TPTP Problem File: ITP077^1.p
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%------------------------------------------------------------------------------
% File : ITP077^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Hilbert_Function problem prob_146__11622158_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Hilbert_Function/prob_146__11622158_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% Syntax : Number of formulae : 540 ( 127 unt; 186 typ; 0 def)
% Number of atoms : 1058 ( 380 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 3365 ( 86 ~; 11 |; 63 &;2650 @)
% ( 0 <=>; 555 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 7 avg)
% Number of types : 44 ( 43 usr)
% Number of type conns : 402 ( 402 >; 0 *; 0 +; 0 <<)
% Number of symbols : 146 ( 143 usr; 8 con; 0-3 aty)
% Number of variables : 1068 ( 87 ^; 916 !; 65 ?;1068 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:37:07.615
%------------------------------------------------------------------------------
% Could-be-implicit typings (43)
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
set_list_list_list_a: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Multiset__Omultiset_It__List__Olist_Itf__a_J_J_J,type,
multis971982480list_a: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
list_set_list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
set_list_set_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__Set__Oset_Itf__a_J_J_J_J,type,
set_list_list_set_a: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
list_set_set_list_a: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__List__Olist_It__Set__Oset_Itf__a_J_J_J_J,type,
list_set_list_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J_J,type,
set_list_set_set_a: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J_J,type,
list_set_set_set_a: $tType ).
thf(ty_n_t__List__Olist_It__Multiset__Omultiset_It__List__Olist_Itf__a_J_J_J,type,
list_multiset_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Multiset__Omultiset_It__List__Olist_Itf__a_J_J_J,type,
set_multiset_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Multiset__Omultiset_Itf__a_J_J_J,type,
set_list_multiset_a: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
multiset_set_list_a: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Multiset__Omultiset_Itf__a_J_J_J,type,
list_set_multiset_a: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
multiset_set_set_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
list_list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
set_list_list_a: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J,type,
multiset_multiset_a: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
list_set_list_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
list_list_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
set_set_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
set_list_set_a: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
list_set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
set_set_set_a: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__List__Olist_Itf__a_J_J,type,
multiset_list_a: $tType ).
thf(ty_n_t__List__Olist_It__Multiset__Omultiset_Itf__a_J_J,type,
list_multiset_a: $tType ).
thf(ty_n_t__Set__Oset_It__Multiset__Omultiset_Itf__a_J_J,type,
set_multiset_a: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
multiset_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
list_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__List__Olist_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__List__Olist_It__Set__Oset_Itf__a_J_J,type,
list_set_a: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Nat__Onat_J,type,
multiset_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Multiset__Omultiset_Itf__a_J,type,
multiset_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 (143)
thf(sy_c_Fun_Obij__betw_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
bij_be94573046list_a: ( list_a > list_a ) > set_list_a > set_list_a > $o ).
thf(sy_c_Fun_Obij__betw_001t__List__Olist_Itf__a_J_001tf__a,type,
bij_betw_list_a_a: ( list_a > a ) > set_list_a > set_a > $o ).
thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).
thf(sy_c_Fun_Obij__betw_001tf__a_001t__List__Olist_Itf__a_J,type,
bij_betw_a_list_a: ( a > list_a ) > set_a > set_list_a > $o ).
thf(sy_c_Fun_Obij__betw_001tf__a_001tf__a,type,
bij_betw_a_a: ( a > a ) > set_a > set_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
inj_on_list_a_list_a: ( list_a > list_a ) > set_list_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_Itf__a_J_001tf__a,type,
inj_on_list_a_a: ( list_a > a ) > set_list_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001t__List__Olist_Itf__a_J,type,
inj_on_a_list_a: ( a > list_a ) > set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001tf__a,type,
inj_on_a_a: ( a > a ) > set_a > $o ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Multiset__Omultiset_It__List__Olist_Itf__a_J_J,type,
groups1271887971list_a: list_multiset_list_a > multiset_list_a ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Multiset__Omultiset_Itf__a_J,type,
groups1592617181iset_a: list_multiset_a > multiset_a ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Nat__Onat,type,
groups921905271st_nat: list_nat > nat ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001tf__a,type,
groups1792256535list_a: list_a > a ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
hilber921249868list_a: set_list_a > ( list_a > list_a ) > list_a > list_a ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__List__Olist_Itf__a_J_001tf__a,type,
hilber2125729734st_a_a: set_list_a > ( list_a > a ) > a > list_a ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Nat__Onat_001t__Nat__Onat,type,
hilber815131374at_nat: set_nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001tf__a_001t__List__Olist_Itf__a_J,type,
hilber165648082list_a: set_a > ( a > list_a ) > list_a > a ).
thf(sy_c_Hilbert__Choice_Oinv__into_001tf__a_001tf__a,type,
hilbert_inv_into_a_a: set_a > ( a > a ) > a > a ).
thf(sy_c_Hilbert__Function__Mirabelle__sksumwvvhs_Odirect__decomp_001t__Multiset__Omultiset_Itf__a_J,type,
hilber2024005914iset_a: set_multiset_a > list_set_multiset_a > $o ).
thf(sy_c_Hilbert__Function__Mirabelle__sksumwvvhs_Odirect__decomp_001t__Nat__Onat,type,
hilber1014902394mp_nat: set_nat > list_set_nat > $o ).
thf(sy_c_Hilbert__Function__Mirabelle__sksumwvvhs_Odirect__decomp_001tf__a,type,
hilber2037636820comp_a: set_a > list_set_a > $o ).
thf(sy_c_List_Olist__ex_001t__Set__Oset_Itf__a_J,type,
list_ex_set_a: ( set_a > $o ) > list_set_a > $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__Set__Oset_Itf__a_J,type,
list_update_set_a: list_set_a > nat > set_a > list_set_a ).
thf(sy_c_List_Olist__update_001tf__a,type,
list_update_a: list_a > nat > a > list_a ).
thf(sy_c_List_Olistset_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
listset_list_list_a: list_set_list_list_a > set_list_list_list_a ).
thf(sy_c_List_Olistset_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
listset_list_set_a: list_set_list_set_a > set_list_list_set_a ).
thf(sy_c_List_Olistset_001t__List__Olist_Itf__a_J,type,
listset_list_a: list_set_list_a > set_list_list_a ).
thf(sy_c_List_Olistset_001t__Multiset__Omultiset_Itf__a_J,type,
listset_multiset_a: list_set_multiset_a > set_list_multiset_a ).
thf(sy_c_List_Olistset_001t__Nat__Onat,type,
listset_nat: list_set_nat > set_list_nat ).
thf(sy_c_List_Olistset_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
listset_set_list_a: list_set_set_list_a > set_list_set_list_a ).
thf(sy_c_List_Olistset_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
listset_set_set_a: list_set_set_set_a > set_list_set_set_a ).
thf(sy_c_List_Olistset_001t__Set__Oset_Itf__a_J,type,
listset_set_a: list_set_set_a > set_list_set_a ).
thf(sy_c_List_Olistset_001tf__a,type,
listset_a: list_set_a > set_list_a ).
thf(sy_c_List_Onth_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
nth_list_list_a: list_list_list_a > nat > list_list_a ).
thf(sy_c_List_Onth_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
nth_list_set_a: list_list_set_a > nat > list_set_a ).
thf(sy_c_List_Onth_001t__List__Olist_Itf__a_J,type,
nth_list_a: list_list_a > nat > list_a ).
thf(sy_c_List_Onth_001t__Multiset__Omultiset_Itf__a_J,type,
nth_multiset_a: list_multiset_a > nat > multiset_a ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
nth_set_list_list_a: list_set_list_list_a > nat > set_list_list_a ).
thf(sy_c_List_Onth_001t__Set__Oset_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
nth_set_list_set_a: list_set_list_set_a > nat > set_list_set_a ).
thf(sy_c_List_Onth_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
nth_set_list_a: list_set_list_a > nat > set_list_a ).
thf(sy_c_List_Onth_001t__Set__Oset_It__Multiset__Omultiset_Itf__a_J_J,type,
nth_set_multiset_a: list_set_multiset_a > nat > set_multiset_a ).
thf(sy_c_List_Onth_001t__Set__Oset_It__Nat__Onat_J,type,
nth_set_nat: list_set_nat > nat > set_nat ).
thf(sy_c_List_Onth_001t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
nth_set_set_list_a: list_set_set_list_a > nat > set_set_list_a ).
thf(sy_c_List_Onth_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
nth_set_set_set_a: list_set_set_set_a > nat > set_set_set_a ).
thf(sy_c_List_Onth_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
nth_set_set_a: list_set_set_a > nat > set_set_a ).
thf(sy_c_List_Onth_001t__Set__Oset_Itf__a_J,type,
nth_set_a: list_set_a > nat > set_a ).
thf(sy_c_List_Onth_001tf__a,type,
nth_a: list_a > nat > a ).
thf(sy_c_List_Orev_001t__List__Olist_Itf__a_J,type,
rev_list_a: list_list_a > list_list_a ).
thf(sy_c_List_Orev_001t__Nat__Onat,type,
rev_nat: list_nat > list_nat ).
thf(sy_c_List_Orev_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
rev_set_list_a: list_set_list_a > list_set_list_a ).
thf(sy_c_List_Orev_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
rev_set_set_a: list_set_set_a > list_set_set_a ).
thf(sy_c_List_Orev_001t__Set__Oset_Itf__a_J,type,
rev_set_a: list_set_a > list_set_a ).
thf(sy_c_List_Orev_001tf__a,type,
rev_a: list_a > list_a ).
thf(sy_c_Multiset_Ocomm__monoid__add__class_Osum__mset_001t__Multiset__Omultiset_It__List__Olist_Itf__a_J_J,type,
comm_m324867663list_a: multis971982480list_a > multiset_list_a ).
thf(sy_c_Multiset_Ocomm__monoid__add__class_Osum__mset_001t__Multiset__Omultiset_Itf__a_J,type,
comm_m2145643721iset_a: multiset_multiset_a > multiset_a ).
thf(sy_c_Multiset_Ocomm__monoid__add__class_Osum__mset_001t__Nat__Onat,type,
comm_m1100186507et_nat: multiset_nat > nat ).
thf(sy_c_Multiset_Ocomm__monoid__add__class_Osum__mset_001tf__a,type,
comm_m543484931mset_a: multiset_a > a ).
thf(sy_c_Multiset_Omset_001t__List__Olist_Itf__a_J,type,
mset_list_a: list_list_a > multiset_list_a ).
thf(sy_c_Multiset_Omset_001t__Multiset__Omultiset_It__List__Olist_Itf__a_J_J,type,
mset_multiset_list_a: list_multiset_list_a > multis971982480list_a ).
thf(sy_c_Multiset_Omset_001t__Multiset__Omultiset_Itf__a_J,type,
mset_multiset_a: list_multiset_a > multiset_multiset_a ).
thf(sy_c_Multiset_Omset_001t__Nat__Onat,type,
mset_nat: list_nat > multiset_nat ).
thf(sy_c_Multiset_Omset_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
mset_set_list_a: list_set_list_a > multiset_set_list_a ).
thf(sy_c_Multiset_Omset_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
mset_set_set_a: list_set_set_a > multiset_set_set_a ).
thf(sy_c_Multiset_Omset_001t__Set__Oset_Itf__a_J,type,
mset_set_a: list_set_a > multiset_set_a ).
thf(sy_c_Multiset_Omset_001tf__a,type,
mset_a: list_a > multiset_a ).
thf(sy_c_Multiset_Oset__mset_001t__List__Olist_Itf__a_J,type,
set_mset_list_a: multiset_list_a > set_list_a ).
thf(sy_c_Multiset_Oset__mset_001t__Multiset__Omultiset_It__List__Olist_Itf__a_J_J,type,
set_ms1660427399list_a: multis971982480list_a > set_multiset_list_a ).
thf(sy_c_Multiset_Oset__mset_001t__Multiset__Omultiset_Itf__a_J,type,
set_mset_multiset_a: multiset_multiset_a > set_multiset_a ).
thf(sy_c_Multiset_Oset__mset_001t__Set__Oset_Itf__a_J,type,
set_mset_set_a: multiset_set_a > set_set_a ).
thf(sy_c_Multiset_Oset__mset_001tf__a,type,
set_mset_a: multiset_a > set_a ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
size_s575106428list_a: list_list_list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
size_s341332310_set_a: list_list_set_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
size_s1427607542list_a: list_list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Multiset__Omultiset_Itf__a_J_J,type,
size_s1657263798iset_a: list_multiset_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
size_s240404444list_a: list_set_list_list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__List__Olist_It__Set__Oset_Itf__a_J_J_J_J,type,
size_s926492982_set_a: list_set_list_set_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
size_s1635937238list_a: list_set_list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Multiset__Omultiset_Itf__a_J_J_J,type,
size_s802269334iset_a: list_set_multiset_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
size_s577819178et_nat: list_set_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
size_s469631926list_a: list_set_set_list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J_J,type,
size_s726512144_set_a: list_set_set_set_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
size_s739728560_set_a: list_set_set_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
size_size_list_set_a: list_set_a > 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_Osize__class_Osize_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
size_s657334288_set_a: multiset_set_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_Itf__a_J,type,
size_size_multiset_a: multiset_a > nat ).
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_Itf__a_J_J,type,
ord_less_set_list_a: set_list_a > set_list_a > $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_I_Eo_Mt__Nat__Onat_J,type,
ord_less_eq_o_nat: ( $o > nat ) > ( $o > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_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_Itf__a_J_J,type,
ord_le1301786372list_a: set_list_a > set_list_a > $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_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Permutation_Operm_001t__Set__Oset_Itf__a_J,type,
perm_set_a: list_set_a > list_set_a > $o ).
thf(sy_c_Permutation_Operm_001tf__a,type,
perm_a: list_a > list_a > $o ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
collect_list_list_a: ( list_list_a > $o ) > set_list_list_a ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
collect_list_set_a: ( list_set_a > $o ) > set_list_set_a ).
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__Multiset__Omultiset_Itf__a_J,type,
collect_multiset_a: ( multiset_a > $o ) > set_multiset_a ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > 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_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
image_list_nat_nat: ( list_nat > nat ) > set_list_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
image_list_a_list_a: ( list_a > list_a ) > set_list_a > set_list_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001tf__a,type,
image_list_a_a: ( list_a > a ) > set_list_a > set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__List__Olist_Itf__a_J,type,
image_a_list_a: ( a > list_a ) > set_a > set_list_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_or1578766727list_a: set_list_a > set_set_list_a ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_or1597314339et_nat: set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_Itf__a_J,type,
set_or1164043265_set_a: set_a > set_set_a ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member1511395513list_a: list_list_list_a > set_list_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
member954073875_set_a: list_list_set_a > set_list_list_set_a > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
member_list_list_a: list_list_a > set_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Multiset__Omultiset_Itf__a_J_J,type,
member518144627iset_a: list_multiset_a > set_list_multiset_a > $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__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
member101195155list_a: list_set_list_a > set_list_set_list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
member814963949_set_a: list_set_set_a > set_list_set_set_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
member_list_set_a: list_set_a > set_list_set_a > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Multiset__Omultiset_It__List__Olist_Itf__a_J_J,type,
member528267379list_a: multiset_list_a > set_multiset_list_a > $o ).
thf(sy_c_member_001t__Multiset__Omultiset_Itf__a_J,type,
member_multiset_a: multiset_a > set_multiset_a > $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_Itf__a_J_J,type,
member_set_list_a: set_list_a > set_set_list_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_a > $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_a____,type,
a3: a ).
thf(sy_v_f____,type,
f: nat > nat ).
thf(sy_v_g____,type,
g: nat > nat ).
thf(sy_v_qs2____,type,
qs2: list_a ).
thf(sy_v_qs____,type,
qs: list_a ).
thf(sy_v_ss1,type,
ss1: list_set_a ).
thf(sy_v_ss2,type,
ss2: list_set_a ).
% Relevant facts (353)
thf(fact_0__092_060open_062mset_Aqs2_A_061_Amset_Aqs_092_060close_062,axiom,
( ( mset_a @ qs2 )
= ( mset_a @ qs ) ) ).
% \<open>mset qs2 = mset qs\<close>
thf(fact_1__092_060open_062a_A_092_060in_062_AA_092_060close_062,axiom,
member_a @ a3 @ a2 ).
% \<open>a \<in> A\<close>
thf(fact_2_a,axiom,
( a3
= ( groups1792256535list_a @ qs ) ) ).
% a
thf(fact_3_len__qs2,axiom,
( ( size_size_list_a @ qs2 )
= ( size_size_list_a @ qs ) ) ).
% len_qs2
thf(fact_4__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062qs_O_A_092_060lbrakk_062a_A_061_Asum__list_Aqs_059_Aqs_A_092_060in_062_Alistset_Ass1_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Qs: list_a] :
( ( a3
= ( groups1792256535list_a @ Qs ) )
=> ~ ( member_list_a @ Qs @ ( listset_a @ ss1 ) ) ) ).
% \<open>\<And>thesis. (\<And>qs. \<lbrakk>a = sum_list qs; qs \<in> listset ss1\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_5_qs2__in,axiom,
member_list_a @ qs2 @ ( listset_a @ ss2 ) ).
% qs2_in
thf(fact_6_direct__decompD_I1_J,axiom,
! [A: set_multiset_a,Ss: list_set_multiset_a,Qs2: list_multiset_a] :
( ( hilber2024005914iset_a @ A @ Ss )
=> ( ( member518144627iset_a @ Qs2 @ ( listset_multiset_a @ Ss ) )
=> ( member_multiset_a @ ( groups1592617181iset_a @ Qs2 ) @ A ) ) ) ).
% direct_decompD(1)
thf(fact_7_direct__decompD_I1_J,axiom,
! [A: set_nat,Ss: list_set_nat,Qs2: list_nat] :
( ( hilber1014902394mp_nat @ A @ Ss )
=> ( ( member_list_nat @ Qs2 @ ( listset_nat @ Ss ) )
=> ( member_nat @ ( groups921905271st_nat @ Qs2 ) @ A ) ) ) ).
% direct_decompD(1)
thf(fact_8_direct__decompD_I1_J,axiom,
! [A: set_a,Ss: list_set_a,Qs2: list_a] :
( ( hilber2037636820comp_a @ A @ Ss )
=> ( ( member_list_a @ Qs2 @ ( listset_a @ Ss ) )
=> ( member_a @ ( groups1792256535list_a @ Qs2 ) @ A ) ) ) ).
% direct_decompD(1)
thf(fact_9_direct__decompE,axiom,
! [A: set_multiset_a,Ss: list_set_multiset_a,A2: multiset_a] :
( ( hilber2024005914iset_a @ A @ Ss )
=> ( ( member_multiset_a @ A2 @ A )
=> ~ ! [Qs: list_multiset_a] :
( ( member518144627iset_a @ Qs @ ( listset_multiset_a @ Ss ) )
=> ( A2
!= ( groups1592617181iset_a @ Qs ) ) ) ) ) ).
% direct_decompE
thf(fact_10_direct__decompE,axiom,
! [A: set_nat,Ss: list_set_nat,A2: nat] :
( ( hilber1014902394mp_nat @ A @ Ss )
=> ( ( member_nat @ A2 @ A )
=> ~ ! [Qs: list_nat] :
( ( member_list_nat @ Qs @ ( listset_nat @ Ss ) )
=> ( A2
!= ( groups921905271st_nat @ Qs ) ) ) ) ) ).
% direct_decompE
thf(fact_11_direct__decompE,axiom,
! [A: set_a,Ss: list_set_a,A2: a] :
( ( hilber2037636820comp_a @ A @ Ss )
=> ( ( member_a @ A2 @ A )
=> ~ ! [Qs: list_a] :
( ( member_list_a @ Qs @ ( listset_a @ Ss ) )
=> ( A2
!= ( groups1792256535list_a @ Qs ) ) ) ) ) ).
% direct_decompE
thf(fact_12_direct__decompI__alt,axiom,
! [Ss: list_set_multiset_a,A: set_multiset_a] :
( ! [Qs: list_multiset_a] :
( ( member518144627iset_a @ Qs @ ( listset_multiset_a @ Ss ) )
=> ( member_multiset_a @ ( groups1592617181iset_a @ Qs ) @ A ) )
=> ( ! [A3: multiset_a] :
( ( member_multiset_a @ A3 @ A )
=> ? [X: list_multiset_a] :
( ( member518144627iset_a @ X @ ( listset_multiset_a @ Ss ) )
& ( A3
= ( groups1592617181iset_a @ X ) )
& ! [Y: list_multiset_a] :
( ( ( member518144627iset_a @ Y @ ( listset_multiset_a @ Ss ) )
& ( A3
= ( groups1592617181iset_a @ Y ) ) )
=> ( Y = X ) ) ) )
=> ( hilber2024005914iset_a @ A @ Ss ) ) ) ).
% direct_decompI_alt
thf(fact_13_direct__decompI__alt,axiom,
! [Ss: list_set_nat,A: set_nat] :
( ! [Qs: list_nat] :
( ( member_list_nat @ Qs @ ( listset_nat @ Ss ) )
=> ( member_nat @ ( groups921905271st_nat @ Qs ) @ A ) )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ? [X: list_nat] :
( ( member_list_nat @ X @ ( listset_nat @ Ss ) )
& ( A3
= ( groups921905271st_nat @ X ) )
& ! [Y: list_nat] :
( ( ( member_list_nat @ Y @ ( listset_nat @ Ss ) )
& ( A3
= ( groups921905271st_nat @ Y ) ) )
=> ( Y = X ) ) ) )
=> ( hilber1014902394mp_nat @ A @ Ss ) ) ) ).
% direct_decompI_alt
thf(fact_14_direct__decompI__alt,axiom,
! [Ss: list_set_a,A: set_a] :
( ! [Qs: list_a] :
( ( member_list_a @ Qs @ ( listset_a @ Ss ) )
=> ( member_a @ ( groups1792256535list_a @ Qs ) @ A ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A )
=> ? [X: list_a] :
( ( member_list_a @ X @ ( listset_a @ Ss ) )
& ( A3
= ( groups1792256535list_a @ X ) )
& ! [Y: list_a] :
( ( ( member_list_a @ Y @ ( listset_a @ Ss ) )
& ( A3
= ( groups1792256535list_a @ Y ) ) )
=> ( Y = X ) ) ) )
=> ( hilber2037636820comp_a @ A @ Ss ) ) ) ).
% direct_decompI_alt
thf(fact_15_direct__decomp__unique,axiom,
! [A: set_nat,Ss: list_set_nat,Qs2: list_nat,Qs3: list_nat] :
( ( hilber1014902394mp_nat @ A @ Ss )
=> ( ( member_list_nat @ Qs2 @ ( listset_nat @ Ss ) )
=> ( ( member_list_nat @ Qs3 @ ( listset_nat @ Ss ) )
=> ( ( ( groups921905271st_nat @ Qs2 )
= ( groups921905271st_nat @ Qs3 ) )
=> ( Qs2 = Qs3 ) ) ) ) ) ).
% direct_decomp_unique
thf(fact_16_direct__decomp__unique,axiom,
! [A: set_a,Ss: list_set_a,Qs2: list_a,Qs3: list_a] :
( ( hilber2037636820comp_a @ A @ Ss )
=> ( ( member_list_a @ Qs2 @ ( listset_a @ Ss ) )
=> ( ( member_list_a @ Qs3 @ ( listset_a @ Ss ) )
=> ( ( ( groups1792256535list_a @ Qs2 )
= ( groups1792256535list_a @ Qs3 ) )
=> ( Qs2 = Qs3 ) ) ) ) ) ).
% direct_decomp_unique
thf(fact_17_sum__list_Orev,axiom,
! [Xs: list_nat] :
( ( groups921905271st_nat @ ( rev_nat @ Xs ) )
= ( groups921905271st_nat @ Xs ) ) ).
% sum_list.rev
thf(fact_18_sum__list_Orev,axiom,
! [Xs: list_a] :
( ( groups1792256535list_a @ ( rev_a @ Xs ) )
= ( groups1792256535list_a @ Xs ) ) ).
% sum_list.rev
thf(fact_19__C1_C,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ qs ) )
=> ( ( nth_a @ qs2 @ I )
= ( nth_a @ qs @ ( g @ I ) ) ) ) ).
% "1"
thf(fact_20_g__bij2,axiom,
bij_betw_nat_nat @ g @ ( set_ord_lessThan_nat @ ( size_size_list_a @ qs2 ) ) @ ( set_ord_lessThan_nat @ ( size_size_list_a @ qs ) ) ).
% g_bij2
thf(fact_21_sum__mset__sum__list,axiom,
! [Xs: list_nat] :
( ( comm_m1100186507et_nat @ ( mset_nat @ Xs ) )
= ( groups921905271st_nat @ Xs ) ) ).
% sum_mset_sum_list
thf(fact_22_sum__mset__sum__list,axiom,
! [Xs: list_multiset_list_a] :
( ( comm_m324867663list_a @ ( mset_multiset_list_a @ Xs ) )
= ( groups1271887971list_a @ Xs ) ) ).
% sum_mset_sum_list
thf(fact_23_sum__mset__sum__list,axiom,
! [Xs: list_multiset_a] :
( ( comm_m2145643721iset_a @ ( mset_multiset_a @ Xs ) )
= ( groups1592617181iset_a @ Xs ) ) ).
% sum_mset_sum_list
thf(fact_24_sum__mset__sum__list,axiom,
! [Xs: list_a] :
( ( comm_m543484931mset_a @ ( mset_a @ Xs ) )
= ( groups1792256535list_a @ Xs ) ) ).
% sum_mset_sum_list
thf(fact_25_direct__decompD_I3_J,axiom,
! [A: set_nat,Ss: list_set_nat] :
( ( hilber1014902394mp_nat @ A @ Ss )
=> ( ( image_list_nat_nat @ groups921905271st_nat @ ( listset_nat @ Ss ) )
= A ) ) ).
% direct_decompD(3)
thf(fact_26_direct__decompD_I3_J,axiom,
! [A: set_a,Ss: list_set_a] :
( ( hilber2037636820comp_a @ A @ Ss )
=> ( ( image_list_a_a @ groups1792256535list_a @ ( listset_a @ Ss ) )
= A ) ) ).
% direct_decompD(3)
thf(fact_27_assms_I1_J,axiom,
hilber2037636820comp_a @ a2 @ ss1 ).
% assms(1)
thf(fact_28_qs__in,axiom,
member_list_a @ qs @ ( listset_a @ ss1 ) ).
% qs_in
thf(fact_29_mset__rev,axiom,
! [Xs: list_set_a] :
( ( mset_set_a @ ( rev_set_a @ Xs ) )
= ( mset_set_a @ Xs ) ) ).
% mset_rev
thf(fact_30_mset__rev,axiom,
! [Xs: list_list_a] :
( ( mset_list_a @ ( rev_list_a @ Xs ) )
= ( mset_list_a @ Xs ) ) ).
% mset_rev
thf(fact_31_mset__rev,axiom,
! [Xs: list_a] :
( ( mset_a @ ( rev_a @ Xs ) )
= ( mset_a @ Xs ) ) ).
% mset_rev
thf(fact_32__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062qs2_O_A_092_060lbrakk_062qs2_A_092_060in_062_Alistset_Ass2_059_Alength_Aqs2_A_061_Alength_Aqs_059_A_092_060And_062i_O_Ai_A_060_Alength_Aqs_A_092_060Longrightarrow_062_Aqs2_A_B_Ai_A_061_Aqs_A_B_Ag_Ai_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Qs22: list_a] :
( ( member_list_a @ Qs22 @ ( listset_a @ ss2 ) )
=> ( ( ( size_size_list_a @ Qs22 )
= ( size_size_list_a @ qs ) )
=> ~ ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ qs ) )
=> ( ( nth_a @ Qs22 @ I2 )
= ( nth_a @ qs @ ( g @ I2 ) ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>qs2. \<lbrakk>qs2 \<in> listset ss2; length qs2 = length qs; \<And>i. i < length qs \<Longrightarrow> qs2 ! i = qs ! g i\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_33_assms_I2_J,axiom,
perm_set_a @ ss1 @ ss2 ).
% assms(2)
thf(fact_34_ex__mset,axiom,
! [X2: multiset_set_a] :
? [Xs2: list_set_a] :
( ( mset_set_a @ Xs2 )
= X2 ) ).
% ex_mset
thf(fact_35_ex__mset,axiom,
! [X2: multiset_list_a] :
? [Xs2: list_list_a] :
( ( mset_list_a @ Xs2 )
= X2 ) ).
% ex_mset
thf(fact_36_ex__mset,axiom,
! [X2: multiset_a] :
? [Xs2: list_a] :
( ( mset_a @ Xs2 )
= X2 ) ).
% ex_mset
thf(fact_37_mset__eq__length,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( mset_list_a @ Xs )
= ( mset_list_a @ Ys ) )
=> ( ( size_s1427607542list_a @ Xs )
= ( size_s1427607542list_a @ Ys ) ) ) ).
% mset_eq_length
thf(fact_38_mset__eq__length,axiom,
! [Xs: list_set_list_a,Ys: list_set_list_a] :
( ( ( mset_set_list_a @ Xs )
= ( mset_set_list_a @ Ys ) )
=> ( ( size_s1635937238list_a @ Xs )
= ( size_s1635937238list_a @ Ys ) ) ) ).
% mset_eq_length
thf(fact_39_mset__eq__length,axiom,
! [Xs: list_set_set_a,Ys: list_set_set_a] :
( ( ( mset_set_set_a @ Xs )
= ( mset_set_set_a @ Ys ) )
=> ( ( size_s739728560_set_a @ Xs )
= ( size_s739728560_set_a @ Ys ) ) ) ).
% mset_eq_length
thf(fact_40_mset__eq__length,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( mset_nat @ Xs )
= ( mset_nat @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% mset_eq_length
thf(fact_41_mset__eq__length,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( mset_a @ Xs )
= ( mset_a @ Ys ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% mset_eq_length
thf(fact_42_mset__eq__length,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( ( mset_set_a @ Xs )
= ( mset_set_a @ Ys ) )
=> ( ( size_size_list_set_a @ Xs )
= ( size_size_list_set_a @ Ys ) ) ) ).
% mset_eq_length
thf(fact_43_mset__bij,axiom,
! [F: nat > nat,Xs: list_list_a,Ys: list_list_a] :
( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ ( size_s1427607542list_a @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_s1427607542list_a @ Ys ) ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s1427607542list_a @ Xs ) )
=> ( ( nth_list_a @ Xs @ I3 )
= ( nth_list_a @ Ys @ ( F @ I3 ) ) ) )
=> ( ( mset_list_a @ Xs )
= ( mset_list_a @ Ys ) ) ) ) ).
% mset_bij
thf(fact_44_mset__bij,axiom,
! [F: nat > nat,Xs: list_set_list_a,Ys: list_set_list_a] :
( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ ( size_s1635937238list_a @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_s1635937238list_a @ Ys ) ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s1635937238list_a @ Xs ) )
=> ( ( nth_set_list_a @ Xs @ I3 )
= ( nth_set_list_a @ Ys @ ( F @ I3 ) ) ) )
=> ( ( mset_set_list_a @ Xs )
= ( mset_set_list_a @ Ys ) ) ) ) ).
% mset_bij
thf(fact_45_mset__bij,axiom,
! [F: nat > nat,Xs: list_set_set_a,Ys: list_set_set_a] :
( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ ( size_s739728560_set_a @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_s739728560_set_a @ Ys ) ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s739728560_set_a @ Xs ) )
=> ( ( nth_set_set_a @ Xs @ I3 )
= ( nth_set_set_a @ Ys @ ( F @ I3 ) ) ) )
=> ( ( mset_set_set_a @ Xs )
= ( mset_set_set_a @ Ys ) ) ) ) ).
% mset_bij
thf(fact_46_mset__bij,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ ( size_size_list_nat @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_size_list_nat @ Ys ) ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I3 )
= ( nth_nat @ Ys @ ( F @ I3 ) ) ) )
=> ( ( mset_nat @ Xs )
= ( mset_nat @ Ys ) ) ) ) ).
% mset_bij
thf(fact_47_mset__bij,axiom,
! [F: nat > nat,Xs: list_a,Ys: list_a] :
( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ ( size_size_list_a @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_size_list_a @ Ys ) ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ Xs @ I3 )
= ( nth_a @ Ys @ ( F @ I3 ) ) ) )
=> ( ( mset_a @ Xs )
= ( mset_a @ Ys ) ) ) ) ).
% mset_bij
thf(fact_48_mset__bij,axiom,
! [F: nat > nat,Xs: list_set_a,Ys: list_set_a] :
( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ Ys ) ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_set_a @ Xs ) )
=> ( ( nth_set_a @ Xs @ I3 )
= ( nth_set_a @ Ys @ ( F @ I3 ) ) ) )
=> ( ( mset_set_a @ Xs )
= ( mset_set_a @ Ys ) ) ) ) ).
% mset_bij
thf(fact_49_listset__permE,axiom,
! [Ys: list_set_list_a,Xs: list_set_set_list_a,F: nat > nat,Xs3: list_set_set_list_a] :
( ( member101195155list_a @ Ys @ ( listset_set_list_a @ Xs ) )
=> ( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ ( size_s469631926list_a @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_s469631926list_a @ Xs3 ) ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s469631926list_a @ Xs ) )
=> ( ( nth_set_set_list_a @ Xs3 @ I3 )
= ( nth_set_set_list_a @ Xs @ ( F @ I3 ) ) ) )
=> ~ ! [Ys2: list_set_list_a] :
( ( member101195155list_a @ Ys2 @ ( listset_set_list_a @ Xs3 ) )
=> ( ( ( size_s1635937238list_a @ Ys2 )
= ( size_s1635937238list_a @ Ys ) )
=> ~ ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s1635937238list_a @ Ys ) )
=> ( ( nth_set_list_a @ Ys2 @ I2 )
= ( nth_set_list_a @ Ys @ ( F @ I2 ) ) ) ) ) ) ) ) ) ).
% listset_permE
thf(fact_50_listset__permE,axiom,
! [Ys: list_set_set_a,Xs: list_set_set_set_a,F: nat > nat,Xs3: list_set_set_set_a] :
( ( member814963949_set_a @ Ys @ ( listset_set_set_a @ Xs ) )
=> ( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ ( size_s726512144_set_a @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_s726512144_set_a @ Xs3 ) ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s726512144_set_a @ Xs ) )
=> ( ( nth_set_set_set_a @ Xs3 @ I3 )
= ( nth_set_set_set_a @ Xs @ ( F @ I3 ) ) ) )
=> ~ ! [Ys2: list_set_set_a] :
( ( member814963949_set_a @ Ys2 @ ( listset_set_set_a @ Xs3 ) )
=> ( ( ( size_s739728560_set_a @ Ys2 )
= ( size_s739728560_set_a @ Ys ) )
=> ~ ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s739728560_set_a @ Ys ) )
=> ( ( nth_set_set_a @ Ys2 @ I2 )
= ( nth_set_set_a @ Ys @ ( F @ I2 ) ) ) ) ) ) ) ) ) ).
% listset_permE
thf(fact_51_listset__permE,axiom,
! [Ys: list_nat,Xs: list_set_nat,F: nat > nat,Xs3: list_set_nat] :
( ( member_list_nat @ Ys @ ( listset_nat @ Xs ) )
=> ( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ ( size_s577819178et_nat @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_s577819178et_nat @ Xs3 ) ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s577819178et_nat @ Xs ) )
=> ( ( nth_set_nat @ Xs3 @ I3 )
= ( nth_set_nat @ Xs @ ( F @ I3 ) ) ) )
=> ~ ! [Ys2: list_nat] :
( ( member_list_nat @ Ys2 @ ( listset_nat @ Xs3 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Ys ) )
=> ~ ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( ( nth_nat @ Ys2 @ I2 )
= ( nth_nat @ Ys @ ( F @ I2 ) ) ) ) ) ) ) ) ) ).
% listset_permE
thf(fact_52_listset__permE,axiom,
! [Ys: list_list_a,Xs: list_set_list_a,F: nat > nat,Xs3: list_set_list_a] :
( ( member_list_list_a @ Ys @ ( listset_list_a @ Xs ) )
=> ( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ ( size_s1635937238list_a @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_s1635937238list_a @ Xs3 ) ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s1635937238list_a @ Xs ) )
=> ( ( nth_set_list_a @ Xs3 @ I3 )
= ( nth_set_list_a @ Xs @ ( F @ I3 ) ) ) )
=> ~ ! [Ys2: list_list_a] :
( ( member_list_list_a @ Ys2 @ ( listset_list_a @ Xs3 ) )
=> ( ( ( size_s1427607542list_a @ Ys2 )
= ( size_s1427607542list_a @ Ys ) )
=> ~ ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s1427607542list_a @ Ys ) )
=> ( ( nth_list_a @ Ys2 @ I2 )
= ( nth_list_a @ Ys @ ( F @ I2 ) ) ) ) ) ) ) ) ) ).
% listset_permE
thf(fact_53_listset__permE,axiom,
! [Ys: list_set_a,Xs: list_set_set_a,F: nat > nat,Xs3: list_set_set_a] :
( ( member_list_set_a @ Ys @ ( listset_set_a @ Xs ) )
=> ( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ ( size_s739728560_set_a @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_s739728560_set_a @ Xs3 ) ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s739728560_set_a @ Xs ) )
=> ( ( nth_set_set_a @ Xs3 @ I3 )
= ( nth_set_set_a @ Xs @ ( F @ I3 ) ) ) )
=> ~ ! [Ys2: list_set_a] :
( ( member_list_set_a @ Ys2 @ ( listset_set_a @ Xs3 ) )
=> ( ( ( size_size_list_set_a @ Ys2 )
= ( size_size_list_set_a @ Ys ) )
=> ~ ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_set_a @ Ys ) )
=> ( ( nth_set_a @ Ys2 @ I2 )
= ( nth_set_a @ Ys @ ( F @ I2 ) ) ) ) ) ) ) ) ) ).
% listset_permE
thf(fact_54_listset__permE,axiom,
! [Ys: list_a,Xs: list_set_a,F: nat > nat,Xs3: list_set_a] :
( ( member_list_a @ Ys @ ( listset_a @ Xs ) )
=> ( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ Xs3 ) ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_set_a @ Xs ) )
=> ( ( nth_set_a @ Xs3 @ I3 )
= ( nth_set_a @ Xs @ ( F @ I3 ) ) ) )
=> ~ ! [Ys2: list_a] :
( ( member_list_a @ Ys2 @ ( listset_a @ Xs3 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Ys ) )
=> ~ ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Ys ) )
=> ( ( nth_a @ Ys2 @ I2 )
= ( nth_a @ Ys @ ( F @ I2 ) ) ) ) ) ) ) ) ) ).
% listset_permE
thf(fact_55_g__bij,axiom,
bij_betw_nat_nat @ g @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ ss1 ) ) @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ ss2 ) ) ).
% g_bij
thf(fact_56_length__rev,axiom,
! [Xs: list_list_a] :
( ( size_s1427607542list_a @ ( rev_list_a @ Xs ) )
= ( size_s1427607542list_a @ Xs ) ) ).
% length_rev
thf(fact_57_length__rev,axiom,
! [Xs: list_set_list_a] :
( ( size_s1635937238list_a @ ( rev_set_list_a @ Xs ) )
= ( size_s1635937238list_a @ Xs ) ) ).
% length_rev
thf(fact_58_length__rev,axiom,
! [Xs: list_set_set_a] :
( ( size_s739728560_set_a @ ( rev_set_set_a @ Xs ) )
= ( size_s739728560_set_a @ Xs ) ) ).
% length_rev
thf(fact_59_length__rev,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rev_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rev
thf(fact_60_length__rev,axiom,
! [Xs: list_a] :
( ( size_size_list_a @ ( rev_a @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_rev
thf(fact_61_length__rev,axiom,
! [Xs: list_set_a] :
( ( size_size_list_set_a @ ( rev_set_a @ Xs ) )
= ( size_size_list_set_a @ Xs ) ) ).
% length_rev
thf(fact_62_lessThan__iff,axiom,
! [I: set_nat,K: set_nat] :
( ( member_set_nat @ I @ ( set_or1597314339et_nat @ K ) )
= ( ord_less_set_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_63_lessThan__iff,axiom,
! [I: set_list_a,K: set_list_a] :
( ( member_set_list_a @ I @ ( set_or1578766727list_a @ K ) )
= ( ord_less_set_list_a @ I @ K ) ) ).
% lessThan_iff
thf(fact_64_lessThan__iff,axiom,
! [I: set_a,K: set_a] :
( ( member_set_a @ I @ ( set_or1164043265_set_a @ K ) )
= ( ord_less_set_a @ I @ K ) ) ).
% lessThan_iff
thf(fact_65_lessThan__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
= ( ord_less_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_66_listsetI,axiom,
! [Ys: list_multiset_a,Xs: list_set_multiset_a] :
( ( ( size_s1657263798iset_a @ Ys )
= ( size_s802269334iset_a @ Xs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s802269334iset_a @ Xs ) )
=> ( member_multiset_a @ ( nth_multiset_a @ Ys @ I3 ) @ ( nth_set_multiset_a @ Xs @ I3 ) ) )
=> ( member518144627iset_a @ Ys @ ( listset_multiset_a @ Xs ) ) ) ) ).
% listsetI
thf(fact_67_listsetI,axiom,
! [Ys: list_list_list_a,Xs: list_set_list_list_a] :
( ( ( size_s575106428list_a @ Ys )
= ( size_s240404444list_a @ Xs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s240404444list_a @ Xs ) )
=> ( member_list_list_a @ ( nth_list_list_a @ Ys @ I3 ) @ ( nth_set_list_list_a @ Xs @ I3 ) ) )
=> ( member1511395513list_a @ Ys @ ( listset_list_list_a @ Xs ) ) ) ) ).
% listsetI
thf(fact_68_listsetI,axiom,
! [Ys: list_list_set_a,Xs: list_set_list_set_a] :
( ( ( size_s341332310_set_a @ Ys )
= ( size_s926492982_set_a @ Xs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s926492982_set_a @ Xs ) )
=> ( member_list_set_a @ ( nth_list_set_a @ Ys @ I3 ) @ ( nth_set_list_set_a @ Xs @ I3 ) ) )
=> ( member954073875_set_a @ Ys @ ( listset_list_set_a @ Xs ) ) ) ) ).
% listsetI
thf(fact_69_listsetI,axiom,
! [Ys: list_set_list_a,Xs: list_set_set_list_a] :
( ( ( size_s1635937238list_a @ Ys )
= ( size_s469631926list_a @ Xs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s469631926list_a @ Xs ) )
=> ( member_set_list_a @ ( nth_set_list_a @ Ys @ I3 ) @ ( nth_set_set_list_a @ Xs @ I3 ) ) )
=> ( member101195155list_a @ Ys @ ( listset_set_list_a @ Xs ) ) ) ) ).
% listsetI
thf(fact_70_listsetI,axiom,
! [Ys: list_set_set_a,Xs: list_set_set_set_a] :
( ( ( size_s739728560_set_a @ Ys )
= ( size_s726512144_set_a @ Xs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s726512144_set_a @ Xs ) )
=> ( member_set_set_a @ ( nth_set_set_a @ Ys @ I3 ) @ ( nth_set_set_set_a @ Xs @ I3 ) ) )
=> ( member814963949_set_a @ Ys @ ( listset_set_set_a @ Xs ) ) ) ) ).
% listsetI
thf(fact_71_listsetI,axiom,
! [Ys: list_nat,Xs: list_set_nat] :
( ( ( size_size_list_nat @ Ys )
= ( size_s577819178et_nat @ Xs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s577819178et_nat @ Xs ) )
=> ( member_nat @ ( nth_nat @ Ys @ I3 ) @ ( nth_set_nat @ Xs @ I3 ) ) )
=> ( member_list_nat @ Ys @ ( listset_nat @ Xs ) ) ) ) ).
% listsetI
thf(fact_72_listsetI,axiom,
! [Ys: list_list_a,Xs: list_set_list_a] :
( ( ( size_s1427607542list_a @ Ys )
= ( size_s1635937238list_a @ Xs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s1635937238list_a @ Xs ) )
=> ( member_list_a @ ( nth_list_a @ Ys @ I3 ) @ ( nth_set_list_a @ Xs @ I3 ) ) )
=> ( member_list_list_a @ Ys @ ( listset_list_a @ Xs ) ) ) ) ).
% listsetI
thf(fact_73_listsetI,axiom,
! [Ys: list_a,Xs: list_set_a] :
( ( ( size_size_list_a @ Ys )
= ( size_size_list_set_a @ Xs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_set_a @ Xs ) )
=> ( member_a @ ( nth_a @ Ys @ I3 ) @ ( nth_set_a @ Xs @ I3 ) ) )
=> ( member_list_a @ Ys @ ( listset_a @ Xs ) ) ) ) ).
% listsetI
thf(fact_74_listsetI,axiom,
! [Ys: list_set_a,Xs: list_set_set_a] :
( ( ( size_size_list_set_a @ Ys )
= ( size_s739728560_set_a @ Xs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s739728560_set_a @ Xs ) )
=> ( member_set_a @ ( nth_set_a @ Ys @ I3 ) @ ( nth_set_set_a @ Xs @ I3 ) ) )
=> ( member_list_set_a @ Ys @ ( listset_set_a @ Xs ) ) ) ) ).
% listsetI
thf(fact_75_len__qs,axiom,
( ( size_size_list_a @ qs )
= ( size_size_list_set_a @ ss1 ) ) ).
% len_qs
thf(fact_76_f__bij,axiom,
bij_betw_nat_nat @ f @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ ss2 ) ) @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ ss1 ) ) ).
% f_bij
thf(fact_77_nth__equalityI,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( size_s1427607542list_a @ Xs )
= ( size_s1427607542list_a @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s1427607542list_a @ Xs ) )
=> ( ( nth_list_a @ Xs @ I3 )
= ( nth_list_a @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_78_nth__equalityI,axiom,
! [Xs: list_set_list_a,Ys: list_set_list_a] :
( ( ( size_s1635937238list_a @ Xs )
= ( size_s1635937238list_a @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s1635937238list_a @ Xs ) )
=> ( ( nth_set_list_a @ Xs @ I3 )
= ( nth_set_list_a @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_79_nth__equalityI,axiom,
! [Xs: list_set_set_a,Ys: list_set_set_a] :
( ( ( size_s739728560_set_a @ Xs )
= ( size_s739728560_set_a @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s739728560_set_a @ Xs ) )
=> ( ( nth_set_set_a @ Xs @ I3 )
= ( nth_set_set_a @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_80_nth__equalityI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I3 )
= ( nth_nat @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_81_nth__equalityI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ Xs @ I3 )
= ( nth_a @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_82_nth__equalityI,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( ( size_size_list_set_a @ Xs )
= ( size_size_list_set_a @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_set_a @ Xs ) )
=> ( ( nth_set_a @ Xs @ I3 )
= ( nth_set_a @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_83_Skolem__list__nth,axiom,
! [K: nat,P: nat > list_a > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X3: list_a] : ( P @ I4 @ X3 ) ) )
= ( ? [Xs4: list_list_a] :
( ( ( size_s1427607542list_a @ Xs4 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P @ I4 @ ( nth_list_a @ Xs4 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_84_Skolem__list__nth,axiom,
! [K: nat,P: nat > set_list_a > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X3: set_list_a] : ( P @ I4 @ X3 ) ) )
= ( ? [Xs4: list_set_list_a] :
( ( ( size_s1635937238list_a @ Xs4 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P @ I4 @ ( nth_set_list_a @ Xs4 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_85_Skolem__list__nth,axiom,
! [K: nat,P: nat > set_set_a > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X3: set_set_a] : ( P @ I4 @ X3 ) ) )
= ( ? [Xs4: list_set_set_a] :
( ( ( size_s739728560_set_a @ Xs4 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P @ I4 @ ( nth_set_set_a @ Xs4 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_86_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X3: nat] : ( P @ I4 @ X3 ) ) )
= ( ? [Xs4: list_nat] :
( ( ( size_size_list_nat @ Xs4 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P @ I4 @ ( nth_nat @ Xs4 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_87_Skolem__list__nth,axiom,
! [K: nat,P: nat > a > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X3: a] : ( P @ I4 @ X3 ) ) )
= ( ? [Xs4: list_a] :
( ( ( size_size_list_a @ Xs4 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P @ I4 @ ( nth_a @ Xs4 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_88_Skolem__list__nth,axiom,
! [K: nat,P: nat > set_a > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X3: set_a] : ( P @ I4 @ X3 ) ) )
= ( ? [Xs4: list_set_a] :
( ( ( size_size_list_set_a @ Xs4 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P @ I4 @ ( nth_set_a @ Xs4 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_89_list__eq__iff__nth__eq,axiom,
( ( ^ [Y2: list_list_a,Z: list_list_a] : Y2 = Z )
= ( ^ [Xs4: list_list_a,Ys3: list_list_a] :
( ( ( size_s1427607542list_a @ Xs4 )
= ( size_s1427607542list_a @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s1427607542list_a @ Xs4 ) )
=> ( ( nth_list_a @ Xs4 @ I4 )
= ( nth_list_a @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_90_list__eq__iff__nth__eq,axiom,
( ( ^ [Y2: list_set_list_a,Z: list_set_list_a] : Y2 = Z )
= ( ^ [Xs4: list_set_list_a,Ys3: list_set_list_a] :
( ( ( size_s1635937238list_a @ Xs4 )
= ( size_s1635937238list_a @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s1635937238list_a @ Xs4 ) )
=> ( ( nth_set_list_a @ Xs4 @ I4 )
= ( nth_set_list_a @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_91_list__eq__iff__nth__eq,axiom,
( ( ^ [Y2: list_set_set_a,Z: list_set_set_a] : Y2 = Z )
= ( ^ [Xs4: list_set_set_a,Ys3: list_set_set_a] :
( ( ( size_s739728560_set_a @ Xs4 )
= ( size_s739728560_set_a @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s739728560_set_a @ Xs4 ) )
=> ( ( nth_set_set_a @ Xs4 @ I4 )
= ( nth_set_set_a @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_92_list__eq__iff__nth__eq,axiom,
( ( ^ [Y2: list_nat,Z: list_nat] : Y2 = Z )
= ( ^ [Xs4: list_nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs4 )
= ( size_size_list_nat @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs4 ) )
=> ( ( nth_nat @ Xs4 @ I4 )
= ( nth_nat @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_93_list__eq__iff__nth__eq,axiom,
( ( ^ [Y2: list_a,Z: list_a] : Y2 = Z )
= ( ^ [Xs4: list_a,Ys3: list_a] :
( ( ( size_size_list_a @ Xs4 )
= ( size_size_list_a @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs4 ) )
=> ( ( nth_a @ Xs4 @ I4 )
= ( nth_a @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_94_list__eq__iff__nth__eq,axiom,
( ( ^ [Y2: list_set_a,Z: list_set_a] : Y2 = Z )
= ( ^ [Xs4: list_set_a,Ys3: list_set_a] :
( ( ( size_size_list_set_a @ Xs4 )
= ( size_size_list_set_a @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_set_a @ Xs4 ) )
=> ( ( nth_set_a @ Xs4 @ I4 )
= ( nth_set_a @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_95_lessThan__eq__iff,axiom,
! [X4: nat,Y3: nat] :
( ( ( set_ord_lessThan_nat @ X4 )
= ( set_ord_lessThan_nat @ Y3 ) )
= ( X4 = Y3 ) ) ).
% lessThan_eq_iff
thf(fact_96_rev__is__rev__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( rev_a @ Xs )
= ( rev_a @ Ys ) )
= ( Xs = Ys ) ) ).
% rev_is_rev_conv
thf(fact_97_rev__is__rev__conv,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( ( rev_set_a @ Xs )
= ( rev_set_a @ Ys ) )
= ( Xs = Ys ) ) ).
% rev_is_rev_conv
thf(fact_98_rev__rev__ident,axiom,
! [Xs: list_a] :
( ( rev_a @ ( rev_a @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_99_rev__rev__ident,axiom,
! [Xs: list_set_a] :
( ( rev_set_a @ ( rev_set_a @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_100_len__ss1,axiom,
( ( size_size_list_set_a @ ss1 )
= ( size_size_list_set_a @ ss2 ) ) ).
% len_ss1
thf(fact_101_g,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_set_a @ ss1 ) )
=> ( ( nth_set_a @ ss2 @ I )
= ( nth_set_a @ ss1 @ ( g @ I ) ) ) ) ).
% g
thf(fact_102_f,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_set_a @ ss2 ) )
=> ( ( nth_set_a @ ss1 @ I )
= ( nth_set_a @ ss2 @ ( f @ I ) ) ) ) ).
% f
thf(fact_103__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062f_O_A_092_060lbrakk_062bij__betw_Af_A_123_O_O_060length_Ass2_125_A_123_O_O_060length_Ass1_125_059_A_092_060And_062i_O_Ai_A_060_Alength_Ass2_A_092_060Longrightarrow_062_Ass1_A_B_Ai_A_061_Ass2_A_B_Af_Ai_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [F2: nat > nat] :
( ( bij_betw_nat_nat @ F2 @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ ss2 ) ) @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ ss1 ) ) )
=> ~ ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_set_a @ ss2 ) )
=> ( ( nth_set_a @ ss1 @ I2 )
= ( nth_set_a @ ss2 @ ( F2 @ I2 ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>f. \<lbrakk>bij_betw f {..<length ss2} {..<length ss1}; \<And>i. i < length ss2 \<Longrightarrow> ss1 ! i = ss2 ! f i\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_104__092_060open_062_092_060exists_062f_O_Abij__betw_Af_A_123_O_O_060length_Ass1_125_A_123_O_O_060length_Ass2_125_A_092_060and_062_A_I_092_060forall_062i_060length_Ass1_O_Ass1_A_B_Ai_A_061_Ass2_A_B_Af_Ai_J_092_060close_062,axiom,
? [F2: nat > nat] :
( ( bij_betw_nat_nat @ F2 @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ ss1 ) ) @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ ss2 ) ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_set_a @ ss1 ) )
=> ( ( nth_set_a @ ss1 @ I2 )
= ( nth_set_a @ ss2 @ ( F2 @ I2 ) ) ) ) ) ).
% \<open>\<exists>f. bij_betw f {..<length ss1} {..<length ss2} \<and> (\<forall>i<length ss1. ss1 ! i = ss2 ! f i)\<close>
thf(fact_105_f__g,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_set_a @ ss1 ) )
=> ( ( f @ ( g @ I ) )
= I ) ) ).
% f_g
thf(fact_106_g__f,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_set_a @ ss2 ) )
=> ( ( g @ ( f @ I ) )
= I ) ) ).
% g_f
thf(fact_107_listsetD_I2_J,axiom,
! [Ys: list_set_set_a,Xs: list_set_set_set_a,I: nat] :
( ( member814963949_set_a @ Ys @ ( listset_set_set_a @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s726512144_set_a @ Xs ) )
=> ( member_set_set_a @ ( nth_set_set_a @ Ys @ I ) @ ( nth_set_set_set_a @ Xs @ I ) ) ) ) ).
% listsetD(2)
thf(fact_108_listsetD_I2_J,axiom,
! [Ys: list_set_list_a,Xs: list_set_set_list_a,I: nat] :
( ( member101195155list_a @ Ys @ ( listset_set_list_a @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s469631926list_a @ Xs ) )
=> ( member_set_list_a @ ( nth_set_list_a @ Ys @ I ) @ ( nth_set_set_list_a @ Xs @ I ) ) ) ) ).
% listsetD(2)
thf(fact_109_listsetD_I2_J,axiom,
! [Ys: list_multiset_a,Xs: list_set_multiset_a,I: nat] :
( ( member518144627iset_a @ Ys @ ( listset_multiset_a @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s802269334iset_a @ Xs ) )
=> ( member_multiset_a @ ( nth_multiset_a @ Ys @ I ) @ ( nth_set_multiset_a @ Xs @ I ) ) ) ) ).
% listsetD(2)
thf(fact_110_listsetD_I2_J,axiom,
! [Ys: list_list_list_a,Xs: list_set_list_list_a,I: nat] :
( ( member1511395513list_a @ Ys @ ( listset_list_list_a @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s240404444list_a @ Xs ) )
=> ( member_list_list_a @ ( nth_list_list_a @ Ys @ I ) @ ( nth_set_list_list_a @ Xs @ I ) ) ) ) ).
% listsetD(2)
thf(fact_111_listsetD_I2_J,axiom,
! [Ys: list_list_set_a,Xs: list_set_list_set_a,I: nat] :
( ( member954073875_set_a @ Ys @ ( listset_list_set_a @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s926492982_set_a @ Xs ) )
=> ( member_list_set_a @ ( nth_list_set_a @ Ys @ I ) @ ( nth_set_list_set_a @ Xs @ I ) ) ) ) ).
% listsetD(2)
thf(fact_112_listsetD_I2_J,axiom,
! [Ys: list_nat,Xs: list_set_nat,I: nat] :
( ( member_list_nat @ Ys @ ( listset_nat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s577819178et_nat @ Xs ) )
=> ( member_nat @ ( nth_nat @ Ys @ I ) @ ( nth_set_nat @ Xs @ I ) ) ) ) ).
% listsetD(2)
thf(fact_113_listsetD_I2_J,axiom,
! [Ys: list_set_a,Xs: list_set_set_a,I: nat] :
( ( member_list_set_a @ Ys @ ( listset_set_a @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s739728560_set_a @ Xs ) )
=> ( member_set_a @ ( nth_set_a @ Ys @ I ) @ ( nth_set_set_a @ Xs @ I ) ) ) ) ).
% listsetD(2)
thf(fact_114_listsetD_I2_J,axiom,
! [Ys: list_list_a,Xs: list_set_list_a,I: nat] :
( ( member_list_list_a @ Ys @ ( listset_list_a @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s1635937238list_a @ Xs ) )
=> ( member_list_a @ ( nth_list_a @ Ys @ I ) @ ( nth_set_list_a @ Xs @ I ) ) ) ) ).
% listsetD(2)
thf(fact_115_listsetD_I2_J,axiom,
! [Ys: list_a,Xs: list_set_a,I: nat] :
( ( member_list_a @ Ys @ ( listset_a @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_set_a @ Xs ) )
=> ( member_a @ ( nth_a @ Ys @ I ) @ ( nth_set_a @ Xs @ I ) ) ) ) ).
% listsetD(2)
thf(fact_116_neq__if__length__neq,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( size_s1427607542list_a @ Xs )
!= ( size_s1427607542list_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_117_neq__if__length__neq,axiom,
! [Xs: list_set_list_a,Ys: list_set_list_a] :
( ( ( size_s1635937238list_a @ Xs )
!= ( size_s1635937238list_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_118_neq__if__length__neq,axiom,
! [Xs: list_set_set_a,Ys: list_set_set_a] :
( ( ( size_s739728560_set_a @ Xs )
!= ( size_s739728560_set_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_119_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_120_neq__if__length__neq,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
!= ( size_size_list_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_121_neq__if__length__neq,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( ( size_size_list_set_a @ Xs )
!= ( size_size_list_set_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_122_mem__Collect__eq,axiom,
! [A2: multiset_a,P: multiset_a > $o] :
( ( member_multiset_a @ A2 @ ( collect_multiset_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_123_mem__Collect__eq,axiom,
! [A2: list_list_a,P: list_list_a > $o] :
( ( member_list_list_a @ A2 @ ( collect_list_list_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_124_mem__Collect__eq,axiom,
! [A2: list_set_a,P: list_set_a > $o] :
( ( member_list_set_a @ A2 @ ( collect_list_set_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_125_mem__Collect__eq,axiom,
! [A2: set_a,P: set_a > $o] :
( ( member_set_a @ A2 @ ( collect_set_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_126_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_127_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_128_mem__Collect__eq,axiom,
! [A2: list_a,P: list_a > $o] :
( ( member_list_a @ A2 @ ( collect_list_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_129_Collect__mem__eq,axiom,
! [A: set_multiset_a] :
( ( collect_multiset_a
@ ^ [X5: multiset_a] : ( member_multiset_a @ X5 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_130_Collect__mem__eq,axiom,
! [A: set_list_list_a] :
( ( collect_list_list_a
@ ^ [X5: list_list_a] : ( member_list_list_a @ X5 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_131_Collect__mem__eq,axiom,
! [A: set_list_set_a] :
( ( collect_list_set_a
@ ^ [X5: list_set_a] : ( member_list_set_a @ X5 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_132_Collect__mem__eq,axiom,
! [A: set_set_a] :
( ( collect_set_a
@ ^ [X5: set_a] : ( member_set_a @ X5 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_133_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X5: nat] : ( member_nat @ X5 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_134_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X5: a] : ( member_a @ X5 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_135_Collect__mem__eq,axiom,
! [A: set_list_a] :
( ( collect_list_a
@ ^ [X5: list_a] : ( member_list_a @ X5 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_136_Collect__cong,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ! [X6: list_a] :
( ( P @ X6 )
= ( Q @ X6 ) )
=> ( ( collect_list_a @ P )
= ( collect_list_a @ Q ) ) ) ).
% Collect_cong
thf(fact_137_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X6: a] :
( ( P @ X6 )
= ( Q @ X6 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_138_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_list_a] :
( ( size_s1427607542list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_139_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_set_list_a] :
( ( size_s1635937238list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_140_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_set_set_a] :
( ( size_s739728560_set_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_141_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_nat] :
( ( size_size_list_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_142_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_a] :
( ( size_size_list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_143_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_set_a] :
( ( size_size_list_set_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_144_rev__swap,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( rev_a @ Xs )
= Ys )
= ( Xs
= ( rev_a @ Ys ) ) ) ).
% rev_swap
thf(fact_145_rev__swap,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( ( rev_set_a @ Xs )
= Ys )
= ( Xs
= ( rev_set_a @ Ys ) ) ) ).
% rev_swap
thf(fact_146_lessThan__strict__subset__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_147_length__induct,axiom,
! [P: list_set_list_a > $o,Xs: list_set_list_a] :
( ! [Xs2: list_set_list_a] :
( ! [Ys4: list_set_list_a] :
( ( ord_less_nat @ ( size_s1635937238list_a @ Ys4 ) @ ( size_s1635937238list_a @ Xs2 ) )
=> ( P @ Ys4 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_148_length__induct,axiom,
! [P: list_set_set_a > $o,Xs: list_set_set_a] :
( ! [Xs2: list_set_set_a] :
( ! [Ys4: list_set_set_a] :
( ( ord_less_nat @ ( size_s739728560_set_a @ Ys4 ) @ ( size_s739728560_set_a @ Xs2 ) )
=> ( P @ Ys4 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_149_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs2: list_nat] :
( ! [Ys4: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys4 ) @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ Ys4 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_150_length__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ! [Xs2: list_a] :
( ! [Ys4: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ Ys4 ) @ ( size_size_list_a @ Xs2 ) )
=> ( P @ Ys4 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_151_length__induct,axiom,
! [P: list_set_a > $o,Xs: list_set_a] :
( ! [Xs2: list_set_a] :
( ! [Ys4: list_set_a] :
( ( ord_less_nat @ ( size_size_list_set_a @ Ys4 ) @ ( size_size_list_set_a @ Xs2 ) )
=> ( P @ Ys4 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_152_listsetD_I1_J,axiom,
! [Ys: list_a,Xs: list_set_a] :
( ( member_list_a @ Ys @ ( listset_a @ Xs ) )
=> ( ( size_size_list_a @ Ys )
= ( size_size_list_set_a @ Xs ) ) ) ).
% listsetD(1)
thf(fact_153_listsetD_I1_J,axiom,
! [Ys: list_set_a,Xs: list_set_set_a] :
( ( member_list_set_a @ Ys @ ( listset_set_a @ Xs ) )
=> ( ( size_size_list_set_a @ Ys )
= ( size_s739728560_set_a @ Xs ) ) ) ).
% listsetD(1)
thf(fact_154_permutation__Ex__bij,axiom,
! [Xs: list_a,Ys: list_a] :
( ( perm_a @ Xs @ Ys )
=> ? [F2: nat > nat] :
( ( bij_betw_nat_nat @ F2 @ ( set_ord_lessThan_nat @ ( size_size_list_a @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_size_list_a @ Ys ) ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ Xs @ I2 )
= ( nth_a @ Ys @ ( F2 @ I2 ) ) ) ) ) ) ).
% permutation_Ex_bij
thf(fact_155_permutation__Ex__bij,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( perm_set_a @ Xs @ Ys )
=> ? [F2: nat > nat] :
( ( bij_betw_nat_nat @ F2 @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ Xs ) ) @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ Ys ) ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_set_a @ Xs ) )
=> ( ( nth_set_a @ Xs @ I2 )
= ( nth_set_a @ Ys @ ( F2 @ I2 ) ) ) ) ) ) ).
% permutation_Ex_bij
thf(fact_156_g__def,axiom,
( g
= ( hilber815131374at_nat @ ( set_ord_lessThan_nat @ ( size_size_list_set_a @ ss2 ) ) @ f ) ) ).
% g_def
thf(fact_157_perm__refl,axiom,
! [L: list_set_a] : ( perm_set_a @ L @ L ) ).
% perm_refl
thf(fact_158_image__eqI,axiom,
! [B: a,F: a > a,X4: a,A: set_a] :
( ( B
= ( F @ X4 ) )
=> ( ( member_a @ X4 @ A )
=> ( member_a @ B @ ( image_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_159_image__eqI,axiom,
! [B: list_a,F: a > list_a,X4: a,A: set_a] :
( ( B
= ( F @ X4 ) )
=> ( ( member_a @ X4 @ A )
=> ( member_list_a @ B @ ( image_a_list_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_160_image__eqI,axiom,
! [B: a,F: list_a > a,X4: list_a,A: set_list_a] :
( ( B
= ( F @ X4 ) )
=> ( ( member_list_a @ X4 @ A )
=> ( member_a @ B @ ( image_list_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_161_image__eqI,axiom,
! [B: list_a,F: list_a > list_a,X4: list_a,A: set_list_a] :
( ( B
= ( F @ X4 ) )
=> ( ( member_list_a @ X4 @ A )
=> ( member_list_a @ B @ ( image_list_a_list_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_162_mset__eq__perm,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( mset_a @ Xs )
= ( mset_a @ Ys ) )
= ( perm_a @ Xs @ Ys ) ) ).
% mset_eq_perm
thf(fact_163_mset__eq__perm,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( ( mset_set_a @ Xs )
= ( mset_set_a @ Ys ) )
= ( perm_set_a @ Xs @ Ys ) ) ).
% mset_eq_perm
thf(fact_164_perm__rev,axiom,
! [Xs: list_set_a] : ( perm_set_a @ ( rev_set_a @ Xs ) @ Xs ) ).
% perm_rev
thf(fact_165_perm__length,axiom,
! [Xs: list_a,Ys: list_a] :
( ( perm_a @ Xs @ Ys )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% perm_length
thf(fact_166_perm__length,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( perm_set_a @ Xs @ Ys )
=> ( ( size_size_list_set_a @ Xs )
= ( size_size_list_set_a @ Ys ) ) ) ).
% perm_length
thf(fact_167_bij__betw__imp__surj__on,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ F @ A @ B2 )
=> ( ( image_nat_nat @ F @ A )
= B2 ) ) ).
% bij_betw_imp_surj_on
thf(fact_168_mbs_Oless__not__eq,axiom,
! [X4: list_a,A: set_list_a,Y3: list_a] :
( ( member_list_a @ X4 @ A )
=> ( ( ord_less_nat @ ( size_size_list_a @ X4 ) @ ( size_size_list_a @ Y3 ) )
=> ( X4 != Y3 ) ) ) ).
% mbs.less_not_eq
thf(fact_169_mbs_Oless__not__eq,axiom,
! [X4: list_set_a,A: set_list_set_a,Y3: list_set_a] :
( ( member_list_set_a @ X4 @ A )
=> ( ( ord_less_nat @ ( size_size_list_set_a @ X4 ) @ ( size_size_list_set_a @ Y3 ) )
=> ( X4 != Y3 ) ) ) ).
% mbs.less_not_eq
thf(fact_170_size__mset,axiom,
! [Xs: list_a] :
( ( size_size_multiset_a @ ( mset_a @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% size_mset
thf(fact_171_size__mset,axiom,
! [Xs: list_set_a] :
( ( size_s657334288_set_a @ ( mset_set_a @ Xs ) )
= ( size_size_list_set_a @ Xs ) ) ).
% size_mset
thf(fact_172_psubsetD,axiom,
! [A: set_a,B2: set_a,C: a] :
( ( ord_less_set_a @ A @ B2 )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_173_psubsetD,axiom,
! [A: set_list_a,B2: set_list_a,C: list_a] :
( ( ord_less_set_list_a @ A @ B2 )
=> ( ( member_list_a @ C @ A )
=> ( member_list_a @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_174_imageI,axiom,
! [X4: a,A: set_a,F: a > a] :
( ( member_a @ X4 @ A )
=> ( member_a @ ( F @ X4 ) @ ( image_a_a @ F @ A ) ) ) ).
% imageI
thf(fact_175_imageI,axiom,
! [X4: a,A: set_a,F: a > list_a] :
( ( member_a @ X4 @ A )
=> ( member_list_a @ ( F @ X4 ) @ ( image_a_list_a @ F @ A ) ) ) ).
% imageI
thf(fact_176_imageI,axiom,
! [X4: list_a,A: set_list_a,F: list_a > a] :
( ( member_list_a @ X4 @ A )
=> ( member_a @ ( F @ X4 ) @ ( image_list_a_a @ F @ A ) ) ) ).
% imageI
thf(fact_177_imageI,axiom,
! [X4: list_a,A: set_list_a,F: list_a > list_a] :
( ( member_list_a @ X4 @ A )
=> ( member_list_a @ ( F @ X4 ) @ ( image_list_a_list_a @ F @ A ) ) ) ).
% imageI
thf(fact_178_rev__image__eqI,axiom,
! [X4: a,A: set_a,B: a,F: a > a] :
( ( member_a @ X4 @ A )
=> ( ( B
= ( F @ X4 ) )
=> ( member_a @ B @ ( image_a_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_179_rev__image__eqI,axiom,
! [X4: a,A: set_a,B: list_a,F: a > list_a] :
( ( member_a @ X4 @ A )
=> ( ( B
= ( F @ X4 ) )
=> ( member_list_a @ B @ ( image_a_list_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_180_rev__image__eqI,axiom,
! [X4: list_a,A: set_list_a,B: a,F: list_a > a] :
( ( member_list_a @ X4 @ A )
=> ( ( B
= ( F @ X4 ) )
=> ( member_a @ B @ ( image_list_a_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_181_rev__image__eqI,axiom,
! [X4: list_a,A: set_list_a,B: list_a,F: list_a > list_a] :
( ( member_list_a @ X4 @ A )
=> ( ( B
= ( F @ X4 ) )
=> ( member_list_a @ B @ ( image_list_a_list_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_182_direct__decomp__def,axiom,
( hilber2037636820comp_a
= ( ^ [A4: set_a,Ss2: list_set_a] : ( bij_betw_list_a_a @ groups1792256535list_a @ ( listset_a @ Ss2 ) @ A4 ) ) ) ).
% direct_decomp_def
thf(fact_183_bij__betwE,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ F @ A @ B2 )
=> ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ B2 ) ) ) ).
% bij_betwE
thf(fact_184_bij__betw__inv,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ F @ A @ B2 )
=> ? [G: nat > nat] : ( bij_betw_nat_nat @ G @ B2 @ A ) ) ).
% bij_betw_inv
thf(fact_185_bij__betw__cong,axiom,
! [A: set_nat,F: nat > nat,G2: nat > nat,A5: set_nat] :
( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ( ( F @ A3 )
= ( G2 @ A3 ) ) )
=> ( ( bij_betw_nat_nat @ F @ A @ A5 )
= ( bij_betw_nat_nat @ G2 @ A @ A5 ) ) ) ).
% bij_betw_cong
thf(fact_186_bij__betw__apply,axiom,
! [F: a > a,A: set_a,B2: set_a,A2: a] :
( ( bij_betw_a_a @ F @ A @ B2 )
=> ( ( member_a @ A2 @ A )
=> ( member_a @ ( F @ A2 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_187_bij__betw__apply,axiom,
! [F: a > list_a,A: set_a,B2: set_list_a,A2: a] :
( ( bij_betw_a_list_a @ F @ A @ B2 )
=> ( ( member_a @ A2 @ A )
=> ( member_list_a @ ( F @ A2 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_188_bij__betw__apply,axiom,
! [F: list_a > a,A: set_list_a,B2: set_a,A2: list_a] :
( ( bij_betw_list_a_a @ F @ A @ B2 )
=> ( ( member_list_a @ A2 @ A )
=> ( member_a @ ( F @ A2 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_189_bij__betw__apply,axiom,
! [F: list_a > list_a,A: set_list_a,B2: set_list_a,A2: list_a] :
( ( bij_be94573046list_a @ F @ A @ B2 )
=> ( ( member_list_a @ A2 @ A )
=> ( member_list_a @ ( F @ A2 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_190_bij__betw__apply,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat,A2: nat] :
( ( bij_betw_nat_nat @ F @ A @ B2 )
=> ( ( member_nat @ A2 @ A )
=> ( member_nat @ ( F @ A2 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_191_bij__betw__iff__bijections,axiom,
( bij_betw_a_a
= ( ^ [F3: a > a,A4: set_a,B3: set_a] :
? [G3: a > a] :
( ! [X5: a] :
( ( member_a @ X5 @ A4 )
=> ( ( member_a @ ( F3 @ X5 ) @ B3 )
& ( ( G3 @ ( F3 @ X5 ) )
= X5 ) ) )
& ! [X5: a] :
( ( member_a @ X5 @ B3 )
=> ( ( member_a @ ( G3 @ X5 ) @ A4 )
& ( ( F3 @ ( G3 @ X5 ) )
= X5 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_192_bij__betw__iff__bijections,axiom,
( bij_betw_list_a_a
= ( ^ [F3: list_a > a,A4: set_list_a,B3: set_a] :
? [G3: a > list_a] :
( ! [X5: list_a] :
( ( member_list_a @ X5 @ A4 )
=> ( ( member_a @ ( F3 @ X5 ) @ B3 )
& ( ( G3 @ ( F3 @ X5 ) )
= X5 ) ) )
& ! [X5: a] :
( ( member_a @ X5 @ B3 )
=> ( ( member_list_a @ ( G3 @ X5 ) @ A4 )
& ( ( F3 @ ( G3 @ X5 ) )
= X5 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_193_bij__betw__iff__bijections,axiom,
( bij_betw_a_list_a
= ( ^ [F3: a > list_a,A4: set_a,B3: set_list_a] :
? [G3: list_a > a] :
( ! [X5: a] :
( ( member_a @ X5 @ A4 )
=> ( ( member_list_a @ ( F3 @ X5 ) @ B3 )
& ( ( G3 @ ( F3 @ X5 ) )
= X5 ) ) )
& ! [X5: list_a] :
( ( member_list_a @ X5 @ B3 )
=> ( ( member_a @ ( G3 @ X5 ) @ A4 )
& ( ( F3 @ ( G3 @ X5 ) )
= X5 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_194_bij__betw__iff__bijections,axiom,
( bij_be94573046list_a
= ( ^ [F3: list_a > list_a,A4: set_list_a,B3: set_list_a] :
? [G3: list_a > list_a] :
( ! [X5: list_a] :
( ( member_list_a @ X5 @ A4 )
=> ( ( member_list_a @ ( F3 @ X5 ) @ B3 )
& ( ( G3 @ ( F3 @ X5 ) )
= X5 ) ) )
& ! [X5: list_a] :
( ( member_list_a @ X5 @ B3 )
=> ( ( member_list_a @ ( G3 @ X5 ) @ A4 )
& ( ( F3 @ ( G3 @ X5 ) )
= X5 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_195_bij__betw__iff__bijections,axiom,
( bij_betw_nat_nat
= ( ^ [F3: nat > nat,A4: set_nat,B3: set_nat] :
? [G3: nat > nat] :
( ! [X5: nat] :
( ( member_nat @ X5 @ A4 )
=> ( ( member_nat @ ( F3 @ X5 ) @ B3 )
& ( ( G3 @ ( F3 @ X5 ) )
= X5 ) ) )
& ! [X5: nat] :
( ( member_nat @ X5 @ B3 )
=> ( ( member_nat @ ( G3 @ X5 ) @ A4 )
& ( ( F3 @ ( G3 @ X5 ) )
= X5 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_196_perm__sym,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( perm_set_a @ Xs @ Ys )
=> ( perm_set_a @ Ys @ Xs ) ) ).
% perm_sym
thf(fact_197_perm_Otrans,axiom,
! [Xs: list_set_a,Ys: list_set_a,Zs: list_set_a] :
( ( perm_set_a @ Xs @ Ys )
=> ( ( perm_set_a @ Ys @ Zs )
=> ( perm_set_a @ Xs @ Zs ) ) ) ).
% perm.trans
thf(fact_198_bij__betw__inv__into__right,axiom,
! [F: nat > nat,A: set_nat,A5: set_nat,A6: nat] :
( ( bij_betw_nat_nat @ F @ A @ A5 )
=> ( ( member_nat @ A6 @ A5 )
=> ( ( F @ ( hilber815131374at_nat @ A @ F @ A6 ) )
= A6 ) ) ) ).
% bij_betw_inv_into_right
thf(fact_199_bij__betw__inv__into__left,axiom,
! [F: nat > nat,A: set_nat,A5: set_nat,A2: nat] :
( ( bij_betw_nat_nat @ F @ A @ A5 )
=> ( ( member_nat @ A2 @ A )
=> ( ( hilber815131374at_nat @ A @ F @ ( F @ A2 ) )
= A2 ) ) ) ).
% bij_betw_inv_into_left
thf(fact_200_inv__into__inv__into__eq,axiom,
! [F: nat > nat,A: set_nat,A5: set_nat,A2: nat] :
( ( bij_betw_nat_nat @ F @ A @ A5 )
=> ( ( member_nat @ A2 @ A )
=> ( ( hilber815131374at_nat @ A5 @ ( hilber815131374at_nat @ A @ F ) @ A2 )
= ( F @ A2 ) ) ) ) ).
% inv_into_inv_into_eq
thf(fact_201_bij__betw__inv__into,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ F @ A @ B2 )
=> ( bij_betw_nat_nat @ ( hilber815131374at_nat @ A @ F ) @ B2 @ A ) ) ).
% bij_betw_inv_into
thf(fact_202_inv__into__injective,axiom,
! [A: set_nat,F: nat > nat,X4: nat,Y3: nat] :
( ( ( hilber815131374at_nat @ A @ F @ X4 )
= ( hilber815131374at_nat @ A @ F @ Y3 ) )
=> ( ( member_nat @ X4 @ ( image_nat_nat @ F @ A ) )
=> ( ( member_nat @ Y3 @ ( image_nat_nat @ F @ A ) )
=> ( X4 = Y3 ) ) ) ) ).
% inv_into_injective
thf(fact_203_inv__into__into,axiom,
! [X4: a,F: a > a,A: set_a] :
( ( member_a @ X4 @ ( image_a_a @ F @ A ) )
=> ( member_a @ ( hilbert_inv_into_a_a @ A @ F @ X4 ) @ A ) ) ).
% inv_into_into
thf(fact_204_inv__into__into,axiom,
! [X4: a,F: list_a > a,A: set_list_a] :
( ( member_a @ X4 @ ( image_list_a_a @ F @ A ) )
=> ( member_list_a @ ( hilber2125729734st_a_a @ A @ F @ X4 ) @ A ) ) ).
% inv_into_into
thf(fact_205_inv__into__into,axiom,
! [X4: list_a,F: a > list_a,A: set_a] :
( ( member_list_a @ X4 @ ( image_a_list_a @ F @ A ) )
=> ( member_a @ ( hilber165648082list_a @ A @ F @ X4 ) @ A ) ) ).
% inv_into_into
thf(fact_206_inv__into__into,axiom,
! [X4: list_a,F: list_a > list_a,A: set_list_a] :
( ( member_list_a @ X4 @ ( image_list_a_list_a @ F @ A ) )
=> ( member_list_a @ ( hilber921249868list_a @ A @ F @ X4 ) @ A ) ) ).
% inv_into_into
thf(fact_207_inv__into__into,axiom,
! [X4: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ X4 @ ( image_nat_nat @ F @ A ) )
=> ( member_nat @ ( hilber815131374at_nat @ A @ F @ X4 ) @ A ) ) ).
% inv_into_into
thf(fact_208_f__inv__into__f,axiom,
! [Y3: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ Y3 @ ( image_nat_nat @ F @ A ) )
=> ( ( F @ ( hilber815131374at_nat @ A @ F @ Y3 ) )
= Y3 ) ) ).
% f_inv_into_f
thf(fact_209_list__ex__length,axiom,
( list_ex_a
= ( ^ [P2: a > $o,Xs4: list_a] :
? [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs4 ) )
& ( P2 @ ( nth_a @ Xs4 @ N2 ) ) ) ) ) ).
% list_ex_length
thf(fact_210_list__ex__length,axiom,
( list_ex_set_a
= ( ^ [P2: set_a > $o,Xs4: list_set_a] :
? [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_set_a @ Xs4 ) )
& ( P2 @ ( nth_set_a @ Xs4 @ N2 ) ) ) ) ) ).
% list_ex_length
thf(fact_211_direct__decompI,axiom,
! [Ss: list_set_a,A: set_a] :
( ( inj_on_list_a_a @ groups1792256535list_a @ ( listset_a @ Ss ) )
=> ( ( ( image_list_a_a @ groups1792256535list_a @ ( listset_a @ Ss ) )
= A )
=> ( hilber2037636820comp_a @ A @ Ss ) ) ) ).
% direct_decompI
thf(fact_212_elem__le__sum__list,axiom,
! [K: nat,Ns: list_nat] :
( ( ord_less_nat @ K @ ( size_size_list_nat @ Ns ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Ns @ K ) @ ( groups921905271st_nat @ Ns ) ) ) ).
% elem_le_sum_list
thf(fact_213_perm__swap,axiom,
! [I: nat,Xs: list_a,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs ) )
=> ( perm_a @ ( list_update_a @ ( list_update_a @ Xs @ I @ ( nth_a @ Xs @ J ) ) @ J @ ( nth_a @ Xs @ I ) ) @ Xs ) ) ) ).
% perm_swap
thf(fact_214_perm__swap,axiom,
! [I: nat,Xs: list_set_a,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_set_a @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_set_a @ Xs ) )
=> ( perm_set_a @ ( list_update_set_a @ ( list_update_set_a @ Xs @ I @ ( nth_set_a @ Xs @ J ) ) @ J @ ( nth_set_a @ Xs @ I ) ) @ Xs ) ) ) ).
% perm_swap
thf(fact_215_mset__swap,axiom,
! [I: nat,Ls: list_a,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Ls ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Ls ) )
=> ( ( mset_a @ ( list_update_a @ ( list_update_a @ Ls @ J @ ( nth_a @ Ls @ I ) ) @ I @ ( nth_a @ Ls @ J ) ) )
= ( mset_a @ Ls ) ) ) ) ).
% mset_swap
thf(fact_216_mset__swap,axiom,
! [I: nat,Ls: list_set_a,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_set_a @ Ls ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_set_a @ Ls ) )
=> ( ( mset_set_a @ ( list_update_set_a @ ( list_update_set_a @ Ls @ J @ ( nth_set_a @ Ls @ I ) ) @ I @ ( nth_set_a @ Ls @ J ) ) )
= ( mset_set_a @ Ls ) ) ) ) ).
% mset_swap
thf(fact_217_nth__mem__mset,axiom,
! [I: nat,Ls: list_list_a] :
( ( ord_less_nat @ I @ ( size_s1427607542list_a @ Ls ) )
=> ( member_list_a @ ( nth_list_a @ Ls @ I ) @ ( set_mset_list_a @ ( mset_list_a @ Ls ) ) ) ) ).
% nth_mem_mset
thf(fact_218_nth__mem__mset,axiom,
! [I: nat,Ls: list_a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Ls ) )
=> ( member_a @ ( nth_a @ Ls @ I ) @ ( set_mset_a @ ( mset_a @ Ls ) ) ) ) ).
% nth_mem_mset
thf(fact_219_nth__mem__mset,axiom,
! [I: nat,Ls: list_set_a] :
( ( ord_less_nat @ I @ ( size_size_list_set_a @ Ls ) )
=> ( member_set_a @ ( nth_set_a @ Ls @ I ) @ ( set_mset_set_a @ ( mset_set_a @ Ls ) ) ) ) ).
% nth_mem_mset
thf(fact_220_in__Union__mset__iff,axiom,
! [X4: a,MM: multiset_multiset_a] :
( ( member_a @ X4 @ ( set_mset_a @ ( comm_m2145643721iset_a @ MM ) ) )
= ( ? [M2: multiset_a] :
( ( member_multiset_a @ M2 @ ( set_mset_multiset_a @ MM ) )
& ( member_a @ X4 @ ( set_mset_a @ M2 ) ) ) ) ) ).
% in_Union_mset_iff
thf(fact_221_in__Union__mset__iff,axiom,
! [X4: list_a,MM: multis971982480list_a] :
( ( member_list_a @ X4 @ ( set_mset_list_a @ ( comm_m324867663list_a @ MM ) ) )
= ( ? [M2: multiset_list_a] :
( ( member528267379list_a @ M2 @ ( set_ms1660427399list_a @ MM ) )
& ( member_list_a @ X4 @ ( set_mset_list_a @ M2 ) ) ) ) ) ).
% in_Union_mset_iff
thf(fact_222_lessThan__subset__iff,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X4 ) @ ( set_ord_lessThan_nat @ Y3 ) )
= ( ord_less_eq_nat @ X4 @ Y3 ) ) ).
% lessThan_subset_iff
thf(fact_223_list__update__beyond,axiom,
! [Xs: list_a,I: nat,X4: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ I )
=> ( ( list_update_a @ Xs @ I @ X4 )
= Xs ) ) ).
% list_update_beyond
thf(fact_224_list__update__beyond,axiom,
! [Xs: list_set_a,I: nat,X4: set_a] :
( ( ord_less_eq_nat @ ( size_size_list_set_a @ Xs ) @ I )
=> ( ( list_update_set_a @ Xs @ I @ X4 )
= Xs ) ) ).
% list_update_beyond
thf(fact_225_length__list__update,axiom,
! [Xs: list_a,I: nat,X4: a] :
( ( size_size_list_a @ ( list_update_a @ Xs @ I @ X4 ) )
= ( size_size_list_a @ Xs ) ) ).
% length_list_update
thf(fact_226_length__list__update,axiom,
! [Xs: list_set_a,I: nat,X4: set_a] :
( ( size_size_list_set_a @ ( list_update_set_a @ Xs @ I @ X4 ) )
= ( size_size_list_set_a @ Xs ) ) ).
% length_list_update
thf(fact_227_inv__into__f__f,axiom,
! [F: nat > nat,A: set_nat,X4: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( member_nat @ X4 @ A )
=> ( ( hilber815131374at_nat @ A @ F @ ( F @ X4 ) )
= X4 ) ) ) ).
% inv_into_f_f
thf(fact_228_list__update__id,axiom,
! [Xs: list_a,I: nat] :
( ( list_update_a @ Xs @ I @ ( nth_a @ Xs @ I ) )
= Xs ) ).
% list_update_id
thf(fact_229_list__update__id,axiom,
! [Xs: list_set_a,I: nat] :
( ( list_update_set_a @ Xs @ I @ ( nth_set_a @ Xs @ I ) )
= Xs ) ).
% list_update_id
thf(fact_230_nth__list__update__neq,axiom,
! [I: nat,J: nat,Xs: list_a,X4: a] :
( ( I != J )
=> ( ( nth_a @ ( list_update_a @ Xs @ I @ X4 ) @ J )
= ( nth_a @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_231_nth__list__update__neq,axiom,
! [I: nat,J: nat,Xs: list_set_a,X4: set_a] :
( ( I != J )
=> ( ( nth_set_a @ ( list_update_set_a @ Xs @ I @ X4 ) @ J )
= ( nth_set_a @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_232_inv__into__image__cancel,axiom,
! [F: nat > nat,A: set_nat,S: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ S @ A )
=> ( ( image_nat_nat @ ( hilber815131374at_nat @ A @ F ) @ ( image_nat_nat @ F @ S ) )
= S ) ) ) ).
% inv_into_image_cancel
thf(fact_233_nth__list__update__eq,axiom,
! [I: nat,Xs: list_a,X4: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( list_update_a @ Xs @ I @ X4 ) @ I )
= X4 ) ) ).
% nth_list_update_eq
thf(fact_234_nth__list__update__eq,axiom,
! [I: nat,Xs: list_set_a,X4: set_a] :
( ( ord_less_nat @ I @ ( size_size_list_set_a @ Xs ) )
=> ( ( nth_set_a @ ( list_update_set_a @ Xs @ I @ X4 ) @ I )
= X4 ) ) ).
% nth_list_update_eq
thf(fact_235_inj__on__inv__into,axiom,
! [B2: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
=> ( inj_on_nat_nat @ ( hilber815131374at_nat @ A @ F ) @ B2 ) ) ).
% inj_on_inv_into
thf(fact_236_inj__on__image__mem__iff,axiom,
! [F: a > a,B2: set_a,A2: a,A: set_a] :
( ( inj_on_a_a @ F @ B2 )
=> ( ( member_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( member_a @ ( F @ A2 ) @ ( image_a_a @ F @ A ) )
= ( member_a @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_237_inj__on__image__mem__iff,axiom,
! [F: a > list_a,B2: set_a,A2: a,A: set_a] :
( ( inj_on_a_list_a @ F @ B2 )
=> ( ( member_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( member_list_a @ ( F @ A2 ) @ ( image_a_list_a @ F @ A ) )
= ( member_a @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_238_inj__on__image__mem__iff,axiom,
! [F: list_a > a,B2: set_list_a,A2: list_a,A: set_list_a] :
( ( inj_on_list_a_a @ F @ B2 )
=> ( ( member_list_a @ A2 @ B2 )
=> ( ( ord_le1301786372list_a @ A @ B2 )
=> ( ( member_a @ ( F @ A2 ) @ ( image_list_a_a @ F @ A ) )
= ( member_list_a @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_239_inj__on__image__mem__iff,axiom,
! [F: list_a > list_a,B2: set_list_a,A2: list_a,A: set_list_a] :
( ( inj_on_list_a_list_a @ F @ B2 )
=> ( ( member_list_a @ A2 @ B2 )
=> ( ( ord_le1301786372list_a @ A @ B2 )
=> ( ( member_list_a @ ( F @ A2 ) @ ( image_list_a_list_a @ F @ A ) )
= ( member_list_a @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_240_bij__betw__imp__inj__on,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ F @ A @ B2 )
=> ( inj_on_nat_nat @ F @ A ) ) ).
% bij_betw_imp_inj_on
thf(fact_241_inv__into__f__eq,axiom,
! [F: nat > nat,A: set_nat,X4: nat,Y3: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( member_nat @ X4 @ A )
=> ( ( ( F @ X4 )
= Y3 )
=> ( ( hilber815131374at_nat @ A @ F @ Y3 )
= X4 ) ) ) ) ).
% inv_into_f_eq
thf(fact_242_image__subsetI,axiom,
! [A: set_a,F: a > a,B2: set_a] :
( ! [X6: a] :
( ( member_a @ X6 @ A )
=> ( member_a @ ( F @ X6 ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_243_image__subsetI,axiom,
! [A: set_a,F: a > list_a,B2: set_list_a] :
( ! [X6: a] :
( ( member_a @ X6 @ A )
=> ( member_list_a @ ( F @ X6 ) @ B2 ) )
=> ( ord_le1301786372list_a @ ( image_a_list_a @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_244_image__subsetI,axiom,
! [A: set_list_a,F: list_a > a,B2: set_a] :
( ! [X6: list_a] :
( ( member_list_a @ X6 @ A )
=> ( member_a @ ( F @ X6 ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_list_a_a @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_245_image__subsetI,axiom,
! [A: set_list_a,F: list_a > list_a,B2: set_list_a] :
( ! [X6: list_a] :
( ( member_list_a @ X6 @ A )
=> ( member_list_a @ ( F @ X6 ) @ B2 ) )
=> ( ord_le1301786372list_a @ ( image_list_a_list_a @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_246_bij__betw__def,axiom,
( bij_betw_nat_nat
= ( ^ [F3: nat > nat,A4: set_nat,B3: set_nat] :
( ( inj_on_nat_nat @ F3 @ A4 )
& ( ( image_nat_nat @ F3 @ A4 )
= B3 ) ) ) ) ).
% bij_betw_def
thf(fact_247_bij__betw__imageI,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ( image_nat_nat @ F @ A )
= B2 )
=> ( bij_betw_nat_nat @ F @ A @ B2 ) ) ) ).
% bij_betw_imageI
thf(fact_248_inj__on__imp__bij__betw,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( bij_betw_nat_nat @ F @ A @ ( image_nat_nat @ F @ A ) ) ) ).
% inj_on_imp_bij_betw
thf(fact_249_mbs_Ominimal,axiom,
! [X4: list_a,A: set_list_a,P: list_a > $o] :
( ( member_list_a @ X4 @ A )
=> ( ( P @ X4 )
=> ? [X6: list_a] :
( ( member_list_a @ X6 @ A )
& ( ord_less_eq_nat @ ( size_size_list_a @ X6 ) @ ( size_size_list_a @ X4 ) )
& ( P @ X6 )
& ! [Xa: list_a] :
( ( member_list_a @ Xa @ A )
=> ( ( ord_less_nat @ ( size_size_list_a @ Xa ) @ ( size_size_list_a @ X6 ) )
=> ~ ( P @ Xa ) ) ) ) ) ) ).
% mbs.minimal
thf(fact_250_mbs_Ominimal,axiom,
! [X4: list_set_a,A: set_list_set_a,P: list_set_a > $o] :
( ( member_list_set_a @ X4 @ A )
=> ( ( P @ X4 )
=> ? [X6: list_set_a] :
( ( member_list_set_a @ X6 @ A )
& ( ord_less_eq_nat @ ( size_size_list_set_a @ X6 ) @ ( size_size_list_set_a @ X4 ) )
& ( P @ X6 )
& ! [Xa: list_set_a] :
( ( member_list_set_a @ Xa @ A )
=> ( ( ord_less_nat @ ( size_size_list_set_a @ Xa ) @ ( size_size_list_set_a @ X6 ) )
=> ~ ( P @ Xa ) ) ) ) ) ) ).
% mbs.minimal
thf(fact_251_bij__betw__subset,axiom,
! [F: nat > nat,A: set_nat,A5: set_nat,B2: set_nat,B4: set_nat] :
( ( bij_betw_nat_nat @ F @ A @ A5 )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( ( image_nat_nat @ F @ B2 )
= B4 )
=> ( bij_betw_nat_nat @ F @ B2 @ B4 ) ) ) ) ).
% bij_betw_subset
thf(fact_252_bij__betw__byWitness,axiom,
! [A: set_nat,F4: nat > nat,F: nat > nat,A5: set_nat] :
( ! [X6: nat] :
( ( member_nat @ X6 @ A )
=> ( ( F4 @ ( F @ X6 ) )
= X6 ) )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A5 )
=> ( ( F @ ( F4 @ X6 ) )
= X6 ) )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ A5 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F4 @ A5 ) @ A )
=> ( bij_betw_nat_nat @ F @ A @ A5 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_253_image__inv__into__cancel,axiom,
! [F: nat > nat,A: set_nat,A5: set_nat,B4: set_nat] :
( ( ( image_nat_nat @ F @ A )
= A5 )
=> ( ( ord_less_eq_set_nat @ B4 @ A5 )
=> ( ( image_nat_nat @ F @ ( image_nat_nat @ ( hilber815131374at_nat @ A @ F ) @ B4 ) )
= B4 ) ) ) ).
% image_inv_into_cancel
thf(fact_254_listset__mono,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( ( size_size_list_set_a @ Xs )
= ( size_size_list_set_a @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_set_a @ Ys ) )
=> ( ord_less_eq_set_a @ ( nth_set_a @ Xs @ I3 ) @ ( nth_set_a @ Ys @ I3 ) ) )
=> ( ord_le1301786372list_a @ ( listset_a @ Xs ) @ ( listset_a @ Ys ) ) ) ) ).
% listset_mono
thf(fact_255_nth__list__update,axiom,
! [I: nat,Xs: list_a,J: nat,X4: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth_a @ ( list_update_a @ Xs @ I @ X4 ) @ J )
= X4 ) )
& ( ( I != J )
=> ( ( nth_a @ ( list_update_a @ Xs @ I @ X4 ) @ J )
= ( nth_a @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_256_nth__list__update,axiom,
! [I: nat,Xs: list_set_a,J: nat,X4: set_a] :
( ( ord_less_nat @ I @ ( size_size_list_set_a @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth_set_a @ ( list_update_set_a @ Xs @ I @ X4 ) @ J )
= X4 ) )
& ( ( I != J )
=> ( ( nth_set_a @ ( list_update_set_a @ Xs @ I @ X4 ) @ J )
= ( nth_set_a @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_257_list__update__same__conv,axiom,
! [I: nat,Xs: list_a,X4: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( ( list_update_a @ Xs @ I @ X4 )
= Xs )
= ( ( nth_a @ Xs @ I )
= X4 ) ) ) ).
% list_update_same_conv
thf(fact_258_list__update__same__conv,axiom,
! [I: nat,Xs: list_set_a,X4: set_a] :
( ( ord_less_nat @ I @ ( size_size_list_set_a @ Xs ) )
=> ( ( ( list_update_set_a @ Xs @ I @ X4 )
= Xs )
= ( ( nth_set_a @ Xs @ I )
= X4 ) ) ) ).
% list_update_same_conv
thf(fact_259_bij__betw__inv__into__subset,axiom,
! [F: nat > nat,A: set_nat,A5: set_nat,B2: set_nat,B4: set_nat] :
( ( bij_betw_nat_nat @ F @ A @ A5 )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( ( image_nat_nat @ F @ B2 )
= B4 )
=> ( bij_betw_nat_nat @ ( hilber815131374at_nat @ A @ F ) @ B4 @ B2 ) ) ) ) ).
% bij_betw_inv_into_subset
thf(fact_260_direct__decompD_I2_J,axiom,
! [A: set_a,Ss: list_set_a] :
( ( hilber2037636820comp_a @ A @ Ss )
=> ( inj_on_list_a_a @ groups1792256535list_a @ ( listset_a @ Ss ) ) ) ).
% direct_decompD(2)
thf(fact_261_Schroeder__Bernstein,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat,G2: nat > nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B2 )
=> ( ( inj_on_nat_nat @ G2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G2 @ B2 ) @ A )
=> ? [H: nat > nat] : ( bij_betw_nat_nat @ H @ A @ B2 ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_262_order__refl,axiom,
! [X4: nat] : ( ord_less_eq_nat @ X4 @ X4 ) ).
% order_refl
thf(fact_263_minf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ T @ X ) ) ).
% minf(8)
thf(fact_264_minf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z2 )
=> ( ord_less_eq_nat @ X @ T ) ) ).
% minf(6)
thf(fact_265_subsetI,axiom,
! [A: set_a,B2: set_a] :
( ! [X6: a] :
( ( member_a @ X6 @ A )
=> ( member_a @ X6 @ B2 ) )
=> ( ord_less_eq_set_a @ A @ B2 ) ) ).
% subsetI
thf(fact_266_subsetI,axiom,
! [A: set_list_a,B2: set_list_a] :
( ! [X6: list_a] :
( ( member_list_a @ X6 @ A )
=> ( member_list_a @ X6 @ B2 ) )
=> ( ord_le1301786372list_a @ A @ B2 ) ) ).
% subsetI
thf(fact_267_bounded__Max__nat,axiom,
! [P: nat > $o,X4: nat,M3: nat] :
( ( P @ X4 )
=> ( ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M3 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X: nat] :
( ( P @ X )
=> ( ord_less_eq_nat @ X @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_268_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A4 )
=> ( member_a @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_269_subset__iff,axiom,
( ord_le1301786372list_a
= ( ^ [A4: set_list_a,B3: set_list_a] :
! [T2: list_a] :
( ( member_list_a @ T2 @ A4 )
=> ( member_list_a @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_270_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
! [X5: a] :
( ( member_a @ X5 @ A4 )
=> ( member_a @ X5 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_271_subset__eq,axiom,
( ord_le1301786372list_a
= ( ^ [A4: set_list_a,B3: set_list_a] :
! [X5: list_a] :
( ( member_list_a @ X5 @ A4 )
=> ( member_list_a @ X5 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_272_subsetD,axiom,
! [A: set_a,B2: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_273_subsetD,axiom,
! [A: set_list_a,B2: set_list_a,C: list_a] :
( ( ord_le1301786372list_a @ A @ B2 )
=> ( ( member_list_a @ C @ A )
=> ( member_list_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_274_in__mono,axiom,
! [A: set_a,B2: set_a,X4: a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( member_a @ X4 @ A )
=> ( member_a @ X4 @ B2 ) ) ) ).
% in_mono
thf(fact_275_in__mono,axiom,
! [A: set_list_a,B2: set_list_a,X4: list_a] :
( ( ord_le1301786372list_a @ A @ B2 )
=> ( ( member_list_a @ X4 @ A )
=> ( member_list_a @ X4 @ B2 ) ) ) ).
% in_mono
thf(fact_276_dual__order_Oantisym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_277_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : Y2 = Z )
= ( ^ [A7: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A7 )
& ( ord_less_eq_nat @ A7 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_278_dual__order_Otrans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_279_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A3: nat,B6: nat] :
( ( ord_less_eq_nat @ A3 @ B6 )
=> ( P @ A3 @ B6 ) )
=> ( ! [A3: nat,B6: nat] :
( ( P @ B6 @ A3 )
=> ( P @ A3 @ B6 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_280_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_281_order__trans,axiom,
! [X4: nat,Y3: nat,Z3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z3 )
=> ( ord_less_eq_nat @ X4 @ Z3 ) ) ) ).
% order_trans
thf(fact_282_order__class_Oorder_Oantisym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% order_class.order.antisym
thf(fact_283_ord__le__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_284_ord__eq__le__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_285_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : Y2 = Z )
= ( ^ [A7: nat,B5: nat] :
( ( ord_less_eq_nat @ A7 @ B5 )
& ( ord_less_eq_nat @ B5 @ A7 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_286_antisym__conv,axiom,
! [Y3: nat,X4: nat] :
( ( ord_less_eq_nat @ Y3 @ X4 )
=> ( ( ord_less_eq_nat @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ).
% antisym_conv
thf(fact_287_le__cases3,axiom,
! [X4: nat,Y3: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X4 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X4 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y3 ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ X4 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X4 ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_288_order_Otrans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_289_le__cases,axiom,
! [X4: nat,Y3: nat] :
( ~ ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X4 ) ) ).
% le_cases
thf(fact_290_eq__refl,axiom,
! [X4: nat,Y3: nat] :
( ( X4 = Y3 )
=> ( ord_less_eq_nat @ X4 @ Y3 ) ) ).
% eq_refl
thf(fact_291_linear,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X4 ) ) ).
% linear
thf(fact_292_antisym,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ X4 )
=> ( X4 = Y3 ) ) ) ).
% antisym
thf(fact_293_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_nat
= ( ^ [X7: $o > nat,Y4: $o > nat] :
( ( ord_less_eq_nat @ ( X7 @ $false ) @ ( Y4 @ $false ) )
& ( ord_less_eq_nat @ ( X7 @ $true ) @ ( Y4 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_294_eq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : Y2 = Z )
= ( ^ [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X5 ) ) ) ) ).
% eq_iff
thf(fact_295_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: nat,Y: nat] :
( ( ord_less_eq_nat @ X6 @ Y )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_296_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X6: nat,Y: nat] :
( ( ord_less_eq_nat @ X6 @ Y )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_297_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X6: nat,Y: nat] :
( ( ord_less_eq_nat @ X6 @ Y )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_298_order__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X6: nat,Y: nat] :
( ( ord_less_eq_nat @ X6 @ Y )
=> ( ord_less_eq_nat @ ( F @ X6 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_299_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_300_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_301_not__less__iff__gr__or__eq,axiom,
! [X4: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X4 @ Y3 ) )
= ( ( ord_less_nat @ Y3 @ X4 )
| ( X4 = Y3 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_302_dual__order_Ostrict__trans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_303_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A3: nat,B6: nat] :
( ( ord_less_nat @ A3 @ B6 )
=> ( P @ A3 @ B6 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B6: nat] :
( ( P @ B6 @ A3 )
=> ( P @ A3 @ B6 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_304_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X3: nat] : ( P3 @ X3 ) )
= ( ^ [P2: nat > $o] :
? [N2: nat] :
( ( P2 @ N2 )
& ! [M5: nat] :
( ( ord_less_nat @ M5 @ N2 )
=> ~ ( P2 @ M5 ) ) ) ) ) ).
% exists_least_iff
thf(fact_305_less__imp__not__less,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X4 ) ) ).
% less_imp_not_less
thf(fact_306_order_Ostrict__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_307_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_308_linorder__cases,axiom,
! [X4: nat,Y3: nat] :
( ~ ( ord_less_nat @ X4 @ Y3 )
=> ( ( X4 != Y3 )
=> ( ord_less_nat @ Y3 @ X4 ) ) ) ).
% linorder_cases
thf(fact_309_less__imp__triv,axiom,
! [X4: nat,Y3: nat,P: $o] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ X4 )
=> P ) ) ).
% less_imp_triv
thf(fact_310_less__imp__not__eq2,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( Y3 != X4 ) ) ).
% less_imp_not_eq2
thf(fact_311_antisym__conv3,axiom,
! [Y3: nat,X4: nat] :
( ~ ( ord_less_nat @ Y3 @ X4 )
=> ( ( ~ ( ord_less_nat @ X4 @ Y3 ) )
= ( X4 = Y3 ) ) ) ).
% antisym_conv3
thf(fact_312_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X6: nat] :
( ! [Y6: nat] :
( ( ord_less_nat @ Y6 @ X6 )
=> ( P @ Y6 ) )
=> ( P @ X6 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_313_less__not__sym,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X4 ) ) ).
% less_not_sym
thf(fact_314_less__imp__not__eq,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( X4 != Y3 ) ) ).
% less_imp_not_eq
thf(fact_315_dual__order_Oasym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ~ ( ord_less_nat @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_316_ord__less__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_317_ord__eq__less__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_318_less__irrefl,axiom,
! [X4: nat] :
~ ( ord_less_nat @ X4 @ X4 ) ).
% less_irrefl
thf(fact_319_less__linear,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
| ( X4 = Y3 )
| ( ord_less_nat @ Y3 @ X4 ) ) ).
% less_linear
thf(fact_320_less__trans,axiom,
! [X4: nat,Y3: nat,Z3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z3 )
=> ( ord_less_nat @ X4 @ Z3 ) ) ) ).
% less_trans
thf(fact_321_less__asym_H,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% less_asym'
thf(fact_322_less__asym,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X4 ) ) ).
% less_asym
thf(fact_323_less__imp__neq,axiom,
! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( X4 != Y3 ) ) ).
% less_imp_neq
thf(fact_324_order_Oasym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order.asym
thf(fact_325_neq__iff,axiom,
! [X4: nat,Y3: nat] :
( ( X4 != Y3 )
= ( ( ord_less_nat @ X4 @ Y3 )
| ( ord_less_nat @ Y3 @ X4 ) ) ) ).
% neq_iff
thf(fact_326_neqE,axiom,
! [X4: nat,Y3: nat] :
( ( X4 != Y3 )
=> ( ~ ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ Y3 @ X4 ) ) ) ).
% neqE
thf(fact_327_gt__ex,axiom,
! [X4: nat] :
? [X_1: nat] : ( ord_less_nat @ X4 @ X_1 ) ).
% gt_ex
thf(fact_328_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X6: nat,Y: nat] :
( ( ord_less_nat @ X6 @ Y )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_329_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X6: nat,Y: nat] :
( ( ord_less_nat @ X6 @ Y )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_330_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X6: nat,Y: nat] :
( ( ord_less_nat @ X6 @ Y )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_331_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X6: nat,Y: nat] :
( ( ord_less_nat @ X6 @ Y )
=> ( ord_less_nat @ ( F @ X6 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_332_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( ( P @ X6 )
= ( P4 @ X6 ) ) )
=> ( ? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( ( Q @ X6 )
= ( Q2 @ X6 ) ) )
=> ? [Z2: nat] :
! [X: nat] :
( ( ord_less_nat @ Z2 @ X )
=> ( ( ( P @ X )
& ( Q @ X ) )
= ( ( P4 @ X )
& ( Q2 @ X ) ) ) ) ) ) ).
% pinf(1)
thf(fact_333_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( ( P @ X6 )
= ( P4 @ X6 ) ) )
=> ( ? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( ( Q @ X6 )
= ( Q2 @ X6 ) ) )
=> ? [Z2: nat] :
! [X: nat] :
( ( ord_less_nat @ Z2 @ X )
=> ( ( ( P @ X )
| ( Q @ X ) )
= ( ( P4 @ X )
| ( Q2 @ X ) ) ) ) ) ) ).
% pinf(2)
thf(fact_334_pinf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X: nat] :
( ( ord_less_nat @ Z2 @ X )
=> ( X != T ) ) ).
% pinf(3)
thf(fact_335_pinf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X: nat] :
( ( ord_less_nat @ Z2 @ X )
=> ( X != T ) ) ).
% pinf(4)
thf(fact_336_pinf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X: nat] :
( ( ord_less_nat @ Z2 @ X )
=> ~ ( ord_less_nat @ X @ T ) ) ).
% pinf(5)
thf(fact_337_pinf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X: nat] :
( ( ord_less_nat @ Z2 @ X )
=> ( ord_less_nat @ T @ X ) ) ).
% pinf(7)
thf(fact_338_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( ( P @ X6 )
= ( P4 @ X6 ) ) )
=> ( ? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( ( Q @ X6 )
= ( Q2 @ X6 ) ) )
=> ? [Z2: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z2 )
=> ( ( ( P @ X )
& ( Q @ X ) )
= ( ( P4 @ X )
& ( Q2 @ X ) ) ) ) ) ) ).
% minf(1)
thf(fact_339_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( ( P @ X6 )
= ( P4 @ X6 ) ) )
=> ( ? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( ( Q @ X6 )
= ( Q2 @ X6 ) ) )
=> ? [Z2: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z2 )
=> ( ( ( P @ X )
| ( Q @ X ) )
= ( ( P4 @ X )
| ( Q2 @ X ) ) ) ) ) ) ).
% minf(2)
thf(fact_340_minf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z2 )
=> ( X != T ) ) ).
% minf(3)
thf(fact_341_minf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z2 )
=> ( X != T ) ) ).
% minf(4)
thf(fact_342_minf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z2 )
=> ( ord_less_nat @ X @ T ) ) ).
% minf(5)
thf(fact_343_minf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z2 )
=> ~ ( ord_less_nat @ T @ X ) ) ).
% minf(7)
thf(fact_344_order_Onot__eq__order__implies__strict,axiom,
! [A2: nat,B: nat] :
( ( A2 != B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_345_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_346_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B5: nat,A7: nat] :
( ( ord_less_eq_nat @ B5 @ A7 )
& ( A7 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_347_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A7: nat] :
( ( ord_less_nat @ B5 @ A7 )
| ( A7 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_348_order_Ostrict__implies__order,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_349_dual__order_Ostrict__trans2,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_350_dual__order_Ostrict__trans1,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_351_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A7: nat,B5: nat] :
( ( ord_less_eq_nat @ A7 @ B5 )
& ( A7 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_352_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A7: nat,B5: nat] :
( ( ord_less_nat @ A7 @ B5 )
| ( A7 = B5 ) ) ) ) ).
% order.order_iff_strict
% Conjectures (1)
thf(conj_0,conjecture,
( a3
= ( groups1792256535list_a @ qs2 ) ) ).
%------------------------------------------------------------------------------