TPTP Problem File: SLH0418^1.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_00755_027838__13551634_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1459 ( 435 unt; 191 typ; 0 def)
% Number of atoms : 3932 (1068 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 11641 ( 434 ~; 62 |; 344 &;8847 @)
% ( 0 <=>;1954 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 16 ( 15 usr)
% Number of type conns : 783 ( 783 >; 0 *; 0 +; 0 <<)
% Number of symbols : 178 ( 176 usr; 20 con; 0-5 aty)
% Number of variables : 3758 ( 369 ^;3202 !; 187 ?;3758 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:15:00.021
%------------------------------------------------------------------------------
% Could-be-implicit typings (15)
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__List__Olist_Itf__a_J_J,type,
list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $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__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (176)
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_Countable__Set_Oto__nat__on_001tf__a,type,
counta3566351752493190365t_on_a: set_a > a > nat ).
thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
finite7325466520557071688st_nat: set_list_list_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Nat__Onat_J,type,
finite_card_list_nat: set_list_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__List__Olist_Itf__a_J,type,
finite_card_list_a: set_list_a > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > 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__List__Olist_Itf__a_J,type,
finite_finite_list_a: set_list_a > $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_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
minus_1139252259498527702_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > list_nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_Itf__a_M_Eo_J,type,
minus_minus_a_o: ( a > $o ) > ( a > $o ) > a > $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_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
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__List_Omonoid__add__class_Osum__list_001t__Nat__Onat,type,
groups4561878855575611511st_nat: list_nat > nat ).
thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate_001t__List__Olist_It__Nat__Onat_J,type,
infini2033088105919815547st_nat: set_list_nat > nat > list_nat ).
thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate_001t__Nat__Onat,type,
infini8530281810654367211te_nat: set_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_001tf__a,type,
khovanskii_a: set_a > ( a > a > a ) > a > set_a > $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_O_092_060alpha_062_001tf__a,type,
alpha_a: set_a > ( a > a > a ) > a > set_a > list_nat > a ).
thf(sy_c_Khovanskii_OKhovanskii_OaA_001t__List__Olist_It__Nat__Onat_J,type,
aA_list_nat: set_list_nat > list_list_nat ).
thf(sy_c_Khovanskii_OKhovanskii_OaA_001t__Nat__Onat,type,
aA_nat: set_nat > list_nat ).
thf(sy_c_Khovanskii_OKhovanskii_OaA_001tf__a,type,
aA_a: set_a > list_a ).
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__Nat__Onat,type,
useless_nat: set_nat > ( nat > nat > nat ) > nat > set_nat > list_nat > $o ).
thf(sy_c_Khovanskii_OKhovanskii_Ouseless_001tf__a,type,
useless_a: set_a > ( a > a > a ) > a > set_a > 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_OKhovanskii__axioms_001tf__a,type,
khovanskii_axioms_a: set_a > set_a > $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_Ofinsets_001tf__a,type,
finsets_a: set_a > nat > set_set_a ).
thf(sy_c_Khovanskii_Omax__pointwise,type,
max_pointwise: nat > set_list_nat > list_nat ).
thf(sy_c_Khovanskii_Omin__pointwise,type,
min_pointwise: nat > set_list_nat > list_nat ).
thf(sy_c_Khovanskii_Opointwise__le,type,
pointwise_le: list_nat > list_nat > $o ).
thf(sy_c_Khovanskii_Opointwise__less,type,
pointwise_less: list_nat > list_nat > $o ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__List__Olist_It__Nat__Onat_J_001t__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_Lattices__Big_Oord__class_Oarg__min__on_001tf__a_001t__Nat__Onat,type,
lattic6340287419671400565_a_nat: ( a > nat ) > set_a > a ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__List__Olist_It__Nat__Onat_J,type,
lattic5191180550204456963st_nat: set_list_nat > list_nat ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Nat__Onat,type,
lattic5238388535129920115in_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
lattic3683530169123051065st_nat: set_set_list_nat > set_list_nat ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Set__Oset_Itf__a_J,type,
lattic8209813465164889211_set_a: set_set_a > set_a ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__List__Olist_It__Nat__Onat_J,type,
lattic6411832703407573737st_nat: set_list_nat > list_nat ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Nat__Onat,type,
lattic1093996805478795353in_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
lattic2169124122975652127st_nat: set_set_list_nat > set_list_nat ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_Itf__a_J,type,
lattic2918178356826803221_set_a: set_set_a > set_a ).
thf(sy_c_List_OBleast_001t__List__Olist_It__Nat__Onat_J,type,
bleast_list_nat: set_list_nat > ( list_nat > $o ) > list_nat ).
thf(sy_c_List_OBleast_001t__Nat__Onat,type,
bleast_nat: set_nat > ( nat > $o ) > nat ).
thf(sy_c_List_Oabort__Bleast_001t__List__Olist_It__Nat__Onat_J,type,
abort_5940734655806358670st_nat: set_list_nat > ( list_nat > $o ) > list_nat ).
thf(sy_c_List_Oabort__Bleast_001t__Nat__Onat,type,
abort_Bleast_nat: set_nat > ( nat > $o ) > nat ).
thf(sy_c_List_Odistinct_001t__List__Olist_It__Nat__Onat_J,type,
distinct_list_nat: list_list_nat > $o ).
thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
distinct_nat: list_nat > $o ).
thf(sy_c_List_Odistinct_001tf__a,type,
distinct_a: list_a > $o ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Nat__Onat_J,type,
cons_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
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__List__Olist_Itf__a_J,type,
set_list_a2: list_list_a > set_list_a ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_List_Olist__ex_001t__List__Olist_It__Nat__Onat_J,type,
list_ex_list_nat: ( list_nat > $o ) > list_list_nat > $o ).
thf(sy_c_List_Olist__ex_001t__Nat__Onat,type,
list_ex_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__ex_001tf__a,type,
list_ex_a: ( a > $o ) > list_a > $o ).
thf(sy_c_List_Olist__update_001t__List__Olist_It__Nat__Onat_J,type,
list_update_list_nat: list_list_nat > nat > list_nat > list_list_nat ).
thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
list_update_nat: list_nat > nat > nat > list_nat ).
thf(sy_c_List_Olist__update_001tf__a,type,
list_update_a: list_a > nat > a > list_a ).
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_On__lists_001tf__a,type,
n_lists_a: nat > list_a > list_list_a ).
thf(sy_c_List_Onth_001t__List__Olist_It__Nat__Onat_J,type,
nth_list_nat: list_list_nat > nat > list_nat ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001tf__a,type,
nth_a: list_a > nat > a ).
thf(sy_c_List_Onull_001t__Nat__Onat,type,
null_nat: list_nat > $o ).
thf(sy_c_List_Onull_001tf__a,type,
null_a: list_a > $o ).
thf(sy_c_List_Oproduct__lists_001t__Nat__Onat,type,
product_lists_nat: list_list_nat > list_list_nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
size_s3023201423986296836st_nat: list_list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
size_size_list_a: list_a > nat ).
thf(sy_c_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
bot_bot_list_nat_o: list_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
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_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_OLeast_001t__List__Olist_It__Nat__Onat_J,type,
ord_Least_list_nat: ( list_nat > $o ) > list_nat ).
thf(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat,type,
ord_Least_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oord__class_OLeast_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
ord_Le6937613917142691334st_nat: ( set_list_nat > $o ) > set_list_nat ).
thf(sy_c_Orderings_Oord__class_OLeast_001t__Set__Oset_Itf__a_J,type,
ord_Least_set_a: ( set_a > $o ) > set_a ).
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_001_062_Itf__a_M_Eo_J,type,
ord_less_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__List__Olist_It__Nat__Onat_J,type,
ord_less_list_nat: list_nat > list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
ord_le1190675801316882794st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_It__Nat__Onat_J_M_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J_J,type,
ord_le6558929396352911974_nat_o: ( list_nat > list_nat > $o ) > ( list_nat > list_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
ord_le1520216061033275535_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_Eo_J,type,
ord_less_eq_a_o: ( a > $o ) > ( a > $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_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
orderi7613023277850353061st_nat: ( set_list_nat > set_list_nat > $o ) > ( set_list_nat > set_list_nat > $o ) > set_list_nat > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_Itf__a_J,type,
ordering_top_set_a: ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > set_a > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
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__List__Olist_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
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_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
insert_list_nat: list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Ois__empty_001t__List__Olist_It__Nat__Onat_J,type,
is_empty_list_nat: set_list_nat > $o ).
thf(sy_c_Set_Ois__empty_001tf__a,type,
is_empty_a: set_a > $o ).
thf(sy_c_Wellfounded_OwfP_001t__List__Olist_It__Nat__Onat_J,type,
wfP_list_nat: ( list_nat > list_nat > $o ) > $o ).
thf(sy_c_Wellfounded_OwfP_001t__Nat__Onat,type,
wfP_nat: ( nat > nat > $o ) > $o ).
thf(sy_c_Wellfounded_OwfP_001tf__a,type,
wfP_a: ( a > a > $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_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_A,type,
a2: set_a ).
thf(sy_v_G,type,
g: set_a ).
thf(sy_v_U,type,
u: set_list_nat ).
thf(sy_v_Ua____,type,
ua: set_list_nat ).
thf(sy_v_VF____,type,
vf: nat > nat > set_list_nat ).
thf(sy_v_V____,type,
v: set_list_nat ).
thf(sy_v_delete____,type,
delete: list_nat > list_nat ).
thf(sy_v_i____,type,
i: nat ).
thf(sy_v_r,type,
r: nat ).
thf(sy_v_ra____,type,
ra: nat ).
thf(sy_v_t____,type,
t: nat ).
thf(sy_v_u____,type,
u2: list_nat ).
thf(sy_v_ua____,type,
ua2: list_nat ).
thf(sy_v_v____,type,
v2: list_nat ).
% Relevant facts (1267)
thf(fact_0_minimal__elementsp_Ocases,axiom,
! [U: list_nat > $o,A: list_nat] :
( ( minimal_elementsp @ U @ A )
=> ~ ( ( U @ A )
=> ~ ! [Y: list_nat] :
( ( U @ Y )
=> ~ ( pointwise_less @ Y @ A ) ) ) ) ).
% minimal_elementsp.cases
thf(fact_1_minimal__elementsp_Ointros,axiom,
! [U: list_nat > $o,X: list_nat] :
( ( U @ X )
=> ( ! [Y2: list_nat] :
( ( U @ Y2 )
=> ~ ( pointwise_less @ Y2 @ X ) )
=> ( minimal_elementsp @ U @ X ) ) ) ).
% minimal_elementsp.intros
thf(fact_2_minimal__elementsp_Osimps,axiom,
( minimal_elementsp
= ( ^ [U2: list_nat > $o,A2: list_nat] :
? [X2: list_nat] :
( ( A2 = X2 )
& ( U2 @ X2 )
& ! [Y3: list_nat] :
( ( U2 @ Y3 )
=> ~ ( pointwise_less @ Y3 @ X2 ) ) ) ) ) ).
% minimal_elementsp.simps
thf(fact_3_minimal__elements_Ocases,axiom,
! [A: list_nat,U: set_list_nat] :
( ( member_list_nat @ A @ ( minimal_elements @ U ) )
=> ~ ( ( member_list_nat @ A @ U )
=> ~ ! [Y: list_nat] :
( ( member_list_nat @ Y @ U )
=> ~ ( pointwise_less @ Y @ A ) ) ) ) ).
% minimal_elements.cases
thf(fact_4_minimal__elements_Ointros,axiom,
! [X: list_nat,U: set_list_nat] :
( ( member_list_nat @ X @ U )
=> ( ! [Y2: list_nat] :
( ( member_list_nat @ Y2 @ U )
=> ~ ( pointwise_less @ Y2 @ X ) )
=> ( member_list_nat @ X @ ( minimal_elements @ U ) ) ) ) ).
% minimal_elements.intros
thf(fact_5_minimal__elements_Osimps,axiom,
! [A: list_nat,U: set_list_nat] :
( ( member_list_nat @ A @ ( minimal_elements @ U ) )
= ( ? [X2: list_nat] :
( ( A = X2 )
& ( member_list_nat @ X2 @ U )
& ! [Y3: list_nat] :
( ( member_list_nat @ Y3 @ U )
=> ~ ( pointwise_less @ Y3 @ X2 ) ) ) ) ) ).
% minimal_elements.simps
thf(fact_6__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062u_O_A_092_060lbrakk_062u_A_092_060in_062_AU_059_A_092_060And_062y_O_Ay_A_092_060lhd_062_Au_A_092_060Longrightarrow_062_Ay_A_092_060notin_062_AU_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [U3: list_nat] :
( ( member_list_nat @ U3 @ ua )
=> ~ ! [Y: list_nat] :
( ( pointwise_less @ Y @ U3 )
=> ~ ( member_list_nat @ Y @ ua ) ) ) ).
% \<open>\<And>thesis. (\<And>u. \<lbrakk>u \<in> U; \<And>y. y \<lhd> u \<Longrightarrow> y \<notin> U\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_7_WFP,axiom,
wfP_list_nat @ pointwise_less ).
% WFP
thf(fact_8_that_I2_J,axiom,
member_list_nat @ v2 @ ( vf @ i @ t ) ).
% that(2)
thf(fact_9_that_I1_J,axiom,
member_list_nat @ ua2 @ ( vf @ i @ t ) ).
% that(1)
thf(fact_10_u,axiom,
member_list_nat @ u2 @ ua ).
% u
thf(fact_11_False,axiom,
ua != bot_bot_set_list_nat ).
% False
thf(fact_12_delete__eq__iff,axiom,
! [U4: list_nat,V: list_nat] :
( ( member_list_nat @ U4 @ ( vf @ i @ t ) )
=> ( ( member_list_nat @ V @ ( vf @ i @ t ) )
=> ( ( ( delete @ U4 )
= ( delete @ V ) )
= ( U4 = V ) ) ) ) ).
% delete_eq_iff
thf(fact_13_zmin,axiom,
! [Y4: list_nat] :
( ( pointwise_less @ Y4 @ u2 )
=> ~ ( member_list_nat @ Y4 @ ua ) ) ).
% zmin
thf(fact_14_delete__le__iff,axiom,
! [U4: list_nat,V: list_nat] :
( ( member_list_nat @ U4 @ ( vf @ i @ t ) )
=> ( ( member_list_nat @ V @ ( vf @ i @ t ) )
=> ( ( pointwise_le @ ( delete @ U4 ) @ ( delete @ V ) )
= ( pointwise_le @ U4 @ V ) ) ) ) ).
% delete_le_iff
thf(fact_15_len__delete,axiom,
! [U4: list_nat] :
( ( member_list_nat @ U4 @ ( vf @ i @ t ) )
=> ( ( size_size_list_nat @ ( delete @ U4 ) )
= ra ) ) ).
% len_delete
thf(fact_16_wfPUNIVI,axiom,
! [R: list_nat > list_nat > $o] :
( ! [P: list_nat > $o,X3: list_nat] :
( ! [Xa: list_nat] :
( ! [Y2: list_nat] :
( ( R @ Y2 @ Xa )
=> ( P @ Y2 ) )
=> ( P @ Xa ) )
=> ( P @ X3 ) )
=> ( wfP_list_nat @ R ) ) ).
% wfPUNIVI
thf(fact_17_wfP__induct,axiom,
! [R: list_nat > list_nat > $o,P2: list_nat > $o,A: list_nat] :
( ( wfP_list_nat @ R )
=> ( ! [X3: list_nat] :
( ! [Y: list_nat] :
( ( R @ Y @ X3 )
=> ( P2 @ Y ) )
=> ( P2 @ X3 ) )
=> ( P2 @ A ) ) ) ).
% wfP_induct
thf(fact_18_wfP__eq__minimal,axiom,
( wfP_nat
= ( ^ [R2: nat > nat > $o] :
! [Q: set_nat] :
( ? [X2: nat] : ( member_nat @ X2 @ Q )
=> ? [X2: nat] :
( ( member_nat @ X2 @ Q )
& ! [Y3: nat] :
( ( R2 @ Y3 @ X2 )
=> ~ ( member_nat @ Y3 @ Q ) ) ) ) ) ) ).
% wfP_eq_minimal
thf(fact_19_wfP__eq__minimal,axiom,
( wfP_a
= ( ^ [R2: a > a > $o] :
! [Q: set_a] :
( ? [X2: a] : ( member_a @ X2 @ Q )
=> ? [X2: a] :
( ( member_a @ X2 @ Q )
& ! [Y3: a] :
( ( R2 @ Y3 @ X2 )
=> ~ ( member_a @ Y3 @ Q ) ) ) ) ) ) ).
% wfP_eq_minimal
thf(fact_20_wfP__eq__minimal,axiom,
( wfP_list_nat
= ( ^ [R2: list_nat > list_nat > $o] :
! [Q: set_list_nat] :
( ? [X2: list_nat] : ( member_list_nat @ X2 @ Q )
=> ? [X2: list_nat] :
( ( member_list_nat @ X2 @ Q )
& ! [Y3: list_nat] :
( ( R2 @ Y3 @ X2 )
=> ~ ( member_list_nat @ Y3 @ Q ) ) ) ) ) ) ).
% wfP_eq_minimal
thf(fact_21_wfP__induct__rule,axiom,
! [R: list_nat > list_nat > $o,P2: list_nat > $o,A: list_nat] :
( ( wfP_list_nat @ R )
=> ( ! [X3: list_nat] :
( ! [Y: list_nat] :
( ( R @ Y @ X3 )
=> ( P2 @ Y ) )
=> ( P2 @ X3 ) )
=> ( P2 @ A ) ) ) ).
% wfP_induct_rule
thf(fact_22_wfP__if__convertible__to__wfP,axiom,
! [S: list_nat > list_nat > $o,R3: list_nat > list_nat > $o,F: list_nat > list_nat] :
( ( wfP_list_nat @ S )
=> ( ! [X3: list_nat,Y2: list_nat] :
( ( R3 @ X3 @ Y2 )
=> ( S @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( wfP_list_nat @ R3 ) ) ) ).
% wfP_if_convertible_to_wfP
thf(fact_23__092_060open_062i_A_092_060le_062_Ar_092_060close_062,axiom,
ord_less_eq_nat @ i @ ra ).
% \<open>i \<le> r\<close>
thf(fact_24__092_060open_062t_A_060_Au_____A_B_Ai_092_060close_062,axiom,
ord_less_nat @ t @ ( nth_nat @ u2 @ i ) ).
% \<open>t < u__ ! i\<close>
thf(fact_25_pointwise__le__refl,axiom,
! [X: list_nat] : ( pointwise_le @ X @ X ) ).
% pointwise_le_refl
thf(fact_26_assms,axiom,
! [X: list_nat] :
( ( member_list_nat @ X @ u )
=> ( ( size_size_list_nat @ X )
= r ) ) ).
% assms
thf(fact_27__C_K_C,axiom,
! [V: list_nat] :
( ( member_list_nat @ V @ v )
=> ? [I: nat] :
( ( ord_less_eq_nat @ I @ ra )
& ( ord_less_nat @ ( nth_nat @ V @ I ) @ ( nth_nat @ u2 @ I ) ) ) ) ).
% "*"
thf(fact_28_pointwise__le__iff__nth,axiom,
( pointwise_le
= ( ^ [X2: list_nat,Y3: list_nat] :
( ( ( size_size_list_nat @ X2 )
= ( size_size_list_nat @ Y3 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ X2 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ X2 @ I2 ) @ ( nth_nat @ Y3 @ I2 ) ) ) ) ) ) ).
% pointwise_le_iff_nth
thf(fact_29_pointwise__le__antisym,axiom,
! [X: list_nat,Y4: list_nat] :
( ( pointwise_le @ X @ Y4 )
=> ( ( pointwise_le @ Y4 @ X )
=> ( X = Y4 ) ) ) ).
% pointwise_le_antisym
thf(fact_30_pointwise__less__iff2,axiom,
( pointwise_less
= ( ^ [X2: list_nat,Y3: list_nat] :
( ( pointwise_le @ X2 @ Y3 )
& ? [K: nat] :
( ( ord_less_nat @ K @ ( size_size_list_nat @ X2 ) )
& ( ord_less_nat @ ( nth_nat @ X2 @ K ) @ ( nth_nat @ Y3 @ K ) ) ) ) ) ) ).
% pointwise_less_iff2
thf(fact_31_pointwise__le__trans,axiom,
! [X: list_nat,Y4: list_nat,Z: list_nat] :
( ( pointwise_le @ X @ Y4 )
=> ( ( pointwise_le @ Y4 @ Z )
=> ( pointwise_le @ X @ Z ) ) ) ).
% pointwise_le_trans
thf(fact_32_Khovanskii_Ononempty,axiom,
! [G: set_list_nat,Addition: list_nat > list_nat > list_nat,Zero: list_nat,A3: set_list_nat] :
( ( khovanskii_list_nat @ G @ Addition @ Zero @ A3 )
=> ( A3 != bot_bot_set_list_nat ) ) ).
% Khovanskii.nonempty
thf(fact_33_Khovanskii_Ononempty,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A3: set_a] :
( ( khovanskii_a @ G @ Addition @ Zero @ A3 )
=> ( A3 != bot_bot_set_a ) ) ).
% Khovanskii.nonempty
thf(fact_34_wfP__less,axiom,
wfP_list_nat @ ord_less_list_nat ).
% wfP_less
thf(fact_35_wfP__less,axiom,
wfP_nat @ ord_less_nat ).
% wfP_less
thf(fact_36_wfP__if__convertible__to__nat,axiom,
! [R3: list_nat > list_nat > $o,F: list_nat > nat] :
( ! [X3: list_nat,Y2: list_nat] :
( ( R3 @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( wfP_list_nat @ R3 ) ) ).
% wfP_if_convertible_to_nat
thf(fact_37_pointwise__less__def,axiom,
( pointwise_less
= ( ^ [X2: list_nat,Y3: list_nat] :
( ( pointwise_le @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% pointwise_less_def
thf(fact_38_pointwise__le__iff__less__equal,axiom,
( pointwise_le
= ( ^ [X2: list_nat,Y3: list_nat] :
( ( pointwise_less @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% pointwise_le_iff_less_equal
thf(fact_39_mem__Collect__eq,axiom,
! [A: a,P2: a > $o] :
( ( member_a @ A @ ( collect_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_40_mem__Collect__eq,axiom,
! [A: list_nat,P2: list_nat > $o] :
( ( member_list_nat @ A @ ( collect_list_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_41_mem__Collect__eq,axiom,
! [A: nat,P2: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_42_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_43_Collect__mem__eq,axiom,
! [A3: set_list_nat] :
( ( collect_list_nat
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A3: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_45_Collect__cong,axiom,
! [P2: list_nat > $o,Q2: list_nat > $o] :
( ! [X3: list_nat] :
( ( P2 @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collect_list_nat @ P2 )
= ( collect_list_nat @ Q2 ) ) ) ).
% Collect_cong
thf(fact_46_Collect__cong,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ! [X3: nat] :
( ( P2 @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collect_nat @ P2 )
= ( collect_nat @ Q2 ) ) ) ).
% Collect_cong
thf(fact_47_nth__delete,axiom,
! [U4: list_nat,K2: nat] :
( ( member_list_nat @ U4 @ ( vf @ i @ t ) )
=> ( ( ord_less_nat @ K2 @ ra )
=> ( ( ( ord_less_nat @ K2 @ i )
=> ( ( nth_nat @ ( delete @ U4 ) @ K2 )
= ( nth_nat @ U4 @ K2 ) ) )
& ( ~ ( ord_less_nat @ K2 @ i )
=> ( ( nth_nat @ ( delete @ U4 ) @ K2 )
= ( nth_nat @ U4 @ ( suc @ K2 ) ) ) ) ) ) ) ).
% nth_delete
thf(fact_48_Suc_Oprems,axiom,
! [X: list_nat] :
( ( member_list_nat @ X @ ua )
=> ( ( size_size_list_nat @ X )
= ( suc @ ra ) ) ) ).
% Suc.prems
thf(fact_49_Suc_OIH,axiom,
! [U: set_list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ U )
=> ( ( size_size_list_nat @ X3 )
= ra ) )
=> ( finite8100373058378681591st_nat @ ( minimal_elements @ U ) ) ) ).
% Suc.IH
thf(fact_50_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_a,Z2: list_a] : ( Y5 = Z2 ) )
= ( ^ [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ Xs @ I2 )
= ( nth_a @ Ys @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_51_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_nat,Z2: list_nat] : ( Y5 = Z2 ) )
= ( ^ [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I2 )
= ( nth_nat @ Ys @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_52_Skolem__list__nth,axiom,
! [K2: nat,P2: nat > a > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K2 )
=> ? [X4: a] : ( P2 @ I2 @ X4 ) ) )
= ( ? [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= K2 )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K2 )
=> ( P2 @ I2 @ ( nth_a @ Xs @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_53_Skolem__list__nth,axiom,
! [K2: nat,P2: nat > nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K2 )
=> ? [X4: nat] : ( P2 @ I2 @ X4 ) ) )
= ( ? [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= K2 )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K2 )
=> ( P2 @ I2 @ ( nth_nat @ Xs @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_54_nth__equalityI,axiom,
! [Xs2: list_a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs2 ) )
=> ( ( nth_a @ Xs2 @ I )
= ( nth_a @ Ys2 @ I ) ) )
=> ( Xs2 = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_55_nth__equalityI,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ Xs2 @ I )
= ( nth_nat @ Ys2 @ I ) ) )
=> ( Xs2 = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_56_empty__Collect__eq,axiom,
! [P2: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P2 ) )
= ( ! [X2: nat] :
~ ( P2 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_57_empty__Collect__eq,axiom,
! [P2: list_nat > $o] :
( ( bot_bot_set_list_nat
= ( collect_list_nat @ P2 ) )
= ( ! [X2: list_nat] :
~ ( P2 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_58_empty__Collect__eq,axiom,
! [P2: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P2 ) )
= ( ! [X2: a] :
~ ( P2 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_59_Collect__empty__eq,axiom,
! [P2: nat > $o] :
( ( ( collect_nat @ P2 )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P2 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_60_Collect__empty__eq,axiom,
! [P2: list_nat > $o] :
( ( ( collect_list_nat @ P2 )
= bot_bot_set_list_nat )
= ( ! [X2: list_nat] :
~ ( P2 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_61_Collect__empty__eq,axiom,
! [P2: a > $o] :
( ( ( collect_a @ P2 )
= bot_bot_set_a )
= ( ! [X2: a] :
~ ( P2 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_62_all__not__in__conv,axiom,
! [A3: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat @ X2 @ A3 ) )
= ( A3 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_63_all__not__in__conv,axiom,
! [A3: set_list_nat] :
( ( ! [X2: list_nat] :
~ ( member_list_nat @ X2 @ A3 ) )
= ( A3 = bot_bot_set_list_nat ) ) ).
% all_not_in_conv
thf(fact_64_all__not__in__conv,axiom,
! [A3: set_a] :
( ( ! [X2: a] :
~ ( member_a @ X2 @ A3 ) )
= ( A3 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_65_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_66_empty__iff,axiom,
! [C: list_nat] :
~ ( member_list_nat @ C @ bot_bot_set_list_nat ) ).
% empty_iff
thf(fact_67_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_68_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_69_order__refl,axiom,
! [X: set_list_nat] : ( ord_le6045566169113846134st_nat @ X @ X ) ).
% order_refl
thf(fact_70_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_71_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_72_dual__order_Orefl,axiom,
! [A: set_list_nat] : ( ord_le6045566169113846134st_nat @ A @ A ) ).
% dual_order.refl
thf(fact_73_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_74__092_060open_062_092_060And_062v_At_Ai_O_Av_A_092_060in_062_AVF_Ai_At_A_092_060Longrightarrow_062_Alength_Av_A_061_ASuc_Ar_092_060close_062,axiom,
! [V: list_nat,I3: nat,T: nat] :
( ( member_list_nat @ V @ ( vf @ I3 @ T ) )
=> ( ( size_size_list_nat @ V )
= ( suc @ ra ) ) ) ).
% \<open>\<And>v t i. v \<in> VF i t \<Longrightarrow> length v = Suc r\<close>
thf(fact_75_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_76_bot__set__def,axiom,
( bot_bot_set_list_nat
= ( collect_list_nat @ bot_bot_list_nat_o ) ) ).
% bot_set_def
thf(fact_77_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_78_not__psubset__empty,axiom,
! [A3: set_list_nat] :
~ ( ord_le1190675801316882794st_nat @ A3 @ bot_bot_set_list_nat ) ).
% not_psubset_empty
thf(fact_79_not__psubset__empty,axiom,
! [A3: set_a] :
~ ( ord_less_set_a @ A3 @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_80_Khovanskii_OfinA,axiom,
! [G: set_list_nat,Addition: list_nat > list_nat > list_nat,Zero: list_nat,A3: set_list_nat] :
( ( khovanskii_list_nat @ G @ Addition @ Zero @ A3 )
=> ( finite8100373058378681591st_nat @ A3 ) ) ).
% Khovanskii.finA
thf(fact_81_Khovanskii_OfinA,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A3: set_nat] :
( ( khovanskii_nat @ G @ Addition @ Zero @ A3 )
=> ( finite_finite_nat @ A3 ) ) ).
% Khovanskii.finA
thf(fact_82_Khovanskii_OfinA,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A3: set_a] :
( ( khovanskii_a @ G @ Addition @ Zero @ A3 )
=> ( finite_finite_a @ A3 ) ) ).
% Khovanskii.finA
thf(fact_83_finite__maxlen,axiom,
! [M: set_list_nat] :
( ( finite8100373058378681591st_nat @ M )
=> ? [N: nat] :
! [X5: list_nat] :
( ( member_list_nat @ X5 @ M )
=> ( ord_less_nat @ ( size_size_list_nat @ X5 ) @ N ) ) ) ).
% finite_maxlen
thf(fact_84_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_85_le__cases3,axiom,
! [X: nat,Y4: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y4 )
=> ~ ( ord_less_eq_nat @ Y4 @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y4 @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y4 ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y4 )
=> ~ ( ord_less_eq_nat @ Y4 @ X ) )
=> ( ( ( ord_less_eq_nat @ Y4 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_86_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_87_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_list_nat,Z2: set_list_nat] : ( Y5 = Z2 ) )
= ( ^ [X2: set_list_nat,Y3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X2 @ Y3 )
& ( ord_le6045566169113846134st_nat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_88_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
& ( ord_less_eq_set_a @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_89_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_90_ord__eq__le__trans,axiom,
! [A: set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( A = B )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ord_le6045566169113846134st_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_91_ord__eq__le__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( A = B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_92_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_93_ord__le__eq__trans,axiom,
! [A: set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( B = C )
=> ( ord_le6045566169113846134st_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_94_ord__le__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_95_order__antisym,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ X )
=> ( X = Y4 ) ) ) ).
% order_antisym
thf(fact_96_order__antisym,axiom,
! [X: set_list_nat,Y4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X @ Y4 )
=> ( ( ord_le6045566169113846134st_nat @ Y4 @ X )
=> ( X = Y4 ) ) ) ).
% order_antisym
thf(fact_97_order__antisym,axiom,
! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ( ord_less_eq_set_a @ Y4 @ X )
=> ( X = Y4 ) ) ) ).
% order_antisym
thf(fact_98_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_99_order_Otrans,axiom,
! [A: set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ord_le6045566169113846134st_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_100_order_Otrans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_101_order__trans,axiom,
! [X: nat,Y4: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_102_order__trans,axiom,
! [X: set_list_nat,Y4: set_list_nat,Z: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X @ Y4 )
=> ( ( ord_le6045566169113846134st_nat @ Y4 @ Z )
=> ( ord_le6045566169113846134st_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_103_order__trans,axiom,
! [X: set_a,Y4: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ( ord_less_eq_set_a @ Y4 @ Z )
=> ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% order_trans
thf(fact_104_linorder__wlog,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( P2 @ A4 @ B2 ) )
=> ( ! [A4: nat,B2: nat] :
( ( P2 @ B2 @ A4 )
=> ( P2 @ A4 @ B2 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_105_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
& ( ord_less_eq_nat @ A2 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_106_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_list_nat,Z2: set_list_nat] : ( Y5 = Z2 ) )
= ( ^ [A2: set_list_nat,B3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B3 @ A2 )
& ( ord_le6045566169113846134st_nat @ A2 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_107_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
& ( ord_less_eq_set_a @ A2 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_108_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_109_dual__order_Oantisym,axiom,
! [B: set_list_nat,A: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B @ A )
=> ( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_110_dual__order_Oantisym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_111_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_112_dual__order_Otrans,axiom,
! [B: set_list_nat,A: set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B @ A )
=> ( ( ord_le6045566169113846134st_nat @ C @ B )
=> ( ord_le6045566169113846134st_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_113_dual__order_Otrans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_114_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_115_antisym,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ord_le6045566169113846134st_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_116_antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_117_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
& ( ord_less_eq_nat @ B3 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_118_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_list_nat,Z2: set_list_nat] : ( Y5 = Z2 ) )
= ( ^ [A2: set_list_nat,B3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B3 )
& ( ord_le6045566169113846134st_nat @ B3 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_119_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_120_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_121_order__subst1,axiom,
! [A: nat,F: set_list_nat > nat,B: set_list_nat,C: set_list_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ! [X3: set_list_nat,Y2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_122_order__subst1,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_123_order__subst1,axiom,
! [A: set_list_nat,F: nat > set_list_nat,B: nat,C: nat] :
( ( ord_le6045566169113846134st_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6045566169113846134st_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_124_order__subst1,axiom,
! [A: set_list_nat,F: set_list_nat > set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ ( F @ B ) )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ! [X3: set_list_nat,Y2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y2 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6045566169113846134st_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_125_order__subst1,axiom,
! [A: set_list_nat,F: set_a > set_list_nat,B: set_a,C: set_a] :
( ( ord_le6045566169113846134st_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6045566169113846134st_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_126_order__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_127_order__subst1,axiom,
! [A: set_a,F: set_list_nat > set_a,B: set_list_nat,C: set_list_nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ! [X3: set_list_nat,Y2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_128_order__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_129_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_130_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_list_nat,C: set_list_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le6045566169113846134st_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6045566169113846134st_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_131_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_132_order__subst2,axiom,
! [A: set_list_nat,B: set_list_nat,F: set_list_nat > nat,C: nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_list_nat,Y2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_133_order__subst2,axiom,
! [A: set_list_nat,B: set_list_nat,F: set_list_nat > set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ord_le6045566169113846134st_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_list_nat,Y2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y2 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6045566169113846134st_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_134_order__subst2,axiom,
! [A: set_list_nat,B: set_list_nat,F: set_list_nat > set_a,C: set_a] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X3: set_list_nat,Y2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_135_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_136_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_list_nat,C: set_list_nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_le6045566169113846134st_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6045566169113846134st_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_137_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_138_order__eq__refl,axiom,
! [X: nat,Y4: nat] :
( ( X = Y4 )
=> ( ord_less_eq_nat @ X @ Y4 ) ) ).
% order_eq_refl
thf(fact_139_order__eq__refl,axiom,
! [X: set_list_nat,Y4: set_list_nat] :
( ( X = Y4 )
=> ( ord_le6045566169113846134st_nat @ X @ Y4 ) ) ).
% order_eq_refl
thf(fact_140_order__eq__refl,axiom,
! [X: set_a,Y4: set_a] :
( ( X = Y4 )
=> ( ord_less_eq_set_a @ X @ Y4 ) ) ).
% order_eq_refl
thf(fact_141_linorder__linear,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
| ( ord_less_eq_nat @ Y4 @ X ) ) ).
% linorder_linear
thf(fact_142_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_143_ord__eq__le__subst,axiom,
! [A: set_list_nat,F: nat > set_list_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6045566169113846134st_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_144_ord__eq__le__subst,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_145_ord__eq__le__subst,axiom,
! [A: nat,F: set_list_nat > nat,B: set_list_nat,C: set_list_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ! [X3: set_list_nat,Y2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_146_ord__eq__le__subst,axiom,
! [A: set_list_nat,F: set_list_nat > set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ! [X3: set_list_nat,Y2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y2 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6045566169113846134st_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_147_ord__eq__le__subst,axiom,
! [A: set_a,F: set_list_nat > set_a,B: set_list_nat,C: set_list_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ! [X3: set_list_nat,Y2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_148_ord__eq__le__subst,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_149_ord__eq__le__subst,axiom,
! [A: set_list_nat,F: set_a > set_list_nat,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6045566169113846134st_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_150_ord__eq__le__subst,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_151_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_152_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_list_nat,C: set_list_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6045566169113846134st_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_153_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_154_ord__le__eq__subst,axiom,
! [A: set_list_nat,B: set_list_nat,F: set_list_nat > nat,C: nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_list_nat,Y2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_155_ord__le__eq__subst,axiom,
! [A: set_list_nat,B: set_list_nat,F: set_list_nat > set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_list_nat,Y2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y2 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6045566169113846134st_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_156_ord__le__eq__subst,axiom,
! [A: set_list_nat,B: set_list_nat,F: set_list_nat > set_a,C: set_a] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_list_nat,Y2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_157_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_158_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_list_nat,C: set_list_nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le6045566169113846134st_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_159_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_160_linorder__le__cases,axiom,
! [X: nat,Y4: nat] :
( ~ ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X ) ) ).
% linorder_le_cases
thf(fact_161_order__antisym__conv,axiom,
! [Y4: nat,X: nat] :
( ( ord_less_eq_nat @ Y4 @ X )
=> ( ( ord_less_eq_nat @ X @ Y4 )
= ( X = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_162_order__antisym__conv,axiom,
! [Y4: set_list_nat,X: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ Y4 @ X )
=> ( ( ord_le6045566169113846134st_nat @ X @ Y4 )
= ( X = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_163_order__antisym__conv,axiom,
! [Y4: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y4 @ X )
=> ( ( ord_less_eq_set_a @ X @ Y4 )
= ( X = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_164_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_165_less__imp__neq,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( X != Y4 ) ) ).
% less_imp_neq
thf(fact_166_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_167_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_168_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_169_less__induct,axiom,
! [P2: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y: nat] :
( ( ord_less_nat @ Y @ X3 )
=> ( P2 @ Y ) )
=> ( P2 @ X3 ) )
=> ( P2 @ A ) ) ).
% less_induct
thf(fact_170_antisym__conv3,axiom,
! [Y4: nat,X: nat] :
( ~ ( ord_less_nat @ Y4 @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y4 ) )
= ( X = Y4 ) ) ) ).
% antisym_conv3
thf(fact_171_linorder__cases,axiom,
! [X: nat,Y4: nat] :
( ~ ( ord_less_nat @ X @ Y4 )
=> ( ( X != Y4 )
=> ( ord_less_nat @ Y4 @ X ) ) ) ).
% linorder_cases
thf(fact_172_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_173_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_174_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X6: nat] : ( P3 @ X6 ) )
= ( ^ [P4: nat > $o] :
? [N2: nat] :
( ( P4 @ N2 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ( P4 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_175_linorder__less__wlog,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B2: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( P2 @ A4 @ B2 ) )
=> ( ! [A4: nat] : ( P2 @ A4 @ A4 )
=> ( ! [A4: nat,B2: nat] :
( ( P2 @ B2 @ A4 )
=> ( P2 @ A4 @ B2 ) )
=> ( P2 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_176_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_177_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y4: nat] :
( ( ~ ( ord_less_nat @ X @ Y4 ) )
= ( ( ord_less_nat @ Y4 @ X )
| ( X = Y4 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_178_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_179_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_180_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_181_linorder__neqE,axiom,
! [X: nat,Y4: nat] :
( ( X != Y4 )
=> ( ~ ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ Y4 @ X ) ) ) ).
% linorder_neqE
thf(fact_182_order__less__asym,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ~ ( ord_less_nat @ Y4 @ X ) ) ).
% order_less_asym
thf(fact_183_linorder__neq__iff,axiom,
! [X: nat,Y4: nat] :
( ( X != Y4 )
= ( ( ord_less_nat @ X @ Y4 )
| ( ord_less_nat @ Y4 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_184_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_185_order__less__trans,axiom,
! [X: nat,Y4: nat,Z: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ( ord_less_nat @ Y4 @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_186_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_187_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_188_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_189_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_190_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_191_order__less__not__sym,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ~ ( ord_less_nat @ Y4 @ X ) ) ).
% order_less_not_sym
thf(fact_192_order__less__imp__triv,axiom,
! [X: nat,Y4: nat,P2: $o] :
( ( ord_less_nat @ X @ Y4 )
=> ( ( ord_less_nat @ Y4 @ X )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_193_linorder__less__linear,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
| ( X = Y4 )
| ( ord_less_nat @ Y4 @ X ) ) ).
% linorder_less_linear
thf(fact_194_order__less__imp__not__eq,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( X != Y4 ) ) ).
% order_less_imp_not_eq
thf(fact_195_order__less__imp__not__eq2,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( Y4 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_196_order__less__imp__not__less,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ~ ( ord_less_nat @ Y4 @ X ) ) ).
% order_less_imp_not_less
thf(fact_197_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_198_emptyE,axiom,
! [A: list_nat] :
~ ( member_list_nat @ A @ bot_bot_set_list_nat ) ).
% emptyE
thf(fact_199_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_200_equals0D,axiom,
! [A3: set_nat,A: nat] :
( ( A3 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A3 ) ) ).
% equals0D
thf(fact_201_equals0D,axiom,
! [A3: set_list_nat,A: list_nat] :
( ( A3 = bot_bot_set_list_nat )
=> ~ ( member_list_nat @ A @ A3 ) ) ).
% equals0D
thf(fact_202_equals0D,axiom,
! [A3: set_a,A: a] :
( ( A3 = bot_bot_set_a )
=> ~ ( member_a @ A @ A3 ) ) ).
% equals0D
thf(fact_203_equals0I,axiom,
! [A3: set_nat] :
( ! [Y2: nat] :
~ ( member_nat @ Y2 @ A3 )
=> ( A3 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_204_equals0I,axiom,
! [A3: set_list_nat] :
( ! [Y2: list_nat] :
~ ( member_list_nat @ Y2 @ A3 )
=> ( A3 = bot_bot_set_list_nat ) ) ).
% equals0I
thf(fact_205_equals0I,axiom,
! [A3: set_a] :
( ! [Y2: a] :
~ ( member_a @ Y2 @ A3 )
=> ( A3 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_206_ex__in__conv,axiom,
! [A3: set_nat] :
( ( ? [X2: nat] : ( member_nat @ X2 @ A3 ) )
= ( A3 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_207_ex__in__conv,axiom,
! [A3: set_list_nat] :
( ( ? [X2: list_nat] : ( member_list_nat @ X2 @ A3 ) )
= ( A3 != bot_bot_set_list_nat ) ) ).
% ex_in_conv
thf(fact_208_ex__in__conv,axiom,
! [A3: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A3 ) )
= ( A3 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_209_Ex__list__of__length,axiom,
! [N3: nat] :
? [Xs3: list_nat] :
( ( size_size_list_nat @ Xs3 )
= N3 ) ).
% Ex_list_of_length
thf(fact_210_neq__if__length__neq,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
!= ( size_size_list_nat @ Ys2 ) )
=> ( Xs2 != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_211_leD,axiom,
! [Y4: nat,X: nat] :
( ( ord_less_eq_nat @ Y4 @ X )
=> ~ ( ord_less_nat @ X @ Y4 ) ) ).
% leD
thf(fact_212_leD,axiom,
! [Y4: set_list_nat,X: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ Y4 @ X )
=> ~ ( ord_le1190675801316882794st_nat @ X @ Y4 ) ) ).
% leD
thf(fact_213_leD,axiom,
! [Y4: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y4 @ X )
=> ~ ( ord_less_set_a @ X @ Y4 ) ) ).
% leD
thf(fact_214_leI,axiom,
! [X: nat,Y4: nat] :
( ~ ( ord_less_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X ) ) ).
% leI
thf(fact_215_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_216_nless__le,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( ~ ( ord_le1190675801316882794st_nat @ A @ B ) )
= ( ~ ( ord_le6045566169113846134st_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_217_nless__le,axiom,
! [A: set_a,B: set_a] :
( ( ~ ( ord_less_set_a @ A @ B ) )
= ( ~ ( ord_less_eq_set_a @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_218_antisym__conv1,axiom,
! [X: nat,Y4: nat] :
( ~ ( ord_less_nat @ X @ Y4 )
=> ( ( ord_less_eq_nat @ X @ Y4 )
= ( X = Y4 ) ) ) ).
% antisym_conv1
thf(fact_219_antisym__conv1,axiom,
! [X: set_list_nat,Y4: set_list_nat] :
( ~ ( ord_le1190675801316882794st_nat @ X @ Y4 )
=> ( ( ord_le6045566169113846134st_nat @ X @ Y4 )
= ( X = Y4 ) ) ) ).
% antisym_conv1
thf(fact_220_antisym__conv1,axiom,
! [X: set_a,Y4: set_a] :
( ~ ( ord_less_set_a @ X @ Y4 )
=> ( ( ord_less_eq_set_a @ X @ Y4 )
= ( X = Y4 ) ) ) ).
% antisym_conv1
thf(fact_221_antisym__conv2,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ( ~ ( ord_less_nat @ X @ Y4 ) )
= ( X = Y4 ) ) ) ).
% antisym_conv2
thf(fact_222_antisym__conv2,axiom,
! [X: set_list_nat,Y4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X @ Y4 )
=> ( ( ~ ( ord_le1190675801316882794st_nat @ X @ Y4 ) )
= ( X = Y4 ) ) ) ).
% antisym_conv2
thf(fact_223_antisym__conv2,axiom,
! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ( ~ ( ord_less_set_a @ X @ Y4 ) )
= ( X = Y4 ) ) ) ).
% antisym_conv2
thf(fact_224_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ~ ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_225_less__le__not__le,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [X2: set_list_nat,Y3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X2 @ Y3 )
& ~ ( ord_le6045566169113846134st_nat @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_226_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
& ~ ( ord_less_eq_set_a @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_227_not__le__imp__less,axiom,
! [Y4: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y4 @ X )
=> ( ord_less_nat @ X @ Y4 ) ) ).
% not_le_imp_less
thf(fact_228_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ B3 )
| ( A2 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_229_order_Oorder__iff__strict,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A2: set_list_nat,B3: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A2 @ B3 )
| ( A2 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_230_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B3: set_a] :
( ( ord_less_set_a @ A2 @ B3 )
| ( A2 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_231_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
& ( A2 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_232_order_Ostrict__iff__order,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [A2: set_list_nat,B3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B3 )
& ( A2 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_233_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
& ( A2 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_234_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_235_order_Ostrict__trans1,axiom,
! [A: set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ord_le1190675801316882794st_nat @ B @ C )
=> ( ord_le1190675801316882794st_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_236_order_Ostrict__trans1,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_237_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_238_order_Ostrict__trans2,axiom,
! [A: set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A @ B )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ord_le1190675801316882794st_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_239_order_Ostrict__trans2,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_240_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_241_order_Ostrict__iff__not,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [A2: set_list_nat,B3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B3 )
& ~ ( ord_le6045566169113846134st_nat @ B3 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_242_order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
& ~ ( ord_less_eq_set_a @ B3 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_243_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A2: nat] :
( ( ord_less_nat @ B3 @ A2 )
| ( A2 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_244_dual__order_Oorder__iff__strict,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [B3: set_list_nat,A2: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ B3 @ A2 )
| ( A2 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_245_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B3: set_a,A2: set_a] :
( ( ord_less_set_a @ B3 @ A2 )
| ( A2 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_246_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
& ( A2 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_247_dual__order_Ostrict__iff__order,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [B3: set_list_nat,A2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B3 @ A2 )
& ( A2 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_248_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
& ( A2 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_249_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_250_dual__order_Ostrict__trans1,axiom,
! [B: set_list_nat,A: set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B @ A )
=> ( ( ord_le1190675801316882794st_nat @ C @ B )
=> ( ord_le1190675801316882794st_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_251_dual__order_Ostrict__trans1,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_252_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_253_dual__order_Ostrict__trans2,axiom,
! [B: set_list_nat,A: set_list_nat,C: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ B @ A )
=> ( ( ord_le6045566169113846134st_nat @ C @ B )
=> ( ord_le1190675801316882794st_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_254_dual__order_Ostrict__trans2,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_255_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A2: nat] :
( ( ord_less_eq_nat @ B3 @ A2 )
& ~ ( ord_less_eq_nat @ A2 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_256_dual__order_Ostrict__iff__not,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [B3: set_list_nat,A2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B3 @ A2 )
& ~ ( ord_le6045566169113846134st_nat @ A2 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_257_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_a
= ( ^ [B3: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
& ~ ( ord_less_eq_set_a @ A2 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_258_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_259_order_Ostrict__implies__order,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A @ B )
=> ( ord_le6045566169113846134st_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_260_order_Ostrict__implies__order,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_261_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_262_dual__order_Ostrict__implies__order,axiom,
! [B: set_list_nat,A: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ B @ A )
=> ( ord_le6045566169113846134st_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_263_dual__order_Ostrict__implies__order,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_264_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_265_order__le__less,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [X2: set_list_nat,Y3: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_266_order__le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y3: set_a] :
( ( ord_less_set_a @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_267_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_268_order__less__le,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [X2: set_list_nat,Y3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_269_order__less__le,axiom,
( ord_less_set_a
= ( ^ [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_270_linorder__not__le,axiom,
! [X: nat,Y4: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y4 ) )
= ( ord_less_nat @ Y4 @ X ) ) ).
% linorder_not_le
thf(fact_271_linorder__not__less,axiom,
! [X: nat,Y4: nat] :
( ( ~ ( ord_less_nat @ X @ Y4 ) )
= ( ord_less_eq_nat @ Y4 @ X ) ) ).
% linorder_not_less
thf(fact_272_order__less__imp__le,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ X @ Y4 ) ) ).
% order_less_imp_le
thf(fact_273_order__less__imp__le,axiom,
! [X: set_list_nat,Y4: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ X @ Y4 )
=> ( ord_le6045566169113846134st_nat @ X @ Y4 ) ) ).
% order_less_imp_le
thf(fact_274_order__less__imp__le,axiom,
! [X: set_a,Y4: set_a] :
( ( ord_less_set_a @ X @ Y4 )
=> ( ord_less_eq_set_a @ X @ Y4 ) ) ).
% order_less_imp_le
thf(fact_275_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_276_order__le__neq__trans,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( A != B )
=> ( ord_le1190675801316882794st_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_277_order__le__neq__trans,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_278_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_279_order__neq__le__trans,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( A != B )
=> ( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ord_le1190675801316882794st_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_280_order__neq__le__trans,axiom,
! [A: set_a,B: set_a] :
( ( A != B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( ord_less_set_a @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_281_order__le__less__trans,axiom,
! [X: nat,Y4: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ( ord_less_nat @ Y4 @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_282_order__le__less__trans,axiom,
! [X: set_list_nat,Y4: set_list_nat,Z: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X @ Y4 )
=> ( ( ord_le1190675801316882794st_nat @ Y4 @ Z )
=> ( ord_le1190675801316882794st_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_283_order__le__less__trans,axiom,
! [X: set_a,Y4: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ( ord_less_set_a @ Y4 @ Z )
=> ( ord_less_set_a @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_284_order__less__le__trans,axiom,
! [X: nat,Y4: nat,Z: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_285_order__less__le__trans,axiom,
! [X: set_list_nat,Y4: set_list_nat,Z: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ X @ Y4 )
=> ( ( ord_le6045566169113846134st_nat @ Y4 @ Z )
=> ( ord_le1190675801316882794st_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_286_order__less__le__trans,axiom,
! [X: set_a,Y4: set_a,Z: set_a] :
( ( ord_less_set_a @ X @ Y4 )
=> ( ( ord_less_eq_set_a @ Y4 @ Z )
=> ( ord_less_set_a @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_287_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_288_order__le__less__subst1,axiom,
! [A: set_list_nat,F: nat > set_list_nat,B: nat,C: nat] :
( ( ord_le6045566169113846134st_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_le1190675801316882794st_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le1190675801316882794st_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_289_order__le__less__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_290_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_291_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_list_nat,C: set_list_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le1190675801316882794st_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le1190675801316882794st_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_292_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_293_order__le__less__subst2,axiom,
! [A: set_list_nat,B: set_list_nat,F: set_list_nat > nat,C: nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_list_nat,Y2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_294_order__le__less__subst2,axiom,
! [A: set_list_nat,B: set_list_nat,F: set_list_nat > set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ord_le1190675801316882794st_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_list_nat,Y2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y2 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le1190675801316882794st_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_295_order__le__less__subst2,axiom,
! [A: set_list_nat,B: set_list_nat,F: set_list_nat > set_a,C: set_a] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X3: set_list_nat,Y2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_296_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_297_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_list_nat,C: set_list_nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_le1190675801316882794st_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le1190675801316882794st_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_298_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_299_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_300_order__less__le__subst1,axiom,
! [A: set_list_nat,F: nat > set_list_nat,B: nat,C: nat] :
( ( ord_le1190675801316882794st_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le1190675801316882794st_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_301_order__less__le__subst1,axiom,
! [A: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_302_order__less__le__subst1,axiom,
! [A: nat,F: set_list_nat > nat,B: set_list_nat,C: set_list_nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ! [X3: set_list_nat,Y2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_303_order__less__le__subst1,axiom,
! [A: set_list_nat,F: set_list_nat > set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A @ ( F @ B ) )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ! [X3: set_list_nat,Y2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y2 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le1190675801316882794st_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_304_order__less__le__subst1,axiom,
! [A: set_a,F: set_list_nat > set_a,B: set_list_nat,C: set_list_nat] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ! [X3: set_list_nat,Y2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_305_order__less__le__subst1,axiom,
! [A: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_306_order__less__le__subst1,axiom,
! [A: set_list_nat,F: set_a > set_list_nat,B: set_a,C: set_a] :
( ( ord_le1190675801316882794st_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le1190675801316882794st_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_307_order__less__le__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_308_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_309_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_list_nat,C: set_list_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le6045566169113846134st_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_le1190675801316882794st_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_le1190675801316882794st_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_310_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_set_a @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_311_linorder__le__less__linear,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
| ( ord_less_nat @ Y4 @ X ) ) ).
% linorder_le_less_linear
thf(fact_312_order__le__imp__less__or__eq,axiom,
! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ( ord_less_nat @ X @ Y4 )
| ( X = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_313_order__le__imp__less__or__eq,axiom,
! [X: set_list_nat,Y4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X @ Y4 )
=> ( ( ord_le1190675801316882794st_nat @ X @ Y4 )
| ( X = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_314_order__le__imp__less__or__eq,axiom,
! [X: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X @ Y4 )
=> ( ( ord_less_set_a @ X @ Y4 )
| ( X = Y4 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_315_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_316_bot_Oextremum,axiom,
! [A: set_list_nat] : ( ord_le6045566169113846134st_nat @ bot_bot_set_list_nat @ A ) ).
% bot.extremum
thf(fact_317_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_318_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_319_bot_Oextremum__unique,axiom,
! [A: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ bot_bot_set_list_nat )
= ( A = bot_bot_set_list_nat ) ) ).
% bot.extremum_unique
thf(fact_320_bot_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_321_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_322_bot_Oextremum__uniqueI,axiom,
! [A: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ bot_bot_set_list_nat )
=> ( A = bot_bot_set_list_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_323_bot_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
=> ( A = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_324_bot_Oextremum__strict,axiom,
! [A: set_list_nat] :
~ ( ord_le1190675801316882794st_nat @ A @ bot_bot_set_list_nat ) ).
% bot.extremum_strict
thf(fact_325_bot_Oextremum__strict,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ bot_bot_set_a ) ).
% bot.extremum_strict
thf(fact_326_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_327_bot_Onot__eq__extremum,axiom,
! [A: set_list_nat] :
( ( A != bot_bot_set_list_nat )
= ( ord_le1190675801316882794st_nat @ bot_bot_set_list_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_328_bot_Onot__eq__extremum,axiom,
! [A: set_a] :
( ( A != bot_bot_set_a )
= ( ord_less_set_a @ bot_bot_set_a @ A ) ) ).
% bot.not_eq_extremum
thf(fact_329_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_330_length__induct,axiom,
! [P2: list_nat > $o,Xs2: list_nat] :
( ! [Xs3: list_nat] :
( ! [Ys3: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys3 ) @ ( size_size_list_nat @ Xs3 ) )
=> ( P2 @ Ys3 ) )
=> ( P2 @ Xs3 ) )
=> ( P2 @ Xs2 ) ) ).
% length_induct
thf(fact_331_lessI,axiom,
! [N3: nat] : ( ord_less_nat @ N3 @ ( suc @ N3 ) ) ).
% lessI
thf(fact_332_Suc__mono,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N3 ) ) ) ).
% Suc_mono
thf(fact_333_Suc__less__eq,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N3 ) )
= ( ord_less_nat @ M3 @ N3 ) ) ).
% Suc_less_eq
thf(fact_334_Suc__le__mono,axiom,
! [N3: nat,M3: nat] :
( ( ord_less_eq_nat @ ( suc @ N3 ) @ ( suc @ M3 ) )
= ( ord_less_eq_nat @ N3 @ M3 ) ) ).
% Suc_le_mono
thf(fact_335_min__pointwise__ge__iff,axiom,
! [U: set_list_nat,R: nat,X: list_nat] :
( ( finite8100373058378681591st_nat @ U )
=> ( ( U != bot_bot_set_list_nat )
=> ( ! [U3: list_nat] :
( ( member_list_nat @ U3 @ U )
=> ( ( size_size_list_nat @ U3 )
= R ) )
=> ( ( ( size_size_list_nat @ X )
= R )
=> ( ( pointwise_le @ X @ ( min_pointwise @ R @ U ) )
= ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ U )
=> ( pointwise_le @ X @ X2 ) ) ) ) ) ) ) ) ).
% min_pointwise_ge_iff
thf(fact_336_max__pointwise__le__iff,axiom,
! [U: set_list_nat,R: nat,X: list_nat] :
( ( finite8100373058378681591st_nat @ U )
=> ( ( U != bot_bot_set_list_nat )
=> ( ! [U3: list_nat] :
( ( member_list_nat @ U3 @ U )
=> ( ( size_size_list_nat @ U3 )
= R ) )
=> ( ( ( size_size_list_nat @ X )
= R )
=> ( ( pointwise_le @ ( max_pointwise @ R @ U ) @ X )
= ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ U )
=> ( pointwise_le @ X2 @ X ) ) ) ) ) ) ) ) ).
% max_pointwise_le_iff
thf(fact_337_min__pointwise__le,axiom,
! [U4: list_nat,U: set_list_nat] :
( ( member_list_nat @ U4 @ U )
=> ( ( finite8100373058378681591st_nat @ U )
=> ( pointwise_le @ ( min_pointwise @ ( size_size_list_nat @ U4 ) @ U ) @ U4 ) ) ) ).
% min_pointwise_le
thf(fact_338_Suc__leI,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 ) ) ).
% Suc_leI
thf(fact_339_Suc__le__eq,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
= ( ord_less_nat @ M3 @ N3 ) ) ).
% Suc_le_eq
thf(fact_340_dec__induct,axiom,
! [I3: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( P2 @ I3 )
=> ( ! [N: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( ord_less_nat @ N @ J )
=> ( ( P2 @ N )
=> ( P2 @ ( suc @ N ) ) ) ) )
=> ( P2 @ J ) ) ) ) ).
% dec_induct
thf(fact_341_inc__induct,axiom,
! [I3: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( P2 @ J )
=> ( ! [N: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( ord_less_nat @ N @ J )
=> ( ( P2 @ ( suc @ N ) )
=> ( P2 @ N ) ) ) )
=> ( P2 @ I3 ) ) ) ) ).
% inc_induct
thf(fact_342_empty__subsetI,axiom,
! [A3: set_list_nat] : ( ord_le6045566169113846134st_nat @ bot_bot_set_list_nat @ A3 ) ).
% empty_subsetI
thf(fact_343_empty__subsetI,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A3 ) ).
% empty_subsetI
thf(fact_344_subset__empty,axiom,
! [A3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ bot_bot_set_list_nat )
= ( A3 = bot_bot_set_list_nat ) ) ).
% subset_empty
thf(fact_345_subset__empty,axiom,
! [A3: set_a] :
( ( ord_less_eq_set_a @ A3 @ bot_bot_set_a )
= ( A3 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_346_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_347_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_348_psubsetI,axiom,
! [A3: set_list_nat,B4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ B4 )
=> ( ( A3 != B4 )
=> ( ord_le1190675801316882794st_nat @ A3 @ B4 ) ) ) ).
% psubsetI
thf(fact_349_psubsetI,axiom,
! [A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( A3 != B4 )
=> ( ord_less_set_a @ A3 @ B4 ) ) ) ).
% psubsetI
thf(fact_350_subset__iff__psubset__eq,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A5: set_list_nat,B5: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_351_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_set_a @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_352_subset__psubset__trans,axiom,
! [A3: set_list_nat,B4: set_list_nat,C2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ B4 )
=> ( ( ord_le1190675801316882794st_nat @ B4 @ C2 )
=> ( ord_le1190675801316882794st_nat @ A3 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_353_subset__psubset__trans,axiom,
! [A3: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( ord_less_set_a @ B4 @ C2 )
=> ( ord_less_set_a @ A3 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_354_subset__not__subset__eq,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [A5: set_list_nat,B5: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A5 @ B5 )
& ~ ( ord_le6045566169113846134st_nat @ B5 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_355_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ~ ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_356_psubset__subset__trans,axiom,
! [A3: set_list_nat,B4: set_list_nat,C2: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A3 @ B4 )
=> ( ( ord_le6045566169113846134st_nat @ B4 @ C2 )
=> ( ord_le1190675801316882794st_nat @ A3 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_357_psubset__subset__trans,axiom,
! [A3: set_a,B4: set_a,C2: set_a] :
( ( ord_less_set_a @ A3 @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ord_less_set_a @ A3 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_358_psubset__imp__subset,axiom,
! [A3: set_list_nat,B4: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A3 @ B4 )
=> ( ord_le6045566169113846134st_nat @ A3 @ B4 ) ) ).
% psubset_imp_subset
thf(fact_359_psubset__imp__subset,axiom,
! [A3: set_a,B4: set_a] :
( ( ord_less_set_a @ A3 @ B4 )
=> ( ord_less_eq_set_a @ A3 @ B4 ) ) ).
% psubset_imp_subset
thf(fact_360_psubset__eq,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [A5: set_list_nat,B5: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_361_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_362_psubsetE,axiom,
! [A3: set_list_nat,B4: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A3 @ B4 )
=> ~ ( ( ord_le6045566169113846134st_nat @ A3 @ B4 )
=> ( ord_le6045566169113846134st_nat @ B4 @ A3 ) ) ) ).
% psubsetE
thf(fact_363_psubsetE,axiom,
! [A3: set_a,B4: set_a] :
( ( ord_less_set_a @ A3 @ B4 )
=> ~ ( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ord_less_eq_set_a @ B4 @ A3 ) ) ) ).
% psubsetE
thf(fact_364_psubsetD,axiom,
! [A3: set_list_nat,B4: set_list_nat,C: list_nat] :
( ( ord_le1190675801316882794st_nat @ A3 @ B4 )
=> ( ( member_list_nat @ C @ A3 )
=> ( member_list_nat @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_365_psubsetD,axiom,
! [A3: set_nat,B4: set_nat,C: nat] :
( ( ord_less_set_nat @ A3 @ B4 )
=> ( ( member_nat @ C @ A3 )
=> ( member_nat @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_366_psubsetD,axiom,
! [A3: set_a,B4: set_a,C: a] :
( ( ord_less_set_a @ A3 @ B4 )
=> ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_367_Khovanskii_OAsubG,axiom,
! [G: set_list_nat,Addition: list_nat > list_nat > list_nat,Zero: list_nat,A3: set_list_nat] :
( ( khovanskii_list_nat @ G @ Addition @ Zero @ A3 )
=> ( ord_le6045566169113846134st_nat @ A3 @ G ) ) ).
% Khovanskii.AsubG
thf(fact_368_Khovanskii_OAsubG,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A3: set_a] :
( ( khovanskii_a @ G @ Addition @ Zero @ A3 )
=> ( ord_less_eq_set_a @ A3 @ G ) ) ).
% Khovanskii.AsubG
thf(fact_369_wfP__subset,axiom,
! [R: list_nat > list_nat > $o,P5: list_nat > list_nat > $o] :
( ( wfP_list_nat @ R )
=> ( ( ord_le6558929396352911974_nat_o @ P5 @ R )
=> ( wfP_list_nat @ P5 ) ) ) ).
% wfP_subset
thf(fact_370_max__pointwise__mono,axiom,
! [X7: set_list_nat,X8: set_list_nat,R: nat] :
( ( ord_le6045566169113846134st_nat @ X7 @ X8 )
=> ( ( finite8100373058378681591st_nat @ X8 )
=> ( ( X7 != bot_bot_set_list_nat )
=> ( pointwise_le @ ( max_pointwise @ R @ X7 ) @ ( max_pointwise @ R @ X8 ) ) ) ) ) ).
% max_pointwise_mono
thf(fact_371_n__not__Suc__n,axiom,
! [N3: nat] :
( N3
!= ( suc @ N3 ) ) ).
% n_not_Suc_n
thf(fact_372_Suc__inject,axiom,
! [X: nat,Y4: nat] :
( ( ( suc @ X )
= ( suc @ Y4 ) )
=> ( X = Y4 ) ) ).
% Suc_inject
thf(fact_373_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K2: nat,B: nat] :
( ( P2 @ K2 )
=> ( ! [Y2: nat] :
( ( P2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X3: nat] :
( ( P2 @ X3 )
& ! [Y: nat] :
( ( P2 @ Y )
=> ( ord_less_eq_nat @ Y @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_374_nat__le__linear,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
| ( ord_less_eq_nat @ N3 @ M3 ) ) ).
% nat_le_linear
thf(fact_375_le__antisym,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ( ord_less_eq_nat @ N3 @ M3 )
=> ( M3 = N3 ) ) ) ).
% le_antisym
thf(fact_376_eq__imp__le,axiom,
! [M3: nat,N3: nat] :
( ( M3 = N3 )
=> ( ord_less_eq_nat @ M3 @ N3 ) ) ).
% eq_imp_le
thf(fact_377_le__trans,axiom,
! [I3: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_eq_nat @ J @ K2 )
=> ( ord_less_eq_nat @ I3 @ K2 ) ) ) ).
% le_trans
thf(fact_378_le__refl,axiom,
! [N3: nat] : ( ord_less_eq_nat @ N3 @ N3 ) ).
% le_refl
thf(fact_379_linorder__neqE__nat,axiom,
! [X: nat,Y4: nat] :
( ( X != Y4 )
=> ( ~ ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ Y4 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_380_infinite__descent,axiom,
! [P2: nat > $o,N3: nat] :
( ! [N: nat] :
( ~ ( P2 @ N )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N )
& ~ ( P2 @ M4 ) ) )
=> ( P2 @ N3 ) ) ).
% infinite_descent
thf(fact_381_nat__less__induct,axiom,
! [P2: nat > $o,N3: nat] :
( ! [N: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N )
=> ( P2 @ M4 ) )
=> ( P2 @ N ) )
=> ( P2 @ N3 ) ) ).
% nat_less_induct
thf(fact_382_less__irrefl__nat,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ N3 ) ).
% less_irrefl_nat
thf(fact_383_less__not__refl3,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( S2 != T ) ) ).
% less_not_refl3
thf(fact_384_less__not__refl2,axiom,
! [N3: nat,M3: nat] :
( ( ord_less_nat @ N3 @ M3 )
=> ( M3 != N3 ) ) ).
% less_not_refl2
thf(fact_385_less__not__refl,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ N3 ) ).
% less_not_refl
thf(fact_386_nat__neq__iff,axiom,
! [M3: nat,N3: nat] :
( ( M3 != N3 )
= ( ( ord_less_nat @ M3 @ N3 )
| ( ord_less_nat @ N3 @ M3 ) ) ) ).
% nat_neq_iff
thf(fact_387_size__neq__size__imp__neq,axiom,
! [X: list_nat,Y4: list_nat] :
( ( ( size_size_list_nat @ X )
!= ( size_size_list_nat @ Y4 ) )
=> ( X != Y4 ) ) ).
% size_neq_size_imp_neq
thf(fact_388_transitive__stepwise__le,axiom,
! [M3: nat,N3: nat,R3: nat > nat > $o] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ! [X3: nat] : ( R3 @ X3 @ X3 )
=> ( ! [X3: nat,Y2: nat,Z3: nat] :
( ( R3 @ X3 @ Y2 )
=> ( ( R3 @ Y2 @ Z3 )
=> ( R3 @ X3 @ Z3 ) ) )
=> ( ! [N: nat] : ( R3 @ N @ ( suc @ N ) )
=> ( R3 @ M3 @ N3 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_389_nat__induct__at__least,axiom,
! [M3: nat,N3: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ( P2 @ M3 )
=> ( ! [N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( P2 @ N )
=> ( P2 @ ( suc @ N ) ) ) )
=> ( P2 @ N3 ) ) ) ) ).
% nat_induct_at_least
thf(fact_390_full__nat__induct,axiom,
! [P2: nat > $o,N3: nat] :
( ! [N: nat] :
( ! [M4: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N )
=> ( P2 @ M4 ) )
=> ( P2 @ N ) )
=> ( P2 @ N3 ) ) ).
% full_nat_induct
thf(fact_391_not__less__eq__eq,axiom,
! [M3: nat,N3: nat] :
( ( ~ ( ord_less_eq_nat @ M3 @ N3 ) )
= ( ord_less_eq_nat @ ( suc @ N3 ) @ M3 ) ) ).
% not_less_eq_eq
thf(fact_392_Suc__n__not__le__n,axiom,
! [N3: nat] :
~ ( ord_less_eq_nat @ ( suc @ N3 ) @ N3 ) ).
% Suc_n_not_le_n
thf(fact_393_le__Suc__eq,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ ( suc @ N3 ) )
= ( ( ord_less_eq_nat @ M3 @ N3 )
| ( M3
= ( suc @ N3 ) ) ) ) ).
% le_Suc_eq
thf(fact_394_Suc__le__D,axiom,
! [N3: nat,M5: nat] :
( ( ord_less_eq_nat @ ( suc @ N3 ) @ M5 )
=> ? [M6: nat] :
( M5
= ( suc @ M6 ) ) ) ).
% Suc_le_D
thf(fact_395_le__SucI,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ord_less_eq_nat @ M3 @ ( suc @ N3 ) ) ) ).
% le_SucI
thf(fact_396_le__SucE,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ ( suc @ N3 ) )
=> ( ~ ( ord_less_eq_nat @ M3 @ N3 )
=> ( M3
= ( suc @ N3 ) ) ) ) ).
% le_SucE
thf(fact_397_Suc__leD,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
=> ( ord_less_eq_nat @ M3 @ N3 ) ) ).
% Suc_leD
thf(fact_398_not__less__less__Suc__eq,axiom,
! [N3: nat,M3: nat] :
( ~ ( ord_less_nat @ N3 @ M3 )
=> ( ( ord_less_nat @ N3 @ ( suc @ M3 ) )
= ( N3 = M3 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_399_strict__inc__induct,axiom,
! [I3: nat,J: nat,P2: nat > $o] :
( ( ord_less_nat @ I3 @ J )
=> ( ! [I: nat] :
( ( J
= ( suc @ I ) )
=> ( P2 @ I ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( P2 @ ( suc @ I ) )
=> ( P2 @ I ) ) )
=> ( P2 @ I3 ) ) ) ) ).
% strict_inc_induct
thf(fact_400_less__Suc__induct,axiom,
! [I3: nat,J: nat,P2: nat > nat > $o] :
( ( ord_less_nat @ I3 @ J )
=> ( ! [I: nat] : ( P2 @ I @ ( suc @ I ) )
=> ( ! [I: nat,J2: nat,K3: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ J2 @ K3 )
=> ( ( P2 @ I @ J2 )
=> ( ( P2 @ J2 @ K3 )
=> ( P2 @ I @ K3 ) ) ) ) )
=> ( P2 @ I3 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_401_less__trans__Suc,axiom,
! [I3: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( suc @ I3 ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_402_Suc__less__SucD,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N3 ) )
=> ( ord_less_nat @ M3 @ N3 ) ) ).
% Suc_less_SucD
thf(fact_403_less__antisym,axiom,
! [N3: nat,M3: nat] :
( ~ ( ord_less_nat @ N3 @ M3 )
=> ( ( ord_less_nat @ N3 @ ( suc @ M3 ) )
=> ( M3 = N3 ) ) ) ).
% less_antisym
thf(fact_404_Suc__less__eq2,axiom,
! [N3: nat,M3: nat] :
( ( ord_less_nat @ ( suc @ N3 ) @ M3 )
= ( ? [M7: nat] :
( ( M3
= ( suc @ M7 ) )
& ( ord_less_nat @ N3 @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_405_All__less__Suc,axiom,
! [N3: nat,P2: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N3 ) )
=> ( P2 @ I2 ) ) )
= ( ( P2 @ N3 )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N3 )
=> ( P2 @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_406_not__less__eq,axiom,
! [M3: nat,N3: nat] :
( ( ~ ( ord_less_nat @ M3 @ N3 ) )
= ( ord_less_nat @ N3 @ ( suc @ M3 ) ) ) ).
% not_less_eq
thf(fact_407_less__Suc__eq,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N3 ) )
= ( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ).
% less_Suc_eq
thf(fact_408_Ex__less__Suc,axiom,
! [N3: nat,P2: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N3 ) )
& ( P2 @ I2 ) ) )
= ( ( P2 @ N3 )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N3 )
& ( P2 @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_409_less__SucI,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ord_less_nat @ M3 @ ( suc @ N3 ) ) ) ).
% less_SucI
thf(fact_410_less__SucE,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N3 ) )
=> ( ~ ( ord_less_nat @ M3 @ N3 )
=> ( M3 = N3 ) ) ) ).
% less_SucE
thf(fact_411_Suc__lessI,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ( ( suc @ M3 )
!= N3 )
=> ( ord_less_nat @ ( suc @ M3 ) @ N3 ) ) ) ).
% Suc_lessI
thf(fact_412_Suc__lessE,axiom,
! [I3: nat,K2: nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ K2 )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_413_Suc__lessD,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ ( suc @ M3 ) @ N3 )
=> ( ord_less_nat @ M3 @ N3 ) ) ).
% Suc_lessD
thf(fact_414_Nat_OlessE,axiom,
! [I3: nat,K2: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ( ( K2
!= ( suc @ I3 ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_415_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I3: nat,J: nat] :
( ! [I: nat,J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ ( F @ I ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_416_le__neq__implies__less,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ( M3 != N3 )
=> ( ord_less_nat @ M3 @ N3 ) ) ) ).
% le_neq_implies_less
thf(fact_417_less__or__eq__imp__le,axiom,
! [M3: nat,N3: nat] :
( ( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) )
=> ( ord_less_eq_nat @ M3 @ N3 ) ) ).
% less_or_eq_imp_le
thf(fact_418_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_419_less__imp__le__nat,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ord_less_eq_nat @ M3 @ N3 ) ) ).
% less_imp_le_nat
thf(fact_420_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_421_max__pointwise__ge,axiom,
! [U4: list_nat,U: set_list_nat] :
( ( member_list_nat @ U4 @ U )
=> ( ( finite8100373058378681591st_nat @ U )
=> ( pointwise_le @ U4 @ ( max_pointwise @ ( size_size_list_nat @ U4 ) @ U ) ) ) ) ).
% max_pointwise_ge
thf(fact_422_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N3: nat,N4: nat] :
( ! [N: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
=> ( ( ord_less_eq_nat @ N3 @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_423_lift__Suc__antimono__le,axiom,
! [F: nat > set_list_nat,N3: nat,N4: nat] :
( ! [N: nat] : ( ord_le6045566169113846134st_nat @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
=> ( ( ord_less_eq_nat @ N3 @ N4 )
=> ( ord_le6045566169113846134st_nat @ ( F @ N4 ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_424_lift__Suc__antimono__le,axiom,
! [F: nat > set_a,N3: nat,N4: nat] :
( ! [N: nat] : ( ord_less_eq_set_a @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
=> ( ( ord_less_eq_nat @ N3 @ N4 )
=> ( ord_less_eq_set_a @ ( F @ N4 ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_425_lift__Suc__mono__le,axiom,
! [F: nat > nat,N3: nat,N4: nat] :
( ! [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ N3 @ N4 )
=> ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_426_lift__Suc__mono__le,axiom,
! [F: nat > set_list_nat,N3: nat,N4: nat] :
( ! [N: nat] : ( ord_le6045566169113846134st_nat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ N3 @ N4 )
=> ( ord_le6045566169113846134st_nat @ ( F @ N3 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_427_lift__Suc__mono__le,axiom,
! [F: nat > set_a,N3: nat,N4: nat] :
( ! [N: nat] : ( ord_less_eq_set_a @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ N3 @ N4 )
=> ( ord_less_eq_set_a @ ( F @ N3 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_428_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N3: nat,M3: nat] :
( ! [N: nat] : ( ord_less_nat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less_nat @ ( F @ N3 ) @ ( F @ M3 ) )
= ( ord_less_nat @ N3 @ M3 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_429_lift__Suc__mono__less,axiom,
! [F: nat > nat,N3: nat,N4: nat] :
( ! [N: nat] : ( ord_less_nat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less_nat @ N3 @ N4 )
=> ( ord_less_nat @ ( F @ N3 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_430_le__imp__less__Suc,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ord_less_nat @ M3 @ ( suc @ N3 ) ) ) ).
% le_imp_less_Suc
thf(fact_431_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_432_less__Suc__eq__le,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N3 ) )
= ( ord_less_eq_nat @ M3 @ N3 ) ) ).
% less_Suc_eq_le
thf(fact_433_le__less__Suc__eq,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ( ord_less_nat @ N3 @ ( suc @ M3 ) )
= ( N3 = M3 ) ) ) ).
% le_less_Suc_eq
thf(fact_434_Suc__le__lessD,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
=> ( ord_less_nat @ M3 @ N3 ) ) ).
% Suc_le_lessD
thf(fact_435_ex__min__if__finite,axiom,
! [S: set_list_nat] :
( ( finite8100373058378681591st_nat @ S )
=> ( ( S != bot_bot_set_list_nat )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ S )
& ~ ? [Xa: list_nat] :
( ( member_list_nat @ Xa @ S )
& ( ord_less_list_nat @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_436_ex__min__if__finite,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ S )
& ~ ? [Xa: nat] :
( ( member_nat @ Xa @ S )
& ( ord_less_nat @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_437_infinite__growing,axiom,
! [X8: set_list_nat] :
( ( X8 != bot_bot_set_list_nat )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ X8 )
=> ? [Xa: list_nat] :
( ( member_list_nat @ Xa @ X8 )
& ( ord_less_list_nat @ X3 @ Xa ) ) )
=> ~ ( finite8100373058378681591st_nat @ X8 ) ) ) ).
% infinite_growing
thf(fact_438_infinite__growing,axiom,
! [X8: set_nat] :
( ( X8 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X8 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X8 )
& ( ord_less_nat @ X3 @ Xa ) ) )
=> ~ ( finite_finite_nat @ X8 ) ) ) ).
% infinite_growing
thf(fact_439_finite__has__minimal,axiom,
! [A3: set_list_nat] :
( ( finite8100373058378681591st_nat @ A3 )
=> ( ( A3 != bot_bot_set_list_nat )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ A3 )
& ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ A3 )
=> ( ( ord_less_eq_list_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_440_finite__has__minimal,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A3 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_441_finite__has__minimal,axiom,
! [A3: set_set_list_nat] :
( ( finite7047420756378620717st_nat @ A3 )
=> ( ( A3 != bot_bo3886227569956363488st_nat )
=> ? [X3: set_list_nat] :
( ( member_set_list_nat @ X3 @ A3 )
& ! [Xa: set_list_nat] :
( ( member_set_list_nat @ Xa @ A3 )
=> ( ( ord_le6045566169113846134st_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_442_finite__has__minimal,axiom,
! [A3: set_set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( A3 != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A3 )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_443_finite__has__maximal,axiom,
! [A3: set_list_nat] :
( ( finite8100373058378681591st_nat @ A3 )
=> ( ( A3 != bot_bot_set_list_nat )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ A3 )
& ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ A3 )
=> ( ( ord_less_eq_list_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_444_finite__has__maximal,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A3 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_445_finite__has__maximal,axiom,
! [A3: set_set_list_nat] :
( ( finite7047420756378620717st_nat @ A3 )
=> ( ( A3 != bot_bo3886227569956363488st_nat )
=> ? [X3: set_list_nat] :
( ( member_set_list_nat @ X3 @ A3 )
& ! [Xa: set_list_nat] :
( ( member_set_list_nat @ Xa @ A3 )
=> ( ( ord_le6045566169113846134st_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_446_finite__has__maximal,axiom,
! [A3: set_set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( A3 != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A3 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_447_Collect__empty__eq__bot,axiom,
! [P2: nat > $o] :
( ( ( collect_nat @ P2 )
= bot_bot_set_nat )
= ( P2 = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_448_Collect__empty__eq__bot,axiom,
! [P2: list_nat > $o] :
( ( ( collect_list_nat @ P2 )
= bot_bot_set_list_nat )
= ( P2 = bot_bot_list_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_449_Collect__empty__eq__bot,axiom,
! [P2: a > $o] :
( ( ( collect_a @ P2 )
= bot_bot_set_a )
= ( P2 = bot_bot_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_450_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X2: nat] : ( member_nat @ X2 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_451_bot__empty__eq,axiom,
( bot_bot_list_nat_o
= ( ^ [X2: list_nat] : ( member_list_nat @ X2 @ bot_bot_set_list_nat ) ) ) ).
% bot_empty_eq
thf(fact_452_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X2: a] : ( member_a @ X2 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_453_finite__length__sum__set,axiom,
! [R: nat,N3: nat] : ( finite8100373058378681591st_nat @ ( length_sum_set @ R @ N3 ) ) ).
% finite_length_sum_set
thf(fact_454_subset__antisym,axiom,
! [A3: set_list_nat,B4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ B4 )
=> ( ( ord_le6045566169113846134st_nat @ B4 @ A3 )
=> ( A3 = B4 ) ) ) ).
% subset_antisym
thf(fact_455_subset__antisym,axiom,
! [A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ A3 )
=> ( A3 = B4 ) ) ) ).
% subset_antisym
thf(fact_456_subsetI,axiom,
! [A3: set_nat,B4: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( member_nat @ X3 @ B4 ) )
=> ( ord_less_eq_set_nat @ A3 @ B4 ) ) ).
% subsetI
thf(fact_457_subsetI,axiom,
! [A3: set_list_nat,B4: set_list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A3 )
=> ( member_list_nat @ X3 @ B4 ) )
=> ( ord_le6045566169113846134st_nat @ A3 @ B4 ) ) ).
% subsetI
thf(fact_458_subsetI,axiom,
! [A3: set_a,B4: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( member_a @ X3 @ B4 ) )
=> ( ord_less_eq_set_a @ A3 @ B4 ) ) ).
% subsetI
thf(fact_459_Collect__mono__iff,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q2 ) )
= ( ! [X2: nat] :
( ( P2 @ X2 )
=> ( Q2 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_460_Collect__mono__iff,axiom,
! [P2: list_nat > $o,Q2: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P2 ) @ ( collect_list_nat @ Q2 ) )
= ( ! [X2: list_nat] :
( ( P2 @ X2 )
=> ( Q2 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_461_Collect__mono__iff,axiom,
! [P2: a > $o,Q2: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q2 ) )
= ( ! [X2: a] :
( ( P2 @ X2 )
=> ( Q2 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_462_set__eq__subset,axiom,
( ( ^ [Y5: set_list_nat,Z2: set_list_nat] : ( Y5 = Z2 ) )
= ( ^ [A5: set_list_nat,B5: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A5 @ B5 )
& ( ord_le6045566169113846134st_nat @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_463_set__eq__subset,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_464_subset__trans,axiom,
! [A3: set_list_nat,B4: set_list_nat,C2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ B4 )
=> ( ( ord_le6045566169113846134st_nat @ B4 @ C2 )
=> ( ord_le6045566169113846134st_nat @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_465_subset__trans,axiom,
! [A3: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ord_less_eq_set_a @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_466_Collect__mono,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ! [X3: nat] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q2 ) ) ) ).
% Collect_mono
thf(fact_467_Collect__mono,axiom,
! [P2: list_nat > $o,Q2: list_nat > $o] :
( ! [X3: list_nat] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P2 ) @ ( collect_list_nat @ Q2 ) ) ) ).
% Collect_mono
thf(fact_468_Collect__mono,axiom,
! [P2: a > $o,Q2: a > $o] :
( ! [X3: a] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q2 ) ) ) ).
% Collect_mono
thf(fact_469_subset__refl,axiom,
! [A3: set_list_nat] : ( ord_le6045566169113846134st_nat @ A3 @ A3 ) ).
% subset_refl
thf(fact_470_subset__refl,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ A3 ) ).
% subset_refl
thf(fact_471_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A5 )
=> ( member_nat @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_472_subset__iff,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A5: set_list_nat,B5: set_list_nat] :
! [T2: list_nat] :
( ( member_list_nat @ T2 @ A5 )
=> ( member_list_nat @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_473_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A5 )
=> ( member_a @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_474_equalityD2,axiom,
! [A3: set_list_nat,B4: set_list_nat] :
( ( A3 = B4 )
=> ( ord_le6045566169113846134st_nat @ B4 @ A3 ) ) ).
% equalityD2
thf(fact_475_equalityD2,axiom,
! [A3: set_a,B4: set_a] :
( ( A3 = B4 )
=> ( ord_less_eq_set_a @ B4 @ A3 ) ) ).
% equalityD2
thf(fact_476_equalityD1,axiom,
! [A3: set_list_nat,B4: set_list_nat] :
( ( A3 = B4 )
=> ( ord_le6045566169113846134st_nat @ A3 @ B4 ) ) ).
% equalityD1
thf(fact_477_equalityD1,axiom,
! [A3: set_a,B4: set_a] :
( ( A3 = B4 )
=> ( ord_less_eq_set_a @ A3 @ B4 ) ) ).
% equalityD1
thf(fact_478_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A5 )
=> ( member_nat @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_479_subset__eq,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A5: set_list_nat,B5: set_list_nat] :
! [X2: list_nat] :
( ( member_list_nat @ X2 @ A5 )
=> ( member_list_nat @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_480_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A5 )
=> ( member_a @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_481_equalityE,axiom,
! [A3: set_list_nat,B4: set_list_nat] :
( ( A3 = B4 )
=> ~ ( ( ord_le6045566169113846134st_nat @ A3 @ B4 )
=> ~ ( ord_le6045566169113846134st_nat @ B4 @ A3 ) ) ) ).
% equalityE
thf(fact_482_equalityE,axiom,
! [A3: set_a,B4: set_a] :
( ( A3 = B4 )
=> ~ ( ( ord_less_eq_set_a @ A3 @ B4 )
=> ~ ( ord_less_eq_set_a @ B4 @ A3 ) ) ) ).
% equalityE
thf(fact_483_subsetD,axiom,
! [A3: set_nat,B4: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ( member_nat @ C @ A3 )
=> ( member_nat @ C @ B4 ) ) ) ).
% subsetD
thf(fact_484_subsetD,axiom,
! [A3: set_list_nat,B4: set_list_nat,C: list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ B4 )
=> ( ( member_list_nat @ C @ A3 )
=> ( member_list_nat @ C @ B4 ) ) ) ).
% subsetD
thf(fact_485_subsetD,axiom,
! [A3: set_a,B4: set_a,C: a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B4 ) ) ) ).
% subsetD
thf(fact_486_in__mono,axiom,
! [A3: set_nat,B4: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ( member_nat @ X @ A3 )
=> ( member_nat @ X @ B4 ) ) ) ).
% in_mono
thf(fact_487_in__mono,axiom,
! [A3: set_list_nat,B4: set_list_nat,X: list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ B4 )
=> ( ( member_list_nat @ X @ A3 )
=> ( member_list_nat @ X @ B4 ) ) ) ).
% in_mono
thf(fact_488_in__mono,axiom,
! [A3: set_a,B4: set_a,X: a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( member_a @ X @ A3 )
=> ( member_a @ X @ B4 ) ) ) ).
% in_mono
thf(fact_489_rev__finite__subset,axiom,
! [B4: set_nat,A3: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( finite_finite_nat @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_490_rev__finite__subset,axiom,
! [B4: set_list_nat,A3: set_list_nat] :
( ( finite8100373058378681591st_nat @ B4 )
=> ( ( ord_le6045566169113846134st_nat @ A3 @ B4 )
=> ( finite8100373058378681591st_nat @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_491_rev__finite__subset,axiom,
! [B4: set_a,A3: set_a] :
( ( finite_finite_a @ B4 )
=> ( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( finite_finite_a @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_492_infinite__super,axiom,
! [S: set_nat,T3: set_nat] :
( ( ord_less_eq_set_nat @ S @ T3 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T3 ) ) ) ).
% infinite_super
thf(fact_493_infinite__super,axiom,
! [S: set_list_nat,T3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ S @ T3 )
=> ( ~ ( finite8100373058378681591st_nat @ S )
=> ~ ( finite8100373058378681591st_nat @ T3 ) ) ) ).
% infinite_super
thf(fact_494_infinite__super,axiom,
! [S: set_a,T3: set_a] :
( ( ord_less_eq_set_a @ S @ T3 )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T3 ) ) ) ).
% infinite_super
thf(fact_495_finite__subset,axiom,
! [A3: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ( finite_finite_nat @ B4 )
=> ( finite_finite_nat @ A3 ) ) ) ).
% finite_subset
thf(fact_496_finite__subset,axiom,
! [A3: set_list_nat,B4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ B4 )
=> ( ( finite8100373058378681591st_nat @ B4 )
=> ( finite8100373058378681591st_nat @ A3 ) ) ) ).
% finite_subset
thf(fact_497_finite__subset,axiom,
! [A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( finite_finite_a @ B4 )
=> ( finite_finite_a @ A3 ) ) ) ).
% finite_subset
thf(fact_498_finite__has__maximal2,axiom,
! [A3: set_list_nat,A: list_nat] :
( ( finite8100373058378681591st_nat @ A3 )
=> ( ( member_list_nat @ A @ A3 )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ A3 )
& ( ord_less_eq_list_nat @ A @ X3 )
& ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ A3 )
=> ( ( ord_less_eq_list_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_499_finite__has__maximal2,axiom,
! [A3: set_nat,A: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ A @ A3 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A3 )
& ( ord_less_eq_nat @ A @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A3 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_500_finite__has__maximal2,axiom,
! [A3: set_set_list_nat,A: set_list_nat] :
( ( finite7047420756378620717st_nat @ A3 )
=> ( ( member_set_list_nat @ A @ A3 )
=> ? [X3: set_list_nat] :
( ( member_set_list_nat @ X3 @ A3 )
& ( ord_le6045566169113846134st_nat @ A @ X3 )
& ! [Xa: set_list_nat] :
( ( member_set_list_nat @ Xa @ A3 )
=> ( ( ord_le6045566169113846134st_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_501_finite__has__maximal2,axiom,
! [A3: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( member_set_a @ A @ A3 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
& ( ord_less_eq_set_a @ A @ X3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A3 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_502_finite__has__minimal2,axiom,
! [A3: set_list_nat,A: list_nat] :
( ( finite8100373058378681591st_nat @ A3 )
=> ( ( member_list_nat @ A @ A3 )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ A3 )
& ( ord_less_eq_list_nat @ X3 @ A )
& ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ A3 )
=> ( ( ord_less_eq_list_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_503_finite__has__minimal2,axiom,
! [A3: set_nat,A: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ A @ A3 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A3 )
& ( ord_less_eq_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A3 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_504_finite__has__minimal2,axiom,
! [A3: set_set_list_nat,A: set_list_nat] :
( ( finite7047420756378620717st_nat @ A3 )
=> ( ( member_set_list_nat @ A @ A3 )
=> ? [X3: set_list_nat] :
( ( member_set_list_nat @ X3 @ A3 )
& ( ord_le6045566169113846134st_nat @ X3 @ A )
& ! [Xa: set_list_nat] :
( ( member_set_list_nat @ Xa @ A3 )
=> ( ( ord_le6045566169113846134st_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_505_finite__has__minimal2,axiom,
! [A3: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( member_set_a @ A @ A3 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A3 )
& ( ord_less_eq_set_a @ X3 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A3 )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_506_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_507_finite_OemptyI,axiom,
finite8100373058378681591st_nat @ bot_bot_set_list_nat ).
% finite.emptyI
thf(fact_508_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_509_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_510_infinite__imp__nonempty,axiom,
! [S: set_list_nat] :
( ~ ( finite8100373058378681591st_nat @ S )
=> ( S != bot_bot_set_list_nat ) ) ).
% infinite_imp_nonempty
thf(fact_511_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_512_finite__psubset__induct,axiom,
! [A3: set_list_nat,P2: set_list_nat > $o] :
( ( finite8100373058378681591st_nat @ A3 )
=> ( ! [A6: set_list_nat] :
( ( finite8100373058378681591st_nat @ A6 )
=> ( ! [B6: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ B6 @ A6 )
=> ( P2 @ B6 ) )
=> ( P2 @ A6 ) ) )
=> ( P2 @ A3 ) ) ) ).
% finite_psubset_induct
thf(fact_513_finite__psubset__induct,axiom,
! [A3: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ A3 )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ! [B6: set_nat] :
( ( ord_less_set_nat @ B6 @ A6 )
=> ( P2 @ B6 ) )
=> ( P2 @ A6 ) ) )
=> ( P2 @ A3 ) ) ) ).
% finite_psubset_induct
thf(fact_514_finite__psubset__induct,axiom,
! [A3: set_a,P2: set_a > $o] :
( ( finite_finite_a @ A3 )
=> ( ! [A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ! [B6: set_a] :
( ( ord_less_set_a @ B6 @ A6 )
=> ( P2 @ B6 ) )
=> ( P2 @ A6 ) ) )
=> ( P2 @ A3 ) ) ) ).
% finite_psubset_induct
thf(fact_515_Khovanskii__axioms__def,axiom,
( khovan4585363760863428690ms_nat
= ( ^ [G2: set_nat,A5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ G2 )
& ( finite_finite_nat @ A5 )
& ( A5 != bot_bot_set_nat ) ) ) ) ).
% Khovanskii_axioms_def
thf(fact_516_Khovanskii__axioms__def,axiom,
( khovan1553326461689229922st_nat
= ( ^ [G2: set_list_nat,A5: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A5 @ G2 )
& ( finite8100373058378681591st_nat @ A5 )
& ( A5 != bot_bot_set_list_nat ) ) ) ) ).
% Khovanskii_axioms_def
thf(fact_517_Khovanskii__axioms__def,axiom,
( khovanskii_axioms_a
= ( ^ [G2: set_a,A5: set_a] :
( ( ord_less_eq_set_a @ A5 @ G2 )
& ( finite_finite_a @ A5 )
& ( A5 != bot_bot_set_a ) ) ) ) ).
% Khovanskii_axioms_def
thf(fact_518_Khovanskii__axioms_Ointro,axiom,
! [A3: set_nat,G: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ G )
=> ( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( khovan4585363760863428690ms_nat @ G @ A3 ) ) ) ) ).
% Khovanskii_axioms.intro
thf(fact_519_Khovanskii__axioms_Ointro,axiom,
! [A3: set_list_nat,G: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ G )
=> ( ( finite8100373058378681591st_nat @ A3 )
=> ( ( A3 != bot_bot_set_list_nat )
=> ( khovan1553326461689229922st_nat @ G @ A3 ) ) ) ) ).
% Khovanskii_axioms.intro
thf(fact_520_Khovanskii__axioms_Ointro,axiom,
! [A3: set_a,G: set_a] :
( ( ord_less_eq_set_a @ A3 @ G )
=> ( ( finite_finite_a @ A3 )
=> ( ( A3 != bot_bot_set_a )
=> ( khovanskii_axioms_a @ G @ A3 ) ) ) ) ).
% Khovanskii_axioms.intro
thf(fact_521_arg__min__if__finite_I2_J,axiom,
! [S: set_nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ~ ? [X5: nat] :
( ( member_nat @ X5 @ S )
& ( ord_less_nat @ ( F @ X5 ) @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_522_arg__min__if__finite_I2_J,axiom,
! [S: set_list_nat,F: list_nat > nat] :
( ( finite8100373058378681591st_nat @ S )
=> ( ( S != bot_bot_set_list_nat )
=> ~ ? [X5: list_nat] :
( ( member_list_nat @ X5 @ S )
& ( ord_less_nat @ ( F @ X5 ) @ ( F @ ( lattic5785867957632790475at_nat @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_523_arg__min__if__finite_I2_J,axiom,
! [S: set_a,F: a > nat] :
( ( finite_finite_a @ S )
=> ( ( S != bot_bot_set_a )
=> ~ ? [X5: a] :
( ( member_a @ X5 @ S )
& ( ord_less_nat @ ( F @ X5 ) @ ( F @ ( lattic6340287419671400565_a_nat @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_524_arg__min__least,axiom,
! [S: set_nat,Y4: nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat @ Y4 @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) @ ( F @ Y4 ) ) ) ) ) ).
% arg_min_least
thf(fact_525_arg__min__least,axiom,
! [S: set_list_nat,Y4: list_nat,F: list_nat > nat] :
( ( finite8100373058378681591st_nat @ S )
=> ( ( S != bot_bot_set_list_nat )
=> ( ( member_list_nat @ Y4 @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic5785867957632790475at_nat @ F @ S ) ) @ ( F @ Y4 ) ) ) ) ) ).
% arg_min_least
thf(fact_526_arg__min__least,axiom,
! [S: set_a,Y4: a,F: a > nat] :
( ( finite_finite_a @ S )
=> ( ( S != bot_bot_set_a )
=> ( ( member_a @ Y4 @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic6340287419671400565_a_nat @ F @ S ) ) @ ( F @ Y4 ) ) ) ) ) ).
% arg_min_least
thf(fact_527_nat__descend__induct,axiom,
! [N3: nat,P2: nat > $o,M3: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N3 @ K3 )
=> ( P2 @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N3 )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K3 @ I4 )
=> ( P2 @ I4 ) )
=> ( P2 @ K3 ) ) )
=> ( P2 @ M3 ) ) ) ).
% nat_descend_induct
thf(fact_528_finite__transitivity__chain,axiom,
! [A3: set_nat,R3: nat > nat > $o] :
( ( finite_finite_nat @ A3 )
=> ( ! [X3: nat] :
~ ( R3 @ X3 @ X3 )
=> ( ! [X3: nat,Y2: nat,Z3: nat] :
( ( R3 @ X3 @ Y2 )
=> ( ( R3 @ Y2 @ Z3 )
=> ( R3 @ X3 @ Z3 ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ? [Y: nat] :
( ( member_nat @ Y @ A3 )
& ( R3 @ X3 @ Y ) ) )
=> ( A3 = bot_bot_set_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_529_finite__transitivity__chain,axiom,
! [A3: set_list_nat,R3: list_nat > list_nat > $o] :
( ( finite8100373058378681591st_nat @ A3 )
=> ( ! [X3: list_nat] :
~ ( R3 @ X3 @ X3 )
=> ( ! [X3: list_nat,Y2: list_nat,Z3: list_nat] :
( ( R3 @ X3 @ Y2 )
=> ( ( R3 @ Y2 @ Z3 )
=> ( R3 @ X3 @ Z3 ) ) )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A3 )
=> ? [Y: list_nat] :
( ( member_list_nat @ Y @ A3 )
& ( R3 @ X3 @ Y ) ) )
=> ( A3 = bot_bot_set_list_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_530_finite__transitivity__chain,axiom,
! [A3: set_a,R3: a > a > $o] :
( ( finite_finite_a @ A3 )
=> ( ! [X3: a] :
~ ( R3 @ X3 @ X3 )
=> ( ! [X3: a,Y2: a,Z3: a] :
( ( R3 @ X3 @ Y2 )
=> ( ( R3 @ Y2 @ Z3 )
=> ( R3 @ X3 @ Z3 ) ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ? [Y: a] :
( ( member_a @ Y @ A3 )
& ( R3 @ X3 @ Y ) ) )
=> ( A3 = bot_bot_set_a ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_531_subset__emptyI,axiom,
! [A3: set_nat] :
( ! [X3: nat] :
~ ( member_nat @ X3 @ A3 )
=> ( ord_less_eq_set_nat @ A3 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_532_subset__emptyI,axiom,
! [A3: set_list_nat] :
( ! [X3: list_nat] :
~ ( member_list_nat @ X3 @ A3 )
=> ( ord_le6045566169113846134st_nat @ A3 @ bot_bot_set_list_nat ) ) ).
% subset_emptyI
thf(fact_533_subset__emptyI,axiom,
! [A3: set_a] :
( ! [X3: a] :
~ ( member_a @ X3 @ A3 )
=> ( ord_less_eq_set_a @ A3 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_534_infinite__nat__iff__unbounded__le,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M2: nat] :
? [N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( member_nat @ N2 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_535_infinite__nat__iff__unbounded,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M2: nat] :
? [N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ( member_nat @ N2 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_536_unbounded__k__infinite,axiom,
! [K2: nat,S: set_nat] :
( ! [M6: nat] :
( ( ord_less_nat @ K2 @ M6 )
=> ? [N5: nat] :
( ( ord_less_nat @ M6 @ N5 )
& ( member_nat @ N5 @ S ) ) )
=> ~ ( finite_finite_nat @ S ) ) ).
% unbounded_k_infinite
thf(fact_537_finite__indexed__bound,axiom,
! [A3: set_list_nat,P2: list_nat > nat > $o] :
( ( finite8100373058378681591st_nat @ A3 )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A3 )
=> ? [X_12: nat] : ( P2 @ X3 @ X_12 ) )
=> ? [M6: nat] :
! [X5: list_nat] :
( ( member_list_nat @ X5 @ A3 )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ M6 )
& ( P2 @ X5 @ K3 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_538_finite__indexed__bound,axiom,
! [A3: set_nat,P2: nat > nat > $o] :
( ( finite_finite_nat @ A3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ? [X_12: nat] : ( P2 @ X3 @ X_12 ) )
=> ? [M6: nat] :
! [X5: nat] :
( ( member_nat @ X5 @ A3 )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ M6 )
& ( P2 @ X5 @ K3 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_539_finite__indexed__bound,axiom,
! [A3: set_a,P2: a > nat > $o] :
( ( finite_finite_a @ A3 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ? [X_12: nat] : ( P2 @ X3 @ X_12 ) )
=> ? [M6: nat] :
! [X5: a] :
( ( member_a @ X5 @ A3 )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ M6 )
& ( P2 @ X5 @ K3 ) ) ) ) ) ).
% finite_indexed_bound
thf(fact_540_complete__interval,axiom,
! [A: nat,B: nat,P2: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P2 @ A )
=> ( ~ ( P2 @ B )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A @ C3 )
& ( ord_less_eq_nat @ C3 @ B )
& ! [X5: nat] :
( ( ( ord_less_eq_nat @ A @ X5 )
& ( ord_less_nat @ X5 @ C3 ) )
=> ( P2 @ X5 ) )
& ! [D: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D ) )
=> ( P2 @ X3 ) )
=> ( ord_less_eq_nat @ D @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_541_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_542_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% pinf(6)
thf(fact_543_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ord_less_eq_nat @ T @ X5 ) ) ).
% pinf(8)
thf(fact_544_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ord_less_eq_nat @ X5 @ T ) ) ).
% minf(6)
thf(fact_545_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_546_verit__comp__simplify1_I2_J,axiom,
! [A: set_list_nat] : ( ord_le6045566169113846134st_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_547_verit__comp__simplify1_I2_J,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_548_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_549_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ~ ( ord_less_nat @ T @ X5 ) ) ).
% minf(7)
thf(fact_550_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ord_less_nat @ X5 @ T ) ) ).
% minf(5)
thf(fact_551_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_552_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_553_minf_I2_J,axiom,
! [P2: nat > $o,P6: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P2 @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ( ( P2 @ X5 )
| ( Q2 @ X5 ) )
= ( ( P6 @ X5 )
| ( Q3 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_554_minf_I1_J,axiom,
! [P2: nat > $o,P6: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P2 @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ( ( P2 @ X5 )
& ( Q2 @ X5 ) )
= ( ( P6 @ X5 )
& ( Q3 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_555_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ord_less_nat @ T @ X5 ) ) ).
% pinf(7)
thf(fact_556_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ~ ( ord_less_nat @ X5 @ T ) ) ).
% pinf(5)
thf(fact_557_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_558_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_559_pinf_I2_J,axiom,
! [P2: nat > $o,P6: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P2 @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ( ( P2 @ X5 )
| ( Q2 @ X5 ) )
= ( ( P6 @ X5 )
| ( Q3 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_560_pinf_I1_J,axiom,
! [P2: nat > $o,P6: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P2 @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ( ( P2 @ X5 )
& ( Q2 @ X5 ) )
= ( ( P6 @ X5 )
& ( Q3 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_561_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_562_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% minf(8)
thf(fact_563_bounded__nat__set__is__finite,axiom,
! [N6: set_nat,N3: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ N6 )
=> ( ord_less_nat @ X3 @ N3 ) )
=> ( finite_finite_nat @ N6 ) ) ).
% bounded_nat_set_is_finite
thf(fact_564_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N7: set_nat] :
? [M2: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N7 )
=> ( ord_less_nat @ X2 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_565_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N7: set_nat] :
? [M2: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N7 )
=> ( ord_less_eq_nat @ X2 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_566_VF__def,axiom,
( vf
= ( ^ [I2: nat,T2: nat] :
( collect_list_nat
@ ^ [V2: list_nat] :
( ( member_list_nat @ V2 @ v )
& ( ( nth_nat @ V2 @ I2 )
= T2 ) ) ) ) ) ).
% VF_def
thf(fact_567_V__def,axiom,
( v
= ( collect_list_nat
@ ^ [V2: list_nat] :
( ( member_list_nat @ V2 @ ua )
& ~ ( pointwise_le @ u2 @ V2 ) ) ) ) ).
% V_def
thf(fact_568_pointwise__less__imp___092_060sigma_062,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( pointwise_less @ Xs2 @ Ys2 )
=> ( ord_less_nat @ ( groups4561878855575611511st_nat @ Xs2 ) @ ( groups4561878855575611511st_nat @ Ys2 ) ) ) ).
% pointwise_less_imp_\<sigma>
thf(fact_569_minimal__elementsp__minimal__elements__eq,axiom,
! [U: set_list_nat] :
( ( minimal_elementsp
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ U ) )
= ( ^ [X2: list_nat] : ( member_list_nat @ X2 @ ( minimal_elements @ U ) ) ) ) ).
% minimal_elementsp_minimal_elements_eq
thf(fact_570_minimal__elements__def,axiom,
( minimal_elements
= ( ^ [U2: set_list_nat] :
( collect_list_nat
@ ( minimal_elementsp
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ U2 ) ) ) ) ) ).
% minimal_elements_def
thf(fact_571_length__sum__set__def,axiom,
( length_sum_set
= ( ^ [R2: nat,N2: nat] :
( collect_list_nat
@ ^ [X2: list_nat] :
( ( ( size_size_list_nat @ X2 )
= R2 )
& ( ( groups4561878855575611511st_nat @ X2 )
= N2 ) ) ) ) ) ).
% length_sum_set_def
thf(fact_572_finite__Collect__disjI,axiom,
! [P2: list_nat > $o,Q2: list_nat > $o] :
( ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [X2: list_nat] :
( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) )
= ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P2 ) )
& ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_573_finite__Collect__disjI,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
& ( finite_finite_nat @ ( collect_nat @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_574_finite__Collect__disjI,axiom,
! [P2: a > $o,Q2: a > $o] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X2: a] :
( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) )
= ( ( finite_finite_a @ ( collect_a @ P2 ) )
& ( finite_finite_a @ ( collect_a @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_575_finite__Collect__conjI,axiom,
! [P2: list_nat > $o,Q2: list_nat > $o] :
( ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P2 ) )
| ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q2 ) ) )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [X2: list_nat] :
( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_576_finite__Collect__conjI,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
| ( finite_finite_nat @ ( collect_nat @ Q2 ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_577_finite__Collect__conjI,axiom,
! [P2: a > $o,Q2: a > $o] :
( ( ( finite_finite_a @ ( collect_a @ P2 ) )
| ( finite_finite_a @ ( collect_a @ Q2 ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [X2: a] :
( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_578_wfP__empty,axiom,
( wfP_list_nat
@ ^ [X2: list_nat,Y3: list_nat] : $false ) ).
% wfP_empty
thf(fact_579_finite__Collect__subsets,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B5: set_nat] : ( ord_less_eq_set_nat @ B5 @ A3 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_580_finite__Collect__subsets,axiom,
! [A3: set_list_nat] :
( ( finite8100373058378681591st_nat @ A3 )
=> ( finite7047420756378620717st_nat
@ ( collect_set_list_nat
@ ^ [B5: set_list_nat] : ( ord_le6045566169113846134st_nat @ B5 @ A3 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_581_finite__Collect__subsets,axiom,
! [A3: set_a] :
( ( finite_finite_a @ A3 )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [B5: set_a] : ( ord_less_eq_set_a @ B5 @ A3 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_582_finite__Collect__le__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K2 ) ) ) ).
% finite_Collect_le_nat
thf(fact_583_finite__Collect__less__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_nat @ N2 @ K2 ) ) ) ).
% finite_Collect_less_nat
thf(fact_584_pigeonhole__infinite__rel,axiom,
! [A3: set_list_nat,B4: set_list_nat,R3: list_nat > list_nat > $o] :
( ~ ( finite8100373058378681591st_nat @ A3 )
=> ( ( finite8100373058378681591st_nat @ B4 )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A3 )
=> ? [Xa: list_nat] :
( ( member_list_nat @ Xa @ B4 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ B4 )
& ~ ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [A2: list_nat] :
( ( member_list_nat @ A2 @ A3 )
& ( R3 @ A2 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_585_pigeonhole__infinite__rel,axiom,
! [A3: set_list_nat,B4: set_nat,R3: list_nat > nat > $o] :
( ~ ( finite8100373058378681591st_nat @ A3 )
=> ( ( finite_finite_nat @ B4 )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A3 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B4 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B4 )
& ~ ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [A2: list_nat] :
( ( member_list_nat @ A2 @ A3 )
& ( R3 @ A2 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_586_pigeonhole__infinite__rel,axiom,
! [A3: set_list_nat,B4: set_a,R3: list_nat > a > $o] :
( ~ ( finite8100373058378681591st_nat @ A3 )
=> ( ( finite_finite_a @ B4 )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A3 )
=> ? [Xa: a] :
( ( member_a @ Xa @ B4 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ B4 )
& ~ ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [A2: list_nat] :
( ( member_list_nat @ A2 @ A3 )
& ( R3 @ A2 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_587_pigeonhole__infinite__rel,axiom,
! [A3: set_nat,B4: set_list_nat,R3: nat > list_nat > $o] :
( ~ ( finite_finite_nat @ A3 )
=> ( ( finite8100373058378681591st_nat @ B4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ? [Xa: list_nat] :
( ( member_list_nat @ Xa @ B4 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ B4 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A2: nat] :
( ( member_nat @ A2 @ A3 )
& ( R3 @ A2 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_588_pigeonhole__infinite__rel,axiom,
! [A3: set_nat,B4: set_nat,R3: nat > nat > $o] :
( ~ ( finite_finite_nat @ A3 )
=> ( ( finite_finite_nat @ B4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B4 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B4 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A2: nat] :
( ( member_nat @ A2 @ A3 )
& ( R3 @ A2 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_589_pigeonhole__infinite__rel,axiom,
! [A3: set_nat,B4: set_a,R3: nat > a > $o] :
( ~ ( finite_finite_nat @ A3 )
=> ( ( finite_finite_a @ B4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ? [Xa: a] :
( ( member_a @ Xa @ B4 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ B4 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A2: nat] :
( ( member_nat @ A2 @ A3 )
& ( R3 @ A2 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_590_pigeonhole__infinite__rel,axiom,
! [A3: set_a,B4: set_list_nat,R3: a > list_nat > $o] :
( ~ ( finite_finite_a @ A3 )
=> ( ( finite8100373058378681591st_nat @ B4 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ? [Xa: list_nat] :
( ( member_list_nat @ Xa @ B4 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: list_nat] :
( ( member_list_nat @ X3 @ B4 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A2: a] :
( ( member_a @ A2 @ A3 )
& ( R3 @ A2 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_591_pigeonhole__infinite__rel,axiom,
! [A3: set_a,B4: set_nat,R3: a > nat > $o] :
( ~ ( finite_finite_a @ A3 )
=> ( ( finite_finite_nat @ B4 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B4 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B4 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A2: a] :
( ( member_a @ A2 @ A3 )
& ( R3 @ A2 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_592_pigeonhole__infinite__rel,axiom,
! [A3: set_a,B4: set_a,R3: a > a > $o] :
( ~ ( finite_finite_a @ A3 )
=> ( ( finite_finite_a @ B4 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ? [Xa: a] :
( ( member_a @ Xa @ B4 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ B4 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A2: a] :
( ( member_a @ A2 @ A3 )
& ( R3 @ A2 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_593_not__finite__existsD,axiom,
! [P2: list_nat > $o] :
( ~ ( finite8100373058378681591st_nat @ ( collect_list_nat @ P2 ) )
=> ? [X_1: list_nat] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_594_not__finite__existsD,axiom,
! [P2: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P2 ) )
=> ? [X_1: nat] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_595_not__finite__existsD,axiom,
! [P2: a > $o] :
( ~ ( finite_finite_a @ ( collect_a @ P2 ) )
=> ? [X_1: a] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_596_finite__less__ub,axiom,
! [F: nat > nat,U4: nat] :
( ! [N: nat] : ( ord_less_eq_nat @ N @ ( F @ N ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U4 ) ) ) ) ).
% finite_less_ub
thf(fact_597_finite__M__bounded__by__nat,axiom,
! [P2: nat > $o,I3: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K: nat] :
( ( P2 @ K )
& ( ord_less_nat @ K @ I3 ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_598_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A5 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_599_less__eq__set__def,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A5: set_list_nat,B5: set_list_nat] :
( ord_le1520216061033275535_nat_o
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ A5 )
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_600_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ord_less_eq_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A5 )
@ ^ [X2: a] : ( member_a @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_601_Collect__subset,axiom,
! [A3: set_nat,P2: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( P2 @ X2 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_602_Collect__subset,axiom,
! [A3: set_list_nat,P2: list_nat > $o] :
( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X2: list_nat] :
( ( member_list_nat @ X2 @ A3 )
& ( P2 @ X2 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_603_Collect__subset,axiom,
! [A3: set_a,P2: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A3 )
& ( P2 @ X2 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_604_less__set__def,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [A5: set_list_nat,B5: set_list_nat] :
( ord_less_list_nat_o
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ A5 )
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ B5 ) ) ) ) ).
% less_set_def
thf(fact_605_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ord_less_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A5 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B5 ) ) ) ) ).
% less_set_def
thf(fact_606_less__set__def,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ord_less_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A5 )
@ ^ [X2: a] : ( member_a @ X2 @ B5 ) ) ) ) ).
% less_set_def
thf(fact_607_pred__subset__eq,axiom,
! [R3: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ R3 )
@ ^ [X2: nat] : ( member_nat @ X2 @ S ) )
= ( ord_less_eq_set_nat @ R3 @ S ) ) ).
% pred_subset_eq
thf(fact_608_pred__subset__eq,axiom,
! [R3: set_list_nat,S: set_list_nat] :
( ( ord_le1520216061033275535_nat_o
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ R3 )
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ S ) )
= ( ord_le6045566169113846134st_nat @ R3 @ S ) ) ).
% pred_subset_eq
thf(fact_609_pred__subset__eq,axiom,
! [R3: set_a,S: set_a] :
( ( ord_less_eq_a_o
@ ^ [X2: a] : ( member_a @ X2 @ R3 )
@ ^ [X2: a] : ( member_a @ X2 @ S ) )
= ( ord_less_eq_set_a @ R3 @ S ) ) ).
% pred_subset_eq
thf(fact_610_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X2: nat] : $false ) ) ).
% empty_def
thf(fact_611_empty__def,axiom,
( bot_bot_set_list_nat
= ( collect_list_nat
@ ^ [X2: list_nat] : $false ) ) ).
% empty_def
thf(fact_612_empty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X2: a] : $false ) ) ).
% empty_def
thf(fact_613_Collect__restrict,axiom,
! [X8: set_nat,P2: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ X8 )
& ( P2 @ X2 ) ) )
@ X8 ) ).
% Collect_restrict
thf(fact_614_Collect__restrict,axiom,
! [X8: set_list_nat,P2: list_nat > $o] :
( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X2: list_nat] :
( ( member_list_nat @ X2 @ X8 )
& ( P2 @ X2 ) ) )
@ X8 ) ).
% Collect_restrict
thf(fact_615_Collect__restrict,axiom,
! [X8: set_a,P2: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ X8 )
& ( P2 @ X2 ) ) )
@ X8 ) ).
% Collect_restrict
thf(fact_616_prop__restrict,axiom,
! [X: nat,Z5: set_nat,X8: set_nat,P2: nat > $o] :
( ( member_nat @ X @ Z5 )
=> ( ( ord_less_eq_set_nat @ Z5
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ X8 )
& ( P2 @ X2 ) ) ) )
=> ( P2 @ X ) ) ) ).
% prop_restrict
thf(fact_617_prop__restrict,axiom,
! [X: list_nat,Z5: set_list_nat,X8: set_list_nat,P2: list_nat > $o] :
( ( member_list_nat @ X @ Z5 )
=> ( ( ord_le6045566169113846134st_nat @ Z5
@ ( collect_list_nat
@ ^ [X2: list_nat] :
( ( member_list_nat @ X2 @ X8 )
& ( P2 @ X2 ) ) ) )
=> ( P2 @ X ) ) ) ).
% prop_restrict
thf(fact_618_prop__restrict,axiom,
! [X: a,Z5: set_a,X8: set_a,P2: a > $o] :
( ( member_a @ X @ Z5 )
=> ( ( ord_less_eq_set_a @ Z5
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ X8 )
& ( P2 @ X2 ) ) ) )
=> ( P2 @ X ) ) ) ).
% prop_restrict
thf(fact_619_pointwise__le__imp___092_060sigma_062,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( pointwise_le @ Xs2 @ Ys2 )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ Xs2 ) @ ( groups4561878855575611511st_nat @ Ys2 ) ) ) ).
% pointwise_le_imp_\<sigma>
thf(fact_620_bounded__Max__nat,axiom,
! [P2: nat > $o,X: nat,M: nat] :
( ( P2 @ X )
=> ( ! [X3: nat] :
( ( P2 @ X3 )
=> ( ord_less_eq_nat @ X3 @ M ) )
=> ~ ! [M6: nat] :
( ( P2 @ M6 )
=> ~ ! [X5: nat] :
( ( P2 @ X5 )
=> ( ord_less_eq_nat @ X5 @ M6 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_621_sum__list__mono2,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ I ) @ ( nth_nat @ Ys2 @ I ) ) )
=> ( ord_less_eq_nat @ ( groups4561878855575611511st_nat @ Xs2 ) @ ( groups4561878855575611511st_nat @ Ys2 ) ) ) ) ).
% sum_list_mono2
thf(fact_622_elem__le__sum__list,axiom,
! [K2: nat,Ns: list_nat] :
( ( ord_less_nat @ K2 @ ( size_size_list_nat @ Ns ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Ns @ K2 ) @ ( groups4561878855575611511st_nat @ Ns ) ) ) ).
% elem_le_sum_list
thf(fact_623_sum__list__incr,axiom,
! [I3: nat,X: list_nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ X ) )
=> ( ( groups4561878855575611511st_nat @ ( list_incr @ I3 @ X ) )
= ( suc @ ( groups4561878855575611511st_nat @ X ) ) ) ) ).
% sum_list_incr
thf(fact_624_finite__lists__length__le,axiom,
! [A3: set_nat,N3: nat] :
( ( finite_finite_nat @ A3 )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A3 )
& ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N3 ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_625_finite__lists__length__le,axiom,
! [A3: set_list_nat,N3: nat] :
( ( finite8100373058378681591st_nat @ A3 )
=> ( finite8170528100393595399st_nat
@ ( collec5989764272469232197st_nat
@ ^ [Xs: list_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ A3 )
& ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ N3 ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_626_finite__lists__length__le,axiom,
! [A3: set_a,N3: nat] :
( ( finite_finite_a @ A3 )
=> ( finite_finite_list_a
@ ( collect_list_a
@ ^ [Xs: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ A3 )
& ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N3 ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_627_bot_Oordering__top__axioms,axiom,
( ordering_top_nat
@ ^ [X2: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ X2 )
@ ^ [X2: nat,Y3: nat] : ( ord_less_nat @ Y3 @ X2 )
@ bot_bot_nat ) ).
% bot.ordering_top_axioms
thf(fact_628_bot_Oordering__top__axioms,axiom,
( orderi7613023277850353061st_nat
@ ^ [X2: set_list_nat,Y3: set_list_nat] : ( ord_le6045566169113846134st_nat @ Y3 @ X2 )
@ ^ [X2: set_list_nat,Y3: set_list_nat] : ( ord_le1190675801316882794st_nat @ Y3 @ X2 )
@ bot_bot_set_list_nat ) ).
% bot.ordering_top_axioms
thf(fact_629_bot_Oordering__top__axioms,axiom,
( ordering_top_set_a
@ ^ [X2: set_a,Y3: set_a] : ( ord_less_eq_set_a @ Y3 @ X2 )
@ ^ [X2: set_a,Y3: set_a] : ( ord_less_set_a @ Y3 @ X2 )
@ bot_bot_set_a ) ).
% bot.ordering_top_axioms
thf(fact_630_List_Ofinite__set,axiom,
! [Xs2: list_list_nat] : ( finite8100373058378681591st_nat @ ( set_list_nat2 @ Xs2 ) ) ).
% List.finite_set
thf(fact_631_List_Ofinite__set,axiom,
! [Xs2: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs2 ) ) ).
% List.finite_set
thf(fact_632_List_Ofinite__set,axiom,
! [Xs2: list_a] : ( finite_finite_a @ ( set_a2 @ Xs2 ) ) ).
% List.finite_set
thf(fact_633_length__list__incr,axiom,
! [I3: nat,X: list_nat] :
( ( size_size_list_nat @ ( list_incr @ I3 @ X ) )
= ( size_size_list_nat @ X ) ) ).
% length_list_incr
thf(fact_634_ordering__top_Oextremum__uniqueI,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A )
=> ( A = Top ) ) ) ).
% ordering_top.extremum_uniqueI
thf(fact_635_ordering__top_Onot__eq__extremum,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( A != Top )
= ( Less @ A @ Top ) ) ) ).
% ordering_top.not_eq_extremum
thf(fact_636_ordering__top_Oextremum__unique,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A )
= ( A = Top ) ) ) ).
% ordering_top.extremum_unique
thf(fact_637_ordering__top_Oextremum__strict,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ~ ( Less @ Top @ A ) ) ).
% ordering_top.extremum_strict
thf(fact_638_ordering__top_Oextremum,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( Less_eq @ A @ Top ) ) ).
% ordering_top.extremum
thf(fact_639_subset__code_I1_J,axiom,
! [Xs2: list_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ B4 )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( member_nat @ X2 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_640_subset__code_I1_J,axiom,
! [Xs2: list_list_nat,B4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs2 ) @ B4 )
= ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs2 ) )
=> ( member_list_nat @ X2 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_641_subset__code_I1_J,axiom,
! [Xs2: list_a,B4: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs2 ) @ B4 )
= ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs2 ) )
=> ( member_a @ X2 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_642_finite__list,axiom,
! [A3: set_list_nat] :
( ( finite8100373058378681591st_nat @ A3 )
=> ? [Xs3: list_list_nat] :
( ( set_list_nat2 @ Xs3 )
= A3 ) ) ).
% finite_list
thf(fact_643_finite__list,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ? [Xs3: list_nat] :
( ( set_nat2 @ Xs3 )
= A3 ) ) ).
% finite_list
thf(fact_644_finite__list,axiom,
! [A3: set_a] :
( ( finite_finite_a @ A3 )
=> ? [Xs3: list_a] :
( ( set_a2 @ Xs3 )
= A3 ) ) ).
% finite_list
thf(fact_645_member__le__sum__list,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( ord_less_eq_nat @ X @ ( groups4561878855575611511st_nat @ Xs2 ) ) ) ).
% member_le_sum_list
thf(fact_646_nth__mem,axiom,
! [N3: nat,Xs2: list_list_nat] :
( ( ord_less_nat @ N3 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( member_list_nat @ ( nth_list_nat @ Xs2 @ N3 ) @ ( set_list_nat2 @ Xs2 ) ) ) ).
% nth_mem
thf(fact_647_nth__mem,axiom,
! [N3: nat,Xs2: list_a] :
( ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs2 ) )
=> ( member_a @ ( nth_a @ Xs2 @ N3 ) @ ( set_a2 @ Xs2 ) ) ) ).
% nth_mem
thf(fact_648_nth__mem,axiom,
! [N3: nat,Xs2: list_nat] :
( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs2 ) )
=> ( member_nat @ ( nth_nat @ Xs2 @ N3 ) @ ( set_nat2 @ Xs2 ) ) ) ).
% nth_mem
thf(fact_649_list__ball__nth,axiom,
! [N3: nat,Xs2: list_a,P2: a > $o] :
( ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs2 ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs2 ) )
=> ( P2 @ X3 ) )
=> ( P2 @ ( nth_a @ Xs2 @ N3 ) ) ) ) ).
% list_ball_nth
thf(fact_650_list__ball__nth,axiom,
! [N3: nat,Xs2: list_nat,P2: nat > $o] :
( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
=> ( P2 @ X3 ) )
=> ( P2 @ ( nth_nat @ Xs2 @ N3 ) ) ) ) ).
% list_ball_nth
thf(fact_651_in__set__conv__nth,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
= ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
& ( ( nth_list_nat @ Xs2 @ I2 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_652_in__set__conv__nth,axiom,
! [X: a,Xs2: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs2 ) )
= ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs2 ) )
& ( ( nth_a @ Xs2 @ I2 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_653_in__set__conv__nth,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
= ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
& ( ( nth_nat @ Xs2 @ I2 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_654_all__nth__imp__all__set,axiom,
! [Xs2: list_list_nat,P2: list_nat > $o,X: list_nat] :
( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( P2 @ ( nth_list_nat @ Xs2 @ I ) ) )
=> ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
=> ( P2 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_655_all__nth__imp__all__set,axiom,
! [Xs2: list_a,P2: a > $o,X: a] :
( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs2 ) )
=> ( P2 @ ( nth_a @ Xs2 @ I ) ) )
=> ( ( member_a @ X @ ( set_a2 @ Xs2 ) )
=> ( P2 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_656_all__nth__imp__all__set,axiom,
! [Xs2: list_nat,P2: nat > $o,X: nat] :
( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( P2 @ ( nth_nat @ Xs2 @ I ) ) )
=> ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( P2 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_657_all__set__conv__all__nth,axiom,
! [Xs2: list_a,P2: a > $o] :
( ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs2 ) )
=> ( P2 @ X2 ) ) )
= ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs2 ) )
=> ( P2 @ ( nth_a @ Xs2 @ I2 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_658_all__set__conv__all__nth,axiom,
! [Xs2: list_nat,P2: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( P2 @ X2 ) ) )
= ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
=> ( P2 @ ( nth_nat @ Xs2 @ I2 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_659_finite__lists__length__eq,axiom,
! [A3: set_nat,N3: nat] :
( ( finite_finite_nat @ A3 )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A3 )
& ( ( size_size_list_nat @ Xs )
= N3 ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_660_finite__lists__length__eq,axiom,
! [A3: set_list_nat,N3: nat] :
( ( finite8100373058378681591st_nat @ A3 )
=> ( finite8170528100393595399st_nat
@ ( collec5989764272469232197st_nat
@ ^ [Xs: list_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ A3 )
& ( ( size_s3023201423986296836st_nat @ Xs )
= N3 ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_661_finite__lists__length__eq,axiom,
! [A3: set_a,N3: nat] :
( ( finite_finite_a @ A3 )
=> ( finite_finite_list_a
@ ( collect_list_a
@ ^ [Xs: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ A3 )
& ( ( size_size_list_a @ Xs )
= N3 ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_662_list__incr__def,axiom,
( list_incr
= ( ^ [I2: nat,X2: list_nat] : ( list_update_nat @ X2 @ I2 @ ( suc @ ( nth_nat @ X2 @ I2 ) ) ) ) ) ).
% list_incr_def
thf(fact_663_set__n__lists,axiom,
! [N3: nat,Xs2: list_nat] :
( ( set_list_nat2 @ ( n_lists_nat @ N3 @ Xs2 ) )
= ( collect_list_nat
@ ^ [Ys: list_nat] :
( ( ( size_size_list_nat @ Ys )
= N3 )
& ( ord_less_eq_set_nat @ ( set_nat2 @ Ys ) @ ( set_nat2 @ Xs2 ) ) ) ) ) ).
% set_n_lists
thf(fact_664_set__n__lists,axiom,
! [N3: nat,Xs2: list_list_nat] :
( ( set_list_list_nat2 @ ( n_lists_list_nat @ N3 @ Xs2 ) )
= ( collec5989764272469232197st_nat
@ ^ [Ys: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Ys )
= N3 )
& ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Ys ) @ ( set_list_nat2 @ Xs2 ) ) ) ) ) ).
% set_n_lists
thf(fact_665_set__n__lists,axiom,
! [N3: nat,Xs2: list_a] :
( ( set_list_a2 @ ( n_lists_a @ N3 @ Xs2 ) )
= ( collect_list_a
@ ^ [Ys: list_a] :
( ( ( size_size_list_a @ Ys )
= N3 )
& ( ord_less_eq_set_a @ ( set_a2 @ Ys ) @ ( set_a2 @ Xs2 ) ) ) ) ) ).
% set_n_lists
thf(fact_666_finite__lists__distinct__length__eq,axiom,
! [A3: set_nat,N3: nat] :
( ( finite_finite_nat @ A3 )
=> ( finite8100373058378681591st_nat
@ ( collect_list_nat
@ ^ [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= N3 )
& ( distinct_nat @ Xs )
& ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A3 ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_667_finite__lists__distinct__length__eq,axiom,
! [A3: set_list_nat,N3: nat] :
( ( finite8100373058378681591st_nat @ A3 )
=> ( finite8170528100393595399st_nat
@ ( collec5989764272469232197st_nat
@ ^ [Xs: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= N3 )
& ( distinct_list_nat @ Xs )
& ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ A3 ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_668_finite__lists__distinct__length__eq,axiom,
! [A3: set_a,N3: nat] :
( ( finite_finite_a @ A3 )
=> ( finite_finite_list_a
@ ( collect_list_a
@ ^ [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= N3 )
& ( distinct_a @ Xs )
& ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ A3 ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_669_list__update__overwrite,axiom,
! [Xs2: list_nat,I3: nat,X: nat,Y4: nat] :
( ( list_update_nat @ ( list_update_nat @ Xs2 @ I3 @ X ) @ I3 @ Y4 )
= ( list_update_nat @ Xs2 @ I3 @ Y4 ) ) ).
% list_update_overwrite
thf(fact_670_length__list__update,axiom,
! [Xs2: list_nat,I3: nat,X: nat] :
( ( size_size_list_nat @ ( list_update_nat @ Xs2 @ I3 @ X ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_list_update
thf(fact_671_nth__list__update__neq,axiom,
! [I3: nat,J: nat,Xs2: list_a,X: a] :
( ( I3 != J )
=> ( ( nth_a @ ( list_update_a @ Xs2 @ I3 @ X ) @ J )
= ( nth_a @ Xs2 @ J ) ) ) ).
% nth_list_update_neq
thf(fact_672_nth__list__update__neq,axiom,
! [I3: nat,J: nat,Xs2: list_nat,X: nat] :
( ( I3 != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs2 @ I3 @ X ) @ J )
= ( nth_nat @ Xs2 @ J ) ) ) ).
% nth_list_update_neq
thf(fact_673_list__update__id,axiom,
! [Xs2: list_a,I3: nat] :
( ( list_update_a @ Xs2 @ I3 @ ( nth_a @ Xs2 @ I3 ) )
= Xs2 ) ).
% list_update_id
thf(fact_674_list__update__id,axiom,
! [Xs2: list_nat,I3: nat] :
( ( list_update_nat @ Xs2 @ I3 @ ( nth_nat @ Xs2 @ I3 ) )
= Xs2 ) ).
% list_update_id
thf(fact_675_list__update__beyond,axiom,
! [Xs2: list_nat,I3: nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ I3 )
=> ( ( list_update_nat @ Xs2 @ I3 @ X )
= Xs2 ) ) ).
% list_update_beyond
thf(fact_676_nth__list__update__eq,axiom,
! [I3: nat,Xs2: list_a,X: a] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs2 ) )
=> ( ( nth_a @ ( list_update_a @ Xs2 @ I3 @ X ) @ I3 )
= X ) ) ).
% nth_list_update_eq
thf(fact_677_nth__list__update__eq,axiom,
! [I3: nat,Xs2: list_nat,X: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ ( list_update_nat @ Xs2 @ I3 @ X ) @ I3 )
= X ) ) ).
% nth_list_update_eq
thf(fact_678_set__swap,axiom,
! [I3: nat,Xs2: list_a,J: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs2 ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs2 ) )
=> ( ( set_a2 @ ( list_update_a @ ( list_update_a @ Xs2 @ I3 @ ( nth_a @ Xs2 @ J ) ) @ J @ ( nth_a @ Xs2 @ I3 ) ) )
= ( set_a2 @ Xs2 ) ) ) ) ).
% set_swap
thf(fact_679_set__swap,axiom,
! [I3: nat,Xs2: list_nat,J: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs2 @ I3 @ ( nth_nat @ Xs2 @ J ) ) @ J @ ( nth_nat @ Xs2 @ I3 ) ) )
= ( set_nat2 @ Xs2 ) ) ) ) ).
% set_swap
thf(fact_680_distinct__swap,axiom,
! [I3: nat,Xs2: list_a,J: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs2 ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs2 ) )
=> ( ( distinct_a @ ( list_update_a @ ( list_update_a @ Xs2 @ I3 @ ( nth_a @ Xs2 @ J ) ) @ J @ ( nth_a @ Xs2 @ I3 ) ) )
= ( distinct_a @ Xs2 ) ) ) ) ).
% distinct_swap
thf(fact_681_distinct__swap,axiom,
! [I3: nat,Xs2: list_nat,J: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ( ( distinct_nat @ ( list_update_nat @ ( list_update_nat @ Xs2 @ I3 @ ( nth_nat @ Xs2 @ J ) ) @ J @ ( nth_nat @ Xs2 @ I3 ) ) )
= ( distinct_nat @ Xs2 ) ) ) ) ).
% distinct_swap
thf(fact_682_list__update__swap,axiom,
! [I3: nat,I5: nat,Xs2: list_nat,X: nat,X9: nat] :
( ( I3 != I5 )
=> ( ( list_update_nat @ ( list_update_nat @ Xs2 @ I3 @ X ) @ I5 @ X9 )
= ( list_update_nat @ ( list_update_nat @ Xs2 @ I5 @ X9 ) @ I3 @ X ) ) ) ).
% list_update_swap
thf(fact_683_finite__distinct__list,axiom,
! [A3: set_list_nat] :
( ( finite8100373058378681591st_nat @ A3 )
=> ? [Xs3: list_list_nat] :
( ( ( set_list_nat2 @ Xs3 )
= A3 )
& ( distinct_list_nat @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_684_finite__distinct__list,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ? [Xs3: list_nat] :
( ( ( set_nat2 @ Xs3 )
= A3 )
& ( distinct_nat @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_685_finite__distinct__list,axiom,
! [A3: set_a] :
( ( finite_finite_a @ A3 )
=> ? [Xs3: list_a] :
( ( ( set_a2 @ Xs3 )
= A3 )
& ( distinct_a @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_686_set__update__subsetI,axiom,
! [Xs2: list_nat,A3: set_nat,X: nat,I3: nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A3 )
=> ( ( member_nat @ X @ A3 )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs2 @ I3 @ X ) ) @ A3 ) ) ) ).
% set_update_subsetI
thf(fact_687_set__update__subsetI,axiom,
! [Xs2: list_list_nat,A3: set_list_nat,X: list_nat,I3: nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs2 ) @ A3 )
=> ( ( member_list_nat @ X @ A3 )
=> ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ ( list_update_list_nat @ Xs2 @ I3 @ X ) ) @ A3 ) ) ) ).
% set_update_subsetI
thf(fact_688_set__update__subsetI,axiom,
! [Xs2: list_a,A3: set_a,X: a,I3: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs2 ) @ A3 )
=> ( ( member_a @ X @ A3 )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( list_update_a @ Xs2 @ I3 @ X ) ) @ A3 ) ) ) ).
% set_update_subsetI
thf(fact_689_length__n__lists__elem,axiom,
! [Ys2: list_nat,N3: nat,Xs2: list_nat] :
( ( member_list_nat @ Ys2 @ ( set_list_nat2 @ ( n_lists_nat @ N3 @ Xs2 ) ) )
=> ( ( size_size_list_nat @ Ys2 )
= N3 ) ) ).
% length_n_lists_elem
thf(fact_690_nth__eq__iff__index__eq,axiom,
! [Xs2: list_a,I3: nat,J: nat] :
( ( distinct_a @ Xs2 )
=> ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs2 ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs2 ) )
=> ( ( ( nth_a @ Xs2 @ I3 )
= ( nth_a @ Xs2 @ J ) )
= ( I3 = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_691_nth__eq__iff__index__eq,axiom,
! [Xs2: list_nat,I3: nat,J: nat] :
( ( distinct_nat @ Xs2 )
=> ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ( nth_nat @ Xs2 @ I3 )
= ( nth_nat @ Xs2 @ J ) )
= ( I3 = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_692_distinct__conv__nth,axiom,
( distinct_a
= ( ^ [Xs: list_a] :
! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ! [J3: nat] :
( ( ord_less_nat @ J3 @ ( size_size_list_a @ Xs ) )
=> ( ( I2 != J3 )
=> ( ( nth_a @ Xs @ I2 )
!= ( nth_a @ Xs @ J3 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_693_distinct__conv__nth,axiom,
( distinct_nat
= ( ^ [Xs: list_nat] :
! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ! [J3: nat] :
( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
=> ( ( I2 != J3 )
=> ( ( nth_nat @ Xs @ I2 )
!= ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_694_set__update__memI,axiom,
! [N3: nat,Xs2: list_list_nat,X: list_nat] :
( ( ord_less_nat @ N3 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( member_list_nat @ X @ ( set_list_nat2 @ ( list_update_list_nat @ Xs2 @ N3 @ X ) ) ) ) ).
% set_update_memI
thf(fact_695_set__update__memI,axiom,
! [N3: nat,Xs2: list_a,X: a] :
( ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs2 ) )
=> ( member_a @ X @ ( set_a2 @ ( list_update_a @ Xs2 @ N3 @ X ) ) ) ) ).
% set_update_memI
thf(fact_696_set__update__memI,axiom,
! [N3: nat,Xs2: list_nat,X: nat] :
( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs2 ) )
=> ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs2 @ N3 @ X ) ) ) ) ).
% set_update_memI
thf(fact_697_list__update__same__conv,axiom,
! [I3: nat,Xs2: list_a,X: a] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs2 ) )
=> ( ( ( list_update_a @ Xs2 @ I3 @ X )
= Xs2 )
= ( ( nth_a @ Xs2 @ I3 )
= X ) ) ) ).
% list_update_same_conv
thf(fact_698_list__update__same__conv,axiom,
! [I3: nat,Xs2: list_nat,X: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ( list_update_nat @ Xs2 @ I3 @ X )
= Xs2 )
= ( ( nth_nat @ Xs2 @ I3 )
= X ) ) ) ).
% list_update_same_conv
thf(fact_699_nth__list__update,axiom,
! [I3: nat,Xs2: list_a,J: nat,X: a] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs2 ) )
=> ( ( ( I3 = J )
=> ( ( nth_a @ ( list_update_a @ Xs2 @ I3 @ X ) @ J )
= X ) )
& ( ( I3 != J )
=> ( ( nth_a @ ( list_update_a @ Xs2 @ I3 @ X ) @ J )
= ( nth_a @ Xs2 @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_700_nth__list__update,axiom,
! [I3: nat,Xs2: list_nat,J: nat,X: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ( I3 = J )
=> ( ( nth_nat @ ( list_update_nat @ Xs2 @ I3 @ X ) @ J )
= X ) )
& ( ( I3 != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs2 @ I3 @ X ) @ J )
= ( nth_nat @ Xs2 @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_701_distinct__Ex1,axiom,
! [Xs2: list_list_nat,X: list_nat] :
( ( distinct_list_nat @ Xs2 )
=> ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
=> ? [X3: nat] :
( ( ord_less_nat @ X3 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
& ( ( nth_list_nat @ Xs2 @ X3 )
= X )
& ! [Y: nat] :
( ( ( ord_less_nat @ Y @ ( size_s3023201423986296836st_nat @ Xs2 ) )
& ( ( nth_list_nat @ Xs2 @ Y )
= X ) )
=> ( Y = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_702_distinct__Ex1,axiom,
! [Xs2: list_a,X: a] :
( ( distinct_a @ Xs2 )
=> ( ( member_a @ X @ ( set_a2 @ Xs2 ) )
=> ? [X3: nat] :
( ( ord_less_nat @ X3 @ ( size_size_list_a @ Xs2 ) )
& ( ( nth_a @ Xs2 @ X3 )
= X )
& ! [Y: nat] :
( ( ( ord_less_nat @ Y @ ( size_size_list_a @ Xs2 ) )
& ( ( nth_a @ Xs2 @ Y )
= X ) )
=> ( Y = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_703_distinct__Ex1,axiom,
! [Xs2: list_nat,X: nat] :
( ( distinct_nat @ Xs2 )
=> ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ? [X3: nat] :
( ( ord_less_nat @ X3 @ ( size_size_list_nat @ Xs2 ) )
& ( ( nth_nat @ Xs2 @ X3 )
= X )
& ! [Y: nat] :
( ( ( ord_less_nat @ Y @ ( size_size_list_nat @ Xs2 ) )
& ( ( nth_nat @ Xs2 @ Y )
= X ) )
=> ( Y = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_704_Set_Ois__empty__def,axiom,
( is_empty_list_nat
= ( ^ [A5: set_list_nat] : ( A5 = bot_bot_set_list_nat ) ) ) ).
% Set.is_empty_def
thf(fact_705_Set_Ois__empty__def,axiom,
( is_empty_a
= ( ^ [A5: set_a] : ( A5 = bot_bot_set_a ) ) ) ).
% Set.is_empty_def
thf(fact_706_list__ex__length,axiom,
( list_ex_a
= ( ^ [P4: a > $o,Xs: list_a] :
? [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
& ( P4 @ ( nth_a @ Xs @ N2 ) ) ) ) ) ).
% list_ex_length
thf(fact_707_list__ex__length,axiom,
( list_ex_nat
= ( ^ [P4: nat > $o,Xs: list_nat] :
? [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
& ( P4 @ ( nth_nat @ Xs @ N2 ) ) ) ) ) ).
% list_ex_length
thf(fact_708_list__ex__cong,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat,F: list_nat > $o,G3: list_nat > $o] :
( ( Xs2 = Ys2 )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Ys2 ) )
=> ( ( F @ X3 )
= ( G3 @ X3 ) ) )
=> ( ( list_ex_list_nat @ F @ Xs2 )
= ( list_ex_list_nat @ G3 @ Ys2 ) ) ) ) ).
% list_ex_cong
thf(fact_709_list__ex__cong,axiom,
! [Xs2: list_nat,Ys2: list_nat,F: nat > $o,G3: nat > $o] :
( ( Xs2 = Ys2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Ys2 ) )
=> ( ( F @ X3 )
= ( G3 @ X3 ) ) )
=> ( ( list_ex_nat @ F @ Xs2 )
= ( list_ex_nat @ G3 @ Ys2 ) ) ) ) ).
% list_ex_cong
thf(fact_710_list__ex__cong,axiom,
! [Xs2: list_a,Ys2: list_a,F: a > $o,G3: a > $o] :
( ( Xs2 = Ys2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Ys2 ) )
=> ( ( F @ X3 )
= ( G3 @ X3 ) ) )
=> ( ( list_ex_a @ F @ Xs2 )
= ( list_ex_a @ G3 @ Ys2 ) ) ) ) ).
% list_ex_cong
thf(fact_711_in__set__product__lists__length,axiom,
! [Xs2: list_nat,Xss: list_list_nat] :
( ( member_list_nat @ Xs2 @ ( set_list_nat2 @ ( product_lists_nat @ Xss ) ) )
=> ( ( size_size_list_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_712_is__empty__set,axiom,
! [Xs2: list_a] :
( ( is_empty_a @ ( set_a2 @ Xs2 ) )
= ( null_a @ Xs2 ) ) ).
% is_empty_set
thf(fact_713_subset__Collect__iff,axiom,
! [B4: set_nat,A3: set_nat,P2: nat > $o] :
( ( ord_less_eq_set_nat @ B4 @ A3 )
=> ( ( ord_less_eq_set_nat @ B4
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( P2 @ X2 ) ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ B4 )
=> ( P2 @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_714_subset__Collect__iff,axiom,
! [B4: set_list_nat,A3: set_list_nat,P2: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ B4 @ A3 )
=> ( ( ord_le6045566169113846134st_nat @ B4
@ ( collect_list_nat
@ ^ [X2: list_nat] :
( ( member_list_nat @ X2 @ A3 )
& ( P2 @ X2 ) ) ) )
= ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ B4 )
=> ( P2 @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_715_subset__Collect__iff,axiom,
! [B4: set_a,A3: set_a,P2: a > $o] :
( ( ord_less_eq_set_a @ B4 @ A3 )
=> ( ( ord_less_eq_set_a @ B4
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A3 )
& ( P2 @ X2 ) ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ B4 )
=> ( P2 @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_716_subset__CollectI,axiom,
! [B4: set_nat,A3: set_nat,Q2: nat > $o,P2: nat > $o] :
( ( ord_less_eq_set_nat @ B4 @ A3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B4 )
=> ( ( Q2 @ X3 )
=> ( P2 @ X3 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ B4 )
& ( Q2 @ X2 ) ) )
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( P2 @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_717_subset__CollectI,axiom,
! [B4: set_list_nat,A3: set_list_nat,Q2: list_nat > $o,P2: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ B4 @ A3 )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ B4 )
=> ( ( Q2 @ X3 )
=> ( P2 @ X3 ) ) )
=> ( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X2: list_nat] :
( ( member_list_nat @ X2 @ B4 )
& ( Q2 @ X2 ) ) )
@ ( collect_list_nat
@ ^ [X2: list_nat] :
( ( member_list_nat @ X2 @ A3 )
& ( P2 @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_718_subset__CollectI,axiom,
! [B4: set_a,A3: set_a,Q2: a > $o,P2: a > $o] :
( ( ord_less_eq_set_a @ B4 @ A3 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ B4 )
=> ( ( Q2 @ X3 )
=> ( P2 @ X3 ) ) )
=> ( ord_less_eq_set_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ B4 )
& ( Q2 @ X2 ) ) )
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A3 )
& ( P2 @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_719_conj__subset__def,axiom,
! [A3: set_nat,P2: nat > $o,Q2: nat > $o] :
( ( ord_less_eq_set_nat @ A3
@ ( collect_nat
@ ^ [X2: nat] :
( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) )
= ( ( ord_less_eq_set_nat @ A3 @ ( collect_nat @ P2 ) )
& ( ord_less_eq_set_nat @ A3 @ ( collect_nat @ Q2 ) ) ) ) ).
% conj_subset_def
thf(fact_720_conj__subset__def,axiom,
! [A3: set_list_nat,P2: list_nat > $o,Q2: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ A3
@ ( collect_list_nat
@ ^ [X2: list_nat] :
( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) )
= ( ( ord_le6045566169113846134st_nat @ A3 @ ( collect_list_nat @ P2 ) )
& ( ord_le6045566169113846134st_nat @ A3 @ ( collect_list_nat @ Q2 ) ) ) ) ).
% conj_subset_def
thf(fact_721_conj__subset__def,axiom,
! [A3: set_a,P2: a > $o,Q2: a > $o] :
( ( ord_less_eq_set_a @ A3
@ ( collect_a
@ ^ [X2: a] :
( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) )
= ( ( ord_less_eq_set_a @ A3 @ ( collect_a @ P2 ) )
& ( ord_less_eq_set_a @ A3 @ ( collect_a @ Q2 ) ) ) ) ).
% conj_subset_def
thf(fact_722_Khovanskii_Oset__aA,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A3: set_a] :
( ( khovanskii_a @ G @ Addition @ Zero @ A3 )
=> ( ( set_a2 @ ( aA_a @ A3 ) )
= A3 ) ) ).
% Khovanskii.set_aA
thf(fact_723_Khovanskii_OaA_Ocong,axiom,
aA_a = aA_a ).
% Khovanskii.aA.cong
thf(fact_724_Khovanskii_OaA__idx__eq,axiom,
! [G: set_list_nat,Addition: list_nat > list_nat > list_nat,Zero: list_nat,A3: set_list_nat,A: list_nat] :
( ( khovanskii_list_nat @ G @ Addition @ Zero @ A3 )
=> ( ( member_list_nat @ A @ A3 )
=> ( ( nth_list_nat @ ( aA_list_nat @ A3 ) @ ( counta9103016383634126529st_nat @ A3 @ A ) )
= A ) ) ) ).
% Khovanskii.aA_idx_eq
thf(fact_725_Khovanskii_OaA__idx__eq,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A3: set_nat,A: nat] :
( ( khovanskii_nat @ G @ Addition @ Zero @ A3 )
=> ( ( member_nat @ A @ A3 )
=> ( ( nth_nat @ ( aA_nat @ A3 ) @ ( counta4844910239362777137on_nat @ A3 @ A ) )
= A ) ) ) ).
% Khovanskii.aA_idx_eq
thf(fact_726_Khovanskii_OaA__idx__eq,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A3: set_a,A: a] :
( ( khovanskii_a @ G @ Addition @ Zero @ A3 )
=> ( ( member_a @ A @ A3 )
=> ( ( nth_a @ ( aA_a @ A3 ) @ ( counta3566351752493190365t_on_a @ A3 @ A ) )
= A ) ) ) ).
% Khovanskii.aA_idx_eq
thf(fact_727_Khovanskii_Onth__aA__in__G,axiom,
! [G: set_list_nat,Addition: list_nat > list_nat > list_nat,Zero: list_nat,A3: set_list_nat,I3: nat] :
( ( khovanskii_list_nat @ G @ Addition @ Zero @ A3 )
=> ( ( ord_less_nat @ I3 @ ( finite_card_list_nat @ A3 ) )
=> ( member_list_nat @ ( nth_list_nat @ ( aA_list_nat @ A3 ) @ I3 ) @ G ) ) ) ).
% Khovanskii.nth_aA_in_G
thf(fact_728_Khovanskii_Onth__aA__in__G,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A3: set_nat,I3: nat] :
( ( khovanskii_nat @ G @ Addition @ Zero @ A3 )
=> ( ( ord_less_nat @ I3 @ ( finite_card_nat @ A3 ) )
=> ( member_nat @ ( nth_nat @ ( aA_nat @ A3 ) @ I3 ) @ G ) ) ) ).
% Khovanskii.nth_aA_in_G
thf(fact_729_Khovanskii_Onth__aA__in__G,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A3: set_a,I3: nat] :
( ( khovanskii_a @ G @ Addition @ Zero @ A3 )
=> ( ( ord_less_nat @ I3 @ ( finite_card_a @ A3 ) )
=> ( member_a @ ( nth_a @ ( aA_a @ A3 ) @ I3 ) @ G ) ) ) ).
% Khovanskii.nth_aA_in_G
thf(fact_730_enumerate__mono__iff,axiom,
! [S: set_list_nat,M3: nat,N3: nat] :
( ~ ( finite8100373058378681591st_nat @ S )
=> ( ( ord_less_list_nat @ ( infini2033088105919815547st_nat @ S @ M3 ) @ ( infini2033088105919815547st_nat @ S @ N3 ) )
= ( ord_less_nat @ M3 @ N3 ) ) ) ).
% enumerate_mono_iff
thf(fact_731_enumerate__mono__iff,axiom,
! [S: set_nat,M3: nat,N3: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M3 ) @ ( infini8530281810654367211te_nat @ S @ N3 ) )
= ( ord_less_nat @ M3 @ N3 ) ) ) ).
% enumerate_mono_iff
thf(fact_732_enumerate__mono__le__iff,axiom,
! [S: set_list_nat,M3: nat,N3: nat] :
( ~ ( finite8100373058378681591st_nat @ S )
=> ( ( ord_less_eq_list_nat @ ( infini2033088105919815547st_nat @ S @ M3 ) @ ( infini2033088105919815547st_nat @ S @ N3 ) )
= ( ord_less_eq_nat @ M3 @ N3 ) ) ) ).
% enumerate_mono_le_iff
thf(fact_733_enumerate__mono__le__iff,axiom,
! [S: set_nat,M3: nat,N3: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ( ord_less_eq_nat @ ( infini8530281810654367211te_nat @ S @ M3 ) @ ( infini8530281810654367211te_nat @ S @ N3 ) )
= ( ord_less_eq_nat @ M3 @ N3 ) ) ) ).
% enumerate_mono_le_iff
thf(fact_734_Sup__fin_Osubset__imp,axiom,
! [A3: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B4 )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A3 ) @ ( lattic1093996805478795353in_nat @ B4 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_735_Sup__fin_Osubset__imp,axiom,
! [A3: set_set_list_nat,B4: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ A3 @ B4 )
=> ( ( A3 != bot_bo3886227569956363488st_nat )
=> ( ( finite7047420756378620717st_nat @ B4 )
=> ( ord_le6045566169113846134st_nat @ ( lattic2169124122975652127st_nat @ A3 ) @ ( lattic2169124122975652127st_nat @ B4 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_736_Sup__fin_Osubset__imp,axiom,
! [A3: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B4 )
=> ( ( A3 != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ B4 )
=> ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A3 ) @ ( lattic2918178356826803221_set_a @ B4 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_737_Sup__fin_Osubset__imp,axiom,
! [A3: set_list_nat,B4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ B4 )
=> ( ( A3 != bot_bot_set_list_nat )
=> ( ( finite8100373058378681591st_nat @ B4 )
=> ( ord_less_eq_list_nat @ ( lattic6411832703407573737st_nat @ A3 ) @ ( lattic6411832703407573737st_nat @ B4 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_738_card__Collect__less__nat,axiom,
! [N3: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_nat @ I2 @ N3 ) ) )
= N3 ) ).
% card_Collect_less_nat
thf(fact_739_card__Collect__le__nat,axiom,
! [N3: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I2: nat] : ( ord_less_eq_nat @ I2 @ N3 ) ) )
= ( suc @ N3 ) ) ).
% card_Collect_le_nat
thf(fact_740_finite__enumerate__mono__iff,axiom,
! [S: set_list_nat,M3: nat,N3: nat] :
( ( finite8100373058378681591st_nat @ S )
=> ( ( ord_less_nat @ M3 @ ( finite_card_list_nat @ S ) )
=> ( ( ord_less_nat @ N3 @ ( finite_card_list_nat @ S ) )
=> ( ( ord_less_list_nat @ ( infini2033088105919815547st_nat @ S @ M3 ) @ ( infini2033088105919815547st_nat @ S @ N3 ) )
= ( ord_less_nat @ M3 @ N3 ) ) ) ) ) ).
% finite_enumerate_mono_iff
thf(fact_741_finite__enumerate__mono__iff,axiom,
! [S: set_nat,M3: nat,N3: nat] :
( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ M3 @ ( finite_card_nat @ S ) )
=> ( ( ord_less_nat @ N3 @ ( finite_card_nat @ S ) )
=> ( ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M3 ) @ ( infini8530281810654367211te_nat @ S @ N3 ) )
= ( ord_less_nat @ M3 @ N3 ) ) ) ) ) ).
% finite_enumerate_mono_iff
thf(fact_742_finite__enum__ext,axiom,
! [X8: set_list_nat,Y6: set_list_nat] :
( ! [I: nat] :
( ( ord_less_nat @ I @ ( finite_card_list_nat @ X8 ) )
=> ( ( infini2033088105919815547st_nat @ X8 @ I )
= ( infini2033088105919815547st_nat @ Y6 @ I ) ) )
=> ( ( finite8100373058378681591st_nat @ X8 )
=> ( ( finite8100373058378681591st_nat @ Y6 )
=> ( ( ( finite_card_list_nat @ X8 )
= ( finite_card_list_nat @ Y6 ) )
=> ( X8 = Y6 ) ) ) ) ) ).
% finite_enum_ext
thf(fact_743_finite__enum__ext,axiom,
! [X8: set_nat,Y6: set_nat] :
( ! [I: nat] :
( ( ord_less_nat @ I @ ( finite_card_nat @ X8 ) )
=> ( ( infini8530281810654367211te_nat @ X8 @ I )
= ( infini8530281810654367211te_nat @ Y6 @ I ) ) )
=> ( ( finite_finite_nat @ X8 )
=> ( ( finite_finite_nat @ Y6 )
=> ( ( ( finite_card_nat @ X8 )
= ( finite_card_nat @ Y6 ) )
=> ( X8 = Y6 ) ) ) ) ) ).
% finite_enum_ext
thf(fact_744_finite__enumerate__Ex,axiom,
! [S: set_list_nat,S2: list_nat] :
( ( finite8100373058378681591st_nat @ S )
=> ( ( member_list_nat @ S2 @ S )
=> ? [N: nat] :
( ( ord_less_nat @ N @ ( finite_card_list_nat @ S ) )
& ( ( infini2033088105919815547st_nat @ S @ N )
= S2 ) ) ) ) ).
% finite_enumerate_Ex
thf(fact_745_finite__enumerate__Ex,axiom,
! [S: set_nat,S2: nat] :
( ( finite_finite_nat @ S )
=> ( ( member_nat @ S2 @ S )
=> ? [N: nat] :
( ( ord_less_nat @ N @ ( finite_card_nat @ S ) )
& ( ( infini8530281810654367211te_nat @ S @ N )
= S2 ) ) ) ) ).
% finite_enumerate_Ex
thf(fact_746_finite__enumerate__in__set,axiom,
! [S: set_list_nat,N3: nat] :
( ( finite8100373058378681591st_nat @ S )
=> ( ( ord_less_nat @ N3 @ ( finite_card_list_nat @ S ) )
=> ( member_list_nat @ ( infini2033088105919815547st_nat @ S @ N3 ) @ S ) ) ) ).
% finite_enumerate_in_set
thf(fact_747_finite__enumerate__in__set,axiom,
! [S: set_nat,N3: nat] :
( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ N3 @ ( finite_card_nat @ S ) )
=> ( member_nat @ ( infini8530281810654367211te_nat @ S @ N3 ) @ S ) ) ) ).
% finite_enumerate_in_set
thf(fact_748_Khovanskii_Oidx__less__cardA,axiom,
! [G: set_list_nat,Addition: list_nat > list_nat > list_nat,Zero: list_nat,A3: set_list_nat,A: list_nat] :
( ( khovanskii_list_nat @ G @ Addition @ Zero @ A3 )
=> ( ( member_list_nat @ A @ A3 )
=> ( ord_less_nat @ ( counta9103016383634126529st_nat @ A3 @ A ) @ ( finite_card_list_nat @ A3 ) ) ) ) ).
% Khovanskii.idx_less_cardA
thf(fact_749_Khovanskii_Oidx__less__cardA,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A3: set_nat,A: nat] :
( ( khovanskii_nat @ G @ Addition @ Zero @ A3 )
=> ( ( member_nat @ A @ A3 )
=> ( ord_less_nat @ ( counta4844910239362777137on_nat @ A3 @ A ) @ ( finite_card_nat @ A3 ) ) ) ) ).
% Khovanskii.idx_less_cardA
thf(fact_750_Khovanskii_Oidx__less__cardA,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A3: set_a,A: a] :
( ( khovanskii_a @ G @ Addition @ Zero @ A3 )
=> ( ( member_a @ A @ A3 )
=> ( ord_less_nat @ ( counta3566351752493190365t_on_a @ A3 @ A ) @ ( finite_card_a @ A3 ) ) ) ) ).
% Khovanskii.idx_less_cardA
thf(fact_751_enumerate__in__set,axiom,
! [S: set_list_nat,N3: nat] :
( ~ ( finite8100373058378681591st_nat @ S )
=> ( member_list_nat @ ( infini2033088105919815547st_nat @ S @ N3 ) @ S ) ) ).
% enumerate_in_set
thf(fact_752_enumerate__in__set,axiom,
! [S: set_nat,N3: nat] :
( ~ ( finite_finite_nat @ S )
=> ( member_nat @ ( infini8530281810654367211te_nat @ S @ N3 ) @ S ) ) ).
% enumerate_in_set
thf(fact_753_finite__enumerate__mono,axiom,
! [M3: nat,N3: nat,S: set_list_nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ( finite8100373058378681591st_nat @ S )
=> ( ( ord_less_nat @ N3 @ ( finite_card_list_nat @ S ) )
=> ( ord_less_list_nat @ ( infini2033088105919815547st_nat @ S @ M3 ) @ ( infini2033088105919815547st_nat @ S @ N3 ) ) ) ) ) ).
% finite_enumerate_mono
thf(fact_754_finite__enumerate__mono,axiom,
! [M3: nat,N3: nat,S: set_nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ N3 @ ( finite_card_nat @ S ) )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M3 ) @ ( infini8530281810654367211te_nat @ S @ N3 ) ) ) ) ) ).
% finite_enumerate_mono
thf(fact_755_finite__le__enumerate,axiom,
! [S: set_nat,N3: nat] :
( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ N3 @ ( finite_card_nat @ S ) )
=> ( ord_less_eq_nat @ N3 @ ( infini8530281810654367211te_nat @ S @ N3 ) ) ) ) ).
% finite_le_enumerate
thf(fact_756_finite__enumerate__step,axiom,
! [S: set_list_nat,N3: nat] :
( ( finite8100373058378681591st_nat @ S )
=> ( ( ord_less_nat @ ( suc @ N3 ) @ ( finite_card_list_nat @ S ) )
=> ( ord_less_list_nat @ ( infini2033088105919815547st_nat @ S @ N3 ) @ ( infini2033088105919815547st_nat @ S @ ( suc @ N3 ) ) ) ) ) ).
% finite_enumerate_step
thf(fact_757_finite__enumerate__step,axiom,
! [S: set_nat,N3: nat] :
( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ ( suc @ N3 ) @ ( finite_card_nat @ S ) )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ N3 ) @ ( infini8530281810654367211te_nat @ S @ ( suc @ N3 ) ) ) ) ) ).
% finite_enumerate_step
thf(fact_758_finite__enum__subset,axiom,
! [X8: set_list_nat,Y6: set_list_nat] :
( ! [I: nat] :
( ( ord_less_nat @ I @ ( finite_card_list_nat @ X8 ) )
=> ( ( infini2033088105919815547st_nat @ X8 @ I )
= ( infini2033088105919815547st_nat @ Y6 @ I ) ) )
=> ( ( finite8100373058378681591st_nat @ X8 )
=> ( ( finite8100373058378681591st_nat @ Y6 )
=> ( ( ord_less_eq_nat @ ( finite_card_list_nat @ X8 ) @ ( finite_card_list_nat @ Y6 ) )
=> ( ord_le6045566169113846134st_nat @ X8 @ Y6 ) ) ) ) ) ).
% finite_enum_subset
thf(fact_759_finite__enum__subset,axiom,
! [X8: set_nat,Y6: set_nat] :
( ! [I: nat] :
( ( ord_less_nat @ I @ ( finite_card_nat @ X8 ) )
=> ( ( infini8530281810654367211te_nat @ X8 @ I )
= ( infini8530281810654367211te_nat @ Y6 @ I ) ) )
=> ( ( finite_finite_nat @ X8 )
=> ( ( finite_finite_nat @ Y6 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ X8 ) @ ( finite_card_nat @ Y6 ) )
=> ( ord_less_eq_set_nat @ X8 @ Y6 ) ) ) ) ) ).
% finite_enum_subset
thf(fact_760_infinite__arbitrarily__large,axiom,
! [A3: set_nat,N3: nat] :
( ~ ( finite_finite_nat @ A3 )
=> ? [B8: set_nat] :
( ( finite_finite_nat @ B8 )
& ( ( finite_card_nat @ B8 )
= N3 )
& ( ord_less_eq_set_nat @ B8 @ A3 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_761_infinite__arbitrarily__large,axiom,
! [A3: set_list_nat,N3: nat] :
( ~ ( finite8100373058378681591st_nat @ A3 )
=> ? [B8: set_list_nat] :
( ( finite8100373058378681591st_nat @ B8 )
& ( ( finite_card_list_nat @ B8 )
= N3 )
& ( ord_le6045566169113846134st_nat @ B8 @ A3 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_762_infinite__arbitrarily__large,axiom,
! [A3: set_a,N3: nat] :
( ~ ( finite_finite_a @ A3 )
=> ? [B8: set_a] :
( ( finite_finite_a @ B8 )
& ( ( finite_card_a @ B8 )
= N3 )
& ( ord_less_eq_set_a @ B8 @ A3 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_763_card__subset__eq,axiom,
! [B4: set_nat,A3: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ( ( finite_card_nat @ A3 )
= ( finite_card_nat @ B4 ) )
=> ( A3 = B4 ) ) ) ) ).
% card_subset_eq
thf(fact_764_card__subset__eq,axiom,
! [B4: set_list_nat,A3: set_list_nat] :
( ( finite8100373058378681591st_nat @ B4 )
=> ( ( ord_le6045566169113846134st_nat @ A3 @ B4 )
=> ( ( ( finite_card_list_nat @ A3 )
= ( finite_card_list_nat @ B4 ) )
=> ( A3 = B4 ) ) ) ) ).
% card_subset_eq
thf(fact_765_card__subset__eq,axiom,
! [B4: set_a,A3: set_a] :
( ( finite_finite_a @ B4 )
=> ( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( ( finite_card_a @ A3 )
= ( finite_card_a @ B4 ) )
=> ( A3 = B4 ) ) ) ) ).
% card_subset_eq
thf(fact_766_card__le__if__inj__on__rel,axiom,
! [B4: set_list_nat,A3: set_list_nat,R: list_nat > list_nat > $o] :
( ( finite8100373058378681591st_nat @ B4 )
=> ( ! [A4: list_nat] :
( ( member_list_nat @ A4 @ A3 )
=> ? [B9: list_nat] :
( ( member_list_nat @ B9 @ B4 )
& ( R @ A4 @ B9 ) ) )
=> ( ! [A1: list_nat,A22: list_nat,B2: list_nat] :
( ( member_list_nat @ A1 @ A3 )
=> ( ( member_list_nat @ A22 @ A3 )
=> ( ( member_list_nat @ B2 @ B4 )
=> ( ( R @ A1 @ B2 )
=> ( ( R @ A22 @ B2 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B4 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_767_card__le__if__inj__on__rel,axiom,
! [B4: set_list_nat,A3: set_nat,R: nat > list_nat > $o] :
( ( finite8100373058378681591st_nat @ B4 )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A3 )
=> ? [B9: list_nat] :
( ( member_list_nat @ B9 @ B4 )
& ( R @ A4 @ B9 ) ) )
=> ( ! [A1: nat,A22: nat,B2: list_nat] :
( ( member_nat @ A1 @ A3 )
=> ( ( member_nat @ A22 @ A3 )
=> ( ( member_list_nat @ B2 @ B4 )
=> ( ( R @ A1 @ B2 )
=> ( ( R @ A22 @ B2 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_list_nat @ B4 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_768_card__le__if__inj__on__rel,axiom,
! [B4: set_list_nat,A3: set_a,R: a > list_nat > $o] :
( ( finite8100373058378681591st_nat @ B4 )
=> ( ! [A4: a] :
( ( member_a @ A4 @ A3 )
=> ? [B9: list_nat] :
( ( member_list_nat @ B9 @ B4 )
& ( R @ A4 @ B9 ) ) )
=> ( ! [A1: a,A22: a,B2: list_nat] :
( ( member_a @ A1 @ A3 )
=> ( ( member_a @ A22 @ A3 )
=> ( ( member_list_nat @ B2 @ B4 )
=> ( ( R @ A1 @ B2 )
=> ( ( R @ A22 @ B2 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A3 ) @ ( finite_card_list_nat @ B4 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_769_card__le__if__inj__on__rel,axiom,
! [B4: set_nat,A3: set_list_nat,R: list_nat > nat > $o] :
( ( finite_finite_nat @ B4 )
=> ( ! [A4: list_nat] :
( ( member_list_nat @ A4 @ A3 )
=> ? [B9: nat] :
( ( member_nat @ B9 @ B4 )
& ( R @ A4 @ B9 ) ) )
=> ( ! [A1: list_nat,A22: list_nat,B2: nat] :
( ( member_list_nat @ A1 @ A3 )
=> ( ( member_list_nat @ A22 @ A3 )
=> ( ( member_nat @ B2 @ B4 )
=> ( ( R @ A1 @ B2 )
=> ( ( R @ A22 @ B2 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_nat @ B4 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_770_card__le__if__inj__on__rel,axiom,
! [B4: set_nat,A3: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ B4 )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A3 )
=> ? [B9: nat] :
( ( member_nat @ B9 @ B4 )
& ( R @ A4 @ B9 ) ) )
=> ( ! [A1: nat,A22: nat,B2: nat] :
( ( member_nat @ A1 @ A3 )
=> ( ( member_nat @ A22 @ A3 )
=> ( ( member_nat @ B2 @ B4 )
=> ( ( R @ A1 @ B2 )
=> ( ( R @ A22 @ B2 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B4 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_771_card__le__if__inj__on__rel,axiom,
! [B4: set_nat,A3: set_a,R: a > nat > $o] :
( ( finite_finite_nat @ B4 )
=> ( ! [A4: a] :
( ( member_a @ A4 @ A3 )
=> ? [B9: nat] :
( ( member_nat @ B9 @ B4 )
& ( R @ A4 @ B9 ) ) )
=> ( ! [A1: a,A22: a,B2: nat] :
( ( member_a @ A1 @ A3 )
=> ( ( member_a @ A22 @ A3 )
=> ( ( member_nat @ B2 @ B4 )
=> ( ( R @ A1 @ B2 )
=> ( ( R @ A22 @ B2 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A3 ) @ ( finite_card_nat @ B4 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_772_card__le__if__inj__on__rel,axiom,
! [B4: set_a,A3: set_list_nat,R: list_nat > a > $o] :
( ( finite_finite_a @ B4 )
=> ( ! [A4: list_nat] :
( ( member_list_nat @ A4 @ A3 )
=> ? [B9: a] :
( ( member_a @ B9 @ B4 )
& ( R @ A4 @ B9 ) ) )
=> ( ! [A1: list_nat,A22: list_nat,B2: a] :
( ( member_list_nat @ A1 @ A3 )
=> ( ( member_list_nat @ A22 @ A3 )
=> ( ( member_a @ B2 @ B4 )
=> ( ( R @ A1 @ B2 )
=> ( ( R @ A22 @ B2 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_a @ B4 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_773_card__le__if__inj__on__rel,axiom,
! [B4: set_a,A3: set_nat,R: nat > a > $o] :
( ( finite_finite_a @ B4 )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A3 )
=> ? [B9: a] :
( ( member_a @ B9 @ B4 )
& ( R @ A4 @ B9 ) ) )
=> ( ! [A1: nat,A22: nat,B2: a] :
( ( member_nat @ A1 @ A3 )
=> ( ( member_nat @ A22 @ A3 )
=> ( ( member_a @ B2 @ B4 )
=> ( ( R @ A1 @ B2 )
=> ( ( R @ A22 @ B2 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_a @ B4 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_774_card__le__if__inj__on__rel,axiom,
! [B4: set_a,A3: set_a,R: a > a > $o] :
( ( finite_finite_a @ B4 )
=> ( ! [A4: a] :
( ( member_a @ A4 @ A3 )
=> ? [B9: a] :
( ( member_a @ B9 @ B4 )
& ( R @ A4 @ B9 ) ) )
=> ( ! [A1: a,A22: a,B2: a] :
( ( member_a @ A1 @ A3 )
=> ( ( member_a @ A22 @ A3 )
=> ( ( member_a @ B2 @ B4 )
=> ( ( R @ A1 @ B2 )
=> ( ( R @ A22 @ B2 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A3 ) @ ( finite_card_a @ B4 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_775_Sup__fin_OcoboundedI,axiom,
! [A3: set_list_nat,A: list_nat] :
( ( finite8100373058378681591st_nat @ A3 )
=> ( ( member_list_nat @ A @ A3 )
=> ( ord_less_eq_list_nat @ A @ ( lattic6411832703407573737st_nat @ A3 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_776_Sup__fin_OcoboundedI,axiom,
! [A3: set_nat,A: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ A @ A3 )
=> ( ord_less_eq_nat @ A @ ( lattic1093996805478795353in_nat @ A3 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_777_Sup__fin_OcoboundedI,axiom,
! [A3: set_set_list_nat,A: set_list_nat] :
( ( finite7047420756378620717st_nat @ A3 )
=> ( ( member_set_list_nat @ A @ A3 )
=> ( ord_le6045566169113846134st_nat @ A @ ( lattic2169124122975652127st_nat @ A3 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_778_Sup__fin_OcoboundedI,axiom,
! [A3: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( member_set_a @ A @ A3 )
=> ( ord_less_eq_set_a @ A @ ( lattic2918178356826803221_set_a @ A3 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_779_le__enumerate,axiom,
! [S: set_nat,N3: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ord_less_eq_nat @ N3 @ ( infini8530281810654367211te_nat @ S @ N3 ) ) ) ).
% le_enumerate
thf(fact_780_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_nat,C2: nat] :
( ! [G4: set_nat] :
( ( ord_less_eq_set_nat @ G4 @ F2 )
=> ( ( finite_finite_nat @ G4 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G4 ) @ C2 ) ) )
=> ( ( finite_finite_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_781_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_list_nat,C2: nat] :
( ! [G4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ G4 @ F2 )
=> ( ( finite8100373058378681591st_nat @ G4 )
=> ( ord_less_eq_nat @ ( finite_card_list_nat @ G4 ) @ C2 ) ) )
=> ( ( finite8100373058378681591st_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_list_nat @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_782_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_a,C2: nat] :
( ! [G4: set_a] :
( ( ord_less_eq_set_a @ G4 @ F2 )
=> ( ( finite_finite_a @ G4 )
=> ( ord_less_eq_nat @ ( finite_card_a @ G4 ) @ C2 ) ) )
=> ( ( finite_finite_a @ F2 )
& ( ord_less_eq_nat @ ( finite_card_a @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_783_obtain__subset__with__card__n,axiom,
! [N3: nat,S: set_nat] :
( ( ord_less_eq_nat @ N3 @ ( finite_card_nat @ S ) )
=> ~ ! [T4: set_nat] :
( ( ord_less_eq_set_nat @ T4 @ S )
=> ( ( ( finite_card_nat @ T4 )
= N3 )
=> ~ ( finite_finite_nat @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_784_obtain__subset__with__card__n,axiom,
! [N3: nat,S: set_list_nat] :
( ( ord_less_eq_nat @ N3 @ ( finite_card_list_nat @ S ) )
=> ~ ! [T4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ T4 @ S )
=> ( ( ( finite_card_list_nat @ T4 )
= N3 )
=> ~ ( finite8100373058378681591st_nat @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_785_obtain__subset__with__card__n,axiom,
! [N3: nat,S: set_a] :
( ( ord_less_eq_nat @ N3 @ ( finite_card_a @ S ) )
=> ~ ! [T4: set_a] :
( ( ord_less_eq_set_a @ T4 @ S )
=> ( ( ( finite_card_a @ T4 )
= N3 )
=> ~ ( finite_finite_a @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_786_exists__subset__between,axiom,
! [A3: set_nat,N3: nat,C2: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ N3 )
=> ( ( ord_less_eq_nat @ N3 @ ( finite_card_nat @ C2 ) )
=> ( ( ord_less_eq_set_nat @ A3 @ C2 )
=> ( ( finite_finite_nat @ C2 )
=> ? [B8: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B8 )
& ( ord_less_eq_set_nat @ B8 @ C2 )
& ( ( finite_card_nat @ B8 )
= N3 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_787_exists__subset__between,axiom,
! [A3: set_list_nat,N3: nat,C2: set_list_nat] :
( ( ord_less_eq_nat @ ( finite_card_list_nat @ A3 ) @ N3 )
=> ( ( ord_less_eq_nat @ N3 @ ( finite_card_list_nat @ C2 ) )
=> ( ( ord_le6045566169113846134st_nat @ A3 @ C2 )
=> ( ( finite8100373058378681591st_nat @ C2 )
=> ? [B8: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ B8 )
& ( ord_le6045566169113846134st_nat @ B8 @ C2 )
& ( ( finite_card_list_nat @ B8 )
= N3 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_788_exists__subset__between,axiom,
! [A3: set_a,N3: nat,C2: set_a] :
( ( ord_less_eq_nat @ ( finite_card_a @ A3 ) @ N3 )
=> ( ( ord_less_eq_nat @ N3 @ ( finite_card_a @ C2 ) )
=> ( ( ord_less_eq_set_a @ A3 @ C2 )
=> ( ( finite_finite_a @ C2 )
=> ? [B8: set_a] :
( ( ord_less_eq_set_a @ A3 @ B8 )
& ( ord_less_eq_set_a @ B8 @ C2 )
& ( ( finite_card_a @ B8 )
= N3 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_789_card__seteq,axiom,
! [B4: set_nat,A3: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B4 ) @ ( finite_card_nat @ A3 ) )
=> ( A3 = B4 ) ) ) ) ).
% card_seteq
thf(fact_790_card__seteq,axiom,
! [B4: set_list_nat,A3: set_list_nat] :
( ( finite8100373058378681591st_nat @ B4 )
=> ( ( ord_le6045566169113846134st_nat @ A3 @ B4 )
=> ( ( ord_less_eq_nat @ ( finite_card_list_nat @ B4 ) @ ( finite_card_list_nat @ A3 ) )
=> ( A3 = B4 ) ) ) ) ).
% card_seteq
thf(fact_791_card__seteq,axiom,
! [B4: set_a,A3: set_a] :
( ( finite_finite_a @ B4 )
=> ( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ B4 ) @ ( finite_card_a @ A3 ) )
=> ( A3 = B4 ) ) ) ) ).
% card_seteq
thf(fact_792_card__mono,axiom,
! [B4: set_nat,A3: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B4 ) ) ) ) ).
% card_mono
thf(fact_793_card__mono,axiom,
! [B4: set_list_nat,A3: set_list_nat] :
( ( finite8100373058378681591st_nat @ B4 )
=> ( ( ord_le6045566169113846134st_nat @ A3 @ B4 )
=> ( ord_less_eq_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B4 ) ) ) ) ).
% card_mono
thf(fact_794_card__mono,axiom,
! [B4: set_a,A3: set_a] :
( ( finite_finite_a @ B4 )
=> ( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ord_less_eq_nat @ ( finite_card_a @ A3 ) @ ( finite_card_a @ B4 ) ) ) ) ).
% card_mono
thf(fact_795_card__length,axiom,
! [Xs2: list_a] : ( ord_less_eq_nat @ ( finite_card_a @ ( set_a2 @ Xs2 ) ) @ ( size_size_list_a @ Xs2 ) ) ).
% card_length
thf(fact_796_card__length,axiom,
! [Xs2: list_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( set_nat2 @ Xs2 ) ) @ ( size_size_list_nat @ Xs2 ) ) ).
% card_length
thf(fact_797_psubset__card__mono,axiom,
! [B4: set_list_nat,A3: set_list_nat] :
( ( finite8100373058378681591st_nat @ B4 )
=> ( ( ord_le1190675801316882794st_nat @ A3 @ B4 )
=> ( ord_less_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B4 ) ) ) ) ).
% psubset_card_mono
thf(fact_798_psubset__card__mono,axiom,
! [B4: set_nat,A3: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_set_nat @ A3 @ B4 )
=> ( ord_less_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B4 ) ) ) ) ).
% psubset_card_mono
thf(fact_799_psubset__card__mono,axiom,
! [B4: set_a,A3: set_a] :
( ( finite_finite_a @ B4 )
=> ( ( ord_less_set_a @ A3 @ B4 )
=> ( ord_less_nat @ ( finite_card_a @ A3 ) @ ( finite_card_a @ B4 ) ) ) ) ).
% psubset_card_mono
thf(fact_800_Khovanskii_Onth__le__list__incr,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A3: set_nat,I3: nat,X: list_nat,A: nat] :
( ( khovanskii_nat @ G @ Addition @ Zero @ A3 )
=> ( ( ord_less_nat @ I3 @ ( finite_card_nat @ A3 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ X @ I3 ) @ ( nth_nat @ ( list_incr @ ( counta4844910239362777137on_nat @ A3 @ A ) @ X ) @ I3 ) ) ) ) ).
% Khovanskii.nth_le_list_incr
thf(fact_801_Khovanskii_Onth__le__list__incr,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A3: set_a,I3: nat,X: list_nat,A: a] :
( ( khovanskii_a @ G @ Addition @ Zero @ A3 )
=> ( ( ord_less_nat @ I3 @ ( finite_card_a @ A3 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ X @ I3 ) @ ( nth_nat @ ( list_incr @ ( counta3566351752493190365t_on_a @ A3 @ A ) @ X ) @ I3 ) ) ) ) ).
% Khovanskii.nth_le_list_incr
thf(fact_802_card__distinct,axiom,
! [Xs2: list_a] :
( ( ( finite_card_a @ ( set_a2 @ Xs2 ) )
= ( size_size_list_a @ Xs2 ) )
=> ( distinct_a @ Xs2 ) ) ).
% card_distinct
thf(fact_803_card__distinct,axiom,
! [Xs2: list_nat] :
( ( ( finite_card_nat @ ( set_nat2 @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) )
=> ( distinct_nat @ Xs2 ) ) ).
% card_distinct
thf(fact_804_distinct__card,axiom,
! [Xs2: list_a] :
( ( distinct_a @ Xs2 )
=> ( ( finite_card_a @ ( set_a2 @ Xs2 ) )
= ( size_size_list_a @ Xs2 ) ) ) ).
% distinct_card
thf(fact_805_distinct__card,axiom,
! [Xs2: list_nat] :
( ( distinct_nat @ Xs2 )
=> ( ( finite_card_nat @ ( set_nat2 @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ) ).
% distinct_card
thf(fact_806_Sup__fin_OboundedE,axiom,
! [A3: set_list_nat,X: list_nat] :
( ( finite8100373058378681591st_nat @ A3 )
=> ( ( A3 != bot_bot_set_list_nat )
=> ( ( ord_less_eq_list_nat @ ( lattic6411832703407573737st_nat @ A3 ) @ X )
=> ! [A8: list_nat] :
( ( member_list_nat @ A8 @ A3 )
=> ( ord_less_eq_list_nat @ A8 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_807_Sup__fin_OboundedE,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A3 ) @ X )
=> ! [A8: nat] :
( ( member_nat @ A8 @ A3 )
=> ( ord_less_eq_nat @ A8 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_808_Sup__fin_OboundedE,axiom,
! [A3: set_set_list_nat,X: set_list_nat] :
( ( finite7047420756378620717st_nat @ A3 )
=> ( ( A3 != bot_bo3886227569956363488st_nat )
=> ( ( ord_le6045566169113846134st_nat @ ( lattic2169124122975652127st_nat @ A3 ) @ X )
=> ! [A8: set_list_nat] :
( ( member_set_list_nat @ A8 @ A3 )
=> ( ord_le6045566169113846134st_nat @ A8 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_809_Sup__fin_OboundedE,axiom,
! [A3: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( A3 != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A3 ) @ X )
=> ! [A8: set_a] :
( ( member_set_a @ A8 @ A3 )
=> ( ord_less_eq_set_a @ A8 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_810_Sup__fin_OboundedI,axiom,
! [A3: set_list_nat,X: list_nat] :
( ( finite8100373058378681591st_nat @ A3 )
=> ( ( A3 != bot_bot_set_list_nat )
=> ( ! [A4: list_nat] :
( ( member_list_nat @ A4 @ A3 )
=> ( ord_less_eq_list_nat @ A4 @ X ) )
=> ( ord_less_eq_list_nat @ ( lattic6411832703407573737st_nat @ A3 ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_811_Sup__fin_OboundedI,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A3 )
=> ( ord_less_eq_nat @ A4 @ X ) )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A3 ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_812_Sup__fin_OboundedI,axiom,
! [A3: set_set_list_nat,X: set_list_nat] :
( ( finite7047420756378620717st_nat @ A3 )
=> ( ( A3 != bot_bo3886227569956363488st_nat )
=> ( ! [A4: set_list_nat] :
( ( member_set_list_nat @ A4 @ A3 )
=> ( ord_le6045566169113846134st_nat @ A4 @ X ) )
=> ( ord_le6045566169113846134st_nat @ ( lattic2169124122975652127st_nat @ A3 ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_813_Sup__fin_OboundedI,axiom,
! [A3: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( A3 != bot_bot_set_set_a )
=> ( ! [A4: set_a] :
( ( member_set_a @ A4 @ A3 )
=> ( ord_less_eq_set_a @ A4 @ X ) )
=> ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A3 ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_814_Sup__fin_Obounded__iff,axiom,
! [A3: set_list_nat,X: list_nat] :
( ( finite8100373058378681591st_nat @ A3 )
=> ( ( A3 != bot_bot_set_list_nat )
=> ( ( ord_less_eq_list_nat @ ( lattic6411832703407573737st_nat @ A3 ) @ X )
= ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ A3 )
=> ( ord_less_eq_list_nat @ X2 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_815_Sup__fin_Obounded__iff,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A3 ) @ X )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( ord_less_eq_nat @ X2 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_816_Sup__fin_Obounded__iff,axiom,
! [A3: set_set_list_nat,X: set_list_nat] :
( ( finite7047420756378620717st_nat @ A3 )
=> ( ( A3 != bot_bo3886227569956363488st_nat )
=> ( ( ord_le6045566169113846134st_nat @ ( lattic2169124122975652127st_nat @ A3 ) @ X )
= ( ! [X2: set_list_nat] :
( ( member_set_list_nat @ X2 @ A3 )
=> ( ord_le6045566169113846134st_nat @ X2 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_817_Sup__fin_Obounded__iff,axiom,
! [A3: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( A3 != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ ( lattic2918178356826803221_set_a @ A3 ) @ X )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A3 )
=> ( ord_less_eq_set_a @ X2 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_818_enumerate__step,axiom,
! [S: set_list_nat,N3: nat] :
( ~ ( finite8100373058378681591st_nat @ S )
=> ( ord_less_list_nat @ ( infini2033088105919815547st_nat @ S @ N3 ) @ ( infini2033088105919815547st_nat @ S @ ( suc @ N3 ) ) ) ) ).
% enumerate_step
thf(fact_819_enumerate__step,axiom,
! [S: set_nat,N3: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ N3 ) @ ( infini8530281810654367211te_nat @ S @ ( suc @ N3 ) ) ) ) ).
% enumerate_step
thf(fact_820_enumerate__mono,axiom,
! [M3: nat,N3: nat,S: set_list_nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ~ ( finite8100373058378681591st_nat @ S )
=> ( ord_less_list_nat @ ( infini2033088105919815547st_nat @ S @ M3 ) @ ( infini2033088105919815547st_nat @ S @ N3 ) ) ) ) ).
% enumerate_mono
thf(fact_821_enumerate__mono,axiom,
! [M3: nat,N3: nat,S: set_nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ~ ( finite_finite_nat @ S )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M3 ) @ ( infini8530281810654367211te_nat @ S @ N3 ) ) ) ) ).
% enumerate_mono
thf(fact_822_card__psubset,axiom,
! [B4: set_nat,A3: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B4 ) )
=> ( ord_less_set_nat @ A3 @ B4 ) ) ) ) ).
% card_psubset
thf(fact_823_card__psubset,axiom,
! [B4: set_list_nat,A3: set_list_nat] :
( ( finite8100373058378681591st_nat @ B4 )
=> ( ( ord_le6045566169113846134st_nat @ A3 @ B4 )
=> ( ( ord_less_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B4 ) )
=> ( ord_le1190675801316882794st_nat @ A3 @ B4 ) ) ) ) ).
% card_psubset
thf(fact_824_card__psubset,axiom,
! [B4: set_a,A3: set_a] :
( ( finite_finite_a @ B4 )
=> ( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( ord_less_nat @ ( finite_card_a @ A3 ) @ ( finite_card_a @ B4 ) )
=> ( ord_less_set_a @ A3 @ B4 ) ) ) ) ).
% card_psubset
thf(fact_825_Khovanskii_Ouseless__iff,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A3: set_nat,X: list_nat] :
( ( khovanskii_nat @ G @ Addition @ Zero @ A3 )
=> ( ( ( size_size_list_nat @ X )
= ( finite_card_nat @ A3 ) )
=> ( ( useless_nat @ G @ Addition @ Zero @ A3 @ X )
= ( ? [X2: list_nat] :
( ( member_list_nat @ X2 @ ( minimal_elements @ ( collect_list_nat @ ( useless_nat @ G @ Addition @ Zero @ A3 ) ) ) )
& ( pointwise_le @ X2 @ X ) ) ) ) ) ) ).
% Khovanskii.useless_iff
thf(fact_826_Khovanskii_Ouseless__iff,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A3: set_a,X: list_nat] :
( ( khovanskii_a @ G @ Addition @ Zero @ A3 )
=> ( ( ( size_size_list_nat @ X )
= ( finite_card_a @ A3 ) )
=> ( ( useless_a @ G @ Addition @ Zero @ A3 @ X )
= ( ? [X2: list_nat] :
( ( member_list_nat @ X2 @ ( minimal_elements @ ( collect_list_nat @ ( useless_a @ G @ Addition @ Zero @ A3 ) ) ) )
& ( pointwise_le @ X2 @ X ) ) ) ) ) ) ).
% Khovanskii.useless_iff
thf(fact_827_ex__card,axiom,
! [N3: nat,A3: set_nat] :
( ( ord_less_eq_nat @ N3 @ ( finite_card_nat @ A3 ) )
=> ? [S3: set_nat] :
( ( ord_less_eq_set_nat @ S3 @ A3 )
& ( ( finite_card_nat @ S3 )
= N3 ) ) ) ).
% ex_card
thf(fact_828_ex__card,axiom,
! [N3: nat,A3: set_list_nat] :
( ( ord_less_eq_nat @ N3 @ ( finite_card_list_nat @ A3 ) )
=> ? [S3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ S3 @ A3 )
& ( ( finite_card_list_nat @ S3 )
= N3 ) ) ) ).
% ex_card
thf(fact_829_ex__card,axiom,
! [N3: nat,A3: set_a] :
( ( ord_less_eq_nat @ N3 @ ( finite_card_a @ A3 ) )
=> ? [S3: set_a] :
( ( ord_less_eq_set_a @ S3 @ A3 )
& ( ( finite_card_a @ S3 )
= N3 ) ) ) ).
% ex_card
thf(fact_830_finsets__def,axiom,
( finsets_nat
= ( ^ [A5: set_nat,N2: nat] :
( collect_set_nat
@ ^ [N7: set_nat] :
( ( ord_less_eq_set_nat @ N7 @ A5 )
& ( ( finite_card_nat @ N7 )
= N2 ) ) ) ) ) ).
% finsets_def
thf(fact_831_finsets__def,axiom,
( finsets_list_nat
= ( ^ [A5: set_list_nat,N2: nat] :
( collect_set_list_nat
@ ^ [N7: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ N7 @ A5 )
& ( ( finite_card_list_nat @ N7 )
= N2 ) ) ) ) ) ).
% finsets_def
thf(fact_832_finsets__def,axiom,
( finsets_a
= ( ^ [A5: set_a,N2: nat] :
( collect_set_a
@ ^ [N7: set_a] :
( ( ord_less_eq_set_a @ N7 @ A5 )
& ( ( finite_card_a @ N7 )
= N2 ) ) ) ) ) ).
% finsets_def
thf(fact_833_Khovanskii_Ouseless__leq__useless,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A3: set_nat,X: list_nat,Y4: list_nat] :
( ( khovanskii_nat @ G @ Addition @ Zero @ A3 )
=> ( ( useless_nat @ G @ Addition @ Zero @ A3 @ X )
=> ( ( pointwise_le @ X @ Y4 )
=> ( ( ( size_size_list_nat @ X )
= ( finite_card_nat @ A3 ) )
=> ( useless_nat @ G @ Addition @ Zero @ A3 @ Y4 ) ) ) ) ) ).
% Khovanskii.useless_leq_useless
thf(fact_834_Khovanskii_Ouseless__leq__useless,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A3: set_a,X: list_nat,Y4: list_nat] :
( ( khovanskii_a @ G @ Addition @ Zero @ A3 )
=> ( ( useless_a @ G @ Addition @ Zero @ A3 @ X )
=> ( ( pointwise_le @ X @ Y4 )
=> ( ( ( size_size_list_nat @ X )
= ( finite_card_a @ A3 ) )
=> ( useless_a @ G @ Addition @ Zero @ A3 @ Y4 ) ) ) ) ) ).
% Khovanskii.useless_leq_useless
thf(fact_835_Khovanskii_Oalpha__list__incr,axiom,
! [G: set_list_nat,Addition: list_nat > list_nat > list_nat,Zero: list_nat,A3: set_list_nat,A: list_nat,X: list_nat,N3: nat] :
( ( khovanskii_list_nat @ G @ Addition @ Zero @ A3 )
=> ( ( member_list_nat @ A @ A3 )
=> ( ( member_list_nat @ X @ ( length_sum_set @ ( finite_card_list_nat @ A3 ) @ N3 ) )
=> ( ( alpha_list_nat @ G @ Addition @ Zero @ A3 @ ( list_incr @ ( counta9103016383634126529st_nat @ A3 @ A ) @ X ) )
= ( Addition @ A @ ( alpha_list_nat @ G @ Addition @ Zero @ A3 @ X ) ) ) ) ) ) ).
% Khovanskii.alpha_list_incr
thf(fact_836_Khovanskii_Oalpha__list__incr,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A3: set_nat,A: nat,X: list_nat,N3: nat] :
( ( khovanskii_nat @ G @ Addition @ Zero @ A3 )
=> ( ( member_nat @ A @ A3 )
=> ( ( member_list_nat @ X @ ( length_sum_set @ ( finite_card_nat @ A3 ) @ N3 ) )
=> ( ( alpha_nat @ G @ Addition @ Zero @ A3 @ ( list_incr @ ( counta4844910239362777137on_nat @ A3 @ A ) @ X ) )
= ( Addition @ A @ ( alpha_nat @ G @ Addition @ Zero @ A3 @ X ) ) ) ) ) ) ).
% Khovanskii.alpha_list_incr
thf(fact_837_Khovanskii_Oalpha__list__incr,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A3: set_a,A: a,X: list_nat,N3: nat] :
( ( khovanskii_a @ G @ Addition @ Zero @ A3 )
=> ( ( member_a @ A @ A3 )
=> ( ( member_list_nat @ X @ ( length_sum_set @ ( finite_card_a @ A3 ) @ N3 ) )
=> ( ( alpha_a @ G @ Addition @ Zero @ A3 @ ( list_incr @ ( counta3566351752493190365t_on_a @ A3 @ A ) @ X ) )
= ( Addition @ A @ ( alpha_a @ G @ Addition @ Zero @ A3 @ X ) ) ) ) ) ) ).
% Khovanskii.alpha_list_incr
thf(fact_838_finite__enumerate__Suc_H_H,axiom,
! [S: set_list_nat,N3: nat] :
( ( finite8100373058378681591st_nat @ S )
=> ( ( ord_less_nat @ ( suc @ N3 ) @ ( finite_card_list_nat @ S ) )
=> ( ( infini2033088105919815547st_nat @ S @ ( suc @ N3 ) )
= ( ord_Least_list_nat
@ ^ [S4: list_nat] :
( ( member_list_nat @ S4 @ S )
& ( ord_less_list_nat @ ( infini2033088105919815547st_nat @ S @ N3 ) @ S4 ) ) ) ) ) ) ).
% finite_enumerate_Suc''
thf(fact_839_finite__enumerate__Suc_H_H,axiom,
! [S: set_nat,N3: nat] :
( ( finite_finite_nat @ S )
=> ( ( ord_less_nat @ ( suc @ N3 ) @ ( finite_card_nat @ S ) )
=> ( ( infini8530281810654367211te_nat @ S @ ( suc @ N3 ) )
= ( ord_Least_nat
@ ^ [S4: nat] :
( ( member_nat @ S4 @ S )
& ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ N3 ) @ S4 ) ) ) ) ) ) ).
% finite_enumerate_Suc''
thf(fact_840_nth__le__list__incr,axiom,
! [I3: nat,X: list_nat,A: a] :
( ( ord_less_nat @ I3 @ ( finite_card_a @ a2 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ X @ I3 ) @ ( nth_nat @ ( list_incr @ ( counta3566351752493190365t_on_a @ a2 @ A ) @ X ) @ I3 ) ) ) ).
% nth_le_list_incr
thf(fact_841_set__aA,axiom,
( ( set_a2 @ ( aA_a @ a2 ) )
= a2 ) ).
% set_aA
thf(fact_842_nonempty,axiom,
a2 != bot_bot_set_a ).
% nonempty
thf(fact_843_finA,axiom,
finite_finite_a @ a2 ).
% finA
thf(fact_844_idx__less__cardA,axiom,
! [A: a] :
( ( member_a @ A @ a2 )
=> ( ord_less_nat @ ( counta3566351752493190365t_on_a @ a2 @ A ) @ ( finite_card_a @ a2 ) ) ) ).
% idx_less_cardA
thf(fact_845_aA__idx__eq,axiom,
! [A: a] :
( ( member_a @ A @ a2 )
=> ( ( nth_a @ ( aA_a @ a2 ) @ ( counta3566351752493190365t_on_a @ a2 @ A ) )
= A ) ) ).
% aA_idx_eq
thf(fact_846_LeastI,axiom,
! [P2: nat > $o,K2: nat] :
( ( P2 @ K2 )
=> ( P2 @ ( ord_Least_nat @ P2 ) ) ) ).
% LeastI
thf(fact_847_LeastI2__ex,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ? [X_12: nat] : ( P2 @ X_12 )
=> ( ! [X3: nat] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) )
=> ( Q2 @ ( ord_Least_nat @ P2 ) ) ) ) ).
% LeastI2_ex
thf(fact_848_LeastI__ex,axiom,
! [P2: nat > $o] :
( ? [X_12: nat] : ( P2 @ X_12 )
=> ( P2 @ ( ord_Least_nat @ P2 ) ) ) ).
% LeastI_ex
thf(fact_849_LeastI2,axiom,
! [P2: nat > $o,A: nat,Q2: nat > $o] :
( ( P2 @ A )
=> ( ! [X3: nat] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) )
=> ( Q2 @ ( ord_Least_nat @ P2 ) ) ) ) ).
% LeastI2
thf(fact_850_Least1I,axiom,
! [P2: nat > $o] :
( ? [X5: nat] :
( ( P2 @ X5 )
& ! [Y2: nat] :
( ( P2 @ Y2 )
=> ( ord_less_eq_nat @ X5 @ Y2 ) )
& ! [Y2: nat] :
( ( ( P2 @ Y2 )
& ! [Ya: nat] :
( ( P2 @ Ya )
=> ( ord_less_eq_nat @ Y2 @ Ya ) ) )
=> ( Y2 = X5 ) ) )
=> ( P2 @ ( ord_Least_nat @ P2 ) ) ) ).
% Least1I
thf(fact_851_Least1I,axiom,
! [P2: set_list_nat > $o] :
( ? [X5: set_list_nat] :
( ( P2 @ X5 )
& ! [Y2: set_list_nat] :
( ( P2 @ Y2 )
=> ( ord_le6045566169113846134st_nat @ X5 @ Y2 ) )
& ! [Y2: set_list_nat] :
( ( ( P2 @ Y2 )
& ! [Ya: set_list_nat] :
( ( P2 @ Ya )
=> ( ord_le6045566169113846134st_nat @ Y2 @ Ya ) ) )
=> ( Y2 = X5 ) ) )
=> ( P2 @ ( ord_Le6937613917142691334st_nat @ P2 ) ) ) ).
% Least1I
thf(fact_852_Least1I,axiom,
! [P2: set_a > $o] :
( ? [X5: set_a] :
( ( P2 @ X5 )
& ! [Y2: set_a] :
( ( P2 @ Y2 )
=> ( ord_less_eq_set_a @ X5 @ Y2 ) )
& ! [Y2: set_a] :
( ( ( P2 @ Y2 )
& ! [Ya: set_a] :
( ( P2 @ Ya )
=> ( ord_less_eq_set_a @ Y2 @ Ya ) ) )
=> ( Y2 = X5 ) ) )
=> ( P2 @ ( ord_Least_set_a @ P2 ) ) ) ).
% Least1I
thf(fact_853_Least1__le,axiom,
! [P2: nat > $o,Z: nat] :
( ? [X5: nat] :
( ( P2 @ X5 )
& ! [Y2: nat] :
( ( P2 @ Y2 )
=> ( ord_less_eq_nat @ X5 @ Y2 ) )
& ! [Y2: nat] :
( ( ( P2 @ Y2 )
& ! [Ya: nat] :
( ( P2 @ Ya )
=> ( ord_less_eq_nat @ Y2 @ Ya ) ) )
=> ( Y2 = X5 ) ) )
=> ( ( P2 @ Z )
=> ( ord_less_eq_nat @ ( ord_Least_nat @ P2 ) @ Z ) ) ) ).
% Least1_le
thf(fact_854_Least1__le,axiom,
! [P2: set_list_nat > $o,Z: set_list_nat] :
( ? [X5: set_list_nat] :
( ( P2 @ X5 )
& ! [Y2: set_list_nat] :
( ( P2 @ Y2 )
=> ( ord_le6045566169113846134st_nat @ X5 @ Y2 ) )
& ! [Y2: set_list_nat] :
( ( ( P2 @ Y2 )
& ! [Ya: set_list_nat] :
( ( P2 @ Ya )
=> ( ord_le6045566169113846134st_nat @ Y2 @ Ya ) ) )
=> ( Y2 = X5 ) ) )
=> ( ( P2 @ Z )
=> ( ord_le6045566169113846134st_nat @ ( ord_Le6937613917142691334st_nat @ P2 ) @ Z ) ) ) ).
% Least1_le
thf(fact_855_Least1__le,axiom,
! [P2: set_a > $o,Z: set_a] :
( ? [X5: set_a] :
( ( P2 @ X5 )
& ! [Y2: set_a] :
( ( P2 @ Y2 )
=> ( ord_less_eq_set_a @ X5 @ Y2 ) )
& ! [Y2: set_a] :
( ( ( P2 @ Y2 )
& ! [Ya: set_a] :
( ( P2 @ Ya )
=> ( ord_less_eq_set_a @ Y2 @ Ya ) ) )
=> ( Y2 = X5 ) ) )
=> ( ( P2 @ Z )
=> ( ord_less_eq_set_a @ ( ord_Least_set_a @ P2 ) @ Z ) ) ) ).
% Least1_le
thf(fact_856_LeastI2__order,axiom,
! [P2: nat > $o,X: nat,Q2: nat > $o] :
( ( P2 @ X )
=> ( ! [Y2: nat] :
( ( P2 @ Y2 )
=> ( ord_less_eq_nat @ X @ Y2 ) )
=> ( ! [X3: nat] :
( ( P2 @ X3 )
=> ( ! [Y: nat] :
( ( P2 @ Y )
=> ( ord_less_eq_nat @ X3 @ Y ) )
=> ( Q2 @ X3 ) ) )
=> ( Q2 @ ( ord_Least_nat @ P2 ) ) ) ) ) ).
% LeastI2_order
thf(fact_857_LeastI2__order,axiom,
! [P2: set_list_nat > $o,X: set_list_nat,Q2: set_list_nat > $o] :
( ( P2 @ X )
=> ( ! [Y2: set_list_nat] :
( ( P2 @ Y2 )
=> ( ord_le6045566169113846134st_nat @ X @ Y2 ) )
=> ( ! [X3: set_list_nat] :
( ( P2 @ X3 )
=> ( ! [Y: set_list_nat] :
( ( P2 @ Y )
=> ( ord_le6045566169113846134st_nat @ X3 @ Y ) )
=> ( Q2 @ X3 ) ) )
=> ( Q2 @ ( ord_Le6937613917142691334st_nat @ P2 ) ) ) ) ) ).
% LeastI2_order
thf(fact_858_LeastI2__order,axiom,
! [P2: set_a > $o,X: set_a,Q2: set_a > $o] :
( ( P2 @ X )
=> ( ! [Y2: set_a] :
( ( P2 @ Y2 )
=> ( ord_less_eq_set_a @ X @ Y2 ) )
=> ( ! [X3: set_a] :
( ( P2 @ X3 )
=> ( ! [Y: set_a] :
( ( P2 @ Y )
=> ( ord_less_eq_set_a @ X3 @ Y ) )
=> ( Q2 @ X3 ) ) )
=> ( Q2 @ ( ord_Least_set_a @ P2 ) ) ) ) ) ).
% LeastI2_order
thf(fact_859_Least__equality,axiom,
! [P2: nat > $o,X: nat] :
( ( P2 @ X )
=> ( ! [Y2: nat] :
( ( P2 @ Y2 )
=> ( ord_less_eq_nat @ X @ Y2 ) )
=> ( ( ord_Least_nat @ P2 )
= X ) ) ) ).
% Least_equality
thf(fact_860_Least__equality,axiom,
! [P2: set_list_nat > $o,X: set_list_nat] :
( ( P2 @ X )
=> ( ! [Y2: set_list_nat] :
( ( P2 @ Y2 )
=> ( ord_le6045566169113846134st_nat @ X @ Y2 ) )
=> ( ( ord_Le6937613917142691334st_nat @ P2 )
= X ) ) ) ).
% Least_equality
thf(fact_861_Least__equality,axiom,
! [P2: set_a > $o,X: set_a] :
( ( P2 @ X )
=> ( ! [Y2: set_a] :
( ( P2 @ Y2 )
=> ( ord_less_eq_set_a @ X @ Y2 ) )
=> ( ( ord_Least_set_a @ P2 )
= X ) ) ) ).
% Least_equality
thf(fact_862_LeastI2__wellorder,axiom,
! [P2: nat > $o,A: nat,Q2: nat > $o] :
( ( P2 @ A )
=> ( ! [A4: nat] :
( ( P2 @ A4 )
=> ( ! [B9: nat] :
( ( P2 @ B9 )
=> ( ord_less_eq_nat @ A4 @ B9 ) )
=> ( Q2 @ A4 ) ) )
=> ( Q2 @ ( ord_Least_nat @ P2 ) ) ) ) ).
% LeastI2_wellorder
thf(fact_863_LeastI2__wellorder__ex,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ? [X_12: nat] : ( P2 @ X_12 )
=> ( ! [A4: nat] :
( ( P2 @ A4 )
=> ( ! [B9: nat] :
( ( P2 @ B9 )
=> ( ord_less_eq_nat @ A4 @ B9 ) )
=> ( Q2 @ A4 ) ) )
=> ( Q2 @ ( ord_Least_nat @ P2 ) ) ) ) ).
% LeastI2_wellorder_ex
thf(fact_864_Khovanskii_Oalpha__in__G,axiom,
! [G: set_list_nat,Addition: list_nat > list_nat > list_nat,Zero: list_nat,A3: set_list_nat,X: list_nat] :
( ( khovanskii_list_nat @ G @ Addition @ Zero @ A3 )
=> ( member_list_nat @ ( alpha_list_nat @ G @ Addition @ Zero @ A3 @ X ) @ G ) ) ).
% Khovanskii.alpha_in_G
thf(fact_865_Khovanskii_Oalpha__in__G,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A3: set_nat,X: list_nat] :
( ( khovanskii_nat @ G @ Addition @ Zero @ A3 )
=> ( member_nat @ ( alpha_nat @ G @ Addition @ Zero @ A3 @ X ) @ G ) ) ).
% Khovanskii.alpha_in_G
thf(fact_866_Khovanskii_Oalpha__in__G,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A3: set_a,X: list_nat] :
( ( khovanskii_a @ G @ Addition @ Zero @ A3 )
=> ( member_a @ ( alpha_a @ G @ Addition @ Zero @ A3 @ X ) @ G ) ) ).
% Khovanskii.alpha_in_G
thf(fact_867_Least__le,axiom,
! [P2: nat > $o,K2: nat] :
( ( P2 @ K2 )
=> ( ord_less_eq_nat @ ( ord_Least_nat @ P2 ) @ K2 ) ) ).
% Least_le
thf(fact_868_not__less__Least,axiom,
! [K2: nat,P2: nat > $o] :
( ( ord_less_nat @ K2 @ ( ord_Least_nat @ P2 ) )
=> ~ ( P2 @ K2 ) ) ).
% not_less_Least
thf(fact_869_enumerate__Suc_H_H,axiom,
! [S: set_list_nat,N3: nat] :
( ~ ( finite8100373058378681591st_nat @ S )
=> ( ( infini2033088105919815547st_nat @ S @ ( suc @ N3 ) )
= ( ord_Least_list_nat
@ ^ [S4: list_nat] :
( ( member_list_nat @ S4 @ S )
& ( ord_less_list_nat @ ( infini2033088105919815547st_nat @ S @ N3 ) @ S4 ) ) ) ) ) ).
% enumerate_Suc''
thf(fact_870_enumerate__Suc_H_H,axiom,
! [S: set_nat,N3: nat] :
( ~ ( finite_finite_nat @ S )
=> ( ( infini8530281810654367211te_nat @ S @ ( suc @ N3 ) )
= ( ord_Least_nat
@ ^ [S4: nat] :
( ( member_nat @ S4 @ S )
& ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ N3 ) @ S4 ) ) ) ) ) ).
% enumerate_Suc''
thf(fact_871_Bleast__def,axiom,
( bleast_list_nat
= ( ^ [S5: set_list_nat,P4: list_nat > $o] :
( ord_Least_list_nat
@ ^ [X2: list_nat] :
( ( member_list_nat @ X2 @ S5 )
& ( P4 @ X2 ) ) ) ) ) ).
% Bleast_def
thf(fact_872_Bleast__def,axiom,
( bleast_nat
= ( ^ [S5: set_nat,P4: nat > $o] :
( ord_Least_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ S5 )
& ( P4 @ X2 ) ) ) ) ) ).
% Bleast_def
thf(fact_873_abort__Bleast__def,axiom,
( abort_5940734655806358670st_nat
= ( ^ [S5: set_list_nat,P4: list_nat > $o] :
( ord_Least_list_nat
@ ^ [X2: list_nat] :
( ( member_list_nat @ X2 @ S5 )
& ( P4 @ X2 ) ) ) ) ) ).
% abort_Bleast_def
thf(fact_874_abort__Bleast__def,axiom,
( abort_Bleast_nat
= ( ^ [S5: set_nat,P4: nat > $o] :
( ord_Least_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ S5 )
& ( P4 @ X2 ) ) ) ) ) ).
% abort_Bleast_def
thf(fact_875_card__lists__length__eq,axiom,
! [A3: set_nat,N3: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [Xs: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A3 )
& ( ( size_size_list_nat @ Xs )
= N3 ) ) ) )
= ( power_power_nat @ ( finite_card_nat @ A3 ) @ N3 ) ) ) ).
% card_lists_length_eq
thf(fact_876_card__lists__length__eq,axiom,
! [A3: set_list_nat,N3: nat] :
( ( finite8100373058378681591st_nat @ A3 )
=> ( ( finite7325466520557071688st_nat
@ ( collec5989764272469232197st_nat
@ ^ [Xs: list_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ A3 )
& ( ( size_s3023201423986296836st_nat @ Xs )
= N3 ) ) ) )
= ( power_power_nat @ ( finite_card_list_nat @ A3 ) @ N3 ) ) ) ).
% card_lists_length_eq
thf(fact_877_card__lists__length__eq,axiom,
! [A3: set_a,N3: nat] :
( ( finite_finite_a @ A3 )
=> ( ( finite_card_list_a
@ ( collect_list_a
@ ^ [Xs: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ A3 )
& ( ( size_size_list_a @ Xs )
= N3 ) ) ) )
= ( power_power_nat @ ( finite_card_a @ A3 ) @ N3 ) ) ) ).
% card_lists_length_eq
thf(fact_878_length__n__lists,axiom,
! [N3: nat,Xs2: list_nat] :
( ( size_s3023201423986296836st_nat @ ( n_lists_nat @ N3 @ Xs2 ) )
= ( power_power_nat @ ( size_size_list_nat @ Xs2 ) @ N3 ) ) ).
% length_n_lists
thf(fact_879_nth__aA__in__G,axiom,
! [I3: nat] :
( ( ord_less_nat @ I3 @ ( finite_card_a @ a2 ) )
=> ( member_a @ ( nth_a @ ( aA_a @ a2 ) @ I3 ) @ g ) ) ).
% nth_aA_in_G
thf(fact_880_Khovanskii_Oalpha__plus,axiom,
! [G: set_nat,Addition: nat > nat > nat,Zero: nat,A3: set_nat,X: list_nat,Y4: list_nat] :
( ( khovanskii_nat @ G @ Addition @ Zero @ A3 )
=> ( ( ( size_size_list_nat @ X )
= ( finite_card_nat @ A3 ) )
=> ( ( ( size_size_list_nat @ Y4 )
= ( finite_card_nat @ A3 ) )
=> ( ( alpha_nat @ G @ Addition @ Zero @ A3 @ ( plus_plus_list_nat @ X @ Y4 ) )
= ( Addition @ ( alpha_nat @ G @ Addition @ Zero @ A3 @ X ) @ ( alpha_nat @ G @ Addition @ Zero @ A3 @ Y4 ) ) ) ) ) ) ).
% Khovanskii.alpha_plus
thf(fact_881_Khovanskii_Oalpha__plus,axiom,
! [G: set_a,Addition: a > a > a,Zero: a,A3: set_a,X: list_nat,Y4: list_nat] :
( ( khovanskii_a @ G @ Addition @ Zero @ A3 )
=> ( ( ( size_size_list_nat @ X )
= ( finite_card_a @ A3 ) )
=> ( ( ( size_size_list_nat @ Y4 )
= ( finite_card_a @ A3 ) )
=> ( ( alpha_a @ G @ Addition @ Zero @ A3 @ ( plus_plus_list_nat @ X @ Y4 ) )
= ( Addition @ ( alpha_a @ G @ Addition @ Zero @ A3 @ X ) @ ( alpha_a @ G @ Addition @ Zero @ A3 @ Y4 ) ) ) ) ) ) ).
% Khovanskii.alpha_plus
thf(fact_882_Inf__fin__le__Sup__fin,axiom,
! [A3: set_list_nat] :
( ( finite8100373058378681591st_nat @ A3 )
=> ( ( A3 != bot_bot_set_list_nat )
=> ( ord_less_eq_list_nat @ ( lattic5191180550204456963st_nat @ A3 ) @ ( lattic6411832703407573737st_nat @ A3 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_883_Inf__fin__le__Sup__fin,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A3 ) @ ( lattic1093996805478795353in_nat @ A3 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_884_Inf__fin__le__Sup__fin,axiom,
! [A3: set_set_list_nat] :
( ( finite7047420756378620717st_nat @ A3 )
=> ( ( A3 != bot_bo3886227569956363488st_nat )
=> ( ord_le6045566169113846134st_nat @ ( lattic3683530169123051065st_nat @ A3 ) @ ( lattic2169124122975652127st_nat @ A3 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_885_Inf__fin__le__Sup__fin,axiom,
! [A3: set_set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( A3 != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ ( lattic8209813465164889211_set_a @ A3 ) @ ( lattic2918178356826803221_set_a @ A3 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_886_Cons__le__Cons,axiom,
! [A: nat,X: list_nat,B: nat,Y4: list_nat] :
( ( ord_less_eq_list_nat @ ( cons_nat @ A @ X ) @ ( cons_nat @ B @ Y4 ) )
= ( ( ord_less_nat @ ( size_size_list_nat @ X ) @ ( size_size_list_nat @ Y4 ) )
| ( ( ( size_size_list_nat @ X )
= ( size_size_list_nat @ Y4 ) )
& ( ( ord_less_nat @ A @ B )
| ( ( A = B )
& ( ord_less_eq_list_nat @ X @ Y4 ) ) ) ) ) ) ).
% Cons_le_Cons
thf(fact_887_AsubG,axiom,
ord_less_eq_set_a @ a2 @ g ).
% AsubG
thf(fact_888_list_Oinject,axiom,
! [X21: nat,X222: list_nat,Y21: nat,Y222: list_nat] :
( ( ( cons_nat @ X21 @ X222 )
= ( cons_nat @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_889_list__incr__Cons,axiom,
! [I3: nat,K2: nat,Ks: list_nat] :
( ( list_incr @ ( suc @ I3 ) @ ( cons_nat @ K2 @ Ks ) )
= ( cons_nat @ K2 @ ( list_incr @ I3 @ Ks ) ) ) ).
% list_incr_Cons
thf(fact_890_list__ex__simps_I1_J,axiom,
! [P2: nat > $o,X: nat,Xs2: list_nat] :
( ( list_ex_nat @ P2 @ ( cons_nat @ X @ Xs2 ) )
= ( ( P2 @ X )
| ( list_ex_nat @ P2 @ Xs2 ) ) ) ).
% list_ex_simps(1)
thf(fact_891_nth__Cons__Suc,axiom,
! [X: a,Xs2: list_a,N3: nat] :
( ( nth_a @ ( cons_a @ X @ Xs2 ) @ ( suc @ N3 ) )
= ( nth_a @ Xs2 @ N3 ) ) ).
% nth_Cons_Suc
thf(fact_892_nth__Cons__Suc,axiom,
! [X: nat,Xs2: list_nat,N3: nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ ( suc @ N3 ) )
= ( nth_nat @ Xs2 @ N3 ) ) ).
% nth_Cons_Suc
thf(fact_893_nth__plus__list,axiom,
! [I3: nat,Xs2: list_list_nat,Ys2: list_list_nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( ( nth_list_nat @ ( plus_p2116291331692525561st_nat @ Xs2 @ Ys2 ) @ I3 )
= ( plus_plus_list_nat @ ( nth_list_nat @ Xs2 @ I3 ) @ ( nth_list_nat @ Ys2 @ I3 ) ) ) ) ) ).
% nth_plus_list
thf(fact_894_nth__plus__list,axiom,
! [I3: nat,Xs2: list_nat,Ys2: list_nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys2 ) )
=> ( ( nth_nat @ ( plus_plus_list_nat @ Xs2 @ Ys2 ) @ I3 )
= ( plus_plus_nat @ ( nth_nat @ Xs2 @ I3 ) @ ( nth_nat @ Ys2 @ I3 ) ) ) ) ) ).
% nth_plus_list
thf(fact_895_set__ConsD,axiom,
! [Y4: list_nat,X: list_nat,Xs2: list_list_nat] :
( ( member_list_nat @ Y4 @ ( set_list_nat2 @ ( cons_list_nat @ X @ Xs2 ) ) )
=> ( ( Y4 = X )
| ( member_list_nat @ Y4 @ ( set_list_nat2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_896_set__ConsD,axiom,
! [Y4: a,X: a,Xs2: list_a] :
( ( member_a @ Y4 @ ( set_a2 @ ( cons_a @ X @ Xs2 ) ) )
=> ( ( Y4 = X )
| ( member_a @ Y4 @ ( set_a2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_897_set__ConsD,axiom,
! [Y4: nat,X: nat,Xs2: list_nat] :
( ( member_nat @ Y4 @ ( set_nat2 @ ( cons_nat @ X @ Xs2 ) ) )
=> ( ( Y4 = X )
| ( member_nat @ Y4 @ ( set_nat2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_898_list_Oset__cases,axiom,
! [E: list_nat,A: list_list_nat] :
( ( member_list_nat @ E @ ( set_list_nat2 @ A ) )
=> ( ! [Z22: list_list_nat] :
( A
!= ( cons_list_nat @ E @ Z22 ) )
=> ~ ! [Z1: list_nat,Z22: list_list_nat] :
( ( A
= ( cons_list_nat @ Z1 @ Z22 ) )
=> ~ ( member_list_nat @ E @ ( set_list_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_899_list_Oset__cases,axiom,
! [E: a,A: list_a] :
( ( member_a @ E @ ( set_a2 @ A ) )
=> ( ! [Z22: list_a] :
( A
!= ( cons_a @ E @ Z22 ) )
=> ~ ! [Z1: a,Z22: list_a] :
( ( A
= ( cons_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( set_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_900_list_Oset__cases,axiom,
! [E: nat,A: list_nat] :
( ( member_nat @ E @ ( set_nat2 @ A ) )
=> ( ! [Z22: list_nat] :
( A
!= ( cons_nat @ E @ Z22 ) )
=> ~ ! [Z1: nat,Z22: list_nat] :
( ( A
= ( cons_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat @ E @ ( set_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_901_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_902_list_Oset__intros_I1_J,axiom,
! [X21: a,X222: list_a] : ( member_a @ X21 @ ( set_a2 @ ( cons_a @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_903_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_904_list_Oset__intros_I2_J,axiom,
! [Y4: list_nat,X222: list_list_nat,X21: list_nat] :
( ( member_list_nat @ Y4 @ ( set_list_nat2 @ X222 ) )
=> ( member_list_nat @ Y4 @ ( set_list_nat2 @ ( cons_list_nat @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_905_list_Oset__intros_I2_J,axiom,
! [Y4: a,X222: list_a,X21: a] :
( ( member_a @ Y4 @ ( set_a2 @ X222 ) )
=> ( member_a @ Y4 @ ( set_a2 @ ( cons_a @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_906_list_Oset__intros_I2_J,axiom,
! [Y4: nat,X222: list_nat,X21: nat] :
( ( member_nat @ Y4 @ ( set_nat2 @ X222 ) )
=> ( member_nat @ Y4 @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_907_sum__list__plus,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( groups4561878855575611511st_nat @ ( plus_plus_list_nat @ Xs2 @ Ys2 ) )
= ( plus_plus_nat @ ( groups4561878855575611511st_nat @ Xs2 ) @ ( groups4561878855575611511st_nat @ Ys2 ) ) ) ) ).
% sum_list_plus
thf(fact_908_plus__Cons,axiom,
! [Y4: list_nat,Ys2: list_list_nat,X: list_nat,Xs2: list_list_nat] :
( ( plus_p2116291331692525561st_nat @ ( cons_list_nat @ Y4 @ Ys2 ) @ ( cons_list_nat @ X @ Xs2 ) )
= ( cons_list_nat @ ( plus_plus_list_nat @ Y4 @ X ) @ ( plus_p2116291331692525561st_nat @ Ys2 @ Xs2 ) ) ) ).
% plus_Cons
thf(fact_909_plus__Cons,axiom,
! [Y4: nat,Ys2: list_nat,X: nat,Xs2: list_nat] :
( ( plus_plus_list_nat @ ( cons_nat @ Y4 @ Ys2 ) @ ( cons_nat @ X @ Xs2 ) )
= ( cons_nat @ ( plus_plus_nat @ Y4 @ X ) @ ( plus_plus_list_nat @ Ys2 @ Xs2 ) ) ) ).
% plus_Cons
thf(fact_910_not__Cons__self2,axiom,
! [X: nat,Xs2: list_nat] :
( ( cons_nat @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_911_distinct__length__2__or__more,axiom,
! [A: nat,B: nat,Xs2: list_nat] :
( ( distinct_nat @ ( cons_nat @ A @ ( cons_nat @ B @ Xs2 ) ) )
= ( ( A != B )
& ( distinct_nat @ ( cons_nat @ A @ Xs2 ) )
& ( distinct_nat @ ( cons_nat @ B @ Xs2 ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_912_null__rec_I1_J,axiom,
! [X: nat,Xs2: list_nat] :
~ ( null_nat @ ( cons_nat @ X @ Xs2 ) ) ).
% null_rec(1)
thf(fact_913_set__subset__Cons,axiom,
! [Xs2: list_nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ ( set_nat2 @ ( cons_nat @ X @ Xs2 ) ) ) ).
% set_subset_Cons
thf(fact_914_set__subset__Cons,axiom,
! [Xs2: list_list_nat,X: list_nat] : ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs2 ) @ ( set_list_nat2 @ ( cons_list_nat @ X @ Xs2 ) ) ) ).
% set_subset_Cons
thf(fact_915_set__subset__Cons,axiom,
! [Xs2: list_a,X: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs2 ) @ ( set_a2 @ ( cons_a @ X @ Xs2 ) ) ) ).
% set_subset_Cons
thf(fact_916_length__Suc__conv,axiom,
! [Xs2: list_nat,N3: nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( suc @ N3 ) )
= ( ? [Y3: nat,Ys: list_nat] :
( ( Xs2
= ( cons_nat @ Y3 @ Ys ) )
& ( ( size_size_list_nat @ Ys )
= N3 ) ) ) ) ).
% length_Suc_conv
thf(fact_917_Suc__length__conv,axiom,
! [N3: nat,Xs2: list_nat] :
( ( ( suc @ N3 )
= ( size_size_list_nat @ Xs2 ) )
= ( ? [Y3: nat,Ys: list_nat] :
( ( Xs2
= ( cons_nat @ Y3 @ Ys ) )
& ( ( size_size_list_nat @ Ys )
= N3 ) ) ) ) ).
% Suc_length_conv
thf(fact_918_length__Cons,axiom,
! [X: nat,Xs2: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X @ Xs2 ) )
= ( suc @ ( size_size_list_nat @ Xs2 ) ) ) ).
% length_Cons
thf(fact_919_impossible__Cons,axiom,
! [Xs2: list_nat,Ys2: list_nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ Ys2 ) )
=> ( Xs2
!= ( cons_nat @ X @ Ys2 ) ) ) ).
% impossible_Cons
thf(fact_920_distinct_Osimps_I2_J,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( distinct_list_nat @ ( cons_list_nat @ X @ Xs2 ) )
= ( ~ ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
& ( distinct_list_nat @ Xs2 ) ) ) ).
% distinct.simps(2)
thf(fact_921_distinct_Osimps_I2_J,axiom,
! [X: a,Xs2: list_a] :
( ( distinct_a @ ( cons_a @ X @ Xs2 ) )
= ( ~ ( member_a @ X @ ( set_a2 @ Xs2 ) )
& ( distinct_a @ Xs2 ) ) ) ).
% distinct.simps(2)
thf(fact_922_distinct_Osimps_I2_J,axiom,
! [X: nat,Xs2: list_nat] :
( ( distinct_nat @ ( cons_nat @ X @ Xs2 ) )
= ( ~ ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
& ( distinct_nat @ Xs2 ) ) ) ).
% distinct.simps(2)
thf(fact_923_list__update__code_I3_J,axiom,
! [X: nat,Xs2: list_nat,I3: nat,Y4: nat] :
( ( list_update_nat @ ( cons_nat @ X @ Xs2 ) @ ( suc @ I3 ) @ Y4 )
= ( cons_nat @ X @ ( list_update_nat @ Xs2 @ I3 @ Y4 ) ) ) ).
% list_update_code(3)
thf(fact_924_Inf__fin_OcoboundedI,axiom,
! [A3: set_list_nat,A: list_nat] :
( ( finite8100373058378681591st_nat @ A3 )
=> ( ( member_list_nat @ A @ A3 )
=> ( ord_less_eq_list_nat @ ( lattic5191180550204456963st_nat @ A3 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_925_Inf__fin_OcoboundedI,axiom,
! [A3: set_nat,A: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( member_nat @ A @ A3 )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A3 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_926_Inf__fin_OcoboundedI,axiom,
! [A3: set_set_list_nat,A: set_list_nat] :
( ( finite7047420756378620717st_nat @ A3 )
=> ( ( member_set_list_nat @ A @ A3 )
=> ( ord_le6045566169113846134st_nat @ ( lattic3683530169123051065st_nat @ A3 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_927_Inf__fin_OcoboundedI,axiom,
! [A3: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( member_set_a @ A @ A3 )
=> ( ord_less_eq_set_a @ ( lattic8209813465164889211_set_a @ A3 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_928_Suc__le__length__iff,axiom,
! [N3: nat,Xs2: list_nat] :
( ( ord_less_eq_nat @ ( suc @ N3 ) @ ( size_size_list_nat @ Xs2 ) )
= ( ? [X2: nat,Ys: list_nat] :
( ( Xs2
= ( cons_nat @ X2 @ Ys ) )
& ( ord_less_eq_nat @ N3 @ ( size_size_list_nat @ Ys ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_929_Inf__fin_OboundedE,axiom,
! [A3: set_list_nat,X: list_nat] :
( ( finite8100373058378681591st_nat @ A3 )
=> ( ( A3 != bot_bot_set_list_nat )
=> ( ( ord_less_eq_list_nat @ X @ ( lattic5191180550204456963st_nat @ A3 ) )
=> ! [A8: list_nat] :
( ( member_list_nat @ A8 @ A3 )
=> ( ord_less_eq_list_nat @ X @ A8 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_930_Inf__fin_OboundedE,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A3 ) )
=> ! [A8: nat] :
( ( member_nat @ A8 @ A3 )
=> ( ord_less_eq_nat @ X @ A8 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_931_Inf__fin_OboundedE,axiom,
! [A3: set_set_list_nat,X: set_list_nat] :
( ( finite7047420756378620717st_nat @ A3 )
=> ( ( A3 != bot_bo3886227569956363488st_nat )
=> ( ( ord_le6045566169113846134st_nat @ X @ ( lattic3683530169123051065st_nat @ A3 ) )
=> ! [A8: set_list_nat] :
( ( member_set_list_nat @ A8 @ A3 )
=> ( ord_le6045566169113846134st_nat @ X @ A8 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_932_Inf__fin_OboundedE,axiom,
! [A3: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( A3 != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ X @ ( lattic8209813465164889211_set_a @ A3 ) )
=> ! [A8: set_a] :
( ( member_set_a @ A8 @ A3 )
=> ( ord_less_eq_set_a @ X @ A8 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_933_Inf__fin_OboundedI,axiom,
! [A3: set_list_nat,X: list_nat] :
( ( finite8100373058378681591st_nat @ A3 )
=> ( ( A3 != bot_bot_set_list_nat )
=> ( ! [A4: list_nat] :
( ( member_list_nat @ A4 @ A3 )
=> ( ord_less_eq_list_nat @ X @ A4 ) )
=> ( ord_less_eq_list_nat @ X @ ( lattic5191180550204456963st_nat @ A3 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_934_Inf__fin_OboundedI,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A3 )
=> ( ord_less_eq_nat @ X @ A4 ) )
=> ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A3 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_935_Inf__fin_OboundedI,axiom,
! [A3: set_set_list_nat,X: set_list_nat] :
( ( finite7047420756378620717st_nat @ A3 )
=> ( ( A3 != bot_bo3886227569956363488st_nat )
=> ( ! [A4: set_list_nat] :
( ( member_set_list_nat @ A4 @ A3 )
=> ( ord_le6045566169113846134st_nat @ X @ A4 ) )
=> ( ord_le6045566169113846134st_nat @ X @ ( lattic3683530169123051065st_nat @ A3 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_936_Inf__fin_OboundedI,axiom,
! [A3: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( A3 != bot_bot_set_set_a )
=> ( ! [A4: set_a] :
( ( member_set_a @ A4 @ A3 )
=> ( ord_less_eq_set_a @ X @ A4 ) )
=> ( ord_less_eq_set_a @ X @ ( lattic8209813465164889211_set_a @ A3 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_937_Inf__fin_Obounded__iff,axiom,
! [A3: set_list_nat,X: list_nat] :
( ( finite8100373058378681591st_nat @ A3 )
=> ( ( A3 != bot_bot_set_list_nat )
=> ( ( ord_less_eq_list_nat @ X @ ( lattic5191180550204456963st_nat @ A3 ) )
= ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ A3 )
=> ( ord_less_eq_list_nat @ X @ X2 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_938_Inf__fin_Obounded__iff,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A3 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( ord_less_eq_nat @ X @ X2 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_939_Inf__fin_Obounded__iff,axiom,
! [A3: set_set_list_nat,X: set_list_nat] :
( ( finite7047420756378620717st_nat @ A3 )
=> ( ( A3 != bot_bo3886227569956363488st_nat )
=> ( ( ord_le6045566169113846134st_nat @ X @ ( lattic3683530169123051065st_nat @ A3 ) )
= ( ! [X2: set_list_nat] :
( ( member_set_list_nat @ X2 @ A3 )
=> ( ord_le6045566169113846134st_nat @ X @ X2 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_940_Inf__fin_Obounded__iff,axiom,
! [A3: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A3 )
=> ( ( A3 != bot_bot_set_set_a )
=> ( ( ord_less_eq_set_a @ X @ ( lattic8209813465164889211_set_a @ A3 ) )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A3 )
=> ( ord_less_eq_set_a @ X @ X2 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_941_pointwise__le__plus,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( pointwise_le @ Xs2 @ Ys2 )
=> ( ( ord_less_eq_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Zs ) )
=> ( pointwise_le @ Xs2 @ ( plus_plus_list_nat @ Ys2 @ Zs ) ) ) ) ).
% pointwise_le_plus
thf(fact_942_Inf__fin_Osubset__imp,axiom,
! [A3: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
=> ( ( A3 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B4 )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ B4 ) @ ( lattic5238388535129920115in_nat @ A3 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_943_Inf__fin_Osubset__imp,axiom,
! [A3: set_set_list_nat,B4: set_set_list_nat] :
( ( ord_le1068707526560357548st_nat @ A3 @ B4 )
=> ( ( A3 != bot_bo3886227569956363488st_nat )
=> ( ( finite7047420756378620717st_nat @ B4 )
=> ( ord_le6045566169113846134st_nat @ ( lattic3683530169123051065st_nat @ B4 ) @ ( lattic3683530169123051065st_nat @ A3 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_944_Inf__fin_Osubset__imp,axiom,
! [A3: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A3 @ B4 )
=> ( ( A3 != bot_bot_set_set_a )
=> ( ( finite_finite_set_a @ B4 )
=> ( ord_less_eq_set_a @ ( lattic8209813465164889211_set_a @ B4 ) @ ( lattic8209813465164889211_set_a @ A3 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_945_Inf__fin_Osubset__imp,axiom,
! [A3: set_list_nat,B4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ B4 )
=> ( ( A3 != bot_bot_set_list_nat )
=> ( ( finite8100373058378681591st_nat @ B4 )
=> ( ord_less_eq_list_nat @ ( lattic5191180550204456963st_nat @ B4 ) @ ( lattic5191180550204456963st_nat @ A3 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_946_Cons__less__Cons,axiom,
! [A: nat,X: list_nat,B: nat,Y4: list_nat] :
( ( ord_less_list_nat @ ( cons_nat @ A @ X ) @ ( cons_nat @ B @ Y4 ) )
= ( ( ord_less_nat @ ( size_size_list_nat @ X ) @ ( size_size_list_nat @ Y4 ) )
| ( ( ( size_size_list_nat @ X )
= ( size_size_list_nat @ Y4 ) )
& ( ( ord_less_nat @ A @ B )
| ( ( A = B )
& ( ord_less_list_nat @ X @ Y4 ) ) ) ) ) ) ).
% Cons_less_Cons
thf(fact_947_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_948_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_949_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_950_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_951_add__Suc__right,axiom,
! [M3: nat,N3: nat] :
( ( plus_plus_nat @ M3 @ ( suc @ N3 ) )
= ( suc @ ( plus_plus_nat @ M3 @ N3 ) ) ) ).
% add_Suc_right
thf(fact_952_nat__add__left__cancel__le,axiom,
! [K2: nat,M3: nat,N3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M3 ) @ ( plus_plus_nat @ K2 @ N3 ) )
= ( ord_less_eq_nat @ M3 @ N3 ) ) ).
% nat_add_left_cancel_le
thf(fact_953_nat__add__left__cancel__less,axiom,
! [K2: nat,M3: nat,N3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M3 ) @ ( plus_plus_nat @ K2 @ N3 ) )
= ( ord_less_nat @ M3 @ N3 ) ) ).
% nat_add_left_cancel_less
thf(fact_954_length__sum__set__Suc,axiom,
! [K2: nat,Ks: list_nat,R: nat,N3: nat] :
( ( member_list_nat @ ( cons_nat @ K2 @ Ks ) @ ( length_sum_set @ ( suc @ R ) @ N3 ) )
= ( ? [M2: nat] :
( ( member_list_nat @ Ks @ ( length_sum_set @ R @ M2 ) )
& ( N3
= ( plus_plus_nat @ M2 @ K2 ) ) ) ) ) ).
% length_sum_set_Suc
thf(fact_955_less__add__eq__less,axiom,
! [K2: nat,L: nat,M3: nat,N3: nat] :
( ( ord_less_nat @ K2 @ L )
=> ( ( ( plus_plus_nat @ M3 @ L )
= ( plus_plus_nat @ K2 @ N3 ) )
=> ( ord_less_nat @ M3 @ N3 ) ) ) ).
% less_add_eq_less
thf(fact_956_trans__less__add2,axiom,
! [I3: nat,J: nat,M3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ M3 @ J ) ) ) ).
% trans_less_add2
thf(fact_957_trans__less__add1,axiom,
! [I3: nat,J: nat,M3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ J @ M3 ) ) ) ).
% trans_less_add1
thf(fact_958_add__less__mono1,axiom,
! [I3: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_less_mono1
thf(fact_959_not__add__less2,axiom,
! [J: nat,I3: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I3 ) @ I3 ) ).
% not_add_less2
thf(fact_960_not__add__less1,axiom,
! [I3: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I3 @ J ) @ I3 ) ).
% not_add_less1
thf(fact_961_add__less__mono,axiom,
! [I3: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( ord_less_nat @ K2 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_962_add__lessD1,axiom,
! [I3: nat,J: nat,K2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I3 @ J ) @ K2 )
=> ( ord_less_nat @ I3 @ K2 ) ) ).
% add_lessD1
thf(fact_963_add__leE,axiom,
! [M3: nat,K2: nat,N3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K2 ) @ N3 )
=> ~ ( ( ord_less_eq_nat @ M3 @ N3 )
=> ~ ( ord_less_eq_nat @ K2 @ N3 ) ) ) ).
% add_leE
thf(fact_964_le__add1,axiom,
! [N3: nat,M3: nat] : ( ord_less_eq_nat @ N3 @ ( plus_plus_nat @ N3 @ M3 ) ) ).
% le_add1
thf(fact_965_le__add2,axiom,
! [N3: nat,M3: nat] : ( ord_less_eq_nat @ N3 @ ( plus_plus_nat @ M3 @ N3 ) ) ).
% le_add2
thf(fact_966_add__leD1,axiom,
! [M3: nat,K2: nat,N3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K2 ) @ N3 )
=> ( ord_less_eq_nat @ M3 @ N3 ) ) ).
% add_leD1
thf(fact_967_add__leD2,axiom,
! [M3: nat,K2: nat,N3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K2 ) @ N3 )
=> ( ord_less_eq_nat @ K2 @ N3 ) ) ).
% add_leD2
thf(fact_968_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq_nat @ K2 @ L )
=> ? [N: nat] :
( L
= ( plus_plus_nat @ K2 @ N ) ) ) ).
% le_Suc_ex
thf(fact_969_add__le__mono,axiom,
! [I3: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_970_add__le__mono1,axiom,
! [I3: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_le_mono1
thf(fact_971_trans__le__add1,axiom,
! [I3: nat,J: nat,M3: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ J @ M3 ) ) ) ).
% trans_le_add1
thf(fact_972_trans__le__add2,axiom,
! [I3: nat,J: nat,M3: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ M3 @ J ) ) ) ).
% trans_le_add2
thf(fact_973_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
? [K: nat] :
( N2
= ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% nat_le_iff_add
thf(fact_974_add__Suc__shift,axiom,
! [M3: nat,N3: nat] :
( ( plus_plus_nat @ ( suc @ M3 ) @ N3 )
= ( plus_plus_nat @ M3 @ ( suc @ N3 ) ) ) ).
% add_Suc_shift
thf(fact_975_add__Suc,axiom,
! [M3: nat,N3: nat] :
( ( plus_plus_nat @ ( suc @ M3 ) @ N3 )
= ( suc @ ( plus_plus_nat @ M3 @ N3 ) ) ) ).
% add_Suc
thf(fact_976_nat__arith_Osuc1,axiom,
! [A3: nat,K2: nat,A: nat] :
( ( A3
= ( plus_plus_nat @ K2 @ A ) )
=> ( ( suc @ A3 )
= ( plus_plus_nat @ K2 @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_977_less__natE,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ~ ! [Q4: nat] :
( N3
!= ( suc @ ( plus_plus_nat @ M3 @ Q4 ) ) ) ) ).
% less_natE
thf(fact_978_less__add__Suc1,axiom,
! [I3: nat,M3: nat] : ( ord_less_nat @ I3 @ ( suc @ ( plus_plus_nat @ I3 @ M3 ) ) ) ).
% less_add_Suc1
thf(fact_979_less__add__Suc2,axiom,
! [I3: nat,M3: nat] : ( ord_less_nat @ I3 @ ( suc @ ( plus_plus_nat @ M3 @ I3 ) ) ) ).
% less_add_Suc2
thf(fact_980_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
? [K: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M2 @ K ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_981_less__imp__Suc__add,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ? [K3: nat] :
( N3
= ( suc @ ( plus_plus_nat @ M3 @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_982_mono__nat__linear__lb,axiom,
! [F: nat > nat,M3: nat,K2: nat] :
( ! [M6: nat,N: nat] :
( ( ord_less_nat @ M6 @ N )
=> ( ord_less_nat @ ( F @ M6 ) @ ( F @ N ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M3 ) @ K2 ) @ ( F @ ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_983_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_984_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_985_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B3: nat] :
? [C4: nat] :
( B3
= ( plus_plus_nat @ A2 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_986_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_987_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A @ C3 ) ) ) ).
% less_eqE
thf(fact_988_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_989_add__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_990_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I3: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I3 @ J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_991_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I3: nat,J: nat,K2: nat,L: nat] :
( ( ( I3 = J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_992_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I3: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I3 @ J )
& ( K2 = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_993_add__mono__thms__linordered__field_I5_J,axiom,
! [I3: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I3 @ J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_994_add__mono__thms__linordered__field_I2_J,axiom,
! [I3: nat,J: nat,K2: nat,L: nat] :
( ( ( I3 = J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_995_add__mono__thms__linordered__field_I1_J,axiom,
! [I3: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I3 @ J )
& ( K2 = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_996_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_997_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_998_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_999_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_1000_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_1001_add__mono__thms__linordered__field_I4_J,axiom,
! [I3: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I3 @ J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1002_add__mono__thms__linordered__field_I3_J,axiom,
! [I3: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I3 @ J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1003_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_1004_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_1005_set__plus__mono2,axiom,
! [C2: set_list_nat,D3: set_list_nat,E2: set_list_nat,F2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ C2 @ D3 )
=> ( ( ord_le6045566169113846134st_nat @ E2 @ F2 )
=> ( ord_le6045566169113846134st_nat @ ( plus_p884110394369815071st_nat @ C2 @ E2 ) @ ( plus_p884110394369815071st_nat @ D3 @ F2 ) ) ) ) ).
% set_plus_mono2
thf(fact_1006_Set__Algebras_Osumset__empty_I2_J,axiom,
! [A3: set_list_nat] :
( ( plus_p884110394369815071st_nat @ bot_bot_set_list_nat @ A3 )
= bot_bot_set_list_nat ) ).
% Set_Algebras.sumset_empty(2)
thf(fact_1007_Set__Algebras_Osumset__empty_I1_J,axiom,
! [A3: set_list_nat] :
( ( plus_p884110394369815071st_nat @ A3 @ bot_bot_set_list_nat )
= bot_bot_set_list_nat ) ).
% Set_Algebras.sumset_empty(1)
thf(fact_1008_sum__list__update,axiom,
! [K2: nat,Xs2: list_nat,X: nat] :
( ( ord_less_nat @ K2 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( groups4561878855575611511st_nat @ ( list_update_nat @ Xs2 @ K2 @ X ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( groups4561878855575611511st_nat @ Xs2 ) @ X ) @ ( nth_nat @ Xs2 @ K2 ) ) ) ) ).
% sum_list_update
thf(fact_1009_Diff__cancel,axiom,
! [A3: set_list_nat] :
( ( minus_7954133019191499631st_nat @ A3 @ A3 )
= bot_bot_set_list_nat ) ).
% Diff_cancel
thf(fact_1010_Diff__cancel,axiom,
! [A3: set_a] :
( ( minus_minus_set_a @ A3 @ A3 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_1011_empty__Diff,axiom,
! [A3: set_list_nat] :
( ( minus_7954133019191499631st_nat @ bot_bot_set_list_nat @ A3 )
= bot_bot_set_list_nat ) ).
% empty_Diff
thf(fact_1012_empty__Diff,axiom,
! [A3: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A3 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_1013_Diff__empty,axiom,
! [A3: set_list_nat] :
( ( minus_7954133019191499631st_nat @ A3 @ bot_bot_set_list_nat )
= A3 ) ).
% Diff_empty
thf(fact_1014_Diff__empty,axiom,
! [A3: set_a] :
( ( minus_minus_set_a @ A3 @ bot_bot_set_a )
= A3 ) ).
% Diff_empty
thf(fact_1015_finite__Diff,axiom,
! [A3: set_list_nat,B4: set_list_nat] :
( ( finite8100373058378681591st_nat @ A3 )
=> ( finite8100373058378681591st_nat @ ( minus_7954133019191499631st_nat @ A3 @ B4 ) ) ) ).
% finite_Diff
thf(fact_1016_finite__Diff,axiom,
! [A3: set_nat,B4: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ B4 ) ) ) ).
% finite_Diff
thf(fact_1017_finite__Diff,axiom,
! [A3: set_a,B4: set_a] :
( ( finite_finite_a @ A3 )
=> ( finite_finite_a @ ( minus_minus_set_a @ A3 @ B4 ) ) ) ).
% finite_Diff
thf(fact_1018_finite__Diff2,axiom,
! [B4: set_list_nat,A3: set_list_nat] :
( ( finite8100373058378681591st_nat @ B4 )
=> ( ( finite8100373058378681591st_nat @ ( minus_7954133019191499631st_nat @ A3 @ B4 ) )
= ( finite8100373058378681591st_nat @ A3 ) ) ) ).
% finite_Diff2
thf(fact_1019_finite__Diff2,axiom,
! [B4: set_nat,A3: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ B4 ) )
= ( finite_finite_nat @ A3 ) ) ) ).
% finite_Diff2
thf(fact_1020_finite__Diff2,axiom,
! [B4: set_a,A3: set_a] :
( ( finite_finite_a @ B4 )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A3 @ B4 ) )
= ( finite_finite_a @ A3 ) ) ) ).
% finite_Diff2
thf(fact_1021_Suc__diff__diff,axiom,
! [M3: nat,N3: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M3 ) @ N3 ) @ ( suc @ K2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M3 @ N3 ) @ K2 ) ) ).
% Suc_diff_diff
thf(fact_1022_diff__Suc__Suc,axiom,
! [M3: nat,N3: nat] :
( ( minus_minus_nat @ ( suc @ M3 ) @ ( suc @ N3 ) )
= ( minus_minus_nat @ M3 @ N3 ) ) ).
% diff_Suc_Suc
thf(fact_1023_diff__diff__cancel,axiom,
! [I3: nat,N3: nat] :
( ( ord_less_eq_nat @ I3 @ N3 )
=> ( ( minus_minus_nat @ N3 @ ( minus_minus_nat @ N3 @ I3 ) )
= I3 ) ) ).
% diff_diff_cancel
thf(fact_1024_diff__diff__left,axiom,
! [I3: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J ) @ K2 )
= ( minus_minus_nat @ I3 @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% diff_diff_left
thf(fact_1025_Diff__eq__empty__iff,axiom,
! [A3: set_list_nat,B4: set_list_nat] :
( ( ( minus_7954133019191499631st_nat @ A3 @ B4 )
= bot_bot_set_list_nat )
= ( ord_le6045566169113846134st_nat @ A3 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_1026_Diff__eq__empty__iff,axiom,
! [A3: set_a,B4: set_a] :
( ( ( minus_minus_set_a @ A3 @ B4 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A3 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_1027_Nat_Odiff__diff__right,axiom,
! [K2: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I3 @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ K2 ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1028_Nat_Oadd__diff__assoc2,axiom,
! [K2: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I3 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I3 ) @ K2 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1029_Nat_Oadd__diff__assoc,axiom,
! [K2: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J ) @ K2 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1030_diff__Suc__diff__eq1,axiom,
! [K2: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I3 @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1031_diff__Suc__diff__eq2,axiom,
! [K2: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) @ I3 )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K2 @ I3 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1032_nth__minus__list,axiom,
! [I3: nat,Xs2: list_list_nat,Ys2: list_list_nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( ( nth_list_nat @ ( minus_3911745200923244873st_nat @ Xs2 @ Ys2 ) @ I3 )
= ( minus_minus_list_nat @ ( nth_list_nat @ Xs2 @ I3 ) @ ( nth_list_nat @ Ys2 @ I3 ) ) ) ) ) ).
% nth_minus_list
thf(fact_1033_nth__minus__list,axiom,
! [I3: nat,Xs2: list_nat,Ys2: list_nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys2 ) )
=> ( ( nth_nat @ ( minus_minus_list_nat @ Xs2 @ Ys2 ) @ I3 )
= ( minus_minus_nat @ ( nth_nat @ Xs2 @ I3 ) @ ( nth_nat @ Ys2 @ I3 ) ) ) ) ) ).
% nth_minus_list
thf(fact_1034_add__diff__inverse__nat,axiom,
! [M3: nat,N3: nat] :
( ~ ( ord_less_nat @ M3 @ N3 )
=> ( ( plus_plus_nat @ N3 @ ( minus_minus_nat @ M3 @ N3 ) )
= M3 ) ) ).
% add_diff_inverse_nat
thf(fact_1035_less__diff__conv,axiom,
! [I3: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I3 @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ I3 @ K2 ) @ J ) ) ).
% less_diff_conv
thf(fact_1036_Nat_Ole__imp__diff__is__add,axiom,
! [I3: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ( minus_minus_nat @ J @ I3 )
= K2 )
= ( J
= ( plus_plus_nat @ K2 @ I3 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1037_Nat_Odiff__add__assoc2,axiom,
! [K2: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I3 ) @ K2 )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I3 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1038_Nat_Odiff__add__assoc,axiom,
! [K2: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J ) @ K2 )
= ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J @ K2 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1039_Nat_Ole__diff__conv2,axiom,
! [K2: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_eq_nat @ I3 @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1040_le__diff__conv,axiom,
! [J: nat,K2: nat,I3: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K2 ) @ I3 )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I3 @ K2 ) ) ) ).
% le_diff_conv
thf(fact_1041_diff__add__inverse2,axiom,
! [M3: nat,N3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M3 @ N3 ) @ N3 )
= M3 ) ).
% diff_add_inverse2
thf(fact_1042_diff__add__inverse,axiom,
! [N3: nat,M3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N3 @ M3 ) @ N3 )
= M3 ) ).
% diff_add_inverse
thf(fact_1043_diff__cancel2,axiom,
! [M3: nat,K2: nat,N3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M3 @ K2 ) @ ( plus_plus_nat @ N3 @ K2 ) )
= ( minus_minus_nat @ M3 @ N3 ) ) ).
% diff_cancel2
thf(fact_1044_Nat_Odiff__cancel,axiom,
! [K2: nat,M3: nat,N3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K2 @ M3 ) @ ( plus_plus_nat @ K2 @ N3 ) )
= ( minus_minus_nat @ M3 @ N3 ) ) ).
% Nat.diff_cancel
thf(fact_1045_minus__Cons,axiom,
! [Y4: list_nat,Ys2: list_list_nat,X: list_nat,Xs2: list_list_nat] :
( ( minus_3911745200923244873st_nat @ ( cons_list_nat @ Y4 @ Ys2 ) @ ( cons_list_nat @ X @ Xs2 ) )
= ( cons_list_nat @ ( minus_minus_list_nat @ Y4 @ X ) @ ( minus_3911745200923244873st_nat @ Ys2 @ Xs2 ) ) ) ).
% minus_Cons
thf(fact_1046_minus__Cons,axiom,
! [Y4: nat,Ys2: list_nat,X: nat,Xs2: list_nat] :
( ( minus_minus_list_nat @ ( cons_nat @ Y4 @ Ys2 ) @ ( cons_nat @ X @ Xs2 ) )
= ( cons_nat @ ( minus_minus_nat @ Y4 @ X ) @ ( minus_minus_list_nat @ Ys2 @ Xs2 ) ) ) ).
% minus_Cons
thf(fact_1047_diff__card__le__card__Diff,axiom,
! [B4: set_list_nat,A3: set_list_nat] :
( ( finite8100373058378681591st_nat @ B4 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B4 ) ) @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ B4 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1048_diff__card__le__card__Diff,axiom,
! [B4: set_nat,A3: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B4 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ B4 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1049_diff__card__le__card__Diff,axiom,
! [B4: set_a,A3: set_a] :
( ( finite_finite_a @ B4 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_a @ A3 ) @ ( finite_card_a @ B4 ) ) @ ( finite_card_a @ ( minus_minus_set_a @ A3 @ B4 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1050_card__Diff__subset,axiom,
! [B4: set_nat,A3: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ A3 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ B4 ) )
= ( minus_minus_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B4 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_1051_card__Diff__subset,axiom,
! [B4: set_list_nat,A3: set_list_nat] :
( ( finite8100373058378681591st_nat @ B4 )
=> ( ( ord_le6045566169113846134st_nat @ B4 @ A3 )
=> ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ B4 ) )
= ( minus_minus_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B4 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_1052_card__Diff__subset,axiom,
! [B4: set_a,A3: set_a] :
( ( finite_finite_a @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ A3 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A3 @ B4 ) )
= ( minus_minus_nat @ ( finite_card_a @ A3 ) @ ( finite_card_a @ B4 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_1053_Diff__mono,axiom,
! [A3: set_list_nat,C2: set_list_nat,D3: set_list_nat,B4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ C2 )
=> ( ( ord_le6045566169113846134st_nat @ D3 @ B4 )
=> ( ord_le6045566169113846134st_nat @ ( minus_7954133019191499631st_nat @ A3 @ B4 ) @ ( minus_7954133019191499631st_nat @ C2 @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_1054_Diff__mono,axiom,
! [A3: set_a,C2: set_a,D3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A3 @ C2 )
=> ( ( ord_less_eq_set_a @ D3 @ B4 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A3 @ B4 ) @ ( minus_minus_set_a @ C2 @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_1055_Diff__subset,axiom,
! [A3: set_list_nat,B4: set_list_nat] : ( ord_le6045566169113846134st_nat @ ( minus_7954133019191499631st_nat @ A3 @ B4 ) @ A3 ) ).
% Diff_subset
thf(fact_1056_Diff__subset,axiom,
! [A3: set_a,B4: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A3 @ B4 ) @ A3 ) ).
% Diff_subset
thf(fact_1057_double__diff,axiom,
! [A3: set_list_nat,B4: set_list_nat,C2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ B4 )
=> ( ( ord_le6045566169113846134st_nat @ B4 @ C2 )
=> ( ( minus_7954133019191499631st_nat @ B4 @ ( minus_7954133019191499631st_nat @ C2 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_1058_double__diff,axiom,
! [A3: set_a,B4: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ( ord_less_eq_set_a @ B4 @ C2 )
=> ( ( minus_minus_set_a @ B4 @ ( minus_minus_set_a @ C2 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_1059_diff__le__mono2,axiom,
! [M3: nat,N3: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N3 ) @ ( minus_minus_nat @ L @ M3 ) ) ) ).
% diff_le_mono2
thf(fact_1060_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1061_diff__le__self,axiom,
! [M3: nat,N3: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ N3 ) @ M3 ) ).
% diff_le_self
thf(fact_1062_diff__le__mono,axiom,
! [M3: nat,N3: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ L ) @ ( minus_minus_nat @ N3 @ L ) ) ) ).
% diff_le_mono
thf(fact_1063_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M3: nat,N3: nat] :
( ( ord_less_eq_nat @ K2 @ M3 )
=> ( ( ord_less_eq_nat @ K2 @ N3 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M3 @ K2 ) @ ( minus_minus_nat @ N3 @ K2 ) )
= ( minus_minus_nat @ M3 @ N3 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1064_le__diff__iff,axiom,
! [K2: nat,M3: nat,N3: nat] :
( ( ord_less_eq_nat @ K2 @ M3 )
=> ( ( ord_less_eq_nat @ K2 @ N3 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ K2 ) @ ( minus_minus_nat @ N3 @ K2 ) )
= ( ord_less_eq_nat @ M3 @ N3 ) ) ) ) ).
% le_diff_iff
thf(fact_1065_eq__diff__iff,axiom,
! [K2: nat,M3: nat,N3: nat] :
( ( ord_less_eq_nat @ K2 @ M3 )
=> ( ( ord_less_eq_nat @ K2 @ N3 )
=> ( ( ( minus_minus_nat @ M3 @ K2 )
= ( minus_minus_nat @ N3 @ K2 ) )
= ( M3 = N3 ) ) ) ) ).
% eq_diff_iff
thf(fact_1066_less__imp__diff__less,axiom,
! [J: nat,K2: nat,N3: nat] :
( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N3 ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_1067_diff__less__mono2,axiom,
! [M3: nat,N3: nat,L: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ( ord_less_nat @ M3 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N3 ) @ ( minus_minus_nat @ L @ M3 ) ) ) ) ).
% diff_less_mono2
thf(fact_1068_psubset__imp__ex__mem,axiom,
! [A3: set_list_nat,B4: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A3 @ B4 )
=> ? [B2: list_nat] : ( member_list_nat @ B2 @ ( minus_7954133019191499631st_nat @ B4 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1069_psubset__imp__ex__mem,axiom,
! [A3: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ A3 @ B4 )
=> ? [B2: nat] : ( member_nat @ B2 @ ( minus_minus_set_nat @ B4 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1070_psubset__imp__ex__mem,axiom,
! [A3: set_a,B4: set_a] :
( ( ord_less_set_a @ A3 @ B4 )
=> ? [B2: a] : ( member_a @ B2 @ ( minus_minus_set_a @ B4 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1071_Diff__infinite__finite,axiom,
! [T3: set_list_nat,S: set_list_nat] :
( ( finite8100373058378681591st_nat @ T3 )
=> ( ~ ( finite8100373058378681591st_nat @ S )
=> ~ ( finite8100373058378681591st_nat @ ( minus_7954133019191499631st_nat @ S @ T3 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1072_Diff__infinite__finite,axiom,
! [T3: set_nat,S: set_nat] :
( ( finite_finite_nat @ T3 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T3 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1073_Diff__infinite__finite,axiom,
! [T3: set_a,S: set_a] :
( ( finite_finite_a @ T3 )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ T3 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1074_pairwise__minus__cancel,axiom,
! [Z: list_nat,X: list_nat,Y4: list_nat] :
( ( pointwise_le @ Z @ X )
=> ( ( pointwise_le @ Z @ Y4 )
=> ( ( ( minus_minus_list_nat @ X @ Z )
= ( minus_minus_list_nat @ Y4 @ Z ) )
=> ( X = Y4 ) ) ) ) ).
% pairwise_minus_cancel
thf(fact_1075_zero__induct__lemma,axiom,
! [P2: nat > $o,K2: nat,I3: nat] :
( ( P2 @ K2 )
=> ( ! [N: nat] :
( ( P2 @ ( suc @ N ) )
=> ( P2 @ N ) )
=> ( P2 @ ( minus_minus_nat @ K2 @ I3 ) ) ) ) ).
% zero_induct_lemma
thf(fact_1076_sum__list__minus,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( pointwise_le @ Xs2 @ Ys2 )
=> ( ( groups4561878855575611511st_nat @ ( minus_minus_list_nat @ Ys2 @ Xs2 ) )
= ( minus_minus_nat @ ( groups4561878855575611511st_nat @ Ys2 ) @ ( groups4561878855575611511st_nat @ Xs2 ) ) ) ) ).
% sum_list_minus
thf(fact_1077_diff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% diff_add
thf(fact_1078_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_1079_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1080_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1081_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1082_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1083_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1084_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1085_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1086_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1087_Suc__diff__le,axiom,
! [N3: nat,M3: nat] :
( ( ord_less_eq_nat @ N3 @ M3 )
=> ( ( minus_minus_nat @ ( suc @ M3 ) @ N3 )
= ( suc @ ( minus_minus_nat @ M3 @ N3 ) ) ) ) ).
% Suc_diff_le
thf(fact_1088_diff__less__Suc,axiom,
! [M3: nat,N3: nat] : ( ord_less_nat @ ( minus_minus_nat @ M3 @ N3 ) @ ( suc @ M3 ) ) ).
% diff_less_Suc
thf(fact_1089_Suc__diff__Suc,axiom,
! [N3: nat,M3: nat] :
( ( ord_less_nat @ N3 @ M3 )
=> ( ( suc @ ( minus_minus_nat @ M3 @ ( suc @ N3 ) ) )
= ( minus_minus_nat @ M3 @ N3 ) ) ) ).
% Suc_diff_Suc
thf(fact_1090_less__diff__iff,axiom,
! [K2: nat,M3: nat,N3: nat] :
( ( ord_less_eq_nat @ K2 @ M3 )
=> ( ( ord_less_eq_nat @ K2 @ N3 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M3 @ K2 ) @ ( minus_minus_nat @ N3 @ K2 ) )
= ( ord_less_nat @ M3 @ N3 ) ) ) ) ).
% less_diff_iff
thf(fact_1091_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1092_less__diff__conv2,axiom,
! [K2: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K2 ) @ I3 )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I3 @ K2 ) ) ) ) ).
% less_diff_conv2
thf(fact_1093_card__le__sym__Diff,axiom,
! [A3: set_list_nat,B4: set_list_nat] :
( ( finite8100373058378681591st_nat @ A3 )
=> ( ( finite8100373058378681591st_nat @ B4 )
=> ( ( ord_less_eq_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B4 ) )
=> ( ord_less_eq_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ B4 ) ) @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ B4 @ A3 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_1094_card__le__sym__Diff,axiom,
! [A3: set_nat,B4: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B4 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ B4 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B4 @ A3 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_1095_card__le__sym__Diff,axiom,
! [A3: set_a,B4: set_a] :
( ( finite_finite_a @ A3 )
=> ( ( finite_finite_a @ B4 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ A3 ) @ ( finite_card_a @ B4 ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_set_a @ A3 @ B4 ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B4 @ A3 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_1096_card__less__sym__Diff,axiom,
! [A3: set_list_nat,B4: set_list_nat] :
( ( finite8100373058378681591st_nat @ A3 )
=> ( ( finite8100373058378681591st_nat @ B4 )
=> ( ( ord_less_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B4 ) )
=> ( ord_less_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ B4 ) ) @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ B4 @ A3 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_1097_card__less__sym__Diff,axiom,
! [A3: set_nat,B4: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( finite_finite_nat @ B4 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B4 ) )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ B4 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B4 @ A3 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_1098_card__less__sym__Diff,axiom,
! [A3: set_a,B4: set_a] :
( ( finite_finite_a @ A3 )
=> ( ( finite_finite_a @ B4 )
=> ( ( ord_less_nat @ ( finite_card_a @ A3 ) @ ( finite_card_a @ B4 ) )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A3 @ B4 ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B4 @ A3 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_1099_finite__set__plus,axiom,
! [S2: set_list_nat,T: set_list_nat] :
( ( finite8100373058378681591st_nat @ S2 )
=> ( ( finite8100373058378681591st_nat @ T )
=> ( finite8100373058378681591st_nat @ ( plus_p884110394369815071st_nat @ S2 @ T ) ) ) ) ).
% finite_set_plus
thf(fact_1100_finite__set__plus,axiom,
! [S2: set_nat,T: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ( finite_finite_nat @ T )
=> ( finite_finite_nat @ ( plus_plus_set_nat @ S2 @ T ) ) ) ) ).
% finite_set_plus
thf(fact_1101_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_1102_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_1103_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1104_Diff__iff,axiom,
! [C: list_nat,A3: set_list_nat,B4: set_list_nat] :
( ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A3 @ B4 ) )
= ( ( member_list_nat @ C @ A3 )
& ~ ( member_list_nat @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_1105_Diff__iff,axiom,
! [C: nat,A3: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B4 ) )
= ( ( member_nat @ C @ A3 )
& ~ ( member_nat @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_1106_Diff__iff,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B4 ) )
= ( ( member_a @ C @ A3 )
& ~ ( member_a @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_1107_DiffI,axiom,
! [C: list_nat,A3: set_list_nat,B4: set_list_nat] :
( ( member_list_nat @ C @ A3 )
=> ( ~ ( member_list_nat @ C @ B4 )
=> ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A3 @ B4 ) ) ) ) ).
% DiffI
thf(fact_1108_DiffI,axiom,
! [C: nat,A3: set_nat,B4: set_nat] :
( ( member_nat @ C @ A3 )
=> ( ~ ( member_nat @ C @ B4 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B4 ) ) ) ) ).
% DiffI
thf(fact_1109_DiffI,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ A3 )
=> ( ~ ( member_a @ C @ B4 )
=> ( member_a @ C @ ( minus_minus_set_a @ A3 @ B4 ) ) ) ) ).
% DiffI
thf(fact_1110_set__diff__eq,axiom,
( minus_minus_set_a
= ( ^ [A5: set_a,B5: set_a] :
( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A5 )
& ~ ( member_a @ X2 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1111_set__diff__eq,axiom,
( minus_7954133019191499631st_nat
= ( ^ [A5: set_list_nat,B5: set_list_nat] :
( collect_list_nat
@ ^ [X2: list_nat] :
( ( member_list_nat @ X2 @ A5 )
& ~ ( member_list_nat @ X2 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1112_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A5 )
& ~ ( member_nat @ X2 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1113_minus__set__def,axiom,
( minus_minus_set_a
= ( ^ [A5: set_a,B5: set_a] :
( collect_a
@ ( minus_minus_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A5 )
@ ^ [X2: a] : ( member_a @ X2 @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_1114_minus__set__def,axiom,
( minus_7954133019191499631st_nat
= ( ^ [A5: set_list_nat,B5: set_list_nat] :
( collect_list_nat
@ ( minus_1139252259498527702_nat_o
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ A5 )
@ ^ [X2: list_nat] : ( member_list_nat @ X2 @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_1115_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A5 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_1116_diff__commute,axiom,
! [I3: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I3 @ K2 ) @ J ) ) ).
% diff_commute
thf(fact_1117_DiffD2,axiom,
! [C: list_nat,A3: set_list_nat,B4: set_list_nat] :
( ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A3 @ B4 ) )
=> ~ ( member_list_nat @ C @ B4 ) ) ).
% DiffD2
thf(fact_1118_DiffD2,axiom,
! [C: nat,A3: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B4 ) )
=> ~ ( member_nat @ C @ B4 ) ) ).
% DiffD2
thf(fact_1119_DiffD2,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B4 ) )
=> ~ ( member_a @ C @ B4 ) ) ).
% DiffD2
thf(fact_1120_DiffD1,axiom,
! [C: list_nat,A3: set_list_nat,B4: set_list_nat] :
( ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A3 @ B4 ) )
=> ( member_list_nat @ C @ A3 ) ) ).
% DiffD1
thf(fact_1121_DiffD1,axiom,
! [C: nat,A3: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B4 ) )
=> ( member_nat @ C @ A3 ) ) ).
% DiffD1
thf(fact_1122_DiffD1,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B4 ) )
=> ( member_a @ C @ A3 ) ) ).
% DiffD1
thf(fact_1123_DiffE,axiom,
! [C: list_nat,A3: set_list_nat,B4: set_list_nat] :
( ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A3 @ B4 ) )
=> ~ ( ( member_list_nat @ C @ A3 )
=> ( member_list_nat @ C @ B4 ) ) ) ).
% DiffE
thf(fact_1124_DiffE,axiom,
! [C: nat,A3: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B4 ) )
=> ~ ( ( member_nat @ C @ A3 )
=> ( member_nat @ C @ B4 ) ) ) ).
% DiffE
thf(fact_1125_DiffE,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B4 ) )
=> ~ ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B4 ) ) ) ).
% DiffE
thf(fact_1126_add__le__imp__le__diff,axiom,
! [I3: nat,K2: nat,N3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ N3 )
=> ( ord_less_eq_nat @ I3 @ ( minus_minus_nat @ N3 @ K2 ) ) ) ).
% add_le_imp_le_diff
thf(fact_1127_add__le__add__imp__diff__le,axiom,
! [I3: nat,K2: nat,N3: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ N3 )
=> ( ( ord_less_eq_nat @ N3 @ ( plus_plus_nat @ J @ K2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ N3 )
=> ( ( ord_less_eq_nat @ N3 @ ( plus_plus_nat @ J @ K2 ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N3 @ K2 ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1128_diff__shunt__var,axiom,
! [X: set_list_nat,Y4: set_list_nat] :
( ( ( minus_7954133019191499631st_nat @ X @ Y4 )
= bot_bot_set_list_nat )
= ( ord_le6045566169113846134st_nat @ X @ Y4 ) ) ).
% diff_shunt_var
thf(fact_1129_diff__shunt__var,axiom,
! [X: set_a,Y4: set_a] :
( ( ( minus_minus_set_a @ X @ Y4 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X @ Y4 ) ) ).
% diff_shunt_var
thf(fact_1130_set__update__distinct,axiom,
! [Xs2: list_nat,N3: nat,X: nat] :
( ( distinct_nat @ Xs2 )
=> ( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( set_nat2 @ ( list_update_nat @ Xs2 @ N3 @ X ) )
= ( insert_nat @ X @ ( minus_minus_set_nat @ ( set_nat2 @ Xs2 ) @ ( insert_nat @ ( nth_nat @ Xs2 @ N3 ) @ bot_bot_set_nat ) ) ) ) ) ) ).
% set_update_distinct
thf(fact_1131_set__update__distinct,axiom,
! [Xs2: list_list_nat,N3: nat,X: list_nat] :
( ( distinct_list_nat @ Xs2 )
=> ( ( ord_less_nat @ N3 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ( set_list_nat2 @ ( list_update_list_nat @ Xs2 @ N3 @ X ) )
= ( insert_list_nat @ X @ ( minus_7954133019191499631st_nat @ ( set_list_nat2 @ Xs2 ) @ ( insert_list_nat @ ( nth_list_nat @ Xs2 @ N3 ) @ bot_bot_set_list_nat ) ) ) ) ) ) ).
% set_update_distinct
thf(fact_1132_set__update__distinct,axiom,
! [Xs2: list_a,N3: nat,X: a] :
( ( distinct_a @ Xs2 )
=> ( ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs2 ) )
=> ( ( set_a2 @ ( list_update_a @ Xs2 @ N3 @ X ) )
= ( insert_a @ X @ ( minus_minus_set_a @ ( set_a2 @ Xs2 ) @ ( insert_a @ ( nth_a @ Xs2 @ N3 ) @ bot_bot_set_a ) ) ) ) ) ) ).
% set_update_distinct
thf(fact_1133_insert__iff,axiom,
! [A: list_nat,B: list_nat,A3: set_list_nat] :
( ( member_list_nat @ A @ ( insert_list_nat @ B @ A3 ) )
= ( ( A = B )
| ( member_list_nat @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_1134_insert__iff,axiom,
! [A: nat,B: nat,A3: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A3 ) )
= ( ( A = B )
| ( member_nat @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_1135_insert__iff,axiom,
! [A: a,B: a,A3: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A3 ) )
= ( ( A = B )
| ( member_a @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_1136_insertCI,axiom,
! [A: list_nat,B4: set_list_nat,B: list_nat] :
( ( ~ ( member_list_nat @ A @ B4 )
=> ( A = B ) )
=> ( member_list_nat @ A @ ( insert_list_nat @ B @ B4 ) ) ) ).
% insertCI
thf(fact_1137_insertCI,axiom,
! [A: nat,B4: set_nat,B: nat] :
( ( ~ ( member_nat @ A @ B4 )
=> ( A = B ) )
=> ( member_nat @ A @ ( insert_nat @ B @ B4 ) ) ) ).
% insertCI
thf(fact_1138_insertCI,axiom,
! [A: a,B4: set_a,B: a] :
( ( ~ ( member_a @ A @ B4 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B4 ) ) ) ).
% insertCI
thf(fact_1139_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_1140_singletonI,axiom,
! [A: list_nat] : ( member_list_nat @ A @ ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ).
% singletonI
thf(fact_1141_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_1142_insert__subset,axiom,
! [X: nat,A3: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A3 ) @ B4 )
= ( ( member_nat @ X @ B4 )
& ( ord_less_eq_set_nat @ A3 @ B4 ) ) ) ).
% insert_subset
thf(fact_1143_insert__subset,axiom,
! [X: list_nat,A3: set_list_nat,B4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( insert_list_nat @ X @ A3 ) @ B4 )
= ( ( member_list_nat @ X @ B4 )
& ( ord_le6045566169113846134st_nat @ A3 @ B4 ) ) ) ).
% insert_subset
thf(fact_1144_insert__subset,axiom,
! [X: a,A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A3 ) @ B4 )
= ( ( member_a @ X @ B4 )
& ( ord_less_eq_set_a @ A3 @ B4 ) ) ) ).
% insert_subset
thf(fact_1145_finite__insert,axiom,
! [A: list_nat,A3: set_list_nat] :
( ( finite8100373058378681591st_nat @ ( insert_list_nat @ A @ A3 ) )
= ( finite8100373058378681591st_nat @ A3 ) ) ).
% finite_insert
thf(fact_1146_finite__insert,axiom,
! [A: nat,A3: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A @ A3 ) )
= ( finite_finite_nat @ A3 ) ) ).
% finite_insert
thf(fact_1147_finite__insert,axiom,
! [A: a,A3: set_a] :
( ( finite_finite_a @ ( insert_a @ A @ A3 ) )
= ( finite_finite_a @ A3 ) ) ).
% finite_insert
thf(fact_1148_insert__Diff1,axiom,
! [X: list_nat,B4: set_list_nat,A3: set_list_nat] :
( ( member_list_nat @ X @ B4 )
=> ( ( minus_7954133019191499631st_nat @ ( insert_list_nat @ X @ A3 ) @ B4 )
= ( minus_7954133019191499631st_nat @ A3 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_1149_insert__Diff1,axiom,
! [X: nat,B4: set_nat,A3: set_nat] :
( ( member_nat @ X @ B4 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A3 ) @ B4 )
= ( minus_minus_set_nat @ A3 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_1150_insert__Diff1,axiom,
! [X: a,B4: set_a,A3: set_a] :
( ( member_a @ X @ B4 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A3 ) @ B4 )
= ( minus_minus_set_a @ A3 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_1151_Diff__insert0,axiom,
! [X: list_nat,A3: set_list_nat,B4: set_list_nat] :
( ~ ( member_list_nat @ X @ A3 )
=> ( ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ X @ B4 ) )
= ( minus_7954133019191499631st_nat @ A3 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_1152_Diff__insert0,axiom,
! [X: nat,A3: set_nat,B4: set_nat] :
( ~ ( member_nat @ X @ A3 )
=> ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ X @ B4 ) )
= ( minus_minus_set_nat @ A3 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_1153_Diff__insert0,axiom,
! [X: a,A3: set_a,B4: set_a] :
( ~ ( member_a @ X @ A3 )
=> ( ( minus_minus_set_a @ A3 @ ( insert_a @ X @ B4 ) )
= ( minus_minus_set_a @ A3 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_1154_singleton__conv,axiom,
! [A: nat] :
( ( collect_nat
@ ^ [X2: nat] : ( X2 = A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_1155_singleton__conv,axiom,
! [A: list_nat] :
( ( collect_list_nat
@ ^ [X2: list_nat] : ( X2 = A ) )
= ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ).
% singleton_conv
thf(fact_1156_singleton__conv,axiom,
! [A: a] :
( ( collect_a
@ ^ [X2: a] : ( X2 = A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv
thf(fact_1157_singleton__conv2,axiom,
! [A: nat] :
( ( collect_nat
@ ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 )
@ A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_1158_singleton__conv2,axiom,
! [A: list_nat] :
( ( collect_list_nat
@ ( ^ [Y5: list_nat,Z2: list_nat] : ( Y5 = Z2 )
@ A ) )
= ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ).
% singleton_conv2
thf(fact_1159_singleton__conv2,axiom,
! [A: a] :
( ( collect_a
@ ( ^ [Y5: a,Z2: a] : ( Y5 = Z2 )
@ A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv2
thf(fact_1160_singleton__insert__inj__eq_H,axiom,
! [A: list_nat,A3: set_list_nat,B: list_nat] :
( ( ( insert_list_nat @ A @ A3 )
= ( insert_list_nat @ B @ bot_bot_set_list_nat ) )
= ( ( A = B )
& ( ord_le6045566169113846134st_nat @ A3 @ ( insert_list_nat @ B @ bot_bot_set_list_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1161_singleton__insert__inj__eq_H,axiom,
! [A: a,A3: set_a,B: a] :
( ( ( insert_a @ A @ A3 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1162_singleton__insert__inj__eq,axiom,
! [B: list_nat,A: list_nat,A3: set_list_nat] :
( ( ( insert_list_nat @ B @ bot_bot_set_list_nat )
= ( insert_list_nat @ A @ A3 ) )
= ( ( A = B )
& ( ord_le6045566169113846134st_nat @ A3 @ ( insert_list_nat @ B @ bot_bot_set_list_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1163_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A3: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A3 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1164_insert__Diff__single,axiom,
! [A: list_nat,A3: set_list_nat] :
( ( insert_list_nat @ A @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) )
= ( insert_list_nat @ A @ A3 ) ) ).
% insert_Diff_single
thf(fact_1165_insert__Diff__single,axiom,
! [A: a,A3: set_a] :
( ( insert_a @ A @ ( minus_minus_set_a @ A3 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( insert_a @ A @ A3 ) ) ).
% insert_Diff_single
thf(fact_1166_finite__Diff__insert,axiom,
! [A3: set_list_nat,A: list_nat,B4: set_list_nat] :
( ( finite8100373058378681591st_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ A @ B4 ) ) )
= ( finite8100373058378681591st_nat @ ( minus_7954133019191499631st_nat @ A3 @ B4 ) ) ) ).
% finite_Diff_insert
thf(fact_1167_finite__Diff__insert,axiom,
! [A3: set_nat,A: nat,B4: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A @ B4 ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ B4 ) ) ) ).
% finite_Diff_insert
thf(fact_1168_finite__Diff__insert,axiom,
! [A3: set_a,A: a,B4: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ A @ B4 ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A3 @ B4 ) ) ) ).
% finite_Diff_insert
thf(fact_1169_list_Osimps_I15_J,axiom,
! [X21: a,X222: list_a] :
( ( set_a2 @ ( cons_a @ X21 @ X222 ) )
= ( insert_a @ X21 @ ( set_a2 @ X222 ) ) ) ).
% list.simps(15)
thf(fact_1170_list_Osimps_I15_J,axiom,
! [X21: nat,X222: list_nat] :
( ( set_nat2 @ ( cons_nat @ X21 @ X222 ) )
= ( insert_nat @ X21 @ ( set_nat2 @ X222 ) ) ) ).
% list.simps(15)
thf(fact_1171_Inf__fin_Osingleton,axiom,
! [X: list_nat] :
( ( lattic5191180550204456963st_nat @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) )
= X ) ).
% Inf_fin.singleton
thf(fact_1172_Sup__fin_Osingleton,axiom,
! [X: list_nat] :
( ( lattic6411832703407573737st_nat @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) )
= X ) ).
% Sup_fin.singleton
thf(fact_1173_card__insert__disjoint,axiom,
! [A3: set_list_nat,X: list_nat] :
( ( finite8100373058378681591st_nat @ A3 )
=> ( ~ ( member_list_nat @ X @ A3 )
=> ( ( finite_card_list_nat @ ( insert_list_nat @ X @ A3 ) )
= ( suc @ ( finite_card_list_nat @ A3 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_1174_card__insert__disjoint,axiom,
! [A3: set_nat,X: nat] :
( ( finite_finite_nat @ A3 )
=> ( ~ ( member_nat @ X @ A3 )
=> ( ( finite_card_nat @ ( insert_nat @ X @ A3 ) )
= ( suc @ ( finite_card_nat @ A3 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_1175_card__insert__disjoint,axiom,
! [A3: set_a,X: a] :
( ( finite_finite_a @ A3 )
=> ( ~ ( member_a @ X @ A3 )
=> ( ( finite_card_a @ ( insert_a @ X @ A3 ) )
= ( suc @ ( finite_card_a @ A3 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_1176_insert__Diff__if,axiom,
! [X: list_nat,B4: set_list_nat,A3: set_list_nat] :
( ( ( member_list_nat @ X @ B4 )
=> ( ( minus_7954133019191499631st_nat @ ( insert_list_nat @ X @ A3 ) @ B4 )
= ( minus_7954133019191499631st_nat @ A3 @ B4 ) ) )
& ( ~ ( member_list_nat @ X @ B4 )
=> ( ( minus_7954133019191499631st_nat @ ( insert_list_nat @ X @ A3 ) @ B4 )
= ( insert_list_nat @ X @ ( minus_7954133019191499631st_nat @ A3 @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1177_insert__Diff__if,axiom,
! [X: nat,B4: set_nat,A3: set_nat] :
( ( ( member_nat @ X @ B4 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A3 ) @ B4 )
= ( minus_minus_set_nat @ A3 @ B4 ) ) )
& ( ~ ( member_nat @ X @ B4 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A3 ) @ B4 )
= ( insert_nat @ X @ ( minus_minus_set_nat @ A3 @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1178_insert__Diff__if,axiom,
! [X: a,B4: set_a,A3: set_a] :
( ( ( member_a @ X @ B4 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A3 ) @ B4 )
= ( minus_minus_set_a @ A3 @ B4 ) ) )
& ( ~ ( member_a @ X @ B4 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A3 ) @ B4 )
= ( insert_a @ X @ ( minus_minus_set_a @ A3 @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1179_insert__subsetI,axiom,
! [X: nat,A3: set_nat,X8: set_nat] :
( ( member_nat @ X @ A3 )
=> ( ( ord_less_eq_set_nat @ X8 @ A3 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ X @ X8 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_1180_insert__subsetI,axiom,
! [X: list_nat,A3: set_list_nat,X8: set_list_nat] :
( ( member_list_nat @ X @ A3 )
=> ( ( ord_le6045566169113846134st_nat @ X8 @ A3 )
=> ( ord_le6045566169113846134st_nat @ ( insert_list_nat @ X @ X8 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_1181_insert__subsetI,axiom,
! [X: a,A3: set_a,X8: set_a] :
( ( member_a @ X @ A3 )
=> ( ( ord_less_eq_set_a @ X8 @ A3 )
=> ( ord_less_eq_set_a @ ( insert_a @ X @ X8 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_1182_insert__mono,axiom,
! [C2: set_list_nat,D3: set_list_nat,A: list_nat] :
( ( ord_le6045566169113846134st_nat @ C2 @ D3 )
=> ( ord_le6045566169113846134st_nat @ ( insert_list_nat @ A @ C2 ) @ ( insert_list_nat @ A @ D3 ) ) ) ).
% insert_mono
thf(fact_1183_insert__mono,axiom,
! [C2: set_a,D3: set_a,A: a] :
( ( ord_less_eq_set_a @ C2 @ D3 )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C2 ) @ ( insert_a @ A @ D3 ) ) ) ).
% insert_mono
thf(fact_1184_subset__insert,axiom,
! [X: nat,A3: set_nat,B4: set_nat] :
( ~ ( member_nat @ X @ A3 )
=> ( ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ X @ B4 ) )
= ( ord_less_eq_set_nat @ A3 @ B4 ) ) ) ).
% subset_insert
thf(fact_1185_subset__insert,axiom,
! [X: list_nat,A3: set_list_nat,B4: set_list_nat] :
( ~ ( member_list_nat @ X @ A3 )
=> ( ( ord_le6045566169113846134st_nat @ A3 @ ( insert_list_nat @ X @ B4 ) )
= ( ord_le6045566169113846134st_nat @ A3 @ B4 ) ) ) ).
% subset_insert
thf(fact_1186_subset__insert,axiom,
! [X: a,A3: set_a,B4: set_a] :
( ~ ( member_a @ X @ A3 )
=> ( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X @ B4 ) )
= ( ord_less_eq_set_a @ A3 @ B4 ) ) ) ).
% subset_insert
thf(fact_1187_subset__insertI,axiom,
! [B4: set_list_nat,A: list_nat] : ( ord_le6045566169113846134st_nat @ B4 @ ( insert_list_nat @ A @ B4 ) ) ).
% subset_insertI
thf(fact_1188_subset__insertI,axiom,
! [B4: set_a,A: a] : ( ord_less_eq_set_a @ B4 @ ( insert_a @ A @ B4 ) ) ).
% subset_insertI
thf(fact_1189_subset__insertI2,axiom,
! [A3: set_list_nat,B4: set_list_nat,B: list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ B4 )
=> ( ord_le6045566169113846134st_nat @ A3 @ ( insert_list_nat @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_1190_subset__insertI2,axiom,
! [A3: set_a,B4: set_a,B: a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
=> ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_1191_Collect__conv__if,axiom,
! [P2: nat > $o,A: nat] :
( ( ( P2 @ A )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( X2 = A )
& ( P2 @ X2 ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P2 @ A )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( X2 = A )
& ( P2 @ X2 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_1192_Collect__conv__if,axiom,
! [P2: list_nat > $o,A: list_nat] :
( ( ( P2 @ A )
=> ( ( collect_list_nat
@ ^ [X2: list_nat] :
( ( X2 = A )
& ( P2 @ X2 ) ) )
= ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) )
& ( ~ ( P2 @ A )
=> ( ( collect_list_nat
@ ^ [X2: list_nat] :
( ( X2 = A )
& ( P2 @ X2 ) ) )
= bot_bot_set_list_nat ) ) ) ).
% Collect_conv_if
thf(fact_1193_Collect__conv__if,axiom,
! [P2: a > $o,A: a] :
( ( ( P2 @ A )
=> ( ( collect_a
@ ^ [X2: a] :
( ( X2 = A )
& ( P2 @ X2 ) ) )
= ( insert_a @ A @ bot_bot_set_a ) ) )
& ( ~ ( P2 @ A )
=> ( ( collect_a
@ ^ [X2: a] :
( ( X2 = A )
& ( P2 @ X2 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if
thf(fact_1194_triangle__Suc,axiom,
! [N3: nat] :
( ( nat_triangle @ ( suc @ N3 ) )
= ( plus_plus_nat @ ( nat_triangle @ N3 ) @ ( suc @ N3 ) ) ) ).
% triangle_Suc
thf(fact_1195_Least__eq__0,axiom,
! [P2: nat > $o] :
( ( P2 @ zero_zero_nat )
=> ( ( ord_Least_nat @ P2 )
= zero_zero_nat ) ) ).
% Least_eq_0
thf(fact_1196_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1197_le0,axiom,
! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N3 ) ).
% le0
thf(fact_1198_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1199_neq0__conv,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).
% neq0_conv
thf(fact_1200_less__nat__zero__code,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1201_Nat_Oadd__0__right,axiom,
! [M3: nat] :
( ( plus_plus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% Nat.add_0_right
thf(fact_1202_add__is__0,axiom,
! [M3: nat,N3: nat] :
( ( ( plus_plus_nat @ M3 @ N3 )
= zero_zero_nat )
= ( ( M3 = zero_zero_nat )
& ( N3 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1203_diff__self__eq__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ M3 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1204_diff__0__eq__0,axiom,
! [N3: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N3 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1205_less__Suc0,axiom,
! [N3: nat] :
( ( ord_less_nat @ N3 @ ( suc @ zero_zero_nat ) )
= ( N3 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1206_zero__less__Suc,axiom,
! [N3: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N3 ) ) ).
% zero_less_Suc
thf(fact_1207_add__gr__0,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M3 @ N3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M3 )
| ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% add_gr_0
thf(fact_1208_diff__is__0__eq,axiom,
! [M3: nat,N3: nat] :
( ( ( minus_minus_nat @ M3 @ N3 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M3 @ N3 ) ) ).
% diff_is_0_eq
thf(fact_1209_diff__is__0__eq_H,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
=> ( ( minus_minus_nat @ M3 @ N3 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1210_zero__less__diff,axiom,
! [N3: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N3 @ M3 ) )
= ( ord_less_nat @ M3 @ N3 ) ) ).
% zero_less_diff
thf(fact_1211_Suc__pred,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( suc @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) )
= N3 ) ) ).
% Suc_pred
thf(fact_1212_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top_nat
@ ^ [X2: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ X2 )
@ ^ [X2: nat,Y3: nat] : ( ord_less_nat @ Y3 @ X2 )
@ zero_zero_nat ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_1213_diffs0__imp__equal,axiom,
! [M3: nat,N3: nat] :
( ( ( minus_minus_nat @ M3 @ N3 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N3 @ M3 )
= zero_zero_nat )
=> ( M3 = N3 ) ) ) ).
% diffs0_imp_equal
thf(fact_1214_minus__nat_Odiff__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% minus_nat.diff_0
thf(fact_1215_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N: nat] :
( X
!= ( suc @ N ) ) ) ).
% list_decode.cases
thf(fact_1216_add__eq__self__zero,axiom,
! [M3: nat,N3: nat] :
( ( ( plus_plus_nat @ M3 @ N3 )
= M3 )
=> ( N3 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1217_plus__nat_Oadd__0,axiom,
! [N3: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N3 )
= N3 ) ).
% plus_nat.add_0
thf(fact_1218_Least__Suc,axiom,
! [P2: nat > $o,N3: nat] :
( ( P2 @ N3 )
=> ( ~ ( P2 @ zero_zero_nat )
=> ( ( ord_Least_nat @ P2 )
= ( suc
@ ( ord_Least_nat
@ ^ [M2: nat] : ( P2 @ ( suc @ M2 ) ) ) ) ) ) ) ).
% Least_Suc
thf(fact_1219_Least__Suc2,axiom,
! [P2: nat > $o,N3: nat,Q2: nat > $o,M3: nat] :
( ( P2 @ N3 )
=> ( ( Q2 @ M3 )
=> ( ~ ( P2 @ zero_zero_nat )
=> ( ! [K3: nat] :
( ( P2 @ ( suc @ K3 ) )
= ( Q2 @ K3 ) )
=> ( ( ord_Least_nat @ P2 )
= ( suc @ ( ord_Least_nat @ Q2 ) ) ) ) ) ) ) ).
% Least_Suc2
thf(fact_1220_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1221_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1222_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1223_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1224_old_Onat_Oexhaust,axiom,
! [Y4: nat] :
( ( Y4 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y4
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1225_nat__induct,axiom,
! [P2: nat > $o,N3: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N: nat] :
( ( P2 @ N )
=> ( P2 @ ( suc @ N ) ) )
=> ( P2 @ N3 ) ) ) ).
% nat_induct
thf(fact_1226_diff__induct,axiom,
! [P2: nat > nat > $o,M3: nat,N3: nat] :
( ! [X3: nat] : ( P2 @ X3 @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P2 @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X3: nat,Y2: nat] :
( ( P2 @ X3 @ Y2 )
=> ( P2 @ ( suc @ X3 ) @ ( suc @ Y2 ) ) )
=> ( P2 @ M3 @ N3 ) ) ) ) ).
% diff_induct
thf(fact_1227_zero__induct,axiom,
! [P2: nat > $o,K2: nat] :
( ( P2 @ K2 )
=> ( ! [N: nat] :
( ( P2 @ ( suc @ N ) )
=> ( P2 @ N ) )
=> ( P2 @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1228_Suc__neq__Zero,axiom,
! [M3: nat] :
( ( suc @ M3 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1229_Zero__neq__Suc,axiom,
! [M3: nat] :
( zero_zero_nat
!= ( suc @ M3 ) ) ).
% Zero_neq_Suc
thf(fact_1230_Zero__not__Suc,axiom,
! [M3: nat] :
( zero_zero_nat
!= ( suc @ M3 ) ) ).
% Zero_not_Suc
thf(fact_1231_not0__implies__Suc,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
=> ? [M6: nat] :
( N3
= ( suc @ M6 ) ) ) ).
% not0_implies_Suc
thf(fact_1232_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1233_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1234_gr0I,axiom,
! [N3: nat] :
( ( N3 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).
% gr0I
thf(fact_1235_not__gr0,axiom,
! [N3: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
= ( N3 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1236_not__less0,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% not_less0
thf(fact_1237_less__zeroE,axiom,
! [N3: nat] :
~ ( ord_less_nat @ N3 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1238_gr__implies__not0,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( N3 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1239_infinite__descent0,axiom,
! [P2: nat > $o,N3: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ~ ( P2 @ N )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N )
& ~ ( P2 @ M4 ) ) ) )
=> ( P2 @ N3 ) ) ) ).
% infinite_descent0
thf(fact_1240_less__eq__nat_Osimps_I1_J,axiom,
! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N3 ) ).
% less_eq_nat.simps(1)
thf(fact_1241_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1242_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1243_le__0__eq,axiom,
! [N3: nat] :
( ( ord_less_eq_nat @ N3 @ zero_zero_nat )
= ( N3 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1244_diff__add__0,axiom,
! [N3: nat,M3: nat] :
( ( minus_minus_nat @ N3 @ ( plus_plus_nat @ N3 @ M3 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1245_diff__less,axiom,
! [N3: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ord_less_nat @ ( minus_minus_nat @ M3 @ N3 ) @ M3 ) ) ) ).
% diff_less
thf(fact_1246_less__imp__add__positive,axiom,
! [I3: nat,J: nat] :
( ( ord_less_nat @ I3 @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I3 @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1247_add__is__1,axiom,
! [M3: nat,N3: nat] :
( ( ( plus_plus_nat @ M3 @ N3 )
= ( suc @ zero_zero_nat ) )
= ( ( ( M3
= ( suc @ zero_zero_nat ) )
& ( N3 = zero_zero_nat ) )
| ( ( M3 = zero_zero_nat )
& ( N3
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1248_one__is__add,axiom,
! [M3: nat,N3: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M3 @ N3 ) )
= ( ( ( M3
= ( suc @ zero_zero_nat ) )
& ( N3 = zero_zero_nat ) )
| ( ( M3 = zero_zero_nat )
& ( N3
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1249_ex__least__nat__le,axiom,
! [P2: nat > $o,N3: nat] :
( ( P2 @ N3 )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N3 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ~ ( P2 @ I4 ) )
& ( P2 @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1250_less__Suc__eq__0__disj,axiom,
! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ ( suc @ N3 ) )
= ( ( M3 = zero_zero_nat )
| ? [J3: nat] :
( ( M3
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N3 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1251_gr0__implies__Suc,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ? [M6: nat] :
( N3
= ( suc @ M6 ) ) ) ).
% gr0_implies_Suc
thf(fact_1252_All__less__Suc2,axiom,
! [N3: nat,P2: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N3 ) )
=> ( P2 @ I2 ) ) )
= ( ( P2 @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N3 )
=> ( P2 @ ( suc @ I2 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1253_gr0__conv__Suc,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
= ( ? [M2: nat] :
( N3
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1254_Ex__less__Suc2,axiom,
! [N3: nat,P2: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N3 ) )
& ( P2 @ I2 ) ) )
= ( ( P2 @ zero_zero_nat )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N3 )
& ( P2 @ ( suc @ I2 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1255_ex__least__nat__less,axiom,
! [P2: nat > $o,N3: nat] :
( ( P2 @ N3 )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N3 )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K3 )
=> ~ ( P2 @ I4 ) )
& ( P2 @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1256_diff__Suc__less,axiom,
! [N3: nat,I3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ord_less_nat @ ( minus_minus_nat @ N3 @ ( suc @ I3 ) ) @ N3 ) ) ).
% diff_Suc_less
thf(fact_1257_nat__diff__split,axiom,
! [P2: nat > $o,A: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P2 @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
=> ( P2 @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_1258_nat__diff__split__asm,axiom,
! [P2: nat > $o,A: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P2 @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
& ~ ( P2 @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1259_card__less,axiom,
! [M: set_nat,I3: nat] :
( ( member_nat @ zero_zero_nat @ M )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K: nat] :
( ( member_nat @ K @ M )
& ( ord_less_nat @ K @ ( suc @ I3 ) ) ) ) )
!= zero_zero_nat ) ) ).
% card_less
thf(fact_1260_card__less__Suc,axiom,
! [M: set_nat,I3: nat] :
( ( member_nat @ zero_zero_nat @ M )
=> ( ( suc
@ ( finite_card_nat
@ ( collect_nat
@ ^ [K: nat] :
( ( member_nat @ ( suc @ K ) @ M )
& ( ord_less_nat @ K @ I3 ) ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K: nat] :
( ( member_nat @ K @ M )
& ( ord_less_nat @ K @ ( suc @ I3 ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_1261_card__less__Suc2,axiom,
! [M: set_nat,I3: nat] :
( ~ ( member_nat @ zero_zero_nat @ M )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K: nat] :
( ( member_nat @ ( suc @ K ) @ M )
& ( ord_less_nat @ K @ I3 ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K: nat] :
( ( member_nat @ K @ M )
& ( ord_less_nat @ K @ ( suc @ I3 ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_1262_power__Suc__0,axiom,
! [N3: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N3 )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_1263_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M3: nat] :
( ( ( power_power_nat @ X @ M3 )
= ( suc @ zero_zero_nat ) )
= ( ( M3 = zero_zero_nat )
| ( X
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_1264_nat__zero__less__power__iff,axiom,
! [X: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N3 = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1265_nat__power__less__imp__less,axiom,
! [I3: nat,M3: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ I3 )
=> ( ( ord_less_nat @ ( power_power_nat @ I3 @ M3 ) @ ( power_power_nat @ I3 @ N3 ) )
=> ( ord_less_nat @ M3 @ N3 ) ) ) ).
% nat_power_less_imp_less
thf(fact_1266_nat__one__le__power,axiom,
! [I3: nat,N3: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I3 )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I3 @ N3 ) ) ) ).
% nat_one_le_power
% Conjectures (1)
thf(conj_0,conjecture,
( ( pointwise_less @ ( delete @ ua2 ) @ ( delete @ v2 ) )
= ( pointwise_less @ ua2 @ v2 ) ) ).
%------------------------------------------------------------------------------