: extended_enat] :
( ( ( ord_le2932123472753598470d_enat @ A @ X5 )
& ( ord_le72135733267957522d_enat @ X5 @ C3 ) )
=> ( P @ X5 ) )
& ! [D2: extended_enat] :
( ! [X3: extended_enat] :
( ( ( ord_le2932123472753598470d_enat @ A @ X3 )
& ( ord_le72135733267957522d_enat @ X3 @ D2 ) )
=> ( P @ X3 ) )
=> ( ord_le2932123472753598470d_enat @ D2 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_764_verit__comp__simplify1_I3_J,axiom,
! [B7: nat,A7: nat] :
( ( ~ ( ord_less_eq_nat @ B7 @ A7 ) )
= ( ord_less_nat @ A7 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_765_verit__comp__simplify1_I3_J,axiom,
! [B7: extended_enat,A7: extended_enat] :
( ( ~ ( ord_le2932123472753598470d_enat @ B7 @ A7 ) )
= ( ord_le72135733267957522d_enat @ A7 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_766_sum__list__update,axiom,
! [K: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ K @ ( size_size_list_nat @ Xs ) )
=> ( ( groups4561878855575611511st_nat @ ( list_update_nat @ Xs @ K @ X ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( groups4561878855575611511st_nat @ Xs ) @ X ) @ ( nth_nat @ Xs @ K ) ) ) ) ).
% sum_list_update
thf(fact_767_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_768_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_769_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_770_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_771_plus__Nil,axiom,
! [Xs: list_nat] :
( ( plus_plus_list_nat @ nil_nat @ Xs )
= nil_nat ) ).
% plus_Nil
thf(fact_772_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_773_drop__drop,axiom,
! [N: nat,M: nat,Xs: list_nat] :
( ( drop_nat @ N @ ( drop_nat @ M @ Xs ) )
= ( drop_nat @ ( plus_plus_nat @ N @ M ) @ Xs ) ) ).
% drop_drop
thf(fact_774_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_775_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_776_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_777_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_778_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_779_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_780_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_781_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_782_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_783_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_784_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_785_length__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( size_size_list_nat @ ( append_nat @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_append
thf(fact_786_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_787_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_788_sum__list_OCons,axiom,
! [X: extended_enat,Xs: list_Extended_enat] :
( ( groups5145338220374282879d_enat @ ( cons_Extended_enat @ X @ Xs ) )
= ( plus_p3455044024723400733d_enat @ X @ ( groups5145338220374282879d_enat @ Xs ) ) ) ).
% sum_list.Cons
thf(fact_789_sum__list_OCons,axiom,
! [X: nat,Xs: list_nat] :
( ( groups4561878855575611511st_nat @ ( cons_nat @ X @ Xs ) )
= ( plus_plus_nat @ X @ ( groups4561878855575611511st_nat @ Xs ) ) ) ).
% sum_list.Cons
thf(fact_790_sum__list__append,axiom,
! [Xs: list_Extended_enat,Ys: list_Extended_enat] :
( ( groups5145338220374282879d_enat @ ( append_Extended_enat @ Xs @ Ys ) )
= ( plus_p3455044024723400733d_enat @ ( groups5145338220374282879d_enat @ Xs ) @ ( groups5145338220374282879d_enat @ Ys ) ) ) ).
% sum_list_append
thf(fact_791_sum__list__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( groups4561878855575611511st_nat @ ( append_nat @ Xs @ Ys ) )
= ( plus_plus_nat @ ( groups4561878855575611511st_nat @ Xs ) @ ( groups4561878855575611511st_nat @ Ys ) ) ) ).
% sum_list_append
thf(fact_792_nth__append__length__plus,axiom,
! [Xs: list_nat,Ys: list_nat,N: nat] :
( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ N ) )
= ( nth_nat @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_793_length__sum__set__Suc,axiom,
! [K: nat,Ks: list_nat,R2: nat,N: nat] :
( ( member_list_nat @ ( cons_nat @ K @ Ks ) @ ( length_sum_set @ ( suc @ R2 ) @ N ) )
= ( ? [M5: nat] :
( ( member_list_nat @ Ks @ ( length_sum_set @ R2 @ M5 ) )
& ( N
= ( plus_plus_nat @ M5 @ K ) ) ) ) ) ).
% length_sum_set_Suc
thf(fact_794_nth__plus__list,axiom,
! [I: nat,Xs: list_list_nat,Ys: list_list_nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( nth_list_nat @ ( plus_p2116291331692525561st_nat @ Xs @ Ys ) @ I )
= ( plus_plus_list_nat @ ( nth_list_nat @ Xs @ I ) @ ( nth_list_nat @ Ys @ I ) ) ) ) ) ).
% nth_plus_list
thf(fact_795_nth__plus__list,axiom,
! [I: nat,Xs: list_Extended_enat,Ys: list_Extended_enat] :
( ( ord_less_nat @ I @ ( size_s3941691890525107288d_enat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s3941691890525107288d_enat @ Ys ) )
=> ( ( nth_Extended_enat @ ( plus_p2879455426997772835d_enat @ Xs @ Ys ) @ I )
= ( plus_p3455044024723400733d_enat @ ( nth_Extended_enat @ Xs @ I ) @ ( nth_Extended_enat @ Ys @ I ) ) ) ) ) ).
% nth_plus_list
thf(fact_796_nth__plus__list,axiom,
! [I: nat,Xs: list_nat,Ys: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
=> ( ( nth_nat @ ( plus_plus_list_nat @ Xs @ Ys ) @ I )
= ( plus_plus_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).
% nth_plus_list
thf(fact_797_nth__drop,axiom,
! [N: nat,Xs: list_nat,I: nat] :
( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( drop_nat @ N @ Xs ) @ I )
= ( nth_nat @ Xs @ ( plus_plus_nat @ N @ I ) ) ) ) ).
% nth_drop
thf(fact_798_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_799_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_800_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_801_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_802_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_803_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_804_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_805_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_806_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_807_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_808_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N4: nat] :
? [K3: nat] :
( N4
= ( plus_plus_nat @ M5 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_809_sum__list__plus,axiom,
! [Xs: list_Extended_enat,Ys: list_Extended_enat] :
( ( ( size_s3941691890525107288d_enat @ Xs )
= ( size_s3941691890525107288d_enat @ Ys ) )
=> ( ( groups5145338220374282879d_enat @ ( plus_p2879455426997772835d_enat @ Xs @ Ys ) )
= ( plus_p3455044024723400733d_enat @ ( groups5145338220374282879d_enat @ Xs ) @ ( groups5145338220374282879d_enat @ Ys ) ) ) ) ).
% sum_list_plus
thf(fact_810_sum__list__plus,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( groups4561878855575611511st_nat @ ( plus_plus_list_nat @ Xs @ Ys ) )
= ( plus_plus_nat @ ( groups4561878855575611511st_nat @ Xs ) @ ( groups4561878855575611511st_nat @ Ys ) ) ) ) ).
% sum_list_plus
thf(fact_811_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_812_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_813_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_814_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_815_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_816_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_817_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_818_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_819_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_820_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_821_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_822_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_823_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_824_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_825_nat__arith_Osuc1,axiom,
! [A3: nat,K: nat,A: nat] :
( ( A3
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A3 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_826_plus__Cons,axiom,
! [Y: list_nat,Ys: list_list_nat,X: list_nat,Xs: list_list_nat] :
( ( plus_p2116291331692525561st_nat @ ( cons_list_nat @ Y @ Ys ) @ ( cons_list_nat @ X @ Xs ) )
= ( cons_list_nat @ ( plus_plus_list_nat @ Y @ X ) @ ( plus_p2116291331692525561st_nat @ Ys @ Xs ) ) ) ).
% plus_Cons
thf(fact_827_plus__Cons,axiom,
! [Y: extended_enat,Ys: list_Extended_enat,X: extended_enat,Xs: list_Extended_enat] :
( ( plus_p2879455426997772835d_enat @ ( cons_Extended_enat @ Y @ Ys ) @ ( cons_Extended_enat @ X @ Xs ) )
= ( cons_Extended_enat @ ( plus_p3455044024723400733d_enat @ Y @ X ) @ ( plus_p2879455426997772835d_enat @ Ys @ Xs ) ) ) ).
% plus_Cons
thf(fact_828_plus__Cons,axiom,
! [Y: nat,Ys: list_nat,X: nat,Xs: list_nat] :
( ( plus_plus_list_nat @ ( cons_nat @ Y @ Ys ) @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ ( plus_plus_nat @ Y @ X ) @ ( plus_plus_list_nat @ Ys @ Xs ) ) ) ).
% plus_Cons
thf(fact_829_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_830_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_831_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_832_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_833_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D3 ) ) ) ) ).
% add_strict_mono
thf(fact_834_add__strict__mono,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat,D3: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( ord_le72135733267957522d_enat @ C @ D3 )
=> ( ord_le72135733267957522d_enat @ ( plus_p3455044024723400733d_enat @ A @ C ) @ ( plus_p3455044024723400733d_enat @ B @ D3 ) ) ) ) ).
% add_strict_mono
thf(fact_835_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_836_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_837_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_838_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_839_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_840_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_841_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_842_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B6: nat] :
? [C4: nat] :
( B6
= ( plus_plus_nat @ A2 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_843_le__iff__add,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [A2: extended_enat,B6: extended_enat] :
? [C4: extended_enat] :
( B6
= ( plus_p3455044024723400733d_enat @ A2 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_844_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_845_add__right__mono,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A @ C ) @ ( plus_p3455044024723400733d_enat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_846_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_847_less__eqE,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ~ ! [C3: extended_enat] :
( B
!= ( plus_p3455044024723400733d_enat @ A @ C3 ) ) ) ).
% less_eqE
thf(fact_848_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_849_add__left__mono,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ C @ A ) @ ( plus_p3455044024723400733d_enat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_850_add__mono,axiom,
! [A: nat,B: nat,C: nat,D3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D3 ) ) ) ) ).
% add_mono
thf(fact_851_add__mono,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat,D3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ C @ D3 )
=> ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A @ C ) @ ( plus_p3455044024723400733d_enat @ B @ D3 ) ) ) ) ).
% add_mono
thf(fact_852_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_853_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: extended_enat,J: extended_enat,K: extended_enat,L: extended_enat] :
( ( ( ord_le2932123472753598470d_enat @ I @ J )
& ( ord_le2932123472753598470d_enat @ K @ L ) )
=> ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ I @ K ) @ ( plus_p3455044024723400733d_enat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_854_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_855_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: extended_enat,J: extended_enat,K: extended_enat,L: extended_enat] :
( ( ( I = J )
& ( ord_le2932123472753598470d_enat @ K @ L ) )
=> ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ I @ K ) @ ( plus_p3455044024723400733d_enat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_856_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_857_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: extended_enat,J: extended_enat,K: extended_enat,L: extended_enat] :
( ( ( ord_le2932123472753598470d_enat @ I @ J )
& ( K = L ) )
=> ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ I @ K ) @ ( plus_p3455044024723400733d_enat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_858_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D3 ) ) ) ) ).
% add_less_le_mono
thf(fact_859_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D3 ) ) ) ) ).
% add_le_less_mono
thf(fact_860_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_861_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_862_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_863_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_864_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_865_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_866_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_867_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_868_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_869_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_870_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_871_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_872_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_873_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M5: nat,N4: nat] :
? [K3: nat] :
( N4
= ( suc @ ( plus_plus_nat @ M5 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_874_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_875_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_876_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).
% less_natE
thf(fact_877_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M7: nat,N3: nat] :
( ( ord_less_nat @ M7 @ N3 )
=> ( ord_less_nat @ ( F @ M7 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_878_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_879_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_880_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_881_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_882_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_883_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_884_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_885_take__drop,axiom,
! [N: nat,M: nat,Xs: list_nat] :
( ( take_nat @ N @ ( drop_nat @ M @ Xs ) )
= ( drop_nat @ M @ ( take_nat @ ( plus_plus_nat @ N @ M ) @ Xs ) ) ) ).
% take_drop
thf(fact_886_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_887_verit__comp__simplify1_I2_J,axiom,
! [A: set_list_nat] : ( ord_le6045566169113846134st_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_888_verit__comp__simplify1_I2_J,axiom,
! [A: extended_enat] : ( ord_le2932123472753598470d_enat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_889_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_890_verit__la__disequality,axiom,
! [A: extended_enat,B: extended_enat] :
( ( A = B )
| ~ ( ord_le2932123472753598470d_enat @ A @ B )
| ~ ( ord_le2932123472753598470d_enat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_891_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_892_verit__comp__simplify1_I1_J,axiom,
! [A: extended_enat] :
~ ( ord_le72135733267957522d_enat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_893_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_894_take__add,axiom,
! [I: nat,J: nat,Xs: list_nat] :
( ( take_nat @ ( plus_plus_nat @ I @ J ) @ Xs )
= ( append_nat @ ( take_nat @ I @ Xs ) @ ( take_nat @ J @ ( drop_nat @ I @ Xs ) ) ) ) ).
% take_add
thf(fact_895_pointwise__le__plus,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( pointwise_le @ Xs @ Ys )
=> ( ( ord_less_eq_nat @ ( size_size_list_nat @ Ys ) @ ( size_size_list_nat @ Zs ) )
=> ( pointwise_le @ Xs @ ( plus_plus_list_nat @ Ys @ Zs ) ) ) ) ).
% pointwise_le_plus
thf(fact_896_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_897_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_898_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_899_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_900_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_901_set__plus__mono2,axiom,
! [C2: set_list_nat,D: set_list_nat,E: set_list_nat,F2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ C2 @ D )
=> ( ( ord_le6045566169113846134st_nat @ E @ F2 )
=> ( ord_le6045566169113846134st_nat @ ( plus_p884110394369815071st_nat @ C2 @ E ) @ ( plus_p884110394369815071st_nat @ D @ F2 ) ) ) ) ).
% set_plus_mono2
thf(fact_902_Set__Algebras_Osumset__empty_I2_J,axiom,
! [A3: set_list_nat] :
( ( plus_p884110394369815071st_nat @ bot_bot_set_list_nat @ A3 )
= bot_bot_set_list_nat ) ).
% Set_Algebras.sumset_empty(2)
thf(fact_903_Set__Algebras_Osumset__empty_I1_J,axiom,
! [A3: set_list_nat] :
( ( plus_p884110394369815071st_nat @ A3 @ bot_bot_set_list_nat )
= bot_bot_set_list_nat ) ).
% Set_Algebras.sumset_empty(1)
thf(fact_904_finite__set__plus,axiom,
! [S: set_list_nat,T: set_list_nat] :
( ( finite8100373058378681591st_nat @ S )
=> ( ( finite8100373058378681591st_nat @ T )
=> ( finite8100373058378681591st_nat @ ( plus_p884110394369815071st_nat @ S @ T ) ) ) ) ).
% finite_set_plus
thf(fact_905_finite__set__plus,axiom,
! [S: set_nat,T: set_nat] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_nat @ T )
=> ( finite_finite_nat @ ( plus_plus_set_nat @ S @ T ) ) ) ) ).
% finite_set_plus
thf(fact_906_triangle__Suc,axiom,
! [N: nat] :
( ( nat_triangle @ ( suc @ N ) )
= ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% triangle_Suc
thf(fact_907_VF__def,axiom,
( vf
= ( ^ [I3: nat,T2: nat] :
( collect_list_nat
@ ^ [V: list_nat] :
( ( member_list_nat @ V @ v )
& ( ( nth_nat @ V @ I3 )
= T2 ) ) ) ) ) ).
% VF_def
thf(fact_908_minimal__elements__def,axiom,
( minimal_elements
= ( ^ [U4: set_list_nat] :
( collect_list_nat
@ ( minimal_elementsp
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ U4 ) ) ) ) ) ).
% minimal_elements_def
thf(fact_909_minimal__elementsp__minimal__elements__eq,axiom,
! [U3: set_list_nat] :
( ( minimal_elementsp
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ U3 ) )
= ( ^ [X2: list_nat] : ( member_list_nat @ X2 @ ( minimal_elements @ U3 ) ) ) ) ).
% minimal_elementsp_minimal_elements_eq
thf(fact_910_length__sum__set__def,axiom,
( length_sum_set
= ( ^ [R3: nat,N4: nat] :
( collect_list_nat
@ ^ [X2: list_nat] :
( ( ( size_size_list_nat @ X2 )
= R3 )
& ( ( groups4561878855575611511st_nat @ X2 )
= N4 ) ) ) ) ) ).
% length_sum_set_def
thf(fact_911_finite__Collect__conjI,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
| ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [X2: list_nat] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_912_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_913_finite__Collect__disjI,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [X2: list_nat] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
& ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_914_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_915_V__def,axiom,
( v
= ( collect_list_nat
@ ^ [V: list_nat] :
( ( member_list_nat @ V @ ua )
& ~ ( pointwise_le @ u2 @ V ) ) ) ) ).
% V_def
thf(fact_916_finite__Collect__subsets,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B4: set_nat] : ( ord_less_eq_set_nat @ B4 @ A3 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_917_finite__Collect__subsets,axiom,
! [A3: set_list_nat] :
( ( finite8100373058378681591st_nat @ A3 )
=> ( finite7047420756378620717st_nat
@ ( collect_set_list_nat
@ ^ [B4: set_list_nat] : ( ord_le6045566169113846134st_nat @ B4 @ A3 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_918_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ N4 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_919_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_nat @ N4 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_920_not__finite__existsD,axiom,
! [P: list_nat > $o] :
( ~ ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
=> ? [X_1: list_nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_921_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_922_pigeonhole__infinite__rel,axiom,
! [A3: set_list_nat,B2: set_list_nat,R: list_nat > list_nat > $o] :
( ~ ( finite8100373058378681591st_nat @ A3 )
=> ( ( finite8100373058378681591st_nat @ B2 )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A3 )
=> ? [Xa: list_nat] :
( ( member_list_nat @ Xa @ B2 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ B2 )
& ~ ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [A2: list_nat] :
( ( member_list_nat @ A2 @ A3 )
& ( R @ A2 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_923_pigeonhole__infinite__rel,axiom,
! [A3: set_list_nat,B2: set_nat,R: list_nat > nat > $o] :
( ~ ( finite8100373058378681591st_nat @ A3 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A3 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B2 )
& ~ ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [A2: list_nat] :
( ( member_list_nat @ A2 @ A3 )
& ( R @ A2 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_924_pigeonhole__infinite__rel,axiom,
! [A3: set_nat,B2: set_list_nat,R: nat > list_nat > $o] :
( ~ ( finite_finite_nat @ A3 )
=> ( ( finite8100373058378681591st_nat @ B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ? [Xa: list_nat] :
( ( member_list_nat @ Xa @ B2 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A2: nat] :
( ( member_nat @ A2 @ A3 )
& ( R @ A2 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_925_pigeonhole__infinite__rel,axiom,
! [A3: set_nat,B2: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A3 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A2: nat] :
( ( member_nat @ A2 @ A3 )
& ( R @ A2 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_926_less__set__def,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [A4: set_list_nat,B4: set_list_nat] :
( ord_less_list_nat_o
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ A4 )
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ B4 ) ) ) ) ).
% less_set_def
thf(fact_927_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ord_less_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A4 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B4 ) ) ) ) ).
% less_set_def
thf(fact_928_set__diff__eq,axiom,
( minus_7954133019191499631st_nat
= ( ^ [A4: set_list_nat,B4: set_list_nat] :
( collect_list_nat
@ ^ [X2: list_nat] :
( ( member_list_nat @ X2 @ A4 )
& ~ ( member_list_nat @ X2 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_929_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A4 )
& ~ ( member_nat @ X2 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_930_minus__set__def,axiom,
( minus_7954133019191499631st_nat
= ( ^ [A4: set_list_nat,B4: set_list_nat] :
( collect_list_nat
@ ( minus_1139252259498527702_nat_o
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ A4 )
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_931_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A4 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_932_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X2: nat] : $false ) ) ).
% empty_def
thf(fact_933_empty__def,axiom,
( bot_bot_set_list_nat
= ( collect_list_nat
@ ^ [X2: list_nat] : $false ) ) ).
% empty_def
thf(fact_934_pred__subset__eq,axiom,
! [R: set_nat,S2: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ R )
@ ^ [X2: nat] : ( member_nat @ X2 @ S2 ) )
= ( ord_less_eq_set_nat @ R @ S2 ) ) ).
% pred_subset_eq
thf(fact_935_pred__subset__eq,axiom,
! [R: set_list_nat,S2: set_list_nat] :
( ( ord_le1520216061033275535_nat_o
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ R )
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ S2 ) )
= ( ord_le6045566169113846134st_nat @ R @ S2 ) ) ).
% pred_subset_eq
thf(fact_936_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A4 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_937_less__eq__set__def,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A4: set_list_nat,B4: set_list_nat] :
( ord_le1520216061033275535_nat_o
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ A4 )
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_938_Collect__subset,axiom,
! [A3: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( P @ X2 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_939_Collect__subset,axiom,
! [A3: set_list_nat,P: list_nat > $o] :
( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X2: list_nat] :
( ( member_list_nat @ X2 @ A3 )
& ( P @ X2 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_940_subset__CollectI,axiom,
! [B2: set_nat,A3: set_nat,Q: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B2 @ A3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ B2 )
& ( Q @ X2 ) ) )
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_941_subset__CollectI,axiom,
! [B2: set_list_nat,A3: set_list_nat,Q: list_nat > $o,P: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ B2 @ A3 )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ B2 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X2: list_nat] :
( ( member_list_nat @ X2 @ B2 )
& ( Q @ X2 ) ) )
@ ( collect_list_nat
@ ^ [X2: list_nat] :
( ( member_list_nat @ X2 @ A3 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_942_subset__Collect__iff,axiom,
! [B2: set_nat,A3: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ B2 @ A3 )
=> ( ( ord_less_eq_set_nat @ B2
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( P @ X2 ) ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_943_subset__Collect__iff,axiom,
! [B2: set_list_nat,A3: set_list_nat,P: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ B2 @ A3 )
=> ( ( ord_le6045566169113846134st_nat @ B2
@ ( collect_list_nat
@ ^ [X2: list_nat] :
( ( member_list_nat @ X2 @ A3 )
& ( P @ X2 ) ) ) )
= ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ B2 )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_944_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( P @ K3 )
& ( ord_less_nat @ K3 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_945_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_946_prop__restrict,axiom,
! [X: nat,Z5: set_nat,X6: set_nat,P: nat > $o] :
( ( member_nat @ X @ Z5 )
=> ( ( ord_less_eq_set_nat @ Z5
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ X6 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_947_prop__restrict,axiom,
! [X: list_nat,Z5: set_list_nat,X6: set_list_nat,P: list_nat > $o] :
( ( member_list_nat @ X @ Z5 )
=> ( ( ord_le6045566169113846134st_nat @ Z5
@ ( collect_list_nat
@ ^ [X2: list_nat] :
( ( member_list_nat @ X2 @ X6 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_948_Collect__restrict,axiom,
! [X6: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ X6 )
& ( P @ X2 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_949_Collect__restrict,axiom,
! [X6: set_list_nat,P: list_nat > $o] :
( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X2: list_nat] :
( ( member_list_nat @ X2 @ X6 )
& ( P @ X2 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_950_Succ__def,axiom,
( bNF_Gr3053708287304744325st_nat
= ( ^ [Kl3: set_list_list_nat,Kl4: list_list_nat] :
( collect_list_nat
@ ^ [K3: list_nat] : ( member_list_list_nat @ ( append_list_nat @ Kl4 @ ( cons_list_nat @ K3 @ nil_list_nat ) ) @ Kl3 ) ) ) ) ).
% Succ_def
thf(fact_951_Succ__def,axiom,
( bNF_Gr6352880689984616693cc_nat
= ( ^ [Kl3: set_list_nat,Kl4: list_nat] :
( collect_nat
@ ^ [K3: nat] : ( member_list_nat @ ( append_nat @ Kl4 @ ( cons_nat @ K3 @ nil_nat ) ) @ Kl3 ) ) ) ) ).
% Succ_def
thf(fact_952_bind__simps_I2_J,axiom,
! [X: nat,Xs: list_nat,F: nat > list_nat] :
( ( bind_nat_nat @ ( cons_nat @ X @ Xs ) @ F )
= ( append_nat @ ( F @ X ) @ ( bind_nat_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_953_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_954_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_955_hd__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( hd_nat @ ( upt @ I @ J ) )
= I ) ) ).
% hd_upt
thf(fact_956_bind__simps_I1_J,axiom,
! [F: nat > list_nat] :
( ( bind_nat_nat @ nil_nat @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_957_length__upt,axiom,
! [I: nat,J: nat] :
( ( size_size_list_nat @ ( upt @ I @ J ) )
= ( minus_minus_nat @ J @ I ) ) ).
% length_upt
thf(fact_958_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( upt @ I @ J )
= nil_nat ) ) ).
% upt_conv_Nil
thf(fact_959_drop__upt,axiom,
! [M: nat,I: nat,J: nat] :
( ( drop_nat @ M @ ( upt @ I @ J ) )
= ( upt @ ( plus_plus_nat @ I @ M ) @ J ) ) ).
% drop_upt
thf(fact_960_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_961_take__upt,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M ) @ N )
=> ( ( take_nat @ M @ ( upt @ I @ N ) )
= ( upt @ I @ ( plus_plus_nat @ I @ M ) ) ) ) ).
% take_upt
thf(fact_962_upt__conv__Cons__Cons,axiom,
! [M: nat,N: nat,Ns: list_nat,Q4: nat] :
( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
= ( upt @ M @ Q4 ) )
= ( ( cons_nat @ N @ Ns )
= ( upt @ ( suc @ M ) @ Q4 ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_963_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_964_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_965_upt__rec,axiom,
( upt
= ( ^ [I3: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I3 @ J3 ) @ ( cons_nat @ I3 @ ( upt @ ( suc @ I3 ) @ J3 ) ) @ nil_nat ) ) ) ).
% upt_rec
thf(fact_966_upt__rec__numeral,axiom,
! [M: num,N: num] :
( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
=> ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
& ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
=> ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= nil_nat ) ) ) ).
% upt_rec_numeral
thf(fact_967_map__upt__eqI,axiom,
! [Xs: list_nat,N: nat,M: nat,F: nat > nat] :
( ( ( size_size_list_nat @ Xs )
= ( minus_minus_nat @ N @ M ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I2 )
= ( F @ ( plus_plus_nat @ M @ I2 ) ) ) )
=> ( ( map_nat_nat @ F @ ( upt @ M @ N ) )
= Xs ) ) ) ).
% map_upt_eqI
thf(fact_968_finite__lists__length__le,axiom,
! [A3: set_nat,N: nat] :
( ( finite_finite_nat @ A3 )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs3: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs3 ) @ A3 )
& ( ord_less_eq_nat @ ( size_size_list_nat @ Xs3 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_969_finite__lists__length__le,axiom,
! [A3: set_list_nat,N: nat] :
( ( finite8100373058378681591st_nat @ A3 )
=> ( finite8170528100393595399st_nat
@ ( collec5989764272469232197st_nat
@ ^ [Xs3: list_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs3 ) @ A3 )
& ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs3 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_970_map__ident,axiom,
( ( map_nat_nat
@ ^ [X2: nat] : X2 )
= ( ^ [Xs3: list_nat] : Xs3 ) ) ).
% map_ident
thf(fact_971_List_Ofinite__set,axiom,
! [Xs: list_list_nat] : ( finite8100373058378681591st_nat @ ( set_list_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_972_List_Ofinite__set,axiom,
! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_973_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_974_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_975_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_976_map__eq__conv,axiom,
! [F: nat > nat,Xs: list_nat,G3: nat > nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G3 @ Xs ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X2 )
= ( G3 @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_977_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_978_map__append,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
( ( map_nat_nat @ F @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( map_nat_nat @ F @ Xs ) @ ( map_nat_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_979_set__empty,axiom,
! [Xs: list_nat] :
( ( ( set_nat2 @ Xs )
= bot_bot_set_nat )
= ( Xs = nil_nat ) ) ).
% set_empty
thf(fact_980_set__empty,axiom,
! [Xs: list_list_nat] :
( ( ( set_list_nat2 @ Xs )
= bot_bot_set_list_nat )
= ( Xs = nil_list_nat ) ) ).
% set_empty
thf(fact_981_set__empty2,axiom,
! [Xs: list_nat] :
( ( bot_bot_set_nat
= ( set_nat2 @ Xs ) )
= ( Xs = nil_nat ) ) ).
% set_empty2
thf(fact_982_set__empty2,axiom,
! [Xs: list_list_nat] :
( ( bot_bot_set_list_nat
= ( set_list_nat2 @ Xs ) )
= ( Xs = nil_list_nat ) ) ).
% set_empty2
thf(fact_983_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_984_set__swap,axiom,
! [I: nat,Xs: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I ) ) )
= ( set_nat2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_985_sum__list__mono,axiom,
! [Xs: list_list_nat,F: list_nat > extended_enat,G3: list_nat > extended_enat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( G3 @ X3 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( groups5145338220374282879d_enat @ ( map_li1708695312048858560d_enat @ F @ Xs ) ) @ ( groups5145338220374282879d_enat @ ( map_li1708695312048858560d_enat @ G3 @ Xs ) ) ) ) ).
% sum_list_mono
thf(fact_986_sum__list__mono,axiom,
! [Xs: list_nat,F: nat > extended_enat,G3: nat > extended_enat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( G3 @ X3 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( groups5145338220374282879d_enat @ ( map_na5247803048364578384d_enat @ F @ Xs ) ) @ ( groups5145338220374282879d_enat @ ( map_na5247803048364578384d_enat @ G3 @ Xs ) ) ) ) ).
% sum_list_mono
thf(fact_987_sum__list__mono,axiom,
! [Xs: list_list_nat,F: list_nat > nat,G3: list_nat > nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( G3 @ X3 ) ) )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ F @ Xs ) ) @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ G3 @ Xs ) ) ) ) ).
% sum_list_mono
thf(fact_988_sum__list__mono,axiom,
! [Xs: list_nat,F: nat > nat,G3: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( G3 @ X3 ) ) )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ F @ Xs ) ) @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ G3 @ Xs ) ) ) ) ).
% sum_list_mono
thf(fact_989_subset__code_I1_J,axiom,
! [Xs: list_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B2 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X2 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_990_subset__code_I1_J,axiom,
! [Xs: list_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ B2 )
= ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( member_list_nat @ X2 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_991_sum__list__strict__mono,axiom,
! [Xs: list_list_nat,F: list_nat > extended_enat,G3: list_nat > extended_enat] :
( ( Xs != nil_list_nat )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( G3 @ X3 ) ) )
=> ( ord_le72135733267957522d_enat @ ( groups5145338220374282879d_enat @ ( map_li1708695312048858560d_enat @ F @ Xs ) ) @ ( groups5145338220374282879d_enat @ ( map_li1708695312048858560d_enat @ G3 @ Xs ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_992_sum__list__strict__mono,axiom,
! [Xs: list_nat,F: nat > extended_enat,G3: nat > extended_enat] :
( ( Xs != nil_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( G3 @ X3 ) ) )
=> ( ord_le72135733267957522d_enat @ ( groups5145338220374282879d_enat @ ( map_na5247803048364578384d_enat @ F @ Xs ) ) @ ( groups5145338220374282879d_enat @ ( map_na5247803048364578384d_enat @ G3 @ Xs ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_993_sum__list__strict__mono,axiom,
! [Xs: list_list_nat,F: list_nat > nat,G3: list_nat > nat] :
( ( Xs != nil_list_nat )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( ord_less_nat @ ( F @ X3 ) @ ( G3 @ X3 ) ) )
=> ( ord_less_nat @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ F @ Xs ) ) @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ G3 @ Xs ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_994_sum__list__strict__mono,axiom,
! [Xs: list_nat,F: nat > nat,G3: nat > nat] :
( ( Xs != nil_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ ( F @ X3 ) @ ( G3 @ X3 ) ) )
=> ( ord_less_nat @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ F @ Xs ) ) @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ G3 @ Xs ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_995_map__Suc__upt,axiom,
! [M: nat,N: nat] :
( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
= ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% map_Suc_upt
thf(fact_996_finite__list,axiom,
! [A3: set_list_nat] :
( ( finite8100373058378681591st_nat @ A3 )
=> ? [Xs2: list_list_nat] :
( ( set_list_nat2 @ Xs2 )
= A3 ) ) ).
% finite_list
thf(fact_997_finite__list,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ? [Xs2: list_nat] :
( ( set_nat2 @ Xs2 )
= A3 ) ) ).
% finite_list
thf(fact_998_list_Osimps_I9_J,axiom,
! [F: nat > nat,X21: nat,X222: list_nat] :
( ( map_nat_nat @ F @ ( cons_nat @ X21 @ X222 ) )
= ( cons_nat @ ( F @ X21 ) @ ( map_nat_nat @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_999_Cons__eq__map__D,axiom,
! [X: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( map_nat_nat @ F @ Ys ) )
=> ? [Z3: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z3 @ Zs2 ) )
& ( X
= ( F @ Z3 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1000_map__eq__Cons__D,axiom,
! [F: nat > nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
=> ? [Z3: nat,Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_nat_nat @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_1001_Cons__eq__map__conv,axiom,
! [X: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( map_nat_nat @ F @ Ys ) )
= ( ? [Z6: nat,Zs3: list_nat] :
( ( Ys
= ( cons_nat @ Z6 @ Zs3 ) )
& ( X
= ( F @ Z6 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1002_map__eq__Cons__conv,axiom,
! [F: nat > nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
= ( ? [Z6: nat,Zs3: list_nat] :
( ( Xs
= ( cons_nat @ Z6 @ Zs3 ) )
& ( ( F @ Z6 )
= Y )
& ( ( map_nat_nat @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1003_list_Osimps_I8_J,axiom,
! [F: nat > nat] :
( ( map_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_1004_map__update,axiom,
! [F: nat > nat,Xs: list_nat,K: nat,Y: nat] :
( ( map_nat_nat @ F @ ( list_update_nat @ Xs @ K @ Y ) )
= ( list_update_nat @ ( map_nat_nat @ F @ Xs ) @ K @ ( F @ Y ) ) ) ).
% map_update
thf(fact_1005_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_1006_set__ConsD,axiom,
! [Y: list_nat,X: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ Y @ ( set_list_nat2 @ ( cons_list_nat @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_list_nat @ Y @ ( set_list_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_1007_set__ConsD,axiom,
! [Y: nat,X: nat,Xs: list_nat] :
( ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_nat @ Y @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_1008_list_Oset__cases,axiom,
! [E2: list_nat,A: list_list_nat] :
( ( member_list_nat @ E2 @ ( set_list_nat2 @ A ) )
=> ( ! [Z22: list_list_nat] :
( A
!= ( cons_list_nat @ E2 @ Z22 ) )
=> ~ ! [Z1: list_nat,Z22: list_list_nat] :
( ( A
= ( cons_list_nat @ Z1 @ Z22 ) )
=> ~ ( member_list_nat @ E2 @ ( set_list_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_1009_list_Oset__cases,axiom,
! [E2: nat,A: list_nat] :
( ( member_nat @ E2 @ ( set_nat2 @ A ) )
=> ( ! [Z22: list_nat] :
( A
!= ( cons_nat @ E2 @ Z22 ) )
=> ~ ! [Z1: nat,Z22: list_nat] :
( ( A
= ( cons_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat @ E2 @ ( set_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_1010_list_Oset__intros_I1_J,axiom,
! [X21: list_nat,X222: list_list_nat] : ( member_list_nat @ X21 @ ( set_list_nat2 @ ( cons_list_nat @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_1011_list_Oset__intros_I1_J,axiom,
! [X21: nat,X222: list_nat] : ( member_nat @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_1012_list_Oset__intros_I2_J,axiom,
! [Y: list_nat,X222: list_list_nat,X21: list_nat] :
( ( member_list_nat @ Y @ ( set_list_nat2 @ X222 ) )
=> ( member_list_nat @ Y @ ( set_list_nat2 @ ( cons_list_nat @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_1013_list_Oset__intros_I2_J,axiom,
! [Y: nat,X222: list_nat,X21: nat] :
( ( member_nat @ Y @ ( set_nat2 @ X222 ) )
=> ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_1014_drop__map,axiom,
! [N: nat,F: nat > nat,Xs: list_nat] :
( ( drop_nat @ N @ ( map_nat_nat @ F @ Xs ) )
= ( map_nat_nat @ F @ ( drop_nat @ N @ Xs ) ) ) ).
% drop_map
thf(fact_1015_in__set__takeD,axiom,
! [X: list_nat,N: nat,Xs: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ ( take_list_nat @ N @ Xs ) ) )
=> ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) ) ) ).
% in_set_takeD
thf(fact_1016_in__set__takeD,axiom,
! [X: nat,N: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ ( take_nat @ N @ Xs ) ) )
=> ( member_nat @ X @ ( set_nat2 @ Xs ) ) ) ).
% in_set_takeD
thf(fact_1017_in__set__dropD,axiom,
! [X: list_nat,N: nat,Xs: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ ( drop_list_nat @ N @ Xs ) ) )
=> ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) ) ) ).
% in_set_dropD
thf(fact_1018_in__set__dropD,axiom,
! [X: nat,N: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ ( drop_nat @ N @ Xs ) ) )
=> ( member_nat @ X @ ( set_nat2 @ Xs ) ) ) ).
% in_set_dropD
thf(fact_1019_list_Omap__cong,axiom,
! [X: list_nat,Ya: list_nat,F: nat > nat,G3: nat > nat] :
( ( X = Ya )
=> ( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ Ya ) )
=> ( ( F @ Z3 )
= ( G3 @ Z3 ) ) )
=> ( ( map_nat_nat @ F @ X )
= ( map_nat_nat @ G3 @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_1020_list_Omap__cong0,axiom,
! [X: list_nat,F: nat > nat,G3: nat > nat] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ X ) )
=> ( ( F @ Z3 )
= ( G3 @ Z3 ) ) )
=> ( ( map_nat_nat @ F @ X )
= ( map_nat_nat @ G3 @ X ) ) ) ).
% list.map_cong0
thf(fact_1021_list_Oinj__map__strong,axiom,
! [X: list_nat,Xa2: list_nat,F: nat > nat,Fa: nat > nat] :
( ! [Z3: nat,Za: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ X ) )
=> ( ( member_nat @ Za @ ( set_nat2 @ Xa2 ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( map_nat_nat @ F @ X )
= ( map_nat_nat @ Fa @ Xa2 ) )
=> ( X = Xa2 ) ) ) ).
% list.inj_map_strong
thf(fact_1022_list_Omap__ident__strong,axiom,
! [T: list_list_nat,F: list_nat > list_nat] :
( ! [Z3: list_nat] :
( ( member_list_nat @ Z3 @ ( set_list_nat2 @ T ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_li7225945977422193158st_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_1023_list_Omap__ident__strong,axiom,
! [T: list_nat,F: nat > nat] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ T ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_nat_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_1024_map__ext,axiom,
! [Xs: list_nat,F: nat > nat,G3: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X3 )
= ( G3 @ X3 ) ) )
=> ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G3 @ Xs ) ) ) ).
% map_ext
thf(fact_1025_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_1026_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_1027_map__cong,axiom,
! [Xs: list_nat,Ys: list_nat,F: nat > nat,G3: nat > nat] :
( ( Xs = Ys )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
=> ( ( F @ X3 )
= ( G3 @ X3 ) ) )
=> ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G3 @ Ys ) ) ) ) ).
% map_cong
thf(fact_1028_ex__map__conv,axiom,
! [Ys: list_nat,F: nat > nat] :
( ( ? [Xs3: list_nat] :
( Ys
= ( map_nat_nat @ F @ Xs3 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Ys ) )
=> ? [Y2: nat] :
( X2
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_1029_append__eq__map__conv,axiom,
! [Ys: list_nat,Zs: list_nat,F: nat > nat,Xs: list_nat] :
( ( ( append_nat @ Ys @ Zs )
= ( map_nat_nat @ F @ Xs ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys
= ( map_nat_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_nat @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_1030_map__eq__append__conv,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( append_nat @ Ys @ Zs ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys
= ( map_nat_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_nat @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_1031_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs: list_nat,G3: nat > nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G3 @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1032_list_Omap__ident,axiom,
! [T: list_nat] :
( ( map_nat_nat
@ ^ [X2: nat] : X2
@ T )
= T ) ).
% list.map_ident
thf(fact_1033_hd__map,axiom,
! [Xs: list_nat,F: nat > nat] :
( ( Xs != nil_nat )
=> ( ( hd_nat @ ( map_nat_nat @ F @ Xs ) )
= ( F @ ( hd_nat @ Xs ) ) ) ) ).
% hd_map
thf(fact_1034_list_Omap__sel_I1_J,axiom,
! [A: list_nat,F: nat > nat] :
( ( A != nil_nat )
=> ( ( hd_nat @ ( map_nat_nat @ F @ A ) )
= ( F @ ( hd_nat @ A ) ) ) ) ).
% list.map_sel(1)
thf(fact_1035_set__subset__Cons,axiom,
! [Xs: list_nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_1036_set__subset__Cons,axiom,
! [Xs: list_list_nat,X: list_nat] : ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ ( set_list_nat2 @ ( cons_list_nat @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_1037_empty__set,axiom,
( bot_bot_set_nat
= ( set_nat2 @ nil_nat ) ) ).
% empty_set
thf(fact_1038_empty__set,axiom,
( bot_bot_set_list_nat
= ( set_list_nat2 @ nil_list_nat ) ) ).
% empty_set
thf(fact_1039_split__list__first__prop__iff,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys2: list_nat,X2: nat] :
( ? [Zs3: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Y2: nat] :
( ( member_nat @ Y2 @ ( set_nat2 @ Ys2 ) )
=> ~ ( P @ Y2 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_1040_split__list__last__prop__iff,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys2: list_nat,X2: nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Y2: nat] :
( ( member_nat @ Y2 @ ( set_nat2 @ Zs3 ) )
=> ~ ( P @ Y2 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_1041_in__set__conv__decomp__first,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
= ( ? [Ys2: list_list_nat,Zs3: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X @ Zs3 ) ) )
& ~ ( member_list_nat @ X @ ( set_list_nat2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1042_in__set__conv__decomp__first,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys2: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs3 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1043_in__set__conv__decomp__last,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
= ( ? [Ys2: list_list_nat,Zs3: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X @ Zs3 ) ) )
& ~ ( member_list_nat @ X @ ( set_list_nat2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1044_in__set__conv__decomp__last,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys2: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs3 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1045_split__list__first__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys4: list_nat,X3: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat2 @ Ys4 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_1046_split__list__last__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys4: list_nat,X3: nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_1047_split__list__first__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys4: list_nat,X3: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat2 @ Ys4 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_1048_split__list__last__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys4: list_nat,X3: nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_1049_in__set__conv__decomp,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
= ( ? [Ys2: list_list_nat,Zs3: list_list_nat] :
( Xs
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1050_in__set__conv__decomp,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys2: list_nat,Zs3: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1051_append__Cons__eq__iff,axiom,
! [X: list_nat,Xs: list_list_nat,Ys: list_list_nat,Xs5: list_list_nat,Ys7: list_list_nat] :
( ~ ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ( ~ ( member_list_nat @ X @ ( set_list_nat2 @ Ys ) )
=> ( ( ( append_list_nat @ Xs @ ( cons_list_nat @ X @ Ys ) )
= ( append_list_nat @ Xs5 @ ( cons_list_nat @ X @ Ys7 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys7 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1052_append__Cons__eq__iff,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat,Xs5: list_nat,Ys7: list_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ~ ( member_nat @ X @ ( set_nat2 @ Ys ) )
=> ( ( ( append_nat @ Xs @ ( cons_nat @ X @ Ys ) )
= ( append_nat @ Xs5 @ ( cons_nat @ X @ Ys7 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys7 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1053_split__list__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys4: list_nat,X3: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X3 @ Zs2 ) ) )
=> ~ ( P @ X3 ) ) ) ).
% split_list_propE
thf(fact_1054_split__list__first,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ? [Ys4: list_list_nat,Zs2: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys4 @ ( cons_list_nat @ X @ Zs2 ) ) )
& ~ ( member_list_nat @ X @ ( set_list_nat2 @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_1055_split__list__first,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys4: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X @ Zs2 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_1056_split__list__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys4: list_nat,X3: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X3 @ Zs2 ) ) )
& ( P @ X3 ) ) ) ).
% split_list_prop
thf(fact_1057_split__list__last,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ? [Ys4: list_list_nat,Zs2: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys4 @ ( cons_list_nat @ X @ Zs2 ) ) )
& ~ ( member_list_nat @ X @ ( set_list_nat2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_1058_split__list__last,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys4: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X @ Zs2 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_1059_split__list,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ? [Ys4: list_list_nat,Zs2: list_list_nat] :
( Xs
= ( append_list_nat @ Ys4 @ ( cons_list_nat @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_1060_split__list,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys4: list_nat,Zs2: list_nat] :
( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_1061_member__le__sum__list,axiom,
! [X: extended_enat,Xs: list_Extended_enat] :
( ( member_Extended_enat @ X @ ( set_Extended_enat2 @ Xs ) )
=> ( ord_le2932123472753598470d_enat @ X @ ( groups5145338220374282879d_enat @ Xs ) ) ) ).
% member_le_sum_list
thf(fact_1062_member__le__sum__list,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ X @ ( groups4561878855575611511st_nat @ Xs ) ) ) ).
% member_le_sum_list
thf(fact_1063_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_1064_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_1065_set__drop__subset,axiom,
! [N: nat,Xs: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( drop_nat @ N @ Xs ) ) @ ( set_nat2 @ Xs ) ) ).
% set_drop_subset
thf(fact_1066_set__drop__subset,axiom,
! [N: nat,Xs: list_list_nat] : ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ ( drop_list_nat @ N @ Xs ) ) @ ( set_list_nat2 @ Xs ) ) ).
% set_drop_subset
thf(fact_1067_set__update__subsetI,axiom,
! [Xs: list_nat,A3: set_nat,X: nat,I: nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A3 )
=> ( ( member_nat @ X @ A3 )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X ) ) @ A3 ) ) ) ).
% set_update_subsetI
thf(fact_1068_set__update__subsetI,axiom,
! [Xs: list_list_nat,A3: set_list_nat,X: list_nat,I: nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ A3 )
=> ( ( member_list_nat @ X @ A3 )
=> ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ ( list_update_list_nat @ Xs @ I @ X ) ) @ A3 ) ) ) ).
% set_update_subsetI
thf(fact_1069_hd__in__set,axiom,
! [Xs: list_list_nat] :
( ( Xs != nil_list_nat )
=> ( member_list_nat @ ( hd_list_nat @ Xs ) @ ( set_list_nat2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_1070_hd__in__set,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( member_nat @ ( hd_nat @ Xs ) @ ( set_nat2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_1071_list_Oset__sel_I1_J,axiom,
! [A: list_list_nat] :
( ( A != nil_list_nat )
=> ( member_list_nat @ ( hd_list_nat @ A ) @ ( set_list_nat2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_1072_list_Oset__sel_I1_J,axiom,
! [A: list_nat] :
( ( A != nil_nat )
=> ( member_nat @ ( hd_nat @ A ) @ ( set_nat2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_1073_sum__list__addf,axiom,
! [F: nat > nat,G3: nat > nat,Xs: list_nat] :
( ( groups4561878855575611511st_nat
@ ( map_nat_nat
@ ^ [X2: nat] : ( plus_plus_nat @ ( F @ X2 ) @ ( G3 @ X2 ) )
@ Xs ) )
= ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ F @ Xs ) ) @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ G3 @ Xs ) ) ) ) ).
% sum_list_addf
thf(fact_1074_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_1075_map__equality__iff,axiom,
! [F: nat > nat,Xs: list_nat,G3: nat > nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G3 @ Ys ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys ) )
=> ( ( F @ ( nth_nat @ Xs @ I3 ) )
= ( G3 @ ( nth_nat @ Ys @ I3 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_1076_all__set__conv__all__nth,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( P @ X2 ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1077_all__nth__imp__all__set,axiom,
! [Xs: list_list_nat,P: list_nat > $o,X: list_nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( P @ ( nth_list_nat @ Xs @ I2 ) ) )
=> ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1078_all__nth__imp__all__set,axiom,
! [Xs: list_nat,P: nat > $o,X: nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I2 ) ) )
=> ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1079_in__set__conv__nth,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs ) )
& ( ( nth_list_nat @ Xs @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1080_in__set__conv__nth,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I3 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1081_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_1082_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_1083_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_1084_set__take__subset__set__take,axiom,
! [M: nat,N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( take_nat @ M @ Xs ) ) @ ( set_nat2 @ ( take_nat @ N @ Xs ) ) ) ) ).
% set_take_subset_set_take
thf(fact_1085_set__take__subset__set__take,axiom,
! [M: nat,N: nat,Xs: list_list_nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ ( take_list_nat @ M @ Xs ) ) @ ( set_list_nat2 @ ( take_list_nat @ N @ Xs ) ) ) ) ).
% set_take_subset_set_take
thf(fact_1086_set__drop__subset__set__drop,axiom,
! [N: nat,M: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( drop_nat @ M @ Xs ) ) @ ( set_nat2 @ ( drop_nat @ N @ Xs ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_1087_set__drop__subset__set__drop,axiom,
! [N: nat,M: nat,Xs: list_list_nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ ( drop_list_nat @ M @ Xs ) ) @ ( set_list_nat2 @ ( drop_list_nat @ N @ Xs ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_1088_set__update__memI,axiom,
! [N: nat,Xs: list_list_nat,X: list_nat] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( member_list_nat @ X @ ( set_list_nat2 @ ( list_update_list_nat @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_1089_set__update__memI,axiom,
! [N: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_1090_finite__lists__length__eq,axiom,
! [A3: set_nat,N: nat] :
( ( finite_finite_nat @ A3 )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs3: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs3 ) @ A3 )
& ( ( size_size_list_nat @ Xs3 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_1091_finite__lists__length__eq,axiom,
! [A3: set_list_nat,N: nat] :
( ( finite8100373058378681591st_nat @ A3 )
=> ( finite8170528100393595399st_nat
@ ( collec5989764272469232197st_nat
@ ^ [Xs3: list_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs3 ) @ A3 )
& ( ( size_s3023201423986296836st_nat @ Xs3 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_1092_nth__map__upt,axiom,
! [I: nat,N: nat,M: nat,F: nat > nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ N @ M ) )
=> ( ( nth_nat @ ( map_nat_nat @ F @ ( upt @ M @ N ) ) @ I )
= ( F @ ( plus_plus_nat @ M @ I ) ) ) ) ).
% nth_map_upt
thf(fact_1093_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_1094_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_1095_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_1096_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_1097_the__elem__set,axiom,
! [X: nat] :
( ( the_elem_nat @ ( set_nat2 @ ( cons_nat @ X @ nil_nat ) ) )
= X ) ).
% the_elem_set
thf(fact_1098_sum__list__Suc,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( groups4561878855575611511st_nat
@ ( map_nat_nat
@ ^ [X2: nat] : ( suc @ ( F @ X2 ) )
@ Xs ) )
= ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ F @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ) ).
% sum_list_Suc
thf(fact_1099_enat__ord__number_I2_J,axiom,
! [M: num,N: num] :
( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% enat_ord_number(2)
thf(fact_1100_enat__ord__number_I1_J,axiom,
! [M: num,N: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% enat_ord_number(1)
thf(fact_1101_add__diff__assoc__enat,axiom,
! [Z: extended_enat,Y: extended_enat,X: extended_enat] :
( ( ord_le2932123472753598470d_enat @ Z @ Y )
=> ( ( plus_p3455044024723400733d_enat @ X @ ( minus_3235023915231533773d_enat @ Y @ Z ) )
= ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y ) @ Z ) ) ) ).
% add_diff_assoc_enat
thf(fact_1102_enat__less__induct,axiom,
! [P: extended_enat > $o,N: extended_enat] :
( ! [N3: extended_enat] :
( ! [M2: extended_enat] :
( ( ord_le72135733267957522d_enat @ M2 @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% enat_less_induct
thf(fact_1103_set__n__lists,axiom,
! [N: nat,Xs: list_nat] :
( ( set_list_nat2 @ ( n_lists_nat @ N @ Xs ) )
= ( collect_list_nat
@ ^ [Ys2: list_nat] :
( ( ( size_size_list_nat @ Ys2 )
= N )
& ( ord_less_eq_set_nat @ ( set_nat2 @ Ys2 ) @ ( set_nat2 @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_1104_set__n__lists,axiom,
! [N: nat,Xs: list_list_nat] :
( ( set_list_list_nat2 @ ( n_lists_list_nat @ N @ Xs ) )
= ( collec5989764272469232197st_nat
@ ^ [Ys2: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Ys2 )
= N )
& ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Ys2 ) @ ( set_list_nat2 @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_1105_size__list__conv__sum__list,axiom,
( size_list_nat
= ( ^ [F3: nat > nat,Xs3: list_nat] : ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ F3 @ Xs3 ) ) @ ( size_size_list_nat @ Xs3 ) ) ) ) ).
% size_list_conv_sum_list
thf(fact_1106_size__list__append,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
( ( size_list_nat @ F @ ( append_nat @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_list_nat @ F @ Xs ) @ ( size_list_nat @ F @ Ys ) ) ) ).
% size_list_append
thf(fact_1107_size__list__pointwise,axiom,
! [Xs: list_list_nat,F: list_nat > nat,G3: list_nat > nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( G3 @ X3 ) ) )
=> ( ord_less_eq_nat @ ( size_list_list_nat @ F @ Xs ) @ ( size_list_list_nat @ G3 @ Xs ) ) ) ).
% size_list_pointwise
thf(fact_1108_size__list__pointwise,axiom,
! [Xs: list_nat,F: nat > nat,G3: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( G3 @ X3 ) ) )
=> ( ord_less_eq_nat @ ( size_list_nat @ F @ Xs ) @ ( size_list_nat @ G3 @ Xs ) ) ) ).
% size_list_pointwise
thf(fact_1109_size__list__estimation_H,axiom,
! [X: list_nat,Xs: list_list_nat,Y: nat,F: list_nat > nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ( ( ord_less_eq_nat @ Y @ ( F @ X ) )
=> ( ord_less_eq_nat @ Y @ ( size_list_list_nat @ F @ Xs ) ) ) ) ).
% size_list_estimation'
thf(fact_1110_size__list__estimation_H,axiom,
! [X: nat,Xs: list_nat,Y: nat,F: nat > nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( ord_less_eq_nat @ Y @ ( F @ X ) )
=> ( ord_less_eq_nat @ Y @ ( size_list_nat @ F @ Xs ) ) ) ) ).
% size_list_estimation'
thf(fact_1111_size__list__estimation,axiom,
! [X: list_nat,Xs: list_list_nat,Y: nat,F: list_nat > nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ( ( ord_less_nat @ Y @ ( F @ X ) )
=> ( ord_less_nat @ Y @ ( size_list_list_nat @ F @ Xs ) ) ) ) ).
% size_list_estimation
thf(fact_1112_size__list__estimation,axiom,
! [X: nat,Xs: list_nat,Y: nat,F: nat > nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( ord_less_nat @ Y @ ( F @ X ) )
=> ( ord_less_nat @ Y @ ( size_list_nat @ F @ Xs ) ) ) ) ).
% size_list_estimation
thf(fact_1113_length__n__lists__elem,axiom,
! [Ys: list_nat,N: nat,Xs: list_nat] :
( ( member_list_nat @ Ys @ ( set_list_nat2 @ ( n_lists_nat @ N @ Xs ) ) )
=> ( ( size_size_list_nat @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_1114_ediff__le__self,axiom,
! [X: extended_enat,Y: extended_enat] : ( ord_le2932123472753598470d_enat @ ( minus_3235023915231533773d_enat @ X @ Y ) @ X ) ).
% ediff_le_self
thf(fact_1115_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_1116_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_1117_le__zero__eq,axiom,
! [N: extended_enat] :
( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
= ( N = zero_z5237406670263579293d_enat ) ) ).
% le_zero_eq
thf(fact_1118_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_1119_not__gr__zero,axiom,
! [N: extended_enat] :
( ( ~ ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) )
= ( N = zero_z5237406670263579293d_enat ) ) ).
% not_gr_zero
thf(fact_1120_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_1121_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_1122_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_1123_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1124_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1125_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1126_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1127_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1128_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1129_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1130_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1131_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1132_drop0,axiom,
( ( drop_nat @ zero_zero_nat )
= ( ^ [X2: list_nat] : X2 ) ) ).
% drop0
thf(fact_1133_triangle__0,axiom,
( ( nat_triangle @ zero_zero_nat )
= zero_zero_nat ) ).
% triangle_0
thf(fact_1134_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_1135_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_1136_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_1137_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_1138_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_1139_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_1140_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_1141_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_1142_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_1143_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1144_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1145_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1146_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1147_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1148_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1149_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_1150_nth__Cons__0,axiom,
! [X: nat,Xs: list_nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_1151_sum__list_ONil,axiom,
( ( groups5145338220374282879d_enat @ nil_Extended_enat )
= zero_z5237406670263579293d_enat ) ).
% sum_list.Nil
thf(fact_1152_sum__list_ONil,axiom,
( ( groups4561878855575611511st_nat @ nil_nat )
= zero_zero_nat ) ).
% sum_list.Nil
thf(fact_1153_sum__list__eq__0__iff,axiom,
! [Ns: list_Extended_enat] :
( ( ( groups5145338220374282879d_enat @ Ns )
= zero_z5237406670263579293d_enat )
= ( ! [X2: extended_enat] :
( ( member_Extended_enat @ X2 @ ( set_Extended_enat2 @ Ns ) )
=> ( X2 = zero_z5237406670263579293d_enat ) ) ) ) ).
% sum_list_eq_0_iff
thf(fact_1154_sum__list__eq__0__iff,axiom,
! [Ns: list_nat] :
( ( ( groups4561878855575611511st_nat @ Ns )
= zero_zero_nat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Ns ) )
=> ( X2 = zero_zero_nat ) ) ) ) ).
% sum_list_eq_0_iff
thf(fact_1155_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_1156_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_1157_take0,axiom,
( ( take_nat @ zero_zero_nat )
= ( ^ [Xs3: list_nat] : nil_nat ) ) ).
% take0
thf(fact_1158_sum__list__0,axiom,
! [Xs: list_nat] :
( ( groups4561878855575611511st_nat
@ ( map_nat_nat
@ ^ [X2: nat] : zero_zero_nat
@ Xs ) )
= zero_zero_nat ) ).
% sum_list_0
thf(fact_1159_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_1160_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_1161_hd__take,axiom,
! [J: nat,Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ J )
=> ( ( hd_nat @ ( take_nat @ J @ Xs ) )
= ( hd_nat @ Xs ) ) ) ).
% hd_take
thf(fact_1162_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_1163_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1164_add__nonpos__eq__0__iff,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ zero_z5237406670263579293d_enat )
=> ( ( ord_le2932123472753598470d_enat @ Y @ zero_z5237406670263579293d_enat )
=> ( ( ( plus_p3455044024723400733d_enat @ X @ Y )
= zero_z5237406670263579293d_enat )
= ( ( X = zero_z5237406670263579293d_enat )
& ( Y = zero_z5237406670263579293d_enat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1165_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1166_add__nonneg__eq__0__iff,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ X )
=> ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ Y )
=> ( ( ( plus_p3455044024723400733d_enat @ X @ Y )
= zero_z5237406670263579293d_enat )
= ( ( X = zero_z5237406670263579293d_enat )
& ( Y = zero_z5237406670263579293d_enat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1167_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_1168_add__nonpos__nonpos,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ zero_z5237406670263579293d_enat )
=> ( ( ord_le2932123472753598470d_enat @ B @ zero_z5237406670263579293d_enat )
=> ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ zero_z5237406670263579293d_enat ) ) ) ).
% add_nonpos_nonpos
thf(fact_1169_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_1170_add__nonneg__nonneg,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ A )
=> ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ B )
=> ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1171_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_1172_add__increasing2,axiom,
! [C: extended_enat,B: extended_enat,A: extended_enat] :
( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ C )
=> ( ( ord_le2932123472753598470d_enat @ B @ A )
=> ( ord_le2932123472753598470d_enat @ B @ ( plus_p3455044024723400733d_enat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_1173_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_1174_add__decreasing2,axiom,
! [C: extended_enat,A: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ C @ zero_z5237406670263579293d_enat )
=> ( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1175_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_1176_add__increasing,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ A )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ord_le2932123472753598470d_enat @ B @ ( plus_p3455044024723400733d_enat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_1177_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_1178_add__decreasing,axiom,
! [A: extended_enat,C: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ zero_z5237406670263579293d_enat )
=> ( ( ord_le2932123472753598470d_enat @ C @ B )
=> ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_1179_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_1180_add__neg__neg,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ zero_z5237406670263579293d_enat )
=> ( ( ord_le72135733267957522d_enat @ B @ zero_z5237406670263579293d_enat )
=> ( ord_le72135733267957522d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ zero_z5237406670263579293d_enat ) ) ) ).
% add_neg_neg
thf(fact_1181_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_1182_add__pos__pos,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ A )
=> ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ B )
=> ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_1183_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_1184_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ~ ! [C3: extended_enat] :
( ( B
= ( plus_p3455044024723400733d_enat @ A @ C3 ) )
=> ( C3 = zero_z5237406670263579293d_enat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_1185_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_1186_pos__add__strict,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat] :
( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ A )
=> ( ( ord_le72135733267957522d_enat @ B @ C )
=> ( ord_le72135733267957522d_enat @ B @ ( plus_p3455044024723400733d_enat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_1187_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1188_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1189_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1190_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1191_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1192_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1193_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1194_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1195_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1196_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_1197_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1198_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N3: nat] :
( X
!= ( suc @ N3 ) ) ) ).
% list_decode.cases
thf(fact_1199_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1200_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1201_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1202_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1203_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1204_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_1205_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
=> ( ! [X3: nat,Y4: nat] :
( ( P @ X3 @ Y4 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y4 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1206_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1207_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1208_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1209_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1210_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M7: nat] :
( N
= ( suc @ M7 ) ) ) ).
% not0_implies_Suc
thf(fact_1211_drop__0,axiom,
! [Xs: list_nat] :
( ( drop_nat @ zero_zero_nat @ Xs )
= Xs ) ).
% drop_0
thf(fact_1212_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K2 )
=> ~ ( P @ I5 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1213_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1214_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1215_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1216_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M7: nat] :
( N
= ( suc @ M7 ) ) ) ).
% gr0_implies_Suc
thf(fact_1217_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1218_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_1219_le__numeral__extra_I3_J,axiom,
ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ zero_z5237406670263579293d_enat ).
% le_numeral_extra(3)
thf(fact_1220_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1221_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_1222_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_1223_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1224_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_1225_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1226_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1227_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1228_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1229_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1230_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1231_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_1232_less__numeral__extra_I3_J,axiom,
~ ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ zero_z5237406670263579293d_enat ) ).
% less_numeral_extra(3)
thf(fact_1233_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_1234_zero__less__iff__neq__zero,axiom,
! [N: extended_enat] :
( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
= ( N != zero_z5237406670263579293d_enat ) ) ).
% zero_less_iff_neq_zero
thf(fact_1235_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_1236_gr__implies__not__zero,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ord_le72135733267957522d_enat @ M @ N )
=> ( N != zero_z5237406670263579293d_enat ) ) ).
% gr_implies_not_zero
thf(fact_1237_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ zero_zero_nat )
= nil_nat ) ).
% upt_0
thf(fact_1238_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I5: nat] :
( ( ord_less_eq_nat @ I5 @ K2 )
=> ~ ( P @ I5 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1239_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1240_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1241_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_1242_map__add__upt,axiom,
! [N: nat,M: nat] :
( ( map_nat_nat
@ ^ [I3: nat] : ( plus_plus_nat @ I3 @ N )
@ ( upt @ zero_zero_nat @ M ) )
= ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).
% map_add_upt
thf(fact_1243_map__decr__upt,axiom,
! [M: nat,N: nat] :
( ( map_nat_nat
@ ^ [N4: nat] : ( minus_minus_nat @ N4 @ ( suc @ zero_zero_nat ) )
@ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
= ( upt @ M @ N ) ) ).
% map_decr_upt
thf(fact_1244_list__incr__nth__diff,axiom,
! [I: nat,X: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ X ) )
=> ( ( ( I = J )
=> ( ( minus_minus_nat @ ( nth_nat @ ( list_incr @ J @ X ) @ I ) @ ( nth_nat @ X @ I ) )
= one_one_nat ) )
& ( ( I != J )
=> ( ( minus_minus_nat @ ( nth_nat @ ( list_incr @ J @ X ) @ I ) @ ( nth_nat @ X @ I ) )
= zero_zero_nat ) ) ) ) ).
% list_incr_nth_diff
thf(fact_1245_i0__less,axiom,
! [N: extended_enat] :
( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
= ( N != zero_z5237406670263579293d_enat ) ) ).
% i0_less
thf(fact_1246_idiff__0__right,axiom,
! [N: extended_enat] :
( ( minus_3235023915231533773d_enat @ N @ zero_z5237406670263579293d_enat )
= N ) ).
% idiff_0_right
thf(fact_1247_idiff__0,axiom,
! [N: extended_enat] :
( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N )
= zero_z5237406670263579293d_enat ) ).
% idiff_0
thf(fact_1248_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1249_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1250_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_1251_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1252_i0__lb,axiom,
! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N ) ).
% i0_lb
thf(fact_1253_ile0__eq,axiom,
! [N: extended_enat] :
( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
= ( N = zero_z5237406670263579293d_enat ) ) ).
% ile0_eq
thf(fact_1254_not__iless0,axiom,
! [N: extended_enat] :
~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).
% not_iless0
thf(fact_1255_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1256_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1257_Suc__eq__plus1,axiom,
( suc
= ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1258_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1259_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1260_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1261_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_1262_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M5: nat,N4: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N4 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% add_eq_if
thf(fact_1263_upt__eq__Cons__conv,axiom,
! [I: nat,J: nat,X: nat,Xs: list_nat] :
( ( ( upt @ I @ J )
= ( cons_nat @ X @ Xs ) )
= ( ( ord_less_nat @ I @ J )
& ( I = X )
& ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
= Xs ) ) ) ).
% upt_eq_Cons_conv
thf(fact_1264_bot__enat__def,axiom,
bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).
% bot_enat_def
thf(fact_1265_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N3: nat] :
( ~ ( P @ N3 )
& ( P @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
% Helper facts (5)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
pointwise_le @ ( delete @ ua2 ) @ ( delete @ v2 ) ).
%------------------------------------------------------------------------------