TPTP Problem File: SLH0271^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Equivalence_Relation_Enumeration/0007_Equivalence_Relation_Enumeration/prob_00501_019199__12153078_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1553 ( 618 unt; 274 typ; 0 def)
% Number of atoms : 3599 (1580 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 9869 ( 304 ~; 44 |; 203 &;7949 @)
% ( 0 <=>;1369 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 6 avg)
% Number of types : 26 ( 25 usr)
% Number of type conns : 1184 (1184 >; 0 *; 0 +; 0 <<)
% Number of symbols : 252 ( 249 usr; 20 con; 0-4 aty)
% Number of variables : 3287 ( 133 ^;3042 !; 112 ?;3287 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:14:53.883
%------------------------------------------------------------------------------
% Could-be-implicit typings (25)
thf(ty_n_t__List__Olist_It__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
list_l3822697302700470509at_nat: $tType ).
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__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
list_set_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
list_set_a: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
list_int: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__List__Olist_Itf__b_J,type,
list_b: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__b_J,type,
set_b: $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_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (249)
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_001tf__a,type,
equiva2867628904822520638l_of_a: list_a > set_Pr1261947904930325089at_nat ).
thf(sy_c_Equivalence__Relation__Enumeration_Okernel__of_001tf__b,type,
equiva2867628904822520639l_of_b: list_b > set_Pr1261947904930325089at_nat ).
thf(sy_c_Equivalence__Relation__Enumeration_Orgf,type,
equiva3371634703666331078on_rgf: list_nat > $o ).
thf(sy_c_Fun_Ofun__upd_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
fun_up3599288762871520703st_nat: ( list_nat > list_nat ) > list_nat > list_nat > list_nat > list_nat ).
thf(sy_c_Fun_Ofun__upd_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
fun_upd_list_nat_nat: ( list_nat > nat ) > list_nat > nat > list_nat > nat ).
thf(sy_c_Fun_Ofun__upd_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
fun_up5216017410020450626at_nat: ( list_nat > set_Pr1261947904930325089at_nat ) > list_nat > set_Pr1261947904930325089at_nat > list_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Fun_Ofun__upd_001tf__a_001t__Nat__Onat,type,
fun_upd_a_nat: ( a > nat ) > a > nat > a > nat ).
thf(sy_c_Fun_Ofun__upd_001tf__a_001tf__a,type,
fun_upd_a_a: ( a > a ) > a > a > a > a ).
thf(sy_c_Fun_Oinj__on_001t__Int__Oint_001t__Int__Oint,type,
inj_on_int_int: ( int > int ) > set_int > $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__List__Olist_It__Nat__Onat_J_J_001t__List__Olist_It__Nat__Onat_J,type,
inj_on8869394816141173315st_nat: ( list_list_nat > list_nat ) > set_list_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_001t__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
inj_on8270031949885980868at_nat: ( list_list_nat > list_s1210847774152347623at_nat ) > set_list_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__Int__Oint,type,
inj_on_list_nat_int: ( list_nat > int ) > set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
inj_on3049792774292151987st_nat: ( list_nat > list_nat ) > set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
inj_on_list_nat_nat: ( list_nat > nat ) > set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__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_It__Nat__Onat_J,type,
inj_on6731145966573583411st_nat: ( list_a > list_nat ) > set_list_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
inj_on_list_a_list_a: ( list_a > list_a ) > set_list_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Int__Oint,type,
inj_on_nat_int: ( nat > int ) > 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_001t__Nat__Onat_001tf__b,type,
inj_on_nat_b: ( nat > b ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001t__Int__Oint,type,
inj_on_a_int: ( a > int ) > set_a > $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_Oinj__on_001tf__a_001tf__b,type,
inj_on_a_b: ( a > b ) > set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__b_001t__Nat__Onat,type,
inj_on_b_nat: ( b > nat ) > set_b > $o ).
thf(sy_c_Fun_Oinj__on_001tf__b_001tf__a,type,
inj_on_b_a: ( b > a ) > set_b > $o ).
thf(sy_c_Fun_Oinj__on_001tf__b_001tf__b,type,
inj_on_b_b: ( b > b ) > set_b > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
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__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
uminus3195874150345416415st_nat: set_list_nat > set_list_nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__a_J,type,
uminus_uminus_set_a: set_a > set_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001t__List__Olist_It__Nat__Onat_J_001t__Int__Oint,type,
groups40136427346340713at_int: ( list_nat > int ) > int > list_list_nat > int ).
thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
groups42626897855390989at_nat: ( list_nat > nat ) > nat > list_list_nat > nat ).
thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001t__Nat__Onat_001t__Int__Oint,type,
groups7485877704341954137at_int: ( nat > int ) > int > list_nat > int ).
thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum_001t__Nat__Onat_001t__Nat__Onat,type,
groups7488368174851004413at_nat: ( nat > nat ) > nat > list_nat > nat ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
ring_1_of_int_int: int > int ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Int__Oint,type,
sup_sup_int: int > int > int ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
sup_sup_set_list_nat: set_list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_List_Ocan__select_001t__List__Olist_It__Nat__Onat_J,type,
can_select_list_nat: ( list_nat > $o ) > set_list_nat > $o ).
thf(sy_c_List_Ocan__select_001tf__a,type,
can_select_a: ( a > $o ) > set_a > $o ).
thf(sy_c_List_Ocoset_001t__List__Olist_It__Nat__Onat_J,type,
coset_list_nat: list_list_nat > set_list_nat ).
thf(sy_c_List_Ocoset_001t__Nat__Onat,type,
coset_nat: list_nat > set_nat ).
thf(sy_c_List_Ocoset_001tf__a,type,
coset_a: list_a > set_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_001t__Int__Oint,type,
distinct_int: list_int > $o ).
thf(sy_c_List_Odistinct_001t__List__Olist_It__Nat__Onat_J,type,
distinct_list_nat: list_list_nat > $o ).
thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
distinct_nat: list_nat > $o ).
thf(sy_c_List_Odistinct_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
distin4912746231890992349st_nat: list_set_list_nat > $o ).
thf(sy_c_List_Odistinct_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
distin8719635941469336154at_nat: list_s1210847774152347623at_nat > $o ).
thf(sy_c_List_Odistinct_001t__Set__Oset_Itf__a_J,type,
distinct_set_a: list_set_a > $o ).
thf(sy_c_List_Odistinct_001tf__a,type,
distinct_a: list_a > $o ).
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_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
foldin1968479528632951399st_nat: ( list_nat > list_nat > $o ) > ( list_nat > list_nat > $o ) > set_list_nat > ( list_nat > list_nat ) > $o ).
thf(sy_c_List_Ofolding__insort__key_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
foldin951631548397865559st_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > set_list_nat > ( list_nat > nat ) > $o ).
thf(sy_c_List_Ofolding__insort__key_001t__Nat__Onat_001tf__a,type,
foldin508877545616633799_nat_a: ( nat > nat > $o ) > ( nat > nat > $o ) > set_a > ( a > nat ) > $o ).
thf(sy_c_List_Ofolding__insort__key_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__List__Olist_It__Nat__Onat_J,type,
foldin5963648469059051996st_nat: ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ) > ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ) > set_list_nat > ( list_nat > set_Pr1261947904930325089at_nat ) > $o ).
thf(sy_c_List_Ofolding__insort__key_001tf__a_001tf__a,type,
foldin4382019238405368997ey_a_a: ( a > a > $o ) > ( a > a > $o ) > set_a > ( a > a ) > $o ).
thf(sy_c_List_Ofolding__insort__key__axioms_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
foldin8125397140283257594at_nat: set_list_nat > ( list_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_Ogen__length_001t__List__Olist_It__Nat__Onat_J,type,
gen_length_list_nat: nat > list_list_nat > nat ).
thf(sy_c_List_Ogen__length_001t__Nat__Onat,type,
gen_length_nat: nat > list_nat > nat ).
thf(sy_c_List_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_Osorted__key__list__of__set_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
sorted56056757137290820st_nat: ( list_nat > list_nat > $o ) > ( list_nat > list_nat ) > set_list_nat > list_list_nat ).
thf(sy_c_List_Olinorder_Osorted__key__list__of__set_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
sorted4005134659417452724st_nat: ( nat > nat > $o ) > ( list_nat > nat ) > set_list_nat > list_list_nat ).
thf(sy_c_List_Olinorder_Osorted__key__list__of__set_001t__Nat__Onat_001tf__a,type,
sorted2884982002246595626_nat_a: ( nat > nat > $o ) > ( a > nat ) > set_a > list_a ).
thf(sy_c_List_Olinorder_Osorted__key__list__of__set_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__List__Olist_It__Nat__Onat_J,type,
sorted2168255887472420671st_nat: ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ) > ( list_nat > set_Pr1261947904930325089at_nat ) > set_list_nat > list_list_nat ).
thf(sy_c_List_Olinorder_Osorted__key__list__of__set_001tf__a_001tf__a,type,
sorted9165639089771494146et_a_a: ( a > a > $o ) > ( a > a ) > set_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_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__Int__Oint,type,
linord2612477271533052124et_int: set_int > list_int ).
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__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
cons_s6881495754146722583at_nat: set_Pr1261947904930325089at_nat > list_s1210847774152347623at_nat > list_s1210847774152347623at_nat ).
thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
nil_int: list_int ).
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__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__List__Olist_It__Nat__Onat_J_J_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
map_li2855073862107769254st_nat: ( list_list_nat > list_list_nat ) > list_list_list_nat > list_list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_001t__List__Olist_It__Nat__Onat_J,type,
map_li1587113026813354006st_nat: ( list_list_nat > list_nat ) > list_list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_001t__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
map_li4356451683108985649at_nat: ( list_list_nat > list_s1210847774152347623at_nat ) > list_list_list_nat > list_l3822697302700470509at_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
map_li2355978560338012748st_nat: ( list_list_nat > set_list_nat ) > list_list_list_nat > list_set_list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__Int__Oint,type,
map_list_nat_int: ( list_nat > int ) > list_list_nat > list_int ).
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__List__Olist_Itf__a_J_001t__List__Olist_It__Nat__Onat_J,type,
map_list_a_list_nat: ( list_a > list_nat ) > list_list_a > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
map_list_a_list_a: ( list_a > list_a ) > list_list_a > list_list_a ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
map_list_a_set_a: ( list_a > set_a ) > list_list_a > list_set_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_001t__Nat__Onat_001tf__b,type,
map_nat_b: ( nat > b ) > list_nat > list_b ).
thf(sy_c_List_Olist_Omap_001tf__a_001t__Int__Oint,type,
map_a_int: ( a > int ) > list_a > list_int ).
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_Omap_001tf__a_001tf__b,type,
map_a_b: ( a > b ) > list_a > list_b ).
thf(sy_c_List_Olist_Omap_001tf__b_001t__Nat__Onat,type,
map_b_nat: ( b > nat ) > list_b > list_nat ).
thf(sy_c_List_Olist_Omap_001tf__b_001tf__a,type,
map_b_a: ( b > a ) > list_b > list_a ).
thf(sy_c_List_Olist_Omap_001tf__b_001tf__b,type,
map_b_b: ( b > b ) > list_b > list_b ).
thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
set_int2: list_int > set_int ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_list_nat2: list_list_list_nat > set_list_list_nat ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
set_list_nat2: list_list_nat > set_list_nat ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_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_Olist_Oset_001tf__b,type,
set_b2: list_b > set_b ).
thf(sy_c_List_Olist__ex1_001t__List__Olist_It__Nat__Onat_J,type,
list_ex1_list_nat: ( list_nat > $o ) > list_list_nat > $o ).
thf(sy_c_List_Olist__ex1_001t__Nat__Onat,type,
list_ex1_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__ex1_001tf__a,type,
list_ex1_a: ( a > $o ) > list_a > $o ).
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__Nat__Onat,type,
map_ta354985537753071680at_nat: ( list_nat > nat ) > list_list_nat > 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_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_Omap__tailrec__rev_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
map_ta5898645294048341815st_nat: ( list_nat > list_nat ) > list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Omap__tailrec__rev_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
map_ta3159490279798172583at_nat: ( list_nat > nat ) > list_list_nat > list_nat > list_nat ).
thf(sy_c_List_Omap__tailrec__rev_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
map_ta920533232635887399st_nat: ( nat > list_nat ) > list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Omap__tailrec__rev_001t__Nat__Onat_001t__Nat__Onat,type,
map_ta7164188454487880599at_nat: ( nat > nat ) > list_nat > list_nat > list_nat ).
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_On__lists_001t__List__Olist_It__Nat__Onat_J,type,
n_lists_list_nat: nat > list_list_nat > list_list_list_nat ).
thf(sy_c_List_On__lists_001t__Nat__Onat,type,
n_lists_nat: nat > list_nat > list_list_nat ).
thf(sy_c_List_Onth_001t__Int__Oint,type,
nth_int: list_int > nat > int ).
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_Onth_001tf__b,type,
nth_b: list_b > nat > b ).
thf(sy_c_List_Oproduct__lists_001t__Nat__Onat,type,
product_lists_nat: list_list_nat > list_list_nat ).
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_List_Orev_001t__Int__Oint,type,
rev_int: list_int > list_int ).
thf(sy_c_List_Orev_001t__List__Olist_It__Nat__Onat_J,type,
rev_list_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Orev_001t__Nat__Onat,type,
rev_nat: list_nat > list_nat ).
thf(sy_c_List_Orev_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
rev_se8765390391748778439at_nat: list_s1210847774152347623at_nat > list_s1210847774152347623at_nat ).
thf(sy_c_List_Orev_001tf__a,type,
rev_a: list_a > list_a ).
thf(sy_c_List_Orotate1_001t__List__Olist_It__Nat__Onat_J,type,
rotate1_list_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Orotate1_001t__Nat__Onat,type,
rotate1_nat: list_nat > list_nat ).
thf(sy_c_List_Orotate1_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
rotate4238613965387346100at_nat: list_s1210847774152347623at_nat > list_s1210847774152347623at_nat ).
thf(sy_c_List_Orotate1_001tf__a,type,
rotate1_a: list_a > list_a ).
thf(sy_c_List_Orotate_001t__List__Olist_It__Nat__Onat_J,type,
rotate_list_nat: nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Orotate_001t__Nat__Onat,type,
rotate_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Orotate_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
rotate1033626827900196251at_nat: nat > list_s1210847774152347623at_nat > list_s1210847774152347623at_nat ).
thf(sy_c_List_Orotate_001tf__a,type,
rotate_a: nat > list_a > list_a ).
thf(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
sorted_wrt_int: ( int > int > $o ) > list_int > $o ).
thf(sy_c_List_Osorted__wrt_001t__List__Olist_It__Nat__Onat_J,type,
sorted_wrt_list_nat: ( list_nat > list_nat > $o ) > list_list_nat > $o ).
thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
thf(sy_c_List_Osorted__wrt_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
sorted7895944382797763278at_nat: ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ) > list_s1210847774152347623at_nat > $o ).
thf(sy_c_List_Osorted__wrt_001tf__a,type,
sorted_wrt_a: ( a > a > $o ) > list_a > $o ).
thf(sy_c_List_Osubseqs_001t__List__Olist_It__Nat__Onat_J,type,
subseqs_list_nat: list_list_nat > list_list_list_nat ).
thf(sy_c_List_Osubseqs_001t__Nat__Onat,type,
subseqs_nat: list_nat > list_list_nat ).
thf(sy_c_List_Osubseqs_001tf__a,type,
subseqs_a: list_a > list_list_a ).
thf(sy_c_List_Otranspose_001t__List__Olist_It__Nat__Onat_J,type,
transpose_list_nat: list_list_list_nat > list_list_list_nat ).
thf(sy_c_List_Otranspose_001t__Nat__Onat,type,
transpose_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Otranspose_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
transp4399329437217503729at_nat: list_l3822697302700470509at_nat > list_l3822697302700470509at_nat ).
thf(sy_c_List_Otranspose_001tf__a,type,
transpose_a: list_list_a > list_list_a ).
thf(sy_c_List_Ounion_001t__List__Olist_It__Nat__Onat_J,type,
union_list_nat: list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Ounion_001tf__a,type,
union_a: list_a > list_a > list_a ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
size_size_list_int: list_int > 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__List__Olist_Itf__b_J,type,
size_size_list_b: list_b > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
neg_nu3811975205180677377ec_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
ord_le1190675801316882794st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
top_top_list_nat_o: list_nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_Itf__a_M_Eo_J,type,
top_top_a_o: a > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
top_to8617644770344506070st_nat: set_list_list_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
top_top_set_list_nat: set_list_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
top_top_set_list_a: set_list_a ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
top_top_set_a: set_a ).
thf(sy_c_Relation_Otransp__on_001t__Nat__Onat,type,
transp_on_nat: set_nat > ( nat > nat > $o ) > $o ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
modulo_modulo_int: int > int > int ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
modulo_modulo_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_001tf__a,type,
collect_a: ( a > $o ) > 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_Oremove_001t__List__Olist_It__Nat__Onat_J,type,
remove_list_nat: list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Set_Oremove_001tf__a,type,
remove_a: a > set_a > set_a ).
thf(sy_c_Stirling_OStirling,type,
stirling: nat > nat > nat ).
thf(sy_c_Stirling_Ostirling,type,
stirling2: nat > nat > 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_member_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
member_list_list_nat: list_list_nat > set_list_list_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_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_thesis____,type,
thesis: $o ).
thf(sy_v_x,type,
x: list_a ).
thf(sy_v_y,type,
y: list_b ).
% Relevant facts (1272)
thf(fact_0_map__list__to__rgf,axiom,
! [X: list_a] :
? [F: a > nat] :
( ( inj_on_a_nat @ F @ ( set_a2 @ X ) )
& ( equiva3371634703666331078on_rgf @ ( map_a_nat @ F @ X ) ) ) ).
% map_list_to_rgf
thf(fact_1_map__list__to__rgf,axiom,
! [X: list_list_nat] :
? [F: list_nat > nat] :
( ( inj_on_list_nat_nat @ F @ ( set_list_nat2 @ X ) )
& ( equiva3371634703666331078on_rgf @ ( map_list_nat_nat @ F @ X ) ) ) ).
% map_list_to_rgf
thf(fact_2_map__eq__conv,axiom,
! [F2: a > a,Xs: list_a,G: a > a] :
( ( ( map_a_a @ F2 @ Xs )
= ( map_a_a @ G @ Xs ) )
= ( ! [X2: a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( ( F2 @ X2 )
= ( G @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_3_map__eq__conv,axiom,
! [F2: list_nat > nat,Xs: list_list_nat,G: list_nat > nat] :
( ( ( map_list_nat_nat @ F2 @ Xs )
= ( map_list_nat_nat @ G @ Xs ) )
= ( ! [X2: list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ( F2 @ X2 )
= ( G @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_4_map__eq__conv,axiom,
! [F2: list_nat > list_nat,Xs: list_list_nat,G: list_nat > list_nat] :
( ( ( map_li7225945977422193158st_nat @ F2 @ Xs )
= ( map_li7225945977422193158st_nat @ G @ Xs ) )
= ( ! [X2: list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ( F2 @ X2 )
= ( G @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_5_map__eq__conv,axiom,
! [F2: a > nat,Xs: list_a,G: a > nat] :
( ( ( map_a_nat @ F2 @ Xs )
= ( map_a_nat @ G @ Xs ) )
= ( ! [X2: a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( ( F2 @ X2 )
= ( G @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_6_map__eq__conv,axiom,
! [F2: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat,G: list_nat > set_Pr1261947904930325089at_nat] :
( ( ( map_li6003994582982014139at_nat @ F2 @ Xs )
= ( map_li6003994582982014139at_nat @ G @ Xs ) )
= ( ! [X2: list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ( F2 @ X2 )
= ( G @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_7_assms,axiom,
( ( equiva2867628904822520638l_of_a @ x )
= ( equiva2867628904822520639l_of_b @ y ) ) ).
% assms
thf(fact_8_list_Omap__cong,axiom,
! [X: list_a,Ya: list_a,F2: a > a,G: a > a] :
( ( X = Ya )
=> ( ! [Z: a] :
( ( member_a2 @ Z @ ( set_a2 @ Ya ) )
=> ( ( F2 @ Z )
= ( G @ Z ) ) )
=> ( ( map_a_a @ F2 @ X )
= ( map_a_a @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_9_list_Omap__cong,axiom,
! [X: list_list_nat,Ya: list_list_nat,F2: list_nat > nat,G: list_nat > nat] :
( ( X = Ya )
=> ( ! [Z: list_nat] :
( ( member_list_nat2 @ Z @ ( set_list_nat2 @ Ya ) )
=> ( ( F2 @ Z )
= ( G @ Z ) ) )
=> ( ( map_list_nat_nat @ F2 @ X )
= ( map_list_nat_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_10_list_Omap__cong,axiom,
! [X: list_list_nat,Ya: list_list_nat,F2: list_nat > list_nat,G: list_nat > list_nat] :
( ( X = Ya )
=> ( ! [Z: list_nat] :
( ( member_list_nat2 @ Z @ ( set_list_nat2 @ Ya ) )
=> ( ( F2 @ Z )
= ( G @ Z ) ) )
=> ( ( map_li7225945977422193158st_nat @ F2 @ X )
= ( map_li7225945977422193158st_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_11_list_Omap__cong,axiom,
! [X: list_a,Ya: list_a,F2: a > nat,G: a > nat] :
( ( X = Ya )
=> ( ! [Z: a] :
( ( member_a2 @ Z @ ( set_a2 @ Ya ) )
=> ( ( F2 @ Z )
= ( G @ Z ) ) )
=> ( ( map_a_nat @ F2 @ X )
= ( map_a_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_12_list_Omap__cong,axiom,
! [X: list_list_nat,Ya: list_list_nat,F2: list_nat > set_Pr1261947904930325089at_nat,G: list_nat > set_Pr1261947904930325089at_nat] :
( ( X = Ya )
=> ( ! [Z: list_nat] :
( ( member_list_nat2 @ Z @ ( set_list_nat2 @ Ya ) )
=> ( ( F2 @ Z )
= ( G @ Z ) ) )
=> ( ( map_li6003994582982014139at_nat @ F2 @ X )
= ( map_li6003994582982014139at_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_13_list_Omap__cong0,axiom,
! [X: list_a,F2: a > a,G: a > a] :
( ! [Z: a] :
( ( member_a2 @ Z @ ( set_a2 @ X ) )
=> ( ( F2 @ Z )
= ( G @ Z ) ) )
=> ( ( map_a_a @ F2 @ X )
= ( map_a_a @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_14_list_Omap__cong0,axiom,
! [X: list_list_nat,F2: list_nat > nat,G: list_nat > nat] :
( ! [Z: list_nat] :
( ( member_list_nat2 @ Z @ ( set_list_nat2 @ X ) )
=> ( ( F2 @ Z )
= ( G @ Z ) ) )
=> ( ( map_list_nat_nat @ F2 @ X )
= ( map_list_nat_nat @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_15_list_Omap__cong0,axiom,
! [X: list_list_nat,F2: list_nat > list_nat,G: list_nat > list_nat] :
( ! [Z: list_nat] :
( ( member_list_nat2 @ Z @ ( set_list_nat2 @ X ) )
=> ( ( F2 @ Z )
= ( G @ Z ) ) )
=> ( ( map_li7225945977422193158st_nat @ F2 @ X )
= ( map_li7225945977422193158st_nat @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_16_list_Omap__cong0,axiom,
! [X: list_a,F2: a > nat,G: a > nat] :
( ! [Z: a] :
( ( member_a2 @ Z @ ( set_a2 @ X ) )
=> ( ( F2 @ Z )
= ( G @ Z ) ) )
=> ( ( map_a_nat @ F2 @ X )
= ( map_a_nat @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_17_list_Omap__cong0,axiom,
! [X: list_list_nat,F2: list_nat > set_Pr1261947904930325089at_nat,G: list_nat > set_Pr1261947904930325089at_nat] :
( ! [Z: list_nat] :
( ( member_list_nat2 @ Z @ ( set_list_nat2 @ X ) )
=> ( ( F2 @ Z )
= ( G @ Z ) ) )
=> ( ( map_li6003994582982014139at_nat @ F2 @ X )
= ( map_li6003994582982014139at_nat @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_18_list_Oinj__map__strong,axiom,
! [X: list_a,Xa: list_a,F2: a > a,Fa: a > a] :
( ! [Z: a,Za: a] :
( ( member_a2 @ Z @ ( set_a2 @ X ) )
=> ( ( member_a2 @ Za @ ( set_a2 @ Xa ) )
=> ( ( ( F2 @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_a_a @ F2 @ X )
= ( map_a_a @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_19_list_Oinj__map__strong,axiom,
! [X: list_list_nat,Xa: list_list_nat,F2: list_nat > nat,Fa: list_nat > nat] :
( ! [Z: list_nat,Za: list_nat] :
( ( member_list_nat2 @ Z @ ( set_list_nat2 @ X ) )
=> ( ( member_list_nat2 @ Za @ ( set_list_nat2 @ Xa ) )
=> ( ( ( F2 @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_list_nat_nat @ F2 @ X )
= ( map_list_nat_nat @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_20_list_Oinj__map__strong,axiom,
! [X: list_list_nat,Xa: list_list_nat,F2: list_nat > list_nat,Fa: list_nat > list_nat] :
( ! [Z: list_nat,Za: list_nat] :
( ( member_list_nat2 @ Z @ ( set_list_nat2 @ X ) )
=> ( ( member_list_nat2 @ Za @ ( set_list_nat2 @ Xa ) )
=> ( ( ( F2 @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_li7225945977422193158st_nat @ F2 @ X )
= ( map_li7225945977422193158st_nat @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_21_list_Oinj__map__strong,axiom,
! [X: list_a,Xa: list_a,F2: a > nat,Fa: a > nat] :
( ! [Z: a,Za: a] :
( ( member_a2 @ Z @ ( set_a2 @ X ) )
=> ( ( member_a2 @ Za @ ( set_a2 @ Xa ) )
=> ( ( ( F2 @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_a_nat @ F2 @ X )
= ( map_a_nat @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_22_list_Oinj__map__strong,axiom,
! [X: list_list_nat,Xa: list_list_nat,F2: list_nat > set_Pr1261947904930325089at_nat,Fa: list_nat > set_Pr1261947904930325089at_nat] :
( ! [Z: list_nat,Za: list_nat] :
( ( member_list_nat2 @ Z @ ( set_list_nat2 @ X ) )
=> ( ( member_list_nat2 @ Za @ ( set_list_nat2 @ Xa ) )
=> ( ( ( F2 @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_li6003994582982014139at_nat @ F2 @ X )
= ( map_li6003994582982014139at_nat @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_23_list_Omap__ident__strong,axiom,
! [T: list_a,F2: a > a] :
( ! [Z: a] :
( ( member_a2 @ Z @ ( set_a2 @ T ) )
=> ( ( F2 @ Z )
= Z ) )
=> ( ( map_a_a @ F2 @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_24_list_Omap__ident__strong,axiom,
! [T: list_list_nat,F2: list_nat > list_nat] :
( ! [Z: list_nat] :
( ( member_list_nat2 @ Z @ ( set_list_nat2 @ T ) )
=> ( ( F2 @ Z )
= Z ) )
=> ( ( map_li7225945977422193158st_nat @ F2 @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_25_map__ext,axiom,
! [Xs: list_a,F2: a > a,G: a > a] :
( ! [X3: a] :
( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_a_a @ F2 @ Xs )
= ( map_a_a @ G @ Xs ) ) ) ).
% map_ext
thf(fact_26_map__ext,axiom,
! [Xs: list_list_nat,F2: list_nat > nat,G: list_nat > nat] :
( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_list_nat_nat @ F2 @ Xs )
= ( map_list_nat_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_27_map__ext,axiom,
! [Xs: list_list_nat,F2: list_nat > list_nat,G: list_nat > list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_li7225945977422193158st_nat @ F2 @ Xs )
= ( map_li7225945977422193158st_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_28_map__ext,axiom,
! [Xs: list_a,F2: a > nat,G: a > nat] :
( ! [X3: a] :
( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_a_nat @ F2 @ Xs )
= ( map_a_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_29_map__ext,axiom,
! [Xs: list_list_nat,F2: list_nat > set_Pr1261947904930325089at_nat,G: list_nat > set_Pr1261947904930325089at_nat] :
( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_li6003994582982014139at_nat @ F2 @ Xs )
= ( map_li6003994582982014139at_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_30_map__idI,axiom,
! [Xs: list_a,F2: a > a] :
( ! [X3: a] :
( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
=> ( ( F2 @ X3 )
= X3 ) )
=> ( ( map_a_a @ F2 @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_31_map__idI,axiom,
! [Xs: list_list_nat,F2: list_nat > list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( ( F2 @ X3 )
= X3 ) )
=> ( ( map_li7225945977422193158st_nat @ F2 @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_32_map__cong,axiom,
! [Xs: list_a,Ys: list_a,F2: a > a,G: a > a] :
( ( Xs = Ys )
=> ( ! [X3: a] :
( ( member_a2 @ X3 @ ( set_a2 @ Ys ) )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_a_a @ F2 @ Xs )
= ( map_a_a @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_33_map__cong,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,F2: list_nat > nat,G: list_nat > nat] :
( ( Xs = Ys )
=> ( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Ys ) )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_list_nat_nat @ F2 @ Xs )
= ( map_list_nat_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_34_map__cong,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,F2: list_nat > list_nat,G: list_nat > list_nat] :
( ( Xs = Ys )
=> ( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Ys ) )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_li7225945977422193158st_nat @ F2 @ Xs )
= ( map_li7225945977422193158st_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_35_map__cong,axiom,
! [Xs: list_a,Ys: list_a,F2: a > nat,G: a > nat] :
( ( Xs = Ys )
=> ( ! [X3: a] :
( ( member_a2 @ X3 @ ( set_a2 @ Ys ) )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_a_nat @ F2 @ Xs )
= ( map_a_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_36_map__cong,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,F2: list_nat > set_Pr1261947904930325089at_nat,G: list_nat > set_Pr1261947904930325089at_nat] :
( ( Xs = Ys )
=> ( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Ys ) )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_li6003994582982014139at_nat @ F2 @ Xs )
= ( map_li6003994582982014139at_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_37_ex__map__conv,axiom,
! [Ys: list_nat,F2: a > nat] :
( ( ? [Xs2: list_a] :
( Ys
= ( map_a_nat @ F2 @ Xs2 ) ) )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Ys ) )
=> ? [Y: a] :
( X2
= ( F2 @ Y ) ) ) ) ) ).
% ex_map_conv
thf(fact_38_ex__map__conv,axiom,
! [Ys: list_s1210847774152347623at_nat,F2: list_nat > set_Pr1261947904930325089at_nat] :
( ( ? [Xs2: list_list_nat] :
( Ys
= ( map_li6003994582982014139at_nat @ F2 @ Xs2 ) ) )
= ( ! [X2: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X2 @ ( set_se5049602875457034614at_nat @ Ys ) )
=> ? [Y: list_nat] :
( X2
= ( F2 @ Y ) ) ) ) ) ).
% ex_map_conv
thf(fact_39_ex__map__conv,axiom,
! [Ys: list_nat,F2: list_nat > nat] :
( ( ? [Xs2: list_list_nat] :
( Ys
= ( map_list_nat_nat @ F2 @ Xs2 ) ) )
= ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Ys ) )
=> ? [Y: list_nat] :
( X2
= ( F2 @ Y ) ) ) ) ) ).
% ex_map_conv
thf(fact_40_ex__map__conv,axiom,
! [Ys: list_a,F2: a > a] :
( ( ? [Xs2: list_a] :
( Ys
= ( map_a_a @ F2 @ Xs2 ) ) )
= ( ! [X2: a] :
( ( member_a2 @ X2 @ ( set_a2 @ Ys ) )
=> ? [Y: a] :
( X2
= ( F2 @ Y ) ) ) ) ) ).
% ex_map_conv
thf(fact_41_ex__map__conv,axiom,
! [Ys: list_list_nat,F2: list_nat > list_nat] :
( ( ? [Xs2: list_list_nat] :
( Ys
= ( map_li7225945977422193158st_nat @ F2 @ Xs2 ) ) )
= ( ! [X2: list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Ys ) )
=> ? [Y: list_nat] :
( X2
= ( F2 @ Y ) ) ) ) ) ).
% ex_map_conv
thf(fact_42_kernel__of__under__inj__map,axiom,
! [F2: a > a,X: list_a] :
( ( inj_on_a_a @ F2 @ ( set_a2 @ X ) )
=> ( ( equiva2867628904822520638l_of_a @ X )
= ( equiva2867628904822520638l_of_a @ ( map_a_a @ F2 @ X ) ) ) ) ).
% kernel_of_under_inj_map
thf(fact_43_kernel__of__under__inj__map,axiom,
! [F2: a > b,X: list_a] :
( ( inj_on_a_b @ F2 @ ( set_a2 @ X ) )
=> ( ( equiva2867628904822520638l_of_a @ X )
= ( equiva2867628904822520639l_of_b @ ( map_a_b @ F2 @ X ) ) ) ) ).
% kernel_of_under_inj_map
thf(fact_44_kernel__of__under__inj__map,axiom,
! [F2: a > nat,X: list_a] :
( ( inj_on_a_nat @ F2 @ ( set_a2 @ X ) )
=> ( ( equiva2867628904822520638l_of_a @ X )
= ( equiva2048684438135499664of_nat @ ( map_a_nat @ F2 @ X ) ) ) ) ).
% kernel_of_under_inj_map
thf(fact_45_kernel__of__under__inj__map,axiom,
! [F2: b > a,X: list_b] :
( ( inj_on_b_a @ F2 @ ( set_b2 @ X ) )
=> ( ( equiva2867628904822520639l_of_b @ X )
= ( equiva2867628904822520638l_of_a @ ( map_b_a @ F2 @ X ) ) ) ) ).
% kernel_of_under_inj_map
thf(fact_46_kernel__of__under__inj__map,axiom,
! [F2: b > b,X: list_b] :
( ( inj_on_b_b @ F2 @ ( set_b2 @ X ) )
=> ( ( equiva2867628904822520639l_of_b @ X )
= ( equiva2867628904822520639l_of_b @ ( map_b_b @ F2 @ X ) ) ) ) ).
% kernel_of_under_inj_map
thf(fact_47_kernel__of__under__inj__map,axiom,
! [F2: b > nat,X: list_b] :
( ( inj_on_b_nat @ F2 @ ( set_b2 @ X ) )
=> ( ( equiva2867628904822520639l_of_b @ X )
= ( equiva2048684438135499664of_nat @ ( map_b_nat @ F2 @ X ) ) ) ) ).
% kernel_of_under_inj_map
thf(fact_48_kernel__of__under__inj__map,axiom,
! [F2: nat > a,X: list_nat] :
( ( inj_on_nat_a @ F2 @ ( set_nat2 @ X ) )
=> ( ( equiva2048684438135499664of_nat @ X )
= ( equiva2867628904822520638l_of_a @ ( map_nat_a @ F2 @ X ) ) ) ) ).
% kernel_of_under_inj_map
thf(fact_49_kernel__of__under__inj__map,axiom,
! [F2: nat > b,X: list_nat] :
( ( inj_on_nat_b @ F2 @ ( set_nat2 @ X ) )
=> ( ( equiva2048684438135499664of_nat @ X )
= ( equiva2867628904822520639l_of_b @ ( map_nat_b @ F2 @ X ) ) ) ) ).
% kernel_of_under_inj_map
thf(fact_50_kernel__of__under__inj__map,axiom,
! [F2: nat > nat,X: list_nat] :
( ( inj_on_nat_nat @ F2 @ ( set_nat2 @ X ) )
=> ( ( equiva2048684438135499664of_nat @ X )
= ( equiva2048684438135499664of_nat @ ( map_nat_nat @ F2 @ X ) ) ) ) ).
% kernel_of_under_inj_map
thf(fact_51_kernel__of__under__inj__map,axiom,
! [F2: list_nat > a,X: list_list_nat] :
( ( inj_on_list_nat_a @ F2 @ ( set_list_nat2 @ X ) )
=> ( ( equiva6490762433048536736st_nat @ X )
= ( equiva2867628904822520638l_of_a @ ( map_list_nat_a @ F2 @ X ) ) ) ) ).
% kernel_of_under_inj_map
thf(fact_52_Equivalence__Relation__Enumeration_Ocount__list__inj__map,axiom,
! [F2: a > nat,X: list_a,Y2: a] :
( ( inj_on_a_nat @ F2 @ ( set_a2 @ X ) )
=> ( ( member_a2 @ Y2 @ ( set_a2 @ X ) )
=> ( ( count_list_nat @ ( map_a_nat @ F2 @ X ) @ ( F2 @ Y2 ) )
= ( count_list_a @ X @ Y2 ) ) ) ) ).
% Equivalence_Relation_Enumeration.count_list_inj_map
thf(fact_53_Equivalence__Relation__Enumeration_Ocount__list__inj__map,axiom,
! [F2: a > a,X: list_a,Y2: a] :
( ( inj_on_a_a @ F2 @ ( set_a2 @ X ) )
=> ( ( member_a2 @ Y2 @ ( set_a2 @ X ) )
=> ( ( count_list_a @ ( map_a_a @ F2 @ X ) @ ( F2 @ Y2 ) )
= ( count_list_a @ X @ Y2 ) ) ) ) ).
% Equivalence_Relation_Enumeration.count_list_inj_map
thf(fact_54_Equivalence__Relation__Enumeration_Ocount__list__inj__map,axiom,
! [F2: list_nat > set_Pr1261947904930325089at_nat,X: list_list_nat,Y2: list_nat] :
( ( inj_on7522185085906380110at_nat @ F2 @ ( set_list_nat2 @ X ) )
=> ( ( member_list_nat2 @ Y2 @ ( set_list_nat2 @ X ) )
=> ( ( count_6440129622255701469at_nat @ ( map_li6003994582982014139at_nat @ F2 @ X ) @ ( F2 @ Y2 ) )
= ( count_list_list_nat @ X @ Y2 ) ) ) ) ).
% Equivalence_Relation_Enumeration.count_list_inj_map
thf(fact_55_Equivalence__Relation__Enumeration_Ocount__list__inj__map,axiom,
! [F2: list_nat > nat,X: list_list_nat,Y2: list_nat] :
( ( inj_on_list_nat_nat @ F2 @ ( set_list_nat2 @ X ) )
=> ( ( member_list_nat2 @ Y2 @ ( set_list_nat2 @ X ) )
=> ( ( count_list_nat @ ( map_list_nat_nat @ F2 @ X ) @ ( F2 @ Y2 ) )
= ( count_list_list_nat @ X @ Y2 ) ) ) ) ).
% Equivalence_Relation_Enumeration.count_list_inj_map
thf(fact_56_Equivalence__Relation__Enumeration_Ocount__list__inj__map,axiom,
! [F2: a > list_nat,X: list_a,Y2: a] :
( ( inj_on_a_list_nat @ F2 @ ( set_a2 @ X ) )
=> ( ( member_a2 @ Y2 @ ( set_a2 @ X ) )
=> ( ( count_list_list_nat @ ( map_a_list_nat @ F2 @ X ) @ ( F2 @ Y2 ) )
= ( count_list_a @ X @ Y2 ) ) ) ) ).
% Equivalence_Relation_Enumeration.count_list_inj_map
thf(fact_57_Equivalence__Relation__Enumeration_Ocount__list__inj__map,axiom,
! [F2: list_nat > list_nat,X: list_list_nat,Y2: list_nat] :
( ( inj_on3049792774292151987st_nat @ F2 @ ( set_list_nat2 @ X ) )
=> ( ( member_list_nat2 @ Y2 @ ( set_list_nat2 @ X ) )
=> ( ( count_list_list_nat @ ( map_li7225945977422193158st_nat @ F2 @ X ) @ ( F2 @ Y2 ) )
= ( count_list_list_nat @ X @ Y2 ) ) ) ) ).
% Equivalence_Relation_Enumeration.count_list_inj_map
thf(fact_58_List_Ocount__list__inj__map,axiom,
! [F2: a > nat,Xs: list_a,X: a] :
( ( inj_on_a_nat @ F2 @ ( set_a2 @ Xs ) )
=> ( ( member_a2 @ X @ ( set_a2 @ Xs ) )
=> ( ( count_list_nat @ ( map_a_nat @ F2 @ Xs ) @ ( F2 @ X ) )
= ( count_list_a @ Xs @ X ) ) ) ) ).
% List.count_list_inj_map
thf(fact_59_List_Ocount__list__inj__map,axiom,
! [F2: a > a,Xs: list_a,X: a] :
( ( inj_on_a_a @ F2 @ ( set_a2 @ Xs ) )
=> ( ( member_a2 @ X @ ( set_a2 @ Xs ) )
=> ( ( count_list_a @ ( map_a_a @ F2 @ Xs ) @ ( F2 @ X ) )
= ( count_list_a @ Xs @ X ) ) ) ) ).
% List.count_list_inj_map
thf(fact_60_List_Ocount__list__inj__map,axiom,
! [F2: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat,X: list_nat] :
( ( inj_on7522185085906380110at_nat @ F2 @ ( set_list_nat2 @ Xs ) )
=> ( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
=> ( ( count_6440129622255701469at_nat @ ( map_li6003994582982014139at_nat @ F2 @ Xs ) @ ( F2 @ X ) )
= ( count_list_list_nat @ Xs @ X ) ) ) ) ).
% List.count_list_inj_map
thf(fact_61_List_Ocount__list__inj__map,axiom,
! [F2: list_nat > nat,Xs: list_list_nat,X: list_nat] :
( ( inj_on_list_nat_nat @ F2 @ ( set_list_nat2 @ Xs ) )
=> ( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
=> ( ( count_list_nat @ ( map_list_nat_nat @ F2 @ Xs ) @ ( F2 @ X ) )
= ( count_list_list_nat @ Xs @ X ) ) ) ) ).
% List.count_list_inj_map
thf(fact_62_List_Ocount__list__inj__map,axiom,
! [F2: a > list_nat,Xs: list_a,X: a] :
( ( inj_on_a_list_nat @ F2 @ ( set_a2 @ Xs ) )
=> ( ( member_a2 @ X @ ( set_a2 @ Xs ) )
=> ( ( count_list_list_nat @ ( map_a_list_nat @ F2 @ Xs ) @ ( F2 @ X ) )
= ( count_list_a @ Xs @ X ) ) ) ) ).
% List.count_list_inj_map
thf(fact_63_List_Ocount__list__inj__map,axiom,
! [F2: list_nat > list_nat,Xs: list_list_nat,X: list_nat] :
( ( inj_on3049792774292151987st_nat @ F2 @ ( set_list_nat2 @ Xs ) )
=> ( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
=> ( ( count_list_list_nat @ ( map_li7225945977422193158st_nat @ F2 @ Xs ) @ ( F2 @ X ) )
= ( count_list_list_nat @ Xs @ X ) ) ) ) ).
% List.count_list_inj_map
thf(fact_64_map__eq__map__tailrec,axiom,
map_a_nat = map_tailrec_a_nat ).
% map_eq_map_tailrec
thf(fact_65_map__eq__map__tailrec,axiom,
map_li6003994582982014139at_nat = map_ta8671482330076047857at_nat ).
% map_eq_map_tailrec
thf(fact_66_map__eq__map__tailrec,axiom,
map_list_nat_nat = map_ta354985537753071680at_nat ).
% map_eq_map_tailrec
thf(fact_67_map__eq__map__tailrec,axiom,
map_li7225945977422193158st_nat = map_ta9145449198693458768st_nat ).
% map_eq_map_tailrec
thf(fact_68_map__eq__map__tailrec,axiom,
map_a_a = map_tailrec_a_a ).
% map_eq_map_tailrec
thf(fact_69_distinct__adj__map__iff,axiom,
! [F2: a > nat,Xs: list_a] :
( ( inj_on_a_nat @ F2 @ ( set_a2 @ Xs ) )
=> ( ( distinct_adj_nat @ ( map_a_nat @ F2 @ Xs ) )
= ( distinct_adj_a @ Xs ) ) ) ).
% distinct_adj_map_iff
thf(fact_70_distinct__adj__map__iff,axiom,
! [F2: a > a,Xs: list_a] :
( ( inj_on_a_a @ F2 @ ( set_a2 @ Xs ) )
=> ( ( distinct_adj_a @ ( map_a_a @ F2 @ Xs ) )
= ( distinct_adj_a @ Xs ) ) ) ).
% distinct_adj_map_iff
thf(fact_71_distinct__adj__map__iff,axiom,
! [F2: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat] :
( ( inj_on7522185085906380110at_nat @ F2 @ ( set_list_nat2 @ Xs ) )
=> ( ( distin3702590604212146495at_nat @ ( map_li6003994582982014139at_nat @ F2 @ Xs ) )
= ( distin876741697294417026st_nat @ Xs ) ) ) ).
% distinct_adj_map_iff
thf(fact_72_distinct__adj__map__iff,axiom,
! [F2: list_nat > nat,Xs: list_list_nat] :
( ( inj_on_list_nat_nat @ F2 @ ( set_list_nat2 @ Xs ) )
=> ( ( distinct_adj_nat @ ( map_list_nat_nat @ F2 @ Xs ) )
= ( distin876741697294417026st_nat @ Xs ) ) ) ).
% distinct_adj_map_iff
thf(fact_73_distinct__adj__map__iff,axiom,
! [F2: list_nat > list_nat,Xs: list_list_nat] :
( ( inj_on3049792774292151987st_nat @ F2 @ ( set_list_nat2 @ Xs ) )
=> ( ( distin876741697294417026st_nat @ ( map_li7225945977422193158st_nat @ F2 @ Xs ) )
= ( distin876741697294417026st_nat @ Xs ) ) ) ).
% distinct_adj_map_iff
thf(fact_74_distinct__adj__mapI,axiom,
! [Xs: list_a,F2: a > nat] :
( ( distinct_adj_a @ Xs )
=> ( ( inj_on_a_nat @ F2 @ ( set_a2 @ Xs ) )
=> ( distinct_adj_nat @ ( map_a_nat @ F2 @ Xs ) ) ) ) ).
% distinct_adj_mapI
thf(fact_75_distinct__adj__mapI,axiom,
! [Xs: list_a,F2: a > a] :
( ( distinct_adj_a @ Xs )
=> ( ( inj_on_a_a @ F2 @ ( set_a2 @ Xs ) )
=> ( distinct_adj_a @ ( map_a_a @ F2 @ Xs ) ) ) ) ).
% distinct_adj_mapI
thf(fact_76_distinct__adj__mapI,axiom,
! [Xs: list_list_nat,F2: list_nat > set_Pr1261947904930325089at_nat] :
( ( distin876741697294417026st_nat @ Xs )
=> ( ( inj_on7522185085906380110at_nat @ F2 @ ( set_list_nat2 @ Xs ) )
=> ( distin3702590604212146495at_nat @ ( map_li6003994582982014139at_nat @ F2 @ Xs ) ) ) ) ).
% distinct_adj_mapI
thf(fact_77_distinct__adj__mapI,axiom,
! [Xs: list_list_nat,F2: list_nat > nat] :
( ( distin876741697294417026st_nat @ Xs )
=> ( ( inj_on_list_nat_nat @ F2 @ ( set_list_nat2 @ Xs ) )
=> ( distinct_adj_nat @ ( map_list_nat_nat @ F2 @ Xs ) ) ) ) ).
% distinct_adj_mapI
thf(fact_78_distinct__adj__mapI,axiom,
! [Xs: list_list_nat,F2: list_nat > list_nat] :
( ( distin876741697294417026st_nat @ Xs )
=> ( ( inj_on3049792774292151987st_nat @ F2 @ ( set_list_nat2 @ Xs ) )
=> ( distin876741697294417026st_nat @ ( map_li7225945977422193158st_nat @ F2 @ Xs ) ) ) ) ).
% distinct_adj_mapI
thf(fact_79_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_80_folding__insort__key__axioms__def,axiom,
( foldin8125397140283257594at_nat
= ( ^ [S: set_list_nat,F3: list_nat > nat] : ( inj_on_list_nat_nat @ F3 @ S ) ) ) ).
% folding_insort_key_axioms_def
thf(fact_81_folding__insort__key__axioms_Ointro,axiom,
! [F2: a > nat,S2: set_a] :
( ( inj_on_a_nat @ F2 @ S2 )
=> ( foldin5162300008545400710_a_nat @ S2 @ F2 ) ) ).
% folding_insort_key_axioms.intro
thf(fact_82_folding__insort__key__axioms_Ointro,axiom,
! [F2: list_nat > nat,S2: set_list_nat] :
( ( inj_on_list_nat_nat @ F2 @ S2 )
=> ( foldin8125397140283257594at_nat @ S2 @ F2 ) ) ).
% folding_insort_key_axioms.intro
thf(fact_83_inj__on__inverseI,axiom,
! [A: set_list_nat,G: nat > list_nat,F2: list_nat > nat] :
( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ A )
=> ( ( G @ ( F2 @ X3 ) )
= X3 ) )
=> ( inj_on_list_nat_nat @ F2 @ A ) ) ).
% inj_on_inverseI
thf(fact_84_inj__on__inverseI,axiom,
! [A: set_a,G: nat > a,F2: a > nat] :
( ! [X3: a] :
( ( member_a2 @ X3 @ A )
=> ( ( G @ ( F2 @ X3 ) )
= X3 ) )
=> ( inj_on_a_nat @ F2 @ A ) ) ).
% inj_on_inverseI
thf(fact_85_inj__on__contraD,axiom,
! [F2: list_nat > nat,A: set_list_nat,X: list_nat,Y2: list_nat] :
( ( inj_on_list_nat_nat @ F2 @ A )
=> ( ( X != Y2 )
=> ( ( member_list_nat2 @ X @ A )
=> ( ( member_list_nat2 @ Y2 @ A )
=> ( ( F2 @ X )
!= ( F2 @ Y2 ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_86_inj__on__contraD,axiom,
! [F2: a > nat,A: set_a,X: a,Y2: a] :
( ( inj_on_a_nat @ F2 @ A )
=> ( ( X != Y2 )
=> ( ( member_a2 @ X @ A )
=> ( ( member_a2 @ Y2 @ A )
=> ( ( F2 @ X )
!= ( F2 @ Y2 ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_87_inj__on__eq__iff,axiom,
! [F2: list_nat > nat,A: set_list_nat,X: list_nat,Y2: list_nat] :
( ( inj_on_list_nat_nat @ F2 @ A )
=> ( ( member_list_nat2 @ X @ A )
=> ( ( member_list_nat2 @ Y2 @ A )
=> ( ( ( F2 @ X )
= ( F2 @ Y2 ) )
= ( X = Y2 ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_88_inj__on__eq__iff,axiom,
! [F2: a > nat,A: set_a,X: a,Y2: a] :
( ( inj_on_a_nat @ F2 @ A )
=> ( ( member_a2 @ X @ A )
=> ( ( member_a2 @ Y2 @ A )
=> ( ( ( F2 @ X )
= ( F2 @ Y2 ) )
= ( X = Y2 ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_89_inj__on__cong,axiom,
! [A: set_list_nat,F2: list_nat > nat,G: list_nat > nat] :
( ! [A2: list_nat] :
( ( member_list_nat2 @ A2 @ A )
=> ( ( F2 @ A2 )
= ( G @ A2 ) ) )
=> ( ( inj_on_list_nat_nat @ F2 @ A )
= ( inj_on_list_nat_nat @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_90_inj__on__cong,axiom,
! [A: set_a,F2: a > nat,G: a > nat] :
( ! [A2: a] :
( ( member_a2 @ A2 @ A )
=> ( ( F2 @ A2 )
= ( G @ A2 ) ) )
=> ( ( inj_on_a_nat @ F2 @ A )
= ( inj_on_a_nat @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_91_inj__on__def,axiom,
( inj_on_a_nat
= ( ^ [F3: a > nat,A3: set_a] :
! [X2: a] :
( ( member_a2 @ X2 @ A3 )
=> ! [Y: a] :
( ( member_a2 @ Y @ A3 )
=> ( ( ( F3 @ X2 )
= ( F3 @ Y ) )
=> ( X2 = Y ) ) ) ) ) ) ).
% inj_on_def
thf(fact_92_inj__on__def,axiom,
( inj_on_list_nat_nat
= ( ^ [F3: list_nat > nat,A3: set_list_nat] :
! [X2: list_nat] :
( ( member_list_nat2 @ X2 @ A3 )
=> ! [Y: list_nat] :
( ( member_list_nat2 @ Y @ A3 )
=> ( ( ( F3 @ X2 )
= ( F3 @ Y ) )
=> ( X2 = Y ) ) ) ) ) ) ).
% inj_on_def
thf(fact_93_inj__onI,axiom,
! [A: set_list_nat,F2: list_nat > nat] :
( ! [X3: list_nat,Y3: list_nat] :
( ( member_list_nat2 @ X3 @ A )
=> ( ( member_list_nat2 @ Y3 @ A )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y3 ) )
=> ( X3 = Y3 ) ) ) )
=> ( inj_on_list_nat_nat @ F2 @ A ) ) ).
% inj_onI
thf(fact_94_inj__onI,axiom,
! [A: set_a,F2: a > nat] :
( ! [X3: a,Y3: a] :
( ( member_a2 @ X3 @ A )
=> ( ( member_a2 @ Y3 @ A )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y3 ) )
=> ( X3 = Y3 ) ) ) )
=> ( inj_on_a_nat @ F2 @ A ) ) ).
% inj_onI
thf(fact_95_inj__onD,axiom,
! [F2: list_nat > nat,A: set_list_nat,X: list_nat,Y2: list_nat] :
( ( inj_on_list_nat_nat @ F2 @ A )
=> ( ( ( F2 @ X )
= ( F2 @ Y2 ) )
=> ( ( member_list_nat2 @ X @ A )
=> ( ( member_list_nat2 @ Y2 @ A )
=> ( X = Y2 ) ) ) ) ) ).
% inj_onD
thf(fact_96_inj__onD,axiom,
! [F2: a > nat,A: set_a,X: a,Y2: a] :
( ( inj_on_a_nat @ F2 @ A )
=> ( ( ( F2 @ X )
= ( F2 @ Y2 ) )
=> ( ( member_a2 @ X @ A )
=> ( ( member_a2 @ Y2 @ A )
=> ( X = Y2 ) ) ) ) ) ).
% inj_onD
thf(fact_97_distinct__adj__mapD,axiom,
! [F2: a > nat,Xs: list_a] :
( ( distinct_adj_nat @ ( map_a_nat @ F2 @ Xs ) )
=> ( distinct_adj_a @ Xs ) ) ).
% distinct_adj_mapD
thf(fact_98_distinct__adj__mapD,axiom,
! [F2: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat] :
( ( distin3702590604212146495at_nat @ ( map_li6003994582982014139at_nat @ F2 @ Xs ) )
=> ( distin876741697294417026st_nat @ Xs ) ) ).
% distinct_adj_mapD
thf(fact_99_distinct__adj__mapD,axiom,
! [F2: list_nat > nat,Xs: list_list_nat] :
( ( distinct_adj_nat @ ( map_list_nat_nat @ F2 @ Xs ) )
=> ( distin876741697294417026st_nat @ Xs ) ) ).
% distinct_adj_mapD
thf(fact_100_distinct__adj__mapD,axiom,
! [F2: list_nat > list_nat,Xs: list_list_nat] :
( ( distin876741697294417026st_nat @ ( map_li7225945977422193158st_nat @ F2 @ Xs ) )
=> ( distin876741697294417026st_nat @ Xs ) ) ).
% distinct_adj_mapD
thf(fact_101_distinct__adj__mapD,axiom,
! [F2: a > a,Xs: list_a] :
( ( distinct_adj_a @ ( map_a_a @ F2 @ Xs ) )
=> ( distinct_adj_a @ Xs ) ) ).
% distinct_adj_mapD
thf(fact_102_in__set__member,axiom,
! [X: a,Xs: list_a] :
( ( member_a2 @ X @ ( set_a2 @ Xs ) )
= ( member_a @ Xs @ X ) ) ).
% in_set_member
thf(fact_103_in__set__member,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
= ( member_list_nat @ Xs @ X ) ) ).
% in_set_member
thf(fact_104_count__notin,axiom,
! [X: a,Xs: list_a] :
( ~ ( member_a2 @ X @ ( set_a2 @ Xs ) )
=> ( ( count_list_a @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_105_count__notin,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ~ ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
=> ( ( count_list_list_nat @ Xs @ X )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_106_map__inj__on,axiom,
! [F2: a > nat,Xs: list_a,Ys: list_a] :
( ( ( map_a_nat @ F2 @ Xs )
= ( map_a_nat @ F2 @ Ys ) )
=> ( ( inj_on_a_nat @ F2 @ ( sup_sup_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) ) )
=> ( Xs = Ys ) ) ) ).
% map_inj_on
thf(fact_107_map__inj__on,axiom,
! [F2: a > a,Xs: list_a,Ys: list_a] :
( ( ( map_a_a @ F2 @ Xs )
= ( map_a_a @ F2 @ Ys ) )
=> ( ( inj_on_a_a @ F2 @ ( sup_sup_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) ) )
=> ( Xs = Ys ) ) ) ).
% map_inj_on
thf(fact_108_map__inj__on,axiom,
! [F2: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat,Ys: list_list_nat] :
( ( ( map_li6003994582982014139at_nat @ F2 @ Xs )
= ( map_li6003994582982014139at_nat @ F2 @ Ys ) )
=> ( ( inj_on7522185085906380110at_nat @ F2 @ ( sup_sup_set_list_nat @ ( set_list_nat2 @ Xs ) @ ( set_list_nat2 @ Ys ) ) )
=> ( Xs = Ys ) ) ) ).
% map_inj_on
thf(fact_109_map__inj__on,axiom,
! [F2: list_nat > nat,Xs: list_list_nat,Ys: list_list_nat] :
( ( ( map_list_nat_nat @ F2 @ Xs )
= ( map_list_nat_nat @ F2 @ Ys ) )
=> ( ( inj_on_list_nat_nat @ F2 @ ( sup_sup_set_list_nat @ ( set_list_nat2 @ Xs ) @ ( set_list_nat2 @ Ys ) ) )
=> ( Xs = Ys ) ) ) ).
% map_inj_on
thf(fact_110_map__inj__on,axiom,
! [F2: list_nat > list_nat,Xs: list_list_nat,Ys: list_list_nat] :
( ( ( map_li7225945977422193158st_nat @ F2 @ Xs )
= ( map_li7225945977422193158st_nat @ F2 @ Ys ) )
=> ( ( inj_on3049792774292151987st_nat @ F2 @ ( sup_sup_set_list_nat @ ( set_list_nat2 @ Xs ) @ ( set_list_nat2 @ Ys ) ) )
=> ( Xs = Ys ) ) ) ).
% map_inj_on
thf(fact_111_inj__on__map__eq__map,axiom,
! [F2: a > nat,Xs: list_a,Ys: list_a] :
( ( inj_on_a_nat @ F2 @ ( sup_sup_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) ) )
=> ( ( ( map_a_nat @ F2 @ Xs )
= ( map_a_nat @ F2 @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_on_map_eq_map
thf(fact_112_inj__on__map__eq__map,axiom,
! [F2: a > a,Xs: list_a,Ys: list_a] :
( ( inj_on_a_a @ F2 @ ( sup_sup_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) ) )
=> ( ( ( map_a_a @ F2 @ Xs )
= ( map_a_a @ F2 @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_on_map_eq_map
thf(fact_113_inj__on__map__eq__map,axiom,
! [F2: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat,Ys: list_list_nat] :
( ( inj_on7522185085906380110at_nat @ F2 @ ( sup_sup_set_list_nat @ ( set_list_nat2 @ Xs ) @ ( set_list_nat2 @ Ys ) ) )
=> ( ( ( map_li6003994582982014139at_nat @ F2 @ Xs )
= ( map_li6003994582982014139at_nat @ F2 @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_on_map_eq_map
thf(fact_114_inj__on__map__eq__map,axiom,
! [F2: list_nat > nat,Xs: list_list_nat,Ys: list_list_nat] :
( ( inj_on_list_nat_nat @ F2 @ ( sup_sup_set_list_nat @ ( set_list_nat2 @ Xs ) @ ( set_list_nat2 @ Ys ) ) )
=> ( ( ( map_list_nat_nat @ F2 @ Xs )
= ( map_list_nat_nat @ F2 @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_on_map_eq_map
thf(fact_115_inj__on__map__eq__map,axiom,
! [F2: list_nat > list_nat,Xs: list_list_nat,Ys: list_list_nat] :
( ( inj_on3049792774292151987st_nat @ F2 @ ( sup_sup_set_list_nat @ ( set_list_nat2 @ Xs ) @ ( set_list_nat2 @ Ys ) ) )
=> ( ( ( map_li7225945977422193158st_nat @ F2 @ Xs )
= ( map_li7225945977422193158st_nat @ F2 @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_on_map_eq_map
thf(fact_116_distinct__map,axiom,
! [F2: a > nat,Xs: list_a] :
( ( distinct_nat @ ( map_a_nat @ F2 @ Xs ) )
= ( ( distinct_a @ Xs )
& ( inj_on_a_nat @ F2 @ ( set_a2 @ Xs ) ) ) ) ).
% distinct_map
thf(fact_117_distinct__map,axiom,
! [F2: a > a,Xs: list_a] :
( ( distinct_a @ ( map_a_a @ F2 @ Xs ) )
= ( ( distinct_a @ Xs )
& ( inj_on_a_a @ F2 @ ( set_a2 @ Xs ) ) ) ) ).
% distinct_map
thf(fact_118_distinct__map,axiom,
! [F2: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat] :
( ( distin8719635941469336154at_nat @ ( map_li6003994582982014139at_nat @ F2 @ Xs ) )
= ( ( distinct_list_nat @ Xs )
& ( inj_on7522185085906380110at_nat @ F2 @ ( set_list_nat2 @ Xs ) ) ) ) ).
% distinct_map
thf(fact_119_distinct__map,axiom,
! [F2: list_nat > nat,Xs: list_list_nat] :
( ( distinct_nat @ ( map_list_nat_nat @ F2 @ Xs ) )
= ( ( distinct_list_nat @ Xs )
& ( inj_on_list_nat_nat @ F2 @ ( set_list_nat2 @ Xs ) ) ) ) ).
% distinct_map
thf(fact_120_distinct__map,axiom,
! [F2: list_nat > list_nat,Xs: list_list_nat] :
( ( distinct_list_nat @ ( map_li7225945977422193158st_nat @ F2 @ Xs ) )
= ( ( distinct_list_nat @ Xs )
& ( inj_on3049792774292151987st_nat @ F2 @ ( set_list_nat2 @ Xs ) ) ) ) ).
% distinct_map
thf(fact_121_inj__map__eq__map,axiom,
! [F2: a > nat,Xs: list_a,Ys: list_a] :
( ( inj_on_a_nat @ F2 @ top_top_set_a )
=> ( ( ( map_a_nat @ F2 @ Xs )
= ( map_a_nat @ F2 @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_map_eq_map
thf(fact_122_inj__map__eq__map,axiom,
! [F2: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat,Ys: list_list_nat] :
( ( inj_on7522185085906380110at_nat @ F2 @ top_top_set_list_nat )
=> ( ( ( map_li6003994582982014139at_nat @ F2 @ Xs )
= ( map_li6003994582982014139at_nat @ F2 @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_map_eq_map
thf(fact_123_inj__map__eq__map,axiom,
! [F2: list_nat > nat,Xs: list_list_nat,Ys: list_list_nat] :
( ( inj_on_list_nat_nat @ F2 @ top_top_set_list_nat )
=> ( ( ( map_list_nat_nat @ F2 @ Xs )
= ( map_list_nat_nat @ F2 @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_map_eq_map
thf(fact_124_inj__map__eq__map,axiom,
! [F2: list_nat > list_nat,Xs: list_list_nat,Ys: list_list_nat] :
( ( inj_on3049792774292151987st_nat @ F2 @ top_top_set_list_nat )
=> ( ( ( map_li7225945977422193158st_nat @ F2 @ Xs )
= ( map_li7225945977422193158st_nat @ F2 @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_map_eq_map
thf(fact_125_inj__map__eq__map,axiom,
! [F2: a > a,Xs: list_a,Ys: list_a] :
( ( inj_on_a_a @ F2 @ top_top_set_a )
=> ( ( ( map_a_a @ F2 @ Xs )
= ( map_a_a @ F2 @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_map_eq_map
thf(fact_126_map__fun__upd,axiom,
! [Y2: a,Xs: list_a,F2: a > nat,V: nat] :
( ~ ( member_a2 @ Y2 @ ( set_a2 @ Xs ) )
=> ( ( map_a_nat @ ( fun_upd_a_nat @ F2 @ Y2 @ V ) @ Xs )
= ( map_a_nat @ F2 @ Xs ) ) ) ).
% map_fun_upd
thf(fact_127_map__fun__upd,axiom,
! [Y2: a,Xs: list_a,F2: a > a,V: a] :
( ~ ( member_a2 @ Y2 @ ( set_a2 @ Xs ) )
=> ( ( map_a_a @ ( fun_upd_a_a @ F2 @ Y2 @ V ) @ Xs )
= ( map_a_a @ F2 @ Xs ) ) ) ).
% map_fun_upd
thf(fact_128_map__fun__upd,axiom,
! [Y2: list_nat,Xs: list_list_nat,F2: list_nat > set_Pr1261947904930325089at_nat,V: set_Pr1261947904930325089at_nat] :
( ~ ( member_list_nat2 @ Y2 @ ( set_list_nat2 @ Xs ) )
=> ( ( map_li6003994582982014139at_nat @ ( fun_up5216017410020450626at_nat @ F2 @ Y2 @ V ) @ Xs )
= ( map_li6003994582982014139at_nat @ F2 @ Xs ) ) ) ).
% map_fun_upd
thf(fact_129_map__fun__upd,axiom,
! [Y2: list_nat,Xs: list_list_nat,F2: list_nat > nat,V: nat] :
( ~ ( member_list_nat2 @ Y2 @ ( set_list_nat2 @ Xs ) )
=> ( ( map_list_nat_nat @ ( fun_upd_list_nat_nat @ F2 @ Y2 @ V ) @ Xs )
= ( map_list_nat_nat @ F2 @ Xs ) ) ) ).
% map_fun_upd
thf(fact_130_map__fun__upd,axiom,
! [Y2: list_nat,Xs: list_list_nat,F2: list_nat > list_nat,V: list_nat] :
( ~ ( member_list_nat2 @ Y2 @ ( set_list_nat2 @ Xs ) )
=> ( ( map_li7225945977422193158st_nat @ ( fun_up3599288762871520703st_nat @ F2 @ Y2 @ V ) @ Xs )
= ( map_li7225945977422193158st_nat @ F2 @ Xs ) ) ) ).
% map_fun_upd
thf(fact_131_in__set__insert,axiom,
! [X: a,Xs: list_a] :
( ( member_a2 @ X @ ( set_a2 @ Xs ) )
=> ( ( insert_a @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_132_in__set__insert,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
=> ( ( insert_list_nat @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_133_list__ex1__iff,axiom,
( list_ex1_a
= ( ^ [P: a > $o,Xs2: list_a] :
? [X2: a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs2 ) )
& ( P @ X2 )
& ! [Y: a] :
( ( ( member_a2 @ Y @ ( set_a2 @ Xs2 ) )
& ( P @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_134_list__ex1__iff,axiom,
( list_ex1_list_nat
= ( ^ [P: list_nat > $o,Xs2: list_list_nat] :
? [X2: list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs2 ) )
& ( P @ X2 )
& ! [Y: list_nat] :
( ( ( member_list_nat2 @ Y @ ( set_list_nat2 @ Xs2 ) )
& ( P @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_135_enum__rgfs__returns__rgfs,axiom,
! [X: list_nat,N: nat] :
( ( member_list_nat2 @ X @ ( set_list_nat2 @ ( equiva7426478223624825838m_rgfs @ N ) ) )
=> ( equiva3371634703666331078on_rgf @ X ) ) ).
% enum_rgfs_returns_rgfs
thf(fact_136_inj__mapI,axiom,
! [F2: a > nat] :
( ( inj_on_a_nat @ F2 @ top_top_set_a )
=> ( inj_on6731145966573583411st_nat @ ( map_a_nat @ F2 ) @ top_top_set_list_a ) ) ).
% inj_mapI
thf(fact_137_inj__mapI,axiom,
! [F2: list_nat > set_Pr1261947904930325089at_nat] :
( ( inj_on7522185085906380110at_nat @ F2 @ top_top_set_list_nat )
=> ( inj_on8270031949885980868at_nat @ ( map_li6003994582982014139at_nat @ F2 ) @ top_to8617644770344506070st_nat ) ) ).
% inj_mapI
thf(fact_138_inj__mapI,axiom,
! [F2: list_nat > nat] :
( ( inj_on_list_nat_nat @ F2 @ top_top_set_list_nat )
=> ( inj_on8869394816141173315st_nat @ ( map_list_nat_nat @ F2 ) @ top_to8617644770344506070st_nat ) ) ).
% inj_mapI
thf(fact_139_inj__mapI,axiom,
! [F2: list_nat > list_nat] :
( ( inj_on3049792774292151987st_nat @ F2 @ top_top_set_list_nat )
=> ( inj_on2300671324199612755st_nat @ ( map_li7225945977422193158st_nat @ F2 ) @ top_to8617644770344506070st_nat ) ) ).
% inj_mapI
thf(fact_140_inj__mapI,axiom,
! [F2: a > a] :
( ( inj_on_a_a @ F2 @ top_top_set_a )
=> ( inj_on_list_a_list_a @ ( map_a_a @ F2 ) @ top_top_set_list_a ) ) ).
% inj_mapI
thf(fact_141_inj__map,axiom,
! [F2: a > nat] :
( ( inj_on6731145966573583411st_nat @ ( map_a_nat @ F2 ) @ top_top_set_list_a )
= ( inj_on_a_nat @ F2 @ top_top_set_a ) ) ).
% inj_map
thf(fact_142_inj__map,axiom,
! [F2: list_nat > set_Pr1261947904930325089at_nat] :
( ( inj_on8270031949885980868at_nat @ ( map_li6003994582982014139at_nat @ F2 ) @ top_to8617644770344506070st_nat )
= ( inj_on7522185085906380110at_nat @ F2 @ top_top_set_list_nat ) ) ).
% inj_map
thf(fact_143_inj__map,axiom,
! [F2: list_nat > nat] :
( ( inj_on8869394816141173315st_nat @ ( map_list_nat_nat @ F2 ) @ top_to8617644770344506070st_nat )
= ( inj_on_list_nat_nat @ F2 @ top_top_set_list_nat ) ) ).
% inj_map
thf(fact_144_inj__map,axiom,
! [F2: list_nat > list_nat] :
( ( inj_on2300671324199612755st_nat @ ( map_li7225945977422193158st_nat @ F2 ) @ top_to8617644770344506070st_nat )
= ( inj_on3049792774292151987st_nat @ F2 @ top_top_set_list_nat ) ) ).
% inj_map
thf(fact_145_inj__map,axiom,
! [F2: a > a] :
( ( inj_on_list_a_list_a @ ( map_a_a @ F2 ) @ top_top_set_list_a )
= ( inj_on_a_a @ F2 @ top_top_set_a ) ) ).
% inj_map
thf(fact_146_mem__Collect__eq,axiom,
! [A4: list_nat,P2: list_nat > $o] :
( ( member_list_nat2 @ A4 @ ( collect_list_nat @ P2 ) )
= ( P2 @ A4 ) ) ).
% mem_Collect_eq
thf(fact_147_mem__Collect__eq,axiom,
! [A4: a,P2: a > $o] :
( ( member_a2 @ A4 @ ( collect_a @ P2 ) )
= ( P2 @ A4 ) ) ).
% mem_Collect_eq
thf(fact_148_Collect__mem__eq,axiom,
! [A: set_list_nat] :
( ( collect_list_nat
@ ^ [X2: list_nat] : ( member_list_nat2 @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_149_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a2 @ X2 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_150_folding__insort__key_Odistinct__if__distinct__map,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S2: set_a,F2: a > nat,Xs: list_a] :
( ( foldin508877545616633799_nat_a @ Less_eq @ Less @ S2 @ F2 )
=> ( ( distinct_nat @ ( map_a_nat @ F2 @ Xs ) )
=> ( distinct_a @ Xs ) ) ) ).
% folding_insort_key.distinct_if_distinct_map
thf(fact_151_folding__insort__key_Odistinct__if__distinct__map,axiom,
! [Less_eq: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,Less: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,S2: set_list_nat,F2: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat] :
( ( foldin5963648469059051996st_nat @ Less_eq @ Less @ S2 @ F2 )
=> ( ( distin8719635941469336154at_nat @ ( map_li6003994582982014139at_nat @ F2 @ Xs ) )
=> ( distinct_list_nat @ Xs ) ) ) ).
% folding_insort_key.distinct_if_distinct_map
thf(fact_152_folding__insort__key_Odistinct__if__distinct__map,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S2: set_list_nat,F2: list_nat > nat,Xs: list_list_nat] :
( ( foldin951631548397865559st_nat @ Less_eq @ Less @ S2 @ F2 )
=> ( ( distinct_nat @ ( map_list_nat_nat @ F2 @ Xs ) )
=> ( distinct_list_nat @ Xs ) ) ) ).
% folding_insort_key.distinct_if_distinct_map
thf(fact_153_folding__insort__key_Odistinct__if__distinct__map,axiom,
! [Less_eq: list_nat > list_nat > $o,Less: list_nat > list_nat > $o,S2: set_list_nat,F2: list_nat > list_nat,Xs: list_list_nat] :
( ( foldin1968479528632951399st_nat @ Less_eq @ Less @ S2 @ F2 )
=> ( ( distinct_list_nat @ ( map_li7225945977422193158st_nat @ F2 @ Xs ) )
=> ( distinct_list_nat @ Xs ) ) ) ).
% folding_insort_key.distinct_if_distinct_map
thf(fact_154_folding__insort__key_Odistinct__if__distinct__map,axiom,
! [Less_eq: a > a > $o,Less: a > a > $o,S2: set_a,F2: a > a,Xs: list_a] :
( ( foldin4382019238405368997ey_a_a @ Less_eq @ Less @ S2 @ F2 )
=> ( ( distinct_a @ ( map_a_a @ F2 @ Xs ) )
=> ( distinct_a @ Xs ) ) ) ).
% folding_insort_key.distinct_if_distinct_map
thf(fact_155_inj__mapD,axiom,
! [F2: a > nat] :
( ( inj_on6731145966573583411st_nat @ ( map_a_nat @ F2 ) @ top_top_set_list_a )
=> ( inj_on_a_nat @ F2 @ top_top_set_a ) ) ).
% inj_mapD
thf(fact_156_inj__mapD,axiom,
! [F2: list_nat > set_Pr1261947904930325089at_nat] :
( ( inj_on8270031949885980868at_nat @ ( map_li6003994582982014139at_nat @ F2 ) @ top_to8617644770344506070st_nat )
=> ( inj_on7522185085906380110at_nat @ F2 @ top_top_set_list_nat ) ) ).
% inj_mapD
thf(fact_157_inj__mapD,axiom,
! [F2: list_nat > nat] :
( ( inj_on8869394816141173315st_nat @ ( map_list_nat_nat @ F2 ) @ top_to8617644770344506070st_nat )
=> ( inj_on_list_nat_nat @ F2 @ top_top_set_list_nat ) ) ).
% inj_mapD
thf(fact_158_inj__mapD,axiom,
! [F2: list_nat > list_nat] :
( ( inj_on2300671324199612755st_nat @ ( map_li7225945977422193158st_nat @ F2 ) @ top_to8617644770344506070st_nat )
=> ( inj_on3049792774292151987st_nat @ F2 @ top_top_set_list_nat ) ) ).
% inj_mapD
thf(fact_159_inj__mapD,axiom,
! [F2: a > a] :
( ( inj_on_list_a_list_a @ ( map_a_a @ F2 ) @ top_top_set_list_a )
=> ( inj_on_a_a @ F2 @ top_top_set_a ) ) ).
% inj_mapD
thf(fact_160_inj__def,axiom,
! [F2: a > nat] :
( ( inj_on_a_nat @ F2 @ top_top_set_a )
= ( ! [X2: a,Y: a] :
( ( ( F2 @ X2 )
= ( F2 @ Y ) )
=> ( X2 = Y ) ) ) ) ).
% inj_def
thf(fact_161_inj__def,axiom,
! [F2: list_nat > nat] :
( ( inj_on_list_nat_nat @ F2 @ top_top_set_list_nat )
= ( ! [X2: list_nat,Y: list_nat] :
( ( ( F2 @ X2 )
= ( F2 @ Y ) )
=> ( X2 = Y ) ) ) ) ).
% inj_def
thf(fact_162_inj__eq,axiom,
! [F2: a > nat,X: a,Y2: a] :
( ( inj_on_a_nat @ F2 @ top_top_set_a )
=> ( ( ( F2 @ X )
= ( F2 @ Y2 ) )
= ( X = Y2 ) ) ) ).
% inj_eq
thf(fact_163_inj__eq,axiom,
! [F2: list_nat > nat,X: list_nat,Y2: list_nat] :
( ( inj_on_list_nat_nat @ F2 @ top_top_set_list_nat )
=> ( ( ( F2 @ X )
= ( F2 @ Y2 ) )
= ( X = Y2 ) ) ) ).
% inj_eq
thf(fact_164_injI,axiom,
! [F2: a > nat] :
( ! [X3: a,Y3: a] :
( ( ( F2 @ X3 )
= ( F2 @ Y3 ) )
=> ( X3 = Y3 ) )
=> ( inj_on_a_nat @ F2 @ top_top_set_a ) ) ).
% injI
thf(fact_165_injI,axiom,
! [F2: list_nat > nat] :
( ! [X3: list_nat,Y3: list_nat] :
( ( ( F2 @ X3 )
= ( F2 @ Y3 ) )
=> ( X3 = Y3 ) )
=> ( inj_on_list_nat_nat @ F2 @ top_top_set_list_nat ) ) ).
% injI
thf(fact_166_injD,axiom,
! [F2: a > nat,X: a,Y2: a] :
( ( inj_on_a_nat @ F2 @ top_top_set_a )
=> ( ( ( F2 @ X )
= ( F2 @ Y2 ) )
=> ( X = Y2 ) ) ) ).
% injD
thf(fact_167_injD,axiom,
! [F2: list_nat > nat,X: list_nat,Y2: list_nat] :
( ( inj_on_list_nat_nat @ F2 @ top_top_set_list_nat )
=> ( ( ( F2 @ X )
= ( F2 @ Y2 ) )
=> ( X = Y2 ) ) ) ).
% injD
thf(fact_168_folding__insort__key_Oinj__on,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S2: set_a,F2: a > nat] :
( ( foldin508877545616633799_nat_a @ Less_eq @ Less @ S2 @ F2 )
=> ( inj_on_a_nat @ F2 @ S2 ) ) ).
% folding_insort_key.inj_on
thf(fact_169_folding__insort__key_Oinj__on,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S2: set_list_nat,F2: list_nat > nat] :
( ( foldin951631548397865559st_nat @ Less_eq @ Less @ S2 @ F2 )
=> ( inj_on_list_nat_nat @ F2 @ S2 ) ) ).
% folding_insort_key.inj_on
thf(fact_170_map__injective,axiom,
! [F2: a > nat,Xs: list_a,Ys: list_a] :
( ( ( map_a_nat @ F2 @ Xs )
= ( map_a_nat @ F2 @ Ys ) )
=> ( ( inj_on_a_nat @ F2 @ top_top_set_a )
=> ( Xs = Ys ) ) ) ).
% map_injective
thf(fact_171_map__injective,axiom,
! [F2: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat,Ys: list_list_nat] :
( ( ( map_li6003994582982014139at_nat @ F2 @ Xs )
= ( map_li6003994582982014139at_nat @ F2 @ Ys ) )
=> ( ( inj_on7522185085906380110at_nat @ F2 @ top_top_set_list_nat )
=> ( Xs = Ys ) ) ) ).
% map_injective
thf(fact_172_map__injective,axiom,
! [F2: list_nat > nat,Xs: list_list_nat,Ys: list_list_nat] :
( ( ( map_list_nat_nat @ F2 @ Xs )
= ( map_list_nat_nat @ F2 @ Ys ) )
=> ( ( inj_on_list_nat_nat @ F2 @ top_top_set_list_nat )
=> ( Xs = Ys ) ) ) ).
% map_injective
thf(fact_173_map__injective,axiom,
! [F2: list_nat > list_nat,Xs: list_list_nat,Ys: list_list_nat] :
( ( ( map_li7225945977422193158st_nat @ F2 @ Xs )
= ( map_li7225945977422193158st_nat @ F2 @ Ys ) )
=> ( ( inj_on3049792774292151987st_nat @ F2 @ top_top_set_list_nat )
=> ( Xs = Ys ) ) ) ).
% map_injective
thf(fact_174_map__injective,axiom,
! [F2: a > a,Xs: list_a,Ys: list_a] :
( ( ( map_a_a @ F2 @ Xs )
= ( map_a_a @ F2 @ Ys ) )
=> ( ( inj_on_a_a @ F2 @ top_top_set_a )
=> ( Xs = Ys ) ) ) ).
% map_injective
thf(fact_175_count__list__0__iff,axiom,
! [Xs: list_a,X: a] :
( ( ( count_list_a @ Xs @ X )
= zero_zero_nat )
= ( ~ ( member_a2 @ X @ ( set_a2 @ Xs ) ) ) ) ).
% count_list_0_iff
thf(fact_176_count__list__0__iff,axiom,
! [Xs: list_list_nat,X: list_nat] :
( ( ( count_list_list_nat @ Xs @ X )
= zero_zero_nat )
= ( ~ ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) ) ) ) ).
% count_list_0_iff
thf(fact_177_set__union,axiom,
! [Xs: list_a,Ys: list_a] :
( ( set_a2 @ ( union_a @ Xs @ Ys ) )
= ( sup_sup_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) ) ) ).
% set_union
thf(fact_178_set__union,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( set_list_nat2 @ ( union_list_nat @ Xs @ Ys ) )
= ( sup_sup_set_list_nat @ ( set_list_nat2 @ Xs ) @ ( set_list_nat2 @ Ys ) ) ) ).
% set_union
thf(fact_179_can__select__set__list__ex1,axiom,
! [P2: a > $o,A: list_a] :
( ( can_select_a @ P2 @ ( set_a2 @ A ) )
= ( list_ex1_a @ P2 @ A ) ) ).
% can_select_set_list_ex1
thf(fact_180_can__select__set__list__ex1,axiom,
! [P2: list_nat > $o,A: list_list_nat] :
( ( can_select_list_nat @ P2 @ ( set_list_nat2 @ A ) )
= ( list_ex1_list_nat @ P2 @ A ) ) ).
% can_select_set_list_ex1
thf(fact_181_UnCI,axiom,
! [C: list_nat,B: set_list_nat,A: set_list_nat] :
( ( ~ ( member_list_nat2 @ C @ B )
=> ( member_list_nat2 @ C @ A ) )
=> ( member_list_nat2 @ C @ ( sup_sup_set_list_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_182_UnCI,axiom,
! [C: a,B: set_a,A: set_a] :
( ( ~ ( member_a2 @ C @ B )
=> ( member_a2 @ C @ A ) )
=> ( member_a2 @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnCI
thf(fact_183_Un__iff,axiom,
! [C: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ C @ ( sup_sup_set_list_nat @ A @ B ) )
= ( ( member_list_nat2 @ C @ A )
| ( member_list_nat2 @ C @ B ) ) ) ).
% Un_iff
thf(fact_184_Un__iff,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a2 @ C @ ( sup_sup_set_a @ A @ B ) )
= ( ( member_a2 @ C @ A )
| ( member_a2 @ C @ B ) ) ) ).
% Un_iff
thf(fact_185_UNIV__I,axiom,
! [X: list_nat] : ( member_list_nat2 @ X @ top_top_set_list_nat ) ).
% UNIV_I
thf(fact_186_UNIV__I,axiom,
! [X: a] : ( member_a2 @ X @ top_top_set_a ) ).
% UNIV_I
thf(fact_187_can__select__def,axiom,
( can_select_list_nat
= ( ^ [P: list_nat > $o,A3: set_list_nat] :
? [X2: list_nat] :
( ( member_list_nat2 @ X2 @ A3 )
& ( P @ X2 )
& ! [Y: list_nat] :
( ( ( member_list_nat2 @ Y @ A3 )
& ( P @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_188_can__select__def,axiom,
( can_select_a
= ( ^ [P: a > $o,A3: set_a] :
? [X2: a] :
( ( member_a2 @ X2 @ A3 )
& ( P @ X2 )
& ! [Y: a] :
( ( ( member_a2 @ Y @ A3 )
& ( P @ Y ) )
=> ( Y = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_189_UNIV__witness,axiom,
? [X3: list_nat] : ( member_list_nat2 @ X3 @ top_top_set_list_nat ) ).
% UNIV_witness
thf(fact_190_UNIV__witness,axiom,
? [X3: a] : ( member_a2 @ X3 @ top_top_set_a ) ).
% UNIV_witness
thf(fact_191_UNIV__eq__I,axiom,
! [A: set_list_nat] :
( ! [X3: list_nat] : ( member_list_nat2 @ X3 @ A )
=> ( top_top_set_list_nat = A ) ) ).
% UNIV_eq_I
thf(fact_192_UNIV__eq__I,axiom,
! [A: set_a] :
( ! [X3: a] : ( member_a2 @ X3 @ A )
=> ( top_top_set_a = A ) ) ).
% UNIV_eq_I
thf(fact_193_UnI2,axiom,
! [C: list_nat,B: set_list_nat,A: set_list_nat] :
( ( member_list_nat2 @ C @ B )
=> ( member_list_nat2 @ C @ ( sup_sup_set_list_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_194_UnI2,axiom,
! [C: a,B: set_a,A: set_a] :
( ( member_a2 @ C @ B )
=> ( member_a2 @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI2
thf(fact_195_UnI1,axiom,
! [C: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ C @ A )
=> ( member_list_nat2 @ C @ ( sup_sup_set_list_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_196_UnI1,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a2 @ C @ A )
=> ( member_a2 @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% UnI1
thf(fact_197_UnE,axiom,
! [C: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ C @ ( sup_sup_set_list_nat @ A @ B ) )
=> ( ~ ( member_list_nat2 @ C @ A )
=> ( member_list_nat2 @ C @ B ) ) ) ).
% UnE
thf(fact_198_UnE,axiom,
! [C: a,A: set_a,B: set_a] :
( ( member_a2 @ C @ ( sup_sup_set_a @ A @ B ) )
=> ( ~ ( member_a2 @ C @ A )
=> ( member_a2 @ C @ B ) ) ) ).
% UnE
thf(fact_199_iso__tuple__UNIV__I,axiom,
! [X: list_nat] : ( member_list_nat2 @ X @ top_top_set_list_nat ) ).
% iso_tuple_UNIV_I
thf(fact_200_iso__tuple__UNIV__I,axiom,
! [X: a] : ( member_a2 @ X @ top_top_set_a ) ).
% iso_tuple_UNIV_I
thf(fact_201_distinct__product__lists,axiom,
! [Xss: list_list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xss ) )
=> ( distinct_nat @ X3 ) )
=> ( distinct_list_nat @ ( product_lists_nat @ Xss ) ) ) ).
% distinct_product_lists
thf(fact_202_equiv__rels__def,axiom,
( equiva8721718519204927301v_rels
= ( ^ [N2: nat] : ( map_li6003994582982014139at_nat @ equiva2048684438135499664of_nat @ ( equiva7426478223624825838m_rgfs @ N2 ) ) ) ) ).
% equiv_rels_def
thf(fact_203_map__removeAll__inj,axiom,
! [F2: a > nat,X: a,Xs: list_a] :
( ( inj_on_a_nat @ F2 @ top_top_set_a )
=> ( ( map_a_nat @ F2 @ ( removeAll_a @ X @ Xs ) )
= ( removeAll_nat @ ( F2 @ X ) @ ( map_a_nat @ F2 @ Xs ) ) ) ) ).
% map_removeAll_inj
thf(fact_204_map__removeAll__inj,axiom,
! [F2: list_nat > set_Pr1261947904930325089at_nat,X: list_nat,Xs: list_list_nat] :
( ( inj_on7522185085906380110at_nat @ F2 @ top_top_set_list_nat )
=> ( ( map_li6003994582982014139at_nat @ F2 @ ( removeAll_list_nat @ X @ Xs ) )
= ( remove5672899571770113645at_nat @ ( F2 @ X ) @ ( map_li6003994582982014139at_nat @ F2 @ Xs ) ) ) ) ).
% map_removeAll_inj
thf(fact_205_map__removeAll__inj,axiom,
! [F2: list_nat > nat,X: list_nat,Xs: list_list_nat] :
( ( inj_on_list_nat_nat @ F2 @ top_top_set_list_nat )
=> ( ( map_list_nat_nat @ F2 @ ( removeAll_list_nat @ X @ Xs ) )
= ( removeAll_nat @ ( F2 @ X ) @ ( map_list_nat_nat @ F2 @ Xs ) ) ) ) ).
% map_removeAll_inj
thf(fact_206_map__removeAll__inj,axiom,
! [F2: list_nat > list_nat,X: list_nat,Xs: list_list_nat] :
( ( inj_on3049792774292151987st_nat @ F2 @ top_top_set_list_nat )
=> ( ( map_li7225945977422193158st_nat @ F2 @ ( removeAll_list_nat @ X @ Xs ) )
= ( removeAll_list_nat @ ( F2 @ X ) @ ( map_li7225945977422193158st_nat @ F2 @ Xs ) ) ) ) ).
% map_removeAll_inj
thf(fact_207_map__removeAll__inj,axiom,
! [F2: a > a,X: a,Xs: list_a] :
( ( inj_on_a_a @ F2 @ top_top_set_a )
=> ( ( map_a_a @ F2 @ ( removeAll_a @ X @ Xs ) )
= ( removeAll_a @ ( F2 @ X ) @ ( map_a_a @ F2 @ Xs ) ) ) ) ).
% map_removeAll_inj
thf(fact_208_subseqs__distinctD,axiom,
! [Ys: list_nat,Xs: list_nat] :
( ( member_list_nat2 @ Ys @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) )
=> ( ( distinct_nat @ Xs )
=> ( distinct_nat @ Ys ) ) ) ).
% subseqs_distinctD
thf(fact_209_enum__rgfs__len,axiom,
! [X: list_nat,N: nat] :
( ( member_list_nat2 @ X @ ( set_list_nat2 @ ( equiva7426478223624825838m_rgfs @ N ) ) )
=> ( ( size_size_list_nat @ X )
= N ) ) ).
% enum_rgfs_len
thf(fact_210_distinct__set__subseqs,axiom,
! [Xs: list_a] :
( ( distinct_a @ Xs )
=> ( distinct_set_a @ ( map_list_a_set_a @ set_a2 @ ( subseqs_a @ Xs ) ) ) ) ).
% distinct_set_subseqs
thf(fact_211_distinct__set__subseqs,axiom,
! [Xs: list_list_nat] :
( ( distinct_list_nat @ Xs )
=> ( distin4912746231890992349st_nat @ ( map_li2355978560338012748st_nat @ set_list_nat2 @ ( subseqs_list_nat @ Xs ) ) ) ) ).
% distinct_set_subseqs
thf(fact_212_length__map,axiom,
! [F2: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat] :
( ( size_s8736152011456118867at_nat @ ( map_li6003994582982014139at_nat @ F2 @ Xs ) )
= ( size_s3023201423986296836st_nat @ Xs ) ) ).
% length_map
thf(fact_213_length__map,axiom,
! [F2: list_nat > list_nat,Xs: list_list_nat] :
( ( size_s3023201423986296836st_nat @ ( map_li7225945977422193158st_nat @ F2 @ Xs ) )
= ( size_s3023201423986296836st_nat @ Xs ) ) ).
% length_map
thf(fact_214_length__map,axiom,
! [F2: a > a,Xs: list_a] :
( ( size_size_list_a @ ( map_a_a @ F2 @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_map
thf(fact_215_length__map,axiom,
! [F2: a > nat,Xs: list_a] :
( ( size_size_list_nat @ ( map_a_nat @ F2 @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_map
thf(fact_216_length__map,axiom,
! [F2: list_nat > nat,Xs: list_list_nat] :
( ( size_size_list_nat @ ( map_list_nat_nat @ F2 @ Xs ) )
= ( size_s3023201423986296836st_nat @ Xs ) ) ).
% length_map
thf(fact_217_length__map,axiom,
! [F2: nat > nat,Xs: list_nat] :
( ( size_size_list_nat @ ( map_nat_nat @ F2 @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_218_removeAll__id,axiom,
! [X: a,Xs: list_a] :
( ~ ( member_a2 @ X @ ( set_a2 @ Xs ) )
=> ( ( removeAll_a @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_219_removeAll__id,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ~ ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
=> ( ( removeAll_list_nat @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_220_set__rotate1,axiom,
! [Xs: list_a] :
( ( set_a2 @ ( rotate1_a @ Xs ) )
= ( set_a2 @ Xs ) ) ).
% set_rotate1
thf(fact_221_set__rotate1,axiom,
! [Xs: list_list_nat] :
( ( set_list_nat2 @ ( rotate1_list_nat @ Xs ) )
= ( set_list_nat2 @ Xs ) ) ).
% set_rotate1
thf(fact_222_length__rotate1,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rotate1_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rotate1
thf(fact_223_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_224_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_nat] :
( ( size_size_list_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_225_in__set__product__lists__length,axiom,
! [Xs: list_nat,Xss: list_list_nat] :
( ( member_list_nat2 @ Xs @ ( set_list_nat2 @ ( product_lists_nat @ Xss ) ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_226_map__eq__imp__length__eq,axiom,
! [F2: a > nat,Xs: list_a,G: a > nat,Ys: list_a] :
( ( ( map_a_nat @ F2 @ Xs )
= ( map_a_nat @ G @ Ys ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_227_map__eq__imp__length__eq,axiom,
! [F2: a > a,Xs: list_a,G: a > a,Ys: list_a] :
( ( ( map_a_a @ F2 @ Xs )
= ( map_a_a @ G @ Ys ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_228_map__eq__imp__length__eq,axiom,
! [F2: a > nat,Xs: list_a,G: nat > nat,Ys: list_nat] :
( ( ( map_a_nat @ F2 @ Xs )
= ( map_nat_nat @ G @ Ys ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_229_map__eq__imp__length__eq,axiom,
! [F2: a > a,Xs: list_a,G: nat > a,Ys: list_nat] :
( ( ( map_a_a @ F2 @ Xs )
= ( map_nat_a @ G @ Ys ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_230_map__eq__imp__length__eq,axiom,
! [F2: nat > nat,Xs: list_nat,G: a > nat,Ys: list_a] :
( ( ( map_nat_nat @ F2 @ Xs )
= ( map_a_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_231_map__eq__imp__length__eq,axiom,
! [F2: nat > a,Xs: list_nat,G: a > a,Ys: list_a] :
( ( ( map_nat_a @ F2 @ Xs )
= ( map_a_a @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_232_map__eq__imp__length__eq,axiom,
! [F2: a > nat,Xs: list_a,G: list_nat > nat,Ys: list_list_nat] :
( ( ( map_a_nat @ F2 @ Xs )
= ( map_list_nat_nat @ G @ Ys ) )
=> ( ( size_size_list_a @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_233_map__eq__imp__length__eq,axiom,
! [F2: list_nat > nat,Xs: list_list_nat,G: a > nat,Ys: list_a] :
( ( ( map_list_nat_nat @ F2 @ Xs )
= ( map_a_nat @ G @ Ys ) )
=> ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_234_map__eq__imp__length__eq,axiom,
! [F2: list_nat > nat,Xs: list_list_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_list_nat_nat @ F2 @ Xs )
= ( map_nat_nat @ G @ Ys ) )
=> ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_235_map__eq__imp__length__eq,axiom,
! [F2: nat > nat,Xs: list_nat,G: list_nat > nat,Ys: list_list_nat] :
( ( ( map_nat_nat @ F2 @ Xs )
= ( map_list_nat_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_236_kernel__of__eq__len,axiom,
! [X: list_a,Y2: list_a] :
( ( ( equiva2867628904822520638l_of_a @ X )
= ( equiva2867628904822520638l_of_a @ Y2 ) )
=> ( ( size_size_list_a @ X )
= ( size_size_list_a @ Y2 ) ) ) ).
% kernel_of_eq_len
thf(fact_237_kernel__of__eq__len,axiom,
! [X: list_a,Y2: list_b] :
( ( ( equiva2867628904822520638l_of_a @ X )
= ( equiva2867628904822520639l_of_b @ Y2 ) )
=> ( ( size_size_list_a @ X )
= ( size_size_list_b @ Y2 ) ) ) ).
% kernel_of_eq_len
thf(fact_238_kernel__of__eq__len,axiom,
! [X: list_a,Y2: list_nat] :
( ( ( equiva2867628904822520638l_of_a @ X )
= ( equiva2048684438135499664of_nat @ Y2 ) )
=> ( ( size_size_list_a @ X )
= ( size_size_list_nat @ Y2 ) ) ) ).
% kernel_of_eq_len
thf(fact_239_kernel__of__eq__len,axiom,
! [X: list_b,Y2: list_a] :
( ( ( equiva2867628904822520639l_of_b @ X )
= ( equiva2867628904822520638l_of_a @ Y2 ) )
=> ( ( size_size_list_b @ X )
= ( size_size_list_a @ Y2 ) ) ) ).
% kernel_of_eq_len
thf(fact_240_kernel__of__eq__len,axiom,
! [X: list_b,Y2: list_b] :
( ( ( equiva2867628904822520639l_of_b @ X )
= ( equiva2867628904822520639l_of_b @ Y2 ) )
=> ( ( size_size_list_b @ X )
= ( size_size_list_b @ Y2 ) ) ) ).
% kernel_of_eq_len
thf(fact_241_kernel__of__eq__len,axiom,
! [X: list_b,Y2: list_nat] :
( ( ( equiva2867628904822520639l_of_b @ X )
= ( equiva2048684438135499664of_nat @ Y2 ) )
=> ( ( size_size_list_b @ X )
= ( size_size_list_nat @ Y2 ) ) ) ).
% kernel_of_eq_len
thf(fact_242_kernel__of__eq__len,axiom,
! [X: list_nat,Y2: list_a] :
( ( ( equiva2048684438135499664of_nat @ X )
= ( equiva2867628904822520638l_of_a @ Y2 ) )
=> ( ( size_size_list_nat @ X )
= ( size_size_list_a @ Y2 ) ) ) ).
% kernel_of_eq_len
thf(fact_243_kernel__of__eq__len,axiom,
! [X: list_nat,Y2: list_b] :
( ( ( equiva2048684438135499664of_nat @ X )
= ( equiva2867628904822520639l_of_b @ Y2 ) )
=> ( ( size_size_list_nat @ X )
= ( size_size_list_b @ Y2 ) ) ) ).
% kernel_of_eq_len
thf(fact_244_kernel__of__eq__len,axiom,
! [X: list_nat,Y2: list_nat] :
( ( ( equiva2048684438135499664of_nat @ X )
= ( equiva2048684438135499664of_nat @ Y2 ) )
=> ( ( size_size_list_nat @ X )
= ( size_size_list_nat @ Y2 ) ) ) ).
% kernel_of_eq_len
thf(fact_245_kernel__of__inj__on__rgfs__aux,axiom,
! [X: list_nat,Y2: list_nat] :
( ( ( size_size_list_nat @ X )
= ( size_size_list_nat @ Y2 ) )
=> ( ( equiva3371634703666331078on_rgf @ X )
=> ( ( equiva3371634703666331078on_rgf @ Y2 )
=> ( ( ( equiva2048684438135499664of_nat @ X )
= ( equiva2048684438135499664of_nat @ Y2 ) )
=> ( X = Y2 ) ) ) ) ) ).
% kernel_of_inj_on_rgfs_aux
thf(fact_246_rotate1__map,axiom,
! [F2: a > nat,Xs: list_a] :
( ( rotate1_nat @ ( map_a_nat @ F2 @ Xs ) )
= ( map_a_nat @ F2 @ ( rotate1_a @ Xs ) ) ) ).
% rotate1_map
thf(fact_247_rotate1__map,axiom,
! [F2: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat] :
( ( rotate4238613965387346100at_nat @ ( map_li6003994582982014139at_nat @ F2 @ Xs ) )
= ( map_li6003994582982014139at_nat @ F2 @ ( rotate1_list_nat @ Xs ) ) ) ).
% rotate1_map
thf(fact_248_rotate1__map,axiom,
! [F2: list_nat > nat,Xs: list_list_nat] :
( ( rotate1_nat @ ( map_list_nat_nat @ F2 @ Xs ) )
= ( map_list_nat_nat @ F2 @ ( rotate1_list_nat @ Xs ) ) ) ).
% rotate1_map
thf(fact_249_rotate1__map,axiom,
! [F2: list_nat > list_nat,Xs: list_list_nat] :
( ( rotate1_list_nat @ ( map_li7225945977422193158st_nat @ F2 @ Xs ) )
= ( map_li7225945977422193158st_nat @ F2 @ ( rotate1_list_nat @ Xs ) ) ) ).
% rotate1_map
thf(fact_250_rotate1__map,axiom,
! [F2: a > a,Xs: list_a] :
( ( rotate1_a @ ( map_a_a @ F2 @ Xs ) )
= ( map_a_a @ F2 @ ( rotate1_a @ Xs ) ) ) ).
% rotate1_map
thf(fact_251_subseqs__refl,axiom,
! [Xs: list_nat] : ( member_list_nat2 @ Xs @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) ) ).
% subseqs_refl
thf(fact_252_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_253_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_254_top__empty__eq,axiom,
( top_top_list_nat_o
= ( ^ [X2: list_nat] : ( member_list_nat2 @ X2 @ top_top_set_list_nat ) ) ) ).
% top_empty_eq
thf(fact_255_top__empty__eq,axiom,
( top_top_a_o
= ( ^ [X2: a] : ( member_a2 @ X2 @ top_top_set_a ) ) ) ).
% top_empty_eq
thf(fact_256_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_257_remove__code_I1_J,axiom,
! [X: a,Xs: list_a] :
( ( remove_a @ X @ ( set_a2 @ Xs ) )
= ( set_a2 @ ( removeAll_a @ X @ Xs ) ) ) ).
% remove_code(1)
thf(fact_258_remove__code_I1_J,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( remove_list_nat @ X @ ( set_list_nat2 @ Xs ) )
= ( set_list_nat2 @ ( removeAll_list_nat @ X @ Xs ) ) ) ).
% remove_code(1)
thf(fact_259_length__n__lists__elem,axiom,
! [Ys: list_nat,N: nat,Xs: list_nat] :
( ( member_list_nat2 @ Ys @ ( set_list_nat2 @ ( n_lists_nat @ N @ Xs ) ) )
=> ( ( size_size_list_nat @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_260_equiv__rels__enum,axiom,
! [X: list_nat] :
( ( equiva3371634703666331078on_rgf @ X )
=> ( ( count_list_list_nat @ ( equiva7426478223624825838m_rgfs @ ( size_size_list_nat @ X ) ) @ X )
= one_one_nat ) ) ).
% equiv_rels_enum
thf(fact_261_map__removeAll__inj__on,axiom,
! [F2: a > nat,X: a,Xs: list_a] :
( ( inj_on_a_nat @ F2 @ ( insert_a2 @ X @ ( set_a2 @ Xs ) ) )
=> ( ( map_a_nat @ F2 @ ( removeAll_a @ X @ Xs ) )
= ( removeAll_nat @ ( F2 @ X ) @ ( map_a_nat @ F2 @ Xs ) ) ) ) ).
% map_removeAll_inj_on
thf(fact_262_map__removeAll__inj__on,axiom,
! [F2: a > a,X: a,Xs: list_a] :
( ( inj_on_a_a @ F2 @ ( insert_a2 @ X @ ( set_a2 @ Xs ) ) )
=> ( ( map_a_a @ F2 @ ( removeAll_a @ X @ Xs ) )
= ( removeAll_a @ ( F2 @ X ) @ ( map_a_a @ F2 @ Xs ) ) ) ) ).
% map_removeAll_inj_on
thf(fact_263_map__removeAll__inj__on,axiom,
! [F2: list_nat > set_Pr1261947904930325089at_nat,X: list_nat,Xs: list_list_nat] :
( ( inj_on7522185085906380110at_nat @ F2 @ ( insert_list_nat2 @ X @ ( set_list_nat2 @ Xs ) ) )
=> ( ( map_li6003994582982014139at_nat @ F2 @ ( removeAll_list_nat @ X @ Xs ) )
= ( remove5672899571770113645at_nat @ ( F2 @ X ) @ ( map_li6003994582982014139at_nat @ F2 @ Xs ) ) ) ) ).
% map_removeAll_inj_on
thf(fact_264_map__removeAll__inj__on,axiom,
! [F2: list_nat > nat,X: list_nat,Xs: list_list_nat] :
( ( inj_on_list_nat_nat @ F2 @ ( insert_list_nat2 @ X @ ( set_list_nat2 @ Xs ) ) )
=> ( ( map_list_nat_nat @ F2 @ ( removeAll_list_nat @ X @ Xs ) )
= ( removeAll_nat @ ( F2 @ X ) @ ( map_list_nat_nat @ F2 @ Xs ) ) ) ) ).
% map_removeAll_inj_on
thf(fact_265_map__removeAll__inj__on,axiom,
! [F2: list_nat > list_nat,X: list_nat,Xs: list_list_nat] :
( ( inj_on3049792774292151987st_nat @ F2 @ ( insert_list_nat2 @ X @ ( set_list_nat2 @ Xs ) ) )
=> ( ( map_li7225945977422193158st_nat @ F2 @ ( removeAll_list_nat @ X @ Xs ) )
= ( removeAll_list_nat @ ( F2 @ X ) @ ( map_li7225945977422193158st_nat @ F2 @ Xs ) ) ) ) ).
% map_removeAll_inj_on
thf(fact_266_length__removeAll__less,axiom,
! [X: a,Xs: list_a] :
( ( member_a2 @ X @ ( set_a2 @ Xs ) )
=> ( ord_less_nat @ ( size_size_list_a @ ( removeAll_a @ X @ Xs ) ) @ ( size_size_list_a @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_267_length__removeAll__less,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
=> ( ord_less_nat @ ( size_s3023201423986296836st_nat @ ( removeAll_list_nat @ X @ Xs ) ) @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_268_length__removeAll__less,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ ( size_size_list_nat @ ( removeAll_nat @ X @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_269_insertCI,axiom,
! [A4: list_nat,B: set_list_nat,B2: list_nat] :
( ( ~ ( member_list_nat2 @ A4 @ B )
=> ( A4 = B2 ) )
=> ( member_list_nat2 @ A4 @ ( insert_list_nat2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_270_insertCI,axiom,
! [A4: a,B: set_a,B2: a] :
( ( ~ ( member_a2 @ A4 @ B )
=> ( A4 = B2 ) )
=> ( member_a2 @ A4 @ ( insert_a2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_271_insert__iff,axiom,
! [A4: list_nat,B2: list_nat,A: set_list_nat] :
( ( member_list_nat2 @ A4 @ ( insert_list_nat2 @ B2 @ A ) )
= ( ( A4 = B2 )
| ( member_list_nat2 @ A4 @ A ) ) ) ).
% insert_iff
thf(fact_272_insert__iff,axiom,
! [A4: a,B2: a,A: set_a] :
( ( member_a2 @ A4 @ ( insert_a2 @ B2 @ A ) )
= ( ( A4 = B2 )
| ( member_a2 @ A4 @ A ) ) ) ).
% insert_iff
thf(fact_273_member__remove,axiom,
! [X: list_nat,Y2: list_nat,A: set_list_nat] :
( ( member_list_nat2 @ X @ ( remove_list_nat @ Y2 @ A ) )
= ( ( member_list_nat2 @ X @ A )
& ( X != Y2 ) ) ) ).
% member_remove
thf(fact_274_member__remove,axiom,
! [X: a,Y2: a,A: set_a] :
( ( member_a2 @ X @ ( remove_a @ Y2 @ A ) )
= ( ( member_a2 @ X @ A )
& ( X != Y2 ) ) ) ).
% member_remove
thf(fact_275_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_276_List_Oset__insert,axiom,
! [X: a,Xs: list_a] :
( ( set_a2 @ ( insert_a @ X @ Xs ) )
= ( insert_a2 @ X @ ( set_a2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_277_List_Oset__insert,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( set_list_nat2 @ ( insert_list_nat @ X @ Xs ) )
= ( insert_list_nat2 @ X @ ( set_list_nat2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_278_lt__ex,axiom,
! [X: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).
% lt_ex
thf(fact_279_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_280_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_281_less__imp__neq,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_neq
thf(fact_282_less__imp__neq,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_neq
thf(fact_283_order_Oasym,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A4 ) ) ).
% order.asym
thf(fact_284_order_Oasym,axiom,
! [A4: int,B2: int] :
( ( ord_less_int @ A4 @ B2 )
=> ~ ( ord_less_int @ B2 @ A4 ) ) ).
% order.asym
thf(fact_285_ord__eq__less__trans,axiom,
! [A4: nat,B2: nat,C: nat] :
( ( A4 = B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A4 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_286_ord__eq__less__trans,axiom,
! [A4: int,B2: int,C: int] :
( ( A4 = B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A4 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_287_ord__less__eq__trans,axiom,
! [A4: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_nat @ A4 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_288_ord__less__eq__trans,axiom,
! [A4: int,B2: int,C: int] :
( ( ord_less_int @ A4 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_int @ A4 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_289_less__induct,axiom,
! [P2: nat > $o,A4: nat] :
( ! [X3: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X3 )
=> ( P2 @ Y4 ) )
=> ( P2 @ X3 ) )
=> ( P2 @ A4 ) ) ).
% less_induct
thf(fact_290_antisym__conv3,axiom,
! [Y2: nat,X: nat] :
( ~ ( ord_less_nat @ Y2 @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv3
thf(fact_291_antisym__conv3,axiom,
! [Y2: int,X: int] :
( ~ ( ord_less_int @ Y2 @ X )
=> ( ( ~ ( ord_less_int @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv3
thf(fact_292_linorder__cases,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_nat @ X @ Y2 )
=> ( ( X != Y2 )
=> ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_cases
thf(fact_293_linorder__cases,axiom,
! [X: int,Y2: int] :
( ~ ( ord_less_int @ X @ Y2 )
=> ( ( X != Y2 )
=> ( ord_less_int @ Y2 @ X ) ) ) ).
% linorder_cases
thf(fact_294_dual__order_Oasym,axiom,
! [B2: nat,A4: nat] :
( ( ord_less_nat @ B2 @ A4 )
=> ~ ( ord_less_nat @ A4 @ B2 ) ) ).
% dual_order.asym
thf(fact_295_dual__order_Oasym,axiom,
! [B2: int,A4: int] :
( ( ord_less_int @ B2 @ A4 )
=> ~ ( ord_less_int @ A4 @ B2 ) ) ).
% dual_order.asym
thf(fact_296_dual__order_Oirrefl,axiom,
! [A4: nat] :
~ ( ord_less_nat @ A4 @ A4 ) ).
% dual_order.irrefl
thf(fact_297_dual__order_Oirrefl,axiom,
! [A4: int] :
~ ( ord_less_int @ A4 @ A4 ) ).
% dual_order.irrefl
thf(fact_298_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X4: nat] : ( P3 @ X4 ) )
= ( ^ [P: nat > $o] :
? [N2: nat] :
( ( P @ N2 )
& ! [M: nat] :
( ( ord_less_nat @ M @ N2 )
=> ~ ( P @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_299_linorder__less__wlog,axiom,
! [P2: nat > nat > $o,A4: nat,B2: nat] :
( ! [A2: nat,B3: nat] :
( ( ord_less_nat @ A2 @ B3 )
=> ( P2 @ A2 @ B3 ) )
=> ( ! [A2: nat] : ( P2 @ A2 @ A2 )
=> ( ! [A2: nat,B3: nat] :
( ( P2 @ B3 @ A2 )
=> ( P2 @ A2 @ B3 ) )
=> ( P2 @ A4 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_300_linorder__less__wlog,axiom,
! [P2: int > int > $o,A4: int,B2: int] :
( ! [A2: int,B3: int] :
( ( ord_less_int @ A2 @ B3 )
=> ( P2 @ A2 @ B3 ) )
=> ( ! [A2: int] : ( P2 @ A2 @ A2 )
=> ( ! [A2: int,B3: int] :
( ( P2 @ B3 @ A2 )
=> ( P2 @ A2 @ B3 ) )
=> ( P2 @ A4 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_301_order_Ostrict__trans,axiom,
! [A4: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A4 @ C ) ) ) ).
% order.strict_trans
thf(fact_302_order_Ostrict__trans,axiom,
! [A4: int,B2: int,C: int] :
( ( ord_less_int @ A4 @ B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A4 @ C ) ) ) ).
% order.strict_trans
thf(fact_303_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( ( ord_less_nat @ Y2 @ X )
| ( X = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_304_not__less__iff__gr__or__eq,axiom,
! [X: int,Y2: int] :
( ( ~ ( ord_less_int @ X @ Y2 ) )
= ( ( ord_less_int @ Y2 @ X )
| ( X = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_305_dual__order_Ostrict__trans,axiom,
! [B2: nat,A4: nat,C: nat] :
( ( ord_less_nat @ B2 @ A4 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A4 ) ) ) ).
% dual_order.strict_trans
thf(fact_306_dual__order_Ostrict__trans,axiom,
! [B2: int,A4: int,C: int] :
( ( ord_less_int @ B2 @ A4 )
=> ( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ A4 ) ) ) ).
% dual_order.strict_trans
thf(fact_307_order_Ostrict__implies__not__eq,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( A4 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_308_order_Ostrict__implies__not__eq,axiom,
! [A4: int,B2: int] :
( ( ord_less_int @ A4 @ B2 )
=> ( A4 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_309_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: nat,A4: nat] :
( ( ord_less_nat @ B2 @ A4 )
=> ( A4 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_310_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: int,A4: int] :
( ( ord_less_int @ B2 @ A4 )
=> ( A4 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_311_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_312_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_313_linorder__neqE,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
=> ( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_neqE
thf(fact_314_linorder__neqE,axiom,
! [X: int,Y2: int] :
( ( X != Y2 )
=> ( ~ ( ord_less_int @ X @ Y2 )
=> ( ord_less_int @ Y2 @ X ) ) ) ).
% linorder_neqE
thf(fact_315_order__less__asym,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X ) ) ).
% order_less_asym
thf(fact_316_order__less__asym,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X ) ) ).
% order_less_asym
thf(fact_317_linorder__neq__iff,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
= ( ( ord_less_nat @ X @ Y2 )
| ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_318_linorder__neq__iff,axiom,
! [X: int,Y2: int] :
( ( X != Y2 )
= ( ( ord_less_int @ X @ Y2 )
| ( ord_less_int @ Y2 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_319_order__less__asym_H,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A4 ) ) ).
% order_less_asym'
thf(fact_320_order__less__asym_H,axiom,
! [A4: int,B2: int] :
( ( ord_less_int @ A4 @ B2 )
=> ~ ( ord_less_int @ B2 @ A4 ) ) ).
% order_less_asym'
thf(fact_321_order__less__trans,axiom,
! [X: nat,Y2: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_322_order__less__trans,axiom,
! [X: int,Y2: int,Z2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_323_ord__eq__less__subst,axiom,
! [A4: nat,F2: nat > nat,B2: nat,C: nat] :
( ( A4
= ( F2 @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A4 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_324_ord__eq__less__subst,axiom,
! [A4: int,F2: nat > int,B2: nat,C: nat] :
( ( A4
= ( F2 @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ A4 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_325_ord__eq__less__subst,axiom,
! [A4: nat,F2: int > nat,B2: int,C: int] :
( ( A4
= ( F2 @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A4 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_326_ord__eq__less__subst,axiom,
! [A4: int,F2: int > int,B2: int,C: int] :
( ( A4
= ( F2 @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ A4 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_327_ord__less__eq__subst,axiom,
! [A4: nat,B2: nat,F2: nat > nat,C: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A4 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_328_ord__less__eq__subst,axiom,
! [A4: nat,B2: nat,F2: nat > int,C: int] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ ( F2 @ A4 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_329_ord__less__eq__subst,axiom,
! [A4: int,B2: int,F2: int > nat,C: nat] :
( ( ord_less_int @ A4 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A4 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_330_ord__less__eq__subst,axiom,
! [A4: int,B2: int,F2: int > int,C: int] :
( ( ord_less_int @ A4 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ ( F2 @ A4 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_331_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_332_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_333_order__less__subst1,axiom,
! [A4: nat,F2: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A4 @ ( F2 @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_334_order__less__subst1,axiom,
! [A4: nat,F2: int > nat,B2: int,C: int] :
( ( ord_less_nat @ A4 @ ( F2 @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_335_order__less__subst1,axiom,
! [A4: int,F2: nat > int,B2: nat,C: nat] :
( ( ord_less_int @ A4 @ ( F2 @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_336_order__less__subst1,axiom,
! [A4: int,F2: int > int,B2: int,C: int] :
( ( ord_less_int @ A4 @ ( F2 @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_337_order__less__subst2,axiom,
! [A4: nat,B2: nat,F2: nat > nat,C: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ord_less_nat @ ( F2 @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_338_order__less__subst2,axiom,
! [A4: nat,B2: nat,F2: nat > int,C: int] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ord_less_int @ ( F2 @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_339_order__less__subst2,axiom,
! [A4: int,B2: int,F2: int > nat,C: nat] :
( ( ord_less_int @ A4 @ B2 )
=> ( ( ord_less_nat @ ( F2 @ B2 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_340_order__less__subst2,axiom,
! [A4: int,B2: int,F2: int > int,C: int] :
( ( ord_less_int @ A4 @ B2 )
=> ( ( ord_less_int @ ( F2 @ B2 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_341_order__less__not__sym,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X ) ) ).
% order_less_not_sym
thf(fact_342_order__less__not__sym,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X ) ) ).
% order_less_not_sym
thf(fact_343_order__less__imp__triv,axiom,
! [X: nat,Y2: nat,P2: $o] :
( ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_nat @ Y2 @ X )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_344_order__less__imp__triv,axiom,
! [X: int,Y2: int,P2: $o] :
( ( ord_less_int @ X @ Y2 )
=> ( ( ord_less_int @ Y2 @ X )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_345_linorder__less__linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
| ( X = Y2 )
| ( ord_less_nat @ Y2 @ X ) ) ).
% linorder_less_linear
thf(fact_346_linorder__less__linear,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
| ( X = Y2 )
| ( ord_less_int @ Y2 @ X ) ) ).
% linorder_less_linear
thf(fact_347_order__less__imp__not__eq,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( X != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_348_order__less__imp__not__eq,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( X != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_349_order__less__imp__not__eq2,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( Y2 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_350_order__less__imp__not__eq2,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( Y2 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_351_order__less__imp__not__less,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X ) ) ).
% order_less_imp_not_less
thf(fact_352_order__less__imp__not__less,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X ) ) ).
% order_less_imp_not_less
thf(fact_353_insertE,axiom,
! [A4: list_nat,B2: list_nat,A: set_list_nat] :
( ( member_list_nat2 @ A4 @ ( insert_list_nat2 @ B2 @ A ) )
=> ( ( A4 != B2 )
=> ( member_list_nat2 @ A4 @ A ) ) ) ).
% insertE
thf(fact_354_insertE,axiom,
! [A4: a,B2: a,A: set_a] :
( ( member_a2 @ A4 @ ( insert_a2 @ B2 @ A ) )
=> ( ( A4 != B2 )
=> ( member_a2 @ A4 @ A ) ) ) ).
% insertE
thf(fact_355_insertI1,axiom,
! [A4: list_nat,B: set_list_nat] : ( member_list_nat2 @ A4 @ ( insert_list_nat2 @ A4 @ B ) ) ).
% insertI1
thf(fact_356_insertI1,axiom,
! [A4: a,B: set_a] : ( member_a2 @ A4 @ ( insert_a2 @ A4 @ B ) ) ).
% insertI1
thf(fact_357_insertI2,axiom,
! [A4: list_nat,B: set_list_nat,B2: list_nat] :
( ( member_list_nat2 @ A4 @ B )
=> ( member_list_nat2 @ A4 @ ( insert_list_nat2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_358_insertI2,axiom,
! [A4: a,B: set_a,B2: a] :
( ( member_a2 @ A4 @ B )
=> ( member_a2 @ A4 @ ( insert_a2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_359_Set_Oset__insert,axiom,
! [X: list_nat,A: set_list_nat] :
( ( member_list_nat2 @ X @ A )
=> ~ ! [B4: set_list_nat] :
( ( A
= ( insert_list_nat2 @ X @ B4 ) )
=> ( member_list_nat2 @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_360_Set_Oset__insert,axiom,
! [X: a,A: set_a] :
( ( member_a2 @ X @ A )
=> ~ ! [B4: set_a] :
( ( A
= ( insert_a2 @ X @ B4 ) )
=> ( member_a2 @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_361_insert__ident,axiom,
! [X: list_nat,A: set_list_nat,B: set_list_nat] :
( ~ ( member_list_nat2 @ X @ A )
=> ( ~ ( member_list_nat2 @ X @ B )
=> ( ( ( insert_list_nat2 @ X @ A )
= ( insert_list_nat2 @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_362_insert__ident,axiom,
! [X: a,A: set_a,B: set_a] :
( ~ ( member_a2 @ X @ A )
=> ( ~ ( member_a2 @ X @ B )
=> ( ( ( insert_a2 @ X @ A )
= ( insert_a2 @ X @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_363_insert__absorb,axiom,
! [A4: list_nat,A: set_list_nat] :
( ( member_list_nat2 @ A4 @ A )
=> ( ( insert_list_nat2 @ A4 @ A )
= A ) ) ).
% insert_absorb
thf(fact_364_insert__absorb,axiom,
! [A4: a,A: set_a] :
( ( member_a2 @ A4 @ A )
=> ( ( insert_a2 @ A4 @ A )
= A ) ) ).
% insert_absorb
thf(fact_365_insert__eq__iff,axiom,
! [A4: list_nat,A: set_list_nat,B2: list_nat,B: set_list_nat] :
( ~ ( member_list_nat2 @ A4 @ A )
=> ( ~ ( member_list_nat2 @ B2 @ B )
=> ( ( ( insert_list_nat2 @ A4 @ A )
= ( insert_list_nat2 @ B2 @ B ) )
= ( ( ( A4 = B2 )
=> ( A = B ) )
& ( ( A4 != B2 )
=> ? [C2: set_list_nat] :
( ( A
= ( insert_list_nat2 @ B2 @ C2 ) )
& ~ ( member_list_nat2 @ B2 @ C2 )
& ( B
= ( insert_list_nat2 @ A4 @ C2 ) )
& ~ ( member_list_nat2 @ A4 @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_366_insert__eq__iff,axiom,
! [A4: a,A: set_a,B2: a,B: set_a] :
( ~ ( member_a2 @ A4 @ A )
=> ( ~ ( member_a2 @ B2 @ B )
=> ( ( ( insert_a2 @ A4 @ A )
= ( insert_a2 @ B2 @ B ) )
= ( ( ( A4 = B2 )
=> ( A = B ) )
& ( ( A4 != B2 )
=> ? [C2: set_a] :
( ( A
= ( insert_a2 @ B2 @ C2 ) )
& ~ ( member_a2 @ B2 @ C2 )
& ( B
= ( insert_a2 @ A4 @ C2 ) )
& ~ ( member_a2 @ A4 @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_367_mk__disjoint__insert,axiom,
! [A4: list_nat,A: set_list_nat] :
( ( member_list_nat2 @ A4 @ A )
=> ? [B4: set_list_nat] :
( ( A
= ( insert_list_nat2 @ A4 @ B4 ) )
& ~ ( member_list_nat2 @ A4 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_368_mk__disjoint__insert,axiom,
! [A4: a,A: set_a] :
( ( member_a2 @ A4 @ A )
=> ? [B4: set_a] :
( ( A
= ( insert_a2 @ A4 @ B4 ) )
& ~ ( member_a2 @ A4 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_369_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_370_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_371_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_372_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_373_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B2: nat,A4: nat] :
( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A4 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_374_sup_Ostrict__coboundedI2,axiom,
! [C: int,B2: int,A4: int] :
( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ ( sup_sup_int @ A4 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_375_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A4: nat,B2: nat] :
( ( ord_less_nat @ C @ A4 )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A4 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_376_sup_Ostrict__coboundedI1,axiom,
! [C: int,A4: int,B2: int] :
( ( ord_less_int @ C @ A4 )
=> ( ord_less_int @ C @ ( sup_sup_int @ A4 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_377_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( A5
= ( sup_sup_nat @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_378_sup_Ostrict__order__iff,axiom,
( ord_less_int
= ( ^ [B5: int,A5: int] :
( ( A5
= ( sup_sup_int @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_379_sup_Ostrict__boundedE,axiom,
! [B2: nat,C: nat,A4: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B2 @ C ) @ A4 )
=> ~ ( ( ord_less_nat @ B2 @ A4 )
=> ~ ( ord_less_nat @ C @ A4 ) ) ) ).
% sup.strict_boundedE
thf(fact_380_sup_Ostrict__boundedE,axiom,
! [B2: int,C: int,A4: int] :
( ( ord_less_int @ ( sup_sup_int @ B2 @ C ) @ A4 )
=> ~ ( ( ord_less_int @ B2 @ A4 )
=> ~ ( ord_less_int @ C @ A4 ) ) ) ).
% sup.strict_boundedE
thf(fact_381_sup_Oabsorb4,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( sup_sup_nat @ A4 @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_382_sup_Oabsorb4,axiom,
! [A4: int,B2: int] :
( ( ord_less_int @ A4 @ B2 )
=> ( ( sup_sup_int @ A4 @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_383_sup_Oabsorb3,axiom,
! [B2: nat,A4: nat] :
( ( ord_less_nat @ B2 @ A4 )
=> ( ( sup_sup_nat @ A4 @ B2 )
= A4 ) ) ).
% sup.absorb3
thf(fact_384_sup_Oabsorb3,axiom,
! [B2: int,A4: int] :
( ( ord_less_int @ B2 @ A4 )
=> ( ( sup_sup_int @ A4 @ B2 )
= A4 ) ) ).
% sup.absorb3
thf(fact_385_less__supI2,axiom,
! [X: nat,B2: nat,A4: nat] :
( ( ord_less_nat @ X @ B2 )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A4 @ B2 ) ) ) ).
% less_supI2
thf(fact_386_less__supI2,axiom,
! [X: int,B2: int,A4: int] :
( ( ord_less_int @ X @ B2 )
=> ( ord_less_int @ X @ ( sup_sup_int @ A4 @ B2 ) ) ) ).
% less_supI2
thf(fact_387_less__supI1,axiom,
! [X: nat,A4: nat,B2: nat] :
( ( ord_less_nat @ X @ A4 )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A4 @ B2 ) ) ) ).
% less_supI1
thf(fact_388_less__supI1,axiom,
! [X: int,A4: int,B2: int] :
( ( ord_less_int @ X @ A4 )
=> ( ord_less_int @ X @ ( sup_sup_int @ A4 @ B2 ) ) ) ).
% less_supI1
thf(fact_389_length__induct,axiom,
! [P2: 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 ) )
=> ( P2 @ Ys2 ) )
=> ( P2 @ Xs3 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_390_length__pos__if__in__set,axiom,
! [X: a,Xs: list_a] :
( ( member_a2 @ X @ ( set_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_391_length__pos__if__in__set,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_392_length__pos__if__in__set,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_393_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_394_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_395_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_396_bot__nat__0_Onot__eq__extremum,axiom,
! [A4: nat] :
( ( A4 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A4 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_397_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_398_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_399_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_400_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_401_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_402_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_403_size__neq__size__imp__neq,axiom,
! [X: list_nat,Y2: list_nat] :
( ( ( size_size_list_nat @ X )
!= ( size_size_list_nat @ Y2 ) )
=> ( X != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_404_size__neq__size__imp__neq,axiom,
! [X: char,Y2: char] :
( ( ( size_size_char @ X )
!= ( size_size_char @ Y2 ) )
=> ( X != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_405_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_406_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_407_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_408_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_409_bot__nat__0_Oextremum__strict,axiom,
! [A4: nat] :
~ ( ord_less_nat @ A4 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_410_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_411_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_412_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_413_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_414_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_415_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P2 @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P2 @ M3 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_416_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_417_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_418_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_419_rotate1__length01,axiom,
! [Xs: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( ( rotate1_nat @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_420_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_421_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_422_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_423_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_424_dual__order_Orefl,axiom,
! [A4: nat] : ( ord_less_eq_nat @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_425_dual__order_Orefl,axiom,
! [A4: int] : ( ord_less_eq_int @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_426_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_427_sup_Obounded__iff,axiom,
! [B2: nat,C: nat,A4: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A4 )
= ( ( ord_less_eq_nat @ B2 @ A4 )
& ( ord_less_eq_nat @ C @ A4 ) ) ) ).
% sup.bounded_iff
thf(fact_428_sup_Obounded__iff,axiom,
! [B2: int,C: int,A4: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ B2 @ C ) @ A4 )
= ( ( ord_less_eq_int @ B2 @ A4 )
& ( ord_less_eq_int @ C @ A4 ) ) ) ).
% sup.bounded_iff
thf(fact_429_le__sup__iff,axiom,
! [X: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y2 ) @ Z2 )
= ( ( ord_less_eq_nat @ X @ Z2 )
& ( ord_less_eq_nat @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_430_le__sup__iff,axiom,
! [X: int,Y2: int,Z2: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ X @ Y2 ) @ Z2 )
= ( ( ord_less_eq_int @ X @ Z2 )
& ( ord_less_eq_int @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_431_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_432_bot__nat__0_Oextremum,axiom,
! [A4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A4 ) ).
% bot_nat_0.extremum
thf(fact_433_list_Omap__disc__iff,axiom,
! [F2: nat > nat,A4: list_nat] :
( ( ( map_nat_nat @ F2 @ A4 )
= nil_nat )
= ( A4 = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_434_list_Omap__disc__iff,axiom,
! [F2: nat > list_nat,A4: list_nat] :
( ( ( map_nat_list_nat @ F2 @ A4 )
= nil_list_nat )
= ( A4 = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_435_list_Omap__disc__iff,axiom,
! [F2: a > nat,A4: list_a] :
( ( ( map_a_nat @ F2 @ A4 )
= nil_nat )
= ( A4 = nil_a ) ) ).
% list.map_disc_iff
thf(fact_436_list_Omap__disc__iff,axiom,
! [F2: list_nat > set_Pr1261947904930325089at_nat,A4: list_list_nat] :
( ( ( map_li6003994582982014139at_nat @ F2 @ A4 )
= nil_se357566008730718055at_nat )
= ( A4 = nil_list_nat ) ) ).
% list.map_disc_iff
thf(fact_437_list_Omap__disc__iff,axiom,
! [F2: list_nat > nat,A4: list_list_nat] :
( ( ( map_list_nat_nat @ F2 @ A4 )
= nil_nat )
= ( A4 = nil_list_nat ) ) ).
% list.map_disc_iff
thf(fact_438_list_Omap__disc__iff,axiom,
! [F2: list_nat > list_nat,A4: list_list_nat] :
( ( ( map_li7225945977422193158st_nat @ F2 @ A4 )
= nil_list_nat )
= ( A4 = nil_list_nat ) ) ).
% list.map_disc_iff
thf(fact_439_list_Omap__disc__iff,axiom,
! [F2: a > a,A4: list_a] :
( ( ( map_a_a @ F2 @ A4 )
= nil_a )
= ( A4 = nil_a ) ) ).
% list.map_disc_iff
thf(fact_440_Nil__is__map__conv,axiom,
! [F2: nat > nat,Xs: list_nat] :
( ( nil_nat
= ( map_nat_nat @ F2 @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_441_Nil__is__map__conv,axiom,
! [F2: nat > list_nat,Xs: list_nat] :
( ( nil_list_nat
= ( map_nat_list_nat @ F2 @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_442_Nil__is__map__conv,axiom,
! [F2: a > nat,Xs: list_a] :
( ( nil_nat
= ( map_a_nat @ F2 @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_map_conv
thf(fact_443_Nil__is__map__conv,axiom,
! [F2: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat] :
( ( nil_se357566008730718055at_nat
= ( map_li6003994582982014139at_nat @ F2 @ Xs ) )
= ( Xs = nil_list_nat ) ) ).
% Nil_is_map_conv
thf(fact_444_Nil__is__map__conv,axiom,
! [F2: list_nat > nat,Xs: list_list_nat] :
( ( nil_nat
= ( map_list_nat_nat @ F2 @ Xs ) )
= ( Xs = nil_list_nat ) ) ).
% Nil_is_map_conv
thf(fact_445_Nil__is__map__conv,axiom,
! [F2: list_nat > list_nat,Xs: list_list_nat] :
( ( nil_list_nat
= ( map_li7225945977422193158st_nat @ F2 @ Xs ) )
= ( Xs = nil_list_nat ) ) ).
% Nil_is_map_conv
thf(fact_446_Nil__is__map__conv,axiom,
! [F2: a > a,Xs: list_a] :
( ( nil_a
= ( map_a_a @ F2 @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_map_conv
thf(fact_447_map__is__Nil__conv,axiom,
! [F2: nat > nat,Xs: list_nat] :
( ( ( map_nat_nat @ F2 @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_448_map__is__Nil__conv,axiom,
! [F2: nat > list_nat,Xs: list_nat] :
( ( ( map_nat_list_nat @ F2 @ Xs )
= nil_list_nat )
= ( Xs = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_449_map__is__Nil__conv,axiom,
! [F2: a > nat,Xs: list_a] :
( ( ( map_a_nat @ F2 @ Xs )
= nil_nat )
= ( Xs = nil_a ) ) ).
% map_is_Nil_conv
thf(fact_450_map__is__Nil__conv,axiom,
! [F2: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat] :
( ( ( map_li6003994582982014139at_nat @ F2 @ Xs )
= nil_se357566008730718055at_nat )
= ( Xs = nil_list_nat ) ) ).
% map_is_Nil_conv
thf(fact_451_map__is__Nil__conv,axiom,
! [F2: list_nat > nat,Xs: list_list_nat] :
( ( ( map_list_nat_nat @ F2 @ Xs )
= nil_nat )
= ( Xs = nil_list_nat ) ) ).
% map_is_Nil_conv
thf(fact_452_map__is__Nil__conv,axiom,
! [F2: list_nat > list_nat,Xs: list_list_nat] :
( ( ( map_li7225945977422193158st_nat @ F2 @ Xs )
= nil_list_nat )
= ( Xs = nil_list_nat ) ) ).
% map_is_Nil_conv
thf(fact_453_map__is__Nil__conv,axiom,
! [F2: a > a,Xs: list_a] :
( ( ( map_a_a @ F2 @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% map_is_Nil_conv
thf(fact_454_rotate1__is__Nil__conv,axiom,
! [Xs: list_nat] :
( ( ( rotate1_nat @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% rotate1_is_Nil_conv
thf(fact_455_rotate1__is__Nil__conv,axiom,
! [Xs: list_list_nat] :
( ( ( rotate1_list_nat @ Xs )
= nil_list_nat )
= ( Xs = nil_list_nat ) ) ).
% rotate1_is_Nil_conv
thf(fact_456_list__ex1__simps_I1_J,axiom,
! [P2: nat > $o] :
~ ( list_ex1_nat @ P2 @ nil_nat ) ).
% list_ex1_simps(1)
thf(fact_457_list__ex1__simps_I1_J,axiom,
! [P2: list_nat > $o] :
~ ( list_ex1_list_nat @ P2 @ nil_list_nat ) ).
% list_ex1_simps(1)
thf(fact_458_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1316708129612266289at_nat @ M2 )
= zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_459_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_460_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_461_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_462_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_463_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_464_length__0__conv,axiom,
! [Xs: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_list_nat ) ) ).
% length_0_conv
thf(fact_465_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_466_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_467_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_468_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_469_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_470_inj__of__nat,axiom,
inj_on_nat_int @ semiri1314217659103216013at_int @ top_top_set_nat ).
% inj_of_nat
thf(fact_471_UNIV__coset,axiom,
( top_top_set_nat
= ( coset_nat @ nil_nat ) ) ).
% UNIV_coset
thf(fact_472_UNIV__coset,axiom,
( top_top_set_list_nat
= ( coset_list_nat @ nil_list_nat ) ) ).
% UNIV_coset
thf(fact_473_nle__le,axiom,
! [A4: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A4 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A4 )
& ( B2 != A4 ) ) ) ).
% nle_le
thf(fact_474_nle__le,axiom,
! [A4: int,B2: int] :
( ( ~ ( ord_less_eq_int @ A4 @ B2 ) )
= ( ( ord_less_eq_int @ B2 @ A4 )
& ( B2 != A4 ) ) ) ).
% nle_le
thf(fact_475_le__cases3,axiom,
! [X: nat,Y2: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_476_le__cases3,axiom,
! [X: int,Y2: int,Z2: int] :
( ( ( ord_less_eq_int @ X @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y2 @ X )
=> ~ ( ord_less_eq_int @ X @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ X ) )
=> ( ( ( ord_less_eq_int @ Y2 @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X )
=> ~ ( ord_less_eq_int @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_477_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
& ( ord_less_eq_nat @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_478_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z3: int] : ( Y5 = Z3 ) )
= ( ^ [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
& ( ord_less_eq_int @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_479_ord__eq__le__trans,axiom,
! [A4: nat,B2: nat,C: nat] :
( ( A4 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A4 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_480_ord__eq__le__trans,axiom,
! [A4: int,B2: int,C: int] :
( ( A4 = B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ A4 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_481_ord__le__eq__trans,axiom,
! [A4: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A4 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_482_ord__le__eq__trans,axiom,
! [A4: int,B2: int,C: int] :
( ( ord_less_eq_int @ A4 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_int @ A4 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_483_order__antisym,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_484_order__antisym,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_485_order_Otrans,axiom,
! [A4: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A4 @ C ) ) ) ).
% order.trans
thf(fact_486_order_Otrans,axiom,
! [A4: int,B2: int,C: int] :
( ( ord_less_eq_int @ A4 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ A4 @ C ) ) ) ).
% order.trans
thf(fact_487_order__trans,axiom,
! [X: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_488_order__trans,axiom,
! [X: int,Y2: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z2 )
=> ( ord_less_eq_int @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_489_linorder__wlog,axiom,
! [P2: nat > nat > $o,A4: nat,B2: nat] :
( ! [A2: nat,B3: nat] :
( ( ord_less_eq_nat @ A2 @ B3 )
=> ( P2 @ A2 @ B3 ) )
=> ( ! [A2: nat,B3: nat] :
( ( P2 @ B3 @ A2 )
=> ( P2 @ A2 @ B3 ) )
=> ( P2 @ A4 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_490_linorder__wlog,axiom,
! [P2: int > int > $o,A4: int,B2: int] :
( ! [A2: int,B3: int] :
( ( ord_less_eq_int @ A2 @ B3 )
=> ( P2 @ A2 @ B3 ) )
=> ( ! [A2: int,B3: int] :
( ( P2 @ B3 @ A2 )
=> ( P2 @ A2 @ B3 ) )
=> ( P2 @ A4 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_491_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_492_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: int,Z3: int] : ( Y5 = Z3 ) )
= ( ^ [A5: int,B5: int] :
( ( ord_less_eq_int @ B5 @ A5 )
& ( ord_less_eq_int @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_493_dual__order_Oantisym,axiom,
! [B2: nat,A4: nat] :
( ( ord_less_eq_nat @ B2 @ A4 )
=> ( ( ord_less_eq_nat @ A4 @ B2 )
=> ( A4 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_494_dual__order_Oantisym,axiom,
! [B2: int,A4: int] :
( ( ord_less_eq_int @ B2 @ A4 )
=> ( ( ord_less_eq_int @ A4 @ B2 )
=> ( A4 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_495_dual__order_Otrans,axiom,
! [B2: nat,A4: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A4 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A4 ) ) ) ).
% dual_order.trans
thf(fact_496_dual__order_Otrans,axiom,
! [B2: int,A4: int,C: int] :
( ( ord_less_eq_int @ B2 @ A4 )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_eq_int @ C @ A4 ) ) ) ).
% dual_order.trans
thf(fact_497_antisym,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A4 )
=> ( A4 = B2 ) ) ) ).
% antisym
thf(fact_498_antisym,axiom,
! [A4: int,B2: int] :
( ( ord_less_eq_int @ A4 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ A4 )
=> ( A4 = B2 ) ) ) ).
% antisym
thf(fact_499_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_500_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z3: int] : ( Y5 = Z3 ) )
= ( ^ [A5: int,B5: int] :
( ( ord_less_eq_int @ A5 @ B5 )
& ( ord_less_eq_int @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_501_order__subst1,axiom,
! [A4: nat,F2: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_502_order__subst1,axiom,
! [A4: nat,F2: int > nat,B2: int,C: int] :
( ( ord_less_eq_nat @ A4 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_503_order__subst1,axiom,
! [A4: int,F2: nat > int,B2: nat,C: nat] :
( ( ord_less_eq_int @ A4 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_504_order__subst1,axiom,
! [A4: int,F2: int > int,B2: int,C: int] :
( ( ord_less_eq_int @ A4 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_505_order__subst2,axiom,
! [A4: nat,B2: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ ( F2 @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_506_order__subst2,axiom,
! [A4: nat,B2: nat,F2: nat > int,C: int] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_less_eq_int @ ( F2 @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_507_order__subst2,axiom,
! [A4: int,B2: int,F2: int > nat,C: nat] :
( ( ord_less_eq_int @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ ( F2 @ B2 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_508_order__subst2,axiom,
! [A4: int,B2: int,F2: int > int,C: int] :
( ( ord_less_eq_int @ A4 @ B2 )
=> ( ( ord_less_eq_int @ ( F2 @ B2 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_509_order__eq__refl,axiom,
! [X: nat,Y2: nat] :
( ( X = Y2 )
=> ( ord_less_eq_nat @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_510_order__eq__refl,axiom,
! [X: int,Y2: int] :
( ( X = Y2 )
=> ( ord_less_eq_int @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_511_linorder__linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_512_linorder__linear,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
| ( ord_less_eq_int @ Y2 @ X ) ) ).
% linorder_linear
thf(fact_513_ord__eq__le__subst,axiom,
! [A4: nat,F2: nat > nat,B2: nat,C: nat] :
( ( A4
= ( F2 @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_514_ord__eq__le__subst,axiom,
! [A4: int,F2: nat > int,B2: nat,C: nat] :
( ( A4
= ( F2 @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ A4 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_515_ord__eq__le__subst,axiom,
! [A4: nat,F2: int > nat,B2: int,C: int] :
( ( A4
= ( F2 @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_516_ord__eq__le__subst,axiom,
! [A4: int,F2: int > int,B2: int,C: int] :
( ( A4
= ( F2 @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ A4 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_517_ord__le__eq__subst,axiom,
! [A4: nat,B2: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_518_ord__le__eq__subst,axiom,
! [A4: nat,B2: nat,F2: nat > int,C: int] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_519_ord__le__eq__subst,axiom,
! [A4: int,B2: int,F2: int > nat,C: nat] :
( ( ord_less_eq_int @ A4 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_520_ord__le__eq__subst,axiom,
! [A4: int,B2: int,F2: int > int,C: int] :
( ( ord_less_eq_int @ A4 @ B2 )
=> ( ( ( F2 @ B2 )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F2 @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_521_linorder__le__cases,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_le_cases
thf(fact_522_linorder__le__cases,axiom,
! [X: int,Y2: int] :
( ~ ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X ) ) ).
% linorder_le_cases
thf(fact_523_order__antisym__conv,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( ord_less_eq_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_524_order__antisym__conv,axiom,
! [Y2: int,X: int] :
( ( ord_less_eq_int @ Y2 @ X )
=> ( ( ord_less_eq_int @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_525_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_526_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_527_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_528_leD,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ~ ( ord_less_nat @ X @ Y2 ) ) ).
% leD
thf(fact_529_leD,axiom,
! [Y2: int,X: int] :
( ( ord_less_eq_int @ Y2 @ X )
=> ~ ( ord_less_int @ X @ Y2 ) ) ).
% leD
thf(fact_530_leI,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) ) ).
% leI
thf(fact_531_leI,axiom,
! [X: int,Y2: int] :
( ~ ( ord_less_int @ X @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X ) ) ).
% leI
thf(fact_532_nless__le,axiom,
! [A4: nat,B2: nat] :
( ( ~ ( ord_less_nat @ A4 @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A4 @ B2 )
| ( A4 = B2 ) ) ) ).
% nless_le
thf(fact_533_nless__le,axiom,
! [A4: int,B2: int] :
( ( ~ ( ord_less_int @ A4 @ B2 ) )
= ( ~ ( ord_less_eq_int @ A4 @ B2 )
| ( A4 = B2 ) ) ) ).
% nless_le
thf(fact_534_antisym__conv1,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_535_antisym__conv1,axiom,
! [X: int,Y2: int] :
( ~ ( ord_less_int @ X @ Y2 )
=> ( ( ord_less_eq_int @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_536_antisym__conv2,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_537_antisym__conv2,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ~ ( ord_less_int @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_538_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
& ~ ( ord_less_eq_nat @ Y @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_539_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
& ~ ( ord_less_eq_int @ Y @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_540_not__le__imp__less,axiom,
! [Y2: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y2 @ X )
=> ( ord_less_nat @ X @ Y2 ) ) ).
% not_le_imp_less
thf(fact_541_not__le__imp__less,axiom,
! [Y2: int,X: int] :
( ~ ( ord_less_eq_int @ Y2 @ X )
=> ( ord_less_int @ X @ Y2 ) ) ).
% not_le_imp_less
thf(fact_542_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_nat @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_543_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B5: int] :
( ( ord_less_int @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_544_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_545_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A5: int,B5: int] :
( ( ord_less_eq_int @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_546_order_Ostrict__trans1,axiom,
! [A4: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A4 @ C ) ) ) ).
% order.strict_trans1
thf(fact_547_order_Ostrict__trans1,axiom,
! [A4: int,B2: int,C: int] :
( ( ord_less_eq_int @ A4 @ B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A4 @ C ) ) ) ).
% order.strict_trans1
thf(fact_548_order_Ostrict__trans2,axiom,
! [A4: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ A4 @ C ) ) ) ).
% order.strict_trans2
thf(fact_549_order_Ostrict__trans2,axiom,
! [A4: int,B2: int,C: int] :
( ( ord_less_int @ A4 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_int @ A4 @ C ) ) ) ).
% order.strict_trans2
thf(fact_550_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ~ ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_551_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A5: int,B5: int] :
( ( ord_less_eq_int @ A5 @ B5 )
& ~ ( ord_less_eq_int @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_552_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_nat @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_553_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B5: int,A5: int] :
( ( ord_less_int @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_554_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_555_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B5: int,A5: int] :
( ( ord_less_eq_int @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_556_dual__order_Ostrict__trans1,axiom,
! [B2: nat,A4: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A4 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A4 ) ) ) ).
% dual_order.strict_trans1
thf(fact_557_dual__order_Ostrict__trans1,axiom,
! [B2: int,A4: int,C: int] :
( ( ord_less_eq_int @ B2 @ A4 )
=> ( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ A4 ) ) ) ).
% dual_order.strict_trans1
thf(fact_558_dual__order_Ostrict__trans2,axiom,
! [B2: nat,A4: nat,C: nat] :
( ( ord_less_nat @ B2 @ A4 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A4 ) ) ) ).
% dual_order.strict_trans2
thf(fact_559_dual__order_Ostrict__trans2,axiom,
! [B2: int,A4: int,C: int] :
( ( ord_less_int @ B2 @ A4 )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_int @ C @ A4 ) ) ) ).
% dual_order.strict_trans2
thf(fact_560_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_561_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B5: int,A5: int] :
( ( ord_less_eq_int @ B5 @ A5 )
& ~ ( ord_less_eq_int @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_562_order_Ostrict__implies__order,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ord_less_eq_nat @ A4 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_563_order_Ostrict__implies__order,axiom,
! [A4: int,B2: int] :
( ( ord_less_int @ A4 @ B2 )
=> ( ord_less_eq_int @ A4 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_564_dual__order_Ostrict__implies__order,axiom,
! [B2: nat,A4: nat] :
( ( ord_less_nat @ B2 @ A4 )
=> ( ord_less_eq_nat @ B2 @ A4 ) ) ).
% dual_order.strict_implies_order
thf(fact_565_dual__order_Ostrict__implies__order,axiom,
! [B2: int,A4: int] :
( ( ord_less_int @ B2 @ A4 )
=> ( ord_less_eq_int @ B2 @ A4 ) ) ).
% dual_order.strict_implies_order
thf(fact_566_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% order_le_less
thf(fact_567_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% order_le_less
thf(fact_568_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
& ( X2 != Y ) ) ) ) ).
% order_less_le
thf(fact_569_order__less__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
& ( X2 != Y ) ) ) ) ).
% order_less_le
thf(fact_570_linorder__not__le,axiom,
! [X: nat,Y2: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y2 ) )
= ( ord_less_nat @ Y2 @ X ) ) ).
% linorder_not_le
thf(fact_571_linorder__not__le,axiom,
! [X: int,Y2: int] :
( ( ~ ( ord_less_eq_int @ X @ Y2 ) )
= ( ord_less_int @ Y2 @ X ) ) ).
% linorder_not_le
thf(fact_572_linorder__not__less,axiom,
! [X: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_not_less
thf(fact_573_linorder__not__less,axiom,
! [X: int,Y2: int] :
( ( ~ ( ord_less_int @ X @ Y2 ) )
= ( ord_less_eq_int @ Y2 @ X ) ) ).
% linorder_not_less
thf(fact_574_order__less__imp__le,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_575_order__less__imp__le,axiom,
! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_eq_int @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_576_order__le__neq__trans,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( A4 != B2 )
=> ( ord_less_nat @ A4 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_577_order__le__neq__trans,axiom,
! [A4: int,B2: int] :
( ( ord_less_eq_int @ A4 @ B2 )
=> ( ( A4 != B2 )
=> ( ord_less_int @ A4 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_578_order__neq__le__trans,axiom,
! [A4: nat,B2: nat] :
( ( A4 != B2 )
=> ( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ord_less_nat @ A4 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_579_order__neq__le__trans,axiom,
! [A4: int,B2: int] :
( ( A4 != B2 )
=> ( ( ord_less_eq_int @ A4 @ B2 )
=> ( ord_less_int @ A4 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_580_order__le__less__trans,axiom,
! [X: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_581_order__le__less__trans,axiom,
! [X: int,Y2: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_582_order__less__le__trans,axiom,
! [X: nat,Y2: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_583_order__less__le__trans,axiom,
! [X: int,Y2: int,Z2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_584_order__le__less__subst1,axiom,
! [A4: nat,F2: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ ( F2 @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_585_order__le__less__subst1,axiom,
! [A4: nat,F2: int > nat,B2: int,C: int] :
( ( ord_less_eq_nat @ A4 @ ( F2 @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_586_order__le__less__subst1,axiom,
! [A4: int,F2: nat > int,B2: nat,C: nat] :
( ( ord_less_eq_int @ A4 @ ( F2 @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_587_order__le__less__subst1,axiom,
! [A4: int,F2: int > int,B2: int,C: int] :
( ( ord_less_eq_int @ A4 @ ( F2 @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_588_order__le__less__subst2,axiom,
! [A4: nat,B2: nat,F2: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_less_nat @ ( F2 @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_589_order__le__less__subst2,axiom,
! [A4: nat,B2: nat,F2: nat > int,C: int] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_less_int @ ( F2 @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_590_order__le__less__subst2,axiom,
! [A4: int,B2: int,F2: int > nat,C: nat] :
( ( ord_less_eq_int @ A4 @ B2 )
=> ( ( ord_less_nat @ ( F2 @ B2 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_591_order__le__less__subst2,axiom,
! [A4: int,B2: int,F2: int > int,C: int] :
( ( ord_less_eq_int @ A4 @ B2 )
=> ( ( ord_less_int @ ( F2 @ B2 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_592_order__less__le__subst1,axiom,
! [A4: nat,F2: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A4 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_593_order__less__le__subst1,axiom,
! [A4: int,F2: nat > int,B2: nat,C: nat] :
( ( ord_less_int @ A4 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_594_order__less__le__subst1,axiom,
! [A4: nat,F2: int > nat,B2: int,C: int] :
( ( ord_less_nat @ A4 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_595_order__less__le__subst1,axiom,
! [A4: int,F2: int > int,B2: int,C: int] :
( ( ord_less_int @ A4 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ A4 @ ( F2 @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_596_order__less__le__subst2,axiom,
! [A4: nat,B2: nat,F2: nat > nat,C: nat] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ ( F2 @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_597_order__less__le__subst2,axiom,
! [A4: int,B2: int,F2: int > nat,C: nat] :
( ( ord_less_int @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ ( F2 @ B2 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_598_order__less__le__subst2,axiom,
! [A4: nat,B2: nat,F2: nat > int,C: int] :
( ( ord_less_nat @ A4 @ B2 )
=> ( ( ord_less_eq_int @ ( F2 @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_599_order__less__le__subst2,axiom,
! [A4: int,B2: int,F2: int > int,C: int] :
( ( ord_less_int @ A4 @ B2 )
=> ( ( ord_less_eq_int @ ( F2 @ B2 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_int @ ( F2 @ A4 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_600_linorder__le__less__linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
| ( ord_less_nat @ Y2 @ X ) ) ).
% linorder_le_less_linear
thf(fact_601_linorder__le__less__linear,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
| ( ord_less_int @ Y2 @ X ) ) ).
% linorder_le_less_linear
thf(fact_602_order__le__imp__less__or__eq,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_nat @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_603_order__le__imp__less__or__eq,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( ord_less_int @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_604_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_605_bot__nat__0_Oextremum__unique,axiom,
! [A4: nat] :
( ( ord_less_eq_nat @ A4 @ zero_zero_nat )
= ( A4 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_606_bot__nat__0_Oextremum__uniqueI,axiom,
! [A4: nat] :
( ( ord_less_eq_nat @ A4 @ zero_zero_nat )
=> ( A4 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_607_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_608_sup_OcoboundedI2,axiom,
! [C: nat,B2: nat,A4: nat] :
( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A4 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_609_sup_OcoboundedI2,axiom,
! [C: int,B2: int,A4: int] :
( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_eq_int @ C @ ( sup_sup_int @ A4 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_610_sup_OcoboundedI1,axiom,
! [C: nat,A4: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A4 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A4 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_611_sup_OcoboundedI1,axiom,
! [C: int,A4: int,B2: int] :
( ( ord_less_eq_int @ C @ A4 )
=> ( ord_less_eq_int @ C @ ( sup_sup_int @ A4 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_612_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( sup_sup_nat @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_613_sup_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B5: int] :
( ( sup_sup_int @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_614_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( sup_sup_nat @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_615_sup_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [B5: int,A5: int] :
( ( sup_sup_int @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_616_sup_Ocobounded2,axiom,
! [B2: nat,A4: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A4 @ B2 ) ) ).
% sup.cobounded2
thf(fact_617_sup_Ocobounded2,axiom,
! [B2: int,A4: int] : ( ord_less_eq_int @ B2 @ ( sup_sup_int @ A4 @ B2 ) ) ).
% sup.cobounded2
thf(fact_618_sup_Ocobounded1,axiom,
! [A4: nat,B2: nat] : ( ord_less_eq_nat @ A4 @ ( sup_sup_nat @ A4 @ B2 ) ) ).
% sup.cobounded1
thf(fact_619_sup_Ocobounded1,axiom,
! [A4: int,B2: int] : ( ord_less_eq_int @ A4 @ ( sup_sup_int @ A4 @ B2 ) ) ).
% sup.cobounded1
thf(fact_620_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( A5
= ( sup_sup_nat @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_621_sup_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [B5: int,A5: int] :
( A5
= ( sup_sup_int @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_622_sup_OboundedI,axiom,
! [B2: nat,A4: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A4 )
=> ( ( ord_less_eq_nat @ C @ A4 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A4 ) ) ) ).
% sup.boundedI
thf(fact_623_sup_OboundedI,axiom,
! [B2: int,A4: int,C: int] :
( ( ord_less_eq_int @ B2 @ A4 )
=> ( ( ord_less_eq_int @ C @ A4 )
=> ( ord_less_eq_int @ ( sup_sup_int @ B2 @ C ) @ A4 ) ) ) ).
% sup.boundedI
thf(fact_624_sup_OboundedE,axiom,
! [B2: nat,C: nat,A4: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A4 )
=> ~ ( ( ord_less_eq_nat @ B2 @ A4 )
=> ~ ( ord_less_eq_nat @ C @ A4 ) ) ) ).
% sup.boundedE
thf(fact_625_sup_OboundedE,axiom,
! [B2: int,C: int,A4: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ B2 @ C ) @ A4 )
=> ~ ( ( ord_less_eq_int @ B2 @ A4 )
=> ~ ( ord_less_eq_int @ C @ A4 ) ) ) ).
% sup.boundedE
thf(fact_626_sup__absorb2,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( sup_sup_nat @ X @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_627_sup__absorb2,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ( sup_sup_int @ X @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_628_sup__absorb1,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( sup_sup_nat @ X @ Y2 )
= X ) ) ).
% sup_absorb1
thf(fact_629_sup__absorb1,axiom,
! [Y2: int,X: int] :
( ( ord_less_eq_int @ Y2 @ X )
=> ( ( sup_sup_int @ X @ Y2 )
= X ) ) ).
% sup_absorb1
thf(fact_630_sup_Oabsorb2,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( sup_sup_nat @ A4 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_631_sup_Oabsorb2,axiom,
! [A4: int,B2: int] :
( ( ord_less_eq_int @ A4 @ B2 )
=> ( ( sup_sup_int @ A4 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_632_sup_Oabsorb1,axiom,
! [B2: nat,A4: nat] :
( ( ord_less_eq_nat @ B2 @ A4 )
=> ( ( sup_sup_nat @ A4 @ B2 )
= A4 ) ) ).
% sup.absorb1
thf(fact_633_sup_Oabsorb1,axiom,
! [B2: int,A4: int] :
( ( ord_less_eq_int @ B2 @ A4 )
=> ( ( sup_sup_int @ A4 @ B2 )
= A4 ) ) ).
% sup.absorb1
thf(fact_634_sup__unique,axiom,
! [F2: nat > nat > nat,X: nat,Y2: nat] :
( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ X3 @ ( F2 @ X3 @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ ( F2 @ X3 @ Y3 ) )
=> ( ! [X3: nat,Y3: nat,Z: nat] :
( ( ord_less_eq_nat @ Y3 @ X3 )
=> ( ( ord_less_eq_nat @ Z @ X3 )
=> ( ord_less_eq_nat @ ( F2 @ Y3 @ Z ) @ X3 ) ) )
=> ( ( sup_sup_nat @ X @ Y2 )
= ( F2 @ X @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_635_sup__unique,axiom,
! [F2: int > int > int,X: int,Y2: int] :
( ! [X3: int,Y3: int] : ( ord_less_eq_int @ X3 @ ( F2 @ X3 @ Y3 ) )
=> ( ! [X3: int,Y3: int] : ( ord_less_eq_int @ Y3 @ ( F2 @ X3 @ Y3 ) )
=> ( ! [X3: int,Y3: int,Z: int] :
( ( ord_less_eq_int @ Y3 @ X3 )
=> ( ( ord_less_eq_int @ Z @ X3 )
=> ( ord_less_eq_int @ ( F2 @ Y3 @ Z ) @ X3 ) ) )
=> ( ( sup_sup_int @ X @ Y2 )
= ( F2 @ X @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_636_sup_OorderI,axiom,
! [A4: nat,B2: nat] :
( ( A4
= ( sup_sup_nat @ A4 @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A4 ) ) ).
% sup.orderI
thf(fact_637_sup_OorderI,axiom,
! [A4: int,B2: int] :
( ( A4
= ( sup_sup_int @ A4 @ B2 ) )
=> ( ord_less_eq_int @ B2 @ A4 ) ) ).
% sup.orderI
thf(fact_638_sup_OorderE,axiom,
! [B2: nat,A4: nat] :
( ( ord_less_eq_nat @ B2 @ A4 )
=> ( A4
= ( sup_sup_nat @ A4 @ B2 ) ) ) ).
% sup.orderE
thf(fact_639_sup_OorderE,axiom,
! [B2: int,A4: int] :
( ( ord_less_eq_int @ B2 @ A4 )
=> ( A4
= ( sup_sup_int @ A4 @ B2 ) ) ) ).
% sup.orderE
thf(fact_640_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y: nat] :
( ( sup_sup_nat @ X2 @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_641_le__iff__sup,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y: int] :
( ( sup_sup_int @ X2 @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_642_sup__least,axiom,
! [Y2: nat,X: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( ord_less_eq_nat @ Z2 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y2 @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_643_sup__least,axiom,
! [Y2: int,X: int,Z2: int] :
( ( ord_less_eq_int @ Y2 @ X )
=> ( ( ord_less_eq_int @ Z2 @ X )
=> ( ord_less_eq_int @ ( sup_sup_int @ Y2 @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_644_sup__mono,axiom,
! [A4: nat,C: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A4 @ C )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A4 @ B2 ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_645_sup__mono,axiom,
! [A4: int,C: int,B2: int,D: int] :
( ( ord_less_eq_int @ A4 @ C )
=> ( ( ord_less_eq_int @ B2 @ D )
=> ( ord_less_eq_int @ ( sup_sup_int @ A4 @ B2 ) @ ( sup_sup_int @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_646_sup_Omono,axiom,
! [C: nat,A4: nat,D: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A4 )
=> ( ( ord_less_eq_nat @ D @ B2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A4 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_647_sup_Omono,axiom,
! [C: int,A4: int,D: int,B2: int] :
( ( ord_less_eq_int @ C @ A4 )
=> ( ( ord_less_eq_int @ D @ B2 )
=> ( ord_less_eq_int @ ( sup_sup_int @ C @ D ) @ ( sup_sup_int @ A4 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_648_le__supI2,axiom,
! [X: nat,B2: nat,A4: nat] :
( ( ord_less_eq_nat @ X @ B2 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A4 @ B2 ) ) ) ).
% le_supI2
thf(fact_649_le__supI2,axiom,
! [X: int,B2: int,A4: int] :
( ( ord_less_eq_int @ X @ B2 )
=> ( ord_less_eq_int @ X @ ( sup_sup_int @ A4 @ B2 ) ) ) ).
% le_supI2
thf(fact_650_le__supI1,axiom,
! [X: nat,A4: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ A4 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A4 @ B2 ) ) ) ).
% le_supI1
thf(fact_651_le__supI1,axiom,
! [X: int,A4: int,B2: int] :
( ( ord_less_eq_int @ X @ A4 )
=> ( ord_less_eq_int @ X @ ( sup_sup_int @ A4 @ B2 ) ) ) ).
% le_supI1
thf(fact_652_sup__ge2,axiom,
! [Y2: nat,X: nat] : ( ord_less_eq_nat @ Y2 @ ( sup_sup_nat @ X @ Y2 ) ) ).
% sup_ge2
thf(fact_653_sup__ge2,axiom,
! [Y2: int,X: int] : ( ord_less_eq_int @ Y2 @ ( sup_sup_int @ X @ Y2 ) ) ).
% sup_ge2
thf(fact_654_sup__ge1,axiom,
! [X: nat,Y2: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y2 ) ) ).
% sup_ge1
thf(fact_655_sup__ge1,axiom,
! [X: int,Y2: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y2 ) ) ).
% sup_ge1
thf(fact_656_le__supI,axiom,
! [A4: nat,X: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ X )
=> ( ( ord_less_eq_nat @ B2 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A4 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_657_le__supI,axiom,
! [A4: int,X: int,B2: int] :
( ( ord_less_eq_int @ A4 @ X )
=> ( ( ord_less_eq_int @ B2 @ X )
=> ( ord_less_eq_int @ ( sup_sup_int @ A4 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_658_le__supE,axiom,
! [A4: nat,B2: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A4 @ B2 ) @ X )
=> ~ ( ( ord_less_eq_nat @ A4 @ X )
=> ~ ( ord_less_eq_nat @ B2 @ X ) ) ) ).
% le_supE
thf(fact_659_le__supE,axiom,
! [A4: int,B2: int,X: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ A4 @ B2 ) @ X )
=> ~ ( ( ord_less_eq_int @ A4 @ X )
=> ~ ( ord_less_eq_int @ B2 @ X ) ) ) ).
% le_supE
thf(fact_660_inf__sup__ord_I3_J,axiom,
! [X: nat,Y2: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_661_inf__sup__ord_I3_J,axiom,
! [X: int,Y2: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_662_inf__sup__ord_I4_J,axiom,
! [Y2: nat,X: nat] : ( ord_less_eq_nat @ Y2 @ ( sup_sup_nat @ X @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_663_inf__sup__ord_I4_J,axiom,
! [Y2: int,X: int] : ( ord_less_eq_int @ Y2 @ ( sup_sup_int @ X @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_664_list_Osimps_I8_J,axiom,
! [F2: nat > nat] :
( ( map_nat_nat @ F2 @ nil_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_665_list_Osimps_I8_J,axiom,
! [F2: nat > list_nat] :
( ( map_nat_list_nat @ F2 @ nil_nat )
= nil_list_nat ) ).
% list.simps(8)
thf(fact_666_list_Osimps_I8_J,axiom,
! [F2: a > nat] :
( ( map_a_nat @ F2 @ nil_a )
= nil_nat ) ).
% list.simps(8)
thf(fact_667_list_Osimps_I8_J,axiom,
! [F2: list_nat > set_Pr1261947904930325089at_nat] :
( ( map_li6003994582982014139at_nat @ F2 @ nil_list_nat )
= nil_se357566008730718055at_nat ) ).
% list.simps(8)
thf(fact_668_list_Osimps_I8_J,axiom,
! [F2: list_nat > nat] :
( ( map_list_nat_nat @ F2 @ nil_list_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_669_list_Osimps_I8_J,axiom,
! [F2: list_nat > list_nat] :
( ( map_li7225945977422193158st_nat @ F2 @ nil_list_nat )
= nil_list_nat ) ).
% list.simps(8)
thf(fact_670_list_Osimps_I8_J,axiom,
! [F2: a > a] :
( ( map_a_a @ F2 @ nil_a )
= nil_a ) ).
% list.simps(8)
thf(fact_671_distinct_Osimps_I1_J,axiom,
distinct_nat @ nil_nat ).
% distinct.simps(1)
thf(fact_672_distinct_Osimps_I1_J,axiom,
distinct_list_nat @ nil_list_nat ).
% distinct.simps(1)
thf(fact_673_distinct__adj__Nil,axiom,
distinct_adj_nat @ nil_nat ).
% distinct_adj_Nil
thf(fact_674_distinct__adj__Nil,axiom,
distin876741697294417026st_nat @ nil_list_nat ).
% distinct_adj_Nil
thf(fact_675_removeAll_Osimps_I1_J,axiom,
! [X: nat] :
( ( removeAll_nat @ X @ nil_nat )
= nil_nat ) ).
% removeAll.simps(1)
thf(fact_676_removeAll_Osimps_I1_J,axiom,
! [X: list_nat] :
( ( removeAll_list_nat @ X @ nil_list_nat )
= nil_list_nat ) ).
% removeAll.simps(1)
thf(fact_677_rotate1_Osimps_I1_J,axiom,
( ( rotate1_nat @ nil_nat )
= nil_nat ) ).
% rotate1.simps(1)
thf(fact_678_rotate1_Osimps_I1_J,axiom,
( ( rotate1_list_nat @ nil_list_nat )
= nil_list_nat ) ).
% rotate1.simps(1)
thf(fact_679_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_nat @ N @ nil_nat )
= N ) ).
% gen_length_code(1)
thf(fact_680_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_list_nat @ N @ nil_list_nat )
= N ) ).
% gen_length_code(1)
thf(fact_681_member__rec_I2_J,axiom,
! [Y2: nat] :
~ ( member_nat @ nil_nat @ Y2 ) ).
% member_rec(2)
thf(fact_682_member__rec_I2_J,axiom,
! [Y2: list_nat] :
~ ( member_list_nat @ nil_list_nat @ Y2 ) ).
% member_rec(2)
thf(fact_683_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_684_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_685_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_686_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_687_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_688_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_689_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_690_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_691_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ N )
& ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ~ ( P2 @ I ) )
& ( P2 @ K ) ) ) ) ).
% ex_least_nat_le
thf(fact_692_list_Osize_I3_J,axiom,
( ( size_s3023201423986296836st_nat @ nil_list_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_693_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_694_count__le__length,axiom,
! [Xs: list_list_nat,X: list_nat] : ( ord_less_eq_nat @ ( count_list_list_nat @ Xs @ X ) @ ( size_s3023201423986296836st_nat @ Xs ) ) ).
% count_le_length
thf(fact_695_count__le__length,axiom,
! [Xs: list_nat,X: nat] : ( ord_less_eq_nat @ ( count_list_nat @ Xs @ X ) @ ( size_size_list_nat @ Xs ) ) ).
% count_le_length
thf(fact_696_length__removeAll__less__eq,axiom,
! [X: nat,Xs: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( removeAll_nat @ X @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% length_removeAll_less_eq
thf(fact_697_count__list__map__ge,axiom,
! [Xs: list_a,X: a,F2: a > nat] : ( ord_less_eq_nat @ ( count_list_a @ Xs @ X ) @ ( count_list_nat @ ( map_a_nat @ F2 @ Xs ) @ ( F2 @ X ) ) ) ).
% count_list_map_ge
thf(fact_698_count__list__map__ge,axiom,
! [Xs: list_a,X: a,F2: a > a] : ( ord_less_eq_nat @ ( count_list_a @ Xs @ X ) @ ( count_list_a @ ( map_a_a @ F2 @ Xs ) @ ( F2 @ X ) ) ) ).
% count_list_map_ge
thf(fact_699_count__list__map__ge,axiom,
! [Xs: list_list_nat,X: list_nat,F2: list_nat > set_Pr1261947904930325089at_nat] : ( ord_less_eq_nat @ ( count_list_list_nat @ Xs @ X ) @ ( count_6440129622255701469at_nat @ ( map_li6003994582982014139at_nat @ F2 @ Xs ) @ ( F2 @ X ) ) ) ).
% count_list_map_ge
thf(fact_700_count__list__map__ge,axiom,
! [Xs: list_list_nat,X: list_nat,F2: list_nat > nat] : ( ord_less_eq_nat @ ( count_list_list_nat @ Xs @ X ) @ ( count_list_nat @ ( map_list_nat_nat @ F2 @ Xs ) @ ( F2 @ X ) ) ) ).
% count_list_map_ge
thf(fact_701_count__list__map__ge,axiom,
! [Xs: list_list_nat,X: list_nat,F2: list_nat > list_nat] : ( ord_less_eq_nat @ ( count_list_list_nat @ Xs @ X ) @ ( count_list_list_nat @ ( map_li7225945977422193158st_nat @ F2 @ Xs ) @ ( F2 @ X ) ) ) ).
% count_list_map_ge
thf(fact_702_count__list_Osimps_I1_J,axiom,
! [Y2: nat] :
( ( count_list_nat @ nil_nat @ Y2 )
= zero_zero_nat ) ).
% count_list.simps(1)
thf(fact_703_count__list_Osimps_I1_J,axiom,
! [Y2: list_nat] :
( ( count_list_list_nat @ nil_list_nat @ Y2 )
= zero_zero_nat ) ).
% count_list.simps(1)
thf(fact_704_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_705_zero__less__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K2
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_706_pos__int__cases,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ~ ! [N3: nat] :
( ( K2
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_707_map__sorted__distinct__set__unique,axiom,
! [F2: a > nat,Xs: list_a,Ys: list_a] :
( ( inj_on_a_nat @ F2 @ ( sup_sup_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) ) )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_a_nat @ F2 @ Xs ) )
=> ( ( distinct_nat @ ( map_a_nat @ F2 @ Xs ) )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_a_nat @ F2 @ Ys ) )
=> ( ( distinct_nat @ ( map_a_nat @ F2 @ Ys ) )
=> ( ( ( set_a2 @ Xs )
= ( set_a2 @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ) ).
% map_sorted_distinct_set_unique
thf(fact_708_map__sorted__distinct__set__unique,axiom,
! [F2: list_nat > nat,Xs: list_list_nat,Ys: list_list_nat] :
( ( inj_on_list_nat_nat @ F2 @ ( sup_sup_set_list_nat @ ( set_list_nat2 @ Xs ) @ ( set_list_nat2 @ Ys ) ) )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_list_nat_nat @ F2 @ Xs ) )
=> ( ( distinct_nat @ ( map_list_nat_nat @ F2 @ Xs ) )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_list_nat_nat @ F2 @ Ys ) )
=> ( ( distinct_nat @ ( map_list_nat_nat @ F2 @ Ys ) )
=> ( ( ( set_list_nat2 @ Xs )
= ( set_list_nat2 @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ) ).
% map_sorted_distinct_set_unique
thf(fact_709_map__sorted__distinct__set__unique,axiom,
! [F2: a > int,Xs: list_a,Ys: list_a] :
( ( inj_on_a_int @ F2 @ ( sup_sup_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) ) )
=> ( ( sorted_wrt_int @ ord_less_eq_int @ ( map_a_int @ F2 @ Xs ) )
=> ( ( distinct_int @ ( map_a_int @ F2 @ Xs ) )
=> ( ( sorted_wrt_int @ ord_less_eq_int @ ( map_a_int @ F2 @ Ys ) )
=> ( ( distinct_int @ ( map_a_int @ F2 @ Ys ) )
=> ( ( ( set_a2 @ Xs )
= ( set_a2 @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ) ).
% map_sorted_distinct_set_unique
thf(fact_710_map__sorted__distinct__set__unique,axiom,
! [F2: list_nat > int,Xs: list_list_nat,Ys: list_list_nat] :
( ( inj_on_list_nat_int @ F2 @ ( sup_sup_set_list_nat @ ( set_list_nat2 @ Xs ) @ ( set_list_nat2 @ Ys ) ) )
=> ( ( sorted_wrt_int @ ord_less_eq_int @ ( map_list_nat_int @ F2 @ Xs ) )
=> ( ( distinct_int @ ( map_list_nat_int @ F2 @ Xs ) )
=> ( ( sorted_wrt_int @ ord_less_eq_int @ ( map_list_nat_int @ F2 @ Ys ) )
=> ( ( distinct_int @ ( map_list_nat_int @ F2 @ Ys ) )
=> ( ( ( set_list_nat2 @ Xs )
= ( set_list_nat2 @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ) ).
% map_sorted_distinct_set_unique
thf(fact_711_rotate__length01,axiom,
! [Xs: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( ( rotate_nat @ N @ Xs )
= Xs ) ) ).
% rotate_length01
thf(fact_712_horner__sum__simps_I1_J,axiom,
! [F2: nat > nat,A4: nat] :
( ( groups7488368174851004413at_nat @ F2 @ A4 @ nil_nat )
= zero_zero_nat ) ).
% horner_sum_simps(1)
thf(fact_713_horner__sum__simps_I1_J,axiom,
! [F2: list_nat > nat,A4: nat] :
( ( groups42626897855390989at_nat @ F2 @ A4 @ nil_list_nat )
= zero_zero_nat ) ).
% horner_sum_simps(1)
thf(fact_714_horner__sum__simps_I1_J,axiom,
! [F2: nat > int,A4: int] :
( ( groups7485877704341954137at_int @ F2 @ A4 @ nil_nat )
= zero_zero_int ) ).
% horner_sum_simps(1)
thf(fact_715_horner__sum__simps_I1_J,axiom,
! [F2: list_nat > int,A4: int] :
( ( groups40136427346340713at_int @ F2 @ A4 @ nil_list_nat )
= zero_zero_int ) ).
% horner_sum_simps(1)
thf(fact_716_insert__subset,axiom,
! [X: list_nat,A: set_list_nat,B: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( insert_list_nat2 @ X @ A ) @ B )
= ( ( member_list_nat2 @ X @ B )
& ( ord_le6045566169113846134st_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_717_insert__subset,axiom,
! [X: a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a2 @ X @ A ) @ B )
= ( ( member_a2 @ X @ B )
& ( ord_less_eq_set_a @ A @ B ) ) ) ).
% insert_subset
thf(fact_718_rotate__is__Nil__conv,axiom,
! [N: nat,Xs: list_nat] :
( ( ( rotate_nat @ N @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% rotate_is_Nil_conv
thf(fact_719_rotate__is__Nil__conv,axiom,
! [N: nat,Xs: list_list_nat] :
( ( ( rotate_list_nat @ N @ Xs )
= nil_list_nat )
= ( Xs = nil_list_nat ) ) ).
% rotate_is_Nil_conv
thf(fact_720_set__rotate,axiom,
! [N: nat,Xs: list_a] :
( ( set_a2 @ ( rotate_a @ N @ Xs ) )
= ( set_a2 @ Xs ) ) ).
% set_rotate
thf(fact_721_set__rotate,axiom,
! [N: nat,Xs: list_list_nat] :
( ( set_list_nat2 @ ( rotate_list_nat @ N @ Xs ) )
= ( set_list_nat2 @ Xs ) ) ).
% set_rotate
thf(fact_722_length__rotate,axiom,
! [N: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( rotate_nat @ N @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rotate
thf(fact_723_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ Z2 )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% int_one_le_iff_zero_less
thf(fact_724_zero__le__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ? [N3: nat] :
( K2
= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_725_nonneg__int__cases,axiom,
! [K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ~ ! [N3: nat] :
( K2
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_726_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_727_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_728_sorted__wrt_Osimps_I1_J,axiom,
! [P2: list_nat > list_nat > $o] : ( sorted_wrt_list_nat @ P2 @ nil_list_nat ) ).
% sorted_wrt.simps(1)
thf(fact_729_sorted__wrt_Osimps_I1_J,axiom,
! [P2: nat > nat > $o] : ( sorted_wrt_nat @ P2 @ nil_nat ) ).
% sorted_wrt.simps(1)
thf(fact_730_sorted__wrt__mono__rel,axiom,
! [Xs: list_a,P2: a > a > $o,Q: a > a > $o] :
( ! [X3: a,Y3: a] :
( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
=> ( ( member_a2 @ Y3 @ ( set_a2 @ Xs ) )
=> ( ( P2 @ X3 @ Y3 )
=> ( Q @ X3 @ Y3 ) ) ) )
=> ( ( sorted_wrt_a @ P2 @ Xs )
=> ( sorted_wrt_a @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_731_sorted__wrt__mono__rel,axiom,
! [Xs: list_list_nat,P2: list_nat > list_nat > $o,Q: list_nat > list_nat > $o] :
( ! [X3: list_nat,Y3: list_nat] :
( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( ( member_list_nat2 @ Y3 @ ( set_list_nat2 @ Xs ) )
=> ( ( P2 @ X3 @ Y3 )
=> ( Q @ X3 @ Y3 ) ) ) )
=> ( ( sorted_wrt_list_nat @ P2 @ Xs )
=> ( sorted_wrt_list_nat @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_732_sorted__wrt__mono__rel,axiom,
! [Xs: list_nat,P2: nat > nat > $o,Q: nat > nat > $o] :
( ! [X3: nat,Y3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( member_nat2 @ Y3 @ ( set_nat2 @ Xs ) )
=> ( ( P2 @ X3 @ Y3 )
=> ( Q @ X3 @ Y3 ) ) ) )
=> ( ( sorted_wrt_nat @ P2 @ Xs )
=> ( sorted_wrt_nat @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_733_subset__insert,axiom,
! [X: list_nat,A: set_list_nat,B: set_list_nat] :
( ~ ( member_list_nat2 @ X @ A )
=> ( ( ord_le6045566169113846134st_nat @ A @ ( insert_list_nat2 @ X @ B ) )
= ( ord_le6045566169113846134st_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_734_subset__insert,axiom,
! [X: a,A: set_a,B: set_a] :
( ~ ( member_a2 @ X @ A )
=> ( ( ord_less_eq_set_a @ A @ ( insert_a2 @ X @ B ) )
= ( ord_less_eq_set_a @ A @ B ) ) ) ).
% subset_insert
thf(fact_735_subset__code_I1_J,axiom,
! [Xs: list_a,B: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B )
= ( ! [X2: a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( member_a2 @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_736_subset__code_I1_J,axiom,
! [Xs: list_list_nat,B: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ B )
= ( ! [X2: list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( member_list_nat2 @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_737_subset__inj__on,axiom,
! [F2: a > nat,B: set_a,A: set_a] :
( ( inj_on_a_nat @ F2 @ B )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( inj_on_a_nat @ F2 @ A ) ) ) ).
% subset_inj_on
thf(fact_738_subset__inj__on,axiom,
! [F2: list_nat > nat,B: set_list_nat,A: set_list_nat] :
( ( inj_on_list_nat_nat @ F2 @ B )
=> ( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( inj_on_list_nat_nat @ F2 @ A ) ) ) ).
% subset_inj_on
thf(fact_739_inj__on__subset,axiom,
! [F2: a > nat,A: set_a,B: set_a] :
( ( inj_on_a_nat @ F2 @ A )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( inj_on_a_nat @ F2 @ B ) ) ) ).
% inj_on_subset
thf(fact_740_inj__on__subset,axiom,
! [F2: list_nat > nat,A: set_list_nat,B: set_list_nat] :
( ( inj_on_list_nat_nat @ F2 @ A )
=> ( ( ord_le6045566169113846134st_nat @ B @ A )
=> ( inj_on_list_nat_nat @ F2 @ B ) ) ) ).
% inj_on_subset
thf(fact_741_rotate__map,axiom,
! [N: nat,F2: a > nat,Xs: list_a] :
( ( rotate_nat @ N @ ( map_a_nat @ F2 @ Xs ) )
= ( map_a_nat @ F2 @ ( rotate_a @ N @ Xs ) ) ) ).
% rotate_map
thf(fact_742_rotate__map,axiom,
! [N: nat,F2: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat] :
( ( rotate1033626827900196251at_nat @ N @ ( map_li6003994582982014139at_nat @ F2 @ Xs ) )
= ( map_li6003994582982014139at_nat @ F2 @ ( rotate_list_nat @ N @ Xs ) ) ) ).
% rotate_map
thf(fact_743_rotate__map,axiom,
! [N: nat,F2: list_nat > nat,Xs: list_list_nat] :
( ( rotate_nat @ N @ ( map_list_nat_nat @ F2 @ Xs ) )
= ( map_list_nat_nat @ F2 @ ( rotate_list_nat @ N @ Xs ) ) ) ).
% rotate_map
thf(fact_744_rotate__map,axiom,
! [N: nat,F2: list_nat > list_nat,Xs: list_list_nat] :
( ( rotate_list_nat @ N @ ( map_li7225945977422193158st_nat @ F2 @ Xs ) )
= ( map_li7225945977422193158st_nat @ F2 @ ( rotate_list_nat @ N @ Xs ) ) ) ).
% rotate_map
thf(fact_745_rotate__map,axiom,
! [N: nat,F2: a > a,Xs: list_a] :
( ( rotate_a @ N @ ( map_a_a @ F2 @ Xs ) )
= ( map_a_a @ F2 @ ( rotate_a @ N @ Xs ) ) ) ).
% rotate_map
thf(fact_746_strict__sorted__imp__sorted,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ Xs ) ) ).
% strict_sorted_imp_sorted
thf(fact_747_strict__sorted__imp__sorted,axiom,
! [Xs: list_int] :
( ( sorted_wrt_int @ ord_less_int @ Xs )
=> ( sorted_wrt_int @ ord_less_eq_int @ Xs ) ) ).
% strict_sorted_imp_sorted
thf(fact_748_sorted0,axiom,
sorted_wrt_nat @ ord_less_eq_nat @ nil_nat ).
% sorted0
thf(fact_749_sorted0,axiom,
sorted_wrt_int @ ord_less_eq_int @ nil_int ).
% sorted0
thf(fact_750_strict__sorted__simps_I1_J,axiom,
sorted_wrt_nat @ ord_less_nat @ nil_nat ).
% strict_sorted_simps(1)
thf(fact_751_strict__sorted__simps_I1_J,axiom,
sorted_wrt_int @ ord_less_int @ nil_int ).
% strict_sorted_simps(1)
thf(fact_752_strict__sorted__equal,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs )
=> ( ( sorted_wrt_nat @ ord_less_nat @ Ys )
=> ( ( ( set_nat2 @ Ys )
= ( set_nat2 @ Xs ) )
=> ( Ys = Xs ) ) ) ) ).
% strict_sorted_equal
thf(fact_753_strict__sorted__equal,axiom,
! [Xs: list_int,Ys: list_int] :
( ( sorted_wrt_int @ ord_less_int @ Xs )
=> ( ( sorted_wrt_int @ ord_less_int @ Ys )
=> ( ( ( set_int2 @ Ys )
= ( set_int2 @ Xs ) )
=> ( Ys = Xs ) ) ) ) ).
% strict_sorted_equal
thf(fact_754_subset__code_I2_J,axiom,
! [A: set_a,Ys: list_a] :
( ( ord_less_eq_set_a @ A @ ( coset_a @ Ys ) )
= ( ! [X2: a] :
( ( member_a2 @ X2 @ ( set_a2 @ Ys ) )
=> ~ ( member_a2 @ X2 @ A ) ) ) ) ).
% subset_code(2)
thf(fact_755_subset__code_I2_J,axiom,
! [A: set_list_nat,Ys: list_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ ( coset_list_nat @ Ys ) )
= ( ! [X2: list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Ys ) )
=> ~ ( member_list_nat2 @ X2 @ A ) ) ) ) ).
% subset_code(2)
thf(fact_756_strict__sorted__iff,axiom,
! [L: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ L )
= ( ( sorted_wrt_nat @ ord_less_eq_nat @ L )
& ( distinct_nat @ L ) ) ) ).
% strict_sorted_iff
thf(fact_757_strict__sorted__iff,axiom,
! [L: list_int] :
( ( sorted_wrt_int @ ord_less_int @ L )
= ( ( sorted_wrt_int @ ord_less_eq_int @ L )
& ( distinct_int @ L ) ) ) ).
% strict_sorted_iff
thf(fact_758_sorted__distinct__set__unique,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( distinct_nat @ Xs )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ys )
=> ( ( distinct_nat @ Ys )
=> ( ( ( set_nat2 @ Xs )
= ( set_nat2 @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ).
% sorted_distinct_set_unique
thf(fact_759_sorted__distinct__set__unique,axiom,
! [Xs: list_int,Ys: list_int] :
( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
=> ( ( distinct_int @ Xs )
=> ( ( sorted_wrt_int @ ord_less_eq_int @ Ys )
=> ( ( distinct_int @ Ys )
=> ( ( ( set_int2 @ Xs )
= ( set_int2 @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ).
% sorted_distinct_set_unique
thf(fact_760_sorted__wrt01,axiom,
! [Xs: list_nat,P2: nat > nat > $o] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( sorted_wrt_nat @ P2 @ Xs ) ) ).
% sorted_wrt01
thf(fact_761_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_nat @ ( coset_nat @ nil_nat ) @ ( set_nat2 @ nil_nat ) ) ).
% subset_code(3)
thf(fact_762_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_a @ ( coset_a @ nil_a ) @ ( set_a2 @ nil_a ) ) ).
% subset_code(3)
thf(fact_763_subset__code_I3_J,axiom,
~ ( ord_le6045566169113846134st_nat @ ( coset_list_nat @ nil_list_nat ) @ ( set_list_nat2 @ nil_list_nat ) ) ).
% subset_code(3)
thf(fact_764_sorted01,axiom,
! [Xs: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ Xs ) ) ).
% sorted01
thf(fact_765_sorted01,axiom,
! [Xs: list_int] :
( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ one_one_nat )
=> ( sorted_wrt_int @ ord_less_eq_int @ Xs ) ) ).
% sorted01
thf(fact_766_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_767_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_768_folding__insort__key_Oidem__if__sorted__distinct,axiom,
! [Less_eq: a > a > $o,Less: a > a > $o,S2: set_a,F2: a > a,Xs: list_a] :
( ( foldin4382019238405368997ey_a_a @ Less_eq @ Less @ S2 @ F2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ S2 )
=> ( ( sorted_wrt_a @ Less_eq @ ( map_a_a @ F2 @ Xs ) )
=> ( ( distinct_a @ Xs )
=> ( ( sorted9165639089771494146et_a_a @ Less_eq @ F2 @ ( set_a2 @ Xs ) )
= Xs ) ) ) ) ) ).
% folding_insort_key.idem_if_sorted_distinct
thf(fact_769_folding__insort__key_Oidem__if__sorted__distinct,axiom,
! [Less_eq: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,Less: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,S2: set_list_nat,F2: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat] :
( ( foldin5963648469059051996st_nat @ Less_eq @ Less @ S2 @ F2 )
=> ( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ S2 )
=> ( ( sorted7895944382797763278at_nat @ Less_eq @ ( map_li6003994582982014139at_nat @ F2 @ Xs ) )
=> ( ( distinct_list_nat @ Xs )
=> ( ( sorted2168255887472420671st_nat @ Less_eq @ F2 @ ( set_list_nat2 @ Xs ) )
= Xs ) ) ) ) ) ).
% folding_insort_key.idem_if_sorted_distinct
thf(fact_770_folding__insort__key_Oidem__if__sorted__distinct,axiom,
! [Less_eq: list_nat > list_nat > $o,Less: list_nat > list_nat > $o,S2: set_list_nat,F2: list_nat > list_nat,Xs: list_list_nat] :
( ( foldin1968479528632951399st_nat @ Less_eq @ Less @ S2 @ F2 )
=> ( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ S2 )
=> ( ( sorted_wrt_list_nat @ Less_eq @ ( map_li7225945977422193158st_nat @ F2 @ Xs ) )
=> ( ( distinct_list_nat @ Xs )
=> ( ( sorted56056757137290820st_nat @ Less_eq @ F2 @ ( set_list_nat2 @ Xs ) )
= Xs ) ) ) ) ) ).
% folding_insort_key.idem_if_sorted_distinct
thf(fact_771_folding__insort__key_Oidem__if__sorted__distinct,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S2: set_a,F2: a > nat,Xs: list_a] :
( ( foldin508877545616633799_nat_a @ Less_eq @ Less @ S2 @ F2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ S2 )
=> ( ( sorted_wrt_nat @ Less_eq @ ( map_a_nat @ F2 @ Xs ) )
=> ( ( distinct_a @ Xs )
=> ( ( sorted2884982002246595626_nat_a @ Less_eq @ F2 @ ( set_a2 @ Xs ) )
= Xs ) ) ) ) ) ).
% folding_insort_key.idem_if_sorted_distinct
thf(fact_772_folding__insort__key_Oidem__if__sorted__distinct,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S2: set_list_nat,F2: list_nat > nat,Xs: list_list_nat] :
( ( foldin951631548397865559st_nat @ Less_eq @ Less @ S2 @ F2 )
=> ( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ S2 )
=> ( ( sorted_wrt_nat @ Less_eq @ ( map_list_nat_nat @ F2 @ Xs ) )
=> ( ( distinct_list_nat @ Xs )
=> ( ( sorted4005134659417452724st_nat @ Less_eq @ F2 @ ( set_list_nat2 @ Xs ) )
= Xs ) ) ) ) ) ).
% folding_insort_key.idem_if_sorted_distinct
thf(fact_773_sorted__insort__insert__key,axiom,
! [F2: a > nat,Xs: list_a,X: a] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_a_nat @ F2 @ Xs ) )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_a_nat @ F2 @ ( linord1046132949341221836_a_nat @ F2 @ X @ Xs ) ) ) ) ).
% sorted_insort_insert_key
thf(fact_774_sorted__insort__insert__key,axiom,
! [F2: list_nat > nat,Xs: list_list_nat,X: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_list_nat_nat @ F2 @ Xs ) )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_list_nat_nat @ F2 @ ( linord3253225449353161780at_nat @ F2 @ X @ Xs ) ) ) ) ).
% sorted_insort_insert_key
thf(fact_775_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( distinct_nat @ Xs )
=> ( ( linord2614967742042102400et_nat @ ( set_nat2 @ Xs ) )
= Xs ) ) ) ).
% sorted_list_of_set.idem_if_sorted_distinct
thf(fact_776_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
! [Xs: list_int] :
( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
=> ( ( distinct_int @ Xs )
=> ( ( linord2612477271533052124et_int @ ( set_int2 @ Xs ) )
= Xs ) ) ) ).
% sorted_list_of_set.idem_if_sorted_distinct
thf(fact_777_subsetI,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ A )
=> ( member_list_nat2 @ X3 @ B ) )
=> ( ord_le6045566169113846134st_nat @ A @ B ) ) ).
% subsetI
thf(fact_778_subsetI,axiom,
! [A: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a2 @ X3 @ A )
=> ( member_a2 @ X3 @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% subsetI
thf(fact_779_in__mono,axiom,
! [A: set_list_nat,B: set_list_nat,X: list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( member_list_nat2 @ X @ A )
=> ( member_list_nat2 @ X @ B ) ) ) ).
% in_mono
thf(fact_780_in__mono,axiom,
! [A: set_a,B: set_a,X: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a2 @ X @ A )
=> ( member_a2 @ X @ B ) ) ) ).
% in_mono
thf(fact_781_subsetD,axiom,
! [A: set_list_nat,B: set_list_nat,C: list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( member_list_nat2 @ C @ A )
=> ( member_list_nat2 @ C @ B ) ) ) ).
% subsetD
thf(fact_782_subsetD,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( member_a2 @ C @ A )
=> ( member_a2 @ C @ B ) ) ) ).
% subsetD
thf(fact_783_subset__eq,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A3: set_list_nat,B6: set_list_nat] :
! [X2: list_nat] :
( ( member_list_nat2 @ X2 @ A3 )
=> ( member_list_nat2 @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_784_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B6: set_a] :
! [X2: a] :
( ( member_a2 @ X2 @ A3 )
=> ( member_a2 @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_785_subset__iff,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A3: set_list_nat,B6: set_list_nat] :
! [T2: list_nat] :
( ( member_list_nat2 @ T2 @ A3 )
=> ( member_list_nat2 @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_786_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B6: set_a] :
! [T2: a] :
( ( member_a2 @ T2 @ A3 )
=> ( member_a2 @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_787_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
! [A: set_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( linord2614967742042102400et_nat @ A ) ) ).
% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_788_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
! [A: set_int] : ( sorted_wrt_int @ ord_less_eq_int @ ( linord2612477271533052124et_int @ A ) ) ).
% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_789_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
! [A: set_nat] : ( sorted_wrt_nat @ ord_less_nat @ ( linord2614967742042102400et_nat @ A ) ) ).
% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_790_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
! [A: set_int] : ( sorted_wrt_int @ ord_less_int @ ( linord2612477271533052124et_int @ A ) ) ).
% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_791_folding__insort__key_Odistinct__sorted__key__list__of__set,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S2: set_a,F2: a > nat,A: set_a] :
( ( foldin508877545616633799_nat_a @ Less_eq @ Less @ S2 @ F2 )
=> ( ( ord_less_eq_set_a @ A @ S2 )
=> ( distinct_nat @ ( map_a_nat @ F2 @ ( sorted2884982002246595626_nat_a @ Less_eq @ F2 @ A ) ) ) ) ) ).
% folding_insort_key.distinct_sorted_key_list_of_set
thf(fact_792_folding__insort__key_Odistinct__sorted__key__list__of__set,axiom,
! [Less_eq: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,Less: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,S2: set_list_nat,F2: list_nat > set_Pr1261947904930325089at_nat,A: set_list_nat] :
( ( foldin5963648469059051996st_nat @ Less_eq @ Less @ S2 @ F2 )
=> ( ( ord_le6045566169113846134st_nat @ A @ S2 )
=> ( distin8719635941469336154at_nat @ ( map_li6003994582982014139at_nat @ F2 @ ( sorted2168255887472420671st_nat @ Less_eq @ F2 @ A ) ) ) ) ) ).
% folding_insort_key.distinct_sorted_key_list_of_set
thf(fact_793_folding__insort__key_Odistinct__sorted__key__list__of__set,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S2: set_list_nat,F2: list_nat > nat,A: set_list_nat] :
( ( foldin951631548397865559st_nat @ Less_eq @ Less @ S2 @ F2 )
=> ( ( ord_le6045566169113846134st_nat @ A @ S2 )
=> ( distinct_nat @ ( map_list_nat_nat @ F2 @ ( sorted4005134659417452724st_nat @ Less_eq @ F2 @ A ) ) ) ) ) ).
% folding_insort_key.distinct_sorted_key_list_of_set
thf(fact_794_folding__insort__key_Odistinct__sorted__key__list__of__set,axiom,
! [Less_eq: list_nat > list_nat > $o,Less: list_nat > list_nat > $o,S2: set_list_nat,F2: list_nat > list_nat,A: set_list_nat] :
( ( foldin1968479528632951399st_nat @ Less_eq @ Less @ S2 @ F2 )
=> ( ( ord_le6045566169113846134st_nat @ A @ S2 )
=> ( distinct_list_nat @ ( map_li7225945977422193158st_nat @ F2 @ ( sorted56056757137290820st_nat @ Less_eq @ F2 @ A ) ) ) ) ) ).
% folding_insort_key.distinct_sorted_key_list_of_set
thf(fact_795_folding__insort__key_Odistinct__sorted__key__list__of__set,axiom,
! [Less_eq: a > a > $o,Less: a > a > $o,S2: set_a,F2: a > a,A: set_a] :
( ( foldin4382019238405368997ey_a_a @ Less_eq @ Less @ S2 @ F2 )
=> ( ( ord_less_eq_set_a @ A @ S2 )
=> ( distinct_a @ ( map_a_a @ F2 @ ( sorted9165639089771494146et_a_a @ Less_eq @ F2 @ A ) ) ) ) ) ).
% folding_insort_key.distinct_sorted_key_list_of_set
thf(fact_796_folding__insort__key_Osorted__sorted__key__list__of__set,axiom,
! [Less_eq: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,Less: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,S2: set_list_nat,F2: list_nat > set_Pr1261947904930325089at_nat,A: set_list_nat] :
( ( foldin5963648469059051996st_nat @ Less_eq @ Less @ S2 @ F2 )
=> ( ( ord_le6045566169113846134st_nat @ A @ S2 )
=> ( sorted7895944382797763278at_nat @ Less_eq @ ( map_li6003994582982014139at_nat @ F2 @ ( sorted2168255887472420671st_nat @ Less_eq @ F2 @ A ) ) ) ) ) ).
% folding_insort_key.sorted_sorted_key_list_of_set
thf(fact_797_folding__insort__key_Osorted__sorted__key__list__of__set,axiom,
! [Less_eq: list_nat > list_nat > $o,Less: list_nat > list_nat > $o,S2: set_list_nat,F2: list_nat > list_nat,A: set_list_nat] :
( ( foldin1968479528632951399st_nat @ Less_eq @ Less @ S2 @ F2 )
=> ( ( ord_le6045566169113846134st_nat @ A @ S2 )
=> ( sorted_wrt_list_nat @ Less_eq @ ( map_li7225945977422193158st_nat @ F2 @ ( sorted56056757137290820st_nat @ Less_eq @ F2 @ A ) ) ) ) ) ).
% folding_insort_key.sorted_sorted_key_list_of_set
thf(fact_798_folding__insort__key_Osorted__sorted__key__list__of__set,axiom,
! [Less_eq: a > a > $o,Less: a > a > $o,S2: set_a,F2: a > a,A: set_a] :
( ( foldin4382019238405368997ey_a_a @ Less_eq @ Less @ S2 @ F2 )
=> ( ( ord_less_eq_set_a @ A @ S2 )
=> ( sorted_wrt_a @ Less_eq @ ( map_a_a @ F2 @ ( sorted9165639089771494146et_a_a @ Less_eq @ F2 @ A ) ) ) ) ) ).
% folding_insort_key.sorted_sorted_key_list_of_set
thf(fact_799_folding__insort__key_Osorted__sorted__key__list__of__set,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S2: set_a,F2: a > nat,A: set_a] :
( ( foldin508877545616633799_nat_a @ Less_eq @ Less @ S2 @ F2 )
=> ( ( ord_less_eq_set_a @ A @ S2 )
=> ( sorted_wrt_nat @ Less_eq @ ( map_a_nat @ F2 @ ( sorted2884982002246595626_nat_a @ Less_eq @ F2 @ A ) ) ) ) ) ).
% folding_insort_key.sorted_sorted_key_list_of_set
thf(fact_800_folding__insort__key_Osorted__sorted__key__list__of__set,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S2: set_list_nat,F2: list_nat > nat,A: set_list_nat] :
( ( foldin951631548397865559st_nat @ Less_eq @ Less @ S2 @ F2 )
=> ( ( ord_le6045566169113846134st_nat @ A @ S2 )
=> ( sorted_wrt_nat @ Less_eq @ ( map_list_nat_nat @ F2 @ ( sorted4005134659417452724st_nat @ Less_eq @ F2 @ A ) ) ) ) ) ).
% folding_insort_key.sorted_sorted_key_list_of_set
thf(fact_801_folding__insort__key_Ostrict__sorted__key__list__of__set,axiom,
! [Less_eq: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,Less: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,S2: set_list_nat,F2: list_nat > set_Pr1261947904930325089at_nat,A: set_list_nat] :
( ( foldin5963648469059051996st_nat @ Less_eq @ Less @ S2 @ F2 )
=> ( ( ord_le6045566169113846134st_nat @ A @ S2 )
=> ( sorted7895944382797763278at_nat @ Less @ ( map_li6003994582982014139at_nat @ F2 @ ( sorted2168255887472420671st_nat @ Less_eq @ F2 @ A ) ) ) ) ) ).
% folding_insort_key.strict_sorted_key_list_of_set
thf(fact_802_folding__insort__key_Ostrict__sorted__key__list__of__set,axiom,
! [Less_eq: list_nat > list_nat > $o,Less: list_nat > list_nat > $o,S2: set_list_nat,F2: list_nat > list_nat,A: set_list_nat] :
( ( foldin1968479528632951399st_nat @ Less_eq @ Less @ S2 @ F2 )
=> ( ( ord_le6045566169113846134st_nat @ A @ S2 )
=> ( sorted_wrt_list_nat @ Less @ ( map_li7225945977422193158st_nat @ F2 @ ( sorted56056757137290820st_nat @ Less_eq @ F2 @ A ) ) ) ) ) ).
% folding_insort_key.strict_sorted_key_list_of_set
thf(fact_803_folding__insort__key_Ostrict__sorted__key__list__of__set,axiom,
! [Less_eq: a > a > $o,Less: a > a > $o,S2: set_a,F2: a > a,A: set_a] :
( ( foldin4382019238405368997ey_a_a @ Less_eq @ Less @ S2 @ F2 )
=> ( ( ord_less_eq_set_a @ A @ S2 )
=> ( sorted_wrt_a @ Less @ ( map_a_a @ F2 @ ( sorted9165639089771494146et_a_a @ Less_eq @ F2 @ A ) ) ) ) ) ).
% folding_insort_key.strict_sorted_key_list_of_set
thf(fact_804_folding__insort__key_Ostrict__sorted__key__list__of__set,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S2: set_a,F2: a > nat,A: set_a] :
( ( foldin508877545616633799_nat_a @ Less_eq @ Less @ S2 @ F2 )
=> ( ( ord_less_eq_set_a @ A @ S2 )
=> ( sorted_wrt_nat @ Less @ ( map_a_nat @ F2 @ ( sorted2884982002246595626_nat_a @ Less_eq @ F2 @ A ) ) ) ) ) ).
% folding_insort_key.strict_sorted_key_list_of_set
thf(fact_805_folding__insort__key_Ostrict__sorted__key__list__of__set,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,S2: set_list_nat,F2: list_nat > nat,A: set_list_nat] :
( ( foldin951631548397865559st_nat @ Less_eq @ Less @ S2 @ F2 )
=> ( ( ord_le6045566169113846134st_nat @ A @ S2 )
=> ( sorted_wrt_nat @ Less @ ( map_list_nat_nat @ F2 @ ( sorted4005134659417452724st_nat @ Less_eq @ F2 @ A ) ) ) ) ) ).
% folding_insort_key.strict_sorted_key_list_of_set
thf(fact_806_imp__le__cong,axiom,
! [X: int,X5: int,P2: $o,P4: $o] :
( ( X = X5 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> ( P2 = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P2 )
= ( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_807_conj__le__cong,axiom,
! [X: int,X5: int,P2: $o,P4: $o] :
( ( X = X5 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> ( P2 = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P2 )
= ( ( ord_less_eq_int @ zero_zero_int @ X5 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_808_neg__int__cases,axiom,
! [K2: int] :
( ( ord_less_int @ K2 @ zero_zero_int )
=> ~ ! [N3: nat] :
( ( K2
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% neg_int_cases
thf(fact_809_nat__less__iff,axiom,
! [W: int,M2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ M2 )
= ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ).
% nat_less_iff
thf(fact_810_sorted__iff__nth__mono,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_811_sorted__iff__nth__mono,axiom,
! [Xs: list_int] :
( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
= ( ! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
=> ( ord_less_eq_int @ ( nth_int @ Xs @ I2 ) @ ( nth_int @ Xs @ J ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_812_neg__equal__iff__equal,axiom,
! [A4: int,B2: int] :
( ( ( uminus_uminus_int @ A4 )
= ( uminus_uminus_int @ B2 ) )
= ( A4 = B2 ) ) ).
% neg_equal_iff_equal
thf(fact_813_add_Oinverse__inverse,axiom,
! [A4: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A4 ) )
= A4 ) ).
% add.inverse_inverse
thf(fact_814_neg__le__iff__le,axiom,
! [B2: int,A4: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A4 ) )
= ( ord_less_eq_int @ A4 @ B2 ) ) ).
% neg_le_iff_le
thf(fact_815_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_816_neg__0__equal__iff__equal,axiom,
! [A4: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A4 ) )
= ( zero_zero_int = A4 ) ) ).
% neg_0_equal_iff_equal
thf(fact_817_neg__equal__0__iff__equal,axiom,
! [A4: int] :
( ( ( uminus_uminus_int @ A4 )
= zero_zero_int )
= ( A4 = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_818_equal__neg__zero,axiom,
! [A4: int] :
( ( A4
= ( uminus_uminus_int @ A4 ) )
= ( A4 = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_819_neg__equal__zero,axiom,
! [A4: int] :
( ( ( uminus_uminus_int @ A4 )
= A4 )
= ( A4 = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_820_neg__less__iff__less,axiom,
! [B2: int,A4: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A4 ) )
= ( ord_less_int @ A4 @ B2 ) ) ).
% neg_less_iff_less
thf(fact_821_inj__uminus,axiom,
! [A: set_int] : ( inj_on_int_int @ uminus_uminus_int @ A ) ).
% inj_uminus
thf(fact_822_neg__0__le__iff__le,axiom,
! [A4: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A4 ) )
= ( ord_less_eq_int @ A4 @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_823_neg__le__0__iff__le,axiom,
! [A4: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A4 ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A4 ) ) ).
% neg_le_0_iff_le
thf(fact_824_less__eq__neg__nonpos,axiom,
! [A4: int] :
( ( ord_less_eq_int @ A4 @ ( uminus_uminus_int @ A4 ) )
= ( ord_less_eq_int @ A4 @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_825_neg__less__eq__nonneg,axiom,
! [A4: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A4 ) @ A4 )
= ( ord_less_eq_int @ zero_zero_int @ A4 ) ) ).
% neg_less_eq_nonneg
thf(fact_826_less__neg__neg,axiom,
! [A4: int] :
( ( ord_less_int @ A4 @ ( uminus_uminus_int @ A4 ) )
= ( ord_less_int @ A4 @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_827_neg__less__pos,axiom,
! [A4: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A4 ) @ A4 )
= ( ord_less_int @ zero_zero_int @ A4 ) ) ).
% neg_less_pos
thf(fact_828_neg__0__less__iff__less,axiom,
! [A4: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A4 ) )
= ( ord_less_int @ A4 @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_829_neg__less__0__iff__less,axiom,
! [A4: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A4 ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A4 ) ) ).
% neg_less_0_iff_less
thf(fact_830_negative__eq__positive,axiom,
! [N: nat,M2: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M2 ) )
= ( ( N = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_831_nth__map,axiom,
! [N: nat,Xs: list_a,F2: a > nat] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( nth_nat @ ( map_a_nat @ F2 @ Xs ) @ N )
= ( F2 @ ( nth_a @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_832_nth__map,axiom,
! [N: nat,Xs: list_list_nat,F2: list_nat > set_Pr1261947904930325089at_nat] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_se5679702624988424552at_nat @ ( map_li6003994582982014139at_nat @ F2 @ Xs ) @ N )
= ( F2 @ ( nth_list_nat @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_833_nth__map,axiom,
! [N: nat,Xs: list_list_nat,F2: list_nat > nat] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_nat @ ( map_list_nat_nat @ F2 @ Xs ) @ N )
= ( F2 @ ( nth_list_nat @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_834_nth__map,axiom,
! [N: nat,Xs: list_list_nat,F2: list_nat > list_nat] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_list_nat @ ( map_li7225945977422193158st_nat @ F2 @ Xs ) @ N )
= ( F2 @ ( nth_list_nat @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_835_nth__map,axiom,
! [N: nat,Xs: list_a,F2: a > a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( map_a_a @ F2 @ Xs ) @ N )
= ( F2 @ ( nth_a @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_836_nth__map,axiom,
! [N: nat,Xs: list_nat,F2: nat > nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( map_nat_nat @ F2 @ Xs ) @ N )
= ( F2 @ ( nth_nat @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_837_nat__0__iff,axiom,
! [I3: int] :
( ( ( nat2 @ I3 )
= zero_zero_nat )
= ( ord_less_eq_int @ I3 @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_838_nat__le__0,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ Z2 @ zero_zero_int )
=> ( ( nat2 @ Z2 )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_839_zless__nat__conj,axiom,
! [W: int,Z2: int] :
( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ( ord_less_int @ zero_zero_int @ Z2 )
& ( ord_less_int @ W @ Z2 ) ) ) ).
% zless_nat_conj
thf(fact_840_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_841_int__nat__eq,axiom,
! [Z2: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= Z2 ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_842_zero__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z2 ) )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% zero_less_nat_eq
thf(fact_843_psubsetD,axiom,
! [A: set_list_nat,B: set_list_nat,C: list_nat] :
( ( ord_le1190675801316882794st_nat @ A @ B )
=> ( ( member_list_nat2 @ C @ A )
=> ( member_list_nat2 @ C @ B ) ) ) ).
% psubsetD
thf(fact_844_psubsetD,axiom,
! [A: set_a,B: set_a,C: a] :
( ( ord_less_set_a @ A @ B )
=> ( ( member_a2 @ C @ A )
=> ( member_a2 @ C @ B ) ) ) ).
% psubsetD
thf(fact_845_le__imp__neg__le,axiom,
! [A4: int,B2: int] :
( ( ord_less_eq_int @ A4 @ B2 )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A4 ) ) ) ).
% le_imp_neg_le
thf(fact_846_minus__le__iff,axiom,
! [A4: int,B2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A4 ) @ B2 )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ A4 ) ) ).
% minus_le_iff
thf(fact_847_le__minus__iff,axiom,
! [A4: int,B2: int] :
( ( ord_less_eq_int @ A4 @ ( uminus_uminus_int @ B2 ) )
= ( ord_less_eq_int @ B2 @ ( uminus_uminus_int @ A4 ) ) ) ).
% le_minus_iff
thf(fact_848_minus__less__iff,axiom,
! [A4: int,B2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A4 ) @ B2 )
= ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ A4 ) ) ).
% minus_less_iff
thf(fact_849_less__minus__iff,axiom,
! [A4: int,B2: int] :
( ( ord_less_int @ A4 @ ( uminus_uminus_int @ B2 ) )
= ( ord_less_int @ B2 @ ( uminus_uminus_int @ A4 ) ) ) ).
% less_minus_iff
thf(fact_850_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_851_minus__equation__iff,axiom,
! [A4: int,B2: int] :
( ( ( uminus_uminus_int @ A4 )
= B2 )
= ( ( uminus_uminus_int @ B2 )
= A4 ) ) ).
% minus_equation_iff
thf(fact_852_equation__minus__iff,axiom,
! [A4: int,B2: int] :
( ( A4
= ( uminus_uminus_int @ B2 ) )
= ( B2
= ( uminus_uminus_int @ A4 ) ) ) ).
% equation_minus_iff
thf(fact_853_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_854_nth__equalityI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I4 )
= ( nth_nat @ Ys @ I4 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_855_Skolem__list__nth,axiom,
! [K2: nat,P2: nat > nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K2 )
=> ? [X6: nat] : ( P2 @ I2 @ X6 ) ) )
= ( ? [Xs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= K2 )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K2 )
=> ( P2 @ I2 @ ( nth_nat @ Xs2 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_856_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_nat,Z3: list_nat] : ( Y5 = Z3 ) )
= ( ^ [Xs2: list_nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys3 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ Xs2 @ I2 )
= ( nth_nat @ Ys3 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_857_ex__nat,axiom,
( ( ^ [P3: nat > $o] :
? [X4: nat] : ( P3 @ X4 ) )
= ( ^ [P: nat > $o] :
? [X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
& ( P @ ( nat2 @ X2 ) ) ) ) ) ).
% ex_nat
thf(fact_858_all__nat,axiom,
( ( ^ [P3: nat > $o] :
! [X4: nat] : ( P3 @ X4 ) )
= ( ^ [P: nat > $o] :
! [X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( P @ ( nat2 @ X2 ) ) ) ) ) ).
% all_nat
thf(fact_859_eq__nat__nat__iff,axiom,
! [Z2: int,Z4: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z4 )
=> ( ( ( nat2 @ Z2 )
= ( nat2 @ Z4 ) )
= ( Z2 = Z4 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_860_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_861_nth__mem,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( member_a2 @ ( nth_a @ Xs @ N ) @ ( set_a2 @ Xs ) ) ) ).
% nth_mem
thf(fact_862_nth__mem,axiom,
! [N: nat,Xs: list_list_nat] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( member_list_nat2 @ ( nth_list_nat @ Xs @ N ) @ ( set_list_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_863_nth__mem,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat2 @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_864_list__ball__nth,axiom,
! [N: nat,Xs: list_a,P2: a > $o] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ! [X3: a] :
( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
=> ( P2 @ X3 ) )
=> ( P2 @ ( nth_a @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_865_list__ball__nth,axiom,
! [N: nat,Xs: list_list_nat,P2: list_nat > $o] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( P2 @ X3 ) )
=> ( P2 @ ( nth_list_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_866_list__ball__nth,axiom,
! [N: nat,Xs: list_nat,P2: nat > $o] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( P2 @ X3 ) )
=> ( P2 @ ( nth_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_867_in__set__conv__nth,axiom,
! [X: a,Xs: list_a] :
( ( member_a2 @ X @ ( set_a2 @ Xs ) )
= ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
& ( ( nth_a @ Xs @ I2 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_868_in__set__conv__nth,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
= ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
& ( ( nth_list_nat @ Xs @ I2 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_869_in__set__conv__nth,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
= ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I2 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_870_all__nth__imp__all__set,axiom,
! [Xs: list_a,P2: a > $o,X: a] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
=> ( P2 @ ( nth_a @ Xs @ I4 ) ) )
=> ( ( member_a2 @ X @ ( set_a2 @ Xs ) )
=> ( P2 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_871_all__nth__imp__all__set,axiom,
! [Xs: list_list_nat,P2: list_nat > $o,X: list_nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( P2 @ ( nth_list_nat @ Xs @ I4 ) ) )
=> ( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
=> ( P2 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_872_all__nth__imp__all__set,axiom,
! [Xs: list_nat,P2: nat > $o,X: nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( P2 @ ( nth_nat @ Xs @ I4 ) ) )
=> ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( P2 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_873_all__set__conv__all__nth,axiom,
! [Xs: list_a,P2: a > $o] :
( ( ! [X2: a] :
( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
=> ( P2 @ X2 ) ) )
= ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( P2 @ ( nth_a @ Xs @ I2 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_874_all__set__conv__all__nth,axiom,
! [Xs: list_list_nat,P2: list_nat > $o] :
( ( ! [X2: list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( P2 @ X2 ) ) )
= ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( P2 @ ( nth_list_nat @ Xs @ I2 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_875_all__set__conv__all__nth,axiom,
! [Xs: list_nat,P2: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( P2 @ X2 ) ) )
= ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( P2 @ ( nth_nat @ Xs @ I2 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_876_map__equality__iff,axiom,
! [F2: a > nat,Xs: list_a,G: a > nat,Ys: list_a] :
( ( ( map_a_nat @ F2 @ Xs )
= ( map_a_nat @ G @ Ys ) )
= ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Ys ) )
=> ( ( F2 @ ( nth_a @ Xs @ I2 ) )
= ( G @ ( nth_a @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_877_map__equality__iff,axiom,
! [F2: a > a,Xs: list_a,G: a > a,Ys: list_a] :
( ( ( map_a_a @ F2 @ Xs )
= ( map_a_a @ G @ Ys ) )
= ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Ys ) )
=> ( ( F2 @ ( nth_a @ Xs @ I2 ) )
= ( G @ ( nth_a @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_878_map__equality__iff,axiom,
! [F2: a > nat,Xs: list_a,G: nat > nat,Ys: list_nat] :
( ( ( map_a_nat @ F2 @ Xs )
= ( map_nat_nat @ G @ Ys ) )
= ( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( ( F2 @ ( nth_a @ Xs @ I2 ) )
= ( G @ ( nth_nat @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_879_map__equality__iff,axiom,
! [F2: a > a,Xs: list_a,G: nat > a,Ys: list_nat] :
( ( ( map_a_a @ F2 @ Xs )
= ( map_nat_a @ G @ Ys ) )
= ( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( ( F2 @ ( nth_a @ Xs @ I2 ) )
= ( G @ ( nth_nat @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_880_map__equality__iff,axiom,
! [F2: nat > nat,Xs: list_nat,G: a > nat,Ys: list_a] :
( ( ( map_nat_nat @ F2 @ Xs )
= ( map_a_nat @ G @ Ys ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Ys ) )
=> ( ( F2 @ ( nth_nat @ Xs @ I2 ) )
= ( G @ ( nth_a @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_881_map__equality__iff,axiom,
! [F2: nat > a,Xs: list_nat,G: a > a,Ys: list_a] :
( ( ( map_nat_a @ F2 @ Xs )
= ( map_a_a @ G @ Ys ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Ys ) )
=> ( ( F2 @ ( nth_nat @ Xs @ I2 ) )
= ( G @ ( nth_a @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_882_map__equality__iff,axiom,
! [F2: a > nat,Xs: list_a,G: list_nat > nat,Ys: list_list_nat] :
( ( ( map_a_nat @ F2 @ Xs )
= ( map_list_nat_nat @ G @ Ys ) )
= ( ( ( size_size_list_a @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( F2 @ ( nth_a @ Xs @ I2 ) )
= ( G @ ( nth_list_nat @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_883_map__equality__iff,axiom,
! [F2: list_nat > nat,Xs: list_list_nat,G: a > nat,Ys: list_a] :
( ( ( map_list_nat_nat @ F2 @ Xs )
= ( map_a_nat @ G @ Ys ) )
= ( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_a @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Ys ) )
=> ( ( F2 @ ( nth_list_nat @ Xs @ I2 ) )
= ( G @ ( nth_a @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_884_map__equality__iff,axiom,
! [F2: list_nat > nat,Xs: list_list_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_list_nat_nat @ F2 @ Xs )
= ( map_nat_nat @ G @ Ys ) )
= ( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( ( F2 @ ( nth_list_nat @ Xs @ I2 ) )
= ( G @ ( nth_nat @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_885_map__equality__iff,axiom,
! [F2: nat > nat,Xs: list_nat,G: list_nat > nat,Ys: list_list_nat] :
( ( ( map_nat_nat @ F2 @ Xs )
= ( map_list_nat_nat @ G @ Ys ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( F2 @ ( nth_nat @ Xs @ I2 ) )
= ( G @ ( nth_list_nat @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_886_nat__mono__iff,axiom,
! [Z2: int,W: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ W @ Z2 ) ) ) ).
% nat_mono_iff
thf(fact_887_distinct__conv__nth,axiom,
( distinct_nat
= ( ^ [Xs2: list_nat] :
! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
=> ! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ( ( I2 != J )
=> ( ( nth_nat @ Xs2 @ I2 )
!= ( nth_nat @ Xs2 @ J ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_888_nth__eq__iff__index__eq,axiom,
! [Xs: list_nat,I3: nat,J2: nat] :
( ( distinct_nat @ Xs )
=> ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ( ( nth_nat @ Xs @ I3 )
= ( nth_nat @ Xs @ J2 ) )
= ( I3 = J2 ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_889_sorted__wrt__nth__less,axiom,
! [P2: nat > nat > $o,Xs: list_nat,I3: nat,J2: nat] :
( ( sorted_wrt_nat @ P2 @ Xs )
=> ( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( P2 @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Xs @ J2 ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_890_sorted__wrt__iff__nth__less,axiom,
( sorted_wrt_nat
= ( ^ [P: nat > nat > $o,Xs2: list_nat] :
! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ ( nth_nat @ Xs2 @ I2 ) @ ( nth_nat @ Xs2 @ J ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_891_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(3)
thf(fact_892_le__minus__one__simps_I1_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% le_minus_one_simps(1)
thf(fact_893_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(3)
thf(fact_894_less__minus__one__simps_I1_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% less_minus_one_simps(1)
thf(fact_895_int__eq__iff,axiom,
! [M2: nat,Z2: int] :
( ( ( semiri1314217659103216013at_int @ M2 )
= Z2 )
= ( ( M2
= ( nat2 @ Z2 ) )
& ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ) ).
% int_eq_iff
thf(fact_896_nat__0__le,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= Z2 ) ) ).
% nat_0_le
thf(fact_897_int__cases4,axiom,
! [M2: int] :
( ! [N3: nat] :
( M2
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( M2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_898_kernel__of__eq,axiom,
! [X: list_a,Y2: list_a] :
( ( ( equiva2867628904822520638l_of_a @ X )
= ( equiva2867628904822520638l_of_a @ Y2 ) )
= ( ( ( size_size_list_a @ X )
= ( size_size_list_a @ Y2 ) )
& ! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_a @ X ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ( nth_a @ X @ I2 )
= ( nth_a @ X @ J ) )
= ( ( nth_a @ Y2 @ I2 )
= ( nth_a @ Y2 @ J ) ) ) ) ) ) ) ).
% kernel_of_eq
thf(fact_899_kernel__of__eq,axiom,
! [X: list_a,Y2: list_b] :
( ( ( equiva2867628904822520638l_of_a @ X )
= ( equiva2867628904822520639l_of_b @ Y2 ) )
= ( ( ( size_size_list_a @ X )
= ( size_size_list_b @ Y2 ) )
& ! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_a @ X ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ( nth_a @ X @ I2 )
= ( nth_a @ X @ J ) )
= ( ( nth_b @ Y2 @ I2 )
= ( nth_b @ Y2 @ J ) ) ) ) ) ) ) ).
% kernel_of_eq
thf(fact_900_kernel__of__eq,axiom,
! [X: list_a,Y2: list_nat] :
( ( ( equiva2867628904822520638l_of_a @ X )
= ( equiva2048684438135499664of_nat @ Y2 ) )
= ( ( ( size_size_list_a @ X )
= ( size_size_list_nat @ Y2 ) )
& ! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_a @ X ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ( nth_a @ X @ I2 )
= ( nth_a @ X @ J ) )
= ( ( nth_nat @ Y2 @ I2 )
= ( nth_nat @ Y2 @ J ) ) ) ) ) ) ) ).
% kernel_of_eq
thf(fact_901_kernel__of__eq,axiom,
! [X: list_b,Y2: list_a] :
( ( ( equiva2867628904822520639l_of_b @ X )
= ( equiva2867628904822520638l_of_a @ Y2 ) )
= ( ( ( size_size_list_b @ X )
= ( size_size_list_a @ Y2 ) )
& ! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_b @ X ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ( nth_b @ X @ I2 )
= ( nth_b @ X @ J ) )
= ( ( nth_a @ Y2 @ I2 )
= ( nth_a @ Y2 @ J ) ) ) ) ) ) ) ).
% kernel_of_eq
thf(fact_902_kernel__of__eq,axiom,
! [X: list_b,Y2: list_b] :
( ( ( equiva2867628904822520639l_of_b @ X )
= ( equiva2867628904822520639l_of_b @ Y2 ) )
= ( ( ( size_size_list_b @ X )
= ( size_size_list_b @ Y2 ) )
& ! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_b @ X ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ( nth_b @ X @ I2 )
= ( nth_b @ X @ J ) )
= ( ( nth_b @ Y2 @ I2 )
= ( nth_b @ Y2 @ J ) ) ) ) ) ) ) ).
% kernel_of_eq
thf(fact_903_kernel__of__eq,axiom,
! [X: list_b,Y2: list_nat] :
( ( ( equiva2867628904822520639l_of_b @ X )
= ( equiva2048684438135499664of_nat @ Y2 ) )
= ( ( ( size_size_list_b @ X )
= ( size_size_list_nat @ Y2 ) )
& ! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_b @ X ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ( nth_b @ X @ I2 )
= ( nth_b @ X @ J ) )
= ( ( nth_nat @ Y2 @ I2 )
= ( nth_nat @ Y2 @ J ) ) ) ) ) ) ) ).
% kernel_of_eq
thf(fact_904_kernel__of__eq,axiom,
! [X: list_nat,Y2: list_a] :
( ( ( equiva2048684438135499664of_nat @ X )
= ( equiva2867628904822520638l_of_a @ Y2 ) )
= ( ( ( size_size_list_nat @ X )
= ( size_size_list_a @ Y2 ) )
& ! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_nat @ X ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ( nth_nat @ X @ I2 )
= ( nth_nat @ X @ J ) )
= ( ( nth_a @ Y2 @ I2 )
= ( nth_a @ Y2 @ J ) ) ) ) ) ) ) ).
% kernel_of_eq
thf(fact_905_kernel__of__eq,axiom,
! [X: list_nat,Y2: list_b] :
( ( ( equiva2048684438135499664of_nat @ X )
= ( equiva2867628904822520639l_of_b @ Y2 ) )
= ( ( ( size_size_list_nat @ X )
= ( size_size_list_b @ Y2 ) )
& ! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_nat @ X ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ( nth_nat @ X @ I2 )
= ( nth_nat @ X @ J ) )
= ( ( nth_b @ Y2 @ I2 )
= ( nth_b @ Y2 @ J ) ) ) ) ) ) ) ).
% kernel_of_eq
thf(fact_906_kernel__of__eq,axiom,
! [X: list_nat,Y2: list_nat] :
( ( ( equiva2048684438135499664of_nat @ X )
= ( equiva2048684438135499664of_nat @ Y2 ) )
= ( ( ( size_size_list_nat @ X )
= ( size_size_list_nat @ Y2 ) )
& ! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_nat @ X ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ( nth_nat @ X @ I2 )
= ( nth_nat @ X @ J ) )
= ( ( nth_nat @ Y2 @ I2 )
= ( nth_nat @ Y2 @ J ) ) ) ) ) ) ) ).
% kernel_of_eq
thf(fact_907_int__zle__neg,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) )
= ( ( N = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_908_nonpos__int__cases,axiom,
! [K2: int] :
( ( ord_less_eq_int @ K2 @ zero_zero_int )
=> ~ ! [N3: nat] :
( K2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% nonpos_int_cases
thf(fact_909_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_910_sorted__wrt__less__idx,axiom,
! [Ns: list_nat,I3: nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Ns )
=> ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ns ) )
=> ( ord_less_eq_nat @ I3 @ ( nth_nat @ Ns @ I3 ) ) ) ) ).
% sorted_wrt_less_idx
thf(fact_911_sorted__iff__nth__mono__less,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_912_sorted__iff__nth__mono__less,axiom,
! [Xs: list_int] :
( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
= ( ! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
=> ( ord_less_eq_int @ ( nth_int @ Xs @ I2 ) @ ( nth_int @ Xs @ J ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_913_distinct__Ex1,axiom,
! [Xs: list_a,X: a] :
( ( distinct_a @ Xs )
=> ( ( member_a2 @ X @ ( set_a2 @ Xs ) )
=> ? [X3: nat] :
( ( ord_less_nat @ X3 @ ( size_size_list_a @ Xs ) )
& ( ( nth_a @ Xs @ X3 )
= X )
& ! [Y4: nat] :
( ( ( ord_less_nat @ Y4 @ ( size_size_list_a @ Xs ) )
& ( ( nth_a @ Xs @ Y4 )
= X ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_914_distinct__Ex1,axiom,
! [Xs: list_list_nat,X: list_nat] :
( ( distinct_list_nat @ Xs )
=> ( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
=> ? [X3: nat] :
( ( ord_less_nat @ X3 @ ( size_s3023201423986296836st_nat @ Xs ) )
& ( ( nth_list_nat @ Xs @ X3 )
= X )
& ! [Y4: nat] :
( ( ( ord_less_nat @ Y4 @ ( size_s3023201423986296836st_nat @ Xs ) )
& ( ( nth_list_nat @ Xs @ Y4 )
= X ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_915_distinct__Ex1,axiom,
! [Xs: list_nat,X: nat] :
( ( distinct_nat @ Xs )
=> ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ? [X3: nat] :
( ( ord_less_nat @ X3 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ X3 )
= X )
& ! [Y4: nat] :
( ( ( ord_less_nat @ Y4 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ Y4 )
= X ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_916_nat__eq__iff2,axiom,
! [M2: nat,W: int] :
( ( M2
= ( nat2 @ W ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_917_nat__eq__iff,axiom,
! [W: int,M2: nat] :
( ( ( nat2 @ W )
= M2 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_918_nat__less__eq__zless,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ W @ Z2 ) ) ) ).
% nat_less_eq_zless
thf(fact_919_split__nat,axiom,
! [P2: nat > $o,I3: int] :
( ( P2 @ ( nat2 @ I3 ) )
= ( ! [N2: nat] :
( ( I3
= ( semiri1314217659103216013at_int @ N2 ) )
=> ( P2 @ N2 ) )
& ( ( ord_less_int @ I3 @ zero_zero_int )
=> ( P2 @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_920_nat__le__eq__zle,axiom,
! [W: int,Z2: int] :
( ( ( ord_less_int @ zero_zero_int @ W )
| ( ord_less_eq_int @ zero_zero_int @ Z2 ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ord_less_eq_int @ W @ Z2 ) ) ) ).
% nat_le_eq_zle
thf(fact_921_le__nat__iff,axiom,
! [K2: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ( ( ord_less_eq_nat @ N @ ( nat2 @ K2 ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K2 ) ) ) ).
% le_nat_iff
thf(fact_922_int__cases3,axiom,
! [K2: int] :
( ( K2 != zero_zero_int )
=> ( ! [N3: nat] :
( ( K2
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
=> ~ ! [N3: nat] :
( ( K2
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% int_cases3
thf(fact_923_inj__on__nth,axiom,
! [Xs: list_nat,I5: set_nat] :
( ( distinct_nat @ Xs )
=> ( ! [X3: nat] :
( ( member_nat2 @ X3 @ I5 )
=> ( ord_less_nat @ X3 @ ( size_size_list_nat @ Xs ) ) )
=> ( inj_on_nat_nat @ ( nth_nat @ Xs ) @ I5 ) ) ) ).
% inj_on_nth
thf(fact_924_sorted__nth__mono,axiom,
! [Xs: list_nat,I3: nat,J2: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Xs @ J2 ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_925_sorted__nth__mono,axiom,
! [Xs: list_int,I3: nat,J2: nat] :
( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
=> ( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_int @ Xs ) )
=> ( ord_less_eq_int @ ( nth_int @ Xs @ I3 ) @ ( nth_int @ Xs @ J2 ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_926_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_927_one__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ one_one_int @ Z2 ) ) ).
% one_less_nat_eq
thf(fact_928_of__int__of__nat,axiom,
( ring_1_of_int_int
= ( ^ [K3: int] : ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri1314217659103216013at_int @ ( nat2 @ K3 ) ) ) ) ) ).
% of_int_of_nat
thf(fact_929_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_930_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_931_of__int__0,axiom,
( ( ring_1_of_int_int @ zero_zero_int )
= zero_zero_int ) ).
% of_int_0
thf(fact_932_of__int__0__eq__iff,axiom,
! [Z2: int] :
( ( zero_zero_int
= ( ring_1_of_int_int @ Z2 ) )
= ( Z2 = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_933_of__int__eq__0__iff,axiom,
! [Z2: int] :
( ( ( ring_1_of_int_int @ Z2 )
= zero_zero_int )
= ( Z2 = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_934_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_935_of__int__le__0__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ zero_zero_int )
= ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_936_of__int__0__le__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).
% of_int_0_le_iff
thf(fact_937_of__int__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_938_of__int__0__less__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% of_int_0_less_iff
thf(fact_939_of__nat__nat,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= ( ring_1_of_int_int @ Z2 ) ) ) ).
% of_nat_nat
thf(fact_940_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_941_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_942_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_943_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_944_zero__induct,axiom,
! [P2: nat > $o,K2: nat] :
( ( P2 @ K2 )
=> ( ! [N3: nat] :
( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_945_diff__induct,axiom,
! [P2: nat > nat > $o,M2: nat,N: nat] :
( ! [X3: nat] : ( P2 @ X3 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P2 @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P2 @ X3 @ Y3 )
=> ( P2 @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P2 @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_946_nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) )
=> ( P2 @ N ) ) ) ).
% nat_induct
thf(fact_947_old_Onat_Oexhaust,axiom,
! [Y2: nat] :
( ( Y2 != zero_zero_nat )
=> ~ ! [Nat: nat] :
( Y2
!= ( suc @ Nat ) ) ) ).
% old.nat.exhaust
thf(fact_948_nat_OdiscI,axiom,
! [Nat2: nat,X22: nat] :
( ( Nat2
= ( suc @ X22 ) )
=> ( Nat2 != zero_zero_nat ) ) ).
% nat.discI
thf(fact_949_old_Onat_Odistinct_I1_J,axiom,
! [Nat3: nat] :
( zero_zero_nat
!= ( suc @ Nat3 ) ) ).
% old.nat.distinct(1)
thf(fact_950_old_Onat_Odistinct_I2_J,axiom,
! [Nat3: nat] :
( ( suc @ Nat3 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_951_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_952_enum__rgfs_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N3: nat] :
( X
!= ( suc @ N3 ) ) ) ).
% enum_rgfs.cases
thf(fact_953_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_954_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_955_less__Suc__eq__0__disj,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( M2 = zero_zero_nat )
| ? [J: nat] :
( ( M2
= ( suc @ J ) )
& ( ord_less_nat @ J @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_956_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_957_All__less__Suc2,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P2 @ I2 ) ) )
= ( ( P2 @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P2 @ ( suc @ I2 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_958_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M: nat] :
( N
= ( suc @ M ) ) ) ) ).
% gr0_conv_Suc
thf(fact_959_Ex__less__Suc2,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P2 @ I2 ) ) )
= ( ( P2 @ zero_zero_nat )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P2 @ ( suc @ I2 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_960_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_961_coset__def,axiom,
( coset_a
= ( ^ [Xs2: list_a] : ( uminus_uminus_set_a @ ( set_a2 @ Xs2 ) ) ) ) ).
% coset_def
thf(fact_962_coset__def,axiom,
( coset_list_nat
= ( ^ [Xs2: list_list_nat] : ( uminus3195874150345416415st_nat @ ( set_list_nat2 @ Xs2 ) ) ) ) ).
% coset_def
thf(fact_963_compl__coset,axiom,
! [Xs: list_a] :
( ( uminus_uminus_set_a @ ( coset_a @ Xs ) )
= ( set_a2 @ Xs ) ) ).
% compl_coset
thf(fact_964_compl__coset,axiom,
! [Xs: list_list_nat] :
( ( uminus3195874150345416415st_nat @ ( coset_list_nat @ Xs ) )
= ( set_list_nat2 @ Xs ) ) ).
% compl_coset
thf(fact_965_ex__least__nat__less,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K: nat] :
( ( ord_less_nat @ K @ N )
& ! [I: nat] :
( ( ord_less_eq_nat @ I @ K )
=> ~ ( P2 @ I ) )
& ( P2 @ ( suc @ K ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_966_nat__induct__non__zero,axiom,
! [N: nat,P2: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P2 @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_967_of__int__nonneg,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% of_int_nonneg
thf(fact_968_of__int__pos,axiom,
! [Z2: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% of_int_pos
thf(fact_969_not__zle__0__negative,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% not_zle_0_negative
thf(fact_970_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N3: nat] :
( X
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% negD
thf(fact_971_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_972_sorted__iff__nth__Suc,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I2: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Xs @ ( suc @ I2 ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_973_sorted__iff__nth__Suc,axiom,
! [Xs: list_int] :
( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
= ( ! [I2: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ ( size_size_list_int @ Xs ) )
=> ( ord_less_eq_int @ ( nth_int @ Xs @ I2 ) @ ( nth_int @ Xs @ ( suc @ I2 ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_974_Compl__iff,axiom,
! [C: list_nat,A: set_list_nat] :
( ( member_list_nat2 @ C @ ( uminus3195874150345416415st_nat @ A ) )
= ( ~ ( member_list_nat2 @ C @ A ) ) ) ).
% Compl_iff
thf(fact_975_Compl__iff,axiom,
! [C: a,A: set_a] :
( ( member_a2 @ C @ ( uminus_uminus_set_a @ A ) )
= ( ~ ( member_a2 @ C @ A ) ) ) ).
% Compl_iff
thf(fact_976_ComplI,axiom,
! [C: list_nat,A: set_list_nat] :
( ~ ( member_list_nat2 @ C @ A )
=> ( member_list_nat2 @ C @ ( uminus3195874150345416415st_nat @ A ) ) ) ).
% ComplI
thf(fact_977_ComplI,axiom,
! [C: a,A: set_a] :
( ~ ( member_a2 @ C @ A )
=> ( member_a2 @ C @ ( uminus_uminus_set_a @ A ) ) ) ).
% ComplI
thf(fact_978_ComplD,axiom,
! [C: list_nat,A: set_list_nat] :
( ( member_list_nat2 @ C @ ( uminus3195874150345416415st_nat @ A ) )
=> ~ ( member_list_nat2 @ C @ A ) ) ).
% ComplD
thf(fact_979_ComplD,axiom,
! [C: a,A: set_a] :
( ( member_a2 @ C @ ( uminus_uminus_set_a @ A ) )
=> ~ ( member_a2 @ C @ A ) ) ).
% ComplD
thf(fact_980_length__stirling__row,axiom,
! [N: nat] :
( ( size_size_list_nat @ ( stirling_row @ N ) )
= ( suc @ N ) ) ).
% length_stirling_row
thf(fact_981_sorted__wrt__iff__nth__Suc__transp,axiom,
! [P2: nat > nat > $o,Xs: list_nat] :
( ( transp_on_nat @ top_top_set_nat @ P2 )
=> ( ( sorted_wrt_nat @ P2 @ Xs )
= ( ! [I2: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ ( size_size_list_nat @ Xs ) )
=> ( P2 @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Xs @ ( suc @ I2 ) ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_Suc_transp
thf(fact_982_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_983_stirling__code,axiom,
( stirling2
= ( ^ [N2: nat,K3: nat] : ( if_nat @ ( K3 = zero_zero_nat ) @ ( if_nat @ ( N2 = zero_zero_nat ) @ one_one_nat @ zero_zero_nat ) @ ( if_nat @ ( ord_less_nat @ N2 @ K3 ) @ zero_zero_nat @ ( if_nat @ ( K3 = N2 ) @ one_one_nat @ ( nth_nat @ ( stirling_row @ N2 ) @ K3 ) ) ) ) ) ) ).
% stirling_code
thf(fact_984_sorted__rev__iff__nth__mono,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
= ( ! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ J ) @ ( nth_nat @ Xs @ I2 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_985_sorted__rev__iff__nth__mono,axiom,
! [Xs: list_int] :
( ( sorted_wrt_int @ ord_less_eq_int @ ( rev_int @ Xs ) )
= ( ! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
=> ( ord_less_eq_int @ ( nth_int @ Xs @ J ) @ ( nth_int @ Xs @ I2 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_986_Nil__is__rev__conv,axiom,
! [Xs: list_nat] :
( ( nil_nat
= ( rev_nat @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_rev_conv
thf(fact_987_Nil__is__rev__conv,axiom,
! [Xs: list_list_nat] :
( ( nil_list_nat
= ( rev_list_nat @ Xs ) )
= ( Xs = nil_list_nat ) ) ).
% Nil_is_rev_conv
thf(fact_988_rev__is__Nil__conv,axiom,
! [Xs: list_nat] :
( ( ( rev_nat @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% rev_is_Nil_conv
thf(fact_989_rev__is__Nil__conv,axiom,
! [Xs: list_list_nat] :
( ( ( rev_list_nat @ Xs )
= nil_list_nat )
= ( Xs = nil_list_nat ) ) ).
% rev_is_Nil_conv
thf(fact_990_set__rev,axiom,
! [Xs: list_a] :
( ( set_a2 @ ( rev_a @ Xs ) )
= ( set_a2 @ Xs ) ) ).
% set_rev
thf(fact_991_set__rev,axiom,
! [Xs: list_list_nat] :
( ( set_list_nat2 @ ( rev_list_nat @ Xs ) )
= ( set_list_nat2 @ Xs ) ) ).
% set_rev
thf(fact_992_length__rev,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rev_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rev
thf(fact_993_count__list__rev,axiom,
! [Xs: list_list_nat,X: list_nat] :
( ( count_list_list_nat @ ( rev_list_nat @ Xs ) @ X )
= ( count_list_list_nat @ Xs @ X ) ) ).
% count_list_rev
thf(fact_994_stirling__0,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( stirling2 @ N @ zero_zero_nat )
= zero_zero_nat ) ) ).
% stirling_0
thf(fact_995_stirling__less,axiom,
! [N: nat,K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( ( stirling2 @ N @ K2 )
= zero_zero_nat ) ) ).
% stirling_less
thf(fact_996_rev__map,axiom,
! [F2: a > nat,Xs: list_a] :
( ( rev_nat @ ( map_a_nat @ F2 @ Xs ) )
= ( map_a_nat @ F2 @ ( rev_a @ Xs ) ) ) ).
% rev_map
thf(fact_997_rev__map,axiom,
! [F2: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat] :
( ( rev_se8765390391748778439at_nat @ ( map_li6003994582982014139at_nat @ F2 @ Xs ) )
= ( map_li6003994582982014139at_nat @ F2 @ ( rev_list_nat @ Xs ) ) ) ).
% rev_map
thf(fact_998_rev__map,axiom,
! [F2: list_nat > nat,Xs: list_list_nat] :
( ( rev_nat @ ( map_list_nat_nat @ F2 @ Xs ) )
= ( map_list_nat_nat @ F2 @ ( rev_list_nat @ Xs ) ) ) ).
% rev_map
thf(fact_999_rev__map,axiom,
! [F2: list_nat > list_nat,Xs: list_list_nat] :
( ( rev_list_nat @ ( map_li7225945977422193158st_nat @ F2 @ Xs ) )
= ( map_li7225945977422193158st_nat @ F2 @ ( rev_list_nat @ Xs ) ) ) ).
% rev_map
thf(fact_1000_rev__map,axiom,
! [F2: a > a,Xs: list_a] :
( ( rev_a @ ( map_a_a @ F2 @ Xs ) )
= ( map_a_a @ F2 @ ( rev_a @ Xs ) ) ) ).
% rev_map
thf(fact_1001_rev_Osimps_I1_J,axiom,
( ( rev_nat @ nil_nat )
= nil_nat ) ).
% rev.simps(1)
thf(fact_1002_rev_Osimps_I1_J,axiom,
( ( rev_list_nat @ nil_list_nat )
= nil_list_nat ) ).
% rev.simps(1)
thf(fact_1003_stirling_Osimps_I3_J,axiom,
! [N: nat] :
( ( stirling2 @ ( suc @ N ) @ zero_zero_nat )
= zero_zero_nat ) ).
% stirling.simps(3)
thf(fact_1004_stirling_Osimps_I2_J,axiom,
! [K2: nat] :
( ( stirling2 @ zero_zero_nat @ ( suc @ K2 ) )
= zero_zero_nat ) ).
% stirling.simps(2)
thf(fact_1005_stirling_Osimps_I1_J,axiom,
( ( stirling2 @ zero_zero_nat @ zero_zero_nat )
= one_one_nat ) ).
% stirling.simps(1)
thf(fact_1006_sorted__rev__iff__nth__Suc,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
= ( ! [I2: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ ( suc @ I2 ) ) @ ( nth_nat @ Xs @ I2 ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_1007_sorted__rev__iff__nth__Suc,axiom,
! [Xs: list_int] :
( ( sorted_wrt_int @ ord_less_eq_int @ ( rev_int @ Xs ) )
= ( ! [I2: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ ( size_size_list_int @ Xs ) )
=> ( ord_less_eq_int @ ( nth_int @ Xs @ ( suc @ I2 ) ) @ ( nth_int @ Xs @ I2 ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_1008_sorted__rev__nth__mono,axiom,
! [Xs: list_nat,I3: nat,J2: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ J2 ) @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_1009_sorted__rev__nth__mono,axiom,
! [Xs: list_int,I3: nat,J2: nat] :
( ( sorted_wrt_int @ ord_less_eq_int @ ( rev_int @ Xs ) )
=> ( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_int @ Xs ) )
=> ( ord_less_eq_int @ ( nth_int @ Xs @ J2 ) @ ( nth_int @ Xs @ I3 ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_1010_length__transpose__sorted,axiom,
! [Xs: list_list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ ( map_list_nat_nat @ size_size_list_nat @ Xs ) ) )
=> ( ( ( Xs = nil_list_nat )
=> ( ( size_s3023201423986296836st_nat @ ( transpose_nat @ Xs ) )
= zero_zero_nat ) )
& ( ( Xs != nil_list_nat )
=> ( ( size_s3023201423986296836st_nat @ ( transpose_nat @ Xs ) )
= ( size_size_list_nat @ ( nth_list_nat @ Xs @ zero_zero_nat ) ) ) ) ) ) ).
% length_transpose_sorted
thf(fact_1011_Stirling__1,axiom,
! [N: nat] :
( ( stirling @ ( suc @ N ) @ ( suc @ zero_zero_nat ) )
= one_one_nat ) ).
% Stirling_1
thf(fact_1012_Stirling__less,axiom,
! [N: nat,K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( ( stirling @ N @ K2 )
= zero_zero_nat ) ) ).
% Stirling_less
thf(fact_1013_transpose_Osimps_I1_J,axiom,
( ( transpose_nat @ nil_list_nat )
= nil_list_nat ) ).
% transpose.simps(1)
thf(fact_1014_transpose__map__map,axiom,
! [F2: a > nat,Xs: list_list_a] :
( ( transpose_nat @ ( map_list_a_list_nat @ ( map_a_nat @ F2 ) @ Xs ) )
= ( map_list_a_list_nat @ ( map_a_nat @ F2 ) @ ( transpose_a @ Xs ) ) ) ).
% transpose_map_map
thf(fact_1015_transpose__map__map,axiom,
! [F2: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_list_nat] :
( ( transp4399329437217503729at_nat @ ( map_li4356451683108985649at_nat @ ( map_li6003994582982014139at_nat @ F2 ) @ Xs ) )
= ( map_li4356451683108985649at_nat @ ( map_li6003994582982014139at_nat @ F2 ) @ ( transpose_list_nat @ Xs ) ) ) ).
% transpose_map_map
thf(fact_1016_transpose__map__map,axiom,
! [F2: list_nat > nat,Xs: list_list_list_nat] :
( ( transpose_nat @ ( map_li1587113026813354006st_nat @ ( map_list_nat_nat @ F2 ) @ Xs ) )
= ( map_li1587113026813354006st_nat @ ( map_list_nat_nat @ F2 ) @ ( transpose_list_nat @ Xs ) ) ) ).
% transpose_map_map
thf(fact_1017_transpose__map__map,axiom,
! [F2: list_nat > list_nat,Xs: list_list_list_nat] :
( ( transpose_list_nat @ ( map_li2855073862107769254st_nat @ ( map_li7225945977422193158st_nat @ F2 ) @ Xs ) )
= ( map_li2855073862107769254st_nat @ ( map_li7225945977422193158st_nat @ F2 ) @ ( transpose_list_nat @ Xs ) ) ) ).
% transpose_map_map
thf(fact_1018_transpose__map__map,axiom,
! [F2: a > a,Xs: list_list_a] :
( ( transpose_a @ ( map_list_a_list_a @ ( map_a_a @ F2 ) @ Xs ) )
= ( map_list_a_list_a @ ( map_a_a @ F2 ) @ ( transpose_a @ Xs ) ) ) ).
% transpose_map_map
thf(fact_1019_transpose__map__map,axiom,
! [F2: nat > nat,Xs: list_list_nat] :
( ( transpose_nat @ ( map_li7225945977422193158st_nat @ ( map_nat_nat @ F2 ) @ Xs ) )
= ( map_li7225945977422193158st_nat @ ( map_nat_nat @ F2 ) @ ( transpose_nat @ Xs ) ) ) ).
% transpose_map_map
thf(fact_1020_transpose__empty,axiom,
! [Xs: list_list_list_nat] :
( ( ( transpose_list_nat @ Xs )
= nil_list_list_nat )
= ( ! [X2: list_list_nat] :
( ( member_list_list_nat @ X2 @ ( set_list_list_nat2 @ Xs ) )
=> ( X2 = nil_list_nat ) ) ) ) ).
% transpose_empty
thf(fact_1021_transpose__empty,axiom,
! [Xs: list_list_nat] :
( ( ( transpose_nat @ Xs )
= nil_list_nat )
= ( ! [X2: list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( X2 = nil_nat ) ) ) ) ).
% transpose_empty
thf(fact_1022_Stirling_Osimps_I3_J,axiom,
! [N: nat] :
( ( stirling @ ( suc @ N ) @ zero_zero_nat )
= zero_zero_nat ) ).
% Stirling.simps(3)
thf(fact_1023_Stirling_Osimps_I2_J,axiom,
! [K2: nat] :
( ( stirling @ zero_zero_nat @ ( suc @ K2 ) )
= zero_zero_nat ) ).
% Stirling.simps(2)
thf(fact_1024_Stirling_Osimps_I1_J,axiom,
( ( stirling @ zero_zero_nat @ zero_zero_nat )
= one_one_nat ) ).
% Stirling.simps(1)
thf(fact_1025_sorted__transpose,axiom,
! [Xs: list_list_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ ( map_list_nat_nat @ size_size_list_nat @ ( transpose_nat @ Xs ) ) ) ) ).
% sorted_transpose
thf(fact_1026_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_list_nat @ N @ nil_list_nat )
= ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_list_nat @ N @ nil_list_nat )
= nil_list_list_nat ) ) ) ).
% n_lists_Nil
thf(fact_1027_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_nat @ N @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_nat @ N @ nil_nat )
= nil_list_nat ) ) ) ).
% n_lists_Nil
thf(fact_1028_Suc__nat__eq__nat__zadd1,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( suc @ ( nat2 @ Z2 ) )
= ( nat2 @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ) ).
% Suc_nat_eq_nat_zadd1
thf(fact_1029_nth__rotate1,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( rotate1_nat @ Xs ) @ N )
= ( nth_nat @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% nth_rotate1
thf(fact_1030_add__right__cancel,axiom,
! [B2: int,A4: int,C: int] :
( ( ( plus_plus_int @ B2 @ A4 )
= ( plus_plus_int @ C @ A4 ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_1031_add__right__cancel,axiom,
! [B2: nat,A4: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A4 )
= ( plus_plus_nat @ C @ A4 ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_1032_add__left__cancel,axiom,
! [A4: int,B2: int,C: int] :
( ( ( plus_plus_int @ A4 @ B2 )
= ( plus_plus_int @ A4 @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_1033_add__left__cancel,axiom,
! [A4: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A4 @ B2 )
= ( plus_plus_nat @ A4 @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_1034_list_Oinject,axiom,
! [X21: list_nat,X222: list_list_nat,Y21: list_nat,Y22: list_list_nat] :
( ( ( cons_list_nat @ X21 @ X222 )
= ( cons_list_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_1035_list_Oinject,axiom,
! [X21: nat,X222: list_nat,Y21: nat,Y22: list_nat] :
( ( ( cons_nat @ X21 @ X222 )
= ( cons_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_1036_add__le__cancel__left,axiom,
! [C: nat,A4: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A4 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ A4 @ B2 ) ) ).
% add_le_cancel_left
thf(fact_1037_add__le__cancel__left,axiom,
! [C: int,A4: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A4 ) @ ( plus_plus_int @ C @ B2 ) )
= ( ord_less_eq_int @ A4 @ B2 ) ) ).
% add_le_cancel_left
thf(fact_1038_add__le__cancel__right,axiom,
! [A4: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A4 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_eq_nat @ A4 @ B2 ) ) ).
% add_le_cancel_right
thf(fact_1039_add__le__cancel__right,axiom,
! [A4: int,C: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A4 @ C ) @ ( plus_plus_int @ B2 @ C ) )
= ( ord_less_eq_int @ A4 @ B2 ) ) ).
% add_le_cancel_right
thf(fact_1040_add__0,axiom,
! [A4: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A4 )
= A4 ) ).
% add_0
thf(fact_1041_add__0,axiom,
! [A4: int] :
( ( plus_plus_int @ zero_zero_int @ A4 )
= A4 ) ).
% add_0
thf(fact_1042_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y2: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y2 ) )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_1043_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y2: nat] :
( ( ( plus_plus_nat @ X @ Y2 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_1044_add__cancel__right__right,axiom,
! [A4: nat,B2: nat] :
( ( A4
= ( plus_plus_nat @ A4 @ B2 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_1045_add__cancel__right__right,axiom,
! [A4: int,B2: int] :
( ( A4
= ( plus_plus_int @ A4 @ B2 ) )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_1046_add__cancel__right__left,axiom,
! [A4: nat,B2: nat] :
( ( A4
= ( plus_plus_nat @ B2 @ A4 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_1047_add__cancel__right__left,axiom,
! [A4: int,B2: int] :
( ( A4
= ( plus_plus_int @ B2 @ A4 ) )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_1048_add__cancel__left__right,axiom,
! [A4: nat,B2: nat] :
( ( ( plus_plus_nat @ A4 @ B2 )
= A4 )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_1049_add__cancel__left__right,axiom,
! [A4: int,B2: int] :
( ( ( plus_plus_int @ A4 @ B2 )
= A4 )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_1050_add__cancel__left__left,axiom,
! [B2: nat,A4: nat] :
( ( ( plus_plus_nat @ B2 @ A4 )
= A4 )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_1051_add__cancel__left__left,axiom,
! [B2: int,A4: int] :
( ( ( plus_plus_int @ B2 @ A4 )
= A4 )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_1052_double__zero__sym,axiom,
! [A4: int] :
( ( zero_zero_int
= ( plus_plus_int @ A4 @ A4 ) )
= ( A4 = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_1053_add_Oright__neutral,axiom,
! [A4: nat] :
( ( plus_plus_nat @ A4 @ zero_zero_nat )
= A4 ) ).
% add.right_neutral
thf(fact_1054_add_Oright__neutral,axiom,
! [A4: int] :
( ( plus_plus_int @ A4 @ zero_zero_int )
= A4 ) ).
% add.right_neutral
thf(fact_1055_double__eq__0__iff,axiom,
! [A4: int] :
( ( ( plus_plus_int @ A4 @ A4 )
= zero_zero_int )
= ( A4 = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_1056_add__less__cancel__left,axiom,
! [C: nat,A4: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A4 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_nat @ A4 @ B2 ) ) ).
% add_less_cancel_left
thf(fact_1057_add__less__cancel__left,axiom,
! [C: int,A4: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A4 ) @ ( plus_plus_int @ C @ B2 ) )
= ( ord_less_int @ A4 @ B2 ) ) ).
% add_less_cancel_left
thf(fact_1058_add__less__cancel__right,axiom,
! [A4: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A4 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_nat @ A4 @ B2 ) ) ).
% add_less_cancel_right
thf(fact_1059_add__less__cancel__right,axiom,
! [A4: int,C: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A4 @ C ) @ ( plus_plus_int @ B2 @ C ) )
= ( ord_less_int @ A4 @ B2 ) ) ).
% add_less_cancel_right
thf(fact_1060_minus__add__distrib,axiom,
! [A4: int,B2: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A4 @ B2 ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A4 ) @ ( uminus_uminus_int @ B2 ) ) ) ).
% minus_add_distrib
thf(fact_1061_minus__add__cancel,axiom,
! [A4: int,B2: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A4 ) @ ( plus_plus_int @ A4 @ B2 ) )
= B2 ) ).
% minus_add_cancel
thf(fact_1062_add__minus__cancel,axiom,
! [A4: int,B2: int] :
( ( plus_plus_int @ A4 @ ( plus_plus_int @ ( uminus_uminus_int @ A4 ) @ B2 ) )
= B2 ) ).
% add_minus_cancel
thf(fact_1063_mod__self,axiom,
! [A4: nat] :
( ( modulo_modulo_nat @ A4 @ A4 )
= zero_zero_nat ) ).
% mod_self
thf(fact_1064_mod__self,axiom,
! [A4: int] :
( ( modulo_modulo_int @ A4 @ A4 )
= zero_zero_int ) ).
% mod_self
thf(fact_1065_mod__by__0,axiom,
! [A4: nat] :
( ( modulo_modulo_nat @ A4 @ zero_zero_nat )
= A4 ) ).
% mod_by_0
thf(fact_1066_mod__by__0,axiom,
! [A4: int] :
( ( modulo_modulo_int @ A4 @ zero_zero_int )
= A4 ) ).
% mod_by_0
thf(fact_1067_mod__0,axiom,
! [A4: nat] :
( ( modulo_modulo_nat @ zero_zero_nat @ A4 )
= zero_zero_nat ) ).
% mod_0
thf(fact_1068_mod__0,axiom,
! [A4: int] :
( ( modulo_modulo_int @ zero_zero_int @ A4 )
= zero_zero_int ) ).
% mod_0
thf(fact_1069_distinct__adj__Cons__Cons,axiom,
! [X: list_nat,Y2: list_nat,Xs: list_list_nat] :
( ( distin876741697294417026st_nat @ ( cons_list_nat @ X @ ( cons_list_nat @ Y2 @ Xs ) ) )
= ( ( X != Y2 )
& ( distin876741697294417026st_nat @ ( cons_list_nat @ Y2 @ Xs ) ) ) ) ).
% distinct_adj_Cons_Cons
thf(fact_1070_distinct__adj__Cons__Cons,axiom,
! [X: nat,Y2: nat,Xs: list_nat] :
( ( distinct_adj_nat @ ( cons_nat @ X @ ( cons_nat @ Y2 @ Xs ) ) )
= ( ( X != Y2 )
& ( distinct_adj_nat @ ( cons_nat @ Y2 @ Xs ) ) ) ) ).
% distinct_adj_Cons_Cons
thf(fact_1071_add__le__same__cancel1,axiom,
! [B2: nat,A4: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A4 ) @ B2 )
= ( ord_less_eq_nat @ A4 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_1072_add__le__same__cancel1,axiom,
! [B2: int,A4: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B2 @ A4 ) @ B2 )
= ( ord_less_eq_int @ A4 @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_1073_add__le__same__cancel2,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A4 @ B2 ) @ B2 )
= ( ord_less_eq_nat @ A4 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_1074_add__le__same__cancel2,axiom,
! [A4: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A4 @ B2 ) @ B2 )
= ( ord_less_eq_int @ A4 @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_1075_le__add__same__cancel1,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ ( plus_plus_nat @ A4 @ B2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_1076_le__add__same__cancel1,axiom,
! [A4: int,B2: int] :
( ( ord_less_eq_int @ A4 @ ( plus_plus_int @ A4 @ B2 ) )
= ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_1077_le__add__same__cancel2,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ ( plus_plus_nat @ B2 @ A4 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_1078_le__add__same__cancel2,axiom,
! [A4: int,B2: int] :
( ( ord_less_eq_int @ A4 @ ( plus_plus_int @ B2 @ A4 ) )
= ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_1079_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A4: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A4 @ A4 ) @ zero_zero_int )
= ( ord_less_eq_int @ A4 @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_1080_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A4: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A4 @ A4 ) )
= ( ord_less_eq_int @ zero_zero_int @ A4 ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_1081_add__less__same__cancel1,axiom,
! [B2: nat,A4: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B2 @ A4 ) @ B2 )
= ( ord_less_nat @ A4 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_1082_add__less__same__cancel1,axiom,
! [B2: int,A4: int] :
( ( ord_less_int @ ( plus_plus_int @ B2 @ A4 ) @ B2 )
= ( ord_less_int @ A4 @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_1083_add__less__same__cancel2,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A4 @ B2 ) @ B2 )
= ( ord_less_nat @ A4 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_1084_add__less__same__cancel2,axiom,
! [A4: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A4 @ B2 ) @ B2 )
= ( ord_less_int @ A4 @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_1085_less__add__same__cancel1,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_nat @ A4 @ ( plus_plus_nat @ A4 @ B2 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_1086_less__add__same__cancel1,axiom,
! [A4: int,B2: int] :
( ( ord_less_int @ A4 @ ( plus_plus_int @ A4 @ B2 ) )
= ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_1087_less__add__same__cancel2,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_nat @ A4 @ ( plus_plus_nat @ B2 @ A4 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_1088_less__add__same__cancel2,axiom,
! [A4: int,B2: int] :
( ( ord_less_int @ A4 @ ( plus_plus_int @ B2 @ A4 ) )
= ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_1089_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A4: int] :
( ( ord_less_int @ ( plus_plus_int @ A4 @ A4 ) @ zero_zero_int )
= ( ord_less_int @ A4 @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_1090_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A4: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A4 @ A4 ) )
= ( ord_less_int @ zero_zero_int @ A4 ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_1091_ab__left__minus,axiom,
! [A4: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A4 ) @ A4 )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_1092_add_Oright__inverse,axiom,
! [A4: int] :
( ( plus_plus_int @ A4 @ ( uminus_uminus_int @ A4 ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_1093_mod__by__1,axiom,
! [A4: nat] :
( ( modulo_modulo_nat @ A4 @ one_one_nat )
= zero_zero_nat ) ).
% mod_by_1
thf(fact_1094_mod__by__1,axiom,
! [A4: int] :
( ( modulo_modulo_int @ A4 @ one_one_int )
= zero_zero_int ) ).
% mod_by_1
thf(fact_1095_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_1096_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_1097_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_1098_nth__Cons__Suc,axiom,
! [X: list_nat,Xs: list_list_nat,N: nat] :
( ( nth_list_nat @ ( cons_list_nat @ X @ Xs ) @ ( suc @ N ) )
= ( nth_list_nat @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_1099_nth__Cons__Suc,axiom,
! [X: nat,Xs: list_nat,N: nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ ( suc @ N ) )
= ( nth_nat @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_1100_nth__Cons__0,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( nth_list_nat @ ( cons_list_nat @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_1101_nth__Cons__0,axiom,
! [X: nat,Xs: list_nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_1102_rev__singleton__conv,axiom,
! [Xs: list_list_nat,X: list_nat] :
( ( ( rev_list_nat @ Xs )
= ( cons_list_nat @ X @ nil_list_nat ) )
= ( Xs
= ( cons_list_nat @ X @ nil_list_nat ) ) ) ).
% rev_singleton_conv
thf(fact_1103_rev__singleton__conv,axiom,
! [Xs: list_nat,X: nat] :
( ( ( rev_nat @ Xs )
= ( cons_nat @ X @ nil_nat ) )
= ( Xs
= ( cons_nat @ X @ nil_nat ) ) ) ).
% rev_singleton_conv
thf(fact_1104_singleton__rev__conv,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( ( cons_list_nat @ X @ nil_list_nat )
= ( rev_list_nat @ Xs ) )
= ( ( cons_list_nat @ X @ nil_list_nat )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_1105_singleton__rev__conv,axiom,
! [X: nat,Xs: list_nat] :
( ( ( cons_nat @ X @ nil_nat )
= ( rev_nat @ Xs ) )
= ( ( cons_nat @ X @ nil_nat )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_1106_insert__Nil,axiom,
! [X: list_nat] :
( ( insert_list_nat @ X @ nil_list_nat )
= ( cons_list_nat @ X @ nil_list_nat ) ) ).
% insert_Nil
thf(fact_1107_insert__Nil,axiom,
! [X: nat] :
( ( insert_nat @ X @ nil_nat )
= ( cons_nat @ X @ nil_nat ) ) ).
% insert_Nil
thf(fact_1108_not__in__set__insert,axiom,
! [X: a,Xs: list_a] :
( ~ ( member_a2 @ X @ ( set_a2 @ Xs ) )
=> ( ( insert_a @ X @ Xs )
= ( cons_a @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_1109_not__in__set__insert,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ~ ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs ) )
=> ( ( insert_list_nat @ X @ Xs )
= ( cons_list_nat @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_1110_not__in__set__insert,axiom,
! [X: nat,Xs: list_nat] :
( ~ ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X @ Xs )
= ( cons_nat @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_1111_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= zero_zero_int ) ).
% add_neg_numeral_special(8)
thf(fact_1112_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% add_neg_numeral_special(7)
thf(fact_1113_rotate__id,axiom,
! [N: nat,Xs: list_nat] :
( ( ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Xs ) )
= zero_zero_nat )
=> ( ( rotate_nat @ N @ Xs )
= Xs ) ) ).
% rotate_id
thf(fact_1114_map__tailrec__rev_Osimps_I2_J,axiom,
! [F2: list_nat > list_nat,A4: list_nat,As: list_list_nat,Bs: list_list_nat] :
( ( map_ta5898645294048341815st_nat @ F2 @ ( cons_list_nat @ A4 @ As ) @ Bs )
= ( map_ta5898645294048341815st_nat @ F2 @ As @ ( cons_list_nat @ ( F2 @ A4 ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_1115_map__tailrec__rev_Osimps_I2_J,axiom,
! [F2: list_nat > nat,A4: list_nat,As: list_list_nat,Bs: list_nat] :
( ( map_ta3159490279798172583at_nat @ F2 @ ( cons_list_nat @ A4 @ As ) @ Bs )
= ( map_ta3159490279798172583at_nat @ F2 @ As @ ( cons_nat @ ( F2 @ A4 ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_1116_map__tailrec__rev_Osimps_I2_J,axiom,
! [F2: nat > list_nat,A4: nat,As: list_nat,Bs: list_list_nat] :
( ( map_ta920533232635887399st_nat @ F2 @ ( cons_nat @ A4 @ As ) @ Bs )
= ( map_ta920533232635887399st_nat @ F2 @ As @ ( cons_list_nat @ ( F2 @ A4 ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_1117_map__tailrec__rev_Osimps_I2_J,axiom,
! [F2: nat > nat,A4: nat,As: list_nat,Bs: list_nat] :
( ( map_ta7164188454487880599at_nat @ F2 @ ( cons_nat @ A4 @ As ) @ Bs )
= ( map_ta7164188454487880599at_nat @ F2 @ As @ ( cons_nat @ ( F2 @ A4 ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_1118_transpose_Ocases,axiom,
! [X: list_list_list_nat] :
( ( X != nil_list_list_nat )
=> ( ! [Xss2: list_list_list_nat] :
( X
!= ( cons_list_list_nat @ nil_list_nat @ Xss2 ) )
=> ~ ! [X3: list_nat,Xs3: list_list_nat,Xss2: list_list_list_nat] :
( X
!= ( cons_list_list_nat @ ( cons_list_nat @ X3 @ Xs3 ) @ Xss2 ) ) ) ) ).
% transpose.cases
thf(fact_1119_transpose_Ocases,axiom,
! [X: list_list_nat] :
( ( X != nil_list_nat )
=> ( ! [Xss2: list_list_nat] :
( X
!= ( cons_list_nat @ nil_nat @ Xss2 ) )
=> ~ ! [X3: nat,Xs3: list_nat,Xss2: list_list_nat] :
( X
!= ( cons_list_nat @ ( cons_nat @ X3 @ Xs3 ) @ Xss2 ) ) ) ) ).
% transpose.cases
thf(fact_1120_map__eq__Cons__conv,axiom,
! [F2: a > a,Xs: list_a,Y2: a,Ys: list_a] :
( ( ( map_a_a @ F2 @ Xs )
= ( cons_a @ Y2 @ Ys ) )
= ( ? [Z5: a,Zs: list_a] :
( ( Xs
= ( cons_a @ Z5 @ Zs ) )
& ( ( F2 @ Z5 )
= Y2 )
& ( ( map_a_a @ F2 @ Zs )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1121_map__eq__Cons__conv,axiom,
! [F2: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat,Y2: set_Pr1261947904930325089at_nat,Ys: list_s1210847774152347623at_nat] :
( ( ( map_li6003994582982014139at_nat @ F2 @ Xs )
= ( cons_s6881495754146722583at_nat @ Y2 @ Ys ) )
= ( ? [Z5: list_nat,Zs: list_list_nat] :
( ( Xs
= ( cons_list_nat @ Z5 @ Zs ) )
& ( ( F2 @ Z5 )
= Y2 )
& ( ( map_li6003994582982014139at_nat @ F2 @ Zs )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1122_map__eq__Cons__conv,axiom,
! [F2: list_nat > list_nat,Xs: list_list_nat,Y2: list_nat,Ys: list_list_nat] :
( ( ( map_li7225945977422193158st_nat @ F2 @ Xs )
= ( cons_list_nat @ Y2 @ Ys ) )
= ( ? [Z5: list_nat,Zs: list_list_nat] :
( ( Xs
= ( cons_list_nat @ Z5 @ Zs ) )
& ( ( F2 @ Z5 )
= Y2 )
& ( ( map_li7225945977422193158st_nat @ F2 @ Zs )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1123_map__eq__Cons__conv,axiom,
! [F2: nat > list_nat,Xs: list_nat,Y2: list_nat,Ys: list_list_nat] :
( ( ( map_nat_list_nat @ F2 @ Xs )
= ( cons_list_nat @ Y2 @ Ys ) )
= ( ? [Z5: nat,Zs: list_nat] :
( ( Xs
= ( cons_nat @ Z5 @ Zs ) )
& ( ( F2 @ Z5 )
= Y2 )
& ( ( map_nat_list_nat @ F2 @ Zs )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1124_map__eq__Cons__conv,axiom,
! [F2: a > nat,Xs: list_a,Y2: nat,Ys: list_nat] :
( ( ( map_a_nat @ F2 @ Xs )
= ( cons_nat @ Y2 @ Ys ) )
= ( ? [Z5: a,Zs: list_a] :
( ( Xs
= ( cons_a @ Z5 @ Zs ) )
& ( ( F2 @ Z5 )
= Y2 )
& ( ( map_a_nat @ F2 @ Zs )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1125_map__eq__Cons__conv,axiom,
! [F2: list_nat > nat,Xs: list_list_nat,Y2: nat,Ys: list_nat] :
( ( ( map_list_nat_nat @ F2 @ Xs )
= ( cons_nat @ Y2 @ Ys ) )
= ( ? [Z5: list_nat,Zs: list_list_nat] :
( ( Xs
= ( cons_list_nat @ Z5 @ Zs ) )
& ( ( F2 @ Z5 )
= Y2 )
& ( ( map_list_nat_nat @ F2 @ Zs )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1126_map__eq__Cons__conv,axiom,
! [F2: nat > nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F2 @ Xs )
= ( cons_nat @ Y2 @ Ys ) )
= ( ? [Z5: nat,Zs: list_nat] :
( ( Xs
= ( cons_nat @ Z5 @ Zs ) )
& ( ( F2 @ Z5 )
= Y2 )
& ( ( map_nat_nat @ F2 @ Zs )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1127_Cons__eq__map__conv,axiom,
! [X: a,Xs: list_a,F2: a > a,Ys: list_a] :
( ( ( cons_a @ X @ Xs )
= ( map_a_a @ F2 @ Ys ) )
= ( ? [Z5: a,Zs: list_a] :
( ( Ys
= ( cons_a @ Z5 @ Zs ) )
& ( X
= ( F2 @ Z5 ) )
& ( Xs
= ( map_a_a @ F2 @ Zs ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1128_Cons__eq__map__conv,axiom,
! [X: set_Pr1261947904930325089at_nat,Xs: list_s1210847774152347623at_nat,F2: list_nat > set_Pr1261947904930325089at_nat,Ys: list_list_nat] :
( ( ( cons_s6881495754146722583at_nat @ X @ Xs )
= ( map_li6003994582982014139at_nat @ F2 @ Ys ) )
= ( ? [Z5: list_nat,Zs: list_list_nat] :
( ( Ys
= ( cons_list_nat @ Z5 @ Zs ) )
& ( X
= ( F2 @ Z5 ) )
& ( Xs
= ( map_li6003994582982014139at_nat @ F2 @ Zs ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1129_Cons__eq__map__conv,axiom,
! [X: list_nat,Xs: list_list_nat,F2: list_nat > list_nat,Ys: list_list_nat] :
( ( ( cons_list_nat @ X @ Xs )
= ( map_li7225945977422193158st_nat @ F2 @ Ys ) )
= ( ? [Z5: list_nat,Zs: list_list_nat] :
( ( Ys
= ( cons_list_nat @ Z5 @ Zs ) )
& ( X
= ( F2 @ Z5 ) )
& ( Xs
= ( map_li7225945977422193158st_nat @ F2 @ Zs ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1130_Cons__eq__map__conv,axiom,
! [X: list_nat,Xs: list_list_nat,F2: nat > list_nat,Ys: list_nat] :
( ( ( cons_list_nat @ X @ Xs )
= ( map_nat_list_nat @ F2 @ Ys ) )
= ( ? [Z5: nat,Zs: list_nat] :
( ( Ys
= ( cons_nat @ Z5 @ Zs ) )
& ( X
= ( F2 @ Z5 ) )
& ( Xs
= ( map_nat_list_nat @ F2 @ Zs ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1131_Cons__eq__map__conv,axiom,
! [X: nat,Xs: list_nat,F2: a > nat,Ys: list_a] :
( ( ( cons_nat @ X @ Xs )
= ( map_a_nat @ F2 @ Ys ) )
= ( ? [Z5: a,Zs: list_a] :
( ( Ys
= ( cons_a @ Z5 @ Zs ) )
& ( X
= ( F2 @ Z5 ) )
& ( Xs
= ( map_a_nat @ F2 @ Zs ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1132_Cons__eq__map__conv,axiom,
! [X: nat,Xs: list_nat,F2: list_nat > nat,Ys: list_list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( map_list_nat_nat @ F2 @ Ys ) )
= ( ? [Z5: list_nat,Zs: list_list_nat] :
( ( Ys
= ( cons_list_nat @ Z5 @ Zs ) )
& ( X
= ( F2 @ Z5 ) )
& ( Xs
= ( map_list_nat_nat @ F2 @ Zs ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1133_Cons__eq__map__conv,axiom,
! [X: nat,Xs: list_nat,F2: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( map_nat_nat @ F2 @ Ys ) )
= ( ? [Z5: nat,Zs: list_nat] :
( ( Ys
= ( cons_nat @ Z5 @ Zs ) )
& ( X
= ( F2 @ Z5 ) )
& ( Xs
= ( map_nat_nat @ F2 @ Zs ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1134_map__eq__Cons__D,axiom,
! [F2: a > a,Xs: list_a,Y2: a,Ys: list_a] :
( ( ( map_a_a @ F2 @ Xs )
= ( cons_a @ Y2 @ Ys ) )
=> ? [Z: a,Zs2: list_a] :
( ( Xs
= ( cons_a @ Z @ Zs2 ) )
& ( ( F2 @ Z )
= Y2 )
& ( ( map_a_a @ F2 @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_1135_map__eq__Cons__D,axiom,
! [F2: list_nat > set_Pr1261947904930325089at_nat,Xs: list_list_nat,Y2: set_Pr1261947904930325089at_nat,Ys: list_s1210847774152347623at_nat] :
( ( ( map_li6003994582982014139at_nat @ F2 @ Xs )
= ( cons_s6881495754146722583at_nat @ Y2 @ Ys ) )
=> ? [Z: list_nat,Zs2: list_list_nat] :
( ( Xs
= ( cons_list_nat @ Z @ Zs2 ) )
& ( ( F2 @ Z )
= Y2 )
& ( ( map_li6003994582982014139at_nat @ F2 @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_1136_map__eq__Cons__D,axiom,
! [F2: list_nat > list_nat,Xs: list_list_nat,Y2: list_nat,Ys: list_list_nat] :
( ( ( map_li7225945977422193158st_nat @ F2 @ Xs )
= ( cons_list_nat @ Y2 @ Ys ) )
=> ? [Z: list_nat,Zs2: list_list_nat] :
( ( Xs
= ( cons_list_nat @ Z @ Zs2 ) )
& ( ( F2 @ Z )
= Y2 )
& ( ( map_li7225945977422193158st_nat @ F2 @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_1137_map__eq__Cons__D,axiom,
! [F2: nat > list_nat,Xs: list_nat,Y2: list_nat,Ys: list_list_nat] :
( ( ( map_nat_list_nat @ F2 @ Xs )
= ( cons_list_nat @ Y2 @ Ys ) )
=> ? [Z: nat,Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Z @ Zs2 ) )
& ( ( F2 @ Z )
= Y2 )
& ( ( map_nat_list_nat @ F2 @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_1138_map__eq__Cons__D,axiom,
! [F2: a > nat,Xs: list_a,Y2: nat,Ys: list_nat] :
( ( ( map_a_nat @ F2 @ Xs )
= ( cons_nat @ Y2 @ Ys ) )
=> ? [Z: a,Zs2: list_a] :
( ( Xs
= ( cons_a @ Z @ Zs2 ) )
& ( ( F2 @ Z )
= Y2 )
& ( ( map_a_nat @ F2 @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_1139_map__eq__Cons__D,axiom,
! [F2: list_nat > nat,Xs: list_list_nat,Y2: nat,Ys: list_nat] :
( ( ( map_list_nat_nat @ F2 @ Xs )
= ( cons_nat @ Y2 @ Ys ) )
=> ? [Z: list_nat,Zs2: list_list_nat] :
( ( Xs
= ( cons_list_nat @ Z @ Zs2 ) )
& ( ( F2 @ Z )
= Y2 )
& ( ( map_list_nat_nat @ F2 @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_1140_map__eq__Cons__D,axiom,
! [F2: nat > nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F2 @ Xs )
= ( cons_nat @ Y2 @ Ys ) )
=> ? [Z: nat,Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Z @ Zs2 ) )
& ( ( F2 @ Z )
= Y2 )
& ( ( map_nat_nat @ F2 @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_1141_Cons__eq__map__D,axiom,
! [X: a,Xs: list_a,F2: a > a,Ys: list_a] :
( ( ( cons_a @ X @ Xs )
= ( map_a_a @ F2 @ Ys ) )
=> ? [Z: a,Zs2: list_a] :
( ( Ys
= ( cons_a @ Z @ Zs2 ) )
& ( X
= ( F2 @ Z ) )
& ( Xs
= ( map_a_a @ F2 @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1142_Cons__eq__map__D,axiom,
! [X: set_Pr1261947904930325089at_nat,Xs: list_s1210847774152347623at_nat,F2: list_nat > set_Pr1261947904930325089at_nat,Ys: list_list_nat] :
( ( ( cons_s6881495754146722583at_nat @ X @ Xs )
= ( map_li6003994582982014139at_nat @ F2 @ Ys ) )
=> ? [Z: list_nat,Zs2: list_list_nat] :
( ( Ys
= ( cons_list_nat @ Z @ Zs2 ) )
& ( X
= ( F2 @ Z ) )
& ( Xs
= ( map_li6003994582982014139at_nat @ F2 @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1143_Cons__eq__map__D,axiom,
! [X: list_nat,Xs: list_list_nat,F2: list_nat > list_nat,Ys: list_list_nat] :
( ( ( cons_list_nat @ X @ Xs )
= ( map_li7225945977422193158st_nat @ F2 @ Ys ) )
=> ? [Z: list_nat,Zs2: list_list_nat] :
( ( Ys
= ( cons_list_nat @ Z @ Zs2 ) )
& ( X
= ( F2 @ Z ) )
& ( Xs
= ( map_li7225945977422193158st_nat @ F2 @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1144_Cons__eq__map__D,axiom,
! [X: list_nat,Xs: list_list_nat,F2: nat > list_nat,Ys: list_nat] :
( ( ( cons_list_nat @ X @ Xs )
= ( map_nat_list_nat @ F2 @ Ys ) )
=> ? [Z: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z @ Zs2 ) )
& ( X
= ( F2 @ Z ) )
& ( Xs
= ( map_nat_list_nat @ F2 @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1145_Cons__eq__map__D,axiom,
! [X: nat,Xs: list_nat,F2: a > nat,Ys: list_a] :
( ( ( cons_nat @ X @ Xs )
= ( map_a_nat @ F2 @ Ys ) )
=> ? [Z: a,Zs2: list_a] :
( ( Ys
= ( cons_a @ Z @ Zs2 ) )
& ( X
= ( F2 @ Z ) )
& ( Xs
= ( map_a_nat @ F2 @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1146_Cons__eq__map__D,axiom,
! [X: nat,Xs: list_nat,F2: list_nat > nat,Ys: list_list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( map_list_nat_nat @ F2 @ Ys ) )
=> ? [Z: list_nat,Zs2: list_list_nat] :
( ( Ys
= ( cons_list_nat @ Z @ Zs2 ) )
& ( X
= ( F2 @ Z ) )
& ( Xs
= ( map_list_nat_nat @ F2 @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1147_Cons__eq__map__D,axiom,
! [X: nat,Xs: list_nat,F2: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( map_nat_nat @ F2 @ Ys ) )
=> ? [Z: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z @ Zs2 ) )
& ( X
= ( F2 @ Z ) )
& ( Xs
= ( map_nat_nat @ F2 @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1148_list_Osimps_I9_J,axiom,
! [F2: a > a,X21: a,X222: list_a] :
( ( map_a_a @ F2 @ ( cons_a @ X21 @ X222 ) )
= ( cons_a @ ( F2 @ X21 ) @ ( map_a_a @ F2 @ X222 ) ) ) ).
% list.simps(9)
thf(fact_1149_list_Osimps_I9_J,axiom,
! [F2: a > nat,X21: a,X222: list_a] :
( ( map_a_nat @ F2 @ ( cons_a @ X21 @ X222 ) )
= ( cons_nat @ ( F2 @ X21 ) @ ( map_a_nat @ F2 @ X222 ) ) ) ).
% list.simps(9)
thf(fact_1150_list_Osimps_I9_J,axiom,
! [F2: list_nat > set_Pr1261947904930325089at_nat,X21: list_nat,X222: list_list_nat] :
( ( map_li6003994582982014139at_nat @ F2 @ ( cons_list_nat @ X21 @ X222 ) )
= ( cons_s6881495754146722583at_nat @ ( F2 @ X21 ) @ ( map_li6003994582982014139at_nat @ F2 @ X222 ) ) ) ).
% list.simps(9)
thf(fact_1151_list_Osimps_I9_J,axiom,
! [F2: list_nat > list_nat,X21: list_nat,X222: list_list_nat] :
( ( map_li7225945977422193158st_nat @ F2 @ ( cons_list_nat @ X21 @ X222 ) )
= ( cons_list_nat @ ( F2 @ X21 ) @ ( map_li7225945977422193158st_nat @ F2 @ X222 ) ) ) ).
% list.simps(9)
thf(fact_1152_list_Osimps_I9_J,axiom,
! [F2: list_nat > nat,X21: list_nat,X222: list_list_nat] :
( ( map_list_nat_nat @ F2 @ ( cons_list_nat @ X21 @ X222 ) )
= ( cons_nat @ ( F2 @ X21 ) @ ( map_list_nat_nat @ F2 @ X222 ) ) ) ).
% list.simps(9)
thf(fact_1153_list_Osimps_I9_J,axiom,
! [F2: nat > list_nat,X21: nat,X222: list_nat] :
( ( map_nat_list_nat @ F2 @ ( cons_nat @ X21 @ X222 ) )
= ( cons_list_nat @ ( F2 @ X21 ) @ ( map_nat_list_nat @ F2 @ X222 ) ) ) ).
% list.simps(9)
thf(fact_1154_list_Osimps_I9_J,axiom,
! [F2: nat > nat,X21: nat,X222: list_nat] :
( ( map_nat_nat @ F2 @ ( cons_nat @ X21 @ X222 ) )
= ( cons_nat @ ( F2 @ X21 ) @ ( map_nat_nat @ F2 @ X222 ) ) ) ).
% list.simps(9)
thf(fact_1155_list__nonempty__induct,axiom,
! [Xs: list_list_nat,P2: list_list_nat > $o] :
( ( Xs != nil_list_nat )
=> ( ! [X3: list_nat] : ( P2 @ ( cons_list_nat @ X3 @ nil_list_nat ) )
=> ( ! [X3: list_nat,Xs3: list_list_nat] :
( ( Xs3 != nil_list_nat )
=> ( ( P2 @ Xs3 )
=> ( P2 @ ( cons_list_nat @ X3 @ Xs3 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_1156_list__nonempty__induct,axiom,
! [Xs: list_nat,P2: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X3: nat] : ( P2 @ ( cons_nat @ X3 @ nil_nat ) )
=> ( ! [X3: nat,Xs3: list_nat] :
( ( Xs3 != nil_nat )
=> ( ( P2 @ Xs3 )
=> ( P2 @ ( cons_nat @ X3 @ Xs3 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_1157_list__induct2_H,axiom,
! [P2: list_list_nat > list_list_nat > $o,Xs: list_list_nat,Ys: list_list_nat] :
( ( P2 @ nil_list_nat @ nil_list_nat )
=> ( ! [X3: list_nat,Xs3: list_list_nat] : ( P2 @ ( cons_list_nat @ X3 @ Xs3 ) @ nil_list_nat )
=> ( ! [Y3: list_nat,Ys4: list_list_nat] : ( P2 @ nil_list_nat @ ( cons_list_nat @ Y3 @ Ys4 ) )
=> ( ! [X3: list_nat,Xs3: list_list_nat,Y3: list_nat,Ys4: list_list_nat] :
( ( P2 @ Xs3 @ Ys4 )
=> ( P2 @ ( cons_list_nat @ X3 @ Xs3 ) @ ( cons_list_nat @ Y3 @ Ys4 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_1158_list__induct2_H,axiom,
! [P2: list_list_nat > list_nat > $o,Xs: list_list_nat,Ys: list_nat] :
( ( P2 @ nil_list_nat @ nil_nat )
=> ( ! [X3: list_nat,Xs3: list_list_nat] : ( P2 @ ( cons_list_nat @ X3 @ Xs3 ) @ nil_nat )
=> ( ! [Y3: nat,Ys4: list_nat] : ( P2 @ nil_list_nat @ ( cons_nat @ Y3 @ Ys4 ) )
=> ( ! [X3: list_nat,Xs3: list_list_nat,Y3: nat,Ys4: list_nat] :
( ( P2 @ Xs3 @ Ys4 )
=> ( P2 @ ( cons_list_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_1159_list__induct2_H,axiom,
! [P2: list_nat > list_list_nat > $o,Xs: list_nat,Ys: list_list_nat] :
( ( P2 @ nil_nat @ nil_list_nat )
=> ( ! [X3: nat,Xs3: list_nat] : ( P2 @ ( cons_nat @ X3 @ Xs3 ) @ nil_list_nat )
=> ( ! [Y3: list_nat,Ys4: list_list_nat] : ( P2 @ nil_nat @ ( cons_list_nat @ Y3 @ Ys4 ) )
=> ( ! [X3: nat,Xs3: list_nat,Y3: list_nat,Ys4: list_list_nat] :
( ( P2 @ Xs3 @ Ys4 )
=> ( P2 @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_list_nat @ Y3 @ Ys4 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_1160_list__induct2_H,axiom,
! [P2: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
( ( P2 @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs3: list_nat] : ( P2 @ ( cons_nat @ X3 @ Xs3 ) @ nil_nat )
=> ( ! [Y3: nat,Ys4: list_nat] : ( P2 @ nil_nat @ ( cons_nat @ Y3 @ Ys4 ) )
=> ( ! [X3: nat,Xs3: list_nat,Y3: nat,Ys4: list_nat] :
( ( P2 @ Xs3 @ Ys4 )
=> ( P2 @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_1161_neq__Nil__conv,axiom,
! [Xs: list_list_nat] :
( ( Xs != nil_list_nat )
= ( ? [Y: list_nat,Ys3: list_list_nat] :
( Xs
= ( cons_list_nat @ Y @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_1162_neq__Nil__conv,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
= ( ? [Y: nat,Ys3: list_nat] :
( Xs
= ( cons_nat @ Y @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_1163_remdups__adj_Ocases,axiom,
! [X: list_list_nat] :
( ( X != nil_list_nat )
=> ( ! [X3: list_nat] :
( X
!= ( cons_list_nat @ X3 @ nil_list_nat ) )
=> ~ ! [X3: list_nat,Y3: list_nat,Xs3: list_list_nat] :
( X
!= ( cons_list_nat @ X3 @ ( cons_list_nat @ Y3 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_1164_remdups__adj_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ( ! [X3: nat] :
( X
!= ( cons_nat @ X3 @ nil_nat ) )
=> ~ ! [X3: nat,Y3: nat,Xs3: list_nat] :
( X
!= ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_1165_min__list_Ocases,axiom,
! [X: list_nat] :
( ! [X3: nat,Xs3: list_nat] :
( X
!= ( cons_nat @ X3 @ Xs3 ) )
=> ( X = nil_nat ) ) ).
% min_list.cases
thf(fact_1166_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_1167_list_Oexhaust,axiom,
! [Y2: list_nat] :
( ( Y2 != nil_nat )
=> ~ ! [X212: nat,X223: list_nat] :
( Y2
!= ( cons_nat @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_1168_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_1169_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X222: list_nat] :
( ( List
= ( cons_nat @ X21 @ X222 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_1170_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_1171_list_Odistinct_I1_J,axiom,
! [X21: nat,X222: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_1172_member__rec_I1_J,axiom,
! [X: list_nat,Xs: list_list_nat,Y2: list_nat] :
( ( member_list_nat @ ( cons_list_nat @ X @ Xs ) @ Y2 )
= ( ( X = Y2 )
| ( member_list_nat @ Xs @ Y2 ) ) ) ).
% member_rec(1)
thf(fact_1173_member__rec_I1_J,axiom,
! [X: nat,Xs: list_nat,Y2: nat] :
( ( member_nat @ ( cons_nat @ X @ Xs ) @ Y2 )
= ( ( X = Y2 )
| ( member_nat @ Xs @ Y2 ) ) ) ).
% member_rec(1)
thf(fact_1174_distinct__adj__ConsD,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( distin876741697294417026st_nat @ ( cons_list_nat @ X @ Xs ) )
=> ( distin876741697294417026st_nat @ Xs ) ) ).
% distinct_adj_ConsD
thf(fact_1175_distinct__adj__ConsD,axiom,
! [X: nat,Xs: list_nat] :
( ( distinct_adj_nat @ ( cons_nat @ X @ Xs ) )
=> ( distinct_adj_nat @ Xs ) ) ).
% distinct_adj_ConsD
thf(fact_1176_removeAll_Osimps_I2_J,axiom,
! [X: list_nat,Y2: list_nat,Xs: list_list_nat] :
( ( ( X = Y2 )
=> ( ( removeAll_list_nat @ X @ ( cons_list_nat @ Y2 @ Xs ) )
= ( removeAll_list_nat @ X @ Xs ) ) )
& ( ( X != Y2 )
=> ( ( removeAll_list_nat @ X @ ( cons_list_nat @ Y2 @ Xs ) )
= ( cons_list_nat @ Y2 @ ( removeAll_list_nat @ X @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_1177_removeAll_Osimps_I2_J,axiom,
! [X: nat,Y2: nat,Xs: list_nat] :
( ( ( X = Y2 )
=> ( ( removeAll_nat @ X @ ( cons_nat @ Y2 @ Xs ) )
= ( removeAll_nat @ X @ Xs ) ) )
& ( ( X != Y2 )
=> ( ( removeAll_nat @ X @ ( cons_nat @ Y2 @ Xs ) )
= ( cons_nat @ Y2 @ ( removeAll_nat @ X @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_1178_inj__on__Cons1,axiom,
! [X: list_nat,A: set_list_list_nat] : ( inj_on2300671324199612755st_nat @ ( cons_list_nat @ X ) @ A ) ).
% inj_on_Cons1
thf(fact_1179_inj__on__Cons1,axiom,
! [X: nat,A: set_list_nat] : ( inj_on3049792774292151987st_nat @ ( cons_nat @ X ) @ A ) ).
% inj_on_Cons1
thf(fact_1180_verit__sum__simplify,axiom,
! [A4: nat] :
( ( plus_plus_nat @ A4 @ zero_zero_nat )
= A4 ) ).
% verit_sum_simplify
thf(fact_1181_verit__sum__simplify,axiom,
! [A4: int] :
( ( plus_plus_int @ A4 @ zero_zero_int )
= A4 ) ).
% verit_sum_simplify
thf(fact_1182_add__le__imp__le__right,axiom,
! [A4: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A4 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_eq_nat @ A4 @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_1183_add__le__imp__le__right,axiom,
! [A4: int,C: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A4 @ C ) @ ( plus_plus_int @ B2 @ C ) )
=> ( ord_less_eq_int @ A4 @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_1184_add__le__imp__le__left,axiom,
! [C: nat,A4: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A4 ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_eq_nat @ A4 @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_1185_add__le__imp__le__left,axiom,
! [C: int,A4: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A4 ) @ ( plus_plus_int @ C @ B2 ) )
=> ( ord_less_eq_int @ A4 @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_1186_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
? [C3: nat] :
( B5
= ( plus_plus_nat @ A5 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_1187_add__right__mono,axiom,
! [A4: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A4 @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_1188_add__right__mono,axiom,
! [A4: int,B2: int,C: int] :
( ( ord_less_eq_int @ A4 @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A4 @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_1189_less__eqE,axiom,
! [A4: nat,B2: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ~ ! [C4: nat] :
( B2
!= ( plus_plus_nat @ A4 @ C4 ) ) ) ).
% less_eqE
thf(fact_1190_add__left__mono,axiom,
! [A4: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A4 ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_1191_add__left__mono,axiom,
! [A4: int,B2: int,C: int] :
( ( ord_less_eq_int @ A4 @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A4 ) @ ( plus_plus_int @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_1192_add__mono,axiom,
! [A4: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A4 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A4 @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_mono
thf(fact_1193_add__mono,axiom,
! [A4: int,B2: int,C: int,D: int] :
( ( ord_less_eq_int @ A4 @ B2 )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A4 @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% add_mono
thf(fact_1194_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I3: nat,J2: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I3 @ J2 )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1195_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I3: int,J2: int,K2: int,L: int] :
( ( ( ord_less_eq_int @ I3 @ J2 )
& ( ord_less_eq_int @ K2 @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I3 @ K2 ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1196_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I3: nat,J2: nat,K2: nat,L: nat] :
( ( ( I3 = J2 )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1197_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I3: int,J2: int,K2: int,L: int] :
( ( ( I3 = J2 )
& ( ord_less_eq_int @ K2 @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I3 @ K2 ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1198_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I3: nat,J2: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I3 @ J2 )
& ( K2 = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1199_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I3: int,J2: int,K2: int,L: int] :
( ( ( ord_less_eq_int @ I3 @ J2 )
& ( K2 = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I3 @ K2 ) @ ( plus_plus_int @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1200_list_Oset__intros_I2_J,axiom,
! [Y2: nat,X222: list_nat,X21: nat] :
( ( member_nat2 @ Y2 @ ( set_nat2 @ X222 ) )
=> ( member_nat2 @ Y2 @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_1201_plus__int__code_I1_J,axiom,
! [K2: int] :
( ( plus_plus_int @ K2 @ zero_zero_int )
= K2 ) ).
% plus_int_code(1)
thf(fact_1202_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1203_odd__nonzero,axiom,
! [Z2: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1204_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_1205_odd__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1206_le__imp__0__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ).
% le_imp_0_less
thf(fact_1207_mod__by__Suc__0,axiom,
! [M2: nat] :
( ( modulo_modulo_nat @ M2 @ ( suc @ zero_zero_nat ) )
= zero_zero_nat ) ).
% mod_by_Suc_0
thf(fact_1208_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_1209_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1210_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1211_mod__neg__neg__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less_eq_int @ K2 @ zero_zero_int )
=> ( ( ord_less_int @ L @ K2 )
=> ( ( modulo_modulo_int @ K2 @ L )
= K2 ) ) ) ).
% mod_neg_neg_trivial
thf(fact_1212_mod__pos__pos__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ( ( ord_less_int @ K2 @ L )
=> ( ( modulo_modulo_int @ K2 @ L )
= K2 ) ) ) ).
% mod_pos_pos_trivial
thf(fact_1213_rgf__limit_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ~ ! [X3: nat,Xs3: list_nat] :
( X
!= ( cons_nat @ X3 @ Xs3 ) ) ) ).
% rgf_limit.cases
thf(fact_1214_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1215_add__eq__self__zero,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= M2 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1216_Euclidean__Division_Opos__mod__bound,axiom,
! [L: int,K2: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ord_less_int @ ( modulo_modulo_int @ K2 @ L ) @ L ) ) ).
% Euclidean_Division.pos_mod_bound
thf(fact_1217_neg__mod__bound,axiom,
! [L: int,K2: int] :
( ( ord_less_int @ L @ zero_zero_int )
=> ( ord_less_int @ L @ ( modulo_modulo_int @ K2 @ L ) ) ) ).
% neg_mod_bound
thf(fact_1218_zmod__le__nonneg__dividend,axiom,
! [M2: int,K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ M2 )
=> ( ord_less_eq_int @ ( modulo_modulo_int @ M2 @ K2 ) @ M2 ) ) ).
% zmod_le_nonneg_dividend
thf(fact_1219_zmod__zminus1__not__zero,axiom,
! [K2: int,L: int] :
( ( ( modulo_modulo_int @ ( uminus_uminus_int @ K2 ) @ L )
!= zero_zero_int )
=> ( ( modulo_modulo_int @ K2 @ L )
!= zero_zero_int ) ) ).
% zmod_zminus1_not_zero
thf(fact_1220_zmod__zminus2__not__zero,axiom,
! [K2: int,L: int] :
( ( ( modulo_modulo_int @ K2 @ ( uminus_uminus_int @ L ) )
!= zero_zero_int )
=> ( ( modulo_modulo_int @ K2 @ L )
!= zero_zero_int ) ) ).
% zmod_zminus2_not_zero
thf(fact_1221_neg__mod__sign,axiom,
! [L: int,K2: int] :
( ( ord_less_int @ L @ zero_zero_int )
=> ( ord_less_eq_int @ ( modulo_modulo_int @ K2 @ L ) @ zero_zero_int ) ) ).
% neg_mod_sign
thf(fact_1222_Euclidean__Division_Opos__mod__sign,axiom,
! [L: int,K2: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K2 @ L ) ) ) ).
% Euclidean_Division.pos_mod_sign
thf(fact_1223_zmod__trivial__iff,axiom,
! [I3: int,K2: int] :
( ( ( modulo_modulo_int @ I3 @ K2 )
= I3 )
= ( ( K2 = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I3 )
& ( ord_less_int @ I3 @ K2 ) )
| ( ( ord_less_eq_int @ I3 @ zero_zero_int )
& ( ord_less_int @ K2 @ I3 ) ) ) ) ).
% zmod_trivial_iff
thf(fact_1224_one__is__add,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M2 @ N ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1225_add__is__1,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1226_less__imp__add__positive,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ? [K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
& ( ( plus_plus_nat @ I3 @ K )
= J2 ) ) ) ).
% less_imp_add_positive
thf(fact_1227_mod__pos__neg__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ K2 @ L ) @ zero_zero_int )
=> ( ( modulo_modulo_int @ K2 @ L )
= ( plus_plus_int @ K2 @ L ) ) ) ) ).
% mod_pos_neg_trivial
thf(fact_1228_nat__mod__distrib,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y2 )
=> ( ( nat2 @ ( modulo_modulo_int @ X @ Y2 ) )
= ( modulo_modulo_nat @ ( nat2 @ X ) @ ( nat2 @ Y2 ) ) ) ) ) ).
% nat_mod_distrib
thf(fact_1229_stirling__row__code_I1_J,axiom,
( ( stirling_row @ zero_zero_nat )
= ( cons_nat @ one_one_nat @ nil_nat ) ) ).
% stirling_row_code(1)
thf(fact_1230_nat__add__distrib,axiom,
! [Z2: int,Z4: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z4 )
=> ( ( nat2 @ ( plus_plus_int @ Z2 @ Z4 ) )
= ( plus_plus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z4 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_1231_mod__Suc,axiom,
! [M2: nat,N: nat] :
( ( ( ( suc @ ( modulo_modulo_nat @ M2 @ N ) )
= N )
=> ( ( modulo_modulo_nat @ ( suc @ M2 ) @ N )
= zero_zero_nat ) )
& ( ( ( suc @ ( modulo_modulo_nat @ M2 @ N ) )
!= N )
=> ( ( modulo_modulo_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( modulo_modulo_nat @ M2 @ N ) ) ) ) ) ).
% mod_Suc
thf(fact_1232_mod__less__divisor,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( modulo_modulo_nat @ M2 @ N ) @ N ) ) ).
% mod_less_divisor
thf(fact_1233_mod__le__divisor,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ ( modulo_modulo_nat @ M2 @ N ) @ N ) ) ).
% mod_le_divisor
thf(fact_1234_gcd__nat__induct,axiom,
! [P2: nat > nat > $o,M2: nat,N: nat] :
( ! [M4: nat] : ( P2 @ M4 @ zero_zero_nat )
=> ( ! [M4: nat,N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P2 @ N3 @ ( modulo_modulo_nat @ M4 @ N3 ) )
=> ( P2 @ M4 @ N3 ) ) )
=> ( P2 @ M2 @ N ) ) ) ).
% gcd_nat_induct
thf(fact_1235_Euclid__induct,axiom,
! [P2: nat > nat > $o,A4: nat,B2: nat] :
( ! [A2: nat,B3: nat] :
( ( P2 @ A2 @ B3 )
= ( P2 @ B3 @ A2 ) )
=> ( ! [A2: nat] : ( P2 @ A2 @ zero_zero_nat )
=> ( ! [A2: nat,B3: nat] :
( ( P2 @ A2 @ B3 )
=> ( P2 @ A2 @ ( plus_plus_nat @ A2 @ B3 ) ) )
=> ( P2 @ A4 @ B2 ) ) ) ) ).
% Euclid_induct
thf(fact_1236_verit__le__mono__div__int,axiom,
! [A: int,B: int,N: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ord_less_eq_int
@ ( plus_plus_int @ ( divide_divide_int @ A @ N )
@ ( if_int
@ ( ( modulo_modulo_int @ B @ N )
= zero_zero_int )
@ one_one_int
@ zero_zero_int ) )
@ ( divide_divide_int @ B @ N ) ) ) ) ).
% verit_le_mono_div_int
thf(fact_1237_div__pos__pos__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ( ( ord_less_int @ K2 @ L )
=> ( ( divide_divide_int @ K2 @ L )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_1238_div__neg__neg__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less_eq_int @ K2 @ zero_zero_int )
=> ( ( ord_less_int @ L @ K2 )
=> ( ( divide_divide_int @ K2 @ L )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_1239_div__neg__pos__less0,axiom,
! [A4: int,B2: int] :
( ( ord_less_int @ A4 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ ( divide_divide_int @ A4 @ B2 ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_1240_neg__imp__zdiv__neg__iff,axiom,
! [B2: int,A4: int] :
( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A4 @ B2 ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A4 ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_1241_pos__imp__zdiv__neg__iff,axiom,
! [B2: int,A4: int] :
( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_int @ ( divide_divide_int @ A4 @ B2 ) @ zero_zero_int )
= ( ord_less_int @ A4 @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_1242_nonneg1__imp__zdiv__pos__iff,axiom,
! [A4: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A4 )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A4 @ B2 ) )
= ( ( ord_less_eq_int @ B2 @ A4 )
& ( ord_less_int @ zero_zero_int @ B2 ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1243_pos__imp__zdiv__nonneg__iff,axiom,
! [B2: int,A4: int] :
( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A4 @ B2 ) )
= ( ord_less_eq_int @ zero_zero_int @ A4 ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1244_neg__imp__zdiv__nonneg__iff,axiom,
! [B2: int,A4: int] :
( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A4 @ B2 ) )
= ( ord_less_eq_int @ A4 @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1245_pos__imp__zdiv__pos__iff,axiom,
! [K2: int,I3: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I3 @ K2 ) )
= ( ord_less_eq_int @ K2 @ I3 ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_1246_div__nonpos__pos__le0,axiom,
! [A4: int,B2: int] :
( ( ord_less_eq_int @ A4 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A4 @ B2 ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1247_div__nonneg__neg__le0,axiom,
! [A4: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A4 )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A4 @ B2 ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1248_div__int__pos__iff,axiom,
! [K2: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K2 @ L ) )
= ( ( K2 = zero_zero_int )
| ( L = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ K2 )
& ( ord_less_eq_int @ zero_zero_int @ L ) )
| ( ( ord_less_int @ K2 @ zero_zero_int )
& ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% div_int_pos_iff
thf(fact_1249_zdiv__mono2__neg,axiom,
! [A4: int,B7: int,B2: int] :
( ( ord_less_int @ A4 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B7 )
=> ( ( ord_less_eq_int @ B7 @ B2 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A4 @ B7 ) @ ( divide_divide_int @ A4 @ B2 ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1250_zdiv__mono1__neg,axiom,
! [A4: int,A6: int,B2: int] :
( ( ord_less_eq_int @ A4 @ A6 )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A6 @ B2 ) @ ( divide_divide_int @ A4 @ B2 ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1251_zdiv__eq__0__iff,axiom,
! [I3: int,K2: int] :
( ( ( divide_divide_int @ I3 @ K2 )
= zero_zero_int )
= ( ( K2 = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I3 )
& ( ord_less_int @ I3 @ K2 ) )
| ( ( ord_less_eq_int @ I3 @ zero_zero_int )
& ( ord_less_int @ K2 @ I3 ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_1252_zdiv__mono2,axiom,
! [A4: int,B7: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A4 )
=> ( ( ord_less_int @ zero_zero_int @ B7 )
=> ( ( ord_less_eq_int @ B7 @ B2 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A4 @ B2 ) @ ( divide_divide_int @ A4 @ B7 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1253_zdiv__mono1,axiom,
! [A4: int,A6: int,B2: int] :
( ( ord_less_eq_int @ A4 @ A6 )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A4 @ B2 ) @ ( divide_divide_int @ A6 @ B2 ) ) ) ) ).
% zdiv_mono1
thf(fact_1254_int__div__less__self,axiom,
! [X: int,K2: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ one_one_int @ K2 )
=> ( ord_less_int @ ( divide_divide_int @ X @ K2 ) @ X ) ) ) ).
% int_div_less_self
thf(fact_1255_zdiv__mono__strict,axiom,
! [A: int,B: int,N: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ( ( modulo_modulo_int @ A @ N )
= zero_zero_int )
=> ( ( ( modulo_modulo_int @ B @ N )
= zero_zero_int )
=> ( ord_less_int @ ( divide_divide_int @ A @ N ) @ ( divide_divide_int @ B @ N ) ) ) ) ) ) ).
% zdiv_mono_strict
thf(fact_1256_verit__less__mono__div__int2,axiom,
! [A: int,B: int,N: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
=> ( ord_less_eq_int @ ( divide_divide_int @ B @ N ) @ ( divide_divide_int @ A @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_1257_div__eq__minus1,axiom,
! [B2: int] :
( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B2 )
= ( uminus_uminus_int @ one_one_int ) ) ) ).
% div_eq_minus1
thf(fact_1258_div__pos__neg__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ K2 @ L ) @ zero_zero_int )
=> ( ( divide_divide_int @ K2 @ L )
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% div_pos_neg_trivial
thf(fact_1259_div__pos__geq,axiom,
! [L: int,K2: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ( ord_less_eq_int @ L @ K2 )
=> ( ( divide_divide_int @ K2 @ L )
= ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K2 @ L ) @ L ) @ one_one_int ) ) ) ) ).
% div_pos_geq
thf(fact_1260_div__by__Suc__0,axiom,
! [M2: nat] :
( ( divide_divide_nat @ M2 @ ( suc @ zero_zero_nat ) )
= M2 ) ).
% div_by_Suc_0
thf(fact_1261_div__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( divide_divide_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1262_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1263_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1264_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_1265_diff__is__0__eq_H,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1266_diff__is__0__eq,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% diff_is_0_eq
thf(fact_1267_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_1268_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_1269_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M2: nat,N: nat] :
( ( ( divide_divide_nat @ M2 @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M2 @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1270_div__if,axiom,
( divide_divide_nat
= ( ^ [M: nat,N2: nat] :
( if_nat
@ ( ( ord_less_nat @ M @ N2 )
| ( N2 = zero_zero_nat ) )
@ zero_zero_nat
@ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ) ) ).
% div_if
thf(fact_1271_le__div__geq,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( ( divide_divide_nat @ M2 @ N )
= ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ) ) ).
% le_div_geq
% Helper facts (5)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y2: int] :
( ( if_int @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y2: int] :
( ( if_int @ $true @ X @ Y2 )
= X ) ).
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y2: nat] :
( ( if_nat @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y2: nat] :
( ( if_nat @ $true @ X @ Y2 )
= X ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
! [X7: a > nat] :
( ( inj_on_a_nat @ X7 @ ( set_a2 @ x ) )
=> ( ( equiva3371634703666331078on_rgf @ ( map_a_nat @ X7 @ x ) )
=> thesis ) ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------