TPTP Problem File: SLH0870^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : FOL_Seq_Calc2/0016_ProverLemmas/prob_00470_016859__12859722_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1563 ( 626 unt; 273 typ; 0 def)
% Number of atoms : 3720 (1900 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 12470 ( 512 ~; 90 |; 406 &;9822 @)
% ( 0 <=>;1640 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 7 avg)
% Number of types : 45 ( 44 usr)
% Number of type conns : 978 ( 978 >; 0 *; 0 +; 0 <<)
% Number of symbols : 232 ( 229 usr; 15 con; 0-4 aty)
% Number of variables : 4470 ( 158 ^;3879 !; 433 ?;4470 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 08:44:38.069
%------------------------------------------------------------------------------
% Could-be-implicit typings (44)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J_J_J,type,
set_Pr8947555413577213383ist_fm: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J_J,type,
produc8729865333125084903ist_fm: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J_J,type,
set_Pr6724411253394807111ist_fm: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
produc7850321100735229287ist_fm: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__List__Olist_It__SeCaV__Ofm_J_M_062_It__List__Olist_It__SeCaV__Ofm_J_M_Eo_J_J_Mt__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_J,type,
produc6864569604571974008ist_fm: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__SeCaV__Otm_Mt__SeCaV__Ofm_J_Mt__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
produc3792783349651969271ist_fm: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__SeCaV__Otm_J_J_Mt__List__Olist_It__List__Olist_It__SeCaV__Otm_J_J_J_J,type,
set_Pr4883004833441821767ist_tm: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_Mt__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_J_J,type,
set_Pr3202885091612042951ist_fm: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__SeCaV__Otm_J_J_Mt__List__Olist_It__List__Olist_It__SeCaV__Otm_J_J_J,type,
produc698371888225778279ist_tm: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_Mt__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_J,type,
produc4386893285136772327ist_fm: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__SeCaV__Otm_M_062_It__SeCaV__Otm_M_Eo_J_J_Mt__List__Olist_It__SeCaV__Otm_J_J,type,
produc2002131169352006116ist_tm: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__SeCaV__Ofm_M_062_It__SeCaV__Ofm_M_Eo_J_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
produc7963324949210141170ist_fm: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
list_P2887561121880082555ist_fm: $tType ).
thf(ty_n_t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
fset_P8989946509869081563ist_fm: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Otm_J_J_J,type,
set_Pr1507011332596240839ist_tm: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
set_Pr5202636777678657877ist_fm: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Ofm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
set_Pr1916137733696349511ist_fm: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Otm_J_J,type,
produc5776448205642668775ist_tm: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
produc6018962875968178549ist_fm: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Ofm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
produc3245234490656042599ist_fm: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__SeCaV__Otm_Mt__SeCaV__Otm_J_J,type,
set_Pr2455929065695642951_tm_tm: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J_J,type,
set_Pr4706815898642364871_fm_fm: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__SeCaV__Otm_J_J_J,type,
list_list_list_tm: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_J,type,
list_list_list_fm: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_J,type,
set_list_list_fm: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__SeCaV__Otm_Mt__SeCaV__Otm_J,type,
product_prod_tm_tm: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J,type,
product_prod_fm_fm: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__SeCaV__Otm_J_J,type,
list_list_tm: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
list_list_fm: $tType ).
thf(ty_n_t__FSet__Ofset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
fset_list_fm: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__SeCaV__Otm_J_J,type,
set_list_tm: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
set_list_fm: $tType ).
thf(ty_n_t__List__Olist_It__SeCaV__Otm_J,type,
list_tm: $tType ).
thf(ty_n_t__List__Olist_It__SeCaV__Ofm_J,type,
list_fm: $tType ).
thf(ty_n_t__FSet__Ofset_It__SeCaV__Otm_J,type,
fset_tm: $tType ).
thf(ty_n_t__FSet__Ofset_It__SeCaV__Ofm_J,type,
fset_fm: $tType ).
thf(ty_n_t__Set__Oset_It__SeCaV__Otm_J,type,
set_tm: $tType ).
thf(ty_n_t__Set__Oset_It__SeCaV__Ofm_J,type,
set_fm: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Prover__Orule,type,
rule: $tType ).
thf(ty_n_t__SeCaV__Otm,type,
tm: $tType ).
thf(ty_n_t__SeCaV__Ofm,type,
fm: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (229)
thf(sy_c_FSet_Ofmember_001t__List__Olist_It__SeCaV__Ofm_J,type,
fmember_list_fm: list_fm > fset_list_fm > $o ).
thf(sy_c_FSet_Ofmember_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
fmembe3754813877001230652ist_fm: produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o ).
thf(sy_c_FSet_Ofmember_001t__SeCaV__Ofm,type,
fmember_fm: fm > fset_fm > $o ).
thf(sy_c_FSet_Ofmember_001t__SeCaV__Otm,type,
fmember_tm: tm > fset_tm > $o ).
thf(sy_c_FSet_Ofset__of__list_001t__List__Olist_It__SeCaV__Ofm_J,type,
fset_of_list_list_fm: list_list_fm > fset_list_fm ).
thf(sy_c_FSet_Ofset__of__list_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
fset_o3222440871817023035ist_fm: list_P2887561121880082555ist_fm > fset_P8989946509869081563ist_fm ).
thf(sy_c_FSet_Ofset__of__list_001t__SeCaV__Ofm,type,
fset_of_list_fm: list_fm > fset_fm ).
thf(sy_c_FSet_Ofset__of__list_001t__SeCaV__Otm,type,
fset_of_list_tm: list_tm > fset_tm ).
thf(sy_c_Fun_Ofun__upd_001t__SeCaV__Ofm_001t__List__Olist_It__SeCaV__Otm_J,type,
fun_upd_fm_list_tm: ( fm > list_tm ) > fm > list_tm > fm > list_tm ).
thf(sy_c_Fun_Ofun__upd_001t__SeCaV__Otm_001t__List__Olist_It__SeCaV__Otm_J,type,
fun_upd_tm_list_tm: ( tm > list_tm ) > tm > list_tm > tm > list_tm ).
thf(sy_c_Fun_Ofun__upd_001t__SeCaV__Otm_001t__SeCaV__Otm,type,
fun_upd_tm_tm: ( tm > tm ) > tm > tm > tm > tm ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > 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__FSet__Ofset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
if_fset_list_fm: $o > fset_list_fm > fset_list_fm > fset_list_fm ).
thf(sy_c_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
if_fse7999432387889793441ist_fm: $o > fset_P8989946509869081563ist_fm > fset_P8989946509869081563ist_fm > fset_P8989946509869081563ist_fm ).
thf(sy_c_If_001t__FSet__Ofset_It__SeCaV__Ofm_J,type,
if_fset_fm: $o > fset_fm > fset_fm > fset_fm ).
thf(sy_c_If_001t__FSet__Ofset_It__SeCaV__Otm_J,type,
if_fset_tm: $o > fset_tm > fset_tm > fset_tm ).
thf(sy_c_If_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
if_list_list_fm: $o > list_list_fm > list_list_fm > list_list_fm ).
thf(sy_c_If_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
if_lis1897046999900794433ist_fm: $o > list_P2887561121880082555ist_fm > list_P2887561121880082555ist_fm > list_P2887561121880082555ist_fm ).
thf(sy_c_If_001t__List__Olist_It__SeCaV__Ofm_J,type,
if_list_fm: $o > list_fm > list_fm > list_fm ).
thf(sy_c_If_001t__List__Olist_It__SeCaV__Otm_J,type,
if_list_tm: $o > list_tm > list_tm > list_tm ).
thf(sy_c_If_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
if_Pro4760001780252510779ist_fm: $o > produc6018962875968178549ist_fm > produc6018962875968178549ist_fm > produc6018962875968178549ist_fm ).
thf(sy_c_If_001t__SeCaV__Ofm,type,
if_fm: $o > fm > fm > fm ).
thf(sy_c_If_001t__SeCaV__Otm,type,
if_tm: $o > tm > tm > tm ).
thf(sy_c_List_Oappend_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
append_list_list_fm: list_list_list_fm > list_list_list_fm > list_list_list_fm ).
thf(sy_c_List_Oappend_001t__List__Olist_It__List__Olist_It__SeCaV__Otm_J_J,type,
append_list_list_tm: list_list_list_tm > list_list_list_tm > list_list_list_tm ).
thf(sy_c_List_Oappend_001t__List__Olist_It__SeCaV__Ofm_J,type,
append_list_fm: list_list_fm > list_list_fm > list_list_fm ).
thf(sy_c_List_Oappend_001t__List__Olist_It__SeCaV__Otm_J,type,
append_list_tm: list_list_tm > list_list_tm > list_list_tm ).
thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
append1476581234975291920ist_fm: list_P2887561121880082555ist_fm > list_P2887561121880082555ist_fm > list_P2887561121880082555ist_fm ).
thf(sy_c_List_Oappend_001t__SeCaV__Ofm,type,
append_fm: list_fm > list_fm > list_fm ).
thf(sy_c_List_Oappend_001t__SeCaV__Otm,type,
append_tm: list_tm > list_tm > list_tm ).
thf(sy_c_List_Obind_001t__List__Olist_It__SeCaV__Ofm_J_001t__List__Olist_It__SeCaV__Ofm_J,type,
bind_list_fm_list_fm: list_list_fm > ( list_fm > list_list_fm ) > list_list_fm ).
thf(sy_c_List_Obind_001t__List__Olist_It__SeCaV__Ofm_J_001t__List__Olist_It__SeCaV__Otm_J,type,
bind_list_fm_list_tm: list_list_fm > ( list_fm > list_list_tm ) > list_list_tm ).
thf(sy_c_List_Obind_001t__List__Olist_It__SeCaV__Ofm_J_001t__SeCaV__Ofm,type,
bind_list_fm_fm: list_list_fm > ( list_fm > list_fm ) > list_fm ).
thf(sy_c_List_Obind_001t__List__Olist_It__SeCaV__Ofm_J_001t__SeCaV__Otm,type,
bind_list_fm_tm: list_list_fm > ( list_fm > list_tm ) > list_tm ).
thf(sy_c_List_Obind_001t__SeCaV__Ofm_001t__List__Olist_It__SeCaV__Ofm_J,type,
bind_fm_list_fm: list_fm > ( fm > list_list_fm ) > list_list_fm ).
thf(sy_c_List_Obind_001t__SeCaV__Ofm_001t__SeCaV__Ofm,type,
bind_fm_fm: list_fm > ( fm > list_fm ) > list_fm ).
thf(sy_c_List_Obind_001t__SeCaV__Ofm_001t__SeCaV__Otm,type,
bind_fm_tm: list_fm > ( fm > list_tm ) > list_tm ).
thf(sy_c_List_Obind_001t__SeCaV__Otm_001t__List__Olist_It__SeCaV__Ofm_J,type,
bind_tm_list_fm: list_tm > ( tm > list_list_fm ) > list_list_fm ).
thf(sy_c_List_Obind_001t__SeCaV__Otm_001t__List__Olist_It__SeCaV__Otm_J,type,
bind_tm_list_tm: list_tm > ( tm > list_list_tm ) > list_list_tm ).
thf(sy_c_List_Obind_001t__SeCaV__Otm_001t__SeCaV__Ofm,type,
bind_tm_fm: list_tm > ( tm > list_fm ) > list_fm ).
thf(sy_c_List_Obind_001t__SeCaV__Otm_001t__SeCaV__Otm,type,
bind_tm_tm: list_tm > ( tm > list_tm ) > list_tm ).
thf(sy_c_List_Oconcat_001t__List__Olist_It__SeCaV__Ofm_J,type,
concat_list_fm: list_list_list_fm > list_list_fm ).
thf(sy_c_List_Oconcat_001t__List__Olist_It__SeCaV__Otm_J,type,
concat_list_tm: list_list_list_tm > list_list_tm ).
thf(sy_c_List_Oconcat_001t__SeCaV__Ofm,type,
concat_fm: list_list_fm > list_fm ).
thf(sy_c_List_Oconcat_001t__SeCaV__Otm,type,
concat_tm: list_list_tm > list_tm ).
thf(sy_c_List_Oinsert_001t__List__Olist_It__SeCaV__Ofm_J,type,
insert_list_fm: list_fm > list_list_fm > list_list_fm ).
thf(sy_c_List_Oinsert_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
insert5272696590961914513ist_fm: produc6018962875968178549ist_fm > list_P2887561121880082555ist_fm > list_P2887561121880082555ist_fm ).
thf(sy_c_List_Oinsert_001t__SeCaV__Ofm,type,
insert_fm: fm > list_fm > list_fm ).
thf(sy_c_List_Oinsert_001t__SeCaV__Otm,type,
insert_tm: tm > list_tm > list_tm ).
thf(sy_c_List_Olenlex_001t__List__Olist_It__SeCaV__Ofm_J,type,
lenlex_list_fm: set_Pr1916137733696349511ist_fm > set_Pr3202885091612042951ist_fm ).
thf(sy_c_List_Olenlex_001t__List__Olist_It__SeCaV__Otm_J,type,
lenlex_list_tm: set_Pr1507011332596240839ist_tm > set_Pr4883004833441821767ist_tm ).
thf(sy_c_List_Olenlex_001t__SeCaV__Ofm,type,
lenlex_fm: set_Pr4706815898642364871_fm_fm > set_Pr1916137733696349511ist_fm ).
thf(sy_c_List_Olenlex_001t__SeCaV__Otm,type,
lenlex_tm: set_Pr2455929065695642951_tm_tm > set_Pr1507011332596240839ist_tm ).
thf(sy_c_List_Olex_001t__List__Olist_It__SeCaV__Ofm_J,type,
lex_list_fm: set_Pr1916137733696349511ist_fm > set_Pr3202885091612042951ist_fm ).
thf(sy_c_List_Olex_001t__List__Olist_It__SeCaV__Otm_J,type,
lex_list_tm: set_Pr1507011332596240839ist_tm > set_Pr4883004833441821767ist_tm ).
thf(sy_c_List_Olex_001t__SeCaV__Ofm,type,
lex_fm: set_Pr4706815898642364871_fm_fm > set_Pr1916137733696349511ist_fm ).
thf(sy_c_List_Olex_001t__SeCaV__Otm,type,
lex_tm: set_Pr2455929065695642951_tm_tm > set_Pr1507011332596240839ist_tm ).
thf(sy_c_List_Olexord_001t__List__Olist_It__SeCaV__Ofm_J,type,
lexord_list_fm: set_Pr1916137733696349511ist_fm > set_Pr3202885091612042951ist_fm ).
thf(sy_c_List_Olexord_001t__List__Olist_It__SeCaV__Otm_J,type,
lexord_list_tm: set_Pr1507011332596240839ist_tm > set_Pr4883004833441821767ist_tm ).
thf(sy_c_List_Olexord_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
lexord4892917800681400680ist_fm: set_Pr6724411253394807111ist_fm > set_Pr8947555413577213383ist_fm ).
thf(sy_c_List_Olexord_001t__SeCaV__Ofm,type,
lexord_fm: set_Pr4706815898642364871_fm_fm > set_Pr1916137733696349511ist_fm ).
thf(sy_c_List_Olexord_001t__SeCaV__Otm,type,
lexord_tm: set_Pr2455929065695642951_tm_tm > set_Pr1507011332596240839ist_tm ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
cons_list_list_fm: list_list_fm > list_list_list_fm > list_list_list_fm ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__List__Olist_It__SeCaV__Otm_J_J,type,
cons_list_list_tm: list_list_tm > list_list_list_tm > list_list_list_tm ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__SeCaV__Ofm_J,type,
cons_list_fm: list_fm > list_list_fm > list_list_fm ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__SeCaV__Otm_J,type,
cons_list_tm: list_tm > list_list_tm > list_list_tm ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
cons_P6586734674047340843ist_fm: produc6018962875968178549ist_fm > list_P2887561121880082555ist_fm > list_P2887561121880082555ist_fm ).
thf(sy_c_List_Olist_OCons_001t__SeCaV__Ofm,type,
cons_fm: fm > list_fm > list_fm ).
thf(sy_c_List_Olist_OCons_001t__SeCaV__Otm,type,
cons_tm: tm > list_tm > list_tm ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
nil_list_list_fm: list_list_list_fm ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__List__Olist_It__SeCaV__Otm_J_J,type,
nil_list_list_tm: list_list_list_tm ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__SeCaV__Ofm_J,type,
nil_list_fm: list_list_fm ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__SeCaV__Otm_J,type,
nil_list_tm: list_list_tm ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
nil_Pr2223031680201799035ist_fm: list_P2887561121880082555ist_fm ).
thf(sy_c_List_Olist_ONil_001t__SeCaV__Ofm,type,
nil_fm: list_fm ).
thf(sy_c_List_Olist_ONil_001t__SeCaV__Otm,type,
nil_tm: list_tm ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
map_li4351931137408529412ist_fm: ( list_list_fm > list_list_fm ) > list_list_list_fm > list_list_list_fm ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__SeCaV__Ofm_J_001t__List__Olist_It__List__Olist_It__SeCaV__Otm_J_J,type,
map_li1108997747876207612ist_tm: ( list_fm > list_list_tm ) > list_list_fm > list_list_list_tm ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__SeCaV__Ofm_J_001t__List__Olist_It__SeCaV__Ofm_J,type,
map_list_fm_list_fm: ( list_fm > list_fm ) > list_list_fm > list_list_fm ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__SeCaV__Ofm_J_001t__SeCaV__Ofm,type,
map_list_fm_fm: ( list_fm > fm ) > list_list_fm > list_fm ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__SeCaV__Ofm_J_001t__SeCaV__Otm,type,
map_list_fm_tm: ( list_fm > tm ) > list_list_fm > list_tm ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__SeCaV__Otm_J_001t__List__Olist_It__List__Olist_It__SeCaV__Otm_J_J,type,
map_li6264597563971819530ist_tm: ( list_tm > list_list_tm ) > list_list_tm > list_list_list_tm ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__SeCaV__Otm_J_001t__List__Olist_It__SeCaV__Otm_J,type,
map_list_tm_list_tm: ( list_tm > list_tm ) > list_list_tm > list_list_tm ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__SeCaV__Otm_J_001t__SeCaV__Ofm,type,
map_list_tm_fm: ( list_tm > fm ) > list_list_tm > list_fm ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__SeCaV__Otm_J_001t__SeCaV__Otm,type,
map_list_tm_tm: ( list_tm > tm ) > list_list_tm > list_tm ).
thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
map_Pr8802613670079714180ist_fm: ( produc6018962875968178549ist_fm > produc6018962875968178549ist_fm ) > list_P2887561121880082555ist_fm > list_P2887561121880082555ist_fm ).
thf(sy_c_List_Olist_Omap_001t__SeCaV__Ofm_001t__List__Olist_It__SeCaV__Ofm_J,type,
map_fm_list_fm: ( fm > list_fm ) > list_fm > list_list_fm ).
thf(sy_c_List_Olist_Omap_001t__SeCaV__Ofm_001t__List__Olist_It__SeCaV__Otm_J,type,
map_fm_list_tm: ( fm > list_tm ) > list_fm > list_list_tm ).
thf(sy_c_List_Olist_Omap_001t__SeCaV__Ofm_001t__SeCaV__Ofm,type,
map_fm_fm: ( fm > fm ) > list_fm > list_fm ).
thf(sy_c_List_Olist_Omap_001t__SeCaV__Ofm_001t__SeCaV__Otm,type,
map_fm_tm: ( fm > tm ) > list_fm > list_tm ).
thf(sy_c_List_Olist_Omap_001t__SeCaV__Otm_001t__List__Olist_It__SeCaV__Ofm_J,type,
map_tm_list_fm: ( tm > list_fm ) > list_tm > list_list_fm ).
thf(sy_c_List_Olist_Omap_001t__SeCaV__Otm_001t__List__Olist_It__SeCaV__Otm_J,type,
map_tm_list_tm: ( tm > list_tm ) > list_tm > list_list_tm ).
thf(sy_c_List_Olist_Omap_001t__SeCaV__Otm_001t__SeCaV__Ofm,type,
map_tm_fm: ( tm > fm ) > list_tm > list_fm ).
thf(sy_c_List_Olist_Omap_001t__SeCaV__Otm_001t__SeCaV__Otm,type,
map_tm_tm: ( tm > tm ) > list_tm > list_tm ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
set_list_list_fm2: list_list_list_fm > set_list_list_fm ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__SeCaV__Ofm_J,type,
set_list_fm2: list_list_fm > set_list_fm ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__SeCaV__Otm_J,type,
set_list_tm2: list_list_tm > set_list_tm ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
set_Pr5051287786238754058ist_fm: list_P2887561121880082555ist_fm > set_Pr5202636777678657877ist_fm ).
thf(sy_c_List_Olist_Oset_001t__SeCaV__Ofm,type,
set_fm2: list_fm > set_fm ).
thf(sy_c_List_Olist_Oset_001t__SeCaV__Otm,type,
set_tm2: list_tm > set_tm ).
thf(sy_c_List_Olistrel1_001t__List__Olist_It__SeCaV__Ofm_J,type,
listrel1_list_fm: set_Pr1916137733696349511ist_fm > set_Pr3202885091612042951ist_fm ).
thf(sy_c_List_Olistrel1_001t__List__Olist_It__SeCaV__Otm_J,type,
listrel1_list_tm: set_Pr1507011332596240839ist_tm > set_Pr4883004833441821767ist_tm ).
thf(sy_c_List_Olistrel1_001t__SeCaV__Ofm,type,
listrel1_fm: set_Pr4706815898642364871_fm_fm > set_Pr1916137733696349511ist_fm ).
thf(sy_c_List_Olistrel1_001t__SeCaV__Otm,type,
listrel1_tm: set_Pr2455929065695642951_tm_tm > set_Pr1507011332596240839ist_tm ).
thf(sy_c_List_Omap__tailrec_001t__SeCaV__Ofm_001t__List__Olist_It__SeCaV__Otm_J,type,
map_ta7425747110069464646ist_tm: ( fm > list_tm ) > list_fm > list_list_tm ).
thf(sy_c_List_Omap__tailrec_001t__SeCaV__Otm_001t__List__Olist_It__SeCaV__Otm_J,type,
map_ta7538967730773405780ist_tm: ( tm > list_tm ) > list_tm > list_list_tm ).
thf(sy_c_List_Omap__tailrec_001t__SeCaV__Otm_001t__SeCaV__Otm,type,
map_tailrec_tm_tm: ( tm > tm ) > list_tm > list_tm ).
thf(sy_c_List_Omaps_001t__List__Olist_It__SeCaV__Ofm_J_001t__List__Olist_It__SeCaV__Ofm_J,type,
maps_list_fm_list_fm: ( list_fm > list_list_fm ) > list_list_fm > list_list_fm ).
thf(sy_c_List_Omaps_001t__List__Olist_It__SeCaV__Ofm_J_001t__List__Olist_It__SeCaV__Otm_J,type,
maps_list_fm_list_tm: ( list_fm > list_list_tm ) > list_list_fm > list_list_tm ).
thf(sy_c_List_Omaps_001t__List__Olist_It__SeCaV__Ofm_J_001t__SeCaV__Ofm,type,
maps_list_fm_fm: ( list_fm > list_fm ) > list_list_fm > list_fm ).
thf(sy_c_List_Omaps_001t__List__Olist_It__SeCaV__Ofm_J_001t__SeCaV__Otm,type,
maps_list_fm_tm: ( list_fm > list_tm ) > list_list_fm > list_tm ).
thf(sy_c_List_Omaps_001t__SeCaV__Ofm_001t__List__Olist_It__SeCaV__Ofm_J,type,
maps_fm_list_fm: ( fm > list_list_fm ) > list_fm > list_list_fm ).
thf(sy_c_List_Omaps_001t__SeCaV__Ofm_001t__SeCaV__Ofm,type,
maps_fm_fm: ( fm > list_fm ) > list_fm > list_fm ).
thf(sy_c_List_Omaps_001t__SeCaV__Ofm_001t__SeCaV__Otm,type,
maps_fm_tm: ( fm > list_tm ) > list_fm > list_tm ).
thf(sy_c_List_Omaps_001t__SeCaV__Otm_001t__List__Olist_It__SeCaV__Ofm_J,type,
maps_tm_list_fm: ( tm > list_list_fm ) > list_tm > list_list_fm ).
thf(sy_c_List_Omaps_001t__SeCaV__Otm_001t__List__Olist_It__SeCaV__Otm_J,type,
maps_tm_list_tm: ( tm > list_list_tm ) > list_tm > list_list_tm ).
thf(sy_c_List_Omaps_001t__SeCaV__Otm_001t__SeCaV__Ofm,type,
maps_tm_fm: ( tm > list_fm ) > list_tm > list_fm ).
thf(sy_c_List_Omaps_001t__SeCaV__Otm_001t__SeCaV__Otm,type,
maps_tm_tm: ( tm > list_tm ) > list_tm > list_tm ).
thf(sy_c_List_Omember_001t__List__Olist_It__SeCaV__Ofm_J,type,
member_list_fm: list_list_fm > list_fm > $o ).
thf(sy_c_List_Omember_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
member9193610466889212688ist_fm: list_P2887561121880082555ist_fm > produc6018962875968178549ist_fm > $o ).
thf(sy_c_List_Omember_001t__SeCaV__Ofm,type,
member_fm: list_fm > fm > $o ).
thf(sy_c_List_Omember_001t__SeCaV__Otm,type,
member_tm: list_tm > tm > $o ).
thf(sy_c_List_On__lists_001t__List__Olist_It__SeCaV__Ofm_J,type,
n_lists_list_fm: nat > list_list_fm > list_list_list_fm ).
thf(sy_c_List_On__lists_001t__SeCaV__Ofm,type,
n_lists_fm: nat > list_fm > list_list_fm ).
thf(sy_c_List_On__lists_001t__SeCaV__Otm,type,
n_lists_tm: nat > list_tm > list_list_tm ).
thf(sy_c_List_Opartition_001t__List__Olist_It__SeCaV__Ofm_J,type,
partition_list_fm: ( list_fm > $o ) > list_list_fm > produc4386893285136772327ist_fm ).
thf(sy_c_List_Opartition_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
partit1016815737701346804ist_fm: ( produc6018962875968178549ist_fm > $o ) > list_P2887561121880082555ist_fm > produc8729865333125084903ist_fm ).
thf(sy_c_List_Opartition_001t__SeCaV__Ofm,type,
partition_fm: ( fm > $o ) > list_fm > produc3245234490656042599ist_fm ).
thf(sy_c_List_Opartition_001t__SeCaV__Otm,type,
partition_tm: ( tm > $o ) > list_tm > produc5776448205642668775ist_tm ).
thf(sy_c_List_Oproduct__lists_001t__List__Olist_It__SeCaV__Ofm_J,type,
produc373462945560358120ist_fm: list_list_list_fm > list_list_list_fm ).
thf(sy_c_List_Oproduct__lists_001t__SeCaV__Ofm,type,
product_lists_fm: list_list_fm > list_list_fm ).
thf(sy_c_List_Oproduct__lists_001t__SeCaV__Otm,type,
product_lists_tm: list_list_tm > list_list_tm ).
thf(sy_c_List_Oremdups_001t__List__Olist_It__SeCaV__Ofm_J,type,
remdups_list_fm: list_list_fm > list_list_fm ).
thf(sy_c_List_Oremdups_001t__List__Olist_It__SeCaV__Otm_J,type,
remdups_list_tm: list_list_tm > list_list_tm ).
thf(sy_c_List_Oremdups_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
remdup8378870680097181200ist_fm: list_P2887561121880082555ist_fm > list_P2887561121880082555ist_fm ).
thf(sy_c_List_Oremdups_001t__SeCaV__Ofm,type,
remdups_fm: list_fm > list_fm ).
thf(sy_c_List_Oremdups_001t__SeCaV__Otm,type,
remdups_tm: list_tm > list_tm ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
size_s115229985653309035ist_fm: list_list_fm > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__SeCaV__Otm_J_J,type,
size_s9096087352182575069ist_tm: list_list_tm > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
size_s3138477486474831591ist_fm: list_P2887561121880082555ist_fm > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__SeCaV__Ofm_J,type,
size_size_list_fm: list_fm > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__SeCaV__Otm_J,type,
size_size_list_tm: list_tm > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__SeCaV__Ofm,type,
size_size_fm: fm > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__FSet__Ofset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
ord_le4401106045268334603ist_fm: fset_list_fm > fset_list_fm > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
ord_le7716396445699002503ist_fm: fset_P8989946509869081563ist_fm > fset_P8989946509869081563ist_fm > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__FSet__Ofset_It__SeCaV__Ofm_J,type,
ord_less_fset_fm: fset_fm > fset_fm > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__FSet__Ofset_It__SeCaV__Otm_J,type,
ord_less_fset_tm: fset_tm > fset_tm > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
ord_less_set_list_fm: set_list_fm > set_list_fm > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
ord_le9141428925376846849ist_fm: set_Pr5202636777678657877ist_fm > set_Pr5202636777678657877ist_fm > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__SeCaV__Ofm_J,type,
ord_less_set_fm: set_fm > set_fm > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__SeCaV__Otm_J,type,
ord_less_set_tm: set_tm > set_tm > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__FSet__Ofset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
ord_le3182162295460067583ist_fm: fset_list_fm > fset_list_fm > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
ord_le3986950534092794747ist_fm: fset_P8989946509869081563ist_fm > fset_P8989946509869081563ist_fm > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__FSet__Ofset_It__SeCaV__Ofm_J,type,
ord_less_eq_fset_fm: fset_fm > fset_fm > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__FSet__Ofset_It__SeCaV__Otm_J,type,
ord_less_eq_fset_tm: fset_tm > fset_tm > $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__SeCaV__Ofm_J_J,type,
ord_le7838213414353715577ist_fm: set_list_fm > set_list_fm > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
ord_le1771420097867575541ist_fm: set_Pr5202636777678657877ist_fm > set_Pr5202636777678657877ist_fm > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__SeCaV__Ofm_J,type,
ord_less_eq_set_fm: set_fm > set_fm > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__SeCaV__Otm_J,type,
ord_less_eq_set_tm: set_tm > set_tm > $o ).
thf(sy_c_Product__Type_OPair_001_062_It__List__Olist_It__SeCaV__Ofm_J_M_062_It__List__Olist_It__SeCaV__Ofm_J_M_Eo_J_J_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
produc3008759835361761000ist_fm: ( list_fm > list_fm > $o ) > list_list_fm > produc6864569604571974008ist_fm ).
thf(sy_c_Product__Type_OPair_001_062_It__SeCaV__Ofm_M_062_It__SeCaV__Ofm_M_Eo_J_J_001t__List__Olist_It__SeCaV__Ofm_J,type,
produc7687578365188660450ist_fm: ( fm > fm > $o ) > list_fm > produc7963324949210141170ist_fm ).
thf(sy_c_Product__Type_OPair_001_062_It__SeCaV__Otm_M_062_It__SeCaV__Otm_M_Eo_J_J_001t__List__Olist_It__SeCaV__Otm_J,type,
produc1972851280741670356ist_tm: ( tm > tm > $o ) > list_tm > produc2002131169352006116ist_tm ).
thf(sy_c_Product__Type_OPair_001_062_It__SeCaV__Otm_Mt__SeCaV__Ofm_J_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
produc4025633523336838569ist_fm: ( tm > fm ) > produc6018962875968178549ist_fm > produc3792783349651969271ist_fm ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
produc8321651870839017815ist_fm: list_list_fm > list_list_fm > produc4386893285136772327ist_fm ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__SeCaV__Otm_J_J_001t__List__Olist_It__List__Olist_It__SeCaV__Otm_J_J,type,
produc8086241248684931031ist_tm: list_list_tm > list_list_tm > produc698371888225778279ist_tm ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
produc8740291170414093911ist_fm: list_P2887561121880082555ist_fm > list_P2887561121880082555ist_fm > produc8729865333125084903ist_fm ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__SeCaV__Ofm_J_001t__List__Olist_It__SeCaV__Ofm_J,type,
produc7863996417982153943ist_fm: list_fm > list_fm > produc3245234490656042599ist_fm ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__SeCaV__Otm_J_001t__List__Olist_It__SeCaV__Ofm_J,type,
produc1414352766439514085ist_fm: list_tm > list_fm > produc6018962875968178549ist_fm ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__SeCaV__Otm_J_001t__List__Olist_It__SeCaV__Otm_J,type,
produc1418304791525149271ist_tm: list_tm > list_tm > produc5776448205642668775ist_tm ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
produc5488031402154692311ist_fm: produc6018962875968178549ist_fm > produc6018962875968178549ist_fm > produc7850321100735229287ist_fm ).
thf(sy_c_Product__Type_OPair_001t__SeCaV__Ofm_001t__SeCaV__Ofm,type,
product_Pair_fm_fm: fm > fm > product_prod_fm_fm ).
thf(sy_c_Product__Type_OPair_001t__SeCaV__Otm_001t__SeCaV__Otm,type,
product_Pair_tm_tm: tm > tm > product_prod_tm_tm ).
thf(sy_c_ProverLemmas_Oaffects,type,
affects: rule > fm > $o ).
thf(sy_c_Prover_ObranchDone,type,
branchDone: list_fm > $o ).
thf(sy_c_Prover_Ochildren,type,
children: list_tm > rule > list_fm > list_list_fm ).
thf(sy_c_Prover_Oeff,type,
eff: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm > $o ).
thf(sy_c_Prover_Oeffect,type,
effect: rule > produc6018962875968178549ist_fm > fset_P8989946509869081563ist_fm ).
thf(sy_c_Prover_Oparts,type,
parts: list_tm > rule > fm > list_list_fm ).
thf(sy_c_Prover_OsubtermFm,type,
subtermFm: fm > list_tm ).
thf(sy_c_Prover_OsubtermTm,type,
subtermTm: tm > list_tm ).
thf(sy_c_SeCaV_Ofm_OPre,type,
pre: nat > list_tm > fm ).
thf(sy_c_SeCaV_Oparamst,type,
paramst: tm > set_nat ).
thf(sy_c_SeCaV_Osub,type,
sub: nat > tm > fm > fm ).
thf(sy_c_SeCaV_Osub__term,type,
sub_term: nat > tm > tm > tm ).
thf(sy_c_SeCaV_Otm_OFun,type,
fun: nat > list_tm > tm ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__SeCaV__Ofm_J,type,
collect_list_fm: ( list_fm > $o ) > set_list_fm ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
collec5845330005437105888ist_fm: ( produc6018962875968178549ist_fm > $o ) > set_Pr5202636777678657877ist_fm ).
thf(sy_c_Set_OCollect_001t__SeCaV__Ofm,type,
collect_fm: ( fm > $o ) > set_fm ).
thf(sy_c_Set_OCollect_001t__SeCaV__Otm,type,
collect_tm: ( tm > $o ) > set_tm ).
thf(sy_c_Set_Othe__elem_001t__List__Olist_It__SeCaV__Ofm_J,type,
the_elem_list_fm: set_list_fm > list_fm ).
thf(sy_c_Set_Othe__elem_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
the_el878164078765530474ist_fm: set_Pr5202636777678657877ist_fm > produc6018962875968178549ist_fm ).
thf(sy_c_Set_Othe__elem_001t__SeCaV__Ofm,type,
the_elem_fm: set_fm > fm ).
thf(sy_c_Set_Othe__elem_001t__SeCaV__Otm,type,
the_elem_tm: set_tm > tm ).
thf(sy_c_Sublist_Oprefixes_001t__List__Olist_It__SeCaV__Ofm_J,type,
prefixes_list_fm: list_list_fm > list_list_list_fm ).
thf(sy_c_Sublist_Oprefixes_001t__List__Olist_It__SeCaV__Otm_J,type,
prefixes_list_tm: list_list_tm > list_list_list_tm ).
thf(sy_c_Sublist_Oprefixes_001t__SeCaV__Ofm,type,
prefixes_fm: list_fm > list_list_fm ).
thf(sy_c_Sublist_Oprefixes_001t__SeCaV__Otm,type,
prefixes_tm: list_tm > list_list_tm ).
thf(sy_c_Sublist_Osublists_001t__List__Olist_It__SeCaV__Ofm_J,type,
sublists_list_fm: list_list_fm > list_list_list_fm ).
thf(sy_c_Sublist_Osublists_001t__SeCaV__Ofm,type,
sublists_fm: list_fm > list_list_fm ).
thf(sy_c_Sublist_Osublists_001t__SeCaV__Otm,type,
sublists_tm: list_tm > list_list_tm ).
thf(sy_c_Sublist_Osuffixes_001t__List__Olist_It__SeCaV__Ofm_J,type,
suffixes_list_fm: list_list_fm > list_list_list_fm ).
thf(sy_c_Sublist_Osuffixes_001t__SeCaV__Ofm,type,
suffixes_fm: list_fm > list_list_fm ).
thf(sy_c_Sublist_Osuffixes_001t__SeCaV__Otm,type,
suffixes_tm: list_tm > list_list_tm ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
member_list_list_fm: list_list_fm > set_list_list_fm > $o ).
thf(sy_c_member_001t__List__Olist_It__SeCaV__Ofm_J,type,
member_list_fm2: list_fm > set_list_fm > $o ).
thf(sy_c_member_001t__List__Olist_It__SeCaV__Otm_J,type,
member_list_tm: list_tm > set_list_tm > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_Mt__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_J,type,
member5102842307574975760ist_fm: produc4386893285136772327ist_fm > set_Pr3202885091612042951ist_fm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__SeCaV__Otm_J_J_Mt__List__Olist_It__List__Olist_It__SeCaV__Otm_J_J_J,type,
member1414320910663981712ist_tm: produc698371888225778279ist_tm > set_Pr4883004833441821767ist_tm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J_J,type,
member6238937682488541200ist_fm: produc8729865333125084903ist_fm > set_Pr8947555413577213383ist_fm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Ofm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
member1926098302810316688ist_fm: produc3245234490656042599ist_fm > set_Pr1916137733696349511ist_fm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
member4699826688122452638ist_fm: produc6018962875968178549ist_fm > set_Pr5202636777678657877ist_fm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Otm_J_J,type,
member4457312017796942864ist_tm: produc5776448205642668775ist_tm > set_Pr1507011332596240839ist_tm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J,type,
member858341428514850704ist_fm: produc7850321100735229287ist_fm > set_Pr6724411253394807111ist_fm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__SeCaV__Ofm_Mt__SeCaV__Ofm_J,type,
member7780952600467998736_fm_fm: product_prod_fm_fm > set_Pr4706815898642364871_fm_fm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__SeCaV__Otm_Mt__SeCaV__Otm_J,type,
member3121616906494481296_tm_tm: product_prod_tm_tm > set_Pr2455929065695642951_tm_tm > $o ).
thf(sy_c_member_001t__SeCaV__Ofm,type,
member_fm2: fm > set_fm > $o ).
thf(sy_c_member_001t__SeCaV__Otm,type,
member_tm2: tm > set_tm > $o ).
thf(sy_v_A,type,
a: list_tm ).
thf(sy_v_r,type,
r: rule ).
thf(sy_v_ss,type,
ss: fset_P8989946509869081563ist_fm ).
thf(sy_v_z,type,
z: list_fm ).
% Relevant facts (1266)
thf(fact_0_fset__of__list__elem,axiom,
! [X: produc6018962875968178549ist_fm,Xs: list_P2887561121880082555ist_fm] :
( ( fmembe3754813877001230652ist_fm @ X @ ( fset_o3222440871817023035ist_fm @ Xs ) )
= ( member4699826688122452638ist_fm @ X @ ( set_Pr5051287786238754058ist_fm @ Xs ) ) ) ).
% fset_of_list_elem
thf(fact_1_fset__of__list__elem,axiom,
! [X: list_fm,Xs: list_list_fm] :
( ( fmember_list_fm @ X @ ( fset_of_list_list_fm @ Xs ) )
= ( member_list_fm2 @ X @ ( set_list_fm2 @ Xs ) ) ) ).
% fset_of_list_elem
thf(fact_2_fset__of__list__elem,axiom,
! [X: fm,Xs: list_fm] :
( ( fmember_fm @ X @ ( fset_of_list_fm @ Xs ) )
= ( member_fm2 @ X @ ( set_fm2 @ Xs ) ) ) ).
% fset_of_list_elem
thf(fact_3_fset__of__list__elem,axiom,
! [X: tm,Xs: list_tm] :
( ( fmember_tm @ X @ ( fset_of_list_tm @ Xs ) )
= ( member_tm2 @ X @ ( set_tm2 @ Xs ) ) ) ).
% fset_of_list_elem
thf(fact_4_assms_I1_J,axiom,
~ ( branchDone @ z ) ).
% assms(1)
thf(fact_5_assms_I2_J,axiom,
eff @ r @ ( produc1414352766439514085ist_fm @ a @ z ) @ ss ).
% assms(2)
thf(fact_6_set__remdups,axiom,
! [Xs: list_P2887561121880082555ist_fm] :
( ( set_Pr5051287786238754058ist_fm @ ( remdup8378870680097181200ist_fm @ Xs ) )
= ( set_Pr5051287786238754058ist_fm @ Xs ) ) ).
% set_remdups
thf(fact_7_set__remdups,axiom,
! [Xs: list_list_fm] :
( ( set_list_fm2 @ ( remdups_list_fm @ Xs ) )
= ( set_list_fm2 @ Xs ) ) ).
% set_remdups
thf(fact_8_set__remdups,axiom,
! [Xs: list_fm] :
( ( set_fm2 @ ( remdups_fm @ Xs ) )
= ( set_fm2 @ Xs ) ) ).
% set_remdups
thf(fact_9_set__remdups,axiom,
! [Xs: list_tm] :
( ( set_tm2 @ ( remdups_tm @ Xs ) )
= ( set_tm2 @ Xs ) ) ).
% set_remdups
thf(fact_10_concat__append,axiom,
! [Xs: list_list_list_tm,Ys: list_list_list_tm] :
( ( concat_list_tm @ ( append_list_list_tm @ Xs @ Ys ) )
= ( append_list_tm @ ( concat_list_tm @ Xs ) @ ( concat_list_tm @ Ys ) ) ) ).
% concat_append
thf(fact_11_concat__append,axiom,
! [Xs: list_list_fm,Ys: list_list_fm] :
( ( concat_fm @ ( append_list_fm @ Xs @ Ys ) )
= ( append_fm @ ( concat_fm @ Xs ) @ ( concat_fm @ Ys ) ) ) ).
% concat_append
thf(fact_12_concat__append,axiom,
! [Xs: list_list_tm,Ys: list_list_tm] :
( ( concat_tm @ ( append_list_tm @ Xs @ Ys ) )
= ( append_tm @ ( concat_tm @ Xs ) @ ( concat_tm @ Ys ) ) ) ).
% concat_append
thf(fact_13_map__append,axiom,
! [F: tm > fm,Xs: list_tm,Ys: list_tm] :
( ( map_tm_fm @ F @ ( append_tm @ Xs @ Ys ) )
= ( append_fm @ ( map_tm_fm @ F @ Xs ) @ ( map_tm_fm @ F @ Ys ) ) ) ).
% map_append
thf(fact_14_map__append,axiom,
! [F: list_tm > tm,Xs: list_list_tm,Ys: list_list_tm] :
( ( map_list_tm_tm @ F @ ( append_list_tm @ Xs @ Ys ) )
= ( append_tm @ ( map_list_tm_tm @ F @ Xs ) @ ( map_list_tm_tm @ F @ Ys ) ) ) ).
% map_append
thf(fact_15_map__append,axiom,
! [F: list_tm > list_tm,Xs: list_list_tm,Ys: list_list_tm] :
( ( map_list_tm_list_tm @ F @ ( append_list_tm @ Xs @ Ys ) )
= ( append_list_tm @ ( map_list_tm_list_tm @ F @ Xs ) @ ( map_list_tm_list_tm @ F @ Ys ) ) ) ).
% map_append
thf(fact_16_map__append,axiom,
! [F: list_tm > fm,Xs: list_list_tm,Ys: list_list_tm] :
( ( map_list_tm_fm @ F @ ( append_list_tm @ Xs @ Ys ) )
= ( append_fm @ ( map_list_tm_fm @ F @ Xs ) @ ( map_list_tm_fm @ F @ Ys ) ) ) ).
% map_append
thf(fact_17_map__append,axiom,
! [F: fm > tm,Xs: list_fm,Ys: list_fm] :
( ( map_fm_tm @ F @ ( append_fm @ Xs @ Ys ) )
= ( append_tm @ ( map_fm_tm @ F @ Xs ) @ ( map_fm_tm @ F @ Ys ) ) ) ).
% map_append
thf(fact_18_map__append,axiom,
! [F: fm > fm,Xs: list_fm,Ys: list_fm] :
( ( map_fm_fm @ F @ ( append_fm @ Xs @ Ys ) )
= ( append_fm @ ( map_fm_fm @ F @ Xs ) @ ( map_fm_fm @ F @ Ys ) ) ) ).
% map_append
thf(fact_19_map__append,axiom,
! [F: tm > tm,Xs: list_tm,Ys: list_tm] :
( ( map_tm_tm @ F @ ( append_tm @ Xs @ Ys ) )
= ( append_tm @ ( map_tm_tm @ F @ Xs ) @ ( map_tm_tm @ F @ Ys ) ) ) ).
% map_append
thf(fact_20_map__append,axiom,
! [F: fm > list_tm,Xs: list_fm,Ys: list_fm] :
( ( map_fm_list_tm @ F @ ( append_fm @ Xs @ Ys ) )
= ( append_list_tm @ ( map_fm_list_tm @ F @ Xs ) @ ( map_fm_list_tm @ F @ Ys ) ) ) ).
% map_append
thf(fact_21_map__append,axiom,
! [F: tm > list_tm,Xs: list_tm,Ys: list_tm] :
( ( map_tm_list_tm @ F @ ( append_tm @ Xs @ Ys ) )
= ( append_list_tm @ ( map_tm_list_tm @ F @ Xs ) @ ( map_tm_list_tm @ F @ Ys ) ) ) ).
% map_append
thf(fact_22_map__eq__conv,axiom,
! [F: tm > tm,Xs: list_tm,G: tm > tm] :
( ( ( map_tm_tm @ F @ Xs )
= ( map_tm_tm @ G @ Xs ) )
= ( ! [X2: tm] :
( ( member_tm2 @ X2 @ ( set_tm2 @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_23_map__eq__conv,axiom,
! [F: fm > list_tm,Xs: list_fm,G: fm > list_tm] :
( ( ( map_fm_list_tm @ F @ Xs )
= ( map_fm_list_tm @ G @ Xs ) )
= ( ! [X2: fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_24_map__eq__conv,axiom,
! [F: tm > list_tm,Xs: list_tm,G: tm > list_tm] :
( ( ( map_tm_list_tm @ F @ Xs )
= ( map_tm_list_tm @ G @ Xs ) )
= ( ! [X2: tm] :
( ( member_tm2 @ X2 @ ( set_tm2 @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_25_append_Oassoc,axiom,
! [A: list_list_tm,B: list_list_tm,C: list_list_tm] :
( ( append_list_tm @ ( append_list_tm @ A @ B ) @ C )
= ( append_list_tm @ A @ ( append_list_tm @ B @ C ) ) ) ).
% append.assoc
thf(fact_26_append_Oassoc,axiom,
! [A: list_fm,B: list_fm,C: list_fm] :
( ( append_fm @ ( append_fm @ A @ B ) @ C )
= ( append_fm @ A @ ( append_fm @ B @ C ) ) ) ).
% append.assoc
thf(fact_27_append_Oassoc,axiom,
! [A: list_tm,B: list_tm,C: list_tm] :
( ( append_tm @ ( append_tm @ A @ B ) @ C )
= ( append_tm @ A @ ( append_tm @ B @ C ) ) ) ).
% append.assoc
thf(fact_28_append__assoc,axiom,
! [Xs: list_list_tm,Ys: list_list_tm,Zs: list_list_tm] :
( ( append_list_tm @ ( append_list_tm @ Xs @ Ys ) @ Zs )
= ( append_list_tm @ Xs @ ( append_list_tm @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_29_append__assoc,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_fm] :
( ( append_fm @ ( append_fm @ Xs @ Ys ) @ Zs )
= ( append_fm @ Xs @ ( append_fm @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_30_append__assoc,axiom,
! [Xs: list_tm,Ys: list_tm,Zs: list_tm] :
( ( append_tm @ ( append_tm @ Xs @ Ys ) @ Zs )
= ( append_tm @ Xs @ ( append_tm @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_31_append__same__eq,axiom,
! [Ys: list_list_tm,Xs: list_list_tm,Zs: list_list_tm] :
( ( ( append_list_tm @ Ys @ Xs )
= ( append_list_tm @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_32_append__same__eq,axiom,
! [Ys: list_fm,Xs: list_fm,Zs: list_fm] :
( ( ( append_fm @ Ys @ Xs )
= ( append_fm @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_33_append__same__eq,axiom,
! [Ys: list_tm,Xs: list_tm,Zs: list_tm] :
( ( ( append_tm @ Ys @ Xs )
= ( append_tm @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_34_same__append__eq,axiom,
! [Xs: list_tm,Ys: list_tm,Zs: list_tm] :
( ( ( append_tm @ Xs @ Ys )
= ( append_tm @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_35_same__append__eq,axiom,
! [Xs: list_list_tm,Ys: list_list_tm,Zs: list_list_tm] :
( ( ( append_list_tm @ Xs @ Ys )
= ( append_list_tm @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_36_same__append__eq,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_fm] :
( ( ( append_fm @ Xs @ Ys )
= ( append_fm @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_37_prod_Oinject,axiom,
! [X1: list_tm,X22: list_fm,Y1: list_tm,Y2: list_fm] :
( ( ( produc1414352766439514085ist_fm @ X1 @ X22 )
= ( produc1414352766439514085ist_fm @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_38_old_Oprod_Oinject,axiom,
! [A: list_tm,B: list_fm,A2: list_tm,B2: list_fm] :
( ( ( produc1414352766439514085ist_fm @ A @ B )
= ( produc1414352766439514085ist_fm @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_39_remdups__map__remdups,axiom,
! [F: fm > tm,Xs: list_fm] :
( ( remdups_tm @ ( map_fm_tm @ F @ ( remdups_fm @ Xs ) ) )
= ( remdups_tm @ ( map_fm_tm @ F @ Xs ) ) ) ).
% remdups_map_remdups
thf(fact_40_remdups__map__remdups,axiom,
! [F: tm > fm,Xs: list_tm] :
( ( remdups_fm @ ( map_tm_fm @ F @ ( remdups_tm @ Xs ) ) )
= ( remdups_fm @ ( map_tm_fm @ F @ Xs ) ) ) ).
% remdups_map_remdups
thf(fact_41_remdups__map__remdups,axiom,
! [F: fm > fm,Xs: list_fm] :
( ( remdups_fm @ ( map_fm_fm @ F @ ( remdups_fm @ Xs ) ) )
= ( remdups_fm @ ( map_fm_fm @ F @ Xs ) ) ) ).
% remdups_map_remdups
thf(fact_42_remdups__map__remdups,axiom,
! [F: tm > tm,Xs: list_tm] :
( ( remdups_tm @ ( map_tm_tm @ F @ ( remdups_tm @ Xs ) ) )
= ( remdups_tm @ ( map_tm_tm @ F @ Xs ) ) ) ).
% remdups_map_remdups
thf(fact_43_remdups__map__remdups,axiom,
! [F: list_fm > tm,Xs: list_list_fm] :
( ( remdups_tm @ ( map_list_fm_tm @ F @ ( remdups_list_fm @ Xs ) ) )
= ( remdups_tm @ ( map_list_fm_tm @ F @ Xs ) ) ) ).
% remdups_map_remdups
thf(fact_44_remdups__map__remdups,axiom,
! [F: list_fm > fm,Xs: list_list_fm] :
( ( remdups_fm @ ( map_list_fm_fm @ F @ ( remdups_list_fm @ Xs ) ) )
= ( remdups_fm @ ( map_list_fm_fm @ F @ Xs ) ) ) ).
% remdups_map_remdups
thf(fact_45_remdups__map__remdups,axiom,
! [F: tm > list_fm,Xs: list_tm] :
( ( remdups_list_fm @ ( map_tm_list_fm @ F @ ( remdups_tm @ Xs ) ) )
= ( remdups_list_fm @ ( map_tm_list_fm @ F @ Xs ) ) ) ).
% remdups_map_remdups
thf(fact_46_remdups__map__remdups,axiom,
! [F: fm > list_fm,Xs: list_fm] :
( ( remdups_list_fm @ ( map_fm_list_fm @ F @ ( remdups_fm @ Xs ) ) )
= ( remdups_list_fm @ ( map_fm_list_fm @ F @ Xs ) ) ) ).
% remdups_map_remdups
thf(fact_47_remdups__map__remdups,axiom,
! [F: fm > list_tm,Xs: list_fm] :
( ( remdups_list_tm @ ( map_fm_list_tm @ F @ ( remdups_fm @ Xs ) ) )
= ( remdups_list_tm @ ( map_fm_list_tm @ F @ Xs ) ) ) ).
% remdups_map_remdups
thf(fact_48_remdups__map__remdups,axiom,
! [F: tm > list_tm,Xs: list_tm] :
( ( remdups_list_tm @ ( map_tm_list_tm @ F @ ( remdups_tm @ Xs ) ) )
= ( remdups_list_tm @ ( map_tm_list_tm @ F @ Xs ) ) ) ).
% remdups_map_remdups
thf(fact_49_remdups__append2,axiom,
! [Xs: list_list_tm,Ys: list_list_tm] :
( ( remdups_list_tm @ ( append_list_tm @ Xs @ ( remdups_list_tm @ Ys ) ) )
= ( remdups_list_tm @ ( append_list_tm @ Xs @ Ys ) ) ) ).
% remdups_append2
thf(fact_50_remdups__append2,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( remdups_tm @ ( append_tm @ Xs @ ( remdups_tm @ Ys ) ) )
= ( remdups_tm @ ( append_tm @ Xs @ Ys ) ) ) ).
% remdups_append2
thf(fact_51_remdups__append2,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( remdups_fm @ ( append_fm @ Xs @ ( remdups_fm @ Ys ) ) )
= ( remdups_fm @ ( append_fm @ Xs @ Ys ) ) ) ).
% remdups_append2
thf(fact_52_remdups__append2,axiom,
! [Xs: list_list_fm,Ys: list_list_fm] :
( ( remdups_list_fm @ ( append_list_fm @ Xs @ ( remdups_list_fm @ Ys ) ) )
= ( remdups_list_fm @ ( append_list_fm @ Xs @ Ys ) ) ) ).
% remdups_append2
thf(fact_53_exists__fset__of__list,axiom,
! [S: fset_tm] :
? [Xs2: list_tm] :
( ( fset_of_list_tm @ Xs2 )
= S ) ).
% exists_fset_of_list
thf(fact_54_exists__fset__of__list,axiom,
! [S: fset_fm] :
? [Xs2: list_fm] :
( ( fset_of_list_fm @ Xs2 )
= S ) ).
% exists_fset_of_list
thf(fact_55_exists__fset__of__list,axiom,
! [S: fset_list_fm] :
? [Xs2: list_list_fm] :
( ( fset_of_list_list_fm @ Xs2 )
= S ) ).
% exists_fset_of_list
thf(fact_56_exists__fset__of__list,axiom,
! [S: fset_P8989946509869081563ist_fm] :
? [Xs2: list_P2887561121880082555ist_fm] :
( ( fset_o3222440871817023035ist_fm @ Xs2 )
= S ) ).
% exists_fset_of_list
thf(fact_57_Pair__inject,axiom,
! [A: list_tm,B: list_fm,A2: list_tm,B2: list_fm] :
( ( ( produc1414352766439514085ist_fm @ A @ B )
= ( produc1414352766439514085ist_fm @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_58_prod__cases,axiom,
! [P: produc6018962875968178549ist_fm > $o,P2: produc6018962875968178549ist_fm] :
( ! [A3: list_tm,B3: list_fm] : ( P @ ( produc1414352766439514085ist_fm @ A3 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_59_surj__pair,axiom,
! [P2: produc6018962875968178549ist_fm] :
? [X3: list_tm,Y: list_fm] :
( P2
= ( produc1414352766439514085ist_fm @ X3 @ Y ) ) ).
% surj_pair
thf(fact_60_old_Oprod_Oexhaust,axiom,
! [Y3: produc6018962875968178549ist_fm] :
~ ! [A3: list_tm,B3: list_fm] :
( Y3
!= ( produc1414352766439514085ist_fm @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_61_append__eq__append__conv2,axiom,
! [Xs: list_tm,Ys: list_tm,Zs: list_tm,Ts: list_tm] :
( ( ( append_tm @ Xs @ Ys )
= ( append_tm @ Zs @ Ts ) )
= ( ? [Us: list_tm] :
( ( ( Xs
= ( append_tm @ Zs @ Us ) )
& ( ( append_tm @ Us @ Ys )
= Ts ) )
| ( ( ( append_tm @ Xs @ Us )
= Zs )
& ( Ys
= ( append_tm @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_62_append__eq__append__conv2,axiom,
! [Xs: list_list_tm,Ys: list_list_tm,Zs: list_list_tm,Ts: list_list_tm] :
( ( ( append_list_tm @ Xs @ Ys )
= ( append_list_tm @ Zs @ Ts ) )
= ( ? [Us: list_list_tm] :
( ( ( Xs
= ( append_list_tm @ Zs @ Us ) )
& ( ( append_list_tm @ Us @ Ys )
= Ts ) )
| ( ( ( append_list_tm @ Xs @ Us )
= Zs )
& ( Ys
= ( append_list_tm @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_63_append__eq__append__conv2,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_fm,Ts: list_fm] :
( ( ( append_fm @ Xs @ Ys )
= ( append_fm @ Zs @ Ts ) )
= ( ? [Us: list_fm] :
( ( ( Xs
= ( append_fm @ Zs @ Us ) )
& ( ( append_fm @ Us @ Ys )
= Ts ) )
| ( ( ( append_fm @ Xs @ Us )
= Zs )
& ( Ys
= ( append_fm @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_64_append__eq__appendI,axiom,
! [Xs: list_tm,Xs1: list_tm,Zs: list_tm,Ys: list_tm,Us2: list_tm] :
( ( ( append_tm @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_tm @ Xs1 @ Us2 ) )
=> ( ( append_tm @ Xs @ Ys )
= ( append_tm @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_65_append__eq__appendI,axiom,
! [Xs: list_list_tm,Xs1: list_list_tm,Zs: list_list_tm,Ys: list_list_tm,Us2: list_list_tm] :
( ( ( append_list_tm @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_list_tm @ Xs1 @ Us2 ) )
=> ( ( append_list_tm @ Xs @ Ys )
= ( append_list_tm @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_66_append__eq__appendI,axiom,
! [Xs: list_fm,Xs1: list_fm,Zs: list_fm,Ys: list_fm,Us2: list_fm] :
( ( ( append_fm @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_fm @ Xs1 @ Us2 ) )
=> ( ( append_fm @ Xs @ Ys )
= ( append_fm @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_67_eqfelem__imp__iff,axiom,
! [X: produc6018962875968178549ist_fm,Y3: produc6018962875968178549ist_fm,A4: fset_P8989946509869081563ist_fm] :
( ( X = Y3 )
=> ( ( fmembe3754813877001230652ist_fm @ X @ A4 )
= ( fmembe3754813877001230652ist_fm @ Y3 @ A4 ) ) ) ).
% eqfelem_imp_iff
thf(fact_68_eqfelem__imp__iff,axiom,
! [X: tm,Y3: tm,A4: fset_tm] :
( ( X = Y3 )
=> ( ( fmember_tm @ X @ A4 )
= ( fmember_tm @ Y3 @ A4 ) ) ) ).
% eqfelem_imp_iff
thf(fact_69_eqfelem__imp__iff,axiom,
! [X: fm,Y3: fm,A4: fset_fm] :
( ( X = Y3 )
=> ( ( fmember_fm @ X @ A4 )
= ( fmember_fm @ Y3 @ A4 ) ) ) ).
% eqfelem_imp_iff
thf(fact_70_eqfelem__imp__iff,axiom,
! [X: list_fm,Y3: list_fm,A4: fset_list_fm] :
( ( X = Y3 )
=> ( ( fmember_list_fm @ X @ A4 )
= ( fmember_list_fm @ Y3 @ A4 ) ) ) ).
% eqfelem_imp_iff
thf(fact_71_if__split__fmem2,axiom,
! [A: produc6018962875968178549ist_fm,Q: $o,X: fset_P8989946509869081563ist_fm,Y3: fset_P8989946509869081563ist_fm] :
( ( fmembe3754813877001230652ist_fm @ A @ ( if_fse7999432387889793441ist_fm @ Q @ X @ Y3 ) )
= ( ( Q
=> ( fmembe3754813877001230652ist_fm @ A @ X ) )
& ( ~ Q
=> ( fmembe3754813877001230652ist_fm @ A @ Y3 ) ) ) ) ).
% if_split_fmem2
thf(fact_72_if__split__fmem2,axiom,
! [A: tm,Q: $o,X: fset_tm,Y3: fset_tm] :
( ( fmember_tm @ A @ ( if_fset_tm @ Q @ X @ Y3 ) )
= ( ( Q
=> ( fmember_tm @ A @ X ) )
& ( ~ Q
=> ( fmember_tm @ A @ Y3 ) ) ) ) ).
% if_split_fmem2
thf(fact_73_if__split__fmem2,axiom,
! [A: fm,Q: $o,X: fset_fm,Y3: fset_fm] :
( ( fmember_fm @ A @ ( if_fset_fm @ Q @ X @ Y3 ) )
= ( ( Q
=> ( fmember_fm @ A @ X ) )
& ( ~ Q
=> ( fmember_fm @ A @ Y3 ) ) ) ) ).
% if_split_fmem2
thf(fact_74_if__split__fmem2,axiom,
! [A: list_fm,Q: $o,X: fset_list_fm,Y3: fset_list_fm] :
( ( fmember_list_fm @ A @ ( if_fset_list_fm @ Q @ X @ Y3 ) )
= ( ( Q
=> ( fmember_list_fm @ A @ X ) )
& ( ~ Q
=> ( fmember_list_fm @ A @ Y3 ) ) ) ) ).
% if_split_fmem2
thf(fact_75_if__split__fmem1,axiom,
! [Q: $o,X: produc6018962875968178549ist_fm,Y3: produc6018962875968178549ist_fm,B: fset_P8989946509869081563ist_fm] :
( ( fmembe3754813877001230652ist_fm @ ( if_Pro4760001780252510779ist_fm @ Q @ X @ Y3 ) @ B )
= ( ( Q
=> ( fmembe3754813877001230652ist_fm @ X @ B ) )
& ( ~ Q
=> ( fmembe3754813877001230652ist_fm @ Y3 @ B ) ) ) ) ).
% if_split_fmem1
thf(fact_76_if__split__fmem1,axiom,
! [Q: $o,X: tm,Y3: tm,B: fset_tm] :
( ( fmember_tm @ ( if_tm @ Q @ X @ Y3 ) @ B )
= ( ( Q
=> ( fmember_tm @ X @ B ) )
& ( ~ Q
=> ( fmember_tm @ Y3 @ B ) ) ) ) ).
% if_split_fmem1
thf(fact_77_if__split__fmem1,axiom,
! [Q: $o,X: fm,Y3: fm,B: fset_fm] :
( ( fmember_fm @ ( if_fm @ Q @ X @ Y3 ) @ B )
= ( ( Q
=> ( fmember_fm @ X @ B ) )
& ( ~ Q
=> ( fmember_fm @ Y3 @ B ) ) ) ) ).
% if_split_fmem1
thf(fact_78_if__split__fmem1,axiom,
! [Q: $o,X: list_fm,Y3: list_fm,B: fset_list_fm] :
( ( fmember_list_fm @ ( if_list_fm @ Q @ X @ Y3 ) @ B )
= ( ( Q
=> ( fmember_list_fm @ X @ B ) )
& ( ~ Q
=> ( fmember_list_fm @ Y3 @ B ) ) ) ) ).
% if_split_fmem1
thf(fact_79_eqfset__imp__iff,axiom,
! [A4: fset_P8989946509869081563ist_fm,B4: fset_P8989946509869081563ist_fm,X: produc6018962875968178549ist_fm] :
( ( A4 = B4 )
=> ( ( fmembe3754813877001230652ist_fm @ X @ A4 )
= ( fmembe3754813877001230652ist_fm @ X @ B4 ) ) ) ).
% eqfset_imp_iff
thf(fact_80_eqfset__imp__iff,axiom,
! [A4: fset_tm,B4: fset_tm,X: tm] :
( ( A4 = B4 )
=> ( ( fmember_tm @ X @ A4 )
= ( fmember_tm @ X @ B4 ) ) ) ).
% eqfset_imp_iff
thf(fact_81_eqfset__imp__iff,axiom,
! [A4: fset_fm,B4: fset_fm,X: fm] :
( ( A4 = B4 )
=> ( ( fmember_fm @ X @ A4 )
= ( fmember_fm @ X @ B4 ) ) ) ).
% eqfset_imp_iff
thf(fact_82_eqfset__imp__iff,axiom,
! [A4: fset_list_fm,B4: fset_list_fm,X: list_fm] :
( ( A4 = B4 )
=> ( ( fmember_list_fm @ X @ A4 )
= ( fmember_list_fm @ X @ B4 ) ) ) ).
% eqfset_imp_iff
thf(fact_83_eq__fmem__trans,axiom,
! [A: produc6018962875968178549ist_fm,B: produc6018962875968178549ist_fm,A4: fset_P8989946509869081563ist_fm] :
( ( A = B )
=> ( ( fmembe3754813877001230652ist_fm @ B @ A4 )
=> ( fmembe3754813877001230652ist_fm @ A @ A4 ) ) ) ).
% eq_fmem_trans
thf(fact_84_eq__fmem__trans,axiom,
! [A: tm,B: tm,A4: fset_tm] :
( ( A = B )
=> ( ( fmember_tm @ B @ A4 )
=> ( fmember_tm @ A @ A4 ) ) ) ).
% eq_fmem_trans
thf(fact_85_eq__fmem__trans,axiom,
! [A: fm,B: fm,A4: fset_fm] :
( ( A = B )
=> ( ( fmember_fm @ B @ A4 )
=> ( fmember_fm @ A @ A4 ) ) ) ).
% eq_fmem_trans
thf(fact_86_eq__fmem__trans,axiom,
! [A: list_fm,B: list_fm,A4: fset_list_fm] :
( ( A = B )
=> ( ( fmember_list_fm @ B @ A4 )
=> ( fmember_list_fm @ A @ A4 ) ) ) ).
% eq_fmem_trans
thf(fact_87_fequalityCE,axiom,
! [A4: fset_P8989946509869081563ist_fm,B4: fset_P8989946509869081563ist_fm,C: produc6018962875968178549ist_fm] :
( ( A4 = B4 )
=> ( ( ( fmembe3754813877001230652ist_fm @ C @ A4 )
=> ~ ( fmembe3754813877001230652ist_fm @ C @ B4 ) )
=> ~ ( ~ ( fmembe3754813877001230652ist_fm @ C @ A4 )
=> ( fmembe3754813877001230652ist_fm @ C @ B4 ) ) ) ) ).
% fequalityCE
thf(fact_88_fequalityCE,axiom,
! [A4: fset_tm,B4: fset_tm,C: tm] :
( ( A4 = B4 )
=> ( ( ( fmember_tm @ C @ A4 )
=> ~ ( fmember_tm @ C @ B4 ) )
=> ~ ( ~ ( fmember_tm @ C @ A4 )
=> ( fmember_tm @ C @ B4 ) ) ) ) ).
% fequalityCE
thf(fact_89_fequalityCE,axiom,
! [A4: fset_fm,B4: fset_fm,C: fm] :
( ( A4 = B4 )
=> ( ( ( fmember_fm @ C @ A4 )
=> ~ ( fmember_fm @ C @ B4 ) )
=> ~ ( ~ ( fmember_fm @ C @ A4 )
=> ( fmember_fm @ C @ B4 ) ) ) ) ).
% fequalityCE
thf(fact_90_fequalityCE,axiom,
! [A4: fset_list_fm,B4: fset_list_fm,C: list_fm] :
( ( A4 = B4 )
=> ( ( ( fmember_list_fm @ C @ A4 )
=> ~ ( fmember_list_fm @ C @ B4 ) )
=> ~ ( ~ ( fmember_list_fm @ C @ A4 )
=> ( fmember_list_fm @ C @ B4 ) ) ) ) ).
% fequalityCE
thf(fact_91_fset__eqI,axiom,
! [A4: fset_P8989946509869081563ist_fm,B4: fset_P8989946509869081563ist_fm] :
( ! [X3: produc6018962875968178549ist_fm] :
( ( fmembe3754813877001230652ist_fm @ X3 @ A4 )
= ( fmembe3754813877001230652ist_fm @ X3 @ B4 ) )
=> ( A4 = B4 ) ) ).
% fset_eqI
thf(fact_92_fset__eqI,axiom,
! [A4: fset_tm,B4: fset_tm] :
( ! [X3: tm] :
( ( fmember_tm @ X3 @ A4 )
= ( fmember_tm @ X3 @ B4 ) )
=> ( A4 = B4 ) ) ).
% fset_eqI
thf(fact_93_fset__eqI,axiom,
! [A4: fset_fm,B4: fset_fm] :
( ! [X3: fm] :
( ( fmember_fm @ X3 @ A4 )
= ( fmember_fm @ X3 @ B4 ) )
=> ( A4 = B4 ) ) ).
% fset_eqI
thf(fact_94_fset__eqI,axiom,
! [A4: fset_list_fm,B4: fset_list_fm] :
( ! [X3: list_fm] :
( ( fmember_list_fm @ X3 @ A4 )
= ( fmember_list_fm @ X3 @ B4 ) )
=> ( A4 = B4 ) ) ).
% fset_eqI
thf(fact_95_remdups__remdups,axiom,
! [Xs: list_tm] :
( ( remdups_tm @ ( remdups_tm @ Xs ) )
= ( remdups_tm @ Xs ) ) ).
% remdups_remdups
thf(fact_96_remdups__remdups,axiom,
! [Xs: list_fm] :
( ( remdups_fm @ ( remdups_fm @ Xs ) )
= ( remdups_fm @ Xs ) ) ).
% remdups_remdups
thf(fact_97_remdups__remdups,axiom,
! [Xs: list_list_fm] :
( ( remdups_list_fm @ ( remdups_list_fm @ Xs ) )
= ( remdups_list_fm @ Xs ) ) ).
% remdups_remdups
thf(fact_98_ex__map__conv,axiom,
! [Ys: list_list_tm,F: fm > list_tm] :
( ( ? [Xs3: list_fm] :
( Ys
= ( map_fm_list_tm @ F @ Xs3 ) ) )
= ( ! [X2: list_tm] :
( ( member_list_tm @ X2 @ ( set_list_tm2 @ Ys ) )
=> ? [Y4: fm] :
( X2
= ( F @ Y4 ) ) ) ) ) ).
% ex_map_conv
thf(fact_99_ex__map__conv,axiom,
! [Ys: list_list_tm,F: tm > list_tm] :
( ( ? [Xs3: list_tm] :
( Ys
= ( map_tm_list_tm @ F @ Xs3 ) ) )
= ( ! [X2: list_tm] :
( ( member_list_tm @ X2 @ ( set_list_tm2 @ Ys ) )
=> ? [Y4: tm] :
( X2
= ( F @ Y4 ) ) ) ) ) ).
% ex_map_conv
thf(fact_100_ex__map__conv,axiom,
! [Ys: list_tm,F: tm > tm] :
( ( ? [Xs3: list_tm] :
( Ys
= ( map_tm_tm @ F @ Xs3 ) ) )
= ( ! [X2: tm] :
( ( member_tm2 @ X2 @ ( set_tm2 @ Ys ) )
=> ? [Y4: tm] :
( X2
= ( F @ Y4 ) ) ) ) ) ).
% ex_map_conv
thf(fact_101_map__cong,axiom,
! [Xs: list_fm,Ys: list_fm,F: fm > list_tm,G: fm > list_tm] :
( ( Xs = Ys )
=> ( ! [X3: fm] :
( ( member_fm2 @ X3 @ ( set_fm2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_fm_list_tm @ F @ Xs )
= ( map_fm_list_tm @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_102_map__cong,axiom,
! [Xs: list_tm,Ys: list_tm,F: tm > list_tm,G: tm > list_tm] :
( ( Xs = Ys )
=> ( ! [X3: tm] :
( ( member_tm2 @ X3 @ ( set_tm2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_tm_list_tm @ F @ Xs )
= ( map_tm_list_tm @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_103_map__cong,axiom,
! [Xs: list_tm,Ys: list_tm,F: tm > tm,G: tm > tm] :
( ( Xs = Ys )
=> ( ! [X3: tm] :
( ( member_tm2 @ X3 @ ( set_tm2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_tm_tm @ F @ Xs )
= ( map_tm_tm @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_104_map__idI,axiom,
! [Xs: list_list_fm,F: list_fm > list_fm] :
( ! [X3: list_fm] :
( ( member_list_fm2 @ X3 @ ( set_list_fm2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_list_fm_list_fm @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_105_map__idI,axiom,
! [Xs: list_fm,F: fm > fm] :
( ! [X3: fm] :
( ( member_fm2 @ X3 @ ( set_fm2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_fm_fm @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_106_map__idI,axiom,
! [Xs: list_tm,F: tm > tm] :
( ! [X3: tm] :
( ( member_tm2 @ X3 @ ( set_tm2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_tm_tm @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_107_map__idI,axiom,
! [Xs: list_P2887561121880082555ist_fm,F: produc6018962875968178549ist_fm > produc6018962875968178549ist_fm] :
( ! [X3: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ X3 @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_Pr8802613670079714180ist_fm @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_108_mem__Collect__eq,axiom,
! [A: fm,P: fm > $o] :
( ( member_fm2 @ A @ ( collect_fm @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_109_mem__Collect__eq,axiom,
! [A: list_fm,P: list_fm > $o] :
( ( member_list_fm2 @ A @ ( collect_list_fm @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_110_mem__Collect__eq,axiom,
! [A: tm,P: tm > $o] :
( ( member_tm2 @ A @ ( collect_tm @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_111_mem__Collect__eq,axiom,
! [A: produc6018962875968178549ist_fm,P: produc6018962875968178549ist_fm > $o] :
( ( member4699826688122452638ist_fm @ A @ ( collec5845330005437105888ist_fm @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_112_Collect__mem__eq,axiom,
! [A4: set_fm] :
( ( collect_fm
@ ^ [X2: fm] : ( member_fm2 @ X2 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_113_Collect__mem__eq,axiom,
! [A4: set_list_fm] :
( ( collect_list_fm
@ ^ [X2: list_fm] : ( member_list_fm2 @ X2 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_114_Collect__mem__eq,axiom,
! [A4: set_tm] :
( ( collect_tm
@ ^ [X2: tm] : ( member_tm2 @ X2 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_115_Collect__mem__eq,axiom,
! [A4: set_Pr5202636777678657877ist_fm] :
( ( collec5845330005437105888ist_fm
@ ^ [X2: produc6018962875968178549ist_fm] : ( member4699826688122452638ist_fm @ X2 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_116_map__ext,axiom,
! [Xs: list_fm,F: fm > list_tm,G: fm > list_tm] :
( ! [X3: fm] :
( ( member_fm2 @ X3 @ ( set_fm2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_fm_list_tm @ F @ Xs )
= ( map_fm_list_tm @ G @ Xs ) ) ) ).
% map_ext
thf(fact_117_map__ext,axiom,
! [Xs: list_tm,F: tm > list_tm,G: tm > list_tm] :
( ! [X3: tm] :
( ( member_tm2 @ X3 @ ( set_tm2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_tm_list_tm @ F @ Xs )
= ( map_tm_list_tm @ G @ Xs ) ) ) ).
% map_ext
thf(fact_118_map__ext,axiom,
! [Xs: list_tm,F: tm > tm,G: tm > tm] :
( ! [X3: tm] :
( ( member_tm2 @ X3 @ ( set_tm2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_tm_tm @ F @ Xs )
= ( map_tm_tm @ G @ Xs ) ) ) ).
% map_ext
thf(fact_119_list_Omap__ident__strong,axiom,
! [T: list_list_fm,F: list_fm > list_fm] :
( ! [Z: list_fm] :
( ( member_list_fm2 @ Z @ ( set_list_fm2 @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_list_fm_list_fm @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_120_list_Omap__ident__strong,axiom,
! [T: list_fm,F: fm > fm] :
( ! [Z: fm] :
( ( member_fm2 @ Z @ ( set_fm2 @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_fm_fm @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_121_list_Omap__ident__strong,axiom,
! [T: list_tm,F: tm > tm] :
( ! [Z: tm] :
( ( member_tm2 @ Z @ ( set_tm2 @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_tm_tm @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_122_list_Omap__ident__strong,axiom,
! [T: list_P2887561121880082555ist_fm,F: produc6018962875968178549ist_fm > produc6018962875968178549ist_fm] :
( ! [Z: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ Z @ ( set_Pr5051287786238754058ist_fm @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_Pr8802613670079714180ist_fm @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_123_list_Oinj__map__strong,axiom,
! [X: list_fm,Xa: list_fm,F: fm > list_tm,Fa: fm > list_tm] :
( ! [Z: fm,Za: fm] :
( ( member_fm2 @ Z @ ( set_fm2 @ X ) )
=> ( ( member_fm2 @ Za @ ( set_fm2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_fm_list_tm @ F @ X )
= ( map_fm_list_tm @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_124_list_Oinj__map__strong,axiom,
! [X: list_tm,Xa: list_tm,F: tm > list_tm,Fa: tm > list_tm] :
( ! [Z: tm,Za: tm] :
( ( member_tm2 @ Z @ ( set_tm2 @ X ) )
=> ( ( member_tm2 @ Za @ ( set_tm2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_tm_list_tm @ F @ X )
= ( map_tm_list_tm @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_125_list_Oinj__map__strong,axiom,
! [X: list_tm,Xa: list_tm,F: tm > tm,Fa: tm > tm] :
( ! [Z: tm,Za: tm] :
( ( member_tm2 @ Z @ ( set_tm2 @ X ) )
=> ( ( member_tm2 @ Za @ ( set_tm2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_tm_tm @ F @ X )
= ( map_tm_tm @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_126_list_Omap__cong0,axiom,
! [X: list_fm,F: fm > list_tm,G: fm > list_tm] :
( ! [Z: fm] :
( ( member_fm2 @ Z @ ( set_fm2 @ X ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_fm_list_tm @ F @ X )
= ( map_fm_list_tm @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_127_list_Omap__cong0,axiom,
! [X: list_tm,F: tm > list_tm,G: tm > list_tm] :
( ! [Z: tm] :
( ( member_tm2 @ Z @ ( set_tm2 @ X ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_tm_list_tm @ F @ X )
= ( map_tm_list_tm @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_128_list_Omap__cong0,axiom,
! [X: list_tm,F: tm > tm,G: tm > tm] :
( ! [Z: tm] :
( ( member_tm2 @ Z @ ( set_tm2 @ X ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_tm_tm @ F @ X )
= ( map_tm_tm @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_129_list_Omap__cong,axiom,
! [X: list_fm,Ya: list_fm,F: fm > list_tm,G: fm > list_tm] :
( ( X = Ya )
=> ( ! [Z: fm] :
( ( member_fm2 @ Z @ ( set_fm2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_fm_list_tm @ F @ X )
= ( map_fm_list_tm @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_130_list_Omap__cong,axiom,
! [X: list_tm,Ya: list_tm,F: tm > list_tm,G: tm > list_tm] :
( ( X = Ya )
=> ( ! [Z: tm] :
( ( member_tm2 @ Z @ ( set_tm2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_tm_list_tm @ F @ X )
= ( map_tm_list_tm @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_131_list_Omap__cong,axiom,
! [X: list_tm,Ya: list_tm,F: tm > tm,G: tm > tm] :
( ( X = Ya )
=> ( ! [Z: tm] :
( ( member_tm2 @ Z @ ( set_tm2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_tm_tm @ F @ X )
= ( map_tm_tm @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_132_map__eq__append__conv,axiom,
! [F: list_tm > tm,Xs: list_list_tm,Ys: list_tm,Zs: list_tm] :
( ( ( map_list_tm_tm @ F @ Xs )
= ( append_tm @ Ys @ Zs ) )
= ( ? [Us: list_list_tm,Vs: list_list_tm] :
( ( Xs
= ( append_list_tm @ Us @ Vs ) )
& ( Ys
= ( map_list_tm_tm @ F @ Us ) )
& ( Zs
= ( map_list_tm_tm @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_133_map__eq__append__conv,axiom,
! [F: fm > tm,Xs: list_fm,Ys: list_tm,Zs: list_tm] :
( ( ( map_fm_tm @ F @ Xs )
= ( append_tm @ Ys @ Zs ) )
= ( ? [Us: list_fm,Vs: list_fm] :
( ( Xs
= ( append_fm @ Us @ Vs ) )
& ( Ys
= ( map_fm_tm @ F @ Us ) )
& ( Zs
= ( map_fm_tm @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_134_map__eq__append__conv,axiom,
! [F: list_tm > list_tm,Xs: list_list_tm,Ys: list_list_tm,Zs: list_list_tm] :
( ( ( map_list_tm_list_tm @ F @ Xs )
= ( append_list_tm @ Ys @ Zs ) )
= ( ? [Us: list_list_tm,Vs: list_list_tm] :
( ( Xs
= ( append_list_tm @ Us @ Vs ) )
& ( Ys
= ( map_list_tm_list_tm @ F @ Us ) )
& ( Zs
= ( map_list_tm_list_tm @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_135_map__eq__append__conv,axiom,
! [F: tm > fm,Xs: list_tm,Ys: list_fm,Zs: list_fm] :
( ( ( map_tm_fm @ F @ Xs )
= ( append_fm @ Ys @ Zs ) )
= ( ? [Us: list_tm,Vs: list_tm] :
( ( Xs
= ( append_tm @ Us @ Vs ) )
& ( Ys
= ( map_tm_fm @ F @ Us ) )
& ( Zs
= ( map_tm_fm @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_136_map__eq__append__conv,axiom,
! [F: list_tm > fm,Xs: list_list_tm,Ys: list_fm,Zs: list_fm] :
( ( ( map_list_tm_fm @ F @ Xs )
= ( append_fm @ Ys @ Zs ) )
= ( ? [Us: list_list_tm,Vs: list_list_tm] :
( ( Xs
= ( append_list_tm @ Us @ Vs ) )
& ( Ys
= ( map_list_tm_fm @ F @ Us ) )
& ( Zs
= ( map_list_tm_fm @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_137_map__eq__append__conv,axiom,
! [F: fm > fm,Xs: list_fm,Ys: list_fm,Zs: list_fm] :
( ( ( map_fm_fm @ F @ Xs )
= ( append_fm @ Ys @ Zs ) )
= ( ? [Us: list_fm,Vs: list_fm] :
( ( Xs
= ( append_fm @ Us @ Vs ) )
& ( Ys
= ( map_fm_fm @ F @ Us ) )
& ( Zs
= ( map_fm_fm @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_138_map__eq__append__conv,axiom,
! [F: fm > list_tm,Xs: list_fm,Ys: list_list_tm,Zs: list_list_tm] :
( ( ( map_fm_list_tm @ F @ Xs )
= ( append_list_tm @ Ys @ Zs ) )
= ( ? [Us: list_fm,Vs: list_fm] :
( ( Xs
= ( append_fm @ Us @ Vs ) )
& ( Ys
= ( map_fm_list_tm @ F @ Us ) )
& ( Zs
= ( map_fm_list_tm @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_139_map__eq__append__conv,axiom,
! [F: tm > list_tm,Xs: list_tm,Ys: list_list_tm,Zs: list_list_tm] :
( ( ( map_tm_list_tm @ F @ Xs )
= ( append_list_tm @ Ys @ Zs ) )
= ( ? [Us: list_tm,Vs: list_tm] :
( ( Xs
= ( append_tm @ Us @ Vs ) )
& ( Ys
= ( map_tm_list_tm @ F @ Us ) )
& ( Zs
= ( map_tm_list_tm @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_140_map__eq__append__conv,axiom,
! [F: tm > tm,Xs: list_tm,Ys: list_tm,Zs: list_tm] :
( ( ( map_tm_tm @ F @ Xs )
= ( append_tm @ Ys @ Zs ) )
= ( ? [Us: list_tm,Vs: list_tm] :
( ( Xs
= ( append_tm @ Us @ Vs ) )
& ( Ys
= ( map_tm_tm @ F @ Us ) )
& ( Zs
= ( map_tm_tm @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_141_append__eq__map__conv,axiom,
! [Ys: list_tm,Zs: list_tm,F: list_tm > tm,Xs: list_list_tm] :
( ( ( append_tm @ Ys @ Zs )
= ( map_list_tm_tm @ F @ Xs ) )
= ( ? [Us: list_list_tm,Vs: list_list_tm] :
( ( Xs
= ( append_list_tm @ Us @ Vs ) )
& ( Ys
= ( map_list_tm_tm @ F @ Us ) )
& ( Zs
= ( map_list_tm_tm @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_142_append__eq__map__conv,axiom,
! [Ys: list_tm,Zs: list_tm,F: fm > tm,Xs: list_fm] :
( ( ( append_tm @ Ys @ Zs )
= ( map_fm_tm @ F @ Xs ) )
= ( ? [Us: list_fm,Vs: list_fm] :
( ( Xs
= ( append_fm @ Us @ Vs ) )
& ( Ys
= ( map_fm_tm @ F @ Us ) )
& ( Zs
= ( map_fm_tm @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_143_append__eq__map__conv,axiom,
! [Ys: list_list_tm,Zs: list_list_tm,F: list_tm > list_tm,Xs: list_list_tm] :
( ( ( append_list_tm @ Ys @ Zs )
= ( map_list_tm_list_tm @ F @ Xs ) )
= ( ? [Us: list_list_tm,Vs: list_list_tm] :
( ( Xs
= ( append_list_tm @ Us @ Vs ) )
& ( Ys
= ( map_list_tm_list_tm @ F @ Us ) )
& ( Zs
= ( map_list_tm_list_tm @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_144_append__eq__map__conv,axiom,
! [Ys: list_fm,Zs: list_fm,F: tm > fm,Xs: list_tm] :
( ( ( append_fm @ Ys @ Zs )
= ( map_tm_fm @ F @ Xs ) )
= ( ? [Us: list_tm,Vs: list_tm] :
( ( Xs
= ( append_tm @ Us @ Vs ) )
& ( Ys
= ( map_tm_fm @ F @ Us ) )
& ( Zs
= ( map_tm_fm @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_145_append__eq__map__conv,axiom,
! [Ys: list_fm,Zs: list_fm,F: list_tm > fm,Xs: list_list_tm] :
( ( ( append_fm @ Ys @ Zs )
= ( map_list_tm_fm @ F @ Xs ) )
= ( ? [Us: list_list_tm,Vs: list_list_tm] :
( ( Xs
= ( append_list_tm @ Us @ Vs ) )
& ( Ys
= ( map_list_tm_fm @ F @ Us ) )
& ( Zs
= ( map_list_tm_fm @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_146_append__eq__map__conv,axiom,
! [Ys: list_fm,Zs: list_fm,F: fm > fm,Xs: list_fm] :
( ( ( append_fm @ Ys @ Zs )
= ( map_fm_fm @ F @ Xs ) )
= ( ? [Us: list_fm,Vs: list_fm] :
( ( Xs
= ( append_fm @ Us @ Vs ) )
& ( Ys
= ( map_fm_fm @ F @ Us ) )
& ( Zs
= ( map_fm_fm @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_147_append__eq__map__conv,axiom,
! [Ys: list_list_tm,Zs: list_list_tm,F: fm > list_tm,Xs: list_fm] :
( ( ( append_list_tm @ Ys @ Zs )
= ( map_fm_list_tm @ F @ Xs ) )
= ( ? [Us: list_fm,Vs: list_fm] :
( ( Xs
= ( append_fm @ Us @ Vs ) )
& ( Ys
= ( map_fm_list_tm @ F @ Us ) )
& ( Zs
= ( map_fm_list_tm @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_148_append__eq__map__conv,axiom,
! [Ys: list_list_tm,Zs: list_list_tm,F: tm > list_tm,Xs: list_tm] :
( ( ( append_list_tm @ Ys @ Zs )
= ( map_tm_list_tm @ F @ Xs ) )
= ( ? [Us: list_tm,Vs: list_tm] :
( ( Xs
= ( append_tm @ Us @ Vs ) )
& ( Ys
= ( map_tm_list_tm @ F @ Us ) )
& ( Zs
= ( map_tm_list_tm @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_149_append__eq__map__conv,axiom,
! [Ys: list_tm,Zs: list_tm,F: tm > tm,Xs: list_tm] :
( ( ( append_tm @ Ys @ Zs )
= ( map_tm_tm @ F @ Xs ) )
= ( ? [Us: list_tm,Vs: list_tm] :
( ( Xs
= ( append_tm @ Us @ Vs ) )
& ( Ys
= ( map_tm_tm @ F @ Us ) )
& ( Zs
= ( map_tm_tm @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_150_map__concat,axiom,
! [F: fm > list_tm,Xs: list_list_fm] :
( ( map_fm_list_tm @ F @ ( concat_fm @ Xs ) )
= ( concat_list_tm @ ( map_li1108997747876207612ist_tm @ ( map_fm_list_tm @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_151_map__concat,axiom,
! [F: tm > list_tm,Xs: list_list_tm] :
( ( map_tm_list_tm @ F @ ( concat_tm @ Xs ) )
= ( concat_list_tm @ ( map_li6264597563971819530ist_tm @ ( map_tm_list_tm @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_152_map__concat,axiom,
! [F: tm > tm,Xs: list_list_tm] :
( ( map_tm_tm @ F @ ( concat_tm @ Xs ) )
= ( concat_tm @ ( map_list_tm_list_tm @ ( map_tm_tm @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_153_ne__effect__not__branchDone,axiom,
! [B4: list_tm,Z2: list_fm,R: rule,A4: list_tm,Z3: list_fm] :
( ( fmembe3754813877001230652ist_fm @ ( produc1414352766439514085ist_fm @ B4 @ Z2 ) @ ( effect @ R @ ( produc1414352766439514085ist_fm @ A4 @ Z3 ) ) )
=> ~ ( branchDone @ Z3 ) ) ).
% ne_effect_not_branchDone
thf(fact_154_children__preserves__unaffected,axiom,
! [P2: fm,Z3: list_fm,R: rule,Z2: list_fm,A4: list_tm] :
( ( member_fm2 @ P2 @ ( set_fm2 @ Z3 ) )
=> ( ~ ( affects @ R @ P2 )
=> ( ( member_list_fm2 @ Z2 @ ( set_list_fm2 @ ( children @ A4 @ R @ Z3 ) ) )
=> ( member_fm2 @ P2 @ ( set_fm2 @ Z2 ) ) ) ) ) ).
% children_preserves_unaffected
thf(fact_155_concat__map__maps,axiom,
! [F: fm > list_tm,Xs: list_fm] :
( ( concat_tm @ ( map_fm_list_tm @ F @ Xs ) )
= ( maps_fm_tm @ F @ Xs ) ) ).
% concat_map_maps
thf(fact_156_concat__map__maps,axiom,
! [F: tm > list_tm,Xs: list_tm] :
( ( concat_tm @ ( map_tm_list_tm @ F @ Xs ) )
= ( maps_tm_tm @ F @ Xs ) ) ).
% concat_map_maps
thf(fact_157_maps__def,axiom,
( maps_fm_tm
= ( ^ [F2: fm > list_tm,Xs3: list_fm] : ( concat_tm @ ( map_fm_list_tm @ F2 @ Xs3 ) ) ) ) ).
% maps_def
thf(fact_158_maps__def,axiom,
( maps_tm_tm
= ( ^ [F2: tm > list_tm,Xs3: list_tm] : ( concat_tm @ ( map_tm_list_tm @ F2 @ Xs3 ) ) ) ) ).
% maps_def
thf(fact_159_List_Obind__def,axiom,
( bind_fm_tm
= ( ^ [Xs3: list_fm,F2: fm > list_tm] : ( concat_tm @ ( map_fm_list_tm @ F2 @ Xs3 ) ) ) ) ).
% List.bind_def
thf(fact_160_List_Obind__def,axiom,
( bind_tm_tm
= ( ^ [Xs3: list_tm,F2: tm > list_tm] : ( concat_tm @ ( map_tm_list_tm @ F2 @ Xs3 ) ) ) ) ).
% List.bind_def
thf(fact_161_map__eq__map__tailrec,axiom,
map_fm_list_tm = map_ta7425747110069464646ist_tm ).
% map_eq_map_tailrec
thf(fact_162_map__eq__map__tailrec,axiom,
map_tm_list_tm = map_ta7538967730773405780ist_tm ).
% map_eq_map_tailrec
thf(fact_163_map__eq__map__tailrec,axiom,
map_tm_tm = map_tailrec_tm_tm ).
% map_eq_map_tailrec
thf(fact_164_lexord__same__pref__iff,axiom,
! [Xs: list_list_tm,Ys: list_list_tm,Zs: list_list_tm,R: set_Pr1507011332596240839ist_tm] :
( ( member1414320910663981712ist_tm @ ( produc8086241248684931031ist_tm @ ( append_list_tm @ Xs @ Ys ) @ ( append_list_tm @ Xs @ Zs ) ) @ ( lexord_list_tm @ R ) )
= ( ? [X2: list_tm] :
( ( member_list_tm @ X2 @ ( set_list_tm2 @ Xs ) )
& ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ X2 @ X2 ) @ R ) )
| ( member1414320910663981712ist_tm @ ( produc8086241248684931031ist_tm @ Ys @ Zs ) @ ( lexord_list_tm @ R ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_165_lexord__same__pref__iff,axiom,
! [Xs: list_list_fm,Ys: list_list_fm,Zs: list_list_fm,R: set_Pr1916137733696349511ist_fm] :
( ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ ( append_list_fm @ Xs @ Ys ) @ ( append_list_fm @ Xs @ Zs ) ) @ ( lexord_list_fm @ R ) )
= ( ? [X2: list_fm] :
( ( member_list_fm2 @ X2 @ ( set_list_fm2 @ Xs ) )
& ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ X2 @ X2 ) @ R ) )
| ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ Ys @ Zs ) @ ( lexord_list_fm @ R ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_166_lexord__same__pref__iff,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_fm,R: set_Pr4706815898642364871_fm_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ ( append_fm @ Xs @ Ys ) @ ( append_fm @ Xs @ Zs ) ) @ ( lexord_fm @ R ) )
= ( ? [X2: fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
& ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X2 @ X2 ) @ R ) )
| ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Ys @ Zs ) @ ( lexord_fm @ R ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_167_lexord__same__pref__iff,axiom,
! [Xs: list_tm,Ys: list_tm,Zs: list_tm,R: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( append_tm @ Xs @ Ys ) @ ( append_tm @ Xs @ Zs ) ) @ ( lexord_tm @ R ) )
= ( ? [X2: tm] :
( ( member_tm2 @ X2 @ ( set_tm2 @ Xs ) )
& ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X2 @ X2 ) @ R ) )
| ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Ys @ Zs ) @ ( lexord_tm @ R ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_168_lexord__same__pref__iff,axiom,
! [Xs: list_P2887561121880082555ist_fm,Ys: list_P2887561121880082555ist_fm,Zs: list_P2887561121880082555ist_fm,R: set_Pr6724411253394807111ist_fm] :
( ( member6238937682488541200ist_fm @ ( produc8740291170414093911ist_fm @ ( append1476581234975291920ist_fm @ Xs @ Ys ) @ ( append1476581234975291920ist_fm @ Xs @ Zs ) ) @ ( lexord4892917800681400680ist_fm @ R ) )
= ( ? [X2: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ X2 @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
& ( member858341428514850704ist_fm @ ( produc5488031402154692311ist_fm @ X2 @ X2 ) @ R ) )
| ( member6238937682488541200ist_fm @ ( produc8740291170414093911ist_fm @ Ys @ Zs ) @ ( lexord4892917800681400680ist_fm @ R ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_169_lex__append__left__iff,axiom,
! [R: set_Pr2455929065695642951_tm_tm,Xs: list_tm,Ys: list_tm,Zs: list_tm] :
( ! [X3: tm] :
~ ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X3 @ X3 ) @ R )
=> ( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( append_tm @ Xs @ Ys ) @ ( append_tm @ Xs @ Zs ) ) @ ( lex_tm @ R ) )
= ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Ys @ Zs ) @ ( lex_tm @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_170_lex__append__left__iff,axiom,
! [R: set_Pr1507011332596240839ist_tm,Xs: list_list_tm,Ys: list_list_tm,Zs: list_list_tm] :
( ! [X3: list_tm] :
~ ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ X3 @ X3 ) @ R )
=> ( ( member1414320910663981712ist_tm @ ( produc8086241248684931031ist_tm @ ( append_list_tm @ Xs @ Ys ) @ ( append_list_tm @ Xs @ Zs ) ) @ ( lex_list_tm @ R ) )
= ( member1414320910663981712ist_tm @ ( produc8086241248684931031ist_tm @ Ys @ Zs ) @ ( lex_list_tm @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_171_lex__append__left__iff,axiom,
! [R: set_Pr4706815898642364871_fm_fm,Xs: list_fm,Ys: list_fm,Zs: list_fm] :
( ! [X3: fm] :
~ ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X3 @ X3 ) @ R )
=> ( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ ( append_fm @ Xs @ Ys ) @ ( append_fm @ Xs @ Zs ) ) @ ( lex_fm @ R ) )
= ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Ys @ Zs ) @ ( lex_fm @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_172_lex__append__leftD,axiom,
! [R: set_Pr2455929065695642951_tm_tm,Xs: list_tm,Ys: list_tm,Zs: list_tm] :
( ! [X3: tm] :
~ ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X3 @ X3 ) @ R )
=> ( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( append_tm @ Xs @ Ys ) @ ( append_tm @ Xs @ Zs ) ) @ ( lex_tm @ R ) )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Ys @ Zs ) @ ( lex_tm @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_173_lex__append__leftD,axiom,
! [R: set_Pr1507011332596240839ist_tm,Xs: list_list_tm,Ys: list_list_tm,Zs: list_list_tm] :
( ! [X3: list_tm] :
~ ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ X3 @ X3 ) @ R )
=> ( ( member1414320910663981712ist_tm @ ( produc8086241248684931031ist_tm @ ( append_list_tm @ Xs @ Ys ) @ ( append_list_tm @ Xs @ Zs ) ) @ ( lex_list_tm @ R ) )
=> ( member1414320910663981712ist_tm @ ( produc8086241248684931031ist_tm @ Ys @ Zs ) @ ( lex_list_tm @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_174_lex__append__leftD,axiom,
! [R: set_Pr4706815898642364871_fm_fm,Xs: list_fm,Ys: list_fm,Zs: list_fm] :
( ! [X3: fm] :
~ ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X3 @ X3 ) @ R )
=> ( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ ( append_fm @ Xs @ Ys ) @ ( append_fm @ Xs @ Zs ) ) @ ( lex_fm @ R ) )
=> ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Ys @ Zs ) @ ( lex_fm @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_175_lexord__append__leftI,axiom,
! [U: list_tm,V: list_tm,R: set_Pr2455929065695642951_tm_tm,X: list_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ U @ V ) @ ( lexord_tm @ R ) )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( append_tm @ X @ U ) @ ( append_tm @ X @ V ) ) @ ( lexord_tm @ R ) ) ) ).
% lexord_append_leftI
thf(fact_176_lexord__append__leftI,axiom,
! [U: list_list_tm,V: list_list_tm,R: set_Pr1507011332596240839ist_tm,X: list_list_tm] :
( ( member1414320910663981712ist_tm @ ( produc8086241248684931031ist_tm @ U @ V ) @ ( lexord_list_tm @ R ) )
=> ( member1414320910663981712ist_tm @ ( produc8086241248684931031ist_tm @ ( append_list_tm @ X @ U ) @ ( append_list_tm @ X @ V ) ) @ ( lexord_list_tm @ R ) ) ) ).
% lexord_append_leftI
thf(fact_177_lexord__append__leftI,axiom,
! [U: list_fm,V: list_fm,R: set_Pr4706815898642364871_fm_fm,X: list_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ U @ V ) @ ( lexord_fm @ R ) )
=> ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ ( append_fm @ X @ U ) @ ( append_fm @ X @ V ) ) @ ( lexord_fm @ R ) ) ) ).
% lexord_append_leftI
thf(fact_178_lex__append__leftI,axiom,
! [Ys: list_tm,Zs: list_tm,R: set_Pr2455929065695642951_tm_tm,Xs: list_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Ys @ Zs ) @ ( lex_tm @ R ) )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( append_tm @ Xs @ Ys ) @ ( append_tm @ Xs @ Zs ) ) @ ( lex_tm @ R ) ) ) ).
% lex_append_leftI
thf(fact_179_lex__append__leftI,axiom,
! [Ys: list_list_tm,Zs: list_list_tm,R: set_Pr1507011332596240839ist_tm,Xs: list_list_tm] :
( ( member1414320910663981712ist_tm @ ( produc8086241248684931031ist_tm @ Ys @ Zs ) @ ( lex_list_tm @ R ) )
=> ( member1414320910663981712ist_tm @ ( produc8086241248684931031ist_tm @ ( append_list_tm @ Xs @ Ys ) @ ( append_list_tm @ Xs @ Zs ) ) @ ( lex_list_tm @ R ) ) ) ).
% lex_append_leftI
thf(fact_180_lex__append__leftI,axiom,
! [Ys: list_fm,Zs: list_fm,R: set_Pr4706815898642364871_fm_fm,Xs: list_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Ys @ Zs ) @ ( lex_fm @ R ) )
=> ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ ( append_fm @ Xs @ Ys ) @ ( append_fm @ Xs @ Zs ) ) @ ( lex_fm @ R ) ) ) ).
% lex_append_leftI
thf(fact_181_lexord__partial__trans,axiom,
! [Xs: list_list_fm,R: set_Pr1916137733696349511ist_fm,Ys: list_list_fm,Zs: list_list_fm] :
( ! [X3: list_fm,Y: list_fm,Z: list_fm] :
( ( member_list_fm2 @ X3 @ ( set_list_fm2 @ Xs ) )
=> ( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ X3 @ Y ) @ R )
=> ( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Y @ Z ) @ R )
=> ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ X3 @ Z ) @ R ) ) ) )
=> ( ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ Xs @ Ys ) @ ( lexord_list_fm @ R ) )
=> ( ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ Ys @ Zs ) @ ( lexord_list_fm @ R ) )
=> ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ Xs @ Zs ) @ ( lexord_list_fm @ R ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_182_lexord__partial__trans,axiom,
! [Xs: list_fm,R: set_Pr4706815898642364871_fm_fm,Ys: list_fm,Zs: list_fm] :
( ! [X3: fm,Y: fm,Z: fm] :
( ( member_fm2 @ X3 @ ( set_fm2 @ Xs ) )
=> ( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X3 @ Y ) @ R )
=> ( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ Y @ Z ) @ R )
=> ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X3 @ Z ) @ R ) ) ) )
=> ( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Xs @ Ys ) @ ( lexord_fm @ R ) )
=> ( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Ys @ Zs ) @ ( lexord_fm @ R ) )
=> ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Xs @ Zs ) @ ( lexord_fm @ R ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_183_lexord__partial__trans,axiom,
! [Xs: list_tm,R: set_Pr2455929065695642951_tm_tm,Ys: list_tm,Zs: list_tm] :
( ! [X3: tm,Y: tm,Z: tm] :
( ( member_tm2 @ X3 @ ( set_tm2 @ Xs ) )
=> ( ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X3 @ Y ) @ R )
=> ( ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ Y @ Z ) @ R )
=> ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X3 @ Z ) @ R ) ) ) )
=> ( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ Ys ) @ ( lexord_tm @ R ) )
=> ( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Ys @ Zs ) @ ( lexord_tm @ R ) )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ Zs ) @ ( lexord_tm @ R ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_184_lexord__partial__trans,axiom,
! [Xs: list_P2887561121880082555ist_fm,R: set_Pr6724411253394807111ist_fm,Ys: list_P2887561121880082555ist_fm,Zs: list_P2887561121880082555ist_fm] :
( ! [X3: produc6018962875968178549ist_fm,Y: produc6018962875968178549ist_fm,Z: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ X3 @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
=> ( ( member858341428514850704ist_fm @ ( produc5488031402154692311ist_fm @ X3 @ Y ) @ R )
=> ( ( member858341428514850704ist_fm @ ( produc5488031402154692311ist_fm @ Y @ Z ) @ R )
=> ( member858341428514850704ist_fm @ ( produc5488031402154692311ist_fm @ X3 @ Z ) @ R ) ) ) )
=> ( ( member6238937682488541200ist_fm @ ( produc8740291170414093911ist_fm @ Xs @ Ys ) @ ( lexord4892917800681400680ist_fm @ R ) )
=> ( ( member6238937682488541200ist_fm @ ( produc8740291170414093911ist_fm @ Ys @ Zs ) @ ( lexord4892917800681400680ist_fm @ R ) )
=> ( member6238937682488541200ist_fm @ ( produc8740291170414093911ist_fm @ Xs @ Zs ) @ ( lexord4892917800681400680ist_fm @ R ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_185_lexord__append__leftD,axiom,
! [X: list_tm,U: list_tm,V: list_tm,R: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( append_tm @ X @ U ) @ ( append_tm @ X @ V ) ) @ ( lexord_tm @ R ) )
=> ( ! [A3: tm] :
~ ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ A3 @ A3 ) @ R )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ U @ V ) @ ( lexord_tm @ R ) ) ) ) ).
% lexord_append_leftD
thf(fact_186_lexord__append__leftD,axiom,
! [X: list_list_tm,U: list_list_tm,V: list_list_tm,R: set_Pr1507011332596240839ist_tm] :
( ( member1414320910663981712ist_tm @ ( produc8086241248684931031ist_tm @ ( append_list_tm @ X @ U ) @ ( append_list_tm @ X @ V ) ) @ ( lexord_list_tm @ R ) )
=> ( ! [A3: list_tm] :
~ ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ A3 @ A3 ) @ R )
=> ( member1414320910663981712ist_tm @ ( produc8086241248684931031ist_tm @ U @ V ) @ ( lexord_list_tm @ R ) ) ) ) ).
% lexord_append_leftD
thf(fact_187_lexord__append__leftD,axiom,
! [X: list_fm,U: list_fm,V: list_fm,R: set_Pr4706815898642364871_fm_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ ( append_fm @ X @ U ) @ ( append_fm @ X @ V ) ) @ ( lexord_fm @ R ) )
=> ( ! [A3: fm] :
~ ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ A3 @ A3 ) @ R )
=> ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ U @ V ) @ ( lexord_fm @ R ) ) ) ) ).
% lexord_append_leftD
thf(fact_188_effect__preserves__unaffected,axiom,
! [P2: fm,Z3: list_fm,R: rule,B4: list_tm,Z2: list_fm,A4: list_tm] :
( ( member_fm2 @ P2 @ ( set_fm2 @ Z3 ) )
=> ( ~ ( affects @ R @ P2 )
=> ( ( fmembe3754813877001230652ist_fm @ ( produc1414352766439514085ist_fm @ B4 @ Z2 ) @ ( effect @ R @ ( produc1414352766439514085ist_fm @ A4 @ Z3 ) ) )
=> ( member_fm2 @ P2 @ ( set_fm2 @ Z2 ) ) ) ) ) ).
% effect_preserves_unaffected
thf(fact_189_parts__preserves__unaffected,axiom,
! [R: rule,P2: fm,Z2: list_fm,A4: list_tm] :
( ~ ( affects @ R @ P2 )
=> ( ( member_list_fm2 @ Z2 @ ( set_list_fm2 @ ( parts @ A4 @ R @ P2 ) ) )
=> ( member_fm2 @ P2 @ ( set_fm2 @ Z2 ) ) ) ) ).
% parts_preserves_unaffected
thf(fact_190_lexord__append__left__rightI,axiom,
! [A: list_tm,B: list_tm,R: set_Pr1507011332596240839ist_tm,U: list_list_tm,X: list_list_tm,Y3: list_list_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ A @ B ) @ R )
=> ( member1414320910663981712ist_tm @ ( produc8086241248684931031ist_tm @ ( append_list_tm @ U @ ( cons_list_tm @ A @ X ) ) @ ( append_list_tm @ U @ ( cons_list_tm @ B @ Y3 ) ) ) @ ( lexord_list_tm @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_191_lexord__append__left__rightI,axiom,
! [A: fm,B: fm,R: set_Pr4706815898642364871_fm_fm,U: list_fm,X: list_fm,Y3: list_fm] :
( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ A @ B ) @ R )
=> ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ ( append_fm @ U @ ( cons_fm @ A @ X ) ) @ ( append_fm @ U @ ( cons_fm @ B @ Y3 ) ) ) @ ( lexord_fm @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_192_lexord__append__left__rightI,axiom,
! [A: list_fm,B: list_fm,R: set_Pr1916137733696349511ist_fm,U: list_list_fm,X: list_list_fm,Y3: list_list_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ A @ B ) @ R )
=> ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ ( append_list_fm @ U @ ( cons_list_fm @ A @ X ) ) @ ( append_list_fm @ U @ ( cons_list_fm @ B @ Y3 ) ) ) @ ( lexord_list_fm @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_193_lexord__append__left__rightI,axiom,
! [A: tm,B: tm,R: set_Pr2455929065695642951_tm_tm,U: list_tm,X: list_tm,Y3: list_tm] :
( ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ A @ B ) @ R )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( append_tm @ U @ ( cons_tm @ A @ X ) ) @ ( append_tm @ U @ ( cons_tm @ B @ Y3 ) ) ) @ ( lexord_tm @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_194_lexord__cons__cons,axiom,
! [A: list_fm,X: list_list_fm,B: list_fm,Y3: list_list_fm,R: set_Pr1916137733696349511ist_fm] :
( ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ ( cons_list_fm @ A @ X ) @ ( cons_list_fm @ B @ Y3 ) ) @ ( lexord_list_fm @ R ) )
= ( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ A @ B ) @ R )
| ( ( A = B )
& ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ X @ Y3 ) @ ( lexord_list_fm @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_195_lexord__cons__cons,axiom,
! [A: tm,X: list_tm,B: tm,Y3: list_tm,R: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( cons_tm @ A @ X ) @ ( cons_tm @ B @ Y3 ) ) @ ( lexord_tm @ R ) )
= ( ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ A @ B ) @ R )
| ( ( A = B )
& ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ X @ Y3 ) @ ( lexord_tm @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_196_partition__P,axiom,
! [P: list_fm > $o,Xs: list_list_fm,Yes: list_list_fm,No: list_list_fm] :
( ( ( partition_list_fm @ P @ Xs )
= ( produc8321651870839017815ist_fm @ Yes @ No ) )
=> ( ! [X4: list_fm] :
( ( member_list_fm2 @ X4 @ ( set_list_fm2 @ Yes ) )
=> ( P @ X4 ) )
& ! [X4: list_fm] :
( ( member_list_fm2 @ X4 @ ( set_list_fm2 @ No ) )
=> ~ ( P @ X4 ) ) ) ) ).
% partition_P
thf(fact_197_partition__P,axiom,
! [P: fm > $o,Xs: list_fm,Yes: list_fm,No: list_fm] :
( ( ( partition_fm @ P @ Xs )
= ( produc7863996417982153943ist_fm @ Yes @ No ) )
=> ( ! [X4: fm] :
( ( member_fm2 @ X4 @ ( set_fm2 @ Yes ) )
=> ( P @ X4 ) )
& ! [X4: fm] :
( ( member_fm2 @ X4 @ ( set_fm2 @ No ) )
=> ~ ( P @ X4 ) ) ) ) ).
% partition_P
thf(fact_198_partition__P,axiom,
! [P: tm > $o,Xs: list_tm,Yes: list_tm,No: list_tm] :
( ( ( partition_tm @ P @ Xs )
= ( produc1418304791525149271ist_tm @ Yes @ No ) )
=> ( ! [X4: tm] :
( ( member_tm2 @ X4 @ ( set_tm2 @ Yes ) )
=> ( P @ X4 ) )
& ! [X4: tm] :
( ( member_tm2 @ X4 @ ( set_tm2 @ No ) )
=> ~ ( P @ X4 ) ) ) ) ).
% partition_P
thf(fact_199_partition__P,axiom,
! [P: produc6018962875968178549ist_fm > $o,Xs: list_P2887561121880082555ist_fm,Yes: list_P2887561121880082555ist_fm,No: list_P2887561121880082555ist_fm] :
( ( ( partit1016815737701346804ist_fm @ P @ Xs )
= ( produc8740291170414093911ist_fm @ Yes @ No ) )
=> ( ! [X4: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ X4 @ ( set_Pr5051287786238754058ist_fm @ Yes ) )
=> ( P @ X4 ) )
& ! [X4: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ X4 @ ( set_Pr5051287786238754058ist_fm @ No ) )
=> ~ ( P @ X4 ) ) ) ) ).
% partition_P
thf(fact_200_lexord__append__rightI,axiom,
! [Y3: list_list_tm,X: list_list_tm,R: set_Pr1507011332596240839ist_tm] :
( ? [B5: list_tm,Z4: list_list_tm] :
( Y3
= ( cons_list_tm @ B5 @ Z4 ) )
=> ( member1414320910663981712ist_tm @ ( produc8086241248684931031ist_tm @ X @ ( append_list_tm @ X @ Y3 ) ) @ ( lexord_list_tm @ R ) ) ) ).
% lexord_append_rightI
thf(fact_201_lexord__append__rightI,axiom,
! [Y3: list_fm,X: list_fm,R: set_Pr4706815898642364871_fm_fm] :
( ? [B5: fm,Z4: list_fm] :
( Y3
= ( cons_fm @ B5 @ Z4 ) )
=> ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ X @ ( append_fm @ X @ Y3 ) ) @ ( lexord_fm @ R ) ) ) ).
% lexord_append_rightI
thf(fact_202_lexord__append__rightI,axiom,
! [Y3: list_list_fm,X: list_list_fm,R: set_Pr1916137733696349511ist_fm] :
( ? [B5: list_fm,Z4: list_list_fm] :
( Y3
= ( cons_list_fm @ B5 @ Z4 ) )
=> ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ X @ ( append_list_fm @ X @ Y3 ) ) @ ( lexord_list_fm @ R ) ) ) ).
% lexord_append_rightI
thf(fact_203_lexord__append__rightI,axiom,
! [Y3: list_tm,X: list_tm,R: set_Pr2455929065695642951_tm_tm] :
( ? [B5: tm,Z4: list_tm] :
( Y3
= ( cons_tm @ B5 @ Z4 ) )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ X @ ( append_tm @ X @ Y3 ) ) @ ( lexord_tm @ R ) ) ) ).
% lexord_append_rightI
thf(fact_204_lex__append__rightI,axiom,
! [Xs: list_tm,Ys: list_tm,R: set_Pr2455929065695642951_tm_tm,Vs2: list_tm,Us2: list_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ Ys ) @ ( lex_tm @ R ) )
=> ( ( ( size_size_list_tm @ Vs2 )
= ( size_size_list_tm @ Us2 ) )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( append_tm @ Xs @ Us2 ) @ ( append_tm @ Ys @ Vs2 ) ) @ ( lex_tm @ R ) ) ) ) ).
% lex_append_rightI
thf(fact_205_lex__append__rightI,axiom,
! [Xs: list_list_tm,Ys: list_list_tm,R: set_Pr1507011332596240839ist_tm,Vs2: list_list_tm,Us2: list_list_tm] :
( ( member1414320910663981712ist_tm @ ( produc8086241248684931031ist_tm @ Xs @ Ys ) @ ( lex_list_tm @ R ) )
=> ( ( ( size_s9096087352182575069ist_tm @ Vs2 )
= ( size_s9096087352182575069ist_tm @ Us2 ) )
=> ( member1414320910663981712ist_tm @ ( produc8086241248684931031ist_tm @ ( append_list_tm @ Xs @ Us2 ) @ ( append_list_tm @ Ys @ Vs2 ) ) @ ( lex_list_tm @ R ) ) ) ) ).
% lex_append_rightI
thf(fact_206_lex__append__rightI,axiom,
! [Xs: list_fm,Ys: list_fm,R: set_Pr4706815898642364871_fm_fm,Vs2: list_fm,Us2: list_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Xs @ Ys ) @ ( lex_fm @ R ) )
=> ( ( ( size_size_list_fm @ Vs2 )
= ( size_size_list_fm @ Us2 ) )
=> ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ ( append_fm @ Xs @ Us2 ) @ ( append_fm @ Ys @ Vs2 ) ) @ ( lex_fm @ R ) ) ) ) ).
% lex_append_rightI
thf(fact_207_map__fun__upd,axiom,
! [Y3: fm,Xs: list_fm,F: fm > list_tm,V: list_tm] :
( ~ ( member_fm2 @ Y3 @ ( set_fm2 @ Xs ) )
=> ( ( map_fm_list_tm @ ( fun_upd_fm_list_tm @ F @ Y3 @ V ) @ Xs )
= ( map_fm_list_tm @ F @ Xs ) ) ) ).
% map_fun_upd
thf(fact_208_map__fun__upd,axiom,
! [Y3: tm,Xs: list_tm,F: tm > list_tm,V: list_tm] :
( ~ ( member_tm2 @ Y3 @ ( set_tm2 @ Xs ) )
=> ( ( map_tm_list_tm @ ( fun_upd_tm_list_tm @ F @ Y3 @ V ) @ Xs )
= ( map_tm_list_tm @ F @ Xs ) ) ) ).
% map_fun_upd
thf(fact_209_map__fun__upd,axiom,
! [Y3: tm,Xs: list_tm,F: tm > tm,V: tm] :
( ~ ( member_tm2 @ Y3 @ ( set_tm2 @ Xs ) )
=> ( ( map_tm_tm @ ( fun_upd_tm_tm @ F @ Y3 @ V ) @ Xs )
= ( map_tm_tm @ F @ Xs ) ) ) ).
% map_fun_upd
thf(fact_210_lexord__sufE,axiom,
! [Xs: list_tm,Zs: list_tm,Ys: list_tm,Qs: list_tm,R: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( append_tm @ Xs @ Zs ) @ ( append_tm @ Ys @ Qs ) ) @ ( lexord_tm @ R ) )
=> ( ( Xs != Ys )
=> ( ( ( size_size_list_tm @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( ( size_size_list_tm @ Zs )
= ( size_size_list_tm @ Qs ) )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ Ys ) @ ( lexord_tm @ R ) ) ) ) ) ) ).
% lexord_sufE
thf(fact_211_lexord__sufE,axiom,
! [Xs: list_list_tm,Zs: list_list_tm,Ys: list_list_tm,Qs: list_list_tm,R: set_Pr1507011332596240839ist_tm] :
( ( member1414320910663981712ist_tm @ ( produc8086241248684931031ist_tm @ ( append_list_tm @ Xs @ Zs ) @ ( append_list_tm @ Ys @ Qs ) ) @ ( lexord_list_tm @ R ) )
=> ( ( Xs != Ys )
=> ( ( ( size_s9096087352182575069ist_tm @ Xs )
= ( size_s9096087352182575069ist_tm @ Ys ) )
=> ( ( ( size_s9096087352182575069ist_tm @ Zs )
= ( size_s9096087352182575069ist_tm @ Qs ) )
=> ( member1414320910663981712ist_tm @ ( produc8086241248684931031ist_tm @ Xs @ Ys ) @ ( lexord_list_tm @ R ) ) ) ) ) ) ).
% lexord_sufE
thf(fact_212_lexord__sufE,axiom,
! [Xs: list_fm,Zs: list_fm,Ys: list_fm,Qs: list_fm,R: set_Pr4706815898642364871_fm_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ ( append_fm @ Xs @ Zs ) @ ( append_fm @ Ys @ Qs ) ) @ ( lexord_fm @ R ) )
=> ( ( Xs != Ys )
=> ( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Zs )
= ( size_size_list_fm @ Qs ) )
=> ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Xs @ Ys ) @ ( lexord_fm @ R ) ) ) ) ) ) ).
% lexord_sufE
thf(fact_213_in__set__member,axiom,
! [X: list_fm,Xs: list_list_fm] :
( ( member_list_fm2 @ X @ ( set_list_fm2 @ Xs ) )
= ( member_list_fm @ Xs @ X ) ) ).
% in_set_member
thf(fact_214_in__set__member,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
= ( member_fm @ Xs @ X ) ) ).
% in_set_member
thf(fact_215_in__set__member,axiom,
! [X: tm,Xs: list_tm] :
( ( member_tm2 @ X @ ( set_tm2 @ Xs ) )
= ( member_tm @ Xs @ X ) ) ).
% in_set_member
thf(fact_216_in__set__member,axiom,
! [X: produc6018962875968178549ist_fm,Xs: list_P2887561121880082555ist_fm] :
( ( member4699826688122452638ist_fm @ X @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
= ( member9193610466889212688ist_fm @ Xs @ X ) ) ).
% in_set_member
thf(fact_217_list_Oinject,axiom,
! [X21: list_fm,X222: list_list_fm,Y21: list_fm,Y22: list_list_fm] :
( ( ( cons_list_fm @ X21 @ X222 )
= ( cons_list_fm @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_218_list_Oinject,axiom,
! [X21: tm,X222: list_tm,Y21: tm,Y22: list_tm] :
( ( ( cons_tm @ X21 @ X222 )
= ( cons_tm @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_219_append__eq__append__conv,axiom,
! [Xs: list_tm,Ys: list_tm,Us2: list_tm,Vs2: list_tm] :
( ( ( ( size_size_list_tm @ Xs )
= ( size_size_list_tm @ Ys ) )
| ( ( size_size_list_tm @ Us2 )
= ( size_size_list_tm @ Vs2 ) ) )
=> ( ( ( append_tm @ Xs @ Us2 )
= ( append_tm @ Ys @ Vs2 ) )
= ( ( Xs = Ys )
& ( Us2 = Vs2 ) ) ) ) ).
% append_eq_append_conv
thf(fact_220_append__eq__append__conv,axiom,
! [Xs: list_list_tm,Ys: list_list_tm,Us2: list_list_tm,Vs2: list_list_tm] :
( ( ( ( size_s9096087352182575069ist_tm @ Xs )
= ( size_s9096087352182575069ist_tm @ Ys ) )
| ( ( size_s9096087352182575069ist_tm @ Us2 )
= ( size_s9096087352182575069ist_tm @ Vs2 ) ) )
=> ( ( ( append_list_tm @ Xs @ Us2 )
= ( append_list_tm @ Ys @ Vs2 ) )
= ( ( Xs = Ys )
& ( Us2 = Vs2 ) ) ) ) ).
% append_eq_append_conv
thf(fact_221_append__eq__append__conv,axiom,
! [Xs: list_fm,Ys: list_fm,Us2: list_fm,Vs2: list_fm] :
( ( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
| ( ( size_size_list_fm @ Us2 )
= ( size_size_list_fm @ Vs2 ) ) )
=> ( ( ( append_fm @ Xs @ Us2 )
= ( append_fm @ Ys @ Vs2 ) )
= ( ( Xs = Ys )
& ( Us2 = Vs2 ) ) ) ) ).
% append_eq_append_conv
thf(fact_222_length__map,axiom,
! [F: fm > list_tm,Xs: list_fm] :
( ( size_s9096087352182575069ist_tm @ ( map_fm_list_tm @ F @ Xs ) )
= ( size_size_list_fm @ Xs ) ) ).
% length_map
thf(fact_223_length__map,axiom,
! [F: tm > list_tm,Xs: list_tm] :
( ( size_s9096087352182575069ist_tm @ ( map_tm_list_tm @ F @ Xs ) )
= ( size_size_list_tm @ Xs ) ) ).
% length_map
thf(fact_224_length__map,axiom,
! [F: tm > tm,Xs: list_tm] :
( ( size_size_list_tm @ ( map_tm_tm @ F @ Xs ) )
= ( size_size_list_tm @ Xs ) ) ).
% length_map
thf(fact_225_length__remdups__eq,axiom,
! [Xs: list_tm] :
( ( ( size_size_list_tm @ ( remdups_tm @ Xs ) )
= ( size_size_list_tm @ Xs ) )
= ( ( remdups_tm @ Xs )
= Xs ) ) ).
% length_remdups_eq
thf(fact_226_length__remdups__eq,axiom,
! [Xs: list_fm] :
( ( ( size_size_list_fm @ ( remdups_fm @ Xs ) )
= ( size_size_list_fm @ Xs ) )
= ( ( remdups_fm @ Xs )
= Xs ) ) ).
% length_remdups_eq
thf(fact_227_length__remdups__eq,axiom,
! [Xs: list_list_fm] :
( ( ( size_s115229985653309035ist_fm @ ( remdups_list_fm @ Xs ) )
= ( size_s115229985653309035ist_fm @ Xs ) )
= ( ( remdups_list_fm @ Xs )
= Xs ) ) ).
% length_remdups_eq
thf(fact_228_bind__simps_I2_J,axiom,
! [X: list_fm,Xs: list_list_fm,F: list_fm > list_tm] :
( ( bind_list_fm_tm @ ( cons_list_fm @ X @ Xs ) @ F )
= ( append_tm @ ( F @ X ) @ ( bind_list_fm_tm @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_229_bind__simps_I2_J,axiom,
! [X: list_fm,Xs: list_list_fm,F: list_fm > list_list_tm] :
( ( bind_list_fm_list_tm @ ( cons_list_fm @ X @ Xs ) @ F )
= ( append_list_tm @ ( F @ X ) @ ( bind_list_fm_list_tm @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_230_bind__simps_I2_J,axiom,
! [X: list_fm,Xs: list_list_fm,F: list_fm > list_fm] :
( ( bind_list_fm_fm @ ( cons_list_fm @ X @ Xs ) @ F )
= ( append_fm @ ( F @ X ) @ ( bind_list_fm_fm @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_231_bind__simps_I2_J,axiom,
! [X: tm,Xs: list_tm,F: tm > list_tm] :
( ( bind_tm_tm @ ( cons_tm @ X @ Xs ) @ F )
= ( append_tm @ ( F @ X ) @ ( bind_tm_tm @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_232_bind__simps_I2_J,axiom,
! [X: tm,Xs: list_tm,F: tm > list_list_tm] :
( ( bind_tm_list_tm @ ( cons_tm @ X @ Xs ) @ F )
= ( append_list_tm @ ( F @ X ) @ ( bind_tm_list_tm @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_233_bind__simps_I2_J,axiom,
! [X: tm,Xs: list_tm,F: tm > list_fm] :
( ( bind_tm_fm @ ( cons_tm @ X @ Xs ) @ F )
= ( append_fm @ ( F @ X ) @ ( bind_tm_fm @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_234_Cons__in__lex,axiom,
! [X: list_fm,Xs: list_list_fm,Y3: list_fm,Ys: list_list_fm,R: set_Pr1916137733696349511ist_fm] :
( ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ ( cons_list_fm @ X @ Xs ) @ ( cons_list_fm @ Y3 @ Ys ) ) @ ( lex_list_fm @ R ) )
= ( ( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ X @ Y3 ) @ R )
& ( ( size_s115229985653309035ist_fm @ Xs )
= ( size_s115229985653309035ist_fm @ Ys ) ) )
| ( ( X = Y3 )
& ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ Xs @ Ys ) @ ( lex_list_fm @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_235_Cons__in__lex,axiom,
! [X: tm,Xs: list_tm,Y3: tm,Ys: list_tm,R: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( cons_tm @ X @ Xs ) @ ( cons_tm @ Y3 @ Ys ) ) @ ( lex_tm @ R ) )
= ( ( ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X @ Y3 ) @ R )
& ( ( size_size_list_tm @ Xs )
= ( size_size_list_tm @ Ys ) ) )
| ( ( X = Y3 )
& ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ Ys ) @ ( lex_tm @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_236_not__Cons__self2,axiom,
! [X: list_fm,Xs: list_list_fm] :
( ( cons_list_fm @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_237_not__Cons__self2,axiom,
! [X: tm,Xs: list_tm] :
( ( cons_tm @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_238_member__rec_I1_J,axiom,
! [X: list_fm,Xs: list_list_fm,Y3: list_fm] :
( ( member_list_fm @ ( cons_list_fm @ X @ Xs ) @ Y3 )
= ( ( X = Y3 )
| ( member_list_fm @ Xs @ Y3 ) ) ) ).
% member_rec(1)
thf(fact_239_member__rec_I1_J,axiom,
! [X: tm,Xs: list_tm,Y3: tm] :
( ( member_tm @ ( cons_tm @ X @ Xs ) @ Y3 )
= ( ( X = Y3 )
| ( member_tm @ Xs @ Y3 ) ) ) ).
% member_rec(1)
thf(fact_240_map__eq__imp__length__eq,axiom,
! [F: fm > list_tm,Xs: list_fm,G: fm > list_tm,Ys: list_fm] :
( ( ( map_fm_list_tm @ F @ Xs )
= ( map_fm_list_tm @ G @ Ys ) )
=> ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_241_map__eq__imp__length__eq,axiom,
! [F: fm > list_tm,Xs: list_fm,G: tm > list_tm,Ys: list_tm] :
( ( ( map_fm_list_tm @ F @ Xs )
= ( map_tm_list_tm @ G @ Ys ) )
=> ( ( size_size_list_fm @ Xs )
= ( size_size_list_tm @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_242_map__eq__imp__length__eq,axiom,
! [F: tm > list_tm,Xs: list_tm,G: fm > list_tm,Ys: list_fm] :
( ( ( map_tm_list_tm @ F @ Xs )
= ( map_fm_list_tm @ G @ Ys ) )
=> ( ( size_size_list_tm @ Xs )
= ( size_size_list_fm @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_243_map__eq__imp__length__eq,axiom,
! [F: tm > list_tm,Xs: list_tm,G: tm > list_tm,Ys: list_tm] :
( ( ( map_tm_list_tm @ F @ Xs )
= ( map_tm_list_tm @ G @ Ys ) )
=> ( ( size_size_list_tm @ Xs )
= ( size_size_list_tm @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_244_map__eq__imp__length__eq,axiom,
! [F: tm > tm,Xs: list_tm,G: tm > tm,Ys: list_tm] :
( ( ( map_tm_tm @ F @ Xs )
= ( map_tm_tm @ G @ Ys ) )
=> ( ( size_size_list_tm @ Xs )
= ( size_size_list_tm @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_245_set__ConsD,axiom,
! [Y3: fm,X: fm,Xs: list_fm] :
( ( member_fm2 @ Y3 @ ( set_fm2 @ ( cons_fm @ X @ Xs ) ) )
=> ( ( Y3 = X )
| ( member_fm2 @ Y3 @ ( set_fm2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_246_set__ConsD,axiom,
! [Y3: produc6018962875968178549ist_fm,X: produc6018962875968178549ist_fm,Xs: list_P2887561121880082555ist_fm] :
( ( member4699826688122452638ist_fm @ Y3 @ ( set_Pr5051287786238754058ist_fm @ ( cons_P6586734674047340843ist_fm @ X @ Xs ) ) )
=> ( ( Y3 = X )
| ( member4699826688122452638ist_fm @ Y3 @ ( set_Pr5051287786238754058ist_fm @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_247_set__ConsD,axiom,
! [Y3: list_fm,X: list_fm,Xs: list_list_fm] :
( ( member_list_fm2 @ Y3 @ ( set_list_fm2 @ ( cons_list_fm @ X @ Xs ) ) )
=> ( ( Y3 = X )
| ( member_list_fm2 @ Y3 @ ( set_list_fm2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_248_set__ConsD,axiom,
! [Y3: tm,X: tm,Xs: list_tm] :
( ( member_tm2 @ Y3 @ ( set_tm2 @ ( cons_tm @ X @ Xs ) ) )
=> ( ( Y3 = X )
| ( member_tm2 @ Y3 @ ( set_tm2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_249_list_Oset__cases,axiom,
! [E: fm,A: list_fm] :
( ( member_fm2 @ E @ ( set_fm2 @ A ) )
=> ( ! [Z22: list_fm] :
( A
!= ( cons_fm @ E @ Z22 ) )
=> ~ ! [Z1: fm,Z22: list_fm] :
( ( A
= ( cons_fm @ Z1 @ Z22 ) )
=> ~ ( member_fm2 @ E @ ( set_fm2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_250_list_Oset__cases,axiom,
! [E: produc6018962875968178549ist_fm,A: list_P2887561121880082555ist_fm] :
( ( member4699826688122452638ist_fm @ E @ ( set_Pr5051287786238754058ist_fm @ A ) )
=> ( ! [Z22: list_P2887561121880082555ist_fm] :
( A
!= ( cons_P6586734674047340843ist_fm @ E @ Z22 ) )
=> ~ ! [Z1: produc6018962875968178549ist_fm,Z22: list_P2887561121880082555ist_fm] :
( ( A
= ( cons_P6586734674047340843ist_fm @ Z1 @ Z22 ) )
=> ~ ( member4699826688122452638ist_fm @ E @ ( set_Pr5051287786238754058ist_fm @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_251_list_Oset__cases,axiom,
! [E: list_fm,A: list_list_fm] :
( ( member_list_fm2 @ E @ ( set_list_fm2 @ A ) )
=> ( ! [Z22: list_list_fm] :
( A
!= ( cons_list_fm @ E @ Z22 ) )
=> ~ ! [Z1: list_fm,Z22: list_list_fm] :
( ( A
= ( cons_list_fm @ Z1 @ Z22 ) )
=> ~ ( member_list_fm2 @ E @ ( set_list_fm2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_252_list_Oset__cases,axiom,
! [E: tm,A: list_tm] :
( ( member_tm2 @ E @ ( set_tm2 @ A ) )
=> ( ! [Z22: list_tm] :
( A
!= ( cons_tm @ E @ Z22 ) )
=> ~ ! [Z1: tm,Z22: list_tm] :
( ( A
= ( cons_tm @ Z1 @ Z22 ) )
=> ~ ( member_tm2 @ E @ ( set_tm2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_253_list_Oset__intros_I1_J,axiom,
! [X21: fm,X222: list_fm] : ( member_fm2 @ X21 @ ( set_fm2 @ ( cons_fm @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_254_list_Oset__intros_I1_J,axiom,
! [X21: produc6018962875968178549ist_fm,X222: list_P2887561121880082555ist_fm] : ( member4699826688122452638ist_fm @ X21 @ ( set_Pr5051287786238754058ist_fm @ ( cons_P6586734674047340843ist_fm @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_255_list_Oset__intros_I1_J,axiom,
! [X21: list_fm,X222: list_list_fm] : ( member_list_fm2 @ X21 @ ( set_list_fm2 @ ( cons_list_fm @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_256_list_Oset__intros_I1_J,axiom,
! [X21: tm,X222: list_tm] : ( member_tm2 @ X21 @ ( set_tm2 @ ( cons_tm @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_257_list_Oset__intros_I2_J,axiom,
! [Y3: fm,X222: list_fm,X21: fm] :
( ( member_fm2 @ Y3 @ ( set_fm2 @ X222 ) )
=> ( member_fm2 @ Y3 @ ( set_fm2 @ ( cons_fm @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_258_list_Oset__intros_I2_J,axiom,
! [Y3: produc6018962875968178549ist_fm,X222: list_P2887561121880082555ist_fm,X21: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ Y3 @ ( set_Pr5051287786238754058ist_fm @ X222 ) )
=> ( member4699826688122452638ist_fm @ Y3 @ ( set_Pr5051287786238754058ist_fm @ ( cons_P6586734674047340843ist_fm @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_259_list_Oset__intros_I2_J,axiom,
! [Y3: list_fm,X222: list_list_fm,X21: list_fm] :
( ( member_list_fm2 @ Y3 @ ( set_list_fm2 @ X222 ) )
=> ( member_list_fm2 @ Y3 @ ( set_list_fm2 @ ( cons_list_fm @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_260_list_Oset__intros_I2_J,axiom,
! [Y3: tm,X222: list_tm,X21: tm] :
( ( member_tm2 @ Y3 @ ( set_tm2 @ X222 ) )
=> ( member_tm2 @ Y3 @ ( set_tm2 @ ( cons_tm @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_261_Cons__eq__appendI,axiom,
! [X: list_tm,Xs1: list_list_tm,Ys: list_list_tm,Xs: list_list_tm,Zs: list_list_tm] :
( ( ( cons_list_tm @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_list_tm @ Xs1 @ Zs ) )
=> ( ( cons_list_tm @ X @ Xs )
= ( append_list_tm @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_262_Cons__eq__appendI,axiom,
! [X: fm,Xs1: list_fm,Ys: list_fm,Xs: list_fm,Zs: list_fm] :
( ( ( cons_fm @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_fm @ Xs1 @ Zs ) )
=> ( ( cons_fm @ X @ Xs )
= ( append_fm @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_263_Cons__eq__appendI,axiom,
! [X: list_fm,Xs1: list_list_fm,Ys: list_list_fm,Xs: list_list_fm,Zs: list_list_fm] :
( ( ( cons_list_fm @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_list_fm @ Xs1 @ Zs ) )
=> ( ( cons_list_fm @ X @ Xs )
= ( append_list_fm @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_264_Cons__eq__appendI,axiom,
! [X: tm,Xs1: list_tm,Ys: list_tm,Xs: list_tm,Zs: list_tm] :
( ( ( cons_tm @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_tm @ Xs1 @ Zs ) )
=> ( ( cons_tm @ X @ Xs )
= ( append_tm @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_265_append__Cons,axiom,
! [X: list_tm,Xs: list_list_tm,Ys: list_list_tm] :
( ( append_list_tm @ ( cons_list_tm @ X @ Xs ) @ Ys )
= ( cons_list_tm @ X @ ( append_list_tm @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_266_append__Cons,axiom,
! [X: fm,Xs: list_fm,Ys: list_fm] :
( ( append_fm @ ( cons_fm @ X @ Xs ) @ Ys )
= ( cons_fm @ X @ ( append_fm @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_267_append__Cons,axiom,
! [X: list_fm,Xs: list_list_fm,Ys: list_list_fm] :
( ( append_list_fm @ ( cons_list_fm @ X @ Xs ) @ Ys )
= ( cons_list_fm @ X @ ( append_list_fm @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_268_append__Cons,axiom,
! [X: tm,Xs: list_tm,Ys: list_tm] :
( ( append_tm @ ( cons_tm @ X @ Xs ) @ Ys )
= ( cons_tm @ X @ ( append_tm @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_269_map__eq__Cons__conv,axiom,
! [F: fm > list_tm,Xs: list_fm,Y3: list_tm,Ys: list_list_tm] :
( ( ( map_fm_list_tm @ F @ Xs )
= ( cons_list_tm @ Y3 @ Ys ) )
= ( ? [Z5: fm,Zs2: list_fm] :
( ( Xs
= ( cons_fm @ Z5 @ Zs2 ) )
& ( ( F @ Z5 )
= Y3 )
& ( ( map_fm_list_tm @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_270_map__eq__Cons__conv,axiom,
! [F: tm > list_tm,Xs: list_tm,Y3: list_tm,Ys: list_list_tm] :
( ( ( map_tm_list_tm @ F @ Xs )
= ( cons_list_tm @ Y3 @ Ys ) )
= ( ? [Z5: tm,Zs2: list_tm] :
( ( Xs
= ( cons_tm @ Z5 @ Zs2 ) )
& ( ( F @ Z5 )
= Y3 )
& ( ( map_tm_list_tm @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_271_map__eq__Cons__conv,axiom,
! [F: list_fm > list_fm,Xs: list_list_fm,Y3: list_fm,Ys: list_list_fm] :
( ( ( map_list_fm_list_fm @ F @ Xs )
= ( cons_list_fm @ Y3 @ Ys ) )
= ( ? [Z5: list_fm,Zs2: list_list_fm] :
( ( Xs
= ( cons_list_fm @ Z5 @ Zs2 ) )
& ( ( F @ Z5 )
= Y3 )
& ( ( map_list_fm_list_fm @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_272_map__eq__Cons__conv,axiom,
! [F: tm > list_fm,Xs: list_tm,Y3: list_fm,Ys: list_list_fm] :
( ( ( map_tm_list_fm @ F @ Xs )
= ( cons_list_fm @ Y3 @ Ys ) )
= ( ? [Z5: tm,Zs2: list_tm] :
( ( Xs
= ( cons_tm @ Z5 @ Zs2 ) )
& ( ( F @ Z5 )
= Y3 )
& ( ( map_tm_list_fm @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_273_map__eq__Cons__conv,axiom,
! [F: list_fm > tm,Xs: list_list_fm,Y3: tm,Ys: list_tm] :
( ( ( map_list_fm_tm @ F @ Xs )
= ( cons_tm @ Y3 @ Ys ) )
= ( ? [Z5: list_fm,Zs2: list_list_fm] :
( ( Xs
= ( cons_list_fm @ Z5 @ Zs2 ) )
& ( ( F @ Z5 )
= Y3 )
& ( ( map_list_fm_tm @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_274_map__eq__Cons__conv,axiom,
! [F: tm > tm,Xs: list_tm,Y3: tm,Ys: list_tm] :
( ( ( map_tm_tm @ F @ Xs )
= ( cons_tm @ Y3 @ Ys ) )
= ( ? [Z5: tm,Zs2: list_tm] :
( ( Xs
= ( cons_tm @ Z5 @ Zs2 ) )
& ( ( F @ Z5 )
= Y3 )
& ( ( map_tm_tm @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_275_Cons__eq__map__conv,axiom,
! [X: list_tm,Xs: list_list_tm,F: fm > list_tm,Ys: list_fm] :
( ( ( cons_list_tm @ X @ Xs )
= ( map_fm_list_tm @ F @ Ys ) )
= ( ? [Z5: fm,Zs2: list_fm] :
( ( Ys
= ( cons_fm @ Z5 @ Zs2 ) )
& ( X
= ( F @ Z5 ) )
& ( Xs
= ( map_fm_list_tm @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_276_Cons__eq__map__conv,axiom,
! [X: list_tm,Xs: list_list_tm,F: tm > list_tm,Ys: list_tm] :
( ( ( cons_list_tm @ X @ Xs )
= ( map_tm_list_tm @ F @ Ys ) )
= ( ? [Z5: tm,Zs2: list_tm] :
( ( Ys
= ( cons_tm @ Z5 @ Zs2 ) )
& ( X
= ( F @ Z5 ) )
& ( Xs
= ( map_tm_list_tm @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_277_Cons__eq__map__conv,axiom,
! [X: list_fm,Xs: list_list_fm,F: list_fm > list_fm,Ys: list_list_fm] :
( ( ( cons_list_fm @ X @ Xs )
= ( map_list_fm_list_fm @ F @ Ys ) )
= ( ? [Z5: list_fm,Zs2: list_list_fm] :
( ( Ys
= ( cons_list_fm @ Z5 @ Zs2 ) )
& ( X
= ( F @ Z5 ) )
& ( Xs
= ( map_list_fm_list_fm @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_278_Cons__eq__map__conv,axiom,
! [X: list_fm,Xs: list_list_fm,F: tm > list_fm,Ys: list_tm] :
( ( ( cons_list_fm @ X @ Xs )
= ( map_tm_list_fm @ F @ Ys ) )
= ( ? [Z5: tm,Zs2: list_tm] :
( ( Ys
= ( cons_tm @ Z5 @ Zs2 ) )
& ( X
= ( F @ Z5 ) )
& ( Xs
= ( map_tm_list_fm @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_279_Cons__eq__map__conv,axiom,
! [X: tm,Xs: list_tm,F: list_fm > tm,Ys: list_list_fm] :
( ( ( cons_tm @ X @ Xs )
= ( map_list_fm_tm @ F @ Ys ) )
= ( ? [Z5: list_fm,Zs2: list_list_fm] :
( ( Ys
= ( cons_list_fm @ Z5 @ Zs2 ) )
& ( X
= ( F @ Z5 ) )
& ( Xs
= ( map_list_fm_tm @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_280_Cons__eq__map__conv,axiom,
! [X: tm,Xs: list_tm,F: tm > tm,Ys: list_tm] :
( ( ( cons_tm @ X @ Xs )
= ( map_tm_tm @ F @ Ys ) )
= ( ? [Z5: tm,Zs2: list_tm] :
( ( Ys
= ( cons_tm @ Z5 @ Zs2 ) )
& ( X
= ( F @ Z5 ) )
& ( Xs
= ( map_tm_tm @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_281_map__eq__Cons__D,axiom,
! [F: fm > list_tm,Xs: list_fm,Y3: list_tm,Ys: list_list_tm] :
( ( ( map_fm_list_tm @ F @ Xs )
= ( cons_list_tm @ Y3 @ Ys ) )
=> ? [Z: fm,Zs3: list_fm] :
( ( Xs
= ( cons_fm @ Z @ Zs3 ) )
& ( ( F @ Z )
= Y3 )
& ( ( map_fm_list_tm @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_282_map__eq__Cons__D,axiom,
! [F: tm > list_tm,Xs: list_tm,Y3: list_tm,Ys: list_list_tm] :
( ( ( map_tm_list_tm @ F @ Xs )
= ( cons_list_tm @ Y3 @ Ys ) )
=> ? [Z: tm,Zs3: list_tm] :
( ( Xs
= ( cons_tm @ Z @ Zs3 ) )
& ( ( F @ Z )
= Y3 )
& ( ( map_tm_list_tm @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_283_map__eq__Cons__D,axiom,
! [F: list_fm > list_fm,Xs: list_list_fm,Y3: list_fm,Ys: list_list_fm] :
( ( ( map_list_fm_list_fm @ F @ Xs )
= ( cons_list_fm @ Y3 @ Ys ) )
=> ? [Z: list_fm,Zs3: list_list_fm] :
( ( Xs
= ( cons_list_fm @ Z @ Zs3 ) )
& ( ( F @ Z )
= Y3 )
& ( ( map_list_fm_list_fm @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_284_map__eq__Cons__D,axiom,
! [F: tm > list_fm,Xs: list_tm,Y3: list_fm,Ys: list_list_fm] :
( ( ( map_tm_list_fm @ F @ Xs )
= ( cons_list_fm @ Y3 @ Ys ) )
=> ? [Z: tm,Zs3: list_tm] :
( ( Xs
= ( cons_tm @ Z @ Zs3 ) )
& ( ( F @ Z )
= Y3 )
& ( ( map_tm_list_fm @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_285_map__eq__Cons__D,axiom,
! [F: list_fm > tm,Xs: list_list_fm,Y3: tm,Ys: list_tm] :
( ( ( map_list_fm_tm @ F @ Xs )
= ( cons_tm @ Y3 @ Ys ) )
=> ? [Z: list_fm,Zs3: list_list_fm] :
( ( Xs
= ( cons_list_fm @ Z @ Zs3 ) )
& ( ( F @ Z )
= Y3 )
& ( ( map_list_fm_tm @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_286_map__eq__Cons__D,axiom,
! [F: tm > tm,Xs: list_tm,Y3: tm,Ys: list_tm] :
( ( ( map_tm_tm @ F @ Xs )
= ( cons_tm @ Y3 @ Ys ) )
=> ? [Z: tm,Zs3: list_tm] :
( ( Xs
= ( cons_tm @ Z @ Zs3 ) )
& ( ( F @ Z )
= Y3 )
& ( ( map_tm_tm @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_287_Cons__eq__map__D,axiom,
! [X: list_tm,Xs: list_list_tm,F: fm > list_tm,Ys: list_fm] :
( ( ( cons_list_tm @ X @ Xs )
= ( map_fm_list_tm @ F @ Ys ) )
=> ? [Z: fm,Zs3: list_fm] :
( ( Ys
= ( cons_fm @ Z @ Zs3 ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_fm_list_tm @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_288_Cons__eq__map__D,axiom,
! [X: list_tm,Xs: list_list_tm,F: tm > list_tm,Ys: list_tm] :
( ( ( cons_list_tm @ X @ Xs )
= ( map_tm_list_tm @ F @ Ys ) )
=> ? [Z: tm,Zs3: list_tm] :
( ( Ys
= ( cons_tm @ Z @ Zs3 ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_tm_list_tm @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_289_Cons__eq__map__D,axiom,
! [X: list_fm,Xs: list_list_fm,F: list_fm > list_fm,Ys: list_list_fm] :
( ( ( cons_list_fm @ X @ Xs )
= ( map_list_fm_list_fm @ F @ Ys ) )
=> ? [Z: list_fm,Zs3: list_list_fm] :
( ( Ys
= ( cons_list_fm @ Z @ Zs3 ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_list_fm_list_fm @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_290_Cons__eq__map__D,axiom,
! [X: list_fm,Xs: list_list_fm,F: tm > list_fm,Ys: list_tm] :
( ( ( cons_list_fm @ X @ Xs )
= ( map_tm_list_fm @ F @ Ys ) )
=> ? [Z: tm,Zs3: list_tm] :
( ( Ys
= ( cons_tm @ Z @ Zs3 ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_tm_list_fm @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_291_Cons__eq__map__D,axiom,
! [X: tm,Xs: list_tm,F: list_fm > tm,Ys: list_list_fm] :
( ( ( cons_tm @ X @ Xs )
= ( map_list_fm_tm @ F @ Ys ) )
=> ? [Z: list_fm,Zs3: list_list_fm] :
( ( Ys
= ( cons_list_fm @ Z @ Zs3 ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_list_fm_tm @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_292_Cons__eq__map__D,axiom,
! [X: tm,Xs: list_tm,F: tm > tm,Ys: list_tm] :
( ( ( cons_tm @ X @ Xs )
= ( map_tm_tm @ F @ Ys ) )
=> ? [Z: tm,Zs3: list_tm] :
( ( Ys
= ( cons_tm @ Z @ Zs3 ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_tm_tm @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_293_list_Osimps_I9_J,axiom,
! [F: fm > list_tm,X21: fm,X222: list_fm] :
( ( map_fm_list_tm @ F @ ( cons_fm @ X21 @ X222 ) )
= ( cons_list_tm @ ( F @ X21 ) @ ( map_fm_list_tm @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_294_list_Osimps_I9_J,axiom,
! [F: list_fm > list_fm,X21: list_fm,X222: list_list_fm] :
( ( map_list_fm_list_fm @ F @ ( cons_list_fm @ X21 @ X222 ) )
= ( cons_list_fm @ ( F @ X21 ) @ ( map_list_fm_list_fm @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_295_list_Osimps_I9_J,axiom,
! [F: list_fm > tm,X21: list_fm,X222: list_list_fm] :
( ( map_list_fm_tm @ F @ ( cons_list_fm @ X21 @ X222 ) )
= ( cons_tm @ ( F @ X21 ) @ ( map_list_fm_tm @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_296_list_Osimps_I9_J,axiom,
! [F: tm > list_tm,X21: tm,X222: list_tm] :
( ( map_tm_list_tm @ F @ ( cons_tm @ X21 @ X222 ) )
= ( cons_list_tm @ ( F @ X21 ) @ ( map_tm_list_tm @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_297_list_Osimps_I9_J,axiom,
! [F: tm > list_fm,X21: tm,X222: list_tm] :
( ( map_tm_list_fm @ F @ ( cons_tm @ X21 @ X222 ) )
= ( cons_list_fm @ ( F @ X21 ) @ ( map_tm_list_fm @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_298_list_Osimps_I9_J,axiom,
! [F: tm > tm,X21: tm,X222: list_tm] :
( ( map_tm_tm @ F @ ( cons_tm @ X21 @ X222 ) )
= ( cons_tm @ ( F @ X21 ) @ ( map_tm_tm @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_299_split__list__first__prop__iff,axiom,
! [Xs: list_list_tm,P: list_tm > $o] :
( ( ? [X2: list_tm] :
( ( member_list_tm @ X2 @ ( set_list_tm2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys2: list_list_tm,X2: list_tm] :
( ? [Zs2: list_list_tm] :
( Xs
= ( append_list_tm @ Ys2 @ ( cons_list_tm @ X2 @ Zs2 ) ) )
& ( P @ X2 )
& ! [Y4: list_tm] :
( ( member_list_tm @ Y4 @ ( set_list_tm2 @ Ys2 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_300_split__list__first__prop__iff,axiom,
! [Xs: list_fm,P: fm > $o] :
( ( ? [X2: fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys2: list_fm,X2: fm] :
( ? [Zs2: list_fm] :
( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X2 @ Zs2 ) ) )
& ( P @ X2 )
& ! [Y4: fm] :
( ( member_fm2 @ Y4 @ ( set_fm2 @ Ys2 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_301_split__list__first__prop__iff,axiom,
! [Xs: list_P2887561121880082555ist_fm,P: produc6018962875968178549ist_fm > $o] :
( ( ? [X2: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ X2 @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys2: list_P2887561121880082555ist_fm,X2: produc6018962875968178549ist_fm] :
( ? [Zs2: list_P2887561121880082555ist_fm] :
( Xs
= ( append1476581234975291920ist_fm @ Ys2 @ ( cons_P6586734674047340843ist_fm @ X2 @ Zs2 ) ) )
& ( P @ X2 )
& ! [Y4: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ Y4 @ ( set_Pr5051287786238754058ist_fm @ Ys2 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_302_split__list__first__prop__iff,axiom,
! [Xs: list_list_fm,P: list_fm > $o] :
( ( ? [X2: list_fm] :
( ( member_list_fm2 @ X2 @ ( set_list_fm2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys2: list_list_fm,X2: list_fm] :
( ? [Zs2: list_list_fm] :
( Xs
= ( append_list_fm @ Ys2 @ ( cons_list_fm @ X2 @ Zs2 ) ) )
& ( P @ X2 )
& ! [Y4: list_fm] :
( ( member_list_fm2 @ Y4 @ ( set_list_fm2 @ Ys2 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_303_split__list__first__prop__iff,axiom,
! [Xs: list_tm,P: tm > $o] :
( ( ? [X2: tm] :
( ( member_tm2 @ X2 @ ( set_tm2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys2: list_tm,X2: tm] :
( ? [Zs2: list_tm] :
( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X2 @ Zs2 ) ) )
& ( P @ X2 )
& ! [Y4: tm] :
( ( member_tm2 @ Y4 @ ( set_tm2 @ Ys2 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_304_split__list__last__prop__iff,axiom,
! [Xs: list_list_tm,P: list_tm > $o] :
( ( ? [X2: list_tm] :
( ( member_list_tm @ X2 @ ( set_list_tm2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys2: list_list_tm,X2: list_tm,Zs2: list_list_tm] :
( ( Xs
= ( append_list_tm @ Ys2 @ ( cons_list_tm @ X2 @ Zs2 ) ) )
& ( P @ X2 )
& ! [Y4: list_tm] :
( ( member_list_tm @ Y4 @ ( set_list_tm2 @ Zs2 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_305_split__list__last__prop__iff,axiom,
! [Xs: list_fm,P: fm > $o] :
( ( ? [X2: fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys2: list_fm,X2: fm,Zs2: list_fm] :
( ( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X2 @ Zs2 ) ) )
& ( P @ X2 )
& ! [Y4: fm] :
( ( member_fm2 @ Y4 @ ( set_fm2 @ Zs2 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_306_split__list__last__prop__iff,axiom,
! [Xs: list_P2887561121880082555ist_fm,P: produc6018962875968178549ist_fm > $o] :
( ( ? [X2: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ X2 @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys2: list_P2887561121880082555ist_fm,X2: produc6018962875968178549ist_fm,Zs2: list_P2887561121880082555ist_fm] :
( ( Xs
= ( append1476581234975291920ist_fm @ Ys2 @ ( cons_P6586734674047340843ist_fm @ X2 @ Zs2 ) ) )
& ( P @ X2 )
& ! [Y4: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ Y4 @ ( set_Pr5051287786238754058ist_fm @ Zs2 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_307_split__list__last__prop__iff,axiom,
! [Xs: list_list_fm,P: list_fm > $o] :
( ( ? [X2: list_fm] :
( ( member_list_fm2 @ X2 @ ( set_list_fm2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys2: list_list_fm,X2: list_fm,Zs2: list_list_fm] :
( ( Xs
= ( append_list_fm @ Ys2 @ ( cons_list_fm @ X2 @ Zs2 ) ) )
& ( P @ X2 )
& ! [Y4: list_fm] :
( ( member_list_fm2 @ Y4 @ ( set_list_fm2 @ Zs2 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_308_split__list__last__prop__iff,axiom,
! [Xs: list_tm,P: tm > $o] :
( ( ? [X2: tm] :
( ( member_tm2 @ X2 @ ( set_tm2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys2: list_tm,X2: tm,Zs2: list_tm] :
( ( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X2 @ Zs2 ) ) )
& ( P @ X2 )
& ! [Y4: tm] :
( ( member_tm2 @ Y4 @ ( set_tm2 @ Zs2 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_309_in__set__conv__decomp__first,axiom,
! [X: list_tm,Xs: list_list_tm] :
( ( member_list_tm @ X @ ( set_list_tm2 @ Xs ) )
= ( ? [Ys2: list_list_tm,Zs2: list_list_tm] :
( ( Xs
= ( append_list_tm @ Ys2 @ ( cons_list_tm @ X @ Zs2 ) ) )
& ~ ( member_list_tm @ X @ ( set_list_tm2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_310_in__set__conv__decomp__first,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
= ( ? [Ys2: list_fm,Zs2: list_fm] :
( ( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X @ Zs2 ) ) )
& ~ ( member_fm2 @ X @ ( set_fm2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_311_in__set__conv__decomp__first,axiom,
! [X: produc6018962875968178549ist_fm,Xs: list_P2887561121880082555ist_fm] :
( ( member4699826688122452638ist_fm @ X @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
= ( ? [Ys2: list_P2887561121880082555ist_fm,Zs2: list_P2887561121880082555ist_fm] :
( ( Xs
= ( append1476581234975291920ist_fm @ Ys2 @ ( cons_P6586734674047340843ist_fm @ X @ Zs2 ) ) )
& ~ ( member4699826688122452638ist_fm @ X @ ( set_Pr5051287786238754058ist_fm @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_312_in__set__conv__decomp__first,axiom,
! [X: list_fm,Xs: list_list_fm] :
( ( member_list_fm2 @ X @ ( set_list_fm2 @ Xs ) )
= ( ? [Ys2: list_list_fm,Zs2: list_list_fm] :
( ( Xs
= ( append_list_fm @ Ys2 @ ( cons_list_fm @ X @ Zs2 ) ) )
& ~ ( member_list_fm2 @ X @ ( set_list_fm2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_313_in__set__conv__decomp__first,axiom,
! [X: tm,Xs: list_tm] :
( ( member_tm2 @ X @ ( set_tm2 @ Xs ) )
= ( ? [Ys2: list_tm,Zs2: list_tm] :
( ( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X @ Zs2 ) ) )
& ~ ( member_tm2 @ X @ ( set_tm2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_314_in__set__conv__decomp__last,axiom,
! [X: list_tm,Xs: list_list_tm] :
( ( member_list_tm @ X @ ( set_list_tm2 @ Xs ) )
= ( ? [Ys2: list_list_tm,Zs2: list_list_tm] :
( ( Xs
= ( append_list_tm @ Ys2 @ ( cons_list_tm @ X @ Zs2 ) ) )
& ~ ( member_list_tm @ X @ ( set_list_tm2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_315_in__set__conv__decomp__last,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
= ( ? [Ys2: list_fm,Zs2: list_fm] :
( ( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X @ Zs2 ) ) )
& ~ ( member_fm2 @ X @ ( set_fm2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_316_in__set__conv__decomp__last,axiom,
! [X: produc6018962875968178549ist_fm,Xs: list_P2887561121880082555ist_fm] :
( ( member4699826688122452638ist_fm @ X @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
= ( ? [Ys2: list_P2887561121880082555ist_fm,Zs2: list_P2887561121880082555ist_fm] :
( ( Xs
= ( append1476581234975291920ist_fm @ Ys2 @ ( cons_P6586734674047340843ist_fm @ X @ Zs2 ) ) )
& ~ ( member4699826688122452638ist_fm @ X @ ( set_Pr5051287786238754058ist_fm @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_317_in__set__conv__decomp__last,axiom,
! [X: list_fm,Xs: list_list_fm] :
( ( member_list_fm2 @ X @ ( set_list_fm2 @ Xs ) )
= ( ? [Ys2: list_list_fm,Zs2: list_list_fm] :
( ( Xs
= ( append_list_fm @ Ys2 @ ( cons_list_fm @ X @ Zs2 ) ) )
& ~ ( member_list_fm2 @ X @ ( set_list_fm2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_318_in__set__conv__decomp__last,axiom,
! [X: tm,Xs: list_tm] :
( ( member_tm2 @ X @ ( set_tm2 @ Xs ) )
= ( ? [Ys2: list_tm,Zs2: list_tm] :
( ( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X @ Zs2 ) ) )
& ~ ( member_tm2 @ X @ ( set_tm2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_319_split__list__first__propE,axiom,
! [Xs: list_list_tm,P: list_tm > $o] :
( ? [X4: list_tm] :
( ( member_list_tm @ X4 @ ( set_list_tm2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_list_tm,X3: list_tm] :
( ? [Zs3: list_list_tm] :
( Xs
= ( append_list_tm @ Ys3 @ ( cons_list_tm @ X3 @ Zs3 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: list_tm] :
( ( member_list_tm @ Xa2 @ ( set_list_tm2 @ Ys3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_320_split__list__first__propE,axiom,
! [Xs: list_fm,P: fm > $o] :
( ? [X4: fm] :
( ( member_fm2 @ X4 @ ( set_fm2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_fm,X3: fm] :
( ? [Zs3: list_fm] :
( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X3 @ Zs3 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: fm] :
( ( member_fm2 @ Xa2 @ ( set_fm2 @ Ys3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_321_split__list__first__propE,axiom,
! [Xs: list_P2887561121880082555ist_fm,P: produc6018962875968178549ist_fm > $o] :
( ? [X4: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ X4 @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_P2887561121880082555ist_fm,X3: produc6018962875968178549ist_fm] :
( ? [Zs3: list_P2887561121880082555ist_fm] :
( Xs
= ( append1476581234975291920ist_fm @ Ys3 @ ( cons_P6586734674047340843ist_fm @ X3 @ Zs3 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ Xa2 @ ( set_Pr5051287786238754058ist_fm @ Ys3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_322_split__list__first__propE,axiom,
! [Xs: list_list_fm,P: list_fm > $o] :
( ? [X4: list_fm] :
( ( member_list_fm2 @ X4 @ ( set_list_fm2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_list_fm,X3: list_fm] :
( ? [Zs3: list_list_fm] :
( Xs
= ( append_list_fm @ Ys3 @ ( cons_list_fm @ X3 @ Zs3 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: list_fm] :
( ( member_list_fm2 @ Xa2 @ ( set_list_fm2 @ Ys3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_323_split__list__first__propE,axiom,
! [Xs: list_tm,P: tm > $o] :
( ? [X4: tm] :
( ( member_tm2 @ X4 @ ( set_tm2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_tm,X3: tm] :
( ? [Zs3: list_tm] :
( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ X3 @ Zs3 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: tm] :
( ( member_tm2 @ Xa2 @ ( set_tm2 @ Ys3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_324_split__list__last__propE,axiom,
! [Xs: list_list_tm,P: list_tm > $o] :
( ? [X4: list_tm] :
( ( member_list_tm @ X4 @ ( set_list_tm2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_list_tm,X3: list_tm,Zs3: list_list_tm] :
( ( Xs
= ( append_list_tm @ Ys3 @ ( cons_list_tm @ X3 @ Zs3 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: list_tm] :
( ( member_list_tm @ Xa2 @ ( set_list_tm2 @ Zs3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_325_split__list__last__propE,axiom,
! [Xs: list_fm,P: fm > $o] :
( ? [X4: fm] :
( ( member_fm2 @ X4 @ ( set_fm2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_fm,X3: fm,Zs3: list_fm] :
( ( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X3 @ Zs3 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: fm] :
( ( member_fm2 @ Xa2 @ ( set_fm2 @ Zs3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_326_split__list__last__propE,axiom,
! [Xs: list_P2887561121880082555ist_fm,P: produc6018962875968178549ist_fm > $o] :
( ? [X4: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ X4 @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_P2887561121880082555ist_fm,X3: produc6018962875968178549ist_fm,Zs3: list_P2887561121880082555ist_fm] :
( ( Xs
= ( append1476581234975291920ist_fm @ Ys3 @ ( cons_P6586734674047340843ist_fm @ X3 @ Zs3 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ Xa2 @ ( set_Pr5051287786238754058ist_fm @ Zs3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_327_split__list__last__propE,axiom,
! [Xs: list_list_fm,P: list_fm > $o] :
( ? [X4: list_fm] :
( ( member_list_fm2 @ X4 @ ( set_list_fm2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_list_fm,X3: list_fm,Zs3: list_list_fm] :
( ( Xs
= ( append_list_fm @ Ys3 @ ( cons_list_fm @ X3 @ Zs3 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: list_fm] :
( ( member_list_fm2 @ Xa2 @ ( set_list_fm2 @ Zs3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_328_split__list__last__propE,axiom,
! [Xs: list_tm,P: tm > $o] :
( ? [X4: tm] :
( ( member_tm2 @ X4 @ ( set_tm2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_tm,X3: tm,Zs3: list_tm] :
( ( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ X3 @ Zs3 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa2: tm] :
( ( member_tm2 @ Xa2 @ ( set_tm2 @ Zs3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_329_split__list__first__prop,axiom,
! [Xs: list_list_tm,P: list_tm > $o] :
( ? [X4: list_tm] :
( ( member_list_tm @ X4 @ ( set_list_tm2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_list_tm,X3: list_tm] :
( ? [Zs3: list_list_tm] :
( Xs
= ( append_list_tm @ Ys3 @ ( cons_list_tm @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Xa2: list_tm] :
( ( member_list_tm @ Xa2 @ ( set_list_tm2 @ Ys3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_330_split__list__first__prop,axiom,
! [Xs: list_fm,P: fm > $o] :
( ? [X4: fm] :
( ( member_fm2 @ X4 @ ( set_fm2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_fm,X3: fm] :
( ? [Zs3: list_fm] :
( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Xa2: fm] :
( ( member_fm2 @ Xa2 @ ( set_fm2 @ Ys3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_331_split__list__first__prop,axiom,
! [Xs: list_P2887561121880082555ist_fm,P: produc6018962875968178549ist_fm > $o] :
( ? [X4: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ X4 @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_P2887561121880082555ist_fm,X3: produc6018962875968178549ist_fm] :
( ? [Zs3: list_P2887561121880082555ist_fm] :
( Xs
= ( append1476581234975291920ist_fm @ Ys3 @ ( cons_P6586734674047340843ist_fm @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Xa2: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ Xa2 @ ( set_Pr5051287786238754058ist_fm @ Ys3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_332_split__list__first__prop,axiom,
! [Xs: list_list_fm,P: list_fm > $o] :
( ? [X4: list_fm] :
( ( member_list_fm2 @ X4 @ ( set_list_fm2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_list_fm,X3: list_fm] :
( ? [Zs3: list_list_fm] :
( Xs
= ( append_list_fm @ Ys3 @ ( cons_list_fm @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Xa2: list_fm] :
( ( member_list_fm2 @ Xa2 @ ( set_list_fm2 @ Ys3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_333_split__list__first__prop,axiom,
! [Xs: list_tm,P: tm > $o] :
( ? [X4: tm] :
( ( member_tm2 @ X4 @ ( set_tm2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_tm,X3: tm] :
( ? [Zs3: list_tm] :
( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Xa2: tm] :
( ( member_tm2 @ Xa2 @ ( set_tm2 @ Ys3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_334_split__list__last__prop,axiom,
! [Xs: list_list_tm,P: list_tm > $o] :
( ? [X4: list_tm] :
( ( member_list_tm @ X4 @ ( set_list_tm2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_list_tm,X3: list_tm,Zs3: list_list_tm] :
( ( Xs
= ( append_list_tm @ Ys3 @ ( cons_list_tm @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Xa2: list_tm] :
( ( member_list_tm @ Xa2 @ ( set_list_tm2 @ Zs3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_335_split__list__last__prop,axiom,
! [Xs: list_fm,P: fm > $o] :
( ? [X4: fm] :
( ( member_fm2 @ X4 @ ( set_fm2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_fm,X3: fm,Zs3: list_fm] :
( ( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Xa2: fm] :
( ( member_fm2 @ Xa2 @ ( set_fm2 @ Zs3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_336_split__list__last__prop,axiom,
! [Xs: list_P2887561121880082555ist_fm,P: produc6018962875968178549ist_fm > $o] :
( ? [X4: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ X4 @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_P2887561121880082555ist_fm,X3: produc6018962875968178549ist_fm,Zs3: list_P2887561121880082555ist_fm] :
( ( Xs
= ( append1476581234975291920ist_fm @ Ys3 @ ( cons_P6586734674047340843ist_fm @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Xa2: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ Xa2 @ ( set_Pr5051287786238754058ist_fm @ Zs3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_337_split__list__last__prop,axiom,
! [Xs: list_list_fm,P: list_fm > $o] :
( ? [X4: list_fm] :
( ( member_list_fm2 @ X4 @ ( set_list_fm2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_list_fm,X3: list_fm,Zs3: list_list_fm] :
( ( Xs
= ( append_list_fm @ Ys3 @ ( cons_list_fm @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Xa2: list_fm] :
( ( member_list_fm2 @ Xa2 @ ( set_list_fm2 @ Zs3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_338_split__list__last__prop,axiom,
! [Xs: list_tm,P: tm > $o] :
( ? [X4: tm] :
( ( member_tm2 @ X4 @ ( set_tm2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_tm,X3: tm,Zs3: list_tm] :
( ( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Xa2: tm] :
( ( member_tm2 @ Xa2 @ ( set_tm2 @ Zs3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_339_in__set__conv__decomp,axiom,
! [X: list_tm,Xs: list_list_tm] :
( ( member_list_tm @ X @ ( set_list_tm2 @ Xs ) )
= ( ? [Ys2: list_list_tm,Zs2: list_list_tm] :
( Xs
= ( append_list_tm @ Ys2 @ ( cons_list_tm @ X @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_340_in__set__conv__decomp,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
= ( ? [Ys2: list_fm,Zs2: list_fm] :
( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_341_in__set__conv__decomp,axiom,
! [X: produc6018962875968178549ist_fm,Xs: list_P2887561121880082555ist_fm] :
( ( member4699826688122452638ist_fm @ X @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
= ( ? [Ys2: list_P2887561121880082555ist_fm,Zs2: list_P2887561121880082555ist_fm] :
( Xs
= ( append1476581234975291920ist_fm @ Ys2 @ ( cons_P6586734674047340843ist_fm @ X @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_342_in__set__conv__decomp,axiom,
! [X: list_fm,Xs: list_list_fm] :
( ( member_list_fm2 @ X @ ( set_list_fm2 @ Xs ) )
= ( ? [Ys2: list_list_fm,Zs2: list_list_fm] :
( Xs
= ( append_list_fm @ Ys2 @ ( cons_list_fm @ X @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_343_in__set__conv__decomp,axiom,
! [X: tm,Xs: list_tm] :
( ( member_tm2 @ X @ ( set_tm2 @ Xs ) )
= ( ? [Ys2: list_tm,Zs2: list_tm] :
( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_344_append__Cons__eq__iff,axiom,
! [X: list_tm,Xs: list_list_tm,Ys: list_list_tm,Xs4: list_list_tm,Ys4: list_list_tm] :
( ~ ( member_list_tm @ X @ ( set_list_tm2 @ Xs ) )
=> ( ~ ( member_list_tm @ X @ ( set_list_tm2 @ Ys ) )
=> ( ( ( append_list_tm @ Xs @ ( cons_list_tm @ X @ Ys ) )
= ( append_list_tm @ Xs4 @ ( cons_list_tm @ X @ Ys4 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_345_append__Cons__eq__iff,axiom,
! [X: fm,Xs: list_fm,Ys: list_fm,Xs4: list_fm,Ys4: list_fm] :
( ~ ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
=> ( ~ ( member_fm2 @ X @ ( set_fm2 @ Ys ) )
=> ( ( ( append_fm @ Xs @ ( cons_fm @ X @ Ys ) )
= ( append_fm @ Xs4 @ ( cons_fm @ X @ Ys4 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_346_append__Cons__eq__iff,axiom,
! [X: produc6018962875968178549ist_fm,Xs: list_P2887561121880082555ist_fm,Ys: list_P2887561121880082555ist_fm,Xs4: list_P2887561121880082555ist_fm,Ys4: list_P2887561121880082555ist_fm] :
( ~ ( member4699826688122452638ist_fm @ X @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
=> ( ~ ( member4699826688122452638ist_fm @ X @ ( set_Pr5051287786238754058ist_fm @ Ys ) )
=> ( ( ( append1476581234975291920ist_fm @ Xs @ ( cons_P6586734674047340843ist_fm @ X @ Ys ) )
= ( append1476581234975291920ist_fm @ Xs4 @ ( cons_P6586734674047340843ist_fm @ X @ Ys4 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_347_append__Cons__eq__iff,axiom,
! [X: list_fm,Xs: list_list_fm,Ys: list_list_fm,Xs4: list_list_fm,Ys4: list_list_fm] :
( ~ ( member_list_fm2 @ X @ ( set_list_fm2 @ Xs ) )
=> ( ~ ( member_list_fm2 @ X @ ( set_list_fm2 @ Ys ) )
=> ( ( ( append_list_fm @ Xs @ ( cons_list_fm @ X @ Ys ) )
= ( append_list_fm @ Xs4 @ ( cons_list_fm @ X @ Ys4 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_348_append__Cons__eq__iff,axiom,
! [X: tm,Xs: list_tm,Ys: list_tm,Xs4: list_tm,Ys4: list_tm] :
( ~ ( member_tm2 @ X @ ( set_tm2 @ Xs ) )
=> ( ~ ( member_tm2 @ X @ ( set_tm2 @ Ys ) )
=> ( ( ( append_tm @ Xs @ ( cons_tm @ X @ Ys ) )
= ( append_tm @ Xs4 @ ( cons_tm @ X @ Ys4 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_349_split__list__propE,axiom,
! [Xs: list_list_tm,P: list_tm > $o] :
( ? [X4: list_tm] :
( ( member_list_tm @ X4 @ ( set_list_tm2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_list_tm,X3: list_tm] :
( ? [Zs3: list_list_tm] :
( Xs
= ( append_list_tm @ Ys3 @ ( cons_list_tm @ X3 @ Zs3 ) ) )
=> ~ ( P @ X3 ) ) ) ).
% split_list_propE
thf(fact_350_split__list__propE,axiom,
! [Xs: list_fm,P: fm > $o] :
( ? [X4: fm] :
( ( member_fm2 @ X4 @ ( set_fm2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_fm,X3: fm] :
( ? [Zs3: list_fm] :
( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X3 @ Zs3 ) ) )
=> ~ ( P @ X3 ) ) ) ).
% split_list_propE
thf(fact_351_split__list__propE,axiom,
! [Xs: list_P2887561121880082555ist_fm,P: produc6018962875968178549ist_fm > $o] :
( ? [X4: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ X4 @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_P2887561121880082555ist_fm,X3: produc6018962875968178549ist_fm] :
( ? [Zs3: list_P2887561121880082555ist_fm] :
( Xs
= ( append1476581234975291920ist_fm @ Ys3 @ ( cons_P6586734674047340843ist_fm @ X3 @ Zs3 ) ) )
=> ~ ( P @ X3 ) ) ) ).
% split_list_propE
thf(fact_352_split__list__propE,axiom,
! [Xs: list_list_fm,P: list_fm > $o] :
( ? [X4: list_fm] :
( ( member_list_fm2 @ X4 @ ( set_list_fm2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_list_fm,X3: list_fm] :
( ? [Zs3: list_list_fm] :
( Xs
= ( append_list_fm @ Ys3 @ ( cons_list_fm @ X3 @ Zs3 ) ) )
=> ~ ( P @ X3 ) ) ) ).
% split_list_propE
thf(fact_353_split__list__propE,axiom,
! [Xs: list_tm,P: tm > $o] :
( ? [X4: tm] :
( ( member_tm2 @ X4 @ ( set_tm2 @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys3: list_tm,X3: tm] :
( ? [Zs3: list_tm] :
( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ X3 @ Zs3 ) ) )
=> ~ ( P @ X3 ) ) ) ).
% split_list_propE
thf(fact_354_split__list__first,axiom,
! [X: list_tm,Xs: list_list_tm] :
( ( member_list_tm @ X @ ( set_list_tm2 @ Xs ) )
=> ? [Ys3: list_list_tm,Zs3: list_list_tm] :
( ( Xs
= ( append_list_tm @ Ys3 @ ( cons_list_tm @ X @ Zs3 ) ) )
& ~ ( member_list_tm @ X @ ( set_list_tm2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_355_split__list__first,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
=> ? [Ys3: list_fm,Zs3: list_fm] :
( ( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X @ Zs3 ) ) )
& ~ ( member_fm2 @ X @ ( set_fm2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_356_split__list__first,axiom,
! [X: produc6018962875968178549ist_fm,Xs: list_P2887561121880082555ist_fm] :
( ( member4699826688122452638ist_fm @ X @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
=> ? [Ys3: list_P2887561121880082555ist_fm,Zs3: list_P2887561121880082555ist_fm] :
( ( Xs
= ( append1476581234975291920ist_fm @ Ys3 @ ( cons_P6586734674047340843ist_fm @ X @ Zs3 ) ) )
& ~ ( member4699826688122452638ist_fm @ X @ ( set_Pr5051287786238754058ist_fm @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_357_split__list__first,axiom,
! [X: list_fm,Xs: list_list_fm] :
( ( member_list_fm2 @ X @ ( set_list_fm2 @ Xs ) )
=> ? [Ys3: list_list_fm,Zs3: list_list_fm] :
( ( Xs
= ( append_list_fm @ Ys3 @ ( cons_list_fm @ X @ Zs3 ) ) )
& ~ ( member_list_fm2 @ X @ ( set_list_fm2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_358_split__list__first,axiom,
! [X: tm,Xs: list_tm] :
( ( member_tm2 @ X @ ( set_tm2 @ Xs ) )
=> ? [Ys3: list_tm,Zs3: list_tm] :
( ( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ X @ Zs3 ) ) )
& ~ ( member_tm2 @ X @ ( set_tm2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_359_split__list__prop,axiom,
! [Xs: list_list_tm,P: list_tm > $o] :
( ? [X4: list_tm] :
( ( member_list_tm @ X4 @ ( set_list_tm2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_list_tm,X3: list_tm] :
( ? [Zs3: list_list_tm] :
( Xs
= ( append_list_tm @ Ys3 @ ( cons_list_tm @ X3 @ Zs3 ) ) )
& ( P @ X3 ) ) ) ).
% split_list_prop
thf(fact_360_split__list__prop,axiom,
! [Xs: list_fm,P: fm > $o] :
( ? [X4: fm] :
( ( member_fm2 @ X4 @ ( set_fm2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_fm,X3: fm] :
( ? [Zs3: list_fm] :
( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X3 @ Zs3 ) ) )
& ( P @ X3 ) ) ) ).
% split_list_prop
thf(fact_361_split__list__prop,axiom,
! [Xs: list_P2887561121880082555ist_fm,P: produc6018962875968178549ist_fm > $o] :
( ? [X4: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ X4 @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_P2887561121880082555ist_fm,X3: produc6018962875968178549ist_fm] :
( ? [Zs3: list_P2887561121880082555ist_fm] :
( Xs
= ( append1476581234975291920ist_fm @ Ys3 @ ( cons_P6586734674047340843ist_fm @ X3 @ Zs3 ) ) )
& ( P @ X3 ) ) ) ).
% split_list_prop
thf(fact_362_split__list__prop,axiom,
! [Xs: list_list_fm,P: list_fm > $o] :
( ? [X4: list_fm] :
( ( member_list_fm2 @ X4 @ ( set_list_fm2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_list_fm,X3: list_fm] :
( ? [Zs3: list_list_fm] :
( Xs
= ( append_list_fm @ Ys3 @ ( cons_list_fm @ X3 @ Zs3 ) ) )
& ( P @ X3 ) ) ) ).
% split_list_prop
thf(fact_363_split__list__prop,axiom,
! [Xs: list_tm,P: tm > $o] :
( ? [X4: tm] :
( ( member_tm2 @ X4 @ ( set_tm2 @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys3: list_tm,X3: tm] :
( ? [Zs3: list_tm] :
( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ X3 @ Zs3 ) ) )
& ( P @ X3 ) ) ) ).
% split_list_prop
thf(fact_364_split__list__last,axiom,
! [X: list_tm,Xs: list_list_tm] :
( ( member_list_tm @ X @ ( set_list_tm2 @ Xs ) )
=> ? [Ys3: list_list_tm,Zs3: list_list_tm] :
( ( Xs
= ( append_list_tm @ Ys3 @ ( cons_list_tm @ X @ Zs3 ) ) )
& ~ ( member_list_tm @ X @ ( set_list_tm2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_365_split__list__last,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
=> ? [Ys3: list_fm,Zs3: list_fm] :
( ( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X @ Zs3 ) ) )
& ~ ( member_fm2 @ X @ ( set_fm2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_366_split__list__last,axiom,
! [X: produc6018962875968178549ist_fm,Xs: list_P2887561121880082555ist_fm] :
( ( member4699826688122452638ist_fm @ X @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
=> ? [Ys3: list_P2887561121880082555ist_fm,Zs3: list_P2887561121880082555ist_fm] :
( ( Xs
= ( append1476581234975291920ist_fm @ Ys3 @ ( cons_P6586734674047340843ist_fm @ X @ Zs3 ) ) )
& ~ ( member4699826688122452638ist_fm @ X @ ( set_Pr5051287786238754058ist_fm @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_367_split__list__last,axiom,
! [X: list_fm,Xs: list_list_fm] :
( ( member_list_fm2 @ X @ ( set_list_fm2 @ Xs ) )
=> ? [Ys3: list_list_fm,Zs3: list_list_fm] :
( ( Xs
= ( append_list_fm @ Ys3 @ ( cons_list_fm @ X @ Zs3 ) ) )
& ~ ( member_list_fm2 @ X @ ( set_list_fm2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_368_split__list__last,axiom,
! [X: tm,Xs: list_tm] :
( ( member_tm2 @ X @ ( set_tm2 @ Xs ) )
=> ? [Ys3: list_tm,Zs3: list_tm] :
( ( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ X @ Zs3 ) ) )
& ~ ( member_tm2 @ X @ ( set_tm2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_369_split__list,axiom,
! [X: list_tm,Xs: list_list_tm] :
( ( member_list_tm @ X @ ( set_list_tm2 @ Xs ) )
=> ? [Ys3: list_list_tm,Zs3: list_list_tm] :
( Xs
= ( append_list_tm @ Ys3 @ ( cons_list_tm @ X @ Zs3 ) ) ) ) ).
% split_list
thf(fact_370_split__list,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
=> ? [Ys3: list_fm,Zs3: list_fm] :
( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X @ Zs3 ) ) ) ) ).
% split_list
thf(fact_371_split__list,axiom,
! [X: produc6018962875968178549ist_fm,Xs: list_P2887561121880082555ist_fm] :
( ( member4699826688122452638ist_fm @ X @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
=> ? [Ys3: list_P2887561121880082555ist_fm,Zs3: list_P2887561121880082555ist_fm] :
( Xs
= ( append1476581234975291920ist_fm @ Ys3 @ ( cons_P6586734674047340843ist_fm @ X @ Zs3 ) ) ) ) ).
% split_list
thf(fact_372_split__list,axiom,
! [X: list_fm,Xs: list_list_fm] :
( ( member_list_fm2 @ X @ ( set_list_fm2 @ Xs ) )
=> ? [Ys3: list_list_fm,Zs3: list_list_fm] :
( Xs
= ( append_list_fm @ Ys3 @ ( cons_list_fm @ X @ Zs3 ) ) ) ) ).
% split_list
thf(fact_373_split__list,axiom,
! [X: tm,Xs: list_tm] :
( ( member_tm2 @ X @ ( set_tm2 @ Xs ) )
=> ? [Ys3: list_tm,Zs3: list_tm] :
( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ X @ Zs3 ) ) ) ) ).
% split_list
thf(fact_374_concat_Osimps_I2_J,axiom,
! [X: list_list_tm,Xs: list_list_list_tm] :
( ( concat_list_tm @ ( cons_list_list_tm @ X @ Xs ) )
= ( append_list_tm @ X @ ( concat_list_tm @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_375_concat_Osimps_I2_J,axiom,
! [X: list_tm,Xs: list_list_tm] :
( ( concat_tm @ ( cons_list_tm @ X @ Xs ) )
= ( append_tm @ X @ ( concat_tm @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_376_concat_Osimps_I2_J,axiom,
! [X: list_fm,Xs: list_list_fm] :
( ( concat_fm @ ( cons_list_fm @ X @ Xs ) )
= ( append_fm @ X @ ( concat_fm @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_377_remdups_Osimps_I2_J,axiom,
! [X: fm,Xs: list_fm] :
( ( ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
=> ( ( remdups_fm @ ( cons_fm @ X @ Xs ) )
= ( remdups_fm @ Xs ) ) )
& ( ~ ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
=> ( ( remdups_fm @ ( cons_fm @ X @ Xs ) )
= ( cons_fm @ X @ ( remdups_fm @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_378_remdups_Osimps_I2_J,axiom,
! [X: produc6018962875968178549ist_fm,Xs: list_P2887561121880082555ist_fm] :
( ( ( member4699826688122452638ist_fm @ X @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
=> ( ( remdup8378870680097181200ist_fm @ ( cons_P6586734674047340843ist_fm @ X @ Xs ) )
= ( remdup8378870680097181200ist_fm @ Xs ) ) )
& ( ~ ( member4699826688122452638ist_fm @ X @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
=> ( ( remdup8378870680097181200ist_fm @ ( cons_P6586734674047340843ist_fm @ X @ Xs ) )
= ( cons_P6586734674047340843ist_fm @ X @ ( remdup8378870680097181200ist_fm @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_379_remdups_Osimps_I2_J,axiom,
! [X: list_fm,Xs: list_list_fm] :
( ( ( member_list_fm2 @ X @ ( set_list_fm2 @ Xs ) )
=> ( ( remdups_list_fm @ ( cons_list_fm @ X @ Xs ) )
= ( remdups_list_fm @ Xs ) ) )
& ( ~ ( member_list_fm2 @ X @ ( set_list_fm2 @ Xs ) )
=> ( ( remdups_list_fm @ ( cons_list_fm @ X @ Xs ) )
= ( cons_list_fm @ X @ ( remdups_list_fm @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_380_remdups_Osimps_I2_J,axiom,
! [X: tm,Xs: list_tm] :
( ( ( member_tm2 @ X @ ( set_tm2 @ Xs ) )
=> ( ( remdups_tm @ ( cons_tm @ X @ Xs ) )
= ( remdups_tm @ Xs ) ) )
& ( ~ ( member_tm2 @ X @ ( set_tm2 @ Xs ) )
=> ( ( remdups_tm @ ( cons_tm @ X @ Xs ) )
= ( cons_tm @ X @ ( remdups_tm @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_381_maps__simps_I1_J,axiom,
! [F: list_fm > list_tm,X: list_fm,Xs: list_list_fm] :
( ( maps_list_fm_tm @ F @ ( cons_list_fm @ X @ Xs ) )
= ( append_tm @ ( F @ X ) @ ( maps_list_fm_tm @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_382_maps__simps_I1_J,axiom,
! [F: list_fm > list_list_tm,X: list_fm,Xs: list_list_fm] :
( ( maps_list_fm_list_tm @ F @ ( cons_list_fm @ X @ Xs ) )
= ( append_list_tm @ ( F @ X ) @ ( maps_list_fm_list_tm @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_383_maps__simps_I1_J,axiom,
! [F: list_fm > list_fm,X: list_fm,Xs: list_list_fm] :
( ( maps_list_fm_fm @ F @ ( cons_list_fm @ X @ Xs ) )
= ( append_fm @ ( F @ X ) @ ( maps_list_fm_fm @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_384_maps__simps_I1_J,axiom,
! [F: tm > list_tm,X: tm,Xs: list_tm] :
( ( maps_tm_tm @ F @ ( cons_tm @ X @ Xs ) )
= ( append_tm @ ( F @ X ) @ ( maps_tm_tm @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_385_maps__simps_I1_J,axiom,
! [F: tm > list_list_tm,X: tm,Xs: list_tm] :
( ( maps_tm_list_tm @ F @ ( cons_tm @ X @ Xs ) )
= ( append_list_tm @ ( F @ X ) @ ( maps_tm_list_tm @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_386_maps__simps_I1_J,axiom,
! [F: tm > list_fm,X: tm,Xs: list_tm] :
( ( maps_tm_fm @ F @ ( cons_tm @ X @ Xs ) )
= ( append_fm @ ( F @ X ) @ ( maps_tm_fm @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_387_not__in__set__insert,axiom,
! [X: fm,Xs: list_fm] :
( ~ ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
=> ( ( insert_fm @ X @ Xs )
= ( cons_fm @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_388_not__in__set__insert,axiom,
! [X: produc6018962875968178549ist_fm,Xs: list_P2887561121880082555ist_fm] :
( ~ ( member4699826688122452638ist_fm @ X @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
=> ( ( insert5272696590961914513ist_fm @ X @ Xs )
= ( cons_P6586734674047340843ist_fm @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_389_not__in__set__insert,axiom,
! [X: list_fm,Xs: list_list_fm] :
( ~ ( member_list_fm2 @ X @ ( set_list_fm2 @ Xs ) )
=> ( ( insert_list_fm @ X @ Xs )
= ( cons_list_fm @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_390_not__in__set__insert,axiom,
! [X: tm,Xs: list_tm] :
( ~ ( member_tm2 @ X @ ( set_tm2 @ Xs ) )
=> ( ( insert_tm @ X @ Xs )
= ( cons_tm @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_391_lexord__sufI,axiom,
! [U: list_tm,W: list_tm,R: set_Pr2455929065695642951_tm_tm,V: list_tm,Z3: list_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ U @ W ) @ ( lexord_tm @ R ) )
=> ( ( ord_less_eq_nat @ ( size_size_list_tm @ W ) @ ( size_size_list_tm @ U ) )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( append_tm @ U @ V ) @ ( append_tm @ W @ Z3 ) ) @ ( lexord_tm @ R ) ) ) ) ).
% lexord_sufI
thf(fact_392_lexord__sufI,axiom,
! [U: list_list_tm,W: list_list_tm,R: set_Pr1507011332596240839ist_tm,V: list_list_tm,Z3: list_list_tm] :
( ( member1414320910663981712ist_tm @ ( produc8086241248684931031ist_tm @ U @ W ) @ ( lexord_list_tm @ R ) )
=> ( ( ord_less_eq_nat @ ( size_s9096087352182575069ist_tm @ W ) @ ( size_s9096087352182575069ist_tm @ U ) )
=> ( member1414320910663981712ist_tm @ ( produc8086241248684931031ist_tm @ ( append_list_tm @ U @ V ) @ ( append_list_tm @ W @ Z3 ) ) @ ( lexord_list_tm @ R ) ) ) ) ).
% lexord_sufI
thf(fact_393_lexord__sufI,axiom,
! [U: list_fm,W: list_fm,R: set_Pr4706815898642364871_fm_fm,V: list_fm,Z3: list_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ U @ W ) @ ( lexord_fm @ R ) )
=> ( ( ord_less_eq_nat @ ( size_size_list_fm @ W ) @ ( size_size_list_fm @ U ) )
=> ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ ( append_fm @ U @ V ) @ ( append_fm @ W @ Z3 ) ) @ ( lexord_fm @ R ) ) ) ) ).
% lexord_sufI
thf(fact_394_length__n__lists__elem,axiom,
! [Ys: list_fm,N: nat,Xs: list_fm] :
( ( member_list_fm2 @ Ys @ ( set_list_fm2 @ ( n_lists_fm @ N @ Xs ) ) )
=> ( ( size_size_list_fm @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_395_lenlex__append1,axiom,
! [Us2: list_tm,Xs: list_tm,R2: set_Pr2455929065695642951_tm_tm,Vs2: list_tm,Ys: list_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Us2 @ Xs ) @ ( lenlex_tm @ R2 ) )
=> ( ( ( size_size_list_tm @ Vs2 )
= ( size_size_list_tm @ Ys ) )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( append_tm @ Us2 @ Vs2 ) @ ( append_tm @ Xs @ Ys ) ) @ ( lenlex_tm @ R2 ) ) ) ) ).
% lenlex_append1
thf(fact_396_lenlex__append1,axiom,
! [Us2: list_list_tm,Xs: list_list_tm,R2: set_Pr1507011332596240839ist_tm,Vs2: list_list_tm,Ys: list_list_tm] :
( ( member1414320910663981712ist_tm @ ( produc8086241248684931031ist_tm @ Us2 @ Xs ) @ ( lenlex_list_tm @ R2 ) )
=> ( ( ( size_s9096087352182575069ist_tm @ Vs2 )
= ( size_s9096087352182575069ist_tm @ Ys ) )
=> ( member1414320910663981712ist_tm @ ( produc8086241248684931031ist_tm @ ( append_list_tm @ Us2 @ Vs2 ) @ ( append_list_tm @ Xs @ Ys ) ) @ ( lenlex_list_tm @ R2 ) ) ) ) ).
% lenlex_append1
thf(fact_397_lenlex__append1,axiom,
! [Us2: list_fm,Xs: list_fm,R2: set_Pr4706815898642364871_fm_fm,Vs2: list_fm,Ys: list_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Us2 @ Xs ) @ ( lenlex_fm @ R2 ) )
=> ( ( ( size_size_list_fm @ Vs2 )
= ( size_size_list_fm @ Ys ) )
=> ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ ( append_fm @ Us2 @ Vs2 ) @ ( append_fm @ Xs @ Ys ) ) @ ( lenlex_fm @ R2 ) ) ) ) ).
% lenlex_append1
thf(fact_398_listrel1I,axiom,
! [X: list_tm,Y3: list_tm,R: set_Pr1507011332596240839ist_tm,Xs: list_list_tm,Us2: list_list_tm,Vs2: list_list_tm,Ys: list_list_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ X @ Y3 ) @ R )
=> ( ( Xs
= ( append_list_tm @ Us2 @ ( cons_list_tm @ X @ Vs2 ) ) )
=> ( ( Ys
= ( append_list_tm @ Us2 @ ( cons_list_tm @ Y3 @ Vs2 ) ) )
=> ( member1414320910663981712ist_tm @ ( produc8086241248684931031ist_tm @ Xs @ Ys ) @ ( listrel1_list_tm @ R ) ) ) ) ) ).
% listrel1I
thf(fact_399_listrel1I,axiom,
! [X: fm,Y3: fm,R: set_Pr4706815898642364871_fm_fm,Xs: list_fm,Us2: list_fm,Vs2: list_fm,Ys: list_fm] :
( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X @ Y3 ) @ R )
=> ( ( Xs
= ( append_fm @ Us2 @ ( cons_fm @ X @ Vs2 ) ) )
=> ( ( Ys
= ( append_fm @ Us2 @ ( cons_fm @ Y3 @ Vs2 ) ) )
=> ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Xs @ Ys ) @ ( listrel1_fm @ R ) ) ) ) ) ).
% listrel1I
thf(fact_400_listrel1I,axiom,
! [X: list_fm,Y3: list_fm,R: set_Pr1916137733696349511ist_fm,Xs: list_list_fm,Us2: list_list_fm,Vs2: list_list_fm,Ys: list_list_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ X @ Y3 ) @ R )
=> ( ( Xs
= ( append_list_fm @ Us2 @ ( cons_list_fm @ X @ Vs2 ) ) )
=> ( ( Ys
= ( append_list_fm @ Us2 @ ( cons_list_fm @ Y3 @ Vs2 ) ) )
=> ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ Xs @ Ys ) @ ( listrel1_list_fm @ R ) ) ) ) ) ).
% listrel1I
thf(fact_401_listrel1I,axiom,
! [X: tm,Y3: tm,R: set_Pr2455929065695642951_tm_tm,Xs: list_tm,Us2: list_tm,Vs2: list_tm,Ys: list_tm] :
( ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X @ Y3 ) @ R )
=> ( ( Xs
= ( append_tm @ Us2 @ ( cons_tm @ X @ Vs2 ) ) )
=> ( ( Ys
= ( append_tm @ Us2 @ ( cons_tm @ Y3 @ Vs2 ) ) )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ Ys ) @ ( listrel1_tm @ R ) ) ) ) ) ).
% listrel1I
thf(fact_402_listrel1E,axiom,
! [Xs: list_list_tm,Ys: list_list_tm,R: set_Pr1507011332596240839ist_tm] :
( ( member1414320910663981712ist_tm @ ( produc8086241248684931031ist_tm @ Xs @ Ys ) @ ( listrel1_list_tm @ R ) )
=> ~ ! [X3: list_tm,Y: list_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ X3 @ Y ) @ R )
=> ! [Us3: list_list_tm,Vs3: list_list_tm] :
( ( Xs
= ( append_list_tm @ Us3 @ ( cons_list_tm @ X3 @ Vs3 ) ) )
=> ( Ys
!= ( append_list_tm @ Us3 @ ( cons_list_tm @ Y @ Vs3 ) ) ) ) ) ) ).
% listrel1E
thf(fact_403_listrel1E,axiom,
! [Xs: list_fm,Ys: list_fm,R: set_Pr4706815898642364871_fm_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Xs @ Ys ) @ ( listrel1_fm @ R ) )
=> ~ ! [X3: fm,Y: fm] :
( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X3 @ Y ) @ R )
=> ! [Us3: list_fm,Vs3: list_fm] :
( ( Xs
= ( append_fm @ Us3 @ ( cons_fm @ X3 @ Vs3 ) ) )
=> ( Ys
!= ( append_fm @ Us3 @ ( cons_fm @ Y @ Vs3 ) ) ) ) ) ) ).
% listrel1E
thf(fact_404_listrel1E,axiom,
! [Xs: list_list_fm,Ys: list_list_fm,R: set_Pr1916137733696349511ist_fm] :
( ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ Xs @ Ys ) @ ( listrel1_list_fm @ R ) )
=> ~ ! [X3: list_fm,Y: list_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ X3 @ Y ) @ R )
=> ! [Us3: list_list_fm,Vs3: list_list_fm] :
( ( Xs
= ( append_list_fm @ Us3 @ ( cons_list_fm @ X3 @ Vs3 ) ) )
=> ( Ys
!= ( append_list_fm @ Us3 @ ( cons_list_fm @ Y @ Vs3 ) ) ) ) ) ) ).
% listrel1E
thf(fact_405_listrel1E,axiom,
! [Xs: list_tm,Ys: list_tm,R: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ Ys ) @ ( listrel1_tm @ R ) )
=> ~ ! [X3: tm,Y: tm] :
( ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X3 @ Y ) @ R )
=> ! [Us3: list_tm,Vs3: list_tm] :
( ( Xs
= ( append_tm @ Us3 @ ( cons_tm @ X3 @ Vs3 ) ) )
=> ( Ys
!= ( append_tm @ Us3 @ ( cons_tm @ Y @ Vs3 ) ) ) ) ) ) ).
% listrel1E
thf(fact_406_in__set__insert,axiom,
! [X: list_fm,Xs: list_list_fm] :
( ( member_list_fm2 @ X @ ( set_list_fm2 @ Xs ) )
=> ( ( insert_list_fm @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_407_in__set__insert,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
=> ( ( insert_fm @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_408_in__set__insert,axiom,
! [X: tm,Xs: list_tm] :
( ( member_tm2 @ X @ ( set_tm2 @ Xs ) )
=> ( ( insert_tm @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_409_in__set__insert,axiom,
! [X: produc6018962875968178549ist_fm,Xs: list_P2887561121880082555ist_fm] :
( ( member4699826688122452638ist_fm @ X @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
=> ( ( insert5272696590961914513ist_fm @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_410_length__remdups__leq,axiom,
! [Xs: list_tm] : ( ord_less_eq_nat @ ( size_size_list_tm @ ( remdups_tm @ Xs ) ) @ ( size_size_list_tm @ Xs ) ) ).
% length_remdups_leq
thf(fact_411_length__remdups__leq,axiom,
! [Xs: list_fm] : ( ord_less_eq_nat @ ( size_size_list_fm @ ( remdups_fm @ Xs ) ) @ ( size_size_list_fm @ Xs ) ) ).
% length_remdups_leq
thf(fact_412_length__remdups__leq,axiom,
! [Xs: list_list_fm] : ( ord_less_eq_nat @ ( size_s115229985653309035ist_fm @ ( remdups_list_fm @ Xs ) ) @ ( size_s115229985653309035ist_fm @ Xs ) ) ).
% length_remdups_leq
thf(fact_413_Cons__listrel1__Cons,axiom,
! [X: list_fm,Xs: list_list_fm,Y3: list_fm,Ys: list_list_fm,R: set_Pr1916137733696349511ist_fm] :
( ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ ( cons_list_fm @ X @ Xs ) @ ( cons_list_fm @ Y3 @ Ys ) ) @ ( listrel1_list_fm @ R ) )
= ( ( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ X @ Y3 ) @ R )
& ( Xs = Ys ) )
| ( ( X = Y3 )
& ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ Xs @ Ys ) @ ( listrel1_list_fm @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_414_Cons__listrel1__Cons,axiom,
! [X: tm,Xs: list_tm,Y3: tm,Ys: list_tm,R: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( cons_tm @ X @ Xs ) @ ( cons_tm @ Y3 @ Ys ) ) @ ( listrel1_tm @ R ) )
= ( ( ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X @ Y3 ) @ R )
& ( Xs = Ys ) )
| ( ( X = Y3 )
& ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ Ys ) @ ( listrel1_tm @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_415_impossible__Cons,axiom,
! [Xs: list_list_fm,Ys: list_list_fm,X: list_fm] :
( ( ord_less_eq_nat @ ( size_s115229985653309035ist_fm @ Xs ) @ ( size_s115229985653309035ist_fm @ Ys ) )
=> ( Xs
!= ( cons_list_fm @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_416_impossible__Cons,axiom,
! [Xs: list_tm,Ys: list_tm,X: tm] :
( ( ord_less_eq_nat @ ( size_size_list_tm @ Xs ) @ ( size_size_list_tm @ Ys ) )
=> ( Xs
!= ( cons_tm @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_417_insert__remdups,axiom,
! [X: tm,Xs: list_tm] :
( ( insert_tm @ X @ ( remdups_tm @ Xs ) )
= ( remdups_tm @ ( insert_tm @ X @ Xs ) ) ) ).
% insert_remdups
thf(fact_418_insert__remdups,axiom,
! [X: fm,Xs: list_fm] :
( ( insert_fm @ X @ ( remdups_fm @ Xs ) )
= ( remdups_fm @ ( insert_fm @ X @ Xs ) ) ) ).
% insert_remdups
thf(fact_419_insert__remdups,axiom,
! [X: list_fm,Xs: list_list_fm] :
( ( insert_list_fm @ X @ ( remdups_list_fm @ Xs ) )
= ( remdups_list_fm @ ( insert_list_fm @ X @ Xs ) ) ) ).
% insert_remdups
thf(fact_420_listrel1I2,axiom,
! [Xs: list_list_fm,Ys: list_list_fm,R: set_Pr1916137733696349511ist_fm,X: list_fm] :
( ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ Xs @ Ys ) @ ( listrel1_list_fm @ R ) )
=> ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ ( cons_list_fm @ X @ Xs ) @ ( cons_list_fm @ X @ Ys ) ) @ ( listrel1_list_fm @ R ) ) ) ).
% listrel1I2
thf(fact_421_listrel1I2,axiom,
! [Xs: list_tm,Ys: list_tm,R: set_Pr2455929065695642951_tm_tm,X: tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ Ys ) @ ( listrel1_tm @ R ) )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( cons_tm @ X @ Xs ) @ ( cons_tm @ X @ Ys ) ) @ ( listrel1_tm @ R ) ) ) ).
% listrel1I2
thf(fact_422_append__listrel1I,axiom,
! [Xs: list_tm,Ys: list_tm,R: set_Pr2455929065695642951_tm_tm,Us2: list_tm,Vs2: list_tm] :
( ( ( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ Ys ) @ ( listrel1_tm @ R ) )
& ( Us2 = Vs2 ) )
| ( ( Xs = Ys )
& ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Us2 @ Vs2 ) @ ( listrel1_tm @ R ) ) ) )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( append_tm @ Xs @ Us2 ) @ ( append_tm @ Ys @ Vs2 ) ) @ ( listrel1_tm @ R ) ) ) ).
% append_listrel1I
thf(fact_423_append__listrel1I,axiom,
! [Xs: list_list_tm,Ys: list_list_tm,R: set_Pr1507011332596240839ist_tm,Us2: list_list_tm,Vs2: list_list_tm] :
( ( ( ( member1414320910663981712ist_tm @ ( produc8086241248684931031ist_tm @ Xs @ Ys ) @ ( listrel1_list_tm @ R ) )
& ( Us2 = Vs2 ) )
| ( ( Xs = Ys )
& ( member1414320910663981712ist_tm @ ( produc8086241248684931031ist_tm @ Us2 @ Vs2 ) @ ( listrel1_list_tm @ R ) ) ) )
=> ( member1414320910663981712ist_tm @ ( produc8086241248684931031ist_tm @ ( append_list_tm @ Xs @ Us2 ) @ ( append_list_tm @ Ys @ Vs2 ) ) @ ( listrel1_list_tm @ R ) ) ) ).
% append_listrel1I
thf(fact_424_append__listrel1I,axiom,
! [Xs: list_fm,Ys: list_fm,R: set_Pr4706815898642364871_fm_fm,Us2: list_fm,Vs2: list_fm] :
( ( ( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Xs @ Ys ) @ ( listrel1_fm @ R ) )
& ( Us2 = Vs2 ) )
| ( ( Xs = Ys )
& ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Us2 @ Vs2 ) @ ( listrel1_fm @ R ) ) ) )
=> ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ ( append_fm @ Xs @ Us2 ) @ ( append_fm @ Ys @ Vs2 ) ) @ ( listrel1_fm @ R ) ) ) ).
% append_listrel1I
thf(fact_425_listrel1I1,axiom,
! [X: list_fm,Y3: list_fm,R: set_Pr1916137733696349511ist_fm,Xs: list_list_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ X @ Y3 ) @ R )
=> ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ ( cons_list_fm @ X @ Xs ) @ ( cons_list_fm @ Y3 @ Xs ) ) @ ( listrel1_list_fm @ R ) ) ) ).
% listrel1I1
thf(fact_426_listrel1I1,axiom,
! [X: tm,Y3: tm,R: set_Pr2455929065695642951_tm_tm,Xs: list_tm] :
( ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X @ Y3 ) @ R )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( cons_tm @ X @ Xs ) @ ( cons_tm @ Y3 @ Xs ) ) @ ( listrel1_tm @ R ) ) ) ).
% listrel1I1
thf(fact_427_Cons__listrel1E1,axiom,
! [X: list_fm,Xs: list_list_fm,Ys: list_list_fm,R: set_Pr1916137733696349511ist_fm] :
( ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ ( cons_list_fm @ X @ Xs ) @ Ys ) @ ( listrel1_list_fm @ R ) )
=> ( ! [Y: list_fm] :
( ( Ys
= ( cons_list_fm @ Y @ Xs ) )
=> ~ ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ X @ Y ) @ R ) )
=> ~ ! [Zs3: list_list_fm] :
( ( Ys
= ( cons_list_fm @ X @ Zs3 ) )
=> ~ ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ Xs @ Zs3 ) @ ( listrel1_list_fm @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_428_Cons__listrel1E1,axiom,
! [X: tm,Xs: list_tm,Ys: list_tm,R: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( cons_tm @ X @ Xs ) @ Ys ) @ ( listrel1_tm @ R ) )
=> ( ! [Y: tm] :
( ( Ys
= ( cons_tm @ Y @ Xs ) )
=> ~ ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X @ Y ) @ R ) )
=> ~ ! [Zs3: list_tm] :
( ( Ys
= ( cons_tm @ X @ Zs3 ) )
=> ~ ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ Zs3 ) @ ( listrel1_tm @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_429_Cons__listrel1E2,axiom,
! [Xs: list_list_fm,Y3: list_fm,Ys: list_list_fm,R: set_Pr1916137733696349511ist_fm] :
( ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ Xs @ ( cons_list_fm @ Y3 @ Ys ) ) @ ( listrel1_list_fm @ R ) )
=> ( ! [X3: list_fm] :
( ( Xs
= ( cons_list_fm @ X3 @ Ys ) )
=> ~ ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ X3 @ Y3 ) @ R ) )
=> ~ ! [Zs3: list_list_fm] :
( ( Xs
= ( cons_list_fm @ Y3 @ Zs3 ) )
=> ~ ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ Zs3 @ Ys ) @ ( listrel1_list_fm @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_430_Cons__listrel1E2,axiom,
! [Xs: list_tm,Y3: tm,Ys: list_tm,R: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ ( cons_tm @ Y3 @ Ys ) ) @ ( listrel1_tm @ R ) )
=> ( ! [X3: tm] :
( ( Xs
= ( cons_tm @ X3 @ Ys ) )
=> ~ ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X3 @ Y3 ) @ R ) )
=> ~ ! [Zs3: list_tm] :
( ( Xs
= ( cons_tm @ Y3 @ Zs3 ) )
=> ~ ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Zs3 @ Ys ) @ ( listrel1_tm @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_431_List_Oinsert__def,axiom,
( insert_fm
= ( ^ [X2: fm,Xs3: list_fm] : ( if_list_fm @ ( member_fm2 @ X2 @ ( set_fm2 @ Xs3 ) ) @ Xs3 @ ( cons_fm @ X2 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_432_List_Oinsert__def,axiom,
( insert5272696590961914513ist_fm
= ( ^ [X2: produc6018962875968178549ist_fm,Xs3: list_P2887561121880082555ist_fm] : ( if_lis1897046999900794433ist_fm @ ( member4699826688122452638ist_fm @ X2 @ ( set_Pr5051287786238754058ist_fm @ Xs3 ) ) @ Xs3 @ ( cons_P6586734674047340843ist_fm @ X2 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_433_List_Oinsert__def,axiom,
( insert_list_fm
= ( ^ [X2: list_fm,Xs3: list_list_fm] : ( if_list_list_fm @ ( member_list_fm2 @ X2 @ ( set_list_fm2 @ Xs3 ) ) @ Xs3 @ ( cons_list_fm @ X2 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_434_List_Oinsert__def,axiom,
( insert_tm
= ( ^ [X2: tm,Xs3: list_tm] : ( if_list_tm @ ( member_tm2 @ X2 @ ( set_tm2 @ Xs3 ) ) @ Xs3 @ ( cons_tm @ X2 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_435_dual__order_Orefl,axiom,
! [A: set_tm] : ( ord_less_eq_set_tm @ A @ A ) ).
% dual_order.refl
thf(fact_436_dual__order_Orefl,axiom,
! [A: set_fm] : ( ord_less_eq_set_fm @ A @ A ) ).
% dual_order.refl
thf(fact_437_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_438_order__refl,axiom,
! [X: set_tm] : ( ord_less_eq_set_tm @ X @ X ) ).
% order_refl
thf(fact_439_order__refl,axiom,
! [X: set_fm] : ( ord_less_eq_set_fm @ X @ X ) ).
% order_refl
thf(fact_440_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_441_Cons__lenlex__iff,axiom,
! [M: list_fm,Ms: list_list_fm,N: list_fm,Ns: list_list_fm,R: set_Pr1916137733696349511ist_fm] :
( ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ ( cons_list_fm @ M @ Ms ) @ ( cons_list_fm @ N @ Ns ) ) @ ( lenlex_list_fm @ R ) )
= ( ( ord_less_nat @ ( size_s115229985653309035ist_fm @ Ms ) @ ( size_s115229985653309035ist_fm @ Ns ) )
| ( ( ( size_s115229985653309035ist_fm @ Ms )
= ( size_s115229985653309035ist_fm @ Ns ) )
& ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ M @ N ) @ R ) )
| ( ( M = N )
& ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ Ms @ Ns ) @ ( lenlex_list_fm @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_442_Cons__lenlex__iff,axiom,
! [M: tm,Ms: list_tm,N: tm,Ns: list_tm,R: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( cons_tm @ M @ Ms ) @ ( cons_tm @ N @ Ns ) ) @ ( lenlex_tm @ R ) )
= ( ( ord_less_nat @ ( size_size_list_tm @ Ms ) @ ( size_size_list_tm @ Ns ) )
| ( ( ( size_size_list_tm @ Ms )
= ( size_size_list_tm @ Ns ) )
& ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ M @ N ) @ R ) )
| ( ( M = N )
& ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Ms @ Ns ) @ ( lenlex_tm @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_443_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_list_tm,X: list_tm,Ys: list_list_tm,Y3: list_tm,R: set_Pr1507011332596240839ist_tm] :
( ( member1414320910663981712ist_tm @ ( produc8086241248684931031ist_tm @ ( append_list_tm @ Xs @ ( cons_list_tm @ X @ nil_list_tm ) ) @ ( append_list_tm @ Ys @ ( cons_list_tm @ Y3 @ nil_list_tm ) ) ) @ ( listrel1_list_tm @ R ) )
= ( ( ( member1414320910663981712ist_tm @ ( produc8086241248684931031ist_tm @ Xs @ Ys ) @ ( listrel1_list_tm @ R ) )
& ( X = Y3 ) )
| ( ( Xs = Ys )
& ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ X @ Y3 ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_444_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_fm,X: fm,Ys: list_fm,Y3: fm,R: set_Pr4706815898642364871_fm_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ ( append_fm @ Xs @ ( cons_fm @ X @ nil_fm ) ) @ ( append_fm @ Ys @ ( cons_fm @ Y3 @ nil_fm ) ) ) @ ( listrel1_fm @ R ) )
= ( ( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Xs @ Ys ) @ ( listrel1_fm @ R ) )
& ( X = Y3 ) )
| ( ( Xs = Ys )
& ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X @ Y3 ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_445_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_list_fm,X: list_fm,Ys: list_list_fm,Y3: list_fm,R: set_Pr1916137733696349511ist_fm] :
( ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ ( append_list_fm @ Xs @ ( cons_list_fm @ X @ nil_list_fm ) ) @ ( append_list_fm @ Ys @ ( cons_list_fm @ Y3 @ nil_list_fm ) ) ) @ ( listrel1_list_fm @ R ) )
= ( ( ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ Xs @ Ys ) @ ( listrel1_list_fm @ R ) )
& ( X = Y3 ) )
| ( ( Xs = Ys )
& ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ X @ Y3 ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_446_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_tm,X: tm,Ys: list_tm,Y3: tm,R: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( append_tm @ Xs @ ( cons_tm @ X @ nil_tm ) ) @ ( append_tm @ Ys @ ( cons_tm @ Y3 @ nil_tm ) ) ) @ ( listrel1_tm @ R ) )
= ( ( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ Ys ) @ ( listrel1_tm @ R ) )
& ( X = Y3 ) )
| ( ( Xs = Ys )
& ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X @ Y3 ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_447_concat__eq__appendD,axiom,
! [Xss: list_list_list_tm,Ys: list_list_tm,Zs: list_list_tm] :
( ( ( concat_list_tm @ Xss )
= ( append_list_tm @ Ys @ Zs ) )
=> ( ( Xss != nil_list_list_tm )
=> ? [Xss1: list_list_list_tm,Xs2: list_list_tm,Xs5: list_list_tm,Xss2: list_list_list_tm] :
( ( Xss
= ( append_list_list_tm @ Xss1 @ ( cons_list_list_tm @ ( append_list_tm @ Xs2 @ Xs5 ) @ Xss2 ) ) )
& ( Ys
= ( append_list_tm @ ( concat_list_tm @ Xss1 ) @ Xs2 ) )
& ( Zs
= ( append_list_tm @ Xs5 @ ( concat_list_tm @ Xss2 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_448_concat__eq__appendD,axiom,
! [Xss: list_list_tm,Ys: list_tm,Zs: list_tm] :
( ( ( concat_tm @ Xss )
= ( append_tm @ Ys @ Zs ) )
=> ( ( Xss != nil_list_tm )
=> ? [Xss1: list_list_tm,Xs2: list_tm,Xs5: list_tm,Xss2: list_list_tm] :
( ( Xss
= ( append_list_tm @ Xss1 @ ( cons_list_tm @ ( append_tm @ Xs2 @ Xs5 ) @ Xss2 ) ) )
& ( Ys
= ( append_tm @ ( concat_tm @ Xss1 ) @ Xs2 ) )
& ( Zs
= ( append_tm @ Xs5 @ ( concat_tm @ Xss2 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_449_concat__eq__appendD,axiom,
! [Xss: list_list_fm,Ys: list_fm,Zs: list_fm] :
( ( ( concat_fm @ Xss )
= ( append_fm @ Ys @ Zs ) )
=> ( ( Xss != nil_list_fm )
=> ? [Xss1: list_list_fm,Xs2: list_fm,Xs5: list_fm,Xss2: list_list_fm] :
( ( Xss
= ( append_list_fm @ Xss1 @ ( cons_list_fm @ ( append_fm @ Xs2 @ Xs5 ) @ Xss2 ) ) )
& ( Ys
= ( append_fm @ ( concat_fm @ Xss1 ) @ Xs2 ) )
& ( Zs
= ( append_fm @ Xs5 @ ( concat_fm @ Xss2 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_450_parts__in__effect,axiom,
! [P2: fm,Z3: list_fm,B4: list_tm,Z2: list_fm,R: rule,A4: list_tm] :
( ( member_fm2 @ P2 @ ( set_fm2 @ Z3 ) )
=> ( ( fmembe3754813877001230652ist_fm @ ( produc1414352766439514085ist_fm @ B4 @ Z2 ) @ ( effect @ R @ ( produc1414352766439514085ist_fm @ A4 @ Z3 ) ) )
=> ? [C2: list_tm,Xs2: list_fm] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ A4 ) @ ( set_tm2 @ C2 ) )
& ( member_list_fm2 @ Xs2 @ ( set_list_fm2 @ ( parts @ C2 @ R @ P2 ) ) )
& ( ord_less_eq_set_fm @ ( set_fm2 @ Xs2 ) @ ( set_fm2 @ Z2 ) ) ) ) ) ).
% parts_in_effect
thf(fact_451_in__set__product__lists__length,axiom,
! [Xs: list_fm,Xss: list_list_fm] :
( ( member_list_fm2 @ Xs @ ( set_list_fm2 @ ( product_lists_fm @ Xss ) ) )
=> ( ( size_size_list_fm @ Xs )
= ( size_s115229985653309035ist_fm @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_452_lexord__Nil__left,axiom,
! [Y3: list_fm,R: set_Pr4706815898642364871_fm_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ nil_fm @ Y3 ) @ ( lexord_fm @ R ) )
= ( ? [A5: fm,X2: list_fm] :
( Y3
= ( cons_fm @ A5 @ X2 ) ) ) ) ).
% lexord_Nil_left
thf(fact_453_lexord__Nil__left,axiom,
! [Y3: list_list_fm,R: set_Pr1916137733696349511ist_fm] :
( ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ nil_list_fm @ Y3 ) @ ( lexord_list_fm @ R ) )
= ( ? [A5: list_fm,X2: list_list_fm] :
( Y3
= ( cons_list_fm @ A5 @ X2 ) ) ) ) ).
% lexord_Nil_left
thf(fact_454_lexord__Nil__left,axiom,
! [Y3: list_tm,R: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ nil_tm @ Y3 ) @ ( lexord_tm @ R ) )
= ( ? [A5: tm,X2: list_tm] :
( Y3
= ( cons_tm @ A5 @ X2 ) ) ) ) ).
% lexord_Nil_left
thf(fact_455_fsubsetI,axiom,
! [A4: fset_P8989946509869081563ist_fm,B4: fset_P8989946509869081563ist_fm] :
( ! [X3: produc6018962875968178549ist_fm] :
( ( fmembe3754813877001230652ist_fm @ X3 @ A4 )
=> ( fmembe3754813877001230652ist_fm @ X3 @ B4 ) )
=> ( ord_le3986950534092794747ist_fm @ A4 @ B4 ) ) ).
% fsubsetI
thf(fact_456_fsubsetI,axiom,
! [A4: fset_tm,B4: fset_tm] :
( ! [X3: tm] :
( ( fmember_tm @ X3 @ A4 )
=> ( fmember_tm @ X3 @ B4 ) )
=> ( ord_less_eq_fset_tm @ A4 @ B4 ) ) ).
% fsubsetI
thf(fact_457_fsubsetI,axiom,
! [A4: fset_fm,B4: fset_fm] :
( ! [X3: fm] :
( ( fmember_fm @ X3 @ A4 )
=> ( fmember_fm @ X3 @ B4 ) )
=> ( ord_less_eq_fset_fm @ A4 @ B4 ) ) ).
% fsubsetI
thf(fact_458_fsubsetI,axiom,
! [A4: fset_list_fm,B4: fset_list_fm] :
( ! [X3: list_fm] :
( ( fmember_list_fm @ X3 @ A4 )
=> ( fmember_list_fm @ X3 @ B4 ) )
=> ( ord_le3182162295460067583ist_fm @ A4 @ B4 ) ) ).
% fsubsetI
thf(fact_459_append__is__Nil__conv,axiom,
! [Xs: list_list_tm,Ys: list_list_tm] :
( ( ( append_list_tm @ Xs @ Ys )
= nil_list_tm )
= ( ( Xs = nil_list_tm )
& ( Ys = nil_list_tm ) ) ) ).
% append_is_Nil_conv
thf(fact_460_append__is__Nil__conv,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( ( append_fm @ Xs @ Ys )
= nil_fm )
= ( ( Xs = nil_fm )
& ( Ys = nil_fm ) ) ) ).
% append_is_Nil_conv
thf(fact_461_append__is__Nil__conv,axiom,
! [Xs: list_list_fm,Ys: list_list_fm] :
( ( ( append_list_fm @ Xs @ Ys )
= nil_list_fm )
= ( ( Xs = nil_list_fm )
& ( Ys = nil_list_fm ) ) ) ).
% append_is_Nil_conv
thf(fact_462_append__is__Nil__conv,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( ( append_tm @ Xs @ Ys )
= nil_tm )
= ( ( Xs = nil_tm )
& ( Ys = nil_tm ) ) ) ).
% append_is_Nil_conv
thf(fact_463_Nil__is__append__conv,axiom,
! [Xs: list_list_tm,Ys: list_list_tm] :
( ( nil_list_tm
= ( append_list_tm @ Xs @ Ys ) )
= ( ( Xs = nil_list_tm )
& ( Ys = nil_list_tm ) ) ) ).
% Nil_is_append_conv
thf(fact_464_Nil__is__append__conv,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( nil_fm
= ( append_fm @ Xs @ Ys ) )
= ( ( Xs = nil_fm )
& ( Ys = nil_fm ) ) ) ).
% Nil_is_append_conv
thf(fact_465_Nil__is__append__conv,axiom,
! [Xs: list_list_fm,Ys: list_list_fm] :
( ( nil_list_fm
= ( append_list_fm @ Xs @ Ys ) )
= ( ( Xs = nil_list_fm )
& ( Ys = nil_list_fm ) ) ) ).
% Nil_is_append_conv
thf(fact_466_Nil__is__append__conv,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( nil_tm
= ( append_tm @ Xs @ Ys ) )
= ( ( Xs = nil_tm )
& ( Ys = nil_tm ) ) ) ).
% Nil_is_append_conv
thf(fact_467_self__append__conv2,axiom,
! [Y3: list_list_tm,Xs: list_list_tm] :
( ( Y3
= ( append_list_tm @ Xs @ Y3 ) )
= ( Xs = nil_list_tm ) ) ).
% self_append_conv2
thf(fact_468_self__append__conv2,axiom,
! [Y3: list_fm,Xs: list_fm] :
( ( Y3
= ( append_fm @ Xs @ Y3 ) )
= ( Xs = nil_fm ) ) ).
% self_append_conv2
thf(fact_469_self__append__conv2,axiom,
! [Y3: list_list_fm,Xs: list_list_fm] :
( ( Y3
= ( append_list_fm @ Xs @ Y3 ) )
= ( Xs = nil_list_fm ) ) ).
% self_append_conv2
thf(fact_470_self__append__conv2,axiom,
! [Y3: list_tm,Xs: list_tm] :
( ( Y3
= ( append_tm @ Xs @ Y3 ) )
= ( Xs = nil_tm ) ) ).
% self_append_conv2
thf(fact_471_append__self__conv2,axiom,
! [Xs: list_list_tm,Ys: list_list_tm] :
( ( ( append_list_tm @ Xs @ Ys )
= Ys )
= ( Xs = nil_list_tm ) ) ).
% append_self_conv2
thf(fact_472_append__self__conv2,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( ( append_fm @ Xs @ Ys )
= Ys )
= ( Xs = nil_fm ) ) ).
% append_self_conv2
thf(fact_473_append__self__conv2,axiom,
! [Xs: list_list_fm,Ys: list_list_fm] :
( ( ( append_list_fm @ Xs @ Ys )
= Ys )
= ( Xs = nil_list_fm ) ) ).
% append_self_conv2
thf(fact_474_append__self__conv2,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( ( append_tm @ Xs @ Ys )
= Ys )
= ( Xs = nil_tm ) ) ).
% append_self_conv2
thf(fact_475_self__append__conv,axiom,
! [Y3: list_list_tm,Ys: list_list_tm] :
( ( Y3
= ( append_list_tm @ Y3 @ Ys ) )
= ( Ys = nil_list_tm ) ) ).
% self_append_conv
thf(fact_476_self__append__conv,axiom,
! [Y3: list_fm,Ys: list_fm] :
( ( Y3
= ( append_fm @ Y3 @ Ys ) )
= ( Ys = nil_fm ) ) ).
% self_append_conv
thf(fact_477_self__append__conv,axiom,
! [Y3: list_list_fm,Ys: list_list_fm] :
( ( Y3
= ( append_list_fm @ Y3 @ Ys ) )
= ( Ys = nil_list_fm ) ) ).
% self_append_conv
thf(fact_478_self__append__conv,axiom,
! [Y3: list_tm,Ys: list_tm] :
( ( Y3
= ( append_tm @ Y3 @ Ys ) )
= ( Ys = nil_tm ) ) ).
% self_append_conv
thf(fact_479_append__self__conv,axiom,
! [Xs: list_list_tm,Ys: list_list_tm] :
( ( ( append_list_tm @ Xs @ Ys )
= Xs )
= ( Ys = nil_list_tm ) ) ).
% append_self_conv
thf(fact_480_append__self__conv,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( ( append_fm @ Xs @ Ys )
= Xs )
= ( Ys = nil_fm ) ) ).
% append_self_conv
thf(fact_481_append__self__conv,axiom,
! [Xs: list_list_fm,Ys: list_list_fm] :
( ( ( append_list_fm @ Xs @ Ys )
= Xs )
= ( Ys = nil_list_fm ) ) ).
% append_self_conv
thf(fact_482_append__self__conv,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( ( append_tm @ Xs @ Ys )
= Xs )
= ( Ys = nil_tm ) ) ).
% append_self_conv
thf(fact_483_append__Nil2,axiom,
! [Xs: list_list_tm] :
( ( append_list_tm @ Xs @ nil_list_tm )
= Xs ) ).
% append_Nil2
thf(fact_484_append__Nil2,axiom,
! [Xs: list_fm] :
( ( append_fm @ Xs @ nil_fm )
= Xs ) ).
% append_Nil2
thf(fact_485_append__Nil2,axiom,
! [Xs: list_list_fm] :
( ( append_list_fm @ Xs @ nil_list_fm )
= Xs ) ).
% append_Nil2
thf(fact_486_append__Nil2,axiom,
! [Xs: list_tm] :
( ( append_tm @ Xs @ nil_tm )
= Xs ) ).
% append_Nil2
thf(fact_487_append_Oright__neutral,axiom,
! [A: list_list_tm] :
( ( append_list_tm @ A @ nil_list_tm )
= A ) ).
% append.right_neutral
thf(fact_488_append_Oright__neutral,axiom,
! [A: list_fm] :
( ( append_fm @ A @ nil_fm )
= A ) ).
% append.right_neutral
thf(fact_489_append_Oright__neutral,axiom,
! [A: list_list_fm] :
( ( append_list_fm @ A @ nil_list_fm )
= A ) ).
% append.right_neutral
thf(fact_490_append_Oright__neutral,axiom,
! [A: list_tm] :
( ( append_tm @ A @ nil_tm )
= A ) ).
% append.right_neutral
thf(fact_491_map__is__Nil__conv,axiom,
! [F: fm > fm,Xs: list_fm] :
( ( ( map_fm_fm @ F @ Xs )
= nil_fm )
= ( Xs = nil_fm ) ) ).
% map_is_Nil_conv
thf(fact_492_map__is__Nil__conv,axiom,
! [F: tm > fm,Xs: list_tm] :
( ( ( map_tm_fm @ F @ Xs )
= nil_fm )
= ( Xs = nil_tm ) ) ).
% map_is_Nil_conv
thf(fact_493_map__is__Nil__conv,axiom,
! [F: fm > tm,Xs: list_fm] :
( ( ( map_fm_tm @ F @ Xs )
= nil_tm )
= ( Xs = nil_fm ) ) ).
% map_is_Nil_conv
thf(fact_494_map__is__Nil__conv,axiom,
! [F: tm > tm,Xs: list_tm] :
( ( ( map_tm_tm @ F @ Xs )
= nil_tm )
= ( Xs = nil_tm ) ) ).
% map_is_Nil_conv
thf(fact_495_map__is__Nil__conv,axiom,
! [F: list_fm > fm,Xs: list_list_fm] :
( ( ( map_list_fm_fm @ F @ Xs )
= nil_fm )
= ( Xs = nil_list_fm ) ) ).
% map_is_Nil_conv
thf(fact_496_map__is__Nil__conv,axiom,
! [F: fm > list_fm,Xs: list_fm] :
( ( ( map_fm_list_fm @ F @ Xs )
= nil_list_fm )
= ( Xs = nil_fm ) ) ).
% map_is_Nil_conv
thf(fact_497_map__is__Nil__conv,axiom,
! [F: tm > list_fm,Xs: list_tm] :
( ( ( map_tm_list_fm @ F @ Xs )
= nil_list_fm )
= ( Xs = nil_tm ) ) ).
% map_is_Nil_conv
thf(fact_498_map__is__Nil__conv,axiom,
! [F: list_fm > tm,Xs: list_list_fm] :
( ( ( map_list_fm_tm @ F @ Xs )
= nil_tm )
= ( Xs = nil_list_fm ) ) ).
% map_is_Nil_conv
thf(fact_499_map__is__Nil__conv,axiom,
! [F: fm > list_tm,Xs: list_fm] :
( ( ( map_fm_list_tm @ F @ Xs )
= nil_list_tm )
= ( Xs = nil_fm ) ) ).
% map_is_Nil_conv
thf(fact_500_map__is__Nil__conv,axiom,
! [F: tm > list_tm,Xs: list_tm] :
( ( ( map_tm_list_tm @ F @ Xs )
= nil_list_tm )
= ( Xs = nil_tm ) ) ).
% map_is_Nil_conv
thf(fact_501_Nil__is__map__conv,axiom,
! [F: fm > fm,Xs: list_fm] :
( ( nil_fm
= ( map_fm_fm @ F @ Xs ) )
= ( Xs = nil_fm ) ) ).
% Nil_is_map_conv
thf(fact_502_Nil__is__map__conv,axiom,
! [F: tm > fm,Xs: list_tm] :
( ( nil_fm
= ( map_tm_fm @ F @ Xs ) )
= ( Xs = nil_tm ) ) ).
% Nil_is_map_conv
thf(fact_503_Nil__is__map__conv,axiom,
! [F: fm > tm,Xs: list_fm] :
( ( nil_tm
= ( map_fm_tm @ F @ Xs ) )
= ( Xs = nil_fm ) ) ).
% Nil_is_map_conv
thf(fact_504_Nil__is__map__conv,axiom,
! [F: tm > tm,Xs: list_tm] :
( ( nil_tm
= ( map_tm_tm @ F @ Xs ) )
= ( Xs = nil_tm ) ) ).
% Nil_is_map_conv
thf(fact_505_Nil__is__map__conv,axiom,
! [F: list_fm > fm,Xs: list_list_fm] :
( ( nil_fm
= ( map_list_fm_fm @ F @ Xs ) )
= ( Xs = nil_list_fm ) ) ).
% Nil_is_map_conv
thf(fact_506_Nil__is__map__conv,axiom,
! [F: fm > list_fm,Xs: list_fm] :
( ( nil_list_fm
= ( map_fm_list_fm @ F @ Xs ) )
= ( Xs = nil_fm ) ) ).
% Nil_is_map_conv
thf(fact_507_Nil__is__map__conv,axiom,
! [F: tm > list_fm,Xs: list_tm] :
( ( nil_list_fm
= ( map_tm_list_fm @ F @ Xs ) )
= ( Xs = nil_tm ) ) ).
% Nil_is_map_conv
thf(fact_508_Nil__is__map__conv,axiom,
! [F: list_fm > tm,Xs: list_list_fm] :
( ( nil_tm
= ( map_list_fm_tm @ F @ Xs ) )
= ( Xs = nil_list_fm ) ) ).
% Nil_is_map_conv
thf(fact_509_Nil__is__map__conv,axiom,
! [F: fm > list_tm,Xs: list_fm] :
( ( nil_list_tm
= ( map_fm_list_tm @ F @ Xs ) )
= ( Xs = nil_fm ) ) ).
% Nil_is_map_conv
thf(fact_510_Nil__is__map__conv,axiom,
! [F: tm > list_tm,Xs: list_tm] :
( ( nil_list_tm
= ( map_tm_list_tm @ F @ Xs ) )
= ( Xs = nil_tm ) ) ).
% Nil_is_map_conv
thf(fact_511_list_Omap__disc__iff,axiom,
! [F: fm > fm,A: list_fm] :
( ( ( map_fm_fm @ F @ A )
= nil_fm )
= ( A = nil_fm ) ) ).
% list.map_disc_iff
thf(fact_512_list_Omap__disc__iff,axiom,
! [F: tm > fm,A: list_tm] :
( ( ( map_tm_fm @ F @ A )
= nil_fm )
= ( A = nil_tm ) ) ).
% list.map_disc_iff
thf(fact_513_list_Omap__disc__iff,axiom,
! [F: fm > tm,A: list_fm] :
( ( ( map_fm_tm @ F @ A )
= nil_tm )
= ( A = nil_fm ) ) ).
% list.map_disc_iff
thf(fact_514_list_Omap__disc__iff,axiom,
! [F: tm > tm,A: list_tm] :
( ( ( map_tm_tm @ F @ A )
= nil_tm )
= ( A = nil_tm ) ) ).
% list.map_disc_iff
thf(fact_515_list_Omap__disc__iff,axiom,
! [F: list_fm > fm,A: list_list_fm] :
( ( ( map_list_fm_fm @ F @ A )
= nil_fm )
= ( A = nil_list_fm ) ) ).
% list.map_disc_iff
thf(fact_516_list_Omap__disc__iff,axiom,
! [F: fm > list_fm,A: list_fm] :
( ( ( map_fm_list_fm @ F @ A )
= nil_list_fm )
= ( A = nil_fm ) ) ).
% list.map_disc_iff
thf(fact_517_list_Omap__disc__iff,axiom,
! [F: tm > list_fm,A: list_tm] :
( ( ( map_tm_list_fm @ F @ A )
= nil_list_fm )
= ( A = nil_tm ) ) ).
% list.map_disc_iff
thf(fact_518_list_Omap__disc__iff,axiom,
! [F: list_fm > tm,A: list_list_fm] :
( ( ( map_list_fm_tm @ F @ A )
= nil_tm )
= ( A = nil_list_fm ) ) ).
% list.map_disc_iff
thf(fact_519_list_Omap__disc__iff,axiom,
! [F: fm > list_tm,A: list_fm] :
( ( ( map_fm_list_tm @ F @ A )
= nil_list_tm )
= ( A = nil_fm ) ) ).
% list.map_disc_iff
thf(fact_520_list_Omap__disc__iff,axiom,
! [F: tm > list_tm,A: list_tm] :
( ( ( map_tm_list_tm @ F @ A )
= nil_list_tm )
= ( A = nil_tm ) ) ).
% list.map_disc_iff
thf(fact_521_remdups__eq__nil__right__iff,axiom,
! [X: list_fm] :
( ( nil_fm
= ( remdups_fm @ X ) )
= ( X = nil_fm ) ) ).
% remdups_eq_nil_right_iff
thf(fact_522_remdups__eq__nil__right__iff,axiom,
! [X: list_list_fm] :
( ( nil_list_fm
= ( remdups_list_fm @ X ) )
= ( X = nil_list_fm ) ) ).
% remdups_eq_nil_right_iff
thf(fact_523_remdups__eq__nil__right__iff,axiom,
! [X: list_tm] :
( ( nil_tm
= ( remdups_tm @ X ) )
= ( X = nil_tm ) ) ).
% remdups_eq_nil_right_iff
thf(fact_524_remdups__eq__nil__iff,axiom,
! [X: list_fm] :
( ( ( remdups_fm @ X )
= nil_fm )
= ( X = nil_fm ) ) ).
% remdups_eq_nil_iff
thf(fact_525_remdups__eq__nil__iff,axiom,
! [X: list_list_fm] :
( ( ( remdups_list_fm @ X )
= nil_list_fm )
= ( X = nil_list_fm ) ) ).
% remdups_eq_nil_iff
thf(fact_526_remdups__eq__nil__iff,axiom,
! [X: list_tm] :
( ( ( remdups_tm @ X )
= nil_tm )
= ( X = nil_tm ) ) ).
% remdups_eq_nil_iff
thf(fact_527_bind__simps_I1_J,axiom,
! [F: fm > list_fm] :
( ( bind_fm_fm @ nil_fm @ F )
= nil_fm ) ).
% bind_simps(1)
thf(fact_528_bind__simps_I1_J,axiom,
! [F: fm > list_list_fm] :
( ( bind_fm_list_fm @ nil_fm @ F )
= nil_list_fm ) ).
% bind_simps(1)
thf(fact_529_bind__simps_I1_J,axiom,
! [F: fm > list_tm] :
( ( bind_fm_tm @ nil_fm @ F )
= nil_tm ) ).
% bind_simps(1)
thf(fact_530_bind__simps_I1_J,axiom,
! [F: list_fm > list_fm] :
( ( bind_list_fm_fm @ nil_list_fm @ F )
= nil_fm ) ).
% bind_simps(1)
thf(fact_531_bind__simps_I1_J,axiom,
! [F: list_fm > list_list_fm] :
( ( bind_list_fm_list_fm @ nil_list_fm @ F )
= nil_list_fm ) ).
% bind_simps(1)
thf(fact_532_bind__simps_I1_J,axiom,
! [F: list_fm > list_tm] :
( ( bind_list_fm_tm @ nil_list_fm @ F )
= nil_tm ) ).
% bind_simps(1)
thf(fact_533_bind__simps_I1_J,axiom,
! [F: tm > list_fm] :
( ( bind_tm_fm @ nil_tm @ F )
= nil_fm ) ).
% bind_simps(1)
thf(fact_534_bind__simps_I1_J,axiom,
! [F: tm > list_list_fm] :
( ( bind_tm_list_fm @ nil_tm @ F )
= nil_list_fm ) ).
% bind_simps(1)
thf(fact_535_bind__simps_I1_J,axiom,
! [F: tm > list_tm] :
( ( bind_tm_tm @ nil_tm @ F )
= nil_tm ) ).
% bind_simps(1)
thf(fact_536_append1__eq__conv,axiom,
! [Xs: list_list_tm,X: list_tm,Ys: list_list_tm,Y3: list_tm] :
( ( ( append_list_tm @ Xs @ ( cons_list_tm @ X @ nil_list_tm ) )
= ( append_list_tm @ Ys @ ( cons_list_tm @ Y3 @ nil_list_tm ) ) )
= ( ( Xs = Ys )
& ( X = Y3 ) ) ) ).
% append1_eq_conv
thf(fact_537_append1__eq__conv,axiom,
! [Xs: list_fm,X: fm,Ys: list_fm,Y3: fm] :
( ( ( append_fm @ Xs @ ( cons_fm @ X @ nil_fm ) )
= ( append_fm @ Ys @ ( cons_fm @ Y3 @ nil_fm ) ) )
= ( ( Xs = Ys )
& ( X = Y3 ) ) ) ).
% append1_eq_conv
thf(fact_538_append1__eq__conv,axiom,
! [Xs: list_list_fm,X: list_fm,Ys: list_list_fm,Y3: list_fm] :
( ( ( append_list_fm @ Xs @ ( cons_list_fm @ X @ nil_list_fm ) )
= ( append_list_fm @ Ys @ ( cons_list_fm @ Y3 @ nil_list_fm ) ) )
= ( ( Xs = Ys )
& ( X = Y3 ) ) ) ).
% append1_eq_conv
thf(fact_539_append1__eq__conv,axiom,
! [Xs: list_tm,X: tm,Ys: list_tm,Y3: tm] :
( ( ( append_tm @ Xs @ ( cons_tm @ X @ nil_tm ) )
= ( append_tm @ Ys @ ( cons_tm @ Y3 @ nil_tm ) ) )
= ( ( Xs = Ys )
& ( X = Y3 ) ) ) ).
% append1_eq_conv
thf(fact_540_fset__of__list__subset,axiom,
! [Xs: list_list_fm,Ys: list_list_fm] :
( ( ord_le7838213414353715577ist_fm @ ( set_list_fm2 @ Xs ) @ ( set_list_fm2 @ Ys ) )
=> ( ord_le3182162295460067583ist_fm @ ( fset_of_list_list_fm @ Xs ) @ ( fset_of_list_list_fm @ Ys ) ) ) ).
% fset_of_list_subset
thf(fact_541_fset__of__list__subset,axiom,
! [Xs: list_P2887561121880082555ist_fm,Ys: list_P2887561121880082555ist_fm] :
( ( ord_le1771420097867575541ist_fm @ ( set_Pr5051287786238754058ist_fm @ Xs ) @ ( set_Pr5051287786238754058ist_fm @ Ys ) )
=> ( ord_le3986950534092794747ist_fm @ ( fset_o3222440871817023035ist_fm @ Xs ) @ ( fset_o3222440871817023035ist_fm @ Ys ) ) ) ).
% fset_of_list_subset
thf(fact_542_fset__of__list__subset,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ Xs ) @ ( set_tm2 @ Ys ) )
=> ( ord_less_eq_fset_tm @ ( fset_of_list_tm @ Xs ) @ ( fset_of_list_tm @ Ys ) ) ) ).
% fset_of_list_subset
thf(fact_543_fset__of__list__subset,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ ( set_fm2 @ Ys ) )
=> ( ord_less_eq_fset_fm @ ( fset_of_list_fm @ Xs ) @ ( fset_of_list_fm @ Ys ) ) ) ).
% fset_of_list_subset
thf(fact_544_concat__eq__Nil__conv,axiom,
! [Xss: list_list_list_fm] :
( ( ( concat_list_fm @ Xss )
= nil_list_fm )
= ( ! [X2: list_list_fm] :
( ( member_list_list_fm @ X2 @ ( set_list_list_fm2 @ Xss ) )
=> ( X2 = nil_list_fm ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_545_concat__eq__Nil__conv,axiom,
! [Xss: list_list_tm] :
( ( ( concat_tm @ Xss )
= nil_tm )
= ( ! [X2: list_tm] :
( ( member_list_tm @ X2 @ ( set_list_tm2 @ Xss ) )
=> ( X2 = nil_tm ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_546_concat__eq__Nil__conv,axiom,
! [Xss: list_list_fm] :
( ( ( concat_fm @ Xss )
= nil_fm )
= ( ! [X2: list_fm] :
( ( member_list_fm2 @ X2 @ ( set_list_fm2 @ Xss ) )
=> ( X2 = nil_fm ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_547_Nil__eq__concat__conv,axiom,
! [Xss: list_list_list_fm] :
( ( nil_list_fm
= ( concat_list_fm @ Xss ) )
= ( ! [X2: list_list_fm] :
( ( member_list_list_fm @ X2 @ ( set_list_list_fm2 @ Xss ) )
=> ( X2 = nil_list_fm ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_548_Nil__eq__concat__conv,axiom,
! [Xss: list_list_tm] :
( ( nil_tm
= ( concat_tm @ Xss ) )
= ( ! [X2: list_tm] :
( ( member_list_tm @ X2 @ ( set_list_tm2 @ Xss ) )
=> ( X2 = nil_tm ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_549_Nil__eq__concat__conv,axiom,
! [Xss: list_list_fm] :
( ( nil_fm
= ( concat_fm @ Xss ) )
= ( ! [X2: list_fm] :
( ( member_list_fm2 @ X2 @ ( set_list_fm2 @ Xss ) )
=> ( X2 = nil_fm ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_550_insert__Nil,axiom,
! [X: fm] :
( ( insert_fm @ X @ nil_fm )
= ( cons_fm @ X @ nil_fm ) ) ).
% insert_Nil
thf(fact_551_insert__Nil,axiom,
! [X: list_fm] :
( ( insert_list_fm @ X @ nil_list_fm )
= ( cons_list_fm @ X @ nil_list_fm ) ) ).
% insert_Nil
thf(fact_552_insert__Nil,axiom,
! [X: tm] :
( ( insert_tm @ X @ nil_tm )
= ( cons_tm @ X @ nil_tm ) ) ).
% insert_Nil
thf(fact_553_Nil__lenlex__iff1,axiom,
! [Ns: list_fm,R: set_Pr4706815898642364871_fm_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ nil_fm @ Ns ) @ ( lenlex_fm @ R ) )
= ( Ns != nil_fm ) ) ).
% Nil_lenlex_iff1
thf(fact_554_Nil__lenlex__iff1,axiom,
! [Ns: list_list_fm,R: set_Pr1916137733696349511ist_fm] :
( ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ nil_list_fm @ Ns ) @ ( lenlex_list_fm @ R ) )
= ( Ns != nil_list_fm ) ) ).
% Nil_lenlex_iff1
thf(fact_555_Nil__lenlex__iff1,axiom,
! [Ns: list_tm,R: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ nil_tm @ Ns ) @ ( lenlex_tm @ R ) )
= ( Ns != nil_tm ) ) ).
% Nil_lenlex_iff1
thf(fact_556_product__lists_Osimps_I1_J,axiom,
( ( produc373462945560358120ist_fm @ nil_list_list_fm )
= ( cons_list_list_fm @ nil_list_fm @ nil_list_list_fm ) ) ).
% product_lists.simps(1)
thf(fact_557_product__lists_Osimps_I1_J,axiom,
( ( product_lists_tm @ nil_list_tm )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) ).
% product_lists.simps(1)
thf(fact_558_product__lists_Osimps_I1_J,axiom,
( ( product_lists_fm @ nil_list_fm )
= ( cons_list_fm @ nil_fm @ nil_list_fm ) ) ).
% product_lists.simps(1)
thf(fact_559_order__le__imp__less__or__eq,axiom,
! [X: set_tm,Y3: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y3 )
=> ( ( ord_less_set_tm @ X @ Y3 )
| ( X = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_560_order__le__imp__less__or__eq,axiom,
! [X: set_fm,Y3: set_fm] :
( ( ord_less_eq_set_fm @ X @ Y3 )
=> ( ( ord_less_set_fm @ X @ Y3 )
| ( X = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_561_order__le__imp__less__or__eq,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ( ord_less_nat @ X @ Y3 )
| ( X = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_562_linorder__le__less__linear,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
| ( ord_less_nat @ Y3 @ X ) ) ).
% linorder_le_less_linear
thf(fact_563_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_tm,C: set_tm] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_set_tm @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_set_tm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_set_tm @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_564_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_fm,C: set_fm] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_set_fm @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_set_fm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_set_fm @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_565_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_566_order__less__le__subst1,axiom,
! [A: set_tm,F: set_tm > set_tm,B: set_tm,C: set_tm] :
( ( ord_less_set_tm @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_tm @ B @ C )
=> ( ! [X3: set_tm,Y: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_set_tm @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_567_order__less__le__subst1,axiom,
! [A: set_fm,F: set_tm > set_fm,B: set_tm,C: set_tm] :
( ( ord_less_set_fm @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_tm @ B @ C )
=> ( ! [X3: set_tm,Y: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_set_fm @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_568_order__less__le__subst1,axiom,
! [A: nat,F: set_tm > nat,B: set_tm,C: set_tm] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_tm @ B @ C )
=> ( ! [X3: set_tm,Y: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_569_order__less__le__subst1,axiom,
! [A: set_tm,F: set_fm > set_tm,B: set_fm,C: set_fm] :
( ( ord_less_set_tm @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_fm @ B @ C )
=> ( ! [X3: set_fm,Y: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_set_tm @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_570_order__less__le__subst1,axiom,
! [A: set_fm,F: set_fm > set_fm,B: set_fm,C: set_fm] :
( ( ord_less_set_fm @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_fm @ B @ C )
=> ( ! [X3: set_fm,Y: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_set_fm @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_571_order__less__le__subst1,axiom,
! [A: nat,F: set_fm > nat,B: set_fm,C: set_fm] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_fm @ B @ C )
=> ( ! [X3: set_fm,Y: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_572_order__less__le__subst1,axiom,
! [A: set_tm,F: nat > set_tm,B: nat,C: nat] :
( ( ord_less_set_tm @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_set_tm @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_573_order__less__le__subst1,axiom,
! [A: set_fm,F: nat > set_fm,B: nat,C: nat] :
( ( ord_less_set_fm @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_set_fm @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_574_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_575_order__le__less__subst2,axiom,
! [A: set_tm,B: set_tm,F: set_tm > set_tm,C: set_tm] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( ord_less_set_tm @ ( F @ B ) @ C )
=> ( ! [X3: set_tm,Y: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_set_tm @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_576_order__le__less__subst2,axiom,
! [A: set_tm,B: set_tm,F: set_tm > set_fm,C: set_fm] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( ord_less_set_fm @ ( F @ B ) @ C )
=> ( ! [X3: set_tm,Y: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_set_fm @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_577_order__le__less__subst2,axiom,
! [A: set_tm,B: set_tm,F: set_tm > nat,C: nat] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_tm,Y: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_578_order__le__less__subst2,axiom,
! [A: set_fm,B: set_fm,F: set_fm > set_tm,C: set_tm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ( ord_less_set_tm @ ( F @ B ) @ C )
=> ( ! [X3: set_fm,Y: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_set_tm @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_579_order__le__less__subst2,axiom,
! [A: set_fm,B: set_fm,F: set_fm > set_fm,C: set_fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ( ord_less_set_fm @ ( F @ B ) @ C )
=> ( ! [X3: set_fm,Y: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_set_fm @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_580_order__le__less__subst2,axiom,
! [A: set_fm,B: set_fm,F: set_fm > nat,C: nat] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_fm,Y: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_581_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_tm,C: set_tm] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_tm @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_set_tm @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_582_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_fm,C: set_fm] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_fm @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_set_fm @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_583_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_584_order__le__less__subst1,axiom,
! [A: set_tm,F: nat > set_tm,B: nat,C: nat] :
( ( ord_less_eq_set_tm @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_set_tm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_set_tm @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_585_order__le__less__subst1,axiom,
! [A: set_fm,F: nat > set_fm,B: nat,C: nat] :
( ( ord_less_eq_set_fm @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_set_fm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_set_fm @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_586_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_587_order__less__le__trans,axiom,
! [X: set_tm,Y3: set_tm,Z3: set_tm] :
( ( ord_less_set_tm @ X @ Y3 )
=> ( ( ord_less_eq_set_tm @ Y3 @ Z3 )
=> ( ord_less_set_tm @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_588_order__less__le__trans,axiom,
! [X: set_fm,Y3: set_fm,Z3: set_fm] :
( ( ord_less_set_fm @ X @ Y3 )
=> ( ( ord_less_eq_set_fm @ Y3 @ Z3 )
=> ( ord_less_set_fm @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_589_order__less__le__trans,axiom,
! [X: nat,Y3: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_590_order__le__less__trans,axiom,
! [X: set_tm,Y3: set_tm,Z3: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y3 )
=> ( ( ord_less_set_tm @ Y3 @ Z3 )
=> ( ord_less_set_tm @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_591_order__le__less__trans,axiom,
! [X: set_fm,Y3: set_fm,Z3: set_fm] :
( ( ord_less_eq_set_fm @ X @ Y3 )
=> ( ( ord_less_set_fm @ Y3 @ Z3 )
=> ( ord_less_set_fm @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_592_order__le__less__trans,axiom,
! [X: nat,Y3: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_593_order__neq__le__trans,axiom,
! [A: set_tm,B: set_tm] :
( ( A != B )
=> ( ( ord_less_eq_set_tm @ A @ B )
=> ( ord_less_set_tm @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_594_order__neq__le__trans,axiom,
! [A: set_fm,B: set_fm] :
( ( A != B )
=> ( ( ord_less_eq_set_fm @ A @ B )
=> ( ord_less_set_fm @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_595_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_596_order__le__neq__trans,axiom,
! [A: set_tm,B: set_tm] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_tm @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_597_order__le__neq__trans,axiom,
! [A: set_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_fm @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_598_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_599_order__less__imp__le,axiom,
! [X: set_tm,Y3: set_tm] :
( ( ord_less_set_tm @ X @ Y3 )
=> ( ord_less_eq_set_tm @ X @ Y3 ) ) ).
% order_less_imp_le
thf(fact_600_order__less__imp__le,axiom,
! [X: set_fm,Y3: set_fm] :
( ( ord_less_set_fm @ X @ Y3 )
=> ( ord_less_eq_set_fm @ X @ Y3 ) ) ).
% order_less_imp_le
thf(fact_601_order__less__imp__le,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ X @ Y3 ) ) ).
% order_less_imp_le
thf(fact_602_linorder__not__less,axiom,
! [X: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X @ Y3 ) )
= ( ord_less_eq_nat @ Y3 @ X ) ) ).
% linorder_not_less
thf(fact_603_linorder__not__le,axiom,
! [X: nat,Y3: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y3 ) )
= ( ord_less_nat @ Y3 @ X ) ) ).
% linorder_not_le
thf(fact_604_order__less__le,axiom,
( ord_less_set_tm
= ( ^ [X2: set_tm,Y4: set_tm] :
( ( ord_less_eq_set_tm @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_605_order__less__le,axiom,
( ord_less_set_fm
= ( ^ [X2: set_fm,Y4: set_fm] :
( ( ord_less_eq_set_fm @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_606_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_607_order__le__less,axiom,
( ord_less_eq_set_tm
= ( ^ [X2: set_tm,Y4: set_tm] :
( ( ord_less_set_tm @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_608_order__le__less,axiom,
( ord_less_eq_set_fm
= ( ^ [X2: set_fm,Y4: set_fm] :
( ( ord_less_set_fm @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_609_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_nat @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_610_dual__order_Ostrict__implies__order,axiom,
! [B: set_tm,A: set_tm] :
( ( ord_less_set_tm @ B @ A )
=> ( ord_less_eq_set_tm @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_611_dual__order_Ostrict__implies__order,axiom,
! [B: set_fm,A: set_fm] :
( ( ord_less_set_fm @ B @ A )
=> ( ord_less_eq_set_fm @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_612_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_613_order_Ostrict__implies__order,axiom,
! [A: set_tm,B: set_tm] :
( ( ord_less_set_tm @ A @ B )
=> ( ord_less_eq_set_tm @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_614_order_Ostrict__implies__order,axiom,
! [A: set_fm,B: set_fm] :
( ( ord_less_set_fm @ A @ B )
=> ( ord_less_eq_set_fm @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_615_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_616_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_tm
= ( ^ [B6: set_tm,A5: set_tm] :
( ( ord_less_eq_set_tm @ B6 @ A5 )
& ~ ( ord_less_eq_set_tm @ A5 @ B6 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_617_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_fm
= ( ^ [B6: set_fm,A5: set_fm] :
( ( ord_less_eq_set_fm @ B6 @ A5 )
& ~ ( ord_less_eq_set_fm @ A5 @ B6 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_618_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B6: nat,A5: nat] :
( ( ord_less_eq_nat @ B6 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B6 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_619_dual__order_Ostrict__trans2,axiom,
! [B: set_tm,A: set_tm,C: set_tm] :
( ( ord_less_set_tm @ B @ A )
=> ( ( ord_less_eq_set_tm @ C @ B )
=> ( ord_less_set_tm @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_620_dual__order_Ostrict__trans2,axiom,
! [B: set_fm,A: set_fm,C: set_fm] :
( ( ord_less_set_fm @ B @ A )
=> ( ( ord_less_eq_set_fm @ C @ B )
=> ( ord_less_set_fm @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_621_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_622_dual__order_Ostrict__trans1,axiom,
! [B: set_tm,A: set_tm,C: set_tm] :
( ( ord_less_eq_set_tm @ B @ A )
=> ( ( ord_less_set_tm @ C @ B )
=> ( ord_less_set_tm @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_623_dual__order_Ostrict__trans1,axiom,
! [B: set_fm,A: set_fm,C: set_fm] :
( ( ord_less_eq_set_fm @ B @ A )
=> ( ( ord_less_set_fm @ C @ B )
=> ( ord_less_set_fm @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_624_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_625_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_tm
= ( ^ [B6: set_tm,A5: set_tm] :
( ( ord_less_eq_set_tm @ B6 @ A5 )
& ( A5 != B6 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_626_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_fm
= ( ^ [B6: set_fm,A5: set_fm] :
( ( ord_less_eq_set_fm @ B6 @ A5 )
& ( A5 != B6 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_627_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B6: nat,A5: nat] :
( ( ord_less_eq_nat @ B6 @ A5 )
& ( A5 != B6 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_628_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_tm
= ( ^ [B6: set_tm,A5: set_tm] :
( ( ord_less_set_tm @ B6 @ A5 )
| ( A5 = B6 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_629_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_fm
= ( ^ [B6: set_fm,A5: set_fm] :
( ( ord_less_set_fm @ B6 @ A5 )
| ( A5 = B6 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_630_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B6: nat,A5: nat] :
( ( ord_less_nat @ B6 @ A5 )
| ( A5 = B6 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_631_order_Ostrict__iff__not,axiom,
( ord_less_set_tm
= ( ^ [A5: set_tm,B6: set_tm] :
( ( ord_less_eq_set_tm @ A5 @ B6 )
& ~ ( ord_less_eq_set_tm @ B6 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_632_order_Ostrict__iff__not,axiom,
( ord_less_set_fm
= ( ^ [A5: set_fm,B6: set_fm] :
( ( ord_less_eq_set_fm @ A5 @ B6 )
& ~ ( ord_less_eq_set_fm @ B6 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_633_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B6: nat] :
( ( ord_less_eq_nat @ A5 @ B6 )
& ~ ( ord_less_eq_nat @ B6 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_634_order_Ostrict__trans2,axiom,
! [A: set_tm,B: set_tm,C: set_tm] :
( ( ord_less_set_tm @ A @ B )
=> ( ( ord_less_eq_set_tm @ B @ C )
=> ( ord_less_set_tm @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_635_order_Ostrict__trans2,axiom,
! [A: set_fm,B: set_fm,C: set_fm] :
( ( ord_less_set_fm @ A @ B )
=> ( ( ord_less_eq_set_fm @ B @ C )
=> ( ord_less_set_fm @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_636_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_637_order_Ostrict__trans1,axiom,
! [A: set_tm,B: set_tm,C: set_tm] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( ord_less_set_tm @ B @ C )
=> ( ord_less_set_tm @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_638_order_Ostrict__trans1,axiom,
! [A: set_fm,B: set_fm,C: set_fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ( ord_less_set_fm @ B @ C )
=> ( ord_less_set_fm @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_639_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_640_order_Ostrict__iff__order,axiom,
( ord_less_set_tm
= ( ^ [A5: set_tm,B6: set_tm] :
( ( ord_less_eq_set_tm @ A5 @ B6 )
& ( A5 != B6 ) ) ) ) ).
% order.strict_iff_order
thf(fact_641_order_Ostrict__iff__order,axiom,
( ord_less_set_fm
= ( ^ [A5: set_fm,B6: set_fm] :
( ( ord_less_eq_set_fm @ A5 @ B6 )
& ( A5 != B6 ) ) ) ) ).
% order.strict_iff_order
thf(fact_642_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B6: nat] :
( ( ord_less_eq_nat @ A5 @ B6 )
& ( A5 != B6 ) ) ) ) ).
% order.strict_iff_order
thf(fact_643_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_tm
= ( ^ [A5: set_tm,B6: set_tm] :
( ( ord_less_set_tm @ A5 @ B6 )
| ( A5 = B6 ) ) ) ) ).
% order.order_iff_strict
thf(fact_644_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_fm
= ( ^ [A5: set_fm,B6: set_fm] :
( ( ord_less_set_fm @ A5 @ B6 )
| ( A5 = B6 ) ) ) ) ).
% order.order_iff_strict
thf(fact_645_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B6: nat] :
( ( ord_less_nat @ A5 @ B6 )
| ( A5 = B6 ) ) ) ) ).
% order.order_iff_strict
thf(fact_646_not__le__imp__less,axiom,
! [Y3: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y3 @ X )
=> ( ord_less_nat @ X @ Y3 ) ) ).
% not_le_imp_less
thf(fact_647_less__le__not__le,axiom,
( ord_less_set_tm
= ( ^ [X2: set_tm,Y4: set_tm] :
( ( ord_less_eq_set_tm @ X2 @ Y4 )
& ~ ( ord_less_eq_set_tm @ Y4 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_648_less__le__not__le,axiom,
( ord_less_set_fm
= ( ^ [X2: set_fm,Y4: set_fm] :
( ( ord_less_eq_set_fm @ X2 @ Y4 )
& ~ ( ord_less_eq_set_fm @ Y4 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_649_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_650_antisym__conv2,axiom,
! [X: set_tm,Y3: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y3 )
=> ( ( ~ ( ord_less_set_tm @ X @ Y3 ) )
= ( X = Y3 ) ) ) ).
% antisym_conv2
thf(fact_651_antisym__conv2,axiom,
! [X: set_fm,Y3: set_fm] :
( ( ord_less_eq_set_fm @ X @ Y3 )
=> ( ( ~ ( ord_less_set_fm @ X @ Y3 ) )
= ( X = Y3 ) ) ) ).
% antisym_conv2
thf(fact_652_antisym__conv2,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ( ~ ( ord_less_nat @ X @ Y3 ) )
= ( X = Y3 ) ) ) ).
% antisym_conv2
thf(fact_653_antisym__conv1,axiom,
! [X: set_tm,Y3: set_tm] :
( ~ ( ord_less_set_tm @ X @ Y3 )
=> ( ( ord_less_eq_set_tm @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% antisym_conv1
thf(fact_654_antisym__conv1,axiom,
! [X: set_fm,Y3: set_fm] :
( ~ ( ord_less_set_fm @ X @ Y3 )
=> ( ( ord_less_eq_set_fm @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% antisym_conv1
thf(fact_655_antisym__conv1,axiom,
! [X: nat,Y3: nat] :
( ~ ( ord_less_nat @ X @ Y3 )
=> ( ( ord_less_eq_nat @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% antisym_conv1
thf(fact_656_nless__le,axiom,
! [A: set_tm,B: set_tm] :
( ( ~ ( ord_less_set_tm @ A @ B ) )
= ( ~ ( ord_less_eq_set_tm @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_657_nless__le,axiom,
! [A: set_fm,B: set_fm] :
( ( ~ ( ord_less_set_fm @ A @ B ) )
= ( ~ ( ord_less_eq_set_fm @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_658_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_659_leI,axiom,
! [X: nat,Y3: nat] :
( ~ ( ord_less_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X ) ) ).
% leI
thf(fact_660_leD,axiom,
! [Y3: set_tm,X: set_tm] :
( ( ord_less_eq_set_tm @ Y3 @ X )
=> ~ ( ord_less_set_tm @ X @ Y3 ) ) ).
% leD
thf(fact_661_leD,axiom,
! [Y3: set_fm,X: set_fm] :
( ( ord_less_eq_set_fm @ Y3 @ X )
=> ~ ( ord_less_set_fm @ X @ Y3 ) ) ).
% leD
thf(fact_662_leD,axiom,
! [Y3: nat,X: nat] :
( ( ord_less_eq_nat @ Y3 @ X )
=> ~ ( ord_less_nat @ X @ Y3 ) ) ).
% leD
thf(fact_663_concat_Osimps_I1_J,axiom,
( ( concat_list_fm @ nil_list_list_fm )
= nil_list_fm ) ).
% concat.simps(1)
thf(fact_664_concat_Osimps_I1_J,axiom,
( ( concat_tm @ nil_list_tm )
= nil_tm ) ).
% concat.simps(1)
thf(fact_665_concat_Osimps_I1_J,axiom,
( ( concat_fm @ nil_list_fm )
= nil_fm ) ).
% concat.simps(1)
thf(fact_666_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_667_less__imp__neq,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( X != Y3 ) ) ).
% less_imp_neq
thf(fact_668_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_669_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_670_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_671_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_672_antisym__conv3,axiom,
! [Y3: nat,X: nat] :
( ~ ( ord_less_nat @ Y3 @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y3 ) )
= ( X = Y3 ) ) ) ).
% antisym_conv3
thf(fact_673_linorder__cases,axiom,
! [X: nat,Y3: nat] :
( ~ ( ord_less_nat @ X @ Y3 )
=> ( ( X != Y3 )
=> ( ord_less_nat @ Y3 @ X ) ) ) ).
% linorder_cases
thf(fact_674_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_675_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_676_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X5: nat] : ( P3 @ X5 ) )
= ( ^ [P4: nat > $o] :
? [N2: nat] :
( ( P4 @ N2 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ( P4 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_677_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_678_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_679_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X @ Y3 ) )
= ( ( ord_less_nat @ Y3 @ X )
| ( X = Y3 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_680_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_681_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_682_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_683_linorder__neqE,axiom,
! [X: nat,Y3: nat] :
( ( X != Y3 )
=> ( ~ ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ Y3 @ X ) ) ) ).
% linorder_neqE
thf(fact_684_order__less__asym,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X ) ) ).
% order_less_asym
thf(fact_685_linorder__neq__iff,axiom,
! [X: nat,Y3: nat] :
( ( X != Y3 )
= ( ( ord_less_nat @ X @ Y3 )
| ( ord_less_nat @ Y3 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_686_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_687_order__less__trans,axiom,
! [X: nat,Y3: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_688_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_689_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_690_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_691_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_692_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_693_order__less__not__sym,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X ) ) ).
% order_less_not_sym
thf(fact_694_order__less__imp__triv,axiom,
! [X: nat,Y3: nat,P: $o] :
( ( ord_less_nat @ X @ Y3 )
=> ( ( ord_less_nat @ Y3 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_695_linorder__less__linear,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
| ( X = Y3 )
| ( ord_less_nat @ Y3 @ X ) ) ).
% linorder_less_linear
thf(fact_696_order__less__imp__not__eq,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( X != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_697_order__less__imp__not__eq2,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( Y3 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_698_order__less__imp__not__less,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X ) ) ).
% order_less_imp_not_less
thf(fact_699_transpose_Ocases,axiom,
! [X: list_list_list_fm] :
( ( X != nil_list_list_fm )
=> ( ! [Xss3: list_list_list_fm] :
( X
!= ( cons_list_list_fm @ nil_list_fm @ Xss3 ) )
=> ~ ! [X3: list_fm,Xs2: list_list_fm,Xss3: list_list_list_fm] :
( X
!= ( cons_list_list_fm @ ( cons_list_fm @ X3 @ Xs2 ) @ Xss3 ) ) ) ) ).
% transpose.cases
thf(fact_700_transpose_Ocases,axiom,
! [X: list_list_tm] :
( ( X != nil_list_tm )
=> ( ! [Xss3: list_list_tm] :
( X
!= ( cons_list_tm @ nil_tm @ Xss3 ) )
=> ~ ! [X3: tm,Xs2: list_tm,Xss3: list_list_tm] :
( X
!= ( cons_list_tm @ ( cons_tm @ X3 @ Xs2 ) @ Xss3 ) ) ) ) ).
% transpose.cases
thf(fact_701_transpose_Ocases,axiom,
! [X: list_list_fm] :
( ( X != nil_list_fm )
=> ( ! [Xss3: list_list_fm] :
( X
!= ( cons_list_fm @ nil_fm @ Xss3 ) )
=> ~ ! [X3: fm,Xs2: list_fm,Xss3: list_list_fm] :
( X
!= ( cons_list_fm @ ( cons_fm @ X3 @ Xs2 ) @ Xss3 ) ) ) ) ).
% transpose.cases
thf(fact_702_sorted__wrt_Ocases,axiom,
! [X: produc7963324949210141170ist_fm] :
( ! [P5: fm > fm > $o] :
( X
!= ( produc7687578365188660450ist_fm @ P5 @ nil_fm ) )
=> ~ ! [P5: fm > fm > $o,X3: fm,Ys3: list_fm] :
( X
!= ( produc7687578365188660450ist_fm @ P5 @ ( cons_fm @ X3 @ Ys3 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_703_sorted__wrt_Ocases,axiom,
! [X: produc6864569604571974008ist_fm] :
( ! [P5: list_fm > list_fm > $o] :
( X
!= ( produc3008759835361761000ist_fm @ P5 @ nil_list_fm ) )
=> ~ ! [P5: list_fm > list_fm > $o,X3: list_fm,Ys3: list_list_fm] :
( X
!= ( produc3008759835361761000ist_fm @ P5 @ ( cons_list_fm @ X3 @ Ys3 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_704_sorted__wrt_Ocases,axiom,
! [X: produc2002131169352006116ist_tm] :
( ! [P5: tm > tm > $o] :
( X
!= ( produc1972851280741670356ist_tm @ P5 @ nil_tm ) )
=> ~ ! [P5: tm > tm > $o,X3: tm,Ys3: list_tm] :
( X
!= ( produc1972851280741670356ist_tm @ P5 @ ( cons_tm @ X3 @ Ys3 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_705_successively_Ocases,axiom,
! [X: produc7963324949210141170ist_fm] :
( ! [P5: fm > fm > $o] :
( X
!= ( produc7687578365188660450ist_fm @ P5 @ nil_fm ) )
=> ( ! [P5: fm > fm > $o,X3: fm] :
( X
!= ( produc7687578365188660450ist_fm @ P5 @ ( cons_fm @ X3 @ nil_fm ) ) )
=> ~ ! [P5: fm > fm > $o,X3: fm,Y: fm,Xs2: list_fm] :
( X
!= ( produc7687578365188660450ist_fm @ P5 @ ( cons_fm @ X3 @ ( cons_fm @ Y @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_706_successively_Ocases,axiom,
! [X: produc6864569604571974008ist_fm] :
( ! [P5: list_fm > list_fm > $o] :
( X
!= ( produc3008759835361761000ist_fm @ P5 @ nil_list_fm ) )
=> ( ! [P5: list_fm > list_fm > $o,X3: list_fm] :
( X
!= ( produc3008759835361761000ist_fm @ P5 @ ( cons_list_fm @ X3 @ nil_list_fm ) ) )
=> ~ ! [P5: list_fm > list_fm > $o,X3: list_fm,Y: list_fm,Xs2: list_list_fm] :
( X
!= ( produc3008759835361761000ist_fm @ P5 @ ( cons_list_fm @ X3 @ ( cons_list_fm @ Y @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_707_successively_Ocases,axiom,
! [X: produc2002131169352006116ist_tm] :
( ! [P5: tm > tm > $o] :
( X
!= ( produc1972851280741670356ist_tm @ P5 @ nil_tm ) )
=> ( ! [P5: tm > tm > $o,X3: tm] :
( X
!= ( produc1972851280741670356ist_tm @ P5 @ ( cons_tm @ X3 @ nil_tm ) ) )
=> ~ ! [P5: tm > tm > $o,X3: tm,Y: tm,Xs2: list_tm] :
( X
!= ( produc1972851280741670356ist_tm @ P5 @ ( cons_tm @ X3 @ ( cons_tm @ Y @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_708_list__nonempty__induct,axiom,
! [Xs: list_fm,P: list_fm > $o] :
( ( Xs != nil_fm )
=> ( ! [X3: fm] : ( P @ ( cons_fm @ X3 @ nil_fm ) )
=> ( ! [X3: fm,Xs2: list_fm] :
( ( Xs2 != nil_fm )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_fm @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_709_list__nonempty__induct,axiom,
! [Xs: list_list_fm,P: list_list_fm > $o] :
( ( Xs != nil_list_fm )
=> ( ! [X3: list_fm] : ( P @ ( cons_list_fm @ X3 @ nil_list_fm ) )
=> ( ! [X3: list_fm,Xs2: list_list_fm] :
( ( Xs2 != nil_list_fm )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_list_fm @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_710_list__nonempty__induct,axiom,
! [Xs: list_tm,P: list_tm > $o] :
( ( Xs != nil_tm )
=> ( ! [X3: tm] : ( P @ ( cons_tm @ X3 @ nil_tm ) )
=> ( ! [X3: tm,Xs2: list_tm] :
( ( Xs2 != nil_tm )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_tm @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_711_list__induct2_H,axiom,
! [P: list_fm > list_fm > $o,Xs: list_fm,Ys: list_fm] :
( ( P @ nil_fm @ nil_fm )
=> ( ! [X3: fm,Xs2: list_fm] : ( P @ ( cons_fm @ X3 @ Xs2 ) @ nil_fm )
=> ( ! [Y: fm,Ys3: list_fm] : ( P @ nil_fm @ ( cons_fm @ Y @ Ys3 ) )
=> ( ! [X3: fm,Xs2: list_fm,Y: fm,Ys3: list_fm] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_fm @ X3 @ Xs2 ) @ ( cons_fm @ Y @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_712_list__induct2_H,axiom,
! [P: list_fm > list_list_fm > $o,Xs: list_fm,Ys: list_list_fm] :
( ( P @ nil_fm @ nil_list_fm )
=> ( ! [X3: fm,Xs2: list_fm] : ( P @ ( cons_fm @ X3 @ Xs2 ) @ nil_list_fm )
=> ( ! [Y: list_fm,Ys3: list_list_fm] : ( P @ nil_fm @ ( cons_list_fm @ Y @ Ys3 ) )
=> ( ! [X3: fm,Xs2: list_fm,Y: list_fm,Ys3: list_list_fm] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_fm @ X3 @ Xs2 ) @ ( cons_list_fm @ Y @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_713_list__induct2_H,axiom,
! [P: list_fm > list_tm > $o,Xs: list_fm,Ys: list_tm] :
( ( P @ nil_fm @ nil_tm )
=> ( ! [X3: fm,Xs2: list_fm] : ( P @ ( cons_fm @ X3 @ Xs2 ) @ nil_tm )
=> ( ! [Y: tm,Ys3: list_tm] : ( P @ nil_fm @ ( cons_tm @ Y @ Ys3 ) )
=> ( ! [X3: fm,Xs2: list_fm,Y: tm,Ys3: list_tm] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_fm @ X3 @ Xs2 ) @ ( cons_tm @ Y @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_714_list__induct2_H,axiom,
! [P: list_list_fm > list_fm > $o,Xs: list_list_fm,Ys: list_fm] :
( ( P @ nil_list_fm @ nil_fm )
=> ( ! [X3: list_fm,Xs2: list_list_fm] : ( P @ ( cons_list_fm @ X3 @ Xs2 ) @ nil_fm )
=> ( ! [Y: fm,Ys3: list_fm] : ( P @ nil_list_fm @ ( cons_fm @ Y @ Ys3 ) )
=> ( ! [X3: list_fm,Xs2: list_list_fm,Y: fm,Ys3: list_fm] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_list_fm @ X3 @ Xs2 ) @ ( cons_fm @ Y @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_715_list__induct2_H,axiom,
! [P: list_list_fm > list_list_fm > $o,Xs: list_list_fm,Ys: list_list_fm] :
( ( P @ nil_list_fm @ nil_list_fm )
=> ( ! [X3: list_fm,Xs2: list_list_fm] : ( P @ ( cons_list_fm @ X3 @ Xs2 ) @ nil_list_fm )
=> ( ! [Y: list_fm,Ys3: list_list_fm] : ( P @ nil_list_fm @ ( cons_list_fm @ Y @ Ys3 ) )
=> ( ! [X3: list_fm,Xs2: list_list_fm,Y: list_fm,Ys3: list_list_fm] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_list_fm @ X3 @ Xs2 ) @ ( cons_list_fm @ Y @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_716_list__induct2_H,axiom,
! [P: list_list_fm > list_tm > $o,Xs: list_list_fm,Ys: list_tm] :
( ( P @ nil_list_fm @ nil_tm )
=> ( ! [X3: list_fm,Xs2: list_list_fm] : ( P @ ( cons_list_fm @ X3 @ Xs2 ) @ nil_tm )
=> ( ! [Y: tm,Ys3: list_tm] : ( P @ nil_list_fm @ ( cons_tm @ Y @ Ys3 ) )
=> ( ! [X3: list_fm,Xs2: list_list_fm,Y: tm,Ys3: list_tm] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_list_fm @ X3 @ Xs2 ) @ ( cons_tm @ Y @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_717_list__induct2_H,axiom,
! [P: list_tm > list_fm > $o,Xs: list_tm,Ys: list_fm] :
( ( P @ nil_tm @ nil_fm )
=> ( ! [X3: tm,Xs2: list_tm] : ( P @ ( cons_tm @ X3 @ Xs2 ) @ nil_fm )
=> ( ! [Y: fm,Ys3: list_fm] : ( P @ nil_tm @ ( cons_fm @ Y @ Ys3 ) )
=> ( ! [X3: tm,Xs2: list_tm,Y: fm,Ys3: list_fm] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_tm @ X3 @ Xs2 ) @ ( cons_fm @ Y @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_718_list__induct2_H,axiom,
! [P: list_tm > list_list_fm > $o,Xs: list_tm,Ys: list_list_fm] :
( ( P @ nil_tm @ nil_list_fm )
=> ( ! [X3: tm,Xs2: list_tm] : ( P @ ( cons_tm @ X3 @ Xs2 ) @ nil_list_fm )
=> ( ! [Y: list_fm,Ys3: list_list_fm] : ( P @ nil_tm @ ( cons_list_fm @ Y @ Ys3 ) )
=> ( ! [X3: tm,Xs2: list_tm,Y: list_fm,Ys3: list_list_fm] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_tm @ X3 @ Xs2 ) @ ( cons_list_fm @ Y @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_719_list__induct2_H,axiom,
! [P: list_tm > list_tm > $o,Xs: list_tm,Ys: list_tm] :
( ( P @ nil_tm @ nil_tm )
=> ( ! [X3: tm,Xs2: list_tm] : ( P @ ( cons_tm @ X3 @ Xs2 ) @ nil_tm )
=> ( ! [Y: tm,Ys3: list_tm] : ( P @ nil_tm @ ( cons_tm @ Y @ Ys3 ) )
=> ( ! [X3: tm,Xs2: list_tm,Y: tm,Ys3: list_tm] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_tm @ X3 @ Xs2 ) @ ( cons_tm @ Y @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_720_neq__Nil__conv,axiom,
! [Xs: list_fm] :
( ( Xs != nil_fm )
= ( ? [Y4: fm,Ys2: list_fm] :
( Xs
= ( cons_fm @ Y4 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_721_neq__Nil__conv,axiom,
! [Xs: list_list_fm] :
( ( Xs != nil_list_fm )
= ( ? [Y4: list_fm,Ys2: list_list_fm] :
( Xs
= ( cons_list_fm @ Y4 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_722_neq__Nil__conv,axiom,
! [Xs: list_tm] :
( ( Xs != nil_tm )
= ( ? [Y4: tm,Ys2: list_tm] :
( Xs
= ( cons_tm @ Y4 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_723_remdups__adj_Ocases,axiom,
! [X: list_fm] :
( ( X != nil_fm )
=> ( ! [X3: fm] :
( X
!= ( cons_fm @ X3 @ nil_fm ) )
=> ~ ! [X3: fm,Y: fm,Xs2: list_fm] :
( X
!= ( cons_fm @ X3 @ ( cons_fm @ Y @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_724_remdups__adj_Ocases,axiom,
! [X: list_list_fm] :
( ( X != nil_list_fm )
=> ( ! [X3: list_fm] :
( X
!= ( cons_list_fm @ X3 @ nil_list_fm ) )
=> ~ ! [X3: list_fm,Y: list_fm,Xs2: list_list_fm] :
( X
!= ( cons_list_fm @ X3 @ ( cons_list_fm @ Y @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_725_remdups__adj_Ocases,axiom,
! [X: list_tm] :
( ( X != nil_tm )
=> ( ! [X3: tm] :
( X
!= ( cons_tm @ X3 @ nil_tm ) )
=> ~ ! [X3: tm,Y: tm,Xs2: list_tm] :
( X
!= ( cons_tm @ X3 @ ( cons_tm @ Y @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_726_list_Oexhaust,axiom,
! [Y3: list_fm] :
( ( Y3 != nil_fm )
=> ~ ! [X212: fm,X223: list_fm] :
( Y3
!= ( cons_fm @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_727_list_Oexhaust,axiom,
! [Y3: list_list_fm] :
( ( Y3 != nil_list_fm )
=> ~ ! [X212: list_fm,X223: list_list_fm] :
( Y3
!= ( cons_list_fm @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_728_list_Oexhaust,axiom,
! [Y3: list_tm] :
( ( Y3 != nil_tm )
=> ~ ! [X212: tm,X223: list_tm] :
( Y3
!= ( cons_tm @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_729_list_OdiscI,axiom,
! [List: list_fm,X21: fm,X222: list_fm] :
( ( List
= ( cons_fm @ X21 @ X222 ) )
=> ( List != nil_fm ) ) ).
% list.discI
thf(fact_730_list_OdiscI,axiom,
! [List: list_list_fm,X21: list_fm,X222: list_list_fm] :
( ( List
= ( cons_list_fm @ X21 @ X222 ) )
=> ( List != nil_list_fm ) ) ).
% list.discI
thf(fact_731_list_OdiscI,axiom,
! [List: list_tm,X21: tm,X222: list_tm] :
( ( List
= ( cons_tm @ X21 @ X222 ) )
=> ( List != nil_tm ) ) ).
% list.discI
thf(fact_732_list_Odistinct_I1_J,axiom,
! [X21: fm,X222: list_fm] :
( nil_fm
!= ( cons_fm @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_733_list_Odistinct_I1_J,axiom,
! [X21: list_fm,X222: list_list_fm] :
( nil_list_fm
!= ( cons_list_fm @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_734_list_Odistinct_I1_J,axiom,
! [X21: tm,X222: list_tm] :
( nil_tm
!= ( cons_tm @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_735_subset__code_I1_J,axiom,
! [Xs: list_list_fm,B4: set_list_fm] :
( ( ord_le7838213414353715577ist_fm @ ( set_list_fm2 @ Xs ) @ B4 )
= ( ! [X2: list_fm] :
( ( member_list_fm2 @ X2 @ ( set_list_fm2 @ Xs ) )
=> ( member_list_fm2 @ X2 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_736_subset__code_I1_J,axiom,
! [Xs: list_P2887561121880082555ist_fm,B4: set_Pr5202636777678657877ist_fm] :
( ( ord_le1771420097867575541ist_fm @ ( set_Pr5051287786238754058ist_fm @ Xs ) @ B4 )
= ( ! [X2: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ X2 @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
=> ( member4699826688122452638ist_fm @ X2 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_737_subset__code_I1_J,axiom,
! [Xs: list_tm,B4: set_tm] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ Xs ) @ B4 )
= ( ! [X2: tm] :
( ( member_tm2 @ X2 @ ( set_tm2 @ Xs ) )
=> ( member_tm2 @ X2 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_738_subset__code_I1_J,axiom,
! [Xs: list_fm,B4: set_fm] :
( ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ B4 )
= ( ! [X2: fm] :
( ( member_fm2 @ X2 @ ( set_fm2 @ Xs ) )
=> ( member_fm2 @ X2 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_739_eq__Nil__appendI,axiom,
! [Xs: list_list_tm,Ys: list_list_tm] :
( ( Xs = Ys )
=> ( Xs
= ( append_list_tm @ nil_list_tm @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_740_eq__Nil__appendI,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( Xs = Ys )
=> ( Xs
= ( append_fm @ nil_fm @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_741_eq__Nil__appendI,axiom,
! [Xs: list_list_fm,Ys: list_list_fm] :
( ( Xs = Ys )
=> ( Xs
= ( append_list_fm @ nil_list_fm @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_742_eq__Nil__appendI,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( Xs = Ys )
=> ( Xs
= ( append_tm @ nil_tm @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_743_append_Oleft__neutral,axiom,
! [A: list_list_tm] :
( ( append_list_tm @ nil_list_tm @ A )
= A ) ).
% append.left_neutral
thf(fact_744_append_Oleft__neutral,axiom,
! [A: list_fm] :
( ( append_fm @ nil_fm @ A )
= A ) ).
% append.left_neutral
thf(fact_745_append_Oleft__neutral,axiom,
! [A: list_list_fm] :
( ( append_list_fm @ nil_list_fm @ A )
= A ) ).
% append.left_neutral
thf(fact_746_append_Oleft__neutral,axiom,
! [A: list_tm] :
( ( append_tm @ nil_tm @ A )
= A ) ).
% append.left_neutral
thf(fact_747_append__Nil,axiom,
! [Ys: list_list_tm] :
( ( append_list_tm @ nil_list_tm @ Ys )
= Ys ) ).
% append_Nil
thf(fact_748_append__Nil,axiom,
! [Ys: list_fm] :
( ( append_fm @ nil_fm @ Ys )
= Ys ) ).
% append_Nil
thf(fact_749_append__Nil,axiom,
! [Ys: list_list_fm] :
( ( append_list_fm @ nil_list_fm @ Ys )
= Ys ) ).
% append_Nil
thf(fact_750_append__Nil,axiom,
! [Ys: list_tm] :
( ( append_tm @ nil_tm @ Ys )
= Ys ) ).
% append_Nil
thf(fact_751_list_Osimps_I8_J,axiom,
! [F: fm > fm] :
( ( map_fm_fm @ F @ nil_fm )
= nil_fm ) ).
% list.simps(8)
thf(fact_752_list_Osimps_I8_J,axiom,
! [F: fm > tm] :
( ( map_fm_tm @ F @ nil_fm )
= nil_tm ) ).
% list.simps(8)
thf(fact_753_list_Osimps_I8_J,axiom,
! [F: tm > fm] :
( ( map_tm_fm @ F @ nil_tm )
= nil_fm ) ).
% list.simps(8)
thf(fact_754_list_Osimps_I8_J,axiom,
! [F: tm > tm] :
( ( map_tm_tm @ F @ nil_tm )
= nil_tm ) ).
% list.simps(8)
thf(fact_755_list_Osimps_I8_J,axiom,
! [F: fm > list_fm] :
( ( map_fm_list_fm @ F @ nil_fm )
= nil_list_fm ) ).
% list.simps(8)
thf(fact_756_list_Osimps_I8_J,axiom,
! [F: list_fm > fm] :
( ( map_list_fm_fm @ F @ nil_list_fm )
= nil_fm ) ).
% list.simps(8)
thf(fact_757_list_Osimps_I8_J,axiom,
! [F: list_fm > tm] :
( ( map_list_fm_tm @ F @ nil_list_fm )
= nil_tm ) ).
% list.simps(8)
thf(fact_758_list_Osimps_I8_J,axiom,
! [F: tm > list_fm] :
( ( map_tm_list_fm @ F @ nil_tm )
= nil_list_fm ) ).
% list.simps(8)
thf(fact_759_list_Osimps_I8_J,axiom,
! [F: fm > list_tm] :
( ( map_fm_list_tm @ F @ nil_fm )
= nil_list_tm ) ).
% list.simps(8)
thf(fact_760_list_Osimps_I8_J,axiom,
! [F: tm > list_tm] :
( ( map_tm_list_tm @ F @ nil_tm )
= nil_list_tm ) ).
% list.simps(8)
thf(fact_761_fsubsetD,axiom,
! [A4: fset_P8989946509869081563ist_fm,B4: fset_P8989946509869081563ist_fm,C: produc6018962875968178549ist_fm] :
( ( ord_le3986950534092794747ist_fm @ A4 @ B4 )
=> ( ( fmembe3754813877001230652ist_fm @ C @ A4 )
=> ( fmembe3754813877001230652ist_fm @ C @ B4 ) ) ) ).
% fsubsetD
thf(fact_762_fsubsetD,axiom,
! [A4: fset_tm,B4: fset_tm,C: tm] :
( ( ord_less_eq_fset_tm @ A4 @ B4 )
=> ( ( fmember_tm @ C @ A4 )
=> ( fmember_tm @ C @ B4 ) ) ) ).
% fsubsetD
thf(fact_763_fsubsetD,axiom,
! [A4: fset_fm,B4: fset_fm,C: fm] :
( ( ord_less_eq_fset_fm @ A4 @ B4 )
=> ( ( fmember_fm @ C @ A4 )
=> ( fmember_fm @ C @ B4 ) ) ) ).
% fsubsetD
thf(fact_764_fsubsetD,axiom,
! [A4: fset_list_fm,B4: fset_list_fm,C: list_fm] :
( ( ord_le3182162295460067583ist_fm @ A4 @ B4 )
=> ( ( fmember_list_fm @ C @ A4 )
=> ( fmember_list_fm @ C @ B4 ) ) ) ).
% fsubsetD
thf(fact_765_fin__mono,axiom,
! [A4: fset_P8989946509869081563ist_fm,B4: fset_P8989946509869081563ist_fm,X: produc6018962875968178549ist_fm] :
( ( ord_le3986950534092794747ist_fm @ A4 @ B4 )
=> ( ( fmembe3754813877001230652ist_fm @ X @ A4 )
=> ( fmembe3754813877001230652ist_fm @ X @ B4 ) ) ) ).
% fin_mono
thf(fact_766_fin__mono,axiom,
! [A4: fset_tm,B4: fset_tm,X: tm] :
( ( ord_less_eq_fset_tm @ A4 @ B4 )
=> ( ( fmember_tm @ X @ A4 )
=> ( fmember_tm @ X @ B4 ) ) ) ).
% fin_mono
thf(fact_767_fin__mono,axiom,
! [A4: fset_fm,B4: fset_fm,X: fm] :
( ( ord_less_eq_fset_fm @ A4 @ B4 )
=> ( ( fmember_fm @ X @ A4 )
=> ( fmember_fm @ X @ B4 ) ) ) ).
% fin_mono
thf(fact_768_fin__mono,axiom,
! [A4: fset_list_fm,B4: fset_list_fm,X: list_fm] :
( ( ord_le3182162295460067583ist_fm @ A4 @ B4 )
=> ( ( fmember_list_fm @ X @ A4 )
=> ( fmember_list_fm @ X @ B4 ) ) ) ).
% fin_mono
thf(fact_769_remdups_Osimps_I1_J,axiom,
( ( remdups_fm @ nil_fm )
= nil_fm ) ).
% remdups.simps(1)
thf(fact_770_remdups_Osimps_I1_J,axiom,
( ( remdups_list_fm @ nil_list_fm )
= nil_list_fm ) ).
% remdups.simps(1)
thf(fact_771_remdups_Osimps_I1_J,axiom,
( ( remdups_tm @ nil_tm )
= nil_tm ) ).
% remdups.simps(1)
thf(fact_772_maps__simps_I2_J,axiom,
! [F: fm > list_fm] :
( ( maps_fm_fm @ F @ nil_fm )
= nil_fm ) ).
% maps_simps(2)
thf(fact_773_maps__simps_I2_J,axiom,
! [F: fm > list_list_fm] :
( ( maps_fm_list_fm @ F @ nil_fm )
= nil_list_fm ) ).
% maps_simps(2)
thf(fact_774_maps__simps_I2_J,axiom,
! [F: fm > list_tm] :
( ( maps_fm_tm @ F @ nil_fm )
= nil_tm ) ).
% maps_simps(2)
thf(fact_775_maps__simps_I2_J,axiom,
! [F: list_fm > list_fm] :
( ( maps_list_fm_fm @ F @ nil_list_fm )
= nil_fm ) ).
% maps_simps(2)
thf(fact_776_maps__simps_I2_J,axiom,
! [F: list_fm > list_list_fm] :
( ( maps_list_fm_list_fm @ F @ nil_list_fm )
= nil_list_fm ) ).
% maps_simps(2)
thf(fact_777_maps__simps_I2_J,axiom,
! [F: list_fm > list_tm] :
( ( maps_list_fm_tm @ F @ nil_list_fm )
= nil_tm ) ).
% maps_simps(2)
thf(fact_778_maps__simps_I2_J,axiom,
! [F: tm > list_fm] :
( ( maps_tm_fm @ F @ nil_tm )
= nil_fm ) ).
% maps_simps(2)
thf(fact_779_maps__simps_I2_J,axiom,
! [F: tm > list_list_fm] :
( ( maps_tm_list_fm @ F @ nil_tm )
= nil_list_fm ) ).
% maps_simps(2)
thf(fact_780_maps__simps_I2_J,axiom,
! [F: tm > list_tm] :
( ( maps_tm_tm @ F @ nil_tm )
= nil_tm ) ).
% maps_simps(2)
thf(fact_781_member__rec_I2_J,axiom,
! [Y3: fm] :
~ ( member_fm @ nil_fm @ Y3 ) ).
% member_rec(2)
thf(fact_782_member__rec_I2_J,axiom,
! [Y3: list_fm] :
~ ( member_list_fm @ nil_list_fm @ Y3 ) ).
% member_rec(2)
thf(fact_783_member__rec_I2_J,axiom,
! [Y3: tm] :
~ ( member_tm @ nil_tm @ Y3 ) ).
% member_rec(2)
thf(fact_784_set__subset__Cons,axiom,
! [Xs: list_P2887561121880082555ist_fm,X: produc6018962875968178549ist_fm] : ( ord_le1771420097867575541ist_fm @ ( set_Pr5051287786238754058ist_fm @ Xs ) @ ( set_Pr5051287786238754058ist_fm @ ( cons_P6586734674047340843ist_fm @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_785_set__subset__Cons,axiom,
! [Xs: list_list_fm,X: list_fm] : ( ord_le7838213414353715577ist_fm @ ( set_list_fm2 @ Xs ) @ ( set_list_fm2 @ ( cons_list_fm @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_786_set__subset__Cons,axiom,
! [Xs: list_tm,X: tm] : ( ord_less_eq_set_tm @ ( set_tm2 @ Xs ) @ ( set_tm2 @ ( cons_tm @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_787_set__subset__Cons,axiom,
! [Xs: list_fm,X: fm] : ( ord_less_eq_set_fm @ ( set_fm2 @ Xs ) @ ( set_fm2 @ ( cons_fm @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_788_list__induct4,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_fm,Ws: list_fm,P: list_fm > list_fm > list_fm > list_fm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_fm @ Zs ) )
=> ( ( ( size_size_list_fm @ Zs )
= ( size_size_list_fm @ Ws ) )
=> ( ( P @ nil_fm @ nil_fm @ nil_fm @ nil_fm )
=> ( ! [X3: fm,Xs2: list_fm,Y: fm,Ys3: list_fm,Z: fm,Zs3: list_fm,W2: fm,Ws2: list_fm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys3 ) )
=> ( ( ( size_size_list_fm @ Ys3 )
= ( size_size_list_fm @ Zs3 ) )
=> ( ( ( size_size_list_fm @ Zs3 )
= ( size_size_list_fm @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs3 @ Ws2 )
=> ( P @ ( cons_fm @ X3 @ Xs2 ) @ ( cons_fm @ Y @ Ys3 ) @ ( cons_fm @ Z @ Zs3 ) @ ( cons_fm @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_789_list__induct4,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_fm,Ws: list_tm,P: list_fm > list_fm > list_fm > list_tm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_fm @ Zs ) )
=> ( ( ( size_size_list_fm @ Zs )
= ( size_size_list_tm @ Ws ) )
=> ( ( P @ nil_fm @ nil_fm @ nil_fm @ nil_tm )
=> ( ! [X3: fm,Xs2: list_fm,Y: fm,Ys3: list_fm,Z: fm,Zs3: list_fm,W2: tm,Ws2: list_tm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys3 ) )
=> ( ( ( size_size_list_fm @ Ys3 )
= ( size_size_list_fm @ Zs3 ) )
=> ( ( ( size_size_list_fm @ Zs3 )
= ( size_size_list_tm @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs3 @ Ws2 )
=> ( P @ ( cons_fm @ X3 @ Xs2 ) @ ( cons_fm @ Y @ Ys3 ) @ ( cons_fm @ Z @ Zs3 ) @ ( cons_tm @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_790_list__induct4,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_tm,Ws: list_fm,P: list_fm > list_fm > list_tm > list_fm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_tm @ Zs ) )
=> ( ( ( size_size_list_tm @ Zs )
= ( size_size_list_fm @ Ws ) )
=> ( ( P @ nil_fm @ nil_fm @ nil_tm @ nil_fm )
=> ( ! [X3: fm,Xs2: list_fm,Y: fm,Ys3: list_fm,Z: tm,Zs3: list_tm,W2: fm,Ws2: list_fm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys3 ) )
=> ( ( ( size_size_list_fm @ Ys3 )
= ( size_size_list_tm @ Zs3 ) )
=> ( ( ( size_size_list_tm @ Zs3 )
= ( size_size_list_fm @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs3 @ Ws2 )
=> ( P @ ( cons_fm @ X3 @ Xs2 ) @ ( cons_fm @ Y @ Ys3 ) @ ( cons_tm @ Z @ Zs3 ) @ ( cons_fm @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_791_list__induct4,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_tm,Ws: list_tm,P: list_fm > list_fm > list_tm > list_tm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_tm @ Zs ) )
=> ( ( ( size_size_list_tm @ Zs )
= ( size_size_list_tm @ Ws ) )
=> ( ( P @ nil_fm @ nil_fm @ nil_tm @ nil_tm )
=> ( ! [X3: fm,Xs2: list_fm,Y: fm,Ys3: list_fm,Z: tm,Zs3: list_tm,W2: tm,Ws2: list_tm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys3 ) )
=> ( ( ( size_size_list_fm @ Ys3 )
= ( size_size_list_tm @ Zs3 ) )
=> ( ( ( size_size_list_tm @ Zs3 )
= ( size_size_list_tm @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs3 @ Ws2 )
=> ( P @ ( cons_fm @ X3 @ Xs2 ) @ ( cons_fm @ Y @ Ys3 ) @ ( cons_tm @ Z @ Zs3 ) @ ( cons_tm @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_792_list__induct4,axiom,
! [Xs: list_fm,Ys: list_tm,Zs: list_fm,Ws: list_fm,P: list_fm > list_tm > list_fm > list_fm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( ( size_size_list_tm @ Ys )
= ( size_size_list_fm @ Zs ) )
=> ( ( ( size_size_list_fm @ Zs )
= ( size_size_list_fm @ Ws ) )
=> ( ( P @ nil_fm @ nil_tm @ nil_fm @ nil_fm )
=> ( ! [X3: fm,Xs2: list_fm,Y: tm,Ys3: list_tm,Z: fm,Zs3: list_fm,W2: fm,Ws2: list_fm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_tm @ Ys3 ) )
=> ( ( ( size_size_list_tm @ Ys3 )
= ( size_size_list_fm @ Zs3 ) )
=> ( ( ( size_size_list_fm @ Zs3 )
= ( size_size_list_fm @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs3 @ Ws2 )
=> ( P @ ( cons_fm @ X3 @ Xs2 ) @ ( cons_tm @ Y @ Ys3 ) @ ( cons_fm @ Z @ Zs3 ) @ ( cons_fm @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_793_list__induct4,axiom,
! [Xs: list_fm,Ys: list_tm,Zs: list_fm,Ws: list_tm,P: list_fm > list_tm > list_fm > list_tm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( ( size_size_list_tm @ Ys )
= ( size_size_list_fm @ Zs ) )
=> ( ( ( size_size_list_fm @ Zs )
= ( size_size_list_tm @ Ws ) )
=> ( ( P @ nil_fm @ nil_tm @ nil_fm @ nil_tm )
=> ( ! [X3: fm,Xs2: list_fm,Y: tm,Ys3: list_tm,Z: fm,Zs3: list_fm,W2: tm,Ws2: list_tm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_tm @ Ys3 ) )
=> ( ( ( size_size_list_tm @ Ys3 )
= ( size_size_list_fm @ Zs3 ) )
=> ( ( ( size_size_list_fm @ Zs3 )
= ( size_size_list_tm @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs3 @ Ws2 )
=> ( P @ ( cons_fm @ X3 @ Xs2 ) @ ( cons_tm @ Y @ Ys3 ) @ ( cons_fm @ Z @ Zs3 ) @ ( cons_tm @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_794_list__induct4,axiom,
! [Xs: list_fm,Ys: list_tm,Zs: list_tm,Ws: list_fm,P: list_fm > list_tm > list_tm > list_fm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( ( size_size_list_tm @ Ys )
= ( size_size_list_tm @ Zs ) )
=> ( ( ( size_size_list_tm @ Zs )
= ( size_size_list_fm @ Ws ) )
=> ( ( P @ nil_fm @ nil_tm @ nil_tm @ nil_fm )
=> ( ! [X3: fm,Xs2: list_fm,Y: tm,Ys3: list_tm,Z: tm,Zs3: list_tm,W2: fm,Ws2: list_fm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_tm @ Ys3 ) )
=> ( ( ( size_size_list_tm @ Ys3 )
= ( size_size_list_tm @ Zs3 ) )
=> ( ( ( size_size_list_tm @ Zs3 )
= ( size_size_list_fm @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs3 @ Ws2 )
=> ( P @ ( cons_fm @ X3 @ Xs2 ) @ ( cons_tm @ Y @ Ys3 ) @ ( cons_tm @ Z @ Zs3 ) @ ( cons_fm @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_795_list__induct4,axiom,
! [Xs: list_fm,Ys: list_tm,Zs: list_tm,Ws: list_tm,P: list_fm > list_tm > list_tm > list_tm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( ( size_size_list_tm @ Ys )
= ( size_size_list_tm @ Zs ) )
=> ( ( ( size_size_list_tm @ Zs )
= ( size_size_list_tm @ Ws ) )
=> ( ( P @ nil_fm @ nil_tm @ nil_tm @ nil_tm )
=> ( ! [X3: fm,Xs2: list_fm,Y: tm,Ys3: list_tm,Z: tm,Zs3: list_tm,W2: tm,Ws2: list_tm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_tm @ Ys3 ) )
=> ( ( ( size_size_list_tm @ Ys3 )
= ( size_size_list_tm @ Zs3 ) )
=> ( ( ( size_size_list_tm @ Zs3 )
= ( size_size_list_tm @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs3 @ Ws2 )
=> ( P @ ( cons_fm @ X3 @ Xs2 ) @ ( cons_tm @ Y @ Ys3 ) @ ( cons_tm @ Z @ Zs3 ) @ ( cons_tm @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_796_list__induct4,axiom,
! [Xs: list_tm,Ys: list_fm,Zs: list_fm,Ws: list_fm,P: list_tm > list_fm > list_fm > list_fm > $o] :
( ( ( size_size_list_tm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_fm @ Zs ) )
=> ( ( ( size_size_list_fm @ Zs )
= ( size_size_list_fm @ Ws ) )
=> ( ( P @ nil_tm @ nil_fm @ nil_fm @ nil_fm )
=> ( ! [X3: tm,Xs2: list_tm,Y: fm,Ys3: list_fm,Z: fm,Zs3: list_fm,W2: fm,Ws2: list_fm] :
( ( ( size_size_list_tm @ Xs2 )
= ( size_size_list_fm @ Ys3 ) )
=> ( ( ( size_size_list_fm @ Ys3 )
= ( size_size_list_fm @ Zs3 ) )
=> ( ( ( size_size_list_fm @ Zs3 )
= ( size_size_list_fm @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs3 @ Ws2 )
=> ( P @ ( cons_tm @ X3 @ Xs2 ) @ ( cons_fm @ Y @ Ys3 ) @ ( cons_fm @ Z @ Zs3 ) @ ( cons_fm @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_797_list__induct4,axiom,
! [Xs: list_tm,Ys: list_fm,Zs: list_fm,Ws: list_tm,P: list_tm > list_fm > list_fm > list_tm > $o] :
( ( ( size_size_list_tm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_fm @ Zs ) )
=> ( ( ( size_size_list_fm @ Zs )
= ( size_size_list_tm @ Ws ) )
=> ( ( P @ nil_tm @ nil_fm @ nil_fm @ nil_tm )
=> ( ! [X3: tm,Xs2: list_tm,Y: fm,Ys3: list_fm,Z: fm,Zs3: list_fm,W2: tm,Ws2: list_tm] :
( ( ( size_size_list_tm @ Xs2 )
= ( size_size_list_fm @ Ys3 ) )
=> ( ( ( size_size_list_fm @ Ys3 )
= ( size_size_list_fm @ Zs3 ) )
=> ( ( ( size_size_list_fm @ Zs3 )
= ( size_size_list_tm @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs3 @ Ws2 )
=> ( P @ ( cons_tm @ X3 @ Xs2 ) @ ( cons_fm @ Y @ Ys3 ) @ ( cons_fm @ Z @ Zs3 ) @ ( cons_tm @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_798_list__induct3,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_fm,P: list_fm > list_fm > list_fm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_fm @ Zs ) )
=> ( ( P @ nil_fm @ nil_fm @ nil_fm )
=> ( ! [X3: fm,Xs2: list_fm,Y: fm,Ys3: list_fm,Z: fm,Zs3: list_fm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys3 ) )
=> ( ( ( size_size_list_fm @ Ys3 )
= ( size_size_list_fm @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs3 )
=> ( P @ ( cons_fm @ X3 @ Xs2 ) @ ( cons_fm @ Y @ Ys3 ) @ ( cons_fm @ Z @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_799_list__induct3,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_tm,P: list_fm > list_fm > list_tm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_tm @ Zs ) )
=> ( ( P @ nil_fm @ nil_fm @ nil_tm )
=> ( ! [X3: fm,Xs2: list_fm,Y: fm,Ys3: list_fm,Z: tm,Zs3: list_tm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys3 ) )
=> ( ( ( size_size_list_fm @ Ys3 )
= ( size_size_list_tm @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs3 )
=> ( P @ ( cons_fm @ X3 @ Xs2 ) @ ( cons_fm @ Y @ Ys3 ) @ ( cons_tm @ Z @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_800_list__induct3,axiom,
! [Xs: list_fm,Ys: list_tm,Zs: list_fm,P: list_fm > list_tm > list_fm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( ( size_size_list_tm @ Ys )
= ( size_size_list_fm @ Zs ) )
=> ( ( P @ nil_fm @ nil_tm @ nil_fm )
=> ( ! [X3: fm,Xs2: list_fm,Y: tm,Ys3: list_tm,Z: fm,Zs3: list_fm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_tm @ Ys3 ) )
=> ( ( ( size_size_list_tm @ Ys3 )
= ( size_size_list_fm @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs3 )
=> ( P @ ( cons_fm @ X3 @ Xs2 ) @ ( cons_tm @ Y @ Ys3 ) @ ( cons_fm @ Z @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_801_list__induct3,axiom,
! [Xs: list_fm,Ys: list_tm,Zs: list_tm,P: list_fm > list_tm > list_tm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( ( size_size_list_tm @ Ys )
= ( size_size_list_tm @ Zs ) )
=> ( ( P @ nil_fm @ nil_tm @ nil_tm )
=> ( ! [X3: fm,Xs2: list_fm,Y: tm,Ys3: list_tm,Z: tm,Zs3: list_tm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_tm @ Ys3 ) )
=> ( ( ( size_size_list_tm @ Ys3 )
= ( size_size_list_tm @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs3 )
=> ( P @ ( cons_fm @ X3 @ Xs2 ) @ ( cons_tm @ Y @ Ys3 ) @ ( cons_tm @ Z @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_802_list__induct3,axiom,
! [Xs: list_tm,Ys: list_fm,Zs: list_fm,P: list_tm > list_fm > list_fm > $o] :
( ( ( size_size_list_tm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_fm @ Zs ) )
=> ( ( P @ nil_tm @ nil_fm @ nil_fm )
=> ( ! [X3: tm,Xs2: list_tm,Y: fm,Ys3: list_fm,Z: fm,Zs3: list_fm] :
( ( ( size_size_list_tm @ Xs2 )
= ( size_size_list_fm @ Ys3 ) )
=> ( ( ( size_size_list_fm @ Ys3 )
= ( size_size_list_fm @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs3 )
=> ( P @ ( cons_tm @ X3 @ Xs2 ) @ ( cons_fm @ Y @ Ys3 ) @ ( cons_fm @ Z @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_803_list__induct3,axiom,
! [Xs: list_tm,Ys: list_fm,Zs: list_tm,P: list_tm > list_fm > list_tm > $o] :
( ( ( size_size_list_tm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_tm @ Zs ) )
=> ( ( P @ nil_tm @ nil_fm @ nil_tm )
=> ( ! [X3: tm,Xs2: list_tm,Y: fm,Ys3: list_fm,Z: tm,Zs3: list_tm] :
( ( ( size_size_list_tm @ Xs2 )
= ( size_size_list_fm @ Ys3 ) )
=> ( ( ( size_size_list_fm @ Ys3 )
= ( size_size_list_tm @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs3 )
=> ( P @ ( cons_tm @ X3 @ Xs2 ) @ ( cons_fm @ Y @ Ys3 ) @ ( cons_tm @ Z @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_804_list__induct3,axiom,
! [Xs: list_tm,Ys: list_tm,Zs: list_fm,P: list_tm > list_tm > list_fm > $o] :
( ( ( size_size_list_tm @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( ( size_size_list_tm @ Ys )
= ( size_size_list_fm @ Zs ) )
=> ( ( P @ nil_tm @ nil_tm @ nil_fm )
=> ( ! [X3: tm,Xs2: list_tm,Y: tm,Ys3: list_tm,Z: fm,Zs3: list_fm] :
( ( ( size_size_list_tm @ Xs2 )
= ( size_size_list_tm @ Ys3 ) )
=> ( ( ( size_size_list_tm @ Ys3 )
= ( size_size_list_fm @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs3 )
=> ( P @ ( cons_tm @ X3 @ Xs2 ) @ ( cons_tm @ Y @ Ys3 ) @ ( cons_fm @ Z @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_805_list__induct3,axiom,
! [Xs: list_tm,Ys: list_tm,Zs: list_tm,P: list_tm > list_tm > list_tm > $o] :
( ( ( size_size_list_tm @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( ( size_size_list_tm @ Ys )
= ( size_size_list_tm @ Zs ) )
=> ( ( P @ nil_tm @ nil_tm @ nil_tm )
=> ( ! [X3: tm,Xs2: list_tm,Y: tm,Ys3: list_tm,Z: tm,Zs3: list_tm] :
( ( ( size_size_list_tm @ Xs2 )
= ( size_size_list_tm @ Ys3 ) )
=> ( ( ( size_size_list_tm @ Ys3 )
= ( size_size_list_tm @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs3 )
=> ( P @ ( cons_tm @ X3 @ Xs2 ) @ ( cons_tm @ Y @ Ys3 ) @ ( cons_tm @ Z @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_806_list__induct3,axiom,
! [Xs: list_fm,Ys: list_fm,Zs: list_list_fm,P: list_fm > list_fm > list_list_fm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_s115229985653309035ist_fm @ Zs ) )
=> ( ( P @ nil_fm @ nil_fm @ nil_list_fm )
=> ( ! [X3: fm,Xs2: list_fm,Y: fm,Ys3: list_fm,Z: list_fm,Zs3: list_list_fm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys3 ) )
=> ( ( ( size_size_list_fm @ Ys3 )
= ( size_s115229985653309035ist_fm @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs3 )
=> ( P @ ( cons_fm @ X3 @ Xs2 ) @ ( cons_fm @ Y @ Ys3 ) @ ( cons_list_fm @ Z @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_807_list__induct3,axiom,
! [Xs: list_fm,Ys: list_list_fm,Zs: list_fm,P: list_fm > list_list_fm > list_fm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_s115229985653309035ist_fm @ Ys ) )
=> ( ( ( size_s115229985653309035ist_fm @ Ys )
= ( size_size_list_fm @ Zs ) )
=> ( ( P @ nil_fm @ nil_list_fm @ nil_fm )
=> ( ! [X3: fm,Xs2: list_fm,Y: list_fm,Ys3: list_list_fm,Z: fm,Zs3: list_fm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_s115229985653309035ist_fm @ Ys3 ) )
=> ( ( ( size_s115229985653309035ist_fm @ Ys3 )
= ( size_size_list_fm @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs3 )
=> ( P @ ( cons_fm @ X3 @ Xs2 ) @ ( cons_list_fm @ Y @ Ys3 ) @ ( cons_fm @ Z @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_808_list__induct2,axiom,
! [Xs: list_fm,Ys: list_fm,P: list_fm > list_fm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( P @ nil_fm @ nil_fm )
=> ( ! [X3: fm,Xs2: list_fm,Y: fm,Ys3: list_fm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys3 ) )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_fm @ X3 @ Xs2 ) @ ( cons_fm @ Y @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_809_list__induct2,axiom,
! [Xs: list_fm,Ys: list_list_fm,P: list_fm > list_list_fm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_s115229985653309035ist_fm @ Ys ) )
=> ( ( P @ nil_fm @ nil_list_fm )
=> ( ! [X3: fm,Xs2: list_fm,Y: list_fm,Ys3: list_list_fm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_s115229985653309035ist_fm @ Ys3 ) )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_fm @ X3 @ Xs2 ) @ ( cons_list_fm @ Y @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_810_list__induct2,axiom,
! [Xs: list_fm,Ys: list_tm,P: list_fm > list_tm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( P @ nil_fm @ nil_tm )
=> ( ! [X3: fm,Xs2: list_fm,Y: tm,Ys3: list_tm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_tm @ Ys3 ) )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_fm @ X3 @ Xs2 ) @ ( cons_tm @ Y @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_811_list__induct2,axiom,
! [Xs: list_list_fm,Ys: list_fm,P: list_list_fm > list_fm > $o] :
( ( ( size_s115229985653309035ist_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( P @ nil_list_fm @ nil_fm )
=> ( ! [X3: list_fm,Xs2: list_list_fm,Y: fm,Ys3: list_fm] :
( ( ( size_s115229985653309035ist_fm @ Xs2 )
= ( size_size_list_fm @ Ys3 ) )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_list_fm @ X3 @ Xs2 ) @ ( cons_fm @ Y @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_812_list__induct2,axiom,
! [Xs: list_list_fm,Ys: list_list_fm,P: list_list_fm > list_list_fm > $o] :
( ( ( size_s115229985653309035ist_fm @ Xs )
= ( size_s115229985653309035ist_fm @ Ys ) )
=> ( ( P @ nil_list_fm @ nil_list_fm )
=> ( ! [X3: list_fm,Xs2: list_list_fm,Y: list_fm,Ys3: list_list_fm] :
( ( ( size_s115229985653309035ist_fm @ Xs2 )
= ( size_s115229985653309035ist_fm @ Ys3 ) )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_list_fm @ X3 @ Xs2 ) @ ( cons_list_fm @ Y @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_813_list__induct2,axiom,
! [Xs: list_list_fm,Ys: list_tm,P: list_list_fm > list_tm > $o] :
( ( ( size_s115229985653309035ist_fm @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( P @ nil_list_fm @ nil_tm )
=> ( ! [X3: list_fm,Xs2: list_list_fm,Y: tm,Ys3: list_tm] :
( ( ( size_s115229985653309035ist_fm @ Xs2 )
= ( size_size_list_tm @ Ys3 ) )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_list_fm @ X3 @ Xs2 ) @ ( cons_tm @ Y @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_814_list__induct2,axiom,
! [Xs: list_tm,Ys: list_fm,P: list_tm > list_fm > $o] :
( ( ( size_size_list_tm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( P @ nil_tm @ nil_fm )
=> ( ! [X3: tm,Xs2: list_tm,Y: fm,Ys3: list_fm] :
( ( ( size_size_list_tm @ Xs2 )
= ( size_size_list_fm @ Ys3 ) )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_tm @ X3 @ Xs2 ) @ ( cons_fm @ Y @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_815_list__induct2,axiom,
! [Xs: list_tm,Ys: list_list_fm,P: list_tm > list_list_fm > $o] :
( ( ( size_size_list_tm @ Xs )
= ( size_s115229985653309035ist_fm @ Ys ) )
=> ( ( P @ nil_tm @ nil_list_fm )
=> ( ! [X3: tm,Xs2: list_tm,Y: list_fm,Ys3: list_list_fm] :
( ( ( size_size_list_tm @ Xs2 )
= ( size_s115229985653309035ist_fm @ Ys3 ) )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_tm @ X3 @ Xs2 ) @ ( cons_list_fm @ Y @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_816_list__induct2,axiom,
! [Xs: list_tm,Ys: list_tm,P: list_tm > list_tm > $o] :
( ( ( size_size_list_tm @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( P @ nil_tm @ nil_tm )
=> ( ! [X3: tm,Xs2: list_tm,Y: tm,Ys3: list_tm] :
( ( ( size_size_list_tm @ Xs2 )
= ( size_size_list_tm @ Ys3 ) )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_tm @ X3 @ Xs2 ) @ ( cons_tm @ Y @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_817_concat__eq__append__conv,axiom,
! [Xss: list_list_list_tm,Ys: list_list_tm,Zs: list_list_tm] :
( ( ( concat_list_tm @ Xss )
= ( append_list_tm @ Ys @ Zs ) )
= ( ( ( Xss = nil_list_list_tm )
=> ( ( Ys = nil_list_tm )
& ( Zs = nil_list_tm ) ) )
& ( ( Xss != nil_list_list_tm )
=> ? [Xss12: list_list_list_tm,Xs3: list_list_tm,Xs6: list_list_tm,Xss22: list_list_list_tm] :
( ( Xss
= ( append_list_list_tm @ Xss12 @ ( cons_list_list_tm @ ( append_list_tm @ Xs3 @ Xs6 ) @ Xss22 ) ) )
& ( Ys
= ( append_list_tm @ ( concat_list_tm @ Xss12 ) @ Xs3 ) )
& ( Zs
= ( append_list_tm @ Xs6 @ ( concat_list_tm @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_818_concat__eq__append__conv,axiom,
! [Xss: list_list_list_fm,Ys: list_list_fm,Zs: list_list_fm] :
( ( ( concat_list_fm @ Xss )
= ( append_list_fm @ Ys @ Zs ) )
= ( ( ( Xss = nil_list_list_fm )
=> ( ( Ys = nil_list_fm )
& ( Zs = nil_list_fm ) ) )
& ( ( Xss != nil_list_list_fm )
=> ? [Xss12: list_list_list_fm,Xs3: list_list_fm,Xs6: list_list_fm,Xss22: list_list_list_fm] :
( ( Xss
= ( append_list_list_fm @ Xss12 @ ( cons_list_list_fm @ ( append_list_fm @ Xs3 @ Xs6 ) @ Xss22 ) ) )
& ( Ys
= ( append_list_fm @ ( concat_list_fm @ Xss12 ) @ Xs3 ) )
& ( Zs
= ( append_list_fm @ Xs6 @ ( concat_list_fm @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_819_concat__eq__append__conv,axiom,
! [Xss: list_list_tm,Ys: list_tm,Zs: list_tm] :
( ( ( concat_tm @ Xss )
= ( append_tm @ Ys @ Zs ) )
= ( ( ( Xss = nil_list_tm )
=> ( ( Ys = nil_tm )
& ( Zs = nil_tm ) ) )
& ( ( Xss != nil_list_tm )
=> ? [Xss12: list_list_tm,Xs3: list_tm,Xs6: list_tm,Xss22: list_list_tm] :
( ( Xss
= ( append_list_tm @ Xss12 @ ( cons_list_tm @ ( append_tm @ Xs3 @ Xs6 ) @ Xss22 ) ) )
& ( Ys
= ( append_tm @ ( concat_tm @ Xss12 ) @ Xs3 ) )
& ( Zs
= ( append_tm @ Xs6 @ ( concat_tm @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_820_concat__eq__append__conv,axiom,
! [Xss: list_list_fm,Ys: list_fm,Zs: list_fm] :
( ( ( concat_fm @ Xss )
= ( append_fm @ Ys @ Zs ) )
= ( ( ( Xss = nil_list_fm )
=> ( ( Ys = nil_fm )
& ( Zs = nil_fm ) ) )
& ( ( Xss != nil_list_fm )
=> ? [Xss12: list_list_fm,Xs3: list_fm,Xs6: list_fm,Xss22: list_list_fm] :
( ( Xss
= ( append_list_fm @ Xss12 @ ( cons_list_fm @ ( append_fm @ Xs3 @ Xs6 ) @ Xss22 ) ) )
& ( Ys
= ( append_fm @ ( concat_fm @ Xss12 ) @ Xs3 ) )
& ( Zs
= ( append_fm @ Xs6 @ ( concat_fm @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_821_rev__induct,axiom,
! [P: list_list_tm > $o,Xs: list_list_tm] :
( ( P @ nil_list_tm )
=> ( ! [X3: list_tm,Xs2: list_list_tm] :
( ( P @ Xs2 )
=> ( P @ ( append_list_tm @ Xs2 @ ( cons_list_tm @ X3 @ nil_list_tm ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_822_rev__induct,axiom,
! [P: list_fm > $o,Xs: list_fm] :
( ( P @ nil_fm )
=> ( ! [X3: fm,Xs2: list_fm] :
( ( P @ Xs2 )
=> ( P @ ( append_fm @ Xs2 @ ( cons_fm @ X3 @ nil_fm ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_823_rev__induct,axiom,
! [P: list_list_fm > $o,Xs: list_list_fm] :
( ( P @ nil_list_fm )
=> ( ! [X3: list_fm,Xs2: list_list_fm] :
( ( P @ Xs2 )
=> ( P @ ( append_list_fm @ Xs2 @ ( cons_list_fm @ X3 @ nil_list_fm ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_824_rev__induct,axiom,
! [P: list_tm > $o,Xs: list_tm] :
( ( P @ nil_tm )
=> ( ! [X3: tm,Xs2: list_tm] :
( ( P @ Xs2 )
=> ( P @ ( append_tm @ Xs2 @ ( cons_tm @ X3 @ nil_tm ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_825_rev__exhaust,axiom,
! [Xs: list_list_tm] :
( ( Xs != nil_list_tm )
=> ~ ! [Ys3: list_list_tm,Y: list_tm] :
( Xs
!= ( append_list_tm @ Ys3 @ ( cons_list_tm @ Y @ nil_list_tm ) ) ) ) ).
% rev_exhaust
thf(fact_826_rev__exhaust,axiom,
! [Xs: list_fm] :
( ( Xs != nil_fm )
=> ~ ! [Ys3: list_fm,Y: fm] :
( Xs
!= ( append_fm @ Ys3 @ ( cons_fm @ Y @ nil_fm ) ) ) ) ).
% rev_exhaust
thf(fact_827_rev__exhaust,axiom,
! [Xs: list_list_fm] :
( ( Xs != nil_list_fm )
=> ~ ! [Ys3: list_list_fm,Y: list_fm] :
( Xs
!= ( append_list_fm @ Ys3 @ ( cons_list_fm @ Y @ nil_list_fm ) ) ) ) ).
% rev_exhaust
thf(fact_828_rev__exhaust,axiom,
! [Xs: list_tm] :
( ( Xs != nil_tm )
=> ~ ! [Ys3: list_tm,Y: tm] :
( Xs
!= ( append_tm @ Ys3 @ ( cons_tm @ Y @ nil_tm ) ) ) ) ).
% rev_exhaust
thf(fact_829_Cons__eq__append__conv,axiom,
! [X: list_tm,Xs: list_list_tm,Ys: list_list_tm,Zs: list_list_tm] :
( ( ( cons_list_tm @ X @ Xs )
= ( append_list_tm @ Ys @ Zs ) )
= ( ( ( Ys = nil_list_tm )
& ( ( cons_list_tm @ X @ Xs )
= Zs ) )
| ? [Ys5: list_list_tm] :
( ( ( cons_list_tm @ X @ Ys5 )
= Ys )
& ( Xs
= ( append_list_tm @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_830_Cons__eq__append__conv,axiom,
! [X: fm,Xs: list_fm,Ys: list_fm,Zs: list_fm] :
( ( ( cons_fm @ X @ Xs )
= ( append_fm @ Ys @ Zs ) )
= ( ( ( Ys = nil_fm )
& ( ( cons_fm @ X @ Xs )
= Zs ) )
| ? [Ys5: list_fm] :
( ( ( cons_fm @ X @ Ys5 )
= Ys )
& ( Xs
= ( append_fm @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_831_Cons__eq__append__conv,axiom,
! [X: list_fm,Xs: list_list_fm,Ys: list_list_fm,Zs: list_list_fm] :
( ( ( cons_list_fm @ X @ Xs )
= ( append_list_fm @ Ys @ Zs ) )
= ( ( ( Ys = nil_list_fm )
& ( ( cons_list_fm @ X @ Xs )
= Zs ) )
| ? [Ys5: list_list_fm] :
( ( ( cons_list_fm @ X @ Ys5 )
= Ys )
& ( Xs
= ( append_list_fm @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_832_Cons__eq__append__conv,axiom,
! [X: tm,Xs: list_tm,Ys: list_tm,Zs: list_tm] :
( ( ( cons_tm @ X @ Xs )
= ( append_tm @ Ys @ Zs ) )
= ( ( ( Ys = nil_tm )
& ( ( cons_tm @ X @ Xs )
= Zs ) )
| ? [Ys5: list_tm] :
( ( ( cons_tm @ X @ Ys5 )
= Ys )
& ( Xs
= ( append_tm @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_833_append__eq__Cons__conv,axiom,
! [Ys: list_list_tm,Zs: list_list_tm,X: list_tm,Xs: list_list_tm] :
( ( ( append_list_tm @ Ys @ Zs )
= ( cons_list_tm @ X @ Xs ) )
= ( ( ( Ys = nil_list_tm )
& ( Zs
= ( cons_list_tm @ X @ Xs ) ) )
| ? [Ys5: list_list_tm] :
( ( Ys
= ( cons_list_tm @ X @ Ys5 ) )
& ( ( append_list_tm @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_834_append__eq__Cons__conv,axiom,
! [Ys: list_fm,Zs: list_fm,X: fm,Xs: list_fm] :
( ( ( append_fm @ Ys @ Zs )
= ( cons_fm @ X @ Xs ) )
= ( ( ( Ys = nil_fm )
& ( Zs
= ( cons_fm @ X @ Xs ) ) )
| ? [Ys5: list_fm] :
( ( Ys
= ( cons_fm @ X @ Ys5 ) )
& ( ( append_fm @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_835_append__eq__Cons__conv,axiom,
! [Ys: list_list_fm,Zs: list_list_fm,X: list_fm,Xs: list_list_fm] :
( ( ( append_list_fm @ Ys @ Zs )
= ( cons_list_fm @ X @ Xs ) )
= ( ( ( Ys = nil_list_fm )
& ( Zs
= ( cons_list_fm @ X @ Xs ) ) )
| ? [Ys5: list_list_fm] :
( ( Ys
= ( cons_list_fm @ X @ Ys5 ) )
& ( ( append_list_fm @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_836_append__eq__Cons__conv,axiom,
! [Ys: list_tm,Zs: list_tm,X: tm,Xs: list_tm] :
( ( ( append_tm @ Ys @ Zs )
= ( cons_tm @ X @ Xs ) )
= ( ( ( Ys = nil_tm )
& ( Zs
= ( cons_tm @ X @ Xs ) ) )
| ? [Ys5: list_tm] :
( ( Ys
= ( cons_tm @ X @ Ys5 ) )
& ( ( append_tm @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_837_rev__nonempty__induct,axiom,
! [Xs: list_list_tm,P: list_list_tm > $o] :
( ( Xs != nil_list_tm )
=> ( ! [X3: list_tm] : ( P @ ( cons_list_tm @ X3 @ nil_list_tm ) )
=> ( ! [X3: list_tm,Xs2: list_list_tm] :
( ( Xs2 != nil_list_tm )
=> ( ( P @ Xs2 )
=> ( P @ ( append_list_tm @ Xs2 @ ( cons_list_tm @ X3 @ nil_list_tm ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_838_rev__nonempty__induct,axiom,
! [Xs: list_fm,P: list_fm > $o] :
( ( Xs != nil_fm )
=> ( ! [X3: fm] : ( P @ ( cons_fm @ X3 @ nil_fm ) )
=> ( ! [X3: fm,Xs2: list_fm] :
( ( Xs2 != nil_fm )
=> ( ( P @ Xs2 )
=> ( P @ ( append_fm @ Xs2 @ ( cons_fm @ X3 @ nil_fm ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_839_rev__nonempty__induct,axiom,
! [Xs: list_list_fm,P: list_list_fm > $o] :
( ( Xs != nil_list_fm )
=> ( ! [X3: list_fm] : ( P @ ( cons_list_fm @ X3 @ nil_list_fm ) )
=> ( ! [X3: list_fm,Xs2: list_list_fm] :
( ( Xs2 != nil_list_fm )
=> ( ( P @ Xs2 )
=> ( P @ ( append_list_fm @ Xs2 @ ( cons_list_fm @ X3 @ nil_list_fm ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_840_rev__nonempty__induct,axiom,
! [Xs: list_tm,P: list_tm > $o] :
( ( Xs != nil_tm )
=> ( ! [X3: tm] : ( P @ ( cons_tm @ X3 @ nil_tm ) )
=> ( ! [X3: tm,Xs2: list_tm] :
( ( Xs2 != nil_tm )
=> ( ( P @ Xs2 )
=> ( P @ ( append_tm @ Xs2 @ ( cons_tm @ X3 @ nil_tm ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_841_splice_Ocases,axiom,
! [X: produc3245234490656042599ist_fm] :
( ! [Ys3: list_fm] :
( X
!= ( produc7863996417982153943ist_fm @ nil_fm @ Ys3 ) )
=> ~ ! [X3: fm,Xs2: list_fm,Ys3: list_fm] :
( X
!= ( produc7863996417982153943ist_fm @ ( cons_fm @ X3 @ Xs2 ) @ Ys3 ) ) ) ).
% splice.cases
thf(fact_842_splice_Ocases,axiom,
! [X: produc4386893285136772327ist_fm] :
( ! [Ys3: list_list_fm] :
( X
!= ( produc8321651870839017815ist_fm @ nil_list_fm @ Ys3 ) )
=> ~ ! [X3: list_fm,Xs2: list_list_fm,Ys3: list_list_fm] :
( X
!= ( produc8321651870839017815ist_fm @ ( cons_list_fm @ X3 @ Xs2 ) @ Ys3 ) ) ) ).
% splice.cases
thf(fact_843_splice_Ocases,axiom,
! [X: produc5776448205642668775ist_tm] :
( ! [Ys3: list_tm] :
( X
!= ( produc1418304791525149271ist_tm @ nil_tm @ Ys3 ) )
=> ~ ! [X3: tm,Xs2: list_tm,Ys3: list_tm] :
( X
!= ( produc1418304791525149271ist_tm @ ( cons_tm @ X3 @ Xs2 ) @ Ys3 ) ) ) ).
% splice.cases
thf(fact_844_shuffles_Ocases,axiom,
! [X: produc3245234490656042599ist_fm] :
( ! [Ys3: list_fm] :
( X
!= ( produc7863996417982153943ist_fm @ nil_fm @ Ys3 ) )
=> ( ! [Xs2: list_fm] :
( X
!= ( produc7863996417982153943ist_fm @ Xs2 @ nil_fm ) )
=> ~ ! [X3: fm,Xs2: list_fm,Y: fm,Ys3: list_fm] :
( X
!= ( produc7863996417982153943ist_fm @ ( cons_fm @ X3 @ Xs2 ) @ ( cons_fm @ Y @ Ys3 ) ) ) ) ) ).
% shuffles.cases
thf(fact_845_shuffles_Ocases,axiom,
! [X: produc4386893285136772327ist_fm] :
( ! [Ys3: list_list_fm] :
( X
!= ( produc8321651870839017815ist_fm @ nil_list_fm @ Ys3 ) )
=> ( ! [Xs2: list_list_fm] :
( X
!= ( produc8321651870839017815ist_fm @ Xs2 @ nil_list_fm ) )
=> ~ ! [X3: list_fm,Xs2: list_list_fm,Y: list_fm,Ys3: list_list_fm] :
( X
!= ( produc8321651870839017815ist_fm @ ( cons_list_fm @ X3 @ Xs2 ) @ ( cons_list_fm @ Y @ Ys3 ) ) ) ) ) ).
% shuffles.cases
thf(fact_846_shuffles_Ocases,axiom,
! [X: produc5776448205642668775ist_tm] :
( ! [Ys3: list_tm] :
( X
!= ( produc1418304791525149271ist_tm @ nil_tm @ Ys3 ) )
=> ( ! [Xs2: list_tm] :
( X
!= ( produc1418304791525149271ist_tm @ Xs2 @ nil_tm ) )
=> ~ ! [X3: tm,Xs2: list_tm,Y: tm,Ys3: list_tm] :
( X
!= ( produc1418304791525149271ist_tm @ ( cons_tm @ X3 @ Xs2 ) @ ( cons_tm @ Y @ Ys3 ) ) ) ) ) ).
% shuffles.cases
thf(fact_847_map__tailrec__rev_Ocases,axiom,
! [X: produc3792783349651969271ist_fm] :
( ! [F3: tm > fm,Bs: list_fm] :
( X
!= ( produc4025633523336838569ist_fm @ F3 @ ( produc1414352766439514085ist_fm @ nil_tm @ Bs ) ) )
=> ~ ! [F3: tm > fm,A3: tm,As: list_tm,Bs: list_fm] :
( X
!= ( produc4025633523336838569ist_fm @ F3 @ ( produc1414352766439514085ist_fm @ ( cons_tm @ A3 @ As ) @ Bs ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_848_not__Nil__listrel1,axiom,
! [Xs: list_fm,R: set_Pr4706815898642364871_fm_fm] :
~ ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ nil_fm @ Xs ) @ ( listrel1_fm @ R ) ) ).
% not_Nil_listrel1
thf(fact_849_not__Nil__listrel1,axiom,
! [Xs: list_list_fm,R: set_Pr1916137733696349511ist_fm] :
~ ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ nil_list_fm @ Xs ) @ ( listrel1_list_fm @ R ) ) ).
% not_Nil_listrel1
thf(fact_850_not__Nil__listrel1,axiom,
! [Xs: list_tm,R: set_Pr2455929065695642951_tm_tm] :
~ ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ nil_tm @ Xs ) @ ( listrel1_tm @ R ) ) ).
% not_Nil_listrel1
thf(fact_851_not__listrel1__Nil,axiom,
! [Xs: list_fm,R: set_Pr4706815898642364871_fm_fm] :
~ ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Xs @ nil_fm ) @ ( listrel1_fm @ R ) ) ).
% not_listrel1_Nil
thf(fact_852_not__listrel1__Nil,axiom,
! [Xs: list_list_fm,R: set_Pr1916137733696349511ist_fm] :
~ ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ Xs @ nil_list_fm ) @ ( listrel1_list_fm @ R ) ) ).
% not_listrel1_Nil
thf(fact_853_not__listrel1__Nil,axiom,
! [Xs: list_tm,R: set_Pr2455929065695642951_tm_tm] :
~ ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ nil_tm ) @ ( listrel1_tm @ R ) ) ).
% not_listrel1_Nil
thf(fact_854_lexord__Nil__right,axiom,
! [X: list_fm,R: set_Pr4706815898642364871_fm_fm] :
~ ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ X @ nil_fm ) @ ( lexord_fm @ R ) ) ).
% lexord_Nil_right
thf(fact_855_lexord__Nil__right,axiom,
! [X: list_list_fm,R: set_Pr1916137733696349511ist_fm] :
~ ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ X @ nil_list_fm ) @ ( lexord_list_fm @ R ) ) ).
% lexord_Nil_right
thf(fact_856_lexord__Nil__right,axiom,
! [X: list_tm,R: set_Pr2455929065695642951_tm_tm] :
~ ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ X @ nil_tm ) @ ( lexord_tm @ R ) ) ).
% lexord_Nil_right
thf(fact_857_Nil__notin__lex,axiom,
! [Ys: list_fm,R: set_Pr4706815898642364871_fm_fm] :
~ ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ nil_fm @ Ys ) @ ( lex_fm @ R ) ) ).
% Nil_notin_lex
thf(fact_858_Nil__notin__lex,axiom,
! [Ys: list_list_fm,R: set_Pr1916137733696349511ist_fm] :
~ ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ nil_list_fm @ Ys ) @ ( lex_list_fm @ R ) ) ).
% Nil_notin_lex
thf(fact_859_Nil__notin__lex,axiom,
! [Ys: list_tm,R: set_Pr2455929065695642951_tm_tm] :
~ ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ nil_tm @ Ys ) @ ( lex_tm @ R ) ) ).
% Nil_notin_lex
thf(fact_860_Nil2__notin__lex,axiom,
! [Xs: list_fm,R: set_Pr4706815898642364871_fm_fm] :
~ ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Xs @ nil_fm ) @ ( lex_fm @ R ) ) ).
% Nil2_notin_lex
thf(fact_861_Nil2__notin__lex,axiom,
! [Xs: list_list_fm,R: set_Pr1916137733696349511ist_fm] :
~ ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ Xs @ nil_list_fm ) @ ( lex_list_fm @ R ) ) ).
% Nil2_notin_lex
thf(fact_862_Nil2__notin__lex,axiom,
! [Xs: list_tm,R: set_Pr2455929065695642951_tm_tm] :
~ ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ nil_tm ) @ ( lex_tm @ R ) ) ).
% Nil2_notin_lex
thf(fact_863_Nil__lenlex__iff2,axiom,
! [Ns: list_fm,R: set_Pr4706815898642364871_fm_fm] :
~ ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Ns @ nil_fm ) @ ( lenlex_fm @ R ) ) ).
% Nil_lenlex_iff2
thf(fact_864_Nil__lenlex__iff2,axiom,
! [Ns: list_list_fm,R: set_Pr1916137733696349511ist_fm] :
~ ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ Ns @ nil_list_fm ) @ ( lenlex_list_fm @ R ) ) ).
% Nil_lenlex_iff2
thf(fact_865_Nil__lenlex__iff2,axiom,
! [Ns: list_tm,R: set_Pr2455929065695642951_tm_tm] :
~ ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Ns @ nil_tm ) @ ( lenlex_tm @ R ) ) ).
% Nil_lenlex_iff2
thf(fact_866_partition_Osimps_I1_J,axiom,
! [P: fm > $o] :
( ( partition_fm @ P @ nil_fm )
= ( produc7863996417982153943ist_fm @ nil_fm @ nil_fm ) ) ).
% partition.simps(1)
thf(fact_867_partition_Osimps_I1_J,axiom,
! [P: list_fm > $o] :
( ( partition_list_fm @ P @ nil_list_fm )
= ( produc8321651870839017815ist_fm @ nil_list_fm @ nil_list_fm ) ) ).
% partition.simps(1)
thf(fact_868_partition_Osimps_I1_J,axiom,
! [P: tm > $o] :
( ( partition_tm @ P @ nil_tm )
= ( produc1418304791525149271ist_tm @ nil_tm @ nil_tm ) ) ).
% partition.simps(1)
thf(fact_869_parts__in__children,axiom,
! [P2: fm,Z3: list_fm,Z2: list_fm,A4: list_tm,R: rule] :
( ( member_fm2 @ P2 @ ( set_fm2 @ Z3 ) )
=> ( ( member_list_fm2 @ Z2 @ ( set_list_fm2 @ ( children @ A4 @ R @ Z3 ) ) )
=> ? [B7: list_tm,Xs2: list_fm] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ A4 ) @ ( set_tm2 @ B7 ) )
& ( member_list_fm2 @ Xs2 @ ( set_list_fm2 @ ( parts @ B7 @ R @ P2 ) ) )
& ( ord_less_eq_set_fm @ ( set_fm2 @ Xs2 ) @ ( set_fm2 @ Z2 ) ) ) ) ) ).
% parts_in_children
thf(fact_870_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_871_le__cases3,axiom,
! [X: nat,Y3: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ X )
=> ~ ( ord_less_eq_nat @ X @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y3 ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ X ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_872_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_tm,Z6: set_tm] : ( Y6 = Z6 ) )
= ( ^ [X2: set_tm,Y4: set_tm] :
( ( ord_less_eq_set_tm @ X2 @ Y4 )
& ( ord_less_eq_set_tm @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_873_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_fm,Z6: set_fm] : ( Y6 = Z6 ) )
= ( ^ [X2: set_fm,Y4: set_fm] :
( ( ord_less_eq_set_fm @ X2 @ Y4 )
& ( ord_less_eq_set_fm @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_874_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z6: nat] : ( Y6 = Z6 ) )
= ( ^ [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_875_ord__eq__le__trans,axiom,
! [A: set_tm,B: set_tm,C: set_tm] :
( ( A = B )
=> ( ( ord_less_eq_set_tm @ B @ C )
=> ( ord_less_eq_set_tm @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_876_ord__eq__le__trans,axiom,
! [A: set_fm,B: set_fm,C: set_fm] :
( ( A = B )
=> ( ( ord_less_eq_set_fm @ B @ C )
=> ( ord_less_eq_set_fm @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_877_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_878_ord__le__eq__trans,axiom,
! [A: set_tm,B: set_tm,C: set_tm] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_tm @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_879_ord__le__eq__trans,axiom,
! [A: set_fm,B: set_fm,C: set_fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_fm @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_880_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_881_order__antisym,axiom,
! [X: set_tm,Y3: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y3 )
=> ( ( ord_less_eq_set_tm @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% order_antisym
thf(fact_882_order__antisym,axiom,
! [X: set_fm,Y3: set_fm] :
( ( ord_less_eq_set_fm @ X @ Y3 )
=> ( ( ord_less_eq_set_fm @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% order_antisym
thf(fact_883_order__antisym,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% order_antisym
thf(fact_884_order_Otrans,axiom,
! [A: set_tm,B: set_tm,C: set_tm] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( ord_less_eq_set_tm @ B @ C )
=> ( ord_less_eq_set_tm @ A @ C ) ) ) ).
% order.trans
thf(fact_885_order_Otrans,axiom,
! [A: set_fm,B: set_fm,C: set_fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ( ord_less_eq_set_fm @ B @ C )
=> ( ord_less_eq_set_fm @ A @ C ) ) ) ).
% order.trans
thf(fact_886_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_887_order__trans,axiom,
! [X: set_tm,Y3: set_tm,Z3: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y3 )
=> ( ( ord_less_eq_set_tm @ Y3 @ Z3 )
=> ( ord_less_eq_set_tm @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_888_order__trans,axiom,
! [X: set_fm,Y3: set_fm,Z3: set_fm] :
( ( ord_less_eq_set_fm @ X @ Y3 )
=> ( ( ord_less_eq_set_fm @ Y3 @ Z3 )
=> ( ord_less_eq_set_fm @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_889_order__trans,axiom,
! [X: nat,Y3: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z3 )
=> ( ord_less_eq_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_890_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_891_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: set_tm,Z6: set_tm] : ( Y6 = Z6 ) )
= ( ^ [A5: set_tm,B6: set_tm] :
( ( ord_less_eq_set_tm @ B6 @ A5 )
& ( ord_less_eq_set_tm @ A5 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_892_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: set_fm,Z6: set_fm] : ( Y6 = Z6 ) )
= ( ^ [A5: set_fm,B6: set_fm] :
( ( ord_less_eq_set_fm @ B6 @ A5 )
& ( ord_less_eq_set_fm @ A5 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_893_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: nat,Z6: nat] : ( Y6 = Z6 ) )
= ( ^ [A5: nat,B6: nat] :
( ( ord_less_eq_nat @ B6 @ A5 )
& ( ord_less_eq_nat @ A5 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_894_dual__order_Oantisym,axiom,
! [B: set_tm,A: set_tm] :
( ( ord_less_eq_set_tm @ B @ A )
=> ( ( ord_less_eq_set_tm @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_895_dual__order_Oantisym,axiom,
! [B: set_fm,A: set_fm] :
( ( ord_less_eq_set_fm @ B @ A )
=> ( ( ord_less_eq_set_fm @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_896_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_897_dual__order_Otrans,axiom,
! [B: set_tm,A: set_tm,C: set_tm] :
( ( ord_less_eq_set_tm @ B @ A )
=> ( ( ord_less_eq_set_tm @ C @ B )
=> ( ord_less_eq_set_tm @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_898_dual__order_Otrans,axiom,
! [B: set_fm,A: set_fm,C: set_fm] :
( ( ord_less_eq_set_fm @ B @ A )
=> ( ( ord_less_eq_set_fm @ C @ B )
=> ( ord_less_eq_set_fm @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_899_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_900_antisym,axiom,
! [A: set_tm,B: set_tm] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( ord_less_eq_set_tm @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_901_antisym,axiom,
! [A: set_fm,B: set_fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ( ord_less_eq_set_fm @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_902_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_903_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_tm,Z6: set_tm] : ( Y6 = Z6 ) )
= ( ^ [A5: set_tm,B6: set_tm] :
( ( ord_less_eq_set_tm @ A5 @ B6 )
& ( ord_less_eq_set_tm @ B6 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_904_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: set_fm,Z6: set_fm] : ( Y6 = Z6 ) )
= ( ^ [A5: set_fm,B6: set_fm] :
( ( ord_less_eq_set_fm @ A5 @ B6 )
& ( ord_less_eq_set_fm @ B6 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_905_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z6: nat] : ( Y6 = Z6 ) )
= ( ^ [A5: nat,B6: nat] :
( ( ord_less_eq_nat @ A5 @ B6 )
& ( ord_less_eq_nat @ B6 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_906_order__subst1,axiom,
! [A: set_tm,F: set_tm > set_tm,B: set_tm,C: set_tm] :
( ( ord_less_eq_set_tm @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_tm @ B @ C )
=> ( ! [X3: set_tm,Y: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_tm @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_907_order__subst1,axiom,
! [A: set_tm,F: set_fm > set_tm,B: set_fm,C: set_fm] :
( ( ord_less_eq_set_tm @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_fm @ B @ C )
=> ( ! [X3: set_fm,Y: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_tm @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_908_order__subst1,axiom,
! [A: set_tm,F: nat > set_tm,B: nat,C: nat] :
( ( ord_less_eq_set_tm @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_tm @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_909_order__subst1,axiom,
! [A: set_fm,F: set_tm > set_fm,B: set_tm,C: set_tm] :
( ( ord_less_eq_set_fm @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_tm @ B @ C )
=> ( ! [X3: set_tm,Y: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_fm @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_910_order__subst1,axiom,
! [A: set_fm,F: set_fm > set_fm,B: set_fm,C: set_fm] :
( ( ord_less_eq_set_fm @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_fm @ B @ C )
=> ( ! [X3: set_fm,Y: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_fm @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_911_order__subst1,axiom,
! [A: set_fm,F: nat > set_fm,B: nat,C: nat] :
( ( ord_less_eq_set_fm @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_fm @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_912_order__subst1,axiom,
! [A: nat,F: set_tm > nat,B: set_tm,C: set_tm] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_tm @ B @ C )
=> ( ! [X3: set_tm,Y: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_913_order__subst1,axiom,
! [A: nat,F: set_fm > nat,B: set_fm,C: set_fm] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_fm @ B @ C )
=> ( ! [X3: set_fm,Y: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_914_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_915_order__subst2,axiom,
! [A: set_tm,B: set_tm,F: set_tm > set_tm,C: set_tm] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( ord_less_eq_set_tm @ ( F @ B ) @ C )
=> ( ! [X3: set_tm,Y: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_tm @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_916_order__subst2,axiom,
! [A: set_tm,B: set_tm,F: set_tm > set_fm,C: set_fm] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( ord_less_eq_set_fm @ ( F @ B ) @ C )
=> ( ! [X3: set_tm,Y: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_fm @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_917_order__subst2,axiom,
! [A: set_tm,B: set_tm,F: set_tm > nat,C: nat] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_tm,Y: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_918_order__subst2,axiom,
! [A: set_fm,B: set_fm,F: set_fm > set_tm,C: set_tm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ( ord_less_eq_set_tm @ ( F @ B ) @ C )
=> ( ! [X3: set_fm,Y: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_tm @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_919_order__subst2,axiom,
! [A: set_fm,B: set_fm,F: set_fm > set_fm,C: set_fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ( ord_less_eq_set_fm @ ( F @ B ) @ C )
=> ( ! [X3: set_fm,Y: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_fm @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_920_order__subst2,axiom,
! [A: set_fm,B: set_fm,F: set_fm > nat,C: nat] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_fm,Y: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_921_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_tm,C: set_tm] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_tm @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_tm @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_922_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_fm,C: set_fm] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_fm @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_fm @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_923_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_924_order__eq__refl,axiom,
! [X: set_tm,Y3: set_tm] :
( ( X = Y3 )
=> ( ord_less_eq_set_tm @ X @ Y3 ) ) ).
% order_eq_refl
thf(fact_925_order__eq__refl,axiom,
! [X: set_fm,Y3: set_fm] :
( ( X = Y3 )
=> ( ord_less_eq_set_fm @ X @ Y3 ) ) ).
% order_eq_refl
thf(fact_926_order__eq__refl,axiom,
! [X: nat,Y3: nat] :
( ( X = Y3 )
=> ( ord_less_eq_nat @ X @ Y3 ) ) ).
% order_eq_refl
thf(fact_927_linorder__linear,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X ) ) ).
% linorder_linear
thf(fact_928_ord__eq__le__subst,axiom,
! [A: set_tm,F: set_tm > set_tm,B: set_tm,C: set_tm] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_tm @ B @ C )
=> ( ! [X3: set_tm,Y: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_tm @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_929_ord__eq__le__subst,axiom,
! [A: set_fm,F: set_tm > set_fm,B: set_tm,C: set_tm] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_tm @ B @ C )
=> ( ! [X3: set_tm,Y: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_fm @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_930_ord__eq__le__subst,axiom,
! [A: nat,F: set_tm > nat,B: set_tm,C: set_tm] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_tm @ B @ C )
=> ( ! [X3: set_tm,Y: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_931_ord__eq__le__subst,axiom,
! [A: set_tm,F: set_fm > set_tm,B: set_fm,C: set_fm] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_fm @ B @ C )
=> ( ! [X3: set_fm,Y: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_tm @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_932_ord__eq__le__subst,axiom,
! [A: set_fm,F: set_fm > set_fm,B: set_fm,C: set_fm] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_fm @ B @ C )
=> ( ! [X3: set_fm,Y: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_fm @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_933_ord__eq__le__subst,axiom,
! [A: nat,F: set_fm > nat,B: set_fm,C: set_fm] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_fm @ B @ C )
=> ( ! [X3: set_fm,Y: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_934_ord__eq__le__subst,axiom,
! [A: set_tm,F: nat > set_tm,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_tm @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_935_ord__eq__le__subst,axiom,
! [A: set_fm,F: nat > set_fm,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_fm @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_936_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_937_ord__le__eq__subst,axiom,
! [A: set_tm,B: set_tm,F: set_tm > set_tm,C: set_tm] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_tm,Y: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_tm @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_938_ord__le__eq__subst,axiom,
! [A: set_tm,B: set_tm,F: set_tm > set_fm,C: set_fm] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_tm,Y: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_fm @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_939_ord__le__eq__subst,axiom,
! [A: set_tm,B: set_tm,F: set_tm > nat,C: nat] :
( ( ord_less_eq_set_tm @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_tm,Y: set_tm] :
( ( ord_less_eq_set_tm @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_940_ord__le__eq__subst,axiom,
! [A: set_fm,B: set_fm,F: set_fm > set_tm,C: set_tm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_fm,Y: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_tm @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_941_ord__le__eq__subst,axiom,
! [A: set_fm,B: set_fm,F: set_fm > set_fm,C: set_fm] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_fm,Y: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_fm @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_942_ord__le__eq__subst,axiom,
! [A: set_fm,B: set_fm,F: set_fm > nat,C: nat] :
( ( ord_less_eq_set_fm @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_fm,Y: set_fm] :
( ( ord_less_eq_set_fm @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_943_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_tm,C: set_tm] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_set_tm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_tm @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_944_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_fm,C: set_fm] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_set_fm @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_fm @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_945_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_946_linorder__le__cases,axiom,
! [X: nat,Y3: nat] :
( ~ ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X ) ) ).
% linorder_le_cases
thf(fact_947_order__antisym__conv,axiom,
! [Y3: set_tm,X: set_tm] :
( ( ord_less_eq_set_tm @ Y3 @ X )
=> ( ( ord_less_eq_set_tm @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_948_order__antisym__conv,axiom,
! [Y3: set_fm,X: set_fm] :
( ( ord_less_eq_set_fm @ Y3 @ X )
=> ( ( ord_less_eq_set_fm @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_949_order__antisym__conv,axiom,
! [Y3: nat,X: nat] :
( ( ord_less_eq_nat @ Y3 @ X )
=> ( ( ord_less_eq_nat @ X @ Y3 )
= ( X = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_950_same__length__different,axiom,
! [Xs: list_list_tm,Ys: list_list_tm] :
( ( Xs != Ys )
=> ( ( ( size_s9096087352182575069ist_tm @ Xs )
= ( size_s9096087352182575069ist_tm @ Ys ) )
=> ? [Pre: list_list_tm,X3: list_tm,Xs5: list_list_tm,Y: list_tm,Ys6: list_list_tm] :
( ( X3 != Y )
& ( Xs
= ( append_list_tm @ Pre @ ( append_list_tm @ ( cons_list_tm @ X3 @ nil_list_tm ) @ Xs5 ) ) )
& ( Ys
= ( append_list_tm @ Pre @ ( append_list_tm @ ( cons_list_tm @ Y @ nil_list_tm ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_951_same__length__different,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( Xs != Ys )
=> ( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ? [Pre: list_fm,X3: fm,Xs5: list_fm,Y: fm,Ys6: list_fm] :
( ( X3 != Y )
& ( Xs
= ( append_fm @ Pre @ ( append_fm @ ( cons_fm @ X3 @ nil_fm ) @ Xs5 ) ) )
& ( Ys
= ( append_fm @ Pre @ ( append_fm @ ( cons_fm @ Y @ nil_fm ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_952_same__length__different,axiom,
! [Xs: list_list_fm,Ys: list_list_fm] :
( ( Xs != Ys )
=> ( ( ( size_s115229985653309035ist_fm @ Xs )
= ( size_s115229985653309035ist_fm @ Ys ) )
=> ? [Pre: list_list_fm,X3: list_fm,Xs5: list_list_fm,Y: list_fm,Ys6: list_list_fm] :
( ( X3 != Y )
& ( Xs
= ( append_list_fm @ Pre @ ( append_list_fm @ ( cons_list_fm @ X3 @ nil_list_fm ) @ Xs5 ) ) )
& ( Ys
= ( append_list_fm @ Pre @ ( append_list_fm @ ( cons_list_fm @ Y @ nil_list_fm ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_953_same__length__different,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( Xs != Ys )
=> ( ( ( size_size_list_tm @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ? [Pre: list_tm,X3: tm,Xs5: list_tm,Y: tm,Ys6: list_tm] :
( ( X3 != Y )
& ( Xs
= ( append_tm @ Pre @ ( append_tm @ ( cons_tm @ X3 @ nil_tm ) @ Xs5 ) ) )
& ( Ys
= ( append_tm @ Pre @ ( append_tm @ ( cons_tm @ Y @ nil_tm ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_954_longest__common__prefix_Ocases,axiom,
! [X: produc3245234490656042599ist_fm] :
( ! [X3: fm,Xs2: list_fm,Y: fm,Ys3: list_fm] :
( X
!= ( produc7863996417982153943ist_fm @ ( cons_fm @ X3 @ Xs2 ) @ ( cons_fm @ Y @ Ys3 ) ) )
=> ( ! [Uv: list_fm] :
( X
!= ( produc7863996417982153943ist_fm @ nil_fm @ Uv ) )
=> ~ ! [Uu: list_fm] :
( X
!= ( produc7863996417982153943ist_fm @ Uu @ nil_fm ) ) ) ) ).
% longest_common_prefix.cases
thf(fact_955_longest__common__prefix_Ocases,axiom,
! [X: produc4386893285136772327ist_fm] :
( ! [X3: list_fm,Xs2: list_list_fm,Y: list_fm,Ys3: list_list_fm] :
( X
!= ( produc8321651870839017815ist_fm @ ( cons_list_fm @ X3 @ Xs2 ) @ ( cons_list_fm @ Y @ Ys3 ) ) )
=> ( ! [Uv: list_list_fm] :
( X
!= ( produc8321651870839017815ist_fm @ nil_list_fm @ Uv ) )
=> ~ ! [Uu: list_list_fm] :
( X
!= ( produc8321651870839017815ist_fm @ Uu @ nil_list_fm ) ) ) ) ).
% longest_common_prefix.cases
thf(fact_956_longest__common__prefix_Ocases,axiom,
! [X: produc5776448205642668775ist_tm] :
( ! [X3: tm,Xs2: list_tm,Y: tm,Ys3: list_tm] :
( X
!= ( produc1418304791525149271ist_tm @ ( cons_tm @ X3 @ Xs2 ) @ ( cons_tm @ Y @ Ys3 ) ) )
=> ( ! [Uv: list_tm] :
( X
!= ( produc1418304791525149271ist_tm @ nil_tm @ Uv ) )
=> ~ ! [Uu: list_tm] :
( X
!= ( produc1418304791525149271ist_tm @ Uu @ nil_tm ) ) ) ) ).
% longest_common_prefix.cases
thf(fact_957_subset__antisym,axiom,
! [A4: set_tm,B4: set_tm] :
( ( ord_less_eq_set_tm @ A4 @ B4 )
=> ( ( ord_less_eq_set_tm @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% subset_antisym
thf(fact_958_subset__antisym,axiom,
! [A4: set_fm,B4: set_fm] :
( ( ord_less_eq_set_fm @ A4 @ B4 )
=> ( ( ord_less_eq_set_fm @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% subset_antisym
thf(fact_959_subsetI,axiom,
! [A4: set_list_fm,B4: set_list_fm] :
( ! [X3: list_fm] :
( ( member_list_fm2 @ X3 @ A4 )
=> ( member_list_fm2 @ X3 @ B4 ) )
=> ( ord_le7838213414353715577ist_fm @ A4 @ B4 ) ) ).
% subsetI
thf(fact_960_subsetI,axiom,
! [A4: set_Pr5202636777678657877ist_fm,B4: set_Pr5202636777678657877ist_fm] :
( ! [X3: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ X3 @ A4 )
=> ( member4699826688122452638ist_fm @ X3 @ B4 ) )
=> ( ord_le1771420097867575541ist_fm @ A4 @ B4 ) ) ).
% subsetI
thf(fact_961_subsetI,axiom,
! [A4: set_tm,B4: set_tm] :
( ! [X3: tm] :
( ( member_tm2 @ X3 @ A4 )
=> ( member_tm2 @ X3 @ B4 ) )
=> ( ord_less_eq_set_tm @ A4 @ B4 ) ) ).
% subsetI
thf(fact_962_subsetI,axiom,
! [A4: set_fm,B4: set_fm] :
( ! [X3: fm] :
( ( member_fm2 @ X3 @ A4 )
=> ( member_fm2 @ X3 @ B4 ) )
=> ( ord_less_eq_set_fm @ A4 @ B4 ) ) ).
% subsetI
thf(fact_963_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_964_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_965_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_966_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_967_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_968_psubsetI,axiom,
! [A4: set_tm,B4: set_tm] :
( ( ord_less_eq_set_tm @ A4 @ B4 )
=> ( ( A4 != B4 )
=> ( ord_less_set_tm @ A4 @ B4 ) ) ) ).
% psubsetI
thf(fact_969_psubsetI,axiom,
! [A4: set_fm,B4: set_fm] :
( ( ord_less_eq_set_fm @ A4 @ B4 )
=> ( ( A4 != B4 )
=> ( ord_less_set_fm @ A4 @ B4 ) ) ) ).
% psubsetI
thf(fact_970_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_tm
= ( ^ [A6: set_tm,B8: set_tm] :
( ( ord_less_set_tm @ A6 @ B8 )
| ( A6 = B8 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_971_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_fm
= ( ^ [A6: set_fm,B8: set_fm] :
( ( ord_less_set_fm @ A6 @ B8 )
| ( A6 = B8 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_972_subset__psubset__trans,axiom,
! [A4: set_tm,B4: set_tm,C3: set_tm] :
( ( ord_less_eq_set_tm @ A4 @ B4 )
=> ( ( ord_less_set_tm @ B4 @ C3 )
=> ( ord_less_set_tm @ A4 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_973_subset__psubset__trans,axiom,
! [A4: set_fm,B4: set_fm,C3: set_fm] :
( ( ord_less_eq_set_fm @ A4 @ B4 )
=> ( ( ord_less_set_fm @ B4 @ C3 )
=> ( ord_less_set_fm @ A4 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_974_subset__not__subset__eq,axiom,
( ord_less_set_tm
= ( ^ [A6: set_tm,B8: set_tm] :
( ( ord_less_eq_set_tm @ A6 @ B8 )
& ~ ( ord_less_eq_set_tm @ B8 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_975_subset__not__subset__eq,axiom,
( ord_less_set_fm
= ( ^ [A6: set_fm,B8: set_fm] :
( ( ord_less_eq_set_fm @ A6 @ B8 )
& ~ ( ord_less_eq_set_fm @ B8 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_976_psubset__subset__trans,axiom,
! [A4: set_tm,B4: set_tm,C3: set_tm] :
( ( ord_less_set_tm @ A4 @ B4 )
=> ( ( ord_less_eq_set_tm @ B4 @ C3 )
=> ( ord_less_set_tm @ A4 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_977_psubset__subset__trans,axiom,
! [A4: set_fm,B4: set_fm,C3: set_fm] :
( ( ord_less_set_fm @ A4 @ B4 )
=> ( ( ord_less_eq_set_fm @ B4 @ C3 )
=> ( ord_less_set_fm @ A4 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_978_psubset__imp__subset,axiom,
! [A4: set_tm,B4: set_tm] :
( ( ord_less_set_tm @ A4 @ B4 )
=> ( ord_less_eq_set_tm @ A4 @ B4 ) ) ).
% psubset_imp_subset
thf(fact_979_psubset__imp__subset,axiom,
! [A4: set_fm,B4: set_fm] :
( ( ord_less_set_fm @ A4 @ B4 )
=> ( ord_less_eq_set_fm @ A4 @ B4 ) ) ).
% psubset_imp_subset
thf(fact_980_psubset__eq,axiom,
( ord_less_set_tm
= ( ^ [A6: set_tm,B8: set_tm] :
( ( ord_less_eq_set_tm @ A6 @ B8 )
& ( A6 != B8 ) ) ) ) ).
% psubset_eq
thf(fact_981_psubset__eq,axiom,
( ord_less_set_fm
= ( ^ [A6: set_fm,B8: set_fm] :
( ( ord_less_eq_set_fm @ A6 @ B8 )
& ( A6 != B8 ) ) ) ) ).
% psubset_eq
thf(fact_982_psubsetE,axiom,
! [A4: set_tm,B4: set_tm] :
( ( ord_less_set_tm @ A4 @ B4 )
=> ~ ( ( ord_less_eq_set_tm @ A4 @ B4 )
=> ( ord_less_eq_set_tm @ B4 @ A4 ) ) ) ).
% psubsetE
thf(fact_983_psubsetE,axiom,
! [A4: set_fm,B4: set_fm] :
( ( ord_less_set_fm @ A4 @ B4 )
=> ~ ( ( ord_less_eq_set_fm @ A4 @ B4 )
=> ( ord_less_eq_set_fm @ B4 @ A4 ) ) ) ).
% psubsetE
thf(fact_984_pfsubsetD,axiom,
! [A4: fset_P8989946509869081563ist_fm,B4: fset_P8989946509869081563ist_fm,C: produc6018962875968178549ist_fm] :
( ( ord_le7716396445699002503ist_fm @ A4 @ B4 )
=> ( ( fmembe3754813877001230652ist_fm @ C @ A4 )
=> ( fmembe3754813877001230652ist_fm @ C @ B4 ) ) ) ).
% pfsubsetD
thf(fact_985_pfsubsetD,axiom,
! [A4: fset_tm,B4: fset_tm,C: tm] :
( ( ord_less_fset_tm @ A4 @ B4 )
=> ( ( fmember_tm @ C @ A4 )
=> ( fmember_tm @ C @ B4 ) ) ) ).
% pfsubsetD
thf(fact_986_pfsubsetD,axiom,
! [A4: fset_fm,B4: fset_fm,C: fm] :
( ( ord_less_fset_fm @ A4 @ B4 )
=> ( ( fmember_fm @ C @ A4 )
=> ( fmember_fm @ C @ B4 ) ) ) ).
% pfsubsetD
thf(fact_987_pfsubsetD,axiom,
! [A4: fset_list_fm,B4: fset_list_fm,C: list_fm] :
( ( ord_le4401106045268334603ist_fm @ A4 @ B4 )
=> ( ( fmember_list_fm @ C @ A4 )
=> ( fmember_list_fm @ C @ B4 ) ) ) ).
% pfsubsetD
thf(fact_988_in__mono,axiom,
! [A4: set_list_fm,B4: set_list_fm,X: list_fm] :
( ( ord_le7838213414353715577ist_fm @ A4 @ B4 )
=> ( ( member_list_fm2 @ X @ A4 )
=> ( member_list_fm2 @ X @ B4 ) ) ) ).
% in_mono
thf(fact_989_in__mono,axiom,
! [A4: set_Pr5202636777678657877ist_fm,B4: set_Pr5202636777678657877ist_fm,X: produc6018962875968178549ist_fm] :
( ( ord_le1771420097867575541ist_fm @ A4 @ B4 )
=> ( ( member4699826688122452638ist_fm @ X @ A4 )
=> ( member4699826688122452638ist_fm @ X @ B4 ) ) ) ).
% in_mono
thf(fact_990_in__mono,axiom,
! [A4: set_tm,B4: set_tm,X: tm] :
( ( ord_less_eq_set_tm @ A4 @ B4 )
=> ( ( member_tm2 @ X @ A4 )
=> ( member_tm2 @ X @ B4 ) ) ) ).
% in_mono
thf(fact_991_in__mono,axiom,
! [A4: set_fm,B4: set_fm,X: fm] :
( ( ord_less_eq_set_fm @ A4 @ B4 )
=> ( ( member_fm2 @ X @ A4 )
=> ( member_fm2 @ X @ B4 ) ) ) ).
% in_mono
thf(fact_992_subsetD,axiom,
! [A4: set_list_fm,B4: set_list_fm,C: list_fm] :
( ( ord_le7838213414353715577ist_fm @ A4 @ B4 )
=> ( ( member_list_fm2 @ C @ A4 )
=> ( member_list_fm2 @ C @ B4 ) ) ) ).
% subsetD
thf(fact_993_subsetD,axiom,
! [A4: set_Pr5202636777678657877ist_fm,B4: set_Pr5202636777678657877ist_fm,C: produc6018962875968178549ist_fm] :
( ( ord_le1771420097867575541ist_fm @ A4 @ B4 )
=> ( ( member4699826688122452638ist_fm @ C @ A4 )
=> ( member4699826688122452638ist_fm @ C @ B4 ) ) ) ).
% subsetD
thf(fact_994_subsetD,axiom,
! [A4: set_tm,B4: set_tm,C: tm] :
( ( ord_less_eq_set_tm @ A4 @ B4 )
=> ( ( member_tm2 @ C @ A4 )
=> ( member_tm2 @ C @ B4 ) ) ) ).
% subsetD
thf(fact_995_subsetD,axiom,
! [A4: set_fm,B4: set_fm,C: fm] :
( ( ord_less_eq_set_fm @ A4 @ B4 )
=> ( ( member_fm2 @ C @ A4 )
=> ( member_fm2 @ C @ B4 ) ) ) ).
% subsetD
thf(fact_996_equalityE,axiom,
! [A4: set_tm,B4: set_tm] :
( ( A4 = B4 )
=> ~ ( ( ord_less_eq_set_tm @ A4 @ B4 )
=> ~ ( ord_less_eq_set_tm @ B4 @ A4 ) ) ) ).
% equalityE
thf(fact_997_equalityE,axiom,
! [A4: set_fm,B4: set_fm] :
( ( A4 = B4 )
=> ~ ( ( ord_less_eq_set_fm @ A4 @ B4 )
=> ~ ( ord_less_eq_set_fm @ B4 @ A4 ) ) ) ).
% equalityE
thf(fact_998_subset__eq,axiom,
( ord_le7838213414353715577ist_fm
= ( ^ [A6: set_list_fm,B8: set_list_fm] :
! [X2: list_fm] :
( ( member_list_fm2 @ X2 @ A6 )
=> ( member_list_fm2 @ X2 @ B8 ) ) ) ) ).
% subset_eq
thf(fact_999_subset__eq,axiom,
( ord_le1771420097867575541ist_fm
= ( ^ [A6: set_Pr5202636777678657877ist_fm,B8: set_Pr5202636777678657877ist_fm] :
! [X2: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ X2 @ A6 )
=> ( member4699826688122452638ist_fm @ X2 @ B8 ) ) ) ) ).
% subset_eq
thf(fact_1000_subset__eq,axiom,
( ord_less_eq_set_tm
= ( ^ [A6: set_tm,B8: set_tm] :
! [X2: tm] :
( ( member_tm2 @ X2 @ A6 )
=> ( member_tm2 @ X2 @ B8 ) ) ) ) ).
% subset_eq
thf(fact_1001_subset__eq,axiom,
( ord_less_eq_set_fm
= ( ^ [A6: set_fm,B8: set_fm] :
! [X2: fm] :
( ( member_fm2 @ X2 @ A6 )
=> ( member_fm2 @ X2 @ B8 ) ) ) ) ).
% subset_eq
thf(fact_1002_equalityD1,axiom,
! [A4: set_tm,B4: set_tm] :
( ( A4 = B4 )
=> ( ord_less_eq_set_tm @ A4 @ B4 ) ) ).
% equalityD1
thf(fact_1003_equalityD1,axiom,
! [A4: set_fm,B4: set_fm] :
( ( A4 = B4 )
=> ( ord_less_eq_set_fm @ A4 @ B4 ) ) ).
% equalityD1
thf(fact_1004_equalityD2,axiom,
! [A4: set_tm,B4: set_tm] :
( ( A4 = B4 )
=> ( ord_less_eq_set_tm @ B4 @ A4 ) ) ).
% equalityD2
thf(fact_1005_equalityD2,axiom,
! [A4: set_fm,B4: set_fm] :
( ( A4 = B4 )
=> ( ord_less_eq_set_fm @ B4 @ A4 ) ) ).
% equalityD2
thf(fact_1006_subset__iff,axiom,
( ord_le7838213414353715577ist_fm
= ( ^ [A6: set_list_fm,B8: set_list_fm] :
! [T2: list_fm] :
( ( member_list_fm2 @ T2 @ A6 )
=> ( member_list_fm2 @ T2 @ B8 ) ) ) ) ).
% subset_iff
thf(fact_1007_subset__iff,axiom,
( ord_le1771420097867575541ist_fm
= ( ^ [A6: set_Pr5202636777678657877ist_fm,B8: set_Pr5202636777678657877ist_fm] :
! [T2: produc6018962875968178549ist_fm] :
( ( member4699826688122452638ist_fm @ T2 @ A6 )
=> ( member4699826688122452638ist_fm @ T2 @ B8 ) ) ) ) ).
% subset_iff
thf(fact_1008_subset__iff,axiom,
( ord_less_eq_set_tm
= ( ^ [A6: set_tm,B8: set_tm] :
! [T2: tm] :
( ( member_tm2 @ T2 @ A6 )
=> ( member_tm2 @ T2 @ B8 ) ) ) ) ).
% subset_iff
thf(fact_1009_subset__iff,axiom,
( ord_less_eq_set_fm
= ( ^ [A6: set_fm,B8: set_fm] :
! [T2: fm] :
( ( member_fm2 @ T2 @ A6 )
=> ( member_fm2 @ T2 @ B8 ) ) ) ) ).
% subset_iff
thf(fact_1010_subset__refl,axiom,
! [A4: set_tm] : ( ord_less_eq_set_tm @ A4 @ A4 ) ).
% subset_refl
thf(fact_1011_subset__refl,axiom,
! [A4: set_fm] : ( ord_less_eq_set_fm @ A4 @ A4 ) ).
% subset_refl
thf(fact_1012_Collect__mono,axiom,
! [P: tm > $o,Q: tm > $o] :
( ! [X3: tm] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_tm @ ( collect_tm @ P ) @ ( collect_tm @ Q ) ) ) ).
% Collect_mono
thf(fact_1013_Collect__mono,axiom,
! [P: fm > $o,Q: fm > $o] :
( ! [X3: fm] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_fm @ ( collect_fm @ P ) @ ( collect_fm @ Q ) ) ) ).
% Collect_mono
thf(fact_1014_subset__trans,axiom,
! [A4: set_tm,B4: set_tm,C3: set_tm] :
( ( ord_less_eq_set_tm @ A4 @ B4 )
=> ( ( ord_less_eq_set_tm @ B4 @ C3 )
=> ( ord_less_eq_set_tm @ A4 @ C3 ) ) ) ).
% subset_trans
thf(fact_1015_subset__trans,axiom,
! [A4: set_fm,B4: set_fm,C3: set_fm] :
( ( ord_less_eq_set_fm @ A4 @ B4 )
=> ( ( ord_less_eq_set_fm @ B4 @ C3 )
=> ( ord_less_eq_set_fm @ A4 @ C3 ) ) ) ).
% subset_trans
thf(fact_1016_set__eq__subset,axiom,
( ( ^ [Y6: set_tm,Z6: set_tm] : ( Y6 = Z6 ) )
= ( ^ [A6: set_tm,B8: set_tm] :
( ( ord_less_eq_set_tm @ A6 @ B8 )
& ( ord_less_eq_set_tm @ B8 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_1017_set__eq__subset,axiom,
( ( ^ [Y6: set_fm,Z6: set_fm] : ( Y6 = Z6 ) )
= ( ^ [A6: set_fm,B8: set_fm] :
( ( ord_less_eq_set_fm @ A6 @ B8 )
& ( ord_less_eq_set_fm @ B8 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_1018_Collect__mono__iff,axiom,
! [P: tm > $o,Q: tm > $o] :
( ( ord_less_eq_set_tm @ ( collect_tm @ P ) @ ( collect_tm @ Q ) )
= ( ! [X2: tm] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_1019_Collect__mono__iff,axiom,
! [P: fm > $o,Q: fm > $o] :
( ( ord_less_eq_set_fm @ ( collect_fm @ P ) @ ( collect_fm @ Q ) )
= ( ! [X2: fm] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_1020_linorder__neqE__nat,axiom,
! [X: nat,Y3: nat] :
( ( X != Y3 )
=> ( ~ ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ Y3 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_1021_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_1022_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_1023_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_1024_less__not__refl3,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( S2 != T ) ) ).
% less_not_refl3
thf(fact_1025_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_1026_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_1027_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_1028_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1029_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_1030_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_1031_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_1032_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_1033_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1034_size__neq__size__imp__neq,axiom,
! [X: char,Y3: char] :
( ( ( size_size_char @ X )
!= ( size_size_char @ Y3 ) )
=> ( X != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_1035_size__neq__size__imp__neq,axiom,
! [X: fm,Y3: fm] :
( ( ( size_size_fm @ X )
!= ( size_size_fm @ Y3 ) )
=> ( X != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_1036_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1037_prefixes__snoc,axiom,
! [Xs: list_list_tm,X: list_tm] :
( ( prefixes_list_tm @ ( append_list_tm @ Xs @ ( cons_list_tm @ X @ nil_list_tm ) ) )
= ( append_list_list_tm @ ( prefixes_list_tm @ Xs ) @ ( cons_list_list_tm @ ( append_list_tm @ Xs @ ( cons_list_tm @ X @ nil_list_tm ) ) @ nil_list_list_tm ) ) ) ).
% prefixes_snoc
thf(fact_1038_prefixes__snoc,axiom,
! [Xs: list_fm,X: fm] :
( ( prefixes_fm @ ( append_fm @ Xs @ ( cons_fm @ X @ nil_fm ) ) )
= ( append_list_fm @ ( prefixes_fm @ Xs ) @ ( cons_list_fm @ ( append_fm @ Xs @ ( cons_fm @ X @ nil_fm ) ) @ nil_list_fm ) ) ) ).
% prefixes_snoc
thf(fact_1039_prefixes__snoc,axiom,
! [Xs: list_list_fm,X: list_fm] :
( ( prefixes_list_fm @ ( append_list_fm @ Xs @ ( cons_list_fm @ X @ nil_list_fm ) ) )
= ( append_list_list_fm @ ( prefixes_list_fm @ Xs ) @ ( cons_list_list_fm @ ( append_list_fm @ Xs @ ( cons_list_fm @ X @ nil_list_fm ) ) @ nil_list_list_fm ) ) ) ).
% prefixes_snoc
thf(fact_1040_prefixes__snoc,axiom,
! [Xs: list_tm,X: tm] :
( ( prefixes_tm @ ( append_tm @ Xs @ ( cons_tm @ X @ nil_tm ) ) )
= ( append_list_tm @ ( prefixes_tm @ Xs ) @ ( cons_list_tm @ ( append_tm @ Xs @ ( cons_tm @ X @ nil_tm ) ) @ nil_list_tm ) ) ) ).
% prefixes_snoc
thf(fact_1041_prefixes__eq__snoc,axiom,
! [Ys: list_list_tm,Xs: list_list_list_tm,X: list_list_tm] :
( ( ( prefixes_list_tm @ Ys )
= ( append_list_list_tm @ Xs @ ( cons_list_list_tm @ X @ nil_list_list_tm ) ) )
= ( ( ( ( Ys = nil_list_tm )
& ( Xs = nil_list_list_tm ) )
| ? [Z5: list_tm,Zs2: list_list_tm] :
( ( Ys
= ( append_list_tm @ Zs2 @ ( cons_list_tm @ Z5 @ nil_list_tm ) ) )
& ( Xs
= ( prefixes_list_tm @ Zs2 ) ) ) )
& ( X = Ys ) ) ) ).
% prefixes_eq_snoc
thf(fact_1042_prefixes__eq__snoc,axiom,
! [Ys: list_list_fm,Xs: list_list_list_fm,X: list_list_fm] :
( ( ( prefixes_list_fm @ Ys )
= ( append_list_list_fm @ Xs @ ( cons_list_list_fm @ X @ nil_list_list_fm ) ) )
= ( ( ( ( Ys = nil_list_fm )
& ( Xs = nil_list_list_fm ) )
| ? [Z5: list_fm,Zs2: list_list_fm] :
( ( Ys
= ( append_list_fm @ Zs2 @ ( cons_list_fm @ Z5 @ nil_list_fm ) ) )
& ( Xs
= ( prefixes_list_fm @ Zs2 ) ) ) )
& ( X = Ys ) ) ) ).
% prefixes_eq_snoc
thf(fact_1043_prefixes__eq__snoc,axiom,
! [Ys: list_tm,Xs: list_list_tm,X: list_tm] :
( ( ( prefixes_tm @ Ys )
= ( append_list_tm @ Xs @ ( cons_list_tm @ X @ nil_list_tm ) ) )
= ( ( ( ( Ys = nil_tm )
& ( Xs = nil_list_tm ) )
| ? [Z5: tm,Zs2: list_tm] :
( ( Ys
= ( append_tm @ Zs2 @ ( cons_tm @ Z5 @ nil_tm ) ) )
& ( Xs
= ( prefixes_tm @ Zs2 ) ) ) )
& ( X = Ys ) ) ) ).
% prefixes_eq_snoc
thf(fact_1044_prefixes__eq__snoc,axiom,
! [Ys: list_fm,Xs: list_list_fm,X: list_fm] :
( ( ( prefixes_fm @ Ys )
= ( append_list_fm @ Xs @ ( cons_list_fm @ X @ nil_list_fm ) ) )
= ( ( ( ( Ys = nil_fm )
& ( Xs = nil_list_fm ) )
| ? [Z5: fm,Zs2: list_fm] :
( ( Ys
= ( append_fm @ Zs2 @ ( cons_fm @ Z5 @ nil_fm ) ) )
& ( Xs
= ( prefixes_fm @ Zs2 ) ) ) )
& ( X = Ys ) ) ) ).
% prefixes_eq_snoc
thf(fact_1045_subrelI,axiom,
! [R: set_Pr5202636777678657877ist_fm,S2: set_Pr5202636777678657877ist_fm] :
( ! [X3: list_tm,Y: list_fm] :
( ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ X3 @ Y ) @ R )
=> ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ X3 @ Y ) @ S2 ) )
=> ( ord_le1771420097867575541ist_fm @ R @ S2 ) ) ).
% subrelI
thf(fact_1046_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K2 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_1047_suffixes__eq__snoc,axiom,
! [Ys: list_list_fm,Xs: list_list_list_fm,X: list_list_fm] :
( ( ( suffixes_list_fm @ Ys )
= ( append_list_list_fm @ Xs @ ( cons_list_list_fm @ X @ nil_list_list_fm ) ) )
= ( ( ( ( Ys = nil_list_fm )
& ( Xs = nil_list_list_fm ) )
| ? [Z5: list_fm,Zs2: list_list_fm] :
( ( Ys
= ( cons_list_fm @ Z5 @ Zs2 ) )
& ( Xs
= ( suffixes_list_fm @ Zs2 ) ) ) )
& ( X = Ys ) ) ) ).
% suffixes_eq_snoc
thf(fact_1048_suffixes__eq__snoc,axiom,
! [Ys: list_tm,Xs: list_list_tm,X: list_tm] :
( ( ( suffixes_tm @ Ys )
= ( append_list_tm @ Xs @ ( cons_list_tm @ X @ nil_list_tm ) ) )
= ( ( ( ( Ys = nil_tm )
& ( Xs = nil_list_tm ) )
| ? [Z5: tm,Zs2: list_tm] :
( ( Ys
= ( cons_tm @ Z5 @ Zs2 ) )
& ( Xs
= ( suffixes_tm @ Zs2 ) ) ) )
& ( X = Ys ) ) ) ).
% suffixes_eq_snoc
thf(fact_1049_suffixes__eq__snoc,axiom,
! [Ys: list_fm,Xs: list_list_fm,X: list_fm] :
( ( ( suffixes_fm @ Ys )
= ( append_list_fm @ Xs @ ( cons_list_fm @ X @ nil_list_fm ) ) )
= ( ( ( ( Ys = nil_fm )
& ( Xs = nil_list_fm ) )
| ? [Z5: fm,Zs2: list_fm] :
( ( Ys
= ( cons_fm @ Z5 @ Zs2 ) )
& ( Xs
= ( suffixes_fm @ Zs2 ) ) ) )
& ( X = Ys ) ) ) ).
% suffixes_eq_snoc
thf(fact_1050_psubsetD,axiom,
! [A4: set_fm,B4: set_fm,C: fm] :
( ( ord_less_set_fm @ A4 @ B4 )
=> ( ( member_fm2 @ C @ A4 )
=> ( member_fm2 @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_1051_psubsetD,axiom,
! [A4: set_list_fm,B4: set_list_fm,C: list_fm] :
( ( ord_less_set_list_fm @ A4 @ B4 )
=> ( ( member_list_fm2 @ C @ A4 )
=> ( member_list_fm2 @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_1052_psubsetD,axiom,
! [A4: set_tm,B4: set_tm,C: tm] :
( ( ord_less_set_tm @ A4 @ B4 )
=> ( ( member_tm2 @ C @ A4 )
=> ( member_tm2 @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_1053_psubsetD,axiom,
! [A4: set_Pr5202636777678657877ist_fm,B4: set_Pr5202636777678657877ist_fm,C: produc6018962875968178549ist_fm] :
( ( ord_le9141428925376846849ist_fm @ A4 @ B4 )
=> ( ( member4699826688122452638ist_fm @ C @ A4 )
=> ( member4699826688122452638ist_fm @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_1054_prefixes__not__Nil,axiom,
! [Xs: list_fm] :
( ( prefixes_fm @ Xs )
!= nil_list_fm ) ).
% prefixes_not_Nil
thf(fact_1055_suffixes__not__Nil,axiom,
! [Xs: list_fm] :
( ( suffixes_fm @ Xs )
!= nil_list_fm ) ).
% suffixes_not_Nil
thf(fact_1056_prefixes_Osimps_I1_J,axiom,
( ( prefixes_list_fm @ nil_list_fm )
= ( cons_list_list_fm @ nil_list_fm @ nil_list_list_fm ) ) ).
% prefixes.simps(1)
thf(fact_1057_prefixes_Osimps_I1_J,axiom,
( ( prefixes_tm @ nil_tm )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) ).
% prefixes.simps(1)
thf(fact_1058_prefixes_Osimps_I1_J,axiom,
( ( prefixes_fm @ nil_fm )
= ( cons_list_fm @ nil_fm @ nil_list_fm ) ) ).
% prefixes.simps(1)
thf(fact_1059_suffixes_Osimps_I1_J,axiom,
( ( suffixes_list_fm @ nil_list_fm )
= ( cons_list_list_fm @ nil_list_fm @ nil_list_list_fm ) ) ).
% suffixes.simps(1)
thf(fact_1060_suffixes_Osimps_I1_J,axiom,
( ( suffixes_tm @ nil_tm )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) ).
% suffixes.simps(1)
thf(fact_1061_suffixes_Osimps_I1_J,axiom,
( ( suffixes_fm @ nil_fm )
= ( cons_list_fm @ nil_fm @ nil_list_fm ) ) ).
% suffixes.simps(1)
thf(fact_1062_prefixes_Osimps_I2_J,axiom,
! [X: fm,Xs: list_fm] :
( ( prefixes_fm @ ( cons_fm @ X @ Xs ) )
= ( cons_list_fm @ nil_fm @ ( map_list_fm_list_fm @ ( cons_fm @ X ) @ ( prefixes_fm @ Xs ) ) ) ) ).
% prefixes.simps(2)
thf(fact_1063_prefixes_Osimps_I2_J,axiom,
! [X: list_fm,Xs: list_list_fm] :
( ( prefixes_list_fm @ ( cons_list_fm @ X @ Xs ) )
= ( cons_list_list_fm @ nil_list_fm @ ( map_li4351931137408529412ist_fm @ ( cons_list_fm @ X ) @ ( prefixes_list_fm @ Xs ) ) ) ) ).
% prefixes.simps(2)
thf(fact_1064_prefixes_Osimps_I2_J,axiom,
! [X: tm,Xs: list_tm] :
( ( prefixes_tm @ ( cons_tm @ X @ Xs ) )
= ( cons_list_tm @ nil_tm @ ( map_list_tm_list_tm @ ( cons_tm @ X ) @ ( prefixes_tm @ Xs ) ) ) ) ).
% prefixes.simps(2)
thf(fact_1065_suffixes_Osimps_I2_J,axiom,
! [X: fm,Xs: list_fm] :
( ( suffixes_fm @ ( cons_fm @ X @ Xs ) )
= ( append_list_fm @ ( suffixes_fm @ Xs ) @ ( cons_list_fm @ ( cons_fm @ X @ Xs ) @ nil_list_fm ) ) ) ).
% suffixes.simps(2)
thf(fact_1066_suffixes_Osimps_I2_J,axiom,
! [X: list_fm,Xs: list_list_fm] :
( ( suffixes_list_fm @ ( cons_list_fm @ X @ Xs ) )
= ( append_list_list_fm @ ( suffixes_list_fm @ Xs ) @ ( cons_list_list_fm @ ( cons_list_fm @ X @ Xs ) @ nil_list_list_fm ) ) ) ).
% suffixes.simps(2)
thf(fact_1067_suffixes_Osimps_I2_J,axiom,
! [X: tm,Xs: list_tm] :
( ( suffixes_tm @ ( cons_tm @ X @ Xs ) )
= ( append_list_tm @ ( suffixes_tm @ Xs ) @ ( cons_list_tm @ ( cons_tm @ X @ Xs ) @ nil_list_tm ) ) ) ).
% suffixes.simps(2)
thf(fact_1068_sublists_Osimps_I2_J,axiom,
! [X: list_fm,Xs: list_list_fm] :
( ( sublists_list_fm @ ( cons_list_fm @ X @ Xs ) )
= ( append_list_list_fm @ ( sublists_list_fm @ Xs ) @ ( map_li4351931137408529412ist_fm @ ( cons_list_fm @ X ) @ ( prefixes_list_fm @ Xs ) ) ) ) ).
% sublists.simps(2)
thf(fact_1069_sublists_Osimps_I2_J,axiom,
! [X: tm,Xs: list_tm] :
( ( sublists_tm @ ( cons_tm @ X @ Xs ) )
= ( append_list_tm @ ( sublists_tm @ Xs ) @ ( map_list_tm_list_tm @ ( cons_tm @ X ) @ ( prefixes_tm @ Xs ) ) ) ) ).
% sublists.simps(2)
thf(fact_1070_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C4: nat] :
( ( ord_less_eq_nat @ A @ C4 )
& ( ord_less_eq_nat @ C4 @ B )
& ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ C4 ) )
=> ( P @ X4 ) )
& ! [D: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1071_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_1072_pinf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ T ) ) ).
% pinf(6)
thf(fact_1073_pinf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z @ X4 )
=> ( ord_less_eq_nat @ T @ X4 ) ) ).
% pinf(8)
thf(fact_1074_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_1075_verit__comp__simplify1_I2_J,axiom,
! [A: set_tm] : ( ord_less_eq_set_tm @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1076_verit__comp__simplify1_I2_J,axiom,
! [A: set_fm] : ( ord_less_eq_set_fm @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1077_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1078_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_1079_pinf_I1_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P6 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1080_pinf_I2_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P6 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1081_pinf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z @ X4 )
=> ( X4 != T ) ) ).
% pinf(3)
thf(fact_1082_pinf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z @ X4 )
=> ( X4 != T ) ) ).
% pinf(4)
thf(fact_1083_pinf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z @ X4 )
=> ~ ( ord_less_nat @ X4 @ T ) ) ).
% pinf(5)
thf(fact_1084_pinf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z @ X4 )
=> ( ord_less_nat @ T @ X4 ) ) ).
% pinf(7)
thf(fact_1085_minf_I1_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P6 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1086_minf_I2_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P6 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1087_minf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z )
=> ( X4 != T ) ) ).
% minf(3)
thf(fact_1088_minf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z )
=> ( X4 != T ) ) ).
% minf(4)
thf(fact_1089_minf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z )
=> ( ord_less_nat @ X4 @ T ) ) ).
% minf(5)
thf(fact_1090_minf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z )
=> ~ ( ord_less_nat @ T @ X4 ) ) ).
% minf(7)
thf(fact_1091_sublists_Osimps_I1_J,axiom,
( ( sublists_list_fm @ nil_list_fm )
= ( cons_list_list_fm @ nil_list_fm @ nil_list_list_fm ) ) ).
% sublists.simps(1)
thf(fact_1092_sublists_Osimps_I1_J,axiom,
( ( sublists_tm @ nil_tm )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) ).
% sublists.simps(1)
thf(fact_1093_sublists_Osimps_I1_J,axiom,
( ( sublists_fm @ nil_fm )
= ( cons_list_fm @ nil_fm @ nil_list_fm ) ) ).
% sublists.simps(1)
thf(fact_1094_minf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z )
=> ~ ( ord_less_eq_nat @ T @ X4 ) ) ).
% minf(8)
thf(fact_1095_minf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z )
=> ( ord_less_eq_nat @ X4 @ T ) ) ).
% minf(6)
thf(fact_1096_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_list_fm @ N @ nil_list_fm )
= ( cons_list_list_fm @ nil_list_fm @ nil_list_list_fm ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_list_fm @ N @ nil_list_fm )
= nil_list_list_fm ) ) ) ).
% n_lists_Nil
thf(fact_1097_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_tm @ N @ nil_tm )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_tm @ N @ nil_tm )
= nil_list_tm ) ) ) ).
% n_lists_Nil
thf(fact_1098_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_fm @ N @ nil_fm )
= ( cons_list_fm @ nil_fm @ nil_list_fm ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_fm @ N @ nil_fm )
= nil_list_fm ) ) ) ).
% n_lists_Nil
thf(fact_1099_the__elem__set,axiom,
! [X: fm] :
( ( the_elem_fm @ ( set_fm2 @ ( cons_fm @ X @ nil_fm ) ) )
= X ) ).
% the_elem_set
thf(fact_1100_the__elem__set,axiom,
! [X: produc6018962875968178549ist_fm] :
( ( the_el878164078765530474ist_fm @ ( set_Pr5051287786238754058ist_fm @ ( cons_P6586734674047340843ist_fm @ X @ nil_Pr2223031680201799035ist_fm ) ) )
= X ) ).
% the_elem_set
thf(fact_1101_the__elem__set,axiom,
! [X: list_fm] :
( ( the_elem_list_fm @ ( set_list_fm2 @ ( cons_list_fm @ X @ nil_list_fm ) ) )
= X ) ).
% the_elem_set
thf(fact_1102_the__elem__set,axiom,
! [X: tm] :
( ( the_elem_tm @ ( set_tm2 @ ( cons_tm @ X @ nil_tm ) ) )
= X ) ).
% the_elem_set
thf(fact_1103_subtermTm__le,axiom,
! [T: tm,S2: tm] :
( ( member_tm2 @ T @ ( set_tm2 @ ( subtermTm @ S2 ) ) )
=> ( ord_less_eq_set_tm @ ( set_tm2 @ ( subtermTm @ T ) ) @ ( set_tm2 @ ( subtermTm @ S2 ) ) ) ) ).
% subtermTm_le
thf(fact_1104_n__lists_Osimps_I1_J,axiom,
! [Xs: list_list_fm] :
( ( n_lists_list_fm @ zero_zero_nat @ Xs )
= ( cons_list_list_fm @ nil_list_fm @ nil_list_list_fm ) ) ).
% n_lists.simps(1)
thf(fact_1105_n__lists_Osimps_I1_J,axiom,
! [Xs: list_tm] :
( ( n_lists_tm @ zero_zero_nat @ Xs )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) ).
% n_lists.simps(1)
thf(fact_1106_n__lists_Osimps_I1_J,axiom,
! [Xs: list_fm] :
( ( n_lists_fm @ zero_zero_nat @ Xs )
= ( cons_list_fm @ nil_fm @ nil_list_fm ) ) ).
% n_lists.simps(1)
thf(fact_1107_children_Osimps_I1_J,axiom,
! [Uu2: list_tm,Uv2: rule] :
( ( children @ Uu2 @ Uv2 @ nil_fm )
= ( cons_list_fm @ nil_fm @ nil_list_fm ) ) ).
% children.simps(1)
thf(fact_1108_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_1109_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1110_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1111_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1112_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1113_length__0__conv,axiom,
! [Xs: list_fm] :
( ( ( size_size_list_fm @ Xs )
= zero_zero_nat )
= ( Xs = nil_fm ) ) ).
% length_0_conv
thf(fact_1114_length__0__conv,axiom,
! [Xs: list_list_fm] :
( ( ( size_s115229985653309035ist_fm @ Xs )
= zero_zero_nat )
= ( Xs = nil_list_fm ) ) ).
% length_0_conv
thf(fact_1115_length__0__conv,axiom,
! [Xs: list_tm] :
( ( ( size_size_list_tm @ Xs )
= zero_zero_nat )
= ( Xs = nil_tm ) ) ).
% length_0_conv
thf(fact_1116_length__greater__0__conv,axiom,
! [Xs: list_fm] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_fm @ Xs ) )
= ( Xs != nil_fm ) ) ).
% length_greater_0_conv
thf(fact_1117_length__greater__0__conv,axiom,
! [Xs: list_list_fm] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s115229985653309035ist_fm @ Xs ) )
= ( Xs != nil_list_fm ) ) ).
% length_greater_0_conv
thf(fact_1118_length__greater__0__conv,axiom,
! [Xs: list_tm] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_tm @ Xs ) )
= ( Xs != nil_tm ) ) ).
% length_greater_0_conv
thf(fact_1119_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_1120_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1121_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1122_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1123_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1124_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1125_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1126_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1127_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_1128_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_1129_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1130_list_Osize_I3_J,axiom,
( ( size_size_list_fm @ nil_fm )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_1131_list_Osize_I3_J,axiom,
( ( size_s115229985653309035ist_fm @ nil_list_fm )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_1132_list_Osize_I3_J,axiom,
( ( size_size_list_tm @ nil_tm )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_1133_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1134_length__pos__if__in__set,axiom,
! [X: list_fm,Xs: list_list_fm] :
( ( member_list_fm2 @ X @ ( set_list_fm2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s115229985653309035ist_fm @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1135_length__pos__if__in__set,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm2 @ X @ ( set_fm2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_fm @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1136_length__pos__if__in__set,axiom,
! [X: tm,Xs: list_tm] :
( ( member_tm2 @ X @ ( set_tm2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_tm @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1137_length__pos__if__in__set,axiom,
! [X: produc6018962875968178549ist_fm,Xs: list_P2887561121880082555ist_fm] :
( ( member4699826688122452638ist_fm @ X @ ( set_Pr5051287786238754058ist_fm @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s3138477486474831591ist_fm @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_1138_branchDone_Osimps_I1_J,axiom,
~ ( branchDone @ nil_fm ) ).
% branchDone.simps(1)
thf(fact_1139_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C5: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_1140_subtermTm_Osimps_I1_J,axiom,
! [N: nat,Ts: list_tm] :
( ( subtermTm @ ( fun @ N @ Ts ) )
= ( cons_tm @ ( fun @ N @ Ts ) @ ( remdups_tm @ ( concat_tm @ ( map_tm_list_tm @ subtermTm @ Ts ) ) ) ) ) ).
% subtermTm.simps(1)
thf(fact_1141_sub__term__const__transfer_I1_J,axiom,
! [M: nat,A: nat,T: tm,S2: tm] :
( ( ( sub_term @ M @ ( fun @ A @ nil_tm ) @ T )
!= ( sub_term @ M @ S2 @ T ) )
=> ( member_tm2 @ ( fun @ A @ nil_tm ) @ ( set_tm2 @ ( subtermTm @ ( sub_term @ M @ ( fun @ A @ nil_tm ) @ T ) ) ) ) ) ).
% sub_term_const_transfer(1)
thf(fact_1142_subtermFm_Osimps_I1_J,axiom,
! [Uu2: nat,Ts: list_tm] :
( ( subtermFm @ ( pre @ Uu2 @ Ts ) )
= ( concat_tm @ ( map_tm_list_tm @ subtermTm @ Ts ) ) ) ).
% subtermFm.simps(1)
thf(fact_1143_fm_Osize_I8_J,axiom,
! [X11: nat,X12: list_tm] :
( ( size_size_fm @ ( pre @ X11 @ X12 ) )
= zero_zero_nat ) ).
% fm.size(8)
thf(fact_1144_paramst__subtermTm_I1_J,axiom,
! [T: tm,X4: nat] :
( ( member_nat @ X4 @ ( paramst @ T ) )
=> ? [L: list_tm] : ( member_tm2 @ ( fun @ X4 @ L ) @ ( set_tm2 @ ( subtermTm @ T ) ) ) ) ).
% paramst_subtermTm(1)
thf(fact_1145_sub__const__transfer,axiom,
! [M: nat,A: nat,P2: fm,T: tm] :
( ( ( sub @ M @ ( fun @ A @ nil_tm ) @ P2 )
!= ( sub @ M @ T @ P2 ) )
=> ( member_tm2 @ ( fun @ A @ nil_tm ) @ ( set_tm2 @ ( subtermFm @ ( sub @ M @ ( fun @ A @ nil_tm ) @ P2 ) ) ) ) ) ).
% sub_const_transfer
thf(fact_1146_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_1147_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1148_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1149_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_1150_size__sub,axiom,
! [I: nat,T: tm,P2: fm] :
( ( size_size_fm @ ( sub @ I @ T @ P2 ) )
= ( size_size_fm @ P2 ) ) ).
% size_sub
thf(fact_1151_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1152_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1153_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R2 @ X3 @ X3 )
=> ( ! [X3: nat,Y: nat,Z: nat] :
( ( R2 @ X3 @ Y )
=> ( ( R2 @ Y @ Z )
=> ( R2 @ X3 @ Z ) ) )
=> ( ! [N3: nat] : ( R2 @ N3 @ ( suc @ N3 ) )
=> ( R2 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1154_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1155_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1156_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_1157_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1158_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1159_Suc__le__D,axiom,
! [N: nat,M4: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M4 )
=> ? [M5: nat] :
( M4
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_1160_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1161_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1162_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1163_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_1164_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_1165_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_1166_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_1167_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1168_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1169_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ N )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1170_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_1171_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1172_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_1173_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M6: nat] :
( ( M
= ( suc @ M6 ) )
& ( ord_less_nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1174_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_1175_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_1176_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_1177_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1178_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_1179_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1180_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1181_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_1182_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1183_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1184_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1185_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1186_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1187_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_1188_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1189_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_1190_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1191_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1192_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_1193_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1194_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1195_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1196_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1197_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1198_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1199_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1200_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
? [K3: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1201_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1202_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1203_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).
% less_natE
thf(fact_1204_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1205_less__add__eq__less,axiom,
! [K: nat,L2: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L2 )
=> ( ( ( plus_plus_nat @ M @ L2 )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_1206_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_1207_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_1208_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1209_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1210_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1211_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ) ).
% add_less_mono
thf(fact_1212_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_1213_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1214_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1215_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1216_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1217_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1218_le__Suc__ex,axiom,
! [K: nat,L2: nat] :
( ( ord_less_eq_nat @ K @ L2 )
=> ? [N3: nat] :
( L2
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1219_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L2 ) ) ) ) ).
% add_le_mono
thf(fact_1220_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1221_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1222_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1223_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1224_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1225_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1226_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_1227_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1228_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1229_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1230_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_1231_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_1232_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_1233_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_1234_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_1235_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_1236_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_1237_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_1238_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1239_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_1240_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_1241_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_1242_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_1243_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_1244_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1245_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1246_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1247_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1248_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1249_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1250_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_1251_diff__less__mono2,axiom,
! [M: nat,N: nat,L2: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L2 )
=> ( ord_less_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1252_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1253_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1254_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1255_diff__le__mono,axiom,
! [M: nat,N: nat,L2: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L2 ) @ ( minus_minus_nat @ N @ L2 ) ) ) ).
% diff_le_mono
thf(fact_1256_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1257_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1258_diff__le__mono2,axiom,
! [M: nat,N: nat,L2: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L2 @ N ) @ ( minus_minus_nat @ L2 @ M ) ) ) ).
% diff_le_mono2
thf(fact_1259_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_1260_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
=> ( P @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_1261_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
& ~ ( P @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1262_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1263_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M7: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M7 ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1264_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1265_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
% Helper facts (23)
thf(help_If_2_1_If_001t__SeCaV__Ofm_T,axiom,
! [X: fm,Y3: fm] :
( ( if_fm @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__SeCaV__Ofm_T,axiom,
! [X: fm,Y3: fm] :
( ( if_fm @ $true @ X @ Y3 )
= X ) ).
thf(help_If_2_1_If_001t__SeCaV__Otm_T,axiom,
! [X: tm,Y3: tm] :
( ( if_tm @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__SeCaV__Otm_T,axiom,
! [X: tm,Y3: tm] :
( ( if_tm @ $true @ X @ Y3 )
= X ) ).
thf(help_If_2_1_If_001t__FSet__Ofset_It__SeCaV__Ofm_J_T,axiom,
! [X: fset_fm,Y3: fset_fm] :
( ( if_fset_fm @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__FSet__Ofset_It__SeCaV__Ofm_J_T,axiom,
! [X: fset_fm,Y3: fset_fm] :
( ( if_fset_fm @ $true @ X @ Y3 )
= X ) ).
thf(help_If_2_1_If_001t__FSet__Ofset_It__SeCaV__Otm_J_T,axiom,
! [X: fset_tm,Y3: fset_tm] :
( ( if_fset_tm @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__FSet__Ofset_It__SeCaV__Otm_J_T,axiom,
! [X: fset_tm,Y3: fset_tm] :
( ( if_fset_tm @ $true @ X @ Y3 )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__SeCaV__Ofm_J_T,axiom,
! [X: list_fm,Y3: list_fm] :
( ( if_list_fm @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__SeCaV__Ofm_J_T,axiom,
! [X: list_fm,Y3: list_fm] :
( ( if_list_fm @ $true @ X @ Y3 )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__SeCaV__Otm_J_T,axiom,
! [X: list_tm,Y3: list_tm] :
( ( if_list_tm @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__SeCaV__Otm_J_T,axiom,
! [X: list_tm,Y3: list_tm] :
( ( if_list_tm @ $true @ X @ Y3 )
= X ) ).
thf(help_If_2_1_If_001t__FSet__Ofset_It__List__Olist_It__SeCaV__Ofm_J_J_T,axiom,
! [X: fset_list_fm,Y3: fset_list_fm] :
( ( if_fset_list_fm @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__FSet__Ofset_It__List__Olist_It__SeCaV__Ofm_J_J_T,axiom,
! [X: fset_list_fm,Y3: fset_list_fm] :
( ( if_fset_list_fm @ $true @ X @ Y3 )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_T,axiom,
! [X: list_list_fm,Y3: list_list_fm] :
( ( if_list_list_fm @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_T,axiom,
! [X: list_list_fm,Y3: list_list_fm] :
( ( if_list_list_fm @ $true @ X @ Y3 )
= X ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_T,axiom,
! [X: produc6018962875968178549ist_fm,Y3: produc6018962875968178549ist_fm] :
( ( if_Pro4760001780252510779ist_fm @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_T,axiom,
! [X: produc6018962875968178549ist_fm,Y3: produc6018962875968178549ist_fm] :
( ( if_Pro4760001780252510779ist_fm @ $true @ X @ Y3 )
= X ) ).
thf(help_If_2_1_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J_T,axiom,
! [X: fset_P8989946509869081563ist_fm,Y3: fset_P8989946509869081563ist_fm] :
( ( if_fse7999432387889793441ist_fm @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__FSet__Ofset_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J_T,axiom,
! [X: fset_P8989946509869081563ist_fm,Y3: fset_P8989946509869081563ist_fm] :
( ( if_fse7999432387889793441ist_fm @ $true @ X @ Y3 )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J_T,axiom,
! [X: list_P2887561121880082555ist_fm,Y3: list_P2887561121880082555ist_fm] :
( ( if_lis1897046999900794433ist_fm @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J_T,axiom,
! [X: list_P2887561121880082555ist_fm,Y3: list_P2887561121880082555ist_fm] :
( ( if_lis1897046999900794433ist_fm @ $true @ X @ Y3 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
! [X3: list_fm] :
( ( member_list_fm2 @ X3 @ ( set_list_fm2 @ ( children @ ( remdups_tm @ ( append_tm @ a @ ( concat_tm @ ( map_fm_list_tm @ subtermFm @ z ) ) ) ) @ r @ z ) ) )
=> ? [B9: list_tm] : ( fmembe3754813877001230652ist_fm @ ( produc1414352766439514085ist_fm @ B9 @ X3 ) @ ss ) ) ).
%------------------------------------------------------------------------------