TPTP Problem File: SLH0607^1.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 : Rewrite_Properties_Reduction/0020_Rewriting_Properties/prob_00142_005715__14099654_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1602 ( 448 unt; 340 typ; 0 def)
% Number of atoms : 3795 (1325 equ; 0 cnn)
% Maximal formula atoms : 23 ( 3 avg)
% Number of connectives : 14959 ( 565 ~; 71 |; 287 &;12130 @)
% ( 0 <=>;1906 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 8 avg)
% Number of types : 67 ( 66 usr)
% Number of type conns : 834 ( 834 >; 0 *; 0 +; 0 <<)
% Number of symbols : 277 ( 274 usr; 27 con; 0-3 aty)
% Number of variables : 4414 ( 168 ^;4027 !; 219 ?;4414 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:19:10.393
%------------------------------------------------------------------------------
% Could-be-implicit typings (66)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
set_Pr553994874890374343at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
produc349518998152878311at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J_J,type,
set_Pr2972776593051762503rm_a_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
produc4523971788519308903rm_a_b: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
set_Pr7047708467701517959at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
set_Pr1542805901266377927at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
produc492020038885415591at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
produc6392793444374437607at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_Itf__b_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
produc5279506192219892694rm_a_b: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
list_P8469869581646625389at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_Pr8693737435421807431at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
produc2732850333517536310rm_a_b: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_J,type,
set_Pr1811044260758604347_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_J_J,type,
set_Pr8564414093027780873rm_a_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc859450856879609959at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
set_Pr5771542735269860976st_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
produc4708774622424448987_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
produc51424535725745577rm_a_b: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_Pr7682762132356531903at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
produc4155638208356863248st_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
list_P8875379029341186191rm_a_b: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
set_Pr4386577575007340137rm_a_b: $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__List__Olist_Itf__a_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
set_Pr5046312416420021961st_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc1932156733058919263at_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
list_P5366131564312172150at_nat: $tType ).
thf(ty_n_t__List__Olist_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
list_P9162950289778280392at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_Pr6370437063884598352at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
set_Pr9093778441882193744at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
produc357393685978478089rm_a_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
produc3697673438841856213st_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
produc7711739908350443733rm_a_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
produc1828647624359046049st_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
list_P2364656488115551307rm_a_b: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_Pr6841066879934740386at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mtf__a_J_J,type,
set_Pr3254233319718257578_nat_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
produc432399132543013523st_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__b_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
set_Pr1357592988741419896rm_a_b: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc85711943791777264at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
produc8199716216217303280at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
produc1234881154892807749rm_a_b: $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__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc551447040318622060at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mtf__a_J,type,
produc2842622564922169716_nat_a: $tType ).
thf(ty_n_t__List__Olist_I_062_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Nat__Onat_J_J,type,
list_term_a_b_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__b_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
produc6551080308567045442rm_a_b: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
list_P3592885314253461005_a_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
list_list_term_a_b: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
set_Pr4934435412358123699_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J_J,type,
set_Pr4193341848836149977_nat_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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
list_nat_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
product_prod_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
product_prod_nat_a: $tType ).
thf(ty_n_t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
list_term_a_b: $tType ).
thf(ty_n_t__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__Term__Oterm_Itf__a_Mtf__b_J,type,
term_a_b: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (274)
thf(sy_c_Abstract__Rewriting_Omeet_001t__Nat__Onat,type,
abstract_meet_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Abstract__Rewriting_Omeet_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
abstra338393719139700030at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat ).
thf(sy_c_Abstract__Rewriting_Omeet_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
abstra1093061187967292567rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Abstract__Rewriting_Oquasi__commute_001t__Nat__Onat,type,
abstra4838182543893801299te_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Abstract__Rewriting_Oquasi__commute_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
abstra4317384144805385751rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_Abstract__Rewriting_Orestrict_001t__Nat__Onat,type,
abstra1219450721576609224ct_nat: set_Pr1261947904930325089at_nat > set_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Fun__Def_Opair__leq,type,
fun_pair_leq: set_Pr8693737435421807431at_nat ).
thf(sy_c_Fun__Def_Opair__less,type,
fun_pair_less: set_Pr8693737435421807431at_nat ).
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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
minus_1356011639430497352at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
minus_8321449233255521966at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
minus_5192120951422937424rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
minus_8630901905721465484_a_nat: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
sup_su6327502436637775413at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
sup_su718114333110466843at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
sup_su6776935440552674877rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
sup_su459911885395995103_a_nat: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat ).
thf(sy_c_List_Oappend_001t__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
append985823374593552924at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
append1152621114427393060rm_a_b: list_P8875379029341186191rm_a_b > list_P8875379029341186191rm_a_b > list_P8875379029341186191rm_a_b ).
thf(sy_c_List_Oappend_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
append_term_a_b: list_term_a_b > list_term_a_b > list_term_a_b ).
thf(sy_c_List_Obutlast_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
butlast_term_a_b: list_term_a_b > list_term_a_b ).
thf(sy_c_List_Oconcat_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
concat_term_a_b: list_list_term_a_b > list_term_a_b ).
thf(sy_c_List_Odistinct_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
distinct_term_a_b: list_term_a_b > $o ).
thf(sy_c_List_Odrop_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
drop_list_term_a_b: nat > list_list_term_a_b > list_list_term_a_b ).
thf(sy_c_List_Odrop_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
drop_term_a_b: nat > list_term_a_b > list_term_a_b ).
thf(sy_c_List_Oenumerate_001t__Nat__Onat,type,
enumerate_nat: nat > list_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Oenumerate_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
enumerate_term_a_b: nat > list_term_a_b > list_P2364656488115551307rm_a_b ).
thf(sy_c_List_Ogen__length_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
gen_length_term_a_b: nat > list_term_a_b > nat ).
thf(sy_c_List_Olast_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
last_nat_nat: list_nat_nat > nat > nat ).
thf(sy_c_List_Olast_001t__Nat__Onat,type,
last_nat: list_nat > nat ).
thf(sy_c_List_Olast_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
last_P1651921108130080911at_nat: list_P5366131564312172150at_nat > produc8199716216217303280at_nat ).
thf(sy_c_List_Olast_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
last_P6484183829340986144at_nat: list_P6011104703257516679at_nat > product_prod_nat_nat ).
thf(sy_c_List_Olast_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
last_P5835599730086418438at_nat: list_P8469869581646625389at_nat > produc859450856879609959at_nat ).
thf(sy_c_List_Olast_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
last_P6746431593072707880rm_a_b: list_P8875379029341186191rm_a_b > produc357393685978478089rm_a_b ).
thf(sy_c_List_Olast_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
last_P2271748490522340894_a_nat: list_P3592885314253461005_a_nat > product_prod_a_nat ).
thf(sy_c_List_Olast_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
last_term_a_b: list_term_a_b > term_a_b ).
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_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
lenlex325483962726685836at_nat: set_Pr8693737435421807431at_nat > set_Pr1542805901266377927at_nat ).
thf(sy_c_List_Olenlex_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
lenlex_term_a_b: set_Pr4386577575007340137rm_a_b > set_Pr8564414093027780873rm_a_b ).
thf(sy_c_List_Olex_001t__Nat__Onat,type,
lex_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_List_Olex_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
lex_Pr8571645452597969515at_nat: set_Pr8693737435421807431at_nat > set_Pr1542805901266377927at_nat ).
thf(sy_c_List_Olex_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
lex_term_a_b: set_Pr4386577575007340137rm_a_b > set_Pr8564414093027780873rm_a_b ).
thf(sy_c_List_Olist_OCons_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
cons_nat_nat: ( nat > nat ) > list_nat_nat > list_nat_nat ).
thf(sy_c_List_Olist_OCons_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
cons_P4861729644591583992at_nat: ( product_prod_nat_nat > nat ) > list_P9162950289778280392at_nat > list_P9162950289778280392at_nat ).
thf(sy_c_List_Olist_OCons_001_062_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Nat__Onat_J,type,
cons_term_a_b_nat: ( term_a_b > nat ) > list_term_a_b_nat > list_term_a_b_nat ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
cons_P4219629788700907686at_nat: produc8199716216217303280at_nat > list_P5366131564312172150at_nat > list_P5366131564312172150at_nat ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
cons_P6512896166579812791at_nat: product_prod_nat_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
cons_P8732206157123786781at_nat: produc859450856879609959at_nat > list_P8469869581646625389at_nat > list_P8469869581646625389at_nat ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
cons_P3536305108106557631rm_a_b: produc357393685978478089rm_a_b > list_P8875379029341186191rm_a_b > list_P8875379029341186191rm_a_b ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
cons_P5205166803686508359_a_nat: product_prod_a_nat > list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat ).
thf(sy_c_List_Olist_OCons_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
cons_term_a_b: term_a_b > list_term_a_b > list_term_a_b ).
thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_ONil_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
nil_nat_nat: list_nat_nat ).
thf(sy_c_List_Olist_ONil_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
nil_Pr2865493887535707976at_nat: list_P9162950289778280392at_nat ).
thf(sy_c_List_Olist_ONil_001_062_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Nat__Onat_J,type,
nil_term_a_b_nat: list_term_a_b_nat ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
nil_Pr5478986624290739719at_nat: list_P6011104703257516679at_nat ).
thf(sy_c_List_Olist_ONil_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
nil_term_a_b: list_term_a_b ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist_Ohd_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
hd_nat_nat: list_nat_nat > nat > nat ).
thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
hd_nat: list_nat > nat ).
thf(sy_c_List_Olist_Ohd_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
hd_Pro5468458281938048987at_nat: list_P5366131564312172150at_nat > produc8199716216217303280at_nat ).
thf(sy_c_List_Olist_Ohd_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
hd_Pro3460610213475200108at_nat: list_P6011104703257516679at_nat > product_prod_nat_nat ).
thf(sy_c_List_Olist_Ohd_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
hd_Pro8462011474880202578at_nat: list_P8469869581646625389at_nat > produc859450856879609959at_nat ).
thf(sy_c_List_Olist_Ohd_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
hd_Pro6001814652670619252rm_a_b: list_P8875379029341186191rm_a_b > produc357393685978478089rm_a_b ).
thf(sy_c_List_Olist_Ohd_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
hd_Pro8935205257713178578_a_nat: list_P3592885314253461005_a_nat > product_prod_a_nat ).
thf(sy_c_List_Olist_Ohd_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
hd_term_a_b: list_term_a_b > term_a_b ).
thf(sy_c_List_Olist_Ohd_001tf__a,type,
hd_a: list_a > a ).
thf(sy_c_List_Olist__ex_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
list_ex_term_a_b: ( term_a_b > $o ) > list_term_a_b > $o ).
thf(sy_c_List_Olist__update_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
list_update_nat_nat: list_nat_nat > nat > ( nat > nat ) > list_nat_nat ).
thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
list_update_nat: list_nat > nat > nat > list_nat ).
thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
list_u5493879213508619102at_nat: list_P5366131564312172150at_nat > nat > produc8199716216217303280at_nat > list_P5366131564312172150at_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_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
list_u5003261594476800725at_nat: list_P8469869581646625389at_nat > nat > produc859450856879609959at_nat > list_P8469869581646625389at_nat ).
thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
list_u4211539095652206455rm_a_b: list_P8875379029341186191rm_a_b > nat > produc357393685978478089rm_a_b > list_P8875379029341186191rm_a_b ).
thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
list_u4318310736832020111_a_nat: list_P3592885314253461005_a_nat > nat > product_prod_a_nat > list_P3592885314253461005_a_nat ).
thf(sy_c_List_Olist__update_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
list_update_term_a_b: list_term_a_b > nat > term_a_b > list_term_a_b ).
thf(sy_c_List_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_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
listre4828114922151135584at_nat: set_Pr8693737435421807431at_nat > set_Pr1542805901266377927at_nat ).
thf(sy_c_List_Olistrel1_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
listrel1_term_a_b: set_Pr4386577575007340137rm_a_b > set_Pr8564414093027780873rm_a_b ).
thf(sy_c_List_Olistrel_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
listrel_nat_nat_nat: set_Pr9093778441882193744at_nat > set_Pr5771542735269860976st_nat ).
thf(sy_c_List_Olistrel_001t__Nat__Onat_001t__Nat__Onat,type,
listrel_nat_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_List_Olistrel_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
listre818007680106770737at_nat: set_Pr8693737435421807431at_nat > set_Pr1542805901266377927at_nat ).
thf(sy_c_List_Olistrel_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
listre1194016999521814427rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr8564414093027780873rm_a_b ).
thf(sy_c_List_Olistrel_001tf__a_001t__Nat__Onat,type,
listrel_a_nat: set_Pr4934435412358123699_a_nat > set_Pr5046312416420021961st_nat ).
thf(sy_c_List_Omeasures_001t__Nat__Onat,type,
measures_nat: list_nat_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_List_Omeasures_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
measur2679027848233739777at_nat: list_P9162950289778280392at_nat > set_Pr8693737435421807431at_nat ).
thf(sy_c_List_Omeasures_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
measures_term_a_b: list_term_a_b_nat > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_List_Onth_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
nth_nat_nat: list_nat_nat > nat > nat > nat ).
thf(sy_c_List_Onth_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
nth_list_term_a_b: list_list_term_a_b > nat > list_term_a_b ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
nth_Pr4952906083868132727at_nat: list_P5366131564312172150at_nat > nat > produc8199716216217303280at_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_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
nth_Pr5300001119514325452rm_a_b: list_P2364656488115551307rm_a_b > nat > produc1234881154892807749rm_a_b ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
nth_Pr6744343527793145070at_nat: list_P8469869581646625389at_nat > nat > produc859450856879609959at_nat ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
nth_Pr1098598711982373392rm_a_b: list_P8875379029341186191rm_a_b > nat > produc357393685978478089rm_a_b ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
nth_Pr8461465654520414006_a_nat: list_P3592885314253461005_a_nat > nat > product_prod_a_nat ).
thf(sy_c_List_Onth_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
nth_term_a_b: list_term_a_b > nat > term_a_b ).
thf(sy_c_List_Onth_001tf__a,type,
nth_a: list_a > nat > a ).
thf(sy_c_List_Osplice_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
splice_term_a_b: list_term_a_b > list_term_a_b > list_term_a_b ).
thf(sy_c_List_Otake_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
take_list_term_a_b: nat > list_list_term_a_b > list_list_term_a_b ).
thf(sy_c_List_Otake_001t__Nat__Onat,type,
take_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Otake_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
take_P2173866234530122223at_nat: nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Otake_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
take_term_a_b: nat > list_term_a_b > list_term_a_b ).
thf(sy_c_List_Ozip_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
zip_nat_nat_nat: list_nat_nat > list_nat > list_P5366131564312172150at_nat ).
thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Nat__Onat,type,
zip_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Ozip_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
zip_Pr4664179122662387191at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > list_P8469869581646625389at_nat ).
thf(sy_c_List_Ozip_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
zip_te2260252107881041761rm_a_b: list_term_a_b > list_term_a_b > list_P8875379029341186191rm_a_b ).
thf(sy_c_List_Ozip_001tf__a_001t__Nat__Onat,type,
zip_a_nat: list_a > list_nat > list_P3592885314253461005_a_nat ).
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__Term__Oterm_Itf__a_Mtf__b_J,type,
missin5417461770552926780rm_a_b: list_term_a_b > ( nat > nat ) > list_term_a_b ).
thf(sy_c_Missing__List_Opermut__aux_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
missin5661735679256376914rm_a_b: list_term_a_b > ( nat > nat ) > list_term_a_b > list_term_a_b ).
thf(sy_c_Missing__List_Oremove__nth_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
missin7590947408886171483rm_a_b: nat > list_term_a_b > list_term_a_b ).
thf(sy_c_Missing__List_Ounion__list__sorted_001t__Nat__Onat,type,
missin8019018944680490243ed_nat: list_nat > list_nat > list_nat ).
thf(sy_c_Missing__List_Ounion__list__sorted__rel_001t__Nat__Onat,type,
missin7611735270441097048el_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_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
size_s8208510060688613859at_nat: list_nat_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
size_s877380706853472072rm_a_b: list_list_term_a_b > 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_It__Nat__Onat_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
size_s3663718155697878711rm_a_b: list_P2364656488115551307rm_a_b > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
size_s984997627204368545_a_nat: list_P3592885314253461005_a_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
size_s8906293707977694520rm_a_b: list_term_a_b > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
size_size_list_a: list_a > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
bot_bo482883023278783056_nat_o: product_prod_nat_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J,type,
bot_bo7573314457883560170_nat_o: produc859450856879609959at_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_M_Eo_J,type,
bot_bo23760184229683272_a_b_o: produc357393685978478089rm_a_b > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_M_Eo_J,type,
bot_bo1388487281569869310_nat_o: product_prod_a_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
bot_bot_set_list_nat: set_list_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
bot_bo2099793752762293965at_nat: set_Pr1261947904930325089at_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
bot_bo5327735625951526323at_nat: set_Pr8693737435421807431at_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
bot_bo197521221353338581rm_a_b: set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
bot_bo9049108969261143879_a_nat: set_Pr4934435412358123699_a_nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__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_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le6428140832669894131at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
ord_le4883060154188259093rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
ord_le2482536469199093511_a_nat: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__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__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
ord_le3678578370064672496at_nat: set_Pr9093778441882193744at_nat > set_Pr9093778441882193744at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mtf__a_J_J,type,
ord_le7986397757451867146_nat_a: set_Pr3254233319718257578_nat_a > set_Pr3254233319718257578_nat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
ord_le8406513867147106209st_nat: set_Pr3451248702717554689st_nat > set_Pr3451248702717554689st_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
ord_le1912520282604406569st_nat: set_Pr5046312416420021961st_nat > set_Pr5046312416420021961st_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le955236992067077104at_nat: set_Pr6370437063884598352at_nat > set_Pr6370437063884598352at_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_It__Nat__Onat_Mtf__a_J_J,type,
ord_le7924913712489149241_nat_a: set_Pr4193341848836149977_nat_a > set_Pr4193341848836149977_nat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le3000389064537975527at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
ord_le118470702582115849rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le2349859280813574146at_nat: set_Pr6841066879934740386at_nat > set_Pr6841066879934740386at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
ord_le8666007276011122963_a_nat: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat > $o ).
thf(sy_c_Orderings_Oord__class_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_It__Product____Type__Oprod_Itf__b_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
ord_le980006117782565848rm_a_b: set_Pr1357592988741419896rm_a_b > set_Pr1357592988741419896rm_a_b > $o ).
thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
produc5770335208449155351at_nat: ( nat > nat ) > ( nat > nat ) > produc1932156733058919263at_nat ).
thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
produc72220940542539688at_nat: ( nat > nat ) > nat > produc8199716216217303280at_nat ).
thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001tf__a,type,
produc3941937540969104678_nat_a: ( nat > nat ) > a > produc2842622564922169716_nat_a ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__List__Olist_It__Nat__Onat_J,type,
produc7978589510830832328st_nat: list_nat_nat > list_nat > produc4155638208356863248st_nat ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
produc2694037385005941721st_nat: list_nat > list_nat > produc1828647624359046049st_nat ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
produc5151171985953862413rm_a_b: list_nat > term_a_b > produc7711739908350443733rm_a_b ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc5943733680697469783at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > produc6392793444374437607at_nat ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
produc4885699992713594593rm_a_b: list_term_a_b > list_term_a_b > produc51424535725745577rm_a_b ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001t__List__Olist_It__Nat__Onat_J,type,
produc4792949784200893581st_nat: list_a > list_nat > produc432399132543013523st_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
produc7839516862119294504at_nat: nat > ( nat > nat ) > produc85711943791777264at_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
produc1516572978046417917rm_a_b: nat > term_a_b > produc1234881154892807749rm_a_b ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001tf__a,type,
product_Pair_nat_a: nat > a > product_prod_nat_a ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
produc2768974569381949847at_nat: produc8199716216217303280at_nat > produc8199716216217303280at_nat > produc492020038885415591at_nat ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
produc6161850002892822231at_nat: product_prod_nat_nat > product_prod_nat_nat > produc859450856879609959at_nat ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc4662710985925991255at_nat: produc859450856879609959at_nat > produc859450856879609959at_nat > produc349518998152878311at_nat ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
produc1763473618796451543rm_a_b: produc357393685978478089rm_a_b > produc357393685978478089rm_a_b > produc4523971788519308903rm_a_b ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
produc2026711137822539155_a_nat: product_prod_a_nat > product_prod_a_nat > produc4708774622424448987_a_nat ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_Itf__b_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
produc8030969961714872974rm_a_b: set_Pr4934435412358123699_a_nat > produc6551080308567045442rm_a_b > produc5279506192219892694rm_a_b ).
thf(sy_c_Product__Type_OPair_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__List__Olist_It__Nat__Onat_J,type,
produc4563063199488751885st_nat: term_a_b > list_nat > produc3697673438841856213st_nat ).
thf(sy_c_Product__Type_OPair_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
produc3812856575676843240rm_a_b: term_a_b > produc7711739908350443733rm_a_b > produc2732850333517536310rm_a_b ).
thf(sy_c_Product__Type_OPair_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
produc7020197800436672577rm_a_b: term_a_b > term_a_b > produc357393685978478089rm_a_b ).
thf(sy_c_Product__Type_OPair_001tf__a_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
produc7514922136058452582at_nat: a > ( nat > nat ) > produc551447040318622060at_nat ).
thf(sy_c_Product__Type_OPair_001tf__a_001t__Nat__Onat,type,
product_Pair_a_nat: a > nat > product_prod_a_nat ).
thf(sy_c_Product__Type_OPair_001tf__a_001tf__a,type,
product_Pair_a_a: a > a > product_prod_a_a ).
thf(sy_c_Product__Type_OPair_001tf__b_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
produc1437816968797971900rm_a_b: b > term_a_b > produc6551080308567045442rm_a_b ).
thf(sy_c_Relation_Orelcomp_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
relcom9054258243621454929at_nat: set_Pr7682762132356531903at_nat > set_Pr9093778441882193744at_nat > set_Pr9093778441882193744at_nat ).
thf(sy_c_Relation_Orelcomp_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
relcom7598182128343433937at_nat: set_Pr9093778441882193744at_nat > set_Pr6370437063884598352at_nat > set_Pr7682762132356531903at_nat ).
thf(sy_c_Relation_Orelcomp_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
relcom1859588284133128290at_nat: set_Pr9093778441882193744at_nat > set_Pr1261947904930325089at_nat > set_Pr9093778441882193744at_nat ).
thf(sy_c_Relation_Orelcomp_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat_001tf__a,type,
relcom5195258687133253356_nat_a: set_Pr9093778441882193744at_nat > set_Pr4193341848836149977_nat_a > set_Pr3254233319718257578_nat_a ).
thf(sy_c_Relation_Orelcomp_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001tf__a_001t__Nat__Onat,type,
relcom1163327185954966926_a_nat: set_Pr3254233319718257578_nat_a > set_Pr4934435412358123699_a_nat > set_Pr9093778441882193744at_nat ).
thf(sy_c_Relation_Orelcomp_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001tf__a_001tf__a,type,
relcomp_nat_nat_a_a: set_Pr3254233319718257578_nat_a > set_Product_prod_a_a > set_Pr3254233319718257578_nat_a ).
thf(sy_c_Relation_Orelcomp_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
relcom896424912281157969at_nat: set_Pr6370437063884598352at_nat > set_Pr7682762132356531903at_nat > set_Pr6370437063884598352at_nat ).
thf(sy_c_Relation_Orelcomp_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
relcom6405311860805859042at_nat: set_Pr6370437063884598352at_nat > set_Pr9093778441882193744at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Relation_Orelcomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
relcomp_nat_nat_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Relation_Orelcomp_001t__Nat__Onat_001t__Nat__Onat_001tf__a,type,
relcomp_nat_nat_a: set_Pr1261947904930325089at_nat > set_Pr4193341848836149977_nat_a > set_Pr4193341848836149977_nat_a ).
thf(sy_c_Relation_Orelcomp_001t__Nat__Onat_001tf__a_001t__Nat__Onat,type,
relcomp_nat_a_nat: set_Pr4193341848836149977_nat_a > set_Pr4934435412358123699_a_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Relation_Orelcomp_001t__Nat__Onat_001tf__a_001tf__a,type,
relcomp_nat_a_a: set_Pr4193341848836149977_nat_a > set_Product_prod_a_a > set_Pr4193341848836149977_nat_a ).
thf(sy_c_Relation_Orelcomp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
relcom7295539661566034944at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat ).
thf(sy_c_Relation_Orelcomp_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
relcom370159955682700863rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Relation_Orelcomp_001tf__a_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
relcom8552328690651882190at_nat: set_Pr6841066879934740386at_nat > set_Pr9093778441882193744at_nat > set_Pr4934435412358123699_a_nat ).
thf(sy_c_Relation_Orelcomp_001tf__a_001t__Nat__Onat_001t__Nat__Onat,type,
relcomp_a_nat_nat: set_Pr4934435412358123699_a_nat > set_Pr1261947904930325089at_nat > set_Pr4934435412358123699_a_nat ).
thf(sy_c_Relation_Orelcomp_001tf__a_001t__Nat__Onat_001tf__a,type,
relcomp_a_nat_a: set_Pr4934435412358123699_a_nat > set_Pr4193341848836149977_nat_a > set_Product_prod_a_a ).
thf(sy_c_Relation_Orelcomp_001tf__a_001tf__a_001t__Nat__Onat,type,
relcomp_a_a_nat: set_Product_prod_a_a > set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat ).
thf(sy_c_Relation_Orelcomp_001tf__a_001tf__a_001tf__a,type,
relcomp_a_a_a: set_Product_prod_a_a > set_Product_prod_a_a > set_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__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
collec2757435396427831370at_nat: ( produc1932156733058919263at_nat > $o ) > set_Pr7682762132356531903at_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mtf__a_J,type,
collec3024486194029540553_nat_a: ( produc2842622564922169716_nat_a > $o ) > set_Pr3254233319718257578_nat_a ).
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_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
collec7088162979684241874at_nat: ( produc859450856879609959at_nat > $o ) > set_Pr8693737435421807431at_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
collec99177395925924084rm_a_b: ( produc357393685978478089rm_a_b > $o ) > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
collec733310669425992897at_nat: ( produc551447040318622060at_nat > $o ) > set_Pr6841066879934740386at_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
collec4464134535221767506_a_nat: ( product_prod_a_nat > $o ) > set_Pr4934435412358123699_a_nat ).
thf(sy_c_Set_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_Term_Oterm_OFun_001tf__a_001tf__b,type,
fun_a_b: a > list_term_a_b > term_a_b ).
thf(sy_c_Term_Oterm_OVar_001tf__b_001tf__a,type,
var_b_a: b > term_a_b ).
thf(sy_c_Term__Context_Odepth_001tf__a_001tf__b,type,
term_depth_a_b: term_a_b > nat ).
thf(sy_c_Term__Context_Ofunas__term_001tf__a_001tf__b,type,
term_funas_term_a_b: term_a_b > set_Pr4934435412358123699_a_nat ).
thf(sy_c_Term__Context_Oreplace__term__at_001tf__a_001tf__b,type,
term_r6860082780075436317at_a_b: term_a_b > list_nat > term_a_b > term_a_b ).
thf(sy_c_Term__Context_Oreplace__term__at__rel_001tf__a_001tf__b,type,
term_r1280879029893354718el_a_b: produc2732850333517536310rm_a_b > produc2732850333517536310rm_a_b > $o ).
thf(sy_c_Terms__Positions_Oall__ctxt__closed_001tf__a_001tf__b,type,
terms_5226143800768910156ed_a_b: set_Pr4934435412358123699_a_nat > set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_Terms__Positions_Oposs__of__term_001tf__a_001tf__b,type,
terms_7168686267159881682rm_a_b: term_a_b > term_a_b > set_list_nat ).
thf(sy_c_Transitive__Closure_Ortrancl_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
transi3031656534272781564at_nat: set_Pr7682762132356531903at_nat > set_Pr7682762132356531903at_nat ).
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_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
transi5735831900260020618at_nat: set_Pr1542805901266377927at_nat > set_Pr1542805901266377927at_nat ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J,type,
transi911495901722478305rm_a_b: set_Pr8564414093027780873rm_a_b > set_Pr8564414093027780873rm_a_b ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Nat__Onat,type,
transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
transi3422749762518297843at_nat: set_Pr7047708467701517959at_nat > set_Pr7047708467701517959at_nat ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
transi8609417484261754244at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
transi5470879363895450730at_nat: set_Pr553994874890374343at_nat > set_Pr553994874890374343at_nat ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
transi2615809358984392588rm_a_b: set_Pr2972776593051762503rm_a_b > set_Pr2972776593051762503rm_a_b ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
transi2726145917338391738_a_nat: set_Pr1811044260758604347_a_nat > set_Pr1811044260758604347_a_nat ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
transi7742714808557438673rm_a_b: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b ).
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_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__Term__Oterm_Itf__a_Mtf__b_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
accp_P2729577386226225901rm_a_b: ( produc2732850333517536310rm_a_b > produc2732850333517536310rm_a_b > $o ) > produc2732850333517536310rm_a_b > $o ).
thf(sy_c_Wellfounded_Omeasure_001t__Nat__Onat,type,
measure_nat: ( nat > nat ) > set_Pr1261947904930325089at_nat ).
thf(sy_c_Wellfounded_Omeasure_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
measur8038558561449204169at_nat: ( product_prod_nat_nat > nat ) > set_Pr8693737435421807431at_nat ).
thf(sy_c_Wellfounded_Omeasure_001t__Term__Oterm_Itf__a_Mtf__b_J,type,
measure_term_a_b: ( term_a_b > nat ) > set_Pr4386577575007340137rm_a_b ).
thf(sy_c_Wellfounded_Opred__nat,type,
pred_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__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member1096249278104964872at_nat: produc1932156733058919263at_nat > set_Pr7682762132356531903at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
member7226740684066999833at_nat: produc8199716216217303280at_nat > set_Pr9093778441882193744at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mtf__a_J,type,
member7754445317155421323_nat_a: produc2842622564922169716_nat_a > set_Pr3254233319718257578_nat_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
member6987746275253522745st_nat: produc4155638208356863248st_nat > set_Pr5771542735269860976st_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_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member6693912407220327184at_nat: produc6392793444374437607at_nat > set_Pr1542805901266377927at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__List__Olist_It__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
member4405265420456397394rm_a_b: produc51424535725745577rm_a_b > set_Pr8564414093027780873rm_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
member4851138774834033962st_nat: produc432399132543013523st_nat > set_Pr5046312416420021961st_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member8336108448496249625at_nat: produc85711943791777264at_nat > set_Pr6370437063884598352at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
member8962352052110095674_nat_a: product_prod_nat_a > set_Pr4193341848836149977_nat_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
member3795392313164512464at_nat: produc492020038885415591at_nat > set_Pr7047708467701517959at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
member8062223511168850704at_nat: produc349518998152878311at_nat > set_Pr553994874890374343at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J_J,type,
member8417600551952982416rm_a_b: produc4523971788519308903rm_a_b > set_Pr2972776593051762503rm_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J_J,type,
member9062615507155100804_a_nat: produc4708774622424448987_a_nat > set_Pr1811044260758604347_a_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Term__Oterm_Itf__a_Mtf__b_J_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
member5869715511025134514rm_a_b: produc357393685978478089rm_a_b > set_Pr4386577575007340137rm_a_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member5463269792551873667at_nat: produc551447040318622060at_nat > set_Pr6841066879934740386at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
member5724188588386418708_a_nat: product_prod_a_nat > set_Pr4934435412358123699_a_nat > $o ).
thf(sy_c_member_001t__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_001t__Product____Type__Oprod_Itf__b_Mt__Term__Oterm_Itf__a_Mtf__b_J_J,type,
member7000133282412810073rm_a_b: produc6551080308567045442rm_a_b > set_Pr1357592988741419896rm_a_b > $o ).
thf(sy_v__092_060F_062,type,
f: set_Pr4934435412358123699_a_nat ).
thf(sy_v__092_060R_062,type,
r: set_Pr4386577575007340137rm_a_b ).
thf(sy_v__092_060S_062,type,
s: set_Pr4386577575007340137rm_a_b ).
thf(sy_v_f____,type,
f2: a ).
thf(sy_v_ss____,type,
ss: list_term_a_b ).
thf(sy_v_ts____,type,
ts: list_term_a_b ).
thf(sy_v_us____,type,
us: list_term_a_b ).
% Relevant facts (1258)
thf(fact_0_ass_I2_J,axiom,
( ( size_s8906293707977694520rm_a_b @ ss )
= ( size_s8906293707977694520rm_a_b @ ts ) ) ).
% ass(2)
thf(fact_1_us_I1_J,axiom,
( ( size_s8906293707977694520rm_a_b @ us )
= ( size_s8906293707977694520rm_a_b @ ts ) ) ).
% us(1)
thf(fact_2_ctxt__cl_I1_J,axiom,
terms_5226143800768910156ed_a_b @ f @ r ).
% ctxt_cl(1)
thf(fact_3_ass_I1_J,axiom,
member5724188588386418708_a_nat @ ( product_Pair_a_nat @ f2 @ ( size_s8906293707977694520rm_a_b @ ss ) ) @ f ).
% ass(1)
thf(fact_4_us_I2_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ ts ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ ss @ I ) @ ( nth_term_a_b @ us @ I ) ) @ r ) ) ).
% us(2)
thf(fact_5_fu,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ us ) )
=> ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( nth_term_a_b @ us @ I ) ) @ f ) ) ).
% fu
thf(fact_6_ass_I5_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ ts ) )
=> ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( nth_term_a_b @ ss @ I ) ) @ f ) ) ).
% ass(5)
thf(fact_7_funas_I1_J,axiom,
! [S: term_a_b,T: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ r )
=> ( ( S != T )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ f )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ f ) ) ) ) ).
% funas(1)
thf(fact_8_term_Oinject_I2_J,axiom,
! [X21: a,X22: list_term_a_b,Y21: a,Y22: list_term_a_b] :
( ( ( fun_a_b @ X21 @ X22 )
= ( fun_a_b @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% term.inject(2)
thf(fact_9_prod_Oinject,axiom,
! [X1: nat > nat,X2: nat > nat,Y1: nat > nat,Y2: nat > nat] :
( ( ( produc5770335208449155351at_nat @ X1 @ X2 )
= ( produc5770335208449155351at_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_10_prod_Oinject,axiom,
! [X1: nat > nat,X2: a,Y1: nat > nat,Y2: a] :
( ( ( produc3941937540969104678_nat_a @ X1 @ X2 )
= ( produc3941937540969104678_nat_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_11_prod_Oinject,axiom,
! [X1: b,X2: term_a_b,Y1: b,Y2: term_a_b] :
( ( ( produc1437816968797971900rm_a_b @ X1 @ X2 )
= ( produc1437816968797971900rm_a_b @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_12_prod_Oinject,axiom,
! [X1: a,X2: nat > nat,Y1: a,Y2: nat > nat] :
( ( ( produc7514922136058452582at_nat @ X1 @ X2 )
= ( produc7514922136058452582at_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_13_prod_Oinject,axiom,
! [X1: a,X2: a,Y1: a,Y2: a] :
( ( ( product_Pair_a_a @ X1 @ X2 )
= ( product_Pair_a_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_14_prod_Oinject,axiom,
! [X1: term_a_b,X2: term_a_b,Y1: term_a_b,Y2: term_a_b] :
( ( ( produc7020197800436672577rm_a_b @ X1 @ X2 )
= ( produc7020197800436672577rm_a_b @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_15_prod_Oinject,axiom,
! [X1: a,X2: nat,Y1: a,Y2: nat] :
( ( ( product_Pair_a_nat @ X1 @ X2 )
= ( product_Pair_a_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_16_prod_Oinject,axiom,
! [X1: nat > nat,X2: nat,Y1: nat > nat,Y2: nat] :
( ( ( produc72220940542539688at_nat @ X1 @ X2 )
= ( produc72220940542539688at_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_17_prod_Oinject,axiom,
! [X1: product_prod_nat_nat,X2: product_prod_nat_nat,Y1: product_prod_nat_nat,Y2: product_prod_nat_nat] :
( ( ( produc6161850002892822231at_nat @ X1 @ X2 )
= ( produc6161850002892822231at_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_18_prod_Oinject,axiom,
! [X1: nat,X2: nat,Y1: nat,Y2: nat] :
( ( ( product_Pair_nat_nat @ X1 @ X2 )
= ( product_Pair_nat_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_19_old_Oprod_Oinject,axiom,
! [A: nat > nat,B: nat > nat,A2: nat > nat,B2: nat > nat] :
( ( ( produc5770335208449155351at_nat @ A @ B )
= ( produc5770335208449155351at_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_20_old_Oprod_Oinject,axiom,
! [A: nat > nat,B: a,A2: nat > nat,B2: a] :
( ( ( produc3941937540969104678_nat_a @ A @ B )
= ( produc3941937540969104678_nat_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_21_old_Oprod_Oinject,axiom,
! [A: b,B: term_a_b,A2: b,B2: term_a_b] :
( ( ( produc1437816968797971900rm_a_b @ A @ B )
= ( produc1437816968797971900rm_a_b @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_22_old_Oprod_Oinject,axiom,
! [A: a,B: nat > nat,A2: a,B2: nat > nat] :
( ( ( produc7514922136058452582at_nat @ A @ B )
= ( produc7514922136058452582at_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_23_old_Oprod_Oinject,axiom,
! [A: a,B: a,A2: a,B2: a] :
( ( ( product_Pair_a_a @ A @ B )
= ( product_Pair_a_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_24_old_Oprod_Oinject,axiom,
! [A: term_a_b,B: term_a_b,A2: term_a_b,B2: term_a_b] :
( ( ( produc7020197800436672577rm_a_b @ A @ B )
= ( produc7020197800436672577rm_a_b @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_25_old_Oprod_Oinject,axiom,
! [A: a,B: nat,A2: a,B2: nat] :
( ( ( product_Pair_a_nat @ A @ B )
= ( product_Pair_a_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_26_old_Oprod_Oinject,axiom,
! [A: nat > nat,B: nat,A2: nat > nat,B2: nat] :
( ( ( produc72220940542539688at_nat @ A @ B )
= ( produc72220940542539688at_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_27_old_Oprod_Oinject,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
( ( ( produc6161850002892822231at_nat @ A @ B )
= ( produc6161850002892822231at_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_28_old_Oprod_Oinject,axiom,
! [A: nat,B: nat,A2: nat,B2: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_29__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062us_O_A_092_060lbrakk_062length_Aus_A_061_Alength_Ats_059_A_092_060And_062i_O_Ai_A_060_Alength_Ats_A_092_060Longrightarrow_062_A_Iss_A_B_Ai_M_Aus_A_B_Ai_J_A_092_060in_062_A_092_060R_062_059_A_092_060And_062i_O_Ai_A_060_Alength_Ats_A_092_060Longrightarrow_062_A_Ius_A_B_Ai_M_Ats_A_B_Ai_J_A_092_060in_062_A_092_060S_062_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Us: list_term_a_b] :
( ( ( size_s8906293707977694520rm_a_b @ Us )
= ( size_s8906293707977694520rm_a_b @ ts ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ ts ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ ss @ I2 ) @ ( nth_term_a_b @ Us @ I2 ) ) @ r ) )
=> ~ ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ ts ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ Us @ I2 ) @ ( nth_term_a_b @ ts @ I2 ) ) @ s ) ) ) ) ).
% \<open>\<And>thesis. (\<And>us. \<lbrakk>length us = length ts; \<And>i. i < length ts \<Longrightarrow> (ss ! i, us ! i) \<in> \<R>; \<And>i. i < length ts \<Longrightarrow> (us ! i, ts ! i) \<in> \<S>\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_30_us_I3_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ ts ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ us @ I ) @ ( nth_term_a_b @ ts @ I ) ) @ s ) ) ).
% us(3)
thf(fact_31_old_Oprod_Oexhaust,axiom,
! [Y: produc1932156733058919263at_nat] :
~ ! [A3: nat > nat,B3: nat > nat] :
( Y
!= ( produc5770335208449155351at_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_32_old_Oprod_Oexhaust,axiom,
! [Y: produc2842622564922169716_nat_a] :
~ ! [A3: nat > nat,B3: a] :
( Y
!= ( produc3941937540969104678_nat_a @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_33_old_Oprod_Oexhaust,axiom,
! [Y: produc6551080308567045442rm_a_b] :
~ ! [A3: b,B3: term_a_b] :
( Y
!= ( produc1437816968797971900rm_a_b @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_34_old_Oprod_Oexhaust,axiom,
! [Y: produc551447040318622060at_nat] :
~ ! [A3: a,B3: nat > nat] :
( Y
!= ( produc7514922136058452582at_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_35_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_a_a] :
~ ! [A3: a,B3: a] :
( Y
!= ( product_Pair_a_a @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_36_old_Oprod_Oexhaust,axiom,
! [Y: produc357393685978478089rm_a_b] :
~ ! [A3: term_a_b,B3: term_a_b] :
( Y
!= ( produc7020197800436672577rm_a_b @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_37_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_a_nat] :
~ ! [A3: a,B3: nat] :
( Y
!= ( product_Pair_a_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_38_old_Oprod_Oexhaust,axiom,
! [Y: produc8199716216217303280at_nat] :
~ ! [A3: nat > nat,B3: nat] :
( Y
!= ( produc72220940542539688at_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_39_old_Oprod_Oexhaust,axiom,
! [Y: produc859450856879609959at_nat] :
~ ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
( Y
!= ( produc6161850002892822231at_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_40_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_nat_nat] :
~ ! [A3: nat,B3: nat] :
( Y
!= ( product_Pair_nat_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_41_surj__pair,axiom,
! [P: produc1932156733058919263at_nat] :
? [X: nat > nat,Y3: nat > nat] :
( P
= ( produc5770335208449155351at_nat @ X @ Y3 ) ) ).
% surj_pair
thf(fact_42_surj__pair,axiom,
! [P: produc2842622564922169716_nat_a] :
? [X: nat > nat,Y3: a] :
( P
= ( produc3941937540969104678_nat_a @ X @ Y3 ) ) ).
% surj_pair
thf(fact_43_surj__pair,axiom,
! [P: produc6551080308567045442rm_a_b] :
? [X: b,Y3: term_a_b] :
( P
= ( produc1437816968797971900rm_a_b @ X @ Y3 ) ) ).
% surj_pair
thf(fact_44_surj__pair,axiom,
! [P: produc551447040318622060at_nat] :
? [X: a,Y3: nat > nat] :
( P
= ( produc7514922136058452582at_nat @ X @ Y3 ) ) ).
% surj_pair
thf(fact_45_surj__pair,axiom,
! [P: product_prod_a_a] :
? [X: a,Y3: a] :
( P
= ( product_Pair_a_a @ X @ Y3 ) ) ).
% surj_pair
thf(fact_46_surj__pair,axiom,
! [P: produc357393685978478089rm_a_b] :
? [X: term_a_b,Y3: term_a_b] :
( P
= ( produc7020197800436672577rm_a_b @ X @ Y3 ) ) ).
% surj_pair
thf(fact_47_surj__pair,axiom,
! [P: product_prod_a_nat] :
? [X: a,Y3: nat] :
( P
= ( product_Pair_a_nat @ X @ Y3 ) ) ).
% surj_pair
thf(fact_48_surj__pair,axiom,
! [P: produc8199716216217303280at_nat] :
? [X: nat > nat,Y3: nat] :
( P
= ( produc72220940542539688at_nat @ X @ Y3 ) ) ).
% surj_pair
thf(fact_49_surj__pair,axiom,
! [P: produc859450856879609959at_nat] :
? [X: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( P
= ( produc6161850002892822231at_nat @ X @ Y3 ) ) ).
% surj_pair
thf(fact_50_surj__pair,axiom,
! [P: product_prod_nat_nat] :
? [X: nat,Y3: nat] :
( P
= ( product_Pair_nat_nat @ X @ Y3 ) ) ).
% surj_pair
thf(fact_51_prod__cases,axiom,
! [P2: produc1932156733058919263at_nat > $o,P: produc1932156733058919263at_nat] :
( ! [A3: nat > nat,B3: nat > nat] : ( P2 @ ( produc5770335208449155351at_nat @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_52_prod__cases,axiom,
! [P2: produc2842622564922169716_nat_a > $o,P: produc2842622564922169716_nat_a] :
( ! [A3: nat > nat,B3: a] : ( P2 @ ( produc3941937540969104678_nat_a @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_53_prod__cases,axiom,
! [P2: produc6551080308567045442rm_a_b > $o,P: produc6551080308567045442rm_a_b] :
( ! [A3: b,B3: term_a_b] : ( P2 @ ( produc1437816968797971900rm_a_b @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_54_prod__cases,axiom,
! [P2: produc551447040318622060at_nat > $o,P: produc551447040318622060at_nat] :
( ! [A3: a,B3: nat > nat] : ( P2 @ ( produc7514922136058452582at_nat @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_55_prod__cases,axiom,
! [P2: product_prod_a_a > $o,P: product_prod_a_a] :
( ! [A3: a,B3: a] : ( P2 @ ( product_Pair_a_a @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_56_prod__cases,axiom,
! [P2: produc357393685978478089rm_a_b > $o,P: produc357393685978478089rm_a_b] :
( ! [A3: term_a_b,B3: term_a_b] : ( P2 @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_57_prod__cases,axiom,
! [P2: product_prod_a_nat > $o,P: product_prod_a_nat] :
( ! [A3: a,B3: nat] : ( P2 @ ( product_Pair_a_nat @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_58_prod__cases,axiom,
! [P2: produc8199716216217303280at_nat > $o,P: produc8199716216217303280at_nat] :
( ! [A3: nat > nat,B3: nat] : ( P2 @ ( produc72220940542539688at_nat @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_59_prod__cases,axiom,
! [P2: produc859450856879609959at_nat > $o,P: produc859450856879609959at_nat] :
( ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] : ( P2 @ ( produc6161850002892822231at_nat @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_60_prod__cases,axiom,
! [P2: product_prod_nat_nat > $o,P: product_prod_nat_nat] :
( ! [A3: nat,B3: nat] : ( P2 @ ( product_Pair_nat_nat @ A3 @ B3 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_61_Pair__inject,axiom,
! [A: nat > nat,B: nat > nat,A2: nat > nat,B2: nat > nat] :
( ( ( produc5770335208449155351at_nat @ A @ B )
= ( produc5770335208449155351at_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_62_Pair__inject,axiom,
! [A: nat > nat,B: a,A2: nat > nat,B2: a] :
( ( ( produc3941937540969104678_nat_a @ A @ B )
= ( produc3941937540969104678_nat_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_63_Pair__inject,axiom,
! [A: b,B: term_a_b,A2: b,B2: term_a_b] :
( ( ( produc1437816968797971900rm_a_b @ A @ B )
= ( produc1437816968797971900rm_a_b @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_64_Pair__inject,axiom,
! [A: a,B: nat > nat,A2: a,B2: nat > nat] :
( ( ( produc7514922136058452582at_nat @ A @ B )
= ( produc7514922136058452582at_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_65_Pair__inject,axiom,
! [A: a,B: a,A2: a,B2: a] :
( ( ( product_Pair_a_a @ A @ B )
= ( product_Pair_a_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_66_Pair__inject,axiom,
! [A: term_a_b,B: term_a_b,A2: term_a_b,B2: term_a_b] :
( ( ( produc7020197800436672577rm_a_b @ A @ B )
= ( produc7020197800436672577rm_a_b @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_67_Pair__inject,axiom,
! [A: a,B: nat,A2: a,B2: nat] :
( ( ( product_Pair_a_nat @ A @ B )
= ( product_Pair_a_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_68_Pair__inject,axiom,
! [A: nat > nat,B: nat,A2: nat > nat,B2: nat] :
( ( ( produc72220940542539688at_nat @ A @ B )
= ( produc72220940542539688at_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_69_Pair__inject,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
( ( ( produc6161850002892822231at_nat @ A @ B )
= ( produc6161850002892822231at_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_70_Pair__inject,axiom,
! [A: nat,B: nat,A2: nat,B2: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_71_ctxt__cl_I2_J,axiom,
terms_5226143800768910156ed_a_b @ f @ s ).
% ctxt_cl(2)
thf(fact_72_funas_I2_J,axiom,
! [S: term_a_b,T: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ s )
=> ( ( S != T )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ S ) @ f )
& ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ f ) ) ) ) ).
% funas(2)
thf(fact_73_ass_I4_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ ts ) )
=> ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( nth_term_a_b @ ts @ I ) ) @ f ) ) ).
% ass(4)
thf(fact_74_all__ctxt__closed__sig__reflE,axiom,
! [F: set_Pr4934435412358123699_a_nat,R: set_Pr4386577575007340137rm_a_b,T: term_a_b] :
( ( terms_5226143800768910156ed_a_b @ F @ R )
=> ( ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ T ) @ F )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ T ) @ R ) ) ) ).
% all_ctxt_closed_sig_reflE
thf(fact_75_prod__induct3,axiom,
! [P2: produc859450856879609959at_nat > $o,X3: produc859450856879609959at_nat] :
( ! [A3: product_prod_nat_nat,B3: nat,C: nat] : ( P2 @ ( produc6161850002892822231at_nat @ A3 @ ( product_Pair_nat_nat @ B3 @ C ) ) )
=> ( P2 @ X3 ) ) ).
% prod_induct3
thf(fact_76_prod__cases3,axiom,
! [Y: produc859450856879609959at_nat] :
~ ! [A3: product_prod_nat_nat,B3: nat,C: nat] :
( Y
!= ( produc6161850002892822231at_nat @ A3 @ ( product_Pair_nat_nat @ B3 @ C ) ) ) ).
% prod_cases3
thf(fact_77_all__ctxt__closedD,axiom,
! [F2: set_Pr4934435412358123699_a_nat,R2: set_Pr4386577575007340137rm_a_b,F3: a,Ss: list_term_a_b,Ts: list_term_a_b] :
( ( terms_5226143800768910156ed_a_b @ F2 @ R2 )
=> ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ F3 @ ( size_s8906293707977694520rm_a_b @ Ss ) ) @ F2 )
=> ( ( ( size_s8906293707977694520rm_a_b @ Ts )
= ( size_s8906293707977694520rm_a_b @ Ss ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s8906293707977694520rm_a_b @ Ts ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ Ts @ I3 ) @ ( nth_term_a_b @ Ss @ I3 ) ) @ R2 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s8906293707977694520rm_a_b @ Ts ) )
=> ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( nth_term_a_b @ Ts @ I3 ) ) @ F2 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s8906293707977694520rm_a_b @ Ts ) )
=> ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( nth_term_a_b @ Ss @ I3 ) ) @ F2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( fun_a_b @ F3 @ Ts ) @ ( fun_a_b @ F3 @ Ss ) ) @ R2 ) ) ) ) ) ) ) ).
% all_ctxt_closedD
thf(fact_78_ass_I3_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ ts ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ ss @ I ) @ ( nth_term_a_b @ ts @ I ) ) @ ( relcom370159955682700863rm_a_b @ r @ s ) ) ) ).
% ass(3)
thf(fact_79_nth__equalityI,axiom,
! [Xs: list_P3592885314253461005_a_nat,Ys: list_P3592885314253461005_a_nat] :
( ( ( size_s984997627204368545_a_nat @ Xs )
= ( size_s984997627204368545_a_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s984997627204368545_a_nat @ Xs ) )
=> ( ( nth_Pr8461465654520414006_a_nat @ Xs @ I3 )
= ( nth_Pr8461465654520414006_a_nat @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_80_nth__equalityI,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs )
= ( size_s5460976970255530739at_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( ( nth_Pr7617993195940197384at_nat @ Xs @ I3 )
= ( nth_Pr7617993195940197384at_nat @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_81_nth__equalityI,axiom,
! [Xs: list_list_term_a_b,Ys: list_list_term_a_b] :
( ( ( size_s877380706853472072rm_a_b @ Xs )
= ( size_s877380706853472072rm_a_b @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s877380706853472072rm_a_b @ Xs ) )
=> ( ( nth_list_term_a_b @ Xs @ I3 )
= ( nth_list_term_a_b @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_82_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_83_nth__equalityI,axiom,
! [Xs: list_nat_nat,Ys: list_nat_nat] :
( ( ( size_s8208510060688613859at_nat @ Xs )
= ( size_s8208510060688613859at_nat @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s8208510060688613859at_nat @ Xs ) )
=> ( ( nth_nat_nat @ Xs @ I3 )
= ( nth_nat_nat @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_84_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_85_nth__equalityI,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b] :
( ( ( size_s8906293707977694520rm_a_b @ Xs )
= ( size_s8906293707977694520rm_a_b @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( nth_term_a_b @ Xs @ I3 )
= ( nth_term_a_b @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_86_Skolem__list__nth,axiom,
! [K: nat,P2: nat > product_prod_a_nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X4: product_prod_a_nat] : ( P2 @ I4 @ X4 ) ) )
= ( ? [Xs2: list_P3592885314253461005_a_nat] :
( ( ( size_s984997627204368545_a_nat @ Xs2 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P2 @ I4 @ ( nth_Pr8461465654520414006_a_nat @ Xs2 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_87_Skolem__list__nth,axiom,
! [K: nat,P2: nat > product_prod_nat_nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X4: product_prod_nat_nat] : ( P2 @ I4 @ X4 ) ) )
= ( ? [Xs2: list_P6011104703257516679at_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs2 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P2 @ I4 @ ( nth_Pr7617993195940197384at_nat @ Xs2 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_88_Skolem__list__nth,axiom,
! [K: nat,P2: nat > list_term_a_b > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X4: list_term_a_b] : ( P2 @ I4 @ X4 ) ) )
= ( ? [Xs2: list_list_term_a_b] :
( ( ( size_s877380706853472072rm_a_b @ Xs2 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P2 @ I4 @ ( nth_list_term_a_b @ Xs2 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_89_Skolem__list__nth,axiom,
! [K: nat,P2: nat > nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X4: nat] : ( P2 @ I4 @ X4 ) ) )
= ( ? [Xs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P2 @ I4 @ ( nth_nat @ Xs2 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_90_Skolem__list__nth,axiom,
! [K: nat,P2: nat > ( nat > nat ) > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X4: nat > nat] : ( P2 @ I4 @ X4 ) ) )
= ( ? [Xs2: list_nat_nat] :
( ( ( size_s8208510060688613859at_nat @ Xs2 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P2 @ I4 @ ( nth_nat_nat @ Xs2 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_91_Skolem__list__nth,axiom,
! [K: nat,P2: nat > a > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X4: a] : ( P2 @ I4 @ X4 ) ) )
= ( ? [Xs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P2 @ I4 @ ( nth_a @ Xs2 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_92_Skolem__list__nth,axiom,
! [K: nat,P2: nat > term_a_b > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X4: term_a_b] : ( P2 @ I4 @ X4 ) ) )
= ( ? [Xs2: list_term_a_b] :
( ( ( size_s8906293707977694520rm_a_b @ Xs2 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P2 @ I4 @ ( nth_term_a_b @ Xs2 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_93_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_P3592885314253461005_a_nat,Z: list_P3592885314253461005_a_nat] : ( Y4 = Z ) )
= ( ^ [Xs2: list_P3592885314253461005_a_nat,Ys2: list_P3592885314253461005_a_nat] :
( ( ( size_s984997627204368545_a_nat @ Xs2 )
= ( size_s984997627204368545_a_nat @ Ys2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s984997627204368545_a_nat @ Xs2 ) )
=> ( ( nth_Pr8461465654520414006_a_nat @ Xs2 @ I4 )
= ( nth_Pr8461465654520414006_a_nat @ Ys2 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_94_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_P6011104703257516679at_nat,Z: list_P6011104703257516679at_nat] : ( Y4 = Z ) )
= ( ^ [Xs2: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs2 )
= ( size_s5460976970255530739at_nat @ Ys2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s5460976970255530739at_nat @ Xs2 ) )
=> ( ( nth_Pr7617993195940197384at_nat @ Xs2 @ I4 )
= ( nth_Pr7617993195940197384at_nat @ Ys2 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_95_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_list_term_a_b,Z: list_list_term_a_b] : ( Y4 = Z ) )
= ( ^ [Xs2: list_list_term_a_b,Ys2: list_list_term_a_b] :
( ( ( size_s877380706853472072rm_a_b @ Xs2 )
= ( size_s877380706853472072rm_a_b @ Ys2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s877380706853472072rm_a_b @ Xs2 ) )
=> ( ( nth_list_term_a_b @ Xs2 @ I4 )
= ( nth_list_term_a_b @ Ys2 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_96_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_nat,Z: list_nat] : ( Y4 = Z ) )
= ( ^ [Xs2: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ Xs2 @ I4 )
= ( nth_nat @ Ys2 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_97_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_nat_nat,Z: list_nat_nat] : ( Y4 = Z ) )
= ( ^ [Xs2: list_nat_nat,Ys2: list_nat_nat] :
( ( ( size_s8208510060688613859at_nat @ Xs2 )
= ( size_s8208510060688613859at_nat @ Ys2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s8208510060688613859at_nat @ Xs2 ) )
=> ( ( nth_nat_nat @ Xs2 @ I4 )
= ( nth_nat_nat @ Ys2 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_98_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_a,Z: list_a] : ( Y4 = Z ) )
= ( ^ [Xs2: list_a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_a @ Xs2 ) )
=> ( ( nth_a @ Xs2 @ I4 )
= ( nth_a @ Ys2 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_99_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_term_a_b,Z: list_term_a_b] : ( Y4 = Z ) )
= ( ^ [Xs2: list_term_a_b,Ys2: list_term_a_b] :
( ( ( size_s8906293707977694520rm_a_b @ Xs2 )
= ( size_s8906293707977694520rm_a_b @ Ys2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
=> ( ( nth_term_a_b @ Xs2 @ I4 )
= ( nth_term_a_b @ Ys2 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_100_Ex__list__of__length__P,axiom,
! [N: nat,P2: product_prod_a_nat > nat > $o] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ? [X5: product_prod_a_nat] : ( P2 @ X5 @ I3 ) )
=> ? [Xs3: list_P3592885314253461005_a_nat] :
( ( ( size_s984997627204368545_a_nat @ Xs3 )
= N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P2 @ ( nth_Pr8461465654520414006_a_nat @ Xs3 @ I2 ) @ I2 ) ) ) ) ).
% Ex_list_of_length_P
thf(fact_101_Ex__list__of__length__P,axiom,
! [N: nat,P2: product_prod_nat_nat > nat > $o] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ? [X5: product_prod_nat_nat] : ( P2 @ X5 @ I3 ) )
=> ? [Xs3: list_P6011104703257516679at_nat] :
( ( ( size_s5460976970255530739at_nat @ Xs3 )
= N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P2 @ ( nth_Pr7617993195940197384at_nat @ Xs3 @ I2 ) @ I2 ) ) ) ) ).
% Ex_list_of_length_P
thf(fact_102_Ex__list__of__length__P,axiom,
! [N: nat,P2: list_term_a_b > nat > $o] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ? [X5: list_term_a_b] : ( P2 @ X5 @ I3 ) )
=> ? [Xs3: list_list_term_a_b] :
( ( ( size_s877380706853472072rm_a_b @ Xs3 )
= N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P2 @ ( nth_list_term_a_b @ Xs3 @ I2 ) @ I2 ) ) ) ) ).
% Ex_list_of_length_P
thf(fact_103_Ex__list__of__length__P,axiom,
! [N: nat,P2: nat > nat > $o] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ? [X5: nat] : ( P2 @ X5 @ I3 ) )
=> ? [Xs3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P2 @ ( nth_nat @ Xs3 @ I2 ) @ I2 ) ) ) ) ).
% Ex_list_of_length_P
thf(fact_104_Ex__list__of__length__P,axiom,
! [N: nat,P2: ( nat > nat ) > nat > $o] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ? [X5: nat > nat] : ( P2 @ X5 @ I3 ) )
=> ? [Xs3: list_nat_nat] :
( ( ( size_s8208510060688613859at_nat @ Xs3 )
= N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P2 @ ( nth_nat_nat @ Xs3 @ I2 ) @ I2 ) ) ) ) ).
% Ex_list_of_length_P
thf(fact_105_Ex__list__of__length__P,axiom,
! [N: nat,P2: a > nat > $o] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ? [X5: a] : ( P2 @ X5 @ I3 ) )
=> ? [Xs3: list_a] :
( ( ( size_size_list_a @ Xs3 )
= N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P2 @ ( nth_a @ Xs3 @ I2 ) @ I2 ) ) ) ) ).
% Ex_list_of_length_P
thf(fact_106_Ex__list__of__length__P,axiom,
! [N: nat,P2: term_a_b > nat > $o] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ? [X5: term_a_b] : ( P2 @ X5 @ I3 ) )
=> ? [Xs3: list_term_a_b] :
( ( ( size_s8906293707977694520rm_a_b @ Xs3 )
= N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P2 @ ( nth_term_a_b @ Xs3 @ I2 ) @ I2 ) ) ) ) ).
% Ex_list_of_length_P
thf(fact_107_subsetI,axiom,
! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
( ! [X: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X @ A4 )
=> ( member8440522571783428010at_nat @ X @ B4 ) )
=> ( ord_le3146513528884898305at_nat @ A4 @ B4 ) ) ).
% subsetI
thf(fact_108_subsetI,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ! [X: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X @ A4 )
=> ( member5724188588386418708_a_nat @ X @ B4 ) )
=> ( ord_le8666007276011122963_a_nat @ A4 @ B4 ) ) ).
% subsetI
thf(fact_109_subsetI,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ! [X: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X @ A4 )
=> ( member5869715511025134514rm_a_b @ X @ B4 ) )
=> ( ord_le118470702582115849rm_a_b @ A4 @ B4 ) ) ).
% subsetI
thf(fact_110_subsetI,axiom,
! [A4: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat] :
( ! [X: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X @ A4 )
=> ( member8206827879206165904at_nat @ X @ B4 ) )
=> ( ord_le3000389064537975527at_nat @ A4 @ B4 ) ) ).
% subsetI
thf(fact_111_subsetI,axiom,
! [A4: set_list_nat,B4: set_list_nat] :
( ! [X: list_nat] :
( ( member_list_nat @ X @ A4 )
=> ( member_list_nat @ X @ B4 ) )
=> ( ord_le6045566169113846134st_nat @ A4 @ B4 ) ) ).
% subsetI
thf(fact_112_subsetI,axiom,
! [A4: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ! [X: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X @ A4 )
=> ( member1426531477525435216od_a_a @ X @ B4 ) )
=> ( ord_le746702958409616551od_a_a @ A4 @ B4 ) ) ).
% subsetI
thf(fact_113_subsetI,axiom,
! [A4: set_Pr3254233319718257578_nat_a,B4: set_Pr3254233319718257578_nat_a] :
( ! [X: produc2842622564922169716_nat_a] :
( ( member7754445317155421323_nat_a @ X @ A4 )
=> ( member7754445317155421323_nat_a @ X @ B4 ) )
=> ( ord_le7986397757451867146_nat_a @ A4 @ B4 ) ) ).
% subsetI
thf(fact_114_subsetI,axiom,
! [A4: set_Pr6841066879934740386at_nat,B4: set_Pr6841066879934740386at_nat] :
( ! [X: produc551447040318622060at_nat] :
( ( member5463269792551873667at_nat @ X @ A4 )
=> ( member5463269792551873667at_nat @ X @ B4 ) )
=> ( ord_le2349859280813574146at_nat @ A4 @ B4 ) ) ).
% subsetI
thf(fact_115_subsetI,axiom,
! [A4: set_Pr3451248702717554689st_nat,B4: set_Pr3451248702717554689st_nat] :
( ! [X: produc1828647624359046049st_nat] :
( ( member7340969449405702474st_nat @ X @ A4 )
=> ( member7340969449405702474st_nat @ X @ B4 ) )
=> ( ord_le8406513867147106209st_nat @ A4 @ B4 ) ) ).
% subsetI
thf(fact_116_subsetI,axiom,
! [A4: set_Pr5046312416420021961st_nat,B4: set_Pr5046312416420021961st_nat] :
( ! [X: produc432399132543013523st_nat] :
( ( member4851138774834033962st_nat @ X @ A4 )
=> ( member4851138774834033962st_nat @ X @ B4 ) )
=> ( ord_le1912520282604406569st_nat @ A4 @ B4 ) ) ).
% subsetI
thf(fact_117_subset__antisym,axiom,
! [A4: set_Pr3451248702717554689st_nat,B4: set_Pr3451248702717554689st_nat] :
( ( ord_le8406513867147106209st_nat @ A4 @ B4 )
=> ( ( ord_le8406513867147106209st_nat @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% subset_antisym
thf(fact_118_subset__antisym,axiom,
! [A4: set_Pr5046312416420021961st_nat,B4: set_Pr5046312416420021961st_nat] :
( ( ord_le1912520282604406569st_nat @ A4 @ B4 )
=> ( ( ord_le1912520282604406569st_nat @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% subset_antisym
thf(fact_119_subset__antisym,axiom,
! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A4 @ B4 )
=> ( ( ord_le3146513528884898305at_nat @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% subset_antisym
thf(fact_120_subset__antisym,axiom,
! [A4: set_Pr9093778441882193744at_nat,B4: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ A4 @ B4 )
=> ( ( ord_le3678578370064672496at_nat @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% subset_antisym
thf(fact_121_subset__antisym,axiom,
! [A4: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A4 @ B4 )
=> ( ( ord_le746702958409616551od_a_a @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% subset_antisym
thf(fact_122_subset__antisym,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A4 @ B4 )
=> ( ( ord_le8666007276011122963_a_nat @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% subset_antisym
thf(fact_123_order__refl,axiom,
! [X3: set_Pr3451248702717554689st_nat] : ( ord_le8406513867147106209st_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_124_order__refl,axiom,
! [X3: set_Pr5046312416420021961st_nat] : ( ord_le1912520282604406569st_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_125_order__refl,axiom,
! [X3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_126_order__refl,axiom,
! [X3: set_Pr9093778441882193744at_nat] : ( ord_le3678578370064672496at_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_127_order__refl,axiom,
! [X3: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ X3 @ X3 ) ).
% order_refl
thf(fact_128_order__refl,axiom,
! [X3: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_129_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_130_dual__order_Orefl,axiom,
! [A: set_Pr3451248702717554689st_nat] : ( ord_le8406513867147106209st_nat @ A @ A ) ).
% dual_order.refl
thf(fact_131_dual__order_Orefl,axiom,
! [A: set_Pr5046312416420021961st_nat] : ( ord_le1912520282604406569st_nat @ A @ A ) ).
% dual_order.refl
thf(fact_132_dual__order_Orefl,axiom,
! [A: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ A @ A ) ).
% dual_order.refl
thf(fact_133_dual__order_Orefl,axiom,
! [A: set_Pr9093778441882193744at_nat] : ( ord_le3678578370064672496at_nat @ A @ A ) ).
% dual_order.refl
thf(fact_134_dual__order_Orefl,axiom,
! [A: set_Product_prod_a_a] : ( ord_le746702958409616551od_a_a @ A @ A ) ).
% dual_order.refl
thf(fact_135_dual__order_Orefl,axiom,
! [A: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A @ A ) ).
% dual_order.refl
thf(fact_136_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_137_subrelI,axiom,
! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ! [X: nat,Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ S ) )
=> ( ord_le3146513528884898305at_nat @ R2 @ S ) ) ).
% subrelI
thf(fact_138_subrelI,axiom,
! [R2: set_Pr4934435412358123699_a_nat,S: set_Pr4934435412358123699_a_nat] :
( ! [X: a,Y3: nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X @ Y3 ) @ R2 )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X @ Y3 ) @ S ) )
=> ( ord_le8666007276011122963_a_nat @ R2 @ S ) ) ).
% subrelI
thf(fact_139_subrelI,axiom,
! [R2: set_Pr9093778441882193744at_nat,S: set_Pr9093778441882193744at_nat] :
( ! [X: nat > nat,Y3: nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X @ Y3 ) @ R2 )
=> ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X @ Y3 ) @ S ) )
=> ( ord_le3678578370064672496at_nat @ R2 @ S ) ) ).
% subrelI
thf(fact_140_subrelI,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ! [X: term_a_b,Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ S ) )
=> ( ord_le118470702582115849rm_a_b @ R2 @ S ) ) ).
% subrelI
thf(fact_141_subrelI,axiom,
! [R2: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
( ! [X: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ R2 )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ S ) )
=> ( ord_le3000389064537975527at_nat @ R2 @ S ) ) ).
% subrelI
thf(fact_142_subrelI,axiom,
! [R2: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ! [X: a,Y3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R2 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ S ) )
=> ( ord_le746702958409616551od_a_a @ R2 @ S ) ) ).
% subrelI
thf(fact_143_subrelI,axiom,
! [R2: set_Pr3254233319718257578_nat_a,S: set_Pr3254233319718257578_nat_a] :
( ! [X: nat > nat,Y3: a] :
( ( member7754445317155421323_nat_a @ ( produc3941937540969104678_nat_a @ X @ Y3 ) @ R2 )
=> ( member7754445317155421323_nat_a @ ( produc3941937540969104678_nat_a @ X @ Y3 ) @ S ) )
=> ( ord_le7986397757451867146_nat_a @ R2 @ S ) ) ).
% subrelI
thf(fact_144_subrelI,axiom,
! [R2: set_Pr1357592988741419896rm_a_b,S: set_Pr1357592988741419896rm_a_b] :
( ! [X: b,Y3: term_a_b] :
( ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ X @ Y3 ) @ R2 )
=> ( member7000133282412810073rm_a_b @ ( produc1437816968797971900rm_a_b @ X @ Y3 ) @ S ) )
=> ( ord_le980006117782565848rm_a_b @ R2 @ S ) ) ).
% subrelI
thf(fact_145_subrelI,axiom,
! [R2: set_Pr6841066879934740386at_nat,S: set_Pr6841066879934740386at_nat] :
( ! [X: a,Y3: nat > nat] :
( ( member5463269792551873667at_nat @ ( produc7514922136058452582at_nat @ X @ Y3 ) @ R2 )
=> ( member5463269792551873667at_nat @ ( produc7514922136058452582at_nat @ X @ Y3 ) @ S ) )
=> ( ord_le2349859280813574146at_nat @ R2 @ S ) ) ).
% subrelI
thf(fact_146_subrelI,axiom,
! [R2: set_Pr3451248702717554689st_nat,S: set_Pr3451248702717554689st_nat] :
( ! [X: list_nat,Y3: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y3 ) @ R2 )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y3 ) @ S ) )
=> ( ord_le8406513867147106209st_nat @ R2 @ S ) ) ).
% subrelI
thf(fact_147_O__assoc,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b,T2: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ ( relcom370159955682700863rm_a_b @ R3 @ S2 ) @ T2 )
= ( relcom370159955682700863rm_a_b @ R3 @ ( relcom370159955682700863rm_a_b @ S2 @ T2 ) ) ) ).
% O_assoc
thf(fact_148_O__assoc,axiom,
! [R3: set_Pr4934435412358123699_a_nat,S2: set_Pr4193341848836149977_nat_a,T2: set_Product_prod_a_a] :
( ( relcomp_a_a_a @ ( relcomp_a_nat_a @ R3 @ S2 ) @ T2 )
= ( relcomp_a_nat_a @ R3 @ ( relcomp_nat_a_a @ S2 @ T2 ) ) ) ).
% O_assoc
thf(fact_149_O__assoc,axiom,
! [R3: set_Pr4934435412358123699_a_nat,S2: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] :
( ( relcomp_a_nat_nat @ ( relcomp_a_nat_nat @ R3 @ S2 ) @ T2 )
= ( relcomp_a_nat_nat @ R3 @ ( relcomp_nat_nat_nat @ S2 @ T2 ) ) ) ).
% O_assoc
thf(fact_150_O__assoc,axiom,
! [R3: set_Product_prod_a_a,S2: set_Pr4934435412358123699_a_nat,T2: set_Pr1261947904930325089at_nat] :
( ( relcomp_a_nat_nat @ ( relcomp_a_a_nat @ R3 @ S2 ) @ T2 )
= ( relcomp_a_a_nat @ R3 @ ( relcomp_a_nat_nat @ S2 @ T2 ) ) ) ).
% O_assoc
thf(fact_151_O__assoc,axiom,
! [R3: set_Pr4934435412358123699_a_nat,S2: set_Pr1261947904930325089at_nat,T2: set_Pr4193341848836149977_nat_a] :
( ( relcomp_a_nat_a @ ( relcomp_a_nat_nat @ R3 @ S2 ) @ T2 )
= ( relcomp_a_nat_a @ R3 @ ( relcomp_nat_nat_a @ S2 @ T2 ) ) ) ).
% O_assoc
thf(fact_152_O__assoc,axiom,
! [R3: set_Product_prod_a_a,S2: set_Pr4934435412358123699_a_nat,T2: set_Pr4193341848836149977_nat_a] :
( ( relcomp_a_nat_a @ ( relcomp_a_a_nat @ R3 @ S2 ) @ T2 )
= ( relcomp_a_a_a @ R3 @ ( relcomp_a_nat_a @ S2 @ T2 ) ) ) ).
% O_assoc
thf(fact_153_O__assoc,axiom,
! [R3: set_Product_prod_a_a,S2: set_Product_prod_a_a,T2: set_Pr4934435412358123699_a_nat] :
( ( relcomp_a_a_nat @ ( relcomp_a_a_a @ R3 @ S2 ) @ T2 )
= ( relcomp_a_a_nat @ R3 @ ( relcomp_a_a_nat @ S2 @ T2 ) ) ) ).
% O_assoc
thf(fact_154_O__assoc,axiom,
! [R3: set_Pr4934435412358123699_a_nat,S2: set_Pr4193341848836149977_nat_a,T2: set_Pr4934435412358123699_a_nat] :
( ( relcomp_a_a_nat @ ( relcomp_a_nat_a @ R3 @ S2 ) @ T2 )
= ( relcomp_a_nat_nat @ R3 @ ( relcomp_nat_a_nat @ S2 @ T2 ) ) ) ).
% O_assoc
thf(fact_155_O__assoc,axiom,
! [R3: set_Pr3254233319718257578_nat_a,S2: set_Pr4934435412358123699_a_nat,T2: set_Pr4193341848836149977_nat_a] :
( ( relcom5195258687133253356_nat_a @ ( relcom1163327185954966926_a_nat @ R3 @ S2 ) @ T2 )
= ( relcomp_nat_nat_a_a @ R3 @ ( relcomp_a_nat_a @ S2 @ T2 ) ) ) ).
% O_assoc
thf(fact_156_O__assoc,axiom,
! [R3: set_Pr3254233319718257578_nat_a,S2: set_Pr4934435412358123699_a_nat,T2: set_Pr1261947904930325089at_nat] :
( ( relcom1859588284133128290at_nat @ ( relcom1163327185954966926_a_nat @ R3 @ S2 ) @ T2 )
= ( relcom1163327185954966926_a_nat @ R3 @ ( relcomp_a_nat_nat @ S2 @ T2 ) ) ) ).
% O_assoc
thf(fact_157_mem__Collect__eq,axiom,
! [A: produc1932156733058919263at_nat,P2: produc1932156733058919263at_nat > $o] :
( ( member1096249278104964872at_nat @ A @ ( collec2757435396427831370at_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_158_mem__Collect__eq,axiom,
! [A: produc2842622564922169716_nat_a,P2: produc2842622564922169716_nat_a > $o] :
( ( member7754445317155421323_nat_a @ A @ ( collec3024486194029540553_nat_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_159_mem__Collect__eq,axiom,
! [A: produc551447040318622060at_nat,P2: produc551447040318622060at_nat > $o] :
( ( member5463269792551873667at_nat @ A @ ( collec733310669425992897at_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_160_mem__Collect__eq,axiom,
! [A: product_prod_a_a,P2: product_prod_a_a > $o] :
( ( member1426531477525435216od_a_a @ A @ ( collec3336397797384452498od_a_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_161_mem__Collect__eq,axiom,
! [A: list_nat,P2: list_nat > $o] :
( ( member_list_nat @ A @ ( collect_list_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_162_mem__Collect__eq,axiom,
! [A: produc357393685978478089rm_a_b,P2: produc357393685978478089rm_a_b > $o] :
( ( member5869715511025134514rm_a_b @ A @ ( collec99177395925924084rm_a_b @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_163_mem__Collect__eq,axiom,
! [A: product_prod_a_nat,P2: product_prod_a_nat > $o] :
( ( member5724188588386418708_a_nat @ A @ ( collec4464134535221767506_a_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_164_mem__Collect__eq,axiom,
! [A: produc859450856879609959at_nat,P2: produc859450856879609959at_nat > $o] :
( ( member8206827879206165904at_nat @ A @ ( collec7088162979684241874at_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_165_mem__Collect__eq,axiom,
! [A: product_prod_nat_nat,P2: product_prod_nat_nat > $o] :
( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_166_Collect__mem__eq,axiom,
! [A4: set_Pr7682762132356531903at_nat] :
( ( collec2757435396427831370at_nat
@ ^ [X6: produc1932156733058919263at_nat] : ( member1096249278104964872at_nat @ X6 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_167_Collect__mem__eq,axiom,
! [A4: set_Pr3254233319718257578_nat_a] :
( ( collec3024486194029540553_nat_a
@ ^ [X6: produc2842622564922169716_nat_a] : ( member7754445317155421323_nat_a @ X6 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_168_Collect__mem__eq,axiom,
! [A4: set_Pr6841066879934740386at_nat] :
( ( collec733310669425992897at_nat
@ ^ [X6: produc551447040318622060at_nat] : ( member5463269792551873667at_nat @ X6 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_169_Collect__mem__eq,axiom,
! [A4: set_Product_prod_a_a] :
( ( collec3336397797384452498od_a_a
@ ^ [X6: product_prod_a_a] : ( member1426531477525435216od_a_a @ X6 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_170_Collect__mem__eq,axiom,
! [A4: set_list_nat] :
( ( collect_list_nat
@ ^ [X6: list_nat] : ( member_list_nat @ X6 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_171_Collect__mem__eq,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ( collec99177395925924084rm_a_b
@ ^ [X6: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X6 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_172_Collect__mem__eq,axiom,
! [A4: set_Pr4934435412358123699_a_nat] :
( ( collec4464134535221767506_a_nat
@ ^ [X6: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X6 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_173_Collect__mem__eq,axiom,
! [A4: set_Pr8693737435421807431at_nat] :
( ( collec7088162979684241874at_nat
@ ^ [X6: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X6 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_174_Collect__mem__eq,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ( collec3392354462482085612at_nat
@ ^ [X6: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X6 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_175_Collect__cong,axiom,
! [P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
( ! [X: product_prod_nat_nat] :
( ( P2 @ X )
= ( Q @ X ) )
=> ( ( collec3392354462482085612at_nat @ P2 )
= ( collec3392354462482085612at_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_176_Collect__cong,axiom,
! [P2: produc859450856879609959at_nat > $o,Q: produc859450856879609959at_nat > $o] :
( ! [X: produc859450856879609959at_nat] :
( ( P2 @ X )
= ( Q @ X ) )
=> ( ( collec7088162979684241874at_nat @ P2 )
= ( collec7088162979684241874at_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_177_Collect__cong,axiom,
! [P2: product_prod_a_nat > $o,Q: product_prod_a_nat > $o] :
( ! [X: product_prod_a_nat] :
( ( P2 @ X )
= ( Q @ X ) )
=> ( ( collec4464134535221767506_a_nat @ P2 )
= ( collec4464134535221767506_a_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_178_Collect__cong,axiom,
! [P2: produc357393685978478089rm_a_b > $o,Q: produc357393685978478089rm_a_b > $o] :
( ! [X: produc357393685978478089rm_a_b] :
( ( P2 @ X )
= ( Q @ X ) )
=> ( ( collec99177395925924084rm_a_b @ P2 )
= ( collec99177395925924084rm_a_b @ Q ) ) ) ).
% Collect_cong
thf(fact_179_relcomp_Ocases,axiom,
! [A1: a,A22: nat,R2: set_Product_prod_a_a,S: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ A22 ) @ ( relcomp_a_a_nat @ R2 @ S ) )
=> ~ ! [B3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A1 @ B3 ) @ R2 )
=> ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B3 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_180_relcomp_Ocases,axiom,
! [A1: a,A22: nat,R2: set_Pr4934435412358123699_a_nat,S: set_Pr1261947904930325089at_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ A22 ) @ ( relcomp_a_nat_nat @ R2 @ S ) )
=> ~ ! [B3: nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ B3 ) @ R2 )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_181_relcomp_Ocases,axiom,
! [A1: nat,A22: nat,R2: set_Pr4193341848836149977_nat_a,S: set_Pr4934435412358123699_a_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( relcomp_nat_a_nat @ R2 @ S ) )
=> ~ ! [B3: a] :
( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ A1 @ B3 ) @ R2 )
=> ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B3 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_182_relcomp_Ocases,axiom,
! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( relcomp_nat_nat_nat @ R2 @ S ) )
=> ~ ! [B3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ B3 ) @ R2 )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_183_relcomp_Ocases,axiom,
! [A1: a,A22: nat,R2: set_Pr6841066879934740386at_nat,S: set_Pr9093778441882193744at_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ A22 ) @ ( relcom8552328690651882190at_nat @ R2 @ S ) )
=> ~ ! [B3: nat > nat] :
( ( member5463269792551873667at_nat @ ( produc7514922136058452582at_nat @ A1 @ B3 ) @ R2 )
=> ~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B3 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_184_relcomp_Ocases,axiom,
! [A1: nat > nat,A22: nat,R2: set_Pr3254233319718257578_nat_a,S: set_Pr4934435412358123699_a_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A1 @ A22 ) @ ( relcom1163327185954966926_a_nat @ R2 @ S ) )
=> ~ ! [B3: a] :
( ( member7754445317155421323_nat_a @ ( produc3941937540969104678_nat_a @ A1 @ B3 ) @ R2 )
=> ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B3 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_185_relcomp_Ocases,axiom,
! [A1: nat > nat,A22: nat,R2: set_Pr9093778441882193744at_nat,S: set_Pr1261947904930325089at_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A1 @ A22 ) @ ( relcom1859588284133128290at_nat @ R2 @ S ) )
=> ~ ! [B3: nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A1 @ B3 ) @ R2 )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_186_relcomp_Ocases,axiom,
! [A1: nat,A22: nat,R2: set_Pr6370437063884598352at_nat,S: set_Pr9093778441882193744at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( relcom6405311860805859042at_nat @ R2 @ S ) )
=> ~ ! [B3: nat > nat] :
( ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ A1 @ B3 ) @ R2 )
=> ~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B3 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_187_relcomp_Ocases,axiom,
! [A1: nat > nat,A22: nat,R2: set_Pr7682762132356531903at_nat,S: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A1 @ A22 ) @ ( relcom9054258243621454929at_nat @ R2 @ S ) )
=> ~ ! [B3: nat > nat] :
( ( member1096249278104964872at_nat @ ( produc5770335208449155351at_nat @ A1 @ B3 ) @ R2 )
=> ~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B3 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_188_relcomp_Ocases,axiom,
! [A1: term_a_b,A22: term_a_b,R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( relcom370159955682700863rm_a_b @ R2 @ S ) )
=> ~ ! [B3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ B3 ) @ R2 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_189_relcomp_Osimps,axiom,
! [A1: a,A22: nat,R2: set_Product_prod_a_a,S: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ A22 ) @ ( relcomp_a_a_nat @ R2 @ S ) )
= ( ? [A5: a,B5: a,C2: nat] :
( ( A1 = A5 )
& ( A22 = C2 )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A5 @ B5 ) @ R2 )
& ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B5 @ C2 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_190_relcomp_Osimps,axiom,
! [A1: a,A22: nat,R2: set_Pr4934435412358123699_a_nat,S: set_Pr1261947904930325089at_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ A22 ) @ ( relcomp_a_nat_nat @ R2 @ S ) )
= ( ? [A5: a,B5: nat,C2: nat] :
( ( A1 = A5 )
& ( A22 = C2 )
& ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A5 @ B5 ) @ R2 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B5 @ C2 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_191_relcomp_Osimps,axiom,
! [A1: nat,A22: nat,R2: set_Pr4193341848836149977_nat_a,S: set_Pr4934435412358123699_a_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( relcomp_nat_a_nat @ R2 @ S ) )
= ( ? [A5: nat,B5: a,C2: nat] :
( ( A1 = A5 )
& ( A22 = C2 )
& ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ A5 @ B5 ) @ R2 )
& ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B5 @ C2 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_192_relcomp_Osimps,axiom,
! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( relcomp_nat_nat_nat @ R2 @ S ) )
= ( ? [A5: nat,B5: nat,C2: nat] :
( ( A1 = A5 )
& ( A22 = C2 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ B5 ) @ R2 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B5 @ C2 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_193_relcomp_Osimps,axiom,
! [A1: a,A22: nat,R2: set_Pr6841066879934740386at_nat,S: set_Pr9093778441882193744at_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A1 @ A22 ) @ ( relcom8552328690651882190at_nat @ R2 @ S ) )
= ( ? [A5: a,B5: nat > nat,C2: nat] :
( ( A1 = A5 )
& ( A22 = C2 )
& ( member5463269792551873667at_nat @ ( produc7514922136058452582at_nat @ A5 @ B5 ) @ R2 )
& ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B5 @ C2 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_194_relcomp_Osimps,axiom,
! [A1: nat > nat,A22: nat,R2: set_Pr3254233319718257578_nat_a,S: set_Pr4934435412358123699_a_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A1 @ A22 ) @ ( relcom1163327185954966926_a_nat @ R2 @ S ) )
= ( ? [A5: nat > nat,B5: a,C2: nat] :
( ( A1 = A5 )
& ( A22 = C2 )
& ( member7754445317155421323_nat_a @ ( produc3941937540969104678_nat_a @ A5 @ B5 ) @ R2 )
& ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B5 @ C2 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_195_relcomp_Osimps,axiom,
! [A1: nat > nat,A22: nat,R2: set_Pr9093778441882193744at_nat,S: set_Pr1261947904930325089at_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A1 @ A22 ) @ ( relcom1859588284133128290at_nat @ R2 @ S ) )
= ( ? [A5: nat > nat,B5: nat,C2: nat] :
( ( A1 = A5 )
& ( A22 = C2 )
& ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A5 @ B5 ) @ R2 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B5 @ C2 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_196_relcomp_Osimps,axiom,
! [A1: nat,A22: nat,R2: set_Pr6370437063884598352at_nat,S: set_Pr9093778441882193744at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( relcom6405311860805859042at_nat @ R2 @ S ) )
= ( ? [A5: nat,B5: nat > nat,C2: nat] :
( ( A1 = A5 )
& ( A22 = C2 )
& ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ A5 @ B5 ) @ R2 )
& ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B5 @ C2 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_197_relcomp_Osimps,axiom,
! [A1: nat > nat,A22: nat,R2: set_Pr7682762132356531903at_nat,S: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A1 @ A22 ) @ ( relcom9054258243621454929at_nat @ R2 @ S ) )
= ( ? [A5: nat > nat,B5: nat > nat,C2: nat] :
( ( A1 = A5 )
& ( A22 = C2 )
& ( member1096249278104964872at_nat @ ( produc5770335208449155351at_nat @ A5 @ B5 ) @ R2 )
& ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B5 @ C2 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_198_relcomp_Osimps,axiom,
! [A1: term_a_b,A22: term_a_b,R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( relcom370159955682700863rm_a_b @ R2 @ S ) )
= ( ? [A5: term_a_b,B5: term_a_b,C2: term_a_b] :
( ( A1 = A5 )
& ( A22 = C2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B5 ) @ R2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B5 @ C2 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_199_relcomp_OrelcompI,axiom,
! [A: a,B: a,R2: set_Product_prod_a_a,C3: nat,S: set_Pr4934435412358123699_a_nat] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ R2 )
=> ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B @ C3 ) @ S )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A @ C3 ) @ ( relcomp_a_a_nat @ R2 @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_200_relcomp_OrelcompI,axiom,
! [A: nat,B: a,R2: set_Pr4193341848836149977_nat_a,C3: nat,S: set_Pr4934435412358123699_a_nat] :
( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ A @ B ) @ R2 )
=> ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B @ C3 ) @ S )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C3 ) @ ( relcomp_nat_a_nat @ R2 @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_201_relcomp_OrelcompI,axiom,
! [A: a,B: nat,R2: set_Pr4934435412358123699_a_nat,C3: nat,S: set_Pr1261947904930325089at_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A @ B ) @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C3 ) @ S )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A @ C3 ) @ ( relcomp_a_nat_nat @ R2 @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_202_relcomp_OrelcompI,axiom,
! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C3: nat,S: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C3 ) @ S )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C3 ) @ ( relcomp_nat_nat_nat @ R2 @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_203_relcomp_OrelcompI,axiom,
! [A: nat > nat,B: a,R2: set_Pr3254233319718257578_nat_a,C3: nat,S: set_Pr4934435412358123699_a_nat] :
( ( member7754445317155421323_nat_a @ ( produc3941937540969104678_nat_a @ A @ B ) @ R2 )
=> ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B @ C3 ) @ S )
=> ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A @ C3 ) @ ( relcom1163327185954966926_a_nat @ R2 @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_204_relcomp_OrelcompI,axiom,
! [A: a,B: nat > nat,R2: set_Pr6841066879934740386at_nat,C3: nat,S: set_Pr9093778441882193744at_nat] :
( ( member5463269792551873667at_nat @ ( produc7514922136058452582at_nat @ A @ B ) @ R2 )
=> ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B @ C3 ) @ S )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A @ C3 ) @ ( relcom8552328690651882190at_nat @ R2 @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_205_relcomp_OrelcompI,axiom,
! [A: nat,B: nat > nat,R2: set_Pr6370437063884598352at_nat,C3: nat,S: set_Pr9093778441882193744at_nat] :
( ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ A @ B ) @ R2 )
=> ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B @ C3 ) @ S )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C3 ) @ ( relcom6405311860805859042at_nat @ R2 @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_206_relcomp_OrelcompI,axiom,
! [A: nat > nat,B: nat,R2: set_Pr9093778441882193744at_nat,C3: nat,S: set_Pr1261947904930325089at_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A @ B ) @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C3 ) @ S )
=> ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A @ C3 ) @ ( relcom1859588284133128290at_nat @ R2 @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_207_relcomp_OrelcompI,axiom,
! [A: nat > nat,B: nat > nat,R2: set_Pr7682762132356531903at_nat,C3: nat,S: set_Pr9093778441882193744at_nat] :
( ( member1096249278104964872at_nat @ ( produc5770335208449155351at_nat @ A @ B ) @ R2 )
=> ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B @ C3 ) @ S )
=> ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A @ C3 ) @ ( relcom9054258243621454929at_nat @ R2 @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_208_relcomp_OrelcompI,axiom,
! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,C3: term_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C3 ) @ S )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C3 ) @ ( relcom370159955682700863rm_a_b @ R2 @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_209_relcompE,axiom,
! [Xz: product_prod_a_nat,R2: set_Product_prod_a_a,S: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ Xz @ ( relcomp_a_a_nat @ R2 @ S ) )
=> ~ ! [X: a,Y3: a,Z2: nat] :
( ( Xz
= ( product_Pair_a_nat @ X @ Z2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y3 ) @ R2 )
=> ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ Y3 @ Z2 ) @ S ) ) ) ) ).
% relcompE
thf(fact_210_relcompE,axiom,
! [Xz: product_prod_a_nat,R2: set_Pr4934435412358123699_a_nat,S: set_Pr1261947904930325089at_nat] :
( ( member5724188588386418708_a_nat @ Xz @ ( relcomp_a_nat_nat @ R2 @ S ) )
=> ~ ! [X: a,Y3: nat,Z2: nat] :
( ( Xz
= ( product_Pair_a_nat @ X @ Z2 ) )
=> ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X @ Y3 ) @ R2 )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z2 ) @ S ) ) ) ) ).
% relcompE
thf(fact_211_relcompE,axiom,
! [Xz: product_prod_nat_nat,R2: set_Pr4193341848836149977_nat_a,S: set_Pr4934435412358123699_a_nat] :
( ( member8440522571783428010at_nat @ Xz @ ( relcomp_nat_a_nat @ R2 @ S ) )
=> ~ ! [X: nat,Y3: a,Z2: nat] :
( ( Xz
= ( product_Pair_nat_nat @ X @ Z2 ) )
=> ( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ X @ Y3 ) @ R2 )
=> ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ Y3 @ Z2 ) @ S ) ) ) ) ).
% relcompE
thf(fact_212_relcompE,axiom,
! [Xz: product_prod_nat_nat,R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ Xz @ ( relcomp_nat_nat_nat @ R2 @ S ) )
=> ~ ! [X: nat,Y3: nat,Z2: nat] :
( ( Xz
= ( product_Pair_nat_nat @ X @ Z2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R2 )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z2 ) @ S ) ) ) ) ).
% relcompE
thf(fact_213_relcompE,axiom,
! [Xz: product_prod_a_nat,R2: set_Pr6841066879934740386at_nat,S: set_Pr9093778441882193744at_nat] :
( ( member5724188588386418708_a_nat @ Xz @ ( relcom8552328690651882190at_nat @ R2 @ S ) )
=> ~ ! [X: a,Y3: nat > nat,Z2: nat] :
( ( Xz
= ( product_Pair_a_nat @ X @ Z2 ) )
=> ( ( member5463269792551873667at_nat @ ( produc7514922136058452582at_nat @ X @ Y3 ) @ R2 )
=> ~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ Y3 @ Z2 ) @ S ) ) ) ) ).
% relcompE
thf(fact_214_relcompE,axiom,
! [Xz: produc8199716216217303280at_nat,R2: set_Pr3254233319718257578_nat_a,S: set_Pr4934435412358123699_a_nat] :
( ( member7226740684066999833at_nat @ Xz @ ( relcom1163327185954966926_a_nat @ R2 @ S ) )
=> ~ ! [X: nat > nat,Y3: a,Z2: nat] :
( ( Xz
= ( produc72220940542539688at_nat @ X @ Z2 ) )
=> ( ( member7754445317155421323_nat_a @ ( produc3941937540969104678_nat_a @ X @ Y3 ) @ R2 )
=> ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ Y3 @ Z2 ) @ S ) ) ) ) ).
% relcompE
thf(fact_215_relcompE,axiom,
! [Xz: produc8199716216217303280at_nat,R2: set_Pr9093778441882193744at_nat,S: set_Pr1261947904930325089at_nat] :
( ( member7226740684066999833at_nat @ Xz @ ( relcom1859588284133128290at_nat @ R2 @ S ) )
=> ~ ! [X: nat > nat,Y3: nat,Z2: nat] :
( ( Xz
= ( produc72220940542539688at_nat @ X @ Z2 ) )
=> ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X @ Y3 ) @ R2 )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z2 ) @ S ) ) ) ) ).
% relcompE
thf(fact_216_relcompE,axiom,
! [Xz: product_prod_nat_nat,R2: set_Pr6370437063884598352at_nat,S: set_Pr9093778441882193744at_nat] :
( ( member8440522571783428010at_nat @ Xz @ ( relcom6405311860805859042at_nat @ R2 @ S ) )
=> ~ ! [X: nat,Y3: nat > nat,Z2: nat] :
( ( Xz
= ( product_Pair_nat_nat @ X @ Z2 ) )
=> ( ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ X @ Y3 ) @ R2 )
=> ~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ Y3 @ Z2 ) @ S ) ) ) ) ).
% relcompE
thf(fact_217_relcompE,axiom,
! [Xz: produc8199716216217303280at_nat,R2: set_Pr7682762132356531903at_nat,S: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ Xz @ ( relcom9054258243621454929at_nat @ R2 @ S ) )
=> ~ ! [X: nat > nat,Y3: nat > nat,Z2: nat] :
( ( Xz
= ( produc72220940542539688at_nat @ X @ Z2 ) )
=> ( ( member1096249278104964872at_nat @ ( produc5770335208449155351at_nat @ X @ Y3 ) @ R2 )
=> ~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ Y3 @ Z2 ) @ S ) ) ) ) ).
% relcompE
thf(fact_218_relcompE,axiom,
! [Xz: produc357393685978478089rm_a_b,R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ Xz @ ( relcom370159955682700863rm_a_b @ R2 @ S ) )
=> ~ ! [X: term_a_b,Y3: term_a_b,Z2: term_a_b] :
( ( Xz
= ( produc7020197800436672577rm_a_b @ X @ Z2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ R2 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ S ) ) ) ) ).
% relcompE
thf(fact_219_relcompEpair,axiom,
! [A: a,C3: nat,R2: set_Product_prod_a_a,S: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A @ C3 ) @ ( relcomp_a_a_nat @ R2 @ S ) )
=> ~ ! [B3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B3 ) @ R2 )
=> ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B3 @ C3 ) @ S ) ) ) ).
% relcompEpair
thf(fact_220_relcompEpair,axiom,
! [A: a,C3: nat,R2: set_Pr4934435412358123699_a_nat,S: set_Pr1261947904930325089at_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A @ C3 ) @ ( relcomp_a_nat_nat @ R2 @ S ) )
=> ~ ! [B3: nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A @ B3 ) @ R2 )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ C3 ) @ S ) ) ) ).
% relcompEpair
thf(fact_221_relcompEpair,axiom,
! [A: nat,C3: nat,R2: set_Pr4193341848836149977_nat_a,S: set_Pr4934435412358123699_a_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C3 ) @ ( relcomp_nat_a_nat @ R2 @ S ) )
=> ~ ! [B3: a] :
( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ A @ B3 ) @ R2 )
=> ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B3 @ C3 ) @ S ) ) ) ).
% relcompEpair
thf(fact_222_relcompEpair,axiom,
! [A: nat,C3: nat,R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C3 ) @ ( relcomp_nat_nat_nat @ R2 @ S ) )
=> ~ ! [B3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B3 ) @ R2 )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ C3 ) @ S ) ) ) ).
% relcompEpair
thf(fact_223_relcompEpair,axiom,
! [A: a,C3: nat,R2: set_Pr6841066879934740386at_nat,S: set_Pr9093778441882193744at_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ A @ C3 ) @ ( relcom8552328690651882190at_nat @ R2 @ S ) )
=> ~ ! [B3: nat > nat] :
( ( member5463269792551873667at_nat @ ( produc7514922136058452582at_nat @ A @ B3 ) @ R2 )
=> ~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B3 @ C3 ) @ S ) ) ) ).
% relcompEpair
thf(fact_224_relcompEpair,axiom,
! [A: nat > nat,C3: nat,R2: set_Pr3254233319718257578_nat_a,S: set_Pr4934435412358123699_a_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A @ C3 ) @ ( relcom1163327185954966926_a_nat @ R2 @ S ) )
=> ~ ! [B3: a] :
( ( member7754445317155421323_nat_a @ ( produc3941937540969104678_nat_a @ A @ B3 ) @ R2 )
=> ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ B3 @ C3 ) @ S ) ) ) ).
% relcompEpair
thf(fact_225_relcompEpair,axiom,
! [A: nat > nat,C3: nat,R2: set_Pr9093778441882193744at_nat,S: set_Pr1261947904930325089at_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A @ C3 ) @ ( relcom1859588284133128290at_nat @ R2 @ S ) )
=> ~ ! [B3: nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A @ B3 ) @ R2 )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ C3 ) @ S ) ) ) ).
% relcompEpair
thf(fact_226_relcompEpair,axiom,
! [A: nat,C3: nat,R2: set_Pr6370437063884598352at_nat,S: set_Pr9093778441882193744at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C3 ) @ ( relcom6405311860805859042at_nat @ R2 @ S ) )
=> ~ ! [B3: nat > nat] :
( ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ A @ B3 ) @ R2 )
=> ~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B3 @ C3 ) @ S ) ) ) ).
% relcompEpair
thf(fact_227_relcompEpair,axiom,
! [A: nat > nat,C3: nat,R2: set_Pr7682762132356531903at_nat,S: set_Pr9093778441882193744at_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A @ C3 ) @ ( relcom9054258243621454929at_nat @ R2 @ S ) )
=> ~ ! [B3: nat > nat] :
( ( member1096249278104964872at_nat @ ( produc5770335208449155351at_nat @ A @ B3 ) @ R2 )
=> ~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ B3 @ C3 ) @ S ) ) ) ).
% relcompEpair
thf(fact_228_relcompEpair,axiom,
! [A: term_a_b,C3: term_a_b,R2: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C3 ) @ ( relcom370159955682700863rm_a_b @ R2 @ S ) )
=> ~ ! [B3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B3 ) @ R2 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ C3 ) @ S ) ) ) ).
% relcompEpair
thf(fact_229_relcomp__mono,axiom,
! [R4: set_Product_prod_a_a,R2: set_Product_prod_a_a,S3: set_Pr4934435412358123699_a_nat,S: set_Pr4934435412358123699_a_nat] :
( ( ord_le746702958409616551od_a_a @ R4 @ R2 )
=> ( ( ord_le8666007276011122963_a_nat @ S3 @ S )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ R4 @ S3 ) @ ( relcomp_a_a_nat @ R2 @ S ) ) ) ) ).
% relcomp_mono
thf(fact_230_relcomp__mono,axiom,
! [R4: set_Pr4934435412358123699_a_nat,R2: set_Pr4934435412358123699_a_nat,S3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( ord_le8666007276011122963_a_nat @ R4 @ R2 )
=> ( ( ord_le3146513528884898305at_nat @ S3 @ S )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_nat_nat @ R4 @ S3 ) @ ( relcomp_a_nat_nat @ R2 @ S ) ) ) ) ).
% relcomp_mono
thf(fact_231_relcomp__mono,axiom,
! [R4: set_Pr4386577575007340137rm_a_b,R2: set_Pr4386577575007340137rm_a_b,S3: set_Pr4386577575007340137rm_a_b,S: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R4 @ R2 )
=> ( ( ord_le118470702582115849rm_a_b @ S3 @ S )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R4 @ S3 ) @ ( relcom370159955682700863rm_a_b @ R2 @ S ) ) ) ) ).
% relcomp_mono
thf(fact_232_relcomp__mono,axiom,
! [R4: set_Pr4193341848836149977_nat_a,R2: set_Pr4193341848836149977_nat_a,S3: set_Pr4934435412358123699_a_nat,S: set_Pr4934435412358123699_a_nat] :
( ( ord_le7924913712489149241_nat_a @ R4 @ R2 )
=> ( ( ord_le8666007276011122963_a_nat @ S3 @ S )
=> ( ord_le3146513528884898305at_nat @ ( relcomp_nat_a_nat @ R4 @ S3 ) @ ( relcomp_nat_a_nat @ R2 @ S ) ) ) ) ).
% relcomp_mono
thf(fact_233_relcomp__mono,axiom,
! [R4: set_Pr4934435412358123699_a_nat,R2: set_Pr4934435412358123699_a_nat,S3: set_Pr4193341848836149977_nat_a,S: set_Pr4193341848836149977_nat_a] :
( ( ord_le8666007276011122963_a_nat @ R4 @ R2 )
=> ( ( ord_le7924913712489149241_nat_a @ S3 @ S )
=> ( ord_le746702958409616551od_a_a @ ( relcomp_a_nat_a @ R4 @ S3 ) @ ( relcomp_a_nat_a @ R2 @ S ) ) ) ) ).
% relcomp_mono
thf(fact_234_relcomp__mono,axiom,
! [R4: set_Pr1261947904930325089at_nat,R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ R4 @ R2 )
=> ( ( ord_le3146513528884898305at_nat @ S3 @ S )
=> ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ R4 @ S3 ) @ ( relcomp_nat_nat_nat @ R2 @ S ) ) ) ) ).
% relcomp_mono
thf(fact_235_relcomp__mono,axiom,
! [R4: set_Product_prod_a_a,R2: set_Product_prod_a_a,S3: set_Product_prod_a_a,S: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ R4 @ R2 )
=> ( ( ord_le746702958409616551od_a_a @ S3 @ S )
=> ( ord_le746702958409616551od_a_a @ ( relcomp_a_a_a @ R4 @ S3 ) @ ( relcomp_a_a_a @ R2 @ S ) ) ) ) ).
% relcomp_mono
thf(fact_236_relcomp__mono,axiom,
! [R4: set_Pr3254233319718257578_nat_a,R2: set_Pr3254233319718257578_nat_a,S3: set_Pr4934435412358123699_a_nat,S: set_Pr4934435412358123699_a_nat] :
( ( ord_le7986397757451867146_nat_a @ R4 @ R2 )
=> ( ( ord_le8666007276011122963_a_nat @ S3 @ S )
=> ( ord_le3678578370064672496at_nat @ ( relcom1163327185954966926_a_nat @ R4 @ S3 ) @ ( relcom1163327185954966926_a_nat @ R2 @ S ) ) ) ) ).
% relcomp_mono
thf(fact_237_relcomp__mono,axiom,
! [R4: set_Pr6841066879934740386at_nat,R2: set_Pr6841066879934740386at_nat,S3: set_Pr9093778441882193744at_nat,S: set_Pr9093778441882193744at_nat] :
( ( ord_le2349859280813574146at_nat @ R4 @ R2 )
=> ( ( ord_le3678578370064672496at_nat @ S3 @ S )
=> ( ord_le8666007276011122963_a_nat @ ( relcom8552328690651882190at_nat @ R4 @ S3 ) @ ( relcom8552328690651882190at_nat @ R2 @ S ) ) ) ) ).
% relcomp_mono
thf(fact_238_relcomp__mono,axiom,
! [R4: set_Pr6370437063884598352at_nat,R2: set_Pr6370437063884598352at_nat,S3: set_Pr9093778441882193744at_nat,S: set_Pr9093778441882193744at_nat] :
( ( ord_le955236992067077104at_nat @ R4 @ R2 )
=> ( ( ord_le3678578370064672496at_nat @ S3 @ S )
=> ( ord_le3146513528884898305at_nat @ ( relcom6405311860805859042at_nat @ R4 @ S3 ) @ ( relcom6405311860805859042at_nat @ R2 @ S ) ) ) ) ).
% relcomp_mono
thf(fact_239_order__antisym__conv,axiom,
! [Y: set_Pr5046312416420021961st_nat,X3: set_Pr5046312416420021961st_nat] :
( ( ord_le1912520282604406569st_nat @ Y @ X3 )
=> ( ( ord_le1912520282604406569st_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_240_order__antisym__conv,axiom,
! [Y: set_Pr1261947904930325089at_nat,X3: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ Y @ X3 )
=> ( ( ord_le3146513528884898305at_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_241_order__antisym__conv,axiom,
! [Y: set_Pr9093778441882193744at_nat,X3: set_Pr9093778441882193744at_nat] :
( ( ord_le3678578370064672496at_nat @ Y @ X3 )
=> ( ( ord_le3678578370064672496at_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_242_order__antisym__conv,axiom,
! [Y: set_Product_prod_a_a,X3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ Y @ X3 )
=> ( ( ord_le746702958409616551od_a_a @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_243_order__antisym__conv,axiom,
! [Y: set_Pr4934435412358123699_a_nat,X3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ Y @ X3 )
=> ( ( ord_le8666007276011122963_a_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_244_order__antisym__conv,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( ord_less_eq_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_245_linorder__le__cases,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_le_cases
thf(fact_246_ord__le__eq__subst,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,F3: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ( F3 @ B )
= C3 )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ X ) @ ( F3 @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ A ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_247_ord__le__eq__subst,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,F3: set_Pr4934435412358123699_a_nat > nat,C3: nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ( F3 @ B )
= C3 )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_248_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F3: nat > set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F3 @ B )
= C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ X ) @ ( F3 @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ A ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_249_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F3: nat > nat,C3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F3 @ B )
= C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A ) @ C3 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_250_ord__eq__le__subst,axiom,
! [A: set_Pr4934435412358123699_a_nat,F3: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
( ( A
= ( F3 @ B ) )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C3 )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ X ) @ ( F3 @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ A @ ( F3 @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_251_ord__eq__le__subst,axiom,
! [A: nat,F3: set_Pr4934435412358123699_a_nat > nat,B: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
( ( A
= ( F3 @ B ) )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C3 )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F3 @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_252_ord__eq__le__subst,axiom,
! [A: set_Pr4934435412358123699_a_nat,F3: nat > set_Pr4934435412358123699_a_nat,B: nat,C3: nat] :
( ( A
= ( F3 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ X ) @ ( F3 @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ A @ ( F3 @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_253_ord__eq__le__subst,axiom,
! [A: nat,F3: nat > nat,B: nat,C3: nat] :
( ( A
= ( F3 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F3 @ C3 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_254_linorder__linear,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
| ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_linear
thf(fact_255_order__eq__refl,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( X3 = Y )
=> ( ord_le8666007276011122963_a_nat @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_256_order__eq__refl,axiom,
! [X3: nat,Y: nat] :
( ( X3 = Y )
=> ( ord_less_eq_nat @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_257_order__subst2,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,F3: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ord_le8666007276011122963_a_nat @ ( F3 @ B ) @ C3 )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ X ) @ ( F3 @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ A ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_258_order__subst2,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,F3: set_Pr4934435412358123699_a_nat > nat,C3: nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F3 @ B ) @ C3 )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_259_order__subst2,axiom,
! [A: nat,B: nat,F3: nat > set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le8666007276011122963_a_nat @ ( F3 @ B ) @ C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ X ) @ ( F3 @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ A ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_260_order__subst2,axiom,
! [A: nat,B: nat,F3: nat > nat,C3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F3 @ B ) @ C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F3 @ A ) @ C3 ) ) ) ) ).
% order_subst2
thf(fact_261_order__subst1,axiom,
! [A: set_Pr4934435412358123699_a_nat,F3: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ ( F3 @ B ) )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C3 )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ X ) @ ( F3 @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ A @ ( F3 @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_262_order__subst1,axiom,
! [A: set_Pr4934435412358123699_a_nat,F3: nat > set_Pr4934435412358123699_a_nat,B: nat,C3: nat] :
( ( ord_le8666007276011122963_a_nat @ A @ ( F3 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ X ) @ ( F3 @ Y3 ) ) )
=> ( ord_le8666007276011122963_a_nat @ A @ ( F3 @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_263_order__subst1,axiom,
! [A: nat,F3: set_Pr4934435412358123699_a_nat > nat,B: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
( ( ord_less_eq_nat @ A @ ( F3 @ B ) )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C3 )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F3 @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_264_order__subst1,axiom,
! [A: nat,F3: nat > nat,B: nat,C3: nat] :
( ( ord_less_eq_nat @ A @ ( F3 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F3 @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_265_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr4934435412358123699_a_nat,Z: set_Pr4934435412358123699_a_nat] : ( Y4 = Z ) )
= ( ^ [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ B5 )
& ( ord_le8666007276011122963_a_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_266_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_267_antisym,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_268_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_269_dual__order_Otrans,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ( ( ord_le8666007276011122963_a_nat @ C3 @ B )
=> ( ord_le8666007276011122963_a_nat @ C3 @ A ) ) ) ).
% dual_order.trans
thf(fact_270_dual__order_Otrans,axiom,
! [B: nat,A: nat,C3: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C3 @ B )
=> ( ord_less_eq_nat @ C3 @ A ) ) ) ).
% dual_order.trans
thf(fact_271_dual__order_Oantisym,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_272_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_273_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Pr4934435412358123699_a_nat,Z: set_Pr4934435412358123699_a_nat] : ( Y4 = Z ) )
= ( ^ [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B5 @ A5 )
& ( ord_le8666007276011122963_a_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_274_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_275_linorder__wlog,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P2 @ B3 @ A3 )
=> ( P2 @ A3 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_276_order__trans,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y )
=> ( ( ord_le8666007276011122963_a_nat @ Y @ Z3 )
=> ( ord_le8666007276011122963_a_nat @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_277_order__trans,axiom,
! [X3: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_eq_nat @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_278_order_Otrans,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C3 )
=> ( ord_le8666007276011122963_a_nat @ A @ C3 ) ) ) ).
% order.trans
thf(fact_279_order_Otrans,axiom,
! [A: nat,B: nat,C3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ord_less_eq_nat @ A @ C3 ) ) ) ).
% order.trans
thf(fact_280_order__antisym,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y )
=> ( ( ord_le8666007276011122963_a_nat @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_281_order__antisym,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_282_ord__le__eq__trans,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( B = C3 )
=> ( ord_le8666007276011122963_a_nat @ A @ C3 ) ) ) ).
% ord_le_eq_trans
thf(fact_283_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C3 )
=> ( ord_less_eq_nat @ A @ C3 ) ) ) ).
% ord_le_eq_trans
thf(fact_284_ord__eq__le__trans,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
( ( A = B )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C3 )
=> ( ord_le8666007276011122963_a_nat @ A @ C3 ) ) ) ).
% ord_eq_le_trans
thf(fact_285_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C3: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ord_less_eq_nat @ A @ C3 ) ) ) ).
% ord_eq_le_trans
thf(fact_286_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_Pr4934435412358123699_a_nat,Z: set_Pr4934435412358123699_a_nat] : ( Y4 = Z ) )
= ( ^ [X6: set_Pr4934435412358123699_a_nat,Y5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X6 @ Y5 )
& ( ord_le8666007276011122963_a_nat @ Y5 @ X6 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_287_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [X6: nat,Y5: nat] :
( ( ord_less_eq_nat @ X6 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X6 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_288_le__cases3,axiom,
! [X3: nat,Y: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X3 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X3 ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_289_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_290_order__less__imp__not__less,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ~ ( ord_less_nat @ Y @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_291_order__less__imp__not__eq2,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( Y != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_292_order__less__imp__not__eq,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( X3 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_293_linorder__less__linear,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
| ( X3 = Y )
| ( ord_less_nat @ Y @ X3 ) ) ).
% linorder_less_linear
thf(fact_294_order__less__imp__triv,axiom,
! [X3: nat,Y: nat,P2: $o] :
( ( ord_less_nat @ X3 @ Y )
=> ( ( ord_less_nat @ Y @ X3 )
=> P2 ) ) ).
% order_less_imp_triv
thf(fact_295_order__less__not__sym,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ~ ( ord_less_nat @ Y @ X3 ) ) ).
% order_less_not_sym
thf(fact_296_order__less__subst2,axiom,
! [A: nat,B: nat,F3: nat > nat,C3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F3 @ B ) @ C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F3 @ X ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ ( F3 @ A ) @ C3 ) ) ) ) ).
% order_less_subst2
thf(fact_297_order__less__subst1,axiom,
! [A: nat,F3: nat > nat,B: nat,C3: nat] :
( ( ord_less_nat @ A @ ( F3 @ B ) )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F3 @ X ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F3 @ C3 ) ) ) ) ) ).
% order_less_subst1
thf(fact_298_order__less__irrefl,axiom,
! [X3: nat] :
~ ( ord_less_nat @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_299_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F3: nat > nat,C3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F3 @ B )
= C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F3 @ X ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ ( F3 @ A ) @ C3 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_300_ord__eq__less__subst,axiom,
! [A: nat,F3: nat > nat,B: nat,C3: nat] :
( ( A
= ( F3 @ B ) )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F3 @ X ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F3 @ C3 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_301_order__less__trans,axiom,
! [X3: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_302_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_303_linorder__neq__iff,axiom,
! [X3: nat,Y: nat] :
( ( X3 != Y )
= ( ( ord_less_nat @ X3 @ Y )
| ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_304_order__less__asym,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ~ ( ord_less_nat @ Y @ X3 ) ) ).
% order_less_asym
thf(fact_305_linorder__neqE,axiom,
! [X3: nat,Y: nat] :
( ( X3 != Y )
=> ( ~ ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_neqE
thf(fact_306_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_307_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_308_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C3: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C3 @ B )
=> ( ord_less_nat @ C3 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_309_not__less__iff__gr__or__eq,axiom,
! [X3: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X3 @ Y ) )
= ( ( ord_less_nat @ Y @ X3 )
| ( X3 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_310_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ord_less_nat @ A @ C3 ) ) ) ).
% order.strict_trans
thf(fact_311_linorder__less__wlog,axiom,
! [P2: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( P2 @ A3 @ B3 ) )
=> ( ! [A3: nat] : ( P2 @ A3 @ A3 )
=> ( ! [A3: nat,B3: nat] :
( ( P2 @ B3 @ A3 )
=> ( P2 @ A3 @ B3 ) )
=> ( P2 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_312_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X7: nat] : ( P3 @ X7 ) )
= ( ^ [P4: nat > $o] :
? [N2: nat] :
( ( P4 @ N2 )
& ! [M: nat] :
( ( ord_less_nat @ M @ N2 )
=> ~ ( P4 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_313_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_314_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_315_linorder__cases,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_nat @ X3 @ Y )
=> ( ( X3 != Y )
=> ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_cases
thf(fact_316_antisym__conv3,axiom,
! [Y: nat,X3: nat] :
( ~ ( ord_less_nat @ Y @ X3 )
=> ( ( ~ ( ord_less_nat @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv3
thf(fact_317_less__induct,axiom,
! [P2: nat > $o,A: nat] :
( ! [X: nat] :
( ! [Y6: nat] :
( ( ord_less_nat @ Y6 @ X )
=> ( P2 @ Y6 ) )
=> ( P2 @ X ) )
=> ( P2 @ A ) ) ).
% less_induct
thf(fact_318_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C3 )
=> ( ord_less_nat @ A @ C3 ) ) ) ).
% ord_less_eq_trans
thf(fact_319_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C3: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ord_less_nat @ A @ C3 ) ) ) ).
% ord_eq_less_trans
thf(fact_320_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_321_less__imp__neq,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( X3 != Y ) ) ).
% less_imp_neq
thf(fact_322_gt__ex,axiom,
! [X3: nat] :
? [X_1: nat] : ( ord_less_nat @ X3 @ X_1 ) ).
% gt_ex
thf(fact_323_Collect__mono__iff,axiom,
! [P2: product_prod_a_nat > $o,Q: product_prod_a_nat > $o] :
( ( ord_le8666007276011122963_a_nat @ ( collec4464134535221767506_a_nat @ P2 ) @ ( collec4464134535221767506_a_nat @ Q ) )
= ( ! [X6: product_prod_a_nat] :
( ( P2 @ X6 )
=> ( Q @ X6 ) ) ) ) ).
% Collect_mono_iff
thf(fact_324_set__eq__subset,axiom,
( ( ^ [Y4: set_Pr4934435412358123699_a_nat,Z: set_Pr4934435412358123699_a_nat] : ( Y4 = Z ) )
= ( ^ [A6: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A6 @ B6 )
& ( ord_le8666007276011122963_a_nat @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_325_subset__trans,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat,C4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A4 @ B4 )
=> ( ( ord_le8666007276011122963_a_nat @ B4 @ C4 )
=> ( ord_le8666007276011122963_a_nat @ A4 @ C4 ) ) ) ).
% subset_trans
thf(fact_326_Collect__mono,axiom,
! [P2: product_prod_a_nat > $o,Q: product_prod_a_nat > $o] :
( ! [X: product_prod_a_nat] :
( ( P2 @ X )
=> ( Q @ X ) )
=> ( ord_le8666007276011122963_a_nat @ ( collec4464134535221767506_a_nat @ P2 ) @ ( collec4464134535221767506_a_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_327_subset__refl,axiom,
! [A4: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A4 @ A4 ) ).
% subset_refl
thf(fact_328_subset__iff,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [A6: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
! [T3: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ T3 @ A6 )
=> ( member5869715511025134514rm_a_b @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_329_subset__iff,axiom,
( ord_le3000389064537975527at_nat
= ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
! [T3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ T3 @ A6 )
=> ( member8206827879206165904at_nat @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_330_subset__iff,axiom,
( ord_le3146513528884898305at_nat
= ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
! [T3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ T3 @ A6 )
=> ( member8440522571783428010at_nat @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_331_subset__iff,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A6: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
! [T3: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ T3 @ A6 )
=> ( member5724188588386418708_a_nat @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_332_equalityD2,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( A4 = B4 )
=> ( ord_le8666007276011122963_a_nat @ B4 @ A4 ) ) ).
% equalityD2
thf(fact_333_equalityD1,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( A4 = B4 )
=> ( ord_le8666007276011122963_a_nat @ A4 @ B4 ) ) ).
% equalityD1
thf(fact_334_subset__eq,axiom,
( ord_le118470702582115849rm_a_b
= ( ^ [A6: set_Pr4386577575007340137rm_a_b,B6: set_Pr4386577575007340137rm_a_b] :
! [X6: produc357393685978478089rm_a_b] :
( ( member5869715511025134514rm_a_b @ X6 @ A6 )
=> ( member5869715511025134514rm_a_b @ X6 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_335_subset__eq,axiom,
( ord_le3000389064537975527at_nat
= ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
! [X6: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X6 @ A6 )
=> ( member8206827879206165904at_nat @ X6 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_336_subset__eq,axiom,
( ord_le3146513528884898305at_nat
= ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
! [X6: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X6 @ A6 )
=> ( member8440522571783428010at_nat @ X6 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_337_subset__eq,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A6: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
! [X6: product_prod_a_nat] :
( ( member5724188588386418708_a_nat @ X6 @ A6 )
=> ( member5724188588386418708_a_nat @ X6 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_338_equalityE,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( A4 = B4 )
=> ~ ( ( ord_le8666007276011122963_a_nat @ A4 @ B4 )
=> ~ ( ord_le8666007276011122963_a_nat @ B4 @ A4 ) ) ) ).
% equalityE
thf(fact_339_subsetD,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b,C3: produc357393685978478089rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A4 @ B4 )
=> ( ( member5869715511025134514rm_a_b @ C3 @ A4 )
=> ( member5869715511025134514rm_a_b @ C3 @ B4 ) ) ) ).
% subsetD
thf(fact_340_subsetD,axiom,
! [A4: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat,C3: produc859450856879609959at_nat] :
( ( ord_le3000389064537975527at_nat @ A4 @ B4 )
=> ( ( member8206827879206165904at_nat @ C3 @ A4 )
=> ( member8206827879206165904at_nat @ C3 @ B4 ) ) ) ).
% subsetD
thf(fact_341_subsetD,axiom,
! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C3: product_prod_nat_nat] :
( ( ord_le3146513528884898305at_nat @ A4 @ B4 )
=> ( ( member8440522571783428010at_nat @ C3 @ A4 )
=> ( member8440522571783428010at_nat @ C3 @ B4 ) ) ) ).
% subsetD
thf(fact_342_subsetD,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat,C3: product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A4 @ B4 )
=> ( ( member5724188588386418708_a_nat @ C3 @ A4 )
=> ( member5724188588386418708_a_nat @ C3 @ B4 ) ) ) ).
% subsetD
thf(fact_343_in__mono,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b,X3: produc357393685978478089rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A4 @ B4 )
=> ( ( member5869715511025134514rm_a_b @ X3 @ A4 )
=> ( member5869715511025134514rm_a_b @ X3 @ B4 ) ) ) ).
% in_mono
thf(fact_344_in__mono,axiom,
! [A4: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat,X3: produc859450856879609959at_nat] :
( ( ord_le3000389064537975527at_nat @ A4 @ B4 )
=> ( ( member8206827879206165904at_nat @ X3 @ A4 )
=> ( member8206827879206165904at_nat @ X3 @ B4 ) ) ) ).
% in_mono
thf(fact_345_in__mono,axiom,
! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat] :
( ( ord_le3146513528884898305at_nat @ A4 @ B4 )
=> ( ( member8440522571783428010at_nat @ X3 @ A4 )
=> ( member8440522571783428010at_nat @ X3 @ B4 ) ) ) ).
% in_mono
thf(fact_346_in__mono,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat,X3: product_prod_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A4 @ B4 )
=> ( ( member5724188588386418708_a_nat @ X3 @ A4 )
=> ( member5724188588386418708_a_nat @ X3 @ B4 ) ) ) ).
% in_mono
thf(fact_347_neq__if__length__neq,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b] :
( ( ( size_s8906293707977694520rm_a_b @ Xs )
!= ( size_s8906293707977694520rm_a_b @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_348_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_term_a_b] :
( ( size_s8906293707977694520rm_a_b @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_349_order__le__imp__less__or__eq,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y )
=> ( ( ord_le2482536469199093511_a_nat @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_350_order__le__imp__less__or__eq,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_nat @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_351_linorder__le__less__linear,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
| ( ord_less_nat @ Y @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_352_order__less__le__subst2,axiom,
! [A: nat,B: nat,F3: nat > set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le8666007276011122963_a_nat @ ( F3 @ B ) @ C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_le2482536469199093511_a_nat @ ( F3 @ X ) @ ( F3 @ Y3 ) ) )
=> ( ord_le2482536469199093511_a_nat @ ( F3 @ A ) @ C3 ) ) ) ) ).
% order_less_le_subst2
thf(fact_353_order__less__le__subst2,axiom,
! [A: nat,B: nat,F3: nat > nat,C3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F3 @ B ) @ C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F3 @ X ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ ( F3 @ A ) @ C3 ) ) ) ) ).
% order_less_le_subst2
thf(fact_354_order__less__le__subst1,axiom,
! [A: set_Pr4934435412358123699_a_nat,F3: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ A @ ( F3 @ B ) )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C3 )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ X ) @ ( F3 @ Y3 ) ) )
=> ( ord_le2482536469199093511_a_nat @ A @ ( F3 @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_355_order__less__le__subst1,axiom,
! [A: nat,F3: set_Pr4934435412358123699_a_nat > nat,B: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
( ( ord_less_nat @ A @ ( F3 @ B ) )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C3 )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F3 @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_356_order__less__le__subst1,axiom,
! [A: set_Pr4934435412358123699_a_nat,F3: nat > set_Pr4934435412358123699_a_nat,B: nat,C3: nat] :
( ( ord_le2482536469199093511_a_nat @ A @ ( F3 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ X ) @ ( F3 @ Y3 ) ) )
=> ( ord_le2482536469199093511_a_nat @ A @ ( F3 @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_357_order__less__le__subst1,axiom,
! [A: nat,F3: nat > nat,B: nat,C3: nat] :
( ( ord_less_nat @ A @ ( F3 @ B ) )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F3 @ C3 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_358_order__le__less__subst2,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,F3: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ord_le2482536469199093511_a_nat @ ( F3 @ B ) @ C3 )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ X ) @ ( F3 @ Y3 ) ) )
=> ( ord_le2482536469199093511_a_nat @ ( F3 @ A ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_359_order__le__less__subst2,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,F3: set_Pr4934435412358123699_a_nat > nat,C3: nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ord_less_nat @ ( F3 @ B ) @ C3 )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ ( F3 @ A ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_360_order__le__less__subst2,axiom,
! [A: nat,B: nat,F3: nat > set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le2482536469199093511_a_nat @ ( F3 @ B ) @ C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ X ) @ ( F3 @ Y3 ) ) )
=> ( ord_le2482536469199093511_a_nat @ ( F3 @ A ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_361_order__le__less__subst2,axiom,
! [A: nat,B: nat,F3: nat > nat,C3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F3 @ B ) @ C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F3 @ X ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ ( F3 @ A ) @ C3 ) ) ) ) ).
% order_le_less_subst2
thf(fact_362_order__le__less__subst1,axiom,
! [A: set_Pr4934435412358123699_a_nat,F3: nat > set_Pr4934435412358123699_a_nat,B: nat,C3: nat] :
( ( ord_le8666007276011122963_a_nat @ A @ ( F3 @ B ) )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_le2482536469199093511_a_nat @ ( F3 @ X ) @ ( F3 @ Y3 ) ) )
=> ( ord_le2482536469199093511_a_nat @ A @ ( F3 @ C3 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_363_order__le__less__subst1,axiom,
! [A: nat,F3: nat > nat,B: nat,C3: nat] :
( ( ord_less_eq_nat @ A @ ( F3 @ B ) )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F3 @ X ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F3 @ C3 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_364_order__less__le__trans,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z3: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ X3 @ Y )
=> ( ( ord_le8666007276011122963_a_nat @ Y @ Z3 )
=> ( ord_le2482536469199093511_a_nat @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_365_order__less__le__trans,axiom,
! [X3: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_366_order__le__less__trans,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y )
=> ( ( ord_le2482536469199093511_a_nat @ Y @ Z3 )
=> ( ord_le2482536469199093511_a_nat @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_367_order__le__less__trans,axiom,
! [X3: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_368_order__neq__le__trans,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( A != B )
=> ( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ord_le2482536469199093511_a_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_369_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_370_order__le__neq__trans,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( A != B )
=> ( ord_le2482536469199093511_a_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_371_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_372_order__less__imp__le,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ X3 @ Y )
=> ( ord_le8666007276011122963_a_nat @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_373_order__less__imp__le,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_374_linorder__not__less,axiom,
! [X3: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X3 @ Y ) )
= ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_not_less
thf(fact_375_linorder__not__le,axiom,
! [X3: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X3 @ Y ) )
= ( ord_less_nat @ Y @ X3 ) ) ).
% linorder_not_le
thf(fact_376_order__less__le,axiom,
( ord_le2482536469199093511_a_nat
= ( ^ [X6: set_Pr4934435412358123699_a_nat,Y5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X6 @ Y5 )
& ( X6 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_377_order__less__le,axiom,
( ord_less_nat
= ( ^ [X6: nat,Y5: nat] :
( ( ord_less_eq_nat @ X6 @ Y5 )
& ( X6 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_378_order__le__less,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [X6: set_Pr4934435412358123699_a_nat,Y5: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ X6 @ Y5 )
| ( X6 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_379_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X6: nat,Y5: nat] :
( ( ord_less_nat @ X6 @ Y5 )
| ( X6 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_380_dual__order_Ostrict__implies__order,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ B @ A )
=> ( ord_le8666007276011122963_a_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_381_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_382_order_Ostrict__implies__order,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ A @ B )
=> ( ord_le8666007276011122963_a_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_383_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_384_dual__order_Ostrict__iff__not,axiom,
( ord_le2482536469199093511_a_nat
= ( ^ [B5: set_Pr4934435412358123699_a_nat,A5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B5 @ A5 )
& ~ ( ord_le8666007276011122963_a_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_385_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_386_dual__order_Ostrict__trans2,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ B @ A )
=> ( ( ord_le8666007276011122963_a_nat @ C3 @ B )
=> ( ord_le2482536469199093511_a_nat @ C3 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_387_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C3: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C3 @ B )
=> ( ord_less_nat @ C3 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_388_dual__order_Ostrict__trans1,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ( ( ord_le2482536469199093511_a_nat @ C3 @ B )
=> ( ord_le2482536469199093511_a_nat @ C3 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_389_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C3: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C3 @ B )
=> ( ord_less_nat @ C3 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_390_dual__order_Ostrict__iff__order,axiom,
( ord_le2482536469199093511_a_nat
= ( ^ [B5: set_Pr4934435412358123699_a_nat,A5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_391_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_392_dual__order_Oorder__iff__strict,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [B5: set_Pr4934435412358123699_a_nat,A5: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_393_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_less_nat @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_394_order_Ostrict__iff__not,axiom,
( ord_le2482536469199093511_a_nat
= ( ^ [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ B5 )
& ~ ( ord_le8666007276011122963_a_nat @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_395_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ~ ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_396_order_Ostrict__trans2,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ A @ B )
=> ( ( ord_le8666007276011122963_a_nat @ B @ C3 )
=> ( ord_le2482536469199093511_a_nat @ A @ C3 ) ) ) ).
% order.strict_trans2
thf(fact_397_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ord_less_nat @ A @ C3 ) ) ) ).
% order.strict_trans2
thf(fact_398_order_Ostrict__trans1,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( ord_le2482536469199093511_a_nat @ B @ C3 )
=> ( ord_le2482536469199093511_a_nat @ A @ C3 ) ) ) ).
% order.strict_trans1
thf(fact_399_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ord_less_nat @ A @ C3 ) ) ) ).
% order.strict_trans1
thf(fact_400_order_Ostrict__iff__order,axiom,
( ord_le2482536469199093511_a_nat
= ( ^ [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_401_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_402_order_Oorder__iff__strict,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_403_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_nat @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_404_not__le__imp__less,axiom,
! [Y: nat,X3: nat] :
( ~ ( ord_less_eq_nat @ Y @ X3 )
=> ( ord_less_nat @ X3 @ Y ) ) ).
% not_le_imp_less
thf(fact_405_less__le__not__le,axiom,
( ord_le2482536469199093511_a_nat
= ( ^ [X6: set_Pr4934435412358123699_a_nat,Y5: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X6 @ Y5 )
& ~ ( ord_le8666007276011122963_a_nat @ Y5 @ X6 ) ) ) ) ).
% less_le_not_le
thf(fact_406_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X6: nat,Y5: nat] :
( ( ord_less_eq_nat @ X6 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X6 ) ) ) ) ).
% less_le_not_le
thf(fact_407_antisym__conv2,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y )
=> ( ( ~ ( ord_le2482536469199093511_a_nat @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv2
thf(fact_408_antisym__conv2,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ~ ( ord_less_nat @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv2
thf(fact_409_antisym__conv1,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ~ ( ord_le2482536469199093511_a_nat @ X3 @ Y )
=> ( ( ord_le8666007276011122963_a_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% antisym_conv1
thf(fact_410_antisym__conv1,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% antisym_conv1
thf(fact_411_nless__le,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ~ ( ord_le2482536469199093511_a_nat @ A @ B ) )
= ( ~ ( ord_le8666007276011122963_a_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_412_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_413_leI,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ Y @ X3 ) ) ).
% leI
thf(fact_414_leD,axiom,
! [Y: set_Pr4934435412358123699_a_nat,X3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ Y @ X3 )
=> ~ ( ord_le2482536469199093511_a_nat @ X3 @ Y ) ) ).
% leD
thf(fact_415_leD,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ~ ( ord_less_nat @ X3 @ Y ) ) ).
% leD
thf(fact_416_length__induct,axiom,
! [P2: list_term_a_b > $o,Xs: list_term_a_b] :
( ! [Xs3: list_term_a_b] :
( ! [Ys3: list_term_a_b] :
( ( ord_less_nat @ ( size_s8906293707977694520rm_a_b @ Ys3 ) @ ( size_s8906293707977694520rm_a_b @ Xs3 ) )
=> ( P2 @ Ys3 ) )
=> ( P2 @ Xs3 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_417_permut__sound,axiom,
! [I: nat,As: list_term_a_b,F3: nat > nat] :
( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ As ) )
=> ( ( nth_term_a_b @ ( missin5417461770552926780rm_a_b @ As @ F3 ) @ I )
= ( nth_term_a_b @ As @ ( F3 @ I ) ) ) ) ).
% permut_sound
thf(fact_418_O__mono2,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,R5: set_Pr4386577575007340137rm_a_b,T2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R3 @ R5 )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ R3 @ T2 ) @ ( relcom370159955682700863rm_a_b @ R5 @ T2 ) ) ) ).
% O_mono2
thf(fact_419_O__mono2,axiom,
! [R3: set_Pr4934435412358123699_a_nat,R5: set_Pr4934435412358123699_a_nat,T2: set_Pr1261947904930325089at_nat] :
( ( ord_le8666007276011122963_a_nat @ R3 @ R5 )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_nat_nat @ R3 @ T2 ) @ ( relcomp_a_nat_nat @ R5 @ T2 ) ) ) ).
% O_mono2
thf(fact_420_O__mono1,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,R5: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ R3 @ R5 )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ S2 @ R3 ) @ ( relcom370159955682700863rm_a_b @ S2 @ R5 ) ) ) ).
% O_mono1
thf(fact_421_O__mono1,axiom,
! [R3: set_Pr4934435412358123699_a_nat,R5: set_Pr4934435412358123699_a_nat,S2: set_Product_prod_a_a] :
( ( ord_le8666007276011122963_a_nat @ R3 @ R5 )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ S2 @ R3 ) @ ( relcomp_a_a_nat @ S2 @ R5 ) ) ) ).
% O_mono1
thf(fact_422_relcomp3__I,axiom,
! [T: nat,U: a,A4: set_Pr4193341848836149977_nat_a,S: a,B4: set_Pr4934435412358123699_a_nat,V: nat] :
( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ T @ U ) @ A4 )
=> ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ S @ T ) @ B4 )
=> ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ U @ V ) @ B4 )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ S @ V ) @ ( relcomp_a_nat_nat @ B4 @ ( relcomp_nat_a_nat @ A4 @ B4 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_423_relcomp3__I,axiom,
! [T: nat,U: nat > nat,A4: set_Pr6370437063884598352at_nat,S: nat > nat,B4: set_Pr9093778441882193744at_nat,V: nat] :
( ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ T @ U ) @ A4 )
=> ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ S @ T ) @ B4 )
=> ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ U @ V ) @ B4 )
=> ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ S @ V ) @ ( relcom1859588284133128290at_nat @ B4 @ ( relcom6405311860805859042at_nat @ A4 @ B4 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_424_relcomp3__I,axiom,
! [T: term_a_b,U: term_a_b,A4: set_Pr4386577575007340137rm_a_b,S: term_a_b,B4: set_Pr4386577575007340137rm_a_b,V: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ A4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ B4 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ U @ V ) @ B4 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ V ) @ ( relcom370159955682700863rm_a_b @ B4 @ ( relcom370159955682700863rm_a_b @ A4 @ B4 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_425_relcomp3__I,axiom,
! [T: a,U: nat,A4: set_Pr4934435412358123699_a_nat,S: nat,B4: set_Pr4193341848836149977_nat_a,V: a] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ T @ U ) @ A4 )
=> ( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ S @ T ) @ B4 )
=> ( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ U @ V ) @ B4 )
=> ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ S @ V ) @ ( relcomp_nat_a_a @ B4 @ ( relcomp_a_nat_a @ A4 @ B4 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_426_relcomp3__I,axiom,
! [T: nat > nat,U: nat,A4: set_Pr9093778441882193744at_nat,S: nat,B4: set_Pr6370437063884598352at_nat,V: nat > nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ T @ U ) @ A4 )
=> ( ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ S @ T ) @ B4 )
=> ( ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ U @ V ) @ B4 )
=> ( member8336108448496249625at_nat @ ( produc7839516862119294504at_nat @ S @ V ) @ ( relcom896424912281157969at_nat @ B4 @ ( relcom7598182128343433937at_nat @ A4 @ B4 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_427_relcomp3__I,axiom,
! [T: product_prod_nat_nat,U: product_prod_nat_nat,A4: set_Pr8693737435421807431at_nat,S: product_prod_nat_nat,B4: set_Pr8693737435421807431at_nat,V: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ T @ U ) @ A4 )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ S @ T ) @ B4 )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ U @ V ) @ B4 )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ S @ V ) @ ( relcom7295539661566034944at_nat @ B4 @ ( relcom7295539661566034944at_nat @ A4 @ B4 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_428_relcomp3__I,axiom,
! [T: nat,U: nat,A4: set_Pr1261947904930325089at_nat,S: nat,B4: set_Pr1261947904930325089at_nat,V: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ T @ U ) @ A4 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ S @ T ) @ B4 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ U @ V ) @ B4 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ S @ V ) @ ( relcomp_nat_nat_nat @ B4 @ ( relcomp_nat_nat_nat @ A4 @ B4 ) ) ) ) ) ) ).
% relcomp3_I
thf(fact_429_complete__interval,axiom,
! [A: nat,B: nat,P2: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P2 @ A )
=> ( ~ ( P2 @ B )
=> ? [C: nat] :
( ( ord_less_eq_nat @ A @ C )
& ( ord_less_eq_nat @ C @ B )
& ! [X5: nat] :
( ( ( ord_less_eq_nat @ A @ X5 )
& ( ord_less_nat @ X5 @ C ) )
=> ( P2 @ X5 ) )
& ! [D: nat] :
( ! [X: nat] :
( ( ( ord_less_eq_nat @ A @ X )
& ( ord_less_nat @ X @ D ) )
=> ( P2 @ X ) )
=> ( ord_less_eq_nat @ D @ C ) ) ) ) ) ) ).
% complete_interval
thf(fact_430_verit__comp__simplify1_I3_J,axiom,
! [B2: nat,A2: nat] :
( ( ~ ( ord_less_eq_nat @ B2 @ A2 ) )
= ( ord_less_nat @ A2 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_431_pinf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% pinf(6)
thf(fact_432_pinf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( ord_less_eq_nat @ T @ X5 ) ) ).
% pinf(8)
thf(fact_433_minf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( ord_less_eq_nat @ X5 @ T ) ) ).
% minf(6)
thf(fact_434_minf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% minf(8)
thf(fact_435_permut__aux__sound,axiom,
! [I: nat,As: list_term_a_b,F3: nat > nat,Bs: list_term_a_b] :
( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ As ) )
=> ( ( nth_term_a_b @ ( missin5661735679256376914rm_a_b @ As @ F3 @ Bs ) @ I )
= ( nth_term_a_b @ Bs @ ( F3 @ I ) ) ) ) ).
% permut_aux_sound
thf(fact_436_psubsetI,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A4 @ B4 )
=> ( ( A4 != B4 )
=> ( ord_le2482536469199093511_a_nat @ A4 @ B4 ) ) ) ).
% psubsetI
thf(fact_437_psubsetE,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ A4 @ B4 )
=> ~ ( ( ord_le8666007276011122963_a_nat @ A4 @ B4 )
=> ( ord_le8666007276011122963_a_nat @ B4 @ A4 ) ) ) ).
% psubsetE
thf(fact_438_psubset__eq,axiom,
( ord_le2482536469199093511_a_nat
= ( ^ [A6: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_439_psubset__imp__subset,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ A4 @ B4 )
=> ( ord_le8666007276011122963_a_nat @ A4 @ B4 ) ) ).
% psubset_imp_subset
thf(fact_440_psubset__subset__trans,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat,C4: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ A4 @ B4 )
=> ( ( ord_le8666007276011122963_a_nat @ B4 @ C4 )
=> ( ord_le2482536469199093511_a_nat @ A4 @ C4 ) ) ) ).
% psubset_subset_trans
thf(fact_441_subset__not__subset__eq,axiom,
( ord_le2482536469199093511_a_nat
= ( ^ [A6: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A6 @ B6 )
& ~ ( ord_le8666007276011122963_a_nat @ B6 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_442_subset__psubset__trans,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat,C4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A4 @ B4 )
=> ( ( ord_le2482536469199093511_a_nat @ B4 @ C4 )
=> ( ord_le2482536469199093511_a_nat @ A4 @ C4 ) ) ) ).
% subset_psubset_trans
thf(fact_443_subset__iff__psubset__eq,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A6: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_444_verit__comp__simplify1_I2_J,axiom,
! [A: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_445_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_446_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_447_minf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ~ ( ord_less_nat @ T @ X5 ) ) ).
% minf(7)
thf(fact_448_minf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( ord_less_nat @ X5 @ T ) ) ).
% minf(5)
thf(fact_449_minf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_450_minf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_451_minf_I2_J,axiom,
! [P2: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z4 )
=> ( ( P2 @ X )
= ( P5 @ X ) ) )
=> ( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( ( ( P2 @ X5 )
| ( Q @ X5 ) )
= ( ( P5 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_452_minf_I1_J,axiom,
! [P2: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z4 )
=> ( ( P2 @ X )
= ( P5 @ X ) ) )
=> ( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z2 )
=> ( ( ( P2 @ X5 )
& ( Q @ X5 ) )
= ( ( P5 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_453_pinf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( ord_less_nat @ T @ X5 ) ) ).
% pinf(7)
thf(fact_454_pinf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ~ ( ord_less_nat @ X5 @ T ) ) ).
% pinf(5)
thf(fact_455_pinf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_456_pinf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_457_pinf_I2_J,axiom,
! [P2: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ Z4 @ X )
=> ( ( P2 @ X )
= ( P5 @ X ) ) )
=> ( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( ( ( P2 @ X5 )
| ( Q @ X5 ) )
= ( ( P5 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_458_pinf_I1_J,axiom,
! [P2: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ Z4 @ X )
=> ( ( P2 @ X )
= ( P5 @ X ) ) )
=> ( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z2 @ X5 )
=> ( ( ( P2 @ X5 )
& ( Q @ X5 ) )
= ( ( P5 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_459_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_460_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_461_remove__nth__P__compat,axiom,
! [As: list_term_a_b,Bs: list_term_a_b,P2: term_a_b > term_a_b > $o,P: nat] :
( ( ( size_s8906293707977694520rm_a_b @ As )
= ( size_s8906293707977694520rm_a_b @ Bs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s8906293707977694520rm_a_b @ As ) )
=> ( P2 @ ( nth_term_a_b @ As @ I3 ) @ ( nth_term_a_b @ Bs @ I3 ) ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ ( missin7590947408886171483rm_a_b @ P @ As ) ) )
=> ( P2 @ ( nth_term_a_b @ ( missin7590947408886171483rm_a_b @ P @ As ) @ I2 ) @ ( nth_term_a_b @ ( missin7590947408886171483rm_a_b @ P @ Bs ) @ I2 ) ) ) ) ) ).
% remove_nth_P_compat
thf(fact_462_all__ctxt__closed__def,axiom,
( terms_5226143800768910156ed_a_b
= ( ^ [F4: set_Pr4934435412358123699_a_nat,R6: set_Pr4386577575007340137rm_a_b] :
( ! [F5: a,Ts2: list_term_a_b,Ss2: list_term_a_b] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ F5 @ ( size_s8906293707977694520rm_a_b @ Ss2 ) ) @ F4 )
=> ( ( ( size_s8906293707977694520rm_a_b @ Ts2 )
= ( size_s8906293707977694520rm_a_b @ Ss2 ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Ts2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ Ts2 @ I4 ) @ ( nth_term_a_b @ Ss2 @ I4 ) ) @ R6 ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Ts2 ) )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ ( term_funas_term_a_b @ ( nth_term_a_b @ Ts2 @ I4 ) ) @ ( term_funas_term_a_b @ ( nth_term_a_b @ Ss2 @ I4 ) ) ) @ F4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( fun_a_b @ F5 @ Ts2 ) @ ( fun_a_b @ F5 @ Ss2 ) ) @ R6 ) ) ) ) )
& ! [X6: b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( var_b_a @ X6 ) @ ( var_b_a @ X6 ) ) @ R6 ) ) ) ) ).
% all_ctxt_closed_def
thf(fact_463_list__ex__length,axiom,
( list_ex_term_a_b
= ( ^ [P4: term_a_b > $o,Xs2: list_term_a_b] :
? [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
& ( P4 @ ( nth_term_a_b @ Xs2 @ N2 ) ) ) ) ) ).
% list_ex_length
thf(fact_464_listrel__iff__nth,axiom,
! [Xs: list_a,Ys: list_nat,R2: set_Pr4934435412358123699_a_nat] :
( ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Xs @ Ys ) @ ( listrel_a_nat @ R2 ) )
= ( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ ( nth_a @ Xs @ N2 ) @ ( nth_nat @ Ys @ N2 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_465_listrel__iff__nth,axiom,
! [Xs: list_nat_nat,Ys: list_nat,R2: set_Pr9093778441882193744at_nat] :
( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs @ Ys ) @ ( listrel_nat_nat_nat @ R2 ) )
= ( ( ( size_s8208510060688613859at_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_s8208510060688613859at_nat @ Xs ) )
=> ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ ( nth_nat_nat @ Xs @ N2 ) @ ( nth_nat @ Ys @ N2 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_466_listrel__iff__nth,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys ) @ ( listre818007680106770737at_nat @ R2 ) )
= ( ( ( size_s5460976970255530739at_nat @ Xs )
= ( size_s5460976970255530739at_nat @ Ys ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ N2 ) @ ( nth_Pr7617993195940197384at_nat @ Ys @ N2 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_467_listrel__iff__nth,axiom,
! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R2 ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N2 ) @ ( nth_nat @ Ys @ N2 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_468_listrel__iff__nth,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( listre1194016999521814427rm_a_b @ R2 ) )
= ( ( ( size_s8906293707977694520rm_a_b @ Xs )
= ( size_s8906293707977694520rm_a_b @ Ys ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ Xs @ N2 ) @ ( nth_term_a_b @ Ys @ N2 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_469_nth__zip,axiom,
! [I: nat,Xs: list_a,Ys: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
=> ( ( nth_Pr8461465654520414006_a_nat @ ( zip_a_nat @ Xs @ Ys ) @ I )
= ( product_Pair_a_nat @ ( nth_a @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).
% nth_zip
thf(fact_470_nth__zip,axiom,
! [I: nat,Xs: list_nat_nat,Ys: list_nat] :
( ( ord_less_nat @ I @ ( size_s8208510060688613859at_nat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
=> ( ( nth_Pr4952906083868132727at_nat @ ( zip_nat_nat_nat @ Xs @ Ys ) @ I )
= ( produc72220940542539688at_nat @ ( nth_nat_nat @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).
% nth_zip
thf(fact_471_nth__zip,axiom,
! [I: nat,Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
( ( ord_less_nat @ I @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s5460976970255530739at_nat @ Ys ) )
=> ( ( nth_Pr6744343527793145070at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs @ Ys ) @ I )
= ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ I ) @ ( nth_Pr7617993195940197384at_nat @ Ys @ I ) ) ) ) ) ).
% nth_zip
thf(fact_472_nth__zip,axiom,
! [I: nat,Xs: list_nat,Ys: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
=> ( ( nth_Pr7617993195940197384at_nat @ ( zip_nat_nat @ Xs @ Ys ) @ I )
= ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).
% nth_zip
thf(fact_473_nth__zip,axiom,
! [I: nat,Xs: list_term_a_b,Ys: list_term_a_b] :
( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Ys ) )
=> ( ( nth_Pr1098598711982373392rm_a_b @ ( zip_te2260252107881041761rm_a_b @ Xs @ Ys ) @ I )
= ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ Xs @ I ) @ ( nth_term_a_b @ Ys @ I ) ) ) ) ) ).
% nth_zip
thf(fact_474_in__measure,axiom,
! [X3: term_a_b,Y: term_a_b,F3: term_a_b > nat] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ ( measure_term_a_b @ F3 ) )
= ( ord_less_nat @ ( F3 @ X3 ) @ ( F3 @ Y ) ) ) ).
% in_measure
thf(fact_475_in__measure,axiom,
! [X3: product_prod_nat_nat,Y: product_prod_nat_nat,F3: product_prod_nat_nat > nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( measur8038558561449204169at_nat @ F3 ) )
= ( ord_less_nat @ ( F3 @ X3 ) @ ( F3 @ Y ) ) ) ).
% in_measure
thf(fact_476_in__measure,axiom,
! [X3: nat,Y: nat,F3: nat > nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( measure_nat @ F3 ) )
= ( ord_less_nat @ ( F3 @ X3 ) @ ( F3 @ Y ) ) ) ).
% in_measure
thf(fact_477_UnCI,axiom,
! [C3: produc357393685978478089rm_a_b,B4: set_Pr4386577575007340137rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( ~ ( member5869715511025134514rm_a_b @ C3 @ B4 )
=> ( member5869715511025134514rm_a_b @ C3 @ A4 ) )
=> ( member5869715511025134514rm_a_b @ C3 @ ( sup_su6776935440552674877rm_a_b @ A4 @ B4 ) ) ) ).
% UnCI
thf(fact_478_UnCI,axiom,
! [C3: product_prod_a_nat,B4: set_Pr4934435412358123699_a_nat,A4: set_Pr4934435412358123699_a_nat] :
( ( ~ ( member5724188588386418708_a_nat @ C3 @ B4 )
=> ( member5724188588386418708_a_nat @ C3 @ A4 ) )
=> ( member5724188588386418708_a_nat @ C3 @ ( sup_su459911885395995103_a_nat @ A4 @ B4 ) ) ) ).
% UnCI
thf(fact_479_UnCI,axiom,
! [C3: produc859450856879609959at_nat,B4: set_Pr8693737435421807431at_nat,A4: set_Pr8693737435421807431at_nat] :
( ( ~ ( member8206827879206165904at_nat @ C3 @ B4 )
=> ( member8206827879206165904at_nat @ C3 @ A4 ) )
=> ( member8206827879206165904at_nat @ C3 @ ( sup_su718114333110466843at_nat @ A4 @ B4 ) ) ) ).
% UnCI
thf(fact_480_UnCI,axiom,
! [C3: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
( ( ~ ( member8440522571783428010at_nat @ C3 @ B4 )
=> ( member8440522571783428010at_nat @ C3 @ A4 ) )
=> ( member8440522571783428010at_nat @ C3 @ ( sup_su6327502436637775413at_nat @ A4 @ B4 ) ) ) ).
% UnCI
thf(fact_481_Un__iff,axiom,
! [C3: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C3 @ ( sup_su6776935440552674877rm_a_b @ A4 @ B4 ) )
= ( ( member5869715511025134514rm_a_b @ C3 @ A4 )
| ( member5869715511025134514rm_a_b @ C3 @ B4 ) ) ) ).
% Un_iff
thf(fact_482_Un__iff,axiom,
! [C3: product_prod_a_nat,A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C3 @ ( sup_su459911885395995103_a_nat @ A4 @ B4 ) )
= ( ( member5724188588386418708_a_nat @ C3 @ A4 )
| ( member5724188588386418708_a_nat @ C3 @ B4 ) ) ) ).
% Un_iff
thf(fact_483_Un__iff,axiom,
! [C3: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ C3 @ ( sup_su718114333110466843at_nat @ A4 @ B4 ) )
= ( ( member8206827879206165904at_nat @ C3 @ A4 )
| ( member8206827879206165904at_nat @ C3 @ B4 ) ) ) ).
% Un_iff
thf(fact_484_Un__iff,axiom,
! [C3: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C3 @ ( sup_su6327502436637775413at_nat @ A4 @ B4 ) )
= ( ( member8440522571783428010at_nat @ C3 @ A4 )
| ( member8440522571783428010at_nat @ C3 @ B4 ) ) ) ).
% Un_iff
thf(fact_485_Un__subset__iff,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat,C4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A4 @ B4 ) @ C4 )
= ( ( ord_le8666007276011122963_a_nat @ A4 @ C4 )
& ( ord_le8666007276011122963_a_nat @ B4 @ C4 ) ) ) ).
% Un_subset_iff
thf(fact_486_relcomp__distrib2,axiom,
! [S2: set_Pr4386577575007340137rm_a_b,T2: set_Pr4386577575007340137rm_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ ( sup_su6776935440552674877rm_a_b @ S2 @ T2 ) @ R3 )
= ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ S2 @ R3 ) @ ( relcom370159955682700863rm_a_b @ T2 @ R3 ) ) ) ).
% relcomp_distrib2
thf(fact_487_relcomp__distrib,axiom,
! [R3: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b,T2: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ R3 @ ( sup_su6776935440552674877rm_a_b @ S2 @ T2 ) )
= ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ R3 @ S2 ) @ ( relcom370159955682700863rm_a_b @ R3 @ T2 ) ) ) ).
% relcomp_distrib
thf(fact_488_psubsetD,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b,C3: produc357393685978478089rm_a_b] :
( ( ord_le4883060154188259093rm_a_b @ A4 @ B4 )
=> ( ( member5869715511025134514rm_a_b @ C3 @ A4 )
=> ( member5869715511025134514rm_a_b @ C3 @ B4 ) ) ) ).
% psubsetD
thf(fact_489_psubsetD,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat,C3: product_prod_a_nat] :
( ( ord_le2482536469199093511_a_nat @ A4 @ B4 )
=> ( ( member5724188588386418708_a_nat @ C3 @ A4 )
=> ( member5724188588386418708_a_nat @ C3 @ B4 ) ) ) ).
% psubsetD
thf(fact_490_psubsetD,axiom,
! [A4: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat,C3: produc859450856879609959at_nat] :
( ( ord_le6428140832669894131at_nat @ A4 @ B4 )
=> ( ( member8206827879206165904at_nat @ C3 @ A4 )
=> ( member8206827879206165904at_nat @ C3 @ B4 ) ) ) ).
% psubsetD
thf(fact_491_psubsetD,axiom,
! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C3: product_prod_nat_nat] :
( ( ord_le7866589430770878221at_nat @ A4 @ B4 )
=> ( ( member8440522571783428010at_nat @ C3 @ A4 )
=> ( member8440522571783428010at_nat @ C3 @ B4 ) ) ) ).
% psubsetD
thf(fact_492_UnE,axiom,
! [C3: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C3 @ ( sup_su6776935440552674877rm_a_b @ A4 @ B4 ) )
=> ( ~ ( member5869715511025134514rm_a_b @ C3 @ A4 )
=> ( member5869715511025134514rm_a_b @ C3 @ B4 ) ) ) ).
% UnE
thf(fact_493_UnE,axiom,
! [C3: product_prod_a_nat,A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C3 @ ( sup_su459911885395995103_a_nat @ A4 @ B4 ) )
=> ( ~ ( member5724188588386418708_a_nat @ C3 @ A4 )
=> ( member5724188588386418708_a_nat @ C3 @ B4 ) ) ) ).
% UnE
thf(fact_494_UnE,axiom,
! [C3: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ C3 @ ( sup_su718114333110466843at_nat @ A4 @ B4 ) )
=> ( ~ ( member8206827879206165904at_nat @ C3 @ A4 )
=> ( member8206827879206165904at_nat @ C3 @ B4 ) ) ) ).
% UnE
thf(fact_495_UnE,axiom,
! [C3: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C3 @ ( sup_su6327502436637775413at_nat @ A4 @ B4 ) )
=> ( ~ ( member8440522571783428010at_nat @ C3 @ A4 )
=> ( member8440522571783428010at_nat @ C3 @ B4 ) ) ) ).
% UnE
thf(fact_496_UnI1,axiom,
! [C3: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C3 @ A4 )
=> ( member5869715511025134514rm_a_b @ C3 @ ( sup_su6776935440552674877rm_a_b @ A4 @ B4 ) ) ) ).
% UnI1
thf(fact_497_UnI1,axiom,
! [C3: product_prod_a_nat,A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C3 @ A4 )
=> ( member5724188588386418708_a_nat @ C3 @ ( sup_su459911885395995103_a_nat @ A4 @ B4 ) ) ) ).
% UnI1
thf(fact_498_UnI1,axiom,
! [C3: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ C3 @ A4 )
=> ( member8206827879206165904at_nat @ C3 @ ( sup_su718114333110466843at_nat @ A4 @ B4 ) ) ) ).
% UnI1
thf(fact_499_UnI1,axiom,
! [C3: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C3 @ A4 )
=> ( member8440522571783428010at_nat @ C3 @ ( sup_su6327502436637775413at_nat @ A4 @ B4 ) ) ) ).
% UnI1
thf(fact_500_UnI2,axiom,
! [C3: produc357393685978478089rm_a_b,B4: set_Pr4386577575007340137rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C3 @ B4 )
=> ( member5869715511025134514rm_a_b @ C3 @ ( sup_su6776935440552674877rm_a_b @ A4 @ B4 ) ) ) ).
% UnI2
thf(fact_501_UnI2,axiom,
! [C3: product_prod_a_nat,B4: set_Pr4934435412358123699_a_nat,A4: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C3 @ B4 )
=> ( member5724188588386418708_a_nat @ C3 @ ( sup_su459911885395995103_a_nat @ A4 @ B4 ) ) ) ).
% UnI2
thf(fact_502_UnI2,axiom,
! [C3: produc859450856879609959at_nat,B4: set_Pr8693737435421807431at_nat,A4: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ C3 @ B4 )
=> ( member8206827879206165904at_nat @ C3 @ ( sup_su718114333110466843at_nat @ A4 @ B4 ) ) ) ).
% UnI2
thf(fact_503_UnI2,axiom,
! [C3: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C3 @ B4 )
=> ( member8440522571783428010at_nat @ C3 @ ( sup_su6327502436637775413at_nat @ A4 @ B4 ) ) ) ).
% UnI2
thf(fact_504_subset__Un__eq,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A6: set_Pr4934435412358123699_a_nat,B6: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ A6 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_505_subset__UnE,axiom,
! [C4: set_Pr4934435412358123699_a_nat,A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C4 @ ( sup_su459911885395995103_a_nat @ A4 @ B4 ) )
=> ~ ! [A7: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A7 @ A4 )
=> ! [B7: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B7 @ B4 )
=> ( C4
!= ( sup_su459911885395995103_a_nat @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_506_Un__absorb2,axiom,
! [B4: set_Pr4934435412358123699_a_nat,A4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B4 @ A4 )
=> ( ( sup_su459911885395995103_a_nat @ A4 @ B4 )
= A4 ) ) ).
% Un_absorb2
thf(fact_507_Un__absorb1,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A4 @ B4 )
=> ( ( sup_su459911885395995103_a_nat @ A4 @ B4 )
= B4 ) ) ).
% Un_absorb1
thf(fact_508_Un__upper2,axiom,
! [B4: set_Pr4934435412358123699_a_nat,A4: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ B4 @ ( sup_su459911885395995103_a_nat @ A4 @ B4 ) ) ).
% Un_upper2
thf(fact_509_Un__upper1,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A4 @ ( sup_su459911885395995103_a_nat @ A4 @ B4 ) ) ).
% Un_upper1
thf(fact_510_Un__least,axiom,
! [A4: set_Pr4934435412358123699_a_nat,C4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A4 @ C4 )
=> ( ( ord_le8666007276011122963_a_nat @ B4 @ C4 )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A4 @ B4 ) @ C4 ) ) ) ).
% Un_least
thf(fact_511_Un__mono,axiom,
! [A4: set_Pr4934435412358123699_a_nat,C4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat,D2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A4 @ C4 )
=> ( ( ord_le8666007276011122963_a_nat @ B4 @ D2 )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A4 @ B4 ) @ ( sup_su459911885395995103_a_nat @ C4 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_512_linear__term_Ocases,axiom,
! [X3: term_a_b] :
( ! [Uu: b] :
( X3
!= ( var_b_a @ Uu ) )
=> ~ ! [Uv: a,Ts3: list_term_a_b] :
( X3
!= ( fun_a_b @ Uv @ Ts3 ) ) ) ).
% linear_term.cases
thf(fact_513_term_Odistinct_I1_J,axiom,
! [X1: b,X21: a,X22: list_term_a_b] :
( ( var_b_a @ X1 )
!= ( fun_a_b @ X21 @ X22 ) ) ).
% term.distinct(1)
thf(fact_514_Term_Oterm_Oexhaust,axiom,
! [Y: term_a_b] :
( ! [X12: b] :
( Y
!= ( var_b_a @ X12 ) )
=> ~ ! [X212: a,X222: list_term_a_b] :
( Y
!= ( fun_a_b @ X212 @ X222 ) ) ) ).
% Term.term.exhaust
thf(fact_515_listrel__eq__len,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( listre1194016999521814427rm_a_b @ R2 ) )
=> ( ( size_s8906293707977694520rm_a_b @ Xs )
= ( size_s8906293707977694520rm_a_b @ Ys ) ) ) ).
% listrel_eq_len
thf(fact_516_listrel__mono,axiom,
! [R2: set_Pr4934435412358123699_a_nat,S: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ R2 @ S )
=> ( ord_le1912520282604406569st_nat @ ( listrel_a_nat @ R2 ) @ ( listrel_a_nat @ S ) ) ) ).
% listrel_mono
thf(fact_517_remove__nth__id,axiom,
! [Xs: list_term_a_b,N: nat] :
( ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Xs ) @ N )
=> ( ( missin7590947408886171483rm_a_b @ N @ Xs )
= Xs ) ) ).
% remove_nth_id
thf(fact_518_term__to__sig_Ocases,axiom,
! [X3: produc5279506192219892694rm_a_b] :
( ! [F6: set_Pr4934435412358123699_a_nat,V2: b,X: b] :
( X3
!= ( produc8030969961714872974rm_a_b @ F6 @ ( produc1437816968797971900rm_a_b @ V2 @ ( var_b_a @ X ) ) ) )
=> ~ ! [F6: set_Pr4934435412358123699_a_nat,V2: b,F7: a,Ts3: list_term_a_b] :
( X3
!= ( produc8030969961714872974rm_a_b @ F6 @ ( produc1437816968797971900rm_a_b @ V2 @ ( fun_a_b @ F7 @ Ts3 ) ) ) ) ) ).
% term_to_sig.cases
thf(fact_519_remove__nth__sound__l,axiom,
! [P: nat,N: nat,Xs: list_term_a_b] :
( ( ord_less_nat @ P @ N )
=> ( ( nth_term_a_b @ ( missin7590947408886171483rm_a_b @ N @ Xs ) @ P )
= ( nth_term_a_b @ Xs @ P ) ) ) ).
% remove_nth_sound_l
thf(fact_520_le__sup__iff,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat,Z3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ X3 @ Y ) @ Z3 )
= ( ( ord_le8666007276011122963_a_nat @ X3 @ Z3 )
& ( ord_le8666007276011122963_a_nat @ Y @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_521_le__sup__iff,axiom,
! [X3: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X3 @ Y ) @ Z3 )
= ( ( ord_less_eq_nat @ X3 @ Z3 )
& ( ord_less_eq_nat @ Y @ Z3 ) ) ) ).
% le_sup_iff
thf(fact_522_sup_Obounded__iff,axiom,
! [B: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ B @ C3 ) @ A )
= ( ( ord_le8666007276011122963_a_nat @ B @ A )
& ( ord_le8666007276011122963_a_nat @ C3 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_523_sup_Obounded__iff,axiom,
! [B: nat,C3: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C3 ) @ A )
= ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ C3 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_524_adjust__idx__rev__nth,axiom,
! [I: nat,Xs: list_term_a_b,J: nat] :
( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( J != I )
=> ( ( nth_term_a_b @ ( missin7590947408886171483rm_a_b @ I @ Xs ) @ ( missin3815256168798769645dx_rev @ I @ J ) )
= ( nth_term_a_b @ Xs @ J ) ) ) ) ).
% adjust_idx_rev_nth
thf(fact_525_adjust__idx__nth,axiom,
! [I: nat,Xs: list_term_a_b,J: nat] :
( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( nth_term_a_b @ ( missin7590947408886171483rm_a_b @ I @ Xs ) @ J )
= ( nth_term_a_b @ Xs @ ( missing_adjust_idx @ I @ J ) ) ) ) ).
% adjust_idx_nth
thf(fact_526_adjust__idx__rev2,axiom,
! [J: nat,I: nat] :
( ( J != I )
=> ( ( missing_adjust_idx @ I @ ( missin3815256168798769645dx_rev @ I @ J ) )
= J ) ) ).
% adjust_idx_rev2
thf(fact_527_adjust__idx__rev1,axiom,
! [I: nat,J: nat] :
( ( missin3815256168798769645dx_rev @ I @ ( missing_adjust_idx @ I @ J ) )
= J ) ).
% adjust_idx_rev1
thf(fact_528_adjust__idx__i,axiom,
! [I: nat,J: nat] :
( ( missing_adjust_idx @ I @ J )
!= I ) ).
% adjust_idx_i
thf(fact_529_sup_OcoboundedI2,axiom,
! [C3: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C3 @ B )
=> ( ord_le8666007276011122963_a_nat @ C3 @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_530_sup_OcoboundedI2,axiom,
! [C3: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ C3 @ B )
=> ( ord_less_eq_nat @ C3 @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_531_sup_OcoboundedI1,axiom,
! [C3: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C3 @ A )
=> ( ord_le8666007276011122963_a_nat @ C3 @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_532_sup_OcoboundedI1,axiom,
! [C3: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C3 @ A )
=> ( ord_less_eq_nat @ C3 @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_533_sup_Oabsorb__iff2,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [A5: set_Pr4934435412358123699_a_nat,B5: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_534_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( sup_sup_nat @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_535_sup_Oabsorb__iff1,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [B5: set_Pr4934435412358123699_a_nat,A5: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_536_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( sup_sup_nat @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_537_sup_Ocobounded2,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ B @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_538_sup_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_539_sup_Ocobounded1,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ A @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_540_sup_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_541_sup_Oorder__iff,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [B5: set_Pr4934435412358123699_a_nat,A5: set_Pr4934435412358123699_a_nat] :
( A5
= ( sup_su459911885395995103_a_nat @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_542_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( A5
= ( sup_sup_nat @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_543_sup_OboundedI,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ( ( ord_le8666007276011122963_a_nat @ C3 @ A )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ B @ C3 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_544_sup_OboundedI,axiom,
! [B: nat,A: nat,C3: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C3 @ A )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C3 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_545_sup_OboundedE,axiom,
! [B: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ B @ C3 ) @ A )
=> ~ ( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ~ ( ord_le8666007276011122963_a_nat @ C3 @ A ) ) ) ).
% sup.boundedE
thf(fact_546_sup_OboundedE,axiom,
! [B: nat,C3: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C3 ) @ A )
=> ~ ( ( ord_less_eq_nat @ B @ A )
=> ~ ( ord_less_eq_nat @ C3 @ A ) ) ) ).
% sup.boundedE
thf(fact_547_sup__absorb2,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ Y )
=> ( ( sup_su459911885395995103_a_nat @ X3 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_548_sup__absorb2,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( sup_sup_nat @ X3 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_549_sup__absorb1,axiom,
! [Y: set_Pr4934435412358123699_a_nat,X3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ Y @ X3 )
=> ( ( sup_su459911885395995103_a_nat @ X3 @ Y )
= X3 ) ) ).
% sup_absorb1
thf(fact_550_sup__absorb1,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( sup_sup_nat @ X3 @ Y )
= X3 ) ) ).
% sup_absorb1
thf(fact_551_sup_Oabsorb2,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ B )
=> ( ( sup_su459911885395995103_a_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_552_sup_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_553_sup_Oabsorb1,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ( ( sup_su459911885395995103_a_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_554_sup_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_555_sup__unique,axiom,
! [F3: set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat > set_Pr4934435412358123699_a_nat,X3: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ! [X: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ X @ ( F3 @ X @ Y3 ) )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ Y3 @ ( F3 @ X @ Y3 ) )
=> ( ! [X: set_Pr4934435412358123699_a_nat,Y3: set_Pr4934435412358123699_a_nat,Z2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ Y3 @ X )
=> ( ( ord_le8666007276011122963_a_nat @ Z2 @ X )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ Y3 @ Z2 ) @ X ) ) )
=> ( ( sup_su459911885395995103_a_nat @ X3 @ Y )
= ( F3 @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_556_sup__unique,axiom,
! [F3: nat > nat > nat,X3: nat,Y: nat] :
( ! [X: nat,Y3: nat] : ( ord_less_eq_nat @ X @ ( F3 @ X @ Y3 ) )
=> ( ! [X: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ ( F3 @ X @ Y3 ) )
=> ( ! [X: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y3 @ X )
=> ( ( ord_less_eq_nat @ Z2 @ X )
=> ( ord_less_eq_nat @ ( F3 @ Y3 @ Z2 ) @ X ) ) )
=> ( ( sup_sup_nat @ X3 @ Y )
= ( F3 @ X3 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_557_sup_OorderI,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( A
= ( sup_su459911885395995103_a_nat @ A @ B ) )
=> ( ord_le8666007276011122963_a_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_558_sup_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( sup_sup_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_559_sup_OorderE,axiom,
! [B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ B @ A )
=> ( A
= ( sup_su459911885395995103_a_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_560_sup_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( sup_sup_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_561_le__iff__sup,axiom,
( ord_le8666007276011122963_a_nat
= ( ^ [X6: set_Pr4934435412358123699_a_nat,Y5: set_Pr4934435412358123699_a_nat] :
( ( sup_su459911885395995103_a_nat @ X6 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_562_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X6: nat,Y5: nat] :
( ( sup_sup_nat @ X6 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_563_sup__least,axiom,
! [Y: set_Pr4934435412358123699_a_nat,X3: set_Pr4934435412358123699_a_nat,Z3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ Y @ X3 )
=> ( ( ord_le8666007276011122963_a_nat @ Z3 @ X3 )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ Y @ Z3 ) @ X3 ) ) ) ).
% sup_least
thf(fact_564_sup__least,axiom,
! [Y: nat,X3: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( ord_less_eq_nat @ Z3 @ X3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z3 ) @ X3 ) ) ) ).
% sup_least
thf(fact_565_sup__mono,axiom,
! [A: set_Pr4934435412358123699_a_nat,C3: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,D3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ C3 )
=> ( ( ord_le8666007276011122963_a_nat @ B @ D3 )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A @ B ) @ ( sup_su459911885395995103_a_nat @ C3 @ D3 ) ) ) ) ).
% sup_mono
thf(fact_566_sup__mono,axiom,
! [A: nat,C3: nat,B: nat,D3: nat] :
( ( ord_less_eq_nat @ A @ C3 )
=> ( ( ord_less_eq_nat @ B @ D3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ ( sup_sup_nat @ C3 @ D3 ) ) ) ) ).
% sup_mono
thf(fact_567_sup_Omono,axiom,
! [C3: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,D3: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ C3 @ A )
=> ( ( ord_le8666007276011122963_a_nat @ D3 @ B )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ C3 @ D3 ) @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_568_sup_Omono,axiom,
! [C3: nat,A: nat,D3: nat,B: nat] :
( ( ord_less_eq_nat @ C3 @ A )
=> ( ( ord_less_eq_nat @ D3 @ B )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C3 @ D3 ) @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_569_le__supI2,axiom,
! [X3: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ B )
=> ( ord_le8666007276011122963_a_nat @ X3 @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_570_le__supI2,axiom,
! [X3: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ X3 @ B )
=> ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_571_le__supI1,axiom,
! [X3: set_Pr4934435412358123699_a_nat,A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ X3 @ A )
=> ( ord_le8666007276011122963_a_nat @ X3 @ ( sup_su459911885395995103_a_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_572_le__supI1,axiom,
! [X3: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X3 @ A )
=> ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_573_sup__ge2,axiom,
! [Y: set_Pr4934435412358123699_a_nat,X3: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ Y @ ( sup_su459911885395995103_a_nat @ X3 @ Y ) ) ).
% sup_ge2
thf(fact_574_sup__ge2,axiom,
! [Y: nat,X3: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X3 @ Y ) ) ).
% sup_ge2
thf(fact_575_sup__ge1,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ X3 @ ( sup_su459911885395995103_a_nat @ X3 @ Y ) ) ).
% sup_ge1
thf(fact_576_sup__ge1,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ X3 @ Y ) ) ).
% sup_ge1
thf(fact_577_le__supI,axiom,
! [A: set_Pr4934435412358123699_a_nat,X3: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ X3 )
=> ( ( ord_le8666007276011122963_a_nat @ B @ X3 )
=> ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A @ B ) @ X3 ) ) ) ).
% le_supI
thf(fact_578_le__supI,axiom,
! [A: nat,X3: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X3 )
=> ( ( ord_less_eq_nat @ B @ X3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X3 ) ) ) ).
% le_supI
thf(fact_579_le__supE,axiom,
! [A: set_Pr4934435412358123699_a_nat,B: set_Pr4934435412358123699_a_nat,X3: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( sup_su459911885395995103_a_nat @ A @ B ) @ X3 )
=> ~ ( ( ord_le8666007276011122963_a_nat @ A @ X3 )
=> ~ ( ord_le8666007276011122963_a_nat @ B @ X3 ) ) ) ).
% le_supE
thf(fact_580_le__supE,axiom,
! [A: nat,B: nat,X3: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X3 )
=> ~ ( ( ord_less_eq_nat @ A @ X3 )
=> ~ ( ord_less_eq_nat @ B @ X3 ) ) ) ).
% le_supE
thf(fact_581_inf__sup__ord_I3_J,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ X3 @ ( sup_su459911885395995103_a_nat @ X3 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_582_inf__sup__ord_I3_J,axiom,
! [X3: nat,Y: nat] : ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ X3 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_583_inf__sup__ord_I4_J,axiom,
! [Y: set_Pr4934435412358123699_a_nat,X3: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ Y @ ( sup_su459911885395995103_a_nat @ X3 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_584_inf__sup__ord_I4_J,axiom,
! [Y: nat,X3: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X3 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_585_sup_Ostrict__coboundedI2,axiom,
! [C3: nat,B: nat,A: nat] :
( ( ord_less_nat @ C3 @ B )
=> ( ord_less_nat @ C3 @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_586_sup_Ostrict__coboundedI1,axiom,
! [C3: nat,A: nat,B: nat] :
( ( ord_less_nat @ C3 @ A )
=> ( ord_less_nat @ C3 @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_587_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( A5
= ( sup_sup_nat @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_588_sup_Ostrict__boundedE,axiom,
! [B: nat,C3: nat,A: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B @ C3 ) @ A )
=> ~ ( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ C3 @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_589_sup_Oabsorb4,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_590_sup_Oabsorb3,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_591_less__supI2,axiom,
! [X3: nat,B: nat,A: nat] :
( ( ord_less_nat @ X3 @ B )
=> ( ord_less_nat @ X3 @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI2
thf(fact_592_less__supI1,axiom,
! [X3: nat,A: nat,B: nat] :
( ( ord_less_nat @ X3 @ A )
=> ( ord_less_nat @ X3 @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI1
thf(fact_593_adjust__idx__length,axiom,
! [I: nat,Xs: list_term_a_b,J: nat] :
( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_s8906293707977694520rm_a_b @ ( missin7590947408886171483rm_a_b @ I @ Xs ) ) )
=> ( ord_less_nat @ ( missing_adjust_idx @ I @ J ) @ ( size_s8906293707977694520rm_a_b @ Xs ) ) ) ) ).
% adjust_idx_length
thf(fact_594_adjust__idx__rev__length,axiom,
! [I: nat,Xs: list_term_a_b,J: nat] :
( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( J != I )
=> ( ord_less_nat @ ( missin3815256168798769645dx_rev @ I @ J ) @ ( size_s8906293707977694520rm_a_b @ ( missin7590947408886171483rm_a_b @ I @ Xs ) ) ) ) ) ) ).
% adjust_idx_rev_length
thf(fact_595_funas__term_Ocases,axiom,
! [X3: term_a_b] :
( ! [X: b] :
( X3
!= ( var_b_a @ X ) )
=> ~ ! [F7: a,Ts3: list_term_a_b] :
( X3
!= ( fun_a_b @ F7 @ Ts3 ) ) ) ).
% funas_term.cases
thf(fact_596_gterm__of__term_Ocases,axiom,
! [X3: term_a_b] :
( ! [F7: a,Ts3: list_term_a_b] :
( X3
!= ( fun_a_b @ F7 @ Ts3 ) )
=> ~ ! [V2: b] :
( X3
!= ( var_b_a @ V2 ) ) ) ).
% gterm_of_term.cases
thf(fact_597_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
& ( M != N2 ) ) ) ) ).
% nat_less_le
thf(fact_598_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_599_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_600_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_601_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_602_linorder__neqE__nat,axiom,
! [X3: nat,Y: nat] :
( ( X3 != Y )
=> ( ~ ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_neqE_nat
thf(fact_603_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P2 @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P2 @ M3 ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_604_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P2 @ M3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_605_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_606_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_607_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_608_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_609_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_610_size__neq__size__imp__neq,axiom,
! [X3: list_term_a_b,Y: list_term_a_b] :
( ( ( size_s8906293707977694520rm_a_b @ X3 )
!= ( size_s8906293707977694520rm_a_b @ Y ) )
=> ( X3 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_611_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_612_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_613_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_614_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_615_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_616_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K: nat,B: nat] :
( ( P2 @ K )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X: nat] :
( ( P2 @ X )
& ! [Y6: nat] :
( ( P2 @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_617_less__mono__imp__le__mono,axiom,
! [F3: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F3 @ I3 ) @ ( F3 @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F3 @ I ) @ ( F3 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_618_inf__pigeonhole__principle,axiom,
! [N: nat,F3: nat > nat > $o] :
( ! [K2: nat] :
? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( F3 @ K2 @ I2 ) )
=> ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ! [K3: nat] :
? [K4: nat] :
( ( ord_less_eq_nat @ K3 @ K4 )
& ( F3 @ K4 @ I3 ) ) ) ) ).
% inf_pigeonhole_principle
thf(fact_619_nat__descend__induct,axiom,
! [N: nat,P2: nat > $o,M2: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P2 @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I2: nat] :
( ( ord_less_nat @ K2 @ I2 )
=> ( P2 @ I2 ) )
=> ( P2 @ K2 ) ) )
=> ( P2 @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_620_remove__nth__sound__r,axiom,
! [N: nat,P: nat,Xs: list_term_a_b] :
( ( ord_less_eq_nat @ N @ P )
=> ( ( ord_less_nat @ P @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( nth_term_a_b @ ( missin7590947408886171483rm_a_b @ N @ Xs ) @ P )
= ( nth_term_a_b @ Xs @ ( suc @ P ) ) ) ) ) ).
% remove_nth_sound_r
thf(fact_621_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_622_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_623_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_624_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_625_nth__Cons__Suc,axiom,
! [X3: term_a_b,Xs: list_term_a_b,N: nat] :
( ( nth_term_a_b @ ( cons_term_a_b @ X3 @ Xs ) @ ( suc @ N ) )
= ( nth_term_a_b @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_626_zip__Cons__Cons,axiom,
! [X3: term_a_b,Xs: list_term_a_b,Y: term_a_b,Ys: list_term_a_b] :
( ( zip_te2260252107881041761rm_a_b @ ( cons_term_a_b @ X3 @ Xs ) @ ( cons_term_a_b @ Y @ Ys ) )
= ( cons_P3536305108106557631rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ ( zip_te2260252107881041761rm_a_b @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_627_zip__Cons__Cons,axiom,
! [X3: a,Xs: list_a,Y: nat,Ys: list_nat] :
( ( zip_a_nat @ ( cons_a @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( cons_P5205166803686508359_a_nat @ ( product_Pair_a_nat @ X3 @ Y ) @ ( zip_a_nat @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_628_zip__Cons__Cons,axiom,
! [X3: nat > nat,Xs: list_nat_nat,Y: nat,Ys: list_nat] :
( ( zip_nat_nat_nat @ ( cons_nat_nat @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( cons_P4219629788700907686at_nat @ ( produc72220940542539688at_nat @ X3 @ Y ) @ ( zip_nat_nat_nat @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_629_zip__Cons__Cons,axiom,
! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] :
( ( zip_Pr4664179122662387191at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys ) )
= ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( zip_Pr4664179122662387191at_nat @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_630_zip__Cons__Cons,axiom,
! [X3: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( zip_nat_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_631_Suc__length__conv,axiom,
! [N: nat,Xs: list_term_a_b] :
( ( ( suc @ N )
= ( size_s8906293707977694520rm_a_b @ Xs ) )
= ( ? [Y5: term_a_b,Ys2: list_term_a_b] :
( ( Xs
= ( cons_term_a_b @ Y5 @ Ys2 ) )
& ( ( size_s8906293707977694520rm_a_b @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_632_length__Suc__conv,axiom,
! [Xs: list_term_a_b,N: nat] :
( ( ( size_s8906293707977694520rm_a_b @ Xs )
= ( suc @ N ) )
= ( ? [Y5: term_a_b,Ys2: list_term_a_b] :
( ( Xs
= ( cons_term_a_b @ Y5 @ Ys2 ) )
& ( ( size_s8906293707977694520rm_a_b @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_633_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_term_a_b] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s8906293707977694520rm_a_b @ Xs ) )
= ( ? [X6: term_a_b,Ys2: list_term_a_b] :
( ( Xs
= ( cons_term_a_b @ X6 @ Ys2 ) )
& ( ord_less_eq_nat @ N @ ( size_s8906293707977694520rm_a_b @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_634_zip__eq__ConsE,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b,Xy: produc357393685978478089rm_a_b,Xys: list_P8875379029341186191rm_a_b] :
( ( ( zip_te2260252107881041761rm_a_b @ Xs @ Ys )
= ( cons_P3536305108106557631rm_a_b @ Xy @ Xys ) )
=> ~ ! [X: term_a_b,Xs4: list_term_a_b] :
( ( Xs
= ( cons_term_a_b @ X @ Xs4 ) )
=> ! [Y3: term_a_b,Ys4: list_term_a_b] :
( ( Ys
= ( cons_term_a_b @ Y3 @ Ys4 ) )
=> ( ( Xy
= ( produc7020197800436672577rm_a_b @ X @ Y3 ) )
=> ( Xys
!= ( zip_te2260252107881041761rm_a_b @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_635_zip__eq__ConsE,axiom,
! [Xs: list_a,Ys: list_nat,Xy: product_prod_a_nat,Xys: list_P3592885314253461005_a_nat] :
( ( ( zip_a_nat @ Xs @ Ys )
= ( cons_P5205166803686508359_a_nat @ Xy @ Xys ) )
=> ~ ! [X: a,Xs4: list_a] :
( ( Xs
= ( cons_a @ X @ Xs4 ) )
=> ! [Y3: nat,Ys4: list_nat] :
( ( Ys
= ( cons_nat @ Y3 @ Ys4 ) )
=> ( ( Xy
= ( product_Pair_a_nat @ X @ Y3 ) )
=> ( Xys
!= ( zip_a_nat @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_636_zip__eq__ConsE,axiom,
! [Xs: list_nat_nat,Ys: list_nat,Xy: produc8199716216217303280at_nat,Xys: list_P5366131564312172150at_nat] :
( ( ( zip_nat_nat_nat @ Xs @ Ys )
= ( cons_P4219629788700907686at_nat @ Xy @ Xys ) )
=> ~ ! [X: nat > nat,Xs4: list_nat_nat] :
( ( Xs
= ( cons_nat_nat @ X @ Xs4 ) )
=> ! [Y3: nat,Ys4: list_nat] :
( ( Ys
= ( cons_nat @ Y3 @ Ys4 ) )
=> ( ( Xy
= ( produc72220940542539688at_nat @ X @ Y3 ) )
=> ( Xys
!= ( zip_nat_nat_nat @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_637_zip__eq__ConsE,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,Xy: produc859450856879609959at_nat,Xys: list_P8469869581646625389at_nat] :
( ( ( zip_Pr4664179122662387191at_nat @ Xs @ Ys )
= ( cons_P8732206157123786781at_nat @ Xy @ Xys ) )
=> ~ ! [X: product_prod_nat_nat,Xs4: list_P6011104703257516679at_nat] :
( ( Xs
= ( cons_P6512896166579812791at_nat @ X @ Xs4 ) )
=> ! [Y3: product_prod_nat_nat,Ys4: list_P6011104703257516679at_nat] :
( ( Ys
= ( cons_P6512896166579812791at_nat @ Y3 @ Ys4 ) )
=> ( ( Xy
= ( produc6161850002892822231at_nat @ X @ Y3 ) )
=> ( Xys
!= ( zip_Pr4664179122662387191at_nat @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_638_zip__eq__ConsE,axiom,
! [Xs: list_nat,Ys: list_nat,Xy: product_prod_nat_nat,Xys: list_P6011104703257516679at_nat] :
( ( ( zip_nat_nat @ Xs @ Ys )
= ( cons_P6512896166579812791at_nat @ Xy @ Xys ) )
=> ~ ! [X: nat,Xs4: list_nat] :
( ( Xs
= ( cons_nat @ X @ Xs4 ) )
=> ! [Y3: nat,Ys4: list_nat] :
( ( Ys
= ( cons_nat @ Y3 @ Ys4 ) )
=> ( ( Xy
= ( product_Pair_nat_nat @ X @ Y3 ) )
=> ( Xys
!= ( zip_nat_nat @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_639_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_640_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_641_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_642_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_643_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_644_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_645_Ex__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P2 @ I4 ) ) )
= ( ( P2 @ N )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P2 @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_646_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_647_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_nat @ M2 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_648_All__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P2 @ I4 ) ) )
= ( ( P2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P2 @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_649_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M2 )
= ( ? [M4: nat] :
( ( M2
= ( suc @ M4 ) )
& ( ord_less_nat @ N @ M4 ) ) ) ) ).
% Suc_less_eq2
thf(fact_650_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_651_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_652_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_653_less__Suc__induct,axiom,
! [I: nat,J: nat,P2: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] : ( P2 @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P2 @ I3 @ J2 )
=> ( ( P2 @ J2 @ K2 )
=> ( P2 @ I3 @ K2 ) ) ) ) )
=> ( P2 @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_654_strict__inc__induct,axiom,
! [I: nat,J: nat,P2: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P2 @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P2 @ ( suc @ I3 ) )
=> ( P2 @ I3 ) ) )
=> ( P2 @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_655_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_656_Suc__leD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_leD
thf(fact_657_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_658_le__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_659_Suc__le__D,axiom,
! [N: nat,M5: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M5 )
=> ? [M6: nat] :
( M5
= ( suc @ M6 ) ) ) ).
% Suc_le_D
thf(fact_660_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_661_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_662_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_663_full__nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
=> ( P2 @ M3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% full_nat_induct
thf(fact_664_nat__induct__at__least,axiom,
! [M2: nat,N: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( P2 @ M2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_665_transitive__stepwise__le,axiom,
! [M2: nat,N: nat,R3: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ! [X: nat] : ( R3 @ X @ X )
=> ( ! [X: nat,Y3: nat,Z2: nat] :
( ( R3 @ X @ Y3 )
=> ( ( R3 @ Y3 @ Z2 )
=> ( R3 @ X @ Z2 ) ) )
=> ( ! [N3: nat] : ( R3 @ N3 @ ( suc @ N3 ) )
=> ( R3 @ M2 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_666_lift__Suc__mono__less,axiom,
! [F3: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F3 @ N3 ) @ ( F3 @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F3 @ N ) @ ( F3 @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_667_lift__Suc__mono__less__iff,axiom,
! [F3: nat > nat,N: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F3 @ N3 ) @ ( F3 @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F3 @ N ) @ ( F3 @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_668_lift__Suc__antimono__le,axiom,
! [F3: nat > set_Pr4934435412358123699_a_nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_le8666007276011122963_a_nat @ ( F3 @ ( suc @ N3 ) ) @ ( F3 @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ N4 ) @ ( F3 @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_669_lift__Suc__antimono__le,axiom,
! [F3: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F3 @ ( suc @ N3 ) ) @ ( F3 @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F3 @ N4 ) @ ( F3 @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_670_lift__Suc__mono__le,axiom,
! [F3: nat > set_Pr4934435412358123699_a_nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_le8666007276011122963_a_nat @ ( F3 @ N3 ) @ ( F3 @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_le8666007276011122963_a_nat @ ( F3 @ N ) @ ( F3 @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_671_lift__Suc__mono__le,axiom,
! [F3: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F3 @ N3 ) @ ( F3 @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F3 @ N ) @ ( F3 @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_672_Abstract__Rewriting_Ochain__mono,axiom,
! [R5: set_Pr4386577575007340137rm_a_b,R3: set_Pr4386577575007340137rm_a_b,Seq: nat > term_a_b] :
( ( ord_le118470702582115849rm_a_b @ R5 @ R3 )
=> ( ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( Seq @ I3 ) @ ( Seq @ ( suc @ I3 ) ) ) @ R5 )
=> ! [I2: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( Seq @ I2 ) @ ( Seq @ ( suc @ I2 ) ) ) @ R3 ) ) ) ).
% Abstract_Rewriting.chain_mono
thf(fact_673_Abstract__Rewriting_Ochain__mono,axiom,
! [R5: set_Pr8693737435421807431at_nat,R3: set_Pr8693737435421807431at_nat,Seq: nat > product_prod_nat_nat] :
( ( ord_le3000389064537975527at_nat @ R5 @ R3 )
=> ( ! [I3: nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( Seq @ I3 ) @ ( Seq @ ( suc @ I3 ) ) ) @ R5 )
=> ! [I2: nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( Seq @ I2 ) @ ( Seq @ ( suc @ I2 ) ) ) @ R3 ) ) ) ).
% Abstract_Rewriting.chain_mono
thf(fact_674_Abstract__Rewriting_Ochain__mono,axiom,
! [R5: set_Pr1261947904930325089at_nat,R3: set_Pr1261947904930325089at_nat,Seq: nat > nat] :
( ( ord_le3146513528884898305at_nat @ R5 @ R3 )
=> ( ! [I3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( Seq @ I3 ) @ ( Seq @ ( suc @ I3 ) ) ) @ R5 )
=> ! [I2: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( Seq @ I2 ) @ ( Seq @ ( suc @ I2 ) ) ) @ R3 ) ) ) ).
% Abstract_Rewriting.chain_mono
thf(fact_675_impossible__Cons,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b,X3: term_a_b] :
( ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Xs ) @ ( size_s8906293707977694520rm_a_b @ Ys ) )
=> ( Xs
!= ( cons_term_a_b @ X3 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_676_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_677_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_678_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_679_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_680_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_681_inc__induct,axiom,
! [I: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P2 @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% inc_induct
thf(fact_682_dec__induct,axiom,
! [I: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P2 @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) ) )
=> ( P2 @ J ) ) ) ) ).
% dec_induct
thf(fact_683_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_684_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_685_adjust__idx__def,axiom,
( missing_adjust_idx
= ( ^ [I4: nat,J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ I4 ) @ J3 @ ( suc @ J3 ) ) ) ) ).
% adjust_idx_def
thf(fact_686_listrel_OCons,axiom,
! [X3: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b,Xs: list_term_a_b,Ys: list_term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ R2 )
=> ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( listre1194016999521814427rm_a_b @ R2 ) )
=> ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ X3 @ Xs ) @ ( cons_term_a_b @ Y @ Ys ) ) @ ( listre1194016999521814427rm_a_b @ R2 ) ) ) ) ).
% listrel.Cons
thf(fact_687_listrel_OCons,axiom,
! [X3: a,Y: nat,R2: set_Pr4934435412358123699_a_nat,Xs: list_a,Ys: list_nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X3 @ Y ) @ R2 )
=> ( ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Xs @ Ys ) @ ( listrel_a_nat @ R2 ) )
=> ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ ( cons_a @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_a_nat @ R2 ) ) ) ) ).
% listrel.Cons
thf(fact_688_listrel_OCons,axiom,
! [X3: nat > nat,Y: nat,R2: set_Pr9093778441882193744at_nat,Xs: list_nat_nat,Ys: list_nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X3 @ Y ) @ R2 )
=> ( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs @ Ys ) @ ( listrel_nat_nat_nat @ R2 ) )
=> ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ ( cons_nat_nat @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_nat_nat_nat @ R2 ) ) ) ) ).
% listrel.Cons
thf(fact_689_listrel_OCons,axiom,
! [X3: product_prod_nat_nat,Y: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ R2 )
=> ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys ) @ ( listre818007680106770737at_nat @ R2 ) )
=> ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys ) ) @ ( listre818007680106770737at_nat @ R2 ) ) ) ) ).
% listrel.Cons
thf(fact_690_listrel_OCons,axiom,
! [X3: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R2 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R2 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_nat_nat @ R2 ) ) ) ) ).
% listrel.Cons
thf(fact_691_listrel__Cons1,axiom,
! [Y: term_a_b,Ys: list_term_a_b,Xs: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ Y @ Ys ) @ Xs ) @ ( listre1194016999521814427rm_a_b @ R2 ) )
=> ~ ! [Y3: term_a_b,Ys5: list_term_a_b] :
( ( Xs
= ( cons_term_a_b @ Y3 @ Ys5 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Y3 ) @ R2 )
=> ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Ys @ Ys5 ) @ ( listre1194016999521814427rm_a_b @ R2 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_692_listrel__Cons1,axiom,
! [Y: a,Ys: list_a,Xs: list_nat,R2: set_Pr4934435412358123699_a_nat] :
( ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ ( cons_a @ Y @ Ys ) @ Xs ) @ ( listrel_a_nat @ R2 ) )
=> ~ ! [Y3: nat,Ys5: list_nat] :
( ( Xs
= ( cons_nat @ Y3 @ Ys5 ) )
=> ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ Y @ Y3 ) @ R2 )
=> ~ ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Ys @ Ys5 ) @ ( listrel_a_nat @ R2 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_693_listrel__Cons1,axiom,
! [Y: nat > nat,Ys: list_nat_nat,Xs: list_nat,R2: set_Pr9093778441882193744at_nat] :
( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ ( cons_nat_nat @ Y @ Ys ) @ Xs ) @ ( listrel_nat_nat_nat @ R2 ) )
=> ~ ! [Y3: nat,Ys5: list_nat] :
( ( Xs
= ( cons_nat @ Y3 @ Ys5 ) )
=> ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ Y @ Y3 ) @ R2 )
=> ~ ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Ys @ Ys5 ) @ ( listrel_nat_nat_nat @ R2 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_694_listrel__Cons1,axiom,
! [Y: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat,Xs: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ Y @ Ys ) @ Xs ) @ ( listre818007680106770737at_nat @ R2 ) )
=> ~ ! [Y3: product_prod_nat_nat,Ys5: list_P6011104703257516679at_nat] :
( ( Xs
= ( cons_P6512896166579812791at_nat @ Y3 @ Ys5 ) )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ Y3 ) @ R2 )
=> ~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Ys @ Ys5 ) @ ( listre818007680106770737at_nat @ R2 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_695_listrel__Cons1,axiom,
! [Y: nat,Ys: list_nat,Xs: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ Y @ Ys ) @ Xs ) @ ( listrel_nat_nat @ R2 ) )
=> ~ ! [Y3: nat,Ys5: list_nat] :
( ( Xs
= ( cons_nat @ Y3 @ Ys5 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Y3 ) @ R2 )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Ys5 ) @ ( listrel_nat_nat @ R2 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_696_listrel__Cons2,axiom,
! [Xs: list_term_a_b,Y: term_a_b,Ys: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ ( cons_term_a_b @ Y @ Ys ) ) @ ( listre1194016999521814427rm_a_b @ R2 ) )
=> ~ ! [X: term_a_b,Xs3: list_term_a_b] :
( ( Xs
= ( cons_term_a_b @ X @ Xs3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 )
=> ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs3 @ Ys ) @ ( listre1194016999521814427rm_a_b @ R2 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_697_listrel__Cons2,axiom,
! [Xs: list_a,Y: nat,Ys: list_nat,R2: set_Pr4934435412358123699_a_nat] :
( ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_a_nat @ R2 ) )
=> ~ ! [X: a,Xs3: list_a] :
( ( Xs
= ( cons_a @ X @ Xs3 ) )
=> ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X @ Y ) @ R2 )
=> ~ ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Xs3 @ Ys ) @ ( listrel_a_nat @ R2 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_698_listrel__Cons2,axiom,
! [Xs: list_nat_nat,Y: nat,Ys: list_nat,R2: set_Pr9093778441882193744at_nat] :
( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_nat_nat_nat @ R2 ) )
=> ~ ! [X: nat > nat,Xs3: list_nat_nat] :
( ( Xs
= ( cons_nat_nat @ X @ Xs3 ) )
=> ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X @ Y ) @ R2 )
=> ~ ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs3 @ Ys ) @ ( listrel_nat_nat_nat @ R2 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_699_listrel__Cons2,axiom,
! [Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ Y @ Ys ) ) @ ( listre818007680106770737at_nat @ R2 ) )
=> ~ ! [X: product_prod_nat_nat,Xs3: list_P6011104703257516679at_nat] :
( ( Xs
= ( cons_P6512896166579812791at_nat @ X @ Xs3 ) )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 )
=> ~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs3 @ Ys ) @ ( listre818007680106770737at_nat @ R2 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_700_listrel__Cons2,axiom,
! [Xs: list_nat,Y: nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_nat_nat @ R2 ) )
=> ~ ! [X: nat,Xs3: list_nat] :
( ( Xs
= ( cons_nat @ X @ Xs3 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys ) @ ( listrel_nat_nat @ R2 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_701_min__list__Cons,axiom,
! [X3: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
( ( ord_less_eq_nat @ X3 @ 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 @ X3 @ Xs ) ) @ ( missing_min_list_nat @ ( cons_nat @ Y @ Ys ) ) ) ) ) ) ).
% min_list_Cons
thf(fact_702_remove__nth__len,axiom,
! [I: nat,Xs: list_term_a_b] :
( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( size_s8906293707977694520rm_a_b @ Xs )
= ( suc @ ( size_s8906293707977694520rm_a_b @ ( missin7590947408886171483rm_a_b @ I @ Xs ) ) ) ) ) ).
% remove_nth_len
thf(fact_703_nth__remove__nth__conv,axiom,
! [I: nat,N: nat,Xs: list_term_a_b] :
( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ ( missin7590947408886171483rm_a_b @ N @ Xs ) ) )
=> ( ( nth_term_a_b @ ( missin7590947408886171483rm_a_b @ N @ Xs ) @ I )
= ( nth_term_a_b @ Xs @ ( if_nat @ ( ord_less_nat @ I @ N ) @ I @ ( suc @ I ) ) ) ) ) ).
% nth_remove_nth_conv
thf(fact_704_length__Cons,axiom,
! [X3: term_a_b,Xs: list_term_a_b] :
( ( size_s8906293707977694520rm_a_b @ ( cons_term_a_b @ X3 @ Xs ) )
= ( suc @ ( size_s8906293707977694520rm_a_b @ Xs ) ) ) ).
% length_Cons
thf(fact_705_Cons__lenlex__iff,axiom,
! [M2: product_prod_nat_nat,Ms: list_P6011104703257516679at_nat,N: product_prod_nat_nat,Ns: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ M2 @ Ms ) @ ( cons_P6512896166579812791at_nat @ N @ Ns ) ) @ ( lenlex325483962726685836at_nat @ R2 ) )
= ( ( ord_less_nat @ ( size_s5460976970255530739at_nat @ Ms ) @ ( size_s5460976970255530739at_nat @ Ns ) )
| ( ( ( size_s5460976970255530739at_nat @ Ms )
= ( size_s5460976970255530739at_nat @ Ns ) )
& ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ M2 @ N ) @ R2 ) )
| ( ( M2 = N )
& ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Ms @ Ns ) @ ( lenlex325483962726685836at_nat @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_706_Cons__lenlex__iff,axiom,
! [M2: nat,Ms: list_nat,N: nat,Ns: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ M2 @ Ms ) @ ( cons_nat @ N @ Ns ) ) @ ( lenlex_nat @ R2 ) )
= ( ( 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 ) @ R2 ) )
| ( ( M2 = N )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_707_Cons__lenlex__iff,axiom,
! [M2: term_a_b,Ms: list_term_a_b,N: term_a_b,Ns: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ M2 @ Ms ) @ ( cons_term_a_b @ N @ Ns ) ) @ ( lenlex_term_a_b @ R2 ) )
= ( ( ord_less_nat @ ( size_s8906293707977694520rm_a_b @ Ms ) @ ( size_s8906293707977694520rm_a_b @ Ns ) )
| ( ( ( size_s8906293707977694520rm_a_b @ Ms )
= ( size_s8906293707977694520rm_a_b @ Ns ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ M2 @ N ) @ R2 ) )
| ( ( M2 = N )
& ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Ms @ Ns ) @ ( lenlex_term_a_b @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_708_Cons__in__lex,axiom,
! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys ) ) @ ( lex_Pr8571645452597969515at_nat @ R2 ) )
= ( ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ R2 )
& ( ( size_s5460976970255530739at_nat @ Xs )
= ( size_s5460976970255530739at_nat @ Ys ) ) )
| ( ( X3 = Y )
& ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys ) @ ( lex_Pr8571645452597969515at_nat @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_709_Cons__in__lex,axiom,
! [X3: nat,Xs: list_nat,Y: nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( lex_nat @ R2 ) )
= ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R2 )
& ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) )
| ( ( X3 = Y )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_710_Cons__in__lex,axiom,
! [X3: term_a_b,Xs: list_term_a_b,Y: term_a_b,Ys: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ X3 @ Xs ) @ ( cons_term_a_b @ Y @ Ys ) ) @ ( lex_term_a_b @ R2 ) )
= ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ R2 )
& ( ( size_s8906293707977694520rm_a_b @ Xs )
= ( size_s8906293707977694520rm_a_b @ Ys ) ) )
| ( ( X3 = Y )
& ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( lex_term_a_b @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_711_list__induct4,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b,Zs: list_term_a_b,Ws: list_term_a_b,P2: list_term_a_b > list_term_a_b > list_term_a_b > list_term_a_b > $o] :
( ( ( size_s8906293707977694520rm_a_b @ Xs )
= ( size_s8906293707977694520rm_a_b @ Ys ) )
=> ( ( ( size_s8906293707977694520rm_a_b @ Ys )
= ( size_s8906293707977694520rm_a_b @ Zs ) )
=> ( ( ( size_s8906293707977694520rm_a_b @ Zs )
= ( size_s8906293707977694520rm_a_b @ Ws ) )
=> ( ( P2 @ nil_term_a_b @ nil_term_a_b @ nil_term_a_b @ nil_term_a_b )
=> ( ! [X: term_a_b,Xs3: list_term_a_b,Y3: term_a_b,Ys5: list_term_a_b,Z2: term_a_b,Zs2: list_term_a_b,W: term_a_b,Ws2: list_term_a_b] :
( ( ( size_s8906293707977694520rm_a_b @ Xs3 )
= ( size_s8906293707977694520rm_a_b @ Ys5 ) )
=> ( ( ( size_s8906293707977694520rm_a_b @ Ys5 )
= ( size_s8906293707977694520rm_a_b @ Zs2 ) )
=> ( ( ( size_s8906293707977694520rm_a_b @ Zs2 )
= ( size_s8906293707977694520rm_a_b @ Ws2 ) )
=> ( ( P2 @ Xs3 @ Ys5 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_term_a_b @ X @ Xs3 ) @ ( cons_term_a_b @ Y3 @ Ys5 ) @ ( cons_term_a_b @ Z2 @ Zs2 ) @ ( cons_term_a_b @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_712_list__induct3,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b,Zs: list_term_a_b,P2: list_term_a_b > list_term_a_b > list_term_a_b > $o] :
( ( ( size_s8906293707977694520rm_a_b @ Xs )
= ( size_s8906293707977694520rm_a_b @ Ys ) )
=> ( ( ( size_s8906293707977694520rm_a_b @ Ys )
= ( size_s8906293707977694520rm_a_b @ Zs ) )
=> ( ( P2 @ nil_term_a_b @ nil_term_a_b @ nil_term_a_b )
=> ( ! [X: term_a_b,Xs3: list_term_a_b,Y3: term_a_b,Ys5: list_term_a_b,Z2: term_a_b,Zs2: list_term_a_b] :
( ( ( size_s8906293707977694520rm_a_b @ Xs3 )
= ( size_s8906293707977694520rm_a_b @ Ys5 ) )
=> ( ( ( size_s8906293707977694520rm_a_b @ Ys5 )
= ( size_s8906293707977694520rm_a_b @ Zs2 ) )
=> ( ( P2 @ Xs3 @ Ys5 @ Zs2 )
=> ( P2 @ ( cons_term_a_b @ X @ Xs3 ) @ ( cons_term_a_b @ Y3 @ Ys5 ) @ ( cons_term_a_b @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_713_list__induct2,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b,P2: list_term_a_b > list_term_a_b > $o] :
( ( ( size_s8906293707977694520rm_a_b @ Xs )
= ( size_s8906293707977694520rm_a_b @ Ys ) )
=> ( ( P2 @ nil_term_a_b @ nil_term_a_b )
=> ( ! [X: term_a_b,Xs3: list_term_a_b,Y3: term_a_b,Ys5: list_term_a_b] :
( ( ( size_s8906293707977694520rm_a_b @ Xs3 )
= ( size_s8906293707977694520rm_a_b @ Ys5 ) )
=> ( ( P2 @ Xs3 @ Ys5 )
=> ( P2 @ ( cons_term_a_b @ X @ Xs3 ) @ ( cons_term_a_b @ Y3 @ Ys5 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_714_lenlex__irreflexive,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,Xs: list_term_a_b] :
( ! [X: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ R2 )
=> ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Xs ) @ ( lenlex_term_a_b @ R2 ) ) ) ).
% lenlex_irreflexive
thf(fact_715_lenlex__irreflexive,axiom,
! [R2: set_Pr8693737435421807431at_nat,Xs: list_P6011104703257516679at_nat] :
( ! [X: product_prod_nat_nat] :
~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ X ) @ R2 )
=> ~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Xs ) @ ( lenlex325483962726685836at_nat @ R2 ) ) ) ).
% lenlex_irreflexive
thf(fact_716_lenlex__irreflexive,axiom,
! [R2: set_Pr1261947904930325089at_nat,Xs: list_nat] :
( ! [X: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X ) @ R2 )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Xs ) @ ( lenlex_nat @ R2 ) ) ) ).
% lenlex_irreflexive
thf(fact_717_replace__term__at_Ocases,axiom,
! [X3: produc2732850333517536310rm_a_b] :
( ! [S4: term_a_b,T4: term_a_b] :
( X3
!= ( produc3812856575676843240rm_a_b @ S4 @ ( produc5151171985953862413rm_a_b @ nil_nat @ T4 ) ) )
=> ( ! [X: b,V2: nat,Va: list_nat,T4: term_a_b] :
( X3
!= ( produc3812856575676843240rm_a_b @ ( var_b_a @ X ) @ ( produc5151171985953862413rm_a_b @ ( cons_nat @ V2 @ Va ) @ T4 ) ) )
=> ~ ! [F7: a,Ts3: list_term_a_b,I3: nat,Ps: list_nat,T4: term_a_b] :
( X3
!= ( produc3812856575676843240rm_a_b @ ( fun_a_b @ F7 @ Ts3 ) @ ( produc5151171985953862413rm_a_b @ ( cons_nat @ I3 @ Ps ) @ T4 ) ) ) ) ) ).
% replace_term_at.cases
thf(fact_718_ctxt__at__pos_Ocases,axiom,
! [X3: produc3697673438841856213st_nat] :
( ! [S4: term_a_b] :
( X3
!= ( produc4563063199488751885st_nat @ S4 @ nil_nat ) )
=> ( ! [F7: a,Ss3: list_term_a_b,I3: nat,P6: list_nat] :
( X3
!= ( produc4563063199488751885st_nat @ ( fun_a_b @ F7 @ Ss3 ) @ ( cons_nat @ I3 @ P6 ) ) )
=> ~ ! [X: b,V2: nat,Va: list_nat] :
( X3
!= ( produc4563063199488751885st_nat @ ( var_b_a @ X ) @ ( cons_nat @ V2 @ Va ) ) ) ) ) ).
% ctxt_at_pos.cases
thf(fact_719_subt__at_Ocases,axiom,
! [X3: produc3697673438841856213st_nat] :
( ! [S4: term_a_b] :
( X3
!= ( produc4563063199488751885st_nat @ S4 @ nil_nat ) )
=> ( ! [F7: a,Ss3: list_term_a_b,I3: nat,P6: list_nat] :
( X3
!= ( produc4563063199488751885st_nat @ ( fun_a_b @ F7 @ Ss3 ) @ ( cons_nat @ I3 @ P6 ) ) )
=> ~ ! [X: b,V2: nat,Va: list_nat] :
( X3
!= ( produc4563063199488751885st_nat @ ( var_b_a @ X ) @ ( cons_nat @ V2 @ Va ) ) ) ) ) ).
% subt_at.cases
thf(fact_720_fun__at_Ocases,axiom,
! [X3: produc3697673438841856213st_nat] :
( ! [X: b] :
( X3
!= ( produc4563063199488751885st_nat @ ( var_b_a @ X ) @ nil_nat ) )
=> ( ! [F7: a,Ts3: list_term_a_b] :
( X3
!= ( produc4563063199488751885st_nat @ ( fun_a_b @ F7 @ Ts3 ) @ nil_nat ) )
=> ( ! [F7: a,Ts3: list_term_a_b,I3: nat,P6: list_nat] :
( X3
!= ( produc4563063199488751885st_nat @ ( fun_a_b @ F7 @ Ts3 ) @ ( cons_nat @ I3 @ P6 ) ) )
=> ~ ! [Vb: b,V2: nat,Va: list_nat] :
( X3
!= ( produc4563063199488751885st_nat @ ( var_b_a @ Vb ) @ ( cons_nat @ V2 @ Va ) ) ) ) ) ) ).
% fun_at.cases
thf(fact_721_lenlex__length,axiom,
! [Ms: list_term_a_b,Ns: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Ms @ Ns ) @ ( lenlex_term_a_b @ R2 ) )
=> ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Ms ) @ ( size_s8906293707977694520rm_a_b @ Ns ) ) ) ).
% lenlex_length
thf(fact_722_listrel_Osimps,axiom,
! [A1: list_term_a_b,A22: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ A1 @ A22 ) @ ( listre1194016999521814427rm_a_b @ R2 ) )
= ( ( ( A1 = nil_term_a_b )
& ( A22 = nil_term_a_b ) )
| ? [X6: term_a_b,Y5: term_a_b,Xs2: list_term_a_b,Ys2: list_term_a_b] :
( ( A1
= ( cons_term_a_b @ X6 @ Xs2 ) )
& ( A22
= ( cons_term_a_b @ Y5 @ Ys2 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X6 @ Y5 ) @ R2 )
& ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs2 @ Ys2 ) @ ( listre1194016999521814427rm_a_b @ R2 ) ) ) ) ) ).
% listrel.simps
thf(fact_723_listrel_Osimps,axiom,
! [A1: list_a,A22: list_nat,R2: set_Pr4934435412358123699_a_nat] :
( ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ A1 @ A22 ) @ ( listrel_a_nat @ R2 ) )
= ( ( ( A1 = nil_a )
& ( A22 = nil_nat ) )
| ? [X6: a,Y5: nat,Xs2: list_a,Ys2: list_nat] :
( ( A1
= ( cons_a @ X6 @ Xs2 ) )
& ( A22
= ( cons_nat @ Y5 @ Ys2 ) )
& ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X6 @ Y5 ) @ R2 )
& ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Xs2 @ Ys2 ) @ ( listrel_a_nat @ R2 ) ) ) ) ) ).
% listrel.simps
thf(fact_724_listrel_Osimps,axiom,
! [A1: list_nat_nat,A22: list_nat,R2: set_Pr9093778441882193744at_nat] :
( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ A1 @ A22 ) @ ( listrel_nat_nat_nat @ R2 ) )
= ( ( ( A1 = nil_nat_nat )
& ( A22 = nil_nat ) )
| ? [X6: nat > nat,Y5: nat,Xs2: list_nat_nat,Ys2: list_nat] :
( ( A1
= ( cons_nat_nat @ X6 @ Xs2 ) )
& ( A22
= ( cons_nat @ Y5 @ Ys2 ) )
& ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X6 @ Y5 ) @ R2 )
& ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs2 @ Ys2 ) @ ( listrel_nat_nat_nat @ R2 ) ) ) ) ) ).
% listrel.simps
thf(fact_725_listrel_Osimps,axiom,
! [A1: list_P6011104703257516679at_nat,A22: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ A1 @ A22 ) @ ( listre818007680106770737at_nat @ R2 ) )
= ( ( ( A1 = nil_Pr5478986624290739719at_nat )
& ( A22 = nil_Pr5478986624290739719at_nat ) )
| ? [X6: product_prod_nat_nat,Y5: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
( ( A1
= ( cons_P6512896166579812791at_nat @ X6 @ Xs2 ) )
& ( A22
= ( cons_P6512896166579812791at_nat @ Y5 @ Ys2 ) )
& ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X6 @ Y5 ) @ R2 )
& ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs2 @ Ys2 ) @ ( listre818007680106770737at_nat @ R2 ) ) ) ) ) ).
% listrel.simps
thf(fact_726_listrel_Osimps,axiom,
! [A1: list_nat,A22: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A22 ) @ ( listrel_nat_nat @ R2 ) )
= ( ( ( A1 = nil_nat )
& ( A22 = nil_nat ) )
| ? [X6: nat,Y5: nat,Xs2: list_nat,Ys2: list_nat] :
( ( A1
= ( cons_nat @ X6 @ Xs2 ) )
& ( A22
= ( cons_nat @ Y5 @ Ys2 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X6 @ Y5 ) @ R2 )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel_nat_nat @ R2 ) ) ) ) ) ).
% listrel.simps
thf(fact_727_listrel_Ocases,axiom,
! [A1: list_term_a_b,A22: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ A1 @ A22 ) @ ( listre1194016999521814427rm_a_b @ R2 ) )
=> ( ( ( A1 = nil_term_a_b )
=> ( A22 != nil_term_a_b ) )
=> ~ ! [X: term_a_b,Y3: term_a_b,Xs3: list_term_a_b] :
( ( A1
= ( cons_term_a_b @ X @ Xs3 ) )
=> ! [Ys5: list_term_a_b] :
( ( A22
= ( cons_term_a_b @ Y3 @ Ys5 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ R2 )
=> ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs3 @ Ys5 ) @ ( listre1194016999521814427rm_a_b @ R2 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_728_listrel_Ocases,axiom,
! [A1: list_a,A22: list_nat,R2: set_Pr4934435412358123699_a_nat] :
( ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ A1 @ A22 ) @ ( listrel_a_nat @ R2 ) )
=> ( ( ( A1 = nil_a )
=> ( A22 != nil_nat ) )
=> ~ ! [X: a,Y3: nat,Xs3: list_a] :
( ( A1
= ( cons_a @ X @ Xs3 ) )
=> ! [Ys5: list_nat] :
( ( A22
= ( cons_nat @ Y3 @ Ys5 ) )
=> ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X @ Y3 ) @ R2 )
=> ~ ( member4851138774834033962st_nat @ ( produc4792949784200893581st_nat @ Xs3 @ Ys5 ) @ ( listrel_a_nat @ R2 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_729_listrel_Ocases,axiom,
! [A1: list_nat_nat,A22: list_nat,R2: set_Pr9093778441882193744at_nat] :
( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ A1 @ A22 ) @ ( listrel_nat_nat_nat @ R2 ) )
=> ( ( ( A1 = nil_nat_nat )
=> ( A22 != nil_nat ) )
=> ~ ! [X: nat > nat,Y3: nat,Xs3: list_nat_nat] :
( ( A1
= ( cons_nat_nat @ X @ Xs3 ) )
=> ! [Ys5: list_nat] :
( ( A22
= ( cons_nat @ Y3 @ Ys5 ) )
=> ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X @ Y3 ) @ R2 )
=> ~ ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs3 @ Ys5 ) @ ( listrel_nat_nat_nat @ R2 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_730_listrel_Ocases,axiom,
! [A1: list_P6011104703257516679at_nat,A22: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ A1 @ A22 ) @ ( listre818007680106770737at_nat @ R2 ) )
=> ( ( ( A1 = nil_Pr5478986624290739719at_nat )
=> ( A22 != nil_Pr5478986624290739719at_nat ) )
=> ~ ! [X: product_prod_nat_nat,Y3: product_prod_nat_nat,Xs3: list_P6011104703257516679at_nat] :
( ( A1
= ( cons_P6512896166579812791at_nat @ X @ Xs3 ) )
=> ! [Ys5: list_P6011104703257516679at_nat] :
( ( A22
= ( cons_P6512896166579812791at_nat @ Y3 @ Ys5 ) )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ R2 )
=> ~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs3 @ Ys5 ) @ ( listre818007680106770737at_nat @ R2 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_731_listrel_Ocases,axiom,
! [A1: list_nat,A22: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A22 ) @ ( listrel_nat_nat @ R2 ) )
=> ( ( ( A1 = nil_nat )
=> ( A22 != nil_nat ) )
=> ~ ! [X: nat,Y3: nat,Xs3: list_nat] :
( ( A1
= ( cons_nat @ X @ Xs3 ) )
=> ! [Ys5: list_nat] :
( ( A22
= ( cons_nat @ Y3 @ Ys5 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R2 )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys5 ) @ ( listrel_nat_nat @ R2 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_732_poss__of__term__Cons,axiom,
! [I: nat,P: list_nat,U: term_a_b,F3: a,Ts: list_term_a_b] :
( ( member_list_nat @ ( cons_nat @ I @ P ) @ ( terms_7168686267159881682rm_a_b @ U @ ( fun_a_b @ F3 @ Ts ) ) )
=> ( member_list_nat @ P @ ( terms_7168686267159881682rm_a_b @ U @ ( nth_term_a_b @ Ts @ I ) ) ) ) ).
% poss_of_term_Cons
thf(fact_733_lex__take__index,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys ) @ ( lex_Pr8571645452597969515at_nat @ R2 ) )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Ys ) )
=> ( ( ( take_P2173866234530122223at_nat @ I3 @ Xs )
= ( take_P2173866234530122223at_nat @ I3 @ Ys ) )
=> ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ I3 ) @ ( nth_Pr7617993195940197384at_nat @ Ys @ I3 ) ) @ R2 ) ) ) ) ) ).
% lex_take_index
thf(fact_734_lex__take__index,axiom,
! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R2 ) )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys ) )
=> ( ( ( take_nat @ I3 @ Xs )
= ( take_nat @ I3 @ Ys ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Ys @ I3 ) ) @ R2 ) ) ) ) ) ).
% lex_take_index
thf(fact_735_lex__take__index,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( lex_term_a_b @ R2 ) )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( ord_less_nat @ I3 @ ( size_s8906293707977694520rm_a_b @ Ys ) )
=> ( ( ( take_term_a_b @ I3 @ Xs )
= ( take_term_a_b @ I3 @ Ys ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ Xs @ I3 ) @ ( nth_term_a_b @ Ys @ I3 ) ) @ R2 ) ) ) ) ) ).
% lex_take_index
thf(fact_736_in__measures_I2_J,axiom,
! [X3: term_a_b,Y: term_a_b,F3: term_a_b > nat,Fs: list_term_a_b_nat] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ ( measures_term_a_b @ ( cons_term_a_b_nat @ F3 @ Fs ) ) )
= ( ( ord_less_nat @ ( F3 @ X3 ) @ ( F3 @ Y ) )
| ( ( ( F3 @ X3 )
= ( F3 @ Y ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ ( measures_term_a_b @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_737_in__measures_I2_J,axiom,
! [X3: product_prod_nat_nat,Y: product_prod_nat_nat,F3: product_prod_nat_nat > nat,Fs: list_P9162950289778280392at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( measur2679027848233739777at_nat @ ( cons_P4861729644591583992at_nat @ F3 @ Fs ) ) )
= ( ( ord_less_nat @ ( F3 @ X3 ) @ ( F3 @ Y ) )
| ( ( ( F3 @ X3 )
= ( F3 @ Y ) )
& ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( measur2679027848233739777at_nat @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_738_in__measures_I2_J,axiom,
! [X3: nat,Y: nat,F3: nat > nat,Fs: list_nat_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( measures_nat @ ( cons_nat_nat @ F3 @ Fs ) ) )
= ( ( ord_less_nat @ ( F3 @ X3 ) @ ( F3 @ Y ) )
| ( ( ( F3 @ X3 )
= ( F3 @ Y ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( measures_nat @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_739_union__list__sorted_Oelims,axiom,
! [X3: list_nat,Xa: list_nat,Y: list_nat] :
( ( ( missin8019018944680490243ed_nat @ X3 @ Xa )
= Y )
=> ( ! [X: nat,Xs3: list_nat] :
( ( X3
= ( cons_nat @ X @ Xs3 ) )
=> ! [Y3: nat,Ys5: list_nat] :
( ( Xa
= ( cons_nat @ Y3 @ Ys5 ) )
=> ~ ( ( ( X = Y3 )
=> ( Y
= ( cons_nat @ X @ ( missin8019018944680490243ed_nat @ Xs3 @ Ys5 ) ) ) )
& ( ( X != Y3 )
=> ( ( ( ord_less_nat @ X @ Y3 )
=> ( Y
= ( cons_nat @ X @ ( missin8019018944680490243ed_nat @ Xs3 @ ( cons_nat @ Y3 @ Ys5 ) ) ) ) )
& ( ~ ( ord_less_nat @ X @ Y3 )
=> ( Y
= ( cons_nat @ Y3 @ ( missin8019018944680490243ed_nat @ ( cons_nat @ X @ Xs3 ) @ Ys5 ) ) ) ) ) ) ) ) )
=> ( ( ( X3 = nil_nat )
=> ( Y != Xa ) )
=> ~ ! [V2: nat,Va: list_nat] :
( ( X3
= ( cons_nat @ V2 @ Va ) )
=> ( ( Xa = nil_nat )
=> ( Y
!= ( cons_nat @ V2 @ Va ) ) ) ) ) ) ) ).
% union_list_sorted.elims
thf(fact_740_in__measures_I1_J,axiom,
! [X3: term_a_b,Y: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ ( measures_term_a_b @ nil_term_a_b_nat ) ) ).
% in_measures(1)
thf(fact_741_in__measures_I1_J,axiom,
! [X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( measur2679027848233739777at_nat @ nil_Pr2865493887535707976at_nat ) ) ).
% in_measures(1)
thf(fact_742_in__measures_I1_J,axiom,
! [X3: nat,Y: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( measures_nat @ nil_nat_nat ) ) ).
% in_measures(1)
thf(fact_743_take__all__iff,axiom,
! [N: nat,Xs: list_term_a_b] :
( ( ( take_term_a_b @ N @ Xs )
= Xs )
= ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Xs ) @ N ) ) ).
% take_all_iff
thf(fact_744_take__all,axiom,
! [Xs: list_term_a_b,N: nat] :
( ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Xs ) @ N )
=> ( ( take_term_a_b @ N @ Xs )
= Xs ) ) ).
% take_all
thf(fact_745_nth__take,axiom,
! [I: nat,N: nat,Xs: list_term_a_b] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_term_a_b @ ( take_term_a_b @ N @ Xs ) @ I )
= ( nth_term_a_b @ Xs @ I ) ) ) ).
% nth_take
thf(fact_746_depth_Ocases,axiom,
! [X3: term_a_b] :
( ! [X: b] :
( X3
!= ( var_b_a @ X ) )
=> ( ! [F7: a] :
( X3
!= ( fun_a_b @ F7 @ nil_term_a_b ) )
=> ~ ! [F7: a,V2: term_a_b,Va: list_term_a_b] :
( X3
!= ( fun_a_b @ F7 @ ( cons_term_a_b @ V2 @ Va ) ) ) ) ) ).
% depth.cases
thf(fact_747_union__list__sorted_Osimps_I1_J,axiom,
! [X3: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
( ( ( X3 = Y )
=> ( ( missin8019018944680490243ed_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( cons_nat @ X3 @ ( missin8019018944680490243ed_nat @ Xs @ Ys ) ) ) )
& ( ( X3 != Y )
=> ( ( ( ord_less_nat @ X3 @ Y )
=> ( ( missin8019018944680490243ed_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( cons_nat @ X3 @ ( missin8019018944680490243ed_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) ) ) )
& ( ~ ( ord_less_nat @ X3 @ Y )
=> ( ( missin8019018944680490243ed_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( cons_nat @ Y @ ( missin8019018944680490243ed_nat @ ( cons_nat @ X3 @ Xs ) @ Ys ) ) ) ) ) ) ) ).
% union_list_sorted.simps(1)
thf(fact_748_nth__take__lemma,axiom,
! [K: nat,Xs: list_term_a_b,Ys: list_term_a_b] :
( ( ord_less_eq_nat @ K @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( ord_less_eq_nat @ K @ ( size_s8906293707977694520rm_a_b @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( ( nth_term_a_b @ Xs @ I3 )
= ( nth_term_a_b @ Ys @ I3 ) ) )
=> ( ( take_term_a_b @ K @ Xs )
= ( take_term_a_b @ K @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_749_nth__take__prefix,axiom,
! [Ys: list_term_a_b,Xs: list_term_a_b] :
( ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Ys ) @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s8906293707977694520rm_a_b @ Ys ) )
=> ( ( nth_term_a_b @ Xs @ I3 )
= ( nth_term_a_b @ Ys @ I3 ) ) )
=> ( ( take_term_a_b @ ( size_s8906293707977694520rm_a_b @ Ys ) @ Xs )
= Ys ) ) ) ).
% nth_take_prefix
thf(fact_750_measures__less,axiom,
! [F3: term_a_b > nat,X3: term_a_b,Y: term_a_b,Fs: list_term_a_b_nat] :
( ( ord_less_nat @ ( F3 @ X3 ) @ ( F3 @ Y ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ ( measures_term_a_b @ ( cons_term_a_b_nat @ F3 @ Fs ) ) ) ) ).
% measures_less
thf(fact_751_measures__less,axiom,
! [F3: product_prod_nat_nat > nat,X3: product_prod_nat_nat,Y: product_prod_nat_nat,Fs: list_P9162950289778280392at_nat] :
( ( ord_less_nat @ ( F3 @ X3 ) @ ( F3 @ Y ) )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( measur2679027848233739777at_nat @ ( cons_P4861729644591583992at_nat @ F3 @ Fs ) ) ) ) ).
% measures_less
thf(fact_752_measures__less,axiom,
! [F3: nat > nat,X3: nat,Y: nat,Fs: list_nat_nat] :
( ( ord_less_nat @ ( F3 @ X3 ) @ ( F3 @ Y ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( measures_nat @ ( cons_nat_nat @ F3 @ Fs ) ) ) ) ).
% measures_less
thf(fact_753_measures__lesseq,axiom,
! [F3: term_a_b > nat,X3: term_a_b,Y: term_a_b,Fs: list_term_a_b_nat] :
( ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ ( measures_term_a_b @ Fs ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ ( measures_term_a_b @ ( cons_term_a_b_nat @ F3 @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_754_measures__lesseq,axiom,
! [F3: product_prod_nat_nat > nat,X3: product_prod_nat_nat,Y: product_prod_nat_nat,Fs: list_P9162950289778280392at_nat] :
( ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y ) )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( measur2679027848233739777at_nat @ Fs ) )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( measur2679027848233739777at_nat @ ( cons_P4861729644591583992at_nat @ F3 @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_755_measures__lesseq,axiom,
! [F3: nat > nat,X3: nat,Y: nat,Fs: list_nat_nat] :
( ( ord_less_eq_nat @ ( F3 @ X3 ) @ ( F3 @ Y ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( measures_nat @ Fs ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( measures_nat @ ( cons_nat_nat @ F3 @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_756_take__Suc__conv__app__nth,axiom,
! [I: nat,Xs: list_term_a_b] :
( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( take_term_a_b @ ( suc @ I ) @ Xs )
= ( append_term_a_b @ ( take_term_a_b @ I @ Xs ) @ ( cons_term_a_b @ ( nth_term_a_b @ Xs @ I ) @ nil_term_a_b ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_757_nth__append__take__is__nth__conv,axiom,
! [I: nat,J: nat,Xs: list_term_a_b,Ys: list_term_a_b] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( nth_term_a_b @ ( append_term_a_b @ ( take_term_a_b @ J @ Xs ) @ Ys ) @ I )
= ( nth_term_a_b @ Xs @ I ) ) ) ) ).
% nth_append_take_is_nth_conv
thf(fact_758_append__eq__append__conv,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b,Us2: list_term_a_b,Vs: list_term_a_b] :
( ( ( ( size_s8906293707977694520rm_a_b @ Xs )
= ( size_s8906293707977694520rm_a_b @ Ys ) )
| ( ( size_s8906293707977694520rm_a_b @ Us2 )
= ( size_s8906293707977694520rm_a_b @ Vs ) ) )
=> ( ( ( append_term_a_b @ Xs @ Us2 )
= ( append_term_a_b @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us2 = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_759_zip__append,axiom,
! [Xs: list_term_a_b,Us2: list_term_a_b,Ys: list_term_a_b,Vs: list_term_a_b] :
( ( ( size_s8906293707977694520rm_a_b @ Xs )
= ( size_s8906293707977694520rm_a_b @ Us2 ) )
=> ( ( zip_te2260252107881041761rm_a_b @ ( append_term_a_b @ Xs @ Ys ) @ ( append_term_a_b @ Us2 @ Vs ) )
= ( append1152621114427393060rm_a_b @ ( zip_te2260252107881041761rm_a_b @ Xs @ Us2 ) @ ( zip_te2260252107881041761rm_a_b @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_760_nth__append__length,axiom,
! [Xs: list_term_a_b,X3: term_a_b,Ys: list_term_a_b] :
( ( nth_term_a_b @ ( append_term_a_b @ Xs @ ( cons_term_a_b @ X3 @ Ys ) ) @ ( size_s8906293707977694520rm_a_b @ Xs ) )
= X3 ) ).
% nth_append_length
thf(fact_761_same__length__different,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b] :
( ( Xs != Ys )
=> ( ( ( size_s8906293707977694520rm_a_b @ Xs )
= ( size_s8906293707977694520rm_a_b @ Ys ) )
=> ? [Pre: list_term_a_b,X: term_a_b,Xs4: list_term_a_b,Y3: term_a_b,Ys4: list_term_a_b] :
( ( X != Y3 )
& ( Xs
= ( append_term_a_b @ Pre @ ( append_term_a_b @ ( cons_term_a_b @ X @ nil_term_a_b ) @ Xs4 ) ) )
& ( Ys
= ( append_term_a_b @ Pre @ ( append_term_a_b @ ( cons_term_a_b @ Y3 @ nil_term_a_b ) @ Ys4 ) ) ) ) ) ) ).
% same_length_different
thf(fact_762_append__Cons__nth__not__middle,axiom,
! [I: nat,Xs: list_term_a_b,U: term_a_b,Ys: list_term_a_b,Z3: term_a_b] :
( ( I
!= ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( nth_term_a_b @ ( append_term_a_b @ Xs @ ( cons_term_a_b @ U @ Ys ) ) @ I )
= ( nth_term_a_b @ ( append_term_a_b @ Xs @ ( cons_term_a_b @ Z3 @ Ys ) ) @ I ) ) ) ).
% append_Cons_nth_not_middle
thf(fact_763_append__Cons__nth__middle,axiom,
! [I: nat,Xs: list_term_a_b,Y: term_a_b,Zs: list_term_a_b] :
( ( I
= ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( nth_term_a_b @ ( append_term_a_b @ Xs @ ( cons_term_a_b @ Y @ Zs ) ) @ I )
= Y ) ) ).
% append_Cons_nth_middle
thf(fact_764_P__as__bs__extend,axiom,
! [As: list_term_a_b,Bs: list_term_a_b,Cs: list_term_a_b,Ds: list_term_a_b,P2: term_a_b > term_a_b > $o] :
( ( ( size_s8906293707977694520rm_a_b @ As )
= ( size_s8906293707977694520rm_a_b @ Bs ) )
=> ( ( ( size_s8906293707977694520rm_a_b @ Cs )
= ( size_s8906293707977694520rm_a_b @ Ds ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s8906293707977694520rm_a_b @ Bs ) )
=> ( P2 @ ( nth_term_a_b @ As @ I3 ) @ ( nth_term_a_b @ Bs @ I3 ) ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s8906293707977694520rm_a_b @ Ds ) )
=> ( P2 @ ( nth_term_a_b @ Cs @ I3 ) @ ( nth_term_a_b @ Ds @ I3 ) ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s8906293707977694520rm_a_b @ ( append_term_a_b @ Bs @ Ds ) ) )
=> ( P2 @ ( nth_term_a_b @ ( append_term_a_b @ As @ Cs ) @ I2 ) @ ( nth_term_a_b @ ( append_term_a_b @ Bs @ Ds ) @ I2 ) ) ) ) ) ) ) ).
% P_as_bs_extend
thf(fact_765_lex__append__left__iff,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,Xs: list_term_a_b,Ys: list_term_a_b,Zs: list_term_a_b] :
( ! [X: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ R2 )
=> ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( append_term_a_b @ Xs @ Ys ) @ ( append_term_a_b @ Xs @ Zs ) ) @ ( lex_term_a_b @ R2 ) )
= ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Ys @ Zs ) @ ( lex_term_a_b @ R2 ) ) ) ) ).
% lex_append_left_iff
thf(fact_766_lex__append__left__iff,axiom,
! [R2: set_Pr8693737435421807431at_nat,Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat] :
( ! [X: product_prod_nat_nat] :
~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ X ) @ R2 )
=> ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( append985823374593552924at_nat @ Xs @ Ys ) @ ( append985823374593552924at_nat @ Xs @ Zs ) ) @ ( lex_Pr8571645452597969515at_nat @ R2 ) )
= ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Ys @ Zs ) @ ( lex_Pr8571645452597969515at_nat @ R2 ) ) ) ) ).
% lex_append_left_iff
thf(fact_767_lex__append__left__iff,axiom,
! [R2: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ! [X: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X ) @ R2 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Xs @ Zs ) ) @ ( lex_nat @ R2 ) )
= ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lex_nat @ R2 ) ) ) ) ).
% lex_append_left_iff
thf(fact_768_lex__append__leftD,axiom,
! [R2: set_Pr4386577575007340137rm_a_b,Xs: list_term_a_b,Ys: list_term_a_b,Zs: list_term_a_b] :
( ! [X: term_a_b] :
~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ X ) @ R2 )
=> ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( append_term_a_b @ Xs @ Ys ) @ ( append_term_a_b @ Xs @ Zs ) ) @ ( lex_term_a_b @ R2 ) )
=> ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Ys @ Zs ) @ ( lex_term_a_b @ R2 ) ) ) ) ).
% lex_append_leftD
thf(fact_769_lex__append__leftD,axiom,
! [R2: set_Pr8693737435421807431at_nat,Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat] :
( ! [X: product_prod_nat_nat] :
~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ X ) @ R2 )
=> ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( append985823374593552924at_nat @ Xs @ Ys ) @ ( append985823374593552924at_nat @ Xs @ Zs ) ) @ ( lex_Pr8571645452597969515at_nat @ R2 ) )
=> ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Ys @ Zs ) @ ( lex_Pr8571645452597969515at_nat @ R2 ) ) ) ) ).
% lex_append_leftD
thf(fact_770_lex__append__leftD,axiom,
! [R2: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ! [X: nat] :
~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X ) @ R2 )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Xs @ Zs ) ) @ ( lex_nat @ R2 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lex_nat @ R2 ) ) ) ) ).
% lex_append_leftD
thf(fact_771_lex__append__rightI,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b,Vs: list_term_a_b,Us2: list_term_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( lex_term_a_b @ R2 ) )
=> ( ( ( size_s8906293707977694520rm_a_b @ Vs )
= ( size_s8906293707977694520rm_a_b @ Us2 ) )
=> ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( append_term_a_b @ Xs @ Us2 ) @ ( append_term_a_b @ Ys @ Vs ) ) @ ( lex_term_a_b @ R2 ) ) ) ) ).
% lex_append_rightI
thf(fact_772_lenlex__append1,axiom,
! [Us2: list_term_a_b,Xs: list_term_a_b,R3: set_Pr4386577575007340137rm_a_b,Vs: list_term_a_b,Ys: list_term_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Us2 @ Xs ) @ ( lenlex_term_a_b @ R3 ) )
=> ( ( ( size_s8906293707977694520rm_a_b @ Vs )
= ( size_s8906293707977694520rm_a_b @ Ys ) )
=> ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( append_term_a_b @ Us2 @ Vs ) @ ( append_term_a_b @ Xs @ Ys ) ) @ ( lenlex_term_a_b @ R3 ) ) ) ) ).
% lenlex_append1
thf(fact_773_length__Suc__conv__rev,axiom,
! [Xs: list_term_a_b,N: nat] :
( ( ( size_s8906293707977694520rm_a_b @ Xs )
= ( suc @ N ) )
= ( ? [Y5: term_a_b,Ys2: list_term_a_b] :
( ( Xs
= ( append_term_a_b @ Ys2 @ ( cons_term_a_b @ Y5 @ nil_term_a_b ) ) )
& ( ( size_s8906293707977694520rm_a_b @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_774_length__append__singleton,axiom,
! [Xs: list_term_a_b,X3: term_a_b] :
( ( size_s8906293707977694520rm_a_b @ ( append_term_a_b @ Xs @ ( cons_term_a_b @ X3 @ nil_term_a_b ) ) )
= ( suc @ ( size_s8906293707977694520rm_a_b @ Xs ) ) ) ).
% length_append_singleton
thf(fact_775_append__Cons__nth__right,axiom,
! [Xs: list_term_a_b,I: nat,U: term_a_b,Ys: list_term_a_b,Z3: term_a_b] :
( ( ord_less_nat @ ( size_s8906293707977694520rm_a_b @ Xs ) @ I )
=> ( ( nth_term_a_b @ ( append_term_a_b @ Xs @ ( cons_term_a_b @ U @ Ys ) ) @ I )
= ( nth_term_a_b @ ( append_term_a_b @ Xs @ ( cons_term_a_b @ Z3 @ Ys ) ) @ I ) ) ) ).
% append_Cons_nth_right
thf(fact_776_append__Cons__nth__left,axiom,
! [I: nat,Xs: list_term_a_b,U: term_a_b,Ys: list_term_a_b] :
( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( nth_term_a_b @ ( append_term_a_b @ Xs @ ( cons_term_a_b @ U @ Ys ) ) @ I )
= ( nth_term_a_b @ Xs @ I ) ) ) ).
% append_Cons_nth_left
thf(fact_777_nth__append__take,axiom,
! [I: nat,Xs: list_term_a_b,Y: term_a_b,Ys: list_term_a_b] :
( ( ord_less_eq_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( nth_term_a_b @ ( append_term_a_b @ ( take_term_a_b @ I @ Xs ) @ ( cons_term_a_b @ Y @ Ys ) ) @ I )
= Y ) ) ).
% nth_append_take
thf(fact_778_nth__append__drop__is__nth__conv,axiom,
! [J: nat,I: nat,Xs: list_term_a_b,Y: term_a_b] :
( ( ord_less_nat @ J @ I )
=> ( ( ord_less_eq_nat @ J @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( ord_less_eq_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( nth_term_a_b @ ( append_term_a_b @ ( take_term_a_b @ J @ Xs ) @ ( cons_term_a_b @ Y @ ( drop_term_a_b @ ( suc @ J ) @ Xs ) ) ) @ I )
= ( nth_term_a_b @ Xs @ I ) ) ) ) ) ).
% nth_append_drop_is_nth_conv
thf(fact_779_nth__append__take__drop__is__nth__conv,axiom,
! [I: nat,Xs: list_term_a_b,J: nat,Y: term_a_b] :
( ( ord_less_eq_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( ord_less_eq_nat @ J @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( I != J )
=> ( ( nth_term_a_b @ ( append_term_a_b @ ( take_term_a_b @ J @ Xs ) @ ( cons_term_a_b @ Y @ ( drop_term_a_b @ ( suc @ J ) @ Xs ) ) ) @ I )
= ( nth_term_a_b @ Xs @ I ) ) ) ) ) ).
% nth_append_take_drop_is_nth_conv
thf(fact_780_id__take__nth__drop,axiom,
! [I: nat,Xs: list_term_a_b] :
( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( Xs
= ( append_term_a_b @ ( take_term_a_b @ I @ Xs ) @ ( cons_term_a_b @ ( nth_term_a_b @ Xs @ I ) @ ( drop_term_a_b @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_781_drop__all,axiom,
! [Xs: list_term_a_b,N: nat] :
( ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Xs ) @ N )
=> ( ( drop_term_a_b @ N @ Xs )
= nil_term_a_b ) ) ).
% drop_all
thf(fact_782_drop__eq__Nil,axiom,
! [N: nat,Xs: list_term_a_b] :
( ( ( drop_term_a_b @ N @ Xs )
= nil_term_a_b )
= ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_783_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_term_a_b] :
( ( nil_term_a_b
= ( drop_term_a_b @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_784_nth__via__drop,axiom,
! [N: nat,Xs: list_term_a_b,Y: term_a_b,Ys: list_term_a_b] :
( ( ( drop_term_a_b @ N @ Xs )
= ( cons_term_a_b @ Y @ Ys ) )
=> ( ( nth_term_a_b @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_785_append__eq__conv__conj,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b,Zs: list_term_a_b] :
( ( ( append_term_a_b @ Xs @ Ys )
= Zs )
= ( ( Xs
= ( take_term_a_b @ ( size_s8906293707977694520rm_a_b @ Xs ) @ Zs ) )
& ( Ys
= ( drop_term_a_b @ ( size_s8906293707977694520rm_a_b @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_786_append__eq__append__conv__if,axiom,
! [Xs_1: list_term_a_b,Xs_2: list_term_a_b,Ys_1: list_term_a_b,Ys_2: list_term_a_b] :
( ( ( append_term_a_b @ Xs_1 @ Xs_2 )
= ( append_term_a_b @ Ys_1 @ Ys_2 ) )
= ( ( ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Xs_1 ) @ ( size_s8906293707977694520rm_a_b @ Ys_1 ) )
=> ( ( Xs_1
= ( take_term_a_b @ ( size_s8906293707977694520rm_a_b @ Xs_1 ) @ Ys_1 ) )
& ( Xs_2
= ( append_term_a_b @ ( drop_term_a_b @ ( size_s8906293707977694520rm_a_b @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Xs_1 ) @ ( size_s8906293707977694520rm_a_b @ Ys_1 ) )
=> ( ( ( take_term_a_b @ ( size_s8906293707977694520rm_a_b @ Ys_1 ) @ Xs_1 )
= Ys_1 )
& ( ( append_term_a_b @ ( drop_term_a_b @ ( size_s8906293707977694520rm_a_b @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
= Ys_2 ) ) ) ) ) ).
% append_eq_append_conv_if
thf(fact_787_Cons__nth__drop__Suc,axiom,
! [I: nat,Xs: list_term_a_b] :
( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( cons_term_a_b @ ( nth_term_a_b @ Xs @ I ) @ ( drop_term_a_b @ ( suc @ I ) @ Xs ) )
= ( drop_term_a_b @ I @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_788_take__drop__imp__nth,axiom,
! [I: nat,Ss: list_term_a_b,X3: term_a_b] :
( ( ( append_term_a_b @ ( take_term_a_b @ I @ Ss ) @ ( cons_term_a_b @ X3 @ ( drop_term_a_b @ ( suc @ I ) @ Ss ) ) )
= Ss )
=> ( X3
= ( nth_term_a_b @ Ss @ I ) ) ) ).
% take_drop_imp_nth
thf(fact_789_take__hd__drop,axiom,
! [N: nat,Xs: list_term_a_b] :
( ( ord_less_nat @ N @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( append_term_a_b @ ( take_term_a_b @ N @ Xs ) @ ( cons_term_a_b @ ( hd_term_a_b @ ( drop_term_a_b @ N @ Xs ) ) @ nil_term_a_b ) )
= ( take_term_a_b @ ( suc @ N ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_790_take__drop__update__first,axiom,
! [J: nat,Ds: list_term_a_b,Cs: list_term_a_b] :
( ( ord_less_nat @ J @ ( size_s8906293707977694520rm_a_b @ Ds ) )
=> ( ( ( size_s8906293707977694520rm_a_b @ Cs )
= ( size_s8906293707977694520rm_a_b @ Ds ) )
=> ( ( list_update_term_a_b @ ( append_term_a_b @ ( take_term_a_b @ J @ Ds ) @ ( drop_term_a_b @ J @ Cs ) ) @ J @ ( nth_term_a_b @ Ds @ J ) )
= ( append_term_a_b @ ( take_term_a_b @ ( suc @ J ) @ Ds ) @ ( drop_term_a_b @ ( suc @ J ) @ Cs ) ) ) ) ) ).
% take_drop_update_first
thf(fact_791_upd__conv__take__nth__drop,axiom,
! [I: nat,Xs: list_term_a_b,A: term_a_b] :
( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( list_update_term_a_b @ Xs @ I @ A )
= ( append_term_a_b @ ( take_term_a_b @ I @ Xs ) @ ( cons_term_a_b @ A @ ( drop_term_a_b @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_792_take__nth__drop__concat,axiom,
! [I: nat,Xss: list_list_term_a_b,Ys: list_term_a_b,J: nat,Z3: term_a_b] :
( ( ord_less_nat @ I @ ( size_s877380706853472072rm_a_b @ Xss ) )
=> ( ( ( nth_list_term_a_b @ Xss @ I )
= Ys )
=> ( ( ord_less_nat @ J @ ( size_s8906293707977694520rm_a_b @ Ys ) )
=> ( ( ( nth_term_a_b @ Ys @ J )
= Z3 )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ ( size_s8906293707977694520rm_a_b @ ( concat_term_a_b @ Xss ) ) )
& ( ( take_term_a_b @ K2 @ ( concat_term_a_b @ Xss ) )
= ( append_term_a_b @ ( concat_term_a_b @ ( take_list_term_a_b @ I @ Xss ) ) @ ( take_term_a_b @ J @ Ys ) ) )
& ( ( nth_term_a_b @ ( concat_term_a_b @ Xss ) @ K2 )
= ( nth_term_a_b @ ( nth_list_term_a_b @ Xss @ I ) @ J ) )
& ( ( drop_term_a_b @ ( suc @ K2 ) @ ( concat_term_a_b @ Xss ) )
= ( append_term_a_b @ ( drop_term_a_b @ ( suc @ J ) @ Ys ) @ ( concat_term_a_b @ ( drop_list_term_a_b @ ( suc @ I ) @ Xss ) ) ) ) ) ) ) ) ) ).
% take_nth_drop_concat
thf(fact_793_length__list__update,axiom,
! [Xs: list_term_a_b,I: nat,X3: term_a_b] :
( ( size_s8906293707977694520rm_a_b @ ( list_update_term_a_b @ Xs @ I @ X3 ) )
= ( size_s8906293707977694520rm_a_b @ Xs ) ) ).
% length_list_update
thf(fact_794_list__update__id,axiom,
! [Xs: list_term_a_b,I: nat] :
( ( list_update_term_a_b @ Xs @ I @ ( nth_term_a_b @ Xs @ I ) )
= Xs ) ).
% list_update_id
thf(fact_795_nth__list__update__neq,axiom,
! [I: nat,J: nat,Xs: list_term_a_b,X3: term_a_b] :
( ( I != J )
=> ( ( nth_term_a_b @ ( list_update_term_a_b @ Xs @ I @ X3 ) @ J )
= ( nth_term_a_b @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_796_list__update__beyond,axiom,
! [Xs: list_term_a_b,I: nat,X3: term_a_b] :
( ( ord_less_eq_nat @ ( size_s8906293707977694520rm_a_b @ Xs ) @ I )
=> ( ( list_update_term_a_b @ Xs @ I @ X3 )
= Xs ) ) ).
% list_update_beyond
thf(fact_797_list__update__length,axiom,
! [Xs: list_term_a_b,X3: term_a_b,Ys: list_term_a_b,Y: term_a_b] :
( ( list_update_term_a_b @ ( append_term_a_b @ Xs @ ( cons_term_a_b @ X3 @ Ys ) ) @ ( size_s8906293707977694520rm_a_b @ Xs ) @ Y )
= ( append_term_a_b @ Xs @ ( cons_term_a_b @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_798_nth__list__update__eq,axiom,
! [I: nat,Xs: list_term_a_b,X3: term_a_b] :
( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( nth_term_a_b @ ( list_update_term_a_b @ Xs @ I @ X3 ) @ I )
= X3 ) ) ).
% nth_list_update_eq
thf(fact_799_zip__update,axiom,
! [Xs: list_term_a_b,I: nat,X3: term_a_b,Ys: list_term_a_b,Y: term_a_b] :
( ( zip_te2260252107881041761rm_a_b @ ( list_update_term_a_b @ Xs @ I @ X3 ) @ ( list_update_term_a_b @ Ys @ I @ Y ) )
= ( list_u4211539095652206455rm_a_b @ ( zip_te2260252107881041761rm_a_b @ Xs @ Ys ) @ I @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) ) ) ).
% zip_update
thf(fact_800_zip__update,axiom,
! [Xs: list_a,I: nat,X3: a,Ys: list_nat,Y: nat] :
( ( zip_a_nat @ ( list_update_a @ Xs @ I @ X3 ) @ ( list_update_nat @ Ys @ I @ Y ) )
= ( list_u4318310736832020111_a_nat @ ( zip_a_nat @ Xs @ Ys ) @ I @ ( product_Pair_a_nat @ X3 @ Y ) ) ) ).
% zip_update
thf(fact_801_zip__update,axiom,
! [Xs: list_nat_nat,I: nat,X3: nat > nat,Ys: list_nat,Y: nat] :
( ( zip_nat_nat_nat @ ( list_update_nat_nat @ Xs @ I @ X3 ) @ ( list_update_nat @ Ys @ I @ Y ) )
= ( list_u5493879213508619102at_nat @ ( zip_nat_nat_nat @ Xs @ Ys ) @ I @ ( produc72220940542539688at_nat @ X3 @ Y ) ) ) ).
% zip_update
thf(fact_802_zip__update,axiom,
! [Xs: list_P6011104703257516679at_nat,I: nat,X3: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat,Y: product_prod_nat_nat] :
( ( zip_Pr4664179122662387191at_nat @ ( list_u6180841689913720943at_nat @ Xs @ I @ X3 ) @ ( list_u6180841689913720943at_nat @ Ys @ I @ Y ) )
= ( list_u5003261594476800725at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs @ Ys ) @ I @ ( produc6161850002892822231at_nat @ X3 @ Y ) ) ) ).
% zip_update
thf(fact_803_zip__update,axiom,
! [Xs: list_nat,I: nat,X3: nat,Ys: list_nat,Y: nat] :
( ( zip_nat_nat @ ( list_update_nat @ Xs @ I @ X3 ) @ ( list_update_nat @ Ys @ I @ Y ) )
= ( list_u6180841689913720943at_nat @ ( zip_nat_nat @ Xs @ Ys ) @ I @ ( product_Pair_nat_nat @ X3 @ Y ) ) ) ).
% zip_update
thf(fact_804_hd__zip,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b] :
( ( Xs != nil_term_a_b )
=> ( ( Ys != nil_term_a_b )
=> ( ( hd_Pro6001814652670619252rm_a_b @ ( zip_te2260252107881041761rm_a_b @ Xs @ Ys ) )
= ( produc7020197800436672577rm_a_b @ ( hd_term_a_b @ Xs ) @ ( hd_term_a_b @ Ys ) ) ) ) ) ).
% hd_zip
thf(fact_805_hd__zip,axiom,
! [Xs: list_a,Ys: list_nat] :
( ( Xs != nil_a )
=> ( ( Ys != nil_nat )
=> ( ( hd_Pro8935205257713178578_a_nat @ ( zip_a_nat @ Xs @ Ys ) )
= ( product_Pair_a_nat @ ( hd_a @ Xs ) @ ( hd_nat @ Ys ) ) ) ) ) ).
% hd_zip
thf(fact_806_hd__zip,axiom,
! [Xs: list_nat_nat,Ys: list_nat] :
( ( Xs != nil_nat_nat )
=> ( ( Ys != nil_nat )
=> ( ( hd_Pro5468458281938048987at_nat @ ( zip_nat_nat_nat @ Xs @ Ys ) )
= ( produc72220940542539688at_nat @ ( hd_nat_nat @ Xs ) @ ( hd_nat @ Ys ) ) ) ) ) ).
% hd_zip
thf(fact_807_hd__zip,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
( ( Xs != nil_Pr5478986624290739719at_nat )
=> ( ( Ys != nil_Pr5478986624290739719at_nat )
=> ( ( hd_Pro8462011474880202578at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs @ Ys ) )
= ( produc6161850002892822231at_nat @ ( hd_Pro3460610213475200108at_nat @ Xs ) @ ( hd_Pro3460610213475200108at_nat @ Ys ) ) ) ) ) ).
% hd_zip
thf(fact_808_hd__zip,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs != nil_nat )
=> ( ( Ys != nil_nat )
=> ( ( hd_Pro3460610213475200108at_nat @ ( zip_nat_nat @ Xs @ Ys ) )
= ( product_Pair_nat_nat @ ( hd_nat @ Xs ) @ ( hd_nat @ Ys ) ) ) ) ) ).
% hd_zip
thf(fact_809_eq__length__concat__nth,axiom,
! [Xs: list_list_term_a_b,Ys: list_list_term_a_b] :
( ( ( size_s877380706853472072rm_a_b @ Xs )
= ( size_s877380706853472072rm_a_b @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s877380706853472072rm_a_b @ Xs ) )
=> ( ( size_s8906293707977694520rm_a_b @ ( nth_list_term_a_b @ Xs @ I3 ) )
= ( size_s8906293707977694520rm_a_b @ ( nth_list_term_a_b @ Ys @ I3 ) ) ) )
=> ( ( size_s8906293707977694520rm_a_b @ ( concat_term_a_b @ Xs ) )
= ( size_s8906293707977694520rm_a_b @ ( concat_term_a_b @ Ys ) ) ) ) ) ).
% eq_length_concat_nth
thf(fact_810_nth__concat__two__lists,axiom,
! [I: nat,Xs: list_list_term_a_b,Ys: list_list_term_a_b] :
( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ ( concat_term_a_b @ Xs ) ) )
=> ( ( ( size_s877380706853472072rm_a_b @ Ys )
= ( size_s877380706853472072rm_a_b @ Xs ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s877380706853472072rm_a_b @ Xs ) )
=> ( ( size_s8906293707977694520rm_a_b @ ( nth_list_term_a_b @ Ys @ I3 ) )
= ( size_s8906293707977694520rm_a_b @ ( nth_list_term_a_b @ Xs @ I3 ) ) ) )
=> ? [J2: nat,K2: nat] :
( ( ord_less_nat @ J2 @ ( size_s877380706853472072rm_a_b @ Xs ) )
& ( ord_less_nat @ K2 @ ( size_s8906293707977694520rm_a_b @ ( nth_list_term_a_b @ Xs @ J2 ) ) )
& ( ( nth_term_a_b @ ( concat_term_a_b @ Xs ) @ I )
= ( nth_term_a_b @ ( nth_list_term_a_b @ Xs @ J2 ) @ K2 ) )
& ( ( nth_term_a_b @ ( concat_term_a_b @ Ys ) @ I )
= ( nth_term_a_b @ ( nth_list_term_a_b @ Ys @ J2 ) @ K2 ) ) ) ) ) ) ).
% nth_concat_two_lists
thf(fact_811_nth__concat__split,axiom,
! [I: nat,Xs: list_list_term_a_b] :
( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ ( concat_term_a_b @ Xs ) ) )
=> ? [J2: nat,K2: nat] :
( ( ord_less_nat @ J2 @ ( size_s877380706853472072rm_a_b @ Xs ) )
& ( ord_less_nat @ K2 @ ( size_s8906293707977694520rm_a_b @ ( nth_list_term_a_b @ Xs @ J2 ) ) )
& ( ( nth_term_a_b @ ( concat_term_a_b @ Xs ) @ I )
= ( nth_term_a_b @ ( nth_list_term_a_b @ Xs @ J2 ) @ K2 ) ) ) ) ).
% nth_concat_split
thf(fact_812_nth__concat__diff,axiom,
! [I1: nat,Xs: list_list_term_a_b,I22: nat] :
( ( ord_less_nat @ I1 @ ( size_s8906293707977694520rm_a_b @ ( concat_term_a_b @ Xs ) ) )
=> ( ( ord_less_nat @ I22 @ ( size_s8906293707977694520rm_a_b @ ( concat_term_a_b @ Xs ) ) )
=> ( ( I1 != I22 )
=> ? [J1: nat,K1: nat,J22: nat,K22: nat] :
( ( ( product_Pair_nat_nat @ J1 @ K1 )
!= ( product_Pair_nat_nat @ J22 @ K22 ) )
& ( ord_less_nat @ J1 @ ( size_s877380706853472072rm_a_b @ Xs ) )
& ( ord_less_nat @ J22 @ ( size_s877380706853472072rm_a_b @ Xs ) )
& ( ord_less_nat @ K1 @ ( size_s8906293707977694520rm_a_b @ ( nth_list_term_a_b @ Xs @ J1 ) ) )
& ( ord_less_nat @ K22 @ ( size_s8906293707977694520rm_a_b @ ( nth_list_term_a_b @ Xs @ J22 ) ) )
& ( ( nth_term_a_b @ ( concat_term_a_b @ Xs ) @ I1 )
= ( nth_term_a_b @ ( nth_list_term_a_b @ Xs @ J1 ) @ K1 ) )
& ( ( nth_term_a_b @ ( concat_term_a_b @ Xs ) @ I22 )
= ( nth_term_a_b @ ( nth_list_term_a_b @ Xs @ J22 ) @ K22 ) ) ) ) ) ) ).
% nth_concat_diff
thf(fact_813_concat__all__nth,axiom,
! [Xs: list_list_term_a_b,Ys: list_list_term_a_b,P2: term_a_b > term_a_b > $o] :
( ( ( size_s877380706853472072rm_a_b @ Xs )
= ( size_s877380706853472072rm_a_b @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s877380706853472072rm_a_b @ Xs ) )
=> ( ( size_s8906293707977694520rm_a_b @ ( nth_list_term_a_b @ Xs @ I3 ) )
= ( size_s8906293707977694520rm_a_b @ ( nth_list_term_a_b @ Ys @ I3 ) ) ) )
=> ( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ ( size_s877380706853472072rm_a_b @ Xs ) )
=> ( ( ord_less_nat @ J2 @ ( size_s8906293707977694520rm_a_b @ ( nth_list_term_a_b @ Xs @ I3 ) ) )
=> ( P2 @ ( nth_term_a_b @ ( nth_list_term_a_b @ Xs @ I3 ) @ J2 ) @ ( nth_term_a_b @ ( nth_list_term_a_b @ Ys @ I3 ) @ J2 ) ) ) )
=> ! [K3: nat] :
( ( ord_less_nat @ K3 @ ( size_s8906293707977694520rm_a_b @ ( concat_term_a_b @ Xs ) ) )
=> ( P2 @ ( nth_term_a_b @ ( concat_term_a_b @ Xs ) @ K3 ) @ ( nth_term_a_b @ ( concat_term_a_b @ Ys ) @ K3 ) ) ) ) ) ) ).
% concat_all_nth
thf(fact_814_list__update__append1,axiom,
! [I: nat,Xs: list_term_a_b,Ys: list_term_a_b,X3: term_a_b] :
( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( list_update_term_a_b @ ( append_term_a_b @ Xs @ Ys ) @ I @ X3 )
= ( append_term_a_b @ ( list_update_term_a_b @ Xs @ I @ X3 ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_815_parallel__list__update,axiom,
! [N: nat,R2: term_a_b > term_a_b > $o,P: list_term_a_b > $o,Xs: list_term_a_b,Ys: list_term_a_b] :
( ! [Xs3: list_term_a_b,I3: nat,Y3: term_a_b] :
( ( ( size_s8906293707977694520rm_a_b @ Xs3 )
= N )
=> ( ( ord_less_nat @ I3 @ N )
=> ( ( R2 @ ( nth_term_a_b @ Xs3 @ I3 ) @ Y3 )
=> ( ( P @ Xs3 )
=> ( P @ ( list_update_term_a_b @ Xs3 @ I3 @ Y3 ) ) ) ) ) )
=> ( ( ( size_s8906293707977694520rm_a_b @ Xs )
= N )
=> ( ( P @ Xs )
=> ( ( ( size_s8906293707977694520rm_a_b @ Ys )
= N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( R2 @ ( nth_term_a_b @ Xs @ I3 ) @ ( nth_term_a_b @ Ys @ I3 ) ) )
=> ( P @ Ys ) ) ) ) ) ) ).
% parallel_list_update
thf(fact_816_list__update__same__conv,axiom,
! [I: nat,Xs: list_term_a_b,X3: term_a_b] :
( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( ( list_update_term_a_b @ Xs @ I @ X3 )
= Xs )
= ( ( nth_term_a_b @ Xs @ I )
= X3 ) ) ) ).
% list_update_same_conv
thf(fact_817_nth__list__update,axiom,
! [I: nat,Xs: list_term_a_b,J: nat,X3: term_a_b] :
( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth_term_a_b @ ( list_update_term_a_b @ Xs @ I @ X3 ) @ J )
= X3 ) )
& ( ( I != J )
=> ( ( nth_term_a_b @ ( list_update_term_a_b @ Xs @ I @ X3 ) @ J )
= ( nth_term_a_b @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_818_hd__drop__conv__nth,axiom,
! [N: nat,Xs: list_term_a_b] :
( ( ord_less_nat @ N @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( hd_term_a_b @ ( drop_term_a_b @ N @ Xs ) )
= ( nth_term_a_b @ Xs @ N ) ) ) ).
% hd_drop_conv_nth
thf(fact_819_take__drop__update__second,axiom,
! [J: nat,Ds: list_term_a_b,Cs: list_term_a_b] :
( ( ord_less_nat @ J @ ( size_s8906293707977694520rm_a_b @ Ds ) )
=> ( ( ( size_s8906293707977694520rm_a_b @ Cs )
= ( size_s8906293707977694520rm_a_b @ Ds ) )
=> ( ( list_update_term_a_b @ ( append_term_a_b @ ( take_term_a_b @ J @ Ds ) @ ( drop_term_a_b @ J @ Cs ) ) @ J @ ( nth_term_a_b @ Cs @ J ) )
= ( append_term_a_b @ ( take_term_a_b @ J @ Ds ) @ ( drop_term_a_b @ J @ Cs ) ) ) ) ) ).
% take_drop_update_second
thf(fact_820_replace__term__at_Oelims,axiom,
! [X3: term_a_b,Xa: list_nat,Xb: term_a_b,Y: term_a_b] :
( ( ( term_r6860082780075436317at_a_b @ X3 @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_nat )
=> ( Y != Xb ) )
=> ( ! [X: b] :
( ( X3
= ( var_b_a @ X ) )
=> ( ? [V2: nat,Va: list_nat] :
( Xa
= ( cons_nat @ V2 @ Va ) )
=> ( Y
!= ( var_b_a @ X ) ) ) )
=> ~ ! [F7: a,Ts3: list_term_a_b] :
( ( X3
= ( fun_a_b @ F7 @ Ts3 ) )
=> ! [I3: nat,Ps: list_nat] :
( ( Xa
= ( cons_nat @ I3 @ Ps ) )
=> ~ ( ( ( ord_less_nat @ I3 @ ( size_s8906293707977694520rm_a_b @ Ts3 ) )
=> ( Y
= ( fun_a_b @ F7 @ ( list_update_term_a_b @ Ts3 @ I3 @ ( term_r6860082780075436317at_a_b @ ( nth_term_a_b @ Ts3 @ I3 ) @ Ps @ Xb ) ) ) ) )
& ( ~ ( ord_less_nat @ I3 @ ( size_s8906293707977694520rm_a_b @ Ts3 ) )
=> ( Y
= ( fun_a_b @ F7 @ Ts3 ) ) ) ) ) ) ) ) ) ).
% replace_term_at.elims
thf(fact_821_listrel1__iff__update,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys ) @ ( listre4828114922151135584at_nat @ R2 ) )
= ( ? [Y5: product_prod_nat_nat,N2: nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ N2 ) @ Y5 ) @ R2 )
& ( ord_less_nat @ N2 @ ( size_s5460976970255530739at_nat @ Xs ) )
& ( Ys
= ( list_u6180841689913720943at_nat @ Xs @ N2 @ Y5 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_822_listrel1__iff__update,axiom,
! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
= ( ? [Y5: nat,N2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N2 ) @ Y5 ) @ R2 )
& ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
& ( Ys
= ( list_update_nat @ Xs @ N2 @ Y5 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_823_listrel1__iff__update,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( listrel1_term_a_b @ R2 ) )
= ( ? [Y5: term_a_b,N2: nat] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ Xs @ N2 ) @ Y5 ) @ R2 )
& ( ord_less_nat @ N2 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
& ( Ys
= ( list_update_term_a_b @ Xs @ N2 @ Y5 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_824_nth__drop__0,axiom,
! [Ss: list_term_a_b] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s8906293707977694520rm_a_b @ Ss ) )
=> ( ( cons_term_a_b @ ( nth_term_a_b @ Ss @ zero_zero_nat ) @ ( drop_term_a_b @ ( suc @ zero_zero_nat ) @ Ss ) )
= Ss ) ) ).
% nth_drop_0
thf(fact_825_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_826_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_827_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_828_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_829_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_830_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_831_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_832_length__0__conv,axiom,
! [Xs: list_term_a_b] :
( ( ( size_s8906293707977694520rm_a_b @ Xs )
= zero_zero_nat )
= ( Xs = nil_term_a_b ) ) ).
% length_0_conv
thf(fact_833_nth__Cons__0,axiom,
! [X3: term_a_b,Xs: list_term_a_b] :
( ( nth_term_a_b @ ( cons_term_a_b @ X3 @ Xs ) @ zero_zero_nat )
= X3 ) ).
% nth_Cons_0
thf(fact_834_length__greater__0__conv,axiom,
! [Xs: list_term_a_b] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s8906293707977694520rm_a_b @ Xs ) )
= ( Xs != nil_term_a_b ) ) ).
% length_greater_0_conv
thf(fact_835_Cons__listrel1__Cons,axiom,
! [X3: term_a_b,Xs: list_term_a_b,Y: term_a_b,Ys: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ X3 @ Xs ) @ ( cons_term_a_b @ Y @ Ys ) ) @ ( listrel1_term_a_b @ R2 ) )
= ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ R2 )
& ( Xs = Ys ) )
| ( ( X3 = Y )
& ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( listrel1_term_a_b @ R2 ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_836_Cons__listrel1__Cons,axiom,
! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys ) ) @ ( listre4828114922151135584at_nat @ R2 ) )
= ( ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ R2 )
& ( Xs = Ys ) )
| ( ( X3 = Y )
& ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys ) @ ( listre4828114922151135584at_nat @ R2 ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_837_Cons__listrel1__Cons,axiom,
! [X3: nat,Xs: list_nat,Y: nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( listrel1_nat @ R2 ) )
= ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R2 )
& ( Xs = Ys ) )
| ( ( X3 = Y )
& ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_838_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_839_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_840_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_841_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_842_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P2 @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P2 @ M3 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_843_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_844_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_845_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_846_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_847_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_848_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_849_inf__concat__simple_Ocases,axiom,
! [X3: produc8199716216217303280at_nat] :
( ! [F7: nat > nat] :
( X3
!= ( produc72220940542539688at_nat @ F7 @ zero_zero_nat ) )
=> ~ ! [F7: nat > nat,N3: nat] :
( X3
!= ( produc72220940542539688at_nat @ F7 @ ( suc @ N3 ) ) ) ) ).
% inf_concat_simple.cases
thf(fact_850_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M: nat] :
( N
= ( suc @ M ) ) ) ) ).
% gr0_conv_Suc
thf(fact_851_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ).
% gr0_implies_Suc
thf(fact_852_less__Suc__eq__0__disj,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( M2 = zero_zero_nat )
| ? [J3: nat] :
( ( M2
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_853_all__less__two,axiom,
! [P2: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ ( suc @ zero_zero_nat ) ) )
=> ( P2 @ I4 ) ) )
= ( ( P2 @ zero_zero_nat )
& ( P2 @ ( suc @ zero_zero_nat ) ) ) ) ).
% all_less_two
thf(fact_854_all__Suc__conv,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P2 @ I4 ) ) )
= ( ( P2 @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P2 @ ( suc @ I4 ) ) ) ) ) ).
% all_Suc_conv
thf(fact_855_ex__Suc__conv,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P2 @ I4 ) ) )
= ( ( P2 @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P2 @ ( suc @ I4 ) ) ) ) ) ).
% ex_Suc_conv
thf(fact_856_list_Osize_I3_J,axiom,
( ( size_s8906293707977694520rm_a_b @ nil_term_a_b )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_857_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K2 )
=> ~ ( P2 @ I2 ) )
& ( P2 @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_858_listrel1__eq__len,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( listrel1_term_a_b @ R2 ) )
=> ( ( size_s8906293707977694520rm_a_b @ Xs )
= ( size_s8906293707977694520rm_a_b @ Ys ) ) ) ).
% listrel1_eq_len
thf(fact_859_length__code,axiom,
( size_s8906293707977694520rm_a_b
= ( gen_length_term_a_b @ zero_zero_nat ) ) ).
% length_code
thf(fact_860_ex__least__nat__less,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I2: nat] :
( ( ord_less_eq_nat @ I2 @ K2 )
=> ~ ( P2 @ I2 ) )
& ( P2 @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_861_listrel1I1,axiom,
! [X3: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b,Xs: list_term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ R2 )
=> ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ X3 @ Xs ) @ ( cons_term_a_b @ Y @ Xs ) ) @ ( listrel1_term_a_b @ R2 ) ) ) ).
% listrel1I1
thf(fact_862_listrel1I1,axiom,
! [X3: product_prod_nat_nat,Y: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,Xs: list_P6011104703257516679at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ R2 )
=> ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Xs ) ) @ ( listre4828114922151135584at_nat @ R2 ) ) ) ).
% listrel1I1
thf(fact_863_listrel1I1,axiom,
! [X3: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,Xs: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R2 )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y @ Xs ) ) @ ( listrel1_nat @ R2 ) ) ) ).
% listrel1I1
thf(fact_864_Cons__listrel1E1,axiom,
! [X3: term_a_b,Xs: list_term_a_b,Ys: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ X3 @ Xs ) @ Ys ) @ ( listrel1_term_a_b @ R2 ) )
=> ( ! [Y3: term_a_b] :
( ( Ys
= ( cons_term_a_b @ Y3 @ Xs ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R2 ) )
=> ~ ! [Zs2: list_term_a_b] :
( ( Ys
= ( cons_term_a_b @ X3 @ Zs2 ) )
=> ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Zs2 ) @ ( listrel1_term_a_b @ R2 ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_865_Cons__listrel1E1,axiom,
! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ Ys ) @ ( listre4828114922151135584at_nat @ R2 ) )
=> ( ! [Y3: product_prod_nat_nat] :
( ( Ys
= ( cons_P6512896166579812791at_nat @ Y3 @ Xs ) )
=> ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ R2 ) )
=> ~ ! [Zs2: list_P6011104703257516679at_nat] :
( ( Ys
= ( cons_P6512896166579812791at_nat @ X3 @ Zs2 ) )
=> ~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Zs2 ) @ ( listre4828114922151135584at_nat @ R2 ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_866_Cons__listrel1E1,axiom,
! [X3: nat,Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs ) @ Ys ) @ ( listrel1_nat @ R2 ) )
=> ( ! [Y3: nat] :
( ( Ys
= ( cons_nat @ Y3 @ Xs ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 ) )
=> ~ ! [Zs2: list_nat] :
( ( Ys
= ( cons_nat @ X3 @ Zs2 ) )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Zs2 ) @ ( listrel1_nat @ R2 ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_867_Cons__listrel1E2,axiom,
! [Xs: list_term_a_b,Y: term_a_b,Ys: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ ( cons_term_a_b @ Y @ Ys ) ) @ ( listrel1_term_a_b @ R2 ) )
=> ( ! [X: term_a_b] :
( ( Xs
= ( cons_term_a_b @ X @ Ys ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y ) @ R2 ) )
=> ~ ! [Zs2: list_term_a_b] :
( ( Xs
= ( cons_term_a_b @ Y @ Zs2 ) )
=> ~ ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Zs2 @ Ys ) @ ( listrel1_term_a_b @ R2 ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_868_Cons__listrel1E2,axiom,
! [Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ Y @ Ys ) ) @ ( listre4828114922151135584at_nat @ R2 ) )
=> ( ! [X: product_prod_nat_nat] :
( ( Xs
= ( cons_P6512896166579812791at_nat @ X @ Ys ) )
=> ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R2 ) )
=> ~ ! [Zs2: list_P6011104703257516679at_nat] :
( ( Xs
= ( cons_P6512896166579812791at_nat @ Y @ Zs2 ) )
=> ~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Zs2 @ Ys ) @ ( listre4828114922151135584at_nat @ R2 ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_869_Cons__listrel1E2,axiom,
! [Xs: list_nat,Y: nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) @ ( listrel1_nat @ R2 ) )
=> ( ! [X: nat] :
( ( Xs
= ( cons_nat @ X @ Ys ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R2 ) )
=> ~ ! [Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Y @ Zs2 ) )
=> ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Zs2 @ Ys ) @ ( listrel1_nat @ R2 ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_870_hd__conv__nth,axiom,
! [Xs: list_term_a_b] :
( ( Xs != nil_term_a_b )
=> ( ( hd_term_a_b @ Xs )
= ( nth_term_a_b @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_871_funas__term__replace__at__upper,axiom,
! [S: term_a_b,P: list_nat,T: term_a_b] : ( ord_le8666007276011122963_a_nat @ ( term_funas_term_a_b @ ( term_r6860082780075436317at_a_b @ S @ P @ T ) ) @ ( sup_su459911885395995103_a_nat @ ( term_funas_term_a_b @ S ) @ ( term_funas_term_a_b @ T ) ) ) ).
% funas_term_replace_at_upper
thf(fact_872_listrel1E,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( listrel1_term_a_b @ R2 ) )
=> ~ ! [X: term_a_b,Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ R2 )
=> ! [Us: list_term_a_b,Vs2: list_term_a_b] :
( ( Xs
= ( append_term_a_b @ Us @ ( cons_term_a_b @ X @ Vs2 ) ) )
=> ( Ys
!= ( append_term_a_b @ Us @ ( cons_term_a_b @ Y3 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_873_listrel1E,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,R2: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys ) @ ( listre4828114922151135584at_nat @ R2 ) )
=> ~ ! [X: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ R2 )
=> ! [Us: list_P6011104703257516679at_nat,Vs2: list_P6011104703257516679at_nat] :
( ( Xs
= ( append985823374593552924at_nat @ Us @ ( cons_P6512896166579812791at_nat @ X @ Vs2 ) ) )
=> ( Ys
!= ( append985823374593552924at_nat @ Us @ ( cons_P6512896166579812791at_nat @ Y3 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_874_listrel1E,axiom,
! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
=> ~ ! [X: nat,Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R2 )
=> ! [Us: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us @ ( cons_nat @ X @ Vs2 ) ) )
=> ( Ys
!= ( append_nat @ Us @ ( cons_nat @ Y3 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_875_listrel1I,axiom,
! [X3: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b,Xs: list_term_a_b,Us2: list_term_a_b,Vs: list_term_a_b,Ys: list_term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ R2 )
=> ( ( Xs
= ( append_term_a_b @ Us2 @ ( cons_term_a_b @ X3 @ Vs ) ) )
=> ( ( Ys
= ( append_term_a_b @ Us2 @ ( cons_term_a_b @ Y @ Vs ) ) )
=> ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( listrel1_term_a_b @ R2 ) ) ) ) ) ).
% listrel1I
thf(fact_876_listrel1I,axiom,
! [X3: product_prod_nat_nat,Y: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,Xs: list_P6011104703257516679at_nat,Us2: list_P6011104703257516679at_nat,Vs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ R2 )
=> ( ( Xs
= ( append985823374593552924at_nat @ Us2 @ ( cons_P6512896166579812791at_nat @ X3 @ Vs ) ) )
=> ( ( Ys
= ( append985823374593552924at_nat @ Us2 @ ( cons_P6512896166579812791at_nat @ Y @ Vs ) ) )
=> ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys ) @ ( listre4828114922151135584at_nat @ R2 ) ) ) ) ) ).
% listrel1I
thf(fact_877_listrel1I,axiom,
! [X3: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,Xs: list_nat,Us2: list_nat,Vs: list_nat,Ys: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R2 )
=> ( ( Xs
= ( append_nat @ Us2 @ ( cons_nat @ X3 @ Vs ) ) )
=> ( ( Ys
= ( append_nat @ Us2 @ ( cons_nat @ Y @ Vs ) ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) ) ) ) ) ).
% listrel1I
thf(fact_878_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_term_a_b,X3: term_a_b,Ys: list_term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( append_term_a_b @ Xs @ ( cons_term_a_b @ X3 @ nil_term_a_b ) ) @ ( append_term_a_b @ Ys @ ( cons_term_a_b @ Y @ nil_term_a_b ) ) ) @ ( listrel1_term_a_b @ R2 ) )
= ( ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( listrel1_term_a_b @ R2 ) )
& ( X3 = Y ) )
| ( ( Xs = Ys )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ R2 ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_879_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_P6011104703257516679at_nat,X3: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( append985823374593552924at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ X3 @ nil_Pr5478986624290739719at_nat ) ) @ ( append985823374593552924at_nat @ Ys @ ( cons_P6512896166579812791at_nat @ Y @ nil_Pr5478986624290739719at_nat ) ) ) @ ( listre4828114922151135584at_nat @ R2 ) )
= ( ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys ) @ ( listre4828114922151135584at_nat @ R2 ) )
& ( X3 = Y ) )
| ( ( Xs = Ys )
& ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ R2 ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_880_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_nat,X3: nat,Ys: list_nat,Y: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ ( cons_nat @ X3 @ nil_nat ) ) @ ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) ) @ ( listrel1_nat @ R2 ) )
= ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
& ( X3 = Y ) )
| ( ( Xs = Ys )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R2 ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_881_replace__term__at_Osimps_I3_J,axiom,
! [I: nat,Ts: list_term_a_b,F3: a,Ps2: list_nat,T: term_a_b] :
( ( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Ts ) )
=> ( ( term_r6860082780075436317at_a_b @ ( fun_a_b @ F3 @ Ts ) @ ( cons_nat @ I @ Ps2 ) @ T )
= ( fun_a_b @ F3 @ ( list_update_term_a_b @ Ts @ I @ ( term_r6860082780075436317at_a_b @ ( nth_term_a_b @ Ts @ I ) @ Ps2 @ T ) ) ) ) )
& ( ~ ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Ts ) )
=> ( ( term_r6860082780075436317at_a_b @ ( fun_a_b @ F3 @ Ts ) @ ( cons_nat @ I @ Ps2 ) @ T )
= ( fun_a_b @ F3 @ Ts ) ) ) ) ).
% replace_term_at.simps(3)
thf(fact_882_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_883_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_884_shift_Ocases,axiom,
! [X3: produc8199716216217303280at_nat] :
~ ! [F7: nat > nat,J2: nat] :
( X3
!= ( produc72220940542539688at_nat @ F7 @ J2 ) ) ).
% shift.cases
thf(fact_885_zero__le,axiom,
! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% zero_le
thf(fact_886_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_887_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_888_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_889_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_890_replace__term__at_Opelims,axiom,
! [X3: term_a_b,Xa: list_nat,Xb: term_a_b,Y: term_a_b] :
( ( ( term_r6860082780075436317at_a_b @ X3 @ Xa @ Xb )
= Y )
=> ( ( accp_P2729577386226225901rm_a_b @ term_r1280879029893354718el_a_b @ ( produc3812856575676843240rm_a_b @ X3 @ ( produc5151171985953862413rm_a_b @ Xa @ Xb ) ) )
=> ( ( ( Xa = nil_nat )
=> ( ( Y = Xb )
=> ~ ( accp_P2729577386226225901rm_a_b @ term_r1280879029893354718el_a_b @ ( produc3812856575676843240rm_a_b @ X3 @ ( produc5151171985953862413rm_a_b @ nil_nat @ Xb ) ) ) ) )
=> ( ! [X: b] :
( ( X3
= ( var_b_a @ X ) )
=> ! [V2: nat,Va: list_nat] :
( ( Xa
= ( cons_nat @ V2 @ Va ) )
=> ( ( Y
= ( var_b_a @ X ) )
=> ~ ( accp_P2729577386226225901rm_a_b @ term_r1280879029893354718el_a_b @ ( produc3812856575676843240rm_a_b @ ( var_b_a @ X ) @ ( produc5151171985953862413rm_a_b @ ( cons_nat @ V2 @ Va ) @ Xb ) ) ) ) ) )
=> ~ ! [F7: a,Ts3: list_term_a_b] :
( ( X3
= ( fun_a_b @ F7 @ Ts3 ) )
=> ! [I3: nat,Ps: list_nat] :
( ( Xa
= ( cons_nat @ I3 @ Ps ) )
=> ( ( ( ( ord_less_nat @ I3 @ ( size_s8906293707977694520rm_a_b @ Ts3 ) )
=> ( Y
= ( fun_a_b @ F7 @ ( list_update_term_a_b @ Ts3 @ I3 @ ( term_r6860082780075436317at_a_b @ ( nth_term_a_b @ Ts3 @ I3 ) @ Ps @ Xb ) ) ) ) )
& ( ~ ( ord_less_nat @ I3 @ ( size_s8906293707977694520rm_a_b @ Ts3 ) )
=> ( Y
= ( fun_a_b @ F7 @ Ts3 ) ) ) )
=> ~ ( accp_P2729577386226225901rm_a_b @ term_r1280879029893354718el_a_b @ ( produc3812856575676843240rm_a_b @ ( fun_a_b @ F7 @ Ts3 ) @ ( produc5151171985953862413rm_a_b @ ( cons_nat @ I3 @ Ps ) @ Xb ) ) ) ) ) ) ) ) ) ) ).
% replace_term_at.pelims
thf(fact_891_constT__nfunas__term__poss__of__term__empty,axiom,
! [C3: a,T: term_a_b] :
( ( ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ C3 @ zero_zero_nat ) @ ( term_funas_term_a_b @ T ) ) )
= ( ( terms_7168686267159881682rm_a_b @ ( fun_a_b @ C3 @ nil_term_a_b ) @ T )
= bot_bot_set_list_nat ) ) ).
% constT_nfunas_term_poss_of_term_empty
thf(fact_892_all__not__in__conv,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ( ! [X6: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ X6 @ A4 ) )
= ( A4 = bot_bo197521221353338581rm_a_b ) ) ).
% all_not_in_conv
thf(fact_893_all__not__in__conv,axiom,
! [A4: set_Pr4934435412358123699_a_nat] :
( ( ! [X6: product_prod_a_nat] :
~ ( member5724188588386418708_a_nat @ X6 @ A4 ) )
= ( A4 = bot_bo9049108969261143879_a_nat ) ) ).
% all_not_in_conv
thf(fact_894_all__not__in__conv,axiom,
! [A4: set_Pr8693737435421807431at_nat] :
( ( ! [X6: produc859450856879609959at_nat] :
~ ( member8206827879206165904at_nat @ X6 @ A4 ) )
= ( A4 = bot_bo5327735625951526323at_nat ) ) ).
% all_not_in_conv
thf(fact_895_all__not__in__conv,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ( ! [X6: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ X6 @ A4 ) )
= ( A4 = bot_bo2099793752762293965at_nat ) ) ).
% all_not_in_conv
thf(fact_896_empty__iff,axiom,
! [C3: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ C3 @ bot_bo197521221353338581rm_a_b ) ).
% empty_iff
thf(fact_897_empty__iff,axiom,
! [C3: product_prod_a_nat] :
~ ( member5724188588386418708_a_nat @ C3 @ bot_bo9049108969261143879_a_nat ) ).
% empty_iff
thf(fact_898_empty__iff,axiom,
! [C3: produc859450856879609959at_nat] :
~ ( member8206827879206165904at_nat @ C3 @ bot_bo5327735625951526323at_nat ) ).
% empty_iff
thf(fact_899_empty__iff,axiom,
! [C3: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ C3 @ bot_bo2099793752762293965at_nat ) ).
% empty_iff
thf(fact_900_empty__subsetI,axiom,
! [A4: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ bot_bo9049108969261143879_a_nat @ A4 ) ).
% empty_subsetI
thf(fact_901_subset__empty,axiom,
! [A4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A4 @ bot_bo9049108969261143879_a_nat )
= ( A4 = bot_bo9049108969261143879_a_nat ) ) ).
% subset_empty
thf(fact_902_bot_Oextremum__uniqueI,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ bot_bo9049108969261143879_a_nat )
=> ( A = bot_bo9049108969261143879_a_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_903_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_904_bot_Oextremum__unique,axiom,
! [A: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A @ bot_bo9049108969261143879_a_nat )
= ( A = bot_bo9049108969261143879_a_nat ) ) ).
% bot.extremum_unique
thf(fact_905_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_906_bot_Oextremum,axiom,
! [A: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ bot_bo9049108969261143879_a_nat @ A ) ).
% bot.extremum
thf(fact_907_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_908_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_909_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_910_ex__in__conv,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ( ? [X6: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X6 @ A4 ) )
= ( A4 != bot_bo197521221353338581rm_a_b ) ) ).
% ex_in_conv
thf(fact_911_ex__in__conv,axiom,
! [A4: set_Pr4934435412358123699_a_nat] :
( ( ? [X6: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X6 @ A4 ) )
= ( A4 != bot_bo9049108969261143879_a_nat ) ) ).
% ex_in_conv
thf(fact_912_ex__in__conv,axiom,
! [A4: set_Pr8693737435421807431at_nat] :
( ( ? [X6: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X6 @ A4 ) )
= ( A4 != bot_bo5327735625951526323at_nat ) ) ).
% ex_in_conv
thf(fact_913_ex__in__conv,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ( ? [X6: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X6 @ A4 ) )
= ( A4 != bot_bo2099793752762293965at_nat ) ) ).
% ex_in_conv
thf(fact_914_equals0I,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ! [Y3: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ Y3 @ A4 )
=> ( A4 = bot_bo197521221353338581rm_a_b ) ) ).
% equals0I
thf(fact_915_equals0I,axiom,
! [A4: set_Pr4934435412358123699_a_nat] :
( ! [Y3: product_prod_a_nat] :
~ ( member5724188588386418708_a_nat @ Y3 @ A4 )
=> ( A4 = bot_bo9049108969261143879_a_nat ) ) ).
% equals0I
thf(fact_916_equals0I,axiom,
! [A4: set_Pr8693737435421807431at_nat] :
( ! [Y3: produc859450856879609959at_nat] :
~ ( member8206827879206165904at_nat @ Y3 @ A4 )
=> ( A4 = bot_bo5327735625951526323at_nat ) ) ).
% equals0I
thf(fact_917_equals0I,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ! [Y3: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ Y3 @ A4 )
=> ( A4 = bot_bo2099793752762293965at_nat ) ) ).
% equals0I
thf(fact_918_equals0D,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,A: produc357393685978478089rm_a_b] :
( ( A4 = bot_bo197521221353338581rm_a_b )
=> ~ ( member5869715511025134514rm_a_b @ A @ A4 ) ) ).
% equals0D
thf(fact_919_equals0D,axiom,
! [A4: set_Pr4934435412358123699_a_nat,A: product_prod_a_nat] :
( ( A4 = bot_bo9049108969261143879_a_nat )
=> ~ ( member5724188588386418708_a_nat @ A @ A4 ) ) ).
% equals0D
thf(fact_920_equals0D,axiom,
! [A4: set_Pr8693737435421807431at_nat,A: produc859450856879609959at_nat] :
( ( A4 = bot_bo5327735625951526323at_nat )
=> ~ ( member8206827879206165904at_nat @ A @ A4 ) ) ).
% equals0D
thf(fact_921_equals0D,axiom,
! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
( ( A4 = bot_bo2099793752762293965at_nat )
=> ~ ( member8440522571783428010at_nat @ A @ A4 ) ) ).
% equals0D
thf(fact_922_emptyE,axiom,
! [A: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ A @ bot_bo197521221353338581rm_a_b ) ).
% emptyE
thf(fact_923_emptyE,axiom,
! [A: product_prod_a_nat] :
~ ( member5724188588386418708_a_nat @ A @ bot_bo9049108969261143879_a_nat ) ).
% emptyE
thf(fact_924_emptyE,axiom,
! [A: produc859450856879609959at_nat] :
~ ( member8206827879206165904at_nat @ A @ bot_bo5327735625951526323at_nat ) ).
% emptyE
thf(fact_925_emptyE,axiom,
! [A: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ A @ bot_bo2099793752762293965at_nat ) ).
% emptyE
thf(fact_926_subset__emptyI,axiom,
! [A4: set_Pr4386577575007340137rm_a_b] :
( ! [X: produc357393685978478089rm_a_b] :
~ ( member5869715511025134514rm_a_b @ X @ A4 )
=> ( ord_le118470702582115849rm_a_b @ A4 @ bot_bo197521221353338581rm_a_b ) ) ).
% subset_emptyI
thf(fact_927_subset__emptyI,axiom,
! [A4: set_Pr8693737435421807431at_nat] :
( ! [X: produc859450856879609959at_nat] :
~ ( member8206827879206165904at_nat @ X @ A4 )
=> ( ord_le3000389064537975527at_nat @ A4 @ bot_bo5327735625951526323at_nat ) ) ).
% subset_emptyI
thf(fact_928_subset__emptyI,axiom,
! [A4: set_Pr1261947904930325089at_nat] :
( ! [X: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ X @ A4 )
=> ( ord_le3146513528884898305at_nat @ A4 @ bot_bo2099793752762293965at_nat ) ) ).
% subset_emptyI
thf(fact_929_subset__emptyI,axiom,
! [A4: set_Pr4934435412358123699_a_nat] :
( ! [X: product_prod_a_nat] :
~ ( member5724188588386418708_a_nat @ X @ A4 )
=> ( ord_le8666007276011122963_a_nat @ A4 @ bot_bo9049108969261143879_a_nat ) ) ).
% subset_emptyI
thf(fact_930_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_931_relcomp__empty2,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ R3 @ bot_bo197521221353338581rm_a_b )
= bot_bo197521221353338581rm_a_b ) ).
% relcomp_empty2
thf(fact_932_relcomp__empty1,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ bot_bo197521221353338581rm_a_b @ R3 )
= bot_bo197521221353338581rm_a_b ) ).
% relcomp_empty1
thf(fact_933_bot__empty__eq,axiom,
( bot_bo23760184229683272_a_b_o
= ( ^ [X6: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ X6 @ bot_bo197521221353338581rm_a_b ) ) ) ).
% bot_empty_eq
thf(fact_934_bot__empty__eq,axiom,
( bot_bo1388487281569869310_nat_o
= ( ^ [X6: product_prod_a_nat] : ( member5724188588386418708_a_nat @ X6 @ bot_bo9049108969261143879_a_nat ) ) ) ).
% bot_empty_eq
thf(fact_935_bot__empty__eq,axiom,
( bot_bo7573314457883560170_nat_o
= ( ^ [X6: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X6 @ bot_bo5327735625951526323at_nat ) ) ) ).
% bot_empty_eq
thf(fact_936_bot__empty__eq,axiom,
( bot_bo482883023278783056_nat_o
= ( ^ [X6: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X6 @ bot_bo2099793752762293965at_nat ) ) ) ).
% bot_empty_eq
thf(fact_937_union__comp__emptyR,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b,C4: set_Pr4386577575007340137rm_a_b] :
( ( ( relcom370159955682700863rm_a_b @ A4 @ B4 )
= bot_bo197521221353338581rm_a_b )
=> ( ( ( relcom370159955682700863rm_a_b @ A4 @ C4 )
= bot_bo197521221353338581rm_a_b )
=> ( ( relcom370159955682700863rm_a_b @ A4 @ ( sup_su6776935440552674877rm_a_b @ B4 @ C4 ) )
= bot_bo197521221353338581rm_a_b ) ) ) ).
% union_comp_emptyR
thf(fact_938_union__comp__emptyL,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,C4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( ( relcom370159955682700863rm_a_b @ A4 @ C4 )
= bot_bo197521221353338581rm_a_b )
=> ( ( ( relcom370159955682700863rm_a_b @ B4 @ C4 )
= bot_bo197521221353338581rm_a_b )
=> ( ( relcom370159955682700863rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A4 @ B4 ) @ C4 )
= bot_bo197521221353338581rm_a_b ) ) ) ).
% union_comp_emptyL
thf(fact_939_funas__term_Osimps_I1_J,axiom,
! [X3: b] :
( ( term_funas_term_a_b @ ( var_b_a @ X3 ) )
= bot_bo9049108969261143879_a_nat ) ).
% funas_term.simps(1)
thf(fact_940_ssubst__Pair__rhs,axiom,
! [R2: term_a_b,S: term_a_b,R3: set_Pr4386577575007340137rm_a_b,S3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ R2 @ S ) @ R3 )
=> ( ( S3 = S )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ R2 @ S3 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_941_ssubst__Pair__rhs,axiom,
! [R2: a,S: nat,R3: set_Pr4934435412358123699_a_nat,S3: nat] :
( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ R2 @ S ) @ R3 )
=> ( ( S3 = S )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ R2 @ S3 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_942_ssubst__Pair__rhs,axiom,
! [R2: nat > nat,S: nat,R3: set_Pr9093778441882193744at_nat,S3: nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ R2 @ S ) @ R3 )
=> ( ( S3 = S )
=> ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ R2 @ S3 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_943_ssubst__Pair__rhs,axiom,
! [R2: product_prod_nat_nat,S: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat,S3: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ R2 @ S ) @ R3 )
=> ( ( S3 = S )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ R2 @ S3 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_944_ssubst__Pair__rhs,axiom,
! [R2: nat,S: nat,R3: set_Pr1261947904930325089at_nat,S3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S ) @ R3 )
=> ( ( S3 = S )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S3 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_945_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_946_union__list__sorted_Opelims,axiom,
! [X3: list_nat,Xa: list_nat,Y: list_nat] :
( ( ( missin8019018944680490243ed_nat @ X3 @ Xa )
= Y )
=> ( ( accp_P8037286306265792042st_nat @ missin7611735270441097048el_nat @ ( produc2694037385005941721st_nat @ X3 @ Xa ) )
=> ( ! [X: nat,Xs3: list_nat] :
( ( X3
= ( cons_nat @ X @ Xs3 ) )
=> ! [Y3: nat,Ys5: list_nat] :
( ( Xa
= ( cons_nat @ Y3 @ Ys5 ) )
=> ( ( ( ( X = Y3 )
=> ( Y
= ( cons_nat @ X @ ( missin8019018944680490243ed_nat @ Xs3 @ Ys5 ) ) ) )
& ( ( X != Y3 )
=> ( ( ( ord_less_nat @ X @ Y3 )
=> ( Y
= ( cons_nat @ X @ ( missin8019018944680490243ed_nat @ Xs3 @ ( cons_nat @ Y3 @ Ys5 ) ) ) ) )
& ( ~ ( ord_less_nat @ X @ Y3 )
=> ( Y
= ( cons_nat @ Y3 @ ( missin8019018944680490243ed_nat @ ( cons_nat @ X @ Xs3 ) @ Ys5 ) ) ) ) ) ) )
=> ~ ( accp_P8037286306265792042st_nat @ missin7611735270441097048el_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs3 ) @ ( cons_nat @ Y3 @ Ys5 ) ) ) ) ) )
=> ( ( ( X3 = nil_nat )
=> ( ( Y = Xa )
=> ~ ( accp_P8037286306265792042st_nat @ missin7611735270441097048el_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xa ) ) ) )
=> ~ ! [V2: nat,Va: list_nat] :
( ( X3
= ( cons_nat @ V2 @ Va ) )
=> ( ( Xa = nil_nat )
=> ( ( Y
= ( cons_nat @ V2 @ Va ) )
=> ~ ( accp_P8037286306265792042st_nat @ missin7611735270441097048el_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ V2 @ Va ) @ nil_nat ) ) ) ) ) ) ) ) ) ).
% union_list_sorted.pelims
thf(fact_947_depth_Osimps_I2_J,axiom,
! [F3: a] :
( ( term_depth_a_b @ ( fun_a_b @ F3 @ nil_term_a_b ) )
= zero_zero_nat ) ).
% depth.simps(2)
thf(fact_948_nth__append__Cons,axiom,
! [I: nat,Xs: list_term_a_b,Y: term_a_b,Zs: list_term_a_b] :
( ( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( nth_term_a_b @ ( append_term_a_b @ Xs @ ( cons_term_a_b @ Y @ Zs ) ) @ I )
= ( nth_term_a_b @ Xs @ I ) ) )
& ( ~ ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( ( I
= ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( nth_term_a_b @ ( append_term_a_b @ Xs @ ( cons_term_a_b @ Y @ Zs ) ) @ I )
= Y ) )
& ( ( I
!= ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( nth_term_a_b @ ( append_term_a_b @ Xs @ ( cons_term_a_b @ Y @ Zs ) ) @ I )
= ( nth_term_a_b @ Zs @ ( minus_minus_nat @ I @ ( suc @ ( size_s8906293707977694520rm_a_b @ Xs ) ) ) ) ) ) ) ) ) ).
% nth_append_Cons
thf(fact_949_enumerate__simps_I2_J,axiom,
! [N: nat,X3: nat,Xs: list_nat] :
( ( enumerate_nat @ N @ ( cons_nat @ X3 @ Xs ) )
= ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ N @ X3 ) @ ( enumerate_nat @ ( suc @ N ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_950_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_951_length__enumerate,axiom,
! [N: nat,Xs: list_term_a_b] :
( ( size_s3663718155697878711rm_a_b @ ( enumerate_term_a_b @ N @ Xs ) )
= ( size_s8906293707977694520rm_a_b @ Xs ) ) ).
% length_enumerate
thf(fact_952_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_953_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_954_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_955_length__drop,axiom,
! [N: nat,Xs: list_term_a_b] :
( ( size_s8906293707977694520rm_a_b @ ( drop_term_a_b @ N @ Xs ) )
= ( minus_minus_nat @ ( size_s8906293707977694520rm_a_b @ Xs ) @ N ) ) ).
% length_drop
thf(fact_956_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_957_take__append,axiom,
! [N: nat,Xs: list_term_a_b,Ys: list_term_a_b] :
( ( take_term_a_b @ N @ ( append_term_a_b @ Xs @ Ys ) )
= ( append_term_a_b @ ( take_term_a_b @ N @ Xs ) @ ( take_term_a_b @ ( minus_minus_nat @ N @ ( size_s8906293707977694520rm_a_b @ Xs ) ) @ Ys ) ) ) ).
% take_append
thf(fact_958_drop__append,axiom,
! [N: nat,Xs: list_term_a_b,Ys: list_term_a_b] :
( ( drop_term_a_b @ N @ ( append_term_a_b @ Xs @ Ys ) )
= ( append_term_a_b @ ( drop_term_a_b @ N @ Xs ) @ ( drop_term_a_b @ ( minus_minus_nat @ N @ ( size_s8906293707977694520rm_a_b @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_959_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_960_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_961_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_962_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_963_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_964_le__diff__iff_H,axiom,
! [A: nat,C3: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C3 )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C3 @ A ) @ ( minus_minus_nat @ C3 @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_965_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_966_diff__less__Suc,axiom,
! [M2: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_967_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_968_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_969_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_970_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_971_diff__less__mono,axiom,
! [A: nat,B: nat,C3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C3 @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C3 ) @ ( minus_minus_nat @ B @ C3 ) ) ) ) ).
% diff_less_mono
thf(fact_972_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_973_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_974_diff__shunt__var,axiom,
! [X3: set_Pr4934435412358123699_a_nat,Y: set_Pr4934435412358123699_a_nat] :
( ( ( minus_8630901905721465484_a_nat @ X3 @ Y )
= bot_bo9049108969261143879_a_nat )
= ( ord_le8666007276011122963_a_nat @ X3 @ Y ) ) ).
% diff_shunt_var
thf(fact_975_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_976_nth__append,axiom,
! [N: nat,Xs: list_term_a_b,Ys: list_term_a_b] :
( ( ( ord_less_nat @ N @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( nth_term_a_b @ ( append_term_a_b @ Xs @ Ys ) @ N )
= ( nth_term_a_b @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( nth_term_a_b @ ( append_term_a_b @ Xs @ Ys ) @ N )
= ( nth_term_a_b @ Ys @ ( minus_minus_nat @ N @ ( size_s8906293707977694520rm_a_b @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_977_list__update__append,axiom,
! [N: nat,Xs: list_term_a_b,Ys: list_term_a_b,X3: term_a_b] :
( ( ( ord_less_nat @ N @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( list_update_term_a_b @ ( append_term_a_b @ Xs @ Ys ) @ N @ X3 )
= ( append_term_a_b @ ( list_update_term_a_b @ Xs @ N @ X3 ) @ Ys ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( list_update_term_a_b @ ( append_term_a_b @ Xs @ Ys ) @ N @ X3 )
= ( append_term_a_b @ Xs @ ( list_update_term_a_b @ Ys @ ( minus_minus_nat @ N @ ( size_s8906293707977694520rm_a_b @ Xs ) ) @ X3 ) ) ) ) ) ).
% list_update_append
thf(fact_978_adjust__idx__rev__def,axiom,
( missin3815256168798769645dx_rev
= ( ^ [I4: nat,J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ I4 ) @ J3 @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) ) ).
% adjust_idx_rev_def
thf(fact_979_nth__Cons__pos,axiom,
! [N: nat,X3: term_a_b,Xs: list_term_a_b] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_term_a_b @ ( cons_term_a_b @ X3 @ Xs ) @ N )
= ( nth_term_a_b @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_980_distinct__take__drop,axiom,
! [Vs: list_term_a_b,I: nat] :
( ( distinct_term_a_b @ Vs )
=> ( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Vs ) )
=> ( distinct_term_a_b @ ( append_term_a_b @ ( take_term_a_b @ I @ Vs ) @ ( drop_term_a_b @ ( suc @ I ) @ Vs ) ) ) ) ) ).
% distinct_take_drop
thf(fact_981_Diff__eq__empty__iff,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ( minus_8630901905721465484_a_nat @ A4 @ B4 )
= bot_bo9049108969261143879_a_nat )
= ( ord_le8666007276011122963_a_nat @ A4 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_982_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_983_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_984_distinct__swap,axiom,
! [I: nat,Xs: list_term_a_b,J: nat] :
( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( distinct_term_a_b @ ( list_update_term_a_b @ ( list_update_term_a_b @ Xs @ I @ ( nth_term_a_b @ Xs @ J ) ) @ J @ ( nth_term_a_b @ Xs @ I ) ) )
= ( distinct_term_a_b @ Xs ) ) ) ) ).
% distinct_swap
thf(fact_985_Diff__mono,axiom,
! [A4: set_Pr4934435412358123699_a_nat,C4: set_Pr4934435412358123699_a_nat,D2: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A4 @ C4 )
=> ( ( ord_le8666007276011122963_a_nat @ D2 @ B4 )
=> ( ord_le8666007276011122963_a_nat @ ( minus_8630901905721465484_a_nat @ A4 @ B4 ) @ ( minus_8630901905721465484_a_nat @ C4 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_986_Diff__subset,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] : ( ord_le8666007276011122963_a_nat @ ( minus_8630901905721465484_a_nat @ A4 @ B4 ) @ A4 ) ).
% Diff_subset
thf(fact_987_double__diff,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat,C4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A4 @ B4 )
=> ( ( ord_le8666007276011122963_a_nat @ B4 @ C4 )
=> ( ( minus_8630901905721465484_a_nat @ B4 @ ( minus_8630901905721465484_a_nat @ C4 @ A4 ) )
= A4 ) ) ) ).
% double_diff
thf(fact_988_Diff__subset__conv,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat,C4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( minus_8630901905721465484_a_nat @ A4 @ B4 ) @ C4 )
= ( ord_le8666007276011122963_a_nat @ A4 @ ( sup_su459911885395995103_a_nat @ B4 @ C4 ) ) ) ).
% Diff_subset_conv
thf(fact_989_Diff__partition,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ A4 @ B4 )
=> ( ( sup_su459911885395995103_a_nat @ A4 @ ( minus_8630901905721465484_a_nat @ B4 @ A4 ) )
= B4 ) ) ).
% Diff_partition
thf(fact_990_psubset__imp__ex__mem,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le4883060154188259093rm_a_b @ A4 @ B4 )
=> ? [B3: produc357393685978478089rm_a_b] : ( member5869715511025134514rm_a_b @ B3 @ ( minus_5192120951422937424rm_a_b @ B4 @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_991_psubset__imp__ex__mem,axiom,
! [A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( ord_le2482536469199093511_a_nat @ A4 @ B4 )
=> ? [B3: product_prod_a_nat] : ( member5724188588386418708_a_nat @ B3 @ ( minus_8630901905721465484_a_nat @ B4 @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_992_psubset__imp__ex__mem,axiom,
! [A4: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat] :
( ( ord_le6428140832669894131at_nat @ A4 @ B4 )
=> ? [B3: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ B3 @ ( minus_8321449233255521966at_nat @ B4 @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_993_psubset__imp__ex__mem,axiom,
! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
( ( ord_le7866589430770878221at_nat @ A4 @ B4 )
=> ? [B3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ B3 @ ( minus_1356011639430497352at_nat @ B4 @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_994_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_995_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_996_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_997_nth__eq__iff__index__eq,axiom,
! [Xs: list_term_a_b,I: nat,J: nat] :
( ( distinct_term_a_b @ Xs )
=> ( ( ord_less_nat @ I @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( ( nth_term_a_b @ Xs @ I )
= ( nth_term_a_b @ Xs @ J ) )
= ( I = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_998_distinct__conv__nth,axiom,
( distinct_term_a_b
= ( ^ [Xs2: list_term_a_b] :
! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
=> ! [J3: nat] :
( ( ord_less_nat @ J3 @ ( size_s8906293707977694520rm_a_b @ Xs2 ) )
=> ( ( I4 != J3 )
=> ( ( nth_term_a_b @ Xs2 @ I4 )
!= ( nth_term_a_b @ Xs2 @ J3 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_999_nat__induct__non__zero,axiom,
! [N: nat,P2: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P2 @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1000_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_1001_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_1002_nth__Cons_H,axiom,
! [N: nat,X3: term_a_b,Xs: list_term_a_b] :
( ( ( N = zero_zero_nat )
=> ( ( nth_term_a_b @ ( cons_term_a_b @ X3 @ Xs ) @ N )
= X3 ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_term_a_b @ ( cons_term_a_b @ X3 @ Xs ) @ N )
= ( nth_term_a_b @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_1003_remove__nth__length,axiom,
! [N: nat,Xs: list_term_a_b] :
( ( ord_less_nat @ N @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( size_s8906293707977694520rm_a_b @ ( missin7590947408886171483rm_a_b @ N @ Xs ) )
= ( minus_minus_nat @ ( size_s8906293707977694520rm_a_b @ Xs ) @ one_one_nat ) ) ) ).
% remove_nth_length
thf(fact_1004_nth__non__equal__first__eq,axiom,
! [X3: term_a_b,Y: term_a_b,Xs: list_term_a_b,N: nat] :
( ( X3 != Y )
=> ( ( ( nth_term_a_b @ ( cons_term_a_b @ X3 @ Xs ) @ N )
= Y )
= ( ( ( nth_term_a_b @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_1005_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_1006_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_1007_Diff__iff,axiom,
! [C3: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C3 @ ( minus_5192120951422937424rm_a_b @ A4 @ B4 ) )
= ( ( member5869715511025134514rm_a_b @ C3 @ A4 )
& ~ ( member5869715511025134514rm_a_b @ C3 @ B4 ) ) ) ).
% Diff_iff
thf(fact_1008_Diff__iff,axiom,
! [C3: product_prod_a_nat,A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C3 @ ( minus_8630901905721465484_a_nat @ A4 @ B4 ) )
= ( ( member5724188588386418708_a_nat @ C3 @ A4 )
& ~ ( member5724188588386418708_a_nat @ C3 @ B4 ) ) ) ).
% Diff_iff
thf(fact_1009_Diff__iff,axiom,
! [C3: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ C3 @ ( minus_8321449233255521966at_nat @ A4 @ B4 ) )
= ( ( member8206827879206165904at_nat @ C3 @ A4 )
& ~ ( member8206827879206165904at_nat @ C3 @ B4 ) ) ) ).
% Diff_iff
thf(fact_1010_Diff__iff,axiom,
! [C3: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C3 @ ( minus_1356011639430497352at_nat @ A4 @ B4 ) )
= ( ( member8440522571783428010at_nat @ C3 @ A4 )
& ~ ( member8440522571783428010at_nat @ C3 @ B4 ) ) ) ).
% Diff_iff
thf(fact_1011_DiffI,axiom,
! [C3: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C3 @ A4 )
=> ( ~ ( member5869715511025134514rm_a_b @ C3 @ B4 )
=> ( member5869715511025134514rm_a_b @ C3 @ ( minus_5192120951422937424rm_a_b @ A4 @ B4 ) ) ) ) ).
% DiffI
thf(fact_1012_DiffI,axiom,
! [C3: product_prod_a_nat,A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C3 @ A4 )
=> ( ~ ( member5724188588386418708_a_nat @ C3 @ B4 )
=> ( member5724188588386418708_a_nat @ C3 @ ( minus_8630901905721465484_a_nat @ A4 @ B4 ) ) ) ) ).
% DiffI
thf(fact_1013_DiffI,axiom,
! [C3: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ C3 @ A4 )
=> ( ~ ( member8206827879206165904at_nat @ C3 @ B4 )
=> ( member8206827879206165904at_nat @ C3 @ ( minus_8321449233255521966at_nat @ A4 @ B4 ) ) ) ) ).
% DiffI
thf(fact_1014_DiffI,axiom,
! [C3: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C3 @ A4 )
=> ( ~ ( member8440522571783428010at_nat @ C3 @ B4 )
=> ( member8440522571783428010at_nat @ C3 @ ( minus_1356011639430497352at_nat @ A4 @ B4 ) ) ) ) ).
% DiffI
thf(fact_1015_DiffD2,axiom,
! [C3: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C3 @ ( minus_5192120951422937424rm_a_b @ A4 @ B4 ) )
=> ~ ( member5869715511025134514rm_a_b @ C3 @ B4 ) ) ).
% DiffD2
thf(fact_1016_DiffD2,axiom,
! [C3: product_prod_a_nat,A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C3 @ ( minus_8630901905721465484_a_nat @ A4 @ B4 ) )
=> ~ ( member5724188588386418708_a_nat @ C3 @ B4 ) ) ).
% DiffD2
thf(fact_1017_DiffD2,axiom,
! [C3: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ C3 @ ( minus_8321449233255521966at_nat @ A4 @ B4 ) )
=> ~ ( member8206827879206165904at_nat @ C3 @ B4 ) ) ).
% DiffD2
thf(fact_1018_DiffD2,axiom,
! [C3: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C3 @ ( minus_1356011639430497352at_nat @ A4 @ B4 ) )
=> ~ ( member8440522571783428010at_nat @ C3 @ B4 ) ) ).
% DiffD2
thf(fact_1019_DiffD1,axiom,
! [C3: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C3 @ ( minus_5192120951422937424rm_a_b @ A4 @ B4 ) )
=> ( member5869715511025134514rm_a_b @ C3 @ A4 ) ) ).
% DiffD1
thf(fact_1020_DiffD1,axiom,
! [C3: product_prod_a_nat,A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C3 @ ( minus_8630901905721465484_a_nat @ A4 @ B4 ) )
=> ( member5724188588386418708_a_nat @ C3 @ A4 ) ) ).
% DiffD1
thf(fact_1021_DiffD1,axiom,
! [C3: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ C3 @ ( minus_8321449233255521966at_nat @ A4 @ B4 ) )
=> ( member8206827879206165904at_nat @ C3 @ A4 ) ) ).
% DiffD1
thf(fact_1022_DiffD1,axiom,
! [C3: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C3 @ ( minus_1356011639430497352at_nat @ A4 @ B4 ) )
=> ( member8440522571783428010at_nat @ C3 @ A4 ) ) ).
% DiffD1
thf(fact_1023_DiffE,axiom,
! [C3: produc357393685978478089rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ C3 @ ( minus_5192120951422937424rm_a_b @ A4 @ B4 ) )
=> ~ ( ( member5869715511025134514rm_a_b @ C3 @ A4 )
=> ( member5869715511025134514rm_a_b @ C3 @ B4 ) ) ) ).
% DiffE
thf(fact_1024_DiffE,axiom,
! [C3: product_prod_a_nat,A4: set_Pr4934435412358123699_a_nat,B4: set_Pr4934435412358123699_a_nat] :
( ( member5724188588386418708_a_nat @ C3 @ ( minus_8630901905721465484_a_nat @ A4 @ B4 ) )
=> ~ ( ( member5724188588386418708_a_nat @ C3 @ A4 )
=> ( member5724188588386418708_a_nat @ C3 @ B4 ) ) ) ).
% DiffE
thf(fact_1025_DiffE,axiom,
! [C3: produc859450856879609959at_nat,A4: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ C3 @ ( minus_8321449233255521966at_nat @ A4 @ B4 ) )
=> ~ ( ( member8206827879206165904at_nat @ C3 @ A4 )
=> ( member8206827879206165904at_nat @ C3 @ B4 ) ) ) ).
% DiffE
thf(fact_1026_DiffE,axiom,
! [C3: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C3 @ ( minus_1356011639430497352at_nat @ A4 @ B4 ) )
=> ~ ( ( member8440522571783428010at_nat @ C3 @ A4 )
=> ( member8440522571783428010at_nat @ C3 @ B4 ) ) ) ).
% DiffE
thf(fact_1027_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1028_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_1029_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_1030_butlast__take,axiom,
! [N: nat,Xs: list_term_a_b] :
( ( ord_less_eq_nat @ N @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( butlast_term_a_b @ ( take_term_a_b @ N @ Xs ) )
= ( take_term_a_b @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ).
% butlast_take
thf(fact_1031_length__butlast,axiom,
! [Xs: list_term_a_b] :
( ( size_s8906293707977694520rm_a_b @ ( butlast_term_a_b @ Xs ) )
= ( minus_minus_nat @ ( size_s8906293707977694520rm_a_b @ Xs ) @ one_one_nat ) ) ).
% length_butlast
thf(fact_1032_nth__butlast,axiom,
! [N: nat,Xs: list_term_a_b] :
( ( ord_less_nat @ N @ ( size_s8906293707977694520rm_a_b @ ( butlast_term_a_b @ Xs ) ) )
=> ( ( nth_term_a_b @ ( butlast_term_a_b @ Xs ) @ N )
= ( nth_term_a_b @ Xs @ N ) ) ) ).
% nth_butlast
thf(fact_1033_take__butlast,axiom,
! [N: nat,Xs: list_term_a_b] :
( ( ord_less_nat @ N @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( take_term_a_b @ N @ ( butlast_term_a_b @ Xs ) )
= ( take_term_a_b @ N @ Xs ) ) ) ).
% take_butlast
thf(fact_1034_butlast__conv__take,axiom,
( butlast_term_a_b
= ( ^ [Xs2: list_term_a_b] : ( take_term_a_b @ ( minus_minus_nat @ ( size_s8906293707977694520rm_a_b @ Xs2 ) @ one_one_nat ) @ Xs2 ) ) ) ).
% butlast_conv_take
thf(fact_1035_butlast__list__update,axiom,
! [K: nat,Xs: list_term_a_b,X3: term_a_b] :
( ( ( K
= ( minus_minus_nat @ ( size_s8906293707977694520rm_a_b @ Xs ) @ one_one_nat ) )
=> ( ( butlast_term_a_b @ ( list_update_term_a_b @ Xs @ K @ X3 ) )
= ( butlast_term_a_b @ Xs ) ) )
& ( ( K
!= ( minus_minus_nat @ ( size_s8906293707977694520rm_a_b @ Xs ) @ one_one_nat ) )
=> ( ( butlast_term_a_b @ ( list_update_term_a_b @ Xs @ K @ X3 ) )
= ( list_update_term_a_b @ ( butlast_term_a_b @ Xs ) @ K @ X3 ) ) ) ) ).
% butlast_list_update
thf(fact_1036_last__list__update,axiom,
! [Xs: list_term_a_b,K: nat,X3: term_a_b] :
( ( Xs != nil_term_a_b )
=> ( ( ( K
= ( minus_minus_nat @ ( size_s8906293707977694520rm_a_b @ Xs ) @ one_one_nat ) )
=> ( ( last_term_a_b @ ( list_update_term_a_b @ Xs @ K @ X3 ) )
= X3 ) )
& ( ( K
!= ( minus_minus_nat @ ( size_s8906293707977694520rm_a_b @ Xs ) @ one_one_nat ) )
=> ( ( last_term_a_b @ ( list_update_term_a_b @ Xs @ K @ X3 ) )
= ( last_term_a_b @ Xs ) ) ) ) ) ).
% last_list_update
thf(fact_1037_last__drop,axiom,
! [N: nat,Xs: list_term_a_b] :
( ( ord_less_nat @ N @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( last_term_a_b @ ( drop_term_a_b @ N @ Xs ) )
= ( last_term_a_b @ Xs ) ) ) ).
% last_drop
thf(fact_1038_last__zip,axiom,
! [Xs: list_a,Ys: list_nat] :
( ( Xs != nil_a )
=> ( ( Ys != nil_nat )
=> ( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( last_P2271748490522340894_a_nat @ ( zip_a_nat @ Xs @ Ys ) )
= ( product_Pair_a_nat @ ( last_a @ Xs ) @ ( last_nat @ Ys ) ) ) ) ) ) ).
% last_zip
thf(fact_1039_last__zip,axiom,
! [Xs: list_nat_nat,Ys: list_nat] :
( ( Xs != nil_nat_nat )
=> ( ( Ys != nil_nat )
=> ( ( ( size_s8208510060688613859at_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( last_P1651921108130080911at_nat @ ( zip_nat_nat_nat @ Xs @ Ys ) )
= ( produc72220940542539688at_nat @ ( last_nat_nat @ Xs ) @ ( last_nat @ Ys ) ) ) ) ) ) ).
% last_zip
thf(fact_1040_last__zip,axiom,
! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
( ( Xs != nil_Pr5478986624290739719at_nat )
=> ( ( Ys != nil_Pr5478986624290739719at_nat )
=> ( ( ( size_s5460976970255530739at_nat @ Xs )
= ( size_s5460976970255530739at_nat @ Ys ) )
=> ( ( last_P5835599730086418438at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs @ Ys ) )
= ( produc6161850002892822231at_nat @ ( last_P6484183829340986144at_nat @ Xs ) @ ( last_P6484183829340986144at_nat @ Ys ) ) ) ) ) ) ).
% last_zip
thf(fact_1041_last__zip,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs != nil_nat )
=> ( ( Ys != nil_nat )
=> ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( last_P6484183829340986144at_nat @ ( zip_nat_nat @ Xs @ Ys ) )
= ( product_Pair_nat_nat @ ( last_nat @ Xs ) @ ( last_nat @ Ys ) ) ) ) ) ) ).
% last_zip
thf(fact_1042_last__zip,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b] :
( ( Xs != nil_term_a_b )
=> ( ( Ys != nil_term_a_b )
=> ( ( ( size_s8906293707977694520rm_a_b @ Xs )
= ( size_s8906293707977694520rm_a_b @ Ys ) )
=> ( ( last_P6746431593072707880rm_a_b @ ( zip_te2260252107881041761rm_a_b @ Xs @ Ys ) )
= ( produc7020197800436672577rm_a_b @ ( last_term_a_b @ Xs ) @ ( last_term_a_b @ Ys ) ) ) ) ) ) ).
% last_zip
thf(fact_1043_last__conv__nth,axiom,
! [Xs: list_term_a_b] :
( ( Xs != nil_term_a_b )
=> ( ( last_term_a_b @ Xs )
= ( nth_term_a_b @ Xs @ ( minus_minus_nat @ ( size_s8906293707977694520rm_a_b @ Xs ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_1044_rtrancl__list__conv,axiom,
! [S: product_prod_nat_nat,T: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ S @ T ) @ ( transi8609417484261754244at_nat @ R3 ) )
= ( ? [Ts2: list_P6011104703257516679at_nat] :
( ( ( last_P6484183829340986144at_nat @ ( cons_P6512896166579812791at_nat @ S @ Ts2 ) )
= T )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s5460976970255530739at_nat @ Ts2 ) )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ ( cons_P6512896166579812791at_nat @ S @ Ts2 ) @ I4 ) @ ( nth_Pr7617993195940197384at_nat @ ( cons_P6512896166579812791at_nat @ S @ Ts2 ) @ ( suc @ I4 ) ) ) @ R3 ) ) ) ) ) ).
% rtrancl_list_conv
thf(fact_1045_rtrancl__list__conv,axiom,
! [S: term_a_b,T: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
= ( ? [Ts2: list_term_a_b] :
( ( ( last_term_a_b @ ( cons_term_a_b @ S @ Ts2 ) )
= T )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s8906293707977694520rm_a_b @ Ts2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( nth_term_a_b @ ( cons_term_a_b @ S @ Ts2 ) @ I4 ) @ ( nth_term_a_b @ ( cons_term_a_b @ S @ Ts2 ) @ ( suc @ I4 ) ) ) @ R3 ) ) ) ) ) ).
% rtrancl_list_conv
thf(fact_1046_rtrancl__list__conv,axiom,
! [S: nat,T: nat,R3: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ S @ T ) @ ( transi2905341329935302413cl_nat @ R3 ) )
= ( ? [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 ) ) ) @ R3 ) ) ) ) ) ).
% rtrancl_list_conv
thf(fact_1047_pair__lessI2,axiom,
! [A: nat,B: nat,S: nat,T: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ S @ T )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ) ).
% pair_lessI2
thf(fact_1048_listrel__rtrancl__refl,axiom,
! [Xs: list_nat,R2: set_Pr1261947904930325089at_nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Xs ) @ ( listrel_nat_nat @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ).
% listrel_rtrancl_refl
thf(fact_1049_pair__less__iff1,axiom,
! [X3: nat,Y: nat,Z3: nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( product_Pair_nat_nat @ X3 @ Z3 ) ) @ fun_pair_less )
= ( ord_less_nat @ Y @ Z3 ) ) ).
% pair_less_iff1
thf(fact_1050_rtrancl__diff__decomp,axiom,
! [X3: term_a_b,Y: term_a_b,A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ ( minus_5192120951422937424rm_a_b @ ( transi7742714808557438673rm_a_b @ A4 ) @ ( transi7742714808557438673rm_a_b @ B4 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A4 ) @ ( relcom370159955682700863rm_a_b @ ( minus_5192120951422937424rm_a_b @ A4 @ B4 ) @ ( transi7742714808557438673rm_a_b @ A4 ) ) ) ) ) ).
% rtrancl_diff_decomp
thf(fact_1051_rtrancl__diff__decomp,axiom,
! [X3: product_prod_nat_nat,Y: product_prod_nat_nat,A4: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( minus_8321449233255521966at_nat @ ( transi8609417484261754244at_nat @ A4 ) @ ( transi8609417484261754244at_nat @ B4 ) ) )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( relcom7295539661566034944at_nat @ ( transi8609417484261754244at_nat @ A4 ) @ ( relcom7295539661566034944at_nat @ ( minus_8321449233255521966at_nat @ A4 @ B4 ) @ ( transi8609417484261754244at_nat @ A4 ) ) ) ) ) ).
% rtrancl_diff_decomp
thf(fact_1052_rtrancl__diff__decomp,axiom,
! [X3: nat,Y: nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( minus_1356011639430497352at_nat @ ( transi2905341329935302413cl_nat @ A4 ) @ ( transi2905341329935302413cl_nat @ B4 ) ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ A4 ) @ ( relcomp_nat_nat_nat @ ( minus_1356011639430497352at_nat @ A4 @ B4 ) @ ( transi2905341329935302413cl_nat @ A4 ) ) ) ) ) ).
% rtrancl_diff_decomp
thf(fact_1053_steps__map,axiom,
! [P2: term_a_b > $o,Q: set_Pr4386577575007340137rm_a_b > $o,F3: term_a_b > term_a_b,G: set_Pr4386577575007340137rm_a_b > set_Pr4386577575007340137rm_a_b,T: term_a_b,R3: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b,U: term_a_b] :
( ! [T4: term_a_b,U2: term_a_b,R7: set_Pr4386577575007340137rm_a_b] :
( ( P2 @ T4 )
=> ( ( Q @ R7 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T4 @ U2 ) @ R7 )
=> ( ( P2 @ U2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ T4 ) @ ( F3 @ U2 ) ) @ ( G @ R7 ) ) ) ) ) )
=> ( ( P2 @ T )
=> ( ( Q @ R3 )
=> ( ( Q @ S2 )
=> ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( transi7742714808557438673rm_a_b @ ( G @ R3 ) ) ) )
& ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R3 ) @ ( relcom370159955682700863rm_a_b @ S2 @ ( transi7742714808557438673rm_a_b @ R3 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( G @ R3 ) ) @ ( relcom370159955682700863rm_a_b @ ( G @ S2 ) @ ( transi7742714808557438673rm_a_b @ ( G @ R3 ) ) ) ) ) ) ) ) ) ) ) ).
% steps_map
thf(fact_1054_steps__map,axiom,
! [P2: term_a_b > $o,Q: set_Pr4386577575007340137rm_a_b > $o,F3: term_a_b > product_prod_nat_nat,G: set_Pr4386577575007340137rm_a_b > set_Pr8693737435421807431at_nat,T: term_a_b,R3: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b,U: term_a_b] :
( ! [T4: term_a_b,U2: term_a_b,R7: set_Pr4386577575007340137rm_a_b] :
( ( P2 @ T4 )
=> ( ( Q @ R7 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T4 @ U2 ) @ R7 )
=> ( ( P2 @ U2 )
& ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F3 @ T4 ) @ ( F3 @ U2 ) ) @ ( G @ R7 ) ) ) ) ) )
=> ( ( P2 @ T )
=> ( ( Q @ R3 )
=> ( ( Q @ S2 )
=> ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( transi8609417484261754244at_nat @ ( G @ R3 ) ) ) )
& ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R3 ) @ ( relcom370159955682700863rm_a_b @ S2 @ ( transi7742714808557438673rm_a_b @ R3 ) ) ) )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( relcom7295539661566034944at_nat @ ( transi8609417484261754244at_nat @ ( G @ R3 ) ) @ ( relcom7295539661566034944at_nat @ ( G @ S2 ) @ ( transi8609417484261754244at_nat @ ( G @ R3 ) ) ) ) ) ) ) ) ) ) ) ).
% steps_map
thf(fact_1055_steps__map,axiom,
! [P2: product_prod_nat_nat > $o,Q: set_Pr8693737435421807431at_nat > $o,F3: product_prod_nat_nat > term_a_b,G: set_Pr8693737435421807431at_nat > set_Pr4386577575007340137rm_a_b,T: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat,S2: set_Pr8693737435421807431at_nat,U: product_prod_nat_nat] :
( ! [T4: product_prod_nat_nat,U2: product_prod_nat_nat,R7: set_Pr8693737435421807431at_nat] :
( ( P2 @ T4 )
=> ( ( Q @ R7 )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ T4 @ U2 ) @ R7 )
=> ( ( P2 @ U2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ T4 ) @ ( F3 @ U2 ) ) @ ( G @ R7 ) ) ) ) ) )
=> ( ( P2 @ T )
=> ( ( Q @ R3 )
=> ( ( Q @ S2 )
=> ( ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ T @ U ) @ ( transi8609417484261754244at_nat @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( transi7742714808557438673rm_a_b @ ( G @ R3 ) ) ) )
& ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ T @ U ) @ ( relcom7295539661566034944at_nat @ ( transi8609417484261754244at_nat @ R3 ) @ ( relcom7295539661566034944at_nat @ S2 @ ( transi8609417484261754244at_nat @ R3 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( G @ R3 ) ) @ ( relcom370159955682700863rm_a_b @ ( G @ S2 ) @ ( transi7742714808557438673rm_a_b @ ( G @ R3 ) ) ) ) ) ) ) ) ) ) ) ).
% steps_map
thf(fact_1056_steps__map,axiom,
! [P2: product_prod_nat_nat > $o,Q: set_Pr8693737435421807431at_nat > $o,F3: product_prod_nat_nat > product_prod_nat_nat,G: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat,T: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat,S2: set_Pr8693737435421807431at_nat,U: product_prod_nat_nat] :
( ! [T4: product_prod_nat_nat,U2: product_prod_nat_nat,R7: set_Pr8693737435421807431at_nat] :
( ( P2 @ T4 )
=> ( ( Q @ R7 )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ T4 @ U2 ) @ R7 )
=> ( ( P2 @ U2 )
& ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F3 @ T4 ) @ ( F3 @ U2 ) ) @ ( G @ R7 ) ) ) ) ) )
=> ( ( P2 @ T )
=> ( ( Q @ R3 )
=> ( ( Q @ S2 )
=> ( ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ T @ U ) @ ( transi8609417484261754244at_nat @ R3 ) )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( transi8609417484261754244at_nat @ ( G @ R3 ) ) ) )
& ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ T @ U ) @ ( relcom7295539661566034944at_nat @ ( transi8609417484261754244at_nat @ R3 ) @ ( relcom7295539661566034944at_nat @ S2 @ ( transi8609417484261754244at_nat @ R3 ) ) ) )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( relcom7295539661566034944at_nat @ ( transi8609417484261754244at_nat @ ( G @ R3 ) ) @ ( relcom7295539661566034944at_nat @ ( G @ S2 ) @ ( transi8609417484261754244at_nat @ ( G @ R3 ) ) ) ) ) ) ) ) ) ) ) ).
% steps_map
thf(fact_1057_steps__map,axiom,
! [P2: term_a_b > $o,Q: set_Pr4386577575007340137rm_a_b > $o,F3: term_a_b > nat,G: set_Pr4386577575007340137rm_a_b > set_Pr1261947904930325089at_nat,T: term_a_b,R3: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b,U: term_a_b] :
( ! [T4: term_a_b,U2: term_a_b,R7: set_Pr4386577575007340137rm_a_b] :
( ( P2 @ T4 )
=> ( ( Q @ R7 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T4 @ U2 ) @ R7 )
=> ( ( P2 @ U2 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F3 @ T4 ) @ ( F3 @ U2 ) ) @ ( G @ R7 ) ) ) ) ) )
=> ( ( P2 @ T )
=> ( ( Q @ R3 )
=> ( ( Q @ S2 )
=> ( ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( transi2905341329935302413cl_nat @ ( G @ R3 ) ) ) )
& ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ T @ U ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R3 ) @ ( relcom370159955682700863rm_a_b @ S2 @ ( transi7742714808557438673rm_a_b @ R3 ) ) ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ ( G @ R3 ) ) @ ( relcomp_nat_nat_nat @ ( G @ S2 ) @ ( transi2905341329935302413cl_nat @ ( G @ R3 ) ) ) ) ) ) ) ) ) ) ) ).
% steps_map
thf(fact_1058_steps__map,axiom,
! [P2: product_prod_nat_nat > $o,Q: set_Pr8693737435421807431at_nat > $o,F3: product_prod_nat_nat > nat,G: set_Pr8693737435421807431at_nat > set_Pr1261947904930325089at_nat,T: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat,S2: set_Pr8693737435421807431at_nat,U: product_prod_nat_nat] :
( ! [T4: product_prod_nat_nat,U2: product_prod_nat_nat,R7: set_Pr8693737435421807431at_nat] :
( ( P2 @ T4 )
=> ( ( Q @ R7 )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ T4 @ U2 ) @ R7 )
=> ( ( P2 @ U2 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F3 @ T4 ) @ ( F3 @ U2 ) ) @ ( G @ R7 ) ) ) ) ) )
=> ( ( P2 @ T )
=> ( ( Q @ R3 )
=> ( ( Q @ S2 )
=> ( ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ T @ U ) @ ( transi8609417484261754244at_nat @ R3 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( transi2905341329935302413cl_nat @ ( G @ R3 ) ) ) )
& ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ T @ U ) @ ( relcom7295539661566034944at_nat @ ( transi8609417484261754244at_nat @ R3 ) @ ( relcom7295539661566034944at_nat @ S2 @ ( transi8609417484261754244at_nat @ R3 ) ) ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ ( G @ R3 ) ) @ ( relcomp_nat_nat_nat @ ( G @ S2 ) @ ( transi2905341329935302413cl_nat @ ( G @ R3 ) ) ) ) ) ) ) ) ) ) ) ).
% steps_map
thf(fact_1059_steps__map,axiom,
! [P2: nat > $o,Q: set_Pr1261947904930325089at_nat > $o,F3: nat > term_a_b,G: set_Pr1261947904930325089at_nat > set_Pr4386577575007340137rm_a_b,T: nat,R3: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat,U: nat] :
( ! [T4: nat,U2: nat,R7: set_Pr1261947904930325089at_nat] :
( ( P2 @ T4 )
=> ( ( Q @ R7 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ T4 @ U2 ) @ R7 )
=> ( ( P2 @ U2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ T4 ) @ ( F3 @ U2 ) ) @ ( G @ R7 ) ) ) ) ) )
=> ( ( P2 @ T )
=> ( ( Q @ R3 )
=> ( ( Q @ S2 )
=> ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ T @ U ) @ ( transi2905341329935302413cl_nat @ R3 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( transi7742714808557438673rm_a_b @ ( G @ R3 ) ) ) )
& ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ T @ U ) @ ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ ( relcomp_nat_nat_nat @ S2 @ ( transi2905341329935302413cl_nat @ R3 ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( G @ R3 ) ) @ ( relcom370159955682700863rm_a_b @ ( G @ S2 ) @ ( transi7742714808557438673rm_a_b @ ( G @ R3 ) ) ) ) ) ) ) ) ) ) ) ).
% steps_map
thf(fact_1060_steps__map,axiom,
! [P2: nat > $o,Q: set_Pr1261947904930325089at_nat > $o,F3: nat > product_prod_nat_nat,G: set_Pr1261947904930325089at_nat > set_Pr8693737435421807431at_nat,T: nat,R3: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat,U: nat] :
( ! [T4: nat,U2: nat,R7: set_Pr1261947904930325089at_nat] :
( ( P2 @ T4 )
=> ( ( Q @ R7 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ T4 @ U2 ) @ R7 )
=> ( ( P2 @ U2 )
& ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F3 @ T4 ) @ ( F3 @ U2 ) ) @ ( G @ R7 ) ) ) ) ) )
=> ( ( P2 @ T )
=> ( ( Q @ R3 )
=> ( ( Q @ S2 )
=> ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ T @ U ) @ ( transi2905341329935302413cl_nat @ R3 ) )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( transi8609417484261754244at_nat @ ( G @ R3 ) ) ) )
& ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ T @ U ) @ ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ ( relcomp_nat_nat_nat @ S2 @ ( transi2905341329935302413cl_nat @ R3 ) ) ) )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( relcom7295539661566034944at_nat @ ( transi8609417484261754244at_nat @ ( G @ R3 ) ) @ ( relcom7295539661566034944at_nat @ ( G @ S2 ) @ ( transi8609417484261754244at_nat @ ( G @ R3 ) ) ) ) ) ) ) ) ) ) ) ).
% steps_map
thf(fact_1061_steps__map,axiom,
! [P2: nat > $o,Q: set_Pr1261947904930325089at_nat > $o,F3: nat > nat,G: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,T: nat,R3: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat,U: nat] :
( ! [T4: nat,U2: nat,R7: set_Pr1261947904930325089at_nat] :
( ( P2 @ T4 )
=> ( ( Q @ R7 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ T4 @ U2 ) @ R7 )
=> ( ( P2 @ U2 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F3 @ T4 ) @ ( F3 @ U2 ) ) @ ( G @ R7 ) ) ) ) ) )
=> ( ( P2 @ T )
=> ( ( Q @ R3 )
=> ( ( Q @ S2 )
=> ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ T @ U ) @ ( transi2905341329935302413cl_nat @ R3 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( transi2905341329935302413cl_nat @ ( G @ R3 ) ) ) )
& ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ T @ U ) @ ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ ( relcomp_nat_nat_nat @ S2 @ ( transi2905341329935302413cl_nat @ R3 ) ) ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F3 @ T ) @ ( F3 @ U ) ) @ ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ ( G @ R3 ) ) @ ( relcomp_nat_nat_nat @ ( G @ S2 ) @ ( transi2905341329935302413cl_nat @ ( G @ R3 ) ) ) ) ) ) ) ) ) ) ) ).
% steps_map
thf(fact_1062_rtrancl__listrel1__ConsI2,axiom,
! [X3: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b,Xs: list_term_a_b,Ys: list_term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ Xs @ Ys ) @ ( transi911495901722478305rm_a_b @ ( listrel1_term_a_b @ R2 ) ) )
=> ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ ( cons_term_a_b @ X3 @ Xs ) @ ( cons_term_a_b @ Y @ Ys ) ) @ ( transi911495901722478305rm_a_b @ ( listrel1_term_a_b @ R2 ) ) ) ) ) ).
% rtrancl_listrel1_ConsI2
thf(fact_1063_rtrancl__listrel1__ConsI2,axiom,
! [X3: product_prod_nat_nat,Y: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( transi8609417484261754244at_nat @ R2 ) )
=> ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys ) @ ( transi5735831900260020618at_nat @ ( listre4828114922151135584at_nat @ R2 ) ) )
=> ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys ) ) @ ( transi5735831900260020618at_nat @ ( listre4828114922151135584at_nat @ R2 ) ) ) ) ) ).
% rtrancl_listrel1_ConsI2
thf(fact_1064_rtrancl__listrel1__ConsI2,axiom,
! [X3: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( transi5285580207609517981st_nat @ ( listrel1_nat @ R2 ) ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( transi5285580207609517981st_nat @ ( listrel1_nat @ R2 ) ) ) ) ) ).
% rtrancl_listrel1_ConsI2
thf(fact_1065_listrel__rtrancl__trans,axiom,
! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat,Zs: list_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ ( transi2905341329935302413cl_nat @ R2 ) ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( listrel_nat_nat @ ( transi2905341329935302413cl_nat @ R2 ) ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Zs ) @ ( listrel_nat_nat @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ) ) ).
% listrel_rtrancl_trans
thf(fact_1066_listrel__rtrancl__eq__rtrancl__listrel1,axiom,
! [R2: set_Pr1261947904930325089at_nat] :
( ( listrel_nat_nat @ ( transi2905341329935302413cl_nat @ R2 ) )
= ( transi5285580207609517981st_nat @ ( listrel1_nat @ R2 ) ) ) ).
% listrel_rtrancl_eq_rtrancl_listrel1
thf(fact_1067_rtrancl__O__shift,axiom,
! [S2: set_Pr4386577575007340137rm_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( relcom370159955682700863rm_a_b @ S2 @ R3 ) ) @ S2 )
= ( relcom370159955682700863rm_a_b @ S2 @ ( transi7742714808557438673rm_a_b @ ( relcom370159955682700863rm_a_b @ R3 @ S2 ) ) ) ) ).
% rtrancl_O_shift
thf(fact_1068_rtrancl__O__shift,axiom,
! [S2: set_Pr1261947904930325089at_nat,R3: set_Pr1261947904930325089at_nat] :
( ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ ( relcomp_nat_nat_nat @ S2 @ R3 ) ) @ S2 )
= ( relcomp_nat_nat_nat @ S2 @ ( transi2905341329935302413cl_nat @ ( relcomp_nat_nat_nat @ R3 @ S2 ) ) ) ) ).
% rtrancl_O_shift
thf(fact_1069_kleene__induct,axiom,
! [A4: set_Pr4386577575007340137rm_a_b,X8: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b,C4: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ A4 @ X8 )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ B4 @ X8 ) @ X8 )
=> ( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ X8 @ C4 ) @ X8 )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ B4 ) @ ( relcom370159955682700863rm_a_b @ A4 @ ( transi7742714808557438673rm_a_b @ C4 ) ) ) @ X8 ) ) ) ) ).
% kleene_induct
thf(fact_1070_kleene__induct,axiom,
! [A4: set_Pr1261947904930325089at_nat,X8: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C4: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ A4 @ X8 )
=> ( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ B4 @ X8 ) @ X8 )
=> ( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ X8 @ C4 ) @ X8 )
=> ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ B4 ) @ ( relcomp_nat_nat_nat @ A4 @ ( transi2905341329935302413cl_nat @ C4 ) ) ) @ X8 ) ) ) ) ).
% kleene_induct
thf(fact_1071_kleene__induct,axiom,
! [A4: set_Pr4934435412358123699_a_nat,X8: set_Pr4934435412358123699_a_nat,B4: set_Product_prod_a_a,C4: set_Pr1261947904930325089at_nat] :
( ( ord_le8666007276011122963_a_nat @ A4 @ X8 )
=> ( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ B4 @ X8 ) @ X8 )
=> ( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_nat_nat @ X8 @ C4 ) @ X8 )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ ( transitive_rtrancl_a @ B4 ) @ ( relcomp_a_nat_nat @ A4 @ ( transi2905341329935302413cl_nat @ C4 ) ) ) @ X8 ) ) ) ) ).
% kleene_induct
thf(fact_1072_compat__tr__compat,axiom,
! [NS: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S2 ) @ S2 )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ NS ) @ S2 ) @ S2 ) ) ).
% compat_tr_compat
thf(fact_1073_compat__tr__compat,axiom,
! [NS: set_Product_prod_a_a,S2: set_Pr4934435412358123699_a_nat] :
( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ NS @ S2 ) @ S2 )
=> ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ ( transitive_rtrancl_a @ NS ) @ S2 ) @ S2 ) ) ).
% compat_tr_compat
thf(fact_1074_rtrancl__O__push,axiom,
! [S2: set_Pr4386577575007340137rm_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ S2 @ R3 ) @ ( relcom370159955682700863rm_a_b @ R3 @ ( transi7742714808557438673rm_a_b @ S2 ) ) )
=> ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S2 ) @ R3 ) @ ( relcom370159955682700863rm_a_b @ R3 @ ( transi7742714808557438673rm_a_b @ S2 ) ) ) ) ).
% rtrancl_O_push
thf(fact_1075_rtrancl__O__push,axiom,
! [S2: set_Pr1261947904930325089at_nat,R3: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ S2 @ R3 ) @ ( relcomp_nat_nat_nat @ R3 @ ( transi2905341329935302413cl_nat @ S2 ) ) )
=> ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ S2 ) @ R3 ) @ ( relcomp_nat_nat_nat @ R3 @ ( transi2905341329935302413cl_nat @ S2 ) ) ) ) ).
% rtrancl_O_push
thf(fact_1076_listrel1__rtrancl__subset__rtrancl__listrel1,axiom,
! [R2: set_Pr1261947904930325089at_nat] : ( ord_le8406513867147106209st_nat @ ( listrel1_nat @ ( transi2905341329935302413cl_nat @ R2 ) ) @ ( transi5285580207609517981st_nat @ ( listrel1_nat @ R2 ) ) ) ).
% listrel1_rtrancl_subset_rtrancl_listrel1
thf(fact_1077_chain__imp__rtrancl,axiom,
! [S2: nat > term_a_b,R2: set_Pr4386577575007340137rm_a_b,I: nat,J: nat] :
( ! [I3: nat] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( S2 @ I3 ) @ ( S2 @ ( suc @ I3 ) ) ) @ R2 )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( S2 @ I ) @ ( S2 @ J ) ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ) ).
% chain_imp_rtrancl
thf(fact_1078_chain__imp__rtrancl,axiom,
! [S2: nat > product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,I: nat,J: nat] :
( ! [I3: nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( S2 @ I3 ) @ ( S2 @ ( suc @ I3 ) ) ) @ R2 )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( S2 @ I ) @ ( S2 @ J ) ) @ ( transi8609417484261754244at_nat @ R2 ) ) ) ) ).
% chain_imp_rtrancl
thf(fact_1079_chain__imp__rtrancl,axiom,
! [S2: nat > nat,R2: set_Pr1261947904930325089at_nat,I: nat,J: nat] :
( ! [I3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( S2 @ I3 ) @ ( S2 @ ( suc @ I3 ) ) ) @ R2 )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( S2 @ I ) @ ( S2 @ J ) ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ) ).
% chain_imp_rtrancl
thf(fact_1080_non__strict__into__strict,axiom,
! [NS: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b,S: term_a_b,T: term_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S2 ) @ S2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ NS ) @ S2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ S2 ) ) ) ).
% non_strict_into_strict
thf(fact_1081_non__strict__into__strict,axiom,
! [NS: set_Pr7682762132356531903at_nat,S2: set_Pr9093778441882193744at_nat,S: nat > nat,T: nat] :
( ( ord_le3678578370064672496at_nat @ ( relcom9054258243621454929at_nat @ NS @ S2 ) @ S2 )
=> ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ S @ T ) @ ( relcom9054258243621454929at_nat @ ( transi3031656534272781564at_nat @ NS ) @ S2 ) )
=> ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ S @ T ) @ S2 ) ) ) ).
% non_strict_into_strict
thf(fact_1082_non__strict__into__strict,axiom,
! [NS: set_Pr8693737435421807431at_nat,S2: set_Pr8693737435421807431at_nat,S: product_prod_nat_nat,T: product_prod_nat_nat] :
( ( ord_le3000389064537975527at_nat @ ( relcom7295539661566034944at_nat @ NS @ S2 ) @ S2 )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ S @ T ) @ ( relcom7295539661566034944at_nat @ ( transi8609417484261754244at_nat @ NS ) @ S2 ) )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ S @ T ) @ S2 ) ) ) ).
% non_strict_into_strict
thf(fact_1083_non__strict__into__strict,axiom,
! [NS: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat,S: nat,T: nat] :
( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ NS @ S2 ) @ S2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ S @ T ) @ ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ NS ) @ S2 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ S @ T ) @ S2 ) ) ) ).
% non_strict_into_strict
thf(fact_1084_non__strict__into__strict,axiom,
! [NS: set_Product_prod_a_a,S2: set_Pr4934435412358123699_a_nat,S: a,T: nat] :
( ( ord_le8666007276011122963_a_nat @ ( relcomp_a_a_nat @ NS @ S2 ) @ S2 )
=> ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ S @ T ) @ ( relcomp_a_a_nat @ ( transitive_rtrancl_a @ NS ) @ S2 ) )
=> ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ S @ T ) @ S2 ) ) ) ).
% non_strict_into_strict
thf(fact_1085_firstStep,axiom,
! [L2: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b,R3: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y: term_a_b] :
( ( L2
= ( sup_su6776935440552674877rm_a_b @ S2 @ R3 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ ( transi7742714808557438673rm_a_b @ L2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
| ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R3 ) @ ( relcom370159955682700863rm_a_b @ S2 @ ( transi7742714808557438673rm_a_b @ L2 ) ) ) ) ) ) ) ).
% firstStep
thf(fact_1086_firstStep,axiom,
! [L2: set_Pr8693737435421807431at_nat,S2: set_Pr8693737435421807431at_nat,R3: set_Pr8693737435421807431at_nat,X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( L2
= ( sup_su718114333110466843at_nat @ S2 @ R3 ) )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( transi8609417484261754244at_nat @ L2 ) )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( transi8609417484261754244at_nat @ R3 ) )
| ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( relcom7295539661566034944at_nat @ ( transi8609417484261754244at_nat @ R3 ) @ ( relcom7295539661566034944at_nat @ S2 @ ( transi8609417484261754244at_nat @ L2 ) ) ) ) ) ) ) ).
% firstStep
thf(fact_1087_firstStep,axiom,
! [L2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat,R3: set_Pr1261947904930325089at_nat,X3: nat,Y: nat] :
( ( L2
= ( sup_su6327502436637775413at_nat @ S2 @ R3 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( transi2905341329935302413cl_nat @ L2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( transi2905341329935302413cl_nat @ R3 ) )
| ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ ( relcomp_nat_nat_nat @ S2 @ ( transi2905341329935302413cl_nat @ L2 ) ) ) ) ) ) ) ).
% firstStep
thf(fact_1088_first__step,axiom,
! [C4: set_Pr4386577575007340137rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y: term_a_b,Z3: term_a_b] :
( ( C4
= ( sup_su6776935440552674877rm_a_b @ A4 @ B4 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ ( transi7742714808557438673rm_a_b @ C4 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z3 ) @ B4 )
=> ? [Y3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A4 ) @ B4 ) ) ) ) ) ).
% first_step
thf(fact_1089_first__step,axiom,
! [C4: set_Pr8693737435421807431at_nat,A4: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat,X3: product_prod_nat_nat,Y: product_prod_nat_nat,Z3: product_prod_nat_nat] :
( ( C4
= ( sup_su718114333110466843at_nat @ A4 @ B4 ) )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( transi8609417484261754244at_nat @ C4 ) )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ Z3 ) @ B4 )
=> ? [Y3: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ ( relcom7295539661566034944at_nat @ ( transi8609417484261754244at_nat @ A4 ) @ B4 ) ) ) ) ) ).
% first_step
thf(fact_1090_first__step,axiom,
! [C4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,X3: nat,Y: nat,Z3: nat] :
( ( C4
= ( sup_su6327502436637775413at_nat @ A4 @ B4 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( transi2905341329935302413cl_nat @ C4 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z3 ) @ B4 )
=> ? [Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ A4 ) @ B4 ) ) ) ) ) ).
% first_step
thf(fact_1091_first__step__O,axiom,
! [C4: set_Pr4386577575007340137rm_a_b,A4: set_Pr4386577575007340137rm_a_b,B4: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y: term_a_b] :
( ( C4
= ( sup_su6776935440552674877rm_a_b @ A4 @ B4 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ C4 ) @ B4 ) )
=> ? [Y3: term_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A4 ) @ B4 ) ) ) ) ).
% first_step_O
thf(fact_1092_first__step__O,axiom,
! [C4: set_Pr8693737435421807431at_nat,A4: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat,X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( C4
= ( sup_su718114333110466843at_nat @ A4 @ B4 ) )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( relcom7295539661566034944at_nat @ ( transi8609417484261754244at_nat @ C4 ) @ B4 ) )
=> ? [Y3: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ ( relcom7295539661566034944at_nat @ ( transi8609417484261754244at_nat @ A4 ) @ B4 ) ) ) ) ).
% first_step_O
thf(fact_1093_first__step__O,axiom,
! [C4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,X3: nat,Y: nat] :
( ( C4
= ( sup_su6327502436637775413at_nat @ A4 @ B4 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ C4 ) @ B4 ) )
=> ? [Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ A4 ) @ B4 ) ) ) ) ).
% first_step_O
thf(fact_1094_right__comp__S,axiom,
! [X3: term_a_b,Y: term_a_b,S2: set_Pr4386577575007340137rm_a_b,NS: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ ( relcom370159955682700863rm_a_b @ S2 @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ S2 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S2 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ ( relcom370159955682700863rm_a_b @ S2 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S2 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) ) ) ).
% right_comp_S
thf(fact_1095_right__comp__S,axiom,
! [X3: product_prod_nat_nat,Y: product_prod_nat_nat,S2: set_Pr8693737435421807431at_nat,NS: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( relcom7295539661566034944at_nat @ S2 @ ( sup_su718114333110466843at_nat @ ( relcom7295539661566034944at_nat @ S2 @ ( relcom7295539661566034944at_nat @ ( transi8609417484261754244at_nat @ S2 ) @ ( transi8609417484261754244at_nat @ NS ) ) ) @ ( transi8609417484261754244at_nat @ NS ) ) ) )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( relcom7295539661566034944at_nat @ S2 @ ( relcom7295539661566034944at_nat @ ( transi8609417484261754244at_nat @ S2 ) @ ( transi8609417484261754244at_nat @ NS ) ) ) ) ) ).
% right_comp_S
thf(fact_1096_right__comp__S,axiom,
! [X3: nat,Y: nat,S2: set_Pr1261947904930325089at_nat,NS: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( relcomp_nat_nat_nat @ S2 @ ( sup_su6327502436637775413at_nat @ ( relcomp_nat_nat_nat @ S2 @ ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ S2 ) @ ( transi2905341329935302413cl_nat @ NS ) ) ) @ ( transi2905341329935302413cl_nat @ NS ) ) ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( relcomp_nat_nat_nat @ S2 @ ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ S2 ) @ ( transi2905341329935302413cl_nat @ NS ) ) ) ) ) ).
% right_comp_S
thf(fact_1097_rtrancl__U__push,axiom,
! [S2: set_Pr4386577575007340137rm_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ S2 @ R3 ) @ ( relcom370159955682700863rm_a_b @ R3 @ ( transi7742714808557438673rm_a_b @ S2 ) ) )
=> ( ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ S2 @ R3 ) )
= ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R3 ) @ ( transi7742714808557438673rm_a_b @ S2 ) ) ) ) ).
% rtrancl_U_push
thf(fact_1098_rtrancl__U__push,axiom,
! [S2: set_Pr1261947904930325089at_nat,R3: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ S2 @ R3 ) @ ( relcomp_nat_nat_nat @ R3 @ ( transi2905341329935302413cl_nat @ S2 ) ) )
=> ( ( transi2905341329935302413cl_nat @ ( sup_su6327502436637775413at_nat @ S2 @ R3 ) )
= ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ ( transi2905341329935302413cl_nat @ S2 ) ) ) ) ).
% rtrancl_U_push
thf(fact_1099_compatible__conv,axiom,
! [NS: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S2 ) @ S2 )
=> ( ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S2 ) ) @ ( relcom370159955682700863rm_a_b @ S2 @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S2 ) ) ) )
= ( relcom370159955682700863rm_a_b @ S2 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S2 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) ) ) ).
% compatible_conv
thf(fact_1100_compatible__conv,axiom,
! [NS: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ NS @ S2 ) @ S2 )
=> ( ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ ( sup_su6327502436637775413at_nat @ NS @ S2 ) ) @ ( relcomp_nat_nat_nat @ S2 @ ( transi2905341329935302413cl_nat @ ( sup_su6327502436637775413at_nat @ NS @ S2 ) ) ) )
= ( relcomp_nat_nat_nat @ S2 @ ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ S2 ) @ ( transi2905341329935302413cl_nat @ NS ) ) ) ) ) ).
% compatible_conv
thf(fact_1101_rtrancl__listrel1__eq__len,axiom,
! [X3: list_term_a_b,Y: list_term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member4405265420456397394rm_a_b @ ( produc4885699992713594593rm_a_b @ X3 @ Y ) @ ( transi911495901722478305rm_a_b @ ( listrel1_term_a_b @ R2 ) ) )
=> ( ( size_s8906293707977694520rm_a_b @ X3 )
= ( size_s8906293707977694520rm_a_b @ Y ) ) ) ).
% rtrancl_listrel1_eq_len
thf(fact_1102_rtrancl__Un2__separatorE,axiom,
! [B4: set_Pr4386577575007340137rm_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( ( relcom370159955682700863rm_a_b @ B4 @ A4 )
= bot_bo197521221353338581rm_a_b )
=> ( ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ A4 @ B4 ) )
= ( sup_su6776935440552674877rm_a_b @ ( transi7742714808557438673rm_a_b @ A4 ) @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ A4 ) @ ( transi7742714808557438673rm_a_b @ B4 ) ) ) ) ) ).
% rtrancl_Un2_separatorE
thf(fact_1103_rtrancl__Un2__separatorE,axiom,
! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
( ( ( relcomp_nat_nat_nat @ B4 @ A4 )
= bot_bo2099793752762293965at_nat )
=> ( ( transi2905341329935302413cl_nat @ ( sup_su6327502436637775413at_nat @ A4 @ B4 ) )
= ( sup_su6327502436637775413at_nat @ ( transi2905341329935302413cl_nat @ A4 ) @ ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ A4 ) @ ( transi2905341329935302413cl_nat @ B4 ) ) ) ) ) ).
% rtrancl_Un2_separatorE
thf(fact_1104_listrel__reflcl__if__listrel1,axiom,
! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ) ).
% listrel_reflcl_if_listrel1
thf(fact_1105_rtrancl__imp__seq,axiom,
! [X3: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ? [F7: nat > term_a_b,N3: nat] :
( ( ( F7 @ zero_zero_nat )
= X3 )
& ( ( F7 @ N3 )
= Y )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N3 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F7 @ I2 ) @ ( F7 @ ( suc @ I2 ) ) ) @ R2 ) ) ) ) ).
% rtrancl_imp_seq
thf(fact_1106_rtrancl__imp__seq,axiom,
! [X3: product_prod_nat_nat,Y: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( transi8609417484261754244at_nat @ R2 ) )
=> ? [F7: nat > product_prod_nat_nat,N3: nat] :
( ( ( F7 @ zero_zero_nat )
= X3 )
& ( ( F7 @ N3 )
= Y )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N3 )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F7 @ I2 ) @ ( F7 @ ( suc @ I2 ) ) ) @ R2 ) ) ) ) ).
% rtrancl_imp_seq
thf(fact_1107_rtrancl__imp__seq,axiom,
! [X3: nat,Y: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ? [F7: nat > nat,N3: nat] :
( ( ( F7 @ zero_zero_nat )
= X3 )
& ( ( F7 @ N3 )
= Y )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N3 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F7 @ I2 ) @ ( F7 @ ( suc @ I2 ) ) ) @ R2 ) ) ) ) ).
% rtrancl_imp_seq
thf(fact_1108_rtrancl__fun__conv,axiom,
! [S: term_a_b,T: term_a_b,R3: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ S @ T ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
= ( ? [F5: nat > term_a_b,N2: nat] :
( ( ( F5 @ zero_zero_nat )
= S )
& ( ( F5 @ N2 )
= T )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( F5 @ I4 ) @ ( F5 @ ( suc @ I4 ) ) ) @ R3 ) ) ) ) ) ).
% rtrancl_fun_conv
thf(fact_1109_rtrancl__fun__conv,axiom,
! [S: product_prod_nat_nat,T: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ S @ T ) @ ( transi8609417484261754244at_nat @ R3 ) )
= ( ? [F5: nat > product_prod_nat_nat,N2: nat] :
( ( ( F5 @ zero_zero_nat )
= S )
& ( ( F5 @ N2 )
= T )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F5 @ I4 ) @ ( F5 @ ( suc @ I4 ) ) ) @ R3 ) ) ) ) ) ).
% rtrancl_fun_conv
thf(fact_1110_rtrancl__fun__conv,axiom,
! [S: nat,T: nat,R3: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ S @ T ) @ ( transi2905341329935302413cl_nat @ R3 ) )
= ( ? [F5: nat > nat,N2: nat] :
( ( ( F5 @ zero_zero_nat )
= S )
& ( ( F5 @ N2 )
= T )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F5 @ I4 ) @ ( F5 @ ( suc @ I4 ) ) ) @ R3 ) ) ) ) ) ).
% rtrancl_fun_conv
thf(fact_1111_pair__lessI1,axiom,
! [A: nat,B: nat,S: nat,T: nat] :
( ( ord_less_nat @ A @ B )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ).
% pair_lessI1
thf(fact_1112_compatible__rtrancl__split,axiom,
! [NS: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b,X3: term_a_b,Y: term_a_b] :
( ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ NS @ S2 ) @ S2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ NS @ S2 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ ( sup_su6776935440552674877rm_a_b @ ( relcom370159955682700863rm_a_b @ S2 @ ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ S2 ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) @ ( transi7742714808557438673rm_a_b @ NS ) ) ) ) ) ).
% compatible_rtrancl_split
thf(fact_1113_compatible__rtrancl__split,axiom,
! [NS: set_Pr8693737435421807431at_nat,S2: set_Pr8693737435421807431at_nat,X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( ord_le3000389064537975527at_nat @ ( relcom7295539661566034944at_nat @ NS @ S2 ) @ S2 )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( transi8609417484261754244at_nat @ ( sup_su718114333110466843at_nat @ NS @ S2 ) ) )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( sup_su718114333110466843at_nat @ ( relcom7295539661566034944at_nat @ S2 @ ( relcom7295539661566034944at_nat @ ( transi8609417484261754244at_nat @ S2 ) @ ( transi8609417484261754244at_nat @ NS ) ) ) @ ( transi8609417484261754244at_nat @ NS ) ) ) ) ) ).
% compatible_rtrancl_split
thf(fact_1114_compatible__rtrancl__split,axiom,
! [NS: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat,X3: nat,Y: nat] :
( ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ NS @ S2 ) @ S2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( transi2905341329935302413cl_nat @ ( sup_su6327502436637775413at_nat @ NS @ S2 ) ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( sup_su6327502436637775413at_nat @ ( relcomp_nat_nat_nat @ S2 @ ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ S2 ) @ ( transi2905341329935302413cl_nat @ NS ) ) ) @ ( transi2905341329935302413cl_nat @ NS ) ) ) ) ) ).
% compatible_rtrancl_split
thf(fact_1115_rtrancl__idemp__self__comp,axiom,
! [R3: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R3 ) @ ( transi7742714808557438673rm_a_b @ R3 ) )
= ( transi7742714808557438673rm_a_b @ R3 ) ) ).
% rtrancl_idemp_self_comp
thf(fact_1116_rtrancl__idemp__self__comp,axiom,
! [R3: set_Pr1261947904930325089at_nat] :
( ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ ( transi2905341329935302413cl_nat @ R3 ) )
= ( transi2905341329935302413cl_nat @ R3 ) ) ).
% rtrancl_idemp_self_comp
thf(fact_1117_quasi__commute__def,axiom,
( abstra4317384144805385751rm_a_b
= ( ^ [R6: set_Pr4386577575007340137rm_a_b,S5: set_Pr4386577575007340137rm_a_b] : ( ord_le118470702582115849rm_a_b @ ( relcom370159955682700863rm_a_b @ S5 @ R6 ) @ ( relcom370159955682700863rm_a_b @ R6 @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R6 @ S5 ) ) ) ) ) ) ).
% quasi_commute_def
thf(fact_1118_quasi__commute__def,axiom,
( abstra4838182543893801299te_nat
= ( ^ [R6: set_Pr1261947904930325089at_nat,S5: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( relcomp_nat_nat_nat @ S5 @ R6 ) @ ( relcomp_nat_nat_nat @ R6 @ ( transi2905341329935302413cl_nat @ ( sup_su6327502436637775413at_nat @ R6 @ S5 ) ) ) ) ) ) ).
% quasi_commute_def
thf(fact_1119_converse__rtrancl__into__rtrancl,axiom,
! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,C3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C3 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C3 ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_1120_converse__rtrancl__into__rtrancl,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,C3: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R2 )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ C3 ) @ ( transi8609417484261754244at_nat @ R2 ) )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ C3 ) @ ( transi8609417484261754244at_nat @ R2 ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_1121_converse__rtrancl__into__rtrancl,axiom,
! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C3 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C3 ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_1122_converse__rtrancl__induct,axiom,
! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( P2 @ B )
=> ( ! [Y3: term_a_b,Z2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ R2 )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Z2 @ B ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( P2 @ Z2 )
=> ( P2 @ Y3 ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_1123_converse__rtrancl__induct,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,P2: product_prod_nat_nat > $o] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( transi8609417484261754244at_nat @ R2 ) )
=> ( ( P2 @ B )
=> ( ! [Y3: product_prod_nat_nat,Z2: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ Z2 ) @ R2 )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Z2 @ B ) @ ( transi8609417484261754244at_nat @ R2 ) )
=> ( ( P2 @ Z2 )
=> ( P2 @ Y3 ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_1124_converse__rtrancl__induct,axiom,
! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,P2: nat > $o] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( ( P2 @ B )
=> ( ! [Y3: nat,Z2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z2 ) @ R2 )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Z2 @ B ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( ( P2 @ Z2 )
=> ( P2 @ Y3 ) ) ) )
=> ( P2 @ A ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_1125_converse__rtranclE,axiom,
! [X3: term_a_b,Z3: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z3 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( X3 != Z3 )
=> ~ ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y3 ) @ R2 )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z3 ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ) ) ).
% converse_rtranclE
thf(fact_1126_converse__rtranclE,axiom,
! [X3: product_prod_nat_nat,Z3: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Z3 ) @ ( transi8609417484261754244at_nat @ R2 ) )
=> ( ( X3 != Z3 )
=> ~ ! [Y3: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ R2 )
=> ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ Z3 ) @ ( transi8609417484261754244at_nat @ R2 ) ) ) ) ) ).
% converse_rtranclE
thf(fact_1127_converse__rtranclE,axiom,
! [X3: nat,Z3: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Z3 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( ( X3 != Z3 )
=> ~ ! [Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z3 ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ) ) ).
% converse_rtranclE
thf(fact_1128_rtrancl__induct,axiom,
! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,P2: term_a_b > $o] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( P2 @ A )
=> ( ! [Y3: term_a_b,Z2: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ Z2 ) @ R2 )
=> ( ( P2 @ Y3 )
=> ( P2 @ Z2 ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% rtrancl_induct
thf(fact_1129_rtrancl__induct,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,P2: product_prod_nat_nat > $o] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( transi8609417484261754244at_nat @ R2 ) )
=> ( ( P2 @ A )
=> ( ! [Y3: product_prod_nat_nat,Z2: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ Y3 ) @ ( transi8609417484261754244at_nat @ R2 ) )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ Z2 ) @ R2 )
=> ( ( P2 @ Y3 )
=> ( P2 @ Z2 ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% rtrancl_induct
thf(fact_1130_rtrancl__induct,axiom,
! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,P2: nat > $o] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( ( P2 @ A )
=> ( ! [Y3: nat,Z2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ Y3 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z2 ) @ R2 )
=> ( ( P2 @ Y3 )
=> ( P2 @ Z2 ) ) ) )
=> ( P2 @ B ) ) ) ) ).
% rtrancl_induct
thf(fact_1131_rtrancl__trans,axiom,
! [X3: term_a_b,Y: term_a_b,R2: set_Pr4386577575007340137rm_a_b,Z3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Y ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y @ Z3 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X3 @ Z3 ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ) ).
% rtrancl_trans
thf(fact_1132_rtrancl__trans,axiom,
! [X3: product_prod_nat_nat,Y: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,Z3: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( transi8609417484261754244at_nat @ R2 ) )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ Z3 ) @ ( transi8609417484261754244at_nat @ R2 ) )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Z3 ) @ ( transi8609417484261754244at_nat @ R2 ) ) ) ) ).
% rtrancl_trans
thf(fact_1133_rtrancl__trans,axiom,
! [X3: nat,Y: nat,R2: set_Pr1261947904930325089at_nat,Z3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Z3 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Z3 ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ) ).
% rtrancl_trans
thf(fact_1134_rtranclE,axiom,
! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( A != B )
=> ~ ! [Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ Y3 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ B ) @ R2 ) ) ) ) ).
% rtranclE
thf(fact_1135_rtranclE,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( transi8609417484261754244at_nat @ R2 ) )
=> ( ( A != B )
=> ~ ! [Y3: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ Y3 ) @ ( transi8609417484261754244at_nat @ R2 ) )
=> ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ B ) @ R2 ) ) ) ) ).
% rtranclE
thf(fact_1136_rtranclE,axiom,
! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( ( A != B )
=> ~ ! [Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ Y3 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ B ) @ R2 ) ) ) ) ).
% rtranclE
thf(fact_1137_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: term_a_b,B: term_a_b,R2: set_Pr4386577575007340137rm_a_b,C3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C3 ) @ R2 )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C3 ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_1138_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,C3: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( transi8609417484261754244at_nat @ R2 ) )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ C3 ) @ R2 )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ C3 ) @ ( transi8609417484261754244at_nat @ R2 ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_1139_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: nat,B: nat,R2: set_Pr1261947904930325089at_nat,C3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C3 ) @ R2 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C3 ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_1140_rtrancl_Ortrancl__refl,axiom,
! [A: term_a_b,R2: set_Pr4386577575007340137rm_a_b] : ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ A ) @ ( transi7742714808557438673rm_a_b @ R2 ) ) ).
% rtrancl.rtrancl_refl
thf(fact_1141_rtrancl_Ortrancl__refl,axiom,
! [A: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A ) @ ( transi8609417484261754244at_nat @ R2 ) ) ).
% rtrancl.rtrancl_refl
thf(fact_1142_rtrancl_Ortrancl__refl,axiom,
! [A: nat,R2: set_Pr1261947904930325089at_nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ).
% rtrancl.rtrancl_refl
thf(fact_1143_rtrancl_Osimps,axiom,
! [A1: term_a_b,A22: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
= ( ? [A5: term_a_b] :
( ( A1 = A5 )
& ( A22 = A5 ) )
| ? [A5: term_a_b,B5: term_a_b,C2: term_a_b] :
( ( A1 = A5 )
& ( A22 = C2 )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A5 @ B5 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B5 @ C2 ) @ R2 ) ) ) ) ).
% rtrancl.simps
thf(fact_1144_rtrancl_Osimps,axiom,
! [A1: product_prod_nat_nat,A22: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A1 @ A22 ) @ ( transi8609417484261754244at_nat @ R2 ) )
= ( ? [A5: product_prod_nat_nat] :
( ( A1 = A5 )
& ( A22 = A5 ) )
| ? [A5: product_prod_nat_nat,B5: product_prod_nat_nat,C2: product_prod_nat_nat] :
( ( A1 = A5 )
& ( A22 = C2 )
& ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A5 @ B5 ) @ ( transi8609417484261754244at_nat @ R2 ) )
& ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B5 @ C2 ) @ R2 ) ) ) ) ).
% rtrancl.simps
thf(fact_1145_rtrancl_Osimps,axiom,
! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
= ( ? [A5: nat] :
( ( A1 = A5 )
& ( A22 = A5 ) )
| ? [A5: nat,B5: nat,C2: nat] :
( ( A1 = A5 )
& ( A22 = C2 )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ B5 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B5 @ C2 ) @ R2 ) ) ) ) ).
% rtrancl.simps
thf(fact_1146_rtrancl_Ocases,axiom,
! [A1: term_a_b,A22: term_a_b,R2: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ A22 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ( ( A22 != A1 )
=> ~ ! [B3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A1 @ B3 ) @ ( transi7742714808557438673rm_a_b @ R2 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B3 @ A22 ) @ R2 ) ) ) ) ).
% rtrancl.cases
thf(fact_1147_rtrancl_Ocases,axiom,
! [A1: product_prod_nat_nat,A22: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A1 @ A22 ) @ ( transi8609417484261754244at_nat @ R2 ) )
=> ( ( A22 != A1 )
=> ~ ! [B3: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A1 @ B3 ) @ ( transi8609417484261754244at_nat @ R2 ) )
=> ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B3 @ A22 ) @ R2 ) ) ) ) ).
% rtrancl.cases
thf(fact_1148_rtrancl_Ocases,axiom,
! [A1: nat,A22: nat,R2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ( ( A22 != A1 )
=> ~ ! [B3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A1 @ B3 ) @ ( transi2905341329935302413cl_nat @ R2 ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ A22 ) @ R2 ) ) ) ) ).
% rtrancl.cases
thf(fact_1149_rtrancl__subset__rtrancl,axiom,
! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ R2 @ ( transi2905341329935302413cl_nat @ S ) )
=> ( ord_le3146513528884898305at_nat @ ( transi2905341329935302413cl_nat @ R2 ) @ ( transi2905341329935302413cl_nat @ S ) ) ) ).
% rtrancl_subset_rtrancl
thf(fact_1150_rtrancl__subset,axiom,
! [R3: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ R3 @ S2 )
=> ( ( ord_le3146513528884898305at_nat @ S2 @ ( transi2905341329935302413cl_nat @ R3 ) )
=> ( ( transi2905341329935302413cl_nat @ S2 )
= ( transi2905341329935302413cl_nat @ R3 ) ) ) ) ).
% rtrancl_subset
thf(fact_1151_rtrancl__mono,axiom,
! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ R2 @ S )
=> ( ord_le3146513528884898305at_nat @ ( transi2905341329935302413cl_nat @ R2 ) @ ( transi2905341329935302413cl_nat @ S ) ) ) ).
% rtrancl_mono
thf(fact_1152_r__comp__rtrancl__eq,axiom,
! [R2: set_Pr4386577575007340137rm_a_b] :
( ( relcom370159955682700863rm_a_b @ R2 @ ( transi7742714808557438673rm_a_b @ R2 ) )
= ( relcom370159955682700863rm_a_b @ ( transi7742714808557438673rm_a_b @ R2 ) @ R2 ) ) ).
% r_comp_rtrancl_eq
thf(fact_1153_r__comp__rtrancl__eq,axiom,
! [R2: set_Pr1261947904930325089at_nat] :
( ( relcomp_nat_nat_nat @ R2 @ ( transi2905341329935302413cl_nat @ R2 ) )
= ( relcomp_nat_nat_nat @ ( transi2905341329935302413cl_nat @ R2 ) @ R2 ) ) ).
% r_comp_rtrancl_eq
thf(fact_1154_in__rtrancl__UnI,axiom,
! [X3: produc357393685978478089rm_a_b,R3: set_Pr4386577575007340137rm_a_b,S2: set_Pr4386577575007340137rm_a_b] :
( ( ( member5869715511025134514rm_a_b @ X3 @ ( transi7742714808557438673rm_a_b @ R3 ) )
| ( member5869715511025134514rm_a_b @ X3 @ ( transi7742714808557438673rm_a_b @ S2 ) ) )
=> ( member5869715511025134514rm_a_b @ X3 @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ R3 @ S2 ) ) ) ) ).
% in_rtrancl_UnI
thf(fact_1155_in__rtrancl__UnI,axiom,
! [X3: produc859450856879609959at_nat,R3: set_Pr8693737435421807431at_nat,S2: set_Pr8693737435421807431at_nat] :
( ( ( member8206827879206165904at_nat @ X3 @ ( transi8609417484261754244at_nat @ R3 ) )
| ( member8206827879206165904at_nat @ X3 @ ( transi8609417484261754244at_nat @ S2 ) ) )
=> ( member8206827879206165904at_nat @ X3 @ ( transi8609417484261754244at_nat @ ( sup_su718114333110466843at_nat @ R3 @ S2 ) ) ) ) ).
% in_rtrancl_UnI
thf(fact_1156_in__rtrancl__UnI,axiom,
! [X3: product_prod_nat_nat,R3: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( ( ( member8440522571783428010at_nat @ X3 @ ( transi2905341329935302413cl_nat @ R3 ) )
| ( member8440522571783428010at_nat @ X3 @ ( transi2905341329935302413cl_nat @ S2 ) ) )
=> ( member8440522571783428010at_nat @ X3 @ ( transi2905341329935302413cl_nat @ ( sup_su6327502436637775413at_nat @ R3 @ S2 ) ) ) ) ).
% in_rtrancl_UnI
thf(fact_1157_rtrancl__Un__rtrancl,axiom,
! [R3: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
( ( transi2905341329935302413cl_nat @ ( sup_su6327502436637775413at_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ ( transi2905341329935302413cl_nat @ S2 ) ) )
= ( transi2905341329935302413cl_nat @ ( sup_su6327502436637775413at_nat @ R3 @ S2 ) ) ) ).
% rtrancl_Un_rtrancl
thf(fact_1158_converse__rtrancl__induct2,axiom,
! [Ax: term_a_b,Ay: term_a_b,Bx: term_a_b,By: term_a_b,R2: set_Pr2972776593051762503rm_a_b,P2: term_a_b > term_a_b > $o] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ Bx @ By ) ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( P2 @ Bx @ By )
=> ( ! [A3: term_a_b,B3: term_a_b,Aa: term_a_b,Ba: term_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) ) @ R2 )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) @ ( produc7020197800436672577rm_a_b @ Bx @ By ) ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( P2 @ Aa @ Ba )
=> ( P2 @ A3 @ B3 ) ) ) )
=> ( P2 @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_1159_converse__rtrancl__induct2,axiom,
! [Ax: a,Ay: nat,Bx: a,By: nat,R2: set_Pr1811044260758604347_a_nat,P2: a > nat > $o] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Ax @ Ay ) @ ( product_Pair_a_nat @ Bx @ By ) ) @ ( transi2726145917338391738_a_nat @ R2 ) )
=> ( ( P2 @ Bx @ By )
=> ( ! [A3: a,B3: nat,Aa: a,Ba: nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ A3 @ B3 ) @ ( product_Pair_a_nat @ Aa @ Ba ) ) @ R2 )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Aa @ Ba ) @ ( product_Pair_a_nat @ Bx @ By ) ) @ ( transi2726145917338391738_a_nat @ R2 ) )
=> ( ( P2 @ Aa @ Ba )
=> ( P2 @ A3 @ B3 ) ) ) )
=> ( P2 @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_1160_converse__rtrancl__induct2,axiom,
! [Ax: nat > nat,Ay: nat,Bx: nat > nat,By: nat,R2: set_Pr7047708467701517959at_nat,P2: ( nat > nat ) > nat > $o] :
( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Ax @ Ay ) @ ( produc72220940542539688at_nat @ Bx @ By ) ) @ ( transi3422749762518297843at_nat @ R2 ) )
=> ( ( P2 @ Bx @ By )
=> ( ! [A3: nat > nat,B3: nat,Aa: nat > nat,Ba: nat] :
( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ A3 @ B3 ) @ ( produc72220940542539688at_nat @ Aa @ Ba ) ) @ R2 )
=> ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Aa @ Ba ) @ ( produc72220940542539688at_nat @ Bx @ By ) ) @ ( transi3422749762518297843at_nat @ R2 ) )
=> ( ( P2 @ Aa @ Ba )
=> ( P2 @ A3 @ B3 ) ) ) )
=> ( P2 @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_1161_converse__rtrancl__induct2,axiom,
! [Ax: product_prod_nat_nat,Ay: product_prod_nat_nat,Bx: product_prod_nat_nat,By: product_prod_nat_nat,R2: set_Pr553994874890374343at_nat,P2: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ Ax @ Ay ) @ ( produc6161850002892822231at_nat @ Bx @ By ) ) @ ( transi5470879363895450730at_nat @ R2 ) )
=> ( ( P2 @ Bx @ By )
=> ( ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat,Aa: product_prod_nat_nat,Ba: product_prod_nat_nat] :
( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ A3 @ B3 ) @ ( produc6161850002892822231at_nat @ Aa @ Ba ) ) @ R2 )
=> ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ Aa @ Ba ) @ ( produc6161850002892822231at_nat @ Bx @ By ) ) @ ( transi5470879363895450730at_nat @ R2 ) )
=> ( ( P2 @ Aa @ Ba )
=> ( P2 @ A3 @ B3 ) ) ) )
=> ( P2 @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_1162_converse__rtrancl__induct2,axiom,
! [Ax: nat,Ay: nat,Bx: nat,By: nat,R2: set_Pr8693737435421807431at_nat,P2: nat > nat > $o] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi8609417484261754244at_nat @ R2 ) )
=> ( ( P2 @ Bx @ By )
=> ( ! [A3: nat,B3: nat,Aa: nat,Ba: nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ ( product_Pair_nat_nat @ Aa @ Ba ) ) @ R2 )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Aa @ Ba ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi8609417484261754244at_nat @ R2 ) )
=> ( ( P2 @ Aa @ Ba )
=> ( P2 @ A3 @ B3 ) ) ) )
=> ( P2 @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_1163_converse__rtranclE2,axiom,
! [Xa: term_a_b,Xb: term_a_b,Za: term_a_b,Zb: term_a_b,R2: set_Pr2972776593051762503rm_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Xa @ Xb ) @ ( produc7020197800436672577rm_a_b @ Za @ Zb ) ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( ( produc7020197800436672577rm_a_b @ Xa @ Xb )
!= ( produc7020197800436672577rm_a_b @ Za @ Zb ) )
=> ~ ! [A3: term_a_b,B3: term_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Xa @ Xb ) @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) ) @ R2 )
=> ~ ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) @ ( produc7020197800436672577rm_a_b @ Za @ Zb ) ) @ ( transi2615809358984392588rm_a_b @ R2 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_1164_converse__rtranclE2,axiom,
! [Xa: a,Xb: nat,Za: a,Zb: nat,R2: set_Pr1811044260758604347_a_nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Xa @ Xb ) @ ( product_Pair_a_nat @ Za @ Zb ) ) @ ( transi2726145917338391738_a_nat @ R2 ) )
=> ( ( ( product_Pair_a_nat @ Xa @ Xb )
!= ( product_Pair_a_nat @ Za @ Zb ) )
=> ~ ! [A3: a,B3: nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Xa @ Xb ) @ ( product_Pair_a_nat @ A3 @ B3 ) ) @ R2 )
=> ~ ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ A3 @ B3 ) @ ( product_Pair_a_nat @ Za @ Zb ) ) @ ( transi2726145917338391738_a_nat @ R2 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_1165_converse__rtranclE2,axiom,
! [Xa: nat > nat,Xb: nat,Za: nat > nat,Zb: nat,R2: set_Pr7047708467701517959at_nat] :
( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Xa @ Xb ) @ ( produc72220940542539688at_nat @ Za @ Zb ) ) @ ( transi3422749762518297843at_nat @ R2 ) )
=> ( ( ( produc72220940542539688at_nat @ Xa @ Xb )
!= ( produc72220940542539688at_nat @ Za @ Zb ) )
=> ~ ! [A3: nat > nat,B3: nat] :
( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Xa @ Xb ) @ ( produc72220940542539688at_nat @ A3 @ B3 ) ) @ R2 )
=> ~ ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ A3 @ B3 ) @ ( produc72220940542539688at_nat @ Za @ Zb ) ) @ ( transi3422749762518297843at_nat @ R2 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_1166_converse__rtranclE2,axiom,
! [Xa: product_prod_nat_nat,Xb: product_prod_nat_nat,Za: product_prod_nat_nat,Zb: product_prod_nat_nat,R2: set_Pr553994874890374343at_nat] :
( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ Xa @ Xb ) @ ( produc6161850002892822231at_nat @ Za @ Zb ) ) @ ( transi5470879363895450730at_nat @ R2 ) )
=> ( ( ( produc6161850002892822231at_nat @ Xa @ Xb )
!= ( produc6161850002892822231at_nat @ Za @ Zb ) )
=> ~ ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ Xa @ Xb ) @ ( produc6161850002892822231at_nat @ A3 @ B3 ) ) @ R2 )
=> ~ ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ A3 @ B3 ) @ ( produc6161850002892822231at_nat @ Za @ Zb ) ) @ ( transi5470879363895450730at_nat @ R2 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_1167_converse__rtranclE2,axiom,
! [Xa: nat,Xb: nat,Za: nat,Zb: nat,R2: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Xa @ Xb ) @ ( product_Pair_nat_nat @ Za @ Zb ) ) @ ( transi8609417484261754244at_nat @ R2 ) )
=> ( ( ( product_Pair_nat_nat @ Xa @ Xb )
!= ( product_Pair_nat_nat @ Za @ Zb ) )
=> ~ ! [A3: nat,B3: nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Xa @ Xb ) @ ( product_Pair_nat_nat @ A3 @ B3 ) ) @ R2 )
=> ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ ( product_Pair_nat_nat @ Za @ Zb ) ) @ ( transi8609417484261754244at_nat @ R2 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_1168_rtrancl__induct2,axiom,
! [Ax: term_a_b,Ay: term_a_b,Bx: term_a_b,By: term_a_b,R2: set_Pr2972776593051762503rm_a_b,P2: term_a_b > term_a_b > $o] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ Bx @ By ) ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( P2 @ Ax @ Ay )
=> ( ! [A3: term_a_b,B3: term_a_b,Aa: term_a_b,Ba: term_a_b] :
( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ Ax @ Ay ) @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) ) @ ( transi2615809358984392588rm_a_b @ R2 ) )
=> ( ( member8417600551952982416rm_a_b @ ( produc1763473618796451543rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B3 ) @ ( produc7020197800436672577rm_a_b @ Aa @ Ba ) ) @ R2 )
=> ( ( P2 @ A3 @ B3 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_1169_rtrancl__induct2,axiom,
! [Ax: a,Ay: nat,Bx: a,By: nat,R2: set_Pr1811044260758604347_a_nat,P2: a > nat > $o] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Ax @ Ay ) @ ( product_Pair_a_nat @ Bx @ By ) ) @ ( transi2726145917338391738_a_nat @ R2 ) )
=> ( ( P2 @ Ax @ Ay )
=> ( ! [A3: a,B3: nat,Aa: a,Ba: nat] :
( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ Ax @ Ay ) @ ( product_Pair_a_nat @ A3 @ B3 ) ) @ ( transi2726145917338391738_a_nat @ R2 ) )
=> ( ( member9062615507155100804_a_nat @ ( produc2026711137822539155_a_nat @ ( product_Pair_a_nat @ A3 @ B3 ) @ ( product_Pair_a_nat @ Aa @ Ba ) ) @ R2 )
=> ( ( P2 @ A3 @ B3 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_1170_rtrancl__induct2,axiom,
! [Ax: nat > nat,Ay: nat,Bx: nat > nat,By: nat,R2: set_Pr7047708467701517959at_nat,P2: ( nat > nat ) > nat > $o] :
( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Ax @ Ay ) @ ( produc72220940542539688at_nat @ Bx @ By ) ) @ ( transi3422749762518297843at_nat @ R2 ) )
=> ( ( P2 @ Ax @ Ay )
=> ( ! [A3: nat > nat,B3: nat,Aa: nat > nat,Ba: nat] :
( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Ax @ Ay ) @ ( produc72220940542539688at_nat @ A3 @ B3 ) ) @ ( transi3422749762518297843at_nat @ R2 ) )
=> ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ A3 @ B3 ) @ ( produc72220940542539688at_nat @ Aa @ Ba ) ) @ R2 )
=> ( ( P2 @ A3 @ B3 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_1171_rtrancl__induct2,axiom,
! [Ax: product_prod_nat_nat,Ay: product_prod_nat_nat,Bx: product_prod_nat_nat,By: product_prod_nat_nat,R2: set_Pr553994874890374343at_nat,P2: product_prod_nat_nat > product_prod_nat_nat > $o] :
( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ Ax @ Ay ) @ ( produc6161850002892822231at_nat @ Bx @ By ) ) @ ( transi5470879363895450730at_nat @ R2 ) )
=> ( ( P2 @ Ax @ Ay )
=> ( ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat,Aa: product_prod_nat_nat,Ba: product_prod_nat_nat] :
( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ Ax @ Ay ) @ ( produc6161850002892822231at_nat @ A3 @ B3 ) ) @ ( transi5470879363895450730at_nat @ R2 ) )
=> ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ A3 @ B3 ) @ ( produc6161850002892822231at_nat @ Aa @ Ba ) ) @ R2 )
=> ( ( P2 @ A3 @ B3 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_1172_rtrancl__induct2,axiom,
! [Ax: nat,Ay: nat,Bx: nat,By: nat,R2: set_Pr8693737435421807431at_nat,P2: nat > nat > $o] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ Bx @ By ) ) @ ( transi8609417484261754244at_nat @ R2 ) )
=> ( ( P2 @ Ax @ Ay )
=> ( ! [A3: nat,B3: nat,Aa: nat,Ba: nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ Ax @ Ay ) @ ( product_Pair_nat_nat @ A3 @ B3 ) ) @ ( transi8609417484261754244at_nat @ R2 ) )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ ( product_Pair_nat_nat @ Aa @ Ba ) ) @ R2 )
=> ( ( P2 @ A3 @ B3 )
=> ( P2 @ Aa @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_1173_rtrancl__Un__separator__converseE,axiom,
! [A: term_a_b,B: term_a_b,P2: set_Pr4386577575007340137rm_a_b,Q: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ P2 @ Q ) ) )
=> ( ! [X: term_a_b,Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ B ) @ ( transi7742714808557438673rm_a_b @ P2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ Y3 @ X ) @ Q )
=> ( Y3 = X ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ P2 ) ) ) ) ).
% rtrancl_Un_separator_converseE
thf(fact_1174_rtrancl__Un__separator__converseE,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,P2: set_Pr8693737435421807431at_nat,Q: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( transi8609417484261754244at_nat @ ( sup_su718114333110466843at_nat @ P2 @ Q ) ) )
=> ( ! [X: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ B ) @ ( transi8609417484261754244at_nat @ P2 ) )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ X ) @ Q )
=> ( Y3 = X ) ) )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( transi8609417484261754244at_nat @ P2 ) ) ) ) ).
% rtrancl_Un_separator_converseE
thf(fact_1175_rtrancl__Un__separator__converseE,axiom,
! [A: nat,B: nat,P2: set_Pr1261947904930325089at_nat,Q: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ ( sup_su6327502436637775413at_nat @ P2 @ Q ) ) )
=> ( ! [X: nat,Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ B ) @ ( transi2905341329935302413cl_nat @ P2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X ) @ Q )
=> ( Y3 = X ) ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ P2 ) ) ) ) ).
% rtrancl_Un_separator_converseE
thf(fact_1176_rtrancl__Un__separatorE,axiom,
! [A: term_a_b,B: term_a_b,P2: set_Pr4386577575007340137rm_a_b,Q: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ ( sup_su6776935440552674877rm_a_b @ P2 @ Q ) ) )
=> ( ! [X: term_a_b,Y3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ X ) @ ( transi7742714808557438673rm_a_b @ P2 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ X @ Y3 ) @ Q )
=> ( X = Y3 ) ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ P2 ) ) ) ) ).
% rtrancl_Un_separatorE
thf(fact_1177_rtrancl__Un__separatorE,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,P2: set_Pr8693737435421807431at_nat,Q: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( transi8609417484261754244at_nat @ ( sup_su718114333110466843at_nat @ P2 @ Q ) ) )
=> ( ! [X: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ X ) @ ( transi8609417484261754244at_nat @ P2 ) )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ Q )
=> ( X = Y3 ) ) )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( transi8609417484261754244at_nat @ P2 ) ) ) ) ).
% rtrancl_Un_separatorE
thf(fact_1178_rtrancl__Un__separatorE,axiom,
! [A: nat,B: nat,P2: set_Pr1261947904930325089at_nat,Q: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ ( sup_su6327502436637775413at_nat @ P2 @ Q ) ) )
=> ( ! [X: nat,Y3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ X ) @ ( transi2905341329935302413cl_nat @ P2 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ Q )
=> ( X = Y3 ) ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ P2 ) ) ) ) ).
% rtrancl_Un_separatorE
thf(fact_1179_rtrancl__Un__subset,axiom,
! [R3: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( sup_su6327502436637775413at_nat @ ( transi2905341329935302413cl_nat @ R3 ) @ ( transi2905341329935302413cl_nat @ S2 ) ) @ ( transi2905341329935302413cl_nat @ ( sup_su6327502436637775413at_nat @ R3 @ S2 ) ) ) ).
% rtrancl_Un_subset
thf(fact_1180_pair__leqI2,axiom,
! [A: nat,B: nat,S: nat,T: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ S @ T )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ) ).
% pair_leqI2
thf(fact_1181_pair__leqI1,axiom,
! [A: nat,B: nat,S: nat,T: nat] :
( ( ord_less_nat @ A @ B )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ).
% pair_leqI1
thf(fact_1182_meetI,axiom,
! [A: term_a_b,B: term_a_b,A4: set_Pr4386577575007340137rm_a_b,C3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ B ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
=> ( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A @ C3 ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
=> ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C3 ) @ ( abstra1093061187967292567rm_a_b @ A4 ) ) ) ) ).
% meetI
thf(fact_1183_meetI,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,A4: set_Pr8693737435421807431at_nat,C3: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( transi8609417484261754244at_nat @ A4 ) )
=> ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ C3 ) @ ( transi8609417484261754244at_nat @ A4 ) )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ C3 ) @ ( abstra338393719139700030at_nat @ A4 ) ) ) ) ).
% meetI
thf(fact_1184_meetI,axiom,
! [A: nat,B: nat,A4: set_Pr1261947904930325089at_nat,C3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( transi2905341329935302413cl_nat @ A4 ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ C3 ) @ ( transi2905341329935302413cl_nat @ A4 ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C3 ) @ ( abstract_meet_nat @ A4 ) ) ) ) ).
% meetI
thf(fact_1185_restrict__rtrancl,axiom,
! [R2: set_Pr1261947904930325089at_nat,S2: set_nat] : ( ord_le3146513528884898305at_nat @ ( transi2905341329935302413cl_nat @ ( abstra1219450721576609224ct_nat @ R2 @ S2 ) ) @ ( transi2905341329935302413cl_nat @ R2 ) ) ).
% restrict_rtrancl
thf(fact_1186_meetE,axiom,
! [B: term_a_b,C3: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C3 ) @ ( abstra1093061187967292567rm_a_b @ A4 ) )
=> ~ ! [A3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
=> ~ ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C3 ) @ ( transi7742714808557438673rm_a_b @ A4 ) ) ) ) ).
% meetE
thf(fact_1187_meetE,axiom,
! [B: product_prod_nat_nat,C3: product_prod_nat_nat,A4: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ C3 ) @ ( abstra338393719139700030at_nat @ A4 ) )
=> ~ ! [A3: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A3 @ B ) @ ( transi8609417484261754244at_nat @ A4 ) )
=> ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A3 @ C3 ) @ ( transi8609417484261754244at_nat @ A4 ) ) ) ) ).
% meetE
thf(fact_1188_meetE,axiom,
! [B: nat,C3: nat,A4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C3 ) @ ( abstract_meet_nat @ A4 ) )
=> ~ ! [A3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B ) @ ( transi2905341329935302413cl_nat @ A4 ) )
=> ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ C3 ) @ ( transi2905341329935302413cl_nat @ A4 ) ) ) ) ).
% meetE
thf(fact_1189_meetD,axiom,
! [B: term_a_b,C3: term_a_b,A4: set_Pr4386577575007340137rm_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ B @ C3 ) @ ( abstra1093061187967292567rm_a_b @ A4 ) )
=> ? [A3: term_a_b] :
( ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ B ) @ ( transi7742714808557438673rm_a_b @ A4 ) )
& ( member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ A3 @ C3 ) @ ( transi7742714808557438673rm_a_b @ A4 ) ) ) ) ).
% meetD
thf(fact_1190_meetD,axiom,
! [B: product_prod_nat_nat,C3: product_prod_nat_nat,A4: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ C3 ) @ ( abstra338393719139700030at_nat @ A4 ) )
=> ? [A3: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A3 @ B ) @ ( transi8609417484261754244at_nat @ A4 ) )
& ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A3 @ C3 ) @ ( transi8609417484261754244at_nat @ A4 ) ) ) ) ).
% meetD
thf(fact_1191_meetD,axiom,
! [B: nat,C3: nat,A4: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ C3 ) @ ( abstract_meet_nat @ A4 ) )
=> ? [A3: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B ) @ ( transi2905341329935302413cl_nat @ A4 ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ C3 ) @ ( transi2905341329935302413cl_nat @ A4 ) ) ) ) ).
% meetD
thf(fact_1192_pred__nat__trancl__eq__le,axiom,
! [M2: nat,N: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M2 @ N ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% pred_nat_trancl_eq_le
thf(fact_1193_nth__enumerate__eq,axiom,
! [M2: nat,Xs: list_nat,N: nat] :
( ( ord_less_nat @ M2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N @ Xs ) @ M2 )
= ( product_Pair_nat_nat @ ( plus_plus_nat @ N @ M2 ) @ ( nth_nat @ Xs @ M2 ) ) ) ) ).
% nth_enumerate_eq
thf(fact_1194_nth__enumerate__eq,axiom,
! [M2: nat,Xs: list_term_a_b,N: nat] :
( ( ord_less_nat @ M2 @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( nth_Pr5300001119514325452rm_a_b @ ( enumerate_term_a_b @ N @ Xs ) @ M2 )
= ( produc1516572978046417917rm_a_b @ ( plus_plus_nat @ N @ M2 ) @ ( nth_term_a_b @ Xs @ M2 ) ) ) ) ).
% nth_enumerate_eq
thf(fact_1195_add__le__cancel__right,axiom,
! [A: nat,C3: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C3 ) @ ( plus_plus_nat @ B @ C3 ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_1196_add__le__cancel__left,axiom,
! [C3: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C3 @ A ) @ ( plus_plus_nat @ C3 @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_1197_add__less__cancel__left,axiom,
! [C3: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C3 @ A ) @ ( plus_plus_nat @ C3 @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_1198_add__less__cancel__right,axiom,
! [A: nat,C3: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C3 ) @ ( plus_plus_nat @ B @ C3 ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_1199_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1200_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1201_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_1202_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_1203_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_1204_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_1205_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_1206_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_1207_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_1208_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_1209_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_1210_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_1211_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1212_length__append,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b] :
( ( size_s8906293707977694520rm_a_b @ ( append_term_a_b @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_s8906293707977694520rm_a_b @ Xs ) @ ( size_s8906293707977694520rm_a_b @ Ys ) ) ) ).
% length_append
thf(fact_1213_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1214_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1215_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1216_length__splice,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b] :
( ( size_s8906293707977694520rm_a_b @ ( splice_term_a_b @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_s8906293707977694520rm_a_b @ Xs ) @ ( size_s8906293707977694520rm_a_b @ Ys ) ) ) ).
% length_splice
thf(fact_1217_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1218_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1219_nth__append__length__plus,axiom,
! [Xs: list_term_a_b,Ys: list_term_a_b,N: nat] :
( ( nth_term_a_b @ ( append_term_a_b @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_s8906293707977694520rm_a_b @ Xs ) @ N ) )
= ( nth_term_a_b @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_1220_nth__drop,axiom,
! [N: nat,Xs: list_term_a_b,I: nat] :
( ( ord_less_eq_nat @ N @ ( size_s8906293707977694520rm_a_b @ Xs ) )
=> ( ( nth_term_a_b @ ( drop_term_a_b @ N @ Xs ) @ I )
= ( nth_term_a_b @ Xs @ ( plus_plus_nat @ N @ I ) ) ) ) ).
% nth_drop
thf(fact_1221_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1222_add__diff__inverse__nat,axiom,
! [M2: nat,N: nat] :
( ~ ( ord_less_nat @ M2 @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M2 @ N ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_1223_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1224_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C3 )
= ( B
= ( plus_plus_nat @ C3 @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1225_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1226_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C3 @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C3 @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1227_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C3 ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C3 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1228_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C3 )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C3 ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1229_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C3 @ B ) @ A )
= ( plus_plus_nat @ C3 @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1230_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C3 @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C3 @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1231_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C3 @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C3 @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1232_le__add__diff,axiom,
! [A: nat,B: nat,C3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C3 @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C3 ) @ A ) ) ) ).
% le_add_diff
thf(fact_1233_diff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% diff_add
thf(fact_1234_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1235_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1236_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1237_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1238_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1239_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1240_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1241_add__strict__increasing2,axiom,
! [A: nat,B: nat,C3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C3 ) ) ) ) ).
% add_strict_increasing2
thf(fact_1242_add__strict__increasing,axiom,
! [A: nat,B: nat,C3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C3 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C3 ) ) ) ) ).
% add_strict_increasing
thf(fact_1243_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1244_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_1245_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1246_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_1247_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_1248_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_1249_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_1250_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_1251_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_1252_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C: nat] :
( ( B
= ( plus_plus_nat @ A @ C ) )
=> ( C = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_1253_pos__add__strict,axiom,
! [A: nat,B: nat,C3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C3 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C3 ) ) ) ) ).
% pos_add_strict
thf(fact_1254_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1255_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1256_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1257_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y: nat] :
( ( if_nat @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y: nat] :
( ( if_nat @ $true @ X3 @ Y )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
member5869715511025134514rm_a_b @ ( produc7020197800436672577rm_a_b @ ( fun_a_b @ f2 @ ss ) @ ( fun_a_b @ f2 @ us ) ) @ r ).
%------------------------------------------------------------------------------