TPTP Problem File: SLH0118^1.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 : Commuting_Hermitian/0002_Commuting_Hermitian/prob_02040_079918__19591676_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1517 ( 638 unt; 324 typ; 0 def)
% Number of atoms : 3127 (2224 equ; 0 cnn)
% Maximal formula atoms : 18 ( 2 avg)
% Number of connectives : 12676 ( 558 ~; 58 |; 428 &;10130 @)
% ( 0 <=>;1502 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 6 avg)
% Number of types : 26 ( 25 usr)
% Number of type conns : 1440 (1440 >; 0 *; 0 +; 0 <<)
% Number of symbols : 302 ( 299 usr; 37 con; 0-3 aty)
% Number of variables : 4167 ( 272 ^;3541 !; 354 ?;4167 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:38:02.944
%------------------------------------------------------------------------------
% Could-be-implicit typings (25)
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Complex__Ocomplex_J_J_J,type,
list_l3981933317855906654omplex: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Real__Oreal_J_J_J,type,
list_list_list_real: $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__List__Olist_It__List__Olist_It__Complex__Ocomplex_J_J,type,
list_list_complex: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Complex__Ocomplex_J_J,type,
set_list_complex: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Complex__Ocomplex_J_J,type,
list_set_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Complex__Ocomplex_J_J,type,
set_set_complex: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Real__Oreal_J_J,type,
list_list_real: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Real__Oreal_J_J,type,
set_list_real: $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__Polynomial__Opoly_It__Complex__Ocomplex_J,type,
poly_complex: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Real__Oreal_J,type,
poly_real: $tType ).
thf(ty_n_t__List__Olist_It__Complex__Ocomplex_J,type,
list_complex: $tType ).
thf(ty_n_t__Set__Oset_It__Complex__Ocomplex_J,type,
set_complex: $tType ).
thf(ty_n_t__Polynomial__Opoly_It__Nat__Onat_J,type,
poly_nat: $tType ).
thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
list_real: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Complex__Ocomplex,type,
complex: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (299)
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Complex__Ocomplex,type,
bNF_Gr1398276426132326479omplex: set_list_complex > complex > set_list_complex ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Nat__Onat,type,
bNF_Gr1872714664788909425ft_nat: set_list_nat > nat > set_list_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Real__Oreal,type,
bNF_Gr3712412480325189581t_real: set_list_real > real > set_list_real ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Complex__Ocomplex,type,
bNF_Gr1374303899511151571omplex: set_list_complex > list_complex > set_complex ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Nat__Onat,type,
bNF_Gr6352880689984616693cc_nat: set_list_nat > list_nat > set_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Real__Oreal,type,
bNF_Gr2087828336424606033c_real: set_list_real > list_real > set_real ).
thf(sy_c_Commuting__Hermitian_Oeq__comps_001t__Complex__Ocomplex,type,
commut93809757773076895omplex: list_complex > list_nat ).
thf(sy_c_Commuting__Hermitian_Oeq__comps_001t__Nat__Onat,type,
commut2436974278740741825ps_nat: list_nat > list_nat ).
thf(sy_c_Commuting__Hermitian_Oeq__comps_001t__Real__Oreal,type,
commut8680161604938074397s_real: list_real > list_nat ).
thf(sy_c_Commuting__Hermitian_Oeq__comps__rel_001t__Complex__Ocomplex,type,
commut5384305104226550776omplex: list_complex > list_complex > $o ).
thf(sy_c_Commuting__Hermitian_Oeq__comps__rel_001t__Nat__Onat,type,
commut1452772284045945626el_nat: list_nat > list_nat > $o ).
thf(sy_c_Commuting__Hermitian_Oeq__comps__rel_001t__Real__Oreal,type,
commut4159206679679027446l_real: list_real > list_real > $o ).
thf(sy_c_Complex_Ocomplex_ORe,type,
re: complex > real ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Complex__Ocomplex_M_Eo_J,type,
minus_8727706125548526216plex_o: ( complex > $o ) > ( complex > $o ) > complex > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex,type,
minus_minus_complex: complex > complex > complex ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Complex__Ocomplex_J,type,
minus_811609699411566653omplex: set_complex > set_complex > set_complex ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
minus_minus_set_real: set_real > set_real > set_real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
one_one_complex: complex ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Set__Oset_It__Complex__Ocomplex_J,type,
one_one_set_complex: set_complex ).
thf(sy_c_Groups_Oone__class_Oone_001t__Set__Oset_It__Nat__Onat_J,type,
one_one_set_nat: set_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Set__Oset_It__Real__Oreal_J,type,
one_one_set_real: set_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex,type,
zero_zero_complex: complex ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
zero_zero_poly_nat: poly_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Set__Oset_It__Complex__Ocomplex_J,type,
zero_z6614145512433583213omplex: set_complex ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Set__Oset_It__Nat__Onat_J,type,
zero_zero_set_nat: set_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Set__Oset_It__Real__Oreal_J,type,
zero_zero_set_real: set_real ).
thf(sy_c_If_001t__List__Olist_It__Complex__Ocomplex_J,type,
if_list_complex: $o > list_complex > list_complex > list_complex ).
thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
if_list_nat: $o > list_nat > list_nat > list_nat ).
thf(sy_c_If_001t__List__Olist_It__Real__Oreal_J,type,
if_list_real: $o > list_real > list_real > list_real ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_List_Oappend_001t__Complex__Ocomplex,type,
append_complex: list_complex > list_complex > list_complex ).
thf(sy_c_List_Oappend_001t__List__Olist_It__Complex__Ocomplex_J,type,
append_list_complex: list_list_complex > list_list_complex > list_list_complex ).
thf(sy_c_List_Oappend_001t__List__Olist_It__Nat__Onat_J,type,
append_list_nat: list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Oappend_001t__List__Olist_It__Real__Oreal_J,type,
append_list_real: list_list_real > list_list_real > list_list_real ).
thf(sy_c_List_Oappend_001t__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Oappend_001t__Real__Oreal,type,
append_real: list_real > list_real > list_real ).
thf(sy_c_List_Obutlast_001t__Complex__Ocomplex,type,
butlast_complex: list_complex > list_complex ).
thf(sy_c_List_Obutlast_001t__Nat__Onat,type,
butlast_nat: list_nat > list_nat ).
thf(sy_c_List_Obutlast_001t__Real__Oreal,type,
butlast_real: list_real > list_real ).
thf(sy_c_List_Ocan__select_001t__Complex__Ocomplex,type,
can_select_complex: ( complex > $o ) > set_complex > $o ).
thf(sy_c_List_Ocan__select_001t__Nat__Onat,type,
can_select_nat: ( nat > $o ) > set_nat > $o ).
thf(sy_c_List_Oconcat_001t__Complex__Ocomplex,type,
concat_complex: list_list_complex > list_complex ).
thf(sy_c_List_Oconcat_001t__List__Olist_It__Complex__Ocomplex_J,type,
concat_list_complex: list_l3981933317855906654omplex > list_list_complex ).
thf(sy_c_List_Oconcat_001t__List__Olist_It__Nat__Onat_J,type,
concat_list_nat: list_list_list_nat > list_list_nat ).
thf(sy_c_List_Oconcat_001t__List__Olist_It__Real__Oreal_J,type,
concat_list_real: list_list_list_real > list_list_real ).
thf(sy_c_List_Oconcat_001t__Nat__Onat,type,
concat_nat: list_list_nat > list_nat ).
thf(sy_c_List_Oconcat_001t__Real__Oreal,type,
concat_real: list_list_real > list_real ).
thf(sy_c_List_Ocoset_001t__Complex__Ocomplex,type,
coset_complex: list_complex > set_complex ).
thf(sy_c_List_Ocoset_001t__Nat__Onat,type,
coset_nat: list_nat > set_nat ).
thf(sy_c_List_Odistinct_001t__Complex__Ocomplex,type,
distinct_complex: list_complex > $o ).
thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
distinct_nat: list_nat > $o ).
thf(sy_c_List_Odistinct_001t__Real__Oreal,type,
distinct_real: list_real > $o ).
thf(sy_c_List_Odistinct_001t__Set__Oset_It__Complex__Ocomplex_J,type,
distinct_set_complex: list_set_complex > $o ).
thf(sy_c_List_Odistinct__adj_001t__Complex__Ocomplex,type,
distinct_adj_complex: list_complex > $o ).
thf(sy_c_List_Odistinct__adj_001t__Nat__Onat,type,
distinct_adj_nat: list_nat > $o ).
thf(sy_c_List_Odistinct__adj_001t__Real__Oreal,type,
distinct_adj_real: list_real > $o ).
thf(sy_c_List_OdropWhile_001t__Complex__Ocomplex,type,
dropWhile_complex: ( complex > $o ) > list_complex > list_complex ).
thf(sy_c_List_OdropWhile_001t__Nat__Onat,type,
dropWhile_nat: ( nat > $o ) > list_nat > list_nat ).
thf(sy_c_List_OdropWhile_001t__Real__Oreal,type,
dropWhile_real: ( real > $o ) > list_real > list_real ).
thf(sy_c_List_Ofilter_001t__Complex__Ocomplex,type,
filter_complex: ( complex > $o ) > list_complex > list_complex ).
thf(sy_c_List_Ofilter_001t__List__Olist_It__Complex__Ocomplex_J,type,
filter_list_complex: ( list_complex > $o ) > list_list_complex > list_list_complex ).
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__List__Olist_It__Real__Oreal_J,type,
filter_list_real: ( list_real > $o ) > list_list_real > list_list_real ).
thf(sy_c_List_Ofilter_001t__Nat__Onat,type,
filter_nat: ( nat > $o ) > list_nat > list_nat ).
thf(sy_c_List_Ofilter_001t__Real__Oreal,type,
filter_real: ( real > $o ) > list_real > list_real ).
thf(sy_c_List_Ogen__length_001t__Complex__Ocomplex,type,
gen_length_complex: nat > list_complex > 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__Real__Oreal,type,
gen_length_real: nat > list_real > nat ).
thf(sy_c_List_Oinsert_001t__Complex__Ocomplex,type,
insert_complex: complex > list_complex > list_complex ).
thf(sy_c_List_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oinsert_001t__Real__Oreal,type,
insert_real: real > list_real > list_real ).
thf(sy_c_List_Olast_001t__Complex__Ocomplex,type,
last_complex: list_complex > complex ).
thf(sy_c_List_Olast_001t__Nat__Onat,type,
last_nat: list_nat > nat ).
thf(sy_c_List_Olast_001t__Real__Oreal,type,
last_real: list_real > real ).
thf(sy_c_List_Olinorder__class_Oinsort__insert__key_001t__Complex__Ocomplex_001t__Nat__Onat,type,
linord1240774035888917890ex_nat: ( complex > nat ) > complex > list_complex > list_complex ).
thf(sy_c_List_Olinorder__class_Oinsort__insert__key_001t__Nat__Onat_001t__Nat__Onat,type,
linord1921536354676448932at_nat: ( nat > nat ) > nat > list_nat > list_nat ).
thf(sy_c_List_Olinorder__class_Oinsort__key_001t__Complex__Ocomplex_001t__Nat__Onat,type,
linord4454476646278009211ex_nat: ( complex > nat ) > complex > list_complex > list_complex ).
thf(sy_c_List_Olinorder__class_Oinsort__key_001t__Complex__Ocomplex_001t__Real__Oreal,type,
linord67127995744734935x_real: ( complex > real ) > complex > list_complex > list_complex ).
thf(sy_c_List_Olinorder__class_Oinsort__key_001t__Nat__Onat_001t__Nat__Onat,type,
linord8961336180081300637at_nat: ( nat > nat ) > nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Complex__Ocomplex,type,
cons_complex: complex > list_complex > list_complex ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Complex__Ocomplex_J,type,
cons_list_complex: list_complex > list_list_complex > list_list_complex ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Nat__Onat_J,type,
cons_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Real__Oreal_J,type,
cons_list_real: list_real > list_list_real > list_list_real ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Real__Oreal,type,
cons_real: real > list_real > list_real ).
thf(sy_c_List_Olist_ONil_001t__Complex__Ocomplex,type,
nil_complex: list_complex ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Complex__Ocomplex_J,type,
nil_list_complex: list_list_complex ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Nat__Onat_J,type,
nil_list_nat: list_list_nat ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Real__Oreal_J,type,
nil_list_real: list_list_real ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001t__Real__Oreal,type,
nil_real: list_real ).
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_Ocase__list_001_Eo_001t__Nat__Onat,type,
case_list_o_nat: $o > ( nat > list_nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist_Ocase__list_001t__List__Olist_It__Complex__Ocomplex_J_001t__Complex__Ocomplex,type,
case_l7337434744184354388omplex: list_complex > ( complex > list_complex > list_complex ) > list_complex > list_complex ).
thf(sy_c_List_Olist_Ocase__list_001t__List__Olist_It__List__Olist_It__Complex__Ocomplex_J_J_001t__Complex__Ocomplex,type,
case_l6761848517213342308omplex: list_list_complex > ( complex > list_complex > list_list_complex ) > list_complex > list_list_complex ).
thf(sy_c_List_Olist_Ocase__list_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_001t__Nat__Onat,type,
case_l3331202209248957608at_nat: list_list_nat > ( nat > list_nat > list_list_nat ) > list_nat > list_list_nat ).
thf(sy_c_List_Olist_Ocase__list_001t__List__Olist_It__List__Olist_It__Real__Oreal_J_J_001t__Real__Oreal,type,
case_l5204074688173521888l_real: list_list_real > ( real > list_real > list_list_real ) > list_real > list_list_real ).
thf(sy_c_List_Olist_Ocase__list_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
case_l2340614614379431832at_nat: list_nat > ( nat > list_nat > list_nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Ocase__list_001t__List__Olist_It__Real__Oreal_J_001t__Real__Oreal,type,
case_l3379708394843211600l_real: list_real > ( real > list_real > list_real ) > list_real > list_real ).
thf(sy_c_List_Olist_Ohd_001t__Complex__Ocomplex,type,
hd_complex: list_complex > complex ).
thf(sy_c_List_Olist_Ohd_001t__List__Olist_It__Nat__Onat_J,type,
hd_list_nat: list_list_nat > list_nat ).
thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
hd_nat: list_nat > nat ).
thf(sy_c_List_Olist_Ohd_001t__Real__Oreal,type,
hd_real: list_real > real ).
thf(sy_c_List_Olist_Olist__all_001t__Complex__Ocomplex,type,
list_all_complex: ( complex > $o ) > list_complex > $o ).
thf(sy_c_List_Olist_Olist__all_001t__Nat__Onat,type,
list_all_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist_Olist__all_001t__Real__Oreal,type,
list_all_real: ( real > $o ) > list_real > $o ).
thf(sy_c_List_Olist_Omap_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
map_complex_complex: ( complex > complex ) > list_complex > list_complex ).
thf(sy_c_List_Olist_Omap_001t__Complex__Ocomplex_001t__List__Olist_It__Complex__Ocomplex_J,type,
map_co8634619192568165682omplex: ( complex > list_complex ) > list_complex > list_list_complex ).
thf(sy_c_List_Olist_Omap_001t__Complex__Ocomplex_001t__List__Olist_It__List__Olist_It__Complex__Ocomplex_J_J,type,
map_co661557373136449218omplex: ( complex > list_list_complex ) > list_complex > list_l3981933317855906654omplex ).
thf(sy_c_List_Olist_Omap_001t__Complex__Ocomplex_001t__List__Olist_It__Nat__Onat_J,type,
map_complex_list_nat: ( complex > list_nat ) > list_complex > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__Complex__Ocomplex_001t__List__Olist_It__Real__Oreal_J,type,
map_co6541394440057479600t_real: ( complex > list_real ) > list_complex > list_list_real ).
thf(sy_c_List_Olist_Omap_001t__Complex__Ocomplex_001t__Nat__Onat,type,
map_complex_nat: ( complex > nat ) > list_complex > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Complex__Ocomplex_001t__Real__Oreal,type,
map_complex_real: ( complex > real ) > list_complex > list_real ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Complex__Ocomplex_J_001t__Complex__Ocomplex,type,
map_li9134918900125190322omplex: ( list_complex > complex ) > list_list_complex > list_complex ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Complex__Ocomplex_J_001t__List__Olist_It__Complex__Ocomplex_J,type,
map_li2870275437539113154omplex: ( list_complex > list_complex ) > list_list_complex > list_list_complex ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Complex__Ocomplex_J_001t__List__Olist_It__List__Olist_It__Complex__Ocomplex_J_J,type,
map_li6028939799916194386omplex: ( list_complex > list_list_complex ) > list_list_complex > list_l3981933317855906654omplex ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Complex__Ocomplex_J_001t__List__Olist_It__Nat__Onat_J,type,
map_li3202499864523910372st_nat: ( list_complex > list_nat ) > list_list_complex > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Complex__Ocomplex_J_001t__List__Olist_It__Real__Oreal_J,type,
map_li971590449312185664t_real: ( list_complex > list_real ) > list_list_complex > list_list_real ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Complex__Ocomplex_J_001t__Set__Oset_It__Complex__Ocomplex_J,type,
map_li645366948578852712omplex: ( list_complex > set_complex ) > list_list_complex > list_set_complex ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Complex__Ocomplex_J,type,
map_li6798605796755630564omplex: ( list_nat > list_complex ) > list_list_nat > list_list_complex ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
map_li960784813134754710st_nat: ( list_nat > list_list_nat ) > list_list_nat > list_list_list_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__Real__Oreal_J_001t__List__Olist_It__List__Olist_It__Real__Oreal_J_J,type,
map_li1716307582283692110t_real: ( list_real > list_list_real ) > list_list_real > list_list_list_real ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Real__Oreal_J_001t__List__Olist_It__Real__Oreal_J,type,
map_li1455663113306559806t_real: ( list_real > list_real ) > list_list_real > list_list_real ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Real__Oreal_J_001t__Real__Oreal,type,
map_list_real_real: ( list_real > real ) > list_list_real > list_real ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Complex__Ocomplex,type,
map_nat_complex: ( nat > complex ) > list_nat > list_complex ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__List__Olist_It__Complex__Ocomplex_J,type,
map_nat_list_complex: ( nat > list_complex ) > list_nat > list_list_complex ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
map_na6205611841492582150st_nat: ( nat > list_list_nat ) > list_nat > list_list_list_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__Real__Oreal,type,
map_nat_real: ( nat > real ) > list_nat > list_real ).
thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Complex__Ocomplex,type,
map_real_complex: ( real > complex ) > list_real > list_complex ).
thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__List__Olist_It__List__Olist_It__Real__Oreal_J_J,type,
map_re7007078575547571262t_real: ( real > list_list_real ) > list_real > list_list_list_real ).
thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__List__Olist_It__Real__Oreal_J,type,
map_real_list_real: ( real > list_real ) > list_real > list_list_real ).
thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Nat__Onat,type,
map_real_nat: ( real > nat ) > list_real > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Real__Oreal_001t__Real__Oreal,type,
map_real_real: ( real > real ) > list_real > list_real ).
thf(sy_c_List_Olist_Orec__list_001t__List__Olist_It__Complex__Ocomplex_J_001t__Complex__Ocomplex,type,
rec_li3990778930367580964omplex: list_complex > ( complex > list_complex > list_complex > list_complex ) > list_complex > list_complex ).
thf(sy_c_List_Olist_Orec__list_001t__List__Olist_It__Complex__Ocomplex_J_001t__Nat__Onat,type,
rec_li5569328198523947590ex_nat: list_complex > ( nat > list_nat > list_complex > list_complex ) > list_nat > list_complex ).
thf(sy_c_List_Olist_Orec__list_001t__List__Olist_It__Nat__Onat_J_001t__Complex__Ocomplex,type,
rec_li5065709429383092550omplex: list_nat > ( complex > list_complex > list_nat > list_nat ) > list_complex > list_nat ).
thf(sy_c_List_Olist_Orec__list_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
rec_li7516600145284979816at_nat: list_nat > ( nat > list_nat > list_nat > list_nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Orec__list_001t__List__Olist_It__Real__Oreal_J_001t__Complex__Ocomplex,type,
rec_li642222640852133922omplex: list_real > ( complex > list_complex > list_real > list_real ) > list_complex > list_real ).
thf(sy_c_List_Olist_Orec__list_001t__Set__Oset_It__Complex__Ocomplex_J_001t__Complex__Ocomplex,type,
rec_li3674993589600700234omplex: set_complex > ( complex > list_complex > set_complex > set_complex ) > list_complex > set_complex ).
thf(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
set_complex2: list_complex > set_complex ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Complex__Ocomplex_J,type,
set_list_complex2: list_list_complex > set_list_complex ).
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__Real__Oreal_J,type,
set_list_real2: list_list_real > set_list_real ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
set_real2: list_real > set_real ).
thf(sy_c_List_Olist_Otl_001t__Complex__Ocomplex,type,
tl_complex: list_complex > list_complex ).
thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
tl_nat: list_nat > list_nat ).
thf(sy_c_List_Olist_Otl_001t__Real__Oreal,type,
tl_real: list_real > list_real ).
thf(sy_c_List_Olist__ex1_001t__Complex__Ocomplex,type,
list_ex1_complex: ( complex > $o ) > list_complex > $o ).
thf(sy_c_List_Olist__ex1_001t__Nat__Onat,type,
list_ex1_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__ex1_001t__Real__Oreal,type,
list_ex1_real: ( real > $o ) > list_real > $o ).
thf(sy_c_List_Olistset_001t__Nat__Onat,type,
listset_nat: list_set_nat > set_list_nat ).
thf(sy_c_List_Omap__tailrec_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
map_ta3177854533886046316omplex: ( complex > complex ) > list_complex > list_complex ).
thf(sy_c_List_Omap__tailrec_001t__Complex__Ocomplex_001t__Nat__Onat,type,
map_ta9198917811265257358ex_nat: ( complex > nat ) > list_complex > list_nat ).
thf(sy_c_List_Omap__tailrec_001t__Complex__Ocomplex_001t__Real__Oreal,type,
map_ta5686879364588479338x_real: ( complex > real ) > list_complex > list_real ).
thf(sy_c_List_Omap__tailrec_001t__Nat__Onat_001t__Complex__Ocomplex,type,
map_ta5579134486830519694omplex: ( nat > complex ) > list_nat > list_complex ).
thf(sy_c_List_Omap__tailrec_001t__Nat__Onat_001t__Nat__Onat,type,
map_tailrec_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Omaps_001t__Nat__Onat_001t__Nat__Onat,type,
maps_nat_nat: ( nat > list_nat ) > list_nat > list_nat ).
thf(sy_c_List_Omember_001t__Complex__Ocomplex,type,
member_complex: list_complex > complex > $o ).
thf(sy_c_List_Omember_001t__Nat__Onat,type,
member_nat: list_nat > nat > $o ).
thf(sy_c_List_Omember_001t__Real__Oreal,type,
member_real: list_real > real > $o ).
thf(sy_c_List_On__lists_001t__Complex__Ocomplex,type,
n_lists_complex: nat > list_complex > list_list_complex ).
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__Real__Oreal,type,
n_lists_real: nat > list_real > list_list_real ).
thf(sy_c_List_Onths_001t__Complex__Ocomplex,type,
nths_complex: list_complex > set_nat > list_complex ).
thf(sy_c_List_Onths_001t__Nat__Onat,type,
nths_nat: list_nat > set_nat > list_nat ).
thf(sy_c_List_Onths_001t__Real__Oreal,type,
nths_real: list_real > set_nat > list_real ).
thf(sy_c_List_Oproduct__lists_001t__Complex__Ocomplex,type,
produc7545014605101902079omplex: list_list_complex > list_list_complex ).
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__Real__Oreal,type,
product_lists_real: list_list_real > list_list_real ).
thf(sy_c_List_Oremdups__adj_001t__Complex__Ocomplex,type,
remdups_adj_complex: list_complex > list_complex ).
thf(sy_c_List_Oremdups__adj_001t__Nat__Onat,type,
remdups_adj_nat: list_nat > list_nat ).
thf(sy_c_List_Oremdups__adj_001t__Real__Oreal,type,
remdups_adj_real: list_real > list_real ).
thf(sy_c_List_Oremdups__adj__rel_001t__Complex__Ocomplex,type,
remdup6092795584463544805omplex: list_complex > list_complex > $o ).
thf(sy_c_List_Oremdups__adj__rel_001t__Nat__Onat,type,
remdups_adj_rel_nat: list_nat > list_nat > $o ).
thf(sy_c_List_Oremdups__adj__rel_001t__Real__Oreal,type,
remdups_adj_rel_real: list_real > list_real > $o ).
thf(sy_c_List_Orotate1_001t__Complex__Ocomplex,type,
rotate1_complex: list_complex > list_complex ).
thf(sy_c_List_Orotate1_001t__Nat__Onat,type,
rotate1_nat: list_nat > list_nat ).
thf(sy_c_List_Orotate1_001t__Real__Oreal,type,
rotate1_real: list_real > list_real ).
thf(sy_c_List_Oset__Cons_001t__Complex__Ocomplex,type,
set_Cons_complex: set_complex > set_list_complex > set_list_complex ).
thf(sy_c_List_Oset__Cons_001t__Nat__Onat,type,
set_Cons_nat: set_nat > set_list_nat > set_list_nat ).
thf(sy_c_List_Oset__Cons_001t__Real__Oreal,type,
set_Cons_real: set_real > set_list_real > set_list_real ).
thf(sy_c_List_Oshuffles_001t__Complex__Ocomplex,type,
shuffles_complex: list_complex > list_complex > set_list_complex ).
thf(sy_c_List_Oshuffles_001t__Nat__Onat,type,
shuffles_nat: list_nat > list_nat > set_list_nat ).
thf(sy_c_List_Oshuffles_001t__Real__Oreal,type,
shuffles_real: list_real > list_real > set_list_real ).
thf(sy_c_List_Osubseqs_001t__Complex__Ocomplex,type,
subseqs_complex: list_complex > list_list_complex ).
thf(sy_c_List_Osubseqs_001t__Nat__Onat,type,
subseqs_nat: list_nat > list_list_nat ).
thf(sy_c_List_Osubseqs_001t__Real__Oreal,type,
subseqs_real: list_real > list_list_real ).
thf(sy_c_List_Osuccessively_001t__Complex__Ocomplex,type,
successively_complex: ( complex > complex > $o ) > list_complex > $o ).
thf(sy_c_List_Osuccessively_001t__Nat__Onat,type,
successively_nat: ( nat > nat > $o ) > list_nat > $o ).
thf(sy_c_List_Osuccessively_001t__Real__Oreal,type,
successively_real: ( real > real > $o ) > list_real > $o ).
thf(sy_c_List_Otake_001t__Complex__Ocomplex,type,
take_complex: nat > list_complex > list_complex ).
thf(sy_c_List_Otake_001t__Nat__Onat,type,
take_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Otake_001t__Real__Oreal,type,
take_real: nat > list_real > list_real ).
thf(sy_c_List_Otranspose_001t__Complex__Ocomplex,type,
transpose_complex: list_list_complex > list_list_complex ).
thf(sy_c_List_Otranspose_001t__Nat__Onat,type,
transpose_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Otranspose_001t__Real__Oreal,type,
transpose_real: list_list_real > list_list_real ).
thf(sy_c_List_Otranspose__rel_001t__Complex__Ocomplex,type,
transp2032975277941382969omplex: list_list_complex > list_list_complex > $o ).
thf(sy_c_List_Otranspose__rel_001t__Nat__Onat,type,
transpose_rel_nat: list_list_nat > list_list_nat > $o ).
thf(sy_c_List_Otranspose__rel_001t__Real__Oreal,type,
transpose_rel_real: list_list_real > list_list_real > $o ).
thf(sy_c_Missing__Unsorted_Omax__list__non__empty_001t__Nat__Onat,type,
missin53001312869816611ty_nat: list_nat > nat ).
thf(sy_c_Missing__Unsorted_Omax__list__non__empty_001t__Real__Oreal,type,
missin3576488506594132607y_real: list_real > real ).
thf(sy_c_Missing__VectorSpace_Ofind__indices_001t__Complex__Ocomplex,type,
missin8834916005246747252omplex: complex > list_complex > list_nat ).
thf(sy_c_Missing__VectorSpace_Ofind__indices_001t__Nat__Onat,type,
missin5050847376130023830es_nat: nat > list_nat > list_nat ).
thf(sy_c_Missing__VectorSpace_Ofind__indices_001t__Real__Oreal,type,
missin4558892344646173042s_real: real > list_real > list_nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
size_s3451745648224563538omplex: list_complex > 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__Real__Oreal_J,type,
size_size_list_real: list_real > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Complex__Ocomplex_J,type,
bot_bot_set_complex: set_complex ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Complex__Ocomplex_J_J,type,
bot_bo6492010485567502472omplex: set_list_complex ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
bot_bot_set_list_nat: set_list_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Real__Oreal_J_J,type,
bot_bo7660656974785640070t_real: set_list_real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
bot_bot_set_real: set_real ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Complex__Ocomplex,type,
ord_less_eq_complex: complex > complex > $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__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Polynomial_OPoly_001t__Complex__Ocomplex,type,
poly_complex2: list_complex > poly_complex ).
thf(sy_c_Polynomial_OPoly_001t__Nat__Onat,type,
poly_nat2: list_nat > poly_nat ).
thf(sy_c_Polynomial_OPoly_001t__Real__Oreal,type,
poly_real2: list_real > poly_real ).
thf(sy_c_Polynomial_Oplus__coeffs_001t__Complex__Ocomplex,type,
plus_coeffs_complex: list_complex > list_complex > list_complex ).
thf(sy_c_Polynomial_Oplus__coeffs_001t__Nat__Onat,type,
plus_coeffs_nat: list_nat > list_nat > list_nat ).
thf(sy_c_Polynomial_Oplus__coeffs_001t__Real__Oreal,type,
plus_coeffs_real: list_real > list_real > list_real ).
thf(sy_c_Real__Vector__Spaces_OReals_001t__Complex__Ocomplex,type,
real_V2521375963428798218omplex: set_complex ).
thf(sy_c_Real__Vector__Spaces_OReals_001t__Real__Oreal,type,
real_V470468836141973256s_real: set_real ).
thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
collect_complex: ( complex > $o ) > set_complex ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Complex__Ocomplex_J,type,
collect_list_complex: ( list_complex > $o ) > set_list_complex ).
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__Real__Oreal_J,type,
collect_list_real: ( list_real > $o ) > set_list_real ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_Set_Oimage_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
image_1468599708987790691omplex: ( complex > complex ) > set_complex > set_complex ).
thf(sy_c_Set_Oimage_001t__Complex__Ocomplex_001t__List__Olist_It__Complex__Ocomplex_J,type,
image_3109120598033070323omplex: ( complex > list_complex ) > set_complex > set_list_complex ).
thf(sy_c_Set_Oimage_001t__Complex__Ocomplex_001t__Nat__Onat,type,
image_complex_nat: ( complex > nat ) > set_complex > set_nat ).
thf(sy_c_Set_Oimage_001t__Complex__Ocomplex_001t__Real__Oreal,type,
image_complex_real: ( complex > real ) > set_complex > set_real ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Complex__Ocomplex,type,
image_nat_complex: ( nat > complex ) > set_nat > set_complex ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
image_nat_list_nat: ( nat > list_nat ) > set_nat > set_list_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__List__Olist_It__Real__Oreal_J,type,
image_real_list_real: ( real > list_real ) > set_real > set_list_real ).
thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
image_real_real: ( real > real ) > set_real > set_real ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Complex__Ocomplex_J_001t__Set__Oset_It__Complex__Ocomplex_J,type,
image_7998606247489673935omplex: ( set_complex > set_complex ) > set_set_complex > set_set_complex ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oinsert_001t__Complex__Ocomplex,type,
insert_complex2: complex > set_complex > set_complex ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__Complex__Ocomplex_J,type,
insert_list_complex: list_complex > set_list_complex > set_list_complex ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
insert_list_nat: list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__Real__Oreal_J,type,
insert_list_real: list_real > set_list_real > set_list_real ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat2: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
insert_real2: real > set_real > set_real ).
thf(sy_c_Set_Othe__elem_001t__Complex__Ocomplex,type,
the_elem_complex: set_complex > complex ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set_Othe__elem_001t__Real__Oreal,type,
the_elem_real: set_real > real ).
thf(sy_c_Sublist_OLongest__common__prefix_001t__Complex__Ocomplex,type,
longes9130274606949010820omplex: set_list_complex > list_complex ).
thf(sy_c_Sublist_OLongest__common__prefix_001t__Nat__Onat,type,
longes514542611558403238ix_nat: set_list_nat > list_nat ).
thf(sy_c_Sublist_OLongest__common__prefix_001t__Real__Oreal,type,
longes7015494920320935042x_real: set_list_real > list_real ).
thf(sy_c_Sublist_Oprefixes_001t__Complex__Ocomplex,type,
prefixes_complex: list_complex > list_list_complex ).
thf(sy_c_Sublist_Oprefixes_001t__Nat__Onat,type,
prefixes_nat: list_nat > list_list_nat ).
thf(sy_c_Sublist_Oprefixes_001t__Real__Oreal,type,
prefixes_real: list_real > list_list_real ).
thf(sy_c_Sublist_Osublists_001t__Complex__Ocomplex,type,
sublists_complex: list_complex > list_list_complex ).
thf(sy_c_Sublist_Osublists_001t__Nat__Onat,type,
sublists_nat: list_nat > list_list_nat ).
thf(sy_c_Sublist_Osublists_001t__Real__Oreal,type,
sublists_real: list_real > list_list_real ).
thf(sy_c_Sublist_Osuffixes_001t__Complex__Ocomplex,type,
suffixes_complex: list_complex > list_list_complex ).
thf(sy_c_Sublist_Osuffixes_001t__Nat__Onat,type,
suffixes_nat: list_nat > list_list_nat ).
thf(sy_c_Sublist_Osuffixes_001t__Real__Oreal,type,
suffixes_real: list_real > list_list_real ).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Complex__Ocomplex_J,type,
accp_list_complex: ( list_complex > list_complex > $o ) > list_complex > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__List__Olist_It__Complex__Ocomplex_J_J,type,
accp_l5771520762016474373omplex: ( list_list_complex > list_list_complex > $o ) > list_list_complex > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
accp_list_list_nat: ( list_list_nat > list_list_nat > $o ) > list_list_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__List__Olist_It__Real__Oreal_J_J,type,
accp_list_list_real: ( list_list_real > list_list_real > $o ) > list_list_real > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
accp_list_nat: ( list_nat > list_nat > $o ) > list_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Real__Oreal_J,type,
accp_list_real: ( list_real > list_real > $o ) > list_real > $o ).
thf(sy_c_member_001t__Complex__Ocomplex,type,
member_complex2: complex > set_complex > $o ).
thf(sy_c_member_001t__List__Olist_It__Complex__Ocomplex_J,type,
member_list_complex: list_complex > set_list_complex > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__Real__Oreal_J,type,
member_list_real: list_real > set_list_real > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat2: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real2: real > set_real > $o ).
thf(sy_c_member_001t__Set__Oset_It__Complex__Ocomplex_J,type,
member_set_complex: set_complex > set_set_complex > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_v_ec____,type,
ec: list_nat ).
thf(sy_v_l,type,
l: list_complex ).
thf(sy_v_la____,type,
la: list_complex ).
thf(sy_v_m,type,
m: list_nat ).
thf(sy_v_ma____,type,
ma: list_nat ).
thf(sy_v_x____,type,
x: complex ).
thf(sy_v_y____,type,
y: complex ).
% Relevant facts (1183)
thf(fact_0_False,axiom,
x != y ).
% False
thf(fact_1__092_060open_062Re_Ax_A_092_060noteq_062_ARe_Ay_092_060close_062,axiom,
( ( re @ x )
!= ( re @ y ) ) ).
% \<open>Re x \<noteq> Re y\<close>
thf(fact_2__C3_C_I3_J,axiom,
( ma
= ( commut93809757773076895omplex @ ( cons_complex @ x @ ( cons_complex @ y @ la ) ) ) ) ).
% "3"(3)
thf(fact_3_ecr,axiom,
( ec
= ( commut8680161604938074397s_real @ ( map_complex_real @ re @ ( cons_complex @ y @ la ) ) ) ) ).
% ecr
thf(fact_4__C3_C_I2_J,axiom,
! [X: complex] :
( ( member_complex2 @ X @ ( set_complex2 @ ( cons_complex @ x @ ( cons_complex @ y @ la ) ) ) )
=> ( member_complex2 @ X @ real_V2521375963428798218omplex ) ) ).
% "3"(2)
thf(fact_5_ec__def,axiom,
( ec
= ( commut93809757773076895omplex @ ( cons_complex @ y @ la ) ) ) ).
% ec_def
thf(fact_6_list_Omap_I2_J,axiom,
! [F: nat > real,X21: nat,X22: list_nat] :
( ( map_nat_real @ F @ ( cons_nat @ X21 @ X22 ) )
= ( cons_real @ ( F @ X21 ) @ ( map_nat_real @ F @ X22 ) ) ) ).
% list.map(2)
thf(fact_7_list_Omap_I2_J,axiom,
! [F: real > complex,X21: real,X22: list_real] :
( ( map_real_complex @ F @ ( cons_real @ X21 @ X22 ) )
= ( cons_complex @ ( F @ X21 ) @ ( map_real_complex @ F @ X22 ) ) ) ).
% list.map(2)
thf(fact_8_list_Omap_I2_J,axiom,
! [F: real > nat,X21: real,X22: list_real] :
( ( map_real_nat @ F @ ( cons_real @ X21 @ X22 ) )
= ( cons_nat @ ( F @ X21 ) @ ( map_real_nat @ F @ X22 ) ) ) ).
% list.map(2)
thf(fact_9_list_Omap_I2_J,axiom,
! [F: real > real,X21: real,X22: list_real] :
( ( map_real_real @ F @ ( cons_real @ X21 @ X22 ) )
= ( cons_real @ ( F @ X21 ) @ ( map_real_real @ F @ X22 ) ) ) ).
% list.map(2)
thf(fact_10_list_Omap_I2_J,axiom,
! [F: complex > real,X21: complex,X22: list_complex] :
( ( map_complex_real @ F @ ( cons_complex @ X21 @ X22 ) )
= ( cons_real @ ( F @ X21 ) @ ( map_complex_real @ F @ X22 ) ) ) ).
% list.map(2)
thf(fact_11_list_Omap_I2_J,axiom,
! [F: complex > complex,X21: complex,X22: list_complex] :
( ( map_complex_complex @ F @ ( cons_complex @ X21 @ X22 ) )
= ( cons_complex @ ( F @ X21 ) @ ( map_complex_complex @ F @ X22 ) ) ) ).
% list.map(2)
thf(fact_12_list_Omap_I2_J,axiom,
! [F: complex > nat,X21: complex,X22: list_complex] :
( ( map_complex_nat @ F @ ( cons_complex @ X21 @ X22 ) )
= ( cons_nat @ ( F @ X21 ) @ ( map_complex_nat @ F @ X22 ) ) ) ).
% list.map(2)
thf(fact_13_list_Omap_I2_J,axiom,
! [F: nat > complex,X21: nat,X22: list_nat] :
( ( map_nat_complex @ F @ ( cons_nat @ X21 @ X22 ) )
= ( cons_complex @ ( F @ X21 ) @ ( map_nat_complex @ F @ X22 ) ) ) ).
% list.map(2)
thf(fact_14_list_Omap_I2_J,axiom,
! [F: nat > nat,X21: nat,X22: list_nat] :
( ( map_nat_nat @ F @ ( cons_nat @ X21 @ X22 ) )
= ( cons_nat @ ( F @ X21 ) @ ( map_nat_nat @ F @ X22 ) ) ) ).
% list.map(2)
thf(fact_15_Cons__eq__map__D,axiom,
! [X2: complex,Xs: list_complex,F: real > complex,Ys: list_real] :
( ( ( cons_complex @ X2 @ Xs )
= ( map_real_complex @ F @ Ys ) )
=> ? [Z: real,Zs: list_real] :
( ( Ys
= ( cons_real @ Z @ Zs ) )
& ( X2
= ( F @ Z ) )
& ( Xs
= ( map_real_complex @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_16_Cons__eq__map__D,axiom,
! [X2: nat,Xs: list_nat,F: real > nat,Ys: list_real] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_real_nat @ F @ Ys ) )
=> ? [Z: real,Zs: list_real] :
( ( Ys
= ( cons_real @ Z @ Zs ) )
& ( X2
= ( F @ Z ) )
& ( Xs
= ( map_real_nat @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_17_Cons__eq__map__D,axiom,
! [X2: real,Xs: list_real,F: nat > real,Ys: list_nat] :
( ( ( cons_real @ X2 @ Xs )
= ( map_nat_real @ F @ Ys ) )
=> ? [Z: nat,Zs: list_nat] :
( ( Ys
= ( cons_nat @ Z @ Zs ) )
& ( X2
= ( F @ Z ) )
& ( Xs
= ( map_nat_real @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_18_Cons__eq__map__D,axiom,
! [X2: real,Xs: list_real,F: real > real,Ys: list_real] :
( ( ( cons_real @ X2 @ Xs )
= ( map_real_real @ F @ Ys ) )
=> ? [Z: real,Zs: list_real] :
( ( Ys
= ( cons_real @ Z @ Zs ) )
& ( X2
= ( F @ Z ) )
& ( Xs
= ( map_real_real @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_19_Cons__eq__map__D,axiom,
! [X2: real,Xs: list_real,F: complex > real,Ys: list_complex] :
( ( ( cons_real @ X2 @ Xs )
= ( map_complex_real @ F @ Ys ) )
=> ? [Z: complex,Zs: list_complex] :
( ( Ys
= ( cons_complex @ Z @ Zs ) )
& ( X2
= ( F @ Z ) )
& ( Xs
= ( map_complex_real @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_20_Cons__eq__map__D,axiom,
! [X2: complex,Xs: list_complex,F: complex > complex,Ys: list_complex] :
( ( ( cons_complex @ X2 @ Xs )
= ( map_complex_complex @ F @ Ys ) )
=> ? [Z: complex,Zs: list_complex] :
( ( Ys
= ( cons_complex @ Z @ Zs ) )
& ( X2
= ( F @ Z ) )
& ( Xs
= ( map_complex_complex @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_21_Cons__eq__map__D,axiom,
! [X2: complex,Xs: list_complex,F: nat > complex,Ys: list_nat] :
( ( ( cons_complex @ X2 @ Xs )
= ( map_nat_complex @ F @ Ys ) )
=> ? [Z: nat,Zs: list_nat] :
( ( Ys
= ( cons_nat @ Z @ Zs ) )
& ( X2
= ( F @ Z ) )
& ( Xs
= ( map_nat_complex @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_22_Cons__eq__map__D,axiom,
! [X2: nat,Xs: list_nat,F: complex > nat,Ys: list_complex] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_complex_nat @ F @ Ys ) )
=> ? [Z: complex,Zs: list_complex] :
( ( Ys
= ( cons_complex @ Z @ Zs ) )
& ( X2
= ( F @ Z ) )
& ( Xs
= ( map_complex_nat @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_23_Cons__eq__map__D,axiom,
! [X2: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_nat_nat @ F @ Ys ) )
=> ? [Z: nat,Zs: list_nat] :
( ( Ys
= ( cons_nat @ Z @ Zs ) )
& ( X2
= ( F @ Z ) )
& ( Xs
= ( map_nat_nat @ F @ Zs ) ) ) ) ).
% Cons_eq_map_D
thf(fact_24_map__eq__Cons__D,axiom,
! [F: real > complex,Xs: list_real,Y: complex,Ys: list_complex] :
( ( ( map_real_complex @ F @ Xs )
= ( cons_complex @ Y @ Ys ) )
=> ? [Z: real,Zs: list_real] :
( ( Xs
= ( cons_real @ Z @ Zs ) )
& ( ( F @ Z )
= Y )
& ( ( map_real_complex @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_25_map__eq__Cons__D,axiom,
! [F: real > nat,Xs: list_real,Y: nat,Ys: list_nat] :
( ( ( map_real_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
=> ? [Z: real,Zs: list_real] :
( ( Xs
= ( cons_real @ Z @ Zs ) )
& ( ( F @ Z )
= Y )
& ( ( map_real_nat @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_26_map__eq__Cons__D,axiom,
! [F: nat > real,Xs: list_nat,Y: real,Ys: list_real] :
( ( ( map_nat_real @ F @ Xs )
= ( cons_real @ Y @ Ys ) )
=> ? [Z: nat,Zs: list_nat] :
( ( Xs
= ( cons_nat @ Z @ Zs ) )
& ( ( F @ Z )
= Y )
& ( ( map_nat_real @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_27_map__eq__Cons__D,axiom,
! [F: real > real,Xs: list_real,Y: real,Ys: list_real] :
( ( ( map_real_real @ F @ Xs )
= ( cons_real @ Y @ Ys ) )
=> ? [Z: real,Zs: list_real] :
( ( Xs
= ( cons_real @ Z @ Zs ) )
& ( ( F @ Z )
= Y )
& ( ( map_real_real @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_28_map__eq__Cons__D,axiom,
! [F: complex > real,Xs: list_complex,Y: real,Ys: list_real] :
( ( ( map_complex_real @ F @ Xs )
= ( cons_real @ Y @ Ys ) )
=> ? [Z: complex,Zs: list_complex] :
( ( Xs
= ( cons_complex @ Z @ Zs ) )
& ( ( F @ Z )
= Y )
& ( ( map_complex_real @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_29_map__eq__Cons__D,axiom,
! [F: complex > complex,Xs: list_complex,Y: complex,Ys: list_complex] :
( ( ( map_complex_complex @ F @ Xs )
= ( cons_complex @ Y @ Ys ) )
=> ? [Z: complex,Zs: list_complex] :
( ( Xs
= ( cons_complex @ Z @ Zs ) )
& ( ( F @ Z )
= Y )
& ( ( map_complex_complex @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_30_map__eq__Cons__D,axiom,
! [F: nat > complex,Xs: list_nat,Y: complex,Ys: list_complex] :
( ( ( map_nat_complex @ F @ Xs )
= ( cons_complex @ Y @ Ys ) )
=> ? [Z: nat,Zs: list_nat] :
( ( Xs
= ( cons_nat @ Z @ Zs ) )
& ( ( F @ Z )
= Y )
& ( ( map_nat_complex @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_31_map__eq__Cons__D,axiom,
! [F: complex > nat,Xs: list_complex,Y: nat,Ys: list_nat] :
( ( ( map_complex_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
=> ? [Z: complex,Zs: list_complex] :
( ( Xs
= ( cons_complex @ Z @ Zs ) )
& ( ( F @ Z )
= Y )
& ( ( map_complex_nat @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_32_map__eq__Cons__D,axiom,
! [F: nat > nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
=> ? [Z: nat,Zs: list_nat] :
( ( Xs
= ( cons_nat @ Z @ Zs ) )
& ( ( F @ Z )
= Y )
& ( ( map_nat_nat @ F @ Zs )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_33_Cons__eq__map__conv,axiom,
! [X2: complex,Xs: list_complex,F: real > complex,Ys: list_real] :
( ( ( cons_complex @ X2 @ Xs )
= ( map_real_complex @ F @ Ys ) )
= ( ? [Z2: real,Zs2: list_real] :
( ( Ys
= ( cons_real @ Z2 @ Zs2 ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs
= ( map_real_complex @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_34_Cons__eq__map__conv,axiom,
! [X2: nat,Xs: list_nat,F: real > nat,Ys: list_real] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_real_nat @ F @ Ys ) )
= ( ? [Z2: real,Zs2: list_real] :
( ( Ys
= ( cons_real @ Z2 @ Zs2 ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs
= ( map_real_nat @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_35_Cons__eq__map__conv,axiom,
! [X2: real,Xs: list_real,F: nat > real,Ys: list_nat] :
( ( ( cons_real @ X2 @ Xs )
= ( map_nat_real @ F @ Ys ) )
= ( ? [Z2: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z2 @ Zs2 ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs
= ( map_nat_real @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_36_Cons__eq__map__conv,axiom,
! [X2: real,Xs: list_real,F: real > real,Ys: list_real] :
( ( ( cons_real @ X2 @ Xs )
= ( map_real_real @ F @ Ys ) )
= ( ? [Z2: real,Zs2: list_real] :
( ( Ys
= ( cons_real @ Z2 @ Zs2 ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs
= ( map_real_real @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_37_Cons__eq__map__conv,axiom,
! [X2: real,Xs: list_real,F: complex > real,Ys: list_complex] :
( ( ( cons_real @ X2 @ Xs )
= ( map_complex_real @ F @ Ys ) )
= ( ? [Z2: complex,Zs2: list_complex] :
( ( Ys
= ( cons_complex @ Z2 @ Zs2 ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs
= ( map_complex_real @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_38_Cons__eq__map__conv,axiom,
! [X2: complex,Xs: list_complex,F: complex > complex,Ys: list_complex] :
( ( ( cons_complex @ X2 @ Xs )
= ( map_complex_complex @ F @ Ys ) )
= ( ? [Z2: complex,Zs2: list_complex] :
( ( Ys
= ( cons_complex @ Z2 @ Zs2 ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs
= ( map_complex_complex @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_39_Cons__eq__map__conv,axiom,
! [X2: complex,Xs: list_complex,F: nat > complex,Ys: list_nat] :
( ( ( cons_complex @ X2 @ Xs )
= ( map_nat_complex @ F @ Ys ) )
= ( ? [Z2: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z2 @ Zs2 ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs
= ( map_nat_complex @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_40_Cons__eq__map__conv,axiom,
! [X2: nat,Xs: list_nat,F: complex > nat,Ys: list_complex] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_complex_nat @ F @ Ys ) )
= ( ? [Z2: complex,Zs2: list_complex] :
( ( Ys
= ( cons_complex @ Z2 @ Zs2 ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs
= ( map_complex_nat @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_41_Cons__eq__map__conv,axiom,
! [X2: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_nat_nat @ F @ Ys ) )
= ( ? [Z2: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z2 @ Zs2 ) )
& ( X2
= ( F @ Z2 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_42_map__eq__Cons__conv,axiom,
! [F: complex > complex,Xs: list_complex,Y: complex,Ys: list_complex] :
( ( ( map_complex_complex @ F @ Xs )
= ( cons_complex @ Y @ Ys ) )
= ( ? [Z2: complex,Zs2: list_complex] :
( ( Xs
= ( cons_complex @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_complex_complex @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_43_map__eq__Cons__conv,axiom,
! [F: nat > complex,Xs: list_nat,Y: complex,Ys: list_complex] :
( ( ( map_nat_complex @ F @ Xs )
= ( cons_complex @ Y @ Ys ) )
= ( ? [Z2: nat,Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_nat_complex @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_44_map__eq__Cons__conv,axiom,
! [F: real > complex,Xs: list_real,Y: complex,Ys: list_complex] :
( ( ( map_real_complex @ F @ Xs )
= ( cons_complex @ Y @ Ys ) )
= ( ? [Z2: real,Zs2: list_real] :
( ( Xs
= ( cons_real @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_real_complex @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_45_map__eq__Cons__conv,axiom,
! [F: complex > nat,Xs: list_complex,Y: nat,Ys: list_nat] :
( ( ( map_complex_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
= ( ? [Z2: complex,Zs2: list_complex] :
( ( Xs
= ( cons_complex @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_complex_nat @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_46_map__eq__Cons__conv,axiom,
! [F: nat > nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
= ( ? [Z2: nat,Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_nat_nat @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_47_map__eq__Cons__conv,axiom,
! [F: real > nat,Xs: list_real,Y: nat,Ys: list_nat] :
( ( ( map_real_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
= ( ? [Z2: real,Zs2: list_real] :
( ( Xs
= ( cons_real @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_real_nat @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_48_map__eq__Cons__conv,axiom,
! [F: complex > real,Xs: list_complex,Y: real,Ys: list_real] :
( ( ( map_complex_real @ F @ Xs )
= ( cons_real @ Y @ Ys ) )
= ( ? [Z2: complex,Zs2: list_complex] :
( ( Xs
= ( cons_complex @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_complex_real @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_49_map__eq__Cons__conv,axiom,
! [F: nat > real,Xs: list_nat,Y: real,Ys: list_real] :
( ( ( map_nat_real @ F @ Xs )
= ( cons_real @ Y @ Ys ) )
= ( ? [Z2: nat,Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_nat_real @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_50_map__eq__Cons__conv,axiom,
! [F: real > real,Xs: list_real,Y: real,Ys: list_real] :
( ( ( map_real_real @ F @ Xs )
= ( cons_real @ Y @ Ys ) )
= ( ? [Z2: real,Zs2: list_real] :
( ( Xs
= ( cons_real @ Z2 @ Zs2 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_real_real @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_51__092_060open_0621_A_D_Aec_A_061_Aeq__comps_A_Imap_ARe_A_Ix_A_D_Ay_A_D_Al_J_J_092_060close_062,axiom,
( ( cons_nat @ one_one_nat @ ec )
= ( commut8680161604938074397s_real @ ( map_complex_real @ re @ ( cons_complex @ x @ ( cons_complex @ y @ la ) ) ) ) ) ).
% \<open>1 # ec = eq_comps (map Re (x # y # l))\<close>
thf(fact_52_list_Omap__ident,axiom,
! [T: list_nat] :
( ( map_nat_nat
@ ^ [X3: nat] : X3
@ T )
= T ) ).
% list.map_ident
thf(fact_53_list_Omap__ident,axiom,
! [T: list_complex] :
( ( map_complex_complex
@ ^ [X3: complex] : X3
@ T )
= T ) ).
% list.map_ident
thf(fact_54_map__ident,axiom,
( ( map_nat_nat
@ ^ [X3: nat] : X3 )
= ( ^ [Xs2: list_nat] : Xs2 ) ) ).
% map_ident
thf(fact_55_map__ident,axiom,
( ( map_complex_complex
@ ^ [X3: complex] : X3 )
= ( ^ [Xs2: list_complex] : Xs2 ) ) ).
% map_ident
thf(fact_56_assms_I2_J,axiom,
( m
= ( commut93809757773076895omplex @ l ) ) ).
% assms(2)
thf(fact_57_assms_I1_J,axiom,
! [X: complex] :
( ( member_complex2 @ X @ ( set_complex2 @ l ) )
=> ( member_complex2 @ X @ real_V2521375963428798218omplex ) ) ).
% assms(1)
thf(fact_58__092_060open_062m_A_061_A1_A_D_Aec_092_060close_062,axiom,
( ma
= ( cons_nat @ one_one_nat @ ec ) ) ).
% \<open>m = 1 # ec\<close>
thf(fact_59__C3_Ohyps_C,axiom,
! [M: list_nat] :
( ! [X4: complex] :
( ( member_complex2 @ X4 @ ( set_complex2 @ ( cons_complex @ y @ la ) ) )
=> ( member_complex2 @ X4 @ real_V2521375963428798218omplex ) )
=> ( ( M
= ( commut93809757773076895omplex @ ( cons_complex @ y @ la ) ) )
=> ( M
= ( commut8680161604938074397s_real @ ( map_complex_real @ re @ ( cons_complex @ y @ la ) ) ) ) ) ) ).
% "3.hyps"
thf(fact_60_set__ConsD,axiom,
! [Y: complex,X2: complex,Xs: list_complex] :
( ( member_complex2 @ Y @ ( set_complex2 @ ( cons_complex @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_complex2 @ Y @ ( set_complex2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_61_set__ConsD,axiom,
! [Y: nat,X2: nat,Xs: list_nat] :
( ( member_nat2 @ Y @ ( set_nat2 @ ( cons_nat @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_nat2 @ Y @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_62_set__ConsD,axiom,
! [Y: real,X2: real,Xs: list_real] :
( ( member_real2 @ Y @ ( set_real2 @ ( cons_real @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_real2 @ Y @ ( set_real2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_63_list_Oset__cases,axiom,
! [E: complex,A: list_complex] :
( ( member_complex2 @ E @ ( set_complex2 @ A ) )
=> ( ! [Z22: list_complex] :
( A
!= ( cons_complex @ E @ Z22 ) )
=> ~ ! [Z1: complex,Z22: list_complex] :
( ( A
= ( cons_complex @ Z1 @ Z22 ) )
=> ~ ( member_complex2 @ E @ ( set_complex2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_64_list_Oset__cases,axiom,
! [E: nat,A: list_nat] :
( ( member_nat2 @ E @ ( set_nat2 @ A ) )
=> ( ! [Z22: list_nat] :
( A
!= ( cons_nat @ E @ Z22 ) )
=> ~ ! [Z1: nat,Z22: list_nat] :
( ( A
= ( cons_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat2 @ E @ ( set_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_65_list_Oset__cases,axiom,
! [E: real,A: list_real] :
( ( member_real2 @ E @ ( set_real2 @ A ) )
=> ( ! [Z22: list_real] :
( A
!= ( cons_real @ E @ Z22 ) )
=> ~ ! [Z1: real,Z22: list_real] :
( ( A
= ( cons_real @ Z1 @ Z22 ) )
=> ~ ( member_real2 @ E @ ( set_real2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_66_list_Oset__intros_I1_J,axiom,
! [X21: complex,X22: list_complex] : ( member_complex2 @ X21 @ ( set_complex2 @ ( cons_complex @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_67_list_Oset__intros_I1_J,axiom,
! [X21: nat,X22: list_nat] : ( member_nat2 @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_68_list_Oset__intros_I1_J,axiom,
! [X21: real,X22: list_real] : ( member_real2 @ X21 @ ( set_real2 @ ( cons_real @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_69_list_Oset__intros_I2_J,axiom,
! [Y: complex,X22: list_complex,X21: complex] :
( ( member_complex2 @ Y @ ( set_complex2 @ X22 ) )
=> ( member_complex2 @ Y @ ( set_complex2 @ ( cons_complex @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_70_list_Oset__intros_I2_J,axiom,
! [Y: nat,X22: list_nat,X21: nat] :
( ( member_nat2 @ Y @ ( set_nat2 @ X22 ) )
=> ( member_nat2 @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_71_list_Oset__intros_I2_J,axiom,
! [Y: real,X22: list_real,X21: real] :
( ( member_real2 @ Y @ ( set_real2 @ X22 ) )
=> ( member_real2 @ Y @ ( set_real2 @ ( cons_real @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_72_map__eq__conv,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Xs ) )
= ( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) ) ) ) ).
% map_eq_conv
thf(fact_73_map__eq__conv,axiom,
! [F: nat > complex,Xs: list_nat,G: nat > complex] :
( ( ( map_nat_complex @ F @ Xs )
= ( map_nat_complex @ G @ Xs ) )
= ( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) ) ) ) ).
% map_eq_conv
thf(fact_74_map__eq__conv,axiom,
! [F: complex > real,Xs: list_complex,G: complex > real] :
( ( ( map_complex_real @ F @ Xs )
= ( map_complex_real @ G @ Xs ) )
= ( ! [X3: complex] :
( ( member_complex2 @ X3 @ ( set_complex2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) ) ) ) ).
% map_eq_conv
thf(fact_75_map__eq__conv,axiom,
! [F: complex > nat,Xs: list_complex,G: complex > nat] :
( ( ( map_complex_nat @ F @ Xs )
= ( map_complex_nat @ G @ Xs ) )
= ( ! [X3: complex] :
( ( member_complex2 @ X3 @ ( set_complex2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) ) ) ) ).
% map_eq_conv
thf(fact_76_map__eq__conv,axiom,
! [F: complex > complex,Xs: list_complex,G: complex > complex] :
( ( ( map_complex_complex @ F @ Xs )
= ( map_complex_complex @ G @ Xs ) )
= ( ! [X3: complex] :
( ( member_complex2 @ X3 @ ( set_complex2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) ) ) ) ).
% map_eq_conv
thf(fact_77_ex__map__conv,axiom,
! [Ys: list_real,F: complex > real] :
( ( ? [Xs2: list_complex] :
( Ys
= ( map_complex_real @ F @ Xs2 ) ) )
= ( ! [X3: real] :
( ( member_real2 @ X3 @ ( set_real2 @ Ys ) )
=> ? [Y2: complex] :
( X3
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_78_ex__map__conv,axiom,
! [Ys: list_nat,F: nat > nat] :
( ( ? [Xs2: list_nat] :
( Ys
= ( map_nat_nat @ F @ Xs2 ) ) )
= ( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Ys ) )
=> ? [Y2: nat] :
( X3
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_79_ex__map__conv,axiom,
! [Ys: list_nat,F: complex > nat] :
( ( ? [Xs2: list_complex] :
( Ys
= ( map_complex_nat @ F @ Xs2 ) ) )
= ( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Ys ) )
=> ? [Y2: complex] :
( X3
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_80_ex__map__conv,axiom,
! [Ys: list_complex,F: nat > complex] :
( ( ? [Xs2: list_nat] :
( Ys
= ( map_nat_complex @ F @ Xs2 ) ) )
= ( ! [X3: complex] :
( ( member_complex2 @ X3 @ ( set_complex2 @ Ys ) )
=> ? [Y2: nat] :
( X3
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_81_ex__map__conv,axiom,
! [Ys: list_complex,F: complex > complex] :
( ( ? [Xs2: list_complex] :
( Ys
= ( map_complex_complex @ F @ Xs2 ) ) )
= ( ! [X3: complex] :
( ( member_complex2 @ X3 @ ( set_complex2 @ Ys ) )
=> ? [Y2: complex] :
( X3
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_82_map__cong,axiom,
! [Xs: list_nat,Ys: list_nat,F: nat > nat,G: nat > nat] :
( ( Xs = Ys )
=> ( ! [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Ys ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_83_map__cong,axiom,
! [Xs: list_nat,Ys: list_nat,F: nat > complex,G: nat > complex] :
( ( Xs = Ys )
=> ( ! [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Ys ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_nat_complex @ F @ Xs )
= ( map_nat_complex @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_84_map__cong,axiom,
! [Xs: list_complex,Ys: list_complex,F: complex > real,G: complex > real] :
( ( Xs = Ys )
=> ( ! [X4: complex] :
( ( member_complex2 @ X4 @ ( set_complex2 @ Ys ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_complex_real @ F @ Xs )
= ( map_complex_real @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_85_map__cong,axiom,
! [Xs: list_complex,Ys: list_complex,F: complex > nat,G: complex > nat] :
( ( Xs = Ys )
=> ( ! [X4: complex] :
( ( member_complex2 @ X4 @ ( set_complex2 @ Ys ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_complex_nat @ F @ Xs )
= ( map_complex_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_86_map__cong,axiom,
! [Xs: list_complex,Ys: list_complex,F: complex > complex,G: complex > complex] :
( ( Xs = Ys )
=> ( ! [X4: complex] :
( ( member_complex2 @ X4 @ ( set_complex2 @ Ys ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_complex_complex @ F @ Xs )
= ( map_complex_complex @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_87_map__idI,axiom,
! [Xs: list_nat,F: nat > nat] :
( ! [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X4 )
= X4 ) )
=> ( ( map_nat_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_88_map__idI,axiom,
! [Xs: list_complex,F: complex > complex] :
( ! [X4: complex] :
( ( member_complex2 @ X4 @ ( set_complex2 @ Xs ) )
=> ( ( F @ X4 )
= X4 ) )
=> ( ( map_complex_complex @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_89_map__ext,axiom,
! [Xs: list_nat,F: nat > nat,G: nat > nat] :
( ! [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_90_map__ext,axiom,
! [Xs: list_nat,F: nat > complex,G: nat > complex] :
( ! [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_nat_complex @ F @ Xs )
= ( map_nat_complex @ G @ Xs ) ) ) ).
% map_ext
thf(fact_91_map__ext,axiom,
! [Xs: list_complex,F: complex > real,G: complex > real] :
( ! [X4: complex] :
( ( member_complex2 @ X4 @ ( set_complex2 @ Xs ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_complex_real @ F @ Xs )
= ( map_complex_real @ G @ Xs ) ) ) ).
% map_ext
thf(fact_92_map__ext,axiom,
! [Xs: list_complex,F: complex > nat,G: complex > nat] :
( ! [X4: complex] :
( ( member_complex2 @ X4 @ ( set_complex2 @ Xs ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_complex_nat @ F @ Xs )
= ( map_complex_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_93_map__ext,axiom,
! [Xs: list_complex,F: complex > complex,G: complex > complex] :
( ! [X4: complex] :
( ( member_complex2 @ X4 @ ( set_complex2 @ Xs ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_complex_complex @ F @ Xs )
= ( map_complex_complex @ G @ Xs ) ) ) ).
% map_ext
thf(fact_94_list_Omap__ident__strong,axiom,
! [T: list_nat,F: nat > nat] :
( ! [Z: nat] :
( ( member_nat2 @ Z @ ( set_nat2 @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_nat_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_95_list_Omap__ident__strong,axiom,
! [T: list_complex,F: complex > complex] :
( ! [Z: complex] :
( ( member_complex2 @ Z @ ( set_complex2 @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_complex_complex @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_96_list_Oinj__map__strong,axiom,
! [X2: list_nat,Xa: list_nat,F: nat > nat,Fa: nat > nat] :
( ! [Z: nat,Za: nat] :
( ( member_nat2 @ Z @ ( set_nat2 @ X2 ) )
=> ( ( member_nat2 @ Za @ ( set_nat2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_nat_nat @ F @ X2 )
= ( map_nat_nat @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_97_list_Oinj__map__strong,axiom,
! [X2: list_nat,Xa: list_nat,F: nat > complex,Fa: nat > complex] :
( ! [Z: nat,Za: nat] :
( ( member_nat2 @ Z @ ( set_nat2 @ X2 ) )
=> ( ( member_nat2 @ Za @ ( set_nat2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_nat_complex @ F @ X2 )
= ( map_nat_complex @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_98_list_Oinj__map__strong,axiom,
! [X2: list_complex,Xa: list_complex,F: complex > real,Fa: complex > real] :
( ! [Z: complex,Za: complex] :
( ( member_complex2 @ Z @ ( set_complex2 @ X2 ) )
=> ( ( member_complex2 @ Za @ ( set_complex2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_complex_real @ F @ X2 )
= ( map_complex_real @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_99_list_Oinj__map__strong,axiom,
! [X2: list_complex,Xa: list_complex,F: complex > nat,Fa: complex > nat] :
( ! [Z: complex,Za: complex] :
( ( member_complex2 @ Z @ ( set_complex2 @ X2 ) )
=> ( ( member_complex2 @ Za @ ( set_complex2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_complex_nat @ F @ X2 )
= ( map_complex_nat @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_100_list_Oinj__map__strong,axiom,
! [X2: list_complex,Xa: list_complex,F: complex > complex,Fa: complex > complex] :
( ! [Z: complex,Za: complex] :
( ( member_complex2 @ Z @ ( set_complex2 @ X2 ) )
=> ( ( member_complex2 @ Za @ ( set_complex2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_complex_complex @ F @ X2 )
= ( map_complex_complex @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_101_list_Omap__cong0,axiom,
! [X2: list_nat,F: nat > nat,G: nat > nat] :
( ! [Z: nat] :
( ( member_nat2 @ Z @ ( set_nat2 @ X2 ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_nat_nat @ F @ X2 )
= ( map_nat_nat @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_102_list_Omap__cong0,axiom,
! [X2: list_nat,F: nat > complex,G: nat > complex] :
( ! [Z: nat] :
( ( member_nat2 @ Z @ ( set_nat2 @ X2 ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_nat_complex @ F @ X2 )
= ( map_nat_complex @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_103_list_Omap__cong0,axiom,
! [X2: list_complex,F: complex > real,G: complex > real] :
( ! [Z: complex] :
( ( member_complex2 @ Z @ ( set_complex2 @ X2 ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_complex_real @ F @ X2 )
= ( map_complex_real @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_104_list_Omap__cong0,axiom,
! [X2: list_complex,F: complex > nat,G: complex > nat] :
( ! [Z: complex] :
( ( member_complex2 @ Z @ ( set_complex2 @ X2 ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_complex_nat @ F @ X2 )
= ( map_complex_nat @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_105_list_Omap__cong0,axiom,
! [X2: list_complex,F: complex > complex,G: complex > complex] :
( ! [Z: complex] :
( ( member_complex2 @ Z @ ( set_complex2 @ X2 ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_complex_complex @ F @ X2 )
= ( map_complex_complex @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_106_list_Omap__cong,axiom,
! [X2: list_nat,Ya: list_nat,F: nat > nat,G: nat > nat] :
( ( X2 = Ya )
=> ( ! [Z: nat] :
( ( member_nat2 @ Z @ ( set_nat2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_nat_nat @ F @ X2 )
= ( map_nat_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_107_list_Omap__cong,axiom,
! [X2: list_nat,Ya: list_nat,F: nat > complex,G: nat > complex] :
( ( X2 = Ya )
=> ( ! [Z: nat] :
( ( member_nat2 @ Z @ ( set_nat2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_nat_complex @ F @ X2 )
= ( map_nat_complex @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_108_list_Omap__cong,axiom,
! [X2: list_complex,Ya: list_complex,F: complex > real,G: complex > real] :
( ( X2 = Ya )
=> ( ! [Z: complex] :
( ( member_complex2 @ Z @ ( set_complex2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_complex_real @ F @ X2 )
= ( map_complex_real @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_109_list_Omap__cong,axiom,
! [X2: list_complex,Ya: list_complex,F: complex > nat,G: complex > nat] :
( ( X2 = Ya )
=> ( ! [Z: complex] :
( ( member_complex2 @ Z @ ( set_complex2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_complex_nat @ F @ X2 )
= ( map_complex_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_110_list_Omap__cong,axiom,
! [X2: list_complex,Ya: list_complex,F: complex > complex,G: complex > complex] :
( ( X2 = Ya )
=> ( ! [Z: complex] :
( ( member_complex2 @ Z @ ( set_complex2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_complex_complex @ F @ X2 )
= ( map_complex_complex @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_111_not__Cons__self2,axiom,
! [X2: complex,Xs: list_complex] :
( ( cons_complex @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_112_not__Cons__self2,axiom,
! [X2: nat,Xs: list_nat] :
( ( cons_nat @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_113_not__Cons__self2,axiom,
! [X2: real,Xs: list_real] :
( ( cons_real @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_114_list_Oinject,axiom,
! [X21: complex,X22: list_complex,Y21: complex,Y22: list_complex] :
( ( ( cons_complex @ X21 @ X22 )
= ( cons_complex @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_115_list_Oinject,axiom,
! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
( ( ( cons_nat @ X21 @ X22 )
= ( cons_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_116_list_Oinject,axiom,
! [X21: real,X22: list_real,Y21: real,Y22: list_real] :
( ( ( cons_real @ X21 @ X22 )
= ( cons_real @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_117_Reals__1,axiom,
member_real2 @ one_one_real @ real_V470468836141973256s_real ).
% Reals_1
thf(fact_118_Reals__1,axiom,
member_complex2 @ one_one_complex @ real_V2521375963428798218omplex ).
% Reals_1
thf(fact_119_map__eq__map__tailrec,axiom,
map_complex_real = map_ta5686879364588479338x_real ).
% map_eq_map_tailrec
thf(fact_120_map__eq__map__tailrec,axiom,
map_nat_nat = map_tailrec_nat_nat ).
% map_eq_map_tailrec
thf(fact_121_map__eq__map__tailrec,axiom,
map_nat_complex = map_ta5579134486830519694omplex ).
% map_eq_map_tailrec
thf(fact_122_map__eq__map__tailrec,axiom,
map_complex_nat = map_ta9198917811265257358ex_nat ).
% map_eq_map_tailrec
thf(fact_123_map__eq__map__tailrec,axiom,
map_complex_complex = map_ta3177854533886046316omplex ).
% map_eq_map_tailrec
thf(fact_124_List_Oinsert__def,axiom,
( insert_complex
= ( ^ [X3: complex,Xs2: list_complex] : ( if_list_complex @ ( member_complex2 @ X3 @ ( set_complex2 @ Xs2 ) ) @ Xs2 @ ( cons_complex @ X3 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_125_List_Oinsert__def,axiom,
( insert_nat
= ( ^ [X3: nat,Xs2: list_nat] : ( if_list_nat @ ( member_nat2 @ X3 @ ( set_nat2 @ Xs2 ) ) @ Xs2 @ ( cons_nat @ X3 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_126_List_Oinsert__def,axiom,
( insert_real
= ( ^ [X3: real,Xs2: list_real] : ( if_list_real @ ( member_real2 @ X3 @ ( set_real2 @ Xs2 ) ) @ Xs2 @ ( cons_real @ X3 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_127_not__in__set__insert,axiom,
! [X2: complex,Xs: list_complex] :
( ~ ( member_complex2 @ X2 @ ( set_complex2 @ Xs ) )
=> ( ( insert_complex @ X2 @ Xs )
= ( cons_complex @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_128_not__in__set__insert,axiom,
! [X2: nat,Xs: list_nat] :
( ~ ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X2 @ Xs )
= ( cons_nat @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_129_not__in__set__insert,axiom,
! [X2: real,Xs: list_real] :
( ~ ( member_real2 @ X2 @ ( set_real2 @ Xs ) )
=> ( ( insert_real @ X2 @ Xs )
= ( cons_real @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_130_one__natural_Orsp,axiom,
one_one_nat = one_one_nat ).
% one_natural.rsp
thf(fact_131_mem__Collect__eq,axiom,
! [A: complex,P: complex > $o] :
( ( member_complex2 @ A @ ( collect_complex @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_132_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat2 @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_133_Collect__mem__eq,axiom,
! [A2: set_complex] :
( ( collect_complex
@ ^ [X3: complex] : ( member_complex2 @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_134_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat2 @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_135_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X4: nat] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_136_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_137_one__reorient,axiom,
! [X2: complex] :
( ( one_one_complex = X2 )
= ( X2 = one_one_complex ) ) ).
% one_reorient
thf(fact_138_one__reorient,axiom,
! [X2: real] :
( ( one_one_real = X2 )
= ( X2 = one_one_real ) ) ).
% one_reorient
thf(fact_139_member__def,axiom,
( member_nat
= ( ^ [Xs2: list_nat,X3: nat] : ( member_nat2 @ X3 @ ( set_nat2 @ Xs2 ) ) ) ) ).
% member_def
thf(fact_140_member__def,axiom,
( member_complex
= ( ^ [Xs2: list_complex,X3: complex] : ( member_complex2 @ X3 @ ( set_complex2 @ Xs2 ) ) ) ) ).
% member_def
thf(fact_141_member__rec_I1_J,axiom,
! [X2: complex,Xs: list_complex,Y: complex] :
( ( member_complex @ ( cons_complex @ X2 @ Xs ) @ Y )
= ( ( X2 = Y )
| ( member_complex @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_142_member__rec_I1_J,axiom,
! [X2: nat,Xs: list_nat,Y: nat] :
( ( member_nat @ ( cons_nat @ X2 @ Xs ) @ Y )
= ( ( X2 = Y )
| ( member_nat @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_143_member__rec_I1_J,axiom,
! [X2: real,Xs: list_real,Y: real] :
( ( member_real @ ( cons_real @ X2 @ Xs ) @ Y )
= ( ( X2 = Y )
| ( member_real @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_144_list__ex1__iff,axiom,
( list_ex1_nat
= ( ^ [P2: nat > $o,Xs2: list_nat] :
? [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs2 ) )
& ( P2 @ X3 )
& ! [Y2: nat] :
( ( ( member_nat2 @ Y2 @ ( set_nat2 @ Xs2 ) )
& ( P2 @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_145_list__ex1__iff,axiom,
( list_ex1_complex
= ( ^ [P2: complex > $o,Xs2: list_complex] :
? [X3: complex] :
( ( member_complex2 @ X3 @ ( set_complex2 @ Xs2 ) )
& ( P2 @ X3 )
& ! [Y2: complex] :
( ( ( member_complex2 @ Y2 @ ( set_complex2 @ Xs2 ) )
& ( P2 @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_146_insort__insert__triv,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( linord1921536354676448932at_nat
@ ^ [X3: nat] : X3
@ X2
@ Xs )
= Xs ) ) ).
% insort_insert_triv
thf(fact_147_in__set__insert,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_148_in__set__insert,axiom,
! [X2: complex,Xs: list_complex] :
( ( member_complex2 @ X2 @ ( set_complex2 @ Xs ) )
=> ( ( insert_complex @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_149_can__select__set__list__ex1,axiom,
! [P: complex > $o,A2: list_complex] :
( ( can_select_complex @ P @ ( set_complex2 @ A2 ) )
= ( list_ex1_complex @ P @ A2 ) ) ).
% can_select_set_list_ex1
thf(fact_150_list__ex1__simps_I2_J,axiom,
! [P: complex > $o,X2: complex,Xs: list_complex] :
( ( list_ex1_complex @ P @ ( cons_complex @ X2 @ Xs ) )
= ( ( ( P @ X2 )
=> ( list_all_complex
@ ^ [Y2: complex] :
( ~ ( P @ Y2 )
| ( X2 = Y2 ) )
@ Xs ) )
& ( ~ ( P @ X2 )
=> ( list_ex1_complex @ P @ Xs ) ) ) ) ).
% list_ex1_simps(2)
thf(fact_151_list__ex1__simps_I2_J,axiom,
! [P: nat > $o,X2: nat,Xs: list_nat] :
( ( list_ex1_nat @ P @ ( cons_nat @ X2 @ Xs ) )
= ( ( ( P @ X2 )
=> ( list_all_nat
@ ^ [Y2: nat] :
( ~ ( P @ Y2 )
| ( X2 = Y2 ) )
@ Xs ) )
& ( ~ ( P @ X2 )
=> ( list_ex1_nat @ P @ Xs ) ) ) ) ).
% list_ex1_simps(2)
thf(fact_152_list__ex1__simps_I2_J,axiom,
! [P: real > $o,X2: real,Xs: list_real] :
( ( list_ex1_real @ P @ ( cons_real @ X2 @ Xs ) )
= ( ( ( P @ X2 )
=> ( list_all_real
@ ^ [Y2: real] :
( ~ ( P @ Y2 )
| ( X2 = Y2 ) )
@ Xs ) )
& ( ~ ( P @ X2 )
=> ( list_ex1_real @ P @ Xs ) ) ) ) ).
% list_ex1_simps(2)
thf(fact_153_one__complex_Osel_I1_J,axiom,
( ( re @ one_one_complex )
= one_one_real ) ).
% one_complex.sel(1)
thf(fact_154_insort__insert__insort,axiom,
! [X2: nat,Xs: list_nat] :
( ~ ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( linord1921536354676448932at_nat
@ ^ [X3: nat] : X3
@ X2
@ Xs )
= ( linord8961336180081300637at_nat
@ ^ [X3: nat] : X3
@ X2
@ Xs ) ) ) ).
% insort_insert_insort
thf(fact_155_eq__comps__neq__0,axiom,
! [A: nat,M: list_nat,L: list_real] :
( ( ( cons_nat @ A @ M )
= ( commut8680161604938074397s_real @ L ) )
=> ( A != zero_zero_nat ) ) ).
% eq_comps_neq_0
thf(fact_156_eq__comps__neq__0,axiom,
! [A: nat,M: list_nat,L: list_complex] :
( ( ( cons_nat @ A @ M )
= ( commut93809757773076895omplex @ L ) )
=> ( A != zero_zero_nat ) ) ).
% eq_comps_neq_0
thf(fact_157_insert__Nil,axiom,
! [X2: complex] :
( ( insert_complex @ X2 @ nil_complex )
= ( cons_complex @ X2 @ nil_complex ) ) ).
% insert_Nil
thf(fact_158_insert__Nil,axiom,
! [X2: nat] :
( ( insert_nat @ X2 @ nil_nat )
= ( cons_nat @ X2 @ nil_nat ) ) ).
% insert_Nil
thf(fact_159_insert__Nil,axiom,
! [X2: real] :
( ( insert_real @ X2 @ nil_real )
= ( cons_real @ X2 @ nil_real ) ) ).
% insert_Nil
thf(fact_160_List_Oset__insert,axiom,
! [X2: complex,Xs: list_complex] :
( ( set_complex2 @ ( insert_complex @ X2 @ Xs ) )
= ( insert_complex2 @ X2 @ ( set_complex2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_161_insort__insert__key__triv,axiom,
! [F: nat > nat,X2: nat,Xs: list_nat] :
( ( member_nat2 @ ( F @ X2 ) @ ( image_nat_nat @ F @ ( set_nat2 @ Xs ) ) )
=> ( ( linord1921536354676448932at_nat @ F @ X2 @ Xs )
= Xs ) ) ).
% insort_insert_key_triv
thf(fact_162_insort__insert__key__triv,axiom,
! [F: complex > nat,X2: complex,Xs: list_complex] :
( ( member_nat2 @ ( F @ X2 ) @ ( image_complex_nat @ F @ ( set_complex2 @ Xs ) ) )
=> ( ( linord1240774035888917890ex_nat @ F @ X2 @ Xs )
= Xs ) ) ).
% insort_insert_key_triv
thf(fact_163_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_164_zero__reorient,axiom,
! [X2: complex] :
( ( zero_zero_complex = X2 )
= ( X2 = zero_zero_complex ) ) ).
% zero_reorient
thf(fact_165_zero__reorient,axiom,
! [X2: real] :
( ( zero_zero_real = X2 )
= ( X2 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_166_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_167_list_Opred__inject_I1_J,axiom,
! [P: nat > $o] : ( list_all_nat @ P @ nil_nat ) ).
% list.pred_inject(1)
thf(fact_168_list__all__simps_I2_J,axiom,
! [P: nat > $o] : ( list_all_nat @ P @ nil_nat ) ).
% list_all_simps(2)
thf(fact_169_List_Otranspose_Ocases,axiom,
! [X2: list_list_complex] :
( ( X2 != nil_list_complex )
=> ( ! [Xss: list_list_complex] :
( X2
!= ( cons_list_complex @ nil_complex @ Xss ) )
=> ~ ! [X4: complex,Xs3: list_complex,Xss: list_list_complex] :
( X2
!= ( cons_list_complex @ ( cons_complex @ X4 @ Xs3 ) @ Xss ) ) ) ) ).
% List.transpose.cases
thf(fact_170_List_Otranspose_Ocases,axiom,
! [X2: list_list_nat] :
( ( X2 != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X2
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X4: nat,Xs3: list_nat,Xss: list_list_nat] :
( X2
!= ( cons_list_nat @ ( cons_nat @ X4 @ Xs3 ) @ Xss ) ) ) ) ).
% List.transpose.cases
thf(fact_171_List_Otranspose_Ocases,axiom,
! [X2: list_list_real] :
( ( X2 != nil_list_real )
=> ( ! [Xss: list_list_real] :
( X2
!= ( cons_list_real @ nil_real @ Xss ) )
=> ~ ! [X4: real,Xs3: list_real,Xss: list_list_real] :
( X2
!= ( cons_list_real @ ( cons_real @ X4 @ Xs3 ) @ Xss ) ) ) ) ).
% List.transpose.cases
thf(fact_172_eq__comps__not__empty,axiom,
! [L: list_nat] :
( ( L != nil_nat )
=> ( ( commut2436974278740741825ps_nat @ L )
!= nil_nat ) ) ).
% eq_comps_not_empty
thf(fact_173_eq__comps__not__empty,axiom,
! [L: list_real] :
( ( L != nil_real )
=> ( ( commut8680161604938074397s_real @ L )
!= nil_nat ) ) ).
% eq_comps_not_empty
thf(fact_174_eq__comps__not__empty,axiom,
! [L: list_complex] :
( ( L != nil_complex )
=> ( ( commut93809757773076895omplex @ L )
!= nil_nat ) ) ).
% eq_comps_not_empty
thf(fact_175_eq__comps__empty__if,axiom,
! [L: list_nat] :
( ( ( commut2436974278740741825ps_nat @ L )
= nil_nat )
=> ( L = nil_nat ) ) ).
% eq_comps_empty_if
thf(fact_176_eq__comps__empty__if,axiom,
! [L: list_real] :
( ( ( commut8680161604938074397s_real @ L )
= nil_nat )
=> ( L = nil_real ) ) ).
% eq_comps_empty_if
thf(fact_177_eq__comps__empty__if,axiom,
! [L: list_complex] :
( ( ( commut93809757773076895omplex @ L )
= nil_nat )
=> ( L = nil_complex ) ) ).
% eq_comps_empty_if
thf(fact_178_eq__comps_Osimps_I1_J,axiom,
( ( commut2436974278740741825ps_nat @ nil_nat )
= nil_nat ) ).
% eq_comps.simps(1)
thf(fact_179_eq__comps_Osimps_I1_J,axiom,
( ( commut8680161604938074397s_real @ nil_real )
= nil_nat ) ).
% eq_comps.simps(1)
thf(fact_180_eq__comps_Osimps_I1_J,axiom,
( ( commut93809757773076895omplex @ nil_complex )
= nil_nat ) ).
% eq_comps.simps(1)
thf(fact_181_insort__insert__insort__key,axiom,
! [F: nat > nat,X2: nat,Xs: list_nat] :
( ~ ( member_nat2 @ ( F @ X2 ) @ ( image_nat_nat @ F @ ( set_nat2 @ Xs ) ) )
=> ( ( linord1921536354676448932at_nat @ F @ X2 @ Xs )
= ( linord8961336180081300637at_nat @ F @ X2 @ Xs ) ) ) ).
% insort_insert_insort_key
thf(fact_182_insort__insert__insort__key,axiom,
! [F: complex > nat,X2: complex,Xs: list_complex] :
( ~ ( member_nat2 @ ( F @ X2 ) @ ( image_complex_nat @ F @ ( set_complex2 @ Xs ) ) )
=> ( ( linord1240774035888917890ex_nat @ F @ X2 @ Xs )
= ( linord4454476646278009211ex_nat @ F @ X2 @ Xs ) ) ) ).
% insort_insert_insort_key
thf(fact_183_insort__insert__key__def,axiom,
( linord1921536354676448932at_nat
= ( ^ [F2: nat > nat,X3: nat,Xs2: list_nat] : ( if_list_nat @ ( member_nat2 @ ( F2 @ X3 ) @ ( image_nat_nat @ F2 @ ( set_nat2 @ Xs2 ) ) ) @ Xs2 @ ( linord8961336180081300637at_nat @ F2 @ X3 @ Xs2 ) ) ) ) ).
% insort_insert_key_def
thf(fact_184_insort__insert__key__def,axiom,
( linord1240774035888917890ex_nat
= ( ^ [F2: complex > nat,X3: complex,Xs2: list_complex] : ( if_list_complex @ ( member_nat2 @ ( F2 @ X3 ) @ ( image_complex_nat @ F2 @ ( set_complex2 @ Xs2 ) ) ) @ Xs2 @ ( linord4454476646278009211ex_nat @ F2 @ X3 @ Xs2 ) ) ) ) ).
% insort_insert_key_def
thf(fact_185_can__select__def,axiom,
( can_select_complex
= ( ^ [P2: complex > $o,A3: set_complex] :
? [X3: complex] :
( ( member_complex2 @ X3 @ A3 )
& ( P2 @ X3 )
& ! [Y2: complex] :
( ( ( member_complex2 @ Y2 @ A3 )
& ( P2 @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ).
% can_select_def
thf(fact_186_can__select__def,axiom,
( can_select_nat
= ( ^ [P2: nat > $o,A3: set_nat] :
? [X3: nat] :
( ( member_nat2 @ X3 @ A3 )
& ( P2 @ X3 )
& ! [Y2: nat] :
( ( ( member_nat2 @ Y2 @ A3 )
& ( P2 @ Y2 ) )
=> ( Y2 = X3 ) ) ) ) ) ).
% can_select_def
thf(fact_187_list__nonempty__induct,axiom,
! [Xs: list_complex,P: list_complex > $o] :
( ( Xs != nil_complex )
=> ( ! [X4: complex] : ( P @ ( cons_complex @ X4 @ nil_complex ) )
=> ( ! [X4: complex,Xs3: list_complex] :
( ( Xs3 != nil_complex )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_complex @ X4 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_188_list__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X4: nat] : ( P @ ( cons_nat @ X4 @ nil_nat ) )
=> ( ! [X4: nat,Xs3: list_nat] :
( ( Xs3 != nil_nat )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_189_list__nonempty__induct,axiom,
! [Xs: list_real,P: list_real > $o] :
( ( Xs != nil_real )
=> ( ! [X4: real] : ( P @ ( cons_real @ X4 @ nil_real ) )
=> ( ! [X4: real,Xs3: list_real] :
( ( Xs3 != nil_real )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_real @ X4 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_190_induct__list012,axiom,
! [P: list_complex > $o,Xs: list_complex] :
( ( P @ nil_complex )
=> ( ! [X4: complex] : ( P @ ( cons_complex @ X4 @ nil_complex ) )
=> ( ! [X4: complex,Y3: complex,Zs: list_complex] :
( ( P @ Zs )
=> ( ( P @ ( cons_complex @ Y3 @ Zs ) )
=> ( P @ ( cons_complex @ X4 @ ( cons_complex @ Y3 @ Zs ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% induct_list012
thf(fact_191_induct__list012,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ( P @ nil_nat )
=> ( ! [X4: nat] : ( P @ ( cons_nat @ X4 @ nil_nat ) )
=> ( ! [X4: nat,Y3: nat,Zs: list_nat] :
( ( P @ Zs )
=> ( ( P @ ( cons_nat @ Y3 @ Zs ) )
=> ( P @ ( cons_nat @ X4 @ ( cons_nat @ Y3 @ Zs ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% induct_list012
thf(fact_192_induct__list012,axiom,
! [P: list_real > $o,Xs: list_real] :
( ( P @ nil_real )
=> ( ! [X4: real] : ( P @ ( cons_real @ X4 @ nil_real ) )
=> ( ! [X4: real,Y3: real,Zs: list_real] :
( ( P @ Zs )
=> ( ( P @ ( cons_real @ Y3 @ Zs ) )
=> ( P @ ( cons_real @ X4 @ ( cons_real @ Y3 @ Zs ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% induct_list012
thf(fact_193_list__induct2_H,axiom,
! [P: list_complex > list_complex > $o,Xs: list_complex,Ys: list_complex] :
( ( P @ nil_complex @ nil_complex )
=> ( ! [X4: complex,Xs3: list_complex] : ( P @ ( cons_complex @ X4 @ Xs3 ) @ nil_complex )
=> ( ! [Y3: complex,Ys2: list_complex] : ( P @ nil_complex @ ( cons_complex @ Y3 @ Ys2 ) )
=> ( ! [X4: complex,Xs3: list_complex,Y3: complex,Ys2: list_complex] :
( ( P @ Xs3 @ Ys2 )
=> ( P @ ( cons_complex @ X4 @ Xs3 ) @ ( cons_complex @ Y3 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_194_list__induct2_H,axiom,
! [P: list_complex > list_nat > $o,Xs: list_complex,Ys: list_nat] :
( ( P @ nil_complex @ nil_nat )
=> ( ! [X4: complex,Xs3: list_complex] : ( P @ ( cons_complex @ X4 @ Xs3 ) @ nil_nat )
=> ( ! [Y3: nat,Ys2: list_nat] : ( P @ nil_complex @ ( cons_nat @ Y3 @ Ys2 ) )
=> ( ! [X4: complex,Xs3: list_complex,Y3: nat,Ys2: list_nat] :
( ( P @ Xs3 @ Ys2 )
=> ( P @ ( cons_complex @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_195_list__induct2_H,axiom,
! [P: list_complex > list_real > $o,Xs: list_complex,Ys: list_real] :
( ( P @ nil_complex @ nil_real )
=> ( ! [X4: complex,Xs3: list_complex] : ( P @ ( cons_complex @ X4 @ Xs3 ) @ nil_real )
=> ( ! [Y3: real,Ys2: list_real] : ( P @ nil_complex @ ( cons_real @ Y3 @ Ys2 ) )
=> ( ! [X4: complex,Xs3: list_complex,Y3: real,Ys2: list_real] :
( ( P @ Xs3 @ Ys2 )
=> ( P @ ( cons_complex @ X4 @ Xs3 ) @ ( cons_real @ Y3 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_196_list__induct2_H,axiom,
! [P: list_nat > list_complex > $o,Xs: list_nat,Ys: list_complex] :
( ( P @ nil_nat @ nil_complex )
=> ( ! [X4: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X4 @ Xs3 ) @ nil_complex )
=> ( ! [Y3: complex,Ys2: list_complex] : ( P @ nil_nat @ ( cons_complex @ Y3 @ Ys2 ) )
=> ( ! [X4: nat,Xs3: list_nat,Y3: complex,Ys2: list_complex] :
( ( P @ Xs3 @ Ys2 )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_complex @ Y3 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_197_list__induct2_H,axiom,
! [P: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [X4: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X4 @ Xs3 ) @ nil_nat )
=> ( ! [Y3: nat,Ys2: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y3 @ Ys2 ) )
=> ( ! [X4: nat,Xs3: list_nat,Y3: nat,Ys2: list_nat] :
( ( P @ Xs3 @ Ys2 )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_198_list__induct2_H,axiom,
! [P: list_nat > list_real > $o,Xs: list_nat,Ys: list_real] :
( ( P @ nil_nat @ nil_real )
=> ( ! [X4: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X4 @ Xs3 ) @ nil_real )
=> ( ! [Y3: real,Ys2: list_real] : ( P @ nil_nat @ ( cons_real @ Y3 @ Ys2 ) )
=> ( ! [X4: nat,Xs3: list_nat,Y3: real,Ys2: list_real] :
( ( P @ Xs3 @ Ys2 )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_real @ Y3 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_199_list__induct2_H,axiom,
! [P: list_real > list_complex > $o,Xs: list_real,Ys: list_complex] :
( ( P @ nil_real @ nil_complex )
=> ( ! [X4: real,Xs3: list_real] : ( P @ ( cons_real @ X4 @ Xs3 ) @ nil_complex )
=> ( ! [Y3: complex,Ys2: list_complex] : ( P @ nil_real @ ( cons_complex @ Y3 @ Ys2 ) )
=> ( ! [X4: real,Xs3: list_real,Y3: complex,Ys2: list_complex] :
( ( P @ Xs3 @ Ys2 )
=> ( P @ ( cons_real @ X4 @ Xs3 ) @ ( cons_complex @ Y3 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_200_list__induct2_H,axiom,
! [P: list_real > list_nat > $o,Xs: list_real,Ys: list_nat] :
( ( P @ nil_real @ nil_nat )
=> ( ! [X4: real,Xs3: list_real] : ( P @ ( cons_real @ X4 @ Xs3 ) @ nil_nat )
=> ( ! [Y3: nat,Ys2: list_nat] : ( P @ nil_real @ ( cons_nat @ Y3 @ Ys2 ) )
=> ( ! [X4: real,Xs3: list_real,Y3: nat,Ys2: list_nat] :
( ( P @ Xs3 @ Ys2 )
=> ( P @ ( cons_real @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_201_list__induct2_H,axiom,
! [P: list_real > list_real > $o,Xs: list_real,Ys: list_real] :
( ( P @ nil_real @ nil_real )
=> ( ! [X4: real,Xs3: list_real] : ( P @ ( cons_real @ X4 @ Xs3 ) @ nil_real )
=> ( ! [Y3: real,Ys2: list_real] : ( P @ nil_real @ ( cons_real @ Y3 @ Ys2 ) )
=> ( ! [X4: real,Xs3: list_real,Y3: real,Ys2: list_real] :
( ( P @ Xs3 @ Ys2 )
=> ( P @ ( cons_real @ X4 @ Xs3 ) @ ( cons_real @ Y3 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_202_neq__Nil__conv,axiom,
! [Xs: list_complex] :
( ( Xs != nil_complex )
= ( ? [Y2: complex,Ys3: list_complex] :
( Xs
= ( cons_complex @ Y2 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_203_neq__Nil__conv,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
= ( ? [Y2: nat,Ys3: list_nat] :
( Xs
= ( cons_nat @ Y2 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_204_neq__Nil__conv,axiom,
! [Xs: list_real] :
( ( Xs != nil_real )
= ( ? [Y2: real,Ys3: list_real] :
( Xs
= ( cons_real @ Y2 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_205_map__tailrec__rev_Oinduct,axiom,
! [P: ( complex > complex ) > list_complex > list_complex > $o,A0: complex > complex,A1: list_complex,A22: list_complex] :
( ! [F3: complex > complex,X_1: list_complex] : ( P @ F3 @ nil_complex @ X_1 )
=> ( ! [F3: complex > complex,A4: complex,As: list_complex,Bs: list_complex] :
( ( P @ F3 @ As @ ( cons_complex @ ( F3 @ A4 ) @ Bs ) )
=> ( P @ F3 @ ( cons_complex @ A4 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_206_map__tailrec__rev_Oinduct,axiom,
! [P: ( nat > complex ) > list_nat > list_complex > $o,A0: nat > complex,A1: list_nat,A22: list_complex] :
( ! [F3: nat > complex,X_1: list_complex] : ( P @ F3 @ nil_nat @ X_1 )
=> ( ! [F3: nat > complex,A4: nat,As: list_nat,Bs: list_complex] :
( ( P @ F3 @ As @ ( cons_complex @ ( F3 @ A4 ) @ Bs ) )
=> ( P @ F3 @ ( cons_nat @ A4 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_207_map__tailrec__rev_Oinduct,axiom,
! [P: ( real > complex ) > list_real > list_complex > $o,A0: real > complex,A1: list_real,A22: list_complex] :
( ! [F3: real > complex,X_1: list_complex] : ( P @ F3 @ nil_real @ X_1 )
=> ( ! [F3: real > complex,A4: real,As: list_real,Bs: list_complex] :
( ( P @ F3 @ As @ ( cons_complex @ ( F3 @ A4 ) @ Bs ) )
=> ( P @ F3 @ ( cons_real @ A4 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_208_map__tailrec__rev_Oinduct,axiom,
! [P: ( complex > nat ) > list_complex > list_nat > $o,A0: complex > nat,A1: list_complex,A22: list_nat] :
( ! [F3: complex > nat,X_1: list_nat] : ( P @ F3 @ nil_complex @ X_1 )
=> ( ! [F3: complex > nat,A4: complex,As: list_complex,Bs: list_nat] :
( ( P @ F3 @ As @ ( cons_nat @ ( F3 @ A4 ) @ Bs ) )
=> ( P @ F3 @ ( cons_complex @ A4 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_209_map__tailrec__rev_Oinduct,axiom,
! [P: ( nat > nat ) > list_nat > list_nat > $o,A0: nat > nat,A1: list_nat,A22: list_nat] :
( ! [F3: nat > nat,X_1: list_nat] : ( P @ F3 @ nil_nat @ X_1 )
=> ( ! [F3: nat > nat,A4: nat,As: list_nat,Bs: list_nat] :
( ( P @ F3 @ As @ ( cons_nat @ ( F3 @ A4 ) @ Bs ) )
=> ( P @ F3 @ ( cons_nat @ A4 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_210_map__tailrec__rev_Oinduct,axiom,
! [P: ( real > nat ) > list_real > list_nat > $o,A0: real > nat,A1: list_real,A22: list_nat] :
( ! [F3: real > nat,X_1: list_nat] : ( P @ F3 @ nil_real @ X_1 )
=> ( ! [F3: real > nat,A4: real,As: list_real,Bs: list_nat] :
( ( P @ F3 @ As @ ( cons_nat @ ( F3 @ A4 ) @ Bs ) )
=> ( P @ F3 @ ( cons_real @ A4 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_211_map__tailrec__rev_Oinduct,axiom,
! [P: ( complex > real ) > list_complex > list_real > $o,A0: complex > real,A1: list_complex,A22: list_real] :
( ! [F3: complex > real,X_1: list_real] : ( P @ F3 @ nil_complex @ X_1 )
=> ( ! [F3: complex > real,A4: complex,As: list_complex,Bs: list_real] :
( ( P @ F3 @ As @ ( cons_real @ ( F3 @ A4 ) @ Bs ) )
=> ( P @ F3 @ ( cons_complex @ A4 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_212_map__tailrec__rev_Oinduct,axiom,
! [P: ( nat > real ) > list_nat > list_real > $o,A0: nat > real,A1: list_nat,A22: list_real] :
( ! [F3: nat > real,X_1: list_real] : ( P @ F3 @ nil_nat @ X_1 )
=> ( ! [F3: nat > real,A4: nat,As: list_nat,Bs: list_real] :
( ( P @ F3 @ As @ ( cons_real @ ( F3 @ A4 ) @ Bs ) )
=> ( P @ F3 @ ( cons_nat @ A4 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_213_map__tailrec__rev_Oinduct,axiom,
! [P: ( real > real ) > list_real > list_real > $o,A0: real > real,A1: list_real,A22: list_real] :
( ! [F3: real > real,X_1: list_real] : ( P @ F3 @ nil_real @ X_1 )
=> ( ! [F3: real > real,A4: real,As: list_real,Bs: list_real] :
( ( P @ F3 @ As @ ( cons_real @ ( F3 @ A4 ) @ Bs ) )
=> ( P @ F3 @ ( cons_real @ A4 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_214_successively_Oinduct,axiom,
! [P: ( complex > complex > $o ) > list_complex > $o,A0: complex > complex > $o,A1: list_complex] :
( ! [P3: complex > complex > $o] : ( P @ P3 @ nil_complex )
=> ( ! [P3: complex > complex > $o,X4: complex] : ( P @ P3 @ ( cons_complex @ X4 @ nil_complex ) )
=> ( ! [P3: complex > complex > $o,X4: complex,Y3: complex,Xs3: list_complex] :
( ( P @ P3 @ ( cons_complex @ Y3 @ Xs3 ) )
=> ( P @ P3 @ ( cons_complex @ X4 @ ( cons_complex @ Y3 @ Xs3 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_215_successively_Oinduct,axiom,
! [P: ( nat > nat > $o ) > list_nat > $o,A0: nat > nat > $o,A1: list_nat] :
( ! [P3: nat > nat > $o] : ( P @ P3 @ nil_nat )
=> ( ! [P3: nat > nat > $o,X4: nat] : ( P @ P3 @ ( cons_nat @ X4 @ nil_nat ) )
=> ( ! [P3: nat > nat > $o,X4: nat,Y3: nat,Xs3: list_nat] :
( ( P @ P3 @ ( cons_nat @ Y3 @ Xs3 ) )
=> ( P @ P3 @ ( cons_nat @ X4 @ ( cons_nat @ Y3 @ Xs3 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_216_successively_Oinduct,axiom,
! [P: ( real > real > $o ) > list_real > $o,A0: real > real > $o,A1: list_real] :
( ! [P3: real > real > $o] : ( P @ P3 @ nil_real )
=> ( ! [P3: real > real > $o,X4: real] : ( P @ P3 @ ( cons_real @ X4 @ nil_real ) )
=> ( ! [P3: real > real > $o,X4: real,Y3: real,Xs3: list_real] :
( ( P @ P3 @ ( cons_real @ Y3 @ Xs3 ) )
=> ( P @ P3 @ ( cons_real @ X4 @ ( cons_real @ Y3 @ Xs3 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_217_remdups__adj_Oinduct,axiom,
! [P: list_complex > $o,A0: list_complex] :
( ( P @ nil_complex )
=> ( ! [X4: complex] : ( P @ ( cons_complex @ X4 @ nil_complex ) )
=> ( ! [X4: complex,Y3: complex,Xs3: list_complex] :
( ( ( X4 = Y3 )
=> ( P @ ( cons_complex @ X4 @ Xs3 ) ) )
=> ( ( ( X4 != Y3 )
=> ( P @ ( cons_complex @ Y3 @ Xs3 ) ) )
=> ( P @ ( cons_complex @ X4 @ ( cons_complex @ Y3 @ Xs3 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_218_remdups__adj_Oinduct,axiom,
! [P: list_nat > $o,A0: list_nat] :
( ( P @ nil_nat )
=> ( ! [X4: nat] : ( P @ ( cons_nat @ X4 @ nil_nat ) )
=> ( ! [X4: nat,Y3: nat,Xs3: list_nat] :
( ( ( X4 = Y3 )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) ) )
=> ( ( ( X4 != Y3 )
=> ( P @ ( cons_nat @ Y3 @ Xs3 ) ) )
=> ( P @ ( cons_nat @ X4 @ ( cons_nat @ Y3 @ Xs3 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_219_remdups__adj_Oinduct,axiom,
! [P: list_real > $o,A0: list_real] :
( ( P @ nil_real )
=> ( ! [X4: real] : ( P @ ( cons_real @ X4 @ nil_real ) )
=> ( ! [X4: real,Y3: real,Xs3: list_real] :
( ( ( X4 = Y3 )
=> ( P @ ( cons_real @ X4 @ Xs3 ) ) )
=> ( ( ( X4 != Y3 )
=> ( P @ ( cons_real @ Y3 @ Xs3 ) ) )
=> ( P @ ( cons_real @ X4 @ ( cons_real @ Y3 @ Xs3 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_220_sorted__wrt_Oinduct,axiom,
! [P: ( complex > complex > $o ) > list_complex > $o,A0: complex > complex > $o,A1: list_complex] :
( ! [P3: complex > complex > $o] : ( P @ P3 @ nil_complex )
=> ( ! [P3: complex > complex > $o,X4: complex,Ys2: list_complex] :
( ( P @ P3 @ Ys2 )
=> ( P @ P3 @ ( cons_complex @ X4 @ Ys2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_221_sorted__wrt_Oinduct,axiom,
! [P: ( nat > nat > $o ) > list_nat > $o,A0: nat > nat > $o,A1: list_nat] :
( ! [P3: nat > nat > $o] : ( P @ P3 @ nil_nat )
=> ( ! [P3: nat > nat > $o,X4: nat,Ys2: list_nat] :
( ( P @ P3 @ Ys2 )
=> ( P @ P3 @ ( cons_nat @ X4 @ Ys2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_222_sorted__wrt_Oinduct,axiom,
! [P: ( real > real > $o ) > list_real > $o,A0: real > real > $o,A1: list_real] :
( ! [P3: real > real > $o] : ( P @ P3 @ nil_real )
=> ( ! [P3: real > real > $o,X4: real,Ys2: list_real] :
( ( P @ P3 @ Ys2 )
=> ( P @ P3 @ ( cons_real @ X4 @ Ys2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_223_remdups__adj_Ocases,axiom,
! [X2: list_complex] :
( ( X2 != nil_complex )
=> ( ! [X4: complex] :
( X2
!= ( cons_complex @ X4 @ nil_complex ) )
=> ~ ! [X4: complex,Y3: complex,Xs3: list_complex] :
( X2
!= ( cons_complex @ X4 @ ( cons_complex @ Y3 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_224_remdups__adj_Ocases,axiom,
! [X2: list_nat] :
( ( X2 != nil_nat )
=> ( ! [X4: nat] :
( X2
!= ( cons_nat @ X4 @ nil_nat ) )
=> ~ ! [X4: nat,Y3: nat,Xs3: list_nat] :
( X2
!= ( cons_nat @ X4 @ ( cons_nat @ Y3 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_225_remdups__adj_Ocases,axiom,
! [X2: list_real] :
( ( X2 != nil_real )
=> ( ! [X4: real] :
( X2
!= ( cons_real @ X4 @ nil_real ) )
=> ~ ! [X4: real,Y3: real,Xs3: list_real] :
( X2
!= ( cons_real @ X4 @ ( cons_real @ Y3 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_226_shuffles_Oinduct,axiom,
! [P: list_complex > list_complex > $o,A0: list_complex,A1: list_complex] :
( ! [X_1: list_complex] : ( P @ nil_complex @ X_1 )
=> ( ! [Xs3: list_complex] : ( P @ Xs3 @ nil_complex )
=> ( ! [X4: complex,Xs3: list_complex,Y3: complex,Ys2: list_complex] :
( ( P @ Xs3 @ ( cons_complex @ Y3 @ Ys2 ) )
=> ( ( P @ ( cons_complex @ X4 @ Xs3 ) @ Ys2 )
=> ( P @ ( cons_complex @ X4 @ Xs3 ) @ ( cons_complex @ Y3 @ Ys2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_227_shuffles_Oinduct,axiom,
! [P: list_nat > list_nat > $o,A0: list_nat,A1: list_nat] :
( ! [X_1: list_nat] : ( P @ nil_nat @ X_1 )
=> ( ! [Xs3: list_nat] : ( P @ Xs3 @ nil_nat )
=> ( ! [X4: nat,Xs3: list_nat,Y3: nat,Ys2: list_nat] :
( ( P @ Xs3 @ ( cons_nat @ Y3 @ Ys2 ) )
=> ( ( P @ ( cons_nat @ X4 @ Xs3 ) @ Ys2 )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_228_shuffles_Oinduct,axiom,
! [P: list_real > list_real > $o,A0: list_real,A1: list_real] :
( ! [X_1: list_real] : ( P @ nil_real @ X_1 )
=> ( ! [Xs3: list_real] : ( P @ Xs3 @ nil_real )
=> ( ! [X4: real,Xs3: list_real,Y3: real,Ys2: list_real] :
( ( P @ Xs3 @ ( cons_real @ Y3 @ Ys2 ) )
=> ( ( P @ ( cons_real @ X4 @ Xs3 ) @ Ys2 )
=> ( P @ ( cons_real @ X4 @ Xs3 ) @ ( cons_real @ Y3 @ Ys2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_229_min__list_Oinduct,axiom,
! [P: list_complex > $o,A0: list_complex] :
( ! [X4: complex,Xs3: list_complex] :
( ! [X212: complex,X222: list_complex] :
( ( Xs3
= ( cons_complex @ X212 @ X222 ) )
=> ( P @ Xs3 ) )
=> ( P @ ( cons_complex @ X4 @ Xs3 ) ) )
=> ( ( P @ nil_complex )
=> ( P @ A0 ) ) ) ).
% min_list.induct
thf(fact_230_min__list_Oinduct,axiom,
! [P: list_nat > $o,A0: list_nat] :
( ! [X4: nat,Xs3: list_nat] :
( ! [X212: nat,X222: list_nat] :
( ( Xs3
= ( cons_nat @ X212 @ X222 ) )
=> ( P @ Xs3 ) )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) ) )
=> ( ( P @ nil_nat )
=> ( P @ A0 ) ) ) ).
% min_list.induct
thf(fact_231_min__list_Oinduct,axiom,
! [P: list_real > $o,A0: list_real] :
( ! [X4: real,Xs3: list_real] :
( ! [X212: real,X222: list_real] :
( ( Xs3
= ( cons_real @ X212 @ X222 ) )
=> ( P @ Xs3 ) )
=> ( P @ ( cons_real @ X4 @ Xs3 ) ) )
=> ( ( P @ nil_real )
=> ( P @ A0 ) ) ) ).
% min_list.induct
thf(fact_232_min__list_Ocases,axiom,
! [X2: list_complex] :
( ! [X4: complex,Xs3: list_complex] :
( X2
!= ( cons_complex @ X4 @ Xs3 ) )
=> ( X2 = nil_complex ) ) ).
% min_list.cases
thf(fact_233_min__list_Ocases,axiom,
! [X2: list_nat] :
( ! [X4: nat,Xs3: list_nat] :
( X2
!= ( cons_nat @ X4 @ Xs3 ) )
=> ( X2 = nil_nat ) ) ).
% min_list.cases
thf(fact_234_min__list_Ocases,axiom,
! [X2: list_real] :
( ! [X4: real,Xs3: list_real] :
( X2
!= ( cons_real @ X4 @ Xs3 ) )
=> ( X2 = nil_real ) ) ).
% min_list.cases
thf(fact_235_splice_Oinduct,axiom,
! [P: list_complex > list_complex > $o,A0: list_complex,A1: list_complex] :
( ! [X_1: list_complex] : ( P @ nil_complex @ X_1 )
=> ( ! [X4: complex,Xs3: list_complex,Ys2: list_complex] :
( ( P @ Ys2 @ Xs3 )
=> ( P @ ( cons_complex @ X4 @ Xs3 ) @ Ys2 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_236_splice_Oinduct,axiom,
! [P: list_nat > list_nat > $o,A0: list_nat,A1: list_nat] :
( ! [X_1: list_nat] : ( P @ nil_nat @ X_1 )
=> ( ! [X4: nat,Xs3: list_nat,Ys2: list_nat] :
( ( P @ Ys2 @ Xs3 )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) @ Ys2 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_237_splice_Oinduct,axiom,
! [P: list_real > list_real > $o,A0: list_real,A1: list_real] :
( ! [X_1: list_real] : ( P @ nil_real @ X_1 )
=> ( ! [X4: real,Xs3: list_real,Ys2: list_real] :
( ( P @ Ys2 @ Xs3 )
=> ( P @ ( cons_real @ X4 @ Xs3 ) @ Ys2 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_238_list_Oinducts,axiom,
! [P: list_complex > $o,List: list_complex] :
( ( P @ nil_complex )
=> ( ! [X1: complex,X23: list_complex] :
( ( P @ X23 )
=> ( P @ ( cons_complex @ X1 @ X23 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_239_list_Oinducts,axiom,
! [P: list_nat > $o,List: list_nat] :
( ( P @ nil_nat )
=> ( ! [X1: nat,X23: list_nat] :
( ( P @ X23 )
=> ( P @ ( cons_nat @ X1 @ X23 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_240_list_Oinducts,axiom,
! [P: list_real > $o,List: list_real] :
( ( P @ nil_real )
=> ( ! [X1: real,X23: list_real] :
( ( P @ X23 )
=> ( P @ ( cons_real @ X1 @ X23 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_241_list_Oexhaust,axiom,
! [Y: list_complex] :
( ( Y != nil_complex )
=> ~ ! [X213: complex,X223: list_complex] :
( Y
!= ( cons_complex @ X213 @ X223 ) ) ) ).
% list.exhaust
thf(fact_242_list_Oexhaust,axiom,
! [Y: list_nat] :
( ( Y != nil_nat )
=> ~ ! [X213: nat,X223: list_nat] :
( Y
!= ( cons_nat @ X213 @ X223 ) ) ) ).
% list.exhaust
thf(fact_243_list_Oexhaust,axiom,
! [Y: list_real] :
( ( Y != nil_real )
=> ~ ! [X213: real,X223: list_real] :
( Y
!= ( cons_real @ X213 @ X223 ) ) ) ).
% list.exhaust
thf(fact_244_list_OdiscI,axiom,
! [List: list_complex,X21: complex,X22: list_complex] :
( ( List
= ( cons_complex @ X21 @ X22 ) )
=> ( List != nil_complex ) ) ).
% list.discI
thf(fact_245_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X22: list_nat] :
( ( List
= ( cons_nat @ X21 @ X22 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_246_list_OdiscI,axiom,
! [List: list_real,X21: real,X22: list_real] :
( ( List
= ( cons_real @ X21 @ X22 ) )
=> ( List != nil_real ) ) ).
% list.discI
thf(fact_247_list_Odistinct_I1_J,axiom,
! [X21: complex,X22: list_complex] :
( nil_complex
!= ( cons_complex @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_248_list_Odistinct_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_249_list_Odistinct_I1_J,axiom,
! [X21: real,X22: list_real] :
( nil_real
!= ( cons_real @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_250_eq__comps_Oinduct,axiom,
! [P: list_complex > $o,A0: list_complex] :
( ( P @ nil_complex )
=> ( ! [X4: complex] : ( P @ ( cons_complex @ X4 @ nil_complex ) )
=> ( ! [X4: complex,Y3: complex,L2: list_complex] :
( ( P @ ( cons_complex @ Y3 @ L2 ) )
=> ( P @ ( cons_complex @ X4 @ ( cons_complex @ Y3 @ L2 ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% eq_comps.induct
thf(fact_251_eq__comps_Oinduct,axiom,
! [P: list_nat > $o,A0: list_nat] :
( ( P @ nil_nat )
=> ( ! [X4: nat] : ( P @ ( cons_nat @ X4 @ nil_nat ) )
=> ( ! [X4: nat,Y3: nat,L2: list_nat] :
( ( P @ ( cons_nat @ Y3 @ L2 ) )
=> ( P @ ( cons_nat @ X4 @ ( cons_nat @ Y3 @ L2 ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% eq_comps.induct
thf(fact_252_eq__comps_Oinduct,axiom,
! [P: list_real > $o,A0: list_real] :
( ( P @ nil_real )
=> ( ! [X4: real] : ( P @ ( cons_real @ X4 @ nil_real ) )
=> ( ! [X4: real,Y3: real,L2: list_real] :
( ( P @ ( cons_real @ Y3 @ L2 ) )
=> ( P @ ( cons_real @ X4 @ ( cons_real @ Y3 @ L2 ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% eq_comps.induct
thf(fact_253_map__is__Nil__conv,axiom,
! [F: complex > real,Xs: list_complex] :
( ( ( map_complex_real @ F @ Xs )
= nil_real )
= ( Xs = nil_complex ) ) ).
% map_is_Nil_conv
thf(fact_254_map__is__Nil__conv,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_255_map__is__Nil__conv,axiom,
! [F: nat > complex,Xs: list_nat] :
( ( ( map_nat_complex @ F @ Xs )
= nil_complex )
= ( Xs = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_256_map__is__Nil__conv,axiom,
! [F: complex > nat,Xs: list_complex] :
( ( ( map_complex_nat @ F @ Xs )
= nil_nat )
= ( Xs = nil_complex ) ) ).
% map_is_Nil_conv
thf(fact_257_map__is__Nil__conv,axiom,
! [F: complex > complex,Xs: list_complex] :
( ( ( map_complex_complex @ F @ Xs )
= nil_complex )
= ( Xs = nil_complex ) ) ).
% map_is_Nil_conv
thf(fact_258_Nil__is__map__conv,axiom,
! [F: complex > real,Xs: list_complex] :
( ( nil_real
= ( map_complex_real @ F @ Xs ) )
= ( Xs = nil_complex ) ) ).
% Nil_is_map_conv
thf(fact_259_Nil__is__map__conv,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( nil_nat
= ( map_nat_nat @ F @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_260_Nil__is__map__conv,axiom,
! [F: nat > complex,Xs: list_nat] :
( ( nil_complex
= ( map_nat_complex @ F @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_261_Nil__is__map__conv,axiom,
! [F: complex > nat,Xs: list_complex] :
( ( nil_nat
= ( map_complex_nat @ F @ Xs ) )
= ( Xs = nil_complex ) ) ).
% Nil_is_map_conv
thf(fact_262_Nil__is__map__conv,axiom,
! [F: complex > complex,Xs: list_complex] :
( ( nil_complex
= ( map_complex_complex @ F @ Xs ) )
= ( Xs = nil_complex ) ) ).
% Nil_is_map_conv
thf(fact_263_list_Omap__disc__iff,axiom,
! [F: complex > real,A: list_complex] :
( ( ( map_complex_real @ F @ A )
= nil_real )
= ( A = nil_complex ) ) ).
% list.map_disc_iff
thf(fact_264_list_Omap__disc__iff,axiom,
! [F: nat > nat,A: list_nat] :
( ( ( map_nat_nat @ F @ A )
= nil_nat )
= ( A = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_265_list_Omap__disc__iff,axiom,
! [F: nat > complex,A: list_nat] :
( ( ( map_nat_complex @ F @ A )
= nil_complex )
= ( A = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_266_list_Omap__disc__iff,axiom,
! [F: complex > nat,A: list_complex] :
( ( ( map_complex_nat @ F @ A )
= nil_nat )
= ( A = nil_complex ) ) ).
% list.map_disc_iff
thf(fact_267_list_Omap__disc__iff,axiom,
! [F: complex > complex,A: list_complex] :
( ( ( map_complex_complex @ F @ A )
= nil_complex )
= ( A = nil_complex ) ) ).
% list.map_disc_iff
thf(fact_268_list_Osimps_I8_J,axiom,
! [F: complex > real] :
( ( map_complex_real @ F @ nil_complex )
= nil_real ) ).
% list.simps(8)
thf(fact_269_list_Osimps_I8_J,axiom,
! [F: nat > nat] :
( ( map_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_270_list_Osimps_I8_J,axiom,
! [F: nat > complex] :
( ( map_nat_complex @ F @ nil_nat )
= nil_complex ) ).
% list.simps(8)
thf(fact_271_list_Osimps_I8_J,axiom,
! [F: complex > nat] :
( ( map_complex_nat @ F @ nil_complex )
= nil_nat ) ).
% list.simps(8)
thf(fact_272_list_Osimps_I8_J,axiom,
! [F: complex > complex] :
( ( map_complex_complex @ F @ nil_complex )
= nil_complex ) ).
% list.simps(8)
thf(fact_273_list_Opred__inject_I2_J,axiom,
! [P: complex > $o,A: complex,Aa: list_complex] :
( ( list_all_complex @ P @ ( cons_complex @ A @ Aa ) )
= ( ( P @ A )
& ( list_all_complex @ P @ Aa ) ) ) ).
% list.pred_inject(2)
thf(fact_274_list_Opred__inject_I2_J,axiom,
! [P: nat > $o,A: nat,Aa: list_nat] :
( ( list_all_nat @ P @ ( cons_nat @ A @ Aa ) )
= ( ( P @ A )
& ( list_all_nat @ P @ Aa ) ) ) ).
% list.pred_inject(2)
thf(fact_275_list_Opred__inject_I2_J,axiom,
! [P: real > $o,A: real,Aa: list_real] :
( ( list_all_real @ P @ ( cons_real @ A @ Aa ) )
= ( ( P @ A )
& ( list_all_real @ P @ Aa ) ) ) ).
% list.pred_inject(2)
thf(fact_276_list__all__simps_I1_J,axiom,
! [P: complex > $o,X2: complex,Xs: list_complex] :
( ( list_all_complex @ P @ ( cons_complex @ X2 @ Xs ) )
= ( ( P @ X2 )
& ( list_all_complex @ P @ Xs ) ) ) ).
% list_all_simps(1)
thf(fact_277_list__all__simps_I1_J,axiom,
! [P: nat > $o,X2: nat,Xs: list_nat] :
( ( list_all_nat @ P @ ( cons_nat @ X2 @ Xs ) )
= ( ( P @ X2 )
& ( list_all_nat @ P @ Xs ) ) ) ).
% list_all_simps(1)
thf(fact_278_list__all__simps_I1_J,axiom,
! [P: real > $o,X2: real,Xs: list_real] :
( ( list_all_real @ P @ ( cons_real @ X2 @ Xs ) )
= ( ( P @ X2 )
& ( list_all_real @ P @ Xs ) ) ) ).
% list_all_simps(1)
thf(fact_279_list_Opred__cong,axiom,
! [X2: list_nat,Ya: list_nat,P: nat > $o,Pa: nat > $o] :
( ( X2 = Ya )
=> ( ! [Z: nat] :
( ( member_nat2 @ Z @ ( set_nat2 @ Ya ) )
=> ( ( P @ Z )
= ( Pa @ Z ) ) )
=> ( ( list_all_nat @ P @ X2 )
= ( list_all_nat @ Pa @ Ya ) ) ) ) ).
% list.pred_cong
thf(fact_280_list_Opred__cong,axiom,
! [X2: list_complex,Ya: list_complex,P: complex > $o,Pa: complex > $o] :
( ( X2 = Ya )
=> ( ! [Z: complex] :
( ( member_complex2 @ Z @ ( set_complex2 @ Ya ) )
=> ( ( P @ Z )
= ( Pa @ Z ) ) )
=> ( ( list_all_complex @ P @ X2 )
= ( list_all_complex @ Pa @ Ya ) ) ) ) ).
% list.pred_cong
thf(fact_281_list_Opred__mono__strong,axiom,
! [P: nat > $o,X2: list_nat,Pa: nat > $o] :
( ( list_all_nat @ P @ X2 )
=> ( ! [Z: nat] :
( ( member_nat2 @ Z @ ( set_nat2 @ X2 ) )
=> ( ( P @ Z )
=> ( Pa @ Z ) ) )
=> ( list_all_nat @ Pa @ X2 ) ) ) ).
% list.pred_mono_strong
thf(fact_282_list_Opred__mono__strong,axiom,
! [P: complex > $o,X2: list_complex,Pa: complex > $o] :
( ( list_all_complex @ P @ X2 )
=> ( ! [Z: complex] :
( ( member_complex2 @ Z @ ( set_complex2 @ X2 ) )
=> ( ( P @ Z )
=> ( Pa @ Z ) ) )
=> ( list_all_complex @ Pa @ X2 ) ) ) ).
% list.pred_mono_strong
thf(fact_283_Reals__0,axiom,
member_real2 @ zero_zero_real @ real_V470468836141973256s_real ).
% Reals_0
thf(fact_284_Reals__0,axiom,
member_complex2 @ zero_zero_complex @ real_V2521375963428798218omplex ).
% Reals_0
thf(fact_285_list_Omap__cong__pred,axiom,
! [X2: list_complex,Ya: list_complex,F: complex > real,G: complex > real] :
( ( X2 = Ya )
=> ( ( list_all_complex
@ ^ [Z2: complex] :
( ( F @ Z2 )
= ( G @ Z2 ) )
@ Ya )
=> ( ( map_complex_real @ F @ X2 )
= ( map_complex_real @ G @ Ya ) ) ) ) ).
% list.map_cong_pred
thf(fact_286_list_Omap__cong__pred,axiom,
! [X2: list_nat,Ya: list_nat,F: nat > nat,G: nat > nat] :
( ( X2 = Ya )
=> ( ( list_all_nat
@ ^ [Z2: nat] :
( ( F @ Z2 )
= ( G @ Z2 ) )
@ Ya )
=> ( ( map_nat_nat @ F @ X2 )
= ( map_nat_nat @ G @ Ya ) ) ) ) ).
% list.map_cong_pred
thf(fact_287_list_Omap__cong__pred,axiom,
! [X2: list_nat,Ya: list_nat,F: nat > complex,G: nat > complex] :
( ( X2 = Ya )
=> ( ( list_all_nat
@ ^ [Z2: nat] :
( ( F @ Z2 )
= ( G @ Z2 ) )
@ Ya )
=> ( ( map_nat_complex @ F @ X2 )
= ( map_nat_complex @ G @ Ya ) ) ) ) ).
% list.map_cong_pred
thf(fact_288_list_Omap__cong__pred,axiom,
! [X2: list_complex,Ya: list_complex,F: complex > nat,G: complex > nat] :
( ( X2 = Ya )
=> ( ( list_all_complex
@ ^ [Z2: complex] :
( ( F @ Z2 )
= ( G @ Z2 ) )
@ Ya )
=> ( ( map_complex_nat @ F @ X2 )
= ( map_complex_nat @ G @ Ya ) ) ) ) ).
% list.map_cong_pred
thf(fact_289_list_Omap__cong__pred,axiom,
! [X2: list_complex,Ya: list_complex,F: complex > complex,G: complex > complex] :
( ( X2 = Ya )
=> ( ( list_all_complex
@ ^ [Z2: complex] :
( ( F @ Z2 )
= ( G @ Z2 ) )
@ Ya )
=> ( ( map_complex_complex @ F @ X2 )
= ( map_complex_complex @ G @ Ya ) ) ) ) ).
% list.map_cong_pred
thf(fact_290_list__ex1__simps_I1_J,axiom,
! [P: nat > $o] :
~ ( list_ex1_nat @ P @ nil_nat ) ).
% list_ex1_simps(1)
thf(fact_291_member__rec_I2_J,axiom,
! [Y: nat] :
~ ( member_nat @ nil_nat @ Y ) ).
% member_rec(2)
thf(fact_292_list_Oset__map,axiom,
! [F: nat > nat,V: list_nat] :
( ( set_nat2 @ ( map_nat_nat @ F @ V ) )
= ( image_nat_nat @ F @ ( set_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_293_list_Oset__map,axiom,
! [F: complex > real,V: list_complex] :
( ( set_real2 @ ( map_complex_real @ F @ V ) )
= ( image_complex_real @ F @ ( set_complex2 @ V ) ) ) ).
% list.set_map
thf(fact_294_list_Oset__map,axiom,
! [F: complex > nat,V: list_complex] :
( ( set_nat2 @ ( map_complex_nat @ F @ V ) )
= ( image_complex_nat @ F @ ( set_complex2 @ V ) ) ) ).
% list.set_map
thf(fact_295_list_Oset__map,axiom,
! [F: nat > complex,V: list_nat] :
( ( set_complex2 @ ( map_nat_complex @ F @ V ) )
= ( image_nat_complex @ F @ ( set_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_296_list_Oset__map,axiom,
! [F: complex > complex,V: list_complex] :
( ( set_complex2 @ ( map_complex_complex @ F @ V ) )
= ( image_1468599708987790691omplex @ F @ ( set_complex2 @ V ) ) ) ).
% list.set_map
thf(fact_297_set__map,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( set_nat2 @ ( map_nat_nat @ F @ Xs ) )
= ( image_nat_nat @ F @ ( set_nat2 @ Xs ) ) ) ).
% set_map
thf(fact_298_set__map,axiom,
! [F: complex > real,Xs: list_complex] :
( ( set_real2 @ ( map_complex_real @ F @ Xs ) )
= ( image_complex_real @ F @ ( set_complex2 @ Xs ) ) ) ).
% set_map
thf(fact_299_set__map,axiom,
! [F: complex > nat,Xs: list_complex] :
( ( set_nat2 @ ( map_complex_nat @ F @ Xs ) )
= ( image_complex_nat @ F @ ( set_complex2 @ Xs ) ) ) ).
% set_map
thf(fact_300_set__map,axiom,
! [F: nat > complex,Xs: list_nat] :
( ( set_complex2 @ ( map_nat_complex @ F @ Xs ) )
= ( image_nat_complex @ F @ ( set_nat2 @ Xs ) ) ) ).
% set_map
thf(fact_301_set__map,axiom,
! [F: complex > complex,Xs: list_complex] :
( ( set_complex2 @ ( map_complex_complex @ F @ Xs ) )
= ( image_1468599708987790691omplex @ F @ ( set_complex2 @ Xs ) ) ) ).
% set_map
thf(fact_302_list_Oset_I2_J,axiom,
! [X21: complex,X22: list_complex] :
( ( set_complex2 @ ( cons_complex @ X21 @ X22 ) )
= ( insert_complex2 @ X21 @ ( set_complex2 @ X22 ) ) ) ).
% list.set(2)
thf(fact_303_list_Oset_I2_J,axiom,
! [X21: nat,X22: list_nat] :
( ( set_nat2 @ ( cons_nat @ X21 @ X22 ) )
= ( insert_nat2 @ X21 @ ( set_nat2 @ X22 ) ) ) ).
% list.set(2)
thf(fact_304_list_Oset_I2_J,axiom,
! [X21: real,X22: list_real] :
( ( set_real2 @ ( cons_real @ X21 @ X22 ) )
= ( insert_real2 @ X21 @ ( set_real2 @ X22 ) ) ) ).
% list.set(2)
thf(fact_305_eq__comps_Osimps_I2_J,axiom,
! [X2: nat] :
( ( commut2436974278740741825ps_nat @ ( cons_nat @ X2 @ nil_nat ) )
= ( cons_nat @ one_one_nat @ nil_nat ) ) ).
% eq_comps.simps(2)
thf(fact_306_eq__comps_Osimps_I2_J,axiom,
! [X2: real] :
( ( commut8680161604938074397s_real @ ( cons_real @ X2 @ nil_real ) )
= ( cons_nat @ one_one_nat @ nil_nat ) ) ).
% eq_comps.simps(2)
thf(fact_307_eq__comps_Osimps_I2_J,axiom,
! [X2: complex] :
( ( commut93809757773076895omplex @ ( cons_complex @ X2 @ nil_complex ) )
= ( cons_nat @ one_one_nat @ nil_nat ) ) ).
% eq_comps.simps(2)
thf(fact_308_max__list__non__empty_Oinduct,axiom,
! [P: list_nat > $o,A0: list_nat] :
( ! [X4: nat] : ( P @ ( cons_nat @ X4 @ nil_nat ) )
=> ( ! [X4: nat,V2: nat,Va: list_nat] :
( ( P @ ( cons_nat @ V2 @ Va ) )
=> ( P @ ( cons_nat @ X4 @ ( cons_nat @ V2 @ Va ) ) ) )
=> ( ( P @ nil_nat )
=> ( P @ A0 ) ) ) ) ).
% max_list_non_empty.induct
thf(fact_309_max__list__non__empty_Oinduct,axiom,
! [P: list_real > $o,A0: list_real] :
( ! [X4: real] : ( P @ ( cons_real @ X4 @ nil_real ) )
=> ( ! [X4: real,V2: real,Va: list_real] :
( ( P @ ( cons_real @ V2 @ Va ) )
=> ( P @ ( cons_real @ X4 @ ( cons_real @ V2 @ Va ) ) ) )
=> ( ( P @ nil_real )
=> ( P @ A0 ) ) ) ) ).
% max_list_non_empty.induct
thf(fact_310_max__list__non__empty_Ocases,axiom,
! [X2: list_nat] :
( ! [X4: nat] :
( X2
!= ( cons_nat @ X4 @ nil_nat ) )
=> ( ! [X4: nat,V2: nat,Va: list_nat] :
( X2
!= ( cons_nat @ X4 @ ( cons_nat @ V2 @ Va ) ) )
=> ( X2 = nil_nat ) ) ) ).
% max_list_non_empty.cases
thf(fact_311_max__list__non__empty_Ocases,axiom,
! [X2: list_real] :
( ! [X4: real] :
( X2
!= ( cons_real @ X4 @ nil_real ) )
=> ( ! [X4: real,V2: real,Va: list_real] :
( X2
!= ( cons_real @ X4 @ ( cons_real @ V2 @ Va ) ) )
=> ( X2 = nil_real ) ) ) ).
% max_list_non_empty.cases
thf(fact_312_minus__poly__rev__list_Oinduct,axiom,
! [P: list_complex > list_complex > $o,A0: list_complex,A1: list_complex] :
( ! [X4: complex,Xs3: list_complex,Y3: complex,Ys2: list_complex] :
( ( P @ Xs3 @ Ys2 )
=> ( P @ ( cons_complex @ X4 @ Xs3 ) @ ( cons_complex @ Y3 @ Ys2 ) ) )
=> ( ! [Xs3: list_complex] : ( P @ Xs3 @ nil_complex )
=> ( ! [Y3: complex,Ys2: list_complex] : ( P @ nil_complex @ ( cons_complex @ Y3 @ Ys2 ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% minus_poly_rev_list.induct
thf(fact_313_minus__poly__rev__list_Oinduct,axiom,
! [P: list_real > list_real > $o,A0: list_real,A1: list_real] :
( ! [X4: real,Xs3: list_real,Y3: real,Ys2: list_real] :
( ( P @ Xs3 @ Ys2 )
=> ( P @ ( cons_real @ X4 @ Xs3 ) @ ( cons_real @ Y3 @ Ys2 ) ) )
=> ( ! [Xs3: list_real] : ( P @ Xs3 @ nil_real )
=> ( ! [Y3: real,Ys2: list_real] : ( P @ nil_real @ ( cons_real @ Y3 @ Ys2 ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% minus_poly_rev_list.induct
thf(fact_314_longest__common__prefix_Oinduct,axiom,
! [P: list_complex > list_complex > $o,A0: list_complex,A1: list_complex] :
( ! [X4: complex,Xs3: list_complex,Y3: complex,Ys2: list_complex] :
( ( ( X4 = Y3 )
=> ( P @ Xs3 @ Ys2 ) )
=> ( P @ ( cons_complex @ X4 @ Xs3 ) @ ( cons_complex @ Y3 @ Ys2 ) ) )
=> ( ! [X_1: list_complex] : ( P @ nil_complex @ X_1 )
=> ( ! [Uu: list_complex] : ( P @ Uu @ nil_complex )
=> ( P @ A0 @ A1 ) ) ) ) ).
% longest_common_prefix.induct
thf(fact_315_longest__common__prefix_Oinduct,axiom,
! [P: list_nat > list_nat > $o,A0: list_nat,A1: list_nat] :
( ! [X4: nat,Xs3: list_nat,Y3: nat,Ys2: list_nat] :
( ( ( X4 = Y3 )
=> ( P @ Xs3 @ Ys2 ) )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys2 ) ) )
=> ( ! [X_1: list_nat] : ( P @ nil_nat @ X_1 )
=> ( ! [Uu: list_nat] : ( P @ Uu @ nil_nat )
=> ( P @ A0 @ A1 ) ) ) ) ).
% longest_common_prefix.induct
thf(fact_316_longest__common__prefix_Oinduct,axiom,
! [P: list_real > list_real > $o,A0: list_real,A1: list_real] :
( ! [X4: real,Xs3: list_real,Y3: real,Ys2: list_real] :
( ( ( X4 = Y3 )
=> ( P @ Xs3 @ Ys2 ) )
=> ( P @ ( cons_real @ X4 @ Xs3 ) @ ( cons_real @ Y3 @ Ys2 ) ) )
=> ( ! [X_1: list_real] : ( P @ nil_real @ X_1 )
=> ( ! [Uu: list_real] : ( P @ Uu @ nil_real )
=> ( P @ A0 @ A1 ) ) ) ) ).
% longest_common_prefix.induct
thf(fact_317_plus__coeffs_Oinduct,axiom,
! [P: list_complex > list_complex > $o,A0: list_complex,A1: list_complex] :
( ! [Xs3: list_complex] : ( P @ Xs3 @ nil_complex )
=> ( ! [V2: complex,Va: list_complex] : ( P @ nil_complex @ ( cons_complex @ V2 @ Va ) )
=> ( ! [X4: complex,Xs3: list_complex,Y3: complex,Ys2: list_complex] :
( ( P @ Xs3 @ Ys2 )
=> ( P @ ( cons_complex @ X4 @ Xs3 ) @ ( cons_complex @ Y3 @ Ys2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% plus_coeffs.induct
thf(fact_318_plus__coeffs_Oinduct,axiom,
! [P: list_nat > list_nat > $o,A0: list_nat,A1: list_nat] :
( ! [Xs3: list_nat] : ( P @ Xs3 @ nil_nat )
=> ( ! [V2: nat,Va: list_nat] : ( P @ nil_nat @ ( cons_nat @ V2 @ Va ) )
=> ( ! [X4: nat,Xs3: list_nat,Y3: nat,Ys2: list_nat] :
( ( P @ Xs3 @ Ys2 )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% plus_coeffs.induct
thf(fact_319_plus__coeffs_Oinduct,axiom,
! [P: list_real > list_real > $o,A0: list_real,A1: list_real] :
( ! [Xs3: list_real] : ( P @ Xs3 @ nil_real )
=> ( ! [V2: real,Va: list_real] : ( P @ nil_real @ ( cons_real @ V2 @ Va ) )
=> ( ! [X4: real,Xs3: list_real,Y3: real,Ys2: list_real] :
( ( P @ Xs3 @ Ys2 )
=> ( P @ ( cons_real @ X4 @ Xs3 ) @ ( cons_real @ Y3 @ Ys2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% plus_coeffs.induct
thf(fact_320_image__insert,axiom,
! [F: nat > nat,A: nat,B: set_nat] :
( ( image_nat_nat @ F @ ( insert_nat2 @ A @ B ) )
= ( insert_nat2 @ ( F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_321_insert__image,axiom,
! [X2: nat,A2: set_nat,F: nat > nat] :
( ( member_nat2 @ X2 @ A2 )
=> ( ( insert_nat2 @ ( F @ X2 ) @ ( image_nat_nat @ F @ A2 ) )
= ( image_nat_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_322_rel__simps_I93_J,axiom,
one_one_nat != zero_zero_nat ).
% rel_simps(93)
thf(fact_323_rel__simps_I93_J,axiom,
one_one_complex != zero_zero_complex ).
% rel_simps(93)
thf(fact_324_rel__simps_I93_J,axiom,
one_one_real != zero_zero_real ).
% rel_simps(93)
thf(fact_325_zero__complex_Osimps_I1_J,axiom,
( ( re @ zero_zero_complex )
= zero_zero_real ) ).
% zero_complex.simps(1)
thf(fact_326_rev__image__eqI,axiom,
! [X2: complex,A2: set_complex,B2: complex,F: complex > complex] :
( ( member_complex2 @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_complex2 @ B2 @ ( image_1468599708987790691omplex @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_327_rev__image__eqI,axiom,
! [X2: complex,A2: set_complex,B2: nat,F: complex > nat] :
( ( member_complex2 @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_nat2 @ B2 @ ( image_complex_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_328_rev__image__eqI,axiom,
! [X2: nat,A2: set_nat,B2: complex,F: nat > complex] :
( ( member_nat2 @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_complex2 @ B2 @ ( image_nat_complex @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_329_rev__image__eqI,axiom,
! [X2: nat,A2: set_nat,B2: nat,F: nat > nat] :
( ( member_nat2 @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_nat2 @ B2 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_330_ball__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ! [X4: nat] :
( ( member_nat2 @ X4 @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ X4 ) )
=> ! [X: nat] :
( ( member_nat2 @ X @ A2 )
=> ( P @ ( F @ X ) ) ) ) ).
% ball_imageD
thf(fact_331_image__cong,axiom,
! [M2: set_nat,N: set_nat,F: nat > nat,G: nat > nat] :
( ( M2 = N )
=> ( ! [X4: nat] :
( ( member_nat2 @ X4 @ N )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_nat_nat @ F @ M2 )
= ( image_nat_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_332_bex__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ? [X: nat] :
( ( member_nat2 @ X @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X ) )
=> ? [X4: nat] :
( ( member_nat2 @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_333_image__iff,axiom,
! [Z3: nat,F: nat > nat,A2: set_nat] :
( ( member_nat2 @ Z3 @ ( image_nat_nat @ F @ A2 ) )
= ( ? [X3: nat] :
( ( member_nat2 @ X3 @ A2 )
& ( Z3
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_334_image__eqI,axiom,
! [B2: complex,F: complex > complex,X2: complex,A2: set_complex] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_complex2 @ X2 @ A2 )
=> ( member_complex2 @ B2 @ ( image_1468599708987790691omplex @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_335_image__eqI,axiom,
! [B2: nat,F: complex > nat,X2: complex,A2: set_complex] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_complex2 @ X2 @ A2 )
=> ( member_nat2 @ B2 @ ( image_complex_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_336_image__eqI,axiom,
! [B2: complex,F: nat > complex,X2: nat,A2: set_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_nat2 @ X2 @ A2 )
=> ( member_complex2 @ B2 @ ( image_nat_complex @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_337_image__eqI,axiom,
! [B2: nat,F: nat > nat,X2: nat,A2: set_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_nat2 @ X2 @ A2 )
=> ( member_nat2 @ B2 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_338_imageI,axiom,
! [X2: complex,A2: set_complex,F: complex > complex] :
( ( member_complex2 @ X2 @ A2 )
=> ( member_complex2 @ ( F @ X2 ) @ ( image_1468599708987790691omplex @ F @ A2 ) ) ) ).
% imageI
thf(fact_339_imageI,axiom,
! [X2: complex,A2: set_complex,F: complex > nat] :
( ( member_complex2 @ X2 @ A2 )
=> ( member_nat2 @ ( F @ X2 ) @ ( image_complex_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_340_imageI,axiom,
! [X2: nat,A2: set_nat,F: nat > complex] :
( ( member_nat2 @ X2 @ A2 )
=> ( member_complex2 @ ( F @ X2 ) @ ( image_nat_complex @ F @ A2 ) ) ) ).
% imageI
thf(fact_341_imageI,axiom,
! [X2: nat,A2: set_nat,F: nat > nat] :
( ( member_nat2 @ X2 @ A2 )
=> ( member_nat2 @ ( F @ X2 ) @ ( image_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_342_mk__disjoint__insert,axiom,
! [A: complex,A2: set_complex] :
( ( member_complex2 @ A @ A2 )
=> ? [B3: set_complex] :
( ( A2
= ( insert_complex2 @ A @ B3 ) )
& ~ ( member_complex2 @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_343_mk__disjoint__insert,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat2 @ A @ A2 )
=> ? [B3: set_nat] :
( ( A2
= ( insert_nat2 @ A @ B3 ) )
& ~ ( member_nat2 @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_344_insert__eq__iff,axiom,
! [A: complex,A2: set_complex,B2: complex,B: set_complex] :
( ~ ( member_complex2 @ A @ A2 )
=> ( ~ ( member_complex2 @ B2 @ B )
=> ( ( ( insert_complex2 @ A @ A2 )
= ( insert_complex2 @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C: set_complex] :
( ( A2
= ( insert_complex2 @ B2 @ C ) )
& ~ ( member_complex2 @ B2 @ C )
& ( B
= ( insert_complex2 @ A @ C ) )
& ~ ( member_complex2 @ A @ C ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_345_insert__eq__iff,axiom,
! [A: nat,A2: set_nat,B2: nat,B: set_nat] :
( ~ ( member_nat2 @ A @ A2 )
=> ( ~ ( member_nat2 @ B2 @ B )
=> ( ( ( insert_nat2 @ A @ A2 )
= ( insert_nat2 @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C: set_nat] :
( ( A2
= ( insert_nat2 @ B2 @ C ) )
& ~ ( member_nat2 @ B2 @ C )
& ( B
= ( insert_nat2 @ A @ C ) )
& ~ ( member_nat2 @ A @ C ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_346_insert__absorb,axiom,
! [A: complex,A2: set_complex] :
( ( member_complex2 @ A @ A2 )
=> ( ( insert_complex2 @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_347_insert__absorb,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat2 @ A @ A2 )
=> ( ( insert_nat2 @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_348_insert__ident,axiom,
! [X2: complex,A2: set_complex,B: set_complex] :
( ~ ( member_complex2 @ X2 @ A2 )
=> ( ~ ( member_complex2 @ X2 @ B )
=> ( ( ( insert_complex2 @ X2 @ A2 )
= ( insert_complex2 @ X2 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_349_insert__ident,axiom,
! [X2: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat2 @ X2 @ A2 )
=> ( ~ ( member_nat2 @ X2 @ B )
=> ( ( ( insert_nat2 @ X2 @ A2 )
= ( insert_nat2 @ X2 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_350_Set_Oset__insert,axiom,
! [X2: complex,A2: set_complex] :
( ( member_complex2 @ X2 @ A2 )
=> ~ ! [B3: set_complex] :
( ( A2
= ( insert_complex2 @ X2 @ B3 ) )
=> ( member_complex2 @ X2 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_351_Set_Oset__insert,axiom,
! [X2: nat,A2: set_nat] :
( ( member_nat2 @ X2 @ A2 )
=> ~ ! [B3: set_nat] :
( ( A2
= ( insert_nat2 @ X2 @ B3 ) )
=> ( member_nat2 @ X2 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_352_insert__iff,axiom,
! [A: complex,B2: complex,A2: set_complex] :
( ( member_complex2 @ A @ ( insert_complex2 @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_complex2 @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_353_insert__iff,axiom,
! [A: nat,B2: nat,A2: set_nat] :
( ( member_nat2 @ A @ ( insert_nat2 @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_nat2 @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_354_insertI2,axiom,
! [A: complex,B: set_complex,B2: complex] :
( ( member_complex2 @ A @ B )
=> ( member_complex2 @ A @ ( insert_complex2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_355_insertI2,axiom,
! [A: nat,B: set_nat,B2: nat] :
( ( member_nat2 @ A @ B )
=> ( member_nat2 @ A @ ( insert_nat2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_356_insertI1,axiom,
! [A: complex,B: set_complex] : ( member_complex2 @ A @ ( insert_complex2 @ A @ B ) ) ).
% insertI1
thf(fact_357_insertI1,axiom,
! [A: nat,B: set_nat] : ( member_nat2 @ A @ ( insert_nat2 @ A @ B ) ) ).
% insertI1
thf(fact_358_insertCI,axiom,
! [A: complex,B: set_complex,B2: complex] :
( ( ~ ( member_complex2 @ A @ B )
=> ( A = B2 ) )
=> ( member_complex2 @ A @ ( insert_complex2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_359_insertCI,axiom,
! [A: nat,B: set_nat,B2: nat] :
( ( ~ ( member_nat2 @ A @ B )
=> ( A = B2 ) )
=> ( member_nat2 @ A @ ( insert_nat2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_360_insertE,axiom,
! [A: complex,B2: complex,A2: set_complex] :
( ( member_complex2 @ A @ ( insert_complex2 @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_complex2 @ A @ A2 ) ) ) ).
% insertE
thf(fact_361_insertE,axiom,
! [A: nat,B2: nat,A2: set_nat] :
( ( member_nat2 @ A @ ( insert_nat2 @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_nat2 @ A @ A2 ) ) ) ).
% insertE
thf(fact_362_Compr__image__eq,axiom,
! [F: complex > complex,A2: set_complex,P: complex > $o] :
( ( collect_complex
@ ^ [X3: complex] :
( ( member_complex2 @ X3 @ ( image_1468599708987790691omplex @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_1468599708987790691omplex @ F
@ ( collect_complex
@ ^ [X3: complex] :
( ( member_complex2 @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_363_Compr__image__eq,axiom,
! [F: nat > complex,A2: set_nat,P: complex > $o] :
( ( collect_complex
@ ^ [X3: complex] :
( ( member_complex2 @ X3 @ ( image_nat_complex @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_nat_complex @ F
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat2 @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_364_Compr__image__eq,axiom,
! [F: complex > nat,A2: set_complex,P: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( member_nat2 @ X3 @ ( image_complex_nat @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_complex_nat @ F
@ ( collect_complex
@ ^ [X3: complex] :
( ( member_complex2 @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_365_Compr__image__eq,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X3: nat] :
( ( member_nat2 @ X3 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X3 ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat2 @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_366_image__image,axiom,
! [F: nat > nat,G: nat > nat,A2: set_nat] :
( ( image_nat_nat @ F @ ( image_nat_nat @ G @ A2 ) )
= ( image_nat_nat
@ ^ [X3: nat] : ( F @ ( G @ X3 ) )
@ A2 ) ) ).
% image_image
thf(fact_367_image__ident,axiom,
! [Y4: set_nat] :
( ( image_nat_nat
@ ^ [X3: nat] : X3
@ Y4 )
= Y4 ) ).
% image_ident
thf(fact_368_imageE,axiom,
! [B2: complex,F: complex > complex,A2: set_complex] :
( ( member_complex2 @ B2 @ ( image_1468599708987790691omplex @ F @ A2 ) )
=> ~ ! [X4: complex] :
( ( B2
= ( F @ X4 ) )
=> ~ ( member_complex2 @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_369_imageE,axiom,
! [B2: complex,F: nat > complex,A2: set_nat] :
( ( member_complex2 @ B2 @ ( image_nat_complex @ F @ A2 ) )
=> ~ ! [X4: nat] :
( ( B2
= ( F @ X4 ) )
=> ~ ( member_nat2 @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_370_imageE,axiom,
! [B2: nat,F: complex > nat,A2: set_complex] :
( ( member_nat2 @ B2 @ ( image_complex_nat @ F @ A2 ) )
=> ~ ! [X4: complex] :
( ( B2
= ( F @ X4 ) )
=> ~ ( member_complex2 @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_371_imageE,axiom,
! [B2: nat,F: nat > nat,A2: set_nat] :
( ( member_nat2 @ B2 @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [X4: nat] :
( ( B2
= ( F @ X4 ) )
=> ~ ( member_nat2 @ X4 @ A2 ) ) ) ).
% imageE
thf(fact_372_insert__Collect,axiom,
! [A: nat,P: nat > $o] :
( ( insert_nat2 @ A @ ( collect_nat @ P ) )
= ( collect_nat
@ ^ [U: nat] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_373_insert__compr,axiom,
( insert_complex2
= ( ^ [A5: complex,B4: set_complex] :
( collect_complex
@ ^ [X3: complex] :
( ( X3 = A5 )
| ( member_complex2 @ X3 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_374_insert__compr,axiom,
( insert_nat2
= ( ^ [A5: nat,B4: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( X3 = A5 )
| ( member_nat2 @ X3 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_375_n__lists_Osimps_I1_J,axiom,
! [Xs: list_nat] :
( ( n_lists_nat @ zero_zero_nat @ Xs )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% n_lists.simps(1)
thf(fact_376_n__lists__Nil,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( n_lists_nat @ N2 @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) )
& ( ( N2 != zero_zero_nat )
=> ( ( n_lists_nat @ N2 @ nil_nat )
= nil_list_nat ) ) ) ).
% n_lists_Nil
thf(fact_377_list__encode_Oinduct,axiom,
! [P: list_nat > $o,A0: list_nat] :
( ( P @ nil_nat )
=> ( ! [X4: nat,Xs3: list_nat] :
( ( P @ Xs3 )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) ) )
=> ( P @ A0 ) ) ) ).
% list_encode.induct
thf(fact_378_list__encode_Ocases,axiom,
! [X2: list_nat] :
( ( X2 != nil_nat )
=> ~ ! [X4: nat,Xs3: list_nat] :
( X2
!= ( cons_nat @ X4 @ Xs3 ) ) ) ).
% list_encode.cases
thf(fact_379_class__field_Oone__not__zero,axiom,
one_one_complex != zero_zero_complex ).
% class_field.one_not_zero
thf(fact_380_class__field_Oone__not__zero,axiom,
one_one_real != zero_zero_real ).
% class_field.one_not_zero
thf(fact_381_sublists_Osimps_I1_J,axiom,
( ( sublists_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% sublists.simps(1)
thf(fact_382_product__lists_Osimps_I1_J,axiom,
( ( product_lists_nat @ nil_list_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% product_lists.simps(1)
thf(fact_383_map__rec,axiom,
( map_nat_complex
= ( ^ [F2: nat > complex] :
( rec_li5569328198523947590ex_nat @ nil_complex
@ ^ [X3: nat,Uu2: list_nat] : ( cons_complex @ ( F2 @ X3 ) ) ) ) ) ).
% map_rec
thf(fact_384_map__rec,axiom,
( map_complex_complex
= ( ^ [F2: complex > complex] :
( rec_li3990778930367580964omplex @ nil_complex
@ ^ [X3: complex,Uu2: list_complex] : ( cons_complex @ ( F2 @ X3 ) ) ) ) ) ).
% map_rec
thf(fact_385_map__rec,axiom,
( map_nat_nat
= ( ^ [F2: nat > nat] :
( rec_li7516600145284979816at_nat @ nil_nat
@ ^ [X3: nat,Uu2: list_nat] : ( cons_nat @ ( F2 @ X3 ) ) ) ) ) ).
% map_rec
thf(fact_386_map__rec,axiom,
( map_complex_nat
= ( ^ [F2: complex > nat] :
( rec_li5065709429383092550omplex @ nil_nat
@ ^ [X3: complex,Uu2: list_complex] : ( cons_nat @ ( F2 @ X3 ) ) ) ) ) ).
% map_rec
thf(fact_387_map__rec,axiom,
( map_complex_real
= ( ^ [F2: complex > real] :
( rec_li642222640852133922omplex @ nil_real
@ ^ [X3: complex,Uu2: list_complex] : ( cons_real @ ( F2 @ X3 ) ) ) ) ) ).
% map_rec
thf(fact_388_subseqs_Osimps_I1_J,axiom,
( ( subseqs_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% subseqs.simps(1)
thf(fact_389_the__elem__set,axiom,
! [X2: complex] :
( ( the_elem_complex @ ( set_complex2 @ ( cons_complex @ X2 @ nil_complex ) ) )
= X2 ) ).
% the_elem_set
thf(fact_390_the__elem__set,axiom,
! [X2: nat] :
( ( the_elem_nat @ ( set_nat2 @ ( cons_nat @ X2 @ nil_nat ) ) )
= X2 ) ).
% the_elem_set
thf(fact_391_the__elem__set,axiom,
! [X2: real] :
( ( the_elem_real @ ( set_real2 @ ( cons_real @ X2 @ nil_real ) ) )
= X2 ) ).
% the_elem_set
thf(fact_392_Sup_OSUP__identity__eq,axiom,
! [Sup: set_nat > nat,A2: set_nat] :
( ( Sup
@ ( image_nat_nat
@ ^ [X3: nat] : X3
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_393_Inf_OINF__identity__eq,axiom,
! [Inf: set_nat > nat,A2: set_nat] :
( ( Inf
@ ( image_nat_nat
@ ^ [X3: nat] : X3
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_394_max__list__non__empty_Osimps_I1_J,axiom,
! [X2: nat] :
( ( missin53001312869816611ty_nat @ ( cons_nat @ X2 @ nil_nat ) )
= X2 ) ).
% max_list_non_empty.simps(1)
thf(fact_395_max__list__non__empty_Osimps_I1_J,axiom,
! [X2: real] :
( ( missin3576488506594132607y_real @ ( cons_real @ X2 @ nil_real ) )
= X2 ) ).
% max_list_non_empty.simps(1)
thf(fact_396_Cons__in__subseqsD,axiom,
! [Y: complex,Ys: list_complex,Xs: list_complex] :
( ( member_list_complex @ ( cons_complex @ Y @ Ys ) @ ( set_list_complex2 @ ( subseqs_complex @ Xs ) ) )
=> ( member_list_complex @ Ys @ ( set_list_complex2 @ ( subseqs_complex @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_397_Cons__in__subseqsD,axiom,
! [Y: nat,Ys: list_nat,Xs: list_nat] :
( ( member_list_nat @ ( cons_nat @ Y @ Ys ) @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) )
=> ( member_list_nat @ Ys @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_398_Cons__in__subseqsD,axiom,
! [Y: real,Ys: list_real,Xs: list_real] :
( ( member_list_real @ ( cons_real @ Y @ Ys ) @ ( set_list_real2 @ ( subseqs_real @ Xs ) ) )
=> ( member_list_real @ Ys @ ( set_list_real2 @ ( subseqs_real @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_399_Inf_OINF__cong,axiom,
! [A2: set_nat,B: set_nat,C2: nat > nat,D: nat > nat,Inf: set_nat > nat] :
( ( A2 = B )
=> ( ! [X4: nat] :
( ( member_nat2 @ X4 @ B )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( Inf @ ( image_nat_nat @ C2 @ A2 ) )
= ( Inf @ ( image_nat_nat @ D @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_400_Sup_OSUP__cong,axiom,
! [A2: set_nat,B: set_nat,C2: nat > nat,D: nat > nat,Sup: set_nat > nat] :
( ( A2 = B )
=> ( ! [X4: nat] :
( ( member_nat2 @ X4 @ B )
=> ( ( C2 @ X4 )
= ( D @ X4 ) ) )
=> ( ( Sup @ ( image_nat_nat @ C2 @ A2 ) )
= ( Sup @ ( image_nat_nat @ D @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_401_suffixes_Osimps_I1_J,axiom,
( ( suffixes_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% suffixes.simps(1)
thf(fact_402_prefixes_Osimps_I1_J,axiom,
( ( prefixes_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% prefixes.simps(1)
thf(fact_403_plus__coeffs_Osimps_I2_J,axiom,
! [V: complex,Va2: list_complex] :
( ( plus_coeffs_complex @ nil_complex @ ( cons_complex @ V @ Va2 ) )
= ( cons_complex @ V @ Va2 ) ) ).
% plus_coeffs.simps(2)
thf(fact_404_plus__coeffs_Osimps_I2_J,axiom,
! [V: nat,Va2: list_nat] :
( ( plus_coeffs_nat @ nil_nat @ ( cons_nat @ V @ Va2 ) )
= ( cons_nat @ V @ Va2 ) ) ).
% plus_coeffs.simps(2)
thf(fact_405_plus__coeffs_Osimps_I2_J,axiom,
! [V: real,Va2: list_real] :
( ( plus_coeffs_real @ nil_real @ ( cons_real @ V @ Va2 ) )
= ( cons_real @ V @ Va2 ) ) ).
% plus_coeffs.simps(2)
thf(fact_406_vec__space_Oappend__insert,axiom,
! [Xs: list_complex,X2: complex] :
( ( set_complex2 @ ( append_complex @ Xs @ ( cons_complex @ X2 @ nil_complex ) ) )
= ( insert_complex2 @ X2 @ ( set_complex2 @ Xs ) ) ) ).
% vec_space.append_insert
thf(fact_407_vec__space_Oappend__insert,axiom,
! [Xs: list_nat,X2: nat] :
( ( set_nat2 @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) )
= ( insert_nat2 @ X2 @ ( set_nat2 @ Xs ) ) ) ).
% vec_space.append_insert
thf(fact_408_vec__space_Oappend__insert,axiom,
! [Xs: list_real,X2: real] :
( ( set_real2 @ ( append_real @ Xs @ ( cons_real @ X2 @ nil_real ) ) )
= ( insert_real2 @ X2 @ ( set_real2 @ Xs ) ) ) ).
% vec_space.append_insert
thf(fact_409_distinct__insort__key,axiom,
! [F: nat > nat,X2: nat,Xs: list_nat] :
( ( distinct_nat @ ( map_nat_nat @ F @ ( linord8961336180081300637at_nat @ F @ X2 @ Xs ) ) )
= ( ~ ( member_nat2 @ ( F @ X2 ) @ ( image_nat_nat @ F @ ( set_nat2 @ Xs ) ) )
& ( distinct_nat @ ( map_nat_nat @ F @ Xs ) ) ) ) ).
% distinct_insort_key
thf(fact_410_distinct__insort__key,axiom,
! [F: complex > real,X2: complex,Xs: list_complex] :
( ( distinct_real @ ( map_complex_real @ F @ ( linord67127995744734935x_real @ F @ X2 @ Xs ) ) )
= ( ~ ( member_real2 @ ( F @ X2 ) @ ( image_complex_real @ F @ ( set_complex2 @ Xs ) ) )
& ( distinct_real @ ( map_complex_real @ F @ Xs ) ) ) ) ).
% distinct_insort_key
thf(fact_411_distinct__insort__key,axiom,
! [F: complex > nat,X2: complex,Xs: list_complex] :
( ( distinct_nat @ ( map_complex_nat @ F @ ( linord4454476646278009211ex_nat @ F @ X2 @ Xs ) ) )
= ( ~ ( member_nat2 @ ( F @ X2 ) @ ( image_complex_nat @ F @ ( set_complex2 @ Xs ) ) )
& ( distinct_nat @ ( map_complex_nat @ F @ Xs ) ) ) ) ).
% distinct_insort_key
thf(fact_412_product__lists_Osimps_I2_J,axiom,
! [Xs: list_complex,Xss2: list_list_complex] :
( ( produc7545014605101902079omplex @ ( cons_list_complex @ Xs @ Xss2 ) )
= ( concat_list_complex
@ ( map_co661557373136449218omplex
@ ^ [X3: complex] : ( map_li2870275437539113154omplex @ ( cons_complex @ X3 ) @ ( produc7545014605101902079omplex @ Xss2 ) )
@ Xs ) ) ) ).
% product_lists.simps(2)
thf(fact_413_product__lists_Osimps_I2_J,axiom,
! [Xs: list_nat,Xss2: list_list_nat] :
( ( product_lists_nat @ ( cons_list_nat @ Xs @ Xss2 ) )
= ( concat_list_nat
@ ( map_na6205611841492582150st_nat
@ ^ [X3: nat] : ( map_li7225945977422193158st_nat @ ( cons_nat @ X3 ) @ ( product_lists_nat @ Xss2 ) )
@ Xs ) ) ) ).
% product_lists.simps(2)
thf(fact_414_product__lists_Osimps_I2_J,axiom,
! [Xs: list_real,Xss2: list_list_real] :
( ( product_lists_real @ ( cons_list_real @ Xs @ Xss2 ) )
= ( concat_list_real
@ ( map_re7007078575547571262t_real
@ ^ [X3: real] : ( map_li1455663113306559806t_real @ ( cons_real @ X3 ) @ ( product_lists_real @ Xss2 ) )
@ Xs ) ) ) ).
% product_lists.simps(2)
thf(fact_415_not__distinct__decomp,axiom,
! [Ws: list_complex] :
( ~ ( distinct_complex @ Ws )
=> ? [Xs3: list_complex,Ys2: list_complex,Zs: list_complex,Y3: complex] :
( Ws
= ( append_complex @ Xs3 @ ( append_complex @ ( cons_complex @ Y3 @ nil_complex ) @ ( append_complex @ Ys2 @ ( append_complex @ ( cons_complex @ Y3 @ nil_complex ) @ Zs ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_416_not__distinct__decomp,axiom,
! [Ws: list_nat] :
( ~ ( distinct_nat @ Ws )
=> ? [Xs3: list_nat,Ys2: list_nat,Zs: list_nat,Y3: nat] :
( Ws
= ( append_nat @ Xs3 @ ( append_nat @ ( cons_nat @ Y3 @ nil_nat ) @ ( append_nat @ Ys2 @ ( append_nat @ ( cons_nat @ Y3 @ nil_nat ) @ Zs ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_417_not__distinct__decomp,axiom,
! [Ws: list_real] :
( ~ ( distinct_real @ Ws )
=> ? [Xs3: list_real,Ys2: list_real,Zs: list_real,Y3: real] :
( Ws
= ( append_real @ Xs3 @ ( append_real @ ( cons_real @ Y3 @ nil_real ) @ ( append_real @ Ys2 @ ( append_real @ ( cons_real @ Y3 @ nil_real ) @ Zs ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_418_not__distinct__conv__prefix,axiom,
! [As2: list_complex] :
( ( ~ ( distinct_complex @ As2 ) )
= ( ? [Xs2: list_complex,Y2: complex,Ys3: list_complex] :
( ( member_complex2 @ Y2 @ ( set_complex2 @ Xs2 ) )
& ( distinct_complex @ Xs2 )
& ( As2
= ( append_complex @ Xs2 @ ( cons_complex @ Y2 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_419_not__distinct__conv__prefix,axiom,
! [As2: list_nat] :
( ( ~ ( distinct_nat @ As2 ) )
= ( ? [Xs2: list_nat,Y2: nat,Ys3: list_nat] :
( ( member_nat2 @ Y2 @ ( set_nat2 @ Xs2 ) )
& ( distinct_nat @ Xs2 )
& ( As2
= ( append_nat @ Xs2 @ ( cons_nat @ Y2 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_420_not__distinct__conv__prefix,axiom,
! [As2: list_real] :
( ( ~ ( distinct_real @ As2 ) )
= ( ? [Xs2: list_real,Y2: real,Ys3: list_real] :
( ( member_real2 @ Y2 @ ( set_real2 @ Xs2 ) )
& ( distinct_real @ Xs2 )
& ( As2
= ( append_real @ Xs2 @ ( cons_real @ Y2 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_421_concat__eq__append__conv,axiom,
! [Xss2: list_list_nat,Ys: list_nat,Zs3: list_nat] :
( ( ( concat_nat @ Xss2 )
= ( append_nat @ Ys @ Zs3 ) )
= ( ( ( Xss2 = nil_list_nat )
=> ( ( Ys = nil_nat )
& ( Zs3 = nil_nat ) ) )
& ( ( Xss2 != nil_list_nat )
=> ? [Xss1: list_list_nat,Xs2: list_nat,Xs4: list_nat,Xss22: list_list_nat] :
( ( Xss2
= ( append_list_nat @ Xss1 @ ( cons_list_nat @ ( append_nat @ Xs2 @ Xs4 ) @ Xss22 ) ) )
& ( Ys
= ( append_nat @ ( concat_nat @ Xss1 ) @ Xs2 ) )
& ( Zs3
= ( append_nat @ Xs4 @ ( concat_nat @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_422_distinct__set__subseqs,axiom,
! [Xs: list_complex] :
( ( distinct_complex @ Xs )
=> ( distinct_set_complex @ ( map_li645366948578852712omplex @ set_complex2 @ ( subseqs_complex @ Xs ) ) ) ) ).
% distinct_set_subseqs
thf(fact_423_distinct__length__2__or__more,axiom,
! [A: complex,B2: complex,Xs: list_complex] :
( ( distinct_complex @ ( cons_complex @ A @ ( cons_complex @ B2 @ Xs ) ) )
= ( ( A != B2 )
& ( distinct_complex @ ( cons_complex @ A @ Xs ) )
& ( distinct_complex @ ( cons_complex @ B2 @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_424_distinct__length__2__or__more,axiom,
! [A: nat,B2: nat,Xs: list_nat] :
( ( distinct_nat @ ( cons_nat @ A @ ( cons_nat @ B2 @ Xs ) ) )
= ( ( A != B2 )
& ( distinct_nat @ ( cons_nat @ A @ Xs ) )
& ( distinct_nat @ ( cons_nat @ B2 @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_425_distinct__length__2__or__more,axiom,
! [A: real,B2: real,Xs: list_real] :
( ( distinct_real @ ( cons_real @ A @ ( cons_real @ B2 @ Xs ) ) )
= ( ( A != B2 )
& ( distinct_real @ ( cons_real @ A @ Xs ) )
& ( distinct_real @ ( cons_real @ B2 @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_426_distinct_Osimps_I1_J,axiom,
distinct_nat @ nil_nat ).
% distinct.simps(1)
thf(fact_427_append_Osimps_I2_J,axiom,
! [X2: complex,Xs: list_complex,Ys: list_complex] :
( ( append_complex @ ( cons_complex @ X2 @ Xs ) @ Ys )
= ( cons_complex @ X2 @ ( append_complex @ Xs @ Ys ) ) ) ).
% append.simps(2)
thf(fact_428_append_Osimps_I2_J,axiom,
! [X2: nat,Xs: list_nat,Ys: list_nat] :
( ( append_nat @ ( cons_nat @ X2 @ Xs ) @ Ys )
= ( cons_nat @ X2 @ ( append_nat @ Xs @ Ys ) ) ) ).
% append.simps(2)
thf(fact_429_append_Osimps_I2_J,axiom,
! [X2: real,Xs: list_real,Ys: list_real] :
( ( append_real @ ( cons_real @ X2 @ Xs ) @ Ys )
= ( cons_real @ X2 @ ( append_real @ Xs @ Ys ) ) ) ).
% append.simps(2)
thf(fact_430_Cons__eq__appendI,axiom,
! [X2: complex,Xs1: list_complex,Ys: list_complex,Xs: list_complex,Zs3: list_complex] :
( ( ( cons_complex @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_complex @ Xs1 @ Zs3 ) )
=> ( ( cons_complex @ X2 @ Xs )
= ( append_complex @ Ys @ Zs3 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_431_Cons__eq__appendI,axiom,
! [X2: nat,Xs1: list_nat,Ys: list_nat,Xs: list_nat,Zs3: list_nat] :
( ( ( cons_nat @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_nat @ Xs1 @ Zs3 ) )
=> ( ( cons_nat @ X2 @ Xs )
= ( append_nat @ Ys @ Zs3 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_432_Cons__eq__appendI,axiom,
! [X2: real,Xs1: list_real,Ys: list_real,Xs: list_real,Zs3: list_real] :
( ( ( cons_real @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_real @ Xs1 @ Zs3 ) )
=> ( ( cons_real @ X2 @ Xs )
= ( append_real @ Ys @ Zs3 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_433_append_Osimps_I1_J,axiom,
! [Ys: list_nat] :
( ( append_nat @ nil_nat @ Ys )
= Ys ) ).
% append.simps(1)
thf(fact_434_append_Oleft__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ nil_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_435_append_Oright__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ A @ nil_nat )
= A ) ).
% append.right_neutral
thf(fact_436_append__Nil2,axiom,
! [Xs: list_nat] :
( ( append_nat @ Xs @ nil_nat )
= Xs ) ).
% append_Nil2
thf(fact_437_eq__Nil__appendI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append_nat @ nil_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_438_append__self__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_nat ) ) ).
% append_self_conv
thf(fact_439_self__append__conv,axiom,
! [Y: list_nat,Ys: list_nat] :
( ( Y
= ( append_nat @ Y @ Ys ) )
= ( Ys = nil_nat ) ) ).
% self_append_conv
thf(fact_440_append__self__conv2,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_nat ) ) ).
% append_self_conv2
thf(fact_441_self__append__conv2,axiom,
! [Y: list_nat,Xs: list_nat] :
( ( Y
= ( append_nat @ Xs @ Y ) )
= ( Xs = nil_nat ) ) ).
% self_append_conv2
thf(fact_442_Nil__is__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( nil_nat
= ( append_nat @ Xs @ Ys ) )
= ( ( Xs = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_443_append__is__Nil__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= nil_nat )
= ( ( Xs = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_444_map__append,axiom,
! [F: complex > real,Xs: list_complex,Ys: list_complex] :
( ( map_complex_real @ F @ ( append_complex @ Xs @ Ys ) )
= ( append_real @ ( map_complex_real @ F @ Xs ) @ ( map_complex_real @ F @ Ys ) ) ) ).
% map_append
thf(fact_445_map__append,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
( ( map_nat_nat @ F @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( map_nat_nat @ F @ Xs ) @ ( map_nat_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_446_map__append,axiom,
! [F: nat > complex,Xs: list_nat,Ys: list_nat] :
( ( map_nat_complex @ F @ ( append_nat @ Xs @ Ys ) )
= ( append_complex @ ( map_nat_complex @ F @ Xs ) @ ( map_nat_complex @ F @ Ys ) ) ) ).
% map_append
thf(fact_447_map__append,axiom,
! [F: complex > nat,Xs: list_complex,Ys: list_complex] :
( ( map_complex_nat @ F @ ( append_complex @ Xs @ Ys ) )
= ( append_nat @ ( map_complex_nat @ F @ Xs ) @ ( map_complex_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_448_map__append,axiom,
! [F: complex > complex,Xs: list_complex,Ys: list_complex] :
( ( map_complex_complex @ F @ ( append_complex @ Xs @ Ys ) )
= ( append_complex @ ( map_complex_complex @ F @ Xs ) @ ( map_complex_complex @ F @ Ys ) ) ) ).
% map_append
thf(fact_449_append__eq__map__conv,axiom,
! [Ys: list_real,Zs3: list_real,F: complex > real,Xs: list_complex] :
( ( ( append_real @ Ys @ Zs3 )
= ( map_complex_real @ F @ Xs ) )
= ( ? [Us: list_complex,Vs: list_complex] :
( ( Xs
= ( append_complex @ Us @ Vs ) )
& ( Ys
= ( map_complex_real @ F @ Us ) )
& ( Zs3
= ( map_complex_real @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_450_append__eq__map__conv,axiom,
! [Ys: list_nat,Zs3: list_nat,F: nat > nat,Xs: list_nat] :
( ( ( append_nat @ Ys @ Zs3 )
= ( map_nat_nat @ F @ Xs ) )
= ( ? [Us: list_nat,Vs: list_nat] :
( ( Xs
= ( append_nat @ Us @ Vs ) )
& ( Ys
= ( map_nat_nat @ F @ Us ) )
& ( Zs3
= ( map_nat_nat @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_451_append__eq__map__conv,axiom,
! [Ys: list_complex,Zs3: list_complex,F: nat > complex,Xs: list_nat] :
( ( ( append_complex @ Ys @ Zs3 )
= ( map_nat_complex @ F @ Xs ) )
= ( ? [Us: list_nat,Vs: list_nat] :
( ( Xs
= ( append_nat @ Us @ Vs ) )
& ( Ys
= ( map_nat_complex @ F @ Us ) )
& ( Zs3
= ( map_nat_complex @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_452_append__eq__map__conv,axiom,
! [Ys: list_nat,Zs3: list_nat,F: complex > nat,Xs: list_complex] :
( ( ( append_nat @ Ys @ Zs3 )
= ( map_complex_nat @ F @ Xs ) )
= ( ? [Us: list_complex,Vs: list_complex] :
( ( Xs
= ( append_complex @ Us @ Vs ) )
& ( Ys
= ( map_complex_nat @ F @ Us ) )
& ( Zs3
= ( map_complex_nat @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_453_append__eq__map__conv,axiom,
! [Ys: list_complex,Zs3: list_complex,F: complex > complex,Xs: list_complex] :
( ( ( append_complex @ Ys @ Zs3 )
= ( map_complex_complex @ F @ Xs ) )
= ( ? [Us: list_complex,Vs: list_complex] :
( ( Xs
= ( append_complex @ Us @ Vs ) )
& ( Ys
= ( map_complex_complex @ F @ Us ) )
& ( Zs3
= ( map_complex_complex @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_454_map__eq__append__conv,axiom,
! [F: complex > real,Xs: list_complex,Ys: list_real,Zs3: list_real] :
( ( ( map_complex_real @ F @ Xs )
= ( append_real @ Ys @ Zs3 ) )
= ( ? [Us: list_complex,Vs: list_complex] :
( ( Xs
= ( append_complex @ Us @ Vs ) )
& ( Ys
= ( map_complex_real @ F @ Us ) )
& ( Zs3
= ( map_complex_real @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_455_map__eq__append__conv,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat,Zs3: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( append_nat @ Ys @ Zs3 ) )
= ( ? [Us: list_nat,Vs: list_nat] :
( ( Xs
= ( append_nat @ Us @ Vs ) )
& ( Ys
= ( map_nat_nat @ F @ Us ) )
& ( Zs3
= ( map_nat_nat @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_456_map__eq__append__conv,axiom,
! [F: nat > complex,Xs: list_nat,Ys: list_complex,Zs3: list_complex] :
( ( ( map_nat_complex @ F @ Xs )
= ( append_complex @ Ys @ Zs3 ) )
= ( ? [Us: list_nat,Vs: list_nat] :
( ( Xs
= ( append_nat @ Us @ Vs ) )
& ( Ys
= ( map_nat_complex @ F @ Us ) )
& ( Zs3
= ( map_nat_complex @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_457_map__eq__append__conv,axiom,
! [F: complex > nat,Xs: list_complex,Ys: list_nat,Zs3: list_nat] :
( ( ( map_complex_nat @ F @ Xs )
= ( append_nat @ Ys @ Zs3 ) )
= ( ? [Us: list_complex,Vs: list_complex] :
( ( Xs
= ( append_complex @ Us @ Vs ) )
& ( Ys
= ( map_complex_nat @ F @ Us ) )
& ( Zs3
= ( map_complex_nat @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_458_map__eq__append__conv,axiom,
! [F: complex > complex,Xs: list_complex,Ys: list_complex,Zs3: list_complex] :
( ( ( map_complex_complex @ F @ Xs )
= ( append_complex @ Ys @ Zs3 ) )
= ( ? [Us: list_complex,Vs: list_complex] :
( ( Xs
= ( append_complex @ Us @ Vs ) )
& ( Ys
= ( map_complex_complex @ F @ Us ) )
& ( Zs3
= ( map_complex_complex @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_459_concat__eq__Nil__conv,axiom,
! [Xss2: list_list_nat] :
( ( ( concat_nat @ Xss2 )
= nil_nat )
= ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xss2 ) )
=> ( X3 = nil_nat ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_460_Nil__eq__concat__conv,axiom,
! [Xss2: list_list_nat] :
( ( nil_nat
= ( concat_nat @ Xss2 ) )
= ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xss2 ) )
=> ( X3 = nil_nat ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_461_map__concat,axiom,
! [F: complex > real,Xs: list_list_complex] :
( ( map_complex_real @ F @ ( concat_complex @ Xs ) )
= ( concat_real @ ( map_li971590449312185664t_real @ ( map_complex_real @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_462_map__concat,axiom,
! [F: nat > nat,Xs: list_list_nat] :
( ( map_nat_nat @ F @ ( concat_nat @ Xs ) )
= ( concat_nat @ ( map_li7225945977422193158st_nat @ ( map_nat_nat @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_463_map__concat,axiom,
! [F: nat > complex,Xs: list_list_nat] :
( ( map_nat_complex @ F @ ( concat_nat @ Xs ) )
= ( concat_complex @ ( map_li6798605796755630564omplex @ ( map_nat_complex @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_464_map__concat,axiom,
! [F: complex > nat,Xs: list_list_complex] :
( ( map_complex_nat @ F @ ( concat_complex @ Xs ) )
= ( concat_nat @ ( map_li3202499864523910372st_nat @ ( map_complex_nat @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_465_map__concat,axiom,
! [F: complex > complex,Xs: list_list_complex] :
( ( map_complex_complex @ F @ ( concat_complex @ Xs ) )
= ( concat_complex @ ( map_li2870275437539113154omplex @ ( map_complex_complex @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_466_sublists_Osimps_I2_J,axiom,
! [X2: complex,Xs: list_complex] :
( ( sublists_complex @ ( cons_complex @ X2 @ Xs ) )
= ( append_list_complex @ ( sublists_complex @ Xs ) @ ( map_li2870275437539113154omplex @ ( cons_complex @ X2 ) @ ( prefixes_complex @ Xs ) ) ) ) ).
% sublists.simps(2)
thf(fact_467_sublists_Osimps_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( sublists_nat @ ( cons_nat @ X2 @ Xs ) )
= ( append_list_nat @ ( sublists_nat @ Xs ) @ ( map_li7225945977422193158st_nat @ ( cons_nat @ X2 ) @ ( prefixes_nat @ Xs ) ) ) ) ).
% sublists.simps(2)
thf(fact_468_sublists_Osimps_I2_J,axiom,
! [X2: real,Xs: list_real] :
( ( sublists_real @ ( cons_real @ X2 @ Xs ) )
= ( append_list_real @ ( sublists_real @ Xs ) @ ( map_li1455663113306559806t_real @ ( cons_real @ X2 ) @ ( prefixes_real @ Xs ) ) ) ) ).
% sublists.simps(2)
thf(fact_469_prefixes__eq__snoc,axiom,
! [Ys: list_complex,Xs: list_list_complex,X2: list_complex] :
( ( ( prefixes_complex @ Ys )
= ( append_list_complex @ Xs @ ( cons_list_complex @ X2 @ nil_list_complex ) ) )
= ( ( ( ( Ys = nil_complex )
& ( Xs = nil_list_complex ) )
| ? [Z2: complex,Zs2: list_complex] :
( ( Ys
= ( append_complex @ Zs2 @ ( cons_complex @ Z2 @ nil_complex ) ) )
& ( Xs
= ( prefixes_complex @ Zs2 ) ) ) )
& ( X2 = Ys ) ) ) ).
% prefixes_eq_snoc
thf(fact_470_prefixes__eq__snoc,axiom,
! [Ys: list_nat,Xs: list_list_nat,X2: list_nat] :
( ( ( prefixes_nat @ Ys )
= ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) ) )
= ( ( ( ( Ys = nil_nat )
& ( Xs = nil_list_nat ) )
| ? [Z2: nat,Zs2: list_nat] :
( ( Ys
= ( append_nat @ Zs2 @ ( cons_nat @ Z2 @ nil_nat ) ) )
& ( Xs
= ( prefixes_nat @ Zs2 ) ) ) )
& ( X2 = Ys ) ) ) ).
% prefixes_eq_snoc
thf(fact_471_prefixes__eq__snoc,axiom,
! [Ys: list_real,Xs: list_list_real,X2: list_real] :
( ( ( prefixes_real @ Ys )
= ( append_list_real @ Xs @ ( cons_list_real @ X2 @ nil_list_real ) ) )
= ( ( ( ( Ys = nil_real )
& ( Xs = nil_list_real ) )
| ? [Z2: real,Zs2: list_real] :
( ( Ys
= ( append_real @ Zs2 @ ( cons_real @ Z2 @ nil_real ) ) )
& ( Xs
= ( prefixes_real @ Zs2 ) ) ) )
& ( X2 = Ys ) ) ) ).
% prefixes_eq_snoc
thf(fact_472_prefixes__snoc,axiom,
! [Xs: list_complex,X2: complex] :
( ( prefixes_complex @ ( append_complex @ Xs @ ( cons_complex @ X2 @ nil_complex ) ) )
= ( append_list_complex @ ( prefixes_complex @ Xs ) @ ( cons_list_complex @ ( append_complex @ Xs @ ( cons_complex @ X2 @ nil_complex ) ) @ nil_list_complex ) ) ) ).
% prefixes_snoc
thf(fact_473_prefixes__snoc,axiom,
! [Xs: list_nat,X2: nat] :
( ( prefixes_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) )
= ( append_list_nat @ ( prefixes_nat @ Xs ) @ ( cons_list_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) @ nil_list_nat ) ) ) ).
% prefixes_snoc
thf(fact_474_prefixes__snoc,axiom,
! [Xs: list_real,X2: real] :
( ( prefixes_real @ ( append_real @ Xs @ ( cons_real @ X2 @ nil_real ) ) )
= ( append_list_real @ ( prefixes_real @ Xs ) @ ( cons_list_real @ ( append_real @ Xs @ ( cons_real @ X2 @ nil_real ) ) @ nil_list_real ) ) ) ).
% prefixes_snoc
thf(fact_475_suffixes_Osimps_I2_J,axiom,
! [X2: complex,Xs: list_complex] :
( ( suffixes_complex @ ( cons_complex @ X2 @ Xs ) )
= ( append_list_complex @ ( suffixes_complex @ Xs ) @ ( cons_list_complex @ ( cons_complex @ X2 @ Xs ) @ nil_list_complex ) ) ) ).
% suffixes.simps(2)
thf(fact_476_suffixes_Osimps_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( suffixes_nat @ ( cons_nat @ X2 @ Xs ) )
= ( append_list_nat @ ( suffixes_nat @ Xs ) @ ( cons_list_nat @ ( cons_nat @ X2 @ Xs ) @ nil_list_nat ) ) ) ).
% suffixes.simps(2)
thf(fact_477_suffixes_Osimps_I2_J,axiom,
! [X2: real,Xs: list_real] :
( ( suffixes_real @ ( cons_real @ X2 @ Xs ) )
= ( append_list_real @ ( suffixes_real @ Xs ) @ ( cons_list_real @ ( cons_real @ X2 @ Xs ) @ nil_list_real ) ) ) ).
% suffixes.simps(2)
thf(fact_478_suffixes__snoc,axiom,
! [Xs: list_complex,X2: complex] :
( ( suffixes_complex @ ( append_complex @ Xs @ ( cons_complex @ X2 @ nil_complex ) ) )
= ( cons_list_complex @ nil_complex
@ ( map_li2870275437539113154omplex
@ ^ [Ys3: list_complex] : ( append_complex @ Ys3 @ ( cons_complex @ X2 @ nil_complex ) )
@ ( suffixes_complex @ Xs ) ) ) ) ).
% suffixes_snoc
thf(fact_479_suffixes__snoc,axiom,
! [Xs: list_nat,X2: nat] :
( ( suffixes_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) )
= ( cons_list_nat @ nil_nat
@ ( map_li7225945977422193158st_nat
@ ^ [Ys3: list_nat] : ( append_nat @ Ys3 @ ( cons_nat @ X2 @ nil_nat ) )
@ ( suffixes_nat @ Xs ) ) ) ) ).
% suffixes_snoc
thf(fact_480_suffixes__snoc,axiom,
! [Xs: list_real,X2: real] :
( ( suffixes_real @ ( append_real @ Xs @ ( cons_real @ X2 @ nil_real ) ) )
= ( cons_list_real @ nil_real
@ ( map_li1455663113306559806t_real
@ ^ [Ys3: list_real] : ( append_real @ Ys3 @ ( cons_real @ X2 @ nil_real ) )
@ ( suffixes_real @ Xs ) ) ) ) ).
% suffixes_snoc
thf(fact_481_distinct__singleton,axiom,
! [X2: complex] : ( distinct_complex @ ( cons_complex @ X2 @ nil_complex ) ) ).
% distinct_singleton
thf(fact_482_distinct__singleton,axiom,
! [X2: nat] : ( distinct_nat @ ( cons_nat @ X2 @ nil_nat ) ) ).
% distinct_singleton
thf(fact_483_distinct__singleton,axiom,
! [X2: real] : ( distinct_real @ ( cons_real @ X2 @ nil_real ) ) ).
% distinct_singleton
thf(fact_484_distinct_Osimps_I2_J,axiom,
! [X2: complex,Xs: list_complex] :
( ( distinct_complex @ ( cons_complex @ X2 @ Xs ) )
= ( ~ ( member_complex2 @ X2 @ ( set_complex2 @ Xs ) )
& ( distinct_complex @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_485_distinct_Osimps_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( distinct_nat @ ( cons_nat @ X2 @ Xs ) )
= ( ~ ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
& ( distinct_nat @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_486_distinct_Osimps_I2_J,axiom,
! [X2: real,Xs: list_real] :
( ( distinct_real @ ( cons_real @ X2 @ Xs ) )
= ( ~ ( member_real2 @ X2 @ ( set_real2 @ Xs ) )
& ( distinct_real @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_487_rev__nonempty__induct,axiom,
! [Xs: list_complex,P: list_complex > $o] :
( ( Xs != nil_complex )
=> ( ! [X4: complex] : ( P @ ( cons_complex @ X4 @ nil_complex ) )
=> ( ! [X4: complex,Xs3: list_complex] :
( ( Xs3 != nil_complex )
=> ( ( P @ Xs3 )
=> ( P @ ( append_complex @ Xs3 @ ( cons_complex @ X4 @ nil_complex ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_488_rev__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X4: nat] : ( P @ ( cons_nat @ X4 @ nil_nat ) )
=> ( ! [X4: nat,Xs3: list_nat] :
( ( Xs3 != nil_nat )
=> ( ( P @ Xs3 )
=> ( P @ ( append_nat @ Xs3 @ ( cons_nat @ X4 @ nil_nat ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_489_rev__nonempty__induct,axiom,
! [Xs: list_real,P: list_real > $o] :
( ( Xs != nil_real )
=> ( ! [X4: real] : ( P @ ( cons_real @ X4 @ nil_real ) )
=> ( ! [X4: real,Xs3: list_real] :
( ( Xs3 != nil_real )
=> ( ( P @ Xs3 )
=> ( P @ ( append_real @ Xs3 @ ( cons_real @ X4 @ nil_real ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_490_append__eq__Cons__conv,axiom,
! [Ys: list_complex,Zs3: list_complex,X2: complex,Xs: list_complex] :
( ( ( append_complex @ Ys @ Zs3 )
= ( cons_complex @ X2 @ Xs ) )
= ( ( ( Ys = nil_complex )
& ( Zs3
= ( cons_complex @ X2 @ Xs ) ) )
| ? [Ys4: list_complex] :
( ( Ys
= ( cons_complex @ X2 @ Ys4 ) )
& ( ( append_complex @ Ys4 @ Zs3 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_491_append__eq__Cons__conv,axiom,
! [Ys: list_nat,Zs3: list_nat,X2: nat,Xs: list_nat] :
( ( ( append_nat @ Ys @ Zs3 )
= ( cons_nat @ X2 @ Xs ) )
= ( ( ( Ys = nil_nat )
& ( Zs3
= ( cons_nat @ X2 @ Xs ) ) )
| ? [Ys4: list_nat] :
( ( Ys
= ( cons_nat @ X2 @ Ys4 ) )
& ( ( append_nat @ Ys4 @ Zs3 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_492_append__eq__Cons__conv,axiom,
! [Ys: list_real,Zs3: list_real,X2: real,Xs: list_real] :
( ( ( append_real @ Ys @ Zs3 )
= ( cons_real @ X2 @ Xs ) )
= ( ( ( Ys = nil_real )
& ( Zs3
= ( cons_real @ X2 @ Xs ) ) )
| ? [Ys4: list_real] :
( ( Ys
= ( cons_real @ X2 @ Ys4 ) )
& ( ( append_real @ Ys4 @ Zs3 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_493_Cons__eq__append__conv,axiom,
! [X2: complex,Xs: list_complex,Ys: list_complex,Zs3: list_complex] :
( ( ( cons_complex @ X2 @ Xs )
= ( append_complex @ Ys @ Zs3 ) )
= ( ( ( Ys = nil_complex )
& ( ( cons_complex @ X2 @ Xs )
= Zs3 ) )
| ? [Ys4: list_complex] :
( ( ( cons_complex @ X2 @ Ys4 )
= Ys )
& ( Xs
= ( append_complex @ Ys4 @ Zs3 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_494_Cons__eq__append__conv,axiom,
! [X2: nat,Xs: list_nat,Ys: list_nat,Zs3: list_nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( append_nat @ Ys @ Zs3 ) )
= ( ( ( Ys = nil_nat )
& ( ( cons_nat @ X2 @ Xs )
= Zs3 ) )
| ? [Ys4: list_nat] :
( ( ( cons_nat @ X2 @ Ys4 )
= Ys )
& ( Xs
= ( append_nat @ Ys4 @ Zs3 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_495_Cons__eq__append__conv,axiom,
! [X2: real,Xs: list_real,Ys: list_real,Zs3: list_real] :
( ( ( cons_real @ X2 @ Xs )
= ( append_real @ Ys @ Zs3 ) )
= ( ( ( Ys = nil_real )
& ( ( cons_real @ X2 @ Xs )
= Zs3 ) )
| ? [Ys4: list_real] :
( ( ( cons_real @ X2 @ Ys4 )
= Ys )
& ( Xs
= ( append_real @ Ys4 @ Zs3 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_496_append1__eq__conv,axiom,
! [Xs: list_complex,X2: complex,Ys: list_complex,Y: complex] :
( ( ( append_complex @ Xs @ ( cons_complex @ X2 @ nil_complex ) )
= ( append_complex @ Ys @ ( cons_complex @ Y @ nil_complex ) ) )
= ( ( Xs = Ys )
& ( X2 = Y ) ) ) ).
% append1_eq_conv
thf(fact_497_append1__eq__conv,axiom,
! [Xs: list_nat,X2: nat,Ys: list_nat,Y: nat] :
( ( ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) )
= ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) )
= ( ( Xs = Ys )
& ( X2 = Y ) ) ) ).
% append1_eq_conv
thf(fact_498_append1__eq__conv,axiom,
! [Xs: list_real,X2: real,Ys: list_real,Y: real] :
( ( ( append_real @ Xs @ ( cons_real @ X2 @ nil_real ) )
= ( append_real @ Ys @ ( cons_real @ Y @ nil_real ) ) )
= ( ( Xs = Ys )
& ( X2 = Y ) ) ) ).
% append1_eq_conv
thf(fact_499_rev__exhaust,axiom,
! [Xs: list_complex] :
( ( Xs != nil_complex )
=> ~ ! [Ys2: list_complex,Y3: complex] :
( Xs
!= ( append_complex @ Ys2 @ ( cons_complex @ Y3 @ nil_complex ) ) ) ) ).
% rev_exhaust
thf(fact_500_rev__exhaust,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ~ ! [Ys2: list_nat,Y3: nat] :
( Xs
!= ( append_nat @ Ys2 @ ( cons_nat @ Y3 @ nil_nat ) ) ) ) ).
% rev_exhaust
thf(fact_501_rev__exhaust,axiom,
! [Xs: list_real] :
( ( Xs != nil_real )
=> ~ ! [Ys2: list_real,Y3: real] :
( Xs
!= ( append_real @ Ys2 @ ( cons_real @ Y3 @ nil_real ) ) ) ) ).
% rev_exhaust
thf(fact_502_rev__induct,axiom,
! [P: list_complex > $o,Xs: list_complex] :
( ( P @ nil_complex )
=> ( ! [X4: complex,Xs3: list_complex] :
( ( P @ Xs3 )
=> ( P @ ( append_complex @ Xs3 @ ( cons_complex @ X4 @ nil_complex ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_503_rev__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ( P @ nil_nat )
=> ( ! [X4: nat,Xs3: list_nat] :
( ( P @ Xs3 )
=> ( P @ ( append_nat @ Xs3 @ ( cons_nat @ X4 @ nil_nat ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_504_rev__induct,axiom,
! [P: list_real > $o,Xs: list_real] :
( ( P @ nil_real )
=> ( ! [X4: real,Xs3: list_real] :
( ( P @ Xs3 )
=> ( P @ ( append_real @ Xs3 @ ( cons_real @ X4 @ nil_real ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_505_split__list__first__prop__iff,axiom,
! [Xs: list_complex,P: complex > $o] :
( ( ? [X3: complex] :
( ( member_complex2 @ X3 @ ( set_complex2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_complex,X3: complex] :
( ? [Zs2: list_complex] :
( Xs
= ( append_complex @ Ys3 @ ( cons_complex @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Y2: complex] :
( ( member_complex2 @ Y2 @ ( set_complex2 @ Ys3 ) )
=> ~ ( P @ Y2 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_506_split__list__first__prop__iff,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_nat,X3: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Y2: nat] :
( ( member_nat2 @ Y2 @ ( set_nat2 @ Ys3 ) )
=> ~ ( P @ Y2 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_507_split__list__first__prop__iff,axiom,
! [Xs: list_real,P: real > $o] :
( ( ? [X3: real] :
( ( member_real2 @ X3 @ ( set_real2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_real,X3: real] :
( ? [Zs2: list_real] :
( Xs
= ( append_real @ Ys3 @ ( cons_real @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Y2: real] :
( ( member_real2 @ Y2 @ ( set_real2 @ Ys3 ) )
=> ~ ( P @ Y2 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_508_split__list__last__prop__iff,axiom,
! [Xs: list_complex,P: complex > $o] :
( ( ? [X3: complex] :
( ( member_complex2 @ X3 @ ( set_complex2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_complex,X3: complex,Zs2: list_complex] :
( ( Xs
= ( append_complex @ Ys3 @ ( cons_complex @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Y2: complex] :
( ( member_complex2 @ Y2 @ ( set_complex2 @ Zs2 ) )
=> ~ ( P @ Y2 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_509_split__list__last__prop__iff,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_nat,X3: nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Y2: nat] :
( ( member_nat2 @ Y2 @ ( set_nat2 @ Zs2 ) )
=> ~ ( P @ Y2 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_510_split__list__last__prop__iff,axiom,
! [Xs: list_real,P: real > $o] :
( ( ? [X3: real] :
( ( member_real2 @ X3 @ ( set_real2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_real,X3: real,Zs2: list_real] :
( ( Xs
= ( append_real @ Ys3 @ ( cons_real @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Y2: real] :
( ( member_real2 @ Y2 @ ( set_real2 @ Zs2 ) )
=> ~ ( P @ Y2 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_511_in__set__conv__decomp__first,axiom,
! [X2: complex,Xs: list_complex] :
( ( member_complex2 @ X2 @ ( set_complex2 @ Xs ) )
= ( ? [Ys3: list_complex,Zs2: list_complex] :
( ( Xs
= ( append_complex @ Ys3 @ ( cons_complex @ X2 @ Zs2 ) ) )
& ~ ( member_complex2 @ X2 @ ( set_complex2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_512_in__set__conv__decomp__first,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [Ys3: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X2 @ Zs2 ) ) )
& ~ ( member_nat2 @ X2 @ ( set_nat2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_513_in__set__conv__decomp__first,axiom,
! [X2: real,Xs: list_real] :
( ( member_real2 @ X2 @ ( set_real2 @ Xs ) )
= ( ? [Ys3: list_real,Zs2: list_real] :
( ( Xs
= ( append_real @ Ys3 @ ( cons_real @ X2 @ Zs2 ) ) )
& ~ ( member_real2 @ X2 @ ( set_real2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_514_in__set__conv__decomp__last,axiom,
! [X2: complex,Xs: list_complex] :
( ( member_complex2 @ X2 @ ( set_complex2 @ Xs ) )
= ( ? [Ys3: list_complex,Zs2: list_complex] :
( ( Xs
= ( append_complex @ Ys3 @ ( cons_complex @ X2 @ Zs2 ) ) )
& ~ ( member_complex2 @ X2 @ ( set_complex2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_515_in__set__conv__decomp__last,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [Ys3: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X2 @ Zs2 ) ) )
& ~ ( member_nat2 @ X2 @ ( set_nat2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_516_in__set__conv__decomp__last,axiom,
! [X2: real,Xs: list_real] :
( ( member_real2 @ X2 @ ( set_real2 @ Xs ) )
= ( ? [Ys3: list_real,Zs2: list_real] :
( ( Xs
= ( append_real @ Ys3 @ ( cons_real @ X2 @ Zs2 ) ) )
& ~ ( member_real2 @ X2 @ ( set_real2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_517_split__list__first__propE,axiom,
! [Xs: list_complex,P: complex > $o] :
( ? [X: complex] :
( ( member_complex2 @ X @ ( set_complex2 @ Xs ) )
& ( P @ X ) )
=> ~ ! [Ys2: list_complex,X4: complex] :
( ? [Zs: list_complex] :
( Xs
= ( append_complex @ Ys2 @ ( cons_complex @ X4 @ Zs ) ) )
=> ( ( P @ X4 )
=> ~ ! [Xa2: complex] :
( ( member_complex2 @ Xa2 @ ( set_complex2 @ Ys2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_518_split__list__first__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X: nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
& ( P @ X ) )
=> ~ ! [Ys2: list_nat,X4: nat] :
( ? [Zs: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X4 @ Zs ) ) )
=> ( ( P @ X4 )
=> ~ ! [Xa2: nat] :
( ( member_nat2 @ Xa2 @ ( set_nat2 @ Ys2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_519_split__list__first__propE,axiom,
! [Xs: list_real,P: real > $o] :
( ? [X: real] :
( ( member_real2 @ X @ ( set_real2 @ Xs ) )
& ( P @ X ) )
=> ~ ! [Ys2: list_real,X4: real] :
( ? [Zs: list_real] :
( Xs
= ( append_real @ Ys2 @ ( cons_real @ X4 @ Zs ) ) )
=> ( ( P @ X4 )
=> ~ ! [Xa2: real] :
( ( member_real2 @ Xa2 @ ( set_real2 @ Ys2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_520_split__list__last__propE,axiom,
! [Xs: list_complex,P: complex > $o] :
( ? [X: complex] :
( ( member_complex2 @ X @ ( set_complex2 @ Xs ) )
& ( P @ X ) )
=> ~ ! [Ys2: list_complex,X4: complex,Zs: list_complex] :
( ( Xs
= ( append_complex @ Ys2 @ ( cons_complex @ X4 @ Zs ) ) )
=> ( ( P @ X4 )
=> ~ ! [Xa2: complex] :
( ( member_complex2 @ Xa2 @ ( set_complex2 @ Zs ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_521_split__list__last__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X: nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
& ( P @ X ) )
=> ~ ! [Ys2: list_nat,X4: nat,Zs: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X4 @ Zs ) ) )
=> ( ( P @ X4 )
=> ~ ! [Xa2: nat] :
( ( member_nat2 @ Xa2 @ ( set_nat2 @ Zs ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_522_split__list__last__propE,axiom,
! [Xs: list_real,P: real > $o] :
( ? [X: real] :
( ( member_real2 @ X @ ( set_real2 @ Xs ) )
& ( P @ X ) )
=> ~ ! [Ys2: list_real,X4: real,Zs: list_real] :
( ( Xs
= ( append_real @ Ys2 @ ( cons_real @ X4 @ Zs ) ) )
=> ( ( P @ X4 )
=> ~ ! [Xa2: real] :
( ( member_real2 @ Xa2 @ ( set_real2 @ Zs ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_523_split__list__first__prop,axiom,
! [Xs: list_complex,P: complex > $o] :
( ? [X: complex] :
( ( member_complex2 @ X @ ( set_complex2 @ Xs ) )
& ( P @ X ) )
=> ? [Ys2: list_complex,X4: complex] :
( ? [Zs: list_complex] :
( Xs
= ( append_complex @ Ys2 @ ( cons_complex @ X4 @ Zs ) ) )
& ( P @ X4 )
& ! [Xa2: complex] :
( ( member_complex2 @ Xa2 @ ( set_complex2 @ Ys2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_524_split__list__first__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X: nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
& ( P @ X ) )
=> ? [Ys2: list_nat,X4: nat] :
( ? [Zs: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X4 @ Zs ) ) )
& ( P @ X4 )
& ! [Xa2: nat] :
( ( member_nat2 @ Xa2 @ ( set_nat2 @ Ys2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_525_split__list__first__prop,axiom,
! [Xs: list_real,P: real > $o] :
( ? [X: real] :
( ( member_real2 @ X @ ( set_real2 @ Xs ) )
& ( P @ X ) )
=> ? [Ys2: list_real,X4: real] :
( ? [Zs: list_real] :
( Xs
= ( append_real @ Ys2 @ ( cons_real @ X4 @ Zs ) ) )
& ( P @ X4 )
& ! [Xa2: real] :
( ( member_real2 @ Xa2 @ ( set_real2 @ Ys2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_526_split__list__last__prop,axiom,
! [Xs: list_complex,P: complex > $o] :
( ? [X: complex] :
( ( member_complex2 @ X @ ( set_complex2 @ Xs ) )
& ( P @ X ) )
=> ? [Ys2: list_complex,X4: complex,Zs: list_complex] :
( ( Xs
= ( append_complex @ Ys2 @ ( cons_complex @ X4 @ Zs ) ) )
& ( P @ X4 )
& ! [Xa2: complex] :
( ( member_complex2 @ Xa2 @ ( set_complex2 @ Zs ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_527_split__list__last__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X: nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
& ( P @ X ) )
=> ? [Ys2: list_nat,X4: nat,Zs: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X4 @ Zs ) ) )
& ( P @ X4 )
& ! [Xa2: nat] :
( ( member_nat2 @ Xa2 @ ( set_nat2 @ Zs ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_528_split__list__last__prop,axiom,
! [Xs: list_real,P: real > $o] :
( ? [X: real] :
( ( member_real2 @ X @ ( set_real2 @ Xs ) )
& ( P @ X ) )
=> ? [Ys2: list_real,X4: real,Zs: list_real] :
( ( Xs
= ( append_real @ Ys2 @ ( cons_real @ X4 @ Zs ) ) )
& ( P @ X4 )
& ! [Xa2: real] :
( ( member_real2 @ Xa2 @ ( set_real2 @ Zs ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_529_in__set__conv__decomp,axiom,
! [X2: complex,Xs: list_complex] :
( ( member_complex2 @ X2 @ ( set_complex2 @ Xs ) )
= ( ? [Ys3: list_complex,Zs2: list_complex] :
( Xs
= ( append_complex @ Ys3 @ ( cons_complex @ X2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_530_in__set__conv__decomp,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [Ys3: list_nat,Zs2: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_531_in__set__conv__decomp,axiom,
! [X2: real,Xs: list_real] :
( ( member_real2 @ X2 @ ( set_real2 @ Xs ) )
= ( ? [Ys3: list_real,Zs2: list_real] :
( Xs
= ( append_real @ Ys3 @ ( cons_real @ X2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_532_append__Cons__eq__iff,axiom,
! [X2: complex,Xs: list_complex,Ys: list_complex,Xs5: list_complex,Ys5: list_complex] :
( ~ ( member_complex2 @ X2 @ ( set_complex2 @ Xs ) )
=> ( ~ ( member_complex2 @ X2 @ ( set_complex2 @ Ys ) )
=> ( ( ( append_complex @ Xs @ ( cons_complex @ X2 @ Ys ) )
= ( append_complex @ Xs5 @ ( cons_complex @ X2 @ Ys5 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_533_append__Cons__eq__iff,axiom,
! [X2: nat,Xs: list_nat,Ys: list_nat,Xs5: list_nat,Ys5: list_nat] :
( ~ ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ~ ( member_nat2 @ X2 @ ( set_nat2 @ Ys ) )
=> ( ( ( append_nat @ Xs @ ( cons_nat @ X2 @ Ys ) )
= ( append_nat @ Xs5 @ ( cons_nat @ X2 @ Ys5 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_534_append__Cons__eq__iff,axiom,
! [X2: real,Xs: list_real,Ys: list_real,Xs5: list_real,Ys5: list_real] :
( ~ ( member_real2 @ X2 @ ( set_real2 @ Xs ) )
=> ( ~ ( member_real2 @ X2 @ ( set_real2 @ Ys ) )
=> ( ( ( append_real @ Xs @ ( cons_real @ X2 @ Ys ) )
= ( append_real @ Xs5 @ ( cons_real @ X2 @ Ys5 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_535_split__list__propE,axiom,
! [Xs: list_complex,P: complex > $o] :
( ? [X: complex] :
( ( member_complex2 @ X @ ( set_complex2 @ Xs ) )
& ( P @ X ) )
=> ~ ! [Ys2: list_complex,X4: complex] :
( ? [Zs: list_complex] :
( Xs
= ( append_complex @ Ys2 @ ( cons_complex @ X4 @ Zs ) ) )
=> ~ ( P @ X4 ) ) ) ).
% split_list_propE
thf(fact_536_split__list__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X: nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
& ( P @ X ) )
=> ~ ! [Ys2: list_nat,X4: nat] :
( ? [Zs: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X4 @ Zs ) ) )
=> ~ ( P @ X4 ) ) ) ).
% split_list_propE
thf(fact_537_split__list__propE,axiom,
! [Xs: list_real,P: real > $o] :
( ? [X: real] :
( ( member_real2 @ X @ ( set_real2 @ Xs ) )
& ( P @ X ) )
=> ~ ! [Ys2: list_real,X4: real] :
( ? [Zs: list_real] :
( Xs
= ( append_real @ Ys2 @ ( cons_real @ X4 @ Zs ) ) )
=> ~ ( P @ X4 ) ) ) ).
% split_list_propE
thf(fact_538_split__list__first,axiom,
! [X2: complex,Xs: list_complex] :
( ( member_complex2 @ X2 @ ( set_complex2 @ Xs ) )
=> ? [Ys2: list_complex,Zs: list_complex] :
( ( Xs
= ( append_complex @ Ys2 @ ( cons_complex @ X2 @ Zs ) ) )
& ~ ( member_complex2 @ X2 @ ( set_complex2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_539_split__list__first,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ? [Ys2: list_nat,Zs: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs ) ) )
& ~ ( member_nat2 @ X2 @ ( set_nat2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_540_split__list__first,axiom,
! [X2: real,Xs: list_real] :
( ( member_real2 @ X2 @ ( set_real2 @ Xs ) )
=> ? [Ys2: list_real,Zs: list_real] :
( ( Xs
= ( append_real @ Ys2 @ ( cons_real @ X2 @ Zs ) ) )
& ~ ( member_real2 @ X2 @ ( set_real2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_541_split__list__prop,axiom,
! [Xs: list_complex,P: complex > $o] :
( ? [X: complex] :
( ( member_complex2 @ X @ ( set_complex2 @ Xs ) )
& ( P @ X ) )
=> ? [Ys2: list_complex,X4: complex] :
( ? [Zs: list_complex] :
( Xs
= ( append_complex @ Ys2 @ ( cons_complex @ X4 @ Zs ) ) )
& ( P @ X4 ) ) ) ).
% split_list_prop
thf(fact_542_split__list__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X: nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
& ( P @ X ) )
=> ? [Ys2: list_nat,X4: nat] :
( ? [Zs: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X4 @ Zs ) ) )
& ( P @ X4 ) ) ) ).
% split_list_prop
thf(fact_543_split__list__prop,axiom,
! [Xs: list_real,P: real > $o] :
( ? [X: real] :
( ( member_real2 @ X @ ( set_real2 @ Xs ) )
& ( P @ X ) )
=> ? [Ys2: list_real,X4: real] :
( ? [Zs: list_real] :
( Xs
= ( append_real @ Ys2 @ ( cons_real @ X4 @ Zs ) ) )
& ( P @ X4 ) ) ) ).
% split_list_prop
thf(fact_544_split__list__last,axiom,
! [X2: complex,Xs: list_complex] :
( ( member_complex2 @ X2 @ ( set_complex2 @ Xs ) )
=> ? [Ys2: list_complex,Zs: list_complex] :
( ( Xs
= ( append_complex @ Ys2 @ ( cons_complex @ X2 @ Zs ) ) )
& ~ ( member_complex2 @ X2 @ ( set_complex2 @ Zs ) ) ) ) ).
% split_list_last
thf(fact_545_split__list__last,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ? [Ys2: list_nat,Zs: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs ) ) )
& ~ ( member_nat2 @ X2 @ ( set_nat2 @ Zs ) ) ) ) ).
% split_list_last
thf(fact_546_split__list__last,axiom,
! [X2: real,Xs: list_real] :
( ( member_real2 @ X2 @ ( set_real2 @ Xs ) )
=> ? [Ys2: list_real,Zs: list_real] :
( ( Xs
= ( append_real @ Ys2 @ ( cons_real @ X2 @ Zs ) ) )
& ~ ( member_real2 @ X2 @ ( set_real2 @ Zs ) ) ) ) ).
% split_list_last
thf(fact_547_split__list,axiom,
! [X2: complex,Xs: list_complex] :
( ( member_complex2 @ X2 @ ( set_complex2 @ Xs ) )
=> ? [Ys2: list_complex,Zs: list_complex] :
( Xs
= ( append_complex @ Ys2 @ ( cons_complex @ X2 @ Zs ) ) ) ) ).
% split_list
thf(fact_548_split__list,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ? [Ys2: list_nat,Zs: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs ) ) ) ) ).
% split_list
thf(fact_549_split__list,axiom,
! [X2: real,Xs: list_real] :
( ( member_real2 @ X2 @ ( set_real2 @ Xs ) )
=> ? [Ys2: list_real,Zs: list_real] :
( Xs
= ( append_real @ Ys2 @ ( cons_real @ X2 @ Zs ) ) ) ) ).
% split_list
thf(fact_550_suffixes__eq__snoc,axiom,
! [Ys: list_complex,Xs: list_list_complex,X2: list_complex] :
( ( ( suffixes_complex @ Ys )
= ( append_list_complex @ Xs @ ( cons_list_complex @ X2 @ nil_list_complex ) ) )
= ( ( ( ( Ys = nil_complex )
& ( Xs = nil_list_complex ) )
| ? [Z2: complex,Zs2: list_complex] :
( ( Ys
= ( cons_complex @ Z2 @ Zs2 ) )
& ( Xs
= ( suffixes_complex @ Zs2 ) ) ) )
& ( X2 = Ys ) ) ) ).
% suffixes_eq_snoc
thf(fact_551_suffixes__eq__snoc,axiom,
! [Ys: list_nat,Xs: list_list_nat,X2: list_nat] :
( ( ( suffixes_nat @ Ys )
= ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) ) )
= ( ( ( ( Ys = nil_nat )
& ( Xs = nil_list_nat ) )
| ? [Z2: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z2 @ Zs2 ) )
& ( Xs
= ( suffixes_nat @ Zs2 ) ) ) )
& ( X2 = Ys ) ) ) ).
% suffixes_eq_snoc
thf(fact_552_suffixes__eq__snoc,axiom,
! [Ys: list_real,Xs: list_list_real,X2: list_real] :
( ( ( suffixes_real @ Ys )
= ( append_list_real @ Xs @ ( cons_list_real @ X2 @ nil_list_real ) ) )
= ( ( ( ( Ys = nil_real )
& ( Xs = nil_list_real ) )
| ? [Z2: real,Zs2: list_real] :
( ( Ys
= ( cons_real @ Z2 @ Zs2 ) )
& ( Xs
= ( suffixes_real @ Zs2 ) ) ) )
& ( X2 = Ys ) ) ) ).
% suffixes_eq_snoc
thf(fact_553_concat_Osimps_I1_J,axiom,
( ( concat_nat @ nil_list_nat )
= nil_nat ) ).
% concat.simps(1)
thf(fact_554_plus__coeffs_Osimps_I1_J,axiom,
! [Xs: list_nat] :
( ( plus_coeffs_nat @ Xs @ nil_nat )
= Xs ) ).
% plus_coeffs.simps(1)
thf(fact_555_subseqs_Osimps_I2_J,axiom,
! [X2: complex,Xs: list_complex] :
( ( subseqs_complex @ ( cons_complex @ X2 @ Xs ) )
= ( append_list_complex @ ( map_li2870275437539113154omplex @ ( cons_complex @ X2 ) @ ( subseqs_complex @ Xs ) ) @ ( subseqs_complex @ Xs ) ) ) ).
% subseqs.simps(2)
thf(fact_556_subseqs_Osimps_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( subseqs_nat @ ( cons_nat @ X2 @ Xs ) )
= ( append_list_nat @ ( map_li7225945977422193158st_nat @ ( cons_nat @ X2 ) @ ( subseqs_nat @ Xs ) ) @ ( subseqs_nat @ Xs ) ) ) ).
% subseqs.simps(2)
thf(fact_557_subseqs_Osimps_I2_J,axiom,
! [X2: real,Xs: list_real] :
( ( subseqs_real @ ( cons_real @ X2 @ Xs ) )
= ( append_list_real @ ( map_li1455663113306559806t_real @ ( cons_real @ X2 ) @ ( subseqs_real @ Xs ) ) @ ( subseqs_real @ Xs ) ) ) ).
% subseqs.simps(2)
thf(fact_558_concat__map__singleton,axiom,
! [F: nat > complex,Xs: list_nat] :
( ( concat_complex
@ ( map_nat_list_complex
@ ^ [X3: nat] : ( cons_complex @ ( F @ X3 ) @ nil_complex )
@ Xs ) )
= ( map_nat_complex @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_559_concat__map__singleton,axiom,
! [F: complex > complex,Xs: list_complex] :
( ( concat_complex
@ ( map_co8634619192568165682omplex
@ ^ [X3: complex] : ( cons_complex @ ( F @ X3 ) @ nil_complex )
@ Xs ) )
= ( map_complex_complex @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_560_concat__map__singleton,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( concat_nat
@ ( map_nat_list_nat
@ ^ [X3: nat] : ( cons_nat @ ( F @ X3 ) @ nil_nat )
@ Xs ) )
= ( map_nat_nat @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_561_concat__map__singleton,axiom,
! [F: complex > nat,Xs: list_complex] :
( ( concat_nat
@ ( map_complex_list_nat
@ ^ [X3: complex] : ( cons_nat @ ( F @ X3 ) @ nil_nat )
@ Xs ) )
= ( map_complex_nat @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_562_concat__map__singleton,axiom,
! [F: complex > real,Xs: list_complex] :
( ( concat_real
@ ( map_co6541394440057479600t_real
@ ^ [X3: complex] : ( cons_real @ ( F @ X3 ) @ nil_real )
@ Xs ) )
= ( map_complex_real @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_563_prefixes_Osimps_I2_J,axiom,
! [X2: complex,Xs: list_complex] :
( ( prefixes_complex @ ( cons_complex @ X2 @ Xs ) )
= ( cons_list_complex @ nil_complex @ ( map_li2870275437539113154omplex @ ( cons_complex @ X2 ) @ ( prefixes_complex @ Xs ) ) ) ) ).
% prefixes.simps(2)
thf(fact_564_prefixes_Osimps_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( prefixes_nat @ ( cons_nat @ X2 @ Xs ) )
= ( cons_list_nat @ nil_nat @ ( map_li7225945977422193158st_nat @ ( cons_nat @ X2 ) @ ( prefixes_nat @ Xs ) ) ) ) ).
% prefixes.simps(2)
thf(fact_565_prefixes_Osimps_I2_J,axiom,
! [X2: real,Xs: list_real] :
( ( prefixes_real @ ( cons_real @ X2 @ Xs ) )
= ( cons_list_real @ nil_real @ ( map_li1455663113306559806t_real @ ( cons_real @ X2 ) @ ( prefixes_real @ Xs ) ) ) ) ).
% prefixes.simps(2)
thf(fact_566_Succ__def,axiom,
( bNF_Gr1374303899511151571omplex
= ( ^ [Kl: set_list_complex,Kl2: list_complex] :
( collect_complex
@ ^ [K: complex] : ( member_list_complex @ ( append_complex @ Kl2 @ ( cons_complex @ K @ nil_complex ) ) @ Kl ) ) ) ) ).
% Succ_def
thf(fact_567_Succ__def,axiom,
( bNF_Gr6352880689984616693cc_nat
= ( ^ [Kl: set_list_nat,Kl2: list_nat] :
( collect_nat
@ ^ [K: nat] : ( member_list_nat @ ( append_nat @ Kl2 @ ( cons_nat @ K @ nil_nat ) ) @ Kl ) ) ) ) ).
% Succ_def
thf(fact_568_Succ__def,axiom,
( bNF_Gr2087828336424606033c_real
= ( ^ [Kl: set_list_real,Kl2: list_real] :
( collect_real
@ ^ [K: real] : ( member_list_real @ ( append_real @ Kl2 @ ( cons_real @ K @ nil_real ) ) @ Kl ) ) ) ) ).
% Succ_def
thf(fact_569_SuccD,axiom,
! [K2: complex,Kl3: set_list_complex,Kl4: list_complex] :
( ( member_complex2 @ K2 @ ( bNF_Gr1374303899511151571omplex @ Kl3 @ Kl4 ) )
=> ( member_list_complex @ ( append_complex @ Kl4 @ ( cons_complex @ K2 @ nil_complex ) ) @ Kl3 ) ) ).
% SuccD
thf(fact_570_SuccD,axiom,
! [K2: nat,Kl3: set_list_nat,Kl4: list_nat] :
( ( member_nat2 @ K2 @ ( bNF_Gr6352880689984616693cc_nat @ Kl3 @ Kl4 ) )
=> ( member_list_nat @ ( append_nat @ Kl4 @ ( cons_nat @ K2 @ nil_nat ) ) @ Kl3 ) ) ).
% SuccD
thf(fact_571_SuccD,axiom,
! [K2: real,Kl3: set_list_real,Kl4: list_real] :
( ( member_real2 @ K2 @ ( bNF_Gr2087828336424606033c_real @ Kl3 @ Kl4 ) )
=> ( member_list_real @ ( append_real @ Kl4 @ ( cons_real @ K2 @ nil_real ) ) @ Kl3 ) ) ).
% SuccD
thf(fact_572_SuccI,axiom,
! [Kl4: list_complex,K2: complex,Kl3: set_list_complex] :
( ( member_list_complex @ ( append_complex @ Kl4 @ ( cons_complex @ K2 @ nil_complex ) ) @ Kl3 )
=> ( member_complex2 @ K2 @ ( bNF_Gr1374303899511151571omplex @ Kl3 @ Kl4 ) ) ) ).
% SuccI
thf(fact_573_SuccI,axiom,
! [Kl4: list_nat,K2: nat,Kl3: set_list_nat] :
( ( member_list_nat @ ( append_nat @ Kl4 @ ( cons_nat @ K2 @ nil_nat ) ) @ Kl3 )
=> ( member_nat2 @ K2 @ ( bNF_Gr6352880689984616693cc_nat @ Kl3 @ Kl4 ) ) ) ).
% SuccI
thf(fact_574_SuccI,axiom,
! [Kl4: list_real,K2: real,Kl3: set_list_real] :
( ( member_list_real @ ( append_real @ Kl4 @ ( cons_real @ K2 @ nil_real ) ) @ Kl3 )
=> ( member_real2 @ K2 @ ( bNF_Gr2087828336424606033c_real @ Kl3 @ Kl4 ) ) ) ).
% SuccI
thf(fact_575_List_Otranspose_Oinduct,axiom,
! [P: list_list_complex > $o,A0: list_list_complex] :
( ( P @ nil_list_complex )
=> ( ! [Xss: list_list_complex] :
( ( P @ Xss )
=> ( P @ ( cons_list_complex @ nil_complex @ Xss ) ) )
=> ( ! [X4: complex,Xs3: list_complex,Xss: list_list_complex] :
( ( P
@ ( cons_list_complex @ Xs3
@ ( concat_list_complex
@ ( map_li6028939799916194386omplex
@ ( case_l6761848517213342308omplex @ nil_list_complex
@ ^ [H: complex,T2: list_complex] : ( cons_list_complex @ T2 @ nil_list_complex ) )
@ Xss ) ) ) )
=> ( P @ ( cons_list_complex @ ( cons_complex @ X4 @ Xs3 ) @ Xss ) ) )
=> ( P @ A0 ) ) ) ) ).
% List.transpose.induct
thf(fact_576_List_Otranspose_Oinduct,axiom,
! [P: list_list_nat > $o,A0: list_list_nat] :
( ( P @ nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( ( P @ Xss )
=> ( P @ ( cons_list_nat @ nil_nat @ Xss ) ) )
=> ( ! [X4: nat,Xs3: list_nat,Xss: list_list_nat] :
( ( P
@ ( cons_list_nat @ Xs3
@ ( concat_list_nat
@ ( map_li960784813134754710st_nat
@ ( case_l3331202209248957608at_nat @ nil_list_nat
@ ^ [H: nat,T2: list_nat] : ( cons_list_nat @ T2 @ nil_list_nat ) )
@ Xss ) ) ) )
=> ( P @ ( cons_list_nat @ ( cons_nat @ X4 @ Xs3 ) @ Xss ) ) )
=> ( P @ A0 ) ) ) ) ).
% List.transpose.induct
thf(fact_577_List_Otranspose_Oinduct,axiom,
! [P: list_list_real > $o,A0: list_list_real] :
( ( P @ nil_list_real )
=> ( ! [Xss: list_list_real] :
( ( P @ Xss )
=> ( P @ ( cons_list_real @ nil_real @ Xss ) ) )
=> ( ! [X4: real,Xs3: list_real,Xss: list_list_real] :
( ( P
@ ( cons_list_real @ Xs3
@ ( concat_list_real
@ ( map_li1716307582283692110t_real
@ ( case_l5204074688173521888l_real @ nil_list_real
@ ^ [H: real,T2: list_real] : ( cons_list_real @ T2 @ nil_list_real ) )
@ Xss ) ) ) )
=> ( P @ ( cons_list_real @ ( cons_real @ X4 @ Xs3 ) @ Xss ) ) )
=> ( P @ A0 ) ) ) ) ).
% List.transpose.induct
thf(fact_578_n__lists_Osimps_I2_J,axiom,
! [N2: nat,Xs: list_complex] :
( ( n_lists_complex @ ( suc @ N2 ) @ Xs )
= ( concat_list_complex
@ ( map_li6028939799916194386omplex
@ ^ [Ys3: list_complex] :
( map_co8634619192568165682omplex
@ ^ [Y2: complex] : ( cons_complex @ Y2 @ Ys3 )
@ Xs )
@ ( n_lists_complex @ N2 @ Xs ) ) ) ) ).
% n_lists.simps(2)
thf(fact_579_n__lists_Osimps_I2_J,axiom,
! [N2: nat,Xs: list_nat] :
( ( n_lists_nat @ ( suc @ N2 ) @ Xs )
= ( concat_list_nat
@ ( map_li960784813134754710st_nat
@ ^ [Ys3: list_nat] :
( map_nat_list_nat
@ ^ [Y2: nat] : ( cons_nat @ Y2 @ Ys3 )
@ Xs )
@ ( n_lists_nat @ N2 @ Xs ) ) ) ) ).
% n_lists.simps(2)
thf(fact_580_n__lists_Osimps_I2_J,axiom,
! [N2: nat,Xs: list_real] :
( ( n_lists_real @ ( suc @ N2 ) @ Xs )
= ( concat_list_real
@ ( map_li1716307582283692110t_real
@ ^ [Ys3: list_real] :
( map_real_list_real
@ ^ [Y2: real] : ( cons_real @ Y2 @ Ys3 )
@ Xs )
@ ( n_lists_real @ N2 @ Xs ) ) ) ) ).
% n_lists.simps(2)
thf(fact_581_list__decode_Ocases,axiom,
! [X2: nat] :
( ( X2 != zero_zero_nat )
=> ~ ! [N3: nat] :
( X2
!= ( suc @ N3 ) ) ) ).
% list_decode.cases
thf(fact_582_numeral__nat_I7_J,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% numeral_nat(7)
thf(fact_583_maps__simps_I2_J,axiom,
! [F: nat > list_nat] :
( ( maps_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% maps_simps(2)
thf(fact_584_List_Otranspose_Osimps_I3_J,axiom,
! [X2: complex,Xs: list_complex,Xss2: list_list_complex] :
( ( transpose_complex @ ( cons_list_complex @ ( cons_complex @ X2 @ Xs ) @ Xss2 ) )
= ( cons_list_complex
@ ( cons_complex @ X2
@ ( concat_complex
@ ( map_li2870275437539113154omplex
@ ( case_l7337434744184354388omplex @ nil_complex
@ ^ [H: complex,T2: list_complex] : ( cons_complex @ H @ nil_complex ) )
@ Xss2 ) ) )
@ ( transpose_complex
@ ( cons_list_complex @ Xs
@ ( concat_list_complex
@ ( map_li6028939799916194386omplex
@ ( case_l6761848517213342308omplex @ nil_list_complex
@ ^ [H: complex,T2: list_complex] : ( cons_list_complex @ T2 @ nil_list_complex ) )
@ Xss2 ) ) ) ) ) ) ).
% List.transpose.simps(3)
thf(fact_585_List_Otranspose_Osimps_I3_J,axiom,
! [X2: nat,Xs: list_nat,Xss2: list_list_nat] :
( ( transpose_nat @ ( cons_list_nat @ ( cons_nat @ X2 @ Xs ) @ Xss2 ) )
= ( cons_list_nat
@ ( cons_nat @ X2
@ ( concat_nat
@ ( map_li7225945977422193158st_nat
@ ( case_l2340614614379431832at_nat @ nil_nat
@ ^ [H: nat,T2: list_nat] : ( cons_nat @ H @ nil_nat ) )
@ Xss2 ) ) )
@ ( transpose_nat
@ ( cons_list_nat @ Xs
@ ( concat_list_nat
@ ( map_li960784813134754710st_nat
@ ( case_l3331202209248957608at_nat @ nil_list_nat
@ ^ [H: nat,T2: list_nat] : ( cons_list_nat @ T2 @ nil_list_nat ) )
@ Xss2 ) ) ) ) ) ) ).
% List.transpose.simps(3)
thf(fact_586_List_Otranspose_Osimps_I3_J,axiom,
! [X2: real,Xs: list_real,Xss2: list_list_real] :
( ( transpose_real @ ( cons_list_real @ ( cons_real @ X2 @ Xs ) @ Xss2 ) )
= ( cons_list_real
@ ( cons_real @ X2
@ ( concat_real
@ ( map_li1455663113306559806t_real
@ ( case_l3379708394843211600l_real @ nil_real
@ ^ [H: real,T2: list_real] : ( cons_real @ H @ nil_real ) )
@ Xss2 ) ) )
@ ( transpose_real
@ ( cons_list_real @ Xs
@ ( concat_list_real
@ ( map_li1716307582283692110t_real
@ ( case_l5204074688173521888l_real @ nil_list_real
@ ^ [H: real,T2: list_real] : ( cons_list_real @ T2 @ nil_list_real ) )
@ Xss2 ) ) ) ) ) ) ).
% List.transpose.simps(3)
thf(fact_587_List_Otranspose_Oelims,axiom,
! [X2: list_list_complex,Y: list_list_complex] :
( ( ( transpose_complex @ X2 )
= Y )
=> ( ( ( X2 = nil_list_complex )
=> ( Y != nil_list_complex ) )
=> ( ! [Xss: list_list_complex] :
( ( X2
= ( cons_list_complex @ nil_complex @ Xss ) )
=> ( Y
!= ( transpose_complex @ Xss ) ) )
=> ~ ! [X4: complex,Xs3: list_complex,Xss: list_list_complex] :
( ( X2
= ( cons_list_complex @ ( cons_complex @ X4 @ Xs3 ) @ Xss ) )
=> ( Y
!= ( cons_list_complex
@ ( cons_complex @ X4
@ ( concat_complex
@ ( map_li2870275437539113154omplex
@ ( case_l7337434744184354388omplex @ nil_complex
@ ^ [H: complex,T2: list_complex] : ( cons_complex @ H @ nil_complex ) )
@ Xss ) ) )
@ ( transpose_complex
@ ( cons_list_complex @ Xs3
@ ( concat_list_complex
@ ( map_li6028939799916194386omplex
@ ( case_l6761848517213342308omplex @ nil_list_complex
@ ^ [H: complex,T2: list_complex] : ( cons_list_complex @ T2 @ nil_list_complex ) )
@ Xss ) ) ) ) ) ) ) ) ) ) ).
% List.transpose.elims
thf(fact_588_List_Otranspose_Oelims,axiom,
! [X2: list_list_nat,Y: list_list_nat] :
( ( ( transpose_nat @ X2 )
= Y )
=> ( ( ( X2 = nil_list_nat )
=> ( Y != nil_list_nat ) )
=> ( ! [Xss: list_list_nat] :
( ( X2
= ( cons_list_nat @ nil_nat @ Xss ) )
=> ( Y
!= ( transpose_nat @ Xss ) ) )
=> ~ ! [X4: nat,Xs3: list_nat,Xss: list_list_nat] :
( ( X2
= ( cons_list_nat @ ( cons_nat @ X4 @ Xs3 ) @ Xss ) )
=> ( Y
!= ( cons_list_nat
@ ( cons_nat @ X4
@ ( concat_nat
@ ( map_li7225945977422193158st_nat
@ ( case_l2340614614379431832at_nat @ nil_nat
@ ^ [H: nat,T2: list_nat] : ( cons_nat @ H @ nil_nat ) )
@ Xss ) ) )
@ ( transpose_nat
@ ( cons_list_nat @ Xs3
@ ( concat_list_nat
@ ( map_li960784813134754710st_nat
@ ( case_l3331202209248957608at_nat @ nil_list_nat
@ ^ [H: nat,T2: list_nat] : ( cons_list_nat @ T2 @ nil_list_nat ) )
@ Xss ) ) ) ) ) ) ) ) ) ) ).
% List.transpose.elims
thf(fact_589_List_Otranspose_Oelims,axiom,
! [X2: list_list_real,Y: list_list_real] :
( ( ( transpose_real @ X2 )
= Y )
=> ( ( ( X2 = nil_list_real )
=> ( Y != nil_list_real ) )
=> ( ! [Xss: list_list_real] :
( ( X2
= ( cons_list_real @ nil_real @ Xss ) )
=> ( Y
!= ( transpose_real @ Xss ) ) )
=> ~ ! [X4: real,Xs3: list_real,Xss: list_list_real] :
( ( X2
= ( cons_list_real @ ( cons_real @ X4 @ Xs3 ) @ Xss ) )
=> ( Y
!= ( cons_list_real
@ ( cons_real @ X4
@ ( concat_real
@ ( map_li1455663113306559806t_real
@ ( case_l3379708394843211600l_real @ nil_real
@ ^ [H: real,T2: list_real] : ( cons_real @ H @ nil_real ) )
@ Xss ) ) )
@ ( transpose_real
@ ( cons_list_real @ Xs3
@ ( concat_list_real
@ ( map_li1716307582283692110t_real
@ ( case_l5204074688173521888l_real @ nil_list_real
@ ^ [H: real,T2: list_real] : ( cons_list_real @ T2 @ nil_list_real ) )
@ Xss ) ) ) ) ) ) ) ) ) ) ).
% List.transpose.elims
thf(fact_590_empty__Shift,axiom,
! [Kl3: set_list_complex,K2: complex] :
( ( member_list_complex @ nil_complex @ Kl3 )
=> ( ( member_complex2 @ K2 @ ( bNF_Gr1374303899511151571omplex @ Kl3 @ nil_complex ) )
=> ( member_list_complex @ nil_complex @ ( bNF_Gr1398276426132326479omplex @ Kl3 @ K2 ) ) ) ) ).
% empty_Shift
thf(fact_591_empty__Shift,axiom,
! [Kl3: set_list_nat,K2: nat] :
( ( member_list_nat @ nil_nat @ Kl3 )
=> ( ( member_nat2 @ K2 @ ( bNF_Gr6352880689984616693cc_nat @ Kl3 @ nil_nat ) )
=> ( member_list_nat @ nil_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl3 @ K2 ) ) ) ) ).
% empty_Shift
thf(fact_592_Succ__Shift,axiom,
! [Kl3: set_list_complex,K2: complex,Kl4: list_complex] :
( ( bNF_Gr1374303899511151571omplex @ ( bNF_Gr1398276426132326479omplex @ Kl3 @ K2 ) @ Kl4 )
= ( bNF_Gr1374303899511151571omplex @ Kl3 @ ( cons_complex @ K2 @ Kl4 ) ) ) ).
% Succ_Shift
thf(fact_593_Succ__Shift,axiom,
! [Kl3: set_list_nat,K2: nat,Kl4: list_nat] :
( ( bNF_Gr6352880689984616693cc_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl3 @ K2 ) @ Kl4 )
= ( bNF_Gr6352880689984616693cc_nat @ Kl3 @ ( cons_nat @ K2 @ Kl4 ) ) ) ).
% Succ_Shift
thf(fact_594_Succ__Shift,axiom,
! [Kl3: set_list_real,K2: real,Kl4: list_real] :
( ( bNF_Gr2087828336424606033c_real @ ( bNF_Gr3712412480325189581t_real @ Kl3 @ K2 ) @ Kl4 )
= ( bNF_Gr2087828336424606033c_real @ Kl3 @ ( cons_real @ K2 @ Kl4 ) ) ) ).
% Succ_Shift
thf(fact_595_nat_Osimps_I3_J,axiom,
! [X24: nat] :
( ( suc @ X24 )
!= zero_zero_nat ) ).
% nat.simps(3)
thf(fact_596_list_Odisc__eq__case_I2_J,axiom,
! [List: list_nat] :
( ( List != nil_nat )
= ( case_list_o_nat @ $false
@ ^ [Uu2: nat,Uv: list_nat] : $true
@ List ) ) ).
% list.disc_eq_case(2)
thf(fact_597_list_Odisc__eq__case_I1_J,axiom,
! [List: list_nat] :
( ( List = nil_nat )
= ( case_list_o_nat @ $true
@ ^ [Uu2: nat,Uv: list_nat] : $false
@ List ) ) ).
% list.disc_eq_case(1)
thf(fact_598_List_Otranspose_Osimps_I2_J,axiom,
! [Xss2: list_list_nat] :
( ( transpose_nat @ ( cons_list_nat @ nil_nat @ Xss2 ) )
= ( transpose_nat @ Xss2 ) ) ).
% List.transpose.simps(2)
thf(fact_599_transpose__map__map,axiom,
! [F: complex > real,Xs: list_list_complex] :
( ( transpose_real @ ( map_li971590449312185664t_real @ ( map_complex_real @ F ) @ Xs ) )
= ( map_li971590449312185664t_real @ ( map_complex_real @ F ) @ ( transpose_complex @ Xs ) ) ) ).
% transpose_map_map
thf(fact_600_transpose__map__map,axiom,
! [F: nat > nat,Xs: list_list_nat] :
( ( transpose_nat @ ( map_li7225945977422193158st_nat @ ( map_nat_nat @ F ) @ Xs ) )
= ( map_li7225945977422193158st_nat @ ( map_nat_nat @ F ) @ ( transpose_nat @ Xs ) ) ) ).
% transpose_map_map
thf(fact_601_transpose__map__map,axiom,
! [F: nat > complex,Xs: list_list_nat] :
( ( transpose_complex @ ( map_li6798605796755630564omplex @ ( map_nat_complex @ F ) @ Xs ) )
= ( map_li6798605796755630564omplex @ ( map_nat_complex @ F ) @ ( transpose_nat @ Xs ) ) ) ).
% transpose_map_map
thf(fact_602_transpose__map__map,axiom,
! [F: complex > nat,Xs: list_list_complex] :
( ( transpose_nat @ ( map_li3202499864523910372st_nat @ ( map_complex_nat @ F ) @ Xs ) )
= ( map_li3202499864523910372st_nat @ ( map_complex_nat @ F ) @ ( transpose_complex @ Xs ) ) ) ).
% transpose_map_map
thf(fact_603_transpose__map__map,axiom,
! [F: complex > complex,Xs: list_list_complex] :
( ( transpose_complex @ ( map_li2870275437539113154omplex @ ( map_complex_complex @ F ) @ Xs ) )
= ( map_li2870275437539113154omplex @ ( map_complex_complex @ F ) @ ( transpose_complex @ Xs ) ) ) ).
% transpose_map_map
thf(fact_604_ShiftD,axiom,
! [Kl4: list_complex,Kl3: set_list_complex,K2: complex] :
( ( member_list_complex @ Kl4 @ ( bNF_Gr1398276426132326479omplex @ Kl3 @ K2 ) )
=> ( member_list_complex @ ( cons_complex @ K2 @ Kl4 ) @ Kl3 ) ) ).
% ShiftD
thf(fact_605_ShiftD,axiom,
! [Kl4: list_nat,Kl3: set_list_nat,K2: nat] :
( ( member_list_nat @ Kl4 @ ( bNF_Gr1872714664788909425ft_nat @ Kl3 @ K2 ) )
=> ( member_list_nat @ ( cons_nat @ K2 @ Kl4 ) @ Kl3 ) ) ).
% ShiftD
thf(fact_606_ShiftD,axiom,
! [Kl4: list_real,Kl3: set_list_real,K2: real] :
( ( member_list_real @ Kl4 @ ( bNF_Gr3712412480325189581t_real @ Kl3 @ K2 ) )
=> ( member_list_real @ ( cons_real @ K2 @ Kl4 ) @ Kl3 ) ) ).
% ShiftD
thf(fact_607_transpose__empty,axiom,
! [Xs: list_list_nat] :
( ( ( transpose_nat @ Xs )
= nil_list_nat )
= ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( X3 = nil_nat ) ) ) ) ).
% transpose_empty
thf(fact_608_Shift__def,axiom,
( bNF_Gr1398276426132326479omplex
= ( ^ [Kl: set_list_complex,K: complex] :
( collect_list_complex
@ ^ [Kl2: list_complex] : ( member_list_complex @ ( cons_complex @ K @ Kl2 ) @ Kl ) ) ) ) ).
% Shift_def
thf(fact_609_Shift__def,axiom,
( bNF_Gr1872714664788909425ft_nat
= ( ^ [Kl: set_list_nat,K: nat] :
( collect_list_nat
@ ^ [Kl2: list_nat] : ( member_list_nat @ ( cons_nat @ K @ Kl2 ) @ Kl ) ) ) ) ).
% Shift_def
thf(fact_610_Shift__def,axiom,
( bNF_Gr3712412480325189581t_real
= ( ^ [Kl: set_list_real,K: real] :
( collect_list_real
@ ^ [Kl2: list_real] : ( member_list_real @ ( cons_real @ K @ Kl2 ) @ Kl ) ) ) ) ).
% Shift_def
thf(fact_611_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ? [M3: nat] :
( N2
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_612_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_613_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_614_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_615_zero__induct,axiom,
! [P: nat > $o,K2: nat] :
( ( P @ K2 )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_616_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N2: nat] :
( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X4: nat,Y3: nat] :
( ( P @ X4 @ Y3 )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N2 ) ) ) ) ).
% diff_induct
thf(fact_617_nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_induct
thf(fact_618_old_Onat_Oinducts,axiom,
! [P: nat > $o,Nat: nat] :
( ( P @ zero_zero_nat )
=> ( ! [Nat2: nat] :
( ( P @ Nat2 )
=> ( P @ ( suc @ Nat2 ) ) )
=> ( P @ Nat ) ) ) ).
% old.nat.inducts
thf(fact_619_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat2: nat] :
( Y
!= ( suc @ Nat2 ) ) ) ).
% old.nat.exhaust
thf(fact_620_nat_OdiscI,axiom,
! [Nat: nat,X24: nat] :
( ( Nat
= ( suc @ X24 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_621_old_Onat_Odistinct_I1_J,axiom,
! [Nat3: nat] :
( zero_zero_nat
!= ( suc @ Nat3 ) ) ).
% old.nat.distinct(1)
thf(fact_622_old_Onat_Odistinct_I2_J,axiom,
! [Nat3: nat] :
( ( suc @ Nat3 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_623_List_Otranspose_Opsimps_I3_J,axiom,
! [X2: complex,Xs: list_complex,Xss2: list_list_complex] :
( ( accp_l5771520762016474373omplex @ transp2032975277941382969omplex @ ( cons_list_complex @ ( cons_complex @ X2 @ Xs ) @ Xss2 ) )
=> ( ( transpose_complex @ ( cons_list_complex @ ( cons_complex @ X2 @ Xs ) @ Xss2 ) )
= ( cons_list_complex
@ ( cons_complex @ X2
@ ( concat_complex
@ ( map_li2870275437539113154omplex
@ ( case_l7337434744184354388omplex @ nil_complex
@ ^ [H: complex,T2: list_complex] : ( cons_complex @ H @ nil_complex ) )
@ Xss2 ) ) )
@ ( transpose_complex
@ ( cons_list_complex @ Xs
@ ( concat_list_complex
@ ( map_li6028939799916194386omplex
@ ( case_l6761848517213342308omplex @ nil_list_complex
@ ^ [H: complex,T2: list_complex] : ( cons_list_complex @ T2 @ nil_list_complex ) )
@ Xss2 ) ) ) ) ) ) ) ).
% List.transpose.psimps(3)
thf(fact_624_List_Otranspose_Opsimps_I3_J,axiom,
! [X2: nat,Xs: list_nat,Xss2: list_list_nat] :
( ( accp_list_list_nat @ transpose_rel_nat @ ( cons_list_nat @ ( cons_nat @ X2 @ Xs ) @ Xss2 ) )
=> ( ( transpose_nat @ ( cons_list_nat @ ( cons_nat @ X2 @ Xs ) @ Xss2 ) )
= ( cons_list_nat
@ ( cons_nat @ X2
@ ( concat_nat
@ ( map_li7225945977422193158st_nat
@ ( case_l2340614614379431832at_nat @ nil_nat
@ ^ [H: nat,T2: list_nat] : ( cons_nat @ H @ nil_nat ) )
@ Xss2 ) ) )
@ ( transpose_nat
@ ( cons_list_nat @ Xs
@ ( concat_list_nat
@ ( map_li960784813134754710st_nat
@ ( case_l3331202209248957608at_nat @ nil_list_nat
@ ^ [H: nat,T2: list_nat] : ( cons_list_nat @ T2 @ nil_list_nat ) )
@ Xss2 ) ) ) ) ) ) ) ).
% List.transpose.psimps(3)
thf(fact_625_List_Otranspose_Opsimps_I3_J,axiom,
! [X2: real,Xs: list_real,Xss2: list_list_real] :
( ( accp_list_list_real @ transpose_rel_real @ ( cons_list_real @ ( cons_real @ X2 @ Xs ) @ Xss2 ) )
=> ( ( transpose_real @ ( cons_list_real @ ( cons_real @ X2 @ Xs ) @ Xss2 ) )
= ( cons_list_real
@ ( cons_real @ X2
@ ( concat_real
@ ( map_li1455663113306559806t_real
@ ( case_l3379708394843211600l_real @ nil_real
@ ^ [H: real,T2: list_real] : ( cons_real @ H @ nil_real ) )
@ Xss2 ) ) )
@ ( transpose_real
@ ( cons_list_real @ Xs
@ ( concat_list_real
@ ( map_li1716307582283692110t_real
@ ( case_l5204074688173521888l_real @ nil_list_real
@ ^ [H: real,T2: list_real] : ( cons_list_real @ T2 @ nil_list_real ) )
@ Xss2 ) ) ) ) ) ) ) ).
% List.transpose.psimps(3)
thf(fact_626_List_Otranspose_Opelims,axiom,
! [X2: list_list_complex,Y: list_list_complex] :
( ( ( transpose_complex @ X2 )
= Y )
=> ( ( accp_l5771520762016474373omplex @ transp2032975277941382969omplex @ X2 )
=> ( ( ( X2 = nil_list_complex )
=> ( ( Y = nil_list_complex )
=> ~ ( accp_l5771520762016474373omplex @ transp2032975277941382969omplex @ nil_list_complex ) ) )
=> ( ! [Xss: list_list_complex] :
( ( X2
= ( cons_list_complex @ nil_complex @ Xss ) )
=> ( ( Y
= ( transpose_complex @ Xss ) )
=> ~ ( accp_l5771520762016474373omplex @ transp2032975277941382969omplex @ ( cons_list_complex @ nil_complex @ Xss ) ) ) )
=> ~ ! [X4: complex,Xs3: list_complex,Xss: list_list_complex] :
( ( X2
= ( cons_list_complex @ ( cons_complex @ X4 @ Xs3 ) @ Xss ) )
=> ( ( Y
= ( cons_list_complex
@ ( cons_complex @ X4
@ ( concat_complex
@ ( map_li2870275437539113154omplex
@ ( case_l7337434744184354388omplex @ nil_complex
@ ^ [H: complex,T2: list_complex] : ( cons_complex @ H @ nil_complex ) )
@ Xss ) ) )
@ ( transpose_complex
@ ( cons_list_complex @ Xs3
@ ( concat_list_complex
@ ( map_li6028939799916194386omplex
@ ( case_l6761848517213342308omplex @ nil_list_complex
@ ^ [H: complex,T2: list_complex] : ( cons_list_complex @ T2 @ nil_list_complex ) )
@ Xss ) ) ) ) ) )
=> ~ ( accp_l5771520762016474373omplex @ transp2032975277941382969omplex @ ( cons_list_complex @ ( cons_complex @ X4 @ Xs3 ) @ Xss ) ) ) ) ) ) ) ) ).
% List.transpose.pelims
thf(fact_627_List_Otranspose_Opelims,axiom,
! [X2: list_list_nat,Y: list_list_nat] :
( ( ( transpose_nat @ X2 )
= Y )
=> ( ( accp_list_list_nat @ transpose_rel_nat @ X2 )
=> ( ( ( X2 = nil_list_nat )
=> ( ( Y = nil_list_nat )
=> ~ ( accp_list_list_nat @ transpose_rel_nat @ nil_list_nat ) ) )
=> ( ! [Xss: list_list_nat] :
( ( X2
= ( cons_list_nat @ nil_nat @ Xss ) )
=> ( ( Y
= ( transpose_nat @ Xss ) )
=> ~ ( accp_list_list_nat @ transpose_rel_nat @ ( cons_list_nat @ nil_nat @ Xss ) ) ) )
=> ~ ! [X4: nat,Xs3: list_nat,Xss: list_list_nat] :
( ( X2
= ( cons_list_nat @ ( cons_nat @ X4 @ Xs3 ) @ Xss ) )
=> ( ( Y
= ( cons_list_nat
@ ( cons_nat @ X4
@ ( concat_nat
@ ( map_li7225945977422193158st_nat
@ ( case_l2340614614379431832at_nat @ nil_nat
@ ^ [H: nat,T2: list_nat] : ( cons_nat @ H @ nil_nat ) )
@ Xss ) ) )
@ ( transpose_nat
@ ( cons_list_nat @ Xs3
@ ( concat_list_nat
@ ( map_li960784813134754710st_nat
@ ( case_l3331202209248957608at_nat @ nil_list_nat
@ ^ [H: nat,T2: list_nat] : ( cons_list_nat @ T2 @ nil_list_nat ) )
@ Xss ) ) ) ) ) )
=> ~ ( accp_list_list_nat @ transpose_rel_nat @ ( cons_list_nat @ ( cons_nat @ X4 @ Xs3 ) @ Xss ) ) ) ) ) ) ) ) ).
% List.transpose.pelims
thf(fact_628_List_Otranspose_Opelims,axiom,
! [X2: list_list_real,Y: list_list_real] :
( ( ( transpose_real @ X2 )
= Y )
=> ( ( accp_list_list_real @ transpose_rel_real @ X2 )
=> ( ( ( X2 = nil_list_real )
=> ( ( Y = nil_list_real )
=> ~ ( accp_list_list_real @ transpose_rel_real @ nil_list_real ) ) )
=> ( ! [Xss: list_list_real] :
( ( X2
= ( cons_list_real @ nil_real @ Xss ) )
=> ( ( Y
= ( transpose_real @ Xss ) )
=> ~ ( accp_list_list_real @ transpose_rel_real @ ( cons_list_real @ nil_real @ Xss ) ) ) )
=> ~ ! [X4: real,Xs3: list_real,Xss: list_list_real] :
( ( X2
= ( cons_list_real @ ( cons_real @ X4 @ Xs3 ) @ Xss ) )
=> ( ( Y
= ( cons_list_real
@ ( cons_real @ X4
@ ( concat_real
@ ( map_li1455663113306559806t_real
@ ( case_l3379708394843211600l_real @ nil_real
@ ^ [H: real,T2: list_real] : ( cons_real @ H @ nil_real ) )
@ Xss ) ) )
@ ( transpose_real
@ ( cons_list_real @ Xs3
@ ( concat_list_real
@ ( map_li1716307582283692110t_real
@ ( case_l5204074688173521888l_real @ nil_list_real
@ ^ [H: real,T2: list_real] : ( cons_list_real @ T2 @ nil_list_real ) )
@ Xss ) ) ) ) ) )
=> ~ ( accp_list_list_real @ transpose_rel_real @ ( cons_list_real @ ( cons_real @ X4 @ Xs3 ) @ Xss ) ) ) ) ) ) ) ) ).
% List.transpose.pelims
thf(fact_629_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N3: nat] :
( ~ ( P @ N3 )
& ( P @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_630_zero__notin__Suc__image,axiom,
! [A2: set_nat] :
~ ( member_nat2 @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).
% zero_notin_Suc_image
thf(fact_631_unit__vecs__first_Oinduct,axiom,
! [P: nat > nat > $o,A0: nat,A1: nat] :
( ! [N3: nat] : ( P @ N3 @ zero_zero_nat )
=> ( ! [N3: nat,I: nat] :
( ( P @ N3 @ I )
=> ( P @ N3 @ ( suc @ I ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% unit_vecs_first.induct
thf(fact_632_List_Otranspose_Opsimps_I2_J,axiom,
! [Xss2: list_list_nat] :
( ( accp_list_list_nat @ transpose_rel_nat @ ( cons_list_nat @ nil_nat @ Xss2 ) )
=> ( ( transpose_nat @ ( cons_list_nat @ nil_nat @ Xss2 ) )
= ( transpose_nat @ Xss2 ) ) ) ).
% List.transpose.psimps(2)
thf(fact_633_List_Otranspose_Opinduct,axiom,
! [A0: list_list_complex,P: list_list_complex > $o] :
( ( accp_l5771520762016474373omplex @ transp2032975277941382969omplex @ A0 )
=> ( ( ( accp_l5771520762016474373omplex @ transp2032975277941382969omplex @ nil_list_complex )
=> ( P @ nil_list_complex ) )
=> ( ! [Xss: list_list_complex] :
( ( accp_l5771520762016474373omplex @ transp2032975277941382969omplex @ ( cons_list_complex @ nil_complex @ Xss ) )
=> ( ( P @ Xss )
=> ( P @ ( cons_list_complex @ nil_complex @ Xss ) ) ) )
=> ( ! [X4: complex,Xs3: list_complex,Xss: list_list_complex] :
( ( accp_l5771520762016474373omplex @ transp2032975277941382969omplex @ ( cons_list_complex @ ( cons_complex @ X4 @ Xs3 ) @ Xss ) )
=> ( ( P
@ ( cons_list_complex @ Xs3
@ ( concat_list_complex
@ ( map_li6028939799916194386omplex
@ ( case_l6761848517213342308omplex @ nil_list_complex
@ ^ [H: complex,T2: list_complex] : ( cons_list_complex @ T2 @ nil_list_complex ) )
@ Xss ) ) ) )
=> ( P @ ( cons_list_complex @ ( cons_complex @ X4 @ Xs3 ) @ Xss ) ) ) )
=> ( P @ A0 ) ) ) ) ) ).
% List.transpose.pinduct
thf(fact_634_List_Otranspose_Opinduct,axiom,
! [A0: list_list_nat,P: list_list_nat > $o] :
( ( accp_list_list_nat @ transpose_rel_nat @ A0 )
=> ( ( ( accp_list_list_nat @ transpose_rel_nat @ nil_list_nat )
=> ( P @ nil_list_nat ) )
=> ( ! [Xss: list_list_nat] :
( ( accp_list_list_nat @ transpose_rel_nat @ ( cons_list_nat @ nil_nat @ Xss ) )
=> ( ( P @ Xss )
=> ( P @ ( cons_list_nat @ nil_nat @ Xss ) ) ) )
=> ( ! [X4: nat,Xs3: list_nat,Xss: list_list_nat] :
( ( accp_list_list_nat @ transpose_rel_nat @ ( cons_list_nat @ ( cons_nat @ X4 @ Xs3 ) @ Xss ) )
=> ( ( P
@ ( cons_list_nat @ Xs3
@ ( concat_list_nat
@ ( map_li960784813134754710st_nat
@ ( case_l3331202209248957608at_nat @ nil_list_nat
@ ^ [H: nat,T2: list_nat] : ( cons_list_nat @ T2 @ nil_list_nat ) )
@ Xss ) ) ) )
=> ( P @ ( cons_list_nat @ ( cons_nat @ X4 @ Xs3 ) @ Xss ) ) ) )
=> ( P @ A0 ) ) ) ) ) ).
% List.transpose.pinduct
thf(fact_635_List_Otranspose_Opinduct,axiom,
! [A0: list_list_real,P: list_list_real > $o] :
( ( accp_list_list_real @ transpose_rel_real @ A0 )
=> ( ( ( accp_list_list_real @ transpose_rel_real @ nil_list_real )
=> ( P @ nil_list_real ) )
=> ( ! [Xss: list_list_real] :
( ( accp_list_list_real @ transpose_rel_real @ ( cons_list_real @ nil_real @ Xss ) )
=> ( ( P @ Xss )
=> ( P @ ( cons_list_real @ nil_real @ Xss ) ) ) )
=> ( ! [X4: real,Xs3: list_real,Xss: list_list_real] :
( ( accp_list_list_real @ transpose_rel_real @ ( cons_list_real @ ( cons_real @ X4 @ Xs3 ) @ Xss ) )
=> ( ( P
@ ( cons_list_real @ Xs3
@ ( concat_list_real
@ ( map_li1716307582283692110t_real
@ ( case_l5204074688173521888l_real @ nil_list_real
@ ^ [H: real,T2: list_real] : ( cons_list_real @ T2 @ nil_list_real ) )
@ Xss ) ) ) )
=> ( P @ ( cons_list_real @ ( cons_real @ X4 @ Xs3 ) @ Xss ) ) ) )
=> ( P @ A0 ) ) ) ) ) ).
% List.transpose.pinduct
thf(fact_636_inf__concat_Oinduct,axiom,
! [P: ( nat > nat ) > nat > $o,A0: nat > nat,A1: nat] :
( ! [N3: nat > nat] : ( P @ N3 @ zero_zero_nat )
=> ( ! [N3: nat > nat,K3: nat] :
( ( P @ N3 @ K3 )
=> ( P @ N3 @ ( suc @ K3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% inf_concat.induct
thf(fact_637_nths__Cons,axiom,
! [X2: complex,L: list_complex,A2: set_nat] :
( ( nths_complex @ ( cons_complex @ X2 @ L ) @ A2 )
= ( append_complex @ ( if_list_complex @ ( member_nat2 @ zero_zero_nat @ A2 ) @ ( cons_complex @ X2 @ nil_complex ) @ nil_complex )
@ ( nths_complex @ L
@ ( collect_nat
@ ^ [J: nat] : ( member_nat2 @ ( suc @ J ) @ A2 ) ) ) ) ) ).
% nths_Cons
thf(fact_638_nths__Cons,axiom,
! [X2: nat,L: list_nat,A2: set_nat] :
( ( nths_nat @ ( cons_nat @ X2 @ L ) @ A2 )
= ( append_nat @ ( if_list_nat @ ( member_nat2 @ zero_zero_nat @ A2 ) @ ( cons_nat @ X2 @ nil_nat ) @ nil_nat )
@ ( nths_nat @ L
@ ( collect_nat
@ ^ [J: nat] : ( member_nat2 @ ( suc @ J ) @ A2 ) ) ) ) ) ).
% nths_Cons
thf(fact_639_nths__Cons,axiom,
! [X2: real,L: list_real,A2: set_nat] :
( ( nths_real @ ( cons_real @ X2 @ L ) @ A2 )
= ( append_real @ ( if_list_real @ ( member_nat2 @ zero_zero_nat @ A2 ) @ ( cons_real @ X2 @ nil_real ) @ nil_real )
@ ( nths_real @ L
@ ( collect_nat
@ ^ [J: nat] : ( member_nat2 @ ( suc @ J ) @ A2 ) ) ) ) ) ).
% nths_Cons
thf(fact_640_gen__length__code_I2_J,axiom,
! [N2: nat,X2: complex,Xs: list_complex] :
( ( gen_length_complex @ N2 @ ( cons_complex @ X2 @ Xs ) )
= ( gen_length_complex @ ( suc @ N2 ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_641_gen__length__code_I2_J,axiom,
! [N2: nat,X2: nat,Xs: list_nat] :
( ( gen_length_nat @ N2 @ ( cons_nat @ X2 @ Xs ) )
= ( gen_length_nat @ ( suc @ N2 ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_642_gen__length__code_I2_J,axiom,
! [N2: nat,X2: real,Xs: list_real] :
( ( gen_length_real @ N2 @ ( cons_real @ X2 @ Xs ) )
= ( gen_length_real @ ( suc @ N2 ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_643_list_Osel_I3_J,axiom,
! [X21: complex,X22: list_complex] :
( ( tl_complex @ ( cons_complex @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_644_list_Osel_I3_J,axiom,
! [X21: nat,X22: list_nat] :
( ( tl_nat @ ( cons_nat @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_645_list_Osel_I3_J,axiom,
! [X21: real,X22: list_real] :
( ( tl_real @ ( cons_real @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_646_list_Osel_I2_J,axiom,
( ( tl_nat @ nil_nat )
= nil_nat ) ).
% list.sel(2)
thf(fact_647_map__tl,axiom,
! [F: complex > real,Xs: list_complex] :
( ( map_complex_real @ F @ ( tl_complex @ Xs ) )
= ( tl_real @ ( map_complex_real @ F @ Xs ) ) ) ).
% map_tl
thf(fact_648_map__tl,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( map_nat_nat @ F @ ( tl_nat @ Xs ) )
= ( tl_nat @ ( map_nat_nat @ F @ Xs ) ) ) ).
% map_tl
thf(fact_649_map__tl,axiom,
! [F: nat > complex,Xs: list_nat] :
( ( map_nat_complex @ F @ ( tl_nat @ Xs ) )
= ( tl_complex @ ( map_nat_complex @ F @ Xs ) ) ) ).
% map_tl
thf(fact_650_map__tl,axiom,
! [F: complex > nat,Xs: list_complex] :
( ( map_complex_nat @ F @ ( tl_complex @ Xs ) )
= ( tl_nat @ ( map_complex_nat @ F @ Xs ) ) ) ).
% map_tl
thf(fact_651_map__tl,axiom,
! [F: complex > complex,Xs: list_complex] :
( ( map_complex_complex @ F @ ( tl_complex @ Xs ) )
= ( tl_complex @ ( map_complex_complex @ F @ Xs ) ) ) ).
% map_tl
thf(fact_652_nths__nil,axiom,
! [A2: set_nat] :
( ( nths_nat @ nil_nat @ A2 )
= nil_nat ) ).
% nths_nil
thf(fact_653_notin__set__nthsI,axiom,
! [X2: nat,Xs: list_nat,I2: set_nat] :
( ~ ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ~ ( member_nat2 @ X2 @ ( set_nat2 @ ( nths_nat @ Xs @ I2 ) ) ) ) ).
% notin_set_nthsI
thf(fact_654_notin__set__nthsI,axiom,
! [X2: complex,Xs: list_complex,I2: set_nat] :
( ~ ( member_complex2 @ X2 @ ( set_complex2 @ Xs ) )
=> ~ ( member_complex2 @ X2 @ ( set_complex2 @ ( nths_complex @ Xs @ I2 ) ) ) ) ).
% notin_set_nthsI
thf(fact_655_in__set__nthsD,axiom,
! [X2: nat,Xs: list_nat,I2: set_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ ( nths_nat @ Xs @ I2 ) ) )
=> ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) ) ) ).
% in_set_nthsD
thf(fact_656_in__set__nthsD,axiom,
! [X2: complex,Xs: list_complex,I2: set_nat] :
( ( member_complex2 @ X2 @ ( set_complex2 @ ( nths_complex @ Xs @ I2 ) ) )
=> ( member_complex2 @ X2 @ ( set_complex2 @ Xs ) ) ) ).
% in_set_nthsD
thf(fact_657_nths__map,axiom,
! [F: complex > real,Xs: list_complex,I2: set_nat] :
( ( nths_real @ ( map_complex_real @ F @ Xs ) @ I2 )
= ( map_complex_real @ F @ ( nths_complex @ Xs @ I2 ) ) ) ).
% nths_map
thf(fact_658_nths__map,axiom,
! [F: nat > nat,Xs: list_nat,I2: set_nat] :
( ( nths_nat @ ( map_nat_nat @ F @ Xs ) @ I2 )
= ( map_nat_nat @ F @ ( nths_nat @ Xs @ I2 ) ) ) ).
% nths_map
thf(fact_659_nths__map,axiom,
! [F: nat > complex,Xs: list_nat,I2: set_nat] :
( ( nths_complex @ ( map_nat_complex @ F @ Xs ) @ I2 )
= ( map_nat_complex @ F @ ( nths_nat @ Xs @ I2 ) ) ) ).
% nths_map
thf(fact_660_nths__map,axiom,
! [F: complex > nat,Xs: list_complex,I2: set_nat] :
( ( nths_nat @ ( map_complex_nat @ F @ Xs ) @ I2 )
= ( map_complex_nat @ F @ ( nths_complex @ Xs @ I2 ) ) ) ).
% nths_map
thf(fact_661_nths__map,axiom,
! [F: complex > complex,Xs: list_complex,I2: set_nat] :
( ( nths_complex @ ( map_complex_complex @ F @ Xs ) @ I2 )
= ( map_complex_complex @ F @ ( nths_complex @ Xs @ I2 ) ) ) ).
% nths_map
thf(fact_662_tl__Nil,axiom,
! [Xs: list_complex] :
( ( ( tl_complex @ Xs )
= nil_complex )
= ( ( Xs = nil_complex )
| ? [X3: complex] :
( Xs
= ( cons_complex @ X3 @ nil_complex ) ) ) ) ).
% tl_Nil
thf(fact_663_tl__Nil,axiom,
! [Xs: list_nat] :
( ( ( tl_nat @ Xs )
= nil_nat )
= ( ( Xs = nil_nat )
| ? [X3: nat] :
( Xs
= ( cons_nat @ X3 @ nil_nat ) ) ) ) ).
% tl_Nil
thf(fact_664_tl__Nil,axiom,
! [Xs: list_real] :
( ( ( tl_real @ Xs )
= nil_real )
= ( ( Xs = nil_real )
| ? [X3: real] :
( Xs
= ( cons_real @ X3 @ nil_real ) ) ) ) ).
% tl_Nil
thf(fact_665_Nil__tl,axiom,
! [Xs: list_complex] :
( ( nil_complex
= ( tl_complex @ Xs ) )
= ( ( Xs = nil_complex )
| ? [X3: complex] :
( Xs
= ( cons_complex @ X3 @ nil_complex ) ) ) ) ).
% Nil_tl
thf(fact_666_Nil__tl,axiom,
! [Xs: list_nat] :
( ( nil_nat
= ( tl_nat @ Xs ) )
= ( ( Xs = nil_nat )
| ? [X3: nat] :
( Xs
= ( cons_nat @ X3 @ nil_nat ) ) ) ) ).
% Nil_tl
thf(fact_667_Nil__tl,axiom,
! [Xs: list_real] :
( ( nil_real
= ( tl_real @ Xs ) )
= ( ( Xs = nil_real )
| ? [X3: real] :
( Xs
= ( cons_real @ X3 @ nil_real ) ) ) ) ).
% Nil_tl
thf(fact_668_list_Oset__sel_I2_J,axiom,
! [A: list_nat,X2: nat] :
( ( A != nil_nat )
=> ( ( member_nat2 @ X2 @ ( set_nat2 @ ( tl_nat @ A ) ) )
=> ( member_nat2 @ X2 @ ( set_nat2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_669_list_Oset__sel_I2_J,axiom,
! [A: list_complex,X2: complex] :
( ( A != nil_complex )
=> ( ( member_complex2 @ X2 @ ( set_complex2 @ ( tl_complex @ A ) ) )
=> ( member_complex2 @ X2 @ ( set_complex2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_670_list_Omap__sel_I2_J,axiom,
! [A: list_complex,F: complex > real] :
( ( A != nil_complex )
=> ( ( tl_real @ ( map_complex_real @ F @ A ) )
= ( map_complex_real @ F @ ( tl_complex @ A ) ) ) ) ).
% list.map_sel(2)
thf(fact_671_list_Omap__sel_I2_J,axiom,
! [A: list_nat,F: nat > nat] :
( ( A != nil_nat )
=> ( ( tl_nat @ ( map_nat_nat @ F @ A ) )
= ( map_nat_nat @ F @ ( tl_nat @ A ) ) ) ) ).
% list.map_sel(2)
thf(fact_672_list_Omap__sel_I2_J,axiom,
! [A: list_nat,F: nat > complex] :
( ( A != nil_nat )
=> ( ( tl_complex @ ( map_nat_complex @ F @ A ) )
= ( map_nat_complex @ F @ ( tl_nat @ A ) ) ) ) ).
% list.map_sel(2)
thf(fact_673_list_Omap__sel_I2_J,axiom,
! [A: list_complex,F: complex > nat] :
( ( A != nil_complex )
=> ( ( tl_nat @ ( map_complex_nat @ F @ A ) )
= ( map_complex_nat @ F @ ( tl_complex @ A ) ) ) ) ).
% list.map_sel(2)
thf(fact_674_list_Omap__sel_I2_J,axiom,
! [A: list_complex,F: complex > complex] :
( ( A != nil_complex )
=> ( ( tl_complex @ ( map_complex_complex @ F @ A ) )
= ( map_complex_complex @ F @ ( tl_complex @ A ) ) ) ) ).
% list.map_sel(2)
thf(fact_675_tl__append2,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs != nil_nat )
=> ( ( tl_nat @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( tl_nat @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_676_tl__append__if,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( Xs = nil_nat )
=> ( ( tl_nat @ ( append_nat @ Xs @ Ys ) )
= ( tl_nat @ Ys ) ) )
& ( ( Xs != nil_nat )
=> ( ( tl_nat @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( tl_nat @ Xs ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_677_tl__def,axiom,
( tl_nat
= ( case_l2340614614379431832at_nat @ nil_nat
@ ^ [X214: nat,X224: list_nat] : X224 ) ) ).
% tl_def
thf(fact_678_gen__length__code_I1_J,axiom,
! [N2: nat] :
( ( gen_length_nat @ N2 @ nil_nat )
= N2 ) ).
% gen_length_code(1)
thf(fact_679_nths__singleton,axiom,
! [A2: set_nat,X2: complex] :
( ( ( member_nat2 @ zero_zero_nat @ A2 )
=> ( ( nths_complex @ ( cons_complex @ X2 @ nil_complex ) @ A2 )
= ( cons_complex @ X2 @ nil_complex ) ) )
& ( ~ ( member_nat2 @ zero_zero_nat @ A2 )
=> ( ( nths_complex @ ( cons_complex @ X2 @ nil_complex ) @ A2 )
= nil_complex ) ) ) ).
% nths_singleton
thf(fact_680_nths__singleton,axiom,
! [A2: set_nat,X2: nat] :
( ( ( member_nat2 @ zero_zero_nat @ A2 )
=> ( ( nths_nat @ ( cons_nat @ X2 @ nil_nat ) @ A2 )
= ( cons_nat @ X2 @ nil_nat ) ) )
& ( ~ ( member_nat2 @ zero_zero_nat @ A2 )
=> ( ( nths_nat @ ( cons_nat @ X2 @ nil_nat ) @ A2 )
= nil_nat ) ) ) ).
% nths_singleton
thf(fact_681_nths__singleton,axiom,
! [A2: set_nat,X2: real] :
( ( ( member_nat2 @ zero_zero_nat @ A2 )
=> ( ( nths_real @ ( cons_real @ X2 @ nil_real ) @ A2 )
= ( cons_real @ X2 @ nil_real ) ) )
& ( ~ ( member_nat2 @ zero_zero_nat @ A2 )
=> ( ( nths_real @ ( cons_real @ X2 @ nil_real ) @ A2 )
= nil_real ) ) ) ).
% nths_singleton
thf(fact_682_transpose__aux__filter__tail,axiom,
! [Xss2: list_list_nat] :
( ( concat_list_nat
@ ( map_li960784813134754710st_nat
@ ( case_l3331202209248957608at_nat @ nil_list_nat
@ ^ [H: nat,T2: list_nat] : ( cons_list_nat @ T2 @ nil_list_nat ) )
@ Xss2 ) )
= ( map_li7225945977422193158st_nat @ tl_nat
@ ( filter_list_nat
@ ^ [Ys3: list_nat] : ( Ys3 != nil_nat )
@ Xss2 ) ) ) ).
% transpose_aux_filter_tail
thf(fact_683_Poly__snoc__zero,axiom,
! [As2: list_nat] :
( ( poly_nat2 @ ( append_nat @ As2 @ ( cons_nat @ zero_zero_nat @ nil_nat ) ) )
= ( poly_nat2 @ As2 ) ) ).
% Poly_snoc_zero
thf(fact_684_Poly__snoc__zero,axiom,
! [As2: list_complex] :
( ( poly_complex2 @ ( append_complex @ As2 @ ( cons_complex @ zero_zero_complex @ nil_complex ) ) )
= ( poly_complex2 @ As2 ) ) ).
% Poly_snoc_zero
thf(fact_685_Poly__snoc__zero,axiom,
! [As2: list_real] :
( ( poly_real2 @ ( append_real @ As2 @ ( cons_real @ zero_zero_real @ nil_real ) ) )
= ( poly_real2 @ As2 ) ) ).
% Poly_snoc_zero
thf(fact_686_rotate1_Osimps_I2_J,axiom,
! [X2: complex,Xs: list_complex] :
( ( rotate1_complex @ ( cons_complex @ X2 @ Xs ) )
= ( append_complex @ Xs @ ( cons_complex @ X2 @ nil_complex ) ) ) ).
% rotate1.simps(2)
thf(fact_687_rotate1_Osimps_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( rotate1_nat @ ( cons_nat @ X2 @ Xs ) )
= ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) ) ).
% rotate1.simps(2)
thf(fact_688_rotate1_Osimps_I2_J,axiom,
! [X2: real,Xs: list_real] :
( ( rotate1_real @ ( cons_real @ X2 @ Xs ) )
= ( append_real @ Xs @ ( cons_real @ X2 @ nil_real ) ) ) ).
% rotate1.simps(2)
thf(fact_689_remdups__adj__append,axiom,
! [Xs_1: list_complex,X2: complex,Xs_2: list_complex] :
( ( remdups_adj_complex @ ( append_complex @ Xs_1 @ ( cons_complex @ X2 @ Xs_2 ) ) )
= ( append_complex @ ( remdups_adj_complex @ ( append_complex @ Xs_1 @ ( cons_complex @ X2 @ nil_complex ) ) ) @ ( tl_complex @ ( remdups_adj_complex @ ( cons_complex @ X2 @ Xs_2 ) ) ) ) ) ).
% remdups_adj_append
thf(fact_690_remdups__adj__append,axiom,
! [Xs_1: list_nat,X2: nat,Xs_2: list_nat] :
( ( remdups_adj_nat @ ( append_nat @ Xs_1 @ ( cons_nat @ X2 @ Xs_2 ) ) )
= ( append_nat @ ( remdups_adj_nat @ ( append_nat @ Xs_1 @ ( cons_nat @ X2 @ nil_nat ) ) ) @ ( tl_nat @ ( remdups_adj_nat @ ( cons_nat @ X2 @ Xs_2 ) ) ) ) ) ).
% remdups_adj_append
thf(fact_691_remdups__adj__append,axiom,
! [Xs_1: list_real,X2: real,Xs_2: list_real] :
( ( remdups_adj_real @ ( append_real @ Xs_1 @ ( cons_real @ X2 @ Xs_2 ) ) )
= ( append_real @ ( remdups_adj_real @ ( append_real @ Xs_1 @ ( cons_real @ X2 @ nil_real ) ) ) @ ( tl_real @ ( remdups_adj_real @ ( cons_real @ X2 @ Xs_2 ) ) ) ) ) ).
% remdups_adj_append
thf(fact_692_filter__id__conv,axiom,
! [P: complex > $o,Xs: list_complex] :
( ( ( filter_complex @ P @ Xs )
= Xs )
= ( ! [X3: complex] :
( ( member_complex2 @ X3 @ ( set_complex2 @ Xs ) )
=> ( P @ X3 ) ) ) ) ).
% filter_id_conv
thf(fact_693_filter__cong,axiom,
! [Xs: list_nat,Ys: list_nat,P: nat > $o,Q: nat > $o] :
( ( Xs = Ys )
=> ( ! [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Ys ) )
=> ( ( P @ X4 )
= ( Q @ X4 ) ) )
=> ( ( filter_nat @ P @ Xs )
= ( filter_nat @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_694_filter__cong,axiom,
! [Xs: list_complex,Ys: list_complex,P: complex > $o,Q: complex > $o] :
( ( Xs = Ys )
=> ( ! [X4: complex] :
( ( member_complex2 @ X4 @ ( set_complex2 @ Ys ) )
=> ( ( P @ X4 )
= ( Q @ X4 ) ) )
=> ( ( filter_complex @ P @ Xs )
= ( filter_complex @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_695_filter__True,axiom,
! [Xs: list_complex,P: complex > $o] :
( ! [X4: complex] :
( ( member_complex2 @ X4 @ ( set_complex2 @ Xs ) )
=> ( P @ X4 ) )
=> ( ( filter_complex @ P @ Xs )
= Xs ) ) ).
% filter_True
thf(fact_696_filter_Osimps_I1_J,axiom,
! [P: nat > $o] :
( ( filter_nat @ P @ nil_nat )
= nil_nat ) ).
% filter.simps(1)
thf(fact_697_filter_Osimps_I2_J,axiom,
! [P: complex > $o,X2: complex,Xs: list_complex] :
( ( ( P @ X2 )
=> ( ( filter_complex @ P @ ( cons_complex @ X2 @ Xs ) )
= ( cons_complex @ X2 @ ( filter_complex @ P @ Xs ) ) ) )
& ( ~ ( P @ X2 )
=> ( ( filter_complex @ P @ ( cons_complex @ X2 @ Xs ) )
= ( filter_complex @ P @ Xs ) ) ) ) ).
% filter.simps(2)
thf(fact_698_filter_Osimps_I2_J,axiom,
! [P: nat > $o,X2: nat,Xs: list_nat] :
( ( ( P @ X2 )
=> ( ( filter_nat @ P @ ( cons_nat @ X2 @ Xs ) )
= ( cons_nat @ X2 @ ( filter_nat @ P @ Xs ) ) ) )
& ( ~ ( P @ X2 )
=> ( ( filter_nat @ P @ ( cons_nat @ X2 @ Xs ) )
= ( filter_nat @ P @ Xs ) ) ) ) ).
% filter.simps(2)
thf(fact_699_filter_Osimps_I2_J,axiom,
! [P: real > $o,X2: real,Xs: list_real] :
( ( ( P @ X2 )
=> ( ( filter_real @ P @ ( cons_real @ X2 @ Xs ) )
= ( cons_real @ X2 @ ( filter_real @ P @ Xs ) ) ) )
& ( ~ ( P @ X2 )
=> ( ( filter_real @ P @ ( cons_real @ X2 @ Xs ) )
= ( filter_real @ P @ Xs ) ) ) ) ).
% filter.simps(2)
thf(fact_700_remdups__adj_Osimps_I3_J,axiom,
! [X2: complex,Y: complex,Xs: list_complex] :
( ( ( X2 = Y )
=> ( ( remdups_adj_complex @ ( cons_complex @ X2 @ ( cons_complex @ Y @ Xs ) ) )
= ( remdups_adj_complex @ ( cons_complex @ X2 @ Xs ) ) ) )
& ( ( X2 != Y )
=> ( ( remdups_adj_complex @ ( cons_complex @ X2 @ ( cons_complex @ Y @ Xs ) ) )
= ( cons_complex @ X2 @ ( remdups_adj_complex @ ( cons_complex @ Y @ Xs ) ) ) ) ) ) ).
% remdups_adj.simps(3)
thf(fact_701_remdups__adj_Osimps_I3_J,axiom,
! [X2: nat,Y: nat,Xs: list_nat] :
( ( ( X2 = Y )
=> ( ( remdups_adj_nat @ ( cons_nat @ X2 @ ( cons_nat @ Y @ Xs ) ) )
= ( remdups_adj_nat @ ( cons_nat @ X2 @ Xs ) ) ) )
& ( ( X2 != Y )
=> ( ( remdups_adj_nat @ ( cons_nat @ X2 @ ( cons_nat @ Y @ Xs ) ) )
= ( cons_nat @ X2 @ ( remdups_adj_nat @ ( cons_nat @ Y @ Xs ) ) ) ) ) ) ).
% remdups_adj.simps(3)
thf(fact_702_remdups__adj_Osimps_I3_J,axiom,
! [X2: real,Y: real,Xs: list_real] :
( ( ( X2 = Y )
=> ( ( remdups_adj_real @ ( cons_real @ X2 @ ( cons_real @ Y @ Xs ) ) )
= ( remdups_adj_real @ ( cons_real @ X2 @ Xs ) ) ) )
& ( ( X2 != Y )
=> ( ( remdups_adj_real @ ( cons_real @ X2 @ ( cons_real @ Y @ Xs ) ) )
= ( cons_real @ X2 @ ( remdups_adj_real @ ( cons_real @ Y @ Xs ) ) ) ) ) ) ).
% remdups_adj.simps(3)
thf(fact_703_remdups__adj__Nil__iff,axiom,
! [Xs: list_nat] :
( ( ( remdups_adj_nat @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% remdups_adj_Nil_iff
thf(fact_704_remdups__adj_Osimps_I1_J,axiom,
( ( remdups_adj_nat @ nil_nat )
= nil_nat ) ).
% remdups_adj.simps(1)
thf(fact_705_remdups__adj__set,axiom,
! [Xs: list_complex] :
( ( set_complex2 @ ( remdups_adj_complex @ Xs ) )
= ( set_complex2 @ Xs ) ) ).
% remdups_adj_set
thf(fact_706_Poly_Osimps_I1_J,axiom,
( ( poly_nat2 @ nil_nat )
= zero_zero_poly_nat ) ).
% Poly.simps(1)
thf(fact_707_set__filter,axiom,
! [P: nat > $o,Xs: list_nat] :
( ( set_nat2 @ ( filter_nat @ P @ Xs ) )
= ( collect_nat
@ ^ [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
& ( P @ X3 ) ) ) ) ).
% set_filter
thf(fact_708_set__filter,axiom,
! [P: complex > $o,Xs: list_complex] :
( ( set_complex2 @ ( filter_complex @ P @ Xs ) )
= ( collect_complex
@ ^ [X3: complex] :
( ( member_complex2 @ X3 @ ( set_complex2 @ Xs ) )
& ( P @ X3 ) ) ) ) ).
% set_filter
thf(fact_709_rotate1__is__Nil__conv,axiom,
! [Xs: list_nat] :
( ( ( rotate1_nat @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% rotate1_is_Nil_conv
thf(fact_710_rotate1_Osimps_I1_J,axiom,
( ( rotate1_nat @ nil_nat )
= nil_nat ) ).
% rotate1.simps(1)
thf(fact_711_set__rotate1,axiom,
! [Xs: list_complex] :
( ( set_complex2 @ ( rotate1_complex @ Xs ) )
= ( set_complex2 @ Xs ) ) ).
% set_rotate1
thf(fact_712_rotate1__map,axiom,
! [F: complex > real,Xs: list_complex] :
( ( rotate1_real @ ( map_complex_real @ F @ Xs ) )
= ( map_complex_real @ F @ ( rotate1_complex @ Xs ) ) ) ).
% rotate1_map
thf(fact_713_rotate1__map,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( rotate1_nat @ ( map_nat_nat @ F @ Xs ) )
= ( map_nat_nat @ F @ ( rotate1_nat @ Xs ) ) ) ).
% rotate1_map
thf(fact_714_rotate1__map,axiom,
! [F: nat > complex,Xs: list_nat] :
( ( rotate1_complex @ ( map_nat_complex @ F @ Xs ) )
= ( map_nat_complex @ F @ ( rotate1_nat @ Xs ) ) ) ).
% rotate1_map
thf(fact_715_rotate1__map,axiom,
! [F: complex > nat,Xs: list_complex] :
( ( rotate1_nat @ ( map_complex_nat @ F @ Xs ) )
= ( map_complex_nat @ F @ ( rotate1_complex @ Xs ) ) ) ).
% rotate1_map
thf(fact_716_rotate1__map,axiom,
! [F: complex > complex,Xs: list_complex] :
( ( rotate1_complex @ ( map_complex_complex @ F @ Xs ) )
= ( map_complex_complex @ F @ ( rotate1_complex @ Xs ) ) ) ).
% rotate1_map
thf(fact_717_filter__False,axiom,
! [Xs: list_nat,P: nat > $o] :
( ! [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Xs ) )
=> ~ ( P @ X4 ) )
=> ( ( filter_nat @ P @ Xs )
= nil_nat ) ) ).
% filter_False
thf(fact_718_filter__False,axiom,
! [Xs: list_complex,P: complex > $o] :
( ! [X4: complex] :
( ( member_complex2 @ X4 @ ( set_complex2 @ Xs ) )
=> ~ ( P @ X4 ) )
=> ( ( filter_complex @ P @ Xs )
= nil_complex ) ) ).
% filter_False
thf(fact_719_empty__filter__conv,axiom,
! [P: nat > $o,Xs: list_nat] :
( ( nil_nat
= ( filter_nat @ P @ Xs ) )
= ( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ~ ( P @ X3 ) ) ) ) ).
% empty_filter_conv
thf(fact_720_empty__filter__conv,axiom,
! [P: complex > $o,Xs: list_complex] :
( ( nil_complex
= ( filter_complex @ P @ Xs ) )
= ( ! [X3: complex] :
( ( member_complex2 @ X3 @ ( set_complex2 @ Xs ) )
=> ~ ( P @ X3 ) ) ) ) ).
% empty_filter_conv
thf(fact_721_filter__empty__conv,axiom,
! [P: nat > $o,Xs: list_nat] :
( ( ( filter_nat @ P @ Xs )
= nil_nat )
= ( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ~ ( P @ X3 ) ) ) ) ).
% filter_empty_conv
thf(fact_722_filter__empty__conv,axiom,
! [P: complex > $o,Xs: list_complex] :
( ( ( filter_complex @ P @ Xs )
= nil_complex )
= ( ! [X3: complex] :
( ( member_complex2 @ X3 @ ( set_complex2 @ Xs ) )
=> ~ ( P @ X3 ) ) ) ) ).
% filter_empty_conv
thf(fact_723_distinct__map__filter,axiom,
! [F: complex > real,Xs: list_complex,P: complex > $o] :
( ( distinct_real @ ( map_complex_real @ F @ Xs ) )
=> ( distinct_real @ ( map_complex_real @ F @ ( filter_complex @ P @ Xs ) ) ) ) ).
% distinct_map_filter
thf(fact_724_distinct__map__filter,axiom,
! [F: nat > nat,Xs: list_nat,P: nat > $o] :
( ( distinct_nat @ ( map_nat_nat @ F @ Xs ) )
=> ( distinct_nat @ ( map_nat_nat @ F @ ( filter_nat @ P @ Xs ) ) ) ) ).
% distinct_map_filter
thf(fact_725_distinct__map__filter,axiom,
! [F: nat > complex,Xs: list_nat,P: nat > $o] :
( ( distinct_complex @ ( map_nat_complex @ F @ Xs ) )
=> ( distinct_complex @ ( map_nat_complex @ F @ ( filter_nat @ P @ Xs ) ) ) ) ).
% distinct_map_filter
thf(fact_726_distinct__map__filter,axiom,
! [F: complex > nat,Xs: list_complex,P: complex > $o] :
( ( distinct_nat @ ( map_complex_nat @ F @ Xs ) )
=> ( distinct_nat @ ( map_complex_nat @ F @ ( filter_complex @ P @ Xs ) ) ) ) ).
% distinct_map_filter
thf(fact_727_distinct__map__filter,axiom,
! [F: complex > complex,Xs: list_complex,P: complex > $o] :
( ( distinct_complex @ ( map_complex_complex @ F @ Xs ) )
=> ( distinct_complex @ ( map_complex_complex @ F @ ( filter_complex @ P @ Xs ) ) ) ) ).
% distinct_map_filter
thf(fact_728_remdups__adj_Osimps_I2_J,axiom,
! [X2: complex] :
( ( remdups_adj_complex @ ( cons_complex @ X2 @ nil_complex ) )
= ( cons_complex @ X2 @ nil_complex ) ) ).
% remdups_adj.simps(2)
thf(fact_729_remdups__adj_Osimps_I2_J,axiom,
! [X2: nat] :
( ( remdups_adj_nat @ ( cons_nat @ X2 @ nil_nat ) )
= ( cons_nat @ X2 @ nil_nat ) ) ).
% remdups_adj.simps(2)
thf(fact_730_remdups__adj_Osimps_I2_J,axiom,
! [X2: real] :
( ( remdups_adj_real @ ( cons_real @ X2 @ nil_real ) )
= ( cons_real @ X2 @ nil_real ) ) ).
% remdups_adj.simps(2)
thf(fact_731_remdups__adj_Oelims,axiom,
! [X2: list_complex,Y: list_complex] :
( ( ( remdups_adj_complex @ X2 )
= Y )
=> ( ( ( X2 = nil_complex )
=> ( Y != nil_complex ) )
=> ( ! [X4: complex] :
( ( X2
= ( cons_complex @ X4 @ nil_complex ) )
=> ( Y
!= ( cons_complex @ X4 @ nil_complex ) ) )
=> ~ ! [X4: complex,Y3: complex,Xs3: list_complex] :
( ( X2
= ( cons_complex @ X4 @ ( cons_complex @ Y3 @ Xs3 ) ) )
=> ~ ( ( ( X4 = Y3 )
=> ( Y
= ( remdups_adj_complex @ ( cons_complex @ X4 @ Xs3 ) ) ) )
& ( ( X4 != Y3 )
=> ( Y
= ( cons_complex @ X4 @ ( remdups_adj_complex @ ( cons_complex @ Y3 @ Xs3 ) ) ) ) ) ) ) ) ) ) ).
% remdups_adj.elims
thf(fact_732_remdups__adj_Oelims,axiom,
! [X2: list_nat,Y: list_nat] :
( ( ( remdups_adj_nat @ X2 )
= Y )
=> ( ( ( X2 = nil_nat )
=> ( Y != nil_nat ) )
=> ( ! [X4: nat] :
( ( X2
= ( cons_nat @ X4 @ nil_nat ) )
=> ( Y
!= ( cons_nat @ X4 @ nil_nat ) ) )
=> ~ ! [X4: nat,Y3: nat,Xs3: list_nat] :
( ( X2
= ( cons_nat @ X4 @ ( cons_nat @ Y3 @ Xs3 ) ) )
=> ~ ( ( ( X4 = Y3 )
=> ( Y
= ( remdups_adj_nat @ ( cons_nat @ X4 @ Xs3 ) ) ) )
& ( ( X4 != Y3 )
=> ( Y
= ( cons_nat @ X4 @ ( remdups_adj_nat @ ( cons_nat @ Y3 @ Xs3 ) ) ) ) ) ) ) ) ) ) ).
% remdups_adj.elims
thf(fact_733_remdups__adj_Oelims,axiom,
! [X2: list_real,Y: list_real] :
( ( ( remdups_adj_real @ X2 )
= Y )
=> ( ( ( X2 = nil_real )
=> ( Y != nil_real ) )
=> ( ! [X4: real] :
( ( X2
= ( cons_real @ X4 @ nil_real ) )
=> ( Y
!= ( cons_real @ X4 @ nil_real ) ) )
=> ~ ! [X4: real,Y3: real,Xs3: list_real] :
( ( X2
= ( cons_real @ X4 @ ( cons_real @ Y3 @ Xs3 ) ) )
=> ~ ( ( ( X4 = Y3 )
=> ( Y
= ( remdups_adj_real @ ( cons_real @ X4 @ Xs3 ) ) ) )
& ( ( X4 != Y3 )
=> ( Y
= ( cons_real @ X4 @ ( remdups_adj_real @ ( cons_real @ Y3 @ Xs3 ) ) ) ) ) ) ) ) ) ) ).
% remdups_adj.elims
thf(fact_734_eq__comps__hd__neq__tl,axiom,
! [X2: nat,Y: nat,L: list_nat] :
( ( X2 != Y )
=> ( ( tl_nat @ ( commut2436974278740741825ps_nat @ ( cons_nat @ X2 @ ( cons_nat @ Y @ L ) ) ) )
= ( commut2436974278740741825ps_nat @ ( cons_nat @ Y @ L ) ) ) ) ).
% eq_comps_hd_neq_tl
thf(fact_735_eq__comps__hd__neq__tl,axiom,
! [X2: real,Y: real,L: list_real] :
( ( X2 != Y )
=> ( ( tl_nat @ ( commut8680161604938074397s_real @ ( cons_real @ X2 @ ( cons_real @ Y @ L ) ) ) )
= ( commut8680161604938074397s_real @ ( cons_real @ Y @ L ) ) ) ) ).
% eq_comps_hd_neq_tl
thf(fact_736_eq__comps__hd__neq__tl,axiom,
! [X2: complex,Y: complex,L: list_complex] :
( ( X2 != Y )
=> ( ( tl_nat @ ( commut93809757773076895omplex @ ( cons_complex @ X2 @ ( cons_complex @ Y @ L ) ) ) )
= ( commut93809757773076895omplex @ ( cons_complex @ Y @ L ) ) ) ) ).
% eq_comps_hd_neq_tl
thf(fact_737_eq__comps__hd__eq__tl,axiom,
! [X2: nat,Y: nat,L: list_nat] :
( ( X2 = Y )
=> ( ( tl_nat @ ( commut2436974278740741825ps_nat @ ( cons_nat @ X2 @ ( cons_nat @ Y @ L ) ) ) )
= ( tl_nat @ ( commut2436974278740741825ps_nat @ ( cons_nat @ Y @ L ) ) ) ) ) ).
% eq_comps_hd_eq_tl
thf(fact_738_eq__comps__hd__eq__tl,axiom,
! [X2: real,Y: real,L: list_real] :
( ( X2 = Y )
=> ( ( tl_nat @ ( commut8680161604938074397s_real @ ( cons_real @ X2 @ ( cons_real @ Y @ L ) ) ) )
= ( tl_nat @ ( commut8680161604938074397s_real @ ( cons_real @ Y @ L ) ) ) ) ) ).
% eq_comps_hd_eq_tl
thf(fact_739_eq__comps__hd__eq__tl,axiom,
! [X2: complex,Y: complex,L: list_complex] :
( ( X2 = Y )
=> ( ( tl_nat @ ( commut93809757773076895omplex @ ( cons_complex @ X2 @ ( cons_complex @ Y @ L ) ) ) )
= ( tl_nat @ ( commut93809757773076895omplex @ ( cons_complex @ Y @ L ) ) ) ) ) ).
% eq_comps_hd_eq_tl
thf(fact_740_remdups__adj__Cons__alt,axiom,
! [X2: complex,Xs: list_complex] :
( ( cons_complex @ X2 @ ( tl_complex @ ( remdups_adj_complex @ ( cons_complex @ X2 @ Xs ) ) ) )
= ( remdups_adj_complex @ ( cons_complex @ X2 @ Xs ) ) ) ).
% remdups_adj_Cons_alt
thf(fact_741_remdups__adj__Cons__alt,axiom,
! [X2: nat,Xs: list_nat] :
( ( cons_nat @ X2 @ ( tl_nat @ ( remdups_adj_nat @ ( cons_nat @ X2 @ Xs ) ) ) )
= ( remdups_adj_nat @ ( cons_nat @ X2 @ Xs ) ) ) ).
% remdups_adj_Cons_alt
thf(fact_742_remdups__adj__Cons__alt,axiom,
! [X2: real,Xs: list_real] :
( ( cons_real @ X2 @ ( tl_real @ ( remdups_adj_real @ ( cons_real @ X2 @ Xs ) ) ) )
= ( remdups_adj_real @ ( cons_real @ X2 @ Xs ) ) ) ).
% remdups_adj_Cons_alt
thf(fact_743_Cons__eq__filterD,axiom,
! [X2: complex,Xs: list_complex,P: complex > $o,Ys: list_complex] :
( ( ( cons_complex @ X2 @ Xs )
= ( filter_complex @ P @ Ys ) )
=> ? [Us2: list_complex,Vs2: list_complex] :
( ( Ys
= ( append_complex @ Us2 @ ( cons_complex @ X2 @ Vs2 ) ) )
& ! [X: complex] :
( ( member_complex2 @ X @ ( set_complex2 @ Us2 ) )
=> ~ ( P @ X ) )
& ( P @ X2 )
& ( Xs
= ( filter_complex @ P @ Vs2 ) ) ) ) ).
% Cons_eq_filterD
thf(fact_744_Cons__eq__filterD,axiom,
! [X2: nat,Xs: list_nat,P: nat > $o,Ys: list_nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( filter_nat @ P @ Ys ) )
=> ? [Us2: list_nat,Vs2: list_nat] :
( ( Ys
= ( append_nat @ Us2 @ ( cons_nat @ X2 @ Vs2 ) ) )
& ! [X: nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Us2 ) )
=> ~ ( P @ X ) )
& ( P @ X2 )
& ( Xs
= ( filter_nat @ P @ Vs2 ) ) ) ) ).
% Cons_eq_filterD
thf(fact_745_Cons__eq__filterD,axiom,
! [X2: real,Xs: list_real,P: real > $o,Ys: list_real] :
( ( ( cons_real @ X2 @ Xs )
= ( filter_real @ P @ Ys ) )
=> ? [Us2: list_real,Vs2: list_real] :
( ( Ys
= ( append_real @ Us2 @ ( cons_real @ X2 @ Vs2 ) ) )
& ! [X: real] :
( ( member_real2 @ X @ ( set_real2 @ Us2 ) )
=> ~ ( P @ X ) )
& ( P @ X2 )
& ( Xs
= ( filter_real @ P @ Vs2 ) ) ) ) ).
% Cons_eq_filterD
thf(fact_746_filter__eq__ConsD,axiom,
! [P: complex > $o,Ys: list_complex,X2: complex,Xs: list_complex] :
( ( ( filter_complex @ P @ Ys )
= ( cons_complex @ X2 @ Xs ) )
=> ? [Us2: list_complex,Vs2: list_complex] :
( ( Ys
= ( append_complex @ Us2 @ ( cons_complex @ X2 @ Vs2 ) ) )
& ! [X: complex] :
( ( member_complex2 @ X @ ( set_complex2 @ Us2 ) )
=> ~ ( P @ X ) )
& ( P @ X2 )
& ( Xs
= ( filter_complex @ P @ Vs2 ) ) ) ) ).
% filter_eq_ConsD
thf(fact_747_filter__eq__ConsD,axiom,
! [P: nat > $o,Ys: list_nat,X2: nat,Xs: list_nat] :
( ( ( filter_nat @ P @ Ys )
= ( cons_nat @ X2 @ Xs ) )
=> ? [Us2: list_nat,Vs2: list_nat] :
( ( Ys
= ( append_nat @ Us2 @ ( cons_nat @ X2 @ Vs2 ) ) )
& ! [X: nat] :
( ( member_nat2 @ X @ ( set_nat2 @ Us2 ) )
=> ~ ( P @ X ) )
& ( P @ X2 )
& ( Xs
= ( filter_nat @ P @ Vs2 ) ) ) ) ).
% filter_eq_ConsD
thf(fact_748_filter__eq__ConsD,axiom,
! [P: real > $o,Ys: list_real,X2: real,Xs: list_real] :
( ( ( filter_real @ P @ Ys )
= ( cons_real @ X2 @ Xs ) )
=> ? [Us2: list_real,Vs2: list_real] :
( ( Ys
= ( append_real @ Us2 @ ( cons_real @ X2 @ Vs2 ) ) )
& ! [X: real] :
( ( member_real2 @ X @ ( set_real2 @ Us2 ) )
=> ~ ( P @ X ) )
& ( P @ X2 )
& ( Xs
= ( filter_real @ P @ Vs2 ) ) ) ) ).
% filter_eq_ConsD
thf(fact_749_Cons__eq__filter__iff,axiom,
! [X2: complex,Xs: list_complex,P: complex > $o,Ys: list_complex] :
( ( ( cons_complex @ X2 @ Xs )
= ( filter_complex @ P @ Ys ) )
= ( ? [Us: list_complex,Vs: list_complex] :
( ( Ys
= ( append_complex @ Us @ ( cons_complex @ X2 @ Vs ) ) )
& ! [X3: complex] :
( ( member_complex2 @ X3 @ ( set_complex2 @ Us ) )
=> ~ ( P @ X3 ) )
& ( P @ X2 )
& ( Xs
= ( filter_complex @ P @ Vs ) ) ) ) ) ).
% Cons_eq_filter_iff
thf(fact_750_Cons__eq__filter__iff,axiom,
! [X2: nat,Xs: list_nat,P: nat > $o,Ys: list_nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( filter_nat @ P @ Ys ) )
= ( ? [Us: list_nat,Vs: list_nat] :
( ( Ys
= ( append_nat @ Us @ ( cons_nat @ X2 @ Vs ) ) )
& ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Us ) )
=> ~ ( P @ X3 ) )
& ( P @ X2 )
& ( Xs
= ( filter_nat @ P @ Vs ) ) ) ) ) ).
% Cons_eq_filter_iff
thf(fact_751_Cons__eq__filter__iff,axiom,
! [X2: real,Xs: list_real,P: real > $o,Ys: list_real] :
( ( ( cons_real @ X2 @ Xs )
= ( filter_real @ P @ Ys ) )
= ( ? [Us: list_real,Vs: list_real] :
( ( Ys
= ( append_real @ Us @ ( cons_real @ X2 @ Vs ) ) )
& ! [X3: real] :
( ( member_real2 @ X3 @ ( set_real2 @ Us ) )
=> ~ ( P @ X3 ) )
& ( P @ X2 )
& ( Xs
= ( filter_real @ P @ Vs ) ) ) ) ) ).
% Cons_eq_filter_iff
thf(fact_752_filter__eq__Cons__iff,axiom,
! [P: complex > $o,Ys: list_complex,X2: complex,Xs: list_complex] :
( ( ( filter_complex @ P @ Ys )
= ( cons_complex @ X2 @ Xs ) )
= ( ? [Us: list_complex,Vs: list_complex] :
( ( Ys
= ( append_complex @ Us @ ( cons_complex @ X2 @ Vs ) ) )
& ! [X3: complex] :
( ( member_complex2 @ X3 @ ( set_complex2 @ Us ) )
=> ~ ( P @ X3 ) )
& ( P @ X2 )
& ( Xs
= ( filter_complex @ P @ Vs ) ) ) ) ) ).
% filter_eq_Cons_iff
thf(fact_753_filter__eq__Cons__iff,axiom,
! [P: nat > $o,Ys: list_nat,X2: nat,Xs: list_nat] :
( ( ( filter_nat @ P @ Ys )
= ( cons_nat @ X2 @ Xs ) )
= ( ? [Us: list_nat,Vs: list_nat] :
( ( Ys
= ( append_nat @ Us @ ( cons_nat @ X2 @ Vs ) ) )
& ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Us ) )
=> ~ ( P @ X3 ) )
& ( P @ X2 )
& ( Xs
= ( filter_nat @ P @ Vs ) ) ) ) ) ).
% filter_eq_Cons_iff
thf(fact_754_filter__eq__Cons__iff,axiom,
! [P: real > $o,Ys: list_real,X2: real,Xs: list_real] :
( ( ( filter_real @ P @ Ys )
= ( cons_real @ X2 @ Xs ) )
= ( ? [Us: list_real,Vs: list_real] :
( ( Ys
= ( append_real @ Us @ ( cons_real @ X2 @ Vs ) ) )
& ! [X3: real] :
( ( member_real2 @ X3 @ ( set_real2 @ Us ) )
=> ~ ( P @ X3 ) )
& ( P @ X2 )
& ( Xs
= ( filter_real @ P @ Vs ) ) ) ) ) ).
% filter_eq_Cons_iff
thf(fact_755_remdups__adj__append__two,axiom,
! [Xs: list_complex,X2: complex,Y: complex] :
( ( remdups_adj_complex @ ( append_complex @ Xs @ ( cons_complex @ X2 @ ( cons_complex @ Y @ nil_complex ) ) ) )
= ( append_complex @ ( remdups_adj_complex @ ( append_complex @ Xs @ ( cons_complex @ X2 @ nil_complex ) ) ) @ ( if_list_complex @ ( X2 = Y ) @ nil_complex @ ( cons_complex @ Y @ nil_complex ) ) ) ) ).
% remdups_adj_append_two
thf(fact_756_remdups__adj__append__two,axiom,
! [Xs: list_nat,X2: nat,Y: nat] :
( ( remdups_adj_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ ( cons_nat @ Y @ nil_nat ) ) ) )
= ( append_nat @ ( remdups_adj_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) ) @ ( if_list_nat @ ( X2 = Y ) @ nil_nat @ ( cons_nat @ Y @ nil_nat ) ) ) ) ).
% remdups_adj_append_two
thf(fact_757_remdups__adj__append__two,axiom,
! [Xs: list_real,X2: real,Y: real] :
( ( remdups_adj_real @ ( append_real @ Xs @ ( cons_real @ X2 @ ( cons_real @ Y @ nil_real ) ) ) )
= ( append_real @ ( remdups_adj_real @ ( append_real @ Xs @ ( cons_real @ X2 @ nil_real ) ) ) @ ( if_list_real @ ( X2 = Y ) @ nil_real @ ( cons_real @ Y @ nil_real ) ) ) ) ).
% remdups_adj_append_two
thf(fact_758_filter__in__nths,axiom,
! [Xs: list_nat,S: set_nat] :
( ( distinct_nat @ Xs )
=> ( ( filter_nat
@ ^ [X3: nat] : ( member_nat2 @ X3 @ ( set_nat2 @ ( nths_nat @ Xs @ S ) ) )
@ Xs )
= ( nths_nat @ Xs @ S ) ) ) ).
% filter_in_nths
thf(fact_759_filter__in__nths,axiom,
! [Xs: list_complex,S: set_nat] :
( ( distinct_complex @ Xs )
=> ( ( filter_complex
@ ^ [X3: complex] : ( member_complex2 @ X3 @ ( set_complex2 @ ( nths_complex @ Xs @ S ) ) )
@ Xs )
= ( nths_complex @ Xs @ S ) ) ) ).
% filter_in_nths
thf(fact_760_remdups__adj__Cons,axiom,
! [X2: complex,Xs: list_complex] :
( ( remdups_adj_complex @ ( cons_complex @ X2 @ Xs ) )
= ( case_l7337434744184354388omplex @ ( cons_complex @ X2 @ nil_complex )
@ ^ [Y2: complex,Xs2: list_complex] : ( if_list_complex @ ( X2 = Y2 ) @ ( cons_complex @ Y2 @ Xs2 ) @ ( cons_complex @ X2 @ ( cons_complex @ Y2 @ Xs2 ) ) )
@ ( remdups_adj_complex @ Xs ) ) ) ).
% remdups_adj_Cons
thf(fact_761_remdups__adj__Cons,axiom,
! [X2: nat,Xs: list_nat] :
( ( remdups_adj_nat @ ( cons_nat @ X2 @ Xs ) )
= ( case_l2340614614379431832at_nat @ ( cons_nat @ X2 @ nil_nat )
@ ^ [Y2: nat,Xs2: list_nat] : ( if_list_nat @ ( X2 = Y2 ) @ ( cons_nat @ Y2 @ Xs2 ) @ ( cons_nat @ X2 @ ( cons_nat @ Y2 @ Xs2 ) ) )
@ ( remdups_adj_nat @ Xs ) ) ) ).
% remdups_adj_Cons
thf(fact_762_remdups__adj__Cons,axiom,
! [X2: real,Xs: list_real] :
( ( remdups_adj_real @ ( cons_real @ X2 @ Xs ) )
= ( case_l3379708394843211600l_real @ ( cons_real @ X2 @ nil_real )
@ ^ [Y2: real,Xs2: list_real] : ( if_list_real @ ( X2 = Y2 ) @ ( cons_real @ Y2 @ Xs2 ) @ ( cons_real @ X2 @ ( cons_real @ Y2 @ Xs2 ) ) )
@ ( remdups_adj_real @ Xs ) ) ) ).
% remdups_adj_Cons
thf(fact_763_remdups__adj_Opelims,axiom,
! [X2: list_complex,Y: list_complex] :
( ( ( remdups_adj_complex @ X2 )
= Y )
=> ( ( accp_list_complex @ remdup6092795584463544805omplex @ X2 )
=> ( ( ( X2 = nil_complex )
=> ( ( Y = nil_complex )
=> ~ ( accp_list_complex @ remdup6092795584463544805omplex @ nil_complex ) ) )
=> ( ! [X4: complex] :
( ( X2
= ( cons_complex @ X4 @ nil_complex ) )
=> ( ( Y
= ( cons_complex @ X4 @ nil_complex ) )
=> ~ ( accp_list_complex @ remdup6092795584463544805omplex @ ( cons_complex @ X4 @ nil_complex ) ) ) )
=> ~ ! [X4: complex,Y3: complex,Xs3: list_complex] :
( ( X2
= ( cons_complex @ X4 @ ( cons_complex @ Y3 @ Xs3 ) ) )
=> ( ( ( ( X4 = Y3 )
=> ( Y
= ( remdups_adj_complex @ ( cons_complex @ X4 @ Xs3 ) ) ) )
& ( ( X4 != Y3 )
=> ( Y
= ( cons_complex @ X4 @ ( remdups_adj_complex @ ( cons_complex @ Y3 @ Xs3 ) ) ) ) ) )
=> ~ ( accp_list_complex @ remdup6092795584463544805omplex @ ( cons_complex @ X4 @ ( cons_complex @ Y3 @ Xs3 ) ) ) ) ) ) ) ) ) ).
% remdups_adj.pelims
thf(fact_764_remdups__adj_Opelims,axiom,
! [X2: list_nat,Y: list_nat] :
( ( ( remdups_adj_nat @ X2 )
= Y )
=> ( ( accp_list_nat @ remdups_adj_rel_nat @ X2 )
=> ( ( ( X2 = nil_nat )
=> ( ( Y = nil_nat )
=> ~ ( accp_list_nat @ remdups_adj_rel_nat @ nil_nat ) ) )
=> ( ! [X4: nat] :
( ( X2
= ( cons_nat @ X4 @ nil_nat ) )
=> ( ( Y
= ( cons_nat @ X4 @ nil_nat ) )
=> ~ ( accp_list_nat @ remdups_adj_rel_nat @ ( cons_nat @ X4 @ nil_nat ) ) ) )
=> ~ ! [X4: nat,Y3: nat,Xs3: list_nat] :
( ( X2
= ( cons_nat @ X4 @ ( cons_nat @ Y3 @ Xs3 ) ) )
=> ( ( ( ( X4 = Y3 )
=> ( Y
= ( remdups_adj_nat @ ( cons_nat @ X4 @ Xs3 ) ) ) )
& ( ( X4 != Y3 )
=> ( Y
= ( cons_nat @ X4 @ ( remdups_adj_nat @ ( cons_nat @ Y3 @ Xs3 ) ) ) ) ) )
=> ~ ( accp_list_nat @ remdups_adj_rel_nat @ ( cons_nat @ X4 @ ( cons_nat @ Y3 @ Xs3 ) ) ) ) ) ) ) ) ) ).
% remdups_adj.pelims
thf(fact_765_remdups__adj_Opelims,axiom,
! [X2: list_real,Y: list_real] :
( ( ( remdups_adj_real @ X2 )
= Y )
=> ( ( accp_list_real @ remdups_adj_rel_real @ X2 )
=> ( ( ( X2 = nil_real )
=> ( ( Y = nil_real )
=> ~ ( accp_list_real @ remdups_adj_rel_real @ nil_real ) ) )
=> ( ! [X4: real] :
( ( X2
= ( cons_real @ X4 @ nil_real ) )
=> ( ( Y
= ( cons_real @ X4 @ nil_real ) )
=> ~ ( accp_list_real @ remdups_adj_rel_real @ ( cons_real @ X4 @ nil_real ) ) ) )
=> ~ ! [X4: real,Y3: real,Xs3: list_real] :
( ( X2
= ( cons_real @ X4 @ ( cons_real @ Y3 @ Xs3 ) ) )
=> ( ( ( ( X4 = Y3 )
=> ( Y
= ( remdups_adj_real @ ( cons_real @ X4 @ Xs3 ) ) ) )
& ( ( X4 != Y3 )
=> ( Y
= ( cons_real @ X4 @ ( remdups_adj_real @ ( cons_real @ Y3 @ Xs3 ) ) ) ) ) )
=> ~ ( accp_list_real @ remdups_adj_rel_real @ ( cons_real @ X4 @ ( cons_real @ Y3 @ Xs3 ) ) ) ) ) ) ) ) ) ).
% remdups_adj.pelims
thf(fact_766_transpose__aux__filter__head,axiom,
! [Xss2: list_list_complex] :
( ( concat_complex
@ ( map_li2870275437539113154omplex
@ ( case_l7337434744184354388omplex @ nil_complex
@ ^ [H: complex,T2: list_complex] : ( cons_complex @ H @ nil_complex ) )
@ Xss2 ) )
= ( map_li9134918900125190322omplex @ hd_complex
@ ( filter_list_complex
@ ^ [Ys3: list_complex] : ( Ys3 != nil_complex )
@ Xss2 ) ) ) ).
% transpose_aux_filter_head
thf(fact_767_transpose__aux__filter__head,axiom,
! [Xss2: list_list_nat] :
( ( concat_nat
@ ( map_li7225945977422193158st_nat
@ ( case_l2340614614379431832at_nat @ nil_nat
@ ^ [H: nat,T2: list_nat] : ( cons_nat @ H @ nil_nat ) )
@ Xss2 ) )
= ( map_list_nat_nat @ hd_nat
@ ( filter_list_nat
@ ^ [Ys3: list_nat] : ( Ys3 != nil_nat )
@ Xss2 ) ) ) ).
% transpose_aux_filter_head
thf(fact_768_transpose__aux__filter__head,axiom,
! [Xss2: list_list_real] :
( ( concat_real
@ ( map_li1455663113306559806t_real
@ ( case_l3379708394843211600l_real @ nil_real
@ ^ [H: real,T2: list_real] : ( cons_real @ H @ nil_real ) )
@ Xss2 ) )
= ( map_list_real_real @ hd_real
@ ( filter_list_real
@ ^ [Ys3: list_real] : ( Ys3 != nil_real )
@ Xss2 ) ) ) ).
% transpose_aux_filter_head
thf(fact_769_eq__comps_Oelims,axiom,
! [X2: list_nat,Y: list_nat] :
( ( ( commut2436974278740741825ps_nat @ X2 )
= Y )
=> ( ( ( X2 = nil_nat )
=> ( Y != nil_nat ) )
=> ( ( ? [X4: nat] :
( X2
= ( cons_nat @ X4 @ nil_nat ) )
=> ( Y
!= ( cons_nat @ one_one_nat @ nil_nat ) ) )
=> ~ ! [X4: nat,Y3: nat,L2: list_nat] :
( ( X2
= ( cons_nat @ X4 @ ( cons_nat @ Y3 @ L2 ) ) )
=> ( Y
!= ( if_list_nat @ ( X4 = Y3 ) @ ( cons_nat @ ( suc @ ( hd_nat @ ( commut2436974278740741825ps_nat @ ( cons_nat @ Y3 @ L2 ) ) ) ) @ ( tl_nat @ ( commut2436974278740741825ps_nat @ ( cons_nat @ Y3 @ L2 ) ) ) ) @ ( cons_nat @ one_one_nat @ ( commut2436974278740741825ps_nat @ ( cons_nat @ Y3 @ L2 ) ) ) ) ) ) ) ) ) ).
% eq_comps.elims
thf(fact_770_eq__comps_Oelims,axiom,
! [X2: list_real,Y: list_nat] :
( ( ( commut8680161604938074397s_real @ X2 )
= Y )
=> ( ( ( X2 = nil_real )
=> ( Y != nil_nat ) )
=> ( ( ? [X4: real] :
( X2
= ( cons_real @ X4 @ nil_real ) )
=> ( Y
!= ( cons_nat @ one_one_nat @ nil_nat ) ) )
=> ~ ! [X4: real,Y3: real,L2: list_real] :
( ( X2
= ( cons_real @ X4 @ ( cons_real @ Y3 @ L2 ) ) )
=> ( Y
!= ( if_list_nat @ ( X4 = Y3 ) @ ( cons_nat @ ( suc @ ( hd_nat @ ( commut8680161604938074397s_real @ ( cons_real @ Y3 @ L2 ) ) ) ) @ ( tl_nat @ ( commut8680161604938074397s_real @ ( cons_real @ Y3 @ L2 ) ) ) ) @ ( cons_nat @ one_one_nat @ ( commut8680161604938074397s_real @ ( cons_real @ Y3 @ L2 ) ) ) ) ) ) ) ) ) ).
% eq_comps.elims
thf(fact_771_eq__comps_Oelims,axiom,
! [X2: list_complex,Y: list_nat] :
( ( ( commut93809757773076895omplex @ X2 )
= Y )
=> ( ( ( X2 = nil_complex )
=> ( Y != nil_nat ) )
=> ( ( ? [X4: complex] :
( X2
= ( cons_complex @ X4 @ nil_complex ) )
=> ( Y
!= ( cons_nat @ one_one_nat @ nil_nat ) ) )
=> ~ ! [X4: complex,Y3: complex,L2: list_complex] :
( ( X2
= ( cons_complex @ X4 @ ( cons_complex @ Y3 @ L2 ) ) )
=> ( Y
!= ( if_list_nat @ ( X4 = Y3 ) @ ( cons_nat @ ( suc @ ( hd_nat @ ( commut93809757773076895omplex @ ( cons_complex @ Y3 @ L2 ) ) ) ) @ ( tl_nat @ ( commut93809757773076895omplex @ ( cons_complex @ Y3 @ L2 ) ) ) ) @ ( cons_nat @ one_one_nat @ ( commut93809757773076895omplex @ ( cons_complex @ Y3 @ L2 ) ) ) ) ) ) ) ) ) ).
% eq_comps.elims
thf(fact_772_eq__comps_Opelims,axiom,
! [X2: list_nat,Y: list_nat] :
( ( ( commut2436974278740741825ps_nat @ X2 )
= Y )
=> ( ( accp_list_nat @ commut1452772284045945626el_nat @ X2 )
=> ( ( ( X2 = nil_nat )
=> ( ( Y = nil_nat )
=> ~ ( accp_list_nat @ commut1452772284045945626el_nat @ nil_nat ) ) )
=> ( ! [X4: nat] :
( ( X2
= ( cons_nat @ X4 @ nil_nat ) )
=> ( ( Y
= ( cons_nat @ one_one_nat @ nil_nat ) )
=> ~ ( accp_list_nat @ commut1452772284045945626el_nat @ ( cons_nat @ X4 @ nil_nat ) ) ) )
=> ~ ! [X4: nat,Y3: nat,L2: list_nat] :
( ( X2
= ( cons_nat @ X4 @ ( cons_nat @ Y3 @ L2 ) ) )
=> ( ( Y
= ( if_list_nat @ ( X4 = Y3 ) @ ( cons_nat @ ( suc @ ( hd_nat @ ( commut2436974278740741825ps_nat @ ( cons_nat @ Y3 @ L2 ) ) ) ) @ ( tl_nat @ ( commut2436974278740741825ps_nat @ ( cons_nat @ Y3 @ L2 ) ) ) ) @ ( cons_nat @ one_one_nat @ ( commut2436974278740741825ps_nat @ ( cons_nat @ Y3 @ L2 ) ) ) ) )
=> ~ ( accp_list_nat @ commut1452772284045945626el_nat @ ( cons_nat @ X4 @ ( cons_nat @ Y3 @ L2 ) ) ) ) ) ) ) ) ) ).
% eq_comps.pelims
thf(fact_773_eq__comps_Opelims,axiom,
! [X2: list_real,Y: list_nat] :
( ( ( commut8680161604938074397s_real @ X2 )
= Y )
=> ( ( accp_list_real @ commut4159206679679027446l_real @ X2 )
=> ( ( ( X2 = nil_real )
=> ( ( Y = nil_nat )
=> ~ ( accp_list_real @ commut4159206679679027446l_real @ nil_real ) ) )
=> ( ! [X4: real] :
( ( X2
= ( cons_real @ X4 @ nil_real ) )
=> ( ( Y
= ( cons_nat @ one_one_nat @ nil_nat ) )
=> ~ ( accp_list_real @ commut4159206679679027446l_real @ ( cons_real @ X4 @ nil_real ) ) ) )
=> ~ ! [X4: real,Y3: real,L2: list_real] :
( ( X2
= ( cons_real @ X4 @ ( cons_real @ Y3 @ L2 ) ) )
=> ( ( Y
= ( if_list_nat @ ( X4 = Y3 ) @ ( cons_nat @ ( suc @ ( hd_nat @ ( commut8680161604938074397s_real @ ( cons_real @ Y3 @ L2 ) ) ) ) @ ( tl_nat @ ( commut8680161604938074397s_real @ ( cons_real @ Y3 @ L2 ) ) ) ) @ ( cons_nat @ one_one_nat @ ( commut8680161604938074397s_real @ ( cons_real @ Y3 @ L2 ) ) ) ) )
=> ~ ( accp_list_real @ commut4159206679679027446l_real @ ( cons_real @ X4 @ ( cons_real @ Y3 @ L2 ) ) ) ) ) ) ) ) ) ).
% eq_comps.pelims
thf(fact_774_eq__comps_Opelims,axiom,
! [X2: list_complex,Y: list_nat] :
( ( ( commut93809757773076895omplex @ X2 )
= Y )
=> ( ( accp_list_complex @ commut5384305104226550776omplex @ X2 )
=> ( ( ( X2 = nil_complex )
=> ( ( Y = nil_nat )
=> ~ ( accp_list_complex @ commut5384305104226550776omplex @ nil_complex ) ) )
=> ( ! [X4: complex] :
( ( X2
= ( cons_complex @ X4 @ nil_complex ) )
=> ( ( Y
= ( cons_nat @ one_one_nat @ nil_nat ) )
=> ~ ( accp_list_complex @ commut5384305104226550776omplex @ ( cons_complex @ X4 @ nil_complex ) ) ) )
=> ~ ! [X4: complex,Y3: complex,L2: list_complex] :
( ( X2
= ( cons_complex @ X4 @ ( cons_complex @ Y3 @ L2 ) ) )
=> ( ( Y
= ( if_list_nat @ ( X4 = Y3 ) @ ( cons_nat @ ( suc @ ( hd_nat @ ( commut93809757773076895omplex @ ( cons_complex @ Y3 @ L2 ) ) ) ) @ ( tl_nat @ ( commut93809757773076895omplex @ ( cons_complex @ Y3 @ L2 ) ) ) ) @ ( cons_nat @ one_one_nat @ ( commut93809757773076895omplex @ ( cons_complex @ Y3 @ L2 ) ) ) ) )
=> ~ ( accp_list_complex @ commut5384305104226550776omplex @ ( cons_complex @ X4 @ ( cons_complex @ Y3 @ L2 ) ) ) ) ) ) ) ) ) ).
% eq_comps.pelims
thf(fact_775_list_Osel_I1_J,axiom,
! [X21: complex,X22: list_complex] :
( ( hd_complex @ ( cons_complex @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_776_list_Osel_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( ( hd_nat @ ( cons_nat @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_777_list_Osel_I1_J,axiom,
! [X21: real,X22: list_real] :
( ( hd_real @ ( cons_real @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_778_hd__concat,axiom,
! [Xs: list_list_nat] :
( ( Xs != nil_list_nat )
=> ( ( ( hd_list_nat @ Xs )
!= nil_nat )
=> ( ( hd_nat @ ( concat_nat @ Xs ) )
= ( hd_nat @ ( hd_list_nat @ Xs ) ) ) ) ) ).
% hd_concat
thf(fact_779_hd__in__set,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( member_nat2 @ ( hd_nat @ Xs ) @ ( set_nat2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_780_hd__in__set,axiom,
! [Xs: list_complex] :
( ( Xs != nil_complex )
=> ( member_complex2 @ ( hd_complex @ Xs ) @ ( set_complex2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_781_list_Oset__sel_I1_J,axiom,
! [A: list_nat] :
( ( A != nil_nat )
=> ( member_nat2 @ ( hd_nat @ A ) @ ( set_nat2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_782_list_Oset__sel_I1_J,axiom,
! [A: list_complex] :
( ( A != nil_complex )
=> ( member_complex2 @ ( hd_complex @ A ) @ ( set_complex2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_783_hd__map,axiom,
! [Xs: list_complex,F: complex > real] :
( ( Xs != nil_complex )
=> ( ( hd_real @ ( map_complex_real @ F @ Xs ) )
= ( F @ ( hd_complex @ Xs ) ) ) ) ).
% hd_map
thf(fact_784_hd__map,axiom,
! [Xs: list_nat,F: nat > nat] :
( ( Xs != nil_nat )
=> ( ( hd_nat @ ( map_nat_nat @ F @ Xs ) )
= ( F @ ( hd_nat @ Xs ) ) ) ) ).
% hd_map
thf(fact_785_hd__map,axiom,
! [Xs: list_nat,F: nat > complex] :
( ( Xs != nil_nat )
=> ( ( hd_complex @ ( map_nat_complex @ F @ Xs ) )
= ( F @ ( hd_nat @ Xs ) ) ) ) ).
% hd_map
thf(fact_786_hd__map,axiom,
! [Xs: list_complex,F: complex > nat] :
( ( Xs != nil_complex )
=> ( ( hd_nat @ ( map_complex_nat @ F @ Xs ) )
= ( F @ ( hd_complex @ Xs ) ) ) ) ).
% hd_map
thf(fact_787_hd__map,axiom,
! [Xs: list_complex,F: complex > complex] :
( ( Xs != nil_complex )
=> ( ( hd_complex @ ( map_complex_complex @ F @ Xs ) )
= ( F @ ( hd_complex @ Xs ) ) ) ) ).
% hd_map
thf(fact_788_list_Omap__sel_I1_J,axiom,
! [A: list_complex,F: complex > real] :
( ( A != nil_complex )
=> ( ( hd_real @ ( map_complex_real @ F @ A ) )
= ( F @ ( hd_complex @ A ) ) ) ) ).
% list.map_sel(1)
thf(fact_789_list_Omap__sel_I1_J,axiom,
! [A: list_nat,F: nat > nat] :
( ( A != nil_nat )
=> ( ( hd_nat @ ( map_nat_nat @ F @ A ) )
= ( F @ ( hd_nat @ A ) ) ) ) ).
% list.map_sel(1)
thf(fact_790_list_Omap__sel_I1_J,axiom,
! [A: list_nat,F: nat > complex] :
( ( A != nil_nat )
=> ( ( hd_complex @ ( map_nat_complex @ F @ A ) )
= ( F @ ( hd_nat @ A ) ) ) ) ).
% list.map_sel(1)
thf(fact_791_list_Omap__sel_I1_J,axiom,
! [A: list_complex,F: complex > nat] :
( ( A != nil_complex )
=> ( ( hd_nat @ ( map_complex_nat @ F @ A ) )
= ( F @ ( hd_complex @ A ) ) ) ) ).
% list.map_sel(1)
thf(fact_792_list_Omap__sel_I1_J,axiom,
! [A: list_complex,F: complex > complex] :
( ( A != nil_complex )
=> ( ( hd_complex @ ( map_complex_complex @ F @ A ) )
= ( F @ ( hd_complex @ A ) ) ) ) ).
% list.map_sel(1)
thf(fact_793_hd__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( Xs = nil_nat )
=> ( ( hd_nat @ ( append_nat @ Xs @ Ys ) )
= ( hd_nat @ Ys ) ) )
& ( ( Xs != nil_nat )
=> ( ( hd_nat @ ( append_nat @ Xs @ Ys ) )
= ( hd_nat @ Xs ) ) ) ) ).
% hd_append
thf(fact_794_hd__append2,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs != nil_nat )
=> ( ( hd_nat @ ( append_nat @ Xs @ Ys ) )
= ( hd_nat @ Xs ) ) ) ).
% hd_append2
thf(fact_795_longest__common__prefix,axiom,
! [Xs: list_nat,Ys: list_nat] :
? [Ps: list_nat,Xs6: list_nat,Ys6: list_nat] :
( ( Xs
= ( append_nat @ Ps @ Xs6 ) )
& ( Ys
= ( append_nat @ Ps @ Ys6 ) )
& ( ( Xs6 = nil_nat )
| ( Ys6 = nil_nat )
| ( ( hd_nat @ Xs6 )
!= ( hd_nat @ Ys6 ) ) ) ) ).
% longest_common_prefix
thf(fact_796_list_Oexpand,axiom,
! [List: list_nat,List2: list_nat] :
( ( ( List = nil_nat )
= ( List2 = nil_nat ) )
=> ( ( ( List != nil_nat )
=> ( ( List2 != nil_nat )
=> ( ( ( hd_nat @ List )
= ( hd_nat @ List2 ) )
& ( ( tl_nat @ List )
= ( tl_nat @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_797_hd__prefixes,axiom,
! [Xs: list_nat] :
( ( hd_list_nat @ ( prefixes_nat @ Xs ) )
= nil_nat ) ).
% hd_prefixes
thf(fact_798_hd__suffixes,axiom,
! [Xs: list_nat] :
( ( hd_list_nat @ ( suffixes_nat @ Xs ) )
= nil_nat ) ).
% hd_suffixes
thf(fact_799_hd__Cons__tl,axiom,
! [Xs: list_complex] :
( ( Xs != nil_complex )
=> ( ( cons_complex @ ( hd_complex @ Xs ) @ ( tl_complex @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_800_hd__Cons__tl,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( cons_nat @ ( hd_nat @ Xs ) @ ( tl_nat @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_801_hd__Cons__tl,axiom,
! [Xs: list_real] :
( ( Xs != nil_real )
=> ( ( cons_real @ ( hd_real @ Xs ) @ ( tl_real @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_802_list_Oexhaust__sel,axiom,
! [List: list_complex] :
( ( List != nil_complex )
=> ( List
= ( cons_complex @ ( hd_complex @ List ) @ ( tl_complex @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_803_list_Oexhaust__sel,axiom,
! [List: list_nat] :
( ( List != nil_nat )
=> ( List
= ( cons_nat @ ( hd_nat @ List ) @ ( tl_nat @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_804_list_Oexhaust__sel,axiom,
! [List: list_real] :
( ( List != nil_real )
=> ( List
= ( cons_real @ ( hd_real @ List ) @ ( tl_real @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_805_list_Ocollapse,axiom,
! [List: list_complex] :
( ( List != nil_complex )
=> ( ( cons_complex @ ( hd_complex @ List ) @ ( tl_complex @ List ) )
= List ) ) ).
% list.collapse
thf(fact_806_list_Ocollapse,axiom,
! [List: list_nat] :
( ( List != nil_nat )
=> ( ( cons_nat @ ( hd_nat @ List ) @ ( tl_nat @ List ) )
= List ) ) ).
% list.collapse
thf(fact_807_list_Ocollapse,axiom,
! [List: list_real] :
( ( List != nil_real )
=> ( ( cons_real @ ( hd_real @ List ) @ ( tl_real @ List ) )
= List ) ) ).
% list.collapse
thf(fact_808_rotate1__hd__tl,axiom,
! [Xs: list_complex] :
( ( Xs != nil_complex )
=> ( ( rotate1_complex @ Xs )
= ( append_complex @ ( tl_complex @ Xs ) @ ( cons_complex @ ( hd_complex @ Xs ) @ nil_complex ) ) ) ) ).
% rotate1_hd_tl
thf(fact_809_rotate1__hd__tl,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( rotate1_nat @ Xs )
= ( append_nat @ ( tl_nat @ Xs ) @ ( cons_nat @ ( hd_nat @ Xs ) @ nil_nat ) ) ) ) ).
% rotate1_hd_tl
thf(fact_810_rotate1__hd__tl,axiom,
! [Xs: list_real] :
( ( Xs != nil_real )
=> ( ( rotate1_real @ Xs )
= ( append_real @ ( tl_real @ Xs ) @ ( cons_real @ ( hd_real @ Xs ) @ nil_real ) ) ) ) ).
% rotate1_hd_tl
thf(fact_811_eq__comps_Osimps_I3_J,axiom,
! [X2: nat,Y: nat,L: list_nat] :
( ( commut2436974278740741825ps_nat @ ( cons_nat @ X2 @ ( cons_nat @ Y @ L ) ) )
= ( if_list_nat @ ( X2 = Y ) @ ( cons_nat @ ( suc @ ( hd_nat @ ( commut2436974278740741825ps_nat @ ( cons_nat @ Y @ L ) ) ) ) @ ( tl_nat @ ( commut2436974278740741825ps_nat @ ( cons_nat @ Y @ L ) ) ) ) @ ( cons_nat @ one_one_nat @ ( commut2436974278740741825ps_nat @ ( cons_nat @ Y @ L ) ) ) ) ) ).
% eq_comps.simps(3)
thf(fact_812_eq__comps_Osimps_I3_J,axiom,
! [X2: real,Y: real,L: list_real] :
( ( commut8680161604938074397s_real @ ( cons_real @ X2 @ ( cons_real @ Y @ L ) ) )
= ( if_list_nat @ ( X2 = Y ) @ ( cons_nat @ ( suc @ ( hd_nat @ ( commut8680161604938074397s_real @ ( cons_real @ Y @ L ) ) ) ) @ ( tl_nat @ ( commut8680161604938074397s_real @ ( cons_real @ Y @ L ) ) ) ) @ ( cons_nat @ one_one_nat @ ( commut8680161604938074397s_real @ ( cons_real @ Y @ L ) ) ) ) ) ).
% eq_comps.simps(3)
thf(fact_813_eq__comps_Osimps_I3_J,axiom,
! [X2: complex,Y: complex,L: list_complex] :
( ( commut93809757773076895omplex @ ( cons_complex @ X2 @ ( cons_complex @ Y @ L ) ) )
= ( if_list_nat @ ( X2 = Y ) @ ( cons_nat @ ( suc @ ( hd_nat @ ( commut93809757773076895omplex @ ( cons_complex @ Y @ L ) ) ) ) @ ( tl_nat @ ( commut93809757773076895omplex @ ( cons_complex @ Y @ L ) ) ) ) @ ( cons_nat @ one_one_nat @ ( commut93809757773076895omplex @ ( cons_complex @ Y @ L ) ) ) ) ) ).
% eq_comps.simps(3)
thf(fact_814_take__Suc,axiom,
! [Xs: list_complex,N2: nat] :
( ( Xs != nil_complex )
=> ( ( take_complex @ ( suc @ N2 ) @ Xs )
= ( cons_complex @ ( hd_complex @ Xs ) @ ( take_complex @ N2 @ ( tl_complex @ Xs ) ) ) ) ) ).
% take_Suc
thf(fact_815_take__Suc,axiom,
! [Xs: list_nat,N2: nat] :
( ( Xs != nil_nat )
=> ( ( take_nat @ ( suc @ N2 ) @ Xs )
= ( cons_nat @ ( hd_nat @ Xs ) @ ( take_nat @ N2 @ ( tl_nat @ Xs ) ) ) ) ) ).
% take_Suc
thf(fact_816_take__Suc,axiom,
! [Xs: list_real,N2: nat] :
( ( Xs != nil_real )
=> ( ( take_real @ ( suc @ N2 ) @ Xs )
= ( cons_real @ ( hd_real @ Xs ) @ ( take_real @ N2 @ ( tl_real @ Xs ) ) ) ) ) ).
% take_Suc
thf(fact_817_tl__remdups__adj,axiom,
! [Ys: list_nat] :
( ( Ys != nil_nat )
=> ( ( tl_nat @ ( remdups_adj_nat @ Ys ) )
= ( remdups_adj_nat
@ ( dropWhile_nat
@ ^ [X3: nat] :
( X3
= ( hd_nat @ Ys ) )
@ ( tl_nat @ Ys ) ) ) ) ) ).
% tl_remdups_adj
thf(fact_818_take__map,axiom,
! [N2: nat,F: complex > real,Xs: list_complex] :
( ( take_real @ N2 @ ( map_complex_real @ F @ Xs ) )
= ( map_complex_real @ F @ ( take_complex @ N2 @ Xs ) ) ) ).
% take_map
thf(fact_819_take__map,axiom,
! [N2: nat,F: nat > nat,Xs: list_nat] :
( ( take_nat @ N2 @ ( map_nat_nat @ F @ Xs ) )
= ( map_nat_nat @ F @ ( take_nat @ N2 @ Xs ) ) ) ).
% take_map
thf(fact_820_take__map,axiom,
! [N2: nat,F: nat > complex,Xs: list_nat] :
( ( take_complex @ N2 @ ( map_nat_complex @ F @ Xs ) )
= ( map_nat_complex @ F @ ( take_nat @ N2 @ Xs ) ) ) ).
% take_map
thf(fact_821_take__map,axiom,
! [N2: nat,F: complex > nat,Xs: list_complex] :
( ( take_nat @ N2 @ ( map_complex_nat @ F @ Xs ) )
= ( map_complex_nat @ F @ ( take_complex @ N2 @ Xs ) ) ) ).
% take_map
thf(fact_822_take__map,axiom,
! [N2: nat,F: complex > complex,Xs: list_complex] :
( ( take_complex @ N2 @ ( map_complex_complex @ F @ Xs ) )
= ( map_complex_complex @ F @ ( take_complex @ N2 @ Xs ) ) ) ).
% take_map
thf(fact_823_dropWhile_Osimps_I2_J,axiom,
! [P: complex > $o,X2: complex,Xs: list_complex] :
( ( ( P @ X2 )
=> ( ( dropWhile_complex @ P @ ( cons_complex @ X2 @ Xs ) )
= ( dropWhile_complex @ P @ Xs ) ) )
& ( ~ ( P @ X2 )
=> ( ( dropWhile_complex @ P @ ( cons_complex @ X2 @ Xs ) )
= ( cons_complex @ X2 @ Xs ) ) ) ) ).
% dropWhile.simps(2)
thf(fact_824_dropWhile_Osimps_I2_J,axiom,
! [P: nat > $o,X2: nat,Xs: list_nat] :
( ( ( P @ X2 )
=> ( ( dropWhile_nat @ P @ ( cons_nat @ X2 @ Xs ) )
= ( dropWhile_nat @ P @ Xs ) ) )
& ( ~ ( P @ X2 )
=> ( ( dropWhile_nat @ P @ ( cons_nat @ X2 @ Xs ) )
= ( cons_nat @ X2 @ Xs ) ) ) ) ).
% dropWhile.simps(2)
thf(fact_825_dropWhile_Osimps_I2_J,axiom,
! [P: real > $o,X2: real,Xs: list_real] :
( ( ( P @ X2 )
=> ( ( dropWhile_real @ P @ ( cons_real @ X2 @ Xs ) )
= ( dropWhile_real @ P @ Xs ) ) )
& ( ~ ( P @ X2 )
=> ( ( dropWhile_real @ P @ ( cons_real @ X2 @ Xs ) )
= ( cons_real @ X2 @ Xs ) ) ) ) ).
% dropWhile.simps(2)
thf(fact_826_dropWhile_Osimps_I1_J,axiom,
! [P: nat > $o] :
( ( dropWhile_nat @ P @ nil_nat )
= nil_nat ) ).
% dropWhile.simps(1)
thf(fact_827_set__dropWhileD,axiom,
! [X2: nat,P: nat > $o,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ ( dropWhile_nat @ P @ Xs ) ) )
=> ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) ) ) ).
% set_dropWhileD
thf(fact_828_set__dropWhileD,axiom,
! [X2: complex,P: complex > $o,Xs: list_complex] :
( ( member_complex2 @ X2 @ ( set_complex2 @ ( dropWhile_complex @ P @ Xs ) ) )
=> ( member_complex2 @ X2 @ ( set_complex2 @ Xs ) ) ) ).
% set_dropWhileD
thf(fact_829_dropWhile__cong,axiom,
! [L: list_nat,K2: list_nat,P: nat > $o,Q: nat > $o] :
( ( L = K2 )
=> ( ! [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ L ) )
=> ( ( P @ X4 )
= ( Q @ X4 ) ) )
=> ( ( dropWhile_nat @ P @ L )
= ( dropWhile_nat @ Q @ K2 ) ) ) ) ).
% dropWhile_cong
thf(fact_830_dropWhile__cong,axiom,
! [L: list_complex,K2: list_complex,P: complex > $o,Q: complex > $o] :
( ( L = K2 )
=> ( ! [X4: complex] :
( ( member_complex2 @ X4 @ ( set_complex2 @ L ) )
=> ( ( P @ X4 )
= ( Q @ X4 ) ) )
=> ( ( dropWhile_complex @ P @ L )
= ( dropWhile_complex @ Q @ K2 ) ) ) ) ).
% dropWhile_cong
thf(fact_831_in__set__takeD,axiom,
! [X2: nat,N2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ ( take_nat @ N2 @ Xs ) ) )
=> ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) ) ) ).
% in_set_takeD
thf(fact_832_in__set__takeD,axiom,
! [X2: complex,N2: nat,Xs: list_complex] :
( ( member_complex2 @ X2 @ ( set_complex2 @ ( take_complex @ N2 @ Xs ) ) )
=> ( member_complex2 @ X2 @ ( set_complex2 @ Xs ) ) ) ).
% in_set_takeD
thf(fact_833_take_Osimps_I1_J,axiom,
! [N2: nat] :
( ( take_nat @ N2 @ nil_nat )
= nil_nat ) ).
% take.simps(1)
thf(fact_834_take__eq__Nil2,axiom,
! [N2: nat,Xs: list_nat] :
( ( nil_nat
= ( take_nat @ N2 @ Xs ) )
= ( ( N2 = zero_zero_nat )
| ( Xs = nil_nat ) ) ) ).
% take_eq_Nil2
thf(fact_835_take__eq__Nil,axiom,
! [N2: nat,Xs: list_nat] :
( ( ( take_nat @ N2 @ Xs )
= nil_nat )
= ( ( N2 = zero_zero_nat )
| ( Xs = nil_nat ) ) ) ).
% take_eq_Nil
thf(fact_836_take__0,axiom,
! [Xs: list_nat] :
( ( take_nat @ zero_zero_nat @ Xs )
= nil_nat ) ).
% take_0
thf(fact_837_take0,axiom,
( ( take_nat @ zero_zero_nat )
= ( ^ [Xs2: list_nat] : nil_nat ) ) ).
% take0
thf(fact_838_take__Suc__Cons,axiom,
! [N2: nat,X2: complex,Xs: list_complex] :
( ( take_complex @ ( suc @ N2 ) @ ( cons_complex @ X2 @ Xs ) )
= ( cons_complex @ X2 @ ( take_complex @ N2 @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_839_take__Suc__Cons,axiom,
! [N2: nat,X2: nat,Xs: list_nat] :
( ( take_nat @ ( suc @ N2 ) @ ( cons_nat @ X2 @ Xs ) )
= ( cons_nat @ X2 @ ( take_nat @ N2 @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_840_take__Suc__Cons,axiom,
! [N2: nat,X2: real,Xs: list_real] :
( ( take_real @ ( suc @ N2 ) @ ( cons_real @ X2 @ Xs ) )
= ( cons_real @ X2 @ ( take_real @ N2 @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_841_dropWhile__eq__Nil__conv,axiom,
! [P: nat > $o,Xs: list_nat] :
( ( ( dropWhile_nat @ P @ Xs )
= nil_nat )
= ( ! [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
=> ( P @ X3 ) ) ) ) ).
% dropWhile_eq_Nil_conv
thf(fact_842_dropWhile__eq__Nil__conv,axiom,
! [P: complex > $o,Xs: list_complex] :
( ( ( dropWhile_complex @ P @ Xs )
= nil_complex )
= ( ! [X3: complex] :
( ( member_complex2 @ X3 @ ( set_complex2 @ Xs ) )
=> ( P @ X3 ) ) ) ) ).
% dropWhile_eq_Nil_conv
thf(fact_843_dropWhile__append3,axiom,
! [P: complex > $o,Y: complex,Xs: list_complex,Ys: list_complex] :
( ~ ( P @ Y )
=> ( ( dropWhile_complex @ P @ ( append_complex @ Xs @ ( cons_complex @ Y @ Ys ) ) )
= ( append_complex @ ( dropWhile_complex @ P @ Xs ) @ ( cons_complex @ Y @ Ys ) ) ) ) ).
% dropWhile_append3
thf(fact_844_dropWhile__append3,axiom,
! [P: nat > $o,Y: nat,Xs: list_nat,Ys: list_nat] :
( ~ ( P @ Y )
=> ( ( dropWhile_nat @ P @ ( append_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) )
= ( append_nat @ ( dropWhile_nat @ P @ Xs ) @ ( cons_nat @ Y @ Ys ) ) ) ) ).
% dropWhile_append3
thf(fact_845_dropWhile__append3,axiom,
! [P: real > $o,Y: real,Xs: list_real,Ys: list_real] :
( ~ ( P @ Y )
=> ( ( dropWhile_real @ P @ ( append_real @ Xs @ ( cons_real @ Y @ Ys ) ) )
= ( append_real @ ( dropWhile_real @ P @ Xs ) @ ( cons_real @ Y @ Ys ) ) ) ) ).
% dropWhile_append3
thf(fact_846_dropWhile__append2,axiom,
! [Xs: list_nat,P: nat > $o,Ys: list_nat] :
( ! [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Xs ) )
=> ( P @ X4 ) )
=> ( ( dropWhile_nat @ P @ ( append_nat @ Xs @ Ys ) )
= ( dropWhile_nat @ P @ Ys ) ) ) ).
% dropWhile_append2
thf(fact_847_dropWhile__append2,axiom,
! [Xs: list_complex,P: complex > $o,Ys: list_complex] :
( ! [X4: complex] :
( ( member_complex2 @ X4 @ ( set_complex2 @ Xs ) )
=> ( P @ X4 ) )
=> ( ( dropWhile_complex @ P @ ( append_complex @ Xs @ Ys ) )
= ( dropWhile_complex @ P @ Ys ) ) ) ).
% dropWhile_append2
thf(fact_848_dropWhile__append1,axiom,
! [X2: nat,Xs: list_nat,P: nat > $o,Ys: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ~ ( P @ X2 )
=> ( ( dropWhile_nat @ P @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( dropWhile_nat @ P @ Xs ) @ Ys ) ) ) ) ).
% dropWhile_append1
thf(fact_849_dropWhile__append1,axiom,
! [X2: complex,Xs: list_complex,P: complex > $o,Ys: list_complex] :
( ( member_complex2 @ X2 @ ( set_complex2 @ Xs ) )
=> ( ~ ( P @ X2 )
=> ( ( dropWhile_complex @ P @ ( append_complex @ Xs @ Ys ) )
= ( append_complex @ ( dropWhile_complex @ P @ Xs ) @ Ys ) ) ) ) ).
% dropWhile_append1
thf(fact_850_dropWhile__eq__self__iff,axiom,
! [P: nat > $o,Xs: list_nat] :
( ( ( dropWhile_nat @ P @ Xs )
= Xs )
= ( ( Xs = nil_nat )
| ~ ( P @ ( hd_nat @ Xs ) ) ) ) ).
% dropWhile_eq_self_iff
thf(fact_851_hd__dropWhile,axiom,
! [P: nat > $o,Xs: list_nat] :
( ( ( dropWhile_nat @ P @ Xs )
!= nil_nat )
=> ~ ( P @ ( hd_nat @ ( dropWhile_nat @ P @ Xs ) ) ) ) ).
% hd_dropWhile
thf(fact_852_remdups__adj__Cons_H,axiom,
! [X2: complex,Xs: list_complex] :
( ( remdups_adj_complex @ ( cons_complex @ X2 @ Xs ) )
= ( cons_complex @ X2
@ ( remdups_adj_complex
@ ( dropWhile_complex
@ ^ [Y2: complex] : ( Y2 = X2 )
@ Xs ) ) ) ) ).
% remdups_adj_Cons'
thf(fact_853_remdups__adj__Cons_H,axiom,
! [X2: nat,Xs: list_nat] :
( ( remdups_adj_nat @ ( cons_nat @ X2 @ Xs ) )
= ( cons_nat @ X2
@ ( remdups_adj_nat
@ ( dropWhile_nat
@ ^ [Y2: nat] : ( Y2 = X2 )
@ Xs ) ) ) ) ).
% remdups_adj_Cons'
thf(fact_854_remdups__adj__Cons_H,axiom,
! [X2: real,Xs: list_real] :
( ( remdups_adj_real @ ( cons_real @ X2 @ Xs ) )
= ( cons_real @ X2
@ ( remdups_adj_real
@ ( dropWhile_real
@ ^ [Y2: real] : ( Y2 = X2 )
@ Xs ) ) ) ) ).
% remdups_adj_Cons'
thf(fact_855_remdups__adj__append__dropWhile,axiom,
! [Xs: list_complex,Y: complex,Ys: list_complex] :
( ( remdups_adj_complex @ ( append_complex @ Xs @ ( cons_complex @ Y @ Ys ) ) )
= ( append_complex @ ( remdups_adj_complex @ ( append_complex @ Xs @ ( cons_complex @ Y @ nil_complex ) ) )
@ ( remdups_adj_complex
@ ( dropWhile_complex
@ ^ [X3: complex] : ( X3 = Y )
@ Ys ) ) ) ) ).
% remdups_adj_append_dropWhile
thf(fact_856_remdups__adj__append__dropWhile,axiom,
! [Xs: list_nat,Y: nat,Ys: list_nat] :
( ( remdups_adj_nat @ ( append_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) )
= ( append_nat @ ( remdups_adj_nat @ ( append_nat @ Xs @ ( cons_nat @ Y @ nil_nat ) ) )
@ ( remdups_adj_nat
@ ( dropWhile_nat
@ ^ [X3: nat] : ( X3 = Y )
@ Ys ) ) ) ) ).
% remdups_adj_append_dropWhile
thf(fact_857_remdups__adj__append__dropWhile,axiom,
! [Xs: list_real,Y: real,Ys: list_real] :
( ( remdups_adj_real @ ( append_real @ Xs @ ( cons_real @ Y @ Ys ) ) )
= ( append_real @ ( remdups_adj_real @ ( append_real @ Xs @ ( cons_real @ Y @ nil_real ) ) )
@ ( remdups_adj_real
@ ( dropWhile_real
@ ^ [X3: real] : ( X3 = Y )
@ Ys ) ) ) ) ).
% remdups_adj_append_dropWhile
thf(fact_858_take__Cons_H,axiom,
! [N2: nat,X2: complex,Xs: list_complex] :
( ( ( N2 = zero_zero_nat )
=> ( ( take_complex @ N2 @ ( cons_complex @ X2 @ Xs ) )
= nil_complex ) )
& ( ( N2 != zero_zero_nat )
=> ( ( take_complex @ N2 @ ( cons_complex @ X2 @ Xs ) )
= ( cons_complex @ X2 @ ( take_complex @ ( minus_minus_nat @ N2 @ one_one_nat ) @ Xs ) ) ) ) ) ).
% take_Cons'
thf(fact_859_take__Cons_H,axiom,
! [N2: nat,X2: nat,Xs: list_nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( take_nat @ N2 @ ( cons_nat @ X2 @ Xs ) )
= nil_nat ) )
& ( ( N2 != zero_zero_nat )
=> ( ( take_nat @ N2 @ ( cons_nat @ X2 @ Xs ) )
= ( cons_nat @ X2 @ ( take_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ Xs ) ) ) ) ) ).
% take_Cons'
thf(fact_860_take__Cons_H,axiom,
! [N2: nat,X2: real,Xs: list_real] :
( ( ( N2 = zero_zero_nat )
=> ( ( take_real @ N2 @ ( cons_real @ X2 @ Xs ) )
= nil_real ) )
& ( ( N2 != zero_zero_nat )
=> ( ( take_real @ N2 @ ( cons_real @ X2 @ Xs ) )
= ( cons_real @ X2 @ ( take_real @ ( minus_minus_nat @ N2 @ one_one_nat ) @ Xs ) ) ) ) ) ).
% take_Cons'
thf(fact_861_remdups__adj__append_H_H,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs != nil_nat )
=> ( ( remdups_adj_nat @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( remdups_adj_nat @ Xs )
@ ( remdups_adj_nat
@ ( dropWhile_nat
@ ^ [Y2: nat] :
( Y2
= ( last_nat @ Xs ) )
@ Ys ) ) ) ) ) ).
% remdups_adj_append''
thf(fact_862_remdups__adj__append_H,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( Xs = nil_nat )
| ( Ys = nil_nat )
| ( ( last_nat @ Xs )
!= ( hd_nat @ Ys ) ) )
=> ( ( remdups_adj_nat @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( remdups_adj_nat @ Xs ) @ ( remdups_adj_nat @ Ys ) ) ) ) ).
% remdups_adj_append'
thf(fact_863_Cons__in__shuffles__iff,axiom,
! [Z3: complex,Zs3: list_complex,Xs: list_complex,Ys: list_complex] :
( ( member_list_complex @ ( cons_complex @ Z3 @ Zs3 ) @ ( shuffles_complex @ Xs @ Ys ) )
= ( ( ( Xs != nil_complex )
& ( ( hd_complex @ Xs )
= Z3 )
& ( member_list_complex @ Zs3 @ ( shuffles_complex @ ( tl_complex @ Xs ) @ Ys ) ) )
| ( ( Ys != nil_complex )
& ( ( hd_complex @ Ys )
= Z3 )
& ( member_list_complex @ Zs3 @ ( shuffles_complex @ Xs @ ( tl_complex @ Ys ) ) ) ) ) ) ).
% Cons_in_shuffles_iff
thf(fact_864_Cons__in__shuffles__iff,axiom,
! [Z3: nat,Zs3: list_nat,Xs: list_nat,Ys: list_nat] :
( ( member_list_nat @ ( cons_nat @ Z3 @ Zs3 ) @ ( shuffles_nat @ Xs @ Ys ) )
= ( ( ( Xs != nil_nat )
& ( ( hd_nat @ Xs )
= Z3 )
& ( member_list_nat @ Zs3 @ ( shuffles_nat @ ( tl_nat @ Xs ) @ Ys ) ) )
| ( ( Ys != nil_nat )
& ( ( hd_nat @ Ys )
= Z3 )
& ( member_list_nat @ Zs3 @ ( shuffles_nat @ Xs @ ( tl_nat @ Ys ) ) ) ) ) ) ).
% Cons_in_shuffles_iff
thf(fact_865_Cons__in__shuffles__iff,axiom,
! [Z3: real,Zs3: list_real,Xs: list_real,Ys: list_real] :
( ( member_list_real @ ( cons_real @ Z3 @ Zs3 ) @ ( shuffles_real @ Xs @ Ys ) )
= ( ( ( Xs != nil_real )
& ( ( hd_real @ Xs )
= Z3 )
& ( member_list_real @ Zs3 @ ( shuffles_real @ ( tl_real @ Xs ) @ Ys ) ) )
| ( ( Ys != nil_real )
& ( ( hd_real @ Ys )
= Z3 )
& ( member_list_real @ Zs3 @ ( shuffles_real @ Xs @ ( tl_real @ Ys ) ) ) ) ) ) ).
% Cons_in_shuffles_iff
thf(fact_866_Reals__diff,axiom,
! [A: real,B2: real] :
( ( member_real2 @ A @ real_V470468836141973256s_real )
=> ( ( member_real2 @ B2 @ real_V470468836141973256s_real )
=> ( member_real2 @ ( minus_minus_real @ A @ B2 ) @ real_V470468836141973256s_real ) ) ) ).
% Reals_diff
thf(fact_867_Reals__diff,axiom,
! [A: complex,B2: complex] :
( ( member_complex2 @ A @ real_V2521375963428798218omplex )
=> ( ( member_complex2 @ B2 @ real_V2521375963428798218omplex )
=> ( member_complex2 @ ( minus_minus_complex @ A @ B2 ) @ real_V2521375963428798218omplex ) ) ) ).
% Reals_diff
thf(fact_868_diff__eq__diff__eq,axiom,
! [A: complex,B2: complex,C3: complex,D2: complex] :
( ( ( minus_minus_complex @ A @ B2 )
= ( minus_minus_complex @ C3 @ D2 ) )
=> ( ( A = B2 )
= ( C3 = D2 ) ) ) ).
% diff_eq_diff_eq
thf(fact_869_diff__eq__diff__eq,axiom,
! [A: real,B2: real,C3: real,D2: real] :
( ( ( minus_minus_real @ A @ B2 )
= ( minus_minus_real @ C3 @ D2 ) )
=> ( ( A = B2 )
= ( C3 = D2 ) ) ) ).
% diff_eq_diff_eq
thf(fact_870_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C3: nat,B2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C3 ) @ B2 )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B2 ) @ C3 ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_871_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: complex,C3: complex,B2: complex] :
( ( minus_minus_complex @ ( minus_minus_complex @ A @ C3 ) @ B2 )
= ( minus_minus_complex @ ( minus_minus_complex @ A @ B2 ) @ C3 ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_872_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: real,C3: real,B2: real] :
( ( minus_minus_real @ ( minus_minus_real @ A @ C3 ) @ B2 )
= ( minus_minus_real @ ( minus_minus_real @ A @ B2 ) @ C3 ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_873_Nil__in__shufflesI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs = nil_nat )
=> ( ( Ys = nil_nat )
=> ( member_list_nat @ nil_nat @ ( shuffles_nat @ Xs @ Ys ) ) ) ) ).
% Nil_in_shufflesI
thf(fact_874_Nil__in__shuffles,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( member_list_nat @ nil_nat @ ( shuffles_nat @ Xs @ Ys ) )
= ( ( Xs = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% Nil_in_shuffles
thf(fact_875_Cons__in__shuffles__rightI,axiom,
! [Zs3: list_complex,Xs: list_complex,Ys: list_complex,Z3: complex] :
( ( member_list_complex @ Zs3 @ ( shuffles_complex @ Xs @ Ys ) )
=> ( member_list_complex @ ( cons_complex @ Z3 @ Zs3 ) @ ( shuffles_complex @ Xs @ ( cons_complex @ Z3 @ Ys ) ) ) ) ).
% Cons_in_shuffles_rightI
thf(fact_876_Cons__in__shuffles__rightI,axiom,
! [Zs3: list_nat,Xs: list_nat,Ys: list_nat,Z3: nat] :
( ( member_list_nat @ Zs3 @ ( shuffles_nat @ Xs @ Ys ) )
=> ( member_list_nat @ ( cons_nat @ Z3 @ Zs3 ) @ ( shuffles_nat @ Xs @ ( cons_nat @ Z3 @ Ys ) ) ) ) ).
% Cons_in_shuffles_rightI
thf(fact_877_Cons__in__shuffles__rightI,axiom,
! [Zs3: list_real,Xs: list_real,Ys: list_real,Z3: real] :
( ( member_list_real @ Zs3 @ ( shuffles_real @ Xs @ Ys ) )
=> ( member_list_real @ ( cons_real @ Z3 @ Zs3 ) @ ( shuffles_real @ Xs @ ( cons_real @ Z3 @ Ys ) ) ) ) ).
% Cons_in_shuffles_rightI
thf(fact_878_Cons__in__shuffles__leftI,axiom,
! [Zs3: list_complex,Xs: list_complex,Ys: list_complex,Z3: complex] :
( ( member_list_complex @ Zs3 @ ( shuffles_complex @ Xs @ Ys ) )
=> ( member_list_complex @ ( cons_complex @ Z3 @ Zs3 ) @ ( shuffles_complex @ ( cons_complex @ Z3 @ Xs ) @ Ys ) ) ) ).
% Cons_in_shuffles_leftI
thf(fact_879_Cons__in__shuffles__leftI,axiom,
! [Zs3: list_nat,Xs: list_nat,Ys: list_nat,Z3: nat] :
( ( member_list_nat @ Zs3 @ ( shuffles_nat @ Xs @ Ys ) )
=> ( member_list_nat @ ( cons_nat @ Z3 @ Zs3 ) @ ( shuffles_nat @ ( cons_nat @ Z3 @ Xs ) @ Ys ) ) ) ).
% Cons_in_shuffles_leftI
thf(fact_880_Cons__in__shuffles__leftI,axiom,
! [Zs3: list_real,Xs: list_real,Ys: list_real,Z3: real] :
( ( member_list_real @ Zs3 @ ( shuffles_real @ Xs @ Ys ) )
=> ( member_list_real @ ( cons_real @ Z3 @ Zs3 ) @ ( shuffles_real @ ( cons_real @ Z3 @ Xs ) @ Ys ) ) ) ).
% Cons_in_shuffles_leftI
thf(fact_881_minus__nat_Osimps_I1_J,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.simps(1)
thf(fact_882_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_883_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_884_diffs0__imp__equal,axiom,
! [M: nat,N2: nat] :
( ( ( minus_minus_nat @ M @ N2 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N2 @ M )
= zero_zero_nat )
=> ( M = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_885_semiring__norm_I58_J,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ zero_zero_complex )
= A ) ).
% semiring_norm(58)
thf(fact_886_semiring__norm_I58_J,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% semiring_norm(58)
thf(fact_887_verit__minus__simplify_I1_J,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% verit_minus_simplify(1)
thf(fact_888_verit__minus__simplify_I1_J,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ A )
= zero_zero_complex ) ).
% verit_minus_simplify(1)
thf(fact_889_verit__minus__simplify_I1_J,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% verit_minus_simplify(1)
thf(fact_890_verit__minus__simplify_I2_J,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_minus_simplify(2)
thf(fact_891_verit__minus__simplify_I2_J,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ zero_zero_complex )
= A ) ).
% verit_minus_simplify(2)
thf(fact_892_verit__minus__simplify_I2_J,axiom,
! [A: real] :
( ( minus_minus_real @ A @ zero_zero_real )
= A ) ).
% verit_minus_simplify(2)
thf(fact_893_diff__self,axiom,
! [A: complex] :
( ( minus_minus_complex @ A @ A )
= zero_zero_complex ) ).
% diff_self
thf(fact_894_diff__self,axiom,
! [A: real] :
( ( minus_minus_real @ A @ A )
= zero_zero_real ) ).
% diff_self
thf(fact_895_right__minus__eq,axiom,
! [A: complex,B2: complex] :
( ( ( minus_minus_complex @ A @ B2 )
= zero_zero_complex )
= ( A = B2 ) ) ).
% right_minus_eq
thf(fact_896_right__minus__eq,axiom,
! [A: real,B2: real] :
( ( ( minus_minus_real @ A @ B2 )
= zero_zero_real )
= ( A = B2 ) ) ).
% right_minus_eq
thf(fact_897_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_898_diff__numeral__special_I9_J,axiom,
( ( minus_minus_complex @ one_one_complex @ one_one_complex )
= zero_zero_complex ) ).
% diff_numeral_special(9)
thf(fact_899_diff__numeral__special_I9_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% diff_numeral_special(9)
thf(fact_900_last__ConsR,axiom,
! [Xs: list_complex,X2: complex] :
( ( Xs != nil_complex )
=> ( ( last_complex @ ( cons_complex @ X2 @ Xs ) )
= ( last_complex @ Xs ) ) ) ).
% last_ConsR
thf(fact_901_last__ConsR,axiom,
! [Xs: list_nat,X2: nat] :
( ( Xs != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X2 @ Xs ) )
= ( last_nat @ Xs ) ) ) ).
% last_ConsR
thf(fact_902_last__ConsR,axiom,
! [Xs: list_real,X2: real] :
( ( Xs != nil_real )
=> ( ( last_real @ ( cons_real @ X2 @ Xs ) )
= ( last_real @ Xs ) ) ) ).
% last_ConsR
thf(fact_903_last__ConsL,axiom,
! [Xs: list_complex,X2: complex] :
( ( Xs = nil_complex )
=> ( ( last_complex @ ( cons_complex @ X2 @ Xs ) )
= X2 ) ) ).
% last_ConsL
thf(fact_904_last__ConsL,axiom,
! [Xs: list_nat,X2: nat] :
( ( Xs = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X2 @ Xs ) )
= X2 ) ) ).
% last_ConsL
thf(fact_905_last__ConsL,axiom,
! [Xs: list_real,X2: real] :
( ( Xs = nil_real )
=> ( ( last_real @ ( cons_real @ X2 @ Xs ) )
= X2 ) ) ).
% last_ConsL
thf(fact_906_last_Osimps,axiom,
! [Xs: list_complex,X2: complex] :
( ( ( Xs = nil_complex )
=> ( ( last_complex @ ( cons_complex @ X2 @ Xs ) )
= X2 ) )
& ( ( Xs != nil_complex )
=> ( ( last_complex @ ( cons_complex @ X2 @ Xs ) )
= ( last_complex @ Xs ) ) ) ) ).
% last.simps
thf(fact_907_last_Osimps,axiom,
! [Xs: list_nat,X2: nat] :
( ( ( Xs = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X2 @ Xs ) )
= X2 ) )
& ( ( Xs != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X2 @ Xs ) )
= ( last_nat @ Xs ) ) ) ) ).
% last.simps
thf(fact_908_last_Osimps,axiom,
! [Xs: list_real,X2: real] :
( ( ( Xs = nil_real )
=> ( ( last_real @ ( cons_real @ X2 @ Xs ) )
= X2 ) )
& ( ( Xs != nil_real )
=> ( ( last_real @ ( cons_real @ X2 @ Xs ) )
= ( last_real @ Xs ) ) ) ) ).
% last.simps
thf(fact_909_last__in__set,axiom,
! [As2: list_nat] :
( ( As2 != nil_nat )
=> ( member_nat2 @ ( last_nat @ As2 ) @ ( set_nat2 @ As2 ) ) ) ).
% last_in_set
thf(fact_910_last__in__set,axiom,
! [As2: list_complex] :
( ( As2 != nil_complex )
=> ( member_complex2 @ ( last_complex @ As2 ) @ ( set_complex2 @ As2 ) ) ) ).
% last_in_set
thf(fact_911_diff__Suc__1,axiom,
! [N2: nat] :
( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
= N2 ) ).
% diff_Suc_1
thf(fact_912_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N2: nat] :
( ( minus_minus_nat @ M @ ( suc @ N2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N2 ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_913_last__map,axiom,
! [Xs: list_complex,F: complex > real] :
( ( Xs != nil_complex )
=> ( ( last_real @ ( map_complex_real @ F @ Xs ) )
= ( F @ ( last_complex @ Xs ) ) ) ) ).
% last_map
thf(fact_914_last__map,axiom,
! [Xs: list_nat,F: nat > nat] :
( ( Xs != nil_nat )
=> ( ( last_nat @ ( map_nat_nat @ F @ Xs ) )
= ( F @ ( last_nat @ Xs ) ) ) ) ).
% last_map
thf(fact_915_last__map,axiom,
! [Xs: list_nat,F: nat > complex] :
( ( Xs != nil_nat )
=> ( ( last_complex @ ( map_nat_complex @ F @ Xs ) )
= ( F @ ( last_nat @ Xs ) ) ) ) ).
% last_map
thf(fact_916_last__map,axiom,
! [Xs: list_complex,F: complex > nat] :
( ( Xs != nil_complex )
=> ( ( last_nat @ ( map_complex_nat @ F @ Xs ) )
= ( F @ ( last_complex @ Xs ) ) ) ) ).
% last_map
thf(fact_917_last__map,axiom,
! [Xs: list_complex,F: complex > complex] :
( ( Xs != nil_complex )
=> ( ( last_complex @ ( map_complex_complex @ F @ Xs ) )
= ( F @ ( last_complex @ Xs ) ) ) ) ).
% last_map
thf(fact_918_longest__common__suffix,axiom,
! [Xs: list_nat,Ys: list_nat] :
? [Ss: list_nat,Xs6: list_nat,Ys6: list_nat] :
( ( Xs
= ( append_nat @ Xs6 @ Ss ) )
& ( Ys
= ( append_nat @ Ys6 @ Ss ) )
& ( ( Xs6 = nil_nat )
| ( Ys6 = nil_nat )
| ( ( last_nat @ Xs6 )
!= ( last_nat @ Ys6 ) ) ) ) ).
% longest_common_suffix
thf(fact_919_last__appendR,axiom,
! [Ys: list_nat,Xs: list_nat] :
( ( Ys != nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
= ( last_nat @ Ys ) ) ) ).
% last_appendR
thf(fact_920_last__appendL,axiom,
! [Ys: list_nat,Xs: list_nat] :
( ( Ys = nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
= ( last_nat @ Xs ) ) ) ).
% last_appendL
thf(fact_921_last__append,axiom,
! [Ys: list_nat,Xs: list_nat] :
( ( ( Ys = nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
= ( last_nat @ Xs ) ) )
& ( ( Ys != nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
= ( last_nat @ Ys ) ) ) ) ).
% last_append
thf(fact_922_shufflesE,axiom,
! [Zs3: list_complex,Xs: list_complex,Ys: list_complex] :
( ( member_list_complex @ Zs3 @ ( shuffles_complex @ Xs @ Ys ) )
=> ( ( ( Zs3 = Xs )
=> ( Ys != nil_complex ) )
=> ( ( ( Zs3 = Ys )
=> ( Xs != nil_complex ) )
=> ( ! [X4: complex,Xs6: list_complex] :
( ( Xs
= ( cons_complex @ X4 @ Xs6 ) )
=> ! [Z: complex,Zs4: list_complex] :
( ( Zs3
= ( cons_complex @ Z @ Zs4 ) )
=> ( ( X4 = Z )
=> ~ ( member_list_complex @ Zs4 @ ( shuffles_complex @ Xs6 @ Ys ) ) ) ) )
=> ~ ! [Y3: complex,Ys6: list_complex] :
( ( Ys
= ( cons_complex @ Y3 @ Ys6 ) )
=> ! [Z: complex,Zs4: list_complex] :
( ( Zs3
= ( cons_complex @ Z @ Zs4 ) )
=> ( ( Y3 = Z )
=> ~ ( member_list_complex @ Zs4 @ ( shuffles_complex @ Xs @ Ys6 ) ) ) ) ) ) ) ) ) ).
% shufflesE
thf(fact_923_shufflesE,axiom,
! [Zs3: list_nat,Xs: list_nat,Ys: list_nat] :
( ( member_list_nat @ Zs3 @ ( shuffles_nat @ Xs @ Ys ) )
=> ( ( ( Zs3 = Xs )
=> ( Ys != nil_nat ) )
=> ( ( ( Zs3 = Ys )
=> ( Xs != nil_nat ) )
=> ( ! [X4: nat,Xs6: list_nat] :
( ( Xs
= ( cons_nat @ X4 @ Xs6 ) )
=> ! [Z: nat,Zs4: list_nat] :
( ( Zs3
= ( cons_nat @ Z @ Zs4 ) )
=> ( ( X4 = Z )
=> ~ ( member_list_nat @ Zs4 @ ( shuffles_nat @ Xs6 @ Ys ) ) ) ) )
=> ~ ! [Y3: nat,Ys6: list_nat] :
( ( Ys
= ( cons_nat @ Y3 @ Ys6 ) )
=> ! [Z: nat,Zs4: list_nat] :
( ( Zs3
= ( cons_nat @ Z @ Zs4 ) )
=> ( ( Y3 = Z )
=> ~ ( member_list_nat @ Zs4 @ ( shuffles_nat @ Xs @ Ys6 ) ) ) ) ) ) ) ) ) ).
% shufflesE
thf(fact_924_shufflesE,axiom,
! [Zs3: list_real,Xs: list_real,Ys: list_real] :
( ( member_list_real @ Zs3 @ ( shuffles_real @ Xs @ Ys ) )
=> ( ( ( Zs3 = Xs )
=> ( Ys != nil_real ) )
=> ( ( ( Zs3 = Ys )
=> ( Xs != nil_real ) )
=> ( ! [X4: real,Xs6: list_real] :
( ( Xs
= ( cons_real @ X4 @ Xs6 ) )
=> ! [Z: real,Zs4: list_real] :
( ( Zs3
= ( cons_real @ Z @ Zs4 ) )
=> ( ( X4 = Z )
=> ~ ( member_list_real @ Zs4 @ ( shuffles_real @ Xs6 @ Ys ) ) ) ) )
=> ~ ! [Y3: real,Ys6: list_real] :
( ( Ys
= ( cons_real @ Y3 @ Ys6 ) )
=> ! [Z: real,Zs4: list_real] :
( ( Zs3
= ( cons_real @ Z @ Zs4 ) )
=> ( ( Y3 = Z )
=> ~ ( member_list_real @ Zs4 @ ( shuffles_real @ Xs @ Ys6 ) ) ) ) ) ) ) ) ) ).
% shufflesE
thf(fact_925_hd__Nil__eq__last,axiom,
( ( hd_nat @ nil_nat )
= ( last_nat @ nil_nat ) ) ).
% hd_Nil_eq_last
thf(fact_926_last__tl,axiom,
! [Xs: list_nat] :
( ( ( Xs = nil_nat )
| ( ( tl_nat @ Xs )
!= nil_nat ) )
=> ( ( last_nat @ ( tl_nat @ Xs ) )
= ( last_nat @ Xs ) ) ) ).
% last_tl
thf(fact_927_dropWhile__last,axiom,
! [X2: nat,Xs: list_nat,P: nat > $o] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ~ ( P @ X2 )
=> ( ( last_nat @ ( dropWhile_nat @ P @ Xs ) )
= ( last_nat @ Xs ) ) ) ) ).
% dropWhile_last
thf(fact_928_dropWhile__last,axiom,
! [X2: complex,Xs: list_complex,P: complex > $o] :
( ( member_complex2 @ X2 @ ( set_complex2 @ Xs ) )
=> ( ~ ( P @ X2 )
=> ( ( last_complex @ ( dropWhile_complex @ P @ Xs ) )
= ( last_complex @ Xs ) ) ) ) ).
% dropWhile_last
thf(fact_929_last__snoc,axiom,
! [Xs: list_complex,X2: complex] :
( ( last_complex @ ( append_complex @ Xs @ ( cons_complex @ X2 @ nil_complex ) ) )
= X2 ) ).
% last_snoc
thf(fact_930_last__snoc,axiom,
! [Xs: list_nat,X2: nat] :
( ( last_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) )
= X2 ) ).
% last_snoc
thf(fact_931_last__snoc,axiom,
! [Xs: list_real,X2: real] :
( ( last_real @ ( append_real @ Xs @ ( cons_real @ X2 @ nil_real ) ) )
= X2 ) ).
% last_snoc
thf(fact_932_translation__subtract__diff,axiom,
! [A: complex,S: set_complex,T: set_complex] :
( ( image_1468599708987790691omplex
@ ^ [X3: complex] : ( minus_minus_complex @ X3 @ A )
@ ( minus_811609699411566653omplex @ S @ T ) )
= ( minus_811609699411566653omplex
@ ( image_1468599708987790691omplex
@ ^ [X3: complex] : ( minus_minus_complex @ X3 @ A )
@ S )
@ ( image_1468599708987790691omplex
@ ^ [X3: complex] : ( minus_minus_complex @ X3 @ A )
@ T ) ) ) ).
% translation_subtract_diff
thf(fact_933_translation__subtract__diff,axiom,
! [A: real,S: set_real,T: set_real] :
( ( image_real_real
@ ^ [X3: real] : ( minus_minus_real @ X3 @ A )
@ ( minus_minus_set_real @ S @ T ) )
= ( minus_minus_set_real
@ ( image_real_real
@ ^ [X3: real] : ( minus_minus_real @ X3 @ A )
@ S )
@ ( image_real_real
@ ^ [X3: real] : ( minus_minus_real @ X3 @ A )
@ T ) ) ) ).
% translation_subtract_diff
thf(fact_934_distinct__adj__append__iff,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( distinct_adj_nat @ ( append_nat @ Xs @ Ys ) )
= ( ( distinct_adj_nat @ Xs )
& ( distinct_adj_nat @ Ys )
& ( ( Xs = nil_nat )
| ( Ys = nil_nat )
| ( ( last_nat @ Xs )
!= ( hd_nat @ Ys ) ) ) ) ) ).
% distinct_adj_append_iff
thf(fact_935_append__butlast__last__id,axiom,
! [Xs: list_complex] :
( ( Xs != nil_complex )
=> ( ( append_complex @ ( butlast_complex @ Xs ) @ ( cons_complex @ ( last_complex @ Xs ) @ nil_complex ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_936_append__butlast__last__id,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( append_nat @ ( butlast_nat @ Xs ) @ ( cons_nat @ ( last_nat @ Xs ) @ nil_nat ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_937_append__butlast__last__id,axiom,
! [Xs: list_real] :
( ( Xs != nil_real )
=> ( ( append_real @ ( butlast_real @ Xs ) @ ( cons_real @ ( last_real @ Xs ) @ nil_real ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_938_minus__complex_Osel_I1_J,axiom,
! [X2: complex,Y: complex] :
( ( re @ ( minus_minus_complex @ X2 @ Y ) )
= ( minus_minus_real @ ( re @ X2 ) @ ( re @ Y ) ) ) ).
% minus_complex.sel(1)
thf(fact_939_DiffE,axiom,
! [C3: complex,A2: set_complex,B: set_complex] :
( ( member_complex2 @ C3 @ ( minus_811609699411566653omplex @ A2 @ B ) )
=> ~ ( ( member_complex2 @ C3 @ A2 )
=> ( member_complex2 @ C3 @ B ) ) ) ).
% DiffE
thf(fact_940_DiffE,axiom,
! [C3: nat,A2: set_nat,B: set_nat] :
( ( member_nat2 @ C3 @ ( minus_minus_set_nat @ A2 @ B ) )
=> ~ ( ( member_nat2 @ C3 @ A2 )
=> ( member_nat2 @ C3 @ B ) ) ) ).
% DiffE
thf(fact_941_DiffI,axiom,
! [C3: complex,A2: set_complex,B: set_complex] :
( ( member_complex2 @ C3 @ A2 )
=> ( ~ ( member_complex2 @ C3 @ B )
=> ( member_complex2 @ C3 @ ( minus_811609699411566653omplex @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_942_DiffI,axiom,
! [C3: nat,A2: set_nat,B: set_nat] :
( ( member_nat2 @ C3 @ A2 )
=> ( ~ ( member_nat2 @ C3 @ B )
=> ( member_nat2 @ C3 @ ( minus_minus_set_nat @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_943_DiffD1,axiom,
! [C3: complex,A2: set_complex,B: set_complex] :
( ( member_complex2 @ C3 @ ( minus_811609699411566653omplex @ A2 @ B ) )
=> ( member_complex2 @ C3 @ A2 ) ) ).
% DiffD1
thf(fact_944_DiffD1,axiom,
! [C3: nat,A2: set_nat,B: set_nat] :
( ( member_nat2 @ C3 @ ( minus_minus_set_nat @ A2 @ B ) )
=> ( member_nat2 @ C3 @ A2 ) ) ).
% DiffD1
thf(fact_945_DiffD2,axiom,
! [C3: complex,A2: set_complex,B: set_complex] :
( ( member_complex2 @ C3 @ ( minus_811609699411566653omplex @ A2 @ B ) )
=> ~ ( member_complex2 @ C3 @ B ) ) ).
% DiffD2
thf(fact_946_DiffD2,axiom,
! [C3: nat,A2: set_nat,B: set_nat] :
( ( member_nat2 @ C3 @ ( minus_minus_set_nat @ A2 @ B ) )
=> ~ ( member_nat2 @ C3 @ B ) ) ).
% DiffD2
thf(fact_947_Diff__iff,axiom,
! [C3: complex,A2: set_complex,B: set_complex] :
( ( member_complex2 @ C3 @ ( minus_811609699411566653omplex @ A2 @ B ) )
= ( ( member_complex2 @ C3 @ A2 )
& ~ ( member_complex2 @ C3 @ B ) ) ) ).
% Diff_iff
thf(fact_948_Diff__iff,axiom,
! [C3: nat,A2: set_nat,B: set_nat] :
( ( member_nat2 @ C3 @ ( minus_minus_set_nat @ A2 @ B ) )
= ( ( member_nat2 @ C3 @ A2 )
& ~ ( member_nat2 @ C3 @ B ) ) ) ).
% Diff_iff
thf(fact_949_set__diff__eq,axiom,
( minus_811609699411566653omplex
= ( ^ [A3: set_complex,B4: set_complex] :
( collect_complex
@ ^ [X3: complex] :
( ( member_complex2 @ X3 @ A3 )
& ~ ( member_complex2 @ X3 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_950_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
( collect_nat
@ ^ [X3: nat] :
( ( member_nat2 @ X3 @ A3 )
& ~ ( member_nat2 @ X3 @ B4 ) ) ) ) ) ).
% set_diff_eq
thf(fact_951_minus__set__def,axiom,
( minus_811609699411566653omplex
= ( ^ [A3: set_complex,B4: set_complex] :
( collect_complex
@ ( minus_8727706125548526216plex_o
@ ^ [X3: complex] : ( member_complex2 @ X3 @ A3 )
@ ^ [X3: complex] : ( member_complex2 @ X3 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_952_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X3: nat] : ( member_nat2 @ X3 @ A3 )
@ ^ [X3: nat] : ( member_nat2 @ X3 @ B4 ) ) ) ) ) ).
% minus_set_def
thf(fact_953_Diff__insert0,axiom,
! [X2: complex,A2: set_complex,B: set_complex] :
( ~ ( member_complex2 @ X2 @ A2 )
=> ( ( minus_811609699411566653omplex @ A2 @ ( insert_complex2 @ X2 @ B ) )
= ( minus_811609699411566653omplex @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_954_Diff__insert0,axiom,
! [X2: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat2 @ X2 @ A2 )
=> ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X2 @ B ) )
= ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_955_insert__Diff1,axiom,
! [X2: complex,B: set_complex,A2: set_complex] :
( ( member_complex2 @ X2 @ B )
=> ( ( minus_811609699411566653omplex @ ( insert_complex2 @ X2 @ A2 ) @ B )
= ( minus_811609699411566653omplex @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_956_insert__Diff1,axiom,
! [X2: nat,B: set_nat,A2: set_nat] :
( ( member_nat2 @ X2 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X2 @ A2 ) @ B )
= ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_957_insert__Diff__if,axiom,
! [X2: complex,B: set_complex,A2: set_complex] :
( ( ( member_complex2 @ X2 @ B )
=> ( ( minus_811609699411566653omplex @ ( insert_complex2 @ X2 @ A2 ) @ B )
= ( minus_811609699411566653omplex @ A2 @ B ) ) )
& ( ~ ( member_complex2 @ X2 @ B )
=> ( ( minus_811609699411566653omplex @ ( insert_complex2 @ X2 @ A2 ) @ B )
= ( insert_complex2 @ X2 @ ( minus_811609699411566653omplex @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_958_insert__Diff__if,axiom,
! [X2: nat,B: set_nat,A2: set_nat] :
( ( ( member_nat2 @ X2 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X2 @ A2 ) @ B )
= ( minus_minus_set_nat @ A2 @ B ) ) )
& ( ~ ( member_nat2 @ X2 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X2 @ A2 ) @ B )
= ( insert_nat2 @ X2 @ ( minus_minus_set_nat @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_959_map__butlast,axiom,
! [F: complex > real,Xs: list_complex] :
( ( map_complex_real @ F @ ( butlast_complex @ Xs ) )
= ( butlast_real @ ( map_complex_real @ F @ Xs ) ) ) ).
% map_butlast
thf(fact_960_map__butlast,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( map_nat_nat @ F @ ( butlast_nat @ Xs ) )
= ( butlast_nat @ ( map_nat_nat @ F @ Xs ) ) ) ).
% map_butlast
thf(fact_961_map__butlast,axiom,
! [F: nat > complex,Xs: list_nat] :
( ( map_nat_complex @ F @ ( butlast_nat @ Xs ) )
= ( butlast_complex @ ( map_nat_complex @ F @ Xs ) ) ) ).
% map_butlast
thf(fact_962_map__butlast,axiom,
! [F: complex > nat,Xs: list_complex] :
( ( map_complex_nat @ F @ ( butlast_complex @ Xs ) )
= ( butlast_nat @ ( map_complex_nat @ F @ Xs ) ) ) ).
% map_butlast
thf(fact_963_map__butlast,axiom,
! [F: complex > complex,Xs: list_complex] :
( ( map_complex_complex @ F @ ( butlast_complex @ Xs ) )
= ( butlast_complex @ ( map_complex_complex @ F @ Xs ) ) ) ).
% map_butlast
thf(fact_964_in__set__butlastD,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ ( butlast_nat @ Xs ) ) )
=> ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_965_in__set__butlastD,axiom,
! [X2: complex,Xs: list_complex] :
( ( member_complex2 @ X2 @ ( set_complex2 @ ( butlast_complex @ Xs ) ) )
=> ( member_complex2 @ X2 @ ( set_complex2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_966_butlast_Osimps_I1_J,axiom,
( ( butlast_nat @ nil_nat )
= nil_nat ) ).
% butlast.simps(1)
thf(fact_967_distinct__adj__Cons__Cons,axiom,
! [X2: complex,Y: complex,Xs: list_complex] :
( ( distinct_adj_complex @ ( cons_complex @ X2 @ ( cons_complex @ Y @ Xs ) ) )
= ( ( X2 != Y )
& ( distinct_adj_complex @ ( cons_complex @ Y @ Xs ) ) ) ) ).
% distinct_adj_Cons_Cons
thf(fact_968_distinct__adj__Cons__Cons,axiom,
! [X2: nat,Y: nat,Xs: list_nat] :
( ( distinct_adj_nat @ ( cons_nat @ X2 @ ( cons_nat @ Y @ Xs ) ) )
= ( ( X2 != Y )
& ( distinct_adj_nat @ ( cons_nat @ Y @ Xs ) ) ) ) ).
% distinct_adj_Cons_Cons
thf(fact_969_distinct__adj__Cons__Cons,axiom,
! [X2: real,Y: real,Xs: list_real] :
( ( distinct_adj_real @ ( cons_real @ X2 @ ( cons_real @ Y @ Xs ) ) )
= ( ( X2 != Y )
& ( distinct_adj_real @ ( cons_real @ Y @ Xs ) ) ) ) ).
% distinct_adj_Cons_Cons
thf(fact_970_distinct__adj__ConsD,axiom,
! [X2: complex,Xs: list_complex] :
( ( distinct_adj_complex @ ( cons_complex @ X2 @ Xs ) )
=> ( distinct_adj_complex @ Xs ) ) ).
% distinct_adj_ConsD
thf(fact_971_distinct__adj__ConsD,axiom,
! [X2: nat,Xs: list_nat] :
( ( distinct_adj_nat @ ( cons_nat @ X2 @ Xs ) )
=> ( distinct_adj_nat @ Xs ) ) ).
% distinct_adj_ConsD
thf(fact_972_distinct__adj__ConsD,axiom,
! [X2: real,Xs: list_real] :
( ( distinct_adj_real @ ( cons_real @ X2 @ Xs ) )
=> ( distinct_adj_real @ Xs ) ) ).
% distinct_adj_ConsD
thf(fact_973_distinct__adj__Nil,axiom,
distinct_adj_nat @ nil_nat ).
% distinct_adj_Nil
thf(fact_974_distinct__adj__mapD,axiom,
! [F: complex > real,Xs: list_complex] :
( ( distinct_adj_real @ ( map_complex_real @ F @ Xs ) )
=> ( distinct_adj_complex @ Xs ) ) ).
% distinct_adj_mapD
thf(fact_975_distinct__adj__mapD,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( distinct_adj_nat @ ( map_nat_nat @ F @ Xs ) )
=> ( distinct_adj_nat @ Xs ) ) ).
% distinct_adj_mapD
thf(fact_976_distinct__adj__mapD,axiom,
! [F: nat > complex,Xs: list_nat] :
( ( distinct_adj_complex @ ( map_nat_complex @ F @ Xs ) )
=> ( distinct_adj_nat @ Xs ) ) ).
% distinct_adj_mapD
thf(fact_977_distinct__adj__mapD,axiom,
! [F: complex > nat,Xs: list_complex] :
( ( distinct_adj_nat @ ( map_complex_nat @ F @ Xs ) )
=> ( distinct_adj_complex @ Xs ) ) ).
% distinct_adj_mapD
thf(fact_978_distinct__adj__mapD,axiom,
! [F: complex > complex,Xs: list_complex] :
( ( distinct_adj_complex @ ( map_complex_complex @ F @ Xs ) )
=> ( distinct_adj_complex @ Xs ) ) ).
% distinct_adj_mapD
thf(fact_979_butlast_Osimps_I2_J,axiom,
! [Xs: list_complex,X2: complex] :
( ( ( Xs = nil_complex )
=> ( ( butlast_complex @ ( cons_complex @ X2 @ Xs ) )
= nil_complex ) )
& ( ( Xs != nil_complex )
=> ( ( butlast_complex @ ( cons_complex @ X2 @ Xs ) )
= ( cons_complex @ X2 @ ( butlast_complex @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_980_butlast_Osimps_I2_J,axiom,
! [Xs: list_nat,X2: nat] :
( ( ( Xs = nil_nat )
=> ( ( butlast_nat @ ( cons_nat @ X2 @ Xs ) )
= nil_nat ) )
& ( ( Xs != nil_nat )
=> ( ( butlast_nat @ ( cons_nat @ X2 @ Xs ) )
= ( cons_nat @ X2 @ ( butlast_nat @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_981_butlast_Osimps_I2_J,axiom,
! [Xs: list_real,X2: real] :
( ( ( Xs = nil_real )
=> ( ( butlast_real @ ( cons_real @ X2 @ Xs ) )
= nil_real ) )
& ( ( Xs != nil_real )
=> ( ( butlast_real @ ( cons_real @ X2 @ Xs ) )
= ( cons_real @ X2 @ ( butlast_real @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_982_butlast__append,axiom,
! [Ys: list_nat,Xs: list_nat] :
( ( ( Ys = nil_nat )
=> ( ( butlast_nat @ ( append_nat @ Xs @ Ys ) )
= ( butlast_nat @ Xs ) ) )
& ( ( Ys != nil_nat )
=> ( ( butlast_nat @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ Xs @ ( butlast_nat @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_983_in__set__butlast__appendI,axiom,
! [X2: nat,Xs: list_nat,Ys: list_nat] :
( ( ( member_nat2 @ X2 @ ( set_nat2 @ ( butlast_nat @ Xs ) ) )
| ( member_nat2 @ X2 @ ( set_nat2 @ ( butlast_nat @ Ys ) ) ) )
=> ( member_nat2 @ X2 @ ( set_nat2 @ ( butlast_nat @ ( append_nat @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_984_in__set__butlast__appendI,axiom,
! [X2: complex,Xs: list_complex,Ys: list_complex] :
( ( ( member_complex2 @ X2 @ ( set_complex2 @ ( butlast_complex @ Xs ) ) )
| ( member_complex2 @ X2 @ ( set_complex2 @ ( butlast_complex @ Ys ) ) ) )
=> ( member_complex2 @ X2 @ ( set_complex2 @ ( butlast_complex @ ( append_complex @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_985_distinct__adj__singleton,axiom,
! [X2: complex] : ( distinct_adj_complex @ ( cons_complex @ X2 @ nil_complex ) ) ).
% distinct_adj_singleton
thf(fact_986_distinct__adj__singleton,axiom,
! [X2: nat] : ( distinct_adj_nat @ ( cons_nat @ X2 @ nil_nat ) ) ).
% distinct_adj_singleton
thf(fact_987_distinct__adj__singleton,axiom,
! [X2: real] : ( distinct_adj_real @ ( cons_real @ X2 @ nil_real ) ) ).
% distinct_adj_singleton
thf(fact_988_butlast__snoc,axiom,
! [Xs: list_complex,X2: complex] :
( ( butlast_complex @ ( append_complex @ Xs @ ( cons_complex @ X2 @ nil_complex ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_989_butlast__snoc,axiom,
! [Xs: list_nat,X2: nat] :
( ( butlast_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_990_butlast__snoc,axiom,
! [Xs: list_real,X2: real] :
( ( butlast_real @ ( append_real @ Xs @ ( cons_real @ X2 @ nil_real ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_991_distinct__adj__Cons,axiom,
! [X2: complex,Xs: list_complex] :
( ( distinct_adj_complex @ ( cons_complex @ X2 @ Xs ) )
= ( ( Xs = nil_complex )
| ( ( X2
!= ( hd_complex @ Xs ) )
& ( distinct_adj_complex @ Xs ) ) ) ) ).
% distinct_adj_Cons
thf(fact_992_distinct__adj__Cons,axiom,
! [X2: nat,Xs: list_nat] :
( ( distinct_adj_nat @ ( cons_nat @ X2 @ Xs ) )
= ( ( Xs = nil_nat )
| ( ( X2
!= ( hd_nat @ Xs ) )
& ( distinct_adj_nat @ Xs ) ) ) ) ).
% distinct_adj_Cons
thf(fact_993_distinct__adj__Cons,axiom,
! [X2: real,Xs: list_real] :
( ( distinct_adj_real @ ( cons_real @ X2 @ Xs ) )
= ( ( Xs = nil_real )
| ( ( X2
!= ( hd_real @ Xs ) )
& ( distinct_adj_real @ Xs ) ) ) ) ).
% distinct_adj_Cons
thf(fact_994_snoc__eq__iff__butlast,axiom,
! [Xs: list_complex,X2: complex,Ys: list_complex] :
( ( ( append_complex @ Xs @ ( cons_complex @ X2 @ nil_complex ) )
= Ys )
= ( ( Ys != nil_complex )
& ( ( butlast_complex @ Ys )
= Xs )
& ( ( last_complex @ Ys )
= X2 ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_995_snoc__eq__iff__butlast,axiom,
! [Xs: list_nat,X2: nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) )
= Ys )
= ( ( Ys != nil_nat )
& ( ( butlast_nat @ Ys )
= Xs )
& ( ( last_nat @ Ys )
= X2 ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_996_snoc__eq__iff__butlast,axiom,
! [Xs: list_real,X2: real,Ys: list_real] :
( ( ( append_real @ Xs @ ( cons_real @ X2 @ nil_real ) )
= Ys )
= ( ( Ys != nil_real )
& ( ( butlast_real @ Ys )
= Xs )
& ( ( last_real @ Ys )
= X2 ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_997_minus__coset__filter,axiom,
! [A2: set_nat,Xs: list_nat] :
( ( minus_minus_set_nat @ A2 @ ( coset_nat @ Xs ) )
= ( set_nat2
@ ( filter_nat
@ ^ [X3: nat] : ( member_nat2 @ X3 @ A2 )
@ Xs ) ) ) ).
% minus_coset_filter
thf(fact_998_minus__coset__filter,axiom,
! [A2: set_complex,Xs: list_complex] :
( ( minus_811609699411566653omplex @ A2 @ ( coset_complex @ Xs ) )
= ( set_complex2
@ ( filter_complex
@ ^ [X3: complex] : ( member_complex2 @ X3 @ A2 )
@ Xs ) ) ) ).
% minus_coset_filter
thf(fact_999_set__minus__filter__out,axiom,
! [Xs: list_complex,Y: complex] :
( ( minus_811609699411566653omplex @ ( set_complex2 @ Xs ) @ ( insert_complex2 @ Y @ bot_bot_set_complex ) )
= ( set_complex2
@ ( filter_complex
@ ^ [X3: complex] : ( X3 != Y )
@ Xs ) ) ) ).
% set_minus_filter_out
thf(fact_1000_successively__append__iff,axiom,
! [P: nat > nat > $o,Xs: list_nat,Ys: list_nat] :
( ( successively_nat @ P @ ( append_nat @ Xs @ Ys ) )
= ( ( successively_nat @ P @ Xs )
& ( successively_nat @ P @ Ys )
& ( ( Xs = nil_nat )
| ( Ys = nil_nat )
| ( P @ ( last_nat @ Xs ) @ ( hd_nat @ Ys ) ) ) ) ) ).
% successively_append_iff
thf(fact_1001_image__is__empty,axiom,
! [F: nat > nat,A2: set_nat] :
( ( ( image_nat_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_1002_empty__is__image,axiom,
! [F: nat > nat,A2: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_1003_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_1004_singletonD,axiom,
! [B2: complex,A: complex] :
( ( member_complex2 @ B2 @ ( insert_complex2 @ A @ bot_bot_set_complex ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_1005_singletonD,axiom,
! [B2: nat,A: nat] :
( ( member_nat2 @ B2 @ ( insert_nat2 @ A @ bot_bot_set_nat ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_1006_singletonI,axiom,
! [A: complex] : ( member_complex2 @ A @ ( insert_complex2 @ A @ bot_bot_set_complex ) ) ).
% singletonI
thf(fact_1007_singletonI,axiom,
! [A: nat] : ( member_nat2 @ A @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_1008_singleton__iff,axiom,
! [B2: complex,A: complex] :
( ( member_complex2 @ B2 @ ( insert_complex2 @ A @ bot_bot_set_complex ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_1009_singleton__iff,axiom,
! [B2: nat,A: nat] :
( ( member_nat2 @ B2 @ ( insert_nat2 @ A @ bot_bot_set_nat ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_1010_Set_Oempty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X3: nat] : $false ) ) ).
% Set.empty_def
thf(fact_1011_emptyE,axiom,
! [A: complex] :
~ ( member_complex2 @ A @ bot_bot_set_complex ) ).
% emptyE
thf(fact_1012_emptyE,axiom,
! [A: nat] :
~ ( member_nat2 @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_1013_equals0D,axiom,
! [A2: set_complex,A: complex] :
( ( A2 = bot_bot_set_complex )
=> ~ ( member_complex2 @ A @ A2 ) ) ).
% equals0D
thf(fact_1014_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat2 @ A @ A2 ) ) ).
% equals0D
thf(fact_1015_equals0I,axiom,
! [A2: set_complex] :
( ! [Y3: complex] :
~ ( member_complex2 @ Y3 @ A2 )
=> ( A2 = bot_bot_set_complex ) ) ).
% equals0I
thf(fact_1016_equals0I,axiom,
! [A2: set_nat] :
( ! [Y3: nat] :
~ ( member_nat2 @ Y3 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_1017_empty__iff,axiom,
! [C3: complex] :
~ ( member_complex2 @ C3 @ bot_bot_set_complex ) ).
% empty_iff
thf(fact_1018_empty__iff,axiom,
! [C3: nat] :
~ ( member_nat2 @ C3 @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_1019_ex__in__conv,axiom,
! [A2: set_complex] :
( ( ? [X3: complex] : ( member_complex2 @ X3 @ A2 ) )
= ( A2 != bot_bot_set_complex ) ) ).
% ex_in_conv
thf(fact_1020_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X3: nat] : ( member_nat2 @ X3 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_1021_all__not__in__conv,axiom,
! [A2: set_complex] :
( ( ! [X3: complex] :
~ ( member_complex2 @ X3 @ A2 ) )
= ( A2 = bot_bot_set_complex ) ) ).
% all_not_in_conv
thf(fact_1022_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat2 @ X3 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_1023_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_1024_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_1025_successively__cong,axiom,
! [Xs: list_nat,P: nat > nat > $o,Q: nat > nat > $o,Ys: list_nat] :
( ! [X4: nat,Y3: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Xs ) )
=> ( ( member_nat2 @ Y3 @ ( set_nat2 @ Xs ) )
=> ( ( P @ X4 @ Y3 )
= ( Q @ X4 @ Y3 ) ) ) )
=> ( ( Xs = Ys )
=> ( ( successively_nat @ P @ Xs )
= ( successively_nat @ Q @ Ys ) ) ) ) ).
% successively_cong
thf(fact_1026_successively__cong,axiom,
! [Xs: list_complex,P: complex > complex > $o,Q: complex > complex > $o,Ys: list_complex] :
( ! [X4: complex,Y3: complex] :
( ( member_complex2 @ X4 @ ( set_complex2 @ Xs ) )
=> ( ( member_complex2 @ Y3 @ ( set_complex2 @ Xs ) )
=> ( ( P @ X4 @ Y3 )
= ( Q @ X4 @ Y3 ) ) ) )
=> ( ( Xs = Ys )
=> ( ( successively_complex @ P @ Xs )
= ( successively_complex @ Q @ Ys ) ) ) ) ).
% successively_cong
thf(fact_1027_successively__mono,axiom,
! [P: nat > nat > $o,Xs: list_nat,Q: nat > nat > $o] :
( ( successively_nat @ P @ Xs )
=> ( ! [X4: nat,Y3: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Xs ) )
=> ( ( member_nat2 @ Y3 @ ( set_nat2 @ Xs ) )
=> ( ( P @ X4 @ Y3 )
=> ( Q @ X4 @ Y3 ) ) ) )
=> ( successively_nat @ Q @ Xs ) ) ) ).
% successively_mono
thf(fact_1028_successively__mono,axiom,
! [P: complex > complex > $o,Xs: list_complex,Q: complex > complex > $o] :
( ( successively_complex @ P @ Xs )
=> ( ! [X4: complex,Y3: complex] :
( ( member_complex2 @ X4 @ ( set_complex2 @ Xs ) )
=> ( ( member_complex2 @ Y3 @ ( set_complex2 @ Xs ) )
=> ( ( P @ X4 @ Y3 )
=> ( Q @ X4 @ Y3 ) ) ) )
=> ( successively_complex @ Q @ Xs ) ) ) ).
% successively_mono
thf(fact_1029_successively_Osimps_I1_J,axiom,
! [P: nat > nat > $o] : ( successively_nat @ P @ nil_nat ) ).
% successively.simps(1)
thf(fact_1030_successively_Osimps_I3_J,axiom,
! [P: complex > complex > $o,X2: complex,Y: complex,Xs: list_complex] :
( ( successively_complex @ P @ ( cons_complex @ X2 @ ( cons_complex @ Y @ Xs ) ) )
= ( ( P @ X2 @ Y )
& ( successively_complex @ P @ ( cons_complex @ Y @ Xs ) ) ) ) ).
% successively.simps(3)
thf(fact_1031_successively_Osimps_I3_J,axiom,
! [P: nat > nat > $o,X2: nat,Y: nat,Xs: list_nat] :
( ( successively_nat @ P @ ( cons_nat @ X2 @ ( cons_nat @ Y @ Xs ) ) )
= ( ( P @ X2 @ Y )
& ( successively_nat @ P @ ( cons_nat @ Y @ Xs ) ) ) ) ).
% successively.simps(3)
thf(fact_1032_successively_Osimps_I3_J,axiom,
! [P: real > real > $o,X2: real,Y: real,Xs: list_real] :
( ( successively_real @ P @ ( cons_real @ X2 @ ( cons_real @ Y @ Xs ) ) )
= ( ( P @ X2 @ Y )
& ( successively_real @ P @ ( cons_real @ Y @ Xs ) ) ) ) ).
% successively.simps(3)
thf(fact_1033_successively_Oelims_I3_J,axiom,
! [X2: complex > complex > $o,Xa: list_complex] :
( ~ ( successively_complex @ X2 @ Xa )
=> ~ ! [X4: complex,Y3: complex,Xs3: list_complex] :
( ( Xa
= ( cons_complex @ X4 @ ( cons_complex @ Y3 @ Xs3 ) ) )
=> ( ( X2 @ X4 @ Y3 )
& ( successively_complex @ X2 @ ( cons_complex @ Y3 @ Xs3 ) ) ) ) ) ).
% successively.elims(3)
thf(fact_1034_successively_Oelims_I3_J,axiom,
! [X2: nat > nat > $o,Xa: list_nat] :
( ~ ( successively_nat @ X2 @ Xa )
=> ~ ! [X4: nat,Y3: nat,Xs3: list_nat] :
( ( Xa
= ( cons_nat @ X4 @ ( cons_nat @ Y3 @ Xs3 ) ) )
=> ( ( X2 @ X4 @ Y3 )
& ( successively_nat @ X2 @ ( cons_nat @ Y3 @ Xs3 ) ) ) ) ) ).
% successively.elims(3)
thf(fact_1035_successively_Oelims_I3_J,axiom,
! [X2: real > real > $o,Xa: list_real] :
( ~ ( successively_real @ X2 @ Xa )
=> ~ ! [X4: real,Y3: real,Xs3: list_real] :
( ( Xa
= ( cons_real @ X4 @ ( cons_real @ Y3 @ Xs3 ) ) )
=> ( ( X2 @ X4 @ Y3 )
& ( successively_real @ X2 @ ( cons_real @ Y3 @ Xs3 ) ) ) ) ) ).
% successively.elims(3)
thf(fact_1036_map__eq__imp__length__eq,axiom,
! [F: complex > real,Xs: list_complex,G: complex > real,Ys: list_complex] :
( ( ( map_complex_real @ F @ Xs )
= ( map_complex_real @ G @ Ys ) )
=> ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1037_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1038_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs: list_nat,G: complex > nat,Ys: list_complex] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_complex_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1039_map__eq__imp__length__eq,axiom,
! [F: nat > complex,Xs: list_nat,G: nat > complex,Ys: list_nat] :
( ( ( map_nat_complex @ F @ Xs )
= ( map_nat_complex @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1040_map__eq__imp__length__eq,axiom,
! [F: nat > complex,Xs: list_nat,G: complex > complex,Ys: list_complex] :
( ( ( map_nat_complex @ F @ Xs )
= ( map_complex_complex @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1041_map__eq__imp__length__eq,axiom,
! [F: complex > nat,Xs: list_complex,G: nat > nat,Ys: list_nat] :
( ( ( map_complex_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) )
=> ( ( size_s3451745648224563538omplex @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1042_map__eq__imp__length__eq,axiom,
! [F: complex > nat,Xs: list_complex,G: complex > nat,Ys: list_complex] :
( ( ( map_complex_nat @ F @ Xs )
= ( map_complex_nat @ G @ Ys ) )
=> ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1043_map__eq__imp__length__eq,axiom,
! [F: complex > complex,Xs: list_complex,G: nat > complex,Ys: list_nat] :
( ( ( map_complex_complex @ F @ Xs )
= ( map_nat_complex @ G @ Ys ) )
=> ( ( size_s3451745648224563538omplex @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1044_map__eq__imp__length__eq,axiom,
! [F: complex > complex,Xs: list_complex,G: complex > complex,Ys: list_complex] :
( ( ( map_complex_complex @ F @ Xs )
= ( map_complex_complex @ G @ Ys ) )
=> ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1045_length__map,axiom,
! [F: complex > real,Xs: list_complex] :
( ( size_size_list_real @ ( map_complex_real @ F @ Xs ) )
= ( size_s3451745648224563538omplex @ Xs ) ) ).
% length_map
thf(fact_1046_length__map,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( size_size_list_nat @ ( map_nat_nat @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_1047_length__map,axiom,
! [F: nat > complex,Xs: list_nat] :
( ( size_s3451745648224563538omplex @ ( map_nat_complex @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_1048_length__map,axiom,
! [F: complex > nat,Xs: list_complex] :
( ( size_size_list_nat @ ( map_complex_nat @ F @ Xs ) )
= ( size_s3451745648224563538omplex @ Xs ) ) ).
% length_map
thf(fact_1049_length__map,axiom,
! [F: complex > complex,Xs: list_complex] :
( ( size_s3451745648224563538omplex @ ( map_complex_complex @ F @ Xs ) )
= ( size_s3451745648224563538omplex @ Xs ) ) ).
% length_map
thf(fact_1050_successively__map,axiom,
! [P: real > real > $o,F: complex > real,Xs: list_complex] :
( ( successively_real @ P @ ( map_complex_real @ F @ Xs ) )
= ( successively_complex
@ ^ [X3: complex,Y2: complex] : ( P @ ( F @ X3 ) @ ( F @ Y2 ) )
@ Xs ) ) ).
% successively_map
thf(fact_1051_successively__map,axiom,
! [P: nat > nat > $o,F: nat > nat,Xs: list_nat] :
( ( successively_nat @ P @ ( map_nat_nat @ F @ Xs ) )
= ( successively_nat
@ ^ [X3: nat,Y2: nat] : ( P @ ( F @ X3 ) @ ( F @ Y2 ) )
@ Xs ) ) ).
% successively_map
thf(fact_1052_successively__map,axiom,
! [P: complex > complex > $o,F: nat > complex,Xs: list_nat] :
( ( successively_complex @ P @ ( map_nat_complex @ F @ Xs ) )
= ( successively_nat
@ ^ [X3: nat,Y2: nat] : ( P @ ( F @ X3 ) @ ( F @ Y2 ) )
@ Xs ) ) ).
% successively_map
thf(fact_1053_successively__map,axiom,
! [P: nat > nat > $o,F: complex > nat,Xs: list_complex] :
( ( successively_nat @ P @ ( map_complex_nat @ F @ Xs ) )
= ( successively_complex
@ ^ [X3: complex,Y2: complex] : ( P @ ( F @ X3 ) @ ( F @ Y2 ) )
@ Xs ) ) ).
% successively_map
thf(fact_1054_successively__map,axiom,
! [P: complex > complex > $o,F: complex > complex,Xs: list_complex] :
( ( successively_complex @ P @ ( map_complex_complex @ F @ Xs ) )
= ( successively_complex
@ ^ [X3: complex,Y2: complex] : ( P @ ( F @ X3 ) @ ( F @ Y2 ) )
@ Xs ) ) ).
% successively_map
thf(fact_1055_Collect__conv__if2,axiom,
! [P: nat > $o,A: nat] :
( ( ( P @ A )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( A = X3 )
& ( P @ X3 ) ) )
= ( insert_nat2 @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( A = X3 )
& ( P @ X3 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_1056_singleton__conv2,axiom,
! [A: nat] :
( ( collect_nat
@ ( ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
@ A ) )
= ( insert_nat2 @ A @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_1057_Collect__conv__if,axiom,
! [P: nat > $o,A: nat] :
( ( ( P @ A )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( X3 = A )
& ( P @ X3 ) ) )
= ( insert_nat2 @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_nat
@ ^ [X3: nat] :
( ( X3 = A )
& ( P @ X3 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_1058_singleton__conv,axiom,
! [A: nat] :
( ( collect_nat
@ ^ [X3: nat] : ( X3 = A ) )
= ( insert_nat2 @ A @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_1059_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_1060_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_1061_Suc__length__conv,axiom,
! [N2: nat,Xs: list_complex] :
( ( ( suc @ N2 )
= ( size_s3451745648224563538omplex @ Xs ) )
= ( ? [Y2: complex,Ys3: list_complex] :
( ( Xs
= ( cons_complex @ Y2 @ Ys3 ) )
& ( ( size_s3451745648224563538omplex @ Ys3 )
= N2 ) ) ) ) ).
% Suc_length_conv
thf(fact_1062_Suc__length__conv,axiom,
! [N2: nat,Xs: list_nat] :
( ( ( suc @ N2 )
= ( size_size_list_nat @ Xs ) )
= ( ? [Y2: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y2 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N2 ) ) ) ) ).
% Suc_length_conv
thf(fact_1063_Suc__length__conv,axiom,
! [N2: nat,Xs: list_real] :
( ( ( suc @ N2 )
= ( size_size_list_real @ Xs ) )
= ( ? [Y2: real,Ys3: list_real] :
( ( Xs
= ( cons_real @ Y2 @ Ys3 ) )
& ( ( size_size_list_real @ Ys3 )
= N2 ) ) ) ) ).
% Suc_length_conv
thf(fact_1064_length__Suc__conv,axiom,
! [Xs: list_complex,N2: nat] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( suc @ N2 ) )
= ( ? [Y2: complex,Ys3: list_complex] :
( ( Xs
= ( cons_complex @ Y2 @ Ys3 ) )
& ( ( size_s3451745648224563538omplex @ Ys3 )
= N2 ) ) ) ) ).
% length_Suc_conv
thf(fact_1065_length__Suc__conv,axiom,
! [Xs: list_nat,N2: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N2 ) )
= ( ? [Y2: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y2 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N2 ) ) ) ) ).
% length_Suc_conv
thf(fact_1066_length__Suc__conv,axiom,
! [Xs: list_real,N2: nat] :
( ( ( size_size_list_real @ Xs )
= ( suc @ N2 ) )
= ( ? [Y2: real,Ys3: list_real] :
( ( Xs
= ( cons_real @ Y2 @ Ys3 ) )
& ( ( size_size_list_real @ Ys3 )
= N2 ) ) ) ) ).
% length_Suc_conv
thf(fact_1067_length__Cons,axiom,
! [X2: complex,Xs: list_complex] :
( ( size_s3451745648224563538omplex @ ( cons_complex @ X2 @ Xs ) )
= ( suc @ ( size_s3451745648224563538omplex @ Xs ) ) ) ).
% length_Cons
thf(fact_1068_length__Cons,axiom,
! [X2: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X2 @ Xs ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_Cons
thf(fact_1069_length__Cons,axiom,
! [X2: real,Xs: list_real] :
( ( size_size_list_real @ ( cons_real @ X2 @ Xs ) )
= ( suc @ ( size_size_list_real @ Xs ) ) ) ).
% length_Cons
thf(fact_1070_list__induct4,axiom,
! [Xs: list_complex,Ys: list_complex,Zs3: list_complex,Ws: list_complex,P: list_complex > list_complex > list_complex > list_complex > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys )
= ( size_s3451745648224563538omplex @ Zs3 ) )
=> ( ( ( size_s3451745648224563538omplex @ Zs3 )
= ( size_s3451745648224563538omplex @ Ws ) )
=> ( ( P @ nil_complex @ nil_complex @ nil_complex @ nil_complex )
=> ( ! [X4: complex,Xs3: list_complex,Y3: complex,Ys2: list_complex,Z: complex,Zs: list_complex,W: complex,Ws2: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= ( size_s3451745648224563538omplex @ Ys2 ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys2 )
= ( size_s3451745648224563538omplex @ Zs ) )
=> ( ( ( size_s3451745648224563538omplex @ Zs )
= ( size_s3451745648224563538omplex @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys2 @ Zs @ Ws2 )
=> ( P @ ( cons_complex @ X4 @ Xs3 ) @ ( cons_complex @ Y3 @ Ys2 ) @ ( cons_complex @ Z @ Zs ) @ ( cons_complex @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1071_list__induct4,axiom,
! [Xs: list_complex,Ys: list_complex,Zs3: list_complex,Ws: list_nat,P: list_complex > list_complex > list_complex > list_nat > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys )
= ( size_s3451745648224563538omplex @ Zs3 ) )
=> ( ( ( size_s3451745648224563538omplex @ Zs3 )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_complex @ nil_complex @ nil_complex @ nil_nat )
=> ( ! [X4: complex,Xs3: list_complex,Y3: complex,Ys2: list_complex,Z: complex,Zs: list_complex,W: nat,Ws2: list_nat] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= ( size_s3451745648224563538omplex @ Ys2 ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys2 )
= ( size_s3451745648224563538omplex @ Zs ) )
=> ( ( ( size_s3451745648224563538omplex @ Zs )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys2 @ Zs @ Ws2 )
=> ( P @ ( cons_complex @ X4 @ Xs3 ) @ ( cons_complex @ Y3 @ Ys2 ) @ ( cons_complex @ Z @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1072_list__induct4,axiom,
! [Xs: list_complex,Ys: list_complex,Zs3: list_complex,Ws: list_real,P: list_complex > list_complex > list_complex > list_real > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys )
= ( size_s3451745648224563538omplex @ Zs3 ) )
=> ( ( ( size_s3451745648224563538omplex @ Zs3 )
= ( size_size_list_real @ Ws ) )
=> ( ( P @ nil_complex @ nil_complex @ nil_complex @ nil_real )
=> ( ! [X4: complex,Xs3: list_complex,Y3: complex,Ys2: list_complex,Z: complex,Zs: list_complex,W: real,Ws2: list_real] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= ( size_s3451745648224563538omplex @ Ys2 ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys2 )
= ( size_s3451745648224563538omplex @ Zs ) )
=> ( ( ( size_s3451745648224563538omplex @ Zs )
= ( size_size_list_real @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys2 @ Zs @ Ws2 )
=> ( P @ ( cons_complex @ X4 @ Xs3 ) @ ( cons_complex @ Y3 @ Ys2 ) @ ( cons_complex @ Z @ Zs ) @ ( cons_real @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1073_list__induct4,axiom,
! [Xs: list_complex,Ys: list_complex,Zs3: list_nat,Ws: list_complex,P: list_complex > list_complex > list_nat > list_complex > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_s3451745648224563538omplex @ Ws ) )
=> ( ( P @ nil_complex @ nil_complex @ nil_nat @ nil_complex )
=> ( ! [X4: complex,Xs3: list_complex,Y3: complex,Ys2: list_complex,Z: nat,Zs: list_nat,W: complex,Ws2: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= ( size_s3451745648224563538omplex @ Ys2 ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_s3451745648224563538omplex @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys2 @ Zs @ Ws2 )
=> ( P @ ( cons_complex @ X4 @ Xs3 ) @ ( cons_complex @ Y3 @ Ys2 ) @ ( cons_nat @ Z @ Zs ) @ ( cons_complex @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1074_list__induct4,axiom,
! [Xs: list_complex,Ys: list_complex,Zs3: list_nat,Ws: list_nat,P: list_complex > list_complex > list_nat > list_nat > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_complex @ nil_complex @ nil_nat @ nil_nat )
=> ( ! [X4: complex,Xs3: list_complex,Y3: complex,Ys2: list_complex,Z: nat,Zs: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= ( size_s3451745648224563538omplex @ Ys2 ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys2 @ Zs @ Ws2 )
=> ( P @ ( cons_complex @ X4 @ Xs3 ) @ ( cons_complex @ Y3 @ Ys2 ) @ ( cons_nat @ Z @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1075_list__induct4,axiom,
! [Xs: list_complex,Ys: list_complex,Zs3: list_nat,Ws: list_real,P: list_complex > list_complex > list_nat > list_real > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_size_list_real @ Ws ) )
=> ( ( P @ nil_complex @ nil_complex @ nil_nat @ nil_real )
=> ( ! [X4: complex,Xs3: list_complex,Y3: complex,Ys2: list_complex,Z: nat,Zs: list_nat,W: real,Ws2: list_real] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= ( size_s3451745648224563538omplex @ Ys2 ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_real @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys2 @ Zs @ Ws2 )
=> ( P @ ( cons_complex @ X4 @ Xs3 ) @ ( cons_complex @ Y3 @ Ys2 ) @ ( cons_nat @ Z @ Zs ) @ ( cons_real @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1076_list__induct4,axiom,
! [Xs: list_complex,Ys: list_complex,Zs3: list_real,Ws: list_complex,P: list_complex > list_complex > list_real > list_complex > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys )
= ( size_size_list_real @ Zs3 ) )
=> ( ( ( size_size_list_real @ Zs3 )
= ( size_s3451745648224563538omplex @ Ws ) )
=> ( ( P @ nil_complex @ nil_complex @ nil_real @ nil_complex )
=> ( ! [X4: complex,Xs3: list_complex,Y3: complex,Ys2: list_complex,Z: real,Zs: list_real,W: complex,Ws2: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= ( size_s3451745648224563538omplex @ Ys2 ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys2 )
= ( size_size_list_real @ Zs ) )
=> ( ( ( size_size_list_real @ Zs )
= ( size_s3451745648224563538omplex @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys2 @ Zs @ Ws2 )
=> ( P @ ( cons_complex @ X4 @ Xs3 ) @ ( cons_complex @ Y3 @ Ys2 ) @ ( cons_real @ Z @ Zs ) @ ( cons_complex @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1077_list__induct4,axiom,
! [Xs: list_complex,Ys: list_complex,Zs3: list_real,Ws: list_nat,P: list_complex > list_complex > list_real > list_nat > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys )
= ( size_size_list_real @ Zs3 ) )
=> ( ( ( size_size_list_real @ Zs3 )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_complex @ nil_complex @ nil_real @ nil_nat )
=> ( ! [X4: complex,Xs3: list_complex,Y3: complex,Ys2: list_complex,Z: real,Zs: list_real,W: nat,Ws2: list_nat] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= ( size_s3451745648224563538omplex @ Ys2 ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys2 )
= ( size_size_list_real @ Zs ) )
=> ( ( ( size_size_list_real @ Zs )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys2 @ Zs @ Ws2 )
=> ( P @ ( cons_complex @ X4 @ Xs3 ) @ ( cons_complex @ Y3 @ Ys2 ) @ ( cons_real @ Z @ Zs ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1078_list__induct4,axiom,
! [Xs: list_complex,Ys: list_complex,Zs3: list_real,Ws: list_real,P: list_complex > list_complex > list_real > list_real > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys )
= ( size_size_list_real @ Zs3 ) )
=> ( ( ( size_size_list_real @ Zs3 )
= ( size_size_list_real @ Ws ) )
=> ( ( P @ nil_complex @ nil_complex @ nil_real @ nil_real )
=> ( ! [X4: complex,Xs3: list_complex,Y3: complex,Ys2: list_complex,Z: real,Zs: list_real,W: real,Ws2: list_real] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= ( size_s3451745648224563538omplex @ Ys2 ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys2 )
= ( size_size_list_real @ Zs ) )
=> ( ( ( size_size_list_real @ Zs )
= ( size_size_list_real @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys2 @ Zs @ Ws2 )
=> ( P @ ( cons_complex @ X4 @ Xs3 ) @ ( cons_complex @ Y3 @ Ys2 ) @ ( cons_real @ Z @ Zs ) @ ( cons_real @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1079_list__induct4,axiom,
! [Xs: list_complex,Ys: list_nat,Zs3: list_complex,Ws: list_complex,P: list_complex > list_nat > list_complex > list_complex > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_s3451745648224563538omplex @ Zs3 ) )
=> ( ( ( size_s3451745648224563538omplex @ Zs3 )
= ( size_s3451745648224563538omplex @ Ws ) )
=> ( ( P @ nil_complex @ nil_nat @ nil_complex @ nil_complex )
=> ( ! [X4: complex,Xs3: list_complex,Y3: nat,Ys2: list_nat,Z: complex,Zs: list_complex,W: complex,Ws2: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_s3451745648224563538omplex @ Zs ) )
=> ( ( ( size_s3451745648224563538omplex @ Zs )
= ( size_s3451745648224563538omplex @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys2 @ Zs @ Ws2 )
=> ( P @ ( cons_complex @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_complex @ Z @ Zs ) @ ( cons_complex @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1080_list__induct3,axiom,
! [Xs: list_complex,Ys: list_complex,Zs3: list_complex,P: list_complex > list_complex > list_complex > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys )
= ( size_s3451745648224563538omplex @ Zs3 ) )
=> ( ( P @ nil_complex @ nil_complex @ nil_complex )
=> ( ! [X4: complex,Xs3: list_complex,Y3: complex,Ys2: list_complex,Z: complex,Zs: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= ( size_s3451745648224563538omplex @ Ys2 ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys2 )
= ( size_s3451745648224563538omplex @ Zs ) )
=> ( ( P @ Xs3 @ Ys2 @ Zs )
=> ( P @ ( cons_complex @ X4 @ Xs3 ) @ ( cons_complex @ Y3 @ Ys2 ) @ ( cons_complex @ Z @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_1081_list__induct3,axiom,
! [Xs: list_complex,Ys: list_complex,Zs3: list_nat,P: list_complex > list_complex > list_nat > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P @ nil_complex @ nil_complex @ nil_nat )
=> ( ! [X4: complex,Xs3: list_complex,Y3: complex,Ys2: list_complex,Z: nat,Zs: list_nat] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= ( size_s3451745648224563538omplex @ Ys2 ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ Xs3 @ Ys2 @ Zs )
=> ( P @ ( cons_complex @ X4 @ Xs3 ) @ ( cons_complex @ Y3 @ Ys2 ) @ ( cons_nat @ Z @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_1082_list__induct3,axiom,
! [Xs: list_complex,Ys: list_complex,Zs3: list_real,P: list_complex > list_complex > list_real > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys )
= ( size_size_list_real @ Zs3 ) )
=> ( ( P @ nil_complex @ nil_complex @ nil_real )
=> ( ! [X4: complex,Xs3: list_complex,Y3: complex,Ys2: list_complex,Z: real,Zs: list_real] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= ( size_s3451745648224563538omplex @ Ys2 ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys2 )
= ( size_size_list_real @ Zs ) )
=> ( ( P @ Xs3 @ Ys2 @ Zs )
=> ( P @ ( cons_complex @ X4 @ Xs3 ) @ ( cons_complex @ Y3 @ Ys2 ) @ ( cons_real @ Z @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_1083_list__induct3,axiom,
! [Xs: list_complex,Ys: list_nat,Zs3: list_complex,P: list_complex > list_nat > list_complex > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_s3451745648224563538omplex @ Zs3 ) )
=> ( ( P @ nil_complex @ nil_nat @ nil_complex )
=> ( ! [X4: complex,Xs3: list_complex,Y3: nat,Ys2: list_nat,Z: complex,Zs: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_s3451745648224563538omplex @ Zs ) )
=> ( ( P @ Xs3 @ Ys2 @ Zs )
=> ( P @ ( cons_complex @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_complex @ Z @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_1084_list__induct3,axiom,
! [Xs: list_complex,Ys: list_nat,Zs3: list_nat,P: list_complex > list_nat > list_nat > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P @ nil_complex @ nil_nat @ nil_nat )
=> ( ! [X4: complex,Xs3: list_complex,Y3: nat,Ys2: list_nat,Z: nat,Zs: list_nat] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ Xs3 @ Ys2 @ Zs )
=> ( P @ ( cons_complex @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_nat @ Z @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_1085_list__induct3,axiom,
! [Xs: list_complex,Ys: list_nat,Zs3: list_real,P: list_complex > list_nat > list_real > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_real @ Zs3 ) )
=> ( ( P @ nil_complex @ nil_nat @ nil_real )
=> ( ! [X4: complex,Xs3: list_complex,Y3: nat,Ys2: list_nat,Z: real,Zs: list_real] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_real @ Zs ) )
=> ( ( P @ Xs3 @ Ys2 @ Zs )
=> ( P @ ( cons_complex @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys2 ) @ ( cons_real @ Z @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_1086_list__induct3,axiom,
! [Xs: list_complex,Ys: list_real,Zs3: list_complex,P: list_complex > list_real > list_complex > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_size_list_real @ Ys ) )
=> ( ( ( size_size_list_real @ Ys )
= ( size_s3451745648224563538omplex @ Zs3 ) )
=> ( ( P @ nil_complex @ nil_real @ nil_complex )
=> ( ! [X4: complex,Xs3: list_complex,Y3: real,Ys2: list_real,Z: complex,Zs: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= ( size_size_list_real @ Ys2 ) )
=> ( ( ( size_size_list_real @ Ys2 )
= ( size_s3451745648224563538omplex @ Zs ) )
=> ( ( P @ Xs3 @ Ys2 @ Zs )
=> ( P @ ( cons_complex @ X4 @ Xs3 ) @ ( cons_real @ Y3 @ Ys2 ) @ ( cons_complex @ Z @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_1087_list__induct3,axiom,
! [Xs: list_complex,Ys: list_real,Zs3: list_nat,P: list_complex > list_real > list_nat > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_size_list_real @ Ys ) )
=> ( ( ( size_size_list_real @ Ys )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P @ nil_complex @ nil_real @ nil_nat )
=> ( ! [X4: complex,Xs3: list_complex,Y3: real,Ys2: list_real,Z: nat,Zs: list_nat] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= ( size_size_list_real @ Ys2 ) )
=> ( ( ( size_size_list_real @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ Xs3 @ Ys2 @ Zs )
=> ( P @ ( cons_complex @ X4 @ Xs3 ) @ ( cons_real @ Y3 @ Ys2 ) @ ( cons_nat @ Z @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_1088_list__induct3,axiom,
! [Xs: list_complex,Ys: list_real,Zs3: list_real,P: list_complex > list_real > list_real > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_size_list_real @ Ys ) )
=> ( ( ( size_size_list_real @ Ys )
= ( size_size_list_real @ Zs3 ) )
=> ( ( P @ nil_complex @ nil_real @ nil_real )
=> ( ! [X4: complex,Xs3: list_complex,Y3: real,Ys2: list_real,Z: real,Zs: list_real] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= ( size_size_list_real @ Ys2 ) )
=> ( ( ( size_size_list_real @ Ys2 )
= ( size_size_list_real @ Zs ) )
=> ( ( P @ Xs3 @ Ys2 @ Zs )
=> ( P @ ( cons_complex @ X4 @ Xs3 ) @ ( cons_real @ Y3 @ Ys2 ) @ ( cons_real @ Z @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_1089_list__induct3,axiom,
! [Xs: list_nat,Ys: list_complex,Zs3: list_complex,P: list_nat > list_complex > list_complex > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys )
= ( size_s3451745648224563538omplex @ Zs3 ) )
=> ( ( P @ nil_nat @ nil_complex @ nil_complex )
=> ( ! [X4: nat,Xs3: list_nat,Y3: complex,Ys2: list_complex,Z: complex,Zs: list_complex] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_s3451745648224563538omplex @ Ys2 ) )
=> ( ( ( size_s3451745648224563538omplex @ Ys2 )
= ( size_s3451745648224563538omplex @ Zs ) )
=> ( ( P @ Xs3 @ Ys2 @ Zs )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_complex @ Y3 @ Ys2 ) @ ( cons_complex @ Z @ Zs ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs3 ) ) ) ) ) ).
% list_induct3
thf(fact_1090_list__induct2,axiom,
! [Xs: list_complex,Ys: list_complex,P: list_complex > list_complex > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( P @ nil_complex @ nil_complex )
=> ( ! [X4: complex,Xs3: list_complex,Y3: complex,Ys2: list_complex] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= ( size_s3451745648224563538omplex @ Ys2 ) )
=> ( ( P @ Xs3 @ Ys2 )
=> ( P @ ( cons_complex @ X4 @ Xs3 ) @ ( cons_complex @ Y3 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1091_list__induct2,axiom,
! [Xs: list_complex,Ys: list_nat,P: list_complex > list_nat > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_complex @ nil_nat )
=> ( ! [X4: complex,Xs3: list_complex,Y3: nat,Ys2: list_nat] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ Xs3 @ Ys2 )
=> ( P @ ( cons_complex @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1092_list__induct2,axiom,
! [Xs: list_complex,Ys: list_real,P: list_complex > list_real > $o] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_size_list_real @ Ys ) )
=> ( ( P @ nil_complex @ nil_real )
=> ( ! [X4: complex,Xs3: list_complex,Y3: real,Ys2: list_real] :
( ( ( size_s3451745648224563538omplex @ Xs3 )
= ( size_size_list_real @ Ys2 ) )
=> ( ( P @ Xs3 @ Ys2 )
=> ( P @ ( cons_complex @ X4 @ Xs3 ) @ ( cons_real @ Y3 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1093_list__induct2,axiom,
! [Xs: list_nat,Ys: list_complex,P: list_nat > list_complex > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( P @ nil_nat @ nil_complex )
=> ( ! [X4: nat,Xs3: list_nat,Y3: complex,Ys2: list_complex] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_s3451745648224563538omplex @ Ys2 ) )
=> ( ( P @ Xs3 @ Ys2 )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_complex @ Y3 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1094_list__induct2,axiom,
! [Xs: list_nat,Ys: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X4: nat,Xs3: list_nat,Y3: nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ Xs3 @ Ys2 )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1095_list__induct2,axiom,
! [Xs: list_nat,Ys: list_real,P: list_nat > list_real > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_real @ Ys ) )
=> ( ( P @ nil_nat @ nil_real )
=> ( ! [X4: nat,Xs3: list_nat,Y3: real,Ys2: list_real] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_real @ Ys2 ) )
=> ( ( P @ Xs3 @ Ys2 )
=> ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_real @ Y3 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1096_list__induct2,axiom,
! [Xs: list_real,Ys: list_complex,P: list_real > list_complex > $o] :
( ( ( size_size_list_real @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ( ( P @ nil_real @ nil_complex )
=> ( ! [X4: real,Xs3: list_real,Y3: complex,Ys2: list_complex] :
( ( ( size_size_list_real @ Xs3 )
= ( size_s3451745648224563538omplex @ Ys2 ) )
=> ( ( P @ Xs3 @ Ys2 )
=> ( P @ ( cons_real @ X4 @ Xs3 ) @ ( cons_complex @ Y3 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1097_list__induct2,axiom,
! [Xs: list_real,Ys: list_nat,P: list_real > list_nat > $o] :
( ( ( size_size_list_real @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_real @ nil_nat )
=> ( ! [X4: real,Xs3: list_real,Y3: nat,Ys2: list_nat] :
( ( ( size_size_list_real @ Xs3 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ Xs3 @ Ys2 )
=> ( P @ ( cons_real @ X4 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1098_list__induct2,axiom,
! [Xs: list_real,Ys: list_real,P: list_real > list_real > $o] :
( ( ( size_size_list_real @ Xs )
= ( size_size_list_real @ Ys ) )
=> ( ( P @ nil_real @ nil_real )
=> ( ! [X4: real,Xs3: list_real,Y3: real,Ys2: list_real] :
( ( ( size_size_list_real @ Xs3 )
= ( size_size_list_real @ Ys2 ) )
=> ( ( P @ Xs3 @ Ys2 )
=> ( P @ ( cons_real @ X4 @ Xs3 ) @ ( cons_real @ Y3 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1099_successively_Osimps_I2_J,axiom,
! [P: complex > complex > $o,X2: complex] : ( successively_complex @ P @ ( cons_complex @ X2 @ nil_complex ) ) ).
% successively.simps(2)
thf(fact_1100_successively_Osimps_I2_J,axiom,
! [P: nat > nat > $o,X2: nat] : ( successively_nat @ P @ ( cons_nat @ X2 @ nil_nat ) ) ).
% successively.simps(2)
thf(fact_1101_successively_Osimps_I2_J,axiom,
! [P: real > real > $o,X2: real] : ( successively_real @ P @ ( cons_real @ X2 @ nil_real ) ) ).
% successively.simps(2)
thf(fact_1102_successively_Oelims_I1_J,axiom,
! [X2: complex > complex > $o,Xa: list_complex,Y: $o] :
( ( ( successively_complex @ X2 @ Xa )
= Y )
=> ( ( ( Xa = nil_complex )
=> ~ Y )
=> ( ( ? [X4: complex] :
( Xa
= ( cons_complex @ X4 @ nil_complex ) )
=> ~ Y )
=> ~ ! [X4: complex,Y3: complex,Xs3: list_complex] :
( ( Xa
= ( cons_complex @ X4 @ ( cons_complex @ Y3 @ Xs3 ) ) )
=> ( Y
= ( ~ ( ( X2 @ X4 @ Y3 )
& ( successively_complex @ X2 @ ( cons_complex @ Y3 @ Xs3 ) ) ) ) ) ) ) ) ) ).
% successively.elims(1)
thf(fact_1103_successively_Oelims_I1_J,axiom,
! [X2: nat > nat > $o,Xa: list_nat,Y: $o] :
( ( ( successively_nat @ X2 @ Xa )
= Y )
=> ( ( ( Xa = nil_nat )
=> ~ Y )
=> ( ( ? [X4: nat] :
( Xa
= ( cons_nat @ X4 @ nil_nat ) )
=> ~ Y )
=> ~ ! [X4: nat,Y3: nat,Xs3: list_nat] :
( ( Xa
= ( cons_nat @ X4 @ ( cons_nat @ Y3 @ Xs3 ) ) )
=> ( Y
= ( ~ ( ( X2 @ X4 @ Y3 )
& ( successively_nat @ X2 @ ( cons_nat @ Y3 @ Xs3 ) ) ) ) ) ) ) ) ) ).
% successively.elims(1)
thf(fact_1104_successively_Oelims_I1_J,axiom,
! [X2: real > real > $o,Xa: list_real,Y: $o] :
( ( ( successively_real @ X2 @ Xa )
= Y )
=> ( ( ( Xa = nil_real )
=> ~ Y )
=> ( ( ? [X4: real] :
( Xa
= ( cons_real @ X4 @ nil_real ) )
=> ~ Y )
=> ~ ! [X4: real,Y3: real,Xs3: list_real] :
( ( Xa
= ( cons_real @ X4 @ ( cons_real @ Y3 @ Xs3 ) ) )
=> ( Y
= ( ~ ( ( X2 @ X4 @ Y3 )
& ( successively_real @ X2 @ ( cons_real @ Y3 @ Xs3 ) ) ) ) ) ) ) ) ) ).
% successively.elims(1)
thf(fact_1105_successively_Oelims_I2_J,axiom,
! [X2: complex > complex > $o,Xa: list_complex] :
( ( successively_complex @ X2 @ Xa )
=> ( ( Xa != nil_complex )
=> ( ! [X4: complex] :
( Xa
!= ( cons_complex @ X4 @ nil_complex ) )
=> ~ ! [X4: complex,Y3: complex,Xs3: list_complex] :
( ( Xa
= ( cons_complex @ X4 @ ( cons_complex @ Y3 @ Xs3 ) ) )
=> ~ ( ( X2 @ X4 @ Y3 )
& ( successively_complex @ X2 @ ( cons_complex @ Y3 @ Xs3 ) ) ) ) ) ) ) ).
% successively.elims(2)
thf(fact_1106_successively_Oelims_I2_J,axiom,
! [X2: nat > nat > $o,Xa: list_nat] :
( ( successively_nat @ X2 @ Xa )
=> ( ( Xa != nil_nat )
=> ( ! [X4: nat] :
( Xa
!= ( cons_nat @ X4 @ nil_nat ) )
=> ~ ! [X4: nat,Y3: nat,Xs3: list_nat] :
( ( Xa
= ( cons_nat @ X4 @ ( cons_nat @ Y3 @ Xs3 ) ) )
=> ~ ( ( X2 @ X4 @ Y3 )
& ( successively_nat @ X2 @ ( cons_nat @ Y3 @ Xs3 ) ) ) ) ) ) ) ).
% successively.elims(2)
thf(fact_1107_successively_Oelims_I2_J,axiom,
! [X2: real > real > $o,Xa: list_real] :
( ( successively_real @ X2 @ Xa )
=> ( ( Xa != nil_real )
=> ( ! [X4: real] :
( Xa
!= ( cons_real @ X4 @ nil_real ) )
=> ~ ! [X4: real,Y3: real,Xs3: list_real] :
( ( Xa
= ( cons_real @ X4 @ ( cons_real @ Y3 @ Xs3 ) ) )
=> ~ ( ( X2 @ X4 @ Y3 )
& ( successively_real @ X2 @ ( cons_real @ Y3 @ Xs3 ) ) ) ) ) ) ) ).
% successively.elims(2)
thf(fact_1108_insert__Diff,axiom,
! [A: complex,A2: set_complex] :
( ( member_complex2 @ A @ A2 )
=> ( ( insert_complex2 @ A @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex2 @ A @ bot_bot_set_complex ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_1109_insert__Diff,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat2 @ A @ A2 )
=> ( ( insert_nat2 @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_1110_Diff__insert__absorb,axiom,
! [X2: complex,A2: set_complex] :
( ~ ( member_complex2 @ X2 @ A2 )
=> ( ( minus_811609699411566653omplex @ ( insert_complex2 @ X2 @ A2 ) @ ( insert_complex2 @ X2 @ bot_bot_set_complex ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_1111_Diff__insert__absorb,axiom,
! [X2: nat,A2: set_nat] :
( ~ ( member_nat2 @ X2 @ A2 )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X2 @ A2 ) @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_1112_in__image__insert__iff,axiom,
! [B: set_set_complex,X2: complex,A2: set_complex] :
( ! [C4: set_complex] :
( ( member_set_complex @ C4 @ B )
=> ~ ( member_complex2 @ X2 @ C4 ) )
=> ( ( member_set_complex @ A2 @ ( image_7998606247489673935omplex @ ( insert_complex2 @ X2 ) @ B ) )
= ( ( member_complex2 @ X2 @ A2 )
& ( member_set_complex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex2 @ X2 @ bot_bot_set_complex ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1113_in__image__insert__iff,axiom,
! [B: set_set_nat,X2: nat,A2: set_nat] :
( ! [C4: set_nat] :
( ( member_set_nat @ C4 @ B )
=> ~ ( member_nat2 @ X2 @ C4 ) )
=> ( ( member_set_nat @ A2 @ ( image_7916887816326733075et_nat @ ( insert_nat2 @ X2 ) @ B ) )
= ( ( member_nat2 @ X2 @ A2 )
& ( member_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_1114_set__empty2,axiom,
! [Xs: list_nat] :
( ( bot_bot_set_nat
= ( set_nat2 @ Xs ) )
= ( Xs = nil_nat ) ) ).
% set_empty2
thf(fact_1115_set__empty2,axiom,
! [Xs: list_complex] :
( ( bot_bot_set_complex
= ( set_complex2 @ Xs ) )
= ( Xs = nil_complex ) ) ).
% set_empty2
thf(fact_1116_set__empty,axiom,
! [Xs: list_nat] :
( ( ( set_nat2 @ Xs )
= bot_bot_set_nat )
= ( Xs = nil_nat ) ) ).
% set_empty
thf(fact_1117_set__empty,axiom,
! [Xs: list_complex] :
( ( ( set_complex2 @ Xs )
= bot_bot_set_complex )
= ( Xs = nil_complex ) ) ).
% set_empty
thf(fact_1118_empty__set,axiom,
( bot_bot_set_nat
= ( set_nat2 @ nil_nat ) ) ).
% empty_set
thf(fact_1119_empty__set,axiom,
( bot_bot_set_complex
= ( set_complex2 @ nil_complex ) ) ).
% empty_set
thf(fact_1120_successively__remdups__adj__iff,axiom,
! [Xs: list_nat,P: nat > nat > $o] :
( ! [X4: nat] :
( ( member_nat2 @ X4 @ ( set_nat2 @ Xs ) )
=> ( P @ X4 @ X4 ) )
=> ( ( successively_nat @ P @ ( remdups_adj_nat @ Xs ) )
= ( successively_nat @ P @ Xs ) ) ) ).
% successively_remdups_adj_iff
thf(fact_1121_successively__remdups__adj__iff,axiom,
! [Xs: list_complex,P: complex > complex > $o] :
( ! [X4: complex] :
( ( member_complex2 @ X4 @ ( set_complex2 @ Xs ) )
=> ( P @ X4 @ X4 ) )
=> ( ( successively_complex @ P @ ( remdups_adj_complex @ Xs ) )
= ( successively_complex @ P @ Xs ) ) ) ).
% successively_remdups_adj_iff
thf(fact_1122_shuffles_Osimps_I1_J,axiom,
! [Ys: list_nat] :
( ( shuffles_nat @ nil_nat @ Ys )
= ( insert_list_nat @ Ys @ bot_bot_set_list_nat ) ) ).
% shuffles.simps(1)
thf(fact_1123_shuffles_Osimps_I2_J,axiom,
! [Xs: list_nat] :
( ( shuffles_nat @ Xs @ nil_nat )
= ( insert_list_nat @ Xs @ bot_bot_set_list_nat ) ) ).
% shuffles.simps(2)
thf(fact_1124_nths__empty,axiom,
! [Xs: list_nat] :
( ( nths_nat @ Xs @ bot_bot_set_nat )
= nil_nat ) ).
% nths_empty
thf(fact_1125_the__elem__image__unique,axiom,
! [A2: set_nat,F: nat > nat,X2: nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [Y3: nat] :
( ( member_nat2 @ Y3 @ A2 )
=> ( ( F @ Y3 )
= ( F @ X2 ) ) )
=> ( ( the_elem_nat @ ( image_nat_nat @ F @ A2 ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_1126_image__constant__conv,axiom,
! [A2: set_nat,C3: nat] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( image_nat_nat
@ ^ [X3: nat] : C3
@ A2 )
= bot_bot_set_nat ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( image_nat_nat
@ ^ [X3: nat] : C3
@ A2 )
= ( insert_nat2 @ C3 @ bot_bot_set_nat ) ) ) ) ).
% image_constant_conv
thf(fact_1127_image__constant,axiom,
! [X2: nat,A2: set_nat,C3: nat] :
( ( member_nat2 @ X2 @ A2 )
=> ( ( image_nat_nat
@ ^ [X3: nat] : C3
@ A2 )
= ( insert_nat2 @ C3 @ bot_bot_set_nat ) ) ) ).
% image_constant
thf(fact_1128_set__rec,axiom,
( set_complex2
= ( rec_li3674993589600700234omplex @ bot_bot_set_complex
@ ^ [X3: complex,Uu2: list_complex] : ( insert_complex2 @ X3 ) ) ) ).
% set_rec
thf(fact_1129_same__length__different,axiom,
! [Xs: list_complex,Ys: list_complex] :
( ( Xs != Ys )
=> ( ( ( size_s3451745648224563538omplex @ Xs )
= ( size_s3451745648224563538omplex @ Ys ) )
=> ? [Pre: list_complex,X4: complex,Xs6: list_complex,Y3: complex,Ys6: list_complex] :
( ( X4 != Y3 )
& ( Xs
= ( append_complex @ Pre @ ( append_complex @ ( cons_complex @ X4 @ nil_complex ) @ Xs6 ) ) )
& ( Ys
= ( append_complex @ Pre @ ( append_complex @ ( cons_complex @ Y3 @ nil_complex ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_1130_same__length__different,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs != Ys )
=> ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ? [Pre: list_nat,X4: nat,Xs6: list_nat,Y3: nat,Ys6: list_nat] :
( ( X4 != Y3 )
& ( Xs
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X4 @ nil_nat ) @ Xs6 ) ) )
& ( Ys
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y3 @ nil_nat ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_1131_same__length__different,axiom,
! [Xs: list_real,Ys: list_real] :
( ( Xs != Ys )
=> ( ( ( size_size_list_real @ Xs )
= ( size_size_list_real @ Ys ) )
=> ? [Pre: list_real,X4: real,Xs6: list_real,Y3: real,Ys6: list_real] :
( ( X4 != Y3 )
& ( Xs
= ( append_real @ Pre @ ( append_real @ ( cons_real @ X4 @ nil_real ) @ Xs6 ) ) )
& ( Ys
= ( append_real @ Pre @ ( append_real @ ( cons_real @ Y3 @ nil_real ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_1132_successively__Cons,axiom,
! [P: complex > complex > $o,X2: complex,Xs: list_complex] :
( ( successively_complex @ P @ ( cons_complex @ X2 @ Xs ) )
= ( ( Xs = nil_complex )
| ( ( P @ X2 @ ( hd_complex @ Xs ) )
& ( successively_complex @ P @ Xs ) ) ) ) ).
% successively_Cons
thf(fact_1133_successively__Cons,axiom,
! [P: nat > nat > $o,X2: nat,Xs: list_nat] :
( ( successively_nat @ P @ ( cons_nat @ X2 @ Xs ) )
= ( ( Xs = nil_nat )
| ( ( P @ X2 @ ( hd_nat @ Xs ) )
& ( successively_nat @ P @ Xs ) ) ) ) ).
% successively_Cons
thf(fact_1134_successively__Cons,axiom,
! [P: real > real > $o,X2: real,Xs: list_real] :
( ( successively_real @ P @ ( cons_real @ X2 @ Xs ) )
= ( ( Xs = nil_real )
| ( ( P @ X2 @ ( hd_real @ Xs ) )
& ( successively_real @ P @ Xs ) ) ) ) ).
% successively_Cons
thf(fact_1135_eq__comps__singleton,axiom,
! [A: nat,L: list_real] :
( ( ( cons_nat @ A @ nil_nat )
= ( commut8680161604938074397s_real @ L ) )
=> ( A
= ( size_size_list_real @ L ) ) ) ).
% eq_comps_singleton
thf(fact_1136_eq__comps__singleton,axiom,
! [A: nat,L: list_complex] :
( ( ( cons_nat @ A @ nil_nat )
= ( commut93809757773076895omplex @ L ) )
=> ( A
= ( size_s3451745648224563538omplex @ L ) ) ) ).
% eq_comps_singleton
thf(fact_1137_length__Suc__conv__rev,axiom,
! [Xs: list_complex,N2: nat] :
( ( ( size_s3451745648224563538omplex @ Xs )
= ( suc @ N2 ) )
= ( ? [Y2: complex,Ys3: list_complex] :
( ( Xs
= ( append_complex @ Ys3 @ ( cons_complex @ Y2 @ nil_complex ) ) )
& ( ( size_s3451745648224563538omplex @ Ys3 )
= N2 ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_1138_length__Suc__conv__rev,axiom,
! [Xs: list_nat,N2: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N2 ) )
= ( ? [Y2: nat,Ys3: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ Y2 @ nil_nat ) ) )
& ( ( size_size_list_nat @ Ys3 )
= N2 ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_1139_length__Suc__conv__rev,axiom,
! [Xs: list_real,N2: nat] :
( ( ( size_size_list_real @ Xs )
= ( suc @ N2 ) )
= ( ? [Y2: real,Ys3: list_real] :
( ( Xs
= ( append_real @ Ys3 @ ( cons_real @ Y2 @ nil_real ) ) )
& ( ( size_size_list_real @ Ys3 )
= N2 ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_1140_length__append__singleton,axiom,
! [Xs: list_complex,X2: complex] :
( ( size_s3451745648224563538omplex @ ( append_complex @ Xs @ ( cons_complex @ X2 @ nil_complex ) ) )
= ( suc @ ( size_s3451745648224563538omplex @ Xs ) ) ) ).
% length_append_singleton
thf(fact_1141_length__append__singleton,axiom,
! [Xs: list_nat,X2: nat] :
( ( size_size_list_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_append_singleton
thf(fact_1142_length__append__singleton,axiom,
! [Xs: list_real,X2: real] :
( ( size_size_list_real @ ( append_real @ Xs @ ( cons_real @ X2 @ nil_real ) ) )
= ( suc @ ( size_size_list_real @ Xs ) ) ) ).
% length_append_singleton
thf(fact_1143_Nitpick_Osize__list__simp_I2_J,axiom,
( size_size_list_nat
= ( ^ [Xs2: list_nat] : ( if_nat @ ( Xs2 = nil_nat ) @ zero_zero_nat @ ( suc @ ( size_size_list_nat @ ( tl_nat @ Xs2 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_1144_vector__space__over__itself_Ozero__not__in__Basis,axiom,
~ ( member_complex2 @ zero_zero_complex @ ( insert_complex2 @ one_one_complex @ bot_bot_set_complex ) ) ).
% vector_space_over_itself.zero_not_in_Basis
thf(fact_1145_vector__space__over__itself_Ozero__not__in__Basis,axiom,
~ ( member_real2 @ zero_zero_real @ ( insert_real2 @ one_one_real @ bot_bot_set_real ) ) ).
% vector_space_over_itself.zero_not_in_Basis
thf(fact_1146_Diff__not__in,axiom,
! [A: complex,A2: set_complex] :
~ ( member_complex2 @ A @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex2 @ A @ bot_bot_set_complex ) ) ) ).
% Diff_not_in
thf(fact_1147_Diff__not__in,axiom,
! [A: nat,A2: set_nat] :
~ ( member_nat2 @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) ) ).
% Diff_not_in
thf(fact_1148_set__Cons__sing__Nil,axiom,
! [A2: set_complex] :
( ( set_Cons_complex @ A2 @ ( insert_list_complex @ nil_complex @ bot_bo6492010485567502472omplex ) )
= ( image_3109120598033070323omplex
@ ^ [X3: complex] : ( cons_complex @ X3 @ nil_complex )
@ A2 ) ) ).
% set_Cons_sing_Nil
thf(fact_1149_set__Cons__sing__Nil,axiom,
! [A2: set_nat] :
( ( set_Cons_nat @ A2 @ ( insert_list_nat @ nil_nat @ bot_bot_set_list_nat ) )
= ( image_nat_list_nat
@ ^ [X3: nat] : ( cons_nat @ X3 @ nil_nat )
@ A2 ) ) ).
% set_Cons_sing_Nil
thf(fact_1150_set__Cons__sing__Nil,axiom,
! [A2: set_real] :
( ( set_Cons_real @ A2 @ ( insert_list_real @ nil_real @ bot_bo7660656974785640070t_real ) )
= ( image_real_list_real
@ ^ [X3: real] : ( cons_real @ X3 @ nil_real )
@ A2 ) ) ).
% set_Cons_sing_Nil
thf(fact_1151_find__indices__snoc,axiom,
! [X2: complex,Ys: list_complex,Y: complex] :
( ( missin8834916005246747252omplex @ X2 @ ( append_complex @ Ys @ ( cons_complex @ Y @ nil_complex ) ) )
= ( append_nat @ ( missin8834916005246747252omplex @ X2 @ Ys ) @ ( if_list_nat @ ( X2 = Y ) @ ( cons_nat @ ( size_s3451745648224563538omplex @ Ys ) @ nil_nat ) @ nil_nat ) ) ) ).
% find_indices_snoc
thf(fact_1152_find__indices__snoc,axiom,
! [X2: nat,Ys: list_nat,Y: nat] :
( ( missin5050847376130023830es_nat @ X2 @ ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) )
= ( append_nat @ ( missin5050847376130023830es_nat @ X2 @ Ys ) @ ( if_list_nat @ ( X2 = Y ) @ ( cons_nat @ ( size_size_list_nat @ Ys ) @ nil_nat ) @ nil_nat ) ) ) ).
% find_indices_snoc
thf(fact_1153_find__indices__snoc,axiom,
! [X2: real,Ys: list_real,Y: real] :
( ( missin4558892344646173042s_real @ X2 @ ( append_real @ Ys @ ( cons_real @ Y @ nil_real ) ) )
= ( append_nat @ ( missin4558892344646173042s_real @ X2 @ Ys ) @ ( if_list_nat @ ( X2 = Y ) @ ( cons_nat @ ( size_size_list_real @ Ys ) @ nil_nat ) @ nil_nat ) ) ) ).
% find_indices_snoc
thf(fact_1154_set__one,axiom,
( one_one_set_nat
= ( insert_nat2 @ one_one_nat @ bot_bot_set_nat ) ) ).
% set_one
thf(fact_1155_set__one,axiom,
( one_one_set_complex
= ( insert_complex2 @ one_one_complex @ bot_bot_set_complex ) ) ).
% set_one
thf(fact_1156_set__one,axiom,
( one_one_set_real
= ( insert_real2 @ one_one_real @ bot_bot_set_real ) ) ).
% set_one
thf(fact_1157_find__indices__Nil,axiom,
! [X2: nat] :
( ( missin5050847376130023830es_nat @ X2 @ nil_nat )
= nil_nat ) ).
% find_indices_Nil
thf(fact_1158_set__zero,axiom,
( zero_zero_set_nat
= ( insert_nat2 @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% set_zero
thf(fact_1159_set__zero,axiom,
( zero_z6614145512433583213omplex
= ( insert_complex2 @ zero_zero_complex @ bot_bot_set_complex ) ) ).
% set_zero
thf(fact_1160_set__zero,axiom,
( zero_zero_set_real
= ( insert_real2 @ zero_zero_real @ bot_bot_set_real ) ) ).
% set_zero
thf(fact_1161_listset_Osimps_I1_J,axiom,
( ( listset_nat @ nil_set_nat )
= ( insert_list_nat @ nil_nat @ bot_bot_set_list_nat ) ) ).
% listset.simps(1)
thf(fact_1162_Longest__common__prefix__eq__Cons,axiom,
! [L3: set_list_complex,X2: complex] :
( ( L3 != bot_bo6492010485567502472omplex )
=> ( ~ ( member_list_complex @ nil_complex @ L3 )
=> ( ! [X4: list_complex] :
( ( member_list_complex @ X4 @ L3 )
=> ( ( hd_complex @ X4 )
= X2 ) )
=> ( ( longes9130274606949010820omplex @ L3 )
= ( cons_complex @ X2
@ ( longes9130274606949010820omplex
@ ( collect_list_complex
@ ^ [Ys3: list_complex] : ( member_list_complex @ ( cons_complex @ X2 @ Ys3 ) @ L3 ) ) ) ) ) ) ) ) ).
% Longest_common_prefix_eq_Cons
thf(fact_1163_Longest__common__prefix__eq__Cons,axiom,
! [L3: set_list_nat,X2: nat] :
( ( L3 != bot_bot_set_list_nat )
=> ( ~ ( member_list_nat @ nil_nat @ L3 )
=> ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ L3 )
=> ( ( hd_nat @ X4 )
= X2 ) )
=> ( ( longes514542611558403238ix_nat @ L3 )
= ( cons_nat @ X2
@ ( longes514542611558403238ix_nat
@ ( collect_list_nat
@ ^ [Ys3: list_nat] : ( member_list_nat @ ( cons_nat @ X2 @ Ys3 ) @ L3 ) ) ) ) ) ) ) ) ).
% Longest_common_prefix_eq_Cons
thf(fact_1164_Longest__common__prefix__eq__Cons,axiom,
! [L3: set_list_real,X2: real] :
( ( L3 != bot_bo7660656974785640070t_real )
=> ( ~ ( member_list_real @ nil_real @ L3 )
=> ( ! [X4: list_real] :
( ( member_list_real @ X4 @ L3 )
=> ( ( hd_real @ X4 )
= X2 ) )
=> ( ( longes7015494920320935042x_real @ L3 )
= ( cons_real @ X2
@ ( longes7015494920320935042x_real
@ ( collect_list_real
@ ^ [Ys3: list_real] : ( member_list_real @ ( cons_real @ X2 @ Ys3 ) @ L3 ) ) ) ) ) ) ) ) ).
% Longest_common_prefix_eq_Cons
thf(fact_1165_Longest__common__prefix__Nil,axiom,
! [L3: set_list_nat] :
( ( member_list_nat @ nil_nat @ L3 )
=> ( ( longes514542611558403238ix_nat @ L3 )
= nil_nat ) ) ).
% Longest_common_prefix_Nil
thf(fact_1166_diff__is__0__eq_H,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( minus_minus_nat @ M @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1167_diff__is__0__eq,axiom,
! [M: nat,N2: nat] :
( ( ( minus_minus_nat @ M @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% diff_is_0_eq
thf(fact_1168_prod__decode__aux_Oinduct,axiom,
! [P: nat > nat > $o,A0: nat,A1: nat] :
( ! [K3: nat,M3: nat] :
( ( ~ ( ord_less_eq_nat @ M3 @ K3 )
=> ( P @ ( suc @ K3 ) @ ( minus_minus_nat @ M3 @ ( suc @ K3 ) ) ) )
=> ( P @ K3 @ M3 ) )
=> ( P @ A0 @ A1 ) ) ).
% prod_decode_aux.induct
thf(fact_1169_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1170_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_1171_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_1172_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_1173_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1174_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_1175_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q: nat > $o] :
( ! [X4: nat > real] :
( ( P @ X4 )
=> ( P @ ( F @ X4 ) ) )
=> ( ! [X4: nat > real] :
( ( P @ X4 )
=> ! [I: nat] :
( ( Q @ I )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X4 @ I ) )
& ( ord_less_eq_real @ ( X4 @ I ) @ one_one_real ) ) ) )
=> ? [L2: ( nat > real ) > nat > nat] :
( ! [X: nat > real,I3: nat] : ( ord_less_eq_nat @ ( L2 @ X @ I3 ) @ one_one_nat )
& ! [X: nat > real,I3: nat] :
( ( ( P @ X )
& ( Q @ I3 )
& ( ( X @ I3 )
= zero_zero_real ) )
=> ( ( L2 @ X @ I3 )
= zero_zero_nat ) )
& ! [X: nat > real,I3: nat] :
( ( ( P @ X )
& ( Q @ I3 )
& ( ( X @ I3 )
= one_one_real ) )
=> ( ( L2 @ X @ I3 )
= one_one_nat ) )
& ! [X: nat > real,I3: nat] :
( ( ( P @ X )
& ( Q @ I3 )
& ( ( L2 @ X @ I3 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X @ I3 ) @ ( F @ X @ I3 ) ) )
& ! [X: nat > real,I3: nat] :
( ( ( P @ X )
& ( Q @ I3 )
& ( ( L2 @ X @ I3 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X @ I3 ) @ ( X @ I3 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_1176_nonnegative__complex__is__real,axiom,
! [X2: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ X2 )
=> ( member_complex2 @ X2 @ real_V2521375963428798218omplex ) ) ).
% nonnegative_complex_is_real
thf(fact_1177_complex__is__real__iff__compare0,axiom,
! [X2: complex] :
( ( member_complex2 @ X2 @ real_V2521375963428798218omplex )
= ( ( ord_less_eq_complex @ X2 @ zero_zero_complex )
| ( ord_less_eq_complex @ zero_zero_complex @ X2 ) ) ) ).
% complex_is_real_iff_compare0
thf(fact_1178_card__Collect__le__nat,axiom,
! [N2: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I4: nat] : ( ord_less_eq_nat @ I4 @ N2 ) ) )
= ( suc @ N2 ) ) ).
% card_Collect_le_nat
thf(fact_1179_card__Collect__less__nat,axiom,
! [N2: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I4: nat] : ( ord_less_nat @ I4 @ N2 ) ) )
= N2 ) ).
% card_Collect_less_nat
thf(fact_1180_card__less,axiom,
! [M2: set_nat,I5: nat] :
( ( member_nat2 @ zero_zero_nat @ M2 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K: nat] :
( ( member_nat2 @ K @ M2 )
& ( ord_less_nat @ K @ ( suc @ I5 ) ) ) ) )
!= zero_zero_nat ) ) ).
% card_less
thf(fact_1181_card__less__Suc,axiom,
! [M2: set_nat,I5: nat] :
( ( member_nat2 @ zero_zero_nat @ M2 )
=> ( ( suc
@ ( finite_card_nat
@ ( collect_nat
@ ^ [K: nat] :
( ( member_nat2 @ ( suc @ K ) @ M2 )
& ( ord_less_nat @ K @ I5 ) ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K: nat] :
( ( member_nat2 @ K @ M2 )
& ( ord_less_nat @ K @ ( suc @ I5 ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_1182_card__less__Suc2,axiom,
! [M2: set_nat,I5: nat] :
( ~ ( member_nat2 @ zero_zero_nat @ M2 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K: nat] :
( ( member_nat2 @ ( suc @ K ) @ M2 )
& ( ord_less_nat @ K @ I5 ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K: nat] :
( ( member_nat2 @ K @ M2 )
& ( ord_less_nat @ K @ ( suc @ I5 ) ) ) ) ) ) ) ).
% card_less_Suc2
% Helper facts (9)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X2: list_nat,Y: list_nat] :
( ( if_list_nat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X2: list_nat,Y: list_nat] :
( ( if_list_nat @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Real__Oreal_J_T,axiom,
! [X2: list_real,Y: list_real] :
( ( if_list_real @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Real__Oreal_J_T,axiom,
! [X2: list_real,Y: list_real] :
( ( if_list_real @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Complex__Ocomplex_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Complex__Ocomplex_J_T,axiom,
! [X2: list_complex,Y: list_complex] :
( ( if_list_complex @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Complex__Ocomplex_J_T,axiom,
! [X2: list_complex,Y: list_complex] :
( ( if_list_complex @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ma
= ( commut8680161604938074397s_real @ ( map_complex_real @ re @ ( cons_complex @ x @ ( cons_complex @ y @ la ) ) ) ) ) ).
%------------------------------------------------------------------------------