TPTP Problem File: SLH0272^1.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_00421_015997__12138436_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1476 ( 657 unt; 208 typ; 0 def)
% Number of atoms : 3271 (1192 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 9050 ( 405 ~; 71 |; 115 &;7047 @)
% ( 0 <=>;1412 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 22 ( 21 usr)
% Number of type conns : 795 ( 795 >; 0 *; 0 +; 0 <<)
% Number of symbols : 190 ( 187 usr; 14 con; 0-3 aty)
% Number of variables : 3160 ( 233 ^;2810 !; 117 ?;3160 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:14:46.937
%------------------------------------------------------------------------------
% Could-be-implicit typings (21)
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__Set__Oset_It__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
set_li2602615041181522887at_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__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_J,type,
list_list_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_list_set_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__Set__Oset_It__Nat__Onat_J_J,type,
list_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
set_num: $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__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (187)
thf(sy_c_Equiv__Relations_Oequiv_001t__Nat__Onat,type,
equiv_equiv_nat: set_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Equivalence__Relation__Enumeration_Oenum__rgfs,type,
equiva7426478223624825838m_rgfs: nat > list_list_nat ).
thf(sy_c_Equivalence__Relation__Enumeration_Oequiv__rels,type,
equiva8721718519204927301v_rels: nat > list_s1210847774152347623at_nat ).
thf(sy_c_Equivalence__Relation__Enumeration_Okernel__of_001t__List__Olist_It__Nat__Onat_J,type,
equiva6490762433048536736st_nat: list_list_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Equivalence__Relation__Enumeration_Okernel__of_001t__Nat__Onat,type,
equiva2048684438135499664of_nat: list_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Equivalence__Relation__Enumeration_Okernel__of_001t__Set__Oset_It__Nat__Onat_J,type,
equiva50036288466687814et_nat: list_set_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Equivalence__Relation__Enumeration_Okernel__of_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
equiva1173177585473067681at_nat: list_s1210847774152347623at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Equivalence__Relation__Enumeration_Orgf,type,
equiva3371634703666331078on_rgf: list_nat > $o ).
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__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__Num__Onum,type,
inj_on_list_nat_num: ( list_nat > num ) > set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
inj_on1816901372521670873et_nat: ( list_nat > set_nat ) > set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__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__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__List__Olist_It__Nat__Onat_J,type,
inj_on_nat_list_nat: ( nat > list_nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
inj_on_nat_set_nat: ( nat > set_nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
inj_on7202621048659411806at_nat: ( nat > set_Pr1261947904930325089at_nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
inj_on5467128325884351833st_nat: ( set_nat > list_nat ) > set_set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
inj_on_set_nat_nat: ( set_nat > nat ) > set_set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
inj_on4604407203859583615et_nat: ( set_nat > set_nat ) > set_set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
inj_on4888989441411116416at_nat: ( set_Pr1261947904930325089at_nat > nat ) > set_se7855581050983116737at_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
inj_on5220459939087112502et_nat: ( set_Pr1261947904930325089at_nat > set_nat ) > set_se7855581050983116737at_nat > $o ).
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_Oplus__class_Oplus_001t__Num__Onum,type,
plus_plus_num: num > num > num ).
thf(sy_c_Groups_Osgn__class_Osgn_001t__Int__Oint,type,
sgn_sgn_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
uminus5770388063884162150_nat_o: ( list_nat > $o ) > list_nat > $o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
uminus5680096308674742949_nat_o: ( set_Pr1261947904930325089at_nat > $o ) > set_Pr1261947904930325089at_nat > $o ).
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_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
uminus8018730102153342520at_nat: set_se7855581050983116737at_nat > set_se7855581050983116737at_nat ).
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_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__Big_Oord__class_Oarg__min_001t__List__Olist_It__Nat__Onat_J_001t__Int__Oint,type,
lattic340647829947351080at_int: ( list_nat > int ) > ( list_nat > $o ) > list_nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
lattic343138300456401356at_nat: ( list_nat > nat ) > ( list_nat > $o ) > list_nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min_001t__List__Olist_It__Nat__Onat_J_001t__Num__Onum,type,
lattic6123842123466955798at_num: ( list_nat > num ) > ( list_nat > $o ) > list_nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
lattic5017752457218834789at_nat: ( list_nat > set_Pr1261947904930325089at_nat ) > ( list_nat > $o ) > list_nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min_001t__Nat__Onat_001t__Nat__Onat,type,
lattic8739620818006775868at_nat: ( nat > nat ) > ( nat > $o ) > nat ).
thf(sy_c_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__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
coset_4613306324563743060at_nat: list_s1210847774152347623at_nat > set_se7855581050983116737at_nat ).
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__Nat__Onat_J,type,
count_list_set_nat: list_set_nat > set_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_Ofilter_001t__List__Olist_It__Nat__Onat_J,type,
filter_list_nat: ( list_nat > $o ) > list_list_nat > list_list_nat ).
thf(sy_c_List_Ofilter_001t__Nat__Onat,type,
filter_nat: ( nat > $o ) > list_nat > list_nat ).
thf(sy_c_List_Ofilter_001t__Set__Oset_It__Nat__Onat_J,type,
filter_set_nat: ( set_nat > $o ) > list_set_nat > list_set_nat ).
thf(sy_c_List_Ofilter_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
filter4791820933415917918at_nat: ( set_Pr1261947904930325089at_nat > $o ) > list_s1210847774152347623at_nat > list_s1210847774152347623at_nat ).
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_Ogen__length_001t__Set__Oset_It__Nat__Onat_J,type,
gen_length_set_nat: nat > list_set_nat > nat ).
thf(sy_c_List_Ogen__length_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
gen_le5092146751752969972at_nat: nat > list_s1210847774152347623at_nat > 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__Nat__Onat_J,type,
nil_set_nat: list_set_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_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__Nat__Onat_J,type,
map_list_nat_set_nat: ( list_nat > set_nat ) > list_list_nat > list_set_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__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_001t__Set__Oset_It__Nat__Onat_J,type,
map_nat_set_nat: ( nat > set_nat ) > list_nat > list_set_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
map_na6577772983117884747at_nat: ( nat > set_Pr1261947904930325089at_nat ) > list_nat > list_s1210847774152347623at_nat ).
thf(sy_c_List_Olist_Omap_001t__Set__Oset_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
map_set_nat_list_nat: ( set_nat > list_nat ) > list_set_nat > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
map_set_nat_nat: ( set_nat > nat ) > list_set_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
map_set_nat_set_nat: ( set_nat > set_nat ) > list_set_nat > list_set_nat ).
thf(sy_c_List_Olist_Omap_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
map_se1162299115588061717at_nat: ( set_nat > set_Pr1261947904930325089at_nat ) > list_set_nat > list_s1210847774152347623at_nat ).
thf(sy_c_List_Olist_Omap_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
map_se4264141375869589357at_nat: ( set_Pr1261947904930325089at_nat > nat ) > list_s1210847774152347623at_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
map_se493947572549472035et_nat: ( set_Pr1261947904930325089at_nat > set_nat ) > list_s1210847774152347623at_nat > list_set_nat ).
thf(sy_c_List_Olist_Omap_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
map_se4718776794341882500at_nat: ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ) > list_s1210847774152347623at_nat > list_s1210847774152347623at_nat ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_list_nat2: list_list_list_nat > set_list_list_nat ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
set_list_nat2: list_list_nat > set_list_nat ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
set_list_set_nat2: list_list_set_nat > set_list_set_nat ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_li2149138424243681404at_nat: list_l3822697302700470509at_nat > set_li2602615041181522887at_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__Nat__Onat_J,type,
set_set_nat2: list_set_nat > set_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_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_Omember_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_list_nat > list_nat > $o ).
thf(sy_c_List_Omember_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member4653088687621619836at_nat: list_s1210847774152347623at_nat > set_Pr1261947904930325089at_nat > $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_On__lists_001t__Set__Oset_It__Nat__Onat_J,type,
n_lists_set_nat: nat > list_set_nat > list_list_set_nat ).
thf(sy_c_List_On__lists_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
n_list4589530985885001583at_nat: nat > list_s1210847774152347623at_nat > list_l3822697302700470509at_nat ).
thf(sy_c_List_Oproduct__lists_001t__List__Olist_It__Nat__Onat_J,type,
produc6783906451316923569st_nat: list_list_list_nat > list_list_list_nat ).
thf(sy_c_List_Oproduct__lists_001t__Nat__Onat,type,
product_lists_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Oproduct__lists_001t__Set__Oset_It__Nat__Onat_J,type,
produc8109398739672286679et_nat: list_list_set_nat > list_list_set_nat ).
thf(sy_c_List_Oproduct__lists_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc5476187143726666448at_nat: list_l3822697302700470509at_nat > list_l3822697302700470509at_nat ).
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__Nat__Onat_J,type,
rotate1_set_nat: list_set_nat > list_set_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_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__Nat__Onat_J,type,
rotate_set_nat: nat > list_set_nat > list_set_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_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__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
size_s6248950052170075156st_nat: list_list_list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
size_s3023201423986296836st_nat: list_list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_J,type,
size_s8312130442339763642et_nat: list_list_set_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
size_s3876073854528961369at_nat: list_l3822697302700470509at_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__Nat__Onat_J_J,type,
size_s3254054031482475050et_nat: list_set_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__Num__Onum,type,
size_size_num: num > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Num_OBitM,type,
bitM: num > num ).
thf(sy_c_Num_Oinc,type,
inc: num > num ).
thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
neg_numeral_dbl_int: int > int ).
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_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onum_Osize__num,type,
size_num: num > nat ).
thf(sy_c_Num_Onum__of__nat,type,
num_of_nat: nat > num ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
numeral_numeral_int: num > int ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
ord_less_list_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
ord_le220952123948551344_nat_o: ( set_Pr1261947904930325089at_nat > $o ) > ( set_Pr1261947904930325089at_nat > $o ) > $o ).
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__Num__Onum,type,
ord_less_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
ord_le1190675801316882794st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Num__Onum_J,type,
ord_less_set_num: set_num > set_num > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le7866589430770878221at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le3519150613208228973at_nat: set_se7855581050983116737at_nat > set_se7855581050983116737at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
ord_le1520216061033275535_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
ord_le5866676684937102012_nat_o: ( set_Pr1261947904930325089at_nat > $o ) > ( set_Pr1261947904930325089at_nat > $o ) > $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__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Num__Onum_J,type,
ord_less_eq_set_num: set_num > set_num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le3146513528884898305at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le2077887516847798113at_nat: set_se7855581050983116737at_nat > set_se7855581050983116737at_nat > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
collec5989764272469232197st_nat: ( list_list_nat > $o ) > set_list_list_nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
collect_list_set_nat: ( list_set_nat > $o ) > set_list_set_nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
collec861049716225513554at_nat: ( list_s1210847774152347623at_nat > $o ) > set_li2602615041181522887at_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Num__Onum,type,
collect_num: ( num > $o ) > set_num ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
collec5514110066124741708at_nat: ( set_Pr1261947904930325089at_nat > $o ) > set_se7855581050983116737at_nat ).
thf(sy_c_Set_Ofilter_001t__List__Olist_It__Nat__Onat_J,type,
filter_list_nat2: ( list_nat > $o ) > set_list_nat > set_list_nat ).
thf(sy_c_Set_Ofilter_001t__Nat__Onat,type,
filter_nat2: ( nat > $o ) > set_nat > set_nat ).
thf(sy_c_Set_Ofilter_001t__Set__Oset_It__Nat__Onat_J,type,
filter_set_nat2: ( set_nat > $o ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Ofilter_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
filter6836037073761582098at_nat: ( set_Pr1261947904930325089at_nat > $o ) > set_se7855581050983116737at_nat > set_se7855581050983116737at_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
set_ord_lessThan_int: int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Num__Onum,type,
set_ord_lessThan_num: num > set_num ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_or5243998310507771012at_nat: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_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__Int__Oint,type,
member_int: int > set_int > $o ).
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__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
member_list_set_nat: list_set_nat > set_list_set_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member2758946086501665296at_nat: list_s1210847774152347623at_nat > set_li2602615041181522887at_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Num__Onum,type,
member_num: num > set_num > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member2643936169264416010at_nat: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat > $o ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_r,type,
r: set_Pr1261947904930325089at_nat ).
thf(sy_v_y____,type,
y: list_nat ).
% Relevant facts (1262)
thf(fact_0_y__def_I1_J,axiom,
equiva3371634703666331078on_rgf @ y ).
% y_def(1)
thf(fact_1_y__def_I2_J,axiom,
( ( size_size_list_nat @ y )
= n ) ).
% y_def(2)
thf(fact_2_count__list__expand,axiom,
( count_6440129622255701469at_nat
= ( ^ [Xs: list_s1210847774152347623at_nat,X: set_Pr1261947904930325089at_nat] :
( size_s8736152011456118867at_nat
@ ( filter4791820933415917918at_nat
@ ( ^ [Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : ( Y = Z )
@ X )
@ Xs ) ) ) ) ).
% count_list_expand
thf(fact_3_count__list__expand,axiom,
( count_list_list_nat
= ( ^ [Xs: list_list_nat,X: list_nat] :
( size_s3023201423986296836st_nat
@ ( filter_list_nat
@ ( ^ [Y: list_nat,Z: list_nat] : ( Y = Z )
@ X )
@ Xs ) ) ) ) ).
% count_list_expand
thf(fact_4_count__list__expand,axiom,
( count_list_nat
= ( ^ [Xs: list_nat,X: nat] :
( size_size_list_nat
@ ( filter_nat
@ ( ^ [Y: nat,Z: nat] : ( Y = Z )
@ X )
@ Xs ) ) ) ) ).
% count_list_expand
thf(fact_5_count__list__expand,axiom,
( count_list_set_nat
= ( ^ [Xs: list_set_nat,X: set_nat] :
( size_s3254054031482475050et_nat
@ ( filter_set_nat
@ ( ^ [Y: set_nat,Z: set_nat] : ( Y = Z )
@ X )
@ Xs ) ) ) ) ).
% count_list_expand
thf(fact_6__092_060open_062length_A_Ifilter_A_I_092_060lambda_062x_O_Akernel__of_Ay_A_061_Akernel__of_Ax_J_A_Ienum__rgfs_An_J_J_A_061_Alength_A_Ifilter_A_I_I_061_J_Ay_J_A_Ienum__rgfs_An_J_J_092_060close_062,axiom,
( ( size_s3023201423986296836st_nat
@ ( filter_list_nat
@ ^ [X: list_nat] :
( ( equiva2048684438135499664of_nat @ y )
= ( equiva2048684438135499664of_nat @ X ) )
@ ( equiva7426478223624825838m_rgfs @ n ) ) )
= ( size_s3023201423986296836st_nat
@ ( filter_list_nat
@ ( ^ [Y: list_nat,Z: list_nat] : ( Y = Z )
@ y )
@ ( equiva7426478223624825838m_rgfs @ n ) ) ) ) ).
% \<open>length (filter (\<lambda>x. kernel_of y = kernel_of x) (enum_rgfs n)) = length (filter ((=) y) (enum_rgfs n))\<close>
thf(fact_7_filter__filter,axiom,
! [P: set_nat > $o,Q: set_nat > $o,Xs2: list_set_nat] :
( ( filter_set_nat @ P @ ( filter_set_nat @ Q @ Xs2 ) )
= ( filter_set_nat
@ ^ [X: set_nat] :
( ( Q @ X )
& ( P @ X ) )
@ Xs2 ) ) ).
% filter_filter
thf(fact_8_filter__filter,axiom,
! [P: nat > $o,Q: nat > $o,Xs2: list_nat] :
( ( filter_nat @ P @ ( filter_nat @ Q @ Xs2 ) )
= ( filter_nat
@ ^ [X: nat] :
( ( Q @ X )
& ( P @ X ) )
@ Xs2 ) ) ).
% filter_filter
thf(fact_9_filter__filter,axiom,
! [P: set_Pr1261947904930325089at_nat > $o,Q: set_Pr1261947904930325089at_nat > $o,Xs2: list_s1210847774152347623at_nat] :
( ( filter4791820933415917918at_nat @ P @ ( filter4791820933415917918at_nat @ Q @ Xs2 ) )
= ( filter4791820933415917918at_nat
@ ^ [X: set_Pr1261947904930325089at_nat] :
( ( Q @ X )
& ( P @ X ) )
@ Xs2 ) ) ).
% filter_filter
thf(fact_10_filter__filter,axiom,
! [P: list_nat > $o,Q: list_nat > $o,Xs2: list_list_nat] :
( ( filter_list_nat @ P @ ( filter_list_nat @ Q @ Xs2 ) )
= ( filter_list_nat
@ ^ [X: list_nat] :
( ( Q @ X )
& ( P @ X ) )
@ Xs2 ) ) ).
% filter_filter
thf(fact_11__092_060open_062length_A_Ifilter_A_I_092_060lambda_062x_O_Ar_A_061_Akernel__of_Ax_J_A_Ienum__rgfs_An_J_J_A_061_Alength_A_Ifilter_A_I_092_060lambda_062x_O_Akernel__of_Ay_A_061_Akernel__of_Ax_J_A_Ienum__rgfs_An_J_J_092_060close_062,axiom,
( ( size_s3023201423986296836st_nat
@ ( filter_list_nat
@ ^ [X: list_nat] :
( r
= ( equiva2048684438135499664of_nat @ X ) )
@ ( equiva7426478223624825838m_rgfs @ n ) ) )
= ( size_s3023201423986296836st_nat
@ ( filter_list_nat
@ ^ [X: list_nat] :
( ( equiva2048684438135499664of_nat @ y )
= ( equiva2048684438135499664of_nat @ X ) )
@ ( equiva7426478223624825838m_rgfs @ n ) ) ) ) ).
% \<open>length (filter (\<lambda>x. r = kernel_of x) (enum_rgfs n)) = length (filter (\<lambda>x. kernel_of y = kernel_of x) (enum_rgfs n))\<close>
thf(fact_12_y__def_I3_J,axiom,
( ( equiva2048684438135499664of_nat @ y )
= r ) ).
% y_def(3)
thf(fact_13_calculation,axiom,
( ( count_6440129622255701469at_nat @ ( equiva8721718519204927301v_rels @ n ) @ r )
= ( size_s3023201423986296836st_nat
@ ( filter_list_nat
@ ( ^ [Y: list_nat,Z: list_nat] : ( Y = Z )
@ y )
@ ( equiva7426478223624825838m_rgfs @ n ) ) ) ) ).
% calculation
thf(fact_14_a,axiom,
! [X2: list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ ( equiva7426478223624825838m_rgfs @ n ) ) )
=> ( ( ( equiva2048684438135499664of_nat @ y )
= ( equiva2048684438135499664of_nat @ X2 ) )
= ( y = X2 ) ) ) ).
% a
thf(fact_15_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_s1210847774152347623at_nat] :
( ( size_s8736152011456118867at_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_16_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_list_nat] :
( ( size_s3023201423986296836st_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_17_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_nat] :
( ( size_size_list_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_18_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_set_nat] :
( ( size_s3254054031482475050et_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_19_neq__if__length__neq,axiom,
! [Xs2: list_list_nat,Ys: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
!= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_20_neq__if__length__neq,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_21_neq__if__length__neq,axiom,
! [Xs2: list_set_nat,Ys: list_set_nat] :
( ( ( size_s3254054031482475050et_nat @ Xs2 )
!= ( size_s3254054031482475050et_nat @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_22_neq__if__length__neq,axiom,
! [Xs2: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat] :
( ( ( size_s8736152011456118867at_nat @ Xs2 )
!= ( size_s8736152011456118867at_nat @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_23_size__neq__size__imp__neq,axiom,
! [X2: list_list_nat,Y2: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ X2 )
!= ( size_s3023201423986296836st_nat @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_24_size__neq__size__imp__neq,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( ( size_size_list_nat @ X2 )
!= ( size_size_list_nat @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_25_size__neq__size__imp__neq,axiom,
! [X2: list_set_nat,Y2: list_set_nat] :
( ( ( size_s3254054031482475050et_nat @ X2 )
!= ( size_s3254054031482475050et_nat @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_26_size__neq__size__imp__neq,axiom,
! [X2: char,Y2: char] :
( ( ( size_size_char @ X2 )
!= ( size_size_char @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_27_size__neq__size__imp__neq,axiom,
! [X2: num,Y2: num] :
( ( ( size_size_num @ X2 )
!= ( size_size_num @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_28_size__neq__size__imp__neq,axiom,
! [X2: list_s1210847774152347623at_nat,Y2: list_s1210847774152347623at_nat] :
( ( ( size_s8736152011456118867at_nat @ X2 )
!= ( size_s8736152011456118867at_nat @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_29__092_060open_062count__list_A_Iequiv__rels_An_J_Ar_A_061_Alength_A_Ifilter_A_I_092_060lambda_062x_O_Ar_A_061_Akernel__of_Ax_J_A_Ienum__rgfs_An_J_J_092_060close_062,axiom,
( ( count_6440129622255701469at_nat @ ( equiva8721718519204927301v_rels @ n ) @ r )
= ( size_s3023201423986296836st_nat
@ ( filter_list_nat
@ ^ [X: list_nat] :
( r
= ( equiva2048684438135499664of_nat @ X ) )
@ ( equiva7426478223624825838m_rgfs @ n ) ) ) ) ).
% \<open>count_list (equiv_rels n) r = length (filter (\<lambda>x. r = kernel_of x) (enum_rgfs n))\<close>
thf(fact_30_kernel__of__eq__len,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( ( equiva2048684438135499664of_nat @ X2 )
= ( equiva2048684438135499664of_nat @ Y2 ) )
=> ( ( size_size_list_nat @ X2 )
= ( size_size_list_nat @ Y2 ) ) ) ).
% kernel_of_eq_len
thf(fact_31_kernel__of__eq__len,axiom,
! [X2: list_list_nat,Y2: list_nat] :
( ( ( equiva6490762433048536736st_nat @ X2 )
= ( equiva2048684438135499664of_nat @ Y2 ) )
=> ( ( size_s3023201423986296836st_nat @ X2 )
= ( size_size_list_nat @ Y2 ) ) ) ).
% kernel_of_eq_len
thf(fact_32_kernel__of__eq__len,axiom,
! [X2: list_nat,Y2: list_list_nat] :
( ( ( equiva2048684438135499664of_nat @ X2 )
= ( equiva6490762433048536736st_nat @ Y2 ) )
=> ( ( size_size_list_nat @ X2 )
= ( size_s3023201423986296836st_nat @ Y2 ) ) ) ).
% kernel_of_eq_len
thf(fact_33_kernel__of__eq__len,axiom,
! [X2: list_nat,Y2: list_set_nat] :
( ( ( equiva2048684438135499664of_nat @ X2 )
= ( equiva50036288466687814et_nat @ Y2 ) )
=> ( ( size_size_list_nat @ X2 )
= ( size_s3254054031482475050et_nat @ Y2 ) ) ) ).
% kernel_of_eq_len
thf(fact_34_kernel__of__eq__len,axiom,
! [X2: list_set_nat,Y2: list_nat] :
( ( ( equiva50036288466687814et_nat @ X2 )
= ( equiva2048684438135499664of_nat @ Y2 ) )
=> ( ( size_s3254054031482475050et_nat @ X2 )
= ( size_size_list_nat @ Y2 ) ) ) ).
% kernel_of_eq_len
thf(fact_35_kernel__of__eq__len,axiom,
! [X2: list_list_nat,Y2: list_list_nat] :
( ( ( equiva6490762433048536736st_nat @ X2 )
= ( equiva6490762433048536736st_nat @ Y2 ) )
=> ( ( size_s3023201423986296836st_nat @ X2 )
= ( size_s3023201423986296836st_nat @ Y2 ) ) ) ).
% kernel_of_eq_len
thf(fact_36_kernel__of__eq__len,axiom,
! [X2: list_list_nat,Y2: list_set_nat] :
( ( ( equiva6490762433048536736st_nat @ X2 )
= ( equiva50036288466687814et_nat @ Y2 ) )
=> ( ( size_s3023201423986296836st_nat @ X2 )
= ( size_s3254054031482475050et_nat @ Y2 ) ) ) ).
% kernel_of_eq_len
thf(fact_37_kernel__of__eq__len,axiom,
! [X2: list_set_nat,Y2: list_list_nat] :
( ( ( equiva50036288466687814et_nat @ X2 )
= ( equiva6490762433048536736st_nat @ Y2 ) )
=> ( ( size_s3254054031482475050et_nat @ X2 )
= ( size_s3023201423986296836st_nat @ Y2 ) ) ) ).
% kernel_of_eq_len
thf(fact_38_kernel__of__eq__len,axiom,
! [X2: list_set_nat,Y2: list_set_nat] :
( ( ( equiva50036288466687814et_nat @ X2 )
= ( equiva50036288466687814et_nat @ Y2 ) )
=> ( ( size_s3254054031482475050et_nat @ X2 )
= ( size_s3254054031482475050et_nat @ Y2 ) ) ) ).
% kernel_of_eq_len
thf(fact_39_kernel__of__eq__len,axiom,
! [X2: list_s1210847774152347623at_nat,Y2: list_nat] :
( ( ( equiva1173177585473067681at_nat @ X2 )
= ( equiva2048684438135499664of_nat @ Y2 ) )
=> ( ( size_s8736152011456118867at_nat @ X2 )
= ( size_size_list_nat @ Y2 ) ) ) ).
% kernel_of_eq_len
thf(fact_40__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062y_O_A_092_060lbrakk_062rgf_Ay_059_Alength_Ay_A_061_An_059_Akernel__of_Ay_A_061_Ar_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Y3: list_nat] :
( ( equiva3371634703666331078on_rgf @ Y3 )
=> ( ( ( size_size_list_nat @ Y3 )
= n )
=> ( ( equiva2048684438135499664of_nat @ Y3 )
!= r ) ) ) ).
% \<open>\<And>thesis. (\<And>y. \<lbrakk>rgf y; length y = n; kernel_of y = r\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_41_filter__True,axiom,
! [Xs2: list_set_nat,P: set_nat > $o] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs2 ) )
=> ( P @ X3 ) )
=> ( ( filter_set_nat @ P @ Xs2 )
= Xs2 ) ) ).
% filter_True
thf(fact_42_filter__True,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
=> ( P @ X3 ) )
=> ( ( filter_nat @ P @ Xs2 )
= Xs2 ) ) ).
% filter_True
thf(fact_43_filter__True,axiom,
! [Xs2: list_list_nat,P: list_nat > $o] :
( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs2 ) )
=> ( P @ X3 ) )
=> ( ( filter_list_nat @ P @ Xs2 )
= Xs2 ) ) ).
% filter_True
thf(fact_44_filter__True,axiom,
! [Xs2: list_s1210847774152347623at_nat,P: set_Pr1261947904930325089at_nat > $o] :
( ! [X3: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X3 @ ( set_se5049602875457034614at_nat @ Xs2 ) )
=> ( P @ X3 ) )
=> ( ( filter4791820933415917918at_nat @ P @ Xs2 )
= Xs2 ) ) ).
% filter_True
thf(fact_45_set__filter,axiom,
! [P: set_nat > $o,Xs2: list_set_nat] :
( ( set_set_nat2 @ ( filter_set_nat @ P @ Xs2 ) )
= ( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ ( set_set_nat2 @ Xs2 ) )
& ( P @ X ) ) ) ) ).
% set_filter
thf(fact_46_set__filter,axiom,
! [P: nat > $o,Xs2: list_nat] :
( ( set_nat2 @ ( filter_nat @ P @ Xs2 ) )
= ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
& ( P @ X ) ) ) ) ).
% set_filter
thf(fact_47_set__filter,axiom,
! [P: list_nat > $o,Xs2: list_list_nat] :
( ( set_list_nat2 @ ( filter_list_nat @ P @ Xs2 ) )
= ( collect_list_nat
@ ^ [X: list_nat] :
( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs2 ) )
& ( P @ X ) ) ) ) ).
% set_filter
thf(fact_48_set__filter,axiom,
! [P: set_Pr1261947904930325089at_nat > $o,Xs2: list_s1210847774152347623at_nat] :
( ( set_se5049602875457034614at_nat @ ( filter4791820933415917918at_nat @ P @ Xs2 ) )
= ( collec5514110066124741708at_nat
@ ^ [X: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X @ ( set_se5049602875457034614at_nat @ Xs2 ) )
& ( P @ X ) ) ) ) ).
% set_filter
thf(fact_49_kernel__of__inj__on__rgfs__aux,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( ( size_size_list_nat @ X2 )
= ( size_size_list_nat @ Y2 ) )
=> ( ( equiva3371634703666331078on_rgf @ X2 )
=> ( ( equiva3371634703666331078on_rgf @ Y2 )
=> ( ( ( equiva2048684438135499664of_nat @ X2 )
= ( equiva2048684438135499664of_nat @ Y2 ) )
=> ( X2 = Y2 ) ) ) ) ) ).
% kernel_of_inj_on_rgfs_aux
thf(fact_50_enum__rgfs__len,axiom,
! [X2: list_nat,N: nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ ( equiva7426478223624825838m_rgfs @ N ) ) )
=> ( ( size_size_list_nat @ X2 )
= N ) ) ).
% enum_rgfs_len
thf(fact_51_filter__id__conv,axiom,
! [P: set_nat > $o,Xs2: list_set_nat] :
( ( ( filter_set_nat @ P @ Xs2 )
= Xs2 )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ ( set_set_nat2 @ Xs2 ) )
=> ( P @ X ) ) ) ) ).
% filter_id_conv
thf(fact_52_filter__id__conv,axiom,
! [P: nat > $o,Xs2: list_nat] :
( ( ( filter_nat @ P @ Xs2 )
= Xs2 )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( P @ X ) ) ) ) ).
% filter_id_conv
thf(fact_53_filter__id__conv,axiom,
! [P: list_nat > $o,Xs2: list_list_nat] :
( ( ( filter_list_nat @ P @ Xs2 )
= Xs2 )
= ( ! [X: list_nat] :
( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs2 ) )
=> ( P @ X ) ) ) ) ).
% filter_id_conv
thf(fact_54_filter__id__conv,axiom,
! [P: set_Pr1261947904930325089at_nat > $o,Xs2: list_s1210847774152347623at_nat] :
( ( ( filter4791820933415917918at_nat @ P @ Xs2 )
= Xs2 )
= ( ! [X: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X @ ( set_se5049602875457034614at_nat @ Xs2 ) )
=> ( P @ X ) ) ) ) ).
% filter_id_conv
thf(fact_55_filter__cong,axiom,
! [Xs2: list_set_nat,Ys: list_set_nat,P: set_nat > $o,Q: set_nat > $o] :
( ( Xs2 = Ys )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ ( set_set_nat2 @ Ys ) )
=> ( ( P @ X3 )
= ( Q @ X3 ) ) )
=> ( ( filter_set_nat @ P @ Xs2 )
= ( filter_set_nat @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_56_filter__cong,axiom,
! [Xs2: list_nat,Ys: list_nat,P: nat > $o,Q: nat > $o] :
( ( Xs2 = Ys )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
=> ( ( P @ X3 )
= ( Q @ X3 ) ) )
=> ( ( filter_nat @ P @ Xs2 )
= ( filter_nat @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_57_filter__cong,axiom,
! [Xs2: list_list_nat,Ys: list_list_nat,P: list_nat > $o,Q: list_nat > $o] :
( ( Xs2 = Ys )
=> ( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Ys ) )
=> ( ( P @ X3 )
= ( Q @ X3 ) ) )
=> ( ( filter_list_nat @ P @ Xs2 )
= ( filter_list_nat @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_58_filter__cong,axiom,
! [Xs2: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat,P: set_Pr1261947904930325089at_nat > $o,Q: set_Pr1261947904930325089at_nat > $o] :
( ( Xs2 = Ys )
=> ( ! [X3: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X3 @ ( set_se5049602875457034614at_nat @ Ys ) )
=> ( ( P @ X3 )
= ( Q @ X3 ) ) )
=> ( ( filter4791820933415917918at_nat @ P @ Xs2 )
= ( filter4791820933415917918at_nat @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_59_enum__rgfs__returns__rgfs,axiom,
! [X2: list_nat,N: nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ ( equiva7426478223624825838m_rgfs @ N ) ) )
=> ( equiva3371634703666331078on_rgf @ X2 ) ) ).
% enum_rgfs_returns_rgfs
thf(fact_60_equiv__rels__enum,axiom,
! [X2: list_nat] :
( ( equiva3371634703666331078on_rgf @ X2 )
=> ( ( count_list_list_nat @ ( equiva7426478223624825838m_rgfs @ ( size_size_list_nat @ X2 ) ) @ X2 )
= one_one_nat ) ) ).
% equiv_rels_enum
thf(fact_61_in__set__product__lists__length,axiom,
! [Xs2: list_list_nat,Xss: list_list_list_nat] :
( ( member_list_list_nat @ Xs2 @ ( set_list_list_nat2 @ ( produc6783906451316923569st_nat @ Xss ) ) )
=> ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_s6248950052170075156st_nat @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_62_in__set__product__lists__length,axiom,
! [Xs2: list_nat,Xss: list_list_nat] :
( ( member_list_nat2 @ Xs2 @ ( set_list_nat2 @ ( product_lists_nat @ Xss ) ) )
=> ( ( size_size_list_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_63_in__set__product__lists__length,axiom,
! [Xs2: list_set_nat,Xss: list_list_set_nat] :
( ( member_list_set_nat @ Xs2 @ ( set_list_set_nat2 @ ( produc8109398739672286679et_nat @ Xss ) ) )
=> ( ( size_s3254054031482475050et_nat @ Xs2 )
= ( size_s8312130442339763642et_nat @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_64_in__set__product__lists__length,axiom,
! [Xs2: list_s1210847774152347623at_nat,Xss: list_l3822697302700470509at_nat] :
( ( member2758946086501665296at_nat @ Xs2 @ ( set_li2149138424243681404at_nat @ ( produc5476187143726666448at_nat @ Xss ) ) )
=> ( ( size_s8736152011456118867at_nat @ Xs2 )
= ( size_s3876073854528961369at_nat @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_65_length__n__lists__elem,axiom,
! [Ys: list_list_nat,N: nat,Xs2: list_list_nat] :
( ( member_list_list_nat @ Ys @ ( set_list_list_nat2 @ ( n_lists_list_nat @ N @ Xs2 ) ) )
=> ( ( size_s3023201423986296836st_nat @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_66_length__n__lists__elem,axiom,
! [Ys: list_nat,N: nat,Xs2: list_nat] :
( ( member_list_nat2 @ Ys @ ( set_list_nat2 @ ( n_lists_nat @ N @ Xs2 ) ) )
=> ( ( size_size_list_nat @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_67_length__n__lists__elem,axiom,
! [Ys: list_set_nat,N: nat,Xs2: list_set_nat] :
( ( member_list_set_nat @ Ys @ ( set_list_set_nat2 @ ( n_lists_set_nat @ N @ Xs2 ) ) )
=> ( ( size_s3254054031482475050et_nat @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_68_length__n__lists__elem,axiom,
! [Ys: list_s1210847774152347623at_nat,N: nat,Xs2: list_s1210847774152347623at_nat] :
( ( member2758946086501665296at_nat @ Ys @ ( set_li2149138424243681404at_nat @ ( n_list4589530985885001583at_nat @ N @ Xs2 ) ) )
=> ( ( size_s8736152011456118867at_nat @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_69_assms,axiom,
equiv_equiv_nat @ ( set_ord_lessThan_nat @ n ) @ r ).
% assms
thf(fact_70_filter__set,axiom,
! [P: set_nat > $o,Xs2: list_set_nat] :
( ( filter_set_nat2 @ P @ ( set_set_nat2 @ Xs2 ) )
= ( set_set_nat2 @ ( filter_set_nat @ P @ Xs2 ) ) ) ).
% filter_set
thf(fact_71_filter__set,axiom,
! [P: nat > $o,Xs2: list_nat] :
( ( filter_nat2 @ P @ ( set_nat2 @ Xs2 ) )
= ( set_nat2 @ ( filter_nat @ P @ Xs2 ) ) ) ).
% filter_set
thf(fact_72_filter__set,axiom,
! [P: list_nat > $o,Xs2: list_list_nat] :
( ( filter_list_nat2 @ P @ ( set_list_nat2 @ Xs2 ) )
= ( set_list_nat2 @ ( filter_list_nat @ P @ Xs2 ) ) ) ).
% filter_set
thf(fact_73_filter__set,axiom,
! [P: set_Pr1261947904930325089at_nat > $o,Xs2: list_s1210847774152347623at_nat] :
( ( filter6836037073761582098at_nat @ P @ ( set_se5049602875457034614at_nat @ Xs2 ) )
= ( set_se5049602875457034614at_nat @ ( filter4791820933415917918at_nat @ P @ Xs2 ) ) ) ).
% filter_set
thf(fact_74_equiv__rels__def,axiom,
( equiva8721718519204927301v_rels
= ( ^ [N2: nat] : ( map_li6003994582982014139at_nat @ equiva2048684438135499664of_nat @ ( equiva7426478223624825838m_rgfs @ N2 ) ) ) ) ).
% equiv_rels_def
thf(fact_75_in__set__member,axiom,
! [X2: list_nat,Xs2: list_list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs2 ) )
= ( member_list_nat @ Xs2 @ X2 ) ) ).
% in_set_member
thf(fact_76_in__set__member,axiom,
! [X2: set_Pr1261947904930325089at_nat,Xs2: list_s1210847774152347623at_nat] :
( ( member2643936169264416010at_nat @ X2 @ ( set_se5049602875457034614at_nat @ Xs2 ) )
= ( member4653088687621619836at_nat @ Xs2 @ X2 ) ) ).
% in_set_member
thf(fact_77_length__filter__less,axiom,
! [X2: list_nat,Xs2: list_list_nat,P: list_nat > $o] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs2 ) )
=> ( ~ ( P @ X2 )
=> ( ord_less_nat @ ( size_s3023201423986296836st_nat @ ( filter_list_nat @ P @ Xs2 ) ) @ ( size_s3023201423986296836st_nat @ Xs2 ) ) ) ) ).
% length_filter_less
thf(fact_78_length__filter__less,axiom,
! [X2: nat,Xs2: list_nat,P: nat > $o] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ~ ( P @ X2 )
=> ( ord_less_nat @ ( size_size_list_nat @ ( filter_nat @ P @ Xs2 ) ) @ ( size_size_list_nat @ Xs2 ) ) ) ) ).
% length_filter_less
thf(fact_79_length__filter__less,axiom,
! [X2: set_nat,Xs2: list_set_nat,P: set_nat > $o] :
( ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs2 ) )
=> ( ~ ( P @ X2 )
=> ( ord_less_nat @ ( size_s3254054031482475050et_nat @ ( filter_set_nat @ P @ Xs2 ) ) @ ( size_s3254054031482475050et_nat @ Xs2 ) ) ) ) ).
% length_filter_less
thf(fact_80_length__filter__less,axiom,
! [X2: set_Pr1261947904930325089at_nat,Xs2: list_s1210847774152347623at_nat,P: set_Pr1261947904930325089at_nat > $o] :
( ( member2643936169264416010at_nat @ X2 @ ( set_se5049602875457034614at_nat @ Xs2 ) )
=> ( ~ ( P @ X2 )
=> ( ord_less_nat @ ( size_s8736152011456118867at_nat @ ( filter4791820933415917918at_nat @ P @ Xs2 ) ) @ ( size_s8736152011456118867at_nat @ Xs2 ) ) ) ) ).
% length_filter_less
thf(fact_81_count__notin,axiom,
! [X2: list_nat,Xs2: list_list_nat] :
( ~ ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs2 ) )
=> ( ( count_list_list_nat @ Xs2 @ X2 )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_82_count__notin,axiom,
! [X2: set_Pr1261947904930325089at_nat,Xs2: list_s1210847774152347623at_nat] :
( ~ ( member2643936169264416010at_nat @ X2 @ ( set_se5049602875457034614at_nat @ Xs2 ) )
=> ( ( count_6440129622255701469at_nat @ Xs2 @ X2 )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_83_count__notin,axiom,
! [X2: set_nat,Xs2: list_set_nat] :
( ~ ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs2 ) )
=> ( ( count_list_set_nat @ Xs2 @ X2 )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_84_count__notin,axiom,
! [X2: nat,Xs2: list_nat] :
( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ( count_list_nat @ Xs2 @ X2 )
= zero_zero_nat ) ) ).
% count_notin
thf(fact_85_kernel__of__inj__on__rgfs,axiom,
inj_on7522185085906380110at_nat @ equiva2048684438135499664of_nat @ ( collect_list_nat @ equiva3371634703666331078on_rgf ) ).
% kernel_of_inj_on_rgfs
thf(fact_86_filter__False,axiom,
! [Xs2: list_set_nat,P: set_nat > $o] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs2 ) )
=> ~ ( P @ X3 ) )
=> ( ( filter_set_nat @ P @ Xs2 )
= nil_set_nat ) ) ).
% filter_False
thf(fact_87_filter__False,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
=> ~ ( P @ X3 ) )
=> ( ( filter_nat @ P @ Xs2 )
= nil_nat ) ) ).
% filter_False
thf(fact_88_filter__False,axiom,
! [Xs2: list_list_nat,P: list_nat > $o] :
( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs2 ) )
=> ~ ( P @ X3 ) )
=> ( ( filter_list_nat @ P @ Xs2 )
= nil_list_nat ) ) ).
% filter_False
thf(fact_89_filter__False,axiom,
! [Xs2: list_s1210847774152347623at_nat,P: set_Pr1261947904930325089at_nat > $o] :
( ! [X3: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X3 @ ( set_se5049602875457034614at_nat @ Xs2 ) )
=> ~ ( P @ X3 ) )
=> ( ( filter4791820933415917918at_nat @ P @ Xs2 )
= nil_se357566008730718055at_nat ) ) ).
% filter_False
thf(fact_90_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_91_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_92_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_93_list_Omap__disc__iff,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,A: list_list_nat] :
( ( ( map_li6003994582982014139at_nat @ F @ A )
= nil_se357566008730718055at_nat )
= ( A = nil_list_nat ) ) ).
% list.map_disc_iff
thf(fact_94_Nil__is__map__conv,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,Xs2: list_list_nat] :
( ( nil_se357566008730718055at_nat
= ( map_li6003994582982014139at_nat @ F @ Xs2 ) )
= ( Xs2 = nil_list_nat ) ) ).
% Nil_is_map_conv
thf(fact_95_map__is__Nil__conv,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,Xs2: list_list_nat] :
( ( ( map_li6003994582982014139at_nat @ F @ Xs2 )
= nil_se357566008730718055at_nat )
= ( Xs2 = nil_list_nat ) ) ).
% map_is_Nil_conv
thf(fact_96_map__eq__conv,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,Xs2: list_list_nat,G: list_nat > set_Pr1261947904930325089at_nat] :
( ( ( map_li6003994582982014139at_nat @ F @ Xs2 )
= ( map_li6003994582982014139at_nat @ G @ Xs2 ) )
= ( ! [X: list_nat] :
( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs2 ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_97_length__map,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( size_size_list_nat @ ( map_nat_nat @ F @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_map
thf(fact_98_length__map,axiom,
! [F: nat > list_nat,Xs2: list_nat] :
( ( size_s3023201423986296836st_nat @ ( map_nat_list_nat @ F @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_map
thf(fact_99_length__map,axiom,
! [F: list_nat > nat,Xs2: list_list_nat] :
( ( size_size_list_nat @ ( map_list_nat_nat @ F @ Xs2 ) )
= ( size_s3023201423986296836st_nat @ Xs2 ) ) ).
% length_map
thf(fact_100_length__map,axiom,
! [F: set_nat > nat,Xs2: list_set_nat] :
( ( size_size_list_nat @ ( map_set_nat_nat @ F @ Xs2 ) )
= ( size_s3254054031482475050et_nat @ Xs2 ) ) ).
% length_map
thf(fact_101_length__map,axiom,
! [F: nat > set_nat,Xs2: list_nat] :
( ( size_s3254054031482475050et_nat @ ( map_nat_set_nat @ F @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_map
thf(fact_102_length__map,axiom,
! [F: list_nat > list_nat,Xs2: list_list_nat] :
( ( size_s3023201423986296836st_nat @ ( map_li7225945977422193158st_nat @ F @ Xs2 ) )
= ( size_s3023201423986296836st_nat @ Xs2 ) ) ).
% length_map
thf(fact_103_length__map,axiom,
! [F: set_nat > list_nat,Xs2: list_set_nat] :
( ( size_s3023201423986296836st_nat @ ( map_set_nat_list_nat @ F @ Xs2 ) )
= ( size_s3254054031482475050et_nat @ Xs2 ) ) ).
% length_map
thf(fact_104_length__map,axiom,
! [F: list_nat > set_nat,Xs2: list_list_nat] :
( ( size_s3254054031482475050et_nat @ ( map_list_nat_set_nat @ F @ Xs2 ) )
= ( size_s3023201423986296836st_nat @ Xs2 ) ) ).
% length_map
thf(fact_105_length__map,axiom,
! [F: set_nat > set_nat,Xs2: list_set_nat] :
( ( size_s3254054031482475050et_nat @ ( map_set_nat_set_nat @ F @ Xs2 ) )
= ( size_s3254054031482475050et_nat @ Xs2 ) ) ).
% length_map
thf(fact_106_length__map,axiom,
! [F: set_Pr1261947904930325089at_nat > nat,Xs2: list_s1210847774152347623at_nat] :
( ( size_size_list_nat @ ( map_se4264141375869589357at_nat @ F @ Xs2 ) )
= ( size_s8736152011456118867at_nat @ Xs2 ) ) ).
% length_map
thf(fact_107_mem__Collect__eq,axiom,
! [A: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
( ( member2643936169264416010at_nat @ A @ ( collec5514110066124741708at_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_108_mem__Collect__eq,axiom,
! [A: list_nat,P: list_nat > $o] :
( ( member_list_nat2 @ A @ ( collect_list_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_109_Collect__mem__eq,axiom,
! [A2: set_se7855581050983116737at_nat] :
( ( collec5514110066124741708at_nat
@ ^ [X: set_Pr1261947904930325089at_nat] : ( member2643936169264416010at_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_110_Collect__mem__eq,axiom,
! [A2: set_list_nat] :
( ( collect_list_nat
@ ^ [X: list_nat] : ( member_list_nat2 @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_111_Collect__cong,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ! [X3: list_nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_list_nat @ P )
= ( collect_list_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_112_length__0__conv,axiom,
! [Xs2: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= zero_zero_nat )
= ( Xs2 = nil_list_nat ) ) ).
% length_0_conv
thf(fact_113_length__0__conv,axiom,
! [Xs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= zero_zero_nat )
= ( Xs2 = nil_nat ) ) ).
% length_0_conv
thf(fact_114_length__0__conv,axiom,
! [Xs2: list_set_nat] :
( ( ( size_s3254054031482475050et_nat @ Xs2 )
= zero_zero_nat )
= ( Xs2 = nil_set_nat ) ) ).
% length_0_conv
thf(fact_115_length__0__conv,axiom,
! [Xs2: list_s1210847774152347623at_nat] :
( ( ( size_s8736152011456118867at_nat @ Xs2 )
= zero_zero_nat )
= ( Xs2 = nil_se357566008730718055at_nat ) ) ).
% length_0_conv
thf(fact_116_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_117_length__greater__0__conv,axiom,
! [Xs2: list_list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) )
= ( Xs2 != nil_list_nat ) ) ).
% length_greater_0_conv
thf(fact_118_length__greater__0__conv,axiom,
! [Xs2: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) )
= ( Xs2 != nil_nat ) ) ).
% length_greater_0_conv
thf(fact_119_length__greater__0__conv,axiom,
! [Xs2: list_set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ Xs2 ) )
= ( Xs2 != nil_set_nat ) ) ).
% length_greater_0_conv
thf(fact_120_length__greater__0__conv,axiom,
! [Xs2: list_s1210847774152347623at_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s8736152011456118867at_nat @ Xs2 ) )
= ( Xs2 != nil_se357566008730718055at_nat ) ) ).
% length_greater_0_conv
thf(fact_121_list_Osimps_I8_J,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat] :
( ( map_li6003994582982014139at_nat @ F @ nil_list_nat )
= nil_se357566008730718055at_nat ) ).
% list.simps(8)
thf(fact_122_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_123_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_124_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_125_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_126_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_127_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_128_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_129_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_130_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_131_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_132_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_133_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_134_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_135_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_136_linorder__neqE__nat,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_137_all__rels__are__kernels,axiom,
! [N: nat,P2: set_Pr1261947904930325089at_nat] :
( ( equiv_equiv_nat @ ( set_ord_lessThan_nat @ N ) @ P2 )
=> ? [X3: list_set_nat] :
( ( ( equiva50036288466687814et_nat @ X3 )
= P2 )
& ( ( size_s3254054031482475050et_nat @ X3 )
= N ) ) ) ).
% all_rels_are_kernels
thf(fact_138_equiv__on__unique,axiom,
! [A2: set_nat,P2: set_Pr1261947904930325089at_nat,B: set_nat] :
( ( equiv_equiv_nat @ A2 @ P2 )
=> ( ( equiv_equiv_nat @ B @ P2 )
=> ( A2 = B ) ) ) ).
% equiv_on_unique
thf(fact_139_equiv__rels__set,axiom,
! [X2: set_Pr1261947904930325089at_nat,N: nat] :
( ( member2643936169264416010at_nat @ X2 @ ( set_se5049602875457034614at_nat @ ( equiva8721718519204927301v_rels @ N ) ) )
=> ( equiv_equiv_nat @ ( set_ord_lessThan_nat @ N ) @ X2 ) ) ).
% equiv_rels_set
thf(fact_140_list_Osize_I3_J,axiom,
( ( size_s3023201423986296836st_nat @ nil_list_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_141_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_142_list_Osize_I3_J,axiom,
( ( size_s3254054031482475050et_nat @ nil_set_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_143_list_Osize_I3_J,axiom,
( ( size_s8736152011456118867at_nat @ nil_se357566008730718055at_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_144_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_145_count__list_Osimps_I1_J,axiom,
! [Y2: set_Pr1261947904930325089at_nat] :
( ( count_6440129622255701469at_nat @ nil_se357566008730718055at_nat @ Y2 )
= zero_zero_nat ) ).
% count_list.simps(1)
thf(fact_146_count__list_Osimps_I1_J,axiom,
! [Y2: set_nat] :
( ( count_list_set_nat @ nil_set_nat @ Y2 )
= zero_zero_nat ) ).
% count_list.simps(1)
thf(fact_147_count__list_Osimps_I1_J,axiom,
! [Y2: nat] :
( ( count_list_nat @ nil_nat @ Y2 )
= zero_zero_nat ) ).
% count_list.simps(1)
thf(fact_148_kernel__of__under__inj__map,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,X2: list_list_nat] :
( ( inj_on7522185085906380110at_nat @ F @ ( set_list_nat2 @ X2 ) )
=> ( ( equiva6490762433048536736st_nat @ X2 )
= ( equiva1173177585473067681at_nat @ ( map_li6003994582982014139at_nat @ F @ X2 ) ) ) ) ).
% kernel_of_under_inj_map
thf(fact_149_kernel__of__under__inj__map,axiom,
! [F: list_nat > nat,X2: list_list_nat] :
( ( inj_on_list_nat_nat @ F @ ( set_list_nat2 @ X2 ) )
=> ( ( equiva6490762433048536736st_nat @ X2 )
= ( equiva2048684438135499664of_nat @ ( map_list_nat_nat @ F @ X2 ) ) ) ) ).
% kernel_of_under_inj_map
thf(fact_150_kernel__of__under__inj__map,axiom,
! [F: set_Pr1261947904930325089at_nat > nat,X2: list_s1210847774152347623at_nat] :
( ( inj_on4888989441411116416at_nat @ F @ ( set_se5049602875457034614at_nat @ X2 ) )
=> ( ( equiva1173177585473067681at_nat @ X2 )
= ( equiva2048684438135499664of_nat @ ( map_se4264141375869589357at_nat @ F @ X2 ) ) ) ) ).
% kernel_of_under_inj_map
thf(fact_151_kernel__of__under__inj__map,axiom,
! [F: list_nat > set_nat,X2: list_list_nat] :
( ( inj_on1816901372521670873et_nat @ F @ ( set_list_nat2 @ X2 ) )
=> ( ( equiva6490762433048536736st_nat @ X2 )
= ( equiva50036288466687814et_nat @ ( map_list_nat_set_nat @ F @ X2 ) ) ) ) ).
% kernel_of_under_inj_map
thf(fact_152_kernel__of__under__inj__map,axiom,
! [F: set_Pr1261947904930325089at_nat > set_nat,X2: list_s1210847774152347623at_nat] :
( ( inj_on5220459939087112502et_nat @ F @ ( set_se5049602875457034614at_nat @ X2 ) )
=> ( ( equiva1173177585473067681at_nat @ X2 )
= ( equiva50036288466687814et_nat @ ( map_se493947572549472035et_nat @ F @ X2 ) ) ) ) ).
% kernel_of_under_inj_map
thf(fact_153_kernel__of__under__inj__map,axiom,
! [F: nat > nat,X2: list_nat] :
( ( inj_on_nat_nat @ F @ ( set_nat2 @ X2 ) )
=> ( ( equiva2048684438135499664of_nat @ X2 )
= ( equiva2048684438135499664of_nat @ ( map_nat_nat @ F @ X2 ) ) ) ) ).
% kernel_of_under_inj_map
thf(fact_154_kernel__of__under__inj__map,axiom,
! [F: nat > set_nat,X2: list_nat] :
( ( inj_on_nat_set_nat @ F @ ( set_nat2 @ X2 ) )
=> ( ( equiva2048684438135499664of_nat @ X2 )
= ( equiva50036288466687814et_nat @ ( map_nat_set_nat @ F @ X2 ) ) ) ) ).
% kernel_of_under_inj_map
thf(fact_155_kernel__of__under__inj__map,axiom,
! [F: set_nat > nat,X2: list_set_nat] :
( ( inj_on_set_nat_nat @ F @ ( set_set_nat2 @ X2 ) )
=> ( ( equiva50036288466687814et_nat @ X2 )
= ( equiva2048684438135499664of_nat @ ( map_set_nat_nat @ F @ X2 ) ) ) ) ).
% kernel_of_under_inj_map
thf(fact_156_kernel__of__under__inj__map,axiom,
! [F: set_nat > set_nat,X2: list_set_nat] :
( ( inj_on4604407203859583615et_nat @ F @ ( set_set_nat2 @ X2 ) )
=> ( ( equiva50036288466687814et_nat @ X2 )
= ( equiva50036288466687814et_nat @ ( map_set_nat_set_nat @ F @ X2 ) ) ) ) ).
% kernel_of_under_inj_map
thf(fact_157_Equivalence__Relation__Enumeration_Ocount__list__inj__map,axiom,
! [F: nat > nat,X2: list_nat,Y2: nat] :
( ( inj_on_nat_nat @ F @ ( set_nat2 @ X2 ) )
=> ( ( member_nat @ Y2 @ ( set_nat2 @ X2 ) )
=> ( ( count_list_nat @ ( map_nat_nat @ F @ X2 ) @ ( F @ Y2 ) )
= ( count_list_nat @ X2 @ Y2 ) ) ) ) ).
% Equivalence_Relation_Enumeration.count_list_inj_map
thf(fact_158_Equivalence__Relation__Enumeration_Ocount__list__inj__map,axiom,
! [F: nat > list_nat,X2: list_nat,Y2: nat] :
( ( inj_on_nat_list_nat @ F @ ( set_nat2 @ X2 ) )
=> ( ( member_nat @ Y2 @ ( set_nat2 @ X2 ) )
=> ( ( count_list_list_nat @ ( map_nat_list_nat @ F @ X2 ) @ ( F @ Y2 ) )
= ( count_list_nat @ X2 @ Y2 ) ) ) ) ).
% Equivalence_Relation_Enumeration.count_list_inj_map
thf(fact_159_Equivalence__Relation__Enumeration_Ocount__list__inj__map,axiom,
! [F: nat > set_nat,X2: list_nat,Y2: nat] :
( ( inj_on_nat_set_nat @ F @ ( set_nat2 @ X2 ) )
=> ( ( member_nat @ Y2 @ ( set_nat2 @ X2 ) )
=> ( ( count_list_set_nat @ ( map_nat_set_nat @ F @ X2 ) @ ( F @ Y2 ) )
= ( count_list_nat @ X2 @ Y2 ) ) ) ) ).
% Equivalence_Relation_Enumeration.count_list_inj_map
thf(fact_160_Equivalence__Relation__Enumeration_Ocount__list__inj__map,axiom,
! [F: list_nat > nat,X2: list_list_nat,Y2: list_nat] :
( ( inj_on_list_nat_nat @ F @ ( set_list_nat2 @ X2 ) )
=> ( ( member_list_nat2 @ Y2 @ ( set_list_nat2 @ X2 ) )
=> ( ( count_list_nat @ ( map_list_nat_nat @ F @ X2 ) @ ( F @ Y2 ) )
= ( count_list_list_nat @ X2 @ Y2 ) ) ) ) ).
% Equivalence_Relation_Enumeration.count_list_inj_map
thf(fact_161_Equivalence__Relation__Enumeration_Ocount__list__inj__map,axiom,
! [F: set_nat > nat,X2: list_set_nat,Y2: set_nat] :
( ( inj_on_set_nat_nat @ F @ ( set_set_nat2 @ X2 ) )
=> ( ( member_set_nat @ Y2 @ ( set_set_nat2 @ X2 ) )
=> ( ( count_list_nat @ ( map_set_nat_nat @ F @ X2 ) @ ( F @ Y2 ) )
= ( count_list_set_nat @ X2 @ Y2 ) ) ) ) ).
% Equivalence_Relation_Enumeration.count_list_inj_map
thf(fact_162_Equivalence__Relation__Enumeration_Ocount__list__inj__map,axiom,
! [F: list_nat > list_nat,X2: list_list_nat,Y2: list_nat] :
( ( inj_on3049792774292151987st_nat @ F @ ( set_list_nat2 @ X2 ) )
=> ( ( member_list_nat2 @ Y2 @ ( set_list_nat2 @ X2 ) )
=> ( ( count_list_list_nat @ ( map_li7225945977422193158st_nat @ F @ X2 ) @ ( F @ Y2 ) )
= ( count_list_list_nat @ X2 @ Y2 ) ) ) ) ).
% Equivalence_Relation_Enumeration.count_list_inj_map
thf(fact_163_Equivalence__Relation__Enumeration_Ocount__list__inj__map,axiom,
! [F: set_nat > list_nat,X2: list_set_nat,Y2: set_nat] :
( ( inj_on5467128325884351833st_nat @ F @ ( set_set_nat2 @ X2 ) )
=> ( ( member_set_nat @ Y2 @ ( set_set_nat2 @ X2 ) )
=> ( ( count_list_list_nat @ ( map_set_nat_list_nat @ F @ X2 ) @ ( F @ Y2 ) )
= ( count_list_set_nat @ X2 @ Y2 ) ) ) ) ).
% Equivalence_Relation_Enumeration.count_list_inj_map
thf(fact_164_Equivalence__Relation__Enumeration_Ocount__list__inj__map,axiom,
! [F: list_nat > set_nat,X2: list_list_nat,Y2: list_nat] :
( ( inj_on1816901372521670873et_nat @ F @ ( set_list_nat2 @ X2 ) )
=> ( ( member_list_nat2 @ Y2 @ ( set_list_nat2 @ X2 ) )
=> ( ( count_list_set_nat @ ( map_list_nat_set_nat @ F @ X2 ) @ ( F @ Y2 ) )
= ( count_list_list_nat @ X2 @ Y2 ) ) ) ) ).
% Equivalence_Relation_Enumeration.count_list_inj_map
thf(fact_165_Equivalence__Relation__Enumeration_Ocount__list__inj__map,axiom,
! [F: set_nat > set_nat,X2: list_set_nat,Y2: set_nat] :
( ( inj_on4604407203859583615et_nat @ F @ ( set_set_nat2 @ X2 ) )
=> ( ( member_set_nat @ Y2 @ ( set_set_nat2 @ X2 ) )
=> ( ( count_list_set_nat @ ( map_set_nat_set_nat @ F @ X2 ) @ ( F @ Y2 ) )
= ( count_list_set_nat @ X2 @ Y2 ) ) ) ) ).
% Equivalence_Relation_Enumeration.count_list_inj_map
thf(fact_166_Equivalence__Relation__Enumeration_Ocount__list__inj__map,axiom,
! [F: nat > set_Pr1261947904930325089at_nat,X2: list_nat,Y2: nat] :
( ( inj_on7202621048659411806at_nat @ F @ ( set_nat2 @ X2 ) )
=> ( ( member_nat @ Y2 @ ( set_nat2 @ X2 ) )
=> ( ( count_6440129622255701469at_nat @ ( map_na6577772983117884747at_nat @ F @ X2 ) @ ( F @ Y2 ) )
= ( count_list_nat @ X2 @ Y2 ) ) ) ) ).
% Equivalence_Relation_Enumeration.count_list_inj_map
thf(fact_167_List_Ocount__list__inj__map,axiom,
! [F: nat > nat,Xs2: list_nat,X2: nat] :
( ( inj_on_nat_nat @ F @ ( set_nat2 @ Xs2 ) )
=> ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ( count_list_nat @ ( map_nat_nat @ F @ Xs2 ) @ ( F @ X2 ) )
= ( count_list_nat @ Xs2 @ X2 ) ) ) ) ).
% List.count_list_inj_map
thf(fact_168_List_Ocount__list__inj__map,axiom,
! [F: nat > list_nat,Xs2: list_nat,X2: nat] :
( ( inj_on_nat_list_nat @ F @ ( set_nat2 @ Xs2 ) )
=> ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ( count_list_list_nat @ ( map_nat_list_nat @ F @ Xs2 ) @ ( F @ X2 ) )
= ( count_list_nat @ Xs2 @ X2 ) ) ) ) ).
% List.count_list_inj_map
thf(fact_169_List_Ocount__list__inj__map,axiom,
! [F: nat > set_nat,Xs2: list_nat,X2: nat] :
( ( inj_on_nat_set_nat @ F @ ( set_nat2 @ Xs2 ) )
=> ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ( count_list_set_nat @ ( map_nat_set_nat @ F @ Xs2 ) @ ( F @ X2 ) )
= ( count_list_nat @ Xs2 @ X2 ) ) ) ) ).
% List.count_list_inj_map
thf(fact_170_List_Ocount__list__inj__map,axiom,
! [F: list_nat > nat,Xs2: list_list_nat,X2: list_nat] :
( ( inj_on_list_nat_nat @ F @ ( set_list_nat2 @ Xs2 ) )
=> ( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs2 ) )
=> ( ( count_list_nat @ ( map_list_nat_nat @ F @ Xs2 ) @ ( F @ X2 ) )
= ( count_list_list_nat @ Xs2 @ X2 ) ) ) ) ).
% List.count_list_inj_map
thf(fact_171_List_Ocount__list__inj__map,axiom,
! [F: set_nat > nat,Xs2: list_set_nat,X2: set_nat] :
( ( inj_on_set_nat_nat @ F @ ( set_set_nat2 @ Xs2 ) )
=> ( ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs2 ) )
=> ( ( count_list_nat @ ( map_set_nat_nat @ F @ Xs2 ) @ ( F @ X2 ) )
= ( count_list_set_nat @ Xs2 @ X2 ) ) ) ) ).
% List.count_list_inj_map
thf(fact_172_List_Ocount__list__inj__map,axiom,
! [F: list_nat > list_nat,Xs2: list_list_nat,X2: list_nat] :
( ( inj_on3049792774292151987st_nat @ F @ ( set_list_nat2 @ Xs2 ) )
=> ( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs2 ) )
=> ( ( count_list_list_nat @ ( map_li7225945977422193158st_nat @ F @ Xs2 ) @ ( F @ X2 ) )
= ( count_list_list_nat @ Xs2 @ X2 ) ) ) ) ).
% List.count_list_inj_map
thf(fact_173_List_Ocount__list__inj__map,axiom,
! [F: set_nat > list_nat,Xs2: list_set_nat,X2: set_nat] :
( ( inj_on5467128325884351833st_nat @ F @ ( set_set_nat2 @ Xs2 ) )
=> ( ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs2 ) )
=> ( ( count_list_list_nat @ ( map_set_nat_list_nat @ F @ Xs2 ) @ ( F @ X2 ) )
= ( count_list_set_nat @ Xs2 @ X2 ) ) ) ) ).
% List.count_list_inj_map
thf(fact_174_List_Ocount__list__inj__map,axiom,
! [F: list_nat > set_nat,Xs2: list_list_nat,X2: list_nat] :
( ( inj_on1816901372521670873et_nat @ F @ ( set_list_nat2 @ Xs2 ) )
=> ( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs2 ) )
=> ( ( count_list_set_nat @ ( map_list_nat_set_nat @ F @ Xs2 ) @ ( F @ X2 ) )
= ( count_list_list_nat @ Xs2 @ X2 ) ) ) ) ).
% List.count_list_inj_map
thf(fact_175_List_Ocount__list__inj__map,axiom,
! [F: set_nat > set_nat,Xs2: list_set_nat,X2: set_nat] :
( ( inj_on4604407203859583615et_nat @ F @ ( set_set_nat2 @ Xs2 ) )
=> ( ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs2 ) )
=> ( ( count_list_set_nat @ ( map_set_nat_set_nat @ F @ Xs2 ) @ ( F @ X2 ) )
= ( count_list_set_nat @ Xs2 @ X2 ) ) ) ) ).
% List.count_list_inj_map
thf(fact_176_List_Ocount__list__inj__map,axiom,
! [F: nat > set_Pr1261947904930325089at_nat,Xs2: list_nat,X2: nat] :
( ( inj_on7202621048659411806at_nat @ F @ ( set_nat2 @ Xs2 ) )
=> ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ( count_6440129622255701469at_nat @ ( map_na6577772983117884747at_nat @ F @ Xs2 ) @ ( F @ X2 ) )
= ( count_list_nat @ Xs2 @ X2 ) ) ) ) ).
% List.count_list_inj_map
thf(fact_177_kernel__of__equiv,axiom,
! [Xs2: list_list_nat] : ( equiv_equiv_nat @ ( set_ord_lessThan_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) ) @ ( equiva6490762433048536736st_nat @ Xs2 ) ) ).
% kernel_of_equiv
thf(fact_178_kernel__of__equiv,axiom,
! [Xs2: list_s1210847774152347623at_nat] : ( equiv_equiv_nat @ ( set_ord_lessThan_nat @ ( size_s8736152011456118867at_nat @ Xs2 ) ) @ ( equiva1173177585473067681at_nat @ Xs2 ) ) ).
% kernel_of_equiv
thf(fact_179_kernel__of__equiv,axiom,
! [Xs2: list_nat] : ( equiv_equiv_nat @ ( set_ord_lessThan_nat @ ( size_size_list_nat @ Xs2 ) ) @ ( equiva2048684438135499664of_nat @ Xs2 ) ) ).
% kernel_of_equiv
thf(fact_180_kernel__of__equiv,axiom,
! [Xs2: list_set_nat] : ( equiv_equiv_nat @ ( set_ord_lessThan_nat @ ( size_s3254054031482475050et_nat @ Xs2 ) ) @ ( equiva50036288466687814et_nat @ Xs2 ) ) ).
% kernel_of_equiv
thf(fact_181_ex__map__conv,axiom,
! [Ys: list_s1210847774152347623at_nat,F: list_nat > set_Pr1261947904930325089at_nat] :
( ( ? [Xs: list_list_nat] :
( Ys
= ( map_li6003994582982014139at_nat @ F @ Xs ) ) )
= ( ! [X: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X @ ( set_se5049602875457034614at_nat @ Ys ) )
=> ? [Y4: list_nat] :
( X
= ( F @ Y4 ) ) ) ) ) ).
% ex_map_conv
thf(fact_182_map__cong,axiom,
! [Xs2: list_list_nat,Ys: list_list_nat,F: list_nat > set_Pr1261947904930325089at_nat,G: list_nat > set_Pr1261947904930325089at_nat] :
( ( Xs2 = Ys )
=> ( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_li6003994582982014139at_nat @ F @ Xs2 )
= ( map_li6003994582982014139at_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_183_map__idI,axiom,
! [Xs2: list_list_nat,F: list_nat > list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs2 ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_li7225945977422193158st_nat @ F @ Xs2 )
= Xs2 ) ) ).
% map_idI
thf(fact_184_map__idI,axiom,
! [Xs2: list_s1210847774152347623at_nat,F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat] :
( ! [X3: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X3 @ ( set_se5049602875457034614at_nat @ Xs2 ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_se4718776794341882500at_nat @ F @ Xs2 )
= Xs2 ) ) ).
% map_idI
thf(fact_185_map__ext,axiom,
! [Xs2: list_list_nat,F: list_nat > set_Pr1261947904930325089at_nat,G: list_nat > set_Pr1261947904930325089at_nat] :
( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs2 ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_li6003994582982014139at_nat @ F @ Xs2 )
= ( map_li6003994582982014139at_nat @ G @ Xs2 ) ) ) ).
% map_ext
thf(fact_186_list_Omap__ident__strong,axiom,
! [T: list_list_nat,F: list_nat > list_nat] :
( ! [Z2: list_nat] :
( ( member_list_nat2 @ Z2 @ ( set_list_nat2 @ T ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( map_li7225945977422193158st_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_187_list_Omap__ident__strong,axiom,
! [T: list_s1210847774152347623at_nat,F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat] :
( ! [Z2: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ Z2 @ ( set_se5049602875457034614at_nat @ T ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( map_se4718776794341882500at_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_188_list_Oinj__map__strong,axiom,
! [X2: list_list_nat,Xa: list_list_nat,F: list_nat > set_Pr1261947904930325089at_nat,Fa: list_nat > set_Pr1261947904930325089at_nat] :
( ! [Z2: list_nat,Za: list_nat] :
( ( member_list_nat2 @ Z2 @ ( set_list_nat2 @ X2 ) )
=> ( ( member_list_nat2 @ Za @ ( set_list_nat2 @ Xa ) )
=> ( ( ( F @ Z2 )
= ( Fa @ Za ) )
=> ( Z2 = Za ) ) ) )
=> ( ( ( map_li6003994582982014139at_nat @ F @ X2 )
= ( map_li6003994582982014139at_nat @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_189_list_Omap__cong0,axiom,
! [X2: list_list_nat,F: list_nat > set_Pr1261947904930325089at_nat,G: list_nat > set_Pr1261947904930325089at_nat] :
( ! [Z2: list_nat] :
( ( member_list_nat2 @ Z2 @ ( set_list_nat2 @ X2 ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( map_li6003994582982014139at_nat @ F @ X2 )
= ( map_li6003994582982014139at_nat @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_190_list_Omap__cong,axiom,
! [X2: list_list_nat,Ya: list_list_nat,F: list_nat > set_Pr1261947904930325089at_nat,G: list_nat > set_Pr1261947904930325089at_nat] :
( ( X2 = Ya )
=> ( ! [Z2: list_nat] :
( ( member_list_nat2 @ Z2 @ ( set_list_nat2 @ Ya ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( map_li6003994582982014139at_nat @ F @ X2 )
= ( map_li6003994582982014139at_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_191_map__eq__imp__length__eq,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,Xs2: list_list_nat,G: list_nat > set_Pr1261947904930325089at_nat,Ys: list_list_nat] :
( ( ( map_li6003994582982014139at_nat @ F @ Xs2 )
= ( map_li6003994582982014139at_nat @ G @ Ys ) )
=> ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_192_map__eq__imp__length__eq,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,Xs2: list_list_nat,G: nat > set_Pr1261947904930325089at_nat,Ys: list_nat] :
( ( ( map_li6003994582982014139at_nat @ F @ Xs2 )
= ( map_na6577772983117884747at_nat @ G @ Ys ) )
=> ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_193_map__eq__imp__length__eq,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,Xs2: list_list_nat,G: set_nat > set_Pr1261947904930325089at_nat,Ys: list_set_nat] :
( ( ( map_li6003994582982014139at_nat @ F @ Xs2 )
= ( map_se1162299115588061717at_nat @ G @ Ys ) )
=> ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_s3254054031482475050et_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_194_map__eq__imp__length__eq,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,Xs2: list_list_nat,G: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,Ys: list_s1210847774152347623at_nat] :
( ( ( map_li6003994582982014139at_nat @ F @ Xs2 )
= ( map_se4718776794341882500at_nat @ G @ Ys ) )
=> ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_s8736152011456118867at_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_195_map__eq__imp__length__eq,axiom,
! [F: nat > set_Pr1261947904930325089at_nat,Xs2: list_nat,G: list_nat > set_Pr1261947904930325089at_nat,Ys: list_list_nat] :
( ( ( map_na6577772983117884747at_nat @ F @ Xs2 )
= ( map_li6003994582982014139at_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_196_map__eq__imp__length__eq,axiom,
! [F: set_nat > set_Pr1261947904930325089at_nat,Xs2: list_set_nat,G: list_nat > set_Pr1261947904930325089at_nat,Ys: list_list_nat] :
( ( ( map_se1162299115588061717at_nat @ F @ Xs2 )
= ( map_li6003994582982014139at_nat @ G @ Ys ) )
=> ( ( size_s3254054031482475050et_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_197_map__eq__imp__length__eq,axiom,
! [F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,Xs2: list_s1210847774152347623at_nat,G: list_nat > set_Pr1261947904930325089at_nat,Ys: list_list_nat] :
( ( ( map_se4718776794341882500at_nat @ F @ Xs2 )
= ( map_li6003994582982014139at_nat @ G @ Ys ) )
=> ( ( size_s8736152011456118867at_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_198_length__pos__if__in__set,axiom,
! [X2: list_nat,Xs2: list_list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_199_length__pos__if__in__set,axiom,
! [X2: nat,Xs2: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_200_length__pos__if__in__set,axiom,
! [X2: set_nat,Xs2: list_set_nat] :
( ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_201_length__pos__if__in__set,axiom,
! [X2: set_Pr1261947904930325089at_nat,Xs2: list_s1210847774152347623at_nat] :
( ( member2643936169264416010at_nat @ X2 @ ( set_se5049602875457034614at_nat @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s8736152011456118867at_nat @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_202_length__induct,axiom,
! [P: list_list_nat > $o,Xs2: list_list_nat] :
( ! [Xs3: list_list_nat] :
( ! [Ys2: list_list_nat] :
( ( ord_less_nat @ ( size_s3023201423986296836st_nat @ Ys2 ) @ ( size_s3023201423986296836st_nat @ Xs3 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_203_length__induct,axiom,
! [P: list_nat > $o,Xs2: list_nat] :
( ! [Xs3: list_nat] :
( ! [Ys2: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs3 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_204_length__induct,axiom,
! [P: list_set_nat > $o,Xs2: list_set_nat] :
( ! [Xs3: list_set_nat] :
( ! [Ys2: list_set_nat] :
( ( ord_less_nat @ ( size_s3254054031482475050et_nat @ Ys2 ) @ ( size_s3254054031482475050et_nat @ Xs3 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_205_length__induct,axiom,
! [P: list_s1210847774152347623at_nat > $o,Xs2: list_s1210847774152347623at_nat] :
( ! [Xs3: list_s1210847774152347623at_nat] :
( ! [Ys2: list_s1210847774152347623at_nat] :
( ( ord_less_nat @ ( size_s8736152011456118867at_nat @ Ys2 ) @ ( size_s8736152011456118867at_nat @ Xs3 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_206_filter_Osimps_I1_J,axiom,
! [P: list_nat > $o] :
( ( filter_list_nat @ P @ nil_list_nat )
= nil_list_nat ) ).
% filter.simps(1)
thf(fact_207_filter_Osimps_I1_J,axiom,
! [P: set_nat > $o] :
( ( filter_set_nat @ P @ nil_set_nat )
= nil_set_nat ) ).
% filter.simps(1)
thf(fact_208_filter_Osimps_I1_J,axiom,
! [P: nat > $o] :
( ( filter_nat @ P @ nil_nat )
= nil_nat ) ).
% filter.simps(1)
thf(fact_209_filter_Osimps_I1_J,axiom,
! [P: set_Pr1261947904930325089at_nat > $o] :
( ( filter4791820933415917918at_nat @ P @ nil_se357566008730718055at_nat )
= nil_se357566008730718055at_nat ) ).
% filter.simps(1)
thf(fact_210_rgf__exists,axiom,
! [N: nat,R: set_Pr1261947904930325089at_nat] :
( ( equiv_equiv_nat @ ( set_ord_lessThan_nat @ N ) @ R )
=> ? [X3: list_nat] :
( ( equiva3371634703666331078on_rgf @ X3 )
& ( ( size_size_list_nat @ X3 )
= N )
& ( ( equiva2048684438135499664of_nat @ X3 )
= R ) ) ) ).
% rgf_exists
thf(fact_211_filter__empty__conv,axiom,
! [P: set_nat > $o,Xs2: list_set_nat] :
( ( ( filter_set_nat @ P @ Xs2 )
= nil_set_nat )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ ( set_set_nat2 @ Xs2 ) )
=> ~ ( P @ X ) ) ) ) ).
% filter_empty_conv
thf(fact_212_filter__empty__conv,axiom,
! [P: nat > $o,Xs2: list_nat] :
( ( ( filter_nat @ P @ Xs2 )
= nil_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ~ ( P @ X ) ) ) ) ).
% filter_empty_conv
thf(fact_213_filter__empty__conv,axiom,
! [P: list_nat > $o,Xs2: list_list_nat] :
( ( ( filter_list_nat @ P @ Xs2 )
= nil_list_nat )
= ( ! [X: list_nat] :
( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs2 ) )
=> ~ ( P @ X ) ) ) ) ).
% filter_empty_conv
thf(fact_214_filter__empty__conv,axiom,
! [P: set_Pr1261947904930325089at_nat > $o,Xs2: list_s1210847774152347623at_nat] :
( ( ( filter4791820933415917918at_nat @ P @ Xs2 )
= nil_se357566008730718055at_nat )
= ( ! [X: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X @ ( set_se5049602875457034614at_nat @ Xs2 ) )
=> ~ ( P @ X ) ) ) ) ).
% filter_empty_conv
thf(fact_215_empty__filter__conv,axiom,
! [P: set_nat > $o,Xs2: list_set_nat] :
( ( nil_set_nat
= ( filter_set_nat @ P @ Xs2 ) )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ ( set_set_nat2 @ Xs2 ) )
=> ~ ( P @ X ) ) ) ) ).
% empty_filter_conv
thf(fact_216_empty__filter__conv,axiom,
! [P: nat > $o,Xs2: list_nat] :
( ( nil_nat
= ( filter_nat @ P @ Xs2 ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ~ ( P @ X ) ) ) ) ).
% empty_filter_conv
thf(fact_217_empty__filter__conv,axiom,
! [P: list_nat > $o,Xs2: list_list_nat] :
( ( nil_list_nat
= ( filter_list_nat @ P @ Xs2 ) )
= ( ! [X: list_nat] :
( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs2 ) )
=> ~ ( P @ X ) ) ) ) ).
% empty_filter_conv
thf(fact_218_empty__filter__conv,axiom,
! [P: set_Pr1261947904930325089at_nat > $o,Xs2: list_s1210847774152347623at_nat] :
( ( nil_se357566008730718055at_nat
= ( filter4791820933415917918at_nat @ P @ Xs2 ) )
= ( ! [X: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X @ ( set_se5049602875457034614at_nat @ Xs2 ) )
=> ~ ( P @ X ) ) ) ) ).
% empty_filter_conv
thf(fact_219_count__list__0__iff,axiom,
! [Xs2: list_list_nat,X2: list_nat] :
( ( ( count_list_list_nat @ Xs2 @ X2 )
= zero_zero_nat )
= ( ~ ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs2 ) ) ) ) ).
% count_list_0_iff
thf(fact_220_count__list__0__iff,axiom,
! [Xs2: list_s1210847774152347623at_nat,X2: set_Pr1261947904930325089at_nat] :
( ( ( count_6440129622255701469at_nat @ Xs2 @ X2 )
= zero_zero_nat )
= ( ~ ( member2643936169264416010at_nat @ X2 @ ( set_se5049602875457034614at_nat @ Xs2 ) ) ) ) ).
% count_list_0_iff
thf(fact_221_count__list__0__iff,axiom,
! [Xs2: list_set_nat,X2: set_nat] :
( ( ( count_list_set_nat @ Xs2 @ X2 )
= zero_zero_nat )
= ( ~ ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs2 ) ) ) ) ).
% count_list_0_iff
thf(fact_222_count__list__0__iff,axiom,
! [Xs2: list_nat,X2: nat] :
( ( ( count_list_nat @ Xs2 @ X2 )
= zero_zero_nat )
= ( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) ) ) ) ).
% count_list_0_iff
thf(fact_223_lessThan__iff,axiom,
! [I: set_Pr1261947904930325089at_nat,K: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ I @ ( set_or5243998310507771012at_nat @ K ) )
= ( ord_le7866589430770878221at_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_224_lessThan__iff,axiom,
! [I: int,K: int] :
( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
= ( ord_less_int @ I @ K ) ) ).
% lessThan_iff
thf(fact_225_lessThan__iff,axiom,
! [I: num,K: num] :
( ( member_num @ I @ ( set_ord_lessThan_num @ K ) )
= ( ord_less_num @ I @ K ) ) ).
% lessThan_iff
thf(fact_226_lessThan__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
= ( ord_less_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_227_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_228_member__filter,axiom,
! [X2: list_nat,P: list_nat > $o,A2: set_list_nat] :
( ( member_list_nat2 @ X2 @ ( filter_list_nat2 @ P @ A2 ) )
= ( ( member_list_nat2 @ X2 @ A2 )
& ( P @ X2 ) ) ) ).
% member_filter
thf(fact_229_member__filter,axiom,
! [X2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o,A2: set_se7855581050983116737at_nat] :
( ( member2643936169264416010at_nat @ X2 @ ( filter6836037073761582098at_nat @ P @ A2 ) )
= ( ( member2643936169264416010at_nat @ X2 @ A2 )
& ( P @ X2 ) ) ) ).
% member_filter
thf(fact_230_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_231_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_232_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_233_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_234_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_235_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_236_lessThan__eq__iff,axiom,
! [X2: nat,Y2: nat] :
( ( ( set_ord_lessThan_nat @ X2 )
= ( set_ord_lessThan_nat @ Y2 ) )
= ( X2 = Y2 ) ) ).
% lessThan_eq_iff
thf(fact_237_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_238_lessThan__def,axiom,
( set_ord_lessThan_int
= ( ^ [U: int] :
( collect_int
@ ^ [X: int] : ( ord_less_int @ X @ U ) ) ) ) ).
% lessThan_def
thf(fact_239_lessThan__def,axiom,
( set_ord_lessThan_num
= ( ^ [U: num] :
( collect_num
@ ^ [X: num] : ( ord_less_num @ X @ U ) ) ) ) ).
% lessThan_def
thf(fact_240_lessThan__def,axiom,
( set_ord_lessThan_nat
= ( ^ [U: nat] :
( collect_nat
@ ^ [X: nat] : ( ord_less_nat @ X @ U ) ) ) ) ).
% lessThan_def
thf(fact_241_map__list__to__rgf,axiom,
! [X2: list_nat] :
? [F2: nat > nat] :
( ( inj_on_nat_nat @ F2 @ ( set_nat2 @ X2 ) )
& ( equiva3371634703666331078on_rgf @ ( map_nat_nat @ F2 @ X2 ) ) ) ).
% map_list_to_rgf
thf(fact_242_map__list__to__rgf,axiom,
! [X2: list_list_nat] :
? [F2: list_nat > nat] :
( ( inj_on_list_nat_nat @ F2 @ ( set_list_nat2 @ X2 ) )
& ( equiva3371634703666331078on_rgf @ ( map_list_nat_nat @ F2 @ X2 ) ) ) ).
% map_list_to_rgf
thf(fact_243_map__list__to__rgf,axiom,
! [X2: list_s1210847774152347623at_nat] :
? [F2: set_Pr1261947904930325089at_nat > nat] :
( ( inj_on4888989441411116416at_nat @ F2 @ ( set_se5049602875457034614at_nat @ X2 ) )
& ( equiva3371634703666331078on_rgf @ ( map_se4264141375869589357at_nat @ F2 @ X2 ) ) ) ).
% map_list_to_rgf
thf(fact_244_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_245_zero__reorient,axiom,
! [X2: int] :
( ( zero_zero_int = X2 )
= ( X2 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_246_linorder__neqE__linordered__idom,axiom,
! [X2: int,Y2: int] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ Y2 @ X2 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_247_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_248_one__reorient,axiom,
! [X2: int] :
( ( one_one_int = X2 )
= ( X2 = one_one_int ) ) ).
% one_reorient
thf(fact_249_Set_Ofilter__def,axiom,
( filter6836037073761582098at_nat
= ( ^ [P3: set_Pr1261947904930325089at_nat > $o,A3: set_se7855581050983116737at_nat] :
( collec5514110066124741708at_nat
@ ^ [A4: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ A4 @ A3 )
& ( P3 @ A4 ) ) ) ) ) ).
% Set.filter_def
thf(fact_250_Set_Ofilter__def,axiom,
( filter_list_nat2
= ( ^ [P3: list_nat > $o,A3: set_list_nat] :
( collect_list_nat
@ ^ [A4: list_nat] :
( ( member_list_nat2 @ A4 @ A3 )
& ( P3 @ A4 ) ) ) ) ) ).
% Set.filter_def
thf(fact_251_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_252_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_253_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_254_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_255_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_256_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_257_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_258_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_259_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_260_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_261_lessThan__strict__subset__iff,axiom,
! [M: int,N: int] :
( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N ) )
= ( ord_less_int @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_262_lessThan__strict__subset__iff,axiom,
! [M: num,N: num] :
( ( ord_less_set_num @ ( set_ord_lessThan_num @ M ) @ ( set_ord_lessThan_num @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_263_lessThan__strict__subset__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_264_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_265_linorder__inj__onI_H,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [I2: nat,J: nat] :
( ( member_nat @ I2 @ A2 )
=> ( ( member_nat @ J @ A2 )
=> ( ( ord_less_nat @ I2 @ J )
=> ( ( F @ I2 )
!= ( F @ J ) ) ) ) )
=> ( inj_on_nat_nat @ F @ A2 ) ) ).
% linorder_inj_onI'
thf(fact_266_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_267_set__n__lists,axiom,
! [N: nat,Xs2: list_list_nat] :
( ( set_list_list_nat2 @ ( n_lists_list_nat @ N @ Xs2 ) )
= ( collec5989764272469232197st_nat
@ ^ [Ys3: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Ys3 )
= N )
& ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Ys3 ) @ ( set_list_nat2 @ Xs2 ) ) ) ) ) ).
% set_n_lists
thf(fact_268_set__n__lists,axiom,
! [N: nat,Xs2: list_nat] :
( ( set_list_nat2 @ ( n_lists_nat @ N @ Xs2 ) )
= ( collect_list_nat
@ ^ [Ys3: list_nat] :
( ( ( size_size_list_nat @ Ys3 )
= N )
& ( ord_less_eq_set_nat @ ( set_nat2 @ Ys3 ) @ ( set_nat2 @ Xs2 ) ) ) ) ) ).
% set_n_lists
thf(fact_269_set__n__lists,axiom,
! [N: nat,Xs2: list_set_nat] :
( ( set_list_set_nat2 @ ( n_lists_set_nat @ N @ Xs2 ) )
= ( collect_list_set_nat
@ ^ [Ys3: list_set_nat] :
( ( ( size_s3254054031482475050et_nat @ Ys3 )
= N )
& ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Ys3 ) @ ( set_set_nat2 @ Xs2 ) ) ) ) ) ).
% set_n_lists
thf(fact_270_set__n__lists,axiom,
! [N: nat,Xs2: list_s1210847774152347623at_nat] :
( ( set_li2149138424243681404at_nat @ ( n_list4589530985885001583at_nat @ N @ Xs2 ) )
= ( collec861049716225513554at_nat
@ ^ [Ys3: list_s1210847774152347623at_nat] :
( ( ( size_s8736152011456118867at_nat @ Ys3 )
= N )
& ( ord_le2077887516847798113at_nat @ ( set_se5049602875457034614at_nat @ Ys3 ) @ ( set_se5049602875457034614at_nat @ Xs2 ) ) ) ) ) ).
% set_n_lists
thf(fact_271_length__code,axiom,
( size_s3023201423986296836st_nat
= ( gen_length_list_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_272_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_273_length__code,axiom,
( size_s3254054031482475050et_nat
= ( gen_length_set_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_274_length__code,axiom,
( size_s8736152011456118867at_nat
= ( gen_le5092146751752969972at_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_275_inj__on__id2,axiom,
! [A2: set_nat] :
( inj_on_nat_nat
@ ^ [X: nat] : X
@ A2 ) ).
% inj_on_id2
thf(fact_276_map__eq__map__tailrec,axiom,
map_li6003994582982014139at_nat = map_ta8671482330076047857at_nat ).
% map_eq_map_tailrec
thf(fact_277_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_278_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_279_subsetI,axiom,
! [A2: set_list_nat,B: set_list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ A2 )
=> ( member_list_nat2 @ X3 @ B ) )
=> ( ord_le6045566169113846134st_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_280_subsetI,axiom,
! [A2: set_se7855581050983116737at_nat,B: set_se7855581050983116737at_nat] :
( ! [X3: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X3 @ A2 )
=> ( member2643936169264416010at_nat @ X3 @ B ) )
=> ( ord_le2077887516847798113at_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_281_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_282_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_283_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_284_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_285_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_286_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_287_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_288_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_289_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_290_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_291_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_292_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_293_lessThan__subset__iff,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X2 ) @ ( set_ord_lessThan_int @ Y2 ) )
= ( ord_less_eq_int @ X2 @ Y2 ) ) ).
% lessThan_subset_iff
thf(fact_294_lessThan__subset__iff,axiom,
! [X2: num,Y2: num] :
( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X2 ) @ ( set_ord_lessThan_num @ Y2 ) )
= ( ord_less_eq_num @ X2 @ Y2 ) ) ).
% lessThan_subset_iff
thf(fact_295_lessThan__subset__iff,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X2 ) @ ( set_ord_lessThan_nat @ Y2 ) )
= ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% lessThan_subset_iff
thf(fact_296_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_297_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_298_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_299_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_300_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_301_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_302_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_303_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_304_less__set__def,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [A3: set_list_nat,B2: set_list_nat] :
( ord_less_list_nat_o
@ ^ [X: list_nat] : ( member_list_nat2 @ X @ A3 )
@ ^ [X: list_nat] : ( member_list_nat2 @ X @ B2 ) ) ) ) ).
% less_set_def
thf(fact_305_less__set__def,axiom,
( ord_le3519150613208228973at_nat
= ( ^ [A3: set_se7855581050983116737at_nat,B2: set_se7855581050983116737at_nat] :
( ord_le220952123948551344_nat_o
@ ^ [X: set_Pr1261947904930325089at_nat] : ( member2643936169264416010at_nat @ X @ A3 )
@ ^ [X: set_Pr1261947904930325089at_nat] : ( member2643936169264416010at_nat @ X @ B2 ) ) ) ) ).
% less_set_def
thf(fact_306_psubsetD,axiom,
! [A2: set_list_nat,B: set_list_nat,C2: list_nat] :
( ( ord_le1190675801316882794st_nat @ A2 @ B )
=> ( ( member_list_nat2 @ C2 @ A2 )
=> ( member_list_nat2 @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_307_psubsetD,axiom,
! [A2: set_se7855581050983116737at_nat,B: set_se7855581050983116737at_nat,C2: set_Pr1261947904930325089at_nat] :
( ( ord_le3519150613208228973at_nat @ A2 @ B )
=> ( ( member2643936169264416010at_nat @ C2 @ A2 )
=> ( member2643936169264416010at_nat @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_308_inj__on__of__nat,axiom,
! [N4: set_nat] : ( inj_on_nat_nat @ semiri1316708129612266289at_nat @ N4 ) ).
% inj_on_of_nat
thf(fact_309_inj__on__of__nat,axiom,
! [N4: set_nat] : ( inj_on_nat_int @ semiri1314217659103216013at_int @ N4 ) ).
% inj_on_of_nat
thf(fact_310_in__mono,axiom,
! [A2: set_list_nat,B: set_list_nat,X2: list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B )
=> ( ( member_list_nat2 @ X2 @ A2 )
=> ( member_list_nat2 @ X2 @ B ) ) ) ).
% in_mono
thf(fact_311_in__mono,axiom,
! [A2: set_se7855581050983116737at_nat,B: set_se7855581050983116737at_nat,X2: set_Pr1261947904930325089at_nat] :
( ( ord_le2077887516847798113at_nat @ A2 @ B )
=> ( ( member2643936169264416010at_nat @ X2 @ A2 )
=> ( member2643936169264416010at_nat @ X2 @ B ) ) ) ).
% in_mono
thf(fact_312_subsetD,axiom,
! [A2: set_list_nat,B: set_list_nat,C2: list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B )
=> ( ( member_list_nat2 @ C2 @ A2 )
=> ( member_list_nat2 @ C2 @ B ) ) ) ).
% subsetD
thf(fact_313_subsetD,axiom,
! [A2: set_se7855581050983116737at_nat,B: set_se7855581050983116737at_nat,C2: set_Pr1261947904930325089at_nat] :
( ( ord_le2077887516847798113at_nat @ A2 @ B )
=> ( ( member2643936169264416010at_nat @ C2 @ A2 )
=> ( member2643936169264416010at_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_314_subset__eq,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A3: set_list_nat,B2: set_list_nat] :
! [X: list_nat] :
( ( member_list_nat2 @ X @ A3 )
=> ( member_list_nat2 @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_315_subset__eq,axiom,
( ord_le2077887516847798113at_nat
= ( ^ [A3: set_se7855581050983116737at_nat,B2: set_se7855581050983116737at_nat] :
! [X: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X @ A3 )
=> ( member2643936169264416010at_nat @ X @ B2 ) ) ) ) ).
% subset_eq
thf(fact_316_subset__iff,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A3: set_list_nat,B2: set_list_nat] :
! [T2: list_nat] :
( ( member_list_nat2 @ T2 @ A3 )
=> ( member_list_nat2 @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_317_subset__iff,axiom,
( ord_le2077887516847798113at_nat
= ( ^ [A3: set_se7855581050983116737at_nat,B2: set_se7855581050983116737at_nat] :
! [T2: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ T2 @ A3 )
=> ( member2643936169264416010at_nat @ T2 @ B2 ) ) ) ) ).
% subset_iff
thf(fact_318_Collect__mono,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ! [X3: list_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_319_Collect__mono__iff,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) )
= ( ! [X: list_nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_320_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_321_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_322_of__nat__mono,axiom,
! [I: nat,J2: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J2 ) ) ) ).
% of_nat_mono
thf(fact_323_of__nat__mono,axiom,
! [I: nat,J2: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J2 ) ) ) ).
% of_nat_mono
thf(fact_324_Collect__subset,axiom,
! [A2: set_se7855581050983116737at_nat,P: set_Pr1261947904930325089at_nat > $o] :
( ord_le2077887516847798113at_nat
@ ( collec5514110066124741708at_nat
@ ^ [X: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_325_Collect__subset,axiom,
! [A2: set_list_nat,P: list_nat > $o] :
( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X: list_nat] :
( ( member_list_nat2 @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_326_subset__inj__on,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,B: set_list_nat,A2: set_list_nat] :
( ( inj_on7522185085906380110at_nat @ F @ B )
=> ( ( ord_le6045566169113846134st_nat @ A2 @ B )
=> ( inj_on7522185085906380110at_nat @ F @ A2 ) ) ) ).
% subset_inj_on
thf(fact_327_subset__inj__on,axiom,
! [F: nat > nat,B: set_nat,A2: set_nat] :
( ( inj_on_nat_nat @ F @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( inj_on_nat_nat @ F @ A2 ) ) ) ).
% subset_inj_on
thf(fact_328_inj__on__subset,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,A2: set_list_nat,B: set_list_nat] :
( ( inj_on7522185085906380110at_nat @ F @ A2 )
=> ( ( ord_le6045566169113846134st_nat @ B @ A2 )
=> ( inj_on7522185085906380110at_nat @ F @ B ) ) ) ).
% inj_on_subset
thf(fact_329_inj__on__subset,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( inj_on_nat_nat @ F @ B ) ) ) ).
% inj_on_subset
thf(fact_330_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_331_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_332_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_333_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_334_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_335_subset__code_I1_J,axiom,
! [Xs2: list_list_nat,B: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs2 ) @ B )
= ( ! [X: list_nat] :
( ( member_list_nat2 @ X @ ( set_list_nat2 @ Xs2 ) )
=> ( member_list_nat2 @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_336_subset__code_I1_J,axiom,
! [Xs2: list_s1210847774152347623at_nat,B: set_se7855581050983116737at_nat] :
( ( ord_le2077887516847798113at_nat @ ( set_se5049602875457034614at_nat @ Xs2 ) @ B )
= ( ! [X: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X @ ( set_se5049602875457034614at_nat @ Xs2 ) )
=> ( member2643936169264416010at_nat @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_337_linorder__inj__onI,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ( member_nat @ X3 @ A2 )
=> ( ( member_nat @ Y3 @ A2 )
=> ( ( F @ X3 )
!= ( F @ Y3 ) ) ) ) )
=> ( ! [X3: nat,Y3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_nat @ Y3 @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X3 ) ) ) )
=> ( inj_on_nat_nat @ F @ A2 ) ) ) ).
% linorder_inj_onI
thf(fact_338_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_339_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_340_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_341_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_342_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_343_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_344_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_345_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_346_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_347_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_348_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_349_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_350_filter__is__subset,axiom,
! [P: set_nat > $o,Xs2: list_set_nat] : ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ ( filter_set_nat @ P @ Xs2 ) ) @ ( set_set_nat2 @ Xs2 ) ) ).
% filter_is_subset
thf(fact_351_filter__is__subset,axiom,
! [P: nat > $o,Xs2: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( filter_nat @ P @ Xs2 ) ) @ ( set_nat2 @ Xs2 ) ) ).
% filter_is_subset
thf(fact_352_filter__is__subset,axiom,
! [P: list_nat > $o,Xs2: list_list_nat] : ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ ( filter_list_nat @ P @ Xs2 ) ) @ ( set_list_nat2 @ Xs2 ) ) ).
% filter_is_subset
thf(fact_353_filter__is__subset,axiom,
! [P: set_Pr1261947904930325089at_nat > $o,Xs2: list_s1210847774152347623at_nat] : ( ord_le2077887516847798113at_nat @ ( set_se5049602875457034614at_nat @ ( filter4791820933415917918at_nat @ P @ Xs2 ) ) @ ( set_se5049602875457034614at_nat @ Xs2 ) ) ).
% filter_is_subset
thf(fact_354_inj__on__inverseI,axiom,
! [A2: set_nat,G: nat > nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( G @ ( F @ X3 ) )
= X3 ) )
=> ( inj_on_nat_nat @ F @ A2 ) ) ).
% inj_on_inverseI
thf(fact_355_inj__on__inverseI,axiom,
! [A2: set_list_nat,G: set_Pr1261947904930325089at_nat > list_nat,F: list_nat > set_Pr1261947904930325089at_nat] :
( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ A2 )
=> ( ( G @ ( F @ X3 ) )
= X3 ) )
=> ( inj_on7522185085906380110at_nat @ F @ A2 ) ) ).
% inj_on_inverseI
thf(fact_356_inj__on__contraD,axiom,
! [F: nat > nat,A2: set_nat,X2: nat,Y2: nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( ( X2 != Y2 )
=> ( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y2 @ A2 )
=> ( ( F @ X2 )
!= ( F @ Y2 ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_357_inj__on__contraD,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,A2: set_list_nat,X2: list_nat,Y2: list_nat] :
( ( inj_on7522185085906380110at_nat @ F @ A2 )
=> ( ( X2 != Y2 )
=> ( ( member_list_nat2 @ X2 @ A2 )
=> ( ( member_list_nat2 @ Y2 @ A2 )
=> ( ( F @ X2 )
!= ( F @ Y2 ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_358_inj__on__eq__iff,axiom,
! [F: nat > nat,A2: set_nat,X2: nat,Y2: nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y2 @ A2 )
=> ( ( ( F @ X2 )
= ( F @ Y2 ) )
= ( X2 = Y2 ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_359_inj__on__eq__iff,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,A2: set_list_nat,X2: list_nat,Y2: list_nat] :
( ( inj_on7522185085906380110at_nat @ F @ A2 )
=> ( ( member_list_nat2 @ X2 @ A2 )
=> ( ( member_list_nat2 @ Y2 @ A2 )
=> ( ( ( F @ X2 )
= ( F @ Y2 ) )
= ( X2 = Y2 ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_360_inj__on__cong,axiom,
! [A2: set_nat,F: nat > nat,G: nat > nat] :
( ! [A5: nat] :
( ( member_nat @ A5 @ A2 )
=> ( ( F @ A5 )
= ( G @ A5 ) ) )
=> ( ( inj_on_nat_nat @ F @ A2 )
= ( inj_on_nat_nat @ G @ A2 ) ) ) ).
% inj_on_cong
thf(fact_361_inj__on__cong,axiom,
! [A2: set_list_nat,F: list_nat > set_Pr1261947904930325089at_nat,G: list_nat > set_Pr1261947904930325089at_nat] :
( ! [A5: list_nat] :
( ( member_list_nat2 @ A5 @ A2 )
=> ( ( F @ A5 )
= ( G @ A5 ) ) )
=> ( ( inj_on7522185085906380110at_nat @ F @ A2 )
= ( inj_on7522185085906380110at_nat @ G @ A2 ) ) ) ).
% inj_on_cong
thf(fact_362_inj__on__def,axiom,
( inj_on7522185085906380110at_nat
= ( ^ [F3: list_nat > set_Pr1261947904930325089at_nat,A3: set_list_nat] :
! [X: list_nat] :
( ( member_list_nat2 @ X @ A3 )
=> ! [Y4: list_nat] :
( ( member_list_nat2 @ Y4 @ A3 )
=> ( ( ( F3 @ X )
= ( F3 @ Y4 ) )
=> ( X = Y4 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_363_inj__on__def,axiom,
( inj_on_nat_nat
= ( ^ [F3: nat > nat,A3: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A3 )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ A3 )
=> ( ( ( F3 @ X )
= ( F3 @ Y4 ) )
=> ( X = Y4 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_364_inj__onI,axiom,
! [A2: set_nat,F: nat > nat] :
( ! [X3: nat,Y3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_nat @ Y3 @ A2 )
=> ( ( ( F @ X3 )
= ( F @ Y3 ) )
=> ( X3 = Y3 ) ) ) )
=> ( inj_on_nat_nat @ F @ A2 ) ) ).
% inj_onI
thf(fact_365_inj__onI,axiom,
! [A2: set_list_nat,F: list_nat > set_Pr1261947904930325089at_nat] :
( ! [X3: list_nat,Y3: list_nat] :
( ( member_list_nat2 @ X3 @ A2 )
=> ( ( member_list_nat2 @ Y3 @ A2 )
=> ( ( ( F @ X3 )
= ( F @ Y3 ) )
=> ( X3 = Y3 ) ) ) )
=> ( inj_on7522185085906380110at_nat @ F @ A2 ) ) ).
% inj_onI
thf(fact_366_inj__onD,axiom,
! [F: nat > nat,A2: set_nat,X2: nat,Y2: nat] :
( ( inj_on_nat_nat @ F @ A2 )
=> ( ( ( F @ X2 )
= ( F @ Y2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y2 @ A2 )
=> ( X2 = Y2 ) ) ) ) ) ).
% inj_onD
thf(fact_367_inj__onD,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,A2: set_list_nat,X2: list_nat,Y2: list_nat] :
( ( inj_on7522185085906380110at_nat @ F @ A2 )
=> ( ( ( F @ X2 )
= ( F @ Y2 ) )
=> ( ( member_list_nat2 @ X2 @ A2 )
=> ( ( member_list_nat2 @ Y2 @ A2 )
=> ( X2 = Y2 ) ) ) ) ) ).
% inj_onD
thf(fact_368_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_369_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_370_dual__order_Orefl,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% dual_order.refl
thf(fact_371_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_372_order__refl,axiom,
! [X2: int] : ( ord_less_eq_int @ X2 @ X2 ) ).
% order_refl
thf(fact_373_order__refl,axiom,
! [X2: num] : ( ord_less_eq_num @ X2 @ X2 ) ).
% order_refl
thf(fact_374_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_375_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_376_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_377_order__le__imp__less__or__eq,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_int @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_378_order__le__imp__less__or__eq,axiom,
! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ( ord_less_num @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_379_linorder__le__less__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_380_linorder__le__less__linear,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
| ( ord_less_int @ Y2 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_381_linorder__le__less__linear,axiom,
! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
| ( ord_less_num @ Y2 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_382_order__less__le__subst2,axiom,
! [A: nat,B3: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_383_order__less__le__subst2,axiom,
! [A: int,B3: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_384_order__less__le__subst2,axiom,
! [A: num,B3: num,F: num > nat,C2: nat] :
( ( ord_less_num @ A @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_385_order__less__le__subst2,axiom,
! [A: nat,B3: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_386_order__less__le__subst2,axiom,
! [A: int,B3: int,F: int > int,C2: int] :
( ( ord_less_int @ A @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_387_order__less__le__subst2,axiom,
! [A: num,B3: num,F: num > int,C2: int] :
( ( ord_less_num @ A @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_388_order__less__le__subst2,axiom,
! [A: nat,B3: nat,F: nat > num,C2: num] :
( ( ord_less_nat @ A @ B3 )
=> ( ( ord_less_eq_num @ ( F @ B3 ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_389_order__less__le__subst2,axiom,
! [A: int,B3: int,F: int > num,C2: num] :
( ( ord_less_int @ A @ B3 )
=> ( ( ord_less_eq_num @ ( F @ B3 ) @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_390_order__less__le__subst2,axiom,
! [A: num,B3: num,F: num > num,C2: num] :
( ( ord_less_num @ A @ B3 )
=> ( ( ord_less_eq_num @ ( F @ B3 ) @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_391_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_392_order__less__le__subst1,axiom,
! [A: int,F: nat > int,B3: nat,C2: nat] :
( ( ord_less_int @ A @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_393_order__less__le__subst1,axiom,
! [A: num,F: nat > num,B3: nat,C2: nat] :
( ( ord_less_num @ A @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_394_order__less__le__subst1,axiom,
! [A: nat,F: int > nat,B3: int,C2: int] :
( ( ord_less_nat @ A @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_395_order__less__le__subst1,axiom,
! [A: int,F: int > int,B3: int,C2: int] :
( ( ord_less_int @ A @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_396_order__less__le__subst1,axiom,
! [A: num,F: int > num,B3: int,C2: int] :
( ( ord_less_num @ A @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_397_order__less__le__subst1,axiom,
! [A: nat,F: num > nat,B3: num,C2: num] :
( ( ord_less_nat @ A @ ( F @ B3 ) )
=> ( ( ord_less_eq_num @ B3 @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_398_order__less__le__subst1,axiom,
! [A: int,F: num > int,B3: num,C2: num] :
( ( ord_less_int @ A @ ( F @ B3 ) )
=> ( ( ord_less_eq_num @ B3 @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_399_order__less__le__subst1,axiom,
! [A: num,F: num > num,B3: num,C2: num] :
( ( ord_less_num @ A @ ( F @ B3 ) )
=> ( ( ord_less_eq_num @ B3 @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_400_order__le__less__subst2,axiom,
! [A: nat,B3: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_401_order__le__less__subst2,axiom,
! [A: nat,B3: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_402_order__le__less__subst2,axiom,
! [A: nat,B3: nat,F: nat > num,C2: num] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ord_less_num @ ( F @ B3 ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_403_order__le__less__subst2,axiom,
! [A: int,B3: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_404_order__le__less__subst2,axiom,
! [A: int,B3: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_405_order__le__less__subst2,axiom,
! [A: int,B3: int,F: int > num,C2: num] :
( ( ord_less_eq_int @ A @ B3 )
=> ( ( ord_less_num @ ( F @ B3 ) @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_406_order__le__less__subst2,axiom,
! [A: num,B3: num,F: num > nat,C2: nat] :
( ( ord_less_eq_num @ A @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_407_order__le__less__subst2,axiom,
! [A: num,B3: num,F: num > int,C2: int] :
( ( ord_less_eq_num @ A @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_408_order__le__less__subst2,axiom,
! [A: num,B3: num,F: num > num,C2: num] :
( ( ord_less_eq_num @ A @ B3 )
=> ( ( ord_less_num @ ( F @ B3 ) @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_409_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_410_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_411_int__one__le__iff__zero__less,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ one_one_int @ Z3 )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% int_one_le_iff_zero_less
thf(fact_412_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_413_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( K
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_414_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_415_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_416_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M3: nat] :
( ( P @ X2 )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M3 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_417_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_418_le__trans,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_419_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_420_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_421_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_422_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B3 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_423_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_424_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_425_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_426_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_427_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_428_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_429_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J2: nat] :
( ! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J ) ) )
=> ( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_430_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_431_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_432_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N2: nat] :
( ( ord_less_nat @ M5 @ N2 )
| ( M5 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_433_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_434_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N2: nat] :
( ( ord_less_eq_nat @ M5 @ N2 )
& ( M5 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_435_less__eq__set__def,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A3: set_list_nat,B2: set_list_nat] :
( ord_le1520216061033275535_nat_o
@ ^ [X: list_nat] : ( member_list_nat2 @ X @ A3 )
@ ^ [X: list_nat] : ( member_list_nat2 @ X @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_436_less__eq__set__def,axiom,
( ord_le2077887516847798113at_nat
= ( ^ [A3: set_se7855581050983116737at_nat,B2: set_se7855581050983116737at_nat] :
( ord_le5866676684937102012_nat_o
@ ^ [X: set_Pr1261947904930325089at_nat] : ( member2643936169264416010at_nat @ X @ A3 )
@ ^ [X: set_Pr1261947904930325089at_nat] : ( member2643936169264416010at_nat @ X @ B2 ) ) ) ) ).
% less_eq_set_def
thf(fact_437_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_438_length__filter__le,axiom,
! [P: list_nat > $o,Xs2: list_list_nat] : ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ ( filter_list_nat @ P @ Xs2 ) ) @ ( size_s3023201423986296836st_nat @ Xs2 ) ) ).
% length_filter_le
thf(fact_439_length__filter__le,axiom,
! [P: nat > $o,Xs2: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( filter_nat @ P @ Xs2 ) ) @ ( size_size_list_nat @ Xs2 ) ) ).
% length_filter_le
thf(fact_440_length__filter__le,axiom,
! [P: set_nat > $o,Xs2: list_set_nat] : ( ord_less_eq_nat @ ( size_s3254054031482475050et_nat @ ( filter_set_nat @ P @ Xs2 ) ) @ ( size_s3254054031482475050et_nat @ Xs2 ) ) ).
% length_filter_le
thf(fact_441_length__filter__le,axiom,
! [P: set_Pr1261947904930325089at_nat > $o,Xs2: list_s1210847774152347623at_nat] : ( ord_less_eq_nat @ ( size_s8736152011456118867at_nat @ ( filter4791820933415917918at_nat @ P @ Xs2 ) ) @ ( size_s8736152011456118867at_nat @ Xs2 ) ) ).
% length_filter_le
thf(fact_442_count__le__length,axiom,
! [Xs2: list_list_nat,X2: list_nat] : ( ord_less_eq_nat @ ( count_list_list_nat @ Xs2 @ X2 ) @ ( size_s3023201423986296836st_nat @ Xs2 ) ) ).
% count_le_length
thf(fact_443_count__le__length,axiom,
! [Xs2: list_nat,X2: nat] : ( ord_less_eq_nat @ ( count_list_nat @ Xs2 @ X2 ) @ ( size_size_list_nat @ Xs2 ) ) ).
% count_le_length
thf(fact_444_count__le__length,axiom,
! [Xs2: list_set_nat,X2: set_nat] : ( ord_less_eq_nat @ ( count_list_set_nat @ Xs2 @ X2 ) @ ( size_s3254054031482475050et_nat @ Xs2 ) ) ).
% count_le_length
thf(fact_445_count__le__length,axiom,
! [Xs2: list_s1210847774152347623at_nat,X2: set_Pr1261947904930325089at_nat] : ( ord_less_eq_nat @ ( count_6440129622255701469at_nat @ Xs2 @ X2 ) @ ( size_s8736152011456118867at_nat @ Xs2 ) ) ).
% count_le_length
thf(fact_446_count__list__map__ge,axiom,
! [Xs2: list_nat,X2: nat,F: nat > nat] : ( ord_less_eq_nat @ ( count_list_nat @ Xs2 @ X2 ) @ ( count_list_nat @ ( map_nat_nat @ F @ Xs2 ) @ ( F @ X2 ) ) ) ).
% count_list_map_ge
thf(fact_447_count__list__map__ge,axiom,
! [Xs2: list_list_nat,X2: list_nat,F: list_nat > nat] : ( ord_less_eq_nat @ ( count_list_list_nat @ Xs2 @ X2 ) @ ( count_list_nat @ ( map_list_nat_nat @ F @ Xs2 ) @ ( F @ X2 ) ) ) ).
% count_list_map_ge
thf(fact_448_count__list__map__ge,axiom,
! [Xs2: list_set_nat,X2: set_nat,F: set_nat > nat] : ( ord_less_eq_nat @ ( count_list_set_nat @ Xs2 @ X2 ) @ ( count_list_nat @ ( map_set_nat_nat @ F @ Xs2 ) @ ( F @ X2 ) ) ) ).
% count_list_map_ge
thf(fact_449_count__list__map__ge,axiom,
! [Xs2: list_nat,X2: nat,F: nat > list_nat] : ( ord_less_eq_nat @ ( count_list_nat @ Xs2 @ X2 ) @ ( count_list_list_nat @ ( map_nat_list_nat @ F @ Xs2 ) @ ( F @ X2 ) ) ) ).
% count_list_map_ge
thf(fact_450_count__list__map__ge,axiom,
! [Xs2: list_nat,X2: nat,F: nat > set_nat] : ( ord_less_eq_nat @ ( count_list_nat @ Xs2 @ X2 ) @ ( count_list_set_nat @ ( map_nat_set_nat @ F @ Xs2 ) @ ( F @ X2 ) ) ) ).
% count_list_map_ge
thf(fact_451_count__list__map__ge,axiom,
! [Xs2: list_list_nat,X2: list_nat,F: list_nat > list_nat] : ( ord_less_eq_nat @ ( count_list_list_nat @ Xs2 @ X2 ) @ ( count_list_list_nat @ ( map_li7225945977422193158st_nat @ F @ Xs2 ) @ ( F @ X2 ) ) ) ).
% count_list_map_ge
thf(fact_452_count__list__map__ge,axiom,
! [Xs2: list_list_nat,X2: list_nat,F: list_nat > set_nat] : ( ord_less_eq_nat @ ( count_list_list_nat @ Xs2 @ X2 ) @ ( count_list_set_nat @ ( map_list_nat_set_nat @ F @ Xs2 ) @ ( F @ X2 ) ) ) ).
% count_list_map_ge
thf(fact_453_count__list__map__ge,axiom,
! [Xs2: list_set_nat,X2: set_nat,F: set_nat > list_nat] : ( ord_less_eq_nat @ ( count_list_set_nat @ Xs2 @ X2 ) @ ( count_list_list_nat @ ( map_set_nat_list_nat @ F @ Xs2 ) @ ( F @ X2 ) ) ) ).
% count_list_map_ge
thf(fact_454_count__list__map__ge,axiom,
! [Xs2: list_set_nat,X2: set_nat,F: set_nat > set_nat] : ( ord_less_eq_nat @ ( count_list_set_nat @ Xs2 @ X2 ) @ ( count_list_set_nat @ ( map_set_nat_set_nat @ F @ Xs2 ) @ ( F @ X2 ) ) ) ).
% count_list_map_ge
thf(fact_455_count__list__map__ge,axiom,
! [Xs2: list_s1210847774152347623at_nat,X2: set_Pr1261947904930325089at_nat,F: set_Pr1261947904930325089at_nat > nat] : ( ord_less_eq_nat @ ( count_6440129622255701469at_nat @ Xs2 @ X2 ) @ ( count_list_nat @ ( map_se4264141375869589357at_nat @ F @ Xs2 ) @ ( F @ X2 ) ) ) ).
% count_list_map_ge
thf(fact_456_nle__le,axiom,
! [A: nat,B3: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B3 ) )
= ( ( ord_less_eq_nat @ B3 @ A )
& ( B3 != A ) ) ) ).
% nle_le
thf(fact_457_nle__le,axiom,
! [A: int,B3: int] :
( ( ~ ( ord_less_eq_int @ A @ B3 ) )
= ( ( ord_less_eq_int @ B3 @ A )
& ( B3 != A ) ) ) ).
% nle_le
thf(fact_458_nle__le,axiom,
! [A: num,B3: num] :
( ( ~ ( ord_less_eq_num @ A @ B3 ) )
= ( ( ord_less_eq_num @ B3 @ A )
& ( B3 != A ) ) ) ).
% nle_le
thf(fact_459_le__cases3,axiom,
! [X2: nat,Y2: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_460_le__cases3,axiom,
! [X2: int,Y2: int,Z3: int] :
( ( ( ord_less_eq_int @ X2 @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ Z3 ) )
=> ( ( ( ord_less_eq_int @ Y2 @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Z3 ) )
=> ( ( ( ord_less_eq_int @ X2 @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ Y2 ) )
=> ( ( ( ord_less_eq_int @ Z3 @ Y2 )
=> ~ ( ord_less_eq_int @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_int @ Y2 @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ X2 ) )
=> ~ ( ( ord_less_eq_int @ Z3 @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_461_le__cases3,axiom,
! [X2: num,Y2: num,Z3: num] :
( ( ( ord_less_eq_num @ X2 @ Y2 )
=> ~ ( ord_less_eq_num @ Y2 @ Z3 ) )
=> ( ( ( ord_less_eq_num @ Y2 @ X2 )
=> ~ ( ord_less_eq_num @ X2 @ Z3 ) )
=> ( ( ( ord_less_eq_num @ X2 @ Z3 )
=> ~ ( ord_less_eq_num @ Z3 @ Y2 ) )
=> ( ( ( ord_less_eq_num @ Z3 @ Y2 )
=> ~ ( ord_less_eq_num @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_num @ Y2 @ Z3 )
=> ~ ( ord_less_eq_num @ Z3 @ X2 ) )
=> ~ ( ( ord_less_eq_num @ Z3 @ X2 )
=> ~ ( ord_less_eq_num @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_462_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_463_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: int,Z: int] : ( Y = Z ) )
= ( ^ [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
& ( ord_less_eq_int @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_464_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: num,Z: num] : ( Y = Z ) )
= ( ^ [X: num,Y4: num] :
( ( ord_less_eq_num @ X @ Y4 )
& ( ord_less_eq_num @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_465_ord__eq__le__trans,axiom,
! [A: nat,B3: nat,C2: nat] :
( ( A = B3 )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_466_ord__eq__le__trans,axiom,
! [A: int,B3: int,C2: int] :
( ( A = B3 )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ord_less_eq_int @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_467_ord__eq__le__trans,axiom,
! [A: num,B3: num,C2: num] :
( ( A = B3 )
=> ( ( ord_less_eq_num @ B3 @ C2 )
=> ( ord_less_eq_num @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_468_ord__le__eq__trans,axiom,
! [A: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_469_ord__le__eq__trans,axiom,
! [A: int,B3: int,C2: int] :
( ( ord_less_eq_int @ A @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq_int @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_470_ord__le__eq__trans,axiom,
! [A: num,B3: num,C2: num] :
( ( ord_less_eq_num @ A @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq_num @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_471_order__antisym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_472_order__antisym,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_473_order__antisym,axiom,
! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ( ord_less_eq_num @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_474_order_Otrans,axiom,
! [A: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_475_order_Otrans,axiom,
! [A: int,B3: int,C2: int] :
( ( ord_less_eq_int @ A @ B3 )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ord_less_eq_int @ A @ C2 ) ) ) ).
% order.trans
thf(fact_476_order_Otrans,axiom,
! [A: num,B3: num,C2: num] :
( ( ord_less_eq_num @ A @ B3 )
=> ( ( ord_less_eq_num @ B3 @ C2 )
=> ( ord_less_eq_num @ A @ C2 ) ) ) ).
% order.trans
thf(fact_477_order__trans,axiom,
! [X2: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z3 )
=> ( ord_less_eq_nat @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_478_order__trans,axiom,
! [X2: int,Y2: int,Z3: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z3 )
=> ( ord_less_eq_int @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_479_order__trans,axiom,
! [X2: num,Y2: num,Z3: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ( ord_less_eq_num @ Y2 @ Z3 )
=> ( ord_less_eq_num @ X2 @ Z3 ) ) ) ).
% order_trans
thf(fact_480_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B3: nat] :
( ! [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: nat,B4: nat] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A @ B3 ) ) ) ).
% linorder_wlog
thf(fact_481_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B3: int] :
( ! [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: int,B4: int] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A @ B3 ) ) ) ).
% linorder_wlog
thf(fact_482_linorder__wlog,axiom,
! [P: num > num > $o,A: num,B3: num] :
( ! [A5: num,B4: num] :
( ( ord_less_eq_num @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: num,B4: num] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A @ B3 ) ) ) ).
% linorder_wlog
thf(fact_483_dual__order_Oeq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A4 )
& ( ord_less_eq_nat @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_484_dual__order_Oeq__iff,axiom,
( ( ^ [Y: int,Z: int] : ( Y = Z ) )
= ( ^ [A4: int,B5: int] :
( ( ord_less_eq_int @ B5 @ A4 )
& ( ord_less_eq_int @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_485_dual__order_Oeq__iff,axiom,
( ( ^ [Y: num,Z: num] : ( Y = Z ) )
= ( ^ [A4: num,B5: num] :
( ( ord_less_eq_num @ B5 @ A4 )
& ( ord_less_eq_num @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_486_dual__order_Oantisym,axiom,
! [B3: nat,A: nat] :
( ( ord_less_eq_nat @ B3 @ A )
=> ( ( ord_less_eq_nat @ A @ B3 )
=> ( A = B3 ) ) ) ).
% dual_order.antisym
thf(fact_487_dual__order_Oantisym,axiom,
! [B3: int,A: int] :
( ( ord_less_eq_int @ B3 @ A )
=> ( ( ord_less_eq_int @ A @ B3 )
=> ( A = B3 ) ) ) ).
% dual_order.antisym
thf(fact_488_dual__order_Oantisym,axiom,
! [B3: num,A: num] :
( ( ord_less_eq_num @ B3 @ A )
=> ( ( ord_less_eq_num @ A @ B3 )
=> ( A = B3 ) ) ) ).
% dual_order.antisym
thf(fact_489_dual__order_Otrans,axiom,
! [B3: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B3 @ A )
=> ( ( ord_less_eq_nat @ C2 @ B3 )
=> ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_490_dual__order_Otrans,axiom,
! [B3: int,A: int,C2: int] :
( ( ord_less_eq_int @ B3 @ A )
=> ( ( ord_less_eq_int @ C2 @ B3 )
=> ( ord_less_eq_int @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_491_dual__order_Otrans,axiom,
! [B3: num,A: num,C2: num] :
( ( ord_less_eq_num @ B3 @ A )
=> ( ( ord_less_eq_num @ C2 @ B3 )
=> ( ord_less_eq_num @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_492_antisym,axiom,
! [A: nat,B3: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ A )
=> ( A = B3 ) ) ) ).
% antisym
thf(fact_493_antisym,axiom,
! [A: int,B3: int] :
( ( ord_less_eq_int @ A @ B3 )
=> ( ( ord_less_eq_int @ B3 @ A )
=> ( A = B3 ) ) ) ).
% antisym
thf(fact_494_antisym,axiom,
! [A: num,B3: num] :
( ( ord_less_eq_num @ A @ B3 )
=> ( ( ord_less_eq_num @ B3 @ A )
=> ( A = B3 ) ) ) ).
% antisym
thf(fact_495_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ A4 @ B5 )
& ( ord_less_eq_nat @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_496_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: int,Z: int] : ( Y = Z ) )
= ( ^ [A4: int,B5: int] :
( ( ord_less_eq_int @ A4 @ B5 )
& ( ord_less_eq_int @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_497_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: num,Z: num] : ( Y = Z ) )
= ( ^ [A4: num,B5: num] :
( ( ord_less_eq_num @ A4 @ B5 )
& ( ord_less_eq_num @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_498_order__subst1,axiom,
! [A: nat,F: nat > nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_499_order__subst1,axiom,
! [A: nat,F: int > nat,B3: int,C2: int] :
( ( ord_less_eq_nat @ A @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_500_order__subst1,axiom,
! [A: nat,F: num > nat,B3: num,C2: num] :
( ( ord_less_eq_nat @ A @ ( F @ B3 ) )
=> ( ( ord_less_eq_num @ B3 @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_501_order__subst1,axiom,
! [A: int,F: nat > int,B3: nat,C2: nat] :
( ( ord_less_eq_int @ A @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_502_order__subst1,axiom,
! [A: int,F: int > int,B3: int,C2: int] :
( ( ord_less_eq_int @ A @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_503_order__subst1,axiom,
! [A: int,F: num > int,B3: num,C2: num] :
( ( ord_less_eq_int @ A @ ( F @ B3 ) )
=> ( ( ord_less_eq_num @ B3 @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_504_order__subst1,axiom,
! [A: num,F: nat > num,B3: nat,C2: nat] :
( ( ord_less_eq_num @ A @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_505_order__subst1,axiom,
! [A: num,F: int > num,B3: int,C2: int] :
( ( ord_less_eq_num @ A @ ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_506_order__subst1,axiom,
! [A: num,F: num > num,B3: num,C2: num] :
( ( ord_less_eq_num @ A @ ( F @ B3 ) )
=> ( ( ord_less_eq_num @ B3 @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_507_order__subst2,axiom,
! [A: nat,B3: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_508_order__subst2,axiom,
! [A: nat,B3: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_509_order__subst2,axiom,
! [A: nat,B3: nat,F: nat > num,C2: num] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ord_less_eq_num @ ( F @ B3 ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_510_order__subst2,axiom,
! [A: int,B3: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_511_order__subst2,axiom,
! [A: int,B3: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_512_order__subst2,axiom,
! [A: int,B3: int,F: int > num,C2: num] :
( ( ord_less_eq_int @ A @ B3 )
=> ( ( ord_less_eq_num @ ( F @ B3 ) @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_513_order__subst2,axiom,
! [A: num,B3: num,F: num > nat,C2: nat] :
( ( ord_less_eq_num @ A @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_514_order__subst2,axiom,
! [A: num,B3: num,F: num > int,C2: int] :
( ( ord_less_eq_num @ A @ B3 )
=> ( ( ord_less_eq_int @ ( F @ B3 ) @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_515_order__subst2,axiom,
! [A: num,B3: num,F: num > num,C2: num] :
( ( ord_less_eq_num @ A @ B3 )
=> ( ( ord_less_eq_num @ ( F @ B3 ) @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_516_order__eq__refl,axiom,
! [X2: nat,Y2: nat] :
( ( X2 = Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_517_order__eq__refl,axiom,
! [X2: int,Y2: int] :
( ( X2 = Y2 )
=> ( ord_less_eq_int @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_518_order__eq__refl,axiom,
! [X2: num,Y2: num] :
( ( X2 = Y2 )
=> ( ord_less_eq_num @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_519_linorder__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_520_linorder__linear,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
| ( ord_less_eq_int @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_521_linorder__linear,axiom,
! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
| ( ord_less_eq_num @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_522_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B3: nat,C2: nat] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_523_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B3: nat,C2: nat] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_524_ord__eq__le__subst,axiom,
! [A: num,F: nat > num,B3: nat,C2: nat] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_525_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B3: int,C2: int] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_526_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B3: int,C2: int] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_527_ord__eq__le__subst,axiom,
! [A: num,F: int > num,B3: int,C2: int] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_528_ord__eq__le__subst,axiom,
! [A: nat,F: num > nat,B3: num,C2: num] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_less_eq_num @ B3 @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_529_ord__eq__le__subst,axiom,
! [A: int,F: num > int,B3: num,C2: num] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_less_eq_num @ B3 @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_530_ord__eq__le__subst,axiom,
! [A: num,F: num > num,B3: num,C2: num] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_less_eq_num @ B3 @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_531_ord__le__eq__subst,axiom,
! [A: nat,B3: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_532_ord__le__eq__subst,axiom,
! [A: nat,B3: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_533_ord__le__eq__subst,axiom,
! [A: nat,B3: nat,F: nat > num,C2: num] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_534_ord__le__eq__subst,axiom,
! [A: int,B3: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_535_ord__le__eq__subst,axiom,
! [A: int,B3: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_536_ord__le__eq__subst,axiom,
! [A: int,B3: int,F: int > num,C2: num] :
( ( ord_less_eq_int @ A @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_537_ord__le__eq__subst,axiom,
! [A: num,B3: num,F: num > nat,C2: nat] :
( ( ord_less_eq_num @ A @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_538_ord__le__eq__subst,axiom,
! [A: num,B3: num,F: num > int,C2: int] :
( ( ord_less_eq_num @ A @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_539_ord__le__eq__subst,axiom,
! [A: num,B3: num,F: num > num,C2: num] :
( ( ord_less_eq_num @ A @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_540_linorder__le__cases,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_541_linorder__le__cases,axiom,
! [X2: int,Y2: int] :
( ~ ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_542_linorder__le__cases,axiom,
! [X2: num,Y2: num] :
( ~ ( ord_less_eq_num @ X2 @ Y2 )
=> ( ord_less_eq_num @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_543_order__antisym__conv,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_544_order__antisym__conv,axiom,
! [Y2: int,X2: int] :
( ( ord_less_eq_int @ Y2 @ X2 )
=> ( ( ord_less_eq_int @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_545_order__antisym__conv,axiom,
! [Y2: num,X2: num] :
( ( ord_less_eq_num @ Y2 @ X2 )
=> ( ( ord_less_eq_num @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_546_lt__ex,axiom,
! [X2: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X2 ) ).
% lt_ex
thf(fact_547_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_548_gt__ex,axiom,
! [X2: int] :
? [X_1: int] : ( ord_less_int @ X2 @ X_1 ) ).
% gt_ex
thf(fact_549_less__imp__neq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_550_less__imp__neq,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_551_less__imp__neq,axiom,
! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_552_order_Oasym,axiom,
! [A: nat,B3: nat] :
( ( ord_less_nat @ A @ B3 )
=> ~ ( ord_less_nat @ B3 @ A ) ) ).
% order.asym
thf(fact_553_order_Oasym,axiom,
! [A: int,B3: int] :
( ( ord_less_int @ A @ B3 )
=> ~ ( ord_less_int @ B3 @ A ) ) ).
% order.asym
thf(fact_554_order_Oasym,axiom,
! [A: num,B3: num] :
( ( ord_less_num @ A @ B3 )
=> ~ ( ord_less_num @ B3 @ A ) ) ).
% order.asym
thf(fact_555_ord__eq__less__trans,axiom,
! [A: nat,B3: nat,C2: nat] :
( ( A = B3 )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_556_ord__eq__less__trans,axiom,
! [A: int,B3: int,C2: int] :
( ( A = B3 )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_557_ord__eq__less__trans,axiom,
! [A: num,B3: num,C2: num] :
( ( A = B3 )
=> ( ( ord_less_num @ B3 @ C2 )
=> ( ord_less_num @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_558_ord__less__eq__trans,axiom,
! [A: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_559_ord__less__eq__trans,axiom,
! [A: int,B3: int,C2: int] :
( ( ord_less_int @ A @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_560_ord__less__eq__trans,axiom,
! [A: num,B3: num,C2: num] :
( ( ord_less_num @ A @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_num @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_561_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_562_antisym__conv3,axiom,
! [Y2: nat,X2: nat] :
( ~ ( ord_less_nat @ Y2 @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_563_antisym__conv3,axiom,
! [Y2: int,X2: int] :
( ~ ( ord_less_int @ Y2 @ X2 )
=> ( ( ~ ( ord_less_int @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_564_antisym__conv3,axiom,
! [Y2: num,X2: num] :
( ~ ( ord_less_num @ Y2 @ X2 )
=> ( ( ~ ( ord_less_num @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_565_linorder__cases,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_cases
thf(fact_566_linorder__cases,axiom,
! [X2: int,Y2: int] :
( ~ ( ord_less_int @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_int @ Y2 @ X2 ) ) ) ).
% linorder_cases
thf(fact_567_linorder__cases,axiom,
! [X2: num,Y2: num] :
( ~ ( ord_less_num @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_num @ Y2 @ X2 ) ) ) ).
% linorder_cases
thf(fact_568_dual__order_Oasym,axiom,
! [B3: nat,A: nat] :
( ( ord_less_nat @ B3 @ A )
=> ~ ( ord_less_nat @ A @ B3 ) ) ).
% dual_order.asym
thf(fact_569_dual__order_Oasym,axiom,
! [B3: int,A: int] :
( ( ord_less_int @ B3 @ A )
=> ~ ( ord_less_int @ A @ B3 ) ) ).
% dual_order.asym
thf(fact_570_dual__order_Oasym,axiom,
! [B3: num,A: num] :
( ( ord_less_num @ B3 @ A )
=> ~ ( ord_less_num @ A @ B3 ) ) ).
% dual_order.asym
thf(fact_571_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_572_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_573_dual__order_Oirrefl,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% dual_order.irrefl
thf(fact_574_exists__least__iff,axiom,
( ( ^ [P4: nat > $o] :
? [X5: nat] : ( P4 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M5: nat] :
( ( ord_less_nat @ M5 @ N2 )
=> ~ ( P3 @ M5 ) ) ) ) ) ).
% exists_least_iff
thf(fact_575_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B3: nat] :
( ! [A5: nat,B4: nat] :
( ( ord_less_nat @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: nat] : ( P @ A5 @ A5 )
=> ( ! [A5: nat,B4: nat] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_576_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B3: int] :
( ! [A5: int,B4: int] :
( ( ord_less_int @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: int] : ( P @ A5 @ A5 )
=> ( ! [A5: int,B4: int] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_577_linorder__less__wlog,axiom,
! [P: num > num > $o,A: num,B3: num] :
( ! [A5: num,B4: num] :
( ( ord_less_num @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: num] : ( P @ A5 @ A5 )
=> ( ! [A5: num,B4: num] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_578_order_Ostrict__trans,axiom,
! [A: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_579_order_Ostrict__trans,axiom,
! [A: int,B3: int,C2: int] :
( ( ord_less_int @ A @ B3 )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_580_order_Ostrict__trans,axiom,
! [A: num,B3: num,C2: num] :
( ( ord_less_num @ A @ B3 )
=> ( ( ord_less_num @ B3 @ C2 )
=> ( ord_less_num @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_581_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( ( ord_less_nat @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_582_not__less__iff__gr__or__eq,axiom,
! [X2: int,Y2: int] :
( ( ~ ( ord_less_int @ X2 @ Y2 ) )
= ( ( ord_less_int @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_583_not__less__iff__gr__or__eq,axiom,
! [X2: num,Y2: num] :
( ( ~ ( ord_less_num @ X2 @ Y2 ) )
= ( ( ord_less_num @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_584_dual__order_Ostrict__trans,axiom,
! [B3: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B3 @ A )
=> ( ( ord_less_nat @ C2 @ B3 )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_585_dual__order_Ostrict__trans,axiom,
! [B3: int,A: int,C2: int] :
( ( ord_less_int @ B3 @ A )
=> ( ( ord_less_int @ C2 @ B3 )
=> ( ord_less_int @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_586_dual__order_Ostrict__trans,axiom,
! [B3: num,A: num,C2: num] :
( ( ord_less_num @ B3 @ A )
=> ( ( ord_less_num @ C2 @ B3 )
=> ( ord_less_num @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_587_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B3: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( A != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_588_order_Ostrict__implies__not__eq,axiom,
! [A: int,B3: int] :
( ( ord_less_int @ A @ B3 )
=> ( A != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_589_order_Ostrict__implies__not__eq,axiom,
! [A: num,B3: num] :
( ( ord_less_num @ A @ B3 )
=> ( A != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_590_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: nat,A: nat] :
( ( ord_less_nat @ B3 @ A )
=> ( A != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_591_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: int,A: int] :
( ( ord_less_int @ B3 @ A )
=> ( A != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_592_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: num,A: num] :
( ( ord_less_num @ B3 @ A )
=> ( A != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_593_linorder__neqE,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_594_linorder__neqE,axiom,
! [X2: int,Y2: int] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ Y2 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_595_linorder__neqE,axiom,
! [X2: num,Y2: num] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_num @ X2 @ Y2 )
=> ( ord_less_num @ Y2 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_596_order__less__asym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_asym
thf(fact_597_order__less__asym,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X2 ) ) ).
% order_less_asym
thf(fact_598_order__less__asym,axiom,
! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ~ ( ord_less_num @ Y2 @ X2 ) ) ).
% order_less_asym
thf(fact_599_linorder__neq__iff,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
= ( ( ord_less_nat @ X2 @ Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_600_linorder__neq__iff,axiom,
! [X2: int,Y2: int] :
( ( X2 != Y2 )
= ( ( ord_less_int @ X2 @ Y2 )
| ( ord_less_int @ Y2 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_601_linorder__neq__iff,axiom,
! [X2: num,Y2: num] :
( ( X2 != Y2 )
= ( ( ord_less_num @ X2 @ Y2 )
| ( ord_less_num @ Y2 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_602_order__less__asym_H,axiom,
! [A: nat,B3: nat] :
( ( ord_less_nat @ A @ B3 )
=> ~ ( ord_less_nat @ B3 @ A ) ) ).
% order_less_asym'
thf(fact_603_order__less__asym_H,axiom,
! [A: int,B3: int] :
( ( ord_less_int @ A @ B3 )
=> ~ ( ord_less_int @ B3 @ A ) ) ).
% order_less_asym'
thf(fact_604_order__less__asym_H,axiom,
! [A: num,B3: num] :
( ( ord_less_num @ A @ B3 )
=> ~ ( ord_less_num @ B3 @ A ) ) ).
% order_less_asym'
thf(fact_605_order__less__trans,axiom,
! [X2: nat,Y2: nat,Z3: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z3 )
=> ( ord_less_nat @ X2 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_606_order__less__trans,axiom,
! [X2: int,Y2: int,Z3: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z3 )
=> ( ord_less_int @ X2 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_607_order__less__trans,axiom,
! [X2: num,Y2: num,Z3: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( ( ord_less_num @ Y2 @ Z3 )
=> ( ord_less_num @ X2 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_608_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B3: nat,C2: nat] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_609_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B3: nat,C2: nat] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_610_ord__eq__less__subst,axiom,
! [A: num,F: nat > num,B3: nat,C2: nat] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_611_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B3: int,C2: int] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_612_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B3: int,C2: int] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_613_ord__eq__less__subst,axiom,
! [A: num,F: int > num,B3: int,C2: int] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_614_ord__eq__less__subst,axiom,
! [A: nat,F: num > nat,B3: num,C2: num] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_less_num @ B3 @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_615_ord__eq__less__subst,axiom,
! [A: int,F: num > int,B3: num,C2: num] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_less_num @ B3 @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_616_ord__eq__less__subst,axiom,
! [A: num,F: num > num,B3: num,C2: num] :
( ( A
= ( F @ B3 ) )
=> ( ( ord_less_num @ B3 @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_617_ord__less__eq__subst,axiom,
! [A: nat,B3: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_618_ord__less__eq__subst,axiom,
! [A: nat,B3: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_619_ord__less__eq__subst,axiom,
! [A: nat,B3: nat,F: nat > num,C2: num] :
( ( ord_less_nat @ A @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_620_ord__less__eq__subst,axiom,
! [A: int,B3: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_621_ord__less__eq__subst,axiom,
! [A: int,B3: int,F: int > int,C2: int] :
( ( ord_less_int @ A @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_622_ord__less__eq__subst,axiom,
! [A: int,B3: int,F: int > num,C2: num] :
( ( ord_less_int @ A @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_623_ord__less__eq__subst,axiom,
! [A: num,B3: num,F: num > nat,C2: nat] :
( ( ord_less_num @ A @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_624_ord__less__eq__subst,axiom,
! [A: num,B3: num,F: num > int,C2: int] :
( ( ord_less_num @ A @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_625_ord__less__eq__subst,axiom,
! [A: num,B3: num,F: num > num,C2: num] :
( ( ord_less_num @ A @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_626_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_627_order__less__irrefl,axiom,
! [X2: int] :
~ ( ord_less_int @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_628_order__less__irrefl,axiom,
! [X2: num] :
~ ( ord_less_num @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_629_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_630_order__less__subst1,axiom,
! [A: nat,F: int > nat,B3: int,C2: int] :
( ( ord_less_nat @ A @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_631_order__less__subst1,axiom,
! [A: nat,F: num > nat,B3: num,C2: num] :
( ( ord_less_nat @ A @ ( F @ B3 ) )
=> ( ( ord_less_num @ B3 @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_632_order__less__subst1,axiom,
! [A: int,F: nat > int,B3: nat,C2: nat] :
( ( ord_less_int @ A @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_633_order__less__subst1,axiom,
! [A: int,F: int > int,B3: int,C2: int] :
( ( ord_less_int @ A @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_634_order__less__subst1,axiom,
! [A: int,F: num > int,B3: num,C2: num] :
( ( ord_less_int @ A @ ( F @ B3 ) )
=> ( ( ord_less_num @ B3 @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_635_order__less__subst1,axiom,
! [A: num,F: nat > num,B3: nat,C2: nat] :
( ( ord_less_num @ A @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_636_order__less__subst1,axiom,
! [A: num,F: int > num,B3: int,C2: int] :
( ( ord_less_num @ A @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_637_order__less__subst1,axiom,
! [A: num,F: num > num,B3: num,C2: num] :
( ( ord_less_num @ A @ ( F @ B3 ) )
=> ( ( ord_less_num @ B3 @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_638_order__less__subst2,axiom,
! [A: nat,B3: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_639_order__less__subst2,axiom,
! [A: nat,B3: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_640_order__less__subst2,axiom,
! [A: nat,B3: nat,F: nat > num,C2: num] :
( ( ord_less_nat @ A @ B3 )
=> ( ( ord_less_num @ ( F @ B3 ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_641_order__less__subst2,axiom,
! [A: int,B3: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_642_order__less__subst2,axiom,
! [A: int,B3: int,F: int > int,C2: int] :
( ( ord_less_int @ A @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_643_order__less__subst2,axiom,
! [A: int,B3: int,F: int > num,C2: num] :
( ( ord_less_int @ A @ B3 )
=> ( ( ord_less_num @ ( F @ B3 ) @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_644_order__less__subst2,axiom,
! [A: num,B3: num,F: num > nat,C2: nat] :
( ( ord_less_num @ A @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_645_order__less__subst2,axiom,
! [A: num,B3: num,F: num > int,C2: int] :
( ( ord_less_num @ A @ B3 )
=> ( ( ord_less_int @ ( F @ B3 ) @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_646_order__less__subst2,axiom,
! [A: num,B3: num,F: num > num,C2: num] :
( ( ord_less_num @ A @ B3 )
=> ( ( ord_less_num @ ( F @ B3 ) @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_647_order__less__not__sym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_not_sym
thf(fact_648_order__less__not__sym,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X2 ) ) ).
% order_less_not_sym
thf(fact_649_order__less__not__sym,axiom,
! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ~ ( ord_less_num @ Y2 @ X2 ) ) ).
% order_less_not_sym
thf(fact_650_order__less__imp__triv,axiom,
! [X2: nat,Y2: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_651_order__less__imp__triv,axiom,
! [X2: int,Y2: int,P: $o] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ( ord_less_int @ Y2 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_652_order__less__imp__triv,axiom,
! [X2: num,Y2: num,P: $o] :
( ( ord_less_num @ X2 @ Y2 )
=> ( ( ord_less_num @ Y2 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_653_linorder__less__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_less_linear
thf(fact_654_linorder__less__linear,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
| ( X2 = Y2 )
| ( ord_less_int @ Y2 @ X2 ) ) ).
% linorder_less_linear
thf(fact_655_linorder__less__linear,axiom,
! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
| ( X2 = Y2 )
| ( ord_less_num @ Y2 @ X2 ) ) ).
% linorder_less_linear
thf(fact_656_order__less__imp__not__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_657_order__less__imp__not__eq,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_658_order__less__imp__not__eq,axiom,
! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_659_order__less__imp__not__eq2,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_660_order__less__imp__not__eq2,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_661_order__less__imp__not__eq2,axiom,
! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_662_order__less__imp__not__less,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_663_order__less__imp__not__less,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_664_order__less__imp__not__less,axiom,
! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ~ ( ord_less_num @ Y2 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_665_leD,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y2 ) ) ).
% leD
thf(fact_666_leD,axiom,
! [Y2: int,X2: int] :
( ( ord_less_eq_int @ Y2 @ X2 )
=> ~ ( ord_less_int @ X2 @ Y2 ) ) ).
% leD
thf(fact_667_leD,axiom,
! [Y2: num,X2: num] :
( ( ord_less_eq_num @ Y2 @ X2 )
=> ~ ( ord_less_num @ X2 @ Y2 ) ) ).
% leD
thf(fact_668_leI,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% leI
thf(fact_669_leI,axiom,
! [X2: int,Y2: int] :
( ~ ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X2 ) ) ).
% leI
thf(fact_670_leI,axiom,
! [X2: num,Y2: num] :
( ~ ( ord_less_num @ X2 @ Y2 )
=> ( ord_less_eq_num @ Y2 @ X2 ) ) ).
% leI
thf(fact_671_nless__le,axiom,
! [A: nat,B3: nat] :
( ( ~ ( ord_less_nat @ A @ B3 ) )
= ( ~ ( ord_less_eq_nat @ A @ B3 )
| ( A = B3 ) ) ) ).
% nless_le
thf(fact_672_nless__le,axiom,
! [A: int,B3: int] :
( ( ~ ( ord_less_int @ A @ B3 ) )
= ( ~ ( ord_less_eq_int @ A @ B3 )
| ( A = B3 ) ) ) ).
% nless_le
thf(fact_673_nless__le,axiom,
! [A: num,B3: num] :
( ( ~ ( ord_less_num @ A @ B3 ) )
= ( ~ ( ord_less_eq_num @ A @ B3 )
| ( A = B3 ) ) ) ).
% nless_le
thf(fact_674_antisym__conv1,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_675_antisym__conv1,axiom,
! [X2: int,Y2: int] :
( ~ ( ord_less_int @ X2 @ Y2 )
=> ( ( ord_less_eq_int @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_676_antisym__conv1,axiom,
! [X2: num,Y2: num] :
( ~ ( ord_less_num @ X2 @ Y2 )
=> ( ( ord_less_eq_num @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_677_antisym__conv2,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_678_antisym__conv2,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ~ ( ord_less_int @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_679_antisym__conv2,axiom,
! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ( ~ ( ord_less_num @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_680_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_681_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
& ~ ( ord_less_eq_int @ Y4 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_682_less__le__not__le,axiom,
( ord_less_num
= ( ^ [X: num,Y4: num] :
( ( ord_less_eq_num @ X @ Y4 )
& ~ ( ord_less_eq_num @ Y4 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_683_not__le__imp__less,axiom,
! [Y2: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ord_less_nat @ X2 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_684_not__le__imp__less,axiom,
! [Y2: int,X2: int] :
( ~ ( ord_less_eq_int @ Y2 @ X2 )
=> ( ord_less_int @ X2 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_685_not__le__imp__less,axiom,
! [Y2: num,X2: num] :
( ~ ( ord_less_eq_num @ Y2 @ X2 )
=> ( ord_less_num @ X2 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_686_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B5: nat] :
( ( ord_less_nat @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_687_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B5: int] :
( ( ord_less_int @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_688_order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [A4: num,B5: num] :
( ( ord_less_num @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_689_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_690_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A4: int,B5: int] :
( ( ord_less_eq_int @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_691_order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [A4: num,B5: num] :
( ( ord_less_eq_num @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_692_order_Ostrict__trans1,axiom,
! [A: nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_693_order_Ostrict__trans1,axiom,
! [A: int,B3: int,C2: int] :
( ( ord_less_eq_int @ A @ B3 )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_694_order_Ostrict__trans1,axiom,
! [A: num,B3: num,C2: num] :
( ( ord_less_eq_num @ A @ B3 )
=> ( ( ord_less_num @ B3 @ C2 )
=> ( ord_less_num @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_695_order_Ostrict__trans2,axiom,
! [A: nat,B3: nat,C2: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_696_order_Ostrict__trans2,axiom,
! [A: int,B3: int,C2: int] :
( ( ord_less_int @ A @ B3 )
=> ( ( ord_less_eq_int @ B3 @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_697_order_Ostrict__trans2,axiom,
! [A: num,B3: num,C2: num] :
( ( ord_less_num @ A @ B3 )
=> ( ( ord_less_eq_num @ B3 @ C2 )
=> ( ord_less_num @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_698_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ A4 @ B5 )
& ~ ( ord_less_eq_nat @ B5 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_699_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A4: int,B5: int] :
( ( ord_less_eq_int @ A4 @ B5 )
& ~ ( ord_less_eq_int @ B5 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_700_order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [A4: num,B5: num] :
( ( ord_less_eq_num @ A4 @ B5 )
& ~ ( ord_less_eq_num @ B5 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_701_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A4: nat] :
( ( ord_less_nat @ B5 @ A4 )
| ( A4 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_702_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B5: int,A4: int] :
( ( ord_less_int @ B5 @ A4 )
| ( A4 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_703_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [B5: num,A4: num] :
( ( ord_less_num @ B5 @ A4 )
| ( A4 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_704_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B5: nat,A4: nat] :
( ( ord_less_eq_nat @ B5 @ A4 )
& ( A4 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_705_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B5: int,A4: int] :
( ( ord_less_eq_int @ B5 @ A4 )
& ( A4 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_706_dual__order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [B5: num,A4: num] :
( ( ord_less_eq_num @ B5 @ A4 )
& ( A4 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_707_dual__order_Ostrict__trans1,axiom,
! [B3: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B3 @ A )
=> ( ( ord_less_nat @ C2 @ B3 )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_708_dual__order_Ostrict__trans1,axiom,
! [B3: int,A: int,C2: int] :
( ( ord_less_eq_int @ B3 @ A )
=> ( ( ord_less_int @ C2 @ B3 )
=> ( ord_less_int @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_709_dual__order_Ostrict__trans1,axiom,
! [B3: num,A: num,C2: num] :
( ( ord_less_eq_num @ B3 @ A )
=> ( ( ord_less_num @ C2 @ B3 )
=> ( ord_less_num @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_710_dual__order_Ostrict__trans2,axiom,
! [B3: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B3 @ A )
=> ( ( ord_less_eq_nat @ C2 @ B3 )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_711_dual__order_Ostrict__trans2,axiom,
! [B3: int,A: int,C2: int] :
( ( ord_less_int @ B3 @ A )
=> ( ( ord_less_eq_int @ C2 @ B3 )
=> ( ord_less_int @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_712_dual__order_Ostrict__trans2,axiom,
! [B3: num,A: num,C2: num] :
( ( ord_less_num @ B3 @ A )
=> ( ( ord_less_eq_num @ C2 @ B3 )
=> ( ord_less_num @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_713_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B5: nat,A4: nat] :
( ( ord_less_eq_nat @ B5 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_714_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B5: int,A4: int] :
( ( ord_less_eq_int @ B5 @ A4 )
& ~ ( ord_less_eq_int @ A4 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_715_dual__order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [B5: num,A4: num] :
( ( ord_less_eq_num @ B5 @ A4 )
& ~ ( ord_less_eq_num @ A4 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_716_order_Ostrict__implies__order,axiom,
! [A: nat,B3: nat] :
( ( ord_less_nat @ A @ B3 )
=> ( ord_less_eq_nat @ A @ B3 ) ) ).
% order.strict_implies_order
thf(fact_717_order_Ostrict__implies__order,axiom,
! [A: int,B3: int] :
( ( ord_less_int @ A @ B3 )
=> ( ord_less_eq_int @ A @ B3 ) ) ).
% order.strict_implies_order
thf(fact_718_order_Ostrict__implies__order,axiom,
! [A: num,B3: num] :
( ( ord_less_num @ A @ B3 )
=> ( ord_less_eq_num @ A @ B3 ) ) ).
% order.strict_implies_order
thf(fact_719_dual__order_Ostrict__implies__order,axiom,
! [B3: nat,A: nat] :
( ( ord_less_nat @ B3 @ A )
=> ( ord_less_eq_nat @ B3 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_720_dual__order_Ostrict__implies__order,axiom,
! [B3: int,A: int] :
( ( ord_less_int @ B3 @ A )
=> ( ord_less_eq_int @ B3 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_721_dual__order_Ostrict__implies__order,axiom,
! [B3: num,A: num] :
( ( ord_less_num @ B3 @ A )
=> ( ord_less_eq_num @ B3 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_722_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
| ( X = Y4 ) ) ) ) ).
% order_le_less
thf(fact_723_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
| ( X = Y4 ) ) ) ) ).
% order_le_less
thf(fact_724_order__le__less,axiom,
( ord_less_eq_num
= ( ^ [X: num,Y4: num] :
( ( ord_less_num @ X @ Y4 )
| ( X = Y4 ) ) ) ) ).
% order_le_less
thf(fact_725_order__less__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
& ( X != Y4 ) ) ) ) ).
% order_less_le
thf(fact_726_order__less__le,axiom,
( ord_less_int
= ( ^ [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
& ( X != Y4 ) ) ) ) ).
% order_less_le
thf(fact_727_order__less__le,axiom,
( ord_less_num
= ( ^ [X: num,Y4: num] :
( ( ord_less_eq_num @ X @ Y4 )
& ( X != Y4 ) ) ) ) ).
% order_less_le
thf(fact_728_linorder__not__le,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y2 ) )
= ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_not_le
thf(fact_729_linorder__not__le,axiom,
! [X2: int,Y2: int] :
( ( ~ ( ord_less_eq_int @ X2 @ Y2 ) )
= ( ord_less_int @ Y2 @ X2 ) ) ).
% linorder_not_le
thf(fact_730_linorder__not__le,axiom,
! [X2: num,Y2: num] :
( ( ~ ( ord_less_eq_num @ X2 @ Y2 ) )
= ( ord_less_num @ Y2 @ X2 ) ) ).
% linorder_not_le
thf(fact_731_linorder__not__less,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_not_less
thf(fact_732_linorder__not__less,axiom,
! [X2: int,Y2: int] :
( ( ~ ( ord_less_int @ X2 @ Y2 ) )
= ( ord_less_eq_int @ Y2 @ X2 ) ) ).
% linorder_not_less
thf(fact_733_linorder__not__less,axiom,
! [X2: num,Y2: num] :
( ( ~ ( ord_less_num @ X2 @ Y2 ) )
= ( ord_less_eq_num @ Y2 @ X2 ) ) ).
% linorder_not_less
thf(fact_734_order__less__imp__le,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_735_order__less__imp__le,axiom,
! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_eq_int @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_736_order__less__imp__le,axiom,
! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( ord_less_eq_num @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_737_order__le__neq__trans,axiom,
! [A: nat,B3: nat] :
( ( ord_less_eq_nat @ A @ B3 )
=> ( ( A != B3 )
=> ( ord_less_nat @ A @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_738_order__le__neq__trans,axiom,
! [A: int,B3: int] :
( ( ord_less_eq_int @ A @ B3 )
=> ( ( A != B3 )
=> ( ord_less_int @ A @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_739_order__le__neq__trans,axiom,
! [A: num,B3: num] :
( ( ord_less_eq_num @ A @ B3 )
=> ( ( A != B3 )
=> ( ord_less_num @ A @ B3 ) ) ) ).
% order_le_neq_trans
thf(fact_740_order__neq__le__trans,axiom,
! [A: nat,B3: nat] :
( ( A != B3 )
=> ( ( ord_less_eq_nat @ A @ B3 )
=> ( ord_less_nat @ A @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_741_order__neq__le__trans,axiom,
! [A: int,B3: int] :
( ( A != B3 )
=> ( ( ord_less_eq_int @ A @ B3 )
=> ( ord_less_int @ A @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_742_order__neq__le__trans,axiom,
! [A: num,B3: num] :
( ( A != B3 )
=> ( ( ord_less_eq_num @ A @ B3 )
=> ( ord_less_num @ A @ B3 ) ) ) ).
% order_neq_le_trans
thf(fact_743_order__le__less__trans,axiom,
! [X2: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z3 )
=> ( ord_less_nat @ X2 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_744_order__le__less__trans,axiom,
! [X2: int,Y2: int,Z3: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z3 )
=> ( ord_less_int @ X2 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_745_order__le__less__trans,axiom,
! [X2: num,Y2: num,Z3: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ( ord_less_num @ Y2 @ Z3 )
=> ( ord_less_num @ X2 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_746_order__less__le__trans,axiom,
! [X2: nat,Y2: nat,Z3: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z3 )
=> ( ord_less_nat @ X2 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_747_order__less__le__trans,axiom,
! [X2: int,Y2: int,Z3: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ( ord_less_eq_int @ Y2 @ Z3 )
=> ( ord_less_int @ X2 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_748_order__less__le__trans,axiom,
! [X2: num,Y2: num,Z3: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( ( ord_less_eq_num @ Y2 @ Z3 )
=> ( ord_less_num @ X2 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_749_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B3: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_750_order__le__less__subst1,axiom,
! [A: nat,F: int > nat,B3: int,C2: int] :
( ( ord_less_eq_nat @ A @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_751_order__le__less__subst1,axiom,
! [A: nat,F: num > nat,B3: num,C2: num] :
( ( ord_less_eq_nat @ A @ ( F @ B3 ) )
=> ( ( ord_less_num @ B3 @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_752_order__le__less__subst1,axiom,
! [A: int,F: nat > int,B3: nat,C2: nat] :
( ( ord_less_eq_int @ A @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_753_order__le__less__subst1,axiom,
! [A: int,F: int > int,B3: int,C2: int] :
( ( ord_less_eq_int @ A @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_754_order__le__less__subst1,axiom,
! [A: int,F: num > int,B3: num,C2: num] :
( ( ord_less_eq_int @ A @ ( F @ B3 ) )
=> ( ( ord_less_num @ B3 @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_755_order__le__less__subst1,axiom,
! [A: num,F: nat > num,B3: nat,C2: nat] :
( ( ord_less_eq_num @ A @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_756_order__le__less__subst1,axiom,
! [A: num,F: int > num,B3: int,C2: int] :
( ( ord_less_eq_num @ A @ ( F @ B3 ) )
=> ( ( ord_less_int @ B3 @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_757_order__le__less__subst1,axiom,
! [A: num,F: num > num,B3: num,C2: num] :
( ( ord_less_eq_num @ A @ ( F @ B3 ) )
=> ( ( ord_less_num @ B3 @ C2 )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_758_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A4: nat,B5: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% nat_less_as_int
thf(fact_759_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A4: nat,B5: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_760_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_761_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B5: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% nat_leq_as_int
thf(fact_762_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_763_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B5: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_764_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K2 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_765_verit__la__generic,axiom,
! [A: int,X2: int] :
( ( ord_less_eq_int @ A @ X2 )
| ( A = X2 )
| ( ord_less_eq_int @ X2 @ A ) ) ).
% verit_la_generic
thf(fact_766_verit__la__disequality,axiom,
! [A: nat,B3: nat] :
( ( A = B3 )
| ~ ( ord_less_eq_nat @ A @ B3 )
| ~ ( ord_less_eq_nat @ B3 @ A ) ) ).
% verit_la_disequality
thf(fact_767_verit__la__disequality,axiom,
! [A: int,B3: int] :
( ( A = B3 )
| ~ ( ord_less_eq_int @ A @ B3 )
| ~ ( ord_less_eq_int @ B3 @ A ) ) ).
% verit_la_disequality
thf(fact_768_verit__la__disequality,axiom,
! [A: num,B3: num] :
( ( A = B3 )
| ~ ( ord_less_eq_num @ A @ B3 )
| ~ ( ord_less_eq_num @ B3 @ A ) ) ).
% verit_la_disequality
thf(fact_769_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_770_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_771_verit__comp__simplify1_I2_J,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_772_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_773_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_774_verit__comp__simplify1_I1_J,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_775_nat__int__comparison_I1_J,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [A4: nat,B5: nat] :
( ( semiri1314217659103216013at_int @ A4 )
= ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_776_int__if,axiom,
! [P: $o,A: nat,B3: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B3 ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B3 ) )
= ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% int_if
thf(fact_777_verit__comp__simplify1_I3_J,axiom,
! [B6: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
= ( ord_less_nat @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_778_verit__comp__simplify1_I3_J,axiom,
! [B6: int,A6: int] :
( ( ~ ( ord_less_eq_int @ B6 @ A6 ) )
= ( ord_less_int @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_779_verit__comp__simplify1_I3_J,axiom,
! [B6: num,A6: num] :
( ( ~ ( ord_less_eq_num @ B6 @ A6 ) )
= ( ord_less_num @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_780_pred__subset__eq,axiom,
! [R2: set_list_nat,S2: set_list_nat] :
( ( ord_le1520216061033275535_nat_o
@ ^ [X: list_nat] : ( member_list_nat2 @ X @ R2 )
@ ^ [X: list_nat] : ( member_list_nat2 @ X @ S2 ) )
= ( ord_le6045566169113846134st_nat @ R2 @ S2 ) ) ).
% pred_subset_eq
thf(fact_781_pred__subset__eq,axiom,
! [R2: set_se7855581050983116737at_nat,S2: set_se7855581050983116737at_nat] :
( ( ord_le5866676684937102012_nat_o
@ ^ [X: set_Pr1261947904930325089at_nat] : ( member2643936169264416010at_nat @ X @ R2 )
@ ^ [X: set_Pr1261947904930325089at_nat] : ( member2643936169264416010at_nat @ X @ S2 ) )
= ( ord_le2077887516847798113at_nat @ R2 @ S2 ) ) ).
% pred_subset_eq
thf(fact_782_imp__le__cong,axiom,
! [X2: int,X6: int,P: $o,P5: $o] :
( ( X2 = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P5 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> P5 ) ) ) ) ).
% imp_le_cong
thf(fact_783_conj__le__cong,axiom,
! [X2: int,X6: int,P: $o,P5: $o] :
( ( X2 = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P5 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
& P5 ) ) ) ) ).
% conj_le_cong
thf(fact_784_complete__interval,axiom,
! [A: nat,B3: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B3 )
=> ( ( P @ A )
=> ( ~ ( P @ B3 )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A @ C3 )
& ( ord_less_eq_nat @ C3 @ B3 )
& ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ C3 ) )
=> ( P @ X4 ) )
& ! [D: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_785_complete__interval,axiom,
! [A: int,B3: int,P: int > $o] :
( ( ord_less_int @ A @ B3 )
=> ( ( P @ A )
=> ( ~ ( P @ B3 )
=> ? [C3: int] :
( ( ord_less_eq_int @ A @ C3 )
& ( ord_less_eq_int @ C3 @ B3 )
& ! [X4: int] :
( ( ( ord_less_eq_int @ A @ X4 )
& ( ord_less_int @ X4 @ C3 ) )
=> ( P @ X4 ) )
& ! [D: int] :
( ! [X3: int] :
( ( ( ord_less_eq_int @ A @ X3 )
& ( ord_less_int @ X3 @ D ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_int @ D @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_786_pinf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ T ) ) ).
% pinf(6)
thf(fact_787_pinf_I6_J,axiom,
! [T: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ~ ( ord_less_eq_int @ X4 @ T ) ) ).
% pinf(6)
thf(fact_788_pinf_I6_J,axiom,
! [T: num] :
? [Z2: num] :
! [X4: num] :
( ( ord_less_num @ Z2 @ X4 )
=> ~ ( ord_less_eq_num @ X4 @ T ) ) ).
% pinf(6)
thf(fact_789_pinf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ord_less_eq_nat @ T @ X4 ) ) ).
% pinf(8)
thf(fact_790_pinf_I8_J,axiom,
! [T: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ( ord_less_eq_int @ T @ X4 ) ) ).
% pinf(8)
thf(fact_791_pinf_I8_J,axiom,
! [T: num] :
? [Z2: num] :
! [X4: num] :
( ( ord_less_num @ Z2 @ X4 )
=> ( ord_less_eq_num @ T @ X4 ) ) ).
% pinf(8)
thf(fact_792_minf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ~ ( ord_less_nat @ T @ X4 ) ) ).
% minf(7)
thf(fact_793_minf_I7_J,axiom,
! [T: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ~ ( ord_less_int @ T @ X4 ) ) ).
% minf(7)
thf(fact_794_minf_I7_J,axiom,
! [T: num] :
? [Z2: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z2 )
=> ~ ( ord_less_num @ T @ X4 ) ) ).
% minf(7)
thf(fact_795_minf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ord_less_nat @ X4 @ T ) ) ).
% minf(5)
thf(fact_796_minf_I5_J,axiom,
! [T: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ( ord_less_int @ X4 @ T ) ) ).
% minf(5)
thf(fact_797_minf_I5_J,axiom,
! [T: num] :
? [Z2: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z2 )
=> ( ord_less_num @ X4 @ T ) ) ).
% minf(5)
thf(fact_798_minf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( X4 != T ) ) ).
% minf(4)
thf(fact_799_minf_I4_J,axiom,
! [T: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ( X4 != T ) ) ).
% minf(4)
thf(fact_800_minf_I4_J,axiom,
! [T: num] :
? [Z2: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z2 )
=> ( X4 != T ) ) ).
% minf(4)
thf(fact_801_minf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( X4 != T ) ) ).
% minf(3)
thf(fact_802_minf_I3_J,axiom,
! [T: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ( X4 != T ) ) ).
% minf(3)
thf(fact_803_minf_I3_J,axiom,
! [T: num] :
? [Z2: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z2 )
=> ( X4 != T ) ) ).
% minf(3)
thf(fact_804_minf_I2_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P5 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_805_minf_I2_J,axiom,
! [P: int > $o,P5: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P5 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_806_minf_I2_J,axiom,
! [P: num > $o,P5: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z4: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z2 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P5 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_807_minf_I1_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P5 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_808_minf_I1_J,axiom,
! [P: int > $o,P5: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P5 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_809_minf_I1_J,axiom,
! [P: num > $o,P5: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z4: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: num] :
! [X3: num] :
( ( ord_less_num @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z2 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P5 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_810_pinf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ord_less_nat @ T @ X4 ) ) ).
% pinf(7)
thf(fact_811_pinf_I7_J,axiom,
! [T: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ( ord_less_int @ T @ X4 ) ) ).
% pinf(7)
thf(fact_812_pinf_I7_J,axiom,
! [T: num] :
? [Z2: num] :
! [X4: num] :
( ( ord_less_num @ Z2 @ X4 )
=> ( ord_less_num @ T @ X4 ) ) ).
% pinf(7)
thf(fact_813_pinf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ~ ( ord_less_nat @ X4 @ T ) ) ).
% pinf(5)
thf(fact_814_pinf_I5_J,axiom,
! [T: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ~ ( ord_less_int @ X4 @ T ) ) ).
% pinf(5)
thf(fact_815_pinf_I5_J,axiom,
! [T: num] :
? [Z2: num] :
! [X4: num] :
( ( ord_less_num @ Z2 @ X4 )
=> ~ ( ord_less_num @ X4 @ T ) ) ).
% pinf(5)
thf(fact_816_pinf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( X4 != T ) ) ).
% pinf(4)
thf(fact_817_pinf_I4_J,axiom,
! [T: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ( X4 != T ) ) ).
% pinf(4)
thf(fact_818_pinf_I4_J,axiom,
! [T: num] :
? [Z2: num] :
! [X4: num] :
( ( ord_less_num @ Z2 @ X4 )
=> ( X4 != T ) ) ).
% pinf(4)
thf(fact_819_pinf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( X4 != T ) ) ).
% pinf(3)
thf(fact_820_pinf_I3_J,axiom,
! [T: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ( X4 != T ) ) ).
% pinf(3)
thf(fact_821_pinf_I3_J,axiom,
! [T: num] :
? [Z2: num] :
! [X4: num] :
( ( ord_less_num @ Z2 @ X4 )
=> ( X4 != T ) ) ).
% pinf(3)
thf(fact_822_pinf_I2_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P5 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_823_pinf_I2_J,axiom,
! [P: int > $o,P5: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P5 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_824_pinf_I2_J,axiom,
! [P: num > $o,P5: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z4: num] :
! [X3: num] :
( ( ord_less_num @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: num] :
! [X3: num] :
( ( ord_less_num @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: num] :
! [X4: num] :
( ( ord_less_num @ Z2 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P5 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_825_pinf_I1_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z2 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P5 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_826_pinf_I1_J,axiom,
! [P: int > $o,P5: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ Z2 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P5 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_827_pinf_I1_J,axiom,
! [P: num > $o,P5: num > $o,Q: num > $o,Q2: num > $o] :
( ? [Z4: num] :
! [X3: num] :
( ( ord_less_num @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: num] :
! [X3: num] :
( ( ord_less_num @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: num] :
! [X4: num] :
( ( ord_less_num @ Z2 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P5 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_828_minf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ~ ( ord_less_eq_nat @ T @ X4 ) ) ).
% minf(8)
thf(fact_829_minf_I8_J,axiom,
! [T: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ~ ( ord_less_eq_int @ T @ X4 ) ) ).
% minf(8)
thf(fact_830_minf_I8_J,axiom,
! [T: num] :
? [Z2: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z2 )
=> ~ ( ord_less_eq_num @ T @ X4 ) ) ).
% minf(8)
thf(fact_831_minf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z2 )
=> ( ord_less_eq_nat @ X4 @ T ) ) ).
% minf(6)
thf(fact_832_minf_I6_J,axiom,
! [T: int] :
? [Z2: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z2 )
=> ( ord_less_eq_int @ X4 @ T ) ) ).
% minf(6)
thf(fact_833_minf_I6_J,axiom,
! [T: num] :
? [Z2: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z2 )
=> ( ord_less_eq_num @ X4 @ T ) ) ).
% minf(6)
thf(fact_834_arg__min__inj__eq,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,P: list_nat > $o,A: list_nat] :
( ( inj_on7522185085906380110at_nat @ F @ ( collect_list_nat @ P ) )
=> ( ( P @ A )
=> ( ! [Y3: list_nat] :
( ( P @ Y3 )
=> ( ord_le3146513528884898305at_nat @ ( F @ A ) @ ( F @ Y3 ) ) )
=> ( ( lattic5017752457218834789at_nat @ F @ P )
= A ) ) ) ) ).
% arg_min_inj_eq
thf(fact_835_arg__min__inj__eq,axiom,
! [F: nat > nat,P: nat > $o,A: nat] :
( ( inj_on_nat_nat @ F @ ( collect_nat @ P ) )
=> ( ( P @ A )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ ( F @ A ) @ ( F @ Y3 ) ) )
=> ( ( lattic8739620818006775868at_nat @ F @ P )
= A ) ) ) ) ).
% arg_min_inj_eq
thf(fact_836_arg__min__inj__eq,axiom,
! [F: list_nat > nat,P: list_nat > $o,A: list_nat] :
( ( inj_on_list_nat_nat @ F @ ( collect_list_nat @ P ) )
=> ( ( P @ A )
=> ( ! [Y3: list_nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ ( F @ A ) @ ( F @ Y3 ) ) )
=> ( ( lattic343138300456401356at_nat @ F @ P )
= A ) ) ) ) ).
% arg_min_inj_eq
thf(fact_837_arg__min__inj__eq,axiom,
! [F: list_nat > int,P: list_nat > $o,A: list_nat] :
( ( inj_on_list_nat_int @ F @ ( collect_list_nat @ P ) )
=> ( ( P @ A )
=> ( ! [Y3: list_nat] :
( ( P @ Y3 )
=> ( ord_less_eq_int @ ( F @ A ) @ ( F @ Y3 ) ) )
=> ( ( lattic340647829947351080at_int @ F @ P )
= A ) ) ) ) ).
% arg_min_inj_eq
thf(fact_838_arg__min__inj__eq,axiom,
! [F: list_nat > num,P: list_nat > $o,A: list_nat] :
( ( inj_on_list_nat_num @ F @ ( collect_list_nat @ P ) )
=> ( ( P @ A )
=> ( ! [Y3: list_nat] :
( ( P @ Y3 )
=> ( ord_less_eq_num @ ( F @ A ) @ ( F @ Y3 ) ) )
=> ( ( lattic6123842123466955798at_num @ F @ P )
= A ) ) ) ) ).
% arg_min_inj_eq
thf(fact_839_rotate1__length01,axiom,
! [Xs2: list_list_nat] :
( ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ one_one_nat )
=> ( ( rotate1_list_nat @ Xs2 )
= Xs2 ) ) ).
% rotate1_length01
thf(fact_840_rotate1__length01,axiom,
! [Xs2: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat )
=> ( ( rotate1_nat @ Xs2 )
= Xs2 ) ) ).
% rotate1_length01
thf(fact_841_rotate1__length01,axiom,
! [Xs2: list_set_nat] :
( ( ord_less_eq_nat @ ( size_s3254054031482475050et_nat @ Xs2 ) @ one_one_nat )
=> ( ( rotate1_set_nat @ Xs2 )
= Xs2 ) ) ).
% rotate1_length01
thf(fact_842_rotate1__length01,axiom,
! [Xs2: list_s1210847774152347623at_nat] :
( ( ord_less_eq_nat @ ( size_s8736152011456118867at_nat @ Xs2 ) @ one_one_nat )
=> ( ( rotate4238613965387346100at_nat @ Xs2 )
= Xs2 ) ) ).
% rotate1_length01
thf(fact_843_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top_nat
@ ^ [X: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ X )
@ ^ [X: nat,Y4: nat] : ( ord_less_nat @ Y4 @ X )
@ zero_zero_nat ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_844_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% neg_int_cases
thf(fact_845_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_846_subset__code_I3_J,axiom,
~ ( ord_le2077887516847798113at_nat @ ( coset_4613306324563743060at_nat @ nil_se357566008730718055at_nat ) @ ( set_se5049602875457034614at_nat @ nil_se357566008730718055at_nat ) ) ).
% subset_code(3)
thf(fact_847_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_848_neg__equal__iff__equal,axiom,
! [A: int,B3: int] :
( ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B3 ) )
= ( A = B3 ) ) ).
% neg_equal_iff_equal
thf(fact_849_neg__le__iff__le,axiom,
! [B3: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B3 ) ) ).
% neg_le_iff_le
thf(fact_850_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_851_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_852_neg__equal__0__iff__equal,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_853_neg__0__equal__iff__equal,axiom,
! [A: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A ) )
= ( zero_zero_int = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_854_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_855_neg__less__iff__less,axiom,
! [B3: int,A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ B3 ) ) ).
% neg_less_iff_less
thf(fact_856_inj__uminus,axiom,
! [A2: set_int] : ( inj_on_int_int @ uminus_uminus_int @ A2 ) ).
% inj_uminus
thf(fact_857_set__rotate1,axiom,
! [Xs2: list_list_nat] :
( ( set_list_nat2 @ ( rotate1_list_nat @ Xs2 ) )
= ( set_list_nat2 @ Xs2 ) ) ).
% set_rotate1
thf(fact_858_set__rotate1,axiom,
! [Xs2: list_s1210847774152347623at_nat] :
( ( set_se5049602875457034614at_nat @ ( rotate4238613965387346100at_nat @ Xs2 ) )
= ( set_se5049602875457034614at_nat @ Xs2 ) ) ).
% set_rotate1
thf(fact_859_length__rotate1,axiom,
! [Xs2: list_list_nat] :
( ( size_s3023201423986296836st_nat @ ( rotate1_list_nat @ Xs2 ) )
= ( size_s3023201423986296836st_nat @ Xs2 ) ) ).
% length_rotate1
thf(fact_860_length__rotate1,axiom,
! [Xs2: list_nat] :
( ( size_size_list_nat @ ( rotate1_nat @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_rotate1
thf(fact_861_length__rotate1,axiom,
! [Xs2: list_set_nat] :
( ( size_s3254054031482475050et_nat @ ( rotate1_set_nat @ Xs2 ) )
= ( size_s3254054031482475050et_nat @ Xs2 ) ) ).
% length_rotate1
thf(fact_862_length__rotate1,axiom,
! [Xs2: list_s1210847774152347623at_nat] :
( ( size_s8736152011456118867at_nat @ ( rotate4238613965387346100at_nat @ Xs2 ) )
= ( size_s8736152011456118867at_nat @ Xs2 ) ) ).
% length_rotate1
thf(fact_863_neg__less__eq__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_864_less__eq__neg__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_865_neg__le__0__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_le_0_iff_le
thf(fact_866_neg__0__le__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_867_neg__less__0__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_0_iff_less
thf(fact_868_neg__0__less__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_869_neg__less__pos,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_pos
thf(fact_870_less__neg__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_871_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_872_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zle
thf(fact_873_dbl__inc__simps_I4_J,axiom,
( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_inc_simps(4)
thf(fact_874_ordering__top_Oextremum,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( Less_eq @ A @ Top ) ) ).
% ordering_top.extremum
thf(fact_875_ordering__top_Oextremum__strict,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ~ ( Less @ Top @ A ) ) ).
% ordering_top.extremum_strict
thf(fact_876_ordering__top_Oextremum__unique,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A )
= ( A = Top ) ) ) ).
% ordering_top.extremum_unique
thf(fact_877_ordering__top_Onot__eq__extremum,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( A != Top )
= ( Less @ A @ Top ) ) ) ).
% ordering_top.not_eq_extremum
thf(fact_878_ordering__top_Oextremum__uniqueI,axiom,
! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
( ( ordering_top_nat @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A )
=> ( A = Top ) ) ) ).
% ordering_top.extremum_uniqueI
thf(fact_879_equation__minus__iff,axiom,
! [A: int,B3: int] :
( ( A
= ( uminus_uminus_int @ B3 ) )
= ( B3
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_880_minus__equation__iff,axiom,
! [A: int,B3: int] :
( ( ( uminus_uminus_int @ A )
= B3 )
= ( ( uminus_uminus_int @ B3 )
= A ) ) ).
% minus_equation_iff
thf(fact_881_le__minus__iff,axiom,
! [A: int,B3: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B3 ) )
= ( ord_less_eq_int @ B3 @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_882_minus__le__iff,axiom,
! [A: int,B3: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B3 )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B3 ) @ A ) ) ).
% minus_le_iff
thf(fact_883_le__imp__neg__le,axiom,
! [A: int,B3: int] :
( ( ord_less_eq_int @ A @ B3 )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_884_verit__negate__coefficient_I2_J,axiom,
! [A: int,B3: int] :
( ( ord_less_int @ A @ B3 )
=> ( ord_less_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_885_minus__less__iff,axiom,
! [A: int,B3: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B3 )
= ( ord_less_int @ ( uminus_uminus_int @ B3 ) @ A ) ) ).
% minus_less_iff
thf(fact_886_less__minus__iff,axiom,
! [A: int,B3: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ B3 ) )
= ( ord_less_int @ B3 @ ( uminus_uminus_int @ A ) ) ) ).
% less_minus_iff
thf(fact_887_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_888_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_889_int__cases2,axiom,
! [Z3: int] :
( ! [N3: nat] :
( Z3
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z3
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% int_cases2
thf(fact_890_rotate1__map,axiom,
! [F: list_nat > set_Pr1261947904930325089at_nat,Xs2: list_list_nat] :
( ( rotate4238613965387346100at_nat @ ( map_li6003994582982014139at_nat @ F @ Xs2 ) )
= ( map_li6003994582982014139at_nat @ F @ ( rotate1_list_nat @ Xs2 ) ) ) ).
% rotate1_map
thf(fact_891_le__minus__one__simps_I2_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% le_minus_one_simps(2)
thf(fact_892_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(4)
thf(fact_893_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_894_less__minus__one__simps_I2_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% less_minus_one_simps(2)
thf(fact_895_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(4)
thf(fact_896_not__int__zless__negative,axiom,
! [N: nat,M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% not_int_zless_negative
thf(fact_897_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_898_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_899_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_900_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_901_int__cases4,axiom,
! [M: int] :
( ! [N3: nat] :
( M
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( M
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_902_int__zle__neg,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_903_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_904_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% nonpos_int_cases
thf(fact_905_subset__code_I2_J,axiom,
! [A2: set_list_nat,Ys: list_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ ( coset_list_nat @ Ys ) )
= ( ! [X: list_nat] :
( ( member_list_nat2 @ X @ ( set_list_nat2 @ Ys ) )
=> ~ ( member_list_nat2 @ X @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_906_subset__code_I2_J,axiom,
! [A2: set_se7855581050983116737at_nat,Ys: list_s1210847774152347623at_nat] :
( ( ord_le2077887516847798113at_nat @ A2 @ ( coset_4613306324563743060at_nat @ Ys ) )
= ( ! [X: set_Pr1261947904930325089at_nat] :
( ( member2643936169264416010at_nat @ X @ ( set_se5049602875457034614at_nat @ Ys ) )
=> ~ ( member2643936169264416010at_nat @ X @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_907_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% int_cases3
thf(fact_908_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_909_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3811975205180677377ec_int @ one_one_int )
= one_one_int ) ).
% dbl_dec_simps(3)
thf(fact_910_compl__coset,axiom,
! [Xs2: list_list_nat] :
( ( uminus3195874150345416415st_nat @ ( coset_list_nat @ Xs2 ) )
= ( set_list_nat2 @ Xs2 ) ) ).
% compl_coset
thf(fact_911_compl__coset,axiom,
! [Xs2: list_s1210847774152347623at_nat] :
( ( uminus8018730102153342520at_nat @ ( coset_4613306324563743060at_nat @ Xs2 ) )
= ( set_se5049602875457034614at_nat @ Xs2 ) ) ).
% compl_coset
thf(fact_912_coset__def,axiom,
( coset_list_nat
= ( ^ [Xs: list_list_nat] : ( uminus3195874150345416415st_nat @ ( set_list_nat2 @ Xs ) ) ) ) ).
% coset_def
thf(fact_913_coset__def,axiom,
( coset_4613306324563743060at_nat
= ( ^ [Xs: list_s1210847774152347623at_nat] : ( uminus8018730102153342520at_nat @ ( set_se5049602875457034614at_nat @ Xs ) ) ) ) ).
% coset_def
thf(fact_914_sgn__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( sgn_sgn_int @ A )
= ( uminus_uminus_int @ one_one_int ) ) ) ).
% sgn_neg
thf(fact_915_rotate__length01,axiom,
! [Xs2: list_list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ one_one_nat )
=> ( ( rotate_list_nat @ N @ Xs2 )
= Xs2 ) ) ).
% rotate_length01
thf(fact_916_rotate__length01,axiom,
! [Xs2: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat )
=> ( ( rotate_nat @ N @ Xs2 )
= Xs2 ) ) ).
% rotate_length01
thf(fact_917_rotate__length01,axiom,
! [Xs2: list_set_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_s3254054031482475050et_nat @ Xs2 ) @ one_one_nat )
=> ( ( rotate_set_nat @ N @ Xs2 )
= Xs2 ) ) ).
% rotate_length01
thf(fact_918_rotate__length01,axiom,
! [Xs2: list_s1210847774152347623at_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_s8736152011456118867at_nat @ Xs2 ) @ one_one_nat )
=> ( ( rotate1033626827900196251at_nat @ N @ Xs2 )
= Xs2 ) ) ).
% rotate_length01
thf(fact_919_dbl__dec__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus_uminus_int @ ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_920_dbl__inc__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus_uminus_int @ ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_921_nat__less__iff,axiom,
! [W: int,M: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ M )
= ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% nat_less_iff
thf(fact_922_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_int @ M )
= ( numeral_numeral_int @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_923_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_924_int__eq__iff__numeral,axiom,
! [M: nat,V: num] :
( ( ( semiri1314217659103216013at_int @ M )
= ( numeral_numeral_int @ V ) )
= ( M
= ( numeral_numeral_nat @ V ) ) ) ).
% int_eq_iff_numeral
thf(fact_925_nat__numeral,axiom,
! [K: num] :
( ( nat2 @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_nat @ K ) ) ).
% nat_numeral
thf(fact_926_sgn__sgn,axiom,
! [A: int] :
( ( sgn_sgn_int @ ( sgn_sgn_int @ A ) )
= ( sgn_sgn_int @ A ) ) ).
% sgn_sgn
thf(fact_927_ComplI,axiom,
! [C2: list_nat,A2: set_list_nat] :
( ~ ( member_list_nat2 @ C2 @ A2 )
=> ( member_list_nat2 @ C2 @ ( uminus3195874150345416415st_nat @ A2 ) ) ) ).
% ComplI
thf(fact_928_ComplI,axiom,
! [C2: set_Pr1261947904930325089at_nat,A2: set_se7855581050983116737at_nat] :
( ~ ( member2643936169264416010at_nat @ C2 @ A2 )
=> ( member2643936169264416010at_nat @ C2 @ ( uminus8018730102153342520at_nat @ A2 ) ) ) ).
% ComplI
thf(fact_929_Compl__iff,axiom,
! [C2: list_nat,A2: set_list_nat] :
( ( member_list_nat2 @ C2 @ ( uminus3195874150345416415st_nat @ A2 ) )
= ( ~ ( member_list_nat2 @ C2 @ A2 ) ) ) ).
% Compl_iff
thf(fact_930_Compl__iff,axiom,
! [C2: set_Pr1261947904930325089at_nat,A2: set_se7855581050983116737at_nat] :
( ( member2643936169264416010at_nat @ C2 @ ( uminus8018730102153342520at_nat @ A2 ) )
= ( ~ ( member2643936169264416010at_nat @ C2 @ A2 ) ) ) ).
% Compl_iff
thf(fact_931_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_932_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_933_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_934_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_935_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( M = N ) ) ).
% neg_numeral_eq_iff
thf(fact_936_of__nat__numeral,axiom,
! [N: num] :
( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ N ) ) ).
% of_nat_numeral
thf(fact_937_of__nat__numeral,axiom,
! [N: num] :
( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_int @ N ) ) ).
% of_nat_numeral
thf(fact_938_sgn__0,axiom,
( ( sgn_sgn_int @ zero_zero_int )
= zero_zero_int ) ).
% sgn_0
thf(fact_939_sgn__1,axiom,
( ( sgn_sgn_int @ one_one_int )
= one_one_int ) ).
% sgn_1
thf(fact_940_sgn__minus,axiom,
! [A: int] :
( ( sgn_sgn_int @ ( uminus_uminus_int @ A ) )
= ( uminus_uminus_int @ ( sgn_sgn_int @ A ) ) ) ).
% sgn_minus
thf(fact_941_nat__int,axiom,
! [N: nat] :
( ( nat2 @ ( semiri1314217659103216013at_int @ N ) )
= N ) ).
% nat_int
thf(fact_942_set__rotate,axiom,
! [N: nat,Xs2: list_list_nat] :
( ( set_list_nat2 @ ( rotate_list_nat @ N @ Xs2 ) )
= ( set_list_nat2 @ Xs2 ) ) ).
% set_rotate
thf(fact_943_set__rotate,axiom,
! [N: nat,Xs2: list_s1210847774152347623at_nat] :
( ( set_se5049602875457034614at_nat @ ( rotate1033626827900196251at_nat @ N @ Xs2 ) )
= ( set_se5049602875457034614at_nat @ Xs2 ) ) ).
% set_rotate
thf(fact_944_length__rotate,axiom,
! [N: nat,Xs2: list_list_nat] :
( ( size_s3023201423986296836st_nat @ ( rotate_list_nat @ N @ Xs2 ) )
= ( size_s3023201423986296836st_nat @ Xs2 ) ) ).
% length_rotate
thf(fact_945_length__rotate,axiom,
! [N: nat,Xs2: list_nat] :
( ( size_size_list_nat @ ( rotate_nat @ N @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_rotate
thf(fact_946_length__rotate,axiom,
! [N: nat,Xs2: list_set_nat] :
( ( size_s3254054031482475050et_nat @ ( rotate_set_nat @ N @ Xs2 ) )
= ( size_s3254054031482475050et_nat @ Xs2 ) ) ).
% length_rotate
thf(fact_947_length__rotate,axiom,
! [N: nat,Xs2: list_s1210847774152347623at_nat] :
( ( size_s8736152011456118867at_nat @ ( rotate1033626827900196251at_nat @ N @ Xs2 ) )
= ( size_s8736152011456118867at_nat @ Xs2 ) ) ).
% length_rotate
thf(fact_948_neg__numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( ord_less_eq_num @ N @ M ) ) ).
% neg_numeral_le_iff
thf(fact_949_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( ord_less_num @ N @ M ) ) ).
% neg_numeral_less_iff
thf(fact_950_sgn__greater,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( sgn_sgn_int @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% sgn_greater
thf(fact_951_sgn__less,axiom,
! [A: int] :
( ( ord_less_int @ ( sgn_sgn_int @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% sgn_less
thf(fact_952_nat__neg__numeral,axiom,
! [K: num] :
( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= zero_zero_nat ) ).
% nat_neg_numeral
thf(fact_953_sgn__pos,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( sgn_sgn_int @ A )
= one_one_int ) ) ).
% sgn_pos
thf(fact_954_nat__le__0,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ Z3 @ zero_zero_int )
=> ( ( nat2 @ Z3 )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_955_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= zero_zero_nat )
= ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_956_zless__nat__conj,axiom,
! [W: int,Z3: int] :
( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z3 ) )
= ( ( ord_less_int @ zero_zero_int @ Z3 )
& ( ord_less_int @ W @ Z3 ) ) ) ).
% zless_nat_conj
thf(fact_957_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_958_int__nat__eq,axiom,
! [Z3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
= Z3 ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_959_zero__less__nat__eq,axiom,
! [Z3: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z3 ) )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% zero_less_nat_eq
thf(fact_960_ComplD,axiom,
! [C2: list_nat,A2: set_list_nat] :
( ( member_list_nat2 @ C2 @ ( uminus3195874150345416415st_nat @ A2 ) )
=> ~ ( member_list_nat2 @ C2 @ A2 ) ) ).
% ComplD
thf(fact_961_ComplD,axiom,
! [C2: set_Pr1261947904930325089at_nat,A2: set_se7855581050983116737at_nat] :
( ( member2643936169264416010at_nat @ C2 @ ( uminus8018730102153342520at_nat @ A2 ) )
=> ~ ( member2643936169264416010at_nat @ C2 @ A2 ) ) ).
% ComplD
thf(fact_962_uminus__set__def,axiom,
( uminus8018730102153342520at_nat
= ( ^ [A3: set_se7855581050983116737at_nat] :
( collec5514110066124741708at_nat
@ ( uminus5680096308674742949_nat_o
@ ^ [X: set_Pr1261947904930325089at_nat] : ( member2643936169264416010at_nat @ X @ A3 ) ) ) ) ) ).
% uminus_set_def
thf(fact_963_uminus__set__def,axiom,
( uminus3195874150345416415st_nat
= ( ^ [A3: set_list_nat] :
( collect_list_nat
@ ( uminus5770388063884162150_nat_o
@ ^ [X: list_nat] : ( member_list_nat2 @ X @ A3 ) ) ) ) ) ).
% uminus_set_def
thf(fact_964_Compl__eq,axiom,
( uminus8018730102153342520at_nat
= ( ^ [A3: set_se7855581050983116737at_nat] :
( collec5514110066124741708at_nat
@ ^ [X: set_Pr1261947904930325089at_nat] :
~ ( member2643936169264416010at_nat @ X @ A3 ) ) ) ) ).
% Compl_eq
thf(fact_965_Compl__eq,axiom,
( uminus3195874150345416415st_nat
= ( ^ [A3: set_list_nat] :
( collect_list_nat
@ ^ [X: list_nat] :
~ ( member_list_nat2 @ X @ A3 ) ) ) ) ).
% Compl_eq
thf(fact_966_Collect__neg__eq,axiom,
! [P: list_nat > $o] :
( ( collect_list_nat
@ ^ [X: list_nat] :
~ ( P @ X ) )
= ( uminus3195874150345416415st_nat @ ( collect_list_nat @ P ) ) ) ).
% Collect_neg_eq
thf(fact_967_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( numeral_numeral_int @ N ) ) ).
% zero_neq_numeral
thf(fact_968_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_969_sgn__0__0,axiom,
! [A: int] :
( ( ( sgn_sgn_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% sgn_0_0
thf(fact_970_sgn__eq__0__iff,axiom,
! [A: int] :
( ( ( sgn_sgn_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% sgn_eq_0_iff
thf(fact_971_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] :
( ( numeral_numeral_int @ M )
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_972_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] :
( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
!= ( numeral_numeral_int @ N ) ) ).
% neg_numeral_neq_numeral
thf(fact_973_int__ops_I3_J,axiom,
! [N: num] :
( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_int @ N ) ) ).
% int_ops(3)
thf(fact_974_rotate__map,axiom,
! [N: nat,F: list_nat > set_Pr1261947904930325089at_nat,Xs2: list_list_nat] :
( ( rotate1033626827900196251at_nat @ N @ ( map_li6003994582982014139at_nat @ F @ Xs2 ) )
= ( map_li6003994582982014139at_nat @ F @ ( rotate_list_nat @ N @ Xs2 ) ) ) ).
% rotate_map
thf(fact_975_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_976_nat__mono,axiom,
! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y2 ) ) ) ).
% nat_mono
thf(fact_977_ex__nat,axiom,
( ( ^ [P4: nat > $o] :
? [X5: nat] : ( P4 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
& ( P3 @ ( nat2 @ X ) ) ) ) ) ).
% ex_nat
thf(fact_978_all__nat,axiom,
( ( ^ [P4: nat > $o] :
! [X5: nat] : ( P4 @ X5 ) )
= ( ^ [P3: nat > $o] :
! [X: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( P3 @ ( nat2 @ X ) ) ) ) ) ).
% all_nat
thf(fact_979_eq__nat__nat__iff,axiom,
! [Z3: int,Z5: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( ( nat2 @ Z3 )
= ( nat2 @ Z5 ) )
= ( Z3 = Z5 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_980_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_981_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_le_zero
thf(fact_982_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_le_numeral
thf(fact_983_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_le_numeral
thf(fact_984_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_985_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_less_zero
thf(fact_986_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_987_zero__less__numeral,axiom,
! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_less_numeral
thf(fact_988_zero__less__numeral,axiom,
! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_less_numeral
thf(fact_989_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% one_le_numeral
thf(fact_990_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% one_le_numeral
thf(fact_991_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% not_numeral_less_one
thf(fact_992_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_993_not__numeral__le__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_994_neg__numeral__le__numeral,axiom,
! [M: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% neg_numeral_le_numeral
thf(fact_995_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_996_neg__numeral__less__numeral,axiom,
! [M: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% neg_numeral_less_numeral
thf(fact_997_not__numeral__less__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_998_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numeral_numeral_int @ N )
!= ( uminus_uminus_int @ one_one_int ) ) ).
% numeral_neq_neg_one
thf(fact_999_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_1000_sgn__not__eq__imp,axiom,
! [B3: int,A: int] :
( ( ( sgn_sgn_int @ B3 )
!= ( sgn_sgn_int @ A ) )
=> ( ( ( sgn_sgn_int @ A )
!= zero_zero_int )
=> ( ( ( sgn_sgn_int @ B3 )
!= zero_zero_int )
=> ( ( sgn_sgn_int @ A )
= ( uminus_uminus_int @ ( sgn_sgn_int @ B3 ) ) ) ) ) ) ).
% sgn_not_eq_imp
thf(fact_1001_sgn__minus__1,axiom,
( ( sgn_sgn_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% sgn_minus_1
thf(fact_1002_nat__mono__iff,axiom,
! [Z3: int,W: int] :
( ( ord_less_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z3 ) )
= ( ord_less_int @ W @ Z3 ) ) ) ).
% nat_mono_iff
thf(fact_1003_nat__le__iff,axiom,
! [X2: int,N: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X2 ) @ N )
= ( ord_less_eq_int @ X2 @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% nat_le_iff
thf(fact_1004_zless__nat__eq__int__zless,axiom,
! [M: nat,Z3: int] :
( ( ord_less_nat @ M @ ( nat2 @ Z3 ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z3 ) ) ).
% zless_nat_eq_int_zless
thf(fact_1005_nat__0__le,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
= Z3 ) ) ).
% nat_0_le
thf(fact_1006_int__eq__iff,axiom,
! [M: nat,Z3: int] :
( ( ( semiri1314217659103216013at_int @ M )
= Z3 )
= ( ( M
= ( nat2 @ Z3 ) )
& ( ord_less_eq_int @ zero_zero_int @ Z3 ) ) ) ).
% int_eq_iff
thf(fact_1007_neg__numeral__le__zero,axiom,
! [N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% neg_numeral_le_zero
thf(fact_1008_not__zero__le__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_1009_neg__numeral__less__zero,axiom,
! [N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% neg_numeral_less_zero
thf(fact_1010_not__zero__less__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_1011_neg__numeral__le__one,axiom,
! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% neg_numeral_le_one
thf(fact_1012_neg__one__le__numeral,axiom,
! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% neg_one_le_numeral
thf(fact_1013_neg__numeral__le__neg__one,axiom,
! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% neg_numeral_le_neg_one
thf(fact_1014_not__numeral__le__neg__one,axiom,
! [M: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_numeral_le_neg_one
thf(fact_1015_not__one__le__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% not_one_le_neg_numeral
thf(fact_1016_neg__numeral__less__one,axiom,
! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% neg_numeral_less_one
thf(fact_1017_neg__one__less__numeral,axiom,
! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% neg_one_less_numeral
thf(fact_1018_not__numeral__less__neg__one,axiom,
! [M: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_numeral_less_neg_one
thf(fact_1019_not__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% not_one_less_neg_numeral
thf(fact_1020_not__neg__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_1021_sgn__1__pos,axiom,
! [A: int] :
( ( ( sgn_sgn_int @ A )
= one_one_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% sgn_1_pos
thf(fact_1022_zsgn__def,axiom,
( sgn_sgn_int
= ( ^ [I4: int] : ( if_int @ ( I4 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I4 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zsgn_def
thf(fact_1023_nat__eq__iff,axiom,
! [W: int,M: nat] :
( ( ( nat2 @ W )
= M )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_1024_nat__eq__iff2,axiom,
! [M: nat,W: int] :
( ( M
= ( nat2 @ W ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_1025_nat__less__eq__zless,axiom,
! [W: int,Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z3 ) )
= ( ord_less_int @ W @ Z3 ) ) ) ).
% nat_less_eq_zless
thf(fact_1026_split__nat,axiom,
! [P: nat > $o,I: int] :
( ( P @ ( nat2 @ I ) )
= ( ! [N2: nat] :
( ( I
= ( semiri1314217659103216013at_int @ N2 ) )
=> ( P @ N2 ) )
& ( ( ord_less_int @ I @ zero_zero_int )
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_1027_nat__le__eq__zle,axiom,
! [W: int,Z3: int] :
( ( ( ord_less_int @ zero_zero_int @ W )
| ( ord_less_eq_int @ zero_zero_int @ Z3 ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z3 ) )
= ( ord_less_eq_int @ W @ Z3 ) ) ) ).
% nat_le_eq_zle
thf(fact_1028_le__nat__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% le_nat_iff
thf(fact_1029_sgn__1__neg,axiom,
! [A: int] :
( ( ( sgn_sgn_int @ A )
= ( uminus_uminus_int @ one_one_int ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% sgn_1_neg
thf(fact_1030_sgn__if,axiom,
( sgn_sgn_int
= ( ^ [X: int] : ( if_int @ ( X = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ X ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% sgn_if
thf(fact_1031_one__less__nat__eq,axiom,
! [Z3: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z3 ) )
= ( ord_less_int @ one_one_int @ Z3 ) ) ).
% one_less_nat_eq
thf(fact_1032_dbl__simps_I1_J,axiom,
! [K: num] :
( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
= ( uminus_uminus_int @ ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% dbl_simps(1)
thf(fact_1033_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_1034_numeral__num__of__nat__unfold,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
= ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% numeral_num_of_nat_unfold
thf(fact_1035_numeral__num__of__nat__unfold,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( numeral_numeral_int @ ( num_of_nat @ N ) )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( numeral_numeral_int @ ( num_of_nat @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% numeral_num_of_nat_unfold
thf(fact_1036_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1037_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_1038_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_1039_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1040_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1041_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_1042_num__of__nat__numeral__eq,axiom,
! [Q3: num] :
( ( num_of_nat @ ( numeral_numeral_nat @ Q3 ) )
= Q3 ) ).
% num_of_nat_numeral_eq
thf(fact_1043_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_int @ zero_zero_int )
= zero_zero_int ) ).
% dbl_simps(2)
thf(fact_1044_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1045_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1046_of__int__eq__0__iff,axiom,
! [Z3: int] :
( ( ( ring_1_of_int_int @ Z3 )
= zero_zero_int )
= ( Z3 = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_1047_of__int__0__eq__iff,axiom,
! [Z3: int] :
( ( zero_zero_int
= ( ring_1_of_int_int @ Z3 ) )
= ( Z3 = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_1048_of__int__0,axiom,
( ( ring_1_of_int_int @ zero_zero_int )
= zero_zero_int ) ).
% of_int_0
thf(fact_1049_of__int__le__iff,axiom,
! [W: int,Z3: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z3 ) )
= ( ord_less_eq_int @ W @ Z3 ) ) ).
% of_int_le_iff
thf(fact_1050_of__int__numeral,axiom,
! [K: num] :
( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
= ( numeral_numeral_int @ K ) ) ).
% of_int_numeral
thf(fact_1051_of__int__eq__numeral__iff,axiom,
! [Z3: int,N: num] :
( ( ( ring_1_of_int_int @ Z3 )
= ( numeral_numeral_int @ N ) )
= ( Z3
= ( numeral_numeral_int @ N ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_1052_of__int__less__iff,axiom,
! [W: int,Z3: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z3 ) )
= ( ord_less_int @ W @ Z3 ) ) ).
% of_int_less_iff
thf(fact_1053_of__int__1,axiom,
( ( ring_1_of_int_int @ one_one_int )
= one_one_int ) ).
% of_int_1
thf(fact_1054_of__int__eq__1__iff,axiom,
! [Z3: int] :
( ( ( ring_1_of_int_int @ Z3 )
= one_one_int )
= ( Z3 = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_1055_of__int__minus,axiom,
! [Z3: int] :
( ( ring_1_of_int_int @ ( uminus_uminus_int @ Z3 ) )
= ( uminus_uminus_int @ ( ring_1_of_int_int @ Z3 ) ) ) ).
% of_int_minus
thf(fact_1056_of__int__of__nat__eq,axiom,
! [N: nat] :
( ( ring_1_of_int_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% of_int_of_nat_eq
thf(fact_1057_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_1058_negative__zless,axiom,
! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zless
thf(fact_1059_of__int__0__le__iff,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z3 ) )
= ( ord_less_eq_int @ zero_zero_int @ Z3 ) ) ).
% of_int_0_le_iff
thf(fact_1060_of__int__le__0__iff,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z3 ) @ zero_zero_int )
= ( ord_less_eq_int @ Z3 @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_1061_of__int__numeral__le__iff,axiom,
! [N: num,Z3: int] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z3 ) )
= ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z3 ) ) ).
% of_int_numeral_le_iff
thf(fact_1062_of__int__le__numeral__iff,axiom,
! [Z3: int,N: num] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z3 ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_eq_int @ Z3 @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_le_numeral_iff
thf(fact_1063_of__int__0__less__iff,axiom,
! [Z3: int] :
( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z3 ) )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% of_int_0_less_iff
thf(fact_1064_of__int__less__0__iff,axiom,
! [Z3: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z3 ) @ zero_zero_int )
= ( ord_less_int @ Z3 @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_1065_of__int__numeral__less__iff,axiom,
! [N: num,Z3: int] :
( ( ord_less_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z3 ) )
= ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z3 ) ) ).
% of_int_numeral_less_iff
thf(fact_1066_of__int__less__numeral__iff,axiom,
! [Z3: int,N: num] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z3 ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_int @ Z3 @ ( numeral_numeral_int @ N ) ) ) ).
% of_int_less_numeral_iff
thf(fact_1067_of__int__1__le__iff,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z3 ) )
= ( ord_less_eq_int @ one_one_int @ Z3 ) ) ).
% of_int_1_le_iff
thf(fact_1068_of__int__le__1__iff,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z3 ) @ one_one_int )
= ( ord_less_eq_int @ Z3 @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_1069_of__int__1__less__iff,axiom,
! [Z3: int] :
( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z3 ) )
= ( ord_less_int @ one_one_int @ Z3 ) ) ).
% of_int_1_less_iff
thf(fact_1070_of__int__less__1__iff,axiom,
! [Z3: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z3 ) @ one_one_int )
= ( ord_less_int @ Z3 @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_1071_of__nat__nat,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
= ( ring_1_of_int_int @ Z3 ) ) ) ).
% of_nat_nat
thf(fact_1072_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1073_Suc__inject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
=> ( X2 = Y2 ) ) ).
% Suc_inject
thf(fact_1074_inj__Suc,axiom,
! [N4: set_nat] : ( inj_on_nat_nat @ suc @ N4 ) ).
% inj_Suc
thf(fact_1075_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R2 @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z2: nat] :
( ( R2 @ X3 @ Y3 )
=> ( ( R2 @ Y3 @ Z2 )
=> ( R2 @ X3 @ Z2 ) ) )
=> ( ! [N3: nat] : ( R2 @ N3 @ ( suc @ N3 ) )
=> ( R2 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1076_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1077_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1078_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_1079_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1080_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1081_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M4: nat] :
( M6
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_1082_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1083_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1084_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1085_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J: nat] :
( ( ord_less_nat @ I @ J )
=> ( K
!= ( suc @ J ) ) ) ) ) ).
% Nat.lessE
thf(fact_1086_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_1087_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J: nat] :
( ( ord_less_nat @ I @ J )
=> ( K
!= ( suc @ J ) ) ) ) ).
% Suc_lessE
thf(fact_1088_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_1089_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1090_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1091_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ N )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1092_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_1093_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1094_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_1095_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M7: nat] :
( ( M
= ( suc @ M7 ) )
& ( ord_less_nat @ N @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1096_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_1097_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_1098_less__trans__Suc,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1099_less__Suc__induct,axiom,
! [I: nat,J2: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J2 )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ K2 )
=> ( ( P @ I2 @ J )
=> ( ( P @ J @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J2 ) ) ) ) ).
% less_Suc_induct
thf(fact_1100_strict__inc__induct,axiom,
! [I: nat,J2: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J2 )
=> ( ! [I2: nat] :
( ( J2
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_1101_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1102_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1103_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1104_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1105_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1106_old_Onat_Oexhaust,axiom,
! [Y2: nat] :
( ( Y2 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y2
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1107_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_1108_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1109_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1110_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1111_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1112_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1113_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_1114_enum__rgfs_Ocases,axiom,
! [X2: nat] :
( ( X2 != zero_zero_nat )
=> ~ ! [N3: nat] :
( X2
!= ( suc @ N3 ) ) ) ).
% enum_rgfs.cases
thf(fact_1115_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1116_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_1117_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1118_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1119_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1120_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1121_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1122_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_1123_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1124_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_1125_inc__induct,axiom,
! [I: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( P @ J2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J2 )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1126_dec__induct,axiom,
! [I: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J2 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J2 ) ) ) ) ).
% dec_induct
thf(fact_1127_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_1128_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1129_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1130_int__cases,axiom,
! [Z3: int] :
( ! [N3: nat] :
( Z3
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z3
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% int_cases
thf(fact_1131_int__of__nat__induct,axiom,
! [P: int > $o,Z3: int] :
( ! [N3: nat] : ( P @ ( semiri1314217659103216013at_int @ N3 ) )
=> ( ! [N3: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) )
=> ( P @ Z3 ) ) ) ).
% int_of_nat_induct
thf(fact_1132_numeral__num__of__nat,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
= N ) ) ).
% numeral_num_of_nat
thf(fact_1133_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1134_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1135_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_1136_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_1137_negD,axiom,
! [X2: int] :
( ( ord_less_int @ X2 @ zero_zero_int )
=> ? [N3: nat] :
( X2
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% negD
thf(fact_1138_le__num__One__iff,axiom,
! [X2: num] :
( ( ord_less_eq_num @ X2 @ one )
= ( X2 = one ) ) ).
% le_num_One_iff
thf(fact_1139_num_Osize_I4_J,axiom,
( ( size_size_num @ one )
= zero_zero_nat ) ).
% num.size(4)
thf(fact_1140_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_1141_num__of__nat_Osimps_I1_J,axiom,
( ( num_of_nat @ zero_zero_nat )
= one ) ).
% num_of_nat.simps(1)
thf(fact_1142_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral_nat @ one )
= ( suc @ zero_zero_nat ) ) ).
% numeral_1_eq_Suc_0
thf(fact_1143_num__of__nat__One,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ one_one_nat )
=> ( ( num_of_nat @ N )
= one ) ) ).
% num_of_nat_One
thf(fact_1144_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less_num @ M @ one ) ).
% semiring_norm(75)
thf(fact_1145_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% semiring_norm(68)
thf(fact_1146_length__stirling__row,axiom,
! [N: nat] :
( ( size_size_list_nat @ ( stirling_row @ N ) )
= ( suc @ N ) ) ).
% length_stirling_row
thf(fact_1147_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_1148_semiring__norm_I71_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(71)
thf(fact_1149_semiring__norm_I78_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(78)
thf(fact_1150_semiring__norm_I73_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(73)
thf(fact_1151_semiring__norm_I80_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(80)
thf(fact_1152_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% semiring_norm(69)
thf(fact_1153_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_1154_semiring__norm_I72_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(72)
thf(fact_1155_semiring__norm_I81_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(81)
thf(fact_1156_semiring__norm_I70_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% semiring_norm(70)
thf(fact_1157_semiring__norm_I77_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).
% semiring_norm(77)
thf(fact_1158_semiring__norm_I79_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(79)
thf(fact_1159_semiring__norm_I74_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(74)
thf(fact_1160_Suc__1,axiom,
( ( suc @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% Suc_1
thf(fact_1161_eval__nat__numeral_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit1 @ N ) )
= ( suc @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ) ).
% eval_nat_numeral(3)
thf(fact_1162_num_Oexhaust,axiom,
! [Y2: num] :
( ( Y2 != one )
=> ( ! [X23: num] :
( Y2
!= ( bit0 @ X23 ) )
=> ~ ! [X32: num] :
( Y2
!= ( bit1 @ X32 ) ) ) ) ).
% num.exhaust
thf(fact_1163_numeral__3__eq__3,axiom,
( ( numeral_numeral_nat @ ( bit1 @ one ) )
= ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% numeral_3_eq_3
thf(fact_1164_numeral__2__eq__2,axiom,
( ( numeral_numeral_nat @ ( bit0 @ one ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% numeral_2_eq_2
thf(fact_1165_less__2__cases__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ( N = zero_zero_nat )
| ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases_iff
thf(fact_1166_less__2__cases,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( ( N = zero_zero_nat )
| ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases
thf(fact_1167_nat__2,axiom,
( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% nat_2
thf(fact_1168_num__of__nat_Osimps_I2_J,axiom,
! [N: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( num_of_nat @ ( suc @ N ) )
= ( inc @ ( num_of_nat @ N ) ) ) )
& ( ~ ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( num_of_nat @ ( suc @ N ) )
= one ) ) ) ).
% num_of_nat.simps(2)
thf(fact_1169_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_1170_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1171_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1172_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1173_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1174_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1175_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% Suc_numeral
thf(fact_1176_zle__add1__eq__le,axiom,
! [W: int,Z3: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z3 @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z3 ) ) ).
% zle_add1_eq_le
thf(fact_1177_add__2__eq__Suc,axiom,
! [N: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc
thf(fact_1178_add__2__eq__Suc_H,axiom,
! [N: nat] :
( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc'
thf(fact_1179_Suc__nat__number__of__add,axiom,
! [V: num,N: nat] :
( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N ) ) ).
% Suc_nat_number_of_add
thf(fact_1180_add__One,axiom,
! [X2: num] :
( ( plus_plus_num @ X2 @ one )
= ( inc @ X2 ) ) ).
% add_One
thf(fact_1181_num__induct,axiom,
! [P: num > $o,X2: num] :
( ( P @ one )
=> ( ! [X3: num] :
( ( P @ X3 )
=> ( P @ ( inc @ X3 ) ) )
=> ( P @ X2 ) ) ) ).
% num_induct
thf(fact_1182_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_1183_nat__int__add,axiom,
! [A: nat,B3: nat] :
( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B3 ) ) )
= ( plus_plus_nat @ A @ B3 ) ) ).
% nat_int_add
thf(fact_1184_nat__plus__as__int,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B5: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ) ).
% nat_plus_as_int
thf(fact_1185_int__plus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% int_plus
thf(fact_1186_int__ops_I5_J,axiom,
! [A: nat,B3: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B3 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ).
% int_ops(5)
thf(fact_1187_add__inc,axiom,
! [X2: num,Y2: num] :
( ( plus_plus_num @ X2 @ ( inc @ Y2 ) )
= ( inc @ ( plus_plus_num @ X2 @ Y2 ) ) ) ).
% add_inc
thf(fact_1188_zadd__int__left,axiom,
! [M: nat,N: nat,Z3: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z3 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z3 ) ) ).
% zadd_int_left
thf(fact_1189_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_1190_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1191_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1192_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1193_nat__add__distrib,axiom,
! [Z3: int,Z5: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( nat2 @ ( plus_plus_int @ Z3 @ Z5 ) )
= ( plus_plus_nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z5 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_1194_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_1195_trans__less__add2,axiom,
! [I: nat,J2: nat,M: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_less_add2
thf(fact_1196_trans__less__add1,axiom,
! [I: nat,J2: nat,M: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_less_add1
thf(fact_1197_add__less__mono1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_less_mono1
thf(fact_1198_not__add__less2,axiom,
! [J2: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I ) @ I ) ).
% not_add_less2
thf(fact_1199_not__add__less1,axiom,
! [I: nat,J2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J2 ) @ I ) ).
% not_add_less1
thf(fact_1200_add__less__mono,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_less_mono
thf(fact_1201_add__lessD1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J2 ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1202_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1203_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1204_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1205_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1206_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1207_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1208_add__le__mono,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_le_mono
thf(fact_1209_add__le__mono1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_le_mono1
thf(fact_1210_trans__le__add1,axiom,
! [I: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_le_add1
thf(fact_1211_trans__le__add2,axiom,
! [I: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_le_add2
thf(fact_1212_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus_nat @ M5 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1213_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1214_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1215_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1216_num__of__nat__plus__distrib,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( num_of_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N ) ) ) ) ) ).
% num_of_nat_plus_distrib
thf(fact_1217_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1218_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1219_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1220_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M5: nat,N2: nat] :
? [K3: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M5 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1221_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1222_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1223_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q4: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q4 ) ) ) ) ).
% less_natE
thf(fact_1224_less__imp__add__positive,axiom,
! [I: nat,J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J2 ) ) ) ).
% less_imp_add_positive
thf(fact_1225_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1226_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1227_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1228_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1229_odd__nonzero,axiom,
! [Z3: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z3 ) @ Z3 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1230_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1231_zless__add1__eq,axiom,
! [W: int,Z3: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z3 @ one_one_int ) )
= ( ( ord_less_int @ W @ Z3 )
| ( W = Z3 ) ) ) ).
% zless_add1_eq
thf(fact_1232_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_1233_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z6: int] :
? [N2: nat] :
( Z6
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1234_inc_Osimps_I1_J,axiom,
( ( inc @ one )
= ( bit0 @ one ) ) ).
% inc.simps(1)
thf(fact_1235_inc_Osimps_I2_J,axiom,
! [X2: num] :
( ( inc @ ( bit0 @ X2 ) )
= ( bit1 @ X2 ) ) ).
% inc.simps(2)
thf(fact_1236_inc_Osimps_I3_J,axiom,
! [X2: num] :
( ( inc @ ( bit1 @ X2 ) )
= ( bit0 @ ( inc @ X2 ) ) ) ).
% inc.simps(3)
thf(fact_1237_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_1238_int__ops_I4_J,axiom,
! [A: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_1239_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W2: int,Z6: int] :
? [N2: nat] :
( Z6
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1240_odd__less__0__iff,axiom,
! [Z3: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z3 ) @ Z3 ) @ zero_zero_int )
= ( ord_less_int @ Z3 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1241_zless__imp__add1__zle,axiom,
! [W: int,Z3: int] :
( ( ord_less_int @ W @ Z3 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z3 ) ) ).
% zless_imp_add1_zle
thf(fact_1242_add1__zle__eq,axiom,
! [W: int,Z3: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z3 )
= ( ord_less_int @ W @ Z3 ) ) ).
% add1_zle_eq
thf(fact_1243_Suc3__eq__add__3,axiom,
! [N: nat] :
( ( suc @ ( suc @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ).
% Suc3_eq_add_3
thf(fact_1244_nat__1__add__1,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% nat_1_add_1
thf(fact_1245_le__imp__0__less,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z3 ) ) ) ).
% le_imp_0_less
thf(fact_1246_Suc__as__int,axiom,
( suc
= ( ^ [A4: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A4 ) @ one_one_int ) ) ) ) ).
% Suc_as_int
thf(fact_1247_num__of__nat__double,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( num_of_nat @ ( plus_plus_nat @ N @ N ) )
= ( bit0 @ ( num_of_nat @ N ) ) ) ) ).
% num_of_nat_double
thf(fact_1248_num_Osize_I5_J,axiom,
! [X22: num] :
( ( size_size_num @ ( bit0 @ X22 ) )
= ( plus_plus_nat @ ( size_size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size(5)
thf(fact_1249_num_Osize_I6_J,axiom,
! [X33: num] :
( ( size_size_num @ ( bit1 @ X33 ) )
= ( plus_plus_nat @ ( size_size_num @ X33 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size(6)
thf(fact_1250_Suc__nat__eq__nat__zadd1,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( suc @ ( nat2 @ Z3 ) )
= ( nat2 @ ( plus_plus_int @ one_one_int @ Z3 ) ) ) ) ).
% Suc_nat_eq_nat_zadd1
thf(fact_1251_nat__induct2,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct2
thf(fact_1252_num_Osize__gen_I3_J,axiom,
! [X33: num] :
( ( size_num @ ( bit1 @ X33 ) )
= ( plus_plus_nat @ ( size_num @ X33 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size_gen(3)
thf(fact_1253_num_Osize__gen_I2_J,axiom,
! [X22: num] :
( ( size_num @ ( bit0 @ X22 ) )
= ( plus_plus_nat @ ( size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size_gen(2)
thf(fact_1254_num_Osize__gen_I1_J,axiom,
( ( size_num @ one )
= zero_zero_nat ) ).
% num.size_gen(1)
thf(fact_1255_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B3: nat] :
( ! [A5: nat,B4: nat] :
( ( P @ A5 @ B4 )
= ( P @ B4 @ A5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ zero_zero_nat )
=> ( ! [A5: nat,B4: nat] :
( ( P @ A5 @ B4 )
=> ( P @ A5 @ ( plus_plus_nat @ A5 @ B4 ) ) )
=> ( P @ A @ B3 ) ) ) ) ).
% Euclid_induct
thf(fact_1256_inc__BitM__eq,axiom,
! [N: num] :
( ( inc @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% inc_BitM_eq
thf(fact_1257_BitM__inc__eq,axiom,
! [N: num] :
( ( bitM @ ( inc @ N ) )
= ( bit1 @ N ) ) ).
% BitM_inc_eq
thf(fact_1258_eval__nat__numeral_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( suc @ ( numeral_numeral_nat @ ( bitM @ N ) ) ) ) ).
% eval_nat_numeral(2)
thf(fact_1259_one__plus__BitM,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% one_plus_BitM
thf(fact_1260_BitM__plus__one,axiom,
! [N: num] :
( ( plus_plus_num @ ( bitM @ N ) @ one )
= ( bit0 @ N ) ) ).
% BitM_plus_one
thf(fact_1261_Suc__0__div__numeral_I1_J,axiom,
( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ one ) )
= one_one_nat ) ).
% Suc_0_div_numeral(1)
% Helper facts (5)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y2: int] :
( ( if_int @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y2: int] :
( ( if_int @ $true @ X2 @ Y2 )
= X2 ) ).
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y2: nat] :
( ( if_nat @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y2: nat] :
( ( if_nat @ $true @ X2 @ Y2 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( size_s3023201423986296836st_nat
@ ( filter_list_nat
@ ( ^ [Y: list_nat,Z: list_nat] : ( Y = Z )
@ y )
@ ( equiva7426478223624825838m_rgfs @ n ) ) )
= ( count_list_list_nat @ ( equiva7426478223624825838m_rgfs @ n ) @ y ) ) ).
%------------------------------------------------------------------------------