TPTP Problem File: SLH0865^1.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_Calc2/0018_EPathHintikka/prob_00339_013707__13131668_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1807 ( 751 unt; 520 typ; 0 def)
% Number of atoms : 3433 (1963 equ; 0 cnn)
% Maximal formula atoms : 42 ( 2 avg)
% Number of connectives : 13865 ( 738 ~; 151 |; 361 &;11189 @)
% ( 0 <=>;1426 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 6 avg)
% Number of types : 103 ( 102 usr)
% Number of type conns : 978 ( 978 >; 0 *; 0 +; 0 <<)
% Number of symbols : 421 ( 418 usr; 32 con; 0-3 aty)
% Number of variables : 4247 ( 95 ^;3789 !; 363 ?;4247 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 08:45:27.063
%------------------------------------------------------------------------------
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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 (418)
thf(sy_c_EPathHintikka_Opseq,type,
pseq: produc340336539035504054m_rule > list_fm ).
thf(sy_c_EPathHintikka_Otree__fms,type,
tree_fms: stream2709947120125613254m_rule > set_fm ).
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_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_Hintikka_OHintikka,type,
hintikka: set_fm > $o ).
thf(sy_c_Hintikka_Oterms,type,
terms: set_fm > set_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__Nat__Onat_J,type,
if_list_nat: $o > list_nat > list_nat > list_nat ).
thf(sy_c_If_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
if_lis6095641862558746240m_rule: $o > list_P2774625669004309958m_rule > list_P2774625669004309958m_rule > list_P2774625669004309958m_rule ).
thf(sy_c_If_001t__List__Olist_It__Prover__Orule_J,type,
if_list_rule: $o > list_rule > list_rule > list_rule ).
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_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
sup_sup_set_list_fm: set_list_fm > set_list_fm > set_list_fm ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
sup_su6946459741510085528m_rule: set_Pr1822751329126368876m_rule > set_Pr1822751329126368876m_rule > set_Pr1822751329126368876m_rule ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__SeCaV__Ofm_J,type,
sup_sup_set_fm: set_fm > set_fm > set_fm ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__SeCaV__Otm_J,type,
sup_sup_set_tm: set_tm > set_tm > set_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__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
append2714808343334685275m_rule: list_P2774625669004309958m_rule > list_P2774625669004309958m_rule > list_P2774625669004309958m_rule ).
thf(sy_c_List_Oappend_001t__Prover__Orule,type,
append_rule: list_rule > list_rule > list_rule ).
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__SeCaV__Otm,type,
bind_list_fm_tm: list_list_fm > ( list_fm > list_tm ) > list_tm ).
thf(sy_c_List_Obind_001t__Prover__Orule_001t__List__Olist_It__SeCaV__Ofm_J,type,
bind_rule_list_fm: list_rule > ( rule > list_list_fm ) > list_list_fm ).
thf(sy_c_List_Obind_001t__Prover__Orule_001t__Prover__Orule,type,
bind_rule_rule: list_rule > ( rule > list_rule ) > list_rule ).
thf(sy_c_List_Obind_001t__Prover__Orule_001t__SeCaV__Ofm,type,
bind_rule_fm: list_rule > ( rule > list_fm ) > list_fm ).
thf(sy_c_List_Obind_001t__Prover__Orule_001t__SeCaV__Otm,type,
bind_rule_tm: list_rule > ( rule > list_tm ) > list_tm ).
thf(sy_c_List_Obind_001t__SeCaV__Ofm_001t__Prover__Orule,type,
bind_fm_rule: list_fm > ( fm > list_rule ) > list_rule ).
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__Prover__Orule,type,
bind_tm_rule: list_tm > ( tm > list_rule ) > list_rule ).
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_Ocan__select_001t__List__Olist_It__SeCaV__Ofm_J,type,
can_select_list_fm: ( list_fm > $o ) > set_list_fm > $o ).
thf(sy_c_List_Ocan__select_001t__Nat__Onat,type,
can_select_nat: ( nat > $o ) > set_nat > $o ).
thf(sy_c_List_Ocan__select_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
can_se4506553828710032889m_rule: ( produc340336539035504054m_rule > $o ) > set_Pr1822751329126368876m_rule > $o ).
thf(sy_c_List_Ocan__select_001t__SeCaV__Ofm,type,
can_select_fm: ( fm > $o ) > set_fm > $o ).
thf(sy_c_List_Ocan__select_001t__SeCaV__Otm,type,
can_select_tm: ( tm > $o ) > set_tm > $o ).
thf(sy_c_List_Odrop_001t__List__Olist_It__SeCaV__Ofm_J,type,
drop_list_fm: nat > list_list_fm > list_list_fm ).
thf(sy_c_List_Odrop_001t__Nat__Onat,type,
drop_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Odrop_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
drop_P5814889834185618160m_rule: nat > list_P2774625669004309958m_rule > list_P2774625669004309958m_rule ).
thf(sy_c_List_Odrop_001t__Prover__Orule,type,
drop_rule: nat > list_rule > list_rule ).
thf(sy_c_List_Odrop_001t__SeCaV__Ofm,type,
drop_fm: nat > list_fm > list_fm ).
thf(sy_c_List_Odrop_001t__SeCaV__Otm,type,
drop_tm: nat > 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__Nat__Onat,type,
insert_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oinsert_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
insert8371175628248280474m_rule: produc340336539035504054m_rule > list_P2774625669004309958m_rule > list_P2774625669004309958m_rule ).
thf(sy_c_List_Oinsert_001t__Prover__Orule,type,
insert_rule: rule > list_rule > list_rule ).
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__Prover__Orule,type,
lenlex_rule: set_Pr7340090144965549541e_rule > set_Pr5482552573296100997t_rule ).
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__Prover__Orule,type,
lex_rule: set_Pr7340090144965549541e_rule > set_Pr5482552573296100997t_rule ).
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__Nat__Onat,type,
lexord_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
thf(sy_c_List_Olexord_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
lexord6196499238003582211m_rule: set_Pr4971326047967503661m_rule > set_Pr8447737590334104013m_rule ).
thf(sy_c_List_Olexord_001t__Prover__Orule,type,
lexord_rule: set_Pr7340090144965549541e_rule > set_Pr5482552573296100997t_rule ).
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__Prover__Orule_J,type,
cons_list_rule: list_rule > list_list_rule > list_list_rule ).
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__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
cons_P2818572562524668032m_rule: produc340336539035504054m_rule > list_P2774625669004309958m_rule > list_P2774625669004309958m_rule ).
thf(sy_c_List_Olist_OCons_001t__Prover__Orule,type,
cons_rule: rule > list_rule > list_rule ).
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__Prover__Orule_J,type,
nil_list_rule: list_list_rule ).
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__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
nil_Pr2808214839852828720m_rule: list_P2774625669004309958m_rule ).
thf(sy_c_List_Olist_ONil_001t__Prover__Orule,type,
nil_rule: list_rule ).
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_Ohd_001t__List__Olist_It__SeCaV__Ofm_J,type,
hd_list_fm: list_list_fm > list_fm ).
thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
hd_nat: list_nat > nat ).
thf(sy_c_List_Olist_Ohd_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
hd_Pro7241777042969981963m_rule: list_P2774625669004309958m_rule > produc340336539035504054m_rule ).
thf(sy_c_List_Olist_Ohd_001t__Prover__Orule,type,
hd_rule: list_rule > rule ).
thf(sy_c_List_Olist_Ohd_001t__SeCaV__Ofm,type,
hd_fm: list_fm > fm ).
thf(sy_c_List_Olist_Ohd_001t__SeCaV__Otm,type,
hd_tm: list_tm > 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__Prover__Orule_J,type,
set_list_rule2: list_list_rule > set_list_rule ).
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__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
set_Pr4534715572506550497m_rule: list_P2774625669004309958m_rule > set_Pr1822751329126368876m_rule ).
thf(sy_c_List_Olist_Oset_001t__Prover__Orule,type,
set_rule2: list_rule > set_rule ).
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_Olist__ex1_001t__List__Olist_It__SeCaV__Ofm_J,type,
list_ex1_list_fm: ( list_fm > $o ) > list_list_fm > $o ).
thf(sy_c_List_Olist__ex1_001t__Nat__Onat,type,
list_ex1_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__ex1_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
list_e4959145360368724013m_rule: ( produc340336539035504054m_rule > $o ) > list_P2774625669004309958m_rule > $o ).
thf(sy_c_List_Olist__ex1_001t__Prover__Orule,type,
list_ex1_rule: ( rule > $o ) > list_rule > $o ).
thf(sy_c_List_Olist__ex1_001t__SeCaV__Ofm,type,
list_ex1_fm: ( fm > $o ) > list_fm > $o ).
thf(sy_c_List_Olist__ex1_001t__SeCaV__Otm,type,
list_ex1_tm: ( tm > $o ) > list_tm > $o ).
thf(sy_c_List_Olist__update_001t__List__Olist_It__SeCaV__Ofm_J,type,
list_update_list_fm: list_list_fm > nat > list_fm > list_list_fm ).
thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
list_update_nat: list_nat > nat > nat > list_nat ).
thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
list_u1216070999260037064m_rule: list_P2774625669004309958m_rule > nat > produc340336539035504054m_rule > list_P2774625669004309958m_rule ).
thf(sy_c_List_Olist__update_001t__Prover__Orule,type,
list_update_rule: list_rule > nat > rule > list_rule ).
thf(sy_c_List_Olist__update_001t__SeCaV__Ofm,type,
list_update_fm: list_fm > nat > fm > list_fm ).
thf(sy_c_List_Olist__update_001t__SeCaV__Otm,type,
list_update_tm: list_tm > nat > tm > list_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__Prover__Orule,type,
listrel1_rule: set_Pr7340090144965549541e_rule > set_Pr5482552573296100997t_rule ).
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_Olistrel_001t__List__Olist_It__SeCaV__Ofm_J_001t__Prover__Orule,type,
listrel_list_fm_rule: set_Pr2422821757980664944m_rule > set_Pr1649155577491351558t_rule ).
thf(sy_c_List_Olistrel_001t__List__Olist_It__SeCaV__Otm_J_001t__List__Olist_It__SeCaV__Ofm_J,type,
listre5805154493122130495ist_fm: set_Pr5202636777678657877ist_fm > set_Pr7013894833640182997ist_fm ).
thf(sy_c_List_Olistrel_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_001t__Prover__Orule,type,
listre749368180268468182m_rule: set_Pr1822751329126368876m_rule > set_Pr4462384566710528898t_rule ).
thf(sy_c_List_Olistrel_001t__Prover__Orule_001t__List__Olist_It__SeCaV__Ofm_J,type,
listrel_rule_list_fm: set_Pr6384426166747533970ist_fm > set_Pr686609868138099848ist_fm ).
thf(sy_c_List_Olistrel_001t__Prover__Orule_001t__Prover__Orule,type,
listrel_rule_rule: set_Pr7340090144965549541e_rule > set_Pr5482552573296100997t_rule ).
thf(sy_c_List_Olistrel_001t__Prover__Orule_001t__SeCaV__Ofm,type,
listrel_rule_fm: set_Pr8969532344284305932ule_fm > set_Pr1973443253941551234ist_fm ).
thf(sy_c_List_Olistrel_001t__Prover__Orule_001t__SeCaV__Otm,type,
listrel_rule_tm: set_Pr8727017673958796158ule_tm > set_Pr7501189845713910004ist_tm ).
thf(sy_c_List_Olistrel_001t__SeCaV__Ofm_001t__Prover__Orule,type,
listrel_fm_rule: set_Pr7067330020180114026m_rule > set_Pr6455722319876965760t_rule ).
thf(sy_c_List_Olistrel_001t__SeCaV__Ofm_001t__SeCaV__Ofm,type,
listrel_fm_fm: set_Pr4706815898642364871_fm_fm > set_Pr1916137733696349511ist_fm ).
thf(sy_c_List_Olistrel_001t__SeCaV__Ofm_001t__SeCaV__Otm,type,
listrel_fm_tm: set_Pr4464301228316855097_fm_tm > set_Pr7443884325468708281ist_tm ).
thf(sy_c_List_Olistrel_001t__SeCaV__Otm_001t__Prover__Orule,type,
listrel_tm_rule: set_Pr3569831037629835996m_rule > set_Pr6575435784368700402t_rule ).
thf(sy_c_List_Olistrel_001t__SeCaV__Otm_001t__SeCaV__Ofm,type,
listrel_tm_fm: set_Pr2698443736021152725_tm_fm > set_Pr5202636777678657877ist_fm ).
thf(sy_c_List_Olistrel_001t__SeCaV__Otm_001t__SeCaV__Otm,type,
listrel_tm_tm: set_Pr2455929065695642951_tm_tm > set_Pr1507011332596240839ist_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__Nat__Onat,type,
member_nat: list_nat > nat > $o ).
thf(sy_c_List_Omember_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
member4742765151257878875m_rule: list_P2774625669004309958m_rule > produc340336539035504054m_rule > $o ).
thf(sy_c_List_Omember_001t__Prover__Orule,type,
member_rule: list_rule > rule > $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__Prover__Orule,type,
n_lists_rule: nat > list_rule > list_list_rule ).
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_Onth_001t__List__Olist_It__SeCaV__Ofm_J,type,
nth_list_fm: list_list_fm > nat > list_fm ).
thf(sy_c_List_Onth_001t__List__Olist_It__SeCaV__Otm_J,type,
nth_list_tm: list_list_tm > nat > list_tm ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
nth_Pr580027083122244092ist_fm: list_P2887561121880082555ist_fm > nat > produc6018962875968178549ist_fm ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
nth_Pr3936752564452695919m_rule: list_P2774625669004309958m_rule > nat > produc340336539035504054m_rule ).
thf(sy_c_List_Onth_001t__Prover__Orule,type,
nth_rule: list_rule > nat > rule ).
thf(sy_c_List_Onth_001t__SeCaV__Ofm,type,
nth_fm: list_fm > nat > fm ).
thf(sy_c_List_Onth_001t__SeCaV__Otm,type,
nth_tm: list_tm > nat > 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__Nat__Onat,type,
partition_nat: ( nat > $o ) > list_nat > produc1828647624359046049st_nat ).
thf(sy_c_List_Opartition_001t__Prover__Orule,type,
partition_rule: ( rule > $o ) > list_rule > produc1530960993530998309t_rule ).
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__Prover__Orule,type,
product_lists_rule: list_list_rule > list_list_rule ).
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_Orotate1_001t__List__Olist_It__SeCaV__Ofm_J,type,
rotate1_list_fm: list_list_fm > list_list_fm ).
thf(sy_c_List_Orotate1_001t__Prover__Orule,type,
rotate1_rule: list_rule > list_rule ).
thf(sy_c_List_Orotate1_001t__SeCaV__Ofm,type,
rotate1_fm: list_fm > list_fm ).
thf(sy_c_List_Orotate1_001t__SeCaV__Otm,type,
rotate1_tm: list_tm > list_tm ).
thf(sy_c_List_Orotate_001t__List__Olist_It__SeCaV__Ofm_J,type,
rotate_list_fm: nat > list_list_fm > list_list_fm ).
thf(sy_c_List_Orotate_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
rotate8779165051853931260m_rule: nat > list_P2774625669004309958m_rule > list_P2774625669004309958m_rule ).
thf(sy_c_List_Orotate_001t__Prover__Orule,type,
rotate_rule: nat > list_rule > list_rule ).
thf(sy_c_List_Orotate_001t__SeCaV__Ofm,type,
rotate_fm: nat > list_fm > list_fm ).
thf(sy_c_List_Orotate_001t__SeCaV__Otm,type,
rotate_tm: nat > list_tm > list_tm ).
thf(sy_c_List_Osubseqs_001t__List__Olist_It__SeCaV__Ofm_J,type,
subseqs_list_fm: list_list_fm > list_list_list_fm ).
thf(sy_c_List_Osubseqs_001t__Prover__Orule,type,
subseqs_rule: list_rule > list_list_rule ).
thf(sy_c_List_Osubseqs_001t__SeCaV__Ofm,type,
subseqs_fm: list_fm > list_list_fm ).
thf(sy_c_List_Osubseqs_001t__SeCaV__Otm,type,
subseqs_tm: list_tm > list_list_tm ).
thf(sy_c_List_Otake_001t__List__Olist_It__SeCaV__Ofm_J,type,
take_list_fm: nat > list_list_fm > list_list_fm ).
thf(sy_c_List_Otake_001t__Nat__Onat,type,
take_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Otake_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
take_P3067526267515409992m_rule: nat > list_P2774625669004309958m_rule > list_P2774625669004309958m_rule ).
thf(sy_c_List_Otake_001t__Prover__Orule,type,
take_rule: nat > list_rule > list_rule ).
thf(sy_c_List_Otake_001t__SeCaV__Ofm,type,
take_fm: nat > list_fm > list_fm ).
thf(sy_c_List_Otake_001t__SeCaV__Otm,type,
take_tm: nat > list_tm > list_tm ).
thf(sy_c_List_Ounion_001t__List__Olist_It__SeCaV__Ofm_J,type,
union_list_fm: list_list_fm > list_list_fm > list_list_fm ).
thf(sy_c_List_Ounion_001t__Nat__Onat,type,
union_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Ounion_001t__SeCaV__Ofm,type,
union_fm: list_fm > list_fm > list_fm ).
thf(sy_c_List_Ounion_001t__SeCaV__Otm,type,
union_tm: list_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__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__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__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
size_s1575636608424004698m_rule: list_P2774625669004309958m_rule > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Prover__Orule_J,type,
size_size_list_rule: list_rule > 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__SeCaV__Otm,type,
size_size_tm: tm > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__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_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
ord_min_nat: nat > nat > nat ).
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__Prover__Orule_M_062_It__Prover__Orule_M_Eo_J_J_001t__List__Olist_It__Prover__Orule_J,type,
produc9173496138740005677t_rule: ( rule > rule > $o ) > list_rule > produc3824871039275828531t_rule ).
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__Ofm_J_J_001t__List__Olist_It__Prover__Orule_J,type,
produc3733561588192308298t_rule: list_list_fm > list_rule > produc9128558935521592528t_rule ).
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__Ofm_J_J,type,
produc8328755919010440805ist_fm: list_list_tm > list_list_fm > produc4393997333308195317ist_fm ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
produc2694037385005941721st_nat: list_nat > list_nat > produc1828647624359046049st_nat ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J_001t__List__Olist_It__Prover__Orule_J,type,
produc3723918873312807110t_rule: list_P2887561121880082555ist_fm > list_rule > produc1097132844357787852t_rule ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
produc6692282267706994341m_rule: list_P2774625669004309958m_rule > list_P2774625669004309958m_rule > produc7139673775237694317m_rule ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Prover__Orule_J_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
produc7454812661813866372ist_fm: list_rule > list_list_fm > produc3420741554821296722ist_fm ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Prover__Orule_J_001t__List__Olist_It__Prover__Orule_J,type,
produc8979233614864344413t_rule: list_rule > list_rule > produc1530960993530998309t_rule ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Prover__Orule_J_001t__List__Olist_It__SeCaV__Ofm_J,type,
produc9214880830201710206ist_fm: list_rule > list_fm > produc107904248654371276ist_fm ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Prover__Orule_J_001t__List__Olist_It__SeCaV__Otm_J,type,
produc9218832855287345392ist_tm: list_rule > list_tm > produc9088761615183637310ist_tm ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__SeCaV__Ofm_J_001t__List__Olist_It__Prover__Orule_J,type,
produc1396096877967899076t_rule: list_fm > list_rule > produc4505313441960607178t_rule ).
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__Ofm_J_001t__List__Olist_It__SeCaV__Otm_J,type,
produc7867948443067789129ist_tm: list_fm > list_tm > produc3002719820330532825ist_tm ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__SeCaV__Ofm_J_001t__Prover__Orule,type,
produc8901081497008266676m_rule: list_fm > rule > produc7670961829838246714m_rule ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__SeCaV__Otm_J_001t__List__Olist_It__Prover__Orule_J,type,
produc5348893629884664118t_rule: list_tm > list_rule > produc8458110193877372220t_rule ).
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__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_001t__Prover__Orule,type,
produc1733806532565653680m_rule: produc6018962875968178549ist_fm > rule > produc340336539035504054m_rule ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
produc6261311607089640965m_rule: produc340336539035504054m_rule > produc340336539035504054m_rule > produc8828831911945107917m_rule ).
thf(sy_c_Product__Type_OPair_001t__Prover__Orule_001t__List__Olist_It__SeCaV__Ofm_J,type,
produc2087254492244816782ist_fm: rule > list_fm > produc4627698447051649628ist_fm ).
thf(sy_c_Product__Type_OPair_001t__Prover__Orule_001t__Prover__Orule,type,
produc5849431337705160893e_rule: rule > rule > produc7694839378271647877e_rule ).
thf(sy_c_Product__Type_OPair_001t__Prover__Orule_001t__SeCaV__Ofm,type,
product_Pair_rule_fm: rule > fm > product_prod_rule_fm ).
thf(sy_c_Product__Type_OPair_001t__Prover__Orule_001t__SeCaV__Otm,type,
product_Pair_rule_tm: rule > tm > product_prod_rule_tm ).
thf(sy_c_Product__Type_OPair_001t__SeCaV__Ofm_001t__Prover__Orule,type,
product_Pair_fm_rule: fm > rule > product_prod_fm_rule ).
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__Ofm_001t__SeCaV__Otm,type,
product_Pair_fm_tm: fm > tm > product_prod_fm_tm ).
thf(sy_c_Product__Type_OPair_001t__SeCaV__Otm_001t__Prover__Orule,type,
product_Pair_tm_rule: tm > rule > product_prod_tm_rule ).
thf(sy_c_Product__Type_OPair_001t__SeCaV__Otm_001t__SeCaV__Ofm,type,
product_Pair_tm_fm: tm > fm > product_prod_tm_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_Product__Type_Ocurry_001t__List__Olist_It__SeCaV__Otm_J_001t__List__Olist_It__SeCaV__Ofm_J_001_Eo,type,
produc5334687091915550783t_fm_o: ( produc6018962875968178549ist_fm > $o ) > list_tm > list_fm > $o ).
thf(sy_c_ProverLemmas_Oaffects,type,
affects: rule > fm > $o ).
thf(sy_c_ProverLemmas_Onext__rule,type,
next_rule: rule > rule ).
thf(sy_c_Prover_ObranchDone,type,
branchDone: list_fm > $o ).
thf(sy_c_Prover_ObranchDone__rel,type,
branchDone_rel: list_fm > list_fm > $o ).
thf(sy_c_Prover_Ochildren,type,
children: list_tm > rule > list_fm > list_list_fm ).
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_Orule_OAlphaCon,type,
alphaCon: rule ).
thf(sy_c_Prover_Orule_OAlphaDis,type,
alphaDis: rule ).
thf(sy_c_Prover_Orule_OAlphaImp,type,
alphaImp: rule ).
thf(sy_c_Prover_Orule_OBetaCon,type,
betaCon: rule ).
thf(sy_c_Prover_Orule_OBetaDis,type,
betaDis: rule ).
thf(sy_c_Prover_Orule_OBetaImp,type,
betaImp: rule ).
thf(sy_c_Prover_Orule_ODeltaExi,type,
deltaExi: rule ).
thf(sy_c_Prover_Orule_ODeltaUni,type,
deltaUni: rule ).
thf(sy_c_Prover_Orule_OGammaExi,type,
gammaExi: rule ).
thf(sy_c_Prover_Orule_OGammaUni,type,
gammaUni: rule ).
thf(sy_c_Prover_Orule_ONegNeg,type,
negNeg: rule ).
thf(sy_c_Prover_OrulesList,type,
rulesList: list_rule ).
thf(sy_c_Prover_OsubtermFm,type,
subtermFm: fm > list_tm ).
thf(sy_c_Prover_OsubtermTm,type,
subtermTm: tm > list_tm ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
modulo_modulo_nat: nat > nat > nat ).
thf(sy_c_SeCaV_Oext_001t__List__Olist_It__SeCaV__Ofm_J,type,
ext_list_fm: list_list_fm > list_list_fm > $o ).
thf(sy_c_SeCaV_Oext_001t__Prover__Orule,type,
ext_rule: list_rule > list_rule > $o ).
thf(sy_c_SeCaV_Oext_001t__SeCaV__Ofm,type,
ext_fm: list_fm > list_fm > $o ).
thf(sy_c_SeCaV_Oext_001t__SeCaV__Otm,type,
ext_tm: list_tm > list_tm > $o ).
thf(sy_c_SeCaV_Ofm_OCon,type,
con: fm > fm > fm ).
thf(sy_c_SeCaV_Ofm_ODis,type,
dis: fm > fm > fm ).
thf(sy_c_SeCaV_Ofm_OExi,type,
exi: fm > fm ).
thf(sy_c_SeCaV_Ofm_OImp,type,
imp: fm > fm > fm ).
thf(sy_c_SeCaV_Ofm_ONeg,type,
neg: fm > fm ).
thf(sy_c_SeCaV_Ofm_OPre,type,
pre: nat > list_tm > fm ).
thf(sy_c_SeCaV_Ofm_OUni,type,
uni: fm > fm ).
thf(sy_c_SeCaV_Ofm_Osize__fm,type,
size_fm: fm > nat ).
thf(sy_c_SeCaV_Oinc__list,type,
inc_list: list_tm > list_tm ).
thf(sy_c_SeCaV_Oinc__term,type,
inc_term: tm > tm ).
thf(sy_c_SeCaV_Oliftt,type,
liftt: tm > tm ).
thf(sy_c_SeCaV_Oliftts,type,
liftts: list_tm > list_tm ).
thf(sy_c_SeCaV_Omember_001t__List__Olist_It__SeCaV__Ofm_J,type,
member_list_fm2: list_fm > list_list_fm > $o ).
thf(sy_c_SeCaV_Omember_001t__Nat__Onat,type,
member_nat2: nat > list_nat > $o ).
thf(sy_c_SeCaV_Omember_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
member4200870186495857963m_rule: produc340336539035504054m_rule > list_P2774625669004309958m_rule > $o ).
thf(sy_c_SeCaV_Omember_001t__Prover__Orule,type,
member_rule2: rule > list_rule > $o ).
thf(sy_c_SeCaV_Omember_001t__SeCaV__Ofm,type,
member_fm2: fm > list_fm > $o ).
thf(sy_c_SeCaV_Omember_001t__SeCaV__Otm,type,
member_tm2: tm > list_tm > $o ).
thf(sy_c_SeCaV_Onew__list,type,
new_list: nat > list_tm > $o ).
thf(sy_c_SeCaV_Onew__term,type,
new_term: nat > tm > $o ).
thf(sy_c_SeCaV_Onews,type,
news: nat > list_fm > $o ).
thf(sy_c_SeCaV_Oparams,type,
params: fm > set_nat ).
thf(sy_c_SeCaV_Oparamst,type,
paramst: tm > set_nat ).
thf(sy_c_SeCaV_Oparamst_H,type,
paramst2: tm > set_nat ).
thf(sy_c_SeCaV_Oparamsts,type,
paramsts: list_tm > set_nat ).
thf(sy_c_SeCaV_Osequent__calculus,type,
sequent_calculus: list_fm > $o ).
thf(sy_c_SeCaV_Osub,type,
sub: nat > tm > fm > fm ).
thf(sy_c_SeCaV_Osub__list,type,
sub_list: nat > tm > list_tm > list_tm ).
thf(sy_c_SeCaV_Osub__term,type,
sub_term: nat > tm > tm > tm ).
thf(sy_c_SeCaV_Osubst,type,
subst: fm > tm > nat > fm ).
thf(sy_c_SeCaV_Osubstt,type,
substt: tm > tm > nat > tm ).
thf(sy_c_SeCaV_Osubstts,type,
substts: list_tm > tm > nat > list_tm ).
thf(sy_c_SeCaV_Otm_OFun,type,
fun: nat > list_tm > tm ).
thf(sy_c_SeCaV_Otm_OVar,type,
var: nat > tm ).
thf(sy_c_SeCaV_Otm_Osize__tm,type,
size_tm: tm > nat ).
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__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
collec1009276759030335883m_rule: ( produc340336539035504054m_rule > $o ) > set_Pr1822751329126368876m_rule ).
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__Prover__Orule,type,
the_elem_rule: set_rule > rule ).
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_Stream_Ocycle_001t__List__Olist_It__SeCaV__Ofm_J,type,
cycle_list_fm: list_list_fm > stream_list_fm ).
thf(sy_c_Stream_Ocycle_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
cycle_5335117900641983313m_rule: list_P2774625669004309958m_rule > stream2709947120125613254m_rule ).
thf(sy_c_Stream_Ocycle_001t__Prover__Orule,type,
cycle_rule: list_rule > stream_rule ).
thf(sy_c_Stream_Ocycle_001t__SeCaV__Ofm,type,
cycle_fm: list_fm > stream_fm ).
thf(sy_c_Stream_Ocycle_001t__SeCaV__Otm,type,
cycle_tm: list_tm > stream_tm ).
thf(sy_c_Stream_Osdrop_001t__List__Olist_It__SeCaV__Ofm_J,type,
sdrop_list_fm: nat > stream_list_fm > stream_list_fm ).
thf(sy_c_Stream_Osdrop_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
sdrop_9176333610110415838ist_fm: nat > stream4408948924543953275ist_fm > stream4408948924543953275ist_fm ).
thf(sy_c_Stream_Osdrop_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
sdrop_8169176516188972301m_rule: nat > stream2709947120125613254m_rule > stream2709947120125613254m_rule ).
thf(sy_c_Stream_Osdrop_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
sdrop_7192298464603511222m_rule: nat > stream6210534828274662995m_rule > stream6210534828274662995m_rule ).
thf(sy_c_Stream_Osdrop_001t__Prover__Orule,type,
sdrop_rule: nat > stream_rule > stream_rule ).
thf(sy_c_Stream_Osdrop_001t__SeCaV__Ofm,type,
sdrop_fm: nat > stream_fm > stream_fm ).
thf(sy_c_Stream_Osdrop_001t__SeCaV__Otm,type,
sdrop_tm: nat > stream_tm > stream_tm ).
thf(sy_c_Stream_Oshift_001t__List__Olist_It__SeCaV__Ofm_J,type,
shift_list_fm: list_list_fm > stream_list_fm > stream_list_fm ).
thf(sy_c_Stream_Oshift_001t__Nat__Onat,type,
shift_nat: list_nat > stream_nat > stream_nat ).
thf(sy_c_Stream_Oshift_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
shift_2334844276748245581m_rule: list_P2774625669004309958m_rule > stream2709947120125613254m_rule > stream2709947120125613254m_rule ).
thf(sy_c_Stream_Oshift_001t__Prover__Orule,type,
shift_rule: list_rule > stream_rule > stream_rule ).
thf(sy_c_Stream_Oshift_001t__SeCaV__Ofm,type,
shift_fm: list_fm > stream_fm > stream_fm ).
thf(sy_c_Stream_Oshift_001t__SeCaV__Otm,type,
shift_tm: list_tm > stream_tm > stream_tm ).
thf(sy_c_Stream_Osinterleave_001t__Nat__Onat,type,
sinterleave_nat: stream_nat > stream_nat > stream_nat ).
thf(sy_c_Stream_Osinterleave_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
sinter2808979888547679129m_rule: stream2709947120125613254m_rule > stream2709947120125613254m_rule > stream2709947120125613254m_rule ).
thf(sy_c_Stream_Osmember_001t__List__Olist_It__SeCaV__Ofm_J,type,
smember_list_fm: list_fm > stream_list_fm > $o ).
thf(sy_c_Stream_Osmember_001t__Nat__Onat,type,
smember_nat: nat > stream_nat > $o ).
thf(sy_c_Stream_Osmember_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
smembe4439892024482649336m_rule: produc340336539035504054m_rule > stream2709947120125613254m_rule > $o ).
thf(sy_c_Stream_Osmember_001t__SeCaV__Ofm,type,
smember_fm: fm > stream_fm > $o ).
thf(sy_c_Stream_Osmember_001t__SeCaV__Otm,type,
smember_tm: tm > stream_tm > $o ).
thf(sy_c_Stream_Osmerge_001t__List__Olist_It__SeCaV__Ofm_J,type,
smerge_list_fm: stream6017534608192929797ist_fm > stream_list_fm ).
thf(sy_c_Stream_Osmerge_001t__Nat__Onat,type,
smerge_nat: stream_stream_nat > stream_nat ).
thf(sy_c_Stream_Osmerge_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
smerge193809993764105000m_rule: stream3752074346242807894m_rule > stream2709947120125613254m_rule ).
thf(sy_c_Stream_Osmerge_001t__SeCaV__Ofm,type,
smerge_fm: stream_stream_fm > stream_fm ).
thf(sy_c_Stream_Osmerge_001t__SeCaV__Otm,type,
smerge_tm: stream_stream_tm > stream_tm ).
thf(sy_c_Stream_Osnth_001t__List__Olist_It__SeCaV__Ofm_J,type,
snth_list_fm: stream_list_fm > nat > list_fm ).
thf(sy_c_Stream_Osnth_001t__List__Olist_It__SeCaV__Otm_J,type,
snth_list_tm: stream_list_tm > nat > list_tm ).
thf(sy_c_Stream_Osnth_001t__Nat__Onat,type,
snth_nat: stream_nat > nat > nat ).
thf(sy_c_Stream_Osnth_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
snth_P7093566783922538521ist_fm: stream4408948924543953275ist_fm > nat > produc6018962875968178549ist_fm ).
thf(sy_c_Stream_Osnth_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
snth_P6679518042731451922m_rule: stream2709947120125613254m_rule > nat > produc340336539035504054m_rule ).
thf(sy_c_Stream_Osnth_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
snth_P8853764340393315953m_rule: stream6210534828274662995m_rule > nat > produc8828831911945107917m_rule ).
thf(sy_c_Stream_Osnth_001t__Prover__Orule,type,
snth_rule: stream_rule > nat > rule ).
thf(sy_c_Stream_Osnth_001t__SeCaV__Ofm,type,
snth_fm: stream_fm > nat > fm ).
thf(sy_c_Stream_Osnth_001t__SeCaV__Otm,type,
snth_tm: stream_tm > nat > tm ).
thf(sy_c_Stream_Osnth_001t__Stream__Ostream_It__List__Olist_It__SeCaV__Ofm_J_J,type,
snth_stream_list_fm: stream6017534608192929797ist_fm > nat > stream_list_fm ).
thf(sy_c_Stream_Osnth_001t__Stream__Ostream_It__Nat__Onat_J,type,
snth_stream_nat: stream_stream_nat > nat > stream_nat ).
thf(sy_c_Stream_Osnth_001t__Stream__Ostream_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
snth_s6182113952396108578m_rule: stream3752074346242807894m_rule > nat > stream2709947120125613254m_rule ).
thf(sy_c_Stream_Osnth_001t__Stream__Ostream_It__SeCaV__Ofm_J,type,
snth_stream_fm: stream_stream_fm > nat > stream_fm ).
thf(sy_c_Stream_Osnth_001t__Stream__Ostream_It__SeCaV__Otm_J,type,
snth_stream_tm: stream_stream_tm > nat > stream_tm ).
thf(sy_c_Stream_Ostake_001t__List__Olist_It__SeCaV__Ofm_J,type,
stake_list_fm: nat > stream_list_fm > list_list_fm ).
thf(sy_c_Stream_Ostake_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
stake_5421812949518764133m_rule: nat > stream2709947120125613254m_rule > list_P2774625669004309958m_rule ).
thf(sy_c_Stream_Ostake_001t__Prover__Orule,type,
stake_rule: nat > stream_rule > list_rule ).
thf(sy_c_Stream_Ostake_001t__SeCaV__Ofm,type,
stake_fm: nat > stream_fm > list_fm ).
thf(sy_c_Stream_Ostake_001t__SeCaV__Otm,type,
stake_tm: nat > stream_tm > list_tm ).
thf(sy_c_Stream_Ostream_Oshd_001t__List__Olist_It__SeCaV__Ofm_J,type,
shd_list_fm: stream_list_fm > list_fm ).
thf(sy_c_Stream_Ostream_Oshd_001t__List__Olist_It__SeCaV__Otm_J,type,
shd_list_tm: stream_list_tm > list_tm ).
thf(sy_c_Stream_Ostream_Oshd_001t__Nat__Onat,type,
shd_nat: stream_nat > nat ).
thf(sy_c_Stream_Ostream_Oshd_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
shd_Pr3211216682057661985ist_fm: stream4408948924543953275ist_fm > produc6018962875968178549ist_fm ).
thf(sy_c_Stream_Ostream_Oshd_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
shd_Pr4562317740776619530m_rule: stream2709947120125613254m_rule > produc340336539035504054m_rule ).
thf(sy_c_Stream_Ostream_Oshd_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
shd_Pr4461660664618831993m_rule: stream6210534828274662995m_rule > produc8828831911945107917m_rule ).
thf(sy_c_Stream_Ostream_Oshd_001t__Prover__Orule,type,
shd_rule: stream_rule > rule ).
thf(sy_c_Stream_Ostream_Oshd_001t__SeCaV__Ofm,type,
shd_fm: stream_fm > fm ).
thf(sy_c_Stream_Ostream_Oshd_001t__SeCaV__Otm,type,
shd_tm: stream_tm > tm ).
thf(sy_c_Stream_Ostream_Osset_001t__List__Olist_It__SeCaV__Ofm_J,type,
sset_list_fm: stream_list_fm > set_list_fm ).
thf(sy_c_Stream_Ostream_Osset_001t__Nat__Onat,type,
sset_nat: stream_nat > set_nat ).
thf(sy_c_Stream_Ostream_Osset_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
sset_P4484857331586881186m_rule: stream2709947120125613254m_rule > set_Pr1822751329126368876m_rule ).
thf(sy_c_Stream_Ostream_Osset_001t__Prover__Orule,type,
sset_rule: stream_rule > set_rule ).
thf(sy_c_Stream_Ostream_Osset_001t__SeCaV__Ofm,type,
sset_fm: stream_fm > set_fm ).
thf(sy_c_Stream_Ostream_Osset_001t__SeCaV__Otm,type,
sset_tm: stream_tm > set_tm ).
thf(sy_c_Stream_Ostream__all_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
stream2134157564947672044m_rule: ( produc340336539035504054m_rule > $o ) > stream2709947120125613254m_rule > $o ).
thf(sy_c_Stream_Oszip_001t__List__Olist_It__SeCaV__Otm_J_001t__List__Olist_It__SeCaV__Ofm_J,type,
szip_list_tm_list_fm: stream_list_tm > stream_list_fm > stream4408948924543953275ist_fm ).
thf(sy_c_Stream_Oszip_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_001t__Prover__Orule,type,
szip_P2924820683901490861m_rule: stream4408948924543953275ist_fm > stream_rule > stream2709947120125613254m_rule ).
thf(sy_c_Stream_Oszip_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
szip_P811719526838699976m_rule: stream2709947120125613254m_rule > stream2709947120125613254m_rule > stream6210534828274662995m_rule ).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__SeCaV__Ofm_J,type,
accp_list_fm: ( list_fm > list_fm > $o ) > list_fm > $o ).
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__Prover__Orule_J,type,
member_list_rule: list_rule > set_list_rule > $o ).
thf(sy_c_member_001t__List__Olist_It__SeCaV__Ofm_J,type,
member_list_fm3: 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_nat3: 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__Ofm_J_J_Mt__List__Olist_It__Prover__Orule_J_J,type,
member6202700923738835687t_rule: produc9128558935521592528t_rule > set_Pr1649155577491351558t_rule > $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__Ofm_J_J_J,type,
member5109946355746398750ist_fm: produc4393997333308195317ist_fm > set_Pr7013894833640182997ist_fm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
member7340969449405702474st_nat: produc1828647624359046049st_nat > set_Pr3451248702717554689st_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_J_Mt__List__Olist_It__Prover__Orule_J_J,type,
member2627171955076190819t_rule: produc1097132844357787852t_rule > set_Pr4462384566710528898t_rule > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_J,type,
member3128744185454049558m_rule: produc7139673775237694317m_rule > set_Pr8447737590334104013m_rule > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Prover__Orule_J_Mt__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_J,type,
member494883543038539881ist_fm: produc3420741554821296722ist_fm > set_Pr686609868138099848ist_fm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Prover__Orule_J_Mt__List__Olist_It__Prover__Orule_J_J,type,
member695053538577043918t_rule: produc1530960993530998309t_rule > set_Pr5482552573296100997t_rule > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Prover__Orule_J_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
member2905128903584585571ist_fm: produc107904248654371276ist_fm > set_Pr1973443253941551234ist_fm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Prover__Orule_J_Mt__List__Olist_It__SeCaV__Otm_J_J,type,
member2662614233259075797ist_tm: produc9088761615183637310ist_tm > set_Pr7501189845713910004ist_tm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Ofm_J_Mt__List__Olist_It__Prover__Orule_J_J,type,
member7302538096890821473t_rule: produc4505313441960607178t_rule > set_Pr6455722319876965760t_rule > $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__Ofm_J_Mt__List__Olist_It__SeCaV__Otm_J_J,type,
member1683583632484806914ist_tm: produc3002719820330532825ist_tm > set_Pr7443884325468708281ist_tm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Ofm_J_Mt__Prover__Orule_J,type,
member7183523337131018321m_rule: produc7670961829838246714m_rule > set_Pr2422821757980664944m_rule > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__Prover__Orule_J_J,type,
member2031962811952810707t_rule: produc8458110193877372220t_rule > set_Pr6575435784368700402t_rule > $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__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J,type,
member7231649785386036813m_rule: produc340336539035504054m_rule > set_Pr1822751329126368876m_rule > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J,type,
member1411645983481675894m_rule: produc8828831911945107917m_rule > set_Pr4971326047967503661m_rule > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Prover__Orule_Mt__List__Olist_It__SeCaV__Ofm_J_J,type,
member4140259954344421235ist_fm: produc4627698447051649628ist_fm > set_Pr6384426166747533970ist_fm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Prover__Orule_Mt__Prover__Orule_J,type,
member1444691403335511342e_rule: produc7694839378271647877e_rule > set_Pr7340090144965549541e_rule > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Prover__Orule_Mt__SeCaV__Ofm_J,type,
member9168179392308330989ule_fm: product_prod_rule_fm > set_Pr8969532344284305932ule_fm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Prover__Orule_Mt__SeCaV__Otm_J,type,
member9172131417393966175ule_tm: product_prod_rule_tm > set_Pr8727017673958796158ule_tm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__SeCaV__Ofm_Mt__Prover__Orule_J,type,
member6041693456836784459m_rule: product_prod_fm_rule > set_Pr7067330020180114026m_rule > $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__Ofm_Mt__SeCaV__Otm_J,type,
member7784904625553633922_fm_tm: product_prod_fm_tm > set_Pr4464301228316855097_fm_tm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__SeCaV__Otm_Mt__Prover__Orule_J,type,
member5482372101558860477m_rule: product_prod_tm_rule > set_Pr3569831037629835996m_rule > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__SeCaV__Otm_Mt__SeCaV__Ofm_J,type,
member3117664881408846110_tm_fm: product_prod_tm_fm > set_Pr2698443736021152725_tm_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__Prover__Orule,type,
member_rule3: rule > set_rule > $o ).
thf(sy_c_member_001t__SeCaV__Ofm,type,
member_fm3: fm > set_fm > $o ).
thf(sy_c_member_001t__SeCaV__Otm,type,
member_tm3: tm > set_tm > $o ).
thf(sy_v_n____,type,
n: nat ).
thf(sy_v_p____,type,
p: fm ).
thf(sy_v_pre____,type,
pre2: list_P2774625669004309958m_rule ).
thf(sy_v_q____,type,
q: fm ).
thf(sy_v_r____,type,
r: rule ).
thf(sy_v_steps,type,
steps: stream2709947120125613254m_rule ).
thf(sy_v_suf____,type,
suf: stream2709947120125613254m_rule ).
% Relevant facts (1273)
thf(fact_0__092_060open_062Neg_A_ICon_Ap_Aq_J_A_092_060in_062_Atree__fms_Asteps_092_060close_062,axiom,
member_fm3 @ ( neg @ ( con @ p @ q ) ) @ ( tree_fms @ steps ) ).
% \<open>Neg (Con p q) \<in> tree_fms steps\<close>
thf(fact_1_fm_Oinject_I4_J,axiom,
! [X41: fm,X42: fm,Y41: fm,Y42: fm] :
( ( ( con @ X41 @ X42 )
= ( con @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% fm.inject(4)
thf(fact_2_fm_Oinject_I7_J,axiom,
! [X7: fm,Y7: fm] :
( ( ( neg @ X7 )
= ( neg @ Y7 ) )
= ( X7 = Y7 ) ) ).
% fm.inject(7)
thf(fact_3_fm_Odistinct_I35_J,axiom,
! [X41: fm,X42: fm,X7: fm] :
( ( con @ X41 @ X42 )
!= ( neg @ X7 ) ) ).
% fm.distinct(35)
thf(fact_4_Hintikka_OAlphaCon,axiom,
! [H: set_fm,P: fm,Q: fm] :
( ( hintikka @ H )
=> ( ( member_fm3 @ ( neg @ ( con @ P @ Q ) ) @ H )
=> ( ( member_fm3 @ ( neg @ P ) @ H )
& ( member_fm3 @ ( neg @ Q ) @ H ) ) ) ) ).
% Hintikka.AlphaCon
thf(fact_5_n,axiom,
member_fm3 @ ( neg @ ( con @ p @ q ) ) @ ( set_fm2 @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ n @ steps ) ) ) ) ).
% n
thf(fact_6__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062n_O_ANeg_A_ICon_Ap_Aq_J_A_092_060in_062_Aset_A_Ipseq_A_Ishd_A_Isdrop_An_Asteps_J_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [N: nat] :
~ ( member_fm3 @ ( neg @ ( con @ p @ q ) ) @ ( set_fm2 @ ( pseq @ ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N @ steps ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>n. Neg (Con p q) \<in> set (pseq (shd (sdrop n steps))) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_7_rule_Odistinct_I53_J,axiom,
alphaCon != gammaUni ).
% rule.distinct(53)
thf(fact_8_rule_Odistinct_I51_J,axiom,
alphaCon != gammaExi ).
% rule.distinct(51)
thf(fact_9_rule_Odistinct_I49_J,axiom,
alphaCon != negNeg ).
% rule.distinct(49)
thf(fact_10_rule_Odistinct_I47_J,axiom,
alphaCon != deltaExi ).
% rule.distinct(47)
thf(fact_11_rule_Odistinct_I109_J,axiom,
gammaExi != gammaUni ).
% rule.distinct(109)
thf(fact_12_rule_Odistinct_I107_J,axiom,
negNeg != gammaUni ).
% rule.distinct(107)
thf(fact_13_rule_Odistinct_I105_J,axiom,
negNeg != gammaExi ).
% rule.distinct(105)
thf(fact_14_rule_Odistinct_I103_J,axiom,
deltaExi != gammaUni ).
% rule.distinct(103)
thf(fact_15_rule_Odistinct_I101_J,axiom,
deltaExi != gammaExi ).
% rule.distinct(101)
thf(fact_16_rule_Odistinct_I99_J,axiom,
deltaExi != negNeg ).
% rule.distinct(99)
thf(fact_17_Hintikka_ONeg,axiom,
! [H: set_fm,P: fm] :
( ( hintikka @ H )
=> ( ( member_fm3 @ ( neg @ ( neg @ P ) ) @ H )
=> ( member_fm3 @ P @ H ) ) ) ).
% Hintikka.Neg
thf(fact_18_Hintikka_OBetaCon,axiom,
! [H: set_fm,P: fm,Q: fm] :
( ( hintikka @ H )
=> ( ( member_fm3 @ ( con @ P @ Q ) @ H )
=> ( ( member_fm3 @ P @ H )
| ( member_fm3 @ Q @ H ) ) ) ) ).
% Hintikka.BetaCon
thf(fact_19_ori,axiom,
( ( sdrop_8169176516188972301m_rule @ n @ steps )
= ( shift_2334844276748245581m_rule @ pre2 @ suf ) ) ).
% ori
thf(fact_20_member,axiom,
( member4200870186495857963m_rule
= ( ^ [P2: produc340336539035504054m_rule,Z: list_P2774625669004309958m_rule] : ( member7231649785386036813m_rule @ P2 @ ( set_Pr4534715572506550497m_rule @ Z ) ) ) ) ).
% member
thf(fact_21_member,axiom,
( member_nat2
= ( ^ [P2: nat,Z: list_nat] : ( member_nat3 @ P2 @ ( set_nat2 @ Z ) ) ) ) ).
% member
thf(fact_22_member,axiom,
( member_fm2
= ( ^ [P2: fm,Z: list_fm] : ( member_fm3 @ P2 @ ( set_fm2 @ Z ) ) ) ) ).
% member
thf(fact_23_member,axiom,
( member_list_fm2
= ( ^ [P2: list_fm,Z: list_list_fm] : ( member_list_fm3 @ P2 @ ( set_list_fm2 @ Z ) ) ) ) ).
% member
thf(fact_24_member,axiom,
( member_tm2
= ( ^ [P2: tm,Z: list_tm] : ( member_tm3 @ P2 @ ( set_tm2 @ Z ) ) ) ) ).
% member
thf(fact_25_tree__fms__in__pseq,axiom,
! [P: fm,Steps: stream2709947120125613254m_rule] :
( ( member_fm3 @ P @ ( tree_fms @ Steps ) )
=> ? [N: nat] : ( member_fm3 @ P @ ( set_fm2 @ ( pseq @ ( snth_P6679518042731451922m_rule @ Steps @ N ) ) ) ) ) ).
% tree_fms_in_pseq
thf(fact_26_pseq__in__tree__fms,axiom,
! [X: produc340336539035504054m_rule,Steps: stream2709947120125613254m_rule,P: fm] :
( ( member7231649785386036813m_rule @ X @ ( sset_P4484857331586881186m_rule @ Steps ) )
=> ( ( member_fm3 @ P @ ( set_fm2 @ ( pseq @ X ) ) )
=> ( member_fm3 @ P @ ( tree_fms @ Steps ) ) ) ) ).
% pseq_in_tree_fms
thf(fact_27_sdrop__szip,axiom,
! [N2: nat,S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule] :
( ( sdrop_7192298464603511222m_rule @ N2 @ ( szip_P811719526838699976m_rule @ S1 @ S2 ) )
= ( szip_P811719526838699976m_rule @ ( sdrop_8169176516188972301m_rule @ N2 @ S1 ) @ ( sdrop_8169176516188972301m_rule @ N2 @ S2 ) ) ) ).
% sdrop_szip
thf(fact_28_sdrop__szip,axiom,
! [N2: nat,S1: stream4408948924543953275ist_fm,S2: stream_rule] :
( ( sdrop_8169176516188972301m_rule @ N2 @ ( szip_P2924820683901490861m_rule @ S1 @ S2 ) )
= ( szip_P2924820683901490861m_rule @ ( sdrop_9176333610110415838ist_fm @ N2 @ S1 ) @ ( sdrop_rule @ N2 @ S2 ) ) ) ).
% sdrop_szip
thf(fact_29_next__rule_Osimps_I11_J,axiom,
( ( next_rule @ gammaUni )
= negNeg ) ).
% next_rule.simps(11)
thf(fact_30_next__rule_Osimps_I10_J,axiom,
( ( next_rule @ gammaExi )
= gammaUni ) ).
% next_rule.simps(10)
thf(fact_31_next__rule_Osimps_I4_J,axiom,
( ( next_rule @ alphaCon )
= deltaExi ) ).
% next_rule.simps(4)
thf(fact_32_Hintikka_OBasic,axiom,
! [H: set_fm,N2: nat,Ts: list_tm] :
( ( hintikka @ H )
=> ( ( member_fm3 @ ( pre @ N2 @ Ts ) @ H )
=> ~ ( member_fm3 @ ( neg @ ( pre @ N2 @ Ts ) ) @ H ) ) ) ).
% Hintikka.Basic
thf(fact_33_Hintikka_OBetaDis,axiom,
! [H: set_fm,P: fm,Q: fm] :
( ( hintikka @ H )
=> ( ( member_fm3 @ ( neg @ ( dis @ P @ Q ) ) @ H )
=> ( ( member_fm3 @ ( neg @ P ) @ H )
| ( member_fm3 @ ( neg @ Q ) @ H ) ) ) ) ).
% Hintikka.BetaDis
thf(fact_34_Hintikka_OBetaImp,axiom,
! [H: set_fm,P: fm,Q: fm] :
( ( hintikka @ H )
=> ( ( member_fm3 @ ( neg @ ( imp @ P @ Q ) ) @ H )
=> ( ( member_fm3 @ P @ H )
| ( member_fm3 @ ( neg @ Q ) @ H ) ) ) ) ).
% Hintikka.BetaImp
thf(fact_35_Hintikka_OAlphaImp,axiom,
! [H: set_fm,P: fm,Q: fm] :
( ( hintikka @ H )
=> ( ( member_fm3 @ ( imp @ P @ Q ) @ H )
=> ( ( member_fm3 @ ( neg @ P ) @ H )
& ( member_fm3 @ Q @ H ) ) ) ) ).
% Hintikka.AlphaImp
thf(fact_36_fm_Oinject_I3_J,axiom,
! [X31: fm,X32: fm,Y31: fm,Y32: fm] :
( ( ( dis @ X31 @ X32 )
= ( dis @ Y31 @ Y32 ) )
= ( ( X31 = Y31 )
& ( X32 = Y32 ) ) ) ).
% fm.inject(3)
thf(fact_37_fm_Oinject_I2_J,axiom,
! [X21: fm,X22: fm,Y21: fm,Y22: fm] :
( ( ( imp @ X21 @ X22 )
= ( imp @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% fm.inject(2)
thf(fact_38_fm_Oinject_I1_J,axiom,
! [X11: nat,X12: list_tm,Y11: nat,Y12: list_tm] :
( ( ( pre @ X11 @ X12 )
= ( pre @ Y11 @ Y12 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 ) ) ) ).
% fm.inject(1)
thf(fact_39_mem__Collect__eq,axiom,
! [A: fm,P3: fm > $o] :
( ( member_fm3 @ A @ ( collect_fm @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_40_mem__Collect__eq,axiom,
! [A: produc340336539035504054m_rule,P3: produc340336539035504054m_rule > $o] :
( ( member7231649785386036813m_rule @ A @ ( collec1009276759030335883m_rule @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_41_mem__Collect__eq,axiom,
! [A: list_fm,P3: list_fm > $o] :
( ( member_list_fm3 @ A @ ( collect_list_fm @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_42_mem__Collect__eq,axiom,
! [A: tm,P3: tm > $o] :
( ( member_tm3 @ A @ ( collect_tm @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_43_mem__Collect__eq,axiom,
! [A: nat,P3: nat > $o] :
( ( member_nat3 @ A @ ( collect_nat @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A2: set_fm] :
( ( collect_fm
@ ^ [X2: fm] : ( member_fm3 @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A2: set_Pr1822751329126368876m_rule] :
( ( collec1009276759030335883m_rule
@ ^ [X2: produc340336539035504054m_rule] : ( member7231649785386036813m_rule @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A2: set_list_fm] :
( ( collect_list_fm
@ ^ [X2: list_fm] : ( member_list_fm3 @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_Collect__mem__eq,axiom,
! [A2: set_tm] :
( ( collect_tm
@ ^ [X2: tm] : ( member_tm3 @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_48_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat3 @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_49_shift__left__inj,axiom,
! [Xs: list_P2774625669004309958m_rule,S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule] :
( ( ( shift_2334844276748245581m_rule @ Xs @ S1 )
= ( shift_2334844276748245581m_rule @ Xs @ S2 ) )
= ( S1 = S2 ) ) ).
% shift_left_inj
thf(fact_50_sdrop__simps_I1_J,axiom,
! [N2: nat,S: stream2709947120125613254m_rule] :
( ( shd_Pr4562317740776619530m_rule @ ( sdrop_8169176516188972301m_rule @ N2 @ S ) )
= ( snth_P6679518042731451922m_rule @ S @ N2 ) ) ).
% sdrop_simps(1)
thf(fact_51_fm_Odistinct_I13_J,axiom,
! [X21: fm,X22: fm,X31: fm,X32: fm] :
( ( imp @ X21 @ X22 )
!= ( dis @ X31 @ X32 ) ) ).
% fm.distinct(13)
thf(fact_52_fm_Odistinct_I3_J,axiom,
! [X11: nat,X12: list_tm,X31: fm,X32: fm] :
( ( pre @ X11 @ X12 )
!= ( dis @ X31 @ X32 ) ) ).
% fm.distinct(3)
thf(fact_53_fm_Odistinct_I1_J,axiom,
! [X11: nat,X12: list_tm,X21: fm,X22: fm] :
( ( pre @ X11 @ X12 )
!= ( imp @ X21 @ X22 ) ) ).
% fm.distinct(1)
thf(fact_54_snth__sset,axiom,
! [S: stream_fm,N2: nat] : ( member_fm3 @ ( snth_fm @ S @ N2 ) @ ( sset_fm @ S ) ) ).
% snth_sset
thf(fact_55_snth__sset,axiom,
! [S: stream_list_fm,N2: nat] : ( member_list_fm3 @ ( snth_list_fm @ S @ N2 ) @ ( sset_list_fm @ S ) ) ).
% snth_sset
thf(fact_56_snth__sset,axiom,
! [S: stream_tm,N2: nat] : ( member_tm3 @ ( snth_tm @ S @ N2 ) @ ( sset_tm @ S ) ) ).
% snth_sset
thf(fact_57_snth__sset,axiom,
! [S: stream_nat,N2: nat] : ( member_nat3 @ ( snth_nat @ S @ N2 ) @ ( sset_nat @ S ) ) ).
% snth_sset
thf(fact_58_snth__sset,axiom,
! [S: stream2709947120125613254m_rule,N2: nat] : ( member7231649785386036813m_rule @ ( snth_P6679518042731451922m_rule @ S @ N2 ) @ ( sset_P4484857331586881186m_rule @ S ) ) ).
% snth_sset
thf(fact_59_shd__sset,axiom,
! [A: stream_fm] : ( member_fm3 @ ( shd_fm @ A ) @ ( sset_fm @ A ) ) ).
% shd_sset
thf(fact_60_shd__sset,axiom,
! [A: stream_list_fm] : ( member_list_fm3 @ ( shd_list_fm @ A ) @ ( sset_list_fm @ A ) ) ).
% shd_sset
thf(fact_61_shd__sset,axiom,
! [A: stream_tm] : ( member_tm3 @ ( shd_tm @ A ) @ ( sset_tm @ A ) ) ).
% shd_sset
thf(fact_62_shd__sset,axiom,
! [A: stream_nat] : ( member_nat3 @ ( shd_nat @ A ) @ ( sset_nat @ A ) ) ).
% shd_sset
thf(fact_63_shd__sset,axiom,
! [A: stream2709947120125613254m_rule] : ( member7231649785386036813m_rule @ ( shd_Pr4562317740776619530m_rule @ A ) @ ( sset_P4484857331586881186m_rule @ A ) ) ).
% shd_sset
thf(fact_64_fm_Odistinct_I29_J,axiom,
! [X31: fm,X32: fm,X7: fm] :
( ( dis @ X31 @ X32 )
!= ( neg @ X7 ) ) ).
% fm.distinct(29)
thf(fact_65_fm_Odistinct_I21_J,axiom,
! [X21: fm,X22: fm,X7: fm] :
( ( imp @ X21 @ X22 )
!= ( neg @ X7 ) ) ).
% fm.distinct(21)
thf(fact_66_fm_Odistinct_I11_J,axiom,
! [X11: nat,X12: list_tm,X7: fm] :
( ( pre @ X11 @ X12 )
!= ( neg @ X7 ) ) ).
% fm.distinct(11)
thf(fact_67_fm_Odistinct_I23_J,axiom,
! [X31: fm,X32: fm,X41: fm,X42: fm] :
( ( dis @ X31 @ X32 )
!= ( con @ X41 @ X42 ) ) ).
% fm.distinct(23)
thf(fact_68_fm_Odistinct_I15_J,axiom,
! [X21: fm,X22: fm,X41: fm,X42: fm] :
( ( imp @ X21 @ X22 )
!= ( con @ X41 @ X42 ) ) ).
% fm.distinct(15)
thf(fact_69_fm_Odistinct_I5_J,axiom,
! [X11: nat,X12: list_tm,X41: fm,X42: fm] :
( ( pre @ X11 @ X12 )
!= ( con @ X41 @ X42 ) ) ).
% fm.distinct(5)
thf(fact_70_Hintikka_OAlphaDis,axiom,
! [H: set_fm,P: fm,Q: fm] :
( ( hintikka @ H )
=> ( ( member_fm3 @ ( dis @ P @ Q ) @ H )
=> ( ( member_fm3 @ P @ H )
& ( member_fm3 @ Q @ H ) ) ) ) ).
% Hintikka.AlphaDis
thf(fact_71_snth__sset__smerge,axiom,
! [Ss: stream_stream_fm,N2: nat,M: nat] : ( member_fm3 @ ( snth_fm @ ( snth_stream_fm @ Ss @ N2 ) @ M ) @ ( sset_fm @ ( smerge_fm @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_72_snth__sset__smerge,axiom,
! [Ss: stream6017534608192929797ist_fm,N2: nat,M: nat] : ( member_list_fm3 @ ( snth_list_fm @ ( snth_stream_list_fm @ Ss @ N2 ) @ M ) @ ( sset_list_fm @ ( smerge_list_fm @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_73_snth__sset__smerge,axiom,
! [Ss: stream_stream_tm,N2: nat,M: nat] : ( member_tm3 @ ( snth_tm @ ( snth_stream_tm @ Ss @ N2 ) @ M ) @ ( sset_tm @ ( smerge_tm @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_74_snth__sset__smerge,axiom,
! [Ss: stream_stream_nat,N2: nat,M: nat] : ( member_nat3 @ ( snth_nat @ ( snth_stream_nat @ Ss @ N2 ) @ M ) @ ( sset_nat @ ( smerge_nat @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_75_snth__sset__smerge,axiom,
! [Ss: stream3752074346242807894m_rule,N2: nat,M: nat] : ( member7231649785386036813m_rule @ ( snth_P6679518042731451922m_rule @ ( snth_s6182113952396108578m_rule @ Ss @ N2 ) @ M ) @ ( sset_P4484857331586881186m_rule @ ( smerge193809993764105000m_rule @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_76_snth__szip,axiom,
! [S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule,N2: nat] :
( ( snth_P8853764340393315953m_rule @ ( szip_P811719526838699976m_rule @ S1 @ S2 ) @ N2 )
= ( produc6261311607089640965m_rule @ ( snth_P6679518042731451922m_rule @ S1 @ N2 ) @ ( snth_P6679518042731451922m_rule @ S2 @ N2 ) ) ) ).
% snth_szip
thf(fact_77_snth__szip,axiom,
! [S1: stream4408948924543953275ist_fm,S2: stream_rule,N2: nat] :
( ( snth_P6679518042731451922m_rule @ ( szip_P2924820683901490861m_rule @ S1 @ S2 ) @ N2 )
= ( produc1733806532565653680m_rule @ ( snth_P7093566783922538521ist_fm @ S1 @ N2 ) @ ( snth_rule @ S2 @ N2 ) ) ) ).
% snth_szip
thf(fact_78_snth__szip,axiom,
! [S1: stream_list_tm,S2: stream_list_fm,N2: nat] :
( ( snth_P7093566783922538521ist_fm @ ( szip_list_tm_list_fm @ S1 @ S2 ) @ N2 )
= ( produc1414352766439514085ist_fm @ ( snth_list_tm @ S1 @ N2 ) @ ( snth_list_fm @ S2 @ N2 ) ) ) ).
% snth_szip
thf(fact_79_parts__preserves__unaffected,axiom,
! [R: rule,P: fm,Z2: list_fm,A2: list_tm] :
( ~ ( affects @ R @ P )
=> ( ( member_list_fm3 @ Z2 @ ( set_list_fm2 @ ( parts @ A2 @ R @ P ) ) )
=> ( member_fm3 @ P @ ( set_fm2 @ Z2 ) ) ) ) ).
% parts_preserves_unaffected
thf(fact_80_children__preserves__unaffected,axiom,
! [P: fm,Z3: list_fm,R: rule,Z2: list_fm,A2: list_tm] :
( ( member_fm3 @ P @ ( set_fm2 @ Z3 ) )
=> ( ~ ( affects @ R @ P )
=> ( ( member_list_fm3 @ Z2 @ ( set_list_fm2 @ ( children @ A2 @ R @ Z3 ) ) )
=> ( member_fm3 @ P @ ( set_fm2 @ Z2 ) ) ) ) ) ).
% children_preserves_unaffected
thf(fact_81_Stream_Osmember__def,axiom,
( smember_fm
= ( ^ [X2: fm,S3: stream_fm] : ( member_fm3 @ X2 @ ( sset_fm @ S3 ) ) ) ) ).
% Stream.smember_def
thf(fact_82_Stream_Osmember__def,axiom,
( smember_list_fm
= ( ^ [X2: list_fm,S3: stream_list_fm] : ( member_list_fm3 @ X2 @ ( sset_list_fm @ S3 ) ) ) ) ).
% Stream.smember_def
thf(fact_83_Stream_Osmember__def,axiom,
( smember_tm
= ( ^ [X2: tm,S3: stream_tm] : ( member_tm3 @ X2 @ ( sset_tm @ S3 ) ) ) ) ).
% Stream.smember_def
thf(fact_84_Stream_Osmember__def,axiom,
( smember_nat
= ( ^ [X2: nat,S3: stream_nat] : ( member_nat3 @ X2 @ ( sset_nat @ S3 ) ) ) ) ).
% Stream.smember_def
thf(fact_85_Stream_Osmember__def,axiom,
( smembe4439892024482649336m_rule
= ( ^ [X2: produc340336539035504054m_rule,S3: stream2709947120125613254m_rule] : ( member7231649785386036813m_rule @ X2 @ ( sset_P4484857331586881186m_rule @ S3 ) ) ) ) ).
% Stream.smember_def
thf(fact_86_next__rule_Osimps_I1_J,axiom,
( ( next_rule @ negNeg )
= alphaImp ) ).
% next_rule.simps(1)
thf(fact_87_next__rule_Osimps_I5_J,axiom,
( ( next_rule @ deltaExi )
= deltaUni ) ).
% next_rule.simps(5)
thf(fact_88_next__rule_Osimps_I9_J,axiom,
( ( next_rule @ betaCon )
= gammaExi ) ).
% next_rule.simps(9)
thf(fact_89_sset__shift,axiom,
! [Xs: list_fm,S: stream_fm] :
( ( sset_fm @ ( shift_fm @ Xs @ S ) )
= ( sup_sup_set_fm @ ( set_fm2 @ Xs ) @ ( sset_fm @ S ) ) ) ).
% sset_shift
thf(fact_90_sset__shift,axiom,
! [Xs: list_list_fm,S: stream_list_fm] :
( ( sset_list_fm @ ( shift_list_fm @ Xs @ S ) )
= ( sup_sup_set_list_fm @ ( set_list_fm2 @ Xs ) @ ( sset_list_fm @ S ) ) ) ).
% sset_shift
thf(fact_91_sset__shift,axiom,
! [Xs: list_tm,S: stream_tm] :
( ( sset_tm @ ( shift_tm @ Xs @ S ) )
= ( sup_sup_set_tm @ ( set_tm2 @ Xs ) @ ( sset_tm @ S ) ) ) ).
% sset_shift
thf(fact_92_sset__shift,axiom,
! [Xs: list_P2774625669004309958m_rule,S: stream2709947120125613254m_rule] :
( ( sset_P4484857331586881186m_rule @ ( shift_2334844276748245581m_rule @ Xs @ S ) )
= ( sup_su6946459741510085528m_rule @ ( set_Pr4534715572506550497m_rule @ Xs ) @ ( sset_P4484857331586881186m_rule @ S ) ) ) ).
% sset_shift
thf(fact_93_sset__shift,axiom,
! [Xs: list_nat,S: stream_nat] :
( ( sset_nat @ ( shift_nat @ Xs @ S ) )
= ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( sset_nat @ S ) ) ) ).
% sset_shift
thf(fact_94_next__rule_Osimps_I3_J,axiom,
( ( next_rule @ alphaDis )
= alphaCon ) ).
% next_rule.simps(3)
thf(fact_95_szip_Osimps_I1_J,axiom,
! [S1: stream_list_tm,S2: stream_list_fm] :
( ( shd_Pr3211216682057661985ist_fm @ ( szip_list_tm_list_fm @ S1 @ S2 ) )
= ( produc1414352766439514085ist_fm @ ( shd_list_tm @ S1 ) @ ( shd_list_fm @ S2 ) ) ) ).
% szip.simps(1)
thf(fact_96_szip_Osimps_I1_J,axiom,
! [S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule] :
( ( shd_Pr4461660664618831993m_rule @ ( szip_P811719526838699976m_rule @ S1 @ S2 ) )
= ( produc6261311607089640965m_rule @ ( shd_Pr4562317740776619530m_rule @ S1 ) @ ( shd_Pr4562317740776619530m_rule @ S2 ) ) ) ).
% szip.simps(1)
thf(fact_97_szip_Osimps_I1_J,axiom,
! [S1: stream4408948924543953275ist_fm,S2: stream_rule] :
( ( shd_Pr4562317740776619530m_rule @ ( szip_P2924820683901490861m_rule @ S1 @ S2 ) )
= ( produc1733806532565653680m_rule @ ( shd_Pr3211216682057661985ist_fm @ S1 ) @ ( shd_rule @ S2 ) ) ) ).
% szip.simps(1)
thf(fact_98_rule_Odistinct_I1_J,axiom,
alphaDis != alphaImp ).
% rule.distinct(1)
thf(fact_99_rule_Odistinct_I5_J,axiom,
alphaDis != betaCon ).
% rule.distinct(5)
thf(fact_100_rule_Odistinct_I11_J,axiom,
alphaDis != deltaUni ).
% rule.distinct(11)
thf(fact_101_rule_Odistinct_I23_J,axiom,
alphaImp != betaCon ).
% rule.distinct(23)
thf(fact_102_rule_Odistinct_I29_J,axiom,
alphaImp != deltaUni ).
% rule.distinct(29)
thf(fact_103_rule_Odistinct_I59_J,axiom,
betaCon != deltaUni ).
% rule.distinct(59)
thf(fact_104_next__rule_Osimps_I2_J,axiom,
( ( next_rule @ alphaImp )
= alphaDis ) ).
% next_rule.simps(2)
thf(fact_105_rule_Odistinct_I3_J,axiom,
alphaDis != alphaCon ).
% rule.distinct(3)
thf(fact_106_rule_Odistinct_I21_J,axiom,
alphaImp != alphaCon ).
% rule.distinct(21)
thf(fact_107_rule_Odistinct_I39_J,axiom,
alphaCon != betaCon ).
% rule.distinct(39)
thf(fact_108_rule_Odistinct_I45_J,axiom,
alphaCon != deltaUni ).
% rule.distinct(45)
thf(fact_109_rule_Odistinct_I97_J,axiom,
deltaUni != gammaUni ).
% rule.distinct(97)
thf(fact_110_rule_Odistinct_I95_J,axiom,
deltaUni != gammaExi ).
% rule.distinct(95)
thf(fact_111_rule_Odistinct_I93_J,axiom,
deltaUni != negNeg ).
% rule.distinct(93)
thf(fact_112_rule_Odistinct_I91_J,axiom,
deltaUni != deltaExi ).
% rule.distinct(91)
thf(fact_113_rule_Odistinct_I67_J,axiom,
betaCon != gammaUni ).
% rule.distinct(67)
thf(fact_114_rule_Odistinct_I65_J,axiom,
betaCon != gammaExi ).
% rule.distinct(65)
thf(fact_115_rule_Odistinct_I63_J,axiom,
betaCon != negNeg ).
% rule.distinct(63)
thf(fact_116_rule_Odistinct_I61_J,axiom,
betaCon != deltaExi ).
% rule.distinct(61)
thf(fact_117_rule_Odistinct_I37_J,axiom,
alphaImp != gammaUni ).
% rule.distinct(37)
thf(fact_118_rule_Odistinct_I35_J,axiom,
alphaImp != gammaExi ).
% rule.distinct(35)
thf(fact_119_rule_Odistinct_I33_J,axiom,
alphaImp != negNeg ).
% rule.distinct(33)
thf(fact_120_rule_Odistinct_I31_J,axiom,
alphaImp != deltaExi ).
% rule.distinct(31)
thf(fact_121_rule_Odistinct_I19_J,axiom,
alphaDis != gammaUni ).
% rule.distinct(19)
thf(fact_122_rule_Odistinct_I17_J,axiom,
alphaDis != gammaExi ).
% rule.distinct(17)
thf(fact_123_rule_Odistinct_I15_J,axiom,
alphaDis != negNeg ).
% rule.distinct(15)
thf(fact_124_rule_Odistinct_I13_J,axiom,
alphaDis != deltaExi ).
% rule.distinct(13)
thf(fact_125_UnCI,axiom,
! [C: fm,B: set_fm,A2: set_fm] :
( ( ~ ( member_fm3 @ C @ B )
=> ( member_fm3 @ C @ A2 ) )
=> ( member_fm3 @ C @ ( sup_sup_set_fm @ A2 @ B ) ) ) ).
% UnCI
thf(fact_126_UnCI,axiom,
! [C: produc340336539035504054m_rule,B: set_Pr1822751329126368876m_rule,A2: set_Pr1822751329126368876m_rule] :
( ( ~ ( member7231649785386036813m_rule @ C @ B )
=> ( member7231649785386036813m_rule @ C @ A2 ) )
=> ( member7231649785386036813m_rule @ C @ ( sup_su6946459741510085528m_rule @ A2 @ B ) ) ) ).
% UnCI
thf(fact_127_UnCI,axiom,
! [C: list_fm,B: set_list_fm,A2: set_list_fm] :
( ( ~ ( member_list_fm3 @ C @ B )
=> ( member_list_fm3 @ C @ A2 ) )
=> ( member_list_fm3 @ C @ ( sup_sup_set_list_fm @ A2 @ B ) ) ) ).
% UnCI
thf(fact_128_UnCI,axiom,
! [C: tm,B: set_tm,A2: set_tm] :
( ( ~ ( member_tm3 @ C @ B )
=> ( member_tm3 @ C @ A2 ) )
=> ( member_tm3 @ C @ ( sup_sup_set_tm @ A2 @ B ) ) ) ).
% UnCI
thf(fact_129_UnCI,axiom,
! [C: nat,B: set_nat,A2: set_nat] :
( ( ~ ( member_nat3 @ C @ B )
=> ( member_nat3 @ C @ A2 ) )
=> ( member_nat3 @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnCI
thf(fact_130_Un__iff,axiom,
! [C: fm,A2: set_fm,B: set_fm] :
( ( member_fm3 @ C @ ( sup_sup_set_fm @ A2 @ B ) )
= ( ( member_fm3 @ C @ A2 )
| ( member_fm3 @ C @ B ) ) ) ).
% Un_iff
thf(fact_131_Un__iff,axiom,
! [C: produc340336539035504054m_rule,A2: set_Pr1822751329126368876m_rule,B: set_Pr1822751329126368876m_rule] :
( ( member7231649785386036813m_rule @ C @ ( sup_su6946459741510085528m_rule @ A2 @ B ) )
= ( ( member7231649785386036813m_rule @ C @ A2 )
| ( member7231649785386036813m_rule @ C @ B ) ) ) ).
% Un_iff
thf(fact_132_Un__iff,axiom,
! [C: list_fm,A2: set_list_fm,B: set_list_fm] :
( ( member_list_fm3 @ C @ ( sup_sup_set_list_fm @ A2 @ B ) )
= ( ( member_list_fm3 @ C @ A2 )
| ( member_list_fm3 @ C @ B ) ) ) ).
% Un_iff
thf(fact_133_Un__iff,axiom,
! [C: tm,A2: set_tm,B: set_tm] :
( ( member_tm3 @ C @ ( sup_sup_set_tm @ A2 @ B ) )
= ( ( member_tm3 @ C @ A2 )
| ( member_tm3 @ C @ B ) ) ) ).
% Un_iff
thf(fact_134_Un__iff,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat3 @ C @ ( sup_sup_set_nat @ A2 @ B ) )
= ( ( member_nat3 @ C @ A2 )
| ( member_nat3 @ C @ B ) ) ) ).
% Un_iff
thf(fact_135_sup_Oidem,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_136_sup__idem,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_137_sup_Oleft__idem,axiom,
! [A: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B2 ) )
= ( sup_sup_set_nat @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_138_sup__left__idem,axiom,
! [X: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
= ( sup_sup_set_nat @ X @ Y ) ) ).
% sup_left_idem
thf(fact_139_sup_Oright__idem,axiom,
! [A: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ B2 )
= ( sup_sup_set_nat @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_140_prod_Oinject,axiom,
! [X1: list_tm,X23: list_fm,Y1: list_tm,Y2: list_fm] :
( ( ( produc1414352766439514085ist_fm @ X1 @ X23 )
= ( produc1414352766439514085ist_fm @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X23 = Y2 ) ) ) ).
% prod.inject
thf(fact_141_old_Oprod_Oinject,axiom,
! [A: list_tm,B2: list_fm,A3: list_tm,B3: list_fm] :
( ( ( produc1414352766439514085ist_fm @ A @ B2 )
= ( produc1414352766439514085ist_fm @ A3 @ B3 ) )
= ( ( A = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_142_Pair__inject,axiom,
! [A: list_tm,B2: list_fm,A3: list_tm,B3: list_fm] :
( ( ( produc1414352766439514085ist_fm @ A @ B2 )
= ( produc1414352766439514085ist_fm @ A3 @ B3 ) )
=> ~ ( ( A = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_143_prod__cases,axiom,
! [P3: produc6018962875968178549ist_fm > $o,P: produc6018962875968178549ist_fm] :
( ! [A4: list_tm,B4: list_fm] : ( P3 @ ( produc1414352766439514085ist_fm @ A4 @ B4 ) )
=> ( P3 @ P ) ) ).
% prod_cases
thf(fact_144_surj__pair,axiom,
! [P: produc6018962875968178549ist_fm] :
? [X3: list_tm,Y3: list_fm] :
( P
= ( produc1414352766439514085ist_fm @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_145_old_Oprod_Oexhaust,axiom,
! [Y: produc6018962875968178549ist_fm] :
~ ! [A4: list_tm,B4: list_fm] :
( Y
!= ( produc1414352766439514085ist_fm @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_146_sup__left__commute,axiom,
! [X: set_nat,Y: set_nat,Z3: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z3 ) )
= ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X @ Z3 ) ) ) ).
% sup_left_commute
thf(fact_147_sup_Oleft__commute,axiom,
! [B2: set_nat,A: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A @ C ) )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_148_sup__commute,axiom,
( sup_sup_set_nat
= ( ^ [X2: set_nat,Y4: set_nat] : ( sup_sup_set_nat @ Y4 @ X2 ) ) ) ).
% sup_commute
thf(fact_149_sup_Ocommute,axiom,
( sup_sup_set_nat
= ( ^ [A5: set_nat,B5: set_nat] : ( sup_sup_set_nat @ B5 @ A5 ) ) ) ).
% sup.commute
thf(fact_150_sup__assoc,axiom,
! [X: set_nat,Y: set_nat,Z3: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z3 )
= ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z3 ) ) ) ).
% sup_assoc
thf(fact_151_sup_Oassoc,axiom,
! [A: set_nat,B2: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ C )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_152_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat
= ( ^ [X2: set_nat,Y4: set_nat] : ( sup_sup_set_nat @ Y4 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_153_inf__sup__aci_I6_J,axiom,
! [X: set_nat,Y: set_nat,Z3: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z3 )
= ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z3 ) ) ) ).
% inf_sup_aci(6)
thf(fact_154_inf__sup__aci_I7_J,axiom,
! [X: set_nat,Y: set_nat,Z3: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z3 ) )
= ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X @ Z3 ) ) ) ).
% inf_sup_aci(7)
thf(fact_155_inf__sup__aci_I8_J,axiom,
! [X: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
= ( sup_sup_set_nat @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_156_Un__left__commute,axiom,
! [A2: set_nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C2 ) )
= ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_157_Un__left__absorb,axiom,
! [A2: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B ) )
= ( sup_sup_set_nat @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_158_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A6: set_nat,B6: set_nat] : ( sup_sup_set_nat @ B6 @ A6 ) ) ) ).
% Un_commute
thf(fact_159_Un__absorb,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_160_Un__assoc,axiom,
! [A2: set_nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C2 )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_161_ball__Un,axiom,
! [A2: set_nat,B: set_nat,P3: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat3 @ X2 @ ( sup_sup_set_nat @ A2 @ B ) )
=> ( P3 @ X2 ) ) )
= ( ! [X2: nat] :
( ( member_nat3 @ X2 @ A2 )
=> ( P3 @ X2 ) )
& ! [X2: nat] :
( ( member_nat3 @ X2 @ B )
=> ( P3 @ X2 ) ) ) ) ).
% ball_Un
thf(fact_162_bex__Un,axiom,
! [A2: set_nat,B: set_nat,P3: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat3 @ X2 @ ( sup_sup_set_nat @ A2 @ B ) )
& ( P3 @ X2 ) ) )
= ( ? [X2: nat] :
( ( member_nat3 @ X2 @ A2 )
& ( P3 @ X2 ) )
| ? [X2: nat] :
( ( member_nat3 @ X2 @ B )
& ( P3 @ X2 ) ) ) ) ).
% bex_Un
thf(fact_163_UnI2,axiom,
! [C: fm,B: set_fm,A2: set_fm] :
( ( member_fm3 @ C @ B )
=> ( member_fm3 @ C @ ( sup_sup_set_fm @ A2 @ B ) ) ) ).
% UnI2
thf(fact_164_UnI2,axiom,
! [C: produc340336539035504054m_rule,B: set_Pr1822751329126368876m_rule,A2: set_Pr1822751329126368876m_rule] :
( ( member7231649785386036813m_rule @ C @ B )
=> ( member7231649785386036813m_rule @ C @ ( sup_su6946459741510085528m_rule @ A2 @ B ) ) ) ).
% UnI2
thf(fact_165_UnI2,axiom,
! [C: list_fm,B: set_list_fm,A2: set_list_fm] :
( ( member_list_fm3 @ C @ B )
=> ( member_list_fm3 @ C @ ( sup_sup_set_list_fm @ A2 @ B ) ) ) ).
% UnI2
thf(fact_166_UnI2,axiom,
! [C: tm,B: set_tm,A2: set_tm] :
( ( member_tm3 @ C @ B )
=> ( member_tm3 @ C @ ( sup_sup_set_tm @ A2 @ B ) ) ) ).
% UnI2
thf(fact_167_UnI2,axiom,
! [C: nat,B: set_nat,A2: set_nat] :
( ( member_nat3 @ C @ B )
=> ( member_nat3 @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnI2
thf(fact_168_UnI1,axiom,
! [C: fm,A2: set_fm,B: set_fm] :
( ( member_fm3 @ C @ A2 )
=> ( member_fm3 @ C @ ( sup_sup_set_fm @ A2 @ B ) ) ) ).
% UnI1
thf(fact_169_UnI1,axiom,
! [C: produc340336539035504054m_rule,A2: set_Pr1822751329126368876m_rule,B: set_Pr1822751329126368876m_rule] :
( ( member7231649785386036813m_rule @ C @ A2 )
=> ( member7231649785386036813m_rule @ C @ ( sup_su6946459741510085528m_rule @ A2 @ B ) ) ) ).
% UnI1
thf(fact_170_UnI1,axiom,
! [C: list_fm,A2: set_list_fm,B: set_list_fm] :
( ( member_list_fm3 @ C @ A2 )
=> ( member_list_fm3 @ C @ ( sup_sup_set_list_fm @ A2 @ B ) ) ) ).
% UnI1
thf(fact_171_UnI1,axiom,
! [C: tm,A2: set_tm,B: set_tm] :
( ( member_tm3 @ C @ A2 )
=> ( member_tm3 @ C @ ( sup_sup_set_tm @ A2 @ B ) ) ) ).
% UnI1
thf(fact_172_UnI1,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat3 @ C @ A2 )
=> ( member_nat3 @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnI1
thf(fact_173_UnE,axiom,
! [C: fm,A2: set_fm,B: set_fm] :
( ( member_fm3 @ C @ ( sup_sup_set_fm @ A2 @ B ) )
=> ( ~ ( member_fm3 @ C @ A2 )
=> ( member_fm3 @ C @ B ) ) ) ).
% UnE
thf(fact_174_UnE,axiom,
! [C: produc340336539035504054m_rule,A2: set_Pr1822751329126368876m_rule,B: set_Pr1822751329126368876m_rule] :
( ( member7231649785386036813m_rule @ C @ ( sup_su6946459741510085528m_rule @ A2 @ B ) )
=> ( ~ ( member7231649785386036813m_rule @ C @ A2 )
=> ( member7231649785386036813m_rule @ C @ B ) ) ) ).
% UnE
thf(fact_175_UnE,axiom,
! [C: list_fm,A2: set_list_fm,B: set_list_fm] :
( ( member_list_fm3 @ C @ ( sup_sup_set_list_fm @ A2 @ B ) )
=> ( ~ ( member_list_fm3 @ C @ A2 )
=> ( member_list_fm3 @ C @ B ) ) ) ).
% UnE
thf(fact_176_UnE,axiom,
! [C: tm,A2: set_tm,B: set_tm] :
( ( member_tm3 @ C @ ( sup_sup_set_tm @ A2 @ B ) )
=> ( ~ ( member_tm3 @ C @ A2 )
=> ( member_tm3 @ C @ B ) ) ) ).
% UnE
thf(fact_177_UnE,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat3 @ C @ ( sup_sup_set_nat @ A2 @ B ) )
=> ( ~ ( member_nat3 @ C @ A2 )
=> ( member_nat3 @ C @ B ) ) ) ).
% UnE
thf(fact_178_rule_Oexhaust,axiom,
! [Y: rule] :
( ( Y != alphaDis )
=> ( ( Y != alphaImp )
=> ( ( Y != alphaCon )
=> ( ( Y != betaCon )
=> ( ( Y != betaImp )
=> ( ( Y != betaDis )
=> ( ( Y != deltaUni )
=> ( ( Y != deltaExi )
=> ( ( Y != negNeg )
=> ( ( Y != gammaExi )
=> ( Y = gammaUni ) ) ) ) ) ) ) ) ) ) ) ).
% rule.exhaust
thf(fact_179_set__union,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( set_fm2 @ ( union_fm @ Xs @ Ys ) )
= ( sup_sup_set_fm @ ( set_fm2 @ Xs ) @ ( set_fm2 @ Ys ) ) ) ).
% set_union
thf(fact_180_set__union,axiom,
! [Xs: list_list_fm,Ys: list_list_fm] :
( ( set_list_fm2 @ ( union_list_fm @ Xs @ Ys ) )
= ( sup_sup_set_list_fm @ ( set_list_fm2 @ Xs ) @ ( set_list_fm2 @ Ys ) ) ) ).
% set_union
thf(fact_181_set__union,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( set_tm2 @ ( union_tm @ Xs @ Ys ) )
= ( sup_sup_set_tm @ ( set_tm2 @ Xs ) @ ( set_tm2 @ Ys ) ) ) ).
% set_union
thf(fact_182_set__union,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( set_nat2 @ ( union_nat @ Xs @ Ys ) )
= ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) ) ).
% set_union
thf(fact_183_sset__sinterleave,axiom,
! [S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule] :
( ( sset_P4484857331586881186m_rule @ ( sinter2808979888547679129m_rule @ S1 @ S2 ) )
= ( sup_su6946459741510085528m_rule @ ( sset_P4484857331586881186m_rule @ S1 ) @ ( sset_P4484857331586881186m_rule @ S2 ) ) ) ).
% sset_sinterleave
thf(fact_184_sset__sinterleave,axiom,
! [S1: stream_nat,S2: stream_nat] :
( ( sset_nat @ ( sinterleave_nat @ S1 @ S2 ) )
= ( sup_sup_set_nat @ ( sset_nat @ S1 ) @ ( sset_nat @ S2 ) ) ) ).
% sset_sinterleave
thf(fact_185_next__rule_Osimps_I6_J,axiom,
( ( next_rule @ deltaUni )
= betaImp ) ).
% next_rule.simps(6)
thf(fact_186_next__rule_Osimps_I8_J,axiom,
( ( next_rule @ betaDis )
= betaCon ) ).
% next_rule.simps(8)
thf(fact_187_boolean__algebra__cancel_Osup2,axiom,
! [B: set_nat,K: set_nat,B2: set_nat,A: set_nat] :
( ( B
= ( sup_sup_set_nat @ K @ B2 ) )
=> ( ( sup_sup_set_nat @ A @ B )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_188_rule_Odistinct_I69_J,axiom,
betaImp != betaDis ).
% rule.distinct(69)
thf(fact_189_next__rule_Osimps_I7_J,axiom,
( ( next_rule @ betaImp )
= betaDis ) ).
% next_rule.simps(7)
thf(fact_190_rule_Odistinct_I41_J,axiom,
alphaCon != betaImp ).
% rule.distinct(41)
thf(fact_191_rule_Odistinct_I43_J,axiom,
alphaCon != betaDis ).
% rule.distinct(43)
thf(fact_192_rule_Odistinct_I7_J,axiom,
alphaDis != betaImp ).
% rule.distinct(7)
thf(fact_193_rule_Odistinct_I9_J,axiom,
alphaDis != betaDis ).
% rule.distinct(9)
thf(fact_194_rule_Odistinct_I25_J,axiom,
alphaImp != betaImp ).
% rule.distinct(25)
thf(fact_195_rule_Odistinct_I27_J,axiom,
alphaImp != betaDis ).
% rule.distinct(27)
thf(fact_196_rule_Odistinct_I55_J,axiom,
betaCon != betaImp ).
% rule.distinct(55)
thf(fact_197_rule_Odistinct_I57_J,axiom,
betaCon != betaDis ).
% rule.distinct(57)
thf(fact_198_rule_Odistinct_I71_J,axiom,
betaImp != deltaUni ).
% rule.distinct(71)
thf(fact_199_rule_Odistinct_I81_J,axiom,
betaDis != deltaUni ).
% rule.distinct(81)
thf(fact_200_rule_Odistinct_I89_J,axiom,
betaDis != gammaUni ).
% rule.distinct(89)
thf(fact_201_rule_Odistinct_I87_J,axiom,
betaDis != gammaExi ).
% rule.distinct(87)
thf(fact_202_rule_Odistinct_I85_J,axiom,
betaDis != negNeg ).
% rule.distinct(85)
thf(fact_203_rule_Odistinct_I83_J,axiom,
betaDis != deltaExi ).
% rule.distinct(83)
thf(fact_204_rule_Odistinct_I79_J,axiom,
betaImp != gammaUni ).
% rule.distinct(79)
thf(fact_205_rule_Odistinct_I77_J,axiom,
betaImp != gammaExi ).
% rule.distinct(77)
thf(fact_206_rule_Odistinct_I75_J,axiom,
betaImp != negNeg ).
% rule.distinct(75)
thf(fact_207_rule_Odistinct_I73_J,axiom,
betaImp != deltaExi ).
% rule.distinct(73)
thf(fact_208_sinterleave_Osimps_I1_J,axiom,
! [S1: stream2709947120125613254m_rule,S2: stream2709947120125613254m_rule] :
( ( shd_Pr4562317740776619530m_rule @ ( sinter2808979888547679129m_rule @ S1 @ S2 ) )
= ( shd_Pr4562317740776619530m_rule @ S1 ) ) ).
% sinterleave.simps(1)
thf(fact_209_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_nat,K: set_nat,A: set_nat,B2: set_nat] :
( ( A2
= ( sup_sup_set_nat @ K @ A ) )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_210_rulesList__def,axiom,
( rulesList
= ( cons_rule @ negNeg @ ( cons_rule @ alphaImp @ ( cons_rule @ alphaDis @ ( cons_rule @ alphaCon @ ( cons_rule @ deltaExi @ ( cons_rule @ deltaUni @ ( cons_rule @ betaImp @ ( cons_rule @ betaDis @ ( cons_rule @ betaCon @ ( cons_rule @ gammaExi @ ( cons_rule @ gammaUni @ nil_rule ) ) ) ) ) ) ) ) ) ) ) ) ).
% rulesList_def
thf(fact_211_partition__set,axiom,
! [P3: fm > $o,Xs: list_fm,Yes: list_fm,No: list_fm] :
( ( ( partition_fm @ P3 @ Xs )
= ( produc7863996417982153943ist_fm @ Yes @ No ) )
=> ( ( sup_sup_set_fm @ ( set_fm2 @ Yes ) @ ( set_fm2 @ No ) )
= ( set_fm2 @ Xs ) ) ) ).
% partition_set
thf(fact_212_partition__set,axiom,
! [P3: list_fm > $o,Xs: list_list_fm,Yes: list_list_fm,No: list_list_fm] :
( ( ( partition_list_fm @ P3 @ Xs )
= ( produc8321651870839017815ist_fm @ Yes @ No ) )
=> ( ( sup_sup_set_list_fm @ ( set_list_fm2 @ Yes ) @ ( set_list_fm2 @ No ) )
= ( set_list_fm2 @ Xs ) ) ) ).
% partition_set
thf(fact_213_partition__set,axiom,
! [P3: tm > $o,Xs: list_tm,Yes: list_tm,No: list_tm] :
( ( ( partition_tm @ P3 @ Xs )
= ( produc1418304791525149271ist_tm @ Yes @ No ) )
=> ( ( sup_sup_set_tm @ ( set_tm2 @ Yes ) @ ( set_tm2 @ No ) )
= ( set_tm2 @ Xs ) ) ) ).
% partition_set
thf(fact_214_partition__set,axiom,
! [P3: nat > $o,Xs: list_nat,Yes: list_nat,No: list_nat] :
( ( ( partition_nat @ P3 @ Xs )
= ( produc2694037385005941721st_nat @ Yes @ No ) )
=> ( ( sup_sup_set_nat @ ( set_nat2 @ Yes ) @ ( set_nat2 @ No ) )
= ( set_nat2 @ Xs ) ) ) ).
% partition_set
thf(fact_215_in__set__member,axiom,
! [X: produc340336539035504054m_rule,Xs: list_P2774625669004309958m_rule] :
( ( member7231649785386036813m_rule @ X @ ( set_Pr4534715572506550497m_rule @ Xs ) )
= ( member4742765151257878875m_rule @ Xs @ X ) ) ).
% in_set_member
thf(fact_216_in__set__member,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat3 @ X @ ( set_nat2 @ Xs ) )
= ( member_nat @ Xs @ X ) ) ).
% in_set_member
thf(fact_217_in__set__member,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm3 @ X @ ( set_fm2 @ Xs ) )
= ( member_fm @ Xs @ X ) ) ).
% in_set_member
thf(fact_218_in__set__member,axiom,
! [X: list_fm,Xs: list_list_fm] :
( ( member_list_fm3 @ X @ ( set_list_fm2 @ Xs ) )
= ( member_list_fm @ Xs @ X ) ) ).
% in_set_member
thf(fact_219_in__set__member,axiom,
! [X: tm,Xs: list_tm] :
( ( member_tm3 @ X @ ( set_tm2 @ Xs ) )
= ( member_tm @ Xs @ X ) ) ).
% in_set_member
thf(fact_220_ssubst__Pair__rhs,axiom,
! [R: produc6018962875968178549ist_fm,S: rule,R2: set_Pr1822751329126368876m_rule,S4: rule] :
( ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ R @ S ) @ R2 )
=> ( ( S4 = S )
=> ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ R @ S4 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_221_ssubst__Pair__rhs,axiom,
! [R: list_tm,S: list_fm,R2: set_Pr5202636777678657877ist_fm,S4: list_fm] :
( ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ R @ S ) @ R2 )
=> ( ( S4 = S )
=> ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ R @ S4 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_222_parts__exhaust,axiom,
! [R: rule,X: fm] :
( ( ( R = alphaDis )
=> ! [P4: fm,Q2: fm] :
( X
!= ( dis @ P4 @ Q2 ) ) )
=> ( ( ( R = alphaImp )
=> ! [P4: fm,Q2: fm] :
( X
!= ( imp @ P4 @ Q2 ) ) )
=> ( ( ( R = alphaCon )
=> ! [P4: fm,Q2: fm] :
( X
!= ( neg @ ( con @ P4 @ Q2 ) ) ) )
=> ( ( ( R = betaDis )
=> ! [P4: fm,Q2: fm] :
( X
!= ( neg @ ( dis @ P4 @ Q2 ) ) ) )
=> ( ( ( R = betaImp )
=> ! [P4: fm,Q2: fm] :
( X
!= ( neg @ ( imp @ P4 @ Q2 ) ) ) )
=> ( ( ( R = betaCon )
=> ! [P4: fm,Q2: fm] :
( X
!= ( con @ P4 @ Q2 ) ) )
=> ( ( ( R = deltaUni )
=> ! [P4: fm] :
( X
!= ( uni @ P4 ) ) )
=> ( ( ( R = deltaExi )
=> ! [P4: fm] :
( X
!= ( neg @ ( exi @ P4 ) ) ) )
=> ( ( ( R = negNeg )
=> ! [P4: fm] :
( X
!= ( neg @ ( neg @ P4 ) ) ) )
=> ( ( ( R = gammaExi )
=> ! [P4: fm] :
( X
!= ( exi @ P4 ) ) )
=> ( ( ( R = gammaUni )
=> ! [P4: fm] :
( X
!= ( neg @ ( uni @ P4 ) ) ) )
=> ! [A7: list_tm] :
( ( parts @ A7 @ R @ X )
= ( cons_list_fm @ ( cons_fm @ X @ nil_fm ) @ nil_list_fm ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% parts_exhaust
thf(fact_223_stream__all__def,axiom,
( stream2134157564947672044m_rule
= ( ^ [P5: produc340336539035504054m_rule > $o,S3: stream2709947120125613254m_rule] :
! [P2: nat] : ( P5 @ ( snth_P6679518042731451922m_rule @ S3 @ P2 ) ) ) ) ).
% stream_all_def
thf(fact_224_in__set__insert,axiom,
! [X: produc340336539035504054m_rule,Xs: list_P2774625669004309958m_rule] :
( ( member7231649785386036813m_rule @ X @ ( set_Pr4534715572506550497m_rule @ Xs ) )
=> ( ( insert8371175628248280474m_rule @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_225_in__set__insert,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat3 @ X @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_226_in__set__insert,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm3 @ X @ ( set_fm2 @ Xs ) )
=> ( ( insert_fm @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_227_in__set__insert,axiom,
! [X: list_fm,Xs: list_list_fm] :
( ( member_list_fm3 @ X @ ( set_list_fm2 @ Xs ) )
=> ( ( insert_list_fm @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_228_in__set__insert,axiom,
! [X: tm,Xs: list_tm] :
( ( member_tm3 @ X @ ( set_tm2 @ Xs ) )
=> ( ( insert_tm @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_229_curryI,axiom,
! [F: produc6018962875968178549ist_fm > $o,A: list_tm,B2: list_fm] :
( ( F @ ( produc1414352766439514085ist_fm @ A @ B2 ) )
=> ( produc5334687091915550783t_fm_o @ F @ A @ B2 ) ) ).
% curryI
thf(fact_230_list_Oinject,axiom,
! [X21: rule,X22: list_rule,Y21: rule,Y22: list_rule] :
( ( ( cons_rule @ X21 @ X22 )
= ( cons_rule @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_231_list_Oinject,axiom,
! [X21: list_fm,X22: list_list_fm,Y21: list_fm,Y22: list_list_fm] :
( ( ( cons_list_fm @ X21 @ X22 )
= ( cons_list_fm @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_232_list_Oinject,axiom,
! [X21: fm,X22: list_fm,Y21: fm,Y22: list_fm] :
( ( ( cons_fm @ X21 @ X22 )
= ( cons_fm @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_233_list_Oinject,axiom,
! [X21: tm,X22: list_tm,Y21: tm,Y22: list_tm] :
( ( ( cons_tm @ X21 @ X22 )
= ( cons_tm @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_234_fm_Oinject_I6_J,axiom,
! [X6: fm,Y6: fm] :
( ( ( uni @ X6 )
= ( uni @ Y6 ) )
= ( X6 = Y6 ) ) ).
% fm.inject(6)
thf(fact_235_fm_Oinject_I5_J,axiom,
! [X5: fm,Y5: fm] :
( ( ( exi @ X5 )
= ( exi @ Y5 ) )
= ( X5 = Y5 ) ) ).
% fm.inject(5)
thf(fact_236_insert__Nil,axiom,
! [X: rule] :
( ( insert_rule @ X @ nil_rule )
= ( cons_rule @ X @ nil_rule ) ) ).
% insert_Nil
thf(fact_237_insert__Nil,axiom,
! [X: list_fm] :
( ( insert_list_fm @ X @ nil_list_fm )
= ( cons_list_fm @ X @ nil_list_fm ) ) ).
% insert_Nil
thf(fact_238_insert__Nil,axiom,
! [X: fm] :
( ( insert_fm @ X @ nil_fm )
= ( cons_fm @ X @ nil_fm ) ) ).
% insert_Nil
thf(fact_239_insert__Nil,axiom,
! [X: tm] :
( ( insert_tm @ X @ nil_tm )
= ( cons_tm @ X @ nil_tm ) ) ).
% insert_Nil
thf(fact_240_not__in__set__insert,axiom,
! [X: produc340336539035504054m_rule,Xs: list_P2774625669004309958m_rule] :
( ~ ( member7231649785386036813m_rule @ X @ ( set_Pr4534715572506550497m_rule @ Xs ) )
=> ( ( insert8371175628248280474m_rule @ X @ Xs )
= ( cons_P2818572562524668032m_rule @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_241_not__in__set__insert,axiom,
! [X: nat,Xs: list_nat] :
( ~ ( member_nat3 @ X @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X @ Xs )
= ( cons_nat @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_242_not__in__set__insert,axiom,
! [X: rule,Xs: list_rule] :
( ~ ( member_rule3 @ X @ ( set_rule2 @ Xs ) )
=> ( ( insert_rule @ X @ Xs )
= ( cons_rule @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_243_not__in__set__insert,axiom,
! [X: list_fm,Xs: list_list_fm] :
( ~ ( member_list_fm3 @ X @ ( set_list_fm2 @ Xs ) )
=> ( ( insert_list_fm @ X @ Xs )
= ( cons_list_fm @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_244_not__in__set__insert,axiom,
! [X: fm,Xs: list_fm] :
( ~ ( member_fm3 @ X @ ( set_fm2 @ Xs ) )
=> ( ( insert_fm @ X @ Xs )
= ( cons_fm @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_245_not__in__set__insert,axiom,
! [X: tm,Xs: list_tm] :
( ~ ( member_tm3 @ X @ ( set_tm2 @ Xs ) )
=> ( ( insert_tm @ X @ Xs )
= ( cons_tm @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_246_member__rec_I2_J,axiom,
! [Y: rule] :
~ ( member_rule @ nil_rule @ Y ) ).
% member_rec(2)
thf(fact_247_member__rec_I2_J,axiom,
! [Y: fm] :
~ ( member_fm @ nil_fm @ Y ) ).
% member_rec(2)
thf(fact_248_member__rec_I2_J,axiom,
! [Y: list_fm] :
~ ( member_list_fm @ nil_list_fm @ Y ) ).
% member_rec(2)
thf(fact_249_member__rec_I2_J,axiom,
! [Y: tm] :
~ ( member_tm @ nil_tm @ Y ) ).
% member_rec(2)
thf(fact_250_member__rec_I1_J,axiom,
! [X: rule,Xs: list_rule,Y: rule] :
( ( member_rule @ ( cons_rule @ X @ Xs ) @ Y )
= ( ( X = Y )
| ( member_rule @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_251_member__rec_I1_J,axiom,
! [X: list_fm,Xs: list_list_fm,Y: list_fm] :
( ( member_list_fm @ ( cons_list_fm @ X @ Xs ) @ Y )
= ( ( X = Y )
| ( member_list_fm @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_252_member__rec_I1_J,axiom,
! [X: fm,Xs: list_fm,Y: fm] :
( ( member_fm @ ( cons_fm @ X @ Xs ) @ Y )
= ( ( X = Y )
| ( member_fm @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_253_member__rec_I1_J,axiom,
! [X: tm,Xs: list_tm,Y: tm] :
( ( member_tm @ ( cons_tm @ X @ Xs ) @ Y )
= ( ( X = Y )
| ( member_tm @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_254_list_Odistinct_I1_J,axiom,
! [X21: rule,X22: list_rule] :
( nil_rule
!= ( cons_rule @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_255_list_Odistinct_I1_J,axiom,
! [X21: list_fm,X22: list_list_fm] :
( nil_list_fm
!= ( cons_list_fm @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_256_list_Odistinct_I1_J,axiom,
! [X21: fm,X22: list_fm] :
( nil_fm
!= ( cons_fm @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_257_list_Odistinct_I1_J,axiom,
! [X21: tm,X22: list_tm] :
( nil_tm
!= ( cons_tm @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_258_list_OdiscI,axiom,
! [List: list_rule,X21: rule,X22: list_rule] :
( ( List
= ( cons_rule @ X21 @ X22 ) )
=> ( List != nil_rule ) ) ).
% list.discI
thf(fact_259_list_OdiscI,axiom,
! [List: list_list_fm,X21: list_fm,X22: list_list_fm] :
( ( List
= ( cons_list_fm @ X21 @ X22 ) )
=> ( List != nil_list_fm ) ) ).
% list.discI
thf(fact_260_list_OdiscI,axiom,
! [List: list_fm,X21: fm,X22: list_fm] :
( ( List
= ( cons_fm @ X21 @ X22 ) )
=> ( List != nil_fm ) ) ).
% list.discI
thf(fact_261_list_OdiscI,axiom,
! [List: list_tm,X21: tm,X22: list_tm] :
( ( List
= ( cons_tm @ X21 @ X22 ) )
=> ( List != nil_tm ) ) ).
% list.discI
thf(fact_262_list_Oexhaust,axiom,
! [Y: list_rule] :
( ( Y != nil_rule )
=> ~ ! [X212: rule,X222: list_rule] :
( Y
!= ( cons_rule @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_263_list_Oexhaust,axiom,
! [Y: list_list_fm] :
( ( Y != nil_list_fm )
=> ~ ! [X212: list_fm,X222: list_list_fm] :
( Y
!= ( cons_list_fm @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_264_list_Oexhaust,axiom,
! [Y: list_fm] :
( ( Y != nil_fm )
=> ~ ! [X212: fm,X222: list_fm] :
( Y
!= ( cons_fm @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_265_list_Oexhaust,axiom,
! [Y: list_tm] :
( ( Y != nil_tm )
=> ~ ! [X212: tm,X222: list_tm] :
( Y
!= ( cons_tm @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_266_transpose_Ocases,axiom,
! [X: list_list_rule] :
( ( X != nil_list_rule )
=> ( ! [Xss: list_list_rule] :
( X
!= ( cons_list_rule @ nil_rule @ Xss ) )
=> ~ ! [X3: rule,Xs2: list_rule,Xss: list_list_rule] :
( X
!= ( cons_list_rule @ ( cons_rule @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_267_transpose_Ocases,axiom,
! [X: list_list_list_fm] :
( ( X != nil_list_list_fm )
=> ( ! [Xss: list_list_list_fm] :
( X
!= ( cons_list_list_fm @ nil_list_fm @ Xss ) )
=> ~ ! [X3: list_fm,Xs2: list_list_fm,Xss: list_list_list_fm] :
( X
!= ( cons_list_list_fm @ ( cons_list_fm @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_268_transpose_Ocases,axiom,
! [X: list_list_tm] :
( ( X != nil_list_tm )
=> ( ! [Xss: list_list_tm] :
( X
!= ( cons_list_tm @ nil_tm @ Xss ) )
=> ~ ! [X3: tm,Xs2: list_tm,Xss: list_list_tm] :
( X
!= ( cons_list_tm @ ( cons_tm @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_269_transpose_Ocases,axiom,
! [X: list_list_fm] :
( ( X != nil_list_fm )
=> ( ! [Xss: list_list_fm] :
( X
!= ( cons_list_fm @ nil_fm @ Xss ) )
=> ~ ! [X3: fm,Xs2: list_fm,Xss: list_list_fm] :
( X
!= ( cons_list_fm @ ( cons_fm @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_270_remdups__adj_Ocases,axiom,
! [X: list_rule] :
( ( X != nil_rule )
=> ( ! [X3: rule] :
( X
!= ( cons_rule @ X3 @ nil_rule ) )
=> ~ ! [X3: rule,Y3: rule,Xs2: list_rule] :
( X
!= ( cons_rule @ X3 @ ( cons_rule @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_271_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,Y3: list_fm,Xs2: list_list_fm] :
( X
!= ( cons_list_fm @ X3 @ ( cons_list_fm @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_272_remdups__adj_Ocases,axiom,
! [X: list_fm] :
( ( X != nil_fm )
=> ( ! [X3: fm] :
( X
!= ( cons_fm @ X3 @ nil_fm ) )
=> ~ ! [X3: fm,Y3: fm,Xs2: list_fm] :
( X
!= ( cons_fm @ X3 @ ( cons_fm @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_273_remdups__adj_Ocases,axiom,
! [X: list_tm] :
( ( X != nil_tm )
=> ( ! [X3: tm] :
( X
!= ( cons_tm @ X3 @ nil_tm ) )
=> ~ ! [X3: tm,Y3: tm,Xs2: list_tm] :
( X
!= ( cons_tm @ X3 @ ( cons_tm @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_274_neq__Nil__conv,axiom,
! [Xs: list_rule] :
( ( Xs != nil_rule )
= ( ? [Y4: rule,Ys2: list_rule] :
( Xs
= ( cons_rule @ Y4 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_275_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_276_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_277_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_278_list__induct2_H,axiom,
! [P3: list_rule > list_rule > $o,Xs: list_rule,Ys: list_rule] :
( ( P3 @ nil_rule @ nil_rule )
=> ( ! [X3: rule,Xs2: list_rule] : ( P3 @ ( cons_rule @ X3 @ Xs2 ) @ nil_rule )
=> ( ! [Y3: rule,Ys3: list_rule] : ( P3 @ nil_rule @ ( cons_rule @ Y3 @ Ys3 ) )
=> ( ! [X3: rule,Xs2: list_rule,Y3: rule,Ys3: list_rule] :
( ( P3 @ Xs2 @ Ys3 )
=> ( P3 @ ( cons_rule @ X3 @ Xs2 ) @ ( cons_rule @ Y3 @ Ys3 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_279_list__induct2_H,axiom,
! [P3: list_rule > list_fm > $o,Xs: list_rule,Ys: list_fm] :
( ( P3 @ nil_rule @ nil_fm )
=> ( ! [X3: rule,Xs2: list_rule] : ( P3 @ ( cons_rule @ X3 @ Xs2 ) @ nil_fm )
=> ( ! [Y3: fm,Ys3: list_fm] : ( P3 @ nil_rule @ ( cons_fm @ Y3 @ Ys3 ) )
=> ( ! [X3: rule,Xs2: list_rule,Y3: fm,Ys3: list_fm] :
( ( P3 @ Xs2 @ Ys3 )
=> ( P3 @ ( cons_rule @ X3 @ Xs2 ) @ ( cons_fm @ Y3 @ Ys3 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_280_list__induct2_H,axiom,
! [P3: list_rule > list_tm > $o,Xs: list_rule,Ys: list_tm] :
( ( P3 @ nil_rule @ nil_tm )
=> ( ! [X3: rule,Xs2: list_rule] : ( P3 @ ( cons_rule @ X3 @ Xs2 ) @ nil_tm )
=> ( ! [Y3: tm,Ys3: list_tm] : ( P3 @ nil_rule @ ( cons_tm @ Y3 @ Ys3 ) )
=> ( ! [X3: rule,Xs2: list_rule,Y3: tm,Ys3: list_tm] :
( ( P3 @ Xs2 @ Ys3 )
=> ( P3 @ ( cons_rule @ X3 @ Xs2 ) @ ( cons_tm @ Y3 @ Ys3 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_281_list__induct2_H,axiom,
! [P3: list_fm > list_rule > $o,Xs: list_fm,Ys: list_rule] :
( ( P3 @ nil_fm @ nil_rule )
=> ( ! [X3: fm,Xs2: list_fm] : ( P3 @ ( cons_fm @ X3 @ Xs2 ) @ nil_rule )
=> ( ! [Y3: rule,Ys3: list_rule] : ( P3 @ nil_fm @ ( cons_rule @ Y3 @ Ys3 ) )
=> ( ! [X3: fm,Xs2: list_fm,Y3: rule,Ys3: list_rule] :
( ( P3 @ Xs2 @ Ys3 )
=> ( P3 @ ( cons_fm @ X3 @ Xs2 ) @ ( cons_rule @ Y3 @ Ys3 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_282_list__induct2_H,axiom,
! [P3: list_fm > list_fm > $o,Xs: list_fm,Ys: list_fm] :
( ( P3 @ nil_fm @ nil_fm )
=> ( ! [X3: fm,Xs2: list_fm] : ( P3 @ ( cons_fm @ X3 @ Xs2 ) @ nil_fm )
=> ( ! [Y3: fm,Ys3: list_fm] : ( P3 @ nil_fm @ ( cons_fm @ Y3 @ Ys3 ) )
=> ( ! [X3: fm,Xs2: list_fm,Y3: fm,Ys3: list_fm] :
( ( P3 @ Xs2 @ Ys3 )
=> ( P3 @ ( cons_fm @ X3 @ Xs2 ) @ ( cons_fm @ Y3 @ Ys3 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_283_list__induct2_H,axiom,
! [P3: list_fm > list_tm > $o,Xs: list_fm,Ys: list_tm] :
( ( P3 @ nil_fm @ nil_tm )
=> ( ! [X3: fm,Xs2: list_fm] : ( P3 @ ( cons_fm @ X3 @ Xs2 ) @ nil_tm )
=> ( ! [Y3: tm,Ys3: list_tm] : ( P3 @ nil_fm @ ( cons_tm @ Y3 @ Ys3 ) )
=> ( ! [X3: fm,Xs2: list_fm,Y3: tm,Ys3: list_tm] :
( ( P3 @ Xs2 @ Ys3 )
=> ( P3 @ ( cons_fm @ X3 @ Xs2 ) @ ( cons_tm @ Y3 @ Ys3 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_284_list__induct2_H,axiom,
! [P3: list_tm > list_rule > $o,Xs: list_tm,Ys: list_rule] :
( ( P3 @ nil_tm @ nil_rule )
=> ( ! [X3: tm,Xs2: list_tm] : ( P3 @ ( cons_tm @ X3 @ Xs2 ) @ nil_rule )
=> ( ! [Y3: rule,Ys3: list_rule] : ( P3 @ nil_tm @ ( cons_rule @ Y3 @ Ys3 ) )
=> ( ! [X3: tm,Xs2: list_tm,Y3: rule,Ys3: list_rule] :
( ( P3 @ Xs2 @ Ys3 )
=> ( P3 @ ( cons_tm @ X3 @ Xs2 ) @ ( cons_rule @ Y3 @ Ys3 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_285_list__induct2_H,axiom,
! [P3: list_tm > list_fm > $o,Xs: list_tm,Ys: list_fm] :
( ( P3 @ nil_tm @ nil_fm )
=> ( ! [X3: tm,Xs2: list_tm] : ( P3 @ ( cons_tm @ X3 @ Xs2 ) @ nil_fm )
=> ( ! [Y3: fm,Ys3: list_fm] : ( P3 @ nil_tm @ ( cons_fm @ Y3 @ Ys3 ) )
=> ( ! [X3: tm,Xs2: list_tm,Y3: fm,Ys3: list_fm] :
( ( P3 @ Xs2 @ Ys3 )
=> ( P3 @ ( cons_tm @ X3 @ Xs2 ) @ ( cons_fm @ Y3 @ Ys3 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_286_list__induct2_H,axiom,
! [P3: list_tm > list_tm > $o,Xs: list_tm,Ys: list_tm] :
( ( P3 @ nil_tm @ nil_tm )
=> ( ! [X3: tm,Xs2: list_tm] : ( P3 @ ( cons_tm @ X3 @ Xs2 ) @ nil_tm )
=> ( ! [Y3: tm,Ys3: list_tm] : ( P3 @ nil_tm @ ( cons_tm @ Y3 @ Ys3 ) )
=> ( ! [X3: tm,Xs2: list_tm,Y3: tm,Ys3: list_tm] :
( ( P3 @ Xs2 @ Ys3 )
=> ( P3 @ ( cons_tm @ X3 @ Xs2 ) @ ( cons_tm @ Y3 @ Ys3 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_287_list__induct2_H,axiom,
! [P3: list_rule > list_list_fm > $o,Xs: list_rule,Ys: list_list_fm] :
( ( P3 @ nil_rule @ nil_list_fm )
=> ( ! [X3: rule,Xs2: list_rule] : ( P3 @ ( cons_rule @ X3 @ Xs2 ) @ nil_list_fm )
=> ( ! [Y3: list_fm,Ys3: list_list_fm] : ( P3 @ nil_rule @ ( cons_list_fm @ Y3 @ Ys3 ) )
=> ( ! [X3: rule,Xs2: list_rule,Y3: list_fm,Ys3: list_list_fm] :
( ( P3 @ Xs2 @ Ys3 )
=> ( P3 @ ( cons_rule @ X3 @ Xs2 ) @ ( cons_list_fm @ Y3 @ Ys3 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_288_not__Cons__self2,axiom,
! [X: rule,Xs: list_rule] :
( ( cons_rule @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_289_not__Cons__self2,axiom,
! [X: list_fm,Xs: list_list_fm] :
( ( cons_list_fm @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_290_not__Cons__self2,axiom,
! [X: fm,Xs: list_fm] :
( ( cons_fm @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_291_not__Cons__self2,axiom,
! [X: tm,Xs: list_tm] :
( ( cons_tm @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_292_list__nonempty__induct,axiom,
! [Xs: list_rule,P3: list_rule > $o] :
( ( Xs != nil_rule )
=> ( ! [X3: rule] : ( P3 @ ( cons_rule @ X3 @ nil_rule ) )
=> ( ! [X3: rule,Xs2: list_rule] :
( ( Xs2 != nil_rule )
=> ( ( P3 @ Xs2 )
=> ( P3 @ ( cons_rule @ X3 @ Xs2 ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_293_list__nonempty__induct,axiom,
! [Xs: list_list_fm,P3: list_list_fm > $o] :
( ( Xs != nil_list_fm )
=> ( ! [X3: list_fm] : ( P3 @ ( cons_list_fm @ X3 @ nil_list_fm ) )
=> ( ! [X3: list_fm,Xs2: list_list_fm] :
( ( Xs2 != nil_list_fm )
=> ( ( P3 @ Xs2 )
=> ( P3 @ ( cons_list_fm @ X3 @ Xs2 ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_294_list__nonempty__induct,axiom,
! [Xs: list_fm,P3: list_fm > $o] :
( ( Xs != nil_fm )
=> ( ! [X3: fm] : ( P3 @ ( cons_fm @ X3 @ nil_fm ) )
=> ( ! [X3: fm,Xs2: list_fm] :
( ( Xs2 != nil_fm )
=> ( ( P3 @ Xs2 )
=> ( P3 @ ( cons_fm @ X3 @ Xs2 ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_295_list__nonempty__induct,axiom,
! [Xs: list_tm,P3: list_tm > $o] :
( ( Xs != nil_tm )
=> ( ! [X3: tm] : ( P3 @ ( cons_tm @ X3 @ nil_tm ) )
=> ( ! [X3: tm,Xs2: list_tm] :
( ( Xs2 != nil_tm )
=> ( ( P3 @ Xs2 )
=> ( P3 @ ( cons_tm @ X3 @ Xs2 ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_296_map__tailrec__rev_Ocases,axiom,
! [X: produc3792783349651969271ist_fm] :
( ! [F2: tm > fm,Bs: list_fm] :
( X
!= ( produc4025633523336838569ist_fm @ F2 @ ( produc1414352766439514085ist_fm @ nil_tm @ Bs ) ) )
=> ~ ! [F2: tm > fm,A4: tm,As: list_tm,Bs: list_fm] :
( X
!= ( produc4025633523336838569ist_fm @ F2 @ ( produc1414352766439514085ist_fm @ ( cons_tm @ A4 @ As ) @ Bs ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_297_successively_Ocases,axiom,
! [X: produc3824871039275828531t_rule] :
( ! [P6: rule > rule > $o] :
( X
!= ( produc9173496138740005677t_rule @ P6 @ nil_rule ) )
=> ( ! [P6: rule > rule > $o,X3: rule] :
( X
!= ( produc9173496138740005677t_rule @ P6 @ ( cons_rule @ X3 @ nil_rule ) ) )
=> ~ ! [P6: rule > rule > $o,X3: rule,Y3: rule,Xs2: list_rule] :
( X
!= ( produc9173496138740005677t_rule @ P6 @ ( cons_rule @ X3 @ ( cons_rule @ Y3 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_298_successively_Ocases,axiom,
! [X: produc6864569604571974008ist_fm] :
( ! [P6: list_fm > list_fm > $o] :
( X
!= ( produc3008759835361761000ist_fm @ P6 @ nil_list_fm ) )
=> ( ! [P6: list_fm > list_fm > $o,X3: list_fm] :
( X
!= ( produc3008759835361761000ist_fm @ P6 @ ( cons_list_fm @ X3 @ nil_list_fm ) ) )
=> ~ ! [P6: list_fm > list_fm > $o,X3: list_fm,Y3: list_fm,Xs2: list_list_fm] :
( X
!= ( produc3008759835361761000ist_fm @ P6 @ ( cons_list_fm @ X3 @ ( cons_list_fm @ Y3 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_299_successively_Ocases,axiom,
! [X: produc7963324949210141170ist_fm] :
( ! [P6: fm > fm > $o] :
( X
!= ( produc7687578365188660450ist_fm @ P6 @ nil_fm ) )
=> ( ! [P6: fm > fm > $o,X3: fm] :
( X
!= ( produc7687578365188660450ist_fm @ P6 @ ( cons_fm @ X3 @ nil_fm ) ) )
=> ~ ! [P6: fm > fm > $o,X3: fm,Y3: fm,Xs2: list_fm] :
( X
!= ( produc7687578365188660450ist_fm @ P6 @ ( cons_fm @ X3 @ ( cons_fm @ Y3 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_300_successively_Ocases,axiom,
! [X: produc2002131169352006116ist_tm] :
( ! [P6: tm > tm > $o] :
( X
!= ( produc1972851280741670356ist_tm @ P6 @ nil_tm ) )
=> ( ! [P6: tm > tm > $o,X3: tm] :
( X
!= ( produc1972851280741670356ist_tm @ P6 @ ( cons_tm @ X3 @ nil_tm ) ) )
=> ~ ! [P6: tm > tm > $o,X3: tm,Y3: tm,Xs2: list_tm] :
( X
!= ( produc1972851280741670356ist_tm @ P6 @ ( cons_tm @ X3 @ ( cons_tm @ Y3 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_301_sorted__wrt_Ocases,axiom,
! [X: produc3824871039275828531t_rule] :
( ! [P6: rule > rule > $o] :
( X
!= ( produc9173496138740005677t_rule @ P6 @ nil_rule ) )
=> ~ ! [P6: rule > rule > $o,X3: rule,Ys3: list_rule] :
( X
!= ( produc9173496138740005677t_rule @ P6 @ ( cons_rule @ X3 @ Ys3 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_302_sorted__wrt_Ocases,axiom,
! [X: produc6864569604571974008ist_fm] :
( ! [P6: list_fm > list_fm > $o] :
( X
!= ( produc3008759835361761000ist_fm @ P6 @ nil_list_fm ) )
=> ~ ! [P6: list_fm > list_fm > $o,X3: list_fm,Ys3: list_list_fm] :
( X
!= ( produc3008759835361761000ist_fm @ P6 @ ( cons_list_fm @ X3 @ Ys3 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_303_sorted__wrt_Ocases,axiom,
! [X: produc7963324949210141170ist_fm] :
( ! [P6: fm > fm > $o] :
( X
!= ( produc7687578365188660450ist_fm @ P6 @ nil_fm ) )
=> ~ ! [P6: fm > fm > $o,X3: fm,Ys3: list_fm] :
( X
!= ( produc7687578365188660450ist_fm @ P6 @ ( cons_fm @ X3 @ Ys3 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_304_sorted__wrt_Ocases,axiom,
! [X: produc2002131169352006116ist_tm] :
( ! [P6: tm > tm > $o] :
( X
!= ( produc1972851280741670356ist_tm @ P6 @ nil_tm ) )
=> ~ ! [P6: tm > tm > $o,X3: tm,Ys3: list_tm] :
( X
!= ( produc1972851280741670356ist_tm @ P6 @ ( cons_tm @ X3 @ Ys3 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_305_shuffles_Ocases,axiom,
! [X: produc1530960993530998309t_rule] :
( ! [Ys3: list_rule] :
( X
!= ( produc8979233614864344413t_rule @ nil_rule @ Ys3 ) )
=> ( ! [Xs2: list_rule] :
( X
!= ( produc8979233614864344413t_rule @ Xs2 @ nil_rule ) )
=> ~ ! [X3: rule,Xs2: list_rule,Y3: rule,Ys3: list_rule] :
( X
!= ( produc8979233614864344413t_rule @ ( cons_rule @ X3 @ Xs2 ) @ ( cons_rule @ Y3 @ Ys3 ) ) ) ) ) ).
% shuffles.cases
thf(fact_306_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,Y3: list_fm,Ys3: list_list_fm] :
( X
!= ( produc8321651870839017815ist_fm @ ( cons_list_fm @ X3 @ Xs2 ) @ ( cons_list_fm @ Y3 @ Ys3 ) ) ) ) ) ).
% shuffles.cases
thf(fact_307_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,Y3: fm,Ys3: list_fm] :
( X
!= ( produc7863996417982153943ist_fm @ ( cons_fm @ X3 @ Xs2 ) @ ( cons_fm @ Y3 @ Ys3 ) ) ) ) ) ).
% shuffles.cases
thf(fact_308_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,Y3: tm,Ys3: list_tm] :
( X
!= ( produc1418304791525149271ist_tm @ ( cons_tm @ X3 @ Xs2 ) @ ( cons_tm @ Y3 @ Ys3 ) ) ) ) ) ).
% shuffles.cases
thf(fact_309_splice_Ocases,axiom,
! [X: produc1530960993530998309t_rule] :
( ! [Ys3: list_rule] :
( X
!= ( produc8979233614864344413t_rule @ nil_rule @ Ys3 ) )
=> ~ ! [X3: rule,Xs2: list_rule,Ys3: list_rule] :
( X
!= ( produc8979233614864344413t_rule @ ( cons_rule @ X3 @ Xs2 ) @ Ys3 ) ) ) ).
% splice.cases
thf(fact_310_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_311_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_312_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_313_partition_Osimps_I1_J,axiom,
! [P3: rule > $o] :
( ( partition_rule @ P3 @ nil_rule )
= ( produc8979233614864344413t_rule @ nil_rule @ nil_rule ) ) ).
% partition.simps(1)
thf(fact_314_partition_Osimps_I1_J,axiom,
! [P3: fm > $o] :
( ( partition_fm @ P3 @ nil_fm )
= ( produc7863996417982153943ist_fm @ nil_fm @ nil_fm ) ) ).
% partition.simps(1)
thf(fact_315_partition_Osimps_I1_J,axiom,
! [P3: list_fm > $o] :
( ( partition_list_fm @ P3 @ nil_list_fm )
= ( produc8321651870839017815ist_fm @ nil_list_fm @ nil_list_fm ) ) ).
% partition.simps(1)
thf(fact_316_partition_Osimps_I1_J,axiom,
! [P3: tm > $o] :
( ( partition_tm @ P3 @ nil_tm )
= ( produc1418304791525149271ist_tm @ nil_tm @ nil_tm ) ) ).
% partition.simps(1)
thf(fact_317_children_Osimps_I1_J,axiom,
! [Uu: list_tm,Uv: rule] :
( ( children @ Uu @ Uv @ nil_fm )
= ( cons_list_fm @ nil_fm @ nil_list_fm ) ) ).
% children.simps(1)
thf(fact_318_fm_Odistinct_I37_J,axiom,
! [X5: fm,X6: fm] :
( ( exi @ X5 )
!= ( uni @ X6 ) ) ).
% fm.distinct(37)
thf(fact_319_List_Oinsert__def,axiom,
( insert8371175628248280474m_rule
= ( ^ [X2: produc340336539035504054m_rule,Xs3: list_P2774625669004309958m_rule] : ( if_lis6095641862558746240m_rule @ ( member7231649785386036813m_rule @ X2 @ ( set_Pr4534715572506550497m_rule @ Xs3 ) ) @ Xs3 @ ( cons_P2818572562524668032m_rule @ X2 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_320_List_Oinsert__def,axiom,
( insert_nat
= ( ^ [X2: nat,Xs3: list_nat] : ( if_list_nat @ ( member_nat3 @ X2 @ ( set_nat2 @ Xs3 ) ) @ Xs3 @ ( cons_nat @ X2 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_321_List_Oinsert__def,axiom,
( insert_rule
= ( ^ [X2: rule,Xs3: list_rule] : ( if_list_rule @ ( member_rule3 @ X2 @ ( set_rule2 @ Xs3 ) ) @ Xs3 @ ( cons_rule @ X2 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_322_List_Oinsert__def,axiom,
( insert_list_fm
= ( ^ [X2: list_fm,Xs3: list_list_fm] : ( if_list_list_fm @ ( member_list_fm3 @ X2 @ ( set_list_fm2 @ Xs3 ) ) @ Xs3 @ ( cons_list_fm @ X2 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_323_List_Oinsert__def,axiom,
( insert_fm
= ( ^ [X2: fm,Xs3: list_fm] : ( if_list_fm @ ( member_fm3 @ X2 @ ( set_fm2 @ Xs3 ) ) @ Xs3 @ ( cons_fm @ X2 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_324_List_Oinsert__def,axiom,
( insert_tm
= ( ^ [X2: tm,Xs3: list_tm] : ( if_list_tm @ ( member_tm3 @ X2 @ ( set_tm2 @ Xs3 ) ) @ Xs3 @ ( cons_tm @ X2 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_325_set__ConsD,axiom,
! [Y: produc340336539035504054m_rule,X: produc340336539035504054m_rule,Xs: list_P2774625669004309958m_rule] :
( ( member7231649785386036813m_rule @ Y @ ( set_Pr4534715572506550497m_rule @ ( cons_P2818572562524668032m_rule @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member7231649785386036813m_rule @ Y @ ( set_Pr4534715572506550497m_rule @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_326_set__ConsD,axiom,
! [Y: nat,X: nat,Xs: list_nat] :
( ( member_nat3 @ Y @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_nat3 @ Y @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_327_set__ConsD,axiom,
! [Y: rule,X: rule,Xs: list_rule] :
( ( member_rule3 @ Y @ ( set_rule2 @ ( cons_rule @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_rule3 @ Y @ ( set_rule2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_328_set__ConsD,axiom,
! [Y: list_fm,X: list_fm,Xs: list_list_fm] :
( ( member_list_fm3 @ Y @ ( set_list_fm2 @ ( cons_list_fm @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_list_fm3 @ Y @ ( set_list_fm2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_329_set__ConsD,axiom,
! [Y: fm,X: fm,Xs: list_fm] :
( ( member_fm3 @ Y @ ( set_fm2 @ ( cons_fm @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_fm3 @ Y @ ( set_fm2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_330_set__ConsD,axiom,
! [Y: tm,X: tm,Xs: list_tm] :
( ( member_tm3 @ Y @ ( set_tm2 @ ( cons_tm @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_tm3 @ Y @ ( set_tm2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_331_list_Oset__cases,axiom,
! [E: produc340336539035504054m_rule,A: list_P2774625669004309958m_rule] :
( ( member7231649785386036813m_rule @ E @ ( set_Pr4534715572506550497m_rule @ A ) )
=> ( ! [Z22: list_P2774625669004309958m_rule] :
( A
!= ( cons_P2818572562524668032m_rule @ E @ Z22 ) )
=> ~ ! [Z1: produc340336539035504054m_rule,Z22: list_P2774625669004309958m_rule] :
( ( A
= ( cons_P2818572562524668032m_rule @ Z1 @ Z22 ) )
=> ~ ( member7231649785386036813m_rule @ E @ ( set_Pr4534715572506550497m_rule @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_332_list_Oset__cases,axiom,
! [E: nat,A: list_nat] :
( ( member_nat3 @ E @ ( set_nat2 @ A ) )
=> ( ! [Z22: list_nat] :
( A
!= ( cons_nat @ E @ Z22 ) )
=> ~ ! [Z1: nat,Z22: list_nat] :
( ( A
= ( cons_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat3 @ E @ ( set_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_333_list_Oset__cases,axiom,
! [E: rule,A: list_rule] :
( ( member_rule3 @ E @ ( set_rule2 @ A ) )
=> ( ! [Z22: list_rule] :
( A
!= ( cons_rule @ E @ Z22 ) )
=> ~ ! [Z1: rule,Z22: list_rule] :
( ( A
= ( cons_rule @ Z1 @ Z22 ) )
=> ~ ( member_rule3 @ E @ ( set_rule2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_334_list_Oset__cases,axiom,
! [E: list_fm,A: list_list_fm] :
( ( member_list_fm3 @ 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_fm3 @ E @ ( set_list_fm2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_335_list_Oset__cases,axiom,
! [E: fm,A: list_fm] :
( ( member_fm3 @ E @ ( set_fm2 @ A ) )
=> ( ! [Z22: list_fm] :
( A
!= ( cons_fm @ E @ Z22 ) )
=> ~ ! [Z1: fm,Z22: list_fm] :
( ( A
= ( cons_fm @ Z1 @ Z22 ) )
=> ~ ( member_fm3 @ E @ ( set_fm2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_336_list_Oset__cases,axiom,
! [E: tm,A: list_tm] :
( ( member_tm3 @ E @ ( set_tm2 @ A ) )
=> ( ! [Z22: list_tm] :
( A
!= ( cons_tm @ E @ Z22 ) )
=> ~ ! [Z1: tm,Z22: list_tm] :
( ( A
= ( cons_tm @ Z1 @ Z22 ) )
=> ~ ( member_tm3 @ E @ ( set_tm2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_337_list_Oset__intros_I1_J,axiom,
! [X21: produc340336539035504054m_rule,X22: list_P2774625669004309958m_rule] : ( member7231649785386036813m_rule @ X21 @ ( set_Pr4534715572506550497m_rule @ ( cons_P2818572562524668032m_rule @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_338_list_Oset__intros_I1_J,axiom,
! [X21: nat,X22: list_nat] : ( member_nat3 @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_339_list_Oset__intros_I1_J,axiom,
! [X21: rule,X22: list_rule] : ( member_rule3 @ X21 @ ( set_rule2 @ ( cons_rule @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_340_list_Oset__intros_I1_J,axiom,
! [X21: list_fm,X22: list_list_fm] : ( member_list_fm3 @ X21 @ ( set_list_fm2 @ ( cons_list_fm @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_341_list_Oset__intros_I1_J,axiom,
! [X21: fm,X22: list_fm] : ( member_fm3 @ X21 @ ( set_fm2 @ ( cons_fm @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_342_list_Oset__intros_I1_J,axiom,
! [X21: tm,X22: list_tm] : ( member_tm3 @ X21 @ ( set_tm2 @ ( cons_tm @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_343_list_Oset__intros_I2_J,axiom,
! [Y: produc340336539035504054m_rule,X22: list_P2774625669004309958m_rule,X21: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ Y @ ( set_Pr4534715572506550497m_rule @ X22 ) )
=> ( member7231649785386036813m_rule @ Y @ ( set_Pr4534715572506550497m_rule @ ( cons_P2818572562524668032m_rule @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_344_list_Oset__intros_I2_J,axiom,
! [Y: nat,X22: list_nat,X21: nat] :
( ( member_nat3 @ Y @ ( set_nat2 @ X22 ) )
=> ( member_nat3 @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_345_list_Oset__intros_I2_J,axiom,
! [Y: rule,X22: list_rule,X21: rule] :
( ( member_rule3 @ Y @ ( set_rule2 @ X22 ) )
=> ( member_rule3 @ Y @ ( set_rule2 @ ( cons_rule @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_346_list_Oset__intros_I2_J,axiom,
! [Y: list_fm,X22: list_list_fm,X21: list_fm] :
( ( member_list_fm3 @ Y @ ( set_list_fm2 @ X22 ) )
=> ( member_list_fm3 @ Y @ ( set_list_fm2 @ ( cons_list_fm @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_347_list_Oset__intros_I2_J,axiom,
! [Y: fm,X22: list_fm,X21: fm] :
( ( member_fm3 @ Y @ ( set_fm2 @ X22 ) )
=> ( member_fm3 @ Y @ ( set_fm2 @ ( cons_fm @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_348_list_Oset__intros_I2_J,axiom,
! [Y: tm,X22: list_tm,X21: tm] :
( ( member_tm3 @ Y @ ( set_tm2 @ X22 ) )
=> ( member_tm3 @ Y @ ( set_tm2 @ ( cons_tm @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_349_fm_Odistinct_I41_J,axiom,
! [X6: fm,X7: fm] :
( ( uni @ X6 )
!= ( neg @ X7 ) ) ).
% fm.distinct(41)
thf(fact_350_fm_Odistinct_I39_J,axiom,
! [X5: fm,X7: fm] :
( ( exi @ X5 )
!= ( neg @ X7 ) ) ).
% fm.distinct(39)
thf(fact_351_fm_Odistinct_I33_J,axiom,
! [X41: fm,X42: fm,X6: fm] :
( ( con @ X41 @ X42 )
!= ( uni @ X6 ) ) ).
% fm.distinct(33)
thf(fact_352_fm_Odistinct_I31_J,axiom,
! [X41: fm,X42: fm,X5: fm] :
( ( con @ X41 @ X42 )
!= ( exi @ X5 ) ) ).
% fm.distinct(31)
thf(fact_353_fm_Odistinct_I27_J,axiom,
! [X31: fm,X32: fm,X6: fm] :
( ( dis @ X31 @ X32 )
!= ( uni @ X6 ) ) ).
% fm.distinct(27)
thf(fact_354_fm_Odistinct_I25_J,axiom,
! [X31: fm,X32: fm,X5: fm] :
( ( dis @ X31 @ X32 )
!= ( exi @ X5 ) ) ).
% fm.distinct(25)
thf(fact_355_fm_Odistinct_I19_J,axiom,
! [X21: fm,X22: fm,X6: fm] :
( ( imp @ X21 @ X22 )
!= ( uni @ X6 ) ) ).
% fm.distinct(19)
thf(fact_356_fm_Odistinct_I17_J,axiom,
! [X21: fm,X22: fm,X5: fm] :
( ( imp @ X21 @ X22 )
!= ( exi @ X5 ) ) ).
% fm.distinct(17)
thf(fact_357_fm_Odistinct_I9_J,axiom,
! [X11: nat,X12: list_tm,X6: fm] :
( ( pre @ X11 @ X12 )
!= ( uni @ X6 ) ) ).
% fm.distinct(9)
thf(fact_358_fm_Odistinct_I7_J,axiom,
! [X11: nat,X12: list_tm,X5: fm] :
( ( pre @ X11 @ X12 )
!= ( exi @ X5 ) ) ).
% fm.distinct(7)
thf(fact_359_branchDone_Ocases,axiom,
! [X: list_fm] :
( ( X != nil_fm )
=> ( ! [P4: fm,Z4: list_fm] :
( X
!= ( cons_fm @ ( neg @ P4 ) @ Z4 ) )
=> ( ! [V: nat,Va: list_tm,Z4: list_fm] :
( X
!= ( cons_fm @ ( pre @ V @ Va ) @ Z4 ) )
=> ( ! [V: fm,Va: fm,Z4: list_fm] :
( X
!= ( cons_fm @ ( imp @ V @ Va ) @ Z4 ) )
=> ( ! [V: fm,Va: fm,Z4: list_fm] :
( X
!= ( cons_fm @ ( dis @ V @ Va ) @ Z4 ) )
=> ( ! [V: fm,Va: fm,Z4: list_fm] :
( X
!= ( cons_fm @ ( con @ V @ Va ) @ Z4 ) )
=> ( ! [V: fm,Z4: list_fm] :
( X
!= ( cons_fm @ ( exi @ V ) @ Z4 ) )
=> ~ ! [V: fm,Z4: list_fm] :
( X
!= ( cons_fm @ ( uni @ V ) @ Z4 ) ) ) ) ) ) ) ) ) ).
% branchDone.cases
thf(fact_360_shift_Osimps_I1_J,axiom,
! [S: stream2709947120125613254m_rule] :
( ( shift_2334844276748245581m_rule @ nil_Pr2808214839852828720m_rule @ S )
= S ) ).
% shift.simps(1)
thf(fact_361_shift_Osimps_I1_J,axiom,
! [S: stream_rule] :
( ( shift_rule @ nil_rule @ S )
= S ) ).
% shift.simps(1)
thf(fact_362_shift_Osimps_I1_J,axiom,
! [S: stream_fm] :
( ( shift_fm @ nil_fm @ S )
= S ) ).
% shift.simps(1)
thf(fact_363_shift_Osimps_I1_J,axiom,
! [S: stream_list_fm] :
( ( shift_list_fm @ nil_list_fm @ S )
= S ) ).
% shift.simps(1)
thf(fact_364_shift_Osimps_I1_J,axiom,
! [S: stream_tm] :
( ( shift_tm @ nil_tm @ S )
= S ) ).
% shift.simps(1)
thf(fact_365_SeCaV_Omember_Osimps_I2_J,axiom,
! [P: rule,Q: rule,Z3: list_rule] :
( ( member_rule2 @ P @ ( cons_rule @ Q @ Z3 ) )
= ( ( P != Q )
=> ( member_rule2 @ P @ Z3 ) ) ) ).
% SeCaV.member.simps(2)
thf(fact_366_SeCaV_Omember_Osimps_I2_J,axiom,
! [P: list_fm,Q: list_fm,Z3: list_list_fm] :
( ( member_list_fm2 @ P @ ( cons_list_fm @ Q @ Z3 ) )
= ( ( P != Q )
=> ( member_list_fm2 @ P @ Z3 ) ) ) ).
% SeCaV.member.simps(2)
thf(fact_367_SeCaV_Omember_Osimps_I2_J,axiom,
! [P: fm,Q: fm,Z3: list_fm] :
( ( member_fm2 @ P @ ( cons_fm @ Q @ Z3 ) )
= ( ( P != Q )
=> ( member_fm2 @ P @ Z3 ) ) ) ).
% SeCaV.member.simps(2)
thf(fact_368_SeCaV_Omember_Osimps_I2_J,axiom,
! [P: tm,Q: tm,Z3: list_tm] :
( ( member_tm2 @ P @ ( cons_tm @ Q @ Z3 ) )
= ( ( P != Q )
=> ( member_tm2 @ P @ Z3 ) ) ) ).
% SeCaV.member.simps(2)
thf(fact_369_SeCaV_Omember_Osimps_I1_J,axiom,
! [P: rule] :
~ ( member_rule2 @ P @ nil_rule ) ).
% SeCaV.member.simps(1)
thf(fact_370_SeCaV_Omember_Osimps_I1_J,axiom,
! [P: fm] :
~ ( member_fm2 @ P @ nil_fm ) ).
% SeCaV.member.simps(1)
thf(fact_371_SeCaV_Omember_Osimps_I1_J,axiom,
! [P: list_fm] :
~ ( member_list_fm2 @ P @ nil_list_fm ) ).
% SeCaV.member.simps(1)
thf(fact_372_SeCaV_Omember_Osimps_I1_J,axiom,
! [P: tm] :
~ ( member_tm2 @ P @ nil_tm ) ).
% SeCaV.member.simps(1)
thf(fact_373_curryE,axiom,
! [F: produc6018962875968178549ist_fm > $o,A: list_tm,B2: list_fm] :
( ( produc5334687091915550783t_fm_o @ F @ A @ B2 )
=> ( F @ ( produc1414352766439514085ist_fm @ A @ B2 ) ) ) ).
% curryE
thf(fact_374_curryD,axiom,
! [F: produc6018962875968178549ist_fm > $o,A: list_tm,B2: list_fm] :
( ( produc5334687091915550783t_fm_o @ F @ A @ B2 )
=> ( F @ ( produc1414352766439514085ist_fm @ A @ B2 ) ) ) ).
% curryD
thf(fact_375_partition__P,axiom,
! [P3: fm > $o,Xs: list_fm,Yes: list_fm,No: list_fm] :
( ( ( partition_fm @ P3 @ Xs )
= ( produc7863996417982153943ist_fm @ Yes @ No ) )
=> ( ! [X4: fm] :
( ( member_fm3 @ X4 @ ( set_fm2 @ Yes ) )
=> ( P3 @ X4 ) )
& ! [X4: fm] :
( ( member_fm3 @ X4 @ ( set_fm2 @ No ) )
=> ~ ( P3 @ X4 ) ) ) ) ).
% partition_P
thf(fact_376_partition__P,axiom,
! [P3: list_fm > $o,Xs: list_list_fm,Yes: list_list_fm,No: list_list_fm] :
( ( ( partition_list_fm @ P3 @ Xs )
= ( produc8321651870839017815ist_fm @ Yes @ No ) )
=> ( ! [X4: list_fm] :
( ( member_list_fm3 @ X4 @ ( set_list_fm2 @ Yes ) )
=> ( P3 @ X4 ) )
& ! [X4: list_fm] :
( ( member_list_fm3 @ X4 @ ( set_list_fm2 @ No ) )
=> ~ ( P3 @ X4 ) ) ) ) ).
% partition_P
thf(fact_377_partition__P,axiom,
! [P3: tm > $o,Xs: list_tm,Yes: list_tm,No: list_tm] :
( ( ( partition_tm @ P3 @ Xs )
= ( produc1418304791525149271ist_tm @ Yes @ No ) )
=> ( ! [X4: tm] :
( ( member_tm3 @ X4 @ ( set_tm2 @ Yes ) )
=> ( P3 @ X4 ) )
& ! [X4: tm] :
( ( member_tm3 @ X4 @ ( set_tm2 @ No ) )
=> ~ ( P3 @ X4 ) ) ) ) ).
% partition_P
thf(fact_378_fm_Oexhaust,axiom,
! [Y: fm] :
( ! [X112: nat,X122: list_tm] :
( Y
!= ( pre @ X112 @ X122 ) )
=> ( ! [X212: fm,X222: fm] :
( Y
!= ( imp @ X212 @ X222 ) )
=> ( ! [X312: fm,X322: fm] :
( Y
!= ( dis @ X312 @ X322 ) )
=> ( ! [X412: fm,X422: fm] :
( Y
!= ( con @ X412 @ X422 ) )
=> ( ! [X52: fm] :
( Y
!= ( exi @ X52 ) )
=> ( ! [X62: fm] :
( Y
!= ( uni @ X62 ) )
=> ~ ! [X72: fm] :
( Y
!= ( neg @ X72 ) ) ) ) ) ) ) ) ).
% fm.exhaust
thf(fact_379_params_H_H_Ocases,axiom,
! [X: fm] :
( ! [B4: nat,Ts2: list_tm] :
( X
!= ( pre @ B4 @ Ts2 ) )
=> ( ! [P4: fm,Q2: fm] :
( X
!= ( imp @ P4 @ Q2 ) )
=> ( ! [P4: fm,Q2: fm] :
( X
!= ( dis @ P4 @ Q2 ) )
=> ( ! [P4: fm,Q2: fm] :
( X
!= ( con @ P4 @ Q2 ) )
=> ( ! [P4: fm] :
( X
!= ( exi @ P4 ) )
=> ( ! [P4: fm] :
( X
!= ( uni @ P4 ) )
=> ~ ! [P4: fm] :
( X
!= ( neg @ P4 ) ) ) ) ) ) ) ) ).
% params''.cases
thf(fact_380_Neg__exhaust,axiom,
! [X: fm] :
( ! [I: nat,Ts2: list_tm] :
( X
!= ( pre @ I @ Ts2 ) )
=> ( ! [P4: fm,Q2: fm] :
( X
!= ( imp @ P4 @ Q2 ) )
=> ( ! [P4: fm,Q2: fm] :
( X
!= ( dis @ P4 @ Q2 ) )
=> ( ! [P4: fm,Q2: fm] :
( X
!= ( con @ P4 @ Q2 ) )
=> ( ! [P4: fm] :
( X
!= ( exi @ P4 ) )
=> ( ! [P4: fm] :
( X
!= ( uni @ P4 ) )
=> ( ! [I: nat,Ts2: list_tm] :
( X
!= ( neg @ ( pre @ I @ Ts2 ) ) )
=> ( ! [P4: fm,Q2: fm] :
( X
!= ( neg @ ( imp @ P4 @ Q2 ) ) )
=> ( ! [P4: fm,Q2: fm] :
( X
!= ( neg @ ( dis @ P4 @ Q2 ) ) )
=> ( ! [P4: fm,Q2: fm] :
( X
!= ( neg @ ( con @ P4 @ Q2 ) ) )
=> ( ! [P4: fm] :
( X
!= ( neg @ ( exi @ P4 ) ) )
=> ( ! [P4: fm] :
( X
!= ( neg @ ( uni @ P4 ) ) )
=> ~ ! [P4: fm] :
( X
!= ( neg @ ( neg @ P4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Neg_exhaust
thf(fact_381_branchDone_Oelims_I3_J,axiom,
! [X: list_fm] :
( ~ ( branchDone @ X )
=> ( ( X != nil_fm )
=> ( ! [P4: fm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( neg @ P4 ) @ Z4 ) )
=> ( ( member_fm3 @ P4 @ ( set_fm2 @ Z4 ) )
| ( member_fm3 @ ( neg @ ( neg @ P4 ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ( ! [V: nat,Va: list_tm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( pre @ V @ Va ) @ Z4 ) )
=> ( ( member_fm3 @ ( neg @ ( pre @ V @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ( ! [V: fm,Va: fm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( imp @ V @ Va ) @ Z4 ) )
=> ( ( member_fm3 @ ( neg @ ( imp @ V @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ( ! [V: fm,Va: fm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( dis @ V @ Va ) @ Z4 ) )
=> ( ( member_fm3 @ ( neg @ ( dis @ V @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ( ! [V: fm,Va: fm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( con @ V @ Va ) @ Z4 ) )
=> ( ( member_fm3 @ ( neg @ ( con @ V @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ( ! [V: fm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( exi @ V ) @ Z4 ) )
=> ( ( member_fm3 @ ( neg @ ( exi @ V ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ~ ! [V: fm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( uni @ V ) @ Z4 ) )
=> ( ( member_fm3 @ ( neg @ ( uni @ V ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) ) ) ) ) ) ) ) ) ).
% branchDone.elims(3)
thf(fact_382_branchDone_Oelims_I1_J,axiom,
! [X: list_fm,Y: $o] :
( ( ( branchDone @ X )
= Y )
=> ( ( ( X = nil_fm )
=> Y )
=> ( ! [P4: fm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( neg @ P4 ) @ Z4 ) )
=> ( Y
= ( ~ ( ( member_fm3 @ P4 @ ( set_fm2 @ Z4 ) )
| ( member_fm3 @ ( neg @ ( neg @ P4 ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) ) )
=> ( ! [V: nat,Va: list_tm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( pre @ V @ Va ) @ Z4 ) )
=> ( Y
= ( ~ ( ( member_fm3 @ ( neg @ ( pre @ V @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) ) )
=> ( ! [V: fm,Va: fm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( imp @ V @ Va ) @ Z4 ) )
=> ( Y
= ( ~ ( ( member_fm3 @ ( neg @ ( imp @ V @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) ) )
=> ( ! [V: fm,Va: fm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( dis @ V @ Va ) @ Z4 ) )
=> ( Y
= ( ~ ( ( member_fm3 @ ( neg @ ( dis @ V @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) ) )
=> ( ! [V: fm,Va: fm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( con @ V @ Va ) @ Z4 ) )
=> ( Y
= ( ~ ( ( member_fm3 @ ( neg @ ( con @ V @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) ) )
=> ( ! [V: fm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( exi @ V ) @ Z4 ) )
=> ( Y
= ( ~ ( ( member_fm3 @ ( neg @ ( exi @ V ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) ) )
=> ~ ! [V: fm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( uni @ V ) @ Z4 ) )
=> ( Y
= ( ~ ( ( member_fm3 @ ( neg @ ( uni @ V ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% branchDone.elims(1)
thf(fact_383_longest__common__prefix_Ocases,axiom,
! [X: produc1530960993530998309t_rule] :
( ! [X3: rule,Xs2: list_rule,Y3: rule,Ys3: list_rule] :
( X
!= ( produc8979233614864344413t_rule @ ( cons_rule @ X3 @ Xs2 ) @ ( cons_rule @ Y3 @ Ys3 ) ) )
=> ( ! [Uv2: list_rule] :
( X
!= ( produc8979233614864344413t_rule @ nil_rule @ Uv2 ) )
=> ~ ! [Uu2: list_rule] :
( X
!= ( produc8979233614864344413t_rule @ Uu2 @ nil_rule ) ) ) ) ).
% longest_common_prefix.cases
thf(fact_384_longest__common__prefix_Ocases,axiom,
! [X: produc4386893285136772327ist_fm] :
( ! [X3: list_fm,Xs2: list_list_fm,Y3: list_fm,Ys3: list_list_fm] :
( X
!= ( produc8321651870839017815ist_fm @ ( cons_list_fm @ X3 @ Xs2 ) @ ( cons_list_fm @ Y3 @ Ys3 ) ) )
=> ( ! [Uv2: list_list_fm] :
( X
!= ( produc8321651870839017815ist_fm @ nil_list_fm @ Uv2 ) )
=> ~ ! [Uu2: list_list_fm] :
( X
!= ( produc8321651870839017815ist_fm @ Uu2 @ nil_list_fm ) ) ) ) ).
% longest_common_prefix.cases
thf(fact_385_longest__common__prefix_Ocases,axiom,
! [X: produc3245234490656042599ist_fm] :
( ! [X3: fm,Xs2: list_fm,Y3: fm,Ys3: list_fm] :
( X
!= ( produc7863996417982153943ist_fm @ ( cons_fm @ X3 @ Xs2 ) @ ( cons_fm @ Y3 @ Ys3 ) ) )
=> ( ! [Uv2: list_fm] :
( X
!= ( produc7863996417982153943ist_fm @ nil_fm @ Uv2 ) )
=> ~ ! [Uu2: list_fm] :
( X
!= ( produc7863996417982153943ist_fm @ Uu2 @ nil_fm ) ) ) ) ).
% longest_common_prefix.cases
thf(fact_386_longest__common__prefix_Ocases,axiom,
! [X: produc5776448205642668775ist_tm] :
( ! [X3: tm,Xs2: list_tm,Y3: tm,Ys3: list_tm] :
( X
!= ( produc1418304791525149271ist_tm @ ( cons_tm @ X3 @ Xs2 ) @ ( cons_tm @ Y3 @ Ys3 ) ) )
=> ( ! [Uv2: list_tm] :
( X
!= ( produc1418304791525149271ist_tm @ nil_tm @ Uv2 ) )
=> ~ ! [Uu2: list_tm] :
( X
!= ( produc1418304791525149271ist_tm @ Uu2 @ nil_tm ) ) ) ) ).
% longest_common_prefix.cases
thf(fact_387_branchDone_Oelims_I2_J,axiom,
! [X: list_fm] :
( ( branchDone @ X )
=> ( ! [P4: fm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( neg @ P4 ) @ Z4 ) )
=> ~ ( ( member_fm3 @ P4 @ ( set_fm2 @ Z4 ) )
| ( member_fm3 @ ( neg @ ( neg @ P4 ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ( ! [V: nat,Va: list_tm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( pre @ V @ Va ) @ Z4 ) )
=> ~ ( ( member_fm3 @ ( neg @ ( pre @ V @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ( ! [V: fm,Va: fm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( imp @ V @ Va ) @ Z4 ) )
=> ~ ( ( member_fm3 @ ( neg @ ( imp @ V @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ( ! [V: fm,Va: fm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( dis @ V @ Va ) @ Z4 ) )
=> ~ ( ( member_fm3 @ ( neg @ ( dis @ V @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ( ! [V: fm,Va: fm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( con @ V @ Va ) @ Z4 ) )
=> ~ ( ( member_fm3 @ ( neg @ ( con @ V @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ( ! [V: fm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( exi @ V ) @ Z4 ) )
=> ~ ( ( member_fm3 @ ( neg @ ( exi @ V ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ~ ! [V: fm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( uni @ V ) @ Z4 ) )
=> ~ ( ( member_fm3 @ ( neg @ ( uni @ V ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) ) ) ) ) ) ) ) ).
% branchDone.elims(2)
thf(fact_388_ext_Osimps_I2_J,axiom,
! [Y: list_rule,P: rule,Z3: list_rule] :
( ( ext_rule @ Y @ ( cons_rule @ P @ Z3 ) )
= ( ( ( member_rule2 @ P @ Y )
=> ( ext_rule @ Y @ Z3 ) )
& ( member_rule2 @ P @ Y ) ) ) ).
% ext.simps(2)
thf(fact_389_ext_Osimps_I2_J,axiom,
! [Y: list_list_fm,P: list_fm,Z3: list_list_fm] :
( ( ext_list_fm @ Y @ ( cons_list_fm @ P @ Z3 ) )
= ( ( ( member_list_fm2 @ P @ Y )
=> ( ext_list_fm @ Y @ Z3 ) )
& ( member_list_fm2 @ P @ Y ) ) ) ).
% ext.simps(2)
thf(fact_390_ext_Osimps_I2_J,axiom,
! [Y: list_fm,P: fm,Z3: list_fm] :
( ( ext_fm @ Y @ ( cons_fm @ P @ Z3 ) )
= ( ( ( member_fm2 @ P @ Y )
=> ( ext_fm @ Y @ Z3 ) )
& ( member_fm2 @ P @ Y ) ) ) ).
% ext.simps(2)
thf(fact_391_ext_Osimps_I2_J,axiom,
! [Y: list_tm,P: tm,Z3: list_tm] :
( ( ext_tm @ Y @ ( cons_tm @ P @ Z3 ) )
= ( ( ( member_tm2 @ P @ Y )
=> ( ext_tm @ Y @ Z3 ) )
& ( member_tm2 @ P @ Y ) ) ) ).
% ext.simps(2)
thf(fact_392_product__lists_Osimps_I1_J,axiom,
( ( product_lists_rule @ nil_list_rule )
= ( cons_list_rule @ nil_rule @ nil_list_rule ) ) ).
% product_lists.simps(1)
thf(fact_393_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_394_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_395_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_396_sset__cycle,axiom,
! [Xs: list_rule] :
( ( Xs != nil_rule )
=> ( ( sset_rule @ ( cycle_rule @ Xs ) )
= ( set_rule2 @ Xs ) ) ) ).
% sset_cycle
thf(fact_397_sset__cycle,axiom,
! [Xs: list_fm] :
( ( Xs != nil_fm )
=> ( ( sset_fm @ ( cycle_fm @ Xs ) )
= ( set_fm2 @ Xs ) ) ) ).
% sset_cycle
thf(fact_398_sset__cycle,axiom,
! [Xs: list_list_fm] :
( ( Xs != nil_list_fm )
=> ( ( sset_list_fm @ ( cycle_list_fm @ Xs ) )
= ( set_list_fm2 @ Xs ) ) ) ).
% sset_cycle
thf(fact_399_sset__cycle,axiom,
! [Xs: list_tm] :
( ( Xs != nil_tm )
=> ( ( sset_tm @ ( cycle_tm @ Xs ) )
= ( set_tm2 @ Xs ) ) ) ).
% sset_cycle
thf(fact_400_sset__cycle,axiom,
! [Xs: list_P2774625669004309958m_rule] :
( ( Xs != nil_Pr2808214839852828720m_rule )
=> ( ( sset_P4484857331586881186m_rule @ ( cycle_5335117900641983313m_rule @ Xs ) )
= ( set_Pr4534715572506550497m_rule @ Xs ) ) ) ).
% sset_cycle
thf(fact_401_the__elem__set,axiom,
! [X: rule] :
( ( the_elem_rule @ ( set_rule2 @ ( cons_rule @ X @ nil_rule ) ) )
= X ) ).
% the_elem_set
thf(fact_402_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_403_the__elem__set,axiom,
! [X: fm] :
( ( the_elem_fm @ ( set_fm2 @ ( cons_fm @ X @ nil_fm ) ) )
= X ) ).
% the_elem_set
thf(fact_404_the__elem__set,axiom,
! [X: tm] :
( ( the_elem_tm @ ( set_tm2 @ ( cons_tm @ X @ nil_tm ) ) )
= X ) ).
% the_elem_set
thf(fact_405_branchDone_Osimps_I1_J,axiom,
~ ( branchDone @ nil_fm ) ).
% branchDone.simps(1)
thf(fact_406_branchDone__contradiction,axiom,
( branchDone
= ( ^ [Z: list_fm] :
? [P2: fm] :
( ( member_fm3 @ P2 @ ( set_fm2 @ Z ) )
& ( member_fm3 @ ( neg @ P2 ) @ ( set_fm2 @ Z ) ) ) ) ) ).
% branchDone_contradiction
thf(fact_407_cycle__decomp,axiom,
! [U: list_P2774625669004309958m_rule] :
( ( U != nil_Pr2808214839852828720m_rule )
=> ( ( cycle_5335117900641983313m_rule @ U )
= ( shift_2334844276748245581m_rule @ U @ ( cycle_5335117900641983313m_rule @ U ) ) ) ) ).
% cycle_decomp
thf(fact_408_cycle__decomp,axiom,
! [U: list_rule] :
( ( U != nil_rule )
=> ( ( cycle_rule @ U )
= ( shift_rule @ U @ ( cycle_rule @ U ) ) ) ) ).
% cycle_decomp
thf(fact_409_cycle__decomp,axiom,
! [U: list_fm] :
( ( U != nil_fm )
=> ( ( cycle_fm @ U )
= ( shift_fm @ U @ ( cycle_fm @ U ) ) ) ) ).
% cycle_decomp
thf(fact_410_cycle__decomp,axiom,
! [U: list_list_fm] :
( ( U != nil_list_fm )
=> ( ( cycle_list_fm @ U )
= ( shift_list_fm @ U @ ( cycle_list_fm @ U ) ) ) ) ).
% cycle_decomp
thf(fact_411_cycle__decomp,axiom,
! [U: list_tm] :
( ( U != nil_tm )
=> ( ( cycle_tm @ U )
= ( shift_tm @ U @ ( cycle_tm @ U ) ) ) ) ).
% cycle_decomp
thf(fact_412_branchDone_Osimps_I2_J,axiom,
! [P: fm,Z3: list_fm] :
( ( branchDone @ ( cons_fm @ ( neg @ P ) @ Z3 ) )
= ( ( member_fm3 @ P @ ( set_fm2 @ Z3 ) )
| ( member_fm3 @ ( neg @ ( neg @ P ) ) @ ( set_fm2 @ Z3 ) )
| ( branchDone @ Z3 ) ) ) ).
% branchDone.simps(2)
thf(fact_413_ext_Osimps_I1_J,axiom,
! [Y: list_rule] : ( ext_rule @ Y @ nil_rule ) ).
% ext.simps(1)
thf(fact_414_ext_Osimps_I1_J,axiom,
! [Y: list_fm] : ( ext_fm @ Y @ nil_fm ) ).
% ext.simps(1)
thf(fact_415_ext_Osimps_I1_J,axiom,
! [Y: list_list_fm] : ( ext_list_fm @ Y @ nil_list_fm ) ).
% ext.simps(1)
thf(fact_416_ext_Osimps_I1_J,axiom,
! [Y: list_tm] : ( ext_tm @ Y @ nil_tm ) ).
% ext.simps(1)
thf(fact_417_branchDone_Osimps_I6_J,axiom,
! [V2: fm,Va2: fm,Z3: list_fm] :
( ( branchDone @ ( cons_fm @ ( con @ V2 @ Va2 ) @ Z3 ) )
= ( ( member_fm3 @ ( neg @ ( con @ V2 @ Va2 ) ) @ ( set_fm2 @ Z3 ) )
| ( branchDone @ Z3 ) ) ) ).
% branchDone.simps(6)
thf(fact_418_branchDone_Osimps_I7_J,axiom,
! [V2: fm,Z3: list_fm] :
( ( branchDone @ ( cons_fm @ ( exi @ V2 ) @ Z3 ) )
= ( ( member_fm3 @ ( neg @ ( exi @ V2 ) ) @ ( set_fm2 @ Z3 ) )
| ( branchDone @ Z3 ) ) ) ).
% branchDone.simps(7)
thf(fact_419_branchDone_Osimps_I8_J,axiom,
! [V2: fm,Z3: list_fm] :
( ( branchDone @ ( cons_fm @ ( uni @ V2 ) @ Z3 ) )
= ( ( member_fm3 @ ( neg @ ( uni @ V2 ) ) @ ( set_fm2 @ Z3 ) )
| ( branchDone @ Z3 ) ) ) ).
% branchDone.simps(8)
thf(fact_420_branchDone_Osimps_I5_J,axiom,
! [V2: fm,Va2: fm,Z3: list_fm] :
( ( branchDone @ ( cons_fm @ ( dis @ V2 @ Va2 ) @ Z3 ) )
= ( ( member_fm3 @ ( neg @ ( dis @ V2 @ Va2 ) ) @ ( set_fm2 @ Z3 ) )
| ( branchDone @ Z3 ) ) ) ).
% branchDone.simps(5)
thf(fact_421_branchDone_Osimps_I4_J,axiom,
! [V2: fm,Va2: fm,Z3: list_fm] :
( ( branchDone @ ( cons_fm @ ( imp @ V2 @ Va2 ) @ Z3 ) )
= ( ( member_fm3 @ ( neg @ ( imp @ V2 @ Va2 ) ) @ ( set_fm2 @ Z3 ) )
| ( branchDone @ Z3 ) ) ) ).
% branchDone.simps(4)
thf(fact_422_branchDone_Osimps_I3_J,axiom,
! [V2: nat,Va2: list_tm,Z3: list_fm] :
( ( branchDone @ ( cons_fm @ ( pre @ V2 @ Va2 ) @ Z3 ) )
= ( ( member_fm3 @ ( neg @ ( pre @ V2 @ Va2 ) ) @ ( set_fm2 @ Z3 ) )
| ( branchDone @ Z3 ) ) ) ).
% branchDone.simps(3)
thf(fact_423_branchDone_Opelims_I1_J,axiom,
! [X: list_fm,Y: $o] :
( ( ( branchDone @ X )
= Y )
=> ( ( accp_list_fm @ branchDone_rel @ X )
=> ( ( ( X = nil_fm )
=> ( ~ Y
=> ~ ( accp_list_fm @ branchDone_rel @ nil_fm ) ) )
=> ( ! [P4: fm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( neg @ P4 ) @ Z4 ) )
=> ( ( Y
= ( ( member_fm3 @ P4 @ ( set_fm2 @ Z4 ) )
| ( member_fm3 @ ( neg @ ( neg @ P4 ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ~ ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( neg @ P4 ) @ Z4 ) ) ) )
=> ( ! [V: nat,Va: list_tm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( pre @ V @ Va ) @ Z4 ) )
=> ( ( Y
= ( ( member_fm3 @ ( neg @ ( pre @ V @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ~ ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( pre @ V @ Va ) @ Z4 ) ) ) )
=> ( ! [V: fm,Va: fm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( imp @ V @ Va ) @ Z4 ) )
=> ( ( Y
= ( ( member_fm3 @ ( neg @ ( imp @ V @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ~ ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( imp @ V @ Va ) @ Z4 ) ) ) )
=> ( ! [V: fm,Va: fm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( dis @ V @ Va ) @ Z4 ) )
=> ( ( Y
= ( ( member_fm3 @ ( neg @ ( dis @ V @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ~ ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( dis @ V @ Va ) @ Z4 ) ) ) )
=> ( ! [V: fm,Va: fm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( con @ V @ Va ) @ Z4 ) )
=> ( ( Y
= ( ( member_fm3 @ ( neg @ ( con @ V @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ~ ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( con @ V @ Va ) @ Z4 ) ) ) )
=> ( ! [V: fm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( exi @ V ) @ Z4 ) )
=> ( ( Y
= ( ( member_fm3 @ ( neg @ ( exi @ V ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ~ ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( exi @ V ) @ Z4 ) ) ) )
=> ~ ! [V: fm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( uni @ V ) @ Z4 ) )
=> ( ( Y
= ( ( member_fm3 @ ( neg @ ( uni @ V ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) )
=> ~ ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( uni @ V ) @ Z4 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% branchDone.pelims(1)
thf(fact_424_branchDone_Opelims_I3_J,axiom,
! [X: list_fm] :
( ~ ( branchDone @ X )
=> ( ( accp_list_fm @ branchDone_rel @ X )
=> ( ( ( X = nil_fm )
=> ~ ( accp_list_fm @ branchDone_rel @ nil_fm ) )
=> ( ! [P4: fm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( neg @ P4 ) @ Z4 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( neg @ P4 ) @ Z4 ) )
=> ( ( member_fm3 @ P4 @ ( set_fm2 @ Z4 ) )
| ( member_fm3 @ ( neg @ ( neg @ P4 ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) )
=> ( ! [V: nat,Va: list_tm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( pre @ V @ Va ) @ Z4 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( pre @ V @ Va ) @ Z4 ) )
=> ( ( member_fm3 @ ( neg @ ( pre @ V @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) )
=> ( ! [V: fm,Va: fm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( imp @ V @ Va ) @ Z4 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( imp @ V @ Va ) @ Z4 ) )
=> ( ( member_fm3 @ ( neg @ ( imp @ V @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) )
=> ( ! [V: fm,Va: fm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( dis @ V @ Va ) @ Z4 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( dis @ V @ Va ) @ Z4 ) )
=> ( ( member_fm3 @ ( neg @ ( dis @ V @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) )
=> ( ! [V: fm,Va: fm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( con @ V @ Va ) @ Z4 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( con @ V @ Va ) @ Z4 ) )
=> ( ( member_fm3 @ ( neg @ ( con @ V @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) )
=> ( ! [V: fm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( exi @ V ) @ Z4 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( exi @ V ) @ Z4 ) )
=> ( ( member_fm3 @ ( neg @ ( exi @ V ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) )
=> ~ ! [V: fm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( uni @ V ) @ Z4 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( uni @ V ) @ Z4 ) )
=> ( ( member_fm3 @ ( neg @ ( uni @ V ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% branchDone.pelims(3)
thf(fact_425_branchDone_Opelims_I2_J,axiom,
! [X: list_fm] :
( ( branchDone @ X )
=> ( ( accp_list_fm @ branchDone_rel @ X )
=> ( ! [P4: fm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( neg @ P4 ) @ Z4 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( neg @ P4 ) @ Z4 ) )
=> ~ ( ( member_fm3 @ P4 @ ( set_fm2 @ Z4 ) )
| ( member_fm3 @ ( neg @ ( neg @ P4 ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) )
=> ( ! [V: nat,Va: list_tm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( pre @ V @ Va ) @ Z4 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( pre @ V @ Va ) @ Z4 ) )
=> ~ ( ( member_fm3 @ ( neg @ ( pre @ V @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) )
=> ( ! [V: fm,Va: fm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( imp @ V @ Va ) @ Z4 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( imp @ V @ Va ) @ Z4 ) )
=> ~ ( ( member_fm3 @ ( neg @ ( imp @ V @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) )
=> ( ! [V: fm,Va: fm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( dis @ V @ Va ) @ Z4 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( dis @ V @ Va ) @ Z4 ) )
=> ~ ( ( member_fm3 @ ( neg @ ( dis @ V @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) )
=> ( ! [V: fm,Va: fm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( con @ V @ Va ) @ Z4 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( con @ V @ Va ) @ Z4 ) )
=> ~ ( ( member_fm3 @ ( neg @ ( con @ V @ Va ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) )
=> ( ! [V: fm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( exi @ V ) @ Z4 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( exi @ V ) @ Z4 ) )
=> ~ ( ( member_fm3 @ ( neg @ ( exi @ V ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) )
=> ~ ! [V: fm,Z4: list_fm] :
( ( X
= ( cons_fm @ ( uni @ V ) @ Z4 ) )
=> ( ( accp_list_fm @ branchDone_rel @ ( cons_fm @ ( uni @ V ) @ Z4 ) )
=> ~ ( ( member_fm3 @ ( neg @ ( uni @ V ) ) @ ( set_fm2 @ Z4 ) )
| ( branchDone @ Z4 ) ) ) ) ) ) ) ) ) ) ) ) ).
% branchDone.pelims(2)
thf(fact_426_subseqs_Osimps_I1_J,axiom,
( ( subseqs_rule @ nil_rule )
= ( cons_list_rule @ nil_rule @ nil_list_rule ) ) ).
% subseqs.simps(1)
thf(fact_427_subseqs_Osimps_I1_J,axiom,
( ( subseqs_list_fm @ nil_list_fm )
= ( cons_list_list_fm @ nil_list_fm @ nil_list_list_fm ) ) ).
% subseqs.simps(1)
thf(fact_428_subseqs_Osimps_I1_J,axiom,
( ( subseqs_tm @ nil_tm )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) ).
% subseqs.simps(1)
thf(fact_429_subseqs_Osimps_I1_J,axiom,
( ( subseqs_fm @ nil_fm )
= ( cons_list_fm @ nil_fm @ nil_list_fm ) ) ).
% subseqs.simps(1)
thf(fact_430_listrel_Ocases,axiom,
! [A1: list_rule,A22: list_rule,R: set_Pr7340090144965549541e_rule] :
( ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ A1 @ A22 ) @ ( listrel_rule_rule @ R ) )
=> ( ( ( A1 = nil_rule )
=> ( A22 != nil_rule ) )
=> ~ ! [X3: rule,Y3: rule,Xs2: list_rule] :
( ( A1
= ( cons_rule @ X3 @ Xs2 ) )
=> ! [Ys3: list_rule] :
( ( A22
= ( cons_rule @ Y3 @ Ys3 ) )
=> ( ( member1444691403335511342e_rule @ ( produc5849431337705160893e_rule @ X3 @ Y3 ) @ R )
=> ~ ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ Xs2 @ Ys3 ) @ ( listrel_rule_rule @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_431_listrel_Ocases,axiom,
! [A1: list_rule,A22: list_fm,R: set_Pr8969532344284305932ule_fm] :
( ( member2905128903584585571ist_fm @ ( produc9214880830201710206ist_fm @ A1 @ A22 ) @ ( listrel_rule_fm @ R ) )
=> ( ( ( A1 = nil_rule )
=> ( A22 != nil_fm ) )
=> ~ ! [X3: rule,Y3: fm,Xs2: list_rule] :
( ( A1
= ( cons_rule @ X3 @ Xs2 ) )
=> ! [Ys3: list_fm] :
( ( A22
= ( cons_fm @ Y3 @ Ys3 ) )
=> ( ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ X3 @ Y3 ) @ R )
=> ~ ( member2905128903584585571ist_fm @ ( produc9214880830201710206ist_fm @ Xs2 @ Ys3 ) @ ( listrel_rule_fm @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_432_listrel_Ocases,axiom,
! [A1: list_rule,A22: list_tm,R: set_Pr8727017673958796158ule_tm] :
( ( member2662614233259075797ist_tm @ ( produc9218832855287345392ist_tm @ A1 @ A22 ) @ ( listrel_rule_tm @ R ) )
=> ( ( ( A1 = nil_rule )
=> ( A22 != nil_tm ) )
=> ~ ! [X3: rule,Y3: tm,Xs2: list_rule] :
( ( A1
= ( cons_rule @ X3 @ Xs2 ) )
=> ! [Ys3: list_tm] :
( ( A22
= ( cons_tm @ Y3 @ Ys3 ) )
=> ( ( member9172131417393966175ule_tm @ ( product_Pair_rule_tm @ X3 @ Y3 ) @ R )
=> ~ ( member2662614233259075797ist_tm @ ( produc9218832855287345392ist_tm @ Xs2 @ Ys3 ) @ ( listrel_rule_tm @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_433_listrel_Ocases,axiom,
! [A1: list_fm,A22: list_rule,R: set_Pr7067330020180114026m_rule] :
( ( member7302538096890821473t_rule @ ( produc1396096877967899076t_rule @ A1 @ A22 ) @ ( listrel_fm_rule @ R ) )
=> ( ( ( A1 = nil_fm )
=> ( A22 != nil_rule ) )
=> ~ ! [X3: fm,Y3: rule,Xs2: list_fm] :
( ( A1
= ( cons_fm @ X3 @ Xs2 ) )
=> ! [Ys3: list_rule] :
( ( A22
= ( cons_rule @ Y3 @ Ys3 ) )
=> ( ( member6041693456836784459m_rule @ ( product_Pair_fm_rule @ X3 @ Y3 ) @ R )
=> ~ ( member7302538096890821473t_rule @ ( produc1396096877967899076t_rule @ Xs2 @ Ys3 ) @ ( listrel_fm_rule @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_434_listrel_Ocases,axiom,
! [A1: list_fm,A22: list_fm,R: set_Pr4706815898642364871_fm_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ A1 @ A22 ) @ ( listrel_fm_fm @ R ) )
=> ( ( ( A1 = nil_fm )
=> ( A22 != nil_fm ) )
=> ~ ! [X3: fm,Y3: fm,Xs2: list_fm] :
( ( A1
= ( cons_fm @ X3 @ Xs2 ) )
=> ! [Ys3: list_fm] :
( ( A22
= ( cons_fm @ Y3 @ Ys3 ) )
=> ( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X3 @ Y3 ) @ R )
=> ~ ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Xs2 @ Ys3 ) @ ( listrel_fm_fm @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_435_listrel_Ocases,axiom,
! [A1: list_fm,A22: list_tm,R: set_Pr4464301228316855097_fm_tm] :
( ( member1683583632484806914ist_tm @ ( produc7867948443067789129ist_tm @ A1 @ A22 ) @ ( listrel_fm_tm @ R ) )
=> ( ( ( A1 = nil_fm )
=> ( A22 != nil_tm ) )
=> ~ ! [X3: fm,Y3: tm,Xs2: list_fm] :
( ( A1
= ( cons_fm @ X3 @ Xs2 ) )
=> ! [Ys3: list_tm] :
( ( A22
= ( cons_tm @ Y3 @ Ys3 ) )
=> ( ( member7784904625553633922_fm_tm @ ( product_Pair_fm_tm @ X3 @ Y3 ) @ R )
=> ~ ( member1683583632484806914ist_tm @ ( produc7867948443067789129ist_tm @ Xs2 @ Ys3 ) @ ( listrel_fm_tm @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_436_listrel_Ocases,axiom,
! [A1: list_tm,A22: list_rule,R: set_Pr3569831037629835996m_rule] :
( ( member2031962811952810707t_rule @ ( produc5348893629884664118t_rule @ A1 @ A22 ) @ ( listrel_tm_rule @ R ) )
=> ( ( ( A1 = nil_tm )
=> ( A22 != nil_rule ) )
=> ~ ! [X3: tm,Y3: rule,Xs2: list_tm] :
( ( A1
= ( cons_tm @ X3 @ Xs2 ) )
=> ! [Ys3: list_rule] :
( ( A22
= ( cons_rule @ Y3 @ Ys3 ) )
=> ( ( member5482372101558860477m_rule @ ( product_Pair_tm_rule @ X3 @ Y3 ) @ R )
=> ~ ( member2031962811952810707t_rule @ ( produc5348893629884664118t_rule @ Xs2 @ Ys3 ) @ ( listrel_tm_rule @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_437_listrel_Ocases,axiom,
! [A1: list_tm,A22: list_tm,R: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ A1 @ A22 ) @ ( listrel_tm_tm @ R ) )
=> ( ( ( A1 = nil_tm )
=> ( A22 != nil_tm ) )
=> ~ ! [X3: tm,Y3: tm,Xs2: list_tm] :
( ( A1
= ( cons_tm @ X3 @ Xs2 ) )
=> ! [Ys3: list_tm] :
( ( A22
= ( cons_tm @ Y3 @ Ys3 ) )
=> ( ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X3 @ Y3 ) @ R )
=> ~ ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs2 @ Ys3 ) @ ( listrel_tm_tm @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_438_listrel_Ocases,axiom,
! [A1: list_tm,A22: list_fm,R: set_Pr2698443736021152725_tm_fm] :
( ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ A1 @ A22 ) @ ( listrel_tm_fm @ R ) )
=> ( ( ( A1 = nil_tm )
=> ( A22 != nil_fm ) )
=> ~ ! [X3: tm,Y3: fm,Xs2: list_tm] :
( ( A1
= ( cons_tm @ X3 @ Xs2 ) )
=> ! [Ys3: list_fm] :
( ( A22
= ( cons_fm @ Y3 @ Ys3 ) )
=> ( ( member3117664881408846110_tm_fm @ ( product_Pair_tm_fm @ X3 @ Y3 ) @ R )
=> ~ ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ Xs2 @ Ys3 ) @ ( listrel_tm_fm @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_439_listrel_Ocases,axiom,
! [A1: list_rule,A22: list_list_fm,R: set_Pr6384426166747533970ist_fm] :
( ( member494883543038539881ist_fm @ ( produc7454812661813866372ist_fm @ A1 @ A22 ) @ ( listrel_rule_list_fm @ R ) )
=> ( ( ( A1 = nil_rule )
=> ( A22 != nil_list_fm ) )
=> ~ ! [X3: rule,Y3: list_fm,Xs2: list_rule] :
( ( A1
= ( cons_rule @ X3 @ Xs2 ) )
=> ! [Ys3: list_list_fm] :
( ( A22
= ( cons_list_fm @ Y3 @ Ys3 ) )
=> ( ( member4140259954344421235ist_fm @ ( produc2087254492244816782ist_fm @ X3 @ Y3 ) @ R )
=> ~ ( member494883543038539881ist_fm @ ( produc7454812661813866372ist_fm @ Xs2 @ Ys3 ) @ ( listrel_rule_list_fm @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_440_listrel_Osimps,axiom,
! [A1: list_rule,A22: list_rule,R: set_Pr7340090144965549541e_rule] :
( ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ A1 @ A22 ) @ ( listrel_rule_rule @ R ) )
= ( ( ( A1 = nil_rule )
& ( A22 = nil_rule ) )
| ? [X2: rule,Y4: rule,Xs3: list_rule,Ys2: list_rule] :
( ( A1
= ( cons_rule @ X2 @ Xs3 ) )
& ( A22
= ( cons_rule @ Y4 @ Ys2 ) )
& ( member1444691403335511342e_rule @ ( produc5849431337705160893e_rule @ X2 @ Y4 ) @ R )
& ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ Xs3 @ Ys2 ) @ ( listrel_rule_rule @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_441_listrel_Osimps,axiom,
! [A1: list_rule,A22: list_fm,R: set_Pr8969532344284305932ule_fm] :
( ( member2905128903584585571ist_fm @ ( produc9214880830201710206ist_fm @ A1 @ A22 ) @ ( listrel_rule_fm @ R ) )
= ( ( ( A1 = nil_rule )
& ( A22 = nil_fm ) )
| ? [X2: rule,Y4: fm,Xs3: list_rule,Ys2: list_fm] :
( ( A1
= ( cons_rule @ X2 @ Xs3 ) )
& ( A22
= ( cons_fm @ Y4 @ Ys2 ) )
& ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ X2 @ Y4 ) @ R )
& ( member2905128903584585571ist_fm @ ( produc9214880830201710206ist_fm @ Xs3 @ Ys2 ) @ ( listrel_rule_fm @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_442_listrel_Osimps,axiom,
! [A1: list_rule,A22: list_tm,R: set_Pr8727017673958796158ule_tm] :
( ( member2662614233259075797ist_tm @ ( produc9218832855287345392ist_tm @ A1 @ A22 ) @ ( listrel_rule_tm @ R ) )
= ( ( ( A1 = nil_rule )
& ( A22 = nil_tm ) )
| ? [X2: rule,Y4: tm,Xs3: list_rule,Ys2: list_tm] :
( ( A1
= ( cons_rule @ X2 @ Xs3 ) )
& ( A22
= ( cons_tm @ Y4 @ Ys2 ) )
& ( member9172131417393966175ule_tm @ ( product_Pair_rule_tm @ X2 @ Y4 ) @ R )
& ( member2662614233259075797ist_tm @ ( produc9218832855287345392ist_tm @ Xs3 @ Ys2 ) @ ( listrel_rule_tm @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_443_listrel_Osimps,axiom,
! [A1: list_fm,A22: list_rule,R: set_Pr7067330020180114026m_rule] :
( ( member7302538096890821473t_rule @ ( produc1396096877967899076t_rule @ A1 @ A22 ) @ ( listrel_fm_rule @ R ) )
= ( ( ( A1 = nil_fm )
& ( A22 = nil_rule ) )
| ? [X2: fm,Y4: rule,Xs3: list_fm,Ys2: list_rule] :
( ( A1
= ( cons_fm @ X2 @ Xs3 ) )
& ( A22
= ( cons_rule @ Y4 @ Ys2 ) )
& ( member6041693456836784459m_rule @ ( product_Pair_fm_rule @ X2 @ Y4 ) @ R )
& ( member7302538096890821473t_rule @ ( produc1396096877967899076t_rule @ Xs3 @ Ys2 ) @ ( listrel_fm_rule @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_444_listrel_Osimps,axiom,
! [A1: list_fm,A22: list_fm,R: set_Pr4706815898642364871_fm_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ A1 @ A22 ) @ ( listrel_fm_fm @ R ) )
= ( ( ( A1 = nil_fm )
& ( A22 = nil_fm ) )
| ? [X2: fm,Y4: fm,Xs3: list_fm,Ys2: list_fm] :
( ( A1
= ( cons_fm @ X2 @ Xs3 ) )
& ( A22
= ( cons_fm @ Y4 @ Ys2 ) )
& ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X2 @ Y4 ) @ R )
& ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Xs3 @ Ys2 ) @ ( listrel_fm_fm @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_445_listrel_Osimps,axiom,
! [A1: list_fm,A22: list_tm,R: set_Pr4464301228316855097_fm_tm] :
( ( member1683583632484806914ist_tm @ ( produc7867948443067789129ist_tm @ A1 @ A22 ) @ ( listrel_fm_tm @ R ) )
= ( ( ( A1 = nil_fm )
& ( A22 = nil_tm ) )
| ? [X2: fm,Y4: tm,Xs3: list_fm,Ys2: list_tm] :
( ( A1
= ( cons_fm @ X2 @ Xs3 ) )
& ( A22
= ( cons_tm @ Y4 @ Ys2 ) )
& ( member7784904625553633922_fm_tm @ ( product_Pair_fm_tm @ X2 @ Y4 ) @ R )
& ( member1683583632484806914ist_tm @ ( produc7867948443067789129ist_tm @ Xs3 @ Ys2 ) @ ( listrel_fm_tm @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_446_listrel_Osimps,axiom,
! [A1: list_tm,A22: list_rule,R: set_Pr3569831037629835996m_rule] :
( ( member2031962811952810707t_rule @ ( produc5348893629884664118t_rule @ A1 @ A22 ) @ ( listrel_tm_rule @ R ) )
= ( ( ( A1 = nil_tm )
& ( A22 = nil_rule ) )
| ? [X2: tm,Y4: rule,Xs3: list_tm,Ys2: list_rule] :
( ( A1
= ( cons_tm @ X2 @ Xs3 ) )
& ( A22
= ( cons_rule @ Y4 @ Ys2 ) )
& ( member5482372101558860477m_rule @ ( product_Pair_tm_rule @ X2 @ Y4 ) @ R )
& ( member2031962811952810707t_rule @ ( produc5348893629884664118t_rule @ Xs3 @ Ys2 ) @ ( listrel_tm_rule @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_447_listrel_Osimps,axiom,
! [A1: list_tm,A22: list_tm,R: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ A1 @ A22 ) @ ( listrel_tm_tm @ R ) )
= ( ( ( A1 = nil_tm )
& ( A22 = nil_tm ) )
| ? [X2: tm,Y4: tm,Xs3: list_tm,Ys2: list_tm] :
( ( A1
= ( cons_tm @ X2 @ Xs3 ) )
& ( A22
= ( cons_tm @ Y4 @ Ys2 ) )
& ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X2 @ Y4 ) @ R )
& ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs3 @ Ys2 ) @ ( listrel_tm_tm @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_448_listrel_Osimps,axiom,
! [A1: list_tm,A22: list_fm,R: set_Pr2698443736021152725_tm_fm] :
( ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ A1 @ A22 ) @ ( listrel_tm_fm @ R ) )
= ( ( ( A1 = nil_tm )
& ( A22 = nil_fm ) )
| ? [X2: tm,Y4: fm,Xs3: list_tm,Ys2: list_fm] :
( ( A1
= ( cons_tm @ X2 @ Xs3 ) )
& ( A22
= ( cons_fm @ Y4 @ Ys2 ) )
& ( member3117664881408846110_tm_fm @ ( product_Pair_tm_fm @ X2 @ Y4 ) @ R )
& ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ Xs3 @ Ys2 ) @ ( listrel_tm_fm @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_449_listrel_Osimps,axiom,
! [A1: list_rule,A22: list_list_fm,R: set_Pr6384426166747533970ist_fm] :
( ( member494883543038539881ist_fm @ ( produc7454812661813866372ist_fm @ A1 @ A22 ) @ ( listrel_rule_list_fm @ R ) )
= ( ( ( A1 = nil_rule )
& ( A22 = nil_list_fm ) )
| ? [X2: rule,Y4: list_fm,Xs3: list_rule,Ys2: list_list_fm] :
( ( A1
= ( cons_rule @ X2 @ Xs3 ) )
& ( A22
= ( cons_list_fm @ Y4 @ Ys2 ) )
& ( member4140259954344421235ist_fm @ ( produc2087254492244816782ist_fm @ X2 @ Y4 ) @ R )
& ( member494883543038539881ist_fm @ ( produc7454812661813866372ist_fm @ Xs3 @ Ys2 ) @ ( listrel_rule_list_fm @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_450_subseqs__refl,axiom,
! [Xs: list_fm] : ( member_list_fm3 @ Xs @ ( set_list_fm2 @ ( subseqs_fm @ Xs ) ) ) ).
% subseqs_refl
thf(fact_451_listrel__Nil2,axiom,
! [Xs: list_rule,R: set_Pr7340090144965549541e_rule] :
( ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ Xs @ nil_rule ) @ ( listrel_rule_rule @ R ) )
=> ( Xs = nil_rule ) ) ).
% listrel_Nil2
thf(fact_452_listrel__Nil2,axiom,
! [Xs: list_fm,R: set_Pr7067330020180114026m_rule] :
( ( member7302538096890821473t_rule @ ( produc1396096877967899076t_rule @ Xs @ nil_rule ) @ ( listrel_fm_rule @ R ) )
=> ( Xs = nil_fm ) ) ).
% listrel_Nil2
thf(fact_453_listrel__Nil2,axiom,
! [Xs: list_tm,R: set_Pr3569831037629835996m_rule] :
( ( member2031962811952810707t_rule @ ( produc5348893629884664118t_rule @ Xs @ nil_rule ) @ ( listrel_tm_rule @ R ) )
=> ( Xs = nil_tm ) ) ).
% listrel_Nil2
thf(fact_454_listrel__Nil2,axiom,
! [Xs: list_rule,R: set_Pr8969532344284305932ule_fm] :
( ( member2905128903584585571ist_fm @ ( produc9214880830201710206ist_fm @ Xs @ nil_fm ) @ ( listrel_rule_fm @ R ) )
=> ( Xs = nil_rule ) ) ).
% listrel_Nil2
thf(fact_455_listrel__Nil2,axiom,
! [Xs: list_fm,R: set_Pr4706815898642364871_fm_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Xs @ nil_fm ) @ ( listrel_fm_fm @ R ) )
=> ( Xs = nil_fm ) ) ).
% listrel_Nil2
thf(fact_456_listrel__Nil2,axiom,
! [Xs: list_rule,R: set_Pr8727017673958796158ule_tm] :
( ( member2662614233259075797ist_tm @ ( produc9218832855287345392ist_tm @ Xs @ nil_tm ) @ ( listrel_rule_tm @ R ) )
=> ( Xs = nil_rule ) ) ).
% listrel_Nil2
thf(fact_457_listrel__Nil2,axiom,
! [Xs: list_fm,R: set_Pr4464301228316855097_fm_tm] :
( ( member1683583632484806914ist_tm @ ( produc7867948443067789129ist_tm @ Xs @ nil_tm ) @ ( listrel_fm_tm @ R ) )
=> ( Xs = nil_fm ) ) ).
% listrel_Nil2
thf(fact_458_listrel__Nil2,axiom,
! [Xs: list_tm,R: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ nil_tm ) @ ( listrel_tm_tm @ R ) )
=> ( Xs = nil_tm ) ) ).
% listrel_Nil2
thf(fact_459_listrel__Nil2,axiom,
! [Xs: list_tm,R: set_Pr2698443736021152725_tm_fm] :
( ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ Xs @ nil_fm ) @ ( listrel_tm_fm @ R ) )
=> ( Xs = nil_tm ) ) ).
% listrel_Nil2
thf(fact_460_listrel__Nil2,axiom,
! [Xs: list_list_fm,R: set_Pr2422821757980664944m_rule] :
( ( member6202700923738835687t_rule @ ( produc3733561588192308298t_rule @ Xs @ nil_rule ) @ ( listrel_list_fm_rule @ R ) )
=> ( Xs = nil_list_fm ) ) ).
% listrel_Nil2
thf(fact_461_listrel__Nil1,axiom,
! [Xs: list_rule,R: set_Pr7340090144965549541e_rule] :
( ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ nil_rule @ Xs ) @ ( listrel_rule_rule @ R ) )
=> ( Xs = nil_rule ) ) ).
% listrel_Nil1
thf(fact_462_listrel__Nil1,axiom,
! [Xs: list_fm,R: set_Pr8969532344284305932ule_fm] :
( ( member2905128903584585571ist_fm @ ( produc9214880830201710206ist_fm @ nil_rule @ Xs ) @ ( listrel_rule_fm @ R ) )
=> ( Xs = nil_fm ) ) ).
% listrel_Nil1
thf(fact_463_listrel__Nil1,axiom,
! [Xs: list_tm,R: set_Pr8727017673958796158ule_tm] :
( ( member2662614233259075797ist_tm @ ( produc9218832855287345392ist_tm @ nil_rule @ Xs ) @ ( listrel_rule_tm @ R ) )
=> ( Xs = nil_tm ) ) ).
% listrel_Nil1
thf(fact_464_listrel__Nil1,axiom,
! [Xs: list_rule,R: set_Pr7067330020180114026m_rule] :
( ( member7302538096890821473t_rule @ ( produc1396096877967899076t_rule @ nil_fm @ Xs ) @ ( listrel_fm_rule @ R ) )
=> ( Xs = nil_rule ) ) ).
% listrel_Nil1
thf(fact_465_listrel__Nil1,axiom,
! [Xs: list_fm,R: set_Pr4706815898642364871_fm_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ nil_fm @ Xs ) @ ( listrel_fm_fm @ R ) )
=> ( Xs = nil_fm ) ) ).
% listrel_Nil1
thf(fact_466_listrel__Nil1,axiom,
! [Xs: list_tm,R: set_Pr4464301228316855097_fm_tm] :
( ( member1683583632484806914ist_tm @ ( produc7867948443067789129ist_tm @ nil_fm @ Xs ) @ ( listrel_fm_tm @ R ) )
=> ( Xs = nil_tm ) ) ).
% listrel_Nil1
thf(fact_467_listrel__Nil1,axiom,
! [Xs: list_rule,R: set_Pr3569831037629835996m_rule] :
( ( member2031962811952810707t_rule @ ( produc5348893629884664118t_rule @ nil_tm @ Xs ) @ ( listrel_tm_rule @ R ) )
=> ( Xs = nil_rule ) ) ).
% listrel_Nil1
thf(fact_468_listrel__Nil1,axiom,
! [Xs: list_tm,R: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ nil_tm @ Xs ) @ ( listrel_tm_tm @ R ) )
=> ( Xs = nil_tm ) ) ).
% listrel_Nil1
thf(fact_469_listrel__Nil1,axiom,
! [Xs: list_fm,R: set_Pr2698443736021152725_tm_fm] :
( ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ nil_tm @ Xs ) @ ( listrel_tm_fm @ R ) )
=> ( Xs = nil_fm ) ) ).
% listrel_Nil1
thf(fact_470_listrel__Nil1,axiom,
! [Xs: list_list_fm,R: set_Pr6384426166747533970ist_fm] :
( ( member494883543038539881ist_fm @ ( produc7454812661813866372ist_fm @ nil_rule @ Xs ) @ ( listrel_rule_list_fm @ R ) )
=> ( Xs = nil_list_fm ) ) ).
% listrel_Nil1
thf(fact_471_listrel_ONil,axiom,
! [R: set_Pr7340090144965549541e_rule] : ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ nil_rule @ nil_rule ) @ ( listrel_rule_rule @ R ) ) ).
% listrel.Nil
thf(fact_472_listrel_ONil,axiom,
! [R: set_Pr8969532344284305932ule_fm] : ( member2905128903584585571ist_fm @ ( produc9214880830201710206ist_fm @ nil_rule @ nil_fm ) @ ( listrel_rule_fm @ R ) ) ).
% listrel.Nil
thf(fact_473_listrel_ONil,axiom,
! [R: set_Pr8727017673958796158ule_tm] : ( member2662614233259075797ist_tm @ ( produc9218832855287345392ist_tm @ nil_rule @ nil_tm ) @ ( listrel_rule_tm @ R ) ) ).
% listrel.Nil
thf(fact_474_listrel_ONil,axiom,
! [R: set_Pr7067330020180114026m_rule] : ( member7302538096890821473t_rule @ ( produc1396096877967899076t_rule @ nil_fm @ nil_rule ) @ ( listrel_fm_rule @ R ) ) ).
% listrel.Nil
thf(fact_475_listrel_ONil,axiom,
! [R: set_Pr4706815898642364871_fm_fm] : ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ nil_fm @ nil_fm ) @ ( listrel_fm_fm @ R ) ) ).
% listrel.Nil
thf(fact_476_listrel_ONil,axiom,
! [R: set_Pr4464301228316855097_fm_tm] : ( member1683583632484806914ist_tm @ ( produc7867948443067789129ist_tm @ nil_fm @ nil_tm ) @ ( listrel_fm_tm @ R ) ) ).
% listrel.Nil
thf(fact_477_listrel_ONil,axiom,
! [R: set_Pr3569831037629835996m_rule] : ( member2031962811952810707t_rule @ ( produc5348893629884664118t_rule @ nil_tm @ nil_rule ) @ ( listrel_tm_rule @ R ) ) ).
% listrel.Nil
thf(fact_478_listrel_ONil,axiom,
! [R: set_Pr2455929065695642951_tm_tm] : ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ nil_tm @ nil_tm ) @ ( listrel_tm_tm @ R ) ) ).
% listrel.Nil
thf(fact_479_listrel_ONil,axiom,
! [R: set_Pr2698443736021152725_tm_fm] : ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ nil_tm @ nil_fm ) @ ( listrel_tm_fm @ R ) ) ).
% listrel.Nil
thf(fact_480_listrel_ONil,axiom,
! [R: set_Pr6384426166747533970ist_fm] : ( member494883543038539881ist_fm @ ( produc7454812661813866372ist_fm @ nil_rule @ nil_list_fm ) @ ( listrel_rule_list_fm @ R ) ) ).
% listrel.Nil
thf(fact_481_Cons__in__subseqsD,axiom,
! [Y: rule,Ys: list_rule,Xs: list_rule] :
( ( member_list_rule @ ( cons_rule @ Y @ Ys ) @ ( set_list_rule2 @ ( subseqs_rule @ Xs ) ) )
=> ( member_list_rule @ Ys @ ( set_list_rule2 @ ( subseqs_rule @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_482_Cons__in__subseqsD,axiom,
! [Y: list_fm,Ys: list_list_fm,Xs: list_list_fm] :
( ( member_list_list_fm @ ( cons_list_fm @ Y @ Ys ) @ ( set_list_list_fm2 @ ( subseqs_list_fm @ Xs ) ) )
=> ( member_list_list_fm @ Ys @ ( set_list_list_fm2 @ ( subseqs_list_fm @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_483_Cons__in__subseqsD,axiom,
! [Y: fm,Ys: list_fm,Xs: list_fm] :
( ( member_list_fm3 @ ( cons_fm @ Y @ Ys ) @ ( set_list_fm2 @ ( subseqs_fm @ Xs ) ) )
=> ( member_list_fm3 @ Ys @ ( set_list_fm2 @ ( subseqs_fm @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_484_Cons__in__subseqsD,axiom,
! [Y: tm,Ys: list_tm,Xs: list_tm] :
( ( member_list_tm @ ( cons_tm @ Y @ Ys ) @ ( set_list_tm2 @ ( subseqs_tm @ Xs ) ) )
=> ( member_list_tm @ Ys @ ( set_list_tm2 @ ( subseqs_tm @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_485_listrel__Cons2,axiom,
! [Xs: list_rule,Y: rule,Ys: list_rule,R: set_Pr7340090144965549541e_rule] :
( ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ Xs @ ( cons_rule @ Y @ Ys ) ) @ ( listrel_rule_rule @ R ) )
=> ~ ! [X3: rule,Xs2: list_rule] :
( ( Xs
= ( cons_rule @ X3 @ Xs2 ) )
=> ( ( member1444691403335511342e_rule @ ( produc5849431337705160893e_rule @ X3 @ Y ) @ R )
=> ~ ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ Xs2 @ Ys ) @ ( listrel_rule_rule @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_486_listrel__Cons2,axiom,
! [Xs: list_fm,Y: rule,Ys: list_rule,R: set_Pr7067330020180114026m_rule] :
( ( member7302538096890821473t_rule @ ( produc1396096877967899076t_rule @ Xs @ ( cons_rule @ Y @ Ys ) ) @ ( listrel_fm_rule @ R ) )
=> ~ ! [X3: fm,Xs2: list_fm] :
( ( Xs
= ( cons_fm @ X3 @ Xs2 ) )
=> ( ( member6041693456836784459m_rule @ ( product_Pair_fm_rule @ X3 @ Y ) @ R )
=> ~ ( member7302538096890821473t_rule @ ( produc1396096877967899076t_rule @ Xs2 @ Ys ) @ ( listrel_fm_rule @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_487_listrel__Cons2,axiom,
! [Xs: list_tm,Y: rule,Ys: list_rule,R: set_Pr3569831037629835996m_rule] :
( ( member2031962811952810707t_rule @ ( produc5348893629884664118t_rule @ Xs @ ( cons_rule @ Y @ Ys ) ) @ ( listrel_tm_rule @ R ) )
=> ~ ! [X3: tm,Xs2: list_tm] :
( ( Xs
= ( cons_tm @ X3 @ Xs2 ) )
=> ( ( member5482372101558860477m_rule @ ( product_Pair_tm_rule @ X3 @ Y ) @ R )
=> ~ ( member2031962811952810707t_rule @ ( produc5348893629884664118t_rule @ Xs2 @ Ys ) @ ( listrel_tm_rule @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_488_listrel__Cons2,axiom,
! [Xs: list_rule,Y: fm,Ys: list_fm,R: set_Pr8969532344284305932ule_fm] :
( ( member2905128903584585571ist_fm @ ( produc9214880830201710206ist_fm @ Xs @ ( cons_fm @ Y @ Ys ) ) @ ( listrel_rule_fm @ R ) )
=> ~ ! [X3: rule,Xs2: list_rule] :
( ( Xs
= ( cons_rule @ X3 @ Xs2 ) )
=> ( ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ X3 @ Y ) @ R )
=> ~ ( member2905128903584585571ist_fm @ ( produc9214880830201710206ist_fm @ Xs2 @ Ys ) @ ( listrel_rule_fm @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_489_listrel__Cons2,axiom,
! [Xs: list_fm,Y: fm,Ys: list_fm,R: set_Pr4706815898642364871_fm_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Xs @ ( cons_fm @ Y @ Ys ) ) @ ( listrel_fm_fm @ R ) )
=> ~ ! [X3: fm,Xs2: list_fm] :
( ( Xs
= ( cons_fm @ X3 @ Xs2 ) )
=> ( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X3 @ Y ) @ R )
=> ~ ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Xs2 @ Ys ) @ ( listrel_fm_fm @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_490_listrel__Cons2,axiom,
! [Xs: list_rule,Y: tm,Ys: list_tm,R: set_Pr8727017673958796158ule_tm] :
( ( member2662614233259075797ist_tm @ ( produc9218832855287345392ist_tm @ Xs @ ( cons_tm @ Y @ Ys ) ) @ ( listrel_rule_tm @ R ) )
=> ~ ! [X3: rule,Xs2: list_rule] :
( ( Xs
= ( cons_rule @ X3 @ Xs2 ) )
=> ( ( member9172131417393966175ule_tm @ ( product_Pair_rule_tm @ X3 @ Y ) @ R )
=> ~ ( member2662614233259075797ist_tm @ ( produc9218832855287345392ist_tm @ Xs2 @ Ys ) @ ( listrel_rule_tm @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_491_listrel__Cons2,axiom,
! [Xs: list_fm,Y: tm,Ys: list_tm,R: set_Pr4464301228316855097_fm_tm] :
( ( member1683583632484806914ist_tm @ ( produc7867948443067789129ist_tm @ Xs @ ( cons_tm @ Y @ Ys ) ) @ ( listrel_fm_tm @ R ) )
=> ~ ! [X3: fm,Xs2: list_fm] :
( ( Xs
= ( cons_fm @ X3 @ Xs2 ) )
=> ( ( member7784904625553633922_fm_tm @ ( product_Pair_fm_tm @ X3 @ Y ) @ R )
=> ~ ( member1683583632484806914ist_tm @ ( produc7867948443067789129ist_tm @ Xs2 @ Ys ) @ ( listrel_fm_tm @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_492_listrel__Cons2,axiom,
! [Xs: list_tm,Y: tm,Ys: list_tm,R: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ ( cons_tm @ Y @ Ys ) ) @ ( listrel_tm_tm @ R ) )
=> ~ ! [X3: tm,Xs2: list_tm] :
( ( Xs
= ( cons_tm @ X3 @ Xs2 ) )
=> ( ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X3 @ Y ) @ R )
=> ~ ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs2 @ Ys ) @ ( listrel_tm_tm @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_493_listrel__Cons2,axiom,
! [Xs: list_tm,Y: fm,Ys: list_fm,R: set_Pr2698443736021152725_tm_fm] :
( ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ Xs @ ( cons_fm @ Y @ Ys ) ) @ ( listrel_tm_fm @ R ) )
=> ~ ! [X3: tm,Xs2: list_tm] :
( ( Xs
= ( cons_tm @ X3 @ Xs2 ) )
=> ( ( member3117664881408846110_tm_fm @ ( product_Pair_tm_fm @ X3 @ Y ) @ R )
=> ~ ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ Xs2 @ Ys ) @ ( listrel_tm_fm @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_494_listrel__Cons2,axiom,
! [Xs: list_list_fm,Y: rule,Ys: list_rule,R: set_Pr2422821757980664944m_rule] :
( ( member6202700923738835687t_rule @ ( produc3733561588192308298t_rule @ Xs @ ( cons_rule @ Y @ Ys ) ) @ ( listrel_list_fm_rule @ R ) )
=> ~ ! [X3: list_fm,Xs2: list_list_fm] :
( ( Xs
= ( cons_list_fm @ X3 @ Xs2 ) )
=> ( ( member7183523337131018321m_rule @ ( produc8901081497008266676m_rule @ X3 @ Y ) @ R )
=> ~ ( member6202700923738835687t_rule @ ( produc3733561588192308298t_rule @ Xs2 @ Ys ) @ ( listrel_list_fm_rule @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_495_listrel__Cons1,axiom,
! [Y: rule,Ys: list_rule,Xs: list_rule,R: set_Pr7340090144965549541e_rule] :
( ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ ( cons_rule @ Y @ Ys ) @ Xs ) @ ( listrel_rule_rule @ R ) )
=> ~ ! [Y3: rule,Ys3: list_rule] :
( ( Xs
= ( cons_rule @ Y3 @ Ys3 ) )
=> ( ( member1444691403335511342e_rule @ ( produc5849431337705160893e_rule @ Y @ Y3 ) @ R )
=> ~ ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ Ys @ Ys3 ) @ ( listrel_rule_rule @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_496_listrel__Cons1,axiom,
! [Y: rule,Ys: list_rule,Xs: list_fm,R: set_Pr8969532344284305932ule_fm] :
( ( member2905128903584585571ist_fm @ ( produc9214880830201710206ist_fm @ ( cons_rule @ Y @ Ys ) @ Xs ) @ ( listrel_rule_fm @ R ) )
=> ~ ! [Y3: fm,Ys3: list_fm] :
( ( Xs
= ( cons_fm @ Y3 @ Ys3 ) )
=> ( ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ Y @ Y3 ) @ R )
=> ~ ( member2905128903584585571ist_fm @ ( produc9214880830201710206ist_fm @ Ys @ Ys3 ) @ ( listrel_rule_fm @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_497_listrel__Cons1,axiom,
! [Y: rule,Ys: list_rule,Xs: list_tm,R: set_Pr8727017673958796158ule_tm] :
( ( member2662614233259075797ist_tm @ ( produc9218832855287345392ist_tm @ ( cons_rule @ Y @ Ys ) @ Xs ) @ ( listrel_rule_tm @ R ) )
=> ~ ! [Y3: tm,Ys3: list_tm] :
( ( Xs
= ( cons_tm @ Y3 @ Ys3 ) )
=> ( ( member9172131417393966175ule_tm @ ( product_Pair_rule_tm @ Y @ Y3 ) @ R )
=> ~ ( member2662614233259075797ist_tm @ ( produc9218832855287345392ist_tm @ Ys @ Ys3 ) @ ( listrel_rule_tm @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_498_listrel__Cons1,axiom,
! [Y: fm,Ys: list_fm,Xs: list_rule,R: set_Pr7067330020180114026m_rule] :
( ( member7302538096890821473t_rule @ ( produc1396096877967899076t_rule @ ( cons_fm @ Y @ Ys ) @ Xs ) @ ( listrel_fm_rule @ R ) )
=> ~ ! [Y3: rule,Ys3: list_rule] :
( ( Xs
= ( cons_rule @ Y3 @ Ys3 ) )
=> ( ( member6041693456836784459m_rule @ ( product_Pair_fm_rule @ Y @ Y3 ) @ R )
=> ~ ( member7302538096890821473t_rule @ ( produc1396096877967899076t_rule @ Ys @ Ys3 ) @ ( listrel_fm_rule @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_499_listrel__Cons1,axiom,
! [Y: fm,Ys: list_fm,Xs: list_fm,R: set_Pr4706815898642364871_fm_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ ( cons_fm @ Y @ Ys ) @ Xs ) @ ( listrel_fm_fm @ R ) )
=> ~ ! [Y3: fm,Ys3: list_fm] :
( ( Xs
= ( cons_fm @ Y3 @ Ys3 ) )
=> ( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ Y @ Y3 ) @ R )
=> ~ ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Ys @ Ys3 ) @ ( listrel_fm_fm @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_500_listrel__Cons1,axiom,
! [Y: fm,Ys: list_fm,Xs: list_tm,R: set_Pr4464301228316855097_fm_tm] :
( ( member1683583632484806914ist_tm @ ( produc7867948443067789129ist_tm @ ( cons_fm @ Y @ Ys ) @ Xs ) @ ( listrel_fm_tm @ R ) )
=> ~ ! [Y3: tm,Ys3: list_tm] :
( ( Xs
= ( cons_tm @ Y3 @ Ys3 ) )
=> ( ( member7784904625553633922_fm_tm @ ( product_Pair_fm_tm @ Y @ Y3 ) @ R )
=> ~ ( member1683583632484806914ist_tm @ ( produc7867948443067789129ist_tm @ Ys @ Ys3 ) @ ( listrel_fm_tm @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_501_listrel__Cons1,axiom,
! [Y: tm,Ys: list_tm,Xs: list_rule,R: set_Pr3569831037629835996m_rule] :
( ( member2031962811952810707t_rule @ ( produc5348893629884664118t_rule @ ( cons_tm @ Y @ Ys ) @ Xs ) @ ( listrel_tm_rule @ R ) )
=> ~ ! [Y3: rule,Ys3: list_rule] :
( ( Xs
= ( cons_rule @ Y3 @ Ys3 ) )
=> ( ( member5482372101558860477m_rule @ ( product_Pair_tm_rule @ Y @ Y3 ) @ R )
=> ~ ( member2031962811952810707t_rule @ ( produc5348893629884664118t_rule @ Ys @ Ys3 ) @ ( listrel_tm_rule @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_502_listrel__Cons1,axiom,
! [Y: tm,Ys: list_tm,Xs: list_tm,R: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( cons_tm @ Y @ Ys ) @ Xs ) @ ( listrel_tm_tm @ R ) )
=> ~ ! [Y3: tm,Ys3: list_tm] :
( ( Xs
= ( cons_tm @ Y3 @ Ys3 ) )
=> ( ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ Y @ Y3 ) @ R )
=> ~ ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Ys @ Ys3 ) @ ( listrel_tm_tm @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_503_listrel__Cons1,axiom,
! [Y: tm,Ys: list_tm,Xs: list_fm,R: set_Pr2698443736021152725_tm_fm] :
( ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ ( cons_tm @ Y @ Ys ) @ Xs ) @ ( listrel_tm_fm @ R ) )
=> ~ ! [Y3: fm,Ys3: list_fm] :
( ( Xs
= ( cons_fm @ Y3 @ Ys3 ) )
=> ( ( member3117664881408846110_tm_fm @ ( product_Pair_tm_fm @ Y @ Y3 ) @ R )
=> ~ ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ Ys @ Ys3 ) @ ( listrel_tm_fm @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_504_listrel__Cons1,axiom,
! [Y: rule,Ys: list_rule,Xs: list_list_fm,R: set_Pr6384426166747533970ist_fm] :
( ( member494883543038539881ist_fm @ ( produc7454812661813866372ist_fm @ ( cons_rule @ Y @ Ys ) @ Xs ) @ ( listrel_rule_list_fm @ R ) )
=> ~ ! [Y3: list_fm,Ys3: list_list_fm] :
( ( Xs
= ( cons_list_fm @ Y3 @ Ys3 ) )
=> ( ( member4140259954344421235ist_fm @ ( produc2087254492244816782ist_fm @ Y @ Y3 ) @ R )
=> ~ ( member494883543038539881ist_fm @ ( produc7454812661813866372ist_fm @ Ys @ Ys3 ) @ ( listrel_rule_list_fm @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_505_listrel_OCons,axiom,
! [X: rule,Y: rule,R: set_Pr7340090144965549541e_rule,Xs: list_rule,Ys: list_rule] :
( ( member1444691403335511342e_rule @ ( produc5849431337705160893e_rule @ X @ Y ) @ R )
=> ( ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ Xs @ Ys ) @ ( listrel_rule_rule @ R ) )
=> ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ ( cons_rule @ X @ Xs ) @ ( cons_rule @ Y @ Ys ) ) @ ( listrel_rule_rule @ R ) ) ) ) ).
% listrel.Cons
thf(fact_506_listrel_OCons,axiom,
! [X: rule,Y: fm,R: set_Pr8969532344284305932ule_fm,Xs: list_rule,Ys: list_fm] :
( ( member9168179392308330989ule_fm @ ( product_Pair_rule_fm @ X @ Y ) @ R )
=> ( ( member2905128903584585571ist_fm @ ( produc9214880830201710206ist_fm @ Xs @ Ys ) @ ( listrel_rule_fm @ R ) )
=> ( member2905128903584585571ist_fm @ ( produc9214880830201710206ist_fm @ ( cons_rule @ X @ Xs ) @ ( cons_fm @ Y @ Ys ) ) @ ( listrel_rule_fm @ R ) ) ) ) ).
% listrel.Cons
thf(fact_507_listrel_OCons,axiom,
! [X: rule,Y: tm,R: set_Pr8727017673958796158ule_tm,Xs: list_rule,Ys: list_tm] :
( ( member9172131417393966175ule_tm @ ( product_Pair_rule_tm @ X @ Y ) @ R )
=> ( ( member2662614233259075797ist_tm @ ( produc9218832855287345392ist_tm @ Xs @ Ys ) @ ( listrel_rule_tm @ R ) )
=> ( member2662614233259075797ist_tm @ ( produc9218832855287345392ist_tm @ ( cons_rule @ X @ Xs ) @ ( cons_tm @ Y @ Ys ) ) @ ( listrel_rule_tm @ R ) ) ) ) ).
% listrel.Cons
thf(fact_508_listrel_OCons,axiom,
! [X: fm,Y: rule,R: set_Pr7067330020180114026m_rule,Xs: list_fm,Ys: list_rule] :
( ( member6041693456836784459m_rule @ ( product_Pair_fm_rule @ X @ Y ) @ R )
=> ( ( member7302538096890821473t_rule @ ( produc1396096877967899076t_rule @ Xs @ Ys ) @ ( listrel_fm_rule @ R ) )
=> ( member7302538096890821473t_rule @ ( produc1396096877967899076t_rule @ ( cons_fm @ X @ Xs ) @ ( cons_rule @ Y @ Ys ) ) @ ( listrel_fm_rule @ R ) ) ) ) ).
% listrel.Cons
thf(fact_509_listrel_OCons,axiom,
! [X: fm,Y: fm,R: set_Pr4706815898642364871_fm_fm,Xs: list_fm,Ys: list_fm] :
( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X @ Y ) @ R )
=> ( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Xs @ Ys ) @ ( listrel_fm_fm @ R ) )
=> ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ ( cons_fm @ X @ Xs ) @ ( cons_fm @ Y @ Ys ) ) @ ( listrel_fm_fm @ R ) ) ) ) ).
% listrel.Cons
thf(fact_510_listrel_OCons,axiom,
! [X: fm,Y: tm,R: set_Pr4464301228316855097_fm_tm,Xs: list_fm,Ys: list_tm] :
( ( member7784904625553633922_fm_tm @ ( product_Pair_fm_tm @ X @ Y ) @ R )
=> ( ( member1683583632484806914ist_tm @ ( produc7867948443067789129ist_tm @ Xs @ Ys ) @ ( listrel_fm_tm @ R ) )
=> ( member1683583632484806914ist_tm @ ( produc7867948443067789129ist_tm @ ( cons_fm @ X @ Xs ) @ ( cons_tm @ Y @ Ys ) ) @ ( listrel_fm_tm @ R ) ) ) ) ).
% listrel.Cons
thf(fact_511_listrel_OCons,axiom,
! [X: tm,Y: rule,R: set_Pr3569831037629835996m_rule,Xs: list_tm,Ys: list_rule] :
( ( member5482372101558860477m_rule @ ( product_Pair_tm_rule @ X @ Y ) @ R )
=> ( ( member2031962811952810707t_rule @ ( produc5348893629884664118t_rule @ Xs @ Ys ) @ ( listrel_tm_rule @ R ) )
=> ( member2031962811952810707t_rule @ ( produc5348893629884664118t_rule @ ( cons_tm @ X @ Xs ) @ ( cons_rule @ Y @ Ys ) ) @ ( listrel_tm_rule @ R ) ) ) ) ).
% listrel.Cons
thf(fact_512_listrel_OCons,axiom,
! [X: tm,Y: tm,R: set_Pr2455929065695642951_tm_tm,Xs: list_tm,Ys: list_tm] :
( ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X @ Y ) @ R )
=> ( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ Ys ) @ ( listrel_tm_tm @ R ) )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( cons_tm @ X @ Xs ) @ ( cons_tm @ Y @ Ys ) ) @ ( listrel_tm_tm @ R ) ) ) ) ).
% listrel.Cons
thf(fact_513_listrel_OCons,axiom,
! [X: tm,Y: fm,R: set_Pr2698443736021152725_tm_fm,Xs: list_tm,Ys: list_fm] :
( ( member3117664881408846110_tm_fm @ ( product_Pair_tm_fm @ X @ Y ) @ R )
=> ( ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ Xs @ Ys ) @ ( listrel_tm_fm @ R ) )
=> ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ ( cons_tm @ X @ Xs ) @ ( cons_fm @ Y @ Ys ) ) @ ( listrel_tm_fm @ R ) ) ) ) ).
% listrel.Cons
thf(fact_514_listrel_OCons,axiom,
! [X: rule,Y: list_fm,R: set_Pr6384426166747533970ist_fm,Xs: list_rule,Ys: list_list_fm] :
( ( member4140259954344421235ist_fm @ ( produc2087254492244816782ist_fm @ X @ Y ) @ R )
=> ( ( member494883543038539881ist_fm @ ( produc7454812661813866372ist_fm @ Xs @ Ys ) @ ( listrel_rule_list_fm @ R ) )
=> ( member494883543038539881ist_fm @ ( produc7454812661813866372ist_fm @ ( cons_rule @ X @ Xs ) @ ( cons_list_fm @ Y @ Ys ) ) @ ( listrel_rule_list_fm @ R ) ) ) ) ).
% listrel.Cons
thf(fact_515_list__ex1__simps_I1_J,axiom,
! [P3: rule > $o] :
~ ( list_ex1_rule @ P3 @ nil_rule ) ).
% list_ex1_simps(1)
thf(fact_516_list__ex1__simps_I1_J,axiom,
! [P3: fm > $o] :
~ ( list_ex1_fm @ P3 @ nil_fm ) ).
% list_ex1_simps(1)
thf(fact_517_list__ex1__simps_I1_J,axiom,
! [P3: list_fm > $o] :
~ ( list_ex1_list_fm @ P3 @ nil_list_fm ) ).
% list_ex1_simps(1)
thf(fact_518_list__ex1__simps_I1_J,axiom,
! [P3: tm > $o] :
~ ( list_ex1_tm @ P3 @ nil_tm ) ).
% list_ex1_simps(1)
thf(fact_519_lexord__Nil__left,axiom,
! [Y: list_rule,R: set_Pr7340090144965549541e_rule] :
( ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ nil_rule @ Y ) @ ( lexord_rule @ R ) )
= ( ? [A5: rule,X2: list_rule] :
( Y
= ( cons_rule @ A5 @ X2 ) ) ) ) ).
% lexord_Nil_left
thf(fact_520_lexord__Nil__left,axiom,
! [Y: list_list_fm,R: set_Pr1916137733696349511ist_fm] :
( ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ nil_list_fm @ Y ) @ ( lexord_list_fm @ R ) )
= ( ? [A5: list_fm,X2: list_list_fm] :
( Y
= ( cons_list_fm @ A5 @ X2 ) ) ) ) ).
% lexord_Nil_left
thf(fact_521_lexord__Nil__left,axiom,
! [Y: list_fm,R: set_Pr4706815898642364871_fm_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ nil_fm @ Y ) @ ( lexord_fm @ R ) )
= ( ? [A5: fm,X2: list_fm] :
( Y
= ( cons_fm @ A5 @ X2 ) ) ) ) ).
% lexord_Nil_left
thf(fact_522_lexord__Nil__left,axiom,
! [Y: list_tm,R: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ nil_tm @ Y ) @ ( lexord_tm @ R ) )
= ( ? [A5: tm,X2: list_tm] :
( Y
= ( cons_tm @ A5 @ X2 ) ) ) ) ).
% lexord_Nil_left
thf(fact_523_lexord__cons__cons,axiom,
! [A: rule,X: list_rule,B2: rule,Y: list_rule,R: set_Pr7340090144965549541e_rule] :
( ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ ( cons_rule @ A @ X ) @ ( cons_rule @ B2 @ Y ) ) @ ( lexord_rule @ R ) )
= ( ( member1444691403335511342e_rule @ ( produc5849431337705160893e_rule @ A @ B2 ) @ R )
| ( ( A = B2 )
& ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ X @ Y ) @ ( lexord_rule @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_524_lexord__cons__cons,axiom,
! [A: list_fm,X: list_list_fm,B2: list_fm,Y: list_list_fm,R: set_Pr1916137733696349511ist_fm] :
( ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ ( cons_list_fm @ A @ X ) @ ( cons_list_fm @ B2 @ Y ) ) @ ( lexord_list_fm @ R ) )
= ( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ A @ B2 ) @ R )
| ( ( A = B2 )
& ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ X @ Y ) @ ( lexord_list_fm @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_525_lexord__cons__cons,axiom,
! [A: fm,X: list_fm,B2: fm,Y: list_fm,R: set_Pr4706815898642364871_fm_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ ( cons_fm @ A @ X ) @ ( cons_fm @ B2 @ Y ) ) @ ( lexord_fm @ R ) )
= ( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ A @ B2 ) @ R )
| ( ( A = B2 )
& ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ X @ Y ) @ ( lexord_fm @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_526_lexord__cons__cons,axiom,
! [A: tm,X: list_tm,B2: tm,Y: list_tm,R: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( cons_tm @ A @ X ) @ ( cons_tm @ B2 @ Y ) ) @ ( lexord_tm @ R ) )
= ( ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ A @ B2 ) @ R )
| ( ( A = B2 )
& ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ X @ Y ) @ ( lexord_tm @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_527_Cons__listrel1__Cons,axiom,
! [X: rule,Xs: list_rule,Y: rule,Ys: list_rule,R: set_Pr7340090144965549541e_rule] :
( ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ ( cons_rule @ X @ Xs ) @ ( cons_rule @ Y @ Ys ) ) @ ( listrel1_rule @ R ) )
= ( ( ( member1444691403335511342e_rule @ ( produc5849431337705160893e_rule @ X @ Y ) @ R )
& ( Xs = Ys ) )
| ( ( X = Y )
& ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ Xs @ Ys ) @ ( listrel1_rule @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_528_Cons__listrel1__Cons,axiom,
! [X: list_fm,Xs: list_list_fm,Y: list_fm,Ys: list_list_fm,R: set_Pr1916137733696349511ist_fm] :
( ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ ( cons_list_fm @ X @ Xs ) @ ( cons_list_fm @ Y @ Ys ) ) @ ( listrel1_list_fm @ R ) )
= ( ( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ X @ Y ) @ R )
& ( Xs = Ys ) )
| ( ( X = Y )
& ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ Xs @ Ys ) @ ( listrel1_list_fm @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_529_Cons__listrel1__Cons,axiom,
! [X: fm,Xs: list_fm,Y: fm,Ys: list_fm,R: set_Pr4706815898642364871_fm_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ ( cons_fm @ X @ Xs ) @ ( cons_fm @ Y @ Ys ) ) @ ( listrel1_fm @ R ) )
= ( ( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X @ Y ) @ R )
& ( Xs = Ys ) )
| ( ( X = Y )
& ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Xs @ Ys ) @ ( listrel1_fm @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_530_Cons__listrel1__Cons,axiom,
! [X: tm,Xs: list_tm,Y: tm,Ys: list_tm,R: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( cons_tm @ X @ Xs ) @ ( cons_tm @ Y @ Ys ) ) @ ( listrel1_tm @ R ) )
= ( ( ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X @ Y ) @ R )
& ( Xs = Ys ) )
| ( ( X = Y )
& ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ Ys ) @ ( listrel1_tm @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_531_Nil__lenlex__iff1,axiom,
! [Ns: list_rule,R: set_Pr7340090144965549541e_rule] :
( ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ nil_rule @ Ns ) @ ( lenlex_rule @ R ) )
= ( Ns != nil_rule ) ) ).
% Nil_lenlex_iff1
thf(fact_532_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_533_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_534_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_535_sdrop__cycle__eq,axiom,
! [U: list_P2774625669004309958m_rule] :
( ( U != nil_Pr2808214839852828720m_rule )
=> ( ( sdrop_8169176516188972301m_rule @ ( size_s1575636608424004698m_rule @ U ) @ ( cycle_5335117900641983313m_rule @ U ) )
= ( cycle_5335117900641983313m_rule @ U ) ) ) ).
% sdrop_cycle_eq
thf(fact_536_sdrop__cycle__eq,axiom,
! [U: list_rule] :
( ( U != nil_rule )
=> ( ( sdrop_rule @ ( size_size_list_rule @ U ) @ ( cycle_rule @ U ) )
= ( cycle_rule @ U ) ) ) ).
% sdrop_cycle_eq
thf(fact_537_sdrop__cycle__eq,axiom,
! [U: list_fm] :
( ( U != nil_fm )
=> ( ( sdrop_fm @ ( size_size_list_fm @ U ) @ ( cycle_fm @ U ) )
= ( cycle_fm @ U ) ) ) ).
% sdrop_cycle_eq
thf(fact_538_sdrop__cycle__eq,axiom,
! [U: list_list_fm] :
( ( U != nil_list_fm )
=> ( ( sdrop_list_fm @ ( size_s115229985653309035ist_fm @ U ) @ ( cycle_list_fm @ U ) )
= ( cycle_list_fm @ U ) ) ) ).
% sdrop_cycle_eq
thf(fact_539_sdrop__cycle__eq,axiom,
! [U: list_tm] :
( ( U != nil_tm )
=> ( ( sdrop_tm @ ( size_size_list_tm @ U ) @ ( cycle_tm @ U ) )
= ( cycle_tm @ U ) ) ) ).
% sdrop_cycle_eq
thf(fact_540_in__set__product__lists__length,axiom,
! [Xs: list_fm,Xss2: list_list_fm] :
( ( member_list_fm3 @ Xs @ ( set_list_fm2 @ ( product_lists_fm @ Xss2 ) ) )
=> ( ( size_size_list_fm @ Xs )
= ( size_s115229985653309035ist_fm @ Xss2 ) ) ) ).
% in_set_product_lists_length
thf(fact_541_list__induct4,axiom,
! [Xs: list_rule,Ys: list_rule,Zs: list_rule,Ws: list_rule,P3: list_rule > list_rule > list_rule > list_rule > $o] :
( ( ( size_size_list_rule @ Xs )
= ( size_size_list_rule @ Ys ) )
=> ( ( ( size_size_list_rule @ Ys )
= ( size_size_list_rule @ Zs ) )
=> ( ( ( size_size_list_rule @ Zs )
= ( size_size_list_rule @ Ws ) )
=> ( ( P3 @ nil_rule @ nil_rule @ nil_rule @ nil_rule )
=> ( ! [X3: rule,Xs2: list_rule,Y3: rule,Ys3: list_rule,Z4: rule,Zs2: list_rule,W: rule,Ws2: list_rule] :
( ( ( size_size_list_rule @ Xs2 )
= ( size_size_list_rule @ Ys3 ) )
=> ( ( ( size_size_list_rule @ Ys3 )
= ( size_size_list_rule @ Zs2 ) )
=> ( ( ( size_size_list_rule @ Zs2 )
= ( size_size_list_rule @ Ws2 ) )
=> ( ( P3 @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_rule @ X3 @ Xs2 ) @ ( cons_rule @ Y3 @ Ys3 ) @ ( cons_rule @ Z4 @ Zs2 ) @ ( cons_rule @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_542_list__induct4,axiom,
! [Xs: list_rule,Ys: list_rule,Zs: list_rule,Ws: list_fm,P3: list_rule > list_rule > list_rule > list_fm > $o] :
( ( ( size_size_list_rule @ Xs )
= ( size_size_list_rule @ Ys ) )
=> ( ( ( size_size_list_rule @ Ys )
= ( size_size_list_rule @ Zs ) )
=> ( ( ( size_size_list_rule @ Zs )
= ( size_size_list_fm @ Ws ) )
=> ( ( P3 @ nil_rule @ nil_rule @ nil_rule @ nil_fm )
=> ( ! [X3: rule,Xs2: list_rule,Y3: rule,Ys3: list_rule,Z4: rule,Zs2: list_rule,W: fm,Ws2: list_fm] :
( ( ( size_size_list_rule @ Xs2 )
= ( size_size_list_rule @ Ys3 ) )
=> ( ( ( size_size_list_rule @ Ys3 )
= ( size_size_list_rule @ Zs2 ) )
=> ( ( ( size_size_list_rule @ Zs2 )
= ( size_size_list_fm @ Ws2 ) )
=> ( ( P3 @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_rule @ X3 @ Xs2 ) @ ( cons_rule @ Y3 @ Ys3 ) @ ( cons_rule @ Z4 @ Zs2 ) @ ( cons_fm @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_543_list__induct4,axiom,
! [Xs: list_rule,Ys: list_rule,Zs: list_rule,Ws: list_tm,P3: list_rule > list_rule > list_rule > list_tm > $o] :
( ( ( size_size_list_rule @ Xs )
= ( size_size_list_rule @ Ys ) )
=> ( ( ( size_size_list_rule @ Ys )
= ( size_size_list_rule @ Zs ) )
=> ( ( ( size_size_list_rule @ Zs )
= ( size_size_list_tm @ Ws ) )
=> ( ( P3 @ nil_rule @ nil_rule @ nil_rule @ nil_tm )
=> ( ! [X3: rule,Xs2: list_rule,Y3: rule,Ys3: list_rule,Z4: rule,Zs2: list_rule,W: tm,Ws2: list_tm] :
( ( ( size_size_list_rule @ Xs2 )
= ( size_size_list_rule @ Ys3 ) )
=> ( ( ( size_size_list_rule @ Ys3 )
= ( size_size_list_rule @ Zs2 ) )
=> ( ( ( size_size_list_rule @ Zs2 )
= ( size_size_list_tm @ Ws2 ) )
=> ( ( P3 @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_rule @ X3 @ Xs2 ) @ ( cons_rule @ Y3 @ Ys3 ) @ ( cons_rule @ Z4 @ Zs2 ) @ ( cons_tm @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_544_list__induct4,axiom,
! [Xs: list_rule,Ys: list_rule,Zs: list_fm,Ws: list_rule,P3: list_rule > list_rule > list_fm > list_rule > $o] :
( ( ( size_size_list_rule @ Xs )
= ( size_size_list_rule @ Ys ) )
=> ( ( ( size_size_list_rule @ Ys )
= ( size_size_list_fm @ Zs ) )
=> ( ( ( size_size_list_fm @ Zs )
= ( size_size_list_rule @ Ws ) )
=> ( ( P3 @ nil_rule @ nil_rule @ nil_fm @ nil_rule )
=> ( ! [X3: rule,Xs2: list_rule,Y3: rule,Ys3: list_rule,Z4: fm,Zs2: list_fm,W: rule,Ws2: list_rule] :
( ( ( size_size_list_rule @ Xs2 )
= ( size_size_list_rule @ Ys3 ) )
=> ( ( ( size_size_list_rule @ Ys3 )
= ( size_size_list_fm @ Zs2 ) )
=> ( ( ( size_size_list_fm @ Zs2 )
= ( size_size_list_rule @ Ws2 ) )
=> ( ( P3 @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_rule @ X3 @ Xs2 ) @ ( cons_rule @ Y3 @ Ys3 ) @ ( cons_fm @ Z4 @ Zs2 ) @ ( cons_rule @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_545_list__induct4,axiom,
! [Xs: list_rule,Ys: list_rule,Zs: list_fm,Ws: list_fm,P3: list_rule > list_rule > list_fm > list_fm > $o] :
( ( ( size_size_list_rule @ Xs )
= ( size_size_list_rule @ Ys ) )
=> ( ( ( size_size_list_rule @ Ys )
= ( size_size_list_fm @ Zs ) )
=> ( ( ( size_size_list_fm @ Zs )
= ( size_size_list_fm @ Ws ) )
=> ( ( P3 @ nil_rule @ nil_rule @ nil_fm @ nil_fm )
=> ( ! [X3: rule,Xs2: list_rule,Y3: rule,Ys3: list_rule,Z4: fm,Zs2: list_fm,W: fm,Ws2: list_fm] :
( ( ( size_size_list_rule @ Xs2 )
= ( size_size_list_rule @ Ys3 ) )
=> ( ( ( size_size_list_rule @ Ys3 )
= ( size_size_list_fm @ Zs2 ) )
=> ( ( ( size_size_list_fm @ Zs2 )
= ( size_size_list_fm @ Ws2 ) )
=> ( ( P3 @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_rule @ X3 @ Xs2 ) @ ( cons_rule @ Y3 @ Ys3 ) @ ( cons_fm @ Z4 @ Zs2 ) @ ( cons_fm @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_546_list__induct4,axiom,
! [Xs: list_rule,Ys: list_rule,Zs: list_fm,Ws: list_tm,P3: list_rule > list_rule > list_fm > list_tm > $o] :
( ( ( size_size_list_rule @ Xs )
= ( size_size_list_rule @ Ys ) )
=> ( ( ( size_size_list_rule @ Ys )
= ( size_size_list_fm @ Zs ) )
=> ( ( ( size_size_list_fm @ Zs )
= ( size_size_list_tm @ Ws ) )
=> ( ( P3 @ nil_rule @ nil_rule @ nil_fm @ nil_tm )
=> ( ! [X3: rule,Xs2: list_rule,Y3: rule,Ys3: list_rule,Z4: fm,Zs2: list_fm,W: tm,Ws2: list_tm] :
( ( ( size_size_list_rule @ Xs2 )
= ( size_size_list_rule @ Ys3 ) )
=> ( ( ( size_size_list_rule @ Ys3 )
= ( size_size_list_fm @ Zs2 ) )
=> ( ( ( size_size_list_fm @ Zs2 )
= ( size_size_list_tm @ Ws2 ) )
=> ( ( P3 @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_rule @ X3 @ Xs2 ) @ ( cons_rule @ Y3 @ Ys3 ) @ ( cons_fm @ Z4 @ Zs2 ) @ ( cons_tm @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_547_list__induct4,axiom,
! [Xs: list_rule,Ys: list_rule,Zs: list_tm,Ws: list_rule,P3: list_rule > list_rule > list_tm > list_rule > $o] :
( ( ( size_size_list_rule @ Xs )
= ( size_size_list_rule @ Ys ) )
=> ( ( ( size_size_list_rule @ Ys )
= ( size_size_list_tm @ Zs ) )
=> ( ( ( size_size_list_tm @ Zs )
= ( size_size_list_rule @ Ws ) )
=> ( ( P3 @ nil_rule @ nil_rule @ nil_tm @ nil_rule )
=> ( ! [X3: rule,Xs2: list_rule,Y3: rule,Ys3: list_rule,Z4: tm,Zs2: list_tm,W: rule,Ws2: list_rule] :
( ( ( size_size_list_rule @ Xs2 )
= ( size_size_list_rule @ Ys3 ) )
=> ( ( ( size_size_list_rule @ Ys3 )
= ( size_size_list_tm @ Zs2 ) )
=> ( ( ( size_size_list_tm @ Zs2 )
= ( size_size_list_rule @ Ws2 ) )
=> ( ( P3 @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_rule @ X3 @ Xs2 ) @ ( cons_rule @ Y3 @ Ys3 ) @ ( cons_tm @ Z4 @ Zs2 ) @ ( cons_rule @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_548_list__induct4,axiom,
! [Xs: list_rule,Ys: list_rule,Zs: list_tm,Ws: list_fm,P3: list_rule > list_rule > list_tm > list_fm > $o] :
( ( ( size_size_list_rule @ Xs )
= ( size_size_list_rule @ Ys ) )
=> ( ( ( size_size_list_rule @ Ys )
= ( size_size_list_tm @ Zs ) )
=> ( ( ( size_size_list_tm @ Zs )
= ( size_size_list_fm @ Ws ) )
=> ( ( P3 @ nil_rule @ nil_rule @ nil_tm @ nil_fm )
=> ( ! [X3: rule,Xs2: list_rule,Y3: rule,Ys3: list_rule,Z4: tm,Zs2: list_tm,W: fm,Ws2: list_fm] :
( ( ( size_size_list_rule @ Xs2 )
= ( size_size_list_rule @ Ys3 ) )
=> ( ( ( size_size_list_rule @ Ys3 )
= ( size_size_list_tm @ Zs2 ) )
=> ( ( ( size_size_list_tm @ Zs2 )
= ( size_size_list_fm @ Ws2 ) )
=> ( ( P3 @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_rule @ X3 @ Xs2 ) @ ( cons_rule @ Y3 @ Ys3 ) @ ( cons_tm @ Z4 @ Zs2 ) @ ( cons_fm @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_549_list__induct4,axiom,
! [Xs: list_rule,Ys: list_rule,Zs: list_tm,Ws: list_tm,P3: list_rule > list_rule > list_tm > list_tm > $o] :
( ( ( size_size_list_rule @ Xs )
= ( size_size_list_rule @ Ys ) )
=> ( ( ( size_size_list_rule @ Ys )
= ( size_size_list_tm @ Zs ) )
=> ( ( ( size_size_list_tm @ Zs )
= ( size_size_list_tm @ Ws ) )
=> ( ( P3 @ nil_rule @ nil_rule @ nil_tm @ nil_tm )
=> ( ! [X3: rule,Xs2: list_rule,Y3: rule,Ys3: list_rule,Z4: tm,Zs2: list_tm,W: tm,Ws2: list_tm] :
( ( ( size_size_list_rule @ Xs2 )
= ( size_size_list_rule @ Ys3 ) )
=> ( ( ( size_size_list_rule @ Ys3 )
= ( size_size_list_tm @ Zs2 ) )
=> ( ( ( size_size_list_tm @ Zs2 )
= ( size_size_list_tm @ Ws2 ) )
=> ( ( P3 @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_rule @ X3 @ Xs2 ) @ ( cons_rule @ Y3 @ Ys3 ) @ ( cons_tm @ Z4 @ Zs2 ) @ ( cons_tm @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_550_list__induct4,axiom,
! [Xs: list_rule,Ys: list_fm,Zs: list_rule,Ws: list_rule,P3: list_rule > list_fm > list_rule > list_rule > $o] :
( ( ( size_size_list_rule @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_rule @ Zs ) )
=> ( ( ( size_size_list_rule @ Zs )
= ( size_size_list_rule @ Ws ) )
=> ( ( P3 @ nil_rule @ nil_fm @ nil_rule @ nil_rule )
=> ( ! [X3: rule,Xs2: list_rule,Y3: fm,Ys3: list_fm,Z4: rule,Zs2: list_rule,W: rule,Ws2: list_rule] :
( ( ( size_size_list_rule @ Xs2 )
= ( size_size_list_fm @ Ys3 ) )
=> ( ( ( size_size_list_fm @ Ys3 )
= ( size_size_list_rule @ Zs2 ) )
=> ( ( ( size_size_list_rule @ Zs2 )
= ( size_size_list_rule @ Ws2 ) )
=> ( ( P3 @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_rule @ X3 @ Xs2 ) @ ( cons_fm @ Y3 @ Ys3 ) @ ( cons_rule @ Z4 @ Zs2 ) @ ( cons_rule @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_551_list__induct3,axiom,
! [Xs: list_rule,Ys: list_rule,Zs: list_rule,P3: list_rule > list_rule > list_rule > $o] :
( ( ( size_size_list_rule @ Xs )
= ( size_size_list_rule @ Ys ) )
=> ( ( ( size_size_list_rule @ Ys )
= ( size_size_list_rule @ Zs ) )
=> ( ( P3 @ nil_rule @ nil_rule @ nil_rule )
=> ( ! [X3: rule,Xs2: list_rule,Y3: rule,Ys3: list_rule,Z4: rule,Zs2: list_rule] :
( ( ( size_size_list_rule @ Xs2 )
= ( size_size_list_rule @ Ys3 ) )
=> ( ( ( size_size_list_rule @ Ys3 )
= ( size_size_list_rule @ Zs2 ) )
=> ( ( P3 @ Xs2 @ Ys3 @ Zs2 )
=> ( P3 @ ( cons_rule @ X3 @ Xs2 ) @ ( cons_rule @ Y3 @ Ys3 ) @ ( cons_rule @ Z4 @ Zs2 ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_552_list__induct3,axiom,
! [Xs: list_rule,Ys: list_rule,Zs: list_fm,P3: list_rule > list_rule > list_fm > $o] :
( ( ( size_size_list_rule @ Xs )
= ( size_size_list_rule @ Ys ) )
=> ( ( ( size_size_list_rule @ Ys )
= ( size_size_list_fm @ Zs ) )
=> ( ( P3 @ nil_rule @ nil_rule @ nil_fm )
=> ( ! [X3: rule,Xs2: list_rule,Y3: rule,Ys3: list_rule,Z4: fm,Zs2: list_fm] :
( ( ( size_size_list_rule @ Xs2 )
= ( size_size_list_rule @ Ys3 ) )
=> ( ( ( size_size_list_rule @ Ys3 )
= ( size_size_list_fm @ Zs2 ) )
=> ( ( P3 @ Xs2 @ Ys3 @ Zs2 )
=> ( P3 @ ( cons_rule @ X3 @ Xs2 ) @ ( cons_rule @ Y3 @ Ys3 ) @ ( cons_fm @ Z4 @ Zs2 ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_553_list__induct3,axiom,
! [Xs: list_rule,Ys: list_rule,Zs: list_tm,P3: list_rule > list_rule > list_tm > $o] :
( ( ( size_size_list_rule @ Xs )
= ( size_size_list_rule @ Ys ) )
=> ( ( ( size_size_list_rule @ Ys )
= ( size_size_list_tm @ Zs ) )
=> ( ( P3 @ nil_rule @ nil_rule @ nil_tm )
=> ( ! [X3: rule,Xs2: list_rule,Y3: rule,Ys3: list_rule,Z4: tm,Zs2: list_tm] :
( ( ( size_size_list_rule @ Xs2 )
= ( size_size_list_rule @ Ys3 ) )
=> ( ( ( size_size_list_rule @ Ys3 )
= ( size_size_list_tm @ Zs2 ) )
=> ( ( P3 @ Xs2 @ Ys3 @ Zs2 )
=> ( P3 @ ( cons_rule @ X3 @ Xs2 ) @ ( cons_rule @ Y3 @ Ys3 ) @ ( cons_tm @ Z4 @ Zs2 ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_554_list__induct3,axiom,
! [Xs: list_rule,Ys: list_fm,Zs: list_rule,P3: list_rule > list_fm > list_rule > $o] :
( ( ( size_size_list_rule @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_rule @ Zs ) )
=> ( ( P3 @ nil_rule @ nil_fm @ nil_rule )
=> ( ! [X3: rule,Xs2: list_rule,Y3: fm,Ys3: list_fm,Z4: rule,Zs2: list_rule] :
( ( ( size_size_list_rule @ Xs2 )
= ( size_size_list_fm @ Ys3 ) )
=> ( ( ( size_size_list_fm @ Ys3 )
= ( size_size_list_rule @ Zs2 ) )
=> ( ( P3 @ Xs2 @ Ys3 @ Zs2 )
=> ( P3 @ ( cons_rule @ X3 @ Xs2 ) @ ( cons_fm @ Y3 @ Ys3 ) @ ( cons_rule @ Z4 @ Zs2 ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_555_list__induct3,axiom,
! [Xs: list_rule,Ys: list_fm,Zs: list_fm,P3: list_rule > list_fm > list_fm > $o] :
( ( ( size_size_list_rule @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_fm @ Zs ) )
=> ( ( P3 @ nil_rule @ nil_fm @ nil_fm )
=> ( ! [X3: rule,Xs2: list_rule,Y3: fm,Ys3: list_fm,Z4: fm,Zs2: list_fm] :
( ( ( size_size_list_rule @ Xs2 )
= ( size_size_list_fm @ Ys3 ) )
=> ( ( ( size_size_list_fm @ Ys3 )
= ( size_size_list_fm @ Zs2 ) )
=> ( ( P3 @ Xs2 @ Ys3 @ Zs2 )
=> ( P3 @ ( cons_rule @ X3 @ Xs2 ) @ ( cons_fm @ Y3 @ Ys3 ) @ ( cons_fm @ Z4 @ Zs2 ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_556_list__induct3,axiom,
! [Xs: list_rule,Ys: list_fm,Zs: list_tm,P3: list_rule > list_fm > list_tm > $o] :
( ( ( size_size_list_rule @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( ( size_size_list_fm @ Ys )
= ( size_size_list_tm @ Zs ) )
=> ( ( P3 @ nil_rule @ nil_fm @ nil_tm )
=> ( ! [X3: rule,Xs2: list_rule,Y3: fm,Ys3: list_fm,Z4: tm,Zs2: list_tm] :
( ( ( size_size_list_rule @ Xs2 )
= ( size_size_list_fm @ Ys3 ) )
=> ( ( ( size_size_list_fm @ Ys3 )
= ( size_size_list_tm @ Zs2 ) )
=> ( ( P3 @ Xs2 @ Ys3 @ Zs2 )
=> ( P3 @ ( cons_rule @ X3 @ Xs2 ) @ ( cons_fm @ Y3 @ Ys3 ) @ ( cons_tm @ Z4 @ Zs2 ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_557_list__induct3,axiom,
! [Xs: list_rule,Ys: list_tm,Zs: list_rule,P3: list_rule > list_tm > list_rule > $o] :
( ( ( size_size_list_rule @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( ( size_size_list_tm @ Ys )
= ( size_size_list_rule @ Zs ) )
=> ( ( P3 @ nil_rule @ nil_tm @ nil_rule )
=> ( ! [X3: rule,Xs2: list_rule,Y3: tm,Ys3: list_tm,Z4: rule,Zs2: list_rule] :
( ( ( size_size_list_rule @ Xs2 )
= ( size_size_list_tm @ Ys3 ) )
=> ( ( ( size_size_list_tm @ Ys3 )
= ( size_size_list_rule @ Zs2 ) )
=> ( ( P3 @ Xs2 @ Ys3 @ Zs2 )
=> ( P3 @ ( cons_rule @ X3 @ Xs2 ) @ ( cons_tm @ Y3 @ Ys3 ) @ ( cons_rule @ Z4 @ Zs2 ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_558_list__induct3,axiom,
! [Xs: list_rule,Ys: list_tm,Zs: list_fm,P3: list_rule > list_tm > list_fm > $o] :
( ( ( size_size_list_rule @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( ( size_size_list_tm @ Ys )
= ( size_size_list_fm @ Zs ) )
=> ( ( P3 @ nil_rule @ nil_tm @ nil_fm )
=> ( ! [X3: rule,Xs2: list_rule,Y3: tm,Ys3: list_tm,Z4: fm,Zs2: list_fm] :
( ( ( size_size_list_rule @ Xs2 )
= ( size_size_list_tm @ Ys3 ) )
=> ( ( ( size_size_list_tm @ Ys3 )
= ( size_size_list_fm @ Zs2 ) )
=> ( ( P3 @ Xs2 @ Ys3 @ Zs2 )
=> ( P3 @ ( cons_rule @ X3 @ Xs2 ) @ ( cons_tm @ Y3 @ Ys3 ) @ ( cons_fm @ Z4 @ Zs2 ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_559_list__induct3,axiom,
! [Xs: list_rule,Ys: list_tm,Zs: list_tm,P3: list_rule > list_tm > list_tm > $o] :
( ( ( size_size_list_rule @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( ( size_size_list_tm @ Ys )
= ( size_size_list_tm @ Zs ) )
=> ( ( P3 @ nil_rule @ nil_tm @ nil_tm )
=> ( ! [X3: rule,Xs2: list_rule,Y3: tm,Ys3: list_tm,Z4: tm,Zs2: list_tm] :
( ( ( size_size_list_rule @ Xs2 )
= ( size_size_list_tm @ Ys3 ) )
=> ( ( ( size_size_list_tm @ Ys3 )
= ( size_size_list_tm @ Zs2 ) )
=> ( ( P3 @ Xs2 @ Ys3 @ Zs2 )
=> ( P3 @ ( cons_rule @ X3 @ Xs2 ) @ ( cons_tm @ Y3 @ Ys3 ) @ ( cons_tm @ Z4 @ Zs2 ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_560_list__induct3,axiom,
! [Xs: list_fm,Ys: list_rule,Zs: list_rule,P3: list_fm > list_rule > list_rule > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_rule @ Ys ) )
=> ( ( ( size_size_list_rule @ Ys )
= ( size_size_list_rule @ Zs ) )
=> ( ( P3 @ nil_fm @ nil_rule @ nil_rule )
=> ( ! [X3: fm,Xs2: list_fm,Y3: rule,Ys3: list_rule,Z4: rule,Zs2: list_rule] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_rule @ Ys3 ) )
=> ( ( ( size_size_list_rule @ Ys3 )
= ( size_size_list_rule @ Zs2 ) )
=> ( ( P3 @ Xs2 @ Ys3 @ Zs2 )
=> ( P3 @ ( cons_fm @ X3 @ Xs2 ) @ ( cons_rule @ Y3 @ Ys3 ) @ ( cons_rule @ Z4 @ Zs2 ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_561_list__induct2,axiom,
! [Xs: list_rule,Ys: list_rule,P3: list_rule > list_rule > $o] :
( ( ( size_size_list_rule @ Xs )
= ( size_size_list_rule @ Ys ) )
=> ( ( P3 @ nil_rule @ nil_rule )
=> ( ! [X3: rule,Xs2: list_rule,Y3: rule,Ys3: list_rule] :
( ( ( size_size_list_rule @ Xs2 )
= ( size_size_list_rule @ Ys3 ) )
=> ( ( P3 @ Xs2 @ Ys3 )
=> ( P3 @ ( cons_rule @ X3 @ Xs2 ) @ ( cons_rule @ Y3 @ Ys3 ) ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_562_list__induct2,axiom,
! [Xs: list_rule,Ys: list_fm,P3: list_rule > list_fm > $o] :
( ( ( size_size_list_rule @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( P3 @ nil_rule @ nil_fm )
=> ( ! [X3: rule,Xs2: list_rule,Y3: fm,Ys3: list_fm] :
( ( ( size_size_list_rule @ Xs2 )
= ( size_size_list_fm @ Ys3 ) )
=> ( ( P3 @ Xs2 @ Ys3 )
=> ( P3 @ ( cons_rule @ X3 @ Xs2 ) @ ( cons_fm @ Y3 @ Ys3 ) ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_563_list__induct2,axiom,
! [Xs: list_rule,Ys: list_tm,P3: list_rule > list_tm > $o] :
( ( ( size_size_list_rule @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( P3 @ nil_rule @ nil_tm )
=> ( ! [X3: rule,Xs2: list_rule,Y3: tm,Ys3: list_tm] :
( ( ( size_size_list_rule @ Xs2 )
= ( size_size_list_tm @ Ys3 ) )
=> ( ( P3 @ Xs2 @ Ys3 )
=> ( P3 @ ( cons_rule @ X3 @ Xs2 ) @ ( cons_tm @ Y3 @ Ys3 ) ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_564_list__induct2,axiom,
! [Xs: list_fm,Ys: list_rule,P3: list_fm > list_rule > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_rule @ Ys ) )
=> ( ( P3 @ nil_fm @ nil_rule )
=> ( ! [X3: fm,Xs2: list_fm,Y3: rule,Ys3: list_rule] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_rule @ Ys3 ) )
=> ( ( P3 @ Xs2 @ Ys3 )
=> ( P3 @ ( cons_fm @ X3 @ Xs2 ) @ ( cons_rule @ Y3 @ Ys3 ) ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_565_list__induct2,axiom,
! [Xs: list_fm,Ys: list_fm,P3: list_fm > list_fm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( P3 @ nil_fm @ nil_fm )
=> ( ! [X3: fm,Xs2: list_fm,Y3: fm,Ys3: list_fm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_fm @ Ys3 ) )
=> ( ( P3 @ Xs2 @ Ys3 )
=> ( P3 @ ( cons_fm @ X3 @ Xs2 ) @ ( cons_fm @ Y3 @ Ys3 ) ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_566_list__induct2,axiom,
! [Xs: list_fm,Ys: list_tm,P3: list_fm > list_tm > $o] :
( ( ( size_size_list_fm @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( P3 @ nil_fm @ nil_tm )
=> ( ! [X3: fm,Xs2: list_fm,Y3: tm,Ys3: list_tm] :
( ( ( size_size_list_fm @ Xs2 )
= ( size_size_list_tm @ Ys3 ) )
=> ( ( P3 @ Xs2 @ Ys3 )
=> ( P3 @ ( cons_fm @ X3 @ Xs2 ) @ ( cons_tm @ Y3 @ Ys3 ) ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_567_list__induct2,axiom,
! [Xs: list_tm,Ys: list_rule,P3: list_tm > list_rule > $o] :
( ( ( size_size_list_tm @ Xs )
= ( size_size_list_rule @ Ys ) )
=> ( ( P3 @ nil_tm @ nil_rule )
=> ( ! [X3: tm,Xs2: list_tm,Y3: rule,Ys3: list_rule] :
( ( ( size_size_list_tm @ Xs2 )
= ( size_size_list_rule @ Ys3 ) )
=> ( ( P3 @ Xs2 @ Ys3 )
=> ( P3 @ ( cons_tm @ X3 @ Xs2 ) @ ( cons_rule @ Y3 @ Ys3 ) ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_568_list__induct2,axiom,
! [Xs: list_tm,Ys: list_fm,P3: list_tm > list_fm > $o] :
( ( ( size_size_list_tm @ Xs )
= ( size_size_list_fm @ Ys ) )
=> ( ( P3 @ nil_tm @ nil_fm )
=> ( ! [X3: tm,Xs2: list_tm,Y3: fm,Ys3: list_fm] :
( ( ( size_size_list_tm @ Xs2 )
= ( size_size_list_fm @ Ys3 ) )
=> ( ( P3 @ Xs2 @ Ys3 )
=> ( P3 @ ( cons_tm @ X3 @ Xs2 ) @ ( cons_fm @ Y3 @ Ys3 ) ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_569_list__induct2,axiom,
! [Xs: list_tm,Ys: list_tm,P3: list_tm > list_tm > $o] :
( ( ( size_size_list_tm @ Xs )
= ( size_size_list_tm @ Ys ) )
=> ( ( P3 @ nil_tm @ nil_tm )
=> ( ! [X3: tm,Xs2: list_tm,Y3: tm,Ys3: list_tm] :
( ( ( size_size_list_tm @ Xs2 )
= ( size_size_list_tm @ Ys3 ) )
=> ( ( P3 @ Xs2 @ Ys3 )
=> ( P3 @ ( cons_tm @ X3 @ Xs2 ) @ ( cons_tm @ Y3 @ Ys3 ) ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_570_list__induct2,axiom,
! [Xs: list_rule,Ys: list_list_fm,P3: list_rule > list_list_fm > $o] :
( ( ( size_size_list_rule @ Xs )
= ( size_s115229985653309035ist_fm @ Ys ) )
=> ( ( P3 @ nil_rule @ nil_list_fm )
=> ( ! [X3: rule,Xs2: list_rule,Y3: list_fm,Ys3: list_list_fm] :
( ( ( size_size_list_rule @ Xs2 )
= ( size_s115229985653309035ist_fm @ Ys3 ) )
=> ( ( P3 @ Xs2 @ Ys3 )
=> ( P3 @ ( cons_rule @ X3 @ Xs2 ) @ ( cons_list_fm @ Y3 @ Ys3 ) ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_571_listrel__eq__len,axiom,
! [Xs: list_tm,Ys: list_fm,R: set_Pr2698443736021152725_tm_fm] :
( ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ Xs @ Ys ) @ ( listrel_tm_fm @ R ) )
=> ( ( size_size_list_tm @ Xs )
= ( size_size_list_fm @ Ys ) ) ) ).
% listrel_eq_len
thf(fact_572_listrel1I2,axiom,
! [Xs: list_rule,Ys: list_rule,R: set_Pr7340090144965549541e_rule,X: rule] :
( ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ Xs @ Ys ) @ ( listrel1_rule @ R ) )
=> ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ ( cons_rule @ X @ Xs ) @ ( cons_rule @ X @ Ys ) ) @ ( listrel1_rule @ R ) ) ) ).
% listrel1I2
thf(fact_573_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_574_listrel1I2,axiom,
! [Xs: list_fm,Ys: list_fm,R: set_Pr4706815898642364871_fm_fm,X: fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Xs @ Ys ) @ ( listrel1_fm @ R ) )
=> ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ ( cons_fm @ X @ Xs ) @ ( cons_fm @ X @ Ys ) ) @ ( listrel1_fm @ R ) ) ) ).
% listrel1I2
thf(fact_575_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_576_not__Nil__listrel1,axiom,
! [Xs: list_rule,R: set_Pr7340090144965549541e_rule] :
~ ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ nil_rule @ Xs ) @ ( listrel1_rule @ R ) ) ).
% not_Nil_listrel1
thf(fact_577_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_578_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_579_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_580_not__listrel1__Nil,axiom,
! [Xs: list_rule,R: set_Pr7340090144965549541e_rule] :
~ ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ Xs @ nil_rule ) @ ( listrel1_rule @ R ) ) ).
% not_listrel1_Nil
thf(fact_581_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_582_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_583_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_584_lexord__Nil__right,axiom,
! [X: list_rule,R: set_Pr7340090144965549541e_rule] :
~ ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ X @ nil_rule ) @ ( lexord_rule @ R ) ) ).
% lexord_Nil_right
thf(fact_585_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_586_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_587_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_588_list__ex1__iff,axiom,
( list_e4959145360368724013m_rule
= ( ^ [P5: produc340336539035504054m_rule > $o,Xs3: list_P2774625669004309958m_rule] :
? [X2: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X2 @ ( set_Pr4534715572506550497m_rule @ Xs3 ) )
& ( P5 @ X2 )
& ! [Y4: produc340336539035504054m_rule] :
( ( ( member7231649785386036813m_rule @ Y4 @ ( set_Pr4534715572506550497m_rule @ Xs3 ) )
& ( P5 @ Y4 ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_589_list__ex1__iff,axiom,
( list_ex1_nat
= ( ^ [P5: nat > $o,Xs3: list_nat] :
? [X2: nat] :
( ( member_nat3 @ X2 @ ( set_nat2 @ Xs3 ) )
& ( P5 @ X2 )
& ! [Y4: nat] :
( ( ( member_nat3 @ Y4 @ ( set_nat2 @ Xs3 ) )
& ( P5 @ Y4 ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_590_list__ex1__iff,axiom,
( list_ex1_fm
= ( ^ [P5: fm > $o,Xs3: list_fm] :
? [X2: fm] :
( ( member_fm3 @ X2 @ ( set_fm2 @ Xs3 ) )
& ( P5 @ X2 )
& ! [Y4: fm] :
( ( ( member_fm3 @ Y4 @ ( set_fm2 @ Xs3 ) )
& ( P5 @ Y4 ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_591_list__ex1__iff,axiom,
( list_ex1_list_fm
= ( ^ [P5: list_fm > $o,Xs3: list_list_fm] :
? [X2: list_fm] :
( ( member_list_fm3 @ X2 @ ( set_list_fm2 @ Xs3 ) )
& ( P5 @ X2 )
& ! [Y4: list_fm] :
( ( ( member_list_fm3 @ Y4 @ ( set_list_fm2 @ Xs3 ) )
& ( P5 @ Y4 ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_592_list__ex1__iff,axiom,
( list_ex1_tm
= ( ^ [P5: tm > $o,Xs3: list_tm] :
? [X2: tm] :
( ( member_tm3 @ X2 @ ( set_tm2 @ Xs3 ) )
& ( P5 @ X2 )
& ! [Y4: tm] :
( ( ( member_tm3 @ Y4 @ ( set_tm2 @ Xs3 ) )
& ( P5 @ Y4 ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_593_Nil__lenlex__iff2,axiom,
! [Ns: list_rule,R: set_Pr7340090144965549541e_rule] :
~ ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ Ns @ nil_rule ) @ ( lenlex_rule @ R ) ) ).
% Nil_lenlex_iff2
thf(fact_594_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_595_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_596_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_597_listrel1I1,axiom,
! [X: rule,Y: rule,R: set_Pr7340090144965549541e_rule,Xs: list_rule] :
( ( member1444691403335511342e_rule @ ( produc5849431337705160893e_rule @ X @ Y ) @ R )
=> ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ ( cons_rule @ X @ Xs ) @ ( cons_rule @ Y @ Xs ) ) @ ( listrel1_rule @ R ) ) ) ).
% listrel1I1
thf(fact_598_listrel1I1,axiom,
! [X: list_fm,Y: list_fm,R: set_Pr1916137733696349511ist_fm,Xs: list_list_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ X @ Y ) @ R )
=> ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ ( cons_list_fm @ X @ Xs ) @ ( cons_list_fm @ Y @ Xs ) ) @ ( listrel1_list_fm @ R ) ) ) ).
% listrel1I1
thf(fact_599_listrel1I1,axiom,
! [X: fm,Y: fm,R: set_Pr4706815898642364871_fm_fm,Xs: list_fm] :
( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X @ Y ) @ R )
=> ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ ( cons_fm @ X @ Xs ) @ ( cons_fm @ Y @ Xs ) ) @ ( listrel1_fm @ R ) ) ) ).
% listrel1I1
thf(fact_600_listrel1I1,axiom,
! [X: tm,Y: tm,R: set_Pr2455929065695642951_tm_tm,Xs: list_tm] :
( ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X @ Y ) @ R )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( cons_tm @ X @ Xs ) @ ( cons_tm @ Y @ Xs ) ) @ ( listrel1_tm @ R ) ) ) ).
% listrel1I1
thf(fact_601_Cons__listrel1E1,axiom,
! [X: rule,Xs: list_rule,Ys: list_rule,R: set_Pr7340090144965549541e_rule] :
( ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ ( cons_rule @ X @ Xs ) @ Ys ) @ ( listrel1_rule @ R ) )
=> ( ! [Y3: rule] :
( ( Ys
= ( cons_rule @ Y3 @ Xs ) )
=> ~ ( member1444691403335511342e_rule @ ( produc5849431337705160893e_rule @ X @ Y3 ) @ R ) )
=> ~ ! [Zs2: list_rule] :
( ( Ys
= ( cons_rule @ X @ Zs2 ) )
=> ~ ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ Xs @ Zs2 ) @ ( listrel1_rule @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_602_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 ) )
=> ( ! [Y3: list_fm] :
( ( Ys
= ( cons_list_fm @ Y3 @ Xs ) )
=> ~ ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ X @ Y3 ) @ R ) )
=> ~ ! [Zs2: list_list_fm] :
( ( Ys
= ( cons_list_fm @ X @ Zs2 ) )
=> ~ ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ Xs @ Zs2 ) @ ( listrel1_list_fm @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_603_Cons__listrel1E1,axiom,
! [X: fm,Xs: list_fm,Ys: list_fm,R: set_Pr4706815898642364871_fm_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ ( cons_fm @ X @ Xs ) @ Ys ) @ ( listrel1_fm @ R ) )
=> ( ! [Y3: fm] :
( ( Ys
= ( cons_fm @ Y3 @ Xs ) )
=> ~ ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X @ Y3 ) @ R ) )
=> ~ ! [Zs2: list_fm] :
( ( Ys
= ( cons_fm @ X @ Zs2 ) )
=> ~ ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Xs @ Zs2 ) @ ( listrel1_fm @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_604_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 ) )
=> ( ! [Y3: tm] :
( ( Ys
= ( cons_tm @ Y3 @ Xs ) )
=> ~ ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X @ Y3 ) @ R ) )
=> ~ ! [Zs2: list_tm] :
( ( Ys
= ( cons_tm @ X @ Zs2 ) )
=> ~ ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ Zs2 ) @ ( listrel1_tm @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_605_Cons__listrel1E2,axiom,
! [Xs: list_rule,Y: rule,Ys: list_rule,R: set_Pr7340090144965549541e_rule] :
( ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ Xs @ ( cons_rule @ Y @ Ys ) ) @ ( listrel1_rule @ R ) )
=> ( ! [X3: rule] :
( ( Xs
= ( cons_rule @ X3 @ Ys ) )
=> ~ ( member1444691403335511342e_rule @ ( produc5849431337705160893e_rule @ X3 @ Y ) @ R ) )
=> ~ ! [Zs2: list_rule] :
( ( Xs
= ( cons_rule @ Y @ Zs2 ) )
=> ~ ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ Zs2 @ Ys ) @ ( listrel1_rule @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_606_Cons__listrel1E2,axiom,
! [Xs: list_list_fm,Y: list_fm,Ys: list_list_fm,R: set_Pr1916137733696349511ist_fm] :
( ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ Xs @ ( cons_list_fm @ Y @ Ys ) ) @ ( listrel1_list_fm @ R ) )
=> ( ! [X3: list_fm] :
( ( Xs
= ( cons_list_fm @ X3 @ Ys ) )
=> ~ ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ X3 @ Y ) @ R ) )
=> ~ ! [Zs2: list_list_fm] :
( ( Xs
= ( cons_list_fm @ Y @ Zs2 ) )
=> ~ ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ Zs2 @ Ys ) @ ( listrel1_list_fm @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_607_Cons__listrel1E2,axiom,
! [Xs: list_fm,Y: fm,Ys: list_fm,R: set_Pr4706815898642364871_fm_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Xs @ ( cons_fm @ Y @ Ys ) ) @ ( listrel1_fm @ R ) )
=> ( ! [X3: fm] :
( ( Xs
= ( cons_fm @ X3 @ Ys ) )
=> ~ ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X3 @ Y ) @ R ) )
=> ~ ! [Zs2: list_fm] :
( ( Xs
= ( cons_fm @ Y @ Zs2 ) )
=> ~ ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Zs2 @ Ys ) @ ( listrel1_fm @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_608_Cons__listrel1E2,axiom,
! [Xs: list_tm,Y: tm,Ys: list_tm,R: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ ( cons_tm @ Y @ Ys ) ) @ ( listrel1_tm @ R ) )
=> ( ! [X3: tm] :
( ( Xs
= ( cons_tm @ X3 @ Ys ) )
=> ~ ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X3 @ Y ) @ R ) )
=> ~ ! [Zs2: list_tm] :
( ( Xs
= ( cons_tm @ Y @ Zs2 ) )
=> ~ ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Zs2 @ Ys ) @ ( listrel1_tm @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_609_lexord__partial__trans,axiom,
! [Xs: list_P2774625669004309958m_rule,R: set_Pr4971326047967503661m_rule,Ys: list_P2774625669004309958m_rule,Zs: list_P2774625669004309958m_rule] :
( ! [X3: produc340336539035504054m_rule,Y3: produc340336539035504054m_rule,Z4: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X3 @ ( set_Pr4534715572506550497m_rule @ Xs ) )
=> ( ( member1411645983481675894m_rule @ ( produc6261311607089640965m_rule @ X3 @ Y3 ) @ R )
=> ( ( member1411645983481675894m_rule @ ( produc6261311607089640965m_rule @ Y3 @ Z4 ) @ R )
=> ( member1411645983481675894m_rule @ ( produc6261311607089640965m_rule @ X3 @ Z4 ) @ R ) ) ) )
=> ( ( member3128744185454049558m_rule @ ( produc6692282267706994341m_rule @ Xs @ Ys ) @ ( lexord6196499238003582211m_rule @ R ) )
=> ( ( member3128744185454049558m_rule @ ( produc6692282267706994341m_rule @ Ys @ Zs ) @ ( lexord6196499238003582211m_rule @ R ) )
=> ( member3128744185454049558m_rule @ ( produc6692282267706994341m_rule @ Xs @ Zs ) @ ( lexord6196499238003582211m_rule @ R ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_610_lexord__partial__trans,axiom,
! [Xs: list_nat,R: set_Pr1261947904930325089at_nat,Ys: list_nat,Zs: list_nat] :
( ! [X3: nat,Y3: nat,Z4: nat] :
( ( member_nat3 @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z4 ) @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Z4 ) @ R ) ) ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lexord_nat @ R ) )
=> ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lexord_nat @ R ) )
=> ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Zs ) @ ( lexord_nat @ R ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_611_lexord__partial__trans,axiom,
! [Xs: list_fm,R: set_Pr4706815898642364871_fm_fm,Ys: list_fm,Zs: list_fm] :
( ! [X3: fm,Y3: fm,Z4: fm] :
( ( member_fm3 @ X3 @ ( set_fm2 @ Xs ) )
=> ( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X3 @ Y3 ) @ R )
=> ( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ Y3 @ Z4 ) @ R )
=> ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X3 @ Z4 ) @ 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_612_lexord__partial__trans,axiom,
! [Xs: list_list_fm,R: set_Pr1916137733696349511ist_fm,Ys: list_list_fm,Zs: list_list_fm] :
( ! [X3: list_fm,Y3: list_fm,Z4: list_fm] :
( ( member_list_fm3 @ X3 @ ( set_list_fm2 @ Xs ) )
=> ( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ X3 @ Y3 ) @ R )
=> ( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Y3 @ Z4 ) @ R )
=> ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ X3 @ Z4 ) @ 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_613_lexord__partial__trans,axiom,
! [Xs: list_tm,R: set_Pr2455929065695642951_tm_tm,Ys: list_tm,Zs: list_tm] :
( ! [X3: tm,Y3: tm,Z4: tm] :
( ( member_tm3 @ X3 @ ( set_tm2 @ Xs ) )
=> ( ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X3 @ Y3 ) @ R )
=> ( ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ Y3 @ Z4 ) @ R )
=> ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X3 @ Z4 ) @ 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_614_can__select__set__list__ex1,axiom,
! [P3: fm > $o,A2: list_fm] :
( ( can_select_fm @ P3 @ ( set_fm2 @ A2 ) )
= ( list_ex1_fm @ P3 @ A2 ) ) ).
% can_select_set_list_ex1
thf(fact_615_can__select__set__list__ex1,axiom,
! [P3: list_fm > $o,A2: list_list_fm] :
( ( can_select_list_fm @ P3 @ ( set_list_fm2 @ A2 ) )
= ( list_ex1_list_fm @ P3 @ A2 ) ) ).
% can_select_set_list_ex1
thf(fact_616_can__select__set__list__ex1,axiom,
! [P3: tm > $o,A2: list_tm] :
( ( can_select_tm @ P3 @ ( set_tm2 @ A2 ) )
= ( list_ex1_tm @ P3 @ A2 ) ) ).
% can_select_set_list_ex1
thf(fact_617_Cons__in__lex,axiom,
! [X: rule,Xs: list_rule,Y: rule,Ys: list_rule,R: set_Pr7340090144965549541e_rule] :
( ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ ( cons_rule @ X @ Xs ) @ ( cons_rule @ Y @ Ys ) ) @ ( lex_rule @ R ) )
= ( ( ( member1444691403335511342e_rule @ ( produc5849431337705160893e_rule @ X @ Y ) @ R )
& ( ( size_size_list_rule @ Xs )
= ( size_size_list_rule @ Ys ) ) )
| ( ( X = Y )
& ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ Xs @ Ys ) @ ( lex_rule @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_618_Cons__in__lex,axiom,
! [X: list_fm,Xs: list_list_fm,Y: list_fm,Ys: list_list_fm,R: set_Pr1916137733696349511ist_fm] :
( ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ ( cons_list_fm @ X @ Xs ) @ ( cons_list_fm @ Y @ Ys ) ) @ ( lex_list_fm @ R ) )
= ( ( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ X @ Y ) @ R )
& ( ( size_s115229985653309035ist_fm @ Xs )
= ( size_s115229985653309035ist_fm @ Ys ) ) )
| ( ( X = Y )
& ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ Xs @ Ys ) @ ( lex_list_fm @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_619_Cons__in__lex,axiom,
! [X: fm,Xs: list_fm,Y: fm,Ys: list_fm,R: set_Pr4706815898642364871_fm_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ ( cons_fm @ X @ Xs ) @ ( cons_fm @ Y @ Ys ) ) @ ( lex_fm @ R ) )
= ( ( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X @ Y ) @ R )
& ( ( size_size_list_fm @ Xs )
= ( size_size_list_fm @ Ys ) ) )
| ( ( X = Y )
& ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Xs @ Ys ) @ ( lex_fm @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_620_Cons__in__lex,axiom,
! [X: tm,Xs: list_tm,Y: tm,Ys: list_tm,R: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( cons_tm @ X @ Xs ) @ ( cons_tm @ Y @ Ys ) ) @ ( lex_tm @ R ) )
= ( ( ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X @ Y ) @ R )
& ( ( size_size_list_tm @ Xs )
= ( size_size_list_tm @ Ys ) ) )
| ( ( X = Y )
& ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ Ys ) @ ( lex_tm @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_621_Cons__lenlex__iff,axiom,
! [M: rule,Ms: list_rule,N2: rule,Ns: list_rule,R: set_Pr7340090144965549541e_rule] :
( ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ ( cons_rule @ M @ Ms ) @ ( cons_rule @ N2 @ Ns ) ) @ ( lenlex_rule @ R ) )
= ( ( ord_less_nat @ ( size_size_list_rule @ Ms ) @ ( size_size_list_rule @ Ns ) )
| ( ( ( size_size_list_rule @ Ms )
= ( size_size_list_rule @ Ns ) )
& ( member1444691403335511342e_rule @ ( produc5849431337705160893e_rule @ M @ N2 ) @ R ) )
| ( ( M = N2 )
& ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ Ms @ Ns ) @ ( lenlex_rule @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_622_Cons__lenlex__iff,axiom,
! [M: list_fm,Ms: list_list_fm,N2: list_fm,Ns: list_list_fm,R: set_Pr1916137733696349511ist_fm] :
( ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ ( cons_list_fm @ M @ Ms ) @ ( cons_list_fm @ N2 @ 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 @ N2 ) @ R ) )
| ( ( M = N2 )
& ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ Ms @ Ns ) @ ( lenlex_list_fm @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_623_Cons__lenlex__iff,axiom,
! [M: fm,Ms: list_fm,N2: fm,Ns: list_fm,R: set_Pr4706815898642364871_fm_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ ( cons_fm @ M @ Ms ) @ ( cons_fm @ N2 @ Ns ) ) @ ( lenlex_fm @ R ) )
= ( ( ord_less_nat @ ( size_size_list_fm @ Ms ) @ ( size_size_list_fm @ Ns ) )
| ( ( ( size_size_list_fm @ Ms )
= ( size_size_list_fm @ Ns ) )
& ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ M @ N2 ) @ R ) )
| ( ( M = N2 )
& ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Ms @ Ns ) @ ( lenlex_fm @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_624_Cons__lenlex__iff,axiom,
! [M: tm,Ms: list_tm,N2: tm,Ns: list_tm,R: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( cons_tm @ M @ Ms ) @ ( cons_tm @ N2 @ 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 @ N2 ) @ R ) )
| ( ( M = N2 )
& ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Ms @ Ns ) @ ( lenlex_tm @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_625_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_rule,X: rule,Ys: list_rule,Y: rule,R: set_Pr7340090144965549541e_rule] :
( ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ ( append_rule @ Xs @ ( cons_rule @ X @ nil_rule ) ) @ ( append_rule @ Ys @ ( cons_rule @ Y @ nil_rule ) ) ) @ ( listrel1_rule @ R ) )
= ( ( ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ Xs @ Ys ) @ ( listrel1_rule @ R ) )
& ( X = Y ) )
| ( ( Xs = Ys )
& ( member1444691403335511342e_rule @ ( produc5849431337705160893e_rule @ X @ Y ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_626_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_list_fm,X: list_fm,Ys: list_list_fm,Y: 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 @ Y @ nil_list_fm ) ) ) @ ( listrel1_list_fm @ R ) )
= ( ( ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ Xs @ Ys ) @ ( listrel1_list_fm @ R ) )
& ( X = Y ) )
| ( ( Xs = Ys )
& ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ X @ Y ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_627_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_fm,X: fm,Ys: list_fm,Y: fm,R: set_Pr4706815898642364871_fm_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ ( append_fm @ Xs @ ( cons_fm @ X @ nil_fm ) ) @ ( append_fm @ Ys @ ( cons_fm @ Y @ nil_fm ) ) ) @ ( listrel1_fm @ R ) )
= ( ( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Xs @ Ys ) @ ( listrel1_fm @ R ) )
& ( X = Y ) )
| ( ( Xs = Ys )
& ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X @ Y ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_628_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_tm,X: tm,Ys: list_tm,Y: tm,R: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( append_tm @ Xs @ ( cons_tm @ X @ nil_tm ) ) @ ( append_tm @ Ys @ ( cons_tm @ Y @ nil_tm ) ) ) @ ( listrel1_tm @ R ) )
= ( ( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ Ys ) @ ( listrel1_tm @ R ) )
& ( X = Y ) )
| ( ( Xs = Ys )
& ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X @ Y ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_629_stake__cycle__eq,axiom,
! [U: list_rule] :
( ( U != nil_rule )
=> ( ( stake_rule @ ( size_size_list_rule @ U ) @ ( cycle_rule @ U ) )
= U ) ) ).
% stake_cycle_eq
thf(fact_630_stake__cycle__eq,axiom,
! [U: list_fm] :
( ( U != nil_fm )
=> ( ( stake_fm @ ( size_size_list_fm @ U ) @ ( cycle_fm @ U ) )
= U ) ) ).
% stake_cycle_eq
thf(fact_631_stake__cycle__eq,axiom,
! [U: list_list_fm] :
( ( U != nil_list_fm )
=> ( ( stake_list_fm @ ( size_s115229985653309035ist_fm @ U ) @ ( cycle_list_fm @ U ) )
= U ) ) ).
% stake_cycle_eq
thf(fact_632_stake__cycle__eq,axiom,
! [U: list_tm] :
( ( U != nil_tm )
=> ( ( stake_tm @ ( size_size_list_tm @ U ) @ ( cycle_tm @ U ) )
= U ) ) ).
% stake_cycle_eq
thf(fact_633_length__n__lists__elem,axiom,
! [Ys: list_fm,N2: nat,Xs: list_fm] :
( ( member_list_fm3 @ Ys @ ( set_list_fm2 @ ( n_lists_fm @ N2 @ Xs ) ) )
=> ( ( size_size_list_fm @ Ys )
= N2 ) ) ).
% length_n_lists_elem
thf(fact_634_append_Oassoc,axiom,
! [A: list_tm,B2: list_tm,C: list_tm] :
( ( append_tm @ ( append_tm @ A @ B2 ) @ C )
= ( append_tm @ A @ ( append_tm @ B2 @ C ) ) ) ).
% append.assoc
thf(fact_635_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_636_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_637_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_638_append_Oright__neutral,axiom,
! [A: list_rule] :
( ( append_rule @ A @ nil_rule )
= A ) ).
% append.right_neutral
thf(fact_639_append_Oright__neutral,axiom,
! [A: list_fm] :
( ( append_fm @ A @ nil_fm )
= A ) ).
% append.right_neutral
thf(fact_640_append_Oright__neutral,axiom,
! [A: list_list_fm] :
( ( append_list_fm @ A @ nil_list_fm )
= A ) ).
% append.right_neutral
thf(fact_641_append_Oright__neutral,axiom,
! [A: list_tm] :
( ( append_tm @ A @ nil_tm )
= A ) ).
% append.right_neutral
thf(fact_642_append__Nil2,axiom,
! [Xs: list_rule] :
( ( append_rule @ Xs @ nil_rule )
= Xs ) ).
% append_Nil2
thf(fact_643_append__Nil2,axiom,
! [Xs: list_fm] :
( ( append_fm @ Xs @ nil_fm )
= Xs ) ).
% append_Nil2
thf(fact_644_append__Nil2,axiom,
! [Xs: list_list_fm] :
( ( append_list_fm @ Xs @ nil_list_fm )
= Xs ) ).
% append_Nil2
thf(fact_645_append__Nil2,axiom,
! [Xs: list_tm] :
( ( append_tm @ Xs @ nil_tm )
= Xs ) ).
% append_Nil2
thf(fact_646_append__self__conv,axiom,
! [Xs: list_rule,Ys: list_rule] :
( ( ( append_rule @ Xs @ Ys )
= Xs )
= ( Ys = nil_rule ) ) ).
% append_self_conv
thf(fact_647_append__self__conv,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( ( append_fm @ Xs @ Ys )
= Xs )
= ( Ys = nil_fm ) ) ).
% append_self_conv
thf(fact_648_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_649_append__self__conv,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( ( append_tm @ Xs @ Ys )
= Xs )
= ( Ys = nil_tm ) ) ).
% append_self_conv
thf(fact_650_self__append__conv,axiom,
! [Y: list_rule,Ys: list_rule] :
( ( Y
= ( append_rule @ Y @ Ys ) )
= ( Ys = nil_rule ) ) ).
% self_append_conv
thf(fact_651_self__append__conv,axiom,
! [Y: list_fm,Ys: list_fm] :
( ( Y
= ( append_fm @ Y @ Ys ) )
= ( Ys = nil_fm ) ) ).
% self_append_conv
thf(fact_652_self__append__conv,axiom,
! [Y: list_list_fm,Ys: list_list_fm] :
( ( Y
= ( append_list_fm @ Y @ Ys ) )
= ( Ys = nil_list_fm ) ) ).
% self_append_conv
thf(fact_653_self__append__conv,axiom,
! [Y: list_tm,Ys: list_tm] :
( ( Y
= ( append_tm @ Y @ Ys ) )
= ( Ys = nil_tm ) ) ).
% self_append_conv
thf(fact_654_append__self__conv2,axiom,
! [Xs: list_rule,Ys: list_rule] :
( ( ( append_rule @ Xs @ Ys )
= Ys )
= ( Xs = nil_rule ) ) ).
% append_self_conv2
thf(fact_655_append__self__conv2,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( ( append_fm @ Xs @ Ys )
= Ys )
= ( Xs = nil_fm ) ) ).
% append_self_conv2
thf(fact_656_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_657_append__self__conv2,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( ( append_tm @ Xs @ Ys )
= Ys )
= ( Xs = nil_tm ) ) ).
% append_self_conv2
thf(fact_658_self__append__conv2,axiom,
! [Y: list_rule,Xs: list_rule] :
( ( Y
= ( append_rule @ Xs @ Y ) )
= ( Xs = nil_rule ) ) ).
% self_append_conv2
thf(fact_659_self__append__conv2,axiom,
! [Y: list_fm,Xs: list_fm] :
( ( Y
= ( append_fm @ Xs @ Y ) )
= ( Xs = nil_fm ) ) ).
% self_append_conv2
thf(fact_660_self__append__conv2,axiom,
! [Y: list_list_fm,Xs: list_list_fm] :
( ( Y
= ( append_list_fm @ Xs @ Y ) )
= ( Xs = nil_list_fm ) ) ).
% self_append_conv2
thf(fact_661_self__append__conv2,axiom,
! [Y: list_tm,Xs: list_tm] :
( ( Y
= ( append_tm @ Xs @ Y ) )
= ( Xs = nil_tm ) ) ).
% self_append_conv2
thf(fact_662_Nil__is__append__conv,axiom,
! [Xs: list_rule,Ys: list_rule] :
( ( nil_rule
= ( append_rule @ Xs @ Ys ) )
= ( ( Xs = nil_rule )
& ( Ys = nil_rule ) ) ) ).
% Nil_is_append_conv
thf(fact_663_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_664_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_665_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_666_append__is__Nil__conv,axiom,
! [Xs: list_rule,Ys: list_rule] :
( ( ( append_rule @ Xs @ Ys )
= nil_rule )
= ( ( Xs = nil_rule )
& ( Ys = nil_rule ) ) ) ).
% append_is_Nil_conv
thf(fact_667_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_668_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_669_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_670_append__eq__append__conv,axiom,
! [Xs: list_tm,Ys: list_tm,Us: list_tm,Vs: list_tm] :
( ( ( ( size_size_list_tm @ Xs )
= ( size_size_list_tm @ Ys ) )
| ( ( size_size_list_tm @ Us )
= ( size_size_list_tm @ Vs ) ) )
=> ( ( ( append_tm @ Xs @ Us )
= ( append_tm @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_671_shift__append,axiom,
! [Xs: list_tm,Ys: list_tm,S: stream_tm] :
( ( shift_tm @ ( append_tm @ Xs @ Ys ) @ S )
= ( shift_tm @ Xs @ ( shift_tm @ Ys @ S ) ) ) ).
% shift_append
thf(fact_672_shift__append,axiom,
! [Xs: list_P2774625669004309958m_rule,Ys: list_P2774625669004309958m_rule,S: stream2709947120125613254m_rule] :
( ( shift_2334844276748245581m_rule @ ( append2714808343334685275m_rule @ Xs @ Ys ) @ S )
= ( shift_2334844276748245581m_rule @ Xs @ ( shift_2334844276748245581m_rule @ Ys @ S ) ) ) ).
% shift_append
thf(fact_673_append1__eq__conv,axiom,
! [Xs: list_rule,X: rule,Ys: list_rule,Y: rule] :
( ( ( append_rule @ Xs @ ( cons_rule @ X @ nil_rule ) )
= ( append_rule @ Ys @ ( cons_rule @ Y @ nil_rule ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_674_append1__eq__conv,axiom,
! [Xs: list_list_fm,X: list_fm,Ys: list_list_fm,Y: list_fm] :
( ( ( append_list_fm @ Xs @ ( cons_list_fm @ X @ nil_list_fm ) )
= ( append_list_fm @ Ys @ ( cons_list_fm @ Y @ nil_list_fm ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_675_append1__eq__conv,axiom,
! [Xs: list_fm,X: fm,Ys: list_fm,Y: fm] :
( ( ( append_fm @ Xs @ ( cons_fm @ X @ nil_fm ) )
= ( append_fm @ Ys @ ( cons_fm @ Y @ nil_fm ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_676_append1__eq__conv,axiom,
! [Xs: list_tm,X: tm,Ys: list_tm,Y: tm] :
( ( ( append_tm @ Xs @ ( cons_tm @ X @ nil_tm ) )
= ( append_tm @ Ys @ ( cons_tm @ Y @ nil_tm ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_677_set__append,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( set_fm2 @ ( append_fm @ Xs @ Ys ) )
= ( sup_sup_set_fm @ ( set_fm2 @ Xs ) @ ( set_fm2 @ Ys ) ) ) ).
% set_append
thf(fact_678_set__append,axiom,
! [Xs: list_list_fm,Ys: list_list_fm] :
( ( set_list_fm2 @ ( append_list_fm @ Xs @ Ys ) )
= ( sup_sup_set_list_fm @ ( set_list_fm2 @ Xs ) @ ( set_list_fm2 @ Ys ) ) ) ).
% set_append
thf(fact_679_set__append,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( set_tm2 @ ( append_tm @ Xs @ Ys ) )
= ( sup_sup_set_tm @ ( set_tm2 @ Xs ) @ ( set_tm2 @ Ys ) ) ) ).
% set_append
thf(fact_680_set__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( set_nat2 @ ( append_nat @ Xs @ Ys ) )
= ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) ) ).
% set_append
thf(fact_681_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_682_append__eq__appendI,axiom,
! [Xs: list_tm,Xs1: list_tm,Zs: list_tm,Ys: list_tm,Us: list_tm] :
( ( ( append_tm @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_tm @ Xs1 @ Us ) )
=> ( ( append_tm @ Xs @ Ys )
= ( append_tm @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_683_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 ) )
= ( ? [Us2: list_tm] :
( ( ( Xs
= ( append_tm @ Zs @ Us2 ) )
& ( ( append_tm @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_tm @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_tm @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_684_append__Cons,axiom,
! [X: rule,Xs: list_rule,Ys: list_rule] :
( ( append_rule @ ( cons_rule @ X @ Xs ) @ Ys )
= ( cons_rule @ X @ ( append_rule @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_685_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_686_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_687_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_688_Cons__eq__appendI,axiom,
! [X: rule,Xs1: list_rule,Ys: list_rule,Xs: list_rule,Zs: list_rule] :
( ( ( cons_rule @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_rule @ Xs1 @ Zs ) )
=> ( ( cons_rule @ X @ Xs )
= ( append_rule @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_689_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_690_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_691_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_692_append__Nil,axiom,
! [Ys: list_rule] :
( ( append_rule @ nil_rule @ Ys )
= Ys ) ).
% append_Nil
thf(fact_693_append__Nil,axiom,
! [Ys: list_fm] :
( ( append_fm @ nil_fm @ Ys )
= Ys ) ).
% append_Nil
thf(fact_694_append__Nil,axiom,
! [Ys: list_list_fm] :
( ( append_list_fm @ nil_list_fm @ Ys )
= Ys ) ).
% append_Nil
thf(fact_695_append__Nil,axiom,
! [Ys: list_tm] :
( ( append_tm @ nil_tm @ Ys )
= Ys ) ).
% append_Nil
thf(fact_696_append_Oleft__neutral,axiom,
! [A: list_rule] :
( ( append_rule @ nil_rule @ A )
= A ) ).
% append.left_neutral
thf(fact_697_append_Oleft__neutral,axiom,
! [A: list_fm] :
( ( append_fm @ nil_fm @ A )
= A ) ).
% append.left_neutral
thf(fact_698_append_Oleft__neutral,axiom,
! [A: list_list_fm] :
( ( append_list_fm @ nil_list_fm @ A )
= A ) ).
% append.left_neutral
thf(fact_699_append_Oleft__neutral,axiom,
! [A: list_tm] :
( ( append_tm @ nil_tm @ A )
= A ) ).
% append.left_neutral
thf(fact_700_eq__Nil__appendI,axiom,
! [Xs: list_rule,Ys: list_rule] :
( ( Xs = Ys )
=> ( Xs
= ( append_rule @ nil_rule @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_701_eq__Nil__appendI,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( Xs = Ys )
=> ( Xs
= ( append_fm @ nil_fm @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_702_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_703_eq__Nil__appendI,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( Xs = Ys )
=> ( Xs
= ( append_tm @ nil_tm @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_704_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_705_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_706_lex__append__rightI,axiom,
! [Xs: list_tm,Ys: list_tm,R: set_Pr2455929065695642951_tm_tm,Vs: list_tm,Us: list_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ Ys ) @ ( lex_tm @ R ) )
=> ( ( ( size_size_list_tm @ Vs )
= ( size_size_list_tm @ Us ) )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( append_tm @ Xs @ Us ) @ ( append_tm @ Ys @ Vs ) ) @ ( lex_tm @ R ) ) ) ) ).
% lex_append_rightI
thf(fact_707_can__select__def,axiom,
( can_select_fm
= ( ^ [P5: fm > $o,A6: set_fm] :
? [X2: fm] :
( ( member_fm3 @ X2 @ A6 )
& ( P5 @ X2 )
& ! [Y4: fm] :
( ( ( member_fm3 @ Y4 @ A6 )
& ( P5 @ Y4 ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_708_can__select__def,axiom,
( can_se4506553828710032889m_rule
= ( ^ [P5: produc340336539035504054m_rule > $o,A6: set_Pr1822751329126368876m_rule] :
? [X2: produc340336539035504054m_rule] :
( ( member7231649785386036813m_rule @ X2 @ A6 )
& ( P5 @ X2 )
& ! [Y4: produc340336539035504054m_rule] :
( ( ( member7231649785386036813m_rule @ Y4 @ A6 )
& ( P5 @ Y4 ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_709_can__select__def,axiom,
( can_select_list_fm
= ( ^ [P5: list_fm > $o,A6: set_list_fm] :
? [X2: list_fm] :
( ( member_list_fm3 @ X2 @ A6 )
& ( P5 @ X2 )
& ! [Y4: list_fm] :
( ( ( member_list_fm3 @ Y4 @ A6 )
& ( P5 @ Y4 ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_710_can__select__def,axiom,
( can_select_tm
= ( ^ [P5: tm > $o,A6: set_tm] :
? [X2: tm] :
( ( member_tm3 @ X2 @ A6 )
& ( P5 @ X2 )
& ! [Y4: tm] :
( ( ( member_tm3 @ Y4 @ A6 )
& ( P5 @ Y4 ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_711_can__select__def,axiom,
( can_select_nat
= ( ^ [P5: nat > $o,A6: set_nat] :
? [X2: nat] :
( ( member_nat3 @ X2 @ A6 )
& ( P5 @ X2 )
& ! [Y4: nat] :
( ( ( member_nat3 @ Y4 @ A6 )
& ( P5 @ Y4 ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_712_sup_Ostrict__coboundedI2,axiom,
! [C: set_nat,B2: set_nat,A: set_nat] :
( ( ord_less_set_nat @ C @ B2 )
=> ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_713_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B2: nat,A: nat] :
( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_714_sup_Ostrict__coboundedI1,axiom,
! [C: set_nat,A: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ C @ A )
=> ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_715_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_nat @ C @ A )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_716_sup_Ostrict__order__iff,axiom,
( ord_less_set_nat
= ( ^ [B5: set_nat,A5: set_nat] :
( ( A5
= ( sup_sup_set_nat @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_717_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B5: nat,A5: nat] :
( ( A5
= ( sup_sup_nat @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_718_sup_Ostrict__boundedE,axiom,
! [B2: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A )
=> ~ ( ( ord_less_set_nat @ B2 @ A )
=> ~ ( ord_less_set_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_719_sup_Ostrict__boundedE,axiom,
! [B2: nat,C: nat,A: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B2 @ C ) @ A )
=> ~ ( ( ord_less_nat @ B2 @ A )
=> ~ ( ord_less_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_720_sup_Oabsorb4,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ( ( sup_sup_set_nat @ A @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_721_sup_Oabsorb4,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( sup_sup_nat @ A @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_722_sup_Oabsorb3,axiom,
! [B2: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B2 @ A )
=> ( ( sup_sup_set_nat @ A @ B2 )
= A ) ) ).
% sup.absorb3
thf(fact_723_sup_Oabsorb3,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( ( sup_sup_nat @ A @ B2 )
= A ) ) ).
% sup.absorb3
thf(fact_724_less__supI2,axiom,
! [X: set_nat,B2: set_nat,A: set_nat] :
( ( ord_less_set_nat @ X @ B2 )
=> ( ord_less_set_nat @ X @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% less_supI2
thf(fact_725_less__supI2,axiom,
! [X: nat,B2: nat,A: nat] :
( ( ord_less_nat @ X @ B2 )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% less_supI2
thf(fact_726_less__supI1,axiom,
! [X: set_nat,A: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ X @ A )
=> ( ord_less_set_nat @ X @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% less_supI1
thf(fact_727_less__supI1,axiom,
! [X: nat,A: nat,B2: nat] :
( ( ord_less_nat @ X @ A )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% less_supI1
thf(fact_728_rev__induct,axiom,
! [P3: list_rule > $o,Xs: list_rule] :
( ( P3 @ nil_rule )
=> ( ! [X3: rule,Xs2: list_rule] :
( ( P3 @ Xs2 )
=> ( P3 @ ( append_rule @ Xs2 @ ( cons_rule @ X3 @ nil_rule ) ) ) )
=> ( P3 @ Xs ) ) ) ).
% rev_induct
thf(fact_729_rev__induct,axiom,
! [P3: list_list_fm > $o,Xs: list_list_fm] :
( ( P3 @ nil_list_fm )
=> ( ! [X3: list_fm,Xs2: list_list_fm] :
( ( P3 @ Xs2 )
=> ( P3 @ ( append_list_fm @ Xs2 @ ( cons_list_fm @ X3 @ nil_list_fm ) ) ) )
=> ( P3 @ Xs ) ) ) ).
% rev_induct
thf(fact_730_rev__induct,axiom,
! [P3: list_fm > $o,Xs: list_fm] :
( ( P3 @ nil_fm )
=> ( ! [X3: fm,Xs2: list_fm] :
( ( P3 @ Xs2 )
=> ( P3 @ ( append_fm @ Xs2 @ ( cons_fm @ X3 @ nil_fm ) ) ) )
=> ( P3 @ Xs ) ) ) ).
% rev_induct
thf(fact_731_rev__induct,axiom,
! [P3: list_tm > $o,Xs: list_tm] :
( ( P3 @ nil_tm )
=> ( ! [X3: tm,Xs2: list_tm] :
( ( P3 @ Xs2 )
=> ( P3 @ ( append_tm @ Xs2 @ ( cons_tm @ X3 @ nil_tm ) ) ) )
=> ( P3 @ Xs ) ) ) ).
% rev_induct
thf(fact_732_rev__exhaust,axiom,
! [Xs: list_rule] :
( ( Xs != nil_rule )
=> ~ ! [Ys3: list_rule,Y3: rule] :
( Xs
!= ( append_rule @ Ys3 @ ( cons_rule @ Y3 @ nil_rule ) ) ) ) ).
% rev_exhaust
thf(fact_733_rev__exhaust,axiom,
! [Xs: list_list_fm] :
( ( Xs != nil_list_fm )
=> ~ ! [Ys3: list_list_fm,Y3: list_fm] :
( Xs
!= ( append_list_fm @ Ys3 @ ( cons_list_fm @ Y3 @ nil_list_fm ) ) ) ) ).
% rev_exhaust
thf(fact_734_rev__exhaust,axiom,
! [Xs: list_fm] :
( ( Xs != nil_fm )
=> ~ ! [Ys3: list_fm,Y3: fm] :
( Xs
!= ( append_fm @ Ys3 @ ( cons_fm @ Y3 @ nil_fm ) ) ) ) ).
% rev_exhaust
thf(fact_735_rev__exhaust,axiom,
! [Xs: list_tm] :
( ( Xs != nil_tm )
=> ~ ! [Ys3: list_tm,Y3: tm] :
( Xs
!= ( append_tm @ Ys3 @ ( cons_tm @ Y3 @ nil_tm ) ) ) ) ).
% rev_exhaust
thf(fact_736_Cons__eq__append__conv,axiom,
! [X: rule,Xs: list_rule,Ys: list_rule,Zs: list_rule] :
( ( ( cons_rule @ X @ Xs )
= ( append_rule @ Ys @ Zs ) )
= ( ( ( Ys = nil_rule )
& ( ( cons_rule @ X @ Xs )
= Zs ) )
| ? [Ys4: list_rule] :
( ( ( cons_rule @ X @ Ys4 )
= Ys )
& ( Xs
= ( append_rule @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_737_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 ) )
| ? [Ys4: list_list_fm] :
( ( ( cons_list_fm @ X @ Ys4 )
= Ys )
& ( Xs
= ( append_list_fm @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_738_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 ) )
| ? [Ys4: list_fm] :
( ( ( cons_fm @ X @ Ys4 )
= Ys )
& ( Xs
= ( append_fm @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_739_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 ) )
| ? [Ys4: list_tm] :
( ( ( cons_tm @ X @ Ys4 )
= Ys )
& ( Xs
= ( append_tm @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_740_append__eq__Cons__conv,axiom,
! [Ys: list_rule,Zs: list_rule,X: rule,Xs: list_rule] :
( ( ( append_rule @ Ys @ Zs )
= ( cons_rule @ X @ Xs ) )
= ( ( ( Ys = nil_rule )
& ( Zs
= ( cons_rule @ X @ Xs ) ) )
| ? [Ys4: list_rule] :
( ( Ys
= ( cons_rule @ X @ Ys4 ) )
& ( ( append_rule @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_741_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 ) ) )
| ? [Ys4: list_list_fm] :
( ( Ys
= ( cons_list_fm @ X @ Ys4 ) )
& ( ( append_list_fm @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_742_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 ) ) )
| ? [Ys4: list_fm] :
( ( Ys
= ( cons_fm @ X @ Ys4 ) )
& ( ( append_fm @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_743_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 ) ) )
| ? [Ys4: list_tm] :
( ( Ys
= ( cons_tm @ X @ Ys4 ) )
& ( ( append_tm @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_744_rev__nonempty__induct,axiom,
! [Xs: list_rule,P3: list_rule > $o] :
( ( Xs != nil_rule )
=> ( ! [X3: rule] : ( P3 @ ( cons_rule @ X3 @ nil_rule ) )
=> ( ! [X3: rule,Xs2: list_rule] :
( ( Xs2 != nil_rule )
=> ( ( P3 @ Xs2 )
=> ( P3 @ ( append_rule @ Xs2 @ ( cons_rule @ X3 @ nil_rule ) ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_745_rev__nonempty__induct,axiom,
! [Xs: list_list_fm,P3: list_list_fm > $o] :
( ( Xs != nil_list_fm )
=> ( ! [X3: list_fm] : ( P3 @ ( cons_list_fm @ X3 @ nil_list_fm ) )
=> ( ! [X3: list_fm,Xs2: list_list_fm] :
( ( Xs2 != nil_list_fm )
=> ( ( P3 @ Xs2 )
=> ( P3 @ ( append_list_fm @ Xs2 @ ( cons_list_fm @ X3 @ nil_list_fm ) ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_746_rev__nonempty__induct,axiom,
! [Xs: list_fm,P3: list_fm > $o] :
( ( Xs != nil_fm )
=> ( ! [X3: fm] : ( P3 @ ( cons_fm @ X3 @ nil_fm ) )
=> ( ! [X3: fm,Xs2: list_fm] :
( ( Xs2 != nil_fm )
=> ( ( P3 @ Xs2 )
=> ( P3 @ ( append_fm @ Xs2 @ ( cons_fm @ X3 @ nil_fm ) ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_747_rev__nonempty__induct,axiom,
! [Xs: list_tm,P3: list_tm > $o] :
( ( Xs != nil_tm )
=> ( ! [X3: tm] : ( P3 @ ( cons_tm @ X3 @ nil_tm ) )
=> ( ! [X3: tm,Xs2: list_tm] :
( ( Xs2 != nil_tm )
=> ( ( P3 @ Xs2 )
=> ( P3 @ ( append_tm @ Xs2 @ ( cons_tm @ X3 @ nil_tm ) ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_748_split__list__first__prop__iff,axiom,
! [Xs: list_rule,P3: rule > $o] :
( ( ? [X2: rule] :
( ( member_rule3 @ X2 @ ( set_rule2 @ Xs ) )
& ( P3 @ X2 ) ) )
= ( ? [Ys2: list_rule,X2: rule] :
( ? [Zs3: list_rule] :
( Xs
= ( append_rule @ Ys2 @ ( cons_rule @ X2 @ Zs3 ) ) )
& ( P3 @ X2 )
& ! [Y4: rule] :
( ( member_rule3 @ Y4 @ ( set_rule2 @ Ys2 ) )
=> ~ ( P3 @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_749_split__list__first__prop__iff,axiom,
! [Xs: list_list_fm,P3: list_fm > $o] :
( ( ? [X2: list_fm] :
( ( member_list_fm3 @ X2 @ ( set_list_fm2 @ Xs ) )
& ( P3 @ X2 ) ) )
= ( ? [Ys2: list_list_fm,X2: list_fm] :
( ? [Zs3: list_list_fm] :
( Xs
= ( append_list_fm @ Ys2 @ ( cons_list_fm @ X2 @ Zs3 ) ) )
& ( P3 @ X2 )
& ! [Y4: list_fm] :
( ( member_list_fm3 @ Y4 @ ( set_list_fm2 @ Ys2 ) )
=> ~ ( P3 @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_750_split__list__first__prop__iff,axiom,
! [Xs: list_fm,P3: fm > $o] :
( ( ? [X2: fm] :
( ( member_fm3 @ X2 @ ( set_fm2 @ Xs ) )
& ( P3 @ X2 ) ) )
= ( ? [Ys2: list_fm,X2: fm] :
( ? [Zs3: list_fm] :
( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X2 @ Zs3 ) ) )
& ( P3 @ X2 )
& ! [Y4: fm] :
( ( member_fm3 @ Y4 @ ( set_fm2 @ Ys2 ) )
=> ~ ( P3 @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_751_split__list__first__prop__iff,axiom,
! [Xs: list_tm,P3: tm > $o] :
( ( ? [X2: tm] :
( ( member_tm3 @ X2 @ ( set_tm2 @ Xs ) )
& ( P3 @ X2 ) ) )
= ( ? [Ys2: list_tm,X2: tm] :
( ? [Zs3: list_tm] :
( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X2 @ Zs3 ) ) )
& ( P3 @ X2 )
& ! [Y4: tm] :
( ( member_tm3 @ Y4 @ ( set_tm2 @ Ys2 ) )
=> ~ ( P3 @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_752_split__list__last__prop__iff,axiom,
! [Xs: list_rule,P3: rule > $o] :
( ( ? [X2: rule] :
( ( member_rule3 @ X2 @ ( set_rule2 @ Xs ) )
& ( P3 @ X2 ) ) )
= ( ? [Ys2: list_rule,X2: rule,Zs3: list_rule] :
( ( Xs
= ( append_rule @ Ys2 @ ( cons_rule @ X2 @ Zs3 ) ) )
& ( P3 @ X2 )
& ! [Y4: rule] :
( ( member_rule3 @ Y4 @ ( set_rule2 @ Zs3 ) )
=> ~ ( P3 @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_753_split__list__last__prop__iff,axiom,
! [Xs: list_list_fm,P3: list_fm > $o] :
( ( ? [X2: list_fm] :
( ( member_list_fm3 @ X2 @ ( set_list_fm2 @ Xs ) )
& ( P3 @ X2 ) ) )
= ( ? [Ys2: list_list_fm,X2: list_fm,Zs3: list_list_fm] :
( ( Xs
= ( append_list_fm @ Ys2 @ ( cons_list_fm @ X2 @ Zs3 ) ) )
& ( P3 @ X2 )
& ! [Y4: list_fm] :
( ( member_list_fm3 @ Y4 @ ( set_list_fm2 @ Zs3 ) )
=> ~ ( P3 @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_754_split__list__last__prop__iff,axiom,
! [Xs: list_fm,P3: fm > $o] :
( ( ? [X2: fm] :
( ( member_fm3 @ X2 @ ( set_fm2 @ Xs ) )
& ( P3 @ X2 ) ) )
= ( ? [Ys2: list_fm,X2: fm,Zs3: list_fm] :
( ( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X2 @ Zs3 ) ) )
& ( P3 @ X2 )
& ! [Y4: fm] :
( ( member_fm3 @ Y4 @ ( set_fm2 @ Zs3 ) )
=> ~ ( P3 @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_755_split__list__last__prop__iff,axiom,
! [Xs: list_tm,P3: tm > $o] :
( ( ? [X2: tm] :
( ( member_tm3 @ X2 @ ( set_tm2 @ Xs ) )
& ( P3 @ X2 ) ) )
= ( ? [Ys2: list_tm,X2: tm,Zs3: list_tm] :
( ( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X2 @ Zs3 ) ) )
& ( P3 @ X2 )
& ! [Y4: tm] :
( ( member_tm3 @ Y4 @ ( set_tm2 @ Zs3 ) )
=> ~ ( P3 @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_756_in__set__conv__decomp__first,axiom,
! [X: produc340336539035504054m_rule,Xs: list_P2774625669004309958m_rule] :
( ( member7231649785386036813m_rule @ X @ ( set_Pr4534715572506550497m_rule @ Xs ) )
= ( ? [Ys2: list_P2774625669004309958m_rule,Zs3: list_P2774625669004309958m_rule] :
( ( Xs
= ( append2714808343334685275m_rule @ Ys2 @ ( cons_P2818572562524668032m_rule @ X @ Zs3 ) ) )
& ~ ( member7231649785386036813m_rule @ X @ ( set_Pr4534715572506550497m_rule @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_757_in__set__conv__decomp__first,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat3 @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys2: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs3 ) ) )
& ~ ( member_nat3 @ X @ ( set_nat2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_758_in__set__conv__decomp__first,axiom,
! [X: rule,Xs: list_rule] :
( ( member_rule3 @ X @ ( set_rule2 @ Xs ) )
= ( ? [Ys2: list_rule,Zs3: list_rule] :
( ( Xs
= ( append_rule @ Ys2 @ ( cons_rule @ X @ Zs3 ) ) )
& ~ ( member_rule3 @ X @ ( set_rule2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_759_in__set__conv__decomp__first,axiom,
! [X: list_fm,Xs: list_list_fm] :
( ( member_list_fm3 @ X @ ( set_list_fm2 @ Xs ) )
= ( ? [Ys2: list_list_fm,Zs3: list_list_fm] :
( ( Xs
= ( append_list_fm @ Ys2 @ ( cons_list_fm @ X @ Zs3 ) ) )
& ~ ( member_list_fm3 @ X @ ( set_list_fm2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_760_in__set__conv__decomp__first,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm3 @ X @ ( set_fm2 @ Xs ) )
= ( ? [Ys2: list_fm,Zs3: list_fm] :
( ( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X @ Zs3 ) ) )
& ~ ( member_fm3 @ X @ ( set_fm2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_761_in__set__conv__decomp__first,axiom,
! [X: tm,Xs: list_tm] :
( ( member_tm3 @ X @ ( set_tm2 @ Xs ) )
= ( ? [Ys2: list_tm,Zs3: list_tm] :
( ( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X @ Zs3 ) ) )
& ~ ( member_tm3 @ X @ ( set_tm2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_762_in__set__conv__decomp__last,axiom,
! [X: produc340336539035504054m_rule,Xs: list_P2774625669004309958m_rule] :
( ( member7231649785386036813m_rule @ X @ ( set_Pr4534715572506550497m_rule @ Xs ) )
= ( ? [Ys2: list_P2774625669004309958m_rule,Zs3: list_P2774625669004309958m_rule] :
( ( Xs
= ( append2714808343334685275m_rule @ Ys2 @ ( cons_P2818572562524668032m_rule @ X @ Zs3 ) ) )
& ~ ( member7231649785386036813m_rule @ X @ ( set_Pr4534715572506550497m_rule @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_763_in__set__conv__decomp__last,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat3 @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys2: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs3 ) ) )
& ~ ( member_nat3 @ X @ ( set_nat2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_764_in__set__conv__decomp__last,axiom,
! [X: rule,Xs: list_rule] :
( ( member_rule3 @ X @ ( set_rule2 @ Xs ) )
= ( ? [Ys2: list_rule,Zs3: list_rule] :
( ( Xs
= ( append_rule @ Ys2 @ ( cons_rule @ X @ Zs3 ) ) )
& ~ ( member_rule3 @ X @ ( set_rule2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_765_in__set__conv__decomp__last,axiom,
! [X: list_fm,Xs: list_list_fm] :
( ( member_list_fm3 @ X @ ( set_list_fm2 @ Xs ) )
= ( ? [Ys2: list_list_fm,Zs3: list_list_fm] :
( ( Xs
= ( append_list_fm @ Ys2 @ ( cons_list_fm @ X @ Zs3 ) ) )
& ~ ( member_list_fm3 @ X @ ( set_list_fm2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_766_in__set__conv__decomp__last,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm3 @ X @ ( set_fm2 @ Xs ) )
= ( ? [Ys2: list_fm,Zs3: list_fm] :
( ( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X @ Zs3 ) ) )
& ~ ( member_fm3 @ X @ ( set_fm2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_767_in__set__conv__decomp__last,axiom,
! [X: tm,Xs: list_tm] :
( ( member_tm3 @ X @ ( set_tm2 @ Xs ) )
= ( ? [Ys2: list_tm,Zs3: list_tm] :
( ( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X @ Zs3 ) ) )
& ~ ( member_tm3 @ X @ ( set_tm2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_768_split__list__first__propE,axiom,
! [Xs: list_rule,P3: rule > $o] :
( ? [X4: rule] :
( ( member_rule3 @ X4 @ ( set_rule2 @ Xs ) )
& ( P3 @ X4 ) )
=> ~ ! [Ys3: list_rule,X3: rule] :
( ? [Zs2: list_rule] :
( Xs
= ( append_rule @ Ys3 @ ( cons_rule @ X3 @ Zs2 ) ) )
=> ( ( P3 @ X3 )
=> ~ ! [Xa: rule] :
( ( member_rule3 @ Xa @ ( set_rule2 @ Ys3 ) )
=> ~ ( P3 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_769_split__list__first__propE,axiom,
! [Xs: list_list_fm,P3: list_fm > $o] :
( ? [X4: list_fm] :
( ( member_list_fm3 @ X4 @ ( set_list_fm2 @ Xs ) )
& ( P3 @ X4 ) )
=> ~ ! [Ys3: list_list_fm,X3: list_fm] :
( ? [Zs2: list_list_fm] :
( Xs
= ( append_list_fm @ Ys3 @ ( cons_list_fm @ X3 @ Zs2 ) ) )
=> ( ( P3 @ X3 )
=> ~ ! [Xa: list_fm] :
( ( member_list_fm3 @ Xa @ ( set_list_fm2 @ Ys3 ) )
=> ~ ( P3 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_770_split__list__first__propE,axiom,
! [Xs: list_fm,P3: fm > $o] :
( ? [X4: fm] :
( ( member_fm3 @ X4 @ ( set_fm2 @ Xs ) )
& ( P3 @ X4 ) )
=> ~ ! [Ys3: list_fm,X3: fm] :
( ? [Zs2: list_fm] :
( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X3 @ Zs2 ) ) )
=> ( ( P3 @ X3 )
=> ~ ! [Xa: fm] :
( ( member_fm3 @ Xa @ ( set_fm2 @ Ys3 ) )
=> ~ ( P3 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_771_split__list__first__propE,axiom,
! [Xs: list_tm,P3: tm > $o] :
( ? [X4: tm] :
( ( member_tm3 @ X4 @ ( set_tm2 @ Xs ) )
& ( P3 @ X4 ) )
=> ~ ! [Ys3: list_tm,X3: tm] :
( ? [Zs2: list_tm] :
( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ X3 @ Zs2 ) ) )
=> ( ( P3 @ X3 )
=> ~ ! [Xa: tm] :
( ( member_tm3 @ Xa @ ( set_tm2 @ Ys3 ) )
=> ~ ( P3 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_772_split__list__last__propE,axiom,
! [Xs: list_rule,P3: rule > $o] :
( ? [X4: rule] :
( ( member_rule3 @ X4 @ ( set_rule2 @ Xs ) )
& ( P3 @ X4 ) )
=> ~ ! [Ys3: list_rule,X3: rule,Zs2: list_rule] :
( ( Xs
= ( append_rule @ Ys3 @ ( cons_rule @ X3 @ Zs2 ) ) )
=> ( ( P3 @ X3 )
=> ~ ! [Xa: rule] :
( ( member_rule3 @ Xa @ ( set_rule2 @ Zs2 ) )
=> ~ ( P3 @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_773_split__list__last__propE,axiom,
! [Xs: list_list_fm,P3: list_fm > $o] :
( ? [X4: list_fm] :
( ( member_list_fm3 @ X4 @ ( set_list_fm2 @ Xs ) )
& ( P3 @ X4 ) )
=> ~ ! [Ys3: list_list_fm,X3: list_fm,Zs2: list_list_fm] :
( ( Xs
= ( append_list_fm @ Ys3 @ ( cons_list_fm @ X3 @ Zs2 ) ) )
=> ( ( P3 @ X3 )
=> ~ ! [Xa: list_fm] :
( ( member_list_fm3 @ Xa @ ( set_list_fm2 @ Zs2 ) )
=> ~ ( P3 @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_774_split__list__last__propE,axiom,
! [Xs: list_fm,P3: fm > $o] :
( ? [X4: fm] :
( ( member_fm3 @ X4 @ ( set_fm2 @ Xs ) )
& ( P3 @ X4 ) )
=> ~ ! [Ys3: list_fm,X3: fm,Zs2: list_fm] :
( ( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X3 @ Zs2 ) ) )
=> ( ( P3 @ X3 )
=> ~ ! [Xa: fm] :
( ( member_fm3 @ Xa @ ( set_fm2 @ Zs2 ) )
=> ~ ( P3 @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_775_split__list__last__propE,axiom,
! [Xs: list_tm,P3: tm > $o] :
( ? [X4: tm] :
( ( member_tm3 @ X4 @ ( set_tm2 @ Xs ) )
& ( P3 @ X4 ) )
=> ~ ! [Ys3: list_tm,X3: tm,Zs2: list_tm] :
( ( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ X3 @ Zs2 ) ) )
=> ( ( P3 @ X3 )
=> ~ ! [Xa: tm] :
( ( member_tm3 @ Xa @ ( set_tm2 @ Zs2 ) )
=> ~ ( P3 @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_776_split__list__first__prop,axiom,
! [Xs: list_rule,P3: rule > $o] :
( ? [X4: rule] :
( ( member_rule3 @ X4 @ ( set_rule2 @ Xs ) )
& ( P3 @ X4 ) )
=> ? [Ys3: list_rule,X3: rule] :
( ? [Zs2: list_rule] :
( Xs
= ( append_rule @ Ys3 @ ( cons_rule @ X3 @ Zs2 ) ) )
& ( P3 @ X3 )
& ! [Xa: rule] :
( ( member_rule3 @ Xa @ ( set_rule2 @ Ys3 ) )
=> ~ ( P3 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_777_split__list__first__prop,axiom,
! [Xs: list_list_fm,P3: list_fm > $o] :
( ? [X4: list_fm] :
( ( member_list_fm3 @ X4 @ ( set_list_fm2 @ Xs ) )
& ( P3 @ X4 ) )
=> ? [Ys3: list_list_fm,X3: list_fm] :
( ? [Zs2: list_list_fm] :
( Xs
= ( append_list_fm @ Ys3 @ ( cons_list_fm @ X3 @ Zs2 ) ) )
& ( P3 @ X3 )
& ! [Xa: list_fm] :
( ( member_list_fm3 @ Xa @ ( set_list_fm2 @ Ys3 ) )
=> ~ ( P3 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_778_split__list__first__prop,axiom,
! [Xs: list_fm,P3: fm > $o] :
( ? [X4: fm] :
( ( member_fm3 @ X4 @ ( set_fm2 @ Xs ) )
& ( P3 @ X4 ) )
=> ? [Ys3: list_fm,X3: fm] :
( ? [Zs2: list_fm] :
( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X3 @ Zs2 ) ) )
& ( P3 @ X3 )
& ! [Xa: fm] :
( ( member_fm3 @ Xa @ ( set_fm2 @ Ys3 ) )
=> ~ ( P3 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_779_split__list__first__prop,axiom,
! [Xs: list_tm,P3: tm > $o] :
( ? [X4: tm] :
( ( member_tm3 @ X4 @ ( set_tm2 @ Xs ) )
& ( P3 @ X4 ) )
=> ? [Ys3: list_tm,X3: tm] :
( ? [Zs2: list_tm] :
( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ X3 @ Zs2 ) ) )
& ( P3 @ X3 )
& ! [Xa: tm] :
( ( member_tm3 @ Xa @ ( set_tm2 @ Ys3 ) )
=> ~ ( P3 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_780_split__list__last__prop,axiom,
! [Xs: list_rule,P3: rule > $o] :
( ? [X4: rule] :
( ( member_rule3 @ X4 @ ( set_rule2 @ Xs ) )
& ( P3 @ X4 ) )
=> ? [Ys3: list_rule,X3: rule,Zs2: list_rule] :
( ( Xs
= ( append_rule @ Ys3 @ ( cons_rule @ X3 @ Zs2 ) ) )
& ( P3 @ X3 )
& ! [Xa: rule] :
( ( member_rule3 @ Xa @ ( set_rule2 @ Zs2 ) )
=> ~ ( P3 @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_781_split__list__last__prop,axiom,
! [Xs: list_list_fm,P3: list_fm > $o] :
( ? [X4: list_fm] :
( ( member_list_fm3 @ X4 @ ( set_list_fm2 @ Xs ) )
& ( P3 @ X4 ) )
=> ? [Ys3: list_list_fm,X3: list_fm,Zs2: list_list_fm] :
( ( Xs
= ( append_list_fm @ Ys3 @ ( cons_list_fm @ X3 @ Zs2 ) ) )
& ( P3 @ X3 )
& ! [Xa: list_fm] :
( ( member_list_fm3 @ Xa @ ( set_list_fm2 @ Zs2 ) )
=> ~ ( P3 @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_782_split__list__last__prop,axiom,
! [Xs: list_fm,P3: fm > $o] :
( ? [X4: fm] :
( ( member_fm3 @ X4 @ ( set_fm2 @ Xs ) )
& ( P3 @ X4 ) )
=> ? [Ys3: list_fm,X3: fm,Zs2: list_fm] :
( ( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X3 @ Zs2 ) ) )
& ( P3 @ X3 )
& ! [Xa: fm] :
( ( member_fm3 @ Xa @ ( set_fm2 @ Zs2 ) )
=> ~ ( P3 @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_783_split__list__last__prop,axiom,
! [Xs: list_tm,P3: tm > $o] :
( ? [X4: tm] :
( ( member_tm3 @ X4 @ ( set_tm2 @ Xs ) )
& ( P3 @ X4 ) )
=> ? [Ys3: list_tm,X3: tm,Zs2: list_tm] :
( ( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ X3 @ Zs2 ) ) )
& ( P3 @ X3 )
& ! [Xa: tm] :
( ( member_tm3 @ Xa @ ( set_tm2 @ Zs2 ) )
=> ~ ( P3 @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_784_in__set__conv__decomp,axiom,
! [X: produc340336539035504054m_rule,Xs: list_P2774625669004309958m_rule] :
( ( member7231649785386036813m_rule @ X @ ( set_Pr4534715572506550497m_rule @ Xs ) )
= ( ? [Ys2: list_P2774625669004309958m_rule,Zs3: list_P2774625669004309958m_rule] :
( Xs
= ( append2714808343334685275m_rule @ Ys2 @ ( cons_P2818572562524668032m_rule @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_785_in__set__conv__decomp,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat3 @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys2: list_nat,Zs3: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_786_in__set__conv__decomp,axiom,
! [X: rule,Xs: list_rule] :
( ( member_rule3 @ X @ ( set_rule2 @ Xs ) )
= ( ? [Ys2: list_rule,Zs3: list_rule] :
( Xs
= ( append_rule @ Ys2 @ ( cons_rule @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_787_in__set__conv__decomp,axiom,
! [X: list_fm,Xs: list_list_fm] :
( ( member_list_fm3 @ X @ ( set_list_fm2 @ Xs ) )
= ( ? [Ys2: list_list_fm,Zs3: list_list_fm] :
( Xs
= ( append_list_fm @ Ys2 @ ( cons_list_fm @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_788_in__set__conv__decomp,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm3 @ X @ ( set_fm2 @ Xs ) )
= ( ? [Ys2: list_fm,Zs3: list_fm] :
( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_789_in__set__conv__decomp,axiom,
! [X: tm,Xs: list_tm] :
( ( member_tm3 @ X @ ( set_tm2 @ Xs ) )
= ( ? [Ys2: list_tm,Zs3: list_tm] :
( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_790_append__Cons__eq__iff,axiom,
! [X: produc340336539035504054m_rule,Xs: list_P2774625669004309958m_rule,Ys: list_P2774625669004309958m_rule,Xs4: list_P2774625669004309958m_rule,Ys5: list_P2774625669004309958m_rule] :
( ~ ( member7231649785386036813m_rule @ X @ ( set_Pr4534715572506550497m_rule @ Xs ) )
=> ( ~ ( member7231649785386036813m_rule @ X @ ( set_Pr4534715572506550497m_rule @ Ys ) )
=> ( ( ( append2714808343334685275m_rule @ Xs @ ( cons_P2818572562524668032m_rule @ X @ Ys ) )
= ( append2714808343334685275m_rule @ Xs4 @ ( cons_P2818572562524668032m_rule @ X @ Ys5 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_791_append__Cons__eq__iff,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat,Xs4: list_nat,Ys5: list_nat] :
( ~ ( member_nat3 @ X @ ( set_nat2 @ Xs ) )
=> ( ~ ( member_nat3 @ X @ ( set_nat2 @ Ys ) )
=> ( ( ( append_nat @ Xs @ ( cons_nat @ X @ Ys ) )
= ( append_nat @ Xs4 @ ( cons_nat @ X @ Ys5 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_792_append__Cons__eq__iff,axiom,
! [X: rule,Xs: list_rule,Ys: list_rule,Xs4: list_rule,Ys5: list_rule] :
( ~ ( member_rule3 @ X @ ( set_rule2 @ Xs ) )
=> ( ~ ( member_rule3 @ X @ ( set_rule2 @ Ys ) )
=> ( ( ( append_rule @ Xs @ ( cons_rule @ X @ Ys ) )
= ( append_rule @ Xs4 @ ( cons_rule @ X @ Ys5 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_793_append__Cons__eq__iff,axiom,
! [X: list_fm,Xs: list_list_fm,Ys: list_list_fm,Xs4: list_list_fm,Ys5: list_list_fm] :
( ~ ( member_list_fm3 @ X @ ( set_list_fm2 @ Xs ) )
=> ( ~ ( member_list_fm3 @ X @ ( set_list_fm2 @ Ys ) )
=> ( ( ( append_list_fm @ Xs @ ( cons_list_fm @ X @ Ys ) )
= ( append_list_fm @ Xs4 @ ( cons_list_fm @ X @ Ys5 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_794_append__Cons__eq__iff,axiom,
! [X: fm,Xs: list_fm,Ys: list_fm,Xs4: list_fm,Ys5: list_fm] :
( ~ ( member_fm3 @ X @ ( set_fm2 @ Xs ) )
=> ( ~ ( member_fm3 @ X @ ( set_fm2 @ Ys ) )
=> ( ( ( append_fm @ Xs @ ( cons_fm @ X @ Ys ) )
= ( append_fm @ Xs4 @ ( cons_fm @ X @ Ys5 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_795_append__Cons__eq__iff,axiom,
! [X: tm,Xs: list_tm,Ys: list_tm,Xs4: list_tm,Ys5: list_tm] :
( ~ ( member_tm3 @ X @ ( set_tm2 @ Xs ) )
=> ( ~ ( member_tm3 @ X @ ( set_tm2 @ Ys ) )
=> ( ( ( append_tm @ Xs @ ( cons_tm @ X @ Ys ) )
= ( append_tm @ Xs4 @ ( cons_tm @ X @ Ys5 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_796_split__list__propE,axiom,
! [Xs: list_rule,P3: rule > $o] :
( ? [X4: rule] :
( ( member_rule3 @ X4 @ ( set_rule2 @ Xs ) )
& ( P3 @ X4 ) )
=> ~ ! [Ys3: list_rule,X3: rule] :
( ? [Zs2: list_rule] :
( Xs
= ( append_rule @ Ys3 @ ( cons_rule @ X3 @ Zs2 ) ) )
=> ~ ( P3 @ X3 ) ) ) ).
% split_list_propE
thf(fact_797_split__list__propE,axiom,
! [Xs: list_list_fm,P3: list_fm > $o] :
( ? [X4: list_fm] :
( ( member_list_fm3 @ X4 @ ( set_list_fm2 @ Xs ) )
& ( P3 @ X4 ) )
=> ~ ! [Ys3: list_list_fm,X3: list_fm] :
( ? [Zs2: list_list_fm] :
( Xs
= ( append_list_fm @ Ys3 @ ( cons_list_fm @ X3 @ Zs2 ) ) )
=> ~ ( P3 @ X3 ) ) ) ).
% split_list_propE
thf(fact_798_split__list__propE,axiom,
! [Xs: list_fm,P3: fm > $o] :
( ? [X4: fm] :
( ( member_fm3 @ X4 @ ( set_fm2 @ Xs ) )
& ( P3 @ X4 ) )
=> ~ ! [Ys3: list_fm,X3: fm] :
( ? [Zs2: list_fm] :
( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X3 @ Zs2 ) ) )
=> ~ ( P3 @ X3 ) ) ) ).
% split_list_propE
thf(fact_799_split__list__propE,axiom,
! [Xs: list_tm,P3: tm > $o] :
( ? [X4: tm] :
( ( member_tm3 @ X4 @ ( set_tm2 @ Xs ) )
& ( P3 @ X4 ) )
=> ~ ! [Ys3: list_tm,X3: tm] :
( ? [Zs2: list_tm] :
( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ X3 @ Zs2 ) ) )
=> ~ ( P3 @ X3 ) ) ) ).
% split_list_propE
thf(fact_800_split__list__first,axiom,
! [X: produc340336539035504054m_rule,Xs: list_P2774625669004309958m_rule] :
( ( member7231649785386036813m_rule @ X @ ( set_Pr4534715572506550497m_rule @ Xs ) )
=> ? [Ys3: list_P2774625669004309958m_rule,Zs2: list_P2774625669004309958m_rule] :
( ( Xs
= ( append2714808343334685275m_rule @ Ys3 @ ( cons_P2818572562524668032m_rule @ X @ Zs2 ) ) )
& ~ ( member7231649785386036813m_rule @ X @ ( set_Pr4534715572506550497m_rule @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_801_split__list__first,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat3 @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys3: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs2 ) ) )
& ~ ( member_nat3 @ X @ ( set_nat2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_802_split__list__first,axiom,
! [X: rule,Xs: list_rule] :
( ( member_rule3 @ X @ ( set_rule2 @ Xs ) )
=> ? [Ys3: list_rule,Zs2: list_rule] :
( ( Xs
= ( append_rule @ Ys3 @ ( cons_rule @ X @ Zs2 ) ) )
& ~ ( member_rule3 @ X @ ( set_rule2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_803_split__list__first,axiom,
! [X: list_fm,Xs: list_list_fm] :
( ( member_list_fm3 @ X @ ( set_list_fm2 @ Xs ) )
=> ? [Ys3: list_list_fm,Zs2: list_list_fm] :
( ( Xs
= ( append_list_fm @ Ys3 @ ( cons_list_fm @ X @ Zs2 ) ) )
& ~ ( member_list_fm3 @ X @ ( set_list_fm2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_804_split__list__first,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm3 @ X @ ( set_fm2 @ Xs ) )
=> ? [Ys3: list_fm,Zs2: list_fm] :
( ( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X @ Zs2 ) ) )
& ~ ( member_fm3 @ X @ ( set_fm2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_805_split__list__first,axiom,
! [X: tm,Xs: list_tm] :
( ( member_tm3 @ X @ ( set_tm2 @ Xs ) )
=> ? [Ys3: list_tm,Zs2: list_tm] :
( ( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ X @ Zs2 ) ) )
& ~ ( member_tm3 @ X @ ( set_tm2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_806_split__list__prop,axiom,
! [Xs: list_rule,P3: rule > $o] :
( ? [X4: rule] :
( ( member_rule3 @ X4 @ ( set_rule2 @ Xs ) )
& ( P3 @ X4 ) )
=> ? [Ys3: list_rule,X3: rule] :
( ? [Zs2: list_rule] :
( Xs
= ( append_rule @ Ys3 @ ( cons_rule @ X3 @ Zs2 ) ) )
& ( P3 @ X3 ) ) ) ).
% split_list_prop
thf(fact_807_split__list__prop,axiom,
! [Xs: list_list_fm,P3: list_fm > $o] :
( ? [X4: list_fm] :
( ( member_list_fm3 @ X4 @ ( set_list_fm2 @ Xs ) )
& ( P3 @ X4 ) )
=> ? [Ys3: list_list_fm,X3: list_fm] :
( ? [Zs2: list_list_fm] :
( Xs
= ( append_list_fm @ Ys3 @ ( cons_list_fm @ X3 @ Zs2 ) ) )
& ( P3 @ X3 ) ) ) ).
% split_list_prop
thf(fact_808_split__list__prop,axiom,
! [Xs: list_fm,P3: fm > $o] :
( ? [X4: fm] :
( ( member_fm3 @ X4 @ ( set_fm2 @ Xs ) )
& ( P3 @ X4 ) )
=> ? [Ys3: list_fm,X3: fm] :
( ? [Zs2: list_fm] :
( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X3 @ Zs2 ) ) )
& ( P3 @ X3 ) ) ) ).
% split_list_prop
thf(fact_809_split__list__prop,axiom,
! [Xs: list_tm,P3: tm > $o] :
( ? [X4: tm] :
( ( member_tm3 @ X4 @ ( set_tm2 @ Xs ) )
& ( P3 @ X4 ) )
=> ? [Ys3: list_tm,X3: tm] :
( ? [Zs2: list_tm] :
( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ X3 @ Zs2 ) ) )
& ( P3 @ X3 ) ) ) ).
% split_list_prop
thf(fact_810_split__list__last,axiom,
! [X: produc340336539035504054m_rule,Xs: list_P2774625669004309958m_rule] :
( ( member7231649785386036813m_rule @ X @ ( set_Pr4534715572506550497m_rule @ Xs ) )
=> ? [Ys3: list_P2774625669004309958m_rule,Zs2: list_P2774625669004309958m_rule] :
( ( Xs
= ( append2714808343334685275m_rule @ Ys3 @ ( cons_P2818572562524668032m_rule @ X @ Zs2 ) ) )
& ~ ( member7231649785386036813m_rule @ X @ ( set_Pr4534715572506550497m_rule @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_811_split__list__last,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat3 @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys3: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs2 ) ) )
& ~ ( member_nat3 @ X @ ( set_nat2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_812_split__list__last,axiom,
! [X: rule,Xs: list_rule] :
( ( member_rule3 @ X @ ( set_rule2 @ Xs ) )
=> ? [Ys3: list_rule,Zs2: list_rule] :
( ( Xs
= ( append_rule @ Ys3 @ ( cons_rule @ X @ Zs2 ) ) )
& ~ ( member_rule3 @ X @ ( set_rule2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_813_split__list__last,axiom,
! [X: list_fm,Xs: list_list_fm] :
( ( member_list_fm3 @ X @ ( set_list_fm2 @ Xs ) )
=> ? [Ys3: list_list_fm,Zs2: list_list_fm] :
( ( Xs
= ( append_list_fm @ Ys3 @ ( cons_list_fm @ X @ Zs2 ) ) )
& ~ ( member_list_fm3 @ X @ ( set_list_fm2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_814_split__list__last,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm3 @ X @ ( set_fm2 @ Xs ) )
=> ? [Ys3: list_fm,Zs2: list_fm] :
( ( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X @ Zs2 ) ) )
& ~ ( member_fm3 @ X @ ( set_fm2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_815_split__list__last,axiom,
! [X: tm,Xs: list_tm] :
( ( member_tm3 @ X @ ( set_tm2 @ Xs ) )
=> ? [Ys3: list_tm,Zs2: list_tm] :
( ( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ X @ Zs2 ) ) )
& ~ ( member_tm3 @ X @ ( set_tm2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_816_split__list,axiom,
! [X: produc340336539035504054m_rule,Xs: list_P2774625669004309958m_rule] :
( ( member7231649785386036813m_rule @ X @ ( set_Pr4534715572506550497m_rule @ Xs ) )
=> ? [Ys3: list_P2774625669004309958m_rule,Zs2: list_P2774625669004309958m_rule] :
( Xs
= ( append2714808343334685275m_rule @ Ys3 @ ( cons_P2818572562524668032m_rule @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_817_split__list,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat3 @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys3: list_nat,Zs2: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_818_split__list,axiom,
! [X: rule,Xs: list_rule] :
( ( member_rule3 @ X @ ( set_rule2 @ Xs ) )
=> ? [Ys3: list_rule,Zs2: list_rule] :
( Xs
= ( append_rule @ Ys3 @ ( cons_rule @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_819_split__list,axiom,
! [X: list_fm,Xs: list_list_fm] :
( ( member_list_fm3 @ X @ ( set_list_fm2 @ Xs ) )
=> ? [Ys3: list_list_fm,Zs2: list_list_fm] :
( Xs
= ( append_list_fm @ Ys3 @ ( cons_list_fm @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_820_split__list,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm3 @ X @ ( set_fm2 @ Xs ) )
=> ? [Ys3: list_fm,Zs2: list_fm] :
( Xs
= ( append_fm @ Ys3 @ ( cons_fm @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_821_split__list,axiom,
! [X: tm,Xs: list_tm] :
( ( member_tm3 @ X @ ( set_tm2 @ Xs ) )
=> ? [Ys3: list_tm,Zs2: list_tm] :
( Xs
= ( append_tm @ Ys3 @ ( cons_tm @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_822_append__listrel1I,axiom,
! [Xs: list_tm,Ys: list_tm,R: set_Pr2455929065695642951_tm_tm,Us: list_tm,Vs: list_tm] :
( ( ( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ Ys ) @ ( listrel1_tm @ R ) )
& ( Us = Vs ) )
| ( ( Xs = Ys )
& ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Us @ Vs ) @ ( listrel1_tm @ R ) ) ) )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( append_tm @ Xs @ Us ) @ ( append_tm @ Ys @ Vs ) ) @ ( listrel1_tm @ R ) ) ) ).
% append_listrel1I
thf(fact_823_lexord__append__leftI,axiom,
! [U: list_tm,V2: list_tm,R: set_Pr2455929065695642951_tm_tm,X: list_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ U @ V2 ) @ ( lexord_tm @ R ) )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( append_tm @ X @ U ) @ ( append_tm @ X @ V2 ) ) @ ( lexord_tm @ R ) ) ) ).
% lexord_append_leftI
thf(fact_824_stake__sdrop,axiom,
! [N2: nat,S: stream2709947120125613254m_rule] :
( ( shift_2334844276748245581m_rule @ ( stake_5421812949518764133m_rule @ N2 @ S ) @ ( sdrop_8169176516188972301m_rule @ N2 @ S ) )
= S ) ).
% stake_sdrop
thf(fact_825_Nil2__notin__lex,axiom,
! [Xs: list_rule,R: set_Pr7340090144965549541e_rule] :
~ ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ Xs @ nil_rule ) @ ( lex_rule @ R ) ) ).
% Nil2_notin_lex
thf(fact_826_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_827_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_828_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_829_Nil__notin__lex,axiom,
! [Ys: list_rule,R: set_Pr7340090144965549541e_rule] :
~ ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ nil_rule @ Ys ) @ ( lex_rule @ R ) ) ).
% Nil_notin_lex
thf(fact_830_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_831_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_832_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_833_same__length__different,axiom,
! [Xs: list_rule,Ys: list_rule] :
( ( Xs != Ys )
=> ( ( ( size_size_list_rule @ Xs )
= ( size_size_list_rule @ Ys ) )
=> ? [Pre: list_rule,X3: rule,Xs5: list_rule,Y3: rule,Ys6: list_rule] :
( ( X3 != Y3 )
& ( Xs
= ( append_rule @ Pre @ ( append_rule @ ( cons_rule @ X3 @ nil_rule ) @ Xs5 ) ) )
& ( Ys
= ( append_rule @ Pre @ ( append_rule @ ( cons_rule @ Y3 @ nil_rule ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_834_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,Y3: list_fm,Ys6: list_list_fm] :
( ( X3 != Y3 )
& ( 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 @ Y3 @ nil_list_fm ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_835_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,Y3: fm,Ys6: list_fm] :
( ( X3 != Y3 )
& ( Xs
= ( append_fm @ Pre @ ( append_fm @ ( cons_fm @ X3 @ nil_fm ) @ Xs5 ) ) )
& ( Ys
= ( append_fm @ Pre @ ( append_fm @ ( cons_fm @ Y3 @ nil_fm ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_836_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,Y3: tm,Ys6: list_tm] :
( ( X3 != Y3 )
& ( Xs
= ( append_tm @ Pre @ ( append_tm @ ( cons_tm @ X3 @ nil_tm ) @ Xs5 ) ) )
& ( Ys
= ( append_tm @ Pre @ ( append_tm @ ( cons_tm @ Y3 @ nil_tm ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_837_lexord__append__leftD,axiom,
! [X: list_tm,U: list_tm,V2: list_tm,R: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( append_tm @ X @ U ) @ ( append_tm @ X @ V2 ) ) @ ( lexord_tm @ R ) )
=> ( ! [A4: tm] :
~ ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ A4 @ A4 ) @ R )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ U @ V2 ) @ ( lexord_tm @ R ) ) ) ) ).
% lexord_append_leftD
thf(fact_838_lexord__append__rightI,axiom,
! [Y: list_rule,X: list_rule,R: set_Pr7340090144965549541e_rule] :
( ? [B7: rule,Z5: list_rule] :
( Y
= ( cons_rule @ B7 @ Z5 ) )
=> ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ X @ ( append_rule @ X @ Y ) ) @ ( lexord_rule @ R ) ) ) ).
% lexord_append_rightI
thf(fact_839_lexord__append__rightI,axiom,
! [Y: list_list_fm,X: list_list_fm,R: set_Pr1916137733696349511ist_fm] :
( ? [B7: list_fm,Z5: list_list_fm] :
( Y
= ( cons_list_fm @ B7 @ Z5 ) )
=> ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ X @ ( append_list_fm @ X @ Y ) ) @ ( lexord_list_fm @ R ) ) ) ).
% lexord_append_rightI
thf(fact_840_lexord__append__rightI,axiom,
! [Y: list_fm,X: list_fm,R: set_Pr4706815898642364871_fm_fm] :
( ? [B7: fm,Z5: list_fm] :
( Y
= ( cons_fm @ B7 @ Z5 ) )
=> ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ X @ ( append_fm @ X @ Y ) ) @ ( lexord_fm @ R ) ) ) ).
% lexord_append_rightI
thf(fact_841_lexord__append__rightI,axiom,
! [Y: list_tm,X: list_tm,R: set_Pr2455929065695642951_tm_tm] :
( ? [B7: tm,Z5: list_tm] :
( Y
= ( cons_tm @ B7 @ Z5 ) )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ X @ ( append_tm @ X @ Y ) ) @ ( lexord_tm @ R ) ) ) ).
% lexord_append_rightI
thf(fact_842_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_843_lenlex__append1,axiom,
! [Us: list_tm,Xs: list_tm,R2: set_Pr2455929065695642951_tm_tm,Vs: list_tm,Ys: list_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Us @ Xs ) @ ( lenlex_tm @ R2 ) )
=> ( ( ( size_size_list_tm @ Vs )
= ( size_size_list_tm @ Ys ) )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( append_tm @ Us @ Vs ) @ ( append_tm @ Xs @ Ys ) ) @ ( lenlex_tm @ R2 ) ) ) ) ).
% lenlex_append1
thf(fact_844_listrel1I,axiom,
! [X: rule,Y: rule,R: set_Pr7340090144965549541e_rule,Xs: list_rule,Us: list_rule,Vs: list_rule,Ys: list_rule] :
( ( member1444691403335511342e_rule @ ( produc5849431337705160893e_rule @ X @ Y ) @ R )
=> ( ( Xs
= ( append_rule @ Us @ ( cons_rule @ X @ Vs ) ) )
=> ( ( Ys
= ( append_rule @ Us @ ( cons_rule @ Y @ Vs ) ) )
=> ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ Xs @ Ys ) @ ( listrel1_rule @ R ) ) ) ) ) ).
% listrel1I
thf(fact_845_listrel1I,axiom,
! [X: list_fm,Y: list_fm,R: set_Pr1916137733696349511ist_fm,Xs: list_list_fm,Us: list_list_fm,Vs: list_list_fm,Ys: list_list_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ X @ Y ) @ R )
=> ( ( Xs
= ( append_list_fm @ Us @ ( cons_list_fm @ X @ Vs ) ) )
=> ( ( Ys
= ( append_list_fm @ Us @ ( cons_list_fm @ Y @ Vs ) ) )
=> ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ Xs @ Ys ) @ ( listrel1_list_fm @ R ) ) ) ) ) ).
% listrel1I
thf(fact_846_listrel1I,axiom,
! [X: fm,Y: fm,R: set_Pr4706815898642364871_fm_fm,Xs: list_fm,Us: list_fm,Vs: list_fm,Ys: list_fm] :
( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X @ Y ) @ R )
=> ( ( Xs
= ( append_fm @ Us @ ( cons_fm @ X @ Vs ) ) )
=> ( ( Ys
= ( append_fm @ Us @ ( cons_fm @ Y @ Vs ) ) )
=> ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Xs @ Ys ) @ ( listrel1_fm @ R ) ) ) ) ) ).
% listrel1I
thf(fact_847_listrel1I,axiom,
! [X: tm,Y: tm,R: set_Pr2455929065695642951_tm_tm,Xs: list_tm,Us: list_tm,Vs: list_tm,Ys: list_tm] :
( ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X @ Y ) @ R )
=> ( ( Xs
= ( append_tm @ Us @ ( cons_tm @ X @ Vs ) ) )
=> ( ( Ys
= ( append_tm @ Us @ ( cons_tm @ Y @ Vs ) ) )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ Ys ) @ ( listrel1_tm @ R ) ) ) ) ) ).
% listrel1I
thf(fact_848_listrel1E,axiom,
! [Xs: list_rule,Ys: list_rule,R: set_Pr7340090144965549541e_rule] :
( ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ Xs @ Ys ) @ ( listrel1_rule @ R ) )
=> ~ ! [X3: rule,Y3: rule] :
( ( member1444691403335511342e_rule @ ( produc5849431337705160893e_rule @ X3 @ Y3 ) @ R )
=> ! [Us3: list_rule,Vs2: list_rule] :
( ( Xs
= ( append_rule @ Us3 @ ( cons_rule @ X3 @ Vs2 ) ) )
=> ( Ys
!= ( append_rule @ Us3 @ ( cons_rule @ Y3 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_849_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,Y3: list_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ X3 @ Y3 ) @ R )
=> ! [Us3: list_list_fm,Vs2: list_list_fm] :
( ( Xs
= ( append_list_fm @ Us3 @ ( cons_list_fm @ X3 @ Vs2 ) ) )
=> ( Ys
!= ( append_list_fm @ Us3 @ ( cons_list_fm @ Y3 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_850_listrel1E,axiom,
! [Xs: list_fm,Ys: list_fm,R: set_Pr4706815898642364871_fm_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ Xs @ Ys ) @ ( listrel1_fm @ R ) )
=> ~ ! [X3: fm,Y3: fm] :
( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ X3 @ Y3 ) @ R )
=> ! [Us3: list_fm,Vs2: list_fm] :
( ( Xs
= ( append_fm @ Us3 @ ( cons_fm @ X3 @ Vs2 ) ) )
=> ( Ys
!= ( append_fm @ Us3 @ ( cons_fm @ Y3 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_851_listrel1E,axiom,
! [Xs: list_tm,Ys: list_tm,R: set_Pr2455929065695642951_tm_tm] :
( ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ Xs @ Ys ) @ ( listrel1_tm @ R ) )
=> ~ ! [X3: tm,Y3: tm] :
( ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ X3 @ Y3 ) @ R )
=> ! [Us3: list_tm,Vs2: list_tm] :
( ( Xs
= ( append_tm @ Us3 @ ( cons_tm @ X3 @ Vs2 ) ) )
=> ( Ys
!= ( append_tm @ Us3 @ ( cons_tm @ Y3 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_852_lexord__append__left__rightI,axiom,
! [A: rule,B2: rule,R: set_Pr7340090144965549541e_rule,U: list_rule,X: list_rule,Y: list_rule] :
( ( member1444691403335511342e_rule @ ( produc5849431337705160893e_rule @ A @ B2 ) @ R )
=> ( member695053538577043918t_rule @ ( produc8979233614864344413t_rule @ ( append_rule @ U @ ( cons_rule @ A @ X ) ) @ ( append_rule @ U @ ( cons_rule @ B2 @ Y ) ) ) @ ( lexord_rule @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_853_lexord__append__left__rightI,axiom,
! [A: list_fm,B2: list_fm,R: set_Pr1916137733696349511ist_fm,U: list_list_fm,X: list_list_fm,Y: list_list_fm] :
( ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ A @ B2 ) @ R )
=> ( member5102842307574975760ist_fm @ ( produc8321651870839017815ist_fm @ ( append_list_fm @ U @ ( cons_list_fm @ A @ X ) ) @ ( append_list_fm @ U @ ( cons_list_fm @ B2 @ Y ) ) ) @ ( lexord_list_fm @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_854_lexord__append__left__rightI,axiom,
! [A: fm,B2: fm,R: set_Pr4706815898642364871_fm_fm,U: list_fm,X: list_fm,Y: list_fm] :
( ( member7780952600467998736_fm_fm @ ( product_Pair_fm_fm @ A @ B2 ) @ R )
=> ( member1926098302810316688ist_fm @ ( produc7863996417982153943ist_fm @ ( append_fm @ U @ ( cons_fm @ A @ X ) ) @ ( append_fm @ U @ ( cons_fm @ B2 @ Y ) ) ) @ ( lexord_fm @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_855_lexord__append__left__rightI,axiom,
! [A: tm,B2: tm,R: set_Pr2455929065695642951_tm_tm,U: list_tm,X: list_tm,Y: list_tm] :
( ( member3121616906494481296_tm_tm @ ( product_Pair_tm_tm @ A @ B2 ) @ R )
=> ( member4457312017796942864ist_tm @ ( produc1418304791525149271ist_tm @ ( append_tm @ U @ ( cons_tm @ A @ X ) ) @ ( append_tm @ U @ ( cons_tm @ B2 @ Y ) ) ) @ ( lexord_tm @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_856_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_fm3 @ 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_857_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_fm3 @ 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_858_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_tm3 @ 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_859_bind__simps_I2_J,axiom,
! [X: rule,Xs: list_rule,F: rule > list_tm] :
( ( bind_rule_tm @ ( cons_rule @ X @ Xs ) @ F )
= ( append_tm @ ( F @ X ) @ ( bind_rule_tm @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_860_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_861_bind__simps_I2_J,axiom,
! [X: fm,Xs: list_fm,F: fm > list_tm] :
( ( bind_fm_tm @ ( cons_fm @ X @ Xs ) @ F )
= ( append_tm @ ( F @ X ) @ ( bind_fm_tm @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_862_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_863_n__lists__Nil,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( n_lists_rule @ N2 @ nil_rule )
= ( cons_list_rule @ nil_rule @ nil_list_rule ) ) )
& ( ( N2 != zero_zero_nat )
=> ( ( n_lists_rule @ N2 @ nil_rule )
= nil_list_rule ) ) ) ).
% n_lists_Nil
thf(fact_864_n__lists__Nil,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( n_lists_list_fm @ N2 @ nil_list_fm )
= ( cons_list_list_fm @ nil_list_fm @ nil_list_list_fm ) ) )
& ( ( N2 != zero_zero_nat )
=> ( ( n_lists_list_fm @ N2 @ nil_list_fm )
= nil_list_list_fm ) ) ) ).
% n_lists_Nil
thf(fact_865_n__lists__Nil,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( n_lists_tm @ N2 @ nil_tm )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) )
& ( ( N2 != zero_zero_nat )
=> ( ( n_lists_tm @ N2 @ nil_tm )
= nil_list_tm ) ) ) ).
% n_lists_Nil
thf(fact_866_n__lists__Nil,axiom,
! [N2: nat] :
( ( ( N2 = zero_zero_nat )
=> ( ( n_lists_fm @ N2 @ nil_fm )
= ( cons_list_fm @ nil_fm @ nil_list_fm ) ) )
& ( ( N2 != zero_zero_nat )
=> ( ( n_lists_fm @ N2 @ nil_fm )
= nil_list_fm ) ) ) ).
% n_lists_Nil
thf(fact_867_stake__cycle__le,axiom,
! [U: list_rule,N2: nat] :
( ( U != nil_rule )
=> ( ( ord_less_nat @ N2 @ ( size_size_list_rule @ U ) )
=> ( ( stake_rule @ N2 @ ( cycle_rule @ U ) )
= ( take_rule @ N2 @ U ) ) ) ) ).
% stake_cycle_le
thf(fact_868_stake__cycle__le,axiom,
! [U: list_fm,N2: nat] :
( ( U != nil_fm )
=> ( ( ord_less_nat @ N2 @ ( size_size_list_fm @ U ) )
=> ( ( stake_fm @ N2 @ ( cycle_fm @ U ) )
= ( take_fm @ N2 @ U ) ) ) ) ).
% stake_cycle_le
thf(fact_869_stake__cycle__le,axiom,
! [U: list_list_fm,N2: nat] :
( ( U != nil_list_fm )
=> ( ( ord_less_nat @ N2 @ ( size_s115229985653309035ist_fm @ U ) )
=> ( ( stake_list_fm @ N2 @ ( cycle_list_fm @ U ) )
= ( take_list_fm @ N2 @ U ) ) ) ) ).
% stake_cycle_le
thf(fact_870_stake__cycle__le,axiom,
! [U: list_tm,N2: nat] :
( ( U != nil_tm )
=> ( ( ord_less_nat @ N2 @ ( size_size_list_tm @ U ) )
=> ( ( stake_tm @ N2 @ ( cycle_tm @ U ) )
= ( take_tm @ N2 @ U ) ) ) ) ).
% stake_cycle_le
thf(fact_871_stake__Suc,axiom,
! [N2: nat,S: stream2709947120125613254m_rule] :
( ( stake_5421812949518764133m_rule @ ( suc @ N2 ) @ S )
= ( append2714808343334685275m_rule @ ( stake_5421812949518764133m_rule @ N2 @ S ) @ ( cons_P2818572562524668032m_rule @ ( snth_P6679518042731451922m_rule @ S @ N2 ) @ nil_Pr2808214839852828720m_rule ) ) ) ).
% stake_Suc
thf(fact_872_stake__Suc,axiom,
! [N2: nat,S: stream_rule] :
( ( stake_rule @ ( suc @ N2 ) @ S )
= ( append_rule @ ( stake_rule @ N2 @ S ) @ ( cons_rule @ ( snth_rule @ S @ N2 ) @ nil_rule ) ) ) ).
% stake_Suc
thf(fact_873_stake__Suc,axiom,
! [N2: nat,S: stream_list_fm] :
( ( stake_list_fm @ ( suc @ N2 ) @ S )
= ( append_list_fm @ ( stake_list_fm @ N2 @ S ) @ ( cons_list_fm @ ( snth_list_fm @ S @ N2 ) @ nil_list_fm ) ) ) ).
% stake_Suc
thf(fact_874_stake__Suc,axiom,
! [N2: nat,S: stream_fm] :
( ( stake_fm @ ( suc @ N2 ) @ S )
= ( append_fm @ ( stake_fm @ N2 @ S ) @ ( cons_fm @ ( snth_fm @ S @ N2 ) @ nil_fm ) ) ) ).
% stake_Suc
thf(fact_875_stake__Suc,axiom,
! [N2: nat,S: stream_tm] :
( ( stake_tm @ ( suc @ N2 ) @ S )
= ( append_tm @ ( stake_tm @ N2 @ S ) @ ( cons_tm @ ( snth_tm @ S @ N2 ) @ nil_tm ) ) ) ).
% stake_Suc
thf(fact_876_n__lists_Osimps_I1_J,axiom,
! [Xs: list_rule] :
( ( n_lists_rule @ zero_zero_nat @ Xs )
= ( cons_list_rule @ nil_rule @ nil_list_rule ) ) ).
% n_lists.simps(1)
thf(fact_877_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_878_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_879_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_880_bind__simps_I1_J,axiom,
! [F: rule > list_rule] :
( ( bind_rule_rule @ nil_rule @ F )
= nil_rule ) ).
% bind_simps(1)
thf(fact_881_bind__simps_I1_J,axiom,
! [F: rule > list_fm] :
( ( bind_rule_fm @ nil_rule @ F )
= nil_fm ) ).
% bind_simps(1)
thf(fact_882_bind__simps_I1_J,axiom,
! [F: rule > list_tm] :
( ( bind_rule_tm @ nil_rule @ F )
= nil_tm ) ).
% bind_simps(1)
thf(fact_883_bind__simps_I1_J,axiom,
! [F: fm > list_rule] :
( ( bind_fm_rule @ nil_fm @ F )
= nil_rule ) ).
% bind_simps(1)
thf(fact_884_bind__simps_I1_J,axiom,
! [F: fm > list_fm] :
( ( bind_fm_fm @ nil_fm @ F )
= nil_fm ) ).
% bind_simps(1)
thf(fact_885_bind__simps_I1_J,axiom,
! [F: fm > list_tm] :
( ( bind_fm_tm @ nil_fm @ F )
= nil_tm ) ).
% bind_simps(1)
thf(fact_886_bind__simps_I1_J,axiom,
! [F: tm > list_rule] :
( ( bind_tm_rule @ nil_tm @ F )
= nil_rule ) ).
% bind_simps(1)
thf(fact_887_bind__simps_I1_J,axiom,
! [F: tm > list_fm] :
( ( bind_tm_fm @ nil_tm @ F )
= nil_fm ) ).
% bind_simps(1)
thf(fact_888_bind__simps_I1_J,axiom,
! [F: tm > list_tm] :
( ( bind_tm_tm @ nil_tm @ F )
= nil_tm ) ).
% bind_simps(1)
thf(fact_889_bind__simps_I1_J,axiom,
! [F: rule > list_list_fm] :
( ( bind_rule_list_fm @ nil_rule @ F )
= nil_list_fm ) ).
% bind_simps(1)
thf(fact_890_length__0__conv,axiom,
! [Xs: list_rule] :
( ( ( size_size_list_rule @ Xs )
= zero_zero_nat )
= ( Xs = nil_rule ) ) ).
% length_0_conv
thf(fact_891_length__0__conv,axiom,
! [Xs: list_fm] :
( ( ( size_size_list_fm @ Xs )
= zero_zero_nat )
= ( Xs = nil_fm ) ) ).
% length_0_conv
thf(fact_892_length__0__conv,axiom,
! [Xs: list_list_fm] :
( ( ( size_s115229985653309035ist_fm @ Xs )
= zero_zero_nat )
= ( Xs = nil_list_fm ) ) ).
% length_0_conv
thf(fact_893_length__0__conv,axiom,
! [Xs: list_tm] :
( ( ( size_size_list_tm @ Xs )
= zero_zero_nat )
= ( Xs = nil_tm ) ) ).
% length_0_conv
thf(fact_894_take__Suc__Cons,axiom,
! [N2: nat,X: rule,Xs: list_rule] :
( ( take_rule @ ( suc @ N2 ) @ ( cons_rule @ X @ Xs ) )
= ( cons_rule @ X @ ( take_rule @ N2 @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_895_take__Suc__Cons,axiom,
! [N2: nat,X: list_fm,Xs: list_list_fm] :
( ( take_list_fm @ ( suc @ N2 ) @ ( cons_list_fm @ X @ Xs ) )
= ( cons_list_fm @ X @ ( take_list_fm @ N2 @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_896_take__Suc__Cons,axiom,
! [N2: nat,X: fm,Xs: list_fm] :
( ( take_fm @ ( suc @ N2 ) @ ( cons_fm @ X @ Xs ) )
= ( cons_fm @ X @ ( take_fm @ N2 @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_897_take__Suc__Cons,axiom,
! [N2: nat,X: tm,Xs: list_tm] :
( ( take_tm @ ( suc @ N2 ) @ ( cons_tm @ X @ Xs ) )
= ( cons_tm @ X @ ( take_tm @ N2 @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_898_take__eq__Nil2,axiom,
! [N2: nat,Xs: list_rule] :
( ( nil_rule
= ( take_rule @ N2 @ Xs ) )
= ( ( N2 = zero_zero_nat )
| ( Xs = nil_rule ) ) ) ).
% take_eq_Nil2
thf(fact_899_take__eq__Nil2,axiom,
! [N2: nat,Xs: list_fm] :
( ( nil_fm
= ( take_fm @ N2 @ Xs ) )
= ( ( N2 = zero_zero_nat )
| ( Xs = nil_fm ) ) ) ).
% take_eq_Nil2
thf(fact_900_take__eq__Nil2,axiom,
! [N2: nat,Xs: list_list_fm] :
( ( nil_list_fm
= ( take_list_fm @ N2 @ Xs ) )
= ( ( N2 = zero_zero_nat )
| ( Xs = nil_list_fm ) ) ) ).
% take_eq_Nil2
thf(fact_901_take__eq__Nil2,axiom,
! [N2: nat,Xs: list_tm] :
( ( nil_tm
= ( take_tm @ N2 @ Xs ) )
= ( ( N2 = zero_zero_nat )
| ( Xs = nil_tm ) ) ) ).
% take_eq_Nil2
thf(fact_902_take__eq__Nil,axiom,
! [N2: nat,Xs: list_rule] :
( ( ( take_rule @ N2 @ Xs )
= nil_rule )
= ( ( N2 = zero_zero_nat )
| ( Xs = nil_rule ) ) ) ).
% take_eq_Nil
thf(fact_903_take__eq__Nil,axiom,
! [N2: nat,Xs: list_fm] :
( ( ( take_fm @ N2 @ Xs )
= nil_fm )
= ( ( N2 = zero_zero_nat )
| ( Xs = nil_fm ) ) ) ).
% take_eq_Nil
thf(fact_904_take__eq__Nil,axiom,
! [N2: nat,Xs: list_list_fm] :
( ( ( take_list_fm @ N2 @ Xs )
= nil_list_fm )
= ( ( N2 = zero_zero_nat )
| ( Xs = nil_list_fm ) ) ) ).
% take_eq_Nil
thf(fact_905_take__eq__Nil,axiom,
! [N2: nat,Xs: list_tm] :
( ( ( take_tm @ N2 @ Xs )
= nil_tm )
= ( ( N2 = zero_zero_nat )
| ( Xs = nil_tm ) ) ) ).
% take_eq_Nil
thf(fact_906_take0,axiom,
( ( take_rule @ zero_zero_nat )
= ( ^ [Xs3: list_rule] : nil_rule ) ) ).
% take0
thf(fact_907_take0,axiom,
( ( take_fm @ zero_zero_nat )
= ( ^ [Xs3: list_fm] : nil_fm ) ) ).
% take0
thf(fact_908_take0,axiom,
( ( take_list_fm @ zero_zero_nat )
= ( ^ [Xs3: list_list_fm] : nil_list_fm ) ) ).
% take0
thf(fact_909_take0,axiom,
( ( take_tm @ zero_zero_nat )
= ( ^ [Xs3: list_tm] : nil_tm ) ) ).
% take0
thf(fact_910_stake__invert__Nil,axiom,
! [N2: nat,S: stream_rule] :
( ( ( stake_rule @ N2 @ S )
= nil_rule )
= ( N2 = zero_zero_nat ) ) ).
% stake_invert_Nil
thf(fact_911_stake__invert__Nil,axiom,
! [N2: nat,S: stream_fm] :
( ( ( stake_fm @ N2 @ S )
= nil_fm )
= ( N2 = zero_zero_nat ) ) ).
% stake_invert_Nil
thf(fact_912_stake__invert__Nil,axiom,
! [N2: nat,S: stream_list_fm] :
( ( ( stake_list_fm @ N2 @ S )
= nil_list_fm )
= ( N2 = zero_zero_nat ) ) ).
% stake_invert_Nil
thf(fact_913_stake__invert__Nil,axiom,
! [N2: nat,S: stream_tm] :
( ( ( stake_tm @ N2 @ S )
= nil_tm )
= ( N2 = zero_zero_nat ) ) ).
% stake_invert_Nil
thf(fact_914_length__greater__0__conv,axiom,
! [Xs: list_rule] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_rule @ Xs ) )
= ( Xs != nil_rule ) ) ).
% length_greater_0_conv
thf(fact_915_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_916_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_917_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_918_take__0,axiom,
! [Xs: list_rule] :
( ( take_rule @ zero_zero_nat @ Xs )
= nil_rule ) ).
% take_0
thf(fact_919_take__0,axiom,
! [Xs: list_fm] :
( ( take_fm @ zero_zero_nat @ Xs )
= nil_fm ) ).
% take_0
thf(fact_920_take__0,axiom,
! [Xs: list_list_fm] :
( ( take_list_fm @ zero_zero_nat @ Xs )
= nil_list_fm ) ).
% take_0
thf(fact_921_take__0,axiom,
! [Xs: list_tm] :
( ( take_tm @ zero_zero_nat @ Xs )
= nil_tm ) ).
% take_0
thf(fact_922_take__Nil,axiom,
! [N2: nat] :
( ( take_rule @ N2 @ nil_rule )
= nil_rule ) ).
% take_Nil
thf(fact_923_take__Nil,axiom,
! [N2: nat] :
( ( take_fm @ N2 @ nil_fm )
= nil_fm ) ).
% take_Nil
thf(fact_924_take__Nil,axiom,
! [N2: nat] :
( ( take_list_fm @ N2 @ nil_list_fm )
= nil_list_fm ) ).
% take_Nil
thf(fact_925_take__Nil,axiom,
! [N2: nat] :
( ( take_tm @ N2 @ nil_tm )
= nil_tm ) ).
% take_Nil
thf(fact_926_in__set__takeD,axiom,
! [X: produc340336539035504054m_rule,N2: nat,Xs: list_P2774625669004309958m_rule] :
( ( member7231649785386036813m_rule @ X @ ( set_Pr4534715572506550497m_rule @ ( take_P3067526267515409992m_rule @ N2 @ Xs ) ) )
=> ( member7231649785386036813m_rule @ X @ ( set_Pr4534715572506550497m_rule @ Xs ) ) ) ).
% in_set_takeD
thf(fact_927_in__set__takeD,axiom,
! [X: nat,N2: nat,Xs: list_nat] :
( ( member_nat3 @ X @ ( set_nat2 @ ( take_nat @ N2 @ Xs ) ) )
=> ( member_nat3 @ X @ ( set_nat2 @ Xs ) ) ) ).
% in_set_takeD
thf(fact_928_in__set__takeD,axiom,
! [X: fm,N2: nat,Xs: list_fm] :
( ( member_fm3 @ X @ ( set_fm2 @ ( take_fm @ N2 @ Xs ) ) )
=> ( member_fm3 @ X @ ( set_fm2 @ Xs ) ) ) ).
% in_set_takeD
thf(fact_929_in__set__takeD,axiom,
! [X: list_fm,N2: nat,Xs: list_list_fm] :
( ( member_list_fm3 @ X @ ( set_list_fm2 @ ( take_list_fm @ N2 @ Xs ) ) )
=> ( member_list_fm3 @ X @ ( set_list_fm2 @ Xs ) ) ) ).
% in_set_takeD
thf(fact_930_in__set__takeD,axiom,
! [X: tm,N2: nat,Xs: list_tm] :
( ( member_tm3 @ X @ ( set_tm2 @ ( take_tm @ N2 @ Xs ) ) )
=> ( member_tm3 @ X @ ( set_tm2 @ Xs ) ) ) ).
% in_set_takeD
thf(fact_931_sdrop_Osimps_I1_J,axiom,
! [S: stream2709947120125613254m_rule] :
( ( sdrop_8169176516188972301m_rule @ zero_zero_nat @ S )
= S ) ).
% sdrop.simps(1)
thf(fact_932_length__Suc__conv,axiom,
! [Xs: list_rule,N2: nat] :
( ( ( size_size_list_rule @ Xs )
= ( suc @ N2 ) )
= ( ? [Y4: rule,Ys2: list_rule] :
( ( Xs
= ( cons_rule @ Y4 @ Ys2 ) )
& ( ( size_size_list_rule @ Ys2 )
= N2 ) ) ) ) ).
% length_Suc_conv
thf(fact_933_length__Suc__conv,axiom,
! [Xs: list_list_fm,N2: nat] :
( ( ( size_s115229985653309035ist_fm @ Xs )
= ( suc @ N2 ) )
= ( ? [Y4: list_fm,Ys2: list_list_fm] :
( ( Xs
= ( cons_list_fm @ Y4 @ Ys2 ) )
& ( ( size_s115229985653309035ist_fm @ Ys2 )
= N2 ) ) ) ) ).
% length_Suc_conv
thf(fact_934_length__Suc__conv,axiom,
! [Xs: list_fm,N2: nat] :
( ( ( size_size_list_fm @ Xs )
= ( suc @ N2 ) )
= ( ? [Y4: fm,Ys2: list_fm] :
( ( Xs
= ( cons_fm @ Y4 @ Ys2 ) )
& ( ( size_size_list_fm @ Ys2 )
= N2 ) ) ) ) ).
% length_Suc_conv
thf(fact_935_length__Suc__conv,axiom,
! [Xs: list_tm,N2: nat] :
( ( ( size_size_list_tm @ Xs )
= ( suc @ N2 ) )
= ( ? [Y4: tm,Ys2: list_tm] :
( ( Xs
= ( cons_tm @ Y4 @ Ys2 ) )
& ( ( size_size_list_tm @ Ys2 )
= N2 ) ) ) ) ).
% length_Suc_conv
thf(fact_936_Suc__length__conv,axiom,
! [N2: nat,Xs: list_rule] :
( ( ( suc @ N2 )
= ( size_size_list_rule @ Xs ) )
= ( ? [Y4: rule,Ys2: list_rule] :
( ( Xs
= ( cons_rule @ Y4 @ Ys2 ) )
& ( ( size_size_list_rule @ Ys2 )
= N2 ) ) ) ) ).
% Suc_length_conv
thf(fact_937_Suc__length__conv,axiom,
! [N2: nat,Xs: list_list_fm] :
( ( ( suc @ N2 )
= ( size_s115229985653309035ist_fm @ Xs ) )
= ( ? [Y4: list_fm,Ys2: list_list_fm] :
( ( Xs
= ( cons_list_fm @ Y4 @ Ys2 ) )
& ( ( size_s115229985653309035ist_fm @ Ys2 )
= N2 ) ) ) ) ).
% Suc_length_conv
thf(fact_938_Suc__length__conv,axiom,
! [N2: nat,Xs: list_fm] :
( ( ( suc @ N2 )
= ( size_size_list_fm @ Xs ) )
= ( ? [Y4: fm,Ys2: list_fm] :
( ( Xs
= ( cons_fm @ Y4 @ Ys2 ) )
& ( ( size_size_list_fm @ Ys2 )
= N2 ) ) ) ) ).
% Suc_length_conv
thf(fact_939_Suc__length__conv,axiom,
! [N2: nat,Xs: list_tm] :
( ( ( suc @ N2 )
= ( size_size_list_tm @ Xs ) )
= ( ? [Y4: tm,Ys2: list_tm] :
( ( Xs
= ( cons_tm @ Y4 @ Ys2 ) )
& ( ( size_size_list_tm @ Ys2 )
= N2 ) ) ) ) ).
% Suc_length_conv
thf(fact_940_list_Osize_I3_J,axiom,
( ( size_size_list_rule @ nil_rule )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_941_list_Osize_I3_J,axiom,
( ( size_size_list_fm @ nil_fm )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_942_list_Osize_I3_J,axiom,
( ( size_s115229985653309035ist_fm @ nil_list_fm )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_943_list_Osize_I3_J,axiom,
( ( size_size_list_tm @ nil_tm )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_944_stake_Osimps_I1_J,axiom,
! [S: stream_rule] :
( ( stake_rule @ zero_zero_nat @ S )
= nil_rule ) ).
% stake.simps(1)
thf(fact_945_stake_Osimps_I1_J,axiom,
! [S: stream_fm] :
( ( stake_fm @ zero_zero_nat @ S )
= nil_fm ) ).
% stake.simps(1)
thf(fact_946_stake_Osimps_I1_J,axiom,
! [S: stream_list_fm] :
( ( stake_list_fm @ zero_zero_nat @ S )
= nil_list_fm ) ).
% stake.simps(1)
thf(fact_947_stake_Osimps_I1_J,axiom,
! [S: stream_tm] :
( ( stake_tm @ zero_zero_nat @ S )
= nil_tm ) ).
% stake.simps(1)
thf(fact_948_snth_Osimps_I1_J,axiom,
! [S: stream2709947120125613254m_rule] :
( ( snth_P6679518042731451922m_rule @ S @ zero_zero_nat )
= ( shd_Pr4562317740776619530m_rule @ S ) ) ).
% snth.simps(1)
thf(fact_949_length__pos__if__in__set,axiom,
! [X: produc340336539035504054m_rule,Xs: list_P2774625669004309958m_rule] :
( ( member7231649785386036813m_rule @ X @ ( set_Pr4534715572506550497m_rule @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s1575636608424004698m_rule @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_950_length__pos__if__in__set,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat3 @ X @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_951_length__pos__if__in__set,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm3 @ X @ ( set_fm2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_fm @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_952_length__pos__if__in__set,axiom,
! [X: list_fm,Xs: list_list_fm] :
( ( member_list_fm3 @ X @ ( set_list_fm2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s115229985653309035ist_fm @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_953_length__pos__if__in__set,axiom,
! [X: tm,Xs: list_tm] :
( ( member_tm3 @ X @ ( set_tm2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_tm @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_954_length__Suc__conv__rev,axiom,
! [Xs: list_rule,N2: nat] :
( ( ( size_size_list_rule @ Xs )
= ( suc @ N2 ) )
= ( ? [Y4: rule,Ys2: list_rule] :
( ( Xs
= ( append_rule @ Ys2 @ ( cons_rule @ Y4 @ nil_rule ) ) )
& ( ( size_size_list_rule @ Ys2 )
= N2 ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_955_length__Suc__conv__rev,axiom,
! [Xs: list_list_fm,N2: nat] :
( ( ( size_s115229985653309035ist_fm @ Xs )
= ( suc @ N2 ) )
= ( ? [Y4: list_fm,Ys2: list_list_fm] :
( ( Xs
= ( append_list_fm @ Ys2 @ ( cons_list_fm @ Y4 @ nil_list_fm ) ) )
& ( ( size_s115229985653309035ist_fm @ Ys2 )
= N2 ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_956_length__Suc__conv__rev,axiom,
! [Xs: list_fm,N2: nat] :
( ( ( size_size_list_fm @ Xs )
= ( suc @ N2 ) )
= ( ? [Y4: fm,Ys2: list_fm] :
( ( Xs
= ( append_fm @ Ys2 @ ( cons_fm @ Y4 @ nil_fm ) ) )
& ( ( size_size_list_fm @ Ys2 )
= N2 ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_957_length__Suc__conv__rev,axiom,
! [Xs: list_tm,N2: nat] :
( ( ( size_size_list_tm @ Xs )
= ( suc @ N2 ) )
= ( ? [Y4: tm,Ys2: list_tm] :
( ( Xs
= ( append_tm @ Ys2 @ ( cons_tm @ Y4 @ nil_tm ) ) )
& ( ( size_size_list_tm @ Ys2 )
= N2 ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_958_take__Suc__conv__app__nth,axiom,
! [I2: nat,Xs: list_rule] :
( ( ord_less_nat @ I2 @ ( size_size_list_rule @ Xs ) )
=> ( ( take_rule @ ( suc @ I2 ) @ Xs )
= ( append_rule @ ( take_rule @ I2 @ Xs ) @ ( cons_rule @ ( nth_rule @ Xs @ I2 ) @ nil_rule ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_959_take__Suc__conv__app__nth,axiom,
! [I2: nat,Xs: list_list_fm] :
( ( ord_less_nat @ I2 @ ( size_s115229985653309035ist_fm @ Xs ) )
=> ( ( take_list_fm @ ( suc @ I2 ) @ Xs )
= ( append_list_fm @ ( take_list_fm @ I2 @ Xs ) @ ( cons_list_fm @ ( nth_list_fm @ Xs @ I2 ) @ nil_list_fm ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_960_take__Suc__conv__app__nth,axiom,
! [I2: nat,Xs: list_fm] :
( ( ord_less_nat @ I2 @ ( size_size_list_fm @ Xs ) )
=> ( ( take_fm @ ( suc @ I2 ) @ Xs )
= ( append_fm @ ( take_fm @ I2 @ Xs ) @ ( cons_fm @ ( nth_fm @ Xs @ I2 ) @ nil_fm ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_961_take__Suc__conv__app__nth,axiom,
! [I2: nat,Xs: list_tm] :
( ( ord_less_nat @ I2 @ ( size_size_list_tm @ Xs ) )
=> ( ( take_tm @ ( suc @ I2 ) @ Xs )
= ( append_tm @ ( take_tm @ I2 @ Xs ) @ ( cons_tm @ ( nth_tm @ Xs @ I2 ) @ nil_tm ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_962_sdrop__cycle__eq__mod__0,axiom,
! [U: list_P2774625669004309958m_rule,N2: nat] :
( ( U != nil_Pr2808214839852828720m_rule )
=> ( ( ( modulo_modulo_nat @ N2 @ ( size_s1575636608424004698m_rule @ U ) )
= zero_zero_nat )
=> ( ( sdrop_8169176516188972301m_rule @ N2 @ ( cycle_5335117900641983313m_rule @ U ) )
= ( cycle_5335117900641983313m_rule @ U ) ) ) ) ).
% sdrop_cycle_eq_mod_0
thf(fact_963_sdrop__cycle__eq__mod__0,axiom,
! [U: list_rule,N2: nat] :
( ( U != nil_rule )
=> ( ( ( modulo_modulo_nat @ N2 @ ( size_size_list_rule @ U ) )
= zero_zero_nat )
=> ( ( sdrop_rule @ N2 @ ( cycle_rule @ U ) )
= ( cycle_rule @ U ) ) ) ) ).
% sdrop_cycle_eq_mod_0
thf(fact_964_sdrop__cycle__eq__mod__0,axiom,
! [U: list_fm,N2: nat] :
( ( U != nil_fm )
=> ( ( ( modulo_modulo_nat @ N2 @ ( size_size_list_fm @ U ) )
= zero_zero_nat )
=> ( ( sdrop_fm @ N2 @ ( cycle_fm @ U ) )
= ( cycle_fm @ U ) ) ) ) ).
% sdrop_cycle_eq_mod_0
thf(fact_965_sdrop__cycle__eq__mod__0,axiom,
! [U: list_list_fm,N2: nat] :
( ( U != nil_list_fm )
=> ( ( ( modulo_modulo_nat @ N2 @ ( size_s115229985653309035ist_fm @ U ) )
= zero_zero_nat )
=> ( ( sdrop_list_fm @ N2 @ ( cycle_list_fm @ U ) )
= ( cycle_list_fm @ U ) ) ) ) ).
% sdrop_cycle_eq_mod_0
thf(fact_966_sdrop__cycle__eq__mod__0,axiom,
! [U: list_tm,N2: nat] :
( ( U != nil_tm )
=> ( ( ( modulo_modulo_nat @ N2 @ ( size_size_list_tm @ U ) )
= zero_zero_nat )
=> ( ( sdrop_tm @ N2 @ ( cycle_tm @ U ) )
= ( cycle_tm @ U ) ) ) ) ).
% sdrop_cycle_eq_mod_0
thf(fact_967_Hintikka__def,axiom,
( hintikka
= ( ^ [H2: set_fm] :
( ! [N3: nat,Ts3: list_tm] :
( ( member_fm3 @ ( pre @ N3 @ Ts3 ) @ H2 )
=> ~ ( member_fm3 @ ( neg @ ( pre @ N3 @ Ts3 ) ) @ H2 ) )
& ! [P2: fm,Q3: fm] :
( ( member_fm3 @ ( dis @ P2 @ Q3 ) @ H2 )
=> ( ( member_fm3 @ P2 @ H2 )
& ( member_fm3 @ Q3 @ H2 ) ) )
& ! [P2: fm,Q3: fm] :
( ( member_fm3 @ ( imp @ P2 @ Q3 ) @ H2 )
=> ( ( member_fm3 @ ( neg @ P2 ) @ H2 )
& ( member_fm3 @ Q3 @ H2 ) ) )
& ! [P2: fm,Q3: fm] :
( ( member_fm3 @ ( neg @ ( con @ P2 @ Q3 ) ) @ H2 )
=> ( ( member_fm3 @ ( neg @ P2 ) @ H2 )
& ( member_fm3 @ ( neg @ Q3 ) @ H2 ) ) )
& ! [P2: fm,Q3: fm] :
( ( member_fm3 @ ( con @ P2 @ Q3 ) @ H2 )
=> ( ( member_fm3 @ P2 @ H2 )
| ( member_fm3 @ Q3 @ H2 ) ) )
& ! [P2: fm,Q3: fm] :
( ( member_fm3 @ ( neg @ ( imp @ P2 @ Q3 ) ) @ H2 )
=> ( ( member_fm3 @ P2 @ H2 )
| ( member_fm3 @ ( neg @ Q3 ) @ H2 ) ) )
& ! [P2: fm,Q3: fm] :
( ( member_fm3 @ ( neg @ ( dis @ P2 @ Q3 ) ) @ H2 )
=> ( ( member_fm3 @ ( neg @ P2 ) @ H2 )
| ( member_fm3 @ ( neg @ Q3 ) @ H2 ) ) )
& ! [P2: fm] :
( ( member_fm3 @ ( exi @ P2 ) @ H2 )
=> ! [X2: tm] :
( ( member_tm3 @ X2 @ ( terms @ H2 ) )
=> ( member_fm3 @ ( sub @ zero_zero_nat @ X2 @ P2 ) @ H2 ) ) )
& ! [P2: fm] :
( ( member_fm3 @ ( neg @ ( uni @ P2 ) ) @ H2 )
=> ! [X2: tm] :
( ( member_tm3 @ X2 @ ( terms @ H2 ) )
=> ( member_fm3 @ ( neg @ ( sub @ zero_zero_nat @ X2 @ P2 ) ) @ H2 ) ) )
& ! [P2: fm] :
( ( member_fm3 @ ( uni @ P2 ) @ H2 )
=> ? [X2: tm] :
( ( member_tm3 @ X2 @ ( terms @ H2 ) )
& ( member_fm3 @ ( sub @ zero_zero_nat @ X2 @ P2 ) @ H2 ) ) )
& ! [P2: fm] :
( ( member_fm3 @ ( neg @ ( exi @ P2 ) ) @ H2 )
=> ? [X2: tm] :
( ( member_tm3 @ X2 @ ( terms @ H2 ) )
& ( member_fm3 @ ( neg @ ( sub @ zero_zero_nat @ X2 @ P2 ) ) @ H2 ) ) )
& ! [P2: fm] :
( ( member_fm3 @ ( neg @ ( neg @ P2 ) ) @ H2 )
=> ( member_fm3 @ P2 @ H2 ) ) ) ) ) ).
% Hintikka_def
thf(fact_968_Hintikka_Ointro,axiom,
! [H: set_fm] :
( ! [N: nat,Ts2: list_tm] :
( ( member_fm3 @ ( pre @ N @ Ts2 ) @ H )
=> ~ ( member_fm3 @ ( neg @ ( pre @ N @ Ts2 ) ) @ H ) )
=> ( ! [P4: fm,Q2: fm] :
( ( member_fm3 @ ( dis @ P4 @ Q2 ) @ H )
=> ( ( member_fm3 @ P4 @ H )
& ( member_fm3 @ Q2 @ H ) ) )
=> ( ! [P4: fm,Q2: fm] :
( ( member_fm3 @ ( imp @ P4 @ Q2 ) @ H )
=> ( ( member_fm3 @ ( neg @ P4 ) @ H )
& ( member_fm3 @ Q2 @ H ) ) )
=> ( ! [P4: fm,Q2: fm] :
( ( member_fm3 @ ( neg @ ( con @ P4 @ Q2 ) ) @ H )
=> ( ( member_fm3 @ ( neg @ P4 ) @ H )
& ( member_fm3 @ ( neg @ Q2 ) @ H ) ) )
=> ( ! [P4: fm,Q2: fm] :
( ( member_fm3 @ ( con @ P4 @ Q2 ) @ H )
=> ( ( member_fm3 @ P4 @ H )
| ( member_fm3 @ Q2 @ H ) ) )
=> ( ! [P4: fm,Q2: fm] :
( ( member_fm3 @ ( neg @ ( imp @ P4 @ Q2 ) ) @ H )
=> ( ( member_fm3 @ P4 @ H )
| ( member_fm3 @ ( neg @ Q2 ) @ H ) ) )
=> ( ! [P4: fm,Q2: fm] :
( ( member_fm3 @ ( neg @ ( dis @ P4 @ Q2 ) ) @ H )
=> ( ( member_fm3 @ ( neg @ P4 ) @ H )
| ( member_fm3 @ ( neg @ Q2 ) @ H ) ) )
=> ( ! [P4: fm] :
( ( member_fm3 @ ( exi @ P4 ) @ H )
=> ! [X3: tm] :
( ( member_tm3 @ X3 @ ( terms @ H ) )
=> ( member_fm3 @ ( sub @ zero_zero_nat @ X3 @ P4 ) @ H ) ) )
=> ( ! [P4: fm] :
( ( member_fm3 @ ( neg @ ( uni @ P4 ) ) @ H )
=> ! [X3: tm] :
( ( member_tm3 @ X3 @ ( terms @ H ) )
=> ( member_fm3 @ ( neg @ ( sub @ zero_zero_nat @ X3 @ P4 ) ) @ H ) ) )
=> ( ! [P4: fm] :
( ( member_fm3 @ ( uni @ P4 ) @ H )
=> ? [X4: tm] :
( ( member_tm3 @ X4 @ ( terms @ H ) )
& ( member_fm3 @ ( sub @ zero_zero_nat @ X4 @ P4 ) @ H ) ) )
=> ( ! [P4: fm] :
( ( member_fm3 @ ( neg @ ( exi @ P4 ) ) @ H )
=> ? [X4: tm] :
( ( member_tm3 @ X4 @ ( terms @ H ) )
& ( member_fm3 @ ( neg @ ( sub @ zero_zero_nat @ X4 @ P4 ) ) @ H ) ) )
=> ( ! [P4: fm] :
( ( member_fm3 @ ( neg @ ( neg @ P4 ) ) @ H )
=> ( member_fm3 @ P4 @ H ) )
=> ( hintikka @ H ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Hintikka.intro
thf(fact_969_nth__Cons__0,axiom,
! [X: rule,Xs: list_rule] :
( ( nth_rule @ ( cons_rule @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_970_nth__Cons__0,axiom,
! [X: list_fm,Xs: list_list_fm] :
( ( nth_list_fm @ ( cons_list_fm @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_971_nth__Cons__0,axiom,
! [X: fm,Xs: list_fm] :
( ( nth_fm @ ( cons_fm @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_972_nth__Cons__0,axiom,
! [X: tm,Xs: list_tm] :
( ( nth_tm @ ( cons_tm @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_973_nth__Cons__Suc,axiom,
! [X: rule,Xs: list_rule,N2: nat] :
( ( nth_rule @ ( cons_rule @ X @ Xs ) @ ( suc @ N2 ) )
= ( nth_rule @ Xs @ N2 ) ) ).
% nth_Cons_Suc
thf(fact_974_nth__Cons__Suc,axiom,
! [X: list_fm,Xs: list_list_fm,N2: nat] :
( ( nth_list_fm @ ( cons_list_fm @ X @ Xs ) @ ( suc @ N2 ) )
= ( nth_list_fm @ Xs @ N2 ) ) ).
% nth_Cons_Suc
thf(fact_975_nth__Cons__Suc,axiom,
! [X: fm,Xs: list_fm,N2: nat] :
( ( nth_fm @ ( cons_fm @ X @ Xs ) @ ( suc @ N2 ) )
= ( nth_fm @ Xs @ N2 ) ) ).
% nth_Cons_Suc
thf(fact_976_nth__Cons__Suc,axiom,
! [X: tm,Xs: list_tm,N2: nat] :
( ( nth_tm @ ( cons_tm @ X @ Xs ) @ ( suc @ N2 ) )
= ( nth_tm @ Xs @ N2 ) ) ).
% nth_Cons_Suc
thf(fact_977_nth__append__length,axiom,
! [Xs: list_rule,X: rule,Ys: list_rule] :
( ( nth_rule @ ( append_rule @ Xs @ ( cons_rule @ X @ Ys ) ) @ ( size_size_list_rule @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_978_nth__append__length,axiom,
! [Xs: list_list_fm,X: list_fm,Ys: list_list_fm] :
( ( nth_list_fm @ ( append_list_fm @ Xs @ ( cons_list_fm @ X @ Ys ) ) @ ( size_s115229985653309035ist_fm @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_979_nth__append__length,axiom,
! [Xs: list_fm,X: fm,Ys: list_fm] :
( ( nth_fm @ ( append_fm @ Xs @ ( cons_fm @ X @ Ys ) ) @ ( size_size_list_fm @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_980_nth__append__length,axiom,
! [Xs: list_tm,X: tm,Ys: list_tm] :
( ( nth_tm @ ( append_tm @ Xs @ ( cons_tm @ X @ Ys ) ) @ ( size_size_list_tm @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_981_stake__nth,axiom,
! [M: nat,N2: nat,S: stream2709947120125613254m_rule] :
( ( ord_less_nat @ M @ N2 )
=> ( ( nth_Pr3936752564452695919m_rule @ ( stake_5421812949518764133m_rule @ N2 @ S ) @ M )
= ( snth_P6679518042731451922m_rule @ S @ M ) ) ) ).
% stake_nth
thf(fact_982_shift__snth__less,axiom,
! [P: nat,Xs: list_P2774625669004309958m_rule,S: stream2709947120125613254m_rule] :
( ( ord_less_nat @ P @ ( size_s1575636608424004698m_rule @ Xs ) )
=> ( ( snth_P6679518042731451922m_rule @ ( shift_2334844276748245581m_rule @ Xs @ S ) @ P )
= ( nth_Pr3936752564452695919m_rule @ Xs @ P ) ) ) ).
% shift_snth_less
thf(fact_983_sub_Osimps_I7_J,axiom,
! [V2: nat,S: tm,P: fm] :
( ( sub @ V2 @ S @ ( neg @ P ) )
= ( neg @ ( sub @ V2 @ S @ P ) ) ) ).
% sub.simps(7)
thf(fact_984_sub_Osimps_I4_J,axiom,
! [V2: nat,S: tm,P: fm,Q: fm] :
( ( sub @ V2 @ S @ ( con @ P @ Q ) )
= ( con @ ( sub @ V2 @ S @ P ) @ ( sub @ V2 @ S @ Q ) ) ) ).
% sub.simps(4)
thf(fact_985_sub_Osimps_I3_J,axiom,
! [V2: nat,S: tm,P: fm,Q: fm] :
( ( sub @ V2 @ S @ ( dis @ P @ Q ) )
= ( dis @ ( sub @ V2 @ S @ P ) @ ( sub @ V2 @ S @ Q ) ) ) ).
% sub.simps(3)
thf(fact_986_sub_Osimps_I2_J,axiom,
! [V2: nat,S: tm,P: fm,Q: fm] :
( ( sub @ V2 @ S @ ( imp @ P @ Q ) )
= ( imp @ ( sub @ V2 @ S @ P ) @ ( sub @ V2 @ S @ Q ) ) ) ).
% sub.simps(2)
thf(fact_987_cycle__nth,axiom,
! [Xs: list_P2774625669004309958m_rule,N2: nat] :
( ( Xs != nil_Pr2808214839852828720m_rule )
=> ( ( snth_P6679518042731451922m_rule @ ( cycle_5335117900641983313m_rule @ Xs ) @ N2 )
= ( nth_Pr3936752564452695919m_rule @ Xs @ ( modulo_modulo_nat @ N2 @ ( size_s1575636608424004698m_rule @ Xs ) ) ) ) ) ).
% cycle_nth
thf(fact_988_cycle__nth,axiom,
! [Xs: list_rule,N2: nat] :
( ( Xs != nil_rule )
=> ( ( snth_rule @ ( cycle_rule @ Xs ) @ N2 )
= ( nth_rule @ Xs @ ( modulo_modulo_nat @ N2 @ ( size_size_list_rule @ Xs ) ) ) ) ) ).
% cycle_nth
thf(fact_989_cycle__nth,axiom,
! [Xs: list_fm,N2: nat] :
( ( Xs != nil_fm )
=> ( ( snth_fm @ ( cycle_fm @ Xs ) @ N2 )
= ( nth_fm @ Xs @ ( modulo_modulo_nat @ N2 @ ( size_size_list_fm @ Xs ) ) ) ) ) ).
% cycle_nth
thf(fact_990_cycle__nth,axiom,
! [Xs: list_list_fm,N2: nat] :
( ( Xs != nil_list_fm )
=> ( ( snth_list_fm @ ( cycle_list_fm @ Xs ) @ N2 )
= ( nth_list_fm @ Xs @ ( modulo_modulo_nat @ N2 @ ( size_s115229985653309035ist_fm @ Xs ) ) ) ) ) ).
% cycle_nth
thf(fact_991_cycle__nth,axiom,
! [Xs: list_tm,N2: nat] :
( ( Xs != nil_tm )
=> ( ( snth_tm @ ( cycle_tm @ Xs ) @ N2 )
= ( nth_tm @ Xs @ ( modulo_modulo_nat @ N2 @ ( size_size_list_tm @ Xs ) ) ) ) ) ).
% cycle_nth
thf(fact_992_mod__suff,axiom,
! [M: nat,P3: nat > $o,K: nat] :
( ! [N: nat] :
( ( ord_less_nat @ M @ N )
=> ( P3 @ ( modulo_modulo_nat @ N @ K ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P3 @ I3 ) ) ) ) ).
% mod_suff
thf(fact_993_mod__hit,axiom,
! [K: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [N: nat] :
( ( ord_less_nat @ M @ N )
& ( ( modulo_modulo_nat @ N @ K )
= I3 ) ) ) ) ).
% mod_hit
thf(fact_994_Hintikka_OGammaExi,axiom,
! [H: set_fm,P: fm] :
( ( hintikka @ H )
=> ( ( member_fm3 @ ( exi @ P ) @ H )
=> ! [X4: tm] :
( ( member_tm3 @ X4 @ ( terms @ H ) )
=> ( member_fm3 @ ( sub @ zero_zero_nat @ X4 @ P ) @ H ) ) ) ) ).
% Hintikka.GammaExi
thf(fact_995_Hintikka_ODeltaUni,axiom,
! [H: set_fm,P: fm] :
( ( hintikka @ H )
=> ( ( member_fm3 @ ( uni @ P ) @ H )
=> ? [X3: tm] :
( ( member_tm3 @ X3 @ ( terms @ H ) )
& ( member_fm3 @ ( sub @ zero_zero_nat @ X3 @ P ) @ H ) ) ) ) ).
% Hintikka.DeltaUni
thf(fact_996_Hintikka_ODeltaExi,axiom,
! [H: set_fm,P: fm] :
( ( hintikka @ H )
=> ( ( member_fm3 @ ( neg @ ( exi @ P ) ) @ H )
=> ? [X3: tm] :
( ( member_tm3 @ X3 @ ( terms @ H ) )
& ( member_fm3 @ ( neg @ ( sub @ zero_zero_nat @ X3 @ P ) ) @ H ) ) ) ) ).
% Hintikka.DeltaExi
thf(fact_997_Hintikka_OGammaUni,axiom,
! [H: set_fm,P: fm] :
( ( hintikka @ H )
=> ( ( member_fm3 @ ( neg @ ( uni @ P ) ) @ H )
=> ! [X4: tm] :
( ( member_tm3 @ X4 @ ( terms @ H ) )
=> ( member_fm3 @ ( neg @ ( sub @ zero_zero_nat @ X4 @ P ) ) @ H ) ) ) ) ).
% Hintikka.GammaUni
thf(fact_998_all__set__conv__all__nth,axiom,
! [Xs: list_fm,P3: fm > $o] :
( ( ! [X2: fm] :
( ( member_fm3 @ X2 @ ( set_fm2 @ Xs ) )
=> ( P3 @ X2 ) ) )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_fm @ Xs ) )
=> ( P3 @ ( nth_fm @ Xs @ I4 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_999_all__set__conv__all__nth,axiom,
! [Xs: list_list_fm,P3: list_fm > $o] :
( ( ! [X2: list_fm] :
( ( member_list_fm3 @ X2 @ ( set_list_fm2 @ Xs ) )
=> ( P3 @ X2 ) ) )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s115229985653309035ist_fm @ Xs ) )
=> ( P3 @ ( nth_list_fm @ Xs @ I4 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1000_all__set__conv__all__nth,axiom,
! [Xs: list_tm,P3: tm > $o] :
( ( ! [X2: tm] :
( ( member_tm3 @ X2 @ ( set_tm2 @ Xs ) )
=> ( P3 @ X2 ) ) )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_tm @ Xs ) )
=> ( P3 @ ( nth_tm @ Xs @ I4 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_1001_all__nth__imp__all__set,axiom,
! [Xs: list_P2774625669004309958m_rule,P3: produc340336539035504054m_rule > $o,X: produc340336539035504054m_rule] :
( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s1575636608424004698m_rule @ Xs ) )
=> ( P3 @ ( nth_Pr3936752564452695919m_rule @ Xs @ I ) ) )
=> ( ( member7231649785386036813m_rule @ X @ ( set_Pr4534715572506550497m_rule @ Xs ) )
=> ( P3 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1002_all__nth__imp__all__set,axiom,
! [Xs: list_nat,P3: nat > $o,X: nat] :
( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( P3 @ ( nth_nat @ Xs @ I ) ) )
=> ( ( member_nat3 @ X @ ( set_nat2 @ Xs ) )
=> ( P3 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1003_all__nth__imp__all__set,axiom,
! [Xs: list_fm,P3: fm > $o,X: fm] :
( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_fm @ Xs ) )
=> ( P3 @ ( nth_fm @ Xs @ I ) ) )
=> ( ( member_fm3 @ X @ ( set_fm2 @ Xs ) )
=> ( P3 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1004_all__nth__imp__all__set,axiom,
! [Xs: list_list_fm,P3: list_fm > $o,X: list_fm] :
( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s115229985653309035ist_fm @ Xs ) )
=> ( P3 @ ( nth_list_fm @ Xs @ I ) ) )
=> ( ( member_list_fm3 @ X @ ( set_list_fm2 @ Xs ) )
=> ( P3 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1005_all__nth__imp__all__set,axiom,
! [Xs: list_tm,P3: tm > $o,X: tm] :
( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_tm @ Xs ) )
=> ( P3 @ ( nth_tm @ Xs @ I ) ) )
=> ( ( member_tm3 @ X @ ( set_tm2 @ Xs ) )
=> ( P3 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_1006_in__set__conv__nth,axiom,
! [X: produc340336539035504054m_rule,Xs: list_P2774625669004309958m_rule] :
( ( member7231649785386036813m_rule @ X @ ( set_Pr4534715572506550497m_rule @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s1575636608424004698m_rule @ Xs ) )
& ( ( nth_Pr3936752564452695919m_rule @ Xs @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1007_in__set__conv__nth,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat3 @ X @ ( set_nat2 @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1008_in__set__conv__nth,axiom,
! [X: fm,Xs: list_fm] :
( ( member_fm3 @ X @ ( set_fm2 @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_fm @ Xs ) )
& ( ( nth_fm @ Xs @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1009_in__set__conv__nth,axiom,
! [X: list_fm,Xs: list_list_fm] :
( ( member_list_fm3 @ X @ ( set_list_fm2 @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s115229985653309035ist_fm @ Xs ) )
& ( ( nth_list_fm @ Xs @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1010_in__set__conv__nth,axiom,
! [X: tm,Xs: list_tm] :
( ( member_tm3 @ X @ ( set_tm2 @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_tm @ Xs ) )
& ( ( nth_tm @ Xs @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_1011_list__ball__nth,axiom,
! [N2: nat,Xs: list_fm,P3: fm > $o] :
( ( ord_less_nat @ N2 @ ( size_size_list_fm @ Xs ) )
=> ( ! [X3: fm] :
( ( member_fm3 @ X3 @ ( set_fm2 @ Xs ) )
=> ( P3 @ X3 ) )
=> ( P3 @ ( nth_fm @ Xs @ N2 ) ) ) ) ).
% list_ball_nth
thf(fact_1012_list__ball__nth,axiom,
! [N2: nat,Xs: list_list_fm,P3: list_fm > $o] :
( ( ord_less_nat @ N2 @ ( size_s115229985653309035ist_fm @ Xs ) )
=> ( ! [X3: list_fm] :
( ( member_list_fm3 @ X3 @ ( set_list_fm2 @ Xs ) )
=> ( P3 @ X3 ) )
=> ( P3 @ ( nth_list_fm @ Xs @ N2 ) ) ) ) ).
% list_ball_nth
thf(fact_1013_list__ball__nth,axiom,
! [N2: nat,Xs: list_tm,P3: tm > $o] :
( ( ord_less_nat @ N2 @ ( size_size_list_tm @ Xs ) )
=> ( ! [X3: tm] :
( ( member_tm3 @ X3 @ ( set_tm2 @ Xs ) )
=> ( P3 @ X3 ) )
=> ( P3 @ ( nth_tm @ Xs @ N2 ) ) ) ) ).
% list_ball_nth
thf(fact_1014_nth__mem,axiom,
! [N2: nat,Xs: list_P2774625669004309958m_rule] :
( ( ord_less_nat @ N2 @ ( size_s1575636608424004698m_rule @ Xs ) )
=> ( member7231649785386036813m_rule @ ( nth_Pr3936752564452695919m_rule @ Xs @ N2 ) @ ( set_Pr4534715572506550497m_rule @ Xs ) ) ) ).
% nth_mem
thf(fact_1015_nth__mem,axiom,
! [N2: nat,Xs: list_nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( member_nat3 @ ( nth_nat @ Xs @ N2 ) @ ( set_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_1016_nth__mem,axiom,
! [N2: nat,Xs: list_fm] :
( ( ord_less_nat @ N2 @ ( size_size_list_fm @ Xs ) )
=> ( member_fm3 @ ( nth_fm @ Xs @ N2 ) @ ( set_fm2 @ Xs ) ) ) ).
% nth_mem
thf(fact_1017_nth__mem,axiom,
! [N2: nat,Xs: list_list_fm] :
( ( ord_less_nat @ N2 @ ( size_s115229985653309035ist_fm @ Xs ) )
=> ( member_list_fm3 @ ( nth_list_fm @ Xs @ N2 ) @ ( set_list_fm2 @ Xs ) ) ) ).
% nth_mem
thf(fact_1018_nth__mem,axiom,
! [N2: nat,Xs: list_tm] :
( ( ord_less_nat @ N2 @ ( size_size_list_tm @ Xs ) )
=> ( member_tm3 @ ( nth_tm @ Xs @ N2 ) @ ( set_tm2 @ Xs ) ) ) ).
% nth_mem
thf(fact_1019_listrel__iff__nth,axiom,
! [Xs: list_P2887561121880082555ist_fm,Ys: list_rule,R: set_Pr1822751329126368876m_rule] :
( ( member2627171955076190819t_rule @ ( produc3723918873312807110t_rule @ Xs @ Ys ) @ ( listre749368180268468182m_rule @ R ) )
= ( ( ( size_s3138477486474831591ist_fm @ Xs )
= ( size_size_list_rule @ Ys ) )
& ! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_s3138477486474831591ist_fm @ Xs ) )
=> ( member7231649785386036813m_rule @ ( produc1733806532565653680m_rule @ ( nth_Pr580027083122244092ist_fm @ Xs @ N3 ) @ ( nth_rule @ Ys @ N3 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_1020_listrel__iff__nth,axiom,
! [Xs: list_list_tm,Ys: list_list_fm,R: set_Pr5202636777678657877ist_fm] :
( ( member5109946355746398750ist_fm @ ( produc8328755919010440805ist_fm @ Xs @ Ys ) @ ( listre5805154493122130495ist_fm @ R ) )
= ( ( ( size_s9096087352182575069ist_tm @ Xs )
= ( size_s115229985653309035ist_fm @ Ys ) )
& ! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_s9096087352182575069ist_tm @ Xs ) )
=> ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ ( nth_list_tm @ Xs @ N3 ) @ ( nth_list_fm @ Ys @ N3 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_1021_listrel__iff__nth,axiom,
! [Xs: list_tm,Ys: list_fm,R: set_Pr2698443736021152725_tm_fm] :
( ( member4699826688122452638ist_fm @ ( produc1414352766439514085ist_fm @ Xs @ Ys ) @ ( listrel_tm_fm @ R ) )
= ( ( ( size_size_list_tm @ Xs )
= ( size_size_list_fm @ Ys ) )
& ! [N3: nat] :
( ( ord_less_nat @ N3 @ ( size_size_list_tm @ Xs ) )
=> ( member3117664881408846110_tm_fm @ ( product_Pair_tm_fm @ ( nth_tm @ Xs @ N3 ) @ ( nth_fm @ Ys @ N3 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_1022_id__take__nth__drop,axiom,
! [I2: nat,Xs: list_rule] :
( ( ord_less_nat @ I2 @ ( size_size_list_rule @ Xs ) )
=> ( Xs
= ( append_rule @ ( take_rule @ I2 @ Xs ) @ ( cons_rule @ ( nth_rule @ Xs @ I2 ) @ ( drop_rule @ ( suc @ I2 ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_1023_id__take__nth__drop,axiom,
! [I2: nat,Xs: list_list_fm] :
( ( ord_less_nat @ I2 @ ( size_s115229985653309035ist_fm @ Xs ) )
=> ( Xs
= ( append_list_fm @ ( take_list_fm @ I2 @ Xs ) @ ( cons_list_fm @ ( nth_list_fm @ Xs @ I2 ) @ ( drop_list_fm @ ( suc @ I2 ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_1024_id__take__nth__drop,axiom,
! [I2: nat,Xs: list_fm] :
( ( ord_less_nat @ I2 @ ( size_size_list_fm @ Xs ) )
=> ( Xs
= ( append_fm @ ( take_fm @ I2 @ Xs ) @ ( cons_fm @ ( nth_fm @ Xs @ I2 ) @ ( drop_fm @ ( suc @ I2 ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_1025_id__take__nth__drop,axiom,
! [I2: nat,Xs: list_tm] :
( ( ord_less_nat @ I2 @ ( size_size_list_tm @ Xs ) )
=> ( Xs
= ( append_tm @ ( take_tm @ I2 @ Xs ) @ ( cons_tm @ ( nth_tm @ Xs @ I2 ) @ ( drop_tm @ ( suc @ I2 ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_1026_sdrop__cycle,axiom,
! [U: list_P2774625669004309958m_rule,N2: nat] :
( ( U != nil_Pr2808214839852828720m_rule )
=> ( ( sdrop_8169176516188972301m_rule @ N2 @ ( cycle_5335117900641983313m_rule @ U ) )
= ( cycle_5335117900641983313m_rule @ ( rotate8779165051853931260m_rule @ ( modulo_modulo_nat @ N2 @ ( size_s1575636608424004698m_rule @ U ) ) @ U ) ) ) ) ).
% sdrop_cycle
thf(fact_1027_sdrop__cycle,axiom,
! [U: list_rule,N2: nat] :
( ( U != nil_rule )
=> ( ( sdrop_rule @ N2 @ ( cycle_rule @ U ) )
= ( cycle_rule @ ( rotate_rule @ ( modulo_modulo_nat @ N2 @ ( size_size_list_rule @ U ) ) @ U ) ) ) ) ).
% sdrop_cycle
thf(fact_1028_sdrop__cycle,axiom,
! [U: list_fm,N2: nat] :
( ( U != nil_fm )
=> ( ( sdrop_fm @ N2 @ ( cycle_fm @ U ) )
= ( cycle_fm @ ( rotate_fm @ ( modulo_modulo_nat @ N2 @ ( size_size_list_fm @ U ) ) @ U ) ) ) ) ).
% sdrop_cycle
thf(fact_1029_sdrop__cycle,axiom,
! [U: list_list_fm,N2: nat] :
( ( U != nil_list_fm )
=> ( ( sdrop_list_fm @ N2 @ ( cycle_list_fm @ U ) )
= ( cycle_list_fm @ ( rotate_list_fm @ ( modulo_modulo_nat @ N2 @ ( size_s115229985653309035ist_fm @ U ) ) @ U ) ) ) ) ).
% sdrop_cycle
thf(fact_1030_sdrop__cycle,axiom,
! [U: list_tm,N2: nat] :
( ( U != nil_tm )
=> ( ( sdrop_tm @ N2 @ ( cycle_tm @ U ) )
= ( cycle_tm @ ( rotate_tm @ ( modulo_modulo_nat @ N2 @ ( size_size_list_tm @ U ) ) @ U ) ) ) ) ).
% sdrop_cycle
thf(fact_1031_min_Oabsorb3,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_min_nat @ A @ B2 )
= A ) ) ).
% min.absorb3
thf(fact_1032_min_Oabsorb4,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( ( ord_min_nat @ A @ B2 )
= B2 ) ) ).
% min.absorb4
thf(fact_1033_min__less__iff__conj,axiom,
! [Z3: nat,X: nat,Y: nat] :
( ( ord_less_nat @ Z3 @ ( ord_min_nat @ X @ Y ) )
= ( ( ord_less_nat @ Z3 @ X )
& ( ord_less_nat @ Z3 @ Y ) ) ) ).
% min_less_iff_conj
thf(fact_1034_list__update__nonempty,axiom,
! [Xs: list_rule,K: nat,X: rule] :
( ( ( list_update_rule @ Xs @ K @ X )
= nil_rule )
= ( Xs = nil_rule ) ) ).
% list_update_nonempty
thf(fact_1035_list__update__nonempty,axiom,
! [Xs: list_fm,K: nat,X: fm] :
( ( ( list_update_fm @ Xs @ K @ X )
= nil_fm )
= ( Xs = nil_fm ) ) ).
% list_update_nonempty
thf(fact_1036_list__update__nonempty,axiom,
! [Xs: list_list_fm,K: nat,X: list_fm] :
( ( ( list_update_list_fm @ Xs @ K @ X )
= nil_list_fm )
= ( Xs = nil_list_fm ) ) ).
% list_update_nonempty
thf(fact_1037_list__update__nonempty,axiom,
! [Xs: list_tm,K: nat,X: tm] :
( ( ( list_update_tm @ Xs @ K @ X )
= nil_tm )
= ( Xs = nil_tm ) ) ).
% list_update_nonempty
thf(fact_1038_rotate__is__Nil__conv,axiom,
! [N2: nat,Xs: list_rule] :
( ( ( rotate_rule @ N2 @ Xs )
= nil_rule )
= ( Xs = nil_rule ) ) ).
% rotate_is_Nil_conv
thf(fact_1039_rotate__is__Nil__conv,axiom,
! [N2: nat,Xs: list_fm] :
( ( ( rotate_fm @ N2 @ Xs )
= nil_fm )
= ( Xs = nil_fm ) ) ).
% rotate_is_Nil_conv
thf(fact_1040_rotate__is__Nil__conv,axiom,
! [N2: nat,Xs: list_list_fm] :
( ( ( rotate_list_fm @ N2 @ Xs )
= nil_list_fm )
= ( Xs = nil_list_fm ) ) ).
% rotate_is_Nil_conv
thf(fact_1041_rotate__is__Nil__conv,axiom,
! [N2: nat,Xs: list_tm] :
( ( ( rotate_tm @ N2 @ Xs )
= nil_tm )
= ( Xs = nil_tm ) ) ).
% rotate_is_Nil_conv
thf(fact_1042_set__rotate,axiom,
! [N2: nat,Xs: list_fm] :
( ( set_fm2 @ ( rotate_fm @ N2 @ Xs ) )
= ( set_fm2 @ Xs ) ) ).
% set_rotate
thf(fact_1043_set__rotate,axiom,
! [N2: nat,Xs: list_list_fm] :
( ( set_list_fm2 @ ( rotate_list_fm @ N2 @ Xs ) )
= ( set_list_fm2 @ Xs ) ) ).
% set_rotate
thf(fact_1044_set__rotate,axiom,
! [N2: nat,Xs: list_tm] :
( ( set_tm2 @ ( rotate_tm @ N2 @ Xs ) )
= ( set_tm2 @ Xs ) ) ).
% set_rotate
thf(fact_1045_rotate1__is__Nil__conv,axiom,
! [Xs: list_rule] :
( ( ( rotate1_rule @ Xs )
= nil_rule )
= ( Xs = nil_rule ) ) ).
% rotate1_is_Nil_conv
thf(fact_1046_rotate1__is__Nil__conv,axiom,
! [Xs: list_fm] :
( ( ( rotate1_fm @ Xs )
= nil_fm )
= ( Xs = nil_fm ) ) ).
% rotate1_is_Nil_conv
thf(fact_1047_rotate1__is__Nil__conv,axiom,
! [Xs: list_list_fm] :
( ( ( rotate1_list_fm @ Xs )
= nil_list_fm )
= ( Xs = nil_list_fm ) ) ).
% rotate1_is_Nil_conv
thf(fact_1048_rotate1__is__Nil__conv,axiom,
! [Xs: list_tm] :
( ( ( rotate1_tm @ Xs )
= nil_tm )
= ( Xs = nil_tm ) ) ).
% rotate1_is_Nil_conv
thf(fact_1049_set__rotate1,axiom,
! [Xs: list_fm] :
( ( set_fm2 @ ( rotate1_fm @ Xs ) )
= ( set_fm2 @ Xs ) ) ).
% set_rotate1
thf(fact_1050_set__rotate1,axiom,
! [Xs: list_list_fm] :
( ( set_list_fm2 @ ( rotate1_list_fm @ Xs ) )
= ( set_list_fm2 @ Xs ) ) ).
% set_rotate1
thf(fact_1051_set__rotate1,axiom,
! [Xs: list_tm] :
( ( set_tm2 @ ( rotate1_tm @ Xs ) )
= ( set_tm2 @ Xs ) ) ).
% set_rotate1
thf(fact_1052_drop__Suc__Cons,axiom,
! [N2: nat,X: rule,Xs: list_rule] :
( ( drop_rule @ ( suc @ N2 ) @ ( cons_rule @ X @ Xs ) )
= ( drop_rule @ N2 @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_1053_drop__Suc__Cons,axiom,
! [N2: nat,X: list_fm,Xs: list_list_fm] :
( ( drop_list_fm @ ( suc @ N2 ) @ ( cons_list_fm @ X @ Xs ) )
= ( drop_list_fm @ N2 @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_1054_drop__Suc__Cons,axiom,
! [N2: nat,X: fm,Xs: list_fm] :
( ( drop_fm @ ( suc @ N2 ) @ ( cons_fm @ X @ Xs ) )
= ( drop_fm @ N2 @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_1055_drop__Suc__Cons,axiom,
! [N2: nat,X: tm,Xs: list_tm] :
( ( drop_tm @ ( suc @ N2 ) @ ( cons_tm @ X @ Xs ) )
= ( drop_tm @ N2 @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_1056_append__take__drop__id,axiom,
! [N2: nat,Xs: list_tm] :
( ( append_tm @ ( take_tm @ N2 @ Xs ) @ ( drop_tm @ N2 @ Xs ) )
= Xs ) ).
% append_take_drop_id
thf(fact_1057_list__update__length,axiom,
! [Xs: list_rule,X: rule,Ys: list_rule,Y: rule] :
( ( list_update_rule @ ( append_rule @ Xs @ ( cons_rule @ X @ Ys ) ) @ ( size_size_list_rule @ Xs ) @ Y )
= ( append_rule @ Xs @ ( cons_rule @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_1058_list__update__length,axiom,
! [Xs: list_list_fm,X: list_fm,Ys: list_list_fm,Y: list_fm] :
( ( list_update_list_fm @ ( append_list_fm @ Xs @ ( cons_list_fm @ X @ Ys ) ) @ ( size_s115229985653309035ist_fm @ Xs ) @ Y )
= ( append_list_fm @ Xs @ ( cons_list_fm @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_1059_list__update__length,axiom,
! [Xs: list_fm,X: fm,Ys: list_fm,Y: fm] :
( ( list_update_fm @ ( append_fm @ Xs @ ( cons_fm @ X @ Ys ) ) @ ( size_size_list_fm @ Xs ) @ Y )
= ( append_fm @ Xs @ ( cons_fm @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_1060_list__update__length,axiom,
! [Xs: list_tm,X: tm,Ys: list_tm,Y: tm] :
( ( list_update_tm @ ( append_tm @ Xs @ ( cons_tm @ X @ Ys ) ) @ ( size_size_list_tm @ Xs ) @ Y )
= ( append_tm @ Xs @ ( cons_tm @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_1061_set__swap,axiom,
! [I2: nat,Xs: list_fm,J: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_fm @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_fm @ Xs ) )
=> ( ( set_fm2 @ ( list_update_fm @ ( list_update_fm @ Xs @ I2 @ ( nth_fm @ Xs @ J ) ) @ J @ ( nth_fm @ Xs @ I2 ) ) )
= ( set_fm2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_1062_set__swap,axiom,
! [I2: nat,Xs: list_list_fm,J: nat] :
( ( ord_less_nat @ I2 @ ( size_s115229985653309035ist_fm @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_s115229985653309035ist_fm @ Xs ) )
=> ( ( set_list_fm2 @ ( list_update_list_fm @ ( list_update_list_fm @ Xs @ I2 @ ( nth_list_fm @ Xs @ J ) ) @ J @ ( nth_list_fm @ Xs @ I2 ) ) )
= ( set_list_fm2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_1063_set__swap,axiom,
! [I2: nat,Xs: list_tm,J: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_tm @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_tm @ Xs ) )
=> ( ( set_tm2 @ ( list_update_tm @ ( list_update_tm @ Xs @ I2 @ ( nth_tm @ Xs @ J ) ) @ J @ ( nth_tm @ Xs @ I2 ) ) )
= ( set_tm2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_1064_drop__Nil,axiom,
! [N2: nat] :
( ( drop_rule @ N2 @ nil_rule )
= nil_rule ) ).
% drop_Nil
thf(fact_1065_drop__Nil,axiom,
! [N2: nat] :
( ( drop_fm @ N2 @ nil_fm )
= nil_fm ) ).
% drop_Nil
thf(fact_1066_drop__Nil,axiom,
! [N2: nat] :
( ( drop_list_fm @ N2 @ nil_list_fm )
= nil_list_fm ) ).
% drop_Nil
thf(fact_1067_drop__Nil,axiom,
! [N2: nat] :
( ( drop_tm @ N2 @ nil_tm )
= nil_tm ) ).
% drop_Nil
thf(fact_1068_list__update_Osimps_I1_J,axiom,
! [I2: nat,V2: rule] :
( ( list_update_rule @ nil_rule @ I2 @ V2 )
= nil_rule ) ).
% list_update.simps(1)
thf(fact_1069_list__update_Osimps_I1_J,axiom,
! [I2: nat,V2: fm] :
( ( list_update_fm @ nil_fm @ I2 @ V2 )
= nil_fm ) ).
% list_update.simps(1)
thf(fact_1070_list__update_Osimps_I1_J,axiom,
! [I2: nat,V2: list_fm] :
( ( list_update_list_fm @ nil_list_fm @ I2 @ V2 )
= nil_list_fm ) ).
% list_update.simps(1)
thf(fact_1071_list__update_Osimps_I1_J,axiom,
! [I2: nat,V2: tm] :
( ( list_update_tm @ nil_tm @ I2 @ V2 )
= nil_tm ) ).
% list_update.simps(1)
thf(fact_1072_list__update__code_I1_J,axiom,
! [I2: nat,Y: rule] :
( ( list_update_rule @ nil_rule @ I2 @ Y )
= nil_rule ) ).
% list_update_code(1)
thf(fact_1073_list__update__code_I1_J,axiom,
! [I2: nat,Y: fm] :
( ( list_update_fm @ nil_fm @ I2 @ Y )
= nil_fm ) ).
% list_update_code(1)
thf(fact_1074_list__update__code_I1_J,axiom,
! [I2: nat,Y: list_fm] :
( ( list_update_list_fm @ nil_list_fm @ I2 @ Y )
= nil_list_fm ) ).
% list_update_code(1)
thf(fact_1075_list__update__code_I1_J,axiom,
! [I2: nat,Y: tm] :
( ( list_update_tm @ nil_tm @ I2 @ Y )
= nil_tm ) ).
% list_update_code(1)
thf(fact_1076_in__set__dropD,axiom,
! [X: produc340336539035504054m_rule,N2: nat,Xs: list_P2774625669004309958m_rule] :
( ( member7231649785386036813m_rule @ X @ ( set_Pr4534715572506550497m_rule @ ( drop_P5814889834185618160m_rule @ N2 @ Xs ) ) )
=> ( member7231649785386036813m_rule @ X @ ( set_Pr4534715572506550497m_rule @ Xs ) ) ) ).
% in_set_dropD
thf(fact_1077_in__set__dropD,axiom,
! [X: nat,N2: nat,Xs: list_nat] :
( ( member_nat3 @ X @ ( set_nat2 @ ( drop_nat @ N2 @ Xs ) ) )
=> ( member_nat3 @ X @ ( set_nat2 @ Xs ) ) ) ).
% in_set_dropD
thf(fact_1078_in__set__dropD,axiom,
! [X: fm,N2: nat,Xs: list_fm] :
( ( member_fm3 @ X @ ( set_fm2 @ ( drop_fm @ N2 @ Xs ) ) )
=> ( member_fm3 @ X @ ( set_fm2 @ Xs ) ) ) ).
% in_set_dropD
thf(fact_1079_in__set__dropD,axiom,
! [X: list_fm,N2: nat,Xs: list_list_fm] :
( ( member_list_fm3 @ X @ ( set_list_fm2 @ ( drop_list_fm @ N2 @ Xs ) ) )
=> ( member_list_fm3 @ X @ ( set_list_fm2 @ Xs ) ) ) ).
% in_set_dropD
thf(fact_1080_in__set__dropD,axiom,
! [X: tm,N2: nat,Xs: list_tm] :
( ( member_tm3 @ X @ ( set_tm2 @ ( drop_tm @ N2 @ Xs ) ) )
=> ( member_tm3 @ X @ ( set_tm2 @ Xs ) ) ) ).
% in_set_dropD
thf(fact_1081_min__less__iff__disj,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ ( ord_min_nat @ X @ Y ) @ Z3 )
= ( ( ord_less_nat @ X @ Z3 )
| ( ord_less_nat @ Y @ Z3 ) ) ) ).
% min_less_iff_disj
thf(fact_1082_min_Ostrict__boundedE,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ ( ord_min_nat @ B2 @ C ) )
=> ~ ( ( ord_less_nat @ A @ B2 )
=> ~ ( ord_less_nat @ A @ C ) ) ) ).
% min.strict_boundedE
thf(fact_1083_min_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A5: nat,B5: nat] :
( ( A5
= ( ord_min_nat @ A5 @ B5 ) )
& ( A5 != B5 ) ) ) ) ).
% min.strict_order_iff
thf(fact_1084_min_Ostrict__coboundedI1,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_nat @ A @ C )
=> ( ord_less_nat @ ( ord_min_nat @ A @ B2 ) @ C ) ) ).
% min.strict_coboundedI1
thf(fact_1085_min_Ostrict__coboundedI2,axiom,
! [B2: nat,C: nat,A: nat] :
( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ ( ord_min_nat @ A @ B2 ) @ C ) ) ).
% min.strict_coboundedI2
thf(fact_1086_rotate1_Osimps_I1_J,axiom,
( ( rotate1_rule @ nil_rule )
= nil_rule ) ).
% rotate1.simps(1)
thf(fact_1087_rotate1_Osimps_I1_J,axiom,
( ( rotate1_fm @ nil_fm )
= nil_fm ) ).
% rotate1.simps(1)
thf(fact_1088_rotate1_Osimps_I1_J,axiom,
( ( rotate1_list_fm @ nil_list_fm )
= nil_list_fm ) ).
% rotate1.simps(1)
thf(fact_1089_rotate1_Osimps_I1_J,axiom,
( ( rotate1_tm @ nil_tm )
= nil_tm ) ).
% rotate1.simps(1)
thf(fact_1090_nth__via__drop,axiom,
! [N2: nat,Xs: list_rule,Y: rule,Ys: list_rule] :
( ( ( drop_rule @ N2 @ Xs )
= ( cons_rule @ Y @ Ys ) )
=> ( ( nth_rule @ Xs @ N2 )
= Y ) ) ).
% nth_via_drop
thf(fact_1091_nth__via__drop,axiom,
! [N2: nat,Xs: list_list_fm,Y: list_fm,Ys: list_list_fm] :
( ( ( drop_list_fm @ N2 @ Xs )
= ( cons_list_fm @ Y @ Ys ) )
=> ( ( nth_list_fm @ Xs @ N2 )
= Y ) ) ).
% nth_via_drop
thf(fact_1092_nth__via__drop,axiom,
! [N2: nat,Xs: list_fm,Y: fm,Ys: list_fm] :
( ( ( drop_fm @ N2 @ Xs )
= ( cons_fm @ Y @ Ys ) )
=> ( ( nth_fm @ Xs @ N2 )
= Y ) ) ).
% nth_via_drop
thf(fact_1093_nth__via__drop,axiom,
! [N2: nat,Xs: list_tm,Y: tm,Ys: list_tm] :
( ( ( drop_tm @ N2 @ Xs )
= ( cons_tm @ Y @ Ys ) )
=> ( ( nth_tm @ Xs @ N2 )
= Y ) ) ).
% nth_via_drop
thf(fact_1094_list__update__code_I2_J,axiom,
! [X: rule,Xs: list_rule,Y: rule] :
( ( list_update_rule @ ( cons_rule @ X @ Xs ) @ zero_zero_nat @ Y )
= ( cons_rule @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_1095_list__update__code_I2_J,axiom,
! [X: list_fm,Xs: list_list_fm,Y: list_fm] :
( ( list_update_list_fm @ ( cons_list_fm @ X @ Xs ) @ zero_zero_nat @ Y )
= ( cons_list_fm @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_1096_list__update__code_I2_J,axiom,
! [X: fm,Xs: list_fm,Y: fm] :
( ( list_update_fm @ ( cons_fm @ X @ Xs ) @ zero_zero_nat @ Y )
= ( cons_fm @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_1097_list__update__code_I2_J,axiom,
! [X: tm,Xs: list_tm,Y: tm] :
( ( list_update_tm @ ( cons_tm @ X @ Xs ) @ zero_zero_nat @ Y )
= ( cons_tm @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_1098_list__update__code_I3_J,axiom,
! [X: rule,Xs: list_rule,I2: nat,Y: rule] :
( ( list_update_rule @ ( cons_rule @ X @ Xs ) @ ( suc @ I2 ) @ Y )
= ( cons_rule @ X @ ( list_update_rule @ Xs @ I2 @ Y ) ) ) ).
% list_update_code(3)
thf(fact_1099_list__update__code_I3_J,axiom,
! [X: list_fm,Xs: list_list_fm,I2: nat,Y: list_fm] :
( ( list_update_list_fm @ ( cons_list_fm @ X @ Xs ) @ ( suc @ I2 ) @ Y )
= ( cons_list_fm @ X @ ( list_update_list_fm @ Xs @ I2 @ Y ) ) ) ).
% list_update_code(3)
thf(fact_1100_list__update__code_I3_J,axiom,
! [X: fm,Xs: list_fm,I2: nat,Y: fm] :
( ( list_update_fm @ ( cons_fm @ X @ Xs ) @ ( suc @ I2 ) @ Y )
= ( cons_fm @ X @ ( list_update_fm @ Xs @ I2 @ Y ) ) ) ).
% list_update_code(3)
thf(fact_1101_list__update__code_I3_J,axiom,
! [X: tm,Xs: list_tm,I2: nat,Y: tm] :
( ( list_update_tm @ ( cons_tm @ X @ Xs ) @ ( suc @ I2 ) @ Y )
= ( cons_tm @ X @ ( list_update_tm @ Xs @ I2 @ Y ) ) ) ).
% list_update_code(3)
thf(fact_1102_rotate__drop__take,axiom,
( rotate_tm
= ( ^ [N3: nat,Xs3: list_tm] : ( append_tm @ ( drop_tm @ ( modulo_modulo_nat @ N3 @ ( size_size_list_tm @ Xs3 ) ) @ Xs3 ) @ ( take_tm @ ( modulo_modulo_nat @ N3 @ ( size_size_list_tm @ Xs3 ) ) @ Xs3 ) ) ) ) ).
% rotate_drop_take
thf(fact_1103_rotate__append,axiom,
! [L: list_tm,Q: list_tm] :
( ( rotate_tm @ ( size_size_list_tm @ L ) @ ( append_tm @ L @ Q ) )
= ( append_tm @ Q @ L ) ) ).
% rotate_append
thf(fact_1104_upd__conv__take__nth__drop,axiom,
! [I2: nat,Xs: list_rule,A: rule] :
( ( ord_less_nat @ I2 @ ( size_size_list_rule @ Xs ) )
=> ( ( list_update_rule @ Xs @ I2 @ A )
= ( append_rule @ ( take_rule @ I2 @ Xs ) @ ( cons_rule @ A @ ( drop_rule @ ( suc @ I2 ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_1105_upd__conv__take__nth__drop,axiom,
! [I2: nat,Xs: list_list_fm,A: list_fm] :
( ( ord_less_nat @ I2 @ ( size_s115229985653309035ist_fm @ Xs ) )
=> ( ( list_update_list_fm @ Xs @ I2 @ A )
= ( append_list_fm @ ( take_list_fm @ I2 @ Xs ) @ ( cons_list_fm @ A @ ( drop_list_fm @ ( suc @ I2 ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_1106_upd__conv__take__nth__drop,axiom,
! [I2: nat,Xs: list_fm,A: fm] :
( ( ord_less_nat @ I2 @ ( size_size_list_fm @ Xs ) )
=> ( ( list_update_fm @ Xs @ I2 @ A )
= ( append_fm @ ( take_fm @ I2 @ Xs ) @ ( cons_fm @ A @ ( drop_fm @ ( suc @ I2 ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_1107_upd__conv__take__nth__drop,axiom,
! [I2: nat,Xs: list_tm,A: tm] :
( ( ord_less_nat @ I2 @ ( size_size_list_tm @ Xs ) )
=> ( ( list_update_tm @ Xs @ I2 @ A )
= ( append_tm @ ( take_tm @ I2 @ Xs ) @ ( cons_tm @ A @ ( drop_tm @ ( suc @ I2 ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_1108_append__eq__conv__conj,axiom,
! [Xs: list_tm,Ys: list_tm,Zs: list_tm] :
( ( ( append_tm @ Xs @ Ys )
= Zs )
= ( ( Xs
= ( take_tm @ ( size_size_list_tm @ Xs ) @ Zs ) )
& ( Ys
= ( drop_tm @ ( size_size_list_tm @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_1109_set__update__memI,axiom,
! [N2: nat,Xs: list_P2774625669004309958m_rule,X: produc340336539035504054m_rule] :
( ( ord_less_nat @ N2 @ ( size_s1575636608424004698m_rule @ Xs ) )
=> ( member7231649785386036813m_rule @ X @ ( set_Pr4534715572506550497m_rule @ ( list_u1216070999260037064m_rule @ Xs @ N2 @ X ) ) ) ) ).
% set_update_memI
thf(fact_1110_set__update__memI,axiom,
! [N2: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
=> ( member_nat3 @ X @ ( set_nat2 @ ( list_update_nat @ Xs @ N2 @ X ) ) ) ) ).
% set_update_memI
thf(fact_1111_set__update__memI,axiom,
! [N2: nat,Xs: list_fm,X: fm] :
( ( ord_less_nat @ N2 @ ( size_size_list_fm @ Xs ) )
=> ( member_fm3 @ X @ ( set_fm2 @ ( list_update_fm @ Xs @ N2 @ X ) ) ) ) ).
% set_update_memI
thf(fact_1112_set__update__memI,axiom,
! [N2: nat,Xs: list_list_fm,X: list_fm] :
( ( ord_less_nat @ N2 @ ( size_s115229985653309035ist_fm @ Xs ) )
=> ( member_list_fm3 @ X @ ( set_list_fm2 @ ( list_update_list_fm @ Xs @ N2 @ X ) ) ) ) ).
% set_update_memI
thf(fact_1113_set__update__memI,axiom,
! [N2: nat,Xs: list_tm,X: tm] :
( ( ord_less_nat @ N2 @ ( size_size_list_tm @ Xs ) )
=> ( member_tm3 @ X @ ( set_tm2 @ ( list_update_tm @ Xs @ N2 @ X ) ) ) ) ).
% set_update_memI
thf(fact_1114_list__update__append1,axiom,
! [I2: nat,Xs: list_tm,Ys: list_tm,X: tm] :
( ( ord_less_nat @ I2 @ ( size_size_list_tm @ Xs ) )
=> ( ( list_update_tm @ ( append_tm @ Xs @ Ys ) @ I2 @ X )
= ( append_tm @ ( list_update_tm @ Xs @ I2 @ X ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_1115_rotate1_Osimps_I2_J,axiom,
! [X: rule,Xs: list_rule] :
( ( rotate1_rule @ ( cons_rule @ X @ Xs ) )
= ( append_rule @ Xs @ ( cons_rule @ X @ nil_rule ) ) ) ).
% rotate1.simps(2)
thf(fact_1116_rotate1_Osimps_I2_J,axiom,
! [X: list_fm,Xs: list_list_fm] :
( ( rotate1_list_fm @ ( cons_list_fm @ X @ Xs ) )
= ( append_list_fm @ Xs @ ( cons_list_fm @ X @ nil_list_fm ) ) ) ).
% rotate1.simps(2)
thf(fact_1117_rotate1_Osimps_I2_J,axiom,
! [X: fm,Xs: list_fm] :
( ( rotate1_fm @ ( cons_fm @ X @ Xs ) )
= ( append_fm @ Xs @ ( cons_fm @ X @ nil_fm ) ) ) ).
% rotate1.simps(2)
thf(fact_1118_rotate1_Osimps_I2_J,axiom,
! [X: tm,Xs: list_tm] :
( ( rotate1_tm @ ( cons_tm @ X @ Xs ) )
= ( append_tm @ Xs @ ( cons_tm @ X @ nil_tm ) ) ) ).
% rotate1.simps(2)
thf(fact_1119_Cons__nth__drop__Suc,axiom,
! [I2: nat,Xs: list_rule] :
( ( ord_less_nat @ I2 @ ( size_size_list_rule @ Xs ) )
=> ( ( cons_rule @ ( nth_rule @ Xs @ I2 ) @ ( drop_rule @ ( suc @ I2 ) @ Xs ) )
= ( drop_rule @ I2 @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_1120_Cons__nth__drop__Suc,axiom,
! [I2: nat,Xs: list_list_fm] :
( ( ord_less_nat @ I2 @ ( size_s115229985653309035ist_fm @ Xs ) )
=> ( ( cons_list_fm @ ( nth_list_fm @ Xs @ I2 ) @ ( drop_list_fm @ ( suc @ I2 ) @ Xs ) )
= ( drop_list_fm @ I2 @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_1121_Cons__nth__drop__Suc,axiom,
! [I2: nat,Xs: list_fm] :
( ( ord_less_nat @ I2 @ ( size_size_list_fm @ Xs ) )
=> ( ( cons_fm @ ( nth_fm @ Xs @ I2 ) @ ( drop_fm @ ( suc @ I2 ) @ Xs ) )
= ( drop_fm @ I2 @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_1122_Cons__nth__drop__Suc,axiom,
! [I2: nat,Xs: list_tm] :
( ( ord_less_nat @ I2 @ ( size_size_list_tm @ Xs ) )
=> ( ( cons_tm @ ( nth_tm @ Xs @ I2 ) @ ( drop_tm @ ( suc @ I2 ) @ Xs ) )
= ( drop_tm @ I2 @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_1123_take__hd__drop,axiom,
! [N2: nat,Xs: list_rule] :
( ( ord_less_nat @ N2 @ ( size_size_list_rule @ Xs ) )
=> ( ( append_rule @ ( take_rule @ N2 @ Xs ) @ ( cons_rule @ ( hd_rule @ ( drop_rule @ N2 @ Xs ) ) @ nil_rule ) )
= ( take_rule @ ( suc @ N2 ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_1124_take__hd__drop,axiom,
! [N2: nat,Xs: list_list_fm] :
( ( ord_less_nat @ N2 @ ( size_s115229985653309035ist_fm @ Xs ) )
=> ( ( append_list_fm @ ( take_list_fm @ N2 @ Xs ) @ ( cons_list_fm @ ( hd_list_fm @ ( drop_list_fm @ N2 @ Xs ) ) @ nil_list_fm ) )
= ( take_list_fm @ ( suc @ N2 ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_1125_take__hd__drop,axiom,
! [N2: nat,Xs: list_fm] :
( ( ord_less_nat @ N2 @ ( size_size_list_fm @ Xs ) )
=> ( ( append_fm @ ( take_fm @ N2 @ Xs ) @ ( cons_fm @ ( hd_fm @ ( drop_fm @ N2 @ Xs ) ) @ nil_fm ) )
= ( take_fm @ ( suc @ N2 ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_1126_take__hd__drop,axiom,
! [N2: nat,Xs: list_tm] :
( ( ord_less_nat @ N2 @ ( size_size_list_tm @ Xs ) )
=> ( ( append_tm @ ( take_tm @ N2 @ Xs ) @ ( cons_tm @ ( hd_tm @ ( drop_tm @ N2 @ Xs ) ) @ nil_tm ) )
= ( take_tm @ ( suc @ N2 ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_1127_stake__append,axiom,
! [N2: nat,U: list_tm,S: stream_tm] :
( ( stake_tm @ N2 @ ( shift_tm @ U @ S ) )
= ( append_tm @ ( take_tm @ ( ord_min_nat @ ( size_size_list_tm @ U ) @ N2 ) @ U ) @ ( stake_tm @ ( minus_minus_nat @ N2 @ ( size_size_list_tm @ U ) ) @ S ) ) ) ).
% stake_append
thf(fact_1128_stake__append,axiom,
! [N2: nat,U: list_P2774625669004309958m_rule,S: stream2709947120125613254m_rule] :
( ( stake_5421812949518764133m_rule @ N2 @ ( shift_2334844276748245581m_rule @ U @ S ) )
= ( append2714808343334685275m_rule @ ( take_P3067526267515409992m_rule @ ( ord_min_nat @ ( size_s1575636608424004698m_rule @ U ) @ N2 ) @ U ) @ ( stake_5421812949518764133m_rule @ ( minus_minus_nat @ N2 @ ( size_s1575636608424004698m_rule @ U ) ) @ S ) ) ) ).
% stake_append
thf(fact_1129_stake__shift,axiom,
! [I2: nat,W2: list_tm,S: stream_tm] :
( ( stake_tm @ I2 @ ( shift_tm @ W2 @ S ) )
= ( append_tm @ ( take_tm @ I2 @ W2 ) @ ( stake_tm @ ( minus_minus_nat @ I2 @ ( size_size_list_tm @ W2 ) ) @ S ) ) ) ).
% stake_shift
thf(fact_1130_stake__shift,axiom,
! [I2: nat,W2: list_P2774625669004309958m_rule,S: stream2709947120125613254m_rule] :
( ( stake_5421812949518764133m_rule @ I2 @ ( shift_2334844276748245581m_rule @ W2 @ S ) )
= ( append2714808343334685275m_rule @ ( take_P3067526267515409992m_rule @ I2 @ W2 ) @ ( stake_5421812949518764133m_rule @ ( minus_minus_nat @ I2 @ ( size_s1575636608424004698m_rule @ W2 ) ) @ S ) ) ) ).
% stake_shift
thf(fact_1131_sdrop__add,axiom,
! [N2: nat,M: nat,S: stream2709947120125613254m_rule] :
( ( sdrop_8169176516188972301m_rule @ N2 @ ( sdrop_8169176516188972301m_rule @ M @ S ) )
= ( sdrop_8169176516188972301m_rule @ ( plus_plus_nat @ M @ N2 ) @ S ) ) ).
% sdrop_add
thf(fact_1132_length__append,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( size_size_list_tm @ ( append_tm @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_tm @ Xs ) @ ( size_size_list_tm @ Ys ) ) ) ).
% length_append
thf(fact_1133_hd__append2,axiom,
! [Xs: list_rule,Ys: list_rule] :
( ( Xs != nil_rule )
=> ( ( hd_rule @ ( append_rule @ Xs @ Ys ) )
= ( hd_rule @ Xs ) ) ) ).
% hd_append2
thf(fact_1134_hd__append2,axiom,
! [Xs: list_fm,Ys: list_fm] :
( ( Xs != nil_fm )
=> ( ( hd_fm @ ( append_fm @ Xs @ Ys ) )
= ( hd_fm @ Xs ) ) ) ).
% hd_append2
thf(fact_1135_hd__append2,axiom,
! [Xs: list_list_fm,Ys: list_list_fm] :
( ( Xs != nil_list_fm )
=> ( ( hd_list_fm @ ( append_list_fm @ Xs @ Ys ) )
= ( hd_list_fm @ Xs ) ) ) ).
% hd_append2
thf(fact_1136_hd__append2,axiom,
! [Xs: list_tm,Ys: list_tm] :
( ( Xs != nil_tm )
=> ( ( hd_tm @ ( append_tm @ Xs @ Ys ) )
= ( hd_tm @ Xs ) ) ) ).
% hd_append2
thf(fact_1137_nth__append__length__plus,axiom,
! [Xs: list_tm,Ys: list_tm,N2: nat] :
( ( nth_tm @ ( append_tm @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_tm @ Xs ) @ N2 ) )
= ( nth_tm @ Ys @ N2 ) ) ).
% nth_append_length_plus
thf(fact_1138_take__append,axiom,
! [N2: nat,Xs: list_tm,Ys: list_tm] :
( ( take_tm @ N2 @ ( append_tm @ Xs @ Ys ) )
= ( append_tm @ ( take_tm @ N2 @ Xs ) @ ( take_tm @ ( minus_minus_nat @ N2 @ ( size_size_list_tm @ Xs ) ) @ Ys ) ) ) ).
% take_append
thf(fact_1139_drop__append,axiom,
! [N2: nat,Xs: list_tm,Ys: list_tm] :
( ( drop_tm @ N2 @ ( append_tm @ Xs @ Ys ) )
= ( append_tm @ ( drop_tm @ N2 @ Xs ) @ ( drop_tm @ ( minus_minus_nat @ N2 @ ( size_size_list_tm @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_1140_stake__add,axiom,
! [M: nat,S: stream_tm,N2: nat] :
( ( append_tm @ ( stake_tm @ M @ S ) @ ( stake_tm @ N2 @ ( sdrop_tm @ M @ S ) ) )
= ( stake_tm @ ( plus_plus_nat @ M @ N2 ) @ S ) ) ).
% stake_add
thf(fact_1141_stake__add,axiom,
! [M: nat,S: stream2709947120125613254m_rule,N2: nat] :
( ( append2714808343334685275m_rule @ ( stake_5421812949518764133m_rule @ M @ S ) @ ( stake_5421812949518764133m_rule @ N2 @ ( sdrop_8169176516188972301m_rule @ M @ S ) ) )
= ( stake_5421812949518764133m_rule @ ( plus_plus_nat @ M @ N2 ) @ S ) ) ).
% stake_add
thf(fact_1142_shift__simps_I1_J,axiom,
! [Xs: list_rule,S: stream_rule] :
( ( ( Xs = nil_rule )
=> ( ( shd_rule @ ( shift_rule @ Xs @ S ) )
= ( shd_rule @ S ) ) )
& ( ( Xs != nil_rule )
=> ( ( shd_rule @ ( shift_rule @ Xs @ S ) )
= ( hd_rule @ Xs ) ) ) ) ).
% shift_simps(1)
thf(fact_1143_shift__simps_I1_J,axiom,
! [Xs: list_fm,S: stream_fm] :
( ( ( Xs = nil_fm )
=> ( ( shd_fm @ ( shift_fm @ Xs @ S ) )
= ( shd_fm @ S ) ) )
& ( ( Xs != nil_fm )
=> ( ( shd_fm @ ( shift_fm @ Xs @ S ) )
= ( hd_fm @ Xs ) ) ) ) ).
% shift_simps(1)
thf(fact_1144_shift__simps_I1_J,axiom,
! [Xs: list_list_fm,S: stream_list_fm] :
( ( ( Xs = nil_list_fm )
=> ( ( shd_list_fm @ ( shift_list_fm @ Xs @ S ) )
= ( shd_list_fm @ S ) ) )
& ( ( Xs != nil_list_fm )
=> ( ( shd_list_fm @ ( shift_list_fm @ Xs @ S ) )
= ( hd_list_fm @ Xs ) ) ) ) ).
% shift_simps(1)
thf(fact_1145_shift__simps_I1_J,axiom,
! [Xs: list_tm,S: stream_tm] :
( ( ( Xs = nil_tm )
=> ( ( shd_tm @ ( shift_tm @ Xs @ S ) )
= ( shd_tm @ S ) ) )
& ( ( Xs != nil_tm )
=> ( ( shd_tm @ ( shift_tm @ Xs @ S ) )
= ( hd_tm @ Xs ) ) ) ) ).
% shift_simps(1)
thf(fact_1146_shift__simps_I1_J,axiom,
! [Xs: list_P2774625669004309958m_rule,S: stream2709947120125613254m_rule] :
( ( ( Xs = nil_Pr2808214839852828720m_rule )
=> ( ( shd_Pr4562317740776619530m_rule @ ( shift_2334844276748245581m_rule @ Xs @ S ) )
= ( shd_Pr4562317740776619530m_rule @ S ) ) )
& ( ( Xs != nil_Pr2808214839852828720m_rule )
=> ( ( shd_Pr4562317740776619530m_rule @ ( shift_2334844276748245581m_rule @ Xs @ S ) )
= ( hd_Pro7241777042969981963m_rule @ Xs ) ) ) ) ).
% shift_simps(1)
thf(fact_1147_list_Osel_I1_J,axiom,
! [X21: rule,X22: list_rule] :
( ( hd_rule @ ( cons_rule @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_1148_list_Osel_I1_J,axiom,
! [X21: list_fm,X22: list_list_fm] :
( ( hd_list_fm @ ( cons_list_fm @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_1149_list_Osel_I1_J,axiom,
! [X21: fm,X22: list_fm] :
( ( hd_fm @ ( cons_fm @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_1150_list_Osel_I1_J,axiom,
! [X21: tm,X22: list_tm] :
( ( hd_tm @ ( cons_tm @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_1151_hd__in__set,axiom,
! [Xs: list_P2774625669004309958m_rule] :
( ( Xs != nil_Pr2808214839852828720m_rule )
=> ( member7231649785386036813m_rule @ ( hd_Pro7241777042969981963m_rule @ Xs ) @ ( set_Pr4534715572506550497m_rule @ Xs ) ) ) ).
% hd_in_set
thf(fact_1152_hd__in__set,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( member_nat3 @ ( hd_nat @ Xs ) @ ( set_nat2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_1153_hd__in__set,axiom,
! [Xs: list_rule] :
( ( Xs != nil_rule )
=> ( member_rule3 @ ( hd_rule @ Xs ) @ ( set_rule2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_1154_hd__in__set,axiom,
! [Xs: list_fm] :
( ( Xs != nil_fm )
=> ( member_fm3 @ ( hd_fm @ Xs ) @ ( set_fm2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_1155_hd__in__set,axiom,
! [Xs: list_list_fm] :
( ( Xs != nil_list_fm )
=> ( member_list_fm3 @ ( hd_list_fm @ Xs ) @ ( set_list_fm2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_1156_hd__in__set,axiom,
! [Xs: list_tm] :
( ( Xs != nil_tm )
=> ( member_tm3 @ ( hd_tm @ Xs ) @ ( set_tm2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_1157_list_Oset__sel_I1_J,axiom,
! [A: list_nat] :
( ( A != nil_nat )
=> ( member_nat3 @ ( hd_nat @ A ) @ ( set_nat2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_1158_list_Oset__sel_I1_J,axiom,
! [A: list_rule] :
( ( A != nil_rule )
=> ( member_rule3 @ ( hd_rule @ A ) @ ( set_rule2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_1159_list_Oset__sel_I1_J,axiom,
! [A: list_fm] :
( ( A != nil_fm )
=> ( member_fm3 @ ( hd_fm @ A ) @ ( set_fm2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_1160_list_Oset__sel_I1_J,axiom,
! [A: list_list_fm] :
( ( A != nil_list_fm )
=> ( member_list_fm3 @ ( hd_list_fm @ A ) @ ( set_list_fm2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_1161_list_Oset__sel_I1_J,axiom,
! [A: list_tm] :
( ( A != nil_tm )
=> ( member_tm3 @ ( hd_tm @ A ) @ ( set_tm2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_1162_fm_Osize__gen_I2_J,axiom,
! [X21: fm,X22: fm] :
( ( size_fm @ ( imp @ X21 @ X22 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_fm @ X21 ) @ ( size_fm @ X22 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% fm.size_gen(2)
thf(fact_1163_fm_Osize__gen_I3_J,axiom,
! [X31: fm,X32: fm] :
( ( size_fm @ ( dis @ X31 @ X32 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_fm @ X31 ) @ ( size_fm @ X32 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% fm.size_gen(3)
thf(fact_1164_fm_Osize__gen_I1_J,axiom,
! [X11: nat,X12: list_tm] :
( ( size_fm @ ( pre @ X11 @ X12 ) )
= zero_zero_nat ) ).
% fm.size_gen(1)
thf(fact_1165_fm_Osize__gen_I7_J,axiom,
! [X7: fm] :
( ( size_fm @ ( neg @ X7 ) )
= ( plus_plus_nat @ ( size_fm @ X7 ) @ ( suc @ zero_zero_nat ) ) ) ).
% fm.size_gen(7)
thf(fact_1166_fm_Osize__gen_I4_J,axiom,
! [X41: fm,X42: fm] :
( ( size_fm @ ( con @ X41 @ X42 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_fm @ X41 ) @ ( size_fm @ X42 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% fm.size_gen(4)
thf(fact_1167_fm_Osize__gen_I6_J,axiom,
! [X6: fm] :
( ( size_fm @ ( uni @ X6 ) )
= ( plus_plus_nat @ ( size_fm @ X6 ) @ ( suc @ zero_zero_nat ) ) ) ).
% fm.size_gen(6)
thf(fact_1168_fm_Osize__gen_I5_J,axiom,
! [X5: fm] :
( ( size_fm @ ( exi @ X5 ) )
= ( plus_plus_nat @ ( size_fm @ X5 ) @ ( suc @ zero_zero_nat ) ) ) ).
% fm.size_gen(5)
thf(fact_1169_sub_Osimps_I5_J,axiom,
! [V2: nat,S: tm,P: fm] :
( ( sub @ V2 @ S @ ( exi @ P ) )
= ( exi @ ( sub @ ( plus_plus_nat @ V2 @ one_one_nat ) @ ( inc_term @ S ) @ P ) ) ) ).
% sub.simps(5)
thf(fact_1170_sub_Osimps_I6_J,axiom,
! [V2: nat,S: tm,P: fm] :
( ( sub @ V2 @ S @ ( uni @ P ) )
= ( uni @ ( sub @ ( plus_plus_nat @ V2 @ one_one_nat ) @ ( inc_term @ S ) @ P ) ) ) ).
% sub.simps(6)
thf(fact_1171_fm_Osize_I10_J,axiom,
! [X31: fm,X32: fm] :
( ( size_size_fm @ ( dis @ X31 @ X32 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_size_fm @ X31 ) @ ( size_size_fm @ X32 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% fm.size(10)
thf(fact_1172_fm_Osize_I9_J,axiom,
! [X21: fm,X22: fm] :
( ( size_size_fm @ ( imp @ X21 @ X22 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_size_fm @ X21 ) @ ( size_size_fm @ X22 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% fm.size(9)
thf(fact_1173_size__sub,axiom,
! [I2: nat,T: tm,P: fm] :
( ( size_size_fm @ ( sub @ I2 @ T @ P ) )
= ( size_size_fm @ P ) ) ).
% size_sub
thf(fact_1174_fm_Osize_I8_J,axiom,
! [X11: nat,X12: list_tm] :
( ( size_size_fm @ ( pre @ X11 @ X12 ) )
= zero_zero_nat ) ).
% fm.size(8)
thf(fact_1175_fm_Osize_I14_J,axiom,
! [X7: fm] :
( ( size_size_fm @ ( neg @ X7 ) )
= ( plus_plus_nat @ ( size_size_fm @ X7 ) @ ( suc @ zero_zero_nat ) ) ) ).
% fm.size(14)
thf(fact_1176_fm_Osize_I11_J,axiom,
! [X41: fm,X42: fm] :
( ( size_size_fm @ ( con @ X41 @ X42 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_size_fm @ X41 ) @ ( size_size_fm @ X42 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% fm.size(11)
thf(fact_1177_fm_Osize_I12_J,axiom,
! [X5: fm] :
( ( size_size_fm @ ( exi @ X5 ) )
= ( plus_plus_nat @ ( size_size_fm @ X5 ) @ ( suc @ zero_zero_nat ) ) ) ).
% fm.size(12)
thf(fact_1178_fm_Osize_I13_J,axiom,
! [X6: fm] :
( ( size_size_fm @ ( uni @ X6 ) )
= ( plus_plus_nat @ ( size_size_fm @ X6 ) @ ( suc @ zero_zero_nat ) ) ) ).
% fm.size(13)
thf(fact_1179_parts__in__children,axiom,
! [P: fm,Z3: list_fm,Z2: list_fm,A2: list_tm,R: rule] :
( ( member_fm3 @ P @ ( set_fm2 @ Z3 ) )
=> ( ( member_list_fm3 @ Z2 @ ( set_list_fm2 @ ( children @ A2 @ R @ Z3 ) ) )
=> ? [B8: list_tm,Xs2: list_fm] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ A2 ) @ ( set_tm2 @ B8 ) )
& ( member_list_fm3 @ Xs2 @ ( set_list_fm2 @ ( parts @ B8 @ R @ P ) ) )
& ( ord_less_eq_set_fm @ ( set_fm2 @ Xs2 ) @ ( set_fm2 @ Z2 ) ) ) ) ) ).
% parts_in_children
thf(fact_1180_GammaUni,axiom,
! [T: tm,P: fm,Z3: list_fm] :
( ( sequent_calculus @ ( cons_fm @ ( neg @ ( sub @ zero_zero_nat @ T @ P ) ) @ Z3 ) )
=> ( sequent_calculus @ ( cons_fm @ ( neg @ ( uni @ P ) ) @ Z3 ) ) ) ).
% GammaUni
thf(fact_1181_s4_I1_J,axiom,
inc_term = liftt ).
% s4(1)
thf(fact_1182_Neg,axiom,
! [P: fm,Z3: list_fm] :
( ( sequent_calculus @ ( cons_fm @ P @ Z3 ) )
=> ( sequent_calculus @ ( cons_fm @ ( neg @ ( neg @ P ) ) @ Z3 ) ) ) ).
% Neg
thf(fact_1183_Basic,axiom,
! [P: fm,Z3: list_fm] :
( ( member_fm2 @ ( neg @ P ) @ Z3 )
=> ( sequent_calculus @ ( cons_fm @ P @ Z3 ) ) ) ).
% Basic
thf(fact_1184_Ext,axiom,
! [Z3: list_fm,Y: list_fm] :
( ( sequent_calculus @ Z3 )
=> ( ( ext_fm @ Y @ Z3 )
=> ( sequent_calculus @ Y ) ) ) ).
% Ext
thf(fact_1185_AlphaDis,axiom,
! [P: fm,Q: fm,Z3: list_fm] :
( ( sequent_calculus @ ( cons_fm @ P @ ( cons_fm @ Q @ Z3 ) ) )
=> ( sequent_calculus @ ( cons_fm @ ( dis @ P @ Q ) @ Z3 ) ) ) ).
% AlphaDis
thf(fact_1186_BetaCon,axiom,
! [P: fm,Z3: list_fm,Q: fm] :
( ( sequent_calculus @ ( cons_fm @ P @ Z3 ) )
=> ( ( sequent_calculus @ ( cons_fm @ Q @ Z3 ) )
=> ( sequent_calculus @ ( cons_fm @ ( con @ P @ Q ) @ Z3 ) ) ) ) ).
% BetaCon
thf(fact_1187_AlphaCon,axiom,
! [P: fm,Q: fm,Z3: list_fm] :
( ( sequent_calculus @ ( cons_fm @ ( neg @ P ) @ ( cons_fm @ ( neg @ Q ) @ Z3 ) ) )
=> ( sequent_calculus @ ( cons_fm @ ( neg @ ( con @ P @ Q ) ) @ Z3 ) ) ) ).
% AlphaCon
thf(fact_1188_AlphaImp,axiom,
! [P: fm,Q: fm,Z3: list_fm] :
( ( sequent_calculus @ ( cons_fm @ ( neg @ P ) @ ( cons_fm @ Q @ Z3 ) ) )
=> ( sequent_calculus @ ( cons_fm @ ( imp @ P @ Q ) @ Z3 ) ) ) ).
% AlphaImp
thf(fact_1189_BetaImp,axiom,
! [P: fm,Z3: list_fm,Q: fm] :
( ( sequent_calculus @ ( cons_fm @ P @ Z3 ) )
=> ( ( sequent_calculus @ ( cons_fm @ ( neg @ Q ) @ Z3 ) )
=> ( sequent_calculus @ ( cons_fm @ ( neg @ ( imp @ P @ Q ) ) @ Z3 ) ) ) ) ).
% BetaImp
thf(fact_1190_BetaDis,axiom,
! [P: fm,Z3: list_fm,Q: fm] :
( ( sequent_calculus @ ( cons_fm @ ( neg @ P ) @ Z3 ) )
=> ( ( sequent_calculus @ ( cons_fm @ ( neg @ Q ) @ Z3 ) )
=> ( sequent_calculus @ ( cons_fm @ ( neg @ ( dis @ P @ Q ) ) @ Z3 ) ) ) ) ).
% BetaDis
thf(fact_1191_GammaExi,axiom,
! [T: tm,P: fm,Z3: list_fm] :
( ( sequent_calculus @ ( cons_fm @ ( sub @ zero_zero_nat @ T @ P ) @ Z3 ) )
=> ( sequent_calculus @ ( cons_fm @ ( exi @ P ) @ Z3 ) ) ) ).
% GammaExi
thf(fact_1192_tm_Oinject_I1_J,axiom,
! [X11: nat,X12: list_tm,Y11: nat,Y12: list_tm] :
( ( ( fun @ X11 @ X12 )
= ( fun @ Y11 @ Y12 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 ) ) ) ).
% tm.inject(1)
thf(fact_1193_news_Osimps_I1_J,axiom,
! [C: nat] : ( news @ C @ nil_fm ) ).
% news.simps(1)
thf(fact_1194_DeltaUni,axiom,
! [I2: nat,P: fm,Z3: list_fm] :
( ( sequent_calculus @ ( cons_fm @ ( sub @ zero_zero_nat @ ( fun @ I2 @ nil_tm ) @ P ) @ Z3 ) )
=> ( ( news @ I2 @ ( cons_fm @ P @ Z3 ) )
=> ( sequent_calculus @ ( cons_fm @ ( uni @ P ) @ Z3 ) ) ) ) ).
% DeltaUni
thf(fact_1195_DeltaExi,axiom,
! [I2: nat,P: fm,Z3: list_fm] :
( ( sequent_calculus @ ( cons_fm @ ( neg @ ( sub @ zero_zero_nat @ ( fun @ I2 @ nil_tm ) @ P ) ) @ Z3 ) )
=> ( ( news @ I2 @ ( cons_fm @ P @ Z3 ) )
=> ( sequent_calculus @ ( cons_fm @ ( neg @ ( exi @ P ) ) @ Z3 ) ) ) ) ).
% DeltaExi
thf(fact_1196_sequent__calculus_Ocases,axiom,
! [A: list_fm] :
( ( sequent_calculus @ A )
=> ( ! [P4: fm,Z4: list_fm] :
( ( A
= ( cons_fm @ P4 @ Z4 ) )
=> ~ ( member_fm2 @ ( neg @ P4 ) @ Z4 ) )
=> ( ! [P4: fm,Q2: fm,Z4: list_fm] :
( ( A
= ( cons_fm @ ( dis @ P4 @ Q2 ) @ Z4 ) )
=> ~ ( sequent_calculus @ ( cons_fm @ P4 @ ( cons_fm @ Q2 @ Z4 ) ) ) )
=> ( ! [P4: fm,Q2: fm,Z4: list_fm] :
( ( A
= ( cons_fm @ ( imp @ P4 @ Q2 ) @ Z4 ) )
=> ~ ( sequent_calculus @ ( cons_fm @ ( neg @ P4 ) @ ( cons_fm @ Q2 @ Z4 ) ) ) )
=> ( ! [P4: fm,Q2: fm,Z4: list_fm] :
( ( A
= ( cons_fm @ ( neg @ ( con @ P4 @ Q2 ) ) @ Z4 ) )
=> ~ ( sequent_calculus @ ( cons_fm @ ( neg @ P4 ) @ ( cons_fm @ ( neg @ Q2 ) @ Z4 ) ) ) )
=> ( ! [P4: fm,Z4: list_fm,Q2: fm] :
( ( A
= ( cons_fm @ ( con @ P4 @ Q2 ) @ Z4 ) )
=> ( ( sequent_calculus @ ( cons_fm @ P4 @ Z4 ) )
=> ~ ( sequent_calculus @ ( cons_fm @ Q2 @ Z4 ) ) ) )
=> ( ! [P4: fm,Z4: list_fm,Q2: fm] :
( ( A
= ( cons_fm @ ( neg @ ( imp @ P4 @ Q2 ) ) @ Z4 ) )
=> ( ( sequent_calculus @ ( cons_fm @ P4 @ Z4 ) )
=> ~ ( sequent_calculus @ ( cons_fm @ ( neg @ Q2 ) @ Z4 ) ) ) )
=> ( ! [P4: fm,Z4: list_fm,Q2: fm] :
( ( A
= ( cons_fm @ ( neg @ ( dis @ P4 @ Q2 ) ) @ Z4 ) )
=> ( ( sequent_calculus @ ( cons_fm @ ( neg @ P4 ) @ Z4 ) )
=> ~ ( sequent_calculus @ ( cons_fm @ ( neg @ Q2 ) @ Z4 ) ) ) )
=> ( ! [T2: tm,P4: fm,Z4: list_fm] :
( ( A
= ( cons_fm @ ( exi @ P4 ) @ Z4 ) )
=> ~ ( sequent_calculus @ ( cons_fm @ ( sub @ zero_zero_nat @ T2 @ P4 ) @ Z4 ) ) )
=> ( ! [T2: tm,P4: fm,Z4: list_fm] :
( ( A
= ( cons_fm @ ( neg @ ( uni @ P4 ) ) @ Z4 ) )
=> ~ ( sequent_calculus @ ( cons_fm @ ( neg @ ( sub @ zero_zero_nat @ T2 @ P4 ) ) @ Z4 ) ) )
=> ( ! [I: nat,P4: fm,Z4: list_fm] :
( ( A
= ( cons_fm @ ( uni @ P4 ) @ Z4 ) )
=> ( ( sequent_calculus @ ( cons_fm @ ( sub @ zero_zero_nat @ ( fun @ I @ nil_tm ) @ P4 ) @ Z4 ) )
=> ~ ( news @ I @ ( cons_fm @ P4 @ Z4 ) ) ) )
=> ( ! [I: nat,P4: fm,Z4: list_fm] :
( ( A
= ( cons_fm @ ( neg @ ( exi @ P4 ) ) @ Z4 ) )
=> ( ( sequent_calculus @ ( cons_fm @ ( neg @ ( sub @ zero_zero_nat @ ( fun @ I @ nil_tm ) @ P4 ) ) @ Z4 ) )
=> ~ ( news @ I @ ( cons_fm @ P4 @ Z4 ) ) ) )
=> ( ! [P4: fm,Z4: list_fm] :
( ( A
= ( cons_fm @ ( neg @ ( neg @ P4 ) ) @ Z4 ) )
=> ~ ( sequent_calculus @ ( cons_fm @ P4 @ Z4 ) ) )
=> ~ ! [Z4: list_fm] :
( ( sequent_calculus @ Z4 )
=> ~ ( ext_fm @ A @ Z4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% sequent_calculus.cases
thf(fact_1197_sequent__calculus_Osimps,axiom,
( sequent_calculus
= ( ^ [A5: list_fm] :
( ? [P2: fm,Z: list_fm] :
( ( A5
= ( cons_fm @ P2 @ Z ) )
& ( member_fm2 @ ( neg @ P2 ) @ Z ) )
| ? [P2: fm,Q3: fm,Z: list_fm] :
( ( A5
= ( cons_fm @ ( dis @ P2 @ Q3 ) @ Z ) )
& ( sequent_calculus @ ( cons_fm @ P2 @ ( cons_fm @ Q3 @ Z ) ) ) )
| ? [P2: fm,Q3: fm,Z: list_fm] :
( ( A5
= ( cons_fm @ ( imp @ P2 @ Q3 ) @ Z ) )
& ( sequent_calculus @ ( cons_fm @ ( neg @ P2 ) @ ( cons_fm @ Q3 @ Z ) ) ) )
| ? [P2: fm,Q3: fm,Z: list_fm] :
( ( A5
= ( cons_fm @ ( neg @ ( con @ P2 @ Q3 ) ) @ Z ) )
& ( sequent_calculus @ ( cons_fm @ ( neg @ P2 ) @ ( cons_fm @ ( neg @ Q3 ) @ Z ) ) ) )
| ? [P2: fm,Z: list_fm,Q3: fm] :
( ( A5
= ( cons_fm @ ( con @ P2 @ Q3 ) @ Z ) )
& ( sequent_calculus @ ( cons_fm @ P2 @ Z ) )
& ( sequent_calculus @ ( cons_fm @ Q3 @ Z ) ) )
| ? [P2: fm,Z: list_fm,Q3: fm] :
( ( A5
= ( cons_fm @ ( neg @ ( imp @ P2 @ Q3 ) ) @ Z ) )
& ( sequent_calculus @ ( cons_fm @ P2 @ Z ) )
& ( sequent_calculus @ ( cons_fm @ ( neg @ Q3 ) @ Z ) ) )
| ? [P2: fm,Z: list_fm,Q3: fm] :
( ( A5
= ( cons_fm @ ( neg @ ( dis @ P2 @ Q3 ) ) @ Z ) )
& ( sequent_calculus @ ( cons_fm @ ( neg @ P2 ) @ Z ) )
& ( sequent_calculus @ ( cons_fm @ ( neg @ Q3 ) @ Z ) ) )
| ? [T3: tm,P2: fm,Z: list_fm] :
( ( A5
= ( cons_fm @ ( exi @ P2 ) @ Z ) )
& ( sequent_calculus @ ( cons_fm @ ( sub @ zero_zero_nat @ T3 @ P2 ) @ Z ) ) )
| ? [T3: tm,P2: fm,Z: list_fm] :
( ( A5
= ( cons_fm @ ( neg @ ( uni @ P2 ) ) @ Z ) )
& ( sequent_calculus @ ( cons_fm @ ( neg @ ( sub @ zero_zero_nat @ T3 @ P2 ) ) @ Z ) ) )
| ? [I4: nat,P2: fm,Z: list_fm] :
( ( A5
= ( cons_fm @ ( uni @ P2 ) @ Z ) )
& ( sequent_calculus @ ( cons_fm @ ( sub @ zero_zero_nat @ ( fun @ I4 @ nil_tm ) @ P2 ) @ Z ) )
& ( news @ I4 @ ( cons_fm @ P2 @ Z ) ) )
| ? [I4: nat,P2: fm,Z: list_fm] :
( ( A5
= ( cons_fm @ ( neg @ ( exi @ P2 ) ) @ Z ) )
& ( sequent_calculus @ ( cons_fm @ ( neg @ ( sub @ zero_zero_nat @ ( fun @ I4 @ nil_tm ) @ P2 ) ) @ Z ) )
& ( news @ I4 @ ( cons_fm @ P2 @ Z ) ) )
| ? [P2: fm,Z: list_fm] :
( ( A5
= ( cons_fm @ ( neg @ ( neg @ P2 ) ) @ Z ) )
& ( sequent_calculus @ ( cons_fm @ P2 @ Z ) ) )
| ? [Z: list_fm,Y4: list_fm] :
( ( A5 = Y4 )
& ( sequent_calculus @ Z )
& ( ext_fm @ Y4 @ Z ) ) ) ) ) ).
% sequent_calculus.simps
thf(fact_1198_inc__term_Osimps_I2_J,axiom,
! [I2: nat,L: list_tm] :
( ( inc_term @ ( fun @ I2 @ L ) )
= ( fun @ I2 @ ( inc_list @ L ) ) ) ).
% inc_term.simps(2)
thf(fact_1199_subterm__Fun__refl,axiom,
! [Ts: list_tm,N2: nat] : ( ord_less_eq_set_tm @ ( set_tm2 @ Ts ) @ ( set_tm2 @ ( subtermTm @ ( fun @ N2 @ Ts ) ) ) ) ).
% subterm_Fun_refl
thf(fact_1200_subtermTm__refl,axiom,
! [T: tm] : ( member_tm3 @ T @ ( set_tm2 @ ( subtermTm @ T ) ) ) ).
% subtermTm_refl
thf(fact_1201_inc__list_Osimps_I1_J,axiom,
( ( inc_list @ nil_tm )
= nil_tm ) ).
% inc_list.simps(1)
thf(fact_1202_inc__list_Osimps_I2_J,axiom,
! [T: tm,L: list_tm] :
( ( inc_list @ ( cons_tm @ T @ L ) )
= ( cons_tm @ ( inc_term @ T ) @ ( inc_list @ L ) ) ) ).
% inc_list.simps(2)
thf(fact_1203_subtermTm__le,axiom,
! [T: tm,S: tm] :
( ( member_tm3 @ T @ ( set_tm2 @ ( subtermTm @ S ) ) )
=> ( ord_less_eq_set_tm @ ( set_tm2 @ ( subtermTm @ T ) ) @ ( set_tm2 @ ( subtermTm @ S ) ) ) ) ).
% subtermTm_le
thf(fact_1204_sub__term__const__transfer_I1_J,axiom,
! [M: nat,A: nat,T: tm,S: tm] :
( ( ( sub_term @ M @ ( fun @ A @ nil_tm ) @ T )
!= ( sub_term @ M @ S @ T ) )
=> ( member_tm3 @ ( fun @ A @ nil_tm ) @ ( set_tm2 @ ( subtermTm @ ( sub_term @ M @ ( fun @ A @ nil_tm ) @ T ) ) ) ) ) ).
% sub_term_const_transfer(1)
thf(fact_1205_substt_Osimps_I2_J,axiom,
! [A: nat,Ts: list_tm,S: tm,K: nat] :
( ( substt @ ( fun @ A @ Ts ) @ S @ K )
= ( fun @ A @ ( substts @ Ts @ S @ K ) ) ) ).
% substt.simps(2)
thf(fact_1206_s5_I1_J,axiom,
( sub_term
= ( ^ [V3: nat,S3: tm,T3: tm] : ( substt @ T3 @ S3 @ V3 ) ) ) ).
% s5(1)
thf(fact_1207_substts_Osimps_I1_J,axiom,
! [S: tm,K: nat] :
( ( substts @ nil_tm @ S @ K )
= nil_tm ) ).
% substts.simps(1)
thf(fact_1208_substts_Osimps_I2_J,axiom,
! [T: tm,Ts: list_tm,S: tm,K: nat] :
( ( substts @ ( cons_tm @ T @ Ts ) @ S @ K )
= ( cons_tm @ ( substt @ T @ S @ K ) @ ( substts @ Ts @ S @ K ) ) ) ).
% substts.simps(2)
thf(fact_1209_liftts_Osimps_I2_J,axiom,
! [T: tm,Ts: list_tm] :
( ( liftts @ ( cons_tm @ T @ Ts ) )
= ( cons_tm @ ( liftt @ T ) @ ( liftts @ Ts ) ) ) ).
% liftts.simps(2)
thf(fact_1210_s4_I2_J,axiom,
inc_list = liftts ).
% s4(2)
thf(fact_1211_liftts_Osimps_I1_J,axiom,
( ( liftts @ nil_tm )
= nil_tm ) ).
% liftts.simps(1)
thf(fact_1212_liftt_Osimps_I2_J,axiom,
! [A: nat,Ts: list_tm] :
( ( liftt @ ( fun @ A @ Ts ) )
= ( fun @ A @ ( liftts @ Ts ) ) ) ).
% liftt.simps(2)
thf(fact_1213_sub__const__transfer,axiom,
! [M: nat,A: nat,P: fm,T: tm] :
( ( ( sub @ M @ ( fun @ A @ nil_tm ) @ P )
!= ( sub @ M @ T @ P ) )
=> ( member_tm3 @ ( fun @ A @ nil_tm ) @ ( set_tm2 @ ( subtermFm @ ( sub @ M @ ( fun @ A @ nil_tm ) @ P ) ) ) ) ) ).
% sub_const_transfer
thf(fact_1214_parts__in__effect,axiom,
! [P: fm,Z3: list_fm,B: list_tm,Z2: list_fm,R: rule,A2: list_tm] :
( ( member_fm3 @ P @ ( set_fm2 @ Z3 ) )
=> ( ( fmembe3754813877001230652ist_fm @ ( produc1414352766439514085ist_fm @ B @ Z2 ) @ ( effect @ R @ ( produc1414352766439514085ist_fm @ A2 @ Z3 ) ) )
=> ? [C3: list_tm,Xs2: list_fm] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ A2 ) @ ( set_tm2 @ C3 ) )
& ( member_list_fm3 @ Xs2 @ ( set_list_fm2 @ ( parts @ C3 @ R @ P ) ) )
& ( ord_less_eq_set_fm @ ( set_fm2 @ Xs2 ) @ ( set_fm2 @ Z2 ) ) ) ) ) ).
% parts_in_effect
thf(fact_1215_effect__preserves__unaffected,axiom,
! [P: fm,Z3: list_fm,R: rule,B: list_tm,Z2: list_fm,A2: list_tm] :
( ( member_fm3 @ P @ ( set_fm2 @ Z3 ) )
=> ( ~ ( affects @ R @ P )
=> ( ( fmembe3754813877001230652ist_fm @ ( produc1414352766439514085ist_fm @ B @ Z2 ) @ ( effect @ R @ ( produc1414352766439514085ist_fm @ A2 @ Z3 ) ) )
=> ( member_fm3 @ P @ ( set_fm2 @ Z2 ) ) ) ) ) ).
% effect_preserves_unaffected
thf(fact_1216_subtermFm_Osimps_I4_J,axiom,
! [P: fm,Q: fm] :
( ( subtermFm @ ( con @ P @ Q ) )
= ( append_tm @ ( subtermFm @ P ) @ ( subtermFm @ Q ) ) ) ).
% subtermFm.simps(4)
thf(fact_1217_subtermFm_Osimps_I3_J,axiom,
! [P: fm,Q: fm] :
( ( subtermFm @ ( dis @ P @ Q ) )
= ( append_tm @ ( subtermFm @ P ) @ ( subtermFm @ Q ) ) ) ).
% subtermFm.simps(3)
thf(fact_1218_subtermFm_Osimps_I2_J,axiom,
! [P: fm,Q: fm] :
( ( subtermFm @ ( imp @ P @ Q ) )
= ( append_tm @ ( subtermFm @ P ) @ ( subtermFm @ Q ) ) ) ).
% subtermFm.simps(2)
thf(fact_1219_ne__effect__not__branchDone,axiom,
! [B: list_tm,Z2: list_fm,R: rule,A2: list_tm,Z3: list_fm] :
( ( fmembe3754813877001230652ist_fm @ ( produc1414352766439514085ist_fm @ B @ Z2 ) @ ( effect @ R @ ( produc1414352766439514085ist_fm @ A2 @ Z3 ) ) )
=> ~ ( branchDone @ Z3 ) ) ).
% ne_effect_not_branchDone
thf(fact_1220_subtermFm_Osimps_I5_J,axiom,
! [P: fm] :
( ( subtermFm @ ( exi @ P ) )
= ( subtermFm @ P ) ) ).
% subtermFm.simps(5)
thf(fact_1221_subtermFm_Osimps_I6_J,axiom,
! [P: fm] :
( ( subtermFm @ ( uni @ P ) )
= ( subtermFm @ P ) ) ).
% subtermFm.simps(6)
thf(fact_1222_subtermFm_Osimps_I7_J,axiom,
! [P: fm] :
( ( subtermFm @ ( neg @ P ) )
= ( subtermFm @ P ) ) ).
% subtermFm.simps(7)
thf(fact_1223_fun__arguments__subterm,axiom,
! [N2: nat,Ts: list_tm,P: fm] :
( ( member_tm3 @ ( fun @ N2 @ Ts ) @ ( set_tm2 @ ( subtermFm @ P ) ) )
=> ( ord_less_eq_set_tm @ ( set_tm2 @ Ts ) @ ( set_tm2 @ ( subtermFm @ P ) ) ) ) ).
% fun_arguments_subterm
thf(fact_1224_subterm__Pre__refl,axiom,
! [Ts: list_tm,N2: nat] : ( ord_less_eq_set_tm @ ( set_tm2 @ Ts ) @ ( set_tm2 @ ( subtermFm @ ( pre @ N2 @ Ts ) ) ) ) ).
% subterm_Pre_refl
thf(fact_1225_substt_Osimps_I1_J,axiom,
! [K: nat,I2: nat,S: tm] :
( ( ( ord_less_nat @ K @ I2 )
=> ( ( substt @ ( var @ I2 ) @ S @ K )
= ( var @ ( minus_minus_nat @ I2 @ one_one_nat ) ) ) )
& ( ~ ( ord_less_nat @ K @ I2 )
=> ( ( ( I2 = K )
=> ( ( substt @ ( var @ I2 ) @ S @ K )
= S ) )
& ( ( I2 != K )
=> ( ( substt @ ( var @ I2 ) @ S @ K )
= ( var @ I2 ) ) ) ) ) ) ).
% substt.simps(1)
thf(fact_1226_paramst__subtermTm_I1_J,axiom,
! [T: tm,X4: nat] :
( ( member_nat3 @ X4 @ ( paramst @ T ) )
=> ? [L2: list_tm] : ( member_tm3 @ ( fun @ X4 @ L2 ) @ ( set_tm2 @ ( subtermTm @ T ) ) ) ) ).
% paramst_subtermTm(1)
thf(fact_1227_tm_Oinject_I2_J,axiom,
! [X23: nat,Y2: nat] :
( ( ( var @ X23 )
= ( var @ Y2 ) )
= ( X23 = Y2 ) ) ).
% tm.inject(2)
thf(fact_1228_paramst__liftt_I1_J,axiom,
! [T: tm] :
( ( paramst @ ( liftt @ T ) )
= ( paramst @ T ) ) ).
% paramst_liftt(1)
thf(fact_1229_tm_Odistinct_I1_J,axiom,
! [X11: nat,X12: list_tm,X23: nat] :
( ( fun @ X11 @ X12 )
!= ( var @ X23 ) ) ).
% tm.distinct(1)
thf(fact_1230_tm_Oexhaust,axiom,
! [Y: tm] :
( ! [X112: nat,X122: list_tm] :
( Y
!= ( fun @ X112 @ X122 ) )
=> ~ ! [X24: nat] :
( Y
!= ( var @ X24 ) ) ) ).
% tm.exhaust
thf(fact_1231_paramst_H_H_Ocases,axiom,
! [X: tm] :
( ! [N: nat] :
( X
!= ( var @ N ) )
=> ~ ! [A4: nat,Ts2: list_tm] :
( X
!= ( fun @ A4 @ Ts2 ) ) ) ).
% paramst''.cases
thf(fact_1232_paramst__sub__term_I1_J,axiom,
! [M: nat,S: tm,T: tm] : ( ord_less_eq_set_nat @ ( paramst @ ( sub_term @ M @ S @ T ) ) @ ( sup_sup_set_nat @ ( paramst @ S ) @ ( paramst @ T ) ) ) ).
% paramst_sub_term(1)
thf(fact_1233_liftt_Osimps_I1_J,axiom,
! [I2: nat] :
( ( liftt @ ( var @ I2 ) )
= ( var @ ( suc @ I2 ) ) ) ).
% liftt.simps(1)
thf(fact_1234_inc__term_Osimps_I1_J,axiom,
! [N2: nat] :
( ( inc_term @ ( var @ N2 ) )
= ( var @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% inc_term.simps(1)
thf(fact_1235_subtermTm_Osimps_I2_J,axiom,
! [N2: nat] :
( ( subtermTm @ ( var @ N2 ) )
= ( cons_tm @ ( var @ N2 ) @ nil_tm ) ) ).
% subtermTm.simps(2)
thf(fact_1236_sub__term_Osimps_I1_J,axiom,
! [N2: nat,V2: nat,S: tm] :
( ( ( ord_less_nat @ N2 @ V2 )
=> ( ( sub_term @ V2 @ S @ ( var @ N2 ) )
= ( var @ N2 ) ) )
& ( ~ ( ord_less_nat @ N2 @ V2 )
=> ( ( ( N2 = V2 )
=> ( ( sub_term @ V2 @ S @ ( var @ N2 ) )
= S ) )
& ( ( N2 != V2 )
=> ( ( sub_term @ V2 @ S @ ( var @ N2 ) )
= ( var @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ) ).
% sub_term.simps(1)
thf(fact_1237_tm_Osize__gen_I2_J,axiom,
! [X23: nat] :
( ( size_tm @ ( var @ X23 ) )
= zero_zero_nat ) ).
% tm.size_gen(2)
thf(fact_1238_s1_I1_J,axiom,
( new_term
= ( ^ [C4: nat,T3: tm] :
~ ( member_nat3 @ C4 @ ( paramst @ T3 ) ) ) ) ).
% s1(1)
thf(fact_1239_new__term_Osimps_I1_J,axiom,
! [C: nat,N2: nat] : ( new_term @ C @ ( var @ N2 ) ) ).
% new_term.simps(1)
thf(fact_1240_new__list_Osimps_I2_J,axiom,
! [C: nat,T: tm,L: list_tm] :
( ( new_list @ C @ ( cons_tm @ T @ L ) )
= ( ( ( new_term @ C @ T )
=> ( new_list @ C @ L ) )
& ( new_term @ C @ T ) ) ) ).
% new_list.simps(2)
thf(fact_1241_new__term_Osimps_I2_J,axiom,
! [C: nat,I2: nat,L: list_tm] :
( ( new_term @ C @ ( fun @ I2 @ L ) )
= ( ( I2 != C )
& ( ( I2 != C )
=> ( new_list @ C @ L ) ) ) ) ).
% new_term.simps(2)
thf(fact_1242_new__list_Osimps_I1_J,axiom,
! [C: nat] : ( new_list @ C @ nil_tm ) ).
% new_list.simps(1)
thf(fact_1243_params__sub,axiom,
! [M: nat,T: tm,P: fm] : ( ord_less_eq_set_nat @ ( params @ ( sub @ M @ T @ P ) ) @ ( sup_sup_set_nat @ ( paramst @ T ) @ ( params @ P ) ) ) ).
% params_sub
thf(fact_1244_p1,axiom,
paramst2 = paramst ).
% p1
thf(fact_1245_params_Osimps_I7_J,axiom,
! [P: fm] :
( ( params @ ( neg @ P ) )
= ( params @ P ) ) ).
% params.simps(7)
thf(fact_1246_params_Osimps_I5_J,axiom,
! [P: fm] :
( ( params @ ( exi @ P ) )
= ( params @ P ) ) ).
% params.simps(5)
thf(fact_1247_params_Osimps_I6_J,axiom,
! [P: fm] :
( ( params @ ( uni @ P ) )
= ( params @ P ) ) ).
% params.simps(6)
thf(fact_1248_params_Osimps_I4_J,axiom,
! [P: fm,Q: fm] :
( ( params @ ( con @ P @ Q ) )
= ( sup_sup_set_nat @ ( params @ P ) @ ( params @ Q ) ) ) ).
% params.simps(4)
thf(fact_1249_params_Osimps_I2_J,axiom,
! [P: fm,Q: fm] :
( ( params @ ( imp @ P @ Q ) )
= ( sup_sup_set_nat @ ( params @ P ) @ ( params @ Q ) ) ) ).
% params.simps(2)
thf(fact_1250_params_Osimps_I3_J,axiom,
! [P: fm,Q: fm] :
( ( params @ ( dis @ P @ Q ) )
= ( sup_sup_set_nat @ ( params @ P ) @ ( params @ Q ) ) ) ).
% params.simps(3)
thf(fact_1251_params__subtermFm,axiom,
! [P: fm,X4: nat] :
( ( member_nat3 @ X4 @ ( params @ P ) )
=> ? [L2: list_tm] : ( member_tm3 @ ( fun @ X4 @ L2 ) @ ( set_tm2 @ ( subtermFm @ P ) ) ) ) ).
% params_subtermFm
thf(fact_1252_subtermFm__subset__params,axiom,
! [P: fm,A2: list_tm] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ ( subtermFm @ P ) ) @ ( set_tm2 @ A2 ) )
=> ( ord_less_eq_set_nat @ ( params @ P ) @ ( paramsts @ A2 ) ) ) ).
% subtermFm_subset_params
thf(fact_1253_paramsts__subset,axiom,
! [A2: list_tm,B: list_tm] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ A2 ) @ ( set_tm2 @ B ) )
=> ( ord_less_eq_set_nat @ ( paramsts @ A2 ) @ ( paramsts @ B ) ) ) ).
% paramsts_subset
thf(fact_1254_paramst__liftt_I2_J,axiom,
! [Ts: list_tm] :
( ( paramsts @ ( liftts @ Ts ) )
= ( paramsts @ Ts ) ) ).
% paramst_liftt(2)
thf(fact_1255_s1_I2_J,axiom,
( new_list
= ( ^ [C4: nat,L3: list_tm] :
~ ( member_nat3 @ C4 @ ( paramsts @ L3 ) ) ) ) ).
% s1(2)
thf(fact_1256_params_Osimps_I1_J,axiom,
! [B2: nat,Ts: list_tm] :
( ( params @ ( pre @ B2 @ Ts ) )
= ( paramsts @ Ts ) ) ).
% params.simps(1)
thf(fact_1257_paramsts_Osimps_I2_J,axiom,
! [T: tm,Ts: list_tm] :
( ( paramsts @ ( cons_tm @ T @ Ts ) )
= ( sup_sup_set_nat @ ( paramst @ T ) @ ( paramsts @ Ts ) ) ) ).
% paramsts.simps(2)
thf(fact_1258_paramst__sub__term_I2_J,axiom,
! [M: nat,S: tm,L: list_tm] : ( ord_less_eq_set_nat @ ( paramsts @ ( sub_list @ M @ S @ L ) ) @ ( sup_sup_set_nat @ ( paramst @ S ) @ ( paramsts @ L ) ) ) ).
% paramst_sub_term(2)
thf(fact_1259_subst_Osimps_I5_J,axiom,
! [P: fm,S: tm,K: nat] :
( ( subst @ ( exi @ P ) @ S @ K )
= ( exi @ ( subst @ P @ ( liftt @ S ) @ ( suc @ K ) ) ) ) ).
% subst.simps(5)
thf(fact_1260_s6,axiom,
( sub
= ( ^ [V3: nat,S3: tm,P2: fm] : ( subst @ P2 @ S3 @ V3 ) ) ) ).
% s6
thf(fact_1261_s5_I2_J,axiom,
( sub_list
= ( ^ [V3: nat,S3: tm,L3: list_tm] : ( substts @ L3 @ S3 @ V3 ) ) ) ).
% s5(2)
thf(fact_1262_sub__list_Osimps_I1_J,axiom,
! [V2: nat,S: tm] :
( ( sub_list @ V2 @ S @ nil_tm )
= nil_tm ) ).
% sub_list.simps(1)
thf(fact_1263_subst_Osimps_I7_J,axiom,
! [P: fm,S: tm,K: nat] :
( ( subst @ ( neg @ P ) @ S @ K )
= ( neg @ ( subst @ P @ S @ K ) ) ) ).
% subst.simps(7)
thf(fact_1264_subst_Osimps_I2_J,axiom,
! [P: fm,Q: fm,S: tm,K: nat] :
( ( subst @ ( imp @ P @ Q ) @ S @ K )
= ( imp @ ( subst @ P @ S @ K ) @ ( subst @ Q @ S @ K ) ) ) ).
% subst.simps(2)
thf(fact_1265_subst_Osimps_I3_J,axiom,
! [P: fm,Q: fm,S: tm,K: nat] :
( ( subst @ ( dis @ P @ Q ) @ S @ K )
= ( dis @ ( subst @ P @ S @ K ) @ ( subst @ Q @ S @ K ) ) ) ).
% subst.simps(3)
thf(fact_1266_subst_Osimps_I4_J,axiom,
! [P: fm,Q: fm,S: tm,K: nat] :
( ( subst @ ( con @ P @ Q ) @ S @ K )
= ( con @ ( subst @ P @ S @ K ) @ ( subst @ Q @ S @ K ) ) ) ).
% subst.simps(4)
thf(fact_1267_subst_Osimps_I1_J,axiom,
! [B2: nat,Ts: list_tm,S: tm,K: nat] :
( ( subst @ ( pre @ B2 @ Ts ) @ S @ K )
= ( pre @ B2 @ ( substts @ Ts @ S @ K ) ) ) ).
% subst.simps(1)
thf(fact_1268_sub_Osimps_I1_J,axiom,
! [V2: nat,S: tm,I2: nat,L: list_tm] :
( ( sub @ V2 @ S @ ( pre @ I2 @ L ) )
= ( pre @ I2 @ ( sub_list @ V2 @ S @ L ) ) ) ).
% sub.simps(1)
thf(fact_1269_sub__term_Osimps_I2_J,axiom,
! [V2: nat,S: tm,I2: nat,L: list_tm] :
( ( sub_term @ V2 @ S @ ( fun @ I2 @ L ) )
= ( fun @ I2 @ ( sub_list @ V2 @ S @ L ) ) ) ).
% sub_term.simps(2)
thf(fact_1270_sub__list_Osimps_I2_J,axiom,
! [V2: nat,S: tm,T: tm,L: list_tm] :
( ( sub_list @ V2 @ S @ ( cons_tm @ T @ L ) )
= ( cons_tm @ ( sub_term @ V2 @ S @ T ) @ ( sub_list @ V2 @ S @ L ) ) ) ).
% sub_list.simps(2)
thf(fact_1271_subst_Osimps_I6_J,axiom,
! [P: fm,S: tm,K: nat] :
( ( subst @ ( uni @ P ) @ S @ K )
= ( uni @ ( subst @ P @ ( liftt @ S ) @ ( suc @ K ) ) ) ) ).
% subst.simps(6)
thf(fact_1272_tm_Osize_I4_J,axiom,
! [X23: nat] :
( ( size_size_tm @ ( var @ X23 ) )
= zero_zero_nat ) ).
% tm.size(4)
% Helper facts (13)
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__SeCaV__Ofm_J_T,axiom,
! [X: list_fm,Y: list_fm] :
( ( if_list_fm @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__SeCaV__Ofm_J_T,axiom,
! [X: list_fm,Y: list_fm] :
( ( if_list_fm @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__SeCaV__Otm_J_T,axiom,
! [X: list_tm,Y: list_tm] :
( ( if_list_tm @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__SeCaV__Otm_J_T,axiom,
! [X: list_tm,Y: list_tm] :
( ( if_list_tm @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Prover__Orule_J_T,axiom,
! [X: list_rule,Y: list_rule] :
( ( if_list_rule @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Prover__Orule_J_T,axiom,
! [X: list_rule,Y: list_rule] :
( ( if_list_rule @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_T,axiom,
! [X: list_list_fm,Y: list_list_fm] :
( ( if_list_list_fm @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_T,axiom,
! [X: list_list_fm,Y: list_list_fm] :
( ( if_list_list_fm @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_T,axiom,
! [P3: $o] :
( ( P3 = $true )
| ( P3 = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_T,axiom,
! [X: list_P2774625669004309958m_rule,Y: list_P2774625669004309958m_rule] :
( ( if_lis6095641862558746240m_rule @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__SeCaV__Otm_J_Mt__List__Olist_It__SeCaV__Ofm_J_J_Mt__Prover__Orule_J_J_T,axiom,
! [X: list_P2774625669004309958m_rule,Y: list_P2774625669004309958m_rule] :
( ( if_lis6095641862558746240m_rule @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( affects @ r @ ( neg @ ( con @ p @ q ) ) )
= ( r = alphaCon ) ) ).
%------------------------------------------------------------------------------