TPTP Problem File: SLH0378^1.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 : VYDRA_MDL/0000_NFA/prob_00556_022394__16094760_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1547 ( 683 unt; 261 typ; 0 def)
% Number of atoms : 3483 (1711 equ; 0 cnn)
% Maximal formula atoms : 15 ( 2 avg)
% Number of connectives : 11283 ( 564 ~; 49 |; 328 &;8896 @)
% ( 0 <=>;1446 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 22 ( 21 usr)
% Number of type conns : 782 ( 782 >; 0 *; 0 +; 0 <<)
% Number of symbols : 243 ( 240 usr; 19 con; 0-6 aty)
% Number of variables : 3527 ( 164 ^;3152 !; 211 ?;3527 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:47:37.745
%------------------------------------------------------------------------------
% Could-be-implicit typings (21)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_set_set_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_I_Eo_J_J_J,type,
list_list_list_o: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_I_Eo_J_J_J,type,
set_list_list_o: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__NFA__Otransition_J_J,type,
list_list_transition: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_I_Eo_J_J_J,type,
set_set_list_o: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__NFA__Otransition_J_J,type,
set_list_transition: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__NFA__Otransition_J_J,type,
set_set_transition: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_I_Eo_J_J,type,
list_list_o: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_I_Eo_J_J,type,
set_list_o: $tType ).
thf(ty_n_t__List__Olist_It__NFA__Otransition_J,type,
list_transition: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
set_set_o: $tType ).
thf(ty_n_t__Set__Oset_It__NFA__Otransition_J,type,
set_transition: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__List__Olist_I_Eo_J,type,
list_o: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__NFA__Otransition,type,
transition: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (240)
thf(sy_c_Finite__Set_OFpow_001t__NFA__Otransition,type,
finite9040295020840969508sition: set_transition > set_set_transition ).
thf(sy_c_Finite__Set_OFpow_001t__Nat__Onat,type,
finite_Fpow_nat: set_nat > set_set_nat ).
thf(sy_c_Finite__Set_Ofinite_001_Eo,type,
finite_finite_o: set_o > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__NFA__Otransition,type,
finite8165534619950747239sition: set_transition > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__NFA__Otransition_J,type,
minus_8944320859760356485sition: set_transition > set_transition > set_transition ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_If_001t__List__Olist_It__List__Olist_I_Eo_J_J,type,
if_list_list_o: $o > list_list_o > list_list_o > list_list_o ).
thf(sy_c_If_001t__List__Olist_It__NFA__Otransition_J,type,
if_list_transition: $o > list_transition > list_transition > list_transition ).
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__Set__Oset_It__Nat__Onat_J,type,
if_set_nat: $o > set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_Eo_J,type,
inf_inf_set_o: set_o > set_o > set_o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__List__Olist_I_Eo_J_J,type,
inf_inf_set_list_o: set_list_o > set_list_o > set_list_o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__NFA__Otransition_J,type,
inf_in8814773338690644108sition: set_transition > set_transition > set_transition ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
inf_inf_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__List__Olist_I_Eo_J_J,type,
sup_sup_set_list_o: set_list_o > set_list_o > set_list_o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__NFA__Otransition_J,type,
sup_su812053455038985074sition: set_transition > set_transition > set_transition ).
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__Set__Oset_It__NFA__Otransition_J_J,type,
sup_su8198498708765531986sition: set_set_transition > set_set_transition > set_set_transition ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
sup_sup_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Lattices__Big_Olinorder__class_OMin_001_Eo,type,
lattic1973801136483472281_Min_o: set_o > $o ).
thf(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Nat__Onat,type,
lattic8721135487736765967in_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Nat__Onat,type,
lattic5238388535129920115in_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Set__Oset_It__Nat__Onat_J,type,
lattic3014633134055518761et_nat: set_set_nat > set_nat ).
thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__NFA__Otransition,type,
lattic8086405597477538456sition: ( transition > transition > transition ) > ( transition > transition > $o ) > ( transition > transition > $o ) > $o ).
thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__Nat__Onat,type,
lattic6009151579333465974et_nat: ( nat > nat > nat ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__Set__Oset_It__Nat__Onat_J,type,
lattic3109210760196336428et_nat: ( set_nat > set_nat > set_nat ) > ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > $o ).
thf(sy_c_Lattices__Big_Osemilattice__set_001t__NFA__Otransition,type,
lattic5875891794066335564sition: ( transition > transition > transition ) > $o ).
thf(sy_c_Lattices__Big_Osemilattice__set_001t__Nat__Onat,type,
lattic1029310888574255042et_nat: ( nat > nat > nat ) > $o ).
thf(sy_c_Lattices__Big_Osemilattice__set_001t__Set__Oset_It__Nat__Onat_J,type,
lattic6452893353811829624et_nat: ( set_nat > set_nat > set_nat ) > $o ).
thf(sy_c_Lattices__Big_Osemilattice__set_OF_001t__NFA__Otransition,type,
lattic3235374346148272024sition: ( transition > transition > transition ) > set_transition > transition ).
thf(sy_c_Lattices__Big_Osemilattice__set_OF_001t__Nat__Onat,type,
lattic7742739596368939638_F_nat: ( nat > nat > nat ) > set_nat > nat ).
thf(sy_c_Lattices__Big_Osemilattice__set_OF_001t__Set__Oset_It__Nat__Onat_J,type,
lattic4908145837437951532et_nat: ( set_nat > set_nat > set_nat ) > set_set_nat > set_nat ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Nat__Onat,type,
lattic1093996805478795353in_nat: set_nat > nat ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_It__Nat__Onat_J,type,
lattic3835124923745554447et_nat: set_set_nat > set_nat ).
thf(sy_c_List_Oappend_001_Eo,type,
append_o: list_o > list_o > list_o ).
thf(sy_c_List_Oappend_001t__List__Olist_I_Eo_J,type,
append_list_o: list_list_o > list_list_o > list_list_o ).
thf(sy_c_List_Oappend_001t__List__Olist_It__List__Olist_I_Eo_J_J,type,
append_list_list_o: list_list_list_o > list_list_list_o > list_list_list_o ).
thf(sy_c_List_Oappend_001t__List__Olist_It__NFA__Otransition_J,type,
append5844159210978444383sition: list_list_transition > list_list_transition > list_list_transition ).
thf(sy_c_List_Oappend_001t__NFA__Otransition,type,
append_transition: list_transition > list_transition > list_transition ).
thf(sy_c_List_Oappend_001t__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Obind_001_Eo_001_Eo,type,
bind_o_o: list_o > ( $o > list_o ) > list_o ).
thf(sy_c_List_Obind_001_Eo_001t__List__Olist_I_Eo_J,type,
bind_o_list_o: list_o > ( $o > list_list_o ) > list_list_o ).
thf(sy_c_List_Obind_001_Eo_001t__NFA__Otransition,type,
bind_o_transition: list_o > ( $o > list_transition ) > list_transition ).
thf(sy_c_List_Obind_001t__List__Olist_I_Eo_J_001_Eo,type,
bind_list_o_o: list_list_o > ( list_o > list_o ) > list_o ).
thf(sy_c_List_Obind_001t__List__Olist_I_Eo_J_001t__List__Olist_I_Eo_J,type,
bind_list_o_list_o: list_list_o > ( list_o > list_list_o ) > list_list_o ).
thf(sy_c_List_Obind_001t__List__Olist_I_Eo_J_001t__NFA__Otransition,type,
bind_l7118857872559238311sition: list_list_o > ( list_o > list_transition ) > list_transition ).
thf(sy_c_List_Obind_001t__NFA__Otransition_001_Eo,type,
bind_transition_o: list_transition > ( transition > list_o ) > list_o ).
thf(sy_c_List_Obind_001t__NFA__Otransition_001t__List__Olist_I_Eo_J,type,
bind_t133343566971731431list_o: list_transition > ( transition > list_list_o ) > list_list_o ).
thf(sy_c_List_Obind_001t__NFA__Otransition_001t__NFA__Otransition,type,
bind_t1640203714015778183sition: list_transition > ( transition > list_transition ) > list_transition ).
thf(sy_c_List_Obutlast_001_Eo,type,
butlast_o: list_o > list_o ).
thf(sy_c_List_Obutlast_001t__List__Olist_I_Eo_J,type,
butlast_list_o: list_list_o > list_list_o ).
thf(sy_c_List_Obutlast_001t__NFA__Otransition,type,
butlast_transition: list_transition > list_transition ).
thf(sy_c_List_Obutlast_001t__Nat__Onat,type,
butlast_nat: list_nat > list_nat ).
thf(sy_c_List_Ocan__select_001t__NFA__Otransition,type,
can_se3600352496914471099sition: ( transition > $o ) > set_transition > $o ).
thf(sy_c_List_Ocan__select_001t__Nat__Onat,type,
can_select_nat: ( nat > $o ) > set_nat > $o ).
thf(sy_c_List_Oconcat_001_Eo,type,
concat_o: list_list_o > list_o ).
thf(sy_c_List_Oconcat_001t__List__Olist_I_Eo_J,type,
concat_list_o: list_list_list_o > list_list_o ).
thf(sy_c_List_Oconcat_001t__NFA__Otransition,type,
concat_transition: list_list_transition > list_transition ).
thf(sy_c_List_Oconcat_001t__Nat__Onat,type,
concat_nat: list_list_nat > list_nat ).
thf(sy_c_List_Ocoset_001_Eo,type,
coset_o: list_o > set_o ).
thf(sy_c_List_Ocoset_001t__List__Olist_I_Eo_J,type,
coset_list_o: list_list_o > set_list_o ).
thf(sy_c_List_Ocoset_001t__NFA__Otransition,type,
coset_transition: list_transition > set_transition ).
thf(sy_c_List_Ocoset_001t__Nat__Onat,type,
coset_nat: list_nat > set_nat ).
thf(sy_c_List_Odistinct_001_Eo,type,
distinct_o: list_o > $o ).
thf(sy_c_List_Odistinct_001t__List__Olist_I_Eo_J,type,
distinct_list_o: list_list_o > $o ).
thf(sy_c_List_Odistinct_001t__List__Olist_It__List__Olist_I_Eo_J_J,type,
distinct_list_list_o: list_list_list_o > $o ).
thf(sy_c_List_Odistinct_001t__List__Olist_It__NFA__Otransition_J,type,
distin4894176225816993341sition: list_list_transition > $o ).
thf(sy_c_List_Odistinct_001t__List__Olist_It__Nat__Onat_J,type,
distinct_list_nat: list_list_nat > $o ).
thf(sy_c_List_Odistinct_001t__NFA__Otransition,type,
distinct_transition: list_transition > $o ).
thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
distinct_nat: list_nat > $o ).
thf(sy_c_List_Ofoldl_001t__Set__Oset_It__Nat__Onat_J_001t__List__Olist_I_Eo_J,type,
foldl_set_nat_list_o: ( set_nat > list_o > set_nat ) > set_nat > list_list_o > set_nat ).
thf(sy_c_List_Oinsert_001_Eo,type,
insert_o: $o > list_o > list_o ).
thf(sy_c_List_Oinsert_001t__List__Olist_I_Eo_J,type,
insert_list_o: list_o > list_list_o > list_list_o ).
thf(sy_c_List_Oinsert_001t__NFA__Otransition,type,
insert_transition: transition > list_transition > list_transition ).
thf(sy_c_List_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olast_001_Eo,type,
last_o: list_o > $o ).
thf(sy_c_List_Olast_001t__List__Olist_I_Eo_J,type,
last_list_o: list_list_o > list_o ).
thf(sy_c_List_Olast_001t__NFA__Otransition,type,
last_transition: list_transition > transition ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001_Eo,type,
linord3142498349692569832_set_o: set_o > list_o ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
linord2614967742042102400et_nat: set_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001_Eo,type,
cons_o: $o > list_o > list_o ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_I_Eo_J,type,
cons_list_o: list_o > list_list_o > list_list_o ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__List__Olist_I_Eo_J_J,type,
cons_list_list_o: list_list_o > list_list_list_o > list_list_list_o ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__NFA__Otransition_J,type,
cons_list_transition: list_transition > list_list_transition > list_list_transition ).
thf(sy_c_List_Olist_OCons_001t__NFA__Otransition,type,
cons_transition: transition > list_transition > list_transition ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_ONil_001_Eo,type,
nil_o: list_o ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_I_Eo_J,type,
nil_list_o: list_list_o ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__List__Olist_I_Eo_J_J,type,
nil_list_list_o: list_list_list_o ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__NFA__Otransition_J,type,
nil_list_transition: list_list_transition ).
thf(sy_c_List_Olist_ONil_001t__NFA__Otransition,type,
nil_transition: list_transition ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_Oset_001_Eo,type,
set_o2: list_o > set_o ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_I_Eo_J,type,
set_list_o2: list_list_o > set_list_o ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__List__Olist_I_Eo_J_J,type,
set_list_list_o2: list_list_list_o > set_list_list_o ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__NFA__Otransition_J,type,
set_list_transition2: list_list_transition > set_list_transition ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
set_list_nat2: list_list_nat > set_list_nat ).
thf(sy_c_List_Olist_Oset_001t__NFA__Otransition,type,
set_transition2: list_transition > set_transition ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist__ex1_001_Eo,type,
list_ex1_o: ( $o > $o ) > list_o > $o ).
thf(sy_c_List_Olist__ex1_001t__List__Olist_I_Eo_J,type,
list_ex1_list_o: ( list_o > $o ) > list_list_o > $o ).
thf(sy_c_List_Olist__ex1_001t__NFA__Otransition,type,
list_ex1_transition: ( transition > $o ) > list_transition > $o ).
thf(sy_c_List_Olist__ex1_001t__Nat__Onat,type,
list_ex1_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Omaps_001_Eo_001_Eo,type,
maps_o_o: ( $o > list_o ) > list_o > list_o ).
thf(sy_c_List_Omaps_001_Eo_001t__List__Olist_I_Eo_J,type,
maps_o_list_o: ( $o > list_list_o ) > list_o > list_list_o ).
thf(sy_c_List_Omaps_001_Eo_001t__NFA__Otransition,type,
maps_o_transition: ( $o > list_transition ) > list_o > list_transition ).
thf(sy_c_List_Omaps_001t__List__Olist_I_Eo_J_001_Eo,type,
maps_list_o_o: ( list_o > list_o ) > list_list_o > list_o ).
thf(sy_c_List_Omaps_001t__List__Olist_I_Eo_J_001t__List__Olist_I_Eo_J,type,
maps_list_o_list_o: ( list_o > list_list_o ) > list_list_o > list_list_o ).
thf(sy_c_List_Omaps_001t__List__Olist_I_Eo_J_001t__NFA__Otransition,type,
maps_l2767646238508560737sition: ( list_o > list_transition ) > list_list_o > list_transition ).
thf(sy_c_List_Omaps_001t__NFA__Otransition_001_Eo,type,
maps_transition_o: ( transition > list_o ) > list_transition > list_o ).
thf(sy_c_List_Omaps_001t__NFA__Otransition_001t__List__Olist_I_Eo_J,type,
maps_t5005503969775829665list_o: ( transition > list_list_o ) > list_transition > list_list_o ).
thf(sy_c_List_Omaps_001t__NFA__Otransition_001t__NFA__Otransition,type,
maps_t8463952688879884097sition: ( transition > list_transition ) > list_transition > list_transition ).
thf(sy_c_List_Omin__list_001_Eo,type,
min_list_o: list_o > $o ).
thf(sy_c_List_Onull_001_Eo,type,
null_o: list_o > $o ).
thf(sy_c_List_Onull_001t__List__Olist_I_Eo_J,type,
null_list_o: list_list_o > $o ).
thf(sy_c_List_Onull_001t__NFA__Otransition,type,
null_transition: list_transition > $o ).
thf(sy_c_List_Oord__class_Olexordp__eq_001_Eo,type,
ord_lexordp_eq_o: list_o > list_o > $o ).
thf(sy_c_List_Oproduct__lists_001_Eo,type,
product_lists_o: list_list_o > list_list_o ).
thf(sy_c_List_Oproduct__lists_001t__List__Olist_I_Eo_J,type,
product_lists_list_o: list_list_list_o > list_list_list_o ).
thf(sy_c_List_Oproduct__lists_001t__NFA__Otransition,type,
produc6248909823095439149sition: list_list_transition > list_list_transition ).
thf(sy_c_List_Oremove1_001_Eo,type,
remove1_o: $o > list_o > list_o ).
thf(sy_c_List_Oremove1_001t__List__Olist_I_Eo_J,type,
remove1_list_o: list_o > list_list_o > list_list_o ).
thf(sy_c_List_Oremove1_001t__NFA__Otransition,type,
remove1_transition: transition > list_transition > list_transition ).
thf(sy_c_List_Oremove1_001t__Nat__Onat,type,
remove1_nat: nat > list_nat > list_nat ).
thf(sy_c_List_OremoveAll_001_Eo,type,
removeAll_o: $o > list_o > list_o ).
thf(sy_c_List_OremoveAll_001t__List__Olist_I_Eo_J,type,
removeAll_list_o: list_o > list_list_o > list_list_o ).
thf(sy_c_List_OremoveAll_001t__List__Olist_It__List__Olist_I_Eo_J_J,type,
remove3821550480258065712list_o: list_list_o > list_list_list_o > list_list_list_o ).
thf(sy_c_List_OremoveAll_001t__List__Olist_It__NFA__Otransition_J,type,
remove2429998804908088272sition: list_transition > list_list_transition > list_list_transition ).
thf(sy_c_List_OremoveAll_001t__List__Olist_It__Nat__Onat_J,type,
removeAll_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_OremoveAll_001t__NFA__Otransition,type,
removeAll_transition: transition > list_transition > list_transition ).
thf(sy_c_List_OremoveAll_001t__Nat__Onat,type,
removeAll_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Osubseqs_001_Eo,type,
subseqs_o: list_o > list_list_o ).
thf(sy_c_List_Osubseqs_001t__List__Olist_I_Eo_J,type,
subseqs_list_o: list_list_o > list_list_list_o ).
thf(sy_c_List_Osubseqs_001t__NFA__Otransition,type,
subseqs_transition: list_transition > list_list_transition ).
thf(sy_c_List_Osubseqs_001t__Nat__Onat,type,
subseqs_nat: list_nat > list_list_nat ).
thf(sy_c_List_Otranspose_001_Eo,type,
transpose_o: list_list_o > list_list_o ).
thf(sy_c_List_Otranspose_001t__List__Olist_I_Eo_J,type,
transpose_list_o: list_list_list_o > list_list_list_o ).
thf(sy_c_List_Otranspose_001t__NFA__Otransition,type,
transpose_transition: list_list_transition > list_list_transition ).
thf(sy_c_List_Ounion_001t__NFA__Otransition,type,
union_transition: list_transition > list_transition > list_transition ).
thf(sy_c_List_Ounion_001t__Nat__Onat,type,
union_nat: list_nat > list_nat > list_nat ).
thf(sy_c_NFA_OQ,type,
q: nat > nat > list_transition > set_nat ).
thf(sy_c_NFA_OSQ,type,
sq: nat > list_transition > set_nat ).
thf(sy_c_NFA_Oaccept,type,
accept: nat > nat > list_transition > set_nat > $o ).
thf(sy_c_NFA_Oaccept__eps,type,
accept_eps: nat > nat > list_transition > set_nat > list_o > $o ).
thf(sy_c_NFA_Odelta,type,
delta: nat > list_transition > set_nat > list_o > set_nat ).
thf(sy_c_NFA_Ofmla__set,type,
fmla_set: transition > set_nat ).
thf(sy_c_NFA_Ofmla__set__rel,type,
fmla_set_rel: transition > transition > $o ).
thf(sy_c_NFA_Onfa,type,
nfa: nat > nat > list_transition > $o ).
thf(sy_c_NFA_Orun,type,
run: nat > list_transition > set_nat > list_list_o > set_nat ).
thf(sy_c_NFA_Orun__accept,type,
run_accept: nat > nat > list_transition > set_nat > list_list_o > $o ).
thf(sy_c_NFA_Orun__accept__eps,type,
run_accept_eps: nat > nat > list_transition > set_nat > list_list_o > list_o > $o ).
thf(sy_c_NFA_Ostate__set,type,
state_set: transition > set_nat ).
thf(sy_c_NFA_Ostate__set__rel,type,
state_set_rel: transition > transition > $o ).
thf(sy_c_NFA_Ostep__eps,type,
step_eps: nat > list_transition > list_o > nat > nat > $o ).
thf(sy_c_NFA_Ostep__eps__closure,type,
step_eps_closure: nat > list_transition > list_o > nat > nat > $o ).
thf(sy_c_NFA_Ostep__eps__closure__set,type,
step_eps_closure_set: nat > list_transition > set_nat > list_o > set_nat ).
thf(sy_c_NFA_Ostep__eps__set,type,
step_eps_set: nat > list_transition > list_o > set_nat > set_nat ).
thf(sy_c_NFA_Ostep__eps__sucs,type,
step_eps_sucs: nat > list_transition > list_o > nat > set_nat ).
thf(sy_c_NFA_Ostep__symb,type,
step_symb: nat > list_transition > nat > nat > $o ).
thf(sy_c_NFA_Ostep__symb__set,type,
step_symb_set: nat > list_transition > set_nat > set_nat ).
thf(sy_c_NFA_Ostep__symb__sucs,type,
step_symb_sucs: nat > list_transition > nat > set_nat ).
thf(sy_c_NFA_Otransition_Oeps__trans,type,
eps_trans: nat > nat > transition ).
thf(sy_c_NFA_Otransition_Osplit__trans,type,
split_trans: nat > nat > transition ).
thf(sy_c_NFA_Otransition_Osymb__trans,type,
symb_trans: nat > transition ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__NFA__Otransition_M_Eo_J,type,
bot_bot_transition_o: transition > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_Eo_J,type,
bot_bot_set_o: set_o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_I_Eo_J_J,type,
bot_bot_set_list_o: set_list_o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__NFA__Otransition_J,type,
bot_bo301567166201926666sition: set_transition ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
bot_bot_set_set_o: set_set_o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__List__Olist_I_Eo_J_J_J,type,
bot_bo64454365476827594list_o: set_set_list_o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__NFA__Otransition_J_J,type,
bot_bo1233527522848825322sition: set_set_transition ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Oord__class_Oless_001_Eo,type,
ord_less_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__NFA__Otransition_J,type,
ord_le5184432651266358346sition: set_transition > set_transition > $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__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
ord_less_eq_set_o: set_o > set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_I_Eo_J_J,type,
ord_le6901083488122529182list_o: set_list_o > set_list_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__NFA__Otransition_J,type,
ord_le8419162016481440574sition: set_transition > set_transition > $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__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
ord_min_nat: nat > nat > nat ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Set__Oset_It__NFA__Otransition_J,type,
ord_mi6397184166219407237sition: set_transition > set_transition > set_transition ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Set__Oset_It__Nat__Onat_J,type,
ord_min_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Set_OCollect_001t__NFA__Otransition,type,
collect_transition: ( transition > $o ) > set_transition ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OPow_001_Eo,type,
pow_o: set_o > set_set_o ).
thf(sy_c_Set_OPow_001t__List__Olist_I_Eo_J,type,
pow_list_o: set_list_o > set_set_list_o ).
thf(sy_c_Set_OPow_001t__NFA__Otransition,type,
pow_transition: set_transition > set_set_transition ).
thf(sy_c_Set_OPow_001t__Nat__Onat,type,
pow_nat: set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__NFA__Otransition_J_001t__Set__Oset_It__NFA__Otransition_J,type,
image_4748612756971788127sition: ( list_transition > set_transition ) > set_list_transition > set_set_transition ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_1775855109352712557et_nat: ( list_nat > set_nat ) > set_list_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__NFA__Otransition_001t__NFA__Otransition,type,
image_5857460390510121477sition: ( transition > transition ) > set_transition > set_transition ).
thf(sy_c_Set_Oimage_001t__NFA__Otransition_001t__Nat__Onat,type,
image_transition_nat: ( transition > nat ) > set_transition > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__NFA__Otransition,type,
image_nat_transition: ( nat > transition ) > set_nat > set_transition ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__NFA__Otransition_J_001t__Set__Oset_It__NFA__Otransition_J,type,
image_698392052263970309sition: ( set_transition > set_transition ) > set_set_transition > set_set_transition ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oinsert_001t__List__Olist_I_Eo_J,type,
insert_list_o2: list_o > set_list_o > set_list_o ).
thf(sy_c_Set_Oinsert_001t__NFA__Otransition,type,
insert_transition2: transition > set_transition > set_transition ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat2: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_I_Eo_J,type,
insert_set_o: set_o > set_set_o > set_set_o ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__List__Olist_I_Eo_J_J,type,
insert_set_list_o: set_list_o > set_set_list_o > set_set_list_o ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__NFA__Otransition_J,type,
insert8494249028948967790sition: set_transition > set_set_transition > set_set_transition ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
insert_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set_Ois__empty_001t__NFA__Otransition,type,
is_empty_transition: set_transition > $o ).
thf(sy_c_Set_Ois__empty_001t__Nat__Onat,type,
is_empty_nat: set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__NFA__Otransition,type,
is_sin1641930644073461938sition: set_transition > $o ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Opairwise_001t__NFA__Otransition,type,
pairwise_transition: ( transition > transition > $o ) > set_transition > $o ).
thf(sy_c_Set_Opairwise_001t__Nat__Onat,type,
pairwise_nat: ( nat > nat > $o ) > set_nat > $o ).
thf(sy_c_Set_Oremove_001t__NFA__Otransition,type,
remove_transition: transition > set_transition > set_transition ).
thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
remove_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Othe__elem_001_Eo,type,
the_elem_o: set_o > $o ).
thf(sy_c_Set_Othe__elem_001t__List__Olist_I_Eo_J,type,
the_elem_list_o: set_list_o > list_o ).
thf(sy_c_Set_Othe__elem_001t__NFA__Otransition,type,
the_elem_transition: set_transition > transition ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Wellfounded_Oaccp_001t__NFA__Otransition,type,
accp_transition: ( transition > transition > $o ) > transition > $o ).
thf(sy_c_Zorn_Ochains_001t__Nat__Onat,type,
chains_nat: set_set_nat > set_set_set_nat ).
thf(sy_c_member_001t__List__Olist_I_Eo_J,type,
member_list_o: list_o > set_list_o > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_I_Eo_J_J,type,
member_list_list_o: list_list_o > set_list_list_o > $o ).
thf(sy_c_member_001t__List__Olist_It__NFA__Otransition_J,type,
member1473516902542837997sition: list_transition > set_list_transition > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__NFA__Otransition,type,
member_transition: transition > set_transition > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__NFA__Otransition_J,type,
member7318969637299765063sition: set_transition > set_set_transition > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
member_set_set_nat: set_set_nat > set_set_set_nat > $o ).
thf(sy_v_q0,type,
q0: nat ).
thf(sy_v_qf,type,
qf: nat ).
thf(sy_v_transs,type,
transs: list_transition ).
% Relevant facts (1276)
thf(fact_0_nfa__axioms,axiom,
nfa @ q0 @ qf @ transs ).
% nfa_axioms
thf(fact_1_step__symb__set__empty,axiom,
! [Q0: nat,Transs: list_transition] :
( ( step_symb_set @ Q0 @ Transs @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% step_symb_set_empty
thf(fact_2_qf__not__in__SQ,axiom,
~ ( member_nat @ qf @ ( sq @ q0 @ transs ) ) ).
% qf_not_in_SQ
thf(fact_3_singletonI,axiom,
! [A: transition] : ( member_transition @ A @ ( insert_transition2 @ A @ bot_bo301567166201926666sition ) ) ).
% singletonI
thf(fact_4_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_5_step__symb__closed,axiom,
! [Q: nat,Q2: nat] :
( ( step_symb @ q0 @ transs @ Q @ Q2 )
=> ( member_nat @ Q2 @ ( q @ q0 @ qf @ transs ) ) ) ).
% step_symb_closed
thf(fact_6_insertCI,axiom,
! [A: nat,B: set_nat,B2: nat] :
( ( ~ ( member_nat @ A @ B )
=> ( A = B2 ) )
=> ( member_nat @ A @ ( insert_nat2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_7_insertCI,axiom,
! [A: transition,B: set_transition,B2: transition] :
( ( ~ ( member_transition @ A @ B )
=> ( A = B2 ) )
=> ( member_transition @ A @ ( insert_transition2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_8_insert__iff,axiom,
! [A: nat,B2: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat2 @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_9_insert__iff,axiom,
! [A: transition,B2: transition,A2: set_transition] :
( ( member_transition @ A @ ( insert_transition2 @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_transition @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_10_insert__absorb2,axiom,
! [X: transition,A2: set_transition] :
( ( insert_transition2 @ X @ ( insert_transition2 @ X @ A2 ) )
= ( insert_transition2 @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_11_insert__absorb2,axiom,
! [X: nat,A2: set_nat] :
( ( insert_nat2 @ X @ ( insert_nat2 @ X @ A2 ) )
= ( insert_nat2 @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_12_empty__iff,axiom,
! [C: transition] :
~ ( member_transition @ C @ bot_bo301567166201926666sition ) ).
% empty_iff
thf(fact_13_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_14_all__not__in__conv,axiom,
! [A2: set_transition] :
( ( ! [X2: transition] :
~ ( member_transition @ X2 @ A2 ) )
= ( A2 = bot_bo301567166201926666sition ) ) ).
% all_not_in_conv
thf(fact_15_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat @ X2 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_16_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_17_Collect__empty__eq,axiom,
! [P: transition > $o] :
( ( ( collect_transition @ P )
= bot_bo301567166201926666sition )
= ( ! [X2: transition] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_18_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_19_empty__Collect__eq,axiom,
! [P: transition > $o] :
( ( bot_bo301567166201926666sition
= ( collect_transition @ P ) )
= ( ! [X2: transition] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_20_step__eps__closure__closed,axiom,
! [Bs: list_o,Q: nat,Q2: nat] :
( ( step_eps_closure @ q0 @ transs @ Bs @ Q @ Q2 )
=> ( ( Q != Q2 )
=> ( member_nat @ Q2 @ ( q @ q0 @ qf @ transs ) ) ) ) ).
% step_eps_closure_closed
thf(fact_21_nfa_Oqf__not__in__SQ,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ~ ( member_nat @ Qf @ ( sq @ Q0 @ Transs ) ) ) ).
% nfa.qf_not_in_SQ
thf(fact_22_nfa_Ostep__symb__closed,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,Q: nat,Q2: nat] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ( ( step_symb @ Q0 @ Transs @ Q @ Q2 )
=> ( member_nat @ Q2 @ ( q @ Q0 @ Qf @ Transs ) ) ) ) ).
% nfa.step_symb_closed
thf(fact_23_nfa_Ostep__eps__closure__closed,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,Bs: list_o,Q: nat,Q2: nat] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ( ( step_eps_closure @ Q0 @ Transs @ Bs @ Q @ Q2 )
=> ( ( Q != Q2 )
=> ( member_nat @ Q2 @ ( q @ Q0 @ Qf @ Transs ) ) ) ) ) ).
% nfa.step_eps_closure_closed
thf(fact_24_step__symb__dest,axiom,
! [Q0: nat,Transs: list_transition,Q: nat,Q2: nat] :
( ( step_symb @ Q0 @ Transs @ Q @ Q2 )
=> ( member_nat @ Q @ ( sq @ Q0 @ Transs ) ) ) ).
% step_symb_dest
thf(fact_25_step__eps__closure__dest,axiom,
! [Q0: nat,Transs: list_transition,Bs: list_o,Q: nat,Q2: nat] :
( ( step_eps_closure @ Q0 @ Transs @ Bs @ Q @ Q2 )
=> ( ( Q != Q2 )
=> ( member_nat @ Q @ ( sq @ Q0 @ Transs ) ) ) ) ).
% step_eps_closure_dest
thf(fact_26_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_27_bot__set__def,axiom,
( bot_bo301567166201926666sition
= ( collect_transition @ bot_bot_transition_o ) ) ).
% bot_set_def
thf(fact_28_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X2: nat] : ( member_nat @ X2 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_29_ex__in__conv,axiom,
! [A2: set_transition] :
( ( ? [X2: transition] : ( member_transition @ X2 @ A2 ) )
= ( A2 != bot_bo301567166201926666sition ) ) ).
% ex_in_conv
thf(fact_30_equals0I,axiom,
! [A2: set_nat] :
( ! [Y: nat] :
~ ( member_nat @ Y @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_31_equals0I,axiom,
! [A2: set_transition] :
( ! [Y: transition] :
~ ( member_transition @ Y @ A2 )
=> ( A2 = bot_bo301567166201926666sition ) ) ).
% equals0I
thf(fact_32_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_33_equals0D,axiom,
! [A2: set_transition,A: transition] :
( ( A2 = bot_bo301567166201926666sition )
=> ~ ( member_transition @ A @ A2 ) ) ).
% equals0D
thf(fact_34_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_35_emptyE,axiom,
! [A: transition] :
~ ( member_transition @ A @ bot_bo301567166201926666sition ) ).
% emptyE
thf(fact_36_mk__disjoint__insert,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ? [B3: set_nat] :
( ( A2
= ( insert_nat2 @ A @ B3 ) )
& ~ ( member_nat @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_37_mk__disjoint__insert,axiom,
! [A: transition,A2: set_transition] :
( ( member_transition @ A @ A2 )
=> ? [B3: set_transition] :
( ( A2
= ( insert_transition2 @ A @ B3 ) )
& ~ ( member_transition @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_38_insert__commute,axiom,
! [X: nat,Y2: nat,A2: set_nat] :
( ( insert_nat2 @ X @ ( insert_nat2 @ Y2 @ A2 ) )
= ( insert_nat2 @ Y2 @ ( insert_nat2 @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_39_insert__commute,axiom,
! [X: transition,Y2: transition,A2: set_transition] :
( ( insert_transition2 @ X @ ( insert_transition2 @ Y2 @ A2 ) )
= ( insert_transition2 @ Y2 @ ( insert_transition2 @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_40_insert__eq__iff,axiom,
! [A: nat,A2: set_nat,B2: nat,B: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ B2 @ B )
=> ( ( ( insert_nat2 @ A @ A2 )
= ( insert_nat2 @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C2: set_nat] :
( ( A2
= ( insert_nat2 @ B2 @ C2 ) )
& ~ ( member_nat @ B2 @ C2 )
& ( B
= ( insert_nat2 @ A @ C2 ) )
& ~ ( member_nat @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_41_insert__eq__iff,axiom,
! [A: transition,A2: set_transition,B2: transition,B: set_transition] :
( ~ ( member_transition @ A @ A2 )
=> ( ~ ( member_transition @ B2 @ B )
=> ( ( ( insert_transition2 @ A @ A2 )
= ( insert_transition2 @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C2: set_transition] :
( ( A2
= ( insert_transition2 @ B2 @ C2 ) )
& ~ ( member_transition @ B2 @ C2 )
& ( B
= ( insert_transition2 @ A @ C2 ) )
& ~ ( member_transition @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_42_insert__absorb,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat2 @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_43_insert__absorb,axiom,
! [A: transition,A2: set_transition] :
( ( member_transition @ A @ A2 )
=> ( ( insert_transition2 @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_44_insert__ident,axiom,
! [X: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ~ ( member_nat @ X @ B )
=> ( ( ( insert_nat2 @ X @ A2 )
= ( insert_nat2 @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_45_insert__ident,axiom,
! [X: transition,A2: set_transition,B: set_transition] :
( ~ ( member_transition @ X @ A2 )
=> ( ~ ( member_transition @ X @ B )
=> ( ( ( insert_transition2 @ X @ A2 )
= ( insert_transition2 @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_46_Set_Oset__insert,axiom,
! [X: nat,A2: set_nat] :
( ( member_nat @ X @ A2 )
=> ~ ! [B3: set_nat] :
( ( A2
= ( insert_nat2 @ X @ B3 ) )
=> ( member_nat @ X @ B3 ) ) ) ).
% Set.set_insert
thf(fact_47_Set_Oset__insert,axiom,
! [X: transition,A2: set_transition] :
( ( member_transition @ X @ A2 )
=> ~ ! [B3: set_transition] :
( ( A2
= ( insert_transition2 @ X @ B3 ) )
=> ( member_transition @ X @ B3 ) ) ) ).
% Set.set_insert
thf(fact_48_insertI2,axiom,
! [A: nat,B: set_nat,B2: nat] :
( ( member_nat @ A @ B )
=> ( member_nat @ A @ ( insert_nat2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_49_insertI2,axiom,
! [A: transition,B: set_transition,B2: transition] :
( ( member_transition @ A @ B )
=> ( member_transition @ A @ ( insert_transition2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_50_insertI1,axiom,
! [A: nat,B: set_nat] : ( member_nat @ A @ ( insert_nat2 @ A @ B ) ) ).
% insertI1
thf(fact_51_insertI1,axiom,
! [A: transition,B: set_transition] : ( member_transition @ A @ ( insert_transition2 @ A @ B ) ) ).
% insertI1
thf(fact_52_insertE,axiom,
! [A: nat,B2: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat2 @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_53_insertE,axiom,
! [A: transition,B2: transition,A2: set_transition] :
( ( member_transition @ A @ ( insert_transition2 @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_transition @ A @ A2 ) ) ) ).
% insertE
thf(fact_54_singleton__inject,axiom,
! [A: nat,B2: nat] :
( ( ( insert_nat2 @ A @ bot_bot_set_nat )
= ( insert_nat2 @ B2 @ bot_bot_set_nat ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_55_singleton__inject,axiom,
! [A: transition,B2: transition] :
( ( ( insert_transition2 @ A @ bot_bo301567166201926666sition )
= ( insert_transition2 @ B2 @ bot_bo301567166201926666sition ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_56_insert__not__empty,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat2 @ A @ A2 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_57_insert__not__empty,axiom,
! [A: transition,A2: set_transition] :
( ( insert_transition2 @ A @ A2 )
!= bot_bo301567166201926666sition ) ).
% insert_not_empty
thf(fact_58_doubleton__eq__iff,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ( insert_nat2 @ A @ ( insert_nat2 @ B2 @ bot_bot_set_nat ) )
= ( insert_nat2 @ C @ ( insert_nat2 @ D @ bot_bot_set_nat ) ) )
= ( ( ( A = C )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_59_doubleton__eq__iff,axiom,
! [A: transition,B2: transition,C: transition,D: transition] :
( ( ( insert_transition2 @ A @ ( insert_transition2 @ B2 @ bot_bo301567166201926666sition ) )
= ( insert_transition2 @ C @ ( insert_transition2 @ D @ bot_bo301567166201926666sition ) ) )
= ( ( ( A = C )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_60_singleton__iff,axiom,
! [B2: nat,A: nat] :
( ( member_nat @ B2 @ ( insert_nat2 @ A @ bot_bot_set_nat ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_61_singleton__iff,axiom,
! [B2: transition,A: transition] :
( ( member_transition @ B2 @ ( insert_transition2 @ A @ bot_bo301567166201926666sition ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_62_singletonD,axiom,
! [B2: nat,A: nat] :
( ( member_nat @ B2 @ ( insert_nat2 @ A @ bot_bot_set_nat ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_63_singletonD,axiom,
! [B2: transition,A: transition] :
( ( member_transition @ B2 @ ( insert_transition2 @ A @ bot_bo301567166201926666sition ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_64_step__eps__closed,axiom,
! [Bs: list_o,Q: nat,Q2: nat] :
( ( step_eps @ q0 @ transs @ Bs @ Q @ Q2 )
=> ( member_nat @ Q2 @ ( q @ q0 @ qf @ transs ) ) ) ).
% step_eps_closed
thf(fact_65_Q__diff__qf__SQ,axiom,
( ( minus_minus_set_nat @ ( q @ q0 @ qf @ transs ) @ ( insert_nat2 @ qf @ bot_bot_set_nat ) )
= ( sq @ q0 @ transs ) ) ).
% Q_diff_qf_SQ
thf(fact_66_q0__sub__Q,axiom,
ord_less_eq_set_nat @ ( insert_nat2 @ q0 @ bot_bot_set_nat ) @ ( q @ q0 @ qf @ transs ) ).
% q0_sub_Q
thf(fact_67_step__symb__set__closed,axiom,
! [R: set_nat] : ( ord_less_eq_set_nat @ ( step_symb_set @ q0 @ transs @ R ) @ ( q @ q0 @ qf @ transs ) ) ).
% step_symb_set_closed
thf(fact_68_q0__sub__SQ,axiom,
ord_less_eq_set_nat @ ( insert_nat2 @ q0 @ bot_bot_set_nat ) @ ( sq @ q0 @ transs ) ).
% q0_sub_SQ
thf(fact_69_the__elem__eq,axiom,
! [X: nat] :
( ( the_elem_nat @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= X ) ).
% the_elem_eq
thf(fact_70_the__elem__eq,axiom,
! [X: transition] :
( ( the_elem_transition @ ( insert_transition2 @ X @ bot_bo301567166201926666sition ) )
= X ) ).
% the_elem_eq
thf(fact_71_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_72_mem__Collect__eq,axiom,
! [A: transition,P: transition > $o] :
( ( member_transition @ A @ ( collect_transition @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_73_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_74_Collect__mem__eq,axiom,
! [A2: set_transition] :
( ( collect_transition
@ ^ [X2: transition] : ( member_transition @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_75_step__eps__set__closed,axiom,
! [Bs: list_o,R: set_nat] : ( ord_less_eq_set_nat @ ( step_eps_set @ q0 @ transs @ Bs @ R ) @ ( q @ q0 @ qf @ transs ) ) ).
% step_eps_set_closed
thf(fact_76_is__singletonI,axiom,
! [X: nat] : ( is_singleton_nat @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_77_is__singletonI,axiom,
! [X: transition] : ( is_sin1641930644073461938sition @ ( insert_transition2 @ X @ bot_bo301567166201926666sition ) ) ).
% is_singletonI
thf(fact_78_nfa_OQ__diff__qf__SQ,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ( ( minus_minus_set_nat @ ( q @ Q0 @ Qf @ Transs ) @ ( insert_nat2 @ Qf @ bot_bot_set_nat ) )
= ( sq @ Q0 @ Transs ) ) ) ).
% nfa.Q_diff_qf_SQ
thf(fact_79_transs__not__Nil,axiom,
transs != nil_transition ).
% transs_not_Nil
thf(fact_80_step__eps__closure__set__closed,axiom,
! [R: set_nat,Bs: list_o] :
( ( ord_less_eq_set_nat @ R @ ( q @ q0 @ qf @ transs ) )
=> ( ord_less_eq_set_nat @ ( step_eps_closure_set @ q0 @ transs @ R @ Bs ) @ ( q @ q0 @ qf @ transs ) ) ) ).
% step_eps_closure_set_closed
thf(fact_81_nfa_Oq0__sub__SQ,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ( ord_less_eq_set_nat @ ( insert_nat2 @ Q0 @ bot_bot_set_nat ) @ ( sq @ Q0 @ Transs ) ) ) ).
% nfa.q0_sub_SQ
thf(fact_82_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_83_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_84_subsetI,axiom,
! [A2: set_transition,B: set_transition] :
( ! [X3: transition] :
( ( member_transition @ X3 @ A2 )
=> ( member_transition @ X3 @ B ) )
=> ( ord_le8419162016481440574sition @ A2 @ B ) ) ).
% subsetI
thf(fact_85_subsetI,axiom,
! [A2: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ X3 @ B ) )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_86_subset__antisym,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_87_DiffI,axiom,
! [C: transition,A2: set_transition,B: set_transition] :
( ( member_transition @ C @ A2 )
=> ( ~ ( member_transition @ C @ B )
=> ( member_transition @ C @ ( minus_8944320859760356485sition @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_88_DiffI,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_89_Diff__iff,axiom,
! [C: transition,A2: set_transition,B: set_transition] :
( ( member_transition @ C @ ( minus_8944320859760356485sition @ A2 @ B ) )
= ( ( member_transition @ C @ A2 )
& ~ ( member_transition @ C @ B ) ) ) ).
% Diff_iff
thf(fact_90_Diff__iff,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B ) )
= ( ( member_nat @ C @ A2 )
& ~ ( member_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_91_Diff__idemp,axiom,
! [A2: set_nat,B: set_nat] :
( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ B )
= ( minus_minus_set_nat @ A2 @ B ) ) ).
% Diff_idemp
thf(fact_92_subset__empty,axiom,
! [A2: set_transition] :
( ( ord_le8419162016481440574sition @ A2 @ bot_bo301567166201926666sition )
= ( A2 = bot_bo301567166201926666sition ) ) ).
% subset_empty
thf(fact_93_subset__empty,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_94_empty__subsetI,axiom,
! [A2: set_transition] : ( ord_le8419162016481440574sition @ bot_bo301567166201926666sition @ A2 ) ).
% empty_subsetI
thf(fact_95_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_96_insert__subset,axiom,
! [X: transition,A2: set_transition,B: set_transition] :
( ( ord_le8419162016481440574sition @ ( insert_transition2 @ X @ A2 ) @ B )
= ( ( member_transition @ X @ B )
& ( ord_le8419162016481440574sition @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_97_insert__subset,axiom,
! [X: nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat2 @ X @ A2 ) @ B )
= ( ( member_nat @ X @ B )
& ( ord_less_eq_set_nat @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_98_Diff__empty,axiom,
! [A2: set_transition] :
( ( minus_8944320859760356485sition @ A2 @ bot_bo301567166201926666sition )
= A2 ) ).
% Diff_empty
thf(fact_99_Diff__empty,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Diff_empty
thf(fact_100_empty__Diff,axiom,
! [A2: set_transition] :
( ( minus_8944320859760356485sition @ bot_bo301567166201926666sition @ A2 )
= bot_bo301567166201926666sition ) ).
% empty_Diff
thf(fact_101_empty__Diff,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_102_Diff__cancel,axiom,
! [A2: set_transition] :
( ( minus_8944320859760356485sition @ A2 @ A2 )
= bot_bo301567166201926666sition ) ).
% Diff_cancel
thf(fact_103_Diff__cancel,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ A2 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_104_Diff__insert0,axiom,
! [X: transition,A2: set_transition,B: set_transition] :
( ~ ( member_transition @ X @ A2 )
=> ( ( minus_8944320859760356485sition @ A2 @ ( insert_transition2 @ X @ B ) )
= ( minus_8944320859760356485sition @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_105_Diff__insert0,axiom,
! [X: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ B ) )
= ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_106_insert__Diff1,axiom,
! [X: transition,B: set_transition,A2: set_transition] :
( ( member_transition @ X @ B )
=> ( ( minus_8944320859760356485sition @ ( insert_transition2 @ X @ A2 ) @ B )
= ( minus_8944320859760356485sition @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_107_insert__Diff1,axiom,
! [X: nat,B: set_nat,A2: set_nat] :
( ( member_nat @ X @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A2 ) @ B )
= ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_108_singleton__insert__inj__eq,axiom,
! [B2: transition,A: transition,A2: set_transition] :
( ( ( insert_transition2 @ B2 @ bot_bo301567166201926666sition )
= ( insert_transition2 @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_le8419162016481440574sition @ A2 @ ( insert_transition2 @ B2 @ bot_bo301567166201926666sition ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_109_singleton__insert__inj__eq,axiom,
! [B2: nat,A: nat,A2: set_nat] :
( ( ( insert_nat2 @ B2 @ bot_bot_set_nat )
= ( insert_nat2 @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ B2 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_110_singleton__insert__inj__eq_H,axiom,
! [A: transition,A2: set_transition,B2: transition] :
( ( ( insert_transition2 @ A @ A2 )
= ( insert_transition2 @ B2 @ bot_bo301567166201926666sition ) )
= ( ( A = B2 )
& ( ord_le8419162016481440574sition @ A2 @ ( insert_transition2 @ B2 @ bot_bo301567166201926666sition ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_111_singleton__insert__inj__eq_H,axiom,
! [A: nat,A2: set_nat,B2: nat] :
( ( ( insert_nat2 @ A @ A2 )
= ( insert_nat2 @ B2 @ bot_bot_set_nat ) )
= ( ( A = B2 )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ B2 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_112_Diff__eq__empty__iff,axiom,
! [A2: set_transition,B: set_transition] :
( ( ( minus_8944320859760356485sition @ A2 @ B )
= bot_bo301567166201926666sition )
= ( ord_le8419162016481440574sition @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_113_Diff__eq__empty__iff,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( minus_minus_set_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_114_insert__Diff__single,axiom,
! [A: transition,A2: set_transition] :
( ( insert_transition2 @ A @ ( minus_8944320859760356485sition @ A2 @ ( insert_transition2 @ A @ bot_bo301567166201926666sition ) ) )
= ( insert_transition2 @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_115_insert__Diff__single,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat2 @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) )
= ( insert_nat2 @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_116_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [X2: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y4 )
& ( ord_less_eq_set_nat @ Y4 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_117_ord__eq__le__trans,axiom,
! [A: set_nat,B2: set_nat,C: set_nat] :
( ( A = B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_118_ord__le__eq__trans,axiom,
! [A: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_119_order__antisym,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_120_order_Otrans,axiom,
! [A: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_121_order__trans,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ Z2 )
=> ( ord_less_eq_set_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_122_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [A3: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A3 )
& ( ord_less_eq_set_nat @ A3 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_123_dual__order_Oantisym,axiom,
! [B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_124_dual__order_Otrans,axiom,
! [B2: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( ord_less_eq_set_nat @ C @ B2 )
=> ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_125_DiffE,axiom,
! [C: transition,A2: set_transition,B: set_transition] :
( ( member_transition @ C @ ( minus_8944320859760356485sition @ A2 @ B ) )
=> ~ ( ( member_transition @ C @ A2 )
=> ( member_transition @ C @ B ) ) ) ).
% DiffE
thf(fact_126_DiffE,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_127_DiffD1,axiom,
! [C: transition,A2: set_transition,B: set_transition] :
( ( member_transition @ C @ ( minus_8944320859760356485sition @ A2 @ B ) )
=> ( member_transition @ C @ A2 ) ) ).
% DiffD1
thf(fact_128_DiffD1,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B ) )
=> ( member_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_129_DiffD2,axiom,
! [C: transition,A2: set_transition,B: set_transition] :
( ( member_transition @ C @ ( minus_8944320859760356485sition @ A2 @ B ) )
=> ~ ( member_transition @ C @ B ) ) ).
% DiffD2
thf(fact_130_DiffD2,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B ) )
=> ~ ( member_nat @ C @ B ) ) ).
% DiffD2
thf(fact_131_in__mono,axiom,
! [A2: set_transition,B: set_transition,X: transition] :
( ( ord_le8419162016481440574sition @ A2 @ B )
=> ( ( member_transition @ X @ A2 )
=> ( member_transition @ X @ B ) ) ) ).
% in_mono
thf(fact_132_in__mono,axiom,
! [A2: set_nat,B: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_133_subsetD,axiom,
! [A2: set_transition,B: set_transition,C: transition] :
( ( ord_le8419162016481440574sition @ A2 @ B )
=> ( ( member_transition @ C @ A2 )
=> ( member_transition @ C @ B ) ) ) ).
% subsetD
thf(fact_134_subsetD,axiom,
! [A2: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_135_Diff__mono,axiom,
! [A2: set_nat,C3: set_nat,D2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C3 )
=> ( ( ord_less_eq_set_nat @ D2 @ B )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ ( minus_minus_set_nat @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_136_equalityE,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 = B )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A2 ) ) ) ).
% equalityE
thf(fact_137_subset__eq,axiom,
( ord_le8419162016481440574sition
= ( ^ [A4: set_transition,B5: set_transition] :
! [X2: transition] :
( ( member_transition @ X2 @ A4 )
=> ( member_transition @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_138_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A4 )
=> ( member_nat @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_139_equalityD1,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 = B )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% equalityD1
thf(fact_140_equalityD2,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 = B )
=> ( ord_less_eq_set_nat @ B @ A2 ) ) ).
% equalityD2
thf(fact_141_subset__iff,axiom,
( ord_le8419162016481440574sition
= ( ^ [A4: set_transition,B5: set_transition] :
! [T: transition] :
( ( member_transition @ T @ A4 )
=> ( member_transition @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_142_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A4 )
=> ( member_nat @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_143_Diff__subset,axiom,
! [A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_144_double__diff,axiom,
! [A2: set_nat,B: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C3 )
=> ( ( minus_minus_set_nat @ B @ ( minus_minus_set_nat @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_145_subset__refl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_146_Collect__mono,axiom,
! [P: nat > $o,Q3: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q3 @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q3 ) ) ) ).
% Collect_mono
thf(fact_147_subset__trans,axiom,
! [A2: set_nat,B: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C3 )
=> ( ord_less_eq_set_nat @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_148_antisym,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_149_set__eq__subset,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_150_Collect__mono__iff,axiom,
! [P: nat > $o,Q3: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q3 ) )
= ( ! [X2: nat] :
( ( P @ X2 )
=> ( Q3 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_151_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [A3: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_152_order__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_153_order__subst2,axiom,
! [A: set_nat,B2: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_154_order__eq__refl,axiom,
! [X: set_nat,Y2: set_nat] :
( ( X = Y2 )
=> ( ord_less_eq_set_nat @ X @ Y2 ) ) ).
% order_eq_refl
thf(fact_155_ord__eq__le__subst,axiom,
! [A: set_nat,F: set_nat > set_nat,B2: set_nat,C: set_nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_156_ord__le__eq__subst,axiom,
! [A: set_nat,B2: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_157_order__antisym__conv,axiom,
! [Y2: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X )
=> ( ( ord_less_eq_set_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_158_step__eps__closure__set__step__id,axiom,
! [R: set_nat,Q0: nat,Transs: list_transition,Bs: list_o] :
( ! [Q4: nat,Q5: nat] :
( ( member_nat @ Q4 @ R )
=> ~ ( step_eps @ Q0 @ Transs @ Bs @ Q4 @ Q5 ) )
=> ( ( step_eps_closure_set @ Q0 @ Transs @ R @ Bs )
= R ) ) ).
% step_eps_closure_set_step_id
thf(fact_159_step__eps__closure__set__refl,axiom,
! [R: set_nat,Q0: nat,Transs: list_transition,Bs: list_o] : ( ord_less_eq_set_nat @ R @ ( step_eps_closure_set @ Q0 @ Transs @ R @ Bs ) ) ).
% step_eps_closure_set_refl
thf(fact_160_step__eps__closure__set__mono,axiom,
! [R: set_nat,S: set_nat,Q0: nat,Transs: list_transition,Bs: list_o] :
( ( ord_less_eq_set_nat @ R @ S )
=> ( ord_less_eq_set_nat @ ( step_eps_closure_set @ Q0 @ Transs @ R @ Bs ) @ ( step_eps_closure_set @ Q0 @ Transs @ S @ Bs ) ) ) ).
% step_eps_closure_set_mono
thf(fact_161_step__eps__closure__set__idem,axiom,
! [Q0: nat,Transs: list_transition,R: set_nat,Bs: list_o] :
( ( step_eps_closure_set @ Q0 @ Transs @ ( step_eps_closure_set @ Q0 @ Transs @ R @ Bs ) @ Bs )
= ( step_eps_closure_set @ Q0 @ Transs @ R @ Bs ) ) ).
% step_eps_closure_set_idem
thf(fact_162_step__step__eps__closure,axiom,
! [Q0: nat,Transs: list_transition,Bs: list_o,Q: nat,Q2: nat,R: set_nat] :
( ( step_eps @ Q0 @ Transs @ Bs @ Q @ Q2 )
=> ( ( member_nat @ Q @ R )
=> ( member_nat @ Q2 @ ( step_eps_closure_set @ Q0 @ Transs @ R @ Bs ) ) ) ) ).
% step_step_eps_closure
thf(fact_163_step__eps__set__mono,axiom,
! [R: set_nat,S: set_nat,Q0: nat,Transs: list_transition,Bs: list_o] :
( ( ord_less_eq_set_nat @ R @ S )
=> ( ord_less_eq_set_nat @ ( step_eps_set @ Q0 @ Transs @ Bs @ R ) @ ( step_eps_set @ Q0 @ Transs @ Bs @ S ) ) ) ).
% step_eps_set_mono
thf(fact_164_subset__Diff__insert,axiom,
! [A2: set_transition,B: set_transition,X: transition,C3: set_transition] :
( ( ord_le8419162016481440574sition @ A2 @ ( minus_8944320859760356485sition @ B @ ( insert_transition2 @ X @ C3 ) ) )
= ( ( ord_le8419162016481440574sition @ A2 @ ( minus_8944320859760356485sition @ B @ C3 ) )
& ~ ( member_transition @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_165_subset__Diff__insert,axiom,
! [A2: set_nat,B: set_nat,X: nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B @ ( insert_nat2 @ X @ C3 ) ) )
= ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B @ C3 ) )
& ~ ( member_nat @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_166_subset__insert__iff,axiom,
! [A2: set_transition,X: transition,B: set_transition] :
( ( ord_le8419162016481440574sition @ A2 @ ( insert_transition2 @ X @ B ) )
= ( ( ( member_transition @ X @ A2 )
=> ( ord_le8419162016481440574sition @ ( minus_8944320859760356485sition @ A2 @ ( insert_transition2 @ X @ bot_bo301567166201926666sition ) ) @ B ) )
& ( ~ ( member_transition @ X @ A2 )
=> ( ord_le8419162016481440574sition @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_167_subset__insert__iff,axiom,
! [A2: set_nat,X: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ X @ B ) )
= ( ( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ B ) )
& ( ~ ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_168_Diff__single__insert,axiom,
! [A2: set_transition,X: transition,B: set_transition] :
( ( ord_le8419162016481440574sition @ ( minus_8944320859760356485sition @ A2 @ ( insert_transition2 @ X @ bot_bo301567166201926666sition ) ) @ B )
=> ( ord_le8419162016481440574sition @ A2 @ ( insert_transition2 @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_169_Diff__single__insert,axiom,
! [A2: set_nat,X: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ B )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ X @ B ) ) ) ).
% Diff_single_insert
thf(fact_170_insert__Diff__if,axiom,
! [X: transition,B: set_transition,A2: set_transition] :
( ( ( member_transition @ X @ B )
=> ( ( minus_8944320859760356485sition @ ( insert_transition2 @ X @ A2 ) @ B )
= ( minus_8944320859760356485sition @ A2 @ B ) ) )
& ( ~ ( member_transition @ X @ B )
=> ( ( minus_8944320859760356485sition @ ( insert_transition2 @ X @ A2 ) @ B )
= ( insert_transition2 @ X @ ( minus_8944320859760356485sition @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_171_insert__Diff__if,axiom,
! [X: nat,B: set_nat,A2: set_nat] :
( ( ( member_nat @ X @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A2 ) @ B )
= ( minus_minus_set_nat @ A2 @ B ) ) )
& ( ~ ( member_nat @ X @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A2 ) @ B )
= ( insert_nat2 @ X @ ( minus_minus_set_nat @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_172_bot_Oextremum,axiom,
! [A: set_transition] : ( ord_le8419162016481440574sition @ bot_bo301567166201926666sition @ A ) ).
% bot.extremum
thf(fact_173_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_174_bot_Oextremum__unique,axiom,
! [A: set_transition] :
( ( ord_le8419162016481440574sition @ A @ bot_bo301567166201926666sition )
= ( A = bot_bo301567166201926666sition ) ) ).
% bot.extremum_unique
thf(fact_175_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_176_bot_Oextremum__uniqueI,axiom,
! [A: set_transition] :
( ( ord_le8419162016481440574sition @ A @ bot_bo301567166201926666sition )
=> ( A = bot_bo301567166201926666sition ) ) ).
% bot.extremum_uniqueI
thf(fact_177_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_178_insert__mono,axiom,
! [C3: set_transition,D2: set_transition,A: transition] :
( ( ord_le8419162016481440574sition @ C3 @ D2 )
=> ( ord_le8419162016481440574sition @ ( insert_transition2 @ A @ C3 ) @ ( insert_transition2 @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_179_insert__mono,axiom,
! [C3: set_nat,D2: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ C3 @ D2 )
=> ( ord_less_eq_set_nat @ ( insert_nat2 @ A @ C3 ) @ ( insert_nat2 @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_180_subset__insert,axiom,
! [X: transition,A2: set_transition,B: set_transition] :
( ~ ( member_transition @ X @ A2 )
=> ( ( ord_le8419162016481440574sition @ A2 @ ( insert_transition2 @ X @ B ) )
= ( ord_le8419162016481440574sition @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_181_subset__insert,axiom,
! [X: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ X @ B ) )
= ( ord_less_eq_set_nat @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_182_subset__insertI,axiom,
! [B: set_transition,A: transition] : ( ord_le8419162016481440574sition @ B @ ( insert_transition2 @ A @ B ) ) ).
% subset_insertI
thf(fact_183_subset__insertI,axiom,
! [B: set_nat,A: nat] : ( ord_less_eq_set_nat @ B @ ( insert_nat2 @ A @ B ) ) ).
% subset_insertI
thf(fact_184_subset__insertI2,axiom,
! [A2: set_transition,B: set_transition,B2: transition] :
( ( ord_le8419162016481440574sition @ A2 @ B )
=> ( ord_le8419162016481440574sition @ A2 @ ( insert_transition2 @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_185_subset__insertI2,axiom,
! [A2: set_nat,B: set_nat,B2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_186_nfa_Ostep__eps__closure__set__closed,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,R: set_nat,Bs: list_o] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ( ( ord_less_eq_set_nat @ R @ ( q @ Q0 @ Qf @ Transs ) )
=> ( ord_less_eq_set_nat @ ( step_eps_closure_set @ Q0 @ Transs @ R @ Bs ) @ ( q @ Q0 @ Qf @ Transs ) ) ) ) ).
% nfa.step_eps_closure_set_closed
thf(fact_187_step__symb__set__mono,axiom,
! [R: set_nat,S: set_nat,Q0: nat,Transs: list_transition] :
( ( ord_less_eq_set_nat @ R @ S )
=> ( ord_less_eq_set_nat @ ( step_symb_set @ Q0 @ Transs @ R ) @ ( step_symb_set @ Q0 @ Transs @ S ) ) ) ).
% step_symb_set_mono
thf(fact_188_step__eps__closure__set__empty,axiom,
! [Q0: nat,Transs: list_transition,Bs: list_o] :
( ( step_eps_closure_set @ Q0 @ Transs @ bot_bot_set_nat @ Bs )
= bot_bot_set_nat ) ).
% step_eps_closure_set_empty
thf(fact_189_nfa_Otranss__not__Nil,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ( Transs != nil_transition ) ) ).
% nfa.transs_not_Nil
thf(fact_190_nfa_Ostep__eps__set__closed,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,Bs: list_o,R: set_nat] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ( ord_less_eq_set_nat @ ( step_eps_set @ Q0 @ Transs @ Bs @ R ) @ ( q @ Q0 @ Qf @ Transs ) ) ) ).
% nfa.step_eps_set_closed
thf(fact_191_Diff__insert,axiom,
! [A2: set_transition,A: transition,B: set_transition] :
( ( minus_8944320859760356485sition @ A2 @ ( insert_transition2 @ A @ B ) )
= ( minus_8944320859760356485sition @ ( minus_8944320859760356485sition @ A2 @ B ) @ ( insert_transition2 @ A @ bot_bo301567166201926666sition ) ) ) ).
% Diff_insert
thf(fact_192_Diff__insert,axiom,
! [A2: set_nat,A: nat,B: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ B ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_193_insert__Diff,axiom,
! [A: transition,A2: set_transition] :
( ( member_transition @ A @ A2 )
=> ( ( insert_transition2 @ A @ ( minus_8944320859760356485sition @ A2 @ ( insert_transition2 @ A @ bot_bo301567166201926666sition ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_194_insert__Diff,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat2 @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_195_Diff__insert2,axiom,
! [A2: set_transition,A: transition,B: set_transition] :
( ( minus_8944320859760356485sition @ A2 @ ( insert_transition2 @ A @ B ) )
= ( minus_8944320859760356485sition @ ( minus_8944320859760356485sition @ A2 @ ( insert_transition2 @ A @ bot_bo301567166201926666sition ) ) @ B ) ) ).
% Diff_insert2
thf(fact_196_Diff__insert2,axiom,
! [A2: set_nat,A: nat,B: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ B ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) @ B ) ) ).
% Diff_insert2
thf(fact_197_Diff__insert__absorb,axiom,
! [X: transition,A2: set_transition] :
( ~ ( member_transition @ X @ A2 )
=> ( ( minus_8944320859760356485sition @ ( insert_transition2 @ X @ A2 ) @ ( insert_transition2 @ X @ bot_bo301567166201926666sition ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_198_Diff__insert__absorb,axiom,
! [X: nat,A2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A2 ) @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_199_subset__singletonD,axiom,
! [A2: set_transition,X: transition] :
( ( ord_le8419162016481440574sition @ A2 @ ( insert_transition2 @ X @ bot_bo301567166201926666sition ) )
=> ( ( A2 = bot_bo301567166201926666sition )
| ( A2
= ( insert_transition2 @ X @ bot_bo301567166201926666sition ) ) ) ) ).
% subset_singletonD
thf(fact_200_subset__singletonD,axiom,
! [A2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
=> ( ( A2 = bot_bot_set_nat )
| ( A2
= ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_201_subset__singleton__iff,axiom,
! [X4: set_transition,A: transition] :
( ( ord_le8419162016481440574sition @ X4 @ ( insert_transition2 @ A @ bot_bo301567166201926666sition ) )
= ( ( X4 = bot_bo301567166201926666sition )
| ( X4
= ( insert_transition2 @ A @ bot_bo301567166201926666sition ) ) ) ) ).
% subset_singleton_iff
thf(fact_202_subset__singleton__iff,axiom,
! [X4: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X4 @ ( insert_nat2 @ A @ bot_bot_set_nat ) )
= ( ( X4 = bot_bot_set_nat )
| ( X4
= ( insert_nat2 @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_203_step__eps__dest,axiom,
! [Q0: nat,Transs: list_transition,Bs: list_o,Q: nat,Q2: nat] :
( ( step_eps @ Q0 @ Transs @ Bs @ Q @ Q2 )
=> ( member_nat @ Q @ ( sq @ Q0 @ Transs ) ) ) ).
% step_eps_dest
thf(fact_204_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A4: set_nat] :
( A4
= ( insert_nat2 @ ( the_elem_nat @ A4 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_205_is__singleton__the__elem,axiom,
( is_sin1641930644073461938sition
= ( ^ [A4: set_transition] :
( A4
= ( insert_transition2 @ ( the_elem_transition @ A4 ) @ bot_bo301567166201926666sition ) ) ) ) ).
% is_singleton_the_elem
thf(fact_206_step__eps__closure__empty,axiom,
! [Q0: nat,Transs: list_transition,Bs: list_o,Q: nat,Q2: nat] :
( ( step_eps_closure @ Q0 @ Transs @ Bs @ Q @ Q2 )
=> ( ! [Q5: nat] :
~ ( step_eps @ Q0 @ Transs @ Bs @ Q @ Q5 )
=> ( Q = Q2 ) ) ) ).
% step_eps_closure_empty
thf(fact_207_is__singletonI_H,axiom,
! [A2: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X3: nat,Y: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( X3 = Y ) ) )
=> ( is_singleton_nat @ A2 ) ) ) ).
% is_singletonI'
thf(fact_208_is__singletonI_H,axiom,
! [A2: set_transition] :
( ( A2 != bot_bo301567166201926666sition )
=> ( ! [X3: transition,Y: transition] :
( ( member_transition @ X3 @ A2 )
=> ( ( member_transition @ Y @ A2 )
=> ( X3 = Y ) ) )
=> ( is_sin1641930644073461938sition @ A2 ) ) ) ).
% is_singletonI'
thf(fact_209_SQ__sub__Q,axiom,
! [Q0: nat,Transs: list_transition,Qf: nat] : ( ord_less_eq_set_nat @ ( sq @ Q0 @ Transs ) @ ( q @ Q0 @ Qf @ Transs ) ) ).
% SQ_sub_Q
thf(fact_210_nfa_Ostep__symb__set__closed,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,R: set_nat] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ( ord_less_eq_set_nat @ ( step_symb_set @ Q0 @ Transs @ R ) @ ( q @ Q0 @ Qf @ Transs ) ) ) ).
% nfa.step_symb_set_closed
thf(fact_211_nfa_Ostep__eps__closed,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,Bs: list_o,Q: nat,Q2: nat] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ( ( step_eps @ Q0 @ Transs @ Bs @ Q @ Q2 )
=> ( member_nat @ Q2 @ ( q @ Q0 @ Qf @ Transs ) ) ) ) ).
% nfa.step_eps_closed
thf(fact_212_is__singletonE,axiom,
! [A2: set_nat] :
( ( is_singleton_nat @ A2 )
=> ~ ! [X3: nat] :
( A2
!= ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_213_is__singletonE,axiom,
! [A2: set_transition] :
( ( is_sin1641930644073461938sition @ A2 )
=> ~ ! [X3: transition] :
( A2
!= ( insert_transition2 @ X3 @ bot_bo301567166201926666sition ) ) ) ).
% is_singletonE
thf(fact_214_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A4: set_nat] :
? [X2: nat] :
( A4
= ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_215_is__singleton__def,axiom,
( is_sin1641930644073461938sition
= ( ^ [A4: set_transition] :
? [X2: transition] :
( A4
= ( insert_transition2 @ X2 @ bot_bo301567166201926666sition ) ) ) ) ).
% is_singleton_def
thf(fact_216_nfa_Oq0__sub__Q,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ( ord_less_eq_set_nat @ ( insert_nat2 @ Q0 @ bot_bot_set_nat ) @ ( q @ Q0 @ Qf @ Transs ) ) ) ).
% nfa.q0_sub_Q
thf(fact_217_step__eps__closure__set__closed__union,axiom,
! [R: set_nat,Bs: list_o] : ( ord_less_eq_set_nat @ ( step_eps_closure_set @ q0 @ transs @ R @ Bs ) @ ( sup_sup_set_nat @ R @ ( q @ q0 @ qf @ transs ) ) ) ).
% step_eps_closure_set_closed_union
thf(fact_218_diff__shunt__var,axiom,
! [X: set_transition,Y2: set_transition] :
( ( ( minus_8944320859760356485sition @ X @ Y2 )
= bot_bo301567166201926666sition )
= ( ord_le8419162016481440574sition @ X @ Y2 ) ) ).
% diff_shunt_var
thf(fact_219_diff__shunt__var,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ( minus_minus_set_nat @ X @ Y2 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X @ Y2 ) ) ).
% diff_shunt_var
thf(fact_220_state__closed,axiom,
! [T2: transition] :
( ( member_transition @ T2 @ ( set_transition2 @ transs ) )
=> ( ord_less_eq_set_nat @ ( state_set @ T2 ) @ ( q @ q0 @ qf @ transs ) ) ) ).
% state_closed
thf(fact_221_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X2: nat] : ( member_nat @ X2 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_222_bot__empty__eq,axiom,
( bot_bot_transition_o
= ( ^ [X2: transition] : ( member_transition @ X2 @ bot_bo301567166201926666sition ) ) ) ).
% bot_empty_eq
thf(fact_223_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_224_Collect__empty__eq__bot,axiom,
! [P: transition > $o] :
( ( ( collect_transition @ P )
= bot_bo301567166201926666sition )
= ( P = bot_bot_transition_o ) ) ).
% Collect_empty_eq_bot
thf(fact_225_insert__subsetI,axiom,
! [X: transition,A2: set_transition,X4: set_transition] :
( ( member_transition @ X @ A2 )
=> ( ( ord_le8419162016481440574sition @ X4 @ A2 )
=> ( ord_le8419162016481440574sition @ ( insert_transition2 @ X @ X4 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_226_insert__subsetI,axiom,
! [X: nat,A2: set_nat,X4: set_nat] :
( ( member_nat @ X @ A2 )
=> ( ( ord_less_eq_set_nat @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ ( insert_nat2 @ X @ X4 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_227_subset__emptyI,axiom,
! [A2: set_transition] :
( ! [X3: transition] :
~ ( member_transition @ X3 @ A2 )
=> ( ord_le8419162016481440574sition @ A2 @ bot_bo301567166201926666sition ) ) ).
% subset_emptyI
thf(fact_228_subset__emptyI,axiom,
! [A2: set_nat] :
( ! [X3: nat] :
~ ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_229_sup_Oidem,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_230_sup__idem,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_231_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_232_sup__left__idem,axiom,
! [X: set_nat,Y2: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y2 ) )
= ( sup_sup_set_nat @ X @ Y2 ) ) ).
% sup_left_idem
thf(fact_233_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_234_Un__iff,axiom,
! [C: transition,A2: set_transition,B: set_transition] :
( ( member_transition @ C @ ( sup_su812053455038985074sition @ A2 @ B ) )
= ( ( member_transition @ C @ A2 )
| ( member_transition @ C @ B ) ) ) ).
% Un_iff
thf(fact_235_Un__iff,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) )
= ( ( member_nat @ C @ A2 )
| ( member_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_236_UnCI,axiom,
! [C: transition,B: set_transition,A2: set_transition] :
( ( ~ ( member_transition @ C @ B )
=> ( member_transition @ C @ A2 ) )
=> ( member_transition @ C @ ( sup_su812053455038985074sition @ A2 @ B ) ) ) ).
% UnCI
thf(fact_237_UnCI,axiom,
! [C: nat,B: set_nat,A2: set_nat] :
( ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ A2 ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnCI
thf(fact_238_le__sup__iff,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ Y2 ) @ Z2 )
= ( ( ord_less_eq_set_nat @ X @ Z2 )
& ( ord_less_eq_set_nat @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_239_sup_Obounded__iff,axiom,
! [B2: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A )
= ( ( ord_less_eq_set_nat @ B2 @ A )
& ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_240_sup__bot_Oright__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_241_sup__bot_Oright__neutral,axiom,
! [A: set_transition] :
( ( sup_su812053455038985074sition @ A @ bot_bo301567166201926666sition )
= A ) ).
% sup_bot.right_neutral
thf(fact_242_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A @ B2 ) )
= ( ( A = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_243_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_transition,B2: set_transition] :
( ( bot_bo301567166201926666sition
= ( sup_su812053455038985074sition @ A @ B2 ) )
= ( ( A = bot_bo301567166201926666sition )
& ( B2 = bot_bo301567166201926666sition ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_244_sup__bot_Oleft__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_245_sup__bot_Oleft__neutral,axiom,
! [A: set_transition] :
( ( sup_su812053455038985074sition @ bot_bo301567166201926666sition @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_246_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_nat,B2: set_nat] :
( ( ( sup_sup_set_nat @ A @ B2 )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_247_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_transition,B2: set_transition] :
( ( ( sup_su812053455038985074sition @ A @ B2 )
= bot_bo301567166201926666sition )
= ( ( A = bot_bo301567166201926666sition )
& ( B2 = bot_bo301567166201926666sition ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_248_sup__eq__bot__iff,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ( sup_sup_set_nat @ X @ Y2 )
= bot_bot_set_nat )
= ( ( X = bot_bot_set_nat )
& ( Y2 = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_249_sup__eq__bot__iff,axiom,
! [X: set_transition,Y2: set_transition] :
( ( ( sup_su812053455038985074sition @ X @ Y2 )
= bot_bo301567166201926666sition )
= ( ( X = bot_bo301567166201926666sition )
& ( Y2 = bot_bo301567166201926666sition ) ) ) ).
% sup_eq_bot_iff
thf(fact_250_bot__eq__sup__iff,axiom,
! [X: set_nat,Y2: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X @ Y2 ) )
= ( ( X = bot_bot_set_nat )
& ( Y2 = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_251_bot__eq__sup__iff,axiom,
! [X: set_transition,Y2: set_transition] :
( ( bot_bo301567166201926666sition
= ( sup_su812053455038985074sition @ X @ Y2 ) )
= ( ( X = bot_bo301567166201926666sition )
& ( Y2 = bot_bo301567166201926666sition ) ) ) ).
% bot_eq_sup_iff
thf(fact_252_sup__bot__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% sup_bot_right
thf(fact_253_sup__bot__right,axiom,
! [X: set_transition] :
( ( sup_su812053455038985074sition @ X @ bot_bo301567166201926666sition )
= X ) ).
% sup_bot_right
thf(fact_254_sup__bot__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X )
= X ) ).
% sup_bot_left
thf(fact_255_sup__bot__left,axiom,
! [X: set_transition] :
( ( sup_su812053455038985074sition @ bot_bo301567166201926666sition @ X )
= X ) ).
% sup_bot_left
thf(fact_256_Un__empty,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_257_Un__empty,axiom,
! [A2: set_transition,B: set_transition] :
( ( ( sup_su812053455038985074sition @ A2 @ B )
= bot_bo301567166201926666sition )
= ( ( A2 = bot_bo301567166201926666sition )
& ( B = bot_bo301567166201926666sition ) ) ) ).
% Un_empty
thf(fact_258_Un__subset__iff,axiom,
! [A2: set_nat,B: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C3 )
= ( ( ord_less_eq_set_nat @ A2 @ C3 )
& ( ord_less_eq_set_nat @ B @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_259_Un__insert__right,axiom,
! [A2: set_transition,A: transition,B: set_transition] :
( ( sup_su812053455038985074sition @ A2 @ ( insert_transition2 @ A @ B ) )
= ( insert_transition2 @ A @ ( sup_su812053455038985074sition @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_260_Un__insert__right,axiom,
! [A2: set_nat,A: nat,B: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( insert_nat2 @ A @ B ) )
= ( insert_nat2 @ A @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_261_Un__insert__left,axiom,
! [A: transition,B: set_transition,C3: set_transition] :
( ( sup_su812053455038985074sition @ ( insert_transition2 @ A @ B ) @ C3 )
= ( insert_transition2 @ A @ ( sup_su812053455038985074sition @ B @ C3 ) ) ) ).
% Un_insert_left
thf(fact_262_Un__insert__left,axiom,
! [A: nat,B: set_nat,C3: set_nat] :
( ( sup_sup_set_nat @ ( insert_nat2 @ A @ B ) @ C3 )
= ( insert_nat2 @ A @ ( sup_sup_set_nat @ B @ C3 ) ) ) ).
% Un_insert_left
thf(fact_263_Un__Diff__cancel2,axiom,
! [B: set_nat,A2: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B @ A2 ) @ A2 )
= ( sup_sup_set_nat @ B @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_264_Un__Diff__cancel,axiom,
! [A2: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( minus_minus_set_nat @ B @ A2 ) )
= ( sup_sup_set_nat @ A2 @ B ) ) ).
% Un_Diff_cancel
thf(fact_265_inf__sup__aci_I8_J,axiom,
! [X: set_nat,Y2: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y2 ) )
= ( sup_sup_set_nat @ X @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_266_inf__sup__aci_I7_J,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y2 @ Z2 ) )
= ( sup_sup_set_nat @ Y2 @ ( sup_sup_set_nat @ X @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_267_inf__sup__aci_I6_J,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y2 ) @ Z2 )
= ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y2 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_268_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_269_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_270_sup__assoc,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y2 ) @ Z2 )
= ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y2 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_271_sup_Ocommute,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B4: set_nat] : ( sup_sup_set_nat @ B4 @ A3 ) ) ) ).
% sup.commute
thf(fact_272_sup__commute,axiom,
( sup_sup_set_nat
= ( ^ [X2: set_nat,Y4: set_nat] : ( sup_sup_set_nat @ Y4 @ X2 ) ) ) ).
% sup_commute
thf(fact_273_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_274_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_275_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_276_sup__left__commute,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y2 @ Z2 ) )
= ( sup_sup_set_nat @ Y2 @ ( sup_sup_set_nat @ X @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_277_Un__left__commute,axiom,
! [A2: set_nat,B: set_nat,C3: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C3 ) )
= ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A2 @ C3 ) ) ) ).
% Un_left_commute
thf(fact_278_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_279_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B5: set_nat] : ( sup_sup_set_nat @ B5 @ A4 ) ) ) ).
% Un_commute
thf(fact_280_Un__absorb,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_281_Un__assoc,axiom,
! [A2: set_nat,B: set_nat,C3: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C3 )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C3 ) ) ) ).
% Un_assoc
thf(fact_282_ball__Un,axiom,
! [A2: set_nat,B: set_nat,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( sup_sup_set_nat @ A2 @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( P @ X2 ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_283_bex__Un,axiom,
! [A2: set_nat,B: set_nat,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat @ X2 @ ( sup_sup_set_nat @ A2 @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ X2 ) )
| ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_284_UnI2,axiom,
! [C: transition,B: set_transition,A2: set_transition] :
( ( member_transition @ C @ B )
=> ( member_transition @ C @ ( sup_su812053455038985074sition @ A2 @ B ) ) ) ).
% UnI2
thf(fact_285_UnI2,axiom,
! [C: nat,B: set_nat,A2: set_nat] :
( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnI2
thf(fact_286_UnI1,axiom,
! [C: transition,A2: set_transition,B: set_transition] :
( ( member_transition @ C @ A2 )
=> ( member_transition @ C @ ( sup_su812053455038985074sition @ A2 @ B ) ) ) ).
% UnI1
thf(fact_287_UnI1,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnI1
thf(fact_288_UnE,axiom,
! [C: transition,A2: set_transition,B: set_transition] :
( ( member_transition @ C @ ( sup_su812053455038985074sition @ A2 @ B ) )
=> ( ~ ( member_transition @ C @ A2 )
=> ( member_transition @ C @ B ) ) ) ).
% UnE
thf(fact_289_UnE,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) )
=> ( ~ ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B ) ) ) ).
% UnE
thf(fact_290_inf__sup__ord_I4_J,axiom,
! [Y2: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y2 @ ( sup_sup_set_nat @ X @ Y2 ) ) ).
% inf_sup_ord(4)
thf(fact_291_inf__sup__ord_I3_J,axiom,
! [X: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y2 ) ) ).
% inf_sup_ord(3)
thf(fact_292_le__supE,axiom,
! [A: set_nat,B2: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ X )
=> ~ ( ( ord_less_eq_set_nat @ A @ X )
=> ~ ( ord_less_eq_set_nat @ B2 @ X ) ) ) ).
% le_supE
thf(fact_293_le__supI,axiom,
! [A: set_nat,X: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ X )
=> ( ( ord_less_eq_set_nat @ B2 @ X )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_294_sup__ge1,axiom,
! [X: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y2 ) ) ).
% sup_ge1
thf(fact_295_sup__ge2,axiom,
! [Y2: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y2 @ ( sup_sup_set_nat @ X @ Y2 ) ) ).
% sup_ge2
thf(fact_296_le__supI1,axiom,
! [X: set_nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X @ A )
=> ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_297_le__supI2,axiom,
! [X: set_nat,B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ X @ B2 )
=> ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_298_sup_Omono,axiom,
! [C: set_nat,A: set_nat,D: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C @ A )
=> ( ( ord_less_eq_set_nat @ D @ B2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C @ D ) @ ( sup_sup_set_nat @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_299_sup__mono,axiom,
! [A: set_nat,C: set_nat,B2: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ D )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ ( sup_sup_set_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_300_sup__least,axiom,
! [Y2: set_nat,X: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X )
=> ( ( ord_less_eq_set_nat @ Z2 @ X )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y2 @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_301_le__iff__sup,axiom,
( ord_less_eq_set_nat
= ( ^ [X2: set_nat,Y4: set_nat] :
( ( sup_sup_set_nat @ X2 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_302_sup_OorderE,axiom,
! [B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( A
= ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_303_sup_OorderI,axiom,
! [A: set_nat,B2: set_nat] :
( ( A
= ( sup_sup_set_nat @ A @ B2 ) )
=> ( ord_less_eq_set_nat @ B2 @ A ) ) ).
% sup.orderI
thf(fact_304_sup__unique,axiom,
! [F: set_nat > set_nat > set_nat,X: set_nat,Y2: set_nat] :
( ! [X3: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( F @ X3 @ Y ) )
=> ( ! [X3: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ Y @ ( F @ X3 @ Y ) )
=> ( ! [X3: set_nat,Y: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X3 )
=> ( ( ord_less_eq_set_nat @ Z3 @ X3 )
=> ( ord_less_eq_set_nat @ ( F @ Y @ Z3 ) @ X3 ) ) )
=> ( ( sup_sup_set_nat @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% sup_unique
thf(fact_305_sup_Oabsorb1,axiom,
! [B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( sup_sup_set_nat @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_306_sup_Oabsorb2,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( sup_sup_set_nat @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_307_sup__absorb1,axiom,
! [Y2: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X )
=> ( ( sup_sup_set_nat @ X @ Y2 )
= X ) ) ).
% sup_absorb1
thf(fact_308_sup__absorb2,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ( sup_sup_set_nat @ X @ Y2 )
= Y2 ) ) ).
% sup_absorb2
thf(fact_309_sup_OboundedE,axiom,
! [B2: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A )
=> ~ ( ( ord_less_eq_set_nat @ B2 @ A )
=> ~ ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_310_sup_OboundedI,axiom,
! [B2: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( ord_less_eq_set_nat @ C @ A )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_311_sup_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A3: set_nat] :
( A3
= ( sup_sup_set_nat @ A3 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_312_sup_Ocobounded1,axiom,
! [A: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_313_sup_Ocobounded2,axiom,
! [B2: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B2 @ ( sup_sup_set_nat @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_314_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A3: set_nat] :
( ( sup_sup_set_nat @ A3 @ B4 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_315_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
( ( sup_sup_set_nat @ A3 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_316_sup_OcoboundedI1,axiom,
! [C: set_nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C @ A )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_317_sup_OcoboundedI2,axiom,
! [C: set_nat,B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ C @ B2 )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_318_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_319_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_transition] :
( ( sup_su812053455038985074sition @ X @ bot_bo301567166201926666sition )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_320_Un__empty__right,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Un_empty_right
thf(fact_321_Un__empty__right,axiom,
! [A2: set_transition] :
( ( sup_su812053455038985074sition @ A2 @ bot_bo301567166201926666sition )
= A2 ) ).
% Un_empty_right
thf(fact_322_Un__empty__left,axiom,
! [B: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_323_Un__empty__left,axiom,
! [B: set_transition] :
( ( sup_su812053455038985074sition @ bot_bo301567166201926666sition @ B )
= B ) ).
% Un_empty_left
thf(fact_324_subset__Un__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
( ( sup_sup_set_nat @ A4 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_325_subset__UnE,axiom,
! [C3: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ ( sup_sup_set_nat @ A2 @ B ) )
=> ~ ! [A5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ A2 )
=> ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ B )
=> ( C3
!= ( sup_sup_set_nat @ A5 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_326_Un__absorb2,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( sup_sup_set_nat @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_327_Un__absorb1,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( sup_sup_set_nat @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_328_Un__upper2,axiom,
! [B: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A2 @ B ) ) ).
% Un_upper2
thf(fact_329_Un__upper1,axiom,
! [A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B ) ) ).
% Un_upper1
thf(fact_330_Un__least,axiom,
! [A2: set_nat,C3: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C3 )
=> ( ( ord_less_eq_set_nat @ B @ C3 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C3 ) ) ) ).
% Un_least
thf(fact_331_Un__mono,axiom,
! [A2: set_nat,C3: set_nat,B: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C3 )
=> ( ( ord_less_eq_set_nat @ B @ D2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ ( sup_sup_set_nat @ C3 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_332_Un__Diff,axiom,
! [A2: set_nat,B: set_nat,C3: set_nat] :
( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C3 )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A2 @ C3 ) @ ( minus_minus_set_nat @ B @ C3 ) ) ) ).
% Un_Diff
thf(fact_333_step__eps__closure__set__split,axiom,
! [Q0: nat,Transs: list_transition,R: set_nat,S: set_nat,Bs: list_o] :
( ( step_eps_closure_set @ Q0 @ Transs @ ( sup_sup_set_nat @ R @ S ) @ Bs )
= ( sup_sup_set_nat @ ( step_eps_closure_set @ Q0 @ Transs @ R @ Bs ) @ ( step_eps_closure_set @ Q0 @ Transs @ S @ Bs ) ) ) ).
% step_eps_closure_set_split
thf(fact_334_step__symb__set__split,axiom,
! [Q0: nat,Transs: list_transition,R: set_nat,S: set_nat] :
( ( step_symb_set @ Q0 @ Transs @ ( sup_sup_set_nat @ R @ S ) )
= ( sup_sup_set_nat @ ( step_symb_set @ Q0 @ Transs @ R ) @ ( step_symb_set @ Q0 @ Transs @ S ) ) ) ).
% step_symb_set_split
thf(fact_335_singleton__Un__iff,axiom,
! [X: nat,A2: set_nat,B: set_nat] :
( ( ( insert_nat2 @ X @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A2 @ B ) )
= ( ( ( A2 = bot_bot_set_nat )
& ( B
= ( insert_nat2 @ X @ bot_bot_set_nat ) ) )
| ( ( A2
= ( insert_nat2 @ X @ bot_bot_set_nat ) )
& ( B = bot_bot_set_nat ) )
| ( ( A2
= ( insert_nat2 @ X @ bot_bot_set_nat ) )
& ( B
= ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_336_singleton__Un__iff,axiom,
! [X: transition,A2: set_transition,B: set_transition] :
( ( ( insert_transition2 @ X @ bot_bo301567166201926666sition )
= ( sup_su812053455038985074sition @ A2 @ B ) )
= ( ( ( A2 = bot_bo301567166201926666sition )
& ( B
= ( insert_transition2 @ X @ bot_bo301567166201926666sition ) ) )
| ( ( A2
= ( insert_transition2 @ X @ bot_bo301567166201926666sition ) )
& ( B = bot_bo301567166201926666sition ) )
| ( ( A2
= ( insert_transition2 @ X @ bot_bo301567166201926666sition ) )
& ( B
= ( insert_transition2 @ X @ bot_bo301567166201926666sition ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_337_Un__singleton__iff,axiom,
! [A2: set_nat,B: set_nat,X: nat] :
( ( ( sup_sup_set_nat @ A2 @ B )
= ( insert_nat2 @ X @ bot_bot_set_nat ) )
= ( ( ( A2 = bot_bot_set_nat )
& ( B
= ( insert_nat2 @ X @ bot_bot_set_nat ) ) )
| ( ( A2
= ( insert_nat2 @ X @ bot_bot_set_nat ) )
& ( B = bot_bot_set_nat ) )
| ( ( A2
= ( insert_nat2 @ X @ bot_bot_set_nat ) )
& ( B
= ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_338_Un__singleton__iff,axiom,
! [A2: set_transition,B: set_transition,X: transition] :
( ( ( sup_su812053455038985074sition @ A2 @ B )
= ( insert_transition2 @ X @ bot_bo301567166201926666sition ) )
= ( ( ( A2 = bot_bo301567166201926666sition )
& ( B
= ( insert_transition2 @ X @ bot_bo301567166201926666sition ) ) )
| ( ( A2
= ( insert_transition2 @ X @ bot_bo301567166201926666sition ) )
& ( B = bot_bo301567166201926666sition ) )
| ( ( A2
= ( insert_transition2 @ X @ bot_bo301567166201926666sition ) )
& ( B
= ( insert_transition2 @ X @ bot_bo301567166201926666sition ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_339_insert__is__Un,axiom,
( insert_nat2
= ( ^ [A3: nat] : ( sup_sup_set_nat @ ( insert_nat2 @ A3 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_340_insert__is__Un,axiom,
( insert_transition2
= ( ^ [A3: transition] : ( sup_su812053455038985074sition @ ( insert_transition2 @ A3 @ bot_bo301567166201926666sition ) ) ) ) ).
% insert_is_Un
thf(fact_341_Diff__subset__conv,axiom,
! [A2: set_nat,B: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ C3 )
= ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C3 ) ) ) ).
% Diff_subset_conv
thf(fact_342_Diff__partition,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( sup_sup_set_nat @ A2 @ ( minus_minus_set_nat @ B @ A2 ) )
= B ) ) ).
% Diff_partition
thf(fact_343_nfa_Ostate__closed,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,T2: transition] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ( ( member_transition @ T2 @ ( set_transition2 @ Transs ) )
=> ( ord_less_eq_set_nat @ ( state_set @ T2 ) @ ( q @ Q0 @ Qf @ Transs ) ) ) ) ).
% nfa.state_closed
thf(fact_344_step__eps__closure__set__flip,axiom,
! [Q0: nat,Transs: list_transition,R: set_nat,Bs: list_o,S: set_nat] :
( ( ( step_eps_closure_set @ Q0 @ Transs @ R @ Bs )
= ( sup_sup_set_nat @ R @ S ) )
=> ( ord_less_eq_set_nat @ ( step_eps_closure_set @ Q0 @ Transs @ S @ Bs ) @ ( sup_sup_set_nat @ R @ S ) ) ) ).
% step_eps_closure_set_flip
thf(fact_345_nfa_Ointro,axiom,
! [Transs: list_transition,Q0: nat,Qf: nat] :
( ! [T3: transition] :
( ( member_transition @ T3 @ ( set_transition2 @ Transs ) )
=> ( ord_less_eq_set_nat @ ( state_set @ T3 ) @ ( q @ Q0 @ Qf @ Transs ) ) )
=> ( ( Transs != nil_transition )
=> ( ~ ( member_nat @ Qf @ ( sq @ Q0 @ Transs ) )
=> ( nfa @ Q0 @ Qf @ Transs ) ) ) ) ).
% nfa.intro
thf(fact_346_nfa__def,axiom,
( nfa
= ( ^ [Q02: nat,Qf2: nat,Transs2: list_transition] :
( ! [T: transition] :
( ( member_transition @ T @ ( set_transition2 @ Transs2 ) )
=> ( ord_less_eq_set_nat @ ( state_set @ T ) @ ( q @ Q02 @ Qf2 @ Transs2 ) ) )
& ( Transs2 != nil_transition )
& ~ ( member_nat @ Qf2 @ ( sq @ Q02 @ Transs2 ) ) ) ) ) ).
% nfa_def
thf(fact_347_NFA_OQ__def,axiom,
( q
= ( ^ [Q02: nat,Qf2: nat,Transs2: list_transition] : ( sup_sup_set_nat @ ( sq @ Q02 @ Transs2 ) @ ( insert_nat2 @ Qf2 @ bot_bot_set_nat ) ) ) ) ).
% NFA.Q_def
thf(fact_348_step__eps__closure__set__unfold,axiom,
! [Q0: nat,Transs: list_transition,Bs: list_o,Q: nat,X4: set_nat] :
( ! [Q5: nat] :
( ( step_eps @ Q0 @ Transs @ Bs @ Q @ Q5 )
= ( member_nat @ Q5 @ X4 ) )
=> ( ( step_eps_closure_set @ Q0 @ Transs @ ( insert_nat2 @ Q @ bot_bot_set_nat ) @ Bs )
= ( sup_sup_set_nat @ ( insert_nat2 @ Q @ bot_bot_set_nat ) @ ( step_eps_closure_set @ Q0 @ Transs @ X4 @ Bs ) ) ) ) ).
% step_eps_closure_set_unfold
thf(fact_349_nfa_Ostep__eps__closure__set__closed__union,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,R: set_nat,Bs: list_o] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ( ord_less_eq_set_nat @ ( step_eps_closure_set @ Q0 @ Transs @ R @ Bs ) @ ( sup_sup_set_nat @ R @ ( q @ Q0 @ Qf @ Transs ) ) ) ) ).
% nfa.step_eps_closure_set_closed_union
thf(fact_350_set__empty2,axiom,
! [Xs: list_o] :
( ( bot_bot_set_o
= ( set_o2 @ Xs ) )
= ( Xs = nil_o ) ) ).
% set_empty2
thf(fact_351_set__empty2,axiom,
! [Xs: list_list_o] :
( ( bot_bot_set_list_o
= ( set_list_o2 @ Xs ) )
= ( Xs = nil_list_o ) ) ).
% set_empty2
thf(fact_352_set__empty2,axiom,
! [Xs: list_nat] :
( ( bot_bot_set_nat
= ( set_nat2 @ Xs ) )
= ( Xs = nil_nat ) ) ).
% set_empty2
thf(fact_353_set__empty2,axiom,
! [Xs: list_transition] :
( ( bot_bo301567166201926666sition
= ( set_transition2 @ Xs ) )
= ( Xs = nil_transition ) ) ).
% set_empty2
thf(fact_354_set__empty,axiom,
! [Xs: list_o] :
( ( ( set_o2 @ Xs )
= bot_bot_set_o )
= ( Xs = nil_o ) ) ).
% set_empty
thf(fact_355_set__empty,axiom,
! [Xs: list_list_o] :
( ( ( set_list_o2 @ Xs )
= bot_bot_set_list_o )
= ( Xs = nil_list_o ) ) ).
% set_empty
thf(fact_356_set__empty,axiom,
! [Xs: list_nat] :
( ( ( set_nat2 @ Xs )
= bot_bot_set_nat )
= ( Xs = nil_nat ) ) ).
% set_empty
thf(fact_357_set__empty,axiom,
! [Xs: list_transition] :
( ( ( set_transition2 @ Xs )
= bot_bo301567166201926666sition )
= ( Xs = nil_transition ) ) ).
% set_empty
thf(fact_358_empty__set,axiom,
( bot_bot_set_o
= ( set_o2 @ nil_o ) ) ).
% empty_set
thf(fact_359_empty__set,axiom,
( bot_bot_set_list_o
= ( set_list_o2 @ nil_list_o ) ) ).
% empty_set
thf(fact_360_empty__set,axiom,
( bot_bot_set_nat
= ( set_nat2 @ nil_nat ) ) ).
% empty_set
thf(fact_361_empty__set,axiom,
( bot_bo301567166201926666sition
= ( set_transition2 @ nil_transition ) ) ).
% empty_set
thf(fact_362_subset__code_I1_J,axiom,
! [Xs: list_transition,B: set_transition] :
( ( ord_le8419162016481440574sition @ ( set_transition2 @ Xs ) @ B )
= ( ! [X2: transition] :
( ( member_transition @ X2 @ ( set_transition2 @ Xs ) )
=> ( member_transition @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_363_subset__code_I1_J,axiom,
! [Xs: list_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_364_remove__def,axiom,
( remove_transition
= ( ^ [X2: transition,A4: set_transition] : ( minus_8944320859760356485sition @ A4 @ ( insert_transition2 @ X2 @ bot_bo301567166201926666sition ) ) ) ) ).
% remove_def
thf(fact_365_remove__def,axiom,
( remove_nat
= ( ^ [X2: nat,A4: set_nat] : ( minus_minus_set_nat @ A4 @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ) ).
% remove_def
thf(fact_366_state__set_Osimps_I1_J,axiom,
! [S2: nat,Uu: nat] :
( ( state_set @ ( eps_trans @ S2 @ Uu ) )
= ( insert_nat2 @ S2 @ bot_bot_set_nat ) ) ).
% state_set.simps(1)
thf(fact_367_state__set_Osimps_I2_J,axiom,
! [S2: nat] :
( ( state_set @ ( symb_trans @ S2 ) )
= ( insert_nat2 @ S2 @ bot_bot_set_nat ) ) ).
% state_set.simps(2)
thf(fact_368_state__set_Osimps_I3_J,axiom,
! [S2: nat,S3: nat] :
( ( state_set @ ( split_trans @ S2 @ S3 ) )
= ( insert_nat2 @ S2 @ ( insert_nat2 @ S3 @ bot_bot_set_nat ) ) ) ).
% state_set.simps(3)
thf(fact_369_transition_Oinject_I1_J,axiom,
! [X11: nat,X12: nat,Y11: nat,Y12: nat] :
( ( ( eps_trans @ X11 @ X12 )
= ( eps_trans @ Y11 @ Y12 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 ) ) ) ).
% transition.inject(1)
thf(fact_370_transition_Oinject_I2_J,axiom,
! [X22: nat,Y22: nat] :
( ( ( symb_trans @ X22 )
= ( symb_trans @ Y22 ) )
= ( X22 = Y22 ) ) ).
% transition.inject(2)
thf(fact_371_transition_Oinject_I3_J,axiom,
! [X31: nat,X32: nat,Y31: nat,Y32: nat] :
( ( ( split_trans @ X31 @ X32 )
= ( split_trans @ Y31 @ Y32 ) )
= ( ( X31 = Y31 )
& ( X32 = Y32 ) ) ) ).
% transition.inject(3)
thf(fact_372_member__remove,axiom,
! [X: nat,Y2: nat,A2: set_nat] :
( ( member_nat @ X @ ( remove_nat @ Y2 @ A2 ) )
= ( ( member_nat @ X @ A2 )
& ( X != Y2 ) ) ) ).
% member_remove
thf(fact_373_member__remove,axiom,
! [X: transition,Y2: transition,A2: set_transition] :
( ( member_transition @ X @ ( remove_transition @ Y2 @ A2 ) )
= ( ( member_transition @ X @ A2 )
& ( X != Y2 ) ) ) ).
% member_remove
thf(fact_374_transition_Oexhaust,axiom,
! [Y2: transition] :
( ! [X112: nat,X122: nat] :
( Y2
!= ( eps_trans @ X112 @ X122 ) )
=> ( ! [X23: nat] :
( Y2
!= ( symb_trans @ X23 ) )
=> ~ ! [X312: nat,X322: nat] :
( Y2
!= ( split_trans @ X312 @ X322 ) ) ) ) ).
% transition.exhaust
thf(fact_375_state__set_Ocases,axiom,
! [X: transition] :
( ! [S4: nat,Uu2: nat] :
( X
!= ( eps_trans @ S4 @ Uu2 ) )
=> ( ! [S4: nat] :
( X
!= ( symb_trans @ S4 ) )
=> ~ ! [S4: nat,S5: nat] :
( X
!= ( split_trans @ S4 @ S5 ) ) ) ) ).
% state_set.cases
thf(fact_376_transition_Odistinct_I1_J,axiom,
! [X11: nat,X12: nat,X22: nat] :
( ( eps_trans @ X11 @ X12 )
!= ( symb_trans @ X22 ) ) ).
% transition.distinct(1)
thf(fact_377_transition_Odistinct_I3_J,axiom,
! [X11: nat,X12: nat,X31: nat,X32: nat] :
( ( eps_trans @ X11 @ X12 )
!= ( split_trans @ X31 @ X32 ) ) ).
% transition.distinct(3)
thf(fact_378_transition_Odistinct_I5_J,axiom,
! [X22: nat,X31: nat,X32: nat] :
( ( symb_trans @ X22 )
!= ( split_trans @ X31 @ X32 ) ) ).
% transition.distinct(5)
thf(fact_379_step__eps__mono,axiom,
! [Q0: nat,Transs: list_transition,Q: nat,Q2: nat,Bs: list_o] :
( ( step_eps @ Q0 @ Transs @ nil_o @ Q @ Q2 )
=> ( step_eps @ Q0 @ Transs @ Bs @ Q @ Q2 ) ) ).
% step_eps_mono
thf(fact_380_state__set_Oelims,axiom,
! [X: transition,Y2: set_nat] :
( ( ( state_set @ X )
= Y2 )
=> ( ! [S4: nat] :
( ? [Uu2: nat] :
( X
= ( eps_trans @ S4 @ Uu2 ) )
=> ( Y2
!= ( insert_nat2 @ S4 @ bot_bot_set_nat ) ) )
=> ( ! [S4: nat] :
( ( X
= ( symb_trans @ S4 ) )
=> ( Y2
!= ( insert_nat2 @ S4 @ bot_bot_set_nat ) ) )
=> ~ ! [S4: nat,S5: nat] :
( ( X
= ( split_trans @ S4 @ S5 ) )
=> ( Y2
!= ( insert_nat2 @ S4 @ ( insert_nat2 @ S5 @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% state_set.elims
thf(fact_381_step__eps__closure__set__mono_H,axiom,
! [Q0: nat,Transs: list_transition,R: set_nat,Bs: list_o] : ( ord_less_eq_set_nat @ ( step_eps_closure_set @ Q0 @ Transs @ R @ nil_o ) @ ( step_eps_closure_set @ Q0 @ Transs @ R @ Bs ) ) ).
% step_eps_closure_set_mono'
thf(fact_382_set__union,axiom,
! [Xs: list_transition,Ys: list_transition] :
( ( set_transition2 @ ( union_transition @ Xs @ Ys ) )
= ( sup_su812053455038985074sition @ ( set_transition2 @ Xs ) @ ( set_transition2 @ Ys ) ) ) ).
% set_union
thf(fact_383_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_384_fmla__set_Oelims,axiom,
! [X: transition,Y2: set_nat] :
( ( ( fmla_set @ X )
= Y2 )
=> ( ! [Uu2: nat,N: nat] :
( ( X
= ( eps_trans @ Uu2 @ N ) )
=> ( Y2
!= ( insert_nat2 @ N @ bot_bot_set_nat ) ) )
=> ( ( ? [V: nat] :
( X
= ( symb_trans @ V ) )
=> ( Y2 != bot_bot_set_nat ) )
=> ~ ( ? [V: nat,Va: nat] :
( X
= ( split_trans @ V @ Va ) )
=> ( Y2 != bot_bot_set_nat ) ) ) ) ) ).
% fmla_set.elims
thf(fact_385_fmla__set_Osimps_I1_J,axiom,
! [Uu: nat,N2: nat] :
( ( fmla_set @ ( eps_trans @ Uu @ N2 ) )
= ( insert_nat2 @ N2 @ bot_bot_set_nat ) ) ).
% fmla_set.simps(1)
thf(fact_386_set__removeAll,axiom,
! [X: transition,Xs: list_transition] :
( ( set_transition2 @ ( removeAll_transition @ X @ Xs ) )
= ( minus_8944320859760356485sition @ ( set_transition2 @ Xs ) @ ( insert_transition2 @ X @ bot_bo301567166201926666sition ) ) ) ).
% set_removeAll
thf(fact_387_set__removeAll,axiom,
! [X: nat,Xs: list_nat] :
( ( set_nat2 @ ( removeAll_nat @ X @ Xs ) )
= ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ).
% set_removeAll
thf(fact_388_run__accept__eps__Nil__eps__split,axiom,
! [Q0: nat,Transs: list_transition,R: set_nat,Bs: list_o,S: set_nat,Qf: nat] :
( ( ( step_eps_closure_set @ Q0 @ Transs @ R @ Bs )
= ( sup_sup_set_nat @ R @ S ) )
=> ( ( ( step_symb_set @ Q0 @ Transs @ R )
= bot_bot_set_nat )
=> ( ~ ( member_nat @ Qf @ R )
=> ( ( run_accept_eps @ Q0 @ Qf @ Transs @ R @ nil_list_o @ Bs )
= ( run_accept_eps @ Q0 @ Qf @ Transs @ S @ nil_list_o @ Bs ) ) ) ) ) ).
% run_accept_eps_Nil_eps_split
thf(fact_389_state__set_Opelims,axiom,
! [X: transition,Y2: set_nat] :
( ( ( state_set @ X )
= Y2 )
=> ( ( accp_transition @ state_set_rel @ X )
=> ( ! [S4: nat,Uu2: nat] :
( ( X
= ( eps_trans @ S4 @ Uu2 ) )
=> ( ( Y2
= ( insert_nat2 @ S4 @ bot_bot_set_nat ) )
=> ~ ( accp_transition @ state_set_rel @ ( eps_trans @ S4 @ Uu2 ) ) ) )
=> ( ! [S4: nat] :
( ( X
= ( symb_trans @ S4 ) )
=> ( ( Y2
= ( insert_nat2 @ S4 @ bot_bot_set_nat ) )
=> ~ ( accp_transition @ state_set_rel @ ( symb_trans @ S4 ) ) ) )
=> ~ ! [S4: nat,S5: nat] :
( ( X
= ( split_trans @ S4 @ S5 ) )
=> ( ( Y2
= ( insert_nat2 @ S4 @ ( insert_nat2 @ S5 @ bot_bot_set_nat ) ) )
=> ~ ( accp_transition @ state_set_rel @ ( split_trans @ S4 @ S5 ) ) ) ) ) ) ) ) ).
% state_set.pelims
thf(fact_390_fmla__set_Osimps_I3_J,axiom,
! [V2: nat,Va2: nat] :
( ( fmla_set @ ( split_trans @ V2 @ Va2 ) )
= bot_bot_set_nat ) ).
% fmla_set.simps(3)
thf(fact_391_removeAll__id,axiom,
! [X: nat,Xs: list_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( removeAll_nat @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_392_removeAll__id,axiom,
! [X: transition,Xs: list_transition] :
( ~ ( member_transition @ X @ ( set_transition2 @ Xs ) )
=> ( ( removeAll_transition @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_393_removeAll_Osimps_I1_J,axiom,
! [X: transition] :
( ( removeAll_transition @ X @ nil_transition )
= nil_transition ) ).
% removeAll.simps(1)
thf(fact_394_removeAll_Osimps_I1_J,axiom,
! [X: $o] :
( ( removeAll_o @ X @ nil_o )
= nil_o ) ).
% removeAll.simps(1)
thf(fact_395_removeAll_Osimps_I1_J,axiom,
! [X: list_o] :
( ( removeAll_list_o @ X @ nil_list_o )
= nil_list_o ) ).
% removeAll.simps(1)
thf(fact_396_run__accept__eps__empty,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,Bss: list_list_o,Bs: list_o] :
~ ( run_accept_eps @ Q0 @ Qf @ Transs @ bot_bot_set_nat @ Bss @ Bs ) ).
% run_accept_eps_empty
thf(fact_397_run__accept__eps__Nil__eps,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,R: set_nat,Bs: list_o] :
( ( run_accept_eps @ Q0 @ Qf @ Transs @ ( step_eps_closure_set @ Q0 @ Transs @ R @ Bs ) @ nil_list_o @ Bs )
= ( run_accept_eps @ Q0 @ Qf @ Transs @ R @ nil_list_o @ Bs ) ) ).
% run_accept_eps_Nil_eps
thf(fact_398_run__accept__eps__split,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,R: set_nat,S: set_nat,Bss: list_list_o,Bs: list_o] :
( ( run_accept_eps @ Q0 @ Qf @ Transs @ ( sup_sup_set_nat @ R @ S ) @ Bss @ Bs )
= ( ( run_accept_eps @ Q0 @ Qf @ Transs @ R @ Bss @ Bs )
| ( run_accept_eps @ Q0 @ Qf @ Transs @ S @ Bss @ Bs ) ) ) ).
% run_accept_eps_split
thf(fact_399_remove__code_I1_J,axiom,
! [X: transition,Xs: list_transition] :
( ( remove_transition @ X @ ( set_transition2 @ Xs ) )
= ( set_transition2 @ ( removeAll_transition @ X @ Xs ) ) ) ).
% remove_code(1)
thf(fact_400_fmla__set_Osimps_I2_J,axiom,
! [V2: nat] :
( ( fmla_set @ ( symb_trans @ V2 ) )
= bot_bot_set_nat ) ).
% fmla_set.simps(2)
thf(fact_401_fmla__set_Opelims,axiom,
! [X: transition,Y2: set_nat] :
( ( ( fmla_set @ X )
= Y2 )
=> ( ( accp_transition @ fmla_set_rel @ X )
=> ( ! [Uu2: nat,N: nat] :
( ( X
= ( eps_trans @ Uu2 @ N ) )
=> ( ( Y2
= ( insert_nat2 @ N @ bot_bot_set_nat ) )
=> ~ ( accp_transition @ fmla_set_rel @ ( eps_trans @ Uu2 @ N ) ) ) )
=> ( ! [V: nat] :
( ( X
= ( symb_trans @ V ) )
=> ( ( Y2 = bot_bot_set_nat )
=> ~ ( accp_transition @ fmla_set_rel @ ( symb_trans @ V ) ) ) )
=> ~ ! [V: nat,Va: nat] :
( ( X
= ( split_trans @ V @ Va ) )
=> ( ( Y2 = bot_bot_set_nat )
=> ~ ( accp_transition @ fmla_set_rel @ ( split_trans @ V @ Va ) ) ) ) ) ) ) ) ).
% fmla_set.pelims
thf(fact_402_run__accept__eps__Cons__eps__split,axiom,
! [Q0: nat,Transs: list_transition,R: set_nat,Cs: list_o,S: set_nat,Qf: nat,Css: list_list_o,Bs: list_o] :
( ( ( step_eps_closure_set @ Q0 @ Transs @ R @ Cs )
= ( sup_sup_set_nat @ R @ S ) )
=> ( ( ( step_symb_set @ Q0 @ Transs @ R )
= bot_bot_set_nat )
=> ( ~ ( member_nat @ Qf @ R )
=> ( ( run_accept_eps @ Q0 @ Qf @ Transs @ R @ ( cons_list_o @ Cs @ Css ) @ Bs )
= ( run_accept_eps @ Q0 @ Qf @ Transs @ S @ ( cons_list_o @ Cs @ Css ) @ Bs ) ) ) ) ) ).
% run_accept_eps_Cons_eps_split
thf(fact_403_delta__eps__split,axiom,
! [Q0: nat,Transs: list_transition,R: set_nat,Bs: list_o,S: set_nat] :
( ( ( step_eps_closure_set @ Q0 @ Transs @ R @ Bs )
= ( sup_sup_set_nat @ R @ S ) )
=> ( ( delta @ Q0 @ Transs @ R @ Bs )
= ( sup_sup_set_nat @ ( step_symb_set @ Q0 @ Transs @ R ) @ ( delta @ Q0 @ Transs @ S @ Bs ) ) ) ) ).
% delta_eps_split
thf(fact_404_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_o @ ( coset_o @ nil_o ) @ ( set_o2 @ nil_o ) ) ).
% subset_code(3)
thf(fact_405_subset__code_I3_J,axiom,
~ ( ord_le6901083488122529182list_o @ ( coset_list_o @ nil_list_o ) @ ( set_list_o2 @ nil_list_o ) ) ).
% subset_code(3)
thf(fact_406_subset__code_I3_J,axiom,
~ ( ord_le8419162016481440574sition @ ( coset_transition @ nil_transition ) @ ( set_transition2 @ nil_transition ) ) ).
% subset_code(3)
thf(fact_407_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_nat @ ( coset_nat @ nil_nat ) @ ( set_nat2 @ nil_nat ) ) ).
% subset_code(3)
thf(fact_408_List_Oset__insert,axiom,
! [X: nat,Xs: list_nat] :
( ( set_nat2 @ ( insert_nat @ X @ Xs ) )
= ( insert_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_409_List_Oset__insert,axiom,
! [X: transition,Xs: list_transition] :
( ( set_transition2 @ ( insert_transition @ X @ Xs ) )
= ( insert_transition2 @ X @ ( set_transition2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_410_run__accept__eps__Nil,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,R: set_nat,Cs: list_o] :
( ( run_accept_eps @ Q0 @ Qf @ Transs @ R @ nil_list_o @ Cs )
= ( accept_eps @ Q0 @ Qf @ Transs @ R @ Cs ) ) ).
% run_accept_eps_Nil
thf(fact_411_list_Oinject,axiom,
! [X21: list_o,X222: list_list_o,Y21: list_o,Y222: list_list_o] :
( ( ( cons_list_o @ X21 @ X222 )
= ( cons_list_o @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_412_in__set__insert,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_413_in__set__insert,axiom,
! [X: transition,Xs: list_transition] :
( ( member_transition @ X @ ( set_transition2 @ Xs ) )
=> ( ( insert_transition @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_414_list_Osimps_I15_J,axiom,
! [X21: nat,X222: list_nat] :
( ( set_nat2 @ ( cons_nat @ X21 @ X222 ) )
= ( insert_nat2 @ X21 @ ( set_nat2 @ X222 ) ) ) ).
% list.simps(15)
thf(fact_415_list_Osimps_I15_J,axiom,
! [X21: transition,X222: list_transition] :
( ( set_transition2 @ ( cons_transition @ X21 @ X222 ) )
= ( insert_transition2 @ X21 @ ( set_transition2 @ X222 ) ) ) ).
% list.simps(15)
thf(fact_416_list_Osimps_I15_J,axiom,
! [X21: list_o,X222: list_list_o] :
( ( set_list_o2 @ ( cons_list_o @ X21 @ X222 ) )
= ( insert_list_o2 @ X21 @ ( set_list_o2 @ X222 ) ) ) ).
% list.simps(15)
thf(fact_417_insert__Nil,axiom,
! [X: transition] :
( ( insert_transition @ X @ nil_transition )
= ( cons_transition @ X @ nil_transition ) ) ).
% insert_Nil
thf(fact_418_insert__Nil,axiom,
! [X: $o] :
( ( insert_o @ X @ nil_o )
= ( cons_o @ X @ nil_o ) ) ).
% insert_Nil
thf(fact_419_insert__Nil,axiom,
! [X: list_o] :
( ( insert_list_o @ X @ nil_list_o )
= ( cons_list_o @ X @ nil_list_o ) ) ).
% insert_Nil
thf(fact_420_not__in__set__insert,axiom,
! [X: nat,Xs: list_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X @ Xs )
= ( cons_nat @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_421_not__in__set__insert,axiom,
! [X: transition,Xs: list_transition] :
( ~ ( member_transition @ X @ ( set_transition2 @ Xs ) )
=> ( ( insert_transition @ X @ Xs )
= ( cons_transition @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_422_not__in__set__insert,axiom,
! [X: list_o,Xs: list_list_o] :
( ~ ( member_list_o @ X @ ( set_list_o2 @ Xs ) )
=> ( ( insert_list_o @ X @ Xs )
= ( cons_list_o @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_423_run__accept__eps__Cons,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,R: set_nat,Bs: list_o,Bss: list_list_o,Cs: list_o] :
( ( run_accept_eps @ Q0 @ Qf @ Transs @ R @ ( cons_list_o @ Bs @ Bss ) @ Cs )
= ( run_accept_eps @ Q0 @ Qf @ Transs @ ( delta @ Q0 @ Transs @ R @ Bs ) @ Bss @ Cs ) ) ).
% run_accept_eps_Cons
thf(fact_424_run__accept__eps__Cons__delta__cong,axiom,
! [Q0: nat,Transs: list_transition,R: set_nat,Bs: list_o,S: set_nat,Qf: nat,Bss: list_list_o,Cs: list_o] :
( ( ( delta @ Q0 @ Transs @ R @ Bs )
= ( delta @ Q0 @ Transs @ S @ Bs ) )
=> ( ( run_accept_eps @ Q0 @ Qf @ Transs @ R @ ( cons_list_o @ Bs @ Bss ) @ Cs )
= ( run_accept_eps @ Q0 @ Qf @ Transs @ S @ ( cons_list_o @ Bs @ Bss ) @ Cs ) ) ) ).
% run_accept_eps_Cons_delta_cong
thf(fact_425_not__Cons__self2,axiom,
! [X: list_o,Xs: list_list_o] :
( ( cons_list_o @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_426_List_Oinsert__def,axiom,
( insert_nat
= ( ^ [X2: nat,Xs2: list_nat] : ( if_list_nat @ ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) ) @ Xs2 @ ( cons_nat @ X2 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_427_List_Oinsert__def,axiom,
( insert_transition
= ( ^ [X2: transition,Xs2: list_transition] : ( if_list_transition @ ( member_transition @ X2 @ ( set_transition2 @ Xs2 ) ) @ Xs2 @ ( cons_transition @ X2 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_428_List_Oinsert__def,axiom,
( insert_list_o
= ( ^ [X2: list_o,Xs2: list_list_o] : ( if_list_list_o @ ( member_list_o @ X2 @ ( set_list_o2 @ Xs2 ) ) @ Xs2 @ ( cons_list_o @ X2 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_429_transpose_Ocases,axiom,
! [X: list_list_transition] :
( ( X != nil_list_transition )
=> ( ! [Xss: list_list_transition] :
( X
!= ( cons_list_transition @ nil_transition @ Xss ) )
=> ~ ! [X3: transition,Xs3: list_transition,Xss: list_list_transition] :
( X
!= ( cons_list_transition @ ( cons_transition @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_430_transpose_Ocases,axiom,
! [X: list_list_list_o] :
( ( X != nil_list_list_o )
=> ( ! [Xss: list_list_list_o] :
( X
!= ( cons_list_list_o @ nil_list_o @ Xss ) )
=> ~ ! [X3: list_o,Xs3: list_list_o,Xss: list_list_list_o] :
( X
!= ( cons_list_list_o @ ( cons_list_o @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_431_transpose_Ocases,axiom,
! [X: list_list_o] :
( ( X != nil_list_o )
=> ( ! [Xss: list_list_o] :
( X
!= ( cons_list_o @ nil_o @ Xss ) )
=> ~ ! [X3: $o,Xs3: list_o,Xss: list_list_o] :
( X
!= ( cons_list_o @ ( cons_o @ X3 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_432_list_Odistinct_I1_J,axiom,
! [X21: transition,X222: list_transition] :
( nil_transition
!= ( cons_transition @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_433_list_Odistinct_I1_J,axiom,
! [X21: $o,X222: list_o] :
( nil_o
!= ( cons_o @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_434_list_Odistinct_I1_J,axiom,
! [X21: list_o,X222: list_list_o] :
( nil_list_o
!= ( cons_list_o @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_435_list_OdiscI,axiom,
! [List: list_transition,X21: transition,X222: list_transition] :
( ( List
= ( cons_transition @ X21 @ X222 ) )
=> ( List != nil_transition ) ) ).
% list.discI
thf(fact_436_list_OdiscI,axiom,
! [List: list_o,X21: $o,X222: list_o] :
( ( List
= ( cons_o @ X21 @ X222 ) )
=> ( List != nil_o ) ) ).
% list.discI
thf(fact_437_list_OdiscI,axiom,
! [List: list_list_o,X21: list_o,X222: list_list_o] :
( ( List
= ( cons_list_o @ X21 @ X222 ) )
=> ( List != nil_list_o ) ) ).
% list.discI
thf(fact_438_list_Oexhaust,axiom,
! [Y2: list_transition] :
( ( Y2 != nil_transition )
=> ~ ! [X212: transition,X223: list_transition] :
( Y2
!= ( cons_transition @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_439_list_Oexhaust,axiom,
! [Y2: list_o] :
( ( Y2 != nil_o )
=> ~ ! [X212: $o,X223: list_o] :
( Y2
!= ( cons_o @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_440_list_Oexhaust,axiom,
! [Y2: list_list_o] :
( ( Y2 != nil_list_o )
=> ~ ! [X212: list_o,X223: list_list_o] :
( Y2
!= ( cons_list_o @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_441_min__list_Ocases,axiom,
! [X: list_o] :
( ! [X3: $o,Xs3: list_o] :
( X
!= ( cons_o @ X3 @ Xs3 ) )
=> ( X = nil_o ) ) ).
% min_list.cases
thf(fact_442_remdups__adj_Ocases,axiom,
! [X: list_transition] :
( ( X != nil_transition )
=> ( ! [X3: transition] :
( X
!= ( cons_transition @ X3 @ nil_transition ) )
=> ~ ! [X3: transition,Y: transition,Xs3: list_transition] :
( X
!= ( cons_transition @ X3 @ ( cons_transition @ Y @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_443_remdups__adj_Ocases,axiom,
! [X: list_o] :
( ( X != nil_o )
=> ( ! [X3: $o] :
( X
!= ( cons_o @ X3 @ nil_o ) )
=> ~ ! [X3: $o,Y: $o,Xs3: list_o] :
( X
!= ( cons_o @ X3 @ ( cons_o @ Y @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_444_remdups__adj_Ocases,axiom,
! [X: list_list_o] :
( ( X != nil_list_o )
=> ( ! [X3: list_o] :
( X
!= ( cons_list_o @ X3 @ nil_list_o ) )
=> ~ ! [X3: list_o,Y: list_o,Xs3: list_list_o] :
( X
!= ( cons_list_o @ X3 @ ( cons_list_o @ Y @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_445_neq__Nil__conv,axiom,
! [Xs: list_transition] :
( ( Xs != nil_transition )
= ( ? [Y4: transition,Ys2: list_transition] :
( Xs
= ( cons_transition @ Y4 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_446_neq__Nil__conv,axiom,
! [Xs: list_o] :
( ( Xs != nil_o )
= ( ? [Y4: $o,Ys2: list_o] :
( Xs
= ( cons_o @ Y4 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_447_neq__Nil__conv,axiom,
! [Xs: list_list_o] :
( ( Xs != nil_list_o )
= ( ? [Y4: list_o,Ys2: list_list_o] :
( Xs
= ( cons_list_o @ Y4 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_448_list__induct2_H,axiom,
! [P: list_transition > list_transition > $o,Xs: list_transition,Ys: list_transition] :
( ( P @ nil_transition @ nil_transition )
=> ( ! [X3: transition,Xs3: list_transition] : ( P @ ( cons_transition @ X3 @ Xs3 ) @ nil_transition )
=> ( ! [Y: transition,Ys3: list_transition] : ( P @ nil_transition @ ( cons_transition @ Y @ Ys3 ) )
=> ( ! [X3: transition,Xs3: list_transition,Y: transition,Ys3: list_transition] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_transition @ X3 @ Xs3 ) @ ( cons_transition @ Y @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_449_list__induct2_H,axiom,
! [P: list_transition > list_o > $o,Xs: list_transition,Ys: list_o] :
( ( P @ nil_transition @ nil_o )
=> ( ! [X3: transition,Xs3: list_transition] : ( P @ ( cons_transition @ X3 @ Xs3 ) @ nil_o )
=> ( ! [Y: $o,Ys3: list_o] : ( P @ nil_transition @ ( cons_o @ Y @ Ys3 ) )
=> ( ! [X3: transition,Xs3: list_transition,Y: $o,Ys3: list_o] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_transition @ X3 @ Xs3 ) @ ( cons_o @ Y @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_450_list__induct2_H,axiom,
! [P: list_o > list_transition > $o,Xs: list_o,Ys: list_transition] :
( ( P @ nil_o @ nil_transition )
=> ( ! [X3: $o,Xs3: list_o] : ( P @ ( cons_o @ X3 @ Xs3 ) @ nil_transition )
=> ( ! [Y: transition,Ys3: list_transition] : ( P @ nil_o @ ( cons_transition @ Y @ Ys3 ) )
=> ( ! [X3: $o,Xs3: list_o,Y: transition,Ys3: list_transition] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_o @ X3 @ Xs3 ) @ ( cons_transition @ Y @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_451_list__induct2_H,axiom,
! [P: list_o > list_o > $o,Xs: list_o,Ys: list_o] :
( ( P @ nil_o @ nil_o )
=> ( ! [X3: $o,Xs3: list_o] : ( P @ ( cons_o @ X3 @ Xs3 ) @ nil_o )
=> ( ! [Y: $o,Ys3: list_o] : ( P @ nil_o @ ( cons_o @ Y @ Ys3 ) )
=> ( ! [X3: $o,Xs3: list_o,Y: $o,Ys3: list_o] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_o @ X3 @ Xs3 ) @ ( cons_o @ Y @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_452_list__induct2_H,axiom,
! [P: list_transition > list_list_o > $o,Xs: list_transition,Ys: list_list_o] :
( ( P @ nil_transition @ nil_list_o )
=> ( ! [X3: transition,Xs3: list_transition] : ( P @ ( cons_transition @ X3 @ Xs3 ) @ nil_list_o )
=> ( ! [Y: list_o,Ys3: list_list_o] : ( P @ nil_transition @ ( cons_list_o @ Y @ Ys3 ) )
=> ( ! [X3: transition,Xs3: list_transition,Y: list_o,Ys3: list_list_o] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_transition @ X3 @ Xs3 ) @ ( cons_list_o @ Y @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_453_list__induct2_H,axiom,
! [P: list_o > list_list_o > $o,Xs: list_o,Ys: list_list_o] :
( ( P @ nil_o @ nil_list_o )
=> ( ! [X3: $o,Xs3: list_o] : ( P @ ( cons_o @ X3 @ Xs3 ) @ nil_list_o )
=> ( ! [Y: list_o,Ys3: list_list_o] : ( P @ nil_o @ ( cons_list_o @ Y @ Ys3 ) )
=> ( ! [X3: $o,Xs3: list_o,Y: list_o,Ys3: list_list_o] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_o @ X3 @ Xs3 ) @ ( cons_list_o @ Y @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_454_list__induct2_H,axiom,
! [P: list_list_o > list_transition > $o,Xs: list_list_o,Ys: list_transition] :
( ( P @ nil_list_o @ nil_transition )
=> ( ! [X3: list_o,Xs3: list_list_o] : ( P @ ( cons_list_o @ X3 @ Xs3 ) @ nil_transition )
=> ( ! [Y: transition,Ys3: list_transition] : ( P @ nil_list_o @ ( cons_transition @ Y @ Ys3 ) )
=> ( ! [X3: list_o,Xs3: list_list_o,Y: transition,Ys3: list_transition] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_list_o @ X3 @ Xs3 ) @ ( cons_transition @ Y @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_455_list__induct2_H,axiom,
! [P: list_list_o > list_o > $o,Xs: list_list_o,Ys: list_o] :
( ( P @ nil_list_o @ nil_o )
=> ( ! [X3: list_o,Xs3: list_list_o] : ( P @ ( cons_list_o @ X3 @ Xs3 ) @ nil_o )
=> ( ! [Y: $o,Ys3: list_o] : ( P @ nil_list_o @ ( cons_o @ Y @ Ys3 ) )
=> ( ! [X3: list_o,Xs3: list_list_o,Y: $o,Ys3: list_o] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_list_o @ X3 @ Xs3 ) @ ( cons_o @ Y @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_456_list__induct2_H,axiom,
! [P: list_list_o > list_list_o > $o,Xs: list_list_o,Ys: list_list_o] :
( ( P @ nil_list_o @ nil_list_o )
=> ( ! [X3: list_o,Xs3: list_list_o] : ( P @ ( cons_list_o @ X3 @ Xs3 ) @ nil_list_o )
=> ( ! [Y: list_o,Ys3: list_list_o] : ( P @ nil_list_o @ ( cons_list_o @ Y @ Ys3 ) )
=> ( ! [X3: list_o,Xs3: list_list_o,Y: list_o,Ys3: list_list_o] :
( ( P @ Xs3 @ Ys3 )
=> ( P @ ( cons_list_o @ X3 @ Xs3 ) @ ( cons_list_o @ Y @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_457_list__nonempty__induct,axiom,
! [Xs: list_transition,P: list_transition > $o] :
( ( Xs != nil_transition )
=> ( ! [X3: transition] : ( P @ ( cons_transition @ X3 @ nil_transition ) )
=> ( ! [X3: transition,Xs3: list_transition] :
( ( Xs3 != nil_transition )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_transition @ X3 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_458_list__nonempty__induct,axiom,
! [Xs: list_o,P: list_o > $o] :
( ( Xs != nil_o )
=> ( ! [X3: $o] : ( P @ ( cons_o @ X3 @ nil_o ) )
=> ( ! [X3: $o,Xs3: list_o] :
( ( Xs3 != nil_o )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_o @ X3 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_459_list__nonempty__induct,axiom,
! [Xs: list_list_o,P: list_list_o > $o] :
( ( Xs != nil_list_o )
=> ( ! [X3: list_o] : ( P @ ( cons_list_o @ X3 @ nil_list_o ) )
=> ( ! [X3: list_o,Xs3: list_list_o] :
( ( Xs3 != nil_list_o )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_list_o @ X3 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_460_list_Oset__intros_I2_J,axiom,
! [Y2: nat,X222: list_nat,X21: nat] :
( ( member_nat @ Y2 @ ( set_nat2 @ X222 ) )
=> ( member_nat @ Y2 @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_461_list_Oset__intros_I2_J,axiom,
! [Y2: transition,X222: list_transition,X21: transition] :
( ( member_transition @ Y2 @ ( set_transition2 @ X222 ) )
=> ( member_transition @ Y2 @ ( set_transition2 @ ( cons_transition @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_462_list_Oset__intros_I2_J,axiom,
! [Y2: list_o,X222: list_list_o,X21: list_o] :
( ( member_list_o @ Y2 @ ( set_list_o2 @ X222 ) )
=> ( member_list_o @ Y2 @ ( set_list_o2 @ ( cons_list_o @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_463_list_Oset__intros_I1_J,axiom,
! [X21: nat,X222: list_nat] : ( member_nat @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_464_list_Oset__intros_I1_J,axiom,
! [X21: transition,X222: list_transition] : ( member_transition @ X21 @ ( set_transition2 @ ( cons_transition @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_465_list_Oset__intros_I1_J,axiom,
! [X21: list_o,X222: list_list_o] : ( member_list_o @ X21 @ ( set_list_o2 @ ( cons_list_o @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_466_list_Oset__cases,axiom,
! [E: nat,A: list_nat] :
( ( member_nat @ E @ ( set_nat2 @ A ) )
=> ( ! [Z22: list_nat] :
( A
!= ( cons_nat @ E @ Z22 ) )
=> ~ ! [Z1: nat,Z22: list_nat] :
( ( A
= ( cons_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat @ E @ ( set_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_467_list_Oset__cases,axiom,
! [E: transition,A: list_transition] :
( ( member_transition @ E @ ( set_transition2 @ A ) )
=> ( ! [Z22: list_transition] :
( A
!= ( cons_transition @ E @ Z22 ) )
=> ~ ! [Z1: transition,Z22: list_transition] :
( ( A
= ( cons_transition @ Z1 @ Z22 ) )
=> ~ ( member_transition @ E @ ( set_transition2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_468_list_Oset__cases,axiom,
! [E: list_o,A: list_list_o] :
( ( member_list_o @ E @ ( set_list_o2 @ A ) )
=> ( ! [Z22: list_list_o] :
( A
!= ( cons_list_o @ E @ Z22 ) )
=> ~ ! [Z1: list_o,Z22: list_list_o] :
( ( A
= ( cons_list_o @ Z1 @ Z22 ) )
=> ~ ( member_list_o @ E @ ( set_list_o2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_469_set__ConsD,axiom,
! [Y2: nat,X: nat,Xs: list_nat] :
( ( member_nat @ Y2 @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) )
=> ( ( Y2 = X )
| ( member_nat @ Y2 @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_470_set__ConsD,axiom,
! [Y2: transition,X: transition,Xs: list_transition] :
( ( member_transition @ Y2 @ ( set_transition2 @ ( cons_transition @ X @ Xs ) ) )
=> ( ( Y2 = X )
| ( member_transition @ Y2 @ ( set_transition2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_471_set__ConsD,axiom,
! [Y2: list_o,X: list_o,Xs: list_list_o] :
( ( member_list_o @ Y2 @ ( set_list_o2 @ ( cons_list_o @ X @ Xs ) ) )
=> ( ( Y2 = X )
| ( member_list_o @ Y2 @ ( set_list_o2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_472_removeAll_Osimps_I2_J,axiom,
! [X: list_o,Y2: list_o,Xs: list_list_o] :
( ( ( X = Y2 )
=> ( ( removeAll_list_o @ X @ ( cons_list_o @ Y2 @ Xs ) )
= ( removeAll_list_o @ X @ Xs ) ) )
& ( ( X != Y2 )
=> ( ( removeAll_list_o @ X @ ( cons_list_o @ Y2 @ Xs ) )
= ( cons_list_o @ Y2 @ ( removeAll_list_o @ X @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_473_delta__eps,axiom,
! [Q0: nat,Transs: list_transition,R: set_nat,Bs: list_o] :
( ( delta @ Q0 @ Transs @ ( step_eps_closure_set @ Q0 @ Transs @ R @ Bs ) @ Bs )
= ( delta @ Q0 @ Transs @ R @ Bs ) ) ).
% delta_eps
thf(fact_474_delta__split,axiom,
! [Q0: nat,Transs: list_transition,R: set_nat,S: set_nat,Bs: list_o] :
( ( delta @ Q0 @ Transs @ ( sup_sup_set_nat @ R @ S ) @ Bs )
= ( sup_sup_set_nat @ ( delta @ Q0 @ Transs @ R @ Bs ) @ ( delta @ Q0 @ Transs @ S @ Bs ) ) ) ).
% delta_split
thf(fact_475_set__subset__Cons,axiom,
! [Xs: list_transition,X: transition] : ( ord_le8419162016481440574sition @ ( set_transition2 @ Xs ) @ ( set_transition2 @ ( cons_transition @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_476_set__subset__Cons,axiom,
! [Xs: list_list_o,X: list_o] : ( ord_le6901083488122529182list_o @ ( set_list_o2 @ Xs ) @ ( set_list_o2 @ ( cons_list_o @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_477_set__subset__Cons,axiom,
! [Xs: list_nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_478_accept__eps__empty,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,Bs: list_o] :
~ ( accept_eps @ Q0 @ Qf @ Transs @ bot_bot_set_nat @ Bs ) ).
% accept_eps_empty
thf(fact_479_NFA_Oaccept__eps__def,axiom,
( accept_eps
= ( ^ [Q02: nat,Qf2: nat,Transs2: list_transition,R2: set_nat,Bs2: list_o] : ( member_nat @ Qf2 @ ( step_eps_closure_set @ Q02 @ Transs2 @ R2 @ Bs2 ) ) ) ) ).
% NFA.accept_eps_def
thf(fact_480_accept__eps__split,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,R: set_nat,S: set_nat,Bs: list_o] :
( ( accept_eps @ Q0 @ Qf @ Transs @ ( sup_sup_set_nat @ R @ S ) @ Bs )
= ( ( accept_eps @ Q0 @ Qf @ Transs @ R @ Bs )
| ( accept_eps @ Q0 @ Qf @ Transs @ S @ Bs ) ) ) ).
% accept_eps_split
thf(fact_481_step__eps__accept__eps,axiom,
! [Q0: nat,Transs: list_transition,Bs: list_o,Q: nat,Qf: nat,R: set_nat] :
( ( step_eps @ Q0 @ Transs @ Bs @ Q @ Qf )
=> ( ( member_nat @ Q @ R )
=> ( accept_eps @ Q0 @ Qf @ Transs @ R @ Bs ) ) ) ).
% step_eps_accept_eps
thf(fact_482_run__accept__eps__Cons__eps,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,R: set_nat,Cs: list_o,Css: list_list_o,Bs: list_o] :
( ( run_accept_eps @ Q0 @ Qf @ Transs @ ( step_eps_closure_set @ Q0 @ Transs @ R @ Cs ) @ ( cons_list_o @ Cs @ Css ) @ Bs )
= ( run_accept_eps @ Q0 @ Qf @ Transs @ R @ ( cons_list_o @ Cs @ Css ) @ Bs ) ) ).
% run_accept_eps_Cons_eps
thf(fact_483_subset__code_I2_J,axiom,
! [A2: set_transition,Ys: list_transition] :
( ( ord_le8419162016481440574sition @ A2 @ ( coset_transition @ Ys ) )
= ( ! [X2: transition] :
( ( member_transition @ X2 @ ( set_transition2 @ Ys ) )
=> ~ ( member_transition @ X2 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_484_subset__code_I2_J,axiom,
! [A2: set_nat,Ys: list_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( coset_nat @ Ys ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Ys ) )
=> ~ ( member_nat @ X2 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_485_delta__sub__eps__mono,axiom,
! [S: set_nat,Q0: nat,Transs: list_transition,R: set_nat,Bs: list_o] :
( ( ord_less_eq_set_nat @ S @ ( step_eps_closure_set @ Q0 @ Transs @ R @ Bs ) )
=> ( ord_less_eq_set_nat @ ( delta @ Q0 @ Transs @ S @ Bs ) @ ( delta @ Q0 @ Transs @ R @ Bs ) ) ) ).
% delta_sub_eps_mono
thf(fact_486_NFA_Odelta__def,axiom,
( delta
= ( ^ [Q02: nat,Transs2: list_transition,R2: set_nat,Bs2: list_o] : ( step_symb_set @ Q02 @ Transs2 @ ( step_eps_closure_set @ Q02 @ Transs2 @ R2 @ Bs2 ) ) ) ) ).
% NFA.delta_def
thf(fact_487_delta__step__symb__set__absorb,axiom,
( delta
= ( ^ [Q02: nat,Transs2: list_transition,R2: set_nat,Bs2: list_o] : ( sup_sup_set_nat @ ( delta @ Q02 @ Transs2 @ R2 @ Bs2 ) @ ( step_symb_set @ Q02 @ Transs2 @ R2 ) ) ) ) ).
% delta_step_symb_set_absorb
thf(fact_488_the__elem__set,axiom,
! [X: $o] :
( ( the_elem_o @ ( set_o2 @ ( cons_o @ X @ nil_o ) ) )
= X ) ).
% the_elem_set
thf(fact_489_the__elem__set,axiom,
! [X: transition] :
( ( the_elem_transition @ ( set_transition2 @ ( cons_transition @ X @ nil_transition ) ) )
= X ) ).
% the_elem_set
thf(fact_490_the__elem__set,axiom,
! [X: list_o] :
( ( the_elem_list_o @ ( set_list_o2 @ ( cons_list_o @ X @ nil_list_o ) ) )
= X ) ).
% the_elem_set
thf(fact_491_insert__code_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( insert_nat2 @ X @ ( coset_nat @ Xs ) )
= ( coset_nat @ ( removeAll_nat @ X @ Xs ) ) ) ).
% insert_code(2)
thf(fact_492_insert__code_I2_J,axiom,
! [X: transition,Xs: list_transition] :
( ( insert_transition2 @ X @ ( coset_transition @ Xs ) )
= ( coset_transition @ ( removeAll_transition @ X @ Xs ) ) ) ).
% insert_code(2)
thf(fact_493_NFA_Oaccept__def,axiom,
( accept
= ( ^ [Q02: nat,Qf2: nat,Transs2: list_transition,R2: set_nat] : ( accept_eps @ Q02 @ Qf2 @ Transs2 @ R2 @ nil_o ) ) ) ).
% NFA.accept_def
thf(fact_494_run__eps__split,axiom,
! [Q0: nat,Transs: list_transition,R: set_nat,Bs: list_o,S: set_nat,Bss: list_list_o] :
( ( ( step_eps_closure_set @ Q0 @ Transs @ R @ Bs )
= ( sup_sup_set_nat @ R @ S ) )
=> ( ( ( step_symb_set @ Q0 @ Transs @ R )
= bot_bot_set_nat )
=> ( ( run @ Q0 @ Transs @ R @ ( cons_list_o @ Bs @ Bss ) )
= ( run @ Q0 @ Transs @ S @ ( cons_list_o @ Bs @ Bss ) ) ) ) ) ).
% run_eps_split
thf(fact_495_product__lists_Osimps_I1_J,axiom,
( ( produc6248909823095439149sition @ nil_list_transition )
= ( cons_list_transition @ nil_transition @ nil_list_transition ) ) ).
% product_lists.simps(1)
thf(fact_496_product__lists_Osimps_I1_J,axiom,
( ( product_lists_list_o @ nil_list_list_o )
= ( cons_list_list_o @ nil_list_o @ nil_list_list_o ) ) ).
% product_lists.simps(1)
thf(fact_497_product__lists_Osimps_I1_J,axiom,
( ( product_lists_o @ nil_list_o )
= ( cons_list_o @ nil_o @ nil_list_o ) ) ).
% product_lists.simps(1)
thf(fact_498_subseqs_Osimps_I1_J,axiom,
( ( subseqs_transition @ nil_transition )
= ( cons_list_transition @ nil_transition @ nil_list_transition ) ) ).
% subseqs.simps(1)
thf(fact_499_subseqs_Osimps_I1_J,axiom,
( ( subseqs_list_o @ nil_list_o )
= ( cons_list_list_o @ nil_list_o @ nil_list_list_o ) ) ).
% subseqs.simps(1)
thf(fact_500_subseqs_Osimps_I1_J,axiom,
( ( subseqs_o @ nil_o )
= ( cons_list_o @ nil_o @ nil_list_o ) ) ).
% subseqs.simps(1)
thf(fact_501_Set_Ois__empty__def,axiom,
( is_empty_nat
= ( ^ [A4: set_nat] : ( A4 = bot_bot_set_nat ) ) ) ).
% Set.is_empty_def
thf(fact_502_Set_Ois__empty__def,axiom,
( is_empty_transition
= ( ^ [A4: set_transition] : ( A4 = bot_bo301567166201926666sition ) ) ) ).
% Set.is_empty_def
thf(fact_503_step__symb__sucs__sound,axiom,
! [Q2: nat,Q0: nat,Transs: list_transition,Q: nat] :
( ( member_nat @ Q2 @ ( step_symb_sucs @ Q0 @ Transs @ Q ) )
= ( step_symb @ Q0 @ Transs @ Q @ Q2 ) ) ).
% step_symb_sucs_sound
thf(fact_504_run__empty,axiom,
! [Q0: nat,Transs: list_transition,Bss: list_list_o] :
( ( run @ Q0 @ Transs @ bot_bot_set_nat @ Bss )
= bot_bot_set_nat ) ).
% run_empty
thf(fact_505_Cons__in__subseqsD,axiom,
! [Y2: list_o,Ys: list_list_o,Xs: list_list_o] :
( ( member_list_list_o @ ( cons_list_o @ Y2 @ Ys ) @ ( set_list_list_o2 @ ( subseqs_list_o @ Xs ) ) )
=> ( member_list_list_o @ Ys @ ( set_list_list_o2 @ ( subseqs_list_o @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_506_run__split,axiom,
! [Q0: nat,Transs: list_transition,R: set_nat,S: set_nat,Bss: list_list_o] :
( ( run @ Q0 @ Transs @ ( sup_sup_set_nat @ R @ S ) @ Bss )
= ( sup_sup_set_nat @ ( run @ Q0 @ Transs @ R @ Bss ) @ ( run @ Q0 @ Transs @ S @ Bss ) ) ) ).
% run_split
thf(fact_507_run__Nil,axiom,
! [Q0: nat,Transs: list_transition,R: set_nat] :
( ( run @ Q0 @ Transs @ R @ nil_list_o )
= R ) ).
% run_Nil
thf(fact_508_NFA_Orun__accept__eps__def,axiom,
( run_accept_eps
= ( ^ [Q02: nat,Qf2: nat,Transs2: list_transition,R2: set_nat,Bss2: list_list_o] : ( accept_eps @ Q02 @ Qf2 @ Transs2 @ ( run @ Q02 @ Transs2 @ R2 @ Bss2 ) ) ) ) ).
% NFA.run_accept_eps_def
thf(fact_509_run__Cons,axiom,
! [Q0: nat,Transs: list_transition,R: set_nat,Bs: list_o,Bss: list_list_o] :
( ( run @ Q0 @ Transs @ R @ ( cons_list_o @ Bs @ Bss ) )
= ( run @ Q0 @ Transs @ ( delta @ Q0 @ Transs @ R @ Bs ) @ Bss ) ) ).
% run_Cons
thf(fact_510_NFA_Orun__accept__def,axiom,
( run_accept
= ( ^ [Q02: nat,Qf2: nat,Transs2: list_transition,R2: set_nat,Bss2: list_list_o] : ( accept @ Q02 @ Qf2 @ Transs2 @ ( run @ Q02 @ Transs2 @ R2 @ Bss2 ) ) ) ) ).
% NFA.run_accept_def
thf(fact_511_is__empty__set,axiom,
! [Xs: list_transition] :
( ( is_empty_transition @ ( set_transition2 @ Xs ) )
= ( null_transition @ Xs ) ) ).
% is_empty_set
thf(fact_512_list__ex1__simps_I1_J,axiom,
! [P: transition > $o] :
~ ( list_ex1_transition @ P @ nil_transition ) ).
% list_ex1_simps(1)
thf(fact_513_list__ex1__simps_I1_J,axiom,
! [P: $o > $o] :
~ ( list_ex1_o @ P @ nil_o ) ).
% list_ex1_simps(1)
thf(fact_514_list__ex1__simps_I1_J,axiom,
! [P: list_o > $o] :
~ ( list_ex1_list_o @ P @ nil_list_o ) ).
% list_ex1_simps(1)
thf(fact_515_pairwise__alt,axiom,
( pairwise_transition
= ( ^ [R2: transition > transition > $o,S6: set_transition] :
! [X2: transition] :
( ( member_transition @ X2 @ S6 )
=> ! [Y4: transition] :
( ( member_transition @ Y4 @ ( minus_8944320859760356485sition @ S6 @ ( insert_transition2 @ X2 @ bot_bo301567166201926666sition ) ) )
=> ( R2 @ X2 @ Y4 ) ) ) ) ) ).
% pairwise_alt
thf(fact_516_pairwise__alt,axiom,
( pairwise_nat
= ( ^ [R2: nat > nat > $o,S6: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ S6 )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ ( minus_minus_set_nat @ S6 @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) )
=> ( R2 @ X2 @ Y4 ) ) ) ) ) ).
% pairwise_alt
thf(fact_517_psubset__insert__iff,axiom,
! [A2: set_transition,X: transition,B: set_transition] :
( ( ord_le5184432651266358346sition @ A2 @ ( insert_transition2 @ X @ B ) )
= ( ( ( member_transition @ X @ B )
=> ( ord_le5184432651266358346sition @ A2 @ B ) )
& ( ~ ( member_transition @ X @ B )
=> ( ( ( member_transition @ X @ A2 )
=> ( ord_le5184432651266358346sition @ ( minus_8944320859760356485sition @ A2 @ ( insert_transition2 @ X @ bot_bo301567166201926666sition ) ) @ B ) )
& ( ~ ( member_transition @ X @ A2 )
=> ( ord_le8419162016481440574sition @ A2 @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_518_psubset__insert__iff,axiom,
! [A2: set_nat,X: nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ ( insert_nat2 @ X @ B ) )
= ( ( ( member_nat @ X @ B )
=> ( ord_less_set_nat @ A2 @ B ) )
& ( ~ ( member_nat @ X @ B )
=> ( ( ( member_nat @ X @ A2 )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ B ) )
& ( ~ ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_519_psubsetI,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_set_nat @ A2 @ B ) ) ) ).
% psubsetI
thf(fact_520_psubsetD,axiom,
! [A2: set_nat,B: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B ) ) ) ).
% psubsetD
thf(fact_521_psubsetD,axiom,
! [A2: set_transition,B: set_transition,C: transition] :
( ( ord_le5184432651266358346sition @ A2 @ B )
=> ( ( member_transition @ C @ A2 )
=> ( member_transition @ C @ B ) ) ) ).
% psubsetD
thf(fact_522_pairwiseD,axiom,
! [R: nat > nat > $o,S: set_nat,X: nat,Y2: nat] :
( ( pairwise_nat @ R @ S )
=> ( ( member_nat @ X @ S )
=> ( ( member_nat @ Y2 @ S )
=> ( ( X != Y2 )
=> ( R @ X @ Y2 ) ) ) ) ) ).
% pairwiseD
thf(fact_523_pairwiseD,axiom,
! [R: transition > transition > $o,S: set_transition,X: transition,Y2: transition] :
( ( pairwise_transition @ R @ S )
=> ( ( member_transition @ X @ S )
=> ( ( member_transition @ Y2 @ S )
=> ( ( X != Y2 )
=> ( R @ X @ Y2 ) ) ) ) ) ).
% pairwiseD
thf(fact_524_pairwiseI,axiom,
! [S: set_nat,R: nat > nat > $o] :
( ! [X3: nat,Y: nat] :
( ( member_nat @ X3 @ S )
=> ( ( member_nat @ Y @ S )
=> ( ( X3 != Y )
=> ( R @ X3 @ Y ) ) ) )
=> ( pairwise_nat @ R @ S ) ) ).
% pairwiseI
thf(fact_525_pairwiseI,axiom,
! [S: set_transition,R: transition > transition > $o] :
( ! [X3: transition,Y: transition] :
( ( member_transition @ X3 @ S )
=> ( ( member_transition @ Y @ S )
=> ( ( X3 != Y )
=> ( R @ X3 @ Y ) ) ) )
=> ( pairwise_transition @ R @ S ) ) ).
% pairwiseI
thf(fact_526_leD,axiom,
! [Y2: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X )
=> ~ ( ord_less_set_nat @ X @ Y2 ) ) ).
% leD
thf(fact_527_nless__le,axiom,
! [A: set_nat,B2: set_nat] :
( ( ~ ( ord_less_set_nat @ A @ B2 ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B2 )
| ( A = B2 ) ) ) ).
% nless_le
thf(fact_528_antisym__conv1,axiom,
! [X: set_nat,Y2: set_nat] :
( ~ ( ord_less_set_nat @ X @ Y2 )
=> ( ( ord_less_eq_set_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_529_antisym__conv2,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ( ~ ( ord_less_set_nat @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_530_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X2: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y4 )
& ~ ( ord_less_eq_set_nat @ Y4 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_531_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ A3 @ B4 )
| ( A3 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_532_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
& ( A3 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_533_order_Ostrict__trans1,axiom,
! [A: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_set_nat @ B2 @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_534_order_Ostrict__trans2,axiom,
! [A: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_535_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
& ~ ( ord_less_eq_set_nat @ B4 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_536_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A3: set_nat] :
( ( ord_less_set_nat @ B4 @ A3 )
| ( A3 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_537_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B4: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A3 )
& ( A3 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_538_dual__order_Ostrict__trans1,axiom,
! [B2: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( ord_less_set_nat @ C @ B2 )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_539_dual__order_Ostrict__trans2,axiom,
! [B2: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B2 @ A )
=> ( ( ord_less_eq_set_nat @ C @ B2 )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_540_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B4: set_nat,A3: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A3 )
& ~ ( ord_less_eq_set_nat @ A3 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_541_order_Ostrict__implies__order,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ( ord_less_eq_set_nat @ A @ B2 ) ) ).
% order.strict_implies_order
thf(fact_542_dual__order_Ostrict__implies__order,axiom,
! [B2: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B2 @ A )
=> ( ord_less_eq_set_nat @ B2 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_543_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X2: set_nat,Y4: set_nat] :
( ( ord_less_set_nat @ X2 @ Y4 )
| ( X2 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_544_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X2: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y4 )
& ( X2 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_545_order__less__imp__le,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ord_less_eq_set_nat @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_546_order__le__neq__trans,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_set_nat @ A @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_547_order__neq__le__trans,axiom,
! [A: set_nat,B2: set_nat] :
( ( A != B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ord_less_set_nat @ A @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_548_order__le__less__trans,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ( ord_less_set_nat @ Y2 @ Z2 )
=> ( ord_less_set_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_549_order__less__le__trans,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_set_nat @ X @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ Z2 )
=> ( ord_less_set_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_550_order__le__less__subst2,axiom,
! [A: set_nat,B2: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_551_order__less__le__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X3: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_552_order__le__imp__less__or__eq,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ( ord_less_set_nat @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_553_bot_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_554_bot_Oextremum__strict,axiom,
! [A: set_transition] :
~ ( ord_le5184432651266358346sition @ A @ bot_bo301567166201926666sition ) ).
% bot.extremum_strict
thf(fact_555_bot_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_556_bot_Onot__eq__extremum,axiom,
! [A: set_transition] :
( ( A != bot_bo301567166201926666sition )
= ( ord_le5184432651266358346sition @ bot_bo301567166201926666sition @ A ) ) ).
% bot.not_eq_extremum
thf(fact_557_not__psubset__empty,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_558_not__psubset__empty,axiom,
! [A2: set_transition] :
~ ( ord_le5184432651266358346sition @ A2 @ bot_bo301567166201926666sition ) ).
% not_psubset_empty
thf(fact_559_psubsetE,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ B @ A2 ) ) ) ).
% psubsetE
thf(fact_560_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_561_psubset__imp__subset,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% psubset_imp_subset
thf(fact_562_psubset__subset__trans,axiom,
! [A2: set_nat,B: set_nat,C3: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C3 )
=> ( ord_less_set_nat @ A2 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_563_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
& ~ ( ord_less_eq_set_nat @ B5 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_564_subset__psubset__trans,axiom,
! [A2: set_nat,B: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_set_nat @ B @ C3 )
=> ( ord_less_set_nat @ A2 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_565_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
( ( ord_less_set_nat @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_566_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_567_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_568_sup_Ostrict__order__iff,axiom,
( ord_less_set_nat
= ( ^ [B4: set_nat,A3: set_nat] :
( ( A3
= ( sup_sup_set_nat @ A3 @ B4 ) )
& ( A3 != B4 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_569_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_570_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_571_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_572_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_573_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_574_psubset__imp__ex__mem,axiom,
! [A2: set_transition,B: set_transition] :
( ( ord_le5184432651266358346sition @ A2 @ B )
=> ? [B7: transition] : ( member_transition @ B7 @ ( minus_8944320859760356485sition @ B @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_575_psubset__imp__ex__mem,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ? [B7: nat] : ( member_nat @ B7 @ ( minus_minus_set_nat @ B @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_576_pairwise__empty,axiom,
! [P: nat > nat > $o] : ( pairwise_nat @ P @ bot_bot_set_nat ) ).
% pairwise_empty
thf(fact_577_pairwise__empty,axiom,
! [P: transition > transition > $o] : ( pairwise_transition @ P @ bot_bo301567166201926666sition ) ).
% pairwise_empty
thf(fact_578_pairwise__mono,axiom,
! [P: nat > nat > $o,A2: set_nat,Q3: nat > nat > $o,B: set_nat] :
( ( pairwise_nat @ P @ A2 )
=> ( ! [X3: nat,Y: nat] :
( ( P @ X3 @ Y )
=> ( Q3 @ X3 @ Y ) )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( pairwise_nat @ Q3 @ B ) ) ) ) ).
% pairwise_mono
thf(fact_579_pairwise__subset,axiom,
! [P: nat > nat > $o,S: set_nat,T4: set_nat] :
( ( pairwise_nat @ P @ S )
=> ( ( ord_less_eq_set_nat @ T4 @ S )
=> ( pairwise_nat @ P @ T4 ) ) ) ).
% pairwise_subset
thf(fact_580_pairwise__insert,axiom,
! [R3: nat > nat > $o,X: nat,S2: set_nat] :
( ( pairwise_nat @ R3 @ ( insert_nat2 @ X @ S2 ) )
= ( ! [Y4: nat] :
( ( ( member_nat @ Y4 @ S2 )
& ( Y4 != X ) )
=> ( ( R3 @ X @ Y4 )
& ( R3 @ Y4 @ X ) ) )
& ( pairwise_nat @ R3 @ S2 ) ) ) ).
% pairwise_insert
thf(fact_581_pairwise__insert,axiom,
! [R3: transition > transition > $o,X: transition,S2: set_transition] :
( ( pairwise_transition @ R3 @ ( insert_transition2 @ X @ S2 ) )
= ( ! [Y4: transition] :
( ( ( member_transition @ Y4 @ S2 )
& ( Y4 != X ) )
=> ( ( R3 @ X @ Y4 )
& ( R3 @ Y4 @ X ) ) )
& ( pairwise_transition @ R3 @ S2 ) ) ) ).
% pairwise_insert
thf(fact_582_null__rec_I1_J,axiom,
! [X: list_o,Xs: list_list_o] :
~ ( null_list_o @ ( cons_list_o @ X @ Xs ) ) ).
% null_rec(1)
thf(fact_583_eq__Nil__null,axiom,
! [Xs: list_transition] :
( ( Xs = nil_transition )
= ( null_transition @ Xs ) ) ).
% eq_Nil_null
thf(fact_584_eq__Nil__null,axiom,
! [Xs: list_o] :
( ( Xs = nil_o )
= ( null_o @ Xs ) ) ).
% eq_Nil_null
thf(fact_585_eq__Nil__null,axiom,
! [Xs: list_list_o] :
( ( Xs = nil_list_o )
= ( null_list_o @ Xs ) ) ).
% eq_Nil_null
thf(fact_586_null__rec_I2_J,axiom,
null_transition @ nil_transition ).
% null_rec(2)
thf(fact_587_null__rec_I2_J,axiom,
null_o @ nil_o ).
% null_rec(2)
thf(fact_588_null__rec_I2_J,axiom,
null_list_o @ nil_list_o ).
% null_rec(2)
thf(fact_589_list__ex1__iff,axiom,
( list_ex1_nat
= ( ^ [P2: nat > $o,Xs2: list_nat] :
? [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
& ( P2 @ X2 )
& ! [Y4: nat] :
( ( ( member_nat @ Y4 @ ( set_nat2 @ Xs2 ) )
& ( P2 @ Y4 ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_590_list__ex1__iff,axiom,
( list_ex1_transition
= ( ^ [P2: transition > $o,Xs2: list_transition] :
? [X2: transition] :
( ( member_transition @ X2 @ ( set_transition2 @ Xs2 ) )
& ( P2 @ X2 )
& ! [Y4: transition] :
( ( ( member_transition @ Y4 @ ( set_transition2 @ Xs2 ) )
& ( P2 @ Y4 ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_591_pairwise__singleton,axiom,
! [P: nat > nat > $o,A2: nat] : ( pairwise_nat @ P @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) ).
% pairwise_singleton
thf(fact_592_pairwise__singleton,axiom,
! [P: transition > transition > $o,A2: transition] : ( pairwise_transition @ P @ ( insert_transition2 @ A2 @ bot_bo301567166201926666sition ) ) ).
% pairwise_singleton
thf(fact_593_can__select__set__list__ex1,axiom,
! [P: transition > $o,A2: list_transition] :
( ( can_se3600352496914471099sition @ P @ ( set_transition2 @ A2 ) )
= ( list_ex1_transition @ P @ A2 ) ) ).
% can_select_set_list_ex1
thf(fact_594_can__select__def,axiom,
( can_select_nat
= ( ^ [P2: nat > $o,A4: set_nat] :
? [X2: nat] :
( ( member_nat @ X2 @ A4 )
& ( P2 @ X2 )
& ! [Y4: nat] :
( ( ( member_nat @ Y4 @ A4 )
& ( P2 @ Y4 ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_595_can__select__def,axiom,
( can_se3600352496914471099sition
= ( ^ [P2: transition > $o,A4: set_transition] :
? [X2: transition] :
( ( member_transition @ X2 @ A4 )
& ( P2 @ X2 )
& ! [Y4: transition] :
( ( ( member_transition @ Y4 @ A4 )
& ( P2 @ Y4 ) )
=> ( Y4 = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_596_subset__subseqs,axiom,
! [X4: set_transition,Xs: list_transition] :
( ( ord_le8419162016481440574sition @ X4 @ ( set_transition2 @ Xs ) )
=> ( member7318969637299765063sition @ X4 @ ( image_4748612756971788127sition @ set_transition2 @ ( set_list_transition2 @ ( subseqs_transition @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_597_subset__subseqs,axiom,
! [X4: set_nat,Xs: list_nat] :
( ( ord_less_eq_set_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ( member_set_nat @ X4 @ ( image_1775855109352712557et_nat @ set_nat2 @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_598_transpose__empty,axiom,
! [Xs: list_list_transition] :
( ( ( transpose_transition @ Xs )
= nil_list_transition )
= ( ! [X2: list_transition] :
( ( member1473516902542837997sition @ X2 @ ( set_list_transition2 @ Xs ) )
=> ( X2 = nil_transition ) ) ) ) ).
% transpose_empty
thf(fact_599_transpose__empty,axiom,
! [Xs: list_list_list_o] :
( ( ( transpose_list_o @ Xs )
= nil_list_list_o )
= ( ! [X2: list_list_o] :
( ( member_list_list_o @ X2 @ ( set_list_list_o2 @ Xs ) )
=> ( X2 = nil_list_o ) ) ) ) ).
% transpose_empty
thf(fact_600_transpose__empty,axiom,
! [Xs: list_list_o] :
( ( ( transpose_o @ Xs )
= nil_list_o )
= ( ! [X2: list_o] :
( ( member_list_o @ X2 @ ( set_list_o2 @ Xs ) )
=> ( X2 = nil_o ) ) ) ) ).
% transpose_empty
thf(fact_601_image__eqI,axiom,
! [B2: nat,F: nat > nat,X: nat,A2: set_nat] :
( ( B2
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_602_image__eqI,axiom,
! [B2: transition,F: nat > transition,X: nat,A2: set_nat] :
( ( B2
= ( F @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_transition @ B2 @ ( image_nat_transition @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_603_image__eqI,axiom,
! [B2: nat,F: transition > nat,X: transition,A2: set_transition] :
( ( B2
= ( F @ X ) )
=> ( ( member_transition @ X @ A2 )
=> ( member_nat @ B2 @ ( image_transition_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_604_image__eqI,axiom,
! [B2: transition,F: transition > transition,X: transition,A2: set_transition] :
( ( B2
= ( F @ X ) )
=> ( ( member_transition @ X @ A2 )
=> ( member_transition @ B2 @ ( image_5857460390510121477sition @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_605_image__is__empty,axiom,
! [F: nat > nat,A2: set_nat] :
( ( ( image_nat_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_606_image__is__empty,axiom,
! [F: transition > nat,A2: set_transition] :
( ( ( image_transition_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bo301567166201926666sition ) ) ).
% image_is_empty
thf(fact_607_image__is__empty,axiom,
! [F: nat > transition,A2: set_nat] :
( ( ( image_nat_transition @ F @ A2 )
= bot_bo301567166201926666sition )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_608_image__is__empty,axiom,
! [F: transition > transition,A2: set_transition] :
( ( ( image_5857460390510121477sition @ F @ A2 )
= bot_bo301567166201926666sition )
= ( A2 = bot_bo301567166201926666sition ) ) ).
% image_is_empty
thf(fact_609_empty__is__image,axiom,
! [F: nat > nat,A2: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_610_empty__is__image,axiom,
! [F: transition > nat,A2: set_transition] :
( ( bot_bot_set_nat
= ( image_transition_nat @ F @ A2 ) )
= ( A2 = bot_bo301567166201926666sition ) ) ).
% empty_is_image
thf(fact_611_empty__is__image,axiom,
! [F: nat > transition,A2: set_nat] :
( ( bot_bo301567166201926666sition
= ( image_nat_transition @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_612_empty__is__image,axiom,
! [F: transition > transition,A2: set_transition] :
( ( bot_bo301567166201926666sition
= ( image_5857460390510121477sition @ F @ A2 ) )
= ( A2 = bot_bo301567166201926666sition ) ) ).
% empty_is_image
thf(fact_613_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_614_image__empty,axiom,
! [F: nat > transition] :
( ( image_nat_transition @ F @ bot_bot_set_nat )
= bot_bo301567166201926666sition ) ).
% image_empty
thf(fact_615_image__empty,axiom,
! [F: transition > nat] :
( ( image_transition_nat @ F @ bot_bo301567166201926666sition )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_616_image__empty,axiom,
! [F: transition > transition] :
( ( image_5857460390510121477sition @ F @ bot_bo301567166201926666sition )
= bot_bo301567166201926666sition ) ).
% image_empty
thf(fact_617_insert__image,axiom,
! [X: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( insert_nat2 @ ( F @ X ) @ ( image_nat_nat @ F @ A2 ) )
= ( image_nat_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_618_insert__image,axiom,
! [X: nat,A2: set_nat,F: nat > transition] :
( ( member_nat @ X @ A2 )
=> ( ( insert_transition2 @ ( F @ X ) @ ( image_nat_transition @ F @ A2 ) )
= ( image_nat_transition @ F @ A2 ) ) ) ).
% insert_image
thf(fact_619_insert__image,axiom,
! [X: transition,A2: set_transition,F: transition > nat] :
( ( member_transition @ X @ A2 )
=> ( ( insert_nat2 @ ( F @ X ) @ ( image_transition_nat @ F @ A2 ) )
= ( image_transition_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_620_insert__image,axiom,
! [X: transition,A2: set_transition,F: transition > transition] :
( ( member_transition @ X @ A2 )
=> ( ( insert_transition2 @ ( F @ X ) @ ( image_5857460390510121477sition @ F @ A2 ) )
= ( image_5857460390510121477sition @ F @ A2 ) ) ) ).
% insert_image
thf(fact_621_image__insert,axiom,
! [F: nat > nat,A: nat,B: set_nat] :
( ( image_nat_nat @ F @ ( insert_nat2 @ A @ B ) )
= ( insert_nat2 @ ( F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_622_image__insert,axiom,
! [F: nat > transition,A: nat,B: set_nat] :
( ( image_nat_transition @ F @ ( insert_nat2 @ A @ B ) )
= ( insert_transition2 @ ( F @ A ) @ ( image_nat_transition @ F @ B ) ) ) ).
% image_insert
thf(fact_623_image__insert,axiom,
! [F: transition > nat,A: transition,B: set_transition] :
( ( image_transition_nat @ F @ ( insert_transition2 @ A @ B ) )
= ( insert_nat2 @ ( F @ A ) @ ( image_transition_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_624_image__insert,axiom,
! [F: transition > transition,A: transition,B: set_transition] :
( ( image_5857460390510121477sition @ F @ ( insert_transition2 @ A @ B ) )
= ( insert_transition2 @ ( F @ A ) @ ( image_5857460390510121477sition @ F @ B ) ) ) ).
% image_insert
thf(fact_625_image__Un,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat] :
( ( image_nat_nat @ F @ ( sup_sup_set_nat @ A2 @ B ) )
= ( sup_sup_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_626_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_627_imageI,axiom,
! [X: nat,A2: set_nat,F: nat > transition] :
( ( member_nat @ X @ A2 )
=> ( member_transition @ ( F @ X ) @ ( image_nat_transition @ F @ A2 ) ) ) ).
% imageI
thf(fact_628_imageI,axiom,
! [X: transition,A2: set_transition,F: transition > nat] :
( ( member_transition @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ ( image_transition_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_629_imageI,axiom,
! [X: transition,A2: set_transition,F: transition > transition] :
( ( member_transition @ X @ A2 )
=> ( member_transition @ ( F @ X ) @ ( image_5857460390510121477sition @ F @ A2 ) ) ) ).
% imageI
thf(fact_630_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B2: nat,F: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( B2
= ( F @ X ) )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_631_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B2: transition,F: nat > transition] :
( ( member_nat @ X @ A2 )
=> ( ( B2
= ( F @ X ) )
=> ( member_transition @ B2 @ ( image_nat_transition @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_632_rev__image__eqI,axiom,
! [X: transition,A2: set_transition,B2: nat,F: transition > nat] :
( ( member_transition @ X @ A2 )
=> ( ( B2
= ( F @ X ) )
=> ( member_nat @ B2 @ ( image_transition_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_633_rev__image__eqI,axiom,
! [X: transition,A2: set_transition,B2: transition,F: transition > transition] :
( ( member_transition @ X @ A2 )
=> ( ( B2
= ( F @ X ) )
=> ( member_transition @ B2 @ ( image_5857460390510121477sition @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_634_subset__image__iff,axiom,
! [B: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_635_subset__imageE,axiom,
! [B: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A2 )
=> ( B
!= ( image_nat_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_636_image__subsetI,axiom,
! [A2: set_nat,F: nat > transition,B: set_transition] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_transition @ ( F @ X3 ) @ B ) )
=> ( ord_le8419162016481440574sition @ ( image_nat_transition @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_637_image__subsetI,axiom,
! [A2: set_transition,F: transition > transition,B: set_transition] :
( ! [X3: transition] :
( ( member_transition @ X3 @ A2 )
=> ( member_transition @ ( F @ X3 ) @ B ) )
=> ( ord_le8419162016481440574sition @ ( image_5857460390510121477sition @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_638_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_639_image__subsetI,axiom,
! [A2: set_transition,F: transition > nat,B: set_nat] :
( ! [X3: transition] :
( ( member_transition @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_transition_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_640_image__mono,axiom,
! [A2: set_nat,B: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_641_transpose_Osimps_I1_J,axiom,
( ( transpose_o @ nil_list_o )
= nil_list_o ) ).
% transpose.simps(1)
thf(fact_642_image__diff__subset,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B ) ) @ ( image_nat_nat @ F @ ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_643_transpose_Osimps_I2_J,axiom,
! [Xss2: list_list_transition] :
( ( transpose_transition @ ( cons_list_transition @ nil_transition @ Xss2 ) )
= ( transpose_transition @ Xss2 ) ) ).
% transpose.simps(2)
thf(fact_644_transpose_Osimps_I2_J,axiom,
! [Xss2: list_list_list_o] :
( ( transpose_list_o @ ( cons_list_list_o @ nil_list_o @ Xss2 ) )
= ( transpose_list_o @ Xss2 ) ) ).
% transpose.simps(2)
thf(fact_645_transpose_Osimps_I2_J,axiom,
! [Xss2: list_list_o] :
( ( transpose_o @ ( cons_list_o @ nil_o @ Xss2 ) )
= ( transpose_o @ Xss2 ) ) ).
% transpose.simps(2)
thf(fact_646_verit__comp__simplify1_I2_J,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_647_all__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A2 )
=> ( P @ ( image_nat_nat @ F @ B5 ) ) ) ) ) ).
% all_subset_image
thf(fact_648_subseqs__powset,axiom,
! [Xs: list_transition] :
( ( image_4748612756971788127sition @ set_transition2 @ ( set_list_transition2 @ ( subseqs_transition @ Xs ) ) )
= ( pow_transition @ ( set_transition2 @ Xs ) ) ) ).
% subseqs_powset
thf(fact_649_step__eps__sucs__sound,axiom,
! [Q2: nat,Q0: nat,Transs: list_transition,Bs: list_o,Q: nat] :
( ( member_nat @ Q2 @ ( step_eps_sucs @ Q0 @ Transs @ Bs @ Q ) )
= ( step_eps @ Q0 @ Transs @ Bs @ Q @ Q2 ) ) ).
% step_eps_sucs_sound
thf(fact_650_lexordp__eq_Ocases,axiom,
! [A1: list_o,A22: list_o] :
( ( ord_lexordp_eq_o @ A1 @ A22 )
=> ( ( A1 != nil_o )
=> ( ! [X3: $o] :
( ? [Xs3: list_o] :
( A1
= ( cons_o @ X3 @ Xs3 ) )
=> ! [Y: $o] :
( ? [Ys3: list_o] :
( A22
= ( cons_o @ Y @ Ys3 ) )
=> ~ ( ord_less_o @ X3 @ Y ) ) )
=> ~ ! [X3: $o,Y: $o,Xs3: list_o] :
( ( A1
= ( cons_o @ X3 @ Xs3 ) )
=> ! [Ys3: list_o] :
( ( A22
= ( cons_o @ Y @ Ys3 ) )
=> ( ~ ( ord_less_o @ X3 @ Y )
=> ( ~ ( ord_less_o @ Y @ X3 )
=> ~ ( ord_lexordp_eq_o @ Xs3 @ Ys3 ) ) ) ) ) ) ) ) ).
% lexordp_eq.cases
thf(fact_651_Pow__empty,axiom,
( ( pow_nat @ bot_bot_set_nat )
= ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat ) ) ).
% Pow_empty
thf(fact_652_Pow__empty,axiom,
( ( pow_transition @ bot_bo301567166201926666sition )
= ( insert8494249028948967790sition @ bot_bo301567166201926666sition @ bot_bo1233527522848825322sition ) ) ).
% Pow_empty
thf(fact_653_Pow__singleton__iff,axiom,
! [X4: set_nat,Y5: set_nat] :
( ( ( pow_nat @ X4 )
= ( insert_set_nat @ Y5 @ bot_bot_set_set_nat ) )
= ( ( X4 = bot_bot_set_nat )
& ( Y5 = bot_bot_set_nat ) ) ) ).
% Pow_singleton_iff
thf(fact_654_Pow__singleton__iff,axiom,
! [X4: set_transition,Y5: set_transition] :
( ( ( pow_transition @ X4 )
= ( insert8494249028948967790sition @ Y5 @ bot_bo1233527522848825322sition ) )
= ( ( X4 = bot_bo301567166201926666sition )
& ( Y5 = bot_bo301567166201926666sition ) ) ) ).
% Pow_singleton_iff
thf(fact_655_Pow__iff,axiom,
! [A2: set_nat,B: set_nat] :
( ( member_set_nat @ A2 @ ( pow_nat @ B ) )
= ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% Pow_iff
thf(fact_656_PowI,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( member_set_nat @ A2 @ ( pow_nat @ B ) ) ) ).
% PowI
thf(fact_657_lexordp__eq__simps_I2_J,axiom,
! [Xs: list_o] :
( ( ord_lexordp_eq_o @ Xs @ nil_o )
= ( Xs = nil_o ) ) ).
% lexordp_eq_simps(2)
thf(fact_658_lexordp__eq__simps_I1_J,axiom,
! [Ys: list_o] : ( ord_lexordp_eq_o @ nil_o @ Ys ) ).
% lexordp_eq_simps(1)
thf(fact_659_lexordp__eq__simps_I3_J,axiom,
! [X: $o,Xs: list_o] :
~ ( ord_lexordp_eq_o @ ( cons_o @ X @ Xs ) @ nil_o ) ).
% lexordp_eq_simps(3)
thf(fact_660_Pow__insert,axiom,
! [A: nat,A2: set_nat] :
( ( pow_nat @ ( insert_nat2 @ A @ A2 ) )
= ( sup_sup_set_set_nat @ ( pow_nat @ A2 ) @ ( image_7916887816326733075et_nat @ ( insert_nat2 @ A ) @ ( pow_nat @ A2 ) ) ) ) ).
% Pow_insert
thf(fact_661_Pow__insert,axiom,
! [A: transition,A2: set_transition] :
( ( pow_transition @ ( insert_transition2 @ A @ A2 ) )
= ( sup_su8198498708765531986sition @ ( pow_transition @ A2 ) @ ( image_698392052263970309sition @ ( insert_transition2 @ A ) @ ( pow_transition @ A2 ) ) ) ) ).
% Pow_insert
thf(fact_662_Un__Pow__subset,axiom,
! [A2: set_nat,B: set_nat] : ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ ( pow_nat @ A2 ) @ ( pow_nat @ B ) ) @ ( pow_nat @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% Un_Pow_subset
thf(fact_663_PowD,axiom,
! [A2: set_nat,B: set_nat] :
( ( member_set_nat @ A2 @ ( pow_nat @ B ) )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% PowD
thf(fact_664_Pow__mono,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_le6893508408891458716et_nat @ ( pow_nat @ A2 ) @ ( pow_nat @ B ) ) ) ).
% Pow_mono
thf(fact_665_Pow__bottom,axiom,
! [B: set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( pow_nat @ B ) ) ).
% Pow_bottom
thf(fact_666_Pow__bottom,axiom,
! [B: set_transition] : ( member7318969637299765063sition @ bot_bo301567166201926666sition @ ( pow_transition @ B ) ) ).
% Pow_bottom
thf(fact_667_lexordp__eq_ONil,axiom,
! [Ys: list_o] : ( ord_lexordp_eq_o @ nil_o @ Ys ) ).
% lexordp_eq.Nil
thf(fact_668_in__image__insert__iff,axiom,
! [B: set_set_transition,X: transition,A2: set_transition] :
( ! [C4: set_transition] :
( ( member7318969637299765063sition @ C4 @ B )
=> ~ ( member_transition @ X @ C4 ) )
=> ( ( member7318969637299765063sition @ A2 @ ( image_698392052263970309sition @ ( insert_transition2 @ X ) @ B ) )
= ( ( member_transition @ X @ A2 )
& ( member7318969637299765063sition @ ( minus_8944320859760356485sition @ A2 @ ( insert_transition2 @ X @ bot_bo301567166201926666sition ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_669_in__image__insert__iff,axiom,
! [B: set_set_nat,X: nat,A2: set_nat] :
( ! [C4: set_nat] :
( ( member_set_nat @ C4 @ B )
=> ~ ( member_nat @ X @ C4 ) )
=> ( ( member_set_nat @ A2 @ ( image_7916887816326733075et_nat @ ( insert_nat2 @ X ) @ B ) )
= ( ( member_nat @ X @ A2 )
& ( member_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_670_Pow__set_I1_J,axiom,
( ( pow_o @ ( set_o2 @ nil_o ) )
= ( insert_set_o @ bot_bot_set_o @ bot_bot_set_set_o ) ) ).
% Pow_set(1)
thf(fact_671_Pow__set_I1_J,axiom,
( ( pow_list_o @ ( set_list_o2 @ nil_list_o ) )
= ( insert_set_list_o @ bot_bot_set_list_o @ bot_bo64454365476827594list_o ) ) ).
% Pow_set(1)
thf(fact_672_Pow__set_I1_J,axiom,
( ( pow_nat @ ( set_nat2 @ nil_nat ) )
= ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat ) ) ).
% Pow_set(1)
thf(fact_673_Pow__set_I1_J,axiom,
( ( pow_transition @ ( set_transition2 @ nil_transition ) )
= ( insert8494249028948967790sition @ bot_bo301567166201926666sition @ bot_bo1233527522848825322sition ) ) ).
% Pow_set(1)
thf(fact_674_lexordp__eq_Osimps,axiom,
( ord_lexordp_eq_o
= ( ^ [A12: list_o,A23: list_o] :
( ? [Ys2: list_o] :
( ( A12 = nil_o )
& ( A23 = Ys2 ) )
| ? [X2: $o,Y4: $o,Xs2: list_o,Ys2: list_o] :
( ( A12
= ( cons_o @ X2 @ Xs2 ) )
& ( A23
= ( cons_o @ Y4 @ Ys2 ) )
& ( ord_less_o @ X2 @ Y4 ) )
| ? [X2: $o,Y4: $o,Xs2: list_o,Ys2: list_o] :
( ( A12
= ( cons_o @ X2 @ Xs2 ) )
& ( A23
= ( cons_o @ Y4 @ Ys2 ) )
& ~ ( ord_less_o @ X2 @ Y4 )
& ~ ( ord_less_o @ Y4 @ X2 )
& ( ord_lexordp_eq_o @ Xs2 @ Ys2 ) ) ) ) ) ).
% lexordp_eq.simps
thf(fact_675_finite__induct__select,axiom,
! [S: set_transition,P: set_transition > $o] :
( ( finite8165534619950747239sition @ S )
=> ( ( P @ bot_bo301567166201926666sition )
=> ( ! [T5: set_transition] :
( ( ord_le5184432651266358346sition @ T5 @ S )
=> ( ( P @ T5 )
=> ? [X5: transition] :
( ( member_transition @ X5 @ ( minus_8944320859760356485sition @ S @ T5 ) )
& ( P @ ( insert_transition2 @ X5 @ T5 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_676_finite__induct__select,axiom,
! [S: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [T5: set_nat] :
( ( ord_less_set_nat @ T5 @ S )
=> ( ( P @ T5 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ ( minus_minus_set_nat @ S @ T5 ) )
& ( P @ ( insert_nat2 @ X5 @ T5 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_677_finite__remove__induct,axiom,
! [B: set_transition,P: set_transition > $o] :
( ( finite8165534619950747239sition @ B )
=> ( ( P @ bot_bo301567166201926666sition )
=> ( ! [A6: set_transition] :
( ( finite8165534619950747239sition @ A6 )
=> ( ( A6 != bot_bo301567166201926666sition )
=> ( ( ord_le8419162016481440574sition @ A6 @ B )
=> ( ! [X5: transition] :
( ( member_transition @ X5 @ A6 )
=> ( P @ ( minus_8944320859760356485sition @ A6 @ ( insert_transition2 @ X5 @ bot_bo301567166201926666sition ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_678_finite__remove__induct,axiom,
! [B: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( A6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A6 @ B )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat2 @ X5 @ bot_bot_set_nat ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% finite_remove_induct
thf(fact_679_remove__induct,axiom,
! [P: set_transition > $o,B: set_transition] :
( ( P @ bot_bo301567166201926666sition )
=> ( ( ~ ( finite8165534619950747239sition @ B )
=> ( P @ B ) )
=> ( ! [A6: set_transition] :
( ( finite8165534619950747239sition @ A6 )
=> ( ( A6 != bot_bo301567166201926666sition )
=> ( ( ord_le8419162016481440574sition @ A6 @ B )
=> ( ! [X5: transition] :
( ( member_transition @ X5 @ A6 )
=> ( P @ ( minus_8944320859760356485sition @ A6 @ ( insert_transition2 @ X5 @ bot_bo301567166201926666sition ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_680_remove__induct,axiom,
! [P: set_nat > $o,B: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B )
=> ( P @ B ) )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( A6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A6 @ B )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat2 @ X5 @ bot_bot_set_nat ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B ) ) ) ) ).
% remove_induct
thf(fact_681_NFA_Orun__def,axiom,
( run
= ( ^ [Q02: nat,Transs2: list_transition] : ( foldl_set_nat_list_o @ ( delta @ Q02 @ Transs2 ) ) ) ) ).
% NFA.run_def
thf(fact_682_finite__insert,axiom,
! [A: transition,A2: set_transition] :
( ( finite8165534619950747239sition @ ( insert_transition2 @ A @ A2 ) )
= ( finite8165534619950747239sition @ A2 ) ) ).
% finite_insert
thf(fact_683_finite__insert,axiom,
! [A: nat,A2: set_nat] :
( ( finite_finite_nat @ ( insert_nat2 @ A @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ).
% finite_insert
thf(fact_684_List_Ofinite__set,axiom,
! [Xs: list_transition] : ( finite8165534619950747239sition @ ( set_transition2 @ Xs ) ) ).
% List.finite_set
thf(fact_685_List_Ofinite__set,axiom,
! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_686_finite__Un,axiom,
! [F2: set_nat,G: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G ) )
= ( ( finite_finite_nat @ F2 )
& ( finite_finite_nat @ G ) ) ) ).
% finite_Un
thf(fact_687_finite__Diff2,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_688_finite__Diff,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% finite_Diff
thf(fact_689_finite__Diff__insert,axiom,
! [A2: set_transition,A: transition,B: set_transition] :
( ( finite8165534619950747239sition @ ( minus_8944320859760356485sition @ A2 @ ( insert_transition2 @ A @ B ) ) )
= ( finite8165534619950747239sition @ ( minus_8944320859760356485sition @ A2 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_690_finite__Diff__insert,axiom,
! [A2: set_nat,A: nat,B: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ B ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% finite_Diff_insert
thf(fact_691_finite__UnI,axiom,
! [F2: set_nat,G: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( finite_finite_nat @ G )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_692_Un__infinite,axiom,
! [S: set_nat,T4: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T4 ) ) ) ).
% Un_infinite
thf(fact_693_infinite__Un,axiom,
! [S: set_nat,T4: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T4 ) ) )
= ( ~ ( finite_finite_nat @ S )
| ~ ( finite_finite_nat @ T4 ) ) ) ).
% infinite_Un
thf(fact_694_Diff__infinite__finite,axiom,
! [T4: set_nat,S: set_nat] :
( ( finite_finite_nat @ T4 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T4 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_695_finite_OinsertI,axiom,
! [A2: set_transition,A: transition] :
( ( finite8165534619950747239sition @ A2 )
=> ( finite8165534619950747239sition @ ( insert_transition2 @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_696_finite_OinsertI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( insert_nat2 @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_697_rev__finite__subset,axiom,
! [B: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_698_infinite__super,axiom,
! [S: set_nat,T4: set_nat] :
( ( ord_less_eq_set_nat @ S @ T4 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T4 ) ) ) ).
% infinite_super
thf(fact_699_finite__subset,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_700_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_701_finite_OemptyI,axiom,
finite8165534619950747239sition @ bot_bo301567166201926666sition ).
% finite.emptyI
thf(fact_702_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_703_infinite__imp__nonempty,axiom,
! [S: set_transition] :
( ~ ( finite8165534619950747239sition @ S )
=> ( S != bot_bo301567166201926666sition ) ) ).
% infinite_imp_nonempty
thf(fact_704_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_705_finite__has__minimal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ( ord_less_eq_set_nat @ X3 @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_706_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ A @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_707_finite__has__maximal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ( ord_less_eq_set_nat @ A @ X3 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_708_finite__list,axiom,
! [A2: set_transition] :
( ( finite8165534619950747239sition @ A2 )
=> ? [Xs3: list_transition] :
( ( set_transition2 @ Xs3 )
= A2 ) ) ).
% finite_list
thf(fact_709_finite__list,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ? [Xs3: list_nat] :
( ( set_nat2 @ Xs3 )
= A2 ) ) ).
% finite_list
thf(fact_710_finite__Q,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition] : ( finite_finite_nat @ ( q @ Q0 @ Qf @ Transs ) ) ).
% finite_Q
thf(fact_711_finite__SQ,axiom,
! [Q0: nat,Transs: list_transition] : ( finite_finite_nat @ ( sq @ Q0 @ Transs ) ) ).
% finite_SQ
thf(fact_712_foldl__Cons,axiom,
! [F: set_nat > list_o > set_nat,A: set_nat,X: list_o,Xs: list_list_o] :
( ( foldl_set_nat_list_o @ F @ A @ ( cons_list_o @ X @ Xs ) )
= ( foldl_set_nat_list_o @ F @ ( F @ A @ X ) @ Xs ) ) ).
% foldl_Cons
thf(fact_713_foldl__Nil,axiom,
! [F: set_nat > list_o > set_nat,A: set_nat] :
( ( foldl_set_nat_list_o @ F @ A @ nil_list_o )
= A ) ).
% foldl_Nil
thf(fact_714_foldl__cong,axiom,
! [A: set_nat,B2: set_nat,L: list_list_o,K: list_list_o,F: set_nat > list_o > set_nat,G2: set_nat > list_o > set_nat] :
( ( A = B2 )
=> ( ( L = K )
=> ( ! [A7: set_nat,X3: list_o] :
( ( member_list_o @ X3 @ ( set_list_o2 @ L ) )
=> ( ( F @ A7 @ X3 )
= ( G2 @ A7 @ X3 ) ) )
=> ( ( foldl_set_nat_list_o @ F @ A @ L )
= ( foldl_set_nat_list_o @ G2 @ B2 @ K ) ) ) ) ) ).
% foldl_cong
thf(fact_715_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_716_finite__has__minimal,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_717_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_718_finite__has__maximal,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_719_finite__surj,axiom,
! [A2: set_nat,B: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A2 ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_720_finite__subset__image,axiom,
! [B: set_nat,F: nat > nat,A2: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A2 ) )
=> ? [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A2 )
& ( finite_finite_nat @ C4 )
& ( B
= ( image_nat_nat @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_721_ex__finite__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ? [B5: set_nat] :
( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ B5 ) ) )
= ( ? [B5: set_nat] :
( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A2 )
& ( P @ ( image_nat_nat @ F @ B5 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_722_all__finite__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ! [B5: set_nat] :
( ( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A2 ) ) )
=> ( P @ B5 ) ) )
= ( ! [B5: set_nat] :
( ( ( finite_finite_nat @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A2 ) )
=> ( P @ ( image_nat_nat @ F @ B5 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_723_infinite__finite__induct,axiom,
! [P: set_nat > $o,A2: set_nat] :
( ! [A6: set_nat] :
( ~ ( finite_finite_nat @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat2 @ X3 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_724_infinite__finite__induct,axiom,
! [P: set_transition > $o,A2: set_transition] :
( ! [A6: set_transition] :
( ~ ( finite8165534619950747239sition @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bo301567166201926666sition )
=> ( ! [X3: transition,F3: set_transition] :
( ( finite8165534619950747239sition @ F3 )
=> ( ~ ( member_transition @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_transition2 @ X3 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_725_finite__ne__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( F2 != bot_bot_set_nat )
=> ( ! [X3: nat] : ( P @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) )
=> ( ! [X3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat2 @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_726_finite__ne__induct,axiom,
! [F2: set_transition,P: set_transition > $o] :
( ( finite8165534619950747239sition @ F2 )
=> ( ( F2 != bot_bo301567166201926666sition )
=> ( ! [X3: transition] : ( P @ ( insert_transition2 @ X3 @ bot_bo301567166201926666sition ) )
=> ( ! [X3: transition,F3: set_transition] :
( ( finite8165534619950747239sition @ F3 )
=> ( ( F3 != bot_bo301567166201926666sition )
=> ( ~ ( member_transition @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_transition2 @ X3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_727_finite__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat2 @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_728_finite__induct,axiom,
! [F2: set_transition,P: set_transition > $o] :
( ( finite8165534619950747239sition @ F2 )
=> ( ( P @ bot_bo301567166201926666sition )
=> ( ! [X3: transition,F3: set_transition] :
( ( finite8165534619950747239sition @ F3 )
=> ( ~ ( member_transition @ X3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_transition2 @ X3 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_729_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A3: set_nat] :
( ( A3 = bot_bot_set_nat )
| ? [A4: set_nat,B4: nat] :
( ( A3
= ( insert_nat2 @ B4 @ A4 ) )
& ( finite_finite_nat @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_730_finite_Osimps,axiom,
( finite8165534619950747239sition
= ( ^ [A3: set_transition] :
( ( A3 = bot_bo301567166201926666sition )
| ? [A4: set_transition,B4: transition] :
( ( A3
= ( insert_transition2 @ B4 @ A4 ) )
& ( finite8165534619950747239sition @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_731_finite_Ocases,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ~ ! [A6: set_nat] :
( ? [A7: nat] :
( A
= ( insert_nat2 @ A7 @ A6 ) )
=> ~ ( finite_finite_nat @ A6 ) ) ) ) ).
% finite.cases
thf(fact_732_finite_Ocases,axiom,
! [A: set_transition] :
( ( finite8165534619950747239sition @ A )
=> ( ( A != bot_bo301567166201926666sition )
=> ~ ! [A6: set_transition] :
( ? [A7: transition] :
( A
= ( insert_transition2 @ A7 @ A6 ) )
=> ~ ( finite8165534619950747239sition @ A6 ) ) ) ) ).
% finite.cases
thf(fact_733_finite__subset__induct,axiom,
! [F2: set_transition,A2: set_transition,P: set_transition > $o] :
( ( finite8165534619950747239sition @ F2 )
=> ( ( ord_le8419162016481440574sition @ F2 @ A2 )
=> ( ( P @ bot_bo301567166201926666sition )
=> ( ! [A7: transition,F3: set_transition] :
( ( finite8165534619950747239sition @ F3 )
=> ( ( member_transition @ A7 @ A2 )
=> ( ~ ( member_transition @ A7 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_transition2 @ A7 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_734_finite__subset__induct,axiom,
! [F2: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A7: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A7 @ A2 )
=> ( ~ ( member_nat @ A7 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat2 @ A7 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_735_finite__subset__induct_H,axiom,
! [F2: set_transition,A2: set_transition,P: set_transition > $o] :
( ( finite8165534619950747239sition @ F2 )
=> ( ( ord_le8419162016481440574sition @ F2 @ A2 )
=> ( ( P @ bot_bo301567166201926666sition )
=> ( ! [A7: transition,F3: set_transition] :
( ( finite8165534619950747239sition @ F3 )
=> ( ( member_transition @ A7 @ A2 )
=> ( ( ord_le8419162016481440574sition @ F3 @ A2 )
=> ( ~ ( member_transition @ A7 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_transition2 @ A7 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_736_finite__subset__induct_H,axiom,
! [F2: set_nat,A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A7: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A7 @ A2 )
=> ( ( ord_less_eq_set_nat @ F3 @ A2 )
=> ( ~ ( member_nat @ A7 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat2 @ A7 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_737_infinite__remove,axiom,
! [S: set_transition,A: transition] :
( ~ ( finite8165534619950747239sition @ S )
=> ~ ( finite8165534619950747239sition @ ( minus_8944320859760356485sition @ S @ ( insert_transition2 @ A @ bot_bo301567166201926666sition ) ) ) ) ).
% infinite_remove
thf(fact_738_infinite__remove,axiom,
! [S: set_nat,A: nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_739_infinite__coinduct,axiom,
! [X4: set_transition > $o,A2: set_transition] :
( ( X4 @ A2 )
=> ( ! [A6: set_transition] :
( ( X4 @ A6 )
=> ? [X5: transition] :
( ( member_transition @ X5 @ A6 )
& ( ( X4 @ ( minus_8944320859760356485sition @ A6 @ ( insert_transition2 @ X5 @ bot_bo301567166201926666sition ) ) )
| ~ ( finite8165534619950747239sition @ ( minus_8944320859760356485sition @ A6 @ ( insert_transition2 @ X5 @ bot_bo301567166201926666sition ) ) ) ) ) )
=> ~ ( finite8165534619950747239sition @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_740_infinite__coinduct,axiom,
! [X4: set_nat > $o,A2: set_nat] :
( ( X4 @ A2 )
=> ( ! [A6: set_nat] :
( ( X4 @ A6 )
=> ? [X5: nat] :
( ( member_nat @ X5 @ A6 )
& ( ( X4 @ ( minus_minus_set_nat @ A6 @ ( insert_nat2 @ X5 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A6 @ ( insert_nat2 @ X5 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_741_finite__empty__induct,axiom,
! [A2: set_transition,P: set_transition > $o] :
( ( finite8165534619950747239sition @ A2 )
=> ( ( P @ A2 )
=> ( ! [A7: transition,A6: set_transition] :
( ( finite8165534619950747239sition @ A6 )
=> ( ( member_transition @ A7 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_8944320859760356485sition @ A6 @ ( insert_transition2 @ A7 @ bot_bo301567166201926666sition ) ) ) ) ) )
=> ( P @ bot_bo301567166201926666sition ) ) ) ) ).
% finite_empty_induct
thf(fact_742_finite__empty__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ A2 )
=> ( ! [A7: nat,A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( member_nat @ A7 @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat2 @ A7 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_743_finite__linorder__min__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B7: nat,A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A6 )
=> ( ord_less_nat @ B7 @ X5 ) )
=> ( ( P @ A6 )
=> ( P @ ( insert_nat2 @ B7 @ A6 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_744_finite__linorder__max__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B7: nat,A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A6 )
=> ( ord_less_nat @ X5 @ B7 ) )
=> ( ( P @ A6 )
=> ( P @ ( insert_nat2 @ B7 @ A6 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_745_ex__min__if__finite,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ S )
& ~ ? [Xa: nat] :
( ( member_nat @ Xa @ S )
& ( ord_less_nat @ Xa @ X3 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_746_infinite__growing,axiom,
! [X4: set_nat] :
( ( X4 != bot_bot_set_nat )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X4 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X4 )
& ( ord_less_nat @ X3 @ Xa ) ) )
=> ~ ( finite_finite_nat @ X4 ) ) ) ).
% infinite_growing
thf(fact_747_empty__in__Fpow,axiom,
! [A2: set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( finite_Fpow_nat @ A2 ) ) ).
% empty_in_Fpow
thf(fact_748_empty__in__Fpow,axiom,
! [A2: set_transition] : ( member7318969637299765063sition @ bot_bo301567166201926666sition @ ( finite9040295020840969508sition @ A2 ) ) ).
% empty_in_Fpow
thf(fact_749_Fpow__mono,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_le6893508408891458716et_nat @ ( finite_Fpow_nat @ A2 ) @ ( finite_Fpow_nat @ B ) ) ) ).
% Fpow_mono
thf(fact_750_chains__extend,axiom,
! [C: set_set_nat,S: set_set_nat,Z2: set_nat] :
( ( member_set_set_nat @ C @ ( chains_nat @ S ) )
=> ( ( member_set_nat @ Z2 @ S )
=> ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ C )
=> ( ord_less_eq_set_nat @ X3 @ Z2 ) )
=> ( member_set_set_nat @ ( sup_sup_set_set_nat @ ( insert_set_nat @ Z2 @ bot_bot_set_set_nat ) @ C ) @ ( chains_nat @ S ) ) ) ) ) ).
% chains_extend
thf(fact_751_Sup__fin_Oremove,axiom,
! [A2: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ X @ A2 )
=> ( ( ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
= bot_bot_set_set_nat )
=> ( ( lattic3835124923745554447et_nat @ A2 )
= X ) )
& ( ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
!= bot_bot_set_set_nat )
=> ( ( lattic3835124923745554447et_nat @ A2 )
= ( sup_sup_set_nat @ X @ ( lattic3835124923745554447et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_752_Sup__fin_Oremove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A2 )
= X ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A2 )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_753_Sup__fin_Oinsert__remove,axiom,
! [A2: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
= bot_bot_set_set_nat )
=> ( ( lattic3835124923745554447et_nat @ ( insert_set_nat @ X @ A2 ) )
= X ) )
& ( ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
!= bot_bot_set_set_nat )
=> ( ( lattic3835124923745554447et_nat @ ( insert_set_nat @ X @ A2 ) )
= ( sup_sup_set_nat @ X @ ( lattic3835124923745554447et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_754_Sup__fin_Oinsert__remove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat2 @ X @ A2 ) )
= X ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat2 @ X @ A2 ) )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_755_Sup__fin_Osingleton,axiom,
! [X: nat] :
( ( lattic1093996805478795353in_nat @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= X ) ).
% Sup_fin.singleton
thf(fact_756_Sup__fin_Oinsert,axiom,
! [A2: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ( lattic3835124923745554447et_nat @ ( insert_set_nat @ X @ A2 ) )
= ( sup_sup_set_nat @ X @ ( lattic3835124923745554447et_nat @ A2 ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_757_Sup__fin_Oinsert,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat2 @ X @ A2 ) )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_758_chainsD,axiom,
! [C: set_set_nat,S: set_set_nat,X: set_nat,Y2: set_nat] :
( ( member_set_set_nat @ C @ ( chains_nat @ S ) )
=> ( ( member_set_nat @ X @ C )
=> ( ( member_set_nat @ Y2 @ C )
=> ( ( ord_less_eq_set_nat @ X @ Y2 )
| ( ord_less_eq_set_nat @ Y2 @ X ) ) ) ) ) ).
% chainsD
thf(fact_759_Zorn__Lemma2,axiom,
! [A2: set_set_nat] :
( ! [X3: set_set_nat] :
( ( member_set_set_nat @ X3 @ ( chains_nat @ A2 ) )
=> ? [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
& ! [Xb: set_nat] :
( ( member_set_nat @ Xb @ X3 )
=> ( ord_less_eq_set_nat @ Xb @ Xa ) ) ) )
=> ? [X3: set_nat] :
( ( member_set_nat @ X3 @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ X3 @ Xa )
=> ( Xa = X3 ) ) ) ) ) ).
% Zorn_Lemma2
thf(fact_760_Sup__fin_OcoboundedI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ( ord_less_eq_nat @ A @ ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_761_Sup__fin_OcoboundedI,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ( ord_less_eq_set_nat @ A @ ( lattic3835124923745554447et_nat @ A2 ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_762_Sup__fin_Oin__idem,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A2 ) )
= ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_763_Sup__fin_Oin__idem,axiom,
! [A2: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ X @ A2 )
=> ( ( sup_sup_set_nat @ X @ ( lattic3835124923745554447et_nat @ A2 ) )
= ( lattic3835124923745554447et_nat @ A2 ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_764_Sup__fin_OboundedE,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ X )
=> ! [A8: nat] :
( ( member_nat @ A8 @ A2 )
=> ( ord_less_eq_nat @ A8 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_765_Sup__fin_OboundedE,axiom,
! [A2: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A2 ) @ X )
=> ! [A8: set_nat] :
( ( member_set_nat @ A8 @ A2 )
=> ( ord_less_eq_set_nat @ A8 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_766_Sup__fin_OboundedI,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [A7: nat] :
( ( member_nat @ A7 @ A2 )
=> ( ord_less_eq_nat @ A7 @ X ) )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_767_Sup__fin_OboundedI,axiom,
! [A2: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ! [A7: set_nat] :
( ( member_set_nat @ A7 @ A2 )
=> ( ord_less_eq_set_nat @ A7 @ X ) )
=> ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A2 ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_768_Sup__fin_Obounded__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ X )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ X2 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_769_Sup__fin_Obounded__iff,axiom,
! [A2: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A2 ) @ X )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ X2 @ X ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_770_Sup__fin_Osubset__imp,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ( finite1152437895449049373et_nat @ B )
=> ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A2 ) @ ( lattic3835124923745554447et_nat @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_771_Sup__fin_Osubset__imp,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ ( lattic1093996805478795353in_nat @ B ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_772_Sup__fin_Ohom__commute,axiom,
! [H: set_nat > set_nat,N3: set_set_nat] :
( ! [X3: set_nat,Y: set_nat] :
( ( H @ ( sup_sup_set_nat @ X3 @ Y ) )
= ( sup_sup_set_nat @ ( H @ X3 ) @ ( H @ Y ) ) )
=> ( ( finite1152437895449049373et_nat @ N3 )
=> ( ( N3 != bot_bot_set_set_nat )
=> ( ( H @ ( lattic3835124923745554447et_nat @ N3 ) )
= ( lattic3835124923745554447et_nat @ ( image_7916887816326733075et_nat @ H @ N3 ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_773_Sup__fin_Ohom__commute,axiom,
! [H: nat > nat,N3: set_nat] :
( ! [X3: nat,Y: nat] :
( ( H @ ( sup_sup_nat @ X3 @ Y ) )
= ( sup_sup_nat @ ( H @ X3 ) @ ( H @ Y ) ) )
=> ( ( finite_finite_nat @ N3 )
=> ( ( N3 != bot_bot_set_nat )
=> ( ( H @ ( lattic1093996805478795353in_nat @ N3 ) )
= ( lattic1093996805478795353in_nat @ ( image_nat_nat @ H @ N3 ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_774_Sup__fin_Osubset,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( B != bot_bot_set_set_nat )
=> ( ( ord_le6893508408891458716et_nat @ B @ A2 )
=> ( ( sup_sup_set_nat @ ( lattic3835124923745554447et_nat @ B ) @ ( lattic3835124923745554447et_nat @ A2 ) )
= ( lattic3835124923745554447et_nat @ A2 ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_775_Sup__fin_Osubset,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( B != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ B ) @ ( lattic1093996805478795353in_nat @ A2 ) )
= ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_776_Sup__fin_Oinsert__not__elem,axiom,
! [A2: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ~ ( member_set_nat @ X @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ( lattic3835124923745554447et_nat @ ( insert_set_nat @ X @ A2 ) )
= ( sup_sup_set_nat @ X @ ( lattic3835124923745554447et_nat @ A2 ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_777_Sup__fin_Oinsert__not__elem,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat @ X @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat2 @ X @ A2 ) )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_778_Sup__fin_Oclosed,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ! [X3: set_nat,Y: set_nat] : ( member_set_nat @ ( sup_sup_set_nat @ X3 @ Y ) @ ( insert_set_nat @ X3 @ ( insert_set_nat @ Y @ bot_bot_set_set_nat ) ) )
=> ( member_set_nat @ ( lattic3835124923745554447et_nat @ A2 ) @ A2 ) ) ) ) ).
% Sup_fin.closed
thf(fact_779_Sup__fin_Oclosed,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [X3: nat,Y: nat] : ( member_nat @ ( sup_sup_nat @ X3 @ Y ) @ ( insert_nat2 @ X3 @ ( insert_nat2 @ Y @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ A2 ) ) ) ) ).
% Sup_fin.closed
thf(fact_780_Sup__fin_Ounion,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ( finite1152437895449049373et_nat @ B )
=> ( ( B != bot_bot_set_set_nat )
=> ( ( lattic3835124923745554447et_nat @ ( sup_sup_set_set_nat @ A2 @ B ) )
= ( sup_sup_set_nat @ ( lattic3835124923745554447et_nat @ A2 ) @ ( lattic3835124923745554447et_nat @ B ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_781_Sup__fin_Ounion,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ( B != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( sup_sup_set_nat @ A2 @ B ) )
= ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ ( lattic1093996805478795353in_nat @ B ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_782_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A2: set_o] :
( ( finite_finite_o @ A2 )
=> ( ( ( linord3142498349692569832_set_o @ A2 )
= nil_o )
= ( A2 = bot_bot_set_o ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_783_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( linord2614967742042102400et_nat @ A2 )
= nil_nat )
= ( A2 = bot_bot_set_nat ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_784_Inf__fin__le__Sup__fin,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A2 ) @ ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_785_Inf__fin__le__Sup__fin,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ ( lattic3014633134055518761et_nat @ A2 ) @ ( lattic3835124923745554447et_nat @ A2 ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_786_Inf__fin_Osubset__imp,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ( finite1152437895449049373et_nat @ B )
=> ( ord_less_eq_set_nat @ ( lattic3014633134055518761et_nat @ B ) @ ( lattic3014633134055518761et_nat @ A2 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_787_Inf__fin_Osubset__imp,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ B ) @ ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_788_Inf__fin_Osingleton,axiom,
! [X: nat] :
( ( lattic5238388535129920115in_nat @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= X ) ).
% Inf_fin.singleton
thf(fact_789_sup__Inf__absorb,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ( ( sup_sup_nat @ ( lattic5238388535129920115in_nat @ A2 ) @ A )
= A ) ) ) ).
% sup_Inf_absorb
thf(fact_790_sup__Inf__absorb,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ( ( sup_sup_set_nat @ ( lattic3014633134055518761et_nat @ A2 ) @ A )
= A ) ) ) ).
% sup_Inf_absorb
thf(fact_791_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
( ( linord3142498349692569832_set_o @ bot_bot_set_o )
= nil_o ) ).
% sorted_list_of_set.sorted_key_list_of_set_empty
thf(fact_792_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
( ( linord2614967742042102400et_nat @ bot_bot_set_nat )
= nil_nat ) ).
% sorted_list_of_set.sorted_key_list_of_set_empty
thf(fact_793_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
! [A2: set_o] :
( ~ ( finite_finite_o @ A2 )
=> ( ( linord3142498349692569832_set_o @ A2 )
= nil_o ) ) ).
% sorted_list_of_set.fold_insort_key.infinite
thf(fact_794_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( linord2614967742042102400et_nat @ A2 )
= nil_nat ) ) ).
% sorted_list_of_set.fold_insort_key.infinite
thf(fact_795_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( set_nat2 @ ( linord2614967742042102400et_nat @ A2 ) )
= A2 ) ) ).
% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_796_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( linord2614967742042102400et_nat @ A2 )
= ( linord2614967742042102400et_nat @ B ) )
=> ( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B )
=> ( A2 = B ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_797_Inf__fin_OcoboundedI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A2 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_798_Inf__fin_OcoboundedI,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ( ord_less_eq_set_nat @ ( lattic3014633134055518761et_nat @ A2 ) @ A ) ) ) ).
% Inf_fin.coboundedI
thf(fact_799_Inf__fin_Obounded__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A2 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ X @ X2 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_800_Inf__fin_Obounded__iff,axiom,
! [A2: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ( ord_less_eq_set_nat @ X @ ( lattic3014633134055518761et_nat @ A2 ) )
= ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ X @ X2 ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_801_Inf__fin_OboundedI,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [A7: nat] :
( ( member_nat @ A7 @ A2 )
=> ( ord_less_eq_nat @ X @ A7 ) )
=> ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_802_Inf__fin_OboundedI,axiom,
! [A2: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ! [A7: set_nat] :
( ( member_set_nat @ A7 @ A2 )
=> ( ord_less_eq_set_nat @ X @ A7 ) )
=> ( ord_less_eq_set_nat @ X @ ( lattic3014633134055518761et_nat @ A2 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_803_Inf__fin_OboundedE,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A2 ) )
=> ! [A8: nat] :
( ( member_nat @ A8 @ A2 )
=> ( ord_less_eq_nat @ X @ A8 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_804_Inf__fin_OboundedE,axiom,
! [A2: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ( ord_less_eq_set_nat @ X @ ( lattic3014633134055518761et_nat @ A2 ) )
=> ! [A8: set_nat] :
( ( member_set_nat @ A8 @ A2 )
=> ( ord_less_eq_set_nat @ X @ A8 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_805_sorted__list__of__set__nonempty,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( linord2614967742042102400et_nat @ A2 )
= ( cons_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ ( linord2614967742042102400et_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ ( lattic8721135487736765967in_nat @ A2 ) @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% sorted_list_of_set_nonempty
thf(fact_806_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( linord2614967742042102400et_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) )
= ( remove1_nat @ X @ ( linord2614967742042102400et_nat @ A2 ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_807_Inf__fin_Oremove,axiom,
! [A2: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ X @ A2 )
=> ( ( ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
= bot_bot_set_set_nat )
=> ( ( lattic3014633134055518761et_nat @ A2 )
= X ) )
& ( ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
!= bot_bot_set_set_nat )
=> ( ( lattic3014633134055518761et_nat @ A2 )
= ( inf_inf_set_nat @ X @ ( lattic3014633134055518761et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_808_Inf__fin_Oremove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A2 )
= X ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A2 )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_809_Int__subset__iff,axiom,
! [C3: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ ( inf_inf_set_nat @ A2 @ B ) )
= ( ( ord_less_eq_set_nat @ C3 @ A2 )
& ( ord_less_eq_set_nat @ C3 @ B ) ) ) ).
% Int_subset_iff
thf(fact_810_Int__insert__right__if1,axiom,
! [A: transition,A2: set_transition,B: set_transition] :
( ( member_transition @ A @ A2 )
=> ( ( inf_in8814773338690644108sition @ A2 @ ( insert_transition2 @ A @ B ) )
= ( insert_transition2 @ A @ ( inf_in8814773338690644108sition @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_811_Int__insert__right__if1,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat2 @ A @ B ) )
= ( insert_nat2 @ A @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% Int_insert_right_if1
thf(fact_812_Int__insert__right__if0,axiom,
! [A: transition,A2: set_transition,B: set_transition] :
( ~ ( member_transition @ A @ A2 )
=> ( ( inf_in8814773338690644108sition @ A2 @ ( insert_transition2 @ A @ B ) )
= ( inf_in8814773338690644108sition @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_813_Int__insert__right__if0,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat2 @ A @ B ) )
= ( inf_inf_set_nat @ A2 @ B ) ) ) ).
% Int_insert_right_if0
thf(fact_814_insert__inter__insert,axiom,
! [A: transition,A2: set_transition,B: set_transition] :
( ( inf_in8814773338690644108sition @ ( insert_transition2 @ A @ A2 ) @ ( insert_transition2 @ A @ B ) )
= ( insert_transition2 @ A @ ( inf_in8814773338690644108sition @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_815_insert__inter__insert,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( insert_nat2 @ A @ A2 ) @ ( insert_nat2 @ A @ B ) )
= ( insert_nat2 @ A @ ( inf_inf_set_nat @ A2 @ B ) ) ) ).
% insert_inter_insert
thf(fact_816_Int__insert__left__if1,axiom,
! [A: transition,C3: set_transition,B: set_transition] :
( ( member_transition @ A @ C3 )
=> ( ( inf_in8814773338690644108sition @ ( insert_transition2 @ A @ B ) @ C3 )
= ( insert_transition2 @ A @ ( inf_in8814773338690644108sition @ B @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_817_Int__insert__left__if1,axiom,
! [A: nat,C3: set_nat,B: set_nat] :
( ( member_nat @ A @ C3 )
=> ( ( inf_inf_set_nat @ ( insert_nat2 @ A @ B ) @ C3 )
= ( insert_nat2 @ A @ ( inf_inf_set_nat @ B @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_818_Int__insert__left__if0,axiom,
! [A: transition,C3: set_transition,B: set_transition] :
( ~ ( member_transition @ A @ C3 )
=> ( ( inf_in8814773338690644108sition @ ( insert_transition2 @ A @ B ) @ C3 )
= ( inf_in8814773338690644108sition @ B @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_819_Int__insert__left__if0,axiom,
! [A: nat,C3: set_nat,B: set_nat] :
( ~ ( member_nat @ A @ C3 )
=> ( ( inf_inf_set_nat @ ( insert_nat2 @ A @ B ) @ C3 )
= ( inf_inf_set_nat @ B @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_820_inf__right__idem,axiom,
! [X: set_nat,Y2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y2 ) @ Y2 )
= ( inf_inf_set_nat @ X @ Y2 ) ) ).
% inf_right_idem
thf(fact_821_inf_Oright__idem,axiom,
! [A: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ B2 )
= ( inf_inf_set_nat @ A @ B2 ) ) ).
% inf.right_idem
thf(fact_822_inf__left__idem,axiom,
! [X: set_nat,Y2: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ X @ Y2 ) )
= ( inf_inf_set_nat @ X @ Y2 ) ) ).
% inf_left_idem
thf(fact_823_inf_Oleft__idem,axiom,
! [A: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ A @ B2 ) )
= ( inf_inf_set_nat @ A @ B2 ) ) ).
% inf.left_idem
thf(fact_824_inf__idem,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ X )
= X ) ).
% inf_idem
thf(fact_825_inf_Oidem,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ A )
= A ) ).
% inf.idem
thf(fact_826_Un__Int__eq_I1_J,axiom,
! [S: set_nat,T4: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T4 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_827_Un__Int__eq_I2_J,axiom,
! [S: set_nat,T4: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T4 ) @ T4 )
= T4 ) ).
% Un_Int_eq(2)
thf(fact_828_Un__Int__eq_I3_J,axiom,
! [S: set_nat,T4: set_nat] :
( ( inf_inf_set_nat @ S @ ( sup_sup_set_nat @ S @ T4 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_829_Un__Int__eq_I4_J,axiom,
! [T4: set_nat,S: set_nat] :
( ( inf_inf_set_nat @ T4 @ ( sup_sup_set_nat @ S @ T4 ) )
= T4 ) ).
% Un_Int_eq(4)
thf(fact_830_Int__Un__eq_I1_J,axiom,
! [S: set_nat,T4: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T4 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_831_Int__Un__eq_I2_J,axiom,
! [S: set_nat,T4: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T4 ) @ T4 )
= T4 ) ).
% Int_Un_eq(2)
thf(fact_832_Int__Un__eq_I3_J,axiom,
! [S: set_nat,T4: set_nat] :
( ( sup_sup_set_nat @ S @ ( inf_inf_set_nat @ S @ T4 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_833_Int__Un__eq_I4_J,axiom,
! [T4: set_nat,S: set_nat] :
( ( sup_sup_set_nat @ T4 @ ( inf_inf_set_nat @ S @ T4 ) )
= T4 ) ).
% Int_Un_eq(4)
thf(fact_834_Pow__Int__eq,axiom,
! [A2: set_nat,B: set_nat] :
( ( pow_nat @ ( inf_inf_set_nat @ A2 @ B ) )
= ( inf_inf_set_set_nat @ ( pow_nat @ A2 ) @ ( pow_nat @ B ) ) ) ).
% Pow_Int_eq
thf(fact_835_le__inf__iff,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y2 @ Z2 ) )
= ( ( ord_less_eq_set_nat @ X @ Y2 )
& ( ord_less_eq_set_nat @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_836_inf_Obounded__iff,axiom,
! [A: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B2 @ C ) )
= ( ( ord_less_eq_set_nat @ A @ B2 )
& ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_837_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_838_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_transition] :
( ( inf_in8814773338690644108sition @ X @ bot_bo301567166201926666sition )
= bot_bo301567166201926666sition ) ).
% boolean_algebra.conj_zero_right
thf(fact_839_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_840_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_transition] :
( ( inf_in8814773338690644108sition @ bot_bo301567166201926666sition @ X )
= bot_bo301567166201926666sition ) ).
% boolean_algebra.conj_zero_left
thf(fact_841_inf__bot__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_842_inf__bot__right,axiom,
! [X: set_transition] :
( ( inf_in8814773338690644108sition @ X @ bot_bo301567166201926666sition )
= bot_bo301567166201926666sition ) ).
% inf_bot_right
thf(fact_843_inf__bot__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_844_inf__bot__left,axiom,
! [X: set_transition] :
( ( inf_in8814773338690644108sition @ bot_bo301567166201926666sition @ X )
= bot_bo301567166201926666sition ) ).
% inf_bot_left
thf(fact_845_insert__disjoint_I1_J,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat2 @ A @ A2 ) @ B )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B )
& ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_846_insert__disjoint_I1_J,axiom,
! [A: transition,A2: set_transition,B: set_transition] :
( ( ( inf_in8814773338690644108sition @ ( insert_transition2 @ A @ A2 ) @ B )
= bot_bo301567166201926666sition )
= ( ~ ( member_transition @ A @ B )
& ( ( inf_in8814773338690644108sition @ A2 @ B )
= bot_bo301567166201926666sition ) ) ) ).
% insert_disjoint(1)
thf(fact_847_insert__disjoint_I2_J,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat2 @ A @ A2 ) @ B ) )
= ( ~ ( member_nat @ A @ B )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_848_insert__disjoint_I2_J,axiom,
! [A: transition,A2: set_transition,B: set_transition] :
( ( bot_bo301567166201926666sition
= ( inf_in8814773338690644108sition @ ( insert_transition2 @ A @ A2 ) @ B ) )
= ( ~ ( member_transition @ A @ B )
& ( bot_bo301567166201926666sition
= ( inf_in8814773338690644108sition @ A2 @ B ) ) ) ) ).
% insert_disjoint(2)
thf(fact_849_disjoint__insert_I1_J,axiom,
! [B: set_nat,A: nat,A2: set_nat] :
( ( ( inf_inf_set_nat @ B @ ( insert_nat2 @ A @ A2 ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B )
& ( ( inf_inf_set_nat @ B @ A2 )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_850_disjoint__insert_I1_J,axiom,
! [B: set_transition,A: transition,A2: set_transition] :
( ( ( inf_in8814773338690644108sition @ B @ ( insert_transition2 @ A @ A2 ) )
= bot_bo301567166201926666sition )
= ( ~ ( member_transition @ A @ B )
& ( ( inf_in8814773338690644108sition @ B @ A2 )
= bot_bo301567166201926666sition ) ) ) ).
% disjoint_insert(1)
thf(fact_851_disjoint__insert_I2_J,axiom,
! [A2: set_nat,B2: nat,B: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ ( insert_nat2 @ B2 @ B ) ) )
= ( ~ ( member_nat @ B2 @ A2 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_852_disjoint__insert_I2_J,axiom,
! [A2: set_transition,B2: transition,B: set_transition] :
( ( bot_bo301567166201926666sition
= ( inf_in8814773338690644108sition @ A2 @ ( insert_transition2 @ B2 @ B ) ) )
= ( ~ ( member_transition @ B2 @ A2 )
& ( bot_bo301567166201926666sition
= ( inf_in8814773338690644108sition @ A2 @ B ) ) ) ) ).
% disjoint_insert(2)
thf(fact_853_inf__sup__absorb,axiom,
! [X: set_nat,Y2: set_nat] :
( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ X @ Y2 ) )
= X ) ).
% inf_sup_absorb
thf(fact_854_sup__inf__absorb,axiom,
! [X: set_nat,Y2: set_nat] :
( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ X @ Y2 ) )
= X ) ).
% sup_inf_absorb
thf(fact_855_Diff__disjoint,axiom,
! [A2: set_transition,B: set_transition] :
( ( inf_in8814773338690644108sition @ A2 @ ( minus_8944320859760356485sition @ B @ A2 ) )
= bot_bo301567166201926666sition ) ).
% Diff_disjoint
thf(fact_856_Diff__disjoint,axiom,
! [A2: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( minus_minus_set_nat @ B @ A2 ) )
= bot_bot_set_nat ) ).
% Diff_disjoint
thf(fact_857_in__set__remove1,axiom,
! [A: nat,B2: nat,Xs: list_nat] :
( ( A != B2 )
=> ( ( member_nat @ A @ ( set_nat2 @ ( remove1_nat @ B2 @ Xs ) ) )
= ( member_nat @ A @ ( set_nat2 @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_858_in__set__remove1,axiom,
! [A: transition,B2: transition,Xs: list_transition] :
( ( A != B2 )
=> ( ( member_transition @ A @ ( set_transition2 @ ( remove1_transition @ B2 @ Xs ) ) )
= ( member_transition @ A @ ( set_transition2 @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_859_Min__singleton,axiom,
! [X: nat] :
( ( lattic8721135487736765967in_nat @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= X ) ).
% Min_singleton
thf(fact_860_Min_Obounded__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ X @ X2 ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_861_Min__gr__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_nat @ X @ X2 ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_862_Inf__fin_Oinsert,axiom,
! [A2: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X @ A2 ) )
= ( inf_inf_set_nat @ X @ ( lattic3014633134055518761et_nat @ A2 ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_863_Inf__fin_Oinsert,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat2 @ X @ A2 ) )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_864_disjoint__iff__not__equal,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ B )
=> ( X2 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_865_disjoint__iff__not__equal,axiom,
! [A2: set_transition,B: set_transition] :
( ( ( inf_in8814773338690644108sition @ A2 @ B )
= bot_bo301567166201926666sition )
= ( ! [X2: transition] :
( ( member_transition @ X2 @ A2 )
=> ! [Y4: transition] :
( ( member_transition @ Y4 @ B )
=> ( X2 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_866_Int__empty__right,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_867_Int__empty__right,axiom,
! [A2: set_transition] :
( ( inf_in8814773338690644108sition @ A2 @ bot_bo301567166201926666sition )
= bot_bo301567166201926666sition ) ).
% Int_empty_right
thf(fact_868_Int__empty__left,axiom,
! [B: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_869_Int__empty__left,axiom,
! [B: set_transition] :
( ( inf_in8814773338690644108sition @ bot_bo301567166201926666sition @ B )
= bot_bo301567166201926666sition ) ).
% Int_empty_left
thf(fact_870_disjoint__iff,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ~ ( member_nat @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_871_disjoint__iff,axiom,
! [A2: set_transition,B: set_transition] :
( ( ( inf_in8814773338690644108sition @ A2 @ B )
= bot_bo301567166201926666sition )
= ( ! [X2: transition] :
( ( member_transition @ X2 @ A2 )
=> ~ ( member_transition @ X2 @ B ) ) ) ) ).
% disjoint_iff
thf(fact_872_Int__emptyI,axiom,
! [A2: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ~ ( member_nat @ X3 @ B ) )
=> ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_873_Int__emptyI,axiom,
! [A2: set_transition,B: set_transition] :
( ! [X3: transition] :
( ( member_transition @ X3 @ A2 )
=> ~ ( member_transition @ X3 @ B ) )
=> ( ( inf_in8814773338690644108sition @ A2 @ B )
= bot_bo301567166201926666sition ) ) ).
% Int_emptyI
thf(fact_874_Int__Collect__mono,axiom,
! [A2: set_transition,B: set_transition,P: transition > $o,Q3: transition > $o] :
( ( ord_le8419162016481440574sition @ A2 @ B )
=> ( ! [X3: transition] :
( ( member_transition @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q3 @ X3 ) ) )
=> ( ord_le8419162016481440574sition @ ( inf_in8814773338690644108sition @ A2 @ ( collect_transition @ P ) ) @ ( inf_in8814773338690644108sition @ B @ ( collect_transition @ Q3 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_875_Int__Collect__mono,axiom,
! [A2: set_nat,B: set_nat,P: nat > $o,Q3: nat > $o] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( P @ X3 )
=> ( Q3 @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q3 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_876_Int__greatest,axiom,
! [C3: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A2 )
=> ( ( ord_less_eq_set_nat @ C3 @ B )
=> ( ord_less_eq_set_nat @ C3 @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% Int_greatest
thf(fact_877_Int__absorb2,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( inf_inf_set_nat @ A2 @ B )
= A2 ) ) ).
% Int_absorb2
thf(fact_878_Int__absorb1,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ B )
= B ) ) ).
% Int_absorb1
thf(fact_879_Int__lower2,axiom,
! [A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ B ) ).
% Int_lower2
thf(fact_880_Int__lower1,axiom,
! [A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ A2 ) ).
% Int_lower1
thf(fact_881_Int__mono,axiom,
! [A2: set_nat,C3: set_nat,B: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C3 )
=> ( ( ord_less_eq_set_nat @ B @ D2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ ( inf_inf_set_nat @ C3 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_882_Int__insert__right,axiom,
! [A: transition,A2: set_transition,B: set_transition] :
( ( ( member_transition @ A @ A2 )
=> ( ( inf_in8814773338690644108sition @ A2 @ ( insert_transition2 @ A @ B ) )
= ( insert_transition2 @ A @ ( inf_in8814773338690644108sition @ A2 @ B ) ) ) )
& ( ~ ( member_transition @ A @ A2 )
=> ( ( inf_in8814773338690644108sition @ A2 @ ( insert_transition2 @ A @ B ) )
= ( inf_in8814773338690644108sition @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_883_Int__insert__right,axiom,
! [A: nat,A2: set_nat,B: set_nat] :
( ( ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat2 @ A @ B ) )
= ( insert_nat2 @ A @ ( inf_inf_set_nat @ A2 @ B ) ) ) )
& ( ~ ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat2 @ A @ B ) )
= ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% Int_insert_right
thf(fact_884_Int__insert__left,axiom,
! [A: transition,C3: set_transition,B: set_transition] :
( ( ( member_transition @ A @ C3 )
=> ( ( inf_in8814773338690644108sition @ ( insert_transition2 @ A @ B ) @ C3 )
= ( insert_transition2 @ A @ ( inf_in8814773338690644108sition @ B @ C3 ) ) ) )
& ( ~ ( member_transition @ A @ C3 )
=> ( ( inf_in8814773338690644108sition @ ( insert_transition2 @ A @ B ) @ C3 )
= ( inf_in8814773338690644108sition @ B @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_885_Int__insert__left,axiom,
! [A: nat,C3: set_nat,B: set_nat] :
( ( ( member_nat @ A @ C3 )
=> ( ( inf_inf_set_nat @ ( insert_nat2 @ A @ B ) @ C3 )
= ( insert_nat2 @ A @ ( inf_inf_set_nat @ B @ C3 ) ) ) )
& ( ~ ( member_nat @ A @ C3 )
=> ( ( inf_inf_set_nat @ ( insert_nat2 @ A @ B ) @ C3 )
= ( inf_inf_set_nat @ B @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_886_Un__Int__distrib2,axiom,
! [B: set_nat,C3: set_nat,A2: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ B @ C3 ) @ A2 )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ B @ A2 ) @ ( sup_sup_set_nat @ C3 @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_887_Int__Un__distrib2,axiom,
! [B: set_nat,C3: set_nat,A2: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ B @ C3 ) @ A2 )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ B @ A2 ) @ ( inf_inf_set_nat @ C3 @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_888_Un__Int__distrib,axiom,
! [A2: set_nat,B: set_nat,C3: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( inf_inf_set_nat @ B @ C3 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ ( sup_sup_set_nat @ A2 @ C3 ) ) ) ).
% Un_Int_distrib
thf(fact_889_Int__Un__distrib,axiom,
! [A2: set_nat,B: set_nat,C3: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C3 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ ( inf_inf_set_nat @ A2 @ C3 ) ) ) ).
% Int_Un_distrib
thf(fact_890_Un__Int__crazy,axiom,
! [A2: set_nat,B: set_nat,C3: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ ( inf_inf_set_nat @ B @ C3 ) ) @ ( inf_inf_set_nat @ C3 @ A2 ) )
= ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ ( sup_sup_set_nat @ B @ C3 ) ) @ ( sup_sup_set_nat @ C3 @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_891_inf__sup__ord_I2_J,axiom,
! [X: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_892_inf__sup__ord_I1_J,axiom,
! [X: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y2 ) @ X ) ).
% inf_sup_ord(1)
thf(fact_893_inf__le1,axiom,
! [X: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y2 ) @ X ) ).
% inf_le1
thf(fact_894_inf__le2,axiom,
! [X: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_895_le__infE,axiom,
! [X: set_nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ A @ B2 ) )
=> ~ ( ( ord_less_eq_set_nat @ X @ A )
=> ~ ( ord_less_eq_set_nat @ X @ B2 ) ) ) ).
% le_infE
thf(fact_896_le__infI,axiom,
! [X: set_nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X @ A )
=> ( ( ord_less_eq_set_nat @ X @ B2 )
=> ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_897_inf__mono,axiom,
! [A: set_nat,C: set_nat,B2: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ D )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ ( inf_inf_set_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_898_le__infI1,axiom,
! [A: set_nat,X: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ X )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_899_le__infI2,axiom,
! [B2: set_nat,X: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ X )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_900_inf_OorderE,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( A
= ( inf_inf_set_nat @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_901_inf_OorderI,axiom,
! [A: set_nat,B2: set_nat] :
( ( A
= ( inf_inf_set_nat @ A @ B2 ) )
=> ( ord_less_eq_set_nat @ A @ B2 ) ) ).
% inf.orderI
thf(fact_902_inf__unique,axiom,
! [F: set_nat > set_nat > set_nat,X: set_nat,Y2: set_nat] :
( ! [X3: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( F @ X3 @ Y ) @ X3 )
=> ( ! [X3: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( F @ X3 @ Y ) @ Y )
=> ( ! [X3: set_nat,Y: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y )
=> ( ( ord_less_eq_set_nat @ X3 @ Z3 )
=> ( ord_less_eq_set_nat @ X3 @ ( F @ Y @ Z3 ) ) ) )
=> ( ( inf_inf_set_nat @ X @ Y2 )
= ( F @ X @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_903_le__iff__inf,axiom,
( ord_less_eq_set_nat
= ( ^ [X2: set_nat,Y4: set_nat] :
( ( inf_inf_set_nat @ X2 @ Y4 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_904_inf_Oabsorb1,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( inf_inf_set_nat @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_905_inf_Oabsorb2,axiom,
! [B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( inf_inf_set_nat @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_906_inf__absorb1,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ( inf_inf_set_nat @ X @ Y2 )
= X ) ) ).
% inf_absorb1
thf(fact_907_inf__absorb2,axiom,
! [Y2: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X )
=> ( ( inf_inf_set_nat @ X @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_908_inf_OboundedE,axiom,
! [A: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B2 @ C ) )
=> ~ ( ( ord_less_eq_set_nat @ A @ B2 )
=> ~ ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_909_inf_OboundedI,axiom,
! [A: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ C )
=> ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_910_inf__greatest,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ( ord_less_eq_set_nat @ X @ Z2 )
=> ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y2 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_911_inf_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
( A3
= ( inf_inf_set_nat @ A3 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_912_inf_Ocobounded1,axiom,
! [A: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_913_inf_Ocobounded2,axiom,
! [A: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_914_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
( ( inf_inf_set_nat @ A3 @ B4 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_915_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A3: set_nat] :
( ( inf_inf_set_nat @ A3 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_916_inf_OcoboundedI1,axiom,
! [A: set_nat,C: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_917_inf_OcoboundedI2,axiom,
! [B2: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_918_distrib__imp1,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ! [X3: set_nat,Y: set_nat,Z3: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ Y @ Z3 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ ( inf_inf_set_nat @ X3 @ Z3 ) ) )
=> ( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y2 @ Z2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y2 ) @ ( sup_sup_set_nat @ X @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_919_distrib__imp2,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ! [X3: set_nat,Y: set_nat,Z3: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ Y @ Z3 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X3 @ Y ) @ ( sup_sup_set_nat @ X3 @ Z3 ) ) )
=> ( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y2 @ Z2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y2 ) @ ( inf_inf_set_nat @ X @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_920_inf__sup__distrib1,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y2 @ Z2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y2 ) @ ( inf_inf_set_nat @ X @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_921_inf__sup__distrib2,axiom,
! [Y2: set_nat,Z2: set_nat,X: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y2 @ Z2 ) @ X )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y2 @ X ) @ ( inf_inf_set_nat @ Z2 @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_922_sup__inf__distrib1,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y2 @ Z2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y2 ) @ ( sup_sup_set_nat @ X @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_923_sup__inf__distrib2,axiom,
! [Y2: set_nat,Z2: set_nat,X: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y2 @ Z2 ) @ X )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y2 @ X ) @ ( sup_sup_set_nat @ Z2 @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_924_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y2 @ Z2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y2 ) @ ( inf_inf_set_nat @ X @ Z2 ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_925_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y2 @ Z2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y2 ) @ ( sup_sup_set_nat @ X @ Z2 ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_926_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y2: set_nat,Z2: set_nat,X: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y2 @ Z2 ) @ X )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y2 @ X ) @ ( inf_inf_set_nat @ Z2 @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_927_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y2: set_nat,Z2: set_nat,X: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y2 @ Z2 ) @ X )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y2 @ X ) @ ( sup_sup_set_nat @ Z2 @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_928_less__infI1,axiom,
! [A: set_nat,X: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A @ X )
=> ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ X ) ) ).
% less_infI1
thf(fact_929_less__infI2,axiom,
! [B2: set_nat,X: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B2 @ X )
=> ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ X ) ) ).
% less_infI2
thf(fact_930_inf_Oabsorb3,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ( ( inf_inf_set_nat @ A @ B2 )
= A ) ) ).
% inf.absorb3
thf(fact_931_inf_Oabsorb4,axiom,
! [B2: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B2 @ A )
=> ( ( inf_inf_set_nat @ A @ B2 )
= B2 ) ) ).
% inf.absorb4
thf(fact_932_inf_Ostrict__boundedE,axiom,
! [A: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ ( inf_inf_set_nat @ B2 @ C ) )
=> ~ ( ( ord_less_set_nat @ A @ B2 )
=> ~ ( ord_less_set_nat @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_933_inf_Ostrict__order__iff,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
( ( A3
= ( inf_inf_set_nat @ A3 @ B4 ) )
& ( A3 != B4 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_934_inf_Ostrict__coboundedI1,axiom,
! [A: set_nat,C: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A @ C )
=> ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_935_inf_Ostrict__coboundedI2,axiom,
! [B2: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B2 @ C )
=> ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_936_inf__left__commute,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y2 @ Z2 ) )
= ( inf_inf_set_nat @ Y2 @ ( inf_inf_set_nat @ X @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_937_inf_Oleft__commute,axiom,
! [B2: set_nat,A: set_nat,C: set_nat] :
( ( inf_inf_set_nat @ B2 @ ( inf_inf_set_nat @ A @ C ) )
= ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_938_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_nat,K: set_nat,B2: set_nat,A: set_nat] :
( ( B
= ( inf_inf_set_nat @ K @ B2 ) )
=> ( ( inf_inf_set_nat @ A @ B )
= ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_939_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_nat,K: set_nat,A: set_nat,B2: set_nat] :
( ( A2
= ( inf_inf_set_nat @ K @ A ) )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_940_inf__commute,axiom,
( inf_inf_set_nat
= ( ^ [X2: set_nat,Y4: set_nat] : ( inf_inf_set_nat @ Y4 @ X2 ) ) ) ).
% inf_commute
thf(fact_941_inf_Ocommute,axiom,
( inf_inf_set_nat
= ( ^ [A3: set_nat,B4: set_nat] : ( inf_inf_set_nat @ B4 @ A3 ) ) ) ).
% inf.commute
thf(fact_942_inf__assoc,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y2 ) @ Z2 )
= ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y2 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_943_inf_Oassoc,axiom,
! [A: set_nat,B2: set_nat,C: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ C )
= ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_944_inf__sup__aci_I1_J,axiom,
( inf_inf_set_nat
= ( ^ [X2: set_nat,Y4: set_nat] : ( inf_inf_set_nat @ Y4 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_945_inf__sup__aci_I2_J,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y2 ) @ Z2 )
= ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y2 @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_946_inf__sup__aci_I3_J,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y2 @ Z2 ) )
= ( inf_inf_set_nat @ Y2 @ ( inf_inf_set_nat @ X @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_947_inf__sup__aci_I4_J,axiom,
! [X: set_nat,Y2: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ X @ Y2 ) )
= ( inf_inf_set_nat @ X @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_948_notin__set__remove1,axiom,
! [X: nat,Xs: list_nat,Y2: nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ~ ( member_nat @ X @ ( set_nat2 @ ( remove1_nat @ Y2 @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_949_notin__set__remove1,axiom,
! [X: transition,Xs: list_transition,Y2: transition] :
( ~ ( member_transition @ X @ ( set_transition2 @ Xs ) )
=> ~ ( member_transition @ X @ ( set_transition2 @ ( remove1_transition @ Y2 @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_950_remove1__idem,axiom,
! [X: nat,Xs: list_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( remove1_nat @ X @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_951_remove1__idem,axiom,
! [X: transition,Xs: list_transition] :
( ~ ( member_transition @ X @ ( set_transition2 @ Xs ) )
=> ( ( remove1_transition @ X @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_952_remove1_Osimps_I1_J,axiom,
! [X: transition] :
( ( remove1_transition @ X @ nil_transition )
= nil_transition ) ).
% remove1.simps(1)
thf(fact_953_remove1_Osimps_I1_J,axiom,
! [X: $o] :
( ( remove1_o @ X @ nil_o )
= nil_o ) ).
% remove1.simps(1)
thf(fact_954_remove1_Osimps_I1_J,axiom,
! [X: list_o] :
( ( remove1_list_o @ X @ nil_list_o )
= nil_list_o ) ).
% remove1.simps(1)
thf(fact_955_remove1_Osimps_I2_J,axiom,
! [X: list_o,Y2: list_o,Xs: list_list_o] :
( ( ( X = Y2 )
=> ( ( remove1_list_o @ X @ ( cons_list_o @ Y2 @ Xs ) )
= Xs ) )
& ( ( X != Y2 )
=> ( ( remove1_list_o @ X @ ( cons_list_o @ Y2 @ Xs ) )
= ( cons_list_o @ Y2 @ ( remove1_list_o @ X @ Xs ) ) ) ) ) ).
% remove1.simps(2)
thf(fact_956_Int__Diff,axiom,
! [A2: set_nat,B: set_nat,C3: set_nat] :
( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ C3 )
= ( inf_inf_set_nat @ A2 @ ( minus_minus_set_nat @ B @ C3 ) ) ) ).
% Int_Diff
thf(fact_957_Diff__Int2,axiom,
! [A2: set_nat,C3: set_nat,B: set_nat] :
( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A2 @ C3 ) @ ( inf_inf_set_nat @ B @ C3 ) )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ A2 @ C3 ) @ B ) ) ).
% Diff_Int2
thf(fact_958_Diff__Diff__Int,axiom,
! [A2: set_nat,B: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( minus_minus_set_nat @ A2 @ B ) )
= ( inf_inf_set_nat @ A2 @ B ) ) ).
% Diff_Diff_Int
thf(fact_959_Diff__Int__distrib,axiom,
! [C3: set_nat,A2: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ C3 @ ( minus_minus_set_nat @ A2 @ B ) )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ C3 @ A2 ) @ ( inf_inf_set_nat @ C3 @ B ) ) ) ).
% Diff_Int_distrib
thf(fact_960_Diff__Int__distrib2,axiom,
! [A2: set_nat,B: set_nat,C3: set_nat] :
( ( inf_inf_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ C3 )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ A2 @ C3 ) @ ( inf_inf_set_nat @ B @ C3 ) ) ) ).
% Diff_Int_distrib2
thf(fact_961_distrib__sup__le,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y2 @ Z2 ) ) @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y2 ) @ ( sup_sup_set_nat @ X @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_962_distrib__inf__le,axiom,
! [X: set_nat,Y2: set_nat,Z2: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y2 ) @ ( inf_inf_set_nat @ X @ Z2 ) ) @ ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y2 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_963_image__Int__subset,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( inf_inf_set_nat @ A2 @ B ) ) @ ( inf_inf_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_964_Min_OcoboundedI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ A ) ) ) ).
% Min.coboundedI
thf(fact_965_Min__eqI,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ! [Y: nat] :
( ( member_nat @ Y @ A2 )
=> ( ord_less_eq_nat @ X @ Y ) )
=> ( ( member_nat @ X @ A2 )
=> ( ( lattic8721135487736765967in_nat @ A2 )
= X ) ) ) ) ).
% Min_eqI
thf(fact_966_Min__le,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ X ) ) ) ).
% Min_le
thf(fact_967_Diff__triv,axiom,
! [A2: set_transition,B: set_transition] :
( ( ( inf_in8814773338690644108sition @ A2 @ B )
= bot_bo301567166201926666sition )
=> ( ( minus_8944320859760356485sition @ A2 @ B )
= A2 ) ) ).
% Diff_triv
thf(fact_968_Diff__triv,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B )
= bot_bot_set_nat )
=> ( ( minus_minus_set_nat @ A2 @ B )
= A2 ) ) ).
% Diff_triv
thf(fact_969_Int__Diff__disjoint,axiom,
! [A2: set_transition,B: set_transition] :
( ( inf_in8814773338690644108sition @ ( inf_in8814773338690644108sition @ A2 @ B ) @ ( minus_8944320859760356485sition @ A2 @ B ) )
= bot_bo301567166201926666sition ) ).
% Int_Diff_disjoint
thf(fact_970_Int__Diff__disjoint,axiom,
! [A2: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ ( minus_minus_set_nat @ A2 @ B ) )
= bot_bot_set_nat ) ).
% Int_Diff_disjoint
thf(fact_971_Un__Int__assoc__eq,axiom,
! [A2: set_nat,B: set_nat,C3: set_nat] :
( ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ C3 )
= ( inf_inf_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C3 ) ) )
= ( ord_less_eq_set_nat @ C3 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_972_Min__in,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( member_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ A2 ) ) ) ).
% Min_in
thf(fact_973_Un__Diff__Int,axiom,
! [A2: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ ( inf_inf_set_nat @ A2 @ B ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_974_Int__Diff__Un,axiom,
! [A2: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ ( minus_minus_set_nat @ A2 @ B ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_975_Diff__Int,axiom,
! [A2: set_nat,B: set_nat,C3: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( inf_inf_set_nat @ B @ C3 ) )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ ( minus_minus_set_nat @ A2 @ C3 ) ) ) ).
% Diff_Int
thf(fact_976_Diff__Un,axiom,
! [A2: set_nat,B: set_nat,C3: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C3 ) )
= ( inf_inf_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ ( minus_minus_set_nat @ A2 @ C3 ) ) ) ).
% Diff_Un
thf(fact_977_set__remove1__subset,axiom,
! [X: transition,Xs: list_transition] : ( ord_le8419162016481440574sition @ ( set_transition2 @ ( remove1_transition @ X @ Xs ) ) @ ( set_transition2 @ Xs ) ) ).
% set_remove1_subset
thf(fact_978_set__remove1__subset,axiom,
! [X: nat,Xs: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( remove1_nat @ X @ Xs ) ) @ ( set_nat2 @ Xs ) ) ).
% set_remove1_subset
thf(fact_979_step__symb__set__proj,axiom,
( step_symb_set
= ( ^ [Q02: nat,Transs2: list_transition,R2: set_nat] : ( step_symb_set @ Q02 @ Transs2 @ ( inf_inf_set_nat @ R2 @ ( sq @ Q02 @ Transs2 ) ) ) ) ) ).
% step_symb_set_proj
thf(fact_980_Min__eq__iff,axiom,
! [A2: set_nat,M: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ( lattic8721135487736765967in_nat @ A2 )
= M )
= ( ( member_nat @ M @ A2 )
& ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ M @ X2 ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_981_Min__le__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ X )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_eq_nat @ X2 @ X ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_982_eq__Min__iff,axiom,
! [A2: set_nat,M: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( M
= ( lattic8721135487736765967in_nat @ A2 ) )
= ( ( member_nat @ M @ A2 )
& ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ M @ X2 ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_983_Min_OboundedE,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) )
=> ! [A8: nat] :
( ( member_nat @ A8 @ A2 )
=> ( ord_less_eq_nat @ X @ A8 ) ) ) ) ) ).
% Min.boundedE
thf(fact_984_Min_OboundedI,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [A7: nat] :
( ( member_nat @ A7 @ A2 )
=> ( ord_less_eq_nat @ X @ A7 ) )
=> ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) ) ) ) ) ).
% Min.boundedI
thf(fact_985_Min__less__iff,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ X )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_nat @ X2 @ X ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_986_Min__insert2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ! [B7: nat] :
( ( member_nat @ B7 @ A2 )
=> ( ord_less_eq_nat @ A @ B7 ) )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat2 @ A @ A2 ) )
= A ) ) ) ).
% Min_insert2
thf(fact_987_Inf__fin_Ohom__commute,axiom,
! [H: set_nat > set_nat,N3: set_set_nat] :
( ! [X3: set_nat,Y: set_nat] :
( ( H @ ( inf_inf_set_nat @ X3 @ Y ) )
= ( inf_inf_set_nat @ ( H @ X3 ) @ ( H @ Y ) ) )
=> ( ( finite1152437895449049373et_nat @ N3 )
=> ( ( N3 != bot_bot_set_set_nat )
=> ( ( H @ ( lattic3014633134055518761et_nat @ N3 ) )
= ( lattic3014633134055518761et_nat @ ( image_7916887816326733075et_nat @ H @ N3 ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_988_Inf__fin_Ohom__commute,axiom,
! [H: nat > nat,N3: set_nat] :
( ! [X3: nat,Y: nat] :
( ( H @ ( inf_inf_nat @ X3 @ Y ) )
= ( inf_inf_nat @ ( H @ X3 ) @ ( H @ Y ) ) )
=> ( ( finite_finite_nat @ N3 )
=> ( ( N3 != bot_bot_set_nat )
=> ( ( H @ ( lattic5238388535129920115in_nat @ N3 ) )
= ( lattic5238388535129920115in_nat @ ( image_nat_nat @ H @ N3 ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_989_Inf__fin_Osubset,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( B != bot_bot_set_set_nat )
=> ( ( ord_le6893508408891458716et_nat @ B @ A2 )
=> ( ( inf_inf_set_nat @ ( lattic3014633134055518761et_nat @ B ) @ ( lattic3014633134055518761et_nat @ A2 ) )
= ( lattic3014633134055518761et_nat @ A2 ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_990_Inf__fin_Osubset,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( B != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ B ) @ ( lattic5238388535129920115in_nat @ A2 ) )
= ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_991_Inf__fin_Oclosed,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ! [X3: set_nat,Y: set_nat] : ( member_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ ( insert_set_nat @ X3 @ ( insert_set_nat @ Y @ bot_bot_set_set_nat ) ) )
=> ( member_set_nat @ ( lattic3014633134055518761et_nat @ A2 ) @ A2 ) ) ) ) ).
% Inf_fin.closed
thf(fact_992_Inf__fin_Oclosed,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [X3: nat,Y: nat] : ( member_nat @ ( inf_inf_nat @ X3 @ Y ) @ ( insert_nat2 @ X3 @ ( insert_nat2 @ Y @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic5238388535129920115in_nat @ A2 ) @ A2 ) ) ) ) ).
% Inf_fin.closed
thf(fact_993_Inf__fin_Oinsert__not__elem,axiom,
! [A2: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ~ ( member_set_nat @ X @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X @ A2 ) )
= ( inf_inf_set_nat @ X @ ( lattic3014633134055518761et_nat @ A2 ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_994_Inf__fin_Oinsert__not__elem,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat @ X @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat2 @ X @ A2 ) )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_995_Inf__fin_Ounion,axiom,
! [A2: set_set_nat,B: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ( finite1152437895449049373et_nat @ B )
=> ( ( B != bot_bot_set_set_nat )
=> ( ( lattic3014633134055518761et_nat @ ( sup_sup_set_set_nat @ A2 @ B ) )
= ( inf_inf_set_nat @ ( lattic3014633134055518761et_nat @ A2 ) @ ( lattic3014633134055518761et_nat @ B ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_996_Inf__fin_Ounion,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ( B != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( sup_sup_set_nat @ A2 @ B ) )
= ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ A2 ) @ ( lattic5238388535129920115in_nat @ B ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_997_Min__antimono,axiom,
! [M2: set_nat,N3: set_nat] :
( ( ord_less_eq_set_nat @ M2 @ N3 )
=> ( ( M2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ N3 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ N3 ) @ ( lattic8721135487736765967in_nat @ M2 ) ) ) ) ) ).
% Min_antimono
thf(fact_998_Min_Osubset__imp,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ B ) @ ( lattic8721135487736765967in_nat @ A2 ) ) ) ) ) ).
% Min.subset_imp
thf(fact_999_Inf__fin_Oinsert__remove,axiom,
! [A2: set_set_nat,X: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
= bot_bot_set_set_nat )
=> ( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X @ A2 ) )
= X ) )
& ( ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
!= bot_bot_set_set_nat )
=> ( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X @ A2 ) )
= ( inf_inf_set_nat @ X @ ( lattic3014633134055518761et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_1000_Inf__fin_Oinsert__remove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat2 @ X @ A2 ) )
= X ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat2 @ X @ A2 ) )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_1001_min__list__Min,axiom,
! [Xs: list_o] :
( ( Xs != nil_o )
=> ( ( min_list_o @ Xs )
= ( lattic1973801136483472281_Min_o @ ( set_o2 @ Xs ) ) ) ) ).
% min_list_Min
thf(fact_1002_set__remove1__eq,axiom,
! [Xs: list_transition,X: transition] :
( ( distinct_transition @ Xs )
=> ( ( set_transition2 @ ( remove1_transition @ X @ Xs ) )
= ( minus_8944320859760356485sition @ ( set_transition2 @ Xs ) @ ( insert_transition2 @ X @ bot_bo301567166201926666sition ) ) ) ) ).
% set_remove1_eq
thf(fact_1003_set__remove1__eq,axiom,
! [Xs: list_nat,X: nat] :
( ( distinct_nat @ Xs )
=> ( ( set_nat2 @ ( remove1_nat @ X @ Xs ) )
= ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ).
% set_remove1_eq
thf(fact_1004_Int__iff,axiom,
! [C: transition,A2: set_transition,B: set_transition] :
( ( member_transition @ C @ ( inf_in8814773338690644108sition @ A2 @ B ) )
= ( ( member_transition @ C @ A2 )
& ( member_transition @ C @ B ) ) ) ).
% Int_iff
thf(fact_1005_Int__iff,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B ) )
= ( ( member_nat @ C @ A2 )
& ( member_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_1006_IntI,axiom,
! [C: transition,A2: set_transition,B: set_transition] :
( ( member_transition @ C @ A2 )
=> ( ( member_transition @ C @ B )
=> ( member_transition @ C @ ( inf_in8814773338690644108sition @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_1007_IntI,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_1008_Int__left__commute,axiom,
! [A2: set_nat,B: set_nat,C3: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ B @ C3 ) )
= ( inf_inf_set_nat @ B @ ( inf_inf_set_nat @ A2 @ C3 ) ) ) ).
% Int_left_commute
thf(fact_1009_Int__left__absorb,axiom,
! [A2: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ A2 @ B ) )
= ( inf_inf_set_nat @ A2 @ B ) ) ).
% Int_left_absorb
thf(fact_1010_Int__commute,axiom,
( inf_inf_set_nat
= ( ^ [A4: set_nat,B5: set_nat] : ( inf_inf_set_nat @ B5 @ A4 ) ) ) ).
% Int_commute
thf(fact_1011_Int__absorb,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_1012_Int__assoc,axiom,
! [A2: set_nat,B: set_nat,C3: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ C3 )
= ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ B @ C3 ) ) ) ).
% Int_assoc
thf(fact_1013_IntD2,axiom,
! [C: transition,A2: set_transition,B: set_transition] :
( ( member_transition @ C @ ( inf_in8814773338690644108sition @ A2 @ B ) )
=> ( member_transition @ C @ B ) ) ).
% IntD2
thf(fact_1014_IntD2,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B ) )
=> ( member_nat @ C @ B ) ) ).
% IntD2
thf(fact_1015_IntD1,axiom,
! [C: transition,A2: set_transition,B: set_transition] :
( ( member_transition @ C @ ( inf_in8814773338690644108sition @ A2 @ B ) )
=> ( member_transition @ C @ A2 ) ) ).
% IntD1
thf(fact_1016_IntD1,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B ) )
=> ( member_nat @ C @ A2 ) ) ).
% IntD1
thf(fact_1017_IntE,axiom,
! [C: transition,A2: set_transition,B: set_transition] :
( ( member_transition @ C @ ( inf_in8814773338690644108sition @ A2 @ B ) )
=> ~ ( ( member_transition @ C @ A2 )
=> ~ ( member_transition @ C @ B ) ) ) ).
% IntE
thf(fact_1018_IntE,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ~ ( member_nat @ C @ B ) ) ) ).
% IntE
thf(fact_1019_distinct__length__2__or__more,axiom,
! [A: list_o,B2: list_o,Xs: list_list_o] :
( ( distinct_list_o @ ( cons_list_o @ A @ ( cons_list_o @ B2 @ Xs ) ) )
= ( ( A != B2 )
& ( distinct_list_o @ ( cons_list_o @ A @ Xs ) )
& ( distinct_list_o @ ( cons_list_o @ B2 @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_1020_distinct_Osimps_I1_J,axiom,
distinct_transition @ nil_transition ).
% distinct.simps(1)
thf(fact_1021_distinct_Osimps_I1_J,axiom,
distinct_o @ nil_o ).
% distinct.simps(1)
thf(fact_1022_distinct_Osimps_I1_J,axiom,
distinct_list_o @ nil_list_o ).
% distinct.simps(1)
thf(fact_1023_finite__distinct__list,axiom,
! [A2: set_transition] :
( ( finite8165534619950747239sition @ A2 )
=> ? [Xs3: list_transition] :
( ( ( set_transition2 @ Xs3 )
= A2 )
& ( distinct_transition @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_1024_finite__distinct__list,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ? [Xs3: list_nat] :
( ( ( set_nat2 @ Xs3 )
= A2 )
& ( distinct_nat @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_1025_distinct__singleton,axiom,
! [X: transition] : ( distinct_transition @ ( cons_transition @ X @ nil_transition ) ) ).
% distinct_singleton
thf(fact_1026_distinct__singleton,axiom,
! [X: $o] : ( distinct_o @ ( cons_o @ X @ nil_o ) ) ).
% distinct_singleton
thf(fact_1027_distinct__singleton,axiom,
! [X: list_o] : ( distinct_list_o @ ( cons_list_o @ X @ nil_list_o ) ) ).
% distinct_singleton
thf(fact_1028_distinct_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( distinct_nat @ ( cons_nat @ X @ Xs ) )
= ( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
& ( distinct_nat @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_1029_distinct_Osimps_I2_J,axiom,
! [X: transition,Xs: list_transition] :
( ( distinct_transition @ ( cons_transition @ X @ Xs ) )
= ( ~ ( member_transition @ X @ ( set_transition2 @ Xs ) )
& ( distinct_transition @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_1030_distinct_Osimps_I2_J,axiom,
! [X: list_o,Xs: list_list_o] :
( ( distinct_list_o @ ( cons_list_o @ X @ Xs ) )
= ( ~ ( member_list_o @ X @ ( set_list_o2 @ Xs ) )
& ( distinct_list_o @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_1031_distinct__concat__iff,axiom,
! [Xs: list_list_o] :
( ( distinct_o @ ( concat_o @ Xs ) )
= ( ( distinct_list_o @ ( removeAll_list_o @ nil_o @ Xs ) )
& ! [Ys2: list_o] :
( ( member_list_o @ Ys2 @ ( set_list_o2 @ Xs ) )
=> ( distinct_o @ Ys2 ) )
& ! [Ys2: list_o,Zs: list_o] :
( ( ( member_list_o @ Ys2 @ ( set_list_o2 @ Xs ) )
& ( member_list_o @ Zs @ ( set_list_o2 @ Xs ) )
& ( Ys2 != Zs ) )
=> ( ( inf_inf_set_o @ ( set_o2 @ Ys2 ) @ ( set_o2 @ Zs ) )
= bot_bot_set_o ) ) ) ) ).
% distinct_concat_iff
thf(fact_1032_distinct__concat__iff,axiom,
! [Xs: list_list_list_o] :
( ( distinct_list_o @ ( concat_list_o @ Xs ) )
= ( ( distinct_list_list_o @ ( remove3821550480258065712list_o @ nil_list_o @ Xs ) )
& ! [Ys2: list_list_o] :
( ( member_list_list_o @ Ys2 @ ( set_list_list_o2 @ Xs ) )
=> ( distinct_list_o @ Ys2 ) )
& ! [Ys2: list_list_o,Zs: list_list_o] :
( ( ( member_list_list_o @ Ys2 @ ( set_list_list_o2 @ Xs ) )
& ( member_list_list_o @ Zs @ ( set_list_list_o2 @ Xs ) )
& ( Ys2 != Zs ) )
=> ( ( inf_inf_set_list_o @ ( set_list_o2 @ Ys2 ) @ ( set_list_o2 @ Zs ) )
= bot_bot_set_list_o ) ) ) ) ).
% distinct_concat_iff
thf(fact_1033_distinct__concat__iff,axiom,
! [Xs: list_list_nat] :
( ( distinct_nat @ ( concat_nat @ Xs ) )
= ( ( distinct_list_nat @ ( removeAll_list_nat @ nil_nat @ Xs ) )
& ! [Ys2: list_nat] :
( ( member_list_nat @ Ys2 @ ( set_list_nat2 @ Xs ) )
=> ( distinct_nat @ Ys2 ) )
& ! [Ys2: list_nat,Zs: list_nat] :
( ( ( member_list_nat @ Ys2 @ ( set_list_nat2 @ Xs ) )
& ( member_list_nat @ Zs @ ( set_list_nat2 @ Xs ) )
& ( Ys2 != Zs ) )
=> ( ( inf_inf_set_nat @ ( set_nat2 @ Ys2 ) @ ( set_nat2 @ Zs ) )
= bot_bot_set_nat ) ) ) ) ).
% distinct_concat_iff
thf(fact_1034_distinct__concat__iff,axiom,
! [Xs: list_list_transition] :
( ( distinct_transition @ ( concat_transition @ Xs ) )
= ( ( distin4894176225816993341sition @ ( remove2429998804908088272sition @ nil_transition @ Xs ) )
& ! [Ys2: list_transition] :
( ( member1473516902542837997sition @ Ys2 @ ( set_list_transition2 @ Xs ) )
=> ( distinct_transition @ Ys2 ) )
& ! [Ys2: list_transition,Zs: list_transition] :
( ( ( member1473516902542837997sition @ Ys2 @ ( set_list_transition2 @ Xs ) )
& ( member1473516902542837997sition @ Zs @ ( set_list_transition2 @ Xs ) )
& ( Ys2 != Zs ) )
=> ( ( inf_in8814773338690644108sition @ ( set_transition2 @ Ys2 ) @ ( set_transition2 @ Zs ) )
= bot_bo301567166201926666sition ) ) ) ) ).
% distinct_concat_iff
thf(fact_1035_distinct__concat,axiom,
! [Xs: list_list_nat] :
( ( distinct_list_nat @ Xs )
=> ( ! [Ys3: list_nat] :
( ( member_list_nat @ Ys3 @ ( set_list_nat2 @ Xs ) )
=> ( distinct_nat @ Ys3 ) )
=> ( ! [Ys3: list_nat,Zs2: list_nat] :
( ( member_list_nat @ Ys3 @ ( set_list_nat2 @ Xs ) )
=> ( ( member_list_nat @ Zs2 @ ( set_list_nat2 @ Xs ) )
=> ( ( Ys3 != Zs2 )
=> ( ( inf_inf_set_nat @ ( set_nat2 @ Ys3 ) @ ( set_nat2 @ Zs2 ) )
= bot_bot_set_nat ) ) ) )
=> ( distinct_nat @ ( concat_nat @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_1036_distinct__concat,axiom,
! [Xs: list_list_transition] :
( ( distin4894176225816993341sition @ Xs )
=> ( ! [Ys3: list_transition] :
( ( member1473516902542837997sition @ Ys3 @ ( set_list_transition2 @ Xs ) )
=> ( distinct_transition @ Ys3 ) )
=> ( ! [Ys3: list_transition,Zs2: list_transition] :
( ( member1473516902542837997sition @ Ys3 @ ( set_list_transition2 @ Xs ) )
=> ( ( member1473516902542837997sition @ Zs2 @ ( set_list_transition2 @ Xs ) )
=> ( ( Ys3 != Zs2 )
=> ( ( inf_in8814773338690644108sition @ ( set_transition2 @ Ys3 ) @ ( set_transition2 @ Zs2 ) )
= bot_bo301567166201926666sition ) ) ) )
=> ( distinct_transition @ ( concat_transition @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_1037_Nil__eq__concat__conv,axiom,
! [Xss2: list_list_transition] :
( ( nil_transition
= ( concat_transition @ Xss2 ) )
= ( ! [X2: list_transition] :
( ( member1473516902542837997sition @ X2 @ ( set_list_transition2 @ Xss2 ) )
=> ( X2 = nil_transition ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_1038_Nil__eq__concat__conv,axiom,
! [Xss2: list_list_o] :
( ( nil_o
= ( concat_o @ Xss2 ) )
= ( ! [X2: list_o] :
( ( member_list_o @ X2 @ ( set_list_o2 @ Xss2 ) )
=> ( X2 = nil_o ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_1039_Nil__eq__concat__conv,axiom,
! [Xss2: list_list_list_o] :
( ( nil_list_o
= ( concat_list_o @ Xss2 ) )
= ( ! [X2: list_list_o] :
( ( member_list_list_o @ X2 @ ( set_list_list_o2 @ Xss2 ) )
=> ( X2 = nil_list_o ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_1040_concat__eq__Nil__conv,axiom,
! [Xss2: list_list_transition] :
( ( ( concat_transition @ Xss2 )
= nil_transition )
= ( ! [X2: list_transition] :
( ( member1473516902542837997sition @ X2 @ ( set_list_transition2 @ Xss2 ) )
=> ( X2 = nil_transition ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_1041_concat__eq__Nil__conv,axiom,
! [Xss2: list_list_o] :
( ( ( concat_o @ Xss2 )
= nil_o )
= ( ! [X2: list_o] :
( ( member_list_o @ X2 @ ( set_list_o2 @ Xss2 ) )
=> ( X2 = nil_o ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_1042_concat__eq__Nil__conv,axiom,
! [Xss2: list_list_list_o] :
( ( ( concat_list_o @ Xss2 )
= nil_list_o )
= ( ! [X2: list_list_o] :
( ( member_list_list_o @ X2 @ ( set_list_list_o2 @ Xss2 ) )
=> ( X2 = nil_list_o ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_1043_concat_Osimps_I1_J,axiom,
( ( concat_transition @ nil_list_transition )
= nil_transition ) ).
% concat.simps(1)
thf(fact_1044_concat_Osimps_I1_J,axiom,
( ( concat_list_o @ nil_list_list_o )
= nil_list_o ) ).
% concat.simps(1)
thf(fact_1045_concat_Osimps_I1_J,axiom,
( ( concat_o @ nil_list_o )
= nil_o ) ).
% concat.simps(1)
thf(fact_1046_distinct__append,axiom,
! [Xs: list_list_o,Ys: list_list_o] :
( ( distinct_list_o @ ( append_list_o @ Xs @ Ys ) )
= ( ( distinct_list_o @ Xs )
& ( distinct_list_o @ Ys )
& ( ( inf_inf_set_list_o @ ( set_list_o2 @ Xs ) @ ( set_list_o2 @ Ys ) )
= bot_bot_set_list_o ) ) ) ).
% distinct_append
thf(fact_1047_distinct__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( distinct_nat @ ( append_nat @ Xs @ Ys ) )
= ( ( distinct_nat @ Xs )
& ( distinct_nat @ Ys )
& ( ( inf_inf_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) )
= bot_bot_set_nat ) ) ) ).
% distinct_append
thf(fact_1048_distinct__append,axiom,
! [Xs: list_transition,Ys: list_transition] :
( ( distinct_transition @ ( append_transition @ Xs @ Ys ) )
= ( ( distinct_transition @ Xs )
& ( distinct_transition @ Ys )
& ( ( inf_in8814773338690644108sition @ ( set_transition2 @ Xs ) @ ( set_transition2 @ Ys ) )
= bot_bo301567166201926666sition ) ) ) ).
% distinct_append
thf(fact_1049_Min_Oinsert__remove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat2 @ X @ A2 ) )
= X ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat2 @ X @ A2 ) )
= ( ord_min_nat @ X @ ( lattic8721135487736765967in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Min.insert_remove
thf(fact_1050_append_Oassoc,axiom,
! [A: list_list_o,B2: list_list_o,C: list_list_o] :
( ( append_list_o @ ( append_list_o @ A @ B2 ) @ C )
= ( append_list_o @ A @ ( append_list_o @ B2 @ C ) ) ) ).
% append.assoc
thf(fact_1051_append__assoc,axiom,
! [Xs: list_list_o,Ys: list_list_o,Zs3: list_list_o] :
( ( append_list_o @ ( append_list_o @ Xs @ Ys ) @ Zs3 )
= ( append_list_o @ Xs @ ( append_list_o @ Ys @ Zs3 ) ) ) ).
% append_assoc
thf(fact_1052_append__same__eq,axiom,
! [Ys: list_list_o,Xs: list_list_o,Zs3: list_list_o] :
( ( ( append_list_o @ Ys @ Xs )
= ( append_list_o @ Zs3 @ Xs ) )
= ( Ys = Zs3 ) ) ).
% append_same_eq
thf(fact_1053_same__append__eq,axiom,
! [Xs: list_list_o,Ys: list_list_o,Zs3: list_list_o] :
( ( ( append_list_o @ Xs @ Ys )
= ( append_list_o @ Xs @ Zs3 ) )
= ( Ys = Zs3 ) ) ).
% same_append_eq
thf(fact_1054_append_Oright__neutral,axiom,
! [A: list_transition] :
( ( append_transition @ A @ nil_transition )
= A ) ).
% append.right_neutral
thf(fact_1055_append_Oright__neutral,axiom,
! [A: list_o] :
( ( append_o @ A @ nil_o )
= A ) ).
% append.right_neutral
thf(fact_1056_append_Oright__neutral,axiom,
! [A: list_list_o] :
( ( append_list_o @ A @ nil_list_o )
= A ) ).
% append.right_neutral
thf(fact_1057_append__Nil2,axiom,
! [Xs: list_transition] :
( ( append_transition @ Xs @ nil_transition )
= Xs ) ).
% append_Nil2
thf(fact_1058_append__Nil2,axiom,
! [Xs: list_o] :
( ( append_o @ Xs @ nil_o )
= Xs ) ).
% append_Nil2
thf(fact_1059_append__Nil2,axiom,
! [Xs: list_list_o] :
( ( append_list_o @ Xs @ nil_list_o )
= Xs ) ).
% append_Nil2
thf(fact_1060_append__self__conv,axiom,
! [Xs: list_transition,Ys: list_transition] :
( ( ( append_transition @ Xs @ Ys )
= Xs )
= ( Ys = nil_transition ) ) ).
% append_self_conv
thf(fact_1061_append__self__conv,axiom,
! [Xs: list_o,Ys: list_o] :
( ( ( append_o @ Xs @ Ys )
= Xs )
= ( Ys = nil_o ) ) ).
% append_self_conv
thf(fact_1062_append__self__conv,axiom,
! [Xs: list_list_o,Ys: list_list_o] :
( ( ( append_list_o @ Xs @ Ys )
= Xs )
= ( Ys = nil_list_o ) ) ).
% append_self_conv
thf(fact_1063_self__append__conv,axiom,
! [Y2: list_transition,Ys: list_transition] :
( ( Y2
= ( append_transition @ Y2 @ Ys ) )
= ( Ys = nil_transition ) ) ).
% self_append_conv
thf(fact_1064_self__append__conv,axiom,
! [Y2: list_o,Ys: list_o] :
( ( Y2
= ( append_o @ Y2 @ Ys ) )
= ( Ys = nil_o ) ) ).
% self_append_conv
thf(fact_1065_self__append__conv,axiom,
! [Y2: list_list_o,Ys: list_list_o] :
( ( Y2
= ( append_list_o @ Y2 @ Ys ) )
= ( Ys = nil_list_o ) ) ).
% self_append_conv
thf(fact_1066_append__self__conv2,axiom,
! [Xs: list_transition,Ys: list_transition] :
( ( ( append_transition @ Xs @ Ys )
= Ys )
= ( Xs = nil_transition ) ) ).
% append_self_conv2
thf(fact_1067_append__self__conv2,axiom,
! [Xs: list_o,Ys: list_o] :
( ( ( append_o @ Xs @ Ys )
= Ys )
= ( Xs = nil_o ) ) ).
% append_self_conv2
thf(fact_1068_append__self__conv2,axiom,
! [Xs: list_list_o,Ys: list_list_o] :
( ( ( append_list_o @ Xs @ Ys )
= Ys )
= ( Xs = nil_list_o ) ) ).
% append_self_conv2
thf(fact_1069_self__append__conv2,axiom,
! [Y2: list_transition,Xs: list_transition] :
( ( Y2
= ( append_transition @ Xs @ Y2 ) )
= ( Xs = nil_transition ) ) ).
% self_append_conv2
thf(fact_1070_self__append__conv2,axiom,
! [Y2: list_o,Xs: list_o] :
( ( Y2
= ( append_o @ Xs @ Y2 ) )
= ( Xs = nil_o ) ) ).
% self_append_conv2
thf(fact_1071_self__append__conv2,axiom,
! [Y2: list_list_o,Xs: list_list_o] :
( ( Y2
= ( append_list_o @ Xs @ Y2 ) )
= ( Xs = nil_list_o ) ) ).
% self_append_conv2
thf(fact_1072_Nil__is__append__conv,axiom,
! [Xs: list_transition,Ys: list_transition] :
( ( nil_transition
= ( append_transition @ Xs @ Ys ) )
= ( ( Xs = nil_transition )
& ( Ys = nil_transition ) ) ) ).
% Nil_is_append_conv
thf(fact_1073_Nil__is__append__conv,axiom,
! [Xs: list_o,Ys: list_o] :
( ( nil_o
= ( append_o @ Xs @ Ys ) )
= ( ( Xs = nil_o )
& ( Ys = nil_o ) ) ) ).
% Nil_is_append_conv
thf(fact_1074_Nil__is__append__conv,axiom,
! [Xs: list_list_o,Ys: list_list_o] :
( ( nil_list_o
= ( append_list_o @ Xs @ Ys ) )
= ( ( Xs = nil_list_o )
& ( Ys = nil_list_o ) ) ) ).
% Nil_is_append_conv
thf(fact_1075_append__is__Nil__conv,axiom,
! [Xs: list_transition,Ys: list_transition] :
( ( ( append_transition @ Xs @ Ys )
= nil_transition )
= ( ( Xs = nil_transition )
& ( Ys = nil_transition ) ) ) ).
% append_is_Nil_conv
thf(fact_1076_append__is__Nil__conv,axiom,
! [Xs: list_o,Ys: list_o] :
( ( ( append_o @ Xs @ Ys )
= nil_o )
= ( ( Xs = nil_o )
& ( Ys = nil_o ) ) ) ).
% append_is_Nil_conv
thf(fact_1077_append__is__Nil__conv,axiom,
! [Xs: list_list_o,Ys: list_list_o] :
( ( ( append_list_o @ Xs @ Ys )
= nil_list_o )
= ( ( Xs = nil_list_o )
& ( Ys = nil_list_o ) ) ) ).
% append_is_Nil_conv
thf(fact_1078_min__bot,axiom,
! [X: set_nat] :
( ( ord_min_set_nat @ bot_bot_set_nat @ X )
= bot_bot_set_nat ) ).
% min_bot
thf(fact_1079_min__bot,axiom,
! [X: set_transition] :
( ( ord_mi6397184166219407237sition @ bot_bo301567166201926666sition @ X )
= bot_bo301567166201926666sition ) ).
% min_bot
thf(fact_1080_min__bot2,axiom,
! [X: set_nat] :
( ( ord_min_set_nat @ X @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% min_bot2
thf(fact_1081_min__bot2,axiom,
! [X: set_transition] :
( ( ord_mi6397184166219407237sition @ X @ bot_bo301567166201926666sition )
= bot_bo301567166201926666sition ) ).
% min_bot2
thf(fact_1082_concat__append,axiom,
! [Xs: list_list_list_o,Ys: list_list_list_o] :
( ( concat_list_o @ ( append_list_list_o @ Xs @ Ys ) )
= ( append_list_o @ ( concat_list_o @ Xs ) @ ( concat_list_o @ Ys ) ) ) ).
% concat_append
thf(fact_1083_concat__append,axiom,
! [Xs: list_list_o,Ys: list_list_o] :
( ( concat_o @ ( append_list_o @ Xs @ Ys ) )
= ( append_o @ ( concat_o @ Xs ) @ ( concat_o @ Ys ) ) ) ).
% concat_append
thf(fact_1084_removeAll__append,axiom,
! [X: list_o,Xs: list_list_o,Ys: list_list_o] :
( ( removeAll_list_o @ X @ ( append_list_o @ Xs @ Ys ) )
= ( append_list_o @ ( removeAll_list_o @ X @ Xs ) @ ( removeAll_list_o @ X @ Ys ) ) ) ).
% removeAll_append
thf(fact_1085_foldl__append,axiom,
! [F: set_nat > list_o > set_nat,A: set_nat,Xs: list_list_o,Ys: list_list_o] :
( ( foldl_set_nat_list_o @ F @ A @ ( append_list_o @ Xs @ Ys ) )
= ( foldl_set_nat_list_o @ F @ ( foldl_set_nat_list_o @ F @ A @ Xs ) @ Ys ) ) ).
% foldl_append
thf(fact_1086_append1__eq__conv,axiom,
! [Xs: list_transition,X: transition,Ys: list_transition,Y2: transition] :
( ( ( append_transition @ Xs @ ( cons_transition @ X @ nil_transition ) )
= ( append_transition @ Ys @ ( cons_transition @ Y2 @ nil_transition ) ) )
= ( ( Xs = Ys )
& ( X = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_1087_append1__eq__conv,axiom,
! [Xs: list_o,X: $o,Ys: list_o,Y2: $o] :
( ( ( append_o @ Xs @ ( cons_o @ X @ nil_o ) )
= ( append_o @ Ys @ ( cons_o @ Y2 @ nil_o ) ) )
= ( ( Xs = Ys )
& ( X = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_1088_append1__eq__conv,axiom,
! [Xs: list_list_o,X: list_o,Ys: list_list_o,Y2: list_o] :
( ( ( append_list_o @ Xs @ ( cons_list_o @ X @ nil_list_o ) )
= ( append_list_o @ Ys @ ( cons_list_o @ Y2 @ nil_list_o ) ) )
= ( ( Xs = Ys )
& ( X = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_1089_set__append,axiom,
! [Xs: list_list_o,Ys: list_list_o] :
( ( set_list_o2 @ ( append_list_o @ Xs @ Ys ) )
= ( sup_sup_set_list_o @ ( set_list_o2 @ Xs ) @ ( set_list_o2 @ Ys ) ) ) ).
% set_append
thf(fact_1090_set__append,axiom,
! [Xs: list_transition,Ys: list_transition] :
( ( set_transition2 @ ( append_transition @ Xs @ Ys ) )
= ( sup_su812053455038985074sition @ ( set_transition2 @ Xs ) @ ( set_transition2 @ Ys ) ) ) ).
% set_append
thf(fact_1091_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_1092_Min__insert,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat2 @ X @ A2 ) )
= ( ord_min_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) ) ) ) ) ).
% Min_insert
thf(fact_1093_concat__eq__append__conv,axiom,
! [Xss2: list_list_transition,Ys: list_transition,Zs3: list_transition] :
( ( ( concat_transition @ Xss2 )
= ( append_transition @ Ys @ Zs3 ) )
= ( ( ( Xss2 = nil_list_transition )
=> ( ( Ys = nil_transition )
& ( Zs3 = nil_transition ) ) )
& ( ( Xss2 != nil_list_transition )
=> ? [Xss1: list_list_transition,Xs2: list_transition,Xs4: list_transition,Xss22: list_list_transition] :
( ( Xss2
= ( append5844159210978444383sition @ Xss1 @ ( cons_list_transition @ ( append_transition @ Xs2 @ Xs4 ) @ Xss22 ) ) )
& ( Ys
= ( append_transition @ ( concat_transition @ Xss1 ) @ Xs2 ) )
& ( Zs3
= ( append_transition @ Xs4 @ ( concat_transition @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_1094_concat__eq__append__conv,axiom,
! [Xss2: list_list_list_o,Ys: list_list_o,Zs3: list_list_o] :
( ( ( concat_list_o @ Xss2 )
= ( append_list_o @ Ys @ Zs3 ) )
= ( ( ( Xss2 = nil_list_list_o )
=> ( ( Ys = nil_list_o )
& ( Zs3 = nil_list_o ) ) )
& ( ( Xss2 != nil_list_list_o )
=> ? [Xss1: list_list_list_o,Xs2: list_list_o,Xs4: list_list_o,Xss22: list_list_list_o] :
( ( Xss2
= ( append_list_list_o @ Xss1 @ ( cons_list_list_o @ ( append_list_o @ Xs2 @ Xs4 ) @ Xss22 ) ) )
& ( Ys
= ( append_list_o @ ( concat_list_o @ Xss1 ) @ Xs2 ) )
& ( Zs3
= ( append_list_o @ Xs4 @ ( concat_list_o @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_1095_concat__eq__append__conv,axiom,
! [Xss2: list_list_o,Ys: list_o,Zs3: list_o] :
( ( ( concat_o @ Xss2 )
= ( append_o @ Ys @ Zs3 ) )
= ( ( ( Xss2 = nil_list_o )
=> ( ( Ys = nil_o )
& ( Zs3 = nil_o ) ) )
& ( ( Xss2 != nil_list_o )
=> ? [Xss1: list_list_o,Xs2: list_o,Xs4: list_o,Xss22: list_list_o] :
( ( Xss2
= ( append_list_o @ Xss1 @ ( cons_list_o @ ( append_o @ Xs2 @ Xs4 ) @ Xss22 ) ) )
& ( Ys
= ( append_o @ ( concat_o @ Xss1 ) @ Xs2 ) )
& ( Zs3
= ( append_o @ Xs4 @ ( concat_o @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_1096_concat_Osimps_I2_J,axiom,
! [X: list_list_o,Xs: list_list_list_o] :
( ( concat_list_o @ ( cons_list_list_o @ X @ Xs ) )
= ( append_list_o @ X @ ( concat_list_o @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_1097_concat_Osimps_I2_J,axiom,
! [X: list_o,Xs: list_list_o] :
( ( concat_o @ ( cons_list_o @ X @ Xs ) )
= ( append_o @ X @ ( concat_o @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_1098_concat__eq__appendD,axiom,
! [Xss2: list_list_list_o,Ys: list_list_o,Zs3: list_list_o] :
( ( ( concat_list_o @ Xss2 )
= ( append_list_o @ Ys @ Zs3 ) )
=> ( ( Xss2 != nil_list_list_o )
=> ? [Xss12: list_list_list_o,Xs3: list_list_o,Xs5: list_list_o,Xss23: list_list_list_o] :
( ( Xss2
= ( append_list_list_o @ Xss12 @ ( cons_list_list_o @ ( append_list_o @ Xs3 @ Xs5 ) @ Xss23 ) ) )
& ( Ys
= ( append_list_o @ ( concat_list_o @ Xss12 ) @ Xs3 ) )
& ( Zs3
= ( append_list_o @ Xs5 @ ( concat_list_o @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_1099_concat__eq__appendD,axiom,
! [Xss2: list_list_o,Ys: list_o,Zs3: list_o] :
( ( ( concat_o @ Xss2 )
= ( append_o @ Ys @ Zs3 ) )
=> ( ( Xss2 != nil_list_o )
=> ? [Xss12: list_list_o,Xs3: list_o,Xs5: list_o,Xss23: list_list_o] :
( ( Xss2
= ( append_list_o @ Xss12 @ ( cons_list_o @ ( append_o @ Xs3 @ Xs5 ) @ Xss23 ) ) )
& ( Ys
= ( append_o @ ( concat_o @ Xss12 ) @ Xs3 ) )
& ( Zs3
= ( append_o @ Xs5 @ ( concat_o @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_1100_Cons__eq__appendI,axiom,
! [X: list_o,Xs1: list_list_o,Ys: list_list_o,Xs: list_list_o,Zs3: list_list_o] :
( ( ( cons_list_o @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_list_o @ Xs1 @ Zs3 ) )
=> ( ( cons_list_o @ X @ Xs )
= ( append_list_o @ Ys @ Zs3 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_1101_append__Cons,axiom,
! [X: list_o,Xs: list_list_o,Ys: list_list_o] :
( ( append_list_o @ ( cons_list_o @ X @ Xs ) @ Ys )
= ( cons_list_o @ X @ ( append_list_o @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_1102_eq__Nil__appendI,axiom,
! [Xs: list_transition,Ys: list_transition] :
( ( Xs = Ys )
=> ( Xs
= ( append_transition @ nil_transition @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_1103_eq__Nil__appendI,axiom,
! [Xs: list_o,Ys: list_o] :
( ( Xs = Ys )
=> ( Xs
= ( append_o @ nil_o @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_1104_eq__Nil__appendI,axiom,
! [Xs: list_list_o,Ys: list_list_o] :
( ( Xs = Ys )
=> ( Xs
= ( append_list_o @ nil_list_o @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_1105_append_Oleft__neutral,axiom,
! [A: list_transition] :
( ( append_transition @ nil_transition @ A )
= A ) ).
% append.left_neutral
thf(fact_1106_append_Oleft__neutral,axiom,
! [A: list_o] :
( ( append_o @ nil_o @ A )
= A ) ).
% append.left_neutral
thf(fact_1107_append_Oleft__neutral,axiom,
! [A: list_list_o] :
( ( append_list_o @ nil_list_o @ A )
= A ) ).
% append.left_neutral
thf(fact_1108_append__Nil,axiom,
! [Ys: list_transition] :
( ( append_transition @ nil_transition @ Ys )
= Ys ) ).
% append_Nil
thf(fact_1109_append__Nil,axiom,
! [Ys: list_o] :
( ( append_o @ nil_o @ Ys )
= Ys ) ).
% append_Nil
thf(fact_1110_append__Nil,axiom,
! [Ys: list_list_o] :
( ( append_list_o @ nil_list_o @ Ys )
= Ys ) ).
% append_Nil
thf(fact_1111_run__comp,axiom,
! [Q0: nat,Transs: list_transition,R: set_nat,Bss: list_list_o,Css: list_list_o] :
( ( run @ Q0 @ Transs @ R @ ( append_list_o @ Bss @ Css ) )
= ( run @ Q0 @ Transs @ ( run @ Q0 @ Transs @ R @ Bss ) @ Css ) ) ).
% run_comp
thf(fact_1112_append__eq__appendI,axiom,
! [Xs: list_list_o,Xs1: list_list_o,Zs3: list_list_o,Ys: list_list_o,Us: list_list_o] :
( ( ( append_list_o @ Xs @ Xs1 )
= Zs3 )
=> ( ( Ys
= ( append_list_o @ Xs1 @ Us ) )
=> ( ( append_list_o @ Xs @ Ys )
= ( append_list_o @ Zs3 @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_1113_append__eq__append__conv2,axiom,
! [Xs: list_list_o,Ys: list_list_o,Zs3: list_list_o,Ts: list_list_o] :
( ( ( append_list_o @ Xs @ Ys )
= ( append_list_o @ Zs3 @ Ts ) )
= ( ? [Us2: list_list_o] :
( ( ( Xs
= ( append_list_o @ Zs3 @ Us2 ) )
& ( ( append_list_o @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_list_o @ Xs @ Us2 )
= Zs3 )
& ( Ys
= ( append_list_o @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_1114_min__def,axiom,
( ord_min_set_nat
= ( ^ [A3: set_nat,B4: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A3 @ B4 ) @ A3 @ B4 ) ) ) ).
% min_def
thf(fact_1115_min__absorb1,axiom,
! [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
=> ( ( ord_min_set_nat @ X @ Y2 )
= X ) ) ).
% min_absorb1
thf(fact_1116_min__absorb2,axiom,
! [Y2: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X )
=> ( ( ord_min_set_nat @ X @ Y2 )
= Y2 ) ) ).
% min_absorb2
thf(fact_1117_remove1__append,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat] :
( ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( remove1_nat @ X @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( remove1_nat @ X @ Xs ) @ Ys ) ) )
& ( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( remove1_nat @ X @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ Xs @ ( remove1_nat @ X @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_1118_remove1__append,axiom,
! [X: list_o,Xs: list_list_o,Ys: list_list_o] :
( ( ( member_list_o @ X @ ( set_list_o2 @ Xs ) )
=> ( ( remove1_list_o @ X @ ( append_list_o @ Xs @ Ys ) )
= ( append_list_o @ ( remove1_list_o @ X @ Xs ) @ Ys ) ) )
& ( ~ ( member_list_o @ X @ ( set_list_o2 @ Xs ) )
=> ( ( remove1_list_o @ X @ ( append_list_o @ Xs @ Ys ) )
= ( append_list_o @ Xs @ ( remove1_list_o @ X @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_1119_remove1__append,axiom,
! [X: transition,Xs: list_transition,Ys: list_transition] :
( ( ( member_transition @ X @ ( set_transition2 @ Xs ) )
=> ( ( remove1_transition @ X @ ( append_transition @ Xs @ Ys ) )
= ( append_transition @ ( remove1_transition @ X @ Xs ) @ Ys ) ) )
& ( ~ ( member_transition @ X @ ( set_transition2 @ Xs ) )
=> ( ( remove1_transition @ X @ ( append_transition @ Xs @ Ys ) )
= ( append_transition @ Xs @ ( remove1_transition @ X @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_1120_split__list,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys3: list_nat,Zs2: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_1121_split__list,axiom,
! [X: transition,Xs: list_transition] :
( ( member_transition @ X @ ( set_transition2 @ Xs ) )
=> ? [Ys3: list_transition,Zs2: list_transition] :
( Xs
= ( append_transition @ Ys3 @ ( cons_transition @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_1122_split__list,axiom,
! [X: list_o,Xs: list_list_o] :
( ( member_list_o @ X @ ( set_list_o2 @ Xs ) )
=> ? [Ys3: list_list_o,Zs2: list_list_o] :
( Xs
= ( append_list_o @ Ys3 @ ( cons_list_o @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_1123_split__list__last,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys3: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs2 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_1124_split__list__last,axiom,
! [X: transition,Xs: list_transition] :
( ( member_transition @ X @ ( set_transition2 @ Xs ) )
=> ? [Ys3: list_transition,Zs2: list_transition] :
( ( Xs
= ( append_transition @ Ys3 @ ( cons_transition @ X @ Zs2 ) ) )
& ~ ( member_transition @ X @ ( set_transition2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_1125_split__list__last,axiom,
! [X: list_o,Xs: list_list_o] :
( ( member_list_o @ X @ ( set_list_o2 @ Xs ) )
=> ? [Ys3: list_list_o,Zs2: list_list_o] :
( ( Xs
= ( append_list_o @ Ys3 @ ( cons_list_o @ X @ Zs2 ) ) )
& ~ ( member_list_o @ X @ ( set_list_o2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_1126_split__list__prop,axiom,
! [Xs: list_transition,P: transition > $o] :
( ? [X5: transition] :
( ( member_transition @ X5 @ ( set_transition2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys3: list_transition,X3: transition] :
( ? [Zs2: list_transition] :
( Xs
= ( append_transition @ Ys3 @ ( cons_transition @ X3 @ Zs2 ) ) )
& ( P @ X3 ) ) ) ).
% split_list_prop
thf(fact_1127_split__list__prop,axiom,
! [Xs: list_list_o,P: list_o > $o] :
( ? [X5: list_o] :
( ( member_list_o @ X5 @ ( set_list_o2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys3: list_list_o,X3: list_o] :
( ? [Zs2: list_list_o] :
( Xs
= ( append_list_o @ Ys3 @ ( cons_list_o @ X3 @ Zs2 ) ) )
& ( P @ X3 ) ) ) ).
% split_list_prop
thf(fact_1128_split__list__first,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys3: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs2 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_1129_split__list__first,axiom,
! [X: transition,Xs: list_transition] :
( ( member_transition @ X @ ( set_transition2 @ Xs ) )
=> ? [Ys3: list_transition,Zs2: list_transition] :
( ( Xs
= ( append_transition @ Ys3 @ ( cons_transition @ X @ Zs2 ) ) )
& ~ ( member_transition @ X @ ( set_transition2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_1130_split__list__first,axiom,
! [X: list_o,Xs: list_list_o] :
( ( member_list_o @ X @ ( set_list_o2 @ Xs ) )
=> ? [Ys3: list_list_o,Zs2: list_list_o] :
( ( Xs
= ( append_list_o @ Ys3 @ ( cons_list_o @ X @ Zs2 ) ) )
& ~ ( member_list_o @ X @ ( set_list_o2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_1131_split__list__propE,axiom,
! [Xs: list_transition,P: transition > $o] :
( ? [X5: transition] :
( ( member_transition @ X5 @ ( set_transition2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys3: list_transition,X3: transition] :
( ? [Zs2: list_transition] :
( Xs
= ( append_transition @ Ys3 @ ( cons_transition @ X3 @ Zs2 ) ) )
=> ~ ( P @ X3 ) ) ) ).
% split_list_propE
thf(fact_1132_split__list__propE,axiom,
! [Xs: list_list_o,P: list_o > $o] :
( ? [X5: list_o] :
( ( member_list_o @ X5 @ ( set_list_o2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys3: list_list_o,X3: list_o] :
( ? [Zs2: list_list_o] :
( Xs
= ( append_list_o @ Ys3 @ ( cons_list_o @ X3 @ Zs2 ) ) )
=> ~ ( P @ X3 ) ) ) ).
% split_list_propE
thf(fact_1133_append__Cons__eq__iff,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat,Xs6: list_nat,Ys4: list_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ~ ( member_nat @ X @ ( set_nat2 @ Ys ) )
=> ( ( ( append_nat @ Xs @ ( cons_nat @ X @ Ys ) )
= ( append_nat @ Xs6 @ ( cons_nat @ X @ Ys4 ) ) )
= ( ( Xs = Xs6 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1134_append__Cons__eq__iff,axiom,
! [X: transition,Xs: list_transition,Ys: list_transition,Xs6: list_transition,Ys4: list_transition] :
( ~ ( member_transition @ X @ ( set_transition2 @ Xs ) )
=> ( ~ ( member_transition @ X @ ( set_transition2 @ Ys ) )
=> ( ( ( append_transition @ Xs @ ( cons_transition @ X @ Ys ) )
= ( append_transition @ Xs6 @ ( cons_transition @ X @ Ys4 ) ) )
= ( ( Xs = Xs6 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1135_append__Cons__eq__iff,axiom,
! [X: list_o,Xs: list_list_o,Ys: list_list_o,Xs6: list_list_o,Ys4: list_list_o] :
( ~ ( member_list_o @ X @ ( set_list_o2 @ Xs ) )
=> ( ~ ( member_list_o @ X @ ( set_list_o2 @ Ys ) )
=> ( ( ( append_list_o @ Xs @ ( cons_list_o @ X @ Ys ) )
= ( append_list_o @ Xs6 @ ( cons_list_o @ X @ Ys4 ) ) )
= ( ( Xs = Xs6 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_1136_in__set__conv__decomp,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys2: list_nat,Zs: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1137_in__set__conv__decomp,axiom,
! [X: transition,Xs: list_transition] :
( ( member_transition @ X @ ( set_transition2 @ Xs ) )
= ( ? [Ys2: list_transition,Zs: list_transition] :
( Xs
= ( append_transition @ Ys2 @ ( cons_transition @ X @ Zs ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1138_in__set__conv__decomp,axiom,
! [X: list_o,Xs: list_list_o] :
( ( member_list_o @ X @ ( set_list_o2 @ Xs ) )
= ( ? [Ys2: list_list_o,Zs: list_list_o] :
( Xs
= ( append_list_o @ Ys2 @ ( cons_list_o @ X @ Zs ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_1139_split__list__last__prop,axiom,
! [Xs: list_transition,P: transition > $o] :
( ? [X5: transition] :
( ( member_transition @ X5 @ ( set_transition2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys3: list_transition,X3: transition,Zs2: list_transition] :
( ( Xs
= ( append_transition @ Ys3 @ ( cons_transition @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa: transition] :
( ( member_transition @ Xa @ ( set_transition2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_1140_split__list__last__prop,axiom,
! [Xs: list_list_o,P: list_o > $o] :
( ? [X5: list_o] :
( ( member_list_o @ X5 @ ( set_list_o2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys3: list_list_o,X3: list_o,Zs2: list_list_o] :
( ( Xs
= ( append_list_o @ Ys3 @ ( cons_list_o @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa: list_o] :
( ( member_list_o @ Xa @ ( set_list_o2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_1141_split__list__first__prop,axiom,
! [Xs: list_transition,P: transition > $o] :
( ? [X5: transition] :
( ( member_transition @ X5 @ ( set_transition2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys3: list_transition,X3: transition] :
( ? [Zs2: list_transition] :
( Xs
= ( append_transition @ Ys3 @ ( cons_transition @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa: transition] :
( ( member_transition @ Xa @ ( set_transition2 @ Ys3 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_1142_split__list__first__prop,axiom,
! [Xs: list_list_o,P: list_o > $o] :
( ? [X5: list_o] :
( ( member_list_o @ X5 @ ( set_list_o2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys3: list_list_o,X3: list_o] :
( ? [Zs2: list_list_o] :
( Xs
= ( append_list_o @ Ys3 @ ( cons_list_o @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Xa: list_o] :
( ( member_list_o @ Xa @ ( set_list_o2 @ Ys3 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_1143_split__list__last__propE,axiom,
! [Xs: list_transition,P: transition > $o] :
( ? [X5: transition] :
( ( member_transition @ X5 @ ( set_transition2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys3: list_transition,X3: transition,Zs2: list_transition] :
( ( Xs
= ( append_transition @ Ys3 @ ( cons_transition @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa: transition] :
( ( member_transition @ Xa @ ( set_transition2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_1144_split__list__last__propE,axiom,
! [Xs: list_list_o,P: list_o > $o] :
( ? [X5: list_o] :
( ( member_list_o @ X5 @ ( set_list_o2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys3: list_list_o,X3: list_o,Zs2: list_list_o] :
( ( Xs
= ( append_list_o @ Ys3 @ ( cons_list_o @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa: list_o] :
( ( member_list_o @ Xa @ ( set_list_o2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_1145_split__list__first__propE,axiom,
! [Xs: list_transition,P: transition > $o] :
( ? [X5: transition] :
( ( member_transition @ X5 @ ( set_transition2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys3: list_transition,X3: transition] :
( ? [Zs2: list_transition] :
( Xs
= ( append_transition @ Ys3 @ ( cons_transition @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa: transition] :
( ( member_transition @ Xa @ ( set_transition2 @ Ys3 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_1146_split__list__first__propE,axiom,
! [Xs: list_list_o,P: list_o > $o] :
( ? [X5: list_o] :
( ( member_list_o @ X5 @ ( set_list_o2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys3: list_list_o,X3: list_o] :
( ? [Zs2: list_list_o] :
( Xs
= ( append_list_o @ Ys3 @ ( cons_list_o @ X3 @ Zs2 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa: list_o] :
( ( member_list_o @ Xa @ ( set_list_o2 @ Ys3 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_1147_in__set__conv__decomp__last,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys2: list_nat,Zs: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Zs ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1148_in__set__conv__decomp__last,axiom,
! [X: transition,Xs: list_transition] :
( ( member_transition @ X @ ( set_transition2 @ Xs ) )
= ( ? [Ys2: list_transition,Zs: list_transition] :
( ( Xs
= ( append_transition @ Ys2 @ ( cons_transition @ X @ Zs ) ) )
& ~ ( member_transition @ X @ ( set_transition2 @ Zs ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1149_in__set__conv__decomp__last,axiom,
! [X: list_o,Xs: list_list_o] :
( ( member_list_o @ X @ ( set_list_o2 @ Xs ) )
= ( ? [Ys2: list_list_o,Zs: list_list_o] :
( ( Xs
= ( append_list_o @ Ys2 @ ( cons_list_o @ X @ Zs ) ) )
& ~ ( member_list_o @ X @ ( set_list_o2 @ Zs ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_1150_in__set__conv__decomp__first,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys2: list_nat,Zs: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1151_in__set__conv__decomp__first,axiom,
! [X: transition,Xs: list_transition] :
( ( member_transition @ X @ ( set_transition2 @ Xs ) )
= ( ? [Ys2: list_transition,Zs: list_transition] :
( ( Xs
= ( append_transition @ Ys2 @ ( cons_transition @ X @ Zs ) ) )
& ~ ( member_transition @ X @ ( set_transition2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1152_in__set__conv__decomp__first,axiom,
! [X: list_o,Xs: list_list_o] :
( ( member_list_o @ X @ ( set_list_o2 @ Xs ) )
= ( ? [Ys2: list_list_o,Zs: list_list_o] :
( ( Xs
= ( append_list_o @ Ys2 @ ( cons_list_o @ X @ Zs ) ) )
& ~ ( member_list_o @ X @ ( set_list_o2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_1153_split__list__last__prop__iff,axiom,
! [Xs: list_transition,P: transition > $o] :
( ( ? [X2: transition] :
( ( member_transition @ X2 @ ( set_transition2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys2: list_transition,X2: transition,Zs: list_transition] :
( ( Xs
= ( append_transition @ Ys2 @ ( cons_transition @ X2 @ Zs ) ) )
& ( P @ X2 )
& ! [Y4: transition] :
( ( member_transition @ Y4 @ ( set_transition2 @ Zs ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_1154_split__list__last__prop__iff,axiom,
! [Xs: list_list_o,P: list_o > $o] :
( ( ? [X2: list_o] :
( ( member_list_o @ X2 @ ( set_list_o2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys2: list_list_o,X2: list_o,Zs: list_list_o] :
( ( Xs
= ( append_list_o @ Ys2 @ ( cons_list_o @ X2 @ Zs ) ) )
& ( P @ X2 )
& ! [Y4: list_o] :
( ( member_list_o @ Y4 @ ( set_list_o2 @ Zs ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_1155_split__list__first__prop__iff,axiom,
! [Xs: list_transition,P: transition > $o] :
( ( ? [X2: transition] :
( ( member_transition @ X2 @ ( set_transition2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys2: list_transition,X2: transition] :
( ? [Zs: list_transition] :
( Xs
= ( append_transition @ Ys2 @ ( cons_transition @ X2 @ Zs ) ) )
& ( P @ X2 )
& ! [Y4: transition] :
( ( member_transition @ Y4 @ ( set_transition2 @ Ys2 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_1156_split__list__first__prop__iff,axiom,
! [Xs: list_list_o,P: list_o > $o] :
( ( ? [X2: list_o] :
( ( member_list_o @ X2 @ ( set_list_o2 @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys2: list_list_o,X2: list_o] :
( ? [Zs: list_list_o] :
( Xs
= ( append_list_o @ Ys2 @ ( cons_list_o @ X2 @ Zs ) ) )
& ( P @ X2 )
& ! [Y4: list_o] :
( ( member_list_o @ Y4 @ ( set_list_o2 @ Ys2 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_1157_rev__nonempty__induct,axiom,
! [Xs: list_transition,P: list_transition > $o] :
( ( Xs != nil_transition )
=> ( ! [X3: transition] : ( P @ ( cons_transition @ X3 @ nil_transition ) )
=> ( ! [X3: transition,Xs3: list_transition] :
( ( Xs3 != nil_transition )
=> ( ( P @ Xs3 )
=> ( P @ ( append_transition @ Xs3 @ ( cons_transition @ X3 @ nil_transition ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_1158_rev__nonempty__induct,axiom,
! [Xs: list_o,P: list_o > $o] :
( ( Xs != nil_o )
=> ( ! [X3: $o] : ( P @ ( cons_o @ X3 @ nil_o ) )
=> ( ! [X3: $o,Xs3: list_o] :
( ( Xs3 != nil_o )
=> ( ( P @ Xs3 )
=> ( P @ ( append_o @ Xs3 @ ( cons_o @ X3 @ nil_o ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_1159_rev__nonempty__induct,axiom,
! [Xs: list_list_o,P: list_list_o > $o] :
( ( Xs != nil_list_o )
=> ( ! [X3: list_o] : ( P @ ( cons_list_o @ X3 @ nil_list_o ) )
=> ( ! [X3: list_o,Xs3: list_list_o] :
( ( Xs3 != nil_list_o )
=> ( ( P @ Xs3 )
=> ( P @ ( append_list_o @ Xs3 @ ( cons_list_o @ X3 @ nil_list_o ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_1160_append__eq__Cons__conv,axiom,
! [Ys: list_transition,Zs3: list_transition,X: transition,Xs: list_transition] :
( ( ( append_transition @ Ys @ Zs3 )
= ( cons_transition @ X @ Xs ) )
= ( ( ( Ys = nil_transition )
& ( Zs3
= ( cons_transition @ X @ Xs ) ) )
| ? [Ys5: list_transition] :
( ( Ys
= ( cons_transition @ X @ Ys5 ) )
& ( ( append_transition @ Ys5 @ Zs3 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_1161_append__eq__Cons__conv,axiom,
! [Ys: list_o,Zs3: list_o,X: $o,Xs: list_o] :
( ( ( append_o @ Ys @ Zs3 )
= ( cons_o @ X @ Xs ) )
= ( ( ( Ys = nil_o )
& ( Zs3
= ( cons_o @ X @ Xs ) ) )
| ? [Ys5: list_o] :
( ( Ys
= ( cons_o @ X @ Ys5 ) )
& ( ( append_o @ Ys5 @ Zs3 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_1162_append__eq__Cons__conv,axiom,
! [Ys: list_list_o,Zs3: list_list_o,X: list_o,Xs: list_list_o] :
( ( ( append_list_o @ Ys @ Zs3 )
= ( cons_list_o @ X @ Xs ) )
= ( ( ( Ys = nil_list_o )
& ( Zs3
= ( cons_list_o @ X @ Xs ) ) )
| ? [Ys5: list_list_o] :
( ( Ys
= ( cons_list_o @ X @ Ys5 ) )
& ( ( append_list_o @ Ys5 @ Zs3 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_1163_Cons__eq__append__conv,axiom,
! [X: transition,Xs: list_transition,Ys: list_transition,Zs3: list_transition] :
( ( ( cons_transition @ X @ Xs )
= ( append_transition @ Ys @ Zs3 ) )
= ( ( ( Ys = nil_transition )
& ( ( cons_transition @ X @ Xs )
= Zs3 ) )
| ? [Ys5: list_transition] :
( ( ( cons_transition @ X @ Ys5 )
= Ys )
& ( Xs
= ( append_transition @ Ys5 @ Zs3 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_1164_Cons__eq__append__conv,axiom,
! [X: $o,Xs: list_o,Ys: list_o,Zs3: list_o] :
( ( ( cons_o @ X @ Xs )
= ( append_o @ Ys @ Zs3 ) )
= ( ( ( Ys = nil_o )
& ( ( cons_o @ X @ Xs )
= Zs3 ) )
| ? [Ys5: list_o] :
( ( ( cons_o @ X @ Ys5 )
= Ys )
& ( Xs
= ( append_o @ Ys5 @ Zs3 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_1165_Cons__eq__append__conv,axiom,
! [X: list_o,Xs: list_list_o,Ys: list_list_o,Zs3: list_list_o] :
( ( ( cons_list_o @ X @ Xs )
= ( append_list_o @ Ys @ Zs3 ) )
= ( ( ( Ys = nil_list_o )
& ( ( cons_list_o @ X @ Xs )
= Zs3 ) )
| ? [Ys5: list_list_o] :
( ( ( cons_list_o @ X @ Ys5 )
= Ys )
& ( Xs
= ( append_list_o @ Ys5 @ Zs3 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_1166_rev__exhaust,axiom,
! [Xs: list_transition] :
( ( Xs != nil_transition )
=> ~ ! [Ys3: list_transition,Y: transition] :
( Xs
!= ( append_transition @ Ys3 @ ( cons_transition @ Y @ nil_transition ) ) ) ) ).
% rev_exhaust
thf(fact_1167_rev__exhaust,axiom,
! [Xs: list_o] :
( ( Xs != nil_o )
=> ~ ! [Ys3: list_o,Y: $o] :
( Xs
!= ( append_o @ Ys3 @ ( cons_o @ Y @ nil_o ) ) ) ) ).
% rev_exhaust
thf(fact_1168_rev__exhaust,axiom,
! [Xs: list_list_o] :
( ( Xs != nil_list_o )
=> ~ ! [Ys3: list_list_o,Y: list_o] :
( Xs
!= ( append_list_o @ Ys3 @ ( cons_list_o @ Y @ nil_list_o ) ) ) ) ).
% rev_exhaust
thf(fact_1169_rev__induct,axiom,
! [P: list_transition > $o,Xs: list_transition] :
( ( P @ nil_transition )
=> ( ! [X3: transition,Xs3: list_transition] :
( ( P @ Xs3 )
=> ( P @ ( append_transition @ Xs3 @ ( cons_transition @ X3 @ nil_transition ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_1170_rev__induct,axiom,
! [P: list_o > $o,Xs: list_o] :
( ( P @ nil_o )
=> ( ! [X3: $o,Xs3: list_o] :
( ( P @ Xs3 )
=> ( P @ ( append_o @ Xs3 @ ( cons_o @ X3 @ nil_o ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_1171_rev__induct,axiom,
! [P: list_list_o > $o,Xs: list_list_o] :
( ( P @ nil_list_o )
=> ( ! [X3: list_o,Xs3: list_list_o] :
( ( P @ Xs3 )
=> ( P @ ( append_list_o @ Xs3 @ ( cons_list_o @ X3 @ nil_list_o ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_1172_not__distinct__decomp,axiom,
! [Ws: list_transition] :
( ~ ( distinct_transition @ Ws )
=> ? [Xs3: list_transition,Ys3: list_transition,Zs2: list_transition,Y: transition] :
( Ws
= ( append_transition @ Xs3 @ ( append_transition @ ( cons_transition @ Y @ nil_transition ) @ ( append_transition @ Ys3 @ ( append_transition @ ( cons_transition @ Y @ nil_transition ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_1173_not__distinct__decomp,axiom,
! [Ws: list_o] :
( ~ ( distinct_o @ Ws )
=> ? [Xs3: list_o,Ys3: list_o,Zs2: list_o,Y: $o] :
( Ws
= ( append_o @ Xs3 @ ( append_o @ ( cons_o @ Y @ nil_o ) @ ( append_o @ Ys3 @ ( append_o @ ( cons_o @ Y @ nil_o ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_1174_not__distinct__decomp,axiom,
! [Ws: list_list_o] :
( ~ ( distinct_list_o @ Ws )
=> ? [Xs3: list_list_o,Ys3: list_list_o,Zs2: list_list_o,Y: list_o] :
( Ws
= ( append_list_o @ Xs3 @ ( append_list_o @ ( cons_list_o @ Y @ nil_list_o ) @ ( append_list_o @ Ys3 @ ( append_list_o @ ( cons_list_o @ Y @ nil_list_o ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_1175_not__distinct__conv__prefix,axiom,
! [As: list_nat] :
( ( ~ ( distinct_nat @ As ) )
= ( ? [Xs2: list_nat,Y4: nat,Ys2: list_nat] :
( ( member_nat @ Y4 @ ( set_nat2 @ Xs2 ) )
& ( distinct_nat @ Xs2 )
& ( As
= ( append_nat @ Xs2 @ ( cons_nat @ Y4 @ Ys2 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_1176_not__distinct__conv__prefix,axiom,
! [As: list_transition] :
( ( ~ ( distinct_transition @ As ) )
= ( ? [Xs2: list_transition,Y4: transition,Ys2: list_transition] :
( ( member_transition @ Y4 @ ( set_transition2 @ Xs2 ) )
& ( distinct_transition @ Xs2 )
& ( As
= ( append_transition @ Xs2 @ ( cons_transition @ Y4 @ Ys2 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_1177_not__distinct__conv__prefix,axiom,
! [As: list_list_o] :
( ( ~ ( distinct_list_o @ As ) )
= ( ? [Xs2: list_list_o,Y4: list_o,Ys2: list_list_o] :
( ( member_list_o @ Y4 @ ( set_list_o2 @ Xs2 ) )
& ( distinct_list_o @ Xs2 )
& ( As
= ( append_list_o @ Xs2 @ ( cons_list_o @ Y4 @ Ys2 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_1178_remove1__split,axiom,
! [A: nat,Xs: list_nat,Ys: list_nat] :
( ( member_nat @ A @ ( set_nat2 @ Xs ) )
=> ( ( ( remove1_nat @ A @ Xs )
= Ys )
= ( ? [Ls: list_nat,Rs: list_nat] :
( ( Xs
= ( append_nat @ Ls @ ( cons_nat @ A @ Rs ) ) )
& ~ ( member_nat @ A @ ( set_nat2 @ Ls ) )
& ( Ys
= ( append_nat @ Ls @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_1179_remove1__split,axiom,
! [A: transition,Xs: list_transition,Ys: list_transition] :
( ( member_transition @ A @ ( set_transition2 @ Xs ) )
=> ( ( ( remove1_transition @ A @ Xs )
= Ys )
= ( ? [Ls: list_transition,Rs: list_transition] :
( ( Xs
= ( append_transition @ Ls @ ( cons_transition @ A @ Rs ) ) )
& ~ ( member_transition @ A @ ( set_transition2 @ Ls ) )
& ( Ys
= ( append_transition @ Ls @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_1180_remove1__split,axiom,
! [A: list_o,Xs: list_list_o,Ys: list_list_o] :
( ( member_list_o @ A @ ( set_list_o2 @ Xs ) )
=> ( ( ( remove1_list_o @ A @ Xs )
= Ys )
= ( ? [Ls: list_list_o,Rs: list_list_o] :
( ( Xs
= ( append_list_o @ Ls @ ( cons_list_o @ A @ Rs ) ) )
& ~ ( member_list_o @ A @ ( set_list_o2 @ Ls ) )
& ( Ys
= ( append_list_o @ Ls @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_1181_hom__Min__commute,axiom,
! [H: nat > nat,N3: set_nat] :
( ! [X3: nat,Y: nat] :
( ( H @ ( ord_min_nat @ X3 @ Y ) )
= ( ord_min_nat @ ( H @ X3 ) @ ( H @ Y ) ) )
=> ( ( finite_finite_nat @ N3 )
=> ( ( N3 != bot_bot_set_nat )
=> ( ( H @ ( lattic8721135487736765967in_nat @ N3 ) )
= ( lattic8721135487736765967in_nat @ ( image_nat_nat @ H @ N3 ) ) ) ) ) ) ).
% hom_Min_commute
thf(fact_1182_Min_Osubset,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( B != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_min_nat @ ( lattic8721135487736765967in_nat @ B ) @ ( lattic8721135487736765967in_nat @ A2 ) )
= ( lattic8721135487736765967in_nat @ A2 ) ) ) ) ) ).
% Min.subset
thf(fact_1183_Min_Oinsert__not__elem,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat @ X @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ ( insert_nat2 @ X @ A2 ) )
= ( ord_min_nat @ X @ ( lattic8721135487736765967in_nat @ A2 ) ) ) ) ) ) ).
% Min.insert_not_elem
thf(fact_1184_Min_Oclosed,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [X3: nat,Y: nat] : ( member_nat @ ( ord_min_nat @ X3 @ Y ) @ ( insert_nat2 @ X3 @ ( insert_nat2 @ Y @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ A2 ) ) ) ) ).
% Min.closed
thf(fact_1185_Min_Ounion,axiom,
! [A2: set_nat,B: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ( B != bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ ( sup_sup_set_nat @ A2 @ B ) )
= ( ord_min_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ ( lattic8721135487736765967in_nat @ B ) ) ) ) ) ) ) ).
% Min.union
thf(fact_1186_Min_Oremove,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ A2 )
= X ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ A2 )
= ( ord_min_nat @ X @ ( lattic8721135487736765967in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Min.remove
thf(fact_1187_bind__simps_I2_J,axiom,
! [X: list_o,Xs: list_list_o,F: list_o > list_list_o] :
( ( bind_list_o_list_o @ ( cons_list_o @ X @ Xs ) @ F )
= ( append_list_o @ ( F @ X ) @ ( bind_list_o_list_o @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_1188_maps__simps_I1_J,axiom,
! [F: list_o > list_list_o,X: list_o,Xs: list_list_o] :
( ( maps_list_o_list_o @ F @ ( cons_list_o @ X @ Xs ) )
= ( append_list_o @ ( F @ X ) @ ( maps_list_o_list_o @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_1189_bind__simps_I1_J,axiom,
! [F: transition > list_transition] :
( ( bind_t1640203714015778183sition @ nil_transition @ F )
= nil_transition ) ).
% bind_simps(1)
thf(fact_1190_bind__simps_I1_J,axiom,
! [F: transition > list_o] :
( ( bind_transition_o @ nil_transition @ F )
= nil_o ) ).
% bind_simps(1)
thf(fact_1191_bind__simps_I1_J,axiom,
! [F: transition > list_list_o] :
( ( bind_t133343566971731431list_o @ nil_transition @ F )
= nil_list_o ) ).
% bind_simps(1)
thf(fact_1192_bind__simps_I1_J,axiom,
! [F: $o > list_transition] :
( ( bind_o_transition @ nil_o @ F )
= nil_transition ) ).
% bind_simps(1)
thf(fact_1193_bind__simps_I1_J,axiom,
! [F: $o > list_o] :
( ( bind_o_o @ nil_o @ F )
= nil_o ) ).
% bind_simps(1)
thf(fact_1194_bind__simps_I1_J,axiom,
! [F: $o > list_list_o] :
( ( bind_o_list_o @ nil_o @ F )
= nil_list_o ) ).
% bind_simps(1)
thf(fact_1195_bind__simps_I1_J,axiom,
! [F: list_o > list_transition] :
( ( bind_l7118857872559238311sition @ nil_list_o @ F )
= nil_transition ) ).
% bind_simps(1)
thf(fact_1196_bind__simps_I1_J,axiom,
! [F: list_o > list_o] :
( ( bind_list_o_o @ nil_list_o @ F )
= nil_o ) ).
% bind_simps(1)
thf(fact_1197_bind__simps_I1_J,axiom,
! [F: list_o > list_list_o] :
( ( bind_list_o_list_o @ nil_list_o @ F )
= nil_list_o ) ).
% bind_simps(1)
thf(fact_1198_maps__simps_I2_J,axiom,
! [F: transition > list_transition] :
( ( maps_t8463952688879884097sition @ F @ nil_transition )
= nil_transition ) ).
% maps_simps(2)
thf(fact_1199_maps__simps_I2_J,axiom,
! [F: transition > list_o] :
( ( maps_transition_o @ F @ nil_transition )
= nil_o ) ).
% maps_simps(2)
thf(fact_1200_maps__simps_I2_J,axiom,
! [F: transition > list_list_o] :
( ( maps_t5005503969775829665list_o @ F @ nil_transition )
= nil_list_o ) ).
% maps_simps(2)
thf(fact_1201_maps__simps_I2_J,axiom,
! [F: $o > list_transition] :
( ( maps_o_transition @ F @ nil_o )
= nil_transition ) ).
% maps_simps(2)
thf(fact_1202_maps__simps_I2_J,axiom,
! [F: $o > list_o] :
( ( maps_o_o @ F @ nil_o )
= nil_o ) ).
% maps_simps(2)
thf(fact_1203_maps__simps_I2_J,axiom,
! [F: $o > list_list_o] :
( ( maps_o_list_o @ F @ nil_o )
= nil_list_o ) ).
% maps_simps(2)
thf(fact_1204_maps__simps_I2_J,axiom,
! [F: list_o > list_transition] :
( ( maps_l2767646238508560737sition @ F @ nil_list_o )
= nil_transition ) ).
% maps_simps(2)
thf(fact_1205_maps__simps_I2_J,axiom,
! [F: list_o > list_o] :
( ( maps_list_o_o @ F @ nil_list_o )
= nil_o ) ).
% maps_simps(2)
thf(fact_1206_maps__simps_I2_J,axiom,
! [F: list_o > list_list_o] :
( ( maps_list_o_list_o @ F @ nil_list_o )
= nil_list_o ) ).
% maps_simps(2)
thf(fact_1207_Inf__fin_Osemilattice__order__set__axioms,axiom,
lattic3109210760196336428et_nat @ inf_inf_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat ).
% Inf_fin.semilattice_order_set_axioms
thf(fact_1208_semilattice__order__set_Osubset__imp,axiom,
! [F: transition > transition > transition,Less_eq: transition > transition > $o,Less: transition > transition > $o,A2: set_transition,B: set_transition] :
( ( lattic8086405597477538456sition @ F @ Less_eq @ Less )
=> ( ( ord_le8419162016481440574sition @ A2 @ B )
=> ( ( A2 != bot_bo301567166201926666sition )
=> ( ( finite8165534619950747239sition @ B )
=> ( Less_eq @ ( lattic3235374346148272024sition @ F @ B ) @ ( lattic3235374346148272024sition @ F @ A2 ) ) ) ) ) ) ).
% semilattice_order_set.subset_imp
thf(fact_1209_semilattice__order__set_Osubset__imp,axiom,
! [F: nat > nat > nat,Less_eq: nat > nat > $o,Less: nat > nat > $o,A2: set_nat,B: set_nat] :
( ( lattic6009151579333465974et_nat @ F @ Less_eq @ Less )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( Less_eq @ ( lattic7742739596368939638_F_nat @ F @ B ) @ ( lattic7742739596368939638_F_nat @ F @ A2 ) ) ) ) ) ) ).
% semilattice_order_set.subset_imp
thf(fact_1210_butlast__snoc,axiom,
! [Xs: list_transition,X: transition] :
( ( butlast_transition @ ( append_transition @ Xs @ ( cons_transition @ X @ nil_transition ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_1211_butlast__snoc,axiom,
! [Xs: list_o,X: $o] :
( ( butlast_o @ ( append_o @ Xs @ ( cons_o @ X @ nil_o ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_1212_butlast__snoc,axiom,
! [Xs: list_list_o,X: list_o] :
( ( butlast_list_o @ ( append_list_o @ Xs @ ( cons_list_o @ X @ nil_list_o ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_1213_in__set__butlastD,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ Xs ) ) )
=> ( member_nat @ X @ ( set_nat2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_1214_in__set__butlastD,axiom,
! [X: transition,Xs: list_transition] :
( ( member_transition @ X @ ( set_transition2 @ ( butlast_transition @ Xs ) ) )
=> ( member_transition @ X @ ( set_transition2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_1215_butlast_Osimps_I1_J,axiom,
( ( butlast_transition @ nil_transition )
= nil_transition ) ).
% butlast.simps(1)
thf(fact_1216_butlast_Osimps_I1_J,axiom,
( ( butlast_o @ nil_o )
= nil_o ) ).
% butlast.simps(1)
thf(fact_1217_butlast_Osimps_I1_J,axiom,
( ( butlast_list_o @ nil_list_o )
= nil_list_o ) ).
% butlast.simps(1)
thf(fact_1218_Sup__fin__def,axiom,
( lattic3835124923745554447et_nat
= ( lattic4908145837437951532et_nat @ sup_sup_set_nat ) ) ).
% Sup_fin_def
thf(fact_1219_butlast_Osimps_I2_J,axiom,
! [Xs: list_transition,X: transition] :
( ( ( Xs = nil_transition )
=> ( ( butlast_transition @ ( cons_transition @ X @ Xs ) )
= nil_transition ) )
& ( ( Xs != nil_transition )
=> ( ( butlast_transition @ ( cons_transition @ X @ Xs ) )
= ( cons_transition @ X @ ( butlast_transition @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_1220_butlast_Osimps_I2_J,axiom,
! [Xs: list_o,X: $o] :
( ( ( Xs = nil_o )
=> ( ( butlast_o @ ( cons_o @ X @ Xs ) )
= nil_o ) )
& ( ( Xs != nil_o )
=> ( ( butlast_o @ ( cons_o @ X @ Xs ) )
= ( cons_o @ X @ ( butlast_o @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_1221_butlast_Osimps_I2_J,axiom,
! [Xs: list_list_o,X: list_o] :
( ( ( Xs = nil_list_o )
=> ( ( butlast_list_o @ ( cons_list_o @ X @ Xs ) )
= nil_list_o ) )
& ( ( Xs != nil_list_o )
=> ( ( butlast_list_o @ ( cons_list_o @ X @ Xs ) )
= ( cons_list_o @ X @ ( butlast_list_o @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_1222_in__set__butlast__appendI,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat] :
( ( ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ Xs ) ) )
| ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ Ys ) ) ) )
=> ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ ( append_nat @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_1223_in__set__butlast__appendI,axiom,
! [X: list_o,Xs: list_list_o,Ys: list_list_o] :
( ( ( member_list_o @ X @ ( set_list_o2 @ ( butlast_list_o @ Xs ) ) )
| ( member_list_o @ X @ ( set_list_o2 @ ( butlast_list_o @ Ys ) ) ) )
=> ( member_list_o @ X @ ( set_list_o2 @ ( butlast_list_o @ ( append_list_o @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_1224_in__set__butlast__appendI,axiom,
! [X: transition,Xs: list_transition,Ys: list_transition] :
( ( ( member_transition @ X @ ( set_transition2 @ ( butlast_transition @ Xs ) ) )
| ( member_transition @ X @ ( set_transition2 @ ( butlast_transition @ Ys ) ) ) )
=> ( member_transition @ X @ ( set_transition2 @ ( butlast_transition @ ( append_transition @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_1225_butlast__append,axiom,
! [Ys: list_transition,Xs: list_transition] :
( ( ( Ys = nil_transition )
=> ( ( butlast_transition @ ( append_transition @ Xs @ Ys ) )
= ( butlast_transition @ Xs ) ) )
& ( ( Ys != nil_transition )
=> ( ( butlast_transition @ ( append_transition @ Xs @ Ys ) )
= ( append_transition @ Xs @ ( butlast_transition @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_1226_butlast__append,axiom,
! [Ys: list_o,Xs: list_o] :
( ( ( Ys = nil_o )
=> ( ( butlast_o @ ( append_o @ Xs @ Ys ) )
= ( butlast_o @ Xs ) ) )
& ( ( Ys != nil_o )
=> ( ( butlast_o @ ( append_o @ Xs @ Ys ) )
= ( append_o @ Xs @ ( butlast_o @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_1227_butlast__append,axiom,
! [Ys: list_list_o,Xs: list_list_o] :
( ( ( Ys = nil_list_o )
=> ( ( butlast_list_o @ ( append_list_o @ Xs @ Ys ) )
= ( butlast_list_o @ Xs ) ) )
& ( ( Ys != nil_list_o )
=> ( ( butlast_list_o @ ( append_list_o @ Xs @ Ys ) )
= ( append_list_o @ Xs @ ( butlast_list_o @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_1228_semilattice__order__set_OboundedE,axiom,
! [F: nat > nat > nat,Less_eq: nat > nat > $o,Less: nat > nat > $o,A2: set_nat,X: nat] :
( ( lattic6009151579333465974et_nat @ F @ Less_eq @ Less )
=> ( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( Less_eq @ X @ ( lattic7742739596368939638_F_nat @ F @ A2 ) )
=> ! [A8: nat] :
( ( member_nat @ A8 @ A2 )
=> ( Less_eq @ X @ A8 ) ) ) ) ) ) ).
% semilattice_order_set.boundedE
thf(fact_1229_semilattice__order__set_OboundedE,axiom,
! [F: transition > transition > transition,Less_eq: transition > transition > $o,Less: transition > transition > $o,A2: set_transition,X: transition] :
( ( lattic8086405597477538456sition @ F @ Less_eq @ Less )
=> ( ( finite8165534619950747239sition @ A2 )
=> ( ( A2 != bot_bo301567166201926666sition )
=> ( ( Less_eq @ X @ ( lattic3235374346148272024sition @ F @ A2 ) )
=> ! [A8: transition] :
( ( member_transition @ A8 @ A2 )
=> ( Less_eq @ X @ A8 ) ) ) ) ) ) ).
% semilattice_order_set.boundedE
thf(fact_1230_semilattice__order__set_OboundedI,axiom,
! [F: nat > nat > nat,Less_eq: nat > nat > $o,Less: nat > nat > $o,A2: set_nat,X: nat] :
( ( lattic6009151579333465974et_nat @ F @ Less_eq @ Less )
=> ( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [A7: nat] :
( ( member_nat @ A7 @ A2 )
=> ( Less_eq @ X @ A7 ) )
=> ( Less_eq @ X @ ( lattic7742739596368939638_F_nat @ F @ A2 ) ) ) ) ) ) ).
% semilattice_order_set.boundedI
thf(fact_1231_semilattice__order__set_OboundedI,axiom,
! [F: transition > transition > transition,Less_eq: transition > transition > $o,Less: transition > transition > $o,A2: set_transition,X: transition] :
( ( lattic8086405597477538456sition @ F @ Less_eq @ Less )
=> ( ( finite8165534619950747239sition @ A2 )
=> ( ( A2 != bot_bo301567166201926666sition )
=> ( ! [A7: transition] :
( ( member_transition @ A7 @ A2 )
=> ( Less_eq @ X @ A7 ) )
=> ( Less_eq @ X @ ( lattic3235374346148272024sition @ F @ A2 ) ) ) ) ) ) ).
% semilattice_order_set.boundedI
thf(fact_1232_semilattice__order__set_Obounded__iff,axiom,
! [F: nat > nat > nat,Less_eq: nat > nat > $o,Less: nat > nat > $o,A2: set_nat,X: nat] :
( ( lattic6009151579333465974et_nat @ F @ Less_eq @ Less )
=> ( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( Less_eq @ X @ ( lattic7742739596368939638_F_nat @ F @ A2 ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( Less_eq @ X @ X2 ) ) ) ) ) ) ) ).
% semilattice_order_set.bounded_iff
thf(fact_1233_semilattice__order__set_Obounded__iff,axiom,
! [F: transition > transition > transition,Less_eq: transition > transition > $o,Less: transition > transition > $o,A2: set_transition,X: transition] :
( ( lattic8086405597477538456sition @ F @ Less_eq @ Less )
=> ( ( finite8165534619950747239sition @ A2 )
=> ( ( A2 != bot_bo301567166201926666sition )
=> ( ( Less_eq @ X @ ( lattic3235374346148272024sition @ F @ A2 ) )
= ( ! [X2: transition] :
( ( member_transition @ X2 @ A2 )
=> ( Less_eq @ X @ X2 ) ) ) ) ) ) ) ).
% semilattice_order_set.bounded_iff
thf(fact_1234_semilattice__set_Oinsert__remove,axiom,
! [F: transition > transition > transition,A2: set_transition,X: transition] :
( ( lattic5875891794066335564sition @ F )
=> ( ( finite8165534619950747239sition @ A2 )
=> ( ( ( ( minus_8944320859760356485sition @ A2 @ ( insert_transition2 @ X @ bot_bo301567166201926666sition ) )
= bot_bo301567166201926666sition )
=> ( ( lattic3235374346148272024sition @ F @ ( insert_transition2 @ X @ A2 ) )
= X ) )
& ( ( ( minus_8944320859760356485sition @ A2 @ ( insert_transition2 @ X @ bot_bo301567166201926666sition ) )
!= bot_bo301567166201926666sition )
=> ( ( lattic3235374346148272024sition @ F @ ( insert_transition2 @ X @ A2 ) )
= ( F @ X @ ( lattic3235374346148272024sition @ F @ ( minus_8944320859760356485sition @ A2 @ ( insert_transition2 @ X @ bot_bo301567166201926666sition ) ) ) ) ) ) ) ) ) ).
% semilattice_set.insert_remove
thf(fact_1235_semilattice__set_Oinsert__remove,axiom,
! [F: nat > nat > nat,A2: set_nat,X: nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ( finite_finite_nat @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic7742739596368939638_F_nat @ F @ ( insert_nat2 @ X @ A2 ) )
= X ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic7742739596368939638_F_nat @ F @ ( insert_nat2 @ X @ A2 ) )
= ( F @ X @ ( lattic7742739596368939638_F_nat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% semilattice_set.insert_remove
thf(fact_1236_semilattice__set_Oremove,axiom,
! [F: transition > transition > transition,A2: set_transition,X: transition] :
( ( lattic5875891794066335564sition @ F )
=> ( ( finite8165534619950747239sition @ A2 )
=> ( ( member_transition @ X @ A2 )
=> ( ( ( ( minus_8944320859760356485sition @ A2 @ ( insert_transition2 @ X @ bot_bo301567166201926666sition ) )
= bot_bo301567166201926666sition )
=> ( ( lattic3235374346148272024sition @ F @ A2 )
= X ) )
& ( ( ( minus_8944320859760356485sition @ A2 @ ( insert_transition2 @ X @ bot_bo301567166201926666sition ) )
!= bot_bo301567166201926666sition )
=> ( ( lattic3235374346148272024sition @ F @ A2 )
= ( F @ X @ ( lattic3235374346148272024sition @ F @ ( minus_8944320859760356485sition @ A2 @ ( insert_transition2 @ X @ bot_bo301567166201926666sition ) ) ) ) ) ) ) ) ) ) ).
% semilattice_set.remove
thf(fact_1237_semilattice__set_Oremove,axiom,
! [F: nat > nat > nat,A2: set_nat,X: nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic7742739596368939638_F_nat @ F @ A2 )
= X ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic7742739596368939638_F_nat @ F @ A2 )
= ( F @ X @ ( lattic7742739596368939638_F_nat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ) ).
% semilattice_set.remove
thf(fact_1238_Sup__fin_Osemilattice__set__axioms,axiom,
lattic6452893353811829624et_nat @ sup_sup_set_nat ).
% Sup_fin.semilattice_set_axioms
thf(fact_1239_semilattice__set_Osingleton,axiom,
! [F: nat > nat > nat,X: nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ( lattic7742739596368939638_F_nat @ F @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= X ) ) ).
% semilattice_set.singleton
thf(fact_1240_semilattice__set_Osingleton,axiom,
! [F: transition > transition > transition,X: transition] :
( ( lattic5875891794066335564sition @ F )
=> ( ( lattic3235374346148272024sition @ F @ ( insert_transition2 @ X @ bot_bo301567166201926666sition ) )
= X ) ) ).
% semilattice_set.singleton
thf(fact_1241_semilattice__set_Ohom__commute,axiom,
! [F: nat > nat > nat,H: nat > nat,N3: set_nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ! [X3: nat,Y: nat] :
( ( H @ ( F @ X3 @ Y ) )
= ( F @ ( H @ X3 ) @ ( H @ Y ) ) )
=> ( ( finite_finite_nat @ N3 )
=> ( ( N3 != bot_bot_set_nat )
=> ( ( H @ ( lattic7742739596368939638_F_nat @ F @ N3 ) )
= ( lattic7742739596368939638_F_nat @ F @ ( image_nat_nat @ H @ N3 ) ) ) ) ) ) ) ).
% semilattice_set.hom_commute
thf(fact_1242_semilattice__set_Ohom__commute,axiom,
! [F: transition > transition > transition,H: transition > transition,N3: set_transition] :
( ( lattic5875891794066335564sition @ F )
=> ( ! [X3: transition,Y: transition] :
( ( H @ ( F @ X3 @ Y ) )
= ( F @ ( H @ X3 ) @ ( H @ Y ) ) )
=> ( ( finite8165534619950747239sition @ N3 )
=> ( ( N3 != bot_bo301567166201926666sition )
=> ( ( H @ ( lattic3235374346148272024sition @ F @ N3 ) )
= ( lattic3235374346148272024sition @ F @ ( image_5857460390510121477sition @ H @ N3 ) ) ) ) ) ) ) ).
% semilattice_set.hom_commute
thf(fact_1243_semilattice__set_Osubset,axiom,
! [F: transition > transition > transition,A2: set_transition,B: set_transition] :
( ( lattic5875891794066335564sition @ F )
=> ( ( finite8165534619950747239sition @ A2 )
=> ( ( B != bot_bo301567166201926666sition )
=> ( ( ord_le8419162016481440574sition @ B @ A2 )
=> ( ( F @ ( lattic3235374346148272024sition @ F @ B ) @ ( lattic3235374346148272024sition @ F @ A2 ) )
= ( lattic3235374346148272024sition @ F @ A2 ) ) ) ) ) ) ).
% semilattice_set.subset
thf(fact_1244_semilattice__set_Osubset,axiom,
! [F: nat > nat > nat,A2: set_nat,B: set_nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ( finite_finite_nat @ A2 )
=> ( ( B != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( F @ ( lattic7742739596368939638_F_nat @ F @ B ) @ ( lattic7742739596368939638_F_nat @ F @ A2 ) )
= ( lattic7742739596368939638_F_nat @ F @ A2 ) ) ) ) ) ) ).
% semilattice_set.subset
thf(fact_1245_semilattice__set_Oinsert__not__elem,axiom,
! [F: nat > nat > nat,A2: set_nat,X: nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat @ X @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic7742739596368939638_F_nat @ F @ ( insert_nat2 @ X @ A2 ) )
= ( F @ X @ ( lattic7742739596368939638_F_nat @ F @ A2 ) ) ) ) ) ) ) ).
% semilattice_set.insert_not_elem
thf(fact_1246_semilattice__set_Oinsert__not__elem,axiom,
! [F: transition > transition > transition,A2: set_transition,X: transition] :
( ( lattic5875891794066335564sition @ F )
=> ( ( finite8165534619950747239sition @ A2 )
=> ( ~ ( member_transition @ X @ A2 )
=> ( ( A2 != bot_bo301567166201926666sition )
=> ( ( lattic3235374346148272024sition @ F @ ( insert_transition2 @ X @ A2 ) )
= ( F @ X @ ( lattic3235374346148272024sition @ F @ A2 ) ) ) ) ) ) ) ).
% semilattice_set.insert_not_elem
thf(fact_1247_semilattice__set_Oinsert,axiom,
! [F: nat > nat > nat,A2: set_nat,X: nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic7742739596368939638_F_nat @ F @ ( insert_nat2 @ X @ A2 ) )
= ( F @ X @ ( lattic7742739596368939638_F_nat @ F @ A2 ) ) ) ) ) ) ).
% semilattice_set.insert
thf(fact_1248_semilattice__set_Oinsert,axiom,
! [F: transition > transition > transition,A2: set_transition,X: transition] :
( ( lattic5875891794066335564sition @ F )
=> ( ( finite8165534619950747239sition @ A2 )
=> ( ( A2 != bot_bo301567166201926666sition )
=> ( ( lattic3235374346148272024sition @ F @ ( insert_transition2 @ X @ A2 ) )
= ( F @ X @ ( lattic3235374346148272024sition @ F @ A2 ) ) ) ) ) ) ).
% semilattice_set.insert
thf(fact_1249_semilattice__set_Oclosed,axiom,
! [F: nat > nat > nat,A2: set_nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [X3: nat,Y: nat] : ( member_nat @ ( F @ X3 @ Y ) @ ( insert_nat2 @ X3 @ ( insert_nat2 @ Y @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic7742739596368939638_F_nat @ F @ A2 ) @ A2 ) ) ) ) ) ).
% semilattice_set.closed
thf(fact_1250_semilattice__set_Oclosed,axiom,
! [F: transition > transition > transition,A2: set_transition] :
( ( lattic5875891794066335564sition @ F )
=> ( ( finite8165534619950747239sition @ A2 )
=> ( ( A2 != bot_bo301567166201926666sition )
=> ( ! [X3: transition,Y: transition] : ( member_transition @ ( F @ X3 @ Y ) @ ( insert_transition2 @ X3 @ ( insert_transition2 @ Y @ bot_bo301567166201926666sition ) ) )
=> ( member_transition @ ( lattic3235374346148272024sition @ F @ A2 ) @ A2 ) ) ) ) ) ).
% semilattice_set.closed
thf(fact_1251_semilattice__set_Ounion,axiom,
! [F: nat > nat > nat,A2: set_nat,B: set_nat] :
( ( lattic1029310888574255042et_nat @ F )
=> ( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B )
=> ( ( B != bot_bot_set_nat )
=> ( ( lattic7742739596368939638_F_nat @ F @ ( sup_sup_set_nat @ A2 @ B ) )
= ( F @ ( lattic7742739596368939638_F_nat @ F @ A2 ) @ ( lattic7742739596368939638_F_nat @ F @ B ) ) ) ) ) ) ) ) ).
% semilattice_set.union
thf(fact_1252_semilattice__set_Ounion,axiom,
! [F: transition > transition > transition,A2: set_transition,B: set_transition] :
( ( lattic5875891794066335564sition @ F )
=> ( ( finite8165534619950747239sition @ A2 )
=> ( ( A2 != bot_bo301567166201926666sition )
=> ( ( finite8165534619950747239sition @ B )
=> ( ( B != bot_bo301567166201926666sition )
=> ( ( lattic3235374346148272024sition @ F @ ( sup_su812053455038985074sition @ A2 @ B ) )
= ( F @ ( lattic3235374346148272024sition @ F @ A2 ) @ ( lattic3235374346148272024sition @ F @ B ) ) ) ) ) ) ) ) ).
% semilattice_set.union
thf(fact_1253_append__butlast__last__id,axiom,
! [Xs: list_transition] :
( ( Xs != nil_transition )
=> ( ( append_transition @ ( butlast_transition @ Xs ) @ ( cons_transition @ ( last_transition @ Xs ) @ nil_transition ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_1254_append__butlast__last__id,axiom,
! [Xs: list_o] :
( ( Xs != nil_o )
=> ( ( append_o @ ( butlast_o @ Xs ) @ ( cons_o @ ( last_o @ Xs ) @ nil_o ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_1255_append__butlast__last__id,axiom,
! [Xs: list_list_o] :
( ( Xs != nil_list_o )
=> ( ( append_list_o @ ( butlast_list_o @ Xs ) @ ( cons_list_o @ ( last_list_o @ Xs ) @ nil_list_o ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_1256_snoc__eq__iff__butlast,axiom,
! [Xs: list_transition,X: transition,Ys: list_transition] :
( ( ( append_transition @ Xs @ ( cons_transition @ X @ nil_transition ) )
= Ys )
= ( ( Ys != nil_transition )
& ( ( butlast_transition @ Ys )
= Xs )
& ( ( last_transition @ Ys )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_1257_snoc__eq__iff__butlast,axiom,
! [Xs: list_o,X: $o,Ys: list_o] :
( ( ( append_o @ Xs @ ( cons_o @ X @ nil_o ) )
= Ys )
= ( ( Ys != nil_o )
& ( ( butlast_o @ Ys )
= Xs )
& ( ( last_o @ Ys )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_1258_snoc__eq__iff__butlast,axiom,
! [Xs: list_list_o,X: list_o,Ys: list_list_o] :
( ( ( append_list_o @ Xs @ ( cons_list_o @ X @ nil_list_o ) )
= Ys )
= ( ( Ys != nil_list_o )
& ( ( butlast_list_o @ Ys )
= Xs )
& ( ( last_list_o @ Ys )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_1259_last__appendR,axiom,
! [Ys: list_transition,Xs: list_transition] :
( ( Ys != nil_transition )
=> ( ( last_transition @ ( append_transition @ Xs @ Ys ) )
= ( last_transition @ Ys ) ) ) ).
% last_appendR
thf(fact_1260_last__appendR,axiom,
! [Ys: list_o,Xs: list_o] :
( ( Ys != nil_o )
=> ( ( last_o @ ( append_o @ Xs @ Ys ) )
= ( last_o @ Ys ) ) ) ).
% last_appendR
thf(fact_1261_last__appendR,axiom,
! [Ys: list_list_o,Xs: list_list_o] :
( ( Ys != nil_list_o )
=> ( ( last_list_o @ ( append_list_o @ Xs @ Ys ) )
= ( last_list_o @ Ys ) ) ) ).
% last_appendR
thf(fact_1262_last__appendL,axiom,
! [Ys: list_transition,Xs: list_transition] :
( ( Ys = nil_transition )
=> ( ( last_transition @ ( append_transition @ Xs @ Ys ) )
= ( last_transition @ Xs ) ) ) ).
% last_appendL
thf(fact_1263_last__appendL,axiom,
! [Ys: list_o,Xs: list_o] :
( ( Ys = nil_o )
=> ( ( last_o @ ( append_o @ Xs @ Ys ) )
= ( last_o @ Xs ) ) ) ).
% last_appendL
thf(fact_1264_last__appendL,axiom,
! [Ys: list_list_o,Xs: list_list_o] :
( ( Ys = nil_list_o )
=> ( ( last_list_o @ ( append_list_o @ Xs @ Ys ) )
= ( last_list_o @ Xs ) ) ) ).
% last_appendL
thf(fact_1265_last__snoc,axiom,
! [Xs: list_transition,X: transition] :
( ( last_transition @ ( append_transition @ Xs @ ( cons_transition @ X @ nil_transition ) ) )
= X ) ).
% last_snoc
thf(fact_1266_last__snoc,axiom,
! [Xs: list_o,X: $o] :
( ( last_o @ ( append_o @ Xs @ ( cons_o @ X @ nil_o ) ) )
= X ) ).
% last_snoc
thf(fact_1267_last__snoc,axiom,
! [Xs: list_list_o,X: list_o] :
( ( last_list_o @ ( append_list_o @ Xs @ ( cons_list_o @ X @ nil_list_o ) ) )
= X ) ).
% last_snoc
thf(fact_1268_last_Osimps,axiom,
! [Xs: list_transition,X: transition] :
( ( ( Xs = nil_transition )
=> ( ( last_transition @ ( cons_transition @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_transition )
=> ( ( last_transition @ ( cons_transition @ X @ Xs ) )
= ( last_transition @ Xs ) ) ) ) ).
% last.simps
thf(fact_1269_last_Osimps,axiom,
! [X: $o,Xs: list_o] :
( ( last_o @ ( cons_o @ X @ Xs ) )
= ( ( ( Xs = nil_o )
=> X )
& ( ( Xs != nil_o )
=> ( last_o @ Xs ) ) ) ) ).
% last.simps
thf(fact_1270_last_Osimps,axiom,
! [Xs: list_list_o,X: list_o] :
( ( ( Xs = nil_list_o )
=> ( ( last_list_o @ ( cons_list_o @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_list_o )
=> ( ( last_list_o @ ( cons_list_o @ X @ Xs ) )
= ( last_list_o @ Xs ) ) ) ) ).
% last.simps
thf(fact_1271_last__ConsL,axiom,
! [Xs: list_transition,X: transition] :
( ( Xs = nil_transition )
=> ( ( last_transition @ ( cons_transition @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_1272_last__ConsL,axiom,
! [Xs: list_o,X: $o] :
( ( Xs = nil_o )
=> ( ( last_o @ ( cons_o @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_1273_last__ConsL,axiom,
! [Xs: list_list_o,X: list_o] :
( ( Xs = nil_list_o )
=> ( ( last_list_o @ ( cons_list_o @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_1274_last__ConsR,axiom,
! [Xs: list_o,X: $o] :
( ( Xs != nil_o )
=> ( ( last_o @ ( cons_o @ X @ Xs ) )
= ( last_o @ Xs ) ) ) ).
% last_ConsR
thf(fact_1275_last__ConsR,axiom,
! [Xs: list_list_o,X: list_o] :
( ( Xs != nil_list_o )
=> ( ( last_list_o @ ( cons_list_o @ X @ Xs ) )
= ( last_list_o @ Xs ) ) ) ).
% last_ConsR
% Helper facts (9)
thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X: set_nat,Y2: set_nat] :
( ( if_set_nat @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X: set_nat,Y2: set_nat] :
( ( if_set_nat @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y2: list_nat] :
( ( if_list_nat @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y2: list_nat] :
( ( if_list_nat @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__NFA__Otransition_J_T,axiom,
! [X: list_transition,Y2: list_transition] :
( ( if_list_transition @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__NFA__Otransition_J_T,axiom,
! [X: list_transition,Y2: list_transition] :
( ( if_list_transition @ $true @ X @ Y2 )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__List__Olist_I_Eo_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__List__Olist_I_Eo_J_J_T,axiom,
! [X: list_list_o,Y2: list_list_o] :
( ( if_list_list_o @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_I_Eo_J_J_T,axiom,
! [X: list_list_o,Y2: list_list_o] :
( ( if_list_list_o @ $true @ X @ Y2 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( step_symb_set @ q0 @ transs @ ( insert_nat2 @ qf @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ).
%------------------------------------------------------------------------------