TPTP Problem File: SLH0164^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Equivalence_Relation_Enumeration/0007_Equivalence_Relation_Enumeration/prob_00373_014293__12091310_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 964 ( 384 unt; 150 typ; 0 def)
% Number of atoms : 2078 (1256 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 7233 ( 316 ~; 30 |; 161 &;5715 @)
% ( 0 <=>;1011 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 7 avg)
% Number of types : 17 ( 16 usr)
% Number of type conns : 715 ( 715 >; 0 *; 0 +; 0 <<)
% Number of symbols : 137 ( 134 usr; 14 con; 0-3 aty)
% Number of variables : 2522 ( 28 ^;2304 !; 190 ?;2522 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:14:37.599
%------------------------------------------------------------------------------
% Could-be-implicit typings (16)
thf(ty_n_t__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
list_s1210847774152347623at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_se7855581050983116737at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
list_list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
set_list_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__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__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__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (134)
thf(sy_c_Equivalence__Relation__Enumeration_Oenum__rgfs,type,
equiva7426478223624825838m_rgfs: nat > list_list_nat ).
thf(sy_c_Equivalence__Relation__Enumeration_Oequiv__rels,type,
equiva8721718519204927301v_rels: nat > list_s1210847774152347623at_nat ).
thf(sy_c_Equivalence__Relation__Enumeration_Okernel__of_001t__List__Olist_It__Nat__Onat_J,type,
equiva6490762433048536736st_nat: list_list_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Equivalence__Relation__Enumeration_Okernel__of_001t__Nat__Onat,type,
equiva2048684438135499664of_nat: list_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Equivalence__Relation__Enumeration_Okernel__of_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
equiva1173177585473067681at_nat: list_s1210847774152347623at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Equivalence__Relation__Enumeration_Okernel__of_001tf__a,type,
equiva2867628904822520638l_of_a: list_a > set_Pr1261947904930325089at_nat ).
thf(sy_c_Equivalence__Relation__Enumeration_Orgf,type,
equiva3371634703666331078on_rgf: list_nat > $o ).
thf(sy_c_Equivalence__Relation__Enumeration_Orgf__limit,type,
equiva5889994315859557365_limit: list_nat > nat ).
thf(sy_c_Equivalence__Relation__Enumeration_Orgf__limit__rel,type,
equiva5575797544161152836it_rel: list_nat > list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
inj_on2300671324199612755st_nat: ( list_list_nat > list_list_nat ) > set_list_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
inj_on3049792774292151987st_nat: ( list_nat > list_nat ) > set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
inj_on_list_nat_nat: ( list_nat > nat ) > set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
inj_on7522185085906380110at_nat: ( list_nat > set_Pr1261947904930325089at_nat ) > set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001tf__a,type,
inj_on_list_nat_a: ( list_nat > a ) > set_list_nat > $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__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
inj_on_nat_list_nat: ( nat > list_nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001tf__a,type,
inj_on_nat_a: ( nat > a ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001t__List__Olist_It__Nat__Onat_J,type,
inj_on_a_list_nat: ( a > list_nat ) > set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001t__Nat__Onat,type,
inj_on_a_nat: ( a > nat ) > 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_Fun_Othe__inv__into_001t__Nat__Onat_001t__Nat__Onat,type,
the_inv_into_nat_nat: set_nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001tf__a,type,
the_inv_into_nat_a: set_nat > ( nat > a ) > a > nat ).
thf(sy_c_Fun_Othe__inv__into_001tf__a_001t__Nat__Onat,type,
the_inv_into_a_nat: set_a > ( a > nat ) > nat > a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_List_Oappend_001t__List__Olist_It__Nat__Onat_J,type,
append_list_nat: list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Oappend_001t__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Oappend_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
append4192317425040545660at_nat: list_s1210847774152347623at_nat > list_s1210847774152347623at_nat > list_s1210847774152347623at_nat ).
thf(sy_c_List_Oappend_001tf__a,type,
append_a: list_a > list_a > list_a ).
thf(sy_c_List_Ocount__list_001t__List__Olist_It__Nat__Onat_J,type,
count_list_list_nat: list_list_nat > list_nat > nat ).
thf(sy_c_List_Ocount__list_001t__Nat__Onat,type,
count_list_nat: list_nat > nat > nat ).
thf(sy_c_List_Ocount__list_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
count_6440129622255701469at_nat: list_s1210847774152347623at_nat > set_Pr1261947904930325089at_nat > nat ).
thf(sy_c_List_Ocount__list_001tf__a,type,
count_list_a: list_a > a > nat ).
thf(sy_c_List_Odistinct__adj_001t__List__Olist_It__Nat__Onat_J,type,
distin876741697294417026st_nat: list_list_nat > $o ).
thf(sy_c_List_Odistinct__adj_001t__Nat__Onat,type,
distinct_adj_nat: list_nat > $o ).
thf(sy_c_List_Odistinct__adj_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
distin3702590604212146495at_nat: list_s1210847774152347623at_nat > $o ).
thf(sy_c_List_Odistinct__adj_001tf__a,type,
distinct_adj_a: list_a > $o ).
thf(sy_c_List_Ofolding__insort__key__axioms_001t__Nat__Onat_001t__Nat__Onat,type,
foldin1360219024038166634at_nat: set_nat > ( nat > nat ) > $o ).
thf(sy_c_List_Ofolding__insort__key__axioms_001tf__a_001t__Nat__Onat,type,
foldin5162300008545400710_a_nat: set_a > ( a > nat ) > $o ).
thf(sy_c_List_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
insert_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oinsert_001tf__a,type,
insert_a: a > list_a > list_a ).
thf(sy_c_List_Olinorder__class_Oinsort__insert__key_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
linord3253225449353161780at_nat: ( list_nat > nat ) > list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Olinorder__class_Oinsort__insert__key_001t__Nat__Onat_001t__Nat__Onat,type,
linord1921536354676448932at_nat: ( nat > nat ) > nat > list_nat > list_nat ).
thf(sy_c_List_Olinorder__class_Oinsort__insert__key_001tf__a_001t__Nat__Onat,type,
linord1046132949341221836_a_nat: ( a > nat ) > a > list_a > list_a ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
linord2614967742042102400et_nat: set_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
cons_list_list_nat: list_list_nat > list_list_list_nat > list_list_list_nat ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Nat__Onat_J,type,
cons_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_Itf__a_J,type,
cons_list_a: list_a > list_list_a > list_list_a ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
nil_list_list_nat: list_list_list_nat ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Nat__Onat_J,type,
nil_list_nat: list_list_nat ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_Itf__a_J,type,
nil_list_a: list_list_a ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
nil_se357566008730718055at_nat: list_s1210847774152347623at_nat ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
map_li7225945977422193158st_nat: ( list_nat > list_nat ) > list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
map_list_nat_nat: ( list_nat > nat ) > list_list_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
map_li6003994582982014139at_nat: ( list_nat > set_Pr1261947904930325089at_nat ) > list_list_nat > list_s1210847774152347623at_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001tf__a,type,
map_list_nat_a: ( list_nat > a ) > list_list_nat > list_a ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
map_nat_list_nat: ( nat > list_nat ) > list_nat > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001tf__a,type,
map_nat_a: ( nat > a ) > list_nat > list_a ).
thf(sy_c_List_Olist_Omap_001tf__a_001t__List__Olist_It__Nat__Onat_J,type,
map_a_list_nat: ( a > list_nat ) > list_a > list_list_nat ).
thf(sy_c_List_Olist_Omap_001tf__a_001t__Nat__Onat,type,
map_a_nat: ( a > nat ) > list_a > list_nat ).
thf(sy_c_List_Olist_Omap_001tf__a_001tf__a,type,
map_a_a: ( a > a ) > list_a > list_a ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
set_list_nat2: list_list_nat > set_list_nat ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_se5049602875457034614at_nat: list_s1210847774152347623at_nat > set_se7855581050983116737at_nat ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_List_Omap__tailrec_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
map_ta9145449198693458768st_nat: ( list_nat > list_nat ) > list_list_nat > list_list_nat ).
thf(sy_c_List_Omap__tailrec_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
map_ta8671482330076047857at_nat: ( list_nat > set_Pr1261947904930325089at_nat ) > list_list_nat > list_s1210847774152347623at_nat ).
thf(sy_c_List_Omap__tailrec_001t__Nat__Onat_001t__Nat__Onat,type,
map_tailrec_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Omap__tailrec_001tf__a_001t__Nat__Onat,type,
map_tailrec_a_nat: ( a > nat ) > list_a > list_nat ).
thf(sy_c_List_Omap__tailrec_001tf__a_001tf__a,type,
map_tailrec_a_a: ( a > a ) > list_a > list_a ).
thf(sy_c_List_Omember_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_list_nat > list_nat > $o ).
thf(sy_c_List_Omember_001t__Nat__Onat,type,
member_nat: list_nat > nat > $o ).
thf(sy_c_List_Omember_001tf__a,type,
member_a: list_a > a > $o ).
thf(sy_c_List_Onth_001t__List__Olist_It__Nat__Onat_J,type,
nth_list_nat: list_list_nat > nat > list_nat ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
nth_se5679702624988424552at_nat: list_s1210847774152347623at_nat > nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_List_Onth_001tf__a,type,
nth_a: list_a > nat > a ).
thf(sy_c_List_OremoveAll_001t__List__Olist_It__Nat__Onat_J,type,
removeAll_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_OremoveAll_001t__Nat__Onat,type,
removeAll_nat: nat > list_nat > list_nat ).
thf(sy_c_List_OremoveAll_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
remove5672899571770113645at_nat: set_Pr1261947904930325089at_nat > list_s1210847774152347623at_nat > list_s1210847774152347623at_nat ).
thf(sy_c_List_OremoveAll_001tf__a,type,
removeAll_a: a > list_a > list_a ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
size_s3023201423986296836st_nat: list_list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
size_s8736152011456118867at_nat: list_s1210847774152347623at_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
size_size_list_a: list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > 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__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
ord_max_nat: nat > nat > nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
image_7976474329151083847st_nat: ( list_nat > list_nat ) > set_list_nat > set_list_nat ).
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_It__Nat__Onat_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_5284832723445046202at_nat: ( list_nat > set_Pr1261947904930325089at_nat ) > set_list_nat > set_se7855581050983116737at_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001tf__a,type,
image_list_nat_a: ( list_nat > a ) > set_list_nat > set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
image_nat_list_nat: ( nat > list_nat ) > set_nat > set_list_nat ).
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_001t__Nat__Onat_001tf__a,type,
image_nat_a: ( nat > a ) > set_nat > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__List__Olist_It__Nat__Onat_J,type,
image_a_list_nat: ( a > list_nat ) > set_a > set_list_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
image_a_nat: ( a > nat ) > set_a > set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
insert_list_nat2: list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat2: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a2: a > set_a > set_a ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Stirling_Ostirling__row,type,
stirling_row: nat > list_nat ).
thf(sy_c_Stirling_Ostirling__row__aux_001t__Nat__Onat,type,
stirling_row_aux_nat: nat > nat > list_nat > list_nat ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_Sublist_Oprefix_001t__Nat__Onat,type,
prefix_nat: list_nat > list_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
accp_list_nat: ( list_nat > list_nat > $o ) > list_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat2: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat2: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member2643936169264416010at_nat: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat > $o ).
thf(sy_c_member_001tf__a,type,
member_a2: a > set_a > $o ).
thf(sy_v_f_H____,type,
f: a > nat ).
thf(sy_v_f____,type,
f2: a > nat ).
thf(sy_v_n____,type,
n: nat ).
thf(sy_v_x1____,type,
x1: list_a ).
thf(sy_v_x2____,type,
x2: a ).
thf(sy_v_xa____,type,
xa: list_a ).
% Relevant facts (810)
thf(fact_0__092_060open_062rgf_A_Imap_Af_H_Ax_J_092_060close_062,axiom,
equiva3371634703666331078on_rgf @ ( map_a_nat @ f @ xa ) ).
% \<open>rgf (map f' x)\<close>
thf(fact_1_a,axiom,
inj_on_a_nat @ f @ ( set_a2 @ xa ) ).
% a
thf(fact_2_map__eq__conv,axiom,
! [F: a > a,Xs: list_a,G: a > a] :
( ( ( map_a_a @ F @ Xs )
= ( map_a_a @ G @ Xs ) )
= ( ! [X: a] :
( ( member_a2 @ X @ ( set_a2 @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_3_map__eq__conv,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Xs ) )
= ( ! [X: nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_4_map__eq__conv,axiom,
! [F: list_nat > list_nat,Xs: list_list_nat,G: list_nat > list_nat] :
( ( ( map_li7225945977422193158st_nat @ F @ Xs )
= ( map_li7225945977422193158st_nat @ G @ Xs ) )
= ( ! [X: list_nat] :
( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_5_map__eq__conv,axiom,
! [F: a > nat,Xs: list_a,G: a > nat] :
( ( ( map_a_nat @ F @ Xs )
= ( map_a_nat @ G @ Xs ) )
= ( ! [X: a] :
( ( member_a2 @ X @ ( set_a2 @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_6_map__eq__conv,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat,G: list_nat > set_Pr1261947904930325089at_nat] :
( ( ( map_li6003994582982014139at_nat @ F @ Xs )
= ( map_li6003994582982014139at_nat @ G @ Xs ) )
= ( ! [X: list_nat] :
( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_7__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062f_O_A_092_060lbrakk_062inj__on_Af_A_Iset_Ax1_J_059_Argf_A_Imap_Af_Ax1_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [F2: a > nat] :
( ( inj_on_a_nat @ F2 @ ( set_a2 @ x1 ) )
=> ~ ( equiva3371634703666331078on_rgf @ ( map_a_nat @ F2 @ x1 ) ) ) ).
% \<open>\<And>thesis. (\<And>f. \<lbrakk>inj_on f (set x1); rgf (map f x1)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_8_list_Omap__cong,axiom,
! [X2: list_a,Ya: list_a,F: a > a,G: a > a] :
( ( X2 = Ya )
=> ( ! [Z: a] :
( ( member_a2 @ Z @ ( set_a2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_a_a @ F @ X2 )
= ( map_a_a @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_9_list_Omap__cong,axiom,
! [X2: list_nat,Ya: list_nat,F: nat > nat,G: nat > nat] :
( ( X2 = Ya )
=> ( ! [Z: nat] :
( ( member_nat2 @ Z @ ( set_nat2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_nat_nat @ F @ X2 )
= ( map_nat_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_10_list_Omap__cong,axiom,
! [X2: list_list_nat,Ya: list_list_nat,F: list_nat > list_nat,G: list_nat > list_nat] :
( ( X2 = Ya )
=> ( ! [Z: list_nat] :
( ( member_list_nat2 @ Z @ ( set_list_nat2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_li7225945977422193158st_nat @ F @ X2 )
= ( map_li7225945977422193158st_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_11_list_Omap__cong,axiom,
! [X2: list_a,Ya: list_a,F: a > nat,G: a > nat] :
( ( X2 = Ya )
=> ( ! [Z: a] :
( ( member_a2 @ Z @ ( set_a2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_a_nat @ F @ X2 )
= ( map_a_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_12_list_Omap__cong,axiom,
! [X2: list_list_nat,Ya: list_list_nat,F: list_nat > set_Pr1261947904930325089at_nat,G: list_nat > set_Pr1261947904930325089at_nat] :
( ( X2 = Ya )
=> ( ! [Z: list_nat] :
( ( member_list_nat2 @ Z @ ( set_list_nat2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_li6003994582982014139at_nat @ F @ X2 )
= ( map_li6003994582982014139at_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_13_list_Omap__cong0,axiom,
! [X2: list_a,F: a > a,G: a > a] :
( ! [Z: a] :
( ( member_a2 @ Z @ ( set_a2 @ X2 ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_a_a @ F @ X2 )
= ( map_a_a @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_14_list_Omap__cong0,axiom,
! [X2: list_nat,F: nat > nat,G: nat > nat] :
( ! [Z: nat] :
( ( member_nat2 @ Z @ ( set_nat2 @ X2 ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_nat_nat @ F @ X2 )
= ( map_nat_nat @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_15_list_Omap__cong0,axiom,
! [X2: list_list_nat,F: list_nat > list_nat,G: list_nat > list_nat] :
( ! [Z: list_nat] :
( ( member_list_nat2 @ Z @ ( set_list_nat2 @ X2 ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_li7225945977422193158st_nat @ F @ X2 )
= ( map_li7225945977422193158st_nat @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_16_list_Omap__cong0,axiom,
! [X2: list_a,F: a > nat,G: a > nat] :
( ! [Z: a] :
( ( member_a2 @ Z @ ( set_a2 @ X2 ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_a_nat @ F @ X2 )
= ( map_a_nat @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_17_list_Omap__cong0,axiom,
! [X2: list_list_nat,F: list_nat > set_Pr1261947904930325089at_nat,G: list_nat > set_Pr1261947904930325089at_nat] :
( ! [Z: list_nat] :
( ( member_list_nat2 @ Z @ ( set_list_nat2 @ X2 ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_li6003994582982014139at_nat @ F @ X2 )
= ( map_li6003994582982014139at_nat @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_18_list_Oinj__map__strong,axiom,
! [X2: list_a,Xa: list_a,F: a > a,Fa: a > a] :
( ! [Z: a,Za: a] :
( ( member_a2 @ Z @ ( set_a2 @ X2 ) )
=> ( ( member_a2 @ Za @ ( set_a2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_a_a @ F @ X2 )
= ( map_a_a @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_19_list_Oinj__map__strong,axiom,
! [X2: list_nat,Xa: list_nat,F: nat > nat,Fa: nat > nat] :
( ! [Z: nat,Za: nat] :
( ( member_nat2 @ Z @ ( set_nat2 @ X2 ) )
=> ( ( member_nat2 @ Za @ ( set_nat2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_nat_nat @ F @ X2 )
= ( map_nat_nat @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_20_list_Oinj__map__strong,axiom,
! [X2: list_list_nat,Xa: list_list_nat,F: list_nat > list_nat,Fa: list_nat > list_nat] :
( ! [Z: list_nat,Za: list_nat] :
( ( member_list_nat2 @ Z @ ( set_list_nat2 @ X2 ) )
=> ( ( member_list_nat2 @ Za @ ( set_list_nat2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_li7225945977422193158st_nat @ F @ X2 )
= ( map_li7225945977422193158st_nat @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_21_list_Oinj__map__strong,axiom,
! [X2: list_a,Xa: list_a,F: a > nat,Fa: a > nat] :
( ! [Z: a,Za: a] :
( ( member_a2 @ Z @ ( set_a2 @ X2 ) )
=> ( ( member_a2 @ Za @ ( set_a2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_a_nat @ F @ X2 )
= ( map_a_nat @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_22_list_Oinj__map__strong,axiom,
! [X2: list_list_nat,Xa: list_list_nat,F: list_nat > set_Pr1261947904930325089at_nat,Fa: list_nat > set_Pr1261947904930325089at_nat] :
( ! [Z: list_nat,Za: list_nat] :
( ( member_list_nat2 @ Z @ ( set_list_nat2 @ X2 ) )
=> ( ( member_list_nat2 @ Za @ ( set_list_nat2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_li6003994582982014139at_nat @ F @ X2 )
= ( map_li6003994582982014139at_nat @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_23_list_Omap__ident__strong,axiom,
! [T: list_a,F: a > a] :
( ! [Z: a] :
( ( member_a2 @ Z @ ( set_a2 @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_a_a @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_24_list_Omap__ident__strong,axiom,
! [T: list_nat,F: nat > nat] :
( ! [Z: nat] :
( ( member_nat2 @ Z @ ( set_nat2 @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_nat_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_25_list_Omap__ident__strong,axiom,
! [T: list_list_nat,F: list_nat > list_nat] :
( ! [Z: list_nat] :
( ( member_list_nat2 @ Z @ ( set_list_nat2 @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_li7225945977422193158st_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_26_map__ext,axiom,
! [Xs: list_a,F: a > a,G: a > a] :
( ! [X3: a] :
( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_a_a @ F @ Xs )
= ( map_a_a @ G @ Xs ) ) ) ).
% map_ext
thf(fact_27_map__ext,axiom,
! [Xs: list_nat,F: nat > nat,G: nat > nat] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_28_map__ext,axiom,
! [Xs: list_list_nat,F: list_nat > list_nat,G: list_nat > list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_li7225945977422193158st_nat @ F @ Xs )
= ( map_li7225945977422193158st_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_29_map__ext,axiom,
! [Xs: list_a,F: a > nat,G: a > nat] :
( ! [X3: a] :
( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_a_nat @ F @ Xs )
= ( map_a_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_30_map__ext,axiom,
! [Xs: list_list_nat,F: list_nat > set_Pr1261947904930325089at_nat,G: list_nat > set_Pr1261947904930325089at_nat] :
( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_li6003994582982014139at_nat @ F @ Xs )
= ( map_li6003994582982014139at_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_31_map__idI,axiom,
! [Xs: list_a,F: a > a] :
( ! [X3: a] :
( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_a_a @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_32_map__idI,axiom,
! [Xs: list_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_nat_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_33_map__idI,axiom,
! [Xs: list_list_nat,F: list_nat > list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_li7225945977422193158st_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_34_map__cong,axiom,
! [Xs: list_a,Ys: list_a,F: a > nat,G: a > nat] :
( ( Xs = Ys )
=> ( ! [X3: a] :
( ( member_a2 @ X3 @ ( set_a2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_a_nat @ F @ Xs )
= ( map_a_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_35_map__cong,axiom,
! [Xs: list_a,Ys: list_a,F: a > a,G: a > a] :
( ( Xs = Ys )
=> ( ! [X3: a] :
( ( member_a2 @ X3 @ ( set_a2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_a_a @ F @ Xs )
= ( map_a_a @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_36_map__cong,axiom,
! [Xs: list_nat,Ys: list_nat,F: nat > nat,G: nat > nat] :
( ( Xs = Ys )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_37_map__cong,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,F: list_nat > set_Pr1261947904930325089at_nat,G: list_nat > set_Pr1261947904930325089at_nat] :
( ( Xs = Ys )
=> ( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_li6003994582982014139at_nat @ F @ Xs )
= ( map_li6003994582982014139at_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_38_map__cong,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,F: list_nat > list_nat,G: list_nat > list_nat] :
( ( Xs = Ys )
=> ( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_li7225945977422193158st_nat @ F @ Xs )
= ( map_li7225945977422193158st_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_39_ex__map__conv,axiom,
! [Ys: list_s1210847774152347623at_nat,F: list_nat > set_Pr1261947904930325089at_nat] :
( ( ? [Xs2: list_list_nat] :
( Ys
= ( map_li6003994582982014139at_nat @ F @ Xs2 ) ) )
= ( ! [X: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X @ ( set_se5049602875457034614at_nat @ Ys ) )
=> ? [Y: list_nat] :
( X
= ( F @ Y ) ) ) ) ) ).
% ex_map_conv
thf(fact_40_ex__map__conv,axiom,
! [Ys: list_a,F: a > a] :
( ( ? [Xs2: list_a] :
( Ys
= ( map_a_a @ F @ Xs2 ) ) )
= ( ! [X: a] :
( ( member_a2 @ X @ ( set_a2 @ Ys ) )
=> ? [Y: a] :
( X
= ( F @ Y ) ) ) ) ) ).
% ex_map_conv
thf(fact_41_ex__map__conv,axiom,
! [Ys: list_nat,F: a > nat] :
( ( ? [Xs2: list_a] :
( Ys
= ( map_a_nat @ F @ Xs2 ) ) )
= ( ! [X: nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Ys ) )
=> ? [Y: a] :
( X
= ( F @ Y ) ) ) ) ) ).
% ex_map_conv
thf(fact_42_ex__map__conv,axiom,
! [Ys: list_nat,F: nat > nat] :
( ( ? [Xs2: list_nat] :
( Ys
= ( map_nat_nat @ F @ Xs2 ) ) )
= ( ! [X: nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Ys ) )
=> ? [Y: nat] :
( X
= ( F @ Y ) ) ) ) ) ).
% ex_map_conv
thf(fact_43_ex__map__conv,axiom,
! [Ys: list_list_nat,F: list_nat > list_nat] :
( ( ? [Xs2: list_list_nat] :
( Ys
= ( map_li7225945977422193158st_nat @ F @ Xs2 ) ) )
= ( ! [X: list_nat] :
( ( member_list_nat2 @ X @ ( set_list_nat2 @ Ys ) )
=> ? [Y: list_nat] :
( X
= ( F @ Y ) ) ) ) ) ).
% ex_map_conv
thf(fact_44_Suc_Ohyps_I1_J,axiom,
! [X2: list_a] :
( ( n
= ( size_size_list_a @ X2 ) )
=> ? [F2: a > nat] :
( ( inj_on_a_nat @ F2 @ ( set_a2 @ X2 ) )
& ( equiva3371634703666331078on_rgf @ ( map_a_nat @ F2 @ X2 ) ) ) ) ).
% Suc.hyps(1)
thf(fact_45_kernel__of__under__inj__map,axiom,
! [F: a > a,X2: list_a] :
( ( inj_on_a_a @ F @ ( set_a2 @ X2 ) )
=> ( ( equiva2867628904822520638l_of_a @ X2 )
= ( equiva2867628904822520638l_of_a @ ( map_a_a @ F @ X2 ) ) ) ) ).
% kernel_of_under_inj_map
thf(fact_46_kernel__of__under__inj__map,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,X2: list_list_nat] :
( ( inj_on7522185085906380110at_nat @ F @ ( set_list_nat2 @ X2 ) )
=> ( ( equiva6490762433048536736st_nat @ X2 )
= ( equiva1173177585473067681at_nat @ ( map_li6003994582982014139at_nat @ F @ X2 ) ) ) ) ).
% kernel_of_under_inj_map
thf(fact_47_kernel__of__under__inj__map,axiom,
! [F: list_nat > list_nat,X2: list_list_nat] :
( ( inj_on3049792774292151987st_nat @ F @ ( set_list_nat2 @ X2 ) )
=> ( ( equiva6490762433048536736st_nat @ X2 )
= ( equiva6490762433048536736st_nat @ ( map_li7225945977422193158st_nat @ F @ X2 ) ) ) ) ).
% kernel_of_under_inj_map
thf(fact_48_kernel__of__under__inj__map,axiom,
! [F: a > nat,X2: list_a] :
( ( inj_on_a_nat @ F @ ( set_a2 @ X2 ) )
=> ( ( equiva2867628904822520638l_of_a @ X2 )
= ( equiva2048684438135499664of_nat @ ( map_a_nat @ F @ X2 ) ) ) ) ).
% kernel_of_under_inj_map
thf(fact_49_kernel__of__under__inj__map,axiom,
! [F: list_nat > nat,X2: list_list_nat] :
( ( inj_on_list_nat_nat @ F @ ( set_list_nat2 @ X2 ) )
=> ( ( equiva6490762433048536736st_nat @ X2 )
= ( equiva2048684438135499664of_nat @ ( map_list_nat_nat @ F @ X2 ) ) ) ) ).
% kernel_of_under_inj_map
thf(fact_50_kernel__of__under__inj__map,axiom,
! [F: nat > nat,X2: list_nat] :
( ( inj_on_nat_nat @ F @ ( set_nat2 @ X2 ) )
=> ( ( equiva2048684438135499664of_nat @ X2 )
= ( equiva2048684438135499664of_nat @ ( map_nat_nat @ F @ X2 ) ) ) ) ).
% kernel_of_under_inj_map
thf(fact_51_l__x1,axiom,
( ( size_size_list_a @ x1 )
= n ) ).
% l_x1
thf(fact_52_False,axiom,
~ ( member_a2 @ x2 @ ( set_a2 @ x1 ) ) ).
% False
thf(fact_53__092_060open_062inj__on_Af_H_A_Iset_Ax1_J_092_060close_062,axiom,
inj_on_a_nat @ f @ ( set_a2 @ x1 ) ).
% \<open>inj_on f' (set x1)\<close>
thf(fact_54_pc__f,axiom,
equiva3371634703666331078on_rgf @ ( map_a_nat @ f2 @ x1 ) ).
% pc_f
thf(fact_55_inj__f,axiom,
inj_on_a_nat @ f2 @ ( set_a2 @ x1 ) ).
% inj_f
thf(fact_56_length__map,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat] :
( ( size_s8736152011456118867at_nat @ ( map_li6003994582982014139at_nat @ F @ Xs ) )
= ( size_s3023201423986296836st_nat @ Xs ) ) ).
% length_map
thf(fact_57_length__map,axiom,
! [F: list_nat > list_nat,Xs: list_list_nat] :
( ( size_s3023201423986296836st_nat @ ( map_li7225945977422193158st_nat @ F @ Xs ) )
= ( size_s3023201423986296836st_nat @ Xs ) ) ).
% length_map
thf(fact_58_length__map,axiom,
! [F: a > a,Xs: list_a] :
( ( size_size_list_a @ ( map_a_a @ F @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_map
thf(fact_59_length__map,axiom,
! [F: nat > a,Xs: list_nat] :
( ( size_size_list_a @ ( map_nat_a @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_60_length__map,axiom,
! [F: a > nat,Xs: list_a] :
( ( size_size_list_nat @ ( map_a_nat @ F @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_map
thf(fact_61_length__map,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( size_size_list_nat @ ( map_nat_nat @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_62_Suc_Ohyps_I2_J,axiom,
( ( suc @ n )
= ( size_size_list_a @ xa ) ) ).
% Suc.hyps(2)
thf(fact_63_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_a] :
( ( size_size_list_a @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_64_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_nat] :
( ( size_size_list_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_65_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_66_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_67_kernel__of__eq__len,axiom,
! [X2: list_a,Y2: list_a] :
( ( ( equiva2867628904822520638l_of_a @ X2 )
= ( equiva2867628904822520638l_of_a @ Y2 ) )
=> ( ( size_size_list_a @ X2 )
= ( size_size_list_a @ Y2 ) ) ) ).
% kernel_of_eq_len
thf(fact_68_kernel__of__eq__len,axiom,
! [X2: list_a,Y2: list_nat] :
( ( ( equiva2867628904822520638l_of_a @ X2 )
= ( equiva2048684438135499664of_nat @ Y2 ) )
=> ( ( size_size_list_a @ X2 )
= ( size_size_list_nat @ Y2 ) ) ) ).
% kernel_of_eq_len
thf(fact_69_kernel__of__eq__len,axiom,
! [X2: list_nat,Y2: list_a] :
( ( ( equiva2048684438135499664of_nat @ X2 )
= ( equiva2867628904822520638l_of_a @ Y2 ) )
=> ( ( size_size_list_nat @ X2 )
= ( size_size_list_a @ Y2 ) ) ) ).
% kernel_of_eq_len
thf(fact_70_kernel__of__eq__len,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( ( equiva2048684438135499664of_nat @ X2 )
= ( equiva2048684438135499664of_nat @ Y2 ) )
=> ( ( size_size_list_nat @ X2 )
= ( size_size_list_nat @ Y2 ) ) ) ).
% kernel_of_eq_len
thf(fact_71_map__eq__imp__length__eq,axiom,
! [F: a > nat,Xs: list_a,G: a > nat,Ys: list_a] :
( ( ( map_a_nat @ F @ Xs )
= ( map_a_nat @ G @ Ys ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_72_map__eq__imp__length__eq,axiom,
! [F: a > a,Xs: list_a,G: a > a,Ys: list_a] :
( ( ( map_a_a @ F @ Xs )
= ( map_a_a @ G @ Ys ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_73_map__eq__imp__length__eq,axiom,
! [F: a > nat,Xs: list_a,G: nat > nat,Ys: list_nat] :
( ( ( map_a_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_74_map__eq__imp__length__eq,axiom,
! [F: a > a,Xs: list_a,G: nat > a,Ys: list_nat] :
( ( ( map_a_a @ F @ Xs )
= ( map_nat_a @ G @ Ys ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_75_map__eq__imp__length__eq,axiom,
! [F: nat > a,Xs: list_nat,G: a > a,Ys: list_a] :
( ( ( map_nat_a @ F @ Xs )
= ( map_a_a @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_76_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs: list_nat,G: a > nat,Ys: list_a] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_a_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_77_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_78_map__eq__imp__length__eq,axiom,
! [F: list_nat > list_nat,Xs: list_list_nat,G: a > list_nat,Ys: list_a] :
( ( ( map_li7225945977422193158st_nat @ F @ Xs )
= ( map_a_list_nat @ G @ Ys ) )
=> ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_79_map__eq__imp__length__eq,axiom,
! [F: list_nat > list_nat,Xs: list_list_nat,G: nat > list_nat,Ys: list_nat] :
( ( ( map_li7225945977422193158st_nat @ F @ Xs )
= ( map_nat_list_nat @ G @ Ys ) )
=> ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_80_map__eq__imp__length__eq,axiom,
! [F: a > list_nat,Xs: list_a,G: list_nat > list_nat,Ys: list_list_nat] :
( ( ( map_a_list_nat @ F @ Xs )
= ( map_li7225945977422193158st_nat @ G @ Ys ) )
=> ( ( size_size_list_a @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_81_b,axiom,
( ( map_a_nat @ f2 @ x1 )
= ( map_a_nat @ f @ x1 ) ) ).
% b
thf(fact_82_f_H__def,axiom,
( f
= ( ^ [Y: a] : ( if_nat @ ( member_a2 @ Y @ ( set_a2 @ x1 ) ) @ ( f2 @ Y ) @ ( equiva5889994315859557365_limit @ ( map_a_nat @ f2 @ x1 ) ) ) ) ) ).
% f'_def
thf(fact_83__092_060open_062inj__on_Af_H_A_Iinsert_Ax2_A_Iset_Ax1_J_J_092_060close_062,axiom,
inj_on_a_nat @ f @ ( insert_a2 @ x2 @ ( set_a2 @ x1 ) ) ).
% \<open>inj_on f' (insert x2 (set x1))\<close>
thf(fact_84__092_060open_062f_H_Ax2_A_092_060notin_062_Af_H_A_096_Aset_Ax1_092_060close_062,axiom,
~ ( member_nat2 @ ( f @ x2 ) @ ( image_a_nat @ f @ ( set_a2 @ x1 ) ) ) ).
% \<open>f' x2 \<notin> f' ` set x1\<close>
thf(fact_85_map__eq__map__tailrec,axiom,
map_a_nat = map_tailrec_a_nat ).
% map_eq_map_tailrec
thf(fact_86_map__eq__map__tailrec,axiom,
map_li6003994582982014139at_nat = map_ta8671482330076047857at_nat ).
% map_eq_map_tailrec
thf(fact_87_map__eq__map__tailrec,axiom,
map_li7225945977422193158st_nat = map_ta9145449198693458768st_nat ).
% map_eq_map_tailrec
thf(fact_88_map__eq__map__tailrec,axiom,
map_nat_nat = map_tailrec_nat_nat ).
% map_eq_map_tailrec
thf(fact_89_map__eq__map__tailrec,axiom,
map_a_a = map_tailrec_a_a ).
% map_eq_map_tailrec
thf(fact_90_List_Ocount__list__inj__map,axiom,
! [F: a > nat,Xs: list_a,X2: a] :
( ( inj_on_a_nat @ F @ ( set_a2 @ Xs ) )
=> ( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( ( count_list_nat @ ( map_a_nat @ F @ Xs ) @ ( F @ X2 ) )
= ( count_list_a @ Xs @ X2 ) ) ) ) ).
% List.count_list_inj_map
thf(fact_91_List_Ocount__list__inj__map,axiom,
! [F: a > a,Xs: list_a,X2: a] :
( ( inj_on_a_a @ F @ ( set_a2 @ Xs ) )
=> ( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( ( count_list_a @ ( map_a_a @ F @ Xs ) @ ( F @ X2 ) )
= ( count_list_a @ Xs @ X2 ) ) ) ) ).
% List.count_list_inj_map
thf(fact_92_List_Ocount__list__inj__map,axiom,
! [F: nat > nat,Xs: list_nat,X2: nat] :
( ( inj_on_nat_nat @ F @ ( set_nat2 @ Xs ) )
=> ( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( count_list_nat @ ( map_nat_nat @ F @ Xs ) @ ( F @ X2 ) )
= ( count_list_nat @ Xs @ X2 ) ) ) ) ).
% List.count_list_inj_map
thf(fact_93_List_Ocount__list__inj__map,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat,X2: list_nat] :
( ( inj_on7522185085906380110at_nat @ F @ ( set_list_nat2 @ Xs ) )
=> ( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ( count_6440129622255701469at_nat @ ( map_li6003994582982014139at_nat @ F @ Xs ) @ ( F @ X2 ) )
= ( count_list_list_nat @ Xs @ X2 ) ) ) ) ).
% List.count_list_inj_map
thf(fact_94_List_Ocount__list__inj__map,axiom,
! [F: a > list_nat,Xs: list_a,X2: a] :
( ( inj_on_a_list_nat @ F @ ( set_a2 @ Xs ) )
=> ( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( ( count_list_list_nat @ ( map_a_list_nat @ F @ Xs ) @ ( F @ X2 ) )
= ( count_list_a @ Xs @ X2 ) ) ) ) ).
% List.count_list_inj_map
thf(fact_95_List_Ocount__list__inj__map,axiom,
! [F: nat > list_nat,Xs: list_nat,X2: nat] :
( ( inj_on_nat_list_nat @ F @ ( set_nat2 @ Xs ) )
=> ( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( count_list_list_nat @ ( map_nat_list_nat @ F @ Xs ) @ ( F @ X2 ) )
= ( count_list_nat @ Xs @ X2 ) ) ) ) ).
% List.count_list_inj_map
thf(fact_96_List_Ocount__list__inj__map,axiom,
! [F: list_nat > list_nat,Xs: list_list_nat,X2: list_nat] :
( ( inj_on3049792774292151987st_nat @ F @ ( set_list_nat2 @ Xs ) )
=> ( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ( count_list_list_nat @ ( map_li7225945977422193158st_nat @ F @ Xs ) @ ( F @ X2 ) )
= ( count_list_list_nat @ Xs @ X2 ) ) ) ) ).
% List.count_list_inj_map
thf(fact_97_Equivalence__Relation__Enumeration_Ocount__list__inj__map,axiom,
! [F: a > nat,X2: list_a,Y2: a] :
( ( inj_on_a_nat @ F @ ( set_a2 @ X2 ) )
=> ( ( member_a2 @ Y2 @ ( set_a2 @ X2 ) )
=> ( ( count_list_nat @ ( map_a_nat @ F @ X2 ) @ ( F @ Y2 ) )
= ( count_list_a @ X2 @ Y2 ) ) ) ) ).
% Equivalence_Relation_Enumeration.count_list_inj_map
thf(fact_98_Equivalence__Relation__Enumeration_Ocount__list__inj__map,axiom,
! [F: a > a,X2: list_a,Y2: a] :
( ( inj_on_a_a @ F @ ( set_a2 @ X2 ) )
=> ( ( member_a2 @ Y2 @ ( set_a2 @ X2 ) )
=> ( ( count_list_a @ ( map_a_a @ F @ X2 ) @ ( F @ Y2 ) )
= ( count_list_a @ X2 @ Y2 ) ) ) ) ).
% Equivalence_Relation_Enumeration.count_list_inj_map
thf(fact_99_Equivalence__Relation__Enumeration_Ocount__list__inj__map,axiom,
! [F: nat > nat,X2: list_nat,Y2: nat] :
( ( inj_on_nat_nat @ F @ ( set_nat2 @ X2 ) )
=> ( ( member_nat2 @ Y2 @ ( set_nat2 @ X2 ) )
=> ( ( count_list_nat @ ( map_nat_nat @ F @ X2 ) @ ( F @ Y2 ) )
= ( count_list_nat @ X2 @ Y2 ) ) ) ) ).
% Equivalence_Relation_Enumeration.count_list_inj_map
thf(fact_100_Equivalence__Relation__Enumeration_Ocount__list__inj__map,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,X2: list_list_nat,Y2: list_nat] :
( ( inj_on7522185085906380110at_nat @ F @ ( set_list_nat2 @ X2 ) )
=> ( ( member_list_nat2 @ Y2 @ ( set_list_nat2 @ X2 ) )
=> ( ( count_6440129622255701469at_nat @ ( map_li6003994582982014139at_nat @ F @ X2 ) @ ( F @ Y2 ) )
= ( count_list_list_nat @ X2 @ Y2 ) ) ) ) ).
% Equivalence_Relation_Enumeration.count_list_inj_map
thf(fact_101_Equivalence__Relation__Enumeration_Ocount__list__inj__map,axiom,
! [F: a > list_nat,X2: list_a,Y2: a] :
( ( inj_on_a_list_nat @ F @ ( set_a2 @ X2 ) )
=> ( ( member_a2 @ Y2 @ ( set_a2 @ X2 ) )
=> ( ( count_list_list_nat @ ( map_a_list_nat @ F @ X2 ) @ ( F @ Y2 ) )
= ( count_list_a @ X2 @ Y2 ) ) ) ) ).
% Equivalence_Relation_Enumeration.count_list_inj_map
thf(fact_102_Equivalence__Relation__Enumeration_Ocount__list__inj__map,axiom,
! [F: nat > list_nat,X2: list_nat,Y2: nat] :
( ( inj_on_nat_list_nat @ F @ ( set_nat2 @ X2 ) )
=> ( ( member_nat2 @ Y2 @ ( set_nat2 @ X2 ) )
=> ( ( count_list_list_nat @ ( map_nat_list_nat @ F @ X2 ) @ ( F @ Y2 ) )
= ( count_list_nat @ X2 @ Y2 ) ) ) ) ).
% Equivalence_Relation_Enumeration.count_list_inj_map
thf(fact_103_Equivalence__Relation__Enumeration_Ocount__list__inj__map,axiom,
! [F: list_nat > list_nat,X2: list_list_nat,Y2: list_nat] :
( ( inj_on3049792774292151987st_nat @ F @ ( set_list_nat2 @ X2 ) )
=> ( ( member_list_nat2 @ Y2 @ ( set_list_nat2 @ X2 ) )
=> ( ( count_list_list_nat @ ( map_li7225945977422193158st_nat @ F @ X2 ) @ ( F @ Y2 ) )
= ( count_list_list_nat @ X2 @ Y2 ) ) ) ) ).
% Equivalence_Relation_Enumeration.count_list_inj_map
thf(fact_104_distinct__adj__map__iff,axiom,
! [F: a > nat,Xs: list_a] :
( ( inj_on_a_nat @ F @ ( set_a2 @ Xs ) )
=> ( ( distinct_adj_nat @ ( map_a_nat @ F @ Xs ) )
= ( distinct_adj_a @ Xs ) ) ) ).
% distinct_adj_map_iff
thf(fact_105_distinct__adj__map__iff,axiom,
! [F: a > a,Xs: list_a] :
( ( inj_on_a_a @ F @ ( set_a2 @ Xs ) )
=> ( ( distinct_adj_a @ ( map_a_a @ F @ Xs ) )
= ( distinct_adj_a @ Xs ) ) ) ).
% distinct_adj_map_iff
thf(fact_106_distinct__adj__map__iff,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( inj_on_nat_nat @ F @ ( set_nat2 @ Xs ) )
=> ( ( distinct_adj_nat @ ( map_nat_nat @ F @ Xs ) )
= ( distinct_adj_nat @ Xs ) ) ) ).
% distinct_adj_map_iff
thf(fact_107_distinct__adj__map__iff,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat] :
( ( inj_on7522185085906380110at_nat @ F @ ( set_list_nat2 @ Xs ) )
=> ( ( distin3702590604212146495at_nat @ ( map_li6003994582982014139at_nat @ F @ Xs ) )
= ( distin876741697294417026st_nat @ Xs ) ) ) ).
% distinct_adj_map_iff
thf(fact_108_distinct__adj__map__iff,axiom,
! [F: list_nat > list_nat,Xs: list_list_nat] :
( ( inj_on3049792774292151987st_nat @ F @ ( set_list_nat2 @ Xs ) )
=> ( ( distin876741697294417026st_nat @ ( map_li7225945977422193158st_nat @ F @ Xs ) )
= ( distin876741697294417026st_nat @ Xs ) ) ) ).
% distinct_adj_map_iff
thf(fact_109_distinct__adj__mapI,axiom,
! [Xs: list_a,F: a > nat] :
( ( distinct_adj_a @ Xs )
=> ( ( inj_on_a_nat @ F @ ( set_a2 @ Xs ) )
=> ( distinct_adj_nat @ ( map_a_nat @ F @ Xs ) ) ) ) ).
% distinct_adj_mapI
thf(fact_110_distinct__adj__mapI,axiom,
! [Xs: list_a,F: a > a] :
( ( distinct_adj_a @ Xs )
=> ( ( inj_on_a_a @ F @ ( set_a2 @ Xs ) )
=> ( distinct_adj_a @ ( map_a_a @ F @ Xs ) ) ) ) ).
% distinct_adj_mapI
thf(fact_111_distinct__adj__mapI,axiom,
! [Xs: list_nat,F: nat > nat] :
( ( distinct_adj_nat @ Xs )
=> ( ( inj_on_nat_nat @ F @ ( set_nat2 @ Xs ) )
=> ( distinct_adj_nat @ ( map_nat_nat @ F @ Xs ) ) ) ) ).
% distinct_adj_mapI
thf(fact_112_distinct__adj__mapI,axiom,
! [Xs: list_list_nat,F: list_nat > set_Pr1261947904930325089at_nat] :
( ( distin876741697294417026st_nat @ Xs )
=> ( ( inj_on7522185085906380110at_nat @ F @ ( set_list_nat2 @ Xs ) )
=> ( distin3702590604212146495at_nat @ ( map_li6003994582982014139at_nat @ F @ Xs ) ) ) ) ).
% distinct_adj_mapI
thf(fact_113_distinct__adj__mapI,axiom,
! [Xs: list_list_nat,F: list_nat > list_nat] :
( ( distin876741697294417026st_nat @ Xs )
=> ( ( inj_on3049792774292151987st_nat @ F @ ( set_list_nat2 @ Xs ) )
=> ( distin876741697294417026st_nat @ ( map_li7225945977422193158st_nat @ F @ Xs ) ) ) ) ).
% distinct_adj_mapI
thf(fact_114__092_060open_062rgf__limit_A_Imap_Af_Ax1_J_A_092_060notin_062_Aset_A_Imap_Af_Ax1_J_092_060close_062,axiom,
~ ( member_nat2 @ ( equiva5889994315859557365_limit @ ( map_a_nat @ f2 @ x1 ) ) @ ( set_nat2 @ ( map_a_nat @ f2 @ x1 ) ) ) ).
% \<open>rgf_limit (map f x1) \<notin> set (map f x1)\<close>
thf(fact_115_folding__insort__key__axioms__def,axiom,
( foldin5162300008545400710_a_nat
= ( ^ [S: set_a,F3: a > nat] : ( inj_on_a_nat @ F3 @ S ) ) ) ).
% folding_insort_key_axioms_def
thf(fact_116_folding__insort__key__axioms__def,axiom,
( foldin1360219024038166634at_nat
= ( ^ [S: set_nat,F3: nat > nat] : ( inj_on_nat_nat @ F3 @ S ) ) ) ).
% folding_insort_key_axioms_def
thf(fact_117_folding__insort__key__axioms_Ointro,axiom,
! [F: a > nat,S2: set_a] :
( ( inj_on_a_nat @ F @ S2 )
=> ( foldin5162300008545400710_a_nat @ S2 @ F ) ) ).
% folding_insort_key_axioms.intro
thf(fact_118_folding__insort__key__axioms_Ointro,axiom,
! [F: nat > nat,S2: set_nat] :
( ( inj_on_nat_nat @ F @ S2 )
=> ( foldin1360219024038166634at_nat @ S2 @ F ) ) ).
% folding_insort_key_axioms.intro
thf(fact_119_list_Oset__map,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,V: list_list_nat] :
( ( set_se5049602875457034614at_nat @ ( map_li6003994582982014139at_nat @ F @ V ) )
= ( image_5284832723445046202at_nat @ F @ ( set_list_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_120_list_Oset__map,axiom,
! [F: a > a,V: list_a] :
( ( set_a2 @ ( map_a_a @ F @ V ) )
= ( image_a_a @ F @ ( set_a2 @ V ) ) ) ).
% list.set_map
thf(fact_121_list_Oset__map,axiom,
! [F: nat > a,V: list_nat] :
( ( set_a2 @ ( map_nat_a @ F @ V ) )
= ( image_nat_a @ F @ ( set_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_122_list_Oset__map,axiom,
! [F: list_nat > a,V: list_list_nat] :
( ( set_a2 @ ( map_list_nat_a @ F @ V ) )
= ( image_list_nat_a @ F @ ( set_list_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_123_list_Oset__map,axiom,
! [F: a > nat,V: list_a] :
( ( set_nat2 @ ( map_a_nat @ F @ V ) )
= ( image_a_nat @ F @ ( set_a2 @ V ) ) ) ).
% list.set_map
thf(fact_124_list_Oset__map,axiom,
! [F: nat > nat,V: list_nat] :
( ( set_nat2 @ ( map_nat_nat @ F @ V ) )
= ( image_nat_nat @ F @ ( set_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_125_list_Oset__map,axiom,
! [F: list_nat > nat,V: list_list_nat] :
( ( set_nat2 @ ( map_list_nat_nat @ F @ V ) )
= ( image_list_nat_nat @ F @ ( set_list_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_126_list_Oset__map,axiom,
! [F: a > list_nat,V: list_a] :
( ( set_list_nat2 @ ( map_a_list_nat @ F @ V ) )
= ( image_a_list_nat @ F @ ( set_a2 @ V ) ) ) ).
% list.set_map
thf(fact_127_list_Oset__map,axiom,
! [F: nat > list_nat,V: list_nat] :
( ( set_list_nat2 @ ( map_nat_list_nat @ F @ V ) )
= ( image_nat_list_nat @ F @ ( set_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_128_list_Oset__map,axiom,
! [F: list_nat > list_nat,V: list_list_nat] :
( ( set_list_nat2 @ ( map_li7225945977422193158st_nat @ F @ V ) )
= ( image_7976474329151083847st_nat @ F @ ( set_list_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_129_kernel__of__inj__on__rgfs__aux,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( ( size_size_list_nat @ X2 )
= ( size_size_list_nat @ Y2 ) )
=> ( ( equiva3371634703666331078on_rgf @ X2 )
=> ( ( equiva3371634703666331078on_rgf @ Y2 )
=> ( ( ( equiva2048684438135499664of_nat @ X2 )
= ( equiva2048684438135499664of_nat @ Y2 ) )
=> ( X2 = Y2 ) ) ) ) ) ).
% kernel_of_inj_on_rgfs_aux
thf(fact_130_distinct__adj__mapD,axiom,
! [F: a > nat,Xs: list_a] :
( ( distinct_adj_nat @ ( map_a_nat @ F @ Xs ) )
=> ( distinct_adj_a @ Xs ) ) ).
% distinct_adj_mapD
thf(fact_131_distinct__adj__mapD,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat] :
( ( distin3702590604212146495at_nat @ ( map_li6003994582982014139at_nat @ F @ Xs ) )
=> ( distin876741697294417026st_nat @ Xs ) ) ).
% distinct_adj_mapD
thf(fact_132_distinct__adj__mapD,axiom,
! [F: list_nat > list_nat,Xs: list_list_nat] :
( ( distin876741697294417026st_nat @ ( map_li7225945977422193158st_nat @ F @ Xs ) )
=> ( distin876741697294417026st_nat @ Xs ) ) ).
% distinct_adj_mapD
thf(fact_133_distinct__adj__mapD,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( distinct_adj_nat @ ( map_nat_nat @ F @ Xs ) )
=> ( distinct_adj_nat @ Xs ) ) ).
% distinct_adj_mapD
thf(fact_134_distinct__adj__mapD,axiom,
! [F: a > a,Xs: list_a] :
( ( distinct_adj_a @ ( map_a_a @ F @ Xs ) )
=> ( distinct_adj_a @ Xs ) ) ).
% distinct_adj_mapD
thf(fact_135_image__set,axiom,
! [F: a > a,Xs: list_a] :
( ( image_a_a @ F @ ( set_a2 @ Xs ) )
= ( set_a2 @ ( map_a_a @ F @ Xs ) ) ) ).
% image_set
thf(fact_136_image__set,axiom,
! [F: a > nat,Xs: list_a] :
( ( image_a_nat @ F @ ( set_a2 @ Xs ) )
= ( set_nat2 @ ( map_a_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_137_image__set,axiom,
! [F: a > list_nat,Xs: list_a] :
( ( image_a_list_nat @ F @ ( set_a2 @ Xs ) )
= ( set_list_nat2 @ ( map_a_list_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_138_image__set,axiom,
! [F: nat > a,Xs: list_nat] :
( ( image_nat_a @ F @ ( set_nat2 @ Xs ) )
= ( set_a2 @ ( map_nat_a @ F @ Xs ) ) ) ).
% image_set
thf(fact_139_image__set,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( image_nat_nat @ F @ ( set_nat2 @ Xs ) )
= ( set_nat2 @ ( map_nat_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_140_image__set,axiom,
! [F: nat > list_nat,Xs: list_nat] :
( ( image_nat_list_nat @ F @ ( set_nat2 @ Xs ) )
= ( set_list_nat2 @ ( map_nat_list_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_141_image__set,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat] :
( ( image_5284832723445046202at_nat @ F @ ( set_list_nat2 @ Xs ) )
= ( set_se5049602875457034614at_nat @ ( map_li6003994582982014139at_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_142_image__set,axiom,
! [F: list_nat > a,Xs: list_list_nat] :
( ( image_list_nat_a @ F @ ( set_list_nat2 @ Xs ) )
= ( set_a2 @ ( map_list_nat_a @ F @ Xs ) ) ) ).
% image_set
thf(fact_143_image__set,axiom,
! [F: list_nat > nat,Xs: list_list_nat] :
( ( image_list_nat_nat @ F @ ( set_list_nat2 @ Xs ) )
= ( set_nat2 @ ( map_list_nat_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_144_image__set,axiom,
! [F: list_nat > list_nat,Xs: list_list_nat] :
( ( image_7976474329151083847st_nat @ F @ ( set_list_nat2 @ Xs ) )
= ( set_list_nat2 @ ( map_li7225945977422193158st_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_145_c,axiom,
ord_less_nat @ ( f @ x2 ) @ ( suc @ ( equiva5889994315859557365_limit @ ( map_a_nat @ f2 @ x1 ) ) ) ).
% c
thf(fact_146_insert__image,axiom,
! [X2: a,A: set_a,F: a > a] :
( ( member_a2 @ X2 @ A )
=> ( ( insert_a2 @ ( F @ X2 ) @ ( image_a_a @ F @ A ) )
= ( image_a_a @ F @ A ) ) ) ).
% insert_image
thf(fact_147_insert__image,axiom,
! [X2: a,A: set_a,F: a > nat] :
( ( member_a2 @ X2 @ A )
=> ( ( insert_nat2 @ ( F @ X2 ) @ ( image_a_nat @ F @ A ) )
= ( image_a_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_148_insert__image,axiom,
! [X2: nat,A: set_nat,F: nat > a] :
( ( member_nat2 @ X2 @ A )
=> ( ( insert_a2 @ ( F @ X2 ) @ ( image_nat_a @ F @ A ) )
= ( image_nat_a @ F @ A ) ) ) ).
% insert_image
thf(fact_149_insert__image,axiom,
! [X2: nat,A: set_nat,F: nat > nat] :
( ( member_nat2 @ X2 @ A )
=> ( ( insert_nat2 @ ( F @ X2 ) @ ( image_nat_nat @ F @ A ) )
= ( image_nat_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_150_insert__image,axiom,
! [X2: list_nat,A: set_list_nat,F: list_nat > a] :
( ( member_list_nat2 @ X2 @ A )
=> ( ( insert_a2 @ ( F @ X2 ) @ ( image_list_nat_a @ F @ A ) )
= ( image_list_nat_a @ F @ A ) ) ) ).
% insert_image
thf(fact_151_insert__image,axiom,
! [X2: list_nat,A: set_list_nat,F: list_nat > nat] :
( ( member_list_nat2 @ X2 @ A )
=> ( ( insert_nat2 @ ( F @ X2 ) @ ( image_list_nat_nat @ F @ A ) )
= ( image_list_nat_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_152_image__insert,axiom,
! [F: a > a,A2: a,B: set_a] :
( ( image_a_a @ F @ ( insert_a2 @ A2 @ B ) )
= ( insert_a2 @ ( F @ A2 ) @ ( image_a_a @ F @ B ) ) ) ).
% image_insert
thf(fact_153_image__insert,axiom,
! [F: a > nat,A2: a,B: set_a] :
( ( image_a_nat @ F @ ( insert_a2 @ A2 @ B ) )
= ( insert_nat2 @ ( F @ A2 ) @ ( image_a_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_154_image__insert,axiom,
! [F: nat > a,A2: nat,B: set_nat] :
( ( image_nat_a @ F @ ( insert_nat2 @ A2 @ B ) )
= ( insert_a2 @ ( F @ A2 ) @ ( image_nat_a @ F @ B ) ) ) ).
% image_insert
thf(fact_155_image__insert,axiom,
! [F: nat > nat,A2: nat,B: set_nat] :
( ( image_nat_nat @ F @ ( insert_nat2 @ A2 @ B ) )
= ( insert_nat2 @ ( F @ A2 ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_156_inj__img__insertE,axiom,
! [F: a > a,A: set_a,X2: a,B: set_a] :
( ( inj_on_a_a @ F @ A )
=> ( ~ ( member_a2 @ X2 @ B )
=> ( ( ( insert_a2 @ X2 @ B )
= ( image_a_a @ F @ A ) )
=> ~ ! [X4: a,A3: set_a] :
( ~ ( member_a2 @ X4 @ A3 )
=> ( ( A
= ( insert_a2 @ X4 @ A3 ) )
=> ( ( X2
= ( F @ X4 ) )
=> ( B
!= ( image_a_a @ F @ A3 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_157_inj__img__insertE,axiom,
! [F: nat > a,A: set_nat,X2: a,B: set_a] :
( ( inj_on_nat_a @ F @ A )
=> ( ~ ( member_a2 @ X2 @ B )
=> ( ( ( insert_a2 @ X2 @ B )
= ( image_nat_a @ F @ A ) )
=> ~ ! [X4: nat,A3: set_nat] :
( ~ ( member_nat2 @ X4 @ A3 )
=> ( ( A
= ( insert_nat2 @ X4 @ A3 ) )
=> ( ( X2
= ( F @ X4 ) )
=> ( B
!= ( image_nat_a @ F @ A3 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_158_inj__img__insertE,axiom,
! [F: list_nat > a,A: set_list_nat,X2: a,B: set_a] :
( ( inj_on_list_nat_a @ F @ A )
=> ( ~ ( member_a2 @ X2 @ B )
=> ( ( ( insert_a2 @ X2 @ B )
= ( image_list_nat_a @ F @ A ) )
=> ~ ! [X4: list_nat,A3: set_list_nat] :
( ~ ( member_list_nat2 @ X4 @ A3 )
=> ( ( A
= ( insert_list_nat2 @ X4 @ A3 ) )
=> ( ( X2
= ( F @ X4 ) )
=> ( B
!= ( image_list_nat_a @ F @ A3 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_159_inj__img__insertE,axiom,
! [F: a > nat,A: set_a,X2: nat,B: set_nat] :
( ( inj_on_a_nat @ F @ A )
=> ( ~ ( member_nat2 @ X2 @ B )
=> ( ( ( insert_nat2 @ X2 @ B )
= ( image_a_nat @ F @ A ) )
=> ~ ! [X4: a,A3: set_a] :
( ~ ( member_a2 @ X4 @ A3 )
=> ( ( A
= ( insert_a2 @ X4 @ A3 ) )
=> ( ( X2
= ( F @ X4 ) )
=> ( B
!= ( image_a_nat @ F @ A3 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_160_inj__img__insertE,axiom,
! [F: nat > nat,A: set_nat,X2: nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ~ ( member_nat2 @ X2 @ B )
=> ( ( ( insert_nat2 @ X2 @ B )
= ( image_nat_nat @ F @ A ) )
=> ~ ! [X4: nat,A3: set_nat] :
( ~ ( member_nat2 @ X4 @ A3 )
=> ( ( A
= ( insert_nat2 @ X4 @ A3 ) )
=> ( ( X2
= ( F @ X4 ) )
=> ( B
!= ( image_nat_nat @ F @ A3 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_161_inj__img__insertE,axiom,
! [F: list_nat > nat,A: set_list_nat,X2: nat,B: set_nat] :
( ( inj_on_list_nat_nat @ F @ A )
=> ( ~ ( member_nat2 @ X2 @ B )
=> ( ( ( insert_nat2 @ X2 @ B )
= ( image_list_nat_nat @ F @ A ) )
=> ~ ! [X4: list_nat,A3: set_list_nat] :
( ~ ( member_list_nat2 @ X4 @ A3 )
=> ( ( A
= ( insert_list_nat2 @ X4 @ A3 ) )
=> ( ( X2
= ( F @ X4 ) )
=> ( B
!= ( image_list_nat_nat @ F @ A3 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_162_inj__img__insertE,axiom,
! [F: a > list_nat,A: set_a,X2: list_nat,B: set_list_nat] :
( ( inj_on_a_list_nat @ F @ A )
=> ( ~ ( member_list_nat2 @ X2 @ B )
=> ( ( ( insert_list_nat2 @ X2 @ B )
= ( image_a_list_nat @ F @ A ) )
=> ~ ! [X4: a,A3: set_a] :
( ~ ( member_a2 @ X4 @ A3 )
=> ( ( A
= ( insert_a2 @ X4 @ A3 ) )
=> ( ( X2
= ( F @ X4 ) )
=> ( B
!= ( image_a_list_nat @ F @ A3 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_163_inj__img__insertE,axiom,
! [F: nat > list_nat,A: set_nat,X2: list_nat,B: set_list_nat] :
( ( inj_on_nat_list_nat @ F @ A )
=> ( ~ ( member_list_nat2 @ X2 @ B )
=> ( ( ( insert_list_nat2 @ X2 @ B )
= ( image_nat_list_nat @ F @ A ) )
=> ~ ! [X4: nat,A3: set_nat] :
( ~ ( member_nat2 @ X4 @ A3 )
=> ( ( A
= ( insert_nat2 @ X4 @ A3 ) )
=> ( ( X2
= ( F @ X4 ) )
=> ( B
!= ( image_nat_list_nat @ F @ A3 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_164_inj__img__insertE,axiom,
! [F: list_nat > list_nat,A: set_list_nat,X2: list_nat,B: set_list_nat] :
( ( inj_on3049792774292151987st_nat @ F @ A )
=> ( ~ ( member_list_nat2 @ X2 @ B )
=> ( ( ( insert_list_nat2 @ X2 @ B )
= ( image_7976474329151083847st_nat @ F @ A ) )
=> ~ ! [X4: list_nat,A3: set_list_nat] :
( ~ ( member_list_nat2 @ X4 @ A3 )
=> ( ( A
= ( insert_list_nat2 @ X4 @ A3 ) )
=> ( ( X2
= ( F @ X4 ) )
=> ( B
!= ( image_7976474329151083847st_nat @ F @ A3 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_165_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a2 @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_166_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat2 @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_167_mem__Collect__eq,axiom,
! [A2: list_nat,P: list_nat > $o] :
( ( member_list_nat2 @ A2 @ ( collect_list_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_168_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a2 @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_169_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat2 @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_170_Collect__mem__eq,axiom,
! [A: set_list_nat] :
( ( collect_list_nat
@ ^ [X: list_nat] : ( member_list_nat2 @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_171_insert__absorb2,axiom,
! [X2: a,A: set_a] :
( ( insert_a2 @ X2 @ ( insert_a2 @ X2 @ A ) )
= ( insert_a2 @ X2 @ A ) ) ).
% insert_absorb2
thf(fact_172_insert__absorb2,axiom,
! [X2: nat,A: set_nat] :
( ( insert_nat2 @ X2 @ ( insert_nat2 @ X2 @ A ) )
= ( insert_nat2 @ X2 @ A ) ) ).
% insert_absorb2
thf(fact_173_insert__iff,axiom,
! [A2: a,B2: a,A: set_a] :
( ( member_a2 @ A2 @ ( insert_a2 @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_a2 @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_174_insert__iff,axiom,
! [A2: nat,B2: nat,A: set_nat] :
( ( member_nat2 @ A2 @ ( insert_nat2 @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_nat2 @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_175_insert__iff,axiom,
! [A2: list_nat,B2: list_nat,A: set_list_nat] :
( ( member_list_nat2 @ A2 @ ( insert_list_nat2 @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_list_nat2 @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_176_insertCI,axiom,
! [A2: a,B: set_a,B2: a] :
( ( ~ ( member_a2 @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_a2 @ A2 @ ( insert_a2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_177_insertCI,axiom,
! [A2: nat,B: set_nat,B2: nat] :
( ( ~ ( member_nat2 @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_nat2 @ A2 @ ( insert_nat2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_178_insertCI,axiom,
! [A2: list_nat,B: set_list_nat,B2: list_nat] :
( ( ~ ( member_list_nat2 @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_list_nat2 @ A2 @ ( insert_list_nat2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_179_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_180_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_181_image__eqI,axiom,
! [B2: a,F: a > a,X2: a,A: set_a] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_a2 @ X2 @ A )
=> ( member_a2 @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_182_image__eqI,axiom,
! [B2: nat,F: a > nat,X2: a,A: set_a] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_a2 @ X2 @ A )
=> ( member_nat2 @ B2 @ ( image_a_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_183_image__eqI,axiom,
! [B2: list_nat,F: a > list_nat,X2: a,A: set_a] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_a2 @ X2 @ A )
=> ( member_list_nat2 @ B2 @ ( image_a_list_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_184_image__eqI,axiom,
! [B2: a,F: nat > a,X2: nat,A: set_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_nat2 @ X2 @ A )
=> ( member_a2 @ B2 @ ( image_nat_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_185_image__eqI,axiom,
! [B2: nat,F: nat > nat,X2: nat,A: set_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_nat2 @ X2 @ A )
=> ( member_nat2 @ B2 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_186_image__eqI,axiom,
! [B2: list_nat,F: nat > list_nat,X2: nat,A: set_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_nat2 @ X2 @ A )
=> ( member_list_nat2 @ B2 @ ( image_nat_list_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_187_image__eqI,axiom,
! [B2: a,F: list_nat > a,X2: list_nat,A: set_list_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_list_nat2 @ X2 @ A )
=> ( member_a2 @ B2 @ ( image_list_nat_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_188_image__eqI,axiom,
! [B2: nat,F: list_nat > nat,X2: list_nat,A: set_list_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_list_nat2 @ X2 @ A )
=> ( member_nat2 @ B2 @ ( image_list_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_189_image__eqI,axiom,
! [B2: list_nat,F: list_nat > list_nat,X2: list_nat,A: set_list_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_list_nat2 @ X2 @ A )
=> ( member_list_nat2 @ B2 @ ( image_7976474329151083847st_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_190_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_191_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_192_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_193_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_194_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_195_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_196_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_197_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_198_less__not__refl3,axiom,
! [S3: nat,T: nat] :
( ( ord_less_nat @ S3 @ T )
=> ( S3 != T ) ) ).
% less_not_refl3
thf(fact_199_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_200_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_201_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_202_linorder__neqE__nat,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_203_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J: nat] :
( ( ord_less_nat @ I @ J )
=> ( K
!= ( suc @ J ) ) ) ) ) ).
% Nat.lessE
thf(fact_204_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_205_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J: nat] :
( ( ord_less_nat @ I @ J )
=> ( K
!= ( suc @ J ) ) ) ) ).
% Suc_lessE
thf(fact_206_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_207_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_208_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_209_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ N )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_210_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_211_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_212_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_213_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M3: nat] :
( ( M
= ( suc @ M3 ) )
& ( ord_less_nat @ N @ M3 ) ) ) ) ).
% Suc_less_eq2
thf(fact_214_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_215_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_216_less__trans__Suc,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_217_less__Suc__induct,axiom,
! [I: nat,J2: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J2 )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( ord_less_nat @ J @ K2 )
=> ( ( P @ I3 @ J )
=> ( ( P @ J @ K2 )
=> ( P @ I3 @ K2 ) ) ) ) )
=> ( P @ I @ J2 ) ) ) ) ).
% less_Suc_induct
thf(fact_218_strict__inc__induct,axiom,
! [I: nat,J2: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J2 )
=> ( ! [I3: nat] :
( ( J2
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_219_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_220_linorder__inj__onI_H,axiom,
! [A: set_nat,F: nat > nat] :
( ! [I3: nat,J: nat] :
( ( member_nat2 @ I3 @ A )
=> ( ( member_nat2 @ J @ A )
=> ( ( ord_less_nat @ I3 @ J )
=> ( ( F @ I3 )
!= ( F @ J ) ) ) ) )
=> ( inj_on_nat_nat @ F @ A ) ) ).
% linorder_inj_onI'
thf(fact_221_length__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ! [Xs3: list_a] :
( ! [Ys2: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ Ys2 ) @ ( size_size_list_a @ Xs3 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_222_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs3: list_nat] :
( ! [Ys2: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs3 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_223_rgf__limit__ge,axiom,
! [Y2: nat,Xs: list_nat] :
( ( member_nat2 @ Y2 @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ Y2 @ ( equiva5889994315859557365_limit @ Xs ) ) ) ).
% rgf_limit_ge
thf(fact_224_imageI,axiom,
! [X2: a,A: set_a,F: a > a] :
( ( member_a2 @ X2 @ A )
=> ( member_a2 @ ( F @ X2 ) @ ( image_a_a @ F @ A ) ) ) ).
% imageI
thf(fact_225_imageI,axiom,
! [X2: a,A: set_a,F: a > nat] :
( ( member_a2 @ X2 @ A )
=> ( member_nat2 @ ( F @ X2 ) @ ( image_a_nat @ F @ A ) ) ) ).
% imageI
thf(fact_226_imageI,axiom,
! [X2: a,A: set_a,F: a > list_nat] :
( ( member_a2 @ X2 @ A )
=> ( member_list_nat2 @ ( F @ X2 ) @ ( image_a_list_nat @ F @ A ) ) ) ).
% imageI
thf(fact_227_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > a] :
( ( member_nat2 @ X2 @ A )
=> ( member_a2 @ ( F @ X2 ) @ ( image_nat_a @ F @ A ) ) ) ).
% imageI
thf(fact_228_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > nat] :
( ( member_nat2 @ X2 @ A )
=> ( member_nat2 @ ( F @ X2 ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_229_imageI,axiom,
! [X2: nat,A: set_nat,F: nat > list_nat] :
( ( member_nat2 @ X2 @ A )
=> ( member_list_nat2 @ ( F @ X2 ) @ ( image_nat_list_nat @ F @ A ) ) ) ).
% imageI
thf(fact_230_imageI,axiom,
! [X2: list_nat,A: set_list_nat,F: list_nat > a] :
( ( member_list_nat2 @ X2 @ A )
=> ( member_a2 @ ( F @ X2 ) @ ( image_list_nat_a @ F @ A ) ) ) ).
% imageI
thf(fact_231_imageI,axiom,
! [X2: list_nat,A: set_list_nat,F: list_nat > nat] :
( ( member_list_nat2 @ X2 @ A )
=> ( member_nat2 @ ( F @ X2 ) @ ( image_list_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_232_imageI,axiom,
! [X2: list_nat,A: set_list_nat,F: list_nat > list_nat] :
( ( member_list_nat2 @ X2 @ A )
=> ( member_list_nat2 @ ( F @ X2 ) @ ( image_7976474329151083847st_nat @ F @ A ) ) ) ).
% imageI
thf(fact_233_image__iff,axiom,
! [Z2: nat,F: a > nat,A: set_a] :
( ( member_nat2 @ Z2 @ ( image_a_nat @ F @ A ) )
= ( ? [X: a] :
( ( member_a2 @ X @ A )
& ( Z2
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_234_image__iff,axiom,
! [Z2: nat,F: nat > nat,A: set_nat] :
( ( member_nat2 @ Z2 @ ( image_nat_nat @ F @ A ) )
= ( ? [X: nat] :
( ( member_nat2 @ X @ A )
& ( Z2
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_235_bex__imageD,axiom,
! [F: a > nat,A: set_a,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat2 @ X5 @ ( image_a_nat @ F @ A ) )
& ( P @ X5 ) )
=> ? [X3: a] :
( ( member_a2 @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_236_bex__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat2 @ X5 @ ( image_nat_nat @ F @ A ) )
& ( P @ X5 ) )
=> ? [X3: nat] :
( ( member_nat2 @ X3 @ A )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_237_image__cong,axiom,
! [M4: set_a,N4: set_a,F: a > nat,G: a > nat] :
( ( M4 = N4 )
=> ( ! [X3: a] :
( ( member_a2 @ X3 @ N4 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_a_nat @ F @ M4 )
= ( image_a_nat @ G @ N4 ) ) ) ) ).
% image_cong
thf(fact_238_image__cong,axiom,
! [M4: set_nat,N4: set_nat,F: nat > nat,G: nat > nat] :
( ( M4 = N4 )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ N4 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_nat @ F @ M4 )
= ( image_nat_nat @ G @ N4 ) ) ) ) ).
% image_cong
thf(fact_239_ball__imageD,axiom,
! [F: a > nat,A: set_a,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( image_a_nat @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X5: a] :
( ( member_a2 @ X5 @ A )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_240_ball__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( image_nat_nat @ F @ A ) )
=> ( P @ X3 ) )
=> ! [X5: nat] :
( ( member_nat2 @ X5 @ A )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_241_rev__image__eqI,axiom,
! [X2: a,A: set_a,B2: a,F: a > a] :
( ( member_a2 @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_a2 @ B2 @ ( image_a_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_242_rev__image__eqI,axiom,
! [X2: a,A: set_a,B2: nat,F: a > nat] :
( ( member_a2 @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_nat2 @ B2 @ ( image_a_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_243_rev__image__eqI,axiom,
! [X2: a,A: set_a,B2: list_nat,F: a > list_nat] :
( ( member_a2 @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_list_nat2 @ B2 @ ( image_a_list_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_244_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B2: a,F: nat > a] :
( ( member_nat2 @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_a2 @ B2 @ ( image_nat_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_245_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B2: nat,F: nat > nat] :
( ( member_nat2 @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_nat2 @ B2 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_246_rev__image__eqI,axiom,
! [X2: nat,A: set_nat,B2: list_nat,F: nat > list_nat] :
( ( member_nat2 @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_list_nat2 @ B2 @ ( image_nat_list_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_247_rev__image__eqI,axiom,
! [X2: list_nat,A: set_list_nat,B2: a,F: list_nat > a] :
( ( member_list_nat2 @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_a2 @ B2 @ ( image_list_nat_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_248_rev__image__eqI,axiom,
! [X2: list_nat,A: set_list_nat,B2: nat,F: list_nat > nat] :
( ( member_list_nat2 @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_nat2 @ B2 @ ( image_list_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_249_rev__image__eqI,axiom,
! [X2: list_nat,A: set_list_nat,B2: list_nat,F: list_nat > list_nat] :
( ( member_list_nat2 @ X2 @ A )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_list_nat2 @ B2 @ ( image_7976474329151083847st_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_250_Suc__inject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
=> ( X2 = Y2 ) ) ).
% Suc_inject
thf(fact_251_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_252_inj__Suc,axiom,
! [N4: set_nat] : ( inj_on_nat_nat @ suc @ N4 ) ).
% inj_Suc
thf(fact_253_insertE,axiom,
! [A2: a,B2: a,A: set_a] :
( ( member_a2 @ A2 @ ( insert_a2 @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_a2 @ A2 @ A ) ) ) ).
% insertE
thf(fact_254_insertE,axiom,
! [A2: nat,B2: nat,A: set_nat] :
( ( member_nat2 @ A2 @ ( insert_nat2 @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_nat2 @ A2 @ A ) ) ) ).
% insertE
thf(fact_255_insertE,axiom,
! [A2: list_nat,B2: list_nat,A: set_list_nat] :
( ( member_list_nat2 @ A2 @ ( insert_list_nat2 @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_list_nat2 @ A2 @ A ) ) ) ).
% insertE
thf(fact_256_insertI1,axiom,
! [A2: a,B: set_a] : ( member_a2 @ A2 @ ( insert_a2 @ A2 @ B ) ) ).
% insertI1
thf(fact_257_insertI1,axiom,
! [A2: nat,B: set_nat] : ( member_nat2 @ A2 @ ( insert_nat2 @ A2 @ B ) ) ).
% insertI1
thf(fact_258_insertI1,axiom,
! [A2: list_nat,B: set_list_nat] : ( member_list_nat2 @ A2 @ ( insert_list_nat2 @ A2 @ B ) ) ).
% insertI1
thf(fact_259_insertI2,axiom,
! [A2: a,B: set_a,B2: a] :
( ( member_a2 @ A2 @ B )
=> ( member_a2 @ A2 @ ( insert_a2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_260_insertI2,axiom,
! [A2: nat,B: set_nat,B2: nat] :
( ( member_nat2 @ A2 @ B )
=> ( member_nat2 @ A2 @ ( insert_nat2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_261_insertI2,axiom,
! [A2: list_nat,B: set_list_nat,B2: list_nat] :
( ( member_list_nat2 @ A2 @ B )
=> ( member_list_nat2 @ A2 @ ( insert_list_nat2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_262_Set_Oset__insert,axiom,
! [X2: a,A: set_a] :
( ( member_a2 @ X2 @ A )
=> ~ ! [B3: set_a] :
( ( A
= ( insert_a2 @ X2 @ B3 ) )
=> ( member_a2 @ X2 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_263_Set_Oset__insert,axiom,
! [X2: nat,A: set_nat] :
( ( member_nat2 @ X2 @ A )
=> ~ ! [B3: set_nat] :
( ( A
= ( insert_nat2 @ X2 @ B3 ) )
=> ( member_nat2 @ X2 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_264_Set_Oset__insert,axiom,
! [X2: list_nat,A: set_list_nat] :
( ( member_list_nat2 @ X2 @ A )
=> ~ ! [B3: set_list_nat] :
( ( A
= ( insert_list_nat2 @ X2 @ B3 ) )
=> ( member_list_nat2 @ X2 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_265_insert__ident,axiom,
! [X2: a,A: set_a,B: set_a] :
( ~ ( member_a2 @ X2 @ A )
=> ( ~ ( member_a2 @ X2 @ B )
=> ( ( ( insert_a2 @ X2 @ A )
= ( insert_a2 @ X2 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_266_insert__ident,axiom,
! [X2: nat,A: set_nat,B: set_nat] :
( ~ ( member_nat2 @ X2 @ A )
=> ( ~ ( member_nat2 @ X2 @ B )
=> ( ( ( insert_nat2 @ X2 @ A )
= ( insert_nat2 @ X2 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_267_insert__ident,axiom,
! [X2: list_nat,A: set_list_nat,B: set_list_nat] :
( ~ ( member_list_nat2 @ X2 @ A )
=> ( ~ ( member_list_nat2 @ X2 @ B )
=> ( ( ( insert_list_nat2 @ X2 @ A )
= ( insert_list_nat2 @ X2 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_268_insert__absorb,axiom,
! [A2: a,A: set_a] :
( ( member_a2 @ A2 @ A )
=> ( ( insert_a2 @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_269_insert__absorb,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat2 @ A2 @ A )
=> ( ( insert_nat2 @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_270_insert__absorb,axiom,
! [A2: list_nat,A: set_list_nat] :
( ( member_list_nat2 @ A2 @ A )
=> ( ( insert_list_nat2 @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_271_insert__eq__iff,axiom,
! [A2: a,A: set_a,B2: a,B: set_a] :
( ~ ( member_a2 @ A2 @ A )
=> ( ~ ( member_a2 @ B2 @ B )
=> ( ( ( insert_a2 @ A2 @ A )
= ( insert_a2 @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C: set_a] :
( ( A
= ( insert_a2 @ B2 @ C ) )
& ~ ( member_a2 @ B2 @ C )
& ( B
= ( insert_a2 @ A2 @ C ) )
& ~ ( member_a2 @ A2 @ C ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_272_insert__eq__iff,axiom,
! [A2: nat,A: set_nat,B2: nat,B: set_nat] :
( ~ ( member_nat2 @ A2 @ A )
=> ( ~ ( member_nat2 @ B2 @ B )
=> ( ( ( insert_nat2 @ A2 @ A )
= ( insert_nat2 @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C: set_nat] :
( ( A
= ( insert_nat2 @ B2 @ C ) )
& ~ ( member_nat2 @ B2 @ C )
& ( B
= ( insert_nat2 @ A2 @ C ) )
& ~ ( member_nat2 @ A2 @ C ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_273_insert__eq__iff,axiom,
! [A2: list_nat,A: set_list_nat,B2: list_nat,B: set_list_nat] :
( ~ ( member_list_nat2 @ A2 @ A )
=> ( ~ ( member_list_nat2 @ B2 @ B )
=> ( ( ( insert_list_nat2 @ A2 @ A )
= ( insert_list_nat2 @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C: set_list_nat] :
( ( A
= ( insert_list_nat2 @ B2 @ C ) )
& ~ ( member_list_nat2 @ B2 @ C )
& ( B
= ( insert_list_nat2 @ A2 @ C ) )
& ~ ( member_list_nat2 @ A2 @ C ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_274_insert__commute,axiom,
! [X2: a,Y2: a,A: set_a] :
( ( insert_a2 @ X2 @ ( insert_a2 @ Y2 @ A ) )
= ( insert_a2 @ Y2 @ ( insert_a2 @ X2 @ A ) ) ) ).
% insert_commute
thf(fact_275_insert__commute,axiom,
! [X2: nat,Y2: nat,A: set_nat] :
( ( insert_nat2 @ X2 @ ( insert_nat2 @ Y2 @ A ) )
= ( insert_nat2 @ Y2 @ ( insert_nat2 @ X2 @ A ) ) ) ).
% insert_commute
thf(fact_276_mk__disjoint__insert,axiom,
! [A2: a,A: set_a] :
( ( member_a2 @ A2 @ A )
=> ? [B3: set_a] :
( ( A
= ( insert_a2 @ A2 @ B3 ) )
& ~ ( member_a2 @ A2 @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_277_mk__disjoint__insert,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat2 @ A2 @ A )
=> ? [B3: set_nat] :
( ( A
= ( insert_nat2 @ A2 @ B3 ) )
& ~ ( member_nat2 @ A2 @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_278_mk__disjoint__insert,axiom,
! [A2: list_nat,A: set_list_nat] :
( ( member_list_nat2 @ A2 @ A )
=> ? [B3: set_list_nat] :
( ( A
= ( insert_list_nat2 @ A2 @ B3 ) )
& ~ ( member_list_nat2 @ A2 @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_279_size__neq__size__imp__neq,axiom,
! [X2: list_a,Y2: list_a] :
( ( ( size_size_list_a @ X2 )
!= ( size_size_list_a @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_280_size__neq__size__imp__neq,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( ( size_size_list_nat @ X2 )
!= ( size_size_list_nat @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_281_size__neq__size__imp__neq,axiom,
! [X2: char,Y2: char] :
( ( ( size_size_char @ X2 )
!= ( size_size_char @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_282_inj__on__inverseI,axiom,
! [A: set_a,G: nat > a,F: a > nat] :
( ! [X3: a] :
( ( member_a2 @ X3 @ A )
=> ( ( G @ ( F @ X3 ) )
= X3 ) )
=> ( inj_on_a_nat @ F @ A ) ) ).
% inj_on_inverseI
thf(fact_283_inj__on__inverseI,axiom,
! [A: set_nat,G: nat > nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ A )
=> ( ( G @ ( F @ X3 ) )
= X3 ) )
=> ( inj_on_nat_nat @ F @ A ) ) ).
% inj_on_inverseI
thf(fact_284_inj__on__contraD,axiom,
! [F: a > nat,A: set_a,X2: a,Y2: a] :
( ( inj_on_a_nat @ F @ A )
=> ( ( X2 != Y2 )
=> ( ( member_a2 @ X2 @ A )
=> ( ( member_a2 @ Y2 @ A )
=> ( ( F @ X2 )
!= ( F @ Y2 ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_285_inj__on__contraD,axiom,
! [F: nat > nat,A: set_nat,X2: nat,Y2: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( X2 != Y2 )
=> ( ( member_nat2 @ X2 @ A )
=> ( ( member_nat2 @ Y2 @ A )
=> ( ( F @ X2 )
!= ( F @ Y2 ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_286_inj__on__eq__iff,axiom,
! [F: a > nat,A: set_a,X2: a,Y2: a] :
( ( inj_on_a_nat @ F @ A )
=> ( ( member_a2 @ X2 @ A )
=> ( ( member_a2 @ Y2 @ A )
=> ( ( ( F @ X2 )
= ( F @ Y2 ) )
= ( X2 = Y2 ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_287_inj__on__eq__iff,axiom,
! [F: nat > nat,A: set_nat,X2: nat,Y2: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( member_nat2 @ X2 @ A )
=> ( ( member_nat2 @ Y2 @ A )
=> ( ( ( F @ X2 )
= ( F @ Y2 ) )
= ( X2 = Y2 ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_288_inj__on__cong,axiom,
! [A: set_a,F: a > nat,G: a > nat] :
( ! [A4: a] :
( ( member_a2 @ A4 @ A )
=> ( ( F @ A4 )
= ( G @ A4 ) ) )
=> ( ( inj_on_a_nat @ F @ A )
= ( inj_on_a_nat @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_289_inj__on__cong,axiom,
! [A: set_nat,F: nat > nat,G: nat > nat] :
( ! [A4: nat] :
( ( member_nat2 @ A4 @ A )
=> ( ( F @ A4 )
= ( G @ A4 ) ) )
=> ( ( inj_on_nat_nat @ F @ A )
= ( inj_on_nat_nat @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_290_inj__on__def,axiom,
( inj_on_a_nat
= ( ^ [F3: a > nat,A5: set_a] :
! [X: a] :
( ( member_a2 @ X @ A5 )
=> ! [Y: a] :
( ( member_a2 @ Y @ A5 )
=> ( ( ( F3 @ X )
= ( F3 @ Y ) )
=> ( X = Y ) ) ) ) ) ) ).
% inj_on_def
thf(fact_291_inj__on__def,axiom,
( inj_on_nat_nat
= ( ^ [F3: nat > nat,A5: set_nat] :
! [X: nat] :
( ( member_nat2 @ X @ A5 )
=> ! [Y: nat] :
( ( member_nat2 @ Y @ A5 )
=> ( ( ( F3 @ X )
= ( F3 @ Y ) )
=> ( X = Y ) ) ) ) ) ) ).
% inj_on_def
thf(fact_292_inj__onI,axiom,
! [A: set_a,F: a > nat] :
( ! [X3: a,Y3: a] :
( ( member_a2 @ X3 @ A )
=> ( ( member_a2 @ Y3 @ A )
=> ( ( ( F @ X3 )
= ( F @ Y3 ) )
=> ( X3 = Y3 ) ) ) )
=> ( inj_on_a_nat @ F @ A ) ) ).
% inj_onI
thf(fact_293_inj__onI,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X3: nat,Y3: nat] :
( ( member_nat2 @ X3 @ A )
=> ( ( member_nat2 @ Y3 @ A )
=> ( ( ( F @ X3 )
= ( F @ Y3 ) )
=> ( X3 = Y3 ) ) ) )
=> ( inj_on_nat_nat @ F @ A ) ) ).
% inj_onI
thf(fact_294_inj__onD,axiom,
! [F: a > nat,A: set_a,X2: a,Y2: a] :
( ( inj_on_a_nat @ F @ A )
=> ( ( ( F @ X2 )
= ( F @ Y2 ) )
=> ( ( member_a2 @ X2 @ A )
=> ( ( member_a2 @ Y2 @ A )
=> ( X2 = Y2 ) ) ) ) ) ).
% inj_onD
thf(fact_295_inj__onD,axiom,
! [F: nat > nat,A: set_nat,X2: nat,Y2: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ( F @ X2 )
= ( F @ Y2 ) )
=> ( ( member_nat2 @ X2 @ A )
=> ( ( member_nat2 @ Y2 @ A )
=> ( X2 = Y2 ) ) ) ) ) ).
% inj_onD
thf(fact_296_inj__on__image__iff,axiom,
! [A: set_a,G: a > nat,F: a > a] :
( ! [X3: a] :
( ( member_a2 @ X3 @ A )
=> ! [Xa2: a] :
( ( member_a2 @ Xa2 @ A )
=> ( ( ( G @ ( F @ X3 ) )
= ( G @ ( F @ Xa2 ) ) )
= ( ( G @ X3 )
= ( G @ Xa2 ) ) ) ) )
=> ( ( inj_on_a_a @ F @ A )
=> ( ( inj_on_a_nat @ G @ ( image_a_a @ F @ A ) )
= ( inj_on_a_nat @ G @ A ) ) ) ) ).
% inj_on_image_iff
thf(fact_297_inj__on__image__iff,axiom,
! [A: set_nat,G: nat > nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat2 @ X3 @ A )
=> ! [Xa2: nat] :
( ( member_nat2 @ Xa2 @ A )
=> ( ( ( G @ ( F @ X3 ) )
= ( G @ ( F @ Xa2 ) ) )
= ( ( G @ X3 )
= ( G @ Xa2 ) ) ) ) )
=> ( ( inj_on_nat_nat @ F @ A )
=> ( ( inj_on_nat_nat @ G @ ( image_nat_nat @ F @ A ) )
= ( inj_on_nat_nat @ G @ A ) ) ) ) ).
% inj_on_image_iff
thf(fact_298_map__removeAll__inj__on,axiom,
! [F: a > nat,X2: a,Xs: list_a] :
( ( inj_on_a_nat @ F @ ( insert_a2 @ X2 @ ( set_a2 @ Xs ) ) )
=> ( ( map_a_nat @ F @ ( removeAll_a @ X2 @ Xs ) )
= ( removeAll_nat @ ( F @ X2 ) @ ( map_a_nat @ F @ Xs ) ) ) ) ).
% map_removeAll_inj_on
thf(fact_299_map__removeAll__inj__on,axiom,
! [F: a > a,X2: a,Xs: list_a] :
( ( inj_on_a_a @ F @ ( insert_a2 @ X2 @ ( set_a2 @ Xs ) ) )
=> ( ( map_a_a @ F @ ( removeAll_a @ X2 @ Xs ) )
= ( removeAll_a @ ( F @ X2 ) @ ( map_a_a @ F @ Xs ) ) ) ) ).
% map_removeAll_inj_on
thf(fact_300_map__removeAll__inj__on,axiom,
! [F: nat > nat,X2: nat,Xs: list_nat] :
( ( inj_on_nat_nat @ F @ ( insert_nat2 @ X2 @ ( set_nat2 @ Xs ) ) )
=> ( ( map_nat_nat @ F @ ( removeAll_nat @ X2 @ Xs ) )
= ( removeAll_nat @ ( F @ X2 ) @ ( map_nat_nat @ F @ Xs ) ) ) ) ).
% map_removeAll_inj_on
thf(fact_301_map__removeAll__inj__on,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,X2: list_nat,Xs: list_list_nat] :
( ( inj_on7522185085906380110at_nat @ F @ ( insert_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) ) )
=> ( ( map_li6003994582982014139at_nat @ F @ ( removeAll_list_nat @ X2 @ Xs ) )
= ( remove5672899571770113645at_nat @ ( F @ X2 ) @ ( map_li6003994582982014139at_nat @ F @ Xs ) ) ) ) ).
% map_removeAll_inj_on
thf(fact_302_map__removeAll__inj__on,axiom,
! [F: list_nat > list_nat,X2: list_nat,Xs: list_list_nat] :
( ( inj_on3049792774292151987st_nat @ F @ ( insert_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) ) )
=> ( ( map_li7225945977422193158st_nat @ F @ ( removeAll_list_nat @ X2 @ Xs ) )
= ( removeAll_list_nat @ ( F @ X2 ) @ ( map_li7225945977422193158st_nat @ F @ Xs ) ) ) ) ).
% map_removeAll_inj_on
thf(fact_303_List_Oset__insert,axiom,
! [X2: a,Xs: list_a] :
( ( set_a2 @ ( insert_a @ X2 @ Xs ) )
= ( insert_a2 @ X2 @ ( set_a2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_304_List_Oset__insert,axiom,
! [X2: nat,Xs: list_nat] :
( ( set_nat2 @ ( insert_nat @ X2 @ Xs ) )
= ( insert_nat2 @ X2 @ ( set_nat2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_305_List_Oset__insert,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( set_list_nat2 @ ( insert_list_nat @ X2 @ Xs ) )
= ( insert_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_306_length__removeAll__less,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ord_less_nat @ ( size_s3023201423986296836st_nat @ ( removeAll_list_nat @ X2 @ Xs ) ) @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_307_length__removeAll__less,axiom,
! [X2: a,Xs: list_a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( ord_less_nat @ ( size_size_list_a @ ( removeAll_a @ X2 @ Xs ) ) @ ( size_size_list_a @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_308_length__removeAll__less,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ ( size_size_list_nat @ ( removeAll_nat @ X2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_309_the__inv__into__onto,axiom,
! [F: nat > a,A: set_nat] :
( ( inj_on_nat_a @ F @ A )
=> ( ( image_a_nat @ ( the_inv_into_nat_a @ A @ F ) @ ( image_nat_a @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_310_the__inv__into__onto,axiom,
! [F: a > nat,A: set_a] :
( ( inj_on_a_nat @ F @ A )
=> ( ( image_nat_a @ ( the_inv_into_a_nat @ A @ F ) @ ( image_a_nat @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_311_the__inv__into__onto,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( image_nat_nat @ ( the_inv_into_nat_nat @ A @ F ) @ ( image_nat_nat @ F @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_312_insort__insert__key__triv,axiom,
! [F: a > nat,X2: a,Xs: list_a] :
( ( member_nat2 @ ( F @ X2 ) @ ( image_a_nat @ F @ ( set_a2 @ Xs ) ) )
=> ( ( linord1046132949341221836_a_nat @ F @ X2 @ Xs )
= Xs ) ) ).
% insort_insert_key_triv
thf(fact_313_insort__insert__key__triv,axiom,
! [F: nat > nat,X2: nat,Xs: list_nat] :
( ( member_nat2 @ ( F @ X2 ) @ ( image_nat_nat @ F @ ( set_nat2 @ Xs ) ) )
=> ( ( linord1921536354676448932at_nat @ F @ X2 @ Xs )
= Xs ) ) ).
% insort_insert_key_triv
thf(fact_314_insort__insert__key__triv,axiom,
! [F: list_nat > nat,X2: list_nat,Xs: list_list_nat] :
( ( member_nat2 @ ( F @ X2 ) @ ( image_list_nat_nat @ F @ ( set_list_nat2 @ Xs ) ) )
=> ( ( linord3253225449353161780at_nat @ F @ X2 @ Xs )
= Xs ) ) ).
% insort_insert_key_triv
thf(fact_315_rgf__imp__initial__segment,axiom,
! [Xs: list_nat] :
( ( equiva3371634703666331078on_rgf @ Xs )
=> ( ( set_nat2 @ Xs )
= ( set_ord_lessThan_nat @ ( equiva5889994315859557365_limit @ Xs ) ) ) ) ).
% rgf_imp_initial_segment
thf(fact_316_x__def,axiom,
( xa
= ( append_a @ x1 @ ( cons_a @ x2 @ nil_a ) ) ) ).
% x_def
thf(fact_317_nth__map,axiom,
! [N: nat,Xs: list_list_nat,F: list_nat > set_Pr1261947904930325089at_nat] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_se5679702624988424552at_nat @ ( map_li6003994582982014139at_nat @ F @ Xs ) @ N )
= ( F @ ( nth_list_nat @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_318_nth__map,axiom,
! [N: nat,Xs: list_list_nat,F: list_nat > list_nat] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_list_nat @ ( map_li7225945977422193158st_nat @ F @ Xs ) @ N )
= ( F @ ( nth_list_nat @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_319_nth__map,axiom,
! [N: nat,Xs: list_a,F: a > nat] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( nth_nat @ ( map_a_nat @ F @ Xs ) @ N )
= ( F @ ( nth_a @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_320_nth__map,axiom,
! [N: nat,Xs: list_a,F: a > a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( map_a_a @ F @ Xs ) @ N )
= ( F @ ( nth_a @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_321_nth__map,axiom,
! [N: nat,Xs: list_nat,F: nat > nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( map_nat_nat @ F @ Xs ) @ N )
= ( F @ ( nth_nat @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_322_length__stirling__row,axiom,
! [N: nat] :
( ( size_size_list_nat @ ( stirling_row @ N ) )
= ( suc @ N ) ) ).
% length_stirling_row
thf(fact_323_length__pos__if__in__set,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_324_length__pos__if__in__set,axiom,
! [X2: a,Xs: list_a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_325_length__pos__if__in__set,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_326_in__set__member,axiom,
! [X2: a,Xs: list_a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
= ( member_a @ Xs @ X2 ) ) ).
% in_set_member
thf(fact_327_in__set__member,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
= ( member_nat @ Xs @ X2 ) ) ).
% in_set_member
thf(fact_328_in__set__member,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
= ( member_list_nat @ Xs @ X2 ) ) ).
% in_set_member
thf(fact_329_list_Oinject,axiom,
! [X21: a,X222: list_a,Y21: a,Y222: list_a] :
( ( ( cons_a @ X21 @ X222 )
= ( cons_a @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_330_list_Oinject,axiom,
! [X21: nat,X222: list_nat,Y21: nat,Y222: list_nat] :
( ( ( cons_nat @ X21 @ X222 )
= ( cons_nat @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_331_list_Oinject,axiom,
! [X21: list_nat,X222: list_list_nat,Y21: list_nat,Y222: list_list_nat] :
( ( ( cons_list_nat @ X21 @ X222 )
= ( cons_list_nat @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_332_append_Oassoc,axiom,
! [A2: list_a,B2: list_a,C2: list_a] :
( ( append_a @ ( append_a @ A2 @ B2 ) @ C2 )
= ( append_a @ A2 @ ( append_a @ B2 @ C2 ) ) ) ).
% append.assoc
thf(fact_333_append_Oassoc,axiom,
! [A2: list_nat,B2: list_nat,C2: list_nat] :
( ( append_nat @ ( append_nat @ A2 @ B2 ) @ C2 )
= ( append_nat @ A2 @ ( append_nat @ B2 @ C2 ) ) ) ).
% append.assoc
thf(fact_334_append__assoc,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( append_a @ ( append_a @ Xs @ Ys ) @ Zs )
= ( append_a @ Xs @ ( append_a @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_335_append__assoc,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( append_nat @ ( append_nat @ Xs @ Ys ) @ Zs )
= ( append_nat @ Xs @ ( append_nat @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_336_append__same__eq,axiom,
! [Ys: list_a,Xs: list_a,Zs: list_a] :
( ( ( append_a @ Ys @ Xs )
= ( append_a @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_337_append__same__eq,axiom,
! [Ys: list_nat,Xs: list_nat,Zs: list_nat] :
( ( ( append_nat @ Ys @ Xs )
= ( append_nat @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_338_same__append__eq,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( append_a @ Xs @ Ys )
= ( append_a @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_339_same__append__eq,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_340_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_341_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_342_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_343_append_Oright__neutral,axiom,
! [A2: list_a] :
( ( append_a @ A2 @ nil_a )
= A2 ) ).
% append.right_neutral
thf(fact_344_append_Oright__neutral,axiom,
! [A2: list_nat] :
( ( append_nat @ A2 @ nil_nat )
= A2 ) ).
% append.right_neutral
thf(fact_345_append_Oright__neutral,axiom,
! [A2: list_list_nat] :
( ( append_list_nat @ A2 @ nil_list_nat )
= A2 ) ).
% append.right_neutral
thf(fact_346_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_347_append__Nil2,axiom,
! [Xs: list_nat] :
( ( append_nat @ Xs @ nil_nat )
= Xs ) ).
% append_Nil2
thf(fact_348_append__Nil2,axiom,
! [Xs: list_list_nat] :
( ( append_list_nat @ Xs @ nil_list_nat )
= Xs ) ).
% append_Nil2
thf(fact_349_append__self__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_350_append__self__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_nat ) ) ).
% append_self_conv
thf(fact_351_append__self__conv,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( ( append_list_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_list_nat ) ) ).
% append_self_conv
thf(fact_352_self__append__conv,axiom,
! [Y2: list_a,Ys: list_a] :
( ( Y2
= ( append_a @ Y2 @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_353_self__append__conv,axiom,
! [Y2: list_nat,Ys: list_nat] :
( ( Y2
= ( append_nat @ Y2 @ Ys ) )
= ( Ys = nil_nat ) ) ).
% self_append_conv
thf(fact_354_self__append__conv,axiom,
! [Y2: list_list_nat,Ys: list_list_nat] :
( ( Y2
= ( append_list_nat @ Y2 @ Ys ) )
= ( Ys = nil_list_nat ) ) ).
% self_append_conv
thf(fact_355_append__self__conv2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_356_append__self__conv2,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_nat ) ) ).
% append_self_conv2
thf(fact_357_append__self__conv2,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( ( append_list_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_list_nat ) ) ).
% append_self_conv2
thf(fact_358_self__append__conv2,axiom,
! [Y2: list_a,Xs: list_a] :
( ( Y2
= ( append_a @ Xs @ Y2 ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_359_self__append__conv2,axiom,
! [Y2: list_nat,Xs: list_nat] :
( ( Y2
= ( append_nat @ Xs @ Y2 ) )
= ( Xs = nil_nat ) ) ).
% self_append_conv2
thf(fact_360_self__append__conv2,axiom,
! [Y2: list_list_nat,Xs: list_list_nat] :
( ( Y2
= ( append_list_nat @ Xs @ Y2 ) )
= ( Xs = nil_list_nat ) ) ).
% self_append_conv2
thf(fact_361_Nil__is__append__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( nil_a
= ( append_a @ Xs @ Ys ) )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% Nil_is_append_conv
thf(fact_362_Nil__is__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( nil_nat
= ( append_nat @ Xs @ Ys ) )
= ( ( Xs = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_363_Nil__is__append__conv,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( nil_list_nat
= ( append_list_nat @ Xs @ Ys ) )
= ( ( Xs = nil_list_nat )
& ( Ys = nil_list_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_364_append__is__Nil__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= nil_a )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% append_is_Nil_conv
thf(fact_365_append__is__Nil__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= nil_nat )
= ( ( Xs = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_366_append__is__Nil__conv,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( ( append_list_nat @ Xs @ Ys )
= nil_list_nat )
= ( ( Xs = nil_list_nat )
& ( Ys = nil_list_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_367_list_Omap__disc__iff,axiom,
! [F: nat > a,A2: list_nat] :
( ( ( map_nat_a @ F @ A2 )
= nil_a )
= ( A2 = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_368_list_Omap__disc__iff,axiom,
! [F: list_nat > a,A2: list_list_nat] :
( ( ( map_list_nat_a @ F @ A2 )
= nil_a )
= ( A2 = nil_list_nat ) ) ).
% list.map_disc_iff
thf(fact_369_list_Omap__disc__iff,axiom,
! [F: list_nat > nat,A2: list_list_nat] :
( ( ( map_list_nat_nat @ F @ A2 )
= nil_nat )
= ( A2 = nil_list_nat ) ) ).
% list.map_disc_iff
thf(fact_370_list_Omap__disc__iff,axiom,
! [F: a > list_nat,A2: list_a] :
( ( ( map_a_list_nat @ F @ A2 )
= nil_list_nat )
= ( A2 = nil_a ) ) ).
% list.map_disc_iff
thf(fact_371_list_Omap__disc__iff,axiom,
! [F: nat > list_nat,A2: list_nat] :
( ( ( map_nat_list_nat @ F @ A2 )
= nil_list_nat )
= ( A2 = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_372_list_Omap__disc__iff,axiom,
! [F: a > nat,A2: list_a] :
( ( ( map_a_nat @ F @ A2 )
= nil_nat )
= ( A2 = nil_a ) ) ).
% list.map_disc_iff
thf(fact_373_list_Omap__disc__iff,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,A2: list_list_nat] :
( ( ( map_li6003994582982014139at_nat @ F @ A2 )
= nil_se357566008730718055at_nat )
= ( A2 = nil_list_nat ) ) ).
% list.map_disc_iff
thf(fact_374_list_Omap__disc__iff,axiom,
! [F: list_nat > list_nat,A2: list_list_nat] :
( ( ( map_li7225945977422193158st_nat @ F @ A2 )
= nil_list_nat )
= ( A2 = nil_list_nat ) ) ).
% list.map_disc_iff
thf(fact_375_list_Omap__disc__iff,axiom,
! [F: nat > nat,A2: list_nat] :
( ( ( map_nat_nat @ F @ A2 )
= nil_nat )
= ( A2 = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_376_list_Omap__disc__iff,axiom,
! [F: a > a,A2: list_a] :
( ( ( map_a_a @ F @ A2 )
= nil_a )
= ( A2 = nil_a ) ) ).
% list.map_disc_iff
thf(fact_377_Nil__is__map__conv,axiom,
! [F: nat > a,Xs: list_nat] :
( ( nil_a
= ( map_nat_a @ F @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_378_Nil__is__map__conv,axiom,
! [F: list_nat > a,Xs: list_list_nat] :
( ( nil_a
= ( map_list_nat_a @ F @ Xs ) )
= ( Xs = nil_list_nat ) ) ).
% Nil_is_map_conv
thf(fact_379_Nil__is__map__conv,axiom,
! [F: list_nat > nat,Xs: list_list_nat] :
( ( nil_nat
= ( map_list_nat_nat @ F @ Xs ) )
= ( Xs = nil_list_nat ) ) ).
% Nil_is_map_conv
thf(fact_380_Nil__is__map__conv,axiom,
! [F: a > list_nat,Xs: list_a] :
( ( nil_list_nat
= ( map_a_list_nat @ F @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_map_conv
thf(fact_381_Nil__is__map__conv,axiom,
! [F: nat > list_nat,Xs: list_nat] :
( ( nil_list_nat
= ( map_nat_list_nat @ F @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_382_Nil__is__map__conv,axiom,
! [F: a > nat,Xs: list_a] :
( ( nil_nat
= ( map_a_nat @ F @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_map_conv
thf(fact_383_Nil__is__map__conv,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat] :
( ( nil_se357566008730718055at_nat
= ( map_li6003994582982014139at_nat @ F @ Xs ) )
= ( Xs = nil_list_nat ) ) ).
% Nil_is_map_conv
thf(fact_384_Nil__is__map__conv,axiom,
! [F: list_nat > list_nat,Xs: list_list_nat] :
( ( nil_list_nat
= ( map_li7225945977422193158st_nat @ F @ Xs ) )
= ( Xs = nil_list_nat ) ) ).
% Nil_is_map_conv
thf(fact_385_Nil__is__map__conv,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( nil_nat
= ( map_nat_nat @ F @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_386_Nil__is__map__conv,axiom,
! [F: a > a,Xs: list_a] :
( ( nil_a
= ( map_a_a @ F @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_map_conv
thf(fact_387_map__is__Nil__conv,axiom,
! [F: nat > a,Xs: list_nat] :
( ( ( map_nat_a @ F @ Xs )
= nil_a )
= ( Xs = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_388_map__is__Nil__conv,axiom,
! [F: list_nat > a,Xs: list_list_nat] :
( ( ( map_list_nat_a @ F @ Xs )
= nil_a )
= ( Xs = nil_list_nat ) ) ).
% map_is_Nil_conv
thf(fact_389_map__is__Nil__conv,axiom,
! [F: list_nat > nat,Xs: list_list_nat] :
( ( ( map_list_nat_nat @ F @ Xs )
= nil_nat )
= ( Xs = nil_list_nat ) ) ).
% map_is_Nil_conv
thf(fact_390_map__is__Nil__conv,axiom,
! [F: a > list_nat,Xs: list_a] :
( ( ( map_a_list_nat @ F @ Xs )
= nil_list_nat )
= ( Xs = nil_a ) ) ).
% map_is_Nil_conv
thf(fact_391_map__is__Nil__conv,axiom,
! [F: nat > list_nat,Xs: list_nat] :
( ( ( map_nat_list_nat @ F @ Xs )
= nil_list_nat )
= ( Xs = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_392_map__is__Nil__conv,axiom,
! [F: a > nat,Xs: list_a] :
( ( ( map_a_nat @ F @ Xs )
= nil_nat )
= ( Xs = nil_a ) ) ).
% map_is_Nil_conv
thf(fact_393_map__is__Nil__conv,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat] :
( ( ( map_li6003994582982014139at_nat @ F @ Xs )
= nil_se357566008730718055at_nat )
= ( Xs = nil_list_nat ) ) ).
% map_is_Nil_conv
thf(fact_394_map__is__Nil__conv,axiom,
! [F: list_nat > list_nat,Xs: list_list_nat] :
( ( ( map_li7225945977422193158st_nat @ F @ Xs )
= nil_list_nat )
= ( Xs = nil_list_nat ) ) ).
% map_is_Nil_conv
thf(fact_395_map__is__Nil__conv,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_396_map__is__Nil__conv,axiom,
! [F: a > a,Xs: list_a] :
( ( ( map_a_a @ F @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% map_is_Nil_conv
thf(fact_397_append__eq__append__conv,axiom,
! [Xs: list_a,Ys: list_a,Us: list_a,Vs: list_a] :
( ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
| ( ( size_size_list_a @ Us )
= ( size_size_list_a @ Vs ) ) )
=> ( ( ( append_a @ Xs @ Us )
= ( append_a @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_398_append__eq__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat,Us: list_nat,Vs: list_nat] :
( ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
| ( ( size_size_list_nat @ Us )
= ( size_size_list_nat @ Vs ) ) )
=> ( ( ( append_nat @ Xs @ Us )
= ( append_nat @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_399_map__append,axiom,
! [F: nat > a,Xs: list_nat,Ys: list_nat] :
( ( map_nat_a @ F @ ( append_nat @ Xs @ Ys ) )
= ( append_a @ ( map_nat_a @ F @ Xs ) @ ( map_nat_a @ F @ Ys ) ) ) ).
% map_append
thf(fact_400_map__append,axiom,
! [F: a > nat,Xs: list_a,Ys: list_a] :
( ( map_a_nat @ F @ ( append_a @ Xs @ Ys ) )
= ( append_nat @ ( map_a_nat @ F @ Xs ) @ ( map_a_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_401_map__append,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat,Ys: list_list_nat] :
( ( map_li6003994582982014139at_nat @ F @ ( append_list_nat @ Xs @ Ys ) )
= ( append4192317425040545660at_nat @ ( map_li6003994582982014139at_nat @ F @ Xs ) @ ( map_li6003994582982014139at_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_402_map__append,axiom,
! [F: list_nat > list_nat,Xs: list_list_nat,Ys: list_list_nat] :
( ( map_li7225945977422193158st_nat @ F @ ( append_list_nat @ Xs @ Ys ) )
= ( append_list_nat @ ( map_li7225945977422193158st_nat @ F @ Xs ) @ ( map_li7225945977422193158st_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_403_map__append,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
( ( map_nat_nat @ F @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( map_nat_nat @ F @ Xs ) @ ( map_nat_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_404_map__append,axiom,
! [F: a > a,Xs: list_a,Ys: list_a] :
( ( map_a_a @ F @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( map_a_a @ F @ Xs ) @ ( map_a_a @ F @ Ys ) ) ) ).
% map_append
thf(fact_405_distinct__adj__Cons__Cons,axiom,
! [X2: a,Y2: a,Xs: list_a] :
( ( distinct_adj_a @ ( cons_a @ X2 @ ( cons_a @ Y2 @ Xs ) ) )
= ( ( X2 != Y2 )
& ( distinct_adj_a @ ( cons_a @ Y2 @ Xs ) ) ) ) ).
% distinct_adj_Cons_Cons
thf(fact_406_distinct__adj__Cons__Cons,axiom,
! [X2: nat,Y2: nat,Xs: list_nat] :
( ( distinct_adj_nat @ ( cons_nat @ X2 @ ( cons_nat @ Y2 @ Xs ) ) )
= ( ( X2 != Y2 )
& ( distinct_adj_nat @ ( cons_nat @ Y2 @ Xs ) ) ) ) ).
% distinct_adj_Cons_Cons
thf(fact_407_distinct__adj__Cons__Cons,axiom,
! [X2: list_nat,Y2: list_nat,Xs: list_list_nat] :
( ( distin876741697294417026st_nat @ ( cons_list_nat @ X2 @ ( cons_list_nat @ Y2 @ Xs ) ) )
= ( ( X2 != Y2 )
& ( distin876741697294417026st_nat @ ( cons_list_nat @ Y2 @ Xs ) ) ) ) ).
% distinct_adj_Cons_Cons
thf(fact_408_removeAll__id,axiom,
! [X2: a,Xs: list_a] :
( ~ ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( ( removeAll_a @ X2 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_409_removeAll__id,axiom,
! [X2: nat,Xs: list_nat] :
( ~ ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( removeAll_nat @ X2 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_410_removeAll__id,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ~ ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ( removeAll_list_nat @ X2 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_411_removeAll__append,axiom,
! [X2: a,Xs: list_a,Ys: list_a] :
( ( removeAll_a @ X2 @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( removeAll_a @ X2 @ Xs ) @ ( removeAll_a @ X2 @ Ys ) ) ) ).
% removeAll_append
thf(fact_412_removeAll__append,axiom,
! [X2: nat,Xs: list_nat,Ys: list_nat] :
( ( removeAll_nat @ X2 @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( removeAll_nat @ X2 @ Xs ) @ ( removeAll_nat @ X2 @ Ys ) ) ) ).
% removeAll_append
thf(fact_413_in__set__insert,axiom,
! [X2: a,Xs: list_a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( ( insert_a @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_414_in__set__insert,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_415_in__set__insert,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ( insert_list_nat @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_416_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_417_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_418_list_Osimps_I15_J,axiom,
! [X21: a,X222: list_a] :
( ( set_a2 @ ( cons_a @ X21 @ X222 ) )
= ( insert_a2 @ X21 @ ( set_a2 @ X222 ) ) ) ).
% list.simps(15)
thf(fact_419_list_Osimps_I15_J,axiom,
! [X21: nat,X222: list_nat] :
( ( set_nat2 @ ( cons_nat @ X21 @ X222 ) )
= ( insert_nat2 @ X21 @ ( set_nat2 @ X222 ) ) ) ).
% list.simps(15)
thf(fact_420_list_Osimps_I15_J,axiom,
! [X21: list_nat,X222: list_list_nat] :
( ( set_list_nat2 @ ( cons_list_nat @ X21 @ X222 ) )
= ( insert_list_nat2 @ X21 @ ( set_list_nat2 @ X222 ) ) ) ).
% list.simps(15)
thf(fact_421_length__0__conv,axiom,
! [Xs: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_list_nat ) ) ).
% length_0_conv
thf(fact_422_length__0__conv,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_a ) ) ).
% length_0_conv
thf(fact_423_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_424_append1__eq__conv,axiom,
! [Xs: list_a,X2: a,Ys: list_a,Y2: a] :
( ( ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) )
= ( append_a @ Ys @ ( cons_a @ Y2 @ nil_a ) ) )
= ( ( Xs = Ys )
& ( X2 = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_425_append1__eq__conv,axiom,
! [Xs: list_nat,X2: nat,Ys: list_nat,Y2: nat] :
( ( ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) )
= ( append_nat @ Ys @ ( cons_nat @ Y2 @ nil_nat ) ) )
= ( ( Xs = Ys )
& ( X2 = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_426_append1__eq__conv,axiom,
! [Xs: list_list_nat,X2: list_nat,Ys: list_list_nat,Y2: list_nat] :
( ( ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) )
= ( append_list_nat @ Ys @ ( cons_list_nat @ Y2 @ nil_list_nat ) ) )
= ( ( Xs = Ys )
& ( X2 = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_427_nth__Cons__Suc,axiom,
! [X2: a,Xs: list_a,N: nat] :
( ( nth_a @ ( cons_a @ X2 @ Xs ) @ ( suc @ N ) )
= ( nth_a @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_428_nth__Cons__Suc,axiom,
! [X2: nat,Xs: list_nat,N: nat] :
( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ ( suc @ N ) )
= ( nth_nat @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_429_nth__Cons__Suc,axiom,
! [X2: list_nat,Xs: list_list_nat,N: nat] :
( ( nth_list_nat @ ( cons_list_nat @ X2 @ Xs ) @ ( suc @ N ) )
= ( nth_list_nat @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_430_nth__Cons__0,axiom,
! [X2: a,Xs: list_a] :
( ( nth_a @ ( cons_a @ X2 @ Xs ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_431_nth__Cons__0,axiom,
! [X2: nat,Xs: list_nat] :
( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_432_nth__Cons__0,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( nth_list_nat @ ( cons_list_nat @ X2 @ Xs ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_433_count__notin,axiom,
! [X2: a,Xs: list_a] :
( ~ ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( ( count_list_a @ Xs @ X2 )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_434_count__notin,axiom,
! [X2: nat,Xs: list_nat] :
( ~ ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( count_list_nat @ Xs @ X2 )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_435_count__notin,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ~ ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ( count_list_list_nat @ Xs @ X2 )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_436__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062x1_Ax2_O_A_092_060lbrakk_062x_A_061_Ax1_A_064_A_091x2_093_059_Alength_Ax1_A_061_An_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [X1: list_a] :
( ? [X23: a] :
( xa
= ( append_a @ X1 @ ( cons_a @ X23 @ nil_a ) ) )
=> ( ( size_size_list_a @ X1 )
!= n ) ) ).
% \<open>\<And>thesis. (\<And>x1 x2. \<lbrakk>x = x1 @ [x2]; length x1 = n\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_437_insert__Nil,axiom,
! [X2: a] :
( ( insert_a @ X2 @ nil_a )
= ( cons_a @ X2 @ nil_a ) ) ).
% insert_Nil
thf(fact_438_insert__Nil,axiom,
! [X2: nat] :
( ( insert_nat @ X2 @ nil_nat )
= ( cons_nat @ X2 @ nil_nat ) ) ).
% insert_Nil
thf(fact_439_insert__Nil,axiom,
! [X2: list_nat] :
( ( insert_list_nat @ X2 @ nil_list_nat )
= ( cons_list_nat @ X2 @ nil_list_nat ) ) ).
% insert_Nil
thf(fact_440_not__in__set__insert,axiom,
! [X2: a,Xs: list_a] :
( ~ ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( ( insert_a @ X2 @ Xs )
= ( cons_a @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_441_not__in__set__insert,axiom,
! [X2: nat,Xs: list_nat] :
( ~ ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X2 @ Xs )
= ( cons_nat @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_442_not__in__set__insert,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ~ ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ( insert_list_nat @ X2 @ Xs )
= ( cons_list_nat @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_443_length__greater__0__conv,axiom,
! [Xs: list_list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s3023201423986296836st_nat @ Xs ) )
= ( Xs != nil_list_nat ) ) ).
% length_greater_0_conv
thf(fact_444_length__greater__0__conv,axiom,
! [Xs: list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) )
= ( Xs != nil_a ) ) ).
% length_greater_0_conv
thf(fact_445_length__greater__0__conv,axiom,
! [Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) )
= ( Xs != nil_nat ) ) ).
% length_greater_0_conv
thf(fact_446_nth__append__length,axiom,
! [Xs: list_list_nat,X2: list_nat,Ys: list_list_nat] :
( ( nth_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ Ys ) ) @ ( size_s3023201423986296836st_nat @ Xs ) )
= X2 ) ).
% nth_append_length
thf(fact_447_nth__append__length,axiom,
! [Xs: list_a,X2: a,Ys: list_a] :
( ( nth_a @ ( append_a @ Xs @ ( cons_a @ X2 @ Ys ) ) @ ( size_size_list_a @ Xs ) )
= X2 ) ).
% nth_append_length
thf(fact_448_nth__append__length,axiom,
! [Xs: list_nat,X2: nat,Ys: list_nat] :
( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ Ys ) ) @ ( size_size_list_nat @ Xs ) )
= X2 ) ).
% nth_append_length
thf(fact_449_same__length__different,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( Xs != Ys )
=> ( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ? [Pre: list_list_nat,X3: list_nat,Xs4: list_list_nat,Y3: list_nat,Ys3: list_list_nat] :
( ( X3 != Y3 )
& ( Xs
= ( append_list_nat @ Pre @ ( append_list_nat @ ( cons_list_nat @ X3 @ nil_list_nat ) @ Xs4 ) ) )
& ( Ys
= ( append_list_nat @ Pre @ ( append_list_nat @ ( cons_list_nat @ Y3 @ nil_list_nat ) @ Ys3 ) ) ) ) ) ) ).
% same_length_different
thf(fact_450_same__length__different,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != Ys )
=> ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ? [Pre: list_a,X3: a,Xs4: list_a,Y3: a,Ys3: list_a] :
( ( X3 != Y3 )
& ( Xs
= ( append_a @ Pre @ ( append_a @ ( cons_a @ X3 @ nil_a ) @ Xs4 ) ) )
& ( Ys
= ( append_a @ Pre @ ( append_a @ ( cons_a @ Y3 @ nil_a ) @ Ys3 ) ) ) ) ) ) ).
% same_length_different
thf(fact_451_same__length__different,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs != Ys )
=> ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ? [Pre: list_nat,X3: nat,Xs4: list_nat,Y3: nat,Ys3: list_nat] :
( ( X3 != Y3 )
& ( Xs
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X3 @ nil_nat ) @ Xs4 ) ) )
& ( Ys
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y3 @ nil_nat ) @ Ys3 ) ) ) ) ) ) ).
% same_length_different
thf(fact_452_list__induct__2__rev,axiom,
! [X2: list_list_nat,Y2: list_list_nat,P: list_list_nat > list_list_nat > $o] :
( ( ( size_s3023201423986296836st_nat @ X2 )
= ( size_s3023201423986296836st_nat @ Y2 ) )
=> ( ( P @ nil_list_nat @ nil_list_nat )
=> ( ! [X3: list_nat,Xs3: list_list_nat,Y3: list_nat,Ys4: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs3 )
= ( size_s3023201423986296836st_nat @ Ys4 ) )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( append_list_nat @ Xs3 @ ( cons_list_nat @ X3 @ nil_list_nat ) ) @ ( append_list_nat @ Ys4 @ ( cons_list_nat @ Y3 @ nil_list_nat ) ) ) ) )
=> ( P @ X2 @ Y2 ) ) ) ) ).
% list_induct_2_rev
thf(fact_453_list__induct__2__rev,axiom,
! [X2: list_list_nat,Y2: list_a,P: list_list_nat > list_a > $o] :
( ( ( size_s3023201423986296836st_nat @ X2 )
= ( size_size_list_a @ Y2 ) )
=> ( ( P @ nil_list_nat @ nil_a )
=> ( ! [X3: list_nat,Xs3: list_list_nat,Y3: a,Ys4: list_a] :
( ( ( size_s3023201423986296836st_nat @ Xs3 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( append_list_nat @ Xs3 @ ( cons_list_nat @ X3 @ nil_list_nat ) ) @ ( append_a @ Ys4 @ ( cons_a @ Y3 @ nil_a ) ) ) ) )
=> ( P @ X2 @ Y2 ) ) ) ) ).
% list_induct_2_rev
thf(fact_454_list__induct__2__rev,axiom,
! [X2: list_list_nat,Y2: list_nat,P: list_list_nat > list_nat > $o] :
( ( ( size_s3023201423986296836st_nat @ X2 )
= ( size_size_list_nat @ Y2 ) )
=> ( ( P @ nil_list_nat @ nil_nat )
=> ( ! [X3: list_nat,Xs3: list_list_nat,Y3: nat,Ys4: list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( append_list_nat @ Xs3 @ ( cons_list_nat @ X3 @ nil_list_nat ) ) @ ( append_nat @ Ys4 @ ( cons_nat @ Y3 @ nil_nat ) ) ) ) )
=> ( P @ X2 @ Y2 ) ) ) ) ).
% list_induct_2_rev
thf(fact_455_list__induct__2__rev,axiom,
! [X2: list_a,Y2: list_list_nat,P: list_a > list_list_nat > $o] :
( ( ( size_size_list_a @ X2 )
= ( size_s3023201423986296836st_nat @ Y2 ) )
=> ( ( P @ nil_a @ nil_list_nat )
=> ( ! [X3: a,Xs3: list_a,Y3: list_nat,Ys4: list_list_nat] :
( ( ( size_size_list_a @ Xs3 )
= ( size_s3023201423986296836st_nat @ Ys4 ) )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( append_a @ Xs3 @ ( cons_a @ X3 @ nil_a ) ) @ ( append_list_nat @ Ys4 @ ( cons_list_nat @ Y3 @ nil_list_nat ) ) ) ) )
=> ( P @ X2 @ Y2 ) ) ) ) ).
% list_induct_2_rev
thf(fact_456_list__induct__2__rev,axiom,
! [X2: list_a,Y2: list_a,P: list_a > list_a > $o] :
( ( ( size_size_list_a @ X2 )
= ( size_size_list_a @ Y2 ) )
=> ( ( P @ nil_a @ nil_a )
=> ( ! [X3: a,Xs3: list_a,Y3: a,Ys4: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( append_a @ Xs3 @ ( cons_a @ X3 @ nil_a ) ) @ ( append_a @ Ys4 @ ( cons_a @ Y3 @ nil_a ) ) ) ) )
=> ( P @ X2 @ Y2 ) ) ) ) ).
% list_induct_2_rev
thf(fact_457_list__induct__2__rev,axiom,
! [X2: list_a,Y2: list_nat,P: list_a > list_nat > $o] :
( ( ( size_size_list_a @ X2 )
= ( size_size_list_nat @ Y2 ) )
=> ( ( P @ nil_a @ nil_nat )
=> ( ! [X3: a,Xs3: list_a,Y3: nat,Ys4: list_nat] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( append_a @ Xs3 @ ( cons_a @ X3 @ nil_a ) ) @ ( append_nat @ Ys4 @ ( cons_nat @ Y3 @ nil_nat ) ) ) ) )
=> ( P @ X2 @ Y2 ) ) ) ) ).
% list_induct_2_rev
thf(fact_458_list__induct__2__rev,axiom,
! [X2: list_nat,Y2: list_list_nat,P: list_nat > list_list_nat > $o] :
( ( ( size_size_list_nat @ X2 )
= ( size_s3023201423986296836st_nat @ Y2 ) )
=> ( ( P @ nil_nat @ nil_list_nat )
=> ( ! [X3: nat,Xs3: list_nat,Y3: list_nat,Ys4: list_list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_s3023201423986296836st_nat @ Ys4 ) )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( append_nat @ Xs3 @ ( cons_nat @ X3 @ nil_nat ) ) @ ( append_list_nat @ Ys4 @ ( cons_list_nat @ Y3 @ nil_list_nat ) ) ) ) )
=> ( P @ X2 @ Y2 ) ) ) ) ).
% list_induct_2_rev
thf(fact_459_list__induct__2__rev,axiom,
! [X2: list_nat,Y2: list_a,P: list_nat > list_a > $o] :
( ( ( size_size_list_nat @ X2 )
= ( size_size_list_a @ Y2 ) )
=> ( ( P @ nil_nat @ nil_a )
=> ( ! [X3: nat,Xs3: list_nat,Y3: a,Ys4: list_a] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( append_nat @ Xs3 @ ( cons_nat @ X3 @ nil_nat ) ) @ ( append_a @ Ys4 @ ( cons_a @ Y3 @ nil_a ) ) ) ) )
=> ( P @ X2 @ Y2 ) ) ) ) ).
% list_induct_2_rev
thf(fact_460_list__induct__2__rev,axiom,
! [X2: list_nat,Y2: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ X2 )
= ( size_size_list_nat @ Y2 ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs3: list_nat,Y3: nat,Ys4: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( append_nat @ Xs3 @ ( cons_nat @ X3 @ nil_nat ) ) @ ( append_nat @ Ys4 @ ( cons_nat @ Y3 @ nil_nat ) ) ) ) )
=> ( P @ X2 @ Y2 ) ) ) ) ).
% list_induct_2_rev
thf(fact_461_member__rec_I2_J,axiom,
! [Y2: a] :
~ ( member_a @ nil_a @ Y2 ) ).
% member_rec(2)
thf(fact_462_member__rec_I2_J,axiom,
! [Y2: nat] :
~ ( member_nat @ nil_nat @ Y2 ) ).
% member_rec(2)
thf(fact_463_member__rec_I2_J,axiom,
! [Y2: list_nat] :
~ ( member_list_nat @ nil_list_nat @ Y2 ) ).
% member_rec(2)
thf(fact_464_member__rec_I1_J,axiom,
! [X2: a,Xs: list_a,Y2: a] :
( ( member_a @ ( cons_a @ X2 @ Xs ) @ Y2 )
= ( ( X2 = Y2 )
| ( member_a @ Xs @ Y2 ) ) ) ).
% member_rec(1)
thf(fact_465_member__rec_I1_J,axiom,
! [X2: nat,Xs: list_nat,Y2: nat] :
( ( member_nat @ ( cons_nat @ X2 @ Xs ) @ Y2 )
= ( ( X2 = Y2 )
| ( member_nat @ Xs @ Y2 ) ) ) ).
% member_rec(1)
thf(fact_466_member__rec_I1_J,axiom,
! [X2: list_nat,Xs: list_list_nat,Y2: list_nat] :
( ( member_list_nat @ ( cons_list_nat @ X2 @ Xs ) @ Y2 )
= ( ( X2 = Y2 )
| ( member_list_nat @ Xs @ Y2 ) ) ) ).
% member_rec(1)
thf(fact_467_list_Odistinct_I1_J,axiom,
! [X21: a,X222: list_a] :
( nil_a
!= ( cons_a @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_468_list_Odistinct_I1_J,axiom,
! [X21: nat,X222: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_469_list_Odistinct_I1_J,axiom,
! [X21: list_nat,X222: list_list_nat] :
( nil_list_nat
!= ( cons_list_nat @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_470_removeAll_Osimps_I2_J,axiom,
! [X2: a,Y2: a,Xs: list_a] :
( ( ( X2 = Y2 )
=> ( ( removeAll_a @ X2 @ ( cons_a @ Y2 @ Xs ) )
= ( removeAll_a @ X2 @ Xs ) ) )
& ( ( X2 != Y2 )
=> ( ( removeAll_a @ X2 @ ( cons_a @ Y2 @ Xs ) )
= ( cons_a @ Y2 @ ( removeAll_a @ X2 @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_471_removeAll_Osimps_I2_J,axiom,
! [X2: nat,Y2: nat,Xs: list_nat] :
( ( ( X2 = Y2 )
=> ( ( removeAll_nat @ X2 @ ( cons_nat @ Y2 @ Xs ) )
= ( removeAll_nat @ X2 @ Xs ) ) )
& ( ( X2 != Y2 )
=> ( ( removeAll_nat @ X2 @ ( cons_nat @ Y2 @ Xs ) )
= ( cons_nat @ Y2 @ ( removeAll_nat @ X2 @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_472_removeAll_Osimps_I2_J,axiom,
! [X2: list_nat,Y2: list_nat,Xs: list_list_nat] :
( ( ( X2 = Y2 )
=> ( ( removeAll_list_nat @ X2 @ ( cons_list_nat @ Y2 @ Xs ) )
= ( removeAll_list_nat @ X2 @ Xs ) ) )
& ( ( X2 != Y2 )
=> ( ( removeAll_list_nat @ X2 @ ( cons_list_nat @ Y2 @ Xs ) )
= ( cons_list_nat @ Y2 @ ( removeAll_list_nat @ X2 @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_473_removeAll_Osimps_I1_J,axiom,
! [X2: a] :
( ( removeAll_a @ X2 @ nil_a )
= nil_a ) ).
% removeAll.simps(1)
thf(fact_474_removeAll_Osimps_I1_J,axiom,
! [X2: nat] :
( ( removeAll_nat @ X2 @ nil_nat )
= nil_nat ) ).
% removeAll.simps(1)
thf(fact_475_removeAll_Osimps_I1_J,axiom,
! [X2: list_nat] :
( ( removeAll_list_nat @ X2 @ nil_list_nat )
= nil_list_nat ) ).
% removeAll.simps(1)
thf(fact_476_list_OdiscI,axiom,
! [List: list_a,X21: a,X222: list_a] :
( ( List
= ( cons_a @ X21 @ X222 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_477_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X222: list_nat] :
( ( List
= ( cons_nat @ X21 @ X222 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_478_list_OdiscI,axiom,
! [List: list_list_nat,X21: list_nat,X222: list_list_nat] :
( ( List
= ( cons_list_nat @ X21 @ X222 ) )
=> ( List != nil_list_nat ) ) ).
% list.discI
thf(fact_479_list_Oexhaust,axiom,
! [Y2: list_a] :
( ( Y2 != nil_a )
=> ~ ! [X212: a,X223: list_a] :
( Y2
!= ( cons_a @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_480_list_Oexhaust,axiom,
! [Y2: list_nat] :
( ( Y2 != nil_nat )
=> ~ ! [X212: nat,X223: list_nat] :
( Y2
!= ( cons_nat @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_481_list_Oexhaust,axiom,
! [Y2: list_list_nat] :
( ( Y2 != nil_list_nat )
=> ~ ! [X212: list_nat,X223: list_list_nat] :
( Y2
!= ( cons_list_nat @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_482_min__list_Ocases,axiom,
! [X2: list_nat] :
( ! [X3: nat,Xs3: list_nat] :
( X2
!= ( cons_nat @ X3 @ Xs3 ) )
=> ( X2 = nil_nat ) ) ).
% min_list.cases
thf(fact_483_transpose_Ocases,axiom,
! [X2: list_list_a] :
( ( X2 != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X2
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X3: a,Xs3: list_a,Xss: list_list_a] :
( X2
!= ( cons_list_a @ ( cons_a @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_484_transpose_Ocases,axiom,
! [X2: list_list_list_nat] :
( ( X2 != nil_list_list_nat )
=> ( ! [Xss: list_list_list_nat] :
( X2
!= ( cons_list_list_nat @ nil_list_nat @ Xss ) )
=> ~ ! [X3: list_nat,Xs3: list_list_nat,Xss: list_list_list_nat] :
( X2
!= ( cons_list_list_nat @ ( cons_list_nat @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_485_transpose_Ocases,axiom,
! [X2: list_list_nat] :
( ( X2 != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X2
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X3: nat,Xs3: list_nat,Xss: list_list_nat] :
( X2
!= ( cons_list_nat @ ( cons_nat @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_486_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_487_append__Nil,axiom,
! [Ys: list_nat] :
( ( append_nat @ nil_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_488_append__Nil,axiom,
! [Ys: list_list_nat] :
( ( append_list_nat @ nil_list_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_489_remdups__adj_Ocases,axiom,
! [X2: list_a] :
( ( X2 != nil_a )
=> ( ! [X3: a] :
( X2
!= ( cons_a @ X3 @ nil_a ) )
=> ~ ! [X3: a,Y3: a,Xs3: list_a] :
( X2
!= ( cons_a @ X3 @ ( cons_a @ Y3 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_490_remdups__adj_Ocases,axiom,
! [X2: list_nat] :
( ( X2 != nil_nat )
=> ( ! [X3: nat] :
( X2
!= ( cons_nat @ X3 @ nil_nat ) )
=> ~ ! [X3: nat,Y3: nat,Xs3: list_nat] :
( X2
!= ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_491_remdups__adj_Ocases,axiom,
! [X2: list_list_nat] :
( ( X2 != nil_list_nat )
=> ( ! [X3: list_nat] :
( X2
!= ( cons_list_nat @ X3 @ nil_list_nat ) )
=> ~ ! [X3: list_nat,Y3: list_nat,Xs3: list_list_nat] :
( X2
!= ( cons_list_nat @ X3 @ ( cons_list_nat @ Y3 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_492_append__Cons,axiom,
! [X2: a,Xs: list_a,Ys: list_a] :
( ( append_a @ ( cons_a @ X2 @ Xs ) @ Ys )
= ( cons_a @ X2 @ ( append_a @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_493_append__Cons,axiom,
! [X2: nat,Xs: list_nat,Ys: list_nat] :
( ( append_nat @ ( cons_nat @ X2 @ Xs ) @ Ys )
= ( cons_nat @ X2 @ ( append_nat @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_494_append__Cons,axiom,
! [X2: list_nat,Xs: list_list_nat,Ys: list_list_nat] :
( ( append_list_nat @ ( cons_list_nat @ X2 @ Xs ) @ Ys )
= ( cons_list_nat @ X2 @ ( append_list_nat @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_495_append_Oleft__neutral,axiom,
! [A2: list_a] :
( ( append_a @ nil_a @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_496_append_Oleft__neutral,axiom,
! [A2: list_nat] :
( ( append_nat @ nil_nat @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_497_append_Oleft__neutral,axiom,
! [A2: list_list_nat] :
( ( append_list_nat @ nil_list_nat @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_498_rev__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ( P @ nil_a )
=> ( ! [X3: a,Xs3: list_a] :
( ( P @ Xs3 )
=> ( P @ ( append_a @ Xs3 @ ( cons_a @ X3 @ nil_a ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_499_rev__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ( P @ nil_nat )
=> ( ! [X3: nat,Xs3: list_nat] :
( ( P @ Xs3 )
=> ( P @ ( append_nat @ Xs3 @ ( cons_nat @ X3 @ nil_nat ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_500_rev__induct,axiom,
! [P: list_list_nat > $o,Xs: list_list_nat] :
( ( P @ nil_list_nat )
=> ( ! [X3: list_nat,Xs3: list_list_nat] :
( ( P @ Xs3 )
=> ( P @ ( append_list_nat @ Xs3 @ ( cons_list_nat @ X3 @ nil_list_nat ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_501_rev__exhaust,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ~ ! [Ys4: list_a,Y3: a] :
( Xs
!= ( append_a @ Ys4 @ ( cons_a @ Y3 @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_502_rev__exhaust,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ~ ! [Ys4: list_nat,Y3: nat] :
( Xs
!= ( append_nat @ Ys4 @ ( cons_nat @ Y3 @ nil_nat ) ) ) ) ).
% rev_exhaust
thf(fact_503_rev__exhaust,axiom,
! [Xs: list_list_nat] :
( ( Xs != nil_list_nat )
=> ~ ! [Ys4: list_list_nat,Y3: list_nat] :
( Xs
!= ( append_list_nat @ Ys4 @ ( cons_list_nat @ Y3 @ nil_list_nat ) ) ) ) ).
% rev_exhaust
thf(fact_504_inj__on__Cons1,axiom,
! [X2: a,A: set_list_a] : ( inj_on_list_a_list_a @ ( cons_a @ X2 ) @ A ) ).
% inj_on_Cons1
thf(fact_505_inj__on__Cons1,axiom,
! [X2: nat,A: set_list_nat] : ( inj_on3049792774292151987st_nat @ ( cons_nat @ X2 ) @ A ) ).
% inj_on_Cons1
thf(fact_506_inj__on__Cons1,axiom,
! [X2: list_nat,A: set_list_list_nat] : ( inj_on2300671324199612755st_nat @ ( cons_list_nat @ X2 ) @ A ) ).
% inj_on_Cons1
thf(fact_507_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y: a,Ys5: list_a] :
( Xs
= ( cons_a @ Y @ Ys5 ) ) ) ) ).
% neq_Nil_conv
thf(fact_508_neq__Nil__conv,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
= ( ? [Y: nat,Ys5: list_nat] :
( Xs
= ( cons_nat @ Y @ Ys5 ) ) ) ) ).
% neq_Nil_conv
thf(fact_509_neq__Nil__conv,axiom,
! [Xs: list_list_nat] :
( ( Xs != nil_list_nat )
= ( ? [Y: list_nat,Ys5: list_list_nat] :
( Xs
= ( cons_list_nat @ Y @ Ys5 ) ) ) ) ).
% neq_Nil_conv
thf(fact_510_list__induct2_H,axiom,
! [P: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
( ( P @ nil_a @ nil_a )
=> ( ! [X3: a,Xs3: list_a] : ( P @ ( cons_a @ X3 @ Xs3 ) @ nil_a )
=> ( ! [Y3: a,Ys4: list_a] : ( P @ nil_a @ ( cons_a @ Y3 @ Ys4 ) )
=> ( ! [X3: a,Xs3: list_a,Y3: a,Ys4: list_a] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_a @ X3 @ Xs3 ) @ ( cons_a @ Y3 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_511_list__induct2_H,axiom,
! [P: list_a > list_nat > $o,Xs: list_a,Ys: list_nat] :
( ( P @ nil_a @ nil_nat )
=> ( ! [X3: a,Xs3: list_a] : ( P @ ( cons_a @ X3 @ Xs3 ) @ nil_nat )
=> ( ! [Y3: nat,Ys4: list_nat] : ( P @ nil_a @ ( cons_nat @ Y3 @ Ys4 ) )
=> ( ! [X3: a,Xs3: list_a,Y3: nat,Ys4: list_nat] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_a @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_512_list__induct2_H,axiom,
! [P: list_a > list_list_nat > $o,Xs: list_a,Ys: list_list_nat] :
( ( P @ nil_a @ nil_list_nat )
=> ( ! [X3: a,Xs3: list_a] : ( P @ ( cons_a @ X3 @ Xs3 ) @ nil_list_nat )
=> ( ! [Y3: list_nat,Ys4: list_list_nat] : ( P @ nil_a @ ( cons_list_nat @ Y3 @ Ys4 ) )
=> ( ! [X3: a,Xs3: list_a,Y3: list_nat,Ys4: list_list_nat] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_a @ X3 @ Xs3 ) @ ( cons_list_nat @ Y3 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_513_list__induct2_H,axiom,
! [P: list_nat > list_a > $o,Xs: list_nat,Ys: list_a] :
( ( P @ nil_nat @ nil_a )
=> ( ! [X3: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X3 @ Xs3 ) @ nil_a )
=> ( ! [Y3: a,Ys4: list_a] : ( P @ nil_nat @ ( cons_a @ Y3 @ Ys4 ) )
=> ( ! [X3: nat,Xs3: list_nat,Y3: a,Ys4: list_a] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_a @ Y3 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_514_list__induct2_H,axiom,
! [P: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X3 @ Xs3 ) @ nil_nat )
=> ( ! [Y3: nat,Ys4: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y3 @ Ys4 ) )
=> ( ! [X3: nat,Xs3: list_nat,Y3: nat,Ys4: list_nat] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_515_list__induct2_H,axiom,
! [P: list_nat > list_list_nat > $o,Xs: list_nat,Ys: list_list_nat] :
( ( P @ nil_nat @ nil_list_nat )
=> ( ! [X3: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X3 @ Xs3 ) @ nil_list_nat )
=> ( ! [Y3: list_nat,Ys4: list_list_nat] : ( P @ nil_nat @ ( cons_list_nat @ Y3 @ Ys4 ) )
=> ( ! [X3: nat,Xs3: list_nat,Y3: list_nat,Ys4: list_list_nat] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_list_nat @ Y3 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_516_list__induct2_H,axiom,
! [P: list_list_nat > list_a > $o,Xs: list_list_nat,Ys: list_a] :
( ( P @ nil_list_nat @ nil_a )
=> ( ! [X3: list_nat,Xs3: list_list_nat] : ( P @ ( cons_list_nat @ X3 @ Xs3 ) @ nil_a )
=> ( ! [Y3: a,Ys4: list_a] : ( P @ nil_list_nat @ ( cons_a @ Y3 @ Ys4 ) )
=> ( ! [X3: list_nat,Xs3: list_list_nat,Y3: a,Ys4: list_a] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_list_nat @ X3 @ Xs3 ) @ ( cons_a @ Y3 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_517_list__induct2_H,axiom,
! [P: list_list_nat > list_nat > $o,Xs: list_list_nat,Ys: list_nat] :
( ( P @ nil_list_nat @ nil_nat )
=> ( ! [X3: list_nat,Xs3: list_list_nat] : ( P @ ( cons_list_nat @ X3 @ Xs3 ) @ nil_nat )
=> ( ! [Y3: nat,Ys4: list_nat] : ( P @ nil_list_nat @ ( cons_nat @ Y3 @ Ys4 ) )
=> ( ! [X3: list_nat,Xs3: list_list_nat,Y3: nat,Ys4: list_nat] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_list_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_518_list__induct2_H,axiom,
! [P: list_list_nat > list_list_nat > $o,Xs: list_list_nat,Ys: list_list_nat] :
( ( P @ nil_list_nat @ nil_list_nat )
=> ( ! [X3: list_nat,Xs3: list_list_nat] : ( P @ ( cons_list_nat @ X3 @ Xs3 ) @ nil_list_nat )
=> ( ! [Y3: list_nat,Ys4: list_list_nat] : ( P @ nil_list_nat @ ( cons_list_nat @ Y3 @ Ys4 ) )
=> ( ! [X3: list_nat,Xs3: list_list_nat,Y3: list_nat,Ys4: list_list_nat] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_list_nat @ X3 @ Xs3 ) @ ( cons_list_nat @ Y3 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_519_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_520_eq__Nil__appendI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append_nat @ nil_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_521_eq__Nil__appendI,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append_list_nat @ nil_list_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_522_not__Cons__self2,axiom,
! [X2: a,Xs: list_a] :
( ( cons_a @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_523_not__Cons__self2,axiom,
! [X2: nat,Xs: list_nat] :
( ( cons_nat @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_524_not__Cons__self2,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( cons_list_nat @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_525_Cons__eq__appendI,axiom,
! [X2: a,Xs1: list_a,Ys: list_a,Xs: list_a,Zs: list_a] :
( ( ( cons_a @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_a @ Xs1 @ Zs ) )
=> ( ( cons_a @ X2 @ Xs )
= ( append_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_526_Cons__eq__appendI,axiom,
! [X2: nat,Xs1: list_nat,Ys: list_nat,Xs: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_nat @ Xs1 @ Zs ) )
=> ( ( cons_nat @ X2 @ Xs )
= ( append_nat @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_527_Cons__eq__appendI,axiom,
! [X2: list_nat,Xs1: list_list_nat,Ys: list_list_nat,Xs: list_list_nat,Zs: list_list_nat] :
( ( ( cons_list_nat @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_list_nat @ Xs1 @ Zs ) )
=> ( ( cons_list_nat @ X2 @ Xs )
= ( append_list_nat @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_528_append__eq__appendI,axiom,
! [Xs: list_a,Xs1: list_a,Zs: list_a,Ys: list_a,Us: list_a] :
( ( ( append_a @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_a @ Xs1 @ Us ) )
=> ( ( append_a @ Xs @ Ys )
= ( append_a @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_529_append__eq__appendI,axiom,
! [Xs: list_nat,Xs1: list_nat,Zs: list_nat,Ys: list_nat,Us: list_nat] :
( ( ( append_nat @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_nat @ Xs1 @ Us ) )
=> ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_530_Cons__eq__append__conv,axiom,
! [X2: a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( cons_a @ X2 @ Xs )
= ( append_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_a )
& ( ( cons_a @ X2 @ Xs )
= Zs ) )
| ? [Ys6: list_a] :
( ( ( cons_a @ X2 @ Ys6 )
= Ys )
& ( Xs
= ( append_a @ Ys6 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_531_Cons__eq__append__conv,axiom,
! [X2: nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( append_nat @ Ys @ Zs ) )
= ( ( ( Ys = nil_nat )
& ( ( cons_nat @ X2 @ Xs )
= Zs ) )
| ? [Ys6: list_nat] :
( ( ( cons_nat @ X2 @ Ys6 )
= Ys )
& ( Xs
= ( append_nat @ Ys6 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_532_Cons__eq__append__conv,axiom,
! [X2: list_nat,Xs: list_list_nat,Ys: list_list_nat,Zs: list_list_nat] :
( ( ( cons_list_nat @ X2 @ Xs )
= ( append_list_nat @ Ys @ Zs ) )
= ( ( ( Ys = nil_list_nat )
& ( ( cons_list_nat @ X2 @ Xs )
= Zs ) )
| ? [Ys6: list_list_nat] :
( ( ( cons_list_nat @ X2 @ Ys6 )
= Ys )
& ( Xs
= ( append_list_nat @ Ys6 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_533_append__eq__Cons__conv,axiom,
! [Ys: list_a,Zs: list_a,X2: a,Xs: list_a] :
( ( ( append_a @ Ys @ Zs )
= ( cons_a @ X2 @ Xs ) )
= ( ( ( Ys = nil_a )
& ( Zs
= ( cons_a @ X2 @ Xs ) ) )
| ? [Ys6: list_a] :
( ( Ys
= ( cons_a @ X2 @ Ys6 ) )
& ( ( append_a @ Ys6 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_534_append__eq__Cons__conv,axiom,
! [Ys: list_nat,Zs: list_nat,X2: nat,Xs: list_nat] :
( ( ( append_nat @ Ys @ Zs )
= ( cons_nat @ X2 @ Xs ) )
= ( ( ( Ys = nil_nat )
& ( Zs
= ( cons_nat @ X2 @ Xs ) ) )
| ? [Ys6: list_nat] :
( ( Ys
= ( cons_nat @ X2 @ Ys6 ) )
& ( ( append_nat @ Ys6 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_535_append__eq__Cons__conv,axiom,
! [Ys: list_list_nat,Zs: list_list_nat,X2: list_nat,Xs: list_list_nat] :
( ( ( append_list_nat @ Ys @ Zs )
= ( cons_list_nat @ X2 @ Xs ) )
= ( ( ( Ys = nil_list_nat )
& ( Zs
= ( cons_list_nat @ X2 @ Xs ) ) )
| ? [Ys6: list_list_nat] :
( ( Ys
= ( cons_list_nat @ X2 @ Ys6 ) )
& ( ( append_list_nat @ Ys6 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_536_rev__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X3: a] : ( P @ ( cons_a @ X3 @ nil_a ) )
=> ( ! [X3: a,Xs3: list_a] :
( ( Xs3 != nil_a )
=> ( ( P @ Xs3 )
=> ( P @ ( append_a @ Xs3 @ ( cons_a @ X3 @ nil_a ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_537_rev__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X3: nat] : ( P @ ( cons_nat @ X3 @ nil_nat ) )
=> ( ! [X3: nat,Xs3: list_nat] :
( ( Xs3 != nil_nat )
=> ( ( P @ Xs3 )
=> ( P @ ( append_nat @ Xs3 @ ( cons_nat @ X3 @ nil_nat ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_538_rev__nonempty__induct,axiom,
! [Xs: list_list_nat,P: list_list_nat > $o] :
( ( Xs != nil_list_nat )
=> ( ! [X3: list_nat] : ( P @ ( cons_list_nat @ X3 @ nil_list_nat ) )
=> ( ! [X3: list_nat,Xs3: list_list_nat] :
( ( Xs3 != nil_list_nat )
=> ( ( P @ Xs3 )
=> ( P @ ( append_list_nat @ Xs3 @ ( cons_list_nat @ X3 @ nil_list_nat ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_539_list__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X3: a] : ( P @ ( cons_a @ X3 @ nil_a ) )
=> ( ! [X3: a,Xs3: list_a] :
( ( Xs3 != nil_a )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_a @ X3 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_540_list__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X3: nat] : ( P @ ( cons_nat @ X3 @ nil_nat ) )
=> ( ! [X3: nat,Xs3: list_nat] :
( ( Xs3 != nil_nat )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_nat @ X3 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_541_list__nonempty__induct,axiom,
! [Xs: list_list_nat,P: list_list_nat > $o] :
( ( Xs != nil_list_nat )
=> ( ! [X3: list_nat] : ( P @ ( cons_list_nat @ X3 @ nil_list_nat ) )
=> ( ! [X3: list_nat,Xs3: list_list_nat] :
( ( Xs3 != nil_list_nat )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_list_nat @ X3 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_542_append__eq__append__conv2,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ts: list_a] :
( ( ( append_a @ Xs @ Ys )
= ( append_a @ Zs @ Ts ) )
= ( ? [Us2: list_a] :
( ( ( Xs
= ( append_a @ Zs @ Us2 ) )
& ( ( append_a @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_a @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_a @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_543_append__eq__append__conv2,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ts: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Zs @ Ts ) )
= ( ? [Us2: list_nat] :
( ( ( Xs
= ( append_nat @ Zs @ Us2 ) )
& ( ( append_nat @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_nat @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_nat @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_544_split__list__first__prop__iff,axiom,
! [Xs: list_a,P: a > $o] :
( ( ? [X: a] :
( ( member_a2 @ X @ ( set_a2 @ Xs ) )
& ( P @ X ) ) )
= ( ? [Ys5: list_a,X: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys5 @ ( cons_a @ X @ Zs2 ) ) )
& ( P @ X )
& ! [Y: a] :
( ( member_a2 @ Y @ ( set_a2 @ Ys5 ) )
=> ~ ( P @ Y ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_545_split__list__first__prop__iff,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ? [X: nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
& ( P @ X ) ) )
= ( ? [Ys5: list_nat,X: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys5 @ ( cons_nat @ X @ Zs2 ) ) )
& ( P @ X )
& ! [Y: nat] :
( ( member_nat2 @ Y @ ( set_nat2 @ Ys5 ) )
=> ~ ( P @ Y ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_546_split__list__first__prop__iff,axiom,
! [Xs: list_list_nat,P: list_nat > $o] :
( ( ? [X: list_nat] :
( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
& ( P @ X ) ) )
= ( ? [Ys5: list_list_nat,X: list_nat] :
( ? [Zs2: list_list_nat] :
( Xs
= ( append_list_nat @ Ys5 @ ( cons_list_nat @ X @ Zs2 ) ) )
& ( P @ X )
& ! [Y: list_nat] :
( ( member_list_nat2 @ Y @ ( set_list_nat2 @ Ys5 ) )
=> ~ ( P @ Y ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_547_split__list__last__prop__iff,axiom,
! [Xs: list_a,P: a > $o] :
( ( ? [X: a] :
( ( member_a2 @ X @ ( set_a2 @ Xs ) )
& ( P @ X ) ) )
= ( ? [Ys5: list_a,X: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys5 @ ( cons_a @ X @ Zs2 ) ) )
& ( P @ X )
& ! [Y: a] :
( ( member_a2 @ Y @ ( set_a2 @ Zs2 ) )
=> ~ ( P @ Y ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_548_split__list__last__prop__iff,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ? [X: nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
& ( P @ X ) ) )
= ( ? [Ys5: list_nat,X: nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys5 @ ( cons_nat @ X @ Zs2 ) ) )
& ( P @ X )
& ! [Y: nat] :
( ( member_nat2 @ Y @ ( set_nat2 @ Zs2 ) )
=> ~ ( P @ Y ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_549_split__list__last__prop__iff,axiom,
! [Xs: list_list_nat,P: list_nat > $o] :
( ( ? [X: list_nat] :
( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
& ( P @ X ) ) )
= ( ? [Ys5: list_list_nat,X: list_nat,Zs2: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys5 @ ( cons_list_nat @ X @ Zs2 ) ) )
& ( P @ X )
& ! [Y: list_nat] :
( ( member_list_nat2 @ Y @ ( set_list_nat2 @ Zs2 ) )
=> ~ ( P @ Y ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_550_in__set__conv__decomp__first,axiom,
! [X2: a,Xs: list_a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
= ( ? [Ys5: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys5 @ ( cons_a @ X2 @ Zs2 ) ) )
& ~ ( member_a2 @ X2 @ ( set_a2 @ Ys5 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_551_in__set__conv__decomp__first,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [Ys5: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys5 @ ( cons_nat @ X2 @ Zs2 ) ) )
& ~ ( member_nat2 @ X2 @ ( set_nat2 @ Ys5 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_552_in__set__conv__decomp__first,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
= ( ? [Ys5: list_list_nat,Zs2: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys5 @ ( cons_list_nat @ X2 @ Zs2 ) ) )
& ~ ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Ys5 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_553_in__set__conv__decomp__last,axiom,
! [X2: a,Xs: list_a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
= ( ? [Ys5: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys5 @ ( cons_a @ X2 @ Zs2 ) ) )
& ~ ( member_a2 @ X2 @ ( set_a2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_554_in__set__conv__decomp__last,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [Ys5: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys5 @ ( cons_nat @ X2 @ Zs2 ) ) )
& ~ ( member_nat2 @ X2 @ ( set_nat2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_555_in__set__conv__decomp__last,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
= ( ? [Ys5: list_list_nat,Zs2: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys5 @ ( cons_list_nat @ X2 @ Zs2 ) ) )
& ~ ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_556_split__list__first__propE,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X5: a] :
( ( member_a2 @ X5 @ ( set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys4: list_a,X3: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys4 @ ( cons_a @ X3 @ Zs3 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa3: a] :
( ( member_a2 @ Xa3 @ ( set_a2 @ Ys4 ) )
=> ~ ( P @ Xa3 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_557_split__list__first__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat2 @ X5 @ ( set_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys4: list_nat,X3: nat] :
( ? [Zs3: list_nat] :
( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X3 @ Zs3 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa3: nat] :
( ( member_nat2 @ Xa3 @ ( set_nat2 @ Ys4 ) )
=> ~ ( P @ Xa3 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_558_split__list__first__propE,axiom,
! [Xs: list_list_nat,P: list_nat > $o] :
( ? [X5: list_nat] :
( ( member_list_nat2 @ X5 @ ( set_list_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys4: list_list_nat,X3: list_nat] :
( ? [Zs3: list_list_nat] :
( Xs
= ( append_list_nat @ Ys4 @ ( cons_list_nat @ X3 @ Zs3 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa3: list_nat] :
( ( member_list_nat2 @ Xa3 @ ( set_list_nat2 @ Ys4 ) )
=> ~ ( P @ Xa3 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_559_split__list__last__propE,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X5: a] :
( ( member_a2 @ X5 @ ( set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys4: list_a,X3: a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys4 @ ( cons_a @ X3 @ Zs3 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa3: a] :
( ( member_a2 @ Xa3 @ ( set_a2 @ Zs3 ) )
=> ~ ( P @ Xa3 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_560_split__list__last__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat2 @ X5 @ ( set_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys4: list_nat,X3: nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X3 @ Zs3 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa3: nat] :
( ( member_nat2 @ Xa3 @ ( set_nat2 @ Zs3 ) )
=> ~ ( P @ Xa3 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_561_split__list__last__propE,axiom,
! [Xs: list_list_nat,P: list_nat > $o] :
( ? [X5: list_nat] :
( ( member_list_nat2 @ X5 @ ( set_list_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys4: list_list_nat,X3: list_nat,Zs3: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys4 @ ( cons_list_nat @ X3 @ Zs3 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa3: list_nat] :
( ( member_list_nat2 @ Xa3 @ ( set_list_nat2 @ Zs3 ) )
=> ~ ( P @ Xa3 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_562_split__list__first__prop,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X5: a] :
( ( member_a2 @ X5 @ ( set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys4: list_a,X3: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys4 @ ( cons_a @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Xa3: a] :
( ( member_a2 @ Xa3 @ ( set_a2 @ Ys4 ) )
=> ~ ( P @ Xa3 ) ) ) ) ).
% split_list_first_prop
thf(fact_563_split__list__first__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat2 @ X5 @ ( set_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys4: list_nat,X3: nat] :
( ? [Zs3: list_nat] :
( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Xa3: nat] :
( ( member_nat2 @ Xa3 @ ( set_nat2 @ Ys4 ) )
=> ~ ( P @ Xa3 ) ) ) ) ).
% split_list_first_prop
thf(fact_564_split__list__first__prop,axiom,
! [Xs: list_list_nat,P: list_nat > $o] :
( ? [X5: list_nat] :
( ( member_list_nat2 @ X5 @ ( set_list_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys4: list_list_nat,X3: list_nat] :
( ? [Zs3: list_list_nat] :
( Xs
= ( append_list_nat @ Ys4 @ ( cons_list_nat @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Xa3: list_nat] :
( ( member_list_nat2 @ Xa3 @ ( set_list_nat2 @ Ys4 ) )
=> ~ ( P @ Xa3 ) ) ) ) ).
% split_list_first_prop
thf(fact_565_split__list__last__prop,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X5: a] :
( ( member_a2 @ X5 @ ( set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys4: list_a,X3: a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys4 @ ( cons_a @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Xa3: a] :
( ( member_a2 @ Xa3 @ ( set_a2 @ Zs3 ) )
=> ~ ( P @ Xa3 ) ) ) ) ).
% split_list_last_prop
thf(fact_566_split__list__last__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat2 @ X5 @ ( set_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys4: list_nat,X3: nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Xa3: nat] :
( ( member_nat2 @ Xa3 @ ( set_nat2 @ Zs3 ) )
=> ~ ( P @ Xa3 ) ) ) ) ).
% split_list_last_prop
thf(fact_567_split__list__last__prop,axiom,
! [Xs: list_list_nat,P: list_nat > $o] :
( ? [X5: list_nat] :
( ( member_list_nat2 @ X5 @ ( set_list_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys4: list_list_nat,X3: list_nat,Zs3: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys4 @ ( cons_list_nat @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Xa3: list_nat] :
( ( member_list_nat2 @ Xa3 @ ( set_list_nat2 @ Zs3 ) )
=> ~ ( P @ Xa3 ) ) ) ) ).
% split_list_last_prop
thf(fact_568_in__set__conv__decomp,axiom,
! [X2: a,Xs: list_a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
= ( ? [Ys5: list_a,Zs2: list_a] :
( Xs
= ( append_a @ Ys5 @ ( cons_a @ X2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_569_in__set__conv__decomp,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [Ys5: list_nat,Zs2: list_nat] :
( Xs
= ( append_nat @ Ys5 @ ( cons_nat @ X2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_570_in__set__conv__decomp,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
= ( ? [Ys5: list_list_nat,Zs2: list_list_nat] :
( Xs
= ( append_list_nat @ Ys5 @ ( cons_list_nat @ X2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_571_append__Cons__eq__iff,axiom,
! [X2: a,Xs: list_a,Ys: list_a,Xs5: list_a,Ys7: list_a] :
( ~ ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( ~ ( member_a2 @ X2 @ ( set_a2 @ Ys ) )
=> ( ( ( append_a @ Xs @ ( cons_a @ X2 @ Ys ) )
= ( append_a @ Xs5 @ ( cons_a @ X2 @ Ys7 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys7 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_572_append__Cons__eq__iff,axiom,
! [X2: nat,Xs: list_nat,Ys: list_nat,Xs5: list_nat,Ys7: list_nat] :
( ~ ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ~ ( member_nat2 @ X2 @ ( set_nat2 @ Ys ) )
=> ( ( ( append_nat @ Xs @ ( cons_nat @ X2 @ Ys ) )
= ( append_nat @ Xs5 @ ( cons_nat @ X2 @ Ys7 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys7 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_573_append__Cons__eq__iff,axiom,
! [X2: list_nat,Xs: list_list_nat,Ys: list_list_nat,Xs5: list_list_nat,Ys7: list_list_nat] :
( ~ ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ~ ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Ys ) )
=> ( ( ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ Ys ) )
= ( append_list_nat @ Xs5 @ ( cons_list_nat @ X2 @ Ys7 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys7 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_574_split__list__propE,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X5: a] :
( ( member_a2 @ X5 @ ( set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys4: list_a,X3: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys4 @ ( cons_a @ X3 @ Zs3 ) ) )
=> ~ ( P @ X3 ) ) ) ).
% split_list_propE
thf(fact_575_split__list__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat2 @ X5 @ ( set_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys4: list_nat,X3: nat] :
( ? [Zs3: list_nat] :
( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X3 @ Zs3 ) ) )
=> ~ ( P @ X3 ) ) ) ).
% split_list_propE
thf(fact_576_split__list__propE,axiom,
! [Xs: list_list_nat,P: list_nat > $o] :
( ? [X5: list_nat] :
( ( member_list_nat2 @ X5 @ ( set_list_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys4: list_list_nat,X3: list_nat] :
( ? [Zs3: list_list_nat] :
( Xs
= ( append_list_nat @ Ys4 @ ( cons_list_nat @ X3 @ Zs3 ) ) )
=> ~ ( P @ X3 ) ) ) ).
% split_list_propE
thf(fact_577_split__list__first,axiom,
! [X2: a,Xs: list_a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ? [Ys4: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys4 @ ( cons_a @ X2 @ Zs3 ) ) )
& ~ ( member_a2 @ X2 @ ( set_a2 @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_578_split__list__first,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ? [Ys4: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X2 @ Zs3 ) ) )
& ~ ( member_nat2 @ X2 @ ( set_nat2 @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_579_split__list__first,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
=> ? [Ys4: list_list_nat,Zs3: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys4 @ ( cons_list_nat @ X2 @ Zs3 ) ) )
& ~ ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_580_split__list__prop,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X5: a] :
( ( member_a2 @ X5 @ ( set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys4: list_a,X3: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys4 @ ( cons_a @ X3 @ Zs3 ) ) )
& ( P @ X3 ) ) ) ).
% split_list_prop
thf(fact_581_split__list__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat2 @ X5 @ ( set_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys4: list_nat,X3: nat] :
( ? [Zs3: list_nat] :
( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X3 @ Zs3 ) ) )
& ( P @ X3 ) ) ) ).
% split_list_prop
thf(fact_582_split__list__prop,axiom,
! [Xs: list_list_nat,P: list_nat > $o] :
( ? [X5: list_nat] :
( ( member_list_nat2 @ X5 @ ( set_list_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys4: list_list_nat,X3: list_nat] :
( ? [Zs3: list_list_nat] :
( Xs
= ( append_list_nat @ Ys4 @ ( cons_list_nat @ X3 @ Zs3 ) ) )
& ( P @ X3 ) ) ) ).
% split_list_prop
thf(fact_583_split__list__last,axiom,
! [X2: a,Xs: list_a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ? [Ys4: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys4 @ ( cons_a @ X2 @ Zs3 ) ) )
& ~ ( member_a2 @ X2 @ ( set_a2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_584_split__list__last,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ? [Ys4: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X2 @ Zs3 ) ) )
& ~ ( member_nat2 @ X2 @ ( set_nat2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_585_split__list__last,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
=> ? [Ys4: list_list_nat,Zs3: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys4 @ ( cons_list_nat @ X2 @ Zs3 ) ) )
& ~ ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_586_split__list,axiom,
! [X2: a,Xs: list_a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ? [Ys4: list_a,Zs3: list_a] :
( Xs
= ( append_a @ Ys4 @ ( cons_a @ X2 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_587_split__list,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ? [Ys4: list_nat,Zs3: list_nat] :
( Xs
= ( append_nat @ Ys4 @ ( cons_nat @ X2 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_588_split__list,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
=> ? [Ys4: list_list_nat,Zs3: list_list_nat] :
( Xs
= ( append_list_nat @ Ys4 @ ( cons_list_nat @ X2 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_589_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ws: list_a,P: list_a > list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X3: a,Xs3: list_a,Y3: a,Ys4: list_a,Z: a,Zs3: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_a @ Zs3 ) )
=> ( ( ( size_size_list_a @ Zs3 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs3 @ Ws2 )
=> ( P @ ( cons_a @ X3 @ Xs3 ) @ ( cons_a @ Y3 @ Ys4 ) @ ( cons_a @ Z @ Zs3 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_590_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ws: list_nat,P: list_a > list_a > list_a > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_a @ nil_a @ nil_a @ nil_nat )
=> ( ! [X3: a,Xs3: list_a,Y3: a,Ys4: list_a,Z: a,Zs3: list_a,W: nat,Ws2: list_nat] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_a @ Zs3 ) )
=> ( ( ( size_size_list_a @ Zs3 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs3 @ Ws2 )
=> ( P @ ( cons_a @ X3 @ Xs3 ) @ ( cons_a @ Y3 @ Ys4 ) @ ( cons_a @ Z @ Zs3 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_591_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_nat,Ws: list_a,P: list_a > list_a > list_nat > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_a @ nil_nat @ nil_a )
=> ( ! [X3: a,Xs3: list_a,Y3: a,Ys4: list_a,Z: nat,Zs3: list_nat,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs3 @ Ws2 )
=> ( P @ ( cons_a @ X3 @ Xs3 ) @ ( cons_a @ Y3 @ Ys4 ) @ ( cons_nat @ Z @ Zs3 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_592_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_nat,Ws: list_nat,P: list_a > list_a > list_nat > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_a @ nil_a @ nil_nat @ nil_nat )
=> ( ! [X3: a,Xs3: list_a,Y3: a,Ys4: list_a,Z: nat,Zs3: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs3 @ Ws2 )
=> ( P @ ( cons_a @ X3 @ Xs3 ) @ ( cons_a @ Y3 @ Ys4 ) @ ( cons_nat @ Z @ Zs3 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_593_list__induct4,axiom,
! [Xs: list_a,Ys: list_nat,Zs: list_a,Ws: list_a,P: list_a > list_nat > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_nat @ nil_a @ nil_a )
=> ( ! [X3: a,Xs3: list_a,Y3: nat,Ys4: list_nat,Z: a,Zs3: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_a @ Zs3 ) )
=> ( ( ( size_size_list_a @ Zs3 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs3 @ Ws2 )
=> ( P @ ( cons_a @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_a @ Z @ Zs3 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_594_list__induct4,axiom,
! [Xs: list_a,Ys: list_nat,Zs: list_a,Ws: list_nat,P: list_a > list_nat > list_a > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_a @ nil_nat @ nil_a @ nil_nat )
=> ( ! [X3: a,Xs3: list_a,Y3: nat,Ys4: list_nat,Z: a,Zs3: list_a,W: nat,Ws2: list_nat] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_a @ Zs3 ) )
=> ( ( ( size_size_list_a @ Zs3 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs3 @ Ws2 )
=> ( P @ ( cons_a @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_a @ Z @ Zs3 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_595_list__induct4,axiom,
! [Xs: list_a,Ys: list_nat,Zs: list_nat,Ws: list_a,P: list_a > list_nat > list_nat > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_nat @ nil_nat @ nil_a )
=> ( ! [X3: a,Xs3: list_a,Y3: nat,Ys4: list_nat,Z: nat,Zs3: list_nat,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs3 @ Ws2 )
=> ( P @ ( cons_a @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z @ Zs3 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_596_list__induct4,axiom,
! [Xs: list_a,Ys: list_nat,Zs: list_nat,Ws: list_nat,P: list_a > list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_a @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X3: a,Xs3: list_a,Y3: nat,Ys4: list_nat,Z: nat,Zs3: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs3 @ Ws2 )
=> ( P @ ( cons_a @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z @ Zs3 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_597_list__induct4,axiom,
! [Xs: list_nat,Ys: list_a,Zs: list_a,Ws: list_a,P: list_nat > list_a > list_a > list_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_nat @ nil_a @ nil_a @ nil_a )
=> ( ! [X3: nat,Xs3: list_nat,Y3: a,Ys4: list_a,Z: a,Zs3: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_a @ Zs3 ) )
=> ( ( ( size_size_list_a @ Zs3 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs3 @ Ws2 )
=> ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_a @ Y3 @ Ys4 ) @ ( cons_a @ Z @ Zs3 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_598_list__induct4,axiom,
! [Xs: list_nat,Ys: list_a,Zs: list_a,Ws: list_nat,P: list_nat > list_a > list_a > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_a @ nil_a @ nil_nat )
=> ( ! [X3: nat,Xs3: list_nat,Y3: a,Ys4: list_a,Z: a,Zs3: list_a,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_a @ Zs3 ) )
=> ( ( ( size_size_list_a @ Zs3 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs3 @ Ws2 )
=> ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_a @ Y3 @ Ys4 ) @ ( cons_a @ Z @ Zs3 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_599_enum__rgfs_Ocases,axiom,
! [X2: nat] :
( ( X2 != zero_zero_nat )
=> ~ ! [N2: nat] :
( X2
!= ( suc @ N2 ) ) ) ).
% enum_rgfs.cases
thf(fact_600_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_601_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_602_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_603_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_604_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_605_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_606_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_607_old_Onat_Oexhaust,axiom,
! [Y2: nat] :
( ( Y2 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y2
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_608_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_609_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_610_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_611_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_612_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_613_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_614_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_615_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_616_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_617_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_618_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_619_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ ( suc @ I2 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_620_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_621_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_622_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_623_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_624_lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_lessThan_nat @ ( suc @ N ) )
= ( insert_nat2 @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% lessThan_Suc_eq_insert_0
thf(fact_625_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_626_stirling__row__code_I2_J,axiom,
! [N: nat] :
( ( stirling_row @ ( suc @ N ) )
= ( stirling_row_aux_nat @ N @ zero_zero_nat @ ( stirling_row @ N ) ) ) ).
% stirling_row_code(2)
thf(fact_627_stirling__row__nonempty,axiom,
! [N: nat] :
( ( stirling_row @ N )
!= nil_nat ) ).
% stirling_row_nonempty
thf(fact_628_rgf__limit_Osimps_I1_J,axiom,
( ( equiva5889994315859557365_limit @ nil_nat )
= zero_zero_nat ) ).
% rgf_limit.simps(1)
thf(fact_629_zero__notin__Suc__image,axiom,
! [A: set_nat] :
~ ( member_nat2 @ zero_zero_nat @ ( image_nat_nat @ suc @ A ) ) ).
% zero_notin_Suc_image
thf(fact_630_lessThan__Suc,axiom,
! [K: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K ) )
= ( insert_nat2 @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).
% lessThan_Suc
thf(fact_631_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_632_rgf__limit_Ocases,axiom,
! [X2: list_nat] :
( ( X2 != nil_nat )
=> ~ ! [X3: nat,Xs3: list_nat] :
( X2
!= ( cons_nat @ X3 @ Xs3 ) ) ) ).
% rgf_limit.cases
thf(fact_633_sorted__list__of__set__lessThan__Suc,axiom,
! [K: nat] :
( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K ) ) )
= ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K ) ) @ ( cons_nat @ K @ nil_nat ) ) ) ).
% sorted_list_of_set_lessThan_Suc
thf(fact_634_sorted__list__of__set__atMost__Suc,axiom,
! [K: nat] :
( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K ) ) )
= ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K ) ) @ ( cons_nat @ ( suc @ K ) @ nil_nat ) ) ) ).
% sorted_list_of_set_atMost_Suc
thf(fact_635_lessThan__Suc__atMost,axiom,
! [K: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K ) )
= ( set_ord_atMost_nat @ K ) ) ).
% lessThan_Suc_atMost
thf(fact_636_atMost__Suc,axiom,
! [K: nat] :
( ( set_ord_atMost_nat @ ( suc @ K ) )
= ( insert_nat2 @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).
% atMost_Suc
thf(fact_637_atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_atMost_nat @ ( suc @ N ) )
= ( insert_nat2 @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% atMost_Suc_eq_insert_0
thf(fact_638_rgf__def,axiom,
( equiva3371634703666331078on_rgf
= ( ^ [X: list_nat] :
! [Ys5: list_nat,Y: nat] :
( ( prefix_nat @ ( append_nat @ Ys5 @ ( cons_nat @ Y @ nil_nat ) ) @ X )
=> ( ord_less_eq_nat @ Y @ ( equiva5889994315859557365_limit @ Ys5 ) ) ) ) ) ).
% rgf_def
thf(fact_639_stirling__row__code_I1_J,axiom,
( ( stirling_row @ zero_zero_nat )
= ( cons_nat @ one_one_nat @ nil_nat ) ) ).
% stirling_row_code(1)
thf(fact_640_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_641_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_642_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_643_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_644_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M4: nat] :
( ( P @ X2 )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M4 ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_645_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_646_le__trans,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_647_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_648_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_649_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_650_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B2 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_651_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z: nat] :
( ( R @ X3 @ Y3 )
=> ( ( R @ Y3 @ Z )
=> ( R @ X3 @ Z ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_652_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_653_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_654_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_655_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_656_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_657_Suc__le__D,axiom,
! [N: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
=> ? [M5: nat] :
( M7
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_658_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_659_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_660_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_661_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_662_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_663_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_664_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_665_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J2: nat] :
( ! [I3: nat,J: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J ) ) )
=> ( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_666_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_667_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_668_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N5: nat] :
( ( ord_less_nat @ M6 @ N5 )
| ( M6 = N5 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_669_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_670_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M6: nat,N5: nat] :
( ( ord_less_eq_nat @ M6 @ N5 )
& ( M6 != N5 ) ) ) ) ).
% nat_less_le
thf(fact_671_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_672_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_673_dec__induct,axiom,
! [I: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J2 ) ) ) ) ).
% dec_induct
thf(fact_674_inc__induct,axiom,
! [I: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( P @ J2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J2 )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_675_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_676_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_677_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_678_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N5: nat] : ( ord_less_eq_nat @ ( suc @ N5 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_679_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_680_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_681_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_682_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_683_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_684_rgf__snoc,axiom,
! [Xs: list_nat,X2: nat] :
( ( equiva3371634703666331078on_rgf @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) )
= ( ( equiva3371634703666331078on_rgf @ Xs )
& ( ord_less_nat @ X2 @ ( plus_plus_nat @ ( equiva5889994315859557365_limit @ Xs ) @ one_one_nat ) ) ) ) ).
% rgf_snoc
thf(fact_685_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_686_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_687_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_688_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_689_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_690_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_691_image__Suc__atLeastLessThan,axiom,
! [I: nat,J2: nat] :
( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I @ J2 ) )
= ( set_or4665077453230672383an_nat @ ( suc @ I ) @ ( suc @ J2 ) ) ) ).
% image_Suc_atLeastLessThan
thf(fact_692_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_693_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_694_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_695_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_696_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_697_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_698_add__le__mono,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_le_mono
thf(fact_699_add__le__mono1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_le_mono1
thf(fact_700_trans__le__add1,axiom,
! [I: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_le_add1
thf(fact_701_trans__le__add2,axiom,
! [I: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_le_add2
thf(fact_702_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N5: nat] :
? [K3: nat] :
( N5
= ( plus_plus_nat @ M6 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_703_add__lessD1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J2 ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_704_add__less__mono,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_less_mono
thf(fact_705_not__add__less1,axiom,
! [I: nat,J2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J2 ) @ I ) ).
% not_add_less1
thf(fact_706_not__add__less2,axiom,
! [J2: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I ) @ I ) ).
% not_add_less2
thf(fact_707_add__less__mono1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_less_mono1
thf(fact_708_trans__less__add1,axiom,
! [I: nat,J2: nat,M: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_less_add1
thf(fact_709_trans__less__add2,axiom,
! [I: nat,J2: nat,M: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_less_add2
thf(fact_710_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_711_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_712_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_713_nat__arith_Osuc1,axiom,
! [A: nat,K: nat,A2: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( suc @ A )
= ( plus_plus_nat @ K @ ( suc @ A2 ) ) ) ) ).
% nat_arith.suc1
thf(fact_714_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_715_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_716_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M6: nat] :
( ( ord_less_nat @ M6 @ N )
& ( P @ M6 ) ) )
= ( ? [X: nat] :
( ( member_nat2 @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P @ X ) ) ) ) ).
% ex_nat_less_eq
thf(fact_717_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M6: nat] :
( ( ord_less_nat @ M6 @ N )
=> ( P @ M6 ) ) )
= ( ! [X: nat] :
( ( member_nat2 @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P @ X ) ) ) ) ).
% all_nat_less_eq
thf(fact_718_lessThan__atLeast0,axiom,
( set_ord_lessThan_nat
= ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% lessThan_atLeast0
thf(fact_719_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_720_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_721_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q ) ) ) ) ).
% less_natE
thf(fact_722_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_723_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_724_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M6: nat,N5: nat] :
? [K3: nat] :
( N5
= ( suc @ ( plus_plus_nat @ M6 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_725_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_726_less__imp__add__positive,axiom,
! [I: nat,J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J2 ) ) ) ).
% less_imp_add_positive
thf(fact_727_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M5: nat,N2: nat] :
( ( ord_less_nat @ M5 @ N2 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_728_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_729_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_730_Suc__eq__plus1,axiom,
( suc
= ( ^ [N5: nat] : ( plus_plus_nat @ N5 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_731_atLeast0__lessThan__Suc,axiom,
! [N: nat] :
( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
= ( insert_nat2 @ N @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% atLeast0_lessThan_Suc
thf(fact_732_atLeast0__lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
= ( insert_nat2 @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).
% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_733_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_734_Euclid__induct,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
= ( P @ B4 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
=> ( P @ A4 @ ( plus_plus_nat @ A4 @ B4 ) ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% Euclid_induct
thf(fact_735_rgf__limit__snoc,axiom,
! [X2: list_nat,Y2: nat] :
( ( equiva5889994315859557365_limit @ ( append_nat @ X2 @ ( cons_nat @ Y2 @ nil_nat ) ) )
= ( ord_max_nat @ ( plus_plus_nat @ Y2 @ one_one_nat ) @ ( equiva5889994315859557365_limit @ X2 ) ) ) ).
% rgf_limit_snoc
thf(fact_736_max__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( suc @ ( ord_max_nat @ M @ N ) ) ) ).
% max_Suc_Suc
thf(fact_737_max__0R,axiom,
! [N: nat] :
( ( ord_max_nat @ N @ zero_zero_nat )
= N ) ).
% max_0R
thf(fact_738_max__0L,axiom,
! [N: nat] :
( ( ord_max_nat @ zero_zero_nat @ N )
= N ) ).
% max_0L
thf(fact_739_max__nat_Oright__neutral,axiom,
! [A2: nat] :
( ( ord_max_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% max_nat.right_neutral
thf(fact_740_max__nat_Oneutr__eq__iff,axiom,
! [A2: nat,B2: nat] :
( ( zero_zero_nat
= ( ord_max_nat @ A2 @ B2 ) )
= ( ( A2 = zero_zero_nat )
& ( B2 = zero_zero_nat ) ) ) ).
% max_nat.neutr_eq_iff
thf(fact_741_max__nat_Oleft__neutral,axiom,
! [A2: nat] :
( ( ord_max_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% max_nat.left_neutral
thf(fact_742_max__nat_Oeq__neutr__iff,axiom,
! [A2: nat,B2: nat] :
( ( ( ord_max_nat @ A2 @ B2 )
= zero_zero_nat )
= ( ( A2 = zero_zero_nat )
& ( B2 = zero_zero_nat ) ) ) ).
% max_nat.eq_neutr_iff
thf(fact_743_nat__add__max__left,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
= ( ord_max_nat @ ( plus_plus_nat @ M @ Q2 ) @ ( plus_plus_nat @ N @ Q2 ) ) ) ).
% nat_add_max_left
thf(fact_744_nat__add__max__right,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
= ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q2 ) ) ) ).
% nat_add_max_right
thf(fact_745_rgf__limit_Osimps_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( equiva5889994315859557365_limit @ ( cons_nat @ X2 @ Xs ) )
= ( ord_max_nat @ ( plus_plus_nat @ X2 @ one_one_nat ) @ ( equiva5889994315859557365_limit @ Xs ) ) ) ).
% rgf_limit.simps(2)
thf(fact_746_rgf__limit_Oelims,axiom,
! [X2: list_nat,Y2: nat] :
( ( ( equiva5889994315859557365_limit @ X2 )
= Y2 )
=> ( ( ( X2 = nil_nat )
=> ( Y2 != zero_zero_nat ) )
=> ~ ! [X3: nat,Xs3: list_nat] :
( ( X2
= ( cons_nat @ X3 @ Xs3 ) )
=> ( Y2
!= ( ord_max_nat @ ( plus_plus_nat @ X3 @ one_one_nat ) @ ( equiva5889994315859557365_limit @ Xs3 ) ) ) ) ) ) ).
% rgf_limit.elims
thf(fact_747_equiv__rels__enum,axiom,
! [X2: list_nat] :
( ( equiva3371634703666331078on_rgf @ X2 )
=> ( ( count_list_list_nat @ ( equiva7426478223624825838m_rgfs @ ( size_size_list_nat @ X2 ) ) @ X2 )
= one_one_nat ) ) ).
% equiv_rels_enum
thf(fact_748_enum__rgfs__returns__rgfs,axiom,
! [X2: list_nat,N: nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ ( equiva7426478223624825838m_rgfs @ N ) ) )
=> ( equiva3371634703666331078on_rgf @ X2 ) ) ).
% enum_rgfs_returns_rgfs
thf(fact_749_enum__rgfs__len,axiom,
! [X2: list_nat,N: nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ ( equiva7426478223624825838m_rgfs @ N ) ) )
=> ( ( size_size_list_nat @ X2 )
= N ) ) ).
% enum_rgfs_len
thf(fact_750_enum__rgfs_Osimps_I1_J,axiom,
( ( equiva7426478223624825838m_rgfs @ zero_zero_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% enum_rgfs.simps(1)
thf(fact_751_equiv__rels__def,axiom,
( equiva8721718519204927301v_rels
= ( ^ [N5: nat] : ( map_li6003994582982014139at_nat @ equiva2048684438135499664of_nat @ ( equiva7426478223624825838m_rgfs @ N5 ) ) ) ) ).
% equiv_rels_def
thf(fact_752_rgf__limit_Opelims,axiom,
! [X2: list_nat,Y2: nat] :
( ( ( equiva5889994315859557365_limit @ X2 )
= Y2 )
=> ( ( accp_list_nat @ equiva5575797544161152836it_rel @ X2 )
=> ( ( ( X2 = nil_nat )
=> ( ( Y2 = zero_zero_nat )
=> ~ ( accp_list_nat @ equiva5575797544161152836it_rel @ nil_nat ) ) )
=> ~ ! [X3: nat,Xs3: list_nat] :
( ( X2
= ( cons_nat @ X3 @ Xs3 ) )
=> ( ( Y2
= ( ord_max_nat @ ( plus_plus_nat @ X3 @ one_one_nat ) @ ( equiva5889994315859557365_limit @ Xs3 ) ) )
=> ~ ( accp_list_nat @ equiva5575797544161152836it_rel @ ( cons_nat @ X3 @ Xs3 ) ) ) ) ) ) ) ).
% rgf_limit.pelims
thf(fact_753_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_754_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_755_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_756_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_757_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_758_diff__diff__left,axiom,
! [I: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J2 ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% diff_diff_left
thf(fact_759_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_760_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_761_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_762_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J2 ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_763_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_764_Nat_Odiff__diff__right,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J2 ) ) ) ).
% Nat.diff_diff_right
thf(fact_765_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_766_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_767_diff__Suc__diff__eq1,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J2 ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_768_diff__Suc__diff__eq2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J2 ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_769_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_770_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_771_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_772_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_773_less__imp__diff__less,axiom,
! [J2: nat,K: nat,N: nat] :
( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J2 @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_774_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_775_diff__commute,axiom,
! [I: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J2 ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J2 ) ) ).
% diff_commute
thf(fact_776_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_777_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_778_less__diff__conv,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J2 ) ) ).
% less_diff_conv
thf(fact_779_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_780_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_781_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_782_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_783_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_784_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_785_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_786_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_787_nat__minus__add__max,axiom,
! [N: nat,M: nat] :
( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
= ( ord_max_nat @ N @ M ) ) ).
% nat_minus_add_max
thf(fact_788_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_789_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_790_diff__less__mono,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C2 ) @ ( minus_minus_nat @ B2 @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_791_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ( minus_minus_nat @ J2 @ I )
= K )
= ( J2
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_792_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_793_Nat_Odiff__add__assoc,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J2 ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J2 @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_794_Nat_Ole__diff__conv2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J2 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_795_le__diff__conv,axiom,
! [J2: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J2 @ K ) @ I )
= ( ord_less_eq_nat @ J2 @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_796_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_797_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_798_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_799_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_800_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_801_le__diff__iff_H,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A2 ) @ ( minus_minus_nat @ C2 @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_802_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_803_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_804_nat__diff__split,axiom,
! [P: nat > $o,A2: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B2 ) )
= ( ( ( ord_less_nat @ A2 @ B2 )
=> ( P @ zero_zero_nat ) )
& ! [D: nat] :
( ( A2
= ( plus_plus_nat @ B2 @ D ) )
=> ( P @ D ) ) ) ) ).
% nat_diff_split
thf(fact_805_nat__diff__split__asm,axiom,
! [P: nat > $o,A2: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B2 ) )
= ( ~ ( ( ( ord_less_nat @ A2 @ B2 )
& ~ ( P @ zero_zero_nat ) )
| ? [D: nat] :
( ( A2
= ( plus_plus_nat @ B2 @ D ) )
& ~ ( P @ D ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_806_less__diff__conv2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J2 @ K ) @ I )
= ( ord_less_nat @ J2 @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_807_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_808_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_809_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M6: nat,N5: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N5 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N5 ) ) ) ) ) ).
% add_eq_if
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y2: nat] :
( ( if_nat @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y2: nat] :
( ( if_nat @ $true @ X2 @ Y2 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
? [F4: a > nat] :
( ( inj_on_a_nat @ F4 @ ( set_a2 @ xa ) )
& ( equiva3371634703666331078on_rgf @ ( map_a_nat @ F4 @ xa ) ) ) ).
%------------------------------------------------------------------------------