TPTP Problem File: SLH0915^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Khovanskii_Theorem/0008_Khovanskii/prob_00965_035900__13633852_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1397 ( 514 unt; 129 typ; 0 def)
% Number of atoms : 3747 (1346 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 11081 ( 394 ~; 59 |; 308 &;8577 @)
% ( 0 <=>;1743 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 7 avg)
% Number of types : 10 ( 9 usr)
% Number of type conns : 737 ( 737 >; 0 *; 0 +; 0 <<)
% Number of symbols : 123 ( 120 usr; 19 con; 0-5 aty)
% Number of variables : 3641 ( 297 ^;3162 !; 182 ?;3641 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:15:25.558
%------------------------------------------------------------------------------
% Could-be-implicit typings (9)
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
set_list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
set_set_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (120)
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_Fun_Obij__betw_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
bij_be5167025596389125688st_nat: ( list_nat > list_nat ) > set_list_nat > set_list_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
bij_be8532844293280997160at_nat: ( list_nat > nat ) > set_list_nat > set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
bij_be5530832588036223086st_nat: ( list_nat > set_list_nat ) > set_list_nat > set_set_list_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
bij_be6293887246118711976st_nat: ( nat > list_nat ) > set_nat > set_list_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
bij_be4838622947795563742st_nat: ( nat > set_list_nat ) > set_nat > set_set_list_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_001t__List__Olist_It__Nat__Onat_J,type,
bij_be3408842901938555246st_nat: ( set_list_nat > list_nat ) > set_set_list_nat > set_list_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_001t__Nat__Onat,type,
bij_be3430625573890428254at_nat: ( set_list_nat > nat ) > set_set_list_nat > set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
bij_be4730079884550779812st_nat: ( set_list_nat > set_list_nat ) > set_set_list_nat > set_set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
inj_on3049792774292151987st_nat: ( list_nat > list_nat ) > set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
inj_on_list_nat_nat: ( list_nat > nat ) > set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
inj_on8624761805129053417st_nat: ( list_nat > set_list_nat ) > set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
inj_on_nat_list_nat: ( nat > list_nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
inj_on_nat_nat: ( nat > nat ) > 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_It__List__Olist_It__Nat__Onat_J_J,type,
minus_3911745200923244873st_nat: list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__List__Olist_It__Nat__Onat_J,type,
minus_minus_list_nat: list_nat > list_nat > list_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
minus_7954133019191499631st_nat: set_list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Set__Oset_It__Nat__Onat_J,type,
one_one_set_nat: set_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_Oplus__class_Oplus_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
plus_p884110394369815071st_nat: set_list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Set__Oset_It__Nat__Onat_J,type,
plus_plus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Set__Oset_It__Nat__Onat_J,type,
zero_zero_set_nat: set_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_001t__List__Olist_It__Nat__Onat_J,type,
khovanskii_list_nat: set_list_nat > ( list_nat > list_nat > list_nat ) > list_nat > set_list_nat > $o ).
thf(sy_c_Khovanskii_OKhovanskii_001t__Nat__Onat,type,
khovanskii_nat: set_nat > ( nat > nat > nat ) > nat > set_nat > $o ).
thf(sy_c_Khovanskii_OKhovanskii_O_092_060alpha_062_001t__List__Olist_It__Nat__Onat_J,type,
alpha_list_nat: set_list_nat > ( list_nat > list_nat > list_nat ) > list_nat > set_list_nat > list_nat > list_nat ).
thf(sy_c_Khovanskii_OKhovanskii_O_092_060alpha_062_001t__Nat__Onat,type,
alpha_nat: set_nat > ( nat > nat > nat ) > nat > set_nat > list_nat > nat ).
thf(sy_c_Khovanskii_OKhovanskii_OaA_001t__List__Olist_It__Nat__Onat_J,type,
aA_list_nat: set_list_nat > list_list_nat ).
thf(sy_c_Khovanskii_OKhovanskii_OaA_001t__Nat__Onat,type,
aA_nat: set_nat > list_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_OKhovanskii_Ouseless_001t__List__Olist_It__Nat__Onat_J,type,
useless_list_nat: set_list_nat > ( list_nat > list_nat > list_nat ) > list_nat > set_list_nat > list_nat > $o ).
thf(sy_c_Khovanskii_OKhovanskii_Ouseless_001t__Nat__Onat,type,
useless_nat: set_nat > ( nat > nat > nat ) > nat > set_nat > list_nat > $o ).
thf(sy_c_Khovanskii_Omax__pointwise,type,
max_pointwise: nat > set_list_nat > list_nat ).
thf(sy_c_Khovanskii_Omin__pointwise,type,
min_pointwise: nat > set_list_nat > list_nat ).
thf(sy_c_Khovanskii_Opointwise__le,type,
pointwise_le: list_nat > list_nat > $o ).
thf(sy_c_Khovanskii_Opointwise__less,type,
pointwise_less: list_nat > list_nat > $o ).
thf(sy_c_List_Ogen__length_001t__List__Olist_It__Nat__Onat_J,type,
gen_length_list_nat: nat > list_list_nat > nat ).
thf(sy_c_List_Ogen__length_001t__Nat__Onat,type,
gen_length_nat: nat > list_nat > nat ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Nat__Onat_J,type,
cons_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__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__List__Olist_It__Nat__Onat_J,type,
map_nat_list_nat: ( nat > list_nat ) > list_nat > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__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__update_001t__List__Olist_It__Nat__Onat_J,type,
list_update_list_nat: list_list_nat > nat > list_nat > list_list_nat ).
thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
list_update_nat: list_nat > nat > nat > list_nat ).
thf(sy_c_List_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_Oreplicate_001t__List__Olist_It__Nat__Onat_J,type,
replicate_list_nat: nat > list_nat > list_list_nat ).
thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
replicate_nat: nat > nat > list_nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
size_s3023201423986296836st_nat: list_list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
bot_bot_set_list_nat: set_list_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
bot_bo3886227569956363488st_nat: set_set_list_nat ).
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__eq_001_062_It__List__Olist_It__Nat__Onat_J_M_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J_J,type,
ord_le6558929396352911974_nat_o: ( list_nat > list_nat > $o ) > ( list_nat > list_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
ord_le1520216061033275535_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
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_It__List__Olist_It__Nat__Onat_J_J,type,
ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
collec5989764272469232197st_nat: ( list_list_nat > $o ) > set_list_list_nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
collect_set_list_nat: ( set_list_nat > $o ) > set_set_list_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
insert_list_nat: list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_Wellfounded_OwfP_001t__List__Olist_It__Nat__Onat_J,type,
wfP_list_nat: ( list_nat > list_nat > $o ) > $o ).
thf(sy_c_Wellfounded_OwfP_001t__Nat__Onat,type,
wfP_nat: ( nat > nat > $o ) > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
member_list_list_nat: list_list_nat > set_list_list_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
member_set_list_nat: set_list_nat > set_set_list_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_v_S,type,
s: set_list_nat ).
thf(sy_v_T____,type,
t: set_list_nat ).
thf(sy_v_f____,type,
f: list_nat > list_nat ).
thf(sy_v_g_H____,type,
g: list_nat > list_nat ).
thf(sy_v_g____,type,
g2: list_nat > list_nat ).
thf(sy_v_m____,type,
m: nat ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_r,type,
r: nat ).
thf(sy_v_x_H,type,
x: list_nat ).
thf(sy_v_x____,type,
x2: list_nat ).
% Relevant facts (1264)
thf(fact_0_that,axiom,
member_list_nat @ x2 @ t ).
% that
thf(fact_1_g__def,axiom,
( g2
= ( map_nat_nat @ suc ) ) ).
% g_def
thf(fact_2__092_060open_062length_Ax_A_061_Ar_092_060close_062,axiom,
( ( size_size_list_nat @ x2 )
= r ) ).
% \<open>length x = r\<close>
thf(fact_3_g_H__def,axiom,
( g
= ( ^ [X: list_nat] : ( minus_minus_list_nat @ X @ ( replicate_nat @ ( size_size_list_nat @ X ) @ one_one_nat ) ) ) ) ).
% g'_def
thf(fact_4_calculation,axiom,
! [X2: list_nat] :
( ( member_list_nat @ X2 @ ( length_sum_set @ r @ m ) )
=> ( member_list_nat @ ( g2 @ X2 ) @ t ) ) ).
% calculation
thf(fact_5__092_060open_062inj__on_Ag_A_Ilength__sum__set_Ar_Am_J_092_060close_062,axiom,
inj_on3049792774292151987st_nat @ g2 @ ( length_sum_set @ r @ m ) ).
% \<open>inj_on g (length_sum_set r m)\<close>
thf(fact_6_r1__x,axiom,
pointwise_le @ ( replicate_nat @ r @ ( suc @ zero_zero_nat ) ) @ x2 ).
% r1_x
thf(fact_7_Khovanskii_OaA_Ocong,axiom,
aA_list_nat = aA_list_nat ).
% Khovanskii.aA.cong
thf(fact_8_Khovanskii_OaA_Ocong,axiom,
aA_nat = aA_nat ).
% Khovanskii.aA.cong
thf(fact_9_Khovanskii_Ouseless_Ocong,axiom,
useless_list_nat = useless_list_nat ).
% Khovanskii.useless.cong
thf(fact_10_Khovanskii_Ouseless_Ocong,axiom,
useless_nat = useless_nat ).
% Khovanskii.useless.cong
thf(fact_11_lenx_H,axiom,
( ( size_size_list_nat @ x )
= r ) ).
% lenx'
thf(fact_12_pointwise__le__refl,axiom,
! [X2: list_nat] : ( pointwise_le @ X2 @ X2 ) ).
% pointwise_le_refl
thf(fact_13_f__def,axiom,
( f
= ( ^ [X: list_nat] : ( minus_minus_list_nat @ X @ x ) ) ) ).
% f_def
thf(fact_14_assms_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ r ).
% assms(1)
thf(fact_15_zero__pointwise__le__iff,axiom,
! [R: nat,X2: list_nat] :
( ( pointwise_le @ ( replicate_nat @ R @ zero_zero_nat ) @ X2 )
= ( ( size_size_list_nat @ X2 )
= R ) ) ).
% zero_pointwise_le_iff
thf(fact_16_pointwise__le__trans,axiom,
! [X2: list_nat,Y: list_nat,Z: list_nat] :
( ( pointwise_le @ X2 @ Y )
=> ( ( pointwise_le @ Y @ Z )
=> ( pointwise_le @ X2 @ Z ) ) ) ).
% pointwise_le_trans
thf(fact_17_pointwise__le__antisym,axiom,
! [X2: list_nat,Y: list_nat] :
( ( pointwise_le @ X2 @ Y )
=> ( ( pointwise_le @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% pointwise_le_antisym
thf(fact_18_pairwise__minus__cancel,axiom,
! [Z: list_nat,X2: list_nat,Y: list_nat] :
( ( pointwise_le @ Z @ X2 )
=> ( ( pointwise_le @ Z @ Y )
=> ( ( ( minus_minus_list_nat @ X2 @ Z )
= ( minus_minus_list_nat @ Y @ Z ) )
=> ( X2 = Y ) ) ) ) ).
% pairwise_minus_cancel
thf(fact_19_map__replicate,axiom,
! [F: nat > list_nat,N: nat,X2: nat] :
( ( map_nat_list_nat @ F @ ( replicate_nat @ N @ X2 ) )
= ( replicate_list_nat @ N @ ( F @ X2 ) ) ) ).
% map_replicate
thf(fact_20_map__replicate,axiom,
! [F: list_nat > nat,N: nat,X2: list_nat] :
( ( map_list_nat_nat @ F @ ( replicate_list_nat @ N @ X2 ) )
= ( replicate_nat @ N @ ( F @ X2 ) ) ) ).
% map_replicate
thf(fact_21_map__replicate,axiom,
! [F: list_nat > list_nat,N: nat,X2: list_nat] :
( ( map_li7225945977422193158st_nat @ F @ ( replicate_list_nat @ N @ X2 ) )
= ( replicate_list_nat @ N @ ( F @ X2 ) ) ) ).
% map_replicate
thf(fact_22_map__replicate,axiom,
! [F: nat > nat,N: nat,X2: nat] :
( ( map_nat_nat @ F @ ( replicate_nat @ N @ X2 ) )
= ( replicate_nat @ N @ ( F @ X2 ) ) ) ).
% map_replicate
thf(fact_23_length__replicate,axiom,
! [N: nat,X2: list_nat] :
( ( size_s3023201423986296836st_nat @ ( replicate_list_nat @ N @ X2 ) )
= N ) ).
% length_replicate
thf(fact_24_length__replicate,axiom,
! [N: nat,X2: nat] :
( ( size_size_list_nat @ ( replicate_nat @ N @ X2 ) )
= N ) ).
% length_replicate
thf(fact_25_replicate__eq__replicate,axiom,
! [M: nat,X2: list_nat,N: nat,Y: list_nat] :
( ( ( replicate_list_nat @ M @ X2 )
= ( replicate_list_nat @ N @ Y ) )
= ( ( M = N )
& ( ( M != zero_zero_nat )
=> ( X2 = Y ) ) ) ) ).
% replicate_eq_replicate
thf(fact_26_replicate__eq__replicate,axiom,
! [M: nat,X2: nat,N: nat,Y: nat] :
( ( ( replicate_nat @ M @ X2 )
= ( replicate_nat @ N @ Y ) )
= ( ( M = N )
& ( ( M != zero_zero_nat )
=> ( X2 = Y ) ) ) ) ).
% replicate_eq_replicate
thf(fact_27_length__map,axiom,
! [F: list_nat > nat,Xs: list_list_nat] :
( ( size_size_list_nat @ ( map_list_nat_nat @ F @ Xs ) )
= ( size_s3023201423986296836st_nat @ Xs ) ) ).
% length_map
thf(fact_28_length__map,axiom,
! [F: nat > list_nat,Xs: list_nat] :
( ( size_s3023201423986296836st_nat @ ( map_nat_list_nat @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_29_length__map,axiom,
! [F: list_nat > list_nat,Xs: list_list_nat] :
( ( size_s3023201423986296836st_nat @ ( map_li7225945977422193158st_nat @ F @ Xs ) )
= ( size_s3023201423986296836st_nat @ Xs ) ) ).
% length_map
thf(fact_30_length__map,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( size_size_list_nat @ ( map_nat_nat @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_31_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_32_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_33_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_34_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_35_nat_Oinject,axiom,
! [X22: nat,Y2: nat] :
( ( ( suc @ X22 )
= ( suc @ Y2 ) )
= ( X22 = Y2 ) ) ).
% nat.inject
thf(fact_36_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_37_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_38_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_39_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_40_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_41_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_42_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_43_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_44_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_45_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_46_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_47_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_48_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_49_mem__Collect__eq,axiom,
! [A: set_list_nat,P: set_list_nat > $o] :
( ( member_set_list_nat @ A @ ( collect_set_list_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_50_mem__Collect__eq,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_51_mem__Collect__eq,axiom,
! [A: list_list_nat,P: list_list_nat > $o] :
( ( member_list_list_nat @ A @ ( collec5989764272469232197st_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_52_mem__Collect__eq,axiom,
! [A: list_nat,P: list_nat > $o] :
( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_53_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_54_Collect__mem__eq,axiom,
! [A2: set_set_list_nat] :
( ( collect_set_list_nat
@ ^ [X: set_list_nat] : ( member_set_list_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_55_Collect__mem__eq,axiom,
! [A2: set_set_nat] :
( ( collect_set_nat
@ ^ [X: set_nat] : ( member_set_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_56_Collect__mem__eq,axiom,
! [A2: set_list_list_nat] :
( ( collec5989764272469232197st_nat
@ ^ [X: list_list_nat] : ( member_list_list_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_57_Collect__mem__eq,axiom,
! [A2: set_list_nat] :
( ( collect_list_nat
@ ^ [X: list_nat] : ( member_list_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_58_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_59_Collect__cong,axiom,
! [P: set_list_nat > $o,Q: set_list_nat > $o] :
( ! [X3: set_list_nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_set_list_nat @ P )
= ( collect_set_list_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_60_Collect__cong,axiom,
! [P: set_nat > $o,Q: set_nat > $o] :
( ! [X3: set_nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_set_nat @ P )
= ( collect_set_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_61_Collect__cong,axiom,
! [P: list_list_nat > $o,Q: list_list_nat > $o] :
( ! [X3: list_list_nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec5989764272469232197st_nat @ P )
= ( collec5989764272469232197st_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_62_Collect__cong,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ! [X3: list_nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_list_nat @ P )
= ( collect_list_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_63_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_64_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_65_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_66_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_67_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_68_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_69_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_70_lift__Suc__mono__less,axiom,
! [F: nat > list_nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_list_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ord_less_list_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_71_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_72_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_73_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_74_lift__Suc__mono__less__iff,axiom,
! [F: nat > list_nat,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_list_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_list_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_75_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_76_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_77_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_78_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_79_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_80_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_81_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_82_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_83_linorder__neqE__nat,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_84_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_85_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_86_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_87_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_88_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_89_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_90_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_91_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_92_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_93_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_94_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_95_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_96_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_97_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_98_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_99_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_100_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_101_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_102_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_103_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_104_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_105_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_106_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_107_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_108_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_109_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_110_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_111_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_112_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_113_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M3: nat] :
( ( M
= ( suc @ M3 ) )
& ( ord_less_nat @ N @ M3 ) ) ) ) ).
% Suc_less_eq2
thf(fact_114_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_115_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_116_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_117_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ N )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_118_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_119_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_120_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_121_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_122_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_123_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_124_length__induct,axiom,
! [P: list_list_nat > $o,Xs: list_list_nat] :
( ! [Xs2: list_list_nat] :
( ! [Ys: list_list_nat] :
( ( ord_less_nat @ ( size_s3023201423986296836st_nat @ Ys ) @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( P @ Ys ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_125_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs2: list_nat] :
( ! [Ys: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys ) @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ Ys ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_126_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_127_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_128_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_129_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_130_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_131_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_132_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_133_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_134_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_135_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_136_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_137_zero__reorient,axiom,
! [X2: set_nat] :
( ( zero_zero_set_nat = X2 )
= ( X2 = zero_zero_set_nat ) ) ).
% zero_reorient
thf(fact_138_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_139_one__reorient,axiom,
! [X2: set_nat] :
( ( one_one_set_nat = X2 )
= ( X2 = one_one_set_nat ) ) ).
% one_reorient
thf(fact_140_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_141_diff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_142_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_143_Suc__inject,axiom,
! [X2: nat,Y: nat] :
( ( ( suc @ X2 )
= ( suc @ Y ) )
=> ( X2 = Y ) ) ).
% Suc_inject
thf(fact_144_size__neq__size__imp__neq,axiom,
! [X2: list_list_nat,Y: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ X2 )
!= ( size_s3023201423986296836st_nat @ Y ) )
=> ( X2 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_145_size__neq__size__imp__neq,axiom,
! [X2: list_nat,Y: list_nat] :
( ( ( size_size_list_nat @ X2 )
!= ( size_size_list_nat @ Y ) )
=> ( X2 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_146_size__neq__size__imp__neq,axiom,
! [X2: char,Y: char] :
( ( ( size_size_char @ X2 )
!= ( size_size_char @ Y ) )
=> ( X2 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_147_neq__if__length__neq,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
!= ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( Xs != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_148_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys2 ) )
=> ( Xs != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_149_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_list_nat] :
( ( size_s3023201423986296836st_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_150_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_nat] :
( ( size_size_list_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_151_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_152_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_153_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_154_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_155_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_156_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_157_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_158_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_159_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_160_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_161_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_162_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_163_map__eq__imp__length__eq,axiom,
! [F: nat > list_nat,Xs: list_nat,G: list_nat > list_nat,Ys2: list_list_nat] :
( ( ( map_nat_list_nat @ F @ Xs )
= ( map_li7225945977422193158st_nat @ G @ Ys2 ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys2 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_164_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs: list_nat,G: list_nat > nat,Ys2: list_list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_list_nat_nat @ G @ Ys2 ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys2 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_165_map__eq__imp__length__eq,axiom,
! [F: list_nat > list_nat,Xs: list_list_nat,G: nat > list_nat,Ys2: list_nat] :
( ( ( map_li7225945977422193158st_nat @ F @ Xs )
= ( map_nat_list_nat @ G @ Ys2 ) )
=> ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_166_map__eq__imp__length__eq,axiom,
! [F: list_nat > nat,Xs: list_list_nat,G: nat > nat,Ys2: list_nat] :
( ( ( map_list_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys2 ) )
=> ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_167_map__eq__imp__length__eq,axiom,
! [F: list_nat > list_nat,Xs: list_list_nat,G: list_nat > list_nat,Ys2: list_list_nat] :
( ( ( map_li7225945977422193158st_nat @ F @ Xs )
= ( map_li7225945977422193158st_nat @ G @ Ys2 ) )
=> ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys2 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_168_map__eq__imp__length__eq,axiom,
! [F: list_nat > nat,Xs: list_list_nat,G: list_nat > nat,Ys2: list_list_nat] :
( ( ( map_list_nat_nat @ F @ Xs )
= ( map_list_nat_nat @ G @ Ys2 ) )
=> ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys2 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_169_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys2: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys2 ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_170_m__def,axiom,
( m
= ( minus_minus_nat @ n @ ( groups4561878855575611511st_nat @ x ) ) ) ).
% m_def
thf(fact_171_f,axiom,
bij_be5167025596389125688st_nat @ f @ s @ ( length_sum_set @ r @ m ) ).
% f
thf(fact_172_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_173_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_174_minus__apply,axiom,
( minus_minus_nat_o
= ( ^ [A3: nat > $o,B2: nat > $o,X: nat] : ( minus_minus_o @ ( A3 @ X ) @ ( B2 @ X ) ) ) ) ).
% minus_apply
thf(fact_175_minus__apply,axiom,
( minus_1139252259498527702_nat_o
= ( ^ [A3: list_nat > $o,B2: list_nat > $o,X: list_nat] : ( minus_minus_o @ ( A3 @ X ) @ ( B2 @ X ) ) ) ) ).
% minus_apply
thf(fact_176_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_1: nat] : ( P @ X_1 )
=> ? [N3: nat] :
( ~ ( P @ N3 )
& ( P @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_177_list__decode_Ocases,axiom,
! [X2: nat] :
( ( X2 != zero_zero_nat )
=> ~ ! [N3: nat] :
( X2
!= ( suc @ N3 ) ) ) ).
% list_decode.cases
thf(fact_178_linorder__inj__onI_H,axiom,
! [A2: set_nat,F: nat > list_nat] :
( ! [I2: nat,J2: nat] :
( ( member_nat @ I2 @ A2 )
=> ( ( member_nat @ J2 @ A2 )
=> ( ( ord_less_nat @ I2 @ J2 )
=> ( ( F @ I2 )
!= ( F @ J2 ) ) ) ) )
=> ( inj_on_nat_list_nat @ F @ A2 ) ) ).
% linorder_inj_onI'
thf(fact_179_linorder__inj__onI_H,axiom,
! [A2: set_list_nat,F: list_nat > nat] :
( ! [I2: list_nat,J2: list_nat] :
( ( member_list_nat @ I2 @ A2 )
=> ( ( member_list_nat @ J2 @ A2 )
=> ( ( ord_less_list_nat @ I2 @ J2 )
=> ( ( F @ I2 )
!= ( F @ J2 ) ) ) ) )
=> ( inj_on_list_nat_nat @ F @ A2 ) ) ).
% linorder_inj_onI'
thf(fact_180_linorder__inj__onI_H,axiom,
! [A2: set_list_nat,F: list_nat > set_list_nat] :
( ! [I2: list_nat,J2: list_nat] :
( ( member_list_nat @ I2 @ A2 )
=> ( ( member_list_nat @ J2 @ A2 )
=> ( ( ord_less_list_nat @ I2 @ J2 )
=> ( ( F @ I2 )
!= ( F @ J2 ) ) ) ) )
=> ( inj_on8624761805129053417st_nat @ F @ A2 ) ) ).
% linorder_inj_onI'
thf(fact_181_linorder__inj__onI_H,axiom,
! [A2: set_list_nat,F: list_nat > list_nat] :
( ! [I2: list_nat,J2: list_nat] :
( ( member_list_nat @ I2 @ A2 )
=> ( ( member_list_nat @ J2 @ A2 )
=> ( ( ord_less_list_nat @ I2 @ J2 )
=> ( ( F @ I2 )
!= ( F @ J2 ) ) ) ) )
=> ( inj_on3049792774292151987st_nat @ F @ A2 ) ) ).
% linorder_inj_onI'
thf(fact_182_linorder__inj__onI_H,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [I2: nat,J2: nat] :
( ( member_nat @ I2 @ A2 )
=> ( ( member_nat @ J2 @ A2 )
=> ( ( ord_less_nat @ I2 @ J2 )
=> ( ( F @ I2 )
!= ( F @ J2 ) ) ) ) )
=> ( inj_on_nat_nat @ F @ A2 ) ) ).
% linorder_inj_onI'
thf(fact_183_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_184_n,axiom,
ord_less_eq_nat @ ( groups4561878855575611511st_nat @ x ) @ n ).
% n
thf(fact_185_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_186_bij__betwE,axiom,
! [F: nat > nat,A2: set_nat,B3: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B3 )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_nat @ ( F @ X4 ) @ B3 ) ) ) ).
% bij_betwE
thf(fact_187_bij__betwE,axiom,
! [F: nat > list_nat,A2: set_nat,B3: set_list_nat] :
( ( bij_be6293887246118711976st_nat @ F @ A2 @ B3 )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_list_nat @ ( F @ X4 ) @ B3 ) ) ) ).
% bij_betwE
thf(fact_188_bij__betwE,axiom,
! [F: list_nat > nat,A2: set_list_nat,B3: set_nat] :
( ( bij_be8532844293280997160at_nat @ F @ A2 @ B3 )
=> ! [X4: list_nat] :
( ( member_list_nat @ X4 @ A2 )
=> ( member_nat @ ( F @ X4 ) @ B3 ) ) ) ).
% bij_betwE
thf(fact_189_bij__betwE,axiom,
! [F: list_nat > list_nat,A2: set_list_nat,B3: set_list_nat] :
( ( bij_be5167025596389125688st_nat @ F @ A2 @ B3 )
=> ! [X4: list_nat] :
( ( member_list_nat @ X4 @ A2 )
=> ( member_list_nat @ ( F @ X4 ) @ B3 ) ) ) ).
% bij_betwE
thf(fact_190_bij__betw__inv,axiom,
! [F: nat > nat,A2: set_nat,B3: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B3 )
=> ? [G2: nat > nat] : ( bij_betw_nat_nat @ G2 @ B3 @ A2 ) ) ).
% bij_betw_inv
thf(fact_191_bij__betw__inv,axiom,
! [F: nat > list_nat,A2: set_nat,B3: set_list_nat] :
( ( bij_be6293887246118711976st_nat @ F @ A2 @ B3 )
=> ? [G2: list_nat > nat] : ( bij_be8532844293280997160at_nat @ G2 @ B3 @ A2 ) ) ).
% bij_betw_inv
thf(fact_192_bij__betw__inv,axiom,
! [F: list_nat > nat,A2: set_list_nat,B3: set_nat] :
( ( bij_be8532844293280997160at_nat @ F @ A2 @ B3 )
=> ? [G2: nat > list_nat] : ( bij_be6293887246118711976st_nat @ G2 @ B3 @ A2 ) ) ).
% bij_betw_inv
thf(fact_193_bij__betw__inv,axiom,
! [F: list_nat > list_nat,A2: set_list_nat,B3: set_list_nat] :
( ( bij_be5167025596389125688st_nat @ F @ A2 @ B3 )
=> ? [G2: list_nat > list_nat] : ( bij_be5167025596389125688st_nat @ G2 @ B3 @ A2 ) ) ).
% bij_betw_inv
thf(fact_194_bij__betw__ball,axiom,
! [F: nat > nat,A2: set_nat,B3: set_nat,Phi: nat > $o] :
( ( bij_betw_nat_nat @ F @ A2 @ B3 )
=> ( ( ! [X: nat] :
( ( member_nat @ X @ B3 )
=> ( Phi @ X ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( Phi @ ( F @ X ) ) ) ) ) ) ).
% bij_betw_ball
thf(fact_195_bij__betw__ball,axiom,
! [F: nat > list_nat,A2: set_nat,B3: set_list_nat,Phi: list_nat > $o] :
( ( bij_be6293887246118711976st_nat @ F @ A2 @ B3 )
=> ( ( ! [X: list_nat] :
( ( member_list_nat @ X @ B3 )
=> ( Phi @ X ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( Phi @ ( F @ X ) ) ) ) ) ) ).
% bij_betw_ball
thf(fact_196_bij__betw__ball,axiom,
! [F: list_nat > nat,A2: set_list_nat,B3: set_nat,Phi: nat > $o] :
( ( bij_be8532844293280997160at_nat @ F @ A2 @ B3 )
=> ( ( ! [X: nat] :
( ( member_nat @ X @ B3 )
=> ( Phi @ X ) ) )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ A2 )
=> ( Phi @ ( F @ X ) ) ) ) ) ) ).
% bij_betw_ball
thf(fact_197_bij__betw__ball,axiom,
! [F: list_nat > list_nat,A2: set_list_nat,B3: set_list_nat,Phi: list_nat > $o] :
( ( bij_be5167025596389125688st_nat @ F @ A2 @ B3 )
=> ( ( ! [X: list_nat] :
( ( member_list_nat @ X @ B3 )
=> ( Phi @ X ) ) )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ A2 )
=> ( Phi @ ( F @ X ) ) ) ) ) ) ).
% bij_betw_ball
thf(fact_198_bij__betw__cong,axiom,
! [A2: set_nat,F: nat > nat,G: nat > nat,A4: set_nat] :
( ! [A5: nat] :
( ( member_nat @ A5 @ A2 )
=> ( ( F @ A5 )
= ( G @ A5 ) ) )
=> ( ( bij_betw_nat_nat @ F @ A2 @ A4 )
= ( bij_betw_nat_nat @ G @ A2 @ A4 ) ) ) ).
% bij_betw_cong
thf(fact_199_bij__betw__cong,axiom,
! [A2: set_nat,F: nat > list_nat,G: nat > list_nat,A4: set_list_nat] :
( ! [A5: nat] :
( ( member_nat @ A5 @ A2 )
=> ( ( F @ A5 )
= ( G @ A5 ) ) )
=> ( ( bij_be6293887246118711976st_nat @ F @ A2 @ A4 )
= ( bij_be6293887246118711976st_nat @ G @ A2 @ A4 ) ) ) ).
% bij_betw_cong
thf(fact_200_bij__betw__cong,axiom,
! [A2: set_list_nat,F: list_nat > nat,G: list_nat > nat,A4: set_nat] :
( ! [A5: list_nat] :
( ( member_list_nat @ A5 @ A2 )
=> ( ( F @ A5 )
= ( G @ A5 ) ) )
=> ( ( bij_be8532844293280997160at_nat @ F @ A2 @ A4 )
= ( bij_be8532844293280997160at_nat @ G @ A2 @ A4 ) ) ) ).
% bij_betw_cong
thf(fact_201_bij__betw__cong,axiom,
! [A2: set_list_nat,F: list_nat > list_nat,G: list_nat > list_nat,A4: set_list_nat] :
( ! [A5: list_nat] :
( ( member_list_nat @ A5 @ A2 )
=> ( ( F @ A5 )
= ( G @ A5 ) ) )
=> ( ( bij_be5167025596389125688st_nat @ F @ A2 @ A4 )
= ( bij_be5167025596389125688st_nat @ G @ A2 @ A4 ) ) ) ).
% bij_betw_cong
thf(fact_202_bij__betw__apply,axiom,
! [F: list_nat > set_list_nat,A2: set_list_nat,B3: set_set_list_nat,A: list_nat] :
( ( bij_be5530832588036223086st_nat @ F @ A2 @ B3 )
=> ( ( member_list_nat @ A @ A2 )
=> ( member_set_list_nat @ ( F @ A ) @ B3 ) ) ) ).
% bij_betw_apply
thf(fact_203_bij__betw__apply,axiom,
! [F: nat > set_list_nat,A2: set_nat,B3: set_set_list_nat,A: nat] :
( ( bij_be4838622947795563742st_nat @ F @ A2 @ B3 )
=> ( ( member_nat @ A @ A2 )
=> ( member_set_list_nat @ ( F @ A ) @ B3 ) ) ) ).
% bij_betw_apply
thf(fact_204_bij__betw__apply,axiom,
! [F: set_list_nat > list_nat,A2: set_set_list_nat,B3: set_list_nat,A: set_list_nat] :
( ( bij_be3408842901938555246st_nat @ F @ A2 @ B3 )
=> ( ( member_set_list_nat @ A @ A2 )
=> ( member_list_nat @ ( F @ A ) @ B3 ) ) ) ).
% bij_betw_apply
thf(fact_205_bij__betw__apply,axiom,
! [F: set_list_nat > nat,A2: set_set_list_nat,B3: set_nat,A: set_list_nat] :
( ( bij_be3430625573890428254at_nat @ F @ A2 @ B3 )
=> ( ( member_set_list_nat @ A @ A2 )
=> ( member_nat @ ( F @ A ) @ B3 ) ) ) ).
% bij_betw_apply
thf(fact_206_bij__betw__apply,axiom,
! [F: set_list_nat > set_list_nat,A2: set_set_list_nat,B3: set_set_list_nat,A: set_list_nat] :
( ( bij_be4730079884550779812st_nat @ F @ A2 @ B3 )
=> ( ( member_set_list_nat @ A @ A2 )
=> ( member_set_list_nat @ ( F @ A ) @ B3 ) ) ) ).
% bij_betw_apply
thf(fact_207_bij__betw__apply,axiom,
! [F: list_nat > nat,A2: set_list_nat,B3: set_nat,A: list_nat] :
( ( bij_be8532844293280997160at_nat @ F @ A2 @ B3 )
=> ( ( member_list_nat @ A @ A2 )
=> ( member_nat @ ( F @ A ) @ B3 ) ) ) ).
% bij_betw_apply
thf(fact_208_bij__betw__apply,axiom,
! [F: nat > list_nat,A2: set_nat,B3: set_list_nat,A: nat] :
( ( bij_be6293887246118711976st_nat @ F @ A2 @ B3 )
=> ( ( member_nat @ A @ A2 )
=> ( member_list_nat @ ( F @ A ) @ B3 ) ) ) ).
% bij_betw_apply
thf(fact_209_bij__betw__apply,axiom,
! [F: nat > nat,A2: set_nat,B3: set_nat,A: nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B3 )
=> ( ( member_nat @ A @ A2 )
=> ( member_nat @ ( F @ A ) @ B3 ) ) ) ).
% bij_betw_apply
thf(fact_210_bij__betw__apply,axiom,
! [F: list_nat > list_nat,A2: set_list_nat,B3: set_list_nat,A: list_nat] :
( ( bij_be5167025596389125688st_nat @ F @ A2 @ B3 )
=> ( ( member_list_nat @ A @ A2 )
=> ( member_list_nat @ ( F @ A ) @ B3 ) ) ) ).
% bij_betw_apply
thf(fact_211_bij__betw__iff__bijections,axiom,
( bij_be3408842901938555246st_nat
= ( ^ [F2: set_list_nat > list_nat,A3: set_set_list_nat,B2: set_list_nat] :
? [G3: list_nat > set_list_nat] :
( ! [X: set_list_nat] :
( ( member_set_list_nat @ X @ A3 )
=> ( ( member_list_nat @ ( F2 @ X ) @ B2 )
& ( ( G3 @ ( F2 @ X ) )
= X ) ) )
& ! [X: list_nat] :
( ( member_list_nat @ X @ B2 )
=> ( ( member_set_list_nat @ ( G3 @ X ) @ A3 )
& ( ( F2 @ ( G3 @ X ) )
= X ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_212_bij__betw__iff__bijections,axiom,
( bij_be3430625573890428254at_nat
= ( ^ [F2: set_list_nat > nat,A3: set_set_list_nat,B2: set_nat] :
? [G3: nat > set_list_nat] :
( ! [X: set_list_nat] :
( ( member_set_list_nat @ X @ A3 )
=> ( ( member_nat @ ( F2 @ X ) @ B2 )
& ( ( G3 @ ( F2 @ X ) )
= X ) ) )
& ! [X: nat] :
( ( member_nat @ X @ B2 )
=> ( ( member_set_list_nat @ ( G3 @ X ) @ A3 )
& ( ( F2 @ ( G3 @ X ) )
= X ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_213_bij__betw__iff__bijections,axiom,
( bij_be5530832588036223086st_nat
= ( ^ [F2: list_nat > set_list_nat,A3: set_list_nat,B2: set_set_list_nat] :
? [G3: set_list_nat > list_nat] :
( ! [X: list_nat] :
( ( member_list_nat @ X @ A3 )
=> ( ( member_set_list_nat @ ( F2 @ X ) @ B2 )
& ( ( G3 @ ( F2 @ X ) )
= X ) ) )
& ! [X: set_list_nat] :
( ( member_set_list_nat @ X @ B2 )
=> ( ( member_list_nat @ ( G3 @ X ) @ A3 )
& ( ( F2 @ ( G3 @ X ) )
= X ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_214_bij__betw__iff__bijections,axiom,
( bij_be4838622947795563742st_nat
= ( ^ [F2: nat > set_list_nat,A3: set_nat,B2: set_set_list_nat] :
? [G3: set_list_nat > nat] :
( ! [X: nat] :
( ( member_nat @ X @ A3 )
=> ( ( member_set_list_nat @ ( F2 @ X ) @ B2 )
& ( ( G3 @ ( F2 @ X ) )
= X ) ) )
& ! [X: set_list_nat] :
( ( member_set_list_nat @ X @ B2 )
=> ( ( member_nat @ ( G3 @ X ) @ A3 )
& ( ( F2 @ ( G3 @ X ) )
= X ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_215_bij__betw__iff__bijections,axiom,
( bij_be4730079884550779812st_nat
= ( ^ [F2: set_list_nat > set_list_nat,A3: set_set_list_nat,B2: set_set_list_nat] :
? [G3: set_list_nat > set_list_nat] :
( ! [X: set_list_nat] :
( ( member_set_list_nat @ X @ A3 )
=> ( ( member_set_list_nat @ ( F2 @ X ) @ B2 )
& ( ( G3 @ ( F2 @ X ) )
= X ) ) )
& ! [X: set_list_nat] :
( ( member_set_list_nat @ X @ B2 )
=> ( ( member_set_list_nat @ ( G3 @ X ) @ A3 )
& ( ( F2 @ ( G3 @ X ) )
= X ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_216_bij__betw__iff__bijections,axiom,
( bij_be6293887246118711976st_nat
= ( ^ [F2: nat > list_nat,A3: set_nat,B2: set_list_nat] :
? [G3: list_nat > nat] :
( ! [X: nat] :
( ( member_nat @ X @ A3 )
=> ( ( member_list_nat @ ( F2 @ X ) @ B2 )
& ( ( G3 @ ( F2 @ X ) )
= X ) ) )
& ! [X: list_nat] :
( ( member_list_nat @ X @ B2 )
=> ( ( member_nat @ ( G3 @ X ) @ A3 )
& ( ( F2 @ ( G3 @ X ) )
= X ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_217_bij__betw__iff__bijections,axiom,
( bij_be8532844293280997160at_nat
= ( ^ [F2: list_nat > nat,A3: set_list_nat,B2: set_nat] :
? [G3: nat > list_nat] :
( ! [X: list_nat] :
( ( member_list_nat @ X @ A3 )
=> ( ( member_nat @ ( F2 @ X ) @ B2 )
& ( ( G3 @ ( F2 @ X ) )
= X ) ) )
& ! [X: nat] :
( ( member_nat @ X @ B2 )
=> ( ( member_list_nat @ ( G3 @ X ) @ A3 )
& ( ( F2 @ ( G3 @ X ) )
= X ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_218_bij__betw__iff__bijections,axiom,
( bij_betw_nat_nat
= ( ^ [F2: nat > nat,A3: set_nat,B2: set_nat] :
? [G3: nat > nat] :
( ! [X: nat] :
( ( member_nat @ X @ A3 )
=> ( ( member_nat @ ( F2 @ X ) @ B2 )
& ( ( G3 @ ( F2 @ X ) )
= X ) ) )
& ! [X: nat] :
( ( member_nat @ X @ B2 )
=> ( ( member_nat @ ( G3 @ X ) @ A3 )
& ( ( F2 @ ( G3 @ X ) )
= X ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_219_bij__betw__iff__bijections,axiom,
( bij_be5167025596389125688st_nat
= ( ^ [F2: list_nat > list_nat,A3: set_list_nat,B2: set_list_nat] :
? [G3: list_nat > list_nat] :
( ! [X: list_nat] :
( ( member_list_nat @ X @ A3 )
=> ( ( member_list_nat @ ( F2 @ X ) @ B2 )
& ( ( G3 @ ( F2 @ X ) )
= X ) ) )
& ! [X: list_nat] :
( ( member_list_nat @ X @ B2 )
=> ( ( member_list_nat @ ( G3 @ X ) @ A3 )
& ( ( F2 @ ( G3 @ X ) )
= X ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_220_bij__betw__imp__inj__on,axiom,
! [F: list_nat > set_list_nat,A2: set_list_nat,B3: set_set_list_nat] :
( ( bij_be5530832588036223086st_nat @ F @ A2 @ B3 )
=> ( inj_on8624761805129053417st_nat @ F @ A2 ) ) ).
% bij_betw_imp_inj_on
thf(fact_221_bij__betw__imp__inj__on,axiom,
! [F: nat > list_nat,A2: set_nat,B3: set_list_nat] :
( ( bij_be6293887246118711976st_nat @ F @ A2 @ B3 )
=> ( inj_on_nat_list_nat @ F @ A2 ) ) ).
% bij_betw_imp_inj_on
thf(fact_222_bij__betw__imp__inj__on,axiom,
! [F: list_nat > nat,A2: set_list_nat,B3: set_nat] :
( ( bij_be8532844293280997160at_nat @ F @ A2 @ B3 )
=> ( inj_on_list_nat_nat @ F @ A2 ) ) ).
% bij_betw_imp_inj_on
thf(fact_223_bij__betw__imp__inj__on,axiom,
! [F: nat > nat,A2: set_nat,B3: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B3 )
=> ( inj_on_nat_nat @ F @ A2 ) ) ).
% bij_betw_imp_inj_on
thf(fact_224_bij__betw__imp__inj__on,axiom,
! [F: list_nat > list_nat,A2: set_list_nat,B3: set_list_nat] :
( ( bij_be5167025596389125688st_nat @ F @ A2 @ B3 )
=> ( inj_on3049792774292151987st_nat @ F @ A2 ) ) ).
% bij_betw_imp_inj_on
thf(fact_225_inj__Suc,axiom,
! [N4: set_nat] : ( inj_on_nat_nat @ suc @ N4 ) ).
% inj_Suc
thf(fact_226_sum__list__minus,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( pointwise_le @ Xs @ Ys2 )
=> ( ( groups4561878855575611511st_nat @ ( minus_minus_list_nat @ Ys2 @ Xs ) )
= ( minus_minus_nat @ ( groups4561878855575611511st_nat @ Ys2 ) @ ( groups4561878855575611511st_nat @ Xs ) ) ) ) ).
% sum_list_minus
thf(fact_227_fun__diff__def,axiom,
( minus_minus_nat_o
= ( ^ [A3: nat > $o,B2: nat > $o,X: nat] : ( minus_minus_o @ ( A3 @ X ) @ ( B2 @ X ) ) ) ) ).
% fun_diff_def
thf(fact_228_fun__diff__def,axiom,
( minus_1139252259498527702_nat_o
= ( ^ [A3: list_nat > $o,B2: list_nat > $o,X: list_nat] : ( minus_minus_o @ ( A3 @ X ) @ ( B2 @ X ) ) ) ) ).
% fun_diff_def
thf(fact_229_inj__onD,axiom,
! [F: list_nat > nat,A2: set_list_nat,X2: list_nat,Y: list_nat] :
( ( inj_on_list_nat_nat @ F @ A2 )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
=> ( ( member_list_nat @ X2 @ A2 )
=> ( ( member_list_nat @ Y @ A2 )
=> ( X2 = Y ) ) ) ) ) ).
% inj_onD
thf(fact_230_inj__onD,axiom,
! [F: list_nat > set_list_nat,A2: set_list_nat,X2: list_nat,Y: list_nat] :
( ( inj_on8624761805129053417st_nat @ F @ A2 )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
=> ( ( member_list_nat @ X2 @ A2 )
=> ( ( member_list_nat @ Y @ A2 )
=> ( X2 = Y ) ) ) ) ) ).
% inj_onD
thf(fact_231_inj__onD,axiom,
! [F: nat > list_nat,A2: set_nat,X2: nat,Y: nat] :
( ( inj_on_nat_list_nat @ F @ A2 )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( X2 = Y ) ) ) ) ) ).
% inj_onD
thf(fact_232_inj__onD,axiom,
! [F: list_nat > list_nat,A2: set_list_nat,X2: list_nat,Y: list_nat] :
( ( inj_on3049792774292151987st_nat @ F @ A2 )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
=> ( ( member_list_nat @ X2 @ A2 )
=> ( ( member_list_nat @ Y @ A2 )
=> ( X2 = Y ) ) ) ) ) ).
% inj_onD
thf(fact_233_inj__onD,axiom,
! [F: nat > nat,A2: set_nat,X2: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( X2 = Y ) ) ) ) ) ).
% inj_onD
thf(fact_234_inj__onI,axiom,
! [A2: set_list_nat,F: list_nat > nat] :
( ! [X3: list_nat,Y3: list_nat] :
( ( member_list_nat @ X3 @ A2 )
=> ( ( member_list_nat @ Y3 @ A2 )
=> ( ( ( F @ X3 )
= ( F @ Y3 ) )
=> ( X3 = Y3 ) ) ) )
=> ( inj_on_list_nat_nat @ F @ A2 ) ) ).
% inj_onI
thf(fact_235_inj__onI,axiom,
! [A2: set_list_nat,F: list_nat > set_list_nat] :
( ! [X3: list_nat,Y3: list_nat] :
( ( member_list_nat @ X3 @ A2 )
=> ( ( member_list_nat @ Y3 @ A2 )
=> ( ( ( F @ X3 )
= ( F @ Y3 ) )
=> ( X3 = Y3 ) ) ) )
=> ( inj_on8624761805129053417st_nat @ F @ A2 ) ) ).
% inj_onI
thf(fact_236_inj__onI,axiom,
! [A2: set_nat,F: nat > list_nat] :
( ! [X3: nat,Y3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_nat @ Y3 @ A2 )
=> ( ( ( F @ X3 )
= ( F @ Y3 ) )
=> ( X3 = Y3 ) ) ) )
=> ( inj_on_nat_list_nat @ F @ A2 ) ) ).
% inj_onI
thf(fact_237_inj__onI,axiom,
! [A2: set_list_nat,F: list_nat > list_nat] :
( ! [X3: list_nat,Y3: list_nat] :
( ( member_list_nat @ X3 @ A2 )
=> ( ( member_list_nat @ Y3 @ A2 )
=> ( ( ( F @ X3 )
= ( F @ Y3 ) )
=> ( X3 = Y3 ) ) ) )
=> ( inj_on3049792774292151987st_nat @ F @ A2 ) ) ).
% inj_onI
thf(fact_238_inj__onI,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [X3: nat,Y3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_nat @ Y3 @ A2 )
=> ( ( ( F @ X3 )
= ( F @ Y3 ) )
=> ( X3 = Y3 ) ) ) )
=> ( inj_on_nat_nat @ F @ A2 ) ) ).
% inj_onI
thf(fact_239_inj__on__def,axiom,
( inj_on_list_nat_nat
= ( ^ [F2: list_nat > nat,A3: set_list_nat] :
! [X: list_nat] :
( ( member_list_nat @ X @ A3 )
=> ! [Y4: list_nat] :
( ( member_list_nat @ Y4 @ A3 )
=> ( ( ( F2 @ X )
= ( F2 @ Y4 ) )
=> ( X = Y4 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_240_inj__on__def,axiom,
( inj_on_nat_list_nat
= ( ^ [F2: nat > list_nat,A3: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A3 )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ A3 )
=> ( ( ( F2 @ X )
= ( F2 @ Y4 ) )
=> ( X = Y4 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_241_inj__on__def,axiom,
( inj_on8624761805129053417st_nat
= ( ^ [F2: list_nat > set_list_nat,A3: set_list_nat] :
! [X: list_nat] :
( ( member_list_nat @ X @ A3 )
=> ! [Y4: list_nat] :
( ( member_list_nat @ Y4 @ A3 )
=> ( ( ( F2 @ X )
= ( F2 @ Y4 ) )
=> ( X = Y4 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_242_inj__on__def,axiom,
( inj_on3049792774292151987st_nat
= ( ^ [F2: list_nat > list_nat,A3: set_list_nat] :
! [X: list_nat] :
( ( member_list_nat @ X @ A3 )
=> ! [Y4: list_nat] :
( ( member_list_nat @ Y4 @ A3 )
=> ( ( ( F2 @ X )
= ( F2 @ Y4 ) )
=> ( X = Y4 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_243_inj__on__def,axiom,
( inj_on_nat_nat
= ( ^ [F2: nat > nat,A3: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A3 )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ A3 )
=> ( ( ( F2 @ X )
= ( F2 @ Y4 ) )
=> ( X = Y4 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_244_inj__on__cong,axiom,
! [A2: set_list_nat,F: list_nat > nat,G: list_nat > nat] :
( ! [A5: list_nat] :
( ( member_list_nat @ A5 @ A2 )
=> ( ( F @ A5 )
= ( G @ A5 ) ) )
=> ( ( inj_on_list_nat_nat @ F @ A2 )
= ( inj_on_list_nat_nat @ G @ A2 ) ) ) ).
% inj_on_cong
thf(fact_245_inj__on__cong,axiom,
! [A2: set_list_nat,F: list_nat > set_list_nat,G: list_nat > set_list_nat] :
( ! [A5: list_nat] :
( ( member_list_nat @ A5 @ A2 )
=> ( ( F @ A5 )
= ( G @ A5 ) ) )
=> ( ( inj_on8624761805129053417st_nat @ F @ A2 )
= ( inj_on8624761805129053417st_nat @ G @ A2 ) ) ) ).
% inj_on_cong
thf(fact_246_inj__on__cong,axiom,
! [A2: set_nat,F: nat > list_nat,G: nat > list_nat] :
( ! [A5: nat] :
( ( member_nat @ A5 @ A2 )
=> ( ( F @ A5 )
= ( G @ A5 ) ) )
=> ( ( inj_on_nat_list_nat @ F @ A2 )
= ( inj_on_nat_list_nat @ G @ A2 ) ) ) ).
% inj_on_cong
thf(fact_247_inj__on__cong,axiom,
! [A2: set_list_nat,F: list_nat > list_nat,G: list_nat > list_nat] :
( ! [A5: list_nat] :
( ( member_list_nat @ A5 @ A2 )
=> ( ( F @ A5 )
= ( G @ A5 ) ) )
=> ( ( inj_on3049792774292151987st_nat @ F @ A2 )
= ( inj_on3049792774292151987st_nat @ G @ A2 ) ) ) ).
% inj_on_cong
thf(fact_248_inj__on__cong,axiom,
! [A2: set_nat,F: nat > nat,G: nat > nat] :
( ! [A5: nat] :
( ( member_nat @ A5 @ A2 )
=> ( ( F @ A5 )
= ( G @ A5 ) ) )
=> ( ( inj_on_nat_nat @ F @ A2 )
= ( inj_on_nat_nat @ G @ A2 ) ) ) ).
% inj_on_cong
thf(fact_249_inj__on__eq__iff,axiom,
! [F: list_nat > nat,A2: set_list_nat,X2: list_nat,Y: list_nat] :
( ( inj_on_list_nat_nat @ F @ A2 )
=> ( ( member_list_nat @ X2 @ A2 )
=> ( ( member_list_nat @ Y @ A2 )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
= ( X2 = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_250_inj__on__eq__iff,axiom,
! [F: list_nat > set_list_nat,A2: set_list_nat,X2: list_nat,Y: list_nat] :
( ( inj_on8624761805129053417st_nat @ F @ A2 )
=> ( ( member_list_nat @ X2 @ A2 )
=> ( ( member_list_nat @ Y @ A2 )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
= ( X2 = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_251_inj__on__eq__iff,axiom,
! [F: nat > list_nat,A2: set_nat,X2: nat,Y: nat] :
( ( inj_on_nat_list_nat @ F @ A2 )
=> ( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
= ( X2 = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_252_inj__on__eq__iff,axiom,
! [F: list_nat > list_nat,A2: set_list_nat,X2: list_nat,Y: list_nat] :
( ( inj_on3049792774292151987st_nat @ F @ A2 )
=> ( ( member_list_nat @ X2 @ A2 )
=> ( ( member_list_nat @ Y @ A2 )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
= ( X2 = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_253_inj__on__eq__iff,axiom,
! [F: nat > nat,A2: set_nat,X2: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
= ( X2 = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_254_inj__on__contraD,axiom,
! [F: list_nat > nat,A2: set_list_nat,X2: list_nat,Y: list_nat] :
( ( inj_on_list_nat_nat @ F @ A2 )
=> ( ( X2 != Y )
=> ( ( member_list_nat @ X2 @ A2 )
=> ( ( member_list_nat @ Y @ A2 )
=> ( ( F @ X2 )
!= ( F @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_255_inj__on__contraD,axiom,
! [F: list_nat > set_list_nat,A2: set_list_nat,X2: list_nat,Y: list_nat] :
( ( inj_on8624761805129053417st_nat @ F @ A2 )
=> ( ( X2 != Y )
=> ( ( member_list_nat @ X2 @ A2 )
=> ( ( member_list_nat @ Y @ A2 )
=> ( ( F @ X2 )
!= ( F @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_256_inj__on__contraD,axiom,
! [F: nat > list_nat,A2: set_nat,X2: nat,Y: nat] :
( ( inj_on_nat_list_nat @ F @ A2 )
=> ( ( X2 != Y )
=> ( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ( F @ X2 )
!= ( F @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_257_inj__on__contraD,axiom,
! [F: list_nat > list_nat,A2: set_list_nat,X2: list_nat,Y: list_nat] :
( ( inj_on3049792774292151987st_nat @ F @ A2 )
=> ( ( X2 != Y )
=> ( ( member_list_nat @ X2 @ A2 )
=> ( ( member_list_nat @ Y @ A2 )
=> ( ( F @ X2 )
!= ( F @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_258_inj__on__contraD,axiom,
! [F: nat > nat,A2: set_nat,X2: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( ( X2 != Y )
=> ( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ( F @ X2 )
!= ( F @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_259_inj__on__inverseI,axiom,
! [A2: set_list_nat,G: nat > list_nat,F: list_nat > nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A2 )
=> ( ( G @ ( F @ X3 ) )
= X3 ) )
=> ( inj_on_list_nat_nat @ F @ A2 ) ) ).
% inj_on_inverseI
thf(fact_260_inj__on__inverseI,axiom,
! [A2: set_list_nat,G: set_list_nat > list_nat,F: list_nat > set_list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A2 )
=> ( ( G @ ( F @ X3 ) )
= X3 ) )
=> ( inj_on8624761805129053417st_nat @ F @ A2 ) ) ).
% inj_on_inverseI
thf(fact_261_inj__on__inverseI,axiom,
! [A2: set_nat,G: list_nat > nat,F: nat > list_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( G @ ( F @ X3 ) )
= X3 ) )
=> ( inj_on_nat_list_nat @ F @ A2 ) ) ).
% inj_on_inverseI
thf(fact_262_inj__on__inverseI,axiom,
! [A2: set_list_nat,G: list_nat > list_nat,F: list_nat > list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A2 )
=> ( ( G @ ( F @ X3 ) )
= X3 ) )
=> ( inj_on3049792774292151987st_nat @ F @ A2 ) ) ).
% inj_on_inverseI
thf(fact_263_inj__on__inverseI,axiom,
! [A2: set_nat,G: nat > nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( G @ ( F @ X3 ) )
= X3 ) )
=> ( inj_on_nat_nat @ F @ A2 ) ) ).
% inj_on_inverseI
thf(fact_264_inj__on__diff,axiom,
! [F: nat > list_nat,A2: set_nat,B3: set_nat] :
( ( inj_on_nat_list_nat @ F @ A2 )
=> ( inj_on_nat_list_nat @ F @ ( minus_minus_set_nat @ A2 @ B3 ) ) ) ).
% inj_on_diff
thf(fact_265_inj__on__diff,axiom,
! [F: list_nat > nat,A2: set_list_nat,B3: set_list_nat] :
( ( inj_on_list_nat_nat @ F @ A2 )
=> ( inj_on_list_nat_nat @ F @ ( minus_7954133019191499631st_nat @ A2 @ B3 ) ) ) ).
% inj_on_diff
thf(fact_266_inj__on__diff,axiom,
! [F: list_nat > set_list_nat,A2: set_list_nat,B3: set_list_nat] :
( ( inj_on8624761805129053417st_nat @ F @ A2 )
=> ( inj_on8624761805129053417st_nat @ F @ ( minus_7954133019191499631st_nat @ A2 @ B3 ) ) ) ).
% inj_on_diff
thf(fact_267_inj__on__diff,axiom,
! [F: list_nat > list_nat,A2: set_list_nat,B3: set_list_nat] :
( ( inj_on3049792774292151987st_nat @ F @ A2 )
=> ( inj_on3049792774292151987st_nat @ F @ ( minus_7954133019191499631st_nat @ A2 @ B3 ) ) ) ).
% inj_on_diff
thf(fact_268_inj__on__diff,axiom,
! [F: nat > nat,A2: set_nat,B3: set_nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( inj_on_nat_nat @ F @ ( minus_minus_set_nat @ A2 @ B3 ) ) ) ).
% inj_on_diff
thf(fact_269_pointwise__less__imp___092_060sigma_062,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( pointwise_less @ Xs @ Ys2 )
=> ( ord_less_nat @ ( groups4561878855575611511st_nat @ Xs ) @ ( groups4561878855575611511st_nat @ Ys2 ) ) ) ).
% pointwise_less_imp_\<sigma>
thf(fact_270_S__def,axiom,
( s
= ( collect_list_nat
@ ^ [X: list_nat] :
( ( pointwise_le @ x @ X )
& ( ( groups4561878855575611511st_nat @ X )
= n ) ) ) ) ).
% S_def
thf(fact_271_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C2: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_272_sum__list__incr,axiom,
! [I: nat,X2: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ X2 ) )
=> ( ( groups4561878855575611511st_nat @ ( list_incr @ I @ X2 ) )
= ( suc @ ( groups4561878855575611511st_nat @ X2 ) ) ) ) ).
% sum_list_incr
thf(fact_273_length__code,axiom,
( size_s3023201423986296836st_nat
= ( gen_length_list_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_274_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_275_Cons__replicate__eq,axiom,
! [X2: list_nat,Xs: list_list_nat,N: nat,Y: list_nat] :
( ( ( cons_list_nat @ X2 @ Xs )
= ( replicate_list_nat @ N @ Y ) )
= ( ( X2 = Y )
& ( ord_less_nat @ zero_zero_nat @ N )
& ( Xs
= ( replicate_list_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ X2 ) ) ) ) ).
% Cons_replicate_eq
thf(fact_276_Cons__replicate__eq,axiom,
! [X2: nat,Xs: list_nat,N: nat,Y: nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( replicate_nat @ N @ Y ) )
= ( ( X2 = Y )
& ( ord_less_nat @ zero_zero_nat @ N )
& ( Xs
= ( replicate_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ X2 ) ) ) ) ).
% Cons_replicate_eq
thf(fact_277_length__sum__set__def,axiom,
( length_sum_set
= ( ^ [R2: nat,N5: nat] :
( collect_list_nat
@ ^ [X: list_nat] :
( ( ( size_size_list_nat @ X )
= R2 )
& ( ( groups4561878855575611511st_nat @ X )
= N5 ) ) ) ) ) ).
% length_sum_set_def
thf(fact_278_list_Oinject,axiom,
! [X21: list_nat,X222: list_list_nat,Y21: list_nat,Y22: list_list_nat] :
( ( ( cons_list_nat @ X21 @ X222 )
= ( cons_list_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_279_list_Oinject,axiom,
! [X21: nat,X222: list_nat,Y21: nat,Y22: list_nat] :
( ( ( cons_nat @ X21 @ X222 )
= ( cons_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_280_map__ident,axiom,
( ( map_li7225945977422193158st_nat
@ ^ [X: list_nat] : X )
= ( ^ [Xs3: list_list_nat] : Xs3 ) ) ).
% map_ident
thf(fact_281_map__ident,axiom,
( ( map_nat_nat
@ ^ [X: nat] : X )
= ( ^ [Xs3: list_nat] : Xs3 ) ) ).
% map_ident
thf(fact_282_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_283_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_284_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_285_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_286_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_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_length__list__incr,axiom,
! [I: nat,X2: list_nat] :
( ( size_size_list_nat @ ( list_incr @ I @ X2 ) )
= ( size_size_list_nat @ X2 ) ) ).
% length_list_incr
thf(fact_289_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_290_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_291_inj__on__diff__nat,axiom,
! [N4: set_nat,K: nat] :
( ! [N3: nat] :
( ( member_nat @ N3 @ N4 )
=> ( ord_less_eq_nat @ K @ N3 ) )
=> ( inj_on_nat_nat
@ ^ [N5: nat] : ( minus_minus_nat @ N5 @ K )
@ N4 ) ) ).
% inj_on_diff_nat
thf(fact_292_inj__on__id2,axiom,
! [A2: set_list_nat] :
( inj_on3049792774292151987st_nat
@ ^ [X: list_nat] : X
@ A2 ) ).
% inj_on_id2
thf(fact_293_inj__on__id2,axiom,
! [A2: set_nat] :
( inj_on_nat_nat
@ ^ [X: nat] : X
@ A2 ) ).
% inj_on_id2
thf(fact_294_list_Omap__ident,axiom,
! [T: list_nat] :
( ( map_nat_nat
@ ^ [X: nat] : X
@ T )
= T ) ).
% list.map_ident
thf(fact_295_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_296_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_297_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_298_lift__Suc__mono__le,axiom,
! [F: nat > set_list_nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_le6045566169113846134st_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_le6045566169113846134st_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_299_impossible__Cons,axiom,
! [Xs: list_nat,Ys2: list_nat,X2: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys2 ) )
=> ( Xs
!= ( cons_nat @ X2 @ Ys2 ) ) ) ).
% impossible_Cons
thf(fact_300_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_301_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_302_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_303_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_304_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_305_not__Cons__self2,axiom,
! [X2: nat,Xs: list_nat] :
( ( cons_nat @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_306_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_307_gen__length__code_I2_J,axiom,
! [N: nat,X2: nat,Xs: list_nat] :
( ( gen_length_nat @ N @ ( cons_nat @ X2 @ Xs ) )
= ( gen_length_nat @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_308_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
= ( ? [X: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ X @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_309_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_310_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_311_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_312_map__eq__Cons__conv,axiom,
! [F: nat > nat,Xs: list_nat,Y: nat,Ys2: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys2 ) )
= ( ? [Z2: nat,Zs: list_nat] :
( ( Xs
= ( cons_nat @ Z2 @ Zs ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_nat_nat @ F @ Zs )
= Ys2 ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_313_Cons__eq__map__conv,axiom,
! [X2: nat,Xs: list_nat,F: nat > nat,Ys2: list_nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_nat_nat @ F @ Ys2 ) )
= ( ? [Z2: nat,Zs: list_nat] :
( ( Ys2
= ( cons_nat @ Z2 @ Zs ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_314_map__eq__Cons__D,axiom,
! [F: nat > nat,Xs: list_nat,Y: nat,Ys2: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys2 ) )
=> ? [Z3: nat,Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_nat_nat @ F @ Zs2 )
= Ys2 ) ) ) ).
% map_eq_Cons_D
thf(fact_315_Cons__eq__map__D,axiom,
! [X2: nat,Xs: list_nat,F: nat > nat,Ys2: list_nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_nat_nat @ F @ Ys2 ) )
=> ? [Z3: nat,Zs2: list_nat] :
( ( Ys2
= ( cons_nat @ Z3 @ Zs2 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_316_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_317_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_318_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_319_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_320_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_321_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R3: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R3 @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z3: nat] :
( ( R3 @ X3 @ Y3 )
=> ( ( R3 @ Y3 @ Z3 )
=> ( R3 @ X3 @ Z3 ) ) )
=> ( ! [N3: nat] : ( R3 @ N3 @ ( suc @ N3 ) )
=> ( R3 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_322_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_323_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_324_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_325_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_326_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_327_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M4: nat] :
( M6
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_328_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_329_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_330_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_331_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_332_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_333_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_334_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N5: nat] :
( ( ord_less_nat @ M5 @ N5 )
| ( M5 = N5 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_335_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_336_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N5: nat] :
( ( ord_less_eq_nat @ M5 @ N5 )
& ( M5 != N5 ) ) ) ) ).
% nat_less_le
thf(fact_337_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_338_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_339_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_340_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_341_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_342_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_343_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_344_map__replicate__const,axiom,
! [K: nat,Lst: list_nat] :
( ( map_nat_nat
@ ^ [X: nat] : K
@ Lst )
= ( replicate_nat @ ( size_size_list_nat @ Lst ) @ K ) ) ).
% map_replicate_const
thf(fact_345_inj__on__Cons1,axiom,
! [X2: nat,A2: set_list_nat] : ( inj_on3049792774292151987st_nat @ ( cons_nat @ X2 ) @ A2 ) ).
% inj_on_Cons1
thf(fact_346_pointwise__le__iff__less__equal,axiom,
( pointwise_le
= ( ^ [X: list_nat,Y4: list_nat] :
( ( pointwise_less @ X @ Y4 )
| ( X = Y4 ) ) ) ) ).
% pointwise_le_iff_less_equal
thf(fact_347_pointwise__less__def,axiom,
( pointwise_less
= ( ^ [X: list_nat,Y4: list_nat] :
( ( pointwise_le @ X @ Y4 )
& ( X != Y4 ) ) ) ) ).
% pointwise_less_def
thf(fact_348_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_349_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_350_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_351_length__Suc__conv,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y4: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y4 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_352_Suc__length__conv,axiom,
! [N: nat,Xs: list_nat] :
( ( ( suc @ N )
= ( size_size_list_nat @ Xs ) )
= ( ? [Y4: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y4 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_353_linorder__inj__onI,axiom,
! [A2: set_list_nat,F: list_nat > list_nat] :
( ! [X3: list_nat,Y3: list_nat] :
( ( ord_less_list_nat @ X3 @ Y3 )
=> ( ( member_list_nat @ X3 @ A2 )
=> ( ( member_list_nat @ Y3 @ A2 )
=> ( ( F @ X3 )
!= ( F @ Y3 ) ) ) ) )
=> ( ! [X3: list_nat,Y3: list_nat] :
( ( member_list_nat @ X3 @ A2 )
=> ( ( member_list_nat @ Y3 @ A2 )
=> ( ( ord_less_eq_list_nat @ X3 @ Y3 )
| ( ord_less_eq_list_nat @ Y3 @ X3 ) ) ) )
=> ( inj_on3049792774292151987st_nat @ F @ A2 ) ) ) ).
% linorder_inj_onI
thf(fact_354_linorder__inj__onI,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ( member_nat @ X3 @ A2 )
=> ( ( member_nat @ Y3 @ A2 )
=> ( ( F @ X3 )
!= ( F @ Y3 ) ) ) ) )
=> ( ! [X3: nat,Y3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_nat @ Y3 @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X3 ) ) ) )
=> ( inj_on_nat_nat @ F @ A2 ) ) ) ).
% linorder_inj_onI
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_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_357_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_358_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_359_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_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_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_362_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_363_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N5: nat] : ( ord_less_eq_nat @ ( suc @ N5 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_364_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_365_replicate__Suc,axiom,
! [N: nat,X2: nat] :
( ( replicate_nat @ ( suc @ N ) @ X2 )
= ( cons_nat @ X2 @ ( replicate_nat @ N @ X2 ) ) ) ).
% replicate_Suc
thf(fact_366_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_367_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_368_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_369_minus__Cons,axiom,
! [Y: list_nat,Ys2: list_list_nat,X2: list_nat,Xs: list_list_nat] :
( ( minus_3911745200923244873st_nat @ ( cons_list_nat @ Y @ Ys2 ) @ ( cons_list_nat @ X2 @ Xs ) )
= ( cons_list_nat @ ( minus_minus_list_nat @ Y @ X2 ) @ ( minus_3911745200923244873st_nat @ Ys2 @ Xs ) ) ) ).
% minus_Cons
thf(fact_370_minus__Cons,axiom,
! [Y: nat,Ys2: list_nat,X2: nat,Xs: list_nat] :
( ( minus_minus_list_nat @ ( cons_nat @ Y @ Ys2 ) @ ( cons_nat @ X2 @ Xs ) )
= ( cons_nat @ ( minus_minus_nat @ Y @ X2 ) @ ( minus_minus_list_nat @ Ys2 @ Xs ) ) ) ).
% minus_Cons
thf(fact_371_pointwise__le__imp___092_060sigma_062,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( pointwise_le @ Xs @ Ys2 )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ Xs ) @ ( groups4561878855575611511st_nat @ Ys2 ) ) ) ).
% pointwise_le_imp_\<sigma>
thf(fact_372_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_373_dementum__def,axiom,
( dementum
= ( ^ [Xs3: list_nat] : ( minus_minus_list_nat @ Xs3 @ ( cons_nat @ zero_zero_nat @ Xs3 ) ) ) ) ).
% dementum_def
thf(fact_374_minimal__elementsp_Osimps,axiom,
( minimal_elementsp
= ( ^ [U: list_nat > $o,A6: list_nat] :
? [X: list_nat] :
( ( A6 = X )
& ( U @ X )
& ! [Y4: list_nat] :
( ( U @ Y4 )
=> ~ ( pointwise_less @ Y4 @ X ) ) ) ) ) ).
% minimal_elementsp.simps
thf(fact_375_minimal__elementsp_Ointros,axiom,
! [U2: list_nat > $o,X2: list_nat] :
( ( U2 @ X2 )
=> ( ! [Y3: list_nat] :
( ( U2 @ Y3 )
=> ~ ( pointwise_less @ Y3 @ X2 ) )
=> ( minimal_elementsp @ U2 @ X2 ) ) ) ).
% minimal_elementsp.intros
thf(fact_376_minimal__elementsp_Ocases,axiom,
! [U2: list_nat > $o,A: list_nat] :
( ( minimal_elementsp @ U2 @ A )
=> ~ ( ( U2 @ A )
=> ~ ! [Y5: list_nat] :
( ( U2 @ Y5 )
=> ~ ( pointwise_less @ Y5 @ A ) ) ) ) ).
% minimal_elementsp.cases
thf(fact_377_sum__list__0,axiom,
! [Xs: list_nat] :
( ( groups4561878855575611511st_nat
@ ( map_nat_nat
@ ^ [X: nat] : zero_zero_nat
@ Xs ) )
= zero_zero_nat ) ).
% sum_list_0
thf(fact_378_minimal__elements_Osimps,axiom,
! [A: list_nat,U2: set_list_nat] :
( ( member_list_nat @ A @ ( minimal_elements @ U2 ) )
= ( ? [X: list_nat] :
( ( A = X )
& ( member_list_nat @ X @ U2 )
& ! [Y4: list_nat] :
( ( member_list_nat @ Y4 @ U2 )
=> ~ ( pointwise_less @ Y4 @ X ) ) ) ) ) ).
% minimal_elements.simps
thf(fact_379_minimal__elements_Ointros,axiom,
! [X2: list_nat,U2: set_list_nat] :
( ( member_list_nat @ X2 @ U2 )
=> ( ! [Y3: list_nat] :
( ( member_list_nat @ Y3 @ U2 )
=> ~ ( pointwise_less @ Y3 @ X2 ) )
=> ( member_list_nat @ X2 @ ( minimal_elements @ U2 ) ) ) ) ).
% minimal_elements.intros
thf(fact_380_minimal__elements_Ocases,axiom,
! [A: list_nat,U2: set_list_nat] :
( ( member_list_nat @ A @ ( minimal_elements @ U2 ) )
=> ~ ( ( member_list_nat @ A @ U2 )
=> ~ ! [Y5: list_nat] :
( ( member_list_nat @ Y5 @ U2 )
=> ~ ( pointwise_less @ Y5 @ A ) ) ) ) ).
% minimal_elements.cases
thf(fact_381_minimal__elements__def,axiom,
( minimal_elements
= ( ^ [U: set_list_nat] :
( collect_list_nat
@ ( minimal_elementsp
@ ^ [X: list_nat] : ( member_list_nat @ X @ U ) ) ) ) ) ).
% minimal_elements_def
thf(fact_382_minimal__elementsp__minimal__elements__eq,axiom,
! [U2: set_list_nat] :
( ( minimal_elementsp
@ ^ [X: list_nat] : ( member_list_nat @ X @ U2 ) )
= ( ^ [X: list_nat] : ( member_list_nat @ X @ ( minimal_elements @ U2 ) ) ) ) ).
% minimal_elementsp_minimal_elements_eq
thf(fact_383_length__dementum,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( dementum @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_dementum
thf(fact_384_inj__on__subset,axiom,
! [F: nat > nat,A2: set_nat,B3: set_nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( inj_on_nat_nat @ F @ B3 ) ) ) ).
% inj_on_subset
thf(fact_385_inj__on__subset,axiom,
! [F: list_nat > list_nat,A2: set_list_nat,B3: set_list_nat] :
( ( inj_on3049792774292151987st_nat @ F @ A2 )
=> ( ( ord_le6045566169113846134st_nat @ B3 @ A2 )
=> ( inj_on3049792774292151987st_nat @ F @ B3 ) ) ) ).
% inj_on_subset
thf(fact_386_subset__inj__on,axiom,
! [F: nat > nat,B3: set_nat,A2: set_nat] :
( ( inj_on_nat_nat @ F @ B3 )
=> ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( inj_on_nat_nat @ F @ A2 ) ) ) ).
% subset_inj_on
thf(fact_387_subset__inj__on,axiom,
! [F: list_nat > list_nat,B3: set_list_nat,A2: set_list_nat] :
( ( inj_on3049792774292151987st_nat @ F @ B3 )
=> ( ( ord_le6045566169113846134st_nat @ A2 @ B3 )
=> ( inj_on3049792774292151987st_nat @ F @ A2 ) ) ) ).
% subset_inj_on
thf(fact_388_list__incr__nth__diff,axiom,
! [I: nat,X2: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ X2 ) )
=> ( ( ( I = J )
=> ( ( minus_minus_nat @ ( nth_nat @ ( list_incr @ J @ X2 ) @ I ) @ ( nth_nat @ X2 @ I ) )
= one_one_nat ) )
& ( ( I != J )
=> ( ( minus_minus_nat @ ( nth_nat @ ( list_incr @ J @ X2 ) @ I ) @ ( nth_nat @ X2 @ I ) )
= zero_zero_nat ) ) ) ) ).
% list_incr_nth_diff
thf(fact_389_Cons__le__Cons,axiom,
! [A: nat,X2: list_nat,B: nat,Y: list_nat] :
( ( ord_less_eq_list_nat @ ( cons_nat @ A @ X2 ) @ ( cons_nat @ B @ Y ) )
= ( ( ord_less_nat @ ( size_size_list_nat @ X2 ) @ ( size_size_list_nat @ Y ) )
| ( ( ( size_size_list_nat @ X2 )
= ( size_size_list_nat @ Y ) )
& ( ( ord_less_nat @ A @ B )
| ( ( A = B )
& ( ord_less_eq_list_nat @ X2 @ Y ) ) ) ) ) ) ).
% Cons_le_Cons
thf(fact_390_Cons__less__Cons,axiom,
! [A: nat,X2: list_nat,B: nat,Y: list_nat] :
( ( ord_less_list_nat @ ( cons_nat @ A @ X2 ) @ ( cons_nat @ B @ Y ) )
= ( ( ord_less_nat @ ( size_size_list_nat @ X2 ) @ ( size_size_list_nat @ Y ) )
| ( ( ( size_size_list_nat @ X2 )
= ( size_size_list_nat @ Y ) )
& ( ( ord_less_nat @ A @ B )
| ( ( A = B )
& ( ord_less_list_nat @ X2 @ Y ) ) ) ) ) ) ).
% Cons_less_Cons
thf(fact_391_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C2: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_392_length__Cons,axiom,
! [X2: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X2 @ Xs ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_Cons
thf(fact_393_nth__Cons__0,axiom,
! [X2: nat,Xs: list_nat] :
( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_394_nth__Cons__Suc,axiom,
! [X2: nat,Xs: list_nat,N: nat] :
( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ ( suc @ N ) )
= ( nth_nat @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_395_nth__replicate,axiom,
! [I: nat,N: nat,X2: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_nat @ ( replicate_nat @ N @ X2 ) @ I )
= X2 ) ) ).
% nth_replicate
thf(fact_396_nth__map,axiom,
! [N: nat,Xs: list_nat,F: nat > nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( map_nat_nat @ F @ Xs ) @ N )
= ( F @ ( nth_nat @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_397_nth__minus__list,axiom,
! [I: nat,Xs: list_list_nat,Ys2: list_list_nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( ( nth_list_nat @ ( minus_3911745200923244873st_nat @ Xs @ Ys2 ) @ I )
= ( minus_minus_list_nat @ ( nth_list_nat @ Xs @ I ) @ ( nth_list_nat @ Ys2 @ I ) ) ) ) ) ).
% nth_minus_list
thf(fact_398_nth__minus__list,axiom,
! [I: nat,Xs: list_nat,Ys2: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys2 ) )
=> ( ( nth_nat @ ( minus_minus_list_nat @ Xs @ Ys2 ) @ I )
= ( minus_minus_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Ys2 @ I ) ) ) ) ) ).
% nth_minus_list
thf(fact_399_nth__Cons__pos,axiom,
! [N: nat,X2: nat,Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_400_subset__Collect__iff,axiom,
! [B3: set_nat,A2: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( ( ord_less_eq_set_nat @ B3
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ X ) ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ B3 )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_401_subset__Collect__iff,axiom,
! [B3: set_list_nat,A2: set_list_nat,P: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ B3 @ A2 )
=> ( ( ord_le6045566169113846134st_nat @ B3
@ ( collect_list_nat
@ ^ [X: list_nat] :
( ( member_list_nat @ X @ A2 )
& ( P @ X ) ) ) )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ B3 )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_402_subset__CollectI,axiom,
! [B3: set_nat,A2: set_nat,Q: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B3 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ B3 )
& ( Q @ X ) ) )
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_403_subset__CollectI,axiom,
! [B3: set_list_nat,A2: set_list_nat,Q: list_nat > $o,P: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ B3 @ A2 )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ B3 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X: list_nat] :
( ( member_list_nat @ X @ B3 )
& ( Q @ X ) ) )
@ ( collect_list_nat
@ ^ [X: list_nat] :
( ( member_list_nat @ X @ A2 )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_404_nth__equalityI,axiom,
! [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 ) ) )
=> ( Xs = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_405_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X5: nat] : ( P @ I3 @ X5 ) ) )
= ( ? [Xs3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_nat @ Xs3 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_406_list__eq__iff__nth__eq,axiom,
( ( ^ [Y6: list_nat,Z4: list_nat] : ( Y6 = Z4 ) )
= ( ^ [Xs3: list_nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs3 ) )
=> ( ( nth_nat @ Xs3 @ I3 )
= ( nth_nat @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_407_map__equality__iff,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys2: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys2 ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys2 ) )
=> ( ( F @ ( nth_nat @ Xs @ I3 ) )
= ( G @ ( nth_nat @ Ys2 @ I3 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_408_nth__Cons_H,axiom,
! [N: nat,X2: nat,Xs: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
= X2 ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_409_sum__list__mono2,axiom,
! [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 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Ys2 @ I2 ) ) )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ Xs ) @ ( groups4561878855575611511st_nat @ Ys2 ) ) ) ) ).
% sum_list_mono2
thf(fact_410_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_411_pointwise__le__iff__nth,axiom,
( pointwise_le
= ( ^ [X: list_nat,Y4: list_nat] :
( ( ( size_size_list_nat @ X )
= ( size_size_list_nat @ Y4 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ X ) )
=> ( ord_less_eq_nat @ ( nth_nat @ X @ I3 ) @ ( nth_nat @ Y4 @ I3 ) ) ) ) ) ) ).
% pointwise_le_iff_nth
thf(fact_412_pointwise__less__iff2,axiom,
( pointwise_less
= ( ^ [X: list_nat,Y4: list_nat] :
( ( pointwise_le @ X @ Y4 )
& ? [K3: nat] :
( ( ord_less_nat @ K3 @ ( size_size_list_nat @ X ) )
& ( ord_less_nat @ ( nth_nat @ X @ K3 ) @ ( nth_nat @ Y4 @ K3 ) ) ) ) ) ) ).
% pointwise_less_iff2
thf(fact_413_nth__non__equal__first__eq,axiom,
! [X2: nat,Y: nat,Xs: list_nat,N: nat] :
( ( X2 != Y )
=> ( ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
= Y )
= ( ( ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_414_list__incr__def,axiom,
( list_incr
= ( ^ [I3: nat,X: list_nat] : ( list_update_nat @ X @ I3 @ ( suc @ ( nth_nat @ X @ I3 ) ) ) ) ) ).
% list_incr_def
thf(fact_415_DiffI,axiom,
! [C: list_nat,A2: set_list_nat,B3: set_list_nat] :
( ( member_list_nat @ C @ A2 )
=> ( ~ ( member_list_nat @ C @ B3 )
=> ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A2 @ B3 ) ) ) ) ).
% DiffI
thf(fact_416_DiffI,axiom,
! [C: nat,A2: set_nat,B3: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ~ ( member_nat @ C @ B3 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) ) ) ) ).
% DiffI
thf(fact_417_Diff__iff,axiom,
! [C: list_nat,A2: set_list_nat,B3: set_list_nat] :
( ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A2 @ B3 ) )
= ( ( member_list_nat @ C @ A2 )
& ~ ( member_list_nat @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_418_Diff__iff,axiom,
! [C: nat,A2: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) )
= ( ( member_nat @ C @ A2 )
& ~ ( member_nat @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_419_psubsetI,axiom,
! [A2: set_list_nat,B3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_le1190675801316882794st_nat @ A2 @ B3 ) ) ) ).
% psubsetI
thf(fact_420_subset__antisym,axiom,
! [A2: set_list_nat,B3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B3 )
=> ( ( ord_le6045566169113846134st_nat @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_421_subsetI,axiom,
! [A2: set_nat,B3: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ X3 @ B3 ) )
=> ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).
% subsetI
thf(fact_422_subsetI,axiom,
! [A2: set_list_nat,B3: set_list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A2 )
=> ( member_list_nat @ X3 @ B3 ) )
=> ( ord_le6045566169113846134st_nat @ A2 @ B3 ) ) ).
% subsetI
thf(fact_423_list__update__overwrite,axiom,
! [Xs: list_nat,I: nat,X2: nat,Y: nat] :
( ( list_update_nat @ ( list_update_nat @ Xs @ I @ X2 ) @ I @ Y )
= ( list_update_nat @ Xs @ I @ Y ) ) ).
% list_update_overwrite
thf(fact_424_length__list__update,axiom,
! [Xs: list_nat,I: nat,X2: nat] :
( ( size_size_list_nat @ ( list_update_nat @ Xs @ I @ X2 ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_list_update
thf(fact_425_nth__list__update__neq,axiom,
! [I: nat,J: nat,Xs: list_nat,X2: nat] :
( ( I != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X2 ) @ J )
= ( nth_nat @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_426_list__update__id,axiom,
! [Xs: list_nat,I: nat] :
( ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ I ) )
= Xs ) ).
% list_update_id
thf(fact_427_list__update__beyond,axiom,
! [Xs: list_nat,I: nat,X2: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I )
=> ( ( list_update_nat @ Xs @ I @ X2 )
= Xs ) ) ).
% list_update_beyond
thf(fact_428_nth__list__update__eq,axiom,
! [I: nat,Xs: list_nat,X2: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X2 ) @ I )
= X2 ) ) ).
% nth_list_update_eq
thf(fact_429_list__update__swap,axiom,
! [I: nat,I5: nat,Xs: list_nat,X2: nat,X6: nat] :
( ( I != I5 )
=> ( ( list_update_nat @ ( list_update_nat @ Xs @ I @ X2 ) @ I5 @ X6 )
= ( list_update_nat @ ( list_update_nat @ Xs @ I5 @ X6 ) @ I @ X2 ) ) ) ).
% list_update_swap
thf(fact_430_map__update,axiom,
! [F: nat > nat,Xs: list_nat,K: nat,Y: nat] :
( ( map_nat_nat @ F @ ( list_update_nat @ Xs @ K @ Y ) )
= ( list_update_nat @ ( map_nat_nat @ F @ Xs ) @ K @ ( F @ Y ) ) ) ).
% map_update
thf(fact_431_list__update__code_I2_J,axiom,
! [X2: nat,Xs: list_nat,Y: nat] :
( ( list_update_nat @ ( cons_nat @ X2 @ Xs ) @ zero_zero_nat @ Y )
= ( cons_nat @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_432_list__update__code_I3_J,axiom,
! [X2: nat,Xs: list_nat,I: nat,Y: nat] :
( ( list_update_nat @ ( cons_nat @ X2 @ Xs ) @ ( suc @ I ) @ Y )
= ( cons_nat @ X2 @ ( list_update_nat @ Xs @ I @ Y ) ) ) ).
% list_update_code(3)
thf(fact_433_list__update__same__conv,axiom,
! [I: nat,Xs: list_nat,X2: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ( list_update_nat @ Xs @ I @ X2 )
= Xs )
= ( ( nth_nat @ Xs @ I )
= X2 ) ) ) ).
% list_update_same_conv
thf(fact_434_nth__list__update,axiom,
! [I: nat,Xs: list_nat,J: nat,X2: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X2 ) @ J )
= X2 ) )
& ( ( I != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X2 ) @ J )
= ( nth_nat @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_435_in__mono,axiom,
! [A2: set_nat,B3: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_nat @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_436_in__mono,axiom,
! [A2: set_list_nat,B3: set_list_nat,X2: list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B3 )
=> ( ( member_list_nat @ X2 @ A2 )
=> ( member_list_nat @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_437_subsetD,axiom,
! [A2: set_nat,B3: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_438_subsetD,axiom,
! [A2: set_list_nat,B3: set_list_nat,C: list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B3 )
=> ( ( member_list_nat @ C @ A2 )
=> ( member_list_nat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_439_equalityE,axiom,
! [A2: set_list_nat,B3: set_list_nat] :
( ( A2 = B3 )
=> ~ ( ( ord_le6045566169113846134st_nat @ A2 @ B3 )
=> ~ ( ord_le6045566169113846134st_nat @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_440_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A3 )
=> ( member_nat @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_441_subset__eq,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A3: set_list_nat,B2: set_list_nat] :
! [X: list_nat] :
( ( member_list_nat @ X @ A3 )
=> ( member_list_nat @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_442_equalityD1,axiom,
! [A2: set_list_nat,B3: set_list_nat] :
( ( A2 = B3 )
=> ( ord_le6045566169113846134st_nat @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_443_equalityD2,axiom,
! [A2: set_list_nat,B3: set_list_nat] :
( ( A2 = B3 )
=> ( ord_le6045566169113846134st_nat @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_444_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A3 )
=> ( member_nat @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_445_subset__iff,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A3: set_list_nat,B2: set_list_nat] :
! [T2: list_nat] :
( ( member_list_nat @ T2 @ A3 )
=> ( member_list_nat @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_446_subset__refl,axiom,
! [A2: set_list_nat] : ( ord_le6045566169113846134st_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_447_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_448_Collect__mono,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ! [X3: list_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_449_subset__trans,axiom,
! [A2: set_list_nat,B3: set_list_nat,C3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B3 )
=> ( ( ord_le6045566169113846134st_nat @ B3 @ C3 )
=> ( ord_le6045566169113846134st_nat @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_450_set__eq__subset,axiom,
( ( ^ [Y6: set_list_nat,Z4: set_list_nat] : ( Y6 = Z4 ) )
= ( ^ [A3: set_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ B2 )
& ( ord_le6045566169113846134st_nat @ B2 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_451_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_452_Collect__mono__iff,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) )
= ( ! [X: list_nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_453_psubsetE,axiom,
! [A2: set_list_nat,B3: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A2 @ B3 )
=> ~ ( ( ord_le6045566169113846134st_nat @ A2 @ B3 )
=> ( ord_le6045566169113846134st_nat @ B3 @ A2 ) ) ) ).
% psubsetE
thf(fact_454_psubset__eq,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [A3: set_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% psubset_eq
thf(fact_455_psubset__imp__subset,axiom,
! [A2: set_list_nat,B3: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A2 @ B3 )
=> ( ord_le6045566169113846134st_nat @ A2 @ B3 ) ) ).
% psubset_imp_subset
thf(fact_456_psubset__subset__trans,axiom,
! [A2: set_list_nat,B3: set_list_nat,C3: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A2 @ B3 )
=> ( ( ord_le6045566169113846134st_nat @ B3 @ C3 )
=> ( ord_le1190675801316882794st_nat @ A2 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_457_subset__not__subset__eq,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [A3: set_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ B2 )
& ~ ( ord_le6045566169113846134st_nat @ B2 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_458_subset__psubset__trans,axiom,
! [A2: set_list_nat,B3: set_list_nat,C3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B3 )
=> ( ( ord_le1190675801316882794st_nat @ B3 @ C3 )
=> ( ord_le1190675801316882794st_nat @ A2 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_459_subset__iff__psubset__eq,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A3: set_list_nat,B2: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_460_psubset__imp__ex__mem,axiom,
! [A2: set_list_nat,B3: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A2 @ B3 )
=> ? [B4: list_nat] : ( member_list_nat @ B4 @ ( minus_7954133019191499631st_nat @ B3 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_461_psubset__imp__ex__mem,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A2 @ B3 )
=> ? [B4: nat] : ( member_nat @ B4 @ ( minus_minus_set_nat @ B3 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_462_DiffD2,axiom,
! [C: list_nat,A2: set_list_nat,B3: set_list_nat] :
( ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A2 @ B3 ) )
=> ~ ( member_list_nat @ C @ B3 ) ) ).
% DiffD2
thf(fact_463_DiffD2,axiom,
! [C: nat,A2: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) )
=> ~ ( member_nat @ C @ B3 ) ) ).
% DiffD2
thf(fact_464_DiffD1,axiom,
! [C: list_nat,A2: set_list_nat,B3: set_list_nat] :
( ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A2 @ B3 ) )
=> ( member_list_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_465_DiffD1,axiom,
! [C: nat,A2: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) )
=> ( member_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_466_DiffE,axiom,
! [C: list_nat,A2: set_list_nat,B3: set_list_nat] :
( ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A2 @ B3 ) )
=> ~ ( ( member_list_nat @ C @ A2 )
=> ( member_list_nat @ C @ B3 ) ) ) ).
% DiffE
thf(fact_467_DiffE,axiom,
! [C: nat,A2: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B3 ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B3 ) ) ) ).
% DiffE
thf(fact_468_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_469_Collect__subset,axiom,
! [A2: set_list_nat,P: list_nat > $o] :
( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X: list_nat] :
( ( member_list_nat @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_470_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A3 )
@ ^ [X: nat] : ( member_nat @ X @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_471_less__eq__set__def,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A3: set_list_nat,B2: set_list_nat] :
( ord_le1520216061033275535_nat_o
@ ^ [X: list_nat] : ( member_list_nat @ X @ A3 )
@ ^ [X: list_nat] : ( member_list_nat @ X @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_472_minus__set__def,axiom,
( minus_7954133019191499631st_nat
= ( ^ [A3: set_list_nat,B2: set_list_nat] :
( collect_list_nat
@ ( minus_1139252259498527702_nat_o
@ ^ [X: list_nat] : ( member_list_nat @ X @ A3 )
@ ^ [X: list_nat] : ( member_list_nat @ X @ B2 ) ) ) ) ) ).
% minus_set_def
thf(fact_473_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A3 )
@ ^ [X: nat] : ( member_nat @ X @ B2 ) ) ) ) ) ).
% minus_set_def
thf(fact_474_set__diff__eq,axiom,
( minus_7954133019191499631st_nat
= ( ^ [A3: set_list_nat,B2: set_list_nat] :
( collect_list_nat
@ ^ [X: list_nat] :
( ( member_list_nat @ X @ A3 )
& ~ ( member_list_nat @ X @ B2 ) ) ) ) ) ).
% set_diff_eq
thf(fact_475_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A3 )
& ~ ( member_nat @ X @ B2 ) ) ) ) ) ).
% set_diff_eq
thf(fact_476_Diff__mono,axiom,
! [A2: set_list_nat,C3: set_list_nat,D: set_list_nat,B3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ C3 )
=> ( ( ord_le6045566169113846134st_nat @ D @ B3 )
=> ( ord_le6045566169113846134st_nat @ ( minus_7954133019191499631st_nat @ A2 @ B3 ) @ ( minus_7954133019191499631st_nat @ C3 @ D ) ) ) ) ).
% Diff_mono
thf(fact_477_Diff__subset,axiom,
! [A2: set_list_nat,B3: set_list_nat] : ( ord_le6045566169113846134st_nat @ ( minus_7954133019191499631st_nat @ A2 @ B3 ) @ A2 ) ).
% Diff_subset
thf(fact_478_double__diff,axiom,
! [A2: set_list_nat,B3: set_list_nat,C3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B3 )
=> ( ( ord_le6045566169113846134st_nat @ B3 @ C3 )
=> ( ( minus_7954133019191499631st_nat @ B3 @ ( minus_7954133019191499631st_nat @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_479_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_480_order__refl,axiom,
! [X2: set_list_nat] : ( ord_le6045566169113846134st_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_481_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_482_dual__order_Orefl,axiom,
! [A: set_list_nat] : ( ord_le6045566169113846134st_nat @ A @ A ) ).
% dual_order.refl
thf(fact_483_minimal__elements__set__tuples__finite,axiom,
! [U2: set_list_nat,R: nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ U2 )
=> ( ( size_size_list_nat @ X3 )
= R ) )
=> ( finite8100373058378681591st_nat @ ( minimal_elements @ U2 ) ) ) ).
% minimal_elements_set_tuples_finite
thf(fact_484_WFP,axiom,
wfP_list_nat @ pointwise_less ).
% WFP
thf(fact_485_finite__length__sum__set,axiom,
! [R: nat,N: nat] : ( finite8100373058378681591st_nat @ ( length_sum_set @ R @ N ) ) ).
% finite_length_sum_set
thf(fact_486_finite__maxlen,axiom,
! [M7: set_list_nat] :
( ( finite8100373058378681591st_nat @ M7 )
=> ? [N3: nat] :
! [X4: list_nat] :
( ( member_list_nat @ X4 @ M7 )
=> ( ord_less_nat @ ( size_size_list_nat @ X4 ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_487_order__antisym__conv,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_488_order__antisym__conv,axiom,
! [Y: set_list_nat,X2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ Y @ X2 )
=> ( ( ord_le6045566169113846134st_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_489_linorder__le__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_490_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_491_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_list_nat,C: set_list_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le6045566169113846134st_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_492_ord__le__eq__subst,axiom,
! [A: set_list_nat,B: set_list_nat,F: set_list_nat > nat,C: nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_list_nat,Y3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_493_ord__le__eq__subst,axiom,
! [A: set_list_nat,B: set_list_nat,F: set_list_nat > set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_list_nat,Y3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y3 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le6045566169113846134st_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_494_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_495_ord__eq__le__subst,axiom,
! [A: set_list_nat,F: nat > set_list_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le6045566169113846134st_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_496_ord__eq__le__subst,axiom,
! [A: nat,F: set_list_nat > nat,B: set_list_nat,C: set_list_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ! [X3: set_list_nat,Y3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_497_ord__eq__le__subst,axiom,
! [A: set_list_nat,F: set_list_nat > set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ! [X3: set_list_nat,Y3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y3 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le6045566169113846134st_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_498_linorder__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_499_order__eq__refl,axiom,
! [X2: nat,Y: nat] :
( ( X2 = Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_500_order__eq__refl,axiom,
! [X2: set_list_nat,Y: set_list_nat] :
( ( X2 = Y )
=> ( ord_le6045566169113846134st_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_501_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_502_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_list_nat,C: set_list_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le6045566169113846134st_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le6045566169113846134st_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_503_order__subst2,axiom,
! [A: set_list_nat,B: set_list_nat,F: set_list_nat > nat,C: nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_list_nat,Y3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_504_order__subst2,axiom,
! [A: set_list_nat,B: set_list_nat,F: set_list_nat > set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ord_le6045566169113846134st_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_list_nat,Y3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y3 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le6045566169113846134st_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_505_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_506_order__subst1,axiom,
! [A: nat,F: set_list_nat > nat,B: set_list_nat,C: set_list_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ! [X3: set_list_nat,Y3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_507_order__subst1,axiom,
! [A: set_list_nat,F: nat > set_list_nat,B: nat,C: nat] :
( ( ord_le6045566169113846134st_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le6045566169113846134st_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_508_order__subst1,axiom,
! [A: set_list_nat,F: set_list_nat > set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ ( F @ B ) )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ! [X3: set_list_nat,Y3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y3 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le6045566169113846134st_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_509_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
= ( ^ [A6: nat,B5: nat] :
( ( ord_less_eq_nat @ A6 @ B5 )
& ( ord_less_eq_nat @ B5 @ A6 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_510_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_list_nat,Z4: set_list_nat] : ( Y6 = Z4 ) )
= ( ^ [A6: set_list_nat,B5: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A6 @ B5 )
& ( ord_le6045566169113846134st_nat @ B5 @ A6 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_511_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_512_antisym,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ord_le6045566169113846134st_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_513_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_514_dual__order_Otrans,axiom,
! [B: set_list_nat,A: set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B @ A )
=> ( ( ord_le6045566169113846134st_nat @ C @ B )
=> ( ord_le6045566169113846134st_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_515_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_516_dual__order_Oantisym,axiom,
! [B: set_list_nat,A: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B @ A )
=> ( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_517_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
= ( ^ [A6: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A6 )
& ( ord_less_eq_nat @ A6 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_518_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: set_list_nat,Z4: set_list_nat] : ( Y6 = Z4 ) )
= ( ^ [A6: set_list_nat,B5: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B5 @ A6 )
& ( ord_le6045566169113846134st_nat @ A6 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_519_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: nat,B4: nat] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_520_order__trans,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_521_order__trans,axiom,
! [X2: set_list_nat,Y: set_list_nat,Z: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X2 @ Y )
=> ( ( ord_le6045566169113846134st_nat @ Y @ Z )
=> ( ord_le6045566169113846134st_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_522_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_523_order_Otrans,axiom,
! [A: set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ord_le6045566169113846134st_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_524_order__antisym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_525_order__antisym,axiom,
! [X2: set_list_nat,Y: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X2 @ Y )
=> ( ( ord_le6045566169113846134st_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_526_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_527_ord__le__eq__trans,axiom,
! [A: set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( B = C )
=> ( ord_le6045566169113846134st_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_528_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_529_ord__eq__le__trans,axiom,
! [A: set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( A = B )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ord_le6045566169113846134st_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_530_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
= ( ^ [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_531_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_list_nat,Z4: set_list_nat] : ( Y6 = Z4 ) )
= ( ^ [X: set_list_nat,Y4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X @ Y4 )
& ( ord_le6045566169113846134st_nat @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_532_le__cases3,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_533_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_534_order__less__imp__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_535_order__less__imp__not__eq2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_536_order__less__imp__not__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_537_linorder__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_538_order__less__imp__triv,axiom,
! [X2: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_539_order__less__not__sym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_540_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_541_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_542_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_543_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_544_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_545_order__less__trans,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_546_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_547_linorder__neq__iff,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
= ( ( ord_less_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_548_order__less__asym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_549_linorder__neqE,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_550_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_551_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_552_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_553_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ( ord_less_nat @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_554_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_555_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B4: nat] :
( ( ord_less_nat @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: nat] : ( P @ A5 @ A5 )
=> ( ! [A5: nat,B4: nat] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_556_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X7: nat] : ( P2 @ X7 ) )
= ( ^ [P3: nat > $o] :
? [N5: nat] :
( ( P3 @ N5 )
& ! [M5: nat] :
( ( ord_less_nat @ M5 @ N5 )
=> ~ ( P3 @ M5 ) ) ) ) ) ).
% exists_least_iff
thf(fact_557_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_558_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_559_linorder__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_560_antisym__conv3,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_nat @ Y @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_561_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_562_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_563_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_564_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_565_less__imp__neq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_566_gt__ex,axiom,
! [X2: nat] :
? [X_12: nat] : ( ord_less_nat @ X2 @ X_12 ) ).
% gt_ex
thf(fact_567_leD,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y ) ) ).
% leD
thf(fact_568_leD,axiom,
! [Y: set_list_nat,X2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ Y @ X2 )
=> ~ ( ord_le1190675801316882794st_nat @ X2 @ Y ) ) ).
% leD
thf(fact_569_leI,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% leI
thf(fact_570_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_571_nless__le,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( ~ ( ord_le1190675801316882794st_nat @ A @ B ) )
= ( ~ ( ord_le6045566169113846134st_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_572_antisym__conv1,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_573_antisym__conv1,axiom,
! [X2: set_list_nat,Y: set_list_nat] :
( ~ ( ord_le1190675801316882794st_nat @ X2 @ Y )
=> ( ( ord_le6045566169113846134st_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_574_antisym__conv2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_575_antisym__conv2,axiom,
! [X2: set_list_nat,Y: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X2 @ Y )
=> ( ( ~ ( ord_le1190675801316882794st_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_576_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_577_less__le__not__le,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [X: set_list_nat,Y4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X @ Y4 )
& ~ ( ord_le6045566169113846134st_nat @ Y4 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_578_not__le__imp__less,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y @ X2 )
=> ( ord_less_nat @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_579_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A6: nat,B5: nat] :
( ( ord_less_nat @ A6 @ B5 )
| ( A6 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_580_order_Oorder__iff__strict,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A6: set_list_nat,B5: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A6 @ B5 )
| ( A6 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_581_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A6: nat,B5: nat] :
( ( ord_less_eq_nat @ A6 @ B5 )
& ( A6 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_582_order_Ostrict__iff__order,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [A6: set_list_nat,B5: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A6 @ B5 )
& ( A6 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_583_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_584_order_Ostrict__trans1,axiom,
! [A: set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ord_le1190675801316882794st_nat @ B @ C )
=> ( ord_le1190675801316882794st_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_585_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_586_order_Ostrict__trans2,axiom,
! [A: set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A @ B )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ord_le1190675801316882794st_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_587_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A6: nat,B5: nat] :
( ( ord_less_eq_nat @ A6 @ B5 )
& ~ ( ord_less_eq_nat @ B5 @ A6 ) ) ) ) ).
% order.strict_iff_not
thf(fact_588_order_Ostrict__iff__not,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [A6: set_list_nat,B5: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A6 @ B5 )
& ~ ( ord_le6045566169113846134st_nat @ B5 @ A6 ) ) ) ) ).
% order.strict_iff_not
thf(fact_589_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A6: nat] :
( ( ord_less_nat @ B5 @ A6 )
| ( A6 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_590_dual__order_Oorder__iff__strict,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [B5: set_list_nat,A6: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ B5 @ A6 )
| ( A6 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_591_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B5: nat,A6: nat] :
( ( ord_less_eq_nat @ B5 @ A6 )
& ( A6 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_592_dual__order_Ostrict__iff__order,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [B5: set_list_nat,A6: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B5 @ A6 )
& ( A6 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_593_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_594_dual__order_Ostrict__trans1,axiom,
! [B: set_list_nat,A: set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B @ A )
=> ( ( ord_le1190675801316882794st_nat @ C @ B )
=> ( ord_le1190675801316882794st_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_595_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_596_dual__order_Ostrict__trans2,axiom,
! [B: set_list_nat,A: set_list_nat,C: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ B @ A )
=> ( ( ord_le6045566169113846134st_nat @ C @ B )
=> ( ord_le1190675801316882794st_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_597_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B5: nat,A6: nat] :
( ( ord_less_eq_nat @ B5 @ A6 )
& ~ ( ord_less_eq_nat @ A6 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_598_dual__order_Ostrict__iff__not,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [B5: set_list_nat,A6: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B5 @ A6 )
& ~ ( ord_le6045566169113846134st_nat @ A6 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_599_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_600_order_Ostrict__implies__order,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A @ B )
=> ( ord_le6045566169113846134st_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_601_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_602_dual__order_Ostrict__implies__order,axiom,
! [B: set_list_nat,A: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ B @ A )
=> ( ord_le6045566169113846134st_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_603_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
| ( X = Y4 ) ) ) ) ).
% order_le_less
thf(fact_604_order__le__less,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [X: set_list_nat,Y4: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ X @ Y4 )
| ( X = Y4 ) ) ) ) ).
% order_le_less
thf(fact_605_order__less__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
& ( X != Y4 ) ) ) ) ).
% order_less_le
thf(fact_606_order__less__le,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [X: set_list_nat,Y4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X @ Y4 )
& ( X != Y4 ) ) ) ) ).
% order_less_le
thf(fact_607_linorder__not__le,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y ) )
= ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_608_linorder__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_609_order__less__imp__le,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_610_order__less__imp__le,axiom,
! [X2: set_list_nat,Y: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ X2 @ Y )
=> ( ord_le6045566169113846134st_nat @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_611_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_612_order__le__neq__trans,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( A != B )
=> ( ord_le1190675801316882794st_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_613_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_614_order__neq__le__trans,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( A != B )
=> ( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ord_le1190675801316882794st_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_615_order__le__less__trans,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_616_order__le__less__trans,axiom,
! [X2: set_list_nat,Y: set_list_nat,Z: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X2 @ Y )
=> ( ( ord_le1190675801316882794st_nat @ Y @ Z )
=> ( ord_le1190675801316882794st_nat @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_617_order__less__le__trans,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_618_order__less__le__trans,axiom,
! [X2: set_list_nat,Y: set_list_nat,Z: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ X2 @ Y )
=> ( ( ord_le6045566169113846134st_nat @ Y @ Z )
=> ( ord_le1190675801316882794st_nat @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_619_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_620_order__le__less__subst1,axiom,
! [A: set_list_nat,F: nat > set_list_nat,B: nat,C: nat] :
( ( ord_le6045566169113846134st_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_le1190675801316882794st_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1190675801316882794st_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_621_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_622_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_list_nat,C: set_list_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le1190675801316882794st_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1190675801316882794st_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_623_order__le__less__subst2,axiom,
! [A: set_list_nat,B: set_list_nat,F: set_list_nat > nat,C: nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_list_nat,Y3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_624_order__le__less__subst2,axiom,
! [A: set_list_nat,B: set_list_nat,F: set_list_nat > set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ord_le1190675801316882794st_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_list_nat,Y3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y3 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1190675801316882794st_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_625_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_626_order__less__le__subst1,axiom,
! [A: set_list_nat,F: nat > set_list_nat,B: nat,C: nat] :
( ( ord_le1190675801316882794st_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1190675801316882794st_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_627_order__less__le__subst1,axiom,
! [A: nat,F: set_list_nat > nat,B: set_list_nat,C: set_list_nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ! [X3: set_list_nat,Y3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_628_order__less__le__subst1,axiom,
! [A: set_list_nat,F: set_list_nat > set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A @ ( F @ B ) )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ! [X3: set_list_nat,Y3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y3 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1190675801316882794st_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_629_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_630_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_list_nat,C: set_list_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le6045566169113846134st_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_le1190675801316882794st_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1190675801316882794st_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_631_linorder__le__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_632_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_633_order__le__imp__less__or__eq,axiom,
! [X2: set_list_nat,Y: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X2 @ Y )
=> ( ( ord_le1190675801316882794st_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_634_finite__Collect__subsets,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B2: set_nat] : ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_635_finite__Collect__subsets,axiom,
! [A2: set_list_nat] :
( ( finite8100373058378681591st_nat @ A2 )
=> ( finite7047420756378620717st_nat
@ ( collect_set_list_nat
@ ^ [B2: set_list_nat] : ( ord_le6045566169113846134st_nat @ B2 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_636_finite__Diff2,axiom,
! [B3: set_list_nat,A2: set_list_nat] :
( ( finite8100373058378681591st_nat @ B3 )
=> ( ( finite8100373058378681591st_nat @ ( minus_7954133019191499631st_nat @ A2 @ B3 ) )
= ( finite8100373058378681591st_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_637_finite__Diff2,axiom,
! [B3: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B3 ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_638_finite__Diff,axiom,
! [A2: set_list_nat,B3: set_list_nat] :
( ( finite8100373058378681591st_nat @ A2 )
=> ( finite8100373058378681591st_nat @ ( minus_7954133019191499631st_nat @ A2 @ B3 ) ) ) ).
% finite_Diff
thf(fact_639_finite__Diff,axiom,
! [A2: set_nat,B3: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B3 ) ) ) ).
% finite_Diff
thf(fact_640_wfP__empty,axiom,
( wfP_list_nat
@ ^ [X: list_nat,Y4: list_nat] : $false ) ).
% wfP_empty
thf(fact_641_min__pointwise__le,axiom,
! [U3: list_nat,U2: set_list_nat] :
( ( member_list_nat @ U3 @ U2 )
=> ( ( finite8100373058378681591st_nat @ U2 )
=> ( pointwise_le @ ( min_pointwise @ ( size_size_list_nat @ U3 ) @ U2 ) @ U3 ) ) ) ).
% min_pointwise_le
thf(fact_642_finite__Collect__disjI,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [X: list_nat] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
& ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_643_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_644_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
@ ^ [X: list_nat] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_645_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
@ ^ [X: nat] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_646_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N5: nat] : ( ord_less_eq_nat @ N5 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_647_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N5: nat] : ( ord_less_nat @ N5 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_648_wfP__subset,axiom,
! [R: list_nat > list_nat > $o,P4: list_nat > list_nat > $o] :
( ( wfP_list_nat @ R )
=> ( ( ord_le6558929396352911974_nat_o @ P4 @ R )
=> ( wfP_list_nat @ P4 ) ) ) ).
% wfP_subset
thf(fact_649_pigeonhole__infinite__rel,axiom,
! [A2: set_list_nat,B3: set_list_nat,R3: list_nat > list_nat > $o] :
( ~ ( finite8100373058378681591st_nat @ A2 )
=> ( ( finite8100373058378681591st_nat @ B3 )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A2 )
=> ? [Xa: list_nat] :
( ( member_list_nat @ Xa @ B3 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ B3 )
& ~ ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [A6: list_nat] :
( ( member_list_nat @ A6 @ A2 )
& ( R3 @ A6 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_650_pigeonhole__infinite__rel,axiom,
! [A2: set_list_nat,B3: set_nat,R3: list_nat > nat > $o] :
( ~ ( finite8100373058378681591st_nat @ A2 )
=> ( ( finite_finite_nat @ B3 )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B3 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B3 )
& ~ ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [A6: list_nat] :
( ( member_list_nat @ A6 @ A2 )
& ( R3 @ A6 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_651_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B3: set_list_nat,R3: nat > list_nat > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite8100373058378681591st_nat @ B3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa: list_nat] :
( ( member_list_nat @ Xa @ B3 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ B3 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A6: nat] :
( ( member_nat @ A6 @ A2 )
& ( R3 @ A6 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_652_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B3: set_nat,R3: nat > nat > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B3 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B3 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A6: nat] :
( ( member_nat @ A6 @ A2 )
& ( R3 @ A6 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_653_not__finite__existsD,axiom,
! [P: list_nat > $o] :
( ~ ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
=> ? [X_12: list_nat] : ( P @ X_12 ) ) ).
% not_finite_existsD
thf(fact_654_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_12: nat] : ( P @ X_12 ) ) ).
% not_finite_existsD
thf(fact_655_finite__has__minimal2,axiom,
! [A2: set_list_nat,A: list_nat] :
( ( finite8100373058378681591st_nat @ A2 )
=> ( ( member_list_nat @ A @ A2 )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ A2 )
& ( ord_less_eq_list_nat @ X3 @ A )
& ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ A2 )
=> ( ( ord_less_eq_list_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_656_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_657_finite__has__minimal2,axiom,
! [A2: set_set_list_nat,A: set_list_nat] :
( ( finite7047420756378620717st_nat @ A2 )
=> ( ( member_set_list_nat @ A @ A2 )
=> ? [X3: set_list_nat] :
( ( member_set_list_nat @ X3 @ A2 )
& ( ord_le6045566169113846134st_nat @ X3 @ A )
& ! [Xa: set_list_nat] :
( ( member_set_list_nat @ Xa @ A2 )
=> ( ( ord_le6045566169113846134st_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_658_finite__has__maximal2,axiom,
! [A2: set_list_nat,A: list_nat] :
( ( finite8100373058378681591st_nat @ A2 )
=> ( ( member_list_nat @ A @ A2 )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ A2 )
& ( ord_less_eq_list_nat @ A @ X3 )
& ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ A2 )
=> ( ( ord_less_eq_list_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_659_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ A @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_660_finite__has__maximal2,axiom,
! [A2: set_set_list_nat,A: set_list_nat] :
( ( finite7047420756378620717st_nat @ A2 )
=> ( ( member_set_list_nat @ A @ A2 )
=> ? [X3: set_list_nat] :
( ( member_set_list_nat @ X3 @ A2 )
& ( ord_le6045566169113846134st_nat @ A @ X3 )
& ! [Xa: set_list_nat] :
( ( member_set_list_nat @ Xa @ A2 )
=> ( ( ord_le6045566169113846134st_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_661_rev__finite__subset,axiom,
! [B3: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_662_rev__finite__subset,axiom,
! [B3: set_list_nat,A2: set_list_nat] :
( ( finite8100373058378681591st_nat @ B3 )
=> ( ( ord_le6045566169113846134st_nat @ A2 @ B3 )
=> ( finite8100373058378681591st_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_663_infinite__super,axiom,
! [S2: set_nat,T3: set_nat] :
( ( ord_less_eq_set_nat @ S2 @ T3 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ T3 ) ) ) ).
% infinite_super
thf(fact_664_infinite__super,axiom,
! [S2: set_list_nat,T3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ S2 @ T3 )
=> ( ~ ( finite8100373058378681591st_nat @ S2 )
=> ~ ( finite8100373058378681591st_nat @ T3 ) ) ) ).
% infinite_super
thf(fact_665_finite__subset,axiom,
! [A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( finite_finite_nat @ B3 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_666_finite__subset,axiom,
! [A2: set_list_nat,B3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B3 )
=> ( ( finite8100373058378681591st_nat @ B3 )
=> ( finite8100373058378681591st_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_667_Diff__infinite__finite,axiom,
! [T3: set_list_nat,S2: set_list_nat] :
( ( finite8100373058378681591st_nat @ T3 )
=> ( ~ ( finite8100373058378681591st_nat @ S2 )
=> ~ ( finite8100373058378681591st_nat @ ( minus_7954133019191499631st_nat @ S2 @ T3 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_668_Diff__infinite__finite,axiom,
! [T3: set_nat,S2: set_nat] :
( ( finite_finite_nat @ T3 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ T3 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_669_bij__betw__finite,axiom,
! [F: list_nat > list_nat,A2: set_list_nat,B3: set_list_nat] :
( ( bij_be5167025596389125688st_nat @ F @ A2 @ B3 )
=> ( ( finite8100373058378681591st_nat @ A2 )
= ( finite8100373058378681591st_nat @ B3 ) ) ) ).
% bij_betw_finite
thf(fact_670_bij__betw__finite,axiom,
! [F: list_nat > nat,A2: set_list_nat,B3: set_nat] :
( ( bij_be8532844293280997160at_nat @ F @ A2 @ B3 )
=> ( ( finite8100373058378681591st_nat @ A2 )
= ( finite_finite_nat @ B3 ) ) ) ).
% bij_betw_finite
thf(fact_671_bij__betw__finite,axiom,
! [F: nat > list_nat,A2: set_nat,B3: set_list_nat] :
( ( bij_be6293887246118711976st_nat @ F @ A2 @ B3 )
=> ( ( finite_finite_nat @ A2 )
= ( finite8100373058378681591st_nat @ B3 ) ) ) ).
% bij_betw_finite
thf(fact_672_bij__betw__finite,axiom,
! [F: nat > nat,A2: set_nat,B3: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B3 )
=> ( ( finite_finite_nat @ A2 )
= ( finite_finite_nat @ B3 ) ) ) ).
% bij_betw_finite
thf(fact_673_wfP__less,axiom,
wfP_list_nat @ ord_less_list_nat ).
% wfP_less
thf(fact_674_wfP__less,axiom,
wfP_nat @ ord_less_nat ).
% wfP_less
thf(fact_675_wfP__if__convertible__to__nat,axiom,
! [R3: list_nat > list_nat > $o,F: list_nat > nat] :
( ! [X3: list_nat,Y3: list_nat] :
( ( R3 @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( wfP_list_nat @ R3 ) ) ).
% wfP_if_convertible_to_nat
thf(fact_676_finite__inverse__image__gen,axiom,
! [A2: set_list_nat,F: list_nat > list_nat,D: set_list_nat] :
( ( finite8100373058378681591st_nat @ A2 )
=> ( ( inj_on3049792774292151987st_nat @ F @ D )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [J3: list_nat] :
( ( member_list_nat @ J3 @ D )
& ( member_list_nat @ ( F @ J3 ) @ A2 ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_677_finite__inverse__image__gen,axiom,
! [A2: set_list_nat,F: nat > list_nat,D: set_nat] :
( ( finite8100373058378681591st_nat @ A2 )
=> ( ( inj_on_nat_list_nat @ F @ D )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [J3: nat] :
( ( member_nat @ J3 @ D )
& ( member_list_nat @ ( F @ J3 ) @ A2 ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_678_finite__inverse__image__gen,axiom,
! [A2: set_nat,F: list_nat > nat,D: set_list_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( inj_on_list_nat_nat @ F @ D )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [J3: list_nat] :
( ( member_list_nat @ J3 @ D )
& ( member_nat @ ( F @ J3 ) @ A2 ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_679_finite__inverse__image__gen,axiom,
! [A2: set_nat,F: nat > nat,D: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( inj_on_nat_nat @ F @ D )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [J3: nat] :
( ( member_nat @ J3 @ D )
& ( member_nat @ ( F @ J3 ) @ A2 ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_680_max__pointwise__ge,axiom,
! [U3: list_nat,U2: set_list_nat] :
( ( member_list_nat @ U3 @ U2 )
=> ( ( finite8100373058378681591st_nat @ U2 )
=> ( pointwise_le @ U3 @ ( max_pointwise @ ( size_size_list_nat @ U3 ) @ U2 ) ) ) ) ).
% max_pointwise_ge
thf(fact_681_pred__subset__eq,axiom,
! [R3: set_nat,S2: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ R3 )
@ ^ [X: nat] : ( member_nat @ X @ S2 ) )
= ( ord_less_eq_set_nat @ R3 @ S2 ) ) ).
% pred_subset_eq
thf(fact_682_pred__subset__eq,axiom,
! [R3: set_list_nat,S2: set_list_nat] :
( ( ord_le1520216061033275535_nat_o
@ ^ [X: list_nat] : ( member_list_nat @ X @ R3 )
@ ^ [X: list_nat] : ( member_list_nat @ X @ S2 ) )
= ( ord_le6045566169113846134st_nat @ R3 @ S2 ) ) ).
% pred_subset_eq
thf(fact_683_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_684_Khovanskii_OAsubG,axiom,
! [G4: set_list_nat,Addition: list_nat > list_nat > list_nat,Zero: list_nat,A2: set_list_nat] :
( ( khovanskii_list_nat @ G4 @ Addition @ Zero @ A2 )
=> ( ord_le6045566169113846134st_nat @ A2 @ G4 ) ) ).
% Khovanskii.AsubG
thf(fact_685_Khovanskii_OfinA,axiom,
! [G4: set_list_nat,Addition: list_nat > list_nat > list_nat,Zero: list_nat,A2: set_list_nat] :
( ( khovanskii_list_nat @ G4 @ Addition @ Zero @ A2 )
=> ( finite8100373058378681591st_nat @ A2 ) ) ).
% Khovanskii.finA
thf(fact_686_Khovanskii_OfinA,axiom,
! [G4: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat] :
( ( khovanskii_nat @ G4 @ Addition @ Zero @ A2 )
=> ( finite_finite_nat @ A2 ) ) ).
% Khovanskii.finA
thf(fact_687_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( P @ K3 )
& ( ord_less_nat @ K3 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_688_finite__less__ub,axiom,
! [F: nat > nat,U3: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N5: nat] : ( ord_less_eq_nat @ ( F @ N5 ) @ U3 ) ) ) ) ).
% finite_less_ub
thf(fact_689_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N6: set_nat] :
? [M5: nat] :
! [X: nat] :
( ( member_nat @ X @ N6 )
=> ( ord_less_nat @ X @ M5 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_690_infinite__nat__iff__unbounded,axiom,
! [S2: set_nat] :
( ( ~ ( finite_finite_nat @ S2 ) )
= ( ! [M5: nat] :
? [N5: nat] :
( ( ord_less_nat @ M5 @ N5 )
& ( member_nat @ N5 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_691_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M7: nat] :
( ( P @ X2 )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M7 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_692_infinite__nat__iff__unbounded__le,axiom,
! [S2: set_nat] :
( ( ~ ( finite_finite_nat @ S2 ) )
= ( ! [M5: nat] :
? [N5: nat] :
( ( ord_less_eq_nat @ M5 @ N5 )
& ( member_nat @ N5 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_693_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N6: set_nat] :
? [M5: nat] :
! [X: nat] :
( ( member_nat @ X @ N6 )
=> ( ord_less_eq_nat @ X @ M5 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_694_unbounded__k__infinite,axiom,
! [K: nat,S2: set_nat] :
( ! [M4: nat] :
( ( ord_less_nat @ K @ M4 )
=> ? [N7: nat] :
( ( ord_less_nat @ M4 @ N7 )
& ( member_nat @ N7 @ S2 ) ) )
=> ~ ( finite_finite_nat @ S2 ) ) ).
% unbounded_k_infinite
thf(fact_695_bounded__nat__set__is__finite,axiom,
! [N4: set_nat,N: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ N4 )
=> ( ord_less_nat @ X3 @ N ) )
=> ( finite_finite_nat @ N4 ) ) ).
% bounded_nat_set_is_finite
thf(fact_696_finite__indexed__bound,axiom,
! [A2: set_list_nat,P: list_nat > nat > $o] :
( ( finite8100373058378681591st_nat @ A2 )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A2 )
=> ? [X_1: nat] : ( P @ X3 @ X_1 ) )
=> ? [M4: nat] :
! [X4: list_nat] :
( ( member_list_nat @ X4 @ A2 )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ M4 )
& ( P @ X4 @ K2 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_697_finite__indexed__bound,axiom,
! [A2: set_nat,P: nat > nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [X_1: nat] : ( P @ X3 @ X_1 ) )
=> ? [M4: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ M4 )
& ( P @ X4 @ K2 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_698_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C4: nat] :
( ( ord_less_eq_nat @ A @ C4 )
& ( ord_less_eq_nat @ C4 @ B )
& ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ C4 ) )
=> ( P @ X4 ) )
& ! [D2: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D2 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D2 @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_699_verit__comp__simplify1_I3_J,axiom,
! [B6: nat,A7: nat] :
( ( ~ ( ord_less_eq_nat @ B6 @ A7 ) )
= ( ord_less_nat @ A7 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_700_Khovanskii_Oalpha__in__G,axiom,
! [G4: set_list_nat,Addition: list_nat > list_nat > list_nat,Zero: list_nat,A2: set_list_nat,X2: list_nat] :
( ( khovanskii_list_nat @ G4 @ Addition @ Zero @ A2 )
=> ( member_list_nat @ ( alpha_list_nat @ G4 @ Addition @ Zero @ A2 @ X2 ) @ G4 ) ) ).
% Khovanskii.alpha_in_G
thf(fact_701_Khovanskii_Oalpha__in__G,axiom,
! [G4: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,X2: list_nat] :
( ( khovanskii_nat @ G4 @ Addition @ Zero @ A2 )
=> ( member_nat @ ( alpha_nat @ G4 @ Addition @ Zero @ A2 @ X2 ) @ G4 ) ) ).
% Khovanskii.alpha_in_G
thf(fact_702_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_703_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_704_verit__comp__simplify1_I2_J,axiom,
! [A: set_list_nat] : ( ord_le6045566169113846134st_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_705_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_706_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X4 ) ) ).
% minf(8)
thf(fact_707_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ord_less_eq_nat @ X4 @ T ) ) ).
% minf(6)
thf(fact_708_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ord_less_eq_nat @ T @ X4 ) ) ).
% pinf(8)
thf(fact_709_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ T ) ) ).
% pinf(6)
thf(fact_710_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ~ ( ord_less_nat @ T @ X4 ) ) ).
% minf(7)
thf(fact_711_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ord_less_nat @ X4 @ T ) ) ).
% minf(5)
thf(fact_712_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( X4 != T ) ) ).
% minf(4)
thf(fact_713_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( X4 != T ) ) ).
% minf(3)
thf(fact_714_minf_I2_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P5 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_715_minf_I1_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P5 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_716_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ord_less_nat @ T @ X4 ) ) ).
% pinf(7)
thf(fact_717_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ~ ( ord_less_nat @ X4 @ T ) ) ).
% pinf(5)
thf(fact_718_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( X4 != T ) ) ).
% pinf(4)
thf(fact_719_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( X4 != T ) ) ).
% pinf(3)
thf(fact_720_pinf_I2_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P5 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_721_pinf_I1_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P5 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_722_sum__list__update,axiom,
! [K: nat,Xs: list_nat,X2: nat] :
( ( ord_less_nat @ K @ ( size_size_list_nat @ Xs ) )
=> ( ( groups4561878855575611511st_nat @ ( list_update_nat @ Xs @ K @ X2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( groups4561878855575611511st_nat @ Xs ) @ X2 ) @ ( nth_nat @ Xs @ K ) ) ) ) ).
% sum_list_update
thf(fact_723_min__pointwise__ge__iff,axiom,
! [U2: set_list_nat,R: nat,X2: list_nat] :
( ( finite8100373058378681591st_nat @ U2 )
=> ( ( U2 != bot_bot_set_list_nat )
=> ( ! [U4: list_nat] :
( ( member_list_nat @ U4 @ U2 )
=> ( ( size_size_list_nat @ U4 )
= R ) )
=> ( ( ( size_size_list_nat @ X2 )
= R )
=> ( ( pointwise_le @ X2 @ ( min_pointwise @ R @ U2 ) )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ U2 )
=> ( pointwise_le @ X2 @ X ) ) ) ) ) ) ) ) ).
% min_pointwise_ge_iff
thf(fact_724_Khovanskii_Ouseless__iff,axiom,
! [G4: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,X2: list_nat] :
( ( khovanskii_nat @ G4 @ Addition @ Zero @ A2 )
=> ( ( ( size_size_list_nat @ X2 )
= ( finite_card_nat @ A2 ) )
=> ( ( useless_nat @ G4 @ Addition @ Zero @ A2 @ X2 )
= ( ? [X: list_nat] :
( ( member_list_nat @ X @ ( minimal_elements @ ( collect_list_nat @ ( useless_nat @ G4 @ Addition @ Zero @ A2 ) ) ) )
& ( pointwise_le @ X @ X2 ) ) ) ) ) ) ).
% Khovanskii.useless_iff
thf(fact_725_Khovanskii_Ouseless__iff,axiom,
! [G4: set_list_nat,Addition: list_nat > list_nat > list_nat,Zero: list_nat,A2: set_list_nat,X2: list_nat] :
( ( khovanskii_list_nat @ G4 @ Addition @ Zero @ A2 )
=> ( ( ( size_size_list_nat @ X2 )
= ( finite_card_list_nat @ A2 ) )
=> ( ( useless_list_nat @ G4 @ Addition @ Zero @ A2 @ X2 )
= ( ? [X: list_nat] :
( ( member_list_nat @ X @ ( minimal_elements @ ( collect_list_nat @ ( useless_list_nat @ G4 @ Addition @ Zero @ A2 ) ) ) )
& ( pointwise_le @ X @ X2 ) ) ) ) ) ) ).
% Khovanskii.useless_iff
thf(fact_726_max__pointwise__le__iff,axiom,
! [U2: set_list_nat,R: nat,X2: list_nat] :
( ( finite8100373058378681591st_nat @ U2 )
=> ( ( U2 != bot_bot_set_list_nat )
=> ( ! [U4: list_nat] :
( ( member_list_nat @ U4 @ U2 )
=> ( ( size_size_list_nat @ U4 )
= R ) )
=> ( ( ( size_size_list_nat @ X2 )
= R )
=> ( ( pointwise_le @ ( max_pointwise @ R @ U2 ) @ X2 )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ U2 )
=> ( pointwise_le @ X @ X2 ) ) ) ) ) ) ) ) ).
% max_pointwise_le_iff
thf(fact_727_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_728_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_729_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_730_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_731_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_732_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_733_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_734_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_735_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I3: nat] : ( ord_less_nat @ I3 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_736_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_737_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_738_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_739_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_740_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_741_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_742_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_743_add__eq__0__iff__both__eq__0,axiom,
! [X2: nat,Y: nat] :
( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_744_zero__eq__add__iff__both__eq__0,axiom,
! [X2: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X2 @ Y ) )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_745_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_746_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_747_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_748_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_749_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_750_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_751_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_752_empty__subsetI,axiom,
! [A2: set_list_nat] : ( ord_le6045566169113846134st_nat @ bot_bot_set_list_nat @ A2 ) ).
% empty_subsetI
thf(fact_753_subset__empty,axiom,
! [A2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ bot_bot_set_list_nat )
= ( A2 = bot_bot_set_list_nat ) ) ).
% subset_empty
thf(fact_754_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_755_inj__on__empty,axiom,
! [F: nat > nat] : ( inj_on_nat_nat @ F @ bot_bot_set_nat ) ).
% inj_on_empty
thf(fact_756_inj__on__empty,axiom,
! [F: list_nat > list_nat] : ( inj_on3049792774292151987st_nat @ F @ bot_bot_set_list_nat ) ).
% inj_on_empty
thf(fact_757_Diff__empty,axiom,
! [A2: set_list_nat] :
( ( minus_7954133019191499631st_nat @ A2 @ bot_bot_set_list_nat )
= A2 ) ).
% Diff_empty
thf(fact_758_empty__Diff,axiom,
! [A2: set_list_nat] :
( ( minus_7954133019191499631st_nat @ bot_bot_set_list_nat @ A2 )
= bot_bot_set_list_nat ) ).
% empty_Diff
thf(fact_759_Diff__cancel,axiom,
! [A2: set_list_nat] :
( ( minus_7954133019191499631st_nat @ A2 @ A2 )
= bot_bot_set_list_nat ) ).
% Diff_cancel
thf(fact_760_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_761_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_762_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_763_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I3: nat] : ( ord_less_eq_nat @ I3 @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_764_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_765_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_766_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_767_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_768_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_769_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_770_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_771_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_772_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_773_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_774_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_775_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_776_card_Oempty,axiom,
( ( finite_card_list_nat @ bot_bot_set_list_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_777_Diff__eq__empty__iff,axiom,
! [A2: set_list_nat,B3: set_list_nat] :
( ( ( minus_7954133019191499631st_nat @ A2 @ B3 )
= bot_bot_set_list_nat )
= ( ord_le6045566169113846134st_nat @ A2 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_778_card_Oinfinite,axiom,
! [A2: set_list_nat] :
( ~ ( finite8100373058378681591st_nat @ A2 )
=> ( ( finite_card_list_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_779_card_Oinfinite,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_card_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_780_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_781_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_782_sum__list_OCons,axiom,
! [X2: nat,Xs: list_nat] :
( ( groups4561878855575611511st_nat @ ( cons_nat @ X2 @ Xs ) )
= ( plus_plus_nat @ X2 @ ( groups4561878855575611511st_nat @ Xs ) ) ) ).
% sum_list.Cons
thf(fact_783_length__sum__set__Suc,axiom,
! [K: nat,Ks: list_nat,R: nat,N: nat] :
( ( member_list_nat @ ( cons_nat @ K @ Ks ) @ ( length_sum_set @ ( suc @ R ) @ N ) )
= ( ? [M5: nat] :
( ( member_list_nat @ Ks @ ( length_sum_set @ R @ M5 ) )
& ( N
= ( plus_plus_nat @ M5 @ K ) ) ) ) ) ).
% length_sum_set_Suc
thf(fact_784_card__0__eq,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( finite_card_nat @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_nat ) ) ) ).
% card_0_eq
thf(fact_785_card__0__eq,axiom,
! [A2: set_list_nat] :
( ( finite8100373058378681591st_nat @ A2 )
=> ( ( ( finite_card_list_nat @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_list_nat ) ) ) ).
% card_0_eq
thf(fact_786_nth__plus__list,axiom,
! [I: nat,Xs: list_list_nat,Ys2: list_list_nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( ( nth_list_nat @ ( plus_p2116291331692525561st_nat @ Xs @ Ys2 ) @ I )
= ( plus_plus_list_nat @ ( nth_list_nat @ Xs @ I ) @ ( nth_list_nat @ Ys2 @ I ) ) ) ) ) ).
% nth_plus_list
thf(fact_787_nth__plus__list,axiom,
! [I: nat,Xs: list_nat,Ys2: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys2 ) )
=> ( ( nth_nat @ ( plus_plus_list_nat @ Xs @ Ys2 ) @ I )
= ( plus_plus_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Ys2 @ I ) ) ) ) ) ).
% nth_plus_list
thf(fact_788_inj__on__add_H,axiom,
! [A: nat,A2: set_nat] :
( inj_on_nat_nat
@ ^ [B5: nat] : ( plus_plus_nat @ B5 @ A )
@ A2 ) ).
% inj_on_add'
thf(fact_789_bij__betw__same__card,axiom,
! [F: nat > nat,A2: set_nat,B3: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B3 )
=> ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B3 ) ) ) ).
% bij_betw_same_card
thf(fact_790_bij__betw__same__card,axiom,
! [F: nat > list_nat,A2: set_nat,B3: set_list_nat] :
( ( bij_be6293887246118711976st_nat @ F @ A2 @ B3 )
=> ( ( finite_card_nat @ A2 )
= ( finite_card_list_nat @ B3 ) ) ) ).
% bij_betw_same_card
thf(fact_791_bij__betw__same__card,axiom,
! [F: list_nat > nat,A2: set_list_nat,B3: set_nat] :
( ( bij_be8532844293280997160at_nat @ F @ A2 @ B3 )
=> ( ( finite_card_list_nat @ A2 )
= ( finite_card_nat @ B3 ) ) ) ).
% bij_betw_same_card
thf(fact_792_bij__betw__same__card,axiom,
! [F: list_nat > list_nat,A2: set_list_nat,B3: set_list_nat] :
( ( bij_be5167025596389125688st_nat @ F @ A2 @ B3 )
=> ( ( finite_card_list_nat @ A2 )
= ( finite_card_list_nat @ B3 ) ) ) ).
% bij_betw_same_card
thf(fact_793_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_794_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_795_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_796_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_797_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_798_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_799_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_800_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_801_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_802_inj__on__add,axiom,
! [A: nat,A2: set_nat] : ( inj_on_nat_nat @ ( plus_plus_nat @ A ) @ A2 ) ).
% inj_on_add
thf(fact_803_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_804_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_805_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_806_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_807_bij__betw__empty1,axiom,
! [F: list_nat > list_nat,A2: set_list_nat] :
( ( bij_be5167025596389125688st_nat @ F @ bot_bot_set_list_nat @ A2 )
=> ( A2 = bot_bot_set_list_nat ) ) ).
% bij_betw_empty1
thf(fact_808_bij__betw__empty2,axiom,
! [F: list_nat > list_nat,A2: set_list_nat] :
( ( bij_be5167025596389125688st_nat @ F @ A2 @ bot_bot_set_list_nat )
=> ( A2 = bot_bot_set_list_nat ) ) ).
% bij_betw_empty2
thf(fact_809_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X: nat] : $false ) ) ).
% empty_def
thf(fact_810_empty__def,axiom,
( bot_bot_set_list_nat
= ( collect_list_nat
@ ^ [X: list_nat] : $false ) ) ).
% empty_def
thf(fact_811_plus__Cons,axiom,
! [Y: list_nat,Ys2: list_list_nat,X2: list_nat,Xs: list_list_nat] :
( ( plus_p2116291331692525561st_nat @ ( cons_list_nat @ Y @ Ys2 ) @ ( cons_list_nat @ X2 @ Xs ) )
= ( cons_list_nat @ ( plus_plus_list_nat @ Y @ X2 ) @ ( plus_p2116291331692525561st_nat @ Ys2 @ Xs ) ) ) ).
% plus_Cons
thf(fact_812_plus__Cons,axiom,
! [Y: nat,Ys2: list_nat,X2: nat,Xs: list_nat] :
( ( plus_plus_list_nat @ ( cons_nat @ Y @ Ys2 ) @ ( cons_nat @ X2 @ Xs ) )
= ( cons_nat @ ( plus_plus_nat @ Y @ X2 ) @ ( plus_plus_list_nat @ Ys2 @ Xs ) ) ) ).
% plus_Cons
thf(fact_813_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_814_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( plus_plus_list_nat @ ( plus_plus_list_nat @ A @ B ) @ C )
= ( plus_plus_list_nat @ A @ ( plus_plus_list_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_815_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_816_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_817_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B: nat,A: nat] :
( ( B3
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_818_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_819_add_Oassoc,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( plus_plus_list_nat @ ( plus_plus_list_nat @ A @ B ) @ C )
= ( plus_plus_list_nat @ A @ ( plus_plus_list_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_820_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A6: nat,B5: nat] : ( plus_plus_nat @ B5 @ A6 ) ) ) ).
% add.commute
thf(fact_821_add_Ocommute,axiom,
( plus_plus_list_nat
= ( ^ [A6: list_nat,B5: list_nat] : ( plus_plus_list_nat @ B5 @ A6 ) ) ) ).
% add.commute
thf(fact_822_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_823_add_Oleft__commute,axiom,
! [B: list_nat,A: list_nat,C: list_nat] :
( ( plus_plus_list_nat @ B @ ( plus_plus_list_nat @ A @ C ) )
= ( plus_plus_list_nat @ A @ ( plus_plus_list_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_824_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_825_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_826_sum__list__plus,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( groups4561878855575611511st_nat @ ( plus_plus_list_nat @ Xs @ Ys2 ) )
= ( plus_plus_nat @ ( groups4561878855575611511st_nat @ Xs ) @ ( groups4561878855575611511st_nat @ Ys2 ) ) ) ) ).
% sum_list_plus
thf(fact_827_Khovanskii_Ononempty,axiom,
! [G4: set_list_nat,Addition: list_nat > list_nat > list_nat,Zero: list_nat,A2: set_list_nat] :
( ( khovanskii_list_nat @ G4 @ Addition @ Zero @ A2 )
=> ( A2 != bot_bot_set_list_nat ) ) ).
% Khovanskii.nonempty
thf(fact_828_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_829_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_830_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_831_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_832_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_833_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_834_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_835_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_836_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_837_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_838_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N5: nat] :
? [K3: nat] :
( N5
= ( plus_plus_nat @ M5 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_839_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_840_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_841_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_842_add__mono,axiom,
! [A: nat,B: nat,C: nat,D3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D3 ) ) ) ) ).
% add_mono
thf(fact_843_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_844_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C4: nat] :
( B
!= ( plus_plus_nat @ A @ C4 ) ) ) ).
% less_eqE
thf(fact_845_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_846_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A6: nat,B5: nat] :
? [C2: nat] :
( B5
= ( plus_plus_nat @ A6 @ C2 ) ) ) ) ).
% le_iff_add
thf(fact_847_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_848_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_849_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_850_bot_Oextremum__uniqueI,axiom,
! [A: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ bot_bot_set_list_nat )
=> ( A = bot_bot_set_list_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_851_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_852_bot_Oextremum__unique,axiom,
! [A: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ bot_bot_set_list_nat )
= ( A = bot_bot_set_list_nat ) ) ).
% bot.extremum_unique
thf(fact_853_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_854_bot_Oextremum,axiom,
! [A: set_list_nat] : ( ord_le6045566169113846134st_nat @ bot_bot_set_list_nat @ A ) ).
% bot.extremum
thf(fact_855_card__eq__0__iff,axiom,
! [A2: set_nat] :
( ( ( finite_card_nat @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_nat )
| ~ ( finite_finite_nat @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_856_card__eq__0__iff,axiom,
! [A2: set_list_nat] :
( ( ( finite_card_list_nat @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_list_nat )
| ~ ( finite8100373058378681591st_nat @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_857_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_858_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_859_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_860_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_861_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_862_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_863_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_864_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_865_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_866_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_867_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_868_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_869_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_870_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D3 ) ) ) ) ).
% add_strict_mono
thf(fact_871_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_872_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_873_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_874_bot_Onot__eq__extremum,axiom,
! [A: set_list_nat] :
( ( A != bot_bot_set_list_nat )
= ( ord_le1190675801316882794st_nat @ bot_bot_set_list_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_875_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_876_bot_Oextremum__strict,axiom,
! [A: set_list_nat] :
~ ( ord_le1190675801316882794st_nat @ A @ bot_bot_set_list_nat ) ).
% bot.extremum_strict
thf(fact_877_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_878_card__gt__0__iff,axiom,
! [A2: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
= ( ( A2 != bot_bot_set_nat )
& ( finite_finite_nat @ A2 ) ) ) ).
% card_gt_0_iff
thf(fact_879_card__gt__0__iff,axiom,
! [A2: set_list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_list_nat @ A2 ) )
= ( ( A2 != bot_bot_set_list_nat )
& ( finite8100373058378681591st_nat @ A2 ) ) ) ).
% card_gt_0_iff
thf(fact_880_Khovanskii_Oalpha__plus,axiom,
! [G4: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,X2: list_nat,Y: list_nat] :
( ( khovanskii_nat @ G4 @ Addition @ Zero @ A2 )
=> ( ( ( size_size_list_nat @ X2 )
= ( finite_card_nat @ A2 ) )
=> ( ( ( size_size_list_nat @ Y )
= ( finite_card_nat @ A2 ) )
=> ( ( alpha_nat @ G4 @ Addition @ Zero @ A2 @ ( plus_plus_list_nat @ X2 @ Y ) )
= ( Addition @ ( alpha_nat @ G4 @ Addition @ Zero @ A2 @ X2 ) @ ( alpha_nat @ G4 @ Addition @ Zero @ A2 @ Y ) ) ) ) ) ) ).
% Khovanskii.alpha_plus
thf(fact_881_Khovanskii_Oalpha__plus,axiom,
! [G4: set_list_nat,Addition: list_nat > list_nat > list_nat,Zero: list_nat,A2: set_list_nat,X2: list_nat,Y: list_nat] :
( ( khovanskii_list_nat @ G4 @ Addition @ Zero @ A2 )
=> ( ( ( size_size_list_nat @ X2 )
= ( finite_card_list_nat @ A2 ) )
=> ( ( ( size_size_list_nat @ Y )
= ( finite_card_list_nat @ A2 ) )
=> ( ( alpha_list_nat @ G4 @ Addition @ Zero @ A2 @ ( plus_plus_list_nat @ X2 @ Y ) )
= ( Addition @ ( alpha_list_nat @ G4 @ Addition @ Zero @ A2 @ X2 ) @ ( alpha_list_nat @ G4 @ Addition @ Zero @ A2 @ Y ) ) ) ) ) ) ).
% Khovanskii.alpha_plus
thf(fact_882_card__subset__eq,axiom,
! [B3: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B3 ) )
=> ( A2 = B3 ) ) ) ) ).
% card_subset_eq
thf(fact_883_card__subset__eq,axiom,
! [B3: set_list_nat,A2: set_list_nat] :
( ( finite8100373058378681591st_nat @ B3 )
=> ( ( ord_le6045566169113846134st_nat @ A2 @ B3 )
=> ( ( ( finite_card_list_nat @ A2 )
= ( finite_card_list_nat @ B3 ) )
=> ( A2 = B3 ) ) ) ) ).
% card_subset_eq
thf(fact_884_infinite__arbitrarily__large,axiom,
! [A2: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A2 )
=> ? [B7: set_nat] :
( ( finite_finite_nat @ B7 )
& ( ( finite_card_nat @ B7 )
= N )
& ( ord_less_eq_set_nat @ B7 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_885_infinite__arbitrarily__large,axiom,
! [A2: set_list_nat,N: nat] :
( ~ ( finite8100373058378681591st_nat @ A2 )
=> ? [B7: set_list_nat] :
( ( finite8100373058378681591st_nat @ B7 )
& ( ( finite_card_list_nat @ B7 )
= N )
& ( ord_le6045566169113846134st_nat @ B7 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_886_add__nonpos__eq__0__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_887_add__nonneg__eq__0__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_888_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_889_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_890_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_891_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_892_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_893_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_894_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_895_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_896_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D3 ) ) ) ) ).
% add_le_less_mono
thf(fact_897_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D3 ) ) ) ) ).
% add_less_le_mono
thf(fact_898_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_899_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_900_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C4: nat] :
( ( B
= ( plus_plus_nat @ A @ C4 ) )
=> ( C4 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_901_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_902_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_903_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_904_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_905_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_906_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_907_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_908_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_909_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_910_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_911_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_912_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_913_diff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% diff_add
thf(fact_914_card__le__if__inj__on__rel,axiom,
! [B3: set_list_nat,A2: set_nat,R: nat > list_nat > $o] :
( ( finite8100373058378681591st_nat @ B3 )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A2 )
=> ? [B8: list_nat] :
( ( member_list_nat @ B8 @ B3 )
& ( R @ A5 @ B8 ) ) )
=> ( ! [A1: nat,A22: nat,B4: list_nat] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_list_nat @ B4 @ B3 )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_list_nat @ B3 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_915_card__le__if__inj__on__rel,axiom,
! [B3: set_list_nat,A2: set_list_nat,R: list_nat > list_nat > $o] :
( ( finite8100373058378681591st_nat @ B3 )
=> ( ! [A5: list_nat] :
( ( member_list_nat @ A5 @ A2 )
=> ? [B8: list_nat] :
( ( member_list_nat @ B8 @ B3 )
& ( R @ A5 @ B8 ) ) )
=> ( ! [A1: list_nat,A22: list_nat,B4: list_nat] :
( ( member_list_nat @ A1 @ A2 )
=> ( ( member_list_nat @ A22 @ A2 )
=> ( ( member_list_nat @ B4 @ B3 )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_list_nat @ A2 ) @ ( finite_card_list_nat @ B3 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_916_card__le__if__inj__on__rel,axiom,
! [B3: set_nat,A2: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ B3 )
=> ( ! [A5: nat] :
( ( member_nat @ A5 @ A2 )
=> ? [B8: nat] :
( ( member_nat @ B8 @ B3 )
& ( R @ A5 @ B8 ) ) )
=> ( ! [A1: nat,A22: nat,B4: nat] :
( ( member_nat @ A1 @ A2 )
=> ( ( member_nat @ A22 @ A2 )
=> ( ( member_nat @ B4 @ B3 )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B3 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_917_card__le__if__inj__on__rel,axiom,
! [B3: set_nat,A2: set_list_nat,R: list_nat > nat > $o] :
( ( finite_finite_nat @ B3 )
=> ( ! [A5: list_nat] :
( ( member_list_nat @ A5 @ A2 )
=> ? [B8: nat] :
( ( member_nat @ B8 @ B3 )
& ( R @ A5 @ B8 ) ) )
=> ( ! [A1: list_nat,A22: list_nat,B4: nat] :
( ( member_list_nat @ A1 @ A2 )
=> ( ( member_list_nat @ A22 @ A2 )
=> ( ( member_nat @ B4 @ B3 )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_list_nat @ A2 ) @ ( finite_card_nat @ B3 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_918_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_919_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_920_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_921_finite__same__card__bij,axiom,
! [A2: set_list_nat,B3: set_list_nat] :
( ( finite8100373058378681591st_nat @ A2 )
=> ( ( finite8100373058378681591st_nat @ B3 )
=> ( ( ( finite_card_list_nat @ A2 )
= ( finite_card_list_nat @ B3 ) )
=> ? [H: list_nat > list_nat] : ( bij_be5167025596389125688st_nat @ H @ A2 @ B3 ) ) ) ) ).
% finite_same_card_bij
thf(fact_922_finite__same__card__bij,axiom,
! [A2: set_list_nat,B3: set_nat] :
( ( finite8100373058378681591st_nat @ A2 )
=> ( ( finite_finite_nat @ B3 )
=> ( ( ( finite_card_list_nat @ A2 )
= ( finite_card_nat @ B3 ) )
=> ? [H: list_nat > nat] : ( bij_be8532844293280997160at_nat @ H @ A2 @ B3 ) ) ) ) ).
% finite_same_card_bij
thf(fact_923_finite__same__card__bij,axiom,
! [A2: set_nat,B3: set_list_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite8100373058378681591st_nat @ B3 )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_list_nat @ B3 ) )
=> ? [H: nat > list_nat] : ( bij_be6293887246118711976st_nat @ H @ A2 @ B3 ) ) ) ) ).
% finite_same_card_bij
thf(fact_924_finite__same__card__bij,axiom,
! [A2: set_nat,B3: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B3 )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B3 ) )
=> ? [H: nat > nat] : ( bij_betw_nat_nat @ H @ A2 @ B3 ) ) ) ) ).
% finite_same_card_bij
thf(fact_925_bij__betw__iff__card,axiom,
! [A2: set_list_nat,B3: set_list_nat] :
( ( finite8100373058378681591st_nat @ A2 )
=> ( ( finite8100373058378681591st_nat @ B3 )
=> ( ( ? [F2: list_nat > list_nat] : ( bij_be5167025596389125688st_nat @ F2 @ A2 @ B3 ) )
= ( ( finite_card_list_nat @ A2 )
= ( finite_card_list_nat @ B3 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_926_bij__betw__iff__card,axiom,
! [A2: set_list_nat,B3: set_nat] :
( ( finite8100373058378681591st_nat @ A2 )
=> ( ( finite_finite_nat @ B3 )
=> ( ( ? [F2: list_nat > nat] : ( bij_be8532844293280997160at_nat @ F2 @ A2 @ B3 ) )
= ( ( finite_card_list_nat @ A2 )
= ( finite_card_nat @ B3 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_927_bij__betw__iff__card,axiom,
! [A2: set_nat,B3: set_list_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite8100373058378681591st_nat @ B3 )
=> ( ( ? [F2: nat > list_nat] : ( bij_be6293887246118711976st_nat @ F2 @ A2 @ B3 ) )
= ( ( finite_card_nat @ A2 )
= ( finite_card_list_nat @ B3 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_928_bij__betw__iff__card,axiom,
! [A2: set_nat,B3: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B3 )
=> ( ( ? [F2: nat > nat] : ( bij_betw_nat_nat @ F2 @ A2 @ B3 ) )
= ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B3 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_929_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_930_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_931_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_932_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).
% less_natE
thf(fact_933_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_934_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_935_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M5: nat,N5: nat] :
? [K3: nat] :
( N5
= ( suc @ ( plus_plus_nat @ M5 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_936_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_937_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_938_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_939_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_940_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_941_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_942_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_943_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_944_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_945_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_946_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_947_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_948_Suc__eq__plus1,axiom,
( suc
= ( ^ [N5: nat] : ( plus_plus_nat @ N5 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_949_finite__has__maximal,axiom,
! [A2: set_list_nat] :
( ( finite8100373058378681591st_nat @ A2 )
=> ( ( A2 != bot_bot_set_list_nat )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ A2 )
& ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ A2 )
=> ( ( ord_less_eq_list_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_950_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_951_finite__has__maximal,axiom,
! [A2: set_set_list_nat] :
( ( finite7047420756378620717st_nat @ A2 )
=> ( ( A2 != bot_bo3886227569956363488st_nat )
=> ? [X3: set_list_nat] :
( ( member_set_list_nat @ X3 @ A2 )
& ! [Xa: set_list_nat] :
( ( member_set_list_nat @ Xa @ A2 )
=> ( ( ord_le6045566169113846134st_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_952_finite__has__minimal,axiom,
! [A2: set_list_nat] :
( ( finite8100373058378681591st_nat @ A2 )
=> ( ( A2 != bot_bot_set_list_nat )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ A2 )
& ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ A2 )
=> ( ( ord_less_eq_list_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_953_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_954_finite__has__minimal,axiom,
! [A2: set_set_list_nat] :
( ( finite7047420756378620717st_nat @ A2 )
=> ( ( A2 != bot_bo3886227569956363488st_nat )
=> ? [X3: set_list_nat] :
( ( member_set_list_nat @ X3 @ A2 )
& ! [Xa: set_list_nat] :
( ( member_set_list_nat @ Xa @ A2 )
=> ( ( ord_le6045566169113846134st_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_955_ex__min__if__finite,axiom,
! [S2: set_list_nat] :
( ( finite8100373058378681591st_nat @ S2 )
=> ( ( S2 != bot_bot_set_list_nat )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ S2 )
& ~ ? [Xa: list_nat] :
( ( member_list_nat @ Xa @ S2 )
& ( ord_less_list_nat @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_956_ex__min__if__finite,axiom,
! [S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ S2 )
& ~ ? [Xa: nat] :
( ( member_nat @ Xa @ S2 )
& ( ord_less_nat @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_957_infinite__growing,axiom,
! [X8: set_list_nat] :
( ( X8 != bot_bot_set_list_nat )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ X8 )
=> ? [Xa: list_nat] :
( ( member_list_nat @ Xa @ X8 )
& ( ord_less_list_nat @ X3 @ Xa ) ) )
=> ~ ( finite8100373058378681591st_nat @ X8 ) ) ) ).
% infinite_growing
thf(fact_958_infinite__growing,axiom,
! [X8: set_nat] :
( ( X8 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X8 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X8 )
& ( ord_less_nat @ X3 @ Xa ) ) )
=> ~ ( finite_finite_nat @ X8 ) ) ) ).
% infinite_growing
thf(fact_959_sum__list__addf,axiom,
! [F: nat > nat,G: nat > nat,Xs: list_nat] :
( ( groups4561878855575611511st_nat
@ ( map_nat_nat
@ ^ [X: nat] : ( plus_plus_nat @ ( F @ X ) @ ( G @ X ) )
@ Xs ) )
= ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ F @ Xs ) ) @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ G @ Xs ) ) ) ) ).
% sum_list_addf
thf(fact_960_gen__length__def,axiom,
( gen_length_nat
= ( ^ [N5: nat,Xs3: list_nat] : ( plus_plus_nat @ N5 @ ( size_size_list_nat @ Xs3 ) ) ) ) ).
% gen_length_def
thf(fact_961_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_962_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_963_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_964_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_965_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_966_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_967_card__mono,axiom,
! [B3: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B3 ) ) ) ) ).
% card_mono
thf(fact_968_card__mono,axiom,
! [B3: set_list_nat,A2: set_list_nat] :
( ( finite8100373058378681591st_nat @ B3 )
=> ( ( ord_le6045566169113846134st_nat @ A2 @ B3 )
=> ( ord_less_eq_nat @ ( finite_card_list_nat @ A2 ) @ ( finite_card_list_nat @ B3 ) ) ) ) ).
% card_mono
thf(fact_969_card__seteq,axiom,
! [B3: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B3 ) @ ( finite_card_nat @ A2 ) )
=> ( A2 = B3 ) ) ) ) ).
% card_seteq
thf(fact_970_card__seteq,axiom,
! [B3: set_list_nat,A2: set_list_nat] :
( ( finite8100373058378681591st_nat @ B3 )
=> ( ( ord_le6045566169113846134st_nat @ A2 @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_list_nat @ B3 ) @ ( finite_card_list_nat @ A2 ) )
=> ( A2 = B3 ) ) ) ) ).
% card_seteq
thf(fact_971_exists__subset__between,axiom,
! [A2: set_nat,N: nat,C3: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C3 ) )
=> ( ( ord_less_eq_set_nat @ A2 @ C3 )
=> ( ( finite_finite_nat @ C3 )
=> ? [B7: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B7 )
& ( ord_less_eq_set_nat @ B7 @ C3 )
& ( ( finite_card_nat @ B7 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_972_exists__subset__between,axiom,
! [A2: set_list_nat,N: nat,C3: set_list_nat] :
( ( ord_less_eq_nat @ ( finite_card_list_nat @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_list_nat @ C3 ) )
=> ( ( ord_le6045566169113846134st_nat @ A2 @ C3 )
=> ( ( finite8100373058378681591st_nat @ C3 )
=> ? [B7: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B7 )
& ( ord_le6045566169113846134st_nat @ B7 @ C3 )
& ( ( finite_card_list_nat @ B7 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_973_obtain__subset__with__card__n,axiom,
! [N: nat,S2: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S2 ) )
=> ~ ! [T4: set_nat] :
( ( ord_less_eq_set_nat @ T4 @ S2 )
=> ( ( ( finite_card_nat @ T4 )
= N )
=> ~ ( finite_finite_nat @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_974_obtain__subset__with__card__n,axiom,
! [N: nat,S2: set_list_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_list_nat @ S2 ) )
=> ~ ! [T4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ T4 @ S2 )
=> ( ( ( finite_card_list_nat @ T4 )
= N )
=> ~ ( finite8100373058378681591st_nat @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_975_finite__if__finite__subsets__card__bdd,axiom,
! [F3: set_nat,C3: nat] :
( ! [G5: set_nat] :
( ( ord_less_eq_set_nat @ G5 @ F3 )
=> ( ( finite_finite_nat @ G5 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G5 ) @ C3 ) ) )
=> ( ( finite_finite_nat @ F3 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F3 ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_976_finite__if__finite__subsets__card__bdd,axiom,
! [F3: set_list_nat,C3: nat] :
( ! [G5: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ G5 @ F3 )
=> ( ( finite8100373058378681591st_nat @ G5 )
=> ( ord_less_eq_nat @ ( finite_card_list_nat @ G5 ) @ C3 ) ) )
=> ( ( finite8100373058378681591st_nat @ F3 )
& ( ord_less_eq_nat @ ( finite_card_list_nat @ F3 ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_977_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_978_card__ge__0__finite,axiom,
! [A2: set_list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_list_nat @ A2 ) )
=> ( finite8100373058378681591st_nat @ A2 ) ) ).
% card_ge_0_finite
thf(fact_979_card__ge__0__finite,axiom,
! [A2: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
=> ( finite_finite_nat @ A2 ) ) ).
% card_ge_0_finite
thf(fact_980_card__le__sym__Diff,axiom,
! [A2: set_list_nat,B3: set_list_nat] :
( ( finite8100373058378681591st_nat @ A2 )
=> ( ( finite8100373058378681591st_nat @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_list_nat @ A2 ) @ ( finite_card_list_nat @ B3 ) )
=> ( ord_less_eq_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ B3 ) ) @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ B3 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_981_card__le__sym__Diff,axiom,
! [A2: set_nat,B3: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B3 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B3 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B3 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_982_card__less__sym__Diff,axiom,
! [A2: set_list_nat,B3: set_list_nat] :
( ( finite8100373058378681591st_nat @ A2 )
=> ( ( finite8100373058378681591st_nat @ B3 )
=> ( ( ord_less_nat @ ( finite_card_list_nat @ A2 ) @ ( finite_card_list_nat @ B3 ) )
=> ( ord_less_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ B3 ) ) @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ B3 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_983_card__less__sym__Diff,axiom,
! [A2: set_nat,B3: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B3 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B3 ) )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B3 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B3 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_984_psubset__card__mono,axiom,
! [B3: set_list_nat,A2: set_list_nat] :
( ( finite8100373058378681591st_nat @ B3 )
=> ( ( ord_le1190675801316882794st_nat @ A2 @ B3 )
=> ( ord_less_nat @ ( finite_card_list_nat @ A2 ) @ ( finite_card_list_nat @ B3 ) ) ) ) ).
% psubset_card_mono
thf(fact_985_psubset__card__mono,axiom,
! [B3: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_set_nat @ A2 @ B3 )
=> ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B3 ) ) ) ) ).
% psubset_card_mono
thf(fact_986_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_987_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_988_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_989_card__less,axiom,
! [M7: set_nat,I: nat] :
( ( member_nat @ zero_zero_nat @ M7 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ K3 @ M7 )
& ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) )
!= zero_zero_nat ) ) ).
% card_less
thf(fact_990_card__less__Suc,axiom,
! [M7: set_nat,I: nat] :
( ( member_nat @ zero_zero_nat @ M7 )
=> ( ( suc
@ ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ ( suc @ K3 ) @ M7 )
& ( ord_less_nat @ K3 @ I ) ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ K3 @ M7 )
& ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_991_card__less__Suc2,axiom,
! [M7: set_nat,I: nat] :
( ~ ( member_nat @ zero_zero_nat @ M7 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ ( suc @ K3 ) @ M7 )
& ( ord_less_nat @ K3 @ I ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ K3 @ M7 )
& ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_992_pointwise__le__plus,axiom,
! [Xs: list_nat,Ys2: list_nat,Zs3: list_nat] :
( ( pointwise_le @ Xs @ Ys2 )
=> ( ( ord_less_eq_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Zs3 ) )
=> ( pointwise_le @ Xs @ ( plus_plus_list_nat @ Ys2 @ Zs3 ) ) ) ) ).
% pointwise_le_plus
thf(fact_993_card__le__Suc0__iff__eq,axiom,
! [A2: set_list_nat] :
( ( finite8100373058378681591st_nat @ A2 )
=> ( ( ord_less_eq_nat @ ( finite_card_list_nat @ A2 ) @ ( suc @ zero_zero_nat ) )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ A2 )
=> ! [Y4: list_nat] :
( ( member_list_nat @ Y4 @ A2 )
=> ( X = Y4 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_994_card__le__Suc0__iff__eq,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( suc @ zero_zero_nat ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ A2 )
=> ( X = Y4 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_995_card__Diff__subset,axiom,
! [B3: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B3 ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B3 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_996_card__Diff__subset,axiom,
! [B3: set_list_nat,A2: set_list_nat] :
( ( finite8100373058378681591st_nat @ B3 )
=> ( ( ord_le6045566169113846134st_nat @ B3 @ A2 )
=> ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ B3 ) )
= ( minus_minus_nat @ ( finite_card_list_nat @ A2 ) @ ( finite_card_list_nat @ B3 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_997_card__psubset,axiom,
! [B3: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B3 ) )
=> ( ord_less_set_nat @ A2 @ B3 ) ) ) ) ).
% card_psubset
thf(fact_998_card__psubset,axiom,
! [B3: set_list_nat,A2: set_list_nat] :
( ( finite8100373058378681591st_nat @ B3 )
=> ( ( ord_le6045566169113846134st_nat @ A2 @ B3 )
=> ( ( ord_less_nat @ ( finite_card_list_nat @ A2 ) @ ( finite_card_list_nat @ B3 ) )
=> ( ord_le1190675801316882794st_nat @ A2 @ B3 ) ) ) ) ).
% card_psubset
thf(fact_999_diff__card__le__card__Diff,axiom,
! [B3: set_list_nat,A2: set_list_nat] :
( ( finite8100373058378681591st_nat @ B3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_list_nat @ A2 ) @ ( finite_card_list_nat @ B3 ) ) @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ B3 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1000_diff__card__le__card__Diff,axiom,
! [B3: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B3 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B3 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1001_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_1002_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M5: nat,N5: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N5 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N5 ) ) ) ) ) ).
% add_eq_if
thf(fact_1003_Khovanskii_Onth__aA__in__G,axiom,
! [G4: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,I: nat] :
( ( khovanskii_nat @ G4 @ Addition @ Zero @ A2 )
=> ( ( ord_less_nat @ I @ ( finite_card_nat @ A2 ) )
=> ( member_nat @ ( nth_nat @ ( aA_nat @ A2 ) @ I ) @ G4 ) ) ) ).
% Khovanskii.nth_aA_in_G
thf(fact_1004_Khovanskii_Onth__aA__in__G,axiom,
! [G4: set_list_nat,Addition: list_nat > list_nat > list_nat,Zero: list_nat,A2: set_list_nat,I: nat] :
( ( khovanskii_list_nat @ G4 @ Addition @ Zero @ A2 )
=> ( ( ord_less_nat @ I @ ( finite_card_list_nat @ A2 ) )
=> ( member_list_nat @ ( nth_list_nat @ ( aA_list_nat @ A2 ) @ I ) @ G4 ) ) ) ).
% Khovanskii.nth_aA_in_G
thf(fact_1005_max__pointwise__mono,axiom,
! [X9: set_list_nat,X8: set_list_nat,R: nat] :
( ( ord_le6045566169113846134st_nat @ X9 @ X8 )
=> ( ( finite8100373058378681591st_nat @ X8 )
=> ( ( X9 != bot_bot_set_list_nat )
=> ( pointwise_le @ ( max_pointwise @ R @ X9 ) @ ( max_pointwise @ R @ X8 ) ) ) ) ) ).
% max_pointwise_mono
thf(fact_1006_Khovanskii_Ouseless__leq__useless,axiom,
! [G4: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat,X2: list_nat,Y: list_nat] :
( ( khovanskii_nat @ G4 @ Addition @ Zero @ A2 )
=> ( ( useless_nat @ G4 @ Addition @ Zero @ A2 @ X2 )
=> ( ( pointwise_le @ X2 @ Y )
=> ( ( ( size_size_list_nat @ X2 )
= ( finite_card_nat @ A2 ) )
=> ( useless_nat @ G4 @ Addition @ Zero @ A2 @ Y ) ) ) ) ) ).
% Khovanskii.useless_leq_useless
thf(fact_1007_Khovanskii_Ouseless__leq__useless,axiom,
! [G4: set_list_nat,Addition: list_nat > list_nat > list_nat,Zero: list_nat,A2: set_list_nat,X2: list_nat,Y: list_nat] :
( ( khovanskii_list_nat @ G4 @ Addition @ Zero @ A2 )
=> ( ( useless_list_nat @ G4 @ Addition @ Zero @ A2 @ X2 )
=> ( ( pointwise_le @ X2 @ Y )
=> ( ( ( size_size_list_nat @ X2 )
= ( finite_card_list_nat @ A2 ) )
=> ( useless_list_nat @ G4 @ Addition @ Zero @ A2 @ Y ) ) ) ) ) ).
% Khovanskii.useless_leq_useless
thf(fact_1008_Khovanskii_Oalpha__replicate__0,axiom,
! [G4: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat] :
( ( khovanskii_nat @ G4 @ Addition @ Zero @ A2 )
=> ( ( alpha_nat @ G4 @ Addition @ Zero @ A2 @ ( replicate_nat @ ( finite_card_nat @ A2 ) @ zero_zero_nat ) )
= Zero ) ) ).
% Khovanskii.alpha_replicate_0
thf(fact_1009_Khovanskii_Oalpha__replicate__0,axiom,
! [G4: set_list_nat,Addition: list_nat > list_nat > list_nat,Zero: list_nat,A2: set_list_nat] :
( ( khovanskii_list_nat @ G4 @ Addition @ Zero @ A2 )
=> ( ( alpha_list_nat @ G4 @ Addition @ Zero @ A2 @ ( replicate_nat @ ( finite_card_list_nat @ A2 ) @ zero_zero_nat ) )
= Zero ) ) ).
% Khovanskii.alpha_replicate_0
thf(fact_1010_sum__list__Suc,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( groups4561878855575611511st_nat
@ ( map_nat_nat
@ ^ [X: nat] : ( suc @ ( F @ X ) )
@ Xs ) )
= ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ F @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ) ).
% sum_list_Suc
thf(fact_1011_card__length__sum__list__rec,axiom,
! [M: nat,N4: nat] :
( ( ord_less_eq_nat @ one_one_nat @ M )
=> ( ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L2: list_nat] :
( ( ( size_size_list_nat @ L2 )
= M )
& ( ( groups4561878855575611511st_nat @ L2 )
= N4 ) ) ) )
= ( plus_plus_nat
@ ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L2: list_nat] :
( ( ( size_size_list_nat @ L2 )
= ( minus_minus_nat @ M @ one_one_nat ) )
& ( ( groups4561878855575611511st_nat @ L2 )
= N4 ) ) ) )
@ ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L2: list_nat] :
( ( ( size_size_list_nat @ L2 )
= M )
& ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L2 ) @ one_one_nat )
= N4 ) ) ) ) ) ) ) ).
% card_length_sum_list_rec
thf(fact_1012_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_1013_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_1014_length__augmentum,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( augmentum @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_augmentum
thf(fact_1015_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1016_all__zeroes__replicate,axiom,
! [R: nat] :
( ( length_sum_set @ R @ zero_zero_nat )
= ( insert_list_nat @ ( replicate_nat @ R @ zero_zero_nat ) @ bot_bot_set_list_nat ) ) ).
% all_zeroes_replicate
thf(fact_1017_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_1018_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_1019_insert__subset,axiom,
! [X2: nat,A2: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ A2 ) @ B3 )
= ( ( member_nat @ X2 @ B3 )
& ( ord_less_eq_set_nat @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_1020_insert__subset,axiom,
! [X2: list_nat,A2: set_list_nat,B3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( insert_list_nat @ X2 @ A2 ) @ B3 )
= ( ( member_list_nat @ X2 @ B3 )
& ( ord_le6045566169113846134st_nat @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_1021_List_Ofinite__set,axiom,
! [Xs: list_list_nat] : ( finite8100373058378681591st_nat @ ( set_list_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_1022_List_Ofinite__set,axiom,
! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_1023_Diff__insert0,axiom,
! [X2: list_nat,A2: set_list_nat,B3: set_list_nat] :
( ~ ( member_list_nat @ X2 @ A2 )
=> ( ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ X2 @ B3 ) )
= ( minus_7954133019191499631st_nat @ A2 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_1024_Diff__insert0,axiom,
! [X2: nat,A2: set_nat,B3: set_nat] :
( ~ ( member_nat @ X2 @ A2 )
=> ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ B3 ) )
= ( minus_minus_set_nat @ A2 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_1025_insert__Diff1,axiom,
! [X2: list_nat,B3: set_list_nat,A2: set_list_nat] :
( ( member_list_nat @ X2 @ B3 )
=> ( ( minus_7954133019191499631st_nat @ ( insert_list_nat @ X2 @ A2 ) @ B3 )
= ( minus_7954133019191499631st_nat @ A2 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_1026_insert__Diff1,axiom,
! [X2: nat,B3: set_nat,A2: set_nat] :
( ( member_nat @ X2 @ B3 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A2 ) @ B3 )
= ( minus_minus_set_nat @ A2 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_1027_map__eq__conv,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Xs ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_1028_singleton__conv,axiom,
! [A: nat] :
( ( collect_nat
@ ^ [X: nat] : ( X = A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_1029_singleton__conv,axiom,
! [A: list_nat] :
( ( collect_list_nat
@ ^ [X: list_nat] : ( X = A ) )
= ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ).
% singleton_conv
thf(fact_1030_singleton__conv2,axiom,
! [A: nat] :
( ( collect_nat
@ ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 )
@ A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_1031_singleton__conv2,axiom,
! [A: list_nat] :
( ( collect_list_nat
@ ( ^ [Y6: list_nat,Z4: list_nat] : ( Y6 = Z4 )
@ A ) )
= ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ).
% singleton_conv2
thf(fact_1032_singleton__insert__inj__eq_H,axiom,
! [A: list_nat,A2: set_list_nat,B: list_nat] :
( ( ( insert_list_nat @ A @ A2 )
= ( insert_list_nat @ B @ bot_bot_set_list_nat ) )
= ( ( A = B )
& ( ord_le6045566169113846134st_nat @ A2 @ ( insert_list_nat @ B @ bot_bot_set_list_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1033_singleton__insert__inj__eq,axiom,
! [B: list_nat,A: list_nat,A2: set_list_nat] :
( ( ( insert_list_nat @ B @ bot_bot_set_list_nat )
= ( insert_list_nat @ A @ A2 ) )
= ( ( A = B )
& ( ord_le6045566169113846134st_nat @ A2 @ ( insert_list_nat @ B @ bot_bot_set_list_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1034_insert__Diff__single,axiom,
! [A: list_nat,A2: set_list_nat] :
( ( insert_list_nat @ A @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) )
= ( insert_list_nat @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_1035_finite__Diff__insert,axiom,
! [A2: set_list_nat,A: list_nat,B3: set_list_nat] :
( ( finite8100373058378681591st_nat @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ A @ B3 ) ) )
= ( finite8100373058378681591st_nat @ ( minus_7954133019191499631st_nat @ A2 @ B3 ) ) ) ).
% finite_Diff_insert
thf(fact_1036_finite__Diff__insert,axiom,
! [A2: set_nat,A: nat,B3: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B3 ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B3 ) ) ) ).
% finite_Diff_insert
thf(fact_1037_list_Osimps_I15_J,axiom,
! [X21: list_nat,X222: list_list_nat] :
( ( set_list_nat2 @ ( cons_list_nat @ X21 @ X222 ) )
= ( insert_list_nat @ X21 @ ( set_list_nat2 @ X222 ) ) ) ).
% list.simps(15)
thf(fact_1038_list_Osimps_I15_J,axiom,
! [X21: nat,X222: list_nat] :
( ( set_nat2 @ ( cons_nat @ X21 @ X222 ) )
= ( insert_nat @ X21 @ ( set_nat2 @ X222 ) ) ) ).
% list.simps(15)
thf(fact_1039_sum__list__eq__0__iff,axiom,
! [Ns: list_nat] :
( ( ( groups4561878855575611511st_nat @ Ns )
= zero_zero_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Ns ) )
=> ( X = zero_zero_nat ) ) ) ) ).
% sum_list_eq_0_iff
thf(fact_1040_Ball__set__replicate,axiom,
! [N: nat,A: nat,P: nat > $o] :
( ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
=> ( P @ X ) ) )
= ( ( P @ A )
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_1041_Bex__set__replicate,axiom,
! [N: nat,A: nat,P: nat > $o] :
( ( ? [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
& ( P @ X ) ) )
= ( ( P @ A )
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_1042_in__set__replicate,axiom,
! [X2: list_nat,N: nat,Y: list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ ( replicate_list_nat @ N @ Y ) ) )
= ( ( X2 = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_1043_in__set__replicate,axiom,
! [X2: nat,N: nat,Y: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ ( replicate_nat @ N @ Y ) ) )
= ( ( X2 = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_1044_card__insert__disjoint,axiom,
! [A2: set_list_nat,X2: list_nat] :
( ( finite8100373058378681591st_nat @ A2 )
=> ( ~ ( member_list_nat @ X2 @ A2 )
=> ( ( finite_card_list_nat @ ( insert_list_nat @ X2 @ A2 ) )
= ( suc @ ( finite_card_list_nat @ A2 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_1045_card__insert__disjoint,axiom,
! [A2: set_nat,X2: nat] :
( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat @ X2 @ A2 )
=> ( ( finite_card_nat @ ( insert_nat @ X2 @ A2 ) )
= ( suc @ ( finite_card_nat @ A2 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_1046_card__Diff__insert,axiom,
! [A: nat,A2: set_nat,B3: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ A @ B3 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B3 ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B3 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1047_card__Diff__insert,axiom,
! [A: list_nat,A2: set_list_nat,B3: set_list_nat] :
( ( member_list_nat @ A @ A2 )
=> ( ~ ( member_list_nat @ A @ B3 )
=> ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ A @ B3 ) ) )
= ( minus_minus_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ B3 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1048_set__replicate,axiom,
! [N: nat,X2: nat] :
( ( N != zero_zero_nat )
=> ( ( set_nat2 @ ( replicate_nat @ N @ X2 ) )
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ).
% set_replicate
thf(fact_1049_set__replicate,axiom,
! [N: nat,X2: list_nat] :
( ( N != zero_zero_nat )
=> ( ( set_list_nat2 @ ( replicate_list_nat @ N @ X2 ) )
= ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) ) ).
% set_replicate
thf(fact_1050_set__swap,axiom,
! [I: nat,Xs: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I ) ) )
= ( set_nat2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_1051_Collect__conv__if2,axiom,
! [P: nat > $o,A: nat] :
( ( ( P @ A )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( A = X )
& ( P @ X ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( A = X )
& ( P @ X ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_1052_Collect__conv__if2,axiom,
! [P: list_nat > $o,A: list_nat] :
( ( ( P @ A )
=> ( ( collect_list_nat
@ ^ [X: list_nat] :
( ( A = X )
& ( P @ X ) ) )
= ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_list_nat
@ ^ [X: list_nat] :
( ( A = X )
& ( P @ X ) ) )
= bot_bot_set_list_nat ) ) ) ).
% Collect_conv_if2
thf(fact_1053_Collect__conv__if,axiom,
! [P: nat > $o,A: nat] :
( ( ( P @ A )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( X = A )
& ( P @ X ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( X = A )
& ( P @ X ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_1054_Collect__conv__if,axiom,
! [P: list_nat > $o,A: list_nat] :
( ( ( P @ A )
=> ( ( collect_list_nat
@ ^ [X: list_nat] :
( ( X = A )
& ( P @ X ) ) )
= ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_list_nat
@ ^ [X: list_nat] :
( ( X = A )
& ( P @ X ) ) )
= bot_bot_set_list_nat ) ) ) ).
% Collect_conv_if
thf(fact_1055_inj__singleton,axiom,
! [A2: set_list_nat] :
( inj_on8624761805129053417st_nat
@ ^ [X: list_nat] : ( insert_list_nat @ X @ bot_bot_set_list_nat )
@ A2 ) ).
% inj_singleton
thf(fact_1056_set__replicate__Suc,axiom,
! [N: nat,X2: nat] :
( ( set_nat2 @ ( replicate_nat @ ( suc @ N ) @ X2 ) )
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ).
% set_replicate_Suc
thf(fact_1057_set__replicate__Suc,axiom,
! [N: nat,X2: list_nat] :
( ( set_list_nat2 @ ( replicate_list_nat @ ( suc @ N ) @ X2 ) )
= ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) ).
% set_replicate_Suc
thf(fact_1058_set__replicate__conv__if,axiom,
! [N: nat,X2: nat] :
( ( ( N = zero_zero_nat )
=> ( ( set_nat2 @ ( replicate_nat @ N @ X2 ) )
= bot_bot_set_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( set_nat2 @ ( replicate_nat @ N @ X2 ) )
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% set_replicate_conv_if
thf(fact_1059_set__replicate__conv__if,axiom,
! [N: nat,X2: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( set_list_nat2 @ ( replicate_list_nat @ N @ X2 ) )
= bot_bot_set_list_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( set_list_nat2 @ ( replicate_list_nat @ N @ X2 ) )
= ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) ) ) ).
% set_replicate_conv_if
thf(fact_1060_set__update__subset__insert,axiom,
! [Xs: list_nat,I: nat,X2: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X2 ) ) @ ( insert_nat @ X2 @ ( set_nat2 @ Xs ) ) ) ).
% set_update_subset_insert
thf(fact_1061_set__update__subset__insert,axiom,
! [Xs: list_list_nat,I: nat,X2: list_nat] : ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ ( list_update_list_nat @ Xs @ I @ X2 ) ) @ ( insert_list_nat @ X2 @ ( set_list_nat2 @ Xs ) ) ) ).
% set_update_subset_insert
thf(fact_1062_subset__code_I1_J,axiom,
! [Xs: list_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B3 )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_1063_subset__code_I1_J,axiom,
! [Xs: list_list_nat,B3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ B3 )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ( member_list_nat @ X @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_1064_subset__insertI2,axiom,
! [A2: set_list_nat,B3: set_list_nat,B: list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B3 )
=> ( ord_le6045566169113846134st_nat @ A2 @ ( insert_list_nat @ B @ B3 ) ) ) ).
% subset_insertI2
thf(fact_1065_subset__insertI,axiom,
! [B3: set_list_nat,A: list_nat] : ( ord_le6045566169113846134st_nat @ B3 @ ( insert_list_nat @ A @ B3 ) ) ).
% subset_insertI
thf(fact_1066_subset__insert,axiom,
! [X2: nat,A2: set_nat,B3: set_nat] :
( ~ ( member_nat @ X2 @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X2 @ B3 ) )
= ( ord_less_eq_set_nat @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_1067_subset__insert,axiom,
! [X2: list_nat,A2: set_list_nat,B3: set_list_nat] :
( ~ ( member_list_nat @ X2 @ A2 )
=> ( ( ord_le6045566169113846134st_nat @ A2 @ ( insert_list_nat @ X2 @ B3 ) )
= ( ord_le6045566169113846134st_nat @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_1068_insert__mono,axiom,
! [C3: set_list_nat,D: set_list_nat,A: list_nat] :
( ( ord_le6045566169113846134st_nat @ C3 @ D )
=> ( ord_le6045566169113846134st_nat @ ( insert_list_nat @ A @ C3 ) @ ( insert_list_nat @ A @ D ) ) ) ).
% insert_mono
thf(fact_1069_ex__map__conv,axiom,
! [Ys2: list_nat,F: nat > nat] :
( ( ? [Xs3: list_nat] :
( Ys2
= ( map_nat_nat @ F @ Xs3 ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Ys2 ) )
=> ? [Y4: nat] :
( X
= ( F @ Y4 ) ) ) ) ) ).
% ex_map_conv
thf(fact_1070_map__cong,axiom,
! [Xs: list_nat,Ys2: list_nat,F: nat > nat,G: nat > nat] :
( ( Xs = Ys2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Ys2 ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys2 ) ) ) ) ).
% map_cong
thf(fact_1071_map__idI,axiom,
! [Xs: list_list_nat,F: list_nat > list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_li7225945977422193158st_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_1072_map__idI,axiom,
! [Xs: list_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_nat_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_1073_map__ext,axiom,
! [Xs: list_nat,F: nat > nat,G: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_1074_list_Omap__ident__strong,axiom,
! [T: list_list_nat,F: list_nat > list_nat] :
( ! [Z3: list_nat] :
( ( member_list_nat @ Z3 @ ( set_list_nat2 @ T ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_li7225945977422193158st_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_1075_list_Omap__ident__strong,axiom,
! [T: list_nat,F: nat > nat] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ T ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_nat_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_1076_list_Oinj__map__strong,axiom,
! [X2: list_nat,Xa2: list_nat,F: nat > nat,Fa: nat > nat] :
( ! [Z3: nat,Za: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ X2 ) )
=> ( ( member_nat @ Za @ ( set_nat2 @ Xa2 ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( map_nat_nat @ F @ X2 )
= ( map_nat_nat @ Fa @ Xa2 ) )
=> ( X2 = Xa2 ) ) ) ).
% list.inj_map_strong
thf(fact_1077_list_Omap__cong0,axiom,
! [X2: list_nat,F: nat > nat,G: nat > nat] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ X2 ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_nat_nat @ F @ X2 )
= ( map_nat_nat @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_1078_list_Omap__cong,axiom,
! [X2: list_nat,Ya: list_nat,F: nat > nat,G: nat > nat] :
( ( X2 = Ya )
=> ( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ Ya ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_nat_nat @ F @ X2 )
= ( map_nat_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_1079_set__subset__Cons,axiom,
! [Xs: list_nat,X2: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_1080_set__subset__Cons,axiom,
! [Xs: list_list_nat,X2: list_nat] : ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ ( set_list_nat2 @ ( cons_list_nat @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_1081_subset__singleton__iff,axiom,
! [X8: set_list_nat,A: list_nat] :
( ( ord_le6045566169113846134st_nat @ X8 @ ( insert_list_nat @ A @ bot_bot_set_list_nat ) )
= ( ( X8 = bot_bot_set_list_nat )
| ( X8
= ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_1082_subset__singletonD,axiom,
! [A2: set_list_nat,X2: list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) )
=> ( ( A2 = bot_bot_set_list_nat )
| ( A2
= ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) ) ) ).
% subset_singletonD
thf(fact_1083_member__le__sum__list,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ X2 @ ( groups4561878855575611511st_nat @ Xs ) ) ) ).
% member_le_sum_list
thf(fact_1084_Diff__insert__absorb,axiom,
! [X2: nat,A2: set_nat] :
( ~ ( member_nat @ X2 @ A2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A2 ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_1085_Diff__insert__absorb,axiom,
! [X2: list_nat,A2: set_list_nat] :
( ~ ( member_list_nat @ X2 @ A2 )
=> ( ( minus_7954133019191499631st_nat @ ( insert_list_nat @ X2 @ A2 ) @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_1086_Diff__insert2,axiom,
! [A2: set_list_nat,A: list_nat,B3: set_list_nat] :
( ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ A @ B3 ) )
= ( minus_7954133019191499631st_nat @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) @ B3 ) ) ).
% Diff_insert2
thf(fact_1087_insert__Diff,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_1088_insert__Diff,axiom,
! [A: list_nat,A2: set_list_nat] :
( ( member_list_nat @ A @ A2 )
=> ( ( insert_list_nat @ A @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_1089_Diff__insert,axiom,
! [A2: set_list_nat,A: list_nat,B3: set_list_nat] :
( ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ A @ B3 ) )
= ( minus_7954133019191499631st_nat @ ( minus_7954133019191499631st_nat @ A2 @ B3 ) @ ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ) ).
% Diff_insert
thf(fact_1090_subset__Diff__insert,axiom,
! [A2: set_nat,B3: set_nat,X2: nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B3 @ ( insert_nat @ X2 @ C3 ) ) )
= ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B3 @ C3 ) )
& ~ ( member_nat @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1091_subset__Diff__insert,axiom,
! [A2: set_list_nat,B3: set_list_nat,X2: list_nat,C3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ ( minus_7954133019191499631st_nat @ B3 @ ( insert_list_nat @ X2 @ C3 ) ) )
= ( ( ord_le6045566169113846134st_nat @ A2 @ ( minus_7954133019191499631st_nat @ B3 @ C3 ) )
& ~ ( member_list_nat @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1092_card__insert__le,axiom,
! [A2: set_nat,X2: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ ( insert_nat @ X2 @ A2 ) ) ) ).
% card_insert_le
thf(fact_1093_card__insert__le,axiom,
! [A2: set_list_nat,X2: list_nat] : ( ord_less_eq_nat @ ( finite_card_list_nat @ A2 ) @ ( finite_card_list_nat @ ( insert_list_nat @ X2 @ A2 ) ) ) ).
% card_insert_le
thf(fact_1094_set__update__subsetI,axiom,
! [Xs: list_nat,A2: set_nat,X2: nat,I: nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
=> ( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X2 ) ) @ A2 ) ) ) ).
% set_update_subsetI
thf(fact_1095_set__update__subsetI,axiom,
! [Xs: list_list_nat,A2: set_list_nat,X2: list_nat,I: nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ A2 )
=> ( ( member_list_nat @ X2 @ A2 )
=> ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ ( list_update_list_nat @ Xs @ I @ X2 ) ) @ A2 ) ) ) ).
% set_update_subsetI
thf(fact_1096_replicate__eqI,axiom,
! [Xs: list_list_nat,N: nat,X2: list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= N )
=> ( ! [Y3: list_nat] :
( ( member_list_nat @ Y3 @ ( set_list_nat2 @ Xs ) )
=> ( Y3 = X2 ) )
=> ( Xs
= ( replicate_list_nat @ N @ X2 ) ) ) ) ).
% replicate_eqI
thf(fact_1097_replicate__eqI,axiom,
! [Xs: list_nat,N: nat,X2: nat] :
( ( ( size_size_list_nat @ Xs )
= N )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ ( set_nat2 @ Xs ) )
=> ( Y3 = X2 ) )
=> ( Xs
= ( replicate_nat @ N @ X2 ) ) ) ) ).
% replicate_eqI
thf(fact_1098_replicate__length__same,axiom,
! [Xs: list_nat,X2: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( X3 = X2 ) )
=> ( ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X2 )
= Xs ) ) ).
% replicate_length_same
thf(fact_1099_Khovanskii_Oset__aA,axiom,
! [G4: set_nat,Addition: nat > nat > nat,Zero: nat,A2: set_nat] :
( ( khovanskii_nat @ G4 @ Addition @ Zero @ A2 )
=> ( ( set_nat2 @ ( aA_nat @ A2 ) )
= A2 ) ) ).
% Khovanskii.set_aA
thf(fact_1100_set__ConsD,axiom,
! [Y: list_nat,X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ Y @ ( set_list_nat2 @ ( cons_list_nat @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_list_nat @ Y @ ( set_list_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_1101_set__ConsD,axiom,
! [Y: nat,X2: nat,Xs: list_nat] :
( ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_nat @ Y @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_1102_list_Oset__cases,axiom,
! [E: list_nat,A: list_list_nat] :
( ( member_list_nat @ E @ ( set_list_nat2 @ A ) )
=> ( ! [Z22: list_list_nat] :
( A
!= ( cons_list_nat @ E @ Z22 ) )
=> ~ ! [Z1: list_nat,Z22: list_list_nat] :
( ( A
= ( cons_list_nat @ Z1 @ Z22 ) )
=> ~ ( member_list_nat @ E @ ( set_list_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_1103_list_Oset__cases,axiom,
! [E: nat,A: list_nat] :
( ( member_nat @ E @ ( set_nat2 @ A ) )
=> ( ! [Z22: list_nat] :
( A
!= ( cons_nat @ E @ Z22 ) )
=> ~ ! [Z1: nat,Z22: list_nat] :
( ( A
= ( cons_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat @ E @ ( set_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_1104_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_1105_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_1106_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_1107_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_1108_insert__Diff__if,axiom,
! [X2: list_nat,B3: set_list_nat,A2: set_list_nat] :
( ( ( member_list_nat @ X2 @ B3 )
=> ( ( minus_7954133019191499631st_nat @ ( insert_list_nat @ X2 @ A2 ) @ B3 )
= ( minus_7954133019191499631st_nat @ A2 @ B3 ) ) )
& ( ~ ( member_list_nat @ X2 @ B3 )
=> ( ( minus_7954133019191499631st_nat @ ( insert_list_nat @ X2 @ A2 ) @ B3 )
= ( insert_list_nat @ X2 @ ( minus_7954133019191499631st_nat @ A2 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1109_insert__Diff__if,axiom,
! [X2: nat,B3: set_nat,A2: set_nat] :
( ( ( member_nat @ X2 @ B3 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A2 ) @ B3 )
= ( minus_minus_set_nat @ A2 @ B3 ) ) )
& ( ~ ( member_nat @ X2 @ B3 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A2 ) @ B3 )
= ( insert_nat @ X2 @ ( minus_minus_set_nat @ A2 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1110_insert__Collect,axiom,
! [A: list_nat,P: list_nat > $o] :
( ( insert_list_nat @ A @ ( collect_list_nat @ P ) )
= ( collect_list_nat
@ ^ [U5: list_nat] :
( ( U5 != A )
=> ( P @ U5 ) ) ) ) ).
% insert_Collect
thf(fact_1111_insert__Collect,axiom,
! [A: nat,P: nat > $o] :
( ( insert_nat @ A @ ( collect_nat @ P ) )
= ( collect_nat
@ ^ [U5: nat] :
( ( U5 != A )
=> ( P @ U5 ) ) ) ) ).
% insert_Collect
thf(fact_1112_insert__compr,axiom,
( insert_list_nat
= ( ^ [A6: list_nat,B2: set_list_nat] :
( collect_list_nat
@ ^ [X: list_nat] :
( ( X = A6 )
| ( member_list_nat @ X @ B2 ) ) ) ) ) ).
% insert_compr
thf(fact_1113_insert__compr,axiom,
( insert_nat
= ( ^ [A6: nat,B2: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( X = A6 )
| ( member_nat @ X @ B2 ) ) ) ) ) ).
% insert_compr
thf(fact_1114_finite__list,axiom,
! [A2: set_list_nat] :
( ( finite8100373058378681591st_nat @ A2 )
=> ? [Xs2: list_list_nat] :
( ( set_list_nat2 @ Xs2 )
= A2 ) ) ).
% finite_list
thf(fact_1115_finite__list,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ? [Xs2: list_nat] :
( ( set_nat2 @ Xs2 )
= A2 ) ) ).
% finite_list
thf(fact_1116_Groups__List_Osum__list__nonneg,axiom,
! [Xs: list_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ X3 ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups4561878855575611511st_nat @ Xs ) ) ) ).
% Groups_List.sum_list_nonneg
thf(fact_1117_sum__list__nonneg__eq__0__iff,axiom,
! [Xs: list_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ X3 ) )
=> ( ( ( groups4561878855575611511st_nat @ Xs )
= zero_zero_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( X = zero_zero_nat ) ) ) ) ) ).
% sum_list_nonneg_eq_0_iff
thf(fact_1118_sum__list__nonpos,axiom,
! [Xs: list_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ X3 @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ Xs ) @ zero_zero_nat ) ) ).
% sum_list_nonpos
thf(fact_1119_finite__ranking__induct,axiom,
! [S2: set_nat,P: set_nat > $o,F: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ S3 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_nat @ X3 @ S3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_1120_finite__ranking__induct,axiom,
! [S2: set_list_nat,P: set_list_nat > $o,F: list_nat > nat] :
( ( finite8100373058378681591st_nat @ S2 )
=> ( ( P @ bot_bot_set_list_nat )
=> ( ! [X3: list_nat,S3: set_list_nat] :
( ( finite8100373058378681591st_nat @ S3 )
=> ( ! [Y5: list_nat] :
( ( member_list_nat @ Y5 @ S3 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X3 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_list_nat @ X3 @ S3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_1121_length__pos__if__in__set,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1122_length__pos__if__in__set,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1123_finite__linorder__min__induct,axiom,
! [A2: set_list_nat,P: set_list_nat > $o] :
( ( finite8100373058378681591st_nat @ A2 )
=> ( ( P @ bot_bot_set_list_nat )
=> ( ! [B4: list_nat,A8: set_list_nat] :
( ( finite8100373058378681591st_nat @ A8 )
=> ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ A8 )
=> ( ord_less_list_nat @ B4 @ X4 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert_list_nat @ B4 @ A8 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_1124_finite__linorder__min__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B4: nat,A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A8 )
=> ( ord_less_nat @ B4 @ X4 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert_nat @ B4 @ A8 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_1125_finite__linorder__max__induct,axiom,
! [A2: set_list_nat,P: set_list_nat > $o] :
( ( finite8100373058378681591st_nat @ A2 )
=> ( ( P @ bot_bot_set_list_nat )
=> ( ! [B4: list_nat,A8: set_list_nat] :
( ( finite8100373058378681591st_nat @ A8 )
=> ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ A8 )
=> ( ord_less_list_nat @ X4 @ B4 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert_list_nat @ B4 @ A8 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_1126_finite__linorder__max__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B4: nat,A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A8 )
=> ( ord_less_nat @ X4 @ B4 ) )
=> ( ( P @ A8 )
=> ( P @ ( insert_nat @ B4 @ A8 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_1127_card__length,axiom,
! [Xs: list_list_nat] : ( ord_less_eq_nat @ ( finite_card_list_nat @ ( set_list_nat2 @ Xs ) ) @ ( size_s3023201423986296836st_nat @ Xs ) ) ).
% card_length
thf(fact_1128_card__length,axiom,
! [Xs: list_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( set_nat2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% card_length
thf(fact_1129_finite__subset__induct_H,axiom,
! [F3: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( ord_less_eq_set_nat @ F3 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A5: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( member_nat @ A5 @ A2 )
=> ( ( ord_less_eq_set_nat @ F4 @ A2 )
=> ( ~ ( member_nat @ A5 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ A5 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1130_finite__subset__induct_H,axiom,
! [F3: set_list_nat,A2: set_list_nat,P: set_list_nat > $o] :
( ( finite8100373058378681591st_nat @ F3 )
=> ( ( ord_le6045566169113846134st_nat @ F3 @ A2 )
=> ( ( P @ bot_bot_set_list_nat )
=> ( ! [A5: list_nat,F4: set_list_nat] :
( ( finite8100373058378681591st_nat @ F4 )
=> ( ( member_list_nat @ A5 @ A2 )
=> ( ( ord_le6045566169113846134st_nat @ F4 @ A2 )
=> ( ~ ( member_list_nat @ A5 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_list_nat @ A5 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1131_finite__subset__induct,axiom,
! [F3: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( ord_less_eq_set_nat @ F3 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A5: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( member_nat @ A5 @ A2 )
=> ( ~ ( member_nat @ A5 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ A5 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1132_finite__subset__induct,axiom,
! [F3: set_list_nat,A2: set_list_nat,P: set_list_nat > $o] :
( ( finite8100373058378681591st_nat @ F3 )
=> ( ( ord_le6045566169113846134st_nat @ F3 @ A2 )
=> ( ( P @ bot_bot_set_list_nat )
=> ( ! [A5: list_nat,F4: set_list_nat] :
( ( finite8100373058378681591st_nat @ F4 )
=> ( ( member_list_nat @ A5 @ A2 )
=> ( ~ ( member_list_nat @ A5 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_list_nat @ A5 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1133_subset__insert__iff,axiom,
! [A2: set_nat,X2: nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X2 @ B3 ) )
= ( ( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B3 ) )
& ( ~ ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_1134_subset__insert__iff,axiom,
! [A2: set_list_nat,X2: list_nat,B3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ ( insert_list_nat @ X2 @ B3 ) )
= ( ( ( member_list_nat @ X2 @ A2 )
=> ( ord_le6045566169113846134st_nat @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) @ B3 ) )
& ( ~ ( member_list_nat @ X2 @ A2 )
=> ( ord_le6045566169113846134st_nat @ A2 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_1135_Diff__single__insert,axiom,
! [A2: set_list_nat,X2: list_nat,B3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) @ B3 )
=> ( ord_le6045566169113846134st_nat @ A2 @ ( insert_list_nat @ X2 @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_1136_nth__mem,axiom,
! [N: nat,Xs: list_list_nat] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( member_list_nat @ ( nth_list_nat @ Xs @ N ) @ ( set_list_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_1137_nth__mem,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_1138_list__ball__nth,axiom,
! [N: nat,Xs: list_nat,P: nat > $o] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( P @ X3 ) )
=> ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_1139_in__set__conv__nth,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs ) )
& ( ( nth_list_nat @ Xs @ I3 )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_1140_in__set__conv__nth,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I3 )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_1141_all__nth__imp__all__set,axiom,
! [Xs: list_list_nat,P: list_nat > $o,X2: list_nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( P @ ( nth_list_nat @ Xs @ I2 ) ) )
=> ( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_1142_all__nth__imp__all__set,axiom,
! [Xs: list_nat,P: nat > $o,X2: nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I2 ) ) )
=> ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_1143_all__set__conv__all__nth,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( P @ X ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1144_card__Suc__eq__finite,axiom,
! [A2: set_list_nat,K: nat] :
( ( ( finite_card_list_nat @ A2 )
= ( suc @ K ) )
= ( ? [B5: list_nat,B2: set_list_nat] :
( ( A2
= ( insert_list_nat @ B5 @ B2 ) )
& ~ ( member_list_nat @ B5 @ B2 )
& ( ( finite_card_list_nat @ B2 )
= K )
& ( finite8100373058378681591st_nat @ B2 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_1145_card__Suc__eq__finite,axiom,
! [A2: set_nat,K: nat] :
( ( ( finite_card_nat @ A2 )
= ( suc @ K ) )
= ( ? [B5: nat,B2: set_nat] :
( ( A2
= ( insert_nat @ B5 @ B2 ) )
& ~ ( member_nat @ B5 @ B2 )
& ( ( finite_card_nat @ B2 )
= K )
& ( finite_finite_nat @ B2 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_1146_card__insert__if,axiom,
! [A2: set_list_nat,X2: list_nat] :
( ( finite8100373058378681591st_nat @ A2 )
=> ( ( ( member_list_nat @ X2 @ A2 )
=> ( ( finite_card_list_nat @ ( insert_list_nat @ X2 @ A2 ) )
= ( finite_card_list_nat @ A2 ) ) )
& ( ~ ( member_list_nat @ X2 @ A2 )
=> ( ( finite_card_list_nat @ ( insert_list_nat @ X2 @ A2 ) )
= ( suc @ ( finite_card_list_nat @ A2 ) ) ) ) ) ) ).
% card_insert_if
thf(fact_1147_card__insert__if,axiom,
! [A2: set_nat,X2: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( member_nat @ X2 @ A2 )
=> ( ( finite_card_nat @ ( insert_nat @ X2 @ A2 ) )
= ( finite_card_nat @ A2 ) ) )
& ( ~ ( member_nat @ X2 @ A2 )
=> ( ( finite_card_nat @ ( insert_nat @ X2 @ A2 ) )
= ( suc @ ( finite_card_nat @ A2 ) ) ) ) ) ) ).
% card_insert_if
thf(fact_1148_finite__empty__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ A2 )
=> ( ! [A5: nat,A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( member_nat @ A5 @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ A5 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_1149_finite__empty__induct,axiom,
! [A2: set_list_nat,P: set_list_nat > $o] :
( ( finite8100373058378681591st_nat @ A2 )
=> ( ( P @ A2 )
=> ( ! [A5: list_nat,A8: set_list_nat] :
( ( finite8100373058378681591st_nat @ A8 )
=> ( ( member_list_nat @ A5 @ A8 )
=> ( ( P @ A8 )
=> ( P @ ( minus_7954133019191499631st_nat @ A8 @ ( insert_list_nat @ A5 @ bot_bot_set_list_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_list_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_1150_infinite__coinduct,axiom,
! [X8: set_nat > $o,A2: set_nat] :
( ( X8 @ A2 )
=> ( ! [A8: set_nat] :
( ( X8 @ A8 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A8 )
& ( ( X8 @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_1151_infinite__coinduct,axiom,
! [X8: set_list_nat > $o,A2: set_list_nat] :
( ( X8 @ A2 )
=> ( ! [A8: set_list_nat] :
( ( X8 @ A8 )
=> ? [X4: list_nat] :
( ( member_list_nat @ X4 @ A8 )
& ( ( X8 @ ( minus_7954133019191499631st_nat @ A8 @ ( insert_list_nat @ X4 @ bot_bot_set_list_nat ) ) )
| ~ ( finite8100373058378681591st_nat @ ( minus_7954133019191499631st_nat @ A8 @ ( insert_list_nat @ X4 @ bot_bot_set_list_nat ) ) ) ) ) )
=> ~ ( finite8100373058378681591st_nat @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_1152_infinite__remove,axiom,
! [S2: set_nat,A: nat] :
( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_1153_infinite__remove,axiom,
! [S2: set_list_nat,A: list_nat] :
( ~ ( finite8100373058378681591st_nat @ S2 )
=> ~ ( finite8100373058378681591st_nat @ ( minus_7954133019191499631st_nat @ S2 @ ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ) ) ).
% infinite_remove
thf(fact_1154_card__1__singletonE,axiom,
! [A2: set_nat] :
( ( ( finite_card_nat @ A2 )
= one_one_nat )
=> ~ ! [X3: nat] :
( A2
!= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% card_1_singletonE
thf(fact_1155_card__1__singletonE,axiom,
! [A2: set_list_nat] :
( ( ( finite_card_list_nat @ A2 )
= one_one_nat )
=> ~ ! [X3: list_nat] :
( A2
!= ( insert_list_nat @ X3 @ bot_bot_set_list_nat ) ) ) ).
% card_1_singletonE
thf(fact_1156_set__update__memI,axiom,
! [N: nat,Xs: list_list_nat,X2: list_nat] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( member_list_nat @ X2 @ ( set_list_nat2 @ ( list_update_list_nat @ Xs @ N @ X2 ) ) ) ) ).
% set_update_memI
thf(fact_1157_set__update__memI,axiom,
! [N: nat,Xs: list_nat,X2: nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ X2 @ ( set_nat2 @ ( list_update_nat @ Xs @ N @ X2 ) ) ) ) ).
% set_update_memI
thf(fact_1158_sum__list__mono,axiom,
! [Xs: list_list_nat,F: list_nat > nat,G: list_nat > nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ F @ Xs ) ) @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ G @ Xs ) ) ) ) ).
% sum_list_mono
thf(fact_1159_sum__list__mono,axiom,
! [Xs: list_nat,F: nat > nat,G: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ F @ Xs ) ) @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ G @ Xs ) ) ) ) ).
% sum_list_mono
thf(fact_1160_remove__induct,axiom,
! [P: set_nat > $o,B3: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B3 )
=> ( P @ B3 ) )
=> ( ! [A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( A8 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A8 @ B3 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A8 )
=> ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% remove_induct
thf(fact_1161_remove__induct,axiom,
! [P: set_list_nat > $o,B3: set_list_nat] :
( ( P @ bot_bot_set_list_nat )
=> ( ( ~ ( finite8100373058378681591st_nat @ B3 )
=> ( P @ B3 ) )
=> ( ! [A8: set_list_nat] :
( ( finite8100373058378681591st_nat @ A8 )
=> ( ( A8 != bot_bot_set_list_nat )
=> ( ( ord_le6045566169113846134st_nat @ A8 @ B3 )
=> ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ A8 )
=> ( P @ ( minus_7954133019191499631st_nat @ A8 @ ( insert_list_nat @ X4 @ bot_bot_set_list_nat ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% remove_induct
thf(fact_1162_finite__remove__induct,axiom,
! [B3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B3 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ( A8 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A8 @ B3 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A8 )
=> ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% finite_remove_induct
thf(fact_1163_finite__remove__induct,axiom,
! [B3: set_list_nat,P: set_list_nat > $o] :
( ( finite8100373058378681591st_nat @ B3 )
=> ( ( P @ bot_bot_set_list_nat )
=> ( ! [A8: set_list_nat] :
( ( finite8100373058378681591st_nat @ A8 )
=> ( ( A8 != bot_bot_set_list_nat )
=> ( ( ord_le6045566169113846134st_nat @ A8 @ B3 )
=> ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ A8 )
=> ( P @ ( minus_7954133019191499631st_nat @ A8 @ ( insert_list_nat @ X4 @ bot_bot_set_list_nat ) ) ) )
=> ( P @ A8 ) ) ) ) )
=> ( P @ B3 ) ) ) ) ).
% finite_remove_induct
thf(fact_1164_card__1__singleton__iff,axiom,
! [A2: set_nat] :
( ( ( finite_card_nat @ A2 )
= ( suc @ zero_zero_nat ) )
= ( ? [X: nat] :
( A2
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% card_1_singleton_iff
thf(fact_1165_card__1__singleton__iff,axiom,
! [A2: set_list_nat] :
( ( ( finite_card_list_nat @ A2 )
= ( suc @ zero_zero_nat ) )
= ( ? [X: list_nat] :
( A2
= ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) ) ) ).
% card_1_singleton_iff
thf(fact_1166_card__eq__SucD,axiom,
! [A2: set_nat,K: nat] :
( ( ( finite_card_nat @ A2 )
= ( suc @ K ) )
=> ? [B4: nat,B7: set_nat] :
( ( A2
= ( insert_nat @ B4 @ B7 ) )
& ~ ( member_nat @ B4 @ B7 )
& ( ( finite_card_nat @ B7 )
= K )
& ( ( K = zero_zero_nat )
=> ( B7 = bot_bot_set_nat ) ) ) ) ).
% card_eq_SucD
thf(fact_1167_card__eq__SucD,axiom,
! [A2: set_list_nat,K: nat] :
( ( ( finite_card_list_nat @ A2 )
= ( suc @ K ) )
=> ? [B4: list_nat,B7: set_list_nat] :
( ( A2
= ( insert_list_nat @ B4 @ B7 ) )
& ~ ( member_list_nat @ B4 @ B7 )
& ( ( finite_card_list_nat @ B7 )
= K )
& ( ( K = zero_zero_nat )
=> ( B7 = bot_bot_set_list_nat ) ) ) ) ).
% card_eq_SucD
thf(fact_1168_card__Suc__eq,axiom,
! [A2: set_nat,K: nat] :
( ( ( finite_card_nat @ A2 )
= ( suc @ K ) )
= ( ? [B5: nat,B2: set_nat] :
( ( A2
= ( insert_nat @ B5 @ B2 ) )
& ~ ( member_nat @ B5 @ B2 )
& ( ( finite_card_nat @ B2 )
= K )
& ( ( K = zero_zero_nat )
=> ( B2 = bot_bot_set_nat ) ) ) ) ) ).
% card_Suc_eq
thf(fact_1169_card__Suc__eq,axiom,
! [A2: set_list_nat,K: nat] :
( ( ( finite_card_list_nat @ A2 )
= ( suc @ K ) )
= ( ? [B5: list_nat,B2: set_list_nat] :
( ( A2
= ( insert_list_nat @ B5 @ B2 ) )
& ~ ( member_list_nat @ B5 @ B2 )
& ( ( finite_card_list_nat @ B2 )
= K )
& ( ( K = zero_zero_nat )
=> ( B2 = bot_bot_set_list_nat ) ) ) ) ) ).
% card_Suc_eq
thf(fact_1170_card__le__Suc__iff,axiom,
! [N: nat,A2: set_list_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_list_nat @ A2 ) )
= ( ? [A6: list_nat,B2: set_list_nat] :
( ( A2
= ( insert_list_nat @ A6 @ B2 ) )
& ~ ( member_list_nat @ A6 @ B2 )
& ( ord_less_eq_nat @ N @ ( finite_card_list_nat @ B2 ) )
& ( finite8100373058378681591st_nat @ B2 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_1171_card__le__Suc__iff,axiom,
! [N: nat,A2: set_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_nat @ A2 ) )
= ( ? [A6: nat,B2: set_nat] :
( ( A2
= ( insert_nat @ A6 @ B2 ) )
& ~ ( member_nat @ A6 @ B2 )
& ( ord_less_eq_nat @ N @ ( finite_card_nat @ B2 ) )
& ( finite_finite_nat @ B2 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_1172_card__Diff1__le,axiom,
! [A2: set_nat,X2: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ).
% card_Diff1_le
thf(fact_1173_card__Diff1__le,axiom,
! [A2: set_list_nat,X2: list_nat] : ( ord_less_eq_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) ) @ ( finite_card_list_nat @ A2 ) ) ).
% card_Diff1_le
thf(fact_1174_infinite__imp__bij__betw,axiom,
! [A2: set_nat,A: nat] :
( ~ ( finite_finite_nat @ A2 )
=> ? [H: nat > nat] : ( bij_betw_nat_nat @ H @ A2 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% infinite_imp_bij_betw
thf(fact_1175_infinite__imp__bij__betw,axiom,
! [A2: set_list_nat,A: list_nat] :
( ~ ( finite8100373058378681591st_nat @ A2 )
=> ? [H: list_nat > list_nat] : ( bij_be5167025596389125688st_nat @ H @ A2 @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ) ) ).
% infinite_imp_bij_betw
thf(fact_1176_psubset__insert__iff,axiom,
! [A2: set_nat,X2: nat,B3: set_nat] :
( ( ord_less_set_nat @ A2 @ ( insert_nat @ X2 @ B3 ) )
= ( ( ( member_nat @ X2 @ B3 )
=> ( ord_less_set_nat @ A2 @ B3 ) )
& ( ~ ( member_nat @ X2 @ B3 )
=> ( ( ( member_nat @ X2 @ A2 )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B3 ) )
& ( ~ ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B3 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1177_psubset__insert__iff,axiom,
! [A2: set_list_nat,X2: list_nat,B3: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A2 @ ( insert_list_nat @ X2 @ B3 ) )
= ( ( ( member_list_nat @ X2 @ B3 )
=> ( ord_le1190675801316882794st_nat @ A2 @ B3 ) )
& ( ~ ( member_list_nat @ X2 @ B3 )
=> ( ( ( member_list_nat @ X2 @ A2 )
=> ( ord_le1190675801316882794st_nat @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) @ B3 ) )
& ( ~ ( member_list_nat @ X2 @ A2 )
=> ( ord_le6045566169113846134st_nat @ A2 @ B3 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1178_finite__induct__select,axiom,
! [S2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ S2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [T4: set_nat] :
( ( ord_less_set_nat @ T4 @ S2 )
=> ( ( P @ T4 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ S2 @ T4 ) )
& ( P @ ( insert_nat @ X4 @ T4 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_induct_select
thf(fact_1179_finite__induct__select,axiom,
! [S2: set_list_nat,P: set_list_nat > $o] :
( ( finite8100373058378681591st_nat @ S2 )
=> ( ( P @ bot_bot_set_list_nat )
=> ( ! [T4: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ T4 @ S2 )
=> ( ( P @ T4 )
=> ? [X4: list_nat] :
( ( member_list_nat @ X4 @ ( minus_7954133019191499631st_nat @ S2 @ T4 ) )
& ( P @ ( insert_list_nat @ X4 @ T4 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_induct_select
thf(fact_1180_finite__lists__length__eq,axiom,
! [A2: set_nat,N: nat] :
( ( finite_finite_nat @ A2 )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs3: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs3 ) @ A2 )
& ( ( size_size_list_nat @ Xs3 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_1181_finite__lists__length__eq,axiom,
! [A2: set_list_nat,N: nat] :
( ( finite8100373058378681591st_nat @ A2 )
=> ( finite8170528100393595399st_nat
@ ( collec5989764272469232197st_nat
@ ^ [Xs3: list_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs3 ) @ A2 )
& ( ( size_s3023201423986296836st_nat @ Xs3 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_1182_nth__equal__first__eq,axiom,
! [X2: list_nat,Xs: list_list_nat,N: nat] :
( ~ ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( ( nth_list_nat @ ( cons_list_nat @ X2 @ Xs ) @ N )
= X2 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1183_nth__equal__first__eq,axiom,
! [X2: nat,Xs: list_nat,N: nat] :
( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
= X2 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1184_card__Suc__Diff1,axiom,
! [A2: set_nat,X2: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X2 @ A2 )
=> ( ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) )
= ( finite_card_nat @ A2 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_1185_card__Suc__Diff1,axiom,
! [A2: set_list_nat,X2: list_nat] :
( ( finite8100373058378681591st_nat @ A2 )
=> ( ( member_list_nat @ X2 @ A2 )
=> ( ( suc @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) ) )
= ( finite_card_list_nat @ A2 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_1186_card_Oinsert__remove,axiom,
! [A2: set_nat,X2: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_card_nat @ ( insert_nat @ X2 @ A2 ) )
= ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_1187_card_Oinsert__remove,axiom,
! [A2: set_list_nat,X2: list_nat] :
( ( finite8100373058378681591st_nat @ A2 )
=> ( ( finite_card_list_nat @ ( insert_list_nat @ X2 @ A2 ) )
= ( suc @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_1188_card_Oremove,axiom,
! [A2: set_nat,X2: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X2 @ A2 )
=> ( ( finite_card_nat @ A2 )
= ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% card.remove
thf(fact_1189_card_Oremove,axiom,
! [A2: set_list_nat,X2: list_nat] :
( ( finite8100373058378681591st_nat @ A2 )
=> ( ( member_list_nat @ X2 @ A2 )
=> ( ( finite_card_list_nat @ A2 )
= ( suc @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) ) ) ) ) ) ).
% card.remove
thf(fact_1190_card__Diff1__less,axiom,
! [A2: set_nat,X2: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X2 @ A2 )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_1191_card__Diff1__less,axiom,
! [A2: set_list_nat,X2: list_nat] :
( ( finite8100373058378681591st_nat @ A2 )
=> ( ( member_list_nat @ X2 @ A2 )
=> ( ord_less_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) ) @ ( finite_card_list_nat @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_1192_card__Diff2__less,axiom,
! [A2: set_nat,X2: nat,Y: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ ( insert_nat @ Y @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_1193_card__Diff2__less,axiom,
! [A2: set_list_nat,X2: list_nat,Y: list_nat] :
( ( finite8100373058378681591st_nat @ A2 )
=> ( ( member_list_nat @ X2 @ A2 )
=> ( ( member_list_nat @ Y @ A2 )
=> ( ord_less_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) @ ( insert_list_nat @ Y @ bot_bot_set_list_nat ) ) ) @ ( finite_card_list_nat @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_1194_card__Diff1__less__iff,axiom,
! [A2: set_nat,X2: nat] :
( ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) )
= ( ( finite_finite_nat @ A2 )
& ( member_nat @ X2 @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_1195_card__Diff1__less__iff,axiom,
! [A2: set_list_nat,X2: list_nat] :
( ( ord_less_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) ) @ ( finite_card_list_nat @ A2 ) )
= ( ( finite8100373058378681591st_nat @ A2 )
& ( member_list_nat @ X2 @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_1196_card__Diff__singleton__if,axiom,
! [X2: nat,A2: set_nat] :
( ( ( member_nat @ X2 @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_nat @ X2 @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
= ( finite_card_nat @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1197_card__Diff__singleton__if,axiom,
! [X2: list_nat,A2: set_list_nat] :
( ( ( member_list_nat @ X2 @ A2 )
=> ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) )
= ( minus_minus_nat @ ( finite_card_list_nat @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_list_nat @ X2 @ A2 )
=> ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) )
= ( finite_card_list_nat @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1198_card__Diff__singleton,axiom,
! [X2: nat,A2: set_nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1199_card__Diff__singleton,axiom,
! [X2: list_nat,A2: set_list_nat] :
( ( member_list_nat @ X2 @ A2 )
=> ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) )
= ( minus_minus_nat @ ( finite_card_list_nat @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1200_finite__lists__length__le,axiom,
! [A2: set_nat,N: nat] :
( ( finite_finite_nat @ A2 )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs3: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs3 ) @ A2 )
& ( ord_less_eq_nat @ ( size_size_list_nat @ Xs3 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_1201_finite__lists__length__le,axiom,
! [A2: set_list_nat,N: nat] :
( ( finite8100373058378681591st_nat @ A2 )
=> ( finite8170528100393595399st_nat
@ ( collec5989764272469232197st_nat
@ ^ [Xs3: list_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs3 ) @ A2 )
& ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs3 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_1202_card__insert__le__m1,axiom,
! [N: nat,Y: set_nat,X2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( insert_nat @ X2 @ Y ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_1203_card__insert__le__m1,axiom,
! [N: nat,Y: set_list_nat,X2: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( finite_card_list_nat @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite_card_list_nat @ ( insert_list_nat @ X2 @ Y ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_1204_set__add__0__right,axiom,
! [A2: set_nat] :
( ( plus_plus_set_nat @ A2 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) )
= A2 ) ).
% set_add_0_right
thf(fact_1205_set__add__0,axiom,
! [A2: set_nat] :
( ( plus_plus_set_nat @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) @ A2 )
= A2 ) ).
% set_add_0
thf(fact_1206_set__plus__mono2,axiom,
! [C3: set_list_nat,D: set_list_nat,E2: set_list_nat,F3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ C3 @ D )
=> ( ( ord_le6045566169113846134st_nat @ E2 @ F3 )
=> ( ord_le6045566169113846134st_nat @ ( plus_p884110394369815071st_nat @ C3 @ E2 ) @ ( plus_p884110394369815071st_nat @ D @ F3 ) ) ) ) ).
% set_plus_mono2
thf(fact_1207_set__zero__plus2,axiom,
! [A2: set_nat,B3: set_nat] :
( ( member_nat @ zero_zero_nat @ A2 )
=> ( ord_less_eq_set_nat @ B3 @ ( plus_plus_set_nat @ A2 @ B3 ) ) ) ).
% set_zero_plus2
thf(fact_1208_set__zero,axiom,
( zero_zero_set_nat
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% set_zero
thf(fact_1209_set__one,axiom,
( one_one_set_nat
= ( insert_nat @ one_one_nat @ bot_bot_set_nat ) ) ).
% set_one
thf(fact_1210_augmentum_Oelims,axiom,
! [X2: list_nat,Y: list_nat] :
( ( ( augmentum @ X2 )
= Y )
=> ( ( ( X2 = nil_nat )
=> ( Y != nil_nat ) )
=> ~ ! [N3: nat,Ns2: list_nat] :
( ( X2
= ( cons_nat @ N3 @ Ns2 ) )
=> ( Y
!= ( cons_nat @ N3 @ ( map_nat_nat @ ( plus_plus_nat @ N3 ) @ ( augmentum @ Ns2 ) ) ) ) ) ) ) ).
% augmentum.elims
thf(fact_1211_augmentum_Osimps_I1_J,axiom,
( ( augmentum @ nil_nat )
= nil_nat ) ).
% augmentum.simps(1)
thf(fact_1212_map__is__Nil__conv,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_1213_Nil__is__map__conv,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( nil_nat
= ( map_nat_nat @ F @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_1214_list_Omap__disc__iff,axiom,
! [F: nat > nat,A: list_nat] :
( ( ( map_nat_nat @ F @ A )
= nil_nat )
= ( A = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_1215_list__update__nonempty,axiom,
! [Xs: list_nat,K: nat,X2: nat] :
( ( ( list_update_nat @ Xs @ K @ X2 )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% list_update_nonempty
thf(fact_1216_minus__Nil,axiom,
! [Xs: list_nat] :
( ( minus_minus_list_nat @ nil_nat @ Xs )
= nil_nat ) ).
% minus_Nil
thf(fact_1217_pointwise__le__Nil,axiom,
! [X2: list_nat] :
( ( pointwise_le @ nil_nat @ X2 )
= ( X2 = nil_nat ) ) ).
% pointwise_le_Nil
thf(fact_1218_pointwise__le__Nil2,axiom,
! [X2: list_nat] :
( ( pointwise_le @ X2 @ nil_nat )
= ( X2 = nil_nat ) ) ).
% pointwise_le_Nil2
thf(fact_1219_plus__Nil,axiom,
! [Xs: list_nat] :
( ( plus_plus_list_nat @ nil_nat @ Xs )
= nil_nat ) ).
% plus_Nil
thf(fact_1220_list__incr__Nil,axiom,
! [I: nat] :
( ( list_incr @ I @ nil_nat )
= nil_nat ) ).
% list_incr_Nil
thf(fact_1221_dementum__Nil,axiom,
( ( dementum @ nil_nat )
= nil_nat ) ).
% dementum_Nil
thf(fact_1222_le__Nil,axiom,
! [X2: list_nat] :
( ( ord_less_eq_list_nat @ X2 @ nil_nat )
= ( X2 = nil_nat ) ) ).
% le_Nil
thf(fact_1223_set__empty2,axiom,
! [Xs: list_nat] :
( ( bot_bot_set_nat
= ( set_nat2 @ Xs ) )
= ( Xs = nil_nat ) ) ).
% set_empty2
thf(fact_1224_set__empty2,axiom,
! [Xs: list_list_nat] :
( ( bot_bot_set_list_nat
= ( set_list_nat2 @ Xs ) )
= ( Xs = nil_list_nat ) ) ).
% set_empty2
thf(fact_1225_set__empty,axiom,
! [Xs: list_nat] :
( ( ( set_nat2 @ Xs )
= bot_bot_set_nat )
= ( Xs = nil_nat ) ) ).
% set_empty
thf(fact_1226_set__empty,axiom,
! [Xs: list_list_nat] :
( ( ( set_list_nat2 @ Xs )
= bot_bot_set_list_nat )
= ( Xs = nil_list_nat ) ) ).
% set_empty
thf(fact_1227_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_1228_sum__list_ONil,axiom,
( ( groups4561878855575611511st_nat @ nil_nat )
= zero_zero_nat ) ).
% sum_list.Nil
thf(fact_1229_empty__replicate,axiom,
! [N: nat,X2: nat] :
( ( nil_nat
= ( replicate_nat @ N @ X2 ) )
= ( N = zero_zero_nat ) ) ).
% empty_replicate
thf(fact_1230_replicate__empty,axiom,
! [N: nat,X2: nat] :
( ( ( replicate_nat @ N @ X2 )
= nil_nat )
= ( N = zero_zero_nat ) ) ).
% replicate_empty
thf(fact_1231_length__greater__0__conv,axiom,
! [Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) )
= ( Xs != nil_nat ) ) ).
% length_greater_0_conv
thf(fact_1232_length__sum__set__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( length_sum_set @ zero_zero_nat @ N )
= ( insert_list_nat @ nil_nat @ bot_bot_set_list_nat ) ) )
& ( ( N != zero_zero_nat )
=> ( ( length_sum_set @ zero_zero_nat @ N )
= bot_bot_set_list_nat ) ) ) ).
% length_sum_set_Nil
thf(fact_1233_empty__set,axiom,
( bot_bot_set_nat
= ( set_nat2 @ nil_nat ) ) ).
% empty_set
thf(fact_1234_empty__set,axiom,
( bot_bot_set_list_nat
= ( set_list_nat2 @ nil_list_nat ) ) ).
% empty_set
thf(fact_1235_Nil__less__Cons,axiom,
! [A: nat,X2: list_nat] : ( ord_less_list_nat @ nil_nat @ ( cons_nat @ A @ X2 ) ) ).
% Nil_less_Cons
thf(fact_1236_list_Osimps_I8_J,axiom,
! [F: nat > nat] :
( ( map_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_1237_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_nat @ N @ nil_nat )
= N ) ).
% gen_length_code(1)
thf(fact_1238_Nil__le__Cons,axiom,
! [X2: list_nat] : ( ord_less_eq_list_nat @ nil_nat @ X2 ) ).
% Nil_le_Cons
thf(fact_1239_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_1240_list__induct2,axiom,
! [Xs: list_nat,Ys2: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys4: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_1241_list__induct3,axiom,
! [Xs: list_nat,Ys2: list_nat,Zs3: list_nat,P: list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys4: list_nat,Z3: nat,Zs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_1242_list__induct4,axiom,
! [Xs: list_nat,Ys2: list_nat,Zs3: list_nat,Ws: list_nat,P: list_nat > list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys4: list_nat,Z3: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z3 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs3 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1243_replicate__0,axiom,
! [X2: nat] :
( ( replicate_nat @ zero_zero_nat @ X2 )
= nil_nat ) ).
% replicate_0
thf(fact_1244_list__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X3: nat] : ( P @ ( cons_nat @ X3 @ nil_nat ) )
=> ( ! [X3: nat,Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_1245_list__induct2_H,axiom,
! [P: list_nat > list_nat > $o,Xs: list_nat,Ys2: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X3 @ Xs2 ) @ nil_nat )
=> ( ! [Y3: nat,Ys4: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y3 @ Ys4 ) )
=> ( ! [X3: nat,Xs2: list_nat,Y3: nat,Ys4: list_nat] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_nat @ X3 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_1246_neq__Nil__conv,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
= ( ? [Y4: nat,Ys3: list_nat] :
( Xs
= ( cons_nat @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_1247_remdups__adj_Ocases,axiom,
! [X2: list_nat] :
( ( X2 != nil_nat )
=> ( ! [X3: nat] :
( X2
!= ( cons_nat @ X3 @ nil_nat ) )
=> ~ ! [X3: nat,Y3: nat,Xs2: list_nat] :
( X2
!= ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_1248_min__list_Ocases,axiom,
! [X2: list_nat] :
( ! [X3: nat,Xs2: list_nat] :
( X2
!= ( cons_nat @ X3 @ Xs2 ) )
=> ( X2 = nil_nat ) ) ).
% min_list.cases
thf(fact_1249_list_Oexhaust,axiom,
! [Y: list_nat] :
( ( Y != nil_nat )
=> ~ ! [X212: nat,X223: list_nat] :
( Y
!= ( cons_nat @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_1250_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X222: list_nat] :
( ( List
= ( cons_nat @ X21 @ X222 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_1251_list_Odistinct_I1_J,axiom,
! [X21: nat,X222: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_1252_list__update_Osimps_I1_J,axiom,
! [I: nat,V: nat] :
( ( list_update_nat @ nil_nat @ I @ V )
= nil_nat ) ).
% list_update.simps(1)
thf(fact_1253_list__update__code_I1_J,axiom,
! [I: nat,Y: nat] :
( ( list_update_nat @ nil_nat @ I @ Y )
= nil_nat ) ).
% list_update_code(1)
thf(fact_1254_list__encode_Ocases,axiom,
! [X2: list_nat] :
( ( X2 != nil_nat )
=> ~ ! [X3: nat,Xs2: list_nat] :
( X2
!= ( cons_nat @ X3 @ Xs2 ) ) ) ).
% list_encode.cases
thf(fact_1255_transpose_Ocases,axiom,
! [X2: list_list_nat] :
( ( X2 != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X2
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X3: nat,Xs2: list_nat,Xss: list_list_nat] :
( X2
!= ( cons_list_nat @ ( cons_nat @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_1256_pointwise__less__Nil,axiom,
! [X2: list_nat] :
~ ( pointwise_less @ nil_nat @ X2 ) ).
% pointwise_less_Nil
thf(fact_1257_pointwise__less__Nil2,axiom,
! [X2: list_nat] :
~ ( pointwise_less @ X2 @ nil_nat ) ).
% pointwise_less_Nil2
thf(fact_1258_sum__list__strict__mono,axiom,
! [Xs: list_list_nat,F: list_nat > nat,G: list_nat > nat] :
( ( Xs != nil_list_nat )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_nat @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ F @ Xs ) ) @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ G @ Xs ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_1259_sum__list__strict__mono,axiom,
! [Xs: list_nat,F: nat > nat,G: nat > nat] :
( ( Xs != nil_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_nat @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ F @ Xs ) ) @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ G @ Xs ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_1260_card__set__1__iff__replicate,axiom,
! [Xs: list_list_nat] :
( ( ( finite_card_list_nat @ ( set_list_nat2 @ Xs ) )
= ( suc @ zero_zero_nat ) )
= ( ( Xs != nil_list_nat )
& ? [X: list_nat] :
( Xs
= ( replicate_list_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ X ) ) ) ) ).
% card_set_1_iff_replicate
thf(fact_1261_card__set__1__iff__replicate,axiom,
! [Xs: list_nat] :
( ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
= ( suc @ zero_zero_nat ) )
= ( ( Xs != nil_nat )
& ? [X: nat] :
( Xs
= ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X ) ) ) ) ).
% card_set_1_iff_replicate
thf(fact_1262_sum_Ofinite__Collect__op,axiom,
! [I6: set_list_nat,X2: list_nat > nat,Y: list_nat > nat] :
( ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [I3: list_nat] :
( ( member_list_nat @ I3 @ I6 )
& ( ( X2 @ I3 )
!= zero_zero_nat ) ) ) )
=> ( ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [I3: list_nat] :
( ( member_list_nat @ I3 @ I6 )
& ( ( Y @ I3 )
!= zero_zero_nat ) ) ) )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [I3: list_nat] :
( ( member_list_nat @ I3 @ I6 )
& ( ( plus_plus_nat @ ( X2 @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_1263_sum_Ofinite__Collect__op,axiom,
! [I6: set_nat,X2: nat > nat,Y: nat > nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I6 )
& ( ( X2 @ I3 )
!= zero_zero_nat ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I6 )
& ( ( Y @ I3 )
!= zero_zero_nat ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I3: nat] :
( ( member_nat @ I3 @ I6 )
& ( ( plus_plus_nat @ ( X2 @ I3 ) @ ( Y @ I3 ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
% 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,
! [X2: nat,Y: nat] :
( ( if_nat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( x2
= ( g2 @ ( g @ x2 ) ) ) ).
%------------------------------------------------------------------------------