TPTP Problem File: SLH0160^1.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_00170_005599__13396710_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1419 ( 637 unt; 148 typ; 0 def)
% Number of atoms : 3354 (1474 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10016 ( 328 ~; 72 |; 235 &;8036 @)
% ( 0 <=>;1345 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 6 avg)
% Number of types : 9 ( 8 usr)
% Number of type conns : 1182 (1182 >; 0 *; 0 +; 0 <<)
% Number of symbols : 143 ( 140 usr; 12 con; 0-3 aty)
% Number of variables : 3425 ( 222 ^;3077 !; 126 ?;3425 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:13:47.126
%------------------------------------------------------------------------------
% Could-be-implicit typings (8)
thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Extended____Nat__Oenat_J,type,
list_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 (140)
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__Nat__Onat,type,
bNF_Gr6352880689984616693cc_nat: set_list_nat > list_nat > set_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_Omonoid_001t__List__Olist_It__Extended____Nat__Oenat_J,type,
monoid1923560632227316973d_enat: ( list_Extended_enat > list_Extended_enat > list_Extended_enat ) > list_Extended_enat > $o ).
thf(sy_c_Groups_Omonoid_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
monoid_list_list_nat: ( list_list_nat > list_list_nat > list_list_nat ) > list_list_nat > $o ).
thf(sy_c_Groups_Omonoid_001t__List__Olist_It__Nat__Onat_J,type,
monoid_list_nat: ( list_nat > list_nat > list_nat ) > list_nat > $o ).
thf(sy_c_Groups_Omonoid_001t__Nat__Onat,type,
monoid_nat: ( nat > nat > nat ) > nat > $o ).
thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nat__Oenat,type,
one_on7984719198319812577d_enat: extended_enat ).
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__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_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_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_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_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__Extended____Nat__Oenat_001t__Extended____Nat__Oenat,type,
bind_E4971823231018073607d_enat: list_Extended_enat > ( extended_enat > list_Extended_enat ) > list_Extended_enat ).
thf(sy_c_List_Obind_001t__Extended____Nat__Oenat_001t__List__Olist_It__Nat__Onat_J,type,
bind_E2037203682813330047st_nat: list_Extended_enat > ( extended_enat > list_list_nat ) > list_list_nat ).
thf(sy_c_List_Obind_001t__Extended____Nat__Oenat_001t__Nat__Onat,type,
bind_E509905648326050543at_nat: list_Extended_enat > ( extended_enat > list_nat ) > list_nat ).
thf(sy_c_List_Obind_001t__List__Olist_It__Nat__Onat_J_001t__Extended____Nat__Oenat,type,
bind_l3791100948236736061d_enat: list_list_nat > ( list_nat > list_Extended_enat ) > list_Extended_enat ).
thf(sy_c_List_Obind_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
bind_l7796378977173581257st_nat: list_list_nat > ( list_nat > list_list_nat ) > list_list_nat ).
thf(sy_c_List_Obind_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
bind_list_nat_nat: list_list_nat > ( list_nat > list_nat ) > list_nat ).
thf(sy_c_List_Obind_001t__Nat__Onat_001t__Extended____Nat__Oenat,type,
bind_n5590761592452524365d_enat: list_nat > ( nat > list_Extended_enat ) > list_Extended_enat ).
thf(sy_c_List_Obind_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
bind_nat_list_nat: list_nat > ( nat > list_list_nat ) > list_list_nat ).
thf(sy_c_List_Obind_001t__Nat__Onat_001t__Nat__Onat,type,
bind_nat_nat: list_nat > ( nat > list_nat ) > list_nat ).
thf(sy_c_List_Obutlast_001t__Extended____Nat__Oenat,type,
butlas2973130096617243576d_enat: list_Extended_enat > list_Extended_enat ).
thf(sy_c_List_Obutlast_001t__Nat__Onat,type,
butlast_nat: list_nat > list_nat ).
thf(sy_c_List_Oconcat_001t__Nat__Onat,type,
concat_nat: list_list_nat > list_nat ).
thf(sy_c_List_Odistinct__adj_001t__Nat__Onat,type,
distinct_adj_nat: list_nat > $o ).
thf(sy_c_List_Odrop_001t__Extended____Nat__Oenat,type,
drop_Extended_enat: nat > list_Extended_enat > list_Extended_enat ).
thf(sy_c_List_Odrop_001t__Nat__Onat,type,
drop_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Ofold_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
fold_l5850465621530151245st_nat: ( list_nat > list_nat > list_nat ) > list_list_nat > list_nat > list_nat ).
thf(sy_c_List_Ofold_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
fold_nat_list_nat: ( nat > list_nat > list_nat ) > list_nat > list_nat > list_nat ).
thf(sy_c_List_Ofoldr_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
foldr_6871341030409798377st_nat: ( list_nat > list_nat > list_nat ) > list_list_nat > list_nat > list_nat ).
thf(sy_c_List_Ogen__length_001t__Nat__Onat,type,
gen_length_nat: nat > list_nat > nat ).
thf(sy_c_List_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olast_001t__Nat__Onat,type,
last_nat: list_nat > 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_Olist__all2_001t__Extended____Nat__Oenat_001t__Extended____Nat__Oenat,type,
list_a8739900810786033671d_enat: ( extended_enat > extended_enat > $o ) > list_Extended_enat > list_Extended_enat > $o ).
thf(sy_c_List_Olist_Olist__all2_001t__Extended____Nat__Oenat_001t__List__Olist_It__Nat__Onat_J,type,
list_a7409325535056916607st_nat: ( extended_enat > list_nat > $o ) > list_Extended_enat > list_list_nat > $o ).
thf(sy_c_List_Olist_Olist__all2_001t__Extended____Nat__Oenat_001t__Nat__Onat,type,
list_a8195072124553474287at_nat: ( extended_enat > nat > $o ) > list_Extended_enat > list_nat > $o ).
thf(sy_c_List_Olist_Olist__all2_001t__List__Olist_It__Nat__Onat_J_001t__Extended____Nat__Oenat,type,
list_a9163222800480322621d_enat: ( list_nat > extended_enat > $o ) > list_list_nat > list_Extended_enat > $o ).
thf(sy_c_List_Olist_Olist__all2_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
list_a4684442626689269705st_nat: ( list_nat > list_nat > $o ) > list_list_nat > list_list_nat > $o ).
thf(sy_c_List_Olist_Olist__all2_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
list_a7394835724046789945at_nat: ( list_nat > nat > $o ) > list_list_nat > list_nat > $o ).
thf(sy_c_List_Olist_Olist__all2_001t__Nat__Onat_001t__Extended____Nat__Oenat,type,
list_a4052556031825172301d_enat: ( nat > extended_enat > $o ) > list_nat > list_Extended_enat > $o ).
thf(sy_c_List_Olist_Olist__all2_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
list_a5155878676884504761st_nat: ( nat > list_nat > $o ) > list_nat > list_list_nat > $o ).
thf(sy_c_List_Olist_Olist__all2_001t__Nat__Onat_001t__Nat__Onat,type,
list_all2_nat_nat: ( nat > nat > $o ) > list_nat > list_nat > $o ).
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__Nat__Onat_001t__Nat__Onat,type,
map_nat_nat: ( nat > nat ) > list_nat > list_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_Otl_001t__Extended____Nat__Oenat,type,
tl_Extended_enat: list_Extended_enat > list_Extended_enat ).
thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
tl_nat: list_nat > list_nat ).
thf(sy_c_List_Olist__ex1_001t__Extended____Nat__Oenat,type,
list_e2739505282040533167d_enat: ( extended_enat > $o ) > list_Extended_enat > $o ).
thf(sy_c_List_Olist__ex1_001t__List__Olist_It__Nat__Onat_J,type,
list_ex1_list_nat: ( list_nat > $o ) > list_list_nat > $o ).
thf(sy_c_List_Olist__ex1_001t__Nat__Onat,type,
list_ex1_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
list_update_nat: list_nat > nat > nat > list_nat ).
thf(sy_c_List_Olistrelp_001t__Nat__Onat_001t__Nat__Onat,type,
listrelp_nat_nat: ( nat > nat > $o ) > list_nat > list_nat > $o ).
thf(sy_c_List_Omap__tailrec_001t__Nat__Onat_001t__Nat__Onat,type,
map_tailrec_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Omap__tailrec__rev_001t__Nat__Onat_001t__Nat__Onat,type,
map_ta7164188454487880599at_nat: ( nat > nat ) > list_nat > list_nat > list_nat ).
thf(sy_c_List_Omaps_001t__Nat__Onat_001t__Nat__Onat,type,
maps_nat_nat: ( nat > list_nat ) > list_nat > list_nat ).
thf(sy_c_List_Omember_001t__Extended____Nat__Oenat,type,
member_Extended_enat: list_Extended_enat > extended_enat > $o ).
thf(sy_c_List_Omember_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_list_nat > list_nat > $o ).
thf(sy_c_List_Omember_001t__Nat__Onat,type,
member_nat: list_nat > nat > $o ).
thf(sy_c_List_On__lists_001t__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__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Oord_Olexordp__eq_001t__Nat__Onat,type,
lexordp_eq_nat: ( nat > nat > $o ) > list_nat > list_nat > $o ).
thf(sy_c_List_Oord__class_Olexordp__eq_001t__Nat__Onat,type,
ord_lexordp_eq_nat: list_nat > list_nat > $o ).
thf(sy_c_List_Oproduct__lists_001t__Nat__Onat,type,
product_lists_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Oremdups__adj_001t__Extended____Nat__Oenat,type,
remdup6152102037098707618d_enat: list_Extended_enat > list_Extended_enat ).
thf(sy_c_List_Oremdups__adj_001t__Nat__Onat,type,
remdups_adj_nat: list_nat > list_nat ).
thf(sy_c_List_Oreplicate_001t__Extended____Nat__Oenat,type,
replic7216382294607269926d_enat: nat > extended_enat > list_Extended_enat ).
thf(sy_c_List_Oreplicate_001t__List__Olist_It__Nat__Onat_J,type,
replicate_list_nat: nat > list_nat > list_list_nat ).
thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
replicate_nat: nat > nat > list_nat ).
thf(sy_c_List_Orev_001t__Extended____Nat__Oenat,type,
rev_Extended_enat: list_Extended_enat > list_Extended_enat ).
thf(sy_c_List_Orev_001t__List__Olist_It__Nat__Onat_J,type,
rev_list_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Orev_001t__Nat__Onat,type,
rev_nat: list_nat > list_nat ).
thf(sy_c_List_Orotate1_001t__Nat__Onat,type,
rotate1_nat: list_nat > list_nat ).
thf(sy_c_List_Orotate_001t__Nat__Onat,type,
rotate_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Osorted__wrt_001t__Extended____Nat__Oenat,type,
sorted143172755617435219d_enat: ( extended_enat > extended_enat > $o ) > list_Extended_enat > $o ).
thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
thf(sy_c_List_Osplice_001t__Nat__Onat,type,
splice_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Osubseqs_001t__Nat__Onat,type,
subseqs_nat: list_nat > list_list_nat ).
thf(sy_c_List_Otake_001t__Extended____Nat__Oenat,type,
take_Extended_enat: nat > list_Extended_enat > list_Extended_enat ).
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__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
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_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__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__eq_001_062_I_Eo_M_062_I_Eo_Mt__Extended____Nat__Oenat_J_J,type,
ord_le5395741666527326556d_enat: ( $o > $o > extended_enat ) > ( $o > $o > extended_enat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_062_I_Eo_Mt__Nat__Onat_J_J,type,
ord_less_eq_o_o_nat: ( $o > $o > nat ) > ( $o > $o > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_062_It__List__Olist_It__Nat__Onat_J_M_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J_J_J,type,
ord_le286611365431560435_nat_o: ( $o > list_nat > list_nat > $o ) > ( $o > list_nat > list_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_J,type,
ord_le7862453073132658707_nat_o: ( $o > nat > nat > $o ) > ( $o > nat > nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Extended____Nat__Oenat_J,type,
ord_le2787558655864224659d_enat: ( $o > extended_enat ) > ( $o > extended_enat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Nat__Onat_J,type,
ord_less_eq_o_nat: ( $o > nat ) > ( $o > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Num__Onum_J,type,
ord_less_eq_o_num: ( $o > num ) > ( $o > num ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_M_062_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_M_Eo_J_J,type,
ord_le2031171171200246342_nat_o: ( list_list_nat > list_list_nat > $o ) > ( list_list_nat > list_list_nat > $o ) > $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_062_It__Nat__Onat_M_Eo_J_J,type,
ord_le2646555220125990790_nat_o: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
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__List__Olist_It__Nat__Onat_J_J,type,
ord_le6806709344281226192st_nat: list_list_nat > list_list_nat > $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_Oorder__class_OGreatest_001_062_I_Eo_Mt__Extended____Nat__Oenat_J,type,
order_7406502399511290778d_enat: ( ( $o > extended_enat ) > $o ) > $o > extended_enat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001_062_I_Eo_Mt__Nat__Onat_J,type,
order_Greatest_o_nat: ( ( $o > nat ) > $o ) > $o > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001_062_It__List__Olist_It__Nat__Onat_J_M_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J_J,type,
order_4166164572150370591_nat_o: ( ( list_nat > list_nat > $o ) > $o ) > list_nat > list_nat > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
order_1729629668087260991_nat_o: ( ( nat > nat > $o ) > $o ) > nat > nat > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Extended____Nat__Oenat,type,
order_2428742583041560895d_enat: ( extended_enat > $o ) > extended_enat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Num__Onum,type,
order_Greatest_num: ( num > $o ) > num ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
modulo_modulo_nat: nat > nat > nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat2: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat2: nat > set_nat > $o ).
thf(sy_v_x,type,
x: list_nat ).
thf(sy_v_y,type,
y: list_nat ).
% Relevant facts (1265)
thf(fact_0_pointwise__le__refl,axiom,
! [X: list_nat] : ( pointwise_le @ X @ X ) ).
% pointwise_le_refl
thf(fact_1_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_2_pointwise__less__def,axiom,
( pointwise_less
= ( ^ [X2: list_nat,Y2: list_nat] :
( ( pointwise_le @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% pointwise_less_def
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_pointwise__le__Nil,axiom,
! [X: list_nat] :
( ( pointwise_le @ nil_nat @ X )
= ( X = nil_nat ) ) ).
% pointwise_le_Nil
thf(fact_5_pointwise__le__Nil2,axiom,
! [X: list_nat] :
( ( pointwise_le @ X @ nil_nat )
= ( X = nil_nat ) ) ).
% pointwise_le_Nil2
thf(fact_6_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_7_pointwise__le__def,axiom,
( pointwise_le
= ( list_all2_nat_nat @ ord_less_eq_nat ) ) ).
% pointwise_le_def
thf(fact_8_list__all2__Nil,axiom,
! [P: nat > list_nat > $o,Ys: list_list_nat] :
( ( list_a5155878676884504761st_nat @ P @ nil_nat @ Ys )
= ( Ys = nil_list_nat ) ) ).
% list_all2_Nil
thf(fact_9_list__all2__Nil,axiom,
! [P: nat > extended_enat > $o,Ys: list_Extended_enat] :
( ( list_a4052556031825172301d_enat @ P @ nil_nat @ Ys )
= ( Ys = nil_Extended_enat ) ) ).
% list_all2_Nil
thf(fact_10_list__all2__Nil,axiom,
! [P: list_nat > nat > $o,Ys: list_nat] :
( ( list_a7394835724046789945at_nat @ P @ nil_list_nat @ Ys )
= ( Ys = nil_nat ) ) ).
% list_all2_Nil
thf(fact_11_list__all2__Nil,axiom,
! [P: list_nat > list_nat > $o,Ys: list_list_nat] :
( ( list_a4684442626689269705st_nat @ P @ nil_list_nat @ Ys )
= ( Ys = nil_list_nat ) ) ).
% list_all2_Nil
thf(fact_12_list__all2__Nil,axiom,
! [P: list_nat > extended_enat > $o,Ys: list_Extended_enat] :
( ( list_a9163222800480322621d_enat @ P @ nil_list_nat @ Ys )
= ( Ys = nil_Extended_enat ) ) ).
% list_all2_Nil
thf(fact_13_list__all2__Nil,axiom,
! [P: extended_enat > nat > $o,Ys: list_nat] :
( ( list_a8195072124553474287at_nat @ P @ nil_Extended_enat @ Ys )
= ( Ys = nil_nat ) ) ).
% list_all2_Nil
thf(fact_14_list__all2__Nil,axiom,
! [P: extended_enat > list_nat > $o,Ys: list_list_nat] :
( ( list_a7409325535056916607st_nat @ P @ nil_Extended_enat @ Ys )
= ( Ys = nil_list_nat ) ) ).
% list_all2_Nil
thf(fact_15_list__all2__Nil,axiom,
! [P: extended_enat > extended_enat > $o,Ys: list_Extended_enat] :
( ( list_a8739900810786033671d_enat @ P @ nil_Extended_enat @ Ys )
= ( Ys = nil_Extended_enat ) ) ).
% list_all2_Nil
thf(fact_16_list__all2__Nil,axiom,
! [P: nat > nat > $o,Ys: list_nat] :
( ( list_all2_nat_nat @ P @ nil_nat @ Ys )
= ( Ys = nil_nat ) ) ).
% list_all2_Nil
thf(fact_17_list__all2__Nil2,axiom,
! [P: list_nat > nat > $o,Xs: list_list_nat] :
( ( list_a7394835724046789945at_nat @ P @ Xs @ nil_nat )
= ( Xs = nil_list_nat ) ) ).
% list_all2_Nil2
thf(fact_18_list__all2__Nil2,axiom,
! [P: extended_enat > nat > $o,Xs: list_Extended_enat] :
( ( list_a8195072124553474287at_nat @ P @ Xs @ nil_nat )
= ( Xs = nil_Extended_enat ) ) ).
% list_all2_Nil2
thf(fact_19_list__all2__Nil2,axiom,
! [P: nat > list_nat > $o,Xs: list_nat] :
( ( list_a5155878676884504761st_nat @ P @ Xs @ nil_list_nat )
= ( Xs = nil_nat ) ) ).
% list_all2_Nil2
thf(fact_20_list__all2__Nil2,axiom,
! [P: list_nat > list_nat > $o,Xs: list_list_nat] :
( ( list_a4684442626689269705st_nat @ P @ Xs @ nil_list_nat )
= ( Xs = nil_list_nat ) ) ).
% list_all2_Nil2
thf(fact_21_list__all2__Nil2,axiom,
! [P: extended_enat > list_nat > $o,Xs: list_Extended_enat] :
( ( list_a7409325535056916607st_nat @ P @ Xs @ nil_list_nat )
= ( Xs = nil_Extended_enat ) ) ).
% list_all2_Nil2
thf(fact_22_list__all2__Nil2,axiom,
! [P: nat > extended_enat > $o,Xs: list_nat] :
( ( list_a4052556031825172301d_enat @ P @ Xs @ nil_Extended_enat )
= ( Xs = nil_nat ) ) ).
% list_all2_Nil2
thf(fact_23_list__all2__Nil2,axiom,
! [P: list_nat > extended_enat > $o,Xs: list_list_nat] :
( ( list_a9163222800480322621d_enat @ P @ Xs @ nil_Extended_enat )
= ( Xs = nil_list_nat ) ) ).
% list_all2_Nil2
thf(fact_24_list__all2__Nil2,axiom,
! [P: extended_enat > extended_enat > $o,Xs: list_Extended_enat] :
( ( list_a8739900810786033671d_enat @ P @ Xs @ nil_Extended_enat )
= ( Xs = nil_Extended_enat ) ) ).
% list_all2_Nil2
thf(fact_25_list__all2__Nil2,axiom,
! [P: nat > nat > $o,Xs: list_nat] :
( ( list_all2_nat_nat @ P @ Xs @ nil_nat )
= ( Xs = nil_nat ) ) ).
% list_all2_Nil2
thf(fact_26_append_Oright__neutral,axiom,
! [A: list_list_nat] :
( ( append_list_nat @ A @ nil_list_nat )
= A ) ).
% append.right_neutral
thf(fact_27_append_Oright__neutral,axiom,
! [A: list_Extended_enat] :
( ( append_Extended_enat @ A @ nil_Extended_enat )
= A ) ).
% append.right_neutral
thf(fact_28_append_Oright__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ A @ nil_nat )
= A ) ).
% append.right_neutral
thf(fact_29_append__Nil2,axiom,
! [Xs: list_list_nat] :
( ( append_list_nat @ Xs @ nil_list_nat )
= Xs ) ).
% append_Nil2
thf(fact_30_append__Nil2,axiom,
! [Xs: list_Extended_enat] :
( ( append_Extended_enat @ Xs @ nil_Extended_enat )
= Xs ) ).
% append_Nil2
thf(fact_31_append__Nil2,axiom,
! [Xs: list_nat] :
( ( append_nat @ Xs @ nil_nat )
= Xs ) ).
% append_Nil2
thf(fact_32_append__self__conv,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( ( append_list_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_list_nat ) ) ).
% append_self_conv
thf(fact_33_append__self__conv,axiom,
! [Xs: list_Extended_enat,Ys: list_Extended_enat] :
( ( ( append_Extended_enat @ Xs @ Ys )
= Xs )
= ( Ys = nil_Extended_enat ) ) ).
% append_self_conv
thf(fact_34_append__self__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_nat ) ) ).
% append_self_conv
thf(fact_35_self__append__conv,axiom,
! [Y: list_list_nat,Ys: list_list_nat] :
( ( Y
= ( append_list_nat @ Y @ Ys ) )
= ( Ys = nil_list_nat ) ) ).
% self_append_conv
thf(fact_36_self__append__conv,axiom,
! [Y: list_Extended_enat,Ys: list_Extended_enat] :
( ( Y
= ( append_Extended_enat @ Y @ Ys ) )
= ( Ys = nil_Extended_enat ) ) ).
% self_append_conv
thf(fact_37_self__append__conv,axiom,
! [Y: list_nat,Ys: list_nat] :
( ( Y
= ( append_nat @ Y @ Ys ) )
= ( Ys = nil_nat ) ) ).
% self_append_conv
thf(fact_38_append__self__conv2,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( ( append_list_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_list_nat ) ) ).
% append_self_conv2
thf(fact_39_append__self__conv2,axiom,
! [Xs: list_Extended_enat,Ys: list_Extended_enat] :
( ( ( append_Extended_enat @ Xs @ Ys )
= Ys )
= ( Xs = nil_Extended_enat ) ) ).
% append_self_conv2
thf(fact_40_append__self__conv2,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_nat ) ) ).
% append_self_conv2
thf(fact_41_self__append__conv2,axiom,
! [Y: list_list_nat,Xs: list_list_nat] :
( ( Y
= ( append_list_nat @ Xs @ Y ) )
= ( Xs = nil_list_nat ) ) ).
% self_append_conv2
thf(fact_42_self__append__conv2,axiom,
! [Y: list_Extended_enat,Xs: list_Extended_enat] :
( ( Y
= ( append_Extended_enat @ Xs @ Y ) )
= ( Xs = nil_Extended_enat ) ) ).
% self_append_conv2
thf(fact_43_self__append__conv2,axiom,
! [Y: list_nat,Xs: list_nat] :
( ( Y
= ( append_nat @ Xs @ Y ) )
= ( Xs = nil_nat ) ) ).
% self_append_conv2
thf(fact_44_Nil__is__append__conv,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( nil_list_nat
= ( append_list_nat @ Xs @ Ys ) )
= ( ( Xs = nil_list_nat )
& ( Ys = nil_list_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_45_Nil__is__append__conv,axiom,
! [Xs: list_Extended_enat,Ys: list_Extended_enat] :
( ( nil_Extended_enat
= ( append_Extended_enat @ Xs @ Ys ) )
= ( ( Xs = nil_Extended_enat )
& ( Ys = nil_Extended_enat ) ) ) ).
% Nil_is_append_conv
thf(fact_46_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_47_append__is__Nil__conv,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( ( append_list_nat @ Xs @ Ys )
= nil_list_nat )
= ( ( Xs = nil_list_nat )
& ( Ys = nil_list_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_48_append__is__Nil__conv,axiom,
! [Xs: list_Extended_enat,Ys: list_Extended_enat] :
( ( ( append_Extended_enat @ Xs @ Ys )
= nil_Extended_enat )
= ( ( Xs = nil_Extended_enat )
& ( Ys = nil_Extended_enat ) ) ) ).
% append_is_Nil_conv
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_append_Oassoc,axiom,
! [A: list_list_nat,B: list_list_nat,C: list_list_nat] :
( ( append_list_nat @ ( append_list_nat @ A @ B ) @ C )
= ( append_list_nat @ A @ ( append_list_nat @ B @ C ) ) ) ).
% append.assoc
thf(fact_51_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_52_append__assoc,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,Zs: list_list_nat] :
( ( append_list_nat @ ( append_list_nat @ Xs @ Ys ) @ Zs )
= ( append_list_nat @ Xs @ ( append_list_nat @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_53_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_54_same__append__eq,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,Zs: list_list_nat] :
( ( ( append_list_nat @ Xs @ Ys )
= ( append_list_nat @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_55_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_56_append__same__eq,axiom,
! [Ys: list_list_nat,Xs: list_list_nat,Zs: list_list_nat] :
( ( ( append_list_nat @ Ys @ Xs )
= ( append_list_nat @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_57_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_58_append__eq__append__conv2,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,Zs: list_list_nat,Ts: list_list_nat] :
( ( ( append_list_nat @ Xs @ Ys )
= ( append_list_nat @ Zs @ Ts ) )
= ( ? [Us: list_list_nat] :
( ( ( Xs
= ( append_list_nat @ Zs @ Us ) )
& ( ( append_list_nat @ Us @ Ys )
= Ts ) )
| ( ( ( append_list_nat @ Xs @ Us )
= Zs )
& ( Ys
= ( append_list_nat @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_59_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 ) )
= ( ? [Us: list_nat] :
( ( ( Xs
= ( append_nat @ Zs @ Us ) )
& ( ( append_nat @ Us @ Ys )
= Ts ) )
| ( ( ( append_nat @ Xs @ Us )
= Zs )
& ( Ys
= ( append_nat @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_60_append__eq__appendI,axiom,
! [Xs: list_list_nat,Xs1: list_list_nat,Zs: list_list_nat,Ys: list_list_nat,Us2: list_list_nat] :
( ( ( append_list_nat @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_list_nat @ Xs1 @ Us2 ) )
=> ( ( append_list_nat @ Xs @ Ys )
= ( append_list_nat @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_61_append__eq__appendI,axiom,
! [Xs: list_nat,Xs1: list_nat,Zs: list_nat,Ys: list_nat,Us2: list_nat] :
( ( ( append_nat @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_nat @ Xs1 @ Us2 ) )
=> ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_62_list__all2__antisym,axiom,
! [P: nat > nat > $o,Q: nat > nat > $o,Xs: list_nat,Ys: list_nat] :
( ! [X3: nat,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( ( Q @ Y3 @ X3 )
=> ( X3 = Y3 ) ) )
=> ( ( list_all2_nat_nat @ P @ Xs @ Ys )
=> ( ( list_all2_nat_nat @ Q @ Ys @ Xs )
=> ( Xs = Ys ) ) ) ) ).
% list_all2_antisym
thf(fact_63_list__all2__trans,axiom,
! [P1: nat > nat > $o,P2: nat > nat > $o,P3: nat > nat > $o,As: list_nat,Bs: list_nat,Cs: list_nat] :
( ! [A2: nat,B2: nat,C2: nat] :
( ( P1 @ A2 @ B2 )
=> ( ( P2 @ B2 @ C2 )
=> ( P3 @ A2 @ C2 ) ) )
=> ( ( list_all2_nat_nat @ P1 @ As @ Bs )
=> ( ( list_all2_nat_nat @ P2 @ Bs @ Cs )
=> ( list_all2_nat_nat @ P3 @ As @ Cs ) ) ) ) ).
% list_all2_trans
thf(fact_64_list__all2__refl,axiom,
! [P: nat > nat > $o,Xs: list_nat] :
( ! [X3: nat] : ( P @ X3 @ X3 )
=> ( list_all2_nat_nat @ P @ Xs @ Xs ) ) ).
% list_all2_refl
thf(fact_65_list__all2__mono,axiom,
! [P: nat > nat > $o,Xs: list_nat,Ys: list_nat,Q: nat > nat > $o] :
( ( list_all2_nat_nat @ P @ Xs @ Ys )
=> ( ! [Xs2: nat,Ys2: nat] :
( ( P @ Xs2 @ Ys2 )
=> ( Q @ Xs2 @ Ys2 ) )
=> ( list_all2_nat_nat @ Q @ Xs @ Ys ) ) ) ).
% list_all2_mono
thf(fact_66_list__all2__eq,axiom,
( ( ^ [Y4: list_nat,Z2: list_nat] : ( Y4 = Z2 ) )
= ( list_all2_nat_nat
@ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) ) ) ).
% list_all2_eq
thf(fact_67_list_Orel__refl,axiom,
! [Ra: nat > nat > $o,X: list_nat] :
( ! [X3: nat] : ( Ra @ X3 @ X3 )
=> ( list_all2_nat_nat @ Ra @ X @ X ) ) ).
% list.rel_refl
thf(fact_68_list_Orel__eq,axiom,
( ( list_all2_nat_nat
@ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [Y4: list_nat,Z2: list_nat] : ( Y4 = Z2 ) ) ) ).
% list.rel_eq
thf(fact_69_eq__Nil__appendI,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append_list_nat @ nil_list_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_70_eq__Nil__appendI,axiom,
! [Xs: list_Extended_enat,Ys: list_Extended_enat] :
( ( Xs = Ys )
=> ( Xs
= ( append_Extended_enat @ nil_Extended_enat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_71_eq__Nil__appendI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append_nat @ nil_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_72_append_Oleft__neutral,axiom,
! [A: list_list_nat] :
( ( append_list_nat @ nil_list_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_73_append_Oleft__neutral,axiom,
! [A: list_Extended_enat] :
( ( append_Extended_enat @ nil_Extended_enat @ A )
= A ) ).
% append.left_neutral
thf(fact_74_append_Oleft__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ nil_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_75_append__Nil,axiom,
! [Ys: list_list_nat] :
( ( append_list_nat @ nil_list_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_76_append__Nil,axiom,
! [Ys: list_Extended_enat] :
( ( append_Extended_enat @ nil_Extended_enat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_77_append__Nil,axiom,
! [Ys: list_nat] :
( ( append_nat @ nil_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_78_list_Octr__transfer_I1_J,axiom,
! [R: nat > list_nat > $o] : ( list_a5155878676884504761st_nat @ R @ nil_nat @ nil_list_nat ) ).
% list.ctr_transfer(1)
thf(fact_79_list_Octr__transfer_I1_J,axiom,
! [R: nat > extended_enat > $o] : ( list_a4052556031825172301d_enat @ R @ nil_nat @ nil_Extended_enat ) ).
% list.ctr_transfer(1)
thf(fact_80_list_Octr__transfer_I1_J,axiom,
! [R: list_nat > nat > $o] : ( list_a7394835724046789945at_nat @ R @ nil_list_nat @ nil_nat ) ).
% list.ctr_transfer(1)
thf(fact_81_list_Octr__transfer_I1_J,axiom,
! [R: list_nat > list_nat > $o] : ( list_a4684442626689269705st_nat @ R @ nil_list_nat @ nil_list_nat ) ).
% list.ctr_transfer(1)
thf(fact_82_list_Octr__transfer_I1_J,axiom,
! [R: list_nat > extended_enat > $o] : ( list_a9163222800480322621d_enat @ R @ nil_list_nat @ nil_Extended_enat ) ).
% list.ctr_transfer(1)
thf(fact_83_list_Octr__transfer_I1_J,axiom,
! [R: extended_enat > nat > $o] : ( list_a8195072124553474287at_nat @ R @ nil_Extended_enat @ nil_nat ) ).
% list.ctr_transfer(1)
thf(fact_84_list_Octr__transfer_I1_J,axiom,
! [R: extended_enat > list_nat > $o] : ( list_a7409325535056916607st_nat @ R @ nil_Extended_enat @ nil_list_nat ) ).
% list.ctr_transfer(1)
thf(fact_85_list_Octr__transfer_I1_J,axiom,
! [R: extended_enat > extended_enat > $o] : ( list_a8739900810786033671d_enat @ R @ nil_Extended_enat @ nil_Extended_enat ) ).
% list.ctr_transfer(1)
thf(fact_86_list_Octr__transfer_I1_J,axiom,
! [R: nat > nat > $o] : ( list_all2_nat_nat @ R @ nil_nat @ nil_nat ) ).
% list.ctr_transfer(1)
thf(fact_87_list__all2__appendI,axiom,
! [P: nat > list_nat > $o,A: list_nat,B: list_list_nat,C: list_nat,D: list_list_nat] :
( ( list_a5155878676884504761st_nat @ P @ A @ B )
=> ( ( list_a5155878676884504761st_nat @ P @ C @ D )
=> ( list_a5155878676884504761st_nat @ P @ ( append_nat @ A @ C ) @ ( append_list_nat @ B @ D ) ) ) ) ).
% list_all2_appendI
thf(fact_88_list__all2__appendI,axiom,
! [P: list_nat > nat > $o,A: list_list_nat,B: list_nat,C: list_list_nat,D: list_nat] :
( ( list_a7394835724046789945at_nat @ P @ A @ B )
=> ( ( list_a7394835724046789945at_nat @ P @ C @ D )
=> ( list_a7394835724046789945at_nat @ P @ ( append_list_nat @ A @ C ) @ ( append_nat @ B @ D ) ) ) ) ).
% list_all2_appendI
thf(fact_89_list__all2__appendI,axiom,
! [P: list_nat > list_nat > $o,A: list_list_nat,B: list_list_nat,C: list_list_nat,D: list_list_nat] :
( ( list_a4684442626689269705st_nat @ P @ A @ B )
=> ( ( list_a4684442626689269705st_nat @ P @ C @ D )
=> ( list_a4684442626689269705st_nat @ P @ ( append_list_nat @ A @ C ) @ ( append_list_nat @ B @ D ) ) ) ) ).
% list_all2_appendI
thf(fact_90_list__all2__appendI,axiom,
! [P: nat > nat > $o,A: list_nat,B: list_nat,C: list_nat,D: list_nat] :
( ( list_all2_nat_nat @ P @ A @ B )
=> ( ( list_all2_nat_nat @ P @ C @ D )
=> ( list_all2_nat_nat @ P @ ( append_nat @ A @ C ) @ ( append_nat @ B @ D ) ) ) ) ).
% list_all2_appendI
thf(fact_91_dual__order_Orefl,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% dual_order.refl
thf(fact_92_dual__order_Orefl,axiom,
! [A: list_nat > list_nat > $o] : ( ord_le6558929396352911974_nat_o @ A @ A ) ).
% dual_order.refl
thf(fact_93_dual__order_Orefl,axiom,
! [A: $o > extended_enat] : ( ord_le2787558655864224659d_enat @ A @ A ) ).
% dual_order.refl
thf(fact_94_dual__order_Orefl,axiom,
! [A: $o > nat] : ( ord_less_eq_o_nat @ A @ A ) ).
% dual_order.refl
thf(fact_95_dual__order_Orefl,axiom,
! [A: nat > nat > $o] : ( ord_le2646555220125990790_nat_o @ A @ A ) ).
% dual_order.refl
thf(fact_96_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_97_dual__order_Orefl,axiom,
! [A: extended_enat] : ( ord_le2932123472753598470d_enat @ A @ A ) ).
% dual_order.refl
thf(fact_98_order__refl,axiom,
! [X: num] : ( ord_less_eq_num @ X @ X ) ).
% order_refl
thf(fact_99_order__refl,axiom,
! [X: list_nat > list_nat > $o] : ( ord_le6558929396352911974_nat_o @ X @ X ) ).
% order_refl
thf(fact_100_order__refl,axiom,
! [X: $o > extended_enat] : ( ord_le2787558655864224659d_enat @ X @ X ) ).
% order_refl
thf(fact_101_order__refl,axiom,
! [X: $o > nat] : ( ord_less_eq_o_nat @ X @ X ) ).
% order_refl
thf(fact_102_order__refl,axiom,
! [X: nat > nat > $o] : ( ord_le2646555220125990790_nat_o @ X @ X ) ).
% order_refl
thf(fact_103_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_104_order__refl,axiom,
! [X: extended_enat] : ( ord_le2932123472753598470d_enat @ X @ X ) ).
% order_refl
thf(fact_105_list__ex1__simps_I1_J,axiom,
! [P: list_nat > $o] :
~ ( list_ex1_list_nat @ P @ nil_list_nat ) ).
% list_ex1_simps(1)
thf(fact_106_list__ex1__simps_I1_J,axiom,
! [P: extended_enat > $o] :
~ ( list_e2739505282040533167d_enat @ P @ nil_Extended_enat ) ).
% list_ex1_simps(1)
thf(fact_107_list__ex1__simps_I1_J,axiom,
! [P: nat > $o] :
~ ( list_ex1_nat @ P @ nil_nat ) ).
% list_ex1_simps(1)
thf(fact_108_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M ) )
=> ~ ! [M2: nat] :
( ( P @ M2 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M2 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_109_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_110_nat__le__linear,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
| ( ord_less_eq_nat @ N @ M3 ) ) ).
% nat_le_linear
thf(fact_111_le__antisym,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( ord_less_eq_nat @ N @ M3 )
=> ( M3 = N ) ) ) ).
% le_antisym
thf(fact_112_mem__Collect__eq,axiom,
! [A: list_nat,P: list_nat > $o] :
( ( member_list_nat2 @ A @ ( collect_list_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_113_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat2 @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_114_Collect__mem__eq,axiom,
! [A3: set_list_nat] :
( ( collect_list_nat
@ ^ [X2: list_nat] : ( member_list_nat2 @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_115_Collect__mem__eq,axiom,
! [A3: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat2 @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_116_eq__imp__le,axiom,
! [M3: nat,N: nat] :
( ( M3 = N )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% eq_imp_le
thf(fact_117_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_118_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_119_list_Orel__mono,axiom,
! [R: list_nat > list_nat > $o,Ra: list_nat > list_nat > $o] :
( ( ord_le6558929396352911974_nat_o @ R @ Ra )
=> ( ord_le2031171171200246342_nat_o @ ( list_a4684442626689269705st_nat @ R ) @ ( list_a4684442626689269705st_nat @ Ra ) ) ) ).
% list.rel_mono
thf(fact_120_list_Orel__mono,axiom,
! [R: nat > nat > $o,Ra: nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ R @ Ra )
=> ( ord_le6558929396352911974_nat_o @ ( list_all2_nat_nat @ R ) @ ( list_all2_nat_nat @ Ra ) ) ) ).
% list.rel_mono
thf(fact_121_nle__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_eq_num @ A @ B ) )
= ( ( ord_less_eq_num @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_122_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_123_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_124_le__cases3,axiom,
! [X: num,Y: num,Z: num] :
( ( ( ord_less_eq_num @ X @ Y )
=> ~ ( ord_less_eq_num @ Y @ Z ) )
=> ( ( ( ord_less_eq_num @ Y @ X )
=> ~ ( ord_less_eq_num @ X @ Z ) )
=> ( ( ( ord_less_eq_num @ X @ Z )
=> ~ ( ord_less_eq_num @ Z @ Y ) )
=> ( ( ( ord_less_eq_num @ Z @ Y )
=> ~ ( ord_less_eq_num @ Y @ X ) )
=> ( ( ( ord_less_eq_num @ Y @ Z )
=> ~ ( ord_less_eq_num @ Z @ X ) )
=> ~ ( ( ord_less_eq_num @ Z @ X )
=> ~ ( ord_less_eq_num @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_125_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_126_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_127_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: num,Z2: num] : ( Y4 = Z2 ) )
= ( ^ [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
& ( ord_less_eq_num @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_128_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: list_nat > list_nat > $o,Z2: list_nat > list_nat > $o] : ( Y4 = Z2 ) )
= ( ^ [X2: list_nat > list_nat > $o,Y2: list_nat > list_nat > $o] :
( ( ord_le6558929396352911974_nat_o @ X2 @ Y2 )
& ( ord_le6558929396352911974_nat_o @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_129_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: $o > extended_enat,Z2: $o > extended_enat] : ( Y4 = Z2 ) )
= ( ^ [X2: $o > extended_enat,Y2: $o > extended_enat] :
( ( ord_le2787558655864224659d_enat @ X2 @ Y2 )
& ( ord_le2787558655864224659d_enat @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_130_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: $o > nat,Z2: $o > nat] : ( Y4 = Z2 ) )
= ( ^ [X2: $o > nat,Y2: $o > nat] :
( ( ord_less_eq_o_nat @ X2 @ Y2 )
& ( ord_less_eq_o_nat @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_131_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat > nat > $o,Z2: nat > nat > $o] : ( Y4 = Z2 ) )
= ( ^ [X2: nat > nat > $o,Y2: nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ X2 @ Y2 )
& ( ord_le2646555220125990790_nat_o @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_132_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_133_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: extended_enat,Z2: extended_enat] : ( Y4 = Z2 ) )
= ( ^ [X2: extended_enat,Y2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X2 @ Y2 )
& ( ord_le2932123472753598470d_enat @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_134_ord__eq__le__trans,axiom,
! [A: num,B: num,C: num] :
( ( A = B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_135_ord__eq__le__trans,axiom,
! [A: list_nat > list_nat > $o,B: list_nat > list_nat > $o,C: list_nat > list_nat > $o] :
( ( A = B )
=> ( ( ord_le6558929396352911974_nat_o @ B @ C )
=> ( ord_le6558929396352911974_nat_o @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_136_ord__eq__le__trans,axiom,
! [A: $o > extended_enat,B: $o > extended_enat,C: $o > extended_enat] :
( ( A = B )
=> ( ( ord_le2787558655864224659d_enat @ B @ C )
=> ( ord_le2787558655864224659d_enat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_137_ord__eq__le__trans,axiom,
! [A: $o > nat,B: $o > nat,C: $o > nat] :
( ( A = B )
=> ( ( ord_less_eq_o_nat @ B @ C )
=> ( ord_less_eq_o_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_138_ord__eq__le__trans,axiom,
! [A: nat > nat > $o,B: nat > nat > $o,C: nat > nat > $o] :
( ( A = B )
=> ( ( ord_le2646555220125990790_nat_o @ B @ C )
=> ( ord_le2646555220125990790_nat_o @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_139_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_140_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_141_ord__le__eq__trans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_142_ord__le__eq__trans,axiom,
! [A: list_nat > list_nat > $o,B: list_nat > list_nat > $o,C: list_nat > list_nat > $o] :
( ( ord_le6558929396352911974_nat_o @ A @ B )
=> ( ( B = C )
=> ( ord_le6558929396352911974_nat_o @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_143_ord__le__eq__trans,axiom,
! [A: $o > extended_enat,B: $o > extended_enat,C: $o > extended_enat] :
( ( ord_le2787558655864224659d_enat @ A @ B )
=> ( ( B = C )
=> ( ord_le2787558655864224659d_enat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_144_ord__le__eq__trans,axiom,
! [A: $o > nat,B: $o > nat,C: $o > nat] :
( ( ord_less_eq_o_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_o_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_145_ord__le__eq__trans,axiom,
! [A: nat > nat > $o,B: nat > nat > $o,C: nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ A @ B )
=> ( ( B = C )
=> ( ord_le2646555220125990790_nat_o @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_146_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_147_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_148_order__antisym,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_149_order__antisym,axiom,
! [X: list_nat > list_nat > $o,Y: list_nat > list_nat > $o] :
( ( ord_le6558929396352911974_nat_o @ X @ Y )
=> ( ( ord_le6558929396352911974_nat_o @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_150_order__antisym,axiom,
! [X: $o > extended_enat,Y: $o > extended_enat] :
( ( ord_le2787558655864224659d_enat @ X @ Y )
=> ( ( ord_le2787558655864224659d_enat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_151_order__antisym,axiom,
! [X: $o > nat,Y: $o > nat] :
( ( ord_less_eq_o_nat @ X @ Y )
=> ( ( ord_less_eq_o_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_152_order__antisym,axiom,
! [X: nat > nat > $o,Y: nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ X @ Y )
=> ( ( ord_le2646555220125990790_nat_o @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_153_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_154_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_155_order_Otrans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% order.trans
thf(fact_156_order_Otrans,axiom,
! [A: list_nat > list_nat > $o,B: list_nat > list_nat > $o,C: list_nat > list_nat > $o] :
( ( ord_le6558929396352911974_nat_o @ A @ B )
=> ( ( ord_le6558929396352911974_nat_o @ B @ C )
=> ( ord_le6558929396352911974_nat_o @ A @ C ) ) ) ).
% order.trans
thf(fact_157_order_Otrans,axiom,
! [A: $o > extended_enat,B: $o > extended_enat,C: $o > extended_enat] :
( ( ord_le2787558655864224659d_enat @ A @ B )
=> ( ( ord_le2787558655864224659d_enat @ B @ C )
=> ( ord_le2787558655864224659d_enat @ A @ C ) ) ) ).
% order.trans
thf(fact_158_order_Otrans,axiom,
! [A: $o > nat,B: $o > nat,C: $o > nat] :
( ( ord_less_eq_o_nat @ A @ B )
=> ( ( ord_less_eq_o_nat @ B @ C )
=> ( ord_less_eq_o_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_159_order_Otrans,axiom,
! [A: nat > nat > $o,B: nat > nat > $o,C: nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ A @ B )
=> ( ( ord_le2646555220125990790_nat_o @ B @ C )
=> ( ord_le2646555220125990790_nat_o @ A @ C ) ) ) ).
% order.trans
thf(fact_160_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_161_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_162_order__trans,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ Z )
=> ( ord_less_eq_num @ X @ Z ) ) ) ).
% order_trans
thf(fact_163_order__trans,axiom,
! [X: list_nat > list_nat > $o,Y: list_nat > list_nat > $o,Z: list_nat > list_nat > $o] :
( ( ord_le6558929396352911974_nat_o @ X @ Y )
=> ( ( ord_le6558929396352911974_nat_o @ Y @ Z )
=> ( ord_le6558929396352911974_nat_o @ X @ Z ) ) ) ).
% order_trans
thf(fact_164_order__trans,axiom,
! [X: $o > extended_enat,Y: $o > extended_enat,Z: $o > extended_enat] :
( ( ord_le2787558655864224659d_enat @ X @ Y )
=> ( ( ord_le2787558655864224659d_enat @ Y @ Z )
=> ( ord_le2787558655864224659d_enat @ X @ Z ) ) ) ).
% order_trans
thf(fact_165_order__trans,axiom,
! [X: $o > nat,Y: $o > nat,Z: $o > nat] :
( ( ord_less_eq_o_nat @ X @ Y )
=> ( ( ord_less_eq_o_nat @ Y @ Z )
=> ( ord_less_eq_o_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_166_order__trans,axiom,
! [X: nat > nat > $o,Y: nat > nat > $o,Z: nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ X @ Y )
=> ( ( ord_le2646555220125990790_nat_o @ Y @ Z )
=> ( ord_le2646555220125990790_nat_o @ X @ Z ) ) ) ).
% order_trans
thf(fact_167_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_168_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_169_linorder__wlog,axiom,
! [P: num > num > $o,A: num,B: num] :
( ! [A2: num,B2: num] :
( ( ord_less_eq_num @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: num,B2: num] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_170_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: nat,B2: nat] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_171_linorder__wlog,axiom,
! [P: extended_enat > extended_enat > $o,A: extended_enat,B: extended_enat] :
( ! [A2: extended_enat,B2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: extended_enat,B2: extended_enat] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_172_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: num,Z2: num] : ( Y4 = Z2 ) )
= ( ^ [A4: num,B3: num] :
( ( ord_less_eq_num @ B3 @ A4 )
& ( ord_less_eq_num @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_173_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: list_nat > list_nat > $o,Z2: list_nat > list_nat > $o] : ( Y4 = Z2 ) )
= ( ^ [A4: list_nat > list_nat > $o,B3: list_nat > list_nat > $o] :
( ( ord_le6558929396352911974_nat_o @ B3 @ A4 )
& ( ord_le6558929396352911974_nat_o @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_174_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: $o > extended_enat,Z2: $o > extended_enat] : ( Y4 = Z2 ) )
= ( ^ [A4: $o > extended_enat,B3: $o > extended_enat] :
( ( ord_le2787558655864224659d_enat @ B3 @ A4 )
& ( ord_le2787558655864224659d_enat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_175_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: $o > nat,Z2: $o > nat] : ( Y4 = Z2 ) )
= ( ^ [A4: $o > nat,B3: $o > nat] :
( ( ord_less_eq_o_nat @ B3 @ A4 )
& ( ord_less_eq_o_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_176_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat > nat > $o,Z2: nat > nat > $o] : ( Y4 = Z2 ) )
= ( ^ [A4: nat > nat > $o,B3: nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ B3 @ A4 )
& ( ord_le2646555220125990790_nat_o @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_177_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_178_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: extended_enat,Z2: extended_enat] : ( Y4 = Z2 ) )
= ( ^ [A4: extended_enat,B3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B3 @ A4 )
& ( ord_le2932123472753598470d_enat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_179_dual__order_Oantisym,axiom,
! [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_180_dual__order_Oantisym,axiom,
! [B: list_nat > list_nat > $o,A: list_nat > list_nat > $o] :
( ( ord_le6558929396352911974_nat_o @ B @ A )
=> ( ( ord_le6558929396352911974_nat_o @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_181_dual__order_Oantisym,axiom,
! [B: $o > extended_enat,A: $o > extended_enat] :
( ( ord_le2787558655864224659d_enat @ B @ A )
=> ( ( ord_le2787558655864224659d_enat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_182_dual__order_Oantisym,axiom,
! [B: $o > nat,A: $o > nat] :
( ( ord_less_eq_o_nat @ B @ A )
=> ( ( ord_less_eq_o_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_183_dual__order_Oantisym,axiom,
! [B: nat > nat > $o,A: nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ B @ A )
=> ( ( ord_le2646555220125990790_nat_o @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_184_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_185_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_186_dual__order_Otrans,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_eq_num @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_187_dual__order_Otrans,axiom,
! [B: list_nat > list_nat > $o,A: list_nat > list_nat > $o,C: list_nat > list_nat > $o] :
( ( ord_le6558929396352911974_nat_o @ B @ A )
=> ( ( ord_le6558929396352911974_nat_o @ C @ B )
=> ( ord_le6558929396352911974_nat_o @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_188_dual__order_Otrans,axiom,
! [B: $o > extended_enat,A: $o > extended_enat,C: $o > extended_enat] :
( ( ord_le2787558655864224659d_enat @ B @ A )
=> ( ( ord_le2787558655864224659d_enat @ C @ B )
=> ( ord_le2787558655864224659d_enat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_189_dual__order_Otrans,axiom,
! [B: $o > nat,A: $o > nat,C: $o > nat] :
( ( ord_less_eq_o_nat @ B @ A )
=> ( ( ord_less_eq_o_nat @ C @ B )
=> ( ord_less_eq_o_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_190_dual__order_Otrans,axiom,
! [B: nat > nat > $o,A: nat > nat > $o,C: nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ B @ A )
=> ( ( ord_le2646555220125990790_nat_o @ C @ B )
=> ( ord_le2646555220125990790_nat_o @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_191_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_192_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_193_antisym,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_194_antisym,axiom,
! [A: list_nat > list_nat > $o,B: list_nat > list_nat > $o] :
( ( ord_le6558929396352911974_nat_o @ A @ B )
=> ( ( ord_le6558929396352911974_nat_o @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_195_antisym,axiom,
! [A: $o > extended_enat,B: $o > extended_enat] :
( ( ord_le2787558655864224659d_enat @ A @ B )
=> ( ( ord_le2787558655864224659d_enat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_196_antisym,axiom,
! [A: $o > nat,B: $o > nat] :
( ( ord_less_eq_o_nat @ A @ B )
=> ( ( ord_less_eq_o_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_197_antisym,axiom,
! [A: nat > nat > $o,B: nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ A @ B )
=> ( ( ord_le2646555220125990790_nat_o @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_198_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_199_antisym,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_200_le__funD,axiom,
! [F: list_nat > list_nat > $o,G: list_nat > list_nat > $o,X: list_nat] :
( ( ord_le6558929396352911974_nat_o @ F @ G )
=> ( ord_le1520216061033275535_nat_o @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funD
thf(fact_201_le__funD,axiom,
! [F: $o > extended_enat,G: $o > extended_enat,X: $o] :
( ( ord_le2787558655864224659d_enat @ F @ G )
=> ( ord_le2932123472753598470d_enat @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funD
thf(fact_202_le__funD,axiom,
! [F: $o > nat,G: $o > nat,X: $o] :
( ( ord_less_eq_o_nat @ F @ G )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funD
thf(fact_203_le__funD,axiom,
! [F: nat > nat > $o,G: nat > nat > $o,X: nat] :
( ( ord_le2646555220125990790_nat_o @ F @ G )
=> ( ord_less_eq_nat_o @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funD
thf(fact_204_le__funE,axiom,
! [F: list_nat > list_nat > $o,G: list_nat > list_nat > $o,X: list_nat] :
( ( ord_le6558929396352911974_nat_o @ F @ G )
=> ( ord_le1520216061033275535_nat_o @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funE
thf(fact_205_le__funE,axiom,
! [F: $o > extended_enat,G: $o > extended_enat,X: $o] :
( ( ord_le2787558655864224659d_enat @ F @ G )
=> ( ord_le2932123472753598470d_enat @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funE
thf(fact_206_le__funE,axiom,
! [F: $o > nat,G: $o > nat,X: $o] :
( ( ord_less_eq_o_nat @ F @ G )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funE
thf(fact_207_le__funE,axiom,
! [F: nat > nat > $o,G: nat > nat > $o,X: nat] :
( ( ord_le2646555220125990790_nat_o @ F @ G )
=> ( ord_less_eq_nat_o @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funE
thf(fact_208_le__funI,axiom,
! [F: list_nat > list_nat > $o,G: list_nat > list_nat > $o] :
( ! [X3: list_nat] : ( ord_le1520216061033275535_nat_o @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_le6558929396352911974_nat_o @ F @ G ) ) ).
% le_funI
thf(fact_209_le__funI,axiom,
! [F: nat > nat > $o,G: nat > nat > $o] :
( ! [X3: nat] : ( ord_less_eq_nat_o @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_le2646555220125990790_nat_o @ F @ G ) ) ).
% le_funI
thf(fact_210_le__funI,axiom,
! [F: $o > nat,G: $o > nat] :
( ! [X3: $o] : ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq_o_nat @ F @ G ) ) ).
% le_funI
thf(fact_211_le__funI,axiom,
! [F: $o > extended_enat,G: $o > extended_enat] :
( ! [X3: $o] : ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_le2787558655864224659d_enat @ F @ G ) ) ).
% le_funI
thf(fact_212_le__fun__def,axiom,
( ord_le6558929396352911974_nat_o
= ( ^ [F2: list_nat > list_nat > $o,G2: list_nat > list_nat > $o] :
! [X2: list_nat] : ( ord_le1520216061033275535_nat_o @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ).
% le_fun_def
thf(fact_213_le__fun__def,axiom,
( ord_le2787558655864224659d_enat
= ( ^ [F2: $o > extended_enat,G2: $o > extended_enat] :
! [X2: $o] : ( ord_le2932123472753598470d_enat @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ).
% le_fun_def
thf(fact_214_le__fun__def,axiom,
( ord_less_eq_o_nat
= ( ^ [F2: $o > nat,G2: $o > nat] :
! [X2: $o] : ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ).
% le_fun_def
thf(fact_215_le__fun__def,axiom,
( ord_le2646555220125990790_nat_o
= ( ^ [F2: nat > nat > $o,G2: nat > nat > $o] :
! [X2: nat] : ( ord_less_eq_nat_o @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ).
% le_fun_def
thf(fact_216_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: num,Z2: num] : ( Y4 = Z2 ) )
= ( ^ [A4: num,B3: num] :
( ( ord_less_eq_num @ A4 @ B3 )
& ( ord_less_eq_num @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_217_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: list_nat > list_nat > $o,Z2: list_nat > list_nat > $o] : ( Y4 = Z2 ) )
= ( ^ [A4: list_nat > list_nat > $o,B3: list_nat > list_nat > $o] :
( ( ord_le6558929396352911974_nat_o @ A4 @ B3 )
& ( ord_le6558929396352911974_nat_o @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_218_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: $o > extended_enat,Z2: $o > extended_enat] : ( Y4 = Z2 ) )
= ( ^ [A4: $o > extended_enat,B3: $o > extended_enat] :
( ( ord_le2787558655864224659d_enat @ A4 @ B3 )
& ( ord_le2787558655864224659d_enat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_219_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: $o > nat,Z2: $o > nat] : ( Y4 = Z2 ) )
= ( ^ [A4: $o > nat,B3: $o > nat] :
( ( ord_less_eq_o_nat @ A4 @ B3 )
& ( ord_less_eq_o_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_220_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat > nat > $o,Z2: nat > nat > $o] : ( Y4 = Z2 ) )
= ( ^ [A4: nat > nat > $o,B3: nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ A4 @ B3 )
& ( ord_le2646555220125990790_nat_o @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_221_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_222_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: extended_enat,Z2: extended_enat] : ( Y4 = Z2 ) )
= ( ^ [A4: extended_enat,B3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A4 @ B3 )
& ( ord_le2932123472753598470d_enat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_223_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_224_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,Y3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_225_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,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le2932123472753598470d_enat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_226_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,Y3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le2932123472753598470d_enat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_227_order__subst1,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_228_order__subst1,axiom,
! [A: extended_enat,F: num > extended_enat,B: num,C: num] :
( ( ord_le2932123472753598470d_enat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le2932123472753598470d_enat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_229_order__subst1,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_230_order__subst1,axiom,
! [A: num,F: extended_enat > num,B: extended_enat,C: extended_enat] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ! [X3: extended_enat,Y3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_231_order__subst1,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_232_order__subst1,axiom,
! [A: nat,F: ( $o > extended_enat ) > nat,B: $o > extended_enat,C: $o > extended_enat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le2787558655864224659d_enat @ B @ C )
=> ( ! [X3: $o > extended_enat,Y3: $o > extended_enat] :
( ( ord_le2787558655864224659d_enat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_233_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_234_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,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_235_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,Y3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_236_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,Y3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_237_order__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_238_order__subst2,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > num,C: num] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X3: extended_enat,Y3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_239_order__subst2,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_240_order__subst2,axiom,
! [A: num,B: num,F: num > extended_enat,C: extended_enat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_241_order__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_242_order__subst2,axiom,
! [A: nat,B: nat,F: nat > $o > extended_enat,C: $o > extended_enat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le2787558655864224659d_enat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le2787558655864224659d_enat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le2787558655864224659d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_243_order__eq__refl,axiom,
! [X: num,Y: num] :
( ( X = Y )
=> ( ord_less_eq_num @ X @ Y ) ) ).
% order_eq_refl
thf(fact_244_order__eq__refl,axiom,
! [X: list_nat > list_nat > $o,Y: list_nat > list_nat > $o] :
( ( X = Y )
=> ( ord_le6558929396352911974_nat_o @ X @ Y ) ) ).
% order_eq_refl
thf(fact_245_order__eq__refl,axiom,
! [X: $o > extended_enat,Y: $o > extended_enat] :
( ( X = Y )
=> ( ord_le2787558655864224659d_enat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_246_order__eq__refl,axiom,
! [X: $o > nat,Y: $o > nat] :
( ( X = Y )
=> ( ord_less_eq_o_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_247_order__eq__refl,axiom,
! [X: nat > nat > $o,Y: nat > nat > $o] :
( ( X = Y )
=> ( ord_le2646555220125990790_nat_o @ X @ Y ) ) ).
% order_eq_refl
thf(fact_248_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_249_order__eq__refl,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( X = Y )
=> ( ord_le2932123472753598470d_enat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_250_linorder__linear,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
| ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_linear
thf(fact_251_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_252_linorder__linear,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X @ Y )
| ( ord_le2932123472753598470d_enat @ Y @ X ) ) ).
% linorder_linear
thf(fact_253_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_254_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,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le2932123472753598470d_enat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_255_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,Y3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_256_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,Y3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le2932123472753598470d_enat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_257_ord__eq__le__subst,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_258_ord__eq__le__subst,axiom,
! [A: num,F: extended_enat > num,B: extended_enat,C: extended_enat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2932123472753598470d_enat @ B @ C )
=> ( ! [X3: extended_enat,Y3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_259_ord__eq__le__subst,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_260_ord__eq__le__subst,axiom,
! [A: extended_enat,F: num > extended_enat,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le2932123472753598470d_enat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_261_ord__eq__le__subst,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_262_ord__eq__le__subst,axiom,
! [A: $o > extended_enat,F: nat > $o > extended_enat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le2787558655864224659d_enat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le2787558655864224659d_enat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_263_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_264_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,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_265_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,Y3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_266_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,Y3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_267_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_268_ord__le__eq__subst,axiom,
! [A: extended_enat,B: extended_enat,F: extended_enat > num,C: num] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: extended_enat,Y3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_269_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_270_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > extended_enat,C: extended_enat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le2932123472753598470d_enat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_271_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_272_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > $o > extended_enat,C: $o > extended_enat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le2787558655864224659d_enat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le2787558655864224659d_enat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_273_linorder__le__cases,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_274_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_275_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_276_order__antisym__conv,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ( ( ord_less_eq_num @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_277_order__antisym__conv,axiom,
! [Y: list_nat > list_nat > $o,X: list_nat > list_nat > $o] :
( ( ord_le6558929396352911974_nat_o @ Y @ X )
=> ( ( ord_le6558929396352911974_nat_o @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_278_order__antisym__conv,axiom,
! [Y: $o > extended_enat,X: $o > extended_enat] :
( ( ord_le2787558655864224659d_enat @ Y @ X )
=> ( ( ord_le2787558655864224659d_enat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_279_order__antisym__conv,axiom,
! [Y: $o > nat,X: $o > nat] :
( ( ord_less_eq_o_nat @ Y @ X )
=> ( ( ord_less_eq_o_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_280_order__antisym__conv,axiom,
! [Y: nat > nat > $o,X: nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ Y @ X )
=> ( ( ord_le2646555220125990790_nat_o @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_281_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_282_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_283_le__Nil,axiom,
! [X: list_list_nat] :
( ( ord_le6806709344281226192st_nat @ X @ nil_list_nat )
= ( X = nil_list_nat ) ) ).
% le_Nil
thf(fact_284_le__Nil,axiom,
! [X: list_Extended_enat] :
( ( ord_le769749158434378124d_enat @ X @ nil_Extended_enat )
= ( X = nil_Extended_enat ) ) ).
% le_Nil
thf(fact_285_le__Nil,axiom,
! [X: list_nat] :
( ( ord_less_eq_list_nat @ X @ nil_nat )
= ( X = nil_nat ) ) ).
% le_Nil
thf(fact_286_GreatestI2__order,axiom,
! [P: num > $o,X: num,Q: num > $o] :
( ( P @ X )
=> ( ! [Y3: num] :
( ( P @ Y3 )
=> ( ord_less_eq_num @ Y3 @ X ) )
=> ( ! [X3: num] :
( ( P @ X3 )
=> ( ! [Y5: num] :
( ( P @ Y5 )
=> ( ord_less_eq_num @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest_num @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_287_GreatestI2__order,axiom,
! [P: ( list_nat > list_nat > $o ) > $o,X: list_nat > list_nat > $o,Q: ( list_nat > list_nat > $o ) > $o] :
( ( P @ X )
=> ( ! [Y3: list_nat > list_nat > $o] :
( ( P @ Y3 )
=> ( ord_le6558929396352911974_nat_o @ Y3 @ X ) )
=> ( ! [X3: list_nat > list_nat > $o] :
( ( P @ X3 )
=> ( ! [Y5: list_nat > list_nat > $o] :
( ( P @ Y5 )
=> ( ord_le6558929396352911974_nat_o @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_4166164572150370591_nat_o @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_288_GreatestI2__order,axiom,
! [P: ( $o > extended_enat ) > $o,X: $o > extended_enat,Q: ( $o > extended_enat ) > $o] :
( ( P @ X )
=> ( ! [Y3: $o > extended_enat] :
( ( P @ Y3 )
=> ( ord_le2787558655864224659d_enat @ Y3 @ X ) )
=> ( ! [X3: $o > extended_enat] :
( ( P @ X3 )
=> ( ! [Y5: $o > extended_enat] :
( ( P @ Y5 )
=> ( ord_le2787558655864224659d_enat @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_7406502399511290778d_enat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_289_GreatestI2__order,axiom,
! [P: ( $o > nat ) > $o,X: $o > nat,Q: ( $o > nat ) > $o] :
( ( P @ X )
=> ( ! [Y3: $o > nat] :
( ( P @ Y3 )
=> ( ord_less_eq_o_nat @ Y3 @ X ) )
=> ( ! [X3: $o > nat] :
( ( P @ X3 )
=> ( ! [Y5: $o > nat] :
( ( P @ Y5 )
=> ( ord_less_eq_o_nat @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest_o_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_290_GreatestI2__order,axiom,
! [P: ( nat > nat > $o ) > $o,X: nat > nat > $o,Q: ( nat > nat > $o ) > $o] :
( ( P @ X )
=> ( ! [Y3: nat > nat > $o] :
( ( P @ Y3 )
=> ( ord_le2646555220125990790_nat_o @ Y3 @ X ) )
=> ( ! [X3: nat > nat > $o] :
( ( P @ X3 )
=> ( ! [Y5: nat > nat > $o] :
( ( P @ Y5 )
=> ( ord_le2646555220125990790_nat_o @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_1729629668087260991_nat_o @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_291_GreatestI2__order,axiom,
! [P: extended_enat > $o,X: extended_enat,Q: extended_enat > $o] :
( ( P @ X )
=> ( ! [Y3: extended_enat] :
( ( P @ Y3 )
=> ( ord_le2932123472753598470d_enat @ Y3 @ X ) )
=> ( ! [X3: extended_enat] :
( ( P @ X3 )
=> ( ! [Y5: extended_enat] :
( ( P @ Y5 )
=> ( ord_le2932123472753598470d_enat @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_2428742583041560895d_enat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_292_GreatestI2__order,axiom,
! [P: nat > $o,X: nat,Q: nat > $o] :
( ( P @ X )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X ) )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_293_Greatest__equality,axiom,
! [P: num > $o,X: num] :
( ( P @ X )
=> ( ! [Y3: num] :
( ( P @ Y3 )
=> ( ord_less_eq_num @ Y3 @ X ) )
=> ( ( order_Greatest_num @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_294_Greatest__equality,axiom,
! [P: ( list_nat > list_nat > $o ) > $o,X: list_nat > list_nat > $o] :
( ( P @ X )
=> ( ! [Y3: list_nat > list_nat > $o] :
( ( P @ Y3 )
=> ( ord_le6558929396352911974_nat_o @ Y3 @ X ) )
=> ( ( order_4166164572150370591_nat_o @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_295_Greatest__equality,axiom,
! [P: ( $o > extended_enat ) > $o,X: $o > extended_enat] :
( ( P @ X )
=> ( ! [Y3: $o > extended_enat] :
( ( P @ Y3 )
=> ( ord_le2787558655864224659d_enat @ Y3 @ X ) )
=> ( ( order_7406502399511290778d_enat @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_296_Greatest__equality,axiom,
! [P: ( $o > nat ) > $o,X: $o > nat] :
( ( P @ X )
=> ( ! [Y3: $o > nat] :
( ( P @ Y3 )
=> ( ord_less_eq_o_nat @ Y3 @ X ) )
=> ( ( order_Greatest_o_nat @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_297_Greatest__equality,axiom,
! [P: ( nat > nat > $o ) > $o,X: nat > nat > $o] :
( ( P @ X )
=> ( ! [Y3: nat > nat > $o] :
( ( P @ Y3 )
=> ( ord_le2646555220125990790_nat_o @ Y3 @ X ) )
=> ( ( order_1729629668087260991_nat_o @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_298_Greatest__equality,axiom,
! [P: extended_enat > $o,X: extended_enat] :
( ( P @ X )
=> ( ! [Y3: extended_enat] :
( ( P @ Y3 )
=> ( ord_le2932123472753598470d_enat @ Y3 @ X ) )
=> ( ( order_2428742583041560895d_enat @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_299_Greatest__equality,axiom,
! [P: nat > $o,X: nat] :
( ( P @ X )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X ) )
=> ( ( order_Greatest_nat @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_300_Nil__le__Cons,axiom,
! [X: list_list_nat] : ( ord_le6806709344281226192st_nat @ nil_list_nat @ X ) ).
% Nil_le_Cons
thf(fact_301_Nil__le__Cons,axiom,
! [X: list_Extended_enat] : ( ord_le769749158434378124d_enat @ nil_Extended_enat @ X ) ).
% Nil_le_Cons
thf(fact_302_Nil__le__Cons,axiom,
! [X: list_nat] : ( ord_less_eq_list_nat @ nil_nat @ X ) ).
% Nil_le_Cons
thf(fact_303_member__rec_I2_J,axiom,
! [Y: list_nat] :
~ ( member_list_nat @ nil_list_nat @ Y ) ).
% member_rec(2)
thf(fact_304_member__rec_I2_J,axiom,
! [Y: extended_enat] :
~ ( member_Extended_enat @ nil_Extended_enat @ Y ) ).
% member_rec(2)
thf(fact_305_member__rec_I2_J,axiom,
! [Y: nat] :
~ ( member_nat @ nil_nat @ Y ) ).
% member_rec(2)
thf(fact_306_bind__simps_I1_J,axiom,
! [F: nat > list_list_nat] :
( ( bind_nat_list_nat @ nil_nat @ F )
= nil_list_nat ) ).
% bind_simps(1)
thf(fact_307_bind__simps_I1_J,axiom,
! [F: nat > list_Extended_enat] :
( ( bind_n5590761592452524365d_enat @ nil_nat @ F )
= nil_Extended_enat ) ).
% bind_simps(1)
thf(fact_308_bind__simps_I1_J,axiom,
! [F: list_nat > list_nat] :
( ( bind_list_nat_nat @ nil_list_nat @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_309_bind__simps_I1_J,axiom,
! [F: list_nat > list_list_nat] :
( ( bind_l7796378977173581257st_nat @ nil_list_nat @ F )
= nil_list_nat ) ).
% bind_simps(1)
thf(fact_310_bind__simps_I1_J,axiom,
! [F: list_nat > list_Extended_enat] :
( ( bind_l3791100948236736061d_enat @ nil_list_nat @ F )
= nil_Extended_enat ) ).
% bind_simps(1)
thf(fact_311_bind__simps_I1_J,axiom,
! [F: extended_enat > list_nat] :
( ( bind_E509905648326050543at_nat @ nil_Extended_enat @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_312_bind__simps_I1_J,axiom,
! [F: extended_enat > list_list_nat] :
( ( bind_E2037203682813330047st_nat @ nil_Extended_enat @ F )
= nil_list_nat ) ).
% bind_simps(1)
thf(fact_313_bind__simps_I1_J,axiom,
! [F: extended_enat > list_Extended_enat] :
( ( bind_E4971823231018073607d_enat @ nil_Extended_enat @ F )
= nil_Extended_enat ) ).
% bind_simps(1)
thf(fact_314_bind__simps_I1_J,axiom,
! [F: nat > list_nat] :
( ( bind_nat_nat @ nil_nat @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_315_verit__comp__simplify1_I2_J,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_316_verit__comp__simplify1_I2_J,axiom,
! [A: list_nat > list_nat > $o] : ( ord_le6558929396352911974_nat_o @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_317_verit__comp__simplify1_I2_J,axiom,
! [A: $o > extended_enat] : ( ord_le2787558655864224659d_enat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_318_verit__comp__simplify1_I2_J,axiom,
! [A: $o > nat] : ( ord_less_eq_o_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_319_verit__comp__simplify1_I2_J,axiom,
! [A: nat > nat > $o] : ( ord_le2646555220125990790_nat_o @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_320_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_321_verit__comp__simplify1_I2_J,axiom,
! [A: extended_enat] : ( ord_le2932123472753598470d_enat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_322_verit__la__disequality,axiom,
! [A: num,B: num] :
( ( A = B )
| ~ ( ord_less_eq_num @ A @ B )
| ~ ( ord_less_eq_num @ B @ A ) ) ).
% verit_la_disequality
thf(fact_323_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_324_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_325_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_num
= ( ^ [X5: $o > num,Y6: $o > num] :
( ( ord_less_eq_num @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_num @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_326_le__rel__bool__arg__iff,axiom,
( ord_le286611365431560435_nat_o
= ( ^ [X5: $o > list_nat > list_nat > $o,Y6: $o > list_nat > list_nat > $o] :
( ( ord_le6558929396352911974_nat_o @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le6558929396352911974_nat_o @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_327_le__rel__bool__arg__iff,axiom,
( ord_le5395741666527326556d_enat
= ( ^ [X5: $o > $o > extended_enat,Y6: $o > $o > extended_enat] :
( ( ord_le2787558655864224659d_enat @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le2787558655864224659d_enat @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_328_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_o_nat
= ( ^ [X5: $o > $o > nat,Y6: $o > $o > nat] :
( ( ord_less_eq_o_nat @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_o_nat @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_329_le__rel__bool__arg__iff,axiom,
( ord_le7862453073132658707_nat_o
= ( ^ [X5: $o > nat > nat > $o,Y6: $o > nat > nat > $o] :
( ( ord_le2646555220125990790_nat_o @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le2646555220125990790_nat_o @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_330_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_nat
= ( ^ [X5: $o > nat,Y6: $o > nat] :
( ( ord_less_eq_nat @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_nat @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_331_le__rel__bool__arg__iff,axiom,
( ord_le2787558655864224659d_enat
= ( ^ [X5: $o > extended_enat,Y6: $o > extended_enat] :
( ( ord_le2932123472753598470d_enat @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le2932123472753598470d_enat @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_332_append_Omonoid__axioms,axiom,
monoid_list_list_nat @ append_list_nat @ nil_list_nat ).
% append.monoid_axioms
thf(fact_333_append_Omonoid__axioms,axiom,
monoid1923560632227316973d_enat @ append_Extended_enat @ nil_Extended_enat ).
% append.monoid_axioms
thf(fact_334_append_Omonoid__axioms,axiom,
monoid_list_nat @ append_nat @ nil_nat ).
% append.monoid_axioms
thf(fact_335_predicate2I,axiom,
! [P: list_nat > list_nat > $o,Q: list_nat > list_nat > $o] :
( ! [X3: list_nat,Y3: list_nat] :
( ( P @ X3 @ Y3 )
=> ( Q @ X3 @ Y3 ) )
=> ( ord_le6558929396352911974_nat_o @ P @ Q ) ) ).
% predicate2I
thf(fact_336_predicate2I,axiom,
! [P: nat > nat > $o,Q: nat > nat > $o] :
( ! [X3: nat,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( Q @ X3 @ Y3 ) )
=> ( ord_le2646555220125990790_nat_o @ P @ Q ) ) ).
% predicate2I
thf(fact_337_rev__predicate2D,axiom,
! [P: list_nat > list_nat > $o,X: list_nat,Y: list_nat,Q: list_nat > list_nat > $o] :
( ( P @ X @ Y )
=> ( ( ord_le6558929396352911974_nat_o @ P @ Q )
=> ( Q @ X @ Y ) ) ) ).
% rev_predicate2D
thf(fact_338_rev__predicate2D,axiom,
! [P: nat > nat > $o,X: nat,Y: nat,Q: nat > nat > $o] :
( ( P @ X @ Y )
=> ( ( ord_le2646555220125990790_nat_o @ P @ Q )
=> ( Q @ X @ Y ) ) ) ).
% rev_predicate2D
thf(fact_339_predicate2D,axiom,
! [P: list_nat > list_nat > $o,Q: list_nat > list_nat > $o,X: list_nat,Y: list_nat] :
( ( ord_le6558929396352911974_nat_o @ P @ Q )
=> ( ( P @ X @ Y )
=> ( Q @ X @ Y ) ) ) ).
% predicate2D
thf(fact_340_predicate2D,axiom,
! [P: nat > nat > $o,Q: nat > nat > $o,X: nat,Y: nat] :
( ( ord_le2646555220125990790_nat_o @ P @ Q )
=> ( ( P @ X @ Y )
=> ( Q @ X @ Y ) ) ) ).
% predicate2D
thf(fact_341_GreatestI__ex__nat,axiom,
! [P: nat > $o,B: nat] :
( ? [X_1: nat] : ( P @ X_1 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_342_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_343_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_344_monoid_Oright__neutral,axiom,
! [F: list_nat > list_nat > list_nat,Z: list_nat,A: list_nat] :
( ( monoid_list_nat @ F @ Z )
=> ( ( F @ A @ Z )
= A ) ) ).
% monoid.right_neutral
thf(fact_345_monoid_Oright__neutral,axiom,
! [F: nat > nat > nat,Z: nat,A: nat] :
( ( monoid_nat @ F @ Z )
=> ( ( F @ A @ Z )
= A ) ) ).
% monoid.right_neutral
thf(fact_346_monoid_Oleft__neutral,axiom,
! [F: list_nat > list_nat > list_nat,Z: list_nat,A: list_nat] :
( ( monoid_list_nat @ F @ Z )
=> ( ( F @ Z @ A )
= A ) ) ).
% monoid.left_neutral
thf(fact_347_monoid_Oleft__neutral,axiom,
! [F: nat > nat > nat,Z: nat,A: nat] :
( ( monoid_nat @ F @ Z )
=> ( ( F @ Z @ A )
= A ) ) ).
% monoid.left_neutral
thf(fact_348_predicate2D__conj,axiom,
! [P: list_nat > list_nat > $o,Q: list_nat > list_nat > $o,R: $o,X: list_nat,Y: list_nat] :
( ( ( ord_le6558929396352911974_nat_o @ P @ Q )
& R )
=> ( R
& ( ( P @ X @ Y )
=> ( Q @ X @ Y ) ) ) ) ).
% predicate2D_conj
thf(fact_349_predicate2D__conj,axiom,
! [P: nat > nat > $o,Q: nat > nat > $o,R: $o,X: nat,Y: nat] :
( ( ( ord_le2646555220125990790_nat_o @ P @ Q )
& R )
=> ( R
& ( ( P @ X @ Y )
=> ( Q @ X @ Y ) ) ) ) ).
% predicate2D_conj
thf(fact_350_refl__ge__eq,axiom,
! [R: list_nat > list_nat > $o] :
( ! [X3: list_nat] : ( R @ X3 @ X3 )
=> ( ord_le6558929396352911974_nat_o
@ ^ [Y4: list_nat,Z2: list_nat] : ( Y4 = Z2 )
@ R ) ) ).
% refl_ge_eq
thf(fact_351_refl__ge__eq,axiom,
! [R: nat > nat > $o] :
( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ord_le2646555220125990790_nat_o
@ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
@ R ) ) ).
% refl_ge_eq
thf(fact_352_ge__eq__refl,axiom,
! [R: list_nat > list_nat > $o,X: list_nat] :
( ( ord_le6558929396352911974_nat_o
@ ^ [Y4: list_nat,Z2: list_nat] : ( Y4 = Z2 )
@ R )
=> ( R @ X @ X ) ) ).
% ge_eq_refl
thf(fact_353_ge__eq__refl,axiom,
! [R: nat > nat > $o,X: nat] :
( ( ord_le2646555220125990790_nat_o
@ ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 )
@ R )
=> ( R @ X @ X ) ) ).
% ge_eq_refl
thf(fact_354_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_355_last__appendR,axiom,
! [Ys: list_nat,Xs: list_nat] :
( ( Ys != nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
= ( last_nat @ Ys ) ) ) ).
% last_appendR
thf(fact_356_last__appendL,axiom,
! [Ys: list_nat,Xs: list_nat] :
( ( Ys = nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
= ( last_nat @ Xs ) ) ) ).
% last_appendL
thf(fact_357_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_nat @ N @ nil_nat )
= N ) ).
% gen_length_code(1)
thf(fact_358_list_Oinject,axiom,
! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
( ( ( cons_nat @ X21 @ X22 )
= ( cons_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_359_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_360_last__snoc,axiom,
! [Xs: list_nat,X: nat] :
( ( last_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= X ) ).
% last_snoc
thf(fact_361_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_362_not__Cons__self2,axiom,
! [X: nat,Xs: list_nat] :
( ( cons_nat @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_363_last__ConsR,axiom,
! [Xs: list_nat,X: nat] :
( ( Xs != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs ) )
= ( last_nat @ Xs ) ) ) ).
% last_ConsR
thf(fact_364_last__ConsL,axiom,
! [Xs: list_nat,X: nat] :
( ( Xs = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_365_last_Osimps,axiom,
! [Xs: list_nat,X: nat] :
( ( ( Xs = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs ) )
= ( last_nat @ Xs ) ) ) ) ).
% last.simps
thf(fact_366_list_Odistinct_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_367_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X22: list_nat] :
( ( List
= ( cons_nat @ X21 @ X22 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_368_list_Oexhaust,axiom,
! [Y: list_nat] :
( ( Y != nil_nat )
=> ~ ! [X212: nat,X222: list_nat] :
( Y
!= ( cons_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_369_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_370_remdups__adj_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ( ! [X3: nat] :
( X
!= ( cons_nat @ X3 @ nil_nat ) )
=> ~ ! [X3: nat,Y3: nat,Xs2: list_nat] :
( X
!= ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_371_neq__Nil__conv,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
= ( ? [Y2: nat,Ys3: list_nat] :
( Xs
= ( cons_nat @ Y2 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_372_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 )
=> ( ! [Y3: nat,Ys2: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y3 @ Ys2 ) )
=> ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys2: list_nat] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_373_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_374_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_375_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_376_list_Orel__inject_I2_J,axiom,
! [R: nat > nat > $o,X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
( ( list_all2_nat_nat @ R @ ( cons_nat @ X21 @ X22 ) @ ( cons_nat @ Y21 @ Y22 ) )
= ( ( R @ X21 @ Y21 )
& ( list_all2_nat_nat @ R @ X22 @ Y22 ) ) ) ).
% list.rel_inject(2)
thf(fact_377_list_Orel__intros_I2_J,axiom,
! [R: nat > nat > $o,X21: nat,Y21: nat,X22: list_nat,Y22: list_nat] :
( ( R @ X21 @ Y21 )
=> ( ( list_all2_nat_nat @ R @ X22 @ Y22 )
=> ( list_all2_nat_nat @ R @ ( cons_nat @ X21 @ X22 ) @ ( cons_nat @ Y21 @ Y22 ) ) ) ) ).
% list.rel_intros(2)
thf(fact_378_list__all2__Cons,axiom,
! [P: nat > nat > $o,X: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( list_all2_nat_nat @ P @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( ( P @ X @ Y )
& ( list_all2_nat_nat @ P @ Xs @ Ys ) ) ) ).
% list_all2_Cons
thf(fact_379_list__all2__Cons1,axiom,
! [P: nat > nat > $o,X: nat,Xs: list_nat,Ys: list_nat] :
( ( list_all2_nat_nat @ P @ ( cons_nat @ X @ Xs ) @ Ys )
= ( ? [Z3: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z3 @ Zs2 ) )
& ( P @ X @ Z3 )
& ( list_all2_nat_nat @ P @ Xs @ Zs2 ) ) ) ) ).
% list_all2_Cons1
thf(fact_380_list__all2__Cons2,axiom,
! [P: nat > nat > $o,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( list_all2_nat_nat @ P @ Xs @ ( cons_nat @ Y @ Ys ) )
= ( ? [Z3: nat,Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Z3 @ Zs2 ) )
& ( P @ Z3 @ Y )
& ( list_all2_nat_nat @ P @ Zs2 @ Ys ) ) ) ) ).
% list_all2_Cons2
thf(fact_381_member__rec_I1_J,axiom,
! [X: nat,Xs: list_nat,Y: nat] :
( ( member_nat @ ( cons_nat @ X @ Xs ) @ Y )
= ( ( X = Y )
| ( member_nat @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_382_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_383_rev__exhaust,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ~ ! [Ys2: list_nat,Y3: nat] :
( Xs
!= ( append_nat @ Ys2 @ ( cons_nat @ Y3 @ nil_nat ) ) ) ) ).
% rev_exhaust
thf(fact_384_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 ) )
| ? [Ys4: list_nat] :
( ( ( cons_nat @ X @ Ys4 )
= Ys )
& ( Xs
= ( append_nat @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_385_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 ) ) )
| ? [Ys4: list_nat] :
( ( Ys
= ( cons_nat @ X @ Ys4 ) )
& ( ( append_nat @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_386_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_387_list_Orel__distinct_I2_J,axiom,
! [R: nat > nat > $o,Y21: nat,Y22: list_nat] :
~ ( list_all2_nat_nat @ R @ ( cons_nat @ Y21 @ Y22 ) @ nil_nat ) ).
% list.rel_distinct(2)
thf(fact_388_list_Orel__distinct_I1_J,axiom,
! [R: nat > nat > $o,Y21: nat,Y22: list_nat] :
~ ( list_all2_nat_nat @ R @ nil_nat @ ( cons_nat @ Y21 @ Y22 ) ) ).
% list.rel_distinct(1)
thf(fact_389_list_Orel__cases,axiom,
! [R: nat > nat > $o,A: list_nat,B: list_nat] :
( ( list_all2_nat_nat @ R @ A @ B )
=> ( ( ( A = nil_nat )
=> ( B != nil_nat ) )
=> ~ ! [X1: nat,X23: list_nat] :
( ( A
= ( cons_nat @ X1 @ X23 ) )
=> ! [Y1: nat,Y23: list_nat] :
( ( B
= ( cons_nat @ Y1 @ Y23 ) )
=> ( ( R @ X1 @ Y1 )
=> ~ ( list_all2_nat_nat @ R @ X23 @ Y23 ) ) ) ) ) ) ).
% list.rel_cases
thf(fact_390_list_Orel__induct,axiom,
! [R: nat > nat > $o,X: list_nat,Y: list_nat,Q: list_nat > list_nat > $o] :
( ( list_all2_nat_nat @ R @ X @ Y )
=> ( ( Q @ nil_nat @ nil_nat )
=> ( ! [A21: nat,A22: list_nat,B21: nat,B22: list_nat] :
( ( R @ A21 @ B21 )
=> ( ( Q @ A22 @ B22 )
=> ( Q @ ( cons_nat @ A21 @ A22 ) @ ( cons_nat @ B21 @ B22 ) ) ) )
=> ( Q @ X @ Y ) ) ) ) ).
% list.rel_induct
thf(fact_391_list__all2__induct,axiom,
! [P: nat > nat > $o,Xs: list_nat,Ys: list_nat,R: list_nat > list_nat > $o] :
( ( list_all2_nat_nat @ P @ Xs @ Ys )
=> ( ( R @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys2: list_nat] :
( ( P @ X3 @ Y3 )
=> ( ( list_all2_nat_nat @ P @ Xs2 @ Ys2 )
=> ( ( R @ Xs2 @ Ys2 )
=> ( R @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) ) ) ) )
=> ( R @ Xs @ Ys ) ) ) ) ).
% list_all2_induct
thf(fact_392_last__append,axiom,
! [Ys: list_nat,Xs: list_nat] :
( ( ( Ys = nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
= ( last_nat @ Xs ) ) )
& ( ( Ys != nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
= ( last_nat @ Ys ) ) ) ) ).
% last_append
thf(fact_393_longest__common__suffix,axiom,
! [Xs: list_nat,Ys: list_nat] :
? [Ss: list_nat,Xs3: list_nat,Ys5: list_nat] :
( ( Xs
= ( append_nat @ Xs3 @ Ss ) )
& ( Ys
= ( append_nat @ Ys5 @ Ss ) )
& ( ( Xs3 = nil_nat )
| ( Ys5 = nil_nat )
| ( ( last_nat @ Xs3 )
!= ( last_nat @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_394_append__butlast__last__id,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( append_nat @ ( butlast_nat @ Xs ) @ ( cons_nat @ ( last_nat @ Xs ) @ nil_nat ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_395_maps__simps_I1_J,axiom,
! [F: nat > list_nat,X: nat,Xs: list_nat] :
( ( maps_nat_nat @ F @ ( cons_nat @ X @ Xs ) )
= ( append_nat @ ( F @ X ) @ ( maps_nat_nat @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_396_snoc__eq__iff__butlast,axiom,
! [Xs: list_nat,X: nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) )
= Ys )
= ( ( Ys != nil_nat )
& ( ( butlast_nat @ Ys )
= Xs )
& ( ( last_nat @ Ys )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_397_insert__Nil,axiom,
! [X: nat] :
( ( insert_nat @ X @ nil_nat )
= ( cons_nat @ X @ nil_nat ) ) ).
% insert_Nil
thf(fact_398_butlast__snoc,axiom,
! [Xs: list_nat,X: nat] :
( ( butlast_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_399_rotate1_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( rotate1_nat @ ( cons_nat @ X @ Xs ) )
= ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) ) ).
% rotate1.simps(2)
thf(fact_400_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_401_map__tailrec__rev_Oelims,axiom,
! [X: nat > nat,Xa: list_nat,Xb: list_nat,Y: list_nat] :
( ( ( map_ta7164188454487880599at_nat @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_nat )
=> ( Y != Xb ) )
=> ~ ! [A2: nat,As2: list_nat] :
( ( Xa
= ( cons_nat @ A2 @ As2 ) )
=> ( Y
!= ( map_ta7164188454487880599at_nat @ X @ As2 @ ( cons_nat @ ( X @ A2 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_402_rotate1__is__Nil__conv,axiom,
! [Xs: list_nat] :
( ( ( rotate1_nat @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% rotate1_is_Nil_conv
thf(fact_403_butlast_Osimps_I1_J,axiom,
( ( butlast_nat @ nil_nat )
= nil_nat ) ).
% butlast.simps(1)
thf(fact_404_rotate1_Osimps_I1_J,axiom,
( ( rotate1_nat @ nil_nat )
= nil_nat ) ).
% rotate1.simps(1)
thf(fact_405_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: nat > nat,A: nat,As: list_nat,Bs: list_nat] :
( ( map_ta7164188454487880599at_nat @ F @ ( cons_nat @ A @ As ) @ Bs )
= ( map_ta7164188454487880599at_nat @ F @ As @ ( cons_nat @ ( F @ A ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_406_butlast_Osimps_I2_J,axiom,
! [Xs: list_nat,X: nat] :
( ( ( Xs = nil_nat )
=> ( ( butlast_nat @ ( cons_nat @ X @ Xs ) )
= nil_nat ) )
& ( ( Xs != nil_nat )
=> ( ( butlast_nat @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ ( butlast_nat @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_407_butlast__append,axiom,
! [Ys: list_nat,Xs: list_nat] :
( ( ( Ys = nil_nat )
=> ( ( butlast_nat @ ( append_nat @ Xs @ Ys ) )
= ( butlast_nat @ Xs ) ) )
& ( ( Ys != nil_nat )
=> ( ( butlast_nat @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ Xs @ ( butlast_nat @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_408_maps__simps_I2_J,axiom,
! [F: nat > list_nat] :
( ( maps_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% maps_simps(2)
thf(fact_409_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_410_subseqs_Osimps_I1_J,axiom,
( ( subseqs_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% subseqs.simps(1)
thf(fact_411_concat__eq__append__conv,axiom,
! [Xss2: list_list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( concat_nat @ Xss2 )
= ( append_nat @ Ys @ Zs ) )
= ( ( ( Xss2 = nil_list_nat )
=> ( ( Ys = nil_nat )
& ( Zs = nil_nat ) ) )
& ( ( Xss2 != nil_list_nat )
=> ? [Xss1: list_list_nat,Xs4: list_nat,Xs5: list_nat,Xss22: list_list_nat] :
( ( Xss2
= ( append_list_nat @ Xss1 @ ( cons_list_nat @ ( append_nat @ Xs4 @ Xs5 ) @ Xss22 ) ) )
& ( Ys
= ( append_nat @ ( concat_nat @ Xss1 ) @ Xs4 ) )
& ( Zs
= ( append_nat @ Xs5 @ ( concat_nat @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_412_SuccI,axiom,
! [Kl: list_nat,K: nat,Kl2: set_list_nat] :
( ( member_list_nat2 @ ( append_nat @ Kl @ ( cons_nat @ K @ nil_nat ) ) @ Kl2 )
=> ( member_nat2 @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_413_SuccD,axiom,
! [K: nat,Kl2: set_list_nat,Kl: list_nat] :
( ( member_nat2 @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ Kl ) )
=> ( member_list_nat2 @ ( append_nat @ Kl @ ( cons_nat @ K @ nil_nat ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_414_concat__eq__appendD,axiom,
! [Xss2: list_list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( concat_nat @ Xss2 )
= ( append_nat @ Ys @ Zs ) )
=> ( ( Xss2 != nil_list_nat )
=> ? [Xss12: list_list_nat,Xs2: list_nat,Xs3: list_nat,Xss23: list_list_nat] :
( ( Xss2
= ( append_list_nat @ Xss12 @ ( cons_list_nat @ ( append_nat @ Xs2 @ Xs3 ) @ Xss23 ) ) )
& ( Ys
= ( append_nat @ ( concat_nat @ Xss12 ) @ Xs2 ) )
& ( Zs
= ( append_nat @ Xs3 @ ( concat_nat @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_415_ord_Olexordp__eq__simps_I3_J,axiom,
! [Less: nat > nat > $o,X: nat,Xs: list_nat] :
~ ( lexordp_eq_nat @ Less @ ( cons_nat @ X @ Xs ) @ nil_nat ) ).
% ord.lexordp_eq_simps(3)
thf(fact_416_rev__eq__Cons__iff,axiom,
! [Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ( rev_nat @ Xs )
= ( cons_nat @ Y @ Ys ) )
= ( Xs
= ( append_nat @ ( rev_nat @ Ys ) @ ( cons_nat @ Y @ nil_nat ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_417_Nil__is__rev__conv,axiom,
! [Xs: list_nat] :
( ( nil_nat
= ( rev_nat @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_rev_conv
thf(fact_418_rev__is__Nil__conv,axiom,
! [Xs: list_nat] :
( ( ( rev_nat @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% rev_is_Nil_conv
thf(fact_419_rev__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( rev_nat @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( rev_nat @ Ys ) @ ( rev_nat @ Xs ) ) ) ).
% rev_append
thf(fact_420_list__all2__rev,axiom,
! [P: nat > nat > $o,Xs: list_nat,Ys: list_nat] :
( ( list_all2_nat_nat @ P @ ( rev_nat @ Xs ) @ ( rev_nat @ Ys ) )
= ( list_all2_nat_nat @ P @ Xs @ Ys ) ) ).
% list_all2_rev
thf(fact_421_ord_Olexordp__eq__simps_I4_J,axiom,
! [Less: nat > nat > $o,X: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( lexordp_eq_nat @ Less @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( ( Less @ X @ Y )
| ( ~ ( Less @ Y @ X )
& ( lexordp_eq_nat @ Less @ Xs @ Ys ) ) ) ) ).
% ord.lexordp_eq_simps(4)
thf(fact_422_ord_Olexordp__eq__simps_I2_J,axiom,
! [Less: nat > nat > $o,Xs: list_nat] :
( ( lexordp_eq_nat @ Less @ Xs @ nil_nat )
= ( Xs = nil_nat ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_423_ord_Olexordp__eq__simps_I1_J,axiom,
! [Less: nat > nat > $o,Ys: list_nat] : ( lexordp_eq_nat @ Less @ nil_nat @ Ys ) ).
% ord.lexordp_eq_simps(1)
thf(fact_424_singleton__rev__conv,axiom,
! [X: nat,Xs: list_nat] :
( ( ( cons_nat @ X @ nil_nat )
= ( rev_nat @ Xs ) )
= ( ( cons_nat @ X @ nil_nat )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_425_rev__singleton__conv,axiom,
! [Xs: list_nat,X: nat] :
( ( ( rev_nat @ Xs )
= ( cons_nat @ X @ nil_nat ) )
= ( Xs
= ( cons_nat @ X @ nil_nat ) ) ) ).
% rev_singleton_conv
thf(fact_426_concat__append,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( concat_nat @ ( append_list_nat @ Xs @ Ys ) )
= ( append_nat @ ( concat_nat @ Xs ) @ ( concat_nat @ Ys ) ) ) ).
% concat_append
thf(fact_427_rev_Osimps_I1_J,axiom,
( ( rev_nat @ nil_nat )
= nil_nat ) ).
% rev.simps(1)
thf(fact_428_list__all2__rev1,axiom,
! [P: nat > nat > $o,Xs: list_nat,Ys: list_nat] :
( ( list_all2_nat_nat @ P @ ( rev_nat @ Xs ) @ Ys )
= ( list_all2_nat_nat @ P @ Xs @ ( rev_nat @ Ys ) ) ) ).
% list_all2_rev1
thf(fact_429_ord_Olexordp__eq_OCons__eq,axiom,
! [Less: nat > nat > $o,X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
( ~ ( Less @ X @ Y )
=> ( ~ ( Less @ Y @ X )
=> ( ( lexordp_eq_nat @ Less @ Xs @ Ys )
=> ( lexordp_eq_nat @ Less @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) ) ) ) ).
% ord.lexordp_eq.Cons_eq
thf(fact_430_ord_Olexordp__eq_OCons,axiom,
! [Less: nat > nat > $o,X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
( ( Less @ X @ Y )
=> ( lexordp_eq_nat @ Less @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) ) ).
% ord.lexordp_eq.Cons
thf(fact_431_ord_Olexordp__eq_ONil,axiom,
! [Less: nat > nat > $o,Ys: list_nat] : ( lexordp_eq_nat @ Less @ nil_nat @ Ys ) ).
% ord.lexordp_eq.Nil
thf(fact_432_ord_Olexordp__eq__pref,axiom,
! [Less: nat > nat > $o,U: list_nat,V: list_nat] : ( lexordp_eq_nat @ Less @ U @ ( append_nat @ U @ V ) ) ).
% ord.lexordp_eq_pref
thf(fact_433_concat_Osimps_I1_J,axiom,
( ( concat_nat @ nil_list_nat )
= nil_nat ) ).
% concat.simps(1)
thf(fact_434_concat_Osimps_I2_J,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( concat_nat @ ( cons_list_nat @ X @ Xs ) )
= ( append_nat @ X @ ( concat_nat @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_435_ord_Olexordp__eq_Osimps,axiom,
( lexordp_eq_nat
= ( ^ [Less2: nat > nat > $o,A1: list_nat,A23: list_nat] :
( ? [Ys3: list_nat] :
( ( A1 = nil_nat )
& ( A23 = Ys3 ) )
| ? [X2: nat,Y2: nat,Xs4: list_nat,Ys3: list_nat] :
( ( A1
= ( cons_nat @ X2 @ Xs4 ) )
& ( A23
= ( cons_nat @ Y2 @ Ys3 ) )
& ( Less2 @ X2 @ Y2 ) )
| ? [X2: nat,Y2: nat,Xs4: list_nat,Ys3: list_nat] :
( ( A1
= ( cons_nat @ X2 @ Xs4 ) )
& ( A23
= ( cons_nat @ Y2 @ Ys3 ) )
& ~ ( Less2 @ X2 @ Y2 )
& ~ ( Less2 @ Y2 @ X2 )
& ( lexordp_eq_nat @ Less2 @ Xs4 @ Ys3 ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_436_ord_Olexordp__eq_Ocases,axiom,
! [Less: nat > nat > $o,A12: list_nat,A24: list_nat] :
( ( lexordp_eq_nat @ Less @ A12 @ A24 )
=> ( ( A12 != nil_nat )
=> ( ! [X3: nat] :
( ? [Xs2: list_nat] :
( A12
= ( cons_nat @ X3 @ Xs2 ) )
=> ! [Y3: nat] :
( ? [Ys2: list_nat] :
( A24
= ( cons_nat @ Y3 @ Ys2 ) )
=> ~ ( Less @ X3 @ Y3 ) ) )
=> ~ ! [X3: nat,Y3: nat,Xs2: list_nat] :
( ( A12
= ( cons_nat @ X3 @ Xs2 ) )
=> ! [Ys2: list_nat] :
( ( A24
= ( cons_nat @ Y3 @ Ys2 ) )
=> ( ~ ( Less @ X3 @ Y3 )
=> ( ~ ( Less @ Y3 @ X3 )
=> ~ ( lexordp_eq_nat @ Less @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_437_rev_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( rev_nat @ ( cons_nat @ X @ Xs ) )
= ( append_nat @ ( rev_nat @ Xs ) @ ( cons_nat @ X @ nil_nat ) ) ) ).
% rev.simps(2)
thf(fact_438_empty__Shift,axiom,
! [Kl2: set_list_nat,K: nat] :
( ( member_list_nat2 @ nil_nat @ Kl2 )
=> ( ( member_nat2 @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ nil_nat ) )
=> ( member_list_nat2 @ nil_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_439_Succ__Shift,axiom,
! [Kl2: set_list_nat,K: nat,Kl: list_nat] :
( ( bNF_Gr6352880689984616693cc_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl2 @ K ) @ Kl )
= ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ ( cons_nat @ K @ Kl ) ) ) ).
% Succ_Shift
thf(fact_440_concat__conv__foldr,axiom,
( concat_nat
= ( ^ [Xss3: list_list_nat] : ( foldr_6871341030409798377st_nat @ append_nat @ Xss3 @ nil_nat ) ) ) ).
% concat_conv_foldr
thf(fact_441_listrelp_Osimps,axiom,
( listrelp_nat_nat
= ( ^ [R2: nat > nat > $o,A1: list_nat,A23: list_nat] :
( ( ( A1 = nil_nat )
& ( A23 = nil_nat ) )
| ? [X2: nat,Y2: nat,Xs4: list_nat,Ys3: list_nat] :
( ( A1
= ( cons_nat @ X2 @ Xs4 ) )
& ( A23
= ( cons_nat @ Y2 @ Ys3 ) )
& ( R2 @ X2 @ Y2 )
& ( listrelp_nat_nat @ R2 @ Xs4 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_442_listrelp_Ocases,axiom,
! [R3: nat > nat > $o,A12: list_nat,A24: list_nat] :
( ( listrelp_nat_nat @ R3 @ A12 @ A24 )
=> ( ( ( A12 = nil_nat )
=> ( A24 != nil_nat ) )
=> ~ ! [X3: nat,Y3: nat,Xs2: list_nat] :
( ( A12
= ( cons_nat @ X3 @ Xs2 ) )
=> ! [Ys2: list_nat] :
( ( A24
= ( cons_nat @ Y3 @ Ys2 ) )
=> ( ( R3 @ X3 @ Y3 )
=> ~ ( listrelp_nat_nat @ R3 @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_443_listrelp_OCons,axiom,
! [R3: nat > nat > $o,X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
( ( R3 @ X @ Y )
=> ( ( listrelp_nat_nat @ R3 @ Xs @ Ys )
=> ( listrelp_nat_nat @ R3 @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_444_listrelp_ONil,axiom,
! [R3: nat > nat > $o] : ( listrelp_nat_nat @ R3 @ nil_nat @ nil_nat ) ).
% listrelp.Nil
thf(fact_445_ShiftD,axiom,
! [Kl: list_nat,Kl2: set_list_nat,K: nat] :
( ( member_list_nat2 @ Kl @ ( bNF_Gr1872714664788909425ft_nat @ Kl2 @ K ) )
=> ( member_list_nat2 @ ( cons_nat @ K @ Kl ) @ Kl2 ) ) ).
% ShiftD
thf(fact_446_lexordp__eq__simps_I3_J,axiom,
! [X: nat,Xs: list_nat] :
~ ( ord_lexordp_eq_nat @ ( cons_nat @ X @ Xs ) @ nil_nat ) ).
% lexordp_eq_simps(3)
thf(fact_447_map__tailrec__rev,axiom,
( map_ta7164188454487880599at_nat
= ( ^ [F2: nat > nat,As3: list_nat] : ( append_nat @ ( rev_nat @ ( map_nat_nat @ F2 @ As3 ) ) ) ) ) ).
% map_tailrec_rev
thf(fact_448_fold__Cons__rev,axiom,
! [Xs: list_nat] :
( ( fold_nat_list_nat @ cons_nat @ Xs )
= ( append_nat @ ( rev_nat @ Xs ) ) ) ).
% fold_Cons_rev
thf(fact_449_remdups__adj__append__two,axiom,
! [Xs: list_nat,X: nat,Y: nat] :
( ( remdups_adj_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ ( cons_nat @ Y @ nil_nat ) ) ) )
= ( append_nat @ ( remdups_adj_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) ) @ ( if_list_nat @ ( X = Y ) @ nil_nat @ ( cons_nat @ Y @ nil_nat ) ) ) ) ).
% remdups_adj_append_two
thf(fact_450_rev__conv__fold,axiom,
( rev_nat
= ( ^ [Xs4: list_nat] : ( fold_nat_list_nat @ cons_nat @ Xs4 @ nil_nat ) ) ) ).
% rev_conv_fold
thf(fact_451_splice_Oelims,axiom,
! [X: list_nat,Xa: list_nat,Y: list_nat] :
( ( ( splice_nat @ X @ Xa )
= Y )
=> ( ( ( X = nil_nat )
=> ( Y != Xa ) )
=> ~ ! [X3: nat,Xs2: list_nat] :
( ( X
= ( cons_nat @ X3 @ Xs2 ) )
=> ( Y
!= ( cons_nat @ X3 @ ( splice_nat @ Xa @ Xs2 ) ) ) ) ) ) ).
% splice.elims
thf(fact_452_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,Xs3: list_nat,Y3: nat,Ys5: list_nat] :
( ( X3 != Y3 )
& ( Xs
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X3 @ nil_nat ) @ Xs3 ) ) )
& ( Ys
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y3 @ nil_nat ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_453_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_454_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_455_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_456_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_457_append__eq__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat,Us2: list_nat,Vs: list_nat] :
( ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
| ( ( size_size_list_nat @ Us2 )
= ( size_size_list_nat @ Vs ) ) )
=> ( ( ( append_nat @ Xs @ Us2 )
= ( append_nat @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us2 = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_458_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_459_length__rev,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rev_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rev
thf(fact_460_remdups__adj__Nil__iff,axiom,
! [Xs: list_nat] :
( ( ( remdups_adj_nat @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% remdups_adj_Nil_iff
thf(fact_461_length__concat__rev,axiom,
! [Xs: list_list_nat] :
( ( size_size_list_nat @ ( concat_nat @ ( rev_list_nat @ Xs ) ) )
= ( size_size_list_nat @ ( concat_nat @ Xs ) ) ) ).
% length_concat_rev
thf(fact_462_lexordp__eq__simps_I2_J,axiom,
! [Xs: list_nat] :
( ( ord_lexordp_eq_nat @ Xs @ nil_nat )
= ( Xs = nil_nat ) ) ).
% lexordp_eq_simps(2)
thf(fact_463_lexordp__eq__simps_I1_J,axiom,
! [Ys: list_nat] : ( ord_lexordp_eq_nat @ nil_nat @ Ys ) ).
% lexordp_eq_simps(1)
thf(fact_464_length__rotate1,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rotate1_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rotate1
thf(fact_465_splice__Nil2,axiom,
! [Xs: list_nat] :
( ( splice_nat @ Xs @ nil_nat )
= Xs ) ).
% splice_Nil2
thf(fact_466_split__Nil__iff,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( splice_nat @ Xs @ Ys )
= nil_nat )
= ( ( Xs = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% split_Nil_iff
thf(fact_467_last__remdups__adj,axiom,
! [Xs: list_nat] :
( ( last_nat @ ( remdups_adj_nat @ Xs ) )
= ( last_nat @ Xs ) ) ).
% last_remdups_adj
thf(fact_468_remdups__adj__length,axiom,
! [Xs: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( remdups_adj_nat @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% remdups_adj_length
thf(fact_469_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_470_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_471_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_nat] :
( ( size_size_list_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_472_map__concat,axiom,
! [F: nat > nat,Xs: list_list_nat] :
( ( map_nat_nat @ F @ ( concat_nat @ Xs ) )
= ( concat_nat @ ( map_li7225945977422193158st_nat @ ( map_nat_nat @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_473_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_474_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 ) )
= ( ? [Z3: nat,Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_nat_nat @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_475_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 ) )
= ( ? [Z3: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z3 @ Zs2 ) )
& ( X
= ( F @ Z3 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_476_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 ) )
=> ? [Z4: nat,Zs3: list_nat] :
( ( Xs
= ( cons_nat @ Z4 @ Zs3 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_nat_nat @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_477_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 ) )
=> ? [Z4: nat,Zs3: list_nat] :
( ( Ys
= ( cons_nat @ Z4 @ Zs3 ) )
& ( X
= ( F @ Z4 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_478_list_Osimps_I9_J,axiom,
! [F: nat > nat,X21: nat,X22: list_nat] :
( ( map_nat_nat @ F @ ( cons_nat @ X21 @ X22 ) )
= ( cons_nat @ ( F @ X21 ) @ ( map_nat_nat @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_479_list_Osimps_I8_J,axiom,
! [F: nat > nat] :
( ( map_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_480_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 ) )
= ( ? [Us: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us @ Vs2 ) )
& ( Ys
= ( map_nat_nat @ F @ Us ) )
& ( Zs
= ( map_nat_nat @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_481_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 ) )
= ( ? [Us: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us @ Vs2 ) )
& ( Ys
= ( map_nat_nat @ F @ Us ) )
& ( Zs
= ( map_nat_nat @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_482_rev__map,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( rev_nat @ ( map_nat_nat @ F @ Xs ) )
= ( map_nat_nat @ F @ ( rev_nat @ Xs ) ) ) ).
% rev_map
thf(fact_483_remdups__adj_Osimps_I3_J,axiom,
! [X: nat,Y: nat,Xs: list_nat] :
( ( ( X = Y )
=> ( ( remdups_adj_nat @ ( cons_nat @ X @ ( cons_nat @ Y @ Xs ) ) )
= ( remdups_adj_nat @ ( cons_nat @ X @ Xs ) ) ) )
& ( ( X != Y )
=> ( ( remdups_adj_nat @ ( cons_nat @ X @ ( cons_nat @ Y @ Xs ) ) )
= ( cons_nat @ X @ ( remdups_adj_nat @ ( cons_nat @ Y @ Xs ) ) ) ) ) ) ).
% remdups_adj.simps(3)
thf(fact_484_remdups__adj_Osimps_I1_J,axiom,
( ( remdups_adj_nat @ nil_nat )
= nil_nat ) ).
% remdups_adj.simps(1)
thf(fact_485_list__all2__lengthD,axiom,
! [P: nat > nat > $o,Xs: list_nat,Ys: list_nat] :
( ( list_all2_nat_nat @ P @ Xs @ Ys )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% list_all2_lengthD
thf(fact_486_map__butlast,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( map_nat_nat @ F @ ( butlast_nat @ Xs ) )
= ( butlast_nat @ ( map_nat_nat @ F @ Xs ) ) ) ).
% map_butlast
thf(fact_487_rotate1__map,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( rotate1_nat @ ( map_nat_nat @ F @ Xs ) )
= ( map_nat_nat @ F @ ( rotate1_nat @ Xs ) ) ) ).
% rotate1_map
thf(fact_488_lexordp__eq_ONil,axiom,
! [Ys: list_nat] : ( ord_lexordp_eq_nat @ nil_nat @ Ys ) ).
% lexordp_eq.Nil
thf(fact_489_lexordp__eq__pref,axiom,
! [U: list_nat,V: list_nat] : ( ord_lexordp_eq_nat @ U @ ( append_nat @ U @ V ) ) ).
% lexordp_eq_pref
thf(fact_490_splice_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat] :
( ( splice_nat @ ( cons_nat @ X @ Xs ) @ Ys )
= ( cons_nat @ X @ ( splice_nat @ Ys @ Xs ) ) ) ).
% splice.simps(2)
thf(fact_491_splice_Osimps_I1_J,axiom,
! [Ys: list_nat] :
( ( splice_nat @ nil_nat @ Ys )
= Ys ) ).
% splice.simps(1)
thf(fact_492_map__eq__map__tailrec,axiom,
map_nat_nat = map_tailrec_nat_nat ).
% map_eq_map_tailrec
thf(fact_493_last__map,axiom,
! [Xs: list_nat,F: nat > nat] :
( ( Xs != nil_nat )
=> ( ( last_nat @ ( map_nat_nat @ F @ Xs ) )
= ( F @ ( last_nat @ Xs ) ) ) ) ).
% last_map
thf(fact_494_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_495_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,Y3: nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_496_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,Y3: nat,Ys2: list_nat,Z4: nat,Zs3: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs3 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_nat @ Z4 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_497_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,Y3: nat,Ys2: list_nat,Z4: nat,Zs3: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs3 @ Ws2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_nat @ Z4 @ Zs3 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_498_remdups__adj_Osimps_I2_J,axiom,
! [X: nat] :
( ( remdups_adj_nat @ ( cons_nat @ X @ nil_nat ) )
= ( cons_nat @ X @ nil_nat ) ) ).
% remdups_adj.simps(2)
thf(fact_499_remdups__adj_Oelims,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( remdups_adj_nat @ X )
= Y )
=> ( ( ( X = nil_nat )
=> ( Y != nil_nat ) )
=> ( ! [X3: nat] :
( ( X
= ( cons_nat @ X3 @ nil_nat ) )
=> ( Y
!= ( cons_nat @ X3 @ nil_nat ) ) )
=> ~ ! [X3: nat,Y3: nat,Xs2: list_nat] :
( ( X
= ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs2 ) ) )
=> ~ ( ( ( X3 = Y3 )
=> ( Y
= ( remdups_adj_nat @ ( cons_nat @ X3 @ Xs2 ) ) ) )
& ( ( X3 != Y3 )
=> ( Y
= ( cons_nat @ X3 @ ( remdups_adj_nat @ ( cons_nat @ Y3 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% remdups_adj.elims
thf(fact_500_list__all2__append,axiom,
! [Xs: list_nat,Ys: list_nat,P: nat > nat > $o,Us2: list_nat,Vs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( list_all2_nat_nat @ P @ ( append_nat @ Xs @ Us2 ) @ ( append_nat @ Ys @ Vs ) )
= ( ( list_all2_nat_nat @ P @ Xs @ Ys )
& ( list_all2_nat_nat @ P @ Us2 @ Vs ) ) ) ) ).
% list_all2_append
thf(fact_501_list__all2__append1,axiom,
! [P: nat > nat > $o,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( list_all2_nat_nat @ P @ ( append_nat @ Xs @ Ys ) @ Zs )
= ( ? [Us: list_nat,Vs2: list_nat] :
( ( Zs
= ( append_nat @ Us @ Vs2 ) )
& ( ( size_size_list_nat @ Us )
= ( size_size_list_nat @ Xs ) )
& ( ( size_size_list_nat @ Vs2 )
= ( size_size_list_nat @ Ys ) )
& ( list_all2_nat_nat @ P @ Xs @ Us )
& ( list_all2_nat_nat @ P @ Ys @ Vs2 ) ) ) ) ).
% list_all2_append1
thf(fact_502_list__all2__append2,axiom,
! [P: nat > nat > $o,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( list_all2_nat_nat @ P @ Xs @ ( append_nat @ Ys @ Zs ) )
= ( ? [Us: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us @ Vs2 ) )
& ( ( size_size_list_nat @ Us )
= ( size_size_list_nat @ Ys ) )
& ( ( size_size_list_nat @ Vs2 )
= ( size_size_list_nat @ Zs ) )
& ( list_all2_nat_nat @ P @ Us @ Ys )
& ( list_all2_nat_nat @ P @ Vs2 @ Zs ) ) ) ) ).
% list_all2_append2
thf(fact_503_fold__append__concat__rev,axiom,
! [Xss2: list_list_nat] :
( ( fold_l5850465621530151245st_nat @ append_nat @ Xss2 )
= ( append_nat @ ( concat_nat @ ( rev_list_nat @ Xss2 ) ) ) ) ).
% fold_append_concat_rev
thf(fact_504_remdups__adj__append_H,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( Xs = nil_nat )
| ( Ys = nil_nat )
| ( ( last_nat @ Xs )
!= ( hd_nat @ Ys ) ) )
=> ( ( remdups_adj_nat @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( remdups_adj_nat @ Xs ) @ ( remdups_adj_nat @ Ys ) ) ) ) ).
% remdups_adj_append'
thf(fact_505_rotate1__length01,axiom,
! [Xs: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( ( rotate1_nat @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_506_remdups__adj__append,axiom,
! [Xs_1: list_nat,X: nat,Xs_2: list_nat] :
( ( remdups_adj_nat @ ( append_nat @ Xs_1 @ ( cons_nat @ X @ Xs_2 ) ) )
= ( append_nat @ ( remdups_adj_nat @ ( append_nat @ Xs_1 @ ( cons_nat @ X @ nil_nat ) ) ) @ ( tl_nat @ ( remdups_adj_nat @ ( cons_nat @ X @ Xs_2 ) ) ) ) ) ).
% remdups_adj_append
thf(fact_507_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_508_length__Suc__conv__rev,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y2: nat,Ys3: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ Y2 @ nil_nat ) ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_509_remdups__adj__singleton,axiom,
! [Xs: list_nat,X: nat] :
( ( ( remdups_adj_nat @ Xs )
= ( cons_nat @ X @ nil_nat ) )
=> ( Xs
= ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X ) ) ) ).
% remdups_adj_singleton
thf(fact_510_nat_Oinject,axiom,
! [X24: nat,Y24: nat] :
( ( ( suc @ X24 )
= ( suc @ Y24 ) )
= ( X24 = Y24 ) ) ).
% nat.inject
thf(fact_511_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_512_Suc__le__mono,axiom,
! [N: nat,M3: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M3 ) )
= ( ord_less_eq_nat @ N @ M3 ) ) ).
% Suc_le_mono
thf(fact_513_length__replicate,axiom,
! [N: nat,X: nat] :
( ( size_size_list_nat @ ( replicate_nat @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_514_concat__replicate__trivial,axiom,
! [I: nat] :
( ( concat_nat @ ( replicate_list_nat @ I @ nil_nat ) )
= nil_nat ) ).
% concat_replicate_trivial
thf(fact_515_map__replicate,axiom,
! [F: nat > nat,N: nat,X: nat] :
( ( map_nat_nat @ F @ ( replicate_nat @ N @ X ) )
= ( replicate_nat @ N @ ( F @ X ) ) ) ).
% map_replicate
thf(fact_516_hd__remdups__adj,axiom,
! [Xs: list_nat] :
( ( hd_nat @ ( remdups_adj_nat @ Xs ) )
= ( hd_nat @ Xs ) ) ).
% hd_remdups_adj
thf(fact_517_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_518_tl__append2,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs != nil_nat )
=> ( ( tl_nat @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( tl_nat @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_519_remdups__adj__Cons__alt,axiom,
! [X: nat,Xs: list_nat] :
( ( cons_nat @ X @ ( tl_nat @ ( remdups_adj_nat @ ( cons_nat @ X @ Xs ) ) ) )
= ( remdups_adj_nat @ ( cons_nat @ X @ Xs ) ) ) ).
% remdups_adj_Cons_alt
thf(fact_520_butlast__rev,axiom,
! [Xs: list_nat] :
( ( butlast_nat @ ( rev_nat @ Xs ) )
= ( rev_nat @ ( tl_nat @ Xs ) ) ) ).
% butlast_rev
thf(fact_521_hd__Cons__tl,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( cons_nat @ ( hd_nat @ Xs ) @ ( tl_nat @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_522_list_Ocollapse,axiom,
! [List: list_nat] :
( ( List != nil_nat )
=> ( ( cons_nat @ ( hd_nat @ List ) @ ( tl_nat @ List ) )
= List ) ) ).
% list.collapse
thf(fact_523_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_524_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_525_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_526_list_Oexpand,axiom,
! [List: list_nat,List2: list_nat] :
( ( ( List = nil_nat )
= ( List2 = nil_nat ) )
=> ( ( ( List != nil_nat )
=> ( ( List2 != nil_nat )
=> ( ( ( hd_nat @ List )
= ( hd_nat @ List2 ) )
& ( ( tl_nat @ List )
= ( tl_nat @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_527_replicate__Suc,axiom,
! [N: nat,X: nat] :
( ( replicate_nat @ ( suc @ N ) @ X )
= ( cons_nat @ X @ ( replicate_nat @ N @ X ) ) ) ).
% replicate_Suc
thf(fact_528_list_Oexhaust__sel,axiom,
! [List: list_nat] :
( ( List != nil_nat )
=> ( List
= ( cons_nat @ ( hd_nat @ List ) @ ( tl_nat @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_529_list_Orel__sel,axiom,
( list_all2_nat_nat
= ( ^ [R4: nat > nat > $o,A4: list_nat,B3: list_nat] :
( ( ( A4 = nil_nat )
= ( B3 = nil_nat ) )
& ( ( A4 != nil_nat )
=> ( ( B3 != nil_nat )
=> ( ( R4 @ ( hd_nat @ A4 ) @ ( hd_nat @ B3 ) )
& ( list_all2_nat_nat @ R4 @ ( tl_nat @ A4 ) @ ( tl_nat @ B3 ) ) ) ) ) ) ) ) ).
% list.rel_sel
thf(fact_530_append__replicate__commute,axiom,
! [N: nat,X: nat,K: nat] :
( ( append_nat @ ( replicate_nat @ N @ X ) @ ( replicate_nat @ K @ X ) )
= ( append_nat @ ( replicate_nat @ K @ X ) @ ( replicate_nat @ N @ X ) ) ) ).
% append_replicate_commute
thf(fact_531_list_Osel_I3_J,axiom,
! [X21: nat,X22: list_nat] :
( ( tl_nat @ ( cons_nat @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_532_transitive__stepwise__le,axiom,
! [M3: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z4: nat] :
( ( R @ X3 @ Y3 )
=> ( ( R @ Y3 @ Z4 )
=> ( R @ X3 @ Z4 ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M3 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_533_nat__induct__at__least,axiom,
! [M3: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( P @ M3 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_534_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M4: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N2 )
=> ( P @ M4 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_535_not__less__eq__eq,axiom,
! [M3: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M3 @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M3 ) ) ).
% not_less_eq_eq
thf(fact_536_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_537_le__Suc__eq,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M3 @ N )
| ( M3
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_538_Suc__le__D,axiom,
! [N: nat,M5: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M5 )
=> ? [M2: nat] :
( M5
= ( suc @ M2 ) ) ) ).
% Suc_le_D
thf(fact_539_le__SucI,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ord_less_eq_nat @ M3 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_540_le__SucE,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M3 @ N )
=> ( M3
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_541_Suc__leD,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% Suc_leD
thf(fact_542_list_Osel_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( ( hd_nat @ ( cons_nat @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_543_list_Osel_I2_J,axiom,
( ( tl_nat @ nil_nat )
= nil_nat ) ).
% list.sel(2)
thf(fact_544_map__tl,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( map_nat_nat @ F @ ( tl_nat @ Xs ) )
= ( tl_nat @ ( map_nat_nat @ F @ Xs ) ) ) ).
% map_tl
thf(fact_545_butlast__tl,axiom,
! [Xs: list_nat] :
( ( butlast_nat @ ( tl_nat @ Xs ) )
= ( tl_nat @ ( butlast_nat @ Xs ) ) ) ).
% butlast_tl
thf(fact_546_hd__concat,axiom,
! [Xs: list_list_nat] :
( ( Xs != nil_list_nat )
=> ( ( ( hd_list_nat @ Xs )
!= nil_nat )
=> ( ( hd_nat @ ( concat_nat @ Xs ) )
= ( hd_nat @ ( hd_list_nat @ Xs ) ) ) ) ) ).
% hd_concat
thf(fact_547_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_548_lift__Suc__antimono__le,axiom,
! [F: nat > extended_enat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_le2932123472753598470d_enat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_549_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_550_lift__Suc__mono__le,axiom,
! [F: nat > extended_enat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_le2932123472753598470d_enat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_551_replicate__app__Cons__same,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( append_nat @ ( replicate_nat @ N @ X ) @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ ( append_nat @ ( replicate_nat @ N @ X ) @ Xs ) ) ) ).
% replicate_app_Cons_same
thf(fact_552_rotate1__hd__tl,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( rotate1_nat @ Xs )
= ( append_nat @ ( tl_nat @ Xs ) @ ( cons_nat @ ( hd_nat @ Xs ) @ nil_nat ) ) ) ) ).
% rotate1_hd_tl
thf(fact_553_tl__Nil,axiom,
! [Xs: list_nat] :
( ( ( tl_nat @ Xs )
= nil_nat )
= ( ( Xs = nil_nat )
| ? [X2: nat] :
( Xs
= ( cons_nat @ X2 @ nil_nat ) ) ) ) ).
% tl_Nil
thf(fact_554_Nil__tl,axiom,
! [Xs: list_nat] :
( ( nil_nat
= ( tl_nat @ Xs ) )
= ( ( Xs = nil_nat )
| ? [X2: nat] :
( Xs
= ( cons_nat @ X2 @ nil_nat ) ) ) ) ).
% Nil_tl
thf(fact_555_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_556_Suc__length__conv,axiom,
! [N: nat,Xs: list_nat] :
( ( ( suc @ N )
= ( size_size_list_nat @ Xs ) )
= ( ? [Y2: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y2 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_557_length__Suc__conv,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y2: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y2 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_558_list_Omap__sel_I2_J,axiom,
! [A: list_nat,F: nat > nat] :
( ( A != nil_nat )
=> ( ( tl_nat @ ( map_nat_nat @ F @ A ) )
= ( map_nat_nat @ F @ ( tl_nat @ A ) ) ) ) ).
% list.map_sel(2)
thf(fact_559_tl__append__if,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( Xs = nil_nat )
=> ( ( tl_nat @ ( append_nat @ Xs @ Ys ) )
= ( tl_nat @ Ys ) ) )
& ( ( Xs != nil_nat )
=> ( ( tl_nat @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( tl_nat @ Xs ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_560_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_561_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_562_longest__common__prefix,axiom,
! [Xs: list_nat,Ys: list_nat] :
? [Ps: list_nat,Xs3: list_nat,Ys5: list_nat] :
( ( Xs
= ( append_nat @ Ps @ Xs3 ) )
& ( Ys
= ( append_nat @ Ps @ Ys5 ) )
& ( ( Xs3 = nil_nat )
| ( Ys5 = nil_nat )
| ( ( hd_nat @ Xs3 )
!= ( hd_nat @ Ys5 ) ) ) ) ).
% longest_common_prefix
thf(fact_563_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_564_comm__append__are__replicate,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Ys @ Xs ) )
=> ? [M2: nat,N2: nat,Zs3: list_nat] :
( ( ( concat_nat @ ( replicate_list_nat @ M2 @ Zs3 ) )
= Xs )
& ( ( concat_nat @ ( replicate_list_nat @ N2 @ Zs3 ) )
= Ys ) ) ) ).
% comm_append_are_replicate
thf(fact_565_last__tl,axiom,
! [Xs: list_nat] :
( ( ( Xs = nil_nat )
| ( ( tl_nat @ Xs )
!= nil_nat ) )
=> ( ( last_nat @ ( tl_nat @ Xs ) )
= ( last_nat @ Xs ) ) ) ).
% last_tl
thf(fact_566_hd__Nil__eq__last,axiom,
( ( hd_nat @ nil_nat )
= ( last_nat @ nil_nat ) ) ).
% hd_Nil_eq_last
thf(fact_567_last__rev,axiom,
! [Xs: list_nat] :
( ( last_nat @ ( rev_nat @ Xs ) )
= ( hd_nat @ Xs ) ) ).
% last_rev
thf(fact_568_hd__rev,axiom,
! [Xs: list_nat] :
( ( hd_nat @ ( rev_nat @ Xs ) )
= ( last_nat @ Xs ) ) ).
% hd_rev
thf(fact_569_gen__length__code_I2_J,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( gen_length_nat @ N @ ( cons_nat @ X @ Xs ) )
= ( gen_length_nat @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_570_replicate__append__same,axiom,
! [I: nat,X: nat] :
( ( append_nat @ ( replicate_nat @ I @ X ) @ ( cons_nat @ X @ nil_nat ) )
= ( cons_nat @ X @ ( replicate_nat @ I @ X ) ) ) ).
% replicate_append_same
thf(fact_571_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
= ( ? [X2: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ X2 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_572_remdups__adj__singleton__iff,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ ( remdups_adj_nat @ Xs ) )
= ( suc @ zero_zero_nat ) )
= ( ( Xs != nil_nat )
& ( Xs
= ( replicate_nat @ ( size_size_list_nat @ Xs ) @ ( hd_nat @ Xs ) ) ) ) ) ).
% remdups_adj_singleton_iff
thf(fact_573_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_574_le__numeral__extra_I4_J,axiom,
ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat ).
% le_numeral_extra(4)
thf(fact_575_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_576_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_577_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_578_le__zero__eq,axiom,
! [N: extended_enat] :
( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
= ( N = zero_z5237406670263579293d_enat ) ) ).
% le_zero_eq
thf(fact_579_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_580_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_581_tl__upt,axiom,
! [M3: nat,N: nat] :
( ( tl_nat @ ( upt @ M3 @ N ) )
= ( upt @ ( suc @ M3 ) @ N ) ) ).
% tl_upt
thf(fact_582_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_583_empty__replicate,axiom,
! [N: nat,X: nat] :
( ( nil_nat
= ( replicate_nat @ N @ X ) )
= ( N = zero_zero_nat ) ) ).
% empty_replicate
thf(fact_584_replicate__empty,axiom,
! [N: nat,X: nat] :
( ( ( replicate_nat @ N @ X )
= nil_nat )
= ( N = zero_zero_nat ) ) ).
% replicate_empty
thf(fact_585_hd__replicate,axiom,
! [N: nat,X: nat] :
( ( N != zero_zero_nat )
=> ( ( hd_nat @ ( replicate_nat @ N @ X ) )
= X ) ) ).
% hd_replicate
thf(fact_586_last__replicate,axiom,
! [N: nat,X: nat] :
( ( N != zero_zero_nat )
=> ( ( last_nat @ ( replicate_nat @ N @ X ) )
= X ) ) ).
% last_replicate
thf(fact_587_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( upt @ I @ J )
= nil_nat ) ) ).
% upt_conv_Nil
thf(fact_588_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_589_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_590_le__numeral__extra_I3_J,axiom,
ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ zero_z5237406670263579293d_enat ).
% le_numeral_extra(3)
thf(fact_591_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ zero_zero_nat )
= nil_nat ) ).
% upt_0
thf(fact_592_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_593_map__Suc__upt,axiom,
! [M3: nat,N: nat] :
( ( map_nat_nat @ suc @ ( upt @ M3 @ N ) )
= ( upt @ ( suc @ M3 ) @ ( suc @ N ) ) ) ).
% map_Suc_upt
thf(fact_594_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_595_zero__le,axiom,
! [X: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ X ) ).
% zero_le
thf(fact_596_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% not0_implies_Suc
thf(fact_597_Zero__not__Suc,axiom,
! [M3: nat] :
( zero_zero_nat
!= ( suc @ M3 ) ) ).
% Zero_not_Suc
thf(fact_598_Zero__neq__Suc,axiom,
! [M3: nat] :
( zero_zero_nat
!= ( suc @ M3 ) ) ).
% Zero_neq_Suc
thf(fact_599_Suc__neq__Zero,axiom,
! [M3: nat] :
( ( suc @ M3 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_600_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_601_diff__induct,axiom,
! [P: nat > nat > $o,M3: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P @ M3 @ N ) ) ) ) ).
% diff_induct
thf(fact_602_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_603_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_604_nat_OdiscI,axiom,
! [Nat: nat,X24: nat] :
( ( Nat
= ( suc @ X24 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_605_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_606_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_607_nat_Odistinct_I1_J,axiom,
! [X24: nat] :
( zero_zero_nat
!= ( suc @ X24 ) ) ).
% nat.distinct(1)
thf(fact_608_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_609_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_610_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_611_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_612_upt__conv__Cons__Cons,axiom,
! [M3: nat,N: nat,Ns: list_nat,Q2: nat] :
( ( ( cons_nat @ M3 @ ( cons_nat @ N @ Ns ) )
= ( upt @ M3 @ Q2 ) )
= ( ( cons_nat @ N @ Ns )
= ( upt @ ( suc @ M3 ) @ Q2 ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_613_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_614_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_615_replicate__0,axiom,
! [X: nat] :
( ( replicate_nat @ zero_zero_nat @ X )
= nil_nat ) ).
% replicate_0
thf(fact_616_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_617_remdups__adj__replicate,axiom,
! [N: nat,X: nat] :
( ( ( N = zero_zero_nat )
=> ( ( remdups_adj_nat @ ( replicate_nat @ N @ X ) )
= nil_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( remdups_adj_nat @ ( replicate_nat @ N @ X ) )
= ( cons_nat @ X @ nil_nat ) ) ) ) ).
% remdups_adj_replicate
thf(fact_618_remdups__adj__length__ge1,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_size_list_nat @ ( remdups_adj_nat @ Xs ) ) ) ) ).
% remdups_adj_length_ge1
thf(fact_619_Nitpick_Osize__list__simp_I2_J,axiom,
( size_size_list_nat
= ( ^ [Xs4: list_nat] : ( if_nat @ ( Xs4 = nil_nat ) @ zero_zero_nat @ ( suc @ ( size_size_list_nat @ ( tl_nat @ Xs4 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_620_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_621_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_622_not__one__le__zero,axiom,
~ ( ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ zero_z5237406670263579293d_enat ) ).
% not_one_le_zero
thf(fact_623_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_624_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_625_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_626_zero__less__one__class_Ozero__le__one,axiom,
ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ one_on7984719198319812577d_enat ).
% zero_less_one_class.zero_le_one
thf(fact_627_n__lists_Osimps_I1_J,axiom,
! [Xs: list_nat] :
( ( n_lists_nat @ zero_zero_nat @ Xs )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% n_lists.simps(1)
thf(fact_628_comm__append__is__replicate,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs != nil_nat )
=> ( ( Ys != nil_nat )
=> ( ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Ys @ Xs ) )
=> ? [N2: nat,Zs3: list_nat] :
( ( ord_less_nat @ one_one_nat @ N2 )
& ( ( concat_nat @ ( replicate_list_nat @ N2 @ Zs3 ) )
= ( append_nat @ Xs @ Ys ) ) ) ) ) ) ).
% comm_append_is_replicate
thf(fact_629_distinct__adj__append__iff,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( distinct_adj_nat @ ( append_nat @ Xs @ Ys ) )
= ( ( distinct_adj_nat @ Xs )
& ( distinct_adj_nat @ Ys )
& ( ( Xs = nil_nat )
| ( Ys = nil_nat )
| ( ( last_nat @ Xs )
!= ( hd_nat @ Ys ) ) ) ) ) ).
% distinct_adj_append_iff
thf(fact_630_rotate__length01,axiom,
! [Xs: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( ( rotate_nat @ N @ Xs )
= Xs ) ) ).
% rotate_length01
thf(fact_631_upt__rec,axiom,
( upt
= ( ^ [I2: nat,J2: nat] : ( if_list_nat @ ( ord_less_nat @ I2 @ J2 ) @ ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J2 ) ) @ nil_nat ) ) ) ).
% upt_rec
thf(fact_632_take__Suc,axiom,
! [Xs: list_nat,N: nat] :
( ( Xs != nil_nat )
=> ( ( take_nat @ ( suc @ N ) @ Xs )
= ( cons_nat @ ( hd_nat @ Xs ) @ ( take_nat @ N @ ( tl_nat @ Xs ) ) ) ) ) ).
% take_Suc
thf(fact_633_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_634_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_635_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_636_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_637_Suc__less__eq,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N ) )
= ( ord_less_nat @ M3 @ N ) ) ).
% Suc_less_eq
thf(fact_638_Suc__mono,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_639_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_640_hd__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( hd_nat @ ( upt @ I @ J ) )
= I ) ) ).
% hd_upt
thf(fact_641_rotate__is__Nil__conv,axiom,
! [N: nat,Xs: list_nat] :
( ( ( rotate_nat @ N @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% rotate_is_Nil_conv
thf(fact_642_distinct__adj__Cons__Cons,axiom,
! [X: nat,Y: nat,Xs: list_nat] :
( ( distinct_adj_nat @ ( cons_nat @ X @ ( cons_nat @ Y @ Xs ) ) )
= ( ( X != Y )
& ( distinct_adj_nat @ ( cons_nat @ Y @ Xs ) ) ) ) ).
% distinct_adj_Cons_Cons
thf(fact_643_length__rotate,axiom,
! [N: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( rotate_nat @ N @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rotate
thf(fact_644_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_645_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_646_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_647_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_648_take0,axiom,
( ( take_nat @ zero_zero_nat )
= ( ^ [Xs4: list_nat] : nil_nat ) ) ).
% take0
thf(fact_649_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_650_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_651_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_652_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_653_lexordp__eq__simps_I4_J,axiom,
! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ord_lexordp_eq_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( ( ord_less_nat @ X @ Y )
| ( ~ ( ord_less_nat @ Y @ X )
& ( ord_lexordp_eq_nat @ Xs @ Ys ) ) ) ) ).
% lexordp_eq_simps(4)
thf(fact_654_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_655_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_656_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_657_list__all2__takeI,axiom,
! [P: nat > nat > $o,Xs: list_nat,Ys: list_nat,N: nat] :
( ( list_all2_nat_nat @ P @ Xs @ Ys )
=> ( list_all2_nat_nat @ P @ ( take_nat @ N @ Xs ) @ ( take_nat @ N @ Ys ) ) ) ).
% list_all2_takeI
thf(fact_658_verit__comp__simplify1_I3_J,axiom,
! [B4: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B4 @ A5 ) )
= ( ord_less_nat @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_659_verit__comp__simplify1_I3_J,axiom,
! [B4: extended_enat,A5: extended_enat] :
( ( ~ ( ord_le2932123472753598470d_enat @ B4 @ A5 ) )
= ( ord_le72135733267957522d_enat @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_660_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_661_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_662_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_663_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_664_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_665_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,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_666_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_667_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,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_668_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,Y3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_669_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,Y3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_670_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_671_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,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_672_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,Y3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_673_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,Y3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ X3 @ Y3 )
=> ( ord_le2932123472753598470d_enat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le72135733267957522d_enat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_674_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_675_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,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_le72135733267957522d_enat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le72135733267957522d_enat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_676_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_677_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_678_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_679_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_680_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_681_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_682_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_683_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_684_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_685_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_686_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_687_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_688_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_689_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_690_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_691_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_692_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_693_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_694_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_695_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_696_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_697_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_698_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_699_dual__order_Ostrict__iff__not,axiom,
( ord_le72135733267957522d_enat
= ( ^ [B3: extended_enat,A4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B3 @ A4 )
& ~ ( ord_le2932123472753598470d_enat @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_700_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_701_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_702_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_703_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_704_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_705_dual__order_Ostrict__iff__order,axiom,
( ord_le72135733267957522d_enat
= ( ^ [B3: extended_enat,A4: extended_enat] :
( ( ord_le2932123472753598470d_enat @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_706_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_nat @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_707_dual__order_Oorder__iff__strict,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [B3: extended_enat,A4: extended_enat] :
( ( ord_le72135733267957522d_enat @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_708_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_709_order_Ostrict__iff__not,axiom,
( ord_le72135733267957522d_enat
= ( ^ [A4: extended_enat,B3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A4 @ B3 )
& ~ ( ord_le2932123472753598470d_enat @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_710_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_711_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_712_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_713_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_714_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_715_order_Ostrict__iff__order,axiom,
( ord_le72135733267957522d_enat
= ( ^ [A4: extended_enat,B3: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_716_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_717_order_Oorder__iff__strict,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [A4: extended_enat,B3: extended_enat] :
( ( ord_le72135733267957522d_enat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_718_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_719_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_720_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_721_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_722_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_723_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_724_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_725_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_726_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_727_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_728_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_729_leI,axiom,
! [X: extended_enat,Y: extended_enat] :
( ~ ( ord_le72135733267957522d_enat @ X @ Y )
=> ( ord_le2932123472753598470d_enat @ Y @ X ) ) ).
% leI
thf(fact_730_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_731_leD,axiom,
! [Y: extended_enat,X: extended_enat] :
( ( ord_le2932123472753598470d_enat @ Y @ X )
=> ~ ( ord_le72135733267957522d_enat @ X @ Y ) ) ).
% leD
thf(fact_732_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_733_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_734_gr__implies__not__zero,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_735_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_736_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_737_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_738_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_739_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_740_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_741_gr__implies__not0,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_742_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_743_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_744_Suc__lessD,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M3 ) @ N )
=> ( ord_less_nat @ M3 @ N ) ) ).
% Suc_lessD
thf(fact_745_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_746_Suc__lessI,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ( ( suc @ M3 )
!= N )
=> ( ord_less_nat @ ( suc @ M3 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_747_less__SucE,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M3 @ N )
=> ( M3 = N ) ) ) ).
% less_SucE
thf(fact_748_less__SucI,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ord_less_nat @ M3 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_749_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ N )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_750_less__Suc__eq,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N ) )
= ( ( ord_less_nat @ M3 @ N )
| ( M3 = N ) ) ) ).
% less_Suc_eq
thf(fact_751_not__less__eq,axiom,
! [M3: nat,N: nat] :
( ( ~ ( ord_less_nat @ M3 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M3 ) ) ) ).
% not_less_eq
thf(fact_752_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_753_Suc__less__eq2,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M3 )
= ( ? [M6: nat] :
( ( M3
= ( suc @ M6 ) )
& ( ord_less_nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_754_less__antisym,axiom,
! [N: nat,M3: nat] :
( ~ ( ord_less_nat @ N @ M3 )
=> ( ( ord_less_nat @ N @ ( suc @ M3 ) )
=> ( M3 = N ) ) ) ).
% less_antisym
thf(fact_755_Suc__less__SucD,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N ) )
=> ( ord_less_nat @ M3 @ N ) ) ).
% Suc_less_SucD
thf(fact_756_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_757_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J3: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ K2 )
=> ( ( P @ I3 @ J3 )
=> ( ( P @ J3 @ K2 )
=> ( P @ I3 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_758_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_759_not__less__less__Suc__eq,axiom,
! [N: nat,M3: nat] :
( ~ ( ord_less_nat @ N @ M3 )
=> ( ( ord_less_nat @ N @ ( suc @ M3 ) )
= ( N = M3 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_760_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_761_le__neq__implies__less,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( M3 != N )
=> ( ord_less_nat @ M3 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_762_less__or__eq__imp__le,axiom,
! [M3: nat,N: nat] :
( ( ( ord_less_nat @ M3 @ N )
| ( M3 = N ) )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_763_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M7: nat,N4: nat] :
( ( ord_less_nat @ M7 @ N4 )
| ( M7 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_764_less__imp__le__nat,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% less_imp_le_nat
thf(fact_765_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M7: nat,N4: nat] :
( ( ord_less_eq_nat @ M7 @ N4 )
& ( M7 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_766_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_767_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_768_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M3: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M3 ) )
= ( ord_less_nat @ N @ M3 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_769_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_770_take__equalityI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ! [I3: nat] :
( ( take_nat @ I3 @ Xs )
= ( take_nat @ I3 @ Ys ) )
=> ( Xs = Ys ) ) ).
% take_equalityI
thf(fact_771_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_772_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_773_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_774_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_775_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_776_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_777_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_778_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_779_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_780_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_781_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_782_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_783_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_784_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_785_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_786_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_787_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_788_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
& ~ ( P @ M4 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_789_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( P @ M4 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_790_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_791_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_792_less__not__refl2,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ N @ M3 )
=> ( M3 != N ) ) ).
% less_not_refl2
thf(fact_793_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_794_nat__neq__iff,axiom,
! [M3: nat,N: nat] :
( ( M3 != N )
= ( ( ord_less_nat @ M3 @ N )
| ( ord_less_nat @ N @ M3 ) ) ) ).
% nat_neq_iff
thf(fact_795_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_796_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_797_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_798_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_799_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_800_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( P @ A2 @ B2 ) )
=> ( ! [A2: nat] : ( P @ A2 @ A2 )
=> ( ! [A2: nat,B2: nat] :
( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_801_exists__least__iff,axiom,
( ( ^ [P4: nat > $o] :
? [X6: nat] : ( P4 @ X6 ) )
= ( ^ [P5: nat > $o] :
? [N4: nat] :
( ( P5 @ N4 )
& ! [M7: nat] :
( ( ord_less_nat @ M7 @ N4 )
=> ~ ( P5 @ M7 ) ) ) ) ) ).
% exists_least_iff
thf(fact_802_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_803_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_804_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_805_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_806_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_807_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_808_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_809_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_810_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_811_gt__ex,axiom,
! [X: nat] :
? [X_12: nat] : ( ord_less_nat @ X @ X_12 ) ).
% gt_ex
thf(fact_812_take__Nil,axiom,
! [N: nat] :
( ( take_nat @ N @ nil_nat )
= nil_nat ) ).
% take_Nil
thf(fact_813_take__butlast,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( take_nat @ N @ ( butlast_nat @ Xs ) )
= ( take_nat @ N @ Xs ) ) ) ).
% take_butlast
thf(fact_814_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs2: list_nat] :
( ! [Ys6: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys6 ) @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ Ys6 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_815_distinct__adj__ConsD,axiom,
! [X: nat,Xs: list_nat] :
( ( distinct_adj_nat @ ( cons_nat @ X @ Xs ) )
=> ( distinct_adj_nat @ Xs ) ) ).
% distinct_adj_ConsD
thf(fact_816_distinct__adj__Nil,axiom,
distinct_adj_nat @ nil_nat ).
% distinct_adj_Nil
thf(fact_817_rotate__map,axiom,
! [N: nat,F: nat > nat,Xs: list_nat] :
( ( rotate_nat @ N @ ( map_nat_nat @ F @ Xs ) )
= ( map_nat_nat @ F @ ( rotate_nat @ N @ Xs ) ) ) ).
% rotate_map
thf(fact_818_distinct__adj__mapD,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( distinct_adj_nat @ ( map_nat_nat @ F @ Xs ) )
=> ( distinct_adj_nat @ Xs ) ) ).
% distinct_adj_mapD
thf(fact_819_distinct__adj__appendD1,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( distinct_adj_nat @ ( append_nat @ Xs @ Ys ) )
=> ( distinct_adj_nat @ Xs ) ) ).
% distinct_adj_appendD1
thf(fact_820_distinct__adj__appendD2,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( distinct_adj_nat @ ( append_nat @ Xs @ Ys ) )
=> ( distinct_adj_nat @ Ys ) ) ).
% distinct_adj_appendD2
thf(fact_821_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_822_take__0,axiom,
! [Xs: list_nat] :
( ( take_nat @ zero_zero_nat @ Xs )
= nil_nat ) ).
% take_0
thf(fact_823_take__tl,axiom,
! [N: nat,Xs: list_nat] :
( ( take_nat @ N @ ( tl_nat @ Xs ) )
= ( tl_nat @ ( take_nat @ ( suc @ N ) @ Xs ) ) ) ).
% take_tl
thf(fact_824_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ ( suc @ I2 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_825_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M7: nat] :
( N
= ( suc @ M7 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_826_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_827_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% gr0_implies_Suc
thf(fact_828_less__Suc__eq__0__disj,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N ) )
= ( ( M3 = zero_zero_nat )
| ? [J2: nat] :
( ( M3
= ( suc @ J2 ) )
& ( ord_less_nat @ J2 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_829_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_830_Suc__leI,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ord_less_eq_nat @ ( suc @ M3 ) @ N ) ) ).
% Suc_leI
thf(fact_831_Suc__le__eq,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N )
= ( ord_less_nat @ M3 @ N ) ) ).
% Suc_le_eq
thf(fact_832_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_833_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_834_Suc__le__lessD,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N )
=> ( ord_less_nat @ M3 @ N ) ) ).
% Suc_le_lessD
thf(fact_835_le__less__Suc__eq,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M3 ) )
= ( N = M3 ) ) ) ).
% le_less_Suc_eq
thf(fact_836_less__Suc__eq__le,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N ) )
= ( ord_less_eq_nat @ M3 @ N ) ) ).
% less_Suc_eq_le
thf(fact_837_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_838_le__imp__less__Suc,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ord_less_nat @ M3 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_839_lexordp__eq_OCons,axiom,
! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_lexordp_eq_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) ) ).
% lexordp_eq.Cons
thf(fact_840_lexordp__eq_OCons__eq,axiom,
! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ~ ( ord_less_nat @ Y @ X )
=> ( ( ord_lexordp_eq_nat @ Xs @ Ys )
=> ( ord_lexordp_eq_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) ) ) ) ).
% lexordp_eq.Cons_eq
thf(fact_841_distinct__adj__singleton,axiom,
! [X: nat] : ( distinct_adj_nat @ ( cons_nat @ X @ nil_nat ) ) ).
% distinct_adj_singleton
thf(fact_842_rotate__append,axiom,
! [L: list_nat,Q2: list_nat] :
( ( rotate_nat @ ( size_size_list_nat @ L ) @ ( append_nat @ L @ Q2 ) )
= ( append_nat @ Q2 @ L ) ) ).
% rotate_append
thf(fact_843_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_844_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_845_lexordp__eq_Ocases,axiom,
! [A12: list_nat,A24: list_nat] :
( ( ord_lexordp_eq_nat @ A12 @ A24 )
=> ( ( A12 != nil_nat )
=> ( ! [X3: nat] :
( ? [Xs2: list_nat] :
( A12
= ( cons_nat @ X3 @ Xs2 ) )
=> ! [Y3: nat] :
( ? [Ys2: list_nat] :
( A24
= ( cons_nat @ Y3 @ Ys2 ) )
=> ~ ( ord_less_nat @ X3 @ Y3 ) ) )
=> ~ ! [X3: nat,Y3: nat,Xs2: list_nat] :
( ( A12
= ( cons_nat @ X3 @ Xs2 ) )
=> ! [Ys2: list_nat] :
( ( A24
= ( cons_nat @ Y3 @ Ys2 ) )
=> ( ~ ( ord_less_nat @ X3 @ Y3 )
=> ( ~ ( ord_less_nat @ Y3 @ X3 )
=> ~ ( ord_lexordp_eq_nat @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% lexordp_eq.cases
thf(fact_846_lexordp__eq_Osimps,axiom,
( ord_lexordp_eq_nat
= ( ^ [A1: list_nat,A23: list_nat] :
( ? [Ys3: list_nat] :
( ( A1 = nil_nat )
& ( A23 = Ys3 ) )
| ? [X2: nat,Y2: nat,Xs4: list_nat,Ys3: list_nat] :
( ( A1
= ( cons_nat @ X2 @ Xs4 ) )
& ( A23
= ( cons_nat @ Y2 @ Ys3 ) )
& ( ord_less_nat @ X2 @ Y2 ) )
| ? [X2: nat,Y2: nat,Xs4: list_nat,Ys3: list_nat] :
( ( A1
= ( cons_nat @ X2 @ Xs4 ) )
& ( A23
= ( cons_nat @ Y2 @ Ys3 ) )
& ~ ( ord_less_nat @ X2 @ Y2 )
& ~ ( ord_less_nat @ Y2 @ X2 )
& ( ord_lexordp_eq_nat @ Xs4 @ Ys3 ) ) ) ) ) ).
% lexordp_eq.simps
thf(fact_847_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_848_distinct__adj__Cons,axiom,
! [X: nat,Xs: list_nat] :
( ( distinct_adj_nat @ ( cons_nat @ X @ Xs ) )
= ( ( Xs = nil_nat )
| ( ( X
!= ( hd_nat @ Xs ) )
& ( distinct_adj_nat @ Xs ) ) ) ) ).
% distinct_adj_Cons
thf(fact_849_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_850_upt__rec__numeral,axiom,
! [M3: num,N: num] :
( ( ( ord_less_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N ) )
=> ( ( upt @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N ) )
= ( cons_nat @ ( numeral_numeral_nat @ M3 ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M3 ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
& ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N ) )
=> ( ( upt @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N ) )
= nil_nat ) ) ) ).
% upt_rec_numeral
thf(fact_851_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M3: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M3 ) ) ) ).
% nat_descend_induct
thf(fact_852_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_853_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ C2 ) )
=> ( P @ X4 ) )
& ! [D2: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D2 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D2 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_854_complete__interval,axiom,
! [A: extended_enat,B: extended_enat,P: extended_enat > $o] :
( ( ord_le72135733267957522d_enat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ C2 )
& ( ord_le2932123472753598470d_enat @ C2 @ B )
& ! [X4: extended_enat] :
( ( ( ord_le2932123472753598470d_enat @ A @ X4 )
& ( ord_le72135733267957522d_enat @ X4 @ C2 ) )
=> ( P @ X4 ) )
& ! [D2: extended_enat] :
( ! [X3: extended_enat] :
( ( ( ord_le2932123472753598470d_enat @ A @ X3 )
& ( ord_le72135733267957522d_enat @ X3 @ D2 ) )
=> ( P @ X3 ) )
=> ( ord_le2932123472753598470d_enat @ D2 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_855_numeral__le__iff,axiom,
! [M3: num,N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_eq_num @ M3 @ N ) ) ).
% numeral_le_iff
thf(fact_856_numeral__le__iff,axiom,
! [M3: num,N: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M3 ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_eq_num @ M3 @ N ) ) ).
% numeral_le_iff
thf(fact_857_drop0,axiom,
( ( drop_nat @ zero_zero_nat )
= ( ^ [X2: list_nat] : X2 ) ) ).
% drop0
thf(fact_858_nth__Cons__0,axiom,
! [X: nat,Xs: list_nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_859_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_860_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_861_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_862_nth__replicate,axiom,
! [I: nat,N: nat,X: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_nat @ ( replicate_nat @ N @ X ) @ I )
= X ) ) ).
% nth_replicate
thf(fact_863_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_864_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_865_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_866_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_867_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_868_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_869_last__drop,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( last_nat @ ( drop_nat @ N @ Xs ) )
= ( last_nat @ Xs ) ) ) ).
% last_drop
thf(fact_870_not__less__Nil,axiom,
! [X: list_nat] :
~ ( ord_less_list_nat @ X @ nil_nat ) ).
% not_less_Nil
thf(fact_871_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_872_list__all2__dropI,axiom,
! [P: nat > nat > $o,Xs: list_nat,Ys: list_nat,N: nat] :
( ( list_all2_nat_nat @ P @ Xs @ Ys )
=> ( list_all2_nat_nat @ P @ ( drop_nat @ N @ Xs ) @ ( drop_nat @ N @ Ys ) ) ) ).
% list_all2_dropI
thf(fact_873_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_874_drop__Nil,axiom,
! [N: nat] :
( ( drop_nat @ N @ nil_nat )
= nil_nat ) ).
% drop_Nil
thf(fact_875_drop__0,axiom,
! [Xs: list_nat] :
( ( drop_nat @ zero_zero_nat @ Xs )
= Xs ) ).
% drop_0
thf(fact_876_tl__drop,axiom,
! [N: nat,Xs: list_nat] :
( ( tl_nat @ ( drop_nat @ N @ Xs ) )
= ( drop_nat @ N @ ( tl_nat @ Xs ) ) ) ).
% tl_drop
thf(fact_877_drop__butlast,axiom,
! [N: nat,Xs: list_nat] :
( ( drop_nat @ N @ ( butlast_nat @ Xs ) )
= ( butlast_nat @ ( drop_nat @ N @ Xs ) ) ) ).
% drop_butlast
thf(fact_878_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_879_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_880_nth__equalityI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I3 )
= ( nth_nat @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_881_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X5: nat] : ( P @ I2 @ X5 ) ) )
= ( ? [Xs4: list_nat] :
( ( ( size_size_list_nat @ Xs4 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_nat @ Xs4 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_882_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_nat,Z2: list_nat] : ( Y4 = Z2 ) )
= ( ^ [Xs4: list_nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs4 )
= ( size_size_list_nat @ Ys3 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs4 ) )
=> ( ( nth_nat @ Xs4 @ I2 )
= ( nth_nat @ Ys3 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_883_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_le_numeral
thf(fact_884_zero__le__numeral,axiom,
! [N: num] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).
% zero_le_numeral
thf(fact_885_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_886_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N ) @ zero_z5237406670263579293d_enat ) ).
% not_numeral_le_zero
thf(fact_887_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% one_le_numeral
thf(fact_888_one__le__numeral,axiom,
! [N: num] : ( ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).
% one_le_numeral
thf(fact_889_drop__Suc,axiom,
! [N: nat,Xs: list_nat] :
( ( drop_nat @ ( suc @ N ) @ Xs )
= ( drop_nat @ N @ ( tl_nat @ Xs ) ) ) ).
% drop_Suc
thf(fact_890_Nil__less__Cons,axiom,
! [A: nat,X: list_nat] : ( ord_less_list_nat @ nil_nat @ ( cons_nat @ A @ X ) ) ).
% Nil_less_Cons
thf(fact_891_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_892_map__equality__iff,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( ( F @ ( nth_nat @ Xs @ I2 ) )
= ( G @ ( nth_nat @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_893_list__all2__nthD,axiom,
! [P: nat > nat > $o,Xs: list_nat,Ys: list_nat,P6: nat] :
( ( list_all2_nat_nat @ P @ Xs @ Ys )
=> ( ( ord_less_nat @ P6 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ P6 ) @ ( nth_nat @ Ys @ P6 ) ) ) ) ).
% list_all2_nthD
thf(fact_894_list__all2__nthD2,axiom,
! [P: nat > nat > $o,Xs: list_nat,Ys: list_nat,P6: nat] :
( ( list_all2_nat_nat @ P @ Xs @ Ys )
=> ( ( ord_less_nat @ P6 @ ( size_size_list_nat @ Ys ) )
=> ( P @ ( nth_nat @ Xs @ P6 ) @ ( nth_nat @ Ys @ P6 ) ) ) ) ).
% list_all2_nthD2
thf(fact_895_list__all2__all__nthI,axiom,
! [A: list_nat,B: list_nat,P: nat > nat > $o] :
( ( ( size_size_list_nat @ A )
= ( size_size_list_nat @ B ) )
=> ( ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ A ) )
=> ( P @ ( nth_nat @ A @ N2 ) @ ( nth_nat @ B @ N2 ) ) )
=> ( list_all2_nat_nat @ P @ A @ B ) ) ) ).
% list_all2_all_nthI
thf(fact_896_list__all2__conv__all__nth,axiom,
( list_all2_nat_nat
= ( ^ [P5: nat > nat > $o,Xs4: list_nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs4 )
= ( size_size_list_nat @ Ys3 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs4 ) )
=> ( P5 @ ( nth_nat @ Xs4 @ I2 ) @ ( nth_nat @ Ys3 @ I2 ) ) ) ) ) ) ).
% list_all2_conv_all_nth
thf(fact_897_hd__conv__nth,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( hd_nat @ Xs )
= ( nth_nat @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_898_nth__butlast,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ ( butlast_nat @ Xs ) ) )
=> ( ( nth_nat @ ( butlast_nat @ Xs ) @ N )
= ( nth_nat @ Xs @ N ) ) ) ).
% nth_butlast
thf(fact_899_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_900_pointwise__le__iff__nth,axiom,
( pointwise_le
= ( ^ [X2: list_nat,Y2: list_nat] :
( ( ( size_size_list_nat @ X2 )
= ( size_size_list_nat @ Y2 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ X2 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ X2 @ I2 ) @ ( nth_nat @ Y2 @ I2 ) ) ) ) ) ) ).
% pointwise_le_iff_nth
thf(fact_901_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 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( ( nth_nat @ Xs @ I3 )
= ( nth_nat @ Ys @ I3 ) ) )
=> ( ( take_nat @ K @ Xs )
= ( take_nat @ K @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_902_nth__tl,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ ( tl_nat @ Xs ) ) )
=> ( ( nth_nat @ ( tl_nat @ Xs ) @ N )
= ( nth_nat @ Xs @ ( suc @ N ) ) ) ) ).
% nth_tl
thf(fact_903_remdups__adj__adjacent,axiom,
! [I: nat,Xs: list_nat] :
( ( ord_less_nat @ ( suc @ I ) @ ( size_size_list_nat @ ( remdups_adj_nat @ Xs ) ) )
=> ( ( nth_nat @ ( remdups_adj_nat @ Xs ) @ I )
!= ( nth_nat @ ( remdups_adj_nat @ Xs ) @ ( suc @ I ) ) ) ) ).
% remdups_adj_adjacent
thf(fact_904_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_905_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_906_minf_I8_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ~ ( ord_less_eq_nat @ T @ X4 ) ) ).
% minf(8)
thf(fact_907_minf_I8_J,axiom,
! [T: extended_enat] :
? [Z4: extended_enat] :
! [X4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X4 @ Z4 )
=> ~ ( ord_le2932123472753598470d_enat @ T @ X4 ) ) ).
% minf(8)
thf(fact_908_minf_I6_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ord_less_eq_nat @ X4 @ T ) ) ).
% minf(6)
thf(fact_909_minf_I6_J,axiom,
! [T: extended_enat] :
? [Z4: extended_enat] :
! [X4: extended_enat] :
( ( ord_le72135733267957522d_enat @ X4 @ Z4 )
=> ( ord_le2932123472753598470d_enat @ X4 @ T ) ) ).
% minf(6)
thf(fact_910_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_911_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_912_list__update__id,axiom,
! [Xs: list_nat,I: nat] :
( ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ I ) )
= Xs ) ).
% list_update_id
thf(fact_913_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_914_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_915_take__update__cancel,axiom,
! [N: nat,M3: nat,Xs: list_nat,Y: nat] :
( ( ord_less_eq_nat @ N @ M3 )
=> ( ( take_nat @ N @ ( list_update_nat @ Xs @ M3 @ Y ) )
= ( take_nat @ N @ Xs ) ) ) ).
% take_update_cancel
thf(fact_916_drop__update__cancel,axiom,
! [N: nat,M3: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ N @ M3 )
=> ( ( drop_nat @ M3 @ ( list_update_nat @ Xs @ N @ X ) )
= ( drop_nat @ M3 @ Xs ) ) ) ).
% drop_update_cancel
thf(fact_917_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_918_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_919_list__all2__update__cong,axiom,
! [P: nat > nat > $o,Xs: list_nat,Ys: list_nat,X: nat,Y: nat,I: nat] :
( ( list_all2_nat_nat @ P @ Xs @ Ys )
=> ( ( P @ X @ Y )
=> ( list_all2_nat_nat @ P @ ( list_update_nat @ Xs @ I @ X ) @ ( list_update_nat @ Ys @ I @ Y ) ) ) ) ).
% list_all2_update_cong
thf(fact_920_take__update__swap,axiom,
! [M3: nat,Xs: list_nat,N: nat,X: nat] :
( ( take_nat @ M3 @ ( list_update_nat @ Xs @ N @ X ) )
= ( list_update_nat @ ( take_nat @ M3 @ Xs ) @ N @ X ) ) ).
% take_update_swap
thf(fact_921_list__update_Osimps_I1_J,axiom,
! [I: nat,V: nat] :
( ( list_update_nat @ nil_nat @ I @ V )
= nil_nat ) ).
% list_update.simps(1)
thf(fact_922_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_923_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_924_list__update__code_I2_J,axiom,
! [X: nat,Xs: list_nat,Y: nat] :
( ( list_update_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat @ Y )
= ( cons_nat @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_925_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_926_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_927_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_928_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_929_pinf_I6_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ T ) ) ).
% pinf(6)
thf(fact_930_pinf_I6_J,axiom,
! [T: extended_enat] :
? [Z4: extended_enat] :
! [X4: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z4 @ X4 )
=> ~ ( ord_le2932123472753598470d_enat @ X4 @ T ) ) ).
% pinf(6)
thf(fact_931_pinf_I8_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ord_less_eq_nat @ T @ X4 ) ) ).
% pinf(8)
thf(fact_932_pinf_I8_J,axiom,
! [T: extended_enat] :
? [Z4: extended_enat] :
! [X4: extended_enat] :
( ( ord_le72135733267957522d_enat @ Z4 @ X4 )
=> ( ord_le2932123472753598470d_enat @ T @ X4 ) ) ).
% pinf(8)
thf(fact_933_enat__ord__number_I1_J,axiom,
! [M3: num,N: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M3 ) @ ( numera1916890842035813515d_enat @ N ) )
= ( ord_less_eq_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N ) ) ) ).
% enat_ord_number(1)
thf(fact_934_nth__Cons__pos,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_935_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_936_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N ) @ one_on7984719198319812577d_enat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_937_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_938_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_939_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_940_diff__self__eq__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ M3 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_941_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_942_Suc__diff__diff,axiom,
! [M3: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M3 ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M3 @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_943_diff__Suc__Suc,axiom,
! [M3: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M3 ) @ ( suc @ N ) )
= ( minus_minus_nat @ M3 @ N ) ) ).
% diff_Suc_Suc
thf(fact_944_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_945_diff__is__0__eq_H,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( minus_minus_nat @ M3 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_946_diff__is__0__eq,axiom,
! [M3: nat,N: nat] :
( ( ( minus_minus_nat @ M3 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M3 @ N ) ) ).
% diff_is_0_eq
thf(fact_947_zero__less__diff,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M3 ) )
= ( ord_less_nat @ M3 @ N ) ) ).
% zero_less_diff
thf(fact_948_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_949_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_950_drop__replicate,axiom,
! [I: nat,K: nat,X: nat] :
( ( drop_nat @ I @ ( replicate_nat @ K @ X ) )
= ( replicate_nat @ ( minus_minus_nat @ K @ I ) @ X ) ) ).
% drop_replicate
thf(fact_951_length__upt,axiom,
! [I: nat,J: nat] :
( ( size_size_list_nat @ ( upt @ I @ J ) )
= ( minus_minus_nat @ J @ I ) ) ).
% length_upt
thf(fact_952_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_953_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_954_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_955_last__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( last_nat @ ( upt @ I @ J ) )
= ( minus_minus_nat @ J @ one_one_nat ) ) ) ).
% last_upt
thf(fact_956_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_957_length__tl,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( tl_nat @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ).
% length_tl
thf(fact_958_length__butlast,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( butlast_nat @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ).
% length_butlast
thf(fact_959_tl__replicate,axiom,
! [N: nat,X: nat] :
( ( tl_nat @ ( replicate_nat @ N @ X ) )
= ( replicate_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ X ) ) ).
% tl_replicate
thf(fact_960_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_961_nth__Cons__numeral,axiom,
! [X: nat,Xs: list_nat,V: num] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ ( numeral_numeral_nat @ V ) )
= ( nth_nat @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) ) ) ).
% nth_Cons_numeral
thf(fact_962_take__Cons__numeral,axiom,
! [V: num,X: nat,Xs: list_nat] :
( ( take_nat @ ( numeral_numeral_nat @ V ) @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ ( take_nat @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) @ Xs ) ) ) ).
% take_Cons_numeral
thf(fact_963_drop__Cons__numeral,axiom,
! [V: num,X: nat,Xs: list_nat] :
( ( drop_nat @ ( numeral_numeral_nat @ V ) @ ( cons_nat @ X @ Xs ) )
= ( drop_nat @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) @ Xs ) ) ).
% drop_Cons_numeral
thf(fact_964_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_965_diff__less__mono2,axiom,
! [M3: nat,N: nat,L: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ( ord_less_nat @ M3 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M3 ) ) ) ) ).
% diff_less_mono2
thf(fact_966_diff__le__mono2,axiom,
! [M3: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M3 ) ) ) ).
% diff_le_mono2
thf(fact_967_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_968_diff__le__self,axiom,
! [M3: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ N ) @ M3 ) ).
% diff_le_self
thf(fact_969_diff__le__mono,axiom,
! [M3: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_970_Nat_Odiff__diff__eq,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M3 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_971_le__diff__iff,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M3 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_972_eq__diff__iff,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M3 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M3 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_973_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_974_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_975_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_976_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_977_diffs0__imp__equal,axiom,
! [M3: nat,N: nat] :
( ( ( minus_minus_nat @ M3 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M3 )
= zero_zero_nat )
=> ( M3 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_978_minus__nat_Odiff__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% minus_nat.diff_0
thf(fact_979_diff__less,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ord_less_nat @ ( minus_minus_nat @ M3 @ N ) @ M3 ) ) ) ).
% diff_less
thf(fact_980_Suc__diff__le,axiom,
! [N: nat,M3: nat] :
( ( ord_less_eq_nat @ N @ M3 )
=> ( ( minus_minus_nat @ ( suc @ M3 ) @ N )
= ( suc @ ( minus_minus_nat @ M3 @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_981_diff__less__Suc,axiom,
! [M3: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M3 @ N ) @ ( suc @ M3 ) ) ).
% diff_less_Suc
thf(fact_982_Suc__diff__Suc,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ N @ M3 )
=> ( ( suc @ ( minus_minus_nat @ M3 @ ( suc @ N ) ) )
= ( minus_minus_nat @ M3 @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_983_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_984_less__diff__iff,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M3 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M3 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M3 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_985_diff__Suc__eq__diff__pred,axiom,
! [M3: nat,N: nat] :
( ( minus_minus_nat @ M3 @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_986_drop__take,axiom,
! [N: nat,M3: nat,Xs: list_nat] :
( ( drop_nat @ N @ ( take_nat @ M3 @ Xs ) )
= ( take_nat @ ( minus_minus_nat @ M3 @ N ) @ ( drop_nat @ N @ Xs ) ) ) ).
% drop_take
thf(fact_987_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_988_drop__update__swap,axiom,
! [M3: nat,N: nat,Xs: list_nat,X: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( drop_nat @ M3 @ ( list_update_nat @ Xs @ N @ X ) )
= ( list_update_nat @ ( drop_nat @ M3 @ Xs ) @ ( minus_minus_nat @ N @ M3 ) @ X ) ) ) ).
% drop_update_swap
thf(fact_989_tl__take,axiom,
! [N: nat,Xs: list_nat] :
( ( tl_nat @ ( take_nat @ N @ Xs ) )
= ( take_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( tl_nat @ Xs ) ) ) ).
% tl_take
thf(fact_990_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M3 ) @ N )
= ( minus_minus_nat @ M3 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_991_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_992_nth__Cons_H,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_993_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_994_drop__Cons_H,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( drop_nat @ N @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ Xs ) ) )
& ( ( N != zero_zero_nat )
=> ( ( drop_nat @ N @ ( cons_nat @ X @ Xs ) )
= ( drop_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ).
% drop_Cons'
thf(fact_995_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_996_take__rev,axiom,
! [N: nat,Xs: list_nat] :
( ( take_nat @ N @ ( rev_nat @ Xs ) )
= ( rev_nat @ ( drop_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ N ) @ Xs ) ) ) ).
% take_rev
thf(fact_997_rev__take,axiom,
! [I: nat,Xs: list_nat] :
( ( rev_nat @ ( take_nat @ I @ Xs ) )
= ( drop_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ I ) @ ( rev_nat @ Xs ) ) ) ).
% rev_take
thf(fact_998_rev__drop,axiom,
! [I: nat,Xs: list_nat] :
( ( rev_nat @ ( drop_nat @ I @ Xs ) )
= ( take_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ I ) @ ( rev_nat @ Xs ) ) ) ).
% rev_drop
thf(fact_999_drop__rev,axiom,
! [N: nat,Xs: list_nat] :
( ( drop_nat @ N @ ( rev_nat @ Xs ) )
= ( rev_nat @ ( take_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ N ) @ Xs ) ) ) ).
% drop_rev
thf(fact_1000_butlast__conv__take,axiom,
( butlast_nat
= ( ^ [Xs4: list_nat] : ( take_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs4 ) @ one_one_nat ) @ Xs4 ) ) ) ).
% butlast_conv_take
thf(fact_1001_butlast__list__update,axiom,
! [K: nat,Xs: list_nat,X: nat] :
( ( ( K
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
=> ( ( butlast_nat @ ( list_update_nat @ Xs @ K @ X ) )
= ( butlast_nat @ Xs ) ) )
& ( ( K
!= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
=> ( ( butlast_nat @ ( list_update_nat @ Xs @ K @ X ) )
= ( list_update_nat @ ( butlast_nat @ Xs ) @ K @ X ) ) ) ) ).
% butlast_list_update
thf(fact_1002_nth__non__equal__first__eq,axiom,
! [X: nat,Y: nat,Xs: list_nat,N: nat] :
( ( X != Y )
=> ( ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= Y )
= ( ( ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_1003_take__Cons_H,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( take_nat @ N @ ( cons_nat @ X @ Xs ) )
= nil_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( take_nat @ N @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ ( take_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ) ).
% take_Cons'
thf(fact_1004_Cons__replicate__eq,axiom,
! [X: nat,Xs: list_nat,N: nat,Y: nat] :
( ( ( cons_nat @ X @ Xs )
= ( replicate_nat @ N @ Y ) )
= ( ( X = Y )
& ( ord_less_nat @ zero_zero_nat @ N )
& ( Xs
= ( replicate_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ X ) ) ) ) ).
% Cons_replicate_eq
thf(fact_1005_rev__nth,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( rev_nat @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ ( suc @ N ) ) ) ) ) ).
% rev_nth
thf(fact_1006_last__conv__nth,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( last_nat @ Xs )
= ( nth_nat @ Xs @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_1007_rev__update,axiom,
! [K: nat,Xs: list_nat,Y: nat] :
( ( ord_less_nat @ K @ ( size_size_list_nat @ Xs ) )
=> ( ( rev_nat @ ( list_update_nat @ Xs @ K @ Y ) )
= ( list_update_nat @ ( rev_nat @ Xs ) @ ( minus_minus_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ K ) @ one_one_nat ) @ Y ) ) ) ).
% rev_update
thf(fact_1008_butlast__take,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( butlast_nat @ ( take_nat @ N @ Xs ) )
= ( take_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ).
% butlast_take
thf(fact_1009_last__list__update,axiom,
! [Xs: list_nat,K: nat,X: nat] :
( ( Xs != nil_nat )
=> ( ( ( K
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
=> ( ( last_nat @ ( list_update_nat @ Xs @ K @ X ) )
= X ) )
& ( ( K
!= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
=> ( ( last_nat @ ( list_update_nat @ Xs @ K @ X ) )
= ( last_nat @ Xs ) ) ) ) ) ).
% last_list_update
thf(fact_1010_map__upt__eqI,axiom,
! [Xs: list_nat,N: nat,M3: nat,F: nat > nat] :
( ( ( size_size_list_nat @ Xs )
= ( minus_minus_nat @ N @ M3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I3 )
= ( F @ ( plus_plus_nat @ M3 @ I3 ) ) ) )
=> ( ( map_nat_nat @ F @ ( upt @ M3 @ N ) )
= Xs ) ) ) ).
% map_upt_eqI
thf(fact_1011_nth__map__upt,axiom,
! [I: nat,N: nat,M3: nat,F: nat > nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ N @ M3 ) )
=> ( ( nth_nat @ ( map_nat_nat @ F @ ( upt @ M3 @ N ) ) @ I )
= ( F @ ( plus_plus_nat @ M3 @ I ) ) ) ) ).
% nth_map_upt
thf(fact_1012_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_1013_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_1014_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_1015_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_1016_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_1017_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_1018_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_1019_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_1020_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_1021_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_1022_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_1023_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_1024_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_1025_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_1026_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_1027_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_1028_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_1029_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_1030_add__is__0,axiom,
! [M3: nat,N: nat] :
( ( ( plus_plus_nat @ M3 @ N )
= zero_zero_nat )
= ( ( M3 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1031_Nat_Oadd__0__right,axiom,
! [M3: nat] :
( ( plus_plus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% Nat.add_0_right
thf(fact_1032_add__Suc__right,axiom,
! [M3: nat,N: nat] :
( ( plus_plus_nat @ M3 @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M3 @ N ) ) ) ).
% add_Suc_right
thf(fact_1033_nat__add__left__cancel__le,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M3 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M3 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1034_nat__add__left__cancel__less,axiom,
! [K: nat,M3: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M3 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M3 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1035_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_1036_drop__drop,axiom,
! [N: nat,M3: nat,Xs: list_nat] :
( ( drop_nat @ N @ ( drop_nat @ M3 @ Xs ) )
= ( drop_nat @ ( plus_plus_nat @ N @ M3 ) @ Xs ) ) ).
% drop_drop
thf(fact_1037_drop__upt,axiom,
! [M3: nat,I: nat,J: nat] :
( ( drop_nat @ M3 @ ( upt @ I @ J ) )
= ( upt @ ( plus_plus_nat @ I @ M3 ) @ J ) ) ).
% drop_upt
thf(fact_1038_minus__Nil,axiom,
! [Xs: list_nat] :
( ( minus_minus_list_nat @ nil_nat @ Xs )
= nil_nat ) ).
% minus_Nil
thf(fact_1039_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_1040_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_1041_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_1042_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_1043_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_1044_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_1045_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_1046_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_1047_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_1048_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_1049_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_1050_add__gr__0,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M3 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M3 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1051_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_1052_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_1053_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_1054_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_1055_take__upt,axiom,
! [I: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M3 ) @ N )
=> ( ( take_nat @ M3 @ ( upt @ I @ N ) )
= ( upt @ I @ ( plus_plus_nat @ I @ M3 ) ) ) ) ).
% take_upt
thf(fact_1056_length__splice,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( size_size_list_nat @ ( splice_nat @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_splice
thf(fact_1057_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_1058_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_1059_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_1060_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_1061_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_1062_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_1063_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_1064_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_1065_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_1066_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_1067_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
? [C3: nat] :
( B3
= ( plus_plus_nat @ A4 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_1068_le__iff__add,axiom,
( ord_le2932123472753598470d_enat
= ( ^ [A4: extended_enat,B3: extended_enat] :
? [C3: extended_enat] :
( B3
= ( plus_p3455044024723400733d_enat @ A4 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_1069_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_1070_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_1071_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_1072_less__eqE,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ~ ! [C2: extended_enat] :
( B
!= ( plus_p3455044024723400733d_enat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_1073_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_1074_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_1075_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_1076_add__mono,axiom,
! [A: extended_enat,B: extended_enat,C: extended_enat,D: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ B )
=> ( ( ord_le2932123472753598470d_enat @ C @ D )
=> ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A @ C ) @ ( plus_p3455044024723400733d_enat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_1077_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_1078_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_1079_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_1080_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_1081_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_1082_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_1083_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M7: nat,N4: nat] :
? [K3: nat] :
( N4
= ( plus_plus_nat @ M7 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1084_trans__le__add2,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M3 @ J ) ) ) ).
% trans_le_add2
thf(fact_1085_trans__le__add1,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M3 ) ) ) ).
% trans_le_add1
thf(fact_1086_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_1087_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_1088_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_1089_add__leD2,axiom,
! [M3: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1090_add__leD1,axiom,
! [M3: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% add_leD1
thf(fact_1091_le__add2,axiom,
! [N: nat,M3: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M3 @ N ) ) ).
% le_add2
thf(fact_1092_le__add1,axiom,
! [N: nat,M3: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M3 ) ) ).
% le_add1
thf(fact_1093_add__leE,axiom,
! [M3: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M3 @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1094_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_1095_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_1096_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_1097_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_1098_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_1099_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_1100_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_1101_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_1102_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_1103_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_1104_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_1105_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_1106_Nat_Odiff__cancel,axiom,
! [K: nat,M3: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M3 ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M3 @ N ) ) ).
% Nat.diff_cancel
thf(fact_1107_diff__cancel2,axiom,
! [M3: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M3 @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M3 @ N ) ) ).
% diff_cancel2
thf(fact_1108_diff__add__inverse,axiom,
! [N: nat,M3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M3 ) @ N )
= M3 ) ).
% diff_add_inverse
thf(fact_1109_diff__add__inverse2,axiom,
! [M3: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M3 @ N ) @ N )
= M3 ) ).
% diff_add_inverse2
thf(fact_1110_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_1111_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_1112_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_1113_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_1114_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_1115_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_1116_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_1117_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_1118_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_1119_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_1120_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_1121_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_1122_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_1123_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_1124_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_1125_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_1126_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1127_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1128_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_1129_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_1130_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_1131_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_1132_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_1133_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_1134_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_1135_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_1136_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_1137_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1138_add__eq__self__zero,axiom,
! [M3: nat,N: nat] :
( ( ( plus_plus_nat @ M3 @ N )
= M3 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1139_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_1140_add__Suc,axiom,
! [M3: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M3 ) @ N )
= ( suc @ ( plus_plus_nat @ M3 @ N ) ) ) ).
% add_Suc
thf(fact_1141_add__Suc__shift,axiom,
! [M3: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M3 ) @ N )
= ( plus_plus_nat @ M3 @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1142_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_1143_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_1144_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_1145_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_1146_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B3: nat] : ( plus_plus_nat @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_1147_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_1148_group__cancel_Oadd2,axiom,
! [B5: nat,K: nat,B: nat,A: nat] :
( ( B5
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B5 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_1149_group__cancel_Oadd1,axiom,
! [A3: nat,K: nat,A: nat,B: nat] :
( ( A3
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A3 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_1150_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_1151_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_1152_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_1153_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_1154_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_1155_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1156_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_1157_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_1158_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_1159_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_1160_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_1161_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_1162_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1163_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1164_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_1165_trans__less__add1,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M3 ) ) ) ).
% trans_less_add1
thf(fact_1166_trans__less__add2,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M3 @ J ) ) ) ).
% trans_less_add2
thf(fact_1167_less__add__eq__less,axiom,
! [K: nat,L: nat,M3: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M3 @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M3 @ N ) ) ) ).
% less_add_eq_less
thf(fact_1168_add__is__1,axiom,
! [M3: nat,N: nat] :
( ( ( plus_plus_nat @ M3 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M3
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M3 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1169_one__is__add,axiom,
! [M3: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M3 @ N ) )
= ( ( ( M3
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M3 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1170_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_1171_less__natE,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M3 @ Q3 ) ) ) ) ).
% less_natE
thf(fact_1172_less__add__Suc1,axiom,
! [I: nat,M3: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M3 ) ) ) ).
% less_add_Suc1
thf(fact_1173_less__add__Suc2,axiom,
! [I: nat,M3: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M3 @ I ) ) ) ).
% less_add_Suc2
thf(fact_1174_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M7: nat,N4: nat] :
? [K3: nat] :
( N4
= ( suc @ ( plus_plus_nat @ M7 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1175_less__imp__Suc__add,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1176_mono__nat__linear__lb,axiom,
! [F: nat > nat,M3: nat,K: nat] :
( ! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ ( F @ M2 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M3 ) @ K ) @ ( F @ ( plus_plus_nat @ M3 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1177_diff__add__0,axiom,
! [N: nat,M3: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M3 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1178_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_1179_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_1180_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_1181_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_1182_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_1183_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_1184_add__diff__inverse__nat,axiom,
! [M3: nat,N: nat] :
( ~ ( ord_less_nat @ M3 @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M3 @ N ) )
= M3 ) ) ).
% add_diff_inverse_nat
thf(fact_1185_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1186_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1187_Suc__eq__plus1,axiom,
( suc
= ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1188_replicate__add,axiom,
! [N: nat,M3: nat,X: nat] :
( ( replicate_nat @ ( plus_plus_nat @ N @ M3 ) @ X )
= ( append_nat @ ( replicate_nat @ N @ X ) @ ( replicate_nat @ M3 @ X ) ) ) ).
% replicate_add
thf(fact_1189_add_Omonoid__axioms,axiom,
monoid_nat @ plus_plus_nat @ zero_zero_nat ).
% add.monoid_axioms
thf(fact_1190_take__drop,axiom,
! [N: nat,M3: nat,Xs: list_nat] :
( ( take_nat @ N @ ( drop_nat @ M3 @ Xs ) )
= ( drop_nat @ M3 @ ( take_nat @ ( plus_plus_nat @ N @ M3 ) @ Xs ) ) ) ).
% take_drop
thf(fact_1191_gen__length__def,axiom,
( gen_length_nat
= ( ^ [N4: nat,Xs4: list_nat] : ( plus_plus_nat @ N4 @ ( size_size_list_nat @ Xs4 ) ) ) ) ).
% gen_length_def
thf(fact_1192_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_1193_add__neg__nonpos,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le72135733267957522d_enat @ A @ zero_z5237406670263579293d_enat )
=> ( ( ord_le2932123472753598470d_enat @ B @ zero_z5237406670263579293d_enat )
=> ( ord_le72135733267957522d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ zero_z5237406670263579293d_enat ) ) ) ).
% add_neg_nonpos
thf(fact_1194_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1195_add__nonneg__pos,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ A )
=> ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ B )
=> ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1196_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_1197_add__nonpos__neg,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le2932123472753598470d_enat @ A @ zero_z5237406670263579293d_enat )
=> ( ( ord_le72135733267957522d_enat @ B @ zero_z5237406670263579293d_enat )
=> ( ord_le72135733267957522d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ zero_z5237406670263579293d_enat ) ) ) ).
% add_nonpos_neg
thf(fact_1198_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1199_add__pos__nonneg,axiom,
! [A: extended_enat,B: extended_enat] :
( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ A )
=> ( ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ B )
=> ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1200_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1201_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1202_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 ) )
& ! [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_1203_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 ) )
| ? [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1204_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_1205_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_1206_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_1207_list_Osize_I4_J,axiom,
! [X21: nat,X22: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_size_list_nat @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_1208_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M7: nat,N4: nat] : ( if_nat @ ( M7 = zero_zero_nat ) @ N4 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M7 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% add_eq_if
thf(fact_1209_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_1210_Nitpick_Osize__list__simp_I1_J,axiom,
( size_list_nat
= ( ^ [F2: nat > nat,Xs4: list_nat] : ( if_nat @ ( Xs4 = nil_nat ) @ zero_zero_nat @ ( suc @ ( plus_plus_nat @ ( F2 @ ( hd_nat @ Xs4 ) ) @ ( size_list_nat @ F2 @ ( tl_nat @ Xs4 ) ) ) ) ) ) ) ).
% Nitpick.size_list_simp(1)
thf(fact_1211_plus__Nil,axiom,
! [Xs: list_nat] :
( ( plus_plus_list_nat @ nil_nat @ Xs )
= nil_nat ) ).
% plus_Nil
thf(fact_1212_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_1213_list_Osize__gen_I1_J,axiom,
! [X: nat > nat] :
( ( size_list_nat @ X @ nil_nat )
= zero_zero_nat ) ).
% list.size_gen(1)
thf(fact_1214_rotate__drop__take,axiom,
( rotate_nat
= ( ^ [N4: nat,Xs4: list_nat] : ( append_nat @ ( drop_nat @ ( modulo_modulo_nat @ N4 @ ( size_size_list_nat @ Xs4 ) ) @ Xs4 ) @ ( take_nat @ ( modulo_modulo_nat @ N4 @ ( size_size_list_nat @ Xs4 ) ) @ Xs4 ) ) ) ) ).
% rotate_drop_take
thf(fact_1215_sorted__rev__iff__nth__mono,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
= ( ! [I2: nat,J2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ J2 ) @ ( nth_nat @ Xs @ I2 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_1216_sorted__rev__iff__nth__mono,axiom,
! [Xs: list_Extended_enat] :
( ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ ( rev_Extended_enat @ Xs ) )
= ( ! [I2: nat,J2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_s3941691890525107288d_enat @ Xs ) )
=> ( ord_le2932123472753598470d_enat @ ( nth_Extended_enat @ Xs @ J2 ) @ ( nth_Extended_enat @ Xs @ I2 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_1217_sorted__map__plus__iff,axiom,
! [A: nat,Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ ( plus_plus_nat @ A ) @ Xs ) )
= ( sorted_wrt_nat @ ord_less_eq_nat @ Xs ) ) ).
% sorted_map_plus_iff
thf(fact_1218_rotate__id,axiom,
! [N: nat,Xs: list_nat] :
( ( ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Xs ) )
= zero_zero_nat )
=> ( ( rotate_nat @ N @ Xs )
= Xs ) ) ).
% rotate_id
thf(fact_1219_sorted__wrt__take,axiom,
! [F: nat > nat > $o,Xs: list_nat,N: nat] :
( ( sorted_wrt_nat @ F @ Xs )
=> ( sorted_wrt_nat @ F @ ( take_nat @ N @ Xs ) ) ) ).
% sorted_wrt_take
thf(fact_1220_sorted__wrt__drop,axiom,
! [F: nat > nat > $o,Xs: list_nat,N: nat] :
( ( sorted_wrt_nat @ F @ Xs )
=> ( sorted_wrt_nat @ F @ ( drop_nat @ N @ Xs ) ) ) ).
% sorted_wrt_drop
thf(fact_1221_sorted__drop,axiom,
! [Xs: list_nat,N: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( drop_nat @ N @ Xs ) ) ) ).
% sorted_drop
thf(fact_1222_sorted__drop,axiom,
! [Xs: list_Extended_enat,N: nat] :
( ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ Xs )
=> ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ ( drop_Extended_enat @ N @ Xs ) ) ) ).
% sorted_drop
thf(fact_1223_sorted__take,axiom,
! [Xs: list_nat,N: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( take_nat @ N @ Xs ) ) ) ).
% sorted_take
thf(fact_1224_sorted__take,axiom,
! [Xs: list_Extended_enat,N: nat] :
( ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ Xs )
=> ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ ( take_Extended_enat @ N @ Xs ) ) ) ).
% sorted_take
thf(fact_1225_sorted1,axiom,
! [X: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X @ nil_nat ) ) ).
% sorted1
thf(fact_1226_sorted1,axiom,
! [X: extended_enat] : ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ ( cons_Extended_enat @ X @ nil_Extended_enat ) ) ).
% sorted1
thf(fact_1227_sorted__wrt01,axiom,
! [Xs: list_nat,P: nat > nat > $o] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( sorted_wrt_nat @ P @ Xs ) ) ).
% sorted_wrt01
thf(fact_1228_sorted__wrt__iff__nth__less,axiom,
( sorted_wrt_nat
= ( ^ [P5: nat > nat > $o,Xs4: list_nat] :
! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs4 ) )
=> ( P5 @ ( nth_nat @ Xs4 @ I2 ) @ ( nth_nat @ Xs4 @ J2 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_1229_sorted__wrt__nth__less,axiom,
! [P: nat > nat > $o,Xs: list_nat,I: nat,J: nat] :
( ( sorted_wrt_nat @ P @ Xs )
=> ( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_1230_sorted__butlast,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( butlast_nat @ Xs ) ) ) ) ).
% sorted_butlast
thf(fact_1231_sorted__butlast,axiom,
! [Xs: list_Extended_enat] :
( ( Xs != nil_Extended_enat )
=> ( ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ Xs )
=> ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ ( butlas2973130096617243576d_enat @ Xs ) ) ) ) ).
% sorted_butlast
thf(fact_1232_strict__sorted__imp__sorted,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ Xs ) ) ).
% strict_sorted_imp_sorted
thf(fact_1233_strict__sorted__imp__sorted,axiom,
! [Xs: list_Extended_enat] :
( ( sorted143172755617435219d_enat @ ord_le72135733267957522d_enat @ Xs )
=> ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ Xs ) ) ).
% strict_sorted_imp_sorted
thf(fact_1234_sorted2,axiom,
! [X: nat,Y: nat,Zs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X @ ( cons_nat @ Y @ Zs ) ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ Y @ Zs ) ) ) ) ).
% sorted2
thf(fact_1235_sorted2,axiom,
! [X: extended_enat,Y: extended_enat,Zs: list_Extended_enat] :
( ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ ( cons_Extended_enat @ X @ ( cons_Extended_enat @ Y @ Zs ) ) )
= ( ( ord_le2932123472753598470d_enat @ X @ Y )
& ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ ( cons_Extended_enat @ Y @ Zs ) ) ) ) ).
% sorted2
thf(fact_1236_sorted0,axiom,
sorted_wrt_nat @ ord_less_eq_nat @ nil_nat ).
% sorted0
thf(fact_1237_sorted0,axiom,
sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ nil_Extended_enat ).
% sorted0
thf(fact_1238_sorted__replicate,axiom,
! [N: nat,X: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( replicate_nat @ N @ X ) ) ).
% sorted_replicate
thf(fact_1239_sorted__replicate,axiom,
! [N: nat,X: extended_enat] : ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ ( replic7216382294607269926d_enat @ N @ X ) ) ).
% sorted_replicate
thf(fact_1240_sorted__tl,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( tl_nat @ Xs ) ) ) ).
% sorted_tl
thf(fact_1241_sorted__tl,axiom,
! [Xs: list_Extended_enat] :
( ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ Xs )
=> ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ ( tl_Extended_enat @ Xs ) ) ) ).
% sorted_tl
thf(fact_1242_sorted__remdups__adj,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( remdups_adj_nat @ Xs ) ) ) ).
% sorted_remdups_adj
thf(fact_1243_sorted__remdups__adj,axiom,
! [Xs: list_Extended_enat] :
( ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ Xs )
=> ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ ( remdup6152102037098707618d_enat @ Xs ) ) ) ).
% sorted_remdups_adj
thf(fact_1244_sorted__upt,axiom,
! [M3: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M3 @ N ) ) ).
% sorted_upt
thf(fact_1245_sorted__wrt_Osimps_I1_J,axiom,
! [P: nat > nat > $o] : ( sorted_wrt_nat @ P @ nil_nat ) ).
% sorted_wrt.simps(1)
thf(fact_1246_sorted__wrt__upt,axiom,
! [M3: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M3 @ N ) ) ).
% sorted_wrt_upt
thf(fact_1247_sorted__wrt1,axiom,
! [P: nat > nat > $o,X: nat] : ( sorted_wrt_nat @ P @ ( cons_nat @ X @ nil_nat ) ) ).
% sorted_wrt1
thf(fact_1248_strict__sorted__simps_I1_J,axiom,
sorted_wrt_nat @ ord_less_nat @ nil_nat ).
% strict_sorted_simps(1)
thf(fact_1249_rotate__conv__mod,axiom,
( rotate_nat
= ( ^ [N4: nat,Xs4: list_nat] : ( rotate_nat @ ( modulo_modulo_nat @ N4 @ ( size_size_list_nat @ Xs4 ) ) @ Xs4 ) ) ) ).
% rotate_conv_mod
thf(fact_1250_sorted01,axiom,
! [Xs: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ Xs ) ) ).
% sorted01
thf(fact_1251_sorted01,axiom,
! [Xs: list_Extended_enat] :
( ( ord_less_eq_nat @ ( size_s3941691890525107288d_enat @ Xs ) @ one_one_nat )
=> ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ Xs ) ) ).
% sorted01
thf(fact_1252_sorted__iff__nth__mono__less,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Xs @ J2 ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_1253_sorted__iff__nth__mono__less,axiom,
! [Xs: list_Extended_enat] :
( ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ Xs )
= ( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_s3941691890525107288d_enat @ Xs ) )
=> ( ord_le2932123472753598470d_enat @ ( nth_Extended_enat @ Xs @ I2 ) @ ( nth_Extended_enat @ Xs @ J2 ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_1254_rotate__rev,axiom,
! [N: nat,Xs: list_nat] :
( ( rotate_nat @ N @ ( rev_nat @ Xs ) )
= ( rev_nat @ ( rotate_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Xs ) ) ) @ Xs ) ) ) ).
% rotate_rev
thf(fact_1255_sorted__wrt__less__idx,axiom,
! [Ns: list_nat,I: nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Ns )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
=> ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).
% sorted_wrt_less_idx
thf(fact_1256_sorted__iff__nth__Suc,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I2: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Xs @ ( suc @ I2 ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_1257_sorted__iff__nth__Suc,axiom,
! [Xs: list_Extended_enat] :
( ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ Xs )
= ( ! [I2: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ ( size_s3941691890525107288d_enat @ Xs ) )
=> ( ord_le2932123472753598470d_enat @ ( nth_Extended_enat @ Xs @ I2 ) @ ( nth_Extended_enat @ Xs @ ( suc @ I2 ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_1258_sorted__iff__nth__mono,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I2: nat,J2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Xs @ J2 ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_1259_sorted__iff__nth__mono,axiom,
! [Xs: list_Extended_enat] :
( ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ Xs )
= ( ! [I2: nat,J2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_s3941691890525107288d_enat @ Xs ) )
=> ( ord_le2932123472753598470d_enat @ ( nth_Extended_enat @ Xs @ I2 ) @ ( nth_Extended_enat @ Xs @ J2 ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_1260_sorted__nth__mono,axiom,
! [Xs: list_nat,I: nat,J: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_1261_sorted__nth__mono,axiom,
! [Xs: list_Extended_enat,I: nat,J: nat] :
( ( sorted143172755617435219d_enat @ ord_le2932123472753598470d_enat @ Xs )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_s3941691890525107288d_enat @ Xs ) )
=> ( ord_le2932123472753598470d_enat @ ( nth_Extended_enat @ Xs @ I ) @ ( nth_Extended_enat @ Xs @ J ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_1262_nth__rotate,axiom,
! [N: nat,Xs: list_nat,M3: nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( rotate_nat @ M3 @ Xs ) @ N )
= ( nth_nat @ Xs @ ( modulo_modulo_nat @ ( plus_plus_nat @ M3 @ N ) @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% nth_rotate
thf(fact_1263_nth__rotate1,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( rotate1_nat @ Xs ) @ N )
= ( nth_nat @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% nth_rotate1
thf(fact_1264_hd__rotate__conv__nth,axiom,
! [Xs: list_nat,N: nat] :
( ( Xs != nil_nat )
=> ( ( hd_nat @ ( rotate_nat @ N @ Xs ) )
= ( nth_nat @ Xs @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% hd_rotate_conv_nth
% 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 @ x @ y )
= ( ( pointwise_less @ x @ y )
| ( x = y ) ) ) ).
%------------------------------------------------------------------------------