TPTP Problem File: SLH0561^1.p
View Solutions
- 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_00890_032545__13600216_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1417 ( 518 unt; 146 typ; 0 def)
% Number of atoms : 4275 (1585 equ; 0 cnn)
% Maximal formula atoms : 17 ( 3 avg)
% Number of connectives : 13339 ( 374 ~; 71 |; 424 &;10525 @)
% ( 0 <=>;1945 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 7 avg)
% Number of types : 17 ( 16 usr)
% Number of type conns : 932 ( 932 >; 0 *; 0 +; 0 <<)
% Number of symbols : 133 ( 130 usr; 8 con; 0-3 aty)
% Number of variables : 4111 ( 364 ^;3550 !; 197 ?;4111 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:15:15.355
%------------------------------------------------------------------------------
% Could-be-implicit typings (16)
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
list_list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
set_list_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
list_set_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
set_set_list_nat: $tType ).
thf(ty_n_t__List__Olist_I_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J_J,type,
list_list_nat_o: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J_J,type,
set_list_nat_o: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
list_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__List__Olist_I_062_It__Nat__Onat_M_Eo_J_J,type,
list_nat_o: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
set_nat_o: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (130)
thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
finite7325466520557071688st_nat: set_list_list_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Nat__Onat_J,type,
finite_card_list_nat: set_list_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
finite8170528100393595399st_nat: set_list_list_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Nat__Onat_J,type,
finite8100373058378681591st_nat: set_list_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
finite7047420756378620717st_nat: set_set_list_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
minus_1139252259498527702_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > list_nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_Eo,type,
minus_minus_o: $o > $o > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__List__Olist_I_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J_J,type,
minus_6424643995335282652_nat_o: list_list_nat_o > list_list_nat_o > list_list_nat_o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__List__Olist_I_062_It__Nat__Onat_M_Eo_J_J,type,
minus_8437841901652764780_nat_o: list_nat_o > list_nat_o > list_nat_o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
minus_1432716909304059609st_nat: list_list_list_nat > list_list_list_nat > list_list_list_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
minus_3911745200923244873st_nat: list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__List__Olist_It__Nat__Onat_J,type,
minus_minus_list_nat: list_nat > list_nat > list_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__List__Olist_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
minus_4578319062706284671st_nat: list_set_list_nat > list_set_list_nat > list_set_list_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
minus_1998526526692677103et_nat: list_set_nat > list_set_nat > list_set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
minus_7954133019191499631st_nat: set_list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
plus_p2116291331692525561st_nat: list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__List__Olist_It__Nat__Onat_J,type,
plus_plus_list_nat: list_nat > list_nat > list_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
groups4396056296759096172at_nat: ( list_nat > nat ) > set_list_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
groups8294997508430121362at_nat: ( set_nat > nat ) > set_set_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
groups2907647131375434839at_nat: ( list_nat > nat ) > set_list_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_001t__Nat__Onat_001t__Nat__Onat,type,
groups708209901874060359at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Nat__Onat,type,
groups4561878855575611511st_nat: list_nat > nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Khovanskii_OKhovanskii_Oaugmentum,type,
augmentum: list_nat > list_nat ).
thf(sy_c_Khovanskii_OKhovanskii_Odementum,type,
dementum: list_nat > list_nat ).
thf(sy_c_Khovanskii_OKhovanskii_Olength__sum__set,type,
length_sum_set: nat > nat > set_list_nat ).
thf(sy_c_Khovanskii_OKhovanskii_Olist__incr,type,
list_incr: nat > list_nat > list_nat ).
thf(sy_c_Khovanskii_OKhovanskii_Ominimal__elements,type,
minimal_elements: set_list_nat > set_list_nat ).
thf(sy_c_Khovanskii_OKhovanskii_Ominimal__elementsp,type,
minimal_elementsp: ( list_nat > $o ) > list_nat > $o ).
thf(sy_c_Khovanskii_Ofinsets_001t__List__Olist_It__Nat__Onat_J,type,
finsets_list_nat: set_list_nat > nat > set_set_list_nat ).
thf(sy_c_Khovanskii_Ofinsets_001t__Nat__Onat,type,
finsets_nat: set_nat > nat > set_set_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__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_Odistinct_001t__List__Olist_It__Nat__Onat_J,type,
distinct_list_nat: list_list_nat > $o ).
thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
distinct_nat: list_nat > $o ).
thf(sy_c_List_Odistinct_001t__Set__Oset_It__Nat__Onat_J,type,
distinct_set_nat: list_set_nat > $o ).
thf(sy_c_List_Olist_OCons_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
cons_list_nat_o: ( list_nat > $o ) > list_list_nat_o > list_list_nat_o ).
thf(sy_c_List_Olist_OCons_001_062_It__Nat__Onat_M_Eo_J,type,
cons_nat_o: ( nat > $o ) > list_nat_o > list_nat_o ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
cons_list_list_nat: list_list_nat > list_list_list_nat > list_list_list_nat ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Nat__Onat_J,type,
cons_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
cons_set_list_nat: set_list_nat > list_set_list_nat > list_set_list_nat ).
thf(sy_c_List_Olist_OCons_001t__Set__Oset_It__Nat__Onat_J,type,
cons_set_nat: set_nat > list_set_nat > list_set_nat ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Nat__Onat_J,type,
nil_list_nat: list_list_nat ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
map_li7225945977422193158st_nat: ( list_nat > list_nat ) > list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
map_list_nat_nat: ( list_nat > nat ) > list_list_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
set_list_nat2: list_list_nat > set_list_nat ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
set_set_nat2: list_set_nat > set_set_nat ).
thf(sy_c_List_On__lists_001t__Nat__Onat,type,
n_lists_nat: nat > list_nat > list_list_nat ).
thf(sy_c_List_Onth_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
nth_list_nat_o: list_list_nat_o > nat > list_nat > $o ).
thf(sy_c_List_Onth_001_062_It__Nat__Onat_M_Eo_J,type,
nth_nat_o: list_nat_o > nat > nat > $o ).
thf(sy_c_List_Onth_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
nth_list_list_nat: list_list_list_nat > nat > list_list_nat ).
thf(sy_c_List_Onth_001t__List__Olist_It__Nat__Onat_J,type,
nth_list_nat: list_list_nat > nat > list_nat ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
nth_set_list_nat: list_set_list_nat > nat > set_list_nat ).
thf(sy_c_List_Onth_001t__Set__Oset_It__Nat__Onat_J,type,
nth_set_nat: list_set_nat > nat > set_nat ).
thf(sy_c_List_Osorted__wrt_001t__List__Olist_It__Nat__Onat_J,type,
sorted_wrt_list_nat: ( list_nat > list_nat > $o ) > list_list_nat > $o ).
thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J_J,type,
size_s6069988884782743777_nat_o: list_list_nat_o > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_062_It__Nat__Onat_M_Eo_J_J,type,
size_size_list_nat_o: list_nat_o > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
size_s6248950052170075156st_nat: list_list_list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
size_s3023201423986296836st_nat: list_list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
size_s40095690673326906st_nat: list_set_list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
size_s3254054031482475050et_nat: list_set_nat > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
ord_less_list_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__List__Olist_It__Nat__Onat_J,type,
ord_less_list_nat: list_nat > list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
ord_le1190675801316882794st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
ord_le1520216061033275535_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__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__Set__Oset_I_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J_J,type,
ord_le2112173733987654127_nat_o: set_list_nat_o > set_list_nat_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
ord_le6029213668185085951_nat_o: set_nat_o > set_nat_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
ord_le1068707526560357548st_nat: set_set_list_nat > set_set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le9131159989063066194et_nat: set_set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
collec5989764272469232197st_nat: ( list_list_nat > $o ) > set_list_list_nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
collect_set_list_nat: ( set_list_nat > $o ) > set_set_list_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
set_or689571727643897636_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > set_list_nat_o ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001_062_It__Nat__Onat_M_Eo_J,type,
set_or99350221437691188_nat_o: ( nat > $o ) > ( nat > $o ) > set_nat_o ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__List__Olist_It__Nat__Onat_J,type,
set_or6836045993805503595st_nat: list_nat > list_nat > set_list_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_or1270514513317581473st_nat: set_list_nat > set_list_nat > set_set_list_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4548717258645045905et_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_or9137876137106135879et_nat: set_set_nat > set_set_nat > set_set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
set_or3326398383446669622_nat_o: ( list_nat > $o ) > set_list_nat_o ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001_062_It__Nat__Onat_M_Eo_J,type,
set_ord_atMost_nat_o: ( nat > $o ) > set_nat_o ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__List__Olist_It__Nat__Onat_J,type,
set_or4185896845444216793st_nat: list_nat > set_list_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_or2492388921469580815st_nat: set_list_nat > set_set_list_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4236626031148496127et_nat: set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_or7210490968680142261et_nat: set_set_nat > set_set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__List__Olist_It__Nat__Onat_J,type,
set_or3033090826390029821st_nat: list_nat > set_list_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_member_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
member_list_nat_o: ( list_nat > $o ) > set_list_nat_o > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_M_Eo_J,type,
member_nat_o: ( nat > $o ) > set_nat_o > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
member_list_list_nat: list_list_nat > set_list_list_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
member_set_list_nat: set_list_nat > set_set_list_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
member_set_set_nat: set_set_nat > set_set_set_nat > $o ).
thf(sy_v_i____,type,
i: nat ).
thf(sy_v_ns,type,
ns: list_nat ).
% Relevant facts (1267)
thf(fact_0_False,axiom,
i != zero_zero_nat ).
% False
thf(fact_1_assms_I1_J,axiom,
~ ( member_nat @ zero_zero_nat @ ( set_nat2 @ ns ) ) ).
% assms(1)
thf(fact_2_sum_Oneutral__const,axiom,
! [A: set_list_nat] :
( ( groups4396056296759096172at_nat
@ ^ [Uu: list_nat] : zero_zero_nat
@ A )
= zero_zero_nat ) ).
% sum.neutral_const
thf(fact_3_sum_Oneutral__const,axiom,
! [A: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [Uu: nat] : zero_zero_nat
@ A )
= zero_zero_nat ) ).
% sum.neutral_const
thf(fact_4_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_5_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_6_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_7_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_8_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ A2 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_9_calculation,axiom,
( ( nth_nat @ ( augmentum @ ( dementum @ ns ) ) @ i )
= ( groups3542108847815614940at_nat
@ ^ [J: nat] : ( minus_minus_nat @ ( nth_nat @ ns @ J ) @ ( if_nat @ ( J = zero_zero_nat ) @ zero_zero_nat @ ( nth_nat @ ns @ ( minus_minus_nat @ J @ one_one_nat ) ) ) )
@ ( set_ord_atMost_nat @ i ) ) ) ).
% calculation
thf(fact_10_atMost__atLeast0,axiom,
( set_ord_atMost_nat
= ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).
% atMost_atLeast0
thf(fact_11_atMost__eq__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( set_or4236626031148496127et_nat @ X )
= ( set_or4236626031148496127et_nat @ Y ) )
= ( X = Y ) ) ).
% atMost_eq_iff
thf(fact_12_atMost__eq__iff,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( set_or4185896845444216793st_nat @ X )
= ( set_or4185896845444216793st_nat @ Y ) )
= ( X = Y ) ) ).
% atMost_eq_iff
thf(fact_13_atMost__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_atMost_nat @ X )
= ( set_ord_atMost_nat @ Y ) )
= ( X = Y ) ) ).
% atMost_eq_iff
thf(fact_14_zero__notin__augmentum,axiom,
! [Ns: list_nat] :
( ~ ( member_nat @ zero_zero_nat @ ( set_nat2 @ Ns ) )
=> ~ ( member_nat @ zero_zero_nat @ ( set_nat2 @ ( augmentum @ Ns ) ) ) ) ).
% zero_notin_augmentum
thf(fact_15_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_16_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_17_diff__right__commute,axiom,
! [A2: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_18_sum_Oreindex__bij__witness,axiom,
! [S: set_set_nat,I: nat > set_nat,J2: set_nat > nat,T: set_nat,H: nat > nat,G: set_nat > nat] :
( ! [A3: set_nat] :
( ( member_set_nat @ A3 @ S )
=> ( ( I @ ( J2 @ A3 ) )
= A3 ) )
=> ( ! [A3: set_nat] :
( ( member_set_nat @ A3 @ S )
=> ( member_nat @ ( J2 @ A3 ) @ T ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T )
=> ( ( J2 @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T )
=> ( member_set_nat @ ( I @ B2 ) @ S ) )
=> ( ! [A3: set_nat] :
( ( member_set_nat @ A3 @ S )
=> ( ( H @ ( J2 @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups8294997508430121362at_nat @ G @ S )
= ( groups3542108847815614940at_nat @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_19_sum_Oreindex__bij__witness,axiom,
! [S: set_set_nat,I: list_nat > set_nat,J2: set_nat > list_nat,T: set_list_nat,H: list_nat > nat,G: set_nat > nat] :
( ! [A3: set_nat] :
( ( member_set_nat @ A3 @ S )
=> ( ( I @ ( J2 @ A3 ) )
= A3 ) )
=> ( ! [A3: set_nat] :
( ( member_set_nat @ A3 @ S )
=> ( member_list_nat @ ( J2 @ A3 ) @ T ) )
=> ( ! [B2: list_nat] :
( ( member_list_nat @ B2 @ T )
=> ( ( J2 @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: list_nat] :
( ( member_list_nat @ B2 @ T )
=> ( member_set_nat @ ( I @ B2 ) @ S ) )
=> ( ! [A3: set_nat] :
( ( member_set_nat @ A3 @ S )
=> ( ( H @ ( J2 @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups8294997508430121362at_nat @ G @ S )
= ( groups4396056296759096172at_nat @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_20_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I: set_nat > nat,J2: nat > set_nat,T: set_set_nat,H: set_nat > nat,G: nat > nat] :
( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( I @ ( J2 @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( member_set_nat @ ( J2 @ A3 ) @ T ) )
=> ( ! [B2: set_nat] :
( ( member_set_nat @ B2 @ T )
=> ( ( J2 @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: set_nat] :
( ( member_set_nat @ B2 @ T )
=> ( member_nat @ ( I @ B2 ) @ S ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H @ ( J2 @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S )
= ( groups8294997508430121362at_nat @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_21_sum_Oreindex__bij__witness,axiom,
! [S: set_list_nat,I: set_nat > list_nat,J2: list_nat > set_nat,T: set_set_nat,H: set_nat > nat,G: list_nat > nat] :
( ! [A3: list_nat] :
( ( member_list_nat @ A3 @ S )
=> ( ( I @ ( J2 @ A3 ) )
= A3 ) )
=> ( ! [A3: list_nat] :
( ( member_list_nat @ A3 @ S )
=> ( member_set_nat @ ( J2 @ A3 ) @ T ) )
=> ( ! [B2: set_nat] :
( ( member_set_nat @ B2 @ T )
=> ( ( J2 @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: set_nat] :
( ( member_set_nat @ B2 @ T )
=> ( member_list_nat @ ( I @ B2 ) @ S ) )
=> ( ! [A3: list_nat] :
( ( member_list_nat @ A3 @ S )
=> ( ( H @ ( J2 @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups4396056296759096172at_nat @ G @ S )
= ( groups8294997508430121362at_nat @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_22_sum_Oreindex__bij__witness,axiom,
! [S: set_list_nat,I: list_nat > list_nat,J2: list_nat > list_nat,T: set_list_nat,H: list_nat > nat,G: list_nat > nat] :
( ! [A3: list_nat] :
( ( member_list_nat @ A3 @ S )
=> ( ( I @ ( J2 @ A3 ) )
= A3 ) )
=> ( ! [A3: list_nat] :
( ( member_list_nat @ A3 @ S )
=> ( member_list_nat @ ( J2 @ A3 ) @ T ) )
=> ( ! [B2: list_nat] :
( ( member_list_nat @ B2 @ T )
=> ( ( J2 @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: list_nat] :
( ( member_list_nat @ B2 @ T )
=> ( member_list_nat @ ( I @ B2 ) @ S ) )
=> ( ! [A3: list_nat] :
( ( member_list_nat @ A3 @ S )
=> ( ( H @ ( J2 @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups4396056296759096172at_nat @ G @ S )
= ( groups4396056296759096172at_nat @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_23_sum_Oreindex__bij__witness,axiom,
! [S: set_list_nat,I: nat > list_nat,J2: list_nat > nat,T: set_nat,H: nat > nat,G: list_nat > nat] :
( ! [A3: list_nat] :
( ( member_list_nat @ A3 @ S )
=> ( ( I @ ( J2 @ A3 ) )
= A3 ) )
=> ( ! [A3: list_nat] :
( ( member_list_nat @ A3 @ S )
=> ( member_nat @ ( J2 @ A3 ) @ T ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T )
=> ( ( J2 @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T )
=> ( member_list_nat @ ( I @ B2 ) @ S ) )
=> ( ! [A3: list_nat] :
( ( member_list_nat @ A3 @ S )
=> ( ( H @ ( J2 @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups4396056296759096172at_nat @ G @ S )
= ( groups3542108847815614940at_nat @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_24_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I: list_nat > nat,J2: nat > list_nat,T: set_list_nat,H: list_nat > nat,G: nat > nat] :
( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( I @ ( J2 @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( member_list_nat @ ( J2 @ A3 ) @ T ) )
=> ( ! [B2: list_nat] :
( ( member_list_nat @ B2 @ T )
=> ( ( J2 @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: list_nat] :
( ( member_list_nat @ B2 @ T )
=> ( member_nat @ ( I @ B2 ) @ S ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H @ ( J2 @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S )
= ( groups4396056296759096172at_nat @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_25_sum_Oreindex__bij__witness,axiom,
! [S: set_nat,I: nat > nat,J2: nat > nat,T: set_nat,H: nat > nat,G: nat > nat] :
( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( I @ ( J2 @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( member_nat @ ( J2 @ A3 ) @ T ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T )
=> ( ( J2 @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T )
=> ( member_nat @ ( I @ B2 ) @ S ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H @ ( J2 @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S )
= ( groups3542108847815614940at_nat @ H @ T ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_26_sum_Oeq__general__inverses,axiom,
! [B3: set_nat,K: nat > set_nat,A: set_set_nat,H: set_nat > nat,Gamma: nat > nat,Phi: set_nat > nat] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B3 )
=> ( ( member_set_nat @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B3 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8294997508430121362at_nat @ Phi @ A )
= ( groups3542108847815614940at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_27_sum_Oeq__general__inverses,axiom,
! [B3: set_list_nat,K: list_nat > set_nat,A: set_set_nat,H: set_nat > list_nat,Gamma: list_nat > nat,Phi: set_nat > nat] :
( ! [Y2: list_nat] :
( ( member_list_nat @ Y2 @ B3 )
=> ( ( member_set_nat @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ( member_list_nat @ ( H @ X2 ) @ B3 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8294997508430121362at_nat @ Phi @ A )
= ( groups4396056296759096172at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_28_sum_Oeq__general__inverses,axiom,
! [B3: set_set_nat,K: set_nat > nat,A: set_nat,H: nat > set_nat,Gamma: set_nat > nat,Phi: nat > nat] :
( ! [Y2: set_nat] :
( ( member_set_nat @ Y2 @ B3 )
=> ( ( member_nat @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_set_nat @ ( H @ X2 ) @ B3 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A )
= ( groups8294997508430121362at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_29_sum_Oeq__general__inverses,axiom,
! [B3: set_set_nat,K: set_nat > list_nat,A: set_list_nat,H: list_nat > set_nat,Gamma: set_nat > nat,Phi: list_nat > nat] :
( ! [Y2: set_nat] :
( ( member_set_nat @ Y2 @ B3 )
=> ( ( member_list_nat @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ A )
=> ( ( member_set_nat @ ( H @ X2 ) @ B3 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups4396056296759096172at_nat @ Phi @ A )
= ( groups8294997508430121362at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_30_sum_Oeq__general__inverses,axiom,
! [B3: set_list_nat,K: list_nat > list_nat,A: set_list_nat,H: list_nat > list_nat,Gamma: list_nat > nat,Phi: list_nat > nat] :
( ! [Y2: list_nat] :
( ( member_list_nat @ Y2 @ B3 )
=> ( ( member_list_nat @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ A )
=> ( ( member_list_nat @ ( H @ X2 ) @ B3 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups4396056296759096172at_nat @ Phi @ A )
= ( groups4396056296759096172at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_31_sum_Oeq__general__inverses,axiom,
! [B3: set_nat,K: nat > list_nat,A: set_list_nat,H: list_nat > nat,Gamma: nat > nat,Phi: list_nat > nat] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B3 )
=> ( ( member_list_nat @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B3 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups4396056296759096172at_nat @ Phi @ A )
= ( groups3542108847815614940at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_32_sum_Oeq__general__inverses,axiom,
! [B3: set_list_nat,K: list_nat > nat,A: set_nat,H: nat > list_nat,Gamma: list_nat > nat,Phi: nat > nat] :
( ! [Y2: list_nat] :
( ( member_list_nat @ Y2 @ B3 )
=> ( ( member_nat @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_list_nat @ ( H @ X2 ) @ B3 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A )
= ( groups4396056296759096172at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_33_sum_Oeq__general__inverses,axiom,
! [B3: set_nat,K: nat > nat,A: set_nat,H: nat > nat,Gamma: nat > nat,Phi: nat > nat] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B3 )
=> ( ( member_nat @ ( K @ Y2 ) @ A )
& ( ( H @ ( K @ Y2 ) )
= Y2 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B3 )
& ( ( K @ ( H @ X2 ) )
= X2 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A )
= ( groups3542108847815614940at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_34_sum_Oeq__general,axiom,
! [B3: set_nat,A: set_set_nat,H: set_nat > nat,Gamma: nat > nat,Phi: set_nat > nat] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B3 )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
& ( ( H @ X3 )
= Y2 )
& ! [Ya: set_nat] :
( ( ( member_set_nat @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B3 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8294997508430121362at_nat @ Phi @ A )
= ( groups3542108847815614940at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_35_sum_Oeq__general,axiom,
! [B3: set_list_nat,A: set_set_nat,H: set_nat > list_nat,Gamma: list_nat > nat,Phi: set_nat > nat] :
( ! [Y2: list_nat] :
( ( member_list_nat @ Y2 @ B3 )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
& ( ( H @ X3 )
= Y2 )
& ! [Ya: set_nat] :
( ( ( member_set_nat @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ( member_list_nat @ ( H @ X2 ) @ B3 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups8294997508430121362at_nat @ Phi @ A )
= ( groups4396056296759096172at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_36_sum_Oeq__general,axiom,
! [B3: set_set_nat,A: set_nat,H: nat > set_nat,Gamma: set_nat > nat,Phi: nat > nat] :
( ! [Y2: set_nat] :
( ( member_set_nat @ Y2 @ B3 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ( H @ X3 )
= Y2 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_set_nat @ ( H @ X2 ) @ B3 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A )
= ( groups8294997508430121362at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_37_sum_Oeq__general,axiom,
! [B3: set_set_nat,A: set_list_nat,H: list_nat > set_nat,Gamma: set_nat > nat,Phi: list_nat > nat] :
( ! [Y2: set_nat] :
( ( member_set_nat @ Y2 @ B3 )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
& ( ( H @ X3 )
= Y2 )
& ! [Ya: list_nat] :
( ( ( member_list_nat @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ A )
=> ( ( member_set_nat @ ( H @ X2 ) @ B3 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups4396056296759096172at_nat @ Phi @ A )
= ( groups8294997508430121362at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_38_sum_Oeq__general,axiom,
! [B3: set_list_nat,A: set_list_nat,H: list_nat > list_nat,Gamma: list_nat > nat,Phi: list_nat > nat] :
( ! [Y2: list_nat] :
( ( member_list_nat @ Y2 @ B3 )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
& ( ( H @ X3 )
= Y2 )
& ! [Ya: list_nat] :
( ( ( member_list_nat @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ A )
=> ( ( member_list_nat @ ( H @ X2 ) @ B3 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups4396056296759096172at_nat @ Phi @ A )
= ( groups4396056296759096172at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_39_sum_Oeq__general,axiom,
! [B3: set_nat,A: set_list_nat,H: list_nat > nat,Gamma: nat > nat,Phi: list_nat > nat] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B3 )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
& ( ( H @ X3 )
= Y2 )
& ! [Ya: list_nat] :
( ( ( member_list_nat @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B3 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups4396056296759096172at_nat @ Phi @ A )
= ( groups3542108847815614940at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_40_sum_Oeq__general,axiom,
! [B3: set_list_nat,A: set_nat,H: nat > list_nat,Gamma: list_nat > nat,Phi: nat > nat] :
( ! [Y2: list_nat] :
( ( member_list_nat @ Y2 @ B3 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ( H @ X3 )
= Y2 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_list_nat @ ( H @ X2 ) @ B3 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A )
= ( groups4396056296759096172at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_41_sum_Oeq__general,axiom,
! [B3: set_nat,A: set_nat,H: nat > nat,Gamma: nat > nat,Phi: nat > nat] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ B3 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ( H @ X3 )
= Y2 )
& ! [Ya: nat] :
( ( ( member_nat @ Ya @ A )
& ( ( H @ Ya )
= Y2 ) )
=> ( Ya = X3 ) ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ ( H @ X2 ) @ B3 )
& ( ( Gamma @ ( H @ X2 ) )
= ( Phi @ X2 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A )
= ( groups3542108847815614940at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_42_sum_Ocong,axiom,
! [A: set_list_nat,B3: set_list_nat,G: list_nat > nat,H: list_nat > nat] :
( ( A = B3 )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ B3 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups4396056296759096172at_nat @ G @ A )
= ( groups4396056296759096172at_nat @ H @ B3 ) ) ) ) ).
% sum.cong
thf(fact_43_sum_Ocong,axiom,
! [A: set_nat,B3: set_nat,G: nat > nat,H: nat > nat] :
( ( A = B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ A )
= ( groups3542108847815614940at_nat @ H @ B3 ) ) ) ) ).
% sum.cong
thf(fact_44_diff__commute,axiom,
! [I: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J2 ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J2 ) ) ).
% diff_commute
thf(fact_45_sum_Oswap,axiom,
! [G: nat > list_nat > nat,B3: set_list_nat,A: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [I2: nat] : ( groups4396056296759096172at_nat @ ( G @ I2 ) @ B3 )
@ A )
= ( groups4396056296759096172at_nat
@ ^ [J: list_nat] :
( groups3542108847815614940at_nat
@ ^ [I2: nat] : ( G @ I2 @ J )
@ A )
@ B3 ) ) ).
% sum.swap
thf(fact_46_sum_Oswap,axiom,
! [G: list_nat > nat > nat,B3: set_nat,A: set_list_nat] :
( ( groups4396056296759096172at_nat
@ ^ [I2: list_nat] : ( groups3542108847815614940at_nat @ ( G @ I2 ) @ B3 )
@ A )
= ( groups3542108847815614940at_nat
@ ^ [J: nat] :
( groups4396056296759096172at_nat
@ ^ [I2: list_nat] : ( G @ I2 @ J )
@ A )
@ B3 ) ) ).
% sum.swap
thf(fact_47_sum_Oswap,axiom,
! [G: list_nat > list_nat > nat,B3: set_list_nat,A: set_list_nat] :
( ( groups4396056296759096172at_nat
@ ^ [I2: list_nat] : ( groups4396056296759096172at_nat @ ( G @ I2 ) @ B3 )
@ A )
= ( groups4396056296759096172at_nat
@ ^ [J: list_nat] :
( groups4396056296759096172at_nat
@ ^ [I2: list_nat] : ( G @ I2 @ J )
@ A )
@ B3 ) ) ).
% sum.swap
thf(fact_48_sum_Oswap,axiom,
! [G: nat > nat > nat,B3: set_nat,A: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [I2: nat] : ( groups3542108847815614940at_nat @ ( G @ I2 ) @ B3 )
@ A )
= ( groups3542108847815614940at_nat
@ ^ [J: nat] :
( groups3542108847815614940at_nat
@ ^ [I2: nat] : ( G @ I2 @ J )
@ A )
@ B3 ) ) ).
% sum.swap
thf(fact_49_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: set_nat > nat,A: set_set_nat] :
( ( ( groups8294997508430121362at_nat @ G @ A )
!= zero_zero_nat )
=> ~ ! [A3: set_nat] :
( ( member_set_nat @ A3 @ A )
=> ( ( G @ A3 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_50_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: list_nat > nat,A: set_list_nat] :
( ( ( groups4396056296759096172at_nat @ G @ A )
!= zero_zero_nat )
=> ~ ! [A3: list_nat] :
( ( member_list_nat @ A3 @ A )
=> ( ( G @ A3 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_51_sum_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > nat,A: set_nat] :
( ( ( groups3542108847815614940at_nat @ G @ A )
!= zero_zero_nat )
=> ~ ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ( ( G @ A3 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_52_sum_Oneutral,axiom,
! [A: set_list_nat,G: list_nat > nat] :
( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ A )
=> ( ( G @ X2 )
= zero_zero_nat ) )
=> ( ( groups4396056296759096172at_nat @ G @ A )
= zero_zero_nat ) ) ).
% sum.neutral
thf(fact_53_sum_Oneutral,axiom,
! [A: set_nat,G: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( G @ X2 )
= zero_zero_nat ) )
=> ( ( groups3542108847815614940at_nat @ G @ A )
= zero_zero_nat ) ) ).
% sum.neutral
thf(fact_54_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_55_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_56__092_060open_062i_A_060_Alength_A_Iaugmentum_A_Idementum_Ans_J_J_092_060close_062,axiom,
ord_less_nat @ i @ ( size_size_list_nat @ ( augmentum @ ( dementum @ ns ) ) ) ).
% \<open>i < length (augmentum (dementum ns))\<close>
thf(fact_57_minus__apply,axiom,
( minus_minus_nat_o
= ( ^ [A4: nat > $o,B4: nat > $o,X4: nat] : ( minus_minus_o @ ( A4 @ X4 ) @ ( B4 @ X4 ) ) ) ) ).
% minus_apply
thf(fact_58_minus__apply,axiom,
( minus_1139252259498527702_nat_o
= ( ^ [A4: list_nat > $o,B4: list_nat > $o,X4: list_nat] : ( minus_minus_o @ ( A4 @ X4 ) @ ( B4 @ X4 ) ) ) ) ).
% minus_apply
thf(fact_59_distinct__augmentum,axiom,
! [Ns: list_nat] :
( ~ ( member_nat @ zero_zero_nat @ ( set_nat2 @ Ns ) )
=> ( distinct_nat @ ( augmentum @ Ns ) ) ) ).
% distinct_augmentum
thf(fact_60_dementum__def,axiom,
( dementum
= ( ^ [Xs: list_nat] : ( minus_minus_list_nat @ Xs @ ( cons_nat @ zero_zero_nat @ Xs ) ) ) ) ).
% dementum_def
thf(fact_61_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_62_sum__delta__notmem_I1_J,axiom,
! [X: list_nat,S2: set_list_nat,P: list_nat > nat,Q: list_nat > nat] :
( ~ ( member_list_nat @ X @ S2 )
=> ( ( groups4396056296759096172at_nat
@ ^ [Y3: list_nat] : ( if_nat @ ( Y3 = X ) @ ( P @ X ) @ ( Q @ Y3 ) )
@ S2 )
= ( groups4396056296759096172at_nat @ Q @ S2 ) ) ) ).
% sum_delta_notmem(1)
thf(fact_63_sum__delta__notmem_I1_J,axiom,
! [X: nat,S2: set_nat,P: nat > nat,Q: nat > nat] :
( ~ ( member_nat @ X @ S2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [Y3: nat] : ( if_nat @ ( Y3 = X ) @ ( P @ X ) @ ( Q @ Y3 ) )
@ S2 )
= ( groups3542108847815614940at_nat @ Q @ S2 ) ) ) ).
% sum_delta_notmem(1)
thf(fact_64_sum__delta__notmem_I2_J,axiom,
! [X: list_nat,S2: set_list_nat,P: list_nat > nat,Q: list_nat > nat] :
( ~ ( member_list_nat @ X @ S2 )
=> ( ( groups4396056296759096172at_nat
@ ^ [Y3: list_nat] : ( if_nat @ ( X = Y3 ) @ ( P @ X ) @ ( Q @ Y3 ) )
@ S2 )
= ( groups4396056296759096172at_nat @ Q @ S2 ) ) ) ).
% sum_delta_notmem(2)
thf(fact_65_sum__delta__notmem_I2_J,axiom,
! [X: nat,S2: set_nat,P: nat > nat,Q: nat > nat] :
( ~ ( member_nat @ X @ S2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [Y3: nat] : ( if_nat @ ( X = Y3 ) @ ( P @ X ) @ ( Q @ Y3 ) )
@ S2 )
= ( groups3542108847815614940at_nat @ Q @ S2 ) ) ) ).
% sum_delta_notmem(2)
thf(fact_66_sum__delta__notmem_I3_J,axiom,
! [X: list_nat,S2: set_list_nat,P: list_nat > nat,Q: list_nat > nat] :
( ~ ( member_list_nat @ X @ S2 )
=> ( ( groups4396056296759096172at_nat
@ ^ [Y3: list_nat] : ( if_nat @ ( Y3 = X ) @ ( P @ Y3 ) @ ( Q @ Y3 ) )
@ S2 )
= ( groups4396056296759096172at_nat @ Q @ S2 ) ) ) ).
% sum_delta_notmem(3)
thf(fact_67_sum__delta__notmem_I3_J,axiom,
! [X: nat,S2: set_nat,P: nat > nat,Q: nat > nat] :
( ~ ( member_nat @ X @ S2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [Y3: nat] : ( if_nat @ ( Y3 = X ) @ ( P @ Y3 ) @ ( Q @ Y3 ) )
@ S2 )
= ( groups3542108847815614940at_nat @ Q @ S2 ) ) ) ).
% sum_delta_notmem(3)
thf(fact_68_mem__Collect__eq,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A2 @ ( collect_set_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_69_mem__Collect__eq,axiom,
! [A2: list_list_nat,P: list_list_nat > $o] :
( ( member_list_list_nat @ A2 @ ( collec5989764272469232197st_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_70_mem__Collect__eq,axiom,
! [A2: set_list_nat,P: set_list_nat > $o] :
( ( member_set_list_nat @ A2 @ ( collect_set_list_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_71_mem__Collect__eq,axiom,
! [A2: list_nat,P: list_nat > $o] :
( ( member_list_nat @ A2 @ ( collect_list_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_72_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_73_Collect__mem__eq,axiom,
! [A: set_set_nat] :
( ( collect_set_nat
@ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_74_Collect__mem__eq,axiom,
! [A: set_list_list_nat] :
( ( collec5989764272469232197st_nat
@ ^ [X4: list_list_nat] : ( member_list_list_nat @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_75_Collect__mem__eq,axiom,
! [A: set_set_list_nat] :
( ( collect_set_list_nat
@ ^ [X4: set_list_nat] : ( member_set_list_nat @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_76_Collect__mem__eq,axiom,
! [A: set_list_nat] :
( ( collect_list_nat
@ ^ [X4: list_nat] : ( member_list_nat @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_77_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_78_Collect__cong,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X2: set_nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_set_nat @ P )
= ( collect_set_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_79_Collect__cong,axiom,
! [P: list_list_nat > $o,Q: list_list_nat > $o] :
( ! [X2: list_list_nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collec5989764272469232197st_nat @ P )
= ( collec5989764272469232197st_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_80_Collect__cong,axiom,
! [P: set_list_nat > $o,Q: set_list_nat > $o] :
( ! [X2: set_list_nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_set_list_nat @ P )
= ( collect_set_list_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_81_Collect__cong,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ! [X2: list_nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_list_nat @ P )
= ( collect_list_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_82_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_83_sum__delta__notmem_I4_J,axiom,
! [X: list_nat,S2: set_list_nat,P: list_nat > nat,Q: list_nat > nat] :
( ~ ( member_list_nat @ X @ S2 )
=> ( ( groups4396056296759096172at_nat
@ ^ [Y3: list_nat] : ( if_nat @ ( X = Y3 ) @ ( P @ Y3 ) @ ( Q @ Y3 ) )
@ S2 )
= ( groups4396056296759096172at_nat @ Q @ S2 ) ) ) ).
% sum_delta_notmem(4)
thf(fact_84_sum__delta__notmem_I4_J,axiom,
! [X: nat,S2: set_nat,P: nat > nat,Q: nat > nat] :
( ~ ( member_nat @ X @ S2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [Y3: nat] : ( if_nat @ ( X = Y3 ) @ ( P @ Y3 ) @ ( Q @ Y3 ) )
@ S2 )
= ( groups3542108847815614940at_nat @ Q @ S2 ) ) ) ).
% sum_delta_notmem(4)
thf(fact_85_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_86_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_87_i,axiom,
ord_less_nat @ i @ ( size_size_list_nat @ ns ) ).
% i
thf(fact_88_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_89_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_90_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_91_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_92_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_93_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_94_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_95_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_96_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_97_length__augmentum,axiom,
! [Xs2: list_nat] :
( ( size_size_list_nat @ ( augmentum @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_augmentum
thf(fact_98_length__dementum,axiom,
! [Xs2: list_nat] :
( ( size_size_list_nat @ ( dementum @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_dementum
thf(fact_99_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_100_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_101_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_102_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_103_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_104_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_105_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_106_nth__minus__list,axiom,
! [I: nat,Xs2: list_list_list_nat,Ys: list_list_list_nat] :
( ( ord_less_nat @ I @ ( size_s6248950052170075156st_nat @ Xs2 ) )
=> ( ( ord_less_nat @ I @ ( size_s6248950052170075156st_nat @ Ys ) )
=> ( ( nth_list_list_nat @ ( minus_1432716909304059609st_nat @ Xs2 @ Ys ) @ I )
= ( minus_3911745200923244873st_nat @ ( nth_list_list_nat @ Xs2 @ I ) @ ( nth_list_list_nat @ Ys @ I ) ) ) ) ) ).
% nth_minus_list
thf(fact_107_nth__minus__list,axiom,
! [I: nat,Xs2: list_set_nat,Ys: list_set_nat] :
( ( ord_less_nat @ I @ ( size_s3254054031482475050et_nat @ Xs2 ) )
=> ( ( ord_less_nat @ I @ ( size_s3254054031482475050et_nat @ Ys ) )
=> ( ( nth_set_nat @ ( minus_1998526526692677103et_nat @ Xs2 @ Ys ) @ I )
= ( minus_minus_set_nat @ ( nth_set_nat @ Xs2 @ I ) @ ( nth_set_nat @ Ys @ I ) ) ) ) ) ).
% nth_minus_list
thf(fact_108_nth__minus__list,axiom,
! [I: nat,Xs2: list_set_list_nat,Ys: list_set_list_nat] :
( ( ord_less_nat @ I @ ( size_s40095690673326906st_nat @ Xs2 ) )
=> ( ( ord_less_nat @ I @ ( size_s40095690673326906st_nat @ Ys ) )
=> ( ( nth_set_list_nat @ ( minus_4578319062706284671st_nat @ Xs2 @ Ys ) @ I )
= ( minus_7954133019191499631st_nat @ ( nth_set_list_nat @ Xs2 @ I ) @ ( nth_set_list_nat @ Ys @ I ) ) ) ) ) ).
% nth_minus_list
thf(fact_109_nth__minus__list,axiom,
! [I: nat,Xs2: list_nat_o,Ys: list_nat_o] :
( ( ord_less_nat @ I @ ( size_size_list_nat_o @ Xs2 ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat_o @ Ys ) )
=> ( ( nth_nat_o @ ( minus_8437841901652764780_nat_o @ Xs2 @ Ys ) @ I )
= ( minus_minus_nat_o @ ( nth_nat_o @ Xs2 @ I ) @ ( nth_nat_o @ Ys @ I ) ) ) ) ) ).
% nth_minus_list
thf(fact_110_nth__minus__list,axiom,
! [I: nat,Xs2: list_list_nat_o,Ys: list_list_nat_o] :
( ( ord_less_nat @ I @ ( size_s6069988884782743777_nat_o @ Xs2 ) )
=> ( ( ord_less_nat @ I @ ( size_s6069988884782743777_nat_o @ Ys ) )
=> ( ( nth_list_nat_o @ ( minus_6424643995335282652_nat_o @ Xs2 @ Ys ) @ I )
= ( minus_1139252259498527702_nat_o @ ( nth_list_nat_o @ Xs2 @ I ) @ ( nth_list_nat_o @ Ys @ I ) ) ) ) ) ).
% nth_minus_list
thf(fact_111_nth__minus__list,axiom,
! [I: nat,Xs2: list_list_nat,Ys: list_list_nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( nth_list_nat @ ( minus_3911745200923244873st_nat @ Xs2 @ Ys ) @ I )
= ( minus_minus_list_nat @ ( nth_list_nat @ Xs2 @ I ) @ ( nth_list_nat @ Ys @ I ) ) ) ) ) ).
% nth_minus_list
thf(fact_112_nth__minus__list,axiom,
! [I: nat,Xs2: list_nat,Ys: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
=> ( ( nth_nat @ ( minus_minus_list_nat @ Xs2 @ Ys ) @ I )
= ( minus_minus_nat @ ( nth_nat @ Xs2 @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).
% nth_minus_list
thf(fact_113_nth__augmentum,axiom,
! [I: nat,Ns: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
=> ( ( nth_nat @ ( augmentum @ Ns ) @ I )
= ( groups3542108847815614940at_nat @ ( nth_nat @ Ns ) @ ( set_ord_atMost_nat @ I ) ) ) ) ).
% nth_augmentum
thf(fact_114_lift__Suc__mono__less,axiom,
! [F: nat > set_list_nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_le1190675801316882794st_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ord_le1190675801316882794st_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_115_lift__Suc__mono__less,axiom,
! [F: nat > set_set_nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_set_set_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ord_less_set_set_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_116_lift__Suc__mono__less,axiom,
! [F: nat > set_nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_set_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ord_less_set_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_117_lift__Suc__mono__less,axiom,
! [F: nat > list_nat > $o,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_list_nat_o @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ord_less_list_nat_o @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_118_lift__Suc__mono__less,axiom,
! [F: nat > nat > $o,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_nat_o @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ord_less_nat_o @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_119_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_120_lift__Suc__mono__less__iff,axiom,
! [F: nat > set_list_nat,N: nat,M: nat] :
( ! [N3: nat] : ( ord_le1190675801316882794st_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_le1190675801316882794st_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_121_lift__Suc__mono__less__iff,axiom,
! [F: nat > set_set_nat,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_set_set_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_set_set_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_122_lift__Suc__mono__less__iff,axiom,
! [F: nat > set_nat,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_set_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_set_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_123_lift__Suc__mono__less__iff,axiom,
! [F: nat > list_nat > $o,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_list_nat_o @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_list_nat_o @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_124_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat > $o,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_nat_o @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat_o @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_125_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_126_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_127_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_128_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_129_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_130_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_131_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_132_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_133_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_134_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_135_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_136_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_137_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_138_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_139_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M2: nat] :
( ( M
= ( suc @ M2 ) )
& ( ord_less_nat @ N @ M2 ) ) ) ) ).
% Suc_less_eq2
thf(fact_140_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_141_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_142_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_143_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_144_less__not__refl3,axiom,
! [S2: nat,T2: nat] :
( ( ord_less_nat @ S2 @ T2 )
=> ( S2 != T2 ) ) ).
% less_not_refl3
thf(fact_145_less__trans__Suc,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_146_less__Suc__induct,axiom,
! [I: nat,J2: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J2 )
=> ( ! [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 @ J2 ) ) ) ) ).
% less_Suc_induct
thf(fact_147_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_148_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_149_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_150_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_151_strict__inc__induct,axiom,
! [I: nat,J2: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J2 )
=> ( ! [I3: nat] :
( ( J2
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_152_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_153_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J: nat] :
( ( M
= ( suc @ J ) )
& ( ord_less_nat @ J @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_154_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_155_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_156_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_157_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_158_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_159_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_160_size__neq__size__imp__neq,axiom,
! [X: list_list_nat,Y: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ X )
!= ( size_s3023201423986296836st_nat @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_161_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_162_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_163_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_164_sum__SucD,axiom,
! [F: list_nat > nat,A: set_list_nat,N: nat] :
( ( ( groups4396056296759096172at_nat @ F @ A )
= ( suc @ N ) )
=> ? [X2: list_nat] :
( ( member_list_nat @ X2 @ A )
& ( ord_less_nat @ zero_zero_nat @ ( F @ X2 ) ) ) ) ).
% sum_SucD
thf(fact_165_sum__SucD,axiom,
! [F: nat > nat,A: set_nat,N: nat] :
( ( ( groups3542108847815614940at_nat @ F @ A )
= ( suc @ N ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_nat @ zero_zero_nat @ ( F @ X2 ) ) ) ) ).
% sum_SucD
thf(fact_166_minus__Cons,axiom,
! [Y: list_list_nat,Ys: list_list_list_nat,X: list_list_nat,Xs2: list_list_list_nat] :
( ( minus_1432716909304059609st_nat @ ( cons_list_list_nat @ Y @ Ys ) @ ( cons_list_list_nat @ X @ Xs2 ) )
= ( cons_list_list_nat @ ( minus_3911745200923244873st_nat @ Y @ X ) @ ( minus_1432716909304059609st_nat @ Ys @ Xs2 ) ) ) ).
% minus_Cons
thf(fact_167_minus__Cons,axiom,
! [Y: set_nat,Ys: list_set_nat,X: set_nat,Xs2: list_set_nat] :
( ( minus_1998526526692677103et_nat @ ( cons_set_nat @ Y @ Ys ) @ ( cons_set_nat @ X @ Xs2 ) )
= ( cons_set_nat @ ( minus_minus_set_nat @ Y @ X ) @ ( minus_1998526526692677103et_nat @ Ys @ Xs2 ) ) ) ).
% minus_Cons
thf(fact_168_minus__Cons,axiom,
! [Y: set_list_nat,Ys: list_set_list_nat,X: set_list_nat,Xs2: list_set_list_nat] :
( ( minus_4578319062706284671st_nat @ ( cons_set_list_nat @ Y @ Ys ) @ ( cons_set_list_nat @ X @ Xs2 ) )
= ( cons_set_list_nat @ ( minus_7954133019191499631st_nat @ Y @ X ) @ ( minus_4578319062706284671st_nat @ Ys @ Xs2 ) ) ) ).
% minus_Cons
thf(fact_169_minus__Cons,axiom,
! [Y: nat > $o,Ys: list_nat_o,X: nat > $o,Xs2: list_nat_o] :
( ( minus_8437841901652764780_nat_o @ ( cons_nat_o @ Y @ Ys ) @ ( cons_nat_o @ X @ Xs2 ) )
= ( cons_nat_o @ ( minus_minus_nat_o @ Y @ X ) @ ( minus_8437841901652764780_nat_o @ Ys @ Xs2 ) ) ) ).
% minus_Cons
thf(fact_170_minus__Cons,axiom,
! [Y: list_nat > $o,Ys: list_list_nat_o,X: list_nat > $o,Xs2: list_list_nat_o] :
( ( minus_6424643995335282652_nat_o @ ( cons_list_nat_o @ Y @ Ys ) @ ( cons_list_nat_o @ X @ Xs2 ) )
= ( cons_list_nat_o @ ( minus_1139252259498527702_nat_o @ Y @ X ) @ ( minus_6424643995335282652_nat_o @ Ys @ Xs2 ) ) ) ).
% minus_Cons
thf(fact_171_minus__Cons,axiom,
! [Y: list_nat,Ys: list_list_nat,X: list_nat,Xs2: list_list_nat] :
( ( minus_3911745200923244873st_nat @ ( cons_list_nat @ Y @ Ys ) @ ( cons_list_nat @ X @ Xs2 ) )
= ( cons_list_nat @ ( minus_minus_list_nat @ Y @ X ) @ ( minus_3911745200923244873st_nat @ Ys @ Xs2 ) ) ) ).
% minus_Cons
thf(fact_172_minus__Cons,axiom,
! [Y: nat,Ys: list_nat,X: nat,Xs2: list_nat] :
( ( minus_minus_list_nat @ ( cons_nat @ Y @ Ys ) @ ( cons_nat @ X @ Xs2 ) )
= ( cons_nat @ ( minus_minus_nat @ Y @ X ) @ ( minus_minus_list_nat @ Ys @ Xs2 ) ) ) ).
% minus_Cons
thf(fact_173_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_174_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_175_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_176_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_177_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_178_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_179_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_180_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_181_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_182_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_183_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_184_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_185_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X2: nat,Y2: nat] :
( ( P @ X2 @ Y2 )
=> ( P @ ( suc @ X2 ) @ ( suc @ Y2 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_186_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_187_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_188_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_189_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_190_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_191_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_192_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_193_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_194_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_195_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_196_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_197_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_198_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_199_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_200_less__imp__diff__less,axiom,
! [J2: nat,K: nat,N: nat] :
( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J2 @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_201_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_202_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_203_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_204_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_205_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_206_sum__cong__Suc,axiom,
! [A: set_nat,F: nat > nat,G: nat > nat] :
( ~ ( member_nat @ zero_zero_nat @ A )
=> ( ! [X2: nat] :
( ( member_nat @ ( suc @ X2 ) @ A )
=> ( ( F @ ( suc @ X2 ) )
= ( G @ ( suc @ X2 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ F @ A )
= ( groups3542108847815614940at_nat @ G @ A ) ) ) ) ).
% sum_cong_Suc
thf(fact_207_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_208_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_209_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_210_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [G: nat > nat,M: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_211_fun__diff__def,axiom,
( minus_minus_nat_o
= ( ^ [A4: nat > $o,B4: nat > $o,X4: nat] : ( minus_minus_o @ ( A4 @ X4 ) @ ( B4 @ X4 ) ) ) ) ).
% fun_diff_def
thf(fact_212_fun__diff__def,axiom,
( minus_1139252259498527702_nat_o
= ( ^ [A4: list_nat > $o,B4: list_nat > $o,X4: list_nat] : ( minus_minus_o @ ( A4 @ X4 ) @ ( B4 @ X4 ) ) ) ) ).
% fun_diff_def
thf(fact_213_sum__shift__lb__Suc0__0,axiom,
! [F: nat > nat,K: nat] :
( ( ( F @ zero_zero_nat )
= zero_zero_nat )
=> ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
= ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_214_list__incr__nth__diff,axiom,
! [I: nat,X: list_nat,J2: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ X ) )
=> ( ( ( I = J2 )
=> ( ( minus_minus_nat @ ( nth_nat @ ( list_incr @ J2 @ X ) @ I ) @ ( nth_nat @ X @ I ) )
= one_one_nat ) )
& ( ( I != J2 )
=> ( ( minus_minus_nat @ ( nth_nat @ ( list_incr @ J2 @ X ) @ I ) @ ( nth_nat @ X @ I ) )
= zero_zero_nat ) ) ) ) ).
% list_incr_nth_diff
thf(fact_215_nth__Cons__pos,axiom,
! [N: nat,X: list_nat,Xs2: list_list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_list_nat @ ( cons_list_nat @ X @ Xs2 ) @ N )
= ( nth_list_nat @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_216_nth__Cons__pos,axiom,
! [N: nat,X: nat,Xs2: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ N )
= ( nth_nat @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_217_ns__le,axiom,
! [J2: nat] :
( ( ord_less_nat @ zero_zero_nat @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ i )
=> ( ord_less_eq_nat @ ( nth_nat @ ns @ ( minus_minus_nat @ J2 @ ( suc @ zero_zero_nat ) ) ) @ ( nth_nat @ ns @ J2 ) ) ) ) ).
% ns_le
thf(fact_218_nth__Cons__Suc,axiom,
! [X: list_nat,Xs2: list_list_nat,N: nat] :
( ( nth_list_nat @ ( cons_list_nat @ X @ Xs2 ) @ ( suc @ N ) )
= ( nth_list_nat @ Xs2 @ N ) ) ).
% nth_Cons_Suc
thf(fact_219_nth__Cons__Suc,axiom,
! [X: nat,Xs2: list_nat,N: nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ ( suc @ N ) )
= ( nth_nat @ Xs2 @ N ) ) ).
% nth_Cons_Suc
thf(fact_220_nth__Cons__0,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( nth_list_nat @ ( cons_list_nat @ X @ Xs2 ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_221_nth__Cons__0,axiom,
! [X: nat,Xs2: list_nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_222_nth__non__equal__first__eq,axiom,
! [X: list_nat,Y: list_nat,Xs2: list_list_nat,N: nat] :
( ( X != Y )
=> ( ( ( nth_list_nat @ ( cons_list_nat @ X @ Xs2 ) @ N )
= Y )
= ( ( ( nth_list_nat @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_223_nth__non__equal__first__eq,axiom,
! [X: nat,Y: nat,Xs2: list_nat,N: nat] :
( ( X != Y )
=> ( ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ N )
= Y )
= ( ( ( nth_nat @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_224_distinct__Ex1,axiom,
! [Xs2: list_set_nat,X: set_nat] :
( ( distinct_set_nat @ Xs2 )
=> ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs2 ) )
=> ? [X2: nat] :
( ( ord_less_nat @ X2 @ ( size_s3254054031482475050et_nat @ Xs2 ) )
& ( ( nth_set_nat @ Xs2 @ X2 )
= X )
& ! [Y4: nat] :
( ( ( ord_less_nat @ Y4 @ ( size_s3254054031482475050et_nat @ Xs2 ) )
& ( ( nth_set_nat @ Xs2 @ Y4 )
= X ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_225_distinct__Ex1,axiom,
! [Xs2: list_list_nat,X: list_nat] :
( ( distinct_list_nat @ Xs2 )
=> ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
=> ? [X2: nat] :
( ( ord_less_nat @ X2 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
& ( ( nth_list_nat @ Xs2 @ X2 )
= X )
& ! [Y4: nat] :
( ( ( ord_less_nat @ Y4 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
& ( ( nth_list_nat @ Xs2 @ Y4 )
= X ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_226_distinct__Ex1,axiom,
! [Xs2: list_nat,X: nat] :
( ( distinct_nat @ Xs2 )
=> ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ? [X2: nat] :
( ( ord_less_nat @ X2 @ ( size_size_list_nat @ Xs2 ) )
& ( ( nth_nat @ Xs2 @ X2 )
= X )
& ! [Y4: nat] :
( ( ( ord_less_nat @ Y4 @ ( size_size_list_nat @ Xs2 ) )
& ( ( nth_nat @ Xs2 @ Y4 )
= X ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_227_nth__Cons_H,axiom,
! [N: nat,X: list_nat,Xs2: list_list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( nth_list_nat @ ( cons_list_nat @ X @ Xs2 ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_list_nat @ ( cons_list_nat @ X @ Xs2 ) @ N )
= ( nth_list_nat @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_228_nth__Cons_H,axiom,
! [N: nat,X: nat,Xs2: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ N )
= ( nth_nat @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_229_nth__eq__iff__index__eq,axiom,
! [Xs2: list_list_nat,I: nat,J2: nat] :
( ( distinct_list_nat @ Xs2 )
=> ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ( ord_less_nat @ J2 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ( ( nth_list_nat @ Xs2 @ I )
= ( nth_list_nat @ Xs2 @ J2 ) )
= ( I = J2 ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_230_nth__eq__iff__index__eq,axiom,
! [Xs2: list_nat,I: nat,J2: nat] :
( ( distinct_nat @ Xs2 )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ( nth_nat @ Xs2 @ I )
= ( nth_nat @ Xs2 @ J2 ) )
= ( I = J2 ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_231_distinct__conv__nth,axiom,
( distinct_list_nat
= ( ^ [Xs: list_list_nat] :
! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ! [J: nat] :
( ( ord_less_nat @ J @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( I2 != J )
=> ( ( nth_list_nat @ Xs @ I2 )
!= ( nth_list_nat @ Xs @ J ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_232_distinct__conv__nth,axiom,
( distinct_nat
= ( ^ [Xs: list_nat] :
! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( I2 != J )
=> ( ( nth_nat @ Xs @ I2 )
!= ( nth_nat @ Xs @ J ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_233_all__set__conv__all__nth,axiom,
! [Xs2: list_list_nat,P: list_nat > $o] :
( ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xs2 ) )
=> ( P @ X4 ) ) )
= ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( P @ ( nth_list_nat @ Xs2 @ I2 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_234_all__set__conv__all__nth,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
=> ( P @ X4 ) ) )
= ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ ( nth_nat @ Xs2 @ I2 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_235_list_Oinject,axiom,
! [X21: list_nat,X222: list_list_nat,Y21: list_nat,Y222: list_list_nat] :
( ( ( cons_list_nat @ X21 @ X222 )
= ( cons_list_nat @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_236_list_Oinject,axiom,
! [X21: nat,X222: list_nat,Y21: nat,Y222: list_nat] :
( ( ( cons_nat @ X21 @ X222 )
= ( cons_nat @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_237_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_238_atLeastatMost__subset__iff,axiom,
! [A2: set_set_nat,B: set_set_nat,C: set_set_nat,D: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( set_or9137876137106135879et_nat @ A2 @ B ) @ ( set_or9137876137106135879et_nat @ C @ D ) )
= ( ~ ( ord_le6893508408891458716et_nat @ A2 @ B )
| ( ( ord_le6893508408891458716et_nat @ C @ A2 )
& ( ord_le6893508408891458716et_nat @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_239_atLeastatMost__subset__iff,axiom,
! [A2: set_list_nat,B: set_list_nat,C: set_list_nat,D: set_list_nat] :
( ( ord_le1068707526560357548st_nat @ ( set_or1270514513317581473st_nat @ A2 @ B ) @ ( set_or1270514513317581473st_nat @ C @ D ) )
= ( ~ ( ord_le6045566169113846134st_nat @ A2 @ B )
| ( ( ord_le6045566169113846134st_nat @ C @ A2 )
& ( ord_le6045566169113846134st_nat @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_240_atLeastatMost__subset__iff,axiom,
! [A2: nat > $o,B: nat > $o,C: nat > $o,D: nat > $o] :
( ( ord_le6029213668185085951_nat_o @ ( set_or99350221437691188_nat_o @ A2 @ B ) @ ( set_or99350221437691188_nat_o @ C @ D ) )
= ( ~ ( ord_less_eq_nat_o @ A2 @ B )
| ( ( ord_less_eq_nat_o @ C @ A2 )
& ( ord_less_eq_nat_o @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_241_atLeastatMost__subset__iff,axiom,
! [A2: list_nat > $o,B: list_nat > $o,C: list_nat > $o,D: list_nat > $o] :
( ( ord_le2112173733987654127_nat_o @ ( set_or689571727643897636_nat_o @ A2 @ B ) @ ( set_or689571727643897636_nat_o @ C @ D ) )
= ( ~ ( ord_le1520216061033275535_nat_o @ A2 @ B )
| ( ( ord_le1520216061033275535_nat_o @ C @ A2 )
& ( ord_le1520216061033275535_nat_o @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_242_atLeastatMost__subset__iff,axiom,
! [A2: list_nat,B: list_nat,C: list_nat,D: list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_or6836045993805503595st_nat @ A2 @ B ) @ ( set_or6836045993805503595st_nat @ C @ D ) )
= ( ~ ( ord_less_eq_list_nat @ A2 @ B )
| ( ( ord_less_eq_list_nat @ C @ A2 )
& ( ord_less_eq_list_nat @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_243_atLeastatMost__subset__iff,axiom,
! [A2: set_nat,B: set_nat,C: set_nat,D: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ A2 @ B ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
= ( ~ ( ord_less_eq_set_nat @ A2 @ B )
| ( ( ord_less_eq_set_nat @ C @ A2 )
& ( ord_less_eq_set_nat @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_244_atLeastatMost__subset__iff,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A2 @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B )
| ( ( ord_less_eq_nat @ C @ A2 )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_245_atLeastAtMost__iff,axiom,
! [I: set_set_nat,L: set_set_nat,U: set_set_nat] :
( ( member_set_set_nat @ I @ ( set_or9137876137106135879et_nat @ L @ U ) )
= ( ( ord_le6893508408891458716et_nat @ L @ I )
& ( ord_le6893508408891458716et_nat @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_246_atLeastAtMost__iff,axiom,
! [I: set_list_nat,L: set_list_nat,U: set_list_nat] :
( ( member_set_list_nat @ I @ ( set_or1270514513317581473st_nat @ L @ U ) )
= ( ( ord_le6045566169113846134st_nat @ L @ I )
& ( ord_le6045566169113846134st_nat @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_247_atLeastAtMost__iff,axiom,
! [I: nat > $o,L: nat > $o,U: nat > $o] :
( ( member_nat_o @ I @ ( set_or99350221437691188_nat_o @ L @ U ) )
= ( ( ord_less_eq_nat_o @ L @ I )
& ( ord_less_eq_nat_o @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_248_atLeastAtMost__iff,axiom,
! [I: list_nat > $o,L: list_nat > $o,U: list_nat > $o] :
( ( member_list_nat_o @ I @ ( set_or689571727643897636_nat_o @ L @ U ) )
= ( ( ord_le1520216061033275535_nat_o @ L @ I )
& ( ord_le1520216061033275535_nat_o @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_249_atLeastAtMost__iff,axiom,
! [I: list_nat,L: list_nat,U: list_nat] :
( ( member_list_nat @ I @ ( set_or6836045993805503595st_nat @ L @ U ) )
= ( ( ord_less_eq_list_nat @ L @ I )
& ( ord_less_eq_list_nat @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_250_atLeastAtMost__iff,axiom,
! [I: set_nat,L: set_nat,U: set_nat] :
( ( member_set_nat @ I @ ( set_or4548717258645045905et_nat @ L @ U ) )
= ( ( ord_less_eq_set_nat @ L @ I )
& ( ord_less_eq_set_nat @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_251_atLeastAtMost__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I )
& ( ord_less_eq_nat @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_252_Icc__eq__Icc,axiom,
! [L: set_set_nat,H: set_set_nat,L2: set_set_nat,H2: set_set_nat] :
( ( ( set_or9137876137106135879et_nat @ L @ H )
= ( set_or9137876137106135879et_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_le6893508408891458716et_nat @ L @ H )
& ~ ( ord_le6893508408891458716et_nat @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_253_Icc__eq__Icc,axiom,
! [L: set_list_nat,H: set_list_nat,L2: set_list_nat,H2: set_list_nat] :
( ( ( set_or1270514513317581473st_nat @ L @ H )
= ( set_or1270514513317581473st_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_le6045566169113846134st_nat @ L @ H )
& ~ ( ord_le6045566169113846134st_nat @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_254_Icc__eq__Icc,axiom,
! [L: nat > $o,H: nat > $o,L2: nat > $o,H2: nat > $o] :
( ( ( set_or99350221437691188_nat_o @ L @ H )
= ( set_or99350221437691188_nat_o @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_nat_o @ L @ H )
& ~ ( ord_less_eq_nat_o @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_255_Icc__eq__Icc,axiom,
! [L: list_nat > $o,H: list_nat > $o,L2: list_nat > $o,H2: list_nat > $o] :
( ( ( set_or689571727643897636_nat_o @ L @ H )
= ( set_or689571727643897636_nat_o @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_le1520216061033275535_nat_o @ L @ H )
& ~ ( ord_le1520216061033275535_nat_o @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_256_Icc__eq__Icc,axiom,
! [L: list_nat,H: list_nat,L2: list_nat,H2: list_nat] :
( ( ( set_or6836045993805503595st_nat @ L @ H )
= ( set_or6836045993805503595st_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_list_nat @ L @ H )
& ~ ( ord_less_eq_list_nat @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_257_Icc__eq__Icc,axiom,
! [L: set_nat,H: set_nat,L2: set_nat,H2: set_nat] :
( ( ( set_or4548717258645045905et_nat @ L @ H )
= ( set_or4548717258645045905et_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_set_nat @ L @ H )
& ~ ( ord_less_eq_set_nat @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_258_Icc__eq__Icc,axiom,
! [L: nat,H: nat,L2: nat,H2: nat] :
( ( ( set_or1269000886237332187st_nat @ L @ H )
= ( set_or1269000886237332187st_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_nat @ L @ H )
& ~ ( ord_less_eq_nat @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_259_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_260_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_261_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_262_atMost__subset__iff,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( set_or7210490968680142261et_nat @ X ) @ ( set_or7210490968680142261et_nat @ Y ) )
= ( ord_le6893508408891458716et_nat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_263_atMost__subset__iff,axiom,
! [X: set_list_nat,Y: set_list_nat] :
( ( ord_le1068707526560357548st_nat @ ( set_or2492388921469580815st_nat @ X ) @ ( set_or2492388921469580815st_nat @ Y ) )
= ( ord_le6045566169113846134st_nat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_264_atMost__subset__iff,axiom,
! [X: nat > $o,Y: nat > $o] :
( ( ord_le6029213668185085951_nat_o @ ( set_ord_atMost_nat_o @ X ) @ ( set_ord_atMost_nat_o @ Y ) )
= ( ord_less_eq_nat_o @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_265_atMost__subset__iff,axiom,
! [X: list_nat > $o,Y: list_nat > $o] :
( ( ord_le2112173733987654127_nat_o @ ( set_or3326398383446669622_nat_o @ X ) @ ( set_or3326398383446669622_nat_o @ Y ) )
= ( ord_le1520216061033275535_nat_o @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_266_atMost__subset__iff,axiom,
! [X: list_nat,Y: list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_or4185896845444216793st_nat @ X ) @ ( set_or4185896845444216793st_nat @ Y ) )
= ( ord_less_eq_list_nat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_267_atMost__subset__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4236626031148496127et_nat @ X ) @ ( set_or4236626031148496127et_nat @ Y ) )
= ( ord_less_eq_set_nat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_268_atMost__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X ) @ ( set_ord_atMost_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_269_atMost__iff,axiom,
! [I: set_set_nat,K: set_set_nat] :
( ( member_set_set_nat @ I @ ( set_or7210490968680142261et_nat @ K ) )
= ( ord_le6893508408891458716et_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_270_atMost__iff,axiom,
! [I: set_list_nat,K: set_list_nat] :
( ( member_set_list_nat @ I @ ( set_or2492388921469580815st_nat @ K ) )
= ( ord_le6045566169113846134st_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_271_atMost__iff,axiom,
! [I: nat > $o,K: nat > $o] :
( ( member_nat_o @ I @ ( set_ord_atMost_nat_o @ K ) )
= ( ord_less_eq_nat_o @ I @ K ) ) ).
% atMost_iff
thf(fact_272_atMost__iff,axiom,
! [I: list_nat > $o,K: list_nat > $o] :
( ( member_list_nat_o @ I @ ( set_or3326398383446669622_nat_o @ K ) )
= ( ord_le1520216061033275535_nat_o @ I @ K ) ) ).
% atMost_iff
thf(fact_273_atMost__iff,axiom,
! [I: list_nat,K: list_nat] :
( ( member_list_nat @ I @ ( set_or4185896845444216793st_nat @ K ) )
= ( ord_less_eq_list_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_274_atMost__iff,axiom,
! [I: set_nat,K: set_nat] :
( ( member_set_nat @ I @ ( set_or4236626031148496127et_nat @ K ) )
= ( ord_less_eq_set_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_275_atMost__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_atMost_nat @ K ) )
= ( ord_less_eq_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_276_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_277_length__list__incr,axiom,
! [I: nat,X: list_nat] :
( ( size_size_list_nat @ ( list_incr @ I @ X ) )
= ( size_size_list_nat @ X ) ) ).
% length_list_incr
thf(fact_278_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_279_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_280_Icc__subset__Iic__iff,axiom,
! [L: set_set_nat,H: set_set_nat,H2: set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( set_or9137876137106135879et_nat @ L @ H ) @ ( set_or7210490968680142261et_nat @ H2 ) )
= ( ~ ( ord_le6893508408891458716et_nat @ L @ H )
| ( ord_le6893508408891458716et_nat @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_281_Icc__subset__Iic__iff,axiom,
! [L: set_list_nat,H: set_list_nat,H2: set_list_nat] :
( ( ord_le1068707526560357548st_nat @ ( set_or1270514513317581473st_nat @ L @ H ) @ ( set_or2492388921469580815st_nat @ H2 ) )
= ( ~ ( ord_le6045566169113846134st_nat @ L @ H )
| ( ord_le6045566169113846134st_nat @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_282_Icc__subset__Iic__iff,axiom,
! [L: nat > $o,H: nat > $o,H2: nat > $o] :
( ( ord_le6029213668185085951_nat_o @ ( set_or99350221437691188_nat_o @ L @ H ) @ ( set_ord_atMost_nat_o @ H2 ) )
= ( ~ ( ord_less_eq_nat_o @ L @ H )
| ( ord_less_eq_nat_o @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_283_Icc__subset__Iic__iff,axiom,
! [L: list_nat > $o,H: list_nat > $o,H2: list_nat > $o] :
( ( ord_le2112173733987654127_nat_o @ ( set_or689571727643897636_nat_o @ L @ H ) @ ( set_or3326398383446669622_nat_o @ H2 ) )
= ( ~ ( ord_le1520216061033275535_nat_o @ L @ H )
| ( ord_le1520216061033275535_nat_o @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_284_Icc__subset__Iic__iff,axiom,
! [L: list_nat,H: list_nat,H2: list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_or6836045993805503595st_nat @ L @ H ) @ ( set_or4185896845444216793st_nat @ H2 ) )
= ( ~ ( ord_less_eq_list_nat @ L @ H )
| ( ord_less_eq_list_nat @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_285_Icc__subset__Iic__iff,axiom,
! [L: set_nat,H: set_nat,H2: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ L @ H ) @ ( set_or4236626031148496127et_nat @ H2 ) )
= ( ~ ( ord_less_eq_set_nat @ L @ H )
| ( ord_less_eq_set_nat @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_286_Icc__subset__Iic__iff,axiom,
! [L: nat,H: nat,H2: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L @ H ) @ ( set_ord_atMost_nat @ H2 ) )
= ( ~ ( ord_less_eq_nat @ L @ H )
| ( ord_less_eq_nat @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_287_list__incr__Cons,axiom,
! [I: nat,K: nat,Ks: list_nat] :
( ( list_incr @ ( suc @ I ) @ ( cons_nat @ K @ Ks ) )
= ( cons_nat @ K @ ( list_incr @ I @ Ks ) ) ) ).
% list_incr_Cons
thf(fact_288_assms_I2_J,axiom,
sorted_wrt_nat @ ord_less_eq_nat @ ns ).
% assms(2)
thf(fact_289_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M6: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M6 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_290_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_291_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_292_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_293_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_294_le__trans,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_295_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_296_lift__Suc__antimono__le,axiom,
! [F: nat > set_set_nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_le6893508408891458716et_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_le6893508408891458716et_nat @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_297_lift__Suc__antimono__le,axiom,
! [F: nat > list_nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_list_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_list_nat @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_298_lift__Suc__antimono__le,axiom,
! [F: nat > set_list_nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_le6045566169113846134st_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_le6045566169113846134st_nat @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_299_lift__Suc__antimono__le,axiom,
! [F: nat > nat > $o,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat_o @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_nat_o @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_300_lift__Suc__antimono__le,axiom,
! [F: nat > list_nat > $o,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_le1520216061033275535_nat_o @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_le1520216061033275535_nat_o @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_301_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_302_lift__Suc__antimono__le,axiom,
! [F: nat > set_nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_303_lift__Suc__mono__le,axiom,
! [F: nat > nat > $o,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat_o @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_nat_o @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_304_lift__Suc__mono__le,axiom,
! [F: nat > list_nat > $o,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_le1520216061033275535_nat_o @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_le1520216061033275535_nat_o @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_305_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_306_lift__Suc__mono__le,axiom,
! [F: nat > set_nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_set_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_307_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_308_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_309_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_310_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_311_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_312_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_313_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_314_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X2: nat] : ( R @ X2 @ X2 )
=> ( ! [X2: nat,Y2: nat,Z: nat] :
( ( R @ X2 @ Y2 )
=> ( ( R @ Y2 @ Z )
=> ( R @ X2 @ Z ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_315_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_316_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_317_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_318_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_319_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_320_Suc__le__D,axiom,
! [N: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
=> ? [M4: nat] :
( M7
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_321_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_322_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_323_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_324_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J2: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_325_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_326_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_327_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_nat @ M5 @ N4 )
| ( M5 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_328_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_329_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
& ( M5 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_330_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_331_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_332_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_333_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_334_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_335_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_336_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_337_impossible__Cons,axiom,
! [Xs2: list_nat,Ys: list_nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ Ys ) )
=> ( Xs2
!= ( cons_nat @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_338_all__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M5: nat] :
( ( ord_less_eq_nat @ M5 @ N )
=> ( P @ M5 ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
=> ( P @ X4 ) ) ) ) ).
% all_nat_less
thf(fact_339_ex__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M5: nat] :
( ( ord_less_eq_nat @ M5 @ N )
& ( P @ M5 ) ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
& ( P @ X4 ) ) ) ) ).
% ex_nat_less
thf(fact_340_sum__mono,axiom,
! [K3: set_list_nat,F: list_nat > nat,G: list_nat > nat] :
( ! [I3: list_nat] :
( ( member_list_nat @ I3 @ K3 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups4396056296759096172at_nat @ F @ K3 ) @ ( groups4396056296759096172at_nat @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_341_sum__mono,axiom,
! [K3: set_nat,F: nat > nat,G: nat > nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ K3 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ K3 ) @ ( groups3542108847815614940at_nat @ G @ K3 ) ) ) ).
% sum_mono
thf(fact_342_atMost__def,axiom,
( set_or4185896845444216793st_nat
= ( ^ [U2: list_nat] :
( collect_list_nat
@ ^ [X4: list_nat] : ( ord_less_eq_list_nat @ X4 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_343_atMost__def,axiom,
( set_or4236626031148496127et_nat
= ( ^ [U2: set_nat] :
( collect_set_nat
@ ^ [X4: set_nat] : ( ord_less_eq_set_nat @ X4 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_344_atMost__def,axiom,
( set_ord_atMost_nat
= ( ^ [U2: nat] :
( collect_nat
@ ^ [X4: nat] : ( ord_less_eq_nat @ X4 @ U2 ) ) ) ) ).
% atMost_def
thf(fact_345_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_346_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_347_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_348_sum__nonpos,axiom,
! [A: set_list_nat,F: list_nat > nat] :
( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups4396056296759096172at_nat @ F @ A ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_349_sum__nonpos,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_350_sum__nonneg,axiom,
! [A: set_list_nat,F: list_nat > nat] :
( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups4396056296759096172at_nat @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_351_sum__nonneg,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups3542108847815614940at_nat @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_352_Suc__le__length__iff,axiom,
! [N: nat,Xs2: list_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs2 ) )
= ( ? [X4: nat,Ys2: list_nat] :
( ( Xs2
= ( cons_nat @ X4 @ Ys2 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_353_atLeastatMost__psubset__iff,axiom,
! [A2: set_nat,B: set_nat,C: set_nat,D: set_nat] :
( ( ord_less_set_set_nat @ ( set_or4548717258645045905et_nat @ A2 @ B ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
= ( ( ~ ( ord_less_eq_set_nat @ A2 @ B )
| ( ( ord_less_eq_set_nat @ C @ A2 )
& ( ord_less_eq_set_nat @ B @ D )
& ( ( ord_less_set_nat @ C @ A2 )
| ( ord_less_set_nat @ B @ D ) ) ) )
& ( ord_less_eq_set_nat @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_354_atLeastatMost__psubset__iff,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A2 @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ( ~ ( ord_less_eq_nat @ A2 @ B )
| ( ( ord_less_eq_nat @ C @ A2 )
& ( ord_less_eq_nat @ B @ D )
& ( ( ord_less_nat @ C @ A2 )
| ( ord_less_nat @ B @ D ) ) ) )
& ( ord_less_eq_nat @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_355_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_356_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_357_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_358_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_359_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_360_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_361_inc__induct,axiom,
! [I: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( P @ J2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J2 )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_362_dec__induct,axiom,
! [I: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J2 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J2 ) ) ) ) ).
% dec_induct
thf(fact_363_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_364_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_365_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_366_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_367_diff__less__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_368_sum__subtractf__nat,axiom,
! [A: set_list_nat,G: list_nat > nat,F: list_nat > nat] :
( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ A )
=> ( ord_less_eq_nat @ ( G @ X2 ) @ ( F @ X2 ) ) )
=> ( ( groups4396056296759096172at_nat
@ ^ [X4: list_nat] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
@ A )
= ( minus_minus_nat @ ( groups4396056296759096172at_nat @ F @ A ) @ ( groups4396056296759096172at_nat @ G @ A ) ) ) ) ).
% sum_subtractf_nat
thf(fact_369_sum__subtractf__nat,axiom,
! [A: set_nat,G: nat > nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ ( G @ X2 ) @ ( F @ X2 ) ) )
=> ( ( groups3542108847815614940at_nat
@ ^ [X4: nat] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
@ A )
= ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( groups3542108847815614940at_nat @ G @ A ) ) ) ) ).
% sum_subtractf_nat
thf(fact_370_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_371_not__Cons__self2,axiom,
! [X: nat,Xs2: list_nat] :
( ( cons_nat @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_372_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_nat] :
( ( size_size_list_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_373_neq__if__length__neq,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_374_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
! [Xs2: list_nat] :
( ( distinct_nat @ Xs2 )
=> ( distinct_nat @ Xs2 ) ) ).
% sorted_list_of_set.distinct_if_distinct_map
thf(fact_375_nth__equal__first__eq,axiom,
! [X: list_nat,Xs2: list_list_nat,N: nat] :
( ~ ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
=> ( ( ord_less_eq_nat @ N @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ( ( nth_list_nat @ ( cons_list_nat @ X @ Xs2 ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_376_nth__equal__first__eq,axiom,
! [X: nat,Xs2: list_nat,N: nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_377_list_Oset__intros_I2_J,axiom,
! [Y: list_nat,X222: list_list_nat,X21: list_nat] :
( ( member_list_nat @ Y @ ( set_list_nat2 @ X222 ) )
=> ( member_list_nat @ Y @ ( set_list_nat2 @ ( cons_list_nat @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_378_list_Oset__intros_I2_J,axiom,
! [Y: nat,X222: list_nat,X21: nat] :
( ( member_nat @ Y @ ( set_nat2 @ X222 ) )
=> ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_379_list_Oset__intros_I1_J,axiom,
! [X21: list_nat,X222: list_list_nat] : ( member_list_nat @ X21 @ ( set_list_nat2 @ ( cons_list_nat @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_380_list_Oset__intros_I1_J,axiom,
! [X21: nat,X222: list_nat] : ( member_nat @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_381_list_Oset__cases,axiom,
! [E: list_nat,A2: list_list_nat] :
( ( member_list_nat @ E @ ( set_list_nat2 @ A2 ) )
=> ( ! [Z2: list_list_nat] :
( A2
!= ( cons_list_nat @ E @ Z2 ) )
=> ~ ! [Z1: list_nat,Z2: list_list_nat] :
( ( A2
= ( cons_list_nat @ Z1 @ Z2 ) )
=> ~ ( member_list_nat @ E @ ( set_list_nat2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_382_list_Oset__cases,axiom,
! [E: nat,A2: list_nat] :
( ( member_nat @ E @ ( set_nat2 @ A2 ) )
=> ( ! [Z2: list_nat] :
( A2
!= ( cons_nat @ E @ Z2 ) )
=> ~ ! [Z1: nat,Z2: list_nat] :
( ( A2
= ( cons_nat @ Z1 @ Z2 ) )
=> ~ ( member_nat @ E @ ( set_nat2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_383_set__ConsD,axiom,
! [Y: list_nat,X: list_nat,Xs2: list_list_nat] :
( ( member_list_nat @ Y @ ( set_list_nat2 @ ( cons_list_nat @ X @ Xs2 ) ) )
=> ( ( Y = X )
| ( member_list_nat @ Y @ ( set_list_nat2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_384_set__ConsD,axiom,
! [Y: nat,X: nat,Xs2: list_nat] :
( ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X @ Xs2 ) ) )
=> ( ( Y = X )
| ( member_nat @ Y @ ( set_nat2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_385_length__induct,axiom,
! [P: list_nat > $o,Xs2: list_nat] :
( ! [Xs3: list_nat] :
( ! [Ys3: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys3 ) @ ( size_size_list_nat @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_386_distinct__length__2__or__more,axiom,
! [A2: nat,B: nat,Xs2: list_nat] :
( ( distinct_nat @ ( cons_nat @ A2 @ ( cons_nat @ B @ Xs2 ) ) )
= ( ( A2 != B )
& ( distinct_nat @ ( cons_nat @ A2 @ Xs2 ) )
& ( distinct_nat @ ( cons_nat @ B @ Xs2 ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_387_sum_Ozero__middle,axiom,
! [P2: nat,K: nat,G: nat > nat,H: nat > nat] :
( ( ord_less_eq_nat @ one_one_nat @ P2 )
=> ( ( ord_less_eq_nat @ K @ P2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [J: nat] : ( if_nat @ ( ord_less_nat @ J @ K ) @ ( G @ J ) @ ( if_nat @ ( J = K ) @ zero_zero_nat @ ( H @ ( minus_minus_nat @ J @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P2 ) )
= ( groups3542108847815614940at_nat
@ ^ [J: nat] : ( if_nat @ ( ord_less_nat @ J @ K ) @ ( G @ J ) @ ( H @ J ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_388_Suc__length__conv,axiom,
! [N: nat,Xs2: list_nat] :
( ( ( suc @ N )
= ( size_size_list_nat @ Xs2 ) )
= ( ? [Y3: nat,Ys2: list_nat] :
( ( Xs2
= ( cons_nat @ Y3 @ Ys2 ) )
& ( ( size_size_list_nat @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_389_length__Suc__conv,axiom,
! [Xs2: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( suc @ N ) )
= ( ? [Y3: nat,Ys2: list_nat] :
( ( Xs2
= ( cons_nat @ Y3 @ Ys2 ) )
& ( ( size_size_list_nat @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_390_nth__equalityI,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ Xs2 @ I3 )
= ( nth_nat @ Ys @ I3 ) ) )
=> ( Xs2 = Ys ) ) ) ).
% nth_equalityI
thf(fact_391_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X5: nat] : ( P @ I2 @ X5 ) ) )
= ( ? [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_nat @ Xs @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_392_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_nat,Z3: list_nat] : ( Y5 = Z3 ) )
= ( ^ [Xs: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I2 )
= ( nth_nat @ Ys2 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_393_distinct_Osimps_I2_J,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( distinct_list_nat @ ( cons_list_nat @ X @ Xs2 ) )
= ( ~ ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
& ( distinct_list_nat @ Xs2 ) ) ) ).
% distinct.simps(2)
thf(fact_394_distinct_Osimps_I2_J,axiom,
! [X: nat,Xs2: list_nat] :
( ( distinct_nat @ ( cons_nat @ X @ Xs2 ) )
= ( ~ ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
& ( distinct_nat @ Xs2 ) ) ) ).
% distinct.simps(2)
thf(fact_395_length__pos__if__in__set,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_396_length__pos__if__in__set,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_397_nth__mem,axiom,
! [N: nat,Xs2: list_list_nat] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( member_list_nat @ ( nth_list_nat @ Xs2 @ N ) @ ( set_list_nat2 @ Xs2 ) ) ) ).
% nth_mem
thf(fact_398_nth__mem,axiom,
! [N: nat,Xs2: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( member_nat @ ( nth_nat @ Xs2 @ N ) @ ( set_nat2 @ Xs2 ) ) ) ).
% nth_mem
thf(fact_399_list__ball__nth,axiom,
! [N: nat,Xs2: list_nat,P: nat > $o] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( P @ X2 ) )
=> ( P @ ( nth_nat @ Xs2 @ N ) ) ) ) ).
% list_ball_nth
thf(fact_400_in__set__conv__nth,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
= ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
& ( ( nth_list_nat @ Xs2 @ I2 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_401_in__set__conv__nth,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
= ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
& ( ( nth_nat @ Xs2 @ I2 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_402_all__nth__imp__all__set,axiom,
! [Xs2: list_list_nat,P: list_nat > $o,X: list_nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( P @ ( nth_list_nat @ Xs2 @ I3 ) ) )
=> ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_403_all__nth__imp__all__set,axiom,
! [Xs2: list_nat,P: nat > $o,X: nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ ( nth_nat @ Xs2 @ I3 ) ) )
=> ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_404_sum__up__index__split_H,axiom,
! [N5: nat,N: nat,F: nat > nat] :
( ( ord_less_eq_nat @ N5 @ N )
=> ( ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ N5 ) ) ) @ ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ ( minus_minus_nat @ N @ N5 ) ) @ N ) ) ) ) ) ).
% sum_up_index_split'
thf(fact_405_Cons__less__Cons,axiom,
! [A2: nat,X: list_nat,B: nat,Y: list_nat] :
( ( ord_less_list_nat @ ( cons_nat @ A2 @ 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 @ A2 @ B )
| ( ( A2 = B )
& ( ord_less_list_nat @ X @ Y ) ) ) ) ) ) ).
% Cons_less_Cons
thf(fact_406_sum__invert,axiom,
! [N5: nat,N: nat,F: nat > nat] :
( ( ord_less_eq_nat @ N5 @ N )
=> ( ( groups3542108847815614940at_nat
@ ^ [I2: nat] : ( F @ ( minus_minus_nat @ N @ I2 ) )
@ ( set_or1269000886237332187st_nat @ ( suc @ ( minus_minus_nat @ N @ N5 ) ) @ N ) )
= ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N5 ) ) ) ) ).
% sum_invert
thf(fact_407_sum__list__augmentum,axiom,
! [Ns: list_nat] :
( ( member_nat @ ( groups4561878855575611511st_nat @ Ns ) @ ( set_nat2 @ ( augmentum @ Ns ) ) )
= ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Ns ) ) ) ).
% sum_list_augmentum
thf(fact_408_add__left__cancel,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_409_add__right__cancel,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_410_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_411_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_412_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_413_augmentum__subset__sum__list,axiom,
! [Ns: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( augmentum @ Ns ) ) @ ( set_ord_atMost_nat @ ( groups4561878855575611511st_nat @ Ns ) ) ) ).
% augmentum_subset_sum_list
thf(fact_414_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_415_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_416_diff__diff__left,axiom,
! [I: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J2 ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% diff_diff_left
thf(fact_417_lessThan__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_lessThan_nat @ X )
= ( set_ord_lessThan_nat @ Y ) )
= ( X = Y ) ) ).
% lessThan_eq_iff
thf(fact_418_sorted__augmentum,axiom,
! [Ns: list_nat] :
( ~ ( member_nat @ zero_zero_nat @ ( set_nat2 @ Ns ) )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( augmentum @ Ns ) ) ) ).
% sorted_augmentum
thf(fact_419_dementum__nonzero,axiom,
! [Ns: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Ns )
=> ( ~ ( member_nat @ zero_zero_nat @ ( set_nat2 @ Ns ) )
=> ~ ( member_nat @ zero_zero_nat @ ( set_nat2 @ ( dementum @ Ns ) ) ) ) ) ).
% dementum_nonzero
thf(fact_420_add__le__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_421_add__le__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_422_add_Oright__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.right_neutral
thf(fact_423_add__cancel__left__left,axiom,
! [B: nat,A2: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_424_add__cancel__left__right,axiom,
! [A2: nat,B: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_425_add__cancel__right__left,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ B @ A2 ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_426_add__cancel__right__right,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ A2 @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_427_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_428_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_429_add__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% add_0
thf(fact_430_add__less__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_431_add__less__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_432_add__diff__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A2 @ B ) ) ).
% add_diff_cancel_left
thf(fact_433_add__diff__cancel__left_H,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ A2 )
= B ) ).
% add_diff_cancel_left'
thf(fact_434_add__diff__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A2 @ B ) ) ).
% add_diff_cancel_right
thf(fact_435_add__diff__cancel__right_H,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_436_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_437_lessThan__iff,axiom,
! [I: list_nat,K: list_nat] :
( ( member_list_nat @ I @ ( set_or3033090826390029821st_nat @ K ) )
= ( ord_less_list_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_438_lessThan__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
= ( ord_less_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_439_Nat_Odiff__diff__right,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J2 ) ) ) ).
% Nat.diff_diff_right
thf(fact_440_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_441_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J2 ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_442_le__add__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_443_le__add__same__cancel1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_444_add__le__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_445_add__le__same__cancel1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_446_less__add__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_447_less__add__same__cancel1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_448_add__less__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_449_add__less__same__cancel1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_450_le__add__diff__inverse,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A2 @ B ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_451_le__add__diff__inverse2,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A2 @ B ) @ B )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_452_diff__add__zero,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_453_lessThan__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_454_diff__Suc__diff__eq1,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J2 ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_455_diff__Suc__diff__eq2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J2 ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_456_nth__plus__list,axiom,
! [I: nat,Xs2: list_list_nat,Ys: list_list_nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( nth_list_nat @ ( plus_p2116291331692525561st_nat @ Xs2 @ Ys ) @ I )
= ( plus_plus_list_nat @ ( nth_list_nat @ Xs2 @ I ) @ ( nth_list_nat @ Ys @ I ) ) ) ) ) ).
% nth_plus_list
thf(fact_457_nth__plus__list,axiom,
! [I: nat,Xs2: list_nat,Ys: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
=> ( ( nth_nat @ ( plus_plus_list_nat @ Xs2 @ Ys ) @ I )
= ( plus_plus_nat @ ( nth_nat @ Xs2 @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).
% nth_plus_list
thf(fact_458_sum_OlessThan__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) @ ( G @ N ) ) ) ).
% sum.lessThan_Suc
thf(fact_459_sum_OatMost__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atMost_Suc
thf(fact_460_sum__list__incr,axiom,
! [I: nat,X: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ X ) )
=> ( ( groups4561878855575611511st_nat @ ( list_incr @ I @ X ) )
= ( suc @ ( groups4561878855575611511st_nat @ X ) ) ) ) ).
% sum_list_incr
thf(fact_461_sum_Ocl__ivl__Suc,axiom,
! [N: nat,M: nat,G: nat > nat] :
( ( ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= zero_zero_nat ) )
& ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_462_sum__list__plus,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( groups4561878855575611511st_nat @ ( plus_plus_list_nat @ Xs2 @ Ys ) )
= ( plus_plus_nat @ ( groups4561878855575611511st_nat @ Xs2 ) @ ( groups4561878855575611511st_nat @ Ys ) ) ) ) ).
% sum_list_plus
thf(fact_463_sorted__wrt__true,axiom,
! [Xs2: list_nat] :
( sorted_wrt_nat
@ ^ [Uu: nat,Uv: nat] : $true
@ Xs2 ) ).
% sorted_wrt_true
thf(fact_464_plus__Cons,axiom,
! [Y: list_nat,Ys: list_list_nat,X: list_nat,Xs2: list_list_nat] :
( ( plus_p2116291331692525561st_nat @ ( cons_list_nat @ Y @ Ys ) @ ( cons_list_nat @ X @ Xs2 ) )
= ( cons_list_nat @ ( plus_plus_list_nat @ Y @ X ) @ ( plus_p2116291331692525561st_nat @ Ys @ Xs2 ) ) ) ).
% plus_Cons
thf(fact_465_plus__Cons,axiom,
! [Y: nat,Ys: list_nat,X: nat,Xs2: list_nat] :
( ( plus_plus_list_nat @ ( cons_nat @ Y @ Ys ) @ ( cons_nat @ X @ Xs2 ) )
= ( cons_nat @ ( plus_plus_nat @ Y @ X ) @ ( plus_plus_list_nat @ Ys @ Xs2 ) ) ) ).
% plus_Cons
thf(fact_466_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_467_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: list_nat,B: list_nat,C: list_nat] :
( ( plus_plus_list_nat @ ( plus_plus_list_nat @ A2 @ B ) @ C )
= ( plus_plus_list_nat @ A2 @ ( plus_plus_list_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_468_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( I = J2 )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_469_group__cancel_Oadd1,axiom,
! [A: nat,K: nat,A2: nat,B: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_470_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B: nat,A2: nat] :
( ( B3
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A2 @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_471_add_Oassoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_472_add_Oassoc,axiom,
! [A2: list_nat,B: list_nat,C: list_nat] :
( ( plus_plus_list_nat @ ( plus_plus_list_nat @ A2 @ B ) @ C )
= ( plus_plus_list_nat @ A2 @ ( plus_plus_list_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_473_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A5: nat,B5: nat] : ( plus_plus_nat @ B5 @ A5 ) ) ) ).
% add.commute
thf(fact_474_add_Ocommute,axiom,
( plus_plus_list_nat
= ( ^ [A5: list_nat,B5: list_nat] : ( plus_plus_list_nat @ B5 @ A5 ) ) ) ).
% add.commute
thf(fact_475_add_Oleft__commute,axiom,
! [B: nat,A2: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A2 @ C ) )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_476_add_Oleft__commute,axiom,
! [B: list_nat,A2: list_nat,C: list_nat] :
( ( plus_plus_list_nat @ B @ ( plus_plus_list_nat @ A2 @ C ) )
= ( plus_plus_list_nat @ A2 @ ( plus_plus_list_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_477_add__left__imp__eq,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_478_add__right__imp__eq,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_479_Iic__subset__Iio__iff,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ A2 ) @ ( set_ord_lessThan_nat @ B ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% Iic_subset_Iio_iff
thf(fact_480_sorted__wrt__mono__rel,axiom,
! [Xs2: list_list_nat,P: list_nat > list_nat > $o,Q: list_nat > list_nat > $o] :
( ! [X2: list_nat,Y2: list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs2 ) )
=> ( ( member_list_nat @ Y2 @ ( set_list_nat2 @ Xs2 ) )
=> ( ( P @ X2 @ Y2 )
=> ( Q @ X2 @ Y2 ) ) ) )
=> ( ( sorted_wrt_list_nat @ P @ Xs2 )
=> ( sorted_wrt_list_nat @ Q @ Xs2 ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_481_sorted__wrt__mono__rel,axiom,
! [Xs2: list_nat,P: nat > nat > $o,Q: nat > nat > $o] :
( ! [X2: nat,Y2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ( member_nat @ Y2 @ ( set_nat2 @ Xs2 ) )
=> ( ( P @ X2 @ Y2 )
=> ( Q @ X2 @ Y2 ) ) ) )
=> ( ( sorted_wrt_nat @ P @ Xs2 )
=> ( sorted_wrt_nat @ Q @ Xs2 ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_482_lessThan__strict__subset__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_483_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J2 )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_484_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( I = J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_485_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_486_add__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_487_add__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_488_less__eqE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A2 @ C2 ) ) ) ).
% less_eqE
thf(fact_489_add__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_490_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
? [C3: nat] :
( B5
= ( plus_plus_nat @ A5 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_491_add__le__imp__le__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_left
thf(fact_492_add__le__imp__le__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_right
thf(fact_493_comm__monoid__add__class_Oadd__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_494_add_Ocomm__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.comm_neutral
thf(fact_495_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_496_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( I = J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_497_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J2 )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_498_add__strict__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_499_add__strict__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_500_add__strict__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_501_add__less__imp__less__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_left
thf(fact_502_add__less__imp__less__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_right
thf(fact_503_add__implies__diff,axiom,
! [C: nat,B: nat,A2: nat] :
( ( ( plus_plus_nat @ C @ B )
= A2 )
=> ( C
= ( minus_minus_nat @ A2 @ B ) ) ) ).
% add_implies_diff
thf(fact_504_diff__diff__eq,axiom,
! [A2: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C )
= ( minus_minus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_505_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_506_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_507_nat__arith_Osuc1,axiom,
! [A: nat,K: nat,A2: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( suc @ A )
= ( plus_plus_nat @ K @ ( suc @ A2 ) ) ) ) ).
% nat_arith.suc1
thf(fact_508_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_509_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_510_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_511_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_512_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_513_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_514_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_515_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_516_add__le__mono,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_le_mono
thf(fact_517_add__le__mono1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_le_mono1
thf(fact_518_trans__le__add1,axiom,
! [I: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_le_add1
thf(fact_519_trans__le__add2,axiom,
! [I: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_le_add2
thf(fact_520_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N4: nat] :
? [K4: nat] :
( N4
= ( plus_plus_nat @ M5 @ K4 ) ) ) ) ).
% nat_le_iff_add
thf(fact_521_add__lessD1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J2 ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_522_add__less__mono,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_less_mono
thf(fact_523_not__add__less1,axiom,
! [I: nat,J2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J2 ) @ I ) ).
% not_add_less1
thf(fact_524_not__add__less2,axiom,
! [J2: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I ) @ I ) ).
% not_add_less2
thf(fact_525_add__less__mono1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_less_mono1
thf(fact_526_trans__less__add1,axiom,
! [I: nat,J2: nat,M: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_less_add1
thf(fact_527_trans__less__add2,axiom,
! [I: nat,J2: nat,M: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_less_add2
thf(fact_528_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_529_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_530_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_531_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_532_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_533_subset__code_I1_J,axiom,
! [Xs2: list_list_nat,B3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs2 ) @ B3 )
= ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xs2 ) )
=> ( member_list_nat @ X4 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_534_subset__code_I1_J,axiom,
! [Xs2: list_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ B3 )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
=> ( member_nat @ X4 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_535_lessThan__def,axiom,
( set_or3033090826390029821st_nat
= ( ^ [U2: list_nat] :
( collect_list_nat
@ ^ [X4: list_nat] : ( ord_less_list_nat @ X4 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_536_lessThan__def,axiom,
( set_ord_lessThan_nat
= ( ^ [U2: nat] :
( collect_nat
@ ^ [X4: nat] : ( ord_less_nat @ X4 @ U2 ) ) ) ) ).
% lessThan_def
thf(fact_537_sum_Odistrib,axiom,
! [G: nat > nat,H: nat > nat,A: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [X4: nat] : ( plus_plus_nat @ ( G @ X4 ) @ ( H @ X4 ) )
@ A )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A ) @ ( groups3542108847815614940at_nat @ H @ A ) ) ) ).
% sum.distrib
thf(fact_538_sum_OlessThan__Suc__shift,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( G @ zero_zero_nat )
@ ( groups3542108847815614940at_nat
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_539_strict__sorted__imp__sorted,axiom,
! [Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs2 )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 ) ) ).
% strict_sorted_imp_sorted
thf(fact_540_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_541_strict__sorted__equal,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs2 )
=> ( ( sorted_wrt_nat @ ord_less_nat @ Ys )
=> ( ( ( set_nat2 @ Ys )
= ( set_nat2 @ Xs2 ) )
=> ( Ys = Xs2 ) ) ) ) ).
% strict_sorted_equal
thf(fact_542_sum_Oub__add__nat,axiom,
! [M: nat,N: nat,G: nat > nat,P2: nat] :
( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P2 ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_543_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_544_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_545_add__nonpos__nonpos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_546_add__nonneg__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_547_add__increasing2,axiom,
! [C: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_548_add__decreasing2,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_549_add__increasing,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_550_add__decreasing,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_551_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_552_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_553_add__le__less__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_554_add__less__le__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_555_pos__add__strict,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_556_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A2 @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_557_add__pos__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_pos_pos
thf(fact_558_add__neg__neg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_559_lessThan__Suc__atMost,axiom,
! [K: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K ) )
= ( set_ord_atMost_nat @ K ) ) ).
% lessThan_Suc_atMost
thf(fact_560_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_561_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J2 @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J2 @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J2 ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_562_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( minus_minus_nat @ B @ A2 )
= C )
= ( B
= ( plus_plus_nat @ C @ A2 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_563_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ A2 @ ( minus_minus_nat @ B @ A2 ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_564_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_565_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_566_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_567_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A2 )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_568_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_569_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_570_le__add__diff,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 ) ) ) ).
% le_add_diff
thf(fact_571_diff__add,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ A2 )
= B ) ) ).
% diff_add
thf(fact_572_less__add__one,axiom,
! [A2: nat] : ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ).
% less_add_one
thf(fact_573_add__mono1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_574_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A2: nat,B: nat] :
( ~ ( ord_less_nat @ A2 @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A2 @ B ) )
= A2 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_575_sum_OatMost__shift,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_nat @ ( G @ zero_zero_nat )
@ ( groups3542108847815614940at_nat
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum.atMost_shift
thf(fact_576_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_577_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_578_less__imp__add__positive,axiom,
! [I: nat,J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J2 ) ) ) ).
% less_imp_add_positive
thf(fact_579_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q2: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).
% less_natE
thf(fact_580_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_581_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_582_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M5: nat,N4: nat] :
? [K4: nat] :
( N4
= ( suc @ ( plus_plus_nat @ M5 @ K4 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_583_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_584_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_585_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_586_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ( minus_minus_nat @ J2 @ I )
= K )
= ( J2
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_587_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_588_Nat_Odiff__add__assoc,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J2 ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J2 @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_589_Nat_Ole__diff__conv2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J2 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_590_le__diff__conv,axiom,
! [J2: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J2 @ K ) @ I )
= ( ord_less_eq_nat @ J2 @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_591_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_592_less__diff__conv,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J2 ) ) ).
% less_diff_conv
thf(fact_593_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_594_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_595_Suc__eq__plus1,axiom,
( suc
= ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_596_set__subset__Cons,axiom,
! [Xs2: list_nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ ( set_nat2 @ ( cons_nat @ X @ Xs2 ) ) ) ).
% set_subset_Cons
thf(fact_597_sum__up__index__split,axiom,
! [F: nat > nat,M: nat,N: nat] :
( ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N ) ) ) ) ) ).
% sum_up_index_split
thf(fact_598_sorted__simps_I2_J,axiom,
! [X: nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X @ Ys ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Ys ) )
=> ( ord_less_eq_nat @ X @ X4 ) )
& ( sorted_wrt_nat @ ord_less_eq_nat @ Ys ) ) ) ).
% sorted_simps(2)
thf(fact_599_strict__sorted__simps_I2_J,axiom,
! [X: nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ ( cons_nat @ X @ Ys ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Ys ) )
=> ( ord_less_nat @ X @ X4 ) )
& ( sorted_wrt_nat @ ord_less_nat @ Ys ) ) ) ).
% strict_sorted_simps(2)
thf(fact_600_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [G: nat > nat,M: nat,K: nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
= ( groups3542108847815614940at_nat
@ ^ [I2: nat] : ( G @ ( plus_plus_nat @ I2 @ K ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% sum.shift_bounds_cl_nat_ivl
thf(fact_601_strict__sorted__iff,axiom,
! [L: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ L )
= ( ( sorted_wrt_nat @ ord_less_eq_nat @ L )
& ( distinct_nat @ L ) ) ) ).
% strict_sorted_iff
thf(fact_602_sorted__distinct__set__unique,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
=> ( ( distinct_nat @ Xs2 )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ys )
=> ( ( distinct_nat @ Ys )
=> ( ( ( set_nat2 @ Xs2 )
= ( set_nat2 @ Ys ) )
=> ( Xs2 = Ys ) ) ) ) ) ) ).
% sorted_distinct_set_unique
thf(fact_603_sorted__wrt01,axiom,
! [Xs2: list_nat,P: nat > nat > $o] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat )
=> ( sorted_wrt_nat @ P @ Xs2 ) ) ).
% sorted_wrt01
thf(fact_604_sorted__wrt__iff__nth__less,axiom,
( sorted_wrt_nat
= ( ^ [P3: nat > nat > $o,Xs: list_nat] :
! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( P3 @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_605_sorted__wrt__nth__less,axiom,
! [P: nat > nat > $o,Xs2: list_nat,I: nat,J2: nat] :
( ( sorted_wrt_nat @ P @ Xs2 )
=> ( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ ( nth_nat @ Xs2 @ I ) @ ( nth_nat @ Xs2 @ J2 ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_606_Cons__le__Cons,axiom,
! [A2: nat,X: list_nat,B: nat,Y: list_nat] :
( ( ord_less_eq_list_nat @ ( cons_nat @ A2 @ 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 @ A2 @ B )
| ( ( A2 = B )
& ( ord_less_eq_list_nat @ X @ Y ) ) ) ) ) ) ).
% Cons_le_Cons
thf(fact_607_add__neg__nonpos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_608_add__nonneg__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_609_add__nonpos__neg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_610_add__pos__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_611_add__strict__increasing,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_612_add__strict__increasing2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_613_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_614_nat__diff__split,axiom,
! [P: nat > $o,A2: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B ) )
= ( ( ( ord_less_nat @ A2 @ B )
=> ( P @ zero_zero_nat ) )
& ! [D2: nat] :
( ( A2
= ( plus_plus_nat @ B @ D2 ) )
=> ( P @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_615_nat__diff__split__asm,axiom,
! [P: nat > $o,A2: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B ) )
= ( ~ ( ( ( ord_less_nat @ A2 @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D2: nat] :
( ( A2
= ( plus_plus_nat @ B @ D2 ) )
& ~ ( P @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_616_less__diff__conv2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J2 @ K ) @ I )
= ( ord_less_nat @ J2 @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_617_sum_Onat__diff__reindex,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I2: nat] : ( G @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) )
@ ( set_ord_lessThan_nat @ N ) )
= ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.nat_diff_reindex
thf(fact_618_sum__diff__distrib,axiom,
! [Q: nat > nat,P: nat > nat,N: nat] :
( ! [X2: nat] : ( ord_less_eq_nat @ ( Q @ X2 ) @ ( P @ X2 ) )
=> ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P @ ( set_ord_lessThan_nat @ N ) ) @ ( groups3542108847815614940at_nat @ Q @ ( set_ord_lessThan_nat @ N ) ) )
= ( groups3542108847815614940at_nat
@ ^ [X4: nat] : ( minus_minus_nat @ ( P @ X4 ) @ ( Q @ X4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% sum_diff_distrib
thf(fact_619_sorted01,axiom,
! [Xs2: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 ) ) ).
% sorted01
thf(fact_620_sorted__iff__nth__mono__less,axiom,
! [Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
= ( ! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ I2 ) @ ( nth_nat @ Xs2 @ J ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_621_sum_OatLeastAtMost__rev,axiom,
! [G: nat > nat,N: nat,M: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ N @ M ) )
= ( groups3542108847815614940at_nat
@ ^ [I2: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I2 ) )
@ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% sum.atLeastAtMost_rev
thf(fact_622_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_623_sum__unroll,axiom,
! [N: nat,F: nat > nat] :
( ( ( N = zero_zero_nat )
=> ( ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ N ) )
= ( F @ zero_zero_nat ) ) )
& ( ( N != zero_zero_nat )
=> ( ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ N ) )
= ( plus_plus_nat @ ( F @ N ) @ ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% sum_unroll
thf(fact_624_list_Osize_I4_J,axiom,
! [X21: nat,X222: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X21 @ X222 ) )
= ( plus_plus_nat @ ( size_size_list_nat @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_625_sum_OatLeast0__atMost__Suc,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_626_sum_Onat__ivl__Suc_H,axiom,
! [M: nat,N: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( G @ ( suc @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_627_sum_OatLeast__Suc__atMost,axiom,
! [M: nat,N: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) )
= ( plus_plus_nat @ ( G @ M ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_628_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M5: nat,N4: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N4 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% add_eq_if
thf(fact_629_sum_OatLeast1__atMost__eq,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
= ( groups3542108847815614940at_nat
@ ^ [K4: nat] : ( G @ ( suc @ K4 ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_630_sum_Onested__swap_H,axiom,
! [A2: nat > nat > nat,N: nat] :
( ( groups3542108847815614940at_nat
@ ^ [I2: nat] : ( groups3542108847815614940at_nat @ ( A2 @ I2 ) @ ( set_ord_lessThan_nat @ I2 ) )
@ ( set_ord_atMost_nat @ N ) )
= ( groups3542108847815614940at_nat
@ ^ [J: nat] :
( groups3542108847815614940at_nat
@ ^ [I2: nat] : ( A2 @ I2 @ J )
@ ( set_or1269000886237332187st_nat @ ( suc @ J ) @ N ) )
@ ( set_ord_lessThan_nat @ N ) ) ) ).
% sum.nested_swap'
thf(fact_631_sorted__iff__nth__Suc,axiom,
! [Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
= ( ! [I2: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ ( size_size_list_nat @ Xs2 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ I2 ) @ ( nth_nat @ Xs2 @ ( suc @ I2 ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_632_sorted__iff__nth__mono,axiom,
! [Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
= ( ! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ I2 ) @ ( nth_nat @ Xs2 @ J ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_633_sorted__nth__mono,axiom,
! [Xs2: list_nat,I: nat,J2: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
=> ( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs2 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ I ) @ ( nth_nat @ Xs2 @ J2 ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_634_sum_OatMost__Suc__shift,axiom,
! [G: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( G @ zero_zero_nat )
@ ( groups3542108847815614940at_nat
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_ord_atMost_nat @ N ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_635_sum_OSuc__reindex__ivl,axiom,
! [M: nat,N: nat,G: nat > nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( G @ M )
@ ( groups3542108847815614940at_nat
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_636_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N3: nat] :
( X
!= ( suc @ N3 ) ) ) ).
% list_decode.cases
thf(fact_637_sum__list__eq__0__iff,axiom,
! [Ns: list_nat] :
( ( ( groups4561878855575611511st_nat @ Ns )
= zero_zero_nat )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Ns ) )
=> ( X4 = zero_zero_nat ) ) ) ) ).
% sum_list_eq_0_iff
thf(fact_638_sum__list_OCons,axiom,
! [X: nat,Xs2: list_nat] :
( ( groups4561878855575611511st_nat @ ( cons_nat @ X @ Xs2 ) )
= ( plus_plus_nat @ X @ ( groups4561878855575611511st_nat @ Xs2 ) ) ) ).
% sum_list.Cons
thf(fact_639_sum__bounds__lt__plus1,axiom,
! [F: nat > nat,Mm: nat] :
( ( groups3542108847815614940at_nat
@ ^ [K4: nat] : ( F @ ( suc @ K4 ) )
@ ( set_ord_lessThan_nat @ Mm ) )
= ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).
% sum_bounds_lt_plus1
thf(fact_640_sum__list__mono2,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ I3 ) @ ( nth_nat @ Ys @ I3 ) ) )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ Xs2 ) @ ( groups4561878855575611511st_nat @ Ys ) ) ) ) ).
% sum_list_mono2
thf(fact_641_elem__le__sum__list,axiom,
! [K: nat,Ns: list_nat] :
( ( ord_less_nat @ K @ ( size_size_list_nat @ Ns ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Ns @ K ) @ ( groups4561878855575611511st_nat @ Ns ) ) ) ).
% elem_le_sum_list
thf(fact_642_psubsetI,axiom,
! [A: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( A != B3 )
=> ( ord_less_set_nat @ A @ B3 ) ) ) ).
% psubsetI
thf(fact_643_subset__antisym,axiom,
! [A: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ A )
=> ( A = B3 ) ) ) ).
% subset_antisym
thf(fact_644_subsetI,axiom,
! [A: set_list_nat,B3: set_list_nat] :
( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ A )
=> ( member_list_nat @ X2 @ B3 ) )
=> ( ord_le6045566169113846134st_nat @ A @ B3 ) ) ).
% subsetI
thf(fact_645_subsetI,axiom,
! [A: set_nat,B3: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B3 ) )
=> ( ord_less_eq_set_nat @ A @ B3 ) ) ).
% subsetI
thf(fact_646_Collect__mono__iff,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) )
= ( ! [X4: list_nat] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_647_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X4: nat] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_648_set__eq__subset,axiom,
( ( ^ [Y5: set_nat,Z3: set_nat] : ( Y5 = Z3 ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_649_subset__trans,axiom,
! [A: set_nat,B3: set_nat,C4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C4 )
=> ( ord_less_eq_set_nat @ A @ C4 ) ) ) ).
% subset_trans
thf(fact_650_Collect__mono,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ! [X2: list_nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_651_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_652_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_653_subset__iff,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A4: set_list_nat,B4: set_list_nat] :
! [T3: list_nat] :
( ( member_list_nat @ T3 @ A4 )
=> ( member_list_nat @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_654_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
! [T3: nat] :
( ( member_nat @ T3 @ A4 )
=> ( member_nat @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_655_equalityD2,axiom,
! [A: set_nat,B3: set_nat] :
( ( A = B3 )
=> ( ord_less_eq_set_nat @ B3 @ A ) ) ).
% equalityD2
thf(fact_656_equalityD1,axiom,
! [A: set_nat,B3: set_nat] :
( ( A = B3 )
=> ( ord_less_eq_set_nat @ A @ B3 ) ) ).
% equalityD1
thf(fact_657_subset__eq,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A4: set_list_nat,B4: set_list_nat] :
! [X4: list_nat] :
( ( member_list_nat @ X4 @ A4 )
=> ( member_list_nat @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_658_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( member_nat @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_659_equalityE,axiom,
! [A: set_nat,B3: set_nat] :
( ( A = B3 )
=> ~ ( ( ord_less_eq_set_nat @ A @ B3 )
=> ~ ( ord_less_eq_set_nat @ B3 @ A ) ) ) ).
% equalityE
thf(fact_660_subsetD,axiom,
! [A: set_list_nat,B3: set_list_nat,C: list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B3 )
=> ( ( member_list_nat @ C @ A )
=> ( member_list_nat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_661_subsetD,axiom,
! [A: set_nat,B3: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_662_in__mono,axiom,
! [A: set_list_nat,B3: set_list_nat,X: list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B3 )
=> ( ( member_list_nat @ X @ A )
=> ( member_list_nat @ X @ B3 ) ) ) ).
% in_mono
thf(fact_663_in__mono,axiom,
! [A: set_nat,B3: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ X @ B3 ) ) ) ).
% in_mono
thf(fact_664_double__diff,axiom,
! [A: set_nat,B3: set_nat,C4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C4 )
=> ( ( minus_minus_set_nat @ B3 @ ( minus_minus_set_nat @ C4 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_665_Diff__subset,axiom,
! [A: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B3 ) @ A ) ).
% Diff_subset
thf(fact_666_Diff__mono,axiom,
! [A: set_nat,C4: set_nat,D3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A @ C4 )
=> ( ( ord_less_eq_set_nat @ D3 @ B3 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B3 ) @ ( minus_minus_set_nat @ C4 @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_667_psubsetD,axiom,
! [A: set_nat,B3: set_nat,C: nat] :
( ( ord_less_set_nat @ A @ B3 )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_668_psubsetD,axiom,
! [A: set_list_nat,B3: set_list_nat,C: list_nat] :
( ( ord_le1190675801316882794st_nat @ A @ B3 )
=> ( ( member_list_nat @ C @ A )
=> ( member_list_nat @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_669_psubset__imp__ex__mem,axiom,
! [A: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A @ B3 )
=> ? [B2: nat] : ( member_nat @ B2 @ ( minus_minus_set_nat @ B3 @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_670_psubset__imp__ex__mem,axiom,
! [A: set_list_nat,B3: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A @ B3 )
=> ? [B2: list_nat] : ( member_list_nat @ B2 @ ( minus_7954133019191499631st_nat @ B3 @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_671_less__eq__set__def,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A4: set_list_nat,B4: set_list_nat] :
( ord_le1520216061033275535_nat_o
@ ^ [X4: list_nat] : ( member_list_nat @ X4 @ A4 )
@ ^ [X4: list_nat] : ( member_list_nat @ X4 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_672_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ord_less_eq_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ A4 )
@ ^ [X4: nat] : ( member_nat @ X4 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_673_Collect__subset,axiom,
! [A: set_list_nat,P: list_nat > $o] :
( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X4: list_nat] :
( ( member_list_nat @ X4 @ A )
& ( P @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_674_Collect__subset,axiom,
! [A: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ X4 ) ) )
@ A ) ).
% Collect_subset
thf(fact_675_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ord_less_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ A4 )
@ ^ [X4: nat] : ( member_nat @ X4 @ B4 ) ) ) ) ).
% less_set_def
thf(fact_676_less__set__def,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [A4: set_list_nat,B4: set_list_nat] :
( ord_less_list_nat_o
@ ^ [X4: list_nat] : ( member_list_nat @ X4 @ A4 )
@ ^ [X4: list_nat] : ( member_list_nat @ X4 @ B4 ) ) ) ) ).
% less_set_def
thf(fact_677_psubsetE,axiom,
! [A: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A @ B3 )
=> ~ ( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ord_less_eq_set_nat @ B3 @ A ) ) ) ).
% psubsetE
thf(fact_678_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% psubset_eq
thf(fact_679_psubset__imp__subset,axiom,
! [A: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A @ B3 )
=> ( ord_less_eq_set_nat @ A @ B3 ) ) ).
% psubset_imp_subset
thf(fact_680_psubset__subset__trans,axiom,
! [A: set_nat,B3: set_nat,C4: set_nat] :
( ( ord_less_set_nat @ A @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C4 )
=> ( ord_less_set_nat @ A @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_681_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ~ ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_682_subset__psubset__trans,axiom,
! [A: set_nat,B3: set_nat,C4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( ord_less_set_nat @ B3 @ C4 )
=> ( ord_less_set_nat @ A @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_683_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_684_member__le__sum__list,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( ord_less_eq_nat @ X @ ( groups4561878855575611511st_nat @ Xs2 ) ) ) ).
% member_le_sum_list
thf(fact_685_sum__list__nonpos,axiom,
! [Xs2: list_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ord_less_eq_nat @ X2 @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ Xs2 ) @ zero_zero_nat ) ) ).
% sum_list_nonpos
thf(fact_686_sum__list__nonneg__eq__0__iff,axiom,
! [Xs2: list_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ X2 ) )
=> ( ( ( groups4561878855575611511st_nat @ Xs2 )
= zero_zero_nat )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
=> ( X4 = zero_zero_nat ) ) ) ) ) ).
% sum_list_nonneg_eq_0_iff
thf(fact_687_Groups__List_Osum__list__nonneg,axiom,
! [Xs2: list_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ X2 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups4561878855575611511st_nat @ Xs2 ) ) ) ).
% Groups_List.sum_list_nonneg
thf(fact_688_distinct__sum__list__conv__Sum,axiom,
! [Xs2: list_nat] :
( ( distinct_nat @ Xs2 )
=> ( ( groups4561878855575611511st_nat @ Xs2 )
= ( groups3542108847815614940at_nat
@ ^ [X4: nat] : X4
@ ( set_nat2 @ Xs2 ) ) ) ) ).
% distinct_sum_list_conv_Sum
thf(fact_689_pointwise__less__imp___092_060sigma_062,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( pointwise_less @ Xs2 @ Ys )
=> ( ord_less_nat @ ( groups4561878855575611511st_nat @ Xs2 ) @ ( groups4561878855575611511st_nat @ Ys ) ) ) ).
% pointwise_less_imp_\<sigma>
thf(fact_690_length__sum__set__def,axiom,
( length_sum_set
= ( ^ [R2: nat,N4: nat] :
( collect_list_nat
@ ^ [X4: list_nat] :
( ( ( size_size_list_nat @ X4 )
= R2 )
& ( ( groups4561878855575611511st_nat @ X4 )
= N4 ) ) ) ) ) ).
% length_sum_set_def
thf(fact_691_Diff__iff,axiom,
! [C: nat,A: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B3 ) )
= ( ( member_nat @ C @ A )
& ~ ( member_nat @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_692_Diff__iff,axiom,
! [C: list_nat,A: set_list_nat,B3: set_list_nat] :
( ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A @ B3 ) )
= ( ( member_list_nat @ C @ A )
& ~ ( member_list_nat @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_693_DiffI,axiom,
! [C: nat,A: set_nat,B3: set_nat] :
( ( member_nat @ C @ A )
=> ( ~ ( member_nat @ C @ B3 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A @ B3 ) ) ) ) ).
% DiffI
thf(fact_694_DiffI,axiom,
! [C: list_nat,A: set_list_nat,B3: set_list_nat] :
( ( member_list_nat @ C @ A )
=> ( ~ ( member_list_nat @ C @ B3 )
=> ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A @ B3 ) ) ) ) ).
% DiffI
thf(fact_695_length__sum__set__Suc,axiom,
! [K: nat,Ks: list_nat,R3: nat,N: nat] :
( ( member_list_nat @ ( cons_nat @ K @ Ks ) @ ( length_sum_set @ ( suc @ R3 ) @ N ) )
= ( ? [M5: nat] :
( ( member_list_nat @ Ks @ ( length_sum_set @ R3 @ M5 ) )
& ( N
= ( plus_plus_nat @ M5 @ K ) ) ) ) ) ).
% length_sum_set_Suc
thf(fact_696_minus__set__def,axiom,
( minus_7954133019191499631st_nat
= ( ^ [A4: set_list_nat,B4: set_list_nat] :
( collect_list_nat
@ ( minus_1139252259498527702_nat_o
@ ^ [X4: list_nat] : ( member_list_nat @ X4 @ A4 )
@ ^ [X4: list_nat] : ( member_list_nat @ X4 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_697_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X4: nat] : ( member_nat @ X4 @ A4 )
@ ^ [X4: nat] : ( member_nat @ X4 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_698_set__diff__eq,axiom,
( minus_7954133019191499631st_nat
= ( ^ [A4: set_list_nat,B4: set_list_nat] :
( collect_list_nat
@ ^ [X4: list_nat] :
( ( member_list_nat @ X4 @ A4 )
& ~ ( member_list_nat @ X4 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_699_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ~ ( member_nat @ X4 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_700_DiffD2,axiom,
! [C: nat,A: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B3 ) )
=> ~ ( member_nat @ C @ B3 ) ) ).
% DiffD2
thf(fact_701_DiffD2,axiom,
! [C: list_nat,A: set_list_nat,B3: set_list_nat] :
( ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A @ B3 ) )
=> ~ ( member_list_nat @ C @ B3 ) ) ).
% DiffD2
thf(fact_702_DiffD1,axiom,
! [C: nat,A: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B3 ) )
=> ( member_nat @ C @ A ) ) ).
% DiffD1
thf(fact_703_DiffD1,axiom,
! [C: list_nat,A: set_list_nat,B3: set_list_nat] :
( ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A @ B3 ) )
=> ( member_list_nat @ C @ A ) ) ).
% DiffD1
thf(fact_704_DiffE,axiom,
! [C: nat,A: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B3 ) )
=> ~ ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B3 ) ) ) ).
% DiffE
thf(fact_705_DiffE,axiom,
! [C: list_nat,A: set_list_nat,B3: set_list_nat] :
( ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A @ B3 ) )
=> ~ ( ( member_list_nat @ C @ A )
=> ( member_list_nat @ C @ B3 ) ) ) ).
% DiffE
thf(fact_706_minimal__elementsp_Osimps,axiom,
( minimal_elementsp
= ( ^ [U3: list_nat > $o,A5: list_nat] :
? [X4: list_nat] :
( ( A5 = X4 )
& ( U3 @ X4 )
& ! [Y3: list_nat] :
( ( U3 @ Y3 )
=> ~ ( pointwise_less @ Y3 @ X4 ) ) ) ) ) ).
% minimal_elementsp.simps
thf(fact_707_minimal__elementsp_Ointros,axiom,
! [U4: list_nat > $o,X: list_nat] :
( ( U4 @ X )
=> ( ! [Y2: list_nat] :
( ( U4 @ Y2 )
=> ~ ( pointwise_less @ Y2 @ X ) )
=> ( minimal_elementsp @ U4 @ X ) ) ) ).
% minimal_elementsp.intros
thf(fact_708_minimal__elementsp_Ocases,axiom,
! [U4: list_nat > $o,A2: list_nat] :
( ( minimal_elementsp @ U4 @ A2 )
=> ~ ( ( U4 @ A2 )
=> ~ ! [Y4: list_nat] :
( ( U4 @ Y4 )
=> ~ ( pointwise_less @ Y4 @ A2 ) ) ) ) ).
% minimal_elementsp.cases
thf(fact_709_minimal__elements__def,axiom,
( minimal_elements
= ( ^ [U3: set_list_nat] :
( collect_list_nat
@ ( minimal_elementsp
@ ^ [X4: list_nat] : ( member_list_nat @ X4 @ U3 ) ) ) ) ) ).
% minimal_elements_def
thf(fact_710_minimal__elementsp__minimal__elements__eq,axiom,
! [U4: set_list_nat] :
( ( minimal_elementsp
@ ^ [X4: list_nat] : ( member_list_nat @ X4 @ U4 ) )
= ( ^ [X4: list_nat] : ( member_list_nat @ X4 @ ( minimal_elements @ U4 ) ) ) ) ).
% minimal_elementsp_minimal_elements_eq
thf(fact_711_minimal__elements_Ocases,axiom,
! [A2: list_nat,U4: set_list_nat] :
( ( member_list_nat @ A2 @ ( minimal_elements @ U4 ) )
=> ~ ( ( member_list_nat @ A2 @ U4 )
=> ~ ! [Y4: list_nat] :
( ( member_list_nat @ Y4 @ U4 )
=> ~ ( pointwise_less @ Y4 @ A2 ) ) ) ) ).
% minimal_elements.cases
thf(fact_712_minimal__elements_Osimps,axiom,
! [A2: list_nat,U4: set_list_nat] :
( ( member_list_nat @ A2 @ ( minimal_elements @ U4 ) )
= ( ? [X4: list_nat] :
( ( A2 = X4 )
& ( member_list_nat @ X4 @ U4 )
& ! [Y3: list_nat] :
( ( member_list_nat @ Y3 @ U4 )
=> ~ ( pointwise_less @ Y3 @ X4 ) ) ) ) ) ).
% minimal_elements.simps
thf(fact_713_minimal__elements_Ointros,axiom,
! [X: list_nat,U4: set_list_nat] :
( ( member_list_nat @ X @ U4 )
=> ( ! [Y2: list_nat] :
( ( member_list_nat @ Y2 @ U4 )
=> ~ ( pointwise_less @ Y2 @ X ) )
=> ( member_list_nat @ X @ ( minimal_elements @ U4 ) ) ) ) ).
% minimal_elements.intros
thf(fact_714_card__length__sum__set,axiom,
! [R3: nat,N: nat] :
( ( finite_card_list_nat @ ( length_sum_set @ ( suc @ R3 ) @ N ) )
= ( groups3542108847815614940at_nat
@ ^ [I2: nat] : ( finite_card_list_nat @ ( length_sum_set @ R3 @ ( minus_minus_nat @ N @ I2 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% card_length_sum_set
thf(fact_715_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_716_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_717_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_718_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_719_card__eq__sum,axiom,
( finite_card_list_nat
= ( groups4396056296759096172at_nat
@ ^ [X4: list_nat] : one_one_nat ) ) ).
% card_eq_sum
thf(fact_720_card__eq__sum,axiom,
( finite_card_nat
= ( groups3542108847815614940at_nat
@ ^ [X4: nat] : one_one_nat ) ) ).
% card_eq_sum
thf(fact_721_card__length,axiom,
! [Xs2: list_list_nat] : ( ord_less_eq_nat @ ( finite_card_list_nat @ ( set_list_nat2 @ Xs2 ) ) @ ( size_s3023201423986296836st_nat @ Xs2 ) ) ).
% card_length
thf(fact_722_card__length,axiom,
! [Xs2: list_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( set_nat2 @ Xs2 ) ) @ ( size_size_list_nat @ Xs2 ) ) ).
% card_length
thf(fact_723_nle__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B ) )
= ( ( ord_less_eq_nat @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_724_le__cases3,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z4 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z4 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z4 )
=> ~ ( ord_less_eq_nat @ Z4 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z4 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z4 )
=> ~ ( ord_less_eq_nat @ Z4 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z4 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_725_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_726_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat,Z3: set_nat] : ( Y5 = Z3 ) )
= ( ^ [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
& ( ord_less_eq_set_nat @ Y3 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_727_ord__eq__le__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_728_ord__eq__le__trans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( A2 = B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_729_ord__le__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_730_ord__le__eq__trans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_731_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_732_order__antisym,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_733_order_Otrans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_734_order_Otrans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_735_order__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z4 )
=> ( ord_less_eq_nat @ X @ Z4 ) ) ) ).
% order_trans
thf(fact_736_order__trans,axiom,
! [X: set_nat,Y: set_nat,Z4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z4 )
=> ( ord_less_eq_set_nat @ X @ Z4 ) ) ) ).
% order_trans
thf(fact_737_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: nat,B2: nat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_738_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_739_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_nat,Z3: set_nat] : ( Y5 = Z3 ) )
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A5 )
& ( ord_less_eq_set_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_740_dual__order_Oantisym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_741_dual__order_Oantisym,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_742_dual__order_Otrans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_743_dual__order_Otrans,axiom,
! [B: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_744_antisym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_745_antisym,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_746_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_747_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat,Z3: set_nat] : ( Y5 = Z3 ) )
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_748_order__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_749_order__subst1,axiom,
! [A2: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_750_order__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_751_order__subst1,axiom,
! [A2: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_752_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_753_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_754_order__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_755_order__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_756_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_757_order__eq__refl,axiom,
! [X: set_nat,Y: set_nat] :
( ( X = Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_758_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_759_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_760_ord__eq__le__subst,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_761_ord__eq__le__subst,axiom,
! [A2: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_762_ord__eq__le__subst,axiom,
! [A2: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_763_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_764_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_765_ord__le__eq__subst,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_766_ord__le__eq__subst,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_767_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_768_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_769_order__antisym__conv,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_770_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_771_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_772_order_Oasym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order.asym
thf(fact_773_ord__eq__less__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_774_ord__less__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_775_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X2: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X2 )
=> ( P @ Y4 ) )
=> ( P @ X2 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_776_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_777_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_778_dual__order_Oasym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ~ ( ord_less_nat @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_779_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_780_exists__least__iff,axiom,
( ( ^ [P4: nat > $o] :
? [X6: nat] : ( P4 @ X6 ) )
= ( ^ [P3: nat > $o] :
? [N4: nat] :
( ( P3 @ N4 )
& ! [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
=> ~ ( P3 @ M5 ) ) ) ) ) ).
% exists_least_iff
thf(fact_781_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B2: nat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_782_order_Ostrict__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_783_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_784_dual__order_Ostrict__trans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_785_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_786_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_787_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_788_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_789_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_790_order__less__asym_H,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_791_order__less__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z4 )
=> ( ord_less_nat @ X @ Z4 ) ) ) ).
% order_less_trans
thf(fact_792_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_793_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_794_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_795_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_796_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_797_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_798_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_799_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_800_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_801_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_802_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_803_distinct__card,axiom,
! [Xs2: list_list_nat] :
( ( distinct_list_nat @ Xs2 )
=> ( ( finite_card_list_nat @ ( set_list_nat2 @ Xs2 ) )
= ( size_s3023201423986296836st_nat @ Xs2 ) ) ) ).
% distinct_card
thf(fact_804_distinct__card,axiom,
! [Xs2: list_nat] :
( ( distinct_nat @ Xs2 )
=> ( ( finite_card_nat @ ( set_nat2 @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ) ).
% distinct_card
thf(fact_805_card__distinct,axiom,
! [Xs2: list_list_nat] :
( ( ( finite_card_list_nat @ ( set_list_nat2 @ Xs2 ) )
= ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( distinct_list_nat @ Xs2 ) ) ).
% card_distinct
thf(fact_806_card__distinct,axiom,
! [Xs2: list_nat] :
( ( ( finite_card_nat @ ( set_nat2 @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) )
=> ( distinct_nat @ Xs2 ) ) ).
% card_distinct
thf(fact_807_sum__Suc,axiom,
! [F: list_nat > nat,A: set_list_nat] :
( ( groups4396056296759096172at_nat
@ ^ [X4: list_nat] : ( suc @ ( F @ X4 ) )
@ A )
= ( plus_plus_nat @ ( groups4396056296759096172at_nat @ F @ A ) @ ( finite_card_list_nat @ A ) ) ) ).
% sum_Suc
thf(fact_808_sum__Suc,axiom,
! [F: nat > nat,A: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [X4: nat] : ( suc @ ( F @ X4 ) )
@ A )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( finite_card_nat @ A ) ) ) ).
% sum_Suc
thf(fact_809_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_810_leD,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ~ ( ord_less_set_nat @ X @ Y ) ) ).
% leD
thf(fact_811_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_812_nless__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_nat @ A2 @ B ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_813_nless__le,axiom,
! [A2: set_nat,B: set_nat] :
( ( ~ ( ord_less_set_nat @ A2 @ B ) )
= ( ~ ( ord_less_eq_set_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_814_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_815_antisym__conv1,axiom,
! [X: set_nat,Y: set_nat] :
( ~ ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_816_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_817_antisym__conv2,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ~ ( ord_less_set_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_818_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
& ~ ( ord_less_eq_nat @ Y3 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_819_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
& ~ ( ord_less_eq_set_nat @ Y3 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_820_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_821_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_nat @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_822_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_set_nat @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_823_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_824_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_825_order_Ostrict__trans1,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_826_order_Ostrict__trans1,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_827_order_Ostrict__trans2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_828_order_Ostrict__trans2,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_829_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ~ ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_830_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ~ ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_831_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_nat @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_832_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B5: set_nat,A5: set_nat] :
( ( ord_less_set_nat @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_833_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_834_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B5: set_nat,A5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_835_dual__order_Ostrict__trans1,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_836_dual__order_Ostrict__trans1,axiom,
! [B: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_837_dual__order_Ostrict__trans2,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_838_dual__order_Ostrict__trans2,axiom,
! [B: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_839_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_840_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B5: set_nat,A5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A5 )
& ~ ( ord_less_eq_set_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_841_order_Ostrict__implies__order,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_842_order_Ostrict__implies__order,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_843_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_844_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B @ A2 )
=> ( ord_less_eq_set_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_845_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_846_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X4: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_847_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
& ( X4 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_848_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X4: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y3 )
& ( X4 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_849_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_850_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_851_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_852_order__less__imp__le,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_853_order__le__neq__trans,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_854_order__le__neq__trans,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_set_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_855_order__neq__le__trans,axiom,
! [A2: nat,B: nat] :
( ( A2 != B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_856_order__neq__le__trans,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 != B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_set_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_857_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z4 )
=> ( ord_less_nat @ X @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_858_order__le__less__trans,axiom,
! [X: set_nat,Y: set_nat,Z4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ Y @ Z4 )
=> ( ord_less_set_nat @ X @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_859_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z4 )
=> ( ord_less_nat @ X @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_860_order__less__le__trans,axiom,
! [X: set_nat,Y: set_nat,Z4: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z4 )
=> ( ord_less_set_nat @ X @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_861_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_862_order__le__less__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_863_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_864_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_865_order__le__less__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_866_order__le__less__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_867_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_868_order__less__le__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_869_order__less__le__subst1,axiom,
! [A2: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_870_order__less__le__subst1,axiom,
! [A2: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_871_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_872_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_873_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_874_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_875_order__le__imp__less__or__eq,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_876_card__length__sum__list__rec,axiom,
! [M: nat,N5: nat] :
( ( ord_less_eq_nat @ one_one_nat @ M )
=> ( ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L3: list_nat] :
( ( ( size_size_list_nat @ L3 )
= M )
& ( ( groups4561878855575611511st_nat @ L3 )
= N5 ) ) ) )
= ( plus_plus_nat
@ ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L3: list_nat] :
( ( ( size_size_list_nat @ L3 )
= ( minus_minus_nat @ M @ one_one_nat ) )
& ( ( groups4561878855575611511st_nat @ L3 )
= N5 ) ) ) )
@ ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L3: list_nat] :
( ( ( size_size_list_nat @ L3 )
= M )
& ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L3 ) @ one_one_nat )
= N5 ) ) ) ) ) ) ) ).
% card_length_sum_list_rec
thf(fact_877_card__lists__distinct__length__eq_H,axiom,
! [K: nat,A: set_list_nat] :
( ( ord_less_nat @ K @ ( finite_card_list_nat @ A ) )
=> ( ( finite7325466520557071688st_nat
@ ( collec5989764272469232197st_nat
@ ^ [Xs: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= K )
& ( distinct_list_nat @ Xs )
& ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ A ) ) ) )
= ( groups708209901874060359at_nat
@ ^ [X4: nat] : X4
@ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite_card_list_nat @ A ) @ K ) @ one_one_nat ) @ ( finite_card_list_nat @ A ) ) ) ) ) ).
% card_lists_distinct_length_eq'
thf(fact_878_card__lists__distinct__length__eq_H,axiom,
! [K: nat,A: set_nat] :
( ( ord_less_nat @ K @ ( finite_card_nat @ A ) )
=> ( ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= K )
& ( distinct_nat @ Xs )
& ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A ) ) ) )
= ( groups708209901874060359at_nat
@ ^ [X4: nat] : X4
@ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite_card_nat @ A ) @ K ) @ one_one_nat ) @ ( finite_card_nat @ A ) ) ) ) ) ).
% card_lists_distinct_length_eq'
thf(fact_879_length__Cons,axiom,
! [X: nat,Xs2: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X @ Xs2 ) )
= ( suc @ ( size_size_list_nat @ Xs2 ) ) ) ).
% length_Cons
thf(fact_880_ex__card,axiom,
! [N: nat,A: set_list_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_list_nat @ A ) )
=> ? [S3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ S3 @ A )
& ( ( finite_card_list_nat @ S3 )
= N ) ) ) ).
% ex_card
thf(fact_881_ex__card,axiom,
! [N: nat,A: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ A ) )
=> ? [S3: set_nat] :
( ( ord_less_eq_set_nat @ S3 @ A )
& ( ( finite_card_nat @ S3 )
= N ) ) ) ).
% ex_card
thf(fact_882_card__lessThan,axiom,
! [U: nat] :
( ( finite_card_nat @ ( set_ord_lessThan_nat @ U ) )
= U ) ).
% card_lessThan
thf(fact_883_card__atMost,axiom,
! [U: nat] :
( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
= ( suc @ U ) ) ).
% card_atMost
thf(fact_884_card__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L @ U ) )
= ( minus_minus_nat @ ( suc @ U ) @ L ) ) ).
% card_atLeastAtMost
thf(fact_885_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: nat > nat,A: set_nat] :
( ( ( groups708209901874060359at_nat @ G @ A )
!= one_one_nat )
=> ~ ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ( ( G @ A3 )
= one_one_nat ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_886_prod_Onot__neutral__contains__not__neutral,axiom,
! [G: list_nat > nat,A: set_list_nat] :
( ( ( groups2907647131375434839at_nat @ G @ A )
!= one_one_nat )
=> ~ ! [A3: list_nat] :
( ( member_list_nat @ A3 @ A )
=> ( ( G @ A3 )
= one_one_nat ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_887_prod__mono,axiom,
! [A: set_nat,F: nat > nat,G: nat > nat] :
( ! [I3: nat] :
( ( member_nat @ I3 @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
& ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ord_less_eq_nat @ ( groups708209901874060359at_nat @ F @ A ) @ ( groups708209901874060359at_nat @ G @ A ) ) ) ).
% prod_mono
thf(fact_888_prod__mono,axiom,
! [A: set_list_nat,F: list_nat > nat,G: list_nat > nat] :
( ! [I3: list_nat] :
( ( member_list_nat @ I3 @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) )
& ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ord_less_eq_nat @ ( groups2907647131375434839at_nat @ F @ A ) @ ( groups2907647131375434839at_nat @ G @ A ) ) ) ).
% prod_mono
thf(fact_889_prod__ge__1,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( F @ X2 ) ) )
=> ( ord_less_eq_nat @ one_one_nat @ ( groups708209901874060359at_nat @ F @ A ) ) ) ).
% prod_ge_1
thf(fact_890_prod__ge__1,axiom,
! [A: set_list_nat,F: list_nat > nat] :
( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( F @ X2 ) ) )
=> ( ord_less_eq_nat @ one_one_nat @ ( groups2907647131375434839at_nat @ F @ A ) ) ) ).
% prod_ge_1
thf(fact_891_prod__le__1,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) )
& ( ord_less_eq_nat @ ( F @ X2 ) @ one_one_nat ) ) )
=> ( ord_less_eq_nat @ ( groups708209901874060359at_nat @ F @ A ) @ one_one_nat ) ) ).
% prod_le_1
thf(fact_892_prod__le__1,axiom,
! [A: set_list_nat,F: list_nat > nat] :
( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) )
& ( ord_less_eq_nat @ ( F @ X2 ) @ one_one_nat ) ) )
=> ( ord_less_eq_nat @ ( groups2907647131375434839at_nat @ F @ A ) @ one_one_nat ) ) ).
% prod_le_1
thf(fact_893_card__less,axiom,
! [M6: set_nat,I: nat] :
( ( member_nat @ zero_zero_nat @ M6 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K4: nat] :
( ( member_nat @ K4 @ M6 )
& ( ord_less_nat @ K4 @ ( suc @ I ) ) ) ) )
!= zero_zero_nat ) ) ).
% card_less
thf(fact_894_card__less__Suc,axiom,
! [M6: set_nat,I: nat] :
( ( member_nat @ zero_zero_nat @ M6 )
=> ( ( suc
@ ( finite_card_nat
@ ( collect_nat
@ ^ [K4: nat] :
( ( member_nat @ ( suc @ K4 ) @ M6 )
& ( ord_less_nat @ K4 @ I ) ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K4: nat] :
( ( member_nat @ K4 @ M6 )
& ( ord_less_nat @ K4 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_895_card__less__Suc2,axiom,
! [M6: set_nat,I: nat] :
( ~ ( member_nat @ zero_zero_nat @ M6 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K4: nat] :
( ( member_nat @ ( suc @ K4 ) @ M6 )
& ( ord_less_nat @ K4 @ I ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K4: nat] :
( ( member_nat @ K4 @ M6 )
& ( ord_less_nat @ K4 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_896_subset__CollectI,axiom,
! [B3: set_list_nat,A: set_list_nat,Q: list_nat > $o,P: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ B3 @ A )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ B3 )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X4: list_nat] :
( ( member_list_nat @ X4 @ B3 )
& ( Q @ X4 ) ) )
@ ( collect_list_nat
@ ^ [X4: list_nat] :
( ( member_list_nat @ X4 @ A )
& ( P @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_897_subset__CollectI,axiom,
! [B3: set_nat,A: set_nat,Q: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B3 @ A )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ B3 )
& ( Q @ X4 ) ) )
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_898_subset__Collect__iff,axiom,
! [B3: set_list_nat,A: set_list_nat,P: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ B3 @ A )
=> ( ( ord_le6045566169113846134st_nat @ B3
@ ( collect_list_nat
@ ^ [X4: list_nat] :
( ( member_list_nat @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ B3 )
=> ( P @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_899_subset__Collect__iff,axiom,
! [B3: set_nat,A: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ B3 @ A )
=> ( ( ord_less_eq_set_nat @ B3
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ B3 )
=> ( P @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_900_prod_Ozero__middle,axiom,
! [P2: nat,K: nat,G: nat > nat,H: nat > nat] :
( ( ord_less_eq_nat @ one_one_nat @ P2 )
=> ( ( ord_less_eq_nat @ K @ P2 )
=> ( ( groups708209901874060359at_nat
@ ^ [J: nat] : ( if_nat @ ( ord_less_nat @ J @ K ) @ ( G @ J ) @ ( if_nat @ ( J = K ) @ one_one_nat @ ( H @ ( minus_minus_nat @ J @ ( suc @ zero_zero_nat ) ) ) ) )
@ ( set_ord_atMost_nat @ P2 ) )
= ( groups708209901874060359at_nat
@ ^ [J: nat] : ( if_nat @ ( ord_less_nat @ J @ K ) @ ( G @ J ) @ ( H @ J ) )
@ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_901_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_eq_nat @ I2 @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_902_card__lists__distinct__length__eq,axiom,
! [A: set_list_nat,K: nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( ( ord_less_eq_nat @ K @ ( finite_card_list_nat @ A ) )
=> ( ( finite7325466520557071688st_nat
@ ( collec5989764272469232197st_nat
@ ^ [Xs: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= K )
& ( distinct_list_nat @ Xs )
& ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ A ) ) ) )
= ( groups708209901874060359at_nat
@ ^ [X4: nat] : X4
@ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite_card_list_nat @ A ) @ K ) @ one_one_nat ) @ ( finite_card_list_nat @ A ) ) ) ) ) ) ).
% card_lists_distinct_length_eq
thf(fact_903_card__lists__distinct__length__eq,axiom,
! [A: set_nat,K: nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_nat @ K @ ( finite_card_nat @ A ) )
=> ( ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= K )
& ( distinct_nat @ Xs )
& ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A ) ) ) )
= ( groups708209901874060359at_nat
@ ^ [X4: nat] : X4
@ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite_card_nat @ A ) @ K ) @ one_one_nat ) @ ( finite_card_nat @ A ) ) ) ) ) ) ).
% card_lists_distinct_length_eq
thf(fact_904_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_905_finite__length__sum__set,axiom,
! [R3: nat,N: nat] : ( finite8100373058378681591st_nat @ ( length_sum_set @ R3 @ N ) ) ).
% finite_length_sum_set
thf(fact_906_minimal__elements__set__tuples__finite,axiom,
! [U4: set_list_nat,R3: nat] :
( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ U4 )
=> ( ( size_size_list_nat @ X2 )
= R3 ) )
=> ( finite8100373058378681591st_nat @ ( minimal_elements @ U4 ) ) ) ).
% minimal_elements_set_tuples_finite
thf(fact_907_finite__atLeastAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% finite_atLeastAtMost
thf(fact_908_finite__lessThan,axiom,
! [K: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K ) ) ).
% finite_lessThan
thf(fact_909_finite__atMost,axiom,
! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).
% finite_atMost
thf(fact_910_finite__Collect__conjI,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
| ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [X4: list_nat] :
( ( P @ X4 )
& ( Q @ X4 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_911_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( P @ X4 )
& ( Q @ X4 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_912_finite__Collect__disjI,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [X4: list_nat] :
( ( P @ X4 )
| ( Q @ X4 ) ) ) )
= ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
& ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_913_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X4: nat] :
( ( P @ X4 )
| ( Q @ X4 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_914_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ N4 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_915_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_nat @ N4 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_916_List_Ofinite__set,axiom,
! [Xs2: list_list_nat] : ( finite8100373058378681591st_nat @ ( set_list_nat2 @ Xs2 ) ) ).
% List.finite_set
thf(fact_917_List_Ofinite__set,axiom,
! [Xs2: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs2 ) ) ).
% List.finite_set
thf(fact_918_finite__Collect__subsets,axiom,
! [A: set_list_nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( finite7047420756378620717st_nat
@ ( collect_set_list_nat
@ ^ [B4: set_list_nat] : ( ord_le6045566169113846134st_nat @ B4 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_919_finite__Collect__subsets,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B4: set_nat] : ( ord_less_eq_set_nat @ B4 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_920_sum__eq__0__iff,axiom,
! [F2: set_list_nat,F: list_nat > nat] :
( ( finite8100373058378681591st_nat @ F2 )
=> ( ( ( groups4396056296759096172at_nat @ F @ F2 )
= zero_zero_nat )
= ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ F2 )
=> ( ( F @ X4 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_921_sum__eq__0__iff,axiom,
! [F2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ F2 )
=> ( ( ( groups3542108847815614940at_nat @ F @ F2 )
= zero_zero_nat )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ F2 )
=> ( ( F @ X4 )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_922_sum_Oinfinite,axiom,
! [A: set_list_nat,G: list_nat > nat] :
( ~ ( finite8100373058378681591st_nat @ A )
=> ( ( groups4396056296759096172at_nat @ G @ A )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_923_sum_Oinfinite,axiom,
! [A: set_nat,G: nat > nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( groups3542108847815614940at_nat @ G @ A )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_924_prod__zero__iff,axiom,
! [A: set_list_nat,F: list_nat > nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( ( ( groups2907647131375434839at_nat @ F @ A )
= zero_zero_nat )
= ( ? [X4: list_nat] :
( ( member_list_nat @ X4 @ A )
& ( ( F @ X4 )
= zero_zero_nat ) ) ) ) ) ).
% prod_zero_iff
thf(fact_925_prod__zero__iff,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ( groups708209901874060359at_nat @ F @ A )
= zero_zero_nat )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ( F @ X4 )
= zero_zero_nat ) ) ) ) ) ).
% prod_zero_iff
thf(fact_926_card_Oinfinite,axiom,
! [A: set_list_nat] :
( ~ ( finite8100373058378681591st_nat @ A )
=> ( ( finite_card_list_nat @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_927_card_Oinfinite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_card_nat @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_928_prod_Oinfinite,axiom,
! [A: set_list_nat,G: list_nat > nat] :
( ~ ( finite8100373058378681591st_nat @ A )
=> ( ( groups2907647131375434839at_nat @ G @ A )
= one_one_nat ) ) ).
% prod.infinite
thf(fact_929_prod_Oinfinite,axiom,
! [A: set_nat,G: nat > nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( groups708209901874060359at_nat @ G @ A )
= one_one_nat ) ) ).
% prod.infinite
thf(fact_930_sum_Odelta_H,axiom,
! [S: set_list_nat,A2: list_nat,B: list_nat > nat] :
( ( finite8100373058378681591st_nat @ S )
=> ( ( ( member_list_nat @ A2 @ S )
=> ( ( groups4396056296759096172at_nat
@ ^ [K4: list_nat] : ( if_nat @ ( A2 = K4 ) @ ( B @ K4 ) @ zero_zero_nat )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_list_nat @ A2 @ S )
=> ( ( groups4396056296759096172at_nat
@ ^ [K4: list_nat] : ( if_nat @ ( A2 = K4 ) @ ( B @ K4 ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_931_sum_Odelta_H,axiom,
! [S: set_nat,A2: nat,B: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A2 @ S )
=> ( ( groups3542108847815614940at_nat
@ ^ [K4: nat] : ( if_nat @ ( A2 = K4 ) @ ( B @ K4 ) @ zero_zero_nat )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_nat @ A2 @ S )
=> ( ( groups3542108847815614940at_nat
@ ^ [K4: nat] : ( if_nat @ ( A2 = K4 ) @ ( B @ K4 ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_932_sum_Odelta,axiom,
! [S: set_list_nat,A2: list_nat,B: list_nat > nat] :
( ( finite8100373058378681591st_nat @ S )
=> ( ( ( member_list_nat @ A2 @ S )
=> ( ( groups4396056296759096172at_nat
@ ^ [K4: list_nat] : ( if_nat @ ( K4 = A2 ) @ ( B @ K4 ) @ zero_zero_nat )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_list_nat @ A2 @ S )
=> ( ( groups4396056296759096172at_nat
@ ^ [K4: list_nat] : ( if_nat @ ( K4 = A2 ) @ ( B @ K4 ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_933_sum_Odelta,axiom,
! [S: set_nat,A2: nat,B: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A2 @ S )
=> ( ( groups3542108847815614940at_nat
@ ^ [K4: nat] : ( if_nat @ ( K4 = A2 ) @ ( B @ K4 ) @ zero_zero_nat )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_nat @ A2 @ S )
=> ( ( groups3542108847815614940at_nat
@ ^ [K4: nat] : ( if_nat @ ( K4 = A2 ) @ ( B @ K4 ) @ zero_zero_nat )
@ S )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_934_prod__eq__1__iff,axiom,
! [A: set_list_nat,F: list_nat > nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( ( ( groups2907647131375434839at_nat @ F @ A )
= one_one_nat )
= ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ A )
=> ( ( F @ X4 )
= one_one_nat ) ) ) ) ) ).
% prod_eq_1_iff
thf(fact_935_prod__eq__1__iff,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ( groups708209901874060359at_nat @ F @ A )
= one_one_nat )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ( F @ X4 )
= one_one_nat ) ) ) ) ) ).
% prod_eq_1_iff
thf(fact_936_prod_Odelta,axiom,
! [S: set_list_nat,A2: list_nat,B: list_nat > nat] :
( ( finite8100373058378681591st_nat @ S )
=> ( ( ( member_list_nat @ A2 @ S )
=> ( ( groups2907647131375434839at_nat
@ ^ [K4: list_nat] : ( if_nat @ ( K4 = A2 ) @ ( B @ K4 ) @ one_one_nat )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_list_nat @ A2 @ S )
=> ( ( groups2907647131375434839at_nat
@ ^ [K4: list_nat] : ( if_nat @ ( K4 = A2 ) @ ( B @ K4 ) @ one_one_nat )
@ S )
= one_one_nat ) ) ) ) ).
% prod.delta
thf(fact_937_prod_Odelta,axiom,
! [S: set_nat,A2: nat,B: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A2 @ S )
=> ( ( groups708209901874060359at_nat
@ ^ [K4: nat] : ( if_nat @ ( K4 = A2 ) @ ( B @ K4 ) @ one_one_nat )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_nat @ A2 @ S )
=> ( ( groups708209901874060359at_nat
@ ^ [K4: nat] : ( if_nat @ ( K4 = A2 ) @ ( B @ K4 ) @ one_one_nat )
@ S )
= one_one_nat ) ) ) ) ).
% prod.delta
thf(fact_938_prod_Odelta_H,axiom,
! [S: set_list_nat,A2: list_nat,B: list_nat > nat] :
( ( finite8100373058378681591st_nat @ S )
=> ( ( ( member_list_nat @ A2 @ S )
=> ( ( groups2907647131375434839at_nat
@ ^ [K4: list_nat] : ( if_nat @ ( A2 = K4 ) @ ( B @ K4 ) @ one_one_nat )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_list_nat @ A2 @ S )
=> ( ( groups2907647131375434839at_nat
@ ^ [K4: list_nat] : ( if_nat @ ( A2 = K4 ) @ ( B @ K4 ) @ one_one_nat )
@ S )
= one_one_nat ) ) ) ) ).
% prod.delta'
thf(fact_939_prod_Odelta_H,axiom,
! [S: set_nat,A2: nat,B: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( ( member_nat @ A2 @ S )
=> ( ( groups708209901874060359at_nat
@ ^ [K4: nat] : ( if_nat @ ( A2 = K4 ) @ ( B @ K4 ) @ one_one_nat )
@ S )
= ( B @ A2 ) ) )
& ( ~ ( member_nat @ A2 @ S )
=> ( ( groups708209901874060359at_nat
@ ^ [K4: nat] : ( if_nat @ ( A2 = K4 ) @ ( B @ K4 ) @ one_one_nat )
@ S )
= one_one_nat ) ) ) ) ).
% prod.delta'
thf(fact_940_prod__pos__nat__iff,axiom,
! [A: set_list_nat,F: list_nat > nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ ( groups2907647131375434839at_nat @ F @ A ) )
= ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) ) ) ) ) ).
% prod_pos_nat_iff
thf(fact_941_prod__pos__nat__iff,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) ) ) ) ) ).
% prod_pos_nat_iff
thf(fact_942_finite__lists__distinct__length__eq,axiom,
! [A: set_list_nat,N: nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( finite8170528100393595399st_nat
@ ( collec5989764272469232197st_nat
@ ^ [Xs: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= N )
& ( distinct_list_nat @ Xs )
& ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ A ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_943_finite__lists__distinct__length__eq,axiom,
! [A: set_nat,N: nat] :
( ( finite_finite_nat @ A )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= N )
& ( distinct_nat @ Xs )
& ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_944_card__subset__eq,axiom,
! [B3: set_list_nat,A: set_list_nat] :
( ( finite8100373058378681591st_nat @ B3 )
=> ( ( ord_le6045566169113846134st_nat @ A @ B3 )
=> ( ( ( finite_card_list_nat @ A )
= ( finite_card_list_nat @ B3 ) )
=> ( A = B3 ) ) ) ) ).
% card_subset_eq
thf(fact_945_card__subset__eq,axiom,
! [B3: set_nat,A: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( ( finite_card_nat @ A )
= ( finite_card_nat @ B3 ) )
=> ( A = B3 ) ) ) ) ).
% card_subset_eq
thf(fact_946_infinite__arbitrarily__large,axiom,
! [A: set_list_nat,N: nat] :
( ~ ( finite8100373058378681591st_nat @ A )
=> ? [B6: set_list_nat] :
( ( finite8100373058378681591st_nat @ B6 )
& ( ( finite_card_list_nat @ B6 )
= N )
& ( ord_le6045566169113846134st_nat @ B6 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_947_infinite__arbitrarily__large,axiom,
! [A: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A )
=> ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ( finite_card_nat @ B6 )
= N )
& ( ord_less_eq_set_nat @ B6 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_948_prod__zero,axiom,
! [A: set_list_nat,F: list_nat > nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( ? [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
& ( ( F @ X3 )
= zero_zero_nat ) )
=> ( ( groups2907647131375434839at_nat @ F @ A )
= zero_zero_nat ) ) ) ).
% prod_zero
thf(fact_949_prod__zero,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ( F @ X3 )
= zero_zero_nat ) )
=> ( ( groups708209901874060359at_nat @ F @ A )
= zero_zero_nat ) ) ) ).
% prod_zero
thf(fact_950_finite__lists__length__eq,axiom,
! [A: set_list_nat,N: nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( finite8170528100393595399st_nat
@ ( collec5989764272469232197st_nat
@ ^ [Xs: list_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ A )
& ( ( size_s3023201423986296836st_nat @ Xs )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_951_finite__lists__length__eq,axiom,
! [A: set_nat,N: nat] :
( ( finite_finite_nat @ A )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A )
& ( ( size_size_list_nat @ Xs )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_952_finite__psubset__induct,axiom,
! [A: set_list_nat,P: set_list_nat > $o] :
( ( finite8100373058378681591st_nat @ A )
=> ( ! [A6: set_list_nat] :
( ( finite8100373058378681591st_nat @ A6 )
=> ( ! [B7: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ B7 @ A6 )
=> ( P @ B7 ) )
=> ( P @ A6 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_953_finite__psubset__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ! [B7: set_nat] :
( ( ord_less_set_nat @ B7 @ A6 )
=> ( P @ B7 ) )
=> ( P @ A6 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_954_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K4: nat] :
( ( P @ K4 )
& ( ord_less_nat @ K4 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_955_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_956_bounded__nat__set__is__finite,axiom,
! [N5: set_nat,N: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ N5 )
=> ( ord_less_nat @ X2 @ N ) )
=> ( finite_finite_nat @ N5 ) ) ).
% bounded_nat_set_is_finite
thf(fact_957_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N6: set_nat] :
? [M5: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ N6 )
=> ( ord_less_nat @ X4 @ M5 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_958_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N6: set_nat] :
? [M5: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ N6 )
=> ( ord_less_eq_nat @ X4 @ M5 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_959_rev__finite__subset,axiom,
! [B3: set_list_nat,A: set_list_nat] :
( ( finite8100373058378681591st_nat @ B3 )
=> ( ( ord_le6045566169113846134st_nat @ A @ B3 )
=> ( finite8100373058378681591st_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_960_rev__finite__subset,axiom,
! [B3: set_nat,A: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ A @ B3 )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_961_infinite__super,axiom,
! [S: set_list_nat,T: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ S @ T )
=> ( ~ ( finite8100373058378681591st_nat @ S )
=> ~ ( finite8100373058378681591st_nat @ T ) ) ) ).
% infinite_super
thf(fact_962_infinite__super,axiom,
! [S: set_nat,T: set_nat] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_super
thf(fact_963_finite__subset,axiom,
! [A: set_list_nat,B3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B3 )
=> ( ( finite8100373058378681591st_nat @ B3 )
=> ( finite8100373058378681591st_nat @ A ) ) ) ).
% finite_subset
thf(fact_964_finite__subset,axiom,
! [A: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( finite_finite_nat @ B3 )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_965_finite__has__minimal2,axiom,
! [A: set_list_nat,A2: list_nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( ( member_list_nat @ A2 @ A )
=> ? [X2: list_nat] :
( ( member_list_nat @ X2 @ A )
& ( ord_less_eq_list_nat @ X2 @ A2 )
& ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ A )
=> ( ( ord_less_eq_list_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_966_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ X2 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_967_finite__has__minimal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ( ord_less_eq_set_nat @ X2 @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_968_finite__has__maximal2,axiom,
! [A: set_list_nat,A2: list_nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( ( member_list_nat @ A2 @ A )
=> ? [X2: list_nat] :
( ( member_list_nat @ X2 @ A )
& ( ord_less_eq_list_nat @ A2 @ X2 )
& ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ A )
=> ( ( ord_less_eq_list_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_969_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ A2 @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_970_finite__has__maximal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ( ord_less_eq_set_nat @ A2 @ X2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_971_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_list_nat,C4: nat] :
( ! [G2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ G2 @ F2 )
=> ( ( finite8100373058378681591st_nat @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_list_nat @ G2 ) @ C4 ) ) )
=> ( ( finite8100373058378681591st_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_list_nat @ F2 ) @ C4 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_972_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_nat,C4: nat] :
( ! [G2: set_nat] :
( ( ord_less_eq_set_nat @ G2 @ F2 )
=> ( ( finite_finite_nat @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G2 ) @ C4 ) ) )
=> ( ( finite_finite_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C4 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_973_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_list_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_list_nat @ S ) )
=> ~ ! [T4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ T4 @ S )
=> ( ( ( finite_card_list_nat @ T4 )
= N )
=> ~ ( finite8100373058378681591st_nat @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_974_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S ) )
=> ~ ! [T4: set_nat] :
( ( ord_less_eq_set_nat @ T4 @ S )
=> ( ( ( finite_card_nat @ T4 )
= N )
=> ~ ( finite_finite_nat @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_975_exists__subset__between,axiom,
! [A: set_list_nat,N: nat,C4: set_list_nat] :
( ( ord_less_eq_nat @ ( finite_card_list_nat @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_list_nat @ C4 ) )
=> ( ( ord_le6045566169113846134st_nat @ A @ C4 )
=> ( ( finite8100373058378681591st_nat @ C4 )
=> ? [B6: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B6 )
& ( ord_le6045566169113846134st_nat @ B6 @ C4 )
& ( ( finite_card_list_nat @ B6 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_976_exists__subset__between,axiom,
! [A: set_nat,N: nat,C4: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C4 ) )
=> ( ( ord_less_eq_set_nat @ A @ C4 )
=> ( ( finite_finite_nat @ C4 )
=> ? [B6: set_nat] :
( ( ord_less_eq_set_nat @ A @ B6 )
& ( ord_less_eq_set_nat @ B6 @ C4 )
& ( ( finite_card_nat @ B6 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_977_card__seteq,axiom,
! [B3: set_list_nat,A: set_list_nat] :
( ( finite8100373058378681591st_nat @ B3 )
=> ( ( ord_le6045566169113846134st_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_list_nat @ B3 ) @ ( finite_card_list_nat @ A ) )
=> ( A = B3 ) ) ) ) ).
% card_seteq
thf(fact_978_card__seteq,axiom,
! [B3: set_nat,A: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B3 ) @ ( finite_card_nat @ A ) )
=> ( A = B3 ) ) ) ) ).
% card_seteq
thf(fact_979_card__mono,axiom,
! [B3: set_list_nat,A: set_list_nat] :
( ( finite8100373058378681591st_nat @ B3 )
=> ( ( ord_le6045566169113846134st_nat @ A @ B3 )
=> ( ord_less_eq_nat @ ( finite_card_list_nat @ A ) @ ( finite_card_list_nat @ B3 ) ) ) ) ).
% card_mono
thf(fact_980_card__mono,axiom,
! [B3: set_nat,A: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B3 ) ) ) ) ).
% card_mono
thf(fact_981_sum__mono__inv,axiom,
! [F: list_nat > nat,I5: set_list_nat,G: list_nat > nat,I: list_nat] :
( ( ( groups4396056296759096172at_nat @ F @ I5 )
= ( groups4396056296759096172at_nat @ G @ I5 ) )
=> ( ! [I3: list_nat] :
( ( member_list_nat @ I3 @ I5 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_list_nat @ I @ I5 )
=> ( ( finite8100373058378681591st_nat @ I5 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_982_sum__mono__inv,axiom,
! [F: nat > nat,I5: set_nat,G: nat > nat,I: nat] :
( ( ( groups3542108847815614940at_nat @ F @ I5 )
= ( groups3542108847815614940at_nat @ G @ I5 ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member_nat @ I @ I5 )
=> ( ( finite_finite_nat @ I5 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_983_card__ge__0__finite,axiom,
! [A: set_list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_list_nat @ A ) )
=> ( finite8100373058378681591st_nat @ A ) ) ).
% card_ge_0_finite
thf(fact_984_card__ge__0__finite,axiom,
! [A: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A ) )
=> ( finite_finite_nat @ A ) ) ).
% card_ge_0_finite
thf(fact_985_card__le__sym__Diff,axiom,
! [A: set_list_nat,B3: set_list_nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( ( finite8100373058378681591st_nat @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_list_nat @ A ) @ ( finite_card_list_nat @ B3 ) )
=> ( ord_less_eq_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A @ B3 ) ) @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ B3 @ A ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_986_card__le__sym__Diff,axiom,
! [A: set_nat,B3: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B3 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B3 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B3 @ A ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_987_card__less__sym__Diff,axiom,
! [A: set_list_nat,B3: set_list_nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( ( finite8100373058378681591st_nat @ B3 )
=> ( ( ord_less_nat @ ( finite_card_list_nat @ A ) @ ( finite_card_list_nat @ B3 ) )
=> ( ord_less_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A @ B3 ) ) @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ B3 @ A ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_988_card__less__sym__Diff,axiom,
! [A: set_nat,B3: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B3 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B3 ) )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B3 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B3 @ A ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_989_card__le__if__inj__on__rel,axiom,
! [B3: set_list_nat,A: set_list_nat,R3: list_nat > list_nat > $o] :
( ( finite8100373058378681591st_nat @ B3 )
=> ( ! [A3: list_nat] :
( ( member_list_nat @ A3 @ A )
=> ? [B8: list_nat] :
( ( member_list_nat @ B8 @ B3 )
& ( R3 @ A3 @ B8 ) ) )
=> ( ! [A1: list_nat,A22: list_nat,B2: list_nat] :
( ( member_list_nat @ A1 @ A )
=> ( ( member_list_nat @ A22 @ A )
=> ( ( member_list_nat @ B2 @ B3 )
=> ( ( R3 @ A1 @ B2 )
=> ( ( R3 @ A22 @ B2 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_list_nat @ A ) @ ( finite_card_list_nat @ B3 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_990_card__le__if__inj__on__rel,axiom,
! [B3: set_list_nat,A: set_nat,R3: nat > list_nat > $o] :
( ( finite8100373058378681591st_nat @ B3 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ? [B8: list_nat] :
( ( member_list_nat @ B8 @ B3 )
& ( R3 @ A3 @ B8 ) ) )
=> ( ! [A1: nat,A22: nat,B2: list_nat] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_list_nat @ B2 @ B3 )
=> ( ( R3 @ A1 @ B2 )
=> ( ( R3 @ A22 @ B2 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_list_nat @ B3 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_991_card__le__if__inj__on__rel,axiom,
! [B3: set_nat,A: set_list_nat,R3: list_nat > nat > $o] :
( ( finite_finite_nat @ B3 )
=> ( ! [A3: list_nat] :
( ( member_list_nat @ A3 @ A )
=> ? [B8: nat] :
( ( member_nat @ B8 @ B3 )
& ( R3 @ A3 @ B8 ) ) )
=> ( ! [A1: list_nat,A22: list_nat,B2: nat] :
( ( member_list_nat @ A1 @ A )
=> ( ( member_list_nat @ A22 @ A )
=> ( ( member_nat @ B2 @ B3 )
=> ( ( R3 @ A1 @ B2 )
=> ( ( R3 @ A22 @ B2 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_list_nat @ A ) @ ( finite_card_nat @ B3 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_992_card__le__if__inj__on__rel,axiom,
! [B3: set_nat,A: set_nat,R3: nat > nat > $o] :
( ( finite_finite_nat @ B3 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ? [B8: nat] :
( ( member_nat @ B8 @ B3 )
& ( R3 @ A3 @ B8 ) ) )
=> ( ! [A1: nat,A22: nat,B2: nat] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_nat @ B2 @ B3 )
=> ( ( R3 @ A1 @ B2 )
=> ( ( R3 @ A22 @ B2 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B3 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_993_finite__lists__length__le,axiom,
! [A: set_list_nat,N: nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( finite8170528100393595399st_nat
@ ( collec5989764272469232197st_nat
@ ^ [Xs: list_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ A )
& ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_994_finite__lists__length__le,axiom,
! [A: set_nat,N: nat] :
( ( finite_finite_nat @ A )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A )
& ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_995_psubset__card__mono,axiom,
! [B3: set_list_nat,A: set_list_nat] :
( ( finite8100373058378681591st_nat @ B3 )
=> ( ( ord_le1190675801316882794st_nat @ A @ B3 )
=> ( ord_less_nat @ ( finite_card_list_nat @ A ) @ ( finite_card_list_nat @ B3 ) ) ) ) ).
% psubset_card_mono
thf(fact_996_psubset__card__mono,axiom,
! [B3: set_nat,A: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_set_nat @ A @ B3 )
=> ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B3 ) ) ) ) ).
% psubset_card_mono
thf(fact_997_finite__maxlen,axiom,
! [M6: set_list_nat] :
( ( finite8100373058378681591st_nat @ M6 )
=> ? [N3: nat] :
! [X3: list_nat] :
( ( member_list_nat @ X3 @ M6 )
=> ( ord_less_nat @ ( size_size_list_nat @ X3 ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_998_not__finite__existsD,axiom,
! [P: list_nat > $o] :
( ~ ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
=> ? [X_1: list_nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_999_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_1000_pigeonhole__infinite__rel,axiom,
! [A: set_list_nat,B3: set_list_nat,R: list_nat > list_nat > $o] :
( ~ ( finite8100373058378681591st_nat @ A )
=> ( ( finite8100373058378681591st_nat @ B3 )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ A )
=> ? [Xa: list_nat] :
( ( member_list_nat @ Xa @ B3 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: list_nat] :
( ( member_list_nat @ X2 @ B3 )
& ~ ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [A5: list_nat] :
( ( member_list_nat @ A5 @ A )
& ( R @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1001_pigeonhole__infinite__rel,axiom,
! [A: set_list_nat,B3: set_nat,R: list_nat > nat > $o] :
( ~ ( finite8100373058378681591st_nat @ A )
=> ( ( finite_finite_nat @ B3 )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B3 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B3 )
& ~ ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [A5: list_nat] :
( ( member_list_nat @ A5 @ A )
& ( R @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1002_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B3: set_list_nat,R: nat > list_nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite8100373058378681591st_nat @ B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ? [Xa: list_nat] :
( ( member_list_nat @ Xa @ B3 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: list_nat] :
( ( member_list_nat @ X2 @ B3 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A )
& ( R @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1003_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B3: set_nat,R: nat > nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B3 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B3 )
& ( R @ X2 @ Xa ) ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B3 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A )
& ( R @ A5 @ X2 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1004_sum_Oswap__restrict,axiom,
! [A: set_list_nat,B3: set_nat,G: list_nat > nat > nat,R: list_nat > nat > $o] :
( ( finite8100373058378681591st_nat @ A )
=> ( ( finite_finite_nat @ B3 )
=> ( ( groups4396056296759096172at_nat
@ ^ [X4: list_nat] :
( groups3542108847815614940at_nat @ ( G @ X4 )
@ ( collect_nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ B3 )
& ( R @ X4 @ Y3 ) ) ) )
@ A )
= ( groups3542108847815614940at_nat
@ ^ [Y3: nat] :
( groups4396056296759096172at_nat
@ ^ [X4: list_nat] : ( G @ X4 @ Y3 )
@ ( collect_list_nat
@ ^ [X4: list_nat] :
( ( member_list_nat @ X4 @ A )
& ( R @ X4 @ Y3 ) ) ) )
@ B3 ) ) ) ) ).
% sum.swap_restrict
thf(fact_1005_sum_Oswap__restrict,axiom,
! [A: set_nat,B3: set_list_nat,G: nat > list_nat > nat,R: nat > list_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( finite8100373058378681591st_nat @ B3 )
=> ( ( groups3542108847815614940at_nat
@ ^ [X4: nat] :
( groups4396056296759096172at_nat @ ( G @ X4 )
@ ( collect_list_nat
@ ^ [Y3: list_nat] :
( ( member_list_nat @ Y3 @ B3 )
& ( R @ X4 @ Y3 ) ) ) )
@ A )
= ( groups4396056296759096172at_nat
@ ^ [Y3: list_nat] :
( groups3542108847815614940at_nat
@ ^ [X4: nat] : ( G @ X4 @ Y3 )
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( R @ X4 @ Y3 ) ) ) )
@ B3 ) ) ) ) ).
% sum.swap_restrict
thf(fact_1006_sum_Oswap__restrict,axiom,
! [A: set_nat,B3: set_nat,G: nat > nat > nat,R: nat > nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B3 )
=> ( ( groups3542108847815614940at_nat
@ ^ [X4: nat] :
( groups3542108847815614940at_nat @ ( G @ X4 )
@ ( collect_nat
@ ^ [Y3: nat] :
( ( member_nat @ Y3 @ B3 )
& ( R @ X4 @ Y3 ) ) ) )
@ A )
= ( groups3542108847815614940at_nat
@ ^ [Y3: nat] :
( groups3542108847815614940at_nat
@ ^ [X4: nat] : ( G @ X4 @ Y3 )
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( R @ X4 @ Y3 ) ) ) )
@ B3 ) ) ) ) ).
% sum.swap_restrict
thf(fact_1007_finite__list,axiom,
! [A: set_list_nat] :
( ( finite8100373058378681591st_nat @ A )
=> ? [Xs3: list_list_nat] :
( ( set_list_nat2 @ Xs3 )
= A ) ) ).
% finite_list
thf(fact_1008_finite__list,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ? [Xs3: list_nat] :
( ( set_nat2 @ Xs3 )
= A ) ) ).
% finite_list
thf(fact_1009_finite__distinct__list,axiom,
! [A: set_list_nat] :
( ( finite8100373058378681591st_nat @ A )
=> ? [Xs3: list_list_nat] :
( ( ( set_list_nat2 @ Xs3 )
= A )
& ( distinct_list_nat @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_1010_finite__distinct__list,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ? [Xs3: list_nat] :
( ( ( set_nat2 @ Xs3 )
= A )
& ( distinct_nat @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_1011_sum_Ofinite__Collect__op,axiom,
! [I5: set_list_nat,X: list_nat > nat,Y: list_nat > nat] :
( ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [I2: list_nat] :
( ( member_list_nat @ I2 @ I5 )
& ( ( X @ I2 )
!= zero_zero_nat ) ) ) )
=> ( ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [I2: list_nat] :
( ( member_list_nat @ I2 @ I5 )
& ( ( Y @ I2 )
!= zero_zero_nat ) ) ) )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [I2: list_nat] :
( ( member_list_nat @ I2 @ I5 )
& ( ( plus_plus_nat @ ( X @ I2 ) @ ( Y @ I2 ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_1012_sum_Ofinite__Collect__op,axiom,
! [I5: set_nat,X: nat > nat,Y: nat > nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ I5 )
& ( ( X @ I2 )
!= zero_zero_nat ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ I5 )
& ( ( Y @ I2 )
!= zero_zero_nat ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ I5 )
& ( ( plus_plus_nat @ ( X @ I2 ) @ ( Y @ I2 ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_1013_sum_Ointer__filter,axiom,
! [A: set_list_nat,G: list_nat > nat,P: list_nat > $o] :
( ( finite8100373058378681591st_nat @ A )
=> ( ( groups4396056296759096172at_nat @ G
@ ( collect_list_nat
@ ^ [X4: list_nat] :
( ( member_list_nat @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( groups4396056296759096172at_nat
@ ^ [X4: list_nat] : ( if_nat @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_nat )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_1014_sum_Ointer__filter,axiom,
! [A: set_nat,G: nat > nat,P: nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( groups3542108847815614940at_nat @ G
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( groups3542108847815614940at_nat
@ ^ [X4: nat] : ( if_nat @ ( P @ X4 ) @ ( G @ X4 ) @ zero_zero_nat )
@ A ) ) ) ).
% sum.inter_filter
thf(fact_1015_prod_Ointer__filter,axiom,
! [A: set_list_nat,G: list_nat > nat,P: list_nat > $o] :
( ( finite8100373058378681591st_nat @ A )
=> ( ( groups2907647131375434839at_nat @ G
@ ( collect_list_nat
@ ^ [X4: list_nat] :
( ( member_list_nat @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( groups2907647131375434839at_nat
@ ^ [X4: list_nat] : ( if_nat @ ( P @ X4 ) @ ( G @ X4 ) @ one_one_nat )
@ A ) ) ) ).
% prod.inter_filter
thf(fact_1016_prod_Ointer__filter,axiom,
! [A: set_nat,G: nat > nat,P: nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( groups708209901874060359at_nat @ G
@ ( collect_nat
@ ^ [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ X4 ) ) ) )
= ( groups708209901874060359at_nat
@ ^ [X4: nat] : ( if_nat @ ( P @ X4 ) @ ( G @ X4 ) @ one_one_nat )
@ A ) ) ) ).
% prod.inter_filter
thf(fact_1017_sum__multicount__gen,axiom,
! [S2: set_list_nat,T2: set_list_nat,R: list_nat > list_nat > $o,K: list_nat > nat] :
( ( finite8100373058378681591st_nat @ S2 )
=> ( ( finite8100373058378681591st_nat @ T2 )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ T2 )
=> ( ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [I2: list_nat] :
( ( member_list_nat @ I2 @ S2 )
& ( R @ I2 @ X2 ) ) ) )
= ( K @ X2 ) ) )
=> ( ( groups4396056296759096172at_nat
@ ^ [I2: list_nat] :
( finite_card_list_nat
@ ( collect_list_nat
@ ^ [J: list_nat] :
( ( member_list_nat @ J @ T2 )
& ( R @ I2 @ J ) ) ) )
@ S2 )
= ( groups4396056296759096172at_nat @ K @ T2 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_1018_sum__multicount__gen,axiom,
! [S2: set_list_nat,T2: set_nat,R: list_nat > nat > $o,K: nat > nat] :
( ( finite8100373058378681591st_nat @ S2 )
=> ( ( finite_finite_nat @ T2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ T2 )
=> ( ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [I2: list_nat] :
( ( member_list_nat @ I2 @ S2 )
& ( R @ I2 @ X2 ) ) ) )
= ( K @ X2 ) ) )
=> ( ( groups4396056296759096172at_nat
@ ^ [I2: list_nat] :
( finite_card_nat
@ ( collect_nat
@ ^ [J: nat] :
( ( member_nat @ J @ T2 )
& ( R @ I2 @ J ) ) ) )
@ S2 )
= ( groups3542108847815614940at_nat @ K @ T2 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_1019_sum__multicount__gen,axiom,
! [S2: set_nat,T2: set_list_nat,R: nat > list_nat > $o,K: list_nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( finite8100373058378681591st_nat @ T2 )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ T2 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ S2 )
& ( R @ I2 @ X2 ) ) ) )
= ( K @ X2 ) ) )
=> ( ( groups3542108847815614940at_nat
@ ^ [I2: nat] :
( finite_card_list_nat
@ ( collect_list_nat
@ ^ [J: list_nat] :
( ( member_list_nat @ J @ T2 )
& ( R @ I2 @ J ) ) ) )
@ S2 )
= ( groups4396056296759096172at_nat @ K @ T2 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_1020_sum__multicount__gen,axiom,
! [S2: set_nat,T2: set_nat,R: nat > nat > $o,K: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( finite_finite_nat @ T2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ T2 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [I2: nat] :
( ( member_nat @ I2 @ S2 )
& ( R @ I2 @ X2 ) ) ) )
= ( K @ X2 ) ) )
=> ( ( groups3542108847815614940at_nat
@ ^ [I2: nat] :
( finite_card_nat
@ ( collect_nat
@ ^ [J: nat] :
( ( member_nat @ J @ T2 )
& ( R @ I2 @ J ) ) ) )
@ S2 )
= ( groups3542108847815614940at_nat @ K @ T2 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_1021_card__le__Suc0__iff__eq,axiom,
! [A: set_list_nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( ( ord_less_eq_nat @ ( finite_card_list_nat @ A ) @ ( suc @ zero_zero_nat ) )
= ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ A )
=> ! [Y3: list_nat] :
( ( member_list_nat @ Y3 @ A )
=> ( X4 = Y3 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_1022_card__le__Suc0__iff__eq,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( suc @ zero_zero_nat ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ A )
=> ( X4 = Y3 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_1023_card__Diff__subset,axiom,
! [B3: set_list_nat,A: set_list_nat] :
( ( finite8100373058378681591st_nat @ B3 )
=> ( ( ord_le6045566169113846134st_nat @ B3 @ A )
=> ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A @ B3 ) )
= ( minus_minus_nat @ ( finite_card_list_nat @ A ) @ ( finite_card_list_nat @ B3 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_1024_card__Diff__subset,axiom,
! [B3: set_nat,A: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ A )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ B3 ) )
= ( minus_minus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B3 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_1025_card__psubset,axiom,
! [B3: set_list_nat,A: set_list_nat] :
( ( finite8100373058378681591st_nat @ B3 )
=> ( ( ord_le6045566169113846134st_nat @ A @ B3 )
=> ( ( ord_less_nat @ ( finite_card_list_nat @ A ) @ ( finite_card_list_nat @ B3 ) )
=> ( ord_le1190675801316882794st_nat @ A @ B3 ) ) ) ) ).
% card_psubset
thf(fact_1026_card__psubset,axiom,
! [B3: set_nat,A: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B3 ) )
=> ( ord_less_set_nat @ A @ B3 ) ) ) ) ).
% card_psubset
thf(fact_1027_diff__card__le__card__Diff,axiom,
! [B3: set_list_nat,A: set_list_nat] :
( ( finite8100373058378681591st_nat @ B3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_list_nat @ A ) @ ( finite_card_list_nat @ B3 ) ) @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A @ B3 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1028_diff__card__le__card__Diff,axiom,
! [B3: set_nat,A: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B3 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B3 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1029_sum__nonneg__eq__0__iff,axiom,
! [A: set_list_nat,F: list_nat > nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ( ( groups4396056296759096172at_nat @ F @ A )
= zero_zero_nat )
= ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ A )
=> ( ( F @ X4 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_1030_sum__nonneg__eq__0__iff,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ( ( groups3542108847815614940at_nat @ F @ A )
= zero_zero_nat )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ( F @ X4 )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_1031_sum__le__included,axiom,
! [S2: set_list_nat,T2: set_list_nat,G: list_nat > nat,I: list_nat > list_nat,F: list_nat > nat] :
( ( finite8100373058378681591st_nat @ S2 )
=> ( ( finite8100373058378681591st_nat @ T2 )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ T2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X2 ) ) )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ S2 )
=> ? [Xa: list_nat] :
( ( member_list_nat @ Xa @ T2 )
& ( ( I @ Xa )
= X2 )
& ( ord_less_eq_nat @ ( F @ X2 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups4396056296759096172at_nat @ F @ S2 ) @ ( groups4396056296759096172at_nat @ G @ T2 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_1032_sum__le__included,axiom,
! [S2: set_list_nat,T2: set_nat,G: nat > nat,I: nat > list_nat,F: list_nat > nat] :
( ( finite8100373058378681591st_nat @ S2 )
=> ( ( finite_finite_nat @ T2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ T2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X2 ) ) )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ S2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T2 )
& ( ( I @ Xa )
= X2 )
& ( ord_less_eq_nat @ ( F @ X2 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups4396056296759096172at_nat @ F @ S2 ) @ ( groups3542108847815614940at_nat @ G @ T2 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_1033_sum__le__included,axiom,
! [S2: set_nat,T2: set_list_nat,G: list_nat > nat,I: list_nat > nat,F: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( finite8100373058378681591st_nat @ T2 )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ T2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X2 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S2 )
=> ? [Xa: list_nat] :
( ( member_list_nat @ Xa @ T2 )
& ( ( I @ Xa )
= X2 )
& ( ord_less_eq_nat @ ( F @ X2 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ S2 ) @ ( groups4396056296759096172at_nat @ G @ T2 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_1034_sum__le__included,axiom,
! [S2: set_nat,T2: set_nat,G: nat > nat,I: nat > nat,F: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( finite_finite_nat @ T2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ T2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X2 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T2 )
& ( ( I @ Xa )
= X2 )
& ( ord_less_eq_nat @ ( F @ X2 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ S2 ) @ ( groups3542108847815614940at_nat @ G @ T2 ) ) ) ) ) ) ).
% sum_le_included
thf(fact_1035_sum__strict__mono__ex1,axiom,
! [A: set_list_nat,F: list_nat > nat,G: list_nat > nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ? [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
& ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_nat @ ( groups4396056296759096172at_nat @ F @ A ) @ ( groups4396056296759096172at_nat @ G @ A ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_1036_sum__strict__mono__ex1,axiom,
! [A: set_nat,F: nat > nat,G: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( groups3542108847815614940at_nat @ G @ A ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_1037_sum_Orelated,axiom,
! [R: nat > nat > $o,S: set_list_nat,H: list_nat > nat,G: list_nat > nat] :
( ( R @ zero_zero_nat @ zero_zero_nat )
=> ( ! [X1: nat,Y1: nat,X23: nat,Y23: nat] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y23 ) )
=> ( R @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
=> ( ( finite8100373058378681591st_nat @ S )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ S )
=> ( R @ ( H @ X2 ) @ ( G @ X2 ) ) )
=> ( R @ ( groups4396056296759096172at_nat @ H @ S ) @ ( groups4396056296759096172at_nat @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_1038_sum_Orelated,axiom,
! [R: nat > nat > $o,S: set_nat,H: nat > nat,G: nat > nat] :
( ( R @ zero_zero_nat @ zero_zero_nat )
=> ( ! [X1: nat,Y1: nat,X23: nat,Y23: nat] :
( ( ( R @ X1 @ X23 )
& ( R @ Y1 @ Y23 ) )
=> ( R @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( R @ ( H @ X2 ) @ ( G @ X2 ) ) )
=> ( R @ ( groups3542108847815614940at_nat @ H @ S ) @ ( groups3542108847815614940at_nat @ G @ S ) ) ) ) ) ) ).
% sum.related
thf(fact_1039_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_list_nat,T5: set_list_nat,S: set_list_nat,I: list_nat > list_nat,J2: list_nat > list_nat,T: set_list_nat,G: list_nat > nat,H: list_nat > nat] :
( ( finite8100373058378681591st_nat @ S4 )
=> ( ( finite8100373058378681591st_nat @ T5 )
=> ( ! [A3: list_nat] :
( ( member_list_nat @ A3 @ ( minus_7954133019191499631st_nat @ S @ S4 ) )
=> ( ( I @ ( J2 @ A3 ) )
= A3 ) )
=> ( ! [A3: list_nat] :
( ( member_list_nat @ A3 @ ( minus_7954133019191499631st_nat @ S @ S4 ) )
=> ( member_list_nat @ ( J2 @ A3 ) @ ( minus_7954133019191499631st_nat @ T @ T5 ) ) )
=> ( ! [B2: list_nat] :
( ( member_list_nat @ B2 @ ( minus_7954133019191499631st_nat @ T @ T5 ) )
=> ( ( J2 @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: list_nat] :
( ( member_list_nat @ B2 @ ( minus_7954133019191499631st_nat @ T @ T5 ) )
=> ( member_list_nat @ ( I @ B2 ) @ ( minus_7954133019191499631st_nat @ S @ S4 ) ) )
=> ( ! [A3: list_nat] :
( ( member_list_nat @ A3 @ S4 )
=> ( ( G @ A3 )
= zero_zero_nat ) )
=> ( ! [B2: list_nat] :
( ( member_list_nat @ B2 @ T5 )
=> ( ( H @ B2 )
= zero_zero_nat ) )
=> ( ! [A3: list_nat] :
( ( member_list_nat @ A3 @ S )
=> ( ( H @ ( J2 @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups4396056296759096172at_nat @ G @ S )
= ( groups4396056296759096172at_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_1040_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_list_nat,T5: set_nat,S: set_list_nat,I: nat > list_nat,J2: list_nat > nat,T: set_nat,G: list_nat > nat,H: nat > nat] :
( ( finite8100373058378681591st_nat @ S4 )
=> ( ( finite_finite_nat @ T5 )
=> ( ! [A3: list_nat] :
( ( member_list_nat @ A3 @ ( minus_7954133019191499631st_nat @ S @ S4 ) )
=> ( ( I @ ( J2 @ A3 ) )
= A3 ) )
=> ( ! [A3: list_nat] :
( ( member_list_nat @ A3 @ ( minus_7954133019191499631st_nat @ S @ S4 ) )
=> ( member_nat @ ( J2 @ A3 ) @ ( minus_minus_set_nat @ T @ T5 ) ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ T @ T5 ) )
=> ( ( J2 @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ T @ T5 ) )
=> ( member_list_nat @ ( I @ B2 ) @ ( minus_7954133019191499631st_nat @ S @ S4 ) ) )
=> ( ! [A3: list_nat] :
( ( member_list_nat @ A3 @ S4 )
=> ( ( G @ A3 )
= zero_zero_nat ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T5 )
=> ( ( H @ B2 )
= zero_zero_nat ) )
=> ( ! [A3: list_nat] :
( ( member_list_nat @ A3 @ S )
=> ( ( H @ ( J2 @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups4396056296759096172at_nat @ G @ S )
= ( groups3542108847815614940at_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_1041_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_nat,T5: set_list_nat,S: set_nat,I: list_nat > nat,J2: nat > list_nat,T: set_list_nat,G: nat > nat,H: list_nat > nat] :
( ( finite_finite_nat @ S4 )
=> ( ( finite8100373058378681591st_nat @ T5 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S @ S4 ) )
=> ( ( I @ ( J2 @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S @ S4 ) )
=> ( member_list_nat @ ( J2 @ A3 ) @ ( minus_7954133019191499631st_nat @ T @ T5 ) ) )
=> ( ! [B2: list_nat] :
( ( member_list_nat @ B2 @ ( minus_7954133019191499631st_nat @ T @ T5 ) )
=> ( ( J2 @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: list_nat] :
( ( member_list_nat @ B2 @ ( minus_7954133019191499631st_nat @ T @ T5 ) )
=> ( member_nat @ ( I @ B2 ) @ ( minus_minus_set_nat @ S @ S4 ) ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S4 )
=> ( ( G @ A3 )
= zero_zero_nat ) )
=> ( ! [B2: list_nat] :
( ( member_list_nat @ B2 @ T5 )
=> ( ( H @ B2 )
= zero_zero_nat ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H @ ( J2 @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S )
= ( groups4396056296759096172at_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_1042_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_nat,T5: set_nat,S: set_nat,I: nat > nat,J2: nat > nat,T: set_nat,G: nat > nat,H: nat > nat] :
( ( finite_finite_nat @ S4 )
=> ( ( finite_finite_nat @ T5 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S @ S4 ) )
=> ( ( I @ ( J2 @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S @ S4 ) )
=> ( member_nat @ ( J2 @ A3 ) @ ( minus_minus_set_nat @ T @ T5 ) ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ T @ T5 ) )
=> ( ( J2 @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ T @ T5 ) )
=> ( member_nat @ ( I @ B2 ) @ ( minus_minus_set_nat @ S @ S4 ) ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S4 )
=> ( ( G @ A3 )
= zero_zero_nat ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T5 )
=> ( ( H @ B2 )
= zero_zero_nat ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H @ ( J2 @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S )
= ( groups3542108847815614940at_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_1043_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_list_nat,T5: set_list_nat,S: set_list_nat,I: list_nat > list_nat,J2: list_nat > list_nat,T: set_list_nat,G: list_nat > nat,H: list_nat > nat] :
( ( finite8100373058378681591st_nat @ S4 )
=> ( ( finite8100373058378681591st_nat @ T5 )
=> ( ! [A3: list_nat] :
( ( member_list_nat @ A3 @ ( minus_7954133019191499631st_nat @ S @ S4 ) )
=> ( ( I @ ( J2 @ A3 ) )
= A3 ) )
=> ( ! [A3: list_nat] :
( ( member_list_nat @ A3 @ ( minus_7954133019191499631st_nat @ S @ S4 ) )
=> ( member_list_nat @ ( J2 @ A3 ) @ ( minus_7954133019191499631st_nat @ T @ T5 ) ) )
=> ( ! [B2: list_nat] :
( ( member_list_nat @ B2 @ ( minus_7954133019191499631st_nat @ T @ T5 ) )
=> ( ( J2 @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: list_nat] :
( ( member_list_nat @ B2 @ ( minus_7954133019191499631st_nat @ T @ T5 ) )
=> ( member_list_nat @ ( I @ B2 ) @ ( minus_7954133019191499631st_nat @ S @ S4 ) ) )
=> ( ! [A3: list_nat] :
( ( member_list_nat @ A3 @ S4 )
=> ( ( G @ A3 )
= one_one_nat ) )
=> ( ! [B2: list_nat] :
( ( member_list_nat @ B2 @ T5 )
=> ( ( H @ B2 )
= one_one_nat ) )
=> ( ! [A3: list_nat] :
( ( member_list_nat @ A3 @ S )
=> ( ( H @ ( J2 @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups2907647131375434839at_nat @ G @ S )
= ( groups2907647131375434839at_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_1044_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_list_nat,T5: set_nat,S: set_list_nat,I: nat > list_nat,J2: list_nat > nat,T: set_nat,G: list_nat > nat,H: nat > nat] :
( ( finite8100373058378681591st_nat @ S4 )
=> ( ( finite_finite_nat @ T5 )
=> ( ! [A3: list_nat] :
( ( member_list_nat @ A3 @ ( minus_7954133019191499631st_nat @ S @ S4 ) )
=> ( ( I @ ( J2 @ A3 ) )
= A3 ) )
=> ( ! [A3: list_nat] :
( ( member_list_nat @ A3 @ ( minus_7954133019191499631st_nat @ S @ S4 ) )
=> ( member_nat @ ( J2 @ A3 ) @ ( minus_minus_set_nat @ T @ T5 ) ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ T @ T5 ) )
=> ( ( J2 @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ T @ T5 ) )
=> ( member_list_nat @ ( I @ B2 ) @ ( minus_7954133019191499631st_nat @ S @ S4 ) ) )
=> ( ! [A3: list_nat] :
( ( member_list_nat @ A3 @ S4 )
=> ( ( G @ A3 )
= one_one_nat ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T5 )
=> ( ( H @ B2 )
= one_one_nat ) )
=> ( ! [A3: list_nat] :
( ( member_list_nat @ A3 @ S )
=> ( ( H @ ( J2 @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups2907647131375434839at_nat @ G @ S )
= ( groups708209901874060359at_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_1045_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_nat,T5: set_list_nat,S: set_nat,I: list_nat > nat,J2: nat > list_nat,T: set_list_nat,G: nat > nat,H: list_nat > nat] :
( ( finite_finite_nat @ S4 )
=> ( ( finite8100373058378681591st_nat @ T5 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S @ S4 ) )
=> ( ( I @ ( J2 @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S @ S4 ) )
=> ( member_list_nat @ ( J2 @ A3 ) @ ( minus_7954133019191499631st_nat @ T @ T5 ) ) )
=> ( ! [B2: list_nat] :
( ( member_list_nat @ B2 @ ( minus_7954133019191499631st_nat @ T @ T5 ) )
=> ( ( J2 @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: list_nat] :
( ( member_list_nat @ B2 @ ( minus_7954133019191499631st_nat @ T @ T5 ) )
=> ( member_nat @ ( I @ B2 ) @ ( minus_minus_set_nat @ S @ S4 ) ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S4 )
=> ( ( G @ A3 )
= one_one_nat ) )
=> ( ! [B2: list_nat] :
( ( member_list_nat @ B2 @ T5 )
=> ( ( H @ B2 )
= one_one_nat ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H @ ( J2 @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups708209901874060359at_nat @ G @ S )
= ( groups2907647131375434839at_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_1046_prod_Oreindex__bij__witness__not__neutral,axiom,
! [S4: set_nat,T5: set_nat,S: set_nat,I: nat > nat,J2: nat > nat,T: set_nat,G: nat > nat,H: nat > nat] :
( ( finite_finite_nat @ S4 )
=> ( ( finite_finite_nat @ T5 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S @ S4 ) )
=> ( ( I @ ( J2 @ A3 ) )
= A3 ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ S @ S4 ) )
=> ( member_nat @ ( J2 @ A3 ) @ ( minus_minus_set_nat @ T @ T5 ) ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ T @ T5 ) )
=> ( ( J2 @ ( I @ B2 ) )
= B2 ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ T @ T5 ) )
=> ( member_nat @ ( I @ B2 ) @ ( minus_minus_set_nat @ S @ S4 ) ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S4 )
=> ( ( G @ A3 )
= one_one_nat ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ T5 )
=> ( ( H @ B2 )
= one_one_nat ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ S )
=> ( ( H @ ( J2 @ A3 ) )
= ( G @ A3 ) ) )
=> ( ( groups708209901874060359at_nat @ G @ S )
= ( groups708209901874060359at_nat @ H @ T ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_1047_sum__eq__Suc0__iff,axiom,
! [A: set_list_nat,F: list_nat > nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( ( ( groups4396056296759096172at_nat @ F @ A )
= ( suc @ zero_zero_nat ) )
= ( ? [X4: list_nat] :
( ( member_list_nat @ X4 @ A )
& ( ( F @ X4 )
= ( suc @ zero_zero_nat ) )
& ! [Y3: list_nat] :
( ( member_list_nat @ Y3 @ A )
=> ( ( X4 != Y3 )
=> ( ( F @ Y3 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_1048_sum__eq__Suc0__iff,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ( groups3542108847815614940at_nat @ F @ A )
= ( suc @ zero_zero_nat ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ( F @ X4 )
= ( suc @ zero_zero_nat ) )
& ! [Y3: nat] :
( ( member_nat @ Y3 @ A )
=> ( ( X4 != Y3 )
=> ( ( F @ Y3 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_1049_sum__eq__1__iff,axiom,
! [A: set_list_nat,F: list_nat > nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( ( ( groups4396056296759096172at_nat @ F @ A )
= one_one_nat )
= ( ? [X4: list_nat] :
( ( member_list_nat @ X4 @ A )
& ( ( F @ X4 )
= one_one_nat )
& ! [Y3: list_nat] :
( ( member_list_nat @ Y3 @ A )
=> ( ( X4 != Y3 )
=> ( ( F @ Y3 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_1050_sum__eq__1__iff,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ( groups3542108847815614940at_nat @ F @ A )
= one_one_nat )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ( F @ X4 )
= one_one_nat )
& ! [Y3: nat] :
( ( member_nat @ Y3 @ A )
=> ( ( X4 != Y3 )
=> ( ( F @ Y3 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_1051_sum__nonneg__leq__bound,axiom,
! [S2: set_list_nat,F: list_nat > nat,B3: nat,I: list_nat] :
( ( finite8100373058378681591st_nat @ S2 )
=> ( ! [I3: list_nat] :
( ( member_list_nat @ I3 @ S2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ( ( groups4396056296759096172at_nat @ F @ S2 )
= B3 )
=> ( ( member_list_nat @ I @ S2 )
=> ( ord_less_eq_nat @ ( F @ I ) @ B3 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_1052_sum__nonneg__leq__bound,axiom,
! [S2: set_nat,F: nat > nat,B3: nat,I: nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ S2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ( ( groups3542108847815614940at_nat @ F @ S2 )
= B3 )
=> ( ( member_nat @ I @ S2 )
=> ( ord_less_eq_nat @ ( F @ I ) @ B3 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_1053_sum__nonneg__0,axiom,
! [S2: set_list_nat,F: list_nat > nat,I: list_nat] :
( ( finite8100373058378681591st_nat @ S2 )
=> ( ! [I3: list_nat] :
( ( member_list_nat @ I3 @ S2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ( ( groups4396056296759096172at_nat @ F @ S2 )
= zero_zero_nat )
=> ( ( member_list_nat @ I @ S2 )
=> ( ( F @ I )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_1054_sum__nonneg__0,axiom,
! [S2: set_nat,F: nat > nat,I: nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ S2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ( ( groups3542108847815614940at_nat @ F @ S2 )
= zero_zero_nat )
=> ( ( member_nat @ I @ S2 )
=> ( ( F @ I )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_1055_sum_Osetdiff__irrelevant,axiom,
! [A: set_list_nat,G: list_nat > nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( ( groups4396056296759096172at_nat @ G
@ ( minus_7954133019191499631st_nat @ A
@ ( collect_list_nat
@ ^ [X4: list_nat] :
( ( G @ X4 )
= zero_zero_nat ) ) ) )
= ( groups4396056296759096172at_nat @ G @ A ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_1056_sum_Osetdiff__irrelevant,axiom,
! [A: set_nat,G: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( groups3542108847815614940at_nat @ G
@ ( minus_minus_set_nat @ A
@ ( collect_nat
@ ^ [X4: nat] :
( ( G @ X4 )
= zero_zero_nat ) ) ) )
= ( groups3542108847815614940at_nat @ G @ A ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_1057_prod_Osetdiff__irrelevant,axiom,
! [A: set_list_nat,G: list_nat > nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( ( groups2907647131375434839at_nat @ G
@ ( minus_7954133019191499631st_nat @ A
@ ( collect_list_nat
@ ^ [X4: list_nat] :
( ( G @ X4 )
= one_one_nat ) ) ) )
= ( groups2907647131375434839at_nat @ G @ A ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_1058_prod_Osetdiff__irrelevant,axiom,
! [A: set_nat,G: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( groups708209901874060359at_nat @ G
@ ( minus_minus_set_nat @ A
@ ( collect_nat
@ ^ [X4: nat] :
( ( G @ X4 )
= one_one_nat ) ) ) )
= ( groups708209901874060359at_nat @ G @ A ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_1059_subset__eq__atLeast0__atMost__finite,axiom,
! [N5: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N5 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
=> ( finite_finite_nat @ N5 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_1060_sum__pos2,axiom,
! [I5: set_list_nat,I: list_nat,F: list_nat > nat] :
( ( finite8100373058378681591st_nat @ I5 )
=> ( ( member_list_nat @ I @ I5 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
=> ( ! [I3: list_nat] :
( ( member_list_nat @ I3 @ I5 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups4396056296759096172at_nat @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_1061_sum__pos2,axiom,
! [I5: set_nat,I: nat,F: nat > nat] :
( ( finite_finite_nat @ I5 )
=> ( ( member_nat @ I @ I5 )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
=> ( ! [I3: nat] :
( ( member_nat @ I3 @ I5 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( groups3542108847815614940at_nat @ F @ I5 ) ) ) ) ) ) ).
% sum_pos2
thf(fact_1062_sum_Omono__neutral__cong__right,axiom,
! [T: set_list_nat,S: set_list_nat,G: list_nat > nat,H: list_nat > nat] :
( ( finite8100373058378681591st_nat @ T )
=> ( ( ord_le6045566169113846134st_nat @ S @ T )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ ( minus_7954133019191499631st_nat @ T @ S ) )
=> ( ( G @ X2 )
= zero_zero_nat ) )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ S )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups4396056296759096172at_nat @ G @ T )
= ( groups4396056296759096172at_nat @ H @ S ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_1063_sum_Omono__neutral__cong__right,axiom,
! [T: set_nat,S: set_nat,G: nat > nat,H: nat > nat] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G @ X2 )
= zero_zero_nat ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ T )
= ( groups3542108847815614940at_nat @ H @ S ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_1064_sum_Omono__neutral__cong__left,axiom,
! [T: set_list_nat,S: set_list_nat,H: list_nat > nat,G: list_nat > nat] :
( ( finite8100373058378681591st_nat @ T )
=> ( ( ord_le6045566169113846134st_nat @ S @ T )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ ( minus_7954133019191499631st_nat @ T @ S ) )
=> ( ( H @ X2 )
= zero_zero_nat ) )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ S )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups4396056296759096172at_nat @ G @ S )
= ( groups4396056296759096172at_nat @ H @ T ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_1065_sum_Omono__neutral__cong__left,axiom,
! [T: set_nat,S: set_nat,H: nat > nat,G: nat > nat] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( H @ X2 )
= zero_zero_nat ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups3542108847815614940at_nat @ G @ S )
= ( groups3542108847815614940at_nat @ H @ T ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_1066_sum_Omono__neutral__right,axiom,
! [T: set_list_nat,S: set_list_nat,G: list_nat > nat] :
( ( finite8100373058378681591st_nat @ T )
=> ( ( ord_le6045566169113846134st_nat @ S @ T )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ ( minus_7954133019191499631st_nat @ T @ S ) )
=> ( ( G @ X2 )
= zero_zero_nat ) )
=> ( ( groups4396056296759096172at_nat @ G @ T )
= ( groups4396056296759096172at_nat @ G @ S ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_1067_sum_Omono__neutral__right,axiom,
! [T: set_nat,S: set_nat,G: nat > nat] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G @ X2 )
= zero_zero_nat ) )
=> ( ( groups3542108847815614940at_nat @ G @ T )
= ( groups3542108847815614940at_nat @ G @ S ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_1068_sum_Omono__neutral__left,axiom,
! [T: set_list_nat,S: set_list_nat,G: list_nat > nat] :
( ( finite8100373058378681591st_nat @ T )
=> ( ( ord_le6045566169113846134st_nat @ S @ T )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ ( minus_7954133019191499631st_nat @ T @ S ) )
=> ( ( G @ X2 )
= zero_zero_nat ) )
=> ( ( groups4396056296759096172at_nat @ G @ S )
= ( groups4396056296759096172at_nat @ G @ T ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_1069_sum_Omono__neutral__left,axiom,
! [T: set_nat,S: set_nat,G: nat > nat] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G @ X2 )
= zero_zero_nat ) )
=> ( ( groups3542108847815614940at_nat @ G @ S )
= ( groups3542108847815614940at_nat @ G @ T ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_1070_sum_Osame__carrierI,axiom,
! [C4: set_list_nat,A: set_list_nat,B3: set_list_nat,G: list_nat > nat,H: list_nat > nat] :
( ( finite8100373058378681591st_nat @ C4 )
=> ( ( ord_le6045566169113846134st_nat @ A @ C4 )
=> ( ( ord_le6045566169113846134st_nat @ B3 @ C4 )
=> ( ! [A3: list_nat] :
( ( member_list_nat @ A3 @ ( minus_7954133019191499631st_nat @ C4 @ A ) )
=> ( ( G @ A3 )
= zero_zero_nat ) )
=> ( ! [B2: list_nat] :
( ( member_list_nat @ B2 @ ( minus_7954133019191499631st_nat @ C4 @ B3 ) )
=> ( ( H @ B2 )
= zero_zero_nat ) )
=> ( ( ( groups4396056296759096172at_nat @ G @ C4 )
= ( groups4396056296759096172at_nat @ H @ C4 ) )
=> ( ( groups4396056296759096172at_nat @ G @ A )
= ( groups4396056296759096172at_nat @ H @ B3 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_1071_sum_Osame__carrierI,axiom,
! [C4: set_nat,A: set_nat,B3: set_nat,G: nat > nat,H: nat > nat] :
( ( finite_finite_nat @ C4 )
=> ( ( ord_less_eq_set_nat @ A @ C4 )
=> ( ( ord_less_eq_set_nat @ B3 @ C4 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A ) )
=> ( ( G @ A3 )
= zero_zero_nat ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ C4 @ B3 ) )
=> ( ( H @ B2 )
= zero_zero_nat ) )
=> ( ( ( groups3542108847815614940at_nat @ G @ C4 )
= ( groups3542108847815614940at_nat @ H @ C4 ) )
=> ( ( groups3542108847815614940at_nat @ G @ A )
= ( groups3542108847815614940at_nat @ H @ B3 ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_1072_sum_Osame__carrier,axiom,
! [C4: set_list_nat,A: set_list_nat,B3: set_list_nat,G: list_nat > nat,H: list_nat > nat] :
( ( finite8100373058378681591st_nat @ C4 )
=> ( ( ord_le6045566169113846134st_nat @ A @ C4 )
=> ( ( ord_le6045566169113846134st_nat @ B3 @ C4 )
=> ( ! [A3: list_nat] :
( ( member_list_nat @ A3 @ ( minus_7954133019191499631st_nat @ C4 @ A ) )
=> ( ( G @ A3 )
= zero_zero_nat ) )
=> ( ! [B2: list_nat] :
( ( member_list_nat @ B2 @ ( minus_7954133019191499631st_nat @ C4 @ B3 ) )
=> ( ( H @ B2 )
= zero_zero_nat ) )
=> ( ( ( groups4396056296759096172at_nat @ G @ A )
= ( groups4396056296759096172at_nat @ H @ B3 ) )
= ( ( groups4396056296759096172at_nat @ G @ C4 )
= ( groups4396056296759096172at_nat @ H @ C4 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_1073_sum_Osame__carrier,axiom,
! [C4: set_nat,A: set_nat,B3: set_nat,G: nat > nat,H: nat > nat] :
( ( finite_finite_nat @ C4 )
=> ( ( ord_less_eq_set_nat @ A @ C4 )
=> ( ( ord_less_eq_set_nat @ B3 @ C4 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A ) )
=> ( ( G @ A3 )
= zero_zero_nat ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ C4 @ B3 ) )
=> ( ( H @ B2 )
= zero_zero_nat ) )
=> ( ( ( groups3542108847815614940at_nat @ G @ A )
= ( groups3542108847815614940at_nat @ H @ B3 ) )
= ( ( groups3542108847815614940at_nat @ G @ C4 )
= ( groups3542108847815614940at_nat @ H @ C4 ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_1074_sum_Osubset__diff,axiom,
! [B3: set_list_nat,A: set_list_nat,G: list_nat > nat] :
( ( ord_le6045566169113846134st_nat @ B3 @ A )
=> ( ( finite8100373058378681591st_nat @ A )
=> ( ( groups4396056296759096172at_nat @ G @ A )
= ( plus_plus_nat @ ( groups4396056296759096172at_nat @ G @ ( minus_7954133019191499631st_nat @ A @ B3 ) ) @ ( groups4396056296759096172at_nat @ G @ B3 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_1075_sum_Osubset__diff,axiom,
! [B3: set_nat,A: set_nat,G: nat > nat] :
( ( ord_less_eq_set_nat @ B3 @ A )
=> ( ( finite_finite_nat @ A )
=> ( ( groups3542108847815614940at_nat @ G @ A )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( minus_minus_set_nat @ A @ B3 ) ) @ ( groups3542108847815614940at_nat @ G @ B3 ) ) ) ) ) ).
% sum.subset_diff
thf(fact_1076_prod_Omono__neutral__cong__right,axiom,
! [T: set_list_nat,S: set_list_nat,G: list_nat > nat,H: list_nat > nat] :
( ( finite8100373058378681591st_nat @ T )
=> ( ( ord_le6045566169113846134st_nat @ S @ T )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ ( minus_7954133019191499631st_nat @ T @ S ) )
=> ( ( G @ X2 )
= one_one_nat ) )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ S )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups2907647131375434839at_nat @ G @ T )
= ( groups2907647131375434839at_nat @ H @ S ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_1077_prod_Omono__neutral__cong__right,axiom,
! [T: set_nat,S: set_nat,G: nat > nat,H: nat > nat] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G @ X2 )
= one_one_nat ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups708209901874060359at_nat @ G @ T )
= ( groups708209901874060359at_nat @ H @ S ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_1078_prod_Omono__neutral__cong__left,axiom,
! [T: set_list_nat,S: set_list_nat,H: list_nat > nat,G: list_nat > nat] :
( ( finite8100373058378681591st_nat @ T )
=> ( ( ord_le6045566169113846134st_nat @ S @ T )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ ( minus_7954133019191499631st_nat @ T @ S ) )
=> ( ( H @ X2 )
= one_one_nat ) )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ S )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups2907647131375434839at_nat @ G @ S )
= ( groups2907647131375434839at_nat @ H @ T ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_1079_prod_Omono__neutral__cong__left,axiom,
! [T: set_nat,S: set_nat,H: nat > nat,G: nat > nat] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( H @ X2 )
= one_one_nat ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ S )
=> ( ( G @ X2 )
= ( H @ X2 ) ) )
=> ( ( groups708209901874060359at_nat @ G @ S )
= ( groups708209901874060359at_nat @ H @ T ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_1080_prod_Omono__neutral__right,axiom,
! [T: set_list_nat,S: set_list_nat,G: list_nat > nat] :
( ( finite8100373058378681591st_nat @ T )
=> ( ( ord_le6045566169113846134st_nat @ S @ T )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ ( minus_7954133019191499631st_nat @ T @ S ) )
=> ( ( G @ X2 )
= one_one_nat ) )
=> ( ( groups2907647131375434839at_nat @ G @ T )
= ( groups2907647131375434839at_nat @ G @ S ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_1081_prod_Omono__neutral__right,axiom,
! [T: set_nat,S: set_nat,G: nat > nat] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G @ X2 )
= one_one_nat ) )
=> ( ( groups708209901874060359at_nat @ G @ T )
= ( groups708209901874060359at_nat @ G @ S ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_1082_prod_Omono__neutral__left,axiom,
! [T: set_list_nat,S: set_list_nat,G: list_nat > nat] :
( ( finite8100373058378681591st_nat @ T )
=> ( ( ord_le6045566169113846134st_nat @ S @ T )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ ( minus_7954133019191499631st_nat @ T @ S ) )
=> ( ( G @ X2 )
= one_one_nat ) )
=> ( ( groups2907647131375434839at_nat @ G @ S )
= ( groups2907647131375434839at_nat @ G @ T ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_1083_prod_Omono__neutral__left,axiom,
! [T: set_nat,S: set_nat,G: nat > nat] :
( ( finite_finite_nat @ T )
=> ( ( ord_less_eq_set_nat @ S @ T )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( minus_minus_set_nat @ T @ S ) )
=> ( ( G @ X2 )
= one_one_nat ) )
=> ( ( groups708209901874060359at_nat @ G @ S )
= ( groups708209901874060359at_nat @ G @ T ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_1084_prod_Osame__carrierI,axiom,
! [C4: set_list_nat,A: set_list_nat,B3: set_list_nat,G: list_nat > nat,H: list_nat > nat] :
( ( finite8100373058378681591st_nat @ C4 )
=> ( ( ord_le6045566169113846134st_nat @ A @ C4 )
=> ( ( ord_le6045566169113846134st_nat @ B3 @ C4 )
=> ( ! [A3: list_nat] :
( ( member_list_nat @ A3 @ ( minus_7954133019191499631st_nat @ C4 @ A ) )
=> ( ( G @ A3 )
= one_one_nat ) )
=> ( ! [B2: list_nat] :
( ( member_list_nat @ B2 @ ( minus_7954133019191499631st_nat @ C4 @ B3 ) )
=> ( ( H @ B2 )
= one_one_nat ) )
=> ( ( ( groups2907647131375434839at_nat @ G @ C4 )
= ( groups2907647131375434839at_nat @ H @ C4 ) )
=> ( ( groups2907647131375434839at_nat @ G @ A )
= ( groups2907647131375434839at_nat @ H @ B3 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_1085_prod_Osame__carrierI,axiom,
! [C4: set_nat,A: set_nat,B3: set_nat,G: nat > nat,H: nat > nat] :
( ( finite_finite_nat @ C4 )
=> ( ( ord_less_eq_set_nat @ A @ C4 )
=> ( ( ord_less_eq_set_nat @ B3 @ C4 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A ) )
=> ( ( G @ A3 )
= one_one_nat ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ C4 @ B3 ) )
=> ( ( H @ B2 )
= one_one_nat ) )
=> ( ( ( groups708209901874060359at_nat @ G @ C4 )
= ( groups708209901874060359at_nat @ H @ C4 ) )
=> ( ( groups708209901874060359at_nat @ G @ A )
= ( groups708209901874060359at_nat @ H @ B3 ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_1086_prod_Osame__carrier,axiom,
! [C4: set_list_nat,A: set_list_nat,B3: set_list_nat,G: list_nat > nat,H: list_nat > nat] :
( ( finite8100373058378681591st_nat @ C4 )
=> ( ( ord_le6045566169113846134st_nat @ A @ C4 )
=> ( ( ord_le6045566169113846134st_nat @ B3 @ C4 )
=> ( ! [A3: list_nat] :
( ( member_list_nat @ A3 @ ( minus_7954133019191499631st_nat @ C4 @ A ) )
=> ( ( G @ A3 )
= one_one_nat ) )
=> ( ! [B2: list_nat] :
( ( member_list_nat @ B2 @ ( minus_7954133019191499631st_nat @ C4 @ B3 ) )
=> ( ( H @ B2 )
= one_one_nat ) )
=> ( ( ( groups2907647131375434839at_nat @ G @ A )
= ( groups2907647131375434839at_nat @ H @ B3 ) )
= ( ( groups2907647131375434839at_nat @ G @ C4 )
= ( groups2907647131375434839at_nat @ H @ C4 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_1087_prod_Osame__carrier,axiom,
! [C4: set_nat,A: set_nat,B3: set_nat,G: nat > nat,H: nat > nat] :
( ( finite_finite_nat @ C4 )
=> ( ( ord_less_eq_set_nat @ A @ C4 )
=> ( ( ord_less_eq_set_nat @ B3 @ C4 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ ( minus_minus_set_nat @ C4 @ A ) )
=> ( ( G @ A3 )
= one_one_nat ) )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ C4 @ B3 ) )
=> ( ( H @ B2 )
= one_one_nat ) )
=> ( ( ( groups708209901874060359at_nat @ G @ A )
= ( groups708209901874060359at_nat @ H @ B3 ) )
= ( ( groups708209901874060359at_nat @ G @ C4 )
= ( groups708209901874060359at_nat @ H @ C4 ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_1088_finite__sorted__distinct__unique,axiom,
! [A: set_list_nat] :
( ( finite8100373058378681591st_nat @ A )
=> ? [X2: list_list_nat] :
( ( ( set_list_nat2 @ X2 )
= A )
& ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ X2 )
& ( distinct_list_nat @ X2 )
& ! [Y4: list_list_nat] :
( ( ( ( set_list_nat2 @ Y4 )
= A )
& ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ Y4 )
& ( distinct_list_nat @ Y4 ) )
=> ( Y4 = X2 ) ) ) ) ).
% finite_sorted_distinct_unique
thf(fact_1089_finite__sorted__distinct__unique,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ? [X2: list_nat] :
( ( ( set_nat2 @ X2 )
= A )
& ( sorted_wrt_nat @ ord_less_eq_nat @ X2 )
& ( distinct_nat @ X2 )
& ! [Y4: list_nat] :
( ( ( ( set_nat2 @ Y4 )
= A )
& ( sorted_wrt_nat @ ord_less_eq_nat @ Y4 )
& ( distinct_nat @ Y4 ) )
=> ( Y4 = X2 ) ) ) ) ).
% finite_sorted_distinct_unique
thf(fact_1090_sum__diff__nat,axiom,
! [B3: set_list_nat,A: set_list_nat,F: list_nat > nat] :
( ( finite8100373058378681591st_nat @ B3 )
=> ( ( ord_le6045566169113846134st_nat @ B3 @ A )
=> ( ( groups4396056296759096172at_nat @ F @ ( minus_7954133019191499631st_nat @ A @ B3 ) )
= ( minus_minus_nat @ ( groups4396056296759096172at_nat @ F @ A ) @ ( groups4396056296759096172at_nat @ F @ B3 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_1091_sum__diff__nat,axiom,
! [B3: set_nat,A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ A )
=> ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A @ B3 ) )
= ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( groups3542108847815614940at_nat @ F @ B3 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_1092_sum__mono2,axiom,
! [B3: set_list_nat,A: set_list_nat,F: list_nat > nat] :
( ( finite8100373058378681591st_nat @ B3 )
=> ( ( ord_le6045566169113846134st_nat @ A @ B3 )
=> ( ! [B2: list_nat] :
( ( member_list_nat @ B2 @ ( minus_7954133019191499631st_nat @ B3 @ A ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B2 ) ) )
=> ( ord_less_eq_nat @ ( groups4396056296759096172at_nat @ F @ A ) @ ( groups4396056296759096172at_nat @ F @ B3 ) ) ) ) ) ).
% sum_mono2
thf(fact_1093_sum__mono2,axiom,
! [B3: set_nat,A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ! [B2: nat] :
( ( member_nat @ B2 @ ( minus_minus_set_nat @ B3 @ A ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B2 ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( groups3542108847815614940at_nat @ F @ B3 ) ) ) ) ) ).
% sum_mono2
thf(fact_1094_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
! [A: set_list_nat] :
( ( finite8100373058378681591st_nat @ A )
=> ~ ! [L4: list_list_nat] :
( ( sorted_wrt_list_nat @ ord_less_list_nat @ L4 )
=> ( ( ( set_list_nat2 @ L4 )
= A )
=> ( ( size_s3023201423986296836st_nat @ L4 )
!= ( finite_card_list_nat @ A ) ) ) ) ) ).
% sorted_list_of_set.finite_set_strict_sorted
thf(fact_1095_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ~ ! [L4: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ L4 )
=> ( ( ( set_nat2 @ L4 )
= A )
=> ( ( size_size_list_nat @ L4 )
!= ( finite_card_nat @ A ) ) ) ) ) ).
% sorted_list_of_set.finite_set_strict_sorted
thf(fact_1096_sum__strict__mono2,axiom,
! [B3: set_list_nat,A: set_list_nat,B: list_nat,F: list_nat > nat] :
( ( finite8100373058378681591st_nat @ B3 )
=> ( ( ord_le6045566169113846134st_nat @ A @ B3 )
=> ( ( member_list_nat @ B @ ( minus_7954133019191499631st_nat @ B3 @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ B3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ord_less_nat @ ( groups4396056296759096172at_nat @ F @ A ) @ ( groups4396056296759096172at_nat @ F @ B3 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_1097_sum__strict__mono2,axiom,
! [B3: set_nat,A: set_nat,B: nat,F: nat > nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( member_nat @ B @ ( minus_minus_set_nat @ B3 @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X2 ) ) )
=> ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( groups3542108847815614940at_nat @ F @ B3 ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_1098_augmentum__cancel,axiom,
! [K: nat,Ns: list_nat] :
( ( minus_minus_list_nat @ ( map_nat_nat @ ( plus_plus_nat @ K ) @ ( augmentum @ Ns ) ) @ ( cons_nat @ K @ ( map_nat_nat @ ( plus_plus_nat @ K ) @ ( augmentum @ Ns ) ) ) )
= Ns ) ).
% augmentum_cancel
thf(fact_1099_augmentum_Osimps_I2_J,axiom,
! [N: nat,Ns: list_nat] :
( ( augmentum @ ( cons_nat @ N @ Ns ) )
= ( cons_nat @ N @ ( map_nat_nat @ ( plus_plus_nat @ N ) @ ( augmentum @ Ns ) ) ) ) ).
% augmentum.simps(2)
thf(fact_1100_map__ident,axiom,
( ( map_nat_nat
@ ^ [X4: nat] : X4 )
= ( ^ [Xs: list_nat] : Xs ) ) ).
% map_ident
thf(fact_1101_map__eq__conv,axiom,
! [F: nat > nat,Xs2: list_nat,G: nat > nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Xs2 ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) ) ) ) ).
% map_eq_conv
thf(fact_1102_length__map,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( size_size_list_nat @ ( map_nat_nat @ F @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_map
thf(fact_1103_distinct__map__plus__iff,axiom,
! [A2: nat,Xs2: list_nat] :
( ( distinct_nat @ ( map_nat_nat @ ( plus_plus_nat @ A2 ) @ Xs2 ) )
= ( distinct_nat @ Xs2 ) ) ).
% distinct_map_plus_iff
thf(fact_1104_sum__list__0,axiom,
! [Xs2: list_nat] :
( ( groups4561878855575611511st_nat
@ ( map_nat_nat
@ ^ [X4: nat] : zero_zero_nat
@ Xs2 ) )
= zero_zero_nat ) ).
% sum_list_0
thf(fact_1105_sorted__map__plus__iff,axiom,
! [A2: nat,Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ ( plus_plus_nat @ A2 ) @ Xs2 ) )
= ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 ) ) ).
% sorted_map_plus_iff
thf(fact_1106_nth__map,axiom,
! [N: nat,Xs2: list_nat,F: nat > nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ ( map_nat_nat @ F @ Xs2 ) @ N )
= ( F @ ( nth_nat @ Xs2 @ N ) ) ) ) ).
% nth_map
thf(fact_1107_sorted__map,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ Xs2 ) )
= ( sorted_wrt_nat
@ ^ [X4: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) )
@ Xs2 ) ) ).
% sorted_map
thf(fact_1108_sum__list__addf,axiom,
! [F: nat > nat,G: nat > nat,Xs2: list_nat] :
( ( groups4561878855575611511st_nat
@ ( map_nat_nat
@ ^ [X4: nat] : ( plus_plus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
@ Xs2 ) )
= ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ F @ Xs2 ) ) @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ G @ Xs2 ) ) ) ) ).
% sum_list_addf
thf(fact_1109_list_Osimps_I9_J,axiom,
! [F: nat > nat,X21: nat,X222: list_nat] :
( ( map_nat_nat @ F @ ( cons_nat @ X21 @ X222 ) )
= ( cons_nat @ ( F @ X21 ) @ ( map_nat_nat @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_1110_Cons__eq__map__D,axiom,
! [X: nat,Xs2: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X @ Xs2 )
= ( map_nat_nat @ F @ Ys ) )
=> ? [Z: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z @ Zs2 ) )
& ( X
= ( F @ Z ) )
& ( Xs2
= ( map_nat_nat @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1111_map__eq__Cons__D,axiom,
! [F: nat > nat,Xs2: list_nat,Y: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( cons_nat @ Y @ Ys ) )
=> ? [Z: nat,Zs2: list_nat] :
( ( Xs2
= ( cons_nat @ Z @ Zs2 ) )
& ( ( F @ Z )
= Y )
& ( ( map_nat_nat @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_1112_Cons__eq__map__conv,axiom,
! [X: nat,Xs2: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X @ Xs2 )
= ( map_nat_nat @ F @ Ys ) )
= ( ? [Z5: nat,Zs3: list_nat] :
( ( Ys
= ( cons_nat @ Z5 @ Zs3 ) )
& ( X
= ( F @ Z5 ) )
& ( Xs2
= ( map_nat_nat @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1113_map__eq__Cons__conv,axiom,
! [F: nat > nat,Xs2: list_nat,Y: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( cons_nat @ Y @ Ys ) )
= ( ? [Z5: nat,Zs3: list_nat] :
( ( Xs2
= ( cons_nat @ Z5 @ Zs3 ) )
& ( ( F @ Z5 )
= Y )
& ( ( map_nat_nat @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1114_list_Omap__cong,axiom,
! [X: list_nat,Ya2: list_nat,F: nat > nat,G: nat > nat] :
( ( X = Ya2 )
=> ( ! [Z: nat] :
( ( member_nat @ Z @ ( set_nat2 @ Ya2 ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_nat_nat @ F @ X )
= ( map_nat_nat @ G @ Ya2 ) ) ) ) ).
% list.map_cong
thf(fact_1115_list_Omap__cong0,axiom,
! [X: list_nat,F: nat > nat,G: nat > nat] :
( ! [Z: nat] :
( ( member_nat @ Z @ ( set_nat2 @ X ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_nat_nat @ F @ X )
= ( map_nat_nat @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_1116_list_Oinj__map__strong,axiom,
! [X: list_nat,Xa2: list_nat,F: nat > nat,Fa: nat > nat] :
( ! [Z: nat,Za: nat] :
( ( member_nat @ Z @ ( set_nat2 @ X ) )
=> ( ( member_nat @ Za @ ( set_nat2 @ Xa2 ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_nat_nat @ F @ X )
= ( map_nat_nat @ Fa @ Xa2 ) )
=> ( X = Xa2 ) ) ) ).
% list.inj_map_strong
thf(fact_1117_list_Omap__ident__strong,axiom,
! [T2: list_list_nat,F: list_nat > list_nat] :
( ! [Z: list_nat] :
( ( member_list_nat @ Z @ ( set_list_nat2 @ T2 ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_li7225945977422193158st_nat @ F @ T2 )
= T2 ) ) ).
% list.map_ident_strong
thf(fact_1118_list_Omap__ident__strong,axiom,
! [T2: list_nat,F: nat > nat] :
( ! [Z: nat] :
( ( member_nat @ Z @ ( set_nat2 @ T2 ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_nat_nat @ F @ T2 )
= T2 ) ) ).
% list.map_ident_strong
thf(fact_1119_map__ext,axiom,
! [Xs2: list_nat,F: nat > nat,G: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_nat_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Xs2 ) ) ) ).
% map_ext
thf(fact_1120_map__idI,axiom,
! [Xs2: list_list_nat,F: list_nat > list_nat] :
( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs2 ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_li7225945977422193158st_nat @ F @ Xs2 )
= Xs2 ) ) ).
% map_idI
thf(fact_1121_map__idI,axiom,
! [Xs2: list_nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_nat_nat @ F @ Xs2 )
= Xs2 ) ) ).
% map_idI
thf(fact_1122_map__cong,axiom,
! [Xs2: list_nat,Ys: list_nat,F: nat > nat,G: nat > nat] :
( ( Xs2 = Ys )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Ys ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_nat_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_1123_ex__map__conv,axiom,
! [Ys: list_nat,F: nat > nat] :
( ( ? [Xs: list_nat] :
( Ys
= ( map_nat_nat @ F @ Xs ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Ys ) )
=> ? [Y3: nat] :
( X4
= ( F @ Y3 ) ) ) ) ) ).
% ex_map_conv
thf(fact_1124_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs2: list_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1125_sorted__wrt__map,axiom,
! [R: nat > nat > $o,F: nat > nat,Xs2: list_nat] :
( ( sorted_wrt_nat @ R @ ( map_nat_nat @ F @ Xs2 ) )
= ( sorted_wrt_nat
@ ^ [X4: nat,Y3: nat] : ( R @ ( F @ X4 ) @ ( F @ Y3 ) )
@ Xs2 ) ) ).
% sorted_wrt_map
thf(fact_1126_list_Omap__ident,axiom,
! [T2: list_nat] :
( ( map_nat_nat
@ ^ [X4: nat] : X4
@ T2 )
= T2 ) ).
% list.map_ident
thf(fact_1127_map__equality__iff,axiom,
! [F: nat > nat,Xs2: list_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Ys ) )
= ( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( ( F @ ( nth_nat @ Xs2 @ I2 ) )
= ( G @ ( nth_nat @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_1128_sum__list__mono,axiom,
! [Xs2: list_list_nat,F: list_nat > nat,G: list_nat > nat] :
( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs2 ) )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ F @ Xs2 ) ) @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ G @ Xs2 ) ) ) ) ).
% sum_list_mono
thf(fact_1129_sum__list__mono,axiom,
! [Xs2: list_nat,F: nat > nat,G: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ F @ Xs2 ) ) @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ G @ Xs2 ) ) ) ) ).
% sum_list_mono
thf(fact_1130_sum__list__distinct__conv__sum__set,axiom,
! [Xs2: list_nat,F: nat > nat] :
( ( distinct_nat @ Xs2 )
=> ( ( groups4561878855575611511st_nat @ ( map_nat_nat @ F @ Xs2 ) )
= ( groups3542108847815614940at_nat @ F @ ( set_nat2 @ Xs2 ) ) ) ) ).
% sum_list_distinct_conv_sum_set
thf(fact_1131_sum_Odistinct__set__conv__list,axiom,
! [Xs2: list_nat,G: nat > nat] :
( ( distinct_nat @ Xs2 )
=> ( ( groups3542108847815614940at_nat @ G @ ( set_nat2 @ Xs2 ) )
= ( groups4561878855575611511st_nat @ ( map_nat_nat @ G @ Xs2 ) ) ) ) ).
% sum.distinct_set_conv_list
thf(fact_1132_sum__list__Suc,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( groups4561878855575611511st_nat
@ ( map_nat_nat
@ ^ [X4: nat] : ( suc @ ( F @ X4 ) )
@ Xs2 ) )
= ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ F @ Xs2 ) ) @ ( size_size_list_nat @ Xs2 ) ) ) ).
% sum_list_Suc
thf(fact_1133_augmentum_Oelims,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( augmentum @ X )
= Y )
=> ( ( ( X = nil_nat )
=> ( Y != nil_nat ) )
=> ~ ! [N3: nat,Ns2: list_nat] :
( ( X
= ( cons_nat @ N3 @ Ns2 ) )
=> ( Y
!= ( cons_nat @ N3 @ ( map_nat_nat @ ( plus_plus_nat @ N3 ) @ ( augmentum @ Ns2 ) ) ) ) ) ) ) ).
% augmentum.elims
thf(fact_1134_finite__nat__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [S5: set_nat] :
? [K4: nat] : ( ord_less_eq_set_nat @ S5 @ ( set_ord_atMost_nat @ K4 ) ) ) ) ).
% finite_nat_iff_bounded_le
thf(fact_1135_augmentum_Osimps_I1_J,axiom,
( ( augmentum @ nil_nat )
= nil_nat ) ).
% augmentum.simps(1)
thf(fact_1136_list_Omap__disc__iff,axiom,
! [F: nat > nat,A2: list_nat] :
( ( ( map_nat_nat @ F @ A2 )
= nil_nat )
= ( A2 = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_1137_Nil__is__map__conv,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( nil_nat
= ( map_nat_nat @ F @ Xs2 ) )
= ( Xs2 = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_1138_map__is__Nil__conv,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= nil_nat )
= ( Xs2 = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_1139_minus__Nil,axiom,
! [Xs2: list_nat] :
( ( minus_minus_list_nat @ nil_nat @ Xs2 )
= nil_nat ) ).
% minus_Nil
thf(fact_1140_plus__Nil,axiom,
! [Xs2: list_nat] :
( ( plus_plus_list_nat @ nil_nat @ Xs2 )
= nil_nat ) ).
% plus_Nil
thf(fact_1141_list__incr__Nil,axiom,
! [I: nat] :
( ( list_incr @ I @ nil_nat )
= nil_nat ) ).
% list_incr_Nil
thf(fact_1142_dementum__Nil,axiom,
( ( dementum @ nil_nat )
= nil_nat ) ).
% dementum_Nil
thf(fact_1143_le__Nil,axiom,
! [X: list_nat] :
( ( ord_less_eq_list_nat @ X @ nil_nat )
= ( X = nil_nat ) ) ).
% le_Nil
thf(fact_1144_length__0__conv,axiom,
! [Xs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= zero_zero_nat )
= ( Xs2 = nil_nat ) ) ).
% length_0_conv
thf(fact_1145_sum__list_ONil,axiom,
( ( groups4561878855575611511st_nat @ nil_nat )
= zero_zero_nat ) ).
% sum_list.Nil
thf(fact_1146_length__greater__0__conv,axiom,
! [Xs2: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) )
= ( Xs2 != nil_nat ) ) ).
% length_greater_0_conv
thf(fact_1147_list_Osimps_I8_J,axiom,
! [F: nat > nat] :
( ( map_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_1148_Nil__less__Cons,axiom,
! [A2: nat,X: list_nat] : ( ord_less_list_nat @ nil_nat @ ( cons_nat @ A2 @ X ) ) ).
% Nil_less_Cons
thf(fact_1149_not__less__Nil,axiom,
! [X: list_nat] :
~ ( ord_less_list_nat @ X @ nil_nat ) ).
% not_less_Nil
thf(fact_1150_Nil__le__Cons,axiom,
! [X: list_nat] : ( ord_less_eq_list_nat @ nil_nat @ X ) ).
% Nil_le_Cons
thf(fact_1151_pointwise__less__Nil2,axiom,
! [X: list_nat] :
~ ( pointwise_less @ X @ nil_nat ) ).
% pointwise_less_Nil2
thf(fact_1152_pointwise__less__Nil,axiom,
! [X: list_nat] :
~ ( pointwise_less @ nil_nat @ X ) ).
% pointwise_less_Nil
thf(fact_1153_transpose_Ocases,axiom,
! [X: list_list_nat] :
( ( X != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X2: nat,Xs3: list_nat,Xss: list_list_nat] :
( X
!= ( cons_list_nat @ ( cons_nat @ X2 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_1154_list__encode_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ~ ! [X2: nat,Xs3: list_nat] :
( X
!= ( cons_nat @ X2 @ Xs3 ) ) ) ).
% list_encode.cases
thf(fact_1155_distinct_Osimps_I1_J,axiom,
distinct_nat @ nil_nat ).
% distinct.simps(1)
thf(fact_1156_sorted__wrt_Osimps_I1_J,axiom,
! [P: nat > nat > $o] : ( sorted_wrt_nat @ P @ nil_nat ) ).
% sorted_wrt.simps(1)
thf(fact_1157_list__nonempty__induct,axiom,
! [Xs2: list_nat,P: list_nat > $o] :
( ( Xs2 != nil_nat )
=> ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
=> ( ! [X2: nat,Xs3: list_nat] :
( ( Xs3 != nil_nat )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_nat @ X2 @ Xs3 ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_1158_list__induct2_H,axiom,
! [P: list_nat > list_nat > $o,Xs2: list_nat,Ys: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X2 @ Xs3 ) @ nil_nat )
=> ( ! [Y2: nat,Ys4: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y2 @ Ys4 ) )
=> ( ! [X2: nat,Xs3: list_nat,Y2: nat,Ys4: list_nat] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_nat @ X2 @ Xs3 ) @ ( cons_nat @ Y2 @ Ys4 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_1159_neq__Nil__conv,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
= ( ? [Y3: nat,Ys2: list_nat] :
( Xs2
= ( cons_nat @ Y3 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_1160_remdups__adj_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ( ! [X2: nat] :
( X
!= ( cons_nat @ X2 @ nil_nat ) )
=> ~ ! [X2: nat,Y2: nat,Xs3: list_nat] :
( X
!= ( cons_nat @ X2 @ ( cons_nat @ Y2 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_1161_min__list_Ocases,axiom,
! [X: list_nat] :
( ! [X2: nat,Xs3: list_nat] :
( X
!= ( cons_nat @ X2 @ Xs3 ) )
=> ( X = nil_nat ) ) ).
% min_list.cases
thf(fact_1162_list_Oexhaust,axiom,
! [Y: list_nat] :
( ( Y != nil_nat )
=> ~ ! [X212: nat,X223: list_nat] :
( Y
!= ( cons_nat @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_1163_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X222: list_nat] :
( ( List
= ( cons_nat @ X21 @ X222 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_1164_list_Odistinct_I1_J,axiom,
! [X21: nat,X222: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_1165_distinct__singleton,axiom,
! [X: nat] : ( distinct_nat @ ( cons_nat @ X @ nil_nat ) ) ).
% distinct_singleton
thf(fact_1166_sorted__wrt1,axiom,
! [P: nat > nat > $o,X: nat] : ( sorted_wrt_nat @ P @ ( cons_nat @ X @ nil_nat ) ) ).
% sorted_wrt1
thf(fact_1167_strict__sorted__simps_I1_J,axiom,
sorted_wrt_nat @ ord_less_nat @ nil_nat ).
% strict_sorted_simps(1)
thf(fact_1168_sorted0,axiom,
sorted_wrt_nat @ ord_less_eq_nat @ nil_nat ).
% sorted0
thf(fact_1169_list__induct4,axiom,
! [Xs2: 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 @ Xs2 )
= ( 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 )
=> ( ! [X2: nat,Xs3: list_nat,Y2: nat,Ys4: list_nat,Z: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs3 ) @ ( cons_nat @ Y2 @ Ys4 ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1170_list__induct3,axiom,
! [Xs2: list_nat,Ys: list_nat,Zs: list_nat,P: list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs3: list_nat,Y2: nat,Ys4: list_nat,Z: nat,Zs2: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 )
=> ( P @ ( cons_nat @ X2 @ Xs3 ) @ ( cons_nat @ Y2 @ Ys4 ) @ ( cons_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_1171_list__induct2,axiom,
! [Xs2: list_nat,Ys: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs3: list_nat,Y2: nat,Ys4: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_nat @ X2 @ Xs3 ) @ ( cons_nat @ Y2 @ Ys4 ) ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1172_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_1173_sorted1,axiom,
! [X: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X @ nil_nat ) ) ).
% sorted1
thf(fact_1174_sum__list__strict__mono,axiom,
! [Xs2: list_list_nat,F: list_nat > nat,G: list_nat > nat] :
( ( Xs2 != nil_list_nat )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs2 ) )
=> ( ord_less_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_nat @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ F @ Xs2 ) ) @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ G @ Xs2 ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_1175_sum__list__strict__mono,axiom,
! [Xs2: list_nat,F: nat > nat,G: nat > nat] :
( ( Xs2 != nil_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ord_less_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less_nat @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ F @ Xs2 ) ) @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ G @ Xs2 ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_1176_infinite__nat__iff__unbounded__le,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M5: nat] :
? [N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
& ( member_nat @ N4 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_1177_unbounded__k__infinite,axiom,
! [K: nat,S: set_nat] :
( ! [M4: nat] :
( ( ord_less_nat @ K @ M4 )
=> ? [N7: nat] :
( ( ord_less_nat @ M4 @ N7 )
& ( member_nat @ N7 @ S ) ) )
=> ~ ( finite_finite_nat @ S ) ) ).
% unbounded_k_infinite
thf(fact_1178_infinite__nat__iff__unbounded,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M5: nat] :
? [N4: nat] :
( ( ord_less_nat @ M5 @ N4 )
& ( member_nat @ N4 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_1179_finite__nat__bounded,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ? [K2: nat] : ( ord_less_eq_set_nat @ S @ ( set_ord_lessThan_nat @ K2 ) ) ) ).
% finite_nat_bounded
thf(fact_1180_finite__nat__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [S5: set_nat] :
? [K4: nat] : ( ord_less_eq_set_nat @ S5 @ ( set_ord_lessThan_nat @ K4 ) ) ) ) ).
% finite_nat_iff_bounded
thf(fact_1181_sum__list__dementum,axiom,
! [Xs2: list_nat,N: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( append_nat @ Xs2 @ ( cons_nat @ N @ nil_nat ) ) )
=> ( ( groups4561878855575611511st_nat @ ( dementum @ ( append_nat @ Xs2 @ ( cons_nat @ N @ nil_nat ) ) ) )
= N ) ) ).
% sum_list_dementum
thf(fact_1182_finsets__def,axiom,
( finsets_list_nat
= ( ^ [A4: set_list_nat,N4: nat] :
( collect_set_list_nat
@ ^ [N6: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ N6 @ A4 )
& ( ( finite_card_list_nat @ N6 )
= N4 ) ) ) ) ) ).
% finsets_def
thf(fact_1183_finsets__def,axiom,
( finsets_nat
= ( ^ [A4: set_nat,N4: nat] :
( collect_set_nat
@ ^ [N6: set_nat] :
( ( ord_less_eq_set_nat @ N6 @ A4 )
& ( ( finite_card_nat @ N6 )
= N4 ) ) ) ) ) ).
% finsets_def
thf(fact_1184_append_Oassoc,axiom,
! [A2: list_nat,B: list_nat,C: list_nat] :
( ( append_nat @ ( append_nat @ A2 @ B ) @ C )
= ( append_nat @ A2 @ ( append_nat @ B @ C ) ) ) ).
% append.assoc
thf(fact_1185_append__assoc,axiom,
! [Xs2: list_nat,Ys: list_nat,Zs: list_nat] :
( ( append_nat @ ( append_nat @ Xs2 @ Ys ) @ Zs )
= ( append_nat @ Xs2 @ ( append_nat @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_1186_append__same__eq,axiom,
! [Ys: list_nat,Xs2: list_nat,Zs: list_nat] :
( ( ( append_nat @ Ys @ Xs2 )
= ( append_nat @ Zs @ Xs2 ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_1187_same__append__eq,axiom,
! [Xs2: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( append_nat @ Xs2 @ Ys )
= ( append_nat @ Xs2 @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_1188_append__is__Nil__conv,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs2 @ Ys )
= nil_nat )
= ( ( Xs2 = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_1189_Nil__is__append__conv,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( nil_nat
= ( append_nat @ Xs2 @ Ys ) )
= ( ( Xs2 = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_1190_self__append__conv2,axiom,
! [Y: list_nat,Xs2: list_nat] :
( ( Y
= ( append_nat @ Xs2 @ Y ) )
= ( Xs2 = nil_nat ) ) ).
% self_append_conv2
thf(fact_1191_append__self__conv2,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs2 @ Ys )
= Ys )
= ( Xs2 = nil_nat ) ) ).
% append_self_conv2
thf(fact_1192_self__append__conv,axiom,
! [Y: list_nat,Ys: list_nat] :
( ( Y
= ( append_nat @ Y @ Ys ) )
= ( Ys = nil_nat ) ) ).
% self_append_conv
thf(fact_1193_append__self__conv,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs2 @ Ys )
= Xs2 )
= ( Ys = nil_nat ) ) ).
% append_self_conv
thf(fact_1194_append__Nil2,axiom,
! [Xs2: list_nat] :
( ( append_nat @ Xs2 @ nil_nat )
= Xs2 ) ).
% append_Nil2
thf(fact_1195_append_Oright__neutral,axiom,
! [A2: list_nat] :
( ( append_nat @ A2 @ nil_nat )
= A2 ) ).
% append.right_neutral
thf(fact_1196_append__eq__append__conv,axiom,
! [Xs2: list_nat,Ys: list_nat,Us: list_nat,Vs: list_nat] :
( ( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
| ( ( size_size_list_nat @ Us )
= ( size_size_list_nat @ Vs ) ) )
=> ( ( ( append_nat @ Xs2 @ Us )
= ( append_nat @ Ys @ Vs ) )
= ( ( Xs2 = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_1197_map__append,axiom,
! [F: nat > nat,Xs2: list_nat,Ys: list_nat] :
( ( map_nat_nat @ F @ ( append_nat @ Xs2 @ Ys ) )
= ( append_nat @ ( map_nat_nat @ F @ Xs2 ) @ ( map_nat_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_1198_append1__eq__conv,axiom,
! [Xs2: list_nat,X: nat,Ys: list_nat,Y: nat] :
( ( ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) )
= ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) )
= ( ( Xs2 = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_1199_length__append,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( size_size_list_nat @ ( append_nat @ Xs2 @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_append
thf(fact_1200_sum__list__append,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( groups4561878855575611511st_nat @ ( append_nat @ Xs2 @ Ys ) )
= ( plus_plus_nat @ ( groups4561878855575611511st_nat @ Xs2 ) @ ( groups4561878855575611511st_nat @ Ys ) ) ) ).
% sum_list_append
thf(fact_1201_nth__append__length,axiom,
! [Xs2: list_nat,X: nat,Ys: list_nat] :
( ( nth_nat @ ( append_nat @ Xs2 @ ( cons_nat @ X @ Ys ) ) @ ( size_size_list_nat @ Xs2 ) )
= X ) ).
% nth_append_length
thf(fact_1202_nth__append__length__plus,axiom,
! [Xs2: list_nat,Ys: list_nat,N: nat] :
( ( nth_nat @ ( append_nat @ Xs2 @ Ys ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs2 ) @ N ) )
= ( nth_nat @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_1203_append__eq__map__conv,axiom,
! [Ys: list_nat,Zs: list_nat,F: nat > nat,Xs2: list_nat] :
( ( ( append_nat @ Ys @ Zs )
= ( map_nat_nat @ F @ Xs2 ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs2
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys
= ( map_nat_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_nat @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_1204_map__eq__append__conv,axiom,
! [F: nat > nat,Xs2: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( append_nat @ Ys @ Zs ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs2
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys
= ( map_nat_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_nat @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_1205_eq__Nil__appendI,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( Xs2 = Ys )
=> ( Xs2
= ( append_nat @ nil_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_1206_append_Oleft__neutral,axiom,
! [A2: list_nat] :
( ( append_nat @ nil_nat @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_1207_append__Nil,axiom,
! [Ys: list_nat] :
( ( append_nat @ nil_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_1208_rev__induct,axiom,
! [P: list_nat > $o,Xs2: list_nat] :
( ( P @ nil_nat )
=> ( ! [X2: nat,Xs3: list_nat] :
( ( P @ Xs3 )
=> ( P @ ( append_nat @ Xs3 @ ( cons_nat @ X2 @ nil_nat ) ) ) )
=> ( P @ Xs2 ) ) ) ).
% rev_induct
thf(fact_1209_rev__exhaust,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ~ ! [Ys4: list_nat,Y2: nat] :
( Xs2
!= ( append_nat @ Ys4 @ ( cons_nat @ Y2 @ nil_nat ) ) ) ) ).
% rev_exhaust
thf(fact_1210_Cons__eq__append__conv,axiom,
! [X: nat,Xs2: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X @ Xs2 )
= ( append_nat @ Ys @ Zs ) )
= ( ( ( Ys = nil_nat )
& ( ( cons_nat @ X @ Xs2 )
= Zs ) )
| ? [Ys5: list_nat] :
( ( ( cons_nat @ X @ Ys5 )
= Ys )
& ( Xs2
= ( append_nat @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_1211_append__eq__Cons__conv,axiom,
! [Ys: list_nat,Zs: list_nat,X: nat,Xs2: list_nat] :
( ( ( append_nat @ Ys @ Zs )
= ( cons_nat @ X @ Xs2 ) )
= ( ( ( Ys = nil_nat )
& ( Zs
= ( cons_nat @ X @ Xs2 ) ) )
| ? [Ys5: list_nat] :
( ( Ys
= ( cons_nat @ X @ Ys5 ) )
& ( ( append_nat @ Ys5 @ Zs )
= Xs2 ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_1212_rev__nonempty__induct,axiom,
! [Xs2: list_nat,P: list_nat > $o] :
( ( Xs2 != nil_nat )
=> ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
=> ( ! [X2: nat,Xs3: list_nat] :
( ( Xs3 != nil_nat )
=> ( ( P @ Xs3 )
=> ( P @ ( append_nat @ Xs3 @ ( cons_nat @ X2 @ nil_nat ) ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% rev_nonempty_induct
thf(fact_1213_append__Cons,axiom,
! [X: nat,Xs2: list_nat,Ys: list_nat] :
( ( append_nat @ ( cons_nat @ X @ Xs2 ) @ Ys )
= ( cons_nat @ X @ ( append_nat @ Xs2 @ Ys ) ) ) ).
% append_Cons
thf(fact_1214_Cons__eq__appendI,axiom,
! [X: nat,Xs1: list_nat,Ys: list_nat,Xs2: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X @ Xs1 )
= Ys )
=> ( ( Xs2
= ( append_nat @ Xs1 @ Zs ) )
=> ( ( cons_nat @ X @ Xs2 )
= ( append_nat @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_1215_append__eq__appendI,axiom,
! [Xs2: list_nat,Xs1: list_nat,Zs: list_nat,Ys: list_nat,Us: list_nat] :
( ( ( append_nat @ Xs2 @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_nat @ Xs1 @ Us ) )
=> ( ( append_nat @ Xs2 @ Ys )
= ( append_nat @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_1216_append__eq__append__conv2,axiom,
! [Xs2: list_nat,Ys: list_nat,Zs: list_nat,Ts: list_nat] :
( ( ( append_nat @ Xs2 @ Ys )
= ( append_nat @ Zs @ Ts ) )
= ( ? [Us2: list_nat] :
( ( ( Xs2
= ( append_nat @ Zs @ Us2 ) )
& ( ( append_nat @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_nat @ Xs2 @ Us2 )
= Zs )
& ( Ys
= ( append_nat @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_1217_sorted__wrt__append,axiom,
! [P: nat > nat > $o,Xs2: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ P @ ( append_nat @ Xs2 @ Ys ) )
= ( ( sorted_wrt_nat @ P @ Xs2 )
& ( sorted_wrt_nat @ P @ Ys )
& ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ ( set_nat2 @ Ys ) )
=> ( P @ X4 @ Y3 ) ) ) ) ) ).
% sorted_wrt_append
thf(fact_1218_split__list__first__prop__iff,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
& ( P @ X4 ) ) )
= ( ? [Ys2: list_nat,X4: nat] :
( ? [Zs3: list_nat] :
( Xs2
= ( append_nat @ Ys2 @ ( cons_nat @ X4 @ Zs3 ) ) )
& ( P @ X4 )
& ! [Y3: nat] :
( ( member_nat @ Y3 @ ( set_nat2 @ Ys2 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_1219_split__list__last__prop__iff,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
& ( P @ X4 ) ) )
= ( ? [Ys2: list_nat,X4: nat,Zs3: list_nat] :
( ( Xs2
= ( append_nat @ Ys2 @ ( cons_nat @ X4 @ Zs3 ) ) )
& ( P @ X4 )
& ! [Y3: nat] :
( ( member_nat @ Y3 @ ( set_nat2 @ Zs3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_1220_in__set__conv__decomp__first,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
= ( ? [Ys2: list_list_nat,Zs3: list_list_nat] :
( ( Xs2
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X @ Zs3 ) ) )
& ~ ( member_list_nat @ X @ ( set_list_nat2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1221_in__set__conv__decomp__first,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
= ( ? [Ys2: list_nat,Zs3: list_nat] :
( ( Xs2
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs3 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1222_in__set__conv__decomp__last,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
= ( ? [Ys2: list_list_nat,Zs3: list_list_nat] :
( ( Xs2
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X @ Zs3 ) ) )
& ~ ( member_list_nat @ X @ ( set_list_nat2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1223_in__set__conv__decomp__last,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
= ( ? [Ys2: list_nat,Zs3: list_nat] :
( ( Xs2
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs3 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1224_split__list__first__propE,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
& ( P @ X3 ) )
=> ~ ! [Ys4: list_nat,X2: nat] :
( ? [Zs2: list_nat] :
( Xs2
= ( append_nat @ Ys4 @ ( cons_nat @ X2 @ Zs2 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat2 @ Ys4 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_1225_split__list__last__propE,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
& ( P @ X3 ) )
=> ~ ! [Ys4: list_nat,X2: nat,Zs2: list_nat] :
( ( Xs2
= ( append_nat @ Ys4 @ ( cons_nat @ X2 @ Zs2 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_1226_split__list__first__prop,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
& ( P @ X3 ) )
=> ? [Ys4: list_nat,X2: nat] :
( ? [Zs2: list_nat] :
( Xs2
= ( append_nat @ Ys4 @ ( cons_nat @ X2 @ Zs2 ) ) )
& ( P @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat2 @ Ys4 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_1227_split__list__last__prop,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
& ( P @ X3 ) )
=> ? [Ys4: list_nat,X2: nat,Zs2: list_nat] :
( ( Xs2
= ( append_nat @ Ys4 @ ( cons_nat @ X2 @ Zs2 ) ) )
& ( P @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_1228_in__set__conv__decomp,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
= ( ? [Ys2: list_list_nat,Zs3: list_list_nat] :
( Xs2
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1229_in__set__conv__decomp,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
= ( ? [Ys2: list_nat,Zs3: list_nat] :
( Xs2
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1230_append__Cons__eq__iff,axiom,
! [X: list_nat,Xs2: list_list_nat,Ys: list_list_nat,Xs4: list_list_nat,Ys6: list_list_nat] :
( ~ ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
=> ( ~ ( member_list_nat @ X @ ( set_list_nat2 @ Ys ) )
=> ( ( ( append_list_nat @ Xs2 @ ( cons_list_nat @ X @ Ys ) )
= ( append_list_nat @ Xs4 @ ( cons_list_nat @ X @ Ys6 ) ) )
= ( ( Xs2 = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1231_append__Cons__eq__iff,axiom,
! [X: nat,Xs2: list_nat,Ys: list_nat,Xs4: list_nat,Ys6: list_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( ~ ( member_nat @ X @ ( set_nat2 @ Ys ) )
=> ( ( ( append_nat @ Xs2 @ ( cons_nat @ X @ Ys ) )
= ( append_nat @ Xs4 @ ( cons_nat @ X @ Ys6 ) ) )
= ( ( Xs2 = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1232_split__list__propE,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
& ( P @ X3 ) )
=> ~ ! [Ys4: list_nat,X2: nat] :
( ? [Zs2: list_nat] :
( Xs2
= ( append_nat @ Ys4 @ ( cons_nat @ X2 @ Zs2 ) ) )
=> ~ ( P @ X2 ) ) ) ).
% split_list_propE
thf(fact_1233_split__list__first,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
=> ? [Ys4: list_list_nat,Zs2: list_list_nat] :
( ( Xs2
= ( append_list_nat @ Ys4 @ ( cons_list_nat @ X @ Zs2 ) ) )
& ~ ( member_list_nat @ X @ ( set_list_nat2 @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_1234_split__list__first,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ? [Ys4: list_nat,Zs2: list_nat] :
( ( Xs2
= ( append_nat @ Ys4 @ ( cons_nat @ X @ Zs2 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_1235_split__list__prop,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
& ( P @ X3 ) )
=> ? [Ys4: list_nat,X2: nat] :
( ? [Zs2: list_nat] :
( Xs2
= ( append_nat @ Ys4 @ ( cons_nat @ X2 @ Zs2 ) ) )
& ( P @ X2 ) ) ) ).
% split_list_prop
thf(fact_1236_split__list__last,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
=> ? [Ys4: list_list_nat,Zs2: list_list_nat] :
( ( Xs2
= ( append_list_nat @ Ys4 @ ( cons_list_nat @ X @ Zs2 ) ) )
& ~ ( member_list_nat @ X @ ( set_list_nat2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_1237_split__list__last,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ? [Ys4: list_nat,Zs2: list_nat] :
( ( Xs2
= ( append_nat @ Ys4 @ ( cons_nat @ X @ Zs2 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_1238_split__list,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
=> ? [Ys4: list_list_nat,Zs2: list_list_nat] :
( Xs2
= ( append_list_nat @ Ys4 @ ( cons_list_nat @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_1239_split__list,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ? [Ys4: list_nat,Zs2: list_nat] :
( Xs2
= ( append_nat @ Ys4 @ ( cons_nat @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_1240_same__length__different,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( Xs2 != Ys )
=> ( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ? [Pre: list_nat,X2: nat,Xs5: list_nat,Y2: nat,Ys7: list_nat] :
( ( X2 != Y2 )
& ( Xs2
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X2 @ nil_nat ) @ Xs5 ) ) )
& ( Ys
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y2 @ nil_nat ) @ Ys7 ) ) ) ) ) ) ).
% same_length_different
thf(fact_1241_sorted__append,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( append_nat @ Xs2 @ Ys ) )
= ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
& ( sorted_wrt_nat @ ord_less_eq_nat @ Ys )
& ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ ( set_nat2 @ Ys ) )
=> ( ord_less_eq_nat @ X4 @ Y3 ) ) ) ) ) ).
% sorted_append
thf(fact_1242_not__distinct__decomp,axiom,
! [Ws: list_nat] :
( ~ ( distinct_nat @ Ws )
=> ? [Xs3: list_nat,Ys4: list_nat,Zs2: list_nat,Y2: nat] :
( Ws
= ( append_nat @ Xs3 @ ( append_nat @ ( cons_nat @ Y2 @ nil_nat ) @ ( append_nat @ Ys4 @ ( append_nat @ ( cons_nat @ Y2 @ nil_nat ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_1243_not__distinct__conv__prefix,axiom,
! [As: list_list_nat] :
( ( ~ ( distinct_list_nat @ As ) )
= ( ? [Xs: list_list_nat,Y3: list_nat,Ys2: list_list_nat] :
( ( member_list_nat @ Y3 @ ( set_list_nat2 @ Xs ) )
& ( distinct_list_nat @ Xs )
& ( As
= ( append_list_nat @ Xs @ ( cons_list_nat @ Y3 @ Ys2 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_1244_not__distinct__conv__prefix,axiom,
! [As: list_nat] :
( ( ~ ( distinct_nat @ As ) )
= ( ? [Xs: list_nat,Y3: nat,Ys2: list_nat] :
( ( member_nat @ Y3 @ ( set_nat2 @ Xs ) )
& ( distinct_nat @ Xs )
& ( As
= ( append_nat @ Xs @ ( cons_nat @ Y3 @ Ys2 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_1245_length__Suc__conv__rev,axiom,
! [Xs2: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( suc @ N ) )
= ( ? [Y3: nat,Ys2: list_nat] :
( ( Xs2
= ( append_nat @ Ys2 @ ( cons_nat @ Y3 @ nil_nat ) ) )
& ( ( size_size_list_nat @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_1246_length__append__singleton,axiom,
! [Xs2: list_nat,X: nat] :
( ( size_size_list_nat @ ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) ) )
= ( suc @ ( size_size_list_nat @ Xs2 ) ) ) ).
% length_append_singleton
thf(fact_1247_nth__append,axiom,
! [N: nat,Xs2: list_nat,Ys: list_nat] :
( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ ( append_nat @ Xs2 @ Ys ) @ N )
= ( nth_nat @ Xs2 @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ ( append_nat @ Xs2 @ Ys ) @ N )
= ( nth_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs2 ) ) ) ) ) ) ).
% nth_append
thf(fact_1248_sorted__imp__pointwise,axiom,
! [Xs2: list_nat,N: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( append_nat @ Xs2 @ ( cons_nat @ N @ nil_nat ) ) )
=> ( pointwise_le @ ( cons_nat @ zero_zero_nat @ Xs2 ) @ ( append_nat @ Xs2 @ ( cons_nat @ N @ nil_nat ) ) ) ) ).
% sorted_imp_pointwise
thf(fact_1249_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_1250_pointwise__le__refl,axiom,
! [X: list_nat] : ( pointwise_le @ X @ X ) ).
% pointwise_le_refl
thf(fact_1251_pointwise__le__Nil,axiom,
! [X: list_nat] :
( ( pointwise_le @ nil_nat @ X )
= ( X = nil_nat ) ) ).
% pointwise_le_Nil
thf(fact_1252_pointwise__le__Nil2,axiom,
! [X: list_nat] :
( ( pointwise_le @ X @ nil_nat )
= ( X = nil_nat ) ) ).
% pointwise_le_Nil2
thf(fact_1253_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_1254_length__n__lists__elem,axiom,
! [Ys: list_nat,N: nat,Xs2: list_nat] :
( ( member_list_nat @ Ys @ ( set_list_nat2 @ ( n_lists_nat @ N @ Xs2 ) ) )
=> ( ( size_size_list_nat @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_1255_distinct__n__lists,axiom,
! [Xs2: list_nat,N: nat] :
( ( distinct_nat @ Xs2 )
=> ( distinct_list_nat @ ( n_lists_nat @ N @ Xs2 ) ) ) ).
% distinct_n_lists
thf(fact_1256_pointwise__le__plus,axiom,
! [Xs2: list_nat,Ys: list_nat,Zs: list_nat] :
( ( pointwise_le @ Xs2 @ Ys )
=> ( ( ord_less_eq_nat @ ( size_size_list_nat @ Ys ) @ ( size_size_list_nat @ Zs ) )
=> ( pointwise_le @ Xs2 @ ( plus_plus_list_nat @ Ys @ Zs ) ) ) ) ).
% pointwise_le_plus
thf(fact_1257_pointwise__le__imp___092_060sigma_062,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( pointwise_le @ Xs2 @ Ys )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ Xs2 ) @ ( groups4561878855575611511st_nat @ Ys ) ) ) ).
% pointwise_le_imp_\<sigma>
thf(fact_1258_pairwise__minus__cancel,axiom,
! [Z4: list_nat,X: list_nat,Y: list_nat] :
( ( pointwise_le @ Z4 @ X )
=> ( ( pointwise_le @ Z4 @ Y )
=> ( ( ( minus_minus_list_nat @ X @ Z4 )
= ( minus_minus_list_nat @ Y @ Z4 ) )
=> ( X = Y ) ) ) ) ).
% pairwise_minus_cancel
thf(fact_1259_pointwise__le__iff__less__equal,axiom,
( pointwise_le
= ( ^ [X4: list_nat,Y3: list_nat] :
( ( pointwise_less @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ) ).
% pointwise_le_iff_less_equal
thf(fact_1260_pointwise__less__def,axiom,
( pointwise_less
= ( ^ [X4: list_nat,Y3: list_nat] :
( ( pointwise_le @ X4 @ Y3 )
& ( X4 != Y3 ) ) ) ) ).
% pointwise_less_def
thf(fact_1261_pointwise__le__trans,axiom,
! [X: list_nat,Y: list_nat,Z4: list_nat] :
( ( pointwise_le @ X @ Y )
=> ( ( pointwise_le @ Y @ Z4 )
=> ( pointwise_le @ X @ Z4 ) ) ) ).
% pointwise_le_trans
thf(fact_1262_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_1263_sum__list__minus,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( pointwise_le @ Xs2 @ Ys )
=> ( ( groups4561878855575611511st_nat @ ( minus_minus_list_nat @ Ys @ Xs2 ) )
= ( minus_minus_nat @ ( groups4561878855575611511st_nat @ Ys ) @ ( groups4561878855575611511st_nat @ Xs2 ) ) ) ) ).
% sum_list_minus
thf(fact_1264_pointwise__le__iff__nth,axiom,
( pointwise_le
= ( ^ [X4: list_nat,Y3: list_nat] :
( ( ( size_size_list_nat @ X4 )
= ( size_size_list_nat @ Y3 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ X4 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ X4 @ I2 ) @ ( nth_nat @ Y3 @ I2 ) ) ) ) ) ) ).
% pointwise_le_iff_nth
thf(fact_1265_pointwise__less__iff2,axiom,
( pointwise_less
= ( ^ [X4: list_nat,Y3: list_nat] :
( ( pointwise_le @ X4 @ Y3 )
& ? [K4: nat] :
( ( ord_less_nat @ K4 @ ( size_size_list_nat @ X4 ) )
& ( ord_less_nat @ ( nth_nat @ X4 @ K4 ) @ ( nth_nat @ Y3 @ K4 ) ) ) ) ) ) ).
% pointwise_less_iff2
thf(fact_1266_set__n__lists,axiom,
! [N: nat,Xs2: list_nat] :
( ( set_list_nat2 @ ( n_lists_nat @ N @ Xs2 ) )
= ( collect_list_nat
@ ^ [Ys2: list_nat] :
( ( ( size_size_list_nat @ Ys2 )
= N )
& ( ord_less_eq_set_nat @ ( set_nat2 @ Ys2 ) @ ( set_nat2 @ Xs2 ) ) ) ) ) ).
% set_n_lists
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
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 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( groups3542108847815614940at_nat
@ ^ [J: nat] : ( minus_minus_nat @ ( nth_nat @ ns @ J ) @ ( if_nat @ ( J = zero_zero_nat ) @ zero_zero_nat @ ( nth_nat @ ns @ ( minus_minus_nat @ J @ one_one_nat ) ) ) )
@ ( set_ord_atMost_nat @ i ) )
= ( groups3542108847815614940at_nat
@ ^ [J: nat] : ( if_nat @ ( J = zero_zero_nat ) @ ( nth_nat @ ns @ zero_zero_nat ) @ ( minus_minus_nat @ ( nth_nat @ ns @ J ) @ ( nth_nat @ ns @ ( minus_minus_nat @ J @ one_one_nat ) ) ) )
@ ( set_or1269000886237332187st_nat @ zero_zero_nat @ i ) ) ) ).
%------------------------------------------------------------------------------