TPTP Problem File: SLH0670^1.p
View Solutions
- 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 : FO_Theory_Rewriting/0073_TA_Clousure_Const/prob_00414_018954__18767022_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1543 ( 434 unt; 271 typ; 0 def)
% Number of atoms : 4003 (1446 equ; 0 cnn)
% Maximal formula atoms : 23 ( 3 avg)
% Number of connectives : 13322 ( 461 ~; 58 |; 316 &;10377 @)
% ( 0 <=>;2110 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 7 avg)
% Number of types : 35 ( 34 usr)
% Number of type conns : 1017 (1017 >; 0 *; 0 +; 0 <<)
% Number of symbols : 240 ( 237 usr; 22 con; 0-4 aty)
% Number of variables : 3942 ( 180 ^;3556 !; 206 ?;3942 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:02:26.786
%------------------------------------------------------------------------------
% Could-be-implicit typings (34)
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
produc3922691147685405600st_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
produc579188371817701616list_a: $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__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
set_Pr5488025237498180813et_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
set_Pr4048851178543822343list_a: $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_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
produc7819656566062154093et_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
produc9164743771328383783list_a: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
list_P6011104703257516679at_nat: $tType ).
thf(ty_n_t__FSet__Ofset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
fset_P5670320511379867111at_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__FSet__Ofset_It__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J_J,type,
fset_P2289935846317411886_b_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
list_P1396940483166286381od_a_a: $tType ).
thf(ty_n_t__FSet__Ofset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
fset_P9143335661316304269od_a_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
set_Product_prod_a_a: $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__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__FSet__Ofset_It__List__Olist_It__Nat__Onat_J_J,type,
fset_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__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
product_prod_a_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
list_list_a: $tType ).
thf(ty_n_t__FSet__Ofset_It__List__Olist_Itf__a_J_J,type,
fset_list_a: $tType ).
thf(ty_n_t__Tree____Automata__Ota_Itf__a_Mtf__b_J,type,
tree_ta_a_b: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__Term__Oterm_Itf__b_Mtf__a_J,type,
term_b_a: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__FSet__Ofset_It__Nat__Onat_J,type,
fset_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__FSet__Ofset_Itf__b_J,type,
fset_b: $tType ).
thf(ty_n_t__FSet__Ofset_Itf__a_J,type,
fset_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (237)
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_001tf__a,type,
basic_189371981877014951ists_a: a > list_a > list_list_a ).
thf(sy_c_FSet_Ofcard_001tf__a,type,
fcard_a: fset_a > nat ).
thf(sy_c_FSet_Ofinsert_001t__Nat__Onat,type,
finsert_nat: nat > fset_nat > fset_nat ).
thf(sy_c_FSet_Ofinsert_001tf__a,type,
finsert_a: a > fset_a > fset_a ).
thf(sy_c_FSet_Ofmember_001t__List__Olist_It__Nat__Onat_J,type,
fmember_list_nat: list_nat > fset_list_nat > $o ).
thf(sy_c_FSet_Ofmember_001t__List__Olist_Itf__a_J,type,
fmember_list_a: list_a > fset_list_a > $o ).
thf(sy_c_FSet_Ofmember_001t__Nat__Onat,type,
fmember_nat: nat > fset_nat > $o ).
thf(sy_c_FSet_Ofmember_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
fmembe1449479052724974408at_nat: product_prod_nat_nat > fset_P5670320511379867111at_nat > $o ).
thf(sy_c_FSet_Ofmember_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
fmembe7738088516360075246od_a_a: product_prod_a_a > fset_P9143335661316304269od_a_a > $o ).
thf(sy_c_FSet_Ofmember_001tf__a,type,
fmember_a: a > fset_a > $o ).
thf(sy_c_FSet_Ofset__of__list_001t__List__Olist_It__Nat__Onat_J,type,
fset_o8830822285705182106st_nat: list_list_nat > fset_list_nat ).
thf(sy_c_FSet_Ofset__of__list_001t__List__Olist_Itf__a_J,type,
fset_of_list_list_a: list_list_a > fset_list_a ).
thf(sy_c_FSet_Ofset__of__list_001t__Nat__Onat,type,
fset_of_list_nat: list_nat > fset_nat ).
thf(sy_c_FSet_Ofset__of__list_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
fset_o7974655632839724743at_nat: list_P6011104703257516679at_nat > fset_P5670320511379867111at_nat ).
thf(sy_c_FSet_Ofset__of__list_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
fset_o3702720905070231533od_a_a: list_P1396940483166286381od_a_a > fset_P9143335661316304269od_a_a ).
thf(sy_c_FSet_Ofset__of__list_001tf__a,type,
fset_of_list_a: list_a > fset_a ).
thf(sy_c_FSet_Ofthe__elem_001tf__a,type,
fthe_elem_a: fset_a > a ).
thf(sy_c_FSet__Utils_Offuns__term_001tf__b_001tf__a,type,
fSet_ffuns_term_b_a: term_b_a > fset_b ).
thf(sy_c_FSet__Utils_Ofvars__term_001tf__b_001tf__a,type,
fSet_fvars_term_b_a: term_b_a > fset_a ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
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__List__Olist_Itf__a_J,type,
finite_finite_list_a: set_list_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__FSet__Ofset_Itf__a_J,type,
minus_minus_fset_a: fset_a > fset_a > fset_a ).
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__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_HOL_Oundefined_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
undefi3946296454836805481at_nat: product_prod_nat_nat ).
thf(sy_c_If_001t__FSet__Ofset_It__Nat__Onat_J,type,
if_fset_nat: $o > fset_nat > fset_nat > fset_nat ).
thf(sy_c_If_001t__FSet__Ofset_Itf__a_J,type,
if_fset_a: $o > fset_a > fset_a > fset_a ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
thf(sy_c_If_001t__Term__Oterm_Itf__b_Mtf__a_J,type,
if_term_b_a: $o > term_b_a > term_b_a > term_b_a ).
thf(sy_c_If_001tf__a,type,
if_a: $o > a > a > a ).
thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate_001t__Nat__Onat,type,
infini8530281810654367211te_nat: set_nat > nat > 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__Nat__Onat_001t__Nat__Onat,type,
lattic7446932960582359483at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_List_Odistinct_001t__List__Olist_It__Nat__Onat_J,type,
distinct_list_nat: list_list_nat > $o ).
thf(sy_c_List_Odistinct_001t__List__Olist_Itf__a_J,type,
distinct_list_a: list_list_a > $o ).
thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
distinct_nat: list_nat > $o ).
thf(sy_c_List_Odistinct_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
distin6923225563576452346at_nat: list_P6011104703257516679at_nat > $o ).
thf(sy_c_List_Odistinct_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
distin132333870042060960od_a_a: list_P1396940483166286381od_a_a > $o ).
thf(sy_c_List_Odistinct_001tf__a,type,
distinct_a: list_a > $o ).
thf(sy_c_List_Olast_001t__Nat__Onat,type,
last_nat: list_nat > nat ).
thf(sy_c_List_Olast_001tf__a,type,
last_a: list_a > a ).
thf(sy_c_List_Olenlex_001t__Nat__Onat,type,
lenlex_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_List_Olenlex_001tf__a,type,
lenlex_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_List_Olex_001t__Nat__Onat,type,
lex_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_List_Olex_001tf__a,type,
lex_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
linord2614967742042102400et_nat: set_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Nat__Onat_J,type,
cons_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_Itf__a_J,type,
cons_list_a: list_a > list_list_a > list_list_a ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Nat__Onat_J,type,
nil_list_nat: list_list_nat ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_Itf__a_J,type,
nil_list_a: list_list_a ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
hd_nat: list_nat > nat ).
thf(sy_c_List_Olist_Ohd_001tf__a,type,
hd_a: list_a > a ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
set_list_nat2: list_list_nat > set_list_nat ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_Itf__a_J,type,
set_list_a2: list_list_a > set_list_a ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
set_Pr5648618587558075414at_nat: list_P6011104703257516679at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
set_Product_prod_a_a2: list_P1396940483166286381od_a_a > set_Product_prod_a_a ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_List_Olist__ex_001t__List__Olist_It__Nat__Onat_J,type,
list_ex_list_nat: ( list_nat > $o ) > list_list_nat > $o ).
thf(sy_c_List_Olist__ex_001t__List__Olist_Itf__a_J,type,
list_ex_list_a: ( list_a > $o ) > list_list_a > $o ).
thf(sy_c_List_Olist__ex_001t__Nat__Onat,type,
list_ex_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__ex_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
list_e7689525607045846085at_nat: ( product_prod_nat_nat > $o ) > list_P6011104703257516679at_nat > $o ).
thf(sy_c_List_Olist__ex_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
list_e6552556518106193515od_a_a: ( product_prod_a_a > $o ) > list_P1396940483166286381od_a_a > $o ).
thf(sy_c_List_Olist__ex_001tf__a,type,
list_ex_a: ( a > $o ) > list_a > $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__List__Olist_Itf__a_J,type,
list_update_list_a: list_list_a > nat > list_a > list_list_a ).
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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
list_u6180841689913720943at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
list_u4967751623001292181od_a_a: list_P1396940483166286381od_a_a > nat > product_prod_a_a > list_P1396940483166286381od_a_a ).
thf(sy_c_List_Olist__update_001tf__a,type,
list_update_a: list_a > nat > a > list_a ).
thf(sy_c_List_Olistrel1_001t__Nat__Onat,type,
listrel1_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_List_Olistrel1_001tf__a,type,
listrel1_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_List_On__lists_001t__Nat__Onat,type,
n_lists_nat: nat > list_nat > list_list_nat ).
thf(sy_c_List_On__lists_001tf__a,type,
n_lists_a: nat > list_a > list_list_a ).
thf(sy_c_List_Onth_001t__List__Olist_It__Nat__Onat_J,type,
nth_list_nat: list_list_nat > nat > list_nat ).
thf(sy_c_List_Onth_001t__List__Olist_Itf__a_J,type,
nth_list_a: list_list_a > nat > list_a ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
nth_Pr7617993195940197384at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
nth_Product_prod_a_a: list_P1396940483166286381od_a_a > nat > product_prod_a_a ).
thf(sy_c_List_Onth_001tf__a,type,
nth_a: list_a > nat > a ).
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_001tf__a,type,
product_lists_a: list_list_a > list_list_a ).
thf(sy_c_List_Oremdups__adj_001t__Nat__Onat,type,
remdups_adj_nat: list_nat > list_nat ).
thf(sy_c_List_Oremdups__adj_001tf__a,type,
remdups_adj_a: list_a > list_a ).
thf(sy_c_List_Oremove1_001t__Nat__Onat,type,
remove1_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oremove1_001tf__a,type,
remove1_a: a > list_a > list_a ).
thf(sy_c_List_OremoveAll_001t__List__Olist_It__Nat__Onat_J,type,
removeAll_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_OremoveAll_001t__List__Olist_Itf__a_J,type,
removeAll_list_a: list_a > list_list_a > list_list_a ).
thf(sy_c_List_OremoveAll_001t__Nat__Onat,type,
removeAll_nat: nat > list_nat > list_nat ).
thf(sy_c_List_OremoveAll_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
remove3673390508374433037at_nat: product_prod_nat_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_OremoveAll_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
remove3678230896347538227od_a_a: product_prod_a_a > list_P1396940483166286381od_a_a > list_P1396940483166286381od_a_a ).
thf(sy_c_List_OremoveAll_001tf__a,type,
removeAll_a: a > list_a > list_a ).
thf(sy_c_List_Oreplicate_001t__List__Olist_It__Nat__Onat_J,type,
replicate_list_nat: nat > list_nat > list_list_nat ).
thf(sy_c_List_Oreplicate_001t__List__Olist_Itf__a_J,type,
replicate_list_a: nat > list_a > list_list_a ).
thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
replicate_nat: nat > nat > list_nat ).
thf(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
replic4235873036481779905at_nat: nat > product_prod_nat_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Oreplicate_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
replic8880659463222285543od_a_a: nat > product_prod_a_a > list_P1396940483166286381od_a_a ).
thf(sy_c_List_Oreplicate_001tf__a,type,
replicate_a: nat > a > list_a ).
thf(sy_c_List_Orev_001t__Nat__Onat,type,
rev_nat: list_nat > list_nat ).
thf(sy_c_List_Orev_001tf__a,type,
rev_a: list_a > list_a ).
thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
thf(sy_c_List_Osorted__wrt_001tf__a,type,
sorted_wrt_a: ( a > a > $o ) > list_a > $o ).
thf(sy_c_List_Ounion_001t__Nat__Onat,type,
union_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Ounion_001tf__a,type,
union_a: list_a > list_a > list_a ).
thf(sy_c_Missing__List_Oadjust__idx,type,
missing_adjust_idx: nat > nat > nat ).
thf(sy_c_Missing__List_Oadjust__idx__rev,type,
missin3815256168798769645dx_rev: nat > nat > nat ).
thf(sy_c_Missing__List_Omin__list_001t__Nat__Onat,type,
missing_min_list_nat: list_nat > nat ).
thf(sy_c_Missing__List_Opermut_001t__List__Olist_It__Nat__Onat_J,type,
missin2960327169605006088st_nat: list_list_nat > ( nat > nat ) > list_list_nat ).
thf(sy_c_Missing__List_Opermut_001t__List__Olist_Itf__a_J,type,
missin8258838934412359004list_a: list_list_a > ( nat > nat ) > list_list_a ).
thf(sy_c_Missing__List_Opermut_001t__Nat__Onat,type,
missing_permut_nat: list_nat > ( nat > nat ) > list_nat ).
thf(sy_c_Missing__List_Opermut_001tf__a,type,
missing_permut_a: list_a > ( nat > nat ) > list_a ).
thf(sy_c_Missing__List_Opermut__aux_001t__List__Olist_It__Nat__Onat_J,type,
missin3204601078308456222st_nat: list_list_nat > ( nat > nat ) > list_list_nat > list_list_nat ).
thf(sy_c_Missing__List_Opermut__aux_001t__List__Olist_Itf__a_J,type,
missin5389789484201179846list_a: list_list_a > ( nat > nat ) > list_list_a > list_list_a ).
thf(sy_c_Missing__List_Opermut__aux_001t__Nat__Onat,type,
missin1888654203714970382ux_nat: list_nat > ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_Missing__List_Opermut__aux_001tf__a,type,
missing_permut_aux_a: list_a > ( nat > nat ) > list_a > list_a ).
thf(sy_c_Missing__List_Oremdups__sort_001t__List__Olist_It__Nat__Onat_J,type,
missin2640887136080315381st_nat: list_list_nat > list_list_nat ).
thf(sy_c_Missing__List_Oremdups__sort_001t__Nat__Onat,type,
missin6101193410121742181rt_nat: list_nat > list_nat ).
thf(sy_c_Missing__List_Oremove__nth_001t__List__Olist_It__Nat__Onat_J,type,
missin5133812807938250791st_nat: nat > list_list_nat > list_list_nat ).
thf(sy_c_Missing__List_Oremove__nth_001t__List__Olist_Itf__a_J,type,
missin8041159176150063613list_a: nat > list_list_a > list_list_a ).
thf(sy_c_Missing__List_Oremove__nth_001t__Nat__Onat,type,
missin7175274867594579095th_nat: nat > list_nat > list_nat ).
thf(sy_c_Missing__List_Oremove__nth_001tf__a,type,
missing_remove_nth_a: nat > list_a > list_a ).
thf(sy_c_Missing__List_Osubtract__list__sorted_001t__Nat__Onat,type,
missin6424796737333596952ed_nat: list_nat > list_nat > list_nat ).
thf(sy_c_Missing__List_Osubtract__list__sorted__rel_001t__Nat__Onat,type,
missin3644257100165608323el_nat: produc1828647624359046049st_nat > produc1828647624359046049st_nat > $o ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
size_s3023201423986296836st_nat: list_list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
size_s349497388124573686list_a: list_list_a > 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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
size_s3885678630836030617od_a_a: list_P1396940483166286381od_a_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
size_size_list_a: list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Term__Oterm_Itf__b_Mtf__a_J,type,
size_size_term_b_a: term_b_a > nat ).
thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).
thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__FSet__Ofset_It__Nat__Onat_J,type,
bot_bot_fset_nat: fset_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__FSet__Ofset_Itf__a_J,type,
bot_bot_fset_a: fset_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__FSet__Ofset_Itf__b_J,type,
bot_bot_fset_b: fset_b ).
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__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__FSet__Ofset_It__Nat__Onat_J,type,
ord_less_fset_nat: fset_nat > fset_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__FSet__Ofset_Itf__a_J,type,
ord_less_fset_a: fset_a > fset_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__List__Olist_It__Nat__Onat_J,type,
ord_less_list_nat: list_nat > list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
ord_le1203424502768444845at_nat: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
ord_le1190675801316882794st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
ord_less_set_list_a: set_list_a > set_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le7866589430770878221at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
ord_le6819997720685908915od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J,type,
ord_le5604493270027003598_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
ord_le704812498762024988_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_Eo,type,
ord_less_eq_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__FSet__Ofset_It__List__Olist_It__Nat__Onat_J_J,type,
ord_le2534233284047104112st_nat: fset_list_nat > fset_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__FSet__Ofset_It__List__Olist_Itf__a_J_J,type,
ord_le510749213327159946list_a: fset_list_a > fset_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__FSet__Ofset_It__Nat__Onat_J,type,
ord_less_eq_fset_nat: fset_nat > fset_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__FSet__Ofset_Itf__a_J,type,
ord_less_eq_fset_a: fset_a > fset_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__List__Olist_It__Nat__Onat_J,type,
ord_less_eq_list_nat: list_nat > list_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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
ord_le8460144461188290721at_nat: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
ord_le8861187494160871172list_a: set_list_a > set_list_a > $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__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_Mtf__a_J_J,type,
ord_le746702958409616551od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_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_Itf__a_J_001t__List__Olist_Itf__a_J,type,
produc6837034575241423639list_a: list_a > list_a > produc9164743771328383783list_a ).
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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
produc8649917717998259994st_nat: product_prod_nat_nat > list_list_nat > produc3922691147685405600st_nat ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
produc2552019737546482272list_a: product_prod_nat_nat > list_list_a > produc579188371817701616list_a ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
produc4532415448927165861et_nat: set_nat > set_nat > produc7819656566062154093et_nat ).
thf(sy_c_Product__Type_OPair_001tf__a_001tf__a,type,
product_Pair_a_a: a > a > product_prod_a_a ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
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__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_Mtf__a_J,type,
collec3336397797384452498od_a_a: ( product_prod_a_a > $o ) > set_Product_prod_a_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_TA__Clousure__Const_Oreflcl__over__single__ta_001tf__a_001tf__b,type,
tA_Clo5715318072815374994ta_a_b: fset_a > fset_P2289935846317411886_b_nat > a > a > tree_ta_a_b ).
thf(sy_c_Term_Orename__subst__domain__range_001tf__a_001tf__b_001tf__a,type,
rename6228196718907852398_a_b_a: ( a > term_b_a ) > ( a > term_b_a ) > a > term_b_a ).
thf(sy_c_Term_Orestrict__subst__domain_001tf__a_001tf__b,type,
restri2809913845141176626in_a_b: set_a > ( a > term_b_a ) > a > term_b_a ).
thf(sy_c_Term_Oterm_OVar_001tf__a_001tf__b,type,
var_a_b: a > term_b_a ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__List__Olist_It__Nat__Onat_J,type,
transi5285580207609517981st_nat: set_Pr3451248702717554689st_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__List__Olist_Itf__a_J,type,
transi7631188966963710983list_a: set_Pr4048851178543822343list_a > set_Pr4048851178543822343list_a ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Nat__Onat,type,
transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Transitive__Closure_Ortrancl_001tf__a,type,
transitive_rtrancl_a: set_Product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_Tree__Automata_Ota__der_001tf__a_001tf__b,type,
tree_ta_der_a_b: tree_ta_a_b > term_b_a > fset_a ).
thf(sy_c_Utils_Oconcat__index__split_001t__Nat__Onat,type,
concat5568775588498683858it_nat: product_prod_nat_nat > list_list_nat > product_prod_nat_nat ).
thf(sy_c_Utils_Oconcat__index__split_001tf__a,type,
concat_index_split_a: product_prod_nat_nat > list_list_a > product_prod_nat_nat ).
thf(sy_c_Utils_Oconcat__index__split__rel_001t__Nat__Onat,type,
concat1575402509940520777el_nat: produc3922691147685405600st_nat > produc3922691147685405600st_nat > $o ).
thf(sy_c_Utils_Oconcat__index__split__rel_001tf__a,type,
concat3165989775726078917_rel_a: produc579188371817701616list_a > produc579188371817701616list_a > $o ).
thf(sy_c_Utils_Otrancl__list_001t__Nat__Onat,type,
trancl_list_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_Utils_Otrancl__list_001tf__a,type,
trancl_list_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_Utils_Otrancl__listp_001t__List__Olist_It__Nat__Onat_J,type,
trancl267208031524270810st_nat: ( list_nat > list_nat > $o ) > list_list_nat > list_list_nat > $o ).
thf(sy_c_Utils_Otrancl__listp_001t__List__Olist_Itf__a_J,type,
trancl_listp_list_a: ( list_a > list_a > $o ) > list_list_a > list_list_a > $o ).
thf(sy_c_Utils_Otrancl__listp_001t__Nat__Onat,type,
trancl_listp_nat: ( nat > nat > $o ) > list_nat > list_nat > $o ).
thf(sy_c_Utils_Otrancl__listp_001tf__a,type,
trancl_listp_a: ( a > a > $o ) > list_a > list_a > $o ).
thf(sy_c_Utils_Ovars__term__list_001tf__b_001tf__a,type,
vars_term_list_b_a: term_b_a > list_a ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
accp_P8037286306265792042st_nat: ( produc1828647624359046049st_nat > produc1828647624359046049st_nat > $o ) > produc1828647624359046049st_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
accp_P6748712283191960023st_nat: ( produc3922691147685405600st_nat > produc3922691147685405600st_nat > $o ) > produc3922691147685405600st_nat > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
accp_P2033140809068429945list_a: ( produc579188371817701616list_a > produc579188371817701616list_a > $o ) > produc579188371817701616list_a > $o ).
thf(sy_c_Wellfounded_Ofinite__psubset_001t__Nat__Onat,type,
finite_psubset_nat: set_Pr5488025237498180813et_nat ).
thf(sy_c_Wellfounded_Omeasure_001t__Nat__Onat,type,
measure_nat: ( nat > nat ) > set_Pr1261947904930325089at_nat ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: 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__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
member8191768239178080336list_a: produc9164743771328383783list_a > set_Pr4048851178543822343list_a > $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_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
member8277197624267554838et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_Q,type,
q: fset_a ).
thf(sy_v__092_060F_062,type,
f: fset_P2289935846317411886_b_nat ).
thf(sy_v_na____,type,
na: nat ).
thf(sy_v_q_092_060_094sub_062c,type,
q_c: a ).
thf(sy_v_q_092_060_094sub_062f,type,
q_f: a ).
thf(sy_v_x____,type,
x: a ).
% Relevant facts (1258)
thf(fact_0_Var_Oprems_I2_J,axiom,
q_c != q_f ).
% Var.prems(2)
thf(fact_1_Var_Oprems_I1_J,axiom,
~ ( fmember_a @ q_c @ q ) ).
% Var.prems(1)
thf(fact_2_Var_Oprems_I4_J,axiom,
( ( size_size_list_a @ ( vars_term_list_b_a @ ( var_a_b @ x ) ) )
= na ) ).
% Var.prems(4)
thf(fact_3_Var_Oprems_I3_J,axiom,
fmember_a @ q_f @ ( tree_ta_der_a_b @ ( tA_Clo5715318072815374994ta_a_b @ q @ f @ q_c @ q_f ) @ ( var_a_b @ x ) ) ).
% Var.prems(3)
thf(fact_4_finsertCI,axiom,
! [A: nat,B: fset_nat,B2: nat] :
( ( ~ ( fmember_nat @ A @ B )
=> ( A = B2 ) )
=> ( fmember_nat @ A @ ( finsert_nat @ B2 @ B ) ) ) ).
% finsertCI
thf(fact_5_finsertCI,axiom,
! [A: a,B: fset_a,B2: a] :
( ( ~ ( fmember_a @ A @ B )
=> ( A = B2 ) )
=> ( fmember_a @ A @ ( finsert_a @ B2 @ B ) ) ) ).
% finsertCI
thf(fact_6_finsert__iff,axiom,
! [A: nat,B2: nat,A2: fset_nat] :
( ( fmember_nat @ A @ ( finsert_nat @ B2 @ A2 ) )
= ( ( A = B2 )
| ( fmember_nat @ A @ A2 ) ) ) ).
% finsert_iff
thf(fact_7_finsert__iff,axiom,
! [A: a,B2: a,A2: fset_a] :
( ( fmember_a @ A @ ( finsert_a @ B2 @ A2 ) )
= ( ( A = B2 )
| ( fmember_a @ A @ A2 ) ) ) ).
% finsert_iff
thf(fact_8_list__update__overwrite,axiom,
! [Xs: list_nat,I: nat,X: nat,Y: nat] :
( ( list_update_nat @ ( list_update_nat @ Xs @ I @ X ) @ I @ Y )
= ( list_update_nat @ Xs @ I @ Y ) ) ).
% list_update_overwrite
thf(fact_9_list__update__overwrite,axiom,
! [Xs: list_a,I: nat,X: a,Y: a] :
( ( list_update_a @ ( list_update_a @ Xs @ I @ X ) @ I @ Y )
= ( list_update_a @ Xs @ I @ Y ) ) ).
% list_update_overwrite
thf(fact_10_finsert__absorb2,axiom,
! [X: nat,A2: fset_nat] :
( ( finsert_nat @ X @ ( finsert_nat @ X @ A2 ) )
= ( finsert_nat @ X @ A2 ) ) ).
% finsert_absorb2
thf(fact_11_finsert__absorb2,axiom,
! [X: a,A2: fset_a] :
( ( finsert_a @ X @ ( finsert_a @ X @ A2 ) )
= ( finsert_a @ X @ A2 ) ) ).
% finsert_absorb2
thf(fact_12_term_Oinject_I1_J,axiom,
! [X1: a,Y1: a] :
( ( ( var_a_b @ X1 )
= ( var_a_b @ Y1 ) )
= ( X1 = Y1 ) ) ).
% term.inject(1)
thf(fact_13_finsertE,axiom,
! [A: nat,B2: nat,A2: fset_nat] :
( ( fmember_nat @ A @ ( finsert_nat @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( fmember_nat @ A @ A2 ) ) ) ).
% finsertE
thf(fact_14_finsertE,axiom,
! [A: a,B2: a,A2: fset_a] :
( ( fmember_a @ A @ ( finsert_a @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( fmember_a @ A @ A2 ) ) ) ).
% finsertE
thf(fact_15_finsertI1,axiom,
! [A: nat,B: fset_nat] : ( fmember_nat @ A @ ( finsert_nat @ A @ B ) ) ).
% finsertI1
thf(fact_16_finsertI1,axiom,
! [A: a,B: fset_a] : ( fmember_a @ A @ ( finsert_a @ A @ B ) ) ).
% finsertI1
thf(fact_17_finsertI2,axiom,
! [A: nat,B: fset_nat,B2: nat] :
( ( fmember_nat @ A @ B )
=> ( fmember_nat @ A @ ( finsert_nat @ B2 @ B ) ) ) ).
% finsertI2
thf(fact_18_finsertI2,axiom,
! [A: a,B: fset_a,B2: a] :
( ( fmember_a @ A @ B )
=> ( fmember_a @ A @ ( finsert_a @ B2 @ B ) ) ) ).
% finsertI2
thf(fact_19_set__finsert,axiom,
! [X: nat,A2: fset_nat] :
( ( fmember_nat @ X @ A2 )
=> ~ ! [B3: fset_nat] :
( ( A2
= ( finsert_nat @ X @ B3 ) )
=> ( fmember_nat @ X @ B3 ) ) ) ).
% set_finsert
thf(fact_20_set__finsert,axiom,
! [X: a,A2: fset_a] :
( ( fmember_a @ X @ A2 )
=> ~ ! [B3: fset_a] :
( ( A2
= ( finsert_a @ X @ B3 ) )
=> ( fmember_a @ X @ B3 ) ) ) ).
% set_finsert
thf(fact_21_finsert__ident,axiom,
! [X: nat,A2: fset_nat,B: fset_nat] :
( ~ ( fmember_nat @ X @ A2 )
=> ( ~ ( fmember_nat @ X @ B )
=> ( ( ( finsert_nat @ X @ A2 )
= ( finsert_nat @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% finsert_ident
thf(fact_22_finsert__ident,axiom,
! [X: a,A2: fset_a,B: fset_a] :
( ~ ( fmember_a @ X @ A2 )
=> ( ~ ( fmember_a @ X @ B )
=> ( ( ( finsert_a @ X @ A2 )
= ( finsert_a @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% finsert_ident
thf(fact_23_finsert__absorb,axiom,
! [A: nat,A2: fset_nat] :
( ( fmember_nat @ A @ A2 )
=> ( ( finsert_nat @ A @ A2 )
= A2 ) ) ).
% finsert_absorb
thf(fact_24_finsert__absorb,axiom,
! [A: a,A2: fset_a] :
( ( fmember_a @ A @ A2 )
=> ( ( finsert_a @ A @ A2 )
= A2 ) ) ).
% finsert_absorb
thf(fact_25_finsert__eq__iff,axiom,
! [A: nat,A2: fset_nat,B2: nat,B: fset_nat] :
( ~ ( fmember_nat @ A @ A2 )
=> ( ~ ( fmember_nat @ B2 @ B )
=> ( ( ( finsert_nat @ A @ A2 )
= ( finsert_nat @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C: fset_nat] :
( ( A2
= ( finsert_nat @ B2 @ C ) )
& ~ ( fmember_nat @ B2 @ C )
& ( B
= ( finsert_nat @ A @ C ) )
& ~ ( fmember_nat @ A @ C ) ) ) ) ) ) ) ).
% finsert_eq_iff
thf(fact_26_finsert__eq__iff,axiom,
! [A: a,A2: fset_a,B2: a,B: fset_a] :
( ~ ( fmember_a @ A @ A2 )
=> ( ~ ( fmember_a @ B2 @ B )
=> ( ( ( finsert_a @ A @ A2 )
= ( finsert_a @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C: fset_a] :
( ( A2
= ( finsert_a @ B2 @ C ) )
& ~ ( fmember_a @ B2 @ C )
& ( B
= ( finsert_a @ A @ C ) )
& ~ ( fmember_a @ A @ C ) ) ) ) ) ) ) ).
% finsert_eq_iff
thf(fact_27_length__replicate,axiom,
! [N: nat,X: list_nat] :
( ( size_s3023201423986296836st_nat @ ( replicate_list_nat @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_28_length__replicate,axiom,
! [N: nat,X: list_a] :
( ( size_s349497388124573686list_a @ ( replicate_list_a @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_29_length__replicate,axiom,
! [N: nat,X: a] :
( ( size_size_list_a @ ( replicate_a @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_30_length__replicate,axiom,
! [N: nat,X: nat] :
( ( size_size_list_nat @ ( replicate_nat @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_31_length__list__update,axiom,
! [Xs: list_list_nat,I: nat,X: list_nat] :
( ( size_s3023201423986296836st_nat @ ( list_update_list_nat @ Xs @ I @ X ) )
= ( size_s3023201423986296836st_nat @ Xs ) ) ).
% length_list_update
thf(fact_32_length__list__update,axiom,
! [Xs: list_list_a,I: nat,X: list_a] :
( ( size_s349497388124573686list_a @ ( list_update_list_a @ Xs @ I @ X ) )
= ( size_s349497388124573686list_a @ Xs ) ) ).
% length_list_update
thf(fact_33_length__list__update,axiom,
! [Xs: list_a,I: nat,X: a] :
( ( size_size_list_a @ ( list_update_a @ Xs @ I @ X ) )
= ( size_size_list_a @ Xs ) ) ).
% length_list_update
thf(fact_34_length__list__update,axiom,
! [Xs: list_nat,I: nat,X: nat] :
( ( size_size_list_nat @ ( list_update_nat @ Xs @ I @ X ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_list_update
thf(fact_35_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_list_nat] :
( ( size_s3023201423986296836st_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_36_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_list_a] :
( ( size_s349497388124573686list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_37_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_a] :
( ( size_size_list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_38_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_nat] :
( ( size_size_list_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_39_neq__if__length__neq,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
!= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_40_neq__if__length__neq,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
!= ( size_s349497388124573686list_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_41_neq__if__length__neq,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
!= ( size_size_list_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_42_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_43_length__induct,axiom,
! [P: list_list_nat > $o,Xs: list_list_nat] :
( ! [Xs2: list_list_nat] :
( ! [Ys2: list_list_nat] :
( ( ord_less_nat @ ( size_s3023201423986296836st_nat @ Ys2 ) @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_44_length__induct,axiom,
! [P: list_list_a > $o,Xs: list_list_a] :
( ! [Xs2: list_list_a] :
( ! [Ys2: list_list_a] :
( ( ord_less_nat @ ( size_s349497388124573686list_a @ Ys2 ) @ ( size_s349497388124573686list_a @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_45_length__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ! [Xs2: list_a] :
( ! [Ys2: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ Ys2 ) @ ( size_size_list_a @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_46_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs2: list_nat] :
( ! [Ys2: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_47_eqfelem__imp__iff,axiom,
! [X: nat,Y: nat,A2: fset_nat] :
( ( X = Y )
=> ( ( fmember_nat @ X @ A2 )
= ( fmember_nat @ Y @ A2 ) ) ) ).
% eqfelem_imp_iff
thf(fact_48_eqfelem__imp__iff,axiom,
! [X: a,Y: a,A2: fset_a] :
( ( X = Y )
=> ( ( fmember_a @ X @ A2 )
= ( fmember_a @ Y @ A2 ) ) ) ).
% eqfelem_imp_iff
thf(fact_49_if__split__fmem2,axiom,
! [A: nat,Q: $o,X: fset_nat,Y: fset_nat] :
( ( fmember_nat @ A @ ( if_fset_nat @ Q @ X @ Y ) )
= ( ( Q
=> ( fmember_nat @ A @ X ) )
& ( ~ Q
=> ( fmember_nat @ A @ Y ) ) ) ) ).
% if_split_fmem2
thf(fact_50_if__split__fmem2,axiom,
! [A: a,Q: $o,X: fset_a,Y: fset_a] :
( ( fmember_a @ A @ ( if_fset_a @ Q @ X @ Y ) )
= ( ( Q
=> ( fmember_a @ A @ X ) )
& ( ~ Q
=> ( fmember_a @ A @ Y ) ) ) ) ).
% if_split_fmem2
thf(fact_51_if__split__fmem1,axiom,
! [Q: $o,X: nat,Y: nat,B2: fset_nat] :
( ( fmember_nat @ ( if_nat @ Q @ X @ Y ) @ B2 )
= ( ( Q
=> ( fmember_nat @ X @ B2 ) )
& ( ~ Q
=> ( fmember_nat @ Y @ B2 ) ) ) ) ).
% if_split_fmem1
thf(fact_52_if__split__fmem1,axiom,
! [Q: $o,X: a,Y: a,B2: fset_a] :
( ( fmember_a @ ( if_a @ Q @ X @ Y ) @ B2 )
= ( ( Q
=> ( fmember_a @ X @ B2 ) )
& ( ~ Q
=> ( fmember_a @ Y @ B2 ) ) ) ) ).
% if_split_fmem1
thf(fact_53_eqfset__imp__iff,axiom,
! [A2: fset_nat,B: fset_nat,X: nat] :
( ( A2 = B )
=> ( ( fmember_nat @ X @ A2 )
= ( fmember_nat @ X @ B ) ) ) ).
% eqfset_imp_iff
thf(fact_54_eqfset__imp__iff,axiom,
! [A2: fset_a,B: fset_a,X: a] :
( ( A2 = B )
=> ( ( fmember_a @ X @ A2 )
= ( fmember_a @ X @ B ) ) ) ).
% eqfset_imp_iff
thf(fact_55_eq__fmem__trans,axiom,
! [A: nat,B2: nat,A2: fset_nat] :
( ( A = B2 )
=> ( ( fmember_nat @ B2 @ A2 )
=> ( fmember_nat @ A @ A2 ) ) ) ).
% eq_fmem_trans
thf(fact_56_eq__fmem__trans,axiom,
! [A: a,B2: a,A2: fset_a] :
( ( A = B2 )
=> ( ( fmember_a @ B2 @ A2 )
=> ( fmember_a @ A @ A2 ) ) ) ).
% eq_fmem_trans
thf(fact_57_fequalityCE,axiom,
! [A2: fset_nat,B: fset_nat,C2: nat] :
( ( A2 = B )
=> ( ( ( fmember_nat @ C2 @ A2 )
=> ~ ( fmember_nat @ C2 @ B ) )
=> ~ ( ~ ( fmember_nat @ C2 @ A2 )
=> ( fmember_nat @ C2 @ B ) ) ) ) ).
% fequalityCE
thf(fact_58_fequalityCE,axiom,
! [A2: fset_a,B: fset_a,C2: a] :
( ( A2 = B )
=> ( ( ( fmember_a @ C2 @ A2 )
=> ~ ( fmember_a @ C2 @ B ) )
=> ~ ( ~ ( fmember_a @ C2 @ A2 )
=> ( fmember_a @ C2 @ B ) ) ) ) ).
% fequalityCE
thf(fact_59_fset__eqI,axiom,
! [A2: fset_nat,B: fset_nat] :
( ! [X2: nat] :
( ( fmember_nat @ X2 @ A2 )
= ( fmember_nat @ X2 @ B ) )
=> ( A2 = B ) ) ).
% fset_eqI
thf(fact_60_fset__eqI,axiom,
! [A2: fset_a,B: fset_a] :
( ! [X2: a] :
( ( fmember_a @ X2 @ A2 )
= ( fmember_a @ X2 @ B ) )
=> ( A2 = B ) ) ).
% fset_eqI
thf(fact_61_finsert__commute,axiom,
! [X: nat,Y: nat,A2: fset_nat] :
( ( finsert_nat @ X @ ( finsert_nat @ Y @ A2 ) )
= ( finsert_nat @ Y @ ( finsert_nat @ X @ A2 ) ) ) ).
% finsert_commute
thf(fact_62_finsert__commute,axiom,
! [X: a,Y: a,A2: fset_a] :
( ( finsert_a @ X @ ( finsert_a @ Y @ A2 ) )
= ( finsert_a @ Y @ ( finsert_a @ X @ A2 ) ) ) ).
% finsert_commute
thf(fact_63_list__update__swap,axiom,
! [I: nat,I2: nat,Xs: list_nat,X: nat,X3: nat] :
( ( I != I2 )
=> ( ( list_update_nat @ ( list_update_nat @ Xs @ I @ X ) @ I2 @ X3 )
= ( list_update_nat @ ( list_update_nat @ Xs @ I2 @ X3 ) @ I @ X ) ) ) ).
% list_update_swap
thf(fact_64_list__update__swap,axiom,
! [I: nat,I2: nat,Xs: list_a,X: a,X3: a] :
( ( I != I2 )
=> ( ( list_update_a @ ( list_update_a @ Xs @ I @ X ) @ I2 @ X3 )
= ( list_update_a @ ( list_update_a @ Xs @ I2 @ X3 ) @ I @ X ) ) ) ).
% list_update_swap
thf(fact_65_reflcl__over__single__ta__vars__term__q_092_060_094sub_062c,axiom,
! [Q_c: a,Q_f: a,Q: fset_a,F: fset_P2289935846317411886_b_nat,U: term_b_a] :
( ( Q_c != Q_f )
=> ( ( fmember_a @ Q_c @ ( tree_ta_der_a_b @ ( tA_Clo5715318072815374994ta_a_b @ Q @ F @ Q_c @ Q_f ) @ U ) )
=> ( ( vars_term_list_b_a @ U )
= ( replicate_a @ ( size_size_list_a @ ( vars_term_list_b_a @ U ) ) @ Q_c ) ) ) ) ).
% reflcl_over_single_ta_vars_term_q\<^sub>c
thf(fact_66_mk__disjoint__finsert,axiom,
! [A: nat,A2: fset_nat] :
( ( fmember_nat @ A @ A2 )
=> ? [B3: fset_nat] :
( ( A2
= ( finsert_nat @ A @ B3 ) )
& ~ ( fmember_nat @ A @ B3 ) ) ) ).
% mk_disjoint_finsert
thf(fact_67_mk__disjoint__finsert,axiom,
! [A: a,A2: fset_a] :
( ( fmember_a @ A @ A2 )
=> ? [B3: fset_a] :
( ( A2
= ( finsert_a @ A @ B3 ) )
& ~ ( fmember_a @ A @ B3 ) ) ) ).
% mk_disjoint_finsert
thf(fact_68_rename__subst__domain__range__Var__rhs,axiom,
! [Rho: a > term_b_a] :
( ( rename6228196718907852398_a_b_a @ Rho @ var_a_b )
= var_a_b ) ).
% rename_subst_domain_range_Var_rhs
thf(fact_69_rename__subst__domain__range__Var__lhs,axiom,
! [Sigma: a > term_b_a] :
( ( rename6228196718907852398_a_b_a @ var_a_b @ Sigma )
= Sigma ) ).
% rename_subst_domain_range_Var_lhs
thf(fact_70_restrict__subst__domain__Var,axiom,
! [V: set_a] :
( ( restri2809913845141176626in_a_b @ V @ var_a_b )
= var_a_b ) ).
% restrict_subst_domain_Var
thf(fact_71_nth__list__update__eq,axiom,
! [I: nat,Xs: list_list_nat,X: list_nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_list_nat @ ( list_update_list_nat @ Xs @ I @ X ) @ I )
= X ) ) ).
% nth_list_update_eq
thf(fact_72_nth__list__update__eq,axiom,
! [I: nat,Xs: list_list_a,X: list_a] :
( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( nth_list_a @ ( list_update_list_a @ Xs @ I @ X ) @ I )
= X ) ) ).
% nth_list_update_eq
thf(fact_73_nth__list__update__eq,axiom,
! [I: nat,Xs: list_a,X: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( list_update_a @ Xs @ I @ X ) @ I )
= X ) ) ).
% nth_list_update_eq
thf(fact_74_nth__list__update__eq,axiom,
! [I: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ I )
= X ) ) ).
% nth_list_update_eq
thf(fact_75_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_76_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M: nat] :
( ( ord_less_nat @ M @ N2 )
& ~ ( P @ M ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_77_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( P @ M ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_78_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_79_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_80_mem__Collect__eq,axiom,
! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_81_mem__Collect__eq,axiom,
! [A: product_prod_a_a,P: product_prod_a_a > $o] :
( ( member1426531477525435216od_a_a @ A @ ( collec3336397797384452498od_a_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_82_mem__Collect__eq,axiom,
! [A: list_nat,P: list_nat > $o] :
( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_83_mem__Collect__eq,axiom,
! [A: list_a,P: list_a > $o] :
( ( member_list_a @ A @ ( collect_list_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_84_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_85_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_86_Collect__mem__eq,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ( collec3392354462482085612at_nat
@ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_87_Collect__mem__eq,axiom,
! [A2: set_Product_prod_a_a] :
( ( collec3336397797384452498od_a_a
@ ^ [X4: product_prod_a_a] : ( member1426531477525435216od_a_a @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_88_Collect__mem__eq,axiom,
! [A2: set_list_nat] :
( ( collect_list_nat
@ ^ [X4: list_nat] : ( member_list_nat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_89_Collect__mem__eq,axiom,
! [A2: set_list_a] :
( ( collect_list_a
@ ^ [X4: list_a] : ( member_list_a @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_90_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X4: a] : ( member_a @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_91_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_92_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_93_list__update__id,axiom,
! [Xs: list_nat,I: nat] :
( ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ I ) )
= Xs ) ).
% list_update_id
thf(fact_94_list__update__id,axiom,
! [Xs: list_a,I: nat] :
( ( list_update_a @ Xs @ I @ ( nth_a @ Xs @ I ) )
= Xs ) ).
% list_update_id
thf(fact_95_nth__list__update__neq,axiom,
! [I: nat,J: nat,Xs: list_nat,X: nat] :
( ( I != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
= ( nth_nat @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_96_nth__list__update__neq,axiom,
! [I: nat,J: nat,Xs: list_a,X: a] :
( ( I != J )
=> ( ( nth_a @ ( list_update_a @ Xs @ I @ X ) @ J )
= ( nth_a @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_97_nth__replicate,axiom,
! [I: nat,N: nat,X: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_nat @ ( replicate_nat @ N @ X ) @ I )
= X ) ) ).
% nth_replicate
thf(fact_98_nth__replicate,axiom,
! [I: nat,N: nat,X: a] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_a @ ( replicate_a @ N @ X ) @ I )
= X ) ) ).
% nth_replicate
thf(fact_99_nth__equalityI,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_list_nat @ Xs @ I3 )
= ( nth_list_nat @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_100_nth__equalityI,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( nth_list_a @ Xs @ I3 )
= ( nth_list_a @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_101_nth__equalityI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ Xs @ I3 )
= ( nth_a @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_102_nth__equalityI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I3 )
= ( nth_nat @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_103_Skolem__list__nth,axiom,
! [K: nat,P: nat > list_nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X5: list_nat] : ( P @ I4 @ X5 ) ) )
= ( ? [Xs3: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs3 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P @ I4 @ ( nth_list_nat @ Xs3 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_104_Skolem__list__nth,axiom,
! [K: nat,P: nat > list_a > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X5: list_a] : ( P @ I4 @ X5 ) ) )
= ( ? [Xs3: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs3 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P @ I4 @ ( nth_list_a @ Xs3 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_105_Skolem__list__nth,axiom,
! [K: nat,P: nat > a > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X5: a] : ( P @ I4 @ X5 ) ) )
= ( ? [Xs3: list_a] :
( ( ( size_size_list_a @ Xs3 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P @ I4 @ ( nth_a @ Xs3 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_106_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X5: nat] : ( P @ I4 @ X5 ) ) )
= ( ? [Xs3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P @ I4 @ ( nth_nat @ Xs3 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_107_list__eq__iff__nth__eq,axiom,
( ( ^ [Y2: list_list_nat,Z: list_list_nat] : ( Y2 = Z ) )
= ( ^ [Xs3: list_list_nat,Ys3: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs3 )
= ( size_s3023201423986296836st_nat @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Xs3 ) )
=> ( ( nth_list_nat @ Xs3 @ I4 )
= ( nth_list_nat @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_108_list__eq__iff__nth__eq,axiom,
( ( ^ [Y2: list_list_a,Z: list_list_a] : ( Y2 = Z ) )
= ( ^ [Xs3: list_list_a,Ys3: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs3 )
= ( size_s349497388124573686list_a @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s349497388124573686list_a @ Xs3 ) )
=> ( ( nth_list_a @ Xs3 @ I4 )
= ( nth_list_a @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_109_list__eq__iff__nth__eq,axiom,
( ( ^ [Y2: list_a,Z: list_a] : ( Y2 = Z ) )
= ( ^ [Xs3: list_a,Ys3: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs3 ) )
=> ( ( nth_a @ Xs3 @ I4 )
= ( nth_a @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_110_list__eq__iff__nth__eq,axiom,
( ( ^ [Y2: list_nat,Z: list_nat] : ( Y2 = Z ) )
= ( ^ [Xs3: list_nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs3 ) )
=> ( ( nth_nat @ Xs3 @ I4 )
= ( nth_nat @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_111_restrict__subst__domain__def,axiom,
( restri2809913845141176626in_a_b
= ( ^ [V2: set_a,Sigma2: a > term_b_a,X4: a] : ( if_term_b_a @ ( member_a @ X4 @ V2 ) @ ( Sigma2 @ X4 ) @ ( var_a_b @ X4 ) ) ) ) ).
% restrict_subst_domain_def
thf(fact_112_nth__list__update,axiom,
! [I: nat,Xs: list_list_nat,J: nat,X: list_nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth_list_nat @ ( list_update_list_nat @ Xs @ I @ X ) @ J )
= X ) )
& ( ( I != J )
=> ( ( nth_list_nat @ ( list_update_list_nat @ Xs @ I @ X ) @ J )
= ( nth_list_nat @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_113_nth__list__update,axiom,
! [I: nat,Xs: list_list_a,J: nat,X: list_a] :
( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth_list_a @ ( list_update_list_a @ Xs @ I @ X ) @ J )
= X ) )
& ( ( I != J )
=> ( ( nth_list_a @ ( list_update_list_a @ Xs @ I @ X ) @ J )
= ( nth_list_a @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_114_nth__list__update,axiom,
! [I: nat,Xs: list_a,J: nat,X: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth_a @ ( list_update_a @ Xs @ I @ X ) @ J )
= X ) )
& ( ( I != J )
=> ( ( nth_a @ ( list_update_a @ Xs @ I @ X ) @ J )
= ( nth_a @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_115_nth__list__update,axiom,
! [I: nat,Xs: list_nat,J: nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
= X ) )
& ( ( I != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
= ( nth_nat @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_116_list__update__same__conv,axiom,
! [I: nat,Xs: list_list_nat,X: list_nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( ( list_update_list_nat @ Xs @ I @ X )
= Xs )
= ( ( nth_list_nat @ Xs @ I )
= X ) ) ) ).
% list_update_same_conv
thf(fact_117_list__update__same__conv,axiom,
! [I: nat,Xs: list_list_a,X: list_a] :
( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( ( list_update_list_a @ Xs @ I @ X )
= Xs )
= ( ( nth_list_a @ Xs @ I )
= X ) ) ) ).
% list_update_same_conv
thf(fact_118_list__update__same__conv,axiom,
! [I: nat,Xs: list_a,X: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( ( list_update_a @ Xs @ I @ X )
= Xs )
= ( ( nth_a @ Xs @ I )
= X ) ) ) ).
% list_update_same_conv
thf(fact_119_list__update__same__conv,axiom,
! [I: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ( list_update_nat @ Xs @ I @ X )
= Xs )
= ( ( nth_nat @ Xs @ I )
= X ) ) ) ).
% list_update_same_conv
thf(fact_120_size__neq__size__imp__neq,axiom,
! [X: term_b_a,Y: term_b_a] :
( ( ( size_size_term_b_a @ X )
!= ( size_size_term_b_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_121_size__neq__size__imp__neq,axiom,
! [X: list_list_nat,Y: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ X )
!= ( size_s3023201423986296836st_nat @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_122_size__neq__size__imp__neq,axiom,
! [X: list_list_a,Y: list_list_a] :
( ( ( size_s349497388124573686list_a @ X )
!= ( size_s349497388124573686list_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_123_size__neq__size__imp__neq,axiom,
! [X: list_a,Y: list_a] :
( ( ( size_size_list_a @ X )
!= ( size_size_list_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_124_size__neq__size__imp__neq,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( size_size_list_nat @ X )
!= ( size_size_list_nat @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_125_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_126_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_127_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_128_parallel__list__update,axiom,
! [N: nat,R: list_nat > list_nat > $o,P2: list_list_nat > $o,Xs: list_list_nat,Ys: list_list_nat] :
( ! [Xs2: list_list_nat,I3: nat,Y3: list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= N )
=> ( ( ord_less_nat @ I3 @ N )
=> ( ( R @ ( nth_list_nat @ Xs2 @ I3 ) @ Y3 )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( list_update_list_nat @ Xs2 @ I3 @ Y3 ) ) ) ) ) )
=> ( ( ( size_s3023201423986296836st_nat @ Xs )
= N )
=> ( ( P2 @ Xs )
=> ( ( ( size_s3023201423986296836st_nat @ Ys )
= N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( R @ ( nth_list_nat @ Xs @ I3 ) @ ( nth_list_nat @ Ys @ I3 ) ) )
=> ( P2 @ Ys ) ) ) ) ) ) ).
% parallel_list_update
thf(fact_129_parallel__list__update,axiom,
! [N: nat,R: list_a > list_a > $o,P2: list_list_a > $o,Xs: list_list_a,Ys: list_list_a] :
( ! [Xs2: list_list_a,I3: nat,Y3: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= N )
=> ( ( ord_less_nat @ I3 @ N )
=> ( ( R @ ( nth_list_a @ Xs2 @ I3 ) @ Y3 )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( list_update_list_a @ Xs2 @ I3 @ Y3 ) ) ) ) ) )
=> ( ( ( size_s349497388124573686list_a @ Xs )
= N )
=> ( ( P2 @ Xs )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( R @ ( nth_list_a @ Xs @ I3 ) @ ( nth_list_a @ Ys @ I3 ) ) )
=> ( P2 @ Ys ) ) ) ) ) ) ).
% parallel_list_update
thf(fact_130_parallel__list__update,axiom,
! [N: nat,R: a > a > $o,P2: list_a > $o,Xs: list_a,Ys: list_a] :
( ! [Xs2: list_a,I3: nat,Y3: a] :
( ( ( size_size_list_a @ Xs2 )
= N )
=> ( ( ord_less_nat @ I3 @ N )
=> ( ( R @ ( nth_a @ Xs2 @ I3 ) @ Y3 )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( list_update_a @ Xs2 @ I3 @ Y3 ) ) ) ) ) )
=> ( ( ( size_size_list_a @ Xs )
= N )
=> ( ( P2 @ Xs )
=> ( ( ( size_size_list_a @ Ys )
= N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( R @ ( nth_a @ Xs @ I3 ) @ ( nth_a @ Ys @ I3 ) ) )
=> ( P2 @ Ys ) ) ) ) ) ) ).
% parallel_list_update
thf(fact_131_parallel__list__update,axiom,
! [N: nat,R: nat > nat > $o,P2: list_nat > $o,Xs: list_nat,Ys: list_nat] :
( ! [Xs2: list_nat,I3: nat,Y3: nat] :
( ( ( size_size_list_nat @ Xs2 )
= N )
=> ( ( ord_less_nat @ I3 @ N )
=> ( ( R @ ( nth_nat @ Xs2 @ I3 ) @ Y3 )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( list_update_nat @ Xs2 @ I3 @ Y3 ) ) ) ) ) )
=> ( ( ( size_size_list_nat @ Xs )
= N )
=> ( ( P2 @ Xs )
=> ( ( ( size_size_list_nat @ Ys )
= N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( R @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Ys @ I3 ) ) )
=> ( P2 @ Ys ) ) ) ) ) ) ).
% parallel_list_update
thf(fact_132_nth__equalityE,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( Xs = Ys )
=> ~ ( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ~ ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_list_nat @ Xs @ I5 )
= ( nth_list_nat @ Ys @ I5 ) ) ) ) ) ).
% nth_equalityE
thf(fact_133_nth__equalityE,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( Xs = Ys )
=> ~ ( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ~ ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( nth_list_a @ Xs @ I5 )
= ( nth_list_a @ Ys @ I5 ) ) ) ) ) ).
% nth_equalityE
thf(fact_134_nth__equalityE,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = Ys )
=> ~ ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ~ ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ Xs @ I5 )
= ( nth_a @ Ys @ I5 ) ) ) ) ) ).
% nth_equalityE
thf(fact_135_nth__equalityE,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs = Ys )
=> ~ ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ~ ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I5 )
= ( nth_nat @ Ys @ I5 ) ) ) ) ) ).
% nth_equalityE
thf(fact_136_Ex__list__of__length__P,axiom,
! [N: nat,P: list_nat > nat > $o] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ? [X6: list_nat] : ( P @ X6 @ I3 ) )
=> ? [Xs2: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ N )
=> ( P @ ( nth_list_nat @ Xs2 @ I5 ) @ I5 ) ) ) ) ).
% Ex_list_of_length_P
thf(fact_137_Ex__list__of__length__P,axiom,
! [N: nat,P: list_a > nat > $o] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ? [X6: list_a] : ( P @ X6 @ I3 ) )
=> ? [Xs2: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ N )
=> ( P @ ( nth_list_a @ Xs2 @ I5 ) @ I5 ) ) ) ) ).
% Ex_list_of_length_P
thf(fact_138_Ex__list__of__length__P,axiom,
! [N: nat,P: a > nat > $o] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ? [X6: a] : ( P @ X6 @ I3 ) )
=> ? [Xs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ N )
=> ( P @ ( nth_a @ Xs2 @ I5 ) @ I5 ) ) ) ) ).
% Ex_list_of_length_P
thf(fact_139_Ex__list__of__length__P,axiom,
! [N: nat,P: nat > nat > $o] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ? [X6: nat] : ( P @ X6 @ I3 ) )
=> ? [Xs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ N )
=> ( P @ ( nth_nat @ Xs2 @ I5 ) @ I5 ) ) ) ) ).
% Ex_list_of_length_P
thf(fact_140_set__swap,axiom,
! [I: nat,Xs: list_list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( set_list_nat2 @ ( list_update_list_nat @ ( list_update_list_nat @ Xs @ I @ ( nth_list_nat @ Xs @ J ) ) @ J @ ( nth_list_nat @ Xs @ I ) ) )
= ( set_list_nat2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_141_set__swap,axiom,
! [I: nat,Xs: list_list_a,J: nat] :
( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( set_list_a2 @ ( list_update_list_a @ ( list_update_list_a @ Xs @ I @ ( nth_list_a @ Xs @ J ) ) @ J @ ( nth_list_a @ Xs @ I ) ) )
= ( set_list_a2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_142_set__swap,axiom,
! [I: nat,Xs: list_a,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs ) )
=> ( ( set_a2 @ ( list_update_a @ ( list_update_a @ Xs @ I @ ( nth_a @ Xs @ J ) ) @ J @ ( nth_a @ Xs @ I ) ) )
= ( set_a2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_143_set__swap,axiom,
! [I: nat,Xs: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I ) ) )
= ( set_nat2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_144_list__ex__length,axiom,
( list_ex_list_nat
= ( ^ [P3: list_nat > $o,Xs3: list_list_nat] :
? [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_s3023201423986296836st_nat @ Xs3 ) )
& ( P3 @ ( nth_list_nat @ Xs3 @ N3 ) ) ) ) ) ).
% list_ex_length
thf(fact_145_list__ex__length,axiom,
( list_ex_list_a
= ( ^ [P3: list_a > $o,Xs3: list_list_a] :
? [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_s349497388124573686list_a @ Xs3 ) )
& ( P3 @ ( nth_list_a @ Xs3 @ N3 ) ) ) ) ) ).
% list_ex_length
thf(fact_146_list__ex__length,axiom,
( list_ex_a
= ( ^ [P3: a > $o,Xs3: list_a] :
? [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs3 ) )
& ( P3 @ ( nth_a @ Xs3 @ N3 ) ) ) ) ) ).
% list_ex_length
thf(fact_147_list__ex__length,axiom,
( list_ex_nat
= ( ^ [P3: nat > $o,Xs3: list_nat] :
? [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs3 ) )
& ( P3 @ ( nth_nat @ Xs3 @ N3 ) ) ) ) ) ).
% list_ex_length
thf(fact_148_distinct__swap,axiom,
! [I: nat,Xs: list_list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( distinct_list_nat @ ( list_update_list_nat @ ( list_update_list_nat @ Xs @ I @ ( nth_list_nat @ Xs @ J ) ) @ J @ ( nth_list_nat @ Xs @ I ) ) )
= ( distinct_list_nat @ Xs ) ) ) ) ).
% distinct_swap
thf(fact_149_distinct__swap,axiom,
! [I: nat,Xs: list_list_a,J: nat] :
( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( distinct_list_a @ ( list_update_list_a @ ( list_update_list_a @ Xs @ I @ ( nth_list_a @ Xs @ J ) ) @ J @ ( nth_list_a @ Xs @ I ) ) )
= ( distinct_list_a @ Xs ) ) ) ) ).
% distinct_swap
thf(fact_150_distinct__swap,axiom,
! [I: nat,Xs: list_a,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs ) )
=> ( ( distinct_a @ ( list_update_a @ ( list_update_a @ Xs @ I @ ( nth_a @ Xs @ J ) ) @ J @ ( nth_a @ Xs @ I ) ) )
= ( distinct_a @ Xs ) ) ) ) ).
% distinct_swap
thf(fact_151_distinct__swap,axiom,
! [I: nat,Xs: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( distinct_nat @ ( list_update_nat @ ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I ) ) )
= ( distinct_nat @ Xs ) ) ) ) ).
% distinct_swap
thf(fact_152_permut__sound,axiom,
! [I: nat,As: list_list_nat,F2: nat > nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ As ) )
=> ( ( nth_list_nat @ ( missin2960327169605006088st_nat @ As @ F2 ) @ I )
= ( nth_list_nat @ As @ ( F2 @ I ) ) ) ) ).
% permut_sound
thf(fact_153_permut__sound,axiom,
! [I: nat,As: list_list_a,F2: nat > nat] :
( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ As ) )
=> ( ( nth_list_a @ ( missin8258838934412359004list_a @ As @ F2 ) @ I )
= ( nth_list_a @ As @ ( F2 @ I ) ) ) ) ).
% permut_sound
thf(fact_154_permut__sound,axiom,
! [I: nat,As: list_a,F2: nat > nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ As ) )
=> ( ( nth_a @ ( missing_permut_a @ As @ F2 ) @ I )
= ( nth_a @ As @ ( F2 @ I ) ) ) ) ).
% permut_sound
thf(fact_155_permut__sound,axiom,
! [I: nat,As: list_nat,F2: nat > nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ As ) )
=> ( ( nth_nat @ ( missing_permut_nat @ As @ F2 ) @ I )
= ( nth_nat @ As @ ( F2 @ I ) ) ) ) ).
% permut_sound
thf(fact_156_trancl__listp_Ocases,axiom,
! [R2: list_nat > list_nat > $o,A1: list_list_nat,A22: list_list_nat] :
( ( trancl267208031524270810st_nat @ R2 @ A1 @ A22 )
=> ( ( ( ( size_s3023201423986296836st_nat @ A1 )
= ( size_s3023201423986296836st_nat @ A22 ) )
=> ~ ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_s3023201423986296836st_nat @ A22 ) )
=> ( R2 @ ( nth_list_nat @ A1 @ I5 ) @ ( nth_list_nat @ A22 @ I5 ) ) ) )
=> ~ ! [Ys4: list_list_nat,I3: nat,Z2: list_nat] :
( ( A22
= ( list_update_list_nat @ Ys4 @ I3 @ Z2 ) )
=> ( ( trancl267208031524270810st_nat @ R2 @ A1 @ Ys4 )
=> ( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Ys4 ) )
=> ~ ( R2 @ ( nth_list_nat @ Ys4 @ I3 ) @ Z2 ) ) ) ) ) ) ).
% trancl_listp.cases
thf(fact_157_trancl__listp_Ocases,axiom,
! [R2: list_a > list_a > $o,A1: list_list_a,A22: list_list_a] :
( ( trancl_listp_list_a @ R2 @ A1 @ A22 )
=> ( ( ( ( size_s349497388124573686list_a @ A1 )
= ( size_s349497388124573686list_a @ A22 ) )
=> ~ ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_s349497388124573686list_a @ A22 ) )
=> ( R2 @ ( nth_list_a @ A1 @ I5 ) @ ( nth_list_a @ A22 @ I5 ) ) ) )
=> ~ ! [Ys4: list_list_a,I3: nat,Z2: list_a] :
( ( A22
= ( list_update_list_a @ Ys4 @ I3 @ Z2 ) )
=> ( ( trancl_listp_list_a @ R2 @ A1 @ Ys4 )
=> ( ( ord_less_nat @ I3 @ ( size_s349497388124573686list_a @ Ys4 ) )
=> ~ ( R2 @ ( nth_list_a @ Ys4 @ I3 ) @ Z2 ) ) ) ) ) ) ).
% trancl_listp.cases
thf(fact_158_trancl__listp_Ocases,axiom,
! [R2: a > a > $o,A1: list_a,A22: list_a] :
( ( trancl_listp_a @ R2 @ A1 @ A22 )
=> ( ( ( ( size_size_list_a @ A1 )
= ( size_size_list_a @ A22 ) )
=> ~ ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_a @ A22 ) )
=> ( R2 @ ( nth_a @ A1 @ I5 ) @ ( nth_a @ A22 @ I5 ) ) ) )
=> ~ ! [Ys4: list_a,I3: nat,Z2: a] :
( ( A22
= ( list_update_a @ Ys4 @ I3 @ Z2 ) )
=> ( ( trancl_listp_a @ R2 @ A1 @ Ys4 )
=> ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Ys4 ) )
=> ~ ( R2 @ ( nth_a @ Ys4 @ I3 ) @ Z2 ) ) ) ) ) ) ).
% trancl_listp.cases
thf(fact_159_trancl__listp_Ocases,axiom,
! [R2: nat > nat > $o,A1: list_nat,A22: list_nat] :
( ( trancl_listp_nat @ R2 @ A1 @ A22 )
=> ( ( ( ( size_size_list_nat @ A1 )
= ( size_size_list_nat @ A22 ) )
=> ~ ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_nat @ A22 ) )
=> ( R2 @ ( nth_nat @ A1 @ I5 ) @ ( nth_nat @ A22 @ I5 ) ) ) )
=> ~ ! [Ys4: list_nat,I3: nat,Z2: nat] :
( ( A22
= ( list_update_nat @ Ys4 @ I3 @ Z2 ) )
=> ( ( trancl_listp_nat @ R2 @ A1 @ Ys4 )
=> ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys4 ) )
=> ~ ( R2 @ ( nth_nat @ Ys4 @ I3 ) @ Z2 ) ) ) ) ) ) ).
% trancl_listp.cases
thf(fact_160_trancl__listp_Osimps,axiom,
( trancl267208031524270810st_nat
= ( ^ [R3: list_nat > list_nat > $o,A12: list_list_nat,A23: list_list_nat] :
( ? [Xs3: list_list_nat,Ys3: list_list_nat] :
( ( A12 = Xs3 )
& ( A23 = Ys3 )
& ( ( size_s3023201423986296836st_nat @ Xs3 )
= ( size_s3023201423986296836st_nat @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Ys3 ) )
=> ( R3 @ ( nth_list_nat @ Xs3 @ I4 ) @ ( nth_list_nat @ Ys3 @ I4 ) ) ) )
| ? [Xs3: list_list_nat,Ys3: list_list_nat,I4: nat,Z3: list_nat] :
( ( A12 = Xs3 )
& ( A23
= ( list_update_list_nat @ Ys3 @ I4 @ Z3 ) )
& ( trancl267208031524270810st_nat @ R3 @ Xs3 @ Ys3 )
& ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Ys3 ) )
& ( R3 @ ( nth_list_nat @ Ys3 @ I4 ) @ Z3 ) ) ) ) ) ).
% trancl_listp.simps
thf(fact_161_trancl__listp_Osimps,axiom,
( trancl_listp_list_a
= ( ^ [R3: list_a > list_a > $o,A12: list_list_a,A23: list_list_a] :
( ? [Xs3: list_list_a,Ys3: list_list_a] :
( ( A12 = Xs3 )
& ( A23 = Ys3 )
& ( ( size_s349497388124573686list_a @ Xs3 )
= ( size_s349497388124573686list_a @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s349497388124573686list_a @ Ys3 ) )
=> ( R3 @ ( nth_list_a @ Xs3 @ I4 ) @ ( nth_list_a @ Ys3 @ I4 ) ) ) )
| ? [Xs3: list_list_a,Ys3: list_list_a,I4: nat,Z3: list_a] :
( ( A12 = Xs3 )
& ( A23
= ( list_update_list_a @ Ys3 @ I4 @ Z3 ) )
& ( trancl_listp_list_a @ R3 @ Xs3 @ Ys3 )
& ( ord_less_nat @ I4 @ ( size_s349497388124573686list_a @ Ys3 ) )
& ( R3 @ ( nth_list_a @ Ys3 @ I4 ) @ Z3 ) ) ) ) ) ).
% trancl_listp.simps
thf(fact_162_trancl__listp_Osimps,axiom,
( trancl_listp_a
= ( ^ [R3: a > a > $o,A12: list_a,A23: list_a] :
( ? [Xs3: list_a,Ys3: list_a] :
( ( A12 = Xs3 )
& ( A23 = Ys3 )
& ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Ys3 ) )
=> ( R3 @ ( nth_a @ Xs3 @ I4 ) @ ( nth_a @ Ys3 @ I4 ) ) ) )
| ? [Xs3: list_a,Ys3: list_a,I4: nat,Z3: a] :
( ( A12 = Xs3 )
& ( A23
= ( list_update_a @ Ys3 @ I4 @ Z3 ) )
& ( trancl_listp_a @ R3 @ Xs3 @ Ys3 )
& ( ord_less_nat @ I4 @ ( size_size_list_a @ Ys3 ) )
& ( R3 @ ( nth_a @ Ys3 @ I4 ) @ Z3 ) ) ) ) ) ).
% trancl_listp.simps
thf(fact_163_trancl__listp_Osimps,axiom,
( trancl_listp_nat
= ( ^ [R3: nat > nat > $o,A12: list_nat,A23: list_nat] :
( ? [Xs3: list_nat,Ys3: list_nat] :
( ( A12 = Xs3 )
& ( A23 = Ys3 )
& ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys3 ) )
=> ( R3 @ ( nth_nat @ Xs3 @ I4 ) @ ( nth_nat @ Ys3 @ I4 ) ) ) )
| ? [Xs3: list_nat,Ys3: list_nat,I4: nat,Z3: nat] :
( ( A12 = Xs3 )
& ( A23
= ( list_update_nat @ Ys3 @ I4 @ Z3 ) )
& ( trancl_listp_nat @ R3 @ Xs3 @ Ys3 )
& ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys3 ) )
& ( R3 @ ( nth_nat @ Ys3 @ I4 ) @ Z3 ) ) ) ) ) ).
% trancl_listp.simps
thf(fact_164_trancl__listp_Olist__trancl,axiom,
! [R2: list_nat > list_nat > $o,Xs: list_list_nat,Ys: list_list_nat,I: nat,Z4: list_nat] :
( ( trancl267208031524270810st_nat @ R2 @ Xs @ Ys )
=> ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( R2 @ ( nth_list_nat @ Ys @ I ) @ Z4 )
=> ( trancl267208031524270810st_nat @ R2 @ Xs @ ( list_update_list_nat @ Ys @ I @ Z4 ) ) ) ) ) ).
% trancl_listp.list_trancl
thf(fact_165_trancl__listp_Olist__trancl,axiom,
! [R2: list_a > list_a > $o,Xs: list_list_a,Ys: list_list_a,I: nat,Z4: list_a] :
( ( trancl_listp_list_a @ R2 @ Xs @ Ys )
=> ( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ Ys ) )
=> ( ( R2 @ ( nth_list_a @ Ys @ I ) @ Z4 )
=> ( trancl_listp_list_a @ R2 @ Xs @ ( list_update_list_a @ Ys @ I @ Z4 ) ) ) ) ) ).
% trancl_listp.list_trancl
thf(fact_166_trancl__listp_Olist__trancl,axiom,
! [R2: a > a > $o,Xs: list_a,Ys: list_a,I: nat,Z4: a] :
( ( trancl_listp_a @ R2 @ Xs @ Ys )
=> ( ( ord_less_nat @ I @ ( size_size_list_a @ Ys ) )
=> ( ( R2 @ ( nth_a @ Ys @ I ) @ Z4 )
=> ( trancl_listp_a @ R2 @ Xs @ ( list_update_a @ Ys @ I @ Z4 ) ) ) ) ) ).
% trancl_listp.list_trancl
thf(fact_167_trancl__listp_Olist__trancl,axiom,
! [R2: nat > nat > $o,Xs: list_nat,Ys: list_nat,I: nat,Z4: nat] :
( ( trancl_listp_nat @ R2 @ Xs @ Ys )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
=> ( ( R2 @ ( nth_nat @ Ys @ I ) @ Z4 )
=> ( trancl_listp_nat @ R2 @ Xs @ ( list_update_nat @ Ys @ I @ Z4 ) ) ) ) ) ).
% trancl_listp.list_trancl
thf(fact_168_list__ex__cong,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,F2: product_prod_nat_nat > $o,G: product_prod_nat_nat > $o] :
( ( Xs = Ys )
=> ( ! [X2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ Ys ) )
=> ( ( F2 @ X2 )
= ( G @ X2 ) ) )
=> ( ( list_e7689525607045846085at_nat @ F2 @ Xs )
= ( list_e7689525607045846085at_nat @ G @ Ys ) ) ) ) ).
% list_ex_cong
thf(fact_169_list__ex__cong,axiom,
! [Xs: list_P1396940483166286381od_a_a,Ys: list_P1396940483166286381od_a_a,F2: product_prod_a_a > $o,G: product_prod_a_a > $o] :
( ( Xs = Ys )
=> ( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ ( set_Product_prod_a_a2 @ Ys ) )
=> ( ( F2 @ X2 )
= ( G @ X2 ) ) )
=> ( ( list_e6552556518106193515od_a_a @ F2 @ Xs )
= ( list_e6552556518106193515od_a_a @ G @ Ys ) ) ) ) ).
% list_ex_cong
thf(fact_170_list__ex__cong,axiom,
! [Xs: list_a,Ys: list_a,F2: a > $o,G: a > $o] :
( ( Xs = Ys )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Ys ) )
=> ( ( F2 @ X2 )
= ( G @ X2 ) ) )
=> ( ( list_ex_a @ F2 @ Xs )
= ( list_ex_a @ G @ Ys ) ) ) ) ).
% list_ex_cong
thf(fact_171_list__ex__cong,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,F2: list_nat > $o,G: list_nat > $o] :
( ( Xs = Ys )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Ys ) )
=> ( ( F2 @ X2 )
= ( G @ X2 ) ) )
=> ( ( list_ex_list_nat @ F2 @ Xs )
= ( list_ex_list_nat @ G @ Ys ) ) ) ) ).
% list_ex_cong
thf(fact_172_list__ex__cong,axiom,
! [Xs: list_list_a,Ys: list_list_a,F2: list_a > $o,G: list_a > $o] :
( ( Xs = Ys )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Ys ) )
=> ( ( F2 @ X2 )
= ( G @ X2 ) ) )
=> ( ( list_ex_list_a @ F2 @ Xs )
= ( list_ex_list_a @ G @ Ys ) ) ) ) ).
% list_ex_cong
thf(fact_173_list__ex__cong,axiom,
! [Xs: list_nat,Ys: list_nat,F2: nat > $o,G: nat > $o] :
( ( Xs = Ys )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Ys ) )
=> ( ( F2 @ X2 )
= ( G @ X2 ) ) )
=> ( ( list_ex_nat @ F2 @ Xs )
= ( list_ex_nat @ G @ Ys ) ) ) ) ).
% list_ex_cong
thf(fact_174_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ Xs )
=> ( distinct_nat @ Xs ) ) ).
% sorted_list_of_set.distinct_if_distinct_map
thf(fact_175_pfsubsetD,axiom,
! [A2: fset_nat,B: fset_nat,C2: nat] :
( ( ord_less_fset_nat @ A2 @ B )
=> ( ( fmember_nat @ C2 @ A2 )
=> ( fmember_nat @ C2 @ B ) ) ) ).
% pfsubsetD
thf(fact_176_pfsubsetD,axiom,
! [A2: fset_a,B: fset_a,C2: a] :
( ( ord_less_fset_a @ A2 @ B )
=> ( ( fmember_a @ C2 @ A2 )
=> ( fmember_a @ C2 @ B ) ) ) ).
% pfsubsetD
thf(fact_177_distinct__Ex1,axiom,
! [Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
( ( distin6923225563576452346at_nat @ Xs )
=> ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ? [X2: nat] :
( ( ord_less_nat @ X2 @ ( size_s5460976970255530739at_nat @ Xs ) )
& ( ( nth_Pr7617993195940197384at_nat @ Xs @ X2 )
= X )
& ! [Y4: nat] :
( ( ( ord_less_nat @ Y4 @ ( size_s5460976970255530739at_nat @ Xs ) )
& ( ( nth_Pr7617993195940197384at_nat @ Xs @ Y4 )
= X ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_178_distinct__Ex1,axiom,
! [Xs: list_P1396940483166286381od_a_a,X: product_prod_a_a] :
( ( distin132333870042060960od_a_a @ Xs )
=> ( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
=> ? [X2: nat] :
( ( ord_less_nat @ X2 @ ( size_s3885678630836030617od_a_a @ Xs ) )
& ( ( nth_Product_prod_a_a @ Xs @ X2 )
= X )
& ! [Y4: nat] :
( ( ( ord_less_nat @ Y4 @ ( size_s3885678630836030617od_a_a @ Xs ) )
& ( ( nth_Product_prod_a_a @ Xs @ Y4 )
= X ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_179_distinct__Ex1,axiom,
! [Xs: list_list_nat,X: list_nat] :
( ( distinct_list_nat @ Xs )
=> ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ? [X2: nat] :
( ( ord_less_nat @ X2 @ ( size_s3023201423986296836st_nat @ Xs ) )
& ( ( nth_list_nat @ Xs @ X2 )
= X )
& ! [Y4: nat] :
( ( ( ord_less_nat @ Y4 @ ( size_s3023201423986296836st_nat @ Xs ) )
& ( ( nth_list_nat @ Xs @ Y4 )
= X ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_180_distinct__Ex1,axiom,
! [Xs: list_list_a,X: list_a] :
( ( distinct_list_a @ Xs )
=> ( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ? [X2: nat] :
( ( ord_less_nat @ X2 @ ( size_s349497388124573686list_a @ Xs ) )
& ( ( nth_list_a @ Xs @ X2 )
= X )
& ! [Y4: nat] :
( ( ( ord_less_nat @ Y4 @ ( size_s349497388124573686list_a @ Xs ) )
& ( ( nth_list_a @ Xs @ Y4 )
= X ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_181_distinct__Ex1,axiom,
! [Xs: list_a,X: a] :
( ( distinct_a @ Xs )
=> ( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ? [X2: nat] :
( ( ord_less_nat @ X2 @ ( size_size_list_a @ Xs ) )
& ( ( nth_a @ Xs @ X2 )
= X )
& ! [Y4: nat] :
( ( ( ord_less_nat @ Y4 @ ( size_size_list_a @ Xs ) )
& ( ( nth_a @ Xs @ Y4 )
= X ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_182_distinct__Ex1,axiom,
! [Xs: list_nat,X: nat] :
( ( distinct_nat @ Xs )
=> ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ? [X2: nat] :
( ( ord_less_nat @ X2 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ X2 )
= X )
& ! [Y4: nat] :
( ( ( ord_less_nat @ Y4 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ Y4 )
= X ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_183_ex__set__conv__ex__nth,axiom,
! [Xs: list_list_nat,P: list_nat > $o] :
( ( ? [X4: list_nat] :
( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Xs ) )
& ( P @ ( nth_list_nat @ Xs @ I4 ) ) ) ) ) ).
% ex_set_conv_ex_nth
thf(fact_184_ex__set__conv__ex__nth,axiom,
! [Xs: list_list_a,P: list_a > $o] :
( ( ? [X4: list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s349497388124573686list_a @ Xs ) )
& ( P @ ( nth_list_a @ Xs @ I4 ) ) ) ) ) ).
% ex_set_conv_ex_nth
thf(fact_185_ex__set__conv__ex__nth,axiom,
! [Xs: list_a,P: a > $o] :
( ( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
& ( P @ ( nth_a @ Xs @ I4 ) ) ) ) ) ).
% ex_set_conv_ex_nth
thf(fact_186_ex__set__conv__ex__nth,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
& ( P @ X4 ) ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
& ( P @ ( nth_nat @ Xs @ I4 ) ) ) ) ) ).
% ex_set_conv_ex_nth
thf(fact_187_replicate__length__same,axiom,
! [Xs: list_list_nat,X: list_nat] :
( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( X2 = X ) )
=> ( ( replicate_list_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ X )
= Xs ) ) ).
% replicate_length_same
thf(fact_188_replicate__length__same,axiom,
! [Xs: list_list_a,X: list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
=> ( X2 = X ) )
=> ( ( replicate_list_a @ ( size_s349497388124573686list_a @ Xs ) @ X )
= Xs ) ) ).
% replicate_length_same
thf(fact_189_replicate__length__same,axiom,
! [Xs: list_a,X: a] :
( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( X2 = X ) )
=> ( ( replicate_a @ ( size_size_list_a @ Xs ) @ X )
= Xs ) ) ).
% replicate_length_same
thf(fact_190_replicate__length__same,axiom,
! [Xs: list_nat,X: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( X2 = X ) )
=> ( ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X )
= Xs ) ) ).
% replicate_length_same
thf(fact_191_replicate__eqI,axiom,
! [Xs: list_P6011104703257516679at_nat,N: nat,X: product_prod_nat_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs )
= N )
=> ( ! [Y3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Y3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ( Y3 = X ) )
=> ( Xs
= ( replic4235873036481779905at_nat @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_192_replicate__eqI,axiom,
! [Xs: list_P1396940483166286381od_a_a,N: nat,X: product_prod_a_a] :
( ( ( size_s3885678630836030617od_a_a @ Xs )
= N )
=> ( ! [Y3: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y3 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( Y3 = X ) )
=> ( Xs
= ( replic8880659463222285543od_a_a @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_193_replicate__eqI,axiom,
! [Xs: list_list_nat,N: nat,X: list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= N )
=> ( ! [Y3: list_nat] :
( ( member_list_nat @ Y3 @ ( set_list_nat2 @ Xs ) )
=> ( Y3 = X ) )
=> ( Xs
= ( replicate_list_nat @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_194_replicate__eqI,axiom,
! [Xs: list_list_a,N: nat,X: list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
= N )
=> ( ! [Y3: list_a] :
( ( member_list_a @ Y3 @ ( set_list_a2 @ Xs ) )
=> ( Y3 = X ) )
=> ( Xs
= ( replicate_list_a @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_195_replicate__eqI,axiom,
! [Xs: list_a,N: nat,X: a] :
( ( ( size_size_list_a @ Xs )
= N )
=> ( ! [Y3: a] :
( ( member_a @ Y3 @ ( set_a2 @ Xs ) )
=> ( Y3 = X ) )
=> ( Xs
= ( replicate_a @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_196_replicate__eqI,axiom,
! [Xs: list_nat,N: nat,X: nat] :
( ( ( size_size_list_nat @ Xs )
= N )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ ( set_nat2 @ Xs ) )
=> ( Y3 = X ) )
=> ( Xs
= ( replicate_nat @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_197_distinct__conv__nth,axiom,
( distinct_list_nat
= ( ^ [Xs3: list_list_nat] :
! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Xs3 ) )
=> ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_s3023201423986296836st_nat @ Xs3 ) )
=> ( ( I4 != J2 )
=> ( ( nth_list_nat @ Xs3 @ I4 )
!= ( nth_list_nat @ Xs3 @ J2 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_198_distinct__conv__nth,axiom,
( distinct_list_a
= ( ^ [Xs3: list_list_a] :
! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s349497388124573686list_a @ Xs3 ) )
=> ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_s349497388124573686list_a @ Xs3 ) )
=> ( ( I4 != J2 )
=> ( ( nth_list_a @ Xs3 @ I4 )
!= ( nth_list_a @ Xs3 @ J2 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_199_distinct__conv__nth,axiom,
( distinct_a
= ( ^ [Xs3: list_a] :
! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs3 ) )
=> ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_size_list_a @ Xs3 ) )
=> ( ( I4 != J2 )
=> ( ( nth_a @ Xs3 @ I4 )
!= ( nth_a @ Xs3 @ J2 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_200_distinct__conv__nth,axiom,
( distinct_nat
= ( ^ [Xs3: list_nat] :
! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs3 ) )
=> ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs3 ) )
=> ( ( I4 != J2 )
=> ( ( nth_nat @ Xs3 @ I4 )
!= ( nth_nat @ Xs3 @ J2 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_201_nth__eq__iff__index__eq,axiom,
! [Xs: list_list_nat,I: nat,J: nat] :
( ( distinct_list_nat @ Xs )
=> ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( ( nth_list_nat @ Xs @ I )
= ( nth_list_nat @ Xs @ J ) )
= ( I = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_202_nth__eq__iff__index__eq,axiom,
! [Xs: list_list_a,I: nat,J: nat] :
( ( distinct_list_a @ Xs )
=> ( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( ( nth_list_a @ Xs @ I )
= ( nth_list_a @ Xs @ J ) )
= ( I = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_203_nth__eq__iff__index__eq,axiom,
! [Xs: list_a,I: nat,J: nat] :
( ( distinct_a @ Xs )
=> ( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs ) )
=> ( ( ( nth_a @ Xs @ I )
= ( nth_a @ Xs @ J ) )
= ( I = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_204_nth__eq__iff__index__eq,axiom,
! [Xs: list_nat,I: nat,J: nat] :
( ( distinct_nat @ Xs )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( ( nth_nat @ Xs @ I )
= ( nth_nat @ Xs @ J ) )
= ( I = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_205_trancl__listp_Obase,axiom,
! [Xs: list_list_nat,Ys: list_list_nat,R2: list_nat > list_nat > $o] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Ys ) )
=> ( R2 @ ( nth_list_nat @ Xs @ I3 ) @ ( nth_list_nat @ Ys @ I3 ) ) )
=> ( trancl267208031524270810st_nat @ R2 @ Xs @ Ys ) ) ) ).
% trancl_listp.base
thf(fact_206_trancl__listp_Obase,axiom,
! [Xs: list_list_a,Ys: list_list_a,R2: list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s349497388124573686list_a @ Ys ) )
=> ( R2 @ ( nth_list_a @ Xs @ I3 ) @ ( nth_list_a @ Ys @ I3 ) ) )
=> ( trancl_listp_list_a @ R2 @ Xs @ Ys ) ) ) ).
% trancl_listp.base
thf(fact_207_trancl__listp_Obase,axiom,
! [Xs: list_a,Ys: list_a,R2: a > a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Ys ) )
=> ( R2 @ ( nth_a @ Xs @ I3 ) @ ( nth_a @ Ys @ I3 ) ) )
=> ( trancl_listp_a @ R2 @ Xs @ Ys ) ) ) ).
% trancl_listp.base
thf(fact_208_trancl__listp_Obase,axiom,
! [Xs: list_nat,Ys: list_nat,R2: nat > nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys ) )
=> ( R2 @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Ys @ I3 ) ) )
=> ( trancl_listp_nat @ R2 @ Xs @ Ys ) ) ) ).
% trancl_listp.base
thf(fact_209_nth__mem,axiom,
! [N: nat,Xs: list_P6011104703257516679at_nat] :
( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( member8440522571783428010at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ N ) @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).
% nth_mem
thf(fact_210_nth__mem,axiom,
! [N: nat,Xs: list_P1396940483166286381od_a_a] :
( ( ord_less_nat @ N @ ( size_s3885678630836030617od_a_a @ Xs ) )
=> ( member1426531477525435216od_a_a @ ( nth_Product_prod_a_a @ Xs @ N ) @ ( set_Product_prod_a_a2 @ Xs ) ) ) ).
% nth_mem
thf(fact_211_nth__mem,axiom,
! [N: nat,Xs: list_list_nat] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( member_list_nat @ ( nth_list_nat @ Xs @ N ) @ ( set_list_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_212_nth__mem,axiom,
! [N: nat,Xs: list_list_a] :
( ( ord_less_nat @ N @ ( size_s349497388124573686list_a @ Xs ) )
=> ( member_list_a @ ( nth_list_a @ Xs @ N ) @ ( set_list_a2 @ Xs ) ) ) ).
% nth_mem
thf(fact_213_nth__mem,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( member_a @ ( nth_a @ Xs @ N ) @ ( set_a2 @ Xs ) ) ) ).
% nth_mem
thf(fact_214_nth__mem,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_215_list__ball__nth,axiom,
! [N: nat,Xs: list_list_nat,P: list_nat > $o] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( P @ X2 ) )
=> ( P @ ( nth_list_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_216_list__ball__nth,axiom,
! [N: nat,Xs: list_list_a,P: list_a > $o] :
( ( ord_less_nat @ N @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
=> ( P @ X2 ) )
=> ( P @ ( nth_list_a @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_217_list__ball__nth,axiom,
! [N: nat,Xs: list_a,P: a > $o] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( P @ X2 ) )
=> ( P @ ( nth_a @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_218_list__ball__nth,axiom,
! [N: nat,Xs: list_nat,P: nat > $o] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( P @ X2 ) )
=> ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_219_in__set__conv__nth,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s5460976970255530739at_nat @ Xs ) )
& ( ( nth_Pr7617993195940197384at_nat @ Xs @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_220_in__set__conv__nth,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3885678630836030617od_a_a @ Xs ) )
& ( ( nth_Product_prod_a_a @ Xs @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_221_in__set__conv__nth,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Xs ) )
& ( ( nth_list_nat @ Xs @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_222_in__set__conv__nth,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s349497388124573686list_a @ Xs ) )
& ( ( nth_list_a @ Xs @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_223_in__set__conv__nth,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
& ( ( nth_a @ Xs @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_224_in__set__conv__nth,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_225_all__nth__imp__all__set,axiom,
! [Xs: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,X: product_prod_nat_nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ I3 ) ) )
=> ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_226_all__nth__imp__all__set,axiom,
! [Xs: list_P1396940483166286381od_a_a,P: product_prod_a_a > $o,X: product_prod_a_a] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3885678630836030617od_a_a @ Xs ) )
=> ( P @ ( nth_Product_prod_a_a @ Xs @ I3 ) ) )
=> ( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_227_all__nth__imp__all__set,axiom,
! [Xs: list_list_nat,P: list_nat > $o,X: list_nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( P @ ( nth_list_nat @ Xs @ I3 ) ) )
=> ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_228_all__nth__imp__all__set,axiom,
! [Xs: list_list_a,P: list_a > $o,X: list_a] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( P @ ( nth_list_a @ Xs @ I3 ) ) )
=> ( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_229_all__nth__imp__all__set,axiom,
! [Xs: list_a,P: a > $o,X: a] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
=> ( P @ ( nth_a @ Xs @ I3 ) ) )
=> ( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_230_all__nth__imp__all__set,axiom,
! [Xs: list_nat,P: nat > $o,X: nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I3 ) ) )
=> ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_231_all__set__conv__all__nth,axiom,
! [Xs: list_list_nat,P: list_nat > $o] :
( ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xs ) )
=> ( P @ X4 ) ) )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( P @ ( nth_list_nat @ Xs @ I4 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_232_all__set__conv__all__nth,axiom,
! [Xs: list_list_a,P: list_a > $o] :
( ( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ Xs ) )
=> ( P @ X4 ) ) )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( P @ ( nth_list_a @ Xs @ I4 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_233_all__set__conv__all__nth,axiom,
! [Xs: list_a,P: a > $o] :
( ( ! [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
=> ( P @ X4 ) ) )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
=> ( P @ ( nth_a @ Xs @ I4 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_234_all__set__conv__all__nth,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ( P @ X4 ) ) )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I4 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_235_set__update__memI,axiom,
! [N: nat,Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_236_set__update__memI,axiom,
! [N: nat,Xs: list_P1396940483166286381od_a_a,X: product_prod_a_a] :
( ( ord_less_nat @ N @ ( size_s3885678630836030617od_a_a @ Xs ) )
=> ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ ( list_u4967751623001292181od_a_a @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_237_set__update__memI,axiom,
! [N: nat,Xs: list_list_nat,X: list_nat] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( member_list_nat @ X @ ( set_list_nat2 @ ( list_update_list_nat @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_238_set__update__memI,axiom,
! [N: nat,Xs: list_list_a,X: list_a] :
( ( ord_less_nat @ N @ ( size_s349497388124573686list_a @ Xs ) )
=> ( member_list_a @ X @ ( set_list_a2 @ ( list_update_list_a @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_239_set__update__memI,axiom,
! [N: nat,Xs: list_a,X: a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( member_a @ X @ ( set_a2 @ ( list_update_a @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_240_set__update__memI,axiom,
! [N: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_241_in__set__idx,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs ) )
& ( ( nth_Pr7617993195940197384at_nat @ Xs @ I3 )
= X ) ) ) ).
% in_set_idx
thf(fact_242_in__set__idx,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
=> ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3885678630836030617od_a_a @ Xs ) )
& ( ( nth_Product_prod_a_a @ Xs @ I3 )
= X ) ) ) ).
% in_set_idx
thf(fact_243_in__set__idx,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs ) )
& ( ( nth_list_nat @ Xs @ I3 )
= X ) ) ) ).
% in_set_idx
thf(fact_244_in__set__idx,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s349497388124573686list_a @ Xs ) )
& ( ( nth_list_a @ Xs @ I3 )
= X ) ) ) ).
% in_set_idx
thf(fact_245_in__set__idx,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
& ( ( nth_a @ Xs @ I3 )
= X ) ) ) ).
% in_set_idx
thf(fact_246_in__set__idx,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I3 )
= X ) ) ) ).
% in_set_idx
thf(fact_247_distinct__union,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( distinct_nat @ ( union_nat @ Xs @ Ys ) )
= ( distinct_nat @ Ys ) ) ).
% distinct_union
thf(fact_248_distinct__union,axiom,
! [Xs: list_a,Ys: list_a] :
( ( distinct_a @ ( union_a @ Xs @ Ys ) )
= ( distinct_a @ Ys ) ) ).
% distinct_union
thf(fact_249_permut__aux__sound,axiom,
! [I: nat,As: list_list_nat,F2: nat > nat,Bs: list_list_nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ As ) )
=> ( ( nth_list_nat @ ( missin3204601078308456222st_nat @ As @ F2 @ Bs ) @ I )
= ( nth_list_nat @ Bs @ ( F2 @ I ) ) ) ) ).
% permut_aux_sound
thf(fact_250_permut__aux__sound,axiom,
! [I: nat,As: list_list_a,F2: nat > nat,Bs: list_list_a] :
( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ As ) )
=> ( ( nth_list_a @ ( missin5389789484201179846list_a @ As @ F2 @ Bs ) @ I )
= ( nth_list_a @ Bs @ ( F2 @ I ) ) ) ) ).
% permut_aux_sound
thf(fact_251_permut__aux__sound,axiom,
! [I: nat,As: list_a,F2: nat > nat,Bs: list_a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ As ) )
=> ( ( nth_a @ ( missing_permut_aux_a @ As @ F2 @ Bs ) @ I )
= ( nth_a @ Bs @ ( F2 @ I ) ) ) ) ).
% permut_aux_sound
thf(fact_252_permut__aux__sound,axiom,
! [I: nat,As: list_nat,F2: nat > nat,Bs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ As ) )
=> ( ( nth_nat @ ( missin1888654203714970382ux_nat @ As @ F2 @ Bs ) @ I )
= ( nth_nat @ Bs @ ( F2 @ I ) ) ) ) ).
% permut_aux_sound
thf(fact_253_remdups__sort_I2_J,axiom,
! [Xs: list_list_nat] :
( ( set_list_nat2 @ ( missin2640887136080315381st_nat @ Xs ) )
= ( set_list_nat2 @ Xs ) ) ).
% remdups_sort(2)
thf(fact_254_remdups__sort_I2_J,axiom,
! [Xs: list_nat] :
( ( set_nat2 @ ( missin6101193410121742181rt_nat @ Xs ) )
= ( set_nat2 @ Xs ) ) ).
% remdups_sort(2)
thf(fact_255_remove__nth__P__compat,axiom,
! [As: list_a,Bs: list_a,P: a > a > $o,P2: nat] :
( ( ( size_size_list_a @ As )
= ( size_size_list_a @ Bs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ As ) )
=> ( P @ ( nth_a @ As @ I3 ) @ ( nth_a @ Bs @ I3 ) ) )
=> ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_a @ ( missing_remove_nth_a @ P2 @ As ) ) )
=> ( P @ ( nth_a @ ( missing_remove_nth_a @ P2 @ As ) @ I5 ) @ ( nth_a @ ( missing_remove_nth_a @ P2 @ Bs ) @ I5 ) ) ) ) ) ).
% remove_nth_P_compat
thf(fact_256_remove__nth__P__compat,axiom,
! [As: list_a,Bs: list_nat,P: a > nat > $o,P2: nat] :
( ( ( size_size_list_a @ As )
= ( size_size_list_nat @ Bs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ As ) )
=> ( P @ ( nth_a @ As @ I3 ) @ ( nth_nat @ Bs @ I3 ) ) )
=> ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_a @ ( missing_remove_nth_a @ P2 @ As ) ) )
=> ( P @ ( nth_a @ ( missing_remove_nth_a @ P2 @ As ) @ I5 ) @ ( nth_nat @ ( missin7175274867594579095th_nat @ P2 @ Bs ) @ I5 ) ) ) ) ) ).
% remove_nth_P_compat
thf(fact_257_remove__nth__P__compat,axiom,
! [As: list_nat,Bs: list_a,P: nat > a > $o,P2: nat] :
( ( ( size_size_list_nat @ As )
= ( size_size_list_a @ Bs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ As ) )
=> ( P @ ( nth_nat @ As @ I3 ) @ ( nth_a @ Bs @ I3 ) ) )
=> ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ P2 @ As ) ) )
=> ( P @ ( nth_nat @ ( missin7175274867594579095th_nat @ P2 @ As ) @ I5 ) @ ( nth_a @ ( missing_remove_nth_a @ P2 @ Bs ) @ I5 ) ) ) ) ) ).
% remove_nth_P_compat
thf(fact_258_remove__nth__P__compat,axiom,
! [As: list_nat,Bs: list_nat,P: nat > nat > $o,P2: nat] :
( ( ( size_size_list_nat @ As )
= ( size_size_list_nat @ Bs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ As ) )
=> ( P @ ( nth_nat @ As @ I3 ) @ ( nth_nat @ Bs @ I3 ) ) )
=> ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ P2 @ As ) ) )
=> ( P @ ( nth_nat @ ( missin7175274867594579095th_nat @ P2 @ As ) @ I5 ) @ ( nth_nat @ ( missin7175274867594579095th_nat @ P2 @ Bs ) @ I5 ) ) ) ) ) ).
% remove_nth_P_compat
thf(fact_259_remove__nth__P__compat,axiom,
! [As: list_list_nat,Bs: list_nat,P: list_nat > nat > $o,P2: nat] :
( ( ( size_s3023201423986296836st_nat @ As )
= ( size_size_list_nat @ Bs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ As ) )
=> ( P @ ( nth_list_nat @ As @ I3 ) @ ( nth_nat @ Bs @ I3 ) ) )
=> ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_s3023201423986296836st_nat @ ( missin5133812807938250791st_nat @ P2 @ As ) ) )
=> ( P @ ( nth_list_nat @ ( missin5133812807938250791st_nat @ P2 @ As ) @ I5 ) @ ( nth_nat @ ( missin7175274867594579095th_nat @ P2 @ Bs ) @ I5 ) ) ) ) ) ).
% remove_nth_P_compat
thf(fact_260_remove__nth__P__compat,axiom,
! [As: list_list_a,Bs: list_nat,P: list_a > nat > $o,P2: nat] :
( ( ( size_s349497388124573686list_a @ As )
= ( size_size_list_nat @ Bs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s349497388124573686list_a @ As ) )
=> ( P @ ( nth_list_a @ As @ I3 ) @ ( nth_nat @ Bs @ I3 ) ) )
=> ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_s349497388124573686list_a @ ( missin8041159176150063613list_a @ P2 @ As ) ) )
=> ( P @ ( nth_list_a @ ( missin8041159176150063613list_a @ P2 @ As ) @ I5 ) @ ( nth_nat @ ( missin7175274867594579095th_nat @ P2 @ Bs ) @ I5 ) ) ) ) ) ).
% remove_nth_P_compat
thf(fact_261_remove__nth__P__compat,axiom,
! [As: list_list_nat,Bs: list_a,P: list_nat > a > $o,P2: nat] :
( ( ( size_s3023201423986296836st_nat @ As )
= ( size_size_list_a @ Bs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ As ) )
=> ( P @ ( nth_list_nat @ As @ I3 ) @ ( nth_a @ Bs @ I3 ) ) )
=> ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_s3023201423986296836st_nat @ ( missin5133812807938250791st_nat @ P2 @ As ) ) )
=> ( P @ ( nth_list_nat @ ( missin5133812807938250791st_nat @ P2 @ As ) @ I5 ) @ ( nth_a @ ( missing_remove_nth_a @ P2 @ Bs ) @ I5 ) ) ) ) ) ).
% remove_nth_P_compat
thf(fact_262_remove__nth__P__compat,axiom,
! [As: list_list_a,Bs: list_a,P: list_a > a > $o,P2: nat] :
( ( ( size_s349497388124573686list_a @ As )
= ( size_size_list_a @ Bs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s349497388124573686list_a @ As ) )
=> ( P @ ( nth_list_a @ As @ I3 ) @ ( nth_a @ Bs @ I3 ) ) )
=> ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_s349497388124573686list_a @ ( missin8041159176150063613list_a @ P2 @ As ) ) )
=> ( P @ ( nth_list_a @ ( missin8041159176150063613list_a @ P2 @ As ) @ I5 ) @ ( nth_a @ ( missing_remove_nth_a @ P2 @ Bs ) @ I5 ) ) ) ) ) ).
% remove_nth_P_compat
thf(fact_263_remove__nth__P__compat,axiom,
! [As: list_nat,Bs: list_list_nat,P: nat > list_nat > $o,P2: nat] :
( ( ( size_size_list_nat @ As )
= ( size_s3023201423986296836st_nat @ Bs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ As ) )
=> ( P @ ( nth_nat @ As @ I3 ) @ ( nth_list_nat @ Bs @ I3 ) ) )
=> ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ P2 @ As ) ) )
=> ( P @ ( nth_nat @ ( missin7175274867594579095th_nat @ P2 @ As ) @ I5 ) @ ( nth_list_nat @ ( missin5133812807938250791st_nat @ P2 @ Bs ) @ I5 ) ) ) ) ) ).
% remove_nth_P_compat
thf(fact_264_remove__nth__P__compat,axiom,
! [As: list_nat,Bs: list_list_a,P: nat > list_a > $o,P2: nat] :
( ( ( size_size_list_nat @ As )
= ( size_s349497388124573686list_a @ Bs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ As ) )
=> ( P @ ( nth_nat @ As @ I3 ) @ ( nth_list_a @ Bs @ I3 ) ) )
=> ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ P2 @ As ) ) )
=> ( P @ ( nth_nat @ ( missin7175274867594579095th_nat @ P2 @ As ) @ I5 ) @ ( nth_list_a @ ( missin8041159176150063613list_a @ P2 @ Bs ) @ I5 ) ) ) ) ) ).
% remove_nth_P_compat
thf(fact_265_in__fset__idx,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( fmember_list_nat @ X @ ( fset_o8830822285705182106st_nat @ Xs ) )
=> ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs ) )
& ( ( nth_list_nat @ Xs @ I3 )
= X ) ) ) ).
% in_fset_idx
thf(fact_266_in__fset__idx,axiom,
! [X: list_a,Xs: list_list_a] :
( ( fmember_list_a @ X @ ( fset_of_list_list_a @ Xs ) )
=> ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s349497388124573686list_a @ Xs ) )
& ( ( nth_list_a @ Xs @ I3 )
= X ) ) ) ).
% in_fset_idx
thf(fact_267_in__fset__idx,axiom,
! [X: a,Xs: list_a] :
( ( fmember_a @ X @ ( fset_of_list_a @ Xs ) )
=> ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
& ( ( nth_a @ Xs @ I3 )
= X ) ) ) ).
% in_fset_idx
thf(fact_268_in__fset__idx,axiom,
! [X: nat,Xs: list_nat] :
( ( fmember_nat @ X @ ( fset_of_list_nat @ Xs ) )
=> ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I3 )
= X ) ) ) ).
% in_fset_idx
thf(fact_269_permut__def,axiom,
( missing_permut_nat
= ( ^ [As2: list_nat,F3: nat > nat] : ( missin1888654203714970382ux_nat @ As2 @ F3 @ As2 ) ) ) ).
% permut_def
thf(fact_270_permut__def,axiom,
( missing_permut_a
= ( ^ [As2: list_a,F3: nat > nat] : ( missing_permut_aux_a @ As2 @ F3 @ As2 ) ) ) ).
% permut_def
thf(fact_271_subseteq__set__conv__nth,axiom,
! [Ss: list_P6011104703257516679at_nat,T2: set_Pr1261947904930325089at_nat] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s5460976970255530739at_nat @ Ss ) )
=> ( member8440522571783428010at_nat @ ( nth_Pr7617993195940197384at_nat @ Ss @ I4 ) @ T2 ) ) )
= ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Ss ) @ T2 ) ) ).
% subseteq_set_conv_nth
thf(fact_272_subseteq__set__conv__nth,axiom,
! [Ss: list_P1396940483166286381od_a_a,T2: set_Product_prod_a_a] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3885678630836030617od_a_a @ Ss ) )
=> ( member1426531477525435216od_a_a @ ( nth_Product_prod_a_a @ Ss @ I4 ) @ T2 ) ) )
= ( ord_le746702958409616551od_a_a @ ( set_Product_prod_a_a2 @ Ss ) @ T2 ) ) ).
% subseteq_set_conv_nth
thf(fact_273_subseteq__set__conv__nth,axiom,
! [Ss: list_list_nat,T2: set_list_nat] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Ss ) )
=> ( member_list_nat @ ( nth_list_nat @ Ss @ I4 ) @ T2 ) ) )
= ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Ss ) @ T2 ) ) ).
% subseteq_set_conv_nth
thf(fact_274_subseteq__set__conv__nth,axiom,
! [Ss: list_list_a,T2: set_list_a] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s349497388124573686list_a @ Ss ) )
=> ( member_list_a @ ( nth_list_a @ Ss @ I4 ) @ T2 ) ) )
= ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ss ) @ T2 ) ) ).
% subseteq_set_conv_nth
thf(fact_275_subseteq__set__conv__nth,axiom,
! [Ss: list_a,T2: set_a] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Ss ) )
=> ( member_a @ ( nth_a @ Ss @ I4 ) @ T2 ) ) )
= ( ord_less_eq_set_a @ ( set_a2 @ Ss ) @ T2 ) ) ).
% subseteq_set_conv_nth
thf(fact_276_subseteq__set__conv__nth,axiom,
! [Ss: list_nat,T2: set_nat] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ss ) )
=> ( member_nat @ ( nth_nat @ Ss @ I4 ) @ T2 ) ) )
= ( ord_less_eq_set_nat @ ( set_nat2 @ Ss ) @ T2 ) ) ).
% subseteq_set_conv_nth
thf(fact_277_set__list__subset__nth__conv,axiom,
! [Xs: list_P6011104703257516679at_nat,A2: set_Pr1261947904930325089at_nat,I: nat] :
( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ A2 )
=> ( ( ord_less_nat @ I @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( member8440522571783428010at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ I ) @ A2 ) ) ) ).
% set_list_subset_nth_conv
thf(fact_278_set__list__subset__nth__conv,axiom,
! [Xs: list_P1396940483166286381od_a_a,A2: set_Product_prod_a_a,I: nat] :
( ( ord_le746702958409616551od_a_a @ ( set_Product_prod_a_a2 @ Xs ) @ A2 )
=> ( ( ord_less_nat @ I @ ( size_s3885678630836030617od_a_a @ Xs ) )
=> ( member1426531477525435216od_a_a @ ( nth_Product_prod_a_a @ Xs @ I ) @ A2 ) ) ) ).
% set_list_subset_nth_conv
thf(fact_279_set__list__subset__nth__conv,axiom,
! [Xs: list_list_nat,A2: set_list_nat,I: nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ A2 )
=> ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( member_list_nat @ ( nth_list_nat @ Xs @ I ) @ A2 ) ) ) ).
% set_list_subset_nth_conv
thf(fact_280_set__list__subset__nth__conv,axiom,
! [Xs: list_list_a,A2: set_list_a,I: nat] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ A2 )
=> ( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ Xs ) )
=> ( member_list_a @ ( nth_list_a @ Xs @ I ) @ A2 ) ) ) ).
% set_list_subset_nth_conv
thf(fact_281_set__list__subset__nth__conv,axiom,
! [Xs: list_a,A2: set_a,I: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ A2 )
=> ( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( member_a @ ( nth_a @ Xs @ I ) @ A2 ) ) ) ).
% set_list_subset_nth_conv
thf(fact_282_set__list__subset__nth__conv,axiom,
! [Xs: list_nat,A2: set_nat,I: nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ ( nth_nat @ Xs @ I ) @ A2 ) ) ) ).
% set_list_subset_nth_conv
thf(fact_283_length__removeAll__less,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ( ord_less_nat @ ( size_s5460976970255530739at_nat @ ( remove3673390508374433037at_nat @ X @ Xs ) ) @ ( size_s5460976970255530739at_nat @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_284_length__removeAll__less,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( ord_less_nat @ ( size_s3885678630836030617od_a_a @ ( remove3678230896347538227od_a_a @ X @ Xs ) ) @ ( size_s3885678630836030617od_a_a @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_285_length__removeAll__less,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ( ord_less_nat @ ( size_s3023201423986296836st_nat @ ( removeAll_list_nat @ X @ Xs ) ) @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_286_length__removeAll__less,axiom,
! [X: list_a,Xs: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ( ord_less_nat @ ( size_s349497388124573686list_a @ ( removeAll_list_a @ X @ Xs ) ) @ ( size_s349497388124573686list_a @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_287_length__removeAll__less,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ord_less_nat @ ( size_size_list_a @ ( removeAll_a @ X @ Xs ) ) @ ( size_size_list_a @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_288_length__removeAll__less,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ ( size_size_list_nat @ ( removeAll_nat @ X @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_289_removeAll__id,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ( ( remove3673390508374433037at_nat @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_290_removeAll__id,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ~ ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( ( remove3678230896347538227od_a_a @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_291_removeAll__id,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ~ ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ( ( removeAll_list_nat @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_292_removeAll__id,axiom,
! [X: list_a,Xs: list_list_a] :
( ~ ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ( ( removeAll_list_a @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_293_removeAll__id,axiom,
! [X: a,Xs: list_a] :
( ~ ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ( removeAll_a @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_294_removeAll__id,axiom,
! [X: nat,Xs: list_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( removeAll_nat @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_295_pfsubset__trans,axiom,
! [A2: fset_a,B: fset_a,C3: fset_a] :
( ( ord_less_fset_a @ A2 @ B )
=> ( ( ord_less_fset_a @ B @ C3 )
=> ( ord_less_fset_a @ A2 @ C3 ) ) ) ).
% pfsubset_trans
thf(fact_296_exists__fset__of__list,axiom,
! [S2: fset_nat] :
? [Xs2: list_nat] :
( ( fset_of_list_nat @ Xs2 )
= S2 ) ).
% exists_fset_of_list
thf(fact_297_exists__fset__of__list,axiom,
! [S2: fset_a] :
? [Xs2: list_a] :
( ( fset_of_list_a @ Xs2 )
= S2 ) ).
% exists_fset_of_list
thf(fact_298_set__removeAll__subset,axiom,
! [X: list_nat,Xs: list_list_nat] : ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ ( removeAll_list_nat @ X @ Xs ) ) @ ( set_list_nat2 @ Xs ) ) ).
% set_removeAll_subset
thf(fact_299_set__removeAll__subset,axiom,
! [X: list_a,Xs: list_list_a] : ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( removeAll_list_a @ X @ Xs ) ) @ ( set_list_a2 @ Xs ) ) ).
% set_removeAll_subset
thf(fact_300_set__removeAll__subset,axiom,
! [X: nat,Xs: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( removeAll_nat @ X @ Xs ) ) @ ( set_nat2 @ Xs ) ) ).
% set_removeAll_subset
thf(fact_301_set__removeAll__subset,axiom,
! [X: a,Xs: list_a] : ( ord_less_eq_set_a @ ( set_a2 @ ( removeAll_a @ X @ Xs ) ) @ ( set_a2 @ Xs ) ) ).
% set_removeAll_subset
thf(fact_302_subset__code_I1_J,axiom,
! [Xs: list_P6011104703257516679at_nat,B: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ B )
= ( ! [X4: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ( member8440522571783428010at_nat @ X4 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_303_subset__code_I1_J,axiom,
! [Xs: list_P1396940483166286381od_a_a,B: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( set_Product_prod_a_a2 @ Xs ) @ B )
= ( ! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( member1426531477525435216od_a_a @ X4 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_304_subset__code_I1_J,axiom,
! [Xs: list_list_nat,B: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ B )
= ( ! [X4: list_nat] :
( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xs ) )
=> ( member_list_nat @ X4 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_305_subset__code_I1_J,axiom,
! [Xs: list_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ B )
= ( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ Xs ) )
=> ( member_list_a @ X4 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_306_subset__code_I1_J,axiom,
! [Xs: list_a,B: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B )
= ( ! [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
=> ( member_a @ X4 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_307_subset__code_I1_J,axiom,
! [Xs: list_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X4 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_308_distinct__removeAll,axiom,
! [Xs: list_nat,X: nat] :
( ( distinct_nat @ Xs )
=> ( distinct_nat @ ( removeAll_nat @ X @ Xs ) ) ) ).
% distinct_removeAll
thf(fact_309_distinct__removeAll,axiom,
! [Xs: list_a,X: a] :
( ( distinct_a @ Xs )
=> ( distinct_a @ ( removeAll_a @ X @ Xs ) ) ) ).
% distinct_removeAll
thf(fact_310_fset__of__list__elem,axiom,
! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( fmembe1449479052724974408at_nat @ X @ ( fset_o7974655632839724743at_nat @ Xs ) )
= ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).
% fset_of_list_elem
thf(fact_311_fset__of__list__elem,axiom,
! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( fmembe7738088516360075246od_a_a @ X @ ( fset_o3702720905070231533od_a_a @ Xs ) )
= ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) ) ) ).
% fset_of_list_elem
thf(fact_312_fset__of__list__elem,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( fmember_list_nat @ X @ ( fset_o8830822285705182106st_nat @ Xs ) )
= ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) ) ) ).
% fset_of_list_elem
thf(fact_313_fset__of__list__elem,axiom,
! [X: list_a,Xs: list_list_a] :
( ( fmember_list_a @ X @ ( fset_of_list_list_a @ Xs ) )
= ( member_list_a @ X @ ( set_list_a2 @ Xs ) ) ) ).
% fset_of_list_elem
thf(fact_314_fset__of__list__elem,axiom,
! [X: nat,Xs: list_nat] :
( ( fmember_nat @ X @ ( fset_of_list_nat @ Xs ) )
= ( member_nat @ X @ ( set_nat2 @ Xs ) ) ) ).
% fset_of_list_elem
thf(fact_315_fset__of__list__elem,axiom,
! [X: a,Xs: list_a] :
( ( fmember_a @ X @ ( fset_of_list_a @ Xs ) )
= ( member_a @ X @ ( set_a2 @ Xs ) ) ) ).
% fset_of_list_elem
thf(fact_316_set__update__subsetI,axiom,
! [Xs: list_P6011104703257516679at_nat,A2: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,I: nat] :
( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ A2 )
=> ( ( member8440522571783428010at_nat @ X @ A2 )
=> ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs @ I @ X ) ) @ A2 ) ) ) ).
% set_update_subsetI
thf(fact_317_set__update__subsetI,axiom,
! [Xs: list_P1396940483166286381od_a_a,A2: set_Product_prod_a_a,X: product_prod_a_a,I: nat] :
( ( ord_le746702958409616551od_a_a @ ( set_Product_prod_a_a2 @ Xs ) @ A2 )
=> ( ( member1426531477525435216od_a_a @ X @ A2 )
=> ( ord_le746702958409616551od_a_a @ ( set_Product_prod_a_a2 @ ( list_u4967751623001292181od_a_a @ Xs @ I @ X ) ) @ A2 ) ) ) ).
% set_update_subsetI
thf(fact_318_set__update__subsetI,axiom,
! [Xs: list_list_nat,A2: set_list_nat,X: list_nat,I: nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ A2 )
=> ( ( member_list_nat @ X @ A2 )
=> ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ ( list_update_list_nat @ Xs @ I @ X ) ) @ A2 ) ) ) ).
% set_update_subsetI
thf(fact_319_set__update__subsetI,axiom,
! [Xs: list_list_a,A2: set_list_a,X: list_a,I: nat] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ A2 )
=> ( ( member_list_a @ X @ A2 )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( list_update_list_a @ Xs @ I @ X ) ) @ A2 ) ) ) ).
% set_update_subsetI
thf(fact_320_set__update__subsetI,axiom,
! [Xs: list_nat,A2: set_nat,X: nat,I: nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X ) ) @ A2 ) ) ) ).
% set_update_subsetI
thf(fact_321_set__update__subsetI,axiom,
! [Xs: list_a,A2: set_a,X: a,I: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ A2 )
=> ( ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( list_update_a @ Xs @ I @ X ) ) @ A2 ) ) ) ).
% set_update_subsetI
thf(fact_322_set__list__subset__eq__nth__conv,axiom,
! [Xs: list_P6011104703257516679at_nat,A2: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ A2 )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( member8440522571783428010at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ I4 ) @ A2 ) ) ) ) ).
% set_list_subset_eq_nth_conv
thf(fact_323_set__list__subset__eq__nth__conv,axiom,
! [Xs: list_P1396940483166286381od_a_a,A2: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( set_Product_prod_a_a2 @ Xs ) @ A2 )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3885678630836030617od_a_a @ Xs ) )
=> ( member1426531477525435216od_a_a @ ( nth_Product_prod_a_a @ Xs @ I4 ) @ A2 ) ) ) ) ).
% set_list_subset_eq_nth_conv
thf(fact_324_set__list__subset__eq__nth__conv,axiom,
! [Xs: list_list_nat,A2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ A2 )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( member_list_nat @ ( nth_list_nat @ Xs @ I4 ) @ A2 ) ) ) ) ).
% set_list_subset_eq_nth_conv
thf(fact_325_set__list__subset__eq__nth__conv,axiom,
! [Xs: list_list_a,A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ A2 )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( member_list_a @ ( nth_list_a @ Xs @ I4 ) @ A2 ) ) ) ) ).
% set_list_subset_eq_nth_conv
thf(fact_326_set__list__subset__eq__nth__conv,axiom,
! [Xs: list_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ A2 )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
=> ( member_a @ ( nth_a @ Xs @ I4 ) @ A2 ) ) ) ) ).
% set_list_subset_eq_nth_conv
thf(fact_327_set__list__subset__eq__nth__conv,axiom,
! [Xs: list_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ ( nth_nat @ Xs @ I4 ) @ A2 ) ) ) ) ).
% set_list_subset_eq_nth_conv
thf(fact_328_remove__nth__sound__l,axiom,
! [P2: nat,N: nat,Xs: list_a] :
( ( ord_less_nat @ P2 @ N )
=> ( ( nth_a @ ( missing_remove_nth_a @ N @ Xs ) @ P2 )
= ( nth_a @ Xs @ P2 ) ) ) ).
% remove_nth_sound_l
thf(fact_329_remove__nth__sound__l,axiom,
! [P2: nat,N: nat,Xs: list_nat] :
( ( ord_less_nat @ P2 @ N )
=> ( ( nth_nat @ ( missin7175274867594579095th_nat @ N @ Xs ) @ P2 )
= ( nth_nat @ Xs @ P2 ) ) ) ).
% remove_nth_sound_l
thf(fact_330_remdups__sort_I3_J,axiom,
! [Xs: list_nat] : ( distinct_nat @ ( missin6101193410121742181rt_nat @ Xs ) ) ).
% remdups_sort(3)
thf(fact_331_fnth__mem,axiom,
! [N: nat,Xs: list_list_nat] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( fmember_list_nat @ ( nth_list_nat @ Xs @ N ) @ ( fset_o8830822285705182106st_nat @ Xs ) ) ) ).
% fnth_mem
thf(fact_332_fnth__mem,axiom,
! [N: nat,Xs: list_list_a] :
( ( ord_less_nat @ N @ ( size_s349497388124573686list_a @ Xs ) )
=> ( fmember_list_a @ ( nth_list_a @ Xs @ N ) @ ( fset_of_list_list_a @ Xs ) ) ) ).
% fnth_mem
thf(fact_333_fnth__mem,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( fmember_a @ ( nth_a @ Xs @ N ) @ ( fset_of_list_a @ Xs ) ) ) ).
% fnth_mem
thf(fact_334_fnth__mem,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( fmember_nat @ ( nth_nat @ Xs @ N ) @ ( fset_of_list_nat @ Xs ) ) ) ).
% fnth_mem
thf(fact_335_in__fset__conv__nth,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( fmember_list_nat @ X @ ( fset_o8830822285705182106st_nat @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Xs ) )
& ( ( nth_list_nat @ Xs @ I4 )
= X ) ) ) ) ).
% in_fset_conv_nth
thf(fact_336_in__fset__conv__nth,axiom,
! [X: list_a,Xs: list_list_a] :
( ( fmember_list_a @ X @ ( fset_of_list_list_a @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s349497388124573686list_a @ Xs ) )
& ( ( nth_list_a @ Xs @ I4 )
= X ) ) ) ) ).
% in_fset_conv_nth
thf(fact_337_in__fset__conv__nth,axiom,
! [X: a,Xs: list_a] :
( ( fmember_a @ X @ ( fset_of_list_a @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
& ( ( nth_a @ Xs @ I4 )
= X ) ) ) ) ).
% in_fset_conv_nth
thf(fact_338_in__fset__conv__nth,axiom,
! [X: nat,Xs: list_nat] :
( ( fmember_nat @ X @ ( fset_of_list_nat @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I4 )
= X ) ) ) ) ).
% in_fset_conv_nth
thf(fact_339_adjust__idx__rev__nth,axiom,
! [I: nat,Xs: list_list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( J != I )
=> ( ( nth_list_nat @ ( missin5133812807938250791st_nat @ I @ Xs ) @ ( missin3815256168798769645dx_rev @ I @ J ) )
= ( nth_list_nat @ Xs @ J ) ) ) ) ).
% adjust_idx_rev_nth
thf(fact_340_adjust__idx__rev__nth,axiom,
! [I: nat,Xs: list_list_a,J: nat] :
( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( J != I )
=> ( ( nth_list_a @ ( missin8041159176150063613list_a @ I @ Xs ) @ ( missin3815256168798769645dx_rev @ I @ J ) )
= ( nth_list_a @ Xs @ J ) ) ) ) ).
% adjust_idx_rev_nth
thf(fact_341_adjust__idx__rev__nth,axiom,
! [I: nat,Xs: list_a,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( J != I )
=> ( ( nth_a @ ( missing_remove_nth_a @ I @ Xs ) @ ( missin3815256168798769645dx_rev @ I @ J ) )
= ( nth_a @ Xs @ J ) ) ) ) ).
% adjust_idx_rev_nth
thf(fact_342_adjust__idx__rev__nth,axiom,
! [I: nat,Xs: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( J != I )
=> ( ( nth_nat @ ( missin7175274867594579095th_nat @ I @ Xs ) @ ( missin3815256168798769645dx_rev @ I @ J ) )
= ( nth_nat @ Xs @ J ) ) ) ) ).
% adjust_idx_rev_nth
thf(fact_343_adjust__idx__nth,axiom,
! [I: nat,Xs: list_list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_list_nat @ ( missin5133812807938250791st_nat @ I @ Xs ) @ J )
= ( nth_list_nat @ Xs @ ( missing_adjust_idx @ I @ J ) ) ) ) ).
% adjust_idx_nth
thf(fact_344_adjust__idx__nth,axiom,
! [I: nat,Xs: list_list_a,J: nat] :
( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( nth_list_a @ ( missin8041159176150063613list_a @ I @ Xs ) @ J )
= ( nth_list_a @ Xs @ ( missing_adjust_idx @ I @ J ) ) ) ) ).
% adjust_idx_nth
thf(fact_345_adjust__idx__nth,axiom,
! [I: nat,Xs: list_a,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( missing_remove_nth_a @ I @ Xs ) @ J )
= ( nth_a @ Xs @ ( missing_adjust_idx @ I @ J ) ) ) ) ).
% adjust_idx_nth
thf(fact_346_adjust__idx__nth,axiom,
! [I: nat,Xs: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( missin7175274867594579095th_nat @ I @ Xs ) @ J )
= ( nth_nat @ Xs @ ( missing_adjust_idx @ I @ J ) ) ) ) ).
% adjust_idx_nth
thf(fact_347_psubsetI,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_set_nat @ A2 @ B ) ) ) ).
% psubsetI
thf(fact_348_psubsetI,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_set_a @ A2 @ B ) ) ) ).
% psubsetI
thf(fact_349_subset__antisym,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_350_subset__antisym,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_351_subsetI,axiom,
! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ! [X2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X2 @ A2 )
=> ( member8440522571783428010at_nat @ X2 @ B ) )
=> ( ord_le3146513528884898305at_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_352_subsetI,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a] :
( ! [X2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ A2 )
=> ( member1426531477525435216od_a_a @ X2 @ B ) )
=> ( ord_le746702958409616551od_a_a @ A2 @ B ) ) ).
% subsetI
thf(fact_353_subsetI,axiom,
! [A2: set_list_nat,B: set_list_nat] :
( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ A2 )
=> ( member_list_nat @ X2 @ B ) )
=> ( ord_le6045566169113846134st_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_354_subsetI,axiom,
! [A2: set_list_a,B: set_list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
=> ( member_list_a @ X2 @ B ) )
=> ( ord_le8861187494160871172list_a @ A2 @ B ) ) ).
% subsetI
thf(fact_355_subsetI,axiom,
! [A2: set_a,B: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ X2 @ B ) )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% subsetI
thf(fact_356_subsetI,axiom,
! [A2: set_nat,B: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ X2 @ B ) )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_357_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_358_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_359_dual__order_Orefl,axiom,
! [A: product_prod_nat_nat > $o] : ( ord_le704812498762024988_nat_o @ A @ A ) ).
% dual_order.refl
thf(fact_360_dual__order_Orefl,axiom,
! [A: product_prod_nat_nat > product_prod_nat_nat > $o] : ( ord_le5604493270027003598_nat_o @ A @ A ) ).
% dual_order.refl
thf(fact_361_dual__order_Orefl,axiom,
! [A: nat > $o] : ( ord_less_eq_nat_o @ A @ A ) ).
% dual_order.refl
thf(fact_362_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_363_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_364_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_365_order__refl,axiom,
! [X: product_prod_nat_nat > $o] : ( ord_le704812498762024988_nat_o @ X @ X ) ).
% order_refl
thf(fact_366_order__refl,axiom,
! [X: product_prod_nat_nat > product_prod_nat_nat > $o] : ( ord_le5604493270027003598_nat_o @ X @ X ) ).
% order_refl
thf(fact_367_order__refl,axiom,
! [X: nat > $o] : ( ord_less_eq_nat_o @ X @ X ) ).
% order_refl
thf(fact_368_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_369_adjust__idx__rev__length,axiom,
! [I: nat,Xs: list_list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( J != I )
=> ( ord_less_nat @ ( missin3815256168798769645dx_rev @ I @ J ) @ ( size_s3023201423986296836st_nat @ ( missin5133812807938250791st_nat @ I @ Xs ) ) ) ) ) ) ).
% adjust_idx_rev_length
thf(fact_370_adjust__idx__rev__length,axiom,
! [I: nat,Xs: list_list_a,J: nat] :
( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( J != I )
=> ( ord_less_nat @ ( missin3815256168798769645dx_rev @ I @ J ) @ ( size_s349497388124573686list_a @ ( missin8041159176150063613list_a @ I @ Xs ) ) ) ) ) ) ).
% adjust_idx_rev_length
thf(fact_371_adjust__idx__rev__length,axiom,
! [I: nat,Xs: list_a,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs ) )
=> ( ( J != I )
=> ( ord_less_nat @ ( missin3815256168798769645dx_rev @ I @ J ) @ ( size_size_list_a @ ( missing_remove_nth_a @ I @ Xs ) ) ) ) ) ) ).
% adjust_idx_rev_length
thf(fact_372_adjust__idx__rev__length,axiom,
! [I: nat,Xs: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( J != I )
=> ( ord_less_nat @ ( missin3815256168798769645dx_rev @ I @ J ) @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ I @ Xs ) ) ) ) ) ) ).
% adjust_idx_rev_length
thf(fact_373_fsubsetI,axiom,
! [A2: fset_nat,B: fset_nat] :
( ! [X2: nat] :
( ( fmember_nat @ X2 @ A2 )
=> ( fmember_nat @ X2 @ B ) )
=> ( ord_less_eq_fset_nat @ A2 @ B ) ) ).
% fsubsetI
thf(fact_374_fsubsetI,axiom,
! [A2: fset_a,B: fset_a] :
( ! [X2: a] :
( ( fmember_a @ X2 @ A2 )
=> ( fmember_a @ X2 @ B ) )
=> ( ord_less_eq_fset_a @ A2 @ B ) ) ).
% fsubsetI
thf(fact_375_finsert__fsubset,axiom,
! [X: nat,A2: fset_nat,B: fset_nat] :
( ( ord_less_eq_fset_nat @ ( finsert_nat @ X @ A2 ) @ B )
= ( ( fmember_nat @ X @ B )
& ( ord_less_eq_fset_nat @ A2 @ B ) ) ) ).
% finsert_fsubset
thf(fact_376_finsert__fsubset,axiom,
! [X: a,A2: fset_a,B: fset_a] :
( ( ord_less_eq_fset_a @ ( finsert_a @ X @ A2 ) @ B )
= ( ( fmember_a @ X @ B )
& ( ord_less_eq_fset_a @ A2 @ B ) ) ) ).
% finsert_fsubset
thf(fact_377_list__update__beyond,axiom,
! [Xs: list_list_nat,I: nat,X: list_nat] :
( ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ I )
=> ( ( list_update_list_nat @ Xs @ I @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_378_list__update__beyond,axiom,
! [Xs: list_list_a,I: nat,X: list_a] :
( ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Xs ) @ I )
=> ( ( list_update_list_a @ Xs @ I @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_379_list__update__beyond,axiom,
! [Xs: list_a,I: nat,X: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ I )
=> ( ( list_update_a @ Xs @ I @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_380_list__update__beyond,axiom,
! [Xs: list_nat,I: nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I )
=> ( ( list_update_nat @ Xs @ I @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_381_fset__of__list__subset,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ ( set_list_nat2 @ Ys ) )
=> ( ord_le2534233284047104112st_nat @ ( fset_o8830822285705182106st_nat @ Xs ) @ ( fset_o8830822285705182106st_nat @ Ys ) ) ) ).
% fset_of_list_subset
thf(fact_382_fset__of__list__subset,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ ( set_list_a2 @ Ys ) )
=> ( ord_le510749213327159946list_a @ ( fset_of_list_list_a @ Xs ) @ ( fset_of_list_list_a @ Ys ) ) ) ).
% fset_of_list_subset
thf(fact_383_fset__of__list__subset,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) )
=> ( ord_less_eq_fset_nat @ ( fset_of_list_nat @ Xs ) @ ( fset_of_list_nat @ Ys ) ) ) ).
% fset_of_list_subset
thf(fact_384_fset__of__list__subset,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) )
=> ( ord_less_eq_fset_a @ ( fset_of_list_a @ Xs ) @ ( fset_of_list_a @ Ys ) ) ) ).
% fset_of_list_subset
thf(fact_385_psubsetD,axiom,
! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: product_prod_nat_nat] :
( ( ord_le7866589430770878221at_nat @ A2 @ B )
=> ( ( member8440522571783428010at_nat @ C2 @ A2 )
=> ( member8440522571783428010at_nat @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_386_psubsetD,axiom,
! [A2: set_Product_prod_a_a,B: set_Product_prod_a_a,C2: product_prod_a_a] :
( ( ord_le6819997720685908915od_a_a @ A2 @ B )
=> ( ( member1426531477525435216od_a_a @ C2 @ A2 )
=> ( member1426531477525435216od_a_a @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_387_psubsetD,axiom,
! [A2: set_list_nat,B: set_list_nat,C2: list_nat] :
( ( ord_le1190675801316882794st_nat @ A2 @ B )
=> ( ( member_list_nat @ C2 @ A2 )
=> ( member_list_nat @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_388_psubsetD,axiom,
! [A2: set_list_a,B: set_list_a,C2: list_a] :
( ( ord_less_set_list_a @ A2 @ B )
=> ( ( member_list_a @ C2 @ A2 )
=> ( member_list_a @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_389_psubsetD,axiom,
! [A2: set_a,B: set_a,C2: a] :
( ( ord_less_set_a @ A2 @ B )
=> ( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_390_psubsetD,axiom,
! [A2: set_nat,B: set_nat,C2: nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ B ) ) ) ).
% psubsetD
thf(fact_391_psubset__trans,axiom,
! [A2: set_nat,B: set_nat,C3: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( ord_less_set_nat @ B @ C3 )
=> ( ord_less_set_nat @ A2 @ C3 ) ) ) ).
% psubset_trans
thf(fact_392_adjust__idx__rev2,axiom,
! [J: nat,I: nat] :
( ( J != I )
=> ( ( missing_adjust_idx @ I @ ( missin3815256168798769645dx_rev @ I @ J ) )
= J ) ) ).
% adjust_idx_rev2
thf(fact_393_adjust__idx__rev1,axiom,
! [I: nat,J: nat] :
( ( missin3815256168798769645dx_rev @ I @ ( missing_adjust_idx @ I @ J ) )
= J ) ).
% adjust_idx_rev1
thf(fact_394_adjust__idx__i,axiom,
! [I: nat,J: nat] :
( ( missing_adjust_idx @ I @ J )
!= I ) ).
% adjust_idx_i
thf(fact_395_less__mono__imp__le__mono,axiom,
! [F2: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F2 @ I3 ) @ ( F2 @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F2 @ I ) @ ( F2 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_396_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_397_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_398_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_399_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_400_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_401_fsubsetD,axiom,
! [A2: fset_nat,B: fset_nat,C2: nat] :
( ( ord_less_eq_fset_nat @ A2 @ B )
=> ( ( fmember_nat @ C2 @ A2 )
=> ( fmember_nat @ C2 @ B ) ) ) ).
% fsubsetD
thf(fact_402_fsubsetD,axiom,
! [A2: fset_a,B: fset_a,C2: a] :
( ( ord_less_eq_fset_a @ A2 @ B )
=> ( ( fmember_a @ C2 @ A2 )
=> ( fmember_a @ C2 @ B ) ) ) ).
% fsubsetD
thf(fact_403_fin__mono,axiom,
! [A2: fset_nat,B: fset_nat,X: nat] :
( ( ord_less_eq_fset_nat @ A2 @ B )
=> ( ( fmember_nat @ X @ A2 )
=> ( fmember_nat @ X @ B ) ) ) ).
% fin_mono
thf(fact_404_fin__mono,axiom,
! [A2: fset_a,B: fset_a,X: a] :
( ( ord_less_eq_fset_a @ A2 @ B )
=> ( ( fmember_a @ X @ A2 )
=> ( fmember_a @ X @ B ) ) ) ).
% fin_mono
thf(fact_405_fsubset__finsertI2,axiom,
! [A2: fset_nat,B: fset_nat,B2: nat] :
( ( ord_less_eq_fset_nat @ A2 @ B )
=> ( ord_less_eq_fset_nat @ A2 @ ( finsert_nat @ B2 @ B ) ) ) ).
% fsubset_finsertI2
thf(fact_406_fsubset__finsertI2,axiom,
! [A2: fset_a,B: fset_a,B2: a] :
( ( ord_less_eq_fset_a @ A2 @ B )
=> ( ord_less_eq_fset_a @ A2 @ ( finsert_a @ B2 @ B ) ) ) ).
% fsubset_finsertI2
thf(fact_407_fsubset__finsertI,axiom,
! [B: fset_nat,A: nat] : ( ord_less_eq_fset_nat @ B @ ( finsert_nat @ A @ B ) ) ).
% fsubset_finsertI
thf(fact_408_fsubset__finsertI,axiom,
! [B: fset_a,A: a] : ( ord_less_eq_fset_a @ B @ ( finsert_a @ A @ B ) ) ).
% fsubset_finsertI
thf(fact_409_finsert__mono,axiom,
! [C3: fset_nat,D: fset_nat,A: nat] :
( ( ord_less_eq_fset_nat @ C3 @ D )
=> ( ord_less_eq_fset_nat @ ( finsert_nat @ A @ C3 ) @ ( finsert_nat @ A @ D ) ) ) ).
% finsert_mono
thf(fact_410_finsert__mono,axiom,
! [C3: fset_a,D: fset_a,A: a] :
( ( ord_less_eq_fset_a @ C3 @ D )
=> ( ord_less_eq_fset_a @ ( finsert_a @ A @ C3 ) @ ( finsert_a @ A @ D ) ) ) ).
% finsert_mono
thf(fact_411_pfsubset__eq,axiom,
( ord_less_fset_a
= ( ^ [A3: fset_a,B4: fset_a] :
( ( ord_less_eq_fset_a @ A3 @ B4 )
& ( A3 != B4 ) ) ) ) ).
% pfsubset_eq
thf(fact_412_less__fset__def,axiom,
( ord_less_fset_a
= ( ^ [Xs3: fset_a,Ys3: fset_a] :
( ( ord_less_eq_fset_a @ Xs3 @ Ys3 )
& ( Xs3 != Ys3 ) ) ) ) ).
% less_fset_def
thf(fact_413_pfsubset__imp__fsubset,axiom,
! [A2: fset_a,B: fset_a] :
( ( ord_less_fset_a @ A2 @ B )
=> ( ord_less_eq_fset_a @ A2 @ B ) ) ).
% pfsubset_imp_fsubset
thf(fact_414_fsubset__not__fsubset__eq,axiom,
( ord_less_fset_a
= ( ^ [A3: fset_a,B4: fset_a] :
( ( ord_less_eq_fset_a @ A3 @ B4 )
& ~ ( ord_less_eq_fset_a @ B4 @ A3 ) ) ) ) ).
% fsubset_not_fsubset_eq
thf(fact_415_fsubset__pfsubset__trans,axiom,
! [A2: fset_a,B: fset_a,C3: fset_a] :
( ( ord_less_eq_fset_a @ A2 @ B )
=> ( ( ord_less_fset_a @ B @ C3 )
=> ( ord_less_fset_a @ A2 @ C3 ) ) ) ).
% fsubset_pfsubset_trans
thf(fact_416_pfsubset__fsubset__trans,axiom,
! [A2: fset_a,B: fset_a,C3: fset_a] :
( ( ord_less_fset_a @ A2 @ B )
=> ( ( ord_less_eq_fset_a @ B @ C3 )
=> ( ord_less_fset_a @ A2 @ C3 ) ) ) ).
% pfsubset_fsubset_trans
thf(fact_417_fsubset__iff__pfsubset__eq,axiom,
( ord_less_eq_fset_a
= ( ^ [A3: fset_a,B4: fset_a] :
( ( ord_less_fset_a @ A3 @ B4 )
| ( A3 = B4 ) ) ) ) ).
% fsubset_iff_pfsubset_eq
thf(fact_418_fsubset__finsert,axiom,
! [X: nat,A2: fset_nat,B: fset_nat] :
( ~ ( fmember_nat @ X @ A2 )
=> ( ( ord_less_eq_fset_nat @ A2 @ ( finsert_nat @ X @ B ) )
= ( ord_less_eq_fset_nat @ A2 @ B ) ) ) ).
% fsubset_finsert
thf(fact_419_fsubset__finsert,axiom,
! [X: a,A2: fset_a,B: fset_a] :
( ~ ( fmember_a @ X @ A2 )
=> ( ( ord_less_eq_fset_a @ A2 @ ( finsert_a @ X @ B ) )
= ( ord_less_eq_fset_a @ A2 @ B ) ) ) ).
% fsubset_finsert
thf(fact_420_length__removeAll__less__eq,axiom,
! [X: list_nat,Xs: list_list_nat] : ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ ( removeAll_list_nat @ X @ Xs ) ) @ ( size_s3023201423986296836st_nat @ Xs ) ) ).
% length_removeAll_less_eq
thf(fact_421_length__removeAll__less__eq,axiom,
! [X: list_a,Xs: list_list_a] : ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ ( removeAll_list_a @ X @ Xs ) ) @ ( size_s349497388124573686list_a @ Xs ) ) ).
% length_removeAll_less_eq
thf(fact_422_length__removeAll__less__eq,axiom,
! [X: a,Xs: list_a] : ( ord_less_eq_nat @ ( size_size_list_a @ ( removeAll_a @ X @ Xs ) ) @ ( size_size_list_a @ Xs ) ) ).
% length_removeAll_less_eq
thf(fact_423_length__removeAll__less__eq,axiom,
! [X: nat,Xs: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( removeAll_nat @ X @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% length_removeAll_less_eq
thf(fact_424_remove__nth__id,axiom,
! [Xs: list_list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ N )
=> ( ( missin5133812807938250791st_nat @ N @ Xs )
= Xs ) ) ).
% remove_nth_id
thf(fact_425_remove__nth__id,axiom,
! [Xs: list_list_a,N: nat] :
( ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Xs ) @ N )
=> ( ( missin8041159176150063613list_a @ N @ Xs )
= Xs ) ) ).
% remove_nth_id
thf(fact_426_remove__nth__id,axiom,
! [Xs: list_a,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N )
=> ( ( missing_remove_nth_a @ N @ Xs )
= Xs ) ) ).
% remove_nth_id
thf(fact_427_remove__nth__id,axiom,
! [Xs: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
=> ( ( missin7175274867594579095th_nat @ N @ Xs )
= Xs ) ) ).
% remove_nth_id
thf(fact_428_nle__le,axiom,
! [A: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_429_le__cases3,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z4 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z4 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z4 )
=> ~ ( ord_less_eq_nat @ Z4 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z4 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z4 )
=> ~ ( ord_less_eq_nat @ Z4 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z4 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_430_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [X4: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y5 )
& ( ord_less_eq_set_nat @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_431_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
= ( ^ [X4: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y5 )
& ( ord_less_eq_set_a @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_432_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: product_prod_nat_nat > $o,Z: product_prod_nat_nat > $o] : ( Y2 = Z ) )
= ( ^ [X4: product_prod_nat_nat > $o,Y5: product_prod_nat_nat > $o] :
( ( ord_le704812498762024988_nat_o @ X4 @ Y5 )
& ( ord_le704812498762024988_nat_o @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_433_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: product_prod_nat_nat > product_prod_nat_nat > $o,Z: product_prod_nat_nat > product_prod_nat_nat > $o] : ( Y2 = Z ) )
= ( ^ [X4: product_prod_nat_nat > product_prod_nat_nat > $o,Y5: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( ord_le5604493270027003598_nat_o @ X4 @ Y5 )
& ( ord_le5604493270027003598_nat_o @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_434_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat > $o,Z: nat > $o] : ( Y2 = Z ) )
= ( ^ [X4: nat > $o,Y5: nat > $o] :
( ( ord_less_eq_nat_o @ X4 @ Y5 )
& ( ord_less_eq_nat_o @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_435_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_436_ord__eq__le__trans,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( A = B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_437_ord__eq__le__trans,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( A = B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_438_ord__eq__le__trans,axiom,
! [A: product_prod_nat_nat > $o,B2: product_prod_nat_nat > $o,C2: product_prod_nat_nat > $o] :
( ( A = B2 )
=> ( ( ord_le704812498762024988_nat_o @ B2 @ C2 )
=> ( ord_le704812498762024988_nat_o @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_439_ord__eq__le__trans,axiom,
! [A: product_prod_nat_nat > product_prod_nat_nat > $o,B2: product_prod_nat_nat > product_prod_nat_nat > $o,C2: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( A = B2 )
=> ( ( ord_le5604493270027003598_nat_o @ B2 @ C2 )
=> ( ord_le5604493270027003598_nat_o @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_440_ord__eq__le__trans,axiom,
! [A: nat > $o,B2: nat > $o,C2: nat > $o] :
( ( A = B2 )
=> ( ( ord_less_eq_nat_o @ B2 @ C2 )
=> ( ord_less_eq_nat_o @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_441_ord__eq__le__trans,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( A = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_442_ord__le__eq__trans,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_443_ord__le__eq__trans,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_444_ord__le__eq__trans,axiom,
! [A: product_prod_nat_nat > $o,B2: product_prod_nat_nat > $o,C2: product_prod_nat_nat > $o] :
( ( ord_le704812498762024988_nat_o @ A @ B2 )
=> ( ( B2 = C2 )
=> ( ord_le704812498762024988_nat_o @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_445_ord__le__eq__trans,axiom,
! [A: product_prod_nat_nat > product_prod_nat_nat > $o,B2: product_prod_nat_nat > product_prod_nat_nat > $o,C2: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( ord_le5604493270027003598_nat_o @ A @ B2 )
=> ( ( B2 = C2 )
=> ( ord_le5604493270027003598_nat_o @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_446_ord__le__eq__trans,axiom,
! [A: nat > $o,B2: nat > $o,C2: nat > $o] :
( ( ord_less_eq_nat_o @ A @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_nat_o @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_447_ord__le__eq__trans,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_448_order__antisym,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_449_order__antisym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_450_order__antisym,axiom,
! [X: product_prod_nat_nat > $o,Y: product_prod_nat_nat > $o] :
( ( ord_le704812498762024988_nat_o @ X @ Y )
=> ( ( ord_le704812498762024988_nat_o @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_451_order__antisym,axiom,
! [X: product_prod_nat_nat > product_prod_nat_nat > $o,Y: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( ord_le5604493270027003598_nat_o @ X @ Y )
=> ( ( ord_le5604493270027003598_nat_o @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_452_order__antisym,axiom,
! [X: nat > $o,Y: nat > $o] :
( ( ord_less_eq_nat_o @ X @ Y )
=> ( ( ord_less_eq_nat_o @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_453_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_454_order_Otrans,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_455_order_Otrans,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% order.trans
thf(fact_456_order_Otrans,axiom,
! [A: product_prod_nat_nat > $o,B2: product_prod_nat_nat > $o,C2: product_prod_nat_nat > $o] :
( ( ord_le704812498762024988_nat_o @ A @ B2 )
=> ( ( ord_le704812498762024988_nat_o @ B2 @ C2 )
=> ( ord_le704812498762024988_nat_o @ A @ C2 ) ) ) ).
% order.trans
thf(fact_457_order_Otrans,axiom,
! [A: product_prod_nat_nat > product_prod_nat_nat > $o,B2: product_prod_nat_nat > product_prod_nat_nat > $o,C2: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( ord_le5604493270027003598_nat_o @ A @ B2 )
=> ( ( ord_le5604493270027003598_nat_o @ B2 @ C2 )
=> ( ord_le5604493270027003598_nat_o @ A @ C2 ) ) ) ).
% order.trans
thf(fact_458_order_Otrans,axiom,
! [A: nat > $o,B2: nat > $o,C2: nat > $o] :
( ( ord_less_eq_nat_o @ A @ B2 )
=> ( ( ord_less_eq_nat_o @ B2 @ C2 )
=> ( ord_less_eq_nat_o @ A @ C2 ) ) ) ).
% order.trans
thf(fact_459_order_Otrans,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_460_order__trans,axiom,
! [X: set_nat,Y: set_nat,Z4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z4 )
=> ( ord_less_eq_set_nat @ X @ Z4 ) ) ) ).
% order_trans
thf(fact_461_order__trans,axiom,
! [X: set_a,Y: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z4 )
=> ( ord_less_eq_set_a @ X @ Z4 ) ) ) ).
% order_trans
thf(fact_462_order__trans,axiom,
! [X: product_prod_nat_nat > $o,Y: product_prod_nat_nat > $o,Z4: product_prod_nat_nat > $o] :
( ( ord_le704812498762024988_nat_o @ X @ Y )
=> ( ( ord_le704812498762024988_nat_o @ Y @ Z4 )
=> ( ord_le704812498762024988_nat_o @ X @ Z4 ) ) ) ).
% order_trans
thf(fact_463_order__trans,axiom,
! [X: product_prod_nat_nat > product_prod_nat_nat > $o,Y: product_prod_nat_nat > product_prod_nat_nat > $o,Z4: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( ord_le5604493270027003598_nat_o @ X @ Y )
=> ( ( ord_le5604493270027003598_nat_o @ Y @ Z4 )
=> ( ord_le5604493270027003598_nat_o @ X @ Z4 ) ) ) ).
% order_trans
thf(fact_464_order__trans,axiom,
! [X: nat > $o,Y: nat > $o,Z4: nat > $o] :
( ( ord_less_eq_nat_o @ X @ Y )
=> ( ( ord_less_eq_nat_o @ Y @ Z4 )
=> ( ord_less_eq_nat_o @ X @ Z4 ) ) ) ).
% order_trans
thf(fact_465_order__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z4 )
=> ( ord_less_eq_nat @ X @ Z4 ) ) ) ).
% order_trans
thf(fact_466_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B2: nat] :
( ! [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ A4 @ B5 )
=> ( P @ A4 @ B5 ) )
=> ( ! [A4: nat,B5: nat] :
( ( P @ B5 @ A4 )
=> ( P @ A4 @ B5 ) )
=> ( P @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_467_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [A5: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A5 )
& ( ord_less_eq_set_nat @ A5 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_468_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
= ( ^ [A5: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ A5 )
& ( ord_less_eq_set_a @ A5 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_469_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: product_prod_nat_nat > $o,Z: product_prod_nat_nat > $o] : ( Y2 = Z ) )
= ( ^ [A5: product_prod_nat_nat > $o,B6: product_prod_nat_nat > $o] :
( ( ord_le704812498762024988_nat_o @ B6 @ A5 )
& ( ord_le704812498762024988_nat_o @ A5 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_470_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: product_prod_nat_nat > product_prod_nat_nat > $o,Z: product_prod_nat_nat > product_prod_nat_nat > $o] : ( Y2 = Z ) )
= ( ^ [A5: product_prod_nat_nat > product_prod_nat_nat > $o,B6: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( ord_le5604493270027003598_nat_o @ B6 @ A5 )
& ( ord_le5604493270027003598_nat_o @ A5 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_471_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: nat > $o,Z: nat > $o] : ( Y2 = Z ) )
= ( ^ [A5: nat > $o,B6: nat > $o] :
( ( ord_less_eq_nat_o @ B6 @ A5 )
& ( ord_less_eq_nat_o @ A5 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_472_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A5: nat,B6: nat] :
( ( ord_less_eq_nat @ B6 @ A5 )
& ( ord_less_eq_nat @ A5 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_473_dual__order_Oantisym,axiom,
! [B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_474_dual__order_Oantisym,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_475_dual__order_Oantisym,axiom,
! [B2: product_prod_nat_nat > $o,A: product_prod_nat_nat > $o] :
( ( ord_le704812498762024988_nat_o @ B2 @ A )
=> ( ( ord_le704812498762024988_nat_o @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_476_dual__order_Oantisym,axiom,
! [B2: product_prod_nat_nat > product_prod_nat_nat > $o,A: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( ord_le5604493270027003598_nat_o @ B2 @ A )
=> ( ( ord_le5604493270027003598_nat_o @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_477_dual__order_Oantisym,axiom,
! [B2: nat > $o,A: nat > $o] :
( ( ord_less_eq_nat_o @ B2 @ A )
=> ( ( ord_less_eq_nat_o @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_478_dual__order_Oantisym,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_479_dual__order_Otrans,axiom,
! [B2: set_nat,A: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( ord_less_eq_set_nat @ C2 @ B2 )
=> ( ord_less_eq_set_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_480_dual__order_Otrans,axiom,
! [B2: set_a,A: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( ord_less_eq_set_a @ C2 @ B2 )
=> ( ord_less_eq_set_a @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_481_dual__order_Otrans,axiom,
! [B2: product_prod_nat_nat > $o,A: product_prod_nat_nat > $o,C2: product_prod_nat_nat > $o] :
( ( ord_le704812498762024988_nat_o @ B2 @ A )
=> ( ( ord_le704812498762024988_nat_o @ C2 @ B2 )
=> ( ord_le704812498762024988_nat_o @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_482_dual__order_Otrans,axiom,
! [B2: product_prod_nat_nat > product_prod_nat_nat > $o,A: product_prod_nat_nat > product_prod_nat_nat > $o,C2: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( ord_le5604493270027003598_nat_o @ B2 @ A )
=> ( ( ord_le5604493270027003598_nat_o @ C2 @ B2 )
=> ( ord_le5604493270027003598_nat_o @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_483_dual__order_Otrans,axiom,
! [B2: nat > $o,A: nat > $o,C2: nat > $o] :
( ( ord_less_eq_nat_o @ B2 @ A )
=> ( ( ord_less_eq_nat_o @ C2 @ B2 )
=> ( ord_less_eq_nat_o @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_484_dual__order_Otrans,axiom,
! [B2: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C2 @ B2 )
=> ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_485_antisym,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_486_antisym,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_487_antisym,axiom,
! [A: product_prod_nat_nat > $o,B2: product_prod_nat_nat > $o] :
( ( ord_le704812498762024988_nat_o @ A @ B2 )
=> ( ( ord_le704812498762024988_nat_o @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_488_antisym,axiom,
! [A: product_prod_nat_nat > product_prod_nat_nat > $o,B2: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( ord_le5604493270027003598_nat_o @ A @ B2 )
=> ( ( ord_le5604493270027003598_nat_o @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_489_antisym,axiom,
! [A: nat > $o,B2: nat > $o] :
( ( ord_less_eq_nat_o @ A @ B2 )
=> ( ( ord_less_eq_nat_o @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_490_antisym,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_491_le__funD,axiom,
! [F2: product_prod_nat_nat > $o,G: product_prod_nat_nat > $o,X: product_prod_nat_nat] :
( ( ord_le704812498762024988_nat_o @ F2 @ G )
=> ( ord_less_eq_o @ ( F2 @ X ) @ ( G @ X ) ) ) ).
% le_funD
thf(fact_492_le__funD,axiom,
! [F2: product_prod_nat_nat > product_prod_nat_nat > $o,G: product_prod_nat_nat > product_prod_nat_nat > $o,X: product_prod_nat_nat] :
( ( ord_le5604493270027003598_nat_o @ F2 @ G )
=> ( ord_le704812498762024988_nat_o @ ( F2 @ X ) @ ( G @ X ) ) ) ).
% le_funD
thf(fact_493_le__funD,axiom,
! [F2: nat > $o,G: nat > $o,X: nat] :
( ( ord_less_eq_nat_o @ F2 @ G )
=> ( ord_less_eq_o @ ( F2 @ X ) @ ( G @ X ) ) ) ).
% le_funD
thf(fact_494_le__funE,axiom,
! [F2: product_prod_nat_nat > $o,G: product_prod_nat_nat > $o,X: product_prod_nat_nat] :
( ( ord_le704812498762024988_nat_o @ F2 @ G )
=> ( ord_less_eq_o @ ( F2 @ X ) @ ( G @ X ) ) ) ).
% le_funE
thf(fact_495_le__funE,axiom,
! [F2: product_prod_nat_nat > product_prod_nat_nat > $o,G: product_prod_nat_nat > product_prod_nat_nat > $o,X: product_prod_nat_nat] :
( ( ord_le5604493270027003598_nat_o @ F2 @ G )
=> ( ord_le704812498762024988_nat_o @ ( F2 @ X ) @ ( G @ X ) ) ) ).
% le_funE
thf(fact_496_le__funE,axiom,
! [F2: nat > $o,G: nat > $o,X: nat] :
( ( ord_less_eq_nat_o @ F2 @ G )
=> ( ord_less_eq_o @ ( F2 @ X ) @ ( G @ X ) ) ) ).
% le_funE
thf(fact_497_le__funI,axiom,
! [F2: product_prod_nat_nat > $o,G: product_prod_nat_nat > $o] :
( ! [X2: product_prod_nat_nat] : ( ord_less_eq_o @ ( F2 @ X2 ) @ ( G @ X2 ) )
=> ( ord_le704812498762024988_nat_o @ F2 @ G ) ) ).
% le_funI
thf(fact_498_le__funI,axiom,
! [F2: nat > $o,G: nat > $o] :
( ! [X2: nat] : ( ord_less_eq_o @ ( F2 @ X2 ) @ ( G @ X2 ) )
=> ( ord_less_eq_nat_o @ F2 @ G ) ) ).
% le_funI
thf(fact_499_le__funI,axiom,
! [F2: product_prod_nat_nat > product_prod_nat_nat > $o,G: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ! [X2: product_prod_nat_nat] : ( ord_le704812498762024988_nat_o @ ( F2 @ X2 ) @ ( G @ X2 ) )
=> ( ord_le5604493270027003598_nat_o @ F2 @ G ) ) ).
% le_funI
thf(fact_500_le__fun__def,axiom,
( ord_le704812498762024988_nat_o
= ( ^ [F3: product_prod_nat_nat > $o,G2: product_prod_nat_nat > $o] :
! [X4: product_prod_nat_nat] : ( ord_less_eq_o @ ( F3 @ X4 ) @ ( G2 @ X4 ) ) ) ) ).
% le_fun_def
thf(fact_501_le__fun__def,axiom,
( ord_le5604493270027003598_nat_o
= ( ^ [F3: product_prod_nat_nat > product_prod_nat_nat > $o,G2: product_prod_nat_nat > product_prod_nat_nat > $o] :
! [X4: product_prod_nat_nat] : ( ord_le704812498762024988_nat_o @ ( F3 @ X4 ) @ ( G2 @ X4 ) ) ) ) ).
% le_fun_def
thf(fact_502_le__fun__def,axiom,
( ord_less_eq_nat_o
= ( ^ [F3: nat > $o,G2: nat > $o] :
! [X4: nat] : ( ord_less_eq_o @ ( F3 @ X4 ) @ ( G2 @ X4 ) ) ) ) ).
% le_fun_def
thf(fact_503_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [A5: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_504_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
= ( ^ [A5: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A5 @ B6 )
& ( ord_less_eq_set_a @ B6 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_505_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: product_prod_nat_nat > $o,Z: product_prod_nat_nat > $o] : ( Y2 = Z ) )
= ( ^ [A5: product_prod_nat_nat > $o,B6: product_prod_nat_nat > $o] :
( ( ord_le704812498762024988_nat_o @ A5 @ B6 )
& ( ord_le704812498762024988_nat_o @ B6 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_506_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: product_prod_nat_nat > product_prod_nat_nat > $o,Z: product_prod_nat_nat > product_prod_nat_nat > $o] : ( Y2 = Z ) )
= ( ^ [A5: product_prod_nat_nat > product_prod_nat_nat > $o,B6: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( ord_le5604493270027003598_nat_o @ A5 @ B6 )
& ( ord_le5604493270027003598_nat_o @ B6 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_507_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat > $o,Z: nat > $o] : ( Y2 = Z ) )
= ( ^ [A5: nat > $o,B6: nat > $o] :
( ( ord_less_eq_nat_o @ A5 @ B6 )
& ( ord_less_eq_nat_o @ B6 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_508_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A5: nat,B6: nat] :
( ( ord_less_eq_nat @ A5 @ B6 )
& ( ord_less_eq_nat @ B6 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_509_order__subst1,axiom,
! [A: nat,F2: nat > nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_510_order__subst1,axiom,
! [A: nat,F2: set_nat > nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_511_order__subst1,axiom,
! [A: nat,F2: set_a > nat,B2: set_a,C2: set_a] :
( ( ord_less_eq_nat @ A @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_512_order__subst1,axiom,
! [A: set_nat,F2: nat > set_nat,B2: nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_513_order__subst1,axiom,
! [A: set_a,F2: nat > set_a,B2: nat,C2: nat] :
( ( ord_less_eq_set_a @ A @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_514_order__subst1,axiom,
! [A: nat,F2: ( nat > $o ) > nat,B2: nat > $o,C2: nat > $o] :
( ( ord_less_eq_nat @ A @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_nat_o @ B2 @ C2 )
=> ( ! [X2: nat > $o,Y3: nat > $o] :
( ( ord_less_eq_nat_o @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_515_order__subst1,axiom,
! [A: set_nat,F2: set_nat > set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_516_order__subst1,axiom,
! [A: set_nat,F2: set_a > set_nat,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_nat @ A @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_517_order__subst1,axiom,
! [A: set_a,F2: set_nat > set_a,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_a @ A @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_518_order__subst1,axiom,
! [A: set_a,F2: set_a > set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_519_order__subst2,axiom,
! [A: nat,B2: nat,F2: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F2 @ B2 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_520_order__subst2,axiom,
! [A: nat,B2: nat,F2: nat > set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F2 @ B2 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_521_order__subst2,axiom,
! [A: nat,B2: nat,F2: nat > set_a,C2: set_a] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_set_a @ ( F2 @ B2 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_522_order__subst2,axiom,
! [A: set_nat,B2: set_nat,F2: set_nat > nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F2 @ B2 ) @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_523_order__subst2,axiom,
! [A: set_a,B2: set_a,F2: set_a > nat,C2: nat] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F2 @ B2 ) @ C2 )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_524_order__subst2,axiom,
! [A: nat,B2: nat,F2: nat > nat > $o,C2: nat > $o] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat_o @ ( F2 @ B2 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat_o @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat_o @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_525_order__subst2,axiom,
! [A: set_nat,B2: set_nat,F2: set_nat > set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F2 @ B2 ) @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_526_order__subst2,axiom,
! [A: set_nat,B2: set_nat,F2: set_nat > set_a,C2: set_a] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_a @ ( F2 @ B2 ) @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_527_order__subst2,axiom,
! [A: set_a,B2: set_a,F2: set_a > set_nat,C2: set_nat] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F2 @ B2 ) @ C2 )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_528_order__subst2,axiom,
! [A: set_a,B2: set_a,F2: set_a > set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ ( F2 @ B2 ) @ C2 )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_529_order__eq__refl,axiom,
! [X: set_nat,Y: set_nat] :
( ( X = Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_530_order__eq__refl,axiom,
! [X: set_a,Y: set_a] :
( ( X = Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% order_eq_refl
thf(fact_531_order__eq__refl,axiom,
! [X: product_prod_nat_nat > $o,Y: product_prod_nat_nat > $o] :
( ( X = Y )
=> ( ord_le704812498762024988_nat_o @ X @ Y ) ) ).
% order_eq_refl
thf(fact_532_order__eq__refl,axiom,
! [X: product_prod_nat_nat > product_prod_nat_nat > $o,Y: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( X = Y )
=> ( ord_le5604493270027003598_nat_o @ X @ Y ) ) ).
% order_eq_refl
thf(fact_533_order__eq__refl,axiom,
! [X: nat > $o,Y: nat > $o] :
( ( X = Y )
=> ( ord_less_eq_nat_o @ X @ Y ) ) ).
% order_eq_refl
thf(fact_534_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_535_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_536_ord__eq__le__subst,axiom,
! [A: nat,F2: nat > nat,B2: nat,C2: nat] :
( ( A
= ( F2 @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_537_ord__eq__le__subst,axiom,
! [A: set_nat,F2: nat > set_nat,B2: nat,C2: nat] :
( ( A
= ( F2 @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_538_ord__eq__le__subst,axiom,
! [A: set_a,F2: nat > set_a,B2: nat,C2: nat] :
( ( A
= ( F2 @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_539_ord__eq__le__subst,axiom,
! [A: nat,F2: set_nat > nat,B2: set_nat,C2: set_nat] :
( ( A
= ( F2 @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_540_ord__eq__le__subst,axiom,
! [A: nat,F2: set_a > nat,B2: set_a,C2: set_a] :
( ( A
= ( F2 @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_541_ord__eq__le__subst,axiom,
! [A: nat > $o,F2: nat > nat > $o,B2: nat,C2: nat] :
( ( A
= ( F2 @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat_o @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat_o @ A @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_542_ord__eq__le__subst,axiom,
! [A: set_nat,F2: set_nat > set_nat,B2: set_nat,C2: set_nat] :
( ( A
= ( F2 @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_543_ord__eq__le__subst,axiom,
! [A: set_a,F2: set_nat > set_a,B2: set_nat,C2: set_nat] :
( ( A
= ( F2 @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_544_ord__eq__le__subst,axiom,
! [A: set_nat,F2: set_a > set_nat,B2: set_a,C2: set_a] :
( ( A
= ( F2 @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_545_ord__eq__le__subst,axiom,
! [A: set_a,F2: set_a > set_a,B2: set_a,C2: set_a] :
( ( A
= ( F2 @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_546_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F2: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F2 @ B2 )
= C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_547_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F2: nat > set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F2 @ B2 )
= C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_548_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F2: nat > set_a,C2: set_a] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F2 @ B2 )
= C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F2 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_549_ord__le__eq__subst,axiom,
! [A: set_nat,B2: set_nat,F2: set_nat > nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ( F2 @ B2 )
= C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_550_ord__le__eq__subst,axiom,
! [A: set_a,B2: set_a,F2: set_a > nat,C2: nat] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ( F2 @ B2 )
= C2 )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_551_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F2: nat > nat > $o,C2: nat > $o] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F2 @ B2 )
= C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat_o @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_nat_o @ ( F2 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_552_ord__le__eq__subst,axiom,
! [A: set_nat,B2: set_nat,F2: set_nat > set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ( F2 @ B2 )
= C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_553_ord__le__eq__subst,axiom,
! [A: set_nat,B2: set_nat,F2: set_nat > set_a,C2: set_a] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ( F2 @ B2 )
= C2 )
=> ( ! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F2 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_554_ord__le__eq__subst,axiom,
! [A: set_a,B2: set_a,F2: set_a > set_nat,C2: set_nat] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ( F2 @ B2 )
= C2 )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_555_ord__le__eq__subst,axiom,
! [A: set_a,B2: set_a,F2: set_a > set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ( F2 @ B2 )
= C2 )
=> ( ! [X2: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y3 )
=> ( ord_less_eq_set_a @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F2 @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_556_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_557_order__antisym__conv,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_558_order__antisym__conv,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_559_order__antisym__conv,axiom,
! [Y: product_prod_nat_nat > $o,X: product_prod_nat_nat > $o] :
( ( ord_le704812498762024988_nat_o @ Y @ X )
=> ( ( ord_le704812498762024988_nat_o @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_560_order__antisym__conv,axiom,
! [Y: product_prod_nat_nat > product_prod_nat_nat > $o,X: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( ord_le5604493270027003598_nat_o @ Y @ X )
=> ( ( ord_le5604493270027003598_nat_o @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_561_order__antisym__conv,axiom,
! [Y: nat > $o,X: nat > $o] :
( ( ord_less_eq_nat_o @ Y @ X )
=> ( ( ord_less_eq_nat_o @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_562_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_563_order__less__imp__not__less,axiom,
! [X: fset_a,Y: fset_a] :
( ( ord_less_fset_a @ X @ Y )
=> ~ ( ord_less_fset_a @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_564_order__less__imp__not__less,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ~ ( ord_less_set_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_565_order__less__imp__not__less,axiom,
! [X: list_nat,Y: list_nat] :
( ( ord_less_list_nat @ X @ Y )
=> ~ ( ord_less_list_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_566_order__less__imp__not__less,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( ord_le1203424502768444845at_nat @ X @ Y )
=> ~ ( ord_le1203424502768444845at_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_567_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_568_order__less__imp__not__eq2,axiom,
! [X: fset_a,Y: fset_a] :
( ( ord_less_fset_a @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_569_order__less__imp__not__eq2,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_570_order__less__imp__not__eq2,axiom,
! [X: list_nat,Y: list_nat] :
( ( ord_less_list_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_571_order__less__imp__not__eq2,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( ord_le1203424502768444845at_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_572_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_573_order__less__imp__not__eq,axiom,
! [X: fset_a,Y: fset_a] :
( ( ord_less_fset_a @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_574_order__less__imp__not__eq,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_575_order__less__imp__not__eq,axiom,
! [X: list_nat,Y: list_nat] :
( ( ord_less_list_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_576_order__less__imp__not__eq,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( ord_le1203424502768444845at_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_577_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_578_linorder__less__linear,axiom,
! [X: list_nat,Y: list_nat] :
( ( ord_less_list_nat @ X @ Y )
| ( X = Y )
| ( ord_less_list_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_579_linorder__less__linear,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( ord_le1203424502768444845at_nat @ X @ Y )
| ( X = Y )
| ( ord_le1203424502768444845at_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_580_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_581_order__less__imp__triv,axiom,
! [X: fset_a,Y: fset_a,P: $o] :
( ( ord_less_fset_a @ X @ Y )
=> ( ( ord_less_fset_a @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_582_order__less__imp__triv,axiom,
! [X: set_nat,Y: set_nat,P: $o] :
( ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_583_order__less__imp__triv,axiom,
! [X: list_nat,Y: list_nat,P: $o] :
( ( ord_less_list_nat @ X @ Y )
=> ( ( ord_less_list_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_584_order__less__imp__triv,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat,P: $o] :
( ( ord_le1203424502768444845at_nat @ X @ Y )
=> ( ( ord_le1203424502768444845at_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_585_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_586_order__less__not__sym,axiom,
! [X: fset_a,Y: fset_a] :
( ( ord_less_fset_a @ X @ Y )
=> ~ ( ord_less_fset_a @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_587_order__less__not__sym,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ~ ( ord_less_set_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_588_order__less__not__sym,axiom,
! [X: list_nat,Y: list_nat] :
( ( ord_less_list_nat @ X @ Y )
=> ~ ( ord_less_list_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_589_order__less__not__sym,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( ord_le1203424502768444845at_nat @ X @ Y )
=> ~ ( ord_le1203424502768444845at_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_590_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_591_order__less__subst2,axiom,
! [A: product_prod_nat_nat,B2: product_prod_nat_nat,F2: product_prod_nat_nat > nat,C2: nat] :
( ( ord_le1203424502768444845at_nat @ A @ B2 )
=> ( ( ord_less_nat @ ( F2 @ B2 ) @ C2 )
=> ( ! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( ( ord_le1203424502768444845at_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_592_order__less__subst2,axiom,
! [A: product_prod_nat_nat,B2: product_prod_nat_nat,F2: product_prod_nat_nat > fset_a,C2: fset_a] :
( ( ord_le1203424502768444845at_nat @ A @ B2 )
=> ( ( ord_less_fset_a @ ( F2 @ B2 ) @ C2 )
=> ( ! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( ( ord_le1203424502768444845at_nat @ X2 @ Y3 )
=> ( ord_less_fset_a @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_fset_a @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_593_order__less__subst2,axiom,
! [A: product_prod_nat_nat,B2: product_prod_nat_nat,F2: product_prod_nat_nat > set_nat,C2: set_nat] :
( ( ord_le1203424502768444845at_nat @ A @ B2 )
=> ( ( ord_less_set_nat @ ( F2 @ B2 ) @ C2 )
=> ( ! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( ( ord_le1203424502768444845at_nat @ X2 @ Y3 )
=> ( ord_less_set_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_set_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_594_order__less__subst2,axiom,
! [A: product_prod_nat_nat,B2: product_prod_nat_nat,F2: product_prod_nat_nat > list_nat,C2: list_nat] :
( ( ord_le1203424502768444845at_nat @ A @ B2 )
=> ( ( ord_less_list_nat @ ( F2 @ B2 ) @ C2 )
=> ( ! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( ( ord_le1203424502768444845at_nat @ X2 @ Y3 )
=> ( ord_less_list_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_list_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_595_order__less__subst2,axiom,
! [A: product_prod_nat_nat,B2: product_prod_nat_nat,F2: product_prod_nat_nat > product_prod_nat_nat,C2: product_prod_nat_nat] :
( ( ord_le1203424502768444845at_nat @ A @ B2 )
=> ( ( ord_le1203424502768444845at_nat @ ( F2 @ B2 ) @ C2 )
=> ( ! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( ( ord_le1203424502768444845at_nat @ X2 @ Y3 )
=> ( ord_le1203424502768444845at_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_le1203424502768444845at_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_596_order__less__subst2,axiom,
! [A: nat,B2: nat,F2: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ ( F2 @ B2 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_597_order__less__subst1,axiom,
! [A: nat,F2: nat > nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F2 @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_598_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_599_ord__less__eq__subst,axiom,
! [A: nat,B2: nat,F2: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ( F2 @ B2 )
= C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_600_ord__eq__less__subst,axiom,
! [A: nat,F2: nat > nat,B2: nat,C2: nat] :
( ( A
= ( F2 @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_601_order__less__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z4 )
=> ( ord_less_nat @ X @ Z4 ) ) ) ).
% order_less_trans
thf(fact_602_order__less__asym_H,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ~ ( ord_less_nat @ B2 @ A ) ) ).
% order_less_asym'
thf(fact_603_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_604_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_605_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_606_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( A != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_607_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( A != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_608_dual__order_Ostrict__trans,axiom,
! [B2: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( ( ord_less_nat @ C2 @ B2 )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_609_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_610_order_Ostrict__trans,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_611_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B2: nat] :
( ! [A4: nat,B5: nat] :
( ( ord_less_nat @ A4 @ B5 )
=> ( P @ A4 @ B5 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B5: nat] :
( ( P @ B5 @ A4 )
=> ( P @ A4 @ B5 ) )
=> ( P @ A @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_612_exists__least__iff,axiom,
( ( ^ [P4: nat > $o] :
? [X7: nat] : ( P4 @ X7 ) )
= ( ^ [P3: nat > $o] :
? [N3: nat] :
( ( P3 @ N3 )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ~ ( P3 @ M3 ) ) ) ) ) ).
% exists_least_iff
thf(fact_613_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_614_dual__order_Oasym,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ B2 @ A )
=> ~ ( ord_less_nat @ A @ B2 ) ) ).
% dual_order.asym
thf(fact_615_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_616_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_617_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X2: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X2 )
=> ( P @ Y4 ) )
=> ( P @ X2 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_618_ord__less__eq__trans,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_619_ord__eq__less__trans,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( A = B2 )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_620_order_Oasym,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ~ ( ord_less_nat @ B2 @ A ) ) ).
% order.asym
thf(fact_621_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_622_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_623_in__mono,axiom,
! [A2: set_nat,B: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_624_subsetD,axiom,
! [A2: set_nat,B: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_625_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A3 )
=> ( member_nat @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_626_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
! [T3: nat] :
( ( member_nat @ T3 @ A3 )
=> ( member_nat @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_627_fset__list__fsubset__eq__nth__conv,axiom,
! [Xs: list_a,A2: fset_a] :
( ( ord_less_eq_fset_a @ ( fset_of_list_a @ Xs ) @ A2 )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs ) )
=> ( fmember_a @ ( nth_a @ Xs @ I4 ) @ A2 ) ) ) ) ).
% fset_list_fsubset_eq_nth_conv
thf(fact_628_fset__list__fsubset__eq__nth__conv,axiom,
! [Xs: list_nat,A2: fset_nat] :
( ( ord_less_eq_fset_nat @ ( fset_of_list_nat @ Xs ) @ A2 )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( fmember_nat @ ( nth_nat @ Xs @ I4 ) @ A2 ) ) ) ) ).
% fset_list_fsubset_eq_nth_conv
thf(fact_629_fsubseteq__fset__conv__nth,axiom,
! [Ss: list_a,T2: fset_a] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Ss ) )
=> ( fmember_a @ ( nth_a @ Ss @ I4 ) @ T2 ) ) )
= ( ord_less_eq_fset_a @ ( fset_of_list_a @ Ss ) @ T2 ) ) ).
% fsubseteq_fset_conv_nth
thf(fact_630_fsubseteq__fset__conv__nth,axiom,
! [Ss: list_nat,T2: fset_nat] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ss ) )
=> ( fmember_nat @ ( nth_nat @ Ss @ I4 ) @ T2 ) ) )
= ( ord_less_eq_fset_nat @ ( fset_of_list_nat @ Ss ) @ T2 ) ) ).
% fsubseteq_fset_conv_nth
thf(fact_631_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_632_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_633_order__less__le__subst2,axiom,
! [A: nat,B2: nat,F2: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F2 @ B2 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_634_order__less__le__subst1,axiom,
! [A: nat,F2: nat > nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_635_order__le__less__subst2,axiom,
! [A: nat,B2: nat,F2: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_nat @ ( F2 @ B2 ) @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_636_order__le__less__subst1,axiom,
! [A: nat,F2: nat > nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F2 @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_637_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z4 )
=> ( ord_less_nat @ X @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_638_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z4 )
=> ( ord_less_nat @ X @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_639_order__neq__le__trans,axiom,
! [A: nat,B2: nat] :
( ( A != B2 )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_nat @ A @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_640_order__le__neq__trans,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_nat @ A @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_641_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_642_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_643_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_644_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_645_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_646_dual__order_Ostrict__implies__order,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( ord_less_eq_nat @ B2 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_647_order_Ostrict__implies__order,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% order.strict_implies_order
thf(fact_648_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B6: nat,A5: nat] :
( ( ord_less_eq_nat @ B6 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B6 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_649_dual__order_Ostrict__trans2,axiom,
! [B2: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C2 @ B2 )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_650_dual__order_Ostrict__trans1,axiom,
! [B2: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_nat @ C2 @ B2 )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_651_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B6: nat,A5: nat] :
( ( ord_less_eq_nat @ B6 @ A5 )
& ( A5 != B6 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_652_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B6: nat,A5: nat] :
( ( ord_less_nat @ B6 @ A5 )
| ( A5 = B6 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_653_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B6: nat] :
( ( ord_less_eq_nat @ A5 @ B6 )
& ~ ( ord_less_eq_nat @ B6 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_654_order_Ostrict__trans2,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_655_order_Ostrict__trans1,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_nat @ B2 @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_656_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B6: nat] :
( ( ord_less_eq_nat @ A5 @ B6 )
& ( A5 != B6 ) ) ) ) ).
% order.strict_iff_order
thf(fact_657_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B6: nat] :
( ( ord_less_nat @ A5 @ B6 )
| ( A5 = B6 ) ) ) ) ).
% order.order_iff_strict
thf(fact_658_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_659_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_660_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_661_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_662_nless__le,axiom,
! [A: nat,B2: nat] :
( ( ~ ( ord_less_nat @ A @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A @ B2 )
| ( A = B2 ) ) ) ).
% nless_le
thf(fact_663_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_664_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_665_adjust__idx__length,axiom,
! [I: nat,Xs: list_a,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ ( missing_remove_nth_a @ I @ Xs ) ) )
=> ( ord_less_nat @ ( missing_adjust_idx @ I @ J ) @ ( size_size_list_a @ Xs ) ) ) ) ).
% adjust_idx_length
thf(fact_666_adjust__idx__length,axiom,
! [I: nat,Xs: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ I @ Xs ) ) )
=> ( ord_less_nat @ ( missing_adjust_idx @ I @ J ) @ ( size_size_list_nat @ Xs ) ) ) ) ).
% adjust_idx_length
thf(fact_667_complete__interval,axiom,
! [A: nat,B2: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B2 )
=> ( ( P @ A )
=> ( ~ ( P @ B2 )
=> ? [C4: nat] :
( ( ord_less_eq_nat @ A @ C4 )
& ( ord_less_eq_nat @ C4 @ B2 )
& ! [X6: nat] :
( ( ( ord_less_eq_nat @ A @ X6 )
& ( ord_less_nat @ X6 @ C4 ) )
=> ( P @ X6 ) )
& ! [D2: nat] :
( ! [X2: nat] :
( ( ( ord_less_eq_nat @ A @ X2 )
& ( ord_less_nat @ X2 @ D2 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_nat @ D2 @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_668_verit__comp__simplify1_I3_J,axiom,
! [B7: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B7 @ A6 ) )
= ( ord_less_nat @ A6 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_669_pinf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ~ ( ord_less_eq_nat @ X6 @ T ) ) ).
% pinf(6)
thf(fact_670_pinf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( ord_less_eq_nat @ T @ X6 ) ) ).
% pinf(8)
thf(fact_671_minf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( ord_less_eq_nat @ X6 @ T ) ) ).
% minf(6)
thf(fact_672_minf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ~ ( ord_less_eq_nat @ T @ X6 ) ) ).
% minf(8)
thf(fact_673_min__list__nth,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Ys @ I3 ) ) )
=> ( ord_less_eq_nat @ ( missing_min_list_nat @ Xs ) @ ( missing_min_list_nat @ Ys ) ) ) ) ).
% min_list_nth
thf(fact_674_remove__nth__sound__r,axiom,
! [N: nat,P2: nat,Xs: list_a] :
( ( ord_less_eq_nat @ N @ P2 )
=> ( ( ord_less_nat @ P2 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( missing_remove_nth_a @ N @ Xs ) @ P2 )
= ( nth_a @ Xs @ ( suc @ P2 ) ) ) ) ) ).
% remove_nth_sound_r
thf(fact_675_remove__nth__sound__r,axiom,
! [N: nat,P2: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ N @ P2 )
=> ( ( ord_less_nat @ P2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( missin7175274867594579095th_nat @ N @ Xs ) @ P2 )
= ( nth_nat @ Xs @ ( suc @ P2 ) ) ) ) ) ).
% remove_nth_sound_r
thf(fact_676_nth__remove__nth__conv,axiom,
! [I: nat,N: nat,Xs: list_a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ ( missing_remove_nth_a @ N @ Xs ) ) )
=> ( ( nth_a @ ( missing_remove_nth_a @ N @ Xs ) @ I )
= ( nth_a @ Xs @ ( if_nat @ ( ord_less_nat @ I @ N ) @ I @ ( suc @ I ) ) ) ) ) ).
% nth_remove_nth_conv
thf(fact_677_nth__remove__nth__conv,axiom,
! [I: nat,N: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ N @ Xs ) ) )
=> ( ( nth_nat @ ( missin7175274867594579095th_nat @ N @ Xs ) @ I )
= ( nth_nat @ Xs @ ( if_nat @ ( ord_less_nat @ I @ N ) @ I @ ( suc @ I ) ) ) ) ) ).
% nth_remove_nth_conv
thf(fact_678_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_679_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_680_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_681_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_682_Suc__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_eq
thf(fact_683_Suc__le__mono,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ).
% Suc_le_mono
thf(fact_684_Suc__leD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_leD
thf(fact_685_le__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N )
=> ( M2
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_686_le__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_687_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_688_Suc__le__D,axiom,
! [N: nat,M4: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M4 )
=> ? [M5: nat] :
( M4
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_689_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_690_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_691_le__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M2 @ N )
| ( M2
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_692_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_693_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_694_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_695_not__less__eq__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_696_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
=> ( P @ M ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_697_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B2 ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_698_nat__induct__at__least,axiom,
! [M2: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( P @ M2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_699_transitive__stepwise__le,axiom,
! [M2: nat,N: nat,R4: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ! [X2: nat] : ( R4 @ X2 @ X2 )
=> ( ! [X2: nat,Y3: nat,Z2: nat] :
( ( R4 @ X2 @ Y3 )
=> ( ( R4 @ Y3 @ Z2 )
=> ( R4 @ X2 @ Z2 ) ) )
=> ( ! [N2: nat] : ( R4 @ N2 @ ( suc @ N2 ) )
=> ( R4 @ M2 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_700_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_701_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_702_lift__Suc__mono__le,axiom,
! [F2: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F2 @ N ) @ ( F2 @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_703_lift__Suc__antimono__le,axiom,
! [F2: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F2 @ N4 ) @ ( F2 @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_704_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_705_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_706_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_707_Suc__lessI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ( suc @ M2 )
!= N )
=> ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_708_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_709_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_710_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ N )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_711_less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) ) ) ).
% less_Suc_eq
thf(fact_712_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_nat @ M2 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_713_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_714_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M2 )
= ( ? [M6: nat] :
( ( M2
= ( suc @ M6 ) )
& ( ord_less_nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_715_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_716_Suc__less__SucD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_SucD
thf(fact_717_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_718_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J3: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ K2 )
=> ( ( P @ I3 @ J3 )
=> ( ( P @ J3 @ K2 )
=> ( P @ I3 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_719_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_720_not__less__less__Suc__eq,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_721_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_722_Suc__le__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_eq
thf(fact_723_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_724_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_725_Suc__le__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_lessD
thf(fact_726_le__less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_727_less__Suc__eq__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_Suc_eq_le
thf(fact_728_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_729_le__imp__less__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_730_lift__Suc__mono__less,axiom,
! [F2: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F2 @ N ) @ ( F2 @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_731_lift__Suc__mono__less__iff,axiom,
! [F2: nat > nat,N: nat,M2: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F2 @ N ) @ ( F2 @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_732_adjust__idx__def,axiom,
( missing_adjust_idx
= ( ^ [I4: nat,J2: nat] : ( if_nat @ ( ord_less_nat @ J2 @ I4 ) @ J2 @ ( suc @ J2 ) ) ) ) ).
% adjust_idx_def
thf(fact_733_min__list,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ord_less_eq_nat @ ( missing_min_list_nat @ Xs ) @ X ) ) ).
% min_list
thf(fact_734_min__list__subset,axiom,
! [Ys: list_nat,Xs: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Ys ) @ ( set_nat2 @ Xs ) )
=> ( ( member_nat @ ( missing_min_list_nat @ Xs ) @ ( set_nat2 @ Ys ) )
=> ( ( missing_min_list_nat @ Xs )
= ( missing_min_list_nat @ Ys ) ) ) ) ).
% min_list_subset
thf(fact_735_verit__la__disequality,axiom,
! [A: nat,B2: nat] :
( ( A = B2 )
| ~ ( ord_less_eq_nat @ A @ B2 )
| ~ ( ord_less_eq_nat @ B2 @ A ) ) ).
% verit_la_disequality
thf(fact_736_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_737_minf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ~ ( ord_less_nat @ T @ X6 ) ) ).
% minf(7)
thf(fact_738_minf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( ord_less_nat @ X6 @ T ) ) ).
% minf(5)
thf(fact_739_minf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_740_minf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_741_minf_I2_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z5 )
=> ( ( P @ X2 )
= ( P5 @ X2 ) ) )
=> ( ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z5 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P5 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_742_minf_I1_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z5 )
=> ( ( P @ X2 )
= ( P5 @ X2 ) ) )
=> ( ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z5 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P5 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_743_pinf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( ord_less_nat @ T @ X6 ) ) ).
% pinf(7)
thf(fact_744_pinf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ~ ( ord_less_nat @ X6 @ T ) ) ).
% pinf(5)
thf(fact_745_pinf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_746_pinf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_747_pinf_I2_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z5 @ X2 )
=> ( ( P @ X2 )
= ( P5 @ X2 ) ) )
=> ( ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z5 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P5 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_748_pinf_I1_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z5 @ X2 )
=> ( ( P @ X2 )
= ( P5 @ X2 ) ) )
=> ( ? [Z5: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z5 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P5 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_749_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_750_remove__nth__len,axiom,
! [I: nat,Xs: list_a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( size_size_list_a @ Xs )
= ( suc @ ( size_size_list_a @ ( missing_remove_nth_a @ I @ Xs ) ) ) ) ) ).
% remove_nth_len
thf(fact_751_remove__nth__len,axiom,
! [I: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( size_size_list_nat @ Xs )
= ( suc @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ I @ Xs ) ) ) ) ) ).
% remove_nth_len
thf(fact_752_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M2: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I5: nat] :
( ( ord_less_nat @ K2 @ I5 )
=> ( P @ I5 ) )
=> ( P @ K2 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_753_inf__pigeonhole__principle,axiom,
! [N: nat,F2: nat > nat > $o] :
( ! [K2: nat] :
? [I5: nat] :
( ( ord_less_nat @ I5 @ N )
& ( F2 @ K2 @ I5 ) )
=> ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ! [K3: nat] :
? [K4: nat] :
( ( ord_less_eq_nat @ K3 @ K4 )
& ( F2 @ K4 @ I3 ) ) ) ) ).
% inf_pigeonhole_principle
thf(fact_754_sorted__iff__nth__mono,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I4: nat,J2: nat] :
( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I4 ) @ ( nth_nat @ Xs @ J2 ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_755_sorted__nth__mono,axiom,
! [Xs: list_nat,I: nat,J: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_756_sorted__iff__nth__Suc,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I4: nat] :
( ( ord_less_nat @ ( suc @ I4 ) @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I4 ) @ ( nth_nat @ Xs @ ( suc @ I4 ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_757_infinite__imp__many__elems,axiom,
! [A2: set_a,N: nat] :
( ~ ( finite_finite_a @ A2 )
=> ? [Xs2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs2 ) @ A2 )
& ( ( size_size_list_a @ Xs2 )
= N )
& ( distinct_a @ Xs2 ) ) ) ).
% infinite_imp_many_elems
thf(fact_758_infinite__imp__many__elems,axiom,
! [A2: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A2 )
=> ? [Xs2: list_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
& ( ( size_size_list_nat @ Xs2 )
= N )
& ( distinct_nat @ Xs2 ) ) ) ).
% infinite_imp_many_elems
thf(fact_759_remdups__adj__adjacent,axiom,
! [I: nat,Xs: list_a] :
( ( ord_less_nat @ ( suc @ I ) @ ( size_size_list_a @ ( remdups_adj_a @ Xs ) ) )
=> ( ( nth_a @ ( remdups_adj_a @ Xs ) @ I )
!= ( nth_a @ ( remdups_adj_a @ Xs ) @ ( suc @ I ) ) ) ) ).
% remdups_adj_adjacent
thf(fact_760_remdups__adj__adjacent,axiom,
! [I: nat,Xs: list_nat] :
( ( ord_less_nat @ ( suc @ I ) @ ( size_size_list_nat @ ( remdups_adj_nat @ Xs ) ) )
=> ( ( nth_nat @ ( remdups_adj_nat @ Xs ) @ I )
!= ( nth_nat @ ( remdups_adj_nat @ Xs ) @ ( suc @ I ) ) ) ) ).
% remdups_adj_adjacent
thf(fact_761_List_Ofinite__set,axiom,
! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_762_remdups__adj__sorted__distinct,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( distinct_nat @ ( remdups_adj_nat @ Xs ) ) ) ).
% remdups_adj_sorted_distinct
thf(fact_763_sorted__wrt__mono__rel,axiom,
! [Xs: list_nat,P: nat > nat > $o,Q: nat > nat > $o] :
( ! [X2: nat,Y3: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( member_nat @ Y3 @ ( set_nat2 @ Xs ) )
=> ( ( P @ X2 @ Y3 )
=> ( Q @ X2 @ Y3 ) ) ) )
=> ( ( sorted_wrt_nat @ P @ Xs )
=> ( sorted_wrt_nat @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_764_finite__Collect__less__eq,axiom,
! [Q: nat > $o,P: nat > $o] :
( ( ord_less_eq_nat_o @ Q @ P )
=> ( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_less_eq
thf(fact_765_sorted__remdups__adj,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( remdups_adj_nat @ Xs ) ) ) ).
% sorted_remdups_adj
thf(fact_766_finite__list,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ? [Xs2: list_nat] :
( ( set_nat2 @ Xs2 )
= A2 ) ) ).
% finite_list
thf(fact_767_finite__sorted__distinct__unique,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ? [X2: list_nat] :
( ( ( set_nat2 @ X2 )
= A2 )
& ( sorted_wrt_nat @ ord_less_eq_nat @ X2 )
& ( distinct_nat @ X2 )
& ! [Y4: list_nat] :
( ( ( ( set_nat2 @ Y4 )
= A2 )
& ( sorted_wrt_nat @ ord_less_eq_nat @ Y4 )
& ( distinct_nat @ Y4 ) )
=> ( Y4 = X2 ) ) ) ) ).
% finite_sorted_distinct_unique
thf(fact_768_strict__sorted__imp__sorted,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ Xs ) ) ).
% strict_sorted_imp_sorted
thf(fact_769_strict__sorted__equal,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs )
=> ( ( sorted_wrt_nat @ ord_less_nat @ Ys )
=> ( ( ( set_nat2 @ Ys )
= ( set_nat2 @ Xs ) )
=> ( Ys = Xs ) ) ) ) ).
% strict_sorted_equal
thf(fact_770_sorted__wrt__less__idx,axiom,
! [Ns: list_nat,I: nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Ns )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
=> ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).
% sorted_wrt_less_idx
thf(fact_771_sorted__replicate,axiom,
! [N: nat,X: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( replicate_nat @ N @ X ) ) ).
% sorted_replicate
thf(fact_772_finite__maxlen,axiom,
! [M7: set_list_a] :
( ( finite_finite_list_a @ M7 )
=> ? [N2: nat] :
! [X6: list_a] :
( ( member_list_a @ X6 @ M7 )
=> ( ord_less_nat @ ( size_size_list_a @ X6 ) @ N2 ) ) ) ).
% finite_maxlen
thf(fact_773_finite__maxlen,axiom,
! [M7: set_list_nat] :
( ( finite8100373058378681591st_nat @ M7 )
=> ? [N2: nat] :
! [X6: list_nat] :
( ( member_list_nat @ X6 @ M7 )
=> ( ord_less_nat @ ( size_size_list_nat @ X6 ) @ N2 ) ) ) ).
% finite_maxlen
thf(fact_774_remdups__adj__length,axiom,
! [Xs: list_a] : ( ord_less_eq_nat @ ( size_size_list_a @ ( remdups_adj_a @ Xs ) ) @ ( size_size_list_a @ Xs ) ) ).
% remdups_adj_length
thf(fact_775_remdups__adj__length,axiom,
! [Xs: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( remdups_adj_nat @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% remdups_adj_length
thf(fact_776_finite__distinct__list,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ? [Xs2: list_nat] :
( ( ( set_nat2 @ Xs2 )
= A2 )
& ( distinct_nat @ Xs2 ) ) ) ).
% finite_distinct_list
thf(fact_777_remdups__sort_I1_J,axiom,
! [Xs: list_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( missin6101193410121742181rt_nat @ Xs ) ) ).
% remdups_sort(1)
thf(fact_778_sorted__wrt__iff__nth__less,axiom,
( sorted_wrt_a
= ( ^ [P3: a > a > $o,Xs3: list_a] :
! [I4: nat,J2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_a @ Xs3 ) )
=> ( P3 @ ( nth_a @ Xs3 @ I4 ) @ ( nth_a @ Xs3 @ J2 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_779_sorted__wrt__iff__nth__less,axiom,
( sorted_wrt_nat
= ( ^ [P3: nat > nat > $o,Xs3: list_nat] :
! [I4: nat,J2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs3 ) )
=> ( P3 @ ( nth_nat @ Xs3 @ I4 ) @ ( nth_nat @ Xs3 @ J2 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_780_sorted__wrt__nth__less,axiom,
! [P: a > a > $o,Xs: list_a,I: nat,J: nat] :
( ( sorted_wrt_a @ P @ Xs )
=> ( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs ) )
=> ( P @ ( nth_a @ Xs @ I ) @ ( nth_a @ Xs @ J ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_781_sorted__wrt__nth__less,axiom,
! [P: nat > nat > $o,Xs: list_nat,I: nat,J: nat] :
( ( sorted_wrt_nat @ P @ Xs )
=> ( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_782_strict__sorted__iff,axiom,
! [L: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ L )
= ( ( sorted_wrt_nat @ ord_less_eq_nat @ L )
& ( distinct_nat @ L ) ) ) ).
% strict_sorted_iff
thf(fact_783_sorted__distinct__set__unique,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( distinct_nat @ Xs )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ys )
=> ( ( distinct_nat @ Ys )
=> ( ( ( set_nat2 @ Xs )
= ( set_nat2 @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ).
% sorted_distinct_set_unique
thf(fact_784_sorted__iff__nth__mono__less,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I4: nat,J2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I4 ) @ ( nth_nat @ Xs @ J2 ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_785_finite__psubset__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ! [B8: set_nat] :
( ( ord_less_set_nat @ B8 @ A7 )
=> ( P @ B8 ) )
=> ( P @ A7 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_786_finite__subset,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_787_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_788_rev__finite__subset,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_789_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_eq_nat @ X2 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_790_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_eq_nat @ A @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_791_sorted__rev__iff__nth__mono,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
= ( ! [I4: nat,J2: nat] :
( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ J2 ) @ ( nth_nat @ Xs @ I4 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_792_sorted__rev__nth__mono,axiom,
! [Xs: list_nat,I: nat,J: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ J ) @ ( nth_nat @ Xs @ I ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_793_sorted__rev__iff__nth__Suc,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
= ( ! [I4: nat] :
( ( ord_less_nat @ ( suc @ I4 ) @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ ( suc @ I4 ) ) @ ( nth_nat @ Xs @ I4 ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_794_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( distinct_nat @ Xs )
=> ( ( linord2614967742042102400et_nat @ ( set_nat2 @ Xs ) )
= Xs ) ) ) ).
% sorted_list_of_set.idem_if_sorted_distinct
thf(fact_795_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ~ ! [L2: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ L2 )
=> ( ( ( set_nat2 @ L2 )
= A2 )
=> ( ( size_size_list_nat @ L2 )
!= ( finite_card_nat @ A2 ) ) ) ) ) ).
% sorted_list_of_set.finite_set_strict_sorted
thf(fact_796_length__rev,axiom,
! [Xs: list_a] :
( ( size_size_list_a @ ( rev_a @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_rev
thf(fact_797_length__rev,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rev_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rev
thf(fact_798_rev__replicate,axiom,
! [N: nat,X: a] :
( ( rev_a @ ( replicate_a @ N @ X ) )
= ( replicate_a @ N @ X ) ) ).
% rev_replicate
thf(fact_799_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( set_nat2 @ ( linord2614967742042102400et_nat @ A2 ) )
= A2 ) ) ).
% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_800_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
! [A2: set_nat] :
( ( size_size_list_nat @ ( linord2614967742042102400et_nat @ A2 ) )
= ( finite_card_nat @ A2 ) ) ).
% sorted_list_of_set.length_sorted_key_list_of_set
thf(fact_801_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( linord2614967742042102400et_nat @ A2 )
= ( linord2614967742042102400et_nat @ B ) )
=> ( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( A2 = B ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_802_card__subset__eq,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B ) )
=> ( A2 = B ) ) ) ) ).
% card_subset_eq
thf(fact_803_infinite__arbitrarily__large,axiom,
! [A2: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A2 )
=> ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ( finite_card_nat @ B3 )
= N )
& ( ord_less_eq_set_nat @ B3 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_804_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
! [A2: set_nat,L: list_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( sorted_wrt_nat @ ord_less_nat @ L )
& ( ( set_nat2 @ L )
= A2 )
& ( ( size_size_list_nat @ L )
= ( finite_card_nat @ A2 ) ) )
= ( ( linord2614967742042102400et_nat @ A2 )
= L ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_unique
thf(fact_805_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
! [A2: set_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( linord2614967742042102400et_nat @ A2 ) ) ).
% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_806_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
! [A2: set_nat] : ( sorted_wrt_nat @ ord_less_nat @ ( linord2614967742042102400et_nat @ A2 ) ) ).
% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_807_card__mono,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) ) ) ) ).
% card_mono
thf(fact_808_card__seteq,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A2 ) )
=> ( A2 = B ) ) ) ) ).
% card_seteq
thf(fact_809_exists__subset__between,axiom,
! [A2: set_nat,N: nat,C3: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C3 ) )
=> ( ( ord_less_eq_set_nat @ A2 @ C3 )
=> ( ( finite_finite_nat @ C3 )
=> ? [B3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ C3 )
& ( ( finite_card_nat @ B3 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_810_obtain__subset__with__card__n,axiom,
! [N: nat,S2: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S2 ) )
=> ~ ! [T4: set_nat] :
( ( ord_less_eq_set_nat @ T4 @ S2 )
=> ( ( ( finite_card_nat @ T4 )
= N )
=> ~ ( finite_finite_nat @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_811_finite__if__finite__subsets__card__bdd,axiom,
! [F4: set_nat,C3: nat] :
( ! [G3: set_nat] :
( ( ord_less_eq_set_nat @ G3 @ F4 )
=> ( ( finite_finite_nat @ G3 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G3 ) @ C3 ) ) )
=> ( ( finite_finite_nat @ F4 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F4 ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_812_card__length,axiom,
! [Xs: list_a] : ( ord_less_eq_nat @ ( finite_card_a @ ( set_a2 @ Xs ) ) @ ( size_size_list_a @ Xs ) ) ).
% card_length
thf(fact_813_card__length,axiom,
! [Xs: list_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( set_nat2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% card_length
thf(fact_814_psubset__card__mono,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_set_nat @ A2 @ B )
=> ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) ) ) ) ).
% psubset_card_mono
thf(fact_815_distinct__card,axiom,
! [Xs: list_a] :
( ( distinct_a @ Xs )
=> ( ( finite_card_a @ ( set_a2 @ Xs ) )
= ( size_size_list_a @ Xs ) ) ) ).
% distinct_card
thf(fact_816_distinct__card,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ Xs )
=> ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ) ).
% distinct_card
thf(fact_817_card__distinct,axiom,
! [Xs: list_a] :
( ( ( finite_card_a @ ( set_a2 @ Xs ) )
= ( size_size_list_a @ Xs ) )
=> ( distinct_a @ Xs ) ) ).
% card_distinct
thf(fact_818_card__distinct,axiom,
! [Xs: list_nat] :
( ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
= ( size_size_list_nat @ Xs ) )
=> ( distinct_nat @ Xs ) ) ).
% card_distinct
thf(fact_819_card__psubset,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) )
=> ( ord_less_set_nat @ A2 @ B ) ) ) ) ).
% card_psubset
thf(fact_820_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A2: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ? [B9: nat] :
( ( member_nat @ B9 @ B )
& ( R @ A4 @ B9 ) ) )
=> ( ! [A13: nat,A24: nat,B5: nat] :
( ( member_nat @ A13 @ A2 )
=> ( ( member_nat @ A24 @ A2 )
=> ( ( member_nat @ B5 @ B )
=> ( ( R @ A13 @ B5 )
=> ( ( R @ A24 @ B5 )
=> ( A13 = A24 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_821_rev__nth,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( rev_a @ Xs ) @ N )
= ( nth_a @ Xs @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ ( suc @ N ) ) ) ) ) ).
% rev_nth
thf(fact_822_rev__nth,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( rev_nat @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ ( suc @ N ) ) ) ) ) ).
% rev_nth
thf(fact_823_card__le__Suc0__iff__eq,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( suc @ zero_zero_nat ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ! [Y5: nat] :
( ( member_nat @ Y5 @ A2 )
=> ( X4 = Y5 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_824_sorted01,axiom,
! [Xs: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ Xs ) ) ).
% sorted01
thf(fact_825_finite__enum__subset,axiom,
! [X8: set_nat,Y6: set_nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( finite_card_nat @ X8 ) )
=> ( ( infini8530281810654367211te_nat @ X8 @ I3 )
= ( infini8530281810654367211te_nat @ Y6 @ I3 ) ) )
=> ( ( finite_finite_nat @ X8 )
=> ( ( finite_finite_nat @ Y6 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ X8 ) @ ( finite_card_nat @ Y6 ) )
=> ( ord_less_eq_set_nat @ X8 @ Y6 ) ) ) ) ) ).
% finite_enum_subset
thf(fact_826_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_827_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_828_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_829_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_830_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_831_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_832_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_833_Suc__diff__diff,axiom,
! [M2: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_834_diff__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_Suc_Suc
thf(fact_835_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_836_replicate__eq__replicate,axiom,
! [M2: nat,X: a,N: nat,Y: a] :
( ( ( replicate_a @ M2 @ X )
= ( replicate_a @ N @ Y ) )
= ( ( M2 = N )
& ( ( M2 != zero_zero_nat )
=> ( X = Y ) ) ) ) ).
% replicate_eq_replicate
thf(fact_837_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_838_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_839_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_840_diff__is__0__eq_H,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_841_diff__is__0__eq,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% diff_is_0_eq
thf(fact_842_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_843_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_844_in__set__replicate,axiom,
! [X: nat,N: nat,Y: nat] :
( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N @ Y ) ) )
= ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_845_in__set__replicate,axiom,
! [X: a,N: nat,Y: a] :
( ( member_a @ X @ ( set_a2 @ ( replicate_a @ N @ Y ) ) )
= ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_846_Bex__set__replicate,axiom,
! [N: nat,A: a,P: a > $o] :
( ( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ ( replicate_a @ N @ A ) ) )
& ( P @ X4 ) ) )
= ( ( P @ A )
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_847_Ball__set__replicate,axiom,
! [N: nat,A: a,P: a > $o] :
( ( ! [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ ( replicate_a @ N @ A ) ) )
=> ( P @ X4 ) ) )
= ( ( P @ A )
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_848_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_849_enumerate__mono__iff,axiom,
! [S2: set_nat,M2: nat,N: nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( ( ord_less_nat @ ( infini8530281810654367211te_nat @ S2 @ M2 ) @ ( infini8530281810654367211te_nat @ S2 @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ) ).
% enumerate_mono_iff
thf(fact_850_enumerate__mono__le__iff,axiom,
! [S2: set_nat,M2: nat,N: nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( ( ord_less_eq_nat @ ( infini8530281810654367211te_nat @ S2 @ M2 ) @ ( infini8530281810654367211te_nat @ S2 @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% enumerate_mono_le_iff
thf(fact_851_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_852_finite__enumerate__mono__iff,axiom,
! [S2: set_nat,M2: nat,N: nat] :
( ( finite_finite_nat @ S2 )
=> ( ( ord_less_nat @ M2 @ ( finite_card_nat @ S2 ) )
=> ( ( ord_less_nat @ N @ ( finite_card_nat @ S2 ) )
=> ( ( ord_less_nat @ ( infini8530281810654367211te_nat @ S2 @ M2 ) @ ( infini8530281810654367211te_nat @ S2 @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ) ).
% finite_enumerate_mono_iff
thf(fact_853_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_854_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_855_diffs0__imp__equal,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M2 )
= zero_zero_nat )
=> ( M2 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_856_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_857_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_858_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_859_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_860_diff__less,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).
% diff_less
thf(fact_861_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_862_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_863_eq__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_864_le__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_865_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_866_diff__le__mono,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_867_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_868_le__diff__iff_H,axiom,
! [A: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_869_diff__le__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_870_diff__less__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_871_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_872_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_873_le__enumerate,axiom,
! [S2: set_nat,N: nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S2 @ N ) ) ) ).
% le_enumerate
thf(fact_874_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_875_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_876_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_877_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_878_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_879_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N: nat] :
( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X2: nat,Y3: nat] :
( ( P @ X2 @ Y3 )
=> ( P @ ( suc @ X2 ) @ ( suc @ Y3 ) ) )
=> ( P @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_880_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_881_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_882_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_883_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_884_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_885_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_886_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M: nat] :
( ( ord_less_nat @ M @ N2 )
& ~ ( P @ M ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_887_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_888_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_889_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_890_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_891_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_892_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_893_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_894_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_895_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_896_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_897_finite__le__enumerate,axiom,
! [S2: set_nat,N: nat] :
( ( finite_finite_nat @ S2 )
=> ( ( ord_less_nat @ N @ ( finite_card_nat @ S2 ) )
=> ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S2 @ N ) ) ) ) ).
% finite_le_enumerate
thf(fact_898_remove__nth__length,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( size_size_list_a @ ( missing_remove_nth_a @ N @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) ) ) ).
% remove_nth_length
thf(fact_899_remove__nth__length,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( size_size_list_nat @ ( missin7175274867594579095th_nat @ N @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ) ).
% remove_nth_length
thf(fact_900_adjust__idx__rev__def,axiom,
( missin3815256168798769645dx_rev
= ( ^ [I4: nat,J2: nat] : ( if_nat @ ( ord_less_nat @ J2 @ I4 ) @ J2 @ ( minus_minus_nat @ J2 @ ( suc @ zero_zero_nat ) ) ) ) ) ).
% adjust_idx_rev_def
thf(fact_901_rev__update,axiom,
! [K: nat,Xs: list_a,Y: a] :
( ( ord_less_nat @ K @ ( size_size_list_a @ Xs ) )
=> ( ( rev_a @ ( list_update_a @ Xs @ K @ Y ) )
= ( list_update_a @ ( rev_a @ Xs ) @ ( minus_minus_nat @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ K ) @ one_one_nat ) @ Y ) ) ) ).
% rev_update
thf(fact_902_rev__update,axiom,
! [K: nat,Xs: list_nat,Y: nat] :
( ( ord_less_nat @ K @ ( size_size_list_nat @ Xs ) )
=> ( ( rev_nat @ ( list_update_nat @ Xs @ K @ Y ) )
= ( list_update_nat @ ( rev_nat @ Xs ) @ ( minus_minus_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ K ) @ one_one_nat ) @ Y ) ) ) ).
% rev_update
thf(fact_903_Suc__diff__Suc,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N ) ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_904_diff__less__Suc,axiom,
! [M2: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_905_Suc__diff__le,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_906_diff__less__mono,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C2 ) @ ( minus_minus_nat @ B2 @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_907_less__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_908_all__less__two,axiom,
! [P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ ( suc @ zero_zero_nat ) ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ( P @ ( suc @ zero_zero_nat ) ) ) ) ).
% all_less_two
thf(fact_909_all__Suc__conv,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% all_Suc_conv
thf(fact_910_ex__Suc__conv,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% ex_Suc_conv
thf(fact_911_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_912_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_913_less__Suc__eq__0__disj,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( M2 = zero_zero_nat )
| ? [J2: nat] :
( ( M2
= ( suc @ J2 ) )
& ( ord_less_nat @ J2 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_914_enumerate__step,axiom,
! [S2: set_nat,N: nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S2 @ N ) @ ( infini8530281810654367211te_nat @ S2 @ ( suc @ N ) ) ) ) ).
% enumerate_step
thf(fact_915_enumerate__mono,axiom,
! [M2: nat,N: nat,S2: set_nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ~ ( finite_finite_nat @ S2 )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S2 @ M2 ) @ ( infini8530281810654367211te_nat @ S2 @ N ) ) ) ) ).
% enumerate_mono
thf(fact_916_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K2 )
=> ~ ( P @ I5 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_917_term_Osize_I3_J,axiom,
! [X1: a] :
( ( size_size_term_b_a @ ( var_a_b @ X1 ) )
= ( suc @ zero_zero_nat ) ) ).
% term.size(3)
thf(fact_918_finite__enum__ext,axiom,
! [X8: set_nat,Y6: set_nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( finite_card_nat @ X8 ) )
=> ( ( infini8530281810654367211te_nat @ X8 @ I3 )
= ( infini8530281810654367211te_nat @ Y6 @ I3 ) ) )
=> ( ( finite_finite_nat @ X8 )
=> ( ( finite_finite_nat @ Y6 )
=> ( ( ( finite_card_nat @ X8 )
= ( finite_card_nat @ Y6 ) )
=> ( X8 = Y6 ) ) ) ) ) ).
% finite_enum_ext
thf(fact_919_finite__enumerate__Ex,axiom,
! [S2: set_nat,S: nat] :
( ( finite_finite_nat @ S2 )
=> ( ( member_nat @ S @ S2 )
=> ? [N2: nat] :
( ( ord_less_nat @ N2 @ ( finite_card_nat @ S2 ) )
& ( ( infini8530281810654367211te_nat @ S2 @ N2 )
= S ) ) ) ) ).
% finite_enumerate_Ex
thf(fact_920_finite__enumerate__in__set,axiom,
! [S2: set_nat,N: nat] :
( ( finite_finite_nat @ S2 )
=> ( ( ord_less_nat @ N @ ( finite_card_nat @ S2 ) )
=> ( member_nat @ ( infini8530281810654367211te_nat @ S2 @ N ) @ S2 ) ) ) ).
% finite_enumerate_in_set
thf(fact_921_finite__enumerate__mono,axiom,
! [M2: nat,N: nat,S2: set_nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( finite_finite_nat @ S2 )
=> ( ( ord_less_nat @ N @ ( finite_card_nat @ S2 ) )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S2 @ M2 ) @ ( infini8530281810654367211te_nat @ S2 @ N ) ) ) ) ) ).
% finite_enumerate_mono
thf(fact_922_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M3: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ N5 )
=> ( ord_less_nat @ X4 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_923_infinite__nat__iff__unbounded,axiom,
! [S2: set_nat] :
( ( ~ ( finite_finite_nat @ S2 ) )
= ( ! [M3: nat] :
? [N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ( member_nat @ N3 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_924_bounded__nat__set__is__finite,axiom,
! [N6: set_nat,N: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ N6 )
=> ( ord_less_nat @ X2 @ N ) )
=> ( finite_finite_nat @ N6 ) ) ).
% bounded_nat_set_is_finite
thf(fact_925_unbounded__k__infinite,axiom,
! [K: nat,S2: set_nat] :
( ! [M5: nat] :
( ( ord_less_nat @ K @ M5 )
=> ? [N7: nat] :
( ( ord_less_nat @ M5 @ N7 )
& ( member_nat @ N7 @ S2 ) ) )
=> ~ ( finite_finite_nat @ S2 ) ) ).
% unbounded_k_infinite
thf(fact_926_length__pos__if__in__set,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_927_length__pos__if__in__set,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_928_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M7: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M7 ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_929_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N5: set_nat] :
? [M3: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ N5 )
=> ( ord_less_eq_nat @ X4 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_930_infinite__nat__iff__unbounded__le,axiom,
! [S2: set_nat] :
( ( ~ ( finite_finite_nat @ S2 ) )
= ( ! [M3: nat] :
? [N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( member_nat @ N3 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_931_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I5: nat] :
( ( ord_less_eq_nat @ I5 @ K2 )
=> ~ ( P @ I5 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_932_sorted__wrt01,axiom,
! [Xs: list_a,P: a > a > $o] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ one_one_nat )
=> ( sorted_wrt_a @ P @ Xs ) ) ).
% sorted_wrt01
thf(fact_933_sorted__wrt01,axiom,
! [Xs: list_nat,P: nat > nat > $o] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( sorted_wrt_nat @ P @ Xs ) ) ).
% sorted_wrt01
thf(fact_934_card__ge__0__finite,axiom,
! [A2: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
=> ( finite_finite_nat @ A2 ) ) ).
% card_ge_0_finite
thf(fact_935_finite__enumerate__step,axiom,
! [S2: set_nat,N: nat] :
( ( finite_finite_nat @ S2 )
=> ( ( ord_less_nat @ ( suc @ N ) @ ( finite_card_nat @ S2 ) )
=> ( ord_less_nat @ ( infini8530281810654367211te_nat @ S2 @ N ) @ ( infini8530281810654367211te_nat @ S2 @ ( suc @ N ) ) ) ) ) ).
% finite_enumerate_step
thf(fact_936_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_937_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_938_fminusI,axiom,
! [C2: a,A2: fset_a,B: fset_a] :
( ( fmember_a @ C2 @ A2 )
=> ( ~ ( fmember_a @ C2 @ B )
=> ( fmember_a @ C2 @ ( minus_minus_fset_a @ A2 @ B ) ) ) ) ).
% fminusI
thf(fact_939_fminus__iff,axiom,
! [C2: a,A2: fset_a,B: fset_a] :
( ( fmember_a @ C2 @ ( minus_minus_fset_a @ A2 @ B ) )
= ( ( fmember_a @ C2 @ A2 )
& ~ ( fmember_a @ C2 @ B ) ) ) ).
% fminus_iff
thf(fact_940_finsert__fminus1,axiom,
! [X: a,B: fset_a,A2: fset_a] :
( ( fmember_a @ X @ B )
=> ( ( minus_minus_fset_a @ ( finsert_a @ X @ A2 ) @ B )
= ( minus_minus_fset_a @ A2 @ B ) ) ) ).
% finsert_fminus1
thf(fact_941_fminusE,axiom,
! [C2: a,A2: fset_a,B: fset_a] :
( ( fmember_a @ C2 @ ( minus_minus_fset_a @ A2 @ B ) )
=> ~ ( ( fmember_a @ C2 @ A2 )
=> ( fmember_a @ C2 @ B ) ) ) ).
% fminusE
thf(fact_942_fminusD1,axiom,
! [C2: a,A2: fset_a,B: fset_a] :
( ( fmember_a @ C2 @ ( minus_minus_fset_a @ A2 @ B ) )
=> ( fmember_a @ C2 @ A2 ) ) ).
% fminusD1
thf(fact_943_fminusD2,axiom,
! [C2: a,A2: fset_a,B: fset_a] :
( ( fmember_a @ C2 @ ( minus_minus_fset_a @ A2 @ B ) )
=> ~ ( fmember_a @ C2 @ B ) ) ).
% fminusD2
thf(fact_944_psubset__imp__ex__mem,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ? [B5: nat] : ( member_nat @ B5 @ ( minus_minus_set_nat @ B @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_945_finsert__fminus__if,axiom,
! [X: a,B: fset_a,A2: fset_a] :
( ( ( fmember_a @ X @ B )
=> ( ( minus_minus_fset_a @ ( finsert_a @ X @ A2 ) @ B )
= ( minus_minus_fset_a @ A2 @ B ) ) )
& ( ~ ( fmember_a @ X @ B )
=> ( ( minus_minus_fset_a @ ( finsert_a @ X @ A2 ) @ B )
= ( finsert_a @ X @ ( minus_minus_fset_a @ A2 @ B ) ) ) ) ) ).
% finsert_fminus_if
thf(fact_946_pfsubset__imp__ex__fmem,axiom,
! [A2: fset_a,B: fset_a] :
( ( ord_less_fset_a @ A2 @ B )
=> ? [B5: a] : ( fmember_a @ B5 @ ( minus_minus_fset_a @ B @ A2 ) ) ) ).
% pfsubset_imp_ex_fmem
thf(fact_947_card__less__sym__Diff,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_948_card__le__sym__Diff,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_949_card__Diff__subset,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_Diff_subset
thf(fact_950_diff__card__le__card__Diff,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_951_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_952_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_953_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_954_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_955_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_956_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_957_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_958_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_959_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_960_Diff__iff,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B ) )
= ( ( member_nat @ C2 @ A2 )
& ~ ( member_nat @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_961_DiffI,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ A2 )
=> ( ~ ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_962_DiffD2,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B ) )
=> ~ ( member_nat @ C2 @ B ) ) ).
% DiffD2
thf(fact_963_DiffD1,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B ) )
=> ( member_nat @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_964_DiffE,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B ) )
=> ~ ( ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ B ) ) ) ).
% DiffE
thf(fact_965_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_966_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_967_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_968_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_969_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_970_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_971_set__subtract__list__sorted,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ys )
=> ( ( set_nat2 @ ( missin6424796737333596952ed_nat @ Xs @ Ys ) )
= ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) ) ) ) ).
% set_subtract_list_sorted
thf(fact_972_fcard__fminus__finsert,axiom,
! [A: a,A2: fset_a,B: fset_a] :
( ( fmember_a @ A @ A2 )
=> ( ~ ( fmember_a @ A @ B )
=> ( ( fcard_a @ ( minus_minus_fset_a @ A2 @ ( finsert_a @ A @ B ) ) )
= ( minus_minus_nat @ ( fcard_a @ ( minus_minus_fset_a @ A2 @ B ) ) @ one_one_nat ) ) ) ) ).
% fcard_fminus_finsert
thf(fact_973_fcard__finsert__le,axiom,
! [A2: fset_a,X: a] : ( ord_less_eq_nat @ ( fcard_a @ A2 ) @ ( fcard_a @ ( finsert_a @ X @ A2 ) ) ) ).
% fcard_finsert_le
thf(fact_974_fcard__finsert__if,axiom,
! [X: a,A2: fset_a] :
( ( ( fmember_a @ X @ A2 )
=> ( ( fcard_a @ ( finsert_a @ X @ A2 ) )
= ( fcard_a @ A2 ) ) )
& ( ~ ( fmember_a @ X @ A2 )
=> ( ( fcard_a @ ( finsert_a @ X @ A2 ) )
= ( suc @ ( fcard_a @ A2 ) ) ) ) ) ).
% fcard_finsert_if
thf(fact_975_fcard__finsert__disjoint,axiom,
! [X: a,A2: fset_a] :
( ~ ( fmember_a @ X @ A2 )
=> ( ( fcard_a @ ( finsert_a @ X @ A2 ) )
= ( suc @ ( fcard_a @ A2 ) ) ) ) ).
% fcard_finsert_disjoint
thf(fact_976_last__nthI,axiom,
! [I: nat,Ts: list_a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Ts ) )
=> ( ~ ( ord_less_nat @ I @ ( minus_minus_nat @ ( size_size_list_a @ Ts ) @ ( suc @ zero_zero_nat ) ) )
=> ( ( nth_a @ Ts @ I )
= ( last_a @ Ts ) ) ) ) ).
% last_nthI
thf(fact_977_last__nthI,axiom,
! [I: nat,Ts: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Ts ) )
=> ( ~ ( ord_less_nat @ I @ ( minus_minus_nat @ ( size_size_list_nat @ Ts ) @ ( suc @ zero_zero_nat ) ) )
=> ( ( nth_nat @ Ts @ I )
= ( last_nat @ Ts ) ) ) ) ).
% last_nthI
thf(fact_978_fcard__fminus__fsingleton,axiom,
! [X: a,A2: fset_a] :
( ( fmember_a @ X @ A2 )
=> ( ( fcard_a @ ( minus_minus_fset_a @ A2 @ ( finsert_a @ X @ bot_bot_fset_a ) ) )
= ( minus_minus_nat @ ( fcard_a @ A2 ) @ one_one_nat ) ) ) ).
% fcard_fminus_fsingleton
thf(fact_979_fcard__fminus__fsingleton__if,axiom,
! [X: a,A2: fset_a] :
( ( ( fmember_a @ X @ A2 )
=> ( ( fcard_a @ ( minus_minus_fset_a @ A2 @ ( finsert_a @ X @ bot_bot_fset_a ) ) )
= ( minus_minus_nat @ ( fcard_a @ A2 ) @ one_one_nat ) ) )
& ( ~ ( fmember_a @ X @ A2 )
=> ( ( fcard_a @ ( minus_minus_fset_a @ A2 @ ( finsert_a @ X @ bot_bot_fset_a ) ) )
= ( fcard_a @ A2 ) ) ) ) ).
% fcard_fminus_fsingleton_if
thf(fact_980_fempty__iff,axiom,
! [C2: a] :
~ ( fmember_a @ C2 @ bot_bot_fset_a ) ).
% fempty_iff
thf(fact_981_all__not__fin__conv,axiom,
! [A2: fset_a] :
( ( ! [X4: a] :
~ ( fmember_a @ X4 @ A2 ) )
= ( A2 = bot_bot_fset_a ) ) ).
% all_not_fin_conv
thf(fact_982_finsert__fminus__single,axiom,
! [A: a,A2: fset_a] :
( ( finsert_a @ A @ ( minus_minus_fset_a @ A2 @ ( finsert_a @ A @ bot_bot_fset_a ) ) )
= ( finsert_a @ A @ A2 ) ) ).
% finsert_fminus_single
thf(fact_983_last__replicate,axiom,
! [N: nat,X: a] :
( ( N != zero_zero_nat )
=> ( ( last_a @ ( replicate_a @ N @ X ) )
= X ) ) ).
% last_replicate
thf(fact_984_femptyE,axiom,
! [A: a] :
~ ( fmember_a @ A @ bot_bot_fset_a ) ).
% femptyE
thf(fact_985_ex__fin__conv,axiom,
! [A2: fset_a] :
( ( ? [X4: a] : ( fmember_a @ X4 @ A2 ) )
= ( A2 != bot_bot_fset_a ) ) ).
% ex_fin_conv
thf(fact_986_equalsffemptyD,axiom,
! [A2: fset_a,A: a] :
( ( A2 = bot_bot_fset_a )
=> ~ ( fmember_a @ A @ A2 ) ) ).
% equalsffemptyD
thf(fact_987_equalsffemptyI,axiom,
! [A2: fset_a] :
( ! [Y3: a] :
~ ( fmember_a @ Y3 @ A2 )
=> ( A2 = bot_bot_fset_a ) ) ).
% equalsffemptyI
thf(fact_988_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_989_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_990_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_991_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_992_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_993_FSet_Ofset__induct,axiom,
! [P: fset_a > $o,S2: fset_a] :
( ( P @ bot_bot_fset_a )
=> ( ! [X2: a,S3: fset_a] :
( ( P @ S3 )
=> ( P @ ( finsert_a @ X2 @ S3 ) ) )
=> ( P @ S2 ) ) ) ).
% FSet.fset_induct
thf(fact_994_fset__exhaust,axiom,
! [S2: fset_a] :
( ( S2 != bot_bot_fset_a )
=> ~ ! [X2: a,S4: fset_a] :
( S2
!= ( finsert_a @ X2 @ S4 ) ) ) ).
% fset_exhaust
thf(fact_995_fdoubleton__eq__iff,axiom,
! [A: a,B2: a,C2: a,D3: a] :
( ( ( finsert_a @ A @ ( finsert_a @ B2 @ bot_bot_fset_a ) )
= ( finsert_a @ C2 @ ( finsert_a @ D3 @ bot_bot_fset_a ) ) )
= ( ( ( A = C2 )
& ( B2 = D3 ) )
| ( ( A = D3 )
& ( B2 = C2 ) ) ) ) ).
% fdoubleton_eq_iff
thf(fact_996_fsingleton__inject,axiom,
! [A: a,B2: a] :
( ( ( finsert_a @ A @ bot_bot_fset_a )
= ( finsert_a @ B2 @ bot_bot_fset_a ) )
=> ( A = B2 ) ) ).
% fsingleton_inject
thf(fact_997_finsert__not__fempty,axiom,
! [A: a,A2: fset_a] :
( ( finsert_a @ A @ A2 )
!= bot_bot_fset_a ) ).
% finsert_not_fempty
thf(fact_998_fset__induct__stronger,axiom,
! [P: fset_a > $o,S2: fset_a] :
( ( P @ bot_bot_fset_a )
=> ( ! [X2: a,S3: fset_a] :
( ~ ( fmember_a @ X2 @ S3 )
=> ( ( P @ S3 )
=> ( P @ ( finsert_a @ X2 @ S3 ) ) ) )
=> ( P @ S2 ) ) ) ).
% fset_induct_stronger
thf(fact_999_fset__strong__cases,axiom,
! [Xs: fset_a] :
( ( Xs != bot_bot_fset_a )
=> ~ ! [Ys4: fset_a,X2: a] :
( ~ ( fmember_a @ X2 @ Ys4 )
=> ( Xs
!= ( finsert_a @ X2 @ Ys4 ) ) ) ) ).
% fset_strong_cases
thf(fact_1000_fsingleton__iff,axiom,
! [B2: a,A: a] :
( ( fmember_a @ B2 @ ( finsert_a @ A @ bot_bot_fset_a ) )
= ( B2 = A ) ) ).
% fsingleton_iff
thf(fact_1001_fset__induct2,axiom,
! [P: fset_a > fset_a > $o,Xsa: fset_a,Ysa: fset_a] :
( ( P @ bot_bot_fset_a @ bot_bot_fset_a )
=> ( ! [X2: a,Xs2: fset_a] :
( ~ ( fmember_a @ X2 @ Xs2 )
=> ( P @ ( finsert_a @ X2 @ Xs2 ) @ bot_bot_fset_a ) )
=> ( ! [Y3: a,Ys4: fset_a] :
( ~ ( fmember_a @ Y3 @ Ys4 )
=> ( P @ bot_bot_fset_a @ ( finsert_a @ Y3 @ Ys4 ) ) )
=> ( ! [X2: a,Xs2: fset_a,Y3: a,Ys4: fset_a] :
( ( P @ Xs2 @ Ys4 )
=> ( ~ ( fmember_a @ X2 @ Xs2 )
=> ( ~ ( fmember_a @ Y3 @ Ys4 )
=> ( P @ ( finsert_a @ X2 @ Xs2 ) @ ( finsert_a @ Y3 @ Ys4 ) ) ) ) )
=> ( P @ Xsa @ Ysa ) ) ) ) ) ).
% fset_induct2
thf(fact_1002_fsubset__fsingletonD,axiom,
! [A2: fset_a,X: a] :
( ( ord_less_eq_fset_a @ A2 @ ( finsert_a @ X @ bot_bot_fset_a ) )
=> ( ( A2 = bot_bot_fset_a )
| ( A2
= ( finsert_a @ X @ bot_bot_fset_a ) ) ) ) ).
% fsubset_fsingletonD
thf(fact_1003_fminus__finsert2,axiom,
! [A2: fset_a,A: a,B: fset_a] :
( ( minus_minus_fset_a @ A2 @ ( finsert_a @ A @ B ) )
= ( minus_minus_fset_a @ ( minus_minus_fset_a @ A2 @ ( finsert_a @ A @ bot_bot_fset_a ) ) @ B ) ) ).
% fminus_finsert2
thf(fact_1004_fminus__finsert,axiom,
! [A2: fset_a,A: a,B: fset_a] :
( ( minus_minus_fset_a @ A2 @ ( finsert_a @ A @ B ) )
= ( minus_minus_fset_a @ ( minus_minus_fset_a @ A2 @ B ) @ ( finsert_a @ A @ bot_bot_fset_a ) ) ) ).
% fminus_finsert
thf(fact_1005_fset__linorder__min__induct,axiom,
! [P: fset_nat > $o,S2: fset_nat] :
( ( P @ bot_bot_fset_nat )
=> ( ! [X2: nat,S3: fset_nat] :
( ! [Y4: nat] :
( ( fmember_nat @ Y4 @ S3 )
=> ( ord_less_nat @ X2 @ Y4 ) )
=> ( ( P @ S3 )
=> ( P @ ( finsert_nat @ X2 @ S3 ) ) ) )
=> ( P @ S2 ) ) ) ).
% fset_linorder_min_induct
thf(fact_1006_fset__linorder__max__induct,axiom,
! [P: fset_nat > $o,S2: fset_nat] :
( ( P @ bot_bot_fset_nat )
=> ( ! [X2: nat,S3: fset_nat] :
( ! [Y4: nat] :
( ( fmember_nat @ Y4 @ S3 )
=> ( ord_less_nat @ Y4 @ X2 ) )
=> ( ( P @ S3 )
=> ( P @ ( finsert_nat @ X2 @ S3 ) ) ) )
=> ( P @ S2 ) ) ) ).
% fset_linorder_max_induct
thf(fact_1007_fminus__finsert__absorb,axiom,
! [X: a,A2: fset_a] :
( ~ ( fmember_a @ X @ A2 )
=> ( ( minus_minus_fset_a @ ( finsert_a @ X @ A2 ) @ ( finsert_a @ X @ bot_bot_fset_a ) )
= A2 ) ) ).
% fminus_finsert_absorb
thf(fact_1008_finsert__fminus,axiom,
! [A: a,A2: fset_a] :
( ( fmember_a @ A @ A2 )
=> ( ( finsert_a @ A @ ( minus_minus_fset_a @ A2 @ ( finsert_a @ A @ bot_bot_fset_a ) ) )
= A2 ) ) ).
% finsert_fminus
thf(fact_1009_fminus__single__finsert,axiom,
! [A2: fset_a,X: a,B: fset_a] :
( ( ord_less_eq_fset_a @ ( minus_minus_fset_a @ A2 @ ( finsert_a @ X @ bot_bot_fset_a ) ) @ B )
=> ( ord_less_eq_fset_a @ A2 @ ( finsert_a @ X @ B ) ) ) ).
% fminus_single_finsert
thf(fact_1010_fsubset__finsert__iff,axiom,
! [A2: fset_a,X: a,B: fset_a] :
( ( ord_less_eq_fset_a @ A2 @ ( finsert_a @ X @ B ) )
= ( ( ( fmember_a @ X @ A2 )
=> ( ord_less_eq_fset_a @ ( minus_minus_fset_a @ A2 @ ( finsert_a @ X @ bot_bot_fset_a ) ) @ B ) )
& ( ~ ( fmember_a @ X @ A2 )
=> ( ord_less_eq_fset_a @ A2 @ B ) ) ) ) ).
% fsubset_finsert_iff
thf(fact_1011_fcard__finsert,axiom,
! [X: a,A2: fset_a] :
( ( fcard_a @ ( finsert_a @ X @ A2 ) )
= ( suc @ ( fcard_a @ ( minus_minus_fset_a @ A2 @ ( finsert_a @ X @ bot_bot_fset_a ) ) ) ) ) ).
% fcard_finsert
thf(fact_1012_fcard__fminus1__le,axiom,
! [A2: fset_a,X: a] : ( ord_less_eq_nat @ ( fcard_a @ ( minus_minus_fset_a @ A2 @ ( finsert_a @ X @ bot_bot_fset_a ) ) ) @ ( fcard_a @ A2 ) ) ).
% fcard_fminus1_le
thf(fact_1013_fcard__Suc__fminus1,axiom,
! [X: a,A2: fset_a] :
( ( fmember_a @ X @ A2 )
=> ( ( suc @ ( fcard_a @ ( minus_minus_fset_a @ A2 @ ( finsert_a @ X @ bot_bot_fset_a ) ) ) )
= ( fcard_a @ A2 ) ) ) ).
% fcard_Suc_fminus1
thf(fact_1014_fcard__fminus1__less,axiom,
! [X: a,A2: fset_a] :
( ( fmember_a @ X @ A2 )
=> ( ord_less_nat @ ( fcard_a @ ( minus_minus_fset_a @ A2 @ ( finsert_a @ X @ bot_bot_fset_a ) ) ) @ ( fcard_a @ A2 ) ) ) ).
% fcard_fminus1_less
thf(fact_1015_fcard__fminus2__less,axiom,
! [X: a,A2: fset_a,Y: a] :
( ( fmember_a @ X @ A2 )
=> ( ( fmember_a @ Y @ A2 )
=> ( ord_less_nat @ ( fcard_a @ ( minus_minus_fset_a @ ( minus_minus_fset_a @ A2 @ ( finsert_a @ X @ bot_bot_fset_a ) ) @ ( finsert_a @ Y @ bot_bot_fset_a ) ) ) @ ( fcard_a @ A2 ) ) ) ) ).
% fcard_fminus2_less
thf(fact_1016_pfsubset__finsert__iff,axiom,
! [A2: fset_a,X: a,B: fset_a] :
( ( ord_less_fset_a @ A2 @ ( finsert_a @ X @ B ) )
= ( ( ( fmember_a @ X @ B )
=> ( ord_less_fset_a @ A2 @ B ) )
& ( ~ ( fmember_a @ X @ B )
=> ( ( ( fmember_a @ X @ A2 )
=> ( ord_less_fset_a @ ( minus_minus_fset_a @ A2 @ ( finsert_a @ X @ bot_bot_fset_a ) ) @ B ) )
& ( ~ ( fmember_a @ X @ A2 )
=> ( ord_less_eq_fset_a @ A2 @ B ) ) ) ) ) ) ).
% pfsubset_finsert_iff
thf(fact_1017_fthe__felem__eq,axiom,
! [X: a] :
( ( fthe_elem_a @ ( finsert_a @ X @ bot_bot_fset_a ) )
= X ) ).
% fthe_felem_eq
thf(fact_1018_fvars__term__Var,axiom,
! [X: a] :
( ( fSet_fvars_term_b_a @ ( var_a_b @ X ) )
= ( finsert_a @ X @ bot_bot_fset_a ) ) ).
% fvars_term_Var
thf(fact_1019_restrict__subst__domain__empty,axiom,
! [Sigma: a > term_b_a] :
( ( restri2809913845141176626in_a_b @ bot_bot_set_a @ Sigma )
= var_a_b ) ).
% restrict_subst_domain_empty
thf(fact_1020_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1021_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1022_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1023_ex__min__if__finite,axiom,
! [S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ S2 )
& ~ ? [Xa: nat] :
( ( member_nat @ Xa @ S2 )
& ( ord_less_nat @ Xa @ X2 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1024_infinite__growing,axiom,
! [X8: set_nat] :
( ( X8 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X8 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X8 )
& ( ord_less_nat @ X2 @ Xa ) ) )
=> ~ ( finite_finite_nat @ X8 ) ) ) ).
% infinite_growing
thf(fact_1025_card__gt__0__iff,axiom,
! [A2: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
= ( ( A2 != bot_bot_set_nat )
& ( finite_finite_nat @ A2 ) ) ) ).
% card_gt_0_iff
thf(fact_1026_set__update__distinct,axiom,
! [Xs: list_a,N: nat,X: a] :
( ( distinct_a @ Xs )
=> ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( set_a2 @ ( list_update_a @ Xs @ N @ X ) )
= ( insert_a @ X @ ( minus_minus_set_a @ ( set_a2 @ Xs ) @ ( insert_a @ ( nth_a @ Xs @ N ) @ bot_bot_set_a ) ) ) ) ) ) ).
% set_update_distinct
thf(fact_1027_set__update__distinct,axiom,
! [Xs: list_nat,N: nat,X: nat] :
( ( distinct_nat @ Xs )
=> ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( set_nat2 @ ( list_update_nat @ Xs @ N @ X ) )
= ( insert_nat @ X @ ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat @ ( nth_nat @ Xs @ N ) @ bot_bot_set_nat ) ) ) ) ) ) ).
% set_update_distinct
thf(fact_1028_ffuns__term__Var,axiom,
! [X: a] :
( ( fSet_ffuns_term_b_a @ ( var_a_b @ X ) )
= bot_bot_fset_b ) ).
% ffuns_term_Var
thf(fact_1029_card__set__1__iff__replicate,axiom,
! [Xs: list_a] :
( ( ( finite_card_a @ ( set_a2 @ Xs ) )
= ( suc @ zero_zero_nat ) )
= ( ( Xs != nil_a )
& ? [X4: a] :
( Xs
= ( replicate_a @ ( size_size_list_a @ Xs ) @ X4 ) ) ) ) ).
% card_set_1_iff_replicate
thf(fact_1030_card__set__1__iff__replicate,axiom,
! [Xs: list_nat] :
( ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
= ( suc @ zero_zero_nat ) )
= ( ( Xs != nil_nat )
& ? [X4: nat] :
( Xs
= ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X4 ) ) ) ) ).
% card_set_1_iff_replicate
thf(fact_1031_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X4: nat] :
~ ( member_nat @ X4 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_1032_empty__iff,axiom,
! [C2: nat] :
~ ( member_nat @ C2 @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_1033_insert__iff,axiom,
! [A: nat,B2: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_1034_insertCI,axiom,
! [A: nat,B: set_nat,B2: nat] :
( ( ~ ( member_nat @ A @ B )
=> ( A = B2 ) )
=> ( member_nat @ A @ ( insert_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_1035_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_1036_insert__subset,axiom,
! [X: nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A2 ) @ B )
= ( ( member_nat @ X @ B )
& ( ord_less_eq_set_nat @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_1037_Diff__insert0,axiom,
! [X: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ B ) )
= ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_1038_insert__Diff1,axiom,
! [X: nat,B: set_nat,A2: set_nat] :
( ( member_nat @ X @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B )
= ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_1039_list__update__nonempty,axiom,
! [Xs: list_a,K: nat,X: a] :
( ( ( list_update_a @ Xs @ K @ X )
= nil_a )
= ( Xs = nil_a ) ) ).
% list_update_nonempty
thf(fact_1040_length__0__conv,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_a ) ) ).
% length_0_conv
thf(fact_1041_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_1042_replicate__empty,axiom,
! [N: nat,X: a] :
( ( ( replicate_a @ N @ X )
= nil_a )
= ( N = zero_zero_nat ) ) ).
% replicate_empty
thf(fact_1043_empty__replicate,axiom,
! [N: nat,X: a] :
( ( nil_a
= ( replicate_a @ N @ X ) )
= ( N = zero_zero_nat ) ) ).
% empty_replicate
thf(fact_1044_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( linord2614967742042102400et_nat @ A2 )
= nil_nat ) ) ).
% sorted_list_of_set.fold_insort_key.infinite
thf(fact_1045_length__greater__0__conv,axiom,
! [Xs: list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) )
= ( Xs != nil_a ) ) ).
% length_greater_0_conv
thf(fact_1046_length__greater__0__conv,axiom,
! [Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) )
= ( Xs != nil_nat ) ) ).
% length_greater_0_conv
thf(fact_1047_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( linord2614967742042102400et_nat @ A2 )
= nil_nat )
= ( A2 = bot_bot_set_nat ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_1048_set__replicate,axiom,
! [N: nat,X: a] :
( ( N != zero_zero_nat )
=> ( ( set_a2 @ ( replicate_a @ N @ X ) )
= ( insert_a @ X @ bot_bot_set_a ) ) ) ).
% set_replicate
thf(fact_1049_singleton__iff,axiom,
! [B2: nat,A: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_1050_singletonD,axiom,
! [B2: nat,A: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_1051_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X4: nat] : ( member_nat @ X4 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_1052_equals0I,axiom,
! [A2: set_nat] :
( ! [Y3: nat] :
~ ( member_nat @ Y3 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_1053_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_1054_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_1055_insert__Diff,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_1056_Diff__insert__absorb,axiom,
! [X: nat,A2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_1057_sorted__wrt_Osimps_I1_J,axiom,
! [P: nat > nat > $o] : ( sorted_wrt_nat @ P @ nil_nat ) ).
% sorted_wrt.simps(1)
thf(fact_1058_list__update__code_I1_J,axiom,
! [I: nat,Y: a] :
( ( list_update_a @ nil_a @ I @ Y )
= nil_a ) ).
% list_update_code(1)
thf(fact_1059_list__update_Osimps_I1_J,axiom,
! [I: nat,V3: a] :
( ( list_update_a @ nil_a @ I @ V3 )
= nil_a ) ).
% list_update.simps(1)
thf(fact_1060_mk__disjoint__insert,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ? [B3: set_nat] :
( ( A2
= ( insert_nat @ A @ B3 ) )
& ~ ( member_nat @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_1061_insert__eq__iff,axiom,
! [A: nat,A2: set_nat,B2: nat,B: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ B2 @ B )
=> ( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C: set_nat] :
( ( A2
= ( insert_nat @ B2 @ C ) )
& ~ ( member_nat @ B2 @ C )
& ( B
= ( insert_nat @ A @ C ) )
& ~ ( member_nat @ A @ C ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1062_insert__absorb,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_1063_insert__ident,axiom,
! [X: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ~ ( member_nat @ X @ B )
=> ( ( ( insert_nat @ X @ A2 )
= ( insert_nat @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_1064_Set_Oset__insert,axiom,
! [X: nat,A2: set_nat] :
( ( member_nat @ X @ A2 )
=> ~ ! [B3: set_nat] :
( ( A2
= ( insert_nat @ X @ B3 ) )
=> ( member_nat @ X @ B3 ) ) ) ).
% Set.set_insert
thf(fact_1065_insertI2,axiom,
! [A: nat,B: set_nat,B2: nat] :
( ( member_nat @ A @ B )
=> ( member_nat @ A @ ( insert_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_1066_insertI1,axiom,
! [A: nat,B: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B ) ) ).
% insertI1
thf(fact_1067_insertE,axiom,
! [A: nat,B2: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_1068_subset__insert,axiom,
! [X: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B ) )
= ( ord_less_eq_set_nat @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_1069_insert__Diff__if,axiom,
! [X: nat,B: set_nat,A2: set_nat] :
( ( ( member_nat @ X @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B )
= ( minus_minus_set_nat @ A2 @ B ) ) )
& ( ~ ( member_nat @ X @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B )
= ( insert_nat @ X @ ( minus_minus_set_nat @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1070_finite__ranking__induct,axiom,
! [S2: set_nat,P: set_nat > $o,F2: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [Y4: nat] :
( ( member_nat @ Y4 @ S3 )
=> ( ord_less_eq_nat @ ( F2 @ Y4 ) @ ( F2 @ X2 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_nat @ X2 @ S3 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_ranking_induct
thf(fact_1071_finite__linorder__max__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B5: nat,A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A7 )
=> ( ord_less_nat @ X6 @ B5 ) )
=> ( ( P @ A7 )
=> ( P @ ( insert_nat @ B5 @ A7 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_1072_finite__linorder__min__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B5: nat,A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A7 )
=> ( ord_less_nat @ B5 @ X6 ) )
=> ( ( P @ A7 )
=> ( P @ ( insert_nat @ B5 @ A7 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_1073_finite__subset__induct_H,axiom,
! [F4: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( ord_less_eq_set_nat @ F4 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A4: nat,F5: set_nat] :
( ( finite_finite_nat @ F5 )
=> ( ( member_nat @ A4 @ A2 )
=> ( ( ord_less_eq_set_nat @ F5 @ A2 )
=> ( ~ ( member_nat @ A4 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_nat @ A4 @ F5 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1074_finite__subset__induct,axiom,
! [F4: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( ord_less_eq_set_nat @ F4 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A4: nat,F5: set_nat] :
( ( finite_finite_nat @ F5 )
=> ( ( member_nat @ A4 @ A2 )
=> ( ~ ( member_nat @ A4 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_nat @ A4 @ F5 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1075_subset__Diff__insert,axiom,
! [A2: set_nat,B: set_nat,X: nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B @ ( insert_nat @ X @ C3 ) ) )
= ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B @ C3 ) )
& ~ ( member_nat @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1076_subset__insert__iff,axiom,
! [A2: set_nat,X: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B ) )
= ( ( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B ) )
& ( ~ ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_1077_card__insert__le,axiom,
! [A2: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ ( insert_nat @ X @ A2 ) ) ) ).
% card_insert_le
thf(fact_1078_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_1079_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_1080_sorted0,axiom,
sorted_wrt_nat @ ord_less_eq_nat @ nil_nat ).
% sorted0
thf(fact_1081_strict__sorted__simps_I1_J,axiom,
sorted_wrt_nat @ ord_less_nat @ nil_nat ).
% strict_sorted_simps(1)
thf(fact_1082_replicate__0,axiom,
! [X: a] :
( ( replicate_a @ zero_zero_nat @ X )
= nil_a ) ).
% replicate_0
thf(fact_1083_last__in__set,axiom,
! [As: list_nat] :
( ( As != nil_nat )
=> ( member_nat @ ( last_nat @ As ) @ ( set_nat2 @ As ) ) ) ).
% last_in_set
thf(fact_1084_finite__remove__induct,axiom,
! [B: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ( A7 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A7 @ B )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A7 )
=> ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_1085_remove__induct,axiom,
! [P: set_nat > $o,B: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B )
=> ( P @ B ) )
=> ( ! [A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ( A7 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A7 @ B )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A7 )
=> ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_1086_card__Diff1__le,axiom,
! [A2: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ).
% card_Diff1_le
thf(fact_1087_finite__induct__select,axiom,
! [S2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ S2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [T4: set_nat] :
( ( ord_less_set_nat @ T4 @ S2 )
=> ( ( P @ T4 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ ( minus_minus_set_nat @ S2 @ T4 ) )
& ( P @ ( insert_nat @ X6 @ T4 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ).
% finite_induct_select
thf(fact_1088_psubset__insert__iff,axiom,
! [A2: set_nat,X: nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ ( insert_nat @ X @ B ) )
= ( ( ( member_nat @ X @ B )
=> ( ord_less_set_nat @ A2 @ B ) )
& ( ~ ( member_nat @ X @ B )
=> ( ( ( member_nat @ X @ A2 )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B ) )
& ( ~ ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1089_set__replicate__conv__if,axiom,
! [N: nat,X: a] :
( ( ( N = zero_zero_nat )
=> ( ( set_a2 @ ( replicate_a @ N @ X ) )
= bot_bot_set_a ) )
& ( ( N != zero_zero_nat )
=> ( ( set_a2 @ ( replicate_a @ N @ X ) )
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% set_replicate_conv_if
thf(fact_1090_set__replicate__Suc,axiom,
! [N: nat,X: a] :
( ( set_a2 @ ( replicate_a @ ( suc @ N ) @ X ) )
= ( insert_a @ X @ bot_bot_set_a ) ) ).
% set_replicate_Suc
thf(fact_1091_set__update__subset__insert,axiom,
! [Xs: list_a,I: nat,X: a] : ( ord_less_eq_set_a @ ( set_a2 @ ( list_update_a @ Xs @ I @ X ) ) @ ( insert_a @ X @ ( set_a2 @ Xs ) ) ) ).
% set_update_subset_insert
thf(fact_1092_card__le__Suc__iff,axiom,
! [N: nat,A2: set_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_nat @ A2 ) )
= ( ? [A5: nat,B4: set_nat] :
( ( A2
= ( insert_nat @ A5 @ B4 ) )
& ~ ( member_nat @ A5 @ B4 )
& ( ord_less_eq_nat @ N @ ( finite_card_nat @ B4 ) )
& ( finite_finite_nat @ B4 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_1093_card__Diff1__less__iff,axiom,
! [A2: set_nat,X: nat] :
( ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) )
= ( ( finite_finite_nat @ A2 )
& ( member_nat @ X @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_1094_card__Diff2__less,axiom,
! [A2: set_nat,X: nat,Y: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( insert_nat @ Y @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_1095_card__Diff1__less,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_1096_distinct__list__update,axiom,
! [Xs: list_nat,A: nat,I: nat] :
( ( distinct_nat @ Xs )
=> ( ~ ( member_nat @ A @ ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat @ ( nth_nat @ Xs @ I ) @ bot_bot_set_nat ) ) )
=> ( distinct_nat @ ( list_update_nat @ Xs @ I @ A ) ) ) ) ).
% distinct_list_update
thf(fact_1097_distinct__list__update,axiom,
! [Xs: list_a,A: a,I: nat] :
( ( distinct_a @ Xs )
=> ( ~ ( member_a @ A @ ( minus_minus_set_a @ ( set_a2 @ Xs ) @ ( insert_a @ ( nth_a @ Xs @ I ) @ bot_bot_set_a ) ) )
=> ( distinct_a @ ( list_update_a @ Xs @ I @ A ) ) ) ) ).
% distinct_list_update
thf(fact_1098_remdups__adj__length__ge1,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_size_list_a @ ( remdups_adj_a @ Xs ) ) ) ) ).
% remdups_adj_length_ge1
thf(fact_1099_remdups__adj__length__ge1,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_size_list_nat @ ( remdups_adj_nat @ Xs ) ) ) ) ).
% remdups_adj_length_ge1
thf(fact_1100_last__conv__nth,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( last_a @ Xs )
= ( nth_a @ Xs @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_1101_last__conv__nth,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( last_nat @ Xs )
= ( nth_nat @ Xs @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_1102_last__list__update,axiom,
! [Xs: list_a,K: nat,X: a] :
( ( Xs != nil_a )
=> ( ( ( K
= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) )
=> ( ( last_a @ ( list_update_a @ Xs @ K @ X ) )
= X ) )
& ( ( K
!= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) )
=> ( ( last_a @ ( list_update_a @ Xs @ K @ X ) )
= ( last_a @ Xs ) ) ) ) ) ).
% last_list_update
thf(fact_1103_last__list__update,axiom,
! [Xs: list_nat,K: nat,X: nat] :
( ( Xs != nil_nat )
=> ( ( ( K
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
=> ( ( last_nat @ ( list_update_nat @ Xs @ K @ X ) )
= X ) )
& ( ( K
!= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
=> ( ( last_nat @ ( list_update_nat @ Xs @ K @ X ) )
= ( last_nat @ Xs ) ) ) ) ) ).
% last_list_update
thf(fact_1104_card__insert__le__m1,axiom,
! [N: nat,Y: set_nat,X: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( insert_nat @ X @ Y ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_1105_in__set__remove1,axiom,
! [A: nat,B2: nat,Xs: list_nat] :
( ( A != B2 )
=> ( ( member_nat @ A @ ( set_nat2 @ ( remove1_nat @ B2 @ Xs ) ) )
= ( member_nat @ A @ ( set_nat2 @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_1106_notin__set__remove1,axiom,
! [X: nat,Xs: list_nat,Y: nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ~ ( member_nat @ X @ ( set_nat2 @ ( remove1_nat @ Y @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_1107_remove1__idem,axiom,
! [X: nat,Xs: list_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( remove1_nat @ X @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_1108_sorted__remove1,axiom,
! [Xs: list_nat,A: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( remove1_nat @ A @ Xs ) ) ) ).
% sorted_remove1
thf(fact_1109_is__singletonI_H,axiom,
! [A2: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X2: nat,Y3: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y3 @ A2 )
=> ( X2 = Y3 ) ) )
=> ( is_singleton_nat @ A2 ) ) ) ).
% is_singletonI'
thf(fact_1110_length__remove1,axiom,
! [X: a,Xs: list_a] :
( ( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ( size_size_list_a @ ( remove1_a @ X @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) ) )
& ( ~ ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ( size_size_list_a @ ( remove1_a @ X @ Xs ) )
= ( size_size_list_a @ Xs ) ) ) ) ).
% length_remove1
thf(fact_1111_length__remove1,axiom,
! [X: nat,Xs: list_nat] :
( ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( size_size_list_nat @ ( remove1_nat @ X @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) )
& ( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( size_size_list_nat @ ( remove1_nat @ X @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ) ) ).
% length_remove1
thf(fact_1112_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( linord2614967742042102400et_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( remove1_nat @ X @ ( linord2614967742042102400et_nat @ A2 ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_1113_arg__min__if__finite_I2_J,axiom,
! [S2: set_nat,F2: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ~ ? [X6: nat] :
( ( member_nat @ X6 @ S2 )
& ( ord_less_nat @ ( F2 @ X6 ) @ ( F2 @ ( lattic7446932960582359483at_nat @ F2 @ S2 ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_1114_arg__min__least,axiom,
! [S2: set_nat,Y: nat,F2: nat > nat] :
( ( finite_finite_nat @ S2 )
=> ( ( S2 != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S2 )
=> ( ord_less_eq_nat @ ( F2 @ ( lattic7446932960582359483at_nat @ F2 @ S2 ) ) @ ( F2 @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_1115_insert__subsetI,axiom,
! [X: nat,A2: set_nat,X8: set_nat] :
( ( member_nat @ X @ A2 )
=> ( ( ord_less_eq_set_nat @ X8 @ A2 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ X @ X8 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_1116_subset__emptyI,axiom,
! [A2: set_nat] :
( ! [X2: nat] :
~ ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_1117_nth__Cons__pos,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_1118_nth__Cons__0,axiom,
! [X: nat,Xs: list_nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_1119_nth__Cons__Suc,axiom,
! [X: nat,Xs: list_nat,N: nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ ( suc @ N ) )
= ( nth_nat @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_1120_fset__of__list__simps_I2_J,axiom,
! [X21: a,X222: list_a] :
( ( fset_of_list_a @ ( cons_a @ X21 @ X222 ) )
= ( finsert_a @ X21 @ ( fset_of_list_a @ X222 ) ) ) ).
% fset_of_list_simps(2)
thf(fact_1121_Cons__less__Cons,axiom,
! [A: nat,X: list_nat,B2: nat,Y: list_nat] :
( ( ord_less_list_nat @ ( cons_nat @ A @ X ) @ ( cons_nat @ B2 @ Y ) )
= ( ( ord_less_nat @ A @ B2 )
| ( ( A = B2 )
& ( ord_less_list_nat @ X @ Y ) ) ) ) ).
% Cons_less_Cons
thf(fact_1122_Cons__le__Cons,axiom,
! [A: nat,X: list_nat,B2: nat,Y: list_nat] :
( ( ord_less_eq_list_nat @ ( cons_nat @ A @ X ) @ ( cons_nat @ B2 @ Y ) )
= ( ( ord_less_nat @ A @ B2 )
| ( ( A = B2 )
& ( ord_less_eq_list_nat @ X @ Y ) ) ) ) ).
% Cons_le_Cons
thf(fact_1123_list__induct2,axiom,
! [Xs: list_a,Ys: list_a,P: list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y3: a,Ys4: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y3 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1124_list__induct2,axiom,
! [Xs: list_a,Ys: list_nat,P: list_a > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_a @ nil_nat )
=> ( ! [X2: a,Xs2: list_a,Y3: nat,Ys4: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1125_list__induct2,axiom,
! [Xs: list_nat,Ys: list_a,P: list_nat > list_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_nat @ nil_a )
=> ( ! [X2: nat,Xs2: list_nat,Y3: a,Ys4: list_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_a @ Y3 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1126_list__induct2,axiom,
! [Xs: list_nat,Ys: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat,Y3: nat,Ys4: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1127_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,P: list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_a @ nil_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y3: a,Ys4: list_a,Z2: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y3 @ Ys4 ) @ ( cons_a @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_1128_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_nat,P: list_a > list_a > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_a @ nil_a @ nil_nat )
=> ( ! [X2: a,Xs2: list_a,Y3: a,Ys4: list_a,Z2: nat,Zs2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y3 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_1129_list__induct3,axiom,
! [Xs: list_a,Ys: list_nat,Zs: list_a,P: list_a > list_nat > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_a @ nil_nat @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y3: nat,Ys4: list_nat,Z2: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_a @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_1130_list__induct3,axiom,
! [Xs: list_a,Ys: list_nat,Zs: list_nat,P: list_a > list_nat > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_a @ nil_nat @ nil_nat )
=> ( ! [X2: a,Xs2: list_a,Y3: nat,Ys4: list_nat,Z2: nat,Zs2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_1131_list__induct3,axiom,
! [Xs: list_nat,Ys: list_a,Zs: list_a,P: list_nat > list_a > list_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_nat @ nil_a @ nil_a )
=> ( ! [X2: nat,Xs2: list_nat,Y3: a,Ys4: list_a,Z2: a,Zs2: list_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_a @ Y3 @ Ys4 ) @ ( cons_a @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_1132_list__induct3,axiom,
! [Xs: list_nat,Ys: list_a,Zs: list_nat,P: list_nat > list_a > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_a @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat,Y3: a,Ys4: list_a,Z2: nat,Zs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_a @ Y3 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_1133_list__induct3,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_a,P: list_nat > list_nat > list_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_a )
=> ( ! [X2: nat,Xs2: list_nat,Y3: nat,Ys4: list_nat,Z2: a,Zs2: list_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_a @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_1134_list__induct3,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat,P: 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 ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat,Y3: nat,Ys4: list_nat,Z2: nat,Zs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_1135_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ws: list_a,P: list_a > list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y3: a,Ys4: list_a,Z2: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y3 @ Ys4 ) @ ( cons_a @ Z2 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1136_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ws: list_nat,P: list_a > list_a > list_a > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_a @ nil_a @ nil_a @ nil_nat )
=> ( ! [X2: a,Xs2: list_a,Y3: a,Ys4: list_a,Z2: a,Zs2: list_a,W: nat,Ws2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y3 @ Ys4 ) @ ( cons_a @ Z2 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1137_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_nat,Ws: list_a,P: list_a > list_a > list_nat > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_a @ nil_nat @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y3: a,Ys4: list_a,Z2: nat,Zs2: list_nat,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y3 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1138_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_nat,Ws: list_nat,P: list_a > list_a > list_nat > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_a @ nil_a @ nil_nat @ nil_nat )
=> ( ! [X2: a,Xs2: list_a,Y3: a,Ys4: list_a,Z2: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y3 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1139_list__induct4,axiom,
! [Xs: list_a,Ys: list_nat,Zs: list_a,Ws: list_a,P: list_a > list_nat > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_nat @ nil_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y3: nat,Ys4: list_nat,Z2: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_a @ Z2 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1140_list__induct4,axiom,
! [Xs: list_a,Ys: list_nat,Zs: list_a,Ws: list_nat,P: list_a > list_nat > list_a > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_a @ nil_nat @ nil_a @ nil_nat )
=> ( ! [X2: a,Xs2: list_a,Y3: nat,Ys4: list_nat,Z2: a,Zs2: list_a,W: nat,Ws2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_a @ Z2 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1141_list__induct4,axiom,
! [Xs: list_a,Ys: list_nat,Zs: list_nat,Ws: list_a,P: list_a > list_nat > list_nat > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_nat @ nil_nat @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y3: nat,Ys4: list_nat,Z2: nat,Zs2: list_nat,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1142_list__induct4,axiom,
! [Xs: list_a,Ys: list_nat,Zs: list_nat,Ws: list_nat,P: list_a > list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_a @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X2: a,Xs2: list_a,Y3: nat,Ys4: list_nat,Z2: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1143_list__induct4,axiom,
! [Xs: list_nat,Ys: list_a,Zs: list_a,Ws: list_a,P: list_nat > list_a > list_a > list_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_nat @ nil_a @ nil_a @ nil_a )
=> ( ! [X2: nat,Xs2: list_nat,Y3: a,Ys4: list_a,Z2: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_a @ Y3 @ Ys4 ) @ ( cons_a @ Z2 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1144_list__induct4,axiom,
! [Xs: list_nat,Ys: list_a,Zs: list_a,Ws: list_nat,P: list_nat > list_a > list_a > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_a @ nil_a @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat,Y3: a,Ys4: list_a,Z2: a,Zs2: list_a,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_a @ Y3 @ Ys4 ) @ ( cons_a @ Z2 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1145_Suc__length__conv,axiom,
! [N: nat,Xs: list_a] :
( ( ( suc @ N )
= ( size_size_list_a @ Xs ) )
= ( ? [Y5: a,Ys3: list_a] :
( ( Xs
= ( cons_a @ Y5 @ Ys3 ) )
& ( ( size_size_list_a @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_1146_Suc__length__conv,axiom,
! [N: nat,Xs: list_nat] :
( ( ( suc @ N )
= ( size_size_list_nat @ Xs ) )
= ( ? [Y5: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y5 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_1147_length__Suc__conv,axiom,
! [Xs: list_a,N: nat] :
( ( ( size_size_list_a @ Xs )
= ( suc @ N ) )
= ( ? [Y5: a,Ys3: list_a] :
( ( Xs
= ( cons_a @ Y5 @ Ys3 ) )
& ( ( size_size_list_a @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_1148_length__Suc__conv,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y5: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y5 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_1149_impossible__Cons,axiom,
! [Xs: list_a,Ys: list_a,X: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) )
=> ( Xs
!= ( cons_a @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_1150_impossible__Cons,axiom,
! [Xs: list_nat,Ys: list_nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) )
=> ( Xs
!= ( cons_nat @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_1151_sorted2,axiom,
! [X: nat,Y: nat,Zs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X @ ( cons_nat @ Y @ Zs ) ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ Y @ Zs ) ) ) ) ).
% sorted2
thf(fact_1152_sorted__wrt1,axiom,
! [P: nat > nat > $o,X: nat] : ( sorted_wrt_nat @ P @ ( cons_nat @ X @ nil_nat ) ) ).
% sorted_wrt1
thf(fact_1153_distinct_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( distinct_nat @ ( cons_nat @ X @ Xs ) )
= ( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
& ( distinct_nat @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_1154_replicate__Suc,axiom,
! [N: nat,X: a] :
( ( replicate_a @ ( suc @ N ) @ X )
= ( cons_a @ X @ ( replicate_a @ N @ X ) ) ) ).
% replicate_Suc
thf(fact_1155_list__update__code_I2_J,axiom,
! [X: a,Xs: list_a,Y: a] :
( ( list_update_a @ ( cons_a @ X @ Xs ) @ zero_zero_nat @ Y )
= ( cons_a @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_1156_list__update__code_I3_J,axiom,
! [X: a,Xs: list_a,I: nat,Y: a] :
( ( list_update_a @ ( cons_a @ X @ Xs ) @ ( suc @ I ) @ Y )
= ( cons_a @ X @ ( list_update_a @ Xs @ I @ Y ) ) ) ).
% list_update_code(3)
thf(fact_1157_list_Oset__intros_I2_J,axiom,
! [Y: nat,X222: list_nat,X21: nat] :
( ( member_nat @ Y @ ( set_nat2 @ X222 ) )
=> ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_1158_list_Oset__intros_I1_J,axiom,
! [X21: nat,X222: list_nat] : ( member_nat @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_1159_list_Oset__cases,axiom,
! [E: nat,A: list_nat] :
( ( member_nat @ E @ ( set_nat2 @ A ) )
=> ( ! [Z22: list_nat] :
( A
!= ( cons_nat @ E @ Z22 ) )
=> ~ ! [Z1: nat,Z22: list_nat] :
( ( A
= ( cons_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat @ E @ ( set_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_1160_set__ConsD,axiom,
! [Y: nat,X: nat,Xs: list_nat] :
( ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_nat @ Y @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_1161_less__eq__list__code_I3_J,axiom,
! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ord_less_eq_list_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( ( ord_less_nat @ X @ Y )
| ( ( X = Y )
& ( ord_less_eq_list_nat @ Xs @ Ys ) ) ) ) ).
% less_eq_list_code(3)
thf(fact_1162_subtract__list__sorted_Osimps_I1_J,axiom,
! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
( ( ( X = Y )
=> ( ( missin6424796737333596952ed_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( missin6424796737333596952ed_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) ) )
& ( ( X != Y )
=> ( ( ( ord_less_nat @ X @ Y )
=> ( ( missin6424796737333596952ed_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( cons_nat @ X @ ( missin6424796737333596952ed_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) ) ) )
& ( ~ ( ord_less_nat @ X @ Y )
=> ( ( missin6424796737333596952ed_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( missin6424796737333596952ed_nat @ ( cons_nat @ X @ Xs ) @ Ys ) ) ) ) ) ) ).
% subtract_list_sorted.simps(1)
thf(fact_1163_less__list__code_I3_J,axiom,
! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ord_less_list_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( ( ord_less_nat @ X @ Y )
| ( ( X = Y )
& ( ord_less_list_nat @ Xs @ Ys ) ) ) ) ).
% less_list_code(3)
thf(fact_1164_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_a @ Xs ) )
= ( ? [X4: a,Ys3: list_a] :
( ( Xs
= ( cons_a @ X4 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_a @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_1165_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
= ( ? [X4: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ X4 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_1166_sorted1,axiom,
! [X: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X @ nil_nat ) ) ).
% sorted1
thf(fact_1167_sorted__simps_I2_J,axiom,
! [X: nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X @ Ys ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Ys ) )
=> ( ord_less_eq_nat @ X @ X4 ) )
& ( sorted_wrt_nat @ ord_less_eq_nat @ Ys ) ) ) ).
% sorted_simps(2)
thf(fact_1168_strict__sorted__simps_I2_J,axiom,
! [X: nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ ( cons_nat @ X @ Ys ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Ys ) )
=> ( ord_less_nat @ X @ X4 ) )
& ( sorted_wrt_nat @ ord_less_nat @ Ys ) ) ) ).
% strict_sorted_simps(2)
thf(fact_1169_min__list__Cons,axiom,
! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
( ( ord_less_eq_nat @ X @ 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 @ X @ Xs ) ) @ ( missing_min_list_nat @ ( cons_nat @ Y @ Ys ) ) ) ) ) ) ).
% min_list_Cons
thf(fact_1170_vars__term__list_Osimps_I1_J,axiom,
! [X: a] :
( ( vars_term_list_b_a @ ( var_a_b @ X ) )
= ( cons_a @ X @ nil_a ) ) ).
% vars_term_list.simps(1)
thf(fact_1171_subtract__list__sorted_Oelims,axiom,
! [X: list_nat,Xa2: list_nat,Y: list_nat] :
( ( ( missin6424796737333596952ed_nat @ X @ Xa2 )
= Y )
=> ( ! [X2: nat,Xs2: list_nat] :
( ( X
= ( cons_nat @ X2 @ Xs2 ) )
=> ! [Y3: nat,Ys4: list_nat] :
( ( Xa2
= ( cons_nat @ Y3 @ Ys4 ) )
=> ~ ( ( ( X2 = Y3 )
=> ( Y
= ( missin6424796737333596952ed_nat @ Xs2 @ ( cons_nat @ Y3 @ Ys4 ) ) ) )
& ( ( X2 != Y3 )
=> ( ( ( ord_less_nat @ X2 @ Y3 )
=> ( Y
= ( cons_nat @ X2 @ ( missin6424796737333596952ed_nat @ Xs2 @ ( cons_nat @ Y3 @ Ys4 ) ) ) ) )
& ( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( Y
= ( missin6424796737333596952ed_nat @ ( cons_nat @ X2 @ Xs2 ) @ Ys4 ) ) ) ) ) ) ) )
=> ( ( ( X = nil_nat )
=> ( Y != nil_nat ) )
=> ~ ! [V4: nat,Va: list_nat] :
( ( X
= ( cons_nat @ V4 @ Va ) )
=> ( ( Xa2 = nil_nat )
=> ( Y
!= ( cons_nat @ V4 @ Va ) ) ) ) ) ) ) ).
% subtract_list_sorted.elims
thf(fact_1172_nth__Cons_H,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_1173_remdups__adj__singleton,axiom,
! [Xs: list_a,X: a] :
( ( ( remdups_adj_a @ Xs )
= ( cons_a @ X @ nil_a ) )
=> ( Xs
= ( replicate_a @ ( size_size_list_a @ Xs ) @ X ) ) ) ).
% remdups_adj_singleton
thf(fact_1174_remdups__adj__singleton,axiom,
! [Xs: list_nat,X: nat] :
( ( ( remdups_adj_nat @ Xs )
= ( cons_nat @ X @ nil_nat ) )
=> ( Xs
= ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X ) ) ) ).
% remdups_adj_singleton
thf(fact_1175_remdups__adj__replicate,axiom,
! [N: nat,X: a] :
( ( ( N = zero_zero_nat )
=> ( ( remdups_adj_a @ ( replicate_a @ N @ X ) )
= nil_a ) )
& ( ( N != zero_zero_nat )
=> ( ( remdups_adj_a @ ( replicate_a @ N @ X ) )
= ( cons_a @ X @ nil_a ) ) ) ) ).
% remdups_adj_replicate
thf(fact_1176_nth__equal__first__eq,axiom,
! [X: a,Xs: list_a,N: nat] :
( ~ ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( ( nth_a @ ( cons_a @ X @ Xs ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1177_nth__equal__first__eq,axiom,
! [X: nat,Xs: list_nat,N: nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_1178_nth__non__equal__first__eq,axiom,
! [X: nat,Y: nat,Xs: list_nat,N: nat] :
( ( X != Y )
=> ( ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= Y )
= ( ( ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_1179_Cons__replicate__eq,axiom,
! [X: a,Xs: list_a,N: nat,Y: a] :
( ( ( cons_a @ X @ Xs )
= ( replicate_a @ N @ Y ) )
= ( ( X = Y )
& ( ord_less_nat @ zero_zero_nat @ N )
& ( Xs
= ( replicate_a @ ( minus_minus_nat @ N @ one_one_nat ) @ X ) ) ) ) ).
% Cons_replicate_eq
thf(fact_1180_length__Cons,axiom,
! [X: a,Xs: list_a] :
( ( size_size_list_a @ ( cons_a @ X @ Xs ) )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ).
% length_Cons
thf(fact_1181_length__Cons,axiom,
! [X: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X @ Xs ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_Cons
thf(fact_1182_sorted__list__of__set__nonempty,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( linord2614967742042102400et_nat @ A2 )
= ( cons_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ ( linord2614967742042102400et_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% sorted_list_of_set_nonempty
thf(fact_1183_Min_Obounded__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_nat @ X @ X4 ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_1184_Min__gr__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ord_less_nat @ X @ X4 ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_1185_Min_OcoboundedI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ A ) ) ) ).
% Min.coboundedI
thf(fact_1186_Min__eqI,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ A2 )
=> ( ord_less_eq_nat @ X @ Y3 ) )
=> ( ( member_nat @ X @ A2 )
=> ( ( lattic8721135487736765967in_nat @ A2 )
= X ) ) ) ) ).
% Min_eqI
thf(fact_1187_Min__le,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ X ) ) ) ).
% Min_le
thf(fact_1188_Min_OboundedI,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A2 )
=> ( ord_less_eq_nat @ X @ A4 ) )
=> ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) ) ) ) ) ).
% Min.boundedI
thf(fact_1189_Min_OboundedE,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) )
=> ! [A8: nat] :
( ( member_nat @ A8 @ A2 )
=> ( ord_less_eq_nat @ X @ A8 ) ) ) ) ) ).
% Min.boundedE
thf(fact_1190_eq__Min__iff,axiom,
! [A2: set_nat,M2: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( M2
= ( lattic8721135487736765967in_nat @ A2 ) )
= ( ( member_nat @ M2 @ A2 )
& ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_nat @ M2 @ X4 ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_1191_Min__le__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ X )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_eq_nat @ X4 @ X ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_1192_Min__eq__iff,axiom,
! [A2: set_nat,M2: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ( lattic8721135487736765967in_nat @ A2 )
= M2 )
= ( ( member_nat @ M2 @ A2 )
& ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_nat @ M2 @ X4 ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_1193_Min__less__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ X )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A2 )
& ( ord_less_nat @ X4 @ X ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_1194_Min__insert2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ! [B5: nat] :
( ( member_nat @ B5 @ A2 )
=> ( ord_less_eq_nat @ A @ B5 ) )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat @ A @ A2 ) )
= A ) ) ) ).
% Min_insert2
thf(fact_1195_Min_Osubset__imp,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ B ) @ ( lattic8721135487736765967in_nat @ A2 ) ) ) ) ) ).
% Min.subset_imp
thf(fact_1196_Min__antimono,axiom,
! [M7: set_nat,N6: set_nat] :
( ( ord_less_eq_set_nat @ M7 @ N6 )
=> ( ( M7 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ N6 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ N6 ) @ ( lattic8721135487736765967in_nat @ M7 ) ) ) ) ) ).
% Min_antimono
thf(fact_1197_length__n__lists__elem,axiom,
! [Ys: list_a,N: nat,Xs: list_a] :
( ( member_list_a @ Ys @ ( set_list_a2 @ ( n_lists_a @ N @ Xs ) ) )
=> ( ( size_size_list_a @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_1198_length__n__lists__elem,axiom,
! [Ys: list_nat,N: nat,Xs: list_nat] :
( ( member_list_nat @ Ys @ ( set_list_nat2 @ ( n_lists_nat @ N @ Xs ) ) )
=> ( ( size_size_list_nat @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_1199_length__add__elem__list__lists,axiom,
! [Ys: list_a,X: a,Xs: list_a] :
( ( member_list_a @ Ys @ ( set_list_a2 @ ( basic_189371981877014951ists_a @ X @ Xs ) ) )
=> ( ( size_size_list_a @ Ys )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ) ).
% length_add_elem_list_lists
thf(fact_1200_length__add__elem__list__lists,axiom,
! [Ys: list_nat,X: nat,Xs: list_nat] :
( ( member_list_nat @ Ys @ ( set_list_nat2 @ ( basic_4874698711677410535ts_nat @ X @ Xs ) ) )
=> ( ( size_size_list_nat @ Ys )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ) ).
% length_add_elem_list_lists
thf(fact_1201_in__set__product__lists__length,axiom,
! [Xs: list_a,Xss: list_list_a] :
( ( member_list_a @ Xs @ ( set_list_a2 @ ( product_lists_a @ Xss ) ) )
=> ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_1202_in__set__product__lists__length,axiom,
! [Xs: list_nat,Xss: list_list_nat] :
( ( member_list_nat @ Xs @ ( set_list_nat2 @ ( product_lists_nat @ Xss ) ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_1203_remdups__adj__singleton__iff,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ ( remdups_adj_a @ Xs ) )
= ( suc @ zero_zero_nat ) )
= ( ( Xs != nil_a )
& ( Xs
= ( replicate_a @ ( size_size_list_a @ Xs ) @ ( hd_a @ Xs ) ) ) ) ) ).
% remdups_adj_singleton_iff
thf(fact_1204_remdups__adj__singleton__iff,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ ( remdups_adj_nat @ Xs ) )
= ( suc @ zero_zero_nat ) )
= ( ( Xs != nil_nat )
& ( Xs
= ( replicate_nat @ ( size_size_list_nat @ Xs ) @ ( hd_nat @ Xs ) ) ) ) ) ).
% remdups_adj_singleton_iff
thf(fact_1205_hd__replicate,axiom,
! [N: nat,X: a] :
( ( N != zero_zero_nat )
=> ( ( hd_a @ ( replicate_a @ N @ X ) )
= X ) ) ).
% hd_replicate
thf(fact_1206_hd__conv__nth,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( hd_nat @ Xs )
= ( nth_nat @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_1207_list_Oset__sel_I1_J,axiom,
! [A: list_nat] :
( ( A != nil_nat )
=> ( member_nat @ ( hd_nat @ A ) @ ( set_nat2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_1208_hd__in__set,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( member_nat @ ( hd_nat @ Xs ) @ ( set_nat2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_1209_less__eq__prod__simp,axiom,
! [X1: nat,Y1: nat,X22: nat,Y22: nat] :
( ( ord_le8460144461188290721at_nat @ ( product_Pair_nat_nat @ X1 @ Y1 ) @ ( product_Pair_nat_nat @ X22 @ Y22 ) )
= ( ( ord_less_nat @ X1 @ X22 )
| ( ( ord_less_eq_nat @ X1 @ X22 )
& ( ord_less_eq_nat @ Y1 @ Y22 ) ) ) ) ).
% less_eq_prod_simp
thf(fact_1210_less__prod__simp,axiom,
! [X1: nat,Y1: nat,X22: nat,Y22: nat] :
( ( ord_le1203424502768444845at_nat @ ( product_Pair_nat_nat @ X1 @ Y1 ) @ ( product_Pair_nat_nat @ X22 @ Y22 ) )
= ( ( ord_less_nat @ X1 @ X22 )
| ( ( ord_less_eq_nat @ X1 @ X22 )
& ( ord_less_nat @ Y1 @ Y22 ) ) ) ) ).
% less_prod_simp
thf(fact_1211_concat__index__split_Osimps,axiom,
! [I_idx: nat,X: list_a,O_idx: nat,Xs: list_list_a] :
( ( ( ord_less_nat @ I_idx @ ( size_size_list_a @ X ) )
=> ( ( concat_index_split_a @ ( product_Pair_nat_nat @ O_idx @ I_idx ) @ ( cons_list_a @ X @ Xs ) )
= ( product_Pair_nat_nat @ O_idx @ I_idx ) ) )
& ( ~ ( ord_less_nat @ I_idx @ ( size_size_list_a @ X ) )
=> ( ( concat_index_split_a @ ( product_Pair_nat_nat @ O_idx @ I_idx ) @ ( cons_list_a @ X @ Xs ) )
= ( concat_index_split_a @ ( product_Pair_nat_nat @ ( suc @ O_idx ) @ ( minus_minus_nat @ I_idx @ ( size_size_list_a @ X ) ) ) @ Xs ) ) ) ) ).
% concat_index_split.simps
thf(fact_1212_concat__index__split_Osimps,axiom,
! [I_idx: nat,X: list_nat,O_idx: nat,Xs: list_list_nat] :
( ( ( ord_less_nat @ I_idx @ ( size_size_list_nat @ X ) )
=> ( ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ O_idx @ I_idx ) @ ( cons_list_nat @ X @ Xs ) )
= ( product_Pair_nat_nat @ O_idx @ I_idx ) ) )
& ( ~ ( ord_less_nat @ I_idx @ ( size_size_list_nat @ X ) )
=> ( ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ O_idx @ I_idx ) @ ( cons_list_nat @ X @ Xs ) )
= ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ ( suc @ O_idx ) @ ( minus_minus_nat @ I_idx @ ( size_size_list_nat @ X ) ) ) @ Xs ) ) ) ) ).
% concat_index_split.simps
thf(fact_1213_prod__decode__aux_Oelims,axiom,
! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
( ( ( nat_prod_decode_aux @ X @ Xa2 )
= Y )
=> ( ( ( ord_less_eq_nat @ Xa2 @ X )
=> ( Y
= ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ Xa2 @ X )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X ) ) ) ) ) ) ) ).
% prod_decode_aux.elims
thf(fact_1214_prod__decode__aux_Osimps,axiom,
( nat_prod_decode_aux
= ( ^ [K5: nat,M3: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M3 @ K5 ) @ ( product_Pair_nat_nat @ M3 @ ( minus_minus_nat @ K5 @ M3 ) ) @ ( nat_prod_decode_aux @ ( suc @ K5 ) @ ( minus_minus_nat @ M3 @ ( suc @ K5 ) ) ) ) ) ) ).
% prod_decode_aux.simps
thf(fact_1215_trancl__list_Olist__trancl,axiom,
! [Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a,I: nat,Z4: a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( trancl_list_a @ R2 ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_a @ Ys ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Ys @ I ) @ Z4 ) @ R2 )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ ( list_update_a @ Ys @ I @ Z4 ) ) @ ( trancl_list_a @ R2 ) ) ) ) ) ).
% trancl_list.list_trancl
thf(fact_1216_trancl__list_Olist__trancl,axiom,
! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat,I: nat,Z4: nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( trancl_list_nat @ R2 ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Ys @ I ) @ Z4 ) @ R2 )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( list_update_nat @ Ys @ I @ Z4 ) ) @ ( trancl_list_nat @ R2 ) ) ) ) ) ).
% trancl_list.list_trancl
thf(fact_1217_trancl__list__appendI,axiom,
! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat,X: nat,Y: nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( trancl_list_nat @ R2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( trancl_list_nat @ R2 ) ) ) ) ).
% trancl_list_appendI
thf(fact_1218_trancl__list_Obase,axiom,
! [Xs: list_a,Ys: list_a,R2: set_Product_prod_a_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Ys ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ I3 ) @ ( nth_a @ Ys @ I3 ) ) @ R2 ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( trancl_list_a @ R2 ) ) ) ) ).
% trancl_list.base
thf(fact_1219_trancl__list_Obase,axiom,
! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Ys @ I3 ) ) @ R2 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( trancl_list_nat @ R2 ) ) ) ) ).
% trancl_list.base
thf(fact_1220_trancl__list_Ocases,axiom,
! [A1: list_a,A22: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A1 @ A22 ) @ ( trancl_list_a @ R2 ) )
=> ( ( ( ( size_size_list_a @ A1 )
= ( size_size_list_a @ A22 ) )
=> ~ ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_a @ A22 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ A1 @ I5 ) @ ( nth_a @ A22 @ I5 ) ) @ R2 ) ) )
=> ~ ! [Ys4: list_a,I3: nat,Z2: a] :
( ( A22
= ( list_update_a @ Ys4 @ I3 @ Z2 ) )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A1 @ Ys4 ) @ ( trancl_list_a @ R2 ) )
=> ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Ys4 ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Ys4 @ I3 ) @ Z2 ) @ R2 ) ) ) ) ) ) ).
% trancl_list.cases
thf(fact_1221_trancl__list_Ocases,axiom,
! [A1: list_nat,A22: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A22 ) @ ( trancl_list_nat @ R2 ) )
=> ( ( ( ( size_size_list_nat @ A1 )
= ( size_size_list_nat @ A22 ) )
=> ~ ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_nat @ A22 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ A1 @ I5 ) @ ( nth_nat @ A22 @ I5 ) ) @ R2 ) ) )
=> ~ ! [Ys4: list_nat,I3: nat,Z2: nat] :
( ( A22
= ( list_update_nat @ Ys4 @ I3 @ Z2 ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ Ys4 ) @ ( trancl_list_nat @ R2 ) )
=> ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys4 ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Ys4 @ I3 ) @ Z2 ) @ R2 ) ) ) ) ) ) ).
% trancl_list.cases
thf(fact_1222_trancl__list_Osimps,axiom,
! [A1: list_a,A22: list_a,R2: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A1 @ A22 ) @ ( trancl_list_a @ R2 ) )
= ( ? [Xs3: list_a,Ys3: list_a] :
( ( A1 = Xs3 )
& ( A22 = Ys3 )
& ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Ys3 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs3 @ I4 ) @ ( nth_a @ Ys3 @ I4 ) ) @ R2 ) ) )
| ? [Xs3: list_a,Ys3: list_a,I4: nat,Z3: a] :
( ( A1 = Xs3 )
& ( A22
= ( list_update_a @ Ys3 @ I4 @ Z3 ) )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs3 @ Ys3 ) @ ( trancl_list_a @ R2 ) )
& ( ord_less_nat @ I4 @ ( size_size_list_a @ Ys3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Ys3 @ I4 ) @ Z3 ) @ R2 ) ) ) ) ).
% trancl_list.simps
thf(fact_1223_trancl__list_Osimps,axiom,
! [A1: list_nat,A22: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A22 ) @ ( trancl_list_nat @ R2 ) )
= ( ? [Xs3: list_nat,Ys3: list_nat] :
( ( A1 = Xs3 )
& ( A22 = Ys3 )
& ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys3 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs3 @ I4 ) @ ( nth_nat @ Ys3 @ I4 ) ) @ R2 ) ) )
| ? [Xs3: list_nat,Ys3: list_nat,I4: nat,Z3: nat] :
( ( A1 = Xs3 )
& ( A22
= ( list_update_nat @ Ys3 @ I4 @ Z3 ) )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys3 ) @ ( trancl_list_nat @ R2 ) )
& ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys3 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Ys3 @ I4 ) @ Z3 ) @ R2 ) ) ) ) ).
% trancl_list.simps
thf(fact_1224_concat__index__split_Oelims,axiom,
! [X: product_prod_nat_nat,Xa2: list_list_a,Y: product_prod_nat_nat] :
( ( ( concat_index_split_a @ X @ Xa2 )
= Y )
=> ( ! [O_idx2: nat,I_idx2: nat] :
( ( X
= ( product_Pair_nat_nat @ O_idx2 @ I_idx2 ) )
=> ! [X2: list_a,Xs2: list_list_a] :
( ( Xa2
= ( cons_list_a @ X2 @ Xs2 ) )
=> ~ ( ( ( ord_less_nat @ I_idx2 @ ( size_size_list_a @ X2 ) )
=> ( Y
= ( product_Pair_nat_nat @ O_idx2 @ I_idx2 ) ) )
& ( ~ ( ord_less_nat @ I_idx2 @ ( size_size_list_a @ X2 ) )
=> ( Y
= ( concat_index_split_a @ ( product_Pair_nat_nat @ ( suc @ O_idx2 ) @ ( minus_minus_nat @ I_idx2 @ ( size_size_list_a @ X2 ) ) ) @ Xs2 ) ) ) ) ) )
=> ~ ( ( Xa2 = nil_list_a )
=> ( Y != undefi3946296454836805481at_nat ) ) ) ) ).
% concat_index_split.elims
thf(fact_1225_concat__index__split_Oelims,axiom,
! [X: product_prod_nat_nat,Xa2: list_list_nat,Y: product_prod_nat_nat] :
( ( ( concat5568775588498683858it_nat @ X @ Xa2 )
= Y )
=> ( ! [O_idx2: nat,I_idx2: nat] :
( ( X
= ( product_Pair_nat_nat @ O_idx2 @ I_idx2 ) )
=> ! [X2: list_nat,Xs2: list_list_nat] :
( ( Xa2
= ( cons_list_nat @ X2 @ Xs2 ) )
=> ~ ( ( ( ord_less_nat @ I_idx2 @ ( size_size_list_nat @ X2 ) )
=> ( Y
= ( product_Pair_nat_nat @ O_idx2 @ I_idx2 ) ) )
& ( ~ ( ord_less_nat @ I_idx2 @ ( size_size_list_nat @ X2 ) )
=> ( Y
= ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ ( suc @ O_idx2 ) @ ( minus_minus_nat @ I_idx2 @ ( size_size_list_nat @ X2 ) ) ) @ Xs2 ) ) ) ) ) )
=> ~ ( ( Xa2 = nil_list_nat )
=> ( Y != undefi3946296454836805481at_nat ) ) ) ) ).
% concat_index_split.elims
thf(fact_1226_Cons__lenlex__iff,axiom,
! [M2: a,Ms: list_a,N: a,Ns: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ M2 @ Ms ) @ ( cons_a @ N @ Ns ) ) @ ( lenlex_a @ R ) )
= ( ( ord_less_nat @ ( size_size_list_a @ Ms ) @ ( size_size_list_a @ Ns ) )
| ( ( ( size_size_list_a @ Ms )
= ( size_size_list_a @ Ns ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ M2 @ N ) @ R ) )
| ( ( M2 = N )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ms @ Ns ) @ ( lenlex_a @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_1227_Cons__lenlex__iff,axiom,
! [M2: nat,Ms: list_nat,N: nat,Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ M2 @ Ms ) @ ( cons_nat @ N @ Ns ) ) @ ( lenlex_nat @ R ) )
= ( ( ord_less_nat @ ( size_size_list_nat @ Ms ) @ ( size_size_list_nat @ Ns ) )
| ( ( ( size_size_list_nat @ Ms )
= ( size_size_list_nat @ Ns ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M2 @ N ) @ R ) )
| ( ( M2 = N )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_1228_lenlex__irreflexive,axiom,
! [R: set_Pr1261947904930325089at_nat,Xs: list_nat] :
( ! [X2: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ X2 ) @ R )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Xs ) @ ( lenlex_nat @ R ) ) ) ).
% lenlex_irreflexive
thf(fact_1229_lenlex__length,axiom,
! [Ms: list_a,Ns: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ms @ Ns ) @ ( lenlex_a @ R ) )
=> ( ord_less_eq_nat @ ( size_size_list_a @ Ms ) @ ( size_size_list_a @ Ns ) ) ) ).
% lenlex_length
thf(fact_1230_lenlex__length,axiom,
! [Ms: list_nat,Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R ) )
=> ( ord_less_eq_nat @ ( size_size_list_nat @ Ms ) @ ( size_size_list_nat @ Ns ) ) ) ).
% lenlex_length
thf(fact_1231_concat__index__split_Opelims,axiom,
! [X: product_prod_nat_nat,Xa2: list_list_a,Y: product_prod_nat_nat] :
( ( ( concat_index_split_a @ X @ Xa2 )
= Y )
=> ( ( accp_P2033140809068429945list_a @ concat3165989775726078917_rel_a @ ( produc2552019737546482272list_a @ X @ Xa2 ) )
=> ( ! [O_idx2: nat,I_idx2: nat] :
( ( X
= ( product_Pair_nat_nat @ O_idx2 @ I_idx2 ) )
=> ! [X2: list_a,Xs2: list_list_a] :
( ( Xa2
= ( cons_list_a @ X2 @ Xs2 ) )
=> ( ( ( ( ord_less_nat @ I_idx2 @ ( size_size_list_a @ X2 ) )
=> ( Y
= ( product_Pair_nat_nat @ O_idx2 @ I_idx2 ) ) )
& ( ~ ( ord_less_nat @ I_idx2 @ ( size_size_list_a @ X2 ) )
=> ( Y
= ( concat_index_split_a @ ( product_Pair_nat_nat @ ( suc @ O_idx2 ) @ ( minus_minus_nat @ I_idx2 @ ( size_size_list_a @ X2 ) ) ) @ Xs2 ) ) ) )
=> ~ ( accp_P2033140809068429945list_a @ concat3165989775726078917_rel_a @ ( produc2552019737546482272list_a @ ( product_Pair_nat_nat @ O_idx2 @ I_idx2 ) @ ( cons_list_a @ X2 @ Xs2 ) ) ) ) ) )
=> ~ ( ( Xa2 = nil_list_a )
=> ( ( Y = undefi3946296454836805481at_nat )
=> ~ ( accp_P2033140809068429945list_a @ concat3165989775726078917_rel_a @ ( produc2552019737546482272list_a @ X @ nil_list_a ) ) ) ) ) ) ) ).
% concat_index_split.pelims
thf(fact_1232_concat__index__split_Opelims,axiom,
! [X: product_prod_nat_nat,Xa2: list_list_nat,Y: product_prod_nat_nat] :
( ( ( concat5568775588498683858it_nat @ X @ Xa2 )
= Y )
=> ( ( accp_P6748712283191960023st_nat @ concat1575402509940520777el_nat @ ( produc8649917717998259994st_nat @ X @ Xa2 ) )
=> ( ! [O_idx2: nat,I_idx2: nat] :
( ( X
= ( product_Pair_nat_nat @ O_idx2 @ I_idx2 ) )
=> ! [X2: list_nat,Xs2: list_list_nat] :
( ( Xa2
= ( cons_list_nat @ X2 @ Xs2 ) )
=> ( ( ( ( ord_less_nat @ I_idx2 @ ( size_size_list_nat @ X2 ) )
=> ( Y
= ( product_Pair_nat_nat @ O_idx2 @ I_idx2 ) ) )
& ( ~ ( ord_less_nat @ I_idx2 @ ( size_size_list_nat @ X2 ) )
=> ( Y
= ( concat5568775588498683858it_nat @ ( product_Pair_nat_nat @ ( suc @ O_idx2 ) @ ( minus_minus_nat @ I_idx2 @ ( size_size_list_nat @ X2 ) ) ) @ Xs2 ) ) ) )
=> ~ ( accp_P6748712283191960023st_nat @ concat1575402509940520777el_nat @ ( produc8649917717998259994st_nat @ ( product_Pair_nat_nat @ O_idx2 @ I_idx2 ) @ ( cons_list_nat @ X2 @ Xs2 ) ) ) ) ) )
=> ~ ( ( Xa2 = nil_list_nat )
=> ( ( Y = undefi3946296454836805481at_nat )
=> ~ ( accp_P6748712283191960023st_nat @ concat1575402509940520777el_nat @ ( produc8649917717998259994st_nat @ X @ nil_list_nat ) ) ) ) ) ) ) ).
% concat_index_split.pelims
thf(fact_1233_Cons__in__lex,axiom,
! [X: a,Xs: list_a,Y: a,Ys: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) ) @ ( lex_a @ R ) )
= ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
& ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) )
| ( ( X = Y )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( lex_a @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_1234_Cons__in__lex,axiom,
! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( lex_nat @ R ) )
= ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
& ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) )
| ( ( X = Y )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_1235_accp__subset,axiom,
! [R1: product_prod_nat_nat > product_prod_nat_nat > $o,R22: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( ord_le5604493270027003598_nat_o @ R1 @ R22 )
=> ( ord_le704812498762024988_nat_o @ ( accp_P4275260045618599050at_nat @ R22 ) @ ( accp_P4275260045618599050at_nat @ R1 ) ) ) ).
% accp_subset
thf(fact_1236_accp__subset__induct,axiom,
! [D: product_prod_nat_nat > $o,R4: product_prod_nat_nat > product_prod_nat_nat > $o,X: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
( ( ord_le704812498762024988_nat_o @ D @ ( accp_P4275260045618599050at_nat @ R4 ) )
=> ( ! [X2: product_prod_nat_nat,Z2: product_prod_nat_nat] :
( ( D @ X2 )
=> ( ( R4 @ Z2 @ X2 )
=> ( D @ Z2 ) ) )
=> ( ( D @ X )
=> ( ! [X2: product_prod_nat_nat] :
( ( D @ X2 )
=> ( ! [Z5: product_prod_nat_nat] :
( ( R4 @ Z5 @ X2 )
=> ( P @ Z5 ) )
=> ( P @ X2 ) ) )
=> ( P @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_1237_prod__decode__aux_Opelims,axiom,
! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
( ( ( nat_prod_decode_aux @ X @ Xa2 )
= Y )
=> ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
=> ~ ( ( ( ( ord_less_eq_nat @ Xa2 @ X )
=> ( Y
= ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ Xa2 @ X )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X ) ) ) ) ) )
=> ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).
% prod_decode_aux.pelims
thf(fact_1238_subtract__list__sorted_Opelims,axiom,
! [X: list_nat,Xa2: list_nat,Y: list_nat] :
( ( ( missin6424796737333596952ed_nat @ X @ Xa2 )
= Y )
=> ( ( accp_P8037286306265792042st_nat @ missin3644257100165608323el_nat @ ( produc2694037385005941721st_nat @ X @ Xa2 ) )
=> ( ! [X2: nat,Xs2: list_nat] :
( ( X
= ( cons_nat @ X2 @ Xs2 ) )
=> ! [Y3: nat,Ys4: list_nat] :
( ( Xa2
= ( cons_nat @ Y3 @ Ys4 ) )
=> ( ( ( ( X2 = Y3 )
=> ( Y
= ( missin6424796737333596952ed_nat @ Xs2 @ ( cons_nat @ Y3 @ Ys4 ) ) ) )
& ( ( X2 != Y3 )
=> ( ( ( ord_less_nat @ X2 @ Y3 )
=> ( Y
= ( cons_nat @ X2 @ ( missin6424796737333596952ed_nat @ Xs2 @ ( cons_nat @ Y3 @ Ys4 ) ) ) ) )
& ( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( Y
= ( missin6424796737333596952ed_nat @ ( cons_nat @ X2 @ Xs2 ) @ Ys4 ) ) ) ) ) )
=> ~ ( accp_P8037286306265792042st_nat @ missin3644257100165608323el_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys4 ) ) ) ) ) )
=> ( ( ( X = nil_nat )
=> ( ( Y = nil_nat )
=> ~ ( accp_P8037286306265792042st_nat @ missin3644257100165608323el_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xa2 ) ) ) )
=> ~ ! [V4: nat,Va: list_nat] :
( ( X
= ( cons_nat @ V4 @ Va ) )
=> ( ( Xa2 = nil_nat )
=> ( ( Y
= ( cons_nat @ V4 @ Va ) )
=> ~ ( accp_P8037286306265792042st_nat @ missin3644257100165608323el_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ V4 @ Va ) @ nil_nat ) ) ) ) ) ) ) ) ) ).
% subtract_list_sorted.pelims
thf(fact_1239_in__measure,axiom,
! [X: nat,Y: nat,F2: nat > nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measure_nat @ F2 ) )
= ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) ) ) ).
% in_measure
thf(fact_1240_in__finite__psubset,axiom,
! [A2: set_nat,B: set_nat] :
( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A2 @ B ) @ finite_psubset_nat )
= ( ( ord_less_set_nat @ A2 @ B )
& ( finite_finite_nat @ B ) ) ) ).
% in_finite_psubset
thf(fact_1241_listrel1__iff__update,axiom,
! [Xs: list_a,Ys: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R ) )
= ( ? [Y5: a,N3: nat] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ N3 ) @ Y5 ) @ R )
& ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs ) )
& ( Ys
= ( list_update_a @ Xs @ N3 @ Y5 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_1242_listrel1__iff__update,axiom,
! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
= ( ? [Y5: nat,N3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N3 ) @ Y5 ) @ R )
& ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs ) )
& ( Ys
= ( list_update_nat @ Xs @ N3 @ Y5 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_1243_Abstract__Rewriting_Ochain__mono,axiom,
! [R5: set_Pr1261947904930325089at_nat,R4: set_Pr1261947904930325089at_nat,Seq: nat > nat] :
( ( ord_le3146513528884898305at_nat @ R5 @ R4 )
=> ( ! [I3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( Seq @ I3 ) @ ( Seq @ ( suc @ I3 ) ) ) @ R5 )
=> ! [I5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( Seq @ I5 ) @ ( Seq @ ( suc @ I5 ) ) ) @ R4 ) ) ) ).
% Abstract_Rewriting.chain_mono
thf(fact_1244_Cons__listrel1__Cons,axiom,
! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( listrel1_nat @ R ) )
= ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
& ( Xs = Ys ) )
| ( ( X = Y )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_1245_Cons__listrel1E2,axiom,
! [Xs: list_nat,Y: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) @ ( listrel1_nat @ R ) )
=> ( ! [X2: nat] :
( ( Xs
= ( cons_nat @ X2 @ Ys ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ R ) )
=> ~ ! [Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Y @ Zs2 ) )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Zs2 @ Ys ) @ ( listrel1_nat @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_1246_Cons__listrel1E1,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ Ys ) @ ( listrel1_nat @ R ) )
=> ( ! [Y3: nat] :
( ( Ys
= ( cons_nat @ Y3 @ Xs ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R ) )
=> ~ ! [Zs2: list_nat] :
( ( Ys
= ( cons_nat @ X @ Zs2 ) )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Zs2 ) @ ( listrel1_nat @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_1247_listrel1I1,axiom,
! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat,Xs: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Xs ) ) @ ( listrel1_nat @ R ) ) ) ).
% listrel1I1
thf(fact_1248_listrel1__eq__len,axiom,
! [Xs: list_a,Ys: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% listrel1_eq_len
thf(fact_1249_listrel1__eq__len,axiom,
! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% listrel1_eq_len
thf(fact_1250_subrelI,axiom,
! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ! [X2: nat,Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ S ) )
=> ( ord_le3146513528884898305at_nat @ R @ S ) ) ).
% subrelI
thf(fact_1251_rtrancl__list__conv,axiom,
! [S: a,T: a,R4: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ S @ T ) @ ( transitive_rtrancl_a @ R4 ) )
= ( ? [Ts2: list_a] :
( ( ( last_a @ ( cons_a @ S @ Ts2 ) )
= T )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Ts2 ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ ( cons_a @ S @ Ts2 ) @ I4 ) @ ( nth_a @ ( cons_a @ S @ Ts2 ) @ ( suc @ I4 ) ) ) @ R4 ) ) ) ) ) ).
% rtrancl_list_conv
thf(fact_1252_rtrancl__list__conv,axiom,
! [S: nat,T: nat,R4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ S @ T ) @ ( transi2905341329935302413cl_nat @ R4 ) )
= ( ? [Ts2: list_nat] :
( ( ( last_nat @ ( cons_nat @ S @ Ts2 ) )
= T )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ts2 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ ( cons_nat @ S @ Ts2 ) @ I4 ) @ ( nth_nat @ ( cons_nat @ S @ Ts2 ) @ ( suc @ I4 ) ) ) @ R4 ) ) ) ) ) ).
% rtrancl_list_conv
thf(fact_1253_rtrancl__listrel1__ConsI2,axiom,
! [X: nat,Y: nat,R: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( transi2905341329935302413cl_nat @ R ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( transi5285580207609517981st_nat @ ( listrel1_nat @ R ) ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( transi5285580207609517981st_nat @ ( listrel1_nat @ R ) ) ) ) ) ).
% rtrancl_listrel1_ConsI2
thf(fact_1254_rtrancl__listrel1__eq__len,axiom,
! [X: list_a,Y: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( transi7631188966963710983list_a @ ( listrel1_a @ R ) ) )
=> ( ( size_size_list_a @ X )
= ( size_size_list_a @ Y ) ) ) ).
% rtrancl_listrel1_eq_len
thf(fact_1255_rtrancl__listrel1__eq__len,axiom,
! [X: list_nat,Y: list_nat,R: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( transi5285580207609517981st_nat @ ( listrel1_nat @ R ) ) )
=> ( ( size_size_list_nat @ X )
= ( size_size_list_nat @ Y ) ) ) ).
% rtrancl_listrel1_eq_len
thf(fact_1256_chain__imp__rtrancl,axiom,
! [S2: nat > nat,R: set_Pr1261947904930325089at_nat,I: nat,J: nat] :
( ! [I3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( S2 @ I3 ) @ ( S2 @ ( suc @ I3 ) ) ) @ R )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( S2 @ I ) @ ( S2 @ J ) ) @ ( transi2905341329935302413cl_nat @ R ) ) ) ) ).
% chain_imp_rtrancl
thf(fact_1257_rtrancl__fun__conv,axiom,
! [S: nat,T: nat,R4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ S @ T ) @ ( transi2905341329935302413cl_nat @ R4 ) )
= ( ? [F3: nat > nat,N3: nat] :
( ( ( F3 @ zero_zero_nat )
= S )
& ( ( F3 @ N3 )
= T )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N3 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F3 @ I4 ) @ ( F3 @ ( suc @ I4 ) ) ) @ R4 ) ) ) ) ) ).
% rtrancl_fun_conv
% Helper facts (13)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__FSet__Ofset_Itf__a_J_T,axiom,
! [X: fset_a,Y: fset_a] :
( ( if_fset_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__FSet__Ofset_Itf__a_J_T,axiom,
! [X: fset_a,Y: fset_a] :
( ( if_fset_a @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__FSet__Ofset_It__Nat__Onat_J_T,axiom,
! [X: fset_nat,Y: fset_nat] :
( ( if_fset_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__FSet__Ofset_It__Nat__Onat_J_T,axiom,
! [X: fset_nat,Y: fset_nat] :
( ( if_fset_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Term__Oterm_Itf__b_Mtf__a_J_T,axiom,
! [X: term_b_a,Y: term_b_a] :
( ( if_term_b_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Term__Oterm_Itf__b_Mtf__a_J_T,axiom,
! [X: term_b_a,Y: term_b_a] :
( ( if_term_b_a @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
? [I5: nat,Q3: a] :
( ( ord_less_nat @ I5 @ na )
& ( fmember_a @ Q3 @ ( finsert_a @ q_f @ q ) )
& ( ( vars_term_list_b_a @ ( var_a_b @ x ) )
= ( list_update_a @ ( replicate_a @ na @ q_c ) @ I5 @ Q3 ) ) ) ).
%------------------------------------------------------------------------------