TPTP Problem File: SLH0911^1.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_00930_034396__13610870_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1416 ( 482 unt; 147 typ; 0 def)
% Number of atoms : 4017 (1329 equ; 0 cnn)
% Maximal formula atoms : 17 ( 3 avg)
% Number of connectives : 11765 ( 401 ~; 62 |; 338 &;9029 @)
% ( 0 <=>;1935 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 7 avg)
% Number of types : 11 ( 10 usr)
% Number of type conns : 907 ( 907 >; 0 *; 0 +; 0 <<)
% Number of symbols : 140 ( 137 usr; 17 con; 0-5 aty)
% Number of variables : 3924 ( 322 ^;3388 !; 214 ?;3924 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:15:21.068
%------------------------------------------------------------------------------
% Could-be-implicit typings (10)
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
list_list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
set_list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
set_set_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__List__Olist_It__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 (137)
thf(sy_c_Countable__Set_Oto__nat__on_001t__List__Olist_It__Nat__Onat_J,type,
counta9103016383634126529st_nat: set_list_nat > list_nat > nat ).
thf(sy_c_Countable__Set_Oto__nat__on_001t__Nat__Onat,type,
counta4844910239362777137on_nat: set_nat > nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Nat__Onat_J,type,
finite_card_list_nat: set_list_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
finite8170528100393595399st_nat: set_list_list_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Nat__Onat_J,type,
finite8100373058378681591st_nat: set_list_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
finite7047420756378620717st_nat: set_set_list_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_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__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
inj_on1816901372521670873et_nat: ( list_nat > set_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_Fun_Oinj__on_001t__Nat__Onat_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
inj_on2924389301302751961st_nat: ( nat > set_list_nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
inj_on_nat_set_nat: ( nat > set_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_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_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__Big_Ocomm__monoid__add__class_Osum_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
groups4396056296759096172at_nat: ( list_nat > nat ) > set_list_nat > nat ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat,type,
groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_Groups__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__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_OKhovanskii__axioms_001t__List__Olist_It__Nat__Onat_J,type,
khovan1553326461689229922st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Khovanskii_OKhovanskii__axioms_001t__Nat__Onat,type,
khovan4585363760863428690ms_nat: set_nat > set_nat > $o ).
thf(sy_c_Khovanskii_Ofinsets_001t__List__Olist_It__Nat__Onat_J,type,
finsets_list_nat: set_list_nat > nat > set_set_list_nat ).
thf(sy_c_Khovanskii_Ofinsets_001t__Nat__Onat,type,
finsets_nat: set_nat > nat > set_set_nat ).
thf(sy_c_Khovanskii_Omax__pointwise,type,
max_pointwise: nat > set_list_nat > list_nat ).
thf(sy_c_Khovanskii_Omin__pointwise,type,
min_pointwise: nat > set_list_nat > list_nat ).
thf(sy_c_Khovanskii_Opointwise__le,type,
pointwise_le: list_nat > list_nat > $o ).
thf(sy_c_Khovanskii_Opointwise__less,type,
pointwise_less: list_nat > list_nat > $o ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
lattic2707248312394557148st_nat: ( list_nat > list_nat ) > ( list_nat > $o ) > list_nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
lattic343138300456401356at_nat: ( list_nat > nat ) > ( list_nat > $o ) > list_nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
lattic4482942756937193746st_nat: ( list_nat > set_list_nat ) > ( list_nat > $o ) > list_nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
lattic6100888674492331394et_nat: ( list_nat > set_nat ) > ( list_nat > $o ) > list_nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min_001t__Nat__Onat_001t__Nat__Onat,type,
lattic8739620818006775868at_nat: ( nat > nat ) > ( nat > $o ) > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min_001t__Nat__Onat_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
lattic7208376603273412482st_nat: ( nat > set_list_nat ) > ( nat > $o ) > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
lattic44696799612376050et_nat: ( nat > set_nat ) > ( nat > $o ) > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
lattic5785867957632790475at_nat: ( list_nat > nat ) > set_list_nat > list_nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Nat__Onat,type,
lattic7446932960582359483at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_List_Oappend_001t__List__Olist_It__Nat__Onat_J,type,
append_list_nat: list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Oappend_001t__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Ofolding__insort__key_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
foldin1968479528632951399st_nat: ( list_nat > list_nat > $o ) > ( list_nat > list_nat > $o ) > set_list_nat > ( list_nat > list_nat ) > $o ).
thf(sy_c_List_Ofolding__insort__key_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
foldin951631548397865559st_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > set_list_nat > ( list_nat > nat ) > $o ).
thf(sy_c_List_Ofolding__insort__key_001t__Nat__Onat_001t__Nat__Onat,type,
foldin8133931898133206727at_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > set_nat > ( nat > nat ) > $o ).
thf(sy_c_List_Olinorder_Osorted__key__list__of__set_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
sorted4005134659417452724st_nat: ( nat > nat > $o ) > ( list_nat > nat ) > set_list_nat > list_list_nat ).
thf(sy_c_List_Olinorder_Osorted__key__list__of__set_001t__Nat__Onat_001t__Nat__Onat,type,
sorted5905597674102116260at_nat: ( nat > nat > $o ) > ( nat > nat ) > set_nat > list_nat ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__List__Olist_It__Nat__Onat_J,type,
linord2712301520579796368st_nat: set_list_nat > list_list_nat ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
linord2614967742042102400et_nat: set_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__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__Nat__Onat,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_list_nat2: list_list_list_nat > set_list_list_nat ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
set_list_nat2: list_list_nat > set_list_nat ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Osize__list_001t__List__Olist_It__Nat__Onat_J,type,
size_list_list_nat: ( list_nat > nat ) > list_list_nat > nat ).
thf(sy_c_List_Olist_Osize__list_001t__Nat__Onat,type,
size_list_nat: ( nat > nat ) > list_nat > nat ).
thf(sy_c_List_On__lists_001t__List__Olist_It__Nat__Onat_J,type,
n_lists_list_nat: nat > list_list_nat > list_list_list_nat ).
thf(sy_c_List_On__lists_001t__Nat__Onat,type,
n_lists_nat: nat > list_nat > list_list_nat ).
thf(sy_c_List_Osorted__wrt_001t__List__Olist_It__Nat__Onat_J,type,
sorted_wrt_list_nat: ( list_nat > list_nat > $o ) > list_list_nat > $o ).
thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_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_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__List__Olist_It__Nat__Onat_J,type,
bot_bot_list_nat: list_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__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_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
ord_less_list_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__List__Olist_It__Nat__Onat_J,type,
ord_less_list_nat: list_nat > list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
ord_le1190675801316882794st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__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_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
ord_le1068707526560357548st_nat: set_set_list_nat > set_set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
collec5989764272469232197st_nat: ( list_list_nat > $o ) > set_list_list_nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
collect_set_list_nat: ( set_list_nat > $o ) > set_set_list_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__List__Olist_It__Nat__Onat_J,type,
set_or4185896845444216793st_nat: list_nat > set_list_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_or2492388921469580815st_nat: set_list_nat > set_set_list_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4236626031148496127et_nat: set_nat > set_set_nat ).
thf(sy_c_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__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_f____,type,
f: 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,type,
x: list_nat ).
thf(sy_v_x_H,type,
x2: list_nat ).
% Relevant facts (1264)
thf(fact_0__092_060open_062inj__on_Af_AS_092_060close_062,axiom,
inj_on3049792774292151987st_nat @ f @ s ).
% \<open>inj_on f S\<close>
thf(fact_1_f__def,axiom,
( f
= ( ^ [X: list_nat] : ( minus_minus_list_nat @ X @ x2 ) ) ) ).
% f_def
thf(fact_2_lenx_H,axiom,
( ( size_size_list_nat @ x2 )
= r ) ).
% lenx'
thf(fact_3_assms_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ r ).
% assms(1)
thf(fact_4_finite__length__sum__set,axiom,
! [R: nat,N: nat] : ( finite8100373058378681591st_nat @ ( length_sum_set @ R @ N ) ) ).
% finite_length_sum_set
thf(fact_5_m__def,axiom,
( m
= ( minus_minus_nat @ n @ ( groups4561878855575611511st_nat @ x2 ) ) ) ).
% m_def
thf(fact_6_S__def,axiom,
( s
= ( collect_list_nat
@ ^ [X: list_nat] :
( ( pointwise_le @ x2 @ X )
& ( ( groups4561878855575611511st_nat @ X )
= n ) ) ) ) ).
% S_def
thf(fact_7_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 ) )
= ( ? [M: nat] :
( ( member_list_nat @ Ks @ ( length_sum_set @ R @ M ) )
& ( N
= ( plus_plus_nat @ M @ K ) ) ) ) ) ).
% length_sum_set_Suc
thf(fact_8_length__sum__set__def,axiom,
( length_sum_set
= ( ^ [R2: nat,N2: nat] :
( collect_list_nat
@ ^ [X: list_nat] :
( ( ( size_size_list_nat @ X )
= R2 )
& ( ( groups4561878855575611511st_nat @ X )
= N2 ) ) ) ) ) ).
% length_sum_set_def
thf(fact_9_pointwise__le__refl,axiom,
! [X2: list_nat] : ( pointwise_le @ X2 @ X2 ) ).
% pointwise_le_refl
thf(fact_10_n,axiom,
ord_less_eq_nat @ ( groups4561878855575611511st_nat @ x2 ) @ n ).
% n
thf(fact_11_Khovanskii_OfinA,axiom,
! [G: set_list_nat,Addition: list_nat > list_nat > list_nat,Zero: list_nat,A: set_list_nat] :
( ( khovanskii_list_nat @ G @ Addition @ Zero @ A )
=> ( finite8100373058378681591st_nat @ A ) ) ).
% Khovanskii.finA
thf(fact_12_Khovanskii_OfinA,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat] :
( ( khovanskii_nat @ G @ Addition @ Zero @ A )
=> ( finite_finite_nat @ A ) ) ).
% Khovanskii.finA
thf(fact_13_plus__Cons,axiom,
! [Y: list_nat,Ys: list_list_nat,X2: list_nat,Xs: list_list_nat] :
( ( plus_p2116291331692525561st_nat @ ( cons_list_nat @ Y @ Ys ) @ ( cons_list_nat @ X2 @ Xs ) )
= ( cons_list_nat @ ( plus_plus_list_nat @ Y @ X2 ) @ ( plus_p2116291331692525561st_nat @ Ys @ Xs ) ) ) ).
% plus_Cons
thf(fact_14_plus__Cons,axiom,
! [Y: nat,Ys: list_nat,X2: nat,Xs: list_nat] :
( ( plus_plus_list_nat @ ( cons_nat @ Y @ Ys ) @ ( cons_nat @ X2 @ Xs ) )
= ( cons_nat @ ( plus_plus_nat @ Y @ X2 ) @ ( plus_plus_list_nat @ Ys @ Xs ) ) ) ).
% plus_Cons
thf(fact_15_minus__Cons,axiom,
! [Y: list_nat,Ys: list_list_nat,X2: list_nat,Xs: list_list_nat] :
( ( minus_3911745200923244873st_nat @ ( cons_list_nat @ Y @ Ys ) @ ( cons_list_nat @ X2 @ Xs ) )
= ( cons_list_nat @ ( minus_minus_list_nat @ Y @ X2 ) @ ( minus_3911745200923244873st_nat @ Ys @ Xs ) ) ) ).
% minus_Cons
thf(fact_16_minus__Cons,axiom,
! [Y: nat,Ys: list_nat,X2: nat,Xs: list_nat] :
( ( minus_minus_list_nat @ ( cons_nat @ Y @ Ys ) @ ( cons_nat @ X2 @ Xs ) )
= ( cons_nat @ ( minus_minus_nat @ Y @ X2 ) @ ( minus_minus_list_nat @ Ys @ Xs ) ) ) ).
% minus_Cons
thf(fact_17_sum__list__plus,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( groups4561878855575611511st_nat @ ( plus_plus_list_nat @ Xs @ Ys ) )
= ( plus_plus_nat @ ( groups4561878855575611511st_nat @ Xs ) @ ( groups4561878855575611511st_nat @ Ys ) ) ) ) ).
% sum_list_plus
thf(fact_18_sum__list__minus,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( pointwise_le @ Xs @ Ys )
=> ( ( groups4561878855575611511st_nat @ ( minus_minus_list_nat @ Ys @ Xs ) )
= ( minus_minus_nat @ ( groups4561878855575611511st_nat @ Ys ) @ ( groups4561878855575611511st_nat @ Xs ) ) ) ) ).
% sum_list_minus
thf(fact_19_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_20_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_21_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_22_dementum__def,axiom,
( dementum
= ( ^ [Xs2: list_nat] : ( minus_minus_list_nat @ Xs2 @ ( cons_nat @ zero_zero_nat @ Xs2 ) ) ) ) ).
% dementum_def
thf(fact_23_minimal__elements__set__tuples__finite,axiom,
! [U: set_list_nat,R: nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ U )
=> ( ( size_size_list_nat @ X3 )
= R ) )
=> ( finite8100373058378681591st_nat @ ( minimal_elements @ U ) ) ) ).
% minimal_elements_set_tuples_finite
thf(fact_24_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_25_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_26_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_27_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_28_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_29_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_30_diff__add__zero,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_31_add__less__same__cancel1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_32_add__less__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_33_add__right__cancel,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_34_add__left__cancel,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_35_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_36_nat_Oinject,axiom,
! [X22: nat,Y2: nat] :
( ( ( suc @ X22 )
= ( suc @ Y2 ) )
= ( X22 = Y2 ) ) ).
% nat.inject
thf(fact_37_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_38_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_39_add__le__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_40_mem__Collect__eq,axiom,
! [A2: list_nat,P: list_nat > $o] :
( ( member_list_nat @ A2 @ ( collect_list_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_41_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_42_Collect__mem__eq,axiom,
! [A: set_list_nat] :
( ( collect_list_nat
@ ^ [X: list_nat] : ( member_list_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_43_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_44_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_45_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_46_add__le__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_47_add__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% add_0
thf(fact_48_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_49_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_50_add__cancel__right__right,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ A2 @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_51_add__cancel__right__left,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ B @ A2 ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_52_add__cancel__left__right,axiom,
! [A2: nat,B: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_53_add__cancel__left__left,axiom,
! [B: nat,A2: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_54_add_Oright__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.right_neutral
thf(fact_55_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ A2 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_56_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_57_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_58_add__less__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_59_add__less__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_60_add__diff__cancel__right_H,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_61_add__diff__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A2 @ B ) ) ).
% add_diff_cancel_right
thf(fact_62_add__diff__cancel__left_H,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ A2 )
= B ) ).
% add_diff_cancel_left'
thf(fact_63_add__diff__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A2 @ B ) ) ).
% add_diff_cancel_left
thf(fact_64_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_65_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_66_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_67_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_68_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_69_Suc__le__mono,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ).
% Suc_le_mono
thf(fact_70_Suc__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_eq
thf(fact_71_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_72_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_73_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_74_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_75_add__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc_right
thf(fact_76_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_77_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_78_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_79_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_80_Suc__diff__diff,axiom,
! [M2: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_81_diff__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_Suc_Suc
thf(fact_82_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_83_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_84_length__dementum,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( dementum @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_dementum
thf(fact_85_le__add__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_86_le__add__same__cancel1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_87_add__le__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_88_add__le__same__cancel1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_89_less__add__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_90_less__add__same__cancel1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_91_diff__is__0__eq_H,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_92_diff__is__0__eq,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% diff_is_0_eq
thf(fact_93_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_94_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_95_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_96_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_97_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_98_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_99_lift__Suc__mono__le,axiom,
! [F: nat > set_list_nat,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_le6045566169113846134st_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_le6045566169113846134st_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_100_lift__Suc__mono__le,axiom,
! [F: nat > set_nat,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_set_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_set_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_101_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_102_lift__Suc__antimono__le,axiom,
! [F: nat > set_list_nat,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_le6045566169113846134st_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_le6045566169113846134st_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_103_lift__Suc__antimono__le,axiom,
! [F: nat > set_nat,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_104_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 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_105_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_106_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_107_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_108_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_109_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_110_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_111_add__le__imp__le__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_right
thf(fact_112_add__le__imp__le__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_left
thf(fact_113_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
? [C2: nat] :
( B2
= ( plus_plus_nat @ A3 @ C2 ) ) ) ) ).
% le_iff_add
thf(fact_114_add__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_115_less__eqE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A2 @ C3 ) ) ) ).
% less_eqE
thf(fact_116_add__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_117_add__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_118_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_119_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_120_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_121_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_122_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_123_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_124_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_125_transitive__stepwise__le,axiom,
! [M2: nat,N: nat,R3: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ! [X3: nat] : ( R3 @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z2: nat] :
( ( R3 @ X3 @ Y3 )
=> ( ( R3 @ Y3 @ Z2 )
=> ( R3 @ X3 @ Z2 ) ) )
=> ( ! [N4: nat] : ( R3 @ N4 @ ( suc @ N4 ) )
=> ( R3 @ M2 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_126_nat__induct__at__least,axiom,
! [M2: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( P @ M2 )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_127_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N4 )
=> ( P @ M3 ) )
=> ( P @ N4 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_128_not__less__eq__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_129_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_130_le__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M2 @ N )
| ( M2
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_131_Suc__le__D,axiom,
! [N: nat,M4: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M4 )
=> ? [M5: nat] :
( M4
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_132_le__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_133_le__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N )
=> ( M2
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_134_Suc__leD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_leD
thf(fact_135_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_136_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_137_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_138_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_139_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_140_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
& ( M != N2 ) ) ) ) ).
% nat_less_le
thf(fact_141_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
? [K2: nat] :
( N2
= ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_142_trans__le__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_143_trans__le__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_144_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_145_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_146_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N4: nat] :
( L
= ( plus_plus_nat @ K @ N4 ) ) ) ).
% le_Suc_ex
thf(fact_147_add__leD2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_148_add__leD1,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% add_leD1
thf(fact_149_le__add2,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).
% le_add2
thf(fact_150_le__add1,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).
% le_add1
thf(fact_151_add__leE,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M2 @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_152_diff__le__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_153_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_154_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_155_diff__le__mono,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_156_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_157_le__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_158_eq__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_159_pointwise__le__plus,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( pointwise_le @ Xs @ Ys )
=> ( ( ord_less_eq_nat @ ( size_size_list_nat @ Ys ) @ ( size_size_list_nat @ Zs ) )
=> ( pointwise_le @ Xs @ ( plus_plus_list_nat @ Ys @ Zs ) ) ) ) ).
% pointwise_le_plus
thf(fact_160_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_161_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_162_add__nonpos__nonpos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_163_add__nonneg__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_164_add__increasing2,axiom,
! [C: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_165_add__decreasing2,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_166_add__increasing,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_167_add__decreasing,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_168_add__less__le__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_169_add__le__less__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_170_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_171_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_172_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( minus_minus_nat @ B @ A2 )
= C )
= ( B
= ( plus_plus_nat @ C @ A2 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_173_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ A2 @ ( minus_minus_nat @ B @ A2 ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_174_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_175_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_176_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_177_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A2 )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_178_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_179_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_180_le__add__diff,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 ) ) ) ).
% le_add_diff
thf(fact_181_diff__add,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ A2 )
= B ) ) ).
% diff_add
thf(fact_182_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K3 )
=> ~ ( P @ I3 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_183_le__imp__less__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_184_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_185_less__Suc__eq__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_Suc_eq_le
thf(fact_186_le__less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_187_Suc__le__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_lessD
thf(fact_188_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I @ N4 )
=> ( ( ord_less_nat @ N4 @ J )
=> ( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_189_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I @ N4 )
=> ( ( ord_less_nat @ N4 @ J )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_190_Suc__le__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_eq
thf(fact_191_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_192_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M5: nat,N4: nat] :
( ( ord_less_nat @ M5 @ N4 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N4 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_193_Suc__diff__le,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_194_diff__less__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_195_less__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_196_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_197_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_198_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_199_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_200_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_201_add__strict__increasing2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_202_add__strict__increasing,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_203_add__pos__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_204_add__nonpos__neg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_205_add__nonneg__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_206_add__neg__nonpos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_207_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ K3 )
=> ~ ( P @ I3 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_208_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_209_pointwise__le__imp___092_060sigma_062,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( pointwise_le @ Xs @ Ys )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ Xs ) @ ( groups4561878855575611511st_nat @ Ys ) ) ) ).
% pointwise_le_imp_\<sigma>
thf(fact_210_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_211_add__right__imp__eq,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_212_add__left__imp__eq,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_213_add_Oleft__commute,axiom,
! [B: nat,A2: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A2 @ C ) )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_214_add_Oleft__commute,axiom,
! [B: list_nat,A2: list_nat,C: list_nat] :
( ( plus_plus_list_nat @ B @ ( plus_plus_list_nat @ A2 @ C ) )
= ( plus_plus_list_nat @ A2 @ ( plus_plus_list_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_215_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B2: nat] : ( plus_plus_nat @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_216_add_Ocommute,axiom,
( plus_plus_list_nat
= ( ^ [A3: list_nat,B2: list_nat] : ( plus_plus_list_nat @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_217_add_Oassoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_218_add_Oassoc,axiom,
! [A2: list_nat,B: list_nat,C: list_nat] :
( ( plus_plus_list_nat @ ( plus_plus_list_nat @ A2 @ B ) @ C )
= ( plus_plus_list_nat @ A2 @ ( plus_plus_list_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_219_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B: nat,A2: nat] :
( ( B3
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A2 @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_220_group__cancel_Oadd1,axiom,
! [A: nat,K: nat,A2: nat,B: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_221_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_222_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_223_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: list_nat,B: list_nat,C: list_nat] :
( ( plus_plus_list_nat @ ( plus_plus_list_nat @ A2 @ B ) @ C )
= ( plus_plus_list_nat @ A2 @ ( plus_plus_list_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_224_diff__right__commute,axiom,
! [A2: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_225_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_226_Suc__inject,axiom,
! [X2: nat,Y: nat] :
( ( ( suc @ X2 )
= ( suc @ Y ) )
=> ( X2 = Y ) ) ).
% Suc_inject
thf(fact_227_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_228_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ~ ( P @ N4 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N4 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_229_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N4 )
=> ( P @ M3 ) )
=> ( P @ N4 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_230_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_231_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_232_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_233_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_234_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_235_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_236_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_237_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_238_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_239_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_240_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_241_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_242_add_Ocomm__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.comm_neutral
thf(fact_243_comm__monoid__add__class_Oadd__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_244_add__less__imp__less__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_right
thf(fact_245_add__less__imp__less__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_left
thf(fact_246_add__strict__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_247_add__strict__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_248_add__strict__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_249_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_250_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_251_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_252_diff__diff__eq,axiom,
! [A2: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C )
= ( minus_minus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_253_add__implies__diff,axiom,
! [C: nat,B: nat,A2: nat] :
( ( ( plus_plus_nat @ C @ B )
= A2 )
=> ( C
= ( minus_minus_nat @ A2 @ B ) ) ) ).
% add_implies_diff
thf(fact_254_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_255_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_256_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_257_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_258_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_259_diff__induct,axiom,
! [P: nat > nat > $o,M2: 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 @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_260_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_261_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_262_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_263_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_264_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_265_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_266_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ~ ( P @ N4 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N4 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_267_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_268_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_269_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_270_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_271_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_272_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_273_not__less__less__Suc__eq,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_274_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_275_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,K3: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K3 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K3 )
=> ( P @ I2 @ K3 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_276_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_277_Suc__less__SucD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_SucD
thf(fact_278_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_279_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M2 )
= ( ? [M6: nat] :
( ( M2
= ( suc @ M6 ) )
& ( ord_less_nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_280_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_281_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_nat @ M2 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_282_less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) ) ) ).
% less_Suc_eq
thf(fact_283_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ N )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_284_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_285_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_286_Suc__lessI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ( suc @ M2 )
!= N )
=> ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_287_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_288_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_289_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_290_add__eq__self__zero,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= M2 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_291_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_292_add__Suc__shift,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( plus_plus_nat @ M2 @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_293_add__Suc,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc
thf(fact_294_nat__arith_Osuc1,axiom,
! [A: nat,K: nat,A2: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( suc @ A )
= ( plus_plus_nat @ K @ ( suc @ A2 ) ) ) ) ).
% nat_arith.suc1
thf(fact_295_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% less_add_eq_less
thf(fact_296_trans__less__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_297_trans__less__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_298_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_299_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_300_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_301_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_302_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_303_diffs0__imp__equal,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M2 )
= zero_zero_nat )
=> ( M2 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_304_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_305_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_306_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_307_diff__less__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_308_diff__add__inverse2,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ N )
= M2 ) ).
% diff_add_inverse2
thf(fact_309_diff__add__inverse,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M2 ) @ N )
= M2 ) ).
% diff_add_inverse
thf(fact_310_diff__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_cancel2
thf(fact_311_Nat_Odiff__cancel,axiom,
! [K: nat,M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% Nat.diff_cancel
thf(fact_312_pos__add__strict,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_313_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ! [C3: nat] :
( ( B
= ( plus_plus_nat @ A2 @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_314_add__pos__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_pos_pos
thf(fact_315_add__neg__neg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_316_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M2: nat] :
( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_317_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_318_less__Suc__eq__0__disj,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( M2 = zero_zero_nat )
| ? [J3: nat] :
( ( M2
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_319_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_320_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_321_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M: nat] :
( N
= ( suc @ M ) ) ) ) ).
% gr0_conv_Suc
thf(fact_322_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_323_one__is__add,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M2 @ N ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_324_add__is__1,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_325_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_326_less__imp__Suc__add,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ? [K3: nat] :
( N
= ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_327_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M: nat,N2: nat] :
? [K2: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_328_less__add__Suc2,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M2 @ I ) ) ) ).
% less_add_Suc2
thf(fact_329_less__add__Suc1,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_330_less__natE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ~ ! [Q2: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M2 @ Q2 ) ) ) ) ).
% less_natE
thf(fact_331_diff__less,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).
% diff_less
thf(fact_332_diff__less__Suc,axiom,
! [M2: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_333_Suc__diff__Suc,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N ) ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_334_diff__add__0,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_335_add__diff__inverse__nat,axiom,
! [M2: nat,N: nat] :
( ~ ( ord_less_nat @ M2 @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M2 @ N ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_336_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_337_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_338_nat__diff__split__asm,axiom,
! [P: nat > $o,A2: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B ) )
= ( ~ ( ( ( ord_less_nat @ A2 @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D2: nat] :
( ( A2
= ( plus_plus_nat @ B @ D2 ) )
& ~ ( P @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_339_nat__diff__split,axiom,
! [P: nat > $o,A2: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B ) )
= ( ( ( ord_less_nat @ A2 @ B )
=> ( P @ zero_zero_nat ) )
& ! [D2: nat] :
( ( A2
= ( plus_plus_nat @ B @ D2 ) )
=> ( P @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_340_minimal__elementsp__minimal__elements__eq,axiom,
! [U: set_list_nat] :
( ( minimal_elementsp
@ ^ [X: list_nat] : ( member_list_nat @ X @ U ) )
= ( ^ [X: list_nat] : ( member_list_nat @ X @ ( minimal_elements @ U ) ) ) ) ).
% minimal_elementsp_minimal_elements_eq
thf(fact_341_minimal__elements__def,axiom,
( minimal_elements
= ( ^ [U2: set_list_nat] :
( collect_list_nat
@ ( minimal_elementsp
@ ^ [X: list_nat] : ( member_list_nat @ X @ U2 ) ) ) ) ) ).
% minimal_elements_def
thf(fact_342_le__add__diff__inverse2,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A2 @ B ) @ B )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_343_le__add__diff__inverse,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A2 @ B ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_344_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_345_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
= ( ? [X: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ X @ Ys2 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_346_Cons__less__Cons,axiom,
! [A2: nat,X2: list_nat,B: nat,Y: list_nat] :
( ( ord_less_list_nat @ ( cons_nat @ A2 @ 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 @ A2 @ B )
| ( ( A2 = B )
& ( ord_less_list_nat @ X2 @ Y ) ) ) ) ) ) ).
% Cons_less_Cons
thf(fact_347_pointwise__less__imp___092_060sigma_062,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( pointwise_less @ Xs @ Ys )
=> ( ord_less_nat @ ( groups4561878855575611511st_nat @ Xs ) @ ( groups4561878855575611511st_nat @ Ys ) ) ) ).
% pointwise_less_imp_\<sigma>
thf(fact_348_min__pointwise__le,axiom,
! [U3: list_nat,U: set_list_nat] :
( ( member_list_nat @ U3 @ U )
=> ( ( finite8100373058378681591st_nat @ U )
=> ( pointwise_le @ ( min_pointwise @ ( size_size_list_nat @ U3 ) @ U ) @ U3 ) ) ) ).
% min_pointwise_le
thf(fact_349_minimal__elements_Ocases,axiom,
! [A2: list_nat,U: set_list_nat] :
( ( member_list_nat @ A2 @ ( minimal_elements @ U ) )
=> ~ ( ( member_list_nat @ A2 @ U )
=> ~ ! [Y4: list_nat] :
( ( member_list_nat @ Y4 @ U )
=> ~ ( pointwise_less @ Y4 @ A2 ) ) ) ) ).
% minimal_elements.cases
thf(fact_350_minimal__elements_Ointros,axiom,
! [X2: list_nat,U: set_list_nat] :
( ( member_list_nat @ X2 @ U )
=> ( ! [Y3: list_nat] :
( ( member_list_nat @ Y3 @ U )
=> ~ ( pointwise_less @ Y3 @ X2 ) )
=> ( member_list_nat @ X2 @ ( minimal_elements @ U ) ) ) ) ).
% minimal_elements.intros
thf(fact_351_minimal__elements_Osimps,axiom,
! [A2: list_nat,U: set_list_nat] :
( ( member_list_nat @ A2 @ ( minimal_elements @ U ) )
= ( ? [X: list_nat] :
( ( A2 = X )
& ( member_list_nat @ X @ U )
& ! [Y5: list_nat] :
( ( member_list_nat @ Y5 @ U )
=> ~ ( pointwise_less @ Y5 @ X ) ) ) ) ) ).
% minimal_elements.simps
thf(fact_352_minimal__elementsp_Ocases,axiom,
! [U: list_nat > $o,A2: list_nat] :
( ( minimal_elementsp @ U @ A2 )
=> ~ ( ( U @ A2 )
=> ~ ! [Y4: list_nat] :
( ( U @ Y4 )
=> ~ ( pointwise_less @ Y4 @ A2 ) ) ) ) ).
% minimal_elementsp.cases
thf(fact_353_minimal__elementsp_Ointros,axiom,
! [U: list_nat > $o,X2: list_nat] :
( ( U @ X2 )
=> ( ! [Y3: list_nat] :
( ( U @ Y3 )
=> ~ ( pointwise_less @ Y3 @ X2 ) )
=> ( minimal_elementsp @ U @ X2 ) ) ) ).
% minimal_elementsp.intros
thf(fact_354_minimal__elementsp_Osimps,axiom,
( minimal_elementsp
= ( ^ [U2: list_nat > $o,A3: list_nat] :
? [X: list_nat] :
( ( A3 = X )
& ( U2 @ X )
& ! [Y5: list_nat] :
( ( U2 @ Y5 )
=> ~ ( pointwise_less @ Y5 @ X ) ) ) ) ) ).
% minimal_elementsp.simps
thf(fact_355_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_356_Khovanskii_OAsubG,axiom,
! [G: set_list_nat,Addition: list_nat > list_nat > list_nat,Zero: list_nat,A: set_list_nat] :
( ( khovanskii_list_nat @ G @ Addition @ Zero @ A )
=> ( ord_le6045566169113846134st_nat @ A @ G ) ) ).
% Khovanskii.AsubG
thf(fact_357_Khovanskii_OAsubG,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat] :
( ( khovanskii_nat @ G @ Addition @ Zero @ A )
=> ( ord_less_eq_set_nat @ A @ G ) ) ).
% Khovanskii.AsubG
thf(fact_358_pointwise__less__def,axiom,
( pointwise_less
= ( ^ [X: list_nat,Y5: list_nat] :
( ( pointwise_le @ X @ Y5 )
& ( X != Y5 ) ) ) ) ).
% pointwise_less_def
thf(fact_359_pointwise__le__iff__less__equal,axiom,
( pointwise_le
= ( ^ [X: list_nat,Y5: list_nat] :
( ( pointwise_less @ X @ Y5 )
| ( X = Y5 ) ) ) ) ).
% pointwise_le_iff_less_equal
thf(fact_360_Cons__le__Cons,axiom,
! [A2: nat,X2: list_nat,B: nat,Y: list_nat] :
( ( ord_less_eq_list_nat @ ( cons_nat @ A2 @ 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 @ A2 @ B )
| ( ( A2 = B )
& ( ord_less_eq_list_nat @ X2 @ Y ) ) ) ) ) ) ).
% Cons_le_Cons
thf(fact_361_not__Cons__self2,axiom,
! [X2: nat,Xs: list_nat] :
( ( cons_nat @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_362_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_nat] :
( ( size_size_list_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_363_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_364_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs3: list_nat] :
( ! [Ys3: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys3 ) @ ( size_size_list_nat @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_365_inj__on__Cons1,axiom,
! [X2: nat,A: set_list_nat] : ( inj_on3049792774292151987st_nat @ ( cons_nat @ X2 ) @ A ) ).
% inj_on_Cons1
thf(fact_366_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_367_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_368_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A2: nat,B: nat] :
( ~ ( ord_less_nat @ A2 @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A2 @ B ) )
= A2 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_369_Suc__length__conv,axiom,
! [N: nat,Xs: list_nat] :
( ( ( suc @ N )
= ( size_size_list_nat @ Xs ) )
= ( ? [Y5: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ Y5 @ Ys2 ) )
& ( ( size_size_list_nat @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_370_length__Suc__conv,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y5: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ Y5 @ Ys2 ) )
& ( ( size_size_list_nat @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_371_impossible__Cons,axiom,
! [Xs: list_nat,Ys: list_nat,X2: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) )
=> ( Xs
!= ( cons_nat @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_372_finite__maxlen,axiom,
! [M7: set_list_nat] :
( ( finite8100373058378681591st_nat @ M7 )
=> ? [N4: nat] :
! [X4: list_nat] :
( ( member_list_nat @ X4 @ M7 )
=> ( ord_less_nat @ ( size_size_list_nat @ X4 ) @ N4 ) ) ) ).
% finite_maxlen
thf(fact_373_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_374_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_375_finite__Collect__subsets,axiom,
! [A: set_list_nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( finite7047420756378620717st_nat
@ ( collect_set_list_nat
@ ^ [B4: set_list_nat] : ( ord_le6045566169113846134st_nat @ B4 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_376_finite__Collect__subsets,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B4: set_nat] : ( ord_less_eq_set_nat @ B4 @ A ) ) ) ) ).
% finite_Collect_subsets
thf(fact_377_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_378_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_379_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_380_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_381_finite__Diff2,axiom,
! [B3: set_list_nat,A: set_list_nat] :
( ( finite8100373058378681591st_nat @ B3 )
=> ( ( finite8100373058378681591st_nat @ ( minus_7954133019191499631st_nat @ A @ B3 ) )
= ( finite8100373058378681591st_nat @ A ) ) ) ).
% finite_Diff2
thf(fact_382_finite__Diff2,axiom,
! [B3: set_nat,A: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B3 ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_Diff2
thf(fact_383_finite__Diff,axiom,
! [A: set_list_nat,B3: set_list_nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( finite8100373058378681591st_nat @ ( minus_7954133019191499631st_nat @ A @ B3 ) ) ) ).
% finite_Diff
thf(fact_384_finite__Diff,axiom,
! [A: set_nat,B3: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B3 ) ) ) ).
% finite_Diff
thf(fact_385_Diff__infinite__finite,axiom,
! [T2: set_list_nat,S2: set_list_nat] :
( ( finite8100373058378681591st_nat @ T2 )
=> ( ~ ( finite8100373058378681591st_nat @ S2 )
=> ~ ( finite8100373058378681591st_nat @ ( minus_7954133019191499631st_nat @ S2 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_386_Diff__infinite__finite,axiom,
! [T2: set_nat,S2: set_nat] :
( ( finite_finite_nat @ T2 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_387_finite__psubset__induct,axiom,
! [A: set_list_nat,P: set_list_nat > $o] :
( ( finite8100373058378681591st_nat @ A )
=> ( ! [A4: set_list_nat] :
( ( finite8100373058378681591st_nat @ A4 )
=> ( ! [B5: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ B5 @ A4 )
=> ( P @ B5 ) )
=> ( P @ A4 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_388_finite__psubset__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ! [B5: set_nat] :
( ( ord_less_set_nat @ B5 @ A4 )
=> ( P @ B5 ) )
=> ( P @ A4 ) ) )
=> ( P @ A ) ) ) ).
% finite_psubset_induct
thf(fact_389_rev__finite__subset,axiom,
! [B3: set_list_nat,A: set_list_nat] :
( ( finite8100373058378681591st_nat @ B3 )
=> ( ( ord_le6045566169113846134st_nat @ A @ B3 )
=> ( finite8100373058378681591st_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_390_rev__finite__subset,axiom,
! [B3: set_nat,A: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ A @ B3 )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_391_infinite__super,axiom,
! [S2: set_list_nat,T2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ S2 @ T2 )
=> ( ~ ( finite8100373058378681591st_nat @ S2 )
=> ~ ( finite8100373058378681591st_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_392_infinite__super,axiom,
! [S2: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_393_finite__subset,axiom,
! [A: set_list_nat,B3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B3 )
=> ( ( finite8100373058378681591st_nat @ B3 )
=> ( finite8100373058378681591st_nat @ A ) ) ) ).
% finite_subset
thf(fact_394_finite__subset,axiom,
! [A: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( finite_finite_nat @ B3 )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_395_pigeonhole__infinite__rel,axiom,
! [A: set_list_nat,B3: set_list_nat,R3: list_nat > list_nat > $o] :
( ~ ( finite8100373058378681591st_nat @ A )
=> ( ( finite8100373058378681591st_nat @ B3 )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
=> ? [Xa: list_nat] :
( ( member_list_nat @ Xa @ B3 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ B3 )
& ~ ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [A3: list_nat] :
( ( member_list_nat @ A3 @ A )
& ( R3 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_396_pigeonhole__infinite__rel,axiom,
! [A: set_list_nat,B3: set_nat,R3: list_nat > nat > $o] :
( ~ ( finite8100373058378681591st_nat @ A )
=> ( ( finite_finite_nat @ B3 )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B3 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B3 )
& ~ ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [A3: list_nat] :
( ( member_list_nat @ A3 @ A )
& ( R3 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_397_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B3: set_list_nat,R3: nat > list_nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite8100373058378681591st_nat @ B3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ? [Xa: list_nat] :
( ( member_list_nat @ Xa @ B3 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ B3 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A )
& ( R3 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_398_pigeonhole__infinite__rel,axiom,
! [A: set_nat,B3: set_nat,R3: nat > nat > $o] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B3 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B3 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A3: nat] :
( ( member_nat @ A3 @ A )
& ( R3 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_399_not__finite__existsD,axiom,
! [P: list_nat > $o] :
( ~ ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
=> ? [X_1: list_nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_400_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_401_finite__has__minimal2,axiom,
! [A: set_list_nat,A2: list_nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( ( member_list_nat @ A2 @ A )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
& ( ord_less_eq_list_nat @ X3 @ A2 )
& ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ A )
=> ( ( ord_less_eq_list_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_402_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ord_less_eq_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_403_finite__has__minimal2,axiom,
! [A: set_set_list_nat,A2: set_list_nat] :
( ( finite7047420756378620717st_nat @ A )
=> ( ( member_set_list_nat @ A2 @ A )
=> ? [X3: set_list_nat] :
( ( member_set_list_nat @ X3 @ A )
& ( ord_le6045566169113846134st_nat @ X3 @ A2 )
& ! [Xa: set_list_nat] :
( ( member_set_list_nat @ Xa @ A )
=> ( ( ord_le6045566169113846134st_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_404_finite__has__minimal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
& ( ord_less_eq_set_nat @ X3 @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_405_finite__has__maximal2,axiom,
! [A: set_list_nat,A2: list_nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( ( member_list_nat @ A2 @ A )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
& ( ord_less_eq_list_nat @ A2 @ X3 )
& ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ A )
=> ( ( ord_less_eq_list_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_406_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ord_less_eq_nat @ A2 @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_407_finite__has__maximal2,axiom,
! [A: set_set_list_nat,A2: set_list_nat] :
( ( finite7047420756378620717st_nat @ A )
=> ( ( member_set_list_nat @ A2 @ A )
=> ? [X3: set_list_nat] :
( ( member_set_list_nat @ X3 @ A )
& ( ord_le6045566169113846134st_nat @ A2 @ X3 )
& ! [Xa: set_list_nat] :
( ( member_set_list_nat @ Xa @ A )
=> ( ( ord_le6045566169113846134st_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_408_finite__has__maximal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
& ( ord_less_eq_set_nat @ A2 @ X3 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_409_finite__inverse__image__gen,axiom,
! [A: set_list_nat,F: list_nat > list_nat,D3: set_list_nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( ( inj_on3049792774292151987st_nat @ F @ D3 )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [J3: list_nat] :
( ( member_list_nat @ J3 @ D3 )
& ( member_list_nat @ ( F @ J3 ) @ A ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_410_finite__inverse__image__gen,axiom,
! [A: set_list_nat,F: nat > list_nat,D3: set_nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( ( inj_on_nat_list_nat @ F @ D3 )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [J3: nat] :
( ( member_nat @ J3 @ D3 )
& ( member_list_nat @ ( F @ J3 ) @ A ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_411_finite__inverse__image__gen,axiom,
! [A: set_nat,F: list_nat > nat,D3: set_list_nat] :
( ( finite_finite_nat @ A )
=> ( ( inj_on_list_nat_nat @ F @ D3 )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [J3: list_nat] :
( ( member_list_nat @ J3 @ D3 )
& ( member_nat @ ( F @ J3 ) @ A ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_412_finite__inverse__image__gen,axiom,
! [A: set_nat,F: nat > nat,D3: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( inj_on_nat_nat @ F @ D3 )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [J3: nat] :
( ( member_nat @ J3 @ D3 )
& ( member_nat @ ( F @ J3 ) @ A ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_413_sum_Ofinite__Collect__op,axiom,
! [I5: set_list_nat,X2: list_nat > nat,Y: list_nat > nat] :
( ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [I4: list_nat] :
( ( member_list_nat @ I4 @ I5 )
& ( ( X2 @ I4 )
!= zero_zero_nat ) ) ) )
=> ( ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [I4: list_nat] :
( ( member_list_nat @ I4 @ I5 )
& ( ( Y @ I4 )
!= zero_zero_nat ) ) ) )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [I4: list_nat] :
( ( member_list_nat @ I4 @ I5 )
& ( ( plus_plus_nat @ ( X2 @ I4 ) @ ( Y @ I4 ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_414_sum_Ofinite__Collect__op,axiom,
! [I5: set_nat,X2: nat > nat,Y: nat > nat] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I5 )
& ( ( X2 @ I4 )
!= zero_zero_nat ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I5 )
& ( ( Y @ I4 )
!= zero_zero_nat ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ I5 )
& ( ( plus_plus_nat @ ( X2 @ I4 ) @ ( Y @ I4 ) )
!= zero_zero_nat ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_415_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_416_linorder__inj__onI,axiom,
! [A: set_list_nat,F: list_nat > list_nat] :
( ! [X3: list_nat,Y3: list_nat] :
( ( ord_less_list_nat @ X3 @ Y3 )
=> ( ( member_list_nat @ X3 @ A )
=> ( ( member_list_nat @ Y3 @ A )
=> ( ( F @ X3 )
!= ( F @ Y3 ) ) ) ) )
=> ( ! [X3: list_nat,Y3: list_nat] :
( ( member_list_nat @ X3 @ A )
=> ( ( member_list_nat @ Y3 @ A )
=> ( ( ord_less_eq_list_nat @ X3 @ Y3 )
| ( ord_less_eq_list_nat @ Y3 @ X3 ) ) ) )
=> ( inj_on3049792774292151987st_nat @ F @ A ) ) ) ).
% linorder_inj_onI
thf(fact_417_linorder__inj__onI,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y3 @ A )
=> ( ( F @ X3 )
!= ( F @ Y3 ) ) ) ) )
=> ( ! [X3: nat,Y3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y3 @ A )
=> ( ( ord_less_eq_nat @ X3 @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X3 ) ) ) )
=> ( inj_on_nat_nat @ F @ A ) ) ) ).
% linorder_inj_onI
thf(fact_418_inj__Suc,axiom,
! [N5: set_nat] : ( inj_on_nat_nat @ suc @ N5 ) ).
% inj_Suc
thf(fact_419_inj__on__diff__nat,axiom,
! [N5: set_nat,K: nat] :
( ! [N4: nat] :
( ( member_nat @ N4 @ N5 )
=> ( ord_less_eq_nat @ K @ N4 ) )
=> ( inj_on_nat_nat
@ ^ [N2: nat] : ( minus_minus_nat @ N2 @ K )
@ N5 ) ) ).
% inj_on_diff_nat
thf(fact_420_inj__onD,axiom,
! [F: list_nat > list_nat,A: set_list_nat,X2: list_nat,Y: list_nat] :
( ( inj_on3049792774292151987st_nat @ F @ A )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
=> ( ( member_list_nat @ X2 @ A )
=> ( ( member_list_nat @ Y @ A )
=> ( X2 = Y ) ) ) ) ) ).
% inj_onD
thf(fact_421_inj__onD,axiom,
! [F: nat > nat,A: set_nat,X2: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
=> ( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y @ A )
=> ( X2 = Y ) ) ) ) ) ).
% inj_onD
thf(fact_422_inj__onI,axiom,
! [A: set_list_nat,F: list_nat > list_nat] :
( ! [X3: list_nat,Y3: list_nat] :
( ( member_list_nat @ X3 @ A )
=> ( ( member_list_nat @ Y3 @ A )
=> ( ( ( F @ X3 )
= ( F @ Y3 ) )
=> ( X3 = Y3 ) ) ) )
=> ( inj_on3049792774292151987st_nat @ F @ A ) ) ).
% inj_onI
thf(fact_423_inj__onI,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X3: nat,Y3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y3 @ A )
=> ( ( ( F @ X3 )
= ( F @ Y3 ) )
=> ( X3 = Y3 ) ) ) )
=> ( inj_on_nat_nat @ F @ A ) ) ).
% inj_onI
thf(fact_424_inj__on__def,axiom,
( inj_on3049792774292151987st_nat
= ( ^ [F2: list_nat > list_nat,A5: set_list_nat] :
! [X: list_nat] :
( ( member_list_nat @ X @ A5 )
=> ! [Y5: list_nat] :
( ( member_list_nat @ Y5 @ A5 )
=> ( ( ( F2 @ X )
= ( F2 @ Y5 ) )
=> ( X = Y5 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_425_inj__on__def,axiom,
( inj_on_nat_nat
= ( ^ [F2: nat > nat,A5: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A5 )
=> ! [Y5: nat] :
( ( member_nat @ Y5 @ A5 )
=> ( ( ( F2 @ X )
= ( F2 @ Y5 ) )
=> ( X = Y5 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_426_inj__on__cong,axiom,
! [A: set_list_nat,F: list_nat > list_nat,G2: list_nat > list_nat] :
( ! [A6: list_nat] :
( ( member_list_nat @ A6 @ A )
=> ( ( F @ A6 )
= ( G2 @ A6 ) ) )
=> ( ( inj_on3049792774292151987st_nat @ F @ A )
= ( inj_on3049792774292151987st_nat @ G2 @ A ) ) ) ).
% inj_on_cong
thf(fact_427_inj__on__cong,axiom,
! [A: set_nat,F: nat > nat,G2: nat > nat] :
( ! [A6: nat] :
( ( member_nat @ A6 @ A )
=> ( ( F @ A6 )
= ( G2 @ A6 ) ) )
=> ( ( inj_on_nat_nat @ F @ A )
= ( inj_on_nat_nat @ G2 @ A ) ) ) ).
% inj_on_cong
thf(fact_428_inj__on__eq__iff,axiom,
! [F: list_nat > list_nat,A: set_list_nat,X2: list_nat,Y: list_nat] :
( ( inj_on3049792774292151987st_nat @ F @ A )
=> ( ( member_list_nat @ X2 @ A )
=> ( ( member_list_nat @ Y @ A )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
= ( X2 = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_429_inj__on__eq__iff,axiom,
! [F: nat > nat,A: set_nat,X2: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
= ( X2 = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_430_inj__on__contraD,axiom,
! [F: list_nat > list_nat,A: set_list_nat,X2: list_nat,Y: list_nat] :
( ( inj_on3049792774292151987st_nat @ F @ A )
=> ( ( X2 != Y )
=> ( ( member_list_nat @ X2 @ A )
=> ( ( member_list_nat @ Y @ A )
=> ( ( F @ X2 )
!= ( F @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_431_inj__on__contraD,axiom,
! [F: nat > nat,A: set_nat,X2: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( X2 != Y )
=> ( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( F @ X2 )
!= ( F @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_432_inj__on__inverseI,axiom,
! [A: set_list_nat,G2: list_nat > list_nat,F: list_nat > list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
=> ( ( G2 @ ( F @ X3 ) )
= X3 ) )
=> ( inj_on3049792774292151987st_nat @ F @ A ) ) ).
% inj_on_inverseI
thf(fact_433_inj__on__inverseI,axiom,
! [A: set_nat,G2: nat > nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( G2 @ ( F @ X3 ) )
= X3 ) )
=> ( inj_on_nat_nat @ F @ A ) ) ).
% inj_on_inverseI
thf(fact_434_inj__on__diff,axiom,
! [F: list_nat > list_nat,A: set_list_nat,B3: set_list_nat] :
( ( inj_on3049792774292151987st_nat @ F @ A )
=> ( inj_on3049792774292151987st_nat @ F @ ( minus_7954133019191499631st_nat @ A @ B3 ) ) ) ).
% inj_on_diff
thf(fact_435_inj__on__diff,axiom,
! [F: nat > nat,A: set_nat,B3: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( inj_on_nat_nat @ F @ ( minus_minus_set_nat @ A @ B3 ) ) ) ).
% inj_on_diff
thf(fact_436_subset__Collect__iff,axiom,
! [B3: set_list_nat,A: set_list_nat,P: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ B3 @ A )
=> ( ( ord_le6045566169113846134st_nat @ B3
@ ( collect_list_nat
@ ^ [X: list_nat] :
( ( member_list_nat @ X @ A )
& ( P @ X ) ) ) )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ B3 )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_437_subset__Collect__iff,axiom,
! [B3: set_nat,A: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ B3 @ A )
=> ( ( ord_less_eq_set_nat @ B3
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ X ) ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ B3 )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_438_subset__CollectI,axiom,
! [B3: set_list_nat,A: set_list_nat,Q: list_nat > $o,P: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ B3 @ A )
=> ( ! [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 @ A )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_439_subset__CollectI,axiom,
! [B3: set_nat,A: set_nat,Q: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B3 @ A )
=> ( ! [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 @ A )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_440_inj__on__id2,axiom,
! [A: set_list_nat] :
( inj_on3049792774292151987st_nat
@ ^ [X: list_nat] : X
@ A ) ).
% inj_on_id2
thf(fact_441_inj__on__id2,axiom,
! [A: set_nat] :
( inj_on_nat_nat
@ ^ [X: nat] : X
@ A ) ).
% inj_on_id2
thf(fact_442_linorder__inj__onI_H,axiom,
! [A: set_list_nat,F: list_nat > list_nat] :
( ! [I2: list_nat,J2: list_nat] :
( ( member_list_nat @ I2 @ A )
=> ( ( member_list_nat @ J2 @ A )
=> ( ( ord_less_list_nat @ I2 @ J2 )
=> ( ( F @ I2 )
!= ( F @ J2 ) ) ) ) )
=> ( inj_on3049792774292151987st_nat @ F @ A ) ) ).
% linorder_inj_onI'
thf(fact_443_linorder__inj__onI_H,axiom,
! [A: set_nat,F: nat > nat] :
( ! [I2: nat,J2: nat] :
( ( member_nat @ I2 @ A )
=> ( ( member_nat @ J2 @ A )
=> ( ( ord_less_nat @ I2 @ J2 )
=> ( ( F @ I2 )
!= ( F @ J2 ) ) ) ) )
=> ( inj_on_nat_nat @ F @ A ) ) ).
% linorder_inj_onI'
thf(fact_444_list__decode_Ocases,axiom,
! [X2: nat] :
( ( X2 != zero_zero_nat )
=> ~ ! [N4: nat] :
( X2
!= ( suc @ N4 ) ) ) ).
% list_decode.cases
thf(fact_445_inj__on__add,axiom,
! [A2: nat,A: set_nat] : ( inj_on_nat_nat @ ( plus_plus_nat @ A2 ) @ A ) ).
% inj_on_add
thf(fact_446_inj__on__subset,axiom,
! [F: list_nat > list_nat,A: set_list_nat,B3: set_list_nat] :
( ( inj_on3049792774292151987st_nat @ F @ A )
=> ( ( ord_le6045566169113846134st_nat @ B3 @ A )
=> ( inj_on3049792774292151987st_nat @ F @ B3 ) ) ) ).
% inj_on_subset
thf(fact_447_inj__on__subset,axiom,
! [F: nat > nat,A: set_nat,B3: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ B3 @ A )
=> ( inj_on_nat_nat @ F @ B3 ) ) ) ).
% inj_on_subset
thf(fact_448_subset__inj__on,axiom,
! [F: list_nat > list_nat,B3: set_list_nat,A: set_list_nat] :
( ( inj_on3049792774292151987st_nat @ F @ B3 )
=> ( ( ord_le6045566169113846134st_nat @ A @ B3 )
=> ( inj_on3049792774292151987st_nat @ F @ A ) ) ) ).
% subset_inj_on
thf(fact_449_subset__inj__on,axiom,
! [F: nat > nat,B3: set_nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ B3 )
=> ( ( ord_less_eq_set_nat @ A @ B3 )
=> ( inj_on_nat_nat @ F @ A ) ) ) ).
% subset_inj_on
thf(fact_450_inj__on__add_H,axiom,
! [A2: nat,A: set_nat] :
( inj_on_nat_nat
@ ^ [B2: nat] : ( plus_plus_nat @ B2 @ A2 )
@ A ) ).
% inj_on_add'
thf(fact_451_psubsetI,axiom,
! [A: set_list_nat,B3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B3 )
=> ( ( A != B3 )
=> ( ord_le1190675801316882794st_nat @ A @ B3 ) ) ) ).
% psubsetI
thf(fact_452_psubsetI,axiom,
! [A: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( A != B3 )
=> ( ord_less_set_nat @ A @ B3 ) ) ) ).
% psubsetI
thf(fact_453_max__pointwise__ge,axiom,
! [U3: list_nat,U: set_list_nat] :
( ( member_list_nat @ U3 @ U )
=> ( ( finite8100373058378681591st_nat @ U )
=> ( pointwise_le @ U3 @ ( max_pointwise @ ( size_size_list_nat @ U3 ) @ U ) ) ) ) ).
% max_pointwise_ge
thf(fact_454_DiffI,axiom,
! [C: list_nat,A: set_list_nat,B3: set_list_nat] :
( ( member_list_nat @ C @ A )
=> ( ~ ( member_list_nat @ C @ B3 )
=> ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A @ B3 ) ) ) ) ).
% DiffI
thf(fact_455_DiffI,axiom,
! [C: nat,A: set_nat,B3: set_nat] :
( ( member_nat @ C @ A )
=> ( ~ ( member_nat @ C @ B3 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A @ B3 ) ) ) ) ).
% DiffI
thf(fact_456_subset__antisym,axiom,
! [A: set_list_nat,B3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B3 )
=> ( ( ord_le6045566169113846134st_nat @ B3 @ A )
=> ( A = B3 ) ) ) ).
% subset_antisym
thf(fact_457_subset__antisym,axiom,
! [A: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ A )
=> ( A = B3 ) ) ) ).
% subset_antisym
thf(fact_458_subsetI,axiom,
! [A: set_list_nat,B3: set_list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
=> ( member_list_nat @ X3 @ B3 ) )
=> ( ord_le6045566169113846134st_nat @ A @ B3 ) ) ).
% subsetI
thf(fact_459_subsetI,axiom,
! [A: set_nat,B3: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B3 ) )
=> ( ord_less_eq_set_nat @ A @ B3 ) ) ).
% subsetI
thf(fact_460_Diff__iff,axiom,
! [C: list_nat,A: set_list_nat,B3: set_list_nat] :
( ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A @ B3 ) )
= ( ( member_list_nat @ C @ A )
& ~ ( member_list_nat @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_461_Diff__iff,axiom,
! [C: nat,A: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B3 ) )
= ( ( member_nat @ C @ A )
& ~ ( member_nat @ C @ B3 ) ) ) ).
% Diff_iff
thf(fact_462_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_463_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_464_set__eq__subset,axiom,
( ( ^ [Y6: set_list_nat,Z3: set_list_nat] : ( Y6 = Z3 ) )
= ( ^ [A5: set_list_nat,B4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A5 @ B4 )
& ( ord_le6045566169113846134st_nat @ B4 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_465_set__eq__subset,axiom,
( ( ^ [Y6: set_nat,Z3: set_nat] : ( Y6 = Z3 ) )
= ( ^ [A5: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_466_subset__trans,axiom,
! [A: set_list_nat,B3: set_list_nat,C4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B3 )
=> ( ( ord_le6045566169113846134st_nat @ B3 @ C4 )
=> ( ord_le6045566169113846134st_nat @ A @ C4 ) ) ) ).
% subset_trans
thf(fact_467_subset__trans,axiom,
! [A: set_nat,B3: set_nat,C4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C4 )
=> ( ord_less_eq_set_nat @ A @ C4 ) ) ) ).
% subset_trans
thf(fact_468_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_469_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_470_subset__refl,axiom,
! [A: set_list_nat] : ( ord_le6045566169113846134st_nat @ A @ A ) ).
% subset_refl
thf(fact_471_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_472_subset__iff,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A5: set_list_nat,B4: set_list_nat] :
! [T3: list_nat] :
( ( member_list_nat @ T3 @ A5 )
=> ( member_list_nat @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_473_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B4: set_nat] :
! [T3: nat] :
( ( member_nat @ T3 @ A5 )
=> ( member_nat @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_474_equalityD2,axiom,
! [A: set_list_nat,B3: set_list_nat] :
( ( A = B3 )
=> ( ord_le6045566169113846134st_nat @ B3 @ A ) ) ).
% equalityD2
thf(fact_475_equalityD2,axiom,
! [A: set_nat,B3: set_nat] :
( ( A = B3 )
=> ( ord_less_eq_set_nat @ B3 @ A ) ) ).
% equalityD2
thf(fact_476_equalityD1,axiom,
! [A: set_list_nat,B3: set_list_nat] :
( ( A = B3 )
=> ( ord_le6045566169113846134st_nat @ A @ B3 ) ) ).
% equalityD1
thf(fact_477_equalityD1,axiom,
! [A: set_nat,B3: set_nat] :
( ( A = B3 )
=> ( ord_less_eq_set_nat @ A @ B3 ) ) ).
% equalityD1
thf(fact_478_subset__eq,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A5: set_list_nat,B4: set_list_nat] :
! [X: list_nat] :
( ( member_list_nat @ X @ A5 )
=> ( member_list_nat @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_479_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B4: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A5 )
=> ( member_nat @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_480_equalityE,axiom,
! [A: set_list_nat,B3: set_list_nat] :
( ( A = B3 )
=> ~ ( ( ord_le6045566169113846134st_nat @ A @ B3 )
=> ~ ( ord_le6045566169113846134st_nat @ B3 @ A ) ) ) ).
% equalityE
thf(fact_481_equalityE,axiom,
! [A: set_nat,B3: set_nat] :
( ( A = B3 )
=> ~ ( ( ord_less_eq_set_nat @ A @ B3 )
=> ~ ( ord_less_eq_set_nat @ B3 @ A ) ) ) ).
% equalityE
thf(fact_482_subsetD,axiom,
! [A: set_list_nat,B3: set_list_nat,C: list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B3 )
=> ( ( member_list_nat @ C @ A )
=> ( member_list_nat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_483_subsetD,axiom,
! [A: set_nat,B3: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B3 ) ) ) ).
% subsetD
thf(fact_484_in__mono,axiom,
! [A: set_list_nat,B3: set_list_nat,X2: list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B3 )
=> ( ( member_list_nat @ X2 @ A )
=> ( member_list_nat @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_485_in__mono,axiom,
! [A: set_nat,B3: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_486_DiffD2,axiom,
! [C: list_nat,A: set_list_nat,B3: set_list_nat] :
( ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A @ B3 ) )
=> ~ ( member_list_nat @ C @ B3 ) ) ).
% DiffD2
thf(fact_487_DiffD2,axiom,
! [C: nat,A: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B3 ) )
=> ~ ( member_nat @ C @ B3 ) ) ).
% DiffD2
thf(fact_488_DiffD1,axiom,
! [C: list_nat,A: set_list_nat,B3: set_list_nat] :
( ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A @ B3 ) )
=> ( member_list_nat @ C @ A ) ) ).
% DiffD1
thf(fact_489_DiffD1,axiom,
! [C: nat,A: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B3 ) )
=> ( member_nat @ C @ A ) ) ).
% DiffD1
thf(fact_490_DiffE,axiom,
! [C: list_nat,A: set_list_nat,B3: set_list_nat] :
( ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A @ B3 ) )
=> ~ ( ( member_list_nat @ C @ A )
=> ( member_list_nat @ C @ B3 ) ) ) ).
% DiffE
thf(fact_491_DiffE,axiom,
! [C: nat,A: set_nat,B3: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B3 ) )
=> ~ ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B3 ) ) ) ).
% DiffE
thf(fact_492_psubsetD,axiom,
! [A: set_list_nat,B3: set_list_nat,C: list_nat] :
( ( ord_le1190675801316882794st_nat @ A @ B3 )
=> ( ( member_list_nat @ C @ A )
=> ( member_list_nat @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_493_psubsetD,axiom,
! [A: set_nat,B3: set_nat,C: nat] :
( ( ord_less_set_nat @ A @ B3 )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_494_Collect__subset,axiom,
! [A: set_list_nat,P: list_nat > $o] :
( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X: list_nat] :
( ( member_list_nat @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_495_Collect__subset,axiom,
! [A: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_496_less__eq__set__def,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A5: set_list_nat,B4: set_list_nat] :
( ord_le1520216061033275535_nat_o
@ ^ [X: list_nat] : ( member_list_nat @ X @ A5 )
@ ^ [X: list_nat] : ( member_list_nat @ X @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_497_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B4: set_nat] :
( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A5 )
@ ^ [X: nat] : ( member_nat @ X @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_498_minus__set__def,axiom,
( minus_7954133019191499631st_nat
= ( ^ [A5: set_list_nat,B4: set_list_nat] :
( collect_list_nat
@ ( minus_1139252259498527702_nat_o
@ ^ [X: list_nat] : ( member_list_nat @ X @ A5 )
@ ^ [X: list_nat] : ( member_list_nat @ X @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_499_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A5: set_nat,B4: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A5 )
@ ^ [X: nat] : ( member_nat @ X @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_500_set__diff__eq,axiom,
( minus_7954133019191499631st_nat
= ( ^ [A5: set_list_nat,B4: set_list_nat] :
( collect_list_nat
@ ^ [X: list_nat] :
( ( member_list_nat @ X @ A5 )
& ~ ( member_list_nat @ X @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_501_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A5: set_nat,B4: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A5 )
& ~ ( member_nat @ X @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_502_less__set__def,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [A5: set_list_nat,B4: set_list_nat] :
( ord_less_list_nat_o
@ ^ [X: list_nat] : ( member_list_nat @ X @ A5 )
@ ^ [X: list_nat] : ( member_list_nat @ X @ B4 ) ) ) ) ).
% less_set_def
thf(fact_503_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B4: set_nat] :
( ord_less_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A5 )
@ ^ [X: nat] : ( member_nat @ X @ B4 ) ) ) ) ).
% less_set_def
thf(fact_504_Diff__mono,axiom,
! [A: set_list_nat,C4: set_list_nat,D3: set_list_nat,B3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ C4 )
=> ( ( ord_le6045566169113846134st_nat @ D3 @ B3 )
=> ( ord_le6045566169113846134st_nat @ ( minus_7954133019191499631st_nat @ A @ B3 ) @ ( minus_7954133019191499631st_nat @ C4 @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_505_Diff__mono,axiom,
! [A: set_nat,C4: set_nat,D3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A @ C4 )
=> ( ( ord_less_eq_set_nat @ D3 @ B3 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B3 ) @ ( minus_minus_set_nat @ C4 @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_506_Diff__subset,axiom,
! [A: set_list_nat,B3: set_list_nat] : ( ord_le6045566169113846134st_nat @ ( minus_7954133019191499631st_nat @ A @ B3 ) @ A ) ).
% Diff_subset
thf(fact_507_Diff__subset,axiom,
! [A: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B3 ) @ A ) ).
% Diff_subset
thf(fact_508_double__diff,axiom,
! [A: set_list_nat,B3: set_list_nat,C4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B3 )
=> ( ( ord_le6045566169113846134st_nat @ B3 @ C4 )
=> ( ( minus_7954133019191499631st_nat @ B3 @ ( minus_7954133019191499631st_nat @ C4 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_509_double__diff,axiom,
! [A: set_nat,B3: set_nat,C4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C4 )
=> ( ( minus_minus_set_nat @ B3 @ ( minus_minus_set_nat @ C4 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_510_psubsetE,axiom,
! [A: set_list_nat,B3: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A @ B3 )
=> ~ ( ( ord_le6045566169113846134st_nat @ A @ B3 )
=> ( ord_le6045566169113846134st_nat @ B3 @ A ) ) ) ).
% psubsetE
thf(fact_511_psubsetE,axiom,
! [A: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A @ B3 )
=> ~ ( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ord_less_eq_set_nat @ B3 @ A ) ) ) ).
% psubsetE
thf(fact_512_psubset__eq,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [A5: set_list_nat,B4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% psubset_eq
thf(fact_513_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% psubset_eq
thf(fact_514_psubset__imp__subset,axiom,
! [A: set_list_nat,B3: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A @ B3 )
=> ( ord_le6045566169113846134st_nat @ A @ B3 ) ) ).
% psubset_imp_subset
thf(fact_515_psubset__imp__subset,axiom,
! [A: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A @ B3 )
=> ( ord_less_eq_set_nat @ A @ B3 ) ) ).
% psubset_imp_subset
thf(fact_516_psubset__subset__trans,axiom,
! [A: set_list_nat,B3: set_list_nat,C4: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A @ B3 )
=> ( ( ord_le6045566169113846134st_nat @ B3 @ C4 )
=> ( ord_le1190675801316882794st_nat @ A @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_517_psubset__subset__trans,axiom,
! [A: set_nat,B3: set_nat,C4: set_nat] :
( ( ord_less_set_nat @ A @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ C4 )
=> ( ord_less_set_nat @ A @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_518_subset__not__subset__eq,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [A5: set_list_nat,B4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A5 @ B4 )
& ~ ( ord_le6045566169113846134st_nat @ B4 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_519_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B4 )
& ~ ( ord_less_eq_set_nat @ B4 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_520_subset__psubset__trans,axiom,
! [A: set_list_nat,B3: set_list_nat,C4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B3 )
=> ( ( ord_le1190675801316882794st_nat @ B3 @ C4 )
=> ( ord_le1190675801316882794st_nat @ A @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_521_subset__psubset__trans,axiom,
! [A: set_nat,B3: set_nat,C4: set_nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( ord_less_set_nat @ B3 @ C4 )
=> ( ord_less_set_nat @ A @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_522_subset__iff__psubset__eq,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A5: set_list_nat,B4: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_523_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_524_psubset__imp__ex__mem,axiom,
! [A: set_list_nat,B3: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A @ B3 )
=> ? [B6: list_nat] : ( member_list_nat @ B6 @ ( minus_7954133019191499631st_nat @ B3 @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_525_psubset__imp__ex__mem,axiom,
! [A: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A @ B3 )
=> ? [B6: nat] : ( member_nat @ B6 @ ( minus_minus_set_nat @ B3 @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_526_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( P @ K2 )
& ( ord_less_nat @ K2 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_527_finite__less__ub,axiom,
! [F: nat > nat,U3: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ N4 @ ( F @ N4 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U3 ) ) ) ) ).
% finite_less_ub
thf(fact_528_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C2: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_529_set__plus__mono2,axiom,
! [C4: set_list_nat,D3: set_list_nat,E: set_list_nat,F3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ C4 @ D3 )
=> ( ( ord_le6045566169113846134st_nat @ E @ F3 )
=> ( ord_le6045566169113846134st_nat @ ( plus_p884110394369815071st_nat @ C4 @ E ) @ ( plus_p884110394369815071st_nat @ D3 @ F3 ) ) ) ) ).
% set_plus_mono2
thf(fact_530_set__plus__mono2,axiom,
! [C4: set_nat,D3: set_nat,E: set_nat,F3: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ D3 )
=> ( ( ord_less_eq_set_nat @ E @ F3 )
=> ( ord_less_eq_set_nat @ ( plus_plus_set_nat @ C4 @ E ) @ ( plus_plus_set_nat @ D3 @ F3 ) ) ) ) ).
% set_plus_mono2
thf(fact_531_set__plus__intro,axiom,
! [A2: nat,C4: set_nat,B: nat,D3: set_nat] :
( ( member_nat @ A2 @ C4 )
=> ( ( member_nat @ B @ D3 )
=> ( member_nat @ ( plus_plus_nat @ A2 @ B ) @ ( plus_plus_set_nat @ C4 @ D3 ) ) ) ) ).
% set_plus_intro
thf(fact_532_set__plus__intro,axiom,
! [A2: list_nat,C4: set_list_nat,B: list_nat,D3: set_list_nat] :
( ( member_list_nat @ A2 @ C4 )
=> ( ( member_list_nat @ B @ D3 )
=> ( member_list_nat @ ( plus_plus_list_nat @ A2 @ B ) @ ( plus_p884110394369815071st_nat @ C4 @ D3 ) ) ) ) ).
% set_plus_intro
thf(fact_533_set__plus__elim,axiom,
! [X2: nat,A: set_nat,B3: set_nat] :
( ( member_nat @ X2 @ ( plus_plus_set_nat @ A @ B3 ) )
=> ~ ! [A6: nat,B6: nat] :
( ( X2
= ( plus_plus_nat @ A6 @ B6 ) )
=> ( ( member_nat @ A6 @ A )
=> ~ ( member_nat @ B6 @ B3 ) ) ) ) ).
% set_plus_elim
thf(fact_534_set__plus__elim,axiom,
! [X2: list_nat,A: set_list_nat,B3: set_list_nat] :
( ( member_list_nat @ X2 @ ( plus_p884110394369815071st_nat @ A @ B3 ) )
=> ~ ! [A6: list_nat,B6: list_nat] :
( ( X2
= ( plus_plus_list_nat @ A6 @ B6 ) )
=> ( ( member_list_nat @ A6 @ A )
=> ~ ( member_list_nat @ B6 @ B3 ) ) ) ) ).
% set_plus_elim
thf(fact_535_finite__set__plus,axiom,
! [S: set_list_nat,T: set_list_nat] :
( ( finite8100373058378681591st_nat @ S )
=> ( ( finite8100373058378681591st_nat @ T )
=> ( finite8100373058378681591st_nat @ ( plus_p884110394369815071st_nat @ S @ T ) ) ) ) ).
% finite_set_plus
thf(fact_536_finite__set__plus,axiom,
! [S: set_nat,T: set_nat] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_nat @ T )
=> ( finite_finite_nat @ ( plus_plus_set_nat @ S @ T ) ) ) ) ).
% finite_set_plus
thf(fact_537_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M7: nat] :
( ( P @ X2 )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M7 ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_538_set__zero__plus2,axiom,
! [A: set_nat,B3: set_nat] :
( ( member_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_set_nat @ B3 @ ( plus_plus_set_nat @ A @ B3 ) ) ) ).
% set_zero_plus2
thf(fact_539_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N6: set_nat] :
? [M: nat] :
! [X: nat] :
( ( member_nat @ X @ N6 )
=> ( ord_less_eq_nat @ X @ M ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_540_bounded__nat__set__is__finite,axiom,
! [N5: set_nat,N: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ N5 )
=> ( ord_less_nat @ X3 @ N ) )
=> ( finite_finite_nat @ N5 ) ) ).
% bounded_nat_set_is_finite
thf(fact_541_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N6: set_nat] :
? [M: nat] :
! [X: nat] :
( ( member_nat @ X @ N6 )
=> ( ord_less_nat @ X @ M ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_542_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C2: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_543_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_544_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_545_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_546_dual__order_Orefl,axiom,
! [A2: set_list_nat] : ( ord_le6045566169113846134st_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_547_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_548_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_549_order__refl,axiom,
! [X2: set_list_nat] : ( ord_le6045566169113846134st_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_550_order__refl,axiom,
! [X2: set_nat] : ( ord_less_eq_set_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_551_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_552_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_553_order__antisym__conv,axiom,
! [Y: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_554_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_555_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ 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 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_556_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > set_list_nat,C: set_list_nat] :
( ( ord_less_eq_nat @ A2 @ 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 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_557_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_558_ord__le__eq__subst,axiom,
! [A2: set_list_nat,B: set_list_nat,F: set_list_nat > nat,C: nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ 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 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_559_ord__le__eq__subst,axiom,
! [A2: set_list_nat,B: set_list_nat,F: set_list_nat > set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ 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 @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_560_ord__le__eq__subst,axiom,
! [A2: set_list_nat,B: set_list_nat,F: set_list_nat > set_nat,C: set_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_list_nat,Y3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_561_ord__le__eq__subst,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_562_ord__le__eq__subst,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_list_nat,C: set_list_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le6045566169113846134st_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_563_ord__le__eq__subst,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_564_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( 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 @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_565_ord__eq__le__subst,axiom,
! [A2: set_list_nat,F: nat > set_list_nat,B: nat,C: nat] :
( ( A2
= ( 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 @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_566_ord__eq__le__subst,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_567_ord__eq__le__subst,axiom,
! [A2: nat,F: set_list_nat > nat,B: set_list_nat,C: set_list_nat] :
( ( A2
= ( 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 @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_568_ord__eq__le__subst,axiom,
! [A2: set_list_nat,F: set_list_nat > set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( A2
= ( 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 @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_569_ord__eq__le__subst,axiom,
! [A2: set_nat,F: set_list_nat > set_nat,B: set_list_nat,C: set_list_nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ! [X3: set_list_nat,Y3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_570_ord__eq__le__subst,axiom,
! [A2: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_571_ord__eq__le__subst,axiom,
! [A2: set_list_nat,F: set_nat > set_list_nat,B: set_nat,C: set_nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le6045566169113846134st_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_572_ord__eq__le__subst,axiom,
! [A2: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_573_linorder__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_574_order__eq__refl,axiom,
! [X2: nat,Y: nat] :
( ( X2 = Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_575_order__eq__refl,axiom,
! [X2: set_list_nat,Y: set_list_nat] :
( ( X2 = Y )
=> ( ord_le6045566169113846134st_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_576_order__eq__refl,axiom,
! [X2: set_nat,Y: set_nat] :
( ( X2 = Y )
=> ( ord_less_eq_set_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_577_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_578_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_list_nat,C: set_list_nat] :
( ( ord_less_eq_nat @ A2 @ 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 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_579_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_580_order__subst2,axiom,
! [A2: set_list_nat,B: set_list_nat,F: set_list_nat > nat,C: nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ 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 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_581_order__subst2,axiom,
! [A2: set_list_nat,B: set_list_nat,F: set_list_nat > set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ 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 @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_582_order__subst2,axiom,
! [A2: set_list_nat,B: set_list_nat,F: set_list_nat > set_nat,C: set_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_list_nat,Y3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_583_order__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_584_order__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_list_nat,C: set_list_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_le6045566169113846134st_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le6045566169113846134st_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_585_order__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_586_order__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_587_order__subst1,axiom,
! [A2: nat,F: set_list_nat > nat,B: set_list_nat,C: set_list_nat] :
( ( ord_less_eq_nat @ A2 @ ( 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 @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_588_order__subst1,axiom,
! [A2: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_589_order__subst1,axiom,
! [A2: set_list_nat,F: nat > set_list_nat,B: nat,C: nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ ( 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 @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_590_order__subst1,axiom,
! [A2: set_list_nat,F: set_list_nat > set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ ( 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 @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_591_order__subst1,axiom,
! [A2: set_list_nat,F: set_nat > set_list_nat,B: set_nat,C: set_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le6045566169113846134st_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_592_order__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_593_order__subst1,axiom,
! [A2: set_nat,F: set_list_nat > set_nat,B: set_list_nat,C: set_list_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ! [X3: set_list_nat,Y3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_594_order__subst1,axiom,
! [A2: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_595_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_596_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_list_nat,Z3: set_list_nat] : ( Y6 = Z3 ) )
= ( ^ [A3: set_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ B2 )
& ( ord_le6045566169113846134st_nat @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_597_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_nat,Z3: set_nat] : ( Y6 = Z3 ) )
= ( ^ [A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_598_antisym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_599_antisym,axiom,
! [A2: set_list_nat,B: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B )
=> ( ( ord_le6045566169113846134st_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_600_antisym,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_601_dual__order_Otrans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_602_dual__order_Otrans,axiom,
! [B: set_list_nat,A2: set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B @ A2 )
=> ( ( ord_le6045566169113846134st_nat @ C @ B )
=> ( ord_le6045566169113846134st_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_603_dual__order_Otrans,axiom,
! [B: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_604_dual__order_Oantisym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_605_dual__order_Oantisym,axiom,
! [B: set_list_nat,A2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B @ A2 )
=> ( ( ord_le6045566169113846134st_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_606_dual__order_Oantisym,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_607_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_608_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: set_list_nat,Z3: set_list_nat] : ( Y6 = Z3 ) )
= ( ^ [A3: set_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B2 @ A3 )
& ( ord_le6045566169113846134st_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_609_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: set_nat,Z3: set_nat] : ( Y6 = Z3 ) )
= ( ^ [A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A3 )
& ( ord_less_eq_set_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_610_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A6: nat,B6: nat] :
( ( ord_less_eq_nat @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: nat,B6: nat] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_611_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_612_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_613_order__trans,axiom,
! [X2: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z )
=> ( ord_less_eq_set_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_614_order_Otrans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_615_order_Otrans,axiom,
! [A2: set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ord_le6045566169113846134st_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_616_order_Otrans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_617_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_618_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_619_order__antisym,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_620_ord__le__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_621_ord__le__eq__trans,axiom,
! [A2: set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_le6045566169113846134st_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_622_ord__le__eq__trans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_623_ord__eq__le__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_624_ord__eq__le__trans,axiom,
! [A2: set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( A2 = B )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ord_le6045566169113846134st_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_625_ord__eq__le__trans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( A2 = B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_626_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
= ( ^ [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_627_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_list_nat,Z3: set_list_nat] : ( Y6 = Z3 ) )
= ( ^ [X: set_list_nat,Y5: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X @ Y5 )
& ( ord_le6045566169113846134st_nat @ Y5 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_628_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_nat,Z3: set_nat] : ( Y6 = Z3 ) )
= ( ^ [X: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y5 )
& ( ord_less_eq_set_nat @ Y5 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_629_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_630_nle__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B ) )
= ( ( ord_less_eq_nat @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_631_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_632_order__less__imp__not__eq2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_633_order__less__imp__not__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_634_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_635_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_636_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_637_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ 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 @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_638_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( 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 @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_639_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_640_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_641_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( 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 @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_642_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_643_order__less__asym_H,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_644_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_645_order__less__asym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_646_linorder__neqE,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_647_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_648_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_649_dual__order_Ostrict__trans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_650_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_651_order_Ostrict__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_652_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A6: nat,B6: nat] :
( ( ord_less_nat @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: nat] : ( P @ A6 @ A6 )
=> ( ! [A6: nat,B6: nat] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_653_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M: nat] :
( ( ord_less_nat @ M @ N2 )
=> ~ ( P3 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_654_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_655_dual__order_Oasym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ~ ( ord_less_nat @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_656_linorder__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_657_antisym__conv3,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_nat @ Y @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_658_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X3: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X3 )
=> ( P @ Y4 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_659_ord__less__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_660_ord__eq__less__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_661_order_Oasym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order.asym
thf(fact_662_less__imp__neq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_663_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_664_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_665_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_666_order__le__imp__less__or__eq,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ord_less_set_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_667_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_668_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_669_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_list_nat,C: set_list_nat] :
( ( ord_less_nat @ A2 @ 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 @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_670_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_671_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_672_order__less__le__subst1,axiom,
! [A2: set_list_nat,F: nat > set_list_nat,B: nat,C: nat] :
( ( ord_le1190675801316882794st_nat @ A2 @ ( 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 @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_673_order__less__le__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_674_order__less__le__subst1,axiom,
! [A2: nat,F: set_list_nat > nat,B: set_list_nat,C: set_list_nat] :
( ( ord_less_nat @ A2 @ ( 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 @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_675_order__less__le__subst1,axiom,
! [A2: set_list_nat,F: set_list_nat > set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A2 @ ( 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 @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_676_order__less__le__subst1,axiom,
! [A2: set_nat,F: set_list_nat > set_nat,B: set_list_nat,C: set_list_nat] :
( ( ord_less_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ! [X3: set_list_nat,Y3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_677_order__less__le__subst1,axiom,
! [A2: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_678_order__less__le__subst1,axiom,
! [A2: set_list_nat,F: set_nat > set_list_nat,B: set_nat,C: set_nat] :
( ( ord_le1190675801316882794st_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1190675801316882794st_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_679_order__less__le__subst1,axiom,
! [A2: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_680_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_681_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_list_nat,C: set_list_nat] :
( ( ord_less_eq_nat @ A2 @ 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 @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_682_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_683_order__le__less__subst2,axiom,
! [A2: set_list_nat,B: set_list_nat,F: set_list_nat > nat,C: nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ 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 @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_684_order__le__less__subst2,axiom,
! [A2: set_list_nat,B: set_list_nat,F: set_list_nat > set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ 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 @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_685_order__le__less__subst2,axiom,
! [A2: set_list_nat,B: set_list_nat,F: set_list_nat > set_nat,C: set_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_list_nat,Y3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_686_order__le__less__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_687_order__le__less__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_list_nat,C: set_list_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_le1190675801316882794st_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le1190675801316882794st_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_688_order__le__less__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_689_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_690_order__le__less__subst1,axiom,
! [A2: set_list_nat,F: nat > set_list_nat,B: nat,C: nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ ( 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 @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_691_order__le__less__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_692_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_693_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_694_order__less__le__trans,axiom,
! [X2: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_set_nat @ X2 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z )
=> ( ord_less_set_nat @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_695_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_696_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_697_order__le__less__trans,axiom,
! [X2: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ord_less_set_nat @ Y @ Z )
=> ( ord_less_set_nat @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_698_order__neq__le__trans,axiom,
! [A2: nat,B: nat] :
( ( A2 != B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_699_order__neq__le__trans,axiom,
! [A2: set_list_nat,B: set_list_nat] :
( ( A2 != B )
=> ( ( ord_le6045566169113846134st_nat @ A2 @ B )
=> ( ord_le1190675801316882794st_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_700_order__neq__le__trans,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 != B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_set_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_701_order__le__neq__trans,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_702_order__le__neq__trans,axiom,
! [A2: set_list_nat,B: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_le1190675801316882794st_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_703_order__le__neq__trans,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_set_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_704_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_705_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_706_order__less__imp__le,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X2 @ Y )
=> ( ord_less_eq_set_nat @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_707_linorder__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_708_linorder__not__le,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y ) )
= ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_709_order__less__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
& ( X != Y5 ) ) ) ) ).
% order_less_le
thf(fact_710_order__less__le,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [X: set_list_nat,Y5: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X @ Y5 )
& ( X != Y5 ) ) ) ) ).
% order_less_le
thf(fact_711_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y5 )
& ( X != Y5 ) ) ) ) ).
% order_less_le
thf(fact_712_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y5: nat] :
( ( ord_less_nat @ X @ Y5 )
| ( X = Y5 ) ) ) ) ).
% order_le_less
thf(fact_713_order__le__less,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [X: set_list_nat,Y5: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ X @ Y5 )
| ( X = Y5 ) ) ) ) ).
% order_le_less
thf(fact_714_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X: set_nat,Y5: set_nat] :
( ( ord_less_set_nat @ X @ Y5 )
| ( X = Y5 ) ) ) ) ).
% order_le_less
thf(fact_715_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_716_dual__order_Ostrict__implies__order,axiom,
! [B: set_list_nat,A2: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ B @ A2 )
=> ( ord_le6045566169113846134st_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_717_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B @ A2 )
=> ( ord_less_eq_set_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_718_order_Ostrict__implies__order,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_719_order_Ostrict__implies__order,axiom,
! [A2: set_list_nat,B: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A2 @ B )
=> ( ord_le6045566169113846134st_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_720_order_Ostrict__implies__order,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_721_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_722_dual__order_Ostrict__iff__not,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [B2: set_list_nat,A3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B2 @ A3 )
& ~ ( ord_le6045566169113846134st_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_723_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B2: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A3 )
& ~ ( ord_less_eq_set_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_724_dual__order_Ostrict__trans2,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_725_dual__order_Ostrict__trans2,axiom,
! [B: set_list_nat,A2: set_list_nat,C: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ B @ A2 )
=> ( ( ord_le6045566169113846134st_nat @ C @ B )
=> ( ord_le1190675801316882794st_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_726_dual__order_Ostrict__trans2,axiom,
! [B: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_727_dual__order_Ostrict__trans1,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_728_dual__order_Ostrict__trans1,axiom,
! [B: set_list_nat,A2: set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B @ A2 )
=> ( ( ord_le1190675801316882794st_nat @ C @ B )
=> ( ord_le1190675801316882794st_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_729_dual__order_Ostrict__trans1,axiom,
! [B: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_730_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_731_dual__order_Ostrict__iff__order,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [B2: set_list_nat,A3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_732_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B2: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_733_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_nat @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_734_dual__order_Oorder__iff__strict,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [B2: set_list_nat,A3: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_735_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B2: set_nat,A3: set_nat] :
( ( ord_less_set_nat @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_736_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_737_order_Ostrict__iff__not,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [A3: set_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ B2 )
& ~ ( ord_le6045566169113846134st_nat @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_738_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B2 )
& ~ ( ord_less_eq_set_nat @ B2 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_739_order_Ostrict__trans2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_740_order_Ostrict__trans2,axiom,
! [A2: set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A2 @ B )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ord_le1190675801316882794st_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_741_order_Ostrict__trans2,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_742_order_Ostrict__trans1,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_743_order_Ostrict__trans1,axiom,
! [A2: set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B )
=> ( ( ord_le1190675801316882794st_nat @ B @ C )
=> ( ord_le1190675801316882794st_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_744_order_Ostrict__trans1,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_745_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_746_order_Ostrict__iff__order,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [A3: set_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_747_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_748_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_749_order_Oorder__iff__strict,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A3: set_list_nat,B2: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_750_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_751_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_752_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y5: nat] :
( ( ord_less_eq_nat @ X @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_753_less__le__not__le,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [X: set_list_nat,Y5: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X @ Y5 )
& ~ ( ord_le6045566169113846134st_nat @ Y5 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_754_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y5 )
& ~ ( ord_less_eq_set_nat @ Y5 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_755_antisym__conv2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_756_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_757_antisym__conv2,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ~ ( ord_less_set_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_758_antisym__conv1,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_759_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_760_antisym__conv1,axiom,
! [X2: set_nat,Y: set_nat] :
( ~ ( ord_less_set_nat @ X2 @ Y )
=> ( ( ord_less_eq_set_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_761_nless__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_nat @ A2 @ B ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_762_nless__le,axiom,
! [A2: set_list_nat,B: set_list_nat] :
( ( ~ ( ord_le1190675801316882794st_nat @ A2 @ B ) )
= ( ~ ( ord_le6045566169113846134st_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_763_nless__le,axiom,
! [A2: set_nat,B: set_nat] :
( ( ~ ( ord_less_set_nat @ A2 @ B ) )
= ( ~ ( ord_less_eq_set_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_764_leI,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% leI
thf(fact_765_leD,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y ) ) ).
% leD
thf(fact_766_leD,axiom,
! [Y: set_list_nat,X2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ Y @ X2 )
=> ~ ( ord_le1190675801316882794st_nat @ X2 @ Y ) ) ).
% leD
thf(fact_767_leD,axiom,
! [Y: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X2 )
=> ~ ( ord_less_set_nat @ X2 @ Y ) ) ).
% leD
thf(fact_768_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_769_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_770_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_771_unbounded__k__infinite,axiom,
! [K: nat,S2: set_nat] :
( ! [M5: nat] :
( ( ord_less_nat @ K @ M5 )
=> ? [N7: nat] :
( ( ord_less_nat @ M5 @ N7 )
& ( member_nat @ N7 @ S2 ) ) )
=> ~ ( finite_finite_nat @ S2 ) ) ).
% unbounded_k_infinite
thf(fact_772_infinite__nat__iff__unbounded,axiom,
! [S2: set_nat] :
( ( ~ ( finite_finite_nat @ S2 ) )
= ( ! [M: nat] :
? [N2: nat] :
( ( ord_less_nat @ M @ N2 )
& ( member_nat @ N2 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_773_infinite__nat__iff__unbounded__le,axiom,
! [S2: set_nat] :
( ( ~ ( finite_finite_nat @ S2 ) )
= ( ! [M: nat] :
? [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
& ( member_nat @ N2 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_774_WFP,axiom,
wfP_list_nat @ pointwise_less ).
% WFP
thf(fact_775_Euclid__induct,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A6: nat,B6: nat] :
( ( P @ A6 @ B6 )
= ( P @ B6 @ A6 ) )
=> ( ! [A6: nat] : ( P @ A6 @ zero_zero_nat )
=> ( ! [A6: nat,B6: nat] :
( ( P @ A6 @ B6 )
=> ( P @ A6 @ ( plus_plus_nat @ A6 @ B6 ) ) )
=> ( P @ A2 @ B ) ) ) ) ).
% Euclid_induct
thf(fact_776_Khovanskii_Oalpha__in__G,axiom,
! [G: set_list_nat,Addition: list_nat > list_nat > list_nat,Zero: list_nat,A: set_list_nat,X2: list_nat] :
( ( khovanskii_list_nat @ G @ Addition @ Zero @ A )
=> ( member_list_nat @ ( alpha_list_nat @ G @ Addition @ Zero @ A @ X2 ) @ G ) ) ).
% Khovanskii.alpha_in_G
thf(fact_777_Khovanskii_Oalpha__in__G,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,X2: list_nat] :
( ( khovanskii_nat @ G @ Addition @ Zero @ A )
=> ( member_nat @ ( alpha_nat @ G @ Addition @ Zero @ A @ X2 ) @ G ) ) ).
% Khovanskii.alpha_in_G
thf(fact_778_wfP__empty,axiom,
( wfP_list_nat
@ ^ [X: list_nat,Y5: list_nat] : $false ) ).
% wfP_empty
thf(fact_779_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_780_wfP__less,axiom,
wfP_list_nat @ ord_less_list_nat ).
% wfP_less
thf(fact_781_wfP__less,axiom,
wfP_nat @ ord_less_nat ).
% wfP_less
thf(fact_782_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M2: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K3 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K3 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_783_triangle__Suc,axiom,
! [N: nat] :
( ( nat_triangle @ ( suc @ N ) )
= ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% triangle_Suc
thf(fact_784_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N4: nat] :
( ~ ( P @ N4 )
& ( P @ ( suc @ N4 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_785_triangle__0,axiom,
( ( nat_triangle @ zero_zero_nat )
= zero_zero_nat ) ).
% triangle_0
thf(fact_786_finite__indexed__bound,axiom,
! [A: set_list_nat,P: list_nat > nat > $o] :
( ( finite8100373058378681591st_nat @ A )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
=> ? [X_12: nat] : ( P @ X3 @ X_12 ) )
=> ? [M5: nat] :
! [X4: list_nat] :
( ( member_list_nat @ X4 @ A )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ M5 )
& ( P @ X4 @ K3 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_787_finite__indexed__bound,axiom,
! [A: set_nat,P: nat > nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ? [X_12: nat] : ( P @ X3 @ X_12 ) )
=> ? [M5: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ M5 )
& ( P @ X4 @ K3 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_788_verit__sum__simplify,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% verit_sum_simplify
thf(fact_789_add__0__iff,axiom,
! [B: nat,A2: nat] :
( ( B
= ( plus_plus_nat @ B @ A2 ) )
= ( A2 = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_790_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_791_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_792_verit__comp__simplify1_I2_J,axiom,
! [A2: set_list_nat] : ( ord_le6045566169113846134st_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_793_verit__comp__simplify1_I2_J,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_794_verit__la__disequality,axiom,
! [A2: nat,B: nat] :
( ( A2 = B )
| ~ ( ord_less_eq_nat @ A2 @ B )
| ~ ( ord_less_eq_nat @ B @ A2 ) ) ).
% verit_la_disequality
thf(fact_795_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_796_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_797_verit__comp__simplify1_I3_J,axiom,
! [B7: nat,A7: nat] :
( ( ~ ( ord_less_eq_nat @ B7 @ A7 ) )
= ( ord_less_nat @ A7 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_798_minf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ~ ( ord_less_eq_nat @ T @ X4 ) ) ).
% minf(8)
thf(fact_799_minf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ord_less_eq_nat @ X4 @ T ) ) ).
% minf(6)
thf(fact_800_pinf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ord_less_eq_nat @ T @ X4 ) ) ).
% pinf(8)
thf(fact_801_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q3 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_802_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q3 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_803_pinf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( X4 != T ) ) ).
% pinf(3)
thf(fact_804_pinf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( X4 != T ) ) ).
% pinf(4)
thf(fact_805_pinf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ~ ( ord_less_nat @ X4 @ T ) ) ).
% pinf(5)
thf(fact_806_pinf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ord_less_nat @ T @ X4 ) ) ).
% pinf(7)
thf(fact_807_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q3 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_808_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q3 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_809_minf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( X4 != T ) ) ).
% minf(3)
thf(fact_810_minf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( X4 != T ) ) ).
% minf(4)
thf(fact_811_minf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ord_less_nat @ X4 @ T ) ) ).
% minf(5)
thf(fact_812_minf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ~ ( ord_less_nat @ T @ X4 ) ) ).
% minf(7)
thf(fact_813_pinf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ T ) ) ).
% pinf(6)
thf(fact_814_complete__interval,axiom,
! [A2: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A2 @ B )
=> ( ( P @ A2 )
=> ( ~ ( P @ B )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A2 @ C3 )
& ( ord_less_eq_nat @ C3 @ B )
& ! [X4: nat] :
( ( ( ord_less_eq_nat @ A2 @ X4 )
& ( ord_less_nat @ X4 @ C3 ) )
=> ( P @ X4 ) )
& ! [D4: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A2 @ X3 )
& ( ord_less_nat @ X3 @ D4 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D4 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_815_min__pointwise__ge__iff,axiom,
! [U: set_list_nat,R: nat,X2: list_nat] :
( ( finite8100373058378681591st_nat @ U )
=> ( ( U != bot_bot_set_list_nat )
=> ( ! [U4: list_nat] :
( ( member_list_nat @ U4 @ U )
=> ( ( size_size_list_nat @ U4 )
= R ) )
=> ( ( ( size_size_list_nat @ X2 )
= R )
=> ( ( pointwise_le @ X2 @ ( min_pointwise @ R @ U ) )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ U )
=> ( pointwise_le @ X2 @ X ) ) ) ) ) ) ) ) ).
% min_pointwise_ge_iff
thf(fact_816_list_Osize__gen_I2_J,axiom,
! [X2: nat > nat,X21: nat,X222: list_nat] :
( ( size_list_nat @ X2 @ ( cons_nat @ X21 @ X222 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( X2 @ X21 ) @ ( size_list_nat @ X2 @ X222 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size_gen(2)
thf(fact_817_subset__empty,axiom,
! [A: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ bot_bot_set_list_nat )
= ( A = bot_bot_set_list_nat ) ) ).
% subset_empty
thf(fact_818_subset__empty,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_819_empty__subsetI,axiom,
! [A: set_list_nat] : ( ord_le6045566169113846134st_nat @ bot_bot_set_list_nat @ A ) ).
% empty_subsetI
thf(fact_820_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_821_inj__on__empty,axiom,
! [F: nat > nat] : ( inj_on_nat_nat @ F @ bot_bot_set_nat ) ).
% inj_on_empty
thf(fact_822_inj__on__empty,axiom,
! [F: list_nat > list_nat] : ( inj_on3049792774292151987st_nat @ F @ bot_bot_set_list_nat ) ).
% inj_on_empty
thf(fact_823_Diff__empty,axiom,
! [A: set_list_nat] :
( ( minus_7954133019191499631st_nat @ A @ bot_bot_set_list_nat )
= A ) ).
% Diff_empty
thf(fact_824_empty__Diff,axiom,
! [A: set_list_nat] :
( ( minus_7954133019191499631st_nat @ bot_bot_set_list_nat @ A )
= bot_bot_set_list_nat ) ).
% empty_Diff
thf(fact_825_Diff__cancel,axiom,
! [A: set_list_nat] :
( ( minus_7954133019191499631st_nat @ A @ A )
= bot_bot_set_list_nat ) ).
% Diff_cancel
thf(fact_826_Diff__eq__empty__iff,axiom,
! [A: set_list_nat,B3: set_list_nat] :
( ( ( minus_7954133019191499631st_nat @ A @ B3 )
= bot_bot_set_list_nat )
= ( ord_le6045566169113846134st_nat @ A @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_827_Diff__eq__empty__iff,axiom,
! [A: set_nat,B3: set_nat] :
( ( ( minus_minus_set_nat @ A @ B3 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_828_finite__transitivity__chain,axiom,
! [A: set_nat,R3: nat > nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [X3: nat] :
~ ( R3 @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z2: nat] :
( ( R3 @ X3 @ Y3 )
=> ( ( R3 @ Y3 @ Z2 )
=> ( R3 @ X3 @ Z2 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ? [Y4: nat] :
( ( member_nat @ Y4 @ A )
& ( R3 @ X3 @ Y4 ) ) )
=> ( A = bot_bot_set_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_829_finite__transitivity__chain,axiom,
! [A: set_list_nat,R3: list_nat > list_nat > $o] :
( ( finite8100373058378681591st_nat @ A )
=> ( ! [X3: list_nat] :
~ ( R3 @ X3 @ X3 )
=> ( ! [X3: list_nat,Y3: list_nat,Z2: list_nat] :
( ( R3 @ X3 @ Y3 )
=> ( ( R3 @ Y3 @ Z2 )
=> ( R3 @ X3 @ Z2 ) ) )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
=> ? [Y4: list_nat] :
( ( member_list_nat @ Y4 @ A )
& ( R3 @ X3 @ Y4 ) ) )
=> ( A = bot_bot_set_list_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_830_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_831_bot_Oextremum,axiom,
! [A2: set_list_nat] : ( ord_le6045566169113846134st_nat @ bot_bot_set_list_nat @ A2 ) ).
% bot.extremum
thf(fact_832_bot_Oextremum,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% bot.extremum
thf(fact_833_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_834_bot_Oextremum__unique,axiom,
! [A2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ bot_bot_set_list_nat )
= ( A2 = bot_bot_set_list_nat ) ) ).
% bot.extremum_unique
thf(fact_835_bot_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_836_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_837_bot_Oextremum__uniqueI,axiom,
! [A2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ bot_bot_set_list_nat )
=> ( A2 = bot_bot_set_list_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_838_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
=> ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_839_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X: nat] : $false ) ) ).
% empty_def
thf(fact_840_empty__def,axiom,
( bot_bot_set_list_nat
= ( collect_list_nat
@ ^ [X: list_nat] : $false ) ) ).
% empty_def
thf(fact_841_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_842_finite_OemptyI,axiom,
finite8100373058378681591st_nat @ bot_bot_set_list_nat ).
% finite.emptyI
thf(fact_843_infinite__imp__nonempty,axiom,
! [S2: set_nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( S2 != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_844_infinite__imp__nonempty,axiom,
! [S2: set_list_nat] :
( ~ ( finite8100373058378681591st_nat @ S2 )
=> ( S2 != bot_bot_set_list_nat ) ) ).
% infinite_imp_nonempty
thf(fact_845_Khovanskii_Ononempty,axiom,
! [G: set_list_nat,Addition: list_nat > list_nat > list_nat,Zero: list_nat,A: set_list_nat] :
( ( khovanskii_list_nat @ G @ Addition @ Zero @ A )
=> ( A != bot_bot_set_list_nat ) ) ).
% Khovanskii.nonempty
thf(fact_846_bot_Onot__eq__extremum,axiom,
! [A2: set_list_nat] :
( ( A2 != bot_bot_set_list_nat )
= ( ord_le1190675801316882794st_nat @ bot_bot_set_list_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_847_bot_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_848_bot_Oextremum__strict,axiom,
! [A2: set_list_nat] :
~ ( ord_le1190675801316882794st_nat @ A2 @ bot_bot_set_list_nat ) ).
% bot.extremum_strict
thf(fact_849_bot_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_850_not__psubset__empty,axiom,
! [A: set_list_nat] :
~ ( ord_le1190675801316882794st_nat @ A @ bot_bot_set_list_nat ) ).
% not_psubset_empty
thf(fact_851_finite__has__minimal,axiom,
! [A: set_list_nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( ( A != bot_bot_set_list_nat )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
& ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ A )
=> ( ( ord_less_eq_list_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_852_finite__has__minimal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_853_finite__has__minimal,axiom,
! [A: set_set_list_nat] :
( ( finite7047420756378620717st_nat @ A )
=> ( ( A != bot_bo3886227569956363488st_nat )
=> ? [X3: set_list_nat] :
( ( member_set_list_nat @ X3 @ A )
& ! [Xa: set_list_nat] :
( ( member_set_list_nat @ Xa @ A )
=> ( ( ord_le6045566169113846134st_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_854_finite__has__minimal,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_855_finite__has__maximal,axiom,
! [A: set_list_nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( ( A != bot_bot_set_list_nat )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
& ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ A )
=> ( ( ord_less_eq_list_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_856_finite__has__maximal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_857_finite__has__maximal,axiom,
! [A: set_set_list_nat] :
( ( finite7047420756378620717st_nat @ A )
=> ( ( A != bot_bo3886227569956363488st_nat )
=> ? [X3: set_list_nat] :
( ( member_set_list_nat @ X3 @ A )
& ! [Xa: set_list_nat] :
( ( member_set_list_nat @ Xa @ A )
=> ( ( ord_le6045566169113846134st_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_858_finite__has__maximal,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_859_infinite__growing,axiom,
! [X6: set_list_nat] :
( ( X6 != bot_bot_set_list_nat )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ X6 )
=> ? [Xa: list_nat] :
( ( member_list_nat @ Xa @ X6 )
& ( ord_less_list_nat @ X3 @ Xa ) ) )
=> ~ ( finite8100373058378681591st_nat @ X6 ) ) ) ).
% infinite_growing
thf(fact_860_infinite__growing,axiom,
! [X6: set_nat] :
( ( X6 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X6 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X6 )
& ( ord_less_nat @ X3 @ Xa ) ) )
=> ~ ( finite_finite_nat @ X6 ) ) ) ).
% infinite_growing
thf(fact_861_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_862_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_863_max__pointwise__mono,axiom,
! [X7: set_list_nat,X6: set_list_nat,R: nat] :
( ( ord_le6045566169113846134st_nat @ X7 @ X6 )
=> ( ( finite8100373058378681591st_nat @ X6 )
=> ( ( X7 != bot_bot_set_list_nat )
=> ( pointwise_le @ ( max_pointwise @ R @ X7 ) @ ( max_pointwise @ R @ X6 ) ) ) ) ) ).
% max_pointwise_mono
thf(fact_864_max__pointwise__le__iff,axiom,
! [U: set_list_nat,R: nat,X2: list_nat] :
( ( finite8100373058378681591st_nat @ U )
=> ( ( U != bot_bot_set_list_nat )
=> ( ! [U4: list_nat] :
( ( member_list_nat @ U4 @ U )
=> ( ( size_size_list_nat @ U4 )
= R ) )
=> ( ( ( size_size_list_nat @ X2 )
= R )
=> ( ( pointwise_le @ ( max_pointwise @ R @ U ) @ X2 )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ U )
=> ( pointwise_le @ X @ X2 ) ) ) ) ) ) ) ) ).
% max_pointwise_le_iff
thf(fact_865_diff__shunt__var,axiom,
! [X2: set_list_nat,Y: set_list_nat] :
( ( ( minus_7954133019191499631st_nat @ X2 @ Y )
= bot_bot_set_list_nat )
= ( ord_le6045566169113846134st_nat @ X2 @ Y ) ) ).
% diff_shunt_var
thf(fact_866_diff__shunt__var,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ( minus_minus_set_nat @ X2 @ Y )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X2 @ Y ) ) ).
% diff_shunt_var
thf(fact_867_Khovanskii__axioms_Ointro,axiom,
! [A: set_list_nat,G: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ G )
=> ( ( finite8100373058378681591st_nat @ A )
=> ( ( A != bot_bot_set_list_nat )
=> ( khovan1553326461689229922st_nat @ G @ A ) ) ) ) ).
% Khovanskii_axioms.intro
thf(fact_868_Khovanskii__axioms_Ointro,axiom,
! [A: set_nat,G: set_nat] :
( ( ord_less_eq_set_nat @ A @ G )
=> ( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( khovan4585363760863428690ms_nat @ G @ A ) ) ) ) ).
% Khovanskii_axioms.intro
thf(fact_869_Khovanskii__axioms__def,axiom,
( khovan1553326461689229922st_nat
= ( ^ [G3: set_list_nat,A5: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A5 @ G3 )
& ( finite8100373058378681591st_nat @ A5 )
& ( A5 != bot_bot_set_list_nat ) ) ) ) ).
% Khovanskii_axioms_def
thf(fact_870_Khovanskii__axioms__def,axiom,
( khovan4585363760863428690ms_nat
= ( ^ [G3: set_nat,A5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ G3 )
& ( finite_finite_nat @ A5 )
& ( A5 != bot_bot_set_nat ) ) ) ) ).
% Khovanskii_axioms_def
thf(fact_871_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_872_arg__min__if__finite_I2_J,axiom,
! [S2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ~ ? [X4: nat] :
( ( member_nat @ X4 @ S2 )
& ( ord_less_nat @ ( F @ X4 ) @ ( F @ ( lattic7446932960582359483at_nat @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_873_arg__min__if__finite_I2_J,axiom,
! [S2: set_list_nat,F: list_nat > nat] :
( ( finite8100373058378681591st_nat @ S2 )
=> ( ( S2 != bot_bot_set_list_nat )
=> ~ ? [X4: list_nat] :
( ( member_list_nat @ X4 @ S2 )
& ( ord_less_nat @ ( F @ X4 ) @ ( F @ ( lattic5785867957632790475at_nat @ F @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_874_arg__min__least,axiom,
! [S2: set_nat,Y: nat,F: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S2 )
=> ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_875_arg__min__least,axiom,
! [S2: set_list_nat,Y: list_nat,F: list_nat > nat] :
( ( finite8100373058378681591st_nat @ S2 )
=> ( ( S2 != bot_bot_set_list_nat )
=> ( ( member_list_nat @ Y @ S2 )
=> ( ord_less_eq_nat @ ( F @ ( lattic5785867957632790475at_nat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_876_subset__emptyI,axiom,
! [A: set_list_nat] :
( ! [X3: list_nat] :
~ ( member_list_nat @ X3 @ A )
=> ( ord_le6045566169113846134st_nat @ A @ bot_bot_set_list_nat ) ) ).
% subset_emptyI
thf(fact_877_subset__emptyI,axiom,
! [A: set_nat] :
( ! [X3: nat] :
~ ( member_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ A @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_878_Collect__restrict,axiom,
! [X6: set_list_nat,P: list_nat > $o] :
( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X: list_nat] :
( ( member_list_nat @ X @ X6 )
& ( P @ X ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_879_Collect__restrict,axiom,
! [X6: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ X6 )
& ( P @ X ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_880_prop__restrict,axiom,
! [X2: list_nat,Z5: set_list_nat,X6: set_list_nat,P: list_nat > $o] :
( ( member_list_nat @ X2 @ Z5 )
=> ( ( ord_le6045566169113846134st_nat @ Z5
@ ( collect_list_nat
@ ^ [X: list_nat] :
( ( member_list_nat @ X @ X6 )
& ( P @ X ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_881_prop__restrict,axiom,
! [X2: nat,Z5: set_nat,X6: set_nat,P: nat > $o] :
( ( member_nat @ X2 @ Z5 )
=> ( ( ord_less_eq_set_nat @ Z5
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ X6 )
& ( P @ X ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_882_Khovanskii_Ouseless__iff,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,X2: list_nat] :
( ( khovanskii_nat @ G @ Addition @ Zero @ A )
=> ( ( ( size_size_list_nat @ X2 )
= ( finite_card_nat @ A ) )
=> ( ( useless_nat @ G @ Addition @ Zero @ A @ X2 )
= ( ? [X: list_nat] :
( ( member_list_nat @ X @ ( minimal_elements @ ( collect_list_nat @ ( useless_nat @ G @ Addition @ Zero @ A ) ) ) )
& ( pointwise_le @ X @ X2 ) ) ) ) ) ) ).
% Khovanskii.useless_iff
thf(fact_883_Khovanskii_Ouseless__iff,axiom,
! [G: set_list_nat,Addition: list_nat > list_nat > list_nat,Zero: list_nat,A: set_list_nat,X2: list_nat] :
( ( khovanskii_list_nat @ G @ Addition @ Zero @ A )
=> ( ( ( size_size_list_nat @ X2 )
= ( finite_card_list_nat @ A ) )
=> ( ( useless_list_nat @ G @ Addition @ Zero @ A @ X2 )
= ( ? [X: list_nat] :
( ( member_list_nat @ X @ ( minimal_elements @ ( collect_list_nat @ ( useless_list_nat @ G @ Addition @ Zero @ A ) ) ) )
& ( pointwise_le @ X @ X2 ) ) ) ) ) ) ).
% Khovanskii.useless_iff
thf(fact_884_arg__min__inj__eq,axiom,
! [F: list_nat > list_nat,P: list_nat > $o,A2: list_nat] :
( ( inj_on3049792774292151987st_nat @ F @ ( collect_list_nat @ P ) )
=> ( ( P @ A2 )
=> ( ! [Y3: list_nat] :
( ( P @ Y3 )
=> ( ord_less_eq_list_nat @ ( F @ A2 ) @ ( F @ Y3 ) ) )
=> ( ( lattic2707248312394557148st_nat @ F @ P )
= A2 ) ) ) ) ).
% arg_min_inj_eq
thf(fact_885_arg__min__inj__eq,axiom,
! [F: list_nat > nat,P: list_nat > $o,A2: list_nat] :
( ( inj_on_list_nat_nat @ F @ ( collect_list_nat @ P ) )
=> ( ( P @ A2 )
=> ( ! [Y3: list_nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ ( F @ Y3 ) ) )
=> ( ( lattic343138300456401356at_nat @ F @ P )
= A2 ) ) ) ) ).
% arg_min_inj_eq
thf(fact_886_arg__min__inj__eq,axiom,
! [F: nat > nat,P: nat > $o,A2: nat] :
( ( inj_on_nat_nat @ F @ ( collect_nat @ P ) )
=> ( ( P @ A2 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ ( F @ Y3 ) ) )
=> ( ( lattic8739620818006775868at_nat @ F @ P )
= A2 ) ) ) ) ).
% arg_min_inj_eq
thf(fact_887_arg__min__inj__eq,axiom,
! [F: list_nat > set_list_nat,P: list_nat > $o,A2: list_nat] :
( ( inj_on8624761805129053417st_nat @ F @ ( collect_list_nat @ P ) )
=> ( ( P @ A2 )
=> ( ! [Y3: list_nat] :
( ( P @ Y3 )
=> ( ord_le6045566169113846134st_nat @ ( F @ A2 ) @ ( F @ Y3 ) ) )
=> ( ( lattic4482942756937193746st_nat @ F @ P )
= A2 ) ) ) ) ).
% arg_min_inj_eq
thf(fact_888_arg__min__inj__eq,axiom,
! [F: nat > set_list_nat,P: nat > $o,A2: nat] :
( ( inj_on2924389301302751961st_nat @ F @ ( collect_nat @ P ) )
=> ( ( P @ A2 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_le6045566169113846134st_nat @ ( F @ A2 ) @ ( F @ Y3 ) ) )
=> ( ( lattic7208376603273412482st_nat @ F @ P )
= A2 ) ) ) ) ).
% arg_min_inj_eq
thf(fact_889_arg__min__inj__eq,axiom,
! [F: list_nat > set_nat,P: list_nat > $o,A2: list_nat] :
( ( inj_on1816901372521670873et_nat @ F @ ( collect_list_nat @ P ) )
=> ( ( P @ A2 )
=> ( ! [Y3: list_nat] :
( ( P @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ ( F @ Y3 ) ) )
=> ( ( lattic6100888674492331394et_nat @ F @ P )
= A2 ) ) ) ) ).
% arg_min_inj_eq
thf(fact_890_arg__min__inj__eq,axiom,
! [F: nat > set_nat,P: nat > $o,A2: nat] :
( ( inj_on_nat_set_nat @ F @ ( collect_nat @ P ) )
=> ( ( P @ A2 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ ( F @ Y3 ) ) )
=> ( ( lattic44696799612376050et_nat @ F @ P )
= A2 ) ) ) ) ).
% arg_min_inj_eq
thf(fact_891_finite__lists__length__le,axiom,
! [A: set_list_nat,N: nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( finite8170528100393595399st_nat
@ ( collec5989764272469232197st_nat
@ ^ [Xs2: list_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs2 ) @ A )
& ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_892_finite__lists__length__le,axiom,
! [A: set_nat,N: nat] :
( ( finite_finite_nat @ A )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs2: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A )
& ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_893_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I4: nat] : ( ord_less_nat @ I4 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_894_List_Ofinite__set,axiom,
! [Xs: list_list_nat] : ( finite8100373058378681591st_nat @ ( set_list_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_895_List_Ofinite__set,axiom,
! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_896_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I4: nat] : ( ord_less_eq_nat @ I4 @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_897_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_898_card_Oempty,axiom,
( ( finite_card_list_nat @ bot_bot_set_list_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_899_card_Oinfinite,axiom,
! [A: set_list_nat] :
( ~ ( finite8100373058378681591st_nat @ A )
=> ( ( finite_card_list_nat @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_900_card_Oinfinite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_card_nat @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_901_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_902_card__0__eq,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ( finite_card_nat @ A )
= zero_zero_nat )
= ( A = bot_bot_set_nat ) ) ) ).
% card_0_eq
thf(fact_903_card__0__eq,axiom,
! [A: set_list_nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( ( ( finite_card_list_nat @ A )
= zero_zero_nat )
= ( A = bot_bot_set_list_nat ) ) ) ).
% card_0_eq
thf(fact_904_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_905_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_906_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_907_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_908_list_Oset__cases,axiom,
! [E2: list_nat,A2: list_list_nat] :
( ( member_list_nat @ E2 @ ( set_list_nat2 @ A2 ) )
=> ( ! [Z22: list_list_nat] :
( A2
!= ( cons_list_nat @ E2 @ Z22 ) )
=> ~ ! [Z1: list_nat,Z22: list_list_nat] :
( ( A2
= ( cons_list_nat @ Z1 @ Z22 ) )
=> ~ ( member_list_nat @ E2 @ ( set_list_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_909_list_Oset__cases,axiom,
! [E2: nat,A2: list_nat] :
( ( member_nat @ E2 @ ( set_nat2 @ A2 ) )
=> ( ! [Z22: list_nat] :
( A2
!= ( cons_nat @ E2 @ Z22 ) )
=> ~ ! [Z1: nat,Z22: list_nat] :
( ( A2
= ( cons_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat @ E2 @ ( set_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_910_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_911_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_912_finite__list,axiom,
! [A: set_list_nat] :
( ( finite8100373058378681591st_nat @ A )
=> ? [Xs3: list_list_nat] :
( ( set_list_nat2 @ Xs3 )
= A ) ) ).
% finite_list
thf(fact_913_finite__list,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ? [Xs3: list_nat] :
( ( set_nat2 @ Xs3 )
= A ) ) ).
% finite_list
thf(fact_914_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_915_card__le__if__inj__on__rel,axiom,
! [B3: set_list_nat,A: set_nat,R: nat > list_nat > $o] :
( ( finite8100373058378681591st_nat @ B3 )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A )
=> ? [B8: list_nat] :
( ( member_list_nat @ B8 @ B3 )
& ( R @ A6 @ B8 ) ) )
=> ( ! [A1: nat,A22: nat,B6: list_nat] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_list_nat @ B6 @ B3 )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_list_nat @ B3 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_916_card__le__if__inj__on__rel,axiom,
! [B3: set_list_nat,A: set_list_nat,R: list_nat > list_nat > $o] :
( ( finite8100373058378681591st_nat @ B3 )
=> ( ! [A6: list_nat] :
( ( member_list_nat @ A6 @ A )
=> ? [B8: list_nat] :
( ( member_list_nat @ B8 @ B3 )
& ( R @ A6 @ B8 ) ) )
=> ( ! [A1: list_nat,A22: list_nat,B6: list_nat] :
( ( member_list_nat @ A1 @ A )
=> ( ( member_list_nat @ A22 @ A )
=> ( ( member_list_nat @ B6 @ B3 )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_list_nat @ A ) @ ( finite_card_list_nat @ B3 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_917_card__le__if__inj__on__rel,axiom,
! [B3: set_nat,A: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ B3 )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A )
=> ? [B8: nat] :
( ( member_nat @ B8 @ B3 )
& ( R @ A6 @ B8 ) ) )
=> ( ! [A1: nat,A22: nat,B6: nat] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_nat @ B6 @ B3 )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B3 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_918_card__le__if__inj__on__rel,axiom,
! [B3: set_nat,A: set_list_nat,R: list_nat > nat > $o] :
( ( finite_finite_nat @ B3 )
=> ( ! [A6: list_nat] :
( ( member_list_nat @ A6 @ A )
=> ? [B8: nat] :
( ( member_nat @ B8 @ B3 )
& ( R @ A6 @ B8 ) ) )
=> ( ! [A1: list_nat,A22: list_nat,B6: nat] :
( ( member_list_nat @ A1 @ A )
=> ( ( member_list_nat @ A22 @ A )
=> ( ( member_nat @ B6 @ B3 )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_list_nat @ A ) @ ( finite_card_nat @ B3 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_919_ex__card,axiom,
! [N: nat,A: set_list_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_list_nat @ A ) )
=> ? [S3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ S3 @ A )
& ( ( finite_card_list_nat @ S3 )
= N ) ) ) ).
% ex_card
thf(fact_920_ex__card,axiom,
! [N: nat,A: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ A ) )
=> ? [S3: set_nat] :
( ( ord_less_eq_set_nat @ S3 @ A )
& ( ( finite_card_nat @ S3 )
= N ) ) ) ).
% ex_card
thf(fact_921_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_922_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_923_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_924_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_925_card__subset__eq,axiom,
! [B3: set_list_nat,A: set_list_nat] :
( ( finite8100373058378681591st_nat @ B3 )
=> ( ( ord_le6045566169113846134st_nat @ A @ B3 )
=> ( ( ( finite_card_list_nat @ A )
= ( finite_card_list_nat @ B3 ) )
=> ( A = B3 ) ) ) ) ).
% card_subset_eq
thf(fact_926_card__subset__eq,axiom,
! [B3: set_nat,A: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( ( finite_card_nat @ A )
= ( finite_card_nat @ B3 ) )
=> ( A = B3 ) ) ) ) ).
% card_subset_eq
thf(fact_927_infinite__arbitrarily__large,axiom,
! [A: set_list_nat,N: nat] :
( ~ ( finite8100373058378681591st_nat @ A )
=> ? [B9: set_list_nat] :
( ( finite8100373058378681591st_nat @ B9 )
& ( ( finite_card_list_nat @ B9 )
= N )
& ( ord_le6045566169113846134st_nat @ B9 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_928_infinite__arbitrarily__large,axiom,
! [A: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A )
=> ? [B9: set_nat] :
( ( finite_finite_nat @ B9 )
& ( ( finite_card_nat @ B9 )
= N )
& ( ord_less_eq_set_nat @ B9 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_929_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_930_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_931_size__list__pointwise,axiom,
! [Xs: list_list_nat,F: list_nat > nat,G2: list_nat > nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( G2 @ X3 ) ) )
=> ( ord_less_eq_nat @ ( size_list_list_nat @ F @ Xs ) @ ( size_list_list_nat @ G2 @ Xs ) ) ) ).
% size_list_pointwise
thf(fact_932_size__list__pointwise,axiom,
! [Xs: list_nat,F: nat > nat,G2: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( G2 @ X3 ) ) )
=> ( ord_less_eq_nat @ ( size_list_nat @ F @ Xs ) @ ( size_list_nat @ G2 @ Xs ) ) ) ).
% size_list_pointwise
thf(fact_933_size__list__estimation_H,axiom,
! [X2: list_nat,Xs: list_list_nat,Y: nat,F: list_nat > nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ( ord_less_eq_nat @ Y @ ( F @ X2 ) )
=> ( ord_less_eq_nat @ Y @ ( size_list_list_nat @ F @ Xs ) ) ) ) ).
% size_list_estimation'
thf(fact_934_size__list__estimation_H,axiom,
! [X2: nat,Xs: list_nat,Y: nat,F: nat > nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( ord_less_eq_nat @ Y @ ( F @ X2 ) )
=> ( ord_less_eq_nat @ Y @ ( size_list_nat @ F @ Xs ) ) ) ) ).
% size_list_estimation'
thf(fact_935_size__list__estimation,axiom,
! [X2: list_nat,Xs: list_list_nat,Y: nat,F: list_nat > nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ( ord_less_nat @ Y @ ( F @ X2 ) )
=> ( ord_less_nat @ Y @ ( size_list_list_nat @ F @ Xs ) ) ) ) ).
% size_list_estimation
thf(fact_936_size__list__estimation,axiom,
! [X2: nat,Xs: list_nat,Y: nat,F: nat > nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( ord_less_nat @ Y @ ( F @ X2 ) )
=> ( ord_less_nat @ Y @ ( size_list_nat @ F @ Xs ) ) ) ) ).
% size_list_estimation
thf(fact_937_Khovanskii_Oset__aA,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat] :
( ( khovanskii_nat @ G @ Addition @ Zero @ A )
=> ( ( set_nat2 @ ( aA_nat @ A ) )
= A ) ) ).
% Khovanskii.set_aA
thf(fact_938_card__eq__0__iff,axiom,
! [A: set_nat] :
( ( ( finite_card_nat @ A )
= zero_zero_nat )
= ( ( A = bot_bot_set_nat )
| ~ ( finite_finite_nat @ A ) ) ) ).
% card_eq_0_iff
thf(fact_939_card__eq__0__iff,axiom,
! [A: set_list_nat] :
( ( ( finite_card_list_nat @ A )
= zero_zero_nat )
= ( ( A = bot_bot_set_list_nat )
| ~ ( finite8100373058378681591st_nat @ A ) ) ) ).
% card_eq_0_iff
thf(fact_940_card__mono,axiom,
! [B3: set_list_nat,A: set_list_nat] :
( ( finite8100373058378681591st_nat @ B3 )
=> ( ( ord_le6045566169113846134st_nat @ A @ B3 )
=> ( ord_less_eq_nat @ ( finite_card_list_nat @ A ) @ ( finite_card_list_nat @ B3 ) ) ) ) ).
% card_mono
thf(fact_941_card__mono,axiom,
! [B3: set_nat,A: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B3 ) ) ) ) ).
% card_mono
thf(fact_942_card__seteq,axiom,
! [B3: set_list_nat,A: set_list_nat] :
( ( finite8100373058378681591st_nat @ B3 )
=> ( ( ord_le6045566169113846134st_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_list_nat @ B3 ) @ ( finite_card_list_nat @ A ) )
=> ( A = B3 ) ) ) ) ).
% card_seteq
thf(fact_943_card__seteq,axiom,
! [B3: set_nat,A: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B3 ) @ ( finite_card_nat @ A ) )
=> ( A = B3 ) ) ) ) ).
% card_seteq
thf(fact_944_exists__subset__between,axiom,
! [A: set_list_nat,N: nat,C4: set_list_nat] :
( ( ord_less_eq_nat @ ( finite_card_list_nat @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_list_nat @ C4 ) )
=> ( ( ord_le6045566169113846134st_nat @ A @ C4 )
=> ( ( finite8100373058378681591st_nat @ C4 )
=> ? [B9: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B9 )
& ( ord_le6045566169113846134st_nat @ B9 @ C4 )
& ( ( finite_card_list_nat @ B9 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_945_exists__subset__between,axiom,
! [A: set_nat,N: nat,C4: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C4 ) )
=> ( ( ord_less_eq_set_nat @ A @ C4 )
=> ( ( finite_finite_nat @ C4 )
=> ? [B9: set_nat] :
( ( ord_less_eq_set_nat @ A @ B9 )
& ( ord_less_eq_set_nat @ B9 @ C4 )
& ( ( finite_card_nat @ B9 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_946_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_947_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_948_finite__if__finite__subsets__card__bdd,axiom,
! [F3: set_list_nat,C4: nat] :
( ! [G4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ G4 @ F3 )
=> ( ( finite8100373058378681591st_nat @ G4 )
=> ( ord_less_eq_nat @ ( finite_card_list_nat @ G4 ) @ C4 ) ) )
=> ( ( finite8100373058378681591st_nat @ F3 )
& ( ord_less_eq_nat @ ( finite_card_list_nat @ F3 ) @ C4 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_949_finite__if__finite__subsets__card__bdd,axiom,
! [F3: set_nat,C4: nat] :
( ! [G4: set_nat] :
( ( ord_less_eq_set_nat @ G4 @ F3 )
=> ( ( finite_finite_nat @ G4 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G4 ) @ C4 ) ) )
=> ( ( finite_finite_nat @ F3 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F3 ) @ C4 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_950_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_951_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_952_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_953_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_954_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_955_card__ge__0__finite,axiom,
! [A: set_list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_list_nat @ A ) )
=> ( finite8100373058378681591st_nat @ A ) ) ).
% card_ge_0_finite
thf(fact_956_card__ge__0__finite,axiom,
! [A: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A ) )
=> ( finite_finite_nat @ A ) ) ).
% card_ge_0_finite
thf(fact_957_card__le__sym__Diff,axiom,
! [A: set_list_nat,B3: set_list_nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( ( finite8100373058378681591st_nat @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_list_nat @ A ) @ ( finite_card_list_nat @ B3 ) )
=> ( ord_less_eq_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A @ B3 ) ) @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ B3 @ A ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_958_card__le__sym__Diff,axiom,
! [A: set_nat,B3: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B3 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B3 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B3 @ A ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_959_card__less__sym__Diff,axiom,
! [A: set_list_nat,B3: set_list_nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( ( finite8100373058378681591st_nat @ B3 )
=> ( ( ord_less_nat @ ( finite_card_list_nat @ A ) @ ( finite_card_list_nat @ B3 ) )
=> ( ord_less_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A @ B3 ) ) @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ B3 @ A ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_960_card__less__sym__Diff,axiom,
! [A: set_nat,B3: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B3 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B3 ) )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B3 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B3 @ A ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_961_psubset__card__mono,axiom,
! [B3: set_list_nat,A: set_list_nat] :
( ( finite8100373058378681591st_nat @ B3 )
=> ( ( ord_le1190675801316882794st_nat @ A @ B3 )
=> ( ord_less_nat @ ( finite_card_list_nat @ A ) @ ( finite_card_list_nat @ B3 ) ) ) ) ).
% psubset_card_mono
thf(fact_962_psubset__card__mono,axiom,
! [B3: set_nat,A: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_set_nat @ A @ B3 )
=> ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B3 ) ) ) ) ).
% psubset_card_mono
thf(fact_963_card__less__Suc2,axiom,
! [M7: set_nat,I: nat] :
( ~ ( member_nat @ zero_zero_nat @ M7 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( member_nat @ ( suc @ K2 ) @ M7 )
& ( ord_less_nat @ K2 @ I ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( member_nat @ K2 @ M7 )
& ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_964_card__less__Suc,axiom,
! [M7: set_nat,I: nat] :
( ( member_nat @ zero_zero_nat @ M7 )
=> ( ( suc
@ ( finite_card_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( member_nat @ ( suc @ K2 ) @ M7 )
& ( ord_less_nat @ K2 @ I ) ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( member_nat @ K2 @ M7 )
& ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_965_card__less,axiom,
! [M7: set_nat,I: nat] :
( ( member_nat @ zero_zero_nat @ M7 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K2: nat] :
( ( member_nat @ K2 @ M7 )
& ( ord_less_nat @ K2 @ ( suc @ I ) ) ) ) )
!= zero_zero_nat ) ) ).
% card_less
thf(fact_966_card__gt__0__iff,axiom,
! [A: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A ) )
= ( ( A != bot_bot_set_nat )
& ( finite_finite_nat @ A ) ) ) ).
% card_gt_0_iff
thf(fact_967_card__gt__0__iff,axiom,
! [A: set_list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_list_nat @ A ) )
= ( ( A != bot_bot_set_list_nat )
& ( finite8100373058378681591st_nat @ A ) ) ) ).
% card_gt_0_iff
thf(fact_968_card__le__Suc0__iff__eq,axiom,
! [A: set_list_nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( ( ord_less_eq_nat @ ( finite_card_list_nat @ A ) @ ( suc @ zero_zero_nat ) )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ A )
=> ! [Y5: list_nat] :
( ( member_list_nat @ Y5 @ A )
=> ( X = Y5 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_969_card__le__Suc0__iff__eq,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( suc @ zero_zero_nat ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ! [Y5: nat] :
( ( member_nat @ Y5 @ A )
=> ( X = Y5 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_970_card__Diff__subset,axiom,
! [B3: set_list_nat,A: set_list_nat] :
( ( finite8100373058378681591st_nat @ B3 )
=> ( ( ord_le6045566169113846134st_nat @ B3 @ A )
=> ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A @ B3 ) )
= ( minus_minus_nat @ ( finite_card_list_nat @ A ) @ ( finite_card_list_nat @ B3 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_971_card__Diff__subset,axiom,
! [B3: set_nat,A: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ B3 @ A )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ B3 ) )
= ( minus_minus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B3 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_972_card__psubset,axiom,
! [B3: set_list_nat,A: set_list_nat] :
( ( finite8100373058378681591st_nat @ B3 )
=> ( ( ord_le6045566169113846134st_nat @ A @ B3 )
=> ( ( ord_less_nat @ ( finite_card_list_nat @ A ) @ ( finite_card_list_nat @ B3 ) )
=> ( ord_le1190675801316882794st_nat @ A @ B3 ) ) ) ) ).
% card_psubset
thf(fact_973_card__psubset,axiom,
! [B3: set_nat,A: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B3 ) )
=> ( ord_less_set_nat @ A @ B3 ) ) ) ) ).
% card_psubset
thf(fact_974_diff__card__le__card__Diff,axiom,
! [B3: set_list_nat,A: set_list_nat] :
( ( finite8100373058378681591st_nat @ B3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_list_nat @ A ) @ ( finite_card_list_nat @ B3 ) ) @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A @ B3 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_975_diff__card__le__card__Diff,axiom,
! [B3: set_nat,A: set_nat] :
( ( finite_finite_nat @ B3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B3 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B3 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_976_finite__lists__length__eq,axiom,
! [A: set_list_nat,N: nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( finite8170528100393595399st_nat
@ ( collec5989764272469232197st_nat
@ ^ [Xs2: list_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs2 ) @ A )
& ( ( size_s3023201423986296836st_nat @ Xs2 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_977_finite__lists__length__eq,axiom,
! [A: set_nat,N: nat] :
( ( finite_finite_nat @ A )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs2: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A )
& ( ( size_size_list_nat @ Xs2 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_978_Khovanskii_Ouseless__leq__useless,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,X2: list_nat,Y: list_nat] :
( ( khovanskii_nat @ G @ Addition @ Zero @ A )
=> ( ( useless_nat @ G @ Addition @ Zero @ A @ X2 )
=> ( ( pointwise_le @ X2 @ Y )
=> ( ( ( size_size_list_nat @ X2 )
= ( finite_card_nat @ A ) )
=> ( useless_nat @ G @ Addition @ Zero @ A @ Y ) ) ) ) ) ).
% Khovanskii.useless_leq_useless
thf(fact_979_Khovanskii_Ouseless__leq__useless,axiom,
! [G: set_list_nat,Addition: list_nat > list_nat > list_nat,Zero: list_nat,A: set_list_nat,X2: list_nat,Y: list_nat] :
( ( khovanskii_list_nat @ G @ Addition @ Zero @ A )
=> ( ( useless_list_nat @ G @ Addition @ Zero @ A @ X2 )
=> ( ( pointwise_le @ X2 @ Y )
=> ( ( ( size_size_list_nat @ X2 )
= ( finite_card_list_nat @ A ) )
=> ( useless_list_nat @ G @ Addition @ Zero @ A @ Y ) ) ) ) ) ).
% Khovanskii.useless_leq_useless
thf(fact_980_Khovanskii_Oalpha__plus,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,X2: list_nat,Y: list_nat] :
( ( khovanskii_nat @ G @ Addition @ Zero @ A )
=> ( ( ( size_size_list_nat @ X2 )
= ( finite_card_nat @ A ) )
=> ( ( ( size_size_list_nat @ Y )
= ( finite_card_nat @ A ) )
=> ( ( alpha_nat @ G @ Addition @ Zero @ A @ ( plus_plus_list_nat @ X2 @ Y ) )
= ( Addition @ ( alpha_nat @ G @ Addition @ Zero @ A @ X2 ) @ ( alpha_nat @ G @ Addition @ Zero @ A @ Y ) ) ) ) ) ) ).
% Khovanskii.alpha_plus
thf(fact_981_Khovanskii_Oalpha__plus,axiom,
! [G: set_list_nat,Addition: list_nat > list_nat > list_nat,Zero: list_nat,A: set_list_nat,X2: list_nat,Y: list_nat] :
( ( khovanskii_list_nat @ G @ Addition @ Zero @ A )
=> ( ( ( size_size_list_nat @ X2 )
= ( finite_card_list_nat @ A ) )
=> ( ( ( size_size_list_nat @ Y )
= ( finite_card_list_nat @ A ) )
=> ( ( alpha_list_nat @ G @ Addition @ Zero @ A @ ( plus_plus_list_nat @ X2 @ Y ) )
= ( Addition @ ( alpha_list_nat @ G @ Addition @ Zero @ A @ X2 ) @ ( alpha_list_nat @ G @ Addition @ Zero @ A @ Y ) ) ) ) ) ) ).
% Khovanskii.alpha_plus
thf(fact_982_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_983_finsets__def,axiom,
( finsets_list_nat
= ( ^ [A5: set_list_nat,N2: nat] :
( collect_set_list_nat
@ ^ [N6: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ N6 @ A5 )
& ( ( finite_card_list_nat @ N6 )
= N2 ) ) ) ) ) ).
% finsets_def
thf(fact_984_finsets__def,axiom,
( finsets_nat
= ( ^ [A5: set_nat,N2: nat] :
( collect_set_nat
@ ^ [N6: set_nat] :
( ( ord_less_eq_set_nat @ N6 @ A5 )
& ( ( finite_card_nat @ N6 )
= N2 ) ) ) ) ) ).
% finsets_def
thf(fact_985_Khovanskii_Oalpha__list__incr,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,A2: nat,X2: list_nat,N: nat] :
( ( khovanskii_nat @ G @ Addition @ Zero @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ( member_list_nat @ X2 @ ( length_sum_set @ ( finite_card_nat @ A ) @ N ) )
=> ( ( alpha_nat @ G @ Addition @ Zero @ A @ ( list_incr @ ( counta4844910239362777137on_nat @ A @ A2 ) @ X2 ) )
= ( Addition @ A2 @ ( alpha_nat @ G @ Addition @ Zero @ A @ X2 ) ) ) ) ) ) ).
% Khovanskii.alpha_list_incr
thf(fact_986_Khovanskii_Oalpha__list__incr,axiom,
! [G: set_list_nat,Addition: list_nat > list_nat > list_nat,Zero: list_nat,A: set_list_nat,A2: list_nat,X2: list_nat,N: nat] :
( ( khovanskii_list_nat @ G @ Addition @ Zero @ A )
=> ( ( member_list_nat @ A2 @ A )
=> ( ( member_list_nat @ X2 @ ( length_sum_set @ ( finite_card_list_nat @ A ) @ N ) )
=> ( ( alpha_list_nat @ G @ Addition @ Zero @ A @ ( list_incr @ ( counta9103016383634126529st_nat @ A @ A2 ) @ X2 ) )
= ( Addition @ A2 @ ( alpha_list_nat @ G @ Addition @ Zero @ A @ X2 ) ) ) ) ) ) ).
% Khovanskii.alpha_list_incr
thf(fact_987_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_988_length__augmentum,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( augmentum @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_augmentum
thf(fact_989_Khovanskii_Oidx__less__cardA,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A: set_nat,A2: nat] :
( ( khovanskii_nat @ G @ Addition @ Zero @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ord_less_nat @ ( counta4844910239362777137on_nat @ A @ A2 ) @ ( finite_card_nat @ A ) ) ) ) ).
% Khovanskii.idx_less_cardA
thf(fact_990_Khovanskii_Oidx__less__cardA,axiom,
! [G: set_list_nat,Addition: list_nat > list_nat > list_nat,Zero: list_nat,A: set_list_nat,A2: list_nat] :
( ( khovanskii_list_nat @ G @ Addition @ Zero @ A )
=> ( ( member_list_nat @ A2 @ A )
=> ( ord_less_nat @ ( counta9103016383634126529st_nat @ A @ A2 ) @ ( finite_card_list_nat @ A ) ) ) ) ).
% Khovanskii.idx_less_cardA
thf(fact_991_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_992_dementum__nonzero,axiom,
! [Ns: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Ns )
=> ( ~ ( member_nat @ zero_zero_nat @ ( set_nat2 @ Ns ) )
=> ~ ( member_nat @ zero_zero_nat @ ( set_nat2 @ ( dementum @ Ns ) ) ) ) ) ).
% dementum_nonzero
thf(fact_993_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_994_sorted__augmentum,axiom,
! [Ns: list_nat] :
( ~ ( member_nat @ zero_zero_nat @ ( set_nat2 @ Ns ) )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( augmentum @ Ns ) ) ) ).
% sorted_augmentum
thf(fact_995_map__ident,axiom,
( ( map_nat_nat
@ ^ [X: nat] : X )
= ( ^ [Xs2: list_nat] : Xs2 ) ) ).
% map_ident
thf(fact_996_map__eq__conv,axiom,
! [F: nat > nat,Xs: list_nat,G2: nat > nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G2 @ Xs ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( F @ X )
= ( G2 @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_997_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_998_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_999_sorted__map__plus__iff,axiom,
! [A2: nat,Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ ( plus_plus_nat @ A2 ) @ Xs ) )
= ( sorted_wrt_nat @ ord_less_eq_nat @ Xs ) ) ).
% sorted_map_plus_iff
thf(fact_1000_augmentum__dementum,axiom,
! [Ns: list_nat] :
( ~ ( member_nat @ zero_zero_nat @ ( set_nat2 @ Ns ) )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ns )
=> ( ( augmentum @ ( dementum @ Ns ) )
= Ns ) ) ) ).
% augmentum_dementum
thf(fact_1001_strict__sorted__equal,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs )
=> ( ( sorted_wrt_nat @ ord_less_nat @ Ys )
=> ( ( ( set_nat2 @ Ys )
= ( set_nat2 @ Xs ) )
=> ( Ys = Xs ) ) ) ) ).
% strict_sorted_equal
thf(fact_1002_list_Omap__cong,axiom,
! [X2: list_nat,Ya: list_nat,F: nat > nat,G2: nat > nat] :
( ( X2 = Ya )
=> ( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( set_nat2 @ Ya ) )
=> ( ( F @ Z2 )
= ( G2 @ Z2 ) ) )
=> ( ( map_nat_nat @ F @ X2 )
= ( map_nat_nat @ G2 @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_1003_list_Omap__cong0,axiom,
! [X2: list_nat,F: nat > nat,G2: nat > nat] :
( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( set_nat2 @ X2 ) )
=> ( ( F @ Z2 )
= ( G2 @ Z2 ) ) )
=> ( ( map_nat_nat @ F @ X2 )
= ( map_nat_nat @ G2 @ X2 ) ) ) ).
% list.map_cong0
thf(fact_1004_list_Oinj__map__strong,axiom,
! [X2: list_nat,Xa2: list_nat,F: nat > nat,Fa: nat > nat] :
( ! [Z2: nat,Za: nat] :
( ( member_nat @ Z2 @ ( set_nat2 @ X2 ) )
=> ( ( member_nat @ Za @ ( set_nat2 @ Xa2 ) )
=> ( ( ( F @ Z2 )
= ( Fa @ Za ) )
=> ( Z2 = Za ) ) ) )
=> ( ( ( map_nat_nat @ F @ X2 )
= ( map_nat_nat @ Fa @ Xa2 ) )
=> ( X2 = Xa2 ) ) ) ).
% list.inj_map_strong
thf(fact_1005_list_Omap__ident__strong,axiom,
! [T: list_list_nat,F: list_nat > list_nat] :
( ! [Z2: list_nat] :
( ( member_list_nat @ Z2 @ ( set_list_nat2 @ T ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( map_li7225945977422193158st_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_1006_list_Omap__ident__strong,axiom,
! [T: list_nat,F: nat > nat] :
( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( set_nat2 @ T ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( map_nat_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_1007_map__ext,axiom,
! [Xs: list_nat,F: nat > nat,G2: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G2 @ Xs ) ) ) ).
% map_ext
thf(fact_1008_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_1009_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_1010_map__cong,axiom,
! [Xs: list_nat,Ys: list_nat,F: nat > nat,G2: nat > nat] :
( ( Xs = Ys )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
=> ( ( F @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G2 @ Ys ) ) ) ) ).
% map_cong
thf(fact_1011_ex__map__conv,axiom,
! [Ys: list_nat,F: nat > nat] :
( ( ? [Xs2: list_nat] :
( Ys
= ( map_nat_nat @ F @ Xs2 ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Ys ) )
=> ? [Y5: nat] :
( X
= ( F @ Y5 ) ) ) ) ) ).
% ex_map_conv
thf(fact_1012_sorted__wrt__mono__rel,axiom,
! [Xs: list_list_nat,P: list_nat > list_nat > $o,Q: list_nat > list_nat > $o] :
( ! [X3: list_nat,Y3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( ( member_list_nat @ Y3 @ ( set_list_nat2 @ Xs ) )
=> ( ( P @ X3 @ Y3 )
=> ( Q @ X3 @ Y3 ) ) ) )
=> ( ( sorted_wrt_list_nat @ P @ Xs )
=> ( sorted_wrt_list_nat @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_1013_sorted__wrt__mono__rel,axiom,
! [Xs: list_nat,P: nat > nat > $o,Q: nat > nat > $o] :
( ! [X3: nat,Y3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( member_nat @ Y3 @ ( set_nat2 @ Xs ) )
=> ( ( P @ X3 @ Y3 )
=> ( Q @ X3 @ Y3 ) ) ) )
=> ( ( sorted_wrt_nat @ P @ Xs )
=> ( sorted_wrt_nat @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_1014_sum__list__addf,axiom,
! [F: nat > nat,G2: nat > nat,Xs: list_nat] :
( ( groups4561878855575611511st_nat
@ ( map_nat_nat
@ ^ [X: nat] : ( plus_plus_nat @ ( F @ X ) @ ( G2 @ X ) )
@ Xs ) )
= ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ F @ Xs ) ) @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ G2 @ Xs ) ) ) ) ).
% sum_list_addf
thf(fact_1015_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs: list_nat,G2: nat > nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G2 @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1016_sorted__wrt__true,axiom,
! [Xs: list_nat] :
( sorted_wrt_nat
@ ^ [Uu: nat,Uv: nat] : $true
@ Xs ) ).
% sorted_wrt_true
thf(fact_1017_sorted__wrt__map,axiom,
! [R3: nat > nat > $o,F: nat > nat,Xs: list_nat] :
( ( sorted_wrt_nat @ R3 @ ( map_nat_nat @ F @ Xs ) )
= ( sorted_wrt_nat
@ ^ [X: nat,Y5: nat] : ( R3 @ ( F @ X ) @ ( F @ Y5 ) )
@ Xs ) ) ).
% sorted_wrt_map
thf(fact_1018_list_Omap__ident,axiom,
! [T: list_nat] :
( ( map_nat_nat
@ ^ [X: nat] : X
@ T )
= T ) ).
% list.map_ident
thf(fact_1019_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_1020_Cons__eq__map__D,axiom,
! [X2: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_nat_nat @ F @ Ys ) )
=> ? [Z2: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z2 @ Zs2 ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1021_map__eq__Cons__D,axiom,
! [F: nat > nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
=> ? [Z2: nat,Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_nat_nat @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_1022_Cons__eq__map__conv,axiom,
! [X2: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_nat_nat @ F @ Ys ) )
= ( ? [Z6: nat,Zs3: list_nat] :
( ( Ys
= ( cons_nat @ Z6 @ Zs3 ) )
& ( X2
= ( F @ Z6 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1023_map__eq__Cons__conv,axiom,
! [F: nat > nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
= ( ? [Z6: nat,Zs3: list_nat] :
( ( Xs
= ( cons_nat @ Z6 @ Zs3 ) )
& ( ( F @ Z6 )
= Y )
& ( ( map_nat_nat @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1024_sorted__map,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ Xs ) )
= ( sorted_wrt_nat
@ ^ [X: nat,Y5: nat] : ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y5 ) )
@ Xs ) ) ).
% sorted_map
thf(fact_1025_sorted2,axiom,
! [X2: nat,Y: nat,Zs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X2 @ ( cons_nat @ Y @ Zs ) ) )
= ( ( ord_less_eq_nat @ X2 @ Y )
& ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ Y @ Zs ) ) ) ) ).
% sorted2
thf(fact_1026_strict__sorted__imp__sorted,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ Xs ) ) ).
% strict_sorted_imp_sorted
thf(fact_1027_sorted__simps_I2_J,axiom,
! [X2: nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X2 @ Ys ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Ys ) )
=> ( ord_less_eq_nat @ X2 @ X ) )
& ( sorted_wrt_nat @ ord_less_eq_nat @ Ys ) ) ) ).
% sorted_simps(2)
thf(fact_1028_strict__sorted__simps_I2_J,axiom,
! [X2: nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ ( cons_nat @ X2 @ Ys ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Ys ) )
=> ( ord_less_nat @ X2 @ X ) )
& ( sorted_wrt_nat @ ord_less_nat @ Ys ) ) ) ).
% strict_sorted_simps(2)
thf(fact_1029_sum__list__mono,axiom,
! [Xs: list_list_nat,F: list_nat > nat,G2: list_nat > nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( G2 @ X3 ) ) )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ F @ Xs ) ) @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ G2 @ Xs ) ) ) ) ).
% sum_list_mono
thf(fact_1030_sum__list__mono,axiom,
! [Xs: list_nat,F: nat > nat,G2: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( G2 @ X3 ) ) )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ F @ Xs ) ) @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ G2 @ Xs ) ) ) ) ).
% sum_list_mono
thf(fact_1031_size__list__conv__sum__list,axiom,
( size_list_nat
= ( ^ [F2: nat > nat,Xs2: list_nat] : ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ F2 @ Xs2 ) ) @ ( size_size_list_nat @ Xs2 ) ) ) ) ).
% size_list_conv_sum_list
thf(fact_1032_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_1033_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
! [A: set_list_nat] :
( ( finite8100373058378681591st_nat @ A )
=> ~ ! [L2: list_list_nat] :
( ( sorted_wrt_list_nat @ ord_less_list_nat @ L2 )
=> ( ( ( set_list_nat2 @ L2 )
= A )
=> ( ( size_s3023201423986296836st_nat @ L2 )
!= ( finite_card_list_nat @ A ) ) ) ) ) ).
% sorted_list_of_set.finite_set_strict_sorted
thf(fact_1034_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ~ ! [L2: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ L2 )
=> ( ( ( set_nat2 @ L2 )
= A )
=> ( ( size_size_list_nat @ L2 )
!= ( finite_card_nat @ A ) ) ) ) ) ).
% sorted_list_of_set.finite_set_strict_sorted
thf(fact_1035_augmentum_Oelims,axiom,
! [X2: list_nat,Y: list_nat] :
( ( ( augmentum @ X2 )
= Y )
=> ( ( ( X2 = nil_nat )
=> ( Y != nil_nat ) )
=> ~ ! [N4: nat,Ns2: list_nat] :
( ( X2
= ( cons_nat @ N4 @ Ns2 ) )
=> ( Y
!= ( cons_nat @ N4 @ ( map_nat_nat @ ( plus_plus_nat @ N4 ) @ ( augmentum @ Ns2 ) ) ) ) ) ) ) ).
% augmentum.elims
thf(fact_1036_folding__insort__key_Ofinite__set__strict__sorted,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S2: set_list_nat,F: list_nat > nat,A: set_list_nat] :
( ( foldin951631548397865559st_nat @ Less_eq @ Less @ S2 @ F )
=> ( ( ord_le6045566169113846134st_nat @ A @ S2 )
=> ( ( finite8100373058378681591st_nat @ A )
=> ~ ! [L2: list_list_nat] :
( ( sorted_wrt_nat @ Less @ ( map_list_nat_nat @ F @ L2 ) )
=> ( ( ( set_list_nat2 @ L2 )
= A )
=> ( ( size_s3023201423986296836st_nat @ L2 )
!= ( finite_card_list_nat @ A ) ) ) ) ) ) ) ).
% folding_insort_key.finite_set_strict_sorted
thf(fact_1037_folding__insort__key_Ofinite__set__strict__sorted,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S2: set_nat,F: nat > nat,A: set_nat] :
( ( foldin8133931898133206727at_nat @ Less_eq @ Less @ S2 @ F )
=> ( ( ord_less_eq_set_nat @ A @ S2 )
=> ( ( finite_finite_nat @ A )
=> ~ ! [L2: list_nat] :
( ( sorted_wrt_nat @ Less @ ( map_nat_nat @ F @ L2 ) )
=> ( ( ( set_nat2 @ L2 )
= A )
=> ( ( size_size_list_nat @ L2 )
!= ( finite_card_nat @ A ) ) ) ) ) ) ) ).
% folding_insort_key.finite_set_strict_sorted
thf(fact_1038_augmentum_Osimps_I1_J,axiom,
( ( augmentum @ nil_nat )
= nil_nat ) ).
% augmentum.simps(1)
thf(fact_1039_list_Omap__disc__iff,axiom,
! [F: nat > nat,A2: list_nat] :
( ( ( map_nat_nat @ F @ A2 )
= nil_nat )
= ( A2 = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_1040_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_1041_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_1042_minus__Nil,axiom,
! [Xs: list_nat] :
( ( minus_minus_list_nat @ nil_nat @ Xs )
= nil_nat ) ).
% minus_Nil
thf(fact_1043_pointwise__le__Nil2,axiom,
! [X2: list_nat] :
( ( pointwise_le @ X2 @ nil_nat )
= ( X2 = nil_nat ) ) ).
% pointwise_le_Nil2
thf(fact_1044_pointwise__le__Nil,axiom,
! [X2: list_nat] :
( ( pointwise_le @ nil_nat @ X2 )
= ( X2 = nil_nat ) ) ).
% pointwise_le_Nil
thf(fact_1045_plus__Nil,axiom,
! [Xs: list_nat] :
( ( plus_plus_list_nat @ nil_nat @ Xs )
= nil_nat ) ).
% plus_Nil
thf(fact_1046_list__incr__Nil,axiom,
! [I: nat] :
( ( list_incr @ I @ nil_nat )
= nil_nat ) ).
% list_incr_Nil
thf(fact_1047_dementum__Nil,axiom,
( ( dementum @ nil_nat )
= nil_nat ) ).
% dementum_Nil
thf(fact_1048_le__Nil,axiom,
! [X2: list_nat] :
( ( ord_less_eq_list_nat @ X2 @ nil_nat )
= ( X2 = nil_nat ) ) ).
% le_Nil
thf(fact_1049_set__empty,axiom,
! [Xs: list_nat] :
( ( ( set_nat2 @ Xs )
= bot_bot_set_nat )
= ( Xs = nil_nat ) ) ).
% set_empty
thf(fact_1050_set__empty,axiom,
! [Xs: list_list_nat] :
( ( ( set_list_nat2 @ Xs )
= bot_bot_set_list_nat )
= ( Xs = nil_list_nat ) ) ).
% set_empty
thf(fact_1051_set__empty2,axiom,
! [Xs: list_nat] :
( ( bot_bot_set_nat
= ( set_nat2 @ Xs ) )
= ( Xs = nil_nat ) ) ).
% set_empty2
thf(fact_1052_set__empty2,axiom,
! [Xs: list_list_nat] :
( ( bot_bot_set_list_nat
= ( set_list_nat2 @ Xs ) )
= ( Xs = nil_list_nat ) ) ).
% set_empty2
thf(fact_1053_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_1054_sum__list_ONil,axiom,
( ( groups4561878855575611511st_nat @ nil_nat )
= zero_zero_nat ) ).
% sum_list.Nil
thf(fact_1055_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_1056_strict__sorted__simps_I1_J,axiom,
sorted_wrt_nat @ ord_less_nat @ nil_nat ).
% strict_sorted_simps(1)
thf(fact_1057_sorted__wrt1,axiom,
! [P: nat > nat > $o,X2: nat] : ( sorted_wrt_nat @ P @ ( cons_nat @ X2 @ nil_nat ) ) ).
% sorted_wrt1
thf(fact_1058_list_Osimps_I8_J,axiom,
! [F: nat > nat] :
( ( map_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_1059_sorted__wrt_Osimps_I1_J,axiom,
! [P: nat > nat > $o] : ( sorted_wrt_nat @ P @ nil_nat ) ).
% sorted_wrt.simps(1)
thf(fact_1060_sorted0,axiom,
sorted_wrt_nat @ ord_less_eq_nat @ nil_nat ).
% sorted0
thf(fact_1061_Nil__le__Cons,axiom,
! [X2: list_nat] : ( ord_less_eq_list_nat @ nil_nat @ X2 ) ).
% Nil_le_Cons
thf(fact_1062_list__encode_Ocases,axiom,
! [X2: list_nat] :
( ( X2 != nil_nat )
=> ~ ! [X3: nat,Xs3: list_nat] :
( X2
!= ( cons_nat @ X3 @ Xs3 ) ) ) ).
% list_encode.cases
thf(fact_1063_list__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X3: nat] : ( P @ ( cons_nat @ X3 @ nil_nat ) )
=> ( ! [X3: nat,Xs3: list_nat] :
( ( Xs3 != nil_nat )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_nat @ X3 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_1064_list__induct2_H,axiom,
! [P: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X3 @ Xs3 ) @ nil_nat )
=> ( ! [Y3: nat,Ys4: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y3 @ Ys4 ) )
=> ( ! [X3: nat,Xs3: list_nat,Y3: nat,Ys4: list_nat] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_1065_neq__Nil__conv,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
= ( ? [Y5: nat,Ys2: list_nat] :
( Xs
= ( cons_nat @ Y5 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_1066_remdups__adj_Ocases,axiom,
! [X2: list_nat] :
( ( X2 != nil_nat )
=> ( ! [X3: nat] :
( X2
!= ( cons_nat @ X3 @ nil_nat ) )
=> ~ ! [X3: nat,Y3: nat,Xs3: list_nat] :
( X2
!= ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_1067_min__list_Ocases,axiom,
! [X2: list_nat] :
( ! [X3: nat,Xs3: list_nat] :
( X2
!= ( cons_nat @ X3 @ Xs3 ) )
=> ( X2 = nil_nat ) ) ).
% min_list.cases
thf(fact_1068_list_Oexhaust,axiom,
! [Y: list_nat] :
( ( Y != nil_nat )
=> ~ ! [X212: nat,X223: list_nat] :
( Y
!= ( cons_nat @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_1069_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X222: list_nat] :
( ( List
= ( cons_nat @ X21 @ X222 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_1070_list_Odistinct_I1_J,axiom,
! [X21: nat,X222: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_1071_transpose_Ocases,axiom,
! [X2: list_list_nat] :
( ( X2 != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X2
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X3: nat,Xs3: list_nat,Xss: list_list_nat] :
( X2
!= ( cons_list_nat @ ( cons_nat @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_1072_pointwise__less__Nil2,axiom,
! [X2: list_nat] :
~ ( pointwise_less @ X2 @ nil_nat ) ).
% pointwise_less_Nil2
thf(fact_1073_pointwise__less__Nil,axiom,
! [X2: list_nat] :
~ ( pointwise_less @ nil_nat @ X2 ) ).
% pointwise_less_Nil
thf(fact_1074_folding__insort__key_Oinj__on,axiom,
! [Less_eq: list_nat > list_nat > $o,Less: list_nat > list_nat > $o,S2: set_list_nat,F: list_nat > list_nat] :
( ( foldin1968479528632951399st_nat @ Less_eq @ Less @ S2 @ F )
=> ( inj_on3049792774292151987st_nat @ F @ S2 ) ) ).
% folding_insort_key.inj_on
thf(fact_1075_folding__insort__key_Oinj__on,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S2: set_nat,F: nat > nat] :
( ( foldin8133931898133206727at_nat @ Less_eq @ Less @ S2 @ F )
=> ( inj_on_nat_nat @ F @ S2 ) ) ).
% folding_insort_key.inj_on
thf(fact_1076_not__less__Nil,axiom,
! [X2: list_nat] :
~ ( ord_less_list_nat @ X2 @ nil_nat ) ).
% not_less_Nil
thf(fact_1077_bot__list__def,axiom,
bot_bot_list_nat = nil_nat ).
% bot_list_def
thf(fact_1078_list__induct4,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ws: list_nat,P: list_nat > list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs3: list_nat,Y3: nat,Ys4: list_nat,Z2: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1079_list__induct3,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat,P: list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs3: list_nat,Y3: nat,Ys4: list_nat,Z2: nat,Zs2: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 )
=> ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_1080_list__induct2,axiom,
! [Xs: list_nat,Ys: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs3: list_nat,Y3: nat,Ys4: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1081_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_1082_empty__set,axiom,
( bot_bot_set_nat
= ( set_nat2 @ nil_nat ) ) ).
% empty_set
thf(fact_1083_empty__set,axiom,
( bot_bot_set_list_nat
= ( set_list_nat2 @ nil_list_nat ) ) ).
% empty_set
thf(fact_1084_list_Osize__gen_I1_J,axiom,
! [X2: nat > nat] :
( ( size_list_nat @ X2 @ nil_nat )
= zero_zero_nat ) ).
% list.size_gen(1)
thf(fact_1085_Nil__less__Cons,axiom,
! [A2: nat,X2: list_nat] : ( ord_less_list_nat @ nil_nat @ ( cons_nat @ A2 @ X2 ) ) ).
% Nil_less_Cons
thf(fact_1086_sorted1,axiom,
! [X2: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X2 @ nil_nat ) ) ).
% sorted1
thf(fact_1087_sum__list__strict__mono,axiom,
! [Xs: list_list_nat,F: list_nat > nat,G2: list_nat > nat] :
( ( Xs != nil_list_nat )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( ord_less_nat @ ( F @ X3 ) @ ( G2 @ X3 ) ) )
=> ( ord_less_nat @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ F @ Xs ) ) @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ G2 @ Xs ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_1088_sum__list__strict__mono,axiom,
! [Xs: list_nat,F: nat > nat,G2: nat > nat] :
( ( Xs != nil_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ ( F @ X3 ) @ ( G2 @ X3 ) ) )
=> ( ord_less_nat @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ F @ Xs ) ) @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ G2 @ Xs ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_1089_sum__list__dementum,axiom,
! [Xs: list_nat,N: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( append_nat @ Xs @ ( cons_nat @ N @ nil_nat ) ) )
=> ( ( groups4561878855575611511st_nat @ ( dementum @ ( append_nat @ Xs @ ( cons_nat @ N @ nil_nat ) ) ) )
= N ) ) ).
% sum_list_dementum
thf(fact_1090_sorted__imp__pointwise,axiom,
! [Xs: list_nat,N: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( append_nat @ Xs @ ( cons_nat @ N @ nil_nat ) ) )
=> ( pointwise_le @ ( cons_nat @ zero_zero_nat @ Xs ) @ ( append_nat @ Xs @ ( cons_nat @ N @ nil_nat ) ) ) ) ).
% sorted_imp_pointwise
thf(fact_1091_append_Oassoc,axiom,
! [A2: list_nat,B: list_nat,C: list_nat] :
( ( append_nat @ ( append_nat @ A2 @ B ) @ C )
= ( append_nat @ A2 @ ( append_nat @ B @ C ) ) ) ).
% append.assoc
thf(fact_1092_append__assoc,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( append_nat @ ( append_nat @ Xs @ Ys ) @ Zs )
= ( append_nat @ Xs @ ( append_nat @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_1093_append__same__eq,axiom,
! [Ys: list_nat,Xs: list_nat,Zs: list_nat] :
( ( ( append_nat @ Ys @ Xs )
= ( append_nat @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_1094_same__append__eq,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_1095_append__is__Nil__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= nil_nat )
= ( ( Xs = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_1096_Nil__is__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( nil_nat
= ( append_nat @ Xs @ Ys ) )
= ( ( Xs = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_1097_self__append__conv2,axiom,
! [Y: list_nat,Xs: list_nat] :
( ( Y
= ( append_nat @ Xs @ Y ) )
= ( Xs = nil_nat ) ) ).
% self_append_conv2
thf(fact_1098_append__self__conv2,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_nat ) ) ).
% append_self_conv2
thf(fact_1099_self__append__conv,axiom,
! [Y: list_nat,Ys: list_nat] :
( ( Y
= ( append_nat @ Y @ Ys ) )
= ( Ys = nil_nat ) ) ).
% self_append_conv
thf(fact_1100_append__self__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_nat ) ) ).
% append_self_conv
thf(fact_1101_append__Nil2,axiom,
! [Xs: list_nat] :
( ( append_nat @ Xs @ nil_nat )
= Xs ) ).
% append_Nil2
thf(fact_1102_append_Oright__neutral,axiom,
! [A2: list_nat] :
( ( append_nat @ A2 @ nil_nat )
= A2 ) ).
% append.right_neutral
thf(fact_1103_append__eq__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat,Us: list_nat,Vs: list_nat] :
( ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
| ( ( size_size_list_nat @ Us )
= ( size_size_list_nat @ Vs ) ) )
=> ( ( ( append_nat @ Xs @ Us )
= ( append_nat @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_1104_map__append,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
( ( map_nat_nat @ F @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( map_nat_nat @ F @ Xs ) @ ( map_nat_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_1105_pointwise__append__le__iff,axiom,
! [U3: list_nat,X2: list_nat,Y: list_nat] :
( ( pointwise_le @ ( append_nat @ U3 @ X2 ) @ ( append_nat @ U3 @ Y ) )
= ( pointwise_le @ X2 @ Y ) ) ).
% pointwise_append_le_iff
thf(fact_1106_append1__eq__conv,axiom,
! [Xs: list_nat,X2: nat,Ys: list_nat,Y: nat] :
( ( ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) )
= ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) )
= ( ( Xs = Ys )
& ( X2 = Y ) ) ) ).
% append1_eq_conv
thf(fact_1107_length__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( size_size_list_nat @ ( append_nat @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_append
thf(fact_1108_sum__list__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( groups4561878855575611511st_nat @ ( append_nat @ Xs @ Ys ) )
= ( plus_plus_nat @ ( groups4561878855575611511st_nat @ Xs ) @ ( groups4561878855575611511st_nat @ Ys ) ) ) ).
% sum_list_append
thf(fact_1109_size__list__append,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
( ( size_list_nat @ F @ ( append_nat @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_list_nat @ F @ Xs ) @ ( size_list_nat @ F @ Ys ) ) ) ).
% size_list_append
thf(fact_1110_sorted__wrt__append,axiom,
! [P: nat > nat > $o,Xs: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ P @ ( append_nat @ Xs @ Ys ) )
= ( ( sorted_wrt_nat @ P @ Xs )
& ( sorted_wrt_nat @ P @ Ys )
& ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ! [Y5: nat] :
( ( member_nat @ Y5 @ ( set_nat2 @ Ys ) )
=> ( P @ X @ Y5 ) ) ) ) ) ).
% sorted_wrt_append
thf(fact_1111_append__eq__map__conv,axiom,
! [Ys: list_nat,Zs: list_nat,F: nat > nat,Xs: list_nat] :
( ( ( append_nat @ Ys @ Zs )
= ( map_nat_nat @ F @ Xs ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys
= ( map_nat_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_nat @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_1112_map__eq__append__conv,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( append_nat @ Ys @ Zs ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys
= ( map_nat_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_nat @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_1113_eq__Nil__appendI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append_nat @ nil_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_1114_append_Oleft__neutral,axiom,
! [A2: list_nat] :
( ( append_nat @ nil_nat @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_1115_append__Nil,axiom,
! [Ys: list_nat] :
( ( append_nat @ nil_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_1116_rev__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X3: nat] : ( P @ ( cons_nat @ X3 @ nil_nat ) )
=> ( ! [X3: nat,Xs3: list_nat] :
( ( Xs3 != nil_nat )
=> ( ( P @ Xs3 )
=> ( P @ ( append_nat @ Xs3 @ ( cons_nat @ X3 @ nil_nat ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_1117_append__eq__Cons__conv,axiom,
! [Ys: list_nat,Zs: list_nat,X2: nat,Xs: list_nat] :
( ( ( append_nat @ Ys @ Zs )
= ( cons_nat @ X2 @ Xs ) )
= ( ( ( Ys = nil_nat )
& ( Zs
= ( cons_nat @ X2 @ Xs ) ) )
| ? [Ys5: list_nat] :
( ( Ys
= ( cons_nat @ X2 @ Ys5 ) )
& ( ( append_nat @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_1118_Cons__eq__append__conv,axiom,
! [X2: nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( append_nat @ Ys @ Zs ) )
= ( ( ( Ys = nil_nat )
& ( ( cons_nat @ X2 @ Xs )
= Zs ) )
| ? [Ys5: list_nat] :
( ( ( cons_nat @ X2 @ Ys5 )
= Ys )
& ( Xs
= ( append_nat @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_1119_rev__exhaust,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ~ ! [Ys4: list_nat,Y3: nat] :
( Xs
!= ( append_nat @ Ys4 @ ( cons_nat @ Y3 @ nil_nat ) ) ) ) ).
% rev_exhaust
thf(fact_1120_rev__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ( P @ nil_nat )
=> ( ! [X3: nat,Xs3: list_nat] :
( ( P @ Xs3 )
=> ( P @ ( append_nat @ Xs3 @ ( cons_nat @ X3 @ nil_nat ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_1121_split__list,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ? [Ys4: list_list_nat,Zs2: list_list_nat] :
( Xs
= ( append_list_nat @ Ys4 @ ( cons_list_nat @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_1122_split__list,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ? [Ys4: list_nat,Zs2: list_nat] :
( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_1123_split__list__last,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ? [Ys4: list_list_nat,Zs2: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys4 @ ( cons_list_nat @ X2 @ Zs2 ) ) )
& ~ ( member_list_nat @ X2 @ ( set_list_nat2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_1124_split__list__last,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ? [Ys4: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X2 @ Zs2 ) ) )
& ~ ( member_nat @ X2 @ ( set_nat2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_1125_split__list__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys4: list_nat,X3: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X3 @ Zs2 ) ) )
& ( P @ X3 ) ) ) ).
% split_list_prop
thf(fact_1126_split__list__first,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ? [Ys4: list_list_nat,Zs2: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys4 @ ( cons_list_nat @ X2 @ Zs2 ) ) )
& ~ ( member_list_nat @ X2 @ ( set_list_nat2 @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_1127_split__list__first,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ? [Ys4: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X2 @ Zs2 ) ) )
& ~ ( member_nat @ X2 @ ( set_nat2 @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_1128_split__list__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys4: list_nat,X3: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X3 @ Zs2 ) ) )
=> ~ ( P @ X3 ) ) ) ).
% split_list_propE
thf(fact_1129_append__Cons__eq__iff,axiom,
! [X2: list_nat,Xs: list_list_nat,Ys: list_list_nat,Xs4: list_list_nat,Ys6: list_list_nat] :
( ~ ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ~ ( member_list_nat @ X2 @ ( set_list_nat2 @ Ys ) )
=> ( ( ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ Ys ) )
= ( append_list_nat @ Xs4 @ ( cons_list_nat @ X2 @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1130_append__Cons__eq__iff,axiom,
! [X2: nat,Xs: list_nat,Ys: list_nat,Xs4: list_nat,Ys6: list_nat] :
( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ~ ( member_nat @ X2 @ ( set_nat2 @ Ys ) )
=> ( ( ( append_nat @ Xs @ ( cons_nat @ X2 @ Ys ) )
= ( append_nat @ Xs4 @ ( cons_nat @ X2 @ Ys6 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1131_in__set__conv__decomp,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
= ( ? [Ys2: list_list_nat,Zs3: list_list_nat] :
( Xs
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1132_in__set__conv__decomp,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [Ys2: list_nat,Zs3: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1133_split__list__last__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys4: list_nat,X3: nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_1134_split__list__first__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys4: list_nat,X3: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat2 @ Ys4 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_1135_split__list__last__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys4: list_nat,X3: nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_1136_split__list__first__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys4: list_nat,X3: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat2 @ Ys4 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_1137_in__set__conv__decomp__last,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
= ( ? [Ys2: list_list_nat,Zs3: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X2 @ Zs3 ) ) )
& ~ ( member_list_nat @ X2 @ ( set_list_nat2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1138_in__set__conv__decomp__last,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [Ys2: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs3 ) ) )
& ~ ( member_nat @ X2 @ ( set_nat2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1139_in__set__conv__decomp__first,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
= ( ? [Ys2: list_list_nat,Zs3: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X2 @ Zs3 ) ) )
& ~ ( member_list_nat @ X2 @ ( set_list_nat2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1140_in__set__conv__decomp__first,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [Ys2: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs3 ) ) )
& ~ ( member_nat @ X2 @ ( set_nat2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1141_split__list__last__prop__iff,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ? [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
& ( P @ X ) ) )
= ( ? [Ys2: list_nat,X: nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs3 ) ) )
& ( P @ X )
& ! [Y5: nat] :
( ( member_nat @ Y5 @ ( set_nat2 @ Zs3 ) )
=> ~ ( P @ Y5 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_1142_split__list__first__prop__iff,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ? [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
& ( P @ X ) ) )
= ( ? [Ys2: list_nat,X: nat] :
( ? [Zs3: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs3 ) ) )
& ( P @ X )
& ! [Y5: nat] :
( ( member_nat @ Y5 @ ( set_nat2 @ Ys2 ) )
=> ~ ( P @ Y5 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_1143_append__eq__appendI,axiom,
! [Xs: list_nat,Xs1: list_nat,Zs: list_nat,Ys: list_nat,Us: list_nat] :
( ( ( append_nat @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_nat @ Xs1 @ Us ) )
=> ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_1144_append__eq__append__conv2,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ts: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Zs @ Ts ) )
= ( ? [Us2: list_nat] :
( ( ( Xs
= ( append_nat @ Zs @ Us2 ) )
& ( ( append_nat @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_nat @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_nat @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_1145_append__Cons,axiom,
! [X2: nat,Xs: list_nat,Ys: list_nat] :
( ( append_nat @ ( cons_nat @ X2 @ Xs ) @ Ys )
= ( cons_nat @ X2 @ ( append_nat @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_1146_Cons__eq__appendI,axiom,
! [X2: nat,Xs1: list_nat,Ys: list_nat,Xs: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_nat @ Xs1 @ Zs ) )
=> ( ( cons_nat @ X2 @ Xs )
= ( append_nat @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_1147_same__length__different,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs != Ys )
=> ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ? [Pre: list_nat,X3: nat,Xs5: list_nat,Y3: nat,Ys7: list_nat] :
( ( X3 != Y3 )
& ( Xs
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X3 @ nil_nat ) @ Xs5 ) ) )
& ( Ys
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y3 @ nil_nat ) @ Ys7 ) ) ) ) ) ) ).
% same_length_different
thf(fact_1148_sorted__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( append_nat @ Xs @ Ys ) )
= ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
& ( sorted_wrt_nat @ ord_less_eq_nat @ Ys )
& ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ! [Y5: nat] :
( ( member_nat @ Y5 @ ( set_nat2 @ Ys ) )
=> ( ord_less_eq_nat @ X @ Y5 ) ) ) ) ) ).
% sorted_append
thf(fact_1149_length__append__singleton,axiom,
! [Xs: list_nat,X2: nat] :
( ( size_size_list_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_append_singleton
thf(fact_1150_length__Suc__conv__rev,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y5: nat,Ys2: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ Y5 @ nil_nat ) ) )
& ( ( size_size_list_nat @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_1151_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_nat @ N @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_nat @ N @ nil_nat )
= nil_list_nat ) ) ) ).
% n_lists_Nil
thf(fact_1152_folding__insort__key_Osorted__key__list__of__set__unique,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S2: set_list_nat,F: list_nat > nat,A: set_list_nat,L: list_list_nat] :
( ( foldin951631548397865559st_nat @ Less_eq @ Less @ S2 @ F )
=> ( ( ord_le6045566169113846134st_nat @ A @ S2 )
=> ( ( finite8100373058378681591st_nat @ A )
=> ( ( ( sorted_wrt_nat @ Less @ ( map_list_nat_nat @ F @ L ) )
& ( ( set_list_nat2 @ L )
= A )
& ( ( size_s3023201423986296836st_nat @ L )
= ( finite_card_list_nat @ A ) ) )
= ( ( sorted4005134659417452724st_nat @ Less_eq @ F @ A )
= L ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_unique
thf(fact_1153_folding__insort__key_Osorted__key__list__of__set__unique,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S2: set_nat,F: nat > nat,A: set_nat,L: list_nat] :
( ( foldin8133931898133206727at_nat @ Less_eq @ Less @ S2 @ F )
=> ( ( ord_less_eq_set_nat @ A @ S2 )
=> ( ( finite_finite_nat @ A )
=> ( ( ( sorted_wrt_nat @ Less @ ( map_nat_nat @ F @ L ) )
& ( ( set_nat2 @ L )
= A )
& ( ( size_size_list_nat @ L )
= ( finite_card_nat @ A ) ) )
= ( ( sorted5905597674102116260at_nat @ Less_eq @ F @ A )
= L ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_unique
thf(fact_1154_length__n__lists__elem,axiom,
! [Ys: list_nat,N: nat,Xs: list_nat] :
( ( member_list_nat @ Ys @ ( set_list_nat2 @ ( n_lists_nat @ N @ Xs ) ) )
=> ( ( size_size_list_nat @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_1155_folding__insort__key_Ostrict__sorted__key__list__of__set,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S2: set_list_nat,F: list_nat > nat,A: set_list_nat] :
( ( foldin951631548397865559st_nat @ Less_eq @ Less @ S2 @ F )
=> ( ( ord_le6045566169113846134st_nat @ A @ S2 )
=> ( sorted_wrt_nat @ Less @ ( map_list_nat_nat @ F @ ( sorted4005134659417452724st_nat @ Less_eq @ F @ A ) ) ) ) ) ).
% folding_insort_key.strict_sorted_key_list_of_set
thf(fact_1156_folding__insort__key_Ostrict__sorted__key__list__of__set,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S2: set_nat,F: nat > nat,A: set_nat] :
( ( foldin8133931898133206727at_nat @ Less_eq @ Less @ S2 @ F )
=> ( ( ord_less_eq_set_nat @ A @ S2 )
=> ( sorted_wrt_nat @ Less @ ( map_nat_nat @ F @ ( sorted5905597674102116260at_nat @ Less_eq @ F @ A ) ) ) ) ) ).
% folding_insort_key.strict_sorted_key_list_of_set
thf(fact_1157_folding__insort__key_Osorted__sorted__key__list__of__set,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S2: set_list_nat,F: list_nat > nat,A: set_list_nat] :
( ( foldin951631548397865559st_nat @ Less_eq @ Less @ S2 @ F )
=> ( ( ord_le6045566169113846134st_nat @ A @ S2 )
=> ( sorted_wrt_nat @ Less_eq @ ( map_list_nat_nat @ F @ ( sorted4005134659417452724st_nat @ Less_eq @ F @ A ) ) ) ) ) ).
% folding_insort_key.sorted_sorted_key_list_of_set
thf(fact_1158_folding__insort__key_Osorted__sorted__key__list__of__set,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S2: set_nat,F: nat > nat,A: set_nat] :
( ( foldin8133931898133206727at_nat @ Less_eq @ Less @ S2 @ F )
=> ( ( ord_less_eq_set_nat @ A @ S2 )
=> ( sorted_wrt_nat @ Less_eq @ ( map_nat_nat @ F @ ( sorted5905597674102116260at_nat @ Less_eq @ F @ A ) ) ) ) ) ).
% folding_insort_key.sorted_sorted_key_list_of_set
thf(fact_1159_set__n__lists,axiom,
! [N: nat,Xs: list_list_nat] :
( ( set_list_list_nat2 @ ( n_lists_list_nat @ N @ Xs ) )
= ( collec5989764272469232197st_nat
@ ^ [Ys2: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Ys2 )
= N )
& ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Ys2 ) @ ( set_list_nat2 @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_1160_set__n__lists,axiom,
! [N: nat,Xs: list_nat] :
( ( set_list_nat2 @ ( n_lists_nat @ N @ Xs ) )
= ( collect_list_nat
@ ^ [Ys2: list_nat] :
( ( ( size_size_list_nat @ Ys2 )
= N )
& ( ord_less_eq_set_nat @ ( set_nat2 @ Ys2 ) @ ( set_nat2 @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_1161_n__lists_Osimps_I1_J,axiom,
! [Xs: list_nat] :
( ( n_lists_nat @ zero_zero_nat @ Xs )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% n_lists.simps(1)
thf(fact_1162_augmentum__subset__sum__list,axiom,
! [Ns: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( augmentum @ Ns ) ) @ ( set_ord_atMost_nat @ ( groups4561878855575611511st_nat @ Ns ) ) ) ).
% augmentum_subset_sum_list
thf(fact_1163_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
! [A: set_list_nat,L: list_list_nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( ( ( sorted_wrt_list_nat @ ord_less_list_nat @ L )
& ( ( set_list_nat2 @ L )
= A )
& ( ( size_s3023201423986296836st_nat @ L )
= ( finite_card_list_nat @ A ) ) )
= ( ( linord2712301520579796368st_nat @ A )
= L ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_unique
thf(fact_1164_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
! [A: set_nat,L: list_nat] :
( ( finite_finite_nat @ A )
=> ( ( ( sorted_wrt_nat @ ord_less_nat @ L )
& ( ( set_nat2 @ L )
= A )
& ( ( size_size_list_nat @ L )
= ( finite_card_nat @ A ) ) )
= ( ( linord2614967742042102400et_nat @ A )
= L ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_unique
thf(fact_1165_atMost__iff,axiom,
! [I: list_nat,K: list_nat] :
( ( member_list_nat @ I @ ( set_or4185896845444216793st_nat @ K ) )
= ( ord_less_eq_list_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_1166_atMost__iff,axiom,
! [I: set_list_nat,K: set_list_nat] :
( ( member_set_list_nat @ I @ ( set_or2492388921469580815st_nat @ K ) )
= ( ord_le6045566169113846134st_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_1167_atMost__iff,axiom,
! [I: set_nat,K: set_nat] :
( ( member_set_nat @ I @ ( set_or4236626031148496127et_nat @ K ) )
= ( ord_less_eq_set_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_1168_atMost__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_atMost_nat @ K ) )
= ( ord_less_eq_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_1169_finite__atMost,axiom,
! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).
% finite_atMost
thf(fact_1170_atMost__subset__iff,axiom,
! [X2: set_list_nat,Y: set_list_nat] :
( ( ord_le1068707526560357548st_nat @ ( set_or2492388921469580815st_nat @ X2 ) @ ( set_or2492388921469580815st_nat @ Y ) )
= ( ord_le6045566169113846134st_nat @ X2 @ Y ) ) ).
% atMost_subset_iff
thf(fact_1171_atMost__subset__iff,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4236626031148496127et_nat @ X2 ) @ ( set_or4236626031148496127et_nat @ Y ) )
= ( ord_less_eq_set_nat @ X2 @ Y ) ) ).
% atMost_subset_iff
thf(fact_1172_atMost__subset__iff,axiom,
! [X2: list_nat,Y: list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_or4185896845444216793st_nat @ X2 ) @ ( set_or4185896845444216793st_nat @ Y ) )
= ( ord_less_eq_list_nat @ X2 @ Y ) ) ).
% atMost_subset_iff
thf(fact_1173_atMost__subset__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X2 ) @ ( set_ord_atMost_nat @ Y ) )
= ( ord_less_eq_nat @ X2 @ Y ) ) ).
% atMost_subset_iff
thf(fact_1174_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
( ( linord2712301520579796368st_nat @ bot_bot_set_list_nat )
= nil_list_nat ) ).
% sorted_list_of_set.sorted_key_list_of_set_empty
thf(fact_1175_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
( ( linord2614967742042102400et_nat @ bot_bot_set_nat )
= nil_nat ) ).
% sorted_list_of_set.sorted_key_list_of_set_empty
thf(fact_1176_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
! [A: set_list_nat] :
( ~ ( finite8100373058378681591st_nat @ A )
=> ( ( linord2712301520579796368st_nat @ A )
= nil_list_nat ) ) ).
% sorted_list_of_set.fold_insort_key.infinite
thf(fact_1177_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( linord2614967742042102400et_nat @ A )
= nil_nat ) ) ).
% sorted_list_of_set.fold_insort_key.infinite
thf(fact_1178_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
! [A: set_list_nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( ( set_list_nat2 @ ( linord2712301520579796368st_nat @ A ) )
= A ) ) ).
% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_1179_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( set_nat2 @ ( linord2614967742042102400et_nat @ A ) )
= A ) ) ).
% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_1180_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
! [A: set_list_nat] :
( ( size_s3023201423986296836st_nat @ ( linord2712301520579796368st_nat @ A ) )
= ( finite_card_list_nat @ A ) ) ).
% sorted_list_of_set.length_sorted_key_list_of_set
thf(fact_1181_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
! [A: set_nat] :
( ( size_size_list_nat @ ( linord2614967742042102400et_nat @ A ) )
= ( finite_card_nat @ A ) ) ).
% sorted_list_of_set.length_sorted_key_list_of_set
thf(fact_1182_card__atMost,axiom,
! [U3: nat] :
( ( finite_card_nat @ ( set_ord_atMost_nat @ U3 ) )
= ( suc @ U3 ) ) ).
% card_atMost
thf(fact_1183_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A: set_list_nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( ( ( linord2712301520579796368st_nat @ A )
= nil_list_nat )
= ( A = bot_bot_set_list_nat ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_1184_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ( linord2614967742042102400et_nat @ A )
= nil_nat )
= ( A = bot_bot_set_nat ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_1185_sorted__list__of__set__atMost__Suc,axiom,
! [K: nat] :
( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K ) ) )
= ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K ) ) @ ( cons_nat @ ( suc @ K ) @ nil_nat ) ) ) ).
% sorted_list_of_set_atMost_Suc
thf(fact_1186_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
! [A: set_list_nat,B3: set_list_nat] :
( ( ( linord2712301520579796368st_nat @ A )
= ( linord2712301520579796368st_nat @ B3 ) )
=> ( ( finite8100373058378681591st_nat @ A )
=> ( ( finite8100373058378681591st_nat @ B3 )
=> ( A = B3 ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_1187_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
! [A: set_nat,B3: set_nat] :
( ( ( linord2614967742042102400et_nat @ A )
= ( linord2614967742042102400et_nat @ B3 ) )
=> ( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B3 )
=> ( A = B3 ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_1188_finite__nat__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [S4: set_nat] :
? [K2: nat] : ( ord_less_eq_set_nat @ S4 @ ( set_ord_atMost_nat @ K2 ) ) ) ) ).
% finite_nat_iff_bounded_le
thf(fact_1189_atMost__def,axiom,
( set_or4185896845444216793st_nat
= ( ^ [U5: list_nat] :
( collect_list_nat
@ ^ [X: list_nat] : ( ord_less_eq_list_nat @ X @ U5 ) ) ) ) ).
% atMost_def
thf(fact_1190_atMost__def,axiom,
( set_or2492388921469580815st_nat
= ( ^ [U5: set_list_nat] :
( collect_set_list_nat
@ ^ [X: set_list_nat] : ( ord_le6045566169113846134st_nat @ X @ U5 ) ) ) ) ).
% atMost_def
thf(fact_1191_atMost__def,axiom,
( set_or4236626031148496127et_nat
= ( ^ [U5: set_nat] :
( collect_set_nat
@ ^ [X: set_nat] : ( ord_less_eq_set_nat @ X @ U5 ) ) ) ) ).
% atMost_def
thf(fact_1192_atMost__def,axiom,
( set_ord_atMost_nat
= ( ^ [U5: nat] :
( collect_nat
@ ^ [X: nat] : ( ord_less_eq_nat @ X @ U5 ) ) ) ) ).
% atMost_def
thf(fact_1193_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
! [A: set_nat] : ( sorted_wrt_nat @ ord_less_nat @ ( linord2614967742042102400et_nat @ A ) ) ).
% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_1194_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
! [A: set_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( linord2614967742042102400et_nat @ A ) ) ).
% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_1195_card__length__sum__set,axiom,
! [R: nat,N: nat] :
( ( finite_card_list_nat @ ( length_sum_set @ ( suc @ R ) @ N ) )
= ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( finite_card_list_nat @ ( length_sum_set @ R @ ( minus_minus_nat @ N @ I4 ) ) )
@ ( set_ord_atMost_nat @ N ) ) ) ).
% card_length_sum_set
thf(fact_1196_sum_Oneutral__const,axiom,
! [A: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [Uu: nat] : zero_zero_nat
@ A )
= zero_zero_nat ) ).
% sum.neutral_const
thf(fact_1197_sum_Oempty,axiom,
! [G2: list_nat > nat] :
( ( groups4396056296759096172at_nat @ G2 @ bot_bot_set_list_nat )
= zero_zero_nat ) ).
% sum.empty
thf(fact_1198_sum_Oempty,axiom,
! [G2: nat > nat] :
( ( groups3542108847815614940at_nat @ G2 @ bot_bot_set_nat )
= zero_zero_nat ) ).
% sum.empty
thf(fact_1199_sum__clauses_I1_J,axiom,
! [F: list_nat > nat] :
( ( groups4396056296759096172at_nat @ F @ bot_bot_set_list_nat )
= zero_zero_nat ) ).
% sum_clauses(1)
thf(fact_1200_sum__clauses_I1_J,axiom,
! [F: nat > nat] :
( ( groups3542108847815614940at_nat @ F @ bot_bot_set_nat )
= zero_zero_nat ) ).
% sum_clauses(1)
thf(fact_1201_sum__eq__0__iff,axiom,
! [F3: set_list_nat,F: list_nat > nat] :
( ( finite8100373058378681591st_nat @ F3 )
=> ( ( ( groups4396056296759096172at_nat @ F @ F3 )
= zero_zero_nat )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ F3 )
=> ( ( F @ X )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_1202_sum__eq__0__iff,axiom,
! [F3: set_nat,F: nat > nat] :
( ( finite_finite_nat @ F3 )
=> ( ( ( groups3542108847815614940at_nat @ F @ F3 )
= zero_zero_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ F3 )
=> ( ( F @ X )
= zero_zero_nat ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_1203_sum_Oinfinite,axiom,
! [A: set_list_nat,G2: list_nat > nat] :
( ~ ( finite8100373058378681591st_nat @ A )
=> ( ( groups4396056296759096172at_nat @ G2 @ A )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_1204_sum_Oinfinite,axiom,
! [A: set_nat,G2: nat > nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( groups3542108847815614940at_nat @ G2 @ A )
= zero_zero_nat ) ) ).
% sum.infinite
thf(fact_1205_sum_Odelta_H,axiom,
! [S2: set_list_nat,A2: list_nat,B: list_nat > nat] :
( ( finite8100373058378681591st_nat @ S2 )
=> ( ( ( member_list_nat @ A2 @ S2 )
=> ( ( groups4396056296759096172at_nat
@ ^ [K2: list_nat] : ( if_nat @ ( A2 = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= ( B @ A2 ) ) )
& ( ~ ( member_list_nat @ A2 @ S2 )
=> ( ( groups4396056296759096172at_nat
@ ^ [K2: list_nat] : ( if_nat @ ( A2 = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_1206_sum_Odelta_H,axiom,
! [S2: set_nat,A2: nat,B: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A2 @ S2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( if_nat @ ( A2 = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= ( B @ A2 ) ) )
& ( ~ ( member_nat @ A2 @ S2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( if_nat @ ( A2 = K2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta'
thf(fact_1207_sum_Odelta,axiom,
! [S2: set_list_nat,A2: list_nat,B: list_nat > nat] :
( ( finite8100373058378681591st_nat @ S2 )
=> ( ( ( member_list_nat @ A2 @ S2 )
=> ( ( groups4396056296759096172at_nat
@ ^ [K2: list_nat] : ( if_nat @ ( K2 = A2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= ( B @ A2 ) ) )
& ( ~ ( member_list_nat @ A2 @ S2 )
=> ( ( groups4396056296759096172at_nat
@ ^ [K2: list_nat] : ( if_nat @ ( K2 = A2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_1208_sum_Odelta,axiom,
! [S2: set_nat,A2: nat,B: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( ( member_nat @ A2 @ S2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( if_nat @ ( K2 = A2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= ( B @ A2 ) ) )
& ( ~ ( member_nat @ A2 @ S2 )
=> ( ( groups3542108847815614940at_nat
@ ^ [K2: nat] : ( if_nat @ ( K2 = A2 ) @ ( B @ K2 ) @ zero_zero_nat )
@ S2 )
= zero_zero_nat ) ) ) ) ).
% sum.delta
thf(fact_1209_sum_OatMost__Suc,axiom,
! [G2: nat > nat,N: nat] :
( ( groups3542108847815614940at_nat @ G2 @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G2 @ ( set_ord_atMost_nat @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ).
% sum.atMost_Suc
thf(fact_1210_sum__nonneg__0,axiom,
! [S: set_list_nat,F: list_nat > nat,I: list_nat] :
( ( finite8100373058378681591st_nat @ S )
=> ( ! [I2: list_nat] :
( ( member_list_nat @ I2 @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
=> ( ( ( groups4396056296759096172at_nat @ F @ S )
= zero_zero_nat )
=> ( ( member_list_nat @ I @ S )
=> ( ( F @ I )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_1211_sum__nonneg__0,axiom,
! [S: set_nat,F: nat > nat,I: nat] :
( ( finite_finite_nat @ S )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
=> ( ( ( groups3542108847815614940at_nat @ F @ S )
= zero_zero_nat )
=> ( ( member_nat @ I @ S )
=> ( ( F @ I )
= zero_zero_nat ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_1212_sum__nonneg__leq__bound,axiom,
! [S: set_list_nat,F: list_nat > nat,B3: nat,I: list_nat] :
( ( finite8100373058378681591st_nat @ S )
=> ( ! [I2: list_nat] :
( ( member_list_nat @ I2 @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
=> ( ( ( groups4396056296759096172at_nat @ F @ S )
= B3 )
=> ( ( member_list_nat @ I @ S )
=> ( ord_less_eq_nat @ ( F @ I ) @ B3 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_1213_sum__nonneg__leq__bound,axiom,
! [S: set_nat,F: nat > nat,B3: nat,I: nat] :
( ( finite_finite_nat @ S )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ S )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I2 ) ) )
=> ( ( ( groups3542108847815614940at_nat @ F @ S )
= B3 )
=> ( ( member_nat @ I @ S )
=> ( ord_less_eq_nat @ ( F @ I ) @ B3 ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_1214_sum__subtractf__nat,axiom,
! [A: set_list_nat,G2: list_nat > nat,F: list_nat > nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
=> ( ord_less_eq_nat @ ( G2 @ X3 ) @ ( F @ X3 ) ) )
=> ( ( groups4396056296759096172at_nat
@ ^ [X: list_nat] : ( minus_minus_nat @ ( F @ X ) @ ( G2 @ X ) )
@ A )
= ( minus_minus_nat @ ( groups4396056296759096172at_nat @ F @ A ) @ ( groups4396056296759096172at_nat @ G2 @ A ) ) ) ) ).
% sum_subtractf_nat
thf(fact_1215_sum__subtractf__nat,axiom,
! [A: set_nat,G2: nat > nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ ( G2 @ X3 ) @ ( F @ X3 ) ) )
=> ( ( groups3542108847815614940at_nat
@ ^ [X: nat] : ( minus_minus_nat @ ( F @ X ) @ ( G2 @ X ) )
@ A )
= ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( groups3542108847815614940at_nat @ G2 @ A ) ) ) ) ).
% sum_subtractf_nat
thf(fact_1216_sum__nonpos,axiom,
! [A: set_list_nat,F: list_nat > nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups4396056296759096172at_nat @ F @ A ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_1217_sum__nonpos,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ zero_zero_nat ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ zero_zero_nat ) ) ).
% sum_nonpos
thf(fact_1218_sum__nonneg,axiom,
! [A: set_list_nat,F: list_nat > nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups4396056296759096172at_nat @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_1219_sum__nonneg,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( groups3542108847815614940at_nat @ F @ A ) ) ) ).
% sum_nonneg
thf(fact_1220_sum__mono,axiom,
! [K4: set_list_nat,F: list_nat > nat,G2: list_nat > nat] :
( ! [I2: list_nat] :
( ( member_list_nat @ I2 @ K4 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( G2 @ I2 ) ) )
=> ( ord_less_eq_nat @ ( groups4396056296759096172at_nat @ F @ K4 ) @ ( groups4396056296759096172at_nat @ G2 @ K4 ) ) ) ).
% sum_mono
thf(fact_1221_sum__mono,axiom,
! [K4: set_nat,F: nat > nat,G2: nat > nat] :
( ! [I2: nat] :
( ( member_nat @ I2 @ K4 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( G2 @ I2 ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ K4 ) @ ( groups3542108847815614940at_nat @ G2 @ K4 ) ) ) ).
% sum_mono
thf(fact_1222_sum__mono__inv,axiom,
! [F: list_nat > nat,I5: set_list_nat,G2: list_nat > nat,I: list_nat] :
( ( ( groups4396056296759096172at_nat @ F @ I5 )
= ( groups4396056296759096172at_nat @ G2 @ I5 ) )
=> ( ! [I2: list_nat] :
( ( member_list_nat @ I2 @ I5 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( G2 @ I2 ) ) )
=> ( ( member_list_nat @ I @ I5 )
=> ( ( finite8100373058378681591st_nat @ I5 )
=> ( ( F @ I )
= ( G2 @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_1223_sum__mono__inv,axiom,
! [F: nat > nat,I5: set_nat,G2: nat > nat,I: nat] :
( ( ( groups3542108847815614940at_nat @ F @ I5 )
= ( groups3542108847815614940at_nat @ G2 @ I5 ) )
=> ( ! [I2: nat] :
( ( member_nat @ I2 @ I5 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( G2 @ I2 ) ) )
=> ( ( member_nat @ I @ I5 )
=> ( ( finite_finite_nat @ I5 )
=> ( ( F @ I )
= ( G2 @ I ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_1224_sum__strict__mono__ex1,axiom,
! [A: set_list_nat,F: list_nat > nat,G2: list_nat > nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( G2 @ X3 ) ) )
=> ( ? [X4: list_nat] :
( ( member_list_nat @ X4 @ A )
& ( ord_less_nat @ ( F @ X4 ) @ ( G2 @ X4 ) ) )
=> ( ord_less_nat @ ( groups4396056296759096172at_nat @ F @ A ) @ ( groups4396056296759096172at_nat @ G2 @ A ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_1225_sum__strict__mono__ex1,axiom,
! [A: set_nat,F: nat > nat,G2: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( G2 @ X3 ) ) )
=> ( ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ord_less_nat @ ( F @ X4 ) @ ( G2 @ X4 ) ) )
=> ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( groups3542108847815614940at_nat @ G2 @ A ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_1226_sum__le__included,axiom,
! [S: set_list_nat,T: set_list_nat,G2: list_nat > nat,I: list_nat > list_nat,F: list_nat > nat] :
( ( finite8100373058378681591st_nat @ S )
=> ( ( finite8100373058378681591st_nat @ T )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ T )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G2 @ X3 ) ) )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ S )
=> ? [Xa: list_nat] :
( ( member_list_nat @ Xa @ T )
& ( ( I @ Xa )
= X3 )
& ( ord_less_eq_nat @ ( F @ X3 ) @ ( G2 @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups4396056296759096172at_nat @ F @ S ) @ ( groups4396056296759096172at_nat @ G2 @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_1227_sum__le__included,axiom,
! [S: set_list_nat,T: set_nat,G2: nat > nat,I: nat > list_nat,F: list_nat > nat] :
( ( finite8100373058378681591st_nat @ S )
=> ( ( finite_finite_nat @ T )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ T )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G2 @ X3 ) ) )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ S )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T )
& ( ( I @ Xa )
= X3 )
& ( ord_less_eq_nat @ ( F @ X3 ) @ ( G2 @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups4396056296759096172at_nat @ F @ S ) @ ( groups3542108847815614940at_nat @ G2 @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_1228_sum__le__included,axiom,
! [S: set_nat,T: set_list_nat,G2: list_nat > nat,I: list_nat > nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( finite8100373058378681591st_nat @ T )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ T )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G2 @ X3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S )
=> ? [Xa: list_nat] :
( ( member_list_nat @ Xa @ T )
& ( ( I @ Xa )
= X3 )
& ( ord_less_eq_nat @ ( F @ X3 ) @ ( G2 @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ S ) @ ( groups4396056296759096172at_nat @ G2 @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_1229_sum__le__included,axiom,
! [S: set_nat,T: set_nat,G2: nat > nat,I: nat > nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_nat @ T )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ T )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( G2 @ X3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ T )
& ( ( I @ Xa )
= X3 )
& ( ord_less_eq_nat @ ( F @ X3 ) @ ( G2 @ Xa ) ) ) )
=> ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ S ) @ ( groups3542108847815614940at_nat @ G2 @ T ) ) ) ) ) ) ).
% sum_le_included
thf(fact_1230_sum__nonneg__eq__0__iff,axiom,
! [A: set_list_nat,F: list_nat > nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ( ( groups4396056296759096172at_nat @ F @ A )
= zero_zero_nat )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ A )
=> ( ( F @ X )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_1231_sum__nonneg__eq__0__iff,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X3 ) ) )
=> ( ( ( groups3542108847815614940at_nat @ F @ A )
= zero_zero_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ( F @ X )
= zero_zero_nat ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_1232_sum_Orelated,axiom,
! [R3: nat > nat > $o,S2: set_list_nat,H: list_nat > nat,G2: list_nat > nat] :
( ( R3 @ zero_zero_nat @ zero_zero_nat )
=> ( ! [X1: nat,Y1: nat,X23: nat,Y23: nat] :
( ( ( R3 @ X1 @ X23 )
& ( R3 @ Y1 @ Y23 ) )
=> ( R3 @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
=> ( ( finite8100373058378681591st_nat @ S2 )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ S2 )
=> ( R3 @ ( H @ X3 ) @ ( G2 @ X3 ) ) )
=> ( R3 @ ( groups4396056296759096172at_nat @ H @ S2 ) @ ( groups4396056296759096172at_nat @ G2 @ S2 ) ) ) ) ) ) ).
% sum.related
thf(fact_1233_sum_Orelated,axiom,
! [R3: nat > nat > $o,S2: set_nat,H: nat > nat,G2: nat > nat] :
( ( R3 @ zero_zero_nat @ zero_zero_nat )
=> ( ! [X1: nat,Y1: nat,X23: nat,Y23: nat] :
( ( ( R3 @ X1 @ X23 )
& ( R3 @ Y1 @ Y23 ) )
=> ( R3 @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
=> ( ( finite_finite_nat @ S2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ S2 )
=> ( R3 @ ( H @ X3 ) @ ( G2 @ X3 ) ) )
=> ( R3 @ ( groups3542108847815614940at_nat @ H @ S2 ) @ ( groups3542108847815614940at_nat @ G2 @ S2 ) ) ) ) ) ) ).
% sum.related
thf(fact_1234_sum__strict__mono,axiom,
! [A: set_list_nat,F: list_nat > nat,G2: list_nat > nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( ( A != bot_bot_set_list_nat )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
=> ( ord_less_nat @ ( F @ X3 ) @ ( G2 @ X3 ) ) )
=> ( ord_less_nat @ ( groups4396056296759096172at_nat @ F @ A ) @ ( groups4396056296759096172at_nat @ G2 @ A ) ) ) ) ) ).
% sum_strict_mono
thf(fact_1235_sum__strict__mono,axiom,
! [A: set_nat,F: nat > nat,G2: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_nat @ ( F @ X3 ) @ ( G2 @ X3 ) ) )
=> ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ A ) @ ( groups3542108847815614940at_nat @ G2 @ A ) ) ) ) ) ).
% sum_strict_mono
thf(fact_1236_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_list_nat,T5: set_list_nat,S2: set_list_nat,I: list_nat > list_nat,J: list_nat > list_nat,T2: set_list_nat,G2: list_nat > nat,H: list_nat > nat] :
( ( finite8100373058378681591st_nat @ S5 )
=> ( ( finite8100373058378681591st_nat @ T5 )
=> ( ! [A6: list_nat] :
( ( member_list_nat @ A6 @ ( minus_7954133019191499631st_nat @ S2 @ S5 ) )
=> ( ( I @ ( J @ A6 ) )
= A6 ) )
=> ( ! [A6: list_nat] :
( ( member_list_nat @ A6 @ ( minus_7954133019191499631st_nat @ S2 @ S5 ) )
=> ( member_list_nat @ ( J @ A6 ) @ ( minus_7954133019191499631st_nat @ T2 @ T5 ) ) )
=> ( ! [B6: list_nat] :
( ( member_list_nat @ B6 @ ( minus_7954133019191499631st_nat @ T2 @ T5 ) )
=> ( ( J @ ( I @ B6 ) )
= B6 ) )
=> ( ! [B6: list_nat] :
( ( member_list_nat @ B6 @ ( minus_7954133019191499631st_nat @ T2 @ T5 ) )
=> ( member_list_nat @ ( I @ B6 ) @ ( minus_7954133019191499631st_nat @ S2 @ S5 ) ) )
=> ( ! [A6: list_nat] :
( ( member_list_nat @ A6 @ S5 )
=> ( ( G2 @ A6 )
= zero_zero_nat ) )
=> ( ! [B6: list_nat] :
( ( member_list_nat @ B6 @ T5 )
=> ( ( H @ B6 )
= zero_zero_nat ) )
=> ( ! [A6: list_nat] :
( ( member_list_nat @ A6 @ S2 )
=> ( ( H @ ( J @ A6 ) )
= ( G2 @ A6 ) ) )
=> ( ( groups4396056296759096172at_nat @ G2 @ S2 )
= ( groups4396056296759096172at_nat @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_1237_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_list_nat,T5: set_nat,S2: set_list_nat,I: nat > list_nat,J: list_nat > nat,T2: set_nat,G2: list_nat > nat,H: nat > nat] :
( ( finite8100373058378681591st_nat @ S5 )
=> ( ( finite_finite_nat @ T5 )
=> ( ! [A6: list_nat] :
( ( member_list_nat @ A6 @ ( minus_7954133019191499631st_nat @ S2 @ S5 ) )
=> ( ( I @ ( J @ A6 ) )
= A6 ) )
=> ( ! [A6: list_nat] :
( ( member_list_nat @ A6 @ ( minus_7954133019191499631st_nat @ S2 @ S5 ) )
=> ( member_nat @ ( J @ A6 ) @ ( minus_minus_set_nat @ T2 @ T5 ) ) )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ ( minus_minus_set_nat @ T2 @ T5 ) )
=> ( ( J @ ( I @ B6 ) )
= B6 ) )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ ( minus_minus_set_nat @ T2 @ T5 ) )
=> ( member_list_nat @ ( I @ B6 ) @ ( minus_7954133019191499631st_nat @ S2 @ S5 ) ) )
=> ( ! [A6: list_nat] :
( ( member_list_nat @ A6 @ S5 )
=> ( ( G2 @ A6 )
= zero_zero_nat ) )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ T5 )
=> ( ( H @ B6 )
= zero_zero_nat ) )
=> ( ! [A6: list_nat] :
( ( member_list_nat @ A6 @ S2 )
=> ( ( H @ ( J @ A6 ) )
= ( G2 @ A6 ) ) )
=> ( ( groups4396056296759096172at_nat @ G2 @ S2 )
= ( groups3542108847815614940at_nat @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_1238_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_nat,T5: set_list_nat,S2: set_nat,I: list_nat > nat,J: nat > list_nat,T2: set_list_nat,G2: nat > nat,H: list_nat > nat] :
( ( finite_finite_nat @ S5 )
=> ( ( finite8100373058378681591st_nat @ T5 )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ ( minus_minus_set_nat @ S2 @ S5 ) )
=> ( ( I @ ( J @ A6 ) )
= A6 ) )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ ( minus_minus_set_nat @ S2 @ S5 ) )
=> ( member_list_nat @ ( J @ A6 ) @ ( minus_7954133019191499631st_nat @ T2 @ T5 ) ) )
=> ( ! [B6: list_nat] :
( ( member_list_nat @ B6 @ ( minus_7954133019191499631st_nat @ T2 @ T5 ) )
=> ( ( J @ ( I @ B6 ) )
= B6 ) )
=> ( ! [B6: list_nat] :
( ( member_list_nat @ B6 @ ( minus_7954133019191499631st_nat @ T2 @ T5 ) )
=> ( member_nat @ ( I @ B6 ) @ ( minus_minus_set_nat @ S2 @ S5 ) ) )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ S5 )
=> ( ( G2 @ A6 )
= zero_zero_nat ) )
=> ( ! [B6: list_nat] :
( ( member_list_nat @ B6 @ T5 )
=> ( ( H @ B6 )
= zero_zero_nat ) )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ S2 )
=> ( ( H @ ( J @ A6 ) )
= ( G2 @ A6 ) ) )
=> ( ( groups3542108847815614940at_nat @ G2 @ S2 )
= ( groups4396056296759096172at_nat @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_1239_sum_Oreindex__bij__witness__not__neutral,axiom,
! [S5: set_nat,T5: set_nat,S2: set_nat,I: nat > nat,J: nat > nat,T2: set_nat,G2: nat > nat,H: nat > nat] :
( ( finite_finite_nat @ S5 )
=> ( ( finite_finite_nat @ T5 )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ ( minus_minus_set_nat @ S2 @ S5 ) )
=> ( ( I @ ( J @ A6 ) )
= A6 ) )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ ( minus_minus_set_nat @ S2 @ S5 ) )
=> ( member_nat @ ( J @ A6 ) @ ( minus_minus_set_nat @ T2 @ T5 ) ) )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ ( minus_minus_set_nat @ T2 @ T5 ) )
=> ( ( J @ ( I @ B6 ) )
= B6 ) )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ ( minus_minus_set_nat @ T2 @ T5 ) )
=> ( member_nat @ ( I @ B6 ) @ ( minus_minus_set_nat @ S2 @ S5 ) ) )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ S5 )
=> ( ( G2 @ A6 )
= zero_zero_nat ) )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ T5 )
=> ( ( H @ B6 )
= zero_zero_nat ) )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ S2 )
=> ( ( H @ ( J @ A6 ) )
= ( G2 @ A6 ) ) )
=> ( ( groups3542108847815614940at_nat @ G2 @ S2 )
= ( groups3542108847815614940at_nat @ H @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_1240_sum__eq__Suc0__iff,axiom,
! [A: set_list_nat,F: list_nat > nat] :
( ( finite8100373058378681591st_nat @ A )
=> ( ( ( groups4396056296759096172at_nat @ F @ A )
= ( suc @ zero_zero_nat ) )
= ( ? [X: list_nat] :
( ( member_list_nat @ X @ A )
& ( ( F @ X )
= ( suc @ zero_zero_nat ) )
& ! [Y5: list_nat] :
( ( member_list_nat @ Y5 @ A )
=> ( ( X != Y5 )
=> ( ( F @ Y5 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_1241_sum__eq__Suc0__iff,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ( groups3542108847815614940at_nat @ F @ A )
= ( suc @ zero_zero_nat ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A )
& ( ( F @ X )
= ( suc @ zero_zero_nat ) )
& ! [Y5: nat] :
( ( member_nat @ Y5 @ A )
=> ( ( X != Y5 )
=> ( ( F @ Y5 )
= zero_zero_nat ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_1242_sum__SucD,axiom,
! [F: nat > nat,A: set_nat,N: nat] :
( ( ( groups3542108847815614940at_nat @ F @ A )
= ( suc @ N ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( ord_less_nat @ zero_zero_nat @ ( F @ X3 ) ) ) ) ).
% sum_SucD
thf(fact_1243_sum__cong__Suc,axiom,
! [A: set_nat,F: nat > nat,G2: nat > nat] :
( ~ ( member_nat @ zero_zero_nat @ A )
=> ( ! [X3: nat] :
( ( member_nat @ ( suc @ X3 ) @ A )
=> ( ( F @ ( suc @ X3 ) )
= ( G2 @ ( suc @ X3 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ F @ A )
= ( groups3542108847815614940at_nat @ G2 @ A ) ) ) ) ).
% sum_cong_Suc
thf(fact_1244_sum_Oneutral,axiom,
! [A: set_nat,G2: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( G2 @ X3 )
= zero_zero_nat ) )
=> ( ( groups3542108847815614940at_nat @ G2 @ A )
= zero_zero_nat ) ) ).
% sum.neutral
thf(fact_1245_sum_Onot__neutral__contains__not__neutral,axiom,
! [G2: list_nat > nat,A: set_list_nat] :
( ( ( groups4396056296759096172at_nat @ G2 @ A )
!= zero_zero_nat )
=> ~ ! [A6: list_nat] :
( ( member_list_nat @ A6 @ A )
=> ( ( G2 @ A6 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_1246_sum_Onot__neutral__contains__not__neutral,axiom,
! [G2: nat > nat,A: set_nat] :
( ( ( groups3542108847815614940at_nat @ G2 @ A )
!= zero_zero_nat )
=> ~ ! [A6: nat] :
( ( member_nat @ A6 @ A )
=> ( ( G2 @ A6 )
= zero_zero_nat ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_1247_sum_Oswap,axiom,
! [G2: nat > nat > nat,B3: set_nat,A: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( groups3542108847815614940at_nat @ ( G2 @ I4 ) @ B3 )
@ A )
= ( groups3542108847815614940at_nat
@ ^ [J3: nat] :
( groups3542108847815614940at_nat
@ ^ [I4: nat] : ( G2 @ I4 @ J3 )
@ A )
@ B3 ) ) ).
% sum.swap
thf(fact_1248_sum_Oreindex__bij__witness,axiom,
! [S2: set_list_nat,I: nat > list_nat,J: list_nat > nat,T2: set_nat,H: nat > nat,G2: list_nat > nat] :
( ! [A6: list_nat] :
( ( member_list_nat @ A6 @ S2 )
=> ( ( I @ ( J @ A6 ) )
= A6 ) )
=> ( ! [A6: list_nat] :
( ( member_list_nat @ A6 @ S2 )
=> ( member_nat @ ( J @ A6 ) @ T2 ) )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ T2 )
=> ( ( J @ ( I @ B6 ) )
= B6 ) )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ T2 )
=> ( member_list_nat @ ( I @ B6 ) @ S2 ) )
=> ( ! [A6: list_nat] :
( ( member_list_nat @ A6 @ S2 )
=> ( ( H @ ( J @ A6 ) )
= ( G2 @ A6 ) ) )
=> ( ( groups4396056296759096172at_nat @ G2 @ S2 )
= ( groups3542108847815614940at_nat @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_1249_sum_Oreindex__bij__witness,axiom,
! [S2: set_nat,I: list_nat > nat,J: nat > list_nat,T2: set_list_nat,H: list_nat > nat,G2: nat > nat] :
( ! [A6: nat] :
( ( member_nat @ A6 @ S2 )
=> ( ( I @ ( J @ A6 ) )
= A6 ) )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ S2 )
=> ( member_list_nat @ ( J @ A6 ) @ T2 ) )
=> ( ! [B6: list_nat] :
( ( member_list_nat @ B6 @ T2 )
=> ( ( J @ ( I @ B6 ) )
= B6 ) )
=> ( ! [B6: list_nat] :
( ( member_list_nat @ B6 @ T2 )
=> ( member_nat @ ( I @ B6 ) @ S2 ) )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ S2 )
=> ( ( H @ ( J @ A6 ) )
= ( G2 @ A6 ) ) )
=> ( ( groups3542108847815614940at_nat @ G2 @ S2 )
= ( groups4396056296759096172at_nat @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_1250_sum_Oreindex__bij__witness,axiom,
! [S2: set_nat,I: nat > nat,J: nat > nat,T2: set_nat,H: nat > nat,G2: nat > nat] :
( ! [A6: nat] :
( ( member_nat @ A6 @ S2 )
=> ( ( I @ ( J @ A6 ) )
= A6 ) )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ S2 )
=> ( member_nat @ ( J @ A6 ) @ T2 ) )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ T2 )
=> ( ( J @ ( I @ B6 ) )
= B6 ) )
=> ( ! [B6: nat] :
( ( member_nat @ B6 @ T2 )
=> ( member_nat @ ( I @ B6 ) @ S2 ) )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ S2 )
=> ( ( H @ ( J @ A6 ) )
= ( G2 @ A6 ) ) )
=> ( ( groups3542108847815614940at_nat @ G2 @ S2 )
= ( groups3542108847815614940at_nat @ H @ T2 ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness
thf(fact_1251_sum_Oeq__general__inverses,axiom,
! [B3: set_nat,K: nat > list_nat,A: set_list_nat,H: list_nat > nat,Gamma: nat > nat,Phi: list_nat > nat] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B3 )
=> ( ( member_list_nat @ ( K @ Y3 ) @ A )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
=> ( ( member_nat @ ( H @ X3 ) @ B3 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups4396056296759096172at_nat @ Phi @ A )
= ( groups3542108847815614940at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_1252_sum_Oeq__general__inverses,axiom,
! [B3: set_list_nat,K: list_nat > nat,A: set_nat,H: nat > list_nat,Gamma: list_nat > nat,Phi: nat > nat] :
( ! [Y3: list_nat] :
( ( member_list_nat @ Y3 @ B3 )
=> ( ( member_nat @ ( K @ Y3 ) @ A )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_list_nat @ ( H @ X3 ) @ B3 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A )
= ( groups4396056296759096172at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_1253_sum_Oeq__general__inverses,axiom,
! [B3: set_nat,K: nat > nat,A: set_nat,H: nat > nat,Gamma: nat > nat,Phi: nat > nat] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B3 )
=> ( ( member_nat @ ( K @ Y3 ) @ A )
& ( ( H @ ( K @ Y3 ) )
= Y3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ ( H @ X3 ) @ B3 )
& ( ( K @ ( H @ X3 ) )
= X3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A )
= ( groups3542108847815614940at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general_inverses
thf(fact_1254_sum_Oeq__general,axiom,
! [B3: set_nat,A: set_list_nat,H: list_nat > nat,Gamma: nat > nat,Phi: list_nat > nat] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B3 )
=> ? [X4: list_nat] :
( ( member_list_nat @ X4 @ A )
& ( ( H @ X4 )
= Y3 )
& ! [Ya2: list_nat] :
( ( ( member_list_nat @ Ya2 @ A )
& ( ( H @ Ya2 )
= Y3 ) )
=> ( Ya2 = X4 ) ) ) )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
=> ( ( member_nat @ ( H @ X3 ) @ B3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups4396056296759096172at_nat @ Phi @ A )
= ( groups3542108847815614940at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_1255_sum_Oeq__general,axiom,
! [B3: set_list_nat,A: set_nat,H: nat > list_nat,Gamma: list_nat > nat,Phi: nat > nat] :
( ! [Y3: list_nat] :
( ( member_list_nat @ Y3 @ B3 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ( H @ X4 )
= Y3 )
& ! [Ya2: nat] :
( ( ( member_nat @ Ya2 @ A )
& ( ( H @ Ya2 )
= Y3 ) )
=> ( Ya2 = X4 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_list_nat @ ( H @ X3 ) @ B3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A )
= ( groups4396056296759096172at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_1256_sum_Oeq__general,axiom,
! [B3: set_nat,A: set_nat,H: nat > nat,Gamma: nat > nat,Phi: nat > nat] :
( ! [Y3: nat] :
( ( member_nat @ Y3 @ B3 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( ( H @ X4 )
= Y3 )
& ! [Ya2: nat] :
( ( ( member_nat @ Ya2 @ A )
& ( ( H @ Ya2 )
= Y3 ) )
=> ( Ya2 = X4 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ ( H @ X3 ) @ B3 )
& ( ( Gamma @ ( H @ X3 ) )
= ( Phi @ X3 ) ) ) )
=> ( ( groups3542108847815614940at_nat @ Phi @ A )
= ( groups3542108847815614940at_nat @ Gamma @ B3 ) ) ) ) ).
% sum.eq_general
thf(fact_1257_sum_Ocong,axiom,
! [A: set_nat,B3: set_nat,G2: nat > nat,H: nat > nat] :
( ( A = B3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B3 )
=> ( ( G2 @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups3542108847815614940at_nat @ G2 @ A )
= ( groups3542108847815614940at_nat @ H @ B3 ) ) ) ) ).
% sum.cong
thf(fact_1258_sum_Oswap__restrict,axiom,
! [A: set_list_nat,B3: set_nat,G2: list_nat > nat > nat,R3: list_nat > nat > $o] :
( ( finite8100373058378681591st_nat @ A )
=> ( ( finite_finite_nat @ B3 )
=> ( ( groups4396056296759096172at_nat
@ ^ [X: list_nat] :
( groups3542108847815614940at_nat @ ( G2 @ X )
@ ( collect_nat
@ ^ [Y5: nat] :
( ( member_nat @ Y5 @ B3 )
& ( R3 @ X @ Y5 ) ) ) )
@ A )
= ( groups3542108847815614940at_nat
@ ^ [Y5: nat] :
( groups4396056296759096172at_nat
@ ^ [X: list_nat] : ( G2 @ X @ Y5 )
@ ( collect_list_nat
@ ^ [X: list_nat] :
( ( member_list_nat @ X @ A )
& ( R3 @ X @ Y5 ) ) ) )
@ B3 ) ) ) ) ).
% sum.swap_restrict
thf(fact_1259_sum_Oswap__restrict,axiom,
! [A: set_nat,B3: set_list_nat,G2: nat > list_nat > nat,R3: nat > list_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( finite8100373058378681591st_nat @ B3 )
=> ( ( groups3542108847815614940at_nat
@ ^ [X: nat] :
( groups4396056296759096172at_nat @ ( G2 @ X )
@ ( collect_list_nat
@ ^ [Y5: list_nat] :
( ( member_list_nat @ Y5 @ B3 )
& ( R3 @ X @ Y5 ) ) ) )
@ A )
= ( groups4396056296759096172at_nat
@ ^ [Y5: list_nat] :
( groups3542108847815614940at_nat
@ ^ [X: nat] : ( G2 @ X @ Y5 )
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( R3 @ X @ Y5 ) ) ) )
@ B3 ) ) ) ) ).
% sum.swap_restrict
thf(fact_1260_sum_Oswap__restrict,axiom,
! [A: set_nat,B3: set_nat,G2: nat > nat > nat,R3: nat > nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B3 )
=> ( ( groups3542108847815614940at_nat
@ ^ [X: nat] :
( groups3542108847815614940at_nat @ ( G2 @ X )
@ ( collect_nat
@ ^ [Y5: nat] :
( ( member_nat @ Y5 @ B3 )
& ( R3 @ X @ Y5 ) ) ) )
@ A )
= ( groups3542108847815614940at_nat
@ ^ [Y5: nat] :
( groups3542108847815614940at_nat
@ ^ [X: nat] : ( G2 @ X @ Y5 )
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( R3 @ X @ Y5 ) ) ) )
@ B3 ) ) ) ) ).
% sum.swap_restrict
thf(fact_1261_sum_Odistrib,axiom,
! [G2: nat > nat,H: nat > nat,A: set_nat] :
( ( groups3542108847815614940at_nat
@ ^ [X: nat] : ( plus_plus_nat @ ( G2 @ X ) @ ( H @ X ) )
@ A )
= ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G2 @ A ) @ ( groups3542108847815614940at_nat @ H @ A ) ) ) ).
% sum.distrib
thf(fact_1262_sum__multicount__gen,axiom,
! [S: set_nat,T: set_list_nat,R3: nat > list_nat > $o,K: list_nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( finite8100373058378681591st_nat @ T )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ T )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ S )
& ( R3 @ I4 @ X3 ) ) ) )
= ( K @ X3 ) ) )
=> ( ( groups3542108847815614940at_nat
@ ^ [I4: nat] :
( finite_card_list_nat
@ ( collect_list_nat
@ ^ [J3: list_nat] :
( ( member_list_nat @ J3 @ T )
& ( R3 @ I4 @ J3 ) ) ) )
@ S )
= ( groups4396056296759096172at_nat @ K @ T ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_1263_sum__multicount__gen,axiom,
! [S: set_nat,T: set_nat,R3: nat > nat > $o,K: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_nat @ T )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ T )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [I4: nat] :
( ( member_nat @ I4 @ S )
& ( R3 @ I4 @ X3 ) ) ) )
= ( K @ X3 ) ) )
=> ( ( groups3542108847815614940at_nat
@ ^ [I4: nat] :
( finite_card_nat
@ ( collect_nat
@ ^ [J3: nat] :
( ( member_nat @ J3 @ T )
& ( R3 @ I4 @ J3 ) ) ) )
@ S )
= ( groups3542108847815614940at_nat @ K @ T ) ) ) ) ) ).
% sum_multicount_gen
% 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 (2)
thf(conj_0,hypothesis,
member_list_nat @ x @ s ).
thf(conj_1,conjecture,
member_list_nat @ ( f @ x ) @ ( length_sum_set @ r @ m ) ).
%------------------------------------------------------------------------------