TPTP Problem File: SLH0171^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Rewrite_Properties_Reduction/0015_Terms_Positions/prob_00458_015773__13747920_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1533 ( 555 unt; 274 typ; 0 def)
% Number of atoms : 3471 (1439 equ; 0 cnn)
% Maximal formula atoms : 23 ( 2 avg)
% Number of connectives : 11066 ( 546 ~; 52 |; 248 &;8584 @)
% ( 0 <=>;1636 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 7 avg)
% Number of types : 35 ( 34 usr)
% Number of type conns : 899 ( 899 >; 0 *; 0 +; 0 <<)
% Number of symbols : 243 ( 240 usr; 20 con; 0-4 aty)
% Number of variables : 3798 ( 234 ^;3394 !; 170 ?;3798 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:18:19.488
%------------------------------------------------------------------------------
% Could-be-implicit typings (34)
thf(ty_n_t__Product____Type__Oprod_It__Subterm____and____Context__Octxt_Itf__a_Mtf__b_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
produc8210972263561988409rm_a_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
produc4787317212837456354st_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
produc2732850333517536310rm_a_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
produc254973753779126261st_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
set_Pr3451248702717554689st_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
produc3697673438841856213st_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
produc7711739908350443733rm_a_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
produc1828647624359046049st_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_Eo_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
produc4226810134323546766st_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_J,type,
set_se6121441497158405097_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J,type,
produc4575160907756185873st_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
list_list_term_a_b: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
list_list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
set_Pr4934435412358123699_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
set_list_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_J,type,
list_list_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
set_set_list_nat: $tType ).
thf(ty_n_t__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mtf__b_J_J,type,
option_Sum_sum_a_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Subterm____and____Context__Octxt_Itf__a_Mtf__b_J,type,
subterm_and_ctxt_a_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
product_prod_a_nat: $tType ).
thf(ty_n_t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
list_term_a_b: $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__Set__Oset_I_062_I_Eo_Mt__Nat__Onat_J_J,type,
set_o_nat: $tType ).
thf(ty_n_t__Term__Oterm_Itf__a_Mtf__b_J,type,
term_a_b: $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__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (240)
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__List__Olist_It__Nat__Onat_J,type,
bNF_Gr9051742241863529473st_nat: set_list_list_nat > list_nat > set_list_list_nat ).
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_OSucc_001t__List__Olist_It__Nat__Onat_J,type,
bNF_Gr3053708287304744325st_nat: set_list_list_nat > list_list_nat > set_list_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Nat__Onat,type,
bNF_Gr6352880689984616693cc_nat: set_list_nat > list_nat > set_nat ).
thf(sy_c_Basic__Utils_Oadd__elem__list__lists_001t__List__Olist_It__Nat__Onat_J,type,
basic_8359458158062141559st_nat: list_nat > list_list_nat > list_list_list_nat ).
thf(sy_c_Basic__Utils_Oadd__elem__list__lists_001t__Nat__Onat,type,
basic_4874698711677410535ts_nat: nat > list_nat > list_list_nat ).
thf(sy_c_Basic__Utils_Oadd__elem__list__lists_001t__Set__Oset_It__Nat__Onat_J,type,
basic_8313613011175402653et_nat: set_nat > list_set_nat > list_list_set_nat ).
thf(sy_c_Basic__Utils_Oadd__elem__list__lists_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
basic_1593220722155286443rm_a_b: term_a_b > list_term_a_b > list_list_term_a_b ).
thf(sy_c_Finite__Set_Ofinite_001_062_I_Eo_Mt__Nat__Onat_J,type,
finite_finite_o_nat: set_o_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
finite8170528100393595399st_nat: set_list_list_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Nat__Onat_J,type,
finite8100373058378681591st_nat: set_list_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
finite9134613430549597258st_nat: set_Pr3451248702717554689st_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite6177210948735845034at_nat: set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
finite6644898363146130708_a_nat: set_Pr4934435412358123699_a_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
finite7047420756378620717st_nat: set_set_list_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
finite568477829172517706_a_nat: set_se6121441497158405097_a_nat > $o ).
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__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
minus_7954133019191499631st_nat: set_list_nat > set_list_nat > set_list_nat ).
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_HOL_Oundefined_001t__Nat__Onat,type,
undefined_nat: nat ).
thf(sy_c_HOL_Oundefined_001t__Subterm____and____Context__Octxt_Itf__a_Mtf__b_J,type,
undefi3573907640150307307xt_a_b: subterm_and_ctxt_a_b ).
thf(sy_c_HOL_Oundefined_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
undefined_term_a_b: term_a_b ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
inf_inf_set_list_nat: set_list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_I_Eo_Mt__Nat__Onat_J,type,
sup_sup_o_nat: ( $o > nat ) > ( $o > nat ) > $o > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
sup_su5723864310361701714st_nat: set_list_list_nat > set_list_list_nat > set_list_list_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
sup_sup_set_list_nat: set_list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
sup_su5841606568262889429st_nat: set_Pr3451248702717554689st_nat > set_Pr3451248702717554689st_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
sup_su6327502436637775413at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
sup_su459911885395995103_a_nat: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat ).
thf(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Nat__Onat,type,
lattic8721135487736765967in_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
lattic5785867957632790475at_nat: ( list_nat > nat ) > set_list_nat > list_nat ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Nat__Onat,type,
lattic5238388535129920115in_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Nat__Onat,type,
lattic1093996805478795353in_nat: set_nat > nat ).
thf(sy_c_List_Oappend_001t__List__Olist_It__Nat__Onat_J,type,
append_list_nat: list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Oappend_001t__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Oappend_001t__Set__Oset_It__Nat__Onat_J,type,
append_set_nat: list_set_nat > list_set_nat > list_set_nat ).
thf(sy_c_List_Oappend_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
append_term_a_b: list_term_a_b > list_term_a_b > list_term_a_b ).
thf(sy_c_List_Obind_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
bind_l7796378977173581257st_nat: list_list_nat > ( list_nat > list_list_nat ) > list_list_nat ).
thf(sy_c_List_Obind_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
bind_list_nat_nat: list_list_nat > ( list_nat > list_nat ) > list_nat ).
thf(sy_c_List_Obind_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
bind_l3154278341557560047et_nat: list_list_nat > ( list_nat > list_set_nat ) > list_set_nat ).
thf(sy_c_List_Obind_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
bind_nat_list_nat: list_nat > ( nat > list_list_nat ) > list_list_nat ).
thf(sy_c_List_Obind_001t__Nat__Onat_001t__Nat__Onat,type,
bind_nat_nat: list_nat > ( nat > list_nat ) > list_nat ).
thf(sy_c_List_Obind_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
bind_nat_set_nat: list_nat > ( nat > list_set_nat ) > list_set_nat ).
thf(sy_c_List_Obind_001t__Nat__Onat_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
bind_nat_term_a_b: list_nat > ( nat > list_term_a_b ) > list_term_a_b ).
thf(sy_c_List_Obind_001t__Set__Oset_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
bind_s6804505294920241007st_nat: list_set_nat > ( set_nat > list_list_nat ) > list_list_nat ).
thf(sy_c_List_Obind_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
bind_set_nat_nat: list_set_nat > ( set_nat > list_nat ) > list_nat ).
thf(sy_c_List_Obind_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
bind_term_a_b_nat: list_term_a_b > ( term_a_b > list_nat ) > list_nat ).
thf(sy_c_List_Obutlast_001t__List__Olist_It__Nat__Onat_J,type,
butlast_list_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Obutlast_001t__Nat__Onat,type,
butlast_nat: list_nat > list_nat ).
thf(sy_c_List_Obutlast_001t__Set__Oset_It__Nat__Onat_J,type,
butlast_set_nat: list_set_nat > list_set_nat ).
thf(sy_c_List_Obutlast_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
butlast_term_a_b: list_term_a_b > list_term_a_b ).
thf(sy_c_List_Oconcat_001t__Nat__Onat,type,
concat_nat: list_list_nat > list_nat ).
thf(sy_c_List_Odrop_001t__Nat__Onat,type,
drop_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Ofoldr_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
foldr_6871341030409798377st_nat: ( list_nat > list_nat > list_nat ) > list_list_nat > list_nat > list_nat ).
thf(sy_c_List_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
insert_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
insert_set_nat: set_nat > list_set_nat > list_set_nat ).
thf(sy_c_List_Oinsert_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
insert_term_a_b: term_a_b > list_term_a_b > list_term_a_b ).
thf(sy_c_List_Olast_001t__List__Olist_It__Nat__Onat_J,type,
last_list_nat: list_list_nat > list_nat ).
thf(sy_c_List_Olast_001t__Nat__Onat,type,
last_nat: list_nat > nat ).
thf(sy_c_List_Olexord_001t__Nat__Onat,type,
lexord_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
linord2614967742042102400et_nat: set_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
cons_list_list_nat: list_list_nat > list_list_list_nat > list_list_list_nat ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Nat__Onat_J,type,
cons_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
cons_list_set_nat: list_set_nat > list_list_set_nat > list_list_set_nat ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
cons_list_term_a_b: list_term_a_b > list_list_term_a_b > list_list_term_a_b ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Set__Oset_It__Nat__Onat_J,type,
cons_set_nat: set_nat > list_set_nat > list_set_nat ).
thf(sy_c_List_Olist_OCons_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
cons_term_a_b: term_a_b > list_term_a_b > list_term_a_b ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
nil_list_list_nat: list_list_list_nat ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Nat__Onat_J,type,
nil_list_nat: list_list_nat ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
nil_list_set_nat: list_list_set_nat ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
nil_list_term_a_b: list_list_term_a_b ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001t__Set__Oset_It__Nat__Onat_J,type,
nil_set_nat: list_set_nat ).
thf(sy_c_List_Olist_ONil_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
nil_term_a_b: list_term_a_b ).
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_Oset_001t__List__Olist_It__Nat__Onat_J,type,
set_list_nat2: list_list_nat > set_list_nat ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist__ex1_001t__List__Olist_It__Nat__Onat_J,type,
list_ex1_list_nat: ( list_nat > $o ) > list_list_nat > $o ).
thf(sy_c_List_Olist__ex1_001t__Nat__Onat,type,
list_ex1_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__ex1_001t__Set__Oset_It__Nat__Onat_J,type,
list_ex1_set_nat: ( set_nat > $o ) > list_set_nat > $o ).
thf(sy_c_List_Olist__ex1_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
list_ex1_term_a_b: ( term_a_b > $o ) > list_term_a_b > $o ).
thf(sy_c_List_Olist__update_001t__List__Olist_It__Nat__Onat_J,type,
list_update_list_nat: list_list_nat > nat > list_nat > list_list_nat ).
thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
list_update_nat: list_nat > nat > nat > list_nat ).
thf(sy_c_List_Olist__update_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
list_update_term_a_b: list_term_a_b > nat > term_a_b > list_term_a_b ).
thf(sy_c_List_Olistrel1_001t__Nat__Onat,type,
listrel1_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_List_Olistrel_001t__Nat__Onat_001t__Nat__Onat,type,
listrel_nat_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_List_Olists_001t__List__Olist_It__Nat__Onat_J,type,
lists_list_nat: set_list_nat > set_list_list_nat ).
thf(sy_c_List_Olists_001t__Nat__Onat,type,
lists_nat: set_nat > set_list_nat ).
thf(sy_c_List_Olistset_001t__Nat__Onat,type,
listset_nat: list_set_nat > set_list_nat ).
thf(sy_c_List_Omap__tailrec__rev_001t__Nat__Onat_001t__Nat__Onat,type,
map_ta7164188454487880599at_nat: ( nat > nat ) > list_nat > list_nat > list_nat ).
thf(sy_c_List_Omaps_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
maps_l5785965478274863235st_nat: ( list_nat > list_list_nat ) > list_list_nat > list_list_nat ).
thf(sy_c_List_Omaps_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
maps_list_nat_nat: ( list_nat > list_nat ) > list_list_nat > list_nat ).
thf(sy_c_List_Omaps_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
maps_l2310356970806526633et_nat: ( list_nat > list_set_nat ) > list_list_nat > list_set_nat ).
thf(sy_c_List_Omaps_001t__List__Olist_It__Nat__Onat_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
maps_l8243100079222783927rm_a_b: ( list_nat > list_term_a_b ) > list_list_nat > list_term_a_b ).
thf(sy_c_List_Omaps_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
maps_nat_list_nat: ( nat > list_list_nat ) > list_nat > list_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_Omaps_001t__Set__Oset_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
maps_s5960583924169207593st_nat: ( set_nat > list_list_nat ) > list_set_nat > list_list_nat ).
thf(sy_c_List_Omaps_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
maps_set_nat_nat: ( set_nat > list_nat ) > list_set_nat > list_nat ).
thf(sy_c_List_Omaps_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
maps_set_nat_set_nat: ( set_nat > list_set_nat ) > list_set_nat > list_set_nat ).
thf(sy_c_List_Omaps_001t__Set__Oset_It__Nat__Onat_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
maps_s8417718525117128285rm_a_b: ( set_nat > list_term_a_b ) > list_set_nat > list_term_a_b ).
thf(sy_c_List_Omaps_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Set__Oset_It__Nat__Onat_J,type,
maps_t8811637242275172829et_nat: ( term_a_b > list_set_nat ) > list_term_a_b > list_set_nat ).
thf(sy_c_List_Omember_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_list_nat > list_nat > $o ).
thf(sy_c_List_Omember_001t__Nat__Onat,type,
member_nat: list_nat > nat > $o ).
thf(sy_c_List_Omember_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: list_set_nat > set_nat > $o ).
thf(sy_c_List_Omember_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
member_term_a_b: list_term_a_b > term_a_b > $o ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
nth_term_a_b: list_term_a_b > nat > term_a_b ).
thf(sy_c_List_Oord_Olexordp__eq_001t__Nat__Onat,type,
lexordp_eq_nat: ( nat > nat > $o ) > list_nat > list_nat > $o ).
thf(sy_c_List_Oproduct__lists_001t__List__Olist_It__Nat__Onat_J,type,
produc6783906451316923569st_nat: list_list_list_nat > list_list_list_nat ).
thf(sy_c_List_Oproduct__lists_001t__Nat__Onat,type,
product_lists_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Oproduct__lists_001t__Set__Oset_It__Nat__Onat_J,type,
produc8109398739672286679et_nat: list_list_set_nat > list_list_set_nat ).
thf(sy_c_List_Oproduct__lists_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
produc17669015410068453rm_a_b: list_list_term_a_b > list_list_term_a_b ).
thf(sy_c_List_Oremove1_001t__List__Olist_It__Nat__Onat_J,type,
remove1_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Oremove1_001t__Nat__Onat,type,
remove1_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Orotate1_001t__Nat__Onat,type,
rotate1_nat: list_nat > list_nat ).
thf(sy_c_List_Osubseqs_001t__List__Olist_It__Nat__Onat_J,type,
subseqs_list_nat: list_list_nat > list_list_list_nat ).
thf(sy_c_List_Osubseqs_001t__Nat__Onat,type,
subseqs_nat: list_nat > list_list_nat ).
thf(sy_c_List_Osubseqs_001t__Set__Oset_It__Nat__Onat_J,type,
subseqs_set_nat: list_set_nat > list_list_set_nat ).
thf(sy_c_List_Osubseqs_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
subseqs_term_a_b: list_term_a_b > list_list_term_a_b ).
thf(sy_c_List_Otake_001t__Nat__Onat,type,
take_nat: nat > list_nat > list_nat ).
thf(sy_c_Missing__List_Oconcat__lists_001t__List__Olist_It__Nat__Onat_J,type,
missin5603497496030997514st_nat: list_list_list_nat > list_list_list_nat ).
thf(sy_c_Missing__List_Oconcat__lists_001t__Nat__Onat,type,
missin4567272213201432058ts_nat: list_list_nat > list_list_nat ).
thf(sy_c_Missing__List_Oconcat__lists_001t__Set__Oset_It__Nat__Onat_J,type,
missin2626218296463274416et_nat: list_list_set_nat > list_list_set_nat ).
thf(sy_c_Missing__List_Oconcat__lists_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
missin8060632096978918206rm_a_b: list_list_term_a_b > list_list_term_a_b ).
thf(sy_c_Missing__List_Omin__list_001t__Nat__Onat,type,
missing_min_list_nat: list_nat > nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
size_s8906293707977694520rm_a_b: list_term_a_b > nat ).
thf(sy_c_Option_Ooption_ONone_001t__Sum____Type__Osum_Itf__a_Mtf__b_J,type,
none_Sum_sum_a_b: option_Sum_sum_a_b ).
thf(sy_c_Order__Relation_Opreorder__on_001t__List__Olist_It__Nat__Onat_J,type,
order_4596338698039041673st_nat: set_list_nat > set_Pr3451248702717554689st_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
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__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
ord_le1190675801316882794st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_062_I_Eo_Mt__Nat__Onat_J_J,type,
ord_less_eq_o_o_nat: ( $o > $o > nat ) > ( $o > $o > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Nat__Onat_J,type,
ord_less_eq_o_nat: ( $o > nat ) > ( $o > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
ord_le3606317655850047935st_nat: ( $o > set_list_nat ) > ( $o > set_list_nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_It__Nat__Onat_J_J,type,
ord_le7022414076629706543et_nat: ( $o > set_nat ) > ( $o > set_nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_J,type,
ord_le7949481618631132124_a_nat: ( $o > set_Pr4934435412358123699_a_nat ) > ( $o > set_Pr4934435412358123699_a_nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
ord_le4706587664122053766st_nat: set_list_list_nat > set_list_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
ord_le8406513867147106209st_nat: set_Pr3451248702717554689st_nat > set_Pr3451248702717554689st_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le3146513528884898305at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
ord_le8666007276011122963_a_nat: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat > $o ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
ord_min_nat: nat > nat > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001_062_I_Eo_Mt__Nat__Onat_J,type,
order_Greatest_o_nat: ( ( $o > nat ) > $o ) > $o > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
order_3081514539752307581st_nat: ( set_list_nat > $o ) > set_list_nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Nat__Onat_J,type,
order_5724808138429204845et_nat: ( set_nat > $o ) > set_nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
order_1399416596184996762_a_nat: ( set_Pr4934435412358123699_a_nat > $o ) > set_Pr4934435412358123699_a_nat ).
thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_001t__List__Olist_It__Nat__Onat_J,type,
produc4727192421694094319st_nat: ( nat > nat > $o ) > list_nat > produc254973753779126261st_nat ).
thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
produc3127733452865184594st_nat: ( nat > nat > $o ) > produc1828647624359046049st_nat > produc4787317212837456354st_nat ).
thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_Eo_J_001t__List__Olist_It__Nat__Onat_J,type,
produc8587622027977423880st_nat: ( nat > $o ) > list_nat > produc4226810134323546766st_nat ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
produc2694037385005941721st_nat: list_nat > list_nat > produc1828647624359046049st_nat ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
produc5151171985953862413rm_a_b: list_nat > term_a_b > produc7711739908350443733rm_a_b ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
produc8282810413953273033st_nat: nat > list_nat > produc4575160907756185873st_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_OPair_001t__Subterm____and____Context__Octxt_Itf__a_Mtf__b_J_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
produc1096132875450744233rm_a_b: subterm_and_ctxt_a_b > produc7711739908350443733rm_a_b > produc8210972263561988409rm_a_b ).
thf(sy_c_Product__Type_OPair_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__List__Olist_It__Nat__Onat_J,type,
produc4563063199488751885st_nat: term_a_b > list_nat > produc3697673438841856213st_nat ).
thf(sy_c_Product__Type_OPair_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
produc3812856575676843240rm_a_b: term_a_b > produc7711739908350443733rm_a_b > produc2732850333517536310rm_a_b ).
thf(sy_c_Product__Type_OPair_001tf__a_001t__Nat__Onat,type,
product_Pair_a_nat: a > nat > product_prod_a_nat ).
thf(sy_c_Relation_OField_001t__List__Olist_It__Nat__Onat_J,type,
field_list_nat: set_Pr3451248702717554689st_nat > set_list_nat ).
thf(sy_c_Relation_OImage_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
image_2597627202720054805st_nat: set_Pr3451248702717554689st_nat > set_list_nat > set_list_nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
collec5989764272469232197st_nat: ( list_list_nat > $o ) > set_list_list_nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
collec1570431334306492044st_nat: ( produc1828647624359046049st_nat > $o ) > set_Pr3451248702717554689st_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
collec3392354462482085612at_nat: ( product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
collec4464134535221767506_a_nat: ( product_prod_a_nat > $o ) > set_Pr4934435412358123699_a_nat ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
insert_list_nat2: list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat2: nat > set_nat > set_nat ).
thf(sy_c_Set_Ois__singleton_001t__List__Olist_It__Nat__Onat_J,type,
is_sin2641923865335537900st_nat: set_list_nat > $o ).
thf(sy_c_Set_Oremove_001t__List__Olist_It__Nat__Onat_J,type,
remove_list_nat: list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Subterm__and__Context_Octxt_OHole_001tf__a_001tf__b,type,
subterm_and_Hole_a_b: subterm_and_ctxt_a_b ).
thf(sy_c_Subterm__and__Context_Octxt_OMore_001tf__a_001tf__b,type,
subterm_and_More_a_b: a > list_term_a_b > subterm_and_ctxt_a_b > list_term_a_b > subterm_and_ctxt_a_b ).
thf(sy_c_Subterm__and__Context_Octxt__apply__term_001tf__a_001tf__b,type,
subter2376574525758040790rm_a_b: subterm_and_ctxt_a_b > term_a_b > term_a_b ).
thf(sy_c_Term_Oterm_OFun_001tf__a_001tf__b,type,
fun_a_b: a > list_term_a_b > term_a_b ).
thf(sy_c_Term_Oterm_OVar_001tf__b_001tf__a,type,
var_b_a: b > term_a_b ).
thf(sy_c_Term_Oterm_Oargs_001tf__a_001tf__b,type,
args_a_b: term_a_b > list_term_a_b ).
thf(sy_c_Term_Oterm_Omap__term_001tf__a_001tf__a_001tf__b_001tf__b,type,
map_term_a_a_b_b: ( a > a ) > ( b > b ) > term_a_b > term_a_b ).
thf(sy_c_Term__Context_Octxt__at__pos_001tf__a_001tf__b,type,
term_ctxt_at_pos_a_b: term_a_b > list_nat > subterm_and_ctxt_a_b ).
thf(sy_c_Term__Context_Ofun__at_001tf__a_001tf__b,type,
term_fun_at_a_b: term_a_b > list_nat > option_Sum_sum_a_b ).
thf(sy_c_Term__Context_Ofunas__ctxt_001tf__a_001tf__b,type,
term_funas_ctxt_a_b: subterm_and_ctxt_a_b > set_Pr4934435412358123699_a_nat ).
thf(sy_c_Term__Context_Ofunas__term_001tf__a_001tf__b,type,
term_funas_term_a_b: term_a_b > set_Pr4934435412358123699_a_nat ).
thf(sy_c_Term__Context_Oground_001tf__a_001tf__b,type,
term_ground_a_b: term_a_b > $o ).
thf(sy_c_Term__Context_Ohole__pos_001tf__a_001tf__b,type,
term_hole_pos_a_b: subterm_and_ctxt_a_b > list_nat ).
thf(sy_c_Term__Context_Opos__diff_001t__List__Olist_It__Nat__Onat_J,type,
term_p7564741194569991203st_nat: list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_Term__Context_Opos__diff_001t__Nat__Onat,type,
term_pos_diff_nat: list_nat > list_nat > list_nat ).
thf(sy_c_Term__Context_Opos__diff_001t__Set__Oset_It__Nat__Onat_J,type,
term_p572219117888194121et_nat: list_set_nat > list_set_nat > list_set_nat ).
thf(sy_c_Term__Context_Opos__diff_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
term_p798503758663136087rm_a_b: list_term_a_b > list_term_a_b > list_term_a_b ).
thf(sy_c_Term__Context_Oposition__less__eq_001t__List__Olist_It__Nat__Onat_J,type,
term_p5934426891874639750st_nat: list_list_nat > list_list_nat > $o ).
thf(sy_c_Term__Context_Oposition__less__eq_001t__Nat__Onat,type,
term_p3503116865373065078eq_nat: list_nat > list_nat > $o ).
thf(sy_c_Term__Context_Oposition__less__eq_001t__Set__Oset_It__Nat__Onat_J,type,
term_p5979160145136640812et_nat: list_set_nat > list_set_nat > $o ).
thf(sy_c_Term__Context_Oposition__less__eq_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
term_p8391561492822560442rm_a_b: list_term_a_b > list_term_a_b > $o ).
thf(sy_c_Term__Context_Oposition__par_001t__List__Olist_It__Nat__Onat_J,type,
term_p4950861579910180738st_nat: list_list_nat > list_list_nat > $o ).
thf(sy_c_Term__Context_Oposition__par_001t__Nat__Onat,type,
term_p5017330785391824242ar_nat: list_nat > list_nat > $o ).
thf(sy_c_Term__Context_Oposition__par_001t__Set__Oset_It__Nat__Onat_J,type,
term_p7908618117148864808et_nat: list_set_nat > list_set_nat > $o ).
thf(sy_c_Term__Context_Oposition__par_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
term_p7407996180858101430rm_a_b: list_term_a_b > list_term_a_b > $o ).
thf(sy_c_Term__Context_Oposs_001tf__a_001tf__b,type,
term_poss_a_b: term_a_b > set_list_nat ).
thf(sy_c_Term__Context_Oreplace__term__at_001tf__a_001tf__b,type,
term_r6860082780075436317at_a_b: term_a_b > list_nat > term_a_b > term_a_b ).
thf(sy_c_Term__Context_Oreplace__term__at__rel_001tf__a_001tf__b,type,
term_r1280879029893354718el_a_b: produc2732850333517536310rm_a_b > produc2732850333517536310rm_a_b > $o ).
thf(sy_c_Term__Context_Osubt__at_001tf__a_001tf__b,type,
term_subt_at_a_b: term_a_b > list_nat > term_a_b ).
thf(sy_c_Term__Context_Osubt__at__rel_001tf__a_001tf__b,type,
term_subt_at_rel_a_b: produc3697673438841856213st_nat > produc3697673438841856213st_nat > $o ).
thf(sy_c_Terms__Positions_Octxt__well__def__hole__path_001tf__a_001tf__b,type,
terms_8374513854926927137th_a_b: set_Pr4934435412358123699_a_nat > subterm_and_ctxt_a_b > $o ).
thf(sy_c_Terms__Positions_Oinv__const__ctxt_001tf__a_001tf__b,type,
terms_3799363064701517935xt_a_b: set_Pr4934435412358123699_a_nat > b > subterm_and_ctxt_a_b > subterm_and_ctxt_a_b ).
thf(sy_c_Terms__Positions_Oinv__const__ctxt_H_001tf__a_001tf__b,type,
terms_130083692264552600xt_a_b: set_Pr4934435412358123699_a_nat > b > subterm_and_ctxt_a_b > term_a_b ).
thf(sy_c_Terms__Positions_Oposs__of__term_001tf__a_001tf__b,type,
terms_7168686267159881682rm_a_b: term_a_b > term_a_b > set_list_nat ).
thf(sy_c_Terms__Positions_Oreplace__term__context__at_001tf__a_001tf__b,type,
terms_4774307173741787698at_a_b: subterm_and_ctxt_a_b > list_nat > term_a_b > subterm_and_ctxt_a_b ).
thf(sy_c_Terms__Positions_Oreplace__term__context__at__rel_001tf__a_001tf__b,type,
terms_6690452295706263049el_a_b: produc8210972263561988409rm_a_b > produc8210972263561988409rm_a_b > $o ).
thf(sy_c_Terms__Positions_Oterm__to__sig_001tf__a_001tf__b,type,
terms_8519481630511763164ig_a_b: set_Pr4934435412358123699_a_nat > b > term_a_b > term_a_b ).
thf(sy_c_Terms__Positions_Ovarposs_001tf__a_001tf__b,type,
terms_varposs_a_b: term_a_b > set_list_nat ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Subterm____and____Context__Octxt_Itf__a_Mtf__b_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
accp_P6624108184151110466rm_a_b: ( produc8210972263561988409rm_a_b > produc8210972263561988409rm_a_b > $o ) > produc8210972263561988409rm_a_b > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
accp_P682940083893826398st_nat: ( produc3697673438841856213st_nat > produc3697673438841856213st_nat > $o ) > produc3697673438841856213st_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
accp_P2729577386226225901rm_a_b: ( produc2732850333517536310rm_a_b > produc2732850333517536310rm_a_b > $o ) > produc2732850333517536310rm_a_b > $o ).
thf(sy_c_member_001_062_I_Eo_Mt__Nat__Onat_J,type,
member_o_nat: ( $o > nat ) > set_o_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
member_list_list_nat: list_list_nat > set_list_list_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat2: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat2: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
member7340969449405702474st_nat: produc1828647624359046049st_nat > set_Pr3451248702717554689st_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
member5724188588386418708_a_nat: product_prod_a_nat > set_Pr4934435412358123699_a_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
member_set_list_nat: set_list_nat > set_set_list_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat2: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
member8034581882086557258_a_nat: set_Pr4934435412358123699_a_nat > set_se6121441497158405097_a_nat > $o ).
thf(sy_v_p,type,
p: list_nat ).
thf(sy_v_s,type,
s: term_a_b ).
thf(sy_v_u,type,
u: term_a_b ).
% Relevant facts (1254)
thf(fact_0_replace__term__at__same__pos,axiom,
! [S: term_a_b,P: list_nat,U: term_a_b,T: term_a_b] :
( ( term_r6860082780075436317at_a_b @ ( term_r6860082780075436317at_a_b @ S @ P @ U ) @ P @ T )
= ( term_r6860082780075436317at_a_b @ S @ P @ T ) ) ).
% replace_term_at_same_pos
thf(fact_1_pos__replace__at__pres,axiom,
! [P: list_nat,S: term_a_b,T: term_a_b] :
( ( member_list_nat2 @ P @ ( term_poss_a_b @ S ) )
=> ( member_list_nat2 @ P @ ( term_poss_a_b @ ( term_r6860082780075436317at_a_b @ S @ P @ T ) ) ) ) ).
% pos_replace_at_pres
thf(fact_2_replace__term__at__not__poss,axiom,
! [P: list_nat,S: term_a_b,T: term_a_b] :
( ~ ( member_list_nat2 @ P @ ( term_poss_a_b @ S ) )
=> ( ( term_r6860082780075436317at_a_b @ S @ P @ T )
= S ) ) ).
% replace_term_at_not_poss
thf(fact_3_par__pos__replace__pres,axiom,
! [P: list_nat,S: term_a_b,Q: list_nat,T: term_a_b] :
( ( member_list_nat2 @ P @ ( term_poss_a_b @ S ) )
=> ( ( term_p5017330785391824242ar_nat @ P @ Q )
=> ( member_list_nat2 @ P @ ( term_poss_a_b @ ( term_r6860082780075436317at_a_b @ S @ Q @ T ) ) ) ) ) ).
% par_pos_replace_pres
thf(fact_4_poss__of__term__possI,axiom,
! [P: list_nat,S: term_a_b,U: term_a_b] :
( ( member_list_nat2 @ P @ ( term_poss_a_b @ S ) )
=> ( ( ( term_subt_at_a_b @ S @ P )
= U )
=> ( member_list_nat2 @ P @ ( terms_7168686267159881682rm_a_b @ U @ S ) ) ) ) ).
% poss_of_term_possI
thf(fact_5_poss__of__termE,axiom,
! [P: list_nat,U: term_a_b,S: term_a_b] :
( ( member_list_nat2 @ P @ ( terms_7168686267159881682rm_a_b @ U @ S ) )
=> ~ ( ( member_list_nat2 @ P @ ( term_poss_a_b @ S ) )
=> ( ( term_subt_at_a_b @ S @ P )
!= U ) ) ) ).
% poss_of_termE
thf(fact_6_varposs__imp__poss,axiom,
! [P: list_nat,S: term_a_b] :
( ( member_list_nat2 @ P @ ( terms_varposs_a_b @ S ) )
=> ( member_list_nat2 @ P @ ( term_poss_a_b @ S ) ) ) ).
% varposs_imp_poss
thf(fact_7_replace__term__at__subt__at__id,axiom,
! [S: term_a_b,P: list_nat] :
( ( term_r6860082780075436317at_a_b @ S @ P @ ( term_subt_at_a_b @ S @ P ) )
= S ) ).
% replace_term_at_subt_at_id
thf(fact_8_map__term__replace__at__dist,axiom,
! [P: list_nat,S: term_a_b,F: a > a,G: b > b,T: term_a_b] :
( ( member_list_nat2 @ P @ ( term_poss_a_b @ S ) )
=> ( ( term_r6860082780075436317at_a_b @ ( map_term_a_a_b_b @ F @ G @ S ) @ P @ ( map_term_a_a_b_b @ F @ G @ T ) )
= ( map_term_a_a_b_b @ F @ G @ ( term_r6860082780075436317at_a_b @ S @ P @ T ) ) ) ) ).
% map_term_replace_at_dist
thf(fact_9_par__pos__replace__term__at,axiom,
! [P: list_nat,S: term_a_b,Q: list_nat,T: term_a_b] :
( ( member_list_nat2 @ P @ ( term_poss_a_b @ S ) )
=> ( ( term_p5017330785391824242ar_nat @ P @ Q )
=> ( ( term_subt_at_a_b @ ( term_r6860082780075436317at_a_b @ S @ Q @ T ) @ P )
= ( term_subt_at_a_b @ S @ P ) ) ) ) ).
% par_pos_replace_term_at
thf(fact_10_ctxt__of__pos__term__apply__replace__at__ident,axiom,
! [P: list_nat,S: term_a_b,T: term_a_b] :
( ( member_list_nat2 @ P @ ( term_poss_a_b @ S ) )
=> ( ( subter2376574525758040790rm_a_b @ ( term_ctxt_at_pos_a_b @ S @ P ) @ T )
= ( term_r6860082780075436317at_a_b @ S @ P @ T ) ) ) ).
% ctxt_of_pos_term_apply_replace_at_ident
thf(fact_11_finite__poss,axiom,
! [S: term_a_b] : ( finite8100373058378681591st_nat @ ( term_poss_a_b @ S ) ) ).
% finite_poss
thf(fact_12_funas__term__replace__at__lower,axiom,
! [P: list_nat,S: term_a_b,T: term_a_b] :
( ( member_list_nat2 @ P @ ( term_poss_a_b @ S ) )
=> ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ ( term_funas_term_a_b @ ( term_r6860082780075436317at_a_b @ S @ P @ T ) ) ) ) ).
% funas_term_replace_at_lower
thf(fact_13_replace__term__at__above,axiom,
! [P: list_nat,Q: list_nat,S: term_a_b,T: term_a_b,U: term_a_b] :
( ( term_p3503116865373065078eq_nat @ P @ Q )
=> ( ( term_r6860082780075436317at_a_b @ ( term_r6860082780075436317at_a_b @ S @ Q @ T ) @ P @ U )
= ( term_r6860082780075436317at_a_b @ S @ P @ U ) ) ) ).
% replace_term_at_above
thf(fact_14_par__not__refl,axiom,
! [P: list_nat] :
~ ( term_p5017330785391824242ar_nat @ P @ P ) ).
% par_not_refl
thf(fact_15_funas__term__subterm__atI,axiom,
! [P: list_nat,S: term_a_b,F2: set_Pr4934435412358123699_a_nat] :
( ( member_list_nat2 @ P @ ( term_poss_a_b @ S ) )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ F2 )
=> ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( term_subt_at_a_b @ S @ P ) ) @ F2 ) ) ) ).
% funas_term_subterm_atI
thf(fact_16_position__par__def,axiom,
( term_p5017330785391824242ar_nat
= ( ^ [P2: list_nat,Q2: list_nat] :
( ~ ( term_p3503116865373065078eq_nat @ P2 @ Q2 )
& ~ ( term_p3503116865373065078eq_nat @ Q2 @ P2 ) ) ) ) ).
% position_par_def
thf(fact_17_finite__varposs,axiom,
! [S: term_a_b] : ( finite8100373058378681591st_nat @ ( terms_varposs_a_b @ S ) ) ).
% finite_varposs
thf(fact_18_position__less__refl,axiom,
! [P: list_nat] : ( term_p3503116865373065078eq_nat @ P @ P ) ).
% position_less_refl
thf(fact_19_ctxt__at__pos__subt__at__id,axiom,
! [P: list_nat,T: term_a_b] :
( ( member_list_nat2 @ P @ ( term_poss_a_b @ T ) )
=> ( ( subter2376574525758040790rm_a_b @ ( term_ctxt_at_pos_a_b @ T @ P ) @ ( term_subt_at_a_b @ T @ P ) )
= T ) ) ).
% ctxt_at_pos_subt_at_id
thf(fact_20_ctxt__at__pos__subt__at__pos,axiom,
! [P: list_nat,T: term_a_b,U: term_a_b] :
( ( member_list_nat2 @ P @ ( term_poss_a_b @ T ) )
=> ( ( term_subt_at_a_b @ ( subter2376574525758040790rm_a_b @ ( term_ctxt_at_pos_a_b @ T @ P ) @ U ) @ P )
= U ) ) ).
% ctxt_at_pos_subt_at_pos
thf(fact_21_subst__at__ctxt__at__eq__termI,axiom,
! [P: list_nat,S: term_a_b,T: term_a_b] :
( ( member_list_nat2 @ P @ ( term_poss_a_b @ S ) )
=> ( ( member_list_nat2 @ P @ ( term_poss_a_b @ T ) )
=> ( ( ( term_subt_at_a_b @ S @ P )
= ( term_subt_at_a_b @ T @ P ) )
=> ( ( ( term_ctxt_at_pos_a_b @ S @ P )
= ( term_ctxt_at_pos_a_b @ T @ P ) )
=> ( S = T ) ) ) ) ) ).
% subst_at_ctxt_at_eq_termI
thf(fact_22_subst__at__ctxt__at__eq__termD,axiom,
! [S: term_a_b,T: term_a_b,P: list_nat] :
( ( S = T )
=> ( ( member_list_nat2 @ P @ ( term_poss_a_b @ T ) )
=> ( ( ( term_subt_at_a_b @ S @ P )
= ( term_subt_at_a_b @ T @ P ) )
& ( ( term_ctxt_at_pos_a_b @ S @ P )
= ( term_ctxt_at_pos_a_b @ T @ P ) ) ) ) ) ).
% subst_at_ctxt_at_eq_termD
thf(fact_23_parallel__replace__term__commute,axiom,
! [P: list_nat,Q: list_nat,S: term_a_b,T: term_a_b,U: term_a_b] :
( ( term_p5017330785391824242ar_nat @ P @ Q )
=> ( ( term_r6860082780075436317at_a_b @ ( term_r6860082780075436317at_a_b @ S @ P @ T ) @ Q @ U )
= ( term_r6860082780075436317at_a_b @ ( term_r6860082780075436317at_a_b @ S @ Q @ U ) @ P @ T ) ) ) ).
% parallel_replace_term_commute
thf(fact_24_ctxt__eq,axiom,
! [C: subterm_and_ctxt_a_b,S: term_a_b,T: term_a_b] :
( ( ( subter2376574525758040790rm_a_b @ C @ S )
= ( subter2376574525758040790rm_a_b @ C @ T ) )
= ( S = T ) ) ).
% ctxt_eq
thf(fact_25_subsetI,axiom,
! [A: set_list_list_nat,B: set_list_list_nat] :
( ! [X: list_list_nat] :
( ( member_list_list_nat @ X @ A )
=> ( member_list_list_nat @ X @ B ) )
=> ( ord_le4706587664122053766st_nat @ A @ B ) ) ).
% subsetI
thf(fact_26_subsetI,axiom,
! [A: set_Pr3451248702717554689st_nat,B: set_Pr3451248702717554689st_nat] :
( ! [X: produc1828647624359046049st_nat] :
( ( member7340969449405702474st_nat @ X @ A )
=> ( member7340969449405702474st_nat @ X @ B ) )
=> ( ord_le8406513867147106209st_nat @ A @ B ) ) ).
% subsetI
thf(fact_27_subsetI,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ! [X: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X @ A )
=> ( member8440522571783428010at_nat @ X @ B ) )
=> ( ord_le3146513528884898305at_nat @ A @ B ) ) ).
% subsetI
thf(fact_28_subsetI,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ! [X: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X @ A )
=> ( member5724188588386418708_a_nat @ X @ B ) )
=> ( ord_le8666007276011122963_a_nat @ A @ B ) ) ).
% subsetI
thf(fact_29_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X: nat] :
( ( member_nat2 @ X @ A )
=> ( member_nat2 @ X @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_30_subsetI,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ! [X: list_nat] :
( ( member_list_nat2 @ X @ A )
=> ( member_list_nat2 @ X @ B ) )
=> ( ord_le6045566169113846134st_nat @ A @ B ) ) ).
% subsetI
thf(fact_31_subset__antisym,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_32_subset__antisym,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ord_le6045566169113846134st_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_33_subset__antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_34_order__refl,axiom,
! [X2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_35_order__refl,axiom,
! [X2: set_list_nat] : ( ord_le6045566169113846134st_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_36_order__refl,axiom,
! [X2: $o > nat] : ( ord_less_eq_o_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_37_order__refl,axiom,
! [X2: set_nat] : ( ord_less_eq_set_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_38_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_39_dual__order_Orefl,axiom,
! [A2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_40_dual__order_Orefl,axiom,
! [A2: set_list_nat] : ( ord_le6045566169113846134st_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_41_dual__order_Orefl,axiom,
! [A2: $o > nat] : ( ord_less_eq_o_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_42_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_43_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_44_greater__eq__subt__at__replace,axiom,
! [P: list_nat,S: term_a_b,Q: list_nat,T: term_a_b] :
( ( member_list_nat2 @ P @ ( term_poss_a_b @ S ) )
=> ( ( term_p3503116865373065078eq_nat @ Q @ P )
=> ( ( term_subt_at_a_b @ ( term_r6860082780075436317at_a_b @ S @ Q @ T ) @ P )
= ( term_subt_at_a_b @ T @ ( term_pos_diff_nat @ P @ Q ) ) ) ) ) ).
% greater_eq_subt_at_replace
thf(fact_45_less__eq__subt__at__replace,axiom,
! [P: list_nat,S: term_a_b,Q: list_nat,T: term_a_b] :
( ( member_list_nat2 @ P @ ( term_poss_a_b @ S ) )
=> ( ( term_p3503116865373065078eq_nat @ P @ Q )
=> ( ( term_subt_at_a_b @ ( term_r6860082780075436317at_a_b @ S @ Q @ T ) @ P )
= ( term_r6860082780075436317at_a_b @ ( term_subt_at_a_b @ S @ P ) @ ( term_pos_diff_nat @ Q @ P ) @ T ) ) ) ) ).
% less_eq_subt_at_replace
thf(fact_46_finite__subset,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( finite6644898363146130708_a_nat @ B )
=> ( finite6644898363146130708_a_nat @ A ) ) ) ).
% finite_subset
thf(fact_47_finite__subset,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( finite8100373058378681591st_nat @ B )
=> ( finite8100373058378681591st_nat @ A ) ) ) ).
% finite_subset
thf(fact_48_finite__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_49_infinite__super,axiom,
! [S2: set_Pr4934435412358123699_a_nat,T2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ S2 @ T2 )
=> ( ~ ( finite6644898363146130708_a_nat @ S2 )
=> ~ ( finite6644898363146130708_a_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_50_infinite__super,axiom,
! [S2: set_list_nat,T2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ S2 @ T2 )
=> ( ~ ( finite8100373058378681591st_nat @ S2 )
=> ~ ( finite8100373058378681591st_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_51_infinite__super,axiom,
! [S2: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_52_rev__finite__subset,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( finite6644898363146130708_a_nat @ B )
=> ( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( finite6644898363146130708_a_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_53_rev__finite__subset,axiom,
! [B: set_list_nat,A: set_list_nat] :
( ( finite8100373058378681591st_nat @ B )
=> ( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( finite8100373058378681591st_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_54_rev__finite__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_55_finite__has__maximal2,axiom,
! [A: set_se6121441497158405097_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( finite568477829172517706_a_nat @ A )
=> ( ( member8034581882086557258_a_nat @ A2 @ A )
=> ? [X: set_Pr4934435412358123699_a_nat] :
( ( member8034581882086557258_a_nat @ X @ A )
& ( ord_le8666007276011122963_a_nat @ A2 @ X )
& ! [Xa: set_Pr4934435412358123699_a_nat] :
( ( member8034581882086557258_a_nat @ Xa @ A )
=> ( ( ord_le8666007276011122963_a_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_56_finite__has__maximal2,axiom,
! [A: set_set_list_nat,A2: set_list_nat] :
( ( finite7047420756378620717st_nat @ A )
=> ( ( member_set_list_nat @ A2 @ A )
=> ? [X: set_list_nat] :
( ( member_set_list_nat @ X @ A )
& ( ord_le6045566169113846134st_nat @ A2 @ X )
& ! [Xa: set_list_nat] :
( ( member_set_list_nat @ Xa @ A )
=> ( ( ord_le6045566169113846134st_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_57_finite__has__maximal2,axiom,
! [A: set_o_nat,A2: $o > nat] :
( ( finite_finite_o_nat @ A )
=> ( ( member_o_nat @ A2 @ A )
=> ? [X: $o > nat] :
( ( member_o_nat @ X @ A )
& ( ord_less_eq_o_nat @ A2 @ X )
& ! [Xa: $o > nat] :
( ( member_o_nat @ Xa @ A )
=> ( ( ord_less_eq_o_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_58_finite__has__maximal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat2 @ A2 @ A )
=> ? [X: set_nat] :
( ( member_set_nat2 @ X @ A )
& ( ord_less_eq_set_nat @ A2 @ X )
& ! [Xa: set_nat] :
( ( member_set_nat2 @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_59_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat2 @ A2 @ A )
=> ? [X: nat] :
( ( member_nat2 @ X @ A )
& ( ord_less_eq_nat @ A2 @ X )
& ! [Xa: nat] :
( ( member_nat2 @ Xa @ A )
=> ( ( ord_less_eq_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_60_replace__term__at__below,axiom,
! [P: list_nat,Q: list_nat,S: term_a_b,T: term_a_b,U: term_a_b] :
( ( ( P != Q )
& ( term_p3503116865373065078eq_nat @ P @ Q ) )
=> ( ( term_r6860082780075436317at_a_b @ ( term_r6860082780075436317at_a_b @ S @ P @ T ) @ Q @ U )
= ( term_r6860082780075436317at_a_b @ S @ P @ ( term_r6860082780075436317at_a_b @ T @ ( term_pos_diff_nat @ Q @ P ) @ U ) ) ) ) ).
% replace_term_at_below
thf(fact_61_order__antisym__conv,axiom,
! [Y: set_Pr4934435412358123699_a_nat,X2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ Y @ X2 )
=> ( ( ord_le8666007276011122963_a_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_62_order__antisym__conv,axiom,
! [Y: set_list_nat,X2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ Y @ X2 )
=> ( ( ord_le6045566169113846134st_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_63_order__antisym__conv,axiom,
! [Y: $o > nat,X2: $o > nat] :
( ( ord_less_eq_o_nat @ Y @ X2 )
=> ( ( ord_less_eq_o_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_64_order__antisym__conv,axiom,
! [Y: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_65_order__antisym__conv,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_66_linorder__le__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_67_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_68_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_69_ord__le__eq__subst,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > nat,C2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_70_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > set_list_nat,C2: set_list_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le6045566169113846134st_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_71_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > $o > nat,C2: $o > nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_o_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_72_ord__le__eq__subst,axiom,
! [A2: set_list_nat,B2: set_list_nat,F: set_list_nat > nat,C2: nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: set_list_nat,Y2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_73_ord__le__eq__subst,axiom,
! [A2: $o > nat,B2: $o > nat,F: ( $o > nat ) > nat,C2: nat] :
( ( ord_less_eq_o_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: $o > nat,Y2: $o > nat] :
( ( ord_less_eq_o_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_74_ord__le__eq__subst,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_75_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_le8666007276011122963_a_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_76_ord__le__eq__subst,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,F: set_Pr4934435412358123699_a_nat > nat,C2: nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_77_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_78_ord__eq__le__subst,axiom,
! [A2: set_nat,F: nat > set_nat,B2: nat,C2: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_79_ord__eq__le__subst,axiom,
! [A2: nat,F: set_nat > nat,B2: set_nat,C2: set_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_80_ord__eq__le__subst,axiom,
! [A2: set_list_nat,F: nat > set_list_nat,B2: nat,C2: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le6045566169113846134st_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_81_ord__eq__le__subst,axiom,
! [A2: $o > nat,F: nat > $o > nat,B2: nat,C2: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_o_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_82_ord__eq__le__subst,axiom,
! [A2: nat,F: set_list_nat > nat,B2: set_list_nat,C2: set_list_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le6045566169113846134st_nat @ B2 @ C2 )
=> ( ! [X: set_list_nat,Y2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_83_ord__eq__le__subst,axiom,
! [A2: nat,F: ( $o > nat ) > nat,B2: $o > nat,C2: $o > nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_o_nat @ B2 @ C2 )
=> ( ! [X: $o > nat,Y2: $o > nat] :
( ( ord_less_eq_o_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_84_ord__eq__le__subst,axiom,
! [A2: set_nat,F: set_nat > set_nat,B2: set_nat,C2: set_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_85_ord__eq__le__subst,axiom,
! [A2: set_Pr4934435412358123699_a_nat,F: nat > set_Pr4934435412358123699_a_nat,B2: nat,C2: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_le8666007276011122963_a_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le8666007276011122963_a_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_86_ord__eq__le__subst,axiom,
! [A2: nat,F: set_Pr4934435412358123699_a_nat > nat,B2: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ C2 )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_87_linorder__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_88_order__eq__refl,axiom,
! [X2: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( X2 = Y )
=> ( ord_le8666007276011122963_a_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_89_order__eq__refl,axiom,
! [X2: set_list_nat,Y: set_list_nat] :
( ( X2 = Y )
=> ( ord_le6045566169113846134st_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_90_order__eq__refl,axiom,
! [X2: $o > nat,Y: $o > nat] :
( ( X2 = Y )
=> ( ord_less_eq_o_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_91_order__eq__refl,axiom,
! [X2: set_nat,Y: set_nat] :
( ( X2 = Y )
=> ( ord_less_eq_set_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_92_order__eq__refl,axiom,
! [X2: nat,Y: nat] :
( ( X2 = Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_93_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_94_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_95_order__subst2,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > nat,C2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_96_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_list_nat,C2: set_list_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_le6045566169113846134st_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le6045566169113846134st_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_97_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > $o > nat,C2: $o > nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_o_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_o_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_98_order__subst2,axiom,
! [A2: set_list_nat,B2: set_list_nat,F: set_list_nat > nat,C2: nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X: set_list_nat,Y2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_99_order__subst2,axiom,
! [A2: $o > nat,B2: $o > nat,F: ( $o > nat ) > nat,C2: nat] :
( ( ord_less_eq_o_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X: $o > nat,Y2: $o > nat] :
( ( ord_less_eq_o_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_100_order__subst2,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_101_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_le8666007276011122963_a_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_le8666007276011122963_a_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_102_order__subst2,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,F: set_Pr4934435412358123699_a_nat > nat,C2: nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_103_order__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_104_order__subst1,axiom,
! [A2: nat,F: set_nat > nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_105_order__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B2: nat,C2: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_106_order__subst1,axiom,
! [A2: nat,F: set_list_nat > nat,B2: set_list_nat,C2: set_list_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le6045566169113846134st_nat @ B2 @ C2 )
=> ( ! [X: set_list_nat,Y2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_107_order__subst1,axiom,
! [A2: nat,F: ( $o > nat ) > nat,B2: $o > nat,C2: $o > nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_o_nat @ B2 @ C2 )
=> ( ! [X: $o > nat,Y2: $o > nat] :
( ( ord_less_eq_o_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_108_order__subst1,axiom,
! [A2: set_list_nat,F: nat > set_list_nat,B2: nat,C2: nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_le6045566169113846134st_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le6045566169113846134st_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_109_order__subst1,axiom,
! [A2: $o > nat,F: nat > $o > nat,B2: nat,C2: nat] :
( ( ord_less_eq_o_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_o_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_o_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_110_order__subst1,axiom,
! [A2: set_nat,F: set_nat > set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_111_order__subst1,axiom,
! [A2: nat,F: set_Pr4934435412358123699_a_nat > nat,B2: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ C2 )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_112_order__subst1,axiom,
! [A2: set_Pr4934435412358123699_a_nat,F: nat > set_Pr4934435412358123699_a_nat,B2: nat,C2: nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_le8666007276011122963_a_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_le8666007276011122963_a_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_113_mem__Collect__eq,axiom,
! [A2: nat,P3: nat > $o] :
( ( member_nat2 @ A2 @ ( collect_nat @ P3 ) )
= ( P3 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_114_mem__Collect__eq,axiom,
! [A2: list_list_nat,P3: list_list_nat > $o] :
( ( member_list_list_nat @ A2 @ ( collec5989764272469232197st_nat @ P3 ) )
= ( P3 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_115_mem__Collect__eq,axiom,
! [A2: produc1828647624359046049st_nat,P3: produc1828647624359046049st_nat > $o] :
( ( member7340969449405702474st_nat @ A2 @ ( collec1570431334306492044st_nat @ P3 ) )
= ( P3 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_116_mem__Collect__eq,axiom,
! [A2: product_prod_nat_nat,P3: product_prod_nat_nat > $o] :
( ( member8440522571783428010at_nat @ A2 @ ( collec3392354462482085612at_nat @ P3 ) )
= ( P3 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_117_mem__Collect__eq,axiom,
! [A2: product_prod_a_nat,P3: product_prod_a_nat > $o] :
( ( member5724188588386418708_a_nat @ A2 @ ( collec4464134535221767506_a_nat @ P3 ) )
= ( P3 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_118_mem__Collect__eq,axiom,
! [A2: list_nat,P3: list_nat > $o] :
( ( member_list_nat2 @ A2 @ ( collect_list_nat @ P3 ) )
= ( P3 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_119_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat2 @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_120_Collect__mem__eq,axiom,
! [A: set_list_list_nat] :
( ( collec5989764272469232197st_nat
@ ^ [X3: list_list_nat] : ( member_list_list_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_121_Collect__mem__eq,axiom,
! [A: set_Pr3451248702717554689st_nat] :
( ( collec1570431334306492044st_nat
@ ^ [X3: produc1828647624359046049st_nat] : ( member7340969449405702474st_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_122_Collect__mem__eq,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ( collec3392354462482085612at_nat
@ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_123_Collect__mem__eq,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ( collec4464134535221767506_a_nat
@ ^ [X3: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_124_Collect__mem__eq,axiom,
! [A: set_list_nat] :
( ( collect_list_nat
@ ^ [X3: list_nat] : ( member_list_nat2 @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_125_Collect__cong,axiom,
! [P3: list_nat > $o,Q3: list_nat > $o] :
( ! [X: list_nat] :
( ( P3 @ X )
= ( Q3 @ X ) )
=> ( ( collect_list_nat @ P3 )
= ( collect_list_nat @ Q3 ) ) ) ).
% Collect_cong
thf(fact_126_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_Pr4934435412358123699_a_nat,Z: set_Pr4934435412358123699_a_nat] : ( Y3 = Z ) )
= ( ^ [A3: set_Pr4934435412358123699_a_nat,B3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A3 @ B3 )
& ( ord_le8666007276011122963_a_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_127_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_list_nat,Z: set_list_nat] : ( Y3 = Z ) )
= ( ^ [A3: set_list_nat,B3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A3 @ B3 )
& ( ord_le6045566169113846134st_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_128_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: $o > nat,Z: $o > nat] : ( Y3 = Z ) )
= ( ^ [A3: $o > nat,B3: $o > nat] :
( ( ord_less_eq_o_nat @ A3 @ B3 )
& ( ord_less_eq_o_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_129_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_130_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_131_le__fun__def,axiom,
( ord_less_eq_o_nat
= ( ^ [F3: $o > nat,G2: $o > nat] :
! [X3: $o] : ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% le_fun_def
thf(fact_132_le__funI,axiom,
! [F: $o > nat,G: $o > nat] :
( ! [X: $o] : ( ord_less_eq_nat @ ( F @ X ) @ ( G @ X ) )
=> ( ord_less_eq_o_nat @ F @ G ) ) ).
% le_funI
thf(fact_133_le__funE,axiom,
! [F: $o > nat,G: $o > nat,X2: $o] :
( ( ord_less_eq_o_nat @ F @ G )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( G @ X2 ) ) ) ).
% le_funE
thf(fact_134_le__funD,axiom,
! [F: $o > nat,G: $o > nat,X2: $o] :
( ( ord_less_eq_o_nat @ F @ G )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( G @ X2 ) ) ) ).
% le_funD
thf(fact_135_antisym,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_136_antisym,axiom,
! [A2: set_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B2 )
=> ( ( ord_le6045566169113846134st_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_137_antisym,axiom,
! [A2: $o > nat,B2: $o > nat] :
( ( ord_less_eq_o_nat @ A2 @ B2 )
=> ( ( ord_less_eq_o_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_138_antisym,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_139_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_140_dual__order_Otrans,axiom,
! [B2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B2 @ A2 )
=> ( ( ord_le8666007276011122963_a_nat @ C2 @ B2 )
=> ( ord_le8666007276011122963_a_nat @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_141_dual__order_Otrans,axiom,
! [B2: set_list_nat,A2: set_list_nat,C2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B2 @ A2 )
=> ( ( ord_le6045566169113846134st_nat @ C2 @ B2 )
=> ( ord_le6045566169113846134st_nat @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_142_dual__order_Otrans,axiom,
! [B2: $o > nat,A2: $o > nat,C2: $o > nat] :
( ( ord_less_eq_o_nat @ B2 @ A2 )
=> ( ( ord_less_eq_o_nat @ C2 @ B2 )
=> ( ord_less_eq_o_nat @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_143_dual__order_Otrans,axiom,
! [B2: set_nat,A2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ C2 @ B2 )
=> ( ord_less_eq_set_nat @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_144_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C2 @ B2 )
=> ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_145_dual__order_Oantisym,axiom,
! [B2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B2 @ A2 )
=> ( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_146_dual__order_Oantisym,axiom,
! [B2: set_list_nat,A2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B2 @ A2 )
=> ( ( ord_le6045566169113846134st_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_147_dual__order_Oantisym,axiom,
! [B2: $o > nat,A2: $o > nat] :
( ( ord_less_eq_o_nat @ B2 @ A2 )
=> ( ( ord_less_eq_o_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_148_dual__order_Oantisym,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_149_dual__order_Oantisym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_150_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_Pr4934435412358123699_a_nat,Z: set_Pr4934435412358123699_a_nat] : ( Y3 = Z ) )
= ( ^ [A3: set_Pr4934435412358123699_a_nat,B3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B3 @ A3 )
& ( ord_le8666007276011122963_a_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_151_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_list_nat,Z: set_list_nat] : ( Y3 = Z ) )
= ( ^ [A3: set_list_nat,B3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B3 @ A3 )
& ( ord_le6045566169113846134st_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_152_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: $o > nat,Z: $o > nat] : ( Y3 = Z ) )
= ( ^ [A3: $o > nat,B3: $o > nat] :
( ( ord_less_eq_o_nat @ B3 @ A3 )
& ( ord_less_eq_o_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_153_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A3 )
& ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_154_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_155_linorder__wlog,axiom,
! [P3: nat > nat > $o,A2: nat,B2: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P3 @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P3 @ B4 @ A4 )
=> ( P3 @ A4 @ B4 ) )
=> ( P3 @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_156_order__trans,axiom,
! [X2: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X2 @ Y )
=> ( ( ord_le8666007276011122963_a_nat @ Y @ Z2 )
=> ( ord_le8666007276011122963_a_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_157_order__trans,axiom,
! [X2: set_list_nat,Y: set_list_nat,Z2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X2 @ Y )
=> ( ( ord_le6045566169113846134st_nat @ Y @ Z2 )
=> ( ord_le6045566169113846134st_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_158_order__trans,axiom,
! [X2: $o > nat,Y: $o > nat,Z2: $o > nat] :
( ( ord_less_eq_o_nat @ X2 @ Y )
=> ( ( ord_less_eq_o_nat @ Y @ Z2 )
=> ( ord_less_eq_o_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_159_order__trans,axiom,
! [X2: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z2 )
=> ( ord_less_eq_set_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_160_order__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_161_order_Otrans,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ C2 )
=> ( ord_le8666007276011122963_a_nat @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_162_order_Otrans,axiom,
! [A2: set_list_nat,B2: set_list_nat,C2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B2 )
=> ( ( ord_le6045566169113846134st_nat @ B2 @ C2 )
=> ( ord_le6045566169113846134st_nat @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_163_order_Otrans,axiom,
! [A2: $o > nat,B2: $o > nat,C2: $o > nat] :
( ( ord_less_eq_o_nat @ A2 @ B2 )
=> ( ( ord_less_eq_o_nat @ B2 @ C2 )
=> ( ord_less_eq_o_nat @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_164_order_Otrans,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_165_order_Otrans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_166_order__antisym,axiom,
! [X2: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X2 @ Y )
=> ( ( ord_le8666007276011122963_a_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_167_order__antisym,axiom,
! [X2: set_list_nat,Y: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X2 @ Y )
=> ( ( ord_le6045566169113846134st_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_168_order__antisym,axiom,
! [X2: $o > nat,Y: $o > nat] :
( ( ord_less_eq_o_nat @ X2 @ Y )
=> ( ( ord_less_eq_o_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_169_order__antisym,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_170_order__antisym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_171_ord__le__eq__trans,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_le8666007276011122963_a_nat @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_172_ord__le__eq__trans,axiom,
! [A2: set_list_nat,B2: set_list_nat,C2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_le6045566169113846134st_nat @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_173_ord__le__eq__trans,axiom,
! [A2: $o > nat,B2: $o > nat,C2: $o > nat] :
( ( ord_less_eq_o_nat @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_o_nat @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_174_ord__le__eq__trans,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_175_ord__le__eq__trans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_176_ord__eq__le__trans,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( A2 = B2 )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ C2 )
=> ( ord_le8666007276011122963_a_nat @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_177_ord__eq__le__trans,axiom,
! [A2: set_list_nat,B2: set_list_nat,C2: set_list_nat] :
( ( A2 = B2 )
=> ( ( ord_le6045566169113846134st_nat @ B2 @ C2 )
=> ( ord_le6045566169113846134st_nat @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_178_ord__eq__le__trans,axiom,
! [A2: $o > nat,B2: $o > nat,C2: $o > nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_o_nat @ B2 @ C2 )
=> ( ord_less_eq_o_nat @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_179_ord__eq__le__trans,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_180_ord__eq__le__trans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_181_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_Pr4934435412358123699_a_nat,Z: set_Pr4934435412358123699_a_nat] : ( Y3 = Z ) )
= ( ^ [X3: set_Pr4934435412358123699_a_nat,Y4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y4 )
& ( ord_le8666007276011122963_a_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_182_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_list_nat,Z: set_list_nat] : ( Y3 = Z ) )
= ( ^ [X3: set_list_nat,Y4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X3 @ Y4 )
& ( ord_le6045566169113846134st_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_183_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: $o > nat,Z: $o > nat] : ( Y3 = Z ) )
= ( ^ [X3: $o > nat,Y4: $o > nat] :
( ( ord_less_eq_o_nat @ X3 @ Y4 )
& ( ord_less_eq_o_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_184_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
& ( ord_less_eq_set_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_185_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_186_le__cases3,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_187_nle__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_188_Collect__mono__iff,axiom,
! [P3: product_prod_a_nat > $o,Q3: product_prod_a_nat > $o] :
( ( ord_le8666007276011122963_a_nat @ ( collec4464134535221767506_a_nat @ P3 ) @ ( collec4464134535221767506_a_nat @ Q3 ) )
= ( ! [X3: product_prod_a_nat] :
( ( P3 @ X3 )
=> ( Q3 @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_189_Collect__mono__iff,axiom,
! [P3: list_nat > $o,Q3: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P3 ) @ ( collect_list_nat @ Q3 ) )
= ( ! [X3: list_nat] :
( ( P3 @ X3 )
=> ( Q3 @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_190_Collect__mono__iff,axiom,
! [P3: nat > $o,Q3: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P3 ) @ ( collect_nat @ Q3 ) )
= ( ! [X3: nat] :
( ( P3 @ X3 )
=> ( Q3 @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_191_set__eq__subset,axiom,
( ( ^ [Y3: set_Pr4934435412358123699_a_nat,Z: set_Pr4934435412358123699_a_nat] : ( Y3 = Z ) )
= ( ^ [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ B5 )
& ( ord_le8666007276011122963_a_nat @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_192_set__eq__subset,axiom,
( ( ^ [Y3: set_list_nat,Z: set_list_nat] : ( Y3 = Z ) )
= ( ^ [A5: set_list_nat,B5: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A5 @ B5 )
& ( ord_le6045566169113846134st_nat @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_193_set__eq__subset,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_194_subset__trans,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C )
=> ( ord_le8666007276011122963_a_nat @ A @ C ) ) ) ).
% subset_trans
thf(fact_195_subset__trans,axiom,
! [A: set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ord_le6045566169113846134st_nat @ A @ C ) ) ) ).
% subset_trans
thf(fact_196_subset__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% subset_trans
thf(fact_197_Collect__mono,axiom,
! [P3: product_prod_a_nat > $o,Q3: product_prod_a_nat > $o] :
( ! [X: product_prod_a_nat] :
( ( P3 @ X )
=> ( Q3 @ X ) )
=> ( ord_le8666007276011122963_a_nat @ ( collec4464134535221767506_a_nat @ P3 ) @ ( collec4464134535221767506_a_nat @ Q3 ) ) ) ).
% Collect_mono
thf(fact_198_Collect__mono,axiom,
! [P3: list_nat > $o,Q3: list_nat > $o] :
( ! [X: list_nat] :
( ( P3 @ X )
=> ( Q3 @ X ) )
=> ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P3 ) @ ( collect_list_nat @ Q3 ) ) ) ).
% Collect_mono
thf(fact_199_Collect__mono,axiom,
! [P3: nat > $o,Q3: nat > $o] :
( ! [X: nat] :
( ( P3 @ X )
=> ( Q3 @ X ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P3 ) @ ( collect_nat @ Q3 ) ) ) ).
% Collect_mono
thf(fact_200_subset__refl,axiom,
! [A: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A @ A ) ).
% subset_refl
thf(fact_201_subset__refl,axiom,
! [A: set_list_nat] : ( ord_le6045566169113846134st_nat @ A @ A ) ).
% subset_refl
thf(fact_202_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_203_subset__iff,axiom,
( ord_le4706587664122053766st_nat
= ( ^ [A5: set_list_list_nat,B5: set_list_list_nat] :
! [T3: list_list_nat] :
( ( member_list_list_nat @ T3 @ A5 )
=> ( member_list_list_nat @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_204_subset__iff,axiom,
( ord_le8406513867147106209st_nat
= ( ^ [A5: set_Pr3451248702717554689st_nat,B5: set_Pr3451248702717554689st_nat] :
! [T3: produc1828647624359046049st_nat] :
( ( member7340969449405702474st_nat @ T3 @ A5 )
=> ( member7340969449405702474st_nat @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_205_subset__iff,axiom,
( ord_le3146513528884898305at_nat
= ( ^ [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
! [T3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ T3 @ A5 )
=> ( member8440522571783428010at_nat @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_206_subset__iff,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
! [T3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ T3 @ A5 )
=> ( member5724188588386418708_a_nat @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_207_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [T3: nat] :
( ( member_nat2 @ T3 @ A5 )
=> ( member_nat2 @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_208_subset__iff,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A5: set_list_nat,B5: set_list_nat] :
! [T3: list_nat] :
( ( member_list_nat2 @ T3 @ A5 )
=> ( member_list_nat2 @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_209_equalityD2,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( A = B )
=> ( ord_le8666007276011122963_a_nat @ B @ A ) ) ).
% equalityD2
thf(fact_210_equalityD2,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( A = B )
=> ( ord_le6045566169113846134st_nat @ B @ A ) ) ).
% equalityD2
thf(fact_211_equalityD2,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% equalityD2
thf(fact_212_equalityD1,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( A = B )
=> ( ord_le8666007276011122963_a_nat @ A @ B ) ) ).
% equalityD1
thf(fact_213_equalityD1,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( A = B )
=> ( ord_le6045566169113846134st_nat @ A @ B ) ) ).
% equalityD1
thf(fact_214_equalityD1,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% equalityD1
thf(fact_215_subset__eq,axiom,
( ord_le4706587664122053766st_nat
= ( ^ [A5: set_list_list_nat,B5: set_list_list_nat] :
! [X3: list_list_nat] :
( ( member_list_list_nat @ X3 @ A5 )
=> ( member_list_list_nat @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_216_subset__eq,axiom,
( ord_le8406513867147106209st_nat
= ( ^ [A5: set_Pr3451248702717554689st_nat,B5: set_Pr3451248702717554689st_nat] :
! [X3: produc1828647624359046049st_nat] :
( ( member7340969449405702474st_nat @ X3 @ A5 )
=> ( member7340969449405702474st_nat @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_217_subset__eq,axiom,
( ord_le3146513528884898305at_nat
= ( ^ [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A5 )
=> ( member8440522571783428010at_nat @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_218_subset__eq,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
! [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ A5 )
=> ( member5724188588386418708_a_nat @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_219_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [X3: nat] :
( ( member_nat2 @ X3 @ A5 )
=> ( member_nat2 @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_220_subset__eq,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A5: set_list_nat,B5: set_list_nat] :
! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ A5 )
=> ( member_list_nat2 @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_221_equalityE,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( A = B )
=> ~ ( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ~ ( ord_le8666007276011122963_a_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_222_equalityE,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( A = B )
=> ~ ( ( ord_le6045566169113846134st_nat @ A @ B )
=> ~ ( ord_le6045566169113846134st_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_223_equalityE,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_224_subsetD,axiom,
! [A: set_list_list_nat,B: set_list_list_nat,C2: list_list_nat] :
( ( ord_le4706587664122053766st_nat @ A @ B )
=> ( ( member_list_list_nat @ C2 @ A )
=> ( member_list_list_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_225_subsetD,axiom,
! [A: set_Pr3451248702717554689st_nat,B: set_Pr3451248702717554689st_nat,C2: produc1828647624359046049st_nat] :
( ( ord_le8406513867147106209st_nat @ A @ B )
=> ( ( member7340969449405702474st_nat @ C2 @ A )
=> ( member7340969449405702474st_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_226_subsetD,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: product_prod_nat_nat] :
( ( ord_le3146513528884898305at_nat @ A @ B )
=> ( ( member8440522571783428010at_nat @ C2 @ A )
=> ( member8440522571783428010at_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_227_subsetD,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C2: product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( member5724188588386418708_a_nat @ C2 @ A )
=> ( member5724188588386418708_a_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_228_subsetD,axiom,
! [A: set_nat,B: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat2 @ C2 @ A )
=> ( member_nat2 @ C2 @ B ) ) ) ).
% subsetD
thf(fact_229_subsetD,axiom,
! [A: set_list_nat,B: set_list_nat,C2: list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( member_list_nat2 @ C2 @ A )
=> ( member_list_nat2 @ C2 @ B ) ) ) ).
% subsetD
thf(fact_230_in__mono,axiom,
! [A: set_list_list_nat,B: set_list_list_nat,X2: list_list_nat] :
( ( ord_le4706587664122053766st_nat @ A @ B )
=> ( ( member_list_list_nat @ X2 @ A )
=> ( member_list_list_nat @ X2 @ B ) ) ) ).
% in_mono
thf(fact_231_in__mono,axiom,
! [A: set_Pr3451248702717554689st_nat,B: set_Pr3451248702717554689st_nat,X2: produc1828647624359046049st_nat] :
( ( ord_le8406513867147106209st_nat @ A @ B )
=> ( ( member7340969449405702474st_nat @ X2 @ A )
=> ( member7340969449405702474st_nat @ X2 @ B ) ) ) ).
% in_mono
thf(fact_232_in__mono,axiom,
! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
( ( ord_le3146513528884898305at_nat @ A @ B )
=> ( ( member8440522571783428010at_nat @ X2 @ A )
=> ( member8440522571783428010at_nat @ X2 @ B ) ) ) ).
% in_mono
thf(fact_233_in__mono,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,X2: product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( member5724188588386418708_a_nat @ X2 @ A )
=> ( member5724188588386418708_a_nat @ X2 @ B ) ) ) ).
% in_mono
thf(fact_234_in__mono,axiom,
! [A: set_nat,B: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat2 @ X2 @ A )
=> ( member_nat2 @ X2 @ B ) ) ) ).
% in_mono
thf(fact_235_in__mono,axiom,
! [A: set_list_nat,B: set_list_nat,X2: list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( member_list_nat2 @ X2 @ A )
=> ( member_list_nat2 @ X2 @ B ) ) ) ).
% in_mono
thf(fact_236_poss__pos__diffI,axiom,
! [P: list_nat,Q: list_nat,S: term_a_b] :
( ( term_p3503116865373065078eq_nat @ P @ Q )
=> ( ( member_list_nat2 @ Q @ ( term_poss_a_b @ S ) )
=> ( member_list_nat2 @ ( term_pos_diff_nat @ Q @ P ) @ ( term_poss_a_b @ ( term_subt_at_a_b @ S @ P ) ) ) ) ) ).
% poss_pos_diffI
thf(fact_237_finite__has__minimal2,axiom,
! [A: set_se6121441497158405097_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( finite568477829172517706_a_nat @ A )
=> ( ( member8034581882086557258_a_nat @ A2 @ A )
=> ? [X: set_Pr4934435412358123699_a_nat] :
( ( member8034581882086557258_a_nat @ X @ A )
& ( ord_le8666007276011122963_a_nat @ X @ A2 )
& ! [Xa: set_Pr4934435412358123699_a_nat] :
( ( member8034581882086557258_a_nat @ Xa @ A )
=> ( ( ord_le8666007276011122963_a_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_238_finite__has__minimal2,axiom,
! [A: set_set_list_nat,A2: set_list_nat] :
( ( finite7047420756378620717st_nat @ A )
=> ( ( member_set_list_nat @ A2 @ A )
=> ? [X: set_list_nat] :
( ( member_set_list_nat @ X @ A )
& ( ord_le6045566169113846134st_nat @ X @ A2 )
& ! [Xa: set_list_nat] :
( ( member_set_list_nat @ Xa @ A )
=> ( ( ord_le6045566169113846134st_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_239_finite__has__minimal2,axiom,
! [A: set_o_nat,A2: $o > nat] :
( ( finite_finite_o_nat @ A )
=> ( ( member_o_nat @ A2 @ A )
=> ? [X: $o > nat] :
( ( member_o_nat @ X @ A )
& ( ord_less_eq_o_nat @ X @ A2 )
& ! [Xa: $o > nat] :
( ( member_o_nat @ Xa @ A )
=> ( ( ord_less_eq_o_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_240_finite__has__minimal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat2 @ A2 @ A )
=> ? [X: set_nat] :
( ( member_set_nat2 @ X @ A )
& ( ord_less_eq_set_nat @ X @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat2 @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_241_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat2 @ A2 @ A )
=> ? [X: nat] :
( ( member_nat2 @ X @ A )
& ( ord_less_eq_nat @ X @ A2 )
& ! [Xa: nat] :
( ( member_nat2 @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_242_hole__pos__replace__term__at,axiom,
! [C: subterm_and_ctxt_a_b,P: list_nat,S: term_a_b,U: term_a_b] :
( ( term_p3503116865373065078eq_nat @ ( term_hole_pos_a_b @ C ) @ P )
=> ( ( term_r6860082780075436317at_a_b @ ( subter2376574525758040790rm_a_b @ C @ S ) @ P @ U )
= ( subter2376574525758040790rm_a_b @ C @ ( term_r6860082780075436317at_a_b @ S @ ( term_pos_diff_nat @ P @ ( term_hole_pos_a_b @ C ) ) @ U ) ) ) ) ).
% hole_pos_replace_term_at
thf(fact_243_eq__ctxt__at__pos__by__poss,axiom,
! [P: list_nat,S: term_a_b,T: term_a_b] :
( ( member_list_nat2 @ P @ ( term_poss_a_b @ S ) )
=> ( ( member_list_nat2 @ P @ ( term_poss_a_b @ T ) )
=> ( ! [Q4: list_nat] :
( ~ ( term_p3503116865373065078eq_nat @ P @ Q4 )
=> ( ( member_list_nat2 @ Q4 @ ( term_poss_a_b @ S ) )
= ( member_list_nat2 @ Q4 @ ( term_poss_a_b @ T ) ) ) )
=> ( ! [Q4: list_nat] :
( ( member_list_nat2 @ Q4 @ ( term_poss_a_b @ S ) )
=> ( ~ ( term_p3503116865373065078eq_nat @ P @ Q4 )
=> ( ( term_fun_at_a_b @ S @ Q4 )
= ( term_fun_at_a_b @ T @ Q4 ) ) ) )
=> ( ( term_ctxt_at_pos_a_b @ S @ P )
= ( term_ctxt_at_pos_a_b @ T @ P ) ) ) ) ) ) ).
% eq_ctxt_at_pos_by_poss
thf(fact_244_ctxt__at__pos__hole__pos,axiom,
! [C: subterm_and_ctxt_a_b,S: term_a_b] :
( ( term_ctxt_at_pos_a_b @ ( subter2376574525758040790rm_a_b @ C @ S ) @ ( term_hole_pos_a_b @ C ) )
= C ) ).
% ctxt_at_pos_hole_pos
thf(fact_245_ctxt__apply__subt__at__hole__pos,axiom,
! [C: subterm_and_ctxt_a_b,S: term_a_b] :
( ( term_subt_at_a_b @ ( subter2376574525758040790rm_a_b @ C @ S ) @ ( term_hole_pos_a_b @ C ) )
= S ) ).
% ctxt_apply_subt_at_hole_pos
thf(fact_246_replace__at__hole__pos,axiom,
! [C: subterm_and_ctxt_a_b,S: term_a_b,T: term_a_b] :
( ( term_r6860082780075436317at_a_b @ ( subter2376574525758040790rm_a_b @ C @ S ) @ ( term_hole_pos_a_b @ C ) @ T )
= ( subter2376574525758040790rm_a_b @ C @ T ) ) ).
% replace_at_hole_pos
thf(fact_247_term__to__sig__id,axiom,
! [T: term_a_b,F2: set_Pr4934435412358123699_a_nat,V: b] :
( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F2 )
=> ( ( terms_8519481630511763164ig_a_b @ F2 @ V @ T )
= T ) ) ).
% term_to_sig_id
thf(fact_248_funas__term__replace__at__upper,axiom,
! [S: term_a_b,P: list_nat,T: term_a_b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( term_r6860082780075436317at_a_b @ S @ P @ T ) ) @ ( sup_su459911885395995103_a_nat @ ( term_funas_term_a_b @ S ) @ ( term_funas_term_a_b @ T ) ) ) ).
% funas_term_replace_at_upper
thf(fact_249_Greatest__equality,axiom,
! [P3: set_Pr4934435412358123699_a_nat > $o,X2: set_Pr4934435412358123699_a_nat] :
( ( P3 @ X2 )
=> ( ! [Y2: set_Pr4934435412358123699_a_nat] :
( ( P3 @ Y2 )
=> ( ord_le8666007276011122963_a_nat @ Y2 @ X2 ) )
=> ( ( order_1399416596184996762_a_nat @ P3 )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_250_Greatest__equality,axiom,
! [P3: set_list_nat > $o,X2: set_list_nat] :
( ( P3 @ X2 )
=> ( ! [Y2: set_list_nat] :
( ( P3 @ Y2 )
=> ( ord_le6045566169113846134st_nat @ Y2 @ X2 ) )
=> ( ( order_3081514539752307581st_nat @ P3 )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_251_Greatest__equality,axiom,
! [P3: ( $o > nat ) > $o,X2: $o > nat] :
( ( P3 @ X2 )
=> ( ! [Y2: $o > nat] :
( ( P3 @ Y2 )
=> ( ord_less_eq_o_nat @ Y2 @ X2 ) )
=> ( ( order_Greatest_o_nat @ P3 )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_252_Greatest__equality,axiom,
! [P3: set_nat > $o,X2: set_nat] :
( ( P3 @ X2 )
=> ( ! [Y2: set_nat] :
( ( P3 @ Y2 )
=> ( ord_less_eq_set_nat @ Y2 @ X2 ) )
=> ( ( order_5724808138429204845et_nat @ P3 )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_253_Greatest__equality,axiom,
! [P3: nat > $o,X2: nat] :
( ( P3 @ X2 )
=> ( ! [Y2: nat] :
( ( P3 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ( order_Greatest_nat @ P3 )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_254_GreatestI2__order,axiom,
! [P3: set_Pr4934435412358123699_a_nat > $o,X2: set_Pr4934435412358123699_a_nat,Q3: set_Pr4934435412358123699_a_nat > $o] :
( ( P3 @ X2 )
=> ( ! [Y2: set_Pr4934435412358123699_a_nat] :
( ( P3 @ Y2 )
=> ( ord_le8666007276011122963_a_nat @ Y2 @ X2 ) )
=> ( ! [X: set_Pr4934435412358123699_a_nat] :
( ( P3 @ X )
=> ( ! [Y5: set_Pr4934435412358123699_a_nat] :
( ( P3 @ Y5 )
=> ( ord_le8666007276011122963_a_nat @ Y5 @ X ) )
=> ( Q3 @ X ) ) )
=> ( Q3 @ ( order_1399416596184996762_a_nat @ P3 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_255_GreatestI2__order,axiom,
! [P3: set_list_nat > $o,X2: set_list_nat,Q3: set_list_nat > $o] :
( ( P3 @ X2 )
=> ( ! [Y2: set_list_nat] :
( ( P3 @ Y2 )
=> ( ord_le6045566169113846134st_nat @ Y2 @ X2 ) )
=> ( ! [X: set_list_nat] :
( ( P3 @ X )
=> ( ! [Y5: set_list_nat] :
( ( P3 @ Y5 )
=> ( ord_le6045566169113846134st_nat @ Y5 @ X ) )
=> ( Q3 @ X ) ) )
=> ( Q3 @ ( order_3081514539752307581st_nat @ P3 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_256_GreatestI2__order,axiom,
! [P3: ( $o > nat ) > $o,X2: $o > nat,Q3: ( $o > nat ) > $o] :
( ( P3 @ X2 )
=> ( ! [Y2: $o > nat] :
( ( P3 @ Y2 )
=> ( ord_less_eq_o_nat @ Y2 @ X2 ) )
=> ( ! [X: $o > nat] :
( ( P3 @ X )
=> ( ! [Y5: $o > nat] :
( ( P3 @ Y5 )
=> ( ord_less_eq_o_nat @ Y5 @ X ) )
=> ( Q3 @ X ) ) )
=> ( Q3 @ ( order_Greatest_o_nat @ P3 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_257_GreatestI2__order,axiom,
! [P3: set_nat > $o,X2: set_nat,Q3: set_nat > $o] :
( ( P3 @ X2 )
=> ( ! [Y2: set_nat] :
( ( P3 @ Y2 )
=> ( ord_less_eq_set_nat @ Y2 @ X2 ) )
=> ( ! [X: set_nat] :
( ( P3 @ X )
=> ( ! [Y5: set_nat] :
( ( P3 @ Y5 )
=> ( ord_less_eq_set_nat @ Y5 @ X ) )
=> ( Q3 @ X ) ) )
=> ( Q3 @ ( order_5724808138429204845et_nat @ P3 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_258_GreatestI2__order,axiom,
! [P3: nat > $o,X2: nat,Q3: nat > $o] :
( ( P3 @ X2 )
=> ( ! [Y2: nat] :
( ( P3 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ! [X: nat] :
( ( P3 @ X )
=> ( ! [Y5: nat] :
( ( P3 @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X ) )
=> ( Q3 @ X ) ) )
=> ( Q3 @ ( order_Greatest_nat @ P3 ) ) ) ) ) ).
% GreatestI2_order
thf(fact_259_infinite__imp__elem,axiom,
! [A: set_list_list_nat] :
( ~ ( finite8170528100393595399st_nat @ A )
=> ? [X: list_list_nat] : ( member_list_list_nat @ X @ A ) ) ).
% infinite_imp_elem
thf(fact_260_infinite__imp__elem,axiom,
! [A: set_Pr3451248702717554689st_nat] :
( ~ ( finite9134613430549597258st_nat @ A )
=> ? [X: produc1828647624359046049st_nat] : ( member7340969449405702474st_nat @ X @ A ) ) ).
% infinite_imp_elem
thf(fact_261_infinite__imp__elem,axiom,
! [A: set_Pr1261947904930325089at_nat] :
( ~ ( finite6177210948735845034at_nat @ A )
=> ? [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A ) ) ).
% infinite_imp_elem
thf(fact_262_infinite__imp__elem,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ~ ( finite6644898363146130708_a_nat @ A )
=> ? [X: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X @ A ) ) ).
% infinite_imp_elem
thf(fact_263_infinite__imp__elem,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ? [X: nat] : ( member_nat2 @ X @ A ) ) ).
% infinite_imp_elem
thf(fact_264_infinite__imp__elem,axiom,
! [A: set_list_nat] :
( ~ ( finite8100373058378681591st_nat @ A )
=> ? [X: list_nat] : ( member_list_nat2 @ X @ A ) ) ).
% infinite_imp_elem
thf(fact_265_hole__pos__in__ctxt__apply,axiom,
! [C: subterm_and_ctxt_a_b,U: term_a_b] : ( member_list_nat2 @ ( term_hole_pos_a_b @ C ) @ ( term_poss_a_b @ ( subter2376574525758040790rm_a_b @ C @ U ) ) ) ).
% hole_pos_in_ctxt_apply
thf(fact_266_Un__iff,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat2 @ C2 @ ( sup_sup_set_nat @ A @ B ) )
= ( ( member_nat2 @ C2 @ A )
| ( member_nat2 @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_267_Un__iff,axiom,
! [C2: list_list_nat,A: set_list_list_nat,B: set_list_list_nat] :
( ( member_list_list_nat @ C2 @ ( sup_su5723864310361701714st_nat @ A @ B ) )
= ( ( member_list_list_nat @ C2 @ A )
| ( member_list_list_nat @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_268_Un__iff,axiom,
! [C2: produc1828647624359046049st_nat,A: set_Pr3451248702717554689st_nat,B: set_Pr3451248702717554689st_nat] :
( ( member7340969449405702474st_nat @ C2 @ ( sup_su5841606568262889429st_nat @ A @ B ) )
= ( ( member7340969449405702474st_nat @ C2 @ A )
| ( member7340969449405702474st_nat @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_269_Un__iff,axiom,
! [C2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C2 @ ( sup_su6327502436637775413at_nat @ A @ B ) )
= ( ( member8440522571783428010at_nat @ C2 @ A )
| ( member8440522571783428010at_nat @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_270_Un__iff,axiom,
! [C2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C2 @ ( sup_su459911885395995103_a_nat @ A @ B ) )
= ( ( member5724188588386418708_a_nat @ C2 @ A )
| ( member5724188588386418708_a_nat @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_271_Un__iff,axiom,
! [C2: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ C2 @ ( sup_sup_set_list_nat @ A @ B ) )
= ( ( member_list_nat2 @ C2 @ A )
| ( member_list_nat2 @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_272_UnCI,axiom,
! [C2: nat,B: set_nat,A: set_nat] :
( ( ~ ( member_nat2 @ C2 @ B )
=> ( member_nat2 @ C2 @ A ) )
=> ( member_nat2 @ C2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_273_UnCI,axiom,
! [C2: list_list_nat,B: set_list_list_nat,A: set_list_list_nat] :
( ( ~ ( member_list_list_nat @ C2 @ B )
=> ( member_list_list_nat @ C2 @ A ) )
=> ( member_list_list_nat @ C2 @ ( sup_su5723864310361701714st_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_274_UnCI,axiom,
! [C2: produc1828647624359046049st_nat,B: set_Pr3451248702717554689st_nat,A: set_Pr3451248702717554689st_nat] :
( ( ~ ( member7340969449405702474st_nat @ C2 @ B )
=> ( member7340969449405702474st_nat @ C2 @ A ) )
=> ( member7340969449405702474st_nat @ C2 @ ( sup_su5841606568262889429st_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_275_UnCI,axiom,
! [C2: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
( ( ~ ( member8440522571783428010at_nat @ C2 @ B )
=> ( member8440522571783428010at_nat @ C2 @ A ) )
=> ( member8440522571783428010at_nat @ C2 @ ( sup_su6327502436637775413at_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_276_UnCI,axiom,
! [C2: product_prod_a_nat,B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ~ ( member5724188588386418708_a_nat @ C2 @ B )
=> ( member5724188588386418708_a_nat @ C2 @ A ) )
=> ( member5724188588386418708_a_nat @ C2 @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_277_UnCI,axiom,
! [C2: list_nat,B: set_list_nat,A: set_list_nat] :
( ( ~ ( member_list_nat2 @ C2 @ B )
=> ( member_list_nat2 @ C2 @ A ) )
=> ( member_list_nat2 @ C2 @ ( sup_sup_set_list_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_278_finite__Un,axiom,
! [F4: set_nat,G3: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F4 @ G3 ) )
= ( ( finite_finite_nat @ F4 )
& ( finite_finite_nat @ G3 ) ) ) ).
% finite_Un
thf(fact_279_finite__Un,axiom,
! [F4: set_Pr4934435412358123699_a_nat,G3: set_Pr4934435412358123699_a_nat] :
( ( finite6644898363146130708_a_nat @ ( sup_su459911885395995103_a_nat @ F4 @ G3 ) )
= ( ( finite6644898363146130708_a_nat @ F4 )
& ( finite6644898363146130708_a_nat @ G3 ) ) ) ).
% finite_Un
thf(fact_280_finite__Un,axiom,
! [F4: set_list_nat,G3: set_list_nat] :
( ( finite8100373058378681591st_nat @ ( sup_sup_set_list_nat @ F4 @ G3 ) )
= ( ( finite8100373058378681591st_nat @ F4 )
& ( finite8100373058378681591st_nat @ G3 ) ) ) ).
% finite_Un
thf(fact_281_Un__subset__iff,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A @ B ) @ C )
= ( ( ord_le8666007276011122963_a_nat @ A @ C )
& ( ord_le8666007276011122963_a_nat @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_282_Un__subset__iff,axiom,
! [A: set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( sup_sup_set_list_nat @ A @ B ) @ C )
= ( ( ord_le6045566169113846134st_nat @ A @ C )
& ( ord_le6045566169113846134st_nat @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_283_Un__subset__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C )
= ( ( ord_less_eq_set_nat @ A @ C )
& ( ord_less_eq_set_nat @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_284_Un__left__commute,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ A @ ( sup_su459911885395995103_a_nat @ B @ C ) )
= ( sup_su459911885395995103_a_nat @ B @ ( sup_su459911885395995103_a_nat @ A @ C ) ) ) ).
% Un_left_commute
thf(fact_285_Un__left__commute,axiom,
! [A: set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( sup_sup_set_list_nat @ A @ ( sup_sup_set_list_nat @ B @ C ) )
= ( sup_sup_set_list_nat @ B @ ( sup_sup_set_list_nat @ A @ C ) ) ) ).
% Un_left_commute
thf(fact_286_Un__left__absorb,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ A @ ( sup_su459911885395995103_a_nat @ A @ B ) )
= ( sup_su459911885395995103_a_nat @ A @ B ) ) ).
% Un_left_absorb
thf(fact_287_Un__left__absorb,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( sup_sup_set_list_nat @ A @ ( sup_sup_set_list_nat @ A @ B ) )
= ( sup_sup_set_list_nat @ A @ B ) ) ).
% Un_left_absorb
thf(fact_288_Un__commute,axiom,
( sup_su459911885395995103_a_nat
= ( ^ [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] : ( sup_su459911885395995103_a_nat @ B5 @ A5 ) ) ) ).
% Un_commute
thf(fact_289_Un__commute,axiom,
( sup_sup_set_list_nat
= ( ^ [A5: set_list_nat,B5: set_list_nat] : ( sup_sup_set_list_nat @ B5 @ A5 ) ) ) ).
% Un_commute
thf(fact_290_Un__absorb,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_291_Un__absorb,axiom,
! [A: set_list_nat] :
( ( sup_sup_set_list_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_292_Un__assoc,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ ( sup_su459911885395995103_a_nat @ A @ B ) @ C )
= ( sup_su459911885395995103_a_nat @ A @ ( sup_su459911885395995103_a_nat @ B @ C ) ) ) ).
% Un_assoc
thf(fact_293_Un__assoc,axiom,
! [A: set_list_nat,B: set_list_nat,C: set_list_nat] :
( ( sup_sup_set_list_nat @ ( sup_sup_set_list_nat @ A @ B ) @ C )
= ( sup_sup_set_list_nat @ A @ ( sup_sup_set_list_nat @ B @ C ) ) ) ).
% Un_assoc
thf(fact_294_ball__Un,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,P3: product_prod_a_nat > $o] :
( ( ! [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ ( sup_su459911885395995103_a_nat @ A @ B ) )
=> ( P3 @ X3 ) ) )
= ( ! [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ A )
=> ( P3 @ X3 ) )
& ! [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ B )
=> ( P3 @ X3 ) ) ) ) ).
% ball_Un
thf(fact_295_ball__Un,axiom,
! [A: set_list_nat,B: set_list_nat,P3: list_nat > $o] :
( ( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ ( sup_sup_set_list_nat @ A @ B ) )
=> ( P3 @ X3 ) ) )
= ( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ A )
=> ( P3 @ X3 ) )
& ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ B )
=> ( P3 @ X3 ) ) ) ) ).
% ball_Un
thf(fact_296_bex__Un,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,P3: product_prod_a_nat > $o] :
( ( ? [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ ( sup_su459911885395995103_a_nat @ A @ B ) )
& ( P3 @ X3 ) ) )
= ( ? [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ A )
& ( P3 @ X3 ) )
| ? [X3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X3 @ B )
& ( P3 @ X3 ) ) ) ) ).
% bex_Un
thf(fact_297_bex__Un,axiom,
! [A: set_list_nat,B: set_list_nat,P3: list_nat > $o] :
( ( ? [X3: list_nat] :
( ( member_list_nat2 @ X3 @ ( sup_sup_set_list_nat @ A @ B ) )
& ( P3 @ X3 ) ) )
= ( ? [X3: list_nat] :
( ( member_list_nat2 @ X3 @ A )
& ( P3 @ X3 ) )
| ? [X3: list_nat] :
( ( member_list_nat2 @ X3 @ B )
& ( P3 @ X3 ) ) ) ) ).
% bex_Un
thf(fact_298_UnI2,axiom,
! [C2: nat,B: set_nat,A: set_nat] :
( ( member_nat2 @ C2 @ B )
=> ( member_nat2 @ C2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_299_UnI2,axiom,
! [C2: list_list_nat,B: set_list_list_nat,A: set_list_list_nat] :
( ( member_list_list_nat @ C2 @ B )
=> ( member_list_list_nat @ C2 @ ( sup_su5723864310361701714st_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_300_UnI2,axiom,
! [C2: produc1828647624359046049st_nat,B: set_Pr3451248702717554689st_nat,A: set_Pr3451248702717554689st_nat] :
( ( member7340969449405702474st_nat @ C2 @ B )
=> ( member7340969449405702474st_nat @ C2 @ ( sup_su5841606568262889429st_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_301_UnI2,axiom,
! [C2: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C2 @ B )
=> ( member8440522571783428010at_nat @ C2 @ ( sup_su6327502436637775413at_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_302_UnI2,axiom,
! [C2: product_prod_a_nat,B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C2 @ B )
=> ( member5724188588386418708_a_nat @ C2 @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_303_UnI2,axiom,
! [C2: list_nat,B: set_list_nat,A: set_list_nat] :
( ( member_list_nat2 @ C2 @ B )
=> ( member_list_nat2 @ C2 @ ( sup_sup_set_list_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_304_UnI1,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat2 @ C2 @ A )
=> ( member_nat2 @ C2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_305_UnI1,axiom,
! [C2: list_list_nat,A: set_list_list_nat,B: set_list_list_nat] :
( ( member_list_list_nat @ C2 @ A )
=> ( member_list_list_nat @ C2 @ ( sup_su5723864310361701714st_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_306_UnI1,axiom,
! [C2: produc1828647624359046049st_nat,A: set_Pr3451248702717554689st_nat,B: set_Pr3451248702717554689st_nat] :
( ( member7340969449405702474st_nat @ C2 @ A )
=> ( member7340969449405702474st_nat @ C2 @ ( sup_su5841606568262889429st_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_307_UnI1,axiom,
! [C2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C2 @ A )
=> ( member8440522571783428010at_nat @ C2 @ ( sup_su6327502436637775413at_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_308_UnI1,axiom,
! [C2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C2 @ A )
=> ( member5724188588386418708_a_nat @ C2 @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_309_UnI1,axiom,
! [C2: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ C2 @ A )
=> ( member_list_nat2 @ C2 @ ( sup_sup_set_list_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_310_UnE,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat2 @ C2 @ ( sup_sup_set_nat @ A @ B ) )
=> ( ~ ( member_nat2 @ C2 @ A )
=> ( member_nat2 @ C2 @ B ) ) ) ).
% UnE
thf(fact_311_UnE,axiom,
! [C2: list_list_nat,A: set_list_list_nat,B: set_list_list_nat] :
( ( member_list_list_nat @ C2 @ ( sup_su5723864310361701714st_nat @ A @ B ) )
=> ( ~ ( member_list_list_nat @ C2 @ A )
=> ( member_list_list_nat @ C2 @ B ) ) ) ).
% UnE
thf(fact_312_UnE,axiom,
! [C2: produc1828647624359046049st_nat,A: set_Pr3451248702717554689st_nat,B: set_Pr3451248702717554689st_nat] :
( ( member7340969449405702474st_nat @ C2 @ ( sup_su5841606568262889429st_nat @ A @ B ) )
=> ( ~ ( member7340969449405702474st_nat @ C2 @ A )
=> ( member7340969449405702474st_nat @ C2 @ B ) ) ) ).
% UnE
thf(fact_313_UnE,axiom,
! [C2: product_prod_nat_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C2 @ ( sup_su6327502436637775413at_nat @ A @ B ) )
=> ( ~ ( member8440522571783428010at_nat @ C2 @ A )
=> ( member8440522571783428010at_nat @ C2 @ B ) ) ) ).
% UnE
thf(fact_314_UnE,axiom,
! [C2: product_prod_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C2 @ ( sup_su459911885395995103_a_nat @ A @ B ) )
=> ( ~ ( member5724188588386418708_a_nat @ C2 @ A )
=> ( member5724188588386418708_a_nat @ C2 @ B ) ) ) ).
% UnE
thf(fact_315_UnE,axiom,
! [C2: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ C2 @ ( sup_sup_set_list_nat @ A @ B ) )
=> ( ~ ( member_list_nat2 @ C2 @ A )
=> ( member_list_nat2 @ C2 @ B ) ) ) ).
% UnE
thf(fact_316_finite__UnI,axiom,
! [F4: set_nat,G3: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( finite_finite_nat @ G3 )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F4 @ G3 ) ) ) ) ).
% finite_UnI
thf(fact_317_finite__UnI,axiom,
! [F4: set_Pr4934435412358123699_a_nat,G3: set_Pr4934435412358123699_a_nat] :
( ( finite6644898363146130708_a_nat @ F4 )
=> ( ( finite6644898363146130708_a_nat @ G3 )
=> ( finite6644898363146130708_a_nat @ ( sup_su459911885395995103_a_nat @ F4 @ G3 ) ) ) ) ).
% finite_UnI
thf(fact_318_finite__UnI,axiom,
! [F4: set_list_nat,G3: set_list_nat] :
( ( finite8100373058378681591st_nat @ F4 )
=> ( ( finite8100373058378681591st_nat @ G3 )
=> ( finite8100373058378681591st_nat @ ( sup_sup_set_list_nat @ F4 @ G3 ) ) ) ) ).
% finite_UnI
thf(fact_319_Un__infinite,axiom,
! [S2: set_nat,T2: set_nat] :
( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S2 @ T2 ) ) ) ).
% Un_infinite
thf(fact_320_Un__infinite,axiom,
! [S2: set_Pr4934435412358123699_a_nat,T2: set_Pr4934435412358123699_a_nat] :
( ~ ( finite6644898363146130708_a_nat @ S2 )
=> ~ ( finite6644898363146130708_a_nat @ ( sup_su459911885395995103_a_nat @ S2 @ T2 ) ) ) ).
% Un_infinite
thf(fact_321_Un__infinite,axiom,
! [S2: set_list_nat,T2: set_list_nat] :
( ~ ( finite8100373058378681591st_nat @ S2 )
=> ~ ( finite8100373058378681591st_nat @ ( sup_sup_set_list_nat @ S2 @ T2 ) ) ) ).
% Un_infinite
thf(fact_322_infinite__Un,axiom,
! [S2: set_nat,T2: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S2 @ T2 ) ) )
= ( ~ ( finite_finite_nat @ S2 )
| ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_Un
thf(fact_323_infinite__Un,axiom,
! [S2: set_Pr4934435412358123699_a_nat,T2: set_Pr4934435412358123699_a_nat] :
( ( ~ ( finite6644898363146130708_a_nat @ ( sup_su459911885395995103_a_nat @ S2 @ T2 ) ) )
= ( ~ ( finite6644898363146130708_a_nat @ S2 )
| ~ ( finite6644898363146130708_a_nat @ T2 ) ) ) ).
% infinite_Un
thf(fact_324_infinite__Un,axiom,
! [S2: set_list_nat,T2: set_list_nat] :
( ( ~ ( finite8100373058378681591st_nat @ ( sup_sup_set_list_nat @ S2 @ T2 ) ) )
= ( ~ ( finite8100373058378681591st_nat @ S2 )
| ~ ( finite8100373058378681591st_nat @ T2 ) ) ) ).
% infinite_Un
thf(fact_325_Un__mono,axiom,
! [A: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,D: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ C )
=> ( ( ord_le8666007276011122963_a_nat @ B @ D )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A @ B ) @ ( sup_su459911885395995103_a_nat @ C @ D ) ) ) ) ).
% Un_mono
thf(fact_326_Un__mono,axiom,
! [A: set_list_nat,C: set_list_nat,B: set_list_nat,D: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ C )
=> ( ( ord_le6045566169113846134st_nat @ B @ D )
=> ( ord_le6045566169113846134st_nat @ ( sup_sup_set_list_nat @ A @ B ) @ ( sup_sup_set_list_nat @ C @ D ) ) ) ) ).
% Un_mono
thf(fact_327_Un__mono,axiom,
! [A: set_nat,C: set_nat,B: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B @ D )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C @ D ) ) ) ) ).
% Un_mono
thf(fact_328_Un__least,axiom,
! [A: set_Pr4934435412358123699_a_nat,C: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ C )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A @ B ) @ C ) ) ) ).
% Un_least
thf(fact_329_Un__least,axiom,
! [A: set_list_nat,C: set_list_nat,B: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ C )
=> ( ( ord_le6045566169113846134st_nat @ B @ C )
=> ( ord_le6045566169113846134st_nat @ ( sup_sup_set_list_nat @ A @ B ) @ C ) ) ) ).
% Un_least
thf(fact_330_Un__least,axiom,
! [A: set_nat,C: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C ) ) ) ).
% Un_least
thf(fact_331_Un__upper1,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ).
% Un_upper1
thf(fact_332_Un__upper1,axiom,
! [A: set_list_nat,B: set_list_nat] : ( ord_le6045566169113846134st_nat @ A @ ( sup_sup_set_list_nat @ A @ B ) ) ).
% Un_upper1
thf(fact_333_Un__upper1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) ) ).
% Un_upper1
thf(fact_334_Un__upper2,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ B @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ).
% Un_upper2
thf(fact_335_Un__upper2,axiom,
! [B: set_list_nat,A: set_list_nat] : ( ord_le6045566169113846134st_nat @ B @ ( sup_sup_set_list_nat @ A @ B ) ) ).
% Un_upper2
thf(fact_336_Un__upper2,axiom,
! [B: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A @ B ) ) ).
% Un_upper2
thf(fact_337_Un__absorb1,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( sup_su459911885395995103_a_nat @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_338_Un__absorb1,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ( sup_sup_set_list_nat @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_339_Un__absorb1,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_340_Un__absorb2,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ( ( sup_su459911885395995103_a_nat @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_341_Un__absorb2,axiom,
! [B: set_list_nat,A: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B @ A )
=> ( ( sup_sup_set_list_nat @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_342_Un__absorb2,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( sup_sup_set_nat @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_343_subset__UnE,axiom,
! [C: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C @ ( sup_su459911885395995103_a_nat @ A @ B ) )
=> ~ ! [A6: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A6 @ A )
=> ! [B6: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B6 @ B )
=> ( C
!= ( sup_su459911885395995103_a_nat @ A6 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_344_subset__UnE,axiom,
! [C: set_list_nat,A: set_list_nat,B: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ C @ ( sup_sup_set_list_nat @ A @ B ) )
=> ~ ! [A6: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A6 @ A )
=> ! [B6: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B6 @ B )
=> ( C
!= ( sup_sup_set_list_nat @ A6 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_345_subset__UnE,axiom,
! [C: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
=> ~ ! [A6: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ A )
=> ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ B )
=> ( C
!= ( sup_sup_set_nat @ A6 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_346_subset__Un__eq,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ A5 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_347_subset__Un__eq,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A5: set_list_nat,B5: set_list_nat] :
( ( sup_sup_set_list_nat @ A5 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_348_subset__Un__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( sup_sup_set_nat @ A5 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_349_eq__term__by__poss__fun__at,axiom,
! [S: term_a_b,T: term_a_b] :
( ( ( term_poss_a_b @ S )
= ( term_poss_a_b @ T ) )
=> ( ! [P4: list_nat] :
( ( member_list_nat2 @ P4 @ ( term_poss_a_b @ S ) )
=> ( ( term_fun_at_a_b @ S @ P4 )
= ( term_fun_at_a_b @ T @ P4 ) ) )
=> ( S = T ) ) ) ).
% eq_term_by_poss_fun_at
thf(fact_350_fuans__term__term__to__sig,axiom,
! [F2: set_Pr4934435412358123699_a_nat,V: b,T: term_a_b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( terms_8519481630511763164ig_a_b @ F2 @ V @ T ) ) @ F2 ) ).
% fuans_term_term_to_sig
thf(fact_351_le__sup__iff,axiom,
! [X2: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ X2 @ Y ) @ Z2 )
= ( ( ord_le8666007276011122963_a_nat @ X2 @ Z2 )
& ( ord_le8666007276011122963_a_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_352_le__sup__iff,axiom,
! [X2: set_list_nat,Y: set_list_nat,Z2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( sup_sup_set_list_nat @ X2 @ Y ) @ Z2 )
= ( ( ord_le6045566169113846134st_nat @ X2 @ Z2 )
& ( ord_le6045566169113846134st_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_353_le__sup__iff,axiom,
! [X2: $o > nat,Y: $o > nat,Z2: $o > nat] :
( ( ord_less_eq_o_nat @ ( sup_sup_o_nat @ X2 @ Y ) @ Z2 )
= ( ( ord_less_eq_o_nat @ X2 @ Z2 )
& ( ord_less_eq_o_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_354_le__sup__iff,axiom,
! [X2: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X2 @ Y ) @ Z2 )
= ( ( ord_less_eq_set_nat @ X2 @ Z2 )
& ( ord_less_eq_set_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_355_le__sup__iff,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ Y ) @ Z2 )
= ( ( ord_less_eq_nat @ X2 @ Z2 )
& ( ord_less_eq_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_356_sup_Obounded__iff,axiom,
! [B2: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ B2 @ C2 ) @ A2 )
= ( ( ord_le8666007276011122963_a_nat @ B2 @ A2 )
& ( ord_le8666007276011122963_a_nat @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_357_sup_Obounded__iff,axiom,
! [B2: set_list_nat,C2: set_list_nat,A2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( sup_sup_set_list_nat @ B2 @ C2 ) @ A2 )
= ( ( ord_le6045566169113846134st_nat @ B2 @ A2 )
& ( ord_le6045566169113846134st_nat @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_358_sup_Obounded__iff,axiom,
! [B2: $o > nat,C2: $o > nat,A2: $o > nat] :
( ( ord_less_eq_o_nat @ ( sup_sup_o_nat @ B2 @ C2 ) @ A2 )
= ( ( ord_less_eq_o_nat @ B2 @ A2 )
& ( ord_less_eq_o_nat @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_359_sup_Obounded__iff,axiom,
! [B2: set_nat,C2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C2 ) @ A2 )
= ( ( ord_less_eq_set_nat @ B2 @ A2 )
& ( ord_less_eq_set_nat @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_360_sup_Obounded__iff,axiom,
! [B2: nat,C2: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A2 )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_361_par__hole__pos__replace__term__context__at,axiom,
! [P: list_nat,C: subterm_and_ctxt_a_b,S: term_a_b,U: term_a_b] :
( ( term_p5017330785391824242ar_nat @ P @ ( term_hole_pos_a_b @ C ) )
=> ( ( term_r6860082780075436317at_a_b @ ( subter2376574525758040790rm_a_b @ C @ S ) @ P @ U )
= ( subter2376574525758040790rm_a_b @ ( terms_4774307173741787698at_a_b @ C @ P @ U ) @ S ) ) ) ).
% par_hole_pos_replace_term_context_at
thf(fact_362_sup_Oidem,axiom,
! [A2: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_363_sup_Oidem,axiom,
! [A2: set_list_nat] :
( ( sup_sup_set_list_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_364_sup_Oidem,axiom,
! [A2: nat] :
( ( sup_sup_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_365_sup__idem,axiom,
! [X2: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_366_sup__idem,axiom,
! [X2: set_list_nat] :
( ( sup_sup_set_list_nat @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_367_sup__idem,axiom,
! [X2: nat] :
( ( sup_sup_nat @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_368_sup_Oleft__idem,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ A2 @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) )
= ( sup_su459911885395995103_a_nat @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_369_sup_Oleft__idem,axiom,
! [A2: set_list_nat,B2: set_list_nat] :
( ( sup_sup_set_list_nat @ A2 @ ( sup_sup_set_list_nat @ A2 @ B2 ) )
= ( sup_sup_set_list_nat @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_370_sup_Oleft__idem,axiom,
! [A2: nat,B2: nat] :
( ( sup_sup_nat @ A2 @ ( sup_sup_nat @ A2 @ B2 ) )
= ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_371_sup__left__idem,axiom,
! [X2: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ X2 @ ( sup_su459911885395995103_a_nat @ X2 @ Y ) )
= ( sup_su459911885395995103_a_nat @ X2 @ Y ) ) ).
% sup_left_idem
thf(fact_372_sup__left__idem,axiom,
! [X2: set_list_nat,Y: set_list_nat] :
( ( sup_sup_set_list_nat @ X2 @ ( sup_sup_set_list_nat @ X2 @ Y ) )
= ( sup_sup_set_list_nat @ X2 @ Y ) ) ).
% sup_left_idem
thf(fact_373_sup__left__idem,axiom,
! [X2: nat,Y: nat] :
( ( sup_sup_nat @ X2 @ ( sup_sup_nat @ X2 @ Y ) )
= ( sup_sup_nat @ X2 @ Y ) ) ).
% sup_left_idem
thf(fact_374_sup_Oright__idem,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) @ B2 )
= ( sup_su459911885395995103_a_nat @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_375_sup_Oright__idem,axiom,
! [A2: set_list_nat,B2: set_list_nat] :
( ( sup_sup_set_list_nat @ ( sup_sup_set_list_nat @ A2 @ B2 ) @ B2 )
= ( sup_sup_set_list_nat @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_376_sup_Oright__idem,axiom,
! [A2: nat,B2: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ A2 @ B2 ) @ B2 )
= ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_377_funas__ctxt__apply,axiom,
! [C: subterm_and_ctxt_a_b,T: term_a_b] :
( ( term_funas_term_a_b @ ( subter2376574525758040790rm_a_b @ C @ T ) )
= ( sup_su459911885395995103_a_nat @ ( term_funas_ctxt_a_b @ C ) @ ( term_funas_term_a_b @ T ) ) ) ).
% funas_ctxt_apply
thf(fact_378_sup__left__commute,axiom,
! [X2: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ X2 @ ( sup_su459911885395995103_a_nat @ Y @ Z2 ) )
= ( sup_su459911885395995103_a_nat @ Y @ ( sup_su459911885395995103_a_nat @ X2 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_379_sup__left__commute,axiom,
! [X2: set_list_nat,Y: set_list_nat,Z2: set_list_nat] :
( ( sup_sup_set_list_nat @ X2 @ ( sup_sup_set_list_nat @ Y @ Z2 ) )
= ( sup_sup_set_list_nat @ Y @ ( sup_sup_set_list_nat @ X2 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_380_sup__left__commute,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( sup_sup_nat @ X2 @ ( sup_sup_nat @ Y @ Z2 ) )
= ( sup_sup_nat @ Y @ ( sup_sup_nat @ X2 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_381_sup_Oleft__commute,axiom,
! [B2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ B2 @ ( sup_su459911885395995103_a_nat @ A2 @ C2 ) )
= ( sup_su459911885395995103_a_nat @ A2 @ ( sup_su459911885395995103_a_nat @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_382_sup_Oleft__commute,axiom,
! [B2: set_list_nat,A2: set_list_nat,C2: set_list_nat] :
( ( sup_sup_set_list_nat @ B2 @ ( sup_sup_set_list_nat @ A2 @ C2 ) )
= ( sup_sup_set_list_nat @ A2 @ ( sup_sup_set_list_nat @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_383_sup_Oleft__commute,axiom,
! [B2: nat,A2: nat,C2: nat] :
( ( sup_sup_nat @ B2 @ ( sup_sup_nat @ A2 @ C2 ) )
= ( sup_sup_nat @ A2 @ ( sup_sup_nat @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_384_sup__commute,axiom,
( sup_su459911885395995103_a_nat
= ( ^ [X3: set_Pr4934435412358123699_a_nat,Y4: set_Pr4934435412358123699_a_nat] : ( sup_su459911885395995103_a_nat @ Y4 @ X3 ) ) ) ).
% sup_commute
thf(fact_385_sup__commute,axiom,
( sup_sup_set_list_nat
= ( ^ [X3: set_list_nat,Y4: set_list_nat] : ( sup_sup_set_list_nat @ Y4 @ X3 ) ) ) ).
% sup_commute
thf(fact_386_sup__commute,axiom,
( sup_sup_nat
= ( ^ [X3: nat,Y4: nat] : ( sup_sup_nat @ Y4 @ X3 ) ) ) ).
% sup_commute
thf(fact_387_sup_Ocommute,axiom,
( sup_su459911885395995103_a_nat
= ( ^ [A3: set_Pr4934435412358123699_a_nat,B3: set_Pr4934435412358123699_a_nat] : ( sup_su459911885395995103_a_nat @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_388_sup_Ocommute,axiom,
( sup_sup_set_list_nat
= ( ^ [A3: set_list_nat,B3: set_list_nat] : ( sup_sup_set_list_nat @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_389_sup_Ocommute,axiom,
( sup_sup_nat
= ( ^ [A3: nat,B3: nat] : ( sup_sup_nat @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_390_sup__assoc,axiom,
! [X2: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ ( sup_su459911885395995103_a_nat @ X2 @ Y ) @ Z2 )
= ( sup_su459911885395995103_a_nat @ X2 @ ( sup_su459911885395995103_a_nat @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_391_sup__assoc,axiom,
! [X2: set_list_nat,Y: set_list_nat,Z2: set_list_nat] :
( ( sup_sup_set_list_nat @ ( sup_sup_set_list_nat @ X2 @ Y ) @ Z2 )
= ( sup_sup_set_list_nat @ X2 @ ( sup_sup_set_list_nat @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_392_sup__assoc,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ X2 @ Y ) @ Z2 )
= ( sup_sup_nat @ X2 @ ( sup_sup_nat @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_393_sup_Oassoc,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) @ C2 )
= ( sup_su459911885395995103_a_nat @ A2 @ ( sup_su459911885395995103_a_nat @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_394_sup_Oassoc,axiom,
! [A2: set_list_nat,B2: set_list_nat,C2: set_list_nat] :
( ( sup_sup_set_list_nat @ ( sup_sup_set_list_nat @ A2 @ B2 ) @ C2 )
= ( sup_sup_set_list_nat @ A2 @ ( sup_sup_set_list_nat @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_395_sup_Oassoc,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ A2 @ B2 ) @ C2 )
= ( sup_sup_nat @ A2 @ ( sup_sup_nat @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_396_inf__sup__aci_I5_J,axiom,
( sup_su459911885395995103_a_nat
= ( ^ [X3: set_Pr4934435412358123699_a_nat,Y4: set_Pr4934435412358123699_a_nat] : ( sup_su459911885395995103_a_nat @ Y4 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_397_inf__sup__aci_I5_J,axiom,
( sup_sup_set_list_nat
= ( ^ [X3: set_list_nat,Y4: set_list_nat] : ( sup_sup_set_list_nat @ Y4 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_398_inf__sup__aci_I5_J,axiom,
( sup_sup_nat
= ( ^ [X3: nat,Y4: nat] : ( sup_sup_nat @ Y4 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_399_inf__sup__aci_I6_J,axiom,
! [X2: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ ( sup_su459911885395995103_a_nat @ X2 @ Y ) @ Z2 )
= ( sup_su459911885395995103_a_nat @ X2 @ ( sup_su459911885395995103_a_nat @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_400_inf__sup__aci_I6_J,axiom,
! [X2: set_list_nat,Y: set_list_nat,Z2: set_list_nat] :
( ( sup_sup_set_list_nat @ ( sup_sup_set_list_nat @ X2 @ Y ) @ Z2 )
= ( sup_sup_set_list_nat @ X2 @ ( sup_sup_set_list_nat @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_401_inf__sup__aci_I6_J,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ X2 @ Y ) @ Z2 )
= ( sup_sup_nat @ X2 @ ( sup_sup_nat @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_402_inf__sup__aci_I7_J,axiom,
! [X2: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ X2 @ ( sup_su459911885395995103_a_nat @ Y @ Z2 ) )
= ( sup_su459911885395995103_a_nat @ Y @ ( sup_su459911885395995103_a_nat @ X2 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_403_inf__sup__aci_I7_J,axiom,
! [X2: set_list_nat,Y: set_list_nat,Z2: set_list_nat] :
( ( sup_sup_set_list_nat @ X2 @ ( sup_sup_set_list_nat @ Y @ Z2 ) )
= ( sup_sup_set_list_nat @ Y @ ( sup_sup_set_list_nat @ X2 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_404_inf__sup__aci_I7_J,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( sup_sup_nat @ X2 @ ( sup_sup_nat @ Y @ Z2 ) )
= ( sup_sup_nat @ Y @ ( sup_sup_nat @ X2 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_405_inf__sup__aci_I8_J,axiom,
! [X2: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ X2 @ ( sup_su459911885395995103_a_nat @ X2 @ Y ) )
= ( sup_su459911885395995103_a_nat @ X2 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_406_inf__sup__aci_I8_J,axiom,
! [X2: set_list_nat,Y: set_list_nat] :
( ( sup_sup_set_list_nat @ X2 @ ( sup_sup_set_list_nat @ X2 @ Y ) )
= ( sup_sup_set_list_nat @ X2 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_407_inf__sup__aci_I8_J,axiom,
! [X2: nat,Y: nat] :
( ( sup_sup_nat @ X2 @ ( sup_sup_nat @ X2 @ Y ) )
= ( sup_sup_nat @ X2 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_408_sup_OcoboundedI2,axiom,
! [C2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C2 @ B2 )
=> ( ord_le8666007276011122963_a_nat @ C2 @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_409_sup_OcoboundedI2,axiom,
! [C2: set_list_nat,B2: set_list_nat,A2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ C2 @ B2 )
=> ( ord_le6045566169113846134st_nat @ C2 @ ( sup_sup_set_list_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_410_sup_OcoboundedI2,axiom,
! [C2: $o > nat,B2: $o > nat,A2: $o > nat] :
( ( ord_less_eq_o_nat @ C2 @ B2 )
=> ( ord_less_eq_o_nat @ C2 @ ( sup_sup_o_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_411_sup_OcoboundedI2,axiom,
! [C2: set_nat,B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ B2 )
=> ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_412_sup_OcoboundedI2,axiom,
! [C2: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ C2 @ B2 )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_413_sup_OcoboundedI1,axiom,
! [C2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C2 @ A2 )
=> ( ord_le8666007276011122963_a_nat @ C2 @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_414_sup_OcoboundedI1,axiom,
! [C2: set_list_nat,A2: set_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ C2 @ A2 )
=> ( ord_le6045566169113846134st_nat @ C2 @ ( sup_sup_set_list_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_415_sup_OcoboundedI1,axiom,
! [C2: $o > nat,A2: $o > nat,B2: $o > nat] :
( ( ord_less_eq_o_nat @ C2 @ A2 )
=> ( ord_less_eq_o_nat @ C2 @ ( sup_sup_o_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_416_sup_OcoboundedI1,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A2 )
=> ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_417_sup_OcoboundedI1,axiom,
! [C2: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_418_sup_Oabsorb__iff2,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A3: set_Pr4934435412358123699_a_nat,B3: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_419_sup_Oabsorb__iff2,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A3: set_list_nat,B3: set_list_nat] :
( ( sup_sup_set_list_nat @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_420_sup_Oabsorb__iff2,axiom,
( ord_less_eq_o_nat
= ( ^ [A3: $o > nat,B3: $o > nat] :
( ( sup_sup_o_nat @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_421_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( sup_sup_set_nat @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_422_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( sup_sup_nat @ A3 @ B3 )
= B3 ) ) ) ).
% sup.absorb_iff2
thf(fact_423_sup_Oabsorb__iff1,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [B3: set_Pr4934435412358123699_a_nat,A3: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_424_sup_Oabsorb__iff1,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [B3: set_list_nat,A3: set_list_nat] :
( ( sup_sup_set_list_nat @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_425_sup_Oabsorb__iff1,axiom,
( ord_less_eq_o_nat
= ( ^ [B3: $o > nat,A3: $o > nat] :
( ( sup_sup_o_nat @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_426_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [B3: set_nat,A3: set_nat] :
( ( sup_sup_set_nat @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_427_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( sup_sup_nat @ A3 @ B3 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_428_sup_Ocobounded2,axiom,
! [B2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ B2 @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_429_sup_Ocobounded2,axiom,
! [B2: set_list_nat,A2: set_list_nat] : ( ord_le6045566169113846134st_nat @ B2 @ ( sup_sup_set_list_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_430_sup_Ocobounded2,axiom,
! [B2: $o > nat,A2: $o > nat] : ( ord_less_eq_o_nat @ B2 @ ( sup_sup_o_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_431_sup_Ocobounded2,axiom,
! [B2: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_432_sup_Ocobounded2,axiom,
! [B2: nat,A2: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_433_sup_Ocobounded1,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A2 @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_434_sup_Ocobounded1,axiom,
! [A2: set_list_nat,B2: set_list_nat] : ( ord_le6045566169113846134st_nat @ A2 @ ( sup_sup_set_list_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_435_sup_Ocobounded1,axiom,
! [A2: $o > nat,B2: $o > nat] : ( ord_less_eq_o_nat @ A2 @ ( sup_sup_o_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_436_sup_Ocobounded1,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_437_sup_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_438_sup_Oorder__iff,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [B3: set_Pr4934435412358123699_a_nat,A3: set_Pr4934435412358123699_a_nat] :
( A3
= ( sup_su459911885395995103_a_nat @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_439_sup_Oorder__iff,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [B3: set_list_nat,A3: set_list_nat] :
( A3
= ( sup_sup_set_list_nat @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_440_sup_Oorder__iff,axiom,
( ord_less_eq_o_nat
= ( ^ [B3: $o > nat,A3: $o > nat] :
( A3
= ( sup_sup_o_nat @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_441_sup_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [B3: set_nat,A3: set_nat] :
( A3
= ( sup_sup_set_nat @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_442_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( A3
= ( sup_sup_nat @ A3 @ B3 ) ) ) ) ).
% sup.order_iff
thf(fact_443_sup_OboundedI,axiom,
! [B2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B2 @ A2 )
=> ( ( ord_le8666007276011122963_a_nat @ C2 @ A2 )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ B2 @ C2 ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_444_sup_OboundedI,axiom,
! [B2: set_list_nat,A2: set_list_nat,C2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B2 @ A2 )
=> ( ( ord_le6045566169113846134st_nat @ C2 @ A2 )
=> ( ord_le6045566169113846134st_nat @ ( sup_sup_set_list_nat @ B2 @ C2 ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_445_sup_OboundedI,axiom,
! [B2: $o > nat,A2: $o > nat,C2: $o > nat] :
( ( ord_less_eq_o_nat @ B2 @ A2 )
=> ( ( ord_less_eq_o_nat @ C2 @ A2 )
=> ( ord_less_eq_o_nat @ ( sup_sup_o_nat @ B2 @ C2 ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_446_sup_OboundedI,axiom,
! [B2: set_nat,A2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ C2 @ A2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C2 ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_447_sup_OboundedI,axiom,
! [B2: nat,A2: nat,C2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_448_sup_OboundedE,axiom,
! [B2: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_le8666007276011122963_a_nat @ B2 @ A2 )
=> ~ ( ord_le8666007276011122963_a_nat @ C2 @ A2 ) ) ) ).
% sup.boundedE
thf(fact_449_sup_OboundedE,axiom,
! [B2: set_list_nat,C2: set_list_nat,A2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( sup_sup_set_list_nat @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_le6045566169113846134st_nat @ B2 @ A2 )
=> ~ ( ord_le6045566169113846134st_nat @ C2 @ A2 ) ) ) ).
% sup.boundedE
thf(fact_450_sup_OboundedE,axiom,
! [B2: $o > nat,C2: $o > nat,A2: $o > nat] :
( ( ord_less_eq_o_nat @ ( sup_sup_o_nat @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_eq_o_nat @ B2 @ A2 )
=> ~ ( ord_less_eq_o_nat @ C2 @ A2 ) ) ) ).
% sup.boundedE
thf(fact_451_sup_OboundedE,axiom,
! [B2: set_nat,C2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ~ ( ord_less_eq_set_nat @ C2 @ A2 ) ) ) ).
% sup.boundedE
thf(fact_452_sup_OboundedE,axiom,
! [B2: nat,C2: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B2 @ A2 )
=> ~ ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).
% sup.boundedE
thf(fact_453_sup__absorb2,axiom,
! [X2: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X2 @ Y )
=> ( ( sup_su459911885395995103_a_nat @ X2 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_454_sup__absorb2,axiom,
! [X2: set_list_nat,Y: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X2 @ Y )
=> ( ( sup_sup_set_list_nat @ X2 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_455_sup__absorb2,axiom,
! [X2: $o > nat,Y: $o > nat] :
( ( ord_less_eq_o_nat @ X2 @ Y )
=> ( ( sup_sup_o_nat @ X2 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_456_sup__absorb2,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( sup_sup_set_nat @ X2 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_457_sup__absorb2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( sup_sup_nat @ X2 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_458_sup__absorb1,axiom,
! [Y: set_Pr4934435412358123699_a_nat,X2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ Y @ X2 )
=> ( ( sup_su459911885395995103_a_nat @ X2 @ Y )
= X2 ) ) ).
% sup_absorb1
thf(fact_459_sup__absorb1,axiom,
! [Y: set_list_nat,X2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ Y @ X2 )
=> ( ( sup_sup_set_list_nat @ X2 @ Y )
= X2 ) ) ).
% sup_absorb1
thf(fact_460_sup__absorb1,axiom,
! [Y: $o > nat,X2: $o > nat] :
( ( ord_less_eq_o_nat @ Y @ X2 )
=> ( ( sup_sup_o_nat @ X2 @ Y )
= X2 ) ) ).
% sup_absorb1
thf(fact_461_sup__absorb1,axiom,
! [Y: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X2 )
=> ( ( sup_sup_set_nat @ X2 @ Y )
= X2 ) ) ).
% sup_absorb1
thf(fact_462_sup__absorb1,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( sup_sup_nat @ X2 @ Y )
= X2 ) ) ).
% sup_absorb1
thf(fact_463_sup_Oabsorb2,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ B2 )
=> ( ( sup_su459911885395995103_a_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_464_sup_Oabsorb2,axiom,
! [A2: set_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ B2 )
=> ( ( sup_sup_set_list_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_465_sup_Oabsorb2,axiom,
! [A2: $o > nat,B2: $o > nat] :
( ( ord_less_eq_o_nat @ A2 @ B2 )
=> ( ( sup_sup_o_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_466_sup_Oabsorb2,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_467_sup_Oabsorb2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_468_sup_Oabsorb1,axiom,
! [B2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B2 @ A2 )
=> ( ( sup_su459911885395995103_a_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_469_sup_Oabsorb1,axiom,
! [B2: set_list_nat,A2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B2 @ A2 )
=> ( ( sup_sup_set_list_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_470_sup_Oabsorb1,axiom,
! [B2: $o > nat,A2: $o > nat] :
( ( ord_less_eq_o_nat @ B2 @ A2 )
=> ( ( sup_sup_o_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_471_sup_Oabsorb1,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_472_sup_Oabsorb1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_473_sup__unique,axiom,
! [F: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,X2: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ! [X: set_Pr4934435412358123699_a_nat,Y2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ X @ ( F @ X @ Y2 ) )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ Y2 @ ( F @ X @ Y2 ) )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y2: set_Pr4934435412358123699_a_nat,Z3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ Y2 @ X )
=> ( ( ord_le8666007276011122963_a_nat @ Z3 @ X )
=> ( ord_le8666007276011122963_a_nat @ ( F @ Y2 @ Z3 ) @ X ) ) )
=> ( ( sup_su459911885395995103_a_nat @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_474_sup__unique,axiom,
! [F: set_list_nat > set_list_nat > set_list_nat,X2: set_list_nat,Y: set_list_nat] :
( ! [X: set_list_nat,Y2: set_list_nat] : ( ord_le6045566169113846134st_nat @ X @ ( F @ X @ Y2 ) )
=> ( ! [X: set_list_nat,Y2: set_list_nat] : ( ord_le6045566169113846134st_nat @ Y2 @ ( F @ X @ Y2 ) )
=> ( ! [X: set_list_nat,Y2: set_list_nat,Z3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ Y2 @ X )
=> ( ( ord_le6045566169113846134st_nat @ Z3 @ X )
=> ( ord_le6045566169113846134st_nat @ ( F @ Y2 @ Z3 ) @ X ) ) )
=> ( ( sup_sup_set_list_nat @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_475_sup__unique,axiom,
! [F: ( $o > nat ) > ( $o > nat ) > $o > nat,X2: $o > nat,Y: $o > nat] :
( ! [X: $o > nat,Y2: $o > nat] : ( ord_less_eq_o_nat @ X @ ( F @ X @ Y2 ) )
=> ( ! [X: $o > nat,Y2: $o > nat] : ( ord_less_eq_o_nat @ Y2 @ ( F @ X @ Y2 ) )
=> ( ! [X: $o > nat,Y2: $o > nat,Z3: $o > nat] :
( ( ord_less_eq_o_nat @ Y2 @ X )
=> ( ( ord_less_eq_o_nat @ Z3 @ X )
=> ( ord_less_eq_o_nat @ ( F @ Y2 @ Z3 ) @ X ) ) )
=> ( ( sup_sup_o_nat @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_476_sup__unique,axiom,
! [F: set_nat > set_nat > set_nat,X2: set_nat,Y: set_nat] :
( ! [X: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ X @ ( F @ X @ Y2 ) )
=> ( ! [X: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ Y2 @ ( F @ X @ Y2 ) )
=> ( ! [X: set_nat,Y2: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X )
=> ( ( ord_less_eq_set_nat @ Z3 @ X )
=> ( ord_less_eq_set_nat @ ( F @ Y2 @ Z3 ) @ X ) ) )
=> ( ( sup_sup_set_nat @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_477_sup__unique,axiom,
! [F: nat > nat > nat,X2: nat,Y: nat] :
( ! [X: nat,Y2: nat] : ( ord_less_eq_nat @ X @ ( F @ X @ Y2 ) )
=> ( ! [X: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ ( F @ X @ Y2 ) )
=> ( ! [X: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( ord_less_eq_nat @ Z3 @ X )
=> ( ord_less_eq_nat @ ( F @ Y2 @ Z3 ) @ X ) ) )
=> ( ( sup_sup_nat @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_478_sup_OorderI,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( A2
= ( sup_su459911885395995103_a_nat @ A2 @ B2 ) )
=> ( ord_le8666007276011122963_a_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_479_sup_OorderI,axiom,
! [A2: set_list_nat,B2: set_list_nat] :
( ( A2
= ( sup_sup_set_list_nat @ A2 @ B2 ) )
=> ( ord_le6045566169113846134st_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_480_sup_OorderI,axiom,
! [A2: $o > nat,B2: $o > nat] :
( ( A2
= ( sup_sup_o_nat @ A2 @ B2 ) )
=> ( ord_less_eq_o_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_481_sup_OorderI,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2
= ( sup_sup_set_nat @ A2 @ B2 ) )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_482_sup_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_483_sup_OorderE,axiom,
! [B2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B2 @ A2 )
=> ( A2
= ( sup_su459911885395995103_a_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_484_sup_OorderE,axiom,
! [B2: set_list_nat,A2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_set_list_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_485_sup_OorderE,axiom,
! [B2: $o > nat,A2: $o > nat] :
( ( ord_less_eq_o_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_o_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_486_sup_OorderE,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_487_sup_OorderE,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_488_le__iff__sup,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [X3: set_Pr4934435412358123699_a_nat,Y4: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_489_le__iff__sup,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [X3: set_list_nat,Y4: set_list_nat] :
( ( sup_sup_set_list_nat @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_490_le__iff__sup,axiom,
( ord_less_eq_o_nat
= ( ^ [X3: $o > nat,Y4: $o > nat] :
( ( sup_sup_o_nat @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_491_le__iff__sup,axiom,
( ord_less_eq_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( sup_sup_set_nat @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_492_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( sup_sup_nat @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_493_sup__least,axiom,
! [Y: set_Pr4934435412358123699_a_nat,X2: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ Y @ X2 )
=> ( ( ord_le8666007276011122963_a_nat @ Z2 @ X2 )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ Y @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_494_sup__least,axiom,
! [Y: set_list_nat,X2: set_list_nat,Z2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ Y @ X2 )
=> ( ( ord_le6045566169113846134st_nat @ Z2 @ X2 )
=> ( ord_le6045566169113846134st_nat @ ( sup_sup_set_list_nat @ Y @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_495_sup__least,axiom,
! [Y: $o > nat,X2: $o > nat,Z2: $o > nat] :
( ( ord_less_eq_o_nat @ Y @ X2 )
=> ( ( ord_less_eq_o_nat @ Z2 @ X2 )
=> ( ord_less_eq_o_nat @ ( sup_sup_o_nat @ Y @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_496_sup__least,axiom,
! [Y: set_nat,X2: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X2 )
=> ( ( ord_less_eq_set_nat @ Z2 @ X2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_497_sup__least,axiom,
! [Y: nat,X2: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_498_sup__mono,axiom,
! [A2: set_Pr4934435412358123699_a_nat,C2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,D2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ C2 )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ D2 )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) @ ( sup_su459911885395995103_a_nat @ C2 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_499_sup__mono,axiom,
! [A2: set_list_nat,C2: set_list_nat,B2: set_list_nat,D2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ C2 )
=> ( ( ord_le6045566169113846134st_nat @ B2 @ D2 )
=> ( ord_le6045566169113846134st_nat @ ( sup_sup_set_list_nat @ A2 @ B2 ) @ ( sup_sup_set_list_nat @ C2 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_500_sup__mono,axiom,
! [A2: $o > nat,C2: $o > nat,B2: $o > nat,D2: $o > nat] :
( ( ord_less_eq_o_nat @ A2 @ C2 )
=> ( ( ord_less_eq_o_nat @ B2 @ D2 )
=> ( ord_less_eq_o_nat @ ( sup_sup_o_nat @ A2 @ B2 ) @ ( sup_sup_o_nat @ C2 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_501_sup__mono,axiom,
! [A2: set_nat,C2: set_nat,B2: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ D2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ ( sup_sup_set_nat @ C2 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_502_sup__mono,axiom,
! [A2: nat,C2: nat,B2: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ D2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ ( sup_sup_nat @ C2 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_503_sup_Omono,axiom,
! [C2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat,D2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C2 @ A2 )
=> ( ( ord_le8666007276011122963_a_nat @ D2 @ B2 )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ C2 @ D2 ) @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_504_sup_Omono,axiom,
! [C2: set_list_nat,A2: set_list_nat,D2: set_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ C2 @ A2 )
=> ( ( ord_le6045566169113846134st_nat @ D2 @ B2 )
=> ( ord_le6045566169113846134st_nat @ ( sup_sup_set_list_nat @ C2 @ D2 ) @ ( sup_sup_set_list_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_505_sup_Omono,axiom,
! [C2: $o > nat,A2: $o > nat,D2: $o > nat,B2: $o > nat] :
( ( ord_less_eq_o_nat @ C2 @ A2 )
=> ( ( ord_less_eq_o_nat @ D2 @ B2 )
=> ( ord_less_eq_o_nat @ ( sup_sup_o_nat @ C2 @ D2 ) @ ( sup_sup_o_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_506_sup_Omono,axiom,
! [C2: set_nat,A2: set_nat,D2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A2 )
=> ( ( ord_less_eq_set_nat @ D2 @ B2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C2 @ D2 ) @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_507_sup_Omono,axiom,
! [C2: nat,A2: nat,D2: nat,B2: nat] :
( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ( ord_less_eq_nat @ D2 @ B2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C2 @ D2 ) @ ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_508_le__supI2,axiom,
! [X2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X2 @ B2 )
=> ( ord_le8666007276011122963_a_nat @ X2 @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_509_le__supI2,axiom,
! [X2: set_list_nat,B2: set_list_nat,A2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X2 @ B2 )
=> ( ord_le6045566169113846134st_nat @ X2 @ ( sup_sup_set_list_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_510_le__supI2,axiom,
! [X2: $o > nat,B2: $o > nat,A2: $o > nat] :
( ( ord_less_eq_o_nat @ X2 @ B2 )
=> ( ord_less_eq_o_nat @ X2 @ ( sup_sup_o_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_511_le__supI2,axiom,
! [X2: set_nat,B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ B2 )
=> ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_512_le__supI2,axiom,
! [X2: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ X2 @ B2 )
=> ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_513_le__supI1,axiom,
! [X2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X2 @ A2 )
=> ( ord_le8666007276011122963_a_nat @ X2 @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_514_le__supI1,axiom,
! [X2: set_list_nat,A2: set_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ X2 @ A2 )
=> ( ord_le6045566169113846134st_nat @ X2 @ ( sup_sup_set_list_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_515_le__supI1,axiom,
! [X2: $o > nat,A2: $o > nat,B2: $o > nat] :
( ( ord_less_eq_o_nat @ X2 @ A2 )
=> ( ord_less_eq_o_nat @ X2 @ ( sup_sup_o_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_516_le__supI1,axiom,
! [X2: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_517_le__supI1,axiom,
! [X2: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_518_sup__ge2,axiom,
! [Y: set_Pr4934435412358123699_a_nat,X2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ Y @ ( sup_su459911885395995103_a_nat @ X2 @ Y ) ) ).
% sup_ge2
thf(fact_519_sup__ge2,axiom,
! [Y: set_list_nat,X2: set_list_nat] : ( ord_le6045566169113846134st_nat @ Y @ ( sup_sup_set_list_nat @ X2 @ Y ) ) ).
% sup_ge2
thf(fact_520_sup__ge2,axiom,
! [Y: $o > nat,X2: $o > nat] : ( ord_less_eq_o_nat @ Y @ ( sup_sup_o_nat @ X2 @ Y ) ) ).
% sup_ge2
thf(fact_521_sup__ge2,axiom,
! [Y: set_nat,X2: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X2 @ Y ) ) ).
% sup_ge2
thf(fact_522_sup__ge2,axiom,
! [Y: nat,X2: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X2 @ Y ) ) ).
% sup_ge2
thf(fact_523_sup__ge1,axiom,
! [X2: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ X2 @ ( sup_su459911885395995103_a_nat @ X2 @ Y ) ) ).
% sup_ge1
thf(fact_524_sup__ge1,axiom,
! [X2: set_list_nat,Y: set_list_nat] : ( ord_le6045566169113846134st_nat @ X2 @ ( sup_sup_set_list_nat @ X2 @ Y ) ) ).
% sup_ge1
thf(fact_525_sup__ge1,axiom,
! [X2: $o > nat,Y: $o > nat] : ( ord_less_eq_o_nat @ X2 @ ( sup_sup_o_nat @ X2 @ Y ) ) ).
% sup_ge1
thf(fact_526_sup__ge1,axiom,
! [X2: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y ) ) ).
% sup_ge1
thf(fact_527_sup__ge1,axiom,
! [X2: nat,Y: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y ) ) ).
% sup_ge1
thf(fact_528_le__supI,axiom,
! [A2: set_Pr4934435412358123699_a_nat,X2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A2 @ X2 )
=> ( ( ord_le8666007276011122963_a_nat @ B2 @ X2 )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_529_le__supI,axiom,
! [A2: set_list_nat,X2: set_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ X2 )
=> ( ( ord_le6045566169113846134st_nat @ B2 @ X2 )
=> ( ord_le6045566169113846134st_nat @ ( sup_sup_set_list_nat @ A2 @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_530_le__supI,axiom,
! [A2: $o > nat,X2: $o > nat,B2: $o > nat] :
( ( ord_less_eq_o_nat @ A2 @ X2 )
=> ( ( ord_less_eq_o_nat @ B2 @ X2 )
=> ( ord_less_eq_o_nat @ ( sup_sup_o_nat @ A2 @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_531_le__supI,axiom,
! [A2: set_nat,X2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ X2 )
=> ( ( ord_less_eq_set_nat @ B2 @ X2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_532_le__supI,axiom,
! [A2: nat,X2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X2 )
=> ( ( ord_less_eq_nat @ B2 @ X2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_533_le__supE,axiom,
! [A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,X2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) @ X2 )
=> ~ ( ( ord_le8666007276011122963_a_nat @ A2 @ X2 )
=> ~ ( ord_le8666007276011122963_a_nat @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_534_le__supE,axiom,
! [A2: set_list_nat,B2: set_list_nat,X2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( sup_sup_set_list_nat @ A2 @ B2 ) @ X2 )
=> ~ ( ( ord_le6045566169113846134st_nat @ A2 @ X2 )
=> ~ ( ord_le6045566169113846134st_nat @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_535_le__supE,axiom,
! [A2: $o > nat,B2: $o > nat,X2: $o > nat] :
( ( ord_less_eq_o_nat @ ( sup_sup_o_nat @ A2 @ B2 ) @ X2 )
=> ~ ( ( ord_less_eq_o_nat @ A2 @ X2 )
=> ~ ( ord_less_eq_o_nat @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_536_le__supE,axiom,
! [A2: set_nat,B2: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ X2 )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ X2 )
=> ~ ( ord_less_eq_set_nat @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_537_le__supE,axiom,
! [A2: nat,B2: nat,X2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X2 )
=> ~ ( ( ord_less_eq_nat @ A2 @ X2 )
=> ~ ( ord_less_eq_nat @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_538_inf__sup__ord_I3_J,axiom,
! [X2: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ X2 @ ( sup_su459911885395995103_a_nat @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_539_inf__sup__ord_I3_J,axiom,
! [X2: set_list_nat,Y: set_list_nat] : ( ord_le6045566169113846134st_nat @ X2 @ ( sup_sup_set_list_nat @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_540_inf__sup__ord_I3_J,axiom,
! [X2: $o > nat,Y: $o > nat] : ( ord_less_eq_o_nat @ X2 @ ( sup_sup_o_nat @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_541_inf__sup__ord_I3_J,axiom,
! [X2: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_542_inf__sup__ord_I3_J,axiom,
! [X2: nat,Y: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_543_inf__sup__ord_I4_J,axiom,
! [Y: set_Pr4934435412358123699_a_nat,X2: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ Y @ ( sup_su459911885395995103_a_nat @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_544_inf__sup__ord_I4_J,axiom,
! [Y: set_list_nat,X2: set_list_nat] : ( ord_le6045566169113846134st_nat @ Y @ ( sup_sup_set_list_nat @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_545_inf__sup__ord_I4_J,axiom,
! [Y: $o > nat,X2: $o > nat] : ( ord_less_eq_o_nat @ Y @ ( sup_sup_o_nat @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_546_inf__sup__ord_I4_J,axiom,
! [Y: set_nat,X2: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_547_inf__sup__ord_I4_J,axiom,
! [Y: nat,X2: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_548_fun__at__hole__pos__ctxt__apply,axiom,
! [C: subterm_and_ctxt_a_b,T: term_a_b] :
( ( term_fun_at_a_b @ ( subter2376574525758040790rm_a_b @ C @ T ) @ ( term_hole_pos_a_b @ C ) )
= ( term_fun_at_a_b @ T @ nil_nat ) ) ).
% fun_at_hole_pos_ctxt_apply
thf(fact_549_ctxt__apply__term__subt__at__hole__pos,axiom,
! [C: subterm_and_ctxt_a_b,S: term_a_b,Q: list_nat] :
( ( term_subt_at_a_b @ ( subter2376574525758040790rm_a_b @ C @ S ) @ ( append_nat @ ( term_hole_pos_a_b @ C ) @ Q ) )
= ( term_subt_at_a_b @ S @ Q ) ) ).
% ctxt_apply_term_subt_at_hole_pos
thf(fact_550_ctxt__apply__term__replace__term__hole__pos,axiom,
! [C: subterm_and_ctxt_a_b,S: term_a_b,Q: list_nat,U: term_a_b] :
( ( term_r6860082780075436317at_a_b @ ( subter2376574525758040790rm_a_b @ C @ S ) @ ( append_nat @ ( term_hole_pos_a_b @ C ) @ Q ) @ U )
= ( subter2376574525758040790rm_a_b @ C @ ( term_r6860082780075436317at_a_b @ S @ Q @ U ) ) ) ).
% ctxt_apply_term_replace_term_hole_pos
thf(fact_551_poss__ctxt__apply,axiom,
! [C: subterm_and_ctxt_a_b,P: list_nat,S: term_a_b] :
( ( member_list_nat2 @ ( append_nat @ ( term_hole_pos_a_b @ C ) @ P ) @ ( term_poss_a_b @ ( subter2376574525758040790rm_a_b @ C @ S ) ) )
= ( member_list_nat2 @ P @ ( term_poss_a_b @ S ) ) ) ).
% poss_ctxt_apply
thf(fact_552_poss__of__term__const__ctxt__apply,axiom,
! [P: list_nat,C2: a,C: subterm_and_ctxt_a_b,S: term_a_b] :
( ( member_list_nat2 @ P @ ( terms_7168686267159881682rm_a_b @ ( fun_a_b @ C2 @ nil_term_a_b ) @ ( subter2376574525758040790rm_a_b @ C @ S ) ) )
=> ( ( term_p5017330785391824242ar_nat @ P @ ( term_hole_pos_a_b @ C ) )
| ( term_p3503116865373065078eq_nat @ ( term_hole_pos_a_b @ C ) @ P ) ) ) ).
% poss_of_term_const_ctxt_apply
thf(fact_553_boolean__algebra__cancel_Osup2,axiom,
! [B: set_Pr4934435412358123699_a_nat,K: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat] :
( ( B
= ( sup_su459911885395995103_a_nat @ K @ B2 ) )
=> ( ( sup_su459911885395995103_a_nat @ A2 @ B )
= ( sup_su459911885395995103_a_nat @ K @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_554_boolean__algebra__cancel_Osup2,axiom,
! [B: set_list_nat,K: set_list_nat,B2: set_list_nat,A2: set_list_nat] :
( ( B
= ( sup_sup_set_list_nat @ K @ B2 ) )
=> ( ( sup_sup_set_list_nat @ A2 @ B )
= ( sup_sup_set_list_nat @ K @ ( sup_sup_set_list_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_555_boolean__algebra__cancel_Osup2,axiom,
! [B: nat,K: nat,B2: nat,A2: nat] :
( ( B
= ( sup_sup_nat @ K @ B2 ) )
=> ( ( sup_sup_nat @ A2 @ B )
= ( sup_sup_nat @ K @ ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_556_boolean__algebra__cancel_Osup1,axiom,
! [A: set_Pr4934435412358123699_a_nat,K: set_Pr4934435412358123699_a_nat,A2: set_Pr4934435412358123699_a_nat,B2: set_Pr4934435412358123699_a_nat] :
( ( A
= ( sup_su459911885395995103_a_nat @ K @ A2 ) )
=> ( ( sup_su459911885395995103_a_nat @ A @ B2 )
= ( sup_su459911885395995103_a_nat @ K @ ( sup_su459911885395995103_a_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_557_boolean__algebra__cancel_Osup1,axiom,
! [A: set_list_nat,K: set_list_nat,A2: set_list_nat,B2: set_list_nat] :
( ( A
= ( sup_sup_set_list_nat @ K @ A2 ) )
=> ( ( sup_sup_set_list_nat @ A @ B2 )
= ( sup_sup_set_list_nat @ K @ ( sup_sup_set_list_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_558_boolean__algebra__cancel_Osup1,axiom,
! [A: nat,K: nat,A2: nat,B2: nat] :
( ( A
= ( sup_sup_nat @ K @ A2 ) )
=> ( ( sup_sup_nat @ A @ B2 )
= ( sup_sup_nat @ K @ ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_559_position__less__Nil__is__bot2,axiom,
! [P: list_term_a_b] :
( ( term_p8391561492822560442rm_a_b @ P @ nil_term_a_b )
= ( P = nil_term_a_b ) ) ).
% position_less_Nil_is_bot2
thf(fact_560_position__less__Nil__is__bot2,axiom,
! [P: list_list_nat] :
( ( term_p5934426891874639750st_nat @ P @ nil_list_nat )
= ( P = nil_list_nat ) ) ).
% position_less_Nil_is_bot2
thf(fact_561_position__less__Nil__is__bot2,axiom,
! [P: list_set_nat] :
( ( term_p5979160145136640812et_nat @ P @ nil_set_nat )
= ( P = nil_set_nat ) ) ).
% position_less_Nil_is_bot2
thf(fact_562_position__less__Nil__is__bot2,axiom,
! [P: list_nat] :
( ( term_p3503116865373065078eq_nat @ P @ nil_nat )
= ( P = nil_nat ) ) ).
% position_less_Nil_is_bot2
thf(fact_563_Nil__not__par_I2_J,axiom,
! [P: list_term_a_b] :
~ ( term_p7407996180858101430rm_a_b @ P @ nil_term_a_b ) ).
% Nil_not_par(2)
thf(fact_564_Nil__not__par_I2_J,axiom,
! [P: list_list_nat] :
~ ( term_p4950861579910180738st_nat @ P @ nil_list_nat ) ).
% Nil_not_par(2)
thf(fact_565_Nil__not__par_I2_J,axiom,
! [P: list_set_nat] :
~ ( term_p7908618117148864808et_nat @ P @ nil_set_nat ) ).
% Nil_not_par(2)
thf(fact_566_Nil__not__par_I2_J,axiom,
! [P: list_nat] :
~ ( term_p5017330785391824242ar_nat @ P @ nil_nat ) ).
% Nil_not_par(2)
thf(fact_567_Nil__not__par_I1_J,axiom,
! [P: list_term_a_b] :
~ ( term_p7407996180858101430rm_a_b @ nil_term_a_b @ P ) ).
% Nil_not_par(1)
thf(fact_568_Nil__not__par_I1_J,axiom,
! [P: list_list_nat] :
~ ( term_p4950861579910180738st_nat @ nil_list_nat @ P ) ).
% Nil_not_par(1)
thf(fact_569_Nil__not__par_I1_J,axiom,
! [P: list_set_nat] :
~ ( term_p7908618117148864808et_nat @ nil_set_nat @ P ) ).
% Nil_not_par(1)
thf(fact_570_Nil__not__par_I1_J,axiom,
! [P: list_nat] :
~ ( term_p5017330785391824242ar_nat @ nil_nat @ P ) ).
% Nil_not_par(1)
thf(fact_571_pos__diff__itself,axiom,
! [P: list_term_a_b] :
( ( term_p798503758663136087rm_a_b @ P @ P )
= nil_term_a_b ) ).
% pos_diff_itself
thf(fact_572_pos__diff__itself,axiom,
! [P: list_list_nat] :
( ( term_p7564741194569991203st_nat @ P @ P )
= nil_list_nat ) ).
% pos_diff_itself
thf(fact_573_pos__diff__itself,axiom,
! [P: list_set_nat] :
( ( term_p572219117888194121et_nat @ P @ P )
= nil_set_nat ) ).
% pos_diff_itself
thf(fact_574_pos__diff__itself,axiom,
! [P: list_nat] :
( ( term_pos_diff_nat @ P @ P )
= nil_nat ) ).
% pos_diff_itself
thf(fact_575_position__diff__Nil,axiom,
! [Q: list_term_a_b] :
( ( term_p798503758663136087rm_a_b @ Q @ nil_term_a_b )
= Q ) ).
% position_diff_Nil
thf(fact_576_position__diff__Nil,axiom,
! [Q: list_list_nat] :
( ( term_p7564741194569991203st_nat @ Q @ nil_list_nat )
= Q ) ).
% position_diff_Nil
thf(fact_577_position__diff__Nil,axiom,
! [Q: list_set_nat] :
( ( term_p572219117888194121et_nat @ Q @ nil_set_nat )
= Q ) ).
% position_diff_Nil
thf(fact_578_position__diff__Nil,axiom,
! [Q: list_nat] :
( ( term_pos_diff_nat @ Q @ nil_nat )
= Q ) ).
% position_diff_Nil
thf(fact_579_pos__diff__append__itself,axiom,
! [P: list_list_nat,Q: list_list_nat] :
( ( term_p7564741194569991203st_nat @ ( append_list_nat @ P @ Q ) @ P )
= Q ) ).
% pos_diff_append_itself
thf(fact_580_pos__diff__append__itself,axiom,
! [P: list_nat,Q: list_nat] :
( ( term_pos_diff_nat @ ( append_nat @ P @ Q ) @ P )
= Q ) ).
% pos_diff_append_itself
thf(fact_581_pos__les__eq__append__diff,axiom,
! [P: list_list_nat,Q: list_list_nat] :
( ( term_p5934426891874639750st_nat @ P @ Q )
=> ( ( append_list_nat @ P @ ( term_p7564741194569991203st_nat @ Q @ P ) )
= Q ) ) ).
% pos_les_eq_append_diff
thf(fact_582_pos__les__eq__append__diff,axiom,
! [P: list_nat,Q: list_nat] :
( ( term_p3503116865373065078eq_nat @ P @ Q )
=> ( ( append_nat @ P @ ( term_pos_diff_nat @ Q @ P ) )
= Q ) ) ).
% pos_les_eq_append_diff
thf(fact_583_replace__subterm__at__itself,axiom,
! [S: term_a_b,P: list_nat,Q: list_nat,T: term_a_b] :
( ( term_r6860082780075436317at_a_b @ S @ P @ ( term_r6860082780075436317at_a_b @ ( term_subt_at_a_b @ S @ P ) @ Q @ T ) )
= ( term_r6860082780075436317at_a_b @ S @ ( append_nat @ P @ Q ) @ T ) ) ).
% replace_subterm_at_itself
thf(fact_584_term__to__sig__ctxt__apply,axiom,
! [F2: set_Pr4934435412358123699_a_nat,C: subterm_and_ctxt_a_b,V: b,S: term_a_b] :
( ( terms_8374513854926927137th_a_b @ F2 @ C )
=> ( ( terms_8519481630511763164ig_a_b @ F2 @ V @ ( subter2376574525758040790rm_a_b @ C @ S ) )
= ( subter2376574525758040790rm_a_b @ ( terms_3799363064701517935xt_a_b @ F2 @ V @ C ) @ ( terms_8519481630511763164ig_a_b @ F2 @ V @ S ) ) ) ) ).
% term_to_sig_ctxt_apply
thf(fact_585_position__less__eq__def,axiom,
( term_p5934426891874639750st_nat
= ( ^ [P2: list_list_nat,Q2: list_list_nat] :
? [R: list_list_nat] :
( ( append_list_nat @ P2 @ R )
= Q2 ) ) ) ).
% position_less_eq_def
thf(fact_586_position__less__eq__def,axiom,
( term_p3503116865373065078eq_nat
= ( ^ [P2: list_nat,Q2: list_nat] :
? [R: list_nat] :
( ( append_nat @ P2 @ R )
= Q2 ) ) ) ).
% position_less_eq_def
thf(fact_587_position__less__eq__induct,axiom,
! [P: list_list_nat,Q: list_list_nat,P3: list_list_nat > list_list_nat > $o] :
( ( term_p5934426891874639750st_nat @ P @ Q )
=> ( ! [P4: list_list_nat] : ( P3 @ P4 @ P4 )
=> ( ! [P4: list_list_nat,Q4: list_list_nat,R2: list_list_nat] :
( ( term_p5934426891874639750st_nat @ P4 @ Q4 )
=> ( ( P3 @ P4 @ Q4 )
=> ( P3 @ P4 @ ( append_list_nat @ Q4 @ R2 ) ) ) )
=> ( P3 @ P @ Q ) ) ) ) ).
% position_less_eq_induct
thf(fact_588_position__less__eq__induct,axiom,
! [P: list_nat,Q: list_nat,P3: list_nat > list_nat > $o] :
( ( term_p3503116865373065078eq_nat @ P @ Q )
=> ( ! [P4: list_nat] : ( P3 @ P4 @ P4 )
=> ( ! [P4: list_nat,Q4: list_nat,R2: list_nat] :
( ( term_p3503116865373065078eq_nat @ P4 @ Q4 )
=> ( ( P3 @ P4 @ Q4 )
=> ( P3 @ P4 @ ( append_nat @ Q4 @ R2 ) ) ) )
=> ( P3 @ P @ Q ) ) ) ) ).
% position_less_eq_induct
thf(fact_589_less__eq__poss__append__itself,axiom,
! [P: list_list_nat,Q: list_list_nat] : ( term_p5934426891874639750st_nat @ P @ ( append_list_nat @ P @ Q ) ) ).
% less_eq_poss_append_itself
thf(fact_590_less__eq__poss__append__itself,axiom,
! [P: list_nat,Q: list_nat] : ( term_p3503116865373065078eq_nat @ P @ ( append_nat @ P @ Q ) ) ).
% less_eq_poss_append_itself
thf(fact_591_Nil__in__poss,axiom,
! [T: term_a_b] : ( member_list_nat2 @ nil_nat @ ( term_poss_a_b @ T ) ) ).
% Nil_in_poss
thf(fact_592_replace__term__at_Osimps_I1_J,axiom,
! [S: term_a_b,T: term_a_b] :
( ( term_r6860082780075436317at_a_b @ S @ nil_nat @ T )
= T ) ).
% replace_term_at.simps(1)
thf(fact_593_subt__at_Osimps_I1_J,axiom,
! [S: term_a_b] :
( ( term_subt_at_a_b @ S @ nil_nat )
= S ) ).
% subt_at.simps(1)
thf(fact_594_position__less__Nil__is__bot,axiom,
! [P: list_term_a_b] : ( term_p8391561492822560442rm_a_b @ nil_term_a_b @ P ) ).
% position_less_Nil_is_bot
thf(fact_595_position__less__Nil__is__bot,axiom,
! [P: list_list_nat] : ( term_p5934426891874639750st_nat @ nil_list_nat @ P ) ).
% position_less_Nil_is_bot
thf(fact_596_position__less__Nil__is__bot,axiom,
! [P: list_set_nat] : ( term_p5979160145136640812et_nat @ nil_set_nat @ P ) ).
% position_less_Nil_is_bot
thf(fact_597_position__less__Nil__is__bot,axiom,
! [P: list_nat] : ( term_p3503116865373065078eq_nat @ nil_nat @ P ) ).
% position_less_Nil_is_bot
thf(fact_598_poss__append__poss,axiom,
! [P: list_nat,Q: list_nat,T: term_a_b] :
( ( member_list_nat2 @ ( append_nat @ P @ Q ) @ ( term_poss_a_b @ T ) )
= ( ( member_list_nat2 @ P @ ( term_poss_a_b @ T ) )
& ( member_list_nat2 @ Q @ ( term_poss_a_b @ ( term_subt_at_a_b @ T @ P ) ) ) ) ) ).
% poss_append_poss
thf(fact_599_subt__at__append__dist,axiom,
! [P: list_nat,Q: list_nat,S: term_a_b] :
( ( member_list_nat2 @ ( append_nat @ P @ Q ) @ ( term_poss_a_b @ S ) )
=> ( ( term_subt_at_a_b @ S @ ( append_nat @ P @ Q ) )
= ( term_subt_at_a_b @ ( term_subt_at_a_b @ S @ P ) @ Q ) ) ) ).
% subt_at_append_dist
thf(fact_600_append_Oright__neutral,axiom,
! [A2: list_term_a_b] :
( ( append_term_a_b @ A2 @ nil_term_a_b )
= A2 ) ).
% append.right_neutral
thf(fact_601_append_Oright__neutral,axiom,
! [A2: list_list_nat] :
( ( append_list_nat @ A2 @ nil_list_nat )
= A2 ) ).
% append.right_neutral
thf(fact_602_append_Oright__neutral,axiom,
! [A2: list_set_nat] :
( ( append_set_nat @ A2 @ nil_set_nat )
= A2 ) ).
% append.right_neutral
thf(fact_603_append_Oright__neutral,axiom,
! [A2: list_nat] :
( ( append_nat @ A2 @ nil_nat )
= A2 ) ).
% append.right_neutral
thf(fact_604_append__Nil2,axiom,
! [Xs: list_term_a_b] :
( ( append_term_a_b @ Xs @ nil_term_a_b )
= Xs ) ).
% append_Nil2
thf(fact_605_append__Nil2,axiom,
! [Xs: list_list_nat] :
( ( append_list_nat @ Xs @ nil_list_nat )
= Xs ) ).
% append_Nil2
thf(fact_606_append__Nil2,axiom,
! [Xs: list_set_nat] :
( ( append_set_nat @ Xs @ nil_set_nat )
= Xs ) ).
% append_Nil2
thf(fact_607_append__Nil2,axiom,
! [Xs: list_nat] :
( ( append_nat @ Xs @ nil_nat )
= Xs ) ).
% append_Nil2
thf(fact_608_append__self__conv,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b] :
( ( ( append_term_a_b @ Xs @ Ys )
= Xs )
= ( Ys = nil_term_a_b ) ) ).
% append_self_conv
thf(fact_609_append__self__conv,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( ( append_list_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_list_nat ) ) ).
% append_self_conv
thf(fact_610_append__self__conv,axiom,
! [Xs: list_set_nat,Ys: list_set_nat] :
( ( ( append_set_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_set_nat ) ) ).
% append_self_conv
thf(fact_611_append__self__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_nat ) ) ).
% append_self_conv
thf(fact_612_self__append__conv,axiom,
! [Y: list_term_a_b,Ys: list_term_a_b] :
( ( Y
= ( append_term_a_b @ Y @ Ys ) )
= ( Ys = nil_term_a_b ) ) ).
% self_append_conv
thf(fact_613_self__append__conv,axiom,
! [Y: list_list_nat,Ys: list_list_nat] :
( ( Y
= ( append_list_nat @ Y @ Ys ) )
= ( Ys = nil_list_nat ) ) ).
% self_append_conv
thf(fact_614_self__append__conv,axiom,
! [Y: list_set_nat,Ys: list_set_nat] :
( ( Y
= ( append_set_nat @ Y @ Ys ) )
= ( Ys = nil_set_nat ) ) ).
% self_append_conv
thf(fact_615_self__append__conv,axiom,
! [Y: list_nat,Ys: list_nat] :
( ( Y
= ( append_nat @ Y @ Ys ) )
= ( Ys = nil_nat ) ) ).
% self_append_conv
thf(fact_616_append__self__conv2,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b] :
( ( ( append_term_a_b @ Xs @ Ys )
= Ys )
= ( Xs = nil_term_a_b ) ) ).
% append_self_conv2
thf(fact_617_append__self__conv2,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( ( append_list_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_list_nat ) ) ).
% append_self_conv2
thf(fact_618_append__self__conv2,axiom,
! [Xs: list_set_nat,Ys: list_set_nat] :
( ( ( append_set_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_set_nat ) ) ).
% append_self_conv2
thf(fact_619_append__self__conv2,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_nat ) ) ).
% append_self_conv2
thf(fact_620_self__append__conv2,axiom,
! [Y: list_term_a_b,Xs: list_term_a_b] :
( ( Y
= ( append_term_a_b @ Xs @ Y ) )
= ( Xs = nil_term_a_b ) ) ).
% self_append_conv2
thf(fact_621_self__append__conv2,axiom,
! [Y: list_list_nat,Xs: list_list_nat] :
( ( Y
= ( append_list_nat @ Xs @ Y ) )
= ( Xs = nil_list_nat ) ) ).
% self_append_conv2
thf(fact_622_self__append__conv2,axiom,
! [Y: list_set_nat,Xs: list_set_nat] :
( ( Y
= ( append_set_nat @ Xs @ Y ) )
= ( Xs = nil_set_nat ) ) ).
% self_append_conv2
thf(fact_623_self__append__conv2,axiom,
! [Y: list_nat,Xs: list_nat] :
( ( Y
= ( append_nat @ Xs @ Y ) )
= ( Xs = nil_nat ) ) ).
% self_append_conv2
thf(fact_624_Nil__is__append__conv,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b] :
( ( nil_term_a_b
= ( append_term_a_b @ Xs @ Ys ) )
= ( ( Xs = nil_term_a_b )
& ( Ys = nil_term_a_b ) ) ) ).
% Nil_is_append_conv
thf(fact_625_Nil__is__append__conv,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( nil_list_nat
= ( append_list_nat @ Xs @ Ys ) )
= ( ( Xs = nil_list_nat )
& ( Ys = nil_list_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_626_Nil__is__append__conv,axiom,
! [Xs: list_set_nat,Ys: list_set_nat] :
( ( nil_set_nat
= ( append_set_nat @ Xs @ Ys ) )
= ( ( Xs = nil_set_nat )
& ( Ys = nil_set_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_627_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_628_append__is__Nil__conv,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b] :
( ( ( append_term_a_b @ Xs @ Ys )
= nil_term_a_b )
= ( ( Xs = nil_term_a_b )
& ( Ys = nil_term_a_b ) ) ) ).
% append_is_Nil_conv
thf(fact_629_append__is__Nil__conv,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( ( append_list_nat @ Xs @ Ys )
= nil_list_nat )
= ( ( Xs = nil_list_nat )
& ( Ys = nil_list_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_630_append__is__Nil__conv,axiom,
! [Xs: list_set_nat,Ys: list_set_nat] :
( ( ( append_set_nat @ Xs @ Ys )
= nil_set_nat )
= ( ( Xs = nil_set_nat )
& ( Ys = nil_set_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_631_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_632_same__append__eq,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_633_same__append__eq,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,Zs: list_list_nat] :
( ( ( append_list_nat @ Xs @ Ys )
= ( append_list_nat @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_634_append__same__eq,axiom,
! [Ys: list_nat,Xs: list_nat,Zs: list_nat] :
( ( ( append_nat @ Ys @ Xs )
= ( append_nat @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_635_append__same__eq,axiom,
! [Ys: list_list_nat,Xs: list_list_nat,Zs: list_list_nat] :
( ( ( append_list_nat @ Ys @ Xs )
= ( append_list_nat @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_636_append__assoc,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( append_nat @ ( append_nat @ Xs @ Ys ) @ Zs )
= ( append_nat @ Xs @ ( append_nat @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_637_append__assoc,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,Zs: list_list_nat] :
( ( append_list_nat @ ( append_list_nat @ Xs @ Ys ) @ Zs )
= ( append_list_nat @ Xs @ ( append_list_nat @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_638_append_Oassoc,axiom,
! [A2: list_nat,B2: list_nat,C2: list_nat] :
( ( append_nat @ ( append_nat @ A2 @ B2 ) @ C2 )
= ( append_nat @ A2 @ ( append_nat @ B2 @ C2 ) ) ) ).
% append.assoc
thf(fact_639_append_Oassoc,axiom,
! [A2: list_list_nat,B2: list_list_nat,C2: list_list_nat] :
( ( append_list_nat @ ( append_list_nat @ A2 @ B2 ) @ C2 )
= ( append_list_nat @ A2 @ ( append_list_nat @ B2 @ C2 ) ) ) ).
% append.assoc
thf(fact_640_append__eq__append__conv2,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ts: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Zs @ Ts ) )
= ( ? [Us: list_nat] :
( ( ( Xs
= ( append_nat @ Zs @ Us ) )
& ( ( append_nat @ Us @ Ys )
= Ts ) )
| ( ( ( append_nat @ Xs @ Us )
= Zs )
& ( Ys
= ( append_nat @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_641_append__eq__append__conv2,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,Zs: list_list_nat,Ts: list_list_nat] :
( ( ( append_list_nat @ Xs @ Ys )
= ( append_list_nat @ Zs @ Ts ) )
= ( ? [Us: list_list_nat] :
( ( ( Xs
= ( append_list_nat @ Zs @ Us ) )
& ( ( append_list_nat @ Us @ Ys )
= Ts ) )
| ( ( ( append_list_nat @ Xs @ Us )
= Zs )
& ( Ys
= ( append_list_nat @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_642_append__eq__appendI,axiom,
! [Xs: list_nat,Xs1: list_nat,Zs: list_nat,Ys: list_nat,Us2: list_nat] :
( ( ( append_nat @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_nat @ Xs1 @ Us2 ) )
=> ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_643_append__eq__appendI,axiom,
! [Xs: list_list_nat,Xs1: list_list_nat,Zs: list_list_nat,Ys: list_list_nat,Us2: list_list_nat] :
( ( ( append_list_nat @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_list_nat @ Xs1 @ Us2 ) )
=> ( ( append_list_nat @ Xs @ Ys )
= ( append_list_nat @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_644_eq__Nil__appendI,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b] :
( ( Xs = Ys )
=> ( Xs
= ( append_term_a_b @ nil_term_a_b @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_645_eq__Nil__appendI,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append_list_nat @ nil_list_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_646_eq__Nil__appendI,axiom,
! [Xs: list_set_nat,Ys: list_set_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append_set_nat @ nil_set_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_647_eq__Nil__appendI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append_nat @ nil_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_648_append_Oleft__neutral,axiom,
! [A2: list_term_a_b] :
( ( append_term_a_b @ nil_term_a_b @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_649_append_Oleft__neutral,axiom,
! [A2: list_list_nat] :
( ( append_list_nat @ nil_list_nat @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_650_append_Oleft__neutral,axiom,
! [A2: list_set_nat] :
( ( append_set_nat @ nil_set_nat @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_651_append_Oleft__neutral,axiom,
! [A2: list_nat] :
( ( append_nat @ nil_nat @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_652_append__Nil,axiom,
! [Ys: list_term_a_b] :
( ( append_term_a_b @ nil_term_a_b @ Ys )
= Ys ) ).
% append_Nil
thf(fact_653_append__Nil,axiom,
! [Ys: list_list_nat] :
( ( append_list_nat @ nil_list_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_654_append__Nil,axiom,
! [Ys: list_set_nat] :
( ( append_set_nat @ nil_set_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_655_append__Nil,axiom,
! [Ys: list_nat] :
( ( append_nat @ nil_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_656_term__to__sig__ctxt__apply_H,axiom,
! [F2: set_Pr4934435412358123699_a_nat,C: subterm_and_ctxt_a_b,V: b,S: term_a_b] :
( ~ ( terms_8374513854926927137th_a_b @ F2 @ C )
=> ( ( terms_8519481630511763164ig_a_b @ F2 @ V @ ( subter2376574525758040790rm_a_b @ C @ S ) )
= ( terms_130083692264552600xt_a_b @ F2 @ V @ C ) ) ) ).
% term_to_sig_ctxt_apply'
thf(fact_657_term_Oinject_I2_J,axiom,
! [X21: a,X22: list_term_a_b,Y21: a,Y22: list_term_a_b] :
( ( ( fun_a_b @ X21 @ X22 )
= ( fun_a_b @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% term.inject(2)
thf(fact_658_list__ex1__simps_I1_J,axiom,
! [P3: term_a_b > $o] :
~ ( list_ex1_term_a_b @ P3 @ nil_term_a_b ) ).
% list_ex1_simps(1)
thf(fact_659_list__ex1__simps_I1_J,axiom,
! [P3: list_nat > $o] :
~ ( list_ex1_list_nat @ P3 @ nil_list_nat ) ).
% list_ex1_simps(1)
thf(fact_660_list__ex1__simps_I1_J,axiom,
! [P3: set_nat > $o] :
~ ( list_ex1_set_nat @ P3 @ nil_set_nat ) ).
% list_ex1_simps(1)
thf(fact_661_list__ex1__simps_I1_J,axiom,
! [P3: nat > $o] :
~ ( list_ex1_nat @ P3 @ nil_nat ) ).
% list_ex1_simps(1)
thf(fact_662_bind__simps_I1_J,axiom,
! [F: nat > list_nat] :
( ( bind_nat_nat @ nil_nat @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_663_bind__simps_I1_J,axiom,
! [F: nat > list_list_nat] :
( ( bind_nat_list_nat @ nil_nat @ F )
= nil_list_nat ) ).
% bind_simps(1)
thf(fact_664_bind__simps_I1_J,axiom,
! [F: nat > list_set_nat] :
( ( bind_nat_set_nat @ nil_nat @ F )
= nil_set_nat ) ).
% bind_simps(1)
thf(fact_665_bind__simps_I1_J,axiom,
! [F: list_nat > list_nat] :
( ( bind_list_nat_nat @ nil_list_nat @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_666_bind__simps_I1_J,axiom,
! [F: set_nat > list_nat] :
( ( bind_set_nat_nat @ nil_set_nat @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_667_bind__simps_I1_J,axiom,
! [F: nat > list_term_a_b] :
( ( bind_nat_term_a_b @ nil_nat @ F )
= nil_term_a_b ) ).
% bind_simps(1)
thf(fact_668_bind__simps_I1_J,axiom,
! [F: term_a_b > list_nat] :
( ( bind_term_a_b_nat @ nil_term_a_b @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_669_bind__simps_I1_J,axiom,
! [F: list_nat > list_list_nat] :
( ( bind_l7796378977173581257st_nat @ nil_list_nat @ F )
= nil_list_nat ) ).
% bind_simps(1)
thf(fact_670_bind__simps_I1_J,axiom,
! [F: list_nat > list_set_nat] :
( ( bind_l3154278341557560047et_nat @ nil_list_nat @ F )
= nil_set_nat ) ).
% bind_simps(1)
thf(fact_671_bind__simps_I1_J,axiom,
! [F: set_nat > list_list_nat] :
( ( bind_s6804505294920241007st_nat @ nil_set_nat @ F )
= nil_list_nat ) ).
% bind_simps(1)
thf(fact_672_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( linord2614967742042102400et_nat @ A )
= nil_nat ) ) ).
% sorted_list_of_set.fold_insort_key.infinite
thf(fact_673_subt__at__Cons__comp,axiom,
! [I: nat,P: list_nat,S: term_a_b] :
( ( member_list_nat2 @ ( cons_nat @ I @ P ) @ ( term_poss_a_b @ S ) )
=> ( ( term_subt_at_a_b @ ( term_subt_at_a_b @ S @ ( cons_nat @ I @ nil_nat ) ) @ P )
= ( term_subt_at_a_b @ S @ ( cons_nat @ I @ P ) ) ) ) ).
% subt_at_Cons_comp
thf(fact_674_member__rec_I2_J,axiom,
! [Y: term_a_b] :
~ ( member_term_a_b @ nil_term_a_b @ Y ) ).
% member_rec(2)
thf(fact_675_member__rec_I2_J,axiom,
! [Y: list_nat] :
~ ( member_list_nat @ nil_list_nat @ Y ) ).
% member_rec(2)
thf(fact_676_member__rec_I2_J,axiom,
! [Y: set_nat] :
~ ( member_set_nat @ nil_set_nat @ Y ) ).
% member_rec(2)
thf(fact_677_member__rec_I2_J,axiom,
! [Y: nat] :
~ ( member_nat @ nil_nat @ Y ) ).
% member_rec(2)
thf(fact_678_le__rel__bool__arg__iff,axiom,
( ord_le7949481618631132124_a_nat
= ( ^ [X4: $o > set_Pr4934435412358123699_a_nat,Y6: $o > set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le8666007276011122963_a_nat @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_679_le__rel__bool__arg__iff,axiom,
( ord_le3606317655850047935st_nat
= ( ^ [X4: $o > set_list_nat,Y6: $o > set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_le6045566169113846134st_nat @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_680_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_o_nat
= ( ^ [X4: $o > $o > nat,Y6: $o > $o > nat] :
( ( ord_less_eq_o_nat @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_o_nat @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_681_le__rel__bool__arg__iff,axiom,
( ord_le7022414076629706543et_nat
= ( ^ [X4: $o > set_nat,Y6: $o > set_nat] :
( ( ord_less_eq_set_nat @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_set_nat @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_682_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_nat
= ( ^ [X4: $o > nat,Y6: $o > nat] :
( ( ord_less_eq_nat @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq_nat @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_683_position__less__eq__Cons,axiom,
! [I: list_nat,Ps: list_list_nat,J: list_nat,Qs: list_list_nat] :
( ( term_p5934426891874639750st_nat @ ( cons_list_nat @ I @ Ps ) @ ( cons_list_nat @ J @ Qs ) )
= ( ( I = J )
& ( term_p5934426891874639750st_nat @ Ps @ Qs ) ) ) ).
% position_less_eq_Cons
thf(fact_684_position__less__eq__Cons,axiom,
! [I: nat,Ps: list_nat,J: nat,Qs: list_nat] :
( ( term_p3503116865373065078eq_nat @ ( cons_nat @ I @ Ps ) @ ( cons_nat @ J @ Qs ) )
= ( ( I = J )
& ( term_p3503116865373065078eq_nat @ Ps @ Qs ) ) ) ).
% position_less_eq_Cons
thf(fact_685_position__diff__Cons,axiom,
! [I: list_nat,Ps: list_list_nat,Qs: list_list_nat] :
( ( term_p7564741194569991203st_nat @ ( cons_list_nat @ I @ Ps ) @ ( cons_list_nat @ I @ Qs ) )
= ( term_p7564741194569991203st_nat @ Ps @ Qs ) ) ).
% position_diff_Cons
thf(fact_686_position__diff__Cons,axiom,
! [I: nat,Ps: list_nat,Qs: list_nat] :
( ( term_pos_diff_nat @ ( cons_nat @ I @ Ps ) @ ( cons_nat @ I @ Qs ) )
= ( term_pos_diff_nat @ Ps @ Qs ) ) ).
% position_diff_Cons
thf(fact_687_append1__eq__conv,axiom,
! [Xs: list_term_a_b,X2: term_a_b,Ys: list_term_a_b,Y: term_a_b] :
( ( ( append_term_a_b @ Xs @ ( cons_term_a_b @ X2 @ nil_term_a_b ) )
= ( append_term_a_b @ Ys @ ( cons_term_a_b @ Y @ nil_term_a_b ) ) )
= ( ( Xs = Ys )
& ( X2 = Y ) ) ) ).
% append1_eq_conv
thf(fact_688_append1__eq__conv,axiom,
! [Xs: list_set_nat,X2: set_nat,Ys: list_set_nat,Y: set_nat] :
( ( ( append_set_nat @ Xs @ ( cons_set_nat @ X2 @ nil_set_nat ) )
= ( append_set_nat @ Ys @ ( cons_set_nat @ Y @ nil_set_nat ) ) )
= ( ( Xs = Ys )
& ( X2 = Y ) ) ) ).
% append1_eq_conv
thf(fact_689_append1__eq__conv,axiom,
! [Xs: list_list_nat,X2: list_nat,Ys: list_list_nat,Y: list_nat] :
( ( ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) )
= ( append_list_nat @ Ys @ ( cons_list_nat @ Y @ nil_list_nat ) ) )
= ( ( Xs = Ys )
& ( X2 = Y ) ) ) ).
% append1_eq_conv
thf(fact_690_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_691_bind__simps_I2_J,axiom,
! [X2: nat,Xs: list_nat,F: nat > list_nat] :
( ( bind_nat_nat @ ( cons_nat @ X2 @ Xs ) @ F )
= ( append_nat @ ( F @ X2 ) @ ( bind_nat_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_692_bind__simps_I2_J,axiom,
! [X2: nat,Xs: list_nat,F: nat > list_list_nat] :
( ( bind_nat_list_nat @ ( cons_nat @ X2 @ Xs ) @ F )
= ( append_list_nat @ ( F @ X2 ) @ ( bind_nat_list_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_693_bind__simps_I2_J,axiom,
! [X2: list_nat,Xs: list_list_nat,F: list_nat > list_nat] :
( ( bind_list_nat_nat @ ( cons_list_nat @ X2 @ Xs ) @ F )
= ( append_nat @ ( F @ X2 ) @ ( bind_list_nat_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_694_bind__simps_I2_J,axiom,
! [X2: list_nat,Xs: list_list_nat,F: list_nat > list_list_nat] :
( ( bind_l7796378977173581257st_nat @ ( cons_list_nat @ X2 @ Xs ) @ F )
= ( append_list_nat @ ( F @ X2 ) @ ( bind_l7796378977173581257st_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_695_transpose_Ocases,axiom,
! [X2: list_list_term_a_b] :
( ( X2 != nil_list_term_a_b )
=> ( ! [Xss: list_list_term_a_b] :
( X2
!= ( cons_list_term_a_b @ nil_term_a_b @ Xss ) )
=> ~ ! [X: term_a_b,Xs2: list_term_a_b,Xss: list_list_term_a_b] :
( X2
!= ( cons_list_term_a_b @ ( cons_term_a_b @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_696_transpose_Ocases,axiom,
! [X2: list_list_set_nat] :
( ( X2 != nil_list_set_nat )
=> ( ! [Xss: list_list_set_nat] :
( X2
!= ( cons_list_set_nat @ nil_set_nat @ Xss ) )
=> ~ ! [X: set_nat,Xs2: list_set_nat,Xss: list_list_set_nat] :
( X2
!= ( cons_list_set_nat @ ( cons_set_nat @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_697_transpose_Ocases,axiom,
! [X2: list_list_list_nat] :
( ( X2 != nil_list_list_nat )
=> ( ! [Xss: list_list_list_nat] :
( X2
!= ( cons_list_list_nat @ nil_list_nat @ Xss ) )
=> ~ ! [X: list_nat,Xs2: list_list_nat,Xss: list_list_list_nat] :
( X2
!= ( cons_list_list_nat @ ( cons_list_nat @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_698_transpose_Ocases,axiom,
! [X2: list_list_nat] :
( ( X2 != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X2
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X: nat,Xs2: list_nat,Xss: list_list_nat] :
( X2
!= ( cons_list_nat @ ( cons_nat @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_699_list__nonempty__induct,axiom,
! [Xs: list_term_a_b,P3: list_term_a_b > $o] :
( ( Xs != nil_term_a_b )
=> ( ! [X: term_a_b] : ( P3 @ ( cons_term_a_b @ X @ nil_term_a_b ) )
=> ( ! [X: term_a_b,Xs2: list_term_a_b] :
( ( Xs2 != nil_term_a_b )
=> ( ( P3 @ Xs2 )
=> ( P3 @ ( cons_term_a_b @ X @ Xs2 ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_700_list__nonempty__induct,axiom,
! [Xs: list_set_nat,P3: list_set_nat > $o] :
( ( Xs != nil_set_nat )
=> ( ! [X: set_nat] : ( P3 @ ( cons_set_nat @ X @ nil_set_nat ) )
=> ( ! [X: set_nat,Xs2: list_set_nat] :
( ( Xs2 != nil_set_nat )
=> ( ( P3 @ Xs2 )
=> ( P3 @ ( cons_set_nat @ X @ Xs2 ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_701_list__nonempty__induct,axiom,
! [Xs: list_list_nat,P3: list_list_nat > $o] :
( ( Xs != nil_list_nat )
=> ( ! [X: list_nat] : ( P3 @ ( cons_list_nat @ X @ nil_list_nat ) )
=> ( ! [X: list_nat,Xs2: list_list_nat] :
( ( Xs2 != nil_list_nat )
=> ( ( P3 @ Xs2 )
=> ( P3 @ ( cons_list_nat @ X @ Xs2 ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_702_list__nonempty__induct,axiom,
! [Xs: list_nat,P3: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X: nat] : ( P3 @ ( cons_nat @ X @ nil_nat ) )
=> ( ! [X: nat,Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( P3 @ Xs2 )
=> ( P3 @ ( cons_nat @ X @ Xs2 ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_703_list__induct2_H,axiom,
! [P3: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
( ( P3 @ nil_nat @ nil_nat )
=> ( ! [X: nat,Xs2: list_nat] : ( P3 @ ( cons_nat @ X @ Xs2 ) @ nil_nat )
=> ( ! [Y2: nat,Ys2: list_nat] : ( P3 @ nil_nat @ ( cons_nat @ Y2 @ Ys2 ) )
=> ( ! [X: nat,Xs2: list_nat,Y2: nat,Ys2: list_nat] :
( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_704_list__induct2_H,axiom,
! [P3: list_set_nat > list_nat > $o,Xs: list_set_nat,Ys: list_nat] :
( ( P3 @ nil_set_nat @ nil_nat )
=> ( ! [X: set_nat,Xs2: list_set_nat] : ( P3 @ ( cons_set_nat @ X @ Xs2 ) @ nil_nat )
=> ( ! [Y2: nat,Ys2: list_nat] : ( P3 @ nil_set_nat @ ( cons_nat @ Y2 @ Ys2 ) )
=> ( ! [X: set_nat,Xs2: list_set_nat,Y2: nat,Ys2: list_nat] :
( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_set_nat @ X @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_705_list__induct2_H,axiom,
! [P3: list_nat > list_set_nat > $o,Xs: list_nat,Ys: list_set_nat] :
( ( P3 @ nil_nat @ nil_set_nat )
=> ( ! [X: nat,Xs2: list_nat] : ( P3 @ ( cons_nat @ X @ Xs2 ) @ nil_set_nat )
=> ( ! [Y2: set_nat,Ys2: list_set_nat] : ( P3 @ nil_nat @ ( cons_set_nat @ Y2 @ Ys2 ) )
=> ( ! [X: nat,Xs2: list_nat,Y2: set_nat,Ys2: list_set_nat] :
( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_nat @ X @ Xs2 ) @ ( cons_set_nat @ Y2 @ Ys2 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_706_list__induct2_H,axiom,
! [P3: list_nat > list_list_nat > $o,Xs: list_nat,Ys: list_list_nat] :
( ( P3 @ nil_nat @ nil_list_nat )
=> ( ! [X: nat,Xs2: list_nat] : ( P3 @ ( cons_nat @ X @ Xs2 ) @ nil_list_nat )
=> ( ! [Y2: list_nat,Ys2: list_list_nat] : ( P3 @ nil_nat @ ( cons_list_nat @ Y2 @ Ys2 ) )
=> ( ! [X: nat,Xs2: list_nat,Y2: list_nat,Ys2: list_list_nat] :
( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_nat @ X @ Xs2 ) @ ( cons_list_nat @ Y2 @ Ys2 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_707_list__induct2_H,axiom,
! [P3: list_list_nat > list_nat > $o,Xs: list_list_nat,Ys: list_nat] :
( ( P3 @ nil_list_nat @ nil_nat )
=> ( ! [X: list_nat,Xs2: list_list_nat] : ( P3 @ ( cons_list_nat @ X @ Xs2 ) @ nil_nat )
=> ( ! [Y2: nat,Ys2: list_nat] : ( P3 @ nil_list_nat @ ( cons_nat @ Y2 @ Ys2 ) )
=> ( ! [X: list_nat,Xs2: list_list_nat,Y2: nat,Ys2: list_nat] :
( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_list_nat @ X @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_708_list__induct2_H,axiom,
! [P3: list_set_nat > list_set_nat > $o,Xs: list_set_nat,Ys: list_set_nat] :
( ( P3 @ nil_set_nat @ nil_set_nat )
=> ( ! [X: set_nat,Xs2: list_set_nat] : ( P3 @ ( cons_set_nat @ X @ Xs2 ) @ nil_set_nat )
=> ( ! [Y2: set_nat,Ys2: list_set_nat] : ( P3 @ nil_set_nat @ ( cons_set_nat @ Y2 @ Ys2 ) )
=> ( ! [X: set_nat,Xs2: list_set_nat,Y2: set_nat,Ys2: list_set_nat] :
( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_set_nat @ X @ Xs2 ) @ ( cons_set_nat @ Y2 @ Ys2 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_709_list__induct2_H,axiom,
! [P3: list_term_a_b > list_nat > $o,Xs: list_term_a_b,Ys: list_nat] :
( ( P3 @ nil_term_a_b @ nil_nat )
=> ( ! [X: term_a_b,Xs2: list_term_a_b] : ( P3 @ ( cons_term_a_b @ X @ Xs2 ) @ nil_nat )
=> ( ! [Y2: nat,Ys2: list_nat] : ( P3 @ nil_term_a_b @ ( cons_nat @ Y2 @ Ys2 ) )
=> ( ! [X: term_a_b,Xs2: list_term_a_b,Y2: nat,Ys2: list_nat] :
( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_term_a_b @ X @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_710_list__induct2_H,axiom,
! [P3: list_set_nat > list_list_nat > $o,Xs: list_set_nat,Ys: list_list_nat] :
( ( P3 @ nil_set_nat @ nil_list_nat )
=> ( ! [X: set_nat,Xs2: list_set_nat] : ( P3 @ ( cons_set_nat @ X @ Xs2 ) @ nil_list_nat )
=> ( ! [Y2: list_nat,Ys2: list_list_nat] : ( P3 @ nil_set_nat @ ( cons_list_nat @ Y2 @ Ys2 ) )
=> ( ! [X: set_nat,Xs2: list_set_nat,Y2: list_nat,Ys2: list_list_nat] :
( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_set_nat @ X @ Xs2 ) @ ( cons_list_nat @ Y2 @ Ys2 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_711_list__induct2_H,axiom,
! [P3: list_nat > list_term_a_b > $o,Xs: list_nat,Ys: list_term_a_b] :
( ( P3 @ nil_nat @ nil_term_a_b )
=> ( ! [X: nat,Xs2: list_nat] : ( P3 @ ( cons_nat @ X @ Xs2 ) @ nil_term_a_b )
=> ( ! [Y2: term_a_b,Ys2: list_term_a_b] : ( P3 @ nil_nat @ ( cons_term_a_b @ Y2 @ Ys2 ) )
=> ( ! [X: nat,Xs2: list_nat,Y2: term_a_b,Ys2: list_term_a_b] :
( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_nat @ X @ Xs2 ) @ ( cons_term_a_b @ Y2 @ Ys2 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_712_list__induct2_H,axiom,
! [P3: list_list_nat > list_set_nat > $o,Xs: list_list_nat,Ys: list_set_nat] :
( ( P3 @ nil_list_nat @ nil_set_nat )
=> ( ! [X: list_nat,Xs2: list_list_nat] : ( P3 @ ( cons_list_nat @ X @ Xs2 ) @ nil_set_nat )
=> ( ! [Y2: set_nat,Ys2: list_set_nat] : ( P3 @ nil_list_nat @ ( cons_set_nat @ Y2 @ Ys2 ) )
=> ( ! [X: list_nat,Xs2: list_list_nat,Y2: set_nat,Ys2: list_set_nat] :
( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_list_nat @ X @ Xs2 ) @ ( cons_set_nat @ Y2 @ Ys2 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_713_neq__Nil__conv,axiom,
! [Xs: list_term_a_b] :
( ( Xs != nil_term_a_b )
= ( ? [Y4: term_a_b,Ys3: list_term_a_b] :
( Xs
= ( cons_term_a_b @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_714_neq__Nil__conv,axiom,
! [Xs: list_set_nat] :
( ( Xs != nil_set_nat )
= ( ? [Y4: set_nat,Ys3: list_set_nat] :
( Xs
= ( cons_set_nat @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_715_neq__Nil__conv,axiom,
! [Xs: list_list_nat] :
( ( Xs != nil_list_nat )
= ( ? [Y4: list_nat,Ys3: list_list_nat] :
( Xs
= ( cons_list_nat @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_716_neq__Nil__conv,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
= ( ? [Y4: nat,Ys3: list_nat] :
( Xs
= ( cons_nat @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_717_remdups__adj_Ocases,axiom,
! [X2: list_term_a_b] :
( ( X2 != nil_term_a_b )
=> ( ! [X: term_a_b] :
( X2
!= ( cons_term_a_b @ X @ nil_term_a_b ) )
=> ~ ! [X: term_a_b,Y2: term_a_b,Xs2: list_term_a_b] :
( X2
!= ( cons_term_a_b @ X @ ( cons_term_a_b @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_718_remdups__adj_Ocases,axiom,
! [X2: list_set_nat] :
( ( X2 != nil_set_nat )
=> ( ! [X: set_nat] :
( X2
!= ( cons_set_nat @ X @ nil_set_nat ) )
=> ~ ! [X: set_nat,Y2: set_nat,Xs2: list_set_nat] :
( X2
!= ( cons_set_nat @ X @ ( cons_set_nat @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_719_remdups__adj_Ocases,axiom,
! [X2: list_list_nat] :
( ( X2 != nil_list_nat )
=> ( ! [X: list_nat] :
( X2
!= ( cons_list_nat @ X @ nil_list_nat ) )
=> ~ ! [X: list_nat,Y2: list_nat,Xs2: list_list_nat] :
( X2
!= ( cons_list_nat @ X @ ( cons_list_nat @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_720_remdups__adj_Ocases,axiom,
! [X2: list_nat] :
( ( X2 != nil_nat )
=> ( ! [X: nat] :
( X2
!= ( cons_nat @ X @ nil_nat ) )
=> ~ ! [X: nat,Y2: nat,Xs2: list_nat] :
( X2
!= ( cons_nat @ X @ ( cons_nat @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_721_List_Omin__list_Ocases,axiom,
! [X2: list_set_nat] :
( ! [X: set_nat,Xs2: list_set_nat] :
( X2
!= ( cons_set_nat @ X @ Xs2 ) )
=> ( X2 = nil_set_nat ) ) ).
% List.min_list.cases
thf(fact_722_List_Omin__list_Ocases,axiom,
! [X2: list_nat] :
( ! [X: nat,Xs2: list_nat] :
( X2
!= ( cons_nat @ X @ Xs2 ) )
=> ( X2 = nil_nat ) ) ).
% List.min_list.cases
thf(fact_723_list_Oexhaust,axiom,
! [Y: list_term_a_b] :
( ( Y != nil_term_a_b )
=> ~ ! [X212: term_a_b,X222: list_term_a_b] :
( Y
!= ( cons_term_a_b @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_724_list_Oexhaust,axiom,
! [Y: list_set_nat] :
( ( Y != nil_set_nat )
=> ~ ! [X212: set_nat,X222: list_set_nat] :
( Y
!= ( cons_set_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_725_list_Oexhaust,axiom,
! [Y: list_list_nat] :
( ( Y != nil_list_nat )
=> ~ ! [X212: list_nat,X222: list_list_nat] :
( Y
!= ( cons_list_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_726_list_Oexhaust,axiom,
! [Y: list_nat] :
( ( Y != nil_nat )
=> ~ ! [X212: nat,X222: list_nat] :
( Y
!= ( cons_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_727_list_OdiscI,axiom,
! [List: list_term_a_b,X21: term_a_b,X22: list_term_a_b] :
( ( List
= ( cons_term_a_b @ X21 @ X22 ) )
=> ( List != nil_term_a_b ) ) ).
% list.discI
thf(fact_728_list_OdiscI,axiom,
! [List: list_set_nat,X21: set_nat,X22: list_set_nat] :
( ( List
= ( cons_set_nat @ X21 @ X22 ) )
=> ( List != nil_set_nat ) ) ).
% list.discI
thf(fact_729_list_OdiscI,axiom,
! [List: list_list_nat,X21: list_nat,X22: list_list_nat] :
( ( List
= ( cons_list_nat @ X21 @ X22 ) )
=> ( List != nil_list_nat ) ) ).
% list.discI
thf(fact_730_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X22: list_nat] :
( ( List
= ( cons_nat @ X21 @ X22 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_731_list_Odistinct_I1_J,axiom,
! [X21: term_a_b,X22: list_term_a_b] :
( nil_term_a_b
!= ( cons_term_a_b @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_732_list_Odistinct_I1_J,axiom,
! [X21: set_nat,X22: list_set_nat] :
( nil_set_nat
!= ( cons_set_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_733_list_Odistinct_I1_J,axiom,
! [X21: list_nat,X22: list_list_nat] :
( nil_list_nat
!= ( cons_list_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_734_list_Odistinct_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_735_Cons__eq__appendI,axiom,
! [X2: list_nat,Xs1: list_list_nat,Ys: list_list_nat,Xs: list_list_nat,Zs: list_list_nat] :
( ( ( cons_list_nat @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_list_nat @ Xs1 @ Zs ) )
=> ( ( cons_list_nat @ X2 @ Xs )
= ( append_list_nat @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_736_Cons__eq__appendI,axiom,
! [X2: nat,Xs1: list_nat,Ys: list_nat,Xs: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_nat @ Xs1 @ Zs ) )
=> ( ( cons_nat @ X2 @ Xs )
= ( append_nat @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_737_append__Cons,axiom,
! [X2: list_nat,Xs: list_list_nat,Ys: list_list_nat] :
( ( append_list_nat @ ( cons_list_nat @ X2 @ Xs ) @ Ys )
= ( cons_list_nat @ X2 @ ( append_list_nat @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_738_append__Cons,axiom,
! [X2: nat,Xs: list_nat,Ys: list_nat] :
( ( append_nat @ ( cons_nat @ X2 @ Xs ) @ Ys )
= ( cons_nat @ X2 @ ( append_nat @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_739_max__list_Ocases,axiom,
! [X2: list_nat] :
( ( X2 != nil_nat )
=> ~ ! [X: nat,Xs2: list_nat] :
( X2
!= ( cons_nat @ X @ Xs2 ) ) ) ).
% max_list.cases
thf(fact_740_par__Cons__iff,axiom,
! [I: list_nat,Ps: list_list_nat,J: list_nat,Qs: list_list_nat] :
( ( term_p4950861579910180738st_nat @ ( cons_list_nat @ I @ Ps ) @ ( cons_list_nat @ J @ Qs ) )
= ( ( I != J )
| ( term_p4950861579910180738st_nat @ Ps @ Qs ) ) ) ).
% par_Cons_iff
thf(fact_741_par__Cons__iff,axiom,
! [I: nat,Ps: list_nat,J: nat,Qs: list_nat] :
( ( term_p5017330785391824242ar_nat @ ( cons_nat @ I @ Ps ) @ ( cons_nat @ J @ Qs ) )
= ( ( I != J )
| ( term_p5017330785391824242ar_nat @ Ps @ Qs ) ) ) ).
% par_Cons_iff
thf(fact_742_par__pos__prefix,axiom,
! [I: list_nat,P: list_list_nat,Q: list_list_nat] :
( ( term_p4950861579910180738st_nat @ ( cons_list_nat @ I @ P ) @ ( cons_list_nat @ I @ Q ) )
=> ( term_p4950861579910180738st_nat @ P @ Q ) ) ).
% par_pos_prefix
thf(fact_743_par__pos__prefix,axiom,
! [I: nat,P: list_nat,Q: list_nat] :
( ( term_p5017330785391824242ar_nat @ ( cons_nat @ I @ P ) @ ( cons_nat @ I @ Q ) )
=> ( term_p5017330785391824242ar_nat @ P @ Q ) ) ).
% par_pos_prefix
thf(fact_744_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
! [A: set_nat,B: set_nat] :
( ( ( linord2614967742042102400et_nat @ A )
= ( linord2614967742042102400et_nat @ B ) )
=> ( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( A = B ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_745_rev__induct,axiom,
! [P3: list_term_a_b > $o,Xs: list_term_a_b] :
( ( P3 @ nil_term_a_b )
=> ( ! [X: term_a_b,Xs2: list_term_a_b] :
( ( P3 @ Xs2 )
=> ( P3 @ ( append_term_a_b @ Xs2 @ ( cons_term_a_b @ X @ nil_term_a_b ) ) ) )
=> ( P3 @ Xs ) ) ) ).
% rev_induct
thf(fact_746_rev__induct,axiom,
! [P3: list_set_nat > $o,Xs: list_set_nat] :
( ( P3 @ nil_set_nat )
=> ( ! [X: set_nat,Xs2: list_set_nat] :
( ( P3 @ Xs2 )
=> ( P3 @ ( append_set_nat @ Xs2 @ ( cons_set_nat @ X @ nil_set_nat ) ) ) )
=> ( P3 @ Xs ) ) ) ).
% rev_induct
thf(fact_747_rev__induct,axiom,
! [P3: list_list_nat > $o,Xs: list_list_nat] :
( ( P3 @ nil_list_nat )
=> ( ! [X: list_nat,Xs2: list_list_nat] :
( ( P3 @ Xs2 )
=> ( P3 @ ( append_list_nat @ Xs2 @ ( cons_list_nat @ X @ nil_list_nat ) ) ) )
=> ( P3 @ Xs ) ) ) ).
% rev_induct
thf(fact_748_rev__induct,axiom,
! [P3: list_nat > $o,Xs: list_nat] :
( ( P3 @ nil_nat )
=> ( ! [X: nat,Xs2: list_nat] :
( ( P3 @ Xs2 )
=> ( P3 @ ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) ) ) )
=> ( P3 @ Xs ) ) ) ).
% rev_induct
thf(fact_749_rev__exhaust,axiom,
! [Xs: list_term_a_b] :
( ( Xs != nil_term_a_b )
=> ~ ! [Ys2: list_term_a_b,Y2: term_a_b] :
( Xs
!= ( append_term_a_b @ Ys2 @ ( cons_term_a_b @ Y2 @ nil_term_a_b ) ) ) ) ).
% rev_exhaust
thf(fact_750_rev__exhaust,axiom,
! [Xs: list_set_nat] :
( ( Xs != nil_set_nat )
=> ~ ! [Ys2: list_set_nat,Y2: set_nat] :
( Xs
!= ( append_set_nat @ Ys2 @ ( cons_set_nat @ Y2 @ nil_set_nat ) ) ) ) ).
% rev_exhaust
thf(fact_751_rev__exhaust,axiom,
! [Xs: list_list_nat] :
( ( Xs != nil_list_nat )
=> ~ ! [Ys2: list_list_nat,Y2: list_nat] :
( Xs
!= ( append_list_nat @ Ys2 @ ( cons_list_nat @ Y2 @ nil_list_nat ) ) ) ) ).
% rev_exhaust
thf(fact_752_rev__exhaust,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ~ ! [Ys2: list_nat,Y2: nat] :
( Xs
!= ( append_nat @ Ys2 @ ( cons_nat @ Y2 @ nil_nat ) ) ) ) ).
% rev_exhaust
thf(fact_753_Cons__eq__append__conv,axiom,
! [X2: term_a_b,Xs: list_term_a_b,Ys: list_term_a_b,Zs: list_term_a_b] :
( ( ( cons_term_a_b @ X2 @ Xs )
= ( append_term_a_b @ Ys @ Zs ) )
= ( ( ( Ys = nil_term_a_b )
& ( ( cons_term_a_b @ X2 @ Xs )
= Zs ) )
| ? [Ys4: list_term_a_b] :
( ( ( cons_term_a_b @ X2 @ Ys4 )
= Ys )
& ( Xs
= ( append_term_a_b @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_754_Cons__eq__append__conv,axiom,
! [X2: set_nat,Xs: list_set_nat,Ys: list_set_nat,Zs: list_set_nat] :
( ( ( cons_set_nat @ X2 @ Xs )
= ( append_set_nat @ Ys @ Zs ) )
= ( ( ( Ys = nil_set_nat )
& ( ( cons_set_nat @ X2 @ Xs )
= Zs ) )
| ? [Ys4: list_set_nat] :
( ( ( cons_set_nat @ X2 @ Ys4 )
= Ys )
& ( Xs
= ( append_set_nat @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_755_Cons__eq__append__conv,axiom,
! [X2: list_nat,Xs: list_list_nat,Ys: list_list_nat,Zs: list_list_nat] :
( ( ( cons_list_nat @ X2 @ Xs )
= ( append_list_nat @ Ys @ Zs ) )
= ( ( ( Ys = nil_list_nat )
& ( ( cons_list_nat @ X2 @ Xs )
= Zs ) )
| ? [Ys4: list_list_nat] :
( ( ( cons_list_nat @ X2 @ Ys4 )
= Ys )
& ( Xs
= ( append_list_nat @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_756_Cons__eq__append__conv,axiom,
! [X2: nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( append_nat @ Ys @ Zs ) )
= ( ( ( Ys = nil_nat )
& ( ( cons_nat @ X2 @ Xs )
= Zs ) )
| ? [Ys4: list_nat] :
( ( ( cons_nat @ X2 @ Ys4 )
= Ys )
& ( Xs
= ( append_nat @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_757_append__eq__Cons__conv,axiom,
! [Ys: list_term_a_b,Zs: list_term_a_b,X2: term_a_b,Xs: list_term_a_b] :
( ( ( append_term_a_b @ Ys @ Zs )
= ( cons_term_a_b @ X2 @ Xs ) )
= ( ( ( Ys = nil_term_a_b )
& ( Zs
= ( cons_term_a_b @ X2 @ Xs ) ) )
| ? [Ys4: list_term_a_b] :
( ( Ys
= ( cons_term_a_b @ X2 @ Ys4 ) )
& ( ( append_term_a_b @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_758_append__eq__Cons__conv,axiom,
! [Ys: list_set_nat,Zs: list_set_nat,X2: set_nat,Xs: list_set_nat] :
( ( ( append_set_nat @ Ys @ Zs )
= ( cons_set_nat @ X2 @ Xs ) )
= ( ( ( Ys = nil_set_nat )
& ( Zs
= ( cons_set_nat @ X2 @ Xs ) ) )
| ? [Ys4: list_set_nat] :
( ( Ys
= ( cons_set_nat @ X2 @ Ys4 ) )
& ( ( append_set_nat @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_759_append__eq__Cons__conv,axiom,
! [Ys: list_list_nat,Zs: list_list_nat,X2: list_nat,Xs: list_list_nat] :
( ( ( append_list_nat @ Ys @ Zs )
= ( cons_list_nat @ X2 @ Xs ) )
= ( ( ( Ys = nil_list_nat )
& ( Zs
= ( cons_list_nat @ X2 @ Xs ) ) )
| ? [Ys4: list_list_nat] :
( ( Ys
= ( cons_list_nat @ X2 @ Ys4 ) )
& ( ( append_list_nat @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_760_append__eq__Cons__conv,axiom,
! [Ys: list_nat,Zs: list_nat,X2: nat,Xs: list_nat] :
( ( ( append_nat @ Ys @ Zs )
= ( cons_nat @ X2 @ Xs ) )
= ( ( ( Ys = nil_nat )
& ( Zs
= ( cons_nat @ X2 @ Xs ) ) )
| ? [Ys4: list_nat] :
( ( Ys
= ( cons_nat @ X2 @ Ys4 ) )
& ( ( append_nat @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_761_rev__nonempty__induct,axiom,
! [Xs: list_term_a_b,P3: list_term_a_b > $o] :
( ( Xs != nil_term_a_b )
=> ( ! [X: term_a_b] : ( P3 @ ( cons_term_a_b @ X @ nil_term_a_b ) )
=> ( ! [X: term_a_b,Xs2: list_term_a_b] :
( ( Xs2 != nil_term_a_b )
=> ( ( P3 @ Xs2 )
=> ( P3 @ ( append_term_a_b @ Xs2 @ ( cons_term_a_b @ X @ nil_term_a_b ) ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_762_rev__nonempty__induct,axiom,
! [Xs: list_set_nat,P3: list_set_nat > $o] :
( ( Xs != nil_set_nat )
=> ( ! [X: set_nat] : ( P3 @ ( cons_set_nat @ X @ nil_set_nat ) )
=> ( ! [X: set_nat,Xs2: list_set_nat] :
( ( Xs2 != nil_set_nat )
=> ( ( P3 @ Xs2 )
=> ( P3 @ ( append_set_nat @ Xs2 @ ( cons_set_nat @ X @ nil_set_nat ) ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_763_rev__nonempty__induct,axiom,
! [Xs: list_list_nat,P3: list_list_nat > $o] :
( ( Xs != nil_list_nat )
=> ( ! [X: list_nat] : ( P3 @ ( cons_list_nat @ X @ nil_list_nat ) )
=> ( ! [X: list_nat,Xs2: list_list_nat] :
( ( Xs2 != nil_list_nat )
=> ( ( P3 @ Xs2 )
=> ( P3 @ ( append_list_nat @ Xs2 @ ( cons_list_nat @ X @ nil_list_nat ) ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_764_rev__nonempty__induct,axiom,
! [Xs: list_nat,P3: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X: nat] : ( P3 @ ( cons_nat @ X @ nil_nat ) )
=> ( ! [X: nat,Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( P3 @ Xs2 )
=> ( P3 @ ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_765_poss__Cons,axiom,
! [I: nat,P: list_nat,T: term_a_b] :
( ( member_list_nat2 @ ( cons_nat @ I @ P ) @ ( term_poss_a_b @ T ) )
=> ( member_list_nat2 @ ( cons_nat @ I @ nil_nat ) @ ( term_poss_a_b @ T ) ) ) ).
% poss_Cons
thf(fact_766_Missing__List_Omin__list_Ocases,axiom,
! [X2: list_nat] :
( ! [X: nat] :
( X2
!= ( cons_nat @ X @ nil_nat ) )
=> ( ! [X: nat,V2: nat,Va: list_nat] :
( X2
!= ( cons_nat @ X @ ( cons_nat @ V2 @ Va ) ) )
=> ( X2 = nil_nat ) ) ) ).
% Missing_List.min_list.cases
thf(fact_767_concat__lists_Ocases,axiom,
! [X2: list_list_nat] :
( ( X2 != nil_list_nat )
=> ~ ! [As: list_nat,Xs2: list_list_nat] :
( X2
!= ( cons_list_nat @ As @ Xs2 ) ) ) ).
% concat_lists.cases
thf(fact_768_list__4__cases,axiom,
! [Xs: list_term_a_b] :
( ( Xs != nil_term_a_b )
=> ( ! [X: term_a_b] :
( Xs
!= ( cons_term_a_b @ X @ nil_term_a_b ) )
=> ( ! [X: term_a_b,Y2: term_a_b] :
( Xs
!= ( cons_term_a_b @ X @ ( cons_term_a_b @ Y2 @ nil_term_a_b ) ) )
=> ~ ! [X: term_a_b,Y2: term_a_b,Z3: term_a_b,Zs2: list_term_a_b] :
( Xs
!= ( cons_term_a_b @ X @ ( cons_term_a_b @ Y2 @ ( cons_term_a_b @ Z3 @ Zs2 ) ) ) ) ) ) ) ).
% list_4_cases
thf(fact_769_list__4__cases,axiom,
! [Xs: list_set_nat] :
( ( Xs != nil_set_nat )
=> ( ! [X: set_nat] :
( Xs
!= ( cons_set_nat @ X @ nil_set_nat ) )
=> ( ! [X: set_nat,Y2: set_nat] :
( Xs
!= ( cons_set_nat @ X @ ( cons_set_nat @ Y2 @ nil_set_nat ) ) )
=> ~ ! [X: set_nat,Y2: set_nat,Z3: set_nat,Zs2: list_set_nat] :
( Xs
!= ( cons_set_nat @ X @ ( cons_set_nat @ Y2 @ ( cons_set_nat @ Z3 @ Zs2 ) ) ) ) ) ) ) ).
% list_4_cases
thf(fact_770_list__4__cases,axiom,
! [Xs: list_list_nat] :
( ( Xs != nil_list_nat )
=> ( ! [X: list_nat] :
( Xs
!= ( cons_list_nat @ X @ nil_list_nat ) )
=> ( ! [X: list_nat,Y2: list_nat] :
( Xs
!= ( cons_list_nat @ X @ ( cons_list_nat @ Y2 @ nil_list_nat ) ) )
=> ~ ! [X: list_nat,Y2: list_nat,Z3: list_nat,Zs2: list_list_nat] :
( Xs
!= ( cons_list_nat @ X @ ( cons_list_nat @ Y2 @ ( cons_list_nat @ Z3 @ Zs2 ) ) ) ) ) ) ) ).
% list_4_cases
thf(fact_771_list__4__cases,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ! [X: nat] :
( Xs
!= ( cons_nat @ X @ nil_nat ) )
=> ( ! [X: nat,Y2: nat] :
( Xs
!= ( cons_nat @ X @ ( cons_nat @ Y2 @ nil_nat ) ) )
=> ~ ! [X: nat,Y2: nat,Z3: nat,Zs2: list_nat] :
( Xs
!= ( cons_nat @ X @ ( cons_nat @ Y2 @ ( cons_nat @ Z3 @ Zs2 ) ) ) ) ) ) ) ).
% list_4_cases
thf(fact_772_list__3__cases,axiom,
! [Xs: list_term_a_b] :
( ( Xs != nil_term_a_b )
=> ( ! [X: term_a_b] :
( Xs
!= ( cons_term_a_b @ X @ nil_term_a_b ) )
=> ~ ! [X: term_a_b,Y2: term_a_b,Ys2: list_term_a_b] :
( Xs
!= ( cons_term_a_b @ X @ ( cons_term_a_b @ Y2 @ Ys2 ) ) ) ) ) ).
% list_3_cases
thf(fact_773_list__3__cases,axiom,
! [Xs: list_set_nat] :
( ( Xs != nil_set_nat )
=> ( ! [X: set_nat] :
( Xs
!= ( cons_set_nat @ X @ nil_set_nat ) )
=> ~ ! [X: set_nat,Y2: set_nat,Ys2: list_set_nat] :
( Xs
!= ( cons_set_nat @ X @ ( cons_set_nat @ Y2 @ Ys2 ) ) ) ) ) ).
% list_3_cases
thf(fact_774_list__3__cases,axiom,
! [Xs: list_list_nat] :
( ( Xs != nil_list_nat )
=> ( ! [X: list_nat] :
( Xs
!= ( cons_list_nat @ X @ nil_list_nat ) )
=> ~ ! [X: list_nat,Y2: list_nat,Ys2: list_list_nat] :
( Xs
!= ( cons_list_nat @ X @ ( cons_list_nat @ Y2 @ Ys2 ) ) ) ) ) ).
% list_3_cases
thf(fact_775_list__3__cases,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ! [X: nat] :
( Xs
!= ( cons_nat @ X @ nil_nat ) )
=> ~ ! [X: nat,Y2: nat,Ys2: list_nat] :
( Xs
!= ( cons_nat @ X @ ( cons_nat @ Y2 @ Ys2 ) ) ) ) ) ).
% list_3_cases
thf(fact_776_product__lists_Osimps_I1_J,axiom,
( ( produc17669015410068453rm_a_b @ nil_list_term_a_b )
= ( cons_list_term_a_b @ nil_term_a_b @ nil_list_term_a_b ) ) ).
% product_lists.simps(1)
thf(fact_777_product__lists_Osimps_I1_J,axiom,
( ( produc6783906451316923569st_nat @ nil_list_list_nat )
= ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) ).
% product_lists.simps(1)
thf(fact_778_product__lists_Osimps_I1_J,axiom,
( ( produc8109398739672286679et_nat @ nil_list_set_nat )
= ( cons_list_set_nat @ nil_set_nat @ nil_list_set_nat ) ) ).
% product_lists.simps(1)
thf(fact_779_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_780_concat__lists_Osimps_I1_J,axiom,
( ( missin8060632096978918206rm_a_b @ nil_list_term_a_b )
= ( cons_list_term_a_b @ nil_term_a_b @ nil_list_term_a_b ) ) ).
% concat_lists.simps(1)
thf(fact_781_concat__lists_Osimps_I1_J,axiom,
( ( missin5603497496030997514st_nat @ nil_list_list_nat )
= ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) ).
% concat_lists.simps(1)
thf(fact_782_concat__lists_Osimps_I1_J,axiom,
( ( missin2626218296463274416et_nat @ nil_list_set_nat )
= ( cons_list_set_nat @ nil_set_nat @ nil_list_set_nat ) ) ).
% concat_lists.simps(1)
thf(fact_783_concat__lists_Osimps_I1_J,axiom,
( ( missin4567272213201432058ts_nat @ nil_list_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% concat_lists.simps(1)
thf(fact_784_subseqs_Osimps_I1_J,axiom,
( ( subseqs_term_a_b @ nil_term_a_b )
= ( cons_list_term_a_b @ nil_term_a_b @ nil_list_term_a_b ) ) ).
% subseqs.simps(1)
thf(fact_785_subseqs_Osimps_I1_J,axiom,
( ( subseqs_list_nat @ nil_list_nat )
= ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) ).
% subseqs.simps(1)
thf(fact_786_subseqs_Osimps_I1_J,axiom,
( ( subseqs_set_nat @ nil_set_nat )
= ( cons_list_set_nat @ nil_set_nat @ nil_list_set_nat ) ) ).
% subseqs.simps(1)
thf(fact_787_subseqs_Osimps_I1_J,axiom,
( ( subseqs_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% subseqs.simps(1)
thf(fact_788_maps__simps_I1_J,axiom,
! [F: nat > list_nat,X2: nat,Xs: list_nat] :
( ( maps_nat_nat @ F @ ( cons_nat @ X2 @ Xs ) )
= ( append_nat @ ( F @ X2 ) @ ( maps_nat_nat @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_789_maps__simps_I1_J,axiom,
! [F: nat > list_list_nat,X2: nat,Xs: list_nat] :
( ( maps_nat_list_nat @ F @ ( cons_nat @ X2 @ Xs ) )
= ( append_list_nat @ ( F @ X2 ) @ ( maps_nat_list_nat @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_790_maps__simps_I1_J,axiom,
! [F: list_nat > list_nat,X2: list_nat,Xs: list_list_nat] :
( ( maps_list_nat_nat @ F @ ( cons_list_nat @ X2 @ Xs ) )
= ( append_nat @ ( F @ X2 ) @ ( maps_list_nat_nat @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_791_maps__simps_I1_J,axiom,
! [F: list_nat > list_list_nat,X2: list_nat,Xs: list_list_nat] :
( ( maps_l5785965478274863235st_nat @ F @ ( cons_list_nat @ X2 @ Xs ) )
= ( append_list_nat @ ( F @ X2 ) @ ( maps_l5785965478274863235st_nat @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_792_add__elem__list__lists_Osimps_I1_J,axiom,
! [X2: term_a_b] :
( ( basic_1593220722155286443rm_a_b @ X2 @ nil_term_a_b )
= ( cons_list_term_a_b @ ( cons_term_a_b @ X2 @ nil_term_a_b ) @ nil_list_term_a_b ) ) ).
% add_elem_list_lists.simps(1)
thf(fact_793_add__elem__list__lists_Osimps_I1_J,axiom,
! [X2: set_nat] :
( ( basic_8313613011175402653et_nat @ X2 @ nil_set_nat )
= ( cons_list_set_nat @ ( cons_set_nat @ X2 @ nil_set_nat ) @ nil_list_set_nat ) ) ).
% add_elem_list_lists.simps(1)
thf(fact_794_add__elem__list__lists_Osimps_I1_J,axiom,
! [X2: list_nat] :
( ( basic_8359458158062141559st_nat @ X2 @ nil_list_nat )
= ( cons_list_list_nat @ ( cons_list_nat @ X2 @ nil_list_nat ) @ nil_list_list_nat ) ) ).
% add_elem_list_lists.simps(1)
thf(fact_795_add__elem__list__lists_Osimps_I1_J,axiom,
! [X2: nat] :
( ( basic_4874698711677410535ts_nat @ X2 @ nil_nat )
= ( cons_list_nat @ ( cons_nat @ X2 @ nil_nat ) @ nil_list_nat ) ) ).
% add_elem_list_lists.simps(1)
thf(fact_796_insert__Nil,axiom,
! [X2: term_a_b] :
( ( insert_term_a_b @ X2 @ nil_term_a_b )
= ( cons_term_a_b @ X2 @ nil_term_a_b ) ) ).
% insert_Nil
thf(fact_797_insert__Nil,axiom,
! [X2: set_nat] :
( ( insert_set_nat @ X2 @ nil_set_nat )
= ( cons_set_nat @ X2 @ nil_set_nat ) ) ).
% insert_Nil
thf(fact_798_insert__Nil,axiom,
! [X2: list_nat] :
( ( insert_list_nat @ X2 @ nil_list_nat )
= ( cons_list_nat @ X2 @ nil_list_nat ) ) ).
% insert_Nil
thf(fact_799_insert__Nil,axiom,
! [X2: nat] :
( ( insert_nat @ X2 @ nil_nat )
= ( cons_nat @ X2 @ nil_nat ) ) ).
% insert_Nil
thf(fact_800_butlast__snoc,axiom,
! [Xs: list_term_a_b,X2: term_a_b] :
( ( butlast_term_a_b @ ( append_term_a_b @ Xs @ ( cons_term_a_b @ X2 @ nil_term_a_b ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_801_butlast__snoc,axiom,
! [Xs: list_set_nat,X2: set_nat] :
( ( butlast_set_nat @ ( append_set_nat @ Xs @ ( cons_set_nat @ X2 @ nil_set_nat ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_802_butlast__snoc,axiom,
! [Xs: list_list_nat,X2: list_nat] :
( ( butlast_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_803_butlast__snoc,axiom,
! [Xs: list_nat,X2: nat] :
( ( butlast_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_804_butlast_Osimps_I1_J,axiom,
( ( butlast_term_a_b @ nil_term_a_b )
= nil_term_a_b ) ).
% butlast.simps(1)
thf(fact_805_butlast_Osimps_I1_J,axiom,
( ( butlast_list_nat @ nil_list_nat )
= nil_list_nat ) ).
% butlast.simps(1)
thf(fact_806_butlast_Osimps_I1_J,axiom,
( ( butlast_set_nat @ nil_set_nat )
= nil_set_nat ) ).
% butlast.simps(1)
thf(fact_807_butlast_Osimps_I1_J,axiom,
( ( butlast_nat @ nil_nat )
= nil_nat ) ).
% butlast.simps(1)
thf(fact_808_butlast_Osimps_I2_J,axiom,
! [Xs: list_term_a_b,X2: term_a_b] :
( ( ( Xs = nil_term_a_b )
=> ( ( butlast_term_a_b @ ( cons_term_a_b @ X2 @ Xs ) )
= nil_term_a_b ) )
& ( ( Xs != nil_term_a_b )
=> ( ( butlast_term_a_b @ ( cons_term_a_b @ X2 @ Xs ) )
= ( cons_term_a_b @ X2 @ ( butlast_term_a_b @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_809_butlast_Osimps_I2_J,axiom,
! [Xs: list_set_nat,X2: set_nat] :
( ( ( Xs = nil_set_nat )
=> ( ( butlast_set_nat @ ( cons_set_nat @ X2 @ Xs ) )
= nil_set_nat ) )
& ( ( Xs != nil_set_nat )
=> ( ( butlast_set_nat @ ( cons_set_nat @ X2 @ Xs ) )
= ( cons_set_nat @ X2 @ ( butlast_set_nat @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_810_butlast_Osimps_I2_J,axiom,
! [Xs: list_list_nat,X2: list_nat] :
( ( ( Xs = nil_list_nat )
=> ( ( butlast_list_nat @ ( cons_list_nat @ X2 @ Xs ) )
= nil_list_nat ) )
& ( ( Xs != nil_list_nat )
=> ( ( butlast_list_nat @ ( cons_list_nat @ X2 @ Xs ) )
= ( cons_list_nat @ X2 @ ( butlast_list_nat @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_811_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_812_butlast__append,axiom,
! [Ys: list_term_a_b,Xs: list_term_a_b] :
( ( ( Ys = nil_term_a_b )
=> ( ( butlast_term_a_b @ ( append_term_a_b @ Xs @ Ys ) )
= ( butlast_term_a_b @ Xs ) ) )
& ( ( Ys != nil_term_a_b )
=> ( ( butlast_term_a_b @ ( append_term_a_b @ Xs @ Ys ) )
= ( append_term_a_b @ Xs @ ( butlast_term_a_b @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_813_butlast__append,axiom,
! [Ys: list_list_nat,Xs: list_list_nat] :
( ( ( Ys = nil_list_nat )
=> ( ( butlast_list_nat @ ( append_list_nat @ Xs @ Ys ) )
= ( butlast_list_nat @ Xs ) ) )
& ( ( Ys != nil_list_nat )
=> ( ( butlast_list_nat @ ( append_list_nat @ Xs @ Ys ) )
= ( append_list_nat @ Xs @ ( butlast_list_nat @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_814_butlast__append,axiom,
! [Ys: list_set_nat,Xs: list_set_nat] :
( ( ( Ys = nil_set_nat )
=> ( ( butlast_set_nat @ ( append_set_nat @ Xs @ Ys ) )
= ( butlast_set_nat @ Xs ) ) )
& ( ( Ys != nil_set_nat )
=> ( ( butlast_set_nat @ ( append_set_nat @ Xs @ Ys ) )
= ( append_set_nat @ Xs @ ( butlast_set_nat @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_815_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_816_maps__simps_I2_J,axiom,
! [F: term_a_b > list_set_nat] :
( ( maps_t8811637242275172829et_nat @ F @ nil_term_a_b )
= nil_set_nat ) ).
% maps_simps(2)
thf(fact_817_maps__simps_I2_J,axiom,
! [F: list_nat > list_nat] :
( ( maps_list_nat_nat @ F @ nil_list_nat )
= nil_nat ) ).
% maps_simps(2)
thf(fact_818_maps__simps_I2_J,axiom,
! [F: list_nat > list_term_a_b] :
( ( maps_l8243100079222783927rm_a_b @ F @ nil_list_nat )
= nil_term_a_b ) ).
% maps_simps(2)
thf(fact_819_maps__simps_I2_J,axiom,
! [F: list_nat > list_list_nat] :
( ( maps_l5785965478274863235st_nat @ F @ nil_list_nat )
= nil_list_nat ) ).
% maps_simps(2)
thf(fact_820_maps__simps_I2_J,axiom,
! [F: list_nat > list_set_nat] :
( ( maps_l2310356970806526633et_nat @ F @ nil_list_nat )
= nil_set_nat ) ).
% maps_simps(2)
thf(fact_821_maps__simps_I2_J,axiom,
! [F: set_nat > list_nat] :
( ( maps_set_nat_nat @ F @ nil_set_nat )
= nil_nat ) ).
% maps_simps(2)
thf(fact_822_maps__simps_I2_J,axiom,
! [F: set_nat > list_term_a_b] :
( ( maps_s8417718525117128285rm_a_b @ F @ nil_set_nat )
= nil_term_a_b ) ).
% maps_simps(2)
thf(fact_823_maps__simps_I2_J,axiom,
! [F: set_nat > list_list_nat] :
( ( maps_s5960583924169207593st_nat @ F @ nil_set_nat )
= nil_list_nat ) ).
% maps_simps(2)
thf(fact_824_maps__simps_I2_J,axiom,
! [F: set_nat > list_set_nat] :
( ( maps_set_nat_set_nat @ F @ nil_set_nat )
= nil_set_nat ) ).
% maps_simps(2)
thf(fact_825_maps__simps_I2_J,axiom,
! [F: nat > list_nat] :
( ( maps_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% maps_simps(2)
thf(fact_826_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_827_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_828_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ( linord2614967742042102400et_nat @ A )
= nil_nat )
= ( A = bot_bot_set_nat ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_829_last__snoc,axiom,
! [Xs: list_nat,X2: nat] :
( ( last_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) )
= X2 ) ).
% last_snoc
thf(fact_830_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_831_concat__eq__append__conv,axiom,
! [Xss2: list_list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( concat_nat @ Xss2 )
= ( append_nat @ Ys @ Zs ) )
= ( ( ( Xss2 = nil_list_nat )
=> ( ( Ys = nil_nat )
& ( Zs = nil_nat ) ) )
& ( ( Xss2 != nil_list_nat )
=> ? [Xss1: list_list_nat,Xs3: list_nat,Xs4: list_nat,Xss22: list_list_nat] :
( ( Xss2
= ( append_list_nat @ Xss1 @ ( cons_list_nat @ ( append_nat @ Xs3 @ Xs4 ) @ Xss22 ) ) )
& ( Ys
= ( append_nat @ ( concat_nat @ Xss1 ) @ Xs3 ) )
& ( Zs
= ( append_nat @ Xs4 @ ( concat_nat @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_832_empty__iff,axiom,
! [C2: list_nat] :
~ ( member_list_nat2 @ C2 @ bot_bot_set_list_nat ) ).
% empty_iff
thf(fact_833_all__not__in__conv,axiom,
! [A: set_list_nat] :
( ( ! [X3: list_nat] :
~ ( member_list_nat2 @ X3 @ A ) )
= ( A = bot_bot_set_list_nat ) ) ).
% all_not_in_conv
thf(fact_834_rotate1__is__Nil__conv,axiom,
! [Xs: list_nat] :
( ( ( rotate1_nat @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% rotate1_is_Nil_conv
thf(fact_835_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
( ( linord2614967742042102400et_nat @ bot_bot_set_nat )
= nil_nat ) ).
% sorted_list_of_set.sorted_key_list_of_set_empty
thf(fact_836_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_837_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_838_emptyE,axiom,
! [A2: list_nat] :
~ ( member_list_nat2 @ A2 @ bot_bot_set_list_nat ) ).
% emptyE
thf(fact_839_equals0D,axiom,
! [A: set_list_nat,A2: list_nat] :
( ( A = bot_bot_set_list_nat )
=> ~ ( member_list_nat2 @ A2 @ A ) ) ).
% equals0D
thf(fact_840_equals0I,axiom,
! [A: set_list_nat] :
( ! [Y2: list_nat] :
~ ( member_list_nat2 @ Y2 @ A )
=> ( A = bot_bot_set_list_nat ) ) ).
% equals0I
thf(fact_841_ex__in__conv,axiom,
! [A: set_list_nat] :
( ( ? [X3: list_nat] : ( member_list_nat2 @ X3 @ A ) )
= ( A != bot_bot_set_list_nat ) ) ).
% ex_in_conv
thf(fact_842_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_843_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_844_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_845_rotate1_Osimps_I1_J,axiom,
( ( rotate1_nat @ nil_nat )
= nil_nat ) ).
% rotate1.simps(1)
thf(fact_846_concat_Osimps_I1_J,axiom,
( ( concat_nat @ nil_list_nat )
= nil_nat ) ).
% concat.simps(1)
thf(fact_847_finite__has__minimal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X: nat] :
( ( member_nat2 @ X @ A )
& ! [Xa: nat] :
( ( member_nat2 @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_848_finite__has__maximal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X: nat] :
( ( member_nat2 @ X @ A )
& ! [Xa: nat] :
( ( member_nat2 @ Xa @ A )
=> ( ( ord_less_eq_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_849_last__ConsR,axiom,
! [Xs: list_nat,X2: nat] :
( ( Xs != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X2 @ Xs ) )
= ( last_nat @ Xs ) ) ) ).
% last_ConsR
thf(fact_850_last__ConsL,axiom,
! [Xs: list_nat,X2: nat] :
( ( Xs = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X2 @ Xs ) )
= X2 ) ) ).
% last_ConsL
thf(fact_851_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_852_longest__common__suffix,axiom,
! [Xs: list_nat,Ys: list_nat] :
? [Ss: list_nat,Xs5: list_nat,Ys5: list_nat] :
( ( Xs
= ( append_nat @ Xs5 @ Ss ) )
& ( Ys
= ( append_nat @ Ys5 @ Ss ) )
& ( ( Xs5 = nil_nat )
| ( Ys5 = nil_nat )
| ( ( last_nat @ Xs5 )
!= ( last_nat @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_853_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_854_subset__emptyI,axiom,
! [A: set_list_nat] :
( ! [X: list_nat] :
~ ( member_list_nat2 @ X @ A )
=> ( ord_le6045566169113846134st_nat @ A @ bot_bot_set_list_nat ) ) ).
% subset_emptyI
thf(fact_855_finite__transitivity__chain,axiom,
! [A: set_list_nat,R3: list_nat > list_nat > $o] :
( ( finite8100373058378681591st_nat @ A )
=> ( ! [X: list_nat] :
~ ( R3 @ X @ X )
=> ( ! [X: list_nat,Y2: list_nat,Z3: list_nat] :
( ( R3 @ X @ Y2 )
=> ( ( R3 @ Y2 @ Z3 )
=> ( R3 @ X @ Z3 ) ) )
=> ( ! [X: list_nat] :
( ( member_list_nat2 @ X @ A )
=> ? [Y5: list_nat] :
( ( member_list_nat2 @ Y5 @ A )
& ( R3 @ X @ Y5 ) ) )
=> ( A = bot_bot_set_list_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_856_map__tailrec__rev_Oelims,axiom,
! [X2: nat > nat,Xa2: list_nat,Xb: list_nat,Y: list_nat] :
( ( ( map_ta7164188454487880599at_nat @ X2 @ Xa2 @ Xb )
= Y )
=> ( ( ( Xa2 = nil_nat )
=> ( Y != Xb ) )
=> ~ ! [A4: nat,As: list_nat] :
( ( Xa2
= ( cons_nat @ A4 @ As ) )
=> ( Y
!= ( map_ta7164188454487880599at_nat @ X2 @ As @ ( cons_nat @ ( X2 @ A4 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_857_SuccI,axiom,
! [Kl: list_list_nat,K: list_nat,Kl2: set_list_list_nat] :
( ( member_list_list_nat @ ( append_list_nat @ Kl @ ( cons_list_nat @ K @ nil_list_nat ) ) @ Kl2 )
=> ( member_list_nat2 @ K @ ( bNF_Gr3053708287304744325st_nat @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_858_SuccI,axiom,
! [Kl: list_nat,K: nat,Kl2: set_list_nat] :
( ( member_list_nat2 @ ( append_nat @ Kl @ ( cons_nat @ K @ nil_nat ) ) @ Kl2 )
=> ( member_nat2 @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_859_SuccD,axiom,
! [K: list_nat,Kl2: set_list_list_nat,Kl: list_list_nat] :
( ( member_list_nat2 @ K @ ( bNF_Gr3053708287304744325st_nat @ Kl2 @ Kl ) )
=> ( member_list_list_nat @ ( append_list_nat @ Kl @ ( cons_list_nat @ K @ nil_list_nat ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_860_SuccD,axiom,
! [K: nat,Kl2: set_list_nat,Kl: list_nat] :
( ( member_nat2 @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ Kl ) )
=> ( member_list_nat2 @ ( append_nat @ Kl @ ( cons_nat @ K @ nil_nat ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_861_empty__Shift,axiom,
! [Kl2: set_list_list_nat,K: list_nat] :
( ( member_list_list_nat @ nil_list_nat @ Kl2 )
=> ( ( member_list_nat2 @ K @ ( bNF_Gr3053708287304744325st_nat @ Kl2 @ nil_list_nat ) )
=> ( member_list_list_nat @ nil_list_nat @ ( bNF_Gr9051742241863529473st_nat @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_862_empty__Shift,axiom,
! [Kl2: set_list_nat,K: nat] :
( ( member_list_nat2 @ nil_nat @ Kl2 )
=> ( ( member_nat2 @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ nil_nat ) )
=> ( member_list_nat2 @ nil_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_863_arg__min__least,axiom,
! [S2: set_list_nat,Y: list_nat,F: list_nat > nat] :
( ( finite8100373058378681591st_nat @ S2 )
=> ( ( S2 != bot_bot_set_list_nat )
=> ( ( member_list_nat2 @ Y @ S2 )
=> ( ord_less_eq_nat @ ( F @ ( lattic5785867957632790475at_nat @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_864_poss__of__term__Cons,axiom,
! [I: nat,P: list_nat,U: term_a_b,F: a,Ts: list_term_a_b] :
( ( member_list_nat2 @ ( cons_nat @ I @ P ) @ ( terms_7168686267159881682rm_a_b @ U @ ( fun_a_b @ F @ Ts ) ) )
=> ( member_list_nat2 @ P @ ( terms_7168686267159881682rm_a_b @ U @ ( nth_term_a_b @ Ts @ I ) ) ) ) ).
% poss_of_term_Cons
thf(fact_865_ord_Olexordp__eq__simps_I3_J,axiom,
! [Less: nat > nat > $o,X2: nat,Xs: list_nat] :
~ ( lexordp_eq_nat @ Less @ ( cons_nat @ X2 @ Xs ) @ nil_nat ) ).
% ord.lexordp_eq_simps(3)
thf(fact_866_concat__conv__foldr,axiom,
( concat_nat
= ( ^ [Xss3: list_list_nat] : ( foldr_6871341030409798377st_nat @ append_nat @ Xss3 @ nil_nat ) ) ) ).
% concat_conv_foldr
thf(fact_867_ord_Olexordp__eq__simps_I1_J,axiom,
! [Less: nat > nat > $o,Ys: list_nat] : ( lexordp_eq_nat @ Less @ nil_nat @ Ys ) ).
% ord.lexordp_eq_simps(1)
thf(fact_868_ord_Olexordp__eq__simps_I2_J,axiom,
! [Less: nat > nat > $o,Xs: list_nat] :
( ( lexordp_eq_nat @ Less @ Xs @ nil_nat )
= ( Xs = nil_nat ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_869_ord_Olexordp__eq_ONil,axiom,
! [Less: nat > nat > $o,Ys: list_nat] : ( lexordp_eq_nat @ Less @ nil_nat @ Ys ) ).
% ord.lexordp_eq.Nil
thf(fact_870_ord_Olexordp__eq_Osimps,axiom,
( lexordp_eq_nat
= ( ^ [Less2: nat > nat > $o,A1: list_nat,A22: list_nat] :
( ? [Ys3: list_nat] :
( ( A1 = nil_nat )
& ( A22 = Ys3 ) )
| ? [X3: nat,Y4: nat,Xs3: list_nat,Ys3: list_nat] :
( ( A1
= ( cons_nat @ X3 @ Xs3 ) )
& ( A22
= ( cons_nat @ Y4 @ Ys3 ) )
& ( Less2 @ X3 @ Y4 ) )
| ? [X3: nat,Y4: nat,Xs3: list_nat,Ys3: list_nat] :
( ( A1
= ( cons_nat @ X3 @ Xs3 ) )
& ( A22
= ( cons_nat @ Y4 @ Ys3 ) )
& ~ ( Less2 @ X3 @ Y4 )
& ~ ( Less2 @ Y4 @ X3 )
& ( lexordp_eq_nat @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_871_ord_Olexordp__eq_Ocases,axiom,
! [Less: nat > nat > $o,A12: list_nat,A23: list_nat] :
( ( lexordp_eq_nat @ Less @ A12 @ A23 )
=> ( ( A12 != nil_nat )
=> ( ! [X: nat] :
( ? [Xs2: list_nat] :
( A12
= ( cons_nat @ X @ Xs2 ) )
=> ! [Y2: nat] :
( ? [Ys2: list_nat] :
( A23
= ( cons_nat @ Y2 @ Ys2 ) )
=> ~ ( Less @ X @ Y2 ) ) )
=> ~ ! [X: nat,Y2: nat,Xs2: list_nat] :
( ( A12
= ( cons_nat @ X @ Xs2 ) )
=> ! [Ys2: list_nat] :
( ( A23
= ( cons_nat @ Y2 @ Ys2 ) )
=> ( ~ ( Less @ X @ Y2 )
=> ( ~ ( Less @ Y2 @ X )
=> ~ ( lexordp_eq_nat @ Less @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_872_subt__at_Osimps_I2_J,axiom,
! [F: a,Ss2: list_term_a_b,I: nat,P: list_nat] :
( ( term_subt_at_a_b @ ( fun_a_b @ F @ Ss2 ) @ ( cons_nat @ I @ P ) )
= ( term_subt_at_a_b @ ( nth_term_a_b @ Ss2 @ I ) @ P ) ) ).
% subt_at.simps(2)
thf(fact_873_Sup__fin_Osubset__imp,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ ( lattic1093996805478795353in_nat @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_874_Inf__fin_Osubset__imp,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ B ) @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_875_Term_Oterm_Osimps_I9_J,axiom,
! [F1: a > a,F22: b > b,X1: b] :
( ( map_term_a_a_b_b @ F1 @ F22 @ ( var_b_a @ X1 ) )
= ( var_b_a @ ( F22 @ X1 ) ) ) ).
% Term.term.simps(9)
thf(fact_876_Inf__fin__le__Sup__fin,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A ) @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_877_replace__term__at_Osimps_I2_J,axiom,
! [X2: b,V: nat,Va2: list_nat,T: term_a_b] :
( ( term_r6860082780075436317at_a_b @ ( var_b_a @ X2 ) @ ( cons_nat @ V @ Va2 ) @ T )
= ( var_b_a @ X2 ) ) ).
% replace_term_at.simps(2)
thf(fact_878_Inf__fin_OcoboundedI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat2 @ A2 @ A )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A ) @ A2 ) ) ) ).
% Inf_fin.coboundedI
thf(fact_879_Sup__fin_OcoboundedI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat2 @ A2 @ A )
=> ( ord_less_eq_nat @ A2 @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_880_Inf__fin_OboundedE,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X2 @ ( lattic5238388535129920115in_nat @ A ) )
=> ! [A7: nat] :
( ( member_nat2 @ A7 @ A )
=> ( ord_less_eq_nat @ X2 @ A7 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_881_Inf__fin_OboundedI,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A4: nat] :
( ( member_nat2 @ A4 @ A )
=> ( ord_less_eq_nat @ X2 @ A4 ) )
=> ( ord_less_eq_nat @ X2 @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_882_Sup__fin_OboundedE,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X2 )
=> ! [A7: nat] :
( ( member_nat2 @ A7 @ A )
=> ( ord_less_eq_nat @ A7 @ X2 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_883_Sup__fin_OboundedI,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A4: nat] :
( ( member_nat2 @ A4 @ A )
=> ( ord_less_eq_nat @ A4 @ X2 ) )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X2 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_884_Inf__fin_Obounded__iff,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X2 @ ( lattic5238388535129920115in_nat @ A ) )
= ( ! [X3: nat] :
( ( member_nat2 @ X3 @ A )
=> ( ord_less_eq_nat @ X2 @ X3 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_885_Sup__fin_Obounded__iff,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X2 )
= ( ! [X3: nat] :
( ( member_nat2 @ X3 @ A )
=> ( ord_less_eq_nat @ X3 @ X2 ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_886_subt__at_Oelims,axiom,
! [X2: term_a_b,Xa2: list_nat,Y: term_a_b] :
( ( ( term_subt_at_a_b @ X2 @ Xa2 )
= Y )
=> ( ( ( Xa2 = nil_nat )
=> ( Y != X2 ) )
=> ( ! [F5: a,Ss: list_term_a_b] :
( ( X2
= ( fun_a_b @ F5 @ Ss ) )
=> ! [I2: nat,P4: list_nat] :
( ( Xa2
= ( cons_nat @ I2 @ P4 ) )
=> ( Y
!= ( term_subt_at_a_b @ ( nth_term_a_b @ Ss @ I2 ) @ P4 ) ) ) )
=> ~ ( ? [X: b] :
( X2
= ( var_b_a @ X ) )
=> ( ? [V2: nat,Va: list_nat] :
( Xa2
= ( cons_nat @ V2 @ Va ) )
=> ( Y != undefined_term_a_b ) ) ) ) ) ) ).
% subt_at.elims
thf(fact_887_insertCI,axiom,
! [A2: list_nat,B: set_list_nat,B2: list_nat] :
( ( ~ ( member_list_nat2 @ A2 @ B )
=> ( A2 = B2 ) )
=> ( member_list_nat2 @ A2 @ ( insert_list_nat2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_888_insert__iff,axiom,
! [A2: list_nat,B2: list_nat,A: set_list_nat] :
( ( member_list_nat2 @ A2 @ ( insert_list_nat2 @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_list_nat2 @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_889_singletonI,axiom,
! [A2: list_nat] : ( member_list_nat2 @ A2 @ ( insert_list_nat2 @ A2 @ bot_bot_set_list_nat ) ) ).
% singletonI
thf(fact_890_insert__subset,axiom,
! [X2: list_nat,A: set_list_nat,B: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( insert_list_nat2 @ X2 @ A ) @ B )
= ( ( member_list_nat2 @ X2 @ B )
& ( ord_le6045566169113846134st_nat @ A @ B ) ) ) ).
% insert_subset
thf(fact_891_varposs__Var,axiom,
! [X2: b] :
( ( terms_varposs_a_b @ ( var_b_a @ X2 ) )
= ( insert_list_nat2 @ nil_nat @ bot_bot_set_list_nat ) ) ).
% varposs_Var
thf(fact_892_insertE,axiom,
! [A2: list_nat,B2: list_nat,A: set_list_nat] :
( ( member_list_nat2 @ A2 @ ( insert_list_nat2 @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_list_nat2 @ A2 @ A ) ) ) ).
% insertE
thf(fact_893_insertI1,axiom,
! [A2: list_nat,B: set_list_nat] : ( member_list_nat2 @ A2 @ ( insert_list_nat2 @ A2 @ B ) ) ).
% insertI1
thf(fact_894_insertI2,axiom,
! [A2: list_nat,B: set_list_nat,B2: list_nat] :
( ( member_list_nat2 @ A2 @ B )
=> ( member_list_nat2 @ A2 @ ( insert_list_nat2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_895_Set_Oset__insert,axiom,
! [X2: list_nat,A: set_list_nat] :
( ( member_list_nat2 @ X2 @ A )
=> ~ ! [B7: set_list_nat] :
( ( A
= ( insert_list_nat2 @ X2 @ B7 ) )
=> ( member_list_nat2 @ X2 @ B7 ) ) ) ).
% Set.set_insert
thf(fact_896_insert__ident,axiom,
! [X2: list_nat,A: set_list_nat,B: set_list_nat] :
( ~ ( member_list_nat2 @ X2 @ A )
=> ( ~ ( member_list_nat2 @ X2 @ B )
=> ( ( ( insert_list_nat2 @ X2 @ A )
= ( insert_list_nat2 @ X2 @ B ) )
= ( A = B ) ) ) ) ).
% insert_ident
thf(fact_897_insert__absorb,axiom,
! [A2: list_nat,A: set_list_nat] :
( ( member_list_nat2 @ A2 @ A )
=> ( ( insert_list_nat2 @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_898_insert__eq__iff,axiom,
! [A2: list_nat,A: set_list_nat,B2: list_nat,B: set_list_nat] :
( ~ ( member_list_nat2 @ A2 @ A )
=> ( ~ ( member_list_nat2 @ B2 @ B )
=> ( ( ( insert_list_nat2 @ A2 @ A )
= ( insert_list_nat2 @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: set_list_nat] :
( ( A
= ( insert_list_nat2 @ B2 @ C3 ) )
& ~ ( member_list_nat2 @ B2 @ C3 )
& ( B
= ( insert_list_nat2 @ A2 @ C3 ) )
& ~ ( member_list_nat2 @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_899_mk__disjoint__insert,axiom,
! [A2: list_nat,A: set_list_nat] :
( ( member_list_nat2 @ A2 @ A )
=> ? [B7: set_list_nat] :
( ( A
= ( insert_list_nat2 @ A2 @ B7 ) )
& ~ ( member_list_nat2 @ A2 @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_900_singletonD,axiom,
! [B2: list_nat,A2: list_nat] :
( ( member_list_nat2 @ B2 @ ( insert_list_nat2 @ A2 @ bot_bot_set_list_nat ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_901_singleton__iff,axiom,
! [B2: list_nat,A2: list_nat] :
( ( member_list_nat2 @ B2 @ ( insert_list_nat2 @ A2 @ bot_bot_set_list_nat ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_902_subset__insert,axiom,
! [X2: list_nat,A: set_list_nat,B: set_list_nat] :
( ~ ( member_list_nat2 @ X2 @ A )
=> ( ( ord_le6045566169113846134st_nat @ A @ ( insert_list_nat2 @ X2 @ B ) )
= ( ord_le6045566169113846134st_nat @ A @ B ) ) ) ).
% subset_insert
thf(fact_903_insert__subsetI,axiom,
! [X2: list_nat,A: set_list_nat,X5: set_list_nat] :
( ( member_list_nat2 @ X2 @ A )
=> ( ( ord_le6045566169113846134st_nat @ X5 @ A )
=> ( ord_le6045566169113846134st_nat @ ( insert_list_nat2 @ X2 @ X5 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_904_infinite__finite__induct,axiom,
! [P3: set_list_nat > $o,A: set_list_nat] :
( ! [A8: set_list_nat] :
( ~ ( finite8100373058378681591st_nat @ A8 )
=> ( P3 @ A8 ) )
=> ( ( P3 @ bot_bot_set_list_nat )
=> ( ! [X: list_nat,F6: set_list_nat] :
( ( finite8100373058378681591st_nat @ F6 )
=> ( ~ ( member_list_nat2 @ X @ F6 )
=> ( ( P3 @ F6 )
=> ( P3 @ ( insert_list_nat2 @ X @ F6 ) ) ) ) )
=> ( P3 @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_905_finite__ne__induct,axiom,
! [F4: set_list_nat,P3: set_list_nat > $o] :
( ( finite8100373058378681591st_nat @ F4 )
=> ( ( F4 != bot_bot_set_list_nat )
=> ( ! [X: list_nat] : ( P3 @ ( insert_list_nat2 @ X @ bot_bot_set_list_nat ) )
=> ( ! [X: list_nat,F6: set_list_nat] :
( ( finite8100373058378681591st_nat @ F6 )
=> ( ( F6 != bot_bot_set_list_nat )
=> ( ~ ( member_list_nat2 @ X @ F6 )
=> ( ( P3 @ F6 )
=> ( P3 @ ( insert_list_nat2 @ X @ F6 ) ) ) ) ) )
=> ( P3 @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_906_finite__induct,axiom,
! [F4: set_list_nat,P3: set_list_nat > $o] :
( ( finite8100373058378681591st_nat @ F4 )
=> ( ( P3 @ bot_bot_set_list_nat )
=> ( ! [X: list_nat,F6: set_list_nat] :
( ( finite8100373058378681591st_nat @ F6 )
=> ( ~ ( member_list_nat2 @ X @ F6 )
=> ( ( P3 @ F6 )
=> ( P3 @ ( insert_list_nat2 @ X @ F6 ) ) ) ) )
=> ( P3 @ F4 ) ) ) ) ).
% finite_induct
thf(fact_907_mem__idx_Ocases,axiom,
! [X2: produc4575160907756185873st_nat] :
( ! [Uu: nat] :
( X2
!= ( produc8282810413953273033st_nat @ Uu @ nil_nat ) )
=> ~ ! [X: nat,A4: nat,As: list_nat] :
( X2
!= ( produc8282810413953273033st_nat @ X @ ( cons_nat @ A4 @ As ) ) ) ) ).
% mem_idx.cases
thf(fact_908_finite__ranking__induct,axiom,
! [S2: set_list_nat,P3: set_list_nat > $o,F: list_nat > nat] :
( ( finite8100373058378681591st_nat @ S2 )
=> ( ( P3 @ bot_bot_set_list_nat )
=> ( ! [X: list_nat,S3: set_list_nat] :
( ( finite8100373058378681591st_nat @ S3 )
=> ( ! [Y5: list_nat] :
( ( member_list_nat2 @ Y5 @ S3 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X ) ) )
=> ( ( P3 @ S3 )
=> ( P3 @ ( insert_list_nat2 @ X @ S3 ) ) ) ) )
=> ( P3 @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_909_finite__subset__induct_H,axiom,
! [F4: set_list_nat,A: set_list_nat,P3: set_list_nat > $o] :
( ( finite8100373058378681591st_nat @ F4 )
=> ( ( ord_le6045566169113846134st_nat @ F4 @ A )
=> ( ( P3 @ bot_bot_set_list_nat )
=> ( ! [A4: list_nat,F6: set_list_nat] :
( ( finite8100373058378681591st_nat @ F6 )
=> ( ( member_list_nat2 @ A4 @ A )
=> ( ( ord_le6045566169113846134st_nat @ F6 @ A )
=> ( ~ ( member_list_nat2 @ A4 @ F6 )
=> ( ( P3 @ F6 )
=> ( P3 @ ( insert_list_nat2 @ A4 @ F6 ) ) ) ) ) ) )
=> ( P3 @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_910_finite__subset__induct,axiom,
! [F4: set_list_nat,A: set_list_nat,P3: set_list_nat > $o] :
( ( finite8100373058378681591st_nat @ F4 )
=> ( ( ord_le6045566169113846134st_nat @ F4 @ A )
=> ( ( P3 @ bot_bot_set_list_nat )
=> ( ! [A4: list_nat,F6: set_list_nat] :
( ( finite8100373058378681591st_nat @ F6 )
=> ( ( member_list_nat2 @ A4 @ A )
=> ( ~ ( member_list_nat2 @ A4 @ F6 )
=> ( ( P3 @ F6 )
=> ( P3 @ ( insert_list_nat2 @ A4 @ F6 ) ) ) ) ) )
=> ( P3 @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_911_poss_Osimps_I1_J,axiom,
! [X2: b] :
( ( term_poss_a_b @ ( var_b_a @ X2 ) )
= ( insert_list_nat2 @ nil_nat @ bot_bot_set_list_nat ) ) ).
% poss.simps(1)
thf(fact_912_subt__at_Osimps_I3_J,axiom,
! [X2: b,V: nat,Va2: list_nat] :
( ( term_subt_at_a_b @ ( var_b_a @ X2 ) @ ( cons_nat @ V @ Va2 ) )
= undefined_term_a_b ) ).
% subt_at.simps(3)
thf(fact_913_subt__at_Opelims,axiom,
! [X2: term_a_b,Xa2: list_nat,Y: term_a_b] :
( ( ( term_subt_at_a_b @ X2 @ Xa2 )
= Y )
=> ( ( accp_P682940083893826398st_nat @ term_subt_at_rel_a_b @ ( produc4563063199488751885st_nat @ X2 @ Xa2 ) )
=> ( ( ( Xa2 = nil_nat )
=> ( ( Y = X2 )
=> ~ ( accp_P682940083893826398st_nat @ term_subt_at_rel_a_b @ ( produc4563063199488751885st_nat @ X2 @ nil_nat ) ) ) )
=> ( ! [F5: a,Ss: list_term_a_b] :
( ( X2
= ( fun_a_b @ F5 @ Ss ) )
=> ! [I2: nat,P4: list_nat] :
( ( Xa2
= ( cons_nat @ I2 @ P4 ) )
=> ( ( Y
= ( term_subt_at_a_b @ ( nth_term_a_b @ Ss @ I2 ) @ P4 ) )
=> ~ ( accp_P682940083893826398st_nat @ term_subt_at_rel_a_b @ ( produc4563063199488751885st_nat @ ( fun_a_b @ F5 @ Ss ) @ ( cons_nat @ I2 @ P4 ) ) ) ) ) )
=> ~ ! [X: b] :
( ( X2
= ( var_b_a @ X ) )
=> ! [V2: nat,Va: list_nat] :
( ( Xa2
= ( cons_nat @ V2 @ Va ) )
=> ( ( Y = undefined_term_a_b )
=> ~ ( accp_P682940083893826398st_nat @ term_subt_at_rel_a_b @ ( produc4563063199488751885st_nat @ ( var_b_a @ X ) @ ( cons_nat @ V2 @ Va ) ) ) ) ) ) ) ) ) ) ).
% subt_at.pelims
thf(fact_914_filter2_Ocases,axiom,
! [X2: produc4787317212837456354st_nat] :
( ! [P5: nat > nat > $o,Uu: list_nat] :
( X2
!= ( produc3127733452865184594st_nat @ P5 @ ( produc2694037385005941721st_nat @ nil_nat @ Uu ) ) )
=> ( ! [P5: nat > nat > $o,V2: nat,Va: list_nat] :
( X2
!= ( produc3127733452865184594st_nat @ P5 @ ( produc2694037385005941721st_nat @ ( cons_nat @ V2 @ Va ) @ nil_nat ) ) )
=> ~ ! [P5: nat > nat > $o,A4: nat,As: list_nat,B4: nat,Bs: list_nat] :
( X2
!= ( produc3127733452865184594st_nat @ P5 @ ( produc2694037385005941721st_nat @ ( cons_nat @ A4 @ As ) @ ( cons_nat @ B4 @ Bs ) ) ) ) ) ) ).
% filter2.cases
thf(fact_915_span_Ocases,axiom,
! [X2: produc4226810134323546766st_nat] :
( ! [P5: nat > $o,X: nat,Xs2: list_nat] :
( X2
!= ( produc8587622027977423880st_nat @ P5 @ ( cons_nat @ X @ Xs2 ) ) )
=> ~ ! [Uu: nat > $o] :
( X2
!= ( produc8587622027977423880st_nat @ Uu @ nil_nat ) ) ) ).
% span.cases
thf(fact_916_successively_Ocases,axiom,
! [X2: produc254973753779126261st_nat] :
( ! [P5: nat > nat > $o] :
( X2
!= ( produc4727192421694094319st_nat @ P5 @ nil_nat ) )
=> ( ! [P5: nat > nat > $o,X: nat] :
( X2
!= ( produc4727192421694094319st_nat @ P5 @ ( cons_nat @ X @ nil_nat ) ) )
=> ~ ! [P5: nat > nat > $o,X: nat,Y2: nat,Xs2: list_nat] :
( X2
!= ( produc4727192421694094319st_nat @ P5 @ ( cons_nat @ X @ ( cons_nat @ Y2 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_917_distinct__eq_Ocases,axiom,
! [X2: produc254973753779126261st_nat] :
( ! [Uu: nat > nat > $o] :
( X2
!= ( produc4727192421694094319st_nat @ Uu @ nil_nat ) )
=> ~ ! [Eq: nat > nat > $o,X: nat,Xs2: list_nat] :
( X2
!= ( produc4727192421694094319st_nat @ Eq @ ( cons_nat @ X @ Xs2 ) ) ) ) ).
% distinct_eq.cases
thf(fact_918_shuffles_Ocases,axiom,
! [X2: produc1828647624359046049st_nat] :
( ! [Ys2: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ nil_nat @ Ys2 ) )
=> ( ! [Xs2: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ Xs2 @ nil_nat ) )
=> ~ ! [X: nat,Xs2: list_nat,Y2: nat,Ys2: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) ) ) ) ).
% shuffles.cases
thf(fact_919_remove__prefix_Ocases,axiom,
! [X2: produc1828647624359046049st_nat] :
( ! [X: nat,Xs2: list_nat,Y2: nat,Ys2: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) )
=> ( ! [Ys2: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ nil_nat @ Ys2 ) )
=> ~ ! [V2: nat,Va: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ ( cons_nat @ V2 @ Va ) @ nil_nat ) ) ) ) ).
% remove_prefix.cases
thf(fact_920_list__inter_Ocases,axiom,
! [X2: produc1828647624359046049st_nat] :
( ! [Bs: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ nil_nat @ Bs ) )
=> ~ ! [A4: nat,As: list_nat,Bs: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ ( cons_nat @ A4 @ As ) @ Bs ) ) ) ).
% list_inter.cases
thf(fact_921_union__list__sorted_Ocases,axiom,
! [X2: produc1828647624359046049st_nat] :
( ! [X: nat,Xs2: list_nat,Y2: nat,Ys2: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) )
=> ( ! [Ys2: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ nil_nat @ Ys2 ) )
=> ~ ! [V2: nat,Va: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ ( cons_nat @ V2 @ Va ) @ nil_nat ) ) ) ) ).
% union_list_sorted.cases
thf(fact_922_subtract__list__sorted_Ocases,axiom,
! [X2: produc1828647624359046049st_nat] :
( ! [X: nat,Xs2: list_nat,Y2: nat,Ys2: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) )
=> ( ! [Ys2: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ nil_nat @ Ys2 ) )
=> ~ ! [V2: nat,Va: list_nat] :
( X2
!= ( produc2694037385005941721st_nat @ ( cons_nat @ V2 @ Va ) @ nil_nat ) ) ) ) ).
% subtract_list_sorted.cases
thf(fact_923_is__singletonI_H,axiom,
! [A: set_list_nat] :
( ( A != bot_bot_set_list_nat )
=> ( ! [X: list_nat,Y2: list_nat] :
( ( member_list_nat2 @ X @ A )
=> ( ( member_list_nat2 @ Y2 @ A )
=> ( X = Y2 ) ) )
=> ( is_sin2641923865335537900st_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_924_listset_Osimps_I1_J,axiom,
( ( listset_nat @ nil_set_nat )
= ( insert_list_nat2 @ nil_nat @ bot_bot_set_list_nat ) ) ).
% listset.simps(1)
thf(fact_925_lists__empty,axiom,
( ( lists_nat @ bot_bot_set_nat )
= ( insert_list_nat2 @ nil_nat @ bot_bot_set_list_nat ) ) ).
% lists_empty
thf(fact_926_DiffI,axiom,
! [C2: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ C2 @ A )
=> ( ~ ( member_list_nat2 @ C2 @ B )
=> ( member_list_nat2 @ C2 @ ( minus_7954133019191499631st_nat @ A @ B ) ) ) ) ).
% DiffI
thf(fact_927_Diff__iff,axiom,
! [C2: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ C2 @ ( minus_7954133019191499631st_nat @ A @ B ) )
= ( ( member_list_nat2 @ C2 @ A )
& ~ ( member_list_nat2 @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_928_Diff__insert0,axiom,
! [X2: list_nat,A: set_list_nat,B: set_list_nat] :
( ~ ( member_list_nat2 @ X2 @ A )
=> ( ( minus_7954133019191499631st_nat @ A @ ( insert_list_nat2 @ X2 @ B ) )
= ( minus_7954133019191499631st_nat @ A @ B ) ) ) ).
% Diff_insert0
thf(fact_929_insert__Diff1,axiom,
! [X2: list_nat,B: set_list_nat,A: set_list_nat] :
( ( member_list_nat2 @ X2 @ B )
=> ( ( minus_7954133019191499631st_nat @ ( insert_list_nat2 @ X2 @ A ) @ B )
= ( minus_7954133019191499631st_nat @ A @ B ) ) ) ).
% insert_Diff1
thf(fact_930_append__in__lists__conv,axiom,
! [Xs: list_nat,Ys: list_nat,A: set_nat] :
( ( member_list_nat2 @ ( append_nat @ Xs @ Ys ) @ ( lists_nat @ A ) )
= ( ( member_list_nat2 @ Xs @ ( lists_nat @ A ) )
& ( member_list_nat2 @ Ys @ ( lists_nat @ A ) ) ) ) ).
% append_in_lists_conv
thf(fact_931_insert__Diff__if,axiom,
! [X2: list_nat,B: set_list_nat,A: set_list_nat] :
( ( ( member_list_nat2 @ X2 @ B )
=> ( ( minus_7954133019191499631st_nat @ ( insert_list_nat2 @ X2 @ A ) @ B )
= ( minus_7954133019191499631st_nat @ A @ B ) ) )
& ( ~ ( member_list_nat2 @ X2 @ B )
=> ( ( minus_7954133019191499631st_nat @ ( insert_list_nat2 @ X2 @ A ) @ B )
= ( insert_list_nat2 @ X2 @ ( minus_7954133019191499631st_nat @ A @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_932_lists_ONil,axiom,
! [A: set_nat] : ( member_list_nat2 @ nil_nat @ ( lists_nat @ A ) ) ).
% lists.Nil
thf(fact_933_DiffE,axiom,
! [C2: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ C2 @ ( minus_7954133019191499631st_nat @ A @ B ) )
=> ~ ( ( member_list_nat2 @ C2 @ A )
=> ( member_list_nat2 @ C2 @ B ) ) ) ).
% DiffE
thf(fact_934_DiffD1,axiom,
! [C2: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ C2 @ ( minus_7954133019191499631st_nat @ A @ B ) )
=> ( member_list_nat2 @ C2 @ A ) ) ).
% DiffD1
thf(fact_935_DiffD2,axiom,
! [C2: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ C2 @ ( minus_7954133019191499631st_nat @ A @ B ) )
=> ~ ( member_list_nat2 @ C2 @ B ) ) ).
% DiffD2
thf(fact_936_Diff__insert__absorb,axiom,
! [X2: list_nat,A: set_list_nat] :
( ~ ( member_list_nat2 @ X2 @ A )
=> ( ( minus_7954133019191499631st_nat @ ( insert_list_nat2 @ X2 @ A ) @ ( insert_list_nat2 @ X2 @ bot_bot_set_list_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_937_insert__Diff,axiom,
! [A2: list_nat,A: set_list_nat] :
( ( member_list_nat2 @ A2 @ A )
=> ( ( insert_list_nat2 @ A2 @ ( minus_7954133019191499631st_nat @ A @ ( insert_list_nat2 @ A2 @ bot_bot_set_list_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_938_subset__Diff__insert,axiom,
! [A: set_list_nat,B: set_list_nat,X2: list_nat,C: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ ( minus_7954133019191499631st_nat @ B @ ( insert_list_nat2 @ X2 @ C ) ) )
= ( ( ord_le6045566169113846134st_nat @ A @ ( minus_7954133019191499631st_nat @ B @ C ) )
& ~ ( member_list_nat2 @ X2 @ A ) ) ) ).
% subset_Diff_insert
thf(fact_939_lists_Osimps,axiom,
! [A2: list_list_nat,A: set_list_nat] :
( ( member_list_list_nat @ A2 @ ( lists_list_nat @ A ) )
= ( ( A2 = nil_list_nat )
| ? [A3: list_nat,L: list_list_nat] :
( ( A2
= ( cons_list_nat @ A3 @ L ) )
& ( member_list_nat2 @ A3 @ A )
& ( member_list_list_nat @ L @ ( lists_list_nat @ A ) ) ) ) ) ).
% lists.simps
thf(fact_940_lists_Osimps,axiom,
! [A2: list_nat,A: set_nat] :
( ( member_list_nat2 @ A2 @ ( lists_nat @ A ) )
= ( ( A2 = nil_nat )
| ? [A3: nat,L: list_nat] :
( ( A2
= ( cons_nat @ A3 @ L ) )
& ( member_nat2 @ A3 @ A )
& ( member_list_nat2 @ L @ ( lists_nat @ A ) ) ) ) ) ).
% lists.simps
thf(fact_941_lists_Ocases,axiom,
! [A2: list_list_nat,A: set_list_nat] :
( ( member_list_list_nat @ A2 @ ( lists_list_nat @ A ) )
=> ( ( A2 != nil_list_nat )
=> ~ ! [A4: list_nat,L2: list_list_nat] :
( ( A2
= ( cons_list_nat @ A4 @ L2 ) )
=> ( ( member_list_nat2 @ A4 @ A )
=> ~ ( member_list_list_nat @ L2 @ ( lists_list_nat @ A ) ) ) ) ) ) ).
% lists.cases
thf(fact_942_lists_Ocases,axiom,
! [A2: list_nat,A: set_nat] :
( ( member_list_nat2 @ A2 @ ( lists_nat @ A ) )
=> ( ( A2 != nil_nat )
=> ~ ! [A4: nat,L2: list_nat] :
( ( A2
= ( cons_nat @ A4 @ L2 ) )
=> ( ( member_nat2 @ A4 @ A )
=> ~ ( member_list_nat2 @ L2 @ ( lists_nat @ A ) ) ) ) ) ) ).
% lists.cases
thf(fact_943_finite__empty__induct,axiom,
! [A: set_list_nat,P3: set_list_nat > $o] :
( ( finite8100373058378681591st_nat @ A )
=> ( ( P3 @ A )
=> ( ! [A4: list_nat,A8: set_list_nat] :
( ( finite8100373058378681591st_nat @ A8 )
=> ( ( member_list_nat2 @ A4 @ A8 )
=> ( ( P3 @ A8 )
=> ( P3 @ ( minus_7954133019191499631st_nat @ A8 @ ( insert_list_nat2 @ A4 @ bot_bot_set_list_nat ) ) ) ) ) )
=> ( P3 @ bot_bot_set_list_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_944_subset__insert__iff,axiom,
! [A: set_list_nat,X2: list_nat,B: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ ( insert_list_nat2 @ X2 @ B ) )
= ( ( ( member_list_nat2 @ X2 @ A )
=> ( ord_le6045566169113846134st_nat @ ( minus_7954133019191499631st_nat @ A @ ( insert_list_nat2 @ X2 @ bot_bot_set_list_nat ) ) @ B ) )
& ( ~ ( member_list_nat2 @ X2 @ A )
=> ( ord_le6045566169113846134st_nat @ A @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_945_finite__remove__induct,axiom,
! [B: set_list_nat,P3: set_list_nat > $o] :
( ( finite8100373058378681591st_nat @ B )
=> ( ( P3 @ bot_bot_set_list_nat )
=> ( ! [A8: set_list_nat] :
( ( finite8100373058378681591st_nat @ A8 )
=> ( ( A8 != bot_bot_set_list_nat )
=> ( ( ord_le6045566169113846134st_nat @ A8 @ B )
=> ( ! [X6: list_nat] :
( ( member_list_nat2 @ X6 @ A8 )
=> ( P3 @ ( minus_7954133019191499631st_nat @ A8 @ ( insert_list_nat2 @ X6 @ bot_bot_set_list_nat ) ) ) )
=> ( P3 @ A8 ) ) ) ) )
=> ( P3 @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_946_remove__induct,axiom,
! [P3: set_list_nat > $o,B: set_list_nat] :
( ( P3 @ bot_bot_set_list_nat )
=> ( ( ~ ( finite8100373058378681591st_nat @ B )
=> ( P3 @ B ) )
=> ( ! [A8: set_list_nat] :
( ( finite8100373058378681591st_nat @ A8 )
=> ( ( A8 != bot_bot_set_list_nat )
=> ( ( ord_le6045566169113846134st_nat @ A8 @ B )
=> ( ! [X6: list_nat] :
( ( member_list_nat2 @ X6 @ A8 )
=> ( P3 @ ( minus_7954133019191499631st_nat @ A8 @ ( insert_list_nat2 @ X6 @ bot_bot_set_list_nat ) ) ) )
=> ( P3 @ A8 ) ) ) ) )
=> ( P3 @ B ) ) ) ) ).
% remove_induct
thf(fact_947_sorted__list__of__set__nonempty,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( linord2614967742042102400et_nat @ A )
= ( cons_nat @ ( lattic8721135487736765967in_nat @ A ) @ ( linord2614967742042102400et_nat @ ( minus_minus_set_nat @ A @ ( insert_nat2 @ ( lattic8721135487736765967in_nat @ A ) @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% sorted_list_of_set_nonempty
thf(fact_948_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_nat,X2: nat,Ys: list_nat,Y: nat,R4: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) @ ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) ) @ ( listrel1_nat @ R4 ) )
= ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R4 ) )
& ( X2 = Y ) )
| ( ( Xs = Ys )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ R4 ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_949_member__remove,axiom,
! [X2: list_nat,Y: list_nat,A: set_list_nat] :
( ( member_list_nat2 @ X2 @ ( remove_list_nat @ Y @ A ) )
= ( ( member_list_nat2 @ X2 @ A )
& ( X2 != Y ) ) ) ).
% member_remove
thf(fact_950_Min_Obounded__iff,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X2 @ ( lattic8721135487736765967in_nat @ A ) )
= ( ! [X3: nat] :
( ( member_nat2 @ X3 @ A )
=> ( ord_less_eq_nat @ X2 @ X3 ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_951_remove1_Osimps_I1_J,axiom,
! [X2: nat] :
( ( remove1_nat @ X2 @ nil_nat )
= nil_nat ) ).
% remove1.simps(1)
thf(fact_952_Min_OcoboundedI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat2 @ A2 @ A )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A ) @ A2 ) ) ) ).
% Min.coboundedI
thf(fact_953_Min__eqI,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ! [Y2: nat] :
( ( member_nat2 @ Y2 @ A )
=> ( ord_less_eq_nat @ X2 @ Y2 ) )
=> ( ( member_nat2 @ X2 @ A )
=> ( ( lattic8721135487736765967in_nat @ A )
= X2 ) ) ) ) ).
% Min_eqI
thf(fact_954_Min__le,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat2 @ X2 @ A )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A ) @ X2 ) ) ) ).
% Min_le
thf(fact_955_not__Nil__listrel1,axiom,
! [Xs: list_nat,R4: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xs ) @ ( listrel1_nat @ R4 ) ) ).
% not_Nil_listrel1
thf(fact_956_not__listrel1__Nil,axiom,
! [Xs: list_nat,R4: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( listrel1_nat @ R4 ) ) ).
% not_listrel1_Nil
thf(fact_957_Min_OboundedI,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A4: nat] :
( ( member_nat2 @ A4 @ A )
=> ( ord_less_eq_nat @ X2 @ A4 ) )
=> ( ord_less_eq_nat @ X2 @ ( lattic8721135487736765967in_nat @ A ) ) ) ) ) ).
% Min.boundedI
thf(fact_958_Min_OboundedE,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X2 @ ( lattic8721135487736765967in_nat @ A ) )
=> ! [A7: nat] :
( ( member_nat2 @ A7 @ A )
=> ( ord_less_eq_nat @ X2 @ A7 ) ) ) ) ) ).
% Min.boundedE
thf(fact_959_eq__Min__iff,axiom,
! [A: set_nat,M: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( M
= ( lattic8721135487736765967in_nat @ A ) )
= ( ( member_nat2 @ M @ A )
& ! [X3: nat] :
( ( member_nat2 @ X3 @ A )
=> ( ord_less_eq_nat @ M @ X3 ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_960_Min__le__iff,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A ) @ X2 )
= ( ? [X3: nat] :
( ( member_nat2 @ X3 @ A )
& ( ord_less_eq_nat @ X3 @ X2 ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_961_Min__eq__iff,axiom,
! [A: set_nat,M: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ( lattic8721135487736765967in_nat @ A )
= M )
= ( ( member_nat2 @ M @ A )
& ! [X3: nat] :
( ( member_nat2 @ X3 @ A )
=> ( ord_less_eq_nat @ M @ X3 ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_962_Min__insert2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ! [B4: nat] :
( ( member_nat2 @ B4 @ A )
=> ( ord_less_eq_nat @ A2 @ B4 ) )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat2 @ A2 @ A ) )
= A2 ) ) ) ).
% Min_insert2
thf(fact_963_Min_Osubset__imp,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ B ) @ ( lattic8721135487736765967in_nat @ A ) ) ) ) ) ).
% Min.subset_imp
thf(fact_964_Min__antimono,axiom,
! [M2: set_nat,N: set_nat] :
( ( ord_less_eq_set_nat @ M2 @ N )
=> ( ( M2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ N )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ N ) @ ( lattic8721135487736765967in_nat @ M2 ) ) ) ) ) ).
% Min_antimono
thf(fact_965_listrel1E,axiom,
! [Xs: list_nat,Ys: list_nat,R4: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R4 ) )
=> ~ ! [X: nat,Y2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y2 ) @ R4 )
=> ! [Us3: list_nat,Vs: list_nat] :
( ( Xs
= ( append_nat @ Us3 @ ( cons_nat @ X @ Vs ) ) )
=> ( Ys
!= ( append_nat @ Us3 @ ( cons_nat @ Y2 @ Vs ) ) ) ) ) ) ).
% listrel1E
thf(fact_966_listrel1I,axiom,
! [X2: nat,Y: nat,R4: set_Pr1261947904930325089at_nat,Xs: list_nat,Us2: list_nat,Vs2: list_nat,Ys: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ R4 )
=> ( ( Xs
= ( append_nat @ Us2 @ ( cons_nat @ X2 @ Vs2 ) ) )
=> ( ( Ys
= ( append_nat @ Us2 @ ( cons_nat @ Y @ Vs2 ) ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R4 ) ) ) ) ) ).
% listrel1I
thf(fact_967_varposs__ground__replace__at,axiom,
! [P: list_nat,S: term_a_b,U: term_a_b] :
( ( member_list_nat2 @ P @ ( terms_varposs_a_b @ S ) )
=> ( ( term_ground_a_b @ U )
=> ( ( terms_varposs_a_b @ ( term_r6860082780075436317at_a_b @ S @ P @ U ) )
= ( minus_7954133019191499631st_nat @ ( terms_varposs_a_b @ S ) @ ( insert_list_nat2 @ P @ bot_bot_set_list_nat ) ) ) ) ) ).
% varposs_ground_replace_at
thf(fact_968_Int__insert__left__if0,axiom,
! [A2: list_nat,C: set_list_nat,B: set_list_nat] :
( ~ ( member_list_nat2 @ A2 @ C )
=> ( ( inf_inf_set_list_nat @ ( insert_list_nat2 @ A2 @ B ) @ C )
= ( inf_inf_set_list_nat @ B @ C ) ) ) ).
% Int_insert_left_if0
thf(fact_969_Int__insert__left__if1,axiom,
! [A2: list_nat,C: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ A2 @ C )
=> ( ( inf_inf_set_list_nat @ ( insert_list_nat2 @ A2 @ B ) @ C )
= ( insert_list_nat2 @ A2 @ ( inf_inf_set_list_nat @ B @ C ) ) ) ) ).
% Int_insert_left_if1
thf(fact_970_Int__insert__right__if0,axiom,
! [A2: list_nat,A: set_list_nat,B: set_list_nat] :
( ~ ( member_list_nat2 @ A2 @ A )
=> ( ( inf_inf_set_list_nat @ A @ ( insert_list_nat2 @ A2 @ B ) )
= ( inf_inf_set_list_nat @ A @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_971_Int__insert__right__if1,axiom,
! [A2: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ A2 @ A )
=> ( ( inf_inf_set_list_nat @ A @ ( insert_list_nat2 @ A2 @ B ) )
= ( insert_list_nat2 @ A2 @ ( inf_inf_set_list_nat @ A @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_972_le__inf__iff,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y @ Z2 ) )
= ( ( ord_less_eq_nat @ X2 @ Y )
& ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_973_inf_Obounded__iff,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C2 ) )
= ( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_974_disjoint__insert_I2_J,axiom,
! [A: set_list_nat,B2: list_nat,B: set_list_nat] :
( ( bot_bot_set_list_nat
= ( inf_inf_set_list_nat @ A @ ( insert_list_nat2 @ B2 @ B ) ) )
= ( ~ ( member_list_nat2 @ B2 @ A )
& ( bot_bot_set_list_nat
= ( inf_inf_set_list_nat @ A @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_975_disjoint__insert_I1_J,axiom,
! [B: set_list_nat,A2: list_nat,A: set_list_nat] :
( ( ( inf_inf_set_list_nat @ B @ ( insert_list_nat2 @ A2 @ A ) )
= bot_bot_set_list_nat )
= ( ~ ( member_list_nat2 @ A2 @ B )
& ( ( inf_inf_set_list_nat @ B @ A )
= bot_bot_set_list_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_976_insert__disjoint_I2_J,axiom,
! [A2: list_nat,A: set_list_nat,B: set_list_nat] :
( ( bot_bot_set_list_nat
= ( inf_inf_set_list_nat @ ( insert_list_nat2 @ A2 @ A ) @ B ) )
= ( ~ ( member_list_nat2 @ A2 @ B )
& ( bot_bot_set_list_nat
= ( inf_inf_set_list_nat @ A @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_977_insert__disjoint_I1_J,axiom,
! [A2: list_nat,A: set_list_nat,B: set_list_nat] :
( ( ( inf_inf_set_list_nat @ ( insert_list_nat2 @ A2 @ A ) @ B )
= bot_bot_set_list_nat )
= ( ~ ( member_list_nat2 @ A2 @ B )
& ( ( inf_inf_set_list_nat @ A @ B )
= bot_bot_set_list_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_978_ground__map__term,axiom,
! [F: a > a,H: b > b,T: term_a_b] :
( ( term_ground_a_b @ ( map_term_a_a_b_b @ F @ H @ T ) )
= ( term_ground_a_b @ T ) ) ).
% ground_map_term
thf(fact_979_disjoint__iff,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( ( inf_inf_set_list_nat @ A @ B )
= bot_bot_set_list_nat )
= ( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ A )
=> ~ ( member_list_nat2 @ X3 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_980_Int__emptyI,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ! [X: list_nat] :
( ( member_list_nat2 @ X @ A )
=> ~ ( member_list_nat2 @ X @ B ) )
=> ( ( inf_inf_set_list_nat @ A @ B )
= bot_bot_set_list_nat ) ) ).
% Int_emptyI
thf(fact_981_Int__insert__right,axiom,
! [A2: list_nat,A: set_list_nat,B: set_list_nat] :
( ( ( member_list_nat2 @ A2 @ A )
=> ( ( inf_inf_set_list_nat @ A @ ( insert_list_nat2 @ A2 @ B ) )
= ( insert_list_nat2 @ A2 @ ( inf_inf_set_list_nat @ A @ B ) ) ) )
& ( ~ ( member_list_nat2 @ A2 @ A )
=> ( ( inf_inf_set_list_nat @ A @ ( insert_list_nat2 @ A2 @ B ) )
= ( inf_inf_set_list_nat @ A @ B ) ) ) ) ).
% Int_insert_right
thf(fact_982_Int__insert__left,axiom,
! [A2: list_nat,C: set_list_nat,B: set_list_nat] :
( ( ( member_list_nat2 @ A2 @ C )
=> ( ( inf_inf_set_list_nat @ ( insert_list_nat2 @ A2 @ B ) @ C )
= ( insert_list_nat2 @ A2 @ ( inf_inf_set_list_nat @ B @ C ) ) ) )
& ( ~ ( member_list_nat2 @ A2 @ C )
=> ( ( inf_inf_set_list_nat @ ( insert_list_nat2 @ A2 @ B ) @ C )
= ( inf_inf_set_list_nat @ B @ C ) ) ) ) ).
% Int_insert_left
thf(fact_983_Int__Collect__mono,axiom,
! [A: set_list_nat,B: set_list_nat,P3: list_nat > $o,Q3: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ A @ B )
=> ( ! [X: list_nat] :
( ( member_list_nat2 @ X @ A )
=> ( ( P3 @ X )
=> ( Q3 @ X ) ) )
=> ( ord_le6045566169113846134st_nat @ ( inf_inf_set_list_nat @ A @ ( collect_list_nat @ P3 ) ) @ ( inf_inf_set_list_nat @ B @ ( collect_list_nat @ Q3 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_984_inf_OcoboundedI2,axiom,
! [B2: nat,C2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_985_inf_OcoboundedI1,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_986_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( inf_inf_nat @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_987_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( inf_inf_nat @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_988_inf_Ocobounded2,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_989_inf_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_990_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( A3
= ( inf_inf_nat @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_991_inf__greatest,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_992_inf_OboundedI,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_993_inf_OboundedE,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_994_inf__absorb2,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( inf_inf_nat @ X2 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_995_inf__absorb1,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( inf_inf_nat @ X2 @ Y )
= X2 ) ) ).
% inf_absorb1
thf(fact_996_inf_Oabsorb2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_997_inf_Oabsorb1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_998_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( inf_inf_nat @ X3 @ Y4 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_999_inf__unique,axiom,
! [F: nat > nat > nat,X2: nat,Y: nat] :
( ! [X: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X @ Y2 ) @ X )
=> ( ! [X: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X @ Y2 ) @ Y2 )
=> ( ! [X: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ X @ Z3 )
=> ( ord_less_eq_nat @ X @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_1000_inf_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_1001_inf_OorderE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2
= ( inf_inf_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_1002_le__infI2,axiom,
! [B2: nat,X2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ X2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X2 ) ) ).
% le_infI2
thf(fact_1003_le__infI1,axiom,
! [A2: nat,X2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X2 ) ) ).
% le_infI1
thf(fact_1004_inf__mono,axiom,
! [A2: nat,C2: nat,B2: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ D2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ ( inf_inf_nat @ C2 @ D2 ) ) ) ) ).
% inf_mono
thf(fact_1005_le__infI,axiom,
! [X2: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X2 @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ B2 )
=> ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_1006_le__infE,axiom,
! [X2: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_nat @ X2 @ A2 )
=> ~ ( ord_less_eq_nat @ X2 @ B2 ) ) ) ).
% le_infE
thf(fact_1007_inf__le2,axiom,
! [X2: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y ) @ Y ) ).
% inf_le2
thf(fact_1008_inf__le1,axiom,
! [X2: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y ) @ X2 ) ).
% inf_le1
thf(fact_1009_inf__sup__ord_I1_J,axiom,
! [X2: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_1010_inf__sup__ord_I2_J,axiom,
! [X2: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_1011_distrib__inf__le,axiom,
! [X2: nat,Y: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X2 @ Y ) @ ( inf_inf_nat @ X2 @ Z2 ) ) @ ( inf_inf_nat @ X2 @ ( sup_sup_nat @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_1012_distrib__sup__le,axiom,
! [X2: nat,Y: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ ( inf_inf_nat @ Y @ Z2 ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X2 @ Y ) @ ( sup_sup_nat @ X2 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_1013_varposs__empty__gound,axiom,
! [S: term_a_b] :
( ( ( terms_varposs_a_b @ S )
= bot_bot_set_list_nat )
= ( term_ground_a_b @ S ) ) ).
% varposs_empty_gound
thf(fact_1014_lexord__append__left__rightI,axiom,
! [A2: nat,B2: nat,R4: set_Pr1261947904930325089at_nat,U: list_nat,X2: list_nat,Y: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R4 )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ U @ ( cons_nat @ A2 @ X2 ) ) @ ( append_nat @ U @ ( cons_nat @ B2 @ Y ) ) ) @ ( lexord_nat @ R4 ) ) ) ).
% lexord_append_left_rightI
thf(fact_1015_listrel_Ocases,axiom,
! [A12: list_nat,A23: list_nat,R4: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A12 @ A23 ) @ ( listrel_nat_nat @ R4 ) )
=> ( ( ( A12 = nil_nat )
=> ( A23 != nil_nat ) )
=> ~ ! [X: nat,Y2: nat,Xs2: list_nat] :
( ( A12
= ( cons_nat @ X @ Xs2 ) )
=> ! [Ys2: list_nat] :
( ( A23
= ( cons_nat @ Y2 @ Ys2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y2 ) @ R4 )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel_nat_nat @ R4 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_1016_Int__iff,axiom,
! [C2: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ C2 @ ( inf_inf_set_list_nat @ A @ B ) )
= ( ( member_list_nat2 @ C2 @ A )
& ( member_list_nat2 @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_1017_IntI,axiom,
! [C2: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ C2 @ A )
=> ( ( member_list_nat2 @ C2 @ B )
=> ( member_list_nat2 @ C2 @ ( inf_inf_set_list_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_1018_min_Oabsorb1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_min_nat @ A2 @ B2 )
= A2 ) ) ).
% min.absorb1
thf(fact_1019_min_Oabsorb2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_min_nat @ A2 @ B2 )
= B2 ) ) ).
% min.absorb2
thf(fact_1020_min_Obounded__iff,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( ord_min_nat @ B2 @ C2 ) )
= ( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% min.bounded_iff
thf(fact_1021_lexord__Nil__left,axiom,
! [Y: list_nat,R4: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Y ) @ ( lexord_nat @ R4 ) )
= ( ? [A3: nat,X3: list_nat] :
( Y
= ( cons_nat @ A3 @ X3 ) ) ) ) ).
% lexord_Nil_left
thf(fact_1022_IntD2,axiom,
! [C2: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ C2 @ ( inf_inf_set_list_nat @ A @ B ) )
=> ( member_list_nat2 @ C2 @ B ) ) ).
% IntD2
thf(fact_1023_IntD1,axiom,
! [C2: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ C2 @ ( inf_inf_set_list_nat @ A @ B ) )
=> ( member_list_nat2 @ C2 @ A ) ) ).
% IntD1
thf(fact_1024_IntE,axiom,
! [C2: list_nat,A: set_list_nat,B: set_list_nat] :
( ( member_list_nat2 @ C2 @ ( inf_inf_set_list_nat @ A @ B ) )
=> ~ ( ( member_list_nat2 @ C2 @ A )
=> ~ ( member_list_nat2 @ C2 @ B ) ) ) ).
% IntE
thf(fact_1025_min__def,axiom,
( ord_min_nat
= ( ^ [A3: nat,B3: nat] : ( if_nat @ ( ord_less_eq_nat @ A3 @ B3 ) @ A3 @ B3 ) ) ) ).
% min_def
thf(fact_1026_min__absorb1,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_min_nat @ X2 @ Y )
= X2 ) ) ).
% min_absorb1
thf(fact_1027_min__absorb2,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_min_nat @ X2 @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_1028_min__le__iff__disj,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( ord_min_nat @ X2 @ Y ) @ Z2 )
= ( ( ord_less_eq_nat @ X2 @ Z2 )
| ( ord_less_eq_nat @ Y @ Z2 ) ) ) ).
% min_le_iff_disj
thf(fact_1029_min_OcoboundedI2,axiom,
! [B2: nat,C2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B2 ) @ C2 ) ) ).
% min.coboundedI2
thf(fact_1030_min_OcoboundedI1,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B2 ) @ C2 ) ) ).
% min.coboundedI1
thf(fact_1031_min_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_min_nat @ A3 @ B3 )
= B3 ) ) ) ).
% min.absorb_iff2
thf(fact_1032_min_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_min_nat @ A3 @ B3 )
= A3 ) ) ) ).
% min.absorb_iff1
thf(fact_1033_min_Ocobounded2,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B2 ) @ B2 ) ).
% min.cobounded2
thf(fact_1034_min_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B2 ) @ A2 ) ).
% min.cobounded1
thf(fact_1035_min_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( A3
= ( ord_min_nat @ A3 @ B3 ) ) ) ) ).
% min.order_iff
thf(fact_1036_min_OboundedI,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ord_less_eq_nat @ A2 @ ( ord_min_nat @ B2 @ C2 ) ) ) ) ).
% min.boundedI
thf(fact_1037_min_OboundedE,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( ord_min_nat @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% min.boundedE
thf(fact_1038_min_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( ord_min_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% min.orderI
thf(fact_1039_min_OorderE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2
= ( ord_min_nat @ A2 @ B2 ) ) ) ).
% min.orderE
thf(fact_1040_min_Omono,axiom,
! [A2: nat,C2: nat,B2: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ D2 )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B2 ) @ ( ord_min_nat @ C2 @ D2 ) ) ) ) ).
% min.mono
thf(fact_1041_listrel__Nil2,axiom,
! [Xs: list_nat,R4: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( listrel_nat_nat @ R4 ) )
=> ( Xs = nil_nat ) ) ).
% listrel_Nil2
thf(fact_1042_listrel__Nil1,axiom,
! [Xs: list_nat,R4: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xs ) @ ( listrel_nat_nat @ R4 ) )
=> ( Xs = nil_nat ) ) ).
% listrel_Nil1
thf(fact_1043_listrel_ONil,axiom,
! [R4: set_Pr1261947904930325089at_nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) @ ( listrel_nat_nat @ R4 ) ) ).
% listrel.Nil
thf(fact_1044_lexord__Nil__right,axiom,
! [X2: list_nat,R4: set_Pr1261947904930325089at_nat] :
~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X2 @ nil_nat ) @ ( lexord_nat @ R4 ) ) ).
% lexord_Nil_right
thf(fact_1045_lexord__append__rightI,axiom,
! [Y: list_nat,X2: list_nat,R4: set_Pr1261947904930325089at_nat] :
( ? [B8: nat,Z4: list_nat] :
( Y
= ( cons_nat @ B8 @ Z4 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X2 @ ( append_nat @ X2 @ Y ) ) @ ( lexord_nat @ R4 ) ) ) ).
% lexord_append_rightI
thf(fact_1046_listrel_Osimps,axiom,
! [A12: list_nat,A23: list_nat,R4: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A12 @ A23 ) @ ( listrel_nat_nat @ R4 ) )
= ( ( ( A12 = nil_nat )
& ( A23 = nil_nat ) )
| ? [X3: nat,Y4: nat,Xs3: list_nat,Ys3: list_nat] :
( ( A12
= ( cons_nat @ X3 @ Xs3 ) )
& ( A23
= ( cons_nat @ Y4 @ Ys3 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ R4 )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys3 ) @ ( listrel_nat_nat @ R4 ) ) ) ) ) ).
% listrel.simps
thf(fact_1047_Missing__List_Omin__list_Oelims,axiom,
! [X2: list_nat,Y: nat] :
( ( ( missing_min_list_nat @ X2 )
= Y )
=> ( ! [X: nat] :
( ( X2
= ( cons_nat @ X @ nil_nat ) )
=> ( Y != X ) )
=> ( ! [X: nat,V2: nat,Va: list_nat] :
( ( X2
= ( cons_nat @ X @ ( cons_nat @ V2 @ Va ) ) )
=> ( Y
!= ( ord_min_nat @ X @ ( missing_min_list_nat @ ( cons_nat @ V2 @ Va ) ) ) ) )
=> ~ ( ( X2 = nil_nat )
=> ( Y != undefined_nat ) ) ) ) ) ).
% Missing_List.min_list.elims
thf(fact_1048_listrel__Nil,axiom,
! [R4: set_Pr1261947904930325089at_nat] :
( ( image_2597627202720054805st_nat @ ( listrel_nat_nat @ R4 ) @ ( insert_list_nat2 @ nil_nat @ bot_bot_set_list_nat ) )
= ( insert_list_nat2 @ nil_nat @ bot_bot_set_list_nat ) ) ).
% listrel_Nil
thf(fact_1049_funas__term__subt__at,axiom,
! [F: a,N2: nat,T: term_a_b] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ F @ N2 ) @ ( term_funas_term_a_b @ T ) )
=> ? [P4: list_nat,Ts2: list_term_a_b] :
( ( member_list_nat2 @ P4 @ ( term_poss_a_b @ T ) )
& ( ( term_subt_at_a_b @ T @ P4 )
= ( fun_a_b @ F @ Ts2 ) )
& ( ( size_s8906293707977694520rm_a_b @ Ts2 )
= N2 ) ) ) ).
% funas_term_subt_at
thf(fact_1050_nth__append__length,axiom,
! [Xs: list_nat,X2: nat,Ys: list_nat] :
( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ Ys ) ) @ ( size_size_list_nat @ Xs ) )
= X2 ) ).
% nth_append_length
thf(fact_1051_min__list__Cons,axiom,
! [X2: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ord_less_eq_nat @ ( missing_min_list_nat @ Xs ) @ ( missing_min_list_nat @ Ys ) )
=> ( ord_less_eq_nat @ ( missing_min_list_nat @ ( cons_nat @ X2 @ Xs ) ) @ ( missing_min_list_nat @ ( cons_nat @ Y @ Ys ) ) ) ) ) ) ).
% min_list_Cons
thf(fact_1052_list__induct2,axiom,
! [Xs: list_nat,Ys: list_nat,P3: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P3 @ nil_nat @ nil_nat )
=> ( ! [X: nat,Xs2: list_nat,Y2: nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1053_list__induct3,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat,P3: list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P3 @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X: nat,Xs2: list_nat,Y2: nat,Ys2: list_nat,Z3: nat,Zs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P3 @ Xs2 @ Ys2 @ Zs2 )
=> ( P3 @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) @ ( cons_nat @ Z3 @ Zs2 ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_1054_list__induct4,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ws: list_nat,P3: list_nat > list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P3 @ nil_nat @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X: nat,Xs2: list_nat,Y2: nat,Ys2: list_nat,Z3: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P3 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) @ ( cons_nat @ Z3 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1055_Missing__List_Omin__list_Osimps_I1_J,axiom,
! [X2: nat] :
( ( missing_min_list_nat @ ( cons_nat @ X2 @ nil_nat ) )
= X2 ) ).
% Missing_List.min_list.simps(1)
thf(fact_1056_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,X: nat,Xs5: list_nat,Y2: nat,Ys5: list_nat] :
( ( X != Y2 )
& ( Xs
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X @ nil_nat ) @ Xs5 ) ) )
& ( Ys
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y2 @ nil_nat ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_1057_append__Cons__nth__middle,axiom,
! [I: nat,Xs: list_nat,Y: nat,Zs: list_nat] :
( ( I
= ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ Y @ Zs ) ) @ I )
= Y ) ) ).
% append_Cons_nth_middle
thf(fact_1058_append__Cons__nth__not__middle,axiom,
! [I: nat,Xs: list_nat,U: nat,Ys: list_nat,Z2: nat] :
( ( I
!= ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ U @ Ys ) ) @ I )
= ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ Z2 @ Ys ) ) @ I ) ) ) ).
% append_Cons_nth_not_middle
thf(fact_1059_min__less__iff__conj,axiom,
! [Z2: nat,X2: nat,Y: nat] :
( ( ord_less_nat @ Z2 @ ( ord_min_nat @ X2 @ Y ) )
= ( ( ord_less_nat @ Z2 @ X2 )
& ( ord_less_nat @ Z2 @ Y ) ) ) ).
% min_less_iff_conj
thf(fact_1060_min_Oabsorb4,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_min_nat @ A2 @ B2 )
= B2 ) ) ).
% min.absorb4
thf(fact_1061_min_Oabsorb3,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_min_nat @ A2 @ B2 )
= A2 ) ) ).
% min.absorb3
thf(fact_1062_Min__gr__iff,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_nat @ X2 @ ( lattic8721135487736765967in_nat @ A ) )
= ( ! [X3: nat] :
( ( member_nat2 @ X3 @ A )
=> ( ord_less_nat @ X2 @ X3 ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_1063_infinite__growing,axiom,
! [X5: set_nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X: nat] :
( ( member_nat2 @ X @ X5 )
=> ? [Xa: nat] :
( ( member_nat2 @ Xa @ X5 )
& ( ord_less_nat @ X @ Xa ) ) )
=> ~ ( finite_finite_nat @ X5 ) ) ) ).
% infinite_growing
thf(fact_1064_ex__min__if__finite,axiom,
! [S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ? [X: nat] :
( ( member_nat2 @ X @ S2 )
& ~ ? [Xa: nat] :
( ( member_nat2 @ Xa @ S2 )
& ( ord_less_nat @ Xa @ X ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1065_leD,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y ) ) ).
% leD
thf(fact_1066_leI,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% leI
thf(fact_1067_nless__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_nat @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_1068_antisym__conv1,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_1069_antisym__conv2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_1070_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_1071_not__le__imp__less,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y @ X2 )
=> ( ord_less_nat @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_1072_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1073_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1074_order_Ostrict__trans1,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_1075_order_Ostrict__trans2,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_1076_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1077_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1078_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1079_dual__order_Ostrict__trans1,axiom,
! [B2: nat,A2: nat,C2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C2 @ B2 )
=> ( ord_less_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_1080_dual__order_Ostrict__trans2,axiom,
! [B2: nat,A2: nat,C2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C2 @ B2 )
=> ( ord_less_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_1081_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1082_order_Ostrict__implies__order,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_1083_dual__order_Ostrict__implies__order,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_1084_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_1085_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_1086_linorder__not__le,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y ) )
= ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_1087_linorder__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_1088_order__less__imp__le,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_1089_order__le__neq__trans,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_1090_order__neq__le__trans,axiom,
! [A2: nat,B2: nat] :
( ( A2 != B2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_1091_order__le__less__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1092_order__less__le__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1093_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1094_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_1095_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1096_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_1097_linorder__le__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_1098_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1099_inf__pigeonhole__principle,axiom,
! [N2: nat,F: nat > nat > $o] :
( ! [K2: nat] :
? [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
& ( F @ K2 @ I3 ) )
=> ? [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
& ! [K3: nat] :
? [K4: nat] :
( ( ord_less_eq_nat @ K3 @ K4 )
& ( F @ K4 @ I2 ) ) ) ) ).
% inf_pigeonhole_principle
thf(fact_1100_sup_Ostrict__coboundedI2,axiom,
! [C2: nat,B2: nat,A2: nat] :
( ( ord_less_nat @ C2 @ B2 )
=> ( ord_less_nat @ C2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_1101_sup_Ostrict__coboundedI1,axiom,
! [C2: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ C2 @ A2 )
=> ( ord_less_nat @ C2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_1102_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( A3
= ( sup_sup_nat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_1103_sup_Ostrict__boundedE,axiom,
! [B2: nat,C2: nat,A2: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_nat @ B2 @ A2 )
=> ~ ( ord_less_nat @ C2 @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_1104_sup_Oabsorb4,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_1105_sup_Oabsorb3,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb3
thf(fact_1106_less__supI2,axiom,
! [X2: nat,B2: nat,A2: nat] :
( ( ord_less_nat @ X2 @ B2 )
=> ( ord_less_nat @ X2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_1107_less__supI1,axiom,
! [X2: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ X2 @ A2 )
=> ( ord_less_nat @ X2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_1108_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_1109_less__imp__neq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_1110_order_Oasym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order.asym
thf(fact_1111_ord__eq__less__trans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( A2 = B2 )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_1112_ord__less__eq__trans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_1113_less__induct,axiom,
! [P3: nat > $o,A2: nat] :
( ! [X: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X )
=> ( P3 @ Y5 ) )
=> ( P3 @ X ) )
=> ( P3 @ A2 ) ) ).
% less_induct
thf(fact_1114_antisym__conv3,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_nat @ Y @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_1115_linorder__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_1116_dual__order_Oasym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ~ ( ord_less_nat @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_1117_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_1118_exists__least__iff,axiom,
( ( ^ [P6: nat > $o] :
? [X7: nat] : ( P6 @ X7 ) )
= ( ^ [P7: nat > $o] :
? [N3: nat] :
( ( P7 @ N3 )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ~ ( P7 @ M3 ) ) ) ) ) ).
% exists_least_iff
thf(fact_1119_linorder__less__wlog,axiom,
! [P3: nat > nat > $o,A2: nat,B2: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( P3 @ A4 @ B4 ) )
=> ( ! [A4: nat] : ( P3 @ A4 @ A4 )
=> ( ! [A4: nat,B4: nat] :
( ( P3 @ B4 @ A4 )
=> ( P3 @ A4 @ B4 ) )
=> ( P3 @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_1120_order_Ostrict__trans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ord_less_nat @ A2 @ C2 ) ) ) ).
% order.strict_trans
thf(fact_1121_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ( ord_less_nat @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_1122_dual__order_Ostrict__trans,axiom,
! [B2: nat,A2: nat,C2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C2 @ B2 )
=> ( ord_less_nat @ C2 @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_1123_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_1124_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_1125_linorder__neqE,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_1126_order__less__asym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_1127_linorder__neq__iff,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
= ( ( ord_less_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_1128_order__less__asym_H,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_1129_order__less__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_1130_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_1131_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_1132_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_1133_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_1134_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_1135_order__less__not__sym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_1136_order__less__imp__triv,axiom,
! [X2: nat,Y: nat,P3: $o] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ X2 )
=> P3 ) ) ).
% order_less_imp_triv
thf(fact_1137_linorder__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_1138_order__less__imp__not__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_1139_order__less__imp__not__eq2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_1140_order__less__imp__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_1141_bot_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_1142_bot_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_1143_min_Ostrict__coboundedI2,axiom,
! [B2: nat,C2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ C2 )
=> ( ord_less_nat @ ( ord_min_nat @ A2 @ B2 ) @ C2 ) ) ).
% min.strict_coboundedI2
thf(fact_1144_min_Ostrict__coboundedI1,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ C2 )
=> ( ord_less_nat @ ( ord_min_nat @ A2 @ B2 ) @ C2 ) ) ).
% min.strict_coboundedI1
thf(fact_1145_min_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( A3
= ( ord_min_nat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% min.strict_order_iff
thf(fact_1146_min_Ostrict__boundedE,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ ( ord_min_nat @ B2 @ C2 ) )
=> ~ ( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ A2 @ C2 ) ) ) ).
% min.strict_boundedE
thf(fact_1147_min__less__iff__disj,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ ( ord_min_nat @ X2 @ Y ) @ Z2 )
= ( ( ord_less_nat @ X2 @ Z2 )
| ( ord_less_nat @ Y @ Z2 ) ) ) ).
% min_less_iff_disj
thf(fact_1148_less__infI1,axiom,
! [A2: nat,X2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ X2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X2 ) ) ).
% less_infI1
thf(fact_1149_less__infI2,axiom,
! [B2: nat,X2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ X2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X2 ) ) ).
% less_infI2
thf(fact_1150_inf_Oabsorb3,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb3
thf(fact_1151_inf_Oabsorb4,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb4
thf(fact_1152_inf_Ostrict__boundedE,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A2 @ ( inf_inf_nat @ B2 @ C2 ) )
=> ~ ( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ A2 @ C2 ) ) ) ).
% inf.strict_boundedE
thf(fact_1153_inf_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( A3
= ( inf_inf_nat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_1154_inf_Ostrict__coboundedI1,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ C2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.strict_coboundedI1
thf(fact_1155_inf_Ostrict__coboundedI2,axiom,
! [B2: nat,C2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ C2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.strict_coboundedI2
thf(fact_1156_finite__linorder__max__induct,axiom,
! [A: set_nat,P3: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P3 @ bot_bot_set_nat )
=> ( ! [B4: nat,A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ! [X6: nat] :
( ( member_nat2 @ X6 @ A8 )
=> ( ord_less_nat @ X6 @ B4 ) )
=> ( ( P3 @ A8 )
=> ( P3 @ ( insert_nat2 @ B4 @ A8 ) ) ) ) )
=> ( P3 @ A ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_1157_finite__linorder__min__induct,axiom,
! [A: set_nat,P3: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P3 @ bot_bot_set_nat )
=> ( ! [B4: nat,A8: set_nat] :
( ( finite_finite_nat @ A8 )
=> ( ! [X6: nat] :
( ( member_nat2 @ X6 @ A8 )
=> ( ord_less_nat @ B4 @ X6 ) )
=> ( ( P3 @ A8 )
=> ( P3 @ ( insert_nat2 @ B4 @ A8 ) ) ) ) )
=> ( P3 @ A ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_1158_Min__less__iff,axiom,
! [A: set_nat,X2: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_nat @ ( lattic8721135487736765967in_nat @ A ) @ X2 )
= ( ? [X3: nat] :
( ( member_nat2 @ X3 @ A )
& ( ord_less_nat @ X3 @ X2 ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_1159_append__Cons__nth__right,axiom,
! [Xs: list_nat,I: nat,U: nat,Ys: list_nat,Z2: nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Xs ) @ I )
=> ( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ U @ Ys ) ) @ I )
= ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ Z2 @ Ys ) ) @ I ) ) ) ).
% append_Cons_nth_right
thf(fact_1160_append__Cons__nth__left,axiom,
! [I: nat,Xs: list_nat,U: nat,Ys: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ U @ Ys ) ) @ I )
= ( nth_nat @ Xs @ I ) ) ) ).
% append_Cons_nth_left
thf(fact_1161_min__list__nth,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) )
=> ( ord_less_eq_nat @ ( missing_min_list_nat @ Xs ) @ ( missing_min_list_nat @ Ys ) ) ) ) ).
% min_list_nth
thf(fact_1162_replace__term__at_Oelims,axiom,
! [X2: term_a_b,Xa2: list_nat,Xb: term_a_b,Y: term_a_b] :
( ( ( term_r6860082780075436317at_a_b @ X2 @ Xa2 @ Xb )
= Y )
=> ( ( ( Xa2 = nil_nat )
=> ( Y != Xb ) )
=> ( ! [X: b] :
( ( X2
= ( var_b_a @ X ) )
=> ( ? [V2: nat,Va: list_nat] :
( Xa2
= ( cons_nat @ V2 @ Va ) )
=> ( Y
!= ( var_b_a @ X ) ) ) )
=> ~ ! [F5: a,Ts2: list_term_a_b] :
( ( X2
= ( fun_a_b @ F5 @ Ts2 ) )
=> ! [I2: nat,Ps2: list_nat] :
( ( Xa2
= ( cons_nat @ I2 @ Ps2 ) )
=> ~ ( ( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ts2 ) )
=> ( Y
= ( fun_a_b @ F5 @ ( list_update_term_a_b @ Ts2 @ I2 @ ( term_r6860082780075436317at_a_b @ ( nth_term_a_b @ Ts2 @ I2 ) @ Ps2 @ Xb ) ) ) ) )
& ( ~ ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ts2 ) )
=> ( Y
= ( fun_a_b @ F5 @ Ts2 ) ) ) ) ) ) ) ) ) ).
% replace_term_at.elims
thf(fact_1163_poss__Cons__poss,axiom,
! [I: nat,Q: list_nat,T: term_a_b] :
( ( member_list_nat2 @ ( cons_nat @ I @ Q ) @ ( term_poss_a_b @ T ) )
= ( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ ( args_a_b @ T ) ) )
& ( member_list_nat2 @ Q @ ( term_poss_a_b @ ( nth_term_a_b @ ( args_a_b @ T ) @ I ) ) ) ) ) ).
% poss_Cons_poss
thf(fact_1164_list__update__nonempty,axiom,
! [Xs: list_nat,K: nat,X2: nat] :
( ( ( list_update_nat @ Xs @ K @ X2 )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% list_update_nonempty
thf(fact_1165_list__update__length,axiom,
! [Xs: list_nat,X2: nat,Ys: list_nat,Y: nat] :
( ( list_update_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ Ys ) ) @ ( size_size_list_nat @ Xs ) @ Y )
= ( append_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_1166_psubset__imp__ex__mem,axiom,
! [A: set_list_nat,B: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A @ B )
=> ? [B4: list_nat] : ( member_list_nat2 @ B4 @ ( minus_7954133019191499631st_nat @ B @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1167_list__update_Osimps_I1_J,axiom,
! [I: nat,V: nat] :
( ( list_update_nat @ nil_nat @ I @ V )
= nil_nat ) ).
% list_update.simps(1)
thf(fact_1168_list__update__code_I1_J,axiom,
! [I: nat,Y: nat] :
( ( list_update_nat @ nil_nat @ I @ Y )
= nil_nat ) ).
% list_update_code(1)
thf(fact_1169_psubset__insert__iff,axiom,
! [A: set_list_nat,X2: list_nat,B: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A @ ( insert_list_nat2 @ X2 @ B ) )
= ( ( ( member_list_nat2 @ X2 @ B )
=> ( ord_le1190675801316882794st_nat @ A @ B ) )
& ( ~ ( member_list_nat2 @ X2 @ B )
=> ( ( ( member_list_nat2 @ X2 @ A )
=> ( ord_le1190675801316882794st_nat @ ( minus_7954133019191499631st_nat @ A @ ( insert_list_nat2 @ X2 @ bot_bot_set_list_nat ) ) @ B ) )
& ( ~ ( member_list_nat2 @ X2 @ A )
=> ( ord_le6045566169113846134st_nat @ A @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1170_replace__term__at_Osimps_I3_J,axiom,
! [I: nat,Ts: list_term_a_b,F: a,Ps: list_nat,T: term_a_b] :
( ( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Ts ) )
=> ( ( term_r6860082780075436317at_a_b @ ( fun_a_b @ F @ Ts ) @ ( cons_nat @ I @ Ps ) @ T )
= ( fun_a_b @ F @ ( list_update_term_a_b @ Ts @ I @ ( term_r6860082780075436317at_a_b @ ( nth_term_a_b @ Ts @ I ) @ Ps @ T ) ) ) ) )
& ( ~ ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Ts ) )
=> ( ( term_r6860082780075436317at_a_b @ ( fun_a_b @ F @ Ts ) @ ( cons_nat @ I @ Ps ) @ T )
= ( fun_a_b @ F @ Ts ) ) ) ) ).
% replace_term_at.simps(3)
thf(fact_1171_replace__term__at_Opelims,axiom,
! [X2: term_a_b,Xa2: list_nat,Xb: term_a_b,Y: term_a_b] :
( ( ( term_r6860082780075436317at_a_b @ X2 @ Xa2 @ Xb )
= Y )
=> ( ( accp_P2729577386226225901rm_a_b @ term_r1280879029893354718el_a_b @ ( produc3812856575676843240rm_a_b @ X2 @ ( produc5151171985953862413rm_a_b @ Xa2 @ Xb ) ) )
=> ( ( ( Xa2 = nil_nat )
=> ( ( Y = Xb )
=> ~ ( accp_P2729577386226225901rm_a_b @ term_r1280879029893354718el_a_b @ ( produc3812856575676843240rm_a_b @ X2 @ ( produc5151171985953862413rm_a_b @ nil_nat @ Xb ) ) ) ) )
=> ( ! [X: b] :
( ( X2
= ( var_b_a @ X ) )
=> ! [V2: nat,Va: list_nat] :
( ( Xa2
= ( cons_nat @ V2 @ Va ) )
=> ( ( Y
= ( var_b_a @ X ) )
=> ~ ( accp_P2729577386226225901rm_a_b @ term_r1280879029893354718el_a_b @ ( produc3812856575676843240rm_a_b @ ( var_b_a @ X ) @ ( produc5151171985953862413rm_a_b @ ( cons_nat @ V2 @ Va ) @ Xb ) ) ) ) ) )
=> ~ ! [F5: a,Ts2: list_term_a_b] :
( ( X2
= ( fun_a_b @ F5 @ Ts2 ) )
=> ! [I2: nat,Ps2: list_nat] :
( ( Xa2
= ( cons_nat @ I2 @ Ps2 ) )
=> ( ( ( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ts2 ) )
=> ( Y
= ( fun_a_b @ F5 @ ( list_update_term_a_b @ Ts2 @ I2 @ ( term_r6860082780075436317at_a_b @ ( nth_term_a_b @ Ts2 @ I2 ) @ Ps2 @ Xb ) ) ) ) )
& ( ~ ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ts2 ) )
=> ( Y
= ( fun_a_b @ F5 @ Ts2 ) ) ) )
=> ~ ( accp_P2729577386226225901rm_a_b @ term_r1280879029893354718el_a_b @ ( produc3812856575676843240rm_a_b @ ( fun_a_b @ F5 @ Ts2 ) @ ( produc5151171985953862413rm_a_b @ ( cons_nat @ I2 @ Ps2 ) @ Xb ) ) ) ) ) ) ) ) ) ) ).
% replace_term_at.pelims
thf(fact_1172_psubsetD,axiom,
! [A: set_list_nat,B: set_list_nat,C2: list_nat] :
( ( ord_le1190675801316882794st_nat @ A @ B )
=> ( ( member_list_nat2 @ C2 @ A )
=> ( member_list_nat2 @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_1173_fun__at__None__nposs__iff,axiom,
! [T: term_a_b,P: list_nat] :
( ( ( term_fun_at_a_b @ T @ P )
= none_Sum_sum_a_b )
= ( ~ ( member_list_nat2 @ P @ ( term_poss_a_b @ T ) ) ) ) ).
% fun_at_None_nposs_iff
thf(fact_1174_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X6 ) ) ).
% minf(8)
thf(fact_1175_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ord_less_eq_nat @ X6 @ T ) ) ).
% minf(6)
thf(fact_1176_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ~ ( ord_less_eq_nat @ X6 @ T ) ) ).
% pinf(6)
thf(fact_1177_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ord_less_eq_nat @ T @ X6 ) ) ).
% pinf(8)
thf(fact_1178_complete__interval,axiom,
! [A2: nat,B2: nat,P3: nat > $o] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( P3 @ A2 )
=> ( ~ ( P3 @ B2 )
=> ? [C4: nat] :
( ( ord_less_eq_nat @ A2 @ C4 )
& ( ord_less_eq_nat @ C4 @ B2 )
& ! [X6: nat] :
( ( ( ord_less_eq_nat @ A2 @ X6 )
& ( ord_less_nat @ X6 @ C4 ) )
=> ( P3 @ X6 ) )
& ! [D3: nat] :
( ! [X: nat] :
( ( ( ord_less_eq_nat @ A2 @ X )
& ( ord_less_nat @ X @ D3 ) )
=> ( P3 @ X ) )
=> ( ord_less_eq_nat @ D3 @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1179_verit__comp__simplify1_I3_J,axiom,
! [B9: nat,A9: nat] :
( ( ~ ( ord_less_eq_nat @ B9 @ A9 ) )
= ( ord_less_nat @ A9 @ B9 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1180_verit__la__disequality,axiom,
! [A2: nat,B2: nat] :
( ( A2 = B2 )
| ~ ( ord_less_eq_nat @ A2 @ B2 )
| ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_1181_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_1182_subset__Image1__Image1__iff,axiom,
! [R4: set_Pr3451248702717554689st_nat,A2: list_nat,B2: list_nat] :
( ( order_4596338698039041673st_nat @ ( field_list_nat @ R4 ) @ R4 )
=> ( ( member_list_nat2 @ A2 @ ( field_list_nat @ R4 ) )
=> ( ( member_list_nat2 @ B2 @ ( field_list_nat @ R4 ) )
=> ( ( ord_le6045566169113846134st_nat @ ( image_2597627202720054805st_nat @ R4 @ ( insert_list_nat2 @ A2 @ bot_bot_set_list_nat ) ) @ ( image_2597627202720054805st_nat @ R4 @ ( insert_list_nat2 @ B2 @ bot_bot_set_list_nat ) ) )
= ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ B2 @ A2 ) @ R4 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_1183_nth__append__Cons,axiom,
! [I: nat,Xs: list_nat,Y: nat,Zs: list_nat] :
( ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ Y @ Zs ) ) @ I )
= ( nth_nat @ Xs @ I ) ) )
& ( ~ ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ( I
= ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ Y @ Zs ) ) @ I )
= Y ) )
& ( ( I
!= ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ Y @ Zs ) ) @ I )
= ( nth_nat @ Zs @ ( minus_minus_nat @ I @ ( suc @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ) ) ) ).
% nth_append_Cons
thf(fact_1184_length__Suc__conv__rev,axiom,
! [Xs: list_nat,N2: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N2 ) )
= ( ? [Y4: nat,Ys3: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ Y4 @ nil_nat ) ) )
& ( ( size_size_list_nat @ Ys3 )
= N2 ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_1185_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_1186_lift__Suc__mono__le,axiom,
! [F: nat > nat,N2: nat,N4: nat] :
( ! [N5: nat] : ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1187_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N2: nat,N4: nat] :
( ! [N5: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N5 ) ) @ ( F @ N5 ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1188_replace__term__context__at_Osimps_I2_J,axiom,
! [I: nat,Ss2: list_term_a_b,F: a,C: subterm_and_ctxt_a_b,Ts: list_term_a_b,Ps: list_nat,U: term_a_b] :
( ( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Ss2 ) )
=> ( ( terms_4774307173741787698at_a_b @ ( subterm_and_More_a_b @ F @ Ss2 @ C @ Ts ) @ ( cons_nat @ I @ Ps ) @ U )
= ( subterm_and_More_a_b @ F @ ( list_update_term_a_b @ Ss2 @ I @ ( term_r6860082780075436317at_a_b @ ( nth_term_a_b @ Ss2 @ I ) @ Ps @ U ) ) @ C @ Ts ) ) )
& ( ~ ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Ss2 ) )
=> ( ( ( I
= ( size_s8906293707977694520rm_a_b @ Ss2 ) )
=> ( ( terms_4774307173741787698at_a_b @ ( subterm_and_More_a_b @ F @ Ss2 @ C @ Ts ) @ ( cons_nat @ I @ Ps ) @ U )
= ( subterm_and_More_a_b @ F @ Ss2 @ ( terms_4774307173741787698at_a_b @ C @ Ps @ U ) @ Ts ) ) )
& ( ( I
!= ( size_s8906293707977694520rm_a_b @ Ss2 ) )
=> ( ( terms_4774307173741787698at_a_b @ ( subterm_and_More_a_b @ F @ Ss2 @ C @ Ts ) @ ( cons_nat @ I @ Ps ) @ U )
= ( subterm_and_More_a_b @ F @ Ss2 @ C @ ( list_update_term_a_b @ Ts @ ( minus_minus_nat @ I @ ( suc @ ( size_s8906293707977694520rm_a_b @ Ss2 ) ) ) @ ( term_r6860082780075436317at_a_b @ ( nth_term_a_b @ Ts @ ( minus_minus_nat @ I @ ( suc @ ( size_s8906293707977694520rm_a_b @ Ss2 ) ) ) ) @ Ps @ U ) ) ) ) ) ) ) ) ).
% replace_term_context_at.simps(2)
thf(fact_1189_take__Suc__conv__app__nth,axiom,
! [I: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( take_nat @ ( suc @ I ) @ Xs )
= ( append_nat @ ( take_nat @ I @ Xs ) @ ( cons_nat @ ( nth_nat @ Xs @ I ) @ nil_nat ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_1190_take__Nil,axiom,
! [N2: nat] :
( ( take_nat @ N2 @ nil_nat )
= nil_nat ) ).
% take_Nil
thf(fact_1191_nth__append__take,axiom,
! [I: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ord_less_eq_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ ( take_nat @ I @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ I )
= Y ) ) ).
% nth_append_take
thf(fact_1192_replace__term__context__at_Oelims,axiom,
! [X2: subterm_and_ctxt_a_b,Xa2: list_nat,Xb: term_a_b,Y: subterm_and_ctxt_a_b] :
( ( ( terms_4774307173741787698at_a_b @ X2 @ Xa2 @ Xb )
= Y )
=> ( ( ( X2 = subterm_and_Hole_a_b )
=> ( Y != subterm_and_Hole_a_b ) )
=> ( ! [F5: a,Ss: list_term_a_b,C5: subterm_and_ctxt_a_b,Ts2: list_term_a_b] :
( ( X2
= ( subterm_and_More_a_b @ F5 @ Ss @ C5 @ Ts2 ) )
=> ! [I2: nat,Ps2: list_nat] :
( ( Xa2
= ( cons_nat @ I2 @ Ps2 ) )
=> ~ ( ( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ss ) )
=> ( Y
= ( subterm_and_More_a_b @ F5 @ ( list_update_term_a_b @ Ss @ I2 @ ( term_r6860082780075436317at_a_b @ ( nth_term_a_b @ Ss @ I2 ) @ Ps2 @ Xb ) ) @ C5 @ Ts2 ) ) )
& ( ~ ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ss ) )
=> ( ( ( I2
= ( size_s8906293707977694520rm_a_b @ Ss ) )
=> ( Y
= ( subterm_and_More_a_b @ F5 @ Ss @ ( terms_4774307173741787698at_a_b @ C5 @ Ps2 @ Xb ) @ Ts2 ) ) )
& ( ( I2
!= ( size_s8906293707977694520rm_a_b @ Ss ) )
=> ( Y
= ( subterm_and_More_a_b @ F5 @ Ss @ C5 @ ( list_update_term_a_b @ Ts2 @ ( minus_minus_nat @ I2 @ ( suc @ ( size_s8906293707977694520rm_a_b @ Ss ) ) ) @ ( term_r6860082780075436317at_a_b @ ( nth_term_a_b @ Ts2 @ ( minus_minus_nat @ I2 @ ( suc @ ( size_s8906293707977694520rm_a_b @ Ss ) ) ) ) @ Ps2 @ Xb ) ) ) ) ) ) ) ) ) )
=> ~ ( ? [V2: a,Va: list_term_a_b,Vb: subterm_and_ctxt_a_b,Vc: list_term_a_b] :
( X2
= ( subterm_and_More_a_b @ V2 @ Va @ Vb @ Vc ) )
=> ( ( Xa2 = nil_nat )
=> ( Y != undefi3573907640150307307xt_a_b ) ) ) ) ) ) ).
% replace_term_context_at.elims
thf(fact_1193_nth__append__drop__is__nth__conv,axiom,
! [J: nat,I: nat,Xs: list_nat,Y: nat] :
( ( ord_less_nat @ J @ I )
=> ( ( ord_less_eq_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ ( take_nat @ J @ Xs ) @ ( cons_nat @ Y @ ( drop_nat @ ( suc @ J ) @ Xs ) ) ) @ I )
= ( nth_nat @ Xs @ I ) ) ) ) ) ).
% nth_append_drop_is_nth_conv
thf(fact_1194_drop__all,axiom,
! [Xs: list_nat,N2: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N2 )
=> ( ( drop_nat @ N2 @ Xs )
= nil_nat ) ) ).
% drop_all
thf(fact_1195_drop__eq__Nil,axiom,
! [N2: nat,Xs: list_nat] :
( ( ( drop_nat @ N2 @ Xs )
= nil_nat )
= ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N2 ) ) ).
% drop_eq_Nil
thf(fact_1196_drop__eq__Nil2,axiom,
! [N2: nat,Xs: list_nat] :
( ( nil_nat
= ( drop_nat @ N2 @ Xs ) )
= ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N2 ) ) ).
% drop_eq_Nil2
thf(fact_1197_drop__Nil,axiom,
! [N2: nat] :
( ( drop_nat @ N2 @ nil_nat )
= nil_nat ) ).
% drop_Nil
thf(fact_1198_take__drop__imp__nth,axiom,
! [I: nat,Ss2: list_nat,X2: nat] :
( ( ( append_nat @ ( take_nat @ I @ Ss2 ) @ ( cons_nat @ X2 @ ( drop_nat @ ( suc @ I ) @ Ss2 ) ) )
= Ss2 )
=> ( X2
= ( nth_nat @ Ss2 @ I ) ) ) ).
% take_drop_imp_nth
thf(fact_1199_id__take__nth__drop,axiom,
! [I: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( Xs
= ( append_nat @ ( take_nat @ I @ Xs ) @ ( cons_nat @ ( nth_nat @ Xs @ I ) @ ( drop_nat @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_1200_nth__append__take__drop__is__nth__conv,axiom,
! [I: nat,Xs: list_nat,J: nat,Y: nat] :
( ( ord_less_eq_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( I != J )
=> ( ( nth_nat @ ( append_nat @ ( take_nat @ J @ Xs ) @ ( cons_nat @ Y @ ( drop_nat @ ( suc @ J ) @ Xs ) ) ) @ I )
= ( nth_nat @ Xs @ I ) ) ) ) ) ).
% nth_append_take_drop_is_nth_conv
thf(fact_1201_upd__conv__take__nth__drop,axiom,
! [I: nat,Xs: list_nat,A2: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ Xs @ I @ A2 )
= ( append_nat @ ( take_nat @ I @ Xs ) @ ( cons_nat @ A2 @ ( drop_nat @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_1202_replace__term__context__at_Opelims,axiom,
! [X2: subterm_and_ctxt_a_b,Xa2: list_nat,Xb: term_a_b,Y: subterm_and_ctxt_a_b] :
( ( ( terms_4774307173741787698at_a_b @ X2 @ Xa2 @ Xb )
= Y )
=> ( ( accp_P6624108184151110466rm_a_b @ terms_6690452295706263049el_a_b @ ( produc1096132875450744233rm_a_b @ X2 @ ( produc5151171985953862413rm_a_b @ Xa2 @ Xb ) ) )
=> ( ( ( X2 = subterm_and_Hole_a_b )
=> ( ( Y = subterm_and_Hole_a_b )
=> ~ ( accp_P6624108184151110466rm_a_b @ terms_6690452295706263049el_a_b @ ( produc1096132875450744233rm_a_b @ subterm_and_Hole_a_b @ ( produc5151171985953862413rm_a_b @ Xa2 @ Xb ) ) ) ) )
=> ( ! [F5: a,Ss: list_term_a_b,C5: subterm_and_ctxt_a_b,Ts2: list_term_a_b] :
( ( X2
= ( subterm_and_More_a_b @ F5 @ Ss @ C5 @ Ts2 ) )
=> ! [I2: nat,Ps2: list_nat] :
( ( Xa2
= ( cons_nat @ I2 @ Ps2 ) )
=> ( ( ( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ss ) )
=> ( Y
= ( subterm_and_More_a_b @ F5 @ ( list_update_term_a_b @ Ss @ I2 @ ( term_r6860082780075436317at_a_b @ ( nth_term_a_b @ Ss @ I2 ) @ Ps2 @ Xb ) ) @ C5 @ Ts2 ) ) )
& ( ~ ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ Ss ) )
=> ( ( ( I2
= ( size_s8906293707977694520rm_a_b @ Ss ) )
=> ( Y
= ( subterm_and_More_a_b @ F5 @ Ss @ ( terms_4774307173741787698at_a_b @ C5 @ Ps2 @ Xb ) @ Ts2 ) ) )
& ( ( I2
!= ( size_s8906293707977694520rm_a_b @ Ss ) )
=> ( Y
= ( subterm_and_More_a_b @ F5 @ Ss @ C5 @ ( list_update_term_a_b @ Ts2 @ ( minus_minus_nat @ I2 @ ( suc @ ( size_s8906293707977694520rm_a_b @ Ss ) ) ) @ ( term_r6860082780075436317at_a_b @ ( nth_term_a_b @ Ts2 @ ( minus_minus_nat @ I2 @ ( suc @ ( size_s8906293707977694520rm_a_b @ Ss ) ) ) ) @ Ps2 @ Xb ) ) ) ) ) ) ) )
=> ~ ( accp_P6624108184151110466rm_a_b @ terms_6690452295706263049el_a_b @ ( produc1096132875450744233rm_a_b @ ( subterm_and_More_a_b @ F5 @ Ss @ C5 @ Ts2 ) @ ( produc5151171985953862413rm_a_b @ ( cons_nat @ I2 @ Ps2 ) @ Xb ) ) ) ) ) )
=> ~ ! [V2: a,Va: list_term_a_b,Vb: subterm_and_ctxt_a_b,Vc: list_term_a_b] :
( ( X2
= ( subterm_and_More_a_b @ V2 @ Va @ Vb @ Vc ) )
=> ( ( Xa2 = nil_nat )
=> ( ( Y = undefi3573907640150307307xt_a_b )
=> ~ ( accp_P6624108184151110466rm_a_b @ terms_6690452295706263049el_a_b @ ( produc1096132875450744233rm_a_b @ ( subterm_and_More_a_b @ V2 @ Va @ Vb @ Vc ) @ ( produc5151171985953862413rm_a_b @ nil_nat @ Xb ) ) ) ) ) ) ) ) ) ) ).
% replace_term_context_at.pelims
thf(fact_1203_take__hd__drop,axiom,
! [N2: nat,Xs: list_nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( ( append_nat @ ( take_nat @ N2 @ Xs ) @ ( cons_nat @ ( hd_nat @ ( drop_nat @ N2 @ Xs ) ) @ nil_nat ) )
= ( take_nat @ ( suc @ N2 ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_1204_add__elem__list__listsE,axiom,
! [Ys: list_nat,X2: nat,Xs: list_nat] :
( ( member_list_nat2 @ Ys @ ( set_list_nat2 @ ( basic_4874698711677410535ts_nat @ X2 @ Xs ) ) )
=> ? [N5: nat] :
( ( ord_less_eq_nat @ N5 @ ( size_size_list_nat @ Xs ) )
& ( Ys
= ( append_nat @ ( take_nat @ N5 @ Xs ) @ ( cons_nat @ X2 @ ( drop_nat @ N5 @ Xs ) ) ) ) ) ) ).
% add_elem_list_listsE
thf(fact_1205_set__empty2,axiom,
! [Xs: list_nat] :
( ( bot_bot_set_nat
= ( set_nat2 @ Xs ) )
= ( Xs = nil_nat ) ) ).
% set_empty2
thf(fact_1206_set__empty,axiom,
! [Xs: list_nat] :
( ( ( set_nat2 @ Xs )
= bot_bot_set_nat )
= ( Xs = nil_nat ) ) ).
% set_empty
thf(fact_1207_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_1208_concat__eq__Nil__conv,axiom,
! [Xss2: list_list_nat] :
( ( ( concat_nat @ Xss2 )
= nil_nat )
= ( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xss2 ) )
=> ( X3 = nil_nat ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_1209_Nil__eq__concat__conv,axiom,
! [Xss2: list_list_nat] :
( ( nil_nat
= ( concat_nat @ Xss2 ) )
= ( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xss2 ) )
=> ( X3 = nil_nat ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_1210_remove1__split,axiom,
! [A2: list_nat,Xs: list_list_nat,Ys: list_list_nat] :
( ( member_list_nat2 @ A2 @ ( set_list_nat2 @ Xs ) )
=> ( ( ( remove1_list_nat @ A2 @ Xs )
= Ys )
= ( ? [Ls: list_list_nat,Rs: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ls @ ( cons_list_nat @ A2 @ Rs ) ) )
& ~ ( member_list_nat2 @ A2 @ ( set_list_nat2 @ Ls ) )
& ( Ys
= ( append_list_nat @ Ls @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_1211_remove1__split,axiom,
! [A2: nat,Xs: list_nat,Ys: list_nat] :
( ( member_nat2 @ A2 @ ( set_nat2 @ Xs ) )
=> ( ( ( remove1_nat @ A2 @ Xs )
= Ys )
= ( ? [Ls: list_nat,Rs: list_nat] :
( ( Xs
= ( append_nat @ Ls @ ( cons_nat @ A2 @ Rs ) ) )
& ~ ( member_nat2 @ A2 @ ( set_nat2 @ Ls ) )
& ( Ys
= ( append_nat @ Ls @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_1212_min__list,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ ( missing_min_list_nat @ Xs ) @ X2 ) ) ).
% min_list
thf(fact_1213_min__list__ex,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( member_nat2 @ ( missing_min_list_nat @ Xs ) @ ( set_nat2 @ Xs ) ) ) ).
% min_list_ex
thf(fact_1214_empty__subseqs,axiom,
! [Xs: list_nat] : ( member_list_nat2 @ nil_nat @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) ) ).
% empty_subseqs
thf(fact_1215_empty__set,axiom,
( bot_bot_set_nat
= ( set_nat2 @ nil_nat ) ) ).
% empty_set
thf(fact_1216_hd__Nil__eq__last,axiom,
( ( hd_nat @ nil_nat )
= ( last_nat @ nil_nat ) ) ).
% hd_Nil_eq_last
thf(fact_1217_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_1218_hd__in__set,axiom,
! [Xs: list_list_nat] :
( ( Xs != nil_list_nat )
=> ( member_list_nat2 @ ( hd_list_nat @ Xs ) @ ( set_list_nat2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_1219_hd__in__set,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( member_nat2 @ ( hd_nat @ Xs ) @ ( set_nat2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_1220_list_Oset__sel_I1_J,axiom,
! [A2: list_list_nat] :
( ( A2 != nil_list_nat )
=> ( member_list_nat2 @ ( hd_list_nat @ A2 ) @ ( set_list_nat2 @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_1221_list_Oset__sel_I1_J,axiom,
! [A2: list_nat] :
( ( A2 != nil_nat )
=> ( member_nat2 @ ( hd_nat @ A2 ) @ ( set_nat2 @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_1222_last__in__set,axiom,
! [As2: list_list_nat] :
( ( As2 != nil_list_nat )
=> ( member_list_nat2 @ ( last_list_nat @ As2 ) @ ( set_list_nat2 @ As2 ) ) ) ).
% last_in_set
thf(fact_1223_last__in__set,axiom,
! [As2: list_nat] :
( ( As2 != nil_nat )
=> ( member_nat2 @ ( last_nat @ As2 ) @ ( set_nat2 @ As2 ) ) ) ).
% last_in_set
thf(fact_1224_subset__code_I1_J,axiom,
! [Xs: list_list_nat,B: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ B )
= ( ! [X3: list_nat] :
( ( member_list_nat2 @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( member_list_nat2 @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_1225_set__subset__Cons,axiom,
! [Xs: list_nat,X2: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_1226_set__update__subsetI,axiom,
! [Xs: list_list_nat,A: set_list_nat,X2: list_nat,I: nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ A )
=> ( ( member_list_nat2 @ X2 @ A )
=> ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ ( list_update_list_nat @ Xs @ I @ X2 ) ) @ A ) ) ) ).
% set_update_subsetI
thf(fact_1227_split__list__first__prop__iff,axiom,
! [Xs: list_nat,P3: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
& ( P3 @ X3 ) ) )
= ( ? [Ys3: list_nat,X3: nat] :
( ? [Zs3: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs3 ) ) )
& ( P3 @ X3 )
& ! [Y4: nat] :
( ( member_nat2 @ Y4 @ ( set_nat2 @ Ys3 ) )
=> ~ ( P3 @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_1228_split__list__last__prop__iff,axiom,
! [Xs: list_nat,P3: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
& ( P3 @ X3 ) ) )
= ( ? [Ys3: list_nat,X3: nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs3 ) ) )
& ( P3 @ X3 )
& ! [Y4: nat] :
( ( member_nat2 @ Y4 @ ( set_nat2 @ Zs3 ) )
=> ~ ( P3 @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_1229_in__set__conv__decomp__first,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
= ( ? [Ys3: list_list_nat,Zs3: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys3 @ ( cons_list_nat @ X2 @ Zs3 ) ) )
& ~ ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1230_in__set__conv__decomp__first,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [Ys3: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X2 @ Zs3 ) ) )
& ~ ( member_nat2 @ X2 @ ( set_nat2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1231_in__set__conv__decomp__last,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
= ( ? [Ys3: list_list_nat,Zs3: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys3 @ ( cons_list_nat @ X2 @ Zs3 ) ) )
& ~ ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1232_in__set__conv__decomp__last,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [Ys3: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X2 @ Zs3 ) ) )
& ~ ( member_nat2 @ X2 @ ( set_nat2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1233_split__list__first__propE,axiom,
! [Xs: list_nat,P3: nat > $o] :
( ? [X6: nat] :
( ( member_nat2 @ X6 @ ( set_nat2 @ Xs ) )
& ( P3 @ X6 ) )
=> ~ ! [Ys2: list_nat,X: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs2 ) ) )
=> ( ( P3 @ X )
=> ~ ! [Xa: nat] :
( ( member_nat2 @ Xa @ ( set_nat2 @ Ys2 ) )
=> ~ ( P3 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_1234_split__list__last__propE,axiom,
! [Xs: list_nat,P3: nat > $o] :
( ? [X6: nat] :
( ( member_nat2 @ X6 @ ( set_nat2 @ Xs ) )
& ( P3 @ X6 ) )
=> ~ ! [Ys2: list_nat,X: nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs2 ) ) )
=> ( ( P3 @ X )
=> ~ ! [Xa: nat] :
( ( member_nat2 @ Xa @ ( set_nat2 @ Zs2 ) )
=> ~ ( P3 @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_1235_split__list__first__prop,axiom,
! [Xs: list_nat,P3: nat > $o] :
( ? [X6: nat] :
( ( member_nat2 @ X6 @ ( set_nat2 @ Xs ) )
& ( P3 @ X6 ) )
=> ? [Ys2: list_nat,X: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs2 ) ) )
& ( P3 @ X )
& ! [Xa: nat] :
( ( member_nat2 @ Xa @ ( set_nat2 @ Ys2 ) )
=> ~ ( P3 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_1236_split__list__last__prop,axiom,
! [Xs: list_nat,P3: nat > $o] :
( ? [X6: nat] :
( ( member_nat2 @ X6 @ ( set_nat2 @ Xs ) )
& ( P3 @ X6 ) )
=> ? [Ys2: list_nat,X: nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs2 ) ) )
& ( P3 @ X )
& ! [Xa: nat] :
( ( member_nat2 @ Xa @ ( set_nat2 @ Zs2 ) )
=> ~ ( P3 @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_1237_in__set__conv__decomp,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
= ( ? [Ys3: list_list_nat,Zs3: list_list_nat] :
( Xs
= ( append_list_nat @ Ys3 @ ( cons_list_nat @ X2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1238_in__set__conv__decomp,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [Ys3: list_nat,Zs3: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1239_append__Cons__eq__iff,axiom,
! [X2: list_nat,Xs: list_list_nat,Ys: list_list_nat,Xs6: list_list_nat,Ys6: list_list_nat] :
( ~ ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ~ ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Ys ) )
=> ( ( ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ Ys ) )
= ( append_list_nat @ Xs6 @ ( cons_list_nat @ X2 @ Ys6 ) ) )
= ( ( Xs = Xs6 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1240_append__Cons__eq__iff,axiom,
! [X2: nat,Xs: list_nat,Ys: list_nat,Xs6: list_nat,Ys6: list_nat] :
( ~ ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ( ~ ( member_nat2 @ X2 @ ( set_nat2 @ Ys ) )
=> ( ( ( append_nat @ Xs @ ( cons_nat @ X2 @ Ys ) )
= ( append_nat @ Xs6 @ ( cons_nat @ X2 @ Ys6 ) ) )
= ( ( Xs = Xs6 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1241_split__list__propE,axiom,
! [Xs: list_nat,P3: nat > $o] :
( ? [X6: nat] :
( ( member_nat2 @ X6 @ ( set_nat2 @ Xs ) )
& ( P3 @ X6 ) )
=> ~ ! [Ys2: list_nat,X: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs2 ) ) )
=> ~ ( P3 @ X ) ) ) ).
% split_list_propE
thf(fact_1242_split__list__first,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
=> ? [Ys2: list_list_nat,Zs2: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X2 @ Zs2 ) ) )
& ~ ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_1243_split__list__first,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ? [Ys2: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs2 ) ) )
& ~ ( member_nat2 @ X2 @ ( set_nat2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_1244_split__list__prop,axiom,
! [Xs: list_nat,P3: nat > $o] :
( ? [X6: nat] :
( ( member_nat2 @ X6 @ ( set_nat2 @ Xs ) )
& ( P3 @ X6 ) )
=> ? [Ys2: list_nat,X: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs2 ) ) )
& ( P3 @ X ) ) ) ).
% split_list_prop
thf(fact_1245_split__list__last,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
=> ? [Ys2: list_list_nat,Zs2: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X2 @ Zs2 ) ) )
& ~ ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_1246_split__list__last,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ? [Ys2: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs2 ) ) )
& ~ ( member_nat2 @ X2 @ ( set_nat2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_1247_split__list,axiom,
! [X2: list_nat,Xs: list_list_nat] :
( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
=> ? [Ys2: list_list_nat,Zs2: list_list_nat] :
( Xs
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_1248_split__list,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
=> ? [Ys2: list_nat,Zs2: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_1249_longest__common__prefix,axiom,
! [Xs: list_nat,Ys: list_nat] :
? [Ps2: list_nat,Xs5: list_nat,Ys5: list_nat] :
( ( Xs
= ( append_nat @ Ps2 @ Xs5 ) )
& ( Ys
= ( append_nat @ Ps2 @ Ys5 ) )
& ( ( Xs5 = nil_nat )
| ( Ys5 = nil_nat )
| ( ( hd_nat @ Xs5 )
!= ( hd_nat @ Ys5 ) ) ) ) ).
% longest_common_prefix
thf(fact_1250_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_1251_remove1__append,axiom,
! [X2: list_nat,Xs: list_list_nat,Ys: list_list_nat] :
( ( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ( remove1_list_nat @ X2 @ ( append_list_nat @ Xs @ Ys ) )
= ( append_list_nat @ ( remove1_list_nat @ X2 @ Xs ) @ Ys ) ) )
& ( ~ ( member_list_nat2 @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ( remove1_list_nat @ X2 @ ( append_list_nat @ Xs @ Ys ) )
= ( append_list_nat @ Xs @ ( remove1_list_nat @ X2 @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_1252_in__set__butlast__appendI,axiom,
! [X2: list_nat,Xs: list_list_nat,Ys: list_list_nat] :
( ( ( member_list_nat2 @ X2 @ ( set_list_nat2 @ ( butlast_list_nat @ Xs ) ) )
| ( member_list_nat2 @ X2 @ ( set_list_nat2 @ ( butlast_list_nat @ Ys ) ) ) )
=> ( member_list_nat2 @ X2 @ ( set_list_nat2 @ ( butlast_list_nat @ ( append_list_nat @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_1253_the__elem__set,axiom,
! [X2: nat] :
( ( the_elem_nat @ ( set_nat2 @ ( cons_nat @ X2 @ nil_nat ) ) )
= X2 ) ).
% the_elem_set
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P3: $o] :
( ( P3 = $true )
| ( P3 = $false ) ) ).
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 ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
member_list_nat2 @ p @ ( term_poss_a_b @ s ) ).
thf(conj_1,conjecture,
member_list_nat2 @ p @ ( terms_7168686267159881682rm_a_b @ u @ ( term_r6860082780075436317at_a_b @ s @ p @ u ) ) ).
%------------------------------------------------------------------------------