TPTP Problem File: SLH0535^1.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/0010_Temporal/prob_00558_025864__16636092_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1558 ( 641 unt; 273 typ; 0 def)
% Number of atoms : 3510 (1356 equ; 0 cnn)
% Maximal formula atoms : 15 ( 2 avg)
% Number of connectives : 12267 ( 490 ~; 52 |; 242 &;10014 @)
% ( 0 <=>;1469 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 7 avg)
% Number of types : 22 ( 21 usr)
% Number of type conns : 689 ( 689 >; 0 *; 0 +; 0 <<)
% Number of symbols : 255 ( 252 usr; 29 con; 0-9 aty)
% Number of variables : 3846 ( 240 ^;3542 !; 64 ?;3846 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:53:44.226
%------------------------------------------------------------------------------
% 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 (252)
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
comple7399068483239264473et_nat: set_set_nat > set_nat ).
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__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_I_Eo_J_J,type,
minus_8912710245716896613list_o: set_list_o > set_list_o > set_list_o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
minus_7954133019191499631st_nat: set_list_nat > set_list_nat > set_list_nat ).
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__NFA__Otransition_J_J,type,
minus_7806277596913192037sition: set_set_transition > set_set_transition > set_set_transition ).
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_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
minus_2447799839930672331et_nat: set_set_set_nat > set_set_set_nat > set_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_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_001_062_It__NFA__Otransition_M_062_It__NFA__Otransition_M_Eo_J_J,type,
sup_su3408646655489644610tion_o: ( transition > transition > $o ) > ( transition > transition > $o ) > transition > transition > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
sup_sup_nat_nat_o: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > nat > $o ).
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__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__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__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_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_Ogen__length_001_Eo,type,
gen_length_o: nat > list_o > nat ).
thf(sy_c_List_Ogen__length_001t__List__Olist_I_Eo_J,type,
gen_length_list_o: nat > list_list_o > nat ).
thf(sy_c_List_Ogen__length_001t__NFA__Otransition,type,
gen_le2281773382894182225sition: nat > list_transition > 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_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_Olist__update_001_Eo,type,
list_update_o: list_o > nat > $o > list_o ).
thf(sy_c_List_Olist__update_001t__List__Olist_I_Eo_J,type,
list_update_list_o: list_list_o > nat > list_o > list_list_o ).
thf(sy_c_List_Olist__update_001t__NFA__Otransition,type,
list_u2676915870505589036sition: list_transition > nat > transition > list_transition ).
thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
list_update_nat: list_nat > nat > nat > list_nat ).
thf(sy_c_List_Omin__list_001_Eo,type,
min_list_o: list_o > $o ).
thf(sy_c_List_Onth_001t__List__Olist_I_Eo_J,type,
nth_list_o: list_list_o > nat > list_o ).
thf(sy_c_List_Onth_001t__NFA__Otransition,type,
nth_transition: list_transition > nat > transition ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
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_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_Onfa__cong,type,
nfa_cong: nat > nat > nat > nat > list_transition > list_transition > $o ).
thf(sy_c_NFA_Onfa__cong_H,type,
nfa_cong2: nat > nat > nat > nat > list_transition > list_transition > $o ).
thf(sy_c_NFA_Onfa__cong_H__axioms,type,
nfa_cong_axioms: nat > nat > nat > list_transition > list_transition > $o ).
thf(sy_c_NFA_Onfa__cong__Plus,type,
nfa_cong_Plus: nat > nat > nat > nat > nat > nat > list_transition > list_transition > list_transition > $o ).
thf(sy_c_NFA_Onfa__cong__Plus__axioms,type,
nfa_cong_Plus_axioms: nat > nat > nat > list_transition > $o ).
thf(sy_c_NFA_Onfa__cong__Star,type,
nfa_cong_Star: nat > nat > nat > list_transition > list_transition > $o ).
thf(sy_c_NFA_Onfa__cong__Star__axioms,type,
nfa_cong_Star_axioms: nat > nat > nat > list_transition > $o ).
thf(sy_c_NFA_Onfa__cong__Times,type,
nfa_cong_Times: nat > nat > nat > list_transition > list_transition > list_transition > $o ).
thf(sy_c_NFA_Onfa__cong__axioms,type,
nfa_cong_axioms2: nat > nat > nat > nat > list_transition > 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__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_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_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__NFA__Otransition_J,type,
size_s3613142680436377136sition: list_transition > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_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__List__Olist_It__Nat__Onat_J_J,type,
bot_bot_set_list_nat: set_list_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__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_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
bot_bo7198184520161983622et_nat: set_set_set_nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__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__List__Olist_It__Nat__Onat_J_J,type,
ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__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_I_Eo_J_J,type,
ord_le4374716579403074808_set_o: set_set_o > set_set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__List__Olist_I_Eo_J_J_J,type,
ord_le2406146954300516094list_o: set_set_list_o > set_set_list_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__NFA__Otransition_J_J,type,
ord_le882869523442495262sition: set_set_transition > set_set_transition > $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_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
ord_le9131159989063066194et_nat: set_set_set_nat > set_set_set_nat > $o ).
thf(sy_c_Set_OCollect_001t__List__Olist_I_Eo_J,type,
collect_list_o: ( list_o > $o ) > set_list_o ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(sy_c_Set_OCollect_001t__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_OCollect_001t__Set__Oset_I_Eo_J,type,
collect_set_o: ( set_o > $o ) > set_set_o ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__List__Olist_I_Eo_J_J,type,
collect_set_list_o: ( set_list_o > $o ) > set_set_list_o ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__NFA__Otransition_J,type,
collec5580505218370054281sition: ( set_transition > $o ) > set_set_transition ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
collect_set_set_nat: ( set_set_nat > $o ) > set_set_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_001_Eo,type,
insert_o2: $o > set_o > set_o ).
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__List__Olist_It__Nat__Onat_J,type,
insert_list_nat: list_nat > set_list_nat > set_list_nat ).
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_Oinsert_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
insert_set_set_nat: set_set_nat > set_set_set_nat > set_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_Transitive__Closure_Ortranclp_001t__NFA__Otransition,type,
transi7192379879768945417sition: ( transition > transition > $o ) > transition > transition > $o ).
thf(sy_c_Transitive__Closure_Ortranclp_001t__Nat__Onat,type,
transi5422400438309235013lp_nat: ( nat > nat > $o ) > nat > nat > $o ).
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_Zorn_Opred__on_Ochain_001t__NFA__Otransition,type,
pred_c3769398113492517609sition: set_transition > ( transition > transition > $o ) > set_transition > $o ).
thf(sy_c_Zorn_Opred__on_Ochain_001t__Nat__Onat,type,
pred_chain_nat: set_nat > ( nat > nat > $o ) > set_nat > $o ).
thf(sy_c_Zorn_Opred__on_Ochain_001t__Set__Oset_It__Nat__Onat_J,type,
pred_chain_set_nat: set_set_nat > ( set_nat > set_nat > $o ) > set_set_nat > $o ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
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_I_Eo_J,type,
member_set_o: set_o > set_set_o > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_I_Eo_J_J,type,
member_set_list_o: set_list_o > set_set_list_o > $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_bs,type,
bs: list_o ).
thf(sy_v_q,type,
q2: nat ).
thf(sy_v_q0,type,
q0: nat ).
thf(sy_v_q0a____,type,
q0a: nat ).
thf(sy_v_qa____,type,
qa: nat ).
thf(sy_v_qf,type,
qf: nat ).
thf(sy_v_qfa____,type,
qfa: nat ).
thf(sy_v_thesis____,type,
thesis: $o ).
thf(sy_v_transs,type,
transs: list_transition ).
thf(sy_v_transsa____,type,
transsa: list_transition ).
thf(sy_v_ts__l____,type,
ts_l: list_transition ).
% Relevant facts (1276)
thf(fact_0__092_060open_062q0_A_092_060noteq_062_Aqf_092_060close_062,axiom,
q0a != qfa ).
% \<open>q0 \<noteq> qf\<close>
thf(fact_1_cong_Oright_Oqf__eq,axiom,
qfa = qfa ).
% cong.right.qf_eq
thf(fact_2_q__q0,axiom,
qa = q0a ).
% q_q0
thf(fact_3_Plus_Oprems_I3_J,axiom,
step_eps_closure @ q0a @ transsa @ bs @ qa @ qfa ).
% Plus.prems(3)
thf(fact_4_base_Ostep__eps__closure__qf,axiom,
! [Bs: list_o,Q: nat,Q2: nat] :
( ( step_eps_closure @ q0a @ transsa @ Bs @ Q @ Q2 )
=> ( ( Q = qfa )
=> ( Q = Q2 ) ) ) ).
% base.step_eps_closure_qf
thf(fact_5_base_Ostep__eps__qf,axiom,
! [Bs: list_o,Q: nat] :
~ ( step_eps @ q0a @ transsa @ Bs @ qfa @ Q ) ).
% base.step_eps_qf
thf(fact_6_base_Ostep__symb__qf,axiom,
! [Q: nat] :
~ ( step_symb @ q0a @ transsa @ qfa @ Q ) ).
% base.step_symb_qf
thf(fact_7_base_Onfa__axioms,axiom,
nfa @ q0a @ qfa @ transsa ).
% base.nfa_axioms
thf(fact_8_assms_I3_J,axiom,
step_eps_closure @ q0 @ transs @ bs @ q2 @ qf ).
% assms(3)
thf(fact_9_cong_Ostep__symb__q0,axiom,
! [Q: nat] :
~ ( step_symb @ q0a @ transsa @ q0a @ Q ) ).
% cong.step_symb_q0
thf(fact_10_base_Ostep__eps__closed,axiom,
! [Bs: list_o,Q: nat,Q2: nat] :
( ( step_eps @ q0a @ transsa @ Bs @ Q @ Q2 )
=> ( member_nat @ Q2 @ ( q @ q0a @ qfa @ transsa ) ) ) ).
% base.step_eps_closed
thf(fact_11_base_Ostep__eps__closure__closed,axiom,
! [Bs: list_o,Q: nat,Q2: nat] :
( ( step_eps_closure @ q0a @ transsa @ Bs @ Q @ Q2 )
=> ( ( Q != Q2 )
=> ( member_nat @ Q2 @ ( q @ q0a @ qfa @ transsa ) ) ) ) ).
% base.step_eps_closure_closed
thf(fact_12_base_Oqf__not__in__SQ,axiom,
~ ( member_nat @ qfa @ ( sq @ q0a @ transsa ) ) ).
% base.qf_not_in_SQ
thf(fact_13_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 )
=> ( ! [Q3: nat] :
~ ( step_eps @ Q0 @ Transs @ Bs @ Q @ Q3 )
=> ( Q = Q2 ) ) ) ).
% step_eps_closure_empty
thf(fact_14_base_Otranss__not__Nil,axiom,
transsa != nil_transition ).
% base.transs_not_Nil
thf(fact_15_base_Ostep__symb__closed,axiom,
! [Q: nat,Q2: nat] :
( ( step_symb @ q0a @ transsa @ Q @ Q2 )
=> ( member_nat @ Q2 @ ( q @ q0a @ qfa @ transsa ) ) ) ).
% base.step_symb_closed
thf(fact_16_cong_Oqf__not__q0,axiom,
~ ( member_nat @ qfa @ ( insert_nat2 @ q0a @ bot_bot_set_nat ) ) ).
% cong.qf_not_q0
thf(fact_17_base_Ostep__eps__set__closed,axiom,
! [Bs: list_o,R: set_nat] : ( ord_less_eq_set_nat @ ( step_eps_set @ q0a @ transsa @ Bs @ R ) @ ( q @ q0a @ qfa @ transsa ) ) ).
% base.step_eps_set_closed
thf(fact_18_base_Oq0__sub__SQ,axiom,
ord_less_eq_set_nat @ ( insert_nat2 @ q0a @ bot_bot_set_nat ) @ ( sq @ q0a @ transsa ) ).
% base.q0_sub_SQ
thf(fact_19_base_Oq0__sub__Q,axiom,
ord_less_eq_set_nat @ ( insert_nat2 @ q0a @ bot_bot_set_nat ) @ ( q @ q0a @ qfa @ transsa ) ).
% base.q0_sub_Q
thf(fact_20_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_21_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_22_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_23_nfa_Ostep__symb__qf,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,Q: nat] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ~ ( step_symb @ Q0 @ Transs @ Qf @ Q ) ) ).
% nfa.step_symb_qf
thf(fact_24_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_25_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_26_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_27_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_28_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_29_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_30_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_31_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_32_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_33_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_34_nfa_Ostep__eps__closure__qf,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 = Qf )
=> ( Q = Q2 ) ) ) ) ).
% nfa.step_eps_closure_qf
thf(fact_35_nfa_Ostep__eps__qf,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,Bs: list_o,Q: nat] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ~ ( step_eps @ Q0 @ Transs @ Bs @ Qf @ Q ) ) ).
% nfa.step_eps_qf
thf(fact_36_singleton__insert__inj__eq,axiom,
! [B: set_transition,A: set_transition,A2: set_set_transition] :
( ( ( insert8494249028948967790sition @ B @ bot_bo1233527522848825322sition )
= ( insert8494249028948967790sition @ A @ A2 ) )
= ( ( A = B )
& ( ord_le882869523442495262sition @ A2 @ ( insert8494249028948967790sition @ B @ bot_bo1233527522848825322sition ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_37_singleton__insert__inj__eq,axiom,
! [B: set_list_o,A: set_list_o,A2: set_set_list_o] :
( ( ( insert_set_list_o @ B @ bot_bo64454365476827594list_o )
= ( insert_set_list_o @ A @ A2 ) )
= ( ( A = B )
& ( ord_le2406146954300516094list_o @ A2 @ ( insert_set_list_o @ B @ bot_bo64454365476827594list_o ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_38_singleton__insert__inj__eq,axiom,
! [B: set_o,A: set_o,A2: set_set_o] :
( ( ( insert_set_o @ B @ bot_bot_set_set_o )
= ( insert_set_o @ A @ A2 ) )
= ( ( A = B )
& ( ord_le4374716579403074808_set_o @ A2 @ ( insert_set_o @ B @ bot_bot_set_set_o ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_39_singleton__insert__inj__eq,axiom,
! [B: transition,A: transition,A2: set_transition] :
( ( ( insert_transition2 @ B @ bot_bo301567166201926666sition )
= ( insert_transition2 @ A @ A2 ) )
= ( ( A = B )
& ( ord_le8419162016481440574sition @ A2 @ ( insert_transition2 @ B @ bot_bo301567166201926666sition ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_40_singleton__insert__inj__eq,axiom,
! [B: list_o,A: list_o,A2: set_list_o] :
( ( ( insert_list_o2 @ B @ bot_bot_set_list_o )
= ( insert_list_o2 @ A @ A2 ) )
= ( ( A = B )
& ( ord_le6901083488122529182list_o @ A2 @ ( insert_list_o2 @ B @ bot_bot_set_list_o ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_41_singleton__insert__inj__eq,axiom,
! [B: $o,A: $o,A2: set_o] :
( ( ( insert_o2 @ B @ bot_bot_set_o )
= ( insert_o2 @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_o @ A2 @ ( insert_o2 @ B @ bot_bot_set_o ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_42_singleton__insert__inj__eq,axiom,
! [B: set_nat,A: set_nat,A2: set_set_nat] :
( ( ( insert_set_nat @ B @ bot_bot_set_set_nat )
= ( insert_set_nat @ A @ A2 ) )
= ( ( A = B )
& ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_43_singleton__insert__inj__eq,axiom,
! [B: nat,A: nat,A2: set_nat] :
( ( ( insert_nat2 @ B @ bot_bot_set_nat )
= ( insert_nat2 @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_44_singleton__insert__inj__eq_H,axiom,
! [A: set_transition,A2: set_set_transition,B: set_transition] :
( ( ( insert8494249028948967790sition @ A @ A2 )
= ( insert8494249028948967790sition @ B @ bot_bo1233527522848825322sition ) )
= ( ( A = B )
& ( ord_le882869523442495262sition @ A2 @ ( insert8494249028948967790sition @ B @ bot_bo1233527522848825322sition ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_45_singleton__insert__inj__eq_H,axiom,
! [A: set_list_o,A2: set_set_list_o,B: set_list_o] :
( ( ( insert_set_list_o @ A @ A2 )
= ( insert_set_list_o @ B @ bot_bo64454365476827594list_o ) )
= ( ( A = B )
& ( ord_le2406146954300516094list_o @ A2 @ ( insert_set_list_o @ B @ bot_bo64454365476827594list_o ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_46_singleton__insert__inj__eq_H,axiom,
! [A: set_o,A2: set_set_o,B: set_o] :
( ( ( insert_set_o @ A @ A2 )
= ( insert_set_o @ B @ bot_bot_set_set_o ) )
= ( ( A = B )
& ( ord_le4374716579403074808_set_o @ A2 @ ( insert_set_o @ B @ bot_bot_set_set_o ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_47_singleton__insert__inj__eq_H,axiom,
! [A: transition,A2: set_transition,B: transition] :
( ( ( insert_transition2 @ A @ A2 )
= ( insert_transition2 @ B @ bot_bo301567166201926666sition ) )
= ( ( A = B )
& ( ord_le8419162016481440574sition @ A2 @ ( insert_transition2 @ B @ bot_bo301567166201926666sition ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_48_singleton__insert__inj__eq_H,axiom,
! [A: list_o,A2: set_list_o,B: list_o] :
( ( ( insert_list_o2 @ A @ A2 )
= ( insert_list_o2 @ B @ bot_bot_set_list_o ) )
= ( ( A = B )
& ( ord_le6901083488122529182list_o @ A2 @ ( insert_list_o2 @ B @ bot_bot_set_list_o ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_49_singleton__insert__inj__eq_H,axiom,
! [A: $o,A2: set_o,B: $o] :
( ( ( insert_o2 @ A @ A2 )
= ( insert_o2 @ B @ bot_bot_set_o ) )
= ( ( A = B )
& ( ord_less_eq_set_o @ A2 @ ( insert_o2 @ B @ bot_bot_set_o ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_50_singleton__insert__inj__eq_H,axiom,
! [A: set_nat,A2: set_set_nat,B: set_nat] :
( ( ( insert_set_nat @ A @ A2 )
= ( insert_set_nat @ B @ bot_bot_set_set_nat ) )
= ( ( A = B )
& ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_51_singleton__insert__inj__eq_H,axiom,
! [A: nat,A2: set_nat,B: nat] :
( ( ( insert_nat2 @ A @ A2 )
= ( insert_nat2 @ B @ bot_bot_set_nat ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_52_base_Ostep__symb__set__closed,axiom,
! [R: set_nat] : ( ord_less_eq_set_nat @ ( step_symb_set @ q0a @ transsa @ R ) @ ( q @ q0a @ qfa @ transsa ) ) ).
% base.step_symb_set_closed
thf(fact_53_base_Ostep__symb__set__qf,axiom,
( ( step_symb_set @ q0a @ transsa @ ( insert_nat2 @ qfa @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ).
% base.step_symb_set_qf
thf(fact_54_base_OQ__diff__qf__SQ,axiom,
( ( minus_minus_set_nat @ ( q @ q0a @ qfa @ transsa ) @ ( insert_nat2 @ qfa @ bot_bot_set_nat ) )
= ( sq @ q0a @ transsa ) ) ).
% base.Q_diff_qf_SQ
thf(fact_55_base_Orun__closed,axiom,
! [R: set_nat,Bss: list_list_o] :
( ( ord_less_eq_set_nat @ R @ ( q @ q0a @ qfa @ transsa ) )
=> ( ord_less_eq_set_nat @ ( run @ q0a @ transsa @ R @ Bss ) @ ( q @ q0a @ qfa @ transsa ) ) ) ).
% base.run_closed
thf(fact_56_base_Odelta__closed,axiom,
! [R: set_nat,Bs: list_o] : ( ord_less_eq_set_nat @ ( delta @ q0a @ transsa @ R @ Bs ) @ ( q @ q0a @ qfa @ transsa ) ) ).
% base.delta_closed
thf(fact_57_base_Odelta__qf,axiom,
! [Bs: list_o] :
( ( delta @ q0a @ transsa @ ( insert_nat2 @ qfa @ bot_bot_set_nat ) @ Bs )
= bot_bot_set_nat ) ).
% base.delta_qf
thf(fact_58_cong_Ostep__symb__set__q0,axiom,
( ( step_symb_set @ q0a @ transsa @ ( insert_nat2 @ q0a @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ).
% cong.step_symb_set_q0
thf(fact_59_mem__Collect__eq,axiom,
! [A: list_o,P: list_o > $o] :
( ( member_list_o @ A @ ( collect_list_o @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_60_mem__Collect__eq,axiom,
! [A: list_nat,P: list_nat > $o] :
( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_61_mem__Collect__eq,axiom,
! [A: set_transition,P: set_transition > $o] :
( ( member7318969637299765063sition @ A @ ( collec5580505218370054281sition @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_62_mem__Collect__eq,axiom,
! [A: set_set_nat,P: set_set_nat > $o] :
( ( member_set_set_nat @ A @ ( collect_set_set_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_63_mem__Collect__eq,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_64_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_65_mem__Collect__eq,axiom,
! [A: transition,P: transition > $o] :
( ( member_transition @ A @ ( collect_transition @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_66_Collect__mem__eq,axiom,
! [A2: set_list_o] :
( ( collect_list_o
@ ^ [X: list_o] : ( member_list_o @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_67_Collect__mem__eq,axiom,
! [A2: set_list_nat] :
( ( collect_list_nat
@ ^ [X: list_nat] : ( member_list_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_68_Collect__mem__eq,axiom,
! [A2: set_set_transition] :
( ( collec5580505218370054281sition
@ ^ [X: set_transition] : ( member7318969637299765063sition @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_69_Collect__mem__eq,axiom,
! [A2: set_set_set_nat] :
( ( collect_set_set_nat
@ ^ [X: set_set_nat] : ( member_set_set_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_70_Collect__mem__eq,axiom,
! [A2: set_set_nat] :
( ( collect_set_nat
@ ^ [X: set_nat] : ( member_set_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_71_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_72_Collect__mem__eq,axiom,
! [A2: set_transition] :
( ( collect_transition
@ ^ [X: transition] : ( member_transition @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_73_Collect__cong,axiom,
! [P: transition > $o,Q4: transition > $o] :
( ! [X2: transition] :
( ( P @ X2 )
= ( Q4 @ X2 ) )
=> ( ( collect_transition @ P )
= ( collect_transition @ Q4 ) ) ) ).
% Collect_cong
thf(fact_74_Collect__cong,axiom,
! [P: nat > $o,Q4: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
= ( Q4 @ X2 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q4 ) ) ) ).
% Collect_cong
thf(fact_75_insert__subset,axiom,
! [X3: set_list_o,A2: set_set_list_o,B2: set_set_list_o] :
( ( ord_le2406146954300516094list_o @ ( insert_set_list_o @ X3 @ A2 ) @ B2 )
= ( ( member_set_list_o @ X3 @ B2 )
& ( ord_le2406146954300516094list_o @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_76_insert__subset,axiom,
! [X3: set_o,A2: set_set_o,B2: set_set_o] :
( ( ord_le4374716579403074808_set_o @ ( insert_set_o @ X3 @ A2 ) @ B2 )
= ( ( member_set_o @ X3 @ B2 )
& ( ord_le4374716579403074808_set_o @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_77_insert__subset,axiom,
! [X3: list_nat,A2: set_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( insert_list_nat @ X3 @ A2 ) @ B2 )
= ( ( member_list_nat @ X3 @ B2 )
& ( ord_le6045566169113846134st_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_78_insert__subset,axiom,
! [X3: set_transition,A2: set_set_transition,B2: set_set_transition] :
( ( ord_le882869523442495262sition @ ( insert8494249028948967790sition @ X3 @ A2 ) @ B2 )
= ( ( member7318969637299765063sition @ X3 @ B2 )
& ( ord_le882869523442495262sition @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_79_insert__subset,axiom,
! [X3: set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( ord_le9131159989063066194et_nat @ ( insert_set_set_nat @ X3 @ A2 ) @ B2 )
= ( ( member_set_set_nat @ X3 @ B2 )
& ( ord_le9131159989063066194et_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_80_insert__subset,axiom,
! [X3: list_o,A2: set_list_o,B2: set_list_o] :
( ( ord_le6901083488122529182list_o @ ( insert_list_o2 @ X3 @ A2 ) @ B2 )
= ( ( member_list_o @ X3 @ B2 )
& ( ord_le6901083488122529182list_o @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_81_insert__subset,axiom,
! [X3: $o,A2: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ ( insert_o2 @ X3 @ A2 ) @ B2 )
= ( ( member_o @ X3 @ B2 )
& ( ord_less_eq_set_o @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_82_insert__subset,axiom,
! [X3: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X3 @ A2 ) @ B2 )
= ( ( member_set_nat @ X3 @ B2 )
& ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_83_insert__subset,axiom,
! [X3: transition,A2: set_transition,B2: set_transition] :
( ( ord_le8419162016481440574sition @ ( insert_transition2 @ X3 @ A2 ) @ B2 )
= ( ( member_transition @ X3 @ B2 )
& ( ord_le8419162016481440574sition @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_84_insert__subset,axiom,
! [X3: nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat2 @ X3 @ A2 ) @ B2 )
= ( ( member_nat @ X3 @ B2 )
& ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_85_singletonI,axiom,
! [A: list_o] : ( member_list_o @ A @ ( insert_list_o2 @ A @ bot_bot_set_list_o ) ) ).
% singletonI
thf(fact_86_singletonI,axiom,
! [A: list_nat] : ( member_list_nat @ A @ ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ).
% singletonI
thf(fact_87_singletonI,axiom,
! [A: set_transition] : ( member7318969637299765063sition @ A @ ( insert8494249028948967790sition @ A @ bot_bo1233527522848825322sition ) ) ).
% singletonI
thf(fact_88_singletonI,axiom,
! [A: set_set_nat] : ( member_set_set_nat @ A @ ( insert_set_set_nat @ A @ bot_bo7198184520161983622et_nat ) ) ).
% singletonI
thf(fact_89_singletonI,axiom,
! [A: set_list_o] : ( member_set_list_o @ A @ ( insert_set_list_o @ A @ bot_bo64454365476827594list_o ) ) ).
% singletonI
thf(fact_90_singletonI,axiom,
! [A: set_o] : ( member_set_o @ A @ ( insert_set_o @ A @ bot_bot_set_set_o ) ) ).
% singletonI
thf(fact_91_singletonI,axiom,
! [A: set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).
% singletonI
thf(fact_92_singletonI,axiom,
! [A: transition] : ( member_transition @ A @ ( insert_transition2 @ A @ bot_bo301567166201926666sition ) ) ).
% singletonI
thf(fact_93_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_94_subset__empty,axiom,
! [A2: set_set_list_o] :
( ( ord_le2406146954300516094list_o @ A2 @ bot_bo64454365476827594list_o )
= ( A2 = bot_bo64454365476827594list_o ) ) ).
% subset_empty
thf(fact_95_subset__empty,axiom,
! [A2: set_set_o] :
( ( ord_le4374716579403074808_set_o @ A2 @ bot_bot_set_set_o )
= ( A2 = bot_bot_set_set_o ) ) ).
% subset_empty
thf(fact_96_subset__empty,axiom,
! [A2: set_transition] :
( ( ord_le8419162016481440574sition @ A2 @ bot_bo301567166201926666sition )
= ( A2 = bot_bo301567166201926666sition ) ) ).
% subset_empty
thf(fact_97_subset__empty,axiom,
! [A2: set_list_o] :
( ( ord_le6901083488122529182list_o @ A2 @ bot_bot_set_list_o )
= ( A2 = bot_bot_set_list_o ) ) ).
% subset_empty
thf(fact_98_subset__empty,axiom,
! [A2: set_o] :
( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
= ( A2 = bot_bot_set_o ) ) ).
% subset_empty
thf(fact_99_subset__empty,axiom,
! [A2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ bot_bot_set_set_nat )
= ( A2 = bot_bot_set_set_nat ) ) ).
% subset_empty
thf(fact_100_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_101_empty__Collect__eq,axiom,
! [P: transition > $o] :
( ( bot_bo301567166201926666sition
= ( collect_transition @ P ) )
= ( ! [X: transition] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_102_empty__Collect__eq,axiom,
! [P: set_list_o > $o] :
( ( bot_bo64454365476827594list_o
= ( collect_set_list_o @ P ) )
= ( ! [X: set_list_o] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_103_empty__Collect__eq,axiom,
! [P: set_o > $o] :
( ( bot_bot_set_set_o
= ( collect_set_o @ P ) )
= ( ! [X: set_o] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_104_empty__Collect__eq,axiom,
! [P: set_nat > $o] :
( ( bot_bot_set_set_nat
= ( collect_set_nat @ P ) )
= ( ! [X: set_nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_105_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_106_Collect__empty__eq,axiom,
! [P: transition > $o] :
( ( ( collect_transition @ P )
= bot_bo301567166201926666sition )
= ( ! [X: transition] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_107_Collect__empty__eq,axiom,
! [P: set_list_o > $o] :
( ( ( collect_set_list_o @ P )
= bot_bo64454365476827594list_o )
= ( ! [X: set_list_o] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_108_Collect__empty__eq,axiom,
! [P: set_o > $o] :
( ( ( collect_set_o @ P )
= bot_bot_set_set_o )
= ( ! [X: set_o] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_109_Collect__empty__eq,axiom,
! [P: set_nat > $o] :
( ( ( collect_set_nat @ P )
= bot_bot_set_set_nat )
= ( ! [X: set_nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_110_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_111_all__not__in__conv,axiom,
! [A2: set_list_o] :
( ( ! [X: list_o] :
~ ( member_list_o @ X @ A2 ) )
= ( A2 = bot_bot_set_list_o ) ) ).
% all_not_in_conv
thf(fact_112_all__not__in__conv,axiom,
! [A2: set_list_nat] :
( ( ! [X: list_nat] :
~ ( member_list_nat @ X @ A2 ) )
= ( A2 = bot_bot_set_list_nat ) ) ).
% all_not_in_conv
thf(fact_113_all__not__in__conv,axiom,
! [A2: set_set_transition] :
( ( ! [X: set_transition] :
~ ( member7318969637299765063sition @ X @ A2 ) )
= ( A2 = bot_bo1233527522848825322sition ) ) ).
% all_not_in_conv
thf(fact_114_all__not__in__conv,axiom,
! [A2: set_set_set_nat] :
( ( ! [X: set_set_nat] :
~ ( member_set_set_nat @ X @ A2 ) )
= ( A2 = bot_bo7198184520161983622et_nat ) ) ).
% all_not_in_conv
thf(fact_115_all__not__in__conv,axiom,
! [A2: set_set_list_o] :
( ( ! [X: set_list_o] :
~ ( member_set_list_o @ X @ A2 ) )
= ( A2 = bot_bo64454365476827594list_o ) ) ).
% all_not_in_conv
thf(fact_116_all__not__in__conv,axiom,
! [A2: set_set_o] :
( ( ! [X: set_o] :
~ ( member_set_o @ X @ A2 ) )
= ( A2 = bot_bot_set_set_o ) ) ).
% all_not_in_conv
thf(fact_117_all__not__in__conv,axiom,
! [A2: set_set_nat] :
( ( ! [X: set_nat] :
~ ( member_set_nat @ X @ A2 ) )
= ( A2 = bot_bot_set_set_nat ) ) ).
% all_not_in_conv
thf(fact_118_all__not__in__conv,axiom,
! [A2: set_transition] :
( ( ! [X: transition] :
~ ( member_transition @ X @ A2 ) )
= ( A2 = bot_bo301567166201926666sition ) ) ).
% all_not_in_conv
thf(fact_119_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X: nat] :
~ ( member_nat @ X @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_120_empty__iff,axiom,
! [C: list_o] :
~ ( member_list_o @ C @ bot_bot_set_list_o ) ).
% empty_iff
thf(fact_121_empty__iff,axiom,
! [C: list_nat] :
~ ( member_list_nat @ C @ bot_bot_set_list_nat ) ).
% empty_iff
thf(fact_122_empty__iff,axiom,
! [C: set_transition] :
~ ( member7318969637299765063sition @ C @ bot_bo1233527522848825322sition ) ).
% empty_iff
thf(fact_123_empty__iff,axiom,
! [C: set_set_nat] :
~ ( member_set_set_nat @ C @ bot_bo7198184520161983622et_nat ) ).
% empty_iff
thf(fact_124_empty__iff,axiom,
! [C: set_list_o] :
~ ( member_set_list_o @ C @ bot_bo64454365476827594list_o ) ).
% empty_iff
thf(fact_125_empty__iff,axiom,
! [C: set_o] :
~ ( member_set_o @ C @ bot_bot_set_set_o ) ).
% empty_iff
thf(fact_126_empty__iff,axiom,
! [C: set_nat] :
~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).
% empty_iff
thf(fact_127_empty__iff,axiom,
! [C: transition] :
~ ( member_transition @ C @ bot_bo301567166201926666sition ) ).
% empty_iff
thf(fact_128_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_129_subset__antisym,axiom,
! [A2: set_transition,B2: set_transition] :
( ( ord_le8419162016481440574sition @ A2 @ B2 )
=> ( ( ord_le8419162016481440574sition @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_130_subset__antisym,axiom,
! [A2: set_list_o,B2: set_list_o] :
( ( ord_le6901083488122529182list_o @ A2 @ B2 )
=> ( ( ord_le6901083488122529182list_o @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_131_subset__antisym,axiom,
! [A2: set_o,B2: set_o] :
( ( ord_less_eq_set_o @ A2 @ B2 )
=> ( ( ord_less_eq_set_o @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_132_subset__antisym,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_133_subset__antisym,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_134_subsetI,axiom,
! [A2: set_list_nat,B2: set_list_nat] :
( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ A2 )
=> ( member_list_nat @ X2 @ B2 ) )
=> ( ord_le6045566169113846134st_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_135_subsetI,axiom,
! [A2: set_set_transition,B2: set_set_transition] :
( ! [X2: set_transition] :
( ( member7318969637299765063sition @ X2 @ A2 )
=> ( member7318969637299765063sition @ X2 @ B2 ) )
=> ( ord_le882869523442495262sition @ A2 @ B2 ) ) ).
% subsetI
thf(fact_136_subsetI,axiom,
! [A2: set_set_set_nat,B2: set_set_set_nat] :
( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ A2 )
=> ( member_set_set_nat @ X2 @ B2 ) )
=> ( ord_le9131159989063066194et_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_137_subsetI,axiom,
! [A2: set_list_o,B2: set_list_o] :
( ! [X2: list_o] :
( ( member_list_o @ X2 @ A2 )
=> ( member_list_o @ X2 @ B2 ) )
=> ( ord_le6901083488122529182list_o @ A2 @ B2 ) ) ).
% subsetI
thf(fact_138_subsetI,axiom,
! [A2: set_o,B2: set_o] :
( ! [X2: $o] :
( ( member_o @ X2 @ A2 )
=> ( member_o @ X2 @ B2 ) )
=> ( ord_less_eq_set_o @ A2 @ B2 ) ) ).
% subsetI
thf(fact_139_subsetI,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( member_set_nat @ X2 @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_140_subsetI,axiom,
! [A2: set_transition,B2: set_transition] :
( ! [X2: transition] :
( ( member_transition @ X2 @ A2 )
=> ( member_transition @ X2 @ B2 ) )
=> ( ord_le8419162016481440574sition @ A2 @ B2 ) ) ).
% subsetI
thf(fact_141_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ X2 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_142_insert__absorb2,axiom,
! [X3: list_o,A2: set_list_o] :
( ( insert_list_o2 @ X3 @ ( insert_list_o2 @ X3 @ A2 ) )
= ( insert_list_o2 @ X3 @ A2 ) ) ).
% insert_absorb2
thf(fact_143_insert__absorb2,axiom,
! [X3: set_transition,A2: set_set_transition] :
( ( insert8494249028948967790sition @ X3 @ ( insert8494249028948967790sition @ X3 @ A2 ) )
= ( insert8494249028948967790sition @ X3 @ A2 ) ) ).
% insert_absorb2
thf(fact_144_insert__absorb2,axiom,
! [X3: set_list_o,A2: set_set_list_o] :
( ( insert_set_list_o @ X3 @ ( insert_set_list_o @ X3 @ A2 ) )
= ( insert_set_list_o @ X3 @ A2 ) ) ).
% insert_absorb2
thf(fact_145_insert__absorb2,axiom,
! [X3: set_o,A2: set_set_o] :
( ( insert_set_o @ X3 @ ( insert_set_o @ X3 @ A2 ) )
= ( insert_set_o @ X3 @ A2 ) ) ).
% insert_absorb2
thf(fact_146_insert__absorb2,axiom,
! [X3: set_nat,A2: set_set_nat] :
( ( insert_set_nat @ X3 @ ( insert_set_nat @ X3 @ A2 ) )
= ( insert_set_nat @ X3 @ A2 ) ) ).
% insert_absorb2
thf(fact_147_insert__absorb2,axiom,
! [X3: nat,A2: set_nat] :
( ( insert_nat2 @ X3 @ ( insert_nat2 @ X3 @ A2 ) )
= ( insert_nat2 @ X3 @ A2 ) ) ).
% insert_absorb2
thf(fact_148_insert__iff,axiom,
! [A: set_list_o,B: set_list_o,A2: set_set_list_o] :
( ( member_set_list_o @ A @ ( insert_set_list_o @ B @ A2 ) )
= ( ( A = B )
| ( member_set_list_o @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_149_insert__iff,axiom,
! [A: set_o,B: set_o,A2: set_set_o] :
( ( member_set_o @ A @ ( insert_set_o @ B @ A2 ) )
= ( ( A = B )
| ( member_set_o @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_150_insert__iff,axiom,
! [A: list_o,B: list_o,A2: set_list_o] :
( ( member_list_o @ A @ ( insert_list_o2 @ B @ A2 ) )
= ( ( A = B )
| ( member_list_o @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_151_insert__iff,axiom,
! [A: list_nat,B: list_nat,A2: set_list_nat] :
( ( member_list_nat @ A @ ( insert_list_nat @ B @ A2 ) )
= ( ( A = B )
| ( member_list_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_152_insert__iff,axiom,
! [A: set_transition,B: set_transition,A2: set_set_transition] :
( ( member7318969637299765063sition @ A @ ( insert8494249028948967790sition @ B @ A2 ) )
= ( ( A = B )
| ( member7318969637299765063sition @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_153_insert__iff,axiom,
! [A: set_set_nat,B: set_set_nat,A2: set_set_set_nat] :
( ( member_set_set_nat @ A @ ( insert_set_set_nat @ B @ A2 ) )
= ( ( A = B )
| ( member_set_set_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_154_insert__iff,axiom,
! [A: set_nat,B: set_nat,A2: set_set_nat] :
( ( member_set_nat @ A @ ( insert_set_nat @ B @ A2 ) )
= ( ( A = B )
| ( member_set_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_155_insert__iff,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat2 @ B @ A2 ) )
= ( ( A = B )
| ( member_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_156_insert__iff,axiom,
! [A: transition,B: transition,A2: set_transition] :
( ( member_transition @ A @ ( insert_transition2 @ B @ A2 ) )
= ( ( A = B )
| ( member_transition @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_157_insertCI,axiom,
! [A: set_list_o,B2: set_set_list_o,B: set_list_o] :
( ( ~ ( member_set_list_o @ A @ B2 )
=> ( A = B ) )
=> ( member_set_list_o @ A @ ( insert_set_list_o @ B @ B2 ) ) ) ).
% insertCI
thf(fact_158_insertCI,axiom,
! [A: set_o,B2: set_set_o,B: set_o] :
( ( ~ ( member_set_o @ A @ B2 )
=> ( A = B ) )
=> ( member_set_o @ A @ ( insert_set_o @ B @ B2 ) ) ) ).
% insertCI
thf(fact_159_insertCI,axiom,
! [A: list_o,B2: set_list_o,B: list_o] :
( ( ~ ( member_list_o @ A @ B2 )
=> ( A = B ) )
=> ( member_list_o @ A @ ( insert_list_o2 @ B @ B2 ) ) ) ).
% insertCI
thf(fact_160_insertCI,axiom,
! [A: list_nat,B2: set_list_nat,B: list_nat] :
( ( ~ ( member_list_nat @ A @ B2 )
=> ( A = B ) )
=> ( member_list_nat @ A @ ( insert_list_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_161_insertCI,axiom,
! [A: set_transition,B2: set_set_transition,B: set_transition] :
( ( ~ ( member7318969637299765063sition @ A @ B2 )
=> ( A = B ) )
=> ( member7318969637299765063sition @ A @ ( insert8494249028948967790sition @ B @ B2 ) ) ) ).
% insertCI
thf(fact_162_insertCI,axiom,
! [A: set_set_nat,B2: set_set_set_nat,B: set_set_nat] :
( ( ~ ( member_set_set_nat @ A @ B2 )
=> ( A = B ) )
=> ( member_set_set_nat @ A @ ( insert_set_set_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_163_insertCI,axiom,
! [A: set_nat,B2: set_set_nat,B: set_nat] :
( ( ~ ( member_set_nat @ A @ B2 )
=> ( A = B ) )
=> ( member_set_nat @ A @ ( insert_set_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_164_insertCI,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( ~ ( member_nat @ A @ B2 )
=> ( A = B ) )
=> ( member_nat @ A @ ( insert_nat2 @ B @ B2 ) ) ) ).
% insertCI
thf(fact_165_insertCI,axiom,
! [A: transition,B2: set_transition,B: transition] :
( ( ~ ( member_transition @ A @ B2 )
=> ( A = B ) )
=> ( member_transition @ A @ ( insert_transition2 @ B @ B2 ) ) ) ).
% insertCI
thf(fact_166_Diff__idemp,axiom,
! [A2: set_transition,B2: set_transition] :
( ( minus_8944320859760356485sition @ ( minus_8944320859760356485sition @ A2 @ B2 ) @ B2 )
= ( minus_8944320859760356485sition @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_167_Diff__idemp,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) @ B2 )
= ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_168_Diff__idemp,axiom,
! [A2: set_nat,B2: set_nat] :
( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ B2 )
= ( minus_minus_set_nat @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_169_Diff__iff,axiom,
! [C: list_o,A2: set_list_o,B2: set_list_o] :
( ( member_list_o @ C @ ( minus_8912710245716896613list_o @ A2 @ B2 ) )
= ( ( member_list_o @ C @ A2 )
& ~ ( member_list_o @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_170_Diff__iff,axiom,
! [C: list_nat,A2: set_list_nat,B2: set_list_nat] :
( ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A2 @ B2 ) )
= ( ( member_list_nat @ C @ A2 )
& ~ ( member_list_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_171_Diff__iff,axiom,
! [C: set_transition,A2: set_set_transition,B2: set_set_transition] :
( ( member7318969637299765063sition @ C @ ( minus_7806277596913192037sition @ A2 @ B2 ) )
= ( ( member7318969637299765063sition @ C @ A2 )
& ~ ( member7318969637299765063sition @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_172_Diff__iff,axiom,
! [C: set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( member_set_set_nat @ C @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) )
= ( ( member_set_set_nat @ C @ A2 )
& ~ ( member_set_set_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_173_Diff__iff,axiom,
! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
= ( ( member_set_nat @ C @ A2 )
& ~ ( member_set_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_174_Diff__iff,axiom,
! [C: transition,A2: set_transition,B2: set_transition] :
( ( member_transition @ C @ ( minus_8944320859760356485sition @ A2 @ B2 ) )
= ( ( member_transition @ C @ A2 )
& ~ ( member_transition @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_175_Diff__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C @ A2 )
& ~ ( member_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_176_DiffI,axiom,
! [C: list_o,A2: set_list_o,B2: set_list_o] :
( ( member_list_o @ C @ A2 )
=> ( ~ ( member_list_o @ C @ B2 )
=> ( member_list_o @ C @ ( minus_8912710245716896613list_o @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_177_DiffI,axiom,
! [C: list_nat,A2: set_list_nat,B2: set_list_nat] :
( ( member_list_nat @ C @ A2 )
=> ( ~ ( member_list_nat @ C @ B2 )
=> ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_178_DiffI,axiom,
! [C: set_transition,A2: set_set_transition,B2: set_set_transition] :
( ( member7318969637299765063sition @ C @ A2 )
=> ( ~ ( member7318969637299765063sition @ C @ B2 )
=> ( member7318969637299765063sition @ C @ ( minus_7806277596913192037sition @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_179_DiffI,axiom,
! [C: set_set_nat,A2: set_set_set_nat,B2: set_set_set_nat] :
( ( member_set_set_nat @ C @ A2 )
=> ( ~ ( member_set_set_nat @ C @ B2 )
=> ( member_set_set_nat @ C @ ( minus_2447799839930672331et_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_180_DiffI,axiom,
! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ A2 )
=> ( ~ ( member_set_nat @ C @ B2 )
=> ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_181_DiffI,axiom,
! [C: transition,A2: set_transition,B2: set_transition] :
( ( member_transition @ C @ A2 )
=> ( ~ ( member_transition @ C @ B2 )
=> ( member_transition @ C @ ( minus_8944320859760356485sition @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_182_DiffI,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ~ ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_183_empty__subsetI,axiom,
! [A2: set_set_list_o] : ( ord_le2406146954300516094list_o @ bot_bo64454365476827594list_o @ A2 ) ).
% empty_subsetI
thf(fact_184_empty__subsetI,axiom,
! [A2: set_set_o] : ( ord_le4374716579403074808_set_o @ bot_bot_set_set_o @ A2 ) ).
% empty_subsetI
thf(fact_185_empty__subsetI,axiom,
! [A2: set_transition] : ( ord_le8419162016481440574sition @ bot_bo301567166201926666sition @ A2 ) ).
% empty_subsetI
thf(fact_186_empty__subsetI,axiom,
! [A2: set_list_o] : ( ord_le6901083488122529182list_o @ bot_bot_set_list_o @ A2 ) ).
% empty_subsetI
thf(fact_187_empty__subsetI,axiom,
! [A2: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A2 ) ).
% empty_subsetI
thf(fact_188_empty__subsetI,axiom,
! [A2: set_set_nat] : ( ord_le6893508408891458716et_nat @ bot_bot_set_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_189_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_190_Diff__cancel,axiom,
! [A2: set_set_nat] :
( ( minus_2163939370556025621et_nat @ A2 @ A2 )
= bot_bot_set_set_nat ) ).
% Diff_cancel
thf(fact_191_Diff__cancel,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ A2 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_192_empty__Diff,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_193_Diff__empty,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Diff_empty
thf(fact_194_insert__Diff1,axiom,
! [X3: transition,B2: set_transition,A2: set_transition] :
( ( member_transition @ X3 @ B2 )
=> ( ( minus_8944320859760356485sition @ ( insert_transition2 @ X3 @ A2 ) @ B2 )
= ( minus_8944320859760356485sition @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_195_insert__Diff1,axiom,
! [X3: nat,B2: set_nat,A2: set_nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X3 @ A2 ) @ B2 )
= ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_196_Diff__insert0,axiom,
! [X3: transition,A2: set_transition,B2: set_transition] :
( ~ ( member_transition @ X3 @ A2 )
=> ( ( minus_8944320859760356485sition @ A2 @ ( insert_transition2 @ X3 @ B2 ) )
= ( minus_8944320859760356485sition @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_197_Diff__insert0,axiom,
! [X3: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X3 @ A2 )
=> ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X3 @ B2 ) )
= ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_198_Diff__eq__empty__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( minus_minus_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_199_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_200_DiffD2,axiom,
! [C: transition,A2: set_transition,B2: set_transition] :
( ( member_transition @ C @ ( minus_8944320859760356485sition @ A2 @ B2 ) )
=> ~ ( member_transition @ C @ B2 ) ) ).
% DiffD2
thf(fact_201_DiffD2,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( member_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_202_DiffD1,axiom,
! [C: transition,A2: set_transition,B2: set_transition] :
( ( member_transition @ C @ ( minus_8944320859760356485sition @ A2 @ B2 ) )
=> ( member_transition @ C @ A2 ) ) ).
% DiffD1
thf(fact_203_DiffD1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_204_DiffE,axiom,
! [C: transition,A2: set_transition,B2: set_transition] :
( ( member_transition @ C @ ( minus_8944320859760356485sition @ A2 @ B2 ) )
=> ~ ( ( member_transition @ C @ A2 )
=> ( member_transition @ C @ B2 ) ) ) ).
% DiffE
thf(fact_205_DiffE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_206_double__diff,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ( minus_minus_set_nat @ B2 @ ( minus_minus_set_nat @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_207_Diff__subset,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_208_Diff__mono,axiom,
! [A2: set_nat,C2: set_nat,D: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ D @ B2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_209_insert__Diff__if,axiom,
! [X3: transition,B2: set_transition,A2: set_transition] :
( ( ( member_transition @ X3 @ B2 )
=> ( ( minus_8944320859760356485sition @ ( insert_transition2 @ X3 @ A2 ) @ B2 )
= ( minus_8944320859760356485sition @ A2 @ B2 ) ) )
& ( ~ ( member_transition @ X3 @ B2 )
=> ( ( minus_8944320859760356485sition @ ( insert_transition2 @ X3 @ A2 ) @ B2 )
= ( insert_transition2 @ X3 @ ( minus_8944320859760356485sition @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_210_insert__Diff__if,axiom,
! [X3: nat,B2: set_nat,A2: set_nat] :
( ( ( member_nat @ X3 @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X3 @ A2 ) @ B2 )
= ( minus_minus_set_nat @ A2 @ B2 ) ) )
& ( ~ ( member_nat @ X3 @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X3 @ A2 ) @ B2 )
= ( insert_nat2 @ X3 @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_211_Diff__insert__absorb,axiom,
! [X3: transition,A2: set_transition] :
( ~ ( member_transition @ X3 @ A2 )
=> ( ( minus_8944320859760356485sition @ ( insert_transition2 @ X3 @ A2 ) @ ( insert_transition2 @ X3 @ bot_bo301567166201926666sition ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_212_Diff__insert__absorb,axiom,
! [X3: nat,A2: set_nat] :
( ~ ( member_nat @ X3 @ A2 )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X3 @ A2 ) @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_213_Diff__insert2,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ B2 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_214_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_215_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_216_Diff__insert,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ B2 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_217_subset__Diff__insert,axiom,
! [A2: set_transition,B2: set_transition,X3: transition,C2: set_transition] :
( ( ord_le8419162016481440574sition @ A2 @ ( minus_8944320859760356485sition @ B2 @ ( insert_transition2 @ X3 @ C2 ) ) )
= ( ( ord_le8419162016481440574sition @ A2 @ ( minus_8944320859760356485sition @ B2 @ C2 ) )
& ~ ( member_transition @ X3 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_218_subset__Diff__insert,axiom,
! [A2: set_nat,B2: set_nat,X3: nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ ( insert_nat2 @ X3 @ C2 ) ) )
= ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ C2 ) )
& ~ ( member_nat @ X3 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_219_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_220_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_221_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_222_Diff__single__insert,axiom,
! [A2: set_nat,X3: nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) @ B2 )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ X3 @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_223_subset__insert__iff,axiom,
! [A2: set_transition,X3: transition,B2: set_transition] :
( ( ord_le8419162016481440574sition @ A2 @ ( insert_transition2 @ X3 @ B2 ) )
= ( ( ( member_transition @ X3 @ A2 )
=> ( ord_le8419162016481440574sition @ ( minus_8944320859760356485sition @ A2 @ ( insert_transition2 @ X3 @ bot_bo301567166201926666sition ) ) @ B2 ) )
& ( ~ ( member_transition @ X3 @ A2 )
=> ( ord_le8419162016481440574sition @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_224_subset__insert__iff,axiom,
! [A2: set_nat,X3: nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ X3 @ B2 ) )
= ( ( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) @ B2 ) )
& ( ~ ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_225_ex__in__conv,axiom,
! [A2: set_transition] :
( ( ? [X: transition] : ( member_transition @ X @ A2 ) )
= ( A2 != bot_bo301567166201926666sition ) ) ).
% ex_in_conv
thf(fact_226_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X: nat] : ( member_nat @ X @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_227_equals0I,axiom,
! [A2: set_transition] :
( ! [Y: transition] :
~ ( member_transition @ Y @ A2 )
=> ( A2 = bot_bo301567166201926666sition ) ) ).
% equals0I
thf(fact_228_equals0I,axiom,
! [A2: set_nat] :
( ! [Y: nat] :
~ ( member_nat @ Y @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_229_equals0D,axiom,
! [A2: set_transition,A: transition] :
( ( A2 = bot_bo301567166201926666sition )
=> ~ ( member_transition @ A @ A2 ) ) ).
% equals0D
thf(fact_230_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_231_emptyE,axiom,
! [A: transition] :
~ ( member_transition @ A @ bot_bo301567166201926666sition ) ).
% emptyE
thf(fact_232_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_233_Collect__mono__iff,axiom,
! [P: nat > $o,Q4: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q4 ) )
= ( ! [X: nat] :
( ( P @ X )
=> ( Q4 @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_234_set__eq__subset,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_235_subset__trans,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_236_Collect__mono,axiom,
! [P: nat > $o,Q4: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
=> ( Q4 @ X2 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q4 ) ) ) ).
% Collect_mono
thf(fact_237_subset__refl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_238_subset__iff,axiom,
( ord_le8419162016481440574sition
= ( ^ [A3: set_transition,B3: set_transition] :
! [T: transition] :
( ( member_transition @ T @ A3 )
=> ( member_transition @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_239_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A3 )
=> ( member_nat @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_240_equalityD2,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_241_equalityD1,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_242_subset__eq,axiom,
( ord_le8419162016481440574sition
= ( ^ [A3: set_transition,B3: set_transition] :
! [X: transition] :
( ( member_transition @ X @ A3 )
=> ( member_transition @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_243_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A3 )
=> ( member_nat @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_244_equalityE,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_245_subsetD,axiom,
! [A2: set_transition,B2: set_transition,C: transition] :
( ( ord_le8419162016481440574sition @ A2 @ B2 )
=> ( ( member_transition @ C @ A2 )
=> ( member_transition @ C @ B2 ) ) ) ).
% subsetD
thf(fact_246_subsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_247_in__mono,axiom,
! [A2: set_transition,B2: set_transition,X3: transition] :
( ( ord_le8419162016481440574sition @ A2 @ B2 )
=> ( ( member_transition @ X3 @ A2 )
=> ( member_transition @ X3 @ B2 ) ) ) ).
% in_mono
thf(fact_248_in__mono,axiom,
! [A2: set_nat,B2: set_nat,X3: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ X3 @ A2 )
=> ( member_nat @ X3 @ B2 ) ) ) ).
% in_mono
thf(fact_249_mk__disjoint__insert,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ? [B4: set_nat] :
( ( A2
= ( insert_nat2 @ A @ B4 ) )
& ~ ( member_nat @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_250_mk__disjoint__insert,axiom,
! [A: transition,A2: set_transition] :
( ( member_transition @ A @ A2 )
=> ? [B4: set_transition] :
( ( A2
= ( insert_transition2 @ A @ B4 ) )
& ~ ( member_transition @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_251_insert__commute,axiom,
! [X3: nat,Y3: nat,A2: set_nat] :
( ( insert_nat2 @ X3 @ ( insert_nat2 @ Y3 @ A2 ) )
= ( insert_nat2 @ Y3 @ ( insert_nat2 @ X3 @ A2 ) ) ) ).
% insert_commute
thf(fact_252_insert__eq__iff,axiom,
! [A: nat,A2: set_nat,B: nat,B2: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ B @ B2 )
=> ( ( ( insert_nat2 @ A @ A2 )
= ( insert_nat2 @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_nat] :
( ( A2
= ( insert_nat2 @ B @ C3 ) )
& ~ ( member_nat @ B @ C3 )
& ( B2
= ( insert_nat2 @ A @ C3 ) )
& ~ ( member_nat @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_253_insert__eq__iff,axiom,
! [A: transition,A2: set_transition,B: transition,B2: set_transition] :
( ~ ( member_transition @ A @ A2 )
=> ( ~ ( member_transition @ B @ B2 )
=> ( ( ( insert_transition2 @ A @ A2 )
= ( insert_transition2 @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_transition] :
( ( A2
= ( insert_transition2 @ B @ C3 ) )
& ~ ( member_transition @ B @ C3 )
& ( B2
= ( insert_transition2 @ A @ C3 ) )
& ~ ( member_transition @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_254_insert__absorb,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat2 @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_255_insert__absorb,axiom,
! [A: transition,A2: set_transition] :
( ( member_transition @ A @ A2 )
=> ( ( insert_transition2 @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_256_insert__ident,axiom,
! [X3: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X3 @ A2 )
=> ( ~ ( member_nat @ X3 @ B2 )
=> ( ( ( insert_nat2 @ X3 @ A2 )
= ( insert_nat2 @ X3 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_257_insert__ident,axiom,
! [X3: transition,A2: set_transition,B2: set_transition] :
( ~ ( member_transition @ X3 @ A2 )
=> ( ~ ( member_transition @ X3 @ B2 )
=> ( ( ( insert_transition2 @ X3 @ A2 )
= ( insert_transition2 @ X3 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_258_Set_Oset__insert,axiom,
! [X3: nat,A2: set_nat] :
( ( member_nat @ X3 @ A2 )
=> ~ ! [B4: set_nat] :
( ( A2
= ( insert_nat2 @ X3 @ B4 ) )
=> ( member_nat @ X3 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_259_Set_Oset__insert,axiom,
! [X3: transition,A2: set_transition] :
( ( member_transition @ X3 @ A2 )
=> ~ ! [B4: set_transition] :
( ( A2
= ( insert_transition2 @ X3 @ B4 ) )
=> ( member_transition @ X3 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_260_insertI2,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( member_nat @ A @ B2 )
=> ( member_nat @ A @ ( insert_nat2 @ B @ B2 ) ) ) ).
% insertI2
thf(fact_261_insertI2,axiom,
! [A: transition,B2: set_transition,B: transition] :
( ( member_transition @ A @ B2 )
=> ( member_transition @ A @ ( insert_transition2 @ B @ B2 ) ) ) ).
% insertI2
thf(fact_262_insertI1,axiom,
! [A: nat,B2: set_nat] : ( member_nat @ A @ ( insert_nat2 @ A @ B2 ) ) ).
% insertI1
thf(fact_263_insertI1,axiom,
! [A: transition,B2: set_transition] : ( member_transition @ A @ ( insert_transition2 @ A @ B2 ) ) ).
% insertI1
thf(fact_264_insertE,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat2 @ B @ A2 ) )
=> ( ( A != B )
=> ( member_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_265_insertE,axiom,
! [A: transition,B: transition,A2: set_transition] :
( ( member_transition @ A @ ( insert_transition2 @ B @ A2 ) )
=> ( ( A != B )
=> ( member_transition @ A @ A2 ) ) ) ).
% insertE
thf(fact_266_nfa_Odelta__qf,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,Bs: list_o] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ( ( delta @ Q0 @ Transs @ ( insert_nat2 @ Qf @ bot_bot_set_nat ) @ Bs )
= bot_bot_set_nat ) ) ).
% nfa.delta_qf
thf(fact_267_nfa_Odelta__closed,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,R: set_nat,Bs: list_o] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ( ord_less_eq_set_nat @ ( delta @ Q0 @ Transs @ R @ Bs ) @ ( q @ Q0 @ Qf @ Transs ) ) ) ).
% nfa.delta_closed
thf(fact_268_nfa_Orun__closed,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,R: set_nat,Bss: list_list_o] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ( ( ord_less_eq_set_nat @ R @ ( q @ Q0 @ Qf @ Transs ) )
=> ( ord_less_eq_set_nat @ ( run @ Q0 @ Transs @ R @ Bss ) @ ( q @ Q0 @ Qf @ Transs ) ) ) ) ).
% nfa.run_closed
thf(fact_269_nfa_Ostep__symb__set__qf,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ( ( step_symb_set @ Q0 @ Transs @ ( insert_nat2 @ Qf @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ) ).
% nfa.step_symb_set_qf
thf(fact_270_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_271_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_272_singleton__inject,axiom,
! [A: nat,B: nat] :
( ( ( insert_nat2 @ A @ bot_bot_set_nat )
= ( insert_nat2 @ B @ bot_bot_set_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_273_insert__not__empty,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat2 @ A @ A2 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_274_doubleton__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ( insert_nat2 @ A @ ( insert_nat2 @ B @ bot_bot_set_nat ) )
= ( insert_nat2 @ C @ ( insert_nat2 @ D2 @ bot_bot_set_nat ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_275_singleton__iff,axiom,
! [B: transition,A: transition] :
( ( member_transition @ B @ ( insert_transition2 @ A @ bot_bo301567166201926666sition ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_276_singleton__iff,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat2 @ A @ bot_bot_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_277_singletonD,axiom,
! [B: transition,A: transition] :
( ( member_transition @ B @ ( insert_transition2 @ A @ bot_bo301567166201926666sition ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_278_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat2 @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_279_subset__insertI2,axiom,
! [A2: set_nat,B2: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_280_subset__insertI,axiom,
! [B2: set_nat,A: nat] : ( ord_less_eq_set_nat @ B2 @ ( insert_nat2 @ A @ B2 ) ) ).
% subset_insertI
thf(fact_281_subset__insert,axiom,
! [X3: transition,A2: set_transition,B2: set_transition] :
( ~ ( member_transition @ X3 @ A2 )
=> ( ( ord_le8419162016481440574sition @ A2 @ ( insert_transition2 @ X3 @ B2 ) )
= ( ord_le8419162016481440574sition @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_282_subset__insert,axiom,
! [X3: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X3 @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ X3 @ B2 ) )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_283_insert__mono,axiom,
! [C2: set_nat,D: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ C2 @ D )
=> ( ord_less_eq_set_nat @ ( insert_nat2 @ A @ C2 ) @ ( insert_nat2 @ A @ D ) ) ) ).
% insert_mono
thf(fact_284_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_285_subset__singletonD,axiom,
! [A2: set_nat,X3: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) )
=> ( ( A2 = bot_bot_set_nat )
| ( A2
= ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_286_base_Orun__closed__Cons,axiom,
! [R: set_nat,Bs: list_o,Bss: list_list_o] : ( ord_less_eq_set_nat @ ( run @ q0a @ transsa @ R @ ( cons_list_o @ Bs @ Bss ) ) @ ( q @ q0a @ qfa @ transsa ) ) ).
% base.run_closed_Cons
thf(fact_287_base_Orun__qf__many,axiom,
! [Bs: list_o,Bss: list_list_o] :
( ( run @ q0a @ transsa @ ( insert_nat2 @ qfa @ bot_bot_set_nat ) @ ( cons_list_o @ Bs @ Bss ) )
= bot_bot_set_nat ) ).
% base.run_qf_many
thf(fact_288_base_Ostep__eps__closure__set__closed,axiom,
! [R: set_nat,Bs: list_o] :
( ( ord_less_eq_set_nat @ R @ ( q @ q0a @ qfa @ transsa ) )
=> ( ord_less_eq_set_nat @ ( step_eps_closure_set @ q0a @ transsa @ R @ Bs ) @ ( q @ q0a @ qfa @ transsa ) ) ) ).
% base.step_eps_closure_set_closed
thf(fact_289_base_Ostep__eps__closure__set__qf,axiom,
! [Bs: list_o] :
( ( step_eps_closure_set @ q0a @ transsa @ ( insert_nat2 @ qfa @ bot_bot_set_nat ) @ Bs )
= ( insert_nat2 @ qfa @ bot_bot_set_nat ) ) ).
% base.step_eps_closure_set_qf
thf(fact_290_nfa__cong__Star__axioms__def,axiom,
( nfa_cong_Star_axioms
= ( ^ [Q02: nat,Q03: nat,Qf2: nat,Transs2: list_transition] :
( ! [Bs2: list_o,Q5: nat] :
( ( step_eps @ Q02 @ Transs2 @ Bs2 @ Q02 @ Q5 )
= ( member_nat @ Q5 @ ( insert_nat2 @ Q03 @ ( insert_nat2 @ Qf2 @ bot_bot_set_nat ) ) ) )
& ! [Q5: nat] :
~ ( step_symb @ Q02 @ Transs2 @ Q02 @ Q5 ) ) ) ) ).
% nfa_cong_Star_axioms_def
thf(fact_291_nfa__cong__Plus__axioms__def,axiom,
( nfa_cong_Plus_axioms
= ( ^ [Q02: nat,Q03: nat,Q04: nat,Transs2: list_transition] :
( ! [Bs2: list_o,Q5: nat] :
( ( step_eps @ Q02 @ Transs2 @ Bs2 @ Q02 @ Q5 )
= ( member_nat @ Q5 @ ( insert_nat2 @ Q03 @ ( insert_nat2 @ Q04 @ bot_bot_set_nat ) ) ) )
& ! [Q5: nat] :
~ ( step_symb @ Q02 @ Transs2 @ Q02 @ Q5 ) ) ) ) ).
% nfa_cong_Plus_axioms_def
thf(fact_292_nfa__cong__Star__axioms_Ointro,axiom,
! [Q0: nat,Transs: list_transition,Q05: nat,Qf: nat] :
( ! [Bs3: list_o,Q6: nat] :
( ( step_eps @ Q0 @ Transs @ Bs3 @ Q0 @ Q6 )
= ( member_nat @ Q6 @ ( insert_nat2 @ Q05 @ ( insert_nat2 @ Qf @ bot_bot_set_nat ) ) ) )
=> ( ! [Q6: nat] :
~ ( step_symb @ Q0 @ Transs @ Q0 @ Q6 )
=> ( nfa_cong_Star_axioms @ Q0 @ Q05 @ Qf @ Transs ) ) ) ).
% nfa_cong_Star_axioms.intro
thf(fact_293_nfa__cong__Plus__axioms_Ointro,axiom,
! [Q0: nat,Transs: list_transition,Q05: nat,Q06: nat] :
( ! [Bs3: list_o,Q6: nat] :
( ( step_eps @ Q0 @ Transs @ Bs3 @ Q0 @ Q6 )
= ( member_nat @ Q6 @ ( insert_nat2 @ Q05 @ ( insert_nat2 @ Q06 @ bot_bot_set_nat ) ) ) )
=> ( ! [Q6: nat] :
~ ( step_symb @ Q0 @ Transs @ Q0 @ Q6 )
=> ( nfa_cong_Plus_axioms @ Q0 @ Q05 @ Q06 @ Transs ) ) ) ).
% nfa_cong_Plus_axioms.intro
thf(fact_294_diff__shunt__var,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( ( minus_minus_set_nat @ X3 @ Y3 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X3 @ Y3 ) ) ).
% diff_shunt_var
thf(fact_295_the__elem__eq,axiom,
! [X3: nat] :
( ( the_elem_nat @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) )
= X3 ) ).
% the_elem_eq
thf(fact_296_order__refl,axiom,
! [X3: set_nat] : ( ord_less_eq_set_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_297_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_298_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_299_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_300_list__split_Ocases,axiom,
! [X3: list_transition] :
( ( X3 != nil_transition )
=> ~ ! [X2: transition,Xs: list_transition] :
( X3
!= ( cons_transition @ X2 @ Xs ) ) ) ).
% list_split.cases
thf(fact_301_list__split_Ocases,axiom,
! [X3: list_o] :
( ( X3 != nil_o )
=> ~ ! [X2: $o,Xs: list_o] :
( X3
!= ( cons_o @ X2 @ Xs ) ) ) ).
% list_split.cases
thf(fact_302_list__split_Ocases,axiom,
! [X3: list_list_o] :
( ( X3 != nil_list_o )
=> ~ ! [X2: list_o,Xs: list_list_o] :
( X3
!= ( cons_list_o @ X2 @ Xs ) ) ) ).
% list_split.cases
thf(fact_303_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_304_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_305_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_306_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_307_step__eps__closure__set__step__id,axiom,
! [R: set_nat,Q0: nat,Transs: list_transition,Bs: list_o] :
( ! [Q6: nat,Q3: nat] :
( ( member_nat @ Q6 @ R )
=> ~ ( step_eps @ Q0 @ Transs @ Bs @ Q6 @ Q3 ) )
=> ( ( step_eps_closure_set @ Q0 @ Transs @ R @ Bs )
= R ) ) ).
% step_eps_closure_set_step_id
thf(fact_308_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_309_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_310_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_311_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_312_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
& ( ord_less_eq_set_nat @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_313_ord__eq__le__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( A = B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_314_ord__le__eq__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_315_order__antisym,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_set_nat @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% order_antisym
thf(fact_316_order_Otrans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_317_order__trans,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_set_nat @ Y3 @ Z2 )
=> ( ord_less_eq_set_nat @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_318_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
& ( ord_less_eq_set_nat @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_319_dual__order_Oantisym,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_320_dual__order_Otrans,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_321_antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_322_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_323_order__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_324_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_325_order__eq__refl,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( X3 = Y3 )
=> ( ord_less_eq_set_nat @ X3 @ Y3 ) ) ).
% order_eq_refl
thf(fact_326_ord__eq__le__subst,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_327_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_328_order__antisym__conv,axiom,
! [Y3: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X3 )
=> ( ( ord_less_eq_set_nat @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_329_nfa_Ostep__eps__closure__set__qf,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,Bs: list_o] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ( ( step_eps_closure_set @ Q0 @ Transs @ ( insert_nat2 @ Qf @ bot_bot_set_nat ) @ Bs )
= ( insert_nat2 @ Qf @ bot_bot_set_nat ) ) ) ).
% nfa.step_eps_closure_set_qf
thf(fact_330_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_331_nfa_Orun__qf__many,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,Bs: list_o,Bss: list_list_o] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ( ( run @ Q0 @ Transs @ ( insert_nat2 @ Qf @ bot_bot_set_nat ) @ ( cons_list_o @ Bs @ Bss ) )
= bot_bot_set_nat ) ) ).
% nfa.run_qf_many
thf(fact_332_nfa_Orun__closed__Cons,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,R: set_nat,Bs: list_o,Bss: list_list_o] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ( ord_less_eq_set_nat @ ( run @ Q0 @ Transs @ R @ ( cons_list_o @ Bs @ Bss ) ) @ ( q @ Q0 @ Qf @ Transs ) ) ) ).
% nfa.run_closed_Cons
thf(fact_333_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_334_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_335_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_336_base_Ostep__eps__closure__set__closed__union,axiom,
! [R: set_nat,Bs: list_o] : ( ord_less_eq_set_nat @ ( step_eps_closure_set @ q0a @ transsa @ R @ Bs ) @ ( sup_sup_set_nat @ R @ ( q @ q0a @ qfa @ transsa ) ) ) ).
% base.step_eps_closure_set_closed_union
thf(fact_337_base_Orun__accept__eps__qf__many,axiom,
! [Bs: list_o,Bss: list_list_o,Cs: list_o] :
~ ( run_accept_eps @ q0a @ qfa @ transsa @ ( insert_nat2 @ qfa @ bot_bot_set_nat ) @ ( cons_list_o @ Bs @ Bss ) @ Cs ) ).
% base.run_accept_eps_qf_many
thf(fact_338_transpose_Ocases,axiom,
! [X3: list_list_transition] :
( ( X3 != nil_list_transition )
=> ( ! [Xss: list_list_transition] :
( X3
!= ( cons_list_transition @ nil_transition @ Xss ) )
=> ~ ! [X2: transition,Xs: list_transition,Xss: list_list_transition] :
( X3
!= ( cons_list_transition @ ( cons_transition @ X2 @ Xs ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_339_transpose_Ocases,axiom,
! [X3: list_list_list_o] :
( ( X3 != nil_list_list_o )
=> ( ! [Xss: list_list_list_o] :
( X3
!= ( cons_list_list_o @ nil_list_o @ Xss ) )
=> ~ ! [X2: list_o,Xs: list_list_o,Xss: list_list_list_o] :
( X3
!= ( cons_list_list_o @ ( cons_list_o @ X2 @ Xs ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_340_transpose_Ocases,axiom,
! [X3: list_list_o] :
( ( X3 != nil_list_o )
=> ( ! [Xss: list_list_o] :
( X3
!= ( cons_list_o @ nil_o @ Xss ) )
=> ~ ! [X2: $o,Xs: list_o,Xss: list_list_o] :
( X3
!= ( cons_list_o @ ( cons_o @ X2 @ Xs ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_341_list_Oinject,axiom,
! [X21: list_o,X22: list_list_o,Y21: list_o,Y22: list_list_o] :
( ( ( cons_list_o @ X21 @ X22 )
= ( cons_list_o @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_342_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
( A3
= ( insert_nat2 @ ( the_elem_nat @ A3 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_343_is__singletonI,axiom,
! [X3: nat] : ( is_singleton_nat @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_344_nfa__cong__Star_Odelta__sub__nfa_H__delta,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Transs: list_transition,Transs3: list_transition,Bs: list_o] :
( ( nfa_cong_Star @ Q0 @ Q05 @ Qf @ Transs @ Transs3 )
=> ( ord_less_eq_set_nat @ ( delta @ Q0 @ Transs @ ( insert_nat2 @ Q0 @ bot_bot_set_nat ) @ Bs ) @ ( delta @ Q05 @ Transs3 @ ( insert_nat2 @ Q05 @ bot_bot_set_nat ) @ Bs ) ) ) ).
% nfa_cong_Star.delta_sub_nfa'_delta
thf(fact_345_base_Ostate__closed,axiom,
! [T2: transition] :
( ( member_transition @ T2 @ ( set_transition2 @ transsa ) )
=> ( ord_less_eq_set_nat @ ( state_set @ T2 ) @ ( q @ q0a @ qfa @ transsa ) ) ) ).
% base.state_closed
thf(fact_346_sup_Oright__idem,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ B )
= ( sup_sup_set_nat @ A @ B ) ) ).
% sup.right_idem
thf(fact_347_sup__left__idem,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y3 ) )
= ( sup_sup_set_nat @ X3 @ Y3 ) ) ).
% sup_left_idem
thf(fact_348_sup_Oleft__idem,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ A @ B ) ) ).
% sup.left_idem
thf(fact_349_sup__idem,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_350_sup_Oidem,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_351_UnCI,axiom,
! [C: transition,B2: set_transition,A2: set_transition] :
( ( ~ ( member_transition @ C @ B2 )
=> ( member_transition @ C @ A2 ) )
=> ( member_transition @ C @ ( sup_su812053455038985074sition @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_352_UnCI,axiom,
! [C: nat,B2: set_nat,A2: set_nat] :
( ( ~ ( member_nat @ C @ B2 )
=> ( member_nat @ C @ A2 ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_353_Un__iff,axiom,
! [C: transition,A2: set_transition,B2: set_transition] :
( ( member_transition @ C @ ( sup_su812053455038985074sition @ A2 @ B2 ) )
= ( ( member_transition @ C @ A2 )
| ( member_transition @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_354_Un__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C @ A2 )
| ( member_nat @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_355_base_Orun__accept__eps__qf__one,axiom,
! [Bs: list_o] : ( run_accept_eps @ q0a @ qfa @ transsa @ ( insert_nat2 @ qfa @ bot_bot_set_nat ) @ nil_list_o @ Bs ) ).
% base.run_accept_eps_qf_one
thf(fact_356_sup_Obounded__iff,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
= ( ( ord_less_eq_set_nat @ B @ A )
& ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_357_le__sup__iff,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X3 @ Y3 ) @ Z2 )
= ( ( ord_less_eq_set_nat @ X3 @ Z2 )
& ( ord_less_eq_set_nat @ Y3 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_358_sup__bot_Oright__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_359_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A @ B ) )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_360_sup__bot_Oleft__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_361_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_362_sup__eq__bot__iff,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( ( sup_sup_set_nat @ X3 @ Y3 )
= bot_bot_set_nat )
= ( ( X3 = bot_bot_set_nat )
& ( Y3 = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_363_bot__eq__sup__iff,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X3 @ Y3 ) )
= ( ( X3 = bot_bot_set_nat )
& ( Y3 = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_364_sup__bot__right,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ X3 @ bot_bot_set_nat )
= X3 ) ).
% sup_bot_right
thf(fact_365_sup__bot__left,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_366_Un__empty,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_367_Un__subset__iff,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C2 )
= ( ( ord_less_eq_set_nat @ A2 @ C2 )
& ( ord_less_eq_set_nat @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_368_Un__insert__right,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( insert_nat2 @ A @ B2 ) )
= ( insert_nat2 @ A @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_369_Un__insert__left,axiom,
! [A: nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( insert_nat2 @ A @ B2 ) @ C2 )
= ( insert_nat2 @ A @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_370_Un__Diff__cancel2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B2 @ A2 ) @ A2 )
= ( sup_sup_set_nat @ B2 @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_371_Un__Diff__cancel,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ A2 ) )
= ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_372_set__empty2,axiom,
! [Xs2: list_list_o] :
( ( bot_bot_set_list_o
= ( set_list_o2 @ Xs2 ) )
= ( Xs2 = nil_list_o ) ) ).
% set_empty2
thf(fact_373_set__empty2,axiom,
! [Xs2: list_o] :
( ( bot_bot_set_o
= ( set_o2 @ Xs2 ) )
= ( Xs2 = nil_o ) ) ).
% set_empty2
thf(fact_374_set__empty2,axiom,
! [Xs2: list_transition] :
( ( bot_bo301567166201926666sition
= ( set_transition2 @ Xs2 ) )
= ( Xs2 = nil_transition ) ) ).
% set_empty2
thf(fact_375_set__empty2,axiom,
! [Xs2: list_nat] :
( ( bot_bot_set_nat
= ( set_nat2 @ Xs2 ) )
= ( Xs2 = nil_nat ) ) ).
% set_empty2
thf(fact_376_set__empty,axiom,
! [Xs2: list_list_o] :
( ( ( set_list_o2 @ Xs2 )
= bot_bot_set_list_o )
= ( Xs2 = nil_list_o ) ) ).
% set_empty
thf(fact_377_set__empty,axiom,
! [Xs2: list_o] :
( ( ( set_o2 @ Xs2 )
= bot_bot_set_o )
= ( Xs2 = nil_o ) ) ).
% set_empty
thf(fact_378_set__empty,axiom,
! [Xs2: list_transition] :
( ( ( set_transition2 @ Xs2 )
= bot_bo301567166201926666sition )
= ( Xs2 = nil_transition ) ) ).
% set_empty
thf(fact_379_set__empty,axiom,
! [Xs2: list_nat] :
( ( ( set_nat2 @ Xs2 )
= bot_bot_set_nat )
= ( Xs2 = nil_nat ) ) ).
% set_empty
thf(fact_380_list_Osimps_I15_J,axiom,
! [X21: nat,X22: list_nat] :
( ( set_nat2 @ ( cons_nat @ X21 @ X22 ) )
= ( insert_nat2 @ X21 @ ( set_nat2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_381_list_Osimps_I15_J,axiom,
! [X21: transition,X22: list_transition] :
( ( set_transition2 @ ( cons_transition @ X21 @ X22 ) )
= ( insert_transition2 @ X21 @ ( set_transition2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_382_list_Osimps_I15_J,axiom,
! [X21: list_o,X22: list_list_o] :
( ( set_list_o2 @ ( cons_list_o @ X21 @ X22 ) )
= ( insert_list_o2 @ X21 @ ( set_list_o2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_383_list_Oset__intros_I2_J,axiom,
! [Y3: nat,X22: list_nat,X21: nat] :
( ( member_nat @ Y3 @ ( set_nat2 @ X22 ) )
=> ( member_nat @ Y3 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_384_list_Oset__intros_I2_J,axiom,
! [Y3: transition,X22: list_transition,X21: transition] :
( ( member_transition @ Y3 @ ( set_transition2 @ X22 ) )
=> ( member_transition @ Y3 @ ( set_transition2 @ ( cons_transition @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_385_list_Oset__intros_I2_J,axiom,
! [Y3: list_o,X22: list_list_o,X21: list_o] :
( ( member_list_o @ Y3 @ ( set_list_o2 @ X22 ) )
=> ( member_list_o @ Y3 @ ( set_list_o2 @ ( cons_list_o @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_386_list_Oset__intros_I1_J,axiom,
! [X21: nat,X22: list_nat] : ( member_nat @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_387_list_Oset__intros_I1_J,axiom,
! [X21: transition,X22: list_transition] : ( member_transition @ X21 @ ( set_transition2 @ ( cons_transition @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_388_list_Oset__intros_I1_J,axiom,
! [X21: list_o,X22: list_list_o] : ( member_list_o @ X21 @ ( set_list_o2 @ ( cons_list_o @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_389_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_390_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_391_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_392_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_nat,K: set_nat,A: set_nat,B: set_nat] :
( ( A2
= ( sup_sup_set_nat @ K @ A ) )
=> ( ( sup_sup_set_nat @ A2 @ B )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_393_boolean__algebra__cancel_Osup2,axiom,
! [B2: set_nat,K: set_nat,B: set_nat,A: set_nat] :
( ( B2
= ( sup_sup_set_nat @ K @ B ) )
=> ( ( sup_sup_set_nat @ A @ B2 )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_394_set__ConsD,axiom,
! [Y3: nat,X3: nat,Xs2: list_nat] :
( ( member_nat @ Y3 @ ( set_nat2 @ ( cons_nat @ X3 @ Xs2 ) ) )
=> ( ( Y3 = X3 )
| ( member_nat @ Y3 @ ( set_nat2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_395_set__ConsD,axiom,
! [Y3: transition,X3: transition,Xs2: list_transition] :
( ( member_transition @ Y3 @ ( set_transition2 @ ( cons_transition @ X3 @ Xs2 ) ) )
=> ( ( Y3 = X3 )
| ( member_transition @ Y3 @ ( set_transition2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_396_set__ConsD,axiom,
! [Y3: list_o,X3: list_o,Xs2: list_list_o] :
( ( member_list_o @ Y3 @ ( set_list_o2 @ ( cons_list_o @ X3 @ Xs2 ) ) )
=> ( ( Y3 = X3 )
| ( member_list_o @ Y3 @ ( set_list_o2 @ Xs2 ) ) ) ) ).
% set_ConsD
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_inf__sup__ord_I4_J,axiom,
! [Y3: set_nat,X3: set_nat] : ( ord_less_eq_set_nat @ Y3 @ ( sup_sup_set_nat @ X3 @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_399_inf__sup__ord_I3_J,axiom,
! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_400_le__supE,axiom,
! [A: set_nat,B: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ X3 )
=> ~ ( ( ord_less_eq_set_nat @ A @ X3 )
=> ~ ( ord_less_eq_set_nat @ B @ X3 ) ) ) ).
% le_supE
thf(fact_401_le__supI,axiom,
! [A: set_nat,X3: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ X3 )
=> ( ( ord_less_eq_set_nat @ B @ X3 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ X3 ) ) ) ).
% le_supI
thf(fact_402_sup__ge1,axiom,
! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y3 ) ) ).
% sup_ge1
thf(fact_403_sup__ge2,axiom,
! [Y3: set_nat,X3: set_nat] : ( ord_less_eq_set_nat @ Y3 @ ( sup_sup_set_nat @ X3 @ Y3 ) ) ).
% sup_ge2
thf(fact_404_le__supI1,axiom,
! [X3: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_405_le__supI2,axiom,
! [X3: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ B )
=> ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_406_sup_Omono,axiom,
! [C: set_nat,A: set_nat,D2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ A )
=> ( ( ord_less_eq_set_nat @ D2 @ B )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C @ D2 ) @ ( sup_sup_set_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_407_sup__mono,axiom,
! [A: set_nat,C: set_nat,B: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B @ D2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_408_sup__least,axiom,
! [Y3: set_nat,X3: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X3 )
=> ( ( ord_less_eq_set_nat @ Z2 @ X3 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y3 @ Z2 ) @ X3 ) ) ) ).
% sup_least
thf(fact_409_le__iff__sup,axiom,
( ord_less_eq_set_nat
= ( ^ [X: set_nat,Y4: set_nat] :
( ( sup_sup_set_nat @ X @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_410_sup_OorderE,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( A
= ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_411_sup_OorderI,axiom,
! [A: set_nat,B: set_nat] :
( ( A
= ( sup_sup_set_nat @ A @ B ) )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_412_sup__unique,axiom,
! [F: set_nat > set_nat > set_nat,X3: set_nat,Y3: set_nat] :
( ! [X2: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( F @ X2 @ Y ) )
=> ( ! [X2: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ Y @ ( F @ X2 @ Y ) )
=> ( ! [X2: set_nat,Y: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X2 )
=> ( ( ord_less_eq_set_nat @ Z3 @ X2 )
=> ( ord_less_eq_set_nat @ ( F @ Y @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup_set_nat @ X3 @ Y3 )
= ( F @ X3 @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_413_sup_Oabsorb1,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( sup_sup_set_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_414_sup_Oabsorb2,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_415_sup__absorb1,axiom,
! [Y3: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X3 )
=> ( ( sup_sup_set_nat @ X3 @ Y3 )
= X3 ) ) ).
% sup_absorb1
thf(fact_416_sup__absorb2,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ( sup_sup_set_nat @ X3 @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_417_sup_OboundedE,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_set_nat @ B @ A )
=> ~ ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_418_sup_OboundedI,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ A )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_419_sup_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [B5: set_nat,A4: set_nat] :
( A4
= ( sup_sup_set_nat @ A4 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_420_sup_Ocobounded1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_421_sup_Ocobounded2,axiom,
! [B: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_422_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [B5: set_nat,A4: set_nat] :
( ( sup_sup_set_nat @ A4 @ B5 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_423_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
( ( sup_sup_set_nat @ A4 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_424_sup_OcoboundedI1,axiom,
! [C: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ A )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_425_sup_OcoboundedI2,axiom,
! [C: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_426_subset__code_I1_J,axiom,
! [Xs2: list_transition,B2: set_transition] :
( ( ord_le8419162016481440574sition @ ( set_transition2 @ Xs2 ) @ B2 )
= ( ! [X: transition] :
( ( member_transition @ X @ ( set_transition2 @ Xs2 ) )
=> ( member_transition @ X @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_427_subset__code_I1_J,axiom,
! [Xs2: list_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ B2 )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( member_nat @ X @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_428_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_429_sup__left__commute,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ Y3 @ Z2 ) )
= ( sup_sup_set_nat @ Y3 @ ( sup_sup_set_nat @ X3 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_430_sup_Oleft__commute,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A @ C ) )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_431_sup__commute,axiom,
( sup_sup_set_nat
= ( ^ [X: set_nat,Y4: set_nat] : ( sup_sup_set_nat @ Y4 @ X ) ) ) ).
% sup_commute
thf(fact_432_sup_Ocommute,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B5: set_nat] : ( sup_sup_set_nat @ B5 @ A4 ) ) ) ).
% sup.commute
thf(fact_433_sup__assoc,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X3 @ Y3 ) @ Z2 )
= ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ Y3 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_434_sup_Oassoc,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C ) ) ) ).
% sup.assoc
thf(fact_435_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat
= ( ^ [X: set_nat,Y4: set_nat] : ( sup_sup_set_nat @ Y4 @ X ) ) ) ).
% inf_sup_aci(5)
thf(fact_436_inf__sup__aci_I6_J,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X3 @ Y3 ) @ Z2 )
= ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ Y3 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_437_inf__sup__aci_I7_J,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ Y3 @ Z2 ) )
= ( sup_sup_set_nat @ Y3 @ ( sup_sup_set_nat @ X3 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_438_inf__sup__aci_I8_J,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y3 ) )
= ( sup_sup_set_nat @ X3 @ Y3 ) ) ).
% inf_sup_aci(8)
thf(fact_439_UnE,axiom,
! [C: transition,A2: set_transition,B2: set_transition] :
( ( member_transition @ C @ ( sup_su812053455038985074sition @ A2 @ B2 ) )
=> ( ~ ( member_transition @ C @ A2 )
=> ( member_transition @ C @ B2 ) ) ) ).
% UnE
thf(fact_440_UnE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) )
=> ( ~ ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% UnE
thf(fact_441_UnI1,axiom,
! [C: transition,A2: set_transition,B2: set_transition] :
( ( member_transition @ C @ A2 )
=> ( member_transition @ C @ ( sup_su812053455038985074sition @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_442_UnI1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_443_UnI2,axiom,
! [C: transition,B2: set_transition,A2: set_transition] :
( ( member_transition @ C @ B2 )
=> ( member_transition @ C @ ( sup_su812053455038985074sition @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_444_UnI2,axiom,
! [C: nat,B2: set_nat,A2: set_nat] :
( ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_445_bex__Un,axiom,
! [A2: set_nat,B2: set_nat,P: nat > $o] :
( ( ? [X: nat] :
( ( member_nat @ X @ ( sup_sup_set_nat @ A2 @ B2 ) )
& ( P @ X ) ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ X ) )
| ? [X: nat] :
( ( member_nat @ X @ B2 )
& ( P @ X ) ) ) ) ).
% bex_Un
thf(fact_446_ball__Un,axiom,
! [A2: set_nat,B2: set_nat,P: nat > $o] :
( ( ! [X: nat] :
( ( member_nat @ X @ ( sup_sup_set_nat @ A2 @ B2 ) )
=> ( P @ X ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( P @ X ) )
& ! [X: nat] :
( ( member_nat @ X @ B2 )
=> ( P @ X ) ) ) ) ).
% ball_Un
thf(fact_447_Un__assoc,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C2 )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% Un_assoc
thf(fact_448_Un__absorb,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_449_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B3: set_nat] : ( sup_sup_set_nat @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_450_Un__left__absorb,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% Un_left_absorb
thf(fact_451_Un__left__commute,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) )
= ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_452_nfa__cong__Star_Orun__accept__eps__Nil,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Transs: list_transition,Transs3: list_transition,Bs: list_o] :
( ( nfa_cong_Star @ Q0 @ Q05 @ Qf @ Transs @ Transs3 )
=> ( run_accept_eps @ Q0 @ Qf @ Transs @ ( insert_nat2 @ Q0 @ bot_bot_set_nat ) @ nil_list_o @ Bs ) ) ).
% nfa_cong_Star.run_accept_eps_Nil
thf(fact_453_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_454_boolean__algebra_Odisj__zero__right,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ X3 @ bot_bot_set_nat )
= X3 ) ).
% boolean_algebra.disj_zero_right
thf(fact_455_Un__empty__right,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Un_empty_right
thf(fact_456_Un__empty__left,axiom,
! [B2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_457_subset__Un__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( sup_sup_set_nat @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_458_subset__UnE,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
=> ~ ! [A5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ A2 )
=> ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ B2 )
=> ( C2
!= ( sup_sup_set_nat @ A5 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_459_Un__absorb2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_460_Un__absorb1,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_461_Un__upper2,axiom,
! [B2: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_462_Un__upper1,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_463_Un__least,axiom,
! [A2: set_nat,C2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_464_Un__mono,axiom,
! [A2: set_nat,C2: set_nat,B2: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ D )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ ( sup_sup_set_nat @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_465_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_466_Un__Diff,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C2 )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A2 @ C2 ) @ ( minus_minus_set_nat @ B2 @ C2 ) ) ) ).
% Un_Diff
thf(fact_467_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_468_empty__set,axiom,
( bot_bot_set_list_o
= ( set_list_o2 @ nil_list_o ) ) ).
% empty_set
thf(fact_469_empty__set,axiom,
( bot_bot_set_o
= ( set_o2 @ nil_o ) ) ).
% empty_set
thf(fact_470_empty__set,axiom,
( bot_bo301567166201926666sition
= ( set_transition2 @ nil_transition ) ) ).
% empty_set
thf(fact_471_empty__set,axiom,
( bot_bot_set_nat
= ( set_nat2 @ nil_nat ) ) ).
% empty_set
thf(fact_472_set__subset__Cons,axiom,
! [Xs2: list_transition,X3: transition] : ( ord_le8419162016481440574sition @ ( set_transition2 @ Xs2 ) @ ( set_transition2 @ ( cons_transition @ X3 @ Xs2 ) ) ) ).
% set_subset_Cons
thf(fact_473_set__subset__Cons,axiom,
! [Xs2: list_list_o,X3: list_o] : ( ord_le6901083488122529182list_o @ ( set_list_o2 @ Xs2 ) @ ( set_list_o2 @ ( cons_list_o @ X3 @ Xs2 ) ) ) ).
% set_subset_Cons
thf(fact_474_set__subset__Cons,axiom,
! [Xs2: list_nat,X3: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ ( set_nat2 @ ( cons_nat @ X3 @ Xs2 ) ) ) ).
% set_subset_Cons
thf(fact_475_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_476_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_477_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_478_run__Nil,axiom,
! [Q0: nat,Transs: list_transition,R: set_nat] :
( ( run @ Q0 @ Transs @ R @ nil_list_o )
= R ) ).
% run_Nil
thf(fact_479_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_480_nfa_Orun__accept__eps__qf__one,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,Bs: list_o] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ( run_accept_eps @ Q0 @ Qf @ Transs @ ( insert_nat2 @ Qf @ bot_bot_set_nat ) @ nil_list_o @ Bs ) ) ).
% nfa.run_accept_eps_qf_one
thf(fact_481_nfa__cong__Star_Ostep__eps__closure__set__q0__split,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Transs: list_transition,Transs3: list_transition,Bs: list_o] :
( ( nfa_cong_Star @ Q0 @ Q05 @ Qf @ Transs @ Transs3 )
=> ( ( step_eps_closure_set @ Q0 @ Transs @ ( insert_nat2 @ Q0 @ bot_bot_set_nat ) @ Bs )
= ( sup_sup_set_nat @ ( insert_nat2 @ Q0 @ ( insert_nat2 @ Qf @ bot_bot_set_nat ) ) @ ( step_eps_closure_set @ Q0 @ Transs @ ( insert_nat2 @ Q05 @ bot_bot_set_nat ) @ Bs ) ) ) ) ).
% nfa_cong_Star.step_eps_closure_set_q0_split
thf(fact_482_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_483_singleton__Un__iff,axiom,
! [X3: nat,A2: set_nat,B2: set_nat] :
( ( ( insert_nat2 @ X3 @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( ( A2 = bot_bot_set_nat )
& ( B2
= ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) )
| ( ( A2
= ( insert_nat2 @ X3 @ bot_bot_set_nat ) )
& ( B2 = bot_bot_set_nat ) )
| ( ( A2
= ( insert_nat2 @ X3 @ bot_bot_set_nat ) )
& ( B2
= ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_484_Un__singleton__iff,axiom,
! [A2: set_nat,B2: set_nat,X3: nat] :
( ( ( sup_sup_set_nat @ A2 @ B2 )
= ( insert_nat2 @ X3 @ bot_bot_set_nat ) )
= ( ( ( A2 = bot_bot_set_nat )
& ( B2
= ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) )
| ( ( A2
= ( insert_nat2 @ X3 @ bot_bot_set_nat ) )
& ( B2 = bot_bot_set_nat ) )
| ( ( A2
= ( insert_nat2 @ X3 @ bot_bot_set_nat ) )
& ( B2
= ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_485_insert__is__Un,axiom,
( insert_nat2
= ( ^ [A4: nat] : ( sup_sup_set_nat @ ( insert_nat2 @ A4 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_486_Diff__partition,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( sup_sup_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ A2 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_487_Diff__subset__conv,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ C2 )
= ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_488_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_489_nfa__cong__Star_Ostep__symb__q0,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Transs: list_transition,Transs3: list_transition,Q: nat] :
( ( nfa_cong_Star @ Q0 @ Q05 @ Qf @ Transs @ Transs3 )
=> ~ ( step_symb @ Q0 @ Transs @ Q0 @ Q ) ) ).
% nfa_cong_Star.step_symb_q0
thf(fact_490_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_491_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_492_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_493_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_494_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_495_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_496_the__elem__set,axiom,
! [X3: $o] :
( ( the_elem_o @ ( set_o2 @ ( cons_o @ X3 @ nil_o ) ) )
= X3 ) ).
% the_elem_set
thf(fact_497_the__elem__set,axiom,
! [X3: transition] :
( ( the_elem_transition @ ( set_transition2 @ ( cons_transition @ X3 @ nil_transition ) ) )
= X3 ) ).
% the_elem_set
thf(fact_498_the__elem__set,axiom,
! [X3: list_o] :
( ( the_elem_list_o @ ( set_list_o2 @ ( cons_list_o @ X3 @ nil_list_o ) ) )
= X3 ) ).
% the_elem_set
thf(fact_499_is__singletonI_H,axiom,
! [A2: set_transition] :
( ( A2 != bot_bo301567166201926666sition )
=> ( ! [X2: transition,Y: transition] :
( ( member_transition @ X2 @ A2 )
=> ( ( member_transition @ Y @ A2 )
=> ( X2 = Y ) ) )
=> ( is_sin1641930644073461938sition @ A2 ) ) ) ).
% is_singletonI'
thf(fact_500_is__singletonI_H,axiom,
! [A2: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X2: nat,Y: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( X2 = Y ) ) )
=> ( is_singleton_nat @ A2 ) ) ) ).
% is_singletonI'
thf(fact_501_nfa__cong__Star_Oaxioms_I2_J,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Transs: list_transition,Transs3: list_transition] :
( ( nfa_cong_Star @ Q0 @ Q05 @ Qf @ Transs @ Transs3 )
=> ( nfa_cong_Star_axioms @ Q0 @ Q05 @ Qf @ Transs ) ) ).
% nfa_cong_Star.axioms(2)
thf(fact_502_not__Cons__self2,axiom,
! [X3: list_o,Xs2: list_list_o] :
( ( cons_list_o @ X3 @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_503_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_504_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_505_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_506_nfa_Orun__accept__eps__qf__many,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,Bs: list_o,Bss: list_list_o,Cs: list_o] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ~ ( run_accept_eps @ Q0 @ Qf @ Transs @ ( insert_nat2 @ Qf @ bot_bot_set_nat ) @ ( cons_list_o @ Bs @ Bss ) @ Cs ) ) ).
% nfa.run_accept_eps_qf_many
thf(fact_507_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_508_step__eps__closure__set__unfold,axiom,
! [Q0: nat,Transs: list_transition,Bs: list_o,Q: nat,X4: set_nat] :
( ! [Q3: nat] :
( ( step_eps @ Q0 @ Transs @ Bs @ Q @ Q3 )
= ( member_nat @ Q3 @ 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_509_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_510_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
? [X: nat] :
( A3
= ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_511_is__singletonE,axiom,
! [A2: set_nat] :
( ( is_singleton_nat @ A2 )
=> ~ ! [X2: nat] :
( A2
!= ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_512_nfa__cong__Star_Odelta__q0__q0_H,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Transs: list_transition,Transs3: list_transition,Bs: list_o] :
( ( nfa_cong_Star @ Q0 @ Q05 @ Qf @ Transs @ Transs3 )
=> ( ( delta @ Q0 @ Transs @ ( insert_nat2 @ Q0 @ bot_bot_set_nat ) @ Bs )
= ( delta @ Q0 @ Transs @ ( insert_nat2 @ Q05 @ bot_bot_set_nat ) @ Bs ) ) ) ).
% nfa_cong_Star.delta_q0_q0'
thf(fact_513_nfa__cong__Star_Ostep__symb__set__q0,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Transs: list_transition,Transs3: list_transition] :
( ( nfa_cong_Star @ Q0 @ Q05 @ Qf @ Transs @ Transs3 )
=> ( ( step_symb_set @ Q0 @ Transs @ ( insert_nat2 @ Q0 @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ) ).
% nfa_cong_Star.step_symb_set_q0
thf(fact_514_nfa__cong__Star_Ostep__eps__q0,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Transs: list_transition,Transs3: list_transition,Bs: list_o,Q: nat] :
( ( nfa_cong_Star @ Q0 @ Q05 @ Qf @ Transs @ Transs3 )
=> ( ( step_eps @ Q0 @ Transs @ Bs @ Q0 @ Q )
= ( member_nat @ Q @ ( insert_nat2 @ Q05 @ ( insert_nat2 @ Qf @ bot_bot_set_nat ) ) ) ) ) ).
% nfa_cong_Star.step_eps_q0
thf(fact_515_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_516_list__nonempty__induct,axiom,
! [Xs2: list_transition,P: list_transition > $o] :
( ( Xs2 != nil_transition )
=> ( ! [X2: transition] : ( P @ ( cons_transition @ X2 @ nil_transition ) )
=> ( ! [X2: transition,Xs: list_transition] :
( ( Xs != nil_transition )
=> ( ( P @ Xs )
=> ( P @ ( cons_transition @ X2 @ Xs ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_517_list__nonempty__induct,axiom,
! [Xs2: list_o,P: list_o > $o] :
( ( Xs2 != nil_o )
=> ( ! [X2: $o] : ( P @ ( cons_o @ X2 @ nil_o ) )
=> ( ! [X2: $o,Xs: list_o] :
( ( Xs != nil_o )
=> ( ( P @ Xs )
=> ( P @ ( cons_o @ X2 @ Xs ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_518_list__nonempty__induct,axiom,
! [Xs2: list_list_o,P: list_list_o > $o] :
( ( Xs2 != nil_list_o )
=> ( ! [X2: list_o] : ( P @ ( cons_list_o @ X2 @ nil_list_o ) )
=> ( ! [X2: list_o,Xs: list_list_o] :
( ( Xs != nil_list_o )
=> ( ( P @ Xs )
=> ( P @ ( cons_list_o @ X2 @ Xs ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_519_list__induct2_H,axiom,
! [P: list_transition > list_transition > $o,Xs2: list_transition,Ys: list_transition] :
( ( P @ nil_transition @ nil_transition )
=> ( ! [X2: transition,Xs: list_transition] : ( P @ ( cons_transition @ X2 @ Xs ) @ nil_transition )
=> ( ! [Y: transition,Ys2: list_transition] : ( P @ nil_transition @ ( cons_transition @ Y @ Ys2 ) )
=> ( ! [X2: transition,Xs: list_transition,Y: transition,Ys2: list_transition] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_transition @ X2 @ Xs ) @ ( cons_transition @ Y @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_520_list__induct2_H,axiom,
! [P: list_transition > list_o > $o,Xs2: list_transition,Ys: list_o] :
( ( P @ nil_transition @ nil_o )
=> ( ! [X2: transition,Xs: list_transition] : ( P @ ( cons_transition @ X2 @ Xs ) @ nil_o )
=> ( ! [Y: $o,Ys2: list_o] : ( P @ nil_transition @ ( cons_o @ Y @ Ys2 ) )
=> ( ! [X2: transition,Xs: list_transition,Y: $o,Ys2: list_o] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_transition @ X2 @ Xs ) @ ( cons_o @ Y @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_521_list__induct2_H,axiom,
! [P: list_o > list_transition > $o,Xs2: list_o,Ys: list_transition] :
( ( P @ nil_o @ nil_transition )
=> ( ! [X2: $o,Xs: list_o] : ( P @ ( cons_o @ X2 @ Xs ) @ nil_transition )
=> ( ! [Y: transition,Ys2: list_transition] : ( P @ nil_o @ ( cons_transition @ Y @ Ys2 ) )
=> ( ! [X2: $o,Xs: list_o,Y: transition,Ys2: list_transition] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_o @ X2 @ Xs ) @ ( cons_transition @ Y @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_522_list__induct2_H,axiom,
! [P: list_o > list_o > $o,Xs2: list_o,Ys: list_o] :
( ( P @ nil_o @ nil_o )
=> ( ! [X2: $o,Xs: list_o] : ( P @ ( cons_o @ X2 @ Xs ) @ nil_o )
=> ( ! [Y: $o,Ys2: list_o] : ( P @ nil_o @ ( cons_o @ Y @ Ys2 ) )
=> ( ! [X2: $o,Xs: list_o,Y: $o,Ys2: list_o] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_o @ X2 @ Xs ) @ ( cons_o @ Y @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_523_list__induct2_H,axiom,
! [P: list_transition > list_list_o > $o,Xs2: list_transition,Ys: list_list_o] :
( ( P @ nil_transition @ nil_list_o )
=> ( ! [X2: transition,Xs: list_transition] : ( P @ ( cons_transition @ X2 @ Xs ) @ nil_list_o )
=> ( ! [Y: list_o,Ys2: list_list_o] : ( P @ nil_transition @ ( cons_list_o @ Y @ Ys2 ) )
=> ( ! [X2: transition,Xs: list_transition,Y: list_o,Ys2: list_list_o] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_transition @ X2 @ Xs ) @ ( cons_list_o @ Y @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_524_list__induct2_H,axiom,
! [P: list_o > list_list_o > $o,Xs2: list_o,Ys: list_list_o] :
( ( P @ nil_o @ nil_list_o )
=> ( ! [X2: $o,Xs: list_o] : ( P @ ( cons_o @ X2 @ Xs ) @ nil_list_o )
=> ( ! [Y: list_o,Ys2: list_list_o] : ( P @ nil_o @ ( cons_list_o @ Y @ Ys2 ) )
=> ( ! [X2: $o,Xs: list_o,Y: list_o,Ys2: list_list_o] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_o @ X2 @ Xs ) @ ( cons_list_o @ Y @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_525_list__induct2_H,axiom,
! [P: list_list_o > list_transition > $o,Xs2: list_list_o,Ys: list_transition] :
( ( P @ nil_list_o @ nil_transition )
=> ( ! [X2: list_o,Xs: list_list_o] : ( P @ ( cons_list_o @ X2 @ Xs ) @ nil_transition )
=> ( ! [Y: transition,Ys2: list_transition] : ( P @ nil_list_o @ ( cons_transition @ Y @ Ys2 ) )
=> ( ! [X2: list_o,Xs: list_list_o,Y: transition,Ys2: list_transition] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_list_o @ X2 @ Xs ) @ ( cons_transition @ Y @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_526_list__induct2_H,axiom,
! [P: list_list_o > list_o > $o,Xs2: list_list_o,Ys: list_o] :
( ( P @ nil_list_o @ nil_o )
=> ( ! [X2: list_o,Xs: list_list_o] : ( P @ ( cons_list_o @ X2 @ Xs ) @ nil_o )
=> ( ! [Y: $o,Ys2: list_o] : ( P @ nil_list_o @ ( cons_o @ Y @ Ys2 ) )
=> ( ! [X2: list_o,Xs: list_list_o,Y: $o,Ys2: list_o] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_list_o @ X2 @ Xs ) @ ( cons_o @ Y @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_527_list__induct2_H,axiom,
! [P: list_list_o > list_list_o > $o,Xs2: list_list_o,Ys: list_list_o] :
( ( P @ nil_list_o @ nil_list_o )
=> ( ! [X2: list_o,Xs: list_list_o] : ( P @ ( cons_list_o @ X2 @ Xs ) @ nil_list_o )
=> ( ! [Y: list_o,Ys2: list_list_o] : ( P @ nil_list_o @ ( cons_list_o @ Y @ Ys2 ) )
=> ( ! [X2: list_o,Xs: list_list_o,Y: list_o,Ys2: list_list_o] :
( ( P @ Xs @ Ys2 )
=> ( P @ ( cons_list_o @ X2 @ Xs ) @ ( cons_list_o @ Y @ Ys2 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_528_neq__Nil__conv,axiom,
! [Xs2: list_transition] :
( ( Xs2 != nil_transition )
= ( ? [Y4: transition,Ys3: list_transition] :
( Xs2
= ( cons_transition @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_529_neq__Nil__conv,axiom,
! [Xs2: list_o] :
( ( Xs2 != nil_o )
= ( ? [Y4: $o,Ys3: list_o] :
( Xs2
= ( cons_o @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_530_neq__Nil__conv,axiom,
! [Xs2: list_list_o] :
( ( Xs2 != nil_list_o )
= ( ? [Y4: list_o,Ys3: list_list_o] :
( Xs2
= ( cons_list_o @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_531_remdups__adj_Ocases,axiom,
! [X3: list_transition] :
( ( X3 != nil_transition )
=> ( ! [X2: transition] :
( X3
!= ( cons_transition @ X2 @ nil_transition ) )
=> ~ ! [X2: transition,Y: transition,Xs: list_transition] :
( X3
!= ( cons_transition @ X2 @ ( cons_transition @ Y @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_532_remdups__adj_Ocases,axiom,
! [X3: list_o] :
( ( X3 != nil_o )
=> ( ! [X2: $o] :
( X3
!= ( cons_o @ X2 @ nil_o ) )
=> ~ ! [X2: $o,Y: $o,Xs: list_o] :
( X3
!= ( cons_o @ X2 @ ( cons_o @ Y @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_533_remdups__adj_Ocases,axiom,
! [X3: list_list_o] :
( ( X3 != nil_list_o )
=> ( ! [X2: list_o] :
( X3
!= ( cons_list_o @ X2 @ nil_list_o ) )
=> ~ ! [X2: list_o,Y: list_o,Xs: list_list_o] :
( X3
!= ( cons_list_o @ X2 @ ( cons_list_o @ Y @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_534_min__list_Ocases,axiom,
! [X3: list_o] :
( ! [X2: $o,Xs: list_o] :
( X3
!= ( cons_o @ X2 @ Xs ) )
=> ( X3 = nil_o ) ) ).
% min_list.cases
thf(fact_535_list_Oexhaust,axiom,
! [Y3: list_transition] :
( ( Y3 != nil_transition )
=> ~ ! [X212: transition,X222: list_transition] :
( Y3
!= ( cons_transition @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_536_list_Oexhaust,axiom,
! [Y3: list_o] :
( ( Y3 != nil_o )
=> ~ ! [X212: $o,X222: list_o] :
( Y3
!= ( cons_o @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_537_list_Oexhaust,axiom,
! [Y3: list_list_o] :
( ( Y3 != nil_list_o )
=> ~ ! [X212: list_o,X222: list_list_o] :
( Y3
!= ( cons_list_o @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_538_list_OdiscI,axiom,
! [List: list_transition,X21: transition,X22: list_transition] :
( ( List
= ( cons_transition @ X21 @ X22 ) )
=> ( List != nil_transition ) ) ).
% list.discI
thf(fact_539_list_OdiscI,axiom,
! [List: list_o,X21: $o,X22: list_o] :
( ( List
= ( cons_o @ X21 @ X22 ) )
=> ( List != nil_o ) ) ).
% list.discI
thf(fact_540_list_OdiscI,axiom,
! [List: list_list_o,X21: list_o,X22: list_list_o] :
( ( List
= ( cons_list_o @ X21 @ X22 ) )
=> ( List != nil_list_o ) ) ).
% list.discI
thf(fact_541_list_Odistinct_I1_J,axiom,
! [X21: transition,X22: list_transition] :
( nil_transition
!= ( cons_transition @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_542_list_Odistinct_I1_J,axiom,
! [X21: $o,X22: list_o] :
( nil_o
!= ( cons_o @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_543_list_Odistinct_I1_J,axiom,
! [X21: list_o,X22: list_list_o] :
( nil_list_o
!= ( cons_list_o @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_544_set__union,axiom,
! [Xs2: list_transition,Ys: list_transition] :
( ( set_transition2 @ ( union_transition @ Xs2 @ Ys ) )
= ( sup_su812053455038985074sition @ ( set_transition2 @ Xs2 ) @ ( set_transition2 @ Ys ) ) ) ).
% set_union
thf(fact_545_set__union,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( set_nat2 @ ( union_nat @ Xs2 @ Ys ) )
= ( sup_sup_set_nat @ ( set_nat2 @ Xs2 ) @ ( set_nat2 @ Ys ) ) ) ).
% set_union
thf(fact_546_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_547_bot__empty__eq,axiom,
( bot_bot_transition_o
= ( ^ [X: transition] : ( member_transition @ X @ bot_bo301567166201926666sition ) ) ) ).
% bot_empty_eq
thf(fact_548_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X: nat] : ( member_nat @ X @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_549_nfa__cong__Star_Ostep__eps__closure__set__q0,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Transs: list_transition,Transs3: list_transition,Bs: list_o] :
( ( nfa_cong_Star @ Q0 @ Q05 @ Qf @ Transs @ Transs3 )
=> ( ord_less_eq_set_nat @ ( step_eps_closure_set @ Q0 @ Transs @ ( insert_nat2 @ Q0 @ bot_bot_set_nat ) @ Bs ) @ ( sup_sup_set_nat @ ( insert_nat2 @ Q0 @ ( insert_nat2 @ Qf @ bot_bot_set_nat ) ) @ ( inf_inf_set_nat @ ( step_eps_closure_set @ Q05 @ Transs3 @ ( insert_nat2 @ Q05 @ bot_bot_set_nat ) @ Bs ) @ ( sq @ Q05 @ Transs3 ) ) ) ) ) ).
% nfa_cong_Star.step_eps_closure_set_q0
thf(fact_550_nfa__cong_H_Ostep__eps__closure__set__cong__reach,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition,R: set_nat,Bs: list_o] :
( ( nfa_cong2 @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( ( ord_less_eq_set_nat @ R @ ( q @ Q05 @ Qf3 @ Transs3 ) )
=> ( ( member_nat @ Qf3 @ ( step_eps_closure_set @ Q05 @ Transs3 @ R @ Bs ) )
=> ( ( step_eps_closure_set @ Q0 @ Transs @ R @ Bs )
= ( sup_sup_set_nat @ ( step_eps_closure_set @ Q05 @ Transs3 @ R @ Bs ) @ ( step_eps_closure_set @ Q0 @ Transs @ ( insert_nat2 @ Qf3 @ bot_bot_set_nat ) @ Bs ) ) ) ) ) ) ).
% nfa_cong'.step_eps_closure_set_cong_reach
thf(fact_551_nfa__cong__Plus_Ostep__eps__closure__set__q0,axiom,
! [Q0: nat,Q05: nat,Q06: nat,Qf: nat,Qf3: nat,Qf4: nat,Transs: list_transition,Transs3: list_transition,Transs4: list_transition,Bs: list_o] :
( ( nfa_cong_Plus @ Q0 @ Q05 @ Q06 @ Qf @ Qf3 @ Qf4 @ Transs @ Transs3 @ Transs4 )
=> ( ( step_eps_closure_set @ Q0 @ Transs @ ( insert_nat2 @ Q0 @ bot_bot_set_nat ) @ Bs )
= ( sup_sup_set_nat @ ( insert_nat2 @ Q0 @ bot_bot_set_nat ) @ ( sup_sup_set_nat @ ( step_eps_closure_set @ Q05 @ Transs3 @ ( insert_nat2 @ Q05 @ bot_bot_set_nat ) @ Bs ) @ ( step_eps_closure_set @ Q06 @ Transs4 @ ( insert_nat2 @ Q06 @ bot_bot_set_nat ) @ Bs ) ) ) ) ) ).
% nfa_cong_Plus.step_eps_closure_set_q0
thf(fact_552_nfa__cong__Plus_Orun__accept__eps__Cons__cong,axiom,
! [Q0: nat,Q05: nat,Q06: nat,Qf: nat,Qf3: nat,Qf4: nat,Transs: list_transition,Transs3: list_transition,Transs4: list_transition,Cs: list_o,Css: list_list_o,Bs: list_o] :
( ( nfa_cong_Plus @ Q0 @ Q05 @ Q06 @ Qf @ Qf3 @ Qf4 @ Transs @ Transs3 @ Transs4 )
=> ( ( run_accept_eps @ Q0 @ Qf @ Transs @ ( insert_nat2 @ Q0 @ bot_bot_set_nat ) @ ( cons_list_o @ Cs @ Css ) @ Bs )
= ( ( run_accept_eps @ Q05 @ Qf3 @ Transs3 @ ( insert_nat2 @ Q05 @ bot_bot_set_nat ) @ ( cons_list_o @ Cs @ Css ) @ Bs )
| ( run_accept_eps @ Q06 @ Qf4 @ Transs4 @ ( insert_nat2 @ Q06 @ bot_bot_set_nat ) @ ( cons_list_o @ Cs @ Css ) @ Bs ) ) ) ) ).
% nfa_cong_Plus.run_accept_eps_Cons_cong
thf(fact_553_set__removeAll,axiom,
! [X3: transition,Xs2: list_transition] :
( ( set_transition2 @ ( removeAll_transition @ X3 @ Xs2 ) )
= ( minus_8944320859760356485sition @ ( set_transition2 @ Xs2 ) @ ( insert_transition2 @ X3 @ bot_bo301567166201926666sition ) ) ) ).
% set_removeAll
thf(fact_554_set__removeAll,axiom,
! [X3: nat,Xs2: list_nat] :
( ( set_nat2 @ ( removeAll_nat @ X3 @ Xs2 ) )
= ( minus_minus_set_nat @ ( set_nat2 @ Xs2 ) @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) ).
% set_removeAll
thf(fact_555_inf__right__idem,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X3 @ Y3 ) @ Y3 )
= ( inf_inf_set_nat @ X3 @ Y3 ) ) ).
% inf_right_idem
thf(fact_556_inf_Oright__idem,axiom,
! [A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B ) @ B )
= ( inf_inf_set_nat @ A @ B ) ) ).
% inf.right_idem
thf(fact_557_inf__left__idem,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ X3 @ Y3 ) )
= ( inf_inf_set_nat @ X3 @ Y3 ) ) ).
% inf_left_idem
thf(fact_558_inf_Oleft__idem,axiom,
! [A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ A @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ).
% inf.left_idem
thf(fact_559_inf__idem,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ X3 @ X3 )
= X3 ) ).
% inf_idem
thf(fact_560_inf_Oidem,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ A )
= A ) ).
% inf.idem
thf(fact_561_Int__iff,axiom,
! [C: transition,A2: set_transition,B2: set_transition] :
( ( member_transition @ C @ ( inf_in8814773338690644108sition @ A2 @ B2 ) )
= ( ( member_transition @ C @ A2 )
& ( member_transition @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_562_Int__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C @ A2 )
& ( member_nat @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_563_IntI,axiom,
! [C: transition,A2: set_transition,B2: set_transition] :
( ( member_transition @ C @ A2 )
=> ( ( member_transition @ C @ B2 )
=> ( member_transition @ C @ ( inf_in8814773338690644108sition @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_564_IntI,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_565_le__inf__iff,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ Y3 @ Z2 ) )
= ( ( ord_less_eq_set_nat @ X3 @ Y3 )
& ( ord_less_eq_set_nat @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_566_inf_Obounded__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) )
= ( ( ord_less_eq_set_nat @ A @ B )
& ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_567_boolean__algebra_Oconj__zero__left,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X3 )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_568_boolean__algebra_Oconj__zero__right,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ X3 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_569_inf__bot__left,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X3 )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_570_inf__bot__right,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ X3 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_571_inf__sup__absorb,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y3 ) )
= X3 ) ).
% inf_sup_absorb
thf(fact_572_sup__inf__absorb,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ X3 @ Y3 ) )
= X3 ) ).
% sup_inf_absorb
thf(fact_573_Int__subset__iff,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_set_nat @ C2 @ A2 )
& ( ord_less_eq_set_nat @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_574_Int__insert__left__if0,axiom,
! [A: transition,C2: set_transition,B2: set_transition] :
( ~ ( member_transition @ A @ C2 )
=> ( ( inf_in8814773338690644108sition @ ( insert_transition2 @ A @ B2 ) @ C2 )
= ( inf_in8814773338690644108sition @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_575_Int__insert__left__if0,axiom,
! [A: nat,C2: set_nat,B2: set_nat] :
( ~ ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat2 @ A @ B2 ) @ C2 )
= ( inf_inf_set_nat @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_576_Int__insert__left__if1,axiom,
! [A: transition,C2: set_transition,B2: set_transition] :
( ( member_transition @ A @ C2 )
=> ( ( inf_in8814773338690644108sition @ ( insert_transition2 @ A @ B2 ) @ C2 )
= ( insert_transition2 @ A @ ( inf_in8814773338690644108sition @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_577_Int__insert__left__if1,axiom,
! [A: nat,C2: set_nat,B2: set_nat] :
( ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat2 @ A @ B2 ) @ C2 )
= ( insert_nat2 @ A @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_578_insert__inter__insert,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ ( insert_nat2 @ A @ A2 ) @ ( insert_nat2 @ A @ B2 ) )
= ( insert_nat2 @ A @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_579_Int__insert__right__if0,axiom,
! [A: transition,A2: set_transition,B2: set_transition] :
( ~ ( member_transition @ A @ A2 )
=> ( ( inf_in8814773338690644108sition @ A2 @ ( insert_transition2 @ A @ B2 ) )
= ( inf_in8814773338690644108sition @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_580_Int__insert__right__if0,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat2 @ A @ B2 ) )
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_581_Int__insert__right__if1,axiom,
! [A: transition,A2: set_transition,B2: set_transition] :
( ( member_transition @ A @ A2 )
=> ( ( inf_in8814773338690644108sition @ A2 @ ( insert_transition2 @ A @ B2 ) )
= ( insert_transition2 @ A @ ( inf_in8814773338690644108sition @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_582_Int__insert__right__if1,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat2 @ A @ B2 ) )
= ( insert_nat2 @ A @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_583_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_584_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_585_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_586_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_587_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_588_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_589_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_590_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_591_removeAll__id,axiom,
! [X3: nat,Xs2: list_nat] :
( ~ ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
=> ( ( removeAll_nat @ X3 @ Xs2 )
= Xs2 ) ) ).
% removeAll_id
thf(fact_592_removeAll__id,axiom,
! [X3: transition,Xs2: list_transition] :
( ~ ( member_transition @ X3 @ ( set_transition2 @ Xs2 ) )
=> ( ( removeAll_transition @ X3 @ Xs2 )
= Xs2 ) ) ).
% removeAll_id
thf(fact_593_disjoint__insert_I2_J,axiom,
! [A2: set_transition,B: transition,B2: set_transition] :
( ( bot_bo301567166201926666sition
= ( inf_in8814773338690644108sition @ A2 @ ( insert_transition2 @ B @ B2 ) ) )
= ( ~ ( member_transition @ B @ A2 )
& ( bot_bo301567166201926666sition
= ( inf_in8814773338690644108sition @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_594_disjoint__insert_I2_J,axiom,
! [A2: set_nat,B: nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ ( insert_nat2 @ B @ B2 ) ) )
= ( ~ ( member_nat @ B @ A2 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_595_disjoint__insert_I1_J,axiom,
! [B2: set_transition,A: transition,A2: set_transition] :
( ( ( inf_in8814773338690644108sition @ B2 @ ( insert_transition2 @ A @ A2 ) )
= bot_bo301567166201926666sition )
= ( ~ ( member_transition @ A @ B2 )
& ( ( inf_in8814773338690644108sition @ B2 @ A2 )
= bot_bo301567166201926666sition ) ) ) ).
% disjoint_insert(1)
thf(fact_596_disjoint__insert_I1_J,axiom,
! [B2: set_nat,A: nat,A2: set_nat] :
( ( ( inf_inf_set_nat @ B2 @ ( insert_nat2 @ A @ A2 ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B2 )
& ( ( inf_inf_set_nat @ B2 @ A2 )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_597_insert__disjoint_I2_J,axiom,
! [A: transition,A2: set_transition,B2: set_transition] :
( ( bot_bo301567166201926666sition
= ( inf_in8814773338690644108sition @ ( insert_transition2 @ A @ A2 ) @ B2 ) )
= ( ~ ( member_transition @ A @ B2 )
& ( bot_bo301567166201926666sition
= ( inf_in8814773338690644108sition @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_598_insert__disjoint_I2_J,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat2 @ A @ A2 ) @ B2 ) )
= ( ~ ( member_nat @ A @ B2 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_599_insert__disjoint_I1_J,axiom,
! [A: transition,A2: set_transition,B2: set_transition] :
( ( ( inf_in8814773338690644108sition @ ( insert_transition2 @ A @ A2 ) @ B2 )
= bot_bo301567166201926666sition )
= ( ~ ( member_transition @ A @ B2 )
& ( ( inf_in8814773338690644108sition @ A2 @ B2 )
= bot_bo301567166201926666sition ) ) ) ).
% insert_disjoint(1)
thf(fact_600_insert__disjoint_I1_J,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat2 @ A @ A2 ) @ B2 )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B2 )
& ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_601_Diff__disjoint,axiom,
! [A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ A2 ) )
= bot_bot_set_nat ) ).
% Diff_disjoint
thf(fact_602_inf__left__commute,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ Y3 @ Z2 ) )
= ( inf_inf_set_nat @ Y3 @ ( inf_inf_set_nat @ X3 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_603_inf_Oleft__commute,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( inf_inf_set_nat @ B @ ( inf_inf_set_nat @ A @ C ) )
= ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_604_inf__commute,axiom,
( inf_inf_set_nat
= ( ^ [X: set_nat,Y4: set_nat] : ( inf_inf_set_nat @ Y4 @ X ) ) ) ).
% inf_commute
thf(fact_605_inf_Ocommute,axiom,
( inf_inf_set_nat
= ( ^ [A4: set_nat,B5: set_nat] : ( inf_inf_set_nat @ B5 @ A4 ) ) ) ).
% inf.commute
thf(fact_606_inf__assoc,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X3 @ Y3 ) @ Z2 )
= ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ Y3 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_607_inf_Oassoc,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C )
= ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) ) ) ).
% inf.assoc
thf(fact_608_inf__sup__aci_I1_J,axiom,
( inf_inf_set_nat
= ( ^ [X: set_nat,Y4: set_nat] : ( inf_inf_set_nat @ Y4 @ X ) ) ) ).
% inf_sup_aci(1)
thf(fact_609_inf__sup__aci_I2_J,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X3 @ Y3 ) @ Z2 )
= ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ Y3 @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_610_inf__sup__aci_I3_J,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ Y3 @ Z2 ) )
= ( inf_inf_set_nat @ Y3 @ ( inf_inf_set_nat @ X3 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_611_inf__sup__aci_I4_J,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ X3 @ Y3 ) )
= ( inf_inf_set_nat @ X3 @ Y3 ) ) ).
% inf_sup_aci(4)
thf(fact_612_Int__left__commute,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C2 ) )
= ( inf_inf_set_nat @ B2 @ ( inf_inf_set_nat @ A2 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_613_Int__left__absorb,axiom,
! [A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( inf_inf_set_nat @ A2 @ B2 ) ) ).
% Int_left_absorb
thf(fact_614_Int__commute,axiom,
( inf_inf_set_nat
= ( ^ [A3: set_nat,B3: set_nat] : ( inf_inf_set_nat @ B3 @ A3 ) ) ) ).
% Int_commute
thf(fact_615_Int__absorb,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_616_Int__assoc,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) ).
% Int_assoc
thf(fact_617_IntD2,axiom,
! [C: transition,A2: set_transition,B2: set_transition] :
( ( member_transition @ C @ ( inf_in8814773338690644108sition @ A2 @ B2 ) )
=> ( member_transition @ C @ B2 ) ) ).
% IntD2
thf(fact_618_IntD2,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C @ B2 ) ) ).
% IntD2
thf(fact_619_IntD1,axiom,
! [C: transition,A2: set_transition,B2: set_transition] :
( ( member_transition @ C @ ( inf_in8814773338690644108sition @ A2 @ B2 ) )
=> ( member_transition @ C @ A2 ) ) ).
% IntD1
thf(fact_620_IntD1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C @ A2 ) ) ).
% IntD1
thf(fact_621_IntE,axiom,
! [C: transition,A2: set_transition,B2: set_transition] :
( ( member_transition @ C @ ( inf_in8814773338690644108sition @ A2 @ B2 ) )
=> ~ ( ( member_transition @ C @ A2 )
=> ~ ( member_transition @ C @ B2 ) ) ) ).
% IntE
thf(fact_622_IntE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ~ ( member_nat @ C @ B2 ) ) ) ).
% IntE
thf(fact_623_boolean__algebra__cancel_Oinf2,axiom,
! [B2: set_nat,K: set_nat,B: set_nat,A: set_nat] :
( ( B2
= ( inf_inf_set_nat @ K @ B ) )
=> ( ( inf_inf_set_nat @ A @ B2 )
= ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_624_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_nat,K: set_nat,A: set_nat,B: set_nat] :
( ( A2
= ( inf_inf_set_nat @ K @ A ) )
=> ( ( inf_inf_set_nat @ A2 @ B )
= ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_625_inf_OcoboundedI2,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_626_inf_OcoboundedI1,axiom,
! [A: set_nat,C: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_627_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [B5: set_nat,A4: set_nat] :
( ( inf_inf_set_nat @ A4 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_628_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
( ( inf_inf_set_nat @ A4 @ B5 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_629_inf_Ocobounded2,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_630_inf_Ocobounded1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_631_inf_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
( A4
= ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_632_inf__greatest,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_set_nat @ X3 @ Z2 )
=> ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ Y3 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_633_inf_OboundedI,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ A @ C )
=> ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_634_inf_OboundedE,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ~ ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_635_inf__absorb2,axiom,
! [Y3: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X3 )
=> ( ( inf_inf_set_nat @ X3 @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_636_inf__absorb1,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ( inf_inf_set_nat @ X3 @ Y3 )
= X3 ) ) ).
% inf_absorb1
thf(fact_637_inf_Oabsorb2,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( inf_inf_set_nat @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_638_inf_Oabsorb1,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( inf_inf_set_nat @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_639_le__iff__inf,axiom,
( ord_less_eq_set_nat
= ( ^ [X: set_nat,Y4: set_nat] :
( ( inf_inf_set_nat @ X @ Y4 )
= X ) ) ) ).
% le_iff_inf
thf(fact_640_inf__unique,axiom,
! [F: set_nat > set_nat > set_nat,X3: set_nat,Y3: set_nat] :
( ! [X2: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( F @ X2 @ Y ) @ X2 )
=> ( ! [X2: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( F @ X2 @ Y ) @ Y )
=> ( ! [X2: set_nat,Y: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ord_less_eq_set_nat @ X2 @ Z3 )
=> ( ord_less_eq_set_nat @ X2 @ ( F @ Y @ Z3 ) ) ) )
=> ( ( inf_inf_set_nat @ X3 @ Y3 )
= ( F @ X3 @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_641_inf_OorderI,axiom,
! [A: set_nat,B: set_nat] :
( ( A
= ( inf_inf_set_nat @ A @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% inf.orderI
thf(fact_642_inf_OorderE,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( A
= ( inf_inf_set_nat @ A @ B ) ) ) ).
% inf.orderE
thf(fact_643_le__infI2,axiom,
! [B: set_nat,X3: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ X3 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X3 ) ) ).
% le_infI2
thf(fact_644_le__infI1,axiom,
! [A: set_nat,X3: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ X3 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X3 ) ) ).
% le_infI1
thf(fact_645_inf__mono,axiom,
! [A: set_nat,C: set_nat,B: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B @ D2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_646_le__infI,axiom,
! [X3: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ A )
=> ( ( ord_less_eq_set_nat @ X3 @ B )
=> ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% le_infI
thf(fact_647_le__infE,axiom,
! [X3: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ A @ B ) )
=> ~ ( ( ord_less_eq_set_nat @ X3 @ A )
=> ~ ( ord_less_eq_set_nat @ X3 @ B ) ) ) ).
% le_infE
thf(fact_648_inf__le2,axiom,
! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_649_inf__le1,axiom,
! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y3 ) @ X3 ) ).
% inf_le1
thf(fact_650_inf__sup__ord_I1_J,axiom,
! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y3 ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_651_inf__sup__ord_I2_J,axiom,
! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_652_distrib__imp1,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ! [X2: set_nat,Y: set_nat,Z3: set_nat] :
( ( inf_inf_set_nat @ X2 @ ( sup_sup_set_nat @ Y @ Z3 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X2 @ Y ) @ ( inf_inf_set_nat @ X2 @ Z3 ) ) )
=> ( ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ Y3 @ Z2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X3 @ Y3 ) @ ( sup_sup_set_nat @ X3 @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_653_distrib__imp2,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ! [X2: set_nat,Y: set_nat,Z3: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( inf_inf_set_nat @ Y @ Z3 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X2 @ Y ) @ ( sup_sup_set_nat @ X2 @ Z3 ) ) )
=> ( ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ Y3 @ Z2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X3 @ Y3 ) @ ( inf_inf_set_nat @ X3 @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_654_inf__sup__distrib1,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ Y3 @ Z2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X3 @ Y3 ) @ ( inf_inf_set_nat @ X3 @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_655_inf__sup__distrib2,axiom,
! [Y3: set_nat,Z2: set_nat,X3: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y3 @ Z2 ) @ X3 )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y3 @ X3 ) @ ( inf_inf_set_nat @ Z2 @ X3 ) ) ) ).
% inf_sup_distrib2
thf(fact_656_sup__inf__distrib1,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ Y3 @ Z2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X3 @ Y3 ) @ ( sup_sup_set_nat @ X3 @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_657_sup__inf__distrib2,axiom,
! [Y3: set_nat,Z2: set_nat,X3: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y3 @ Z2 ) @ X3 )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y3 @ X3 ) @ ( sup_sup_set_nat @ Z2 @ X3 ) ) ) ).
% sup_inf_distrib2
thf(fact_658_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y3: set_nat,Z2: set_nat,X3: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y3 @ Z2 ) @ X3 )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y3 @ X3 ) @ ( sup_sup_set_nat @ Z2 @ X3 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_659_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y3: set_nat,Z2: set_nat,X3: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y3 @ Z2 ) @ X3 )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y3 @ X3 ) @ ( inf_inf_set_nat @ Z2 @ X3 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_660_boolean__algebra_Odisj__conj__distrib,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ Y3 @ Z2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X3 @ Y3 ) @ ( sup_sup_set_nat @ X3 @ Z2 ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_661_boolean__algebra_Oconj__disj__distrib,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ Y3 @ Z2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X3 @ Y3 ) @ ( inf_inf_set_nat @ X3 @ Z2 ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_662_Int__emptyI,axiom,
! [A2: set_transition,B2: set_transition] :
( ! [X2: transition] :
( ( member_transition @ X2 @ A2 )
=> ~ ( member_transition @ X2 @ B2 ) )
=> ( ( inf_in8814773338690644108sition @ A2 @ B2 )
= bot_bo301567166201926666sition ) ) ).
% Int_emptyI
thf(fact_663_Int__emptyI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ~ ( member_nat @ X2 @ B2 ) )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_664_disjoint__iff,axiom,
! [A2: set_transition,B2: set_transition] :
( ( ( inf_in8814773338690644108sition @ A2 @ B2 )
= bot_bo301567166201926666sition )
= ( ! [X: transition] :
( ( member_transition @ X @ A2 )
=> ~ ( member_transition @ X @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_665_disjoint__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ~ ( member_nat @ X @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_666_Int__empty__left,axiom,
! [B2: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B2 )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_667_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_668_disjoint__iff__not__equal,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ B2 )
=> ( X != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_669_Int__mono,axiom,
! [A2: set_nat,C2: set_nat,B2: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ D )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( inf_inf_set_nat @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_670_Int__lower1,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ A2 ) ).
% Int_lower1
thf(fact_671_Int__lower2,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_672_Int__absorb1,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_673_Int__absorb2,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= A2 ) ) ).
% Int_absorb2
thf(fact_674_Int__greatest,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A2 )
=> ( ( ord_less_eq_set_nat @ C2 @ B2 )
=> ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_675_Int__Collect__mono,axiom,
! [A2: set_transition,B2: set_transition,P: transition > $o,Q4: transition > $o] :
( ( ord_le8419162016481440574sition @ A2 @ B2 )
=> ( ! [X2: transition] :
( ( member_transition @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q4 @ X2 ) ) )
=> ( ord_le8419162016481440574sition @ ( inf_in8814773338690644108sition @ A2 @ ( collect_transition @ P ) ) @ ( inf_in8814773338690644108sition @ B2 @ ( collect_transition @ Q4 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_676_Int__Collect__mono,axiom,
! [A2: set_nat,B2: set_nat,P: nat > $o,Q4: nat > $o] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q4 @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B2 @ ( collect_nat @ Q4 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_677_Int__insert__left,axiom,
! [A: transition,C2: set_transition,B2: set_transition] :
( ( ( member_transition @ A @ C2 )
=> ( ( inf_in8814773338690644108sition @ ( insert_transition2 @ A @ B2 ) @ C2 )
= ( insert_transition2 @ A @ ( inf_in8814773338690644108sition @ B2 @ C2 ) ) ) )
& ( ~ ( member_transition @ A @ C2 )
=> ( ( inf_in8814773338690644108sition @ ( insert_transition2 @ A @ B2 ) @ C2 )
= ( inf_in8814773338690644108sition @ B2 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_678_Int__insert__left,axiom,
! [A: nat,C2: set_nat,B2: set_nat] :
( ( ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat2 @ A @ B2 ) @ C2 )
= ( insert_nat2 @ A @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) )
& ( ~ ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat2 @ A @ B2 ) @ C2 )
= ( inf_inf_set_nat @ B2 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_679_Int__insert__right,axiom,
! [A: transition,A2: set_transition,B2: set_transition] :
( ( ( member_transition @ A @ A2 )
=> ( ( inf_in8814773338690644108sition @ A2 @ ( insert_transition2 @ A @ B2 ) )
= ( insert_transition2 @ A @ ( inf_in8814773338690644108sition @ A2 @ B2 ) ) ) )
& ( ~ ( member_transition @ A @ A2 )
=> ( ( inf_in8814773338690644108sition @ A2 @ ( insert_transition2 @ A @ B2 ) )
= ( inf_in8814773338690644108sition @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_680_Int__insert__right,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat2 @ A @ B2 ) )
= ( insert_nat2 @ A @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) )
& ( ~ ( member_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ ( insert_nat2 @ A @ B2 ) )
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_681_Un__Int__distrib2,axiom,
! [B2: set_nat,C2: set_nat,A2: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ B2 @ C2 ) @ A2 )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ B2 @ A2 ) @ ( sup_sup_set_nat @ C2 @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_682_Int__Un__distrib2,axiom,
! [B2: set_nat,C2: set_nat,A2: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ B2 @ C2 ) @ A2 )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ B2 @ A2 ) @ ( inf_inf_set_nat @ C2 @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_683_Un__Int__distrib,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ ( sup_sup_set_nat @ A2 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_684_Int__Un__distrib,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( inf_inf_set_nat @ A2 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_685_Un__Int__crazy,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( inf_inf_set_nat @ B2 @ C2 ) ) @ ( inf_inf_set_nat @ C2 @ A2 ) )
= ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ ( sup_sup_set_nat @ B2 @ C2 ) ) @ ( sup_sup_set_nat @ C2 @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_686_Int__Diff,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ C2 ) ) ) ).
% Int_Diff
thf(fact_687_Diff__Int2,axiom,
! [A2: set_nat,C2: set_nat,B2: set_nat] :
( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A2 @ C2 ) @ ( inf_inf_set_nat @ B2 @ C2 ) )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ A2 @ C2 ) @ B2 ) ) ).
% Diff_Int2
thf(fact_688_Diff__Diff__Int,axiom,
! [A2: set_nat,B2: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( inf_inf_set_nat @ A2 @ B2 ) ) ).
% Diff_Diff_Int
thf(fact_689_Diff__Int__distrib,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ C2 @ A2 ) @ ( inf_inf_set_nat @ C2 @ B2 ) ) ) ).
% Diff_Int_distrib
thf(fact_690_Diff__Int__distrib2,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ C2 )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ A2 @ C2 ) @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_691_nfa__cong_H_Oqf_H__in__SQ,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition] :
( ( nfa_cong2 @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( member_nat @ Qf3 @ ( sq @ Q0 @ Transs ) ) ) ).
% nfa_cong'.qf'_in_SQ
thf(fact_692_nfa__cong_H_Oq__SQ__SQ__nfa_H__SQ,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition,Q: nat] :
( ( nfa_cong2 @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( ( member_nat @ Q @ ( sq @ Q05 @ Transs3 ) )
=> ( ( member_nat @ Q @ ( sq @ Q0 @ Transs ) )
= ( member_nat @ Q @ ( sq @ Q05 @ Transs3 ) ) ) ) ) ).
% nfa_cong'.q_SQ_SQ_nfa'_SQ
thf(fact_693_removeAll_Osimps_I2_J,axiom,
! [X3: list_o,Y3: list_o,Xs2: list_list_o] :
( ( ( X3 = Y3 )
=> ( ( removeAll_list_o @ X3 @ ( cons_list_o @ Y3 @ Xs2 ) )
= ( removeAll_list_o @ X3 @ Xs2 ) ) )
& ( ( X3 != Y3 )
=> ( ( removeAll_list_o @ X3 @ ( cons_list_o @ Y3 @ Xs2 ) )
= ( cons_list_o @ Y3 @ ( removeAll_list_o @ X3 @ Xs2 ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_694_removeAll_Osimps_I1_J,axiom,
! [X3: transition] :
( ( removeAll_transition @ X3 @ nil_transition )
= nil_transition ) ).
% removeAll.simps(1)
thf(fact_695_removeAll_Osimps_I1_J,axiom,
! [X3: list_o] :
( ( removeAll_list_o @ X3 @ nil_list_o )
= nil_list_o ) ).
% removeAll.simps(1)
thf(fact_696_removeAll_Osimps_I1_J,axiom,
! [X3: $o] :
( ( removeAll_o @ X3 @ nil_o )
= nil_o ) ).
% removeAll.simps(1)
thf(fact_697_nfa__cong_H_Oaxioms_I2_J,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition] :
( ( nfa_cong2 @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( nfa @ Q05 @ Qf3 @ Transs3 ) ) ).
% nfa_cong'.axioms(2)
thf(fact_698_nfa__cong_H_Oaxioms_I1_J,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition] :
( ( nfa_cong2 @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( nfa @ Q0 @ Qf @ Transs ) ) ).
% nfa_cong'.axioms(1)
thf(fact_699_nfa__cong_H_Ostep__symb__cong__Q,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition,Q: nat,Q2: nat] :
( ( nfa_cong2 @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( ( step_symb @ Q05 @ Transs3 @ Q @ Q2 )
=> ( step_symb @ Q0 @ Transs @ Q @ Q2 ) ) ) ).
% nfa_cong'.step_symb_cong_Q
thf(fact_700_nfa__cong__Star_Oaxioms_I1_J,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Transs: list_transition,Transs3: list_transition] :
( ( nfa_cong_Star @ Q0 @ Q05 @ Qf @ Transs @ Transs3 )
=> ( nfa_cong2 @ Q0 @ Q05 @ Qf @ Q0 @ Transs @ Transs3 ) ) ).
% nfa_cong_Star.axioms(1)
thf(fact_701_distrib__sup__le,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ Y3 @ Z2 ) ) @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ X3 @ Y3 ) @ ( sup_sup_set_nat @ X3 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_702_distrib__inf__le,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ X3 @ Y3 ) @ ( inf_inf_set_nat @ X3 @ Z2 ) ) @ ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ Y3 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_703_Un__Int__assoc__eq,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) ) )
= ( ord_less_eq_set_nat @ C2 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_704_Int__Diff__disjoint,axiom,
! [A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ A2 @ B2 ) )
= bot_bot_set_nat ) ).
% Int_Diff_disjoint
thf(fact_705_Diff__triv,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
=> ( ( minus_minus_set_nat @ A2 @ B2 )
= A2 ) ) ).
% Diff_triv
thf(fact_706_Un__Diff__Int,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( inf_inf_set_nat @ A2 @ B2 ) )
= A2 ) ).
% Un_Diff_Int
thf(fact_707_Int__Diff__Un,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ A2 @ B2 ) )
= A2 ) ).
% Int_Diff_Un
thf(fact_708_Diff__Int,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C2 ) )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ A2 @ C2 ) ) ) ).
% Diff_Int
thf(fact_709_Diff__Un,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) )
= ( inf_inf_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ A2 @ C2 ) ) ) ).
% Diff_Un
thf(fact_710_nfa__cong__Plus_Ostep__symb__q0,axiom,
! [Q0: nat,Q05: nat,Q06: nat,Qf: nat,Qf3: nat,Qf4: nat,Transs: list_transition,Transs3: list_transition,Transs4: list_transition,Q: nat] :
( ( nfa_cong_Plus @ Q0 @ Q05 @ Q06 @ Qf @ Qf3 @ Qf4 @ Transs @ Transs3 @ Transs4 )
=> ~ ( step_symb @ Q0 @ Transs @ Q0 @ Q ) ) ).
% nfa_cong_Plus.step_symb_q0
thf(fact_711_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_712_nfa__cong_H_Onfa_H__Q__sub__Q,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition] :
( ( nfa_cong2 @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( ord_less_eq_set_nat @ ( q @ Q05 @ Qf3 @ Transs3 ) @ ( q @ Q0 @ Qf @ Transs ) ) ) ).
% nfa_cong'.nfa'_Q_sub_Q
thf(fact_713_nfa__cong_H_OSQ__sub,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition] :
( ( nfa_cong2 @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( ord_less_eq_set_nat @ ( sq @ Q05 @ Transs3 ) @ ( sq @ Q0 @ Transs ) ) ) ).
% nfa_cong'.SQ_sub
thf(fact_714_nfa__cong_H_Onfa_H__step__eps__closure__cong,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition,Bs: list_o,Q: nat,Q2: nat] :
( ( nfa_cong2 @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( ( step_eps_closure @ Q05 @ Transs3 @ Bs @ Q @ Q2 )
=> ( ( member_nat @ Q @ ( q @ Q05 @ Qf3 @ Transs3 ) )
=> ( step_eps_closure @ Q0 @ Transs @ Bs @ Q @ Q2 ) ) ) ) ).
% nfa_cong'.nfa'_step_eps_closure_cong
thf(fact_715_nfa__cong_H_Oeps__nfa_H__step__eps__closure__cong,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition,Bs: list_o,Q: nat,Q2: nat] :
( ( nfa_cong2 @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( ( step_eps_closure @ Q0 @ Transs @ Bs @ Q @ Q2 )
=> ( ( member_nat @ Q @ ( q @ Q05 @ Qf3 @ Transs3 ) )
=> ( ( ( member_nat @ Q2 @ ( q @ Q05 @ Qf3 @ Transs3 ) )
& ( step_eps_closure @ Q05 @ Transs3 @ Bs @ Q @ Q2 ) )
| ( ( step_eps_closure @ Q05 @ Transs3 @ Bs @ Q @ Qf3 )
& ( step_eps_closure @ Q0 @ Transs @ Bs @ Qf3 @ Q2 ) ) ) ) ) ) ).
% nfa_cong'.eps_nfa'_step_eps_closure_cong
thf(fact_716_nfa__cong_H_Onfa_H__eps__step__eps__closure__cong,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition,Bs: list_o,Q: nat,Q2: nat] :
( ( nfa_cong2 @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( ( step_eps_closure @ Q05 @ Transs3 @ Bs @ Q @ Q2 )
=> ( ( member_nat @ Q @ ( q @ Q05 @ Qf3 @ Transs3 ) )
=> ( ( member_nat @ Q2 @ ( q @ Q05 @ Qf3 @ Transs3 ) )
& ( step_eps_closure @ Q0 @ Transs @ Bs @ Q @ Q2 ) ) ) ) ) ).
% nfa_cong'.nfa'_eps_step_eps_closure_cong
thf(fact_717_nfa__cong_H_Ostep__eps__cong__Q,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition,Q: nat,Bs: list_o,Q2: nat] :
( ( nfa_cong2 @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( ( member_nat @ Q @ ( q @ Q05 @ Qf3 @ Transs3 ) )
=> ( ( step_eps @ Q05 @ Transs3 @ Bs @ Q @ Q2 )
=> ( step_eps @ Q0 @ Transs @ Bs @ Q @ Q2 ) ) ) ) ).
% nfa_cong'.step_eps_cong_Q
thf(fact_718_nfa__cong_H_Ostep__symb__set__cong__Q,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition,R: set_nat] :
( ( nfa_cong2 @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( ord_less_eq_set_nat @ ( step_symb_set @ Q05 @ Transs3 @ R ) @ ( step_symb_set @ Q0 @ Transs @ R ) ) ) ).
% nfa_cong'.step_symb_set_cong_Q
thf(fact_719_nfa__cong_H_Ostep__eps__cong__SQ,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition,Q: nat,Bs: list_o,Q2: nat] :
( ( nfa_cong2 @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( ( member_nat @ Q @ ( sq @ Q05 @ Transs3 ) )
=> ( ( step_eps @ Q0 @ Transs @ Bs @ Q @ Q2 )
= ( step_eps @ Q05 @ Transs3 @ Bs @ Q @ Q2 ) ) ) ) ).
% nfa_cong'.step_eps_cong_SQ
thf(fact_720_nfa__cong__Plus_Oaxioms_I3_J,axiom,
! [Q0: nat,Q05: nat,Q06: nat,Qf: nat,Qf3: nat,Qf4: nat,Transs: list_transition,Transs3: list_transition,Transs4: list_transition] :
( ( nfa_cong_Plus @ Q0 @ Q05 @ Q06 @ Qf @ Qf3 @ Qf4 @ Transs @ Transs3 @ Transs4 )
=> ( nfa_cong_Plus_axioms @ Q0 @ Q05 @ Q06 @ Transs ) ) ).
% nfa_cong_Plus.axioms(3)
thf(fact_721_nfa__cong_H_Ostep__symb__cong__SQ,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition,Q: nat,Q2: nat] :
( ( nfa_cong2 @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( ( member_nat @ Q @ ( sq @ Q05 @ Transs3 ) )
=> ( ( step_symb @ Q0 @ Transs @ Q @ Q2 )
= ( step_symb @ Q05 @ Transs3 @ Q @ Q2 ) ) ) ) ).
% nfa_cong'.step_symb_cong_SQ
thf(fact_722_nfa__cong__Star__def,axiom,
( nfa_cong_Star
= ( ^ [Q02: nat,Q03: nat,Qf2: nat,Transs2: list_transition,Transs5: list_transition] :
( ( nfa_cong2 @ Q02 @ Q03 @ Qf2 @ Q02 @ Transs2 @ Transs5 )
& ( nfa_cong_Star_axioms @ Q02 @ Q03 @ Qf2 @ Transs2 ) ) ) ) ).
% nfa_cong_Star_def
thf(fact_723_nfa__cong__Star_Ointro,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Transs: list_transition,Transs3: list_transition] :
( ( nfa_cong2 @ Q0 @ Q05 @ Qf @ Q0 @ Transs @ Transs3 )
=> ( ( nfa_cong_Star_axioms @ Q0 @ Q05 @ Qf @ Transs )
=> ( nfa_cong_Star @ Q0 @ Q05 @ Qf @ Transs @ Transs3 ) ) ) ).
% nfa_cong_Star.intro
thf(fact_724_nfa__cong__Plus_Oqf__not__q0,axiom,
! [Q0: nat,Q05: nat,Q06: nat,Qf: nat,Qf3: nat,Qf4: nat,Transs: list_transition,Transs3: list_transition,Transs4: list_transition] :
( ( nfa_cong_Plus @ Q0 @ Q05 @ Q06 @ Qf @ Qf3 @ Qf4 @ Transs @ Transs3 @ Transs4 )
=> ~ ( member_nat @ Qf @ ( insert_nat2 @ Q0 @ bot_bot_set_nat ) ) ) ).
% nfa_cong_Plus.qf_not_q0
thf(fact_725_nfa__cong_H_Onfa_H__step__eps__closure__set__sub,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition,R: set_nat,Bs: list_o] :
( ( nfa_cong2 @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( ( ord_less_eq_set_nat @ R @ ( q @ Q05 @ Qf3 @ Transs3 ) )
=> ( ord_less_eq_set_nat @ ( step_eps_closure_set @ Q05 @ Transs3 @ R @ Bs ) @ ( step_eps_closure_set @ Q0 @ Transs @ R @ Bs ) ) ) ) ).
% nfa_cong'.nfa'_step_eps_closure_set_sub
thf(fact_726_nfa__cong_H_Ostep__eps__closure__set__cong__unreach,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition,R: set_nat,Bs: list_o] :
( ( nfa_cong2 @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( ( ord_less_eq_set_nat @ R @ ( q @ Q05 @ Qf3 @ Transs3 ) )
=> ( ~ ( member_nat @ Qf3 @ ( step_eps_closure_set @ Q05 @ Transs3 @ R @ Bs ) )
=> ( ( step_eps_closure_set @ Q0 @ Transs @ R @ Bs )
= ( step_eps_closure_set @ Q05 @ Transs3 @ R @ Bs ) ) ) ) ) ).
% nfa_cong'.step_eps_closure_set_cong_unreach
thf(fact_727_nfa__cong_H_Onfa_H__delta__sub__delta,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition,R: set_nat,Bs: list_o] :
( ( nfa_cong2 @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( ( ord_less_eq_set_nat @ R @ ( q @ Q05 @ Qf3 @ Transs3 ) )
=> ( ord_less_eq_set_nat @ ( delta @ Q05 @ Transs3 @ R @ Bs ) @ ( delta @ Q0 @ Transs @ R @ Bs ) ) ) ) ).
% nfa_cong'.nfa'_delta_sub_delta
thf(fact_728_nfa__cong_H_Ostep__symb__set__cong__SQ,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition,R: set_nat] :
( ( nfa_cong2 @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( ( ord_less_eq_set_nat @ R @ ( sq @ Q05 @ Transs3 ) )
=> ( ( step_symb_set @ Q0 @ Transs @ R )
= ( step_symb_set @ Q05 @ Transs3 @ R ) ) ) ) ).
% nfa_cong'.step_symb_set_cong_SQ
thf(fact_729_nfa__cong__Plus_Orun__accept__eps__cong,axiom,
! [Q0: nat,Q05: nat,Q06: nat,Qf: nat,Qf3: nat,Qf4: nat,Transs: list_transition,Transs3: list_transition,Transs4: list_transition,Bss: list_list_o,Bs: list_o] :
( ( nfa_cong_Plus @ Q0 @ Q05 @ Q06 @ Qf @ Qf3 @ Qf4 @ Transs @ Transs3 @ Transs4 )
=> ( ( run_accept_eps @ Q0 @ Qf @ Transs @ ( insert_nat2 @ Q0 @ bot_bot_set_nat ) @ Bss @ Bs )
= ( ( run_accept_eps @ Q05 @ Qf3 @ Transs3 @ ( insert_nat2 @ Q05 @ bot_bot_set_nat ) @ Bss @ Bs )
| ( run_accept_eps @ Q06 @ Qf4 @ Transs4 @ ( insert_nat2 @ Q06 @ bot_bot_set_nat ) @ Bss @ Bs ) ) ) ) ).
% nfa_cong_Plus.run_accept_eps_cong
thf(fact_730_nfa__cong__Plus_Orun__accept__eps__Nil__cong,axiom,
! [Q0: nat,Q05: nat,Q06: nat,Qf: nat,Qf3: nat,Qf4: nat,Transs: list_transition,Transs3: list_transition,Transs4: list_transition,Bs: list_o] :
( ( nfa_cong_Plus @ Q0 @ Q05 @ Q06 @ Qf @ Qf3 @ Qf4 @ Transs @ Transs3 @ Transs4 )
=> ( ( run_accept_eps @ Q0 @ Qf @ Transs @ ( insert_nat2 @ Q0 @ bot_bot_set_nat ) @ nil_list_o @ Bs )
= ( ( run_accept_eps @ Q05 @ Qf3 @ Transs3 @ ( insert_nat2 @ Q05 @ bot_bot_set_nat ) @ nil_list_o @ Bs )
| ( run_accept_eps @ Q06 @ Qf4 @ Transs4 @ ( insert_nat2 @ Q06 @ bot_bot_set_nat ) @ nil_list_o @ Bs ) ) ) ) ).
% nfa_cong_Plus.run_accept_eps_Nil_cong
thf(fact_731_nfa__cong__Plus_Ostep__eps__q0,axiom,
! [Q0: nat,Q05: nat,Q06: nat,Qf: nat,Qf3: nat,Qf4: nat,Transs: list_transition,Transs3: list_transition,Transs4: list_transition,Bs: list_o,Q: nat] :
( ( nfa_cong_Plus @ Q0 @ Q05 @ Q06 @ Qf @ Qf3 @ Qf4 @ Transs @ Transs3 @ Transs4 )
=> ( ( step_eps @ Q0 @ Transs @ Bs @ Q0 @ Q )
= ( member_nat @ Q @ ( insert_nat2 @ Q05 @ ( insert_nat2 @ Q06 @ bot_bot_set_nat ) ) ) ) ) ).
% nfa_cong_Plus.step_eps_q0
thf(fact_732_nfa__cong__Plus_Ostep__symb__set__q0,axiom,
! [Q0: nat,Q05: nat,Q06: nat,Qf: nat,Qf3: nat,Qf4: nat,Transs: list_transition,Transs3: list_transition,Transs4: list_transition] :
( ( nfa_cong_Plus @ Q0 @ Q05 @ Q06 @ Qf @ Qf3 @ Qf4 @ Transs @ Transs3 @ Transs4 )
=> ( ( step_symb_set @ Q0 @ Transs @ ( insert_nat2 @ Q0 @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ) ).
% nfa_cong_Plus.step_symb_set_q0
thf(fact_733_nfa__cong_H_Oaccept__eps__nfa_H__run,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition,R: set_nat,Bss: list_list_o,Bs: list_o] :
( ( nfa_cong2 @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( ( ord_less_eq_set_nat @ R @ ( q @ Q05 @ Qf3 @ Transs3 ) )
=> ( ( accept_eps @ Q0 @ Qf @ Transs @ ( run @ Q05 @ Transs3 @ R @ Bss ) @ Bs )
= ( ( accept_eps @ Q05 @ Qf3 @ Transs3 @ ( run @ Q05 @ Transs3 @ R @ Bss ) @ Bs )
& ( accept_eps @ Q0 @ Qf @ Transs @ ( run @ Q0 @ Transs @ ( insert_nat2 @ Qf3 @ bot_bot_set_nat ) @ nil_list_o ) @ Bs ) ) ) ) ) ).
% nfa_cong'.accept_eps_nfa'_run
thf(fact_734_nfa__cong_H_Odelta__cong__reach,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition,R: set_nat,Bs: list_o] :
( ( nfa_cong2 @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( ( ord_less_eq_set_nat @ R @ ( q @ Q05 @ Qf3 @ Transs3 ) )
=> ( ( accept_eps @ Q05 @ Qf3 @ Transs3 @ R @ Bs )
=> ( ( delta @ Q0 @ Transs @ R @ Bs )
= ( sup_sup_set_nat @ ( delta @ Q05 @ Transs3 @ R @ Bs ) @ ( delta @ Q0 @ Transs @ ( insert_nat2 @ Qf3 @ bot_bot_set_nat ) @ Bs ) ) ) ) ) ) ).
% nfa_cong'.delta_cong_reach
thf(fact_735_insert__subsetI,axiom,
! [X3: transition,A2: set_transition,X4: set_transition] :
( ( member_transition @ X3 @ A2 )
=> ( ( ord_le8419162016481440574sition @ X4 @ A2 )
=> ( ord_le8419162016481440574sition @ ( insert_transition2 @ X3 @ X4 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_736_insert__subsetI,axiom,
! [X3: nat,A2: set_nat,X4: set_nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( ord_less_eq_set_nat @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ ( insert_nat2 @ X3 @ X4 ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_737_subset__emptyI,axiom,
! [A2: set_transition] :
( ! [X2: transition] :
~ ( member_transition @ X2 @ A2 )
=> ( ord_le8419162016481440574sition @ A2 @ bot_bo301567166201926666sition ) ) ).
% subset_emptyI
thf(fact_738_subset__emptyI,axiom,
! [A2: set_nat] :
( ! [X2: nat] :
~ ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_739_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_740_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_741_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_742_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_743_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_744_diff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_745_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_746_nfa__cong_H_Odelta__cong__unreach,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition,R: set_nat,Bs: list_o] :
( ( nfa_cong2 @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( ( ord_less_eq_set_nat @ R @ ( q @ Q05 @ Qf3 @ Transs3 ) )
=> ( ~ ( accept_eps @ Q05 @ Qf3 @ Transs3 @ R @ Bs )
=> ( ( delta @ Q0 @ Transs @ R @ Bs )
= ( delta @ Q05 @ Transs3 @ R @ Bs ) ) ) ) ) ).
% nfa_cong'.delta_cong_unreach
thf(fact_747_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_748_product__lists_Osimps_I1_J,axiom,
( ( produc6248909823095439149sition @ nil_list_transition )
= ( cons_list_transition @ nil_transition @ nil_list_transition ) ) ).
% product_lists.simps(1)
thf(fact_749_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_750_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_751_nfa__cong_H__def,axiom,
( nfa_cong2
= ( ^ [Q02: nat,Q03: nat,Qf2: nat,Qf5: nat,Transs2: list_transition,Transs5: list_transition] :
( ( nfa @ Q02 @ Qf2 @ Transs2 )
& ( nfa @ Q03 @ Qf5 @ Transs5 )
& ( nfa_cong_axioms @ Q02 @ Q03 @ Qf5 @ Transs2 @ Transs5 ) ) ) ) ).
% nfa_cong'_def
thf(fact_752_nfa__cong_H_Ointro,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,Q05: nat,Qf3: nat,Transs3: list_transition] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ( ( nfa @ Q05 @ Qf3 @ Transs3 )
=> ( ( nfa_cong_axioms @ Q0 @ Q05 @ Qf3 @ Transs @ Transs3 )
=> ( nfa_cong2 @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 ) ) ) ) ).
% nfa_cong'.intro
thf(fact_753_nfa__cong_H_Oaxioms_I3_J,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition] :
( ( nfa_cong2 @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( nfa_cong_axioms @ Q0 @ Q05 @ Qf3 @ Transs @ Transs3 ) ) ).
% nfa_cong'.axioms(3)
thf(fact_754_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_755_remove__def,axiom,
( remove_nat
= ( ^ [X: nat,A3: set_nat] : ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ).
% remove_def
thf(fact_756_case__left,axiom,
! [Q: nat] :
( ( step_eps_closure @ q0a @ transsa @ bs @ Q @ qfa )
=> ( ( member_nat @ Q @ ( q @ ( suc @ q0a ) @ qfa @ ts_l ) )
=> ( step_eps_closure @ q0a @ transsa @ nil_o @ Q @ qfa ) ) ) ).
% case_left
thf(fact_757_subseqs_Osimps_I1_J,axiom,
( ( subseqs_transition @ nil_transition )
= ( cons_list_transition @ nil_transition @ nil_list_transition ) ) ).
% subseqs.simps(1)
thf(fact_758_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_759_subseqs_Osimps_I1_J,axiom,
( ( subseqs_o @ nil_o )
= ( cons_list_o @ nil_o @ nil_list_o ) ) ).
% subseqs.simps(1)
thf(fact_760_left_Otranss__not__Nil,axiom,
ts_l != nil_transition ).
% left.transs_not_Nil
thf(fact_761_ts__nonempty_I1_J,axiom,
ts_l != nil_transition ).
% ts_nonempty(1)
thf(fact_762_member__remove,axiom,
! [X3: nat,Y3: nat,A2: set_nat] :
( ( member_nat @ X3 @ ( remove_nat @ Y3 @ A2 ) )
= ( ( member_nat @ X3 @ A2 )
& ( X3 != Y3 ) ) ) ).
% member_remove
thf(fact_763_member__remove,axiom,
! [X3: transition,Y3: transition,A2: set_transition] :
( ( member_transition @ X3 @ ( remove_transition @ Y3 @ A2 ) )
= ( ( member_transition @ X3 @ A2 )
& ( X3 != Y3 ) ) ) ).
% member_remove
thf(fact_764_left_Oqf__not__in__SQ,axiom,
~ ( member_nat @ qfa @ ( sq @ ( suc @ q0a ) @ ts_l ) ) ).
% left.qf_not_in_SQ
thf(fact_765_left_Ostep__eps__closure__qf,axiom,
! [Bs: list_o,Q: nat,Q2: nat] :
( ( step_eps_closure @ ( suc @ q0a ) @ ts_l @ Bs @ Q @ Q2 )
=> ( ( Q = qfa )
=> ( Q = Q2 ) ) ) ).
% left.step_eps_closure_qf
thf(fact_766_left_Onfa__axioms,axiom,
nfa @ ( suc @ q0a ) @ qfa @ ts_l ).
% left.nfa_axioms
thf(fact_767_left_Ostep__eps__qf,axiom,
! [Bs: list_o,Q: nat] :
~ ( step_eps @ ( suc @ q0a ) @ ts_l @ Bs @ qfa @ Q ) ).
% left.step_eps_qf
thf(fact_768_left_Ostep__symb__qf,axiom,
! [Q: nat] :
~ ( step_symb @ ( suc @ q0a ) @ ts_l @ qfa @ Q ) ).
% left.step_symb_qf
thf(fact_769_left_Ostep__eps__closure__closed,axiom,
! [Bs: list_o,Q: nat,Q2: nat] :
( ( step_eps_closure @ ( suc @ q0a ) @ ts_l @ Bs @ Q @ Q2 )
=> ( ( Q != Q2 )
=> ( member_nat @ Q2 @ ( q @ ( suc @ q0a ) @ qfa @ ts_l ) ) ) ) ).
% left.step_eps_closure_closed
thf(fact_770_left_Ostep__eps__closed,axiom,
! [Bs: list_o,Q: nat,Q2: nat] :
( ( step_eps @ ( suc @ q0a ) @ ts_l @ Bs @ Q @ Q2 )
=> ( member_nat @ Q2 @ ( q @ ( suc @ q0a ) @ qfa @ ts_l ) ) ) ).
% left.step_eps_closed
thf(fact_771_cong_OSQ__sub,axiom,
ord_less_eq_set_nat @ ( sq @ ( suc @ q0a ) @ ts_l ) @ ( sq @ q0a @ transsa ) ).
% cong.SQ_sub
thf(fact_772_left_Ostep__symb__closed,axiom,
! [Q: nat,Q2: nat] :
( ( step_symb @ ( suc @ q0a ) @ ts_l @ Q @ Q2 )
=> ( member_nat @ Q2 @ ( q @ ( suc @ q0a ) @ qfa @ ts_l ) ) ) ).
% left.step_symb_closed
thf(fact_773_left_Oq0__sub__SQ,axiom,
ord_less_eq_set_nat @ ( insert_nat2 @ ( suc @ q0a ) @ bot_bot_set_nat ) @ ( sq @ ( suc @ q0a ) @ ts_l ) ).
% left.q0_sub_SQ
thf(fact_774_left_Ostep__eps__closure__set__qf,axiom,
! [Bs: list_o] :
( ( step_eps_closure_set @ ( suc @ q0a ) @ ts_l @ ( insert_nat2 @ qfa @ bot_bot_set_nat ) @ Bs )
= ( insert_nat2 @ qfa @ bot_bot_set_nat ) ) ).
% left.step_eps_closure_set_qf
thf(fact_775_left_Odelta__qf,axiom,
! [Bs: list_o] :
( ( delta @ ( suc @ q0a ) @ ts_l @ ( insert_nat2 @ qfa @ bot_bot_set_nat ) @ Bs )
= bot_bot_set_nat ) ).
% left.delta_qf
thf(fact_776_left_Ostep__symb__set__qf,axiom,
( ( step_symb_set @ ( suc @ q0a ) @ ts_l @ ( insert_nat2 @ qfa @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ).
% left.step_symb_set_qf
thf(fact_777_left_Ostep__eps__closure__set__closed,axiom,
! [R: set_nat,Bs: list_o] :
( ( ord_less_eq_set_nat @ R @ ( q @ ( suc @ q0a ) @ qfa @ ts_l ) )
=> ( ord_less_eq_set_nat @ ( step_eps_closure_set @ ( suc @ q0a ) @ ts_l @ R @ Bs ) @ ( q @ ( suc @ q0a ) @ qfa @ ts_l ) ) ) ).
% left.step_eps_closure_set_closed
thf(fact_778_left_Odelta__closed,axiom,
! [R: set_nat,Bs: list_o] : ( ord_less_eq_set_nat @ ( delta @ ( suc @ q0a ) @ ts_l @ R @ Bs ) @ ( q @ ( suc @ q0a ) @ qfa @ ts_l ) ) ).
% left.delta_closed
thf(fact_779_left_Orun__closed,axiom,
! [R: set_nat,Bss: list_list_o] :
( ( ord_less_eq_set_nat @ R @ ( q @ ( suc @ q0a ) @ qfa @ ts_l ) )
=> ( ord_less_eq_set_nat @ ( run @ ( suc @ q0a ) @ ts_l @ R @ Bss ) @ ( q @ ( suc @ q0a ) @ qfa @ ts_l ) ) ) ).
% left.run_closed
thf(fact_780_left_Ostep__symb__set__closed,axiom,
! [R: set_nat] : ( ord_less_eq_set_nat @ ( step_symb_set @ ( suc @ q0a ) @ ts_l @ R ) @ ( q @ ( suc @ q0a ) @ qfa @ ts_l ) ) ).
% left.step_symb_set_closed
thf(fact_781_left_Ostep__eps__set__closed,axiom,
! [Bs: list_o,R: set_nat] : ( ord_less_eq_set_nat @ ( step_eps_set @ ( suc @ q0a ) @ ts_l @ Bs @ R ) @ ( q @ ( suc @ q0a ) @ qfa @ ts_l ) ) ).
% left.step_eps_set_closed
thf(fact_782_cong_Oq__Q__SQ__nfa_H__SQ,axiom,
! [Q: nat] :
( ( member_nat @ Q @ ( q @ ( suc @ q0a ) @ qfa @ ts_l ) )
=> ( ( member_nat @ Q @ ( sq @ q0a @ transsa ) )
= ( member_nat @ Q @ ( sq @ ( suc @ q0a ) @ ts_l ) ) ) ) ).
% cong.q_Q_SQ_nfa'_SQ
thf(fact_783_cong_Oeps__nfa_H__step__eps__closure,axiom,
! [Bs: list_o,Q: nat,Q2: nat] :
( ( step_eps_closure @ q0a @ transsa @ Bs @ Q @ Q2 )
=> ( ( member_nat @ Q @ ( q @ ( suc @ q0a ) @ qfa @ ts_l ) )
=> ( ( member_nat @ Q2 @ ( q @ ( suc @ q0a ) @ qfa @ ts_l ) )
& ( step_eps_closure @ ( suc @ q0a ) @ ts_l @ Bs @ Q @ Q2 ) ) ) ) ).
% cong.eps_nfa'_step_eps_closure
thf(fact_784_cong_Onfa_H__eps__step__eps__closure,axiom,
! [Bs: list_o,Q: nat,Q2: nat] :
( ( step_eps_closure @ ( suc @ q0a ) @ ts_l @ Bs @ Q @ Q2 )
=> ( ( member_nat @ Q @ ( q @ ( suc @ q0a ) @ qfa @ ts_l ) )
=> ( ( member_nat @ Q2 @ ( q @ ( suc @ q0a ) @ qfa @ ts_l ) )
& ( step_eps_closure @ q0a @ transsa @ Bs @ Q @ Q2 ) ) ) ) ).
% cong.nfa'_eps_step_eps_closure
thf(fact_785_cong_Ostep__eps__cong,axiom,
! [Q: nat,Bs: list_o,Q2: nat] :
( ( member_nat @ Q @ ( q @ ( suc @ q0a ) @ qfa @ ts_l ) )
=> ( ( step_eps @ q0a @ transsa @ Bs @ Q @ Q2 )
= ( step_eps @ ( suc @ q0a ) @ ts_l @ Bs @ Q @ Q2 ) ) ) ).
% cong.step_eps_cong
thf(fact_786_cong_Ostep__symb__cong,axiom,
! [Q: nat,Q2: nat] :
( ( member_nat @ Q @ ( q @ ( suc @ q0a ) @ qfa @ ts_l ) )
=> ( ( step_symb @ q0a @ transsa @ Q @ Q2 )
= ( step_symb @ ( suc @ q0a ) @ ts_l @ Q @ Q2 ) ) ) ).
% cong.step_symb_cong
thf(fact_787_left_Oq0__sub__Q,axiom,
ord_less_eq_set_nat @ ( insert_nat2 @ ( suc @ q0a ) @ bot_bot_set_nat ) @ ( q @ ( suc @ q0a ) @ qfa @ ts_l ) ).
% left.q0_sub_Q
thf(fact_788_left_Orun__accept__eps__qf__many,axiom,
! [Bs: list_o,Bss: list_list_o,Cs: list_o] :
~ ( run_accept_eps @ ( suc @ q0a ) @ qfa @ ts_l @ ( insert_nat2 @ qfa @ bot_bot_set_nat ) @ ( cons_list_o @ Bs @ Bss ) @ Cs ) ).
% left.run_accept_eps_qf_many
thf(fact_789_left_Orun__qf__many,axiom,
! [Bs: list_o,Bss: list_list_o] :
( ( run @ ( suc @ q0a ) @ ts_l @ ( insert_nat2 @ qfa @ bot_bot_set_nat ) @ ( cons_list_o @ Bs @ Bss ) )
= bot_bot_set_nat ) ).
% left.run_qf_many
thf(fact_790_left_Orun__accept__eps__qf__one,axiom,
! [Bs: list_o] : ( run_accept_eps @ ( suc @ q0a ) @ qfa @ ts_l @ ( insert_nat2 @ qfa @ bot_bot_set_nat ) @ nil_list_o @ Bs ) ).
% left.run_accept_eps_qf_one
thf(fact_791_left_Ostep__eps__closure__set__closed__union,axiom,
! [R: set_nat,Bs: list_o] : ( ord_less_eq_set_nat @ ( step_eps_closure_set @ ( suc @ q0a ) @ ts_l @ R @ Bs ) @ ( sup_sup_set_nat @ R @ ( q @ ( suc @ q0a ) @ qfa @ ts_l ) ) ) ).
% left.step_eps_closure_set_closed_union
thf(fact_792_left_Orun__closed__Cons,axiom,
! [R: set_nat,Bs: list_o,Bss: list_list_o] : ( ord_less_eq_set_nat @ ( run @ ( suc @ q0a ) @ ts_l @ R @ ( cons_list_o @ Bs @ Bss ) ) @ ( q @ ( suc @ q0a ) @ qfa @ ts_l ) ) ).
% left.run_closed_Cons
thf(fact_793_left_Ostate__closed,axiom,
! [T2: transition] :
( ( member_transition @ T2 @ ( set_transition2 @ ts_l ) )
=> ( ord_less_eq_set_nat @ ( state_set @ T2 ) @ ( q @ ( suc @ q0a ) @ qfa @ ts_l ) ) ) ).
% left.state_closed
thf(fact_794_cong_Ostep__eps__closure__set__cong,axiom,
! [R: set_nat,Bs: list_o] :
( ( ord_less_eq_set_nat @ R @ ( q @ ( suc @ q0a ) @ qfa @ ts_l ) )
=> ( ( step_eps_closure_set @ q0a @ transsa @ R @ Bs )
= ( step_eps_closure_set @ ( suc @ q0a ) @ ts_l @ R @ Bs ) ) ) ).
% cong.step_eps_closure_set_cong
thf(fact_795_cong_Odelta__cong,axiom,
! [R: set_nat,Bs: list_o] :
( ( ord_less_eq_set_nat @ R @ ( q @ ( suc @ q0a ) @ qfa @ ts_l ) )
=> ( ( delta @ q0a @ transsa @ R @ Bs )
= ( delta @ ( suc @ q0a ) @ ts_l @ R @ Bs ) ) ) ).
% cong.delta_cong
thf(fact_796_cong_Orun__cong,axiom,
! [R: set_nat,Bss: list_list_o] :
( ( ord_less_eq_set_nat @ R @ ( q @ ( suc @ q0a ) @ qfa @ ts_l ) )
=> ( ( run @ q0a @ transsa @ R @ Bss )
= ( run @ ( suc @ q0a ) @ ts_l @ R @ Bss ) ) ) ).
% cong.run_cong
thf(fact_797_cong_Ostep__symb__set__cong,axiom,
! [R: set_nat] :
( ( ord_less_eq_set_nat @ R @ ( q @ ( suc @ q0a ) @ qfa @ ts_l ) )
=> ( ( step_symb_set @ q0a @ transsa @ R )
= ( step_symb_set @ ( suc @ q0a ) @ ts_l @ R ) ) ) ).
% cong.step_symb_set_cong
thf(fact_798_cong_Oaccept__eps__cong,axiom,
! [R: set_nat,Bs: list_o] :
( ( ord_less_eq_set_nat @ R @ ( q @ ( suc @ q0a ) @ qfa @ ts_l ) )
=> ( ( accept_eps @ q0a @ qfa @ transsa @ R @ Bs )
= ( accept_eps @ ( suc @ q0a ) @ qfa @ ts_l @ R @ Bs ) ) ) ).
% cong.accept_eps_cong
thf(fact_799_left_OQ__diff__qf__SQ,axiom,
( ( minus_minus_set_nat @ ( q @ ( suc @ q0a ) @ qfa @ ts_l ) @ ( insert_nat2 @ qfa @ bot_bot_set_nat ) )
= ( sq @ ( suc @ q0a ) @ ts_l ) ) ).
% left.Q_diff_qf_SQ
thf(fact_800_Cons__in__subseqsD,axiom,
! [Y3: list_o,Ys: list_list_o,Xs2: list_list_o] :
( ( member_list_list_o @ ( cons_list_o @ Y3 @ Ys ) @ ( set_list_list_o2 @ ( subseqs_list_o @ Xs2 ) ) )
=> ( member_list_list_o @ Ys @ ( set_list_list_o2 @ ( subseqs_list_o @ Xs2 ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_801_remove__code_I1_J,axiom,
! [X3: transition,Xs2: list_transition] :
( ( remove_transition @ X3 @ ( set_transition2 @ Xs2 ) )
= ( set_transition2 @ ( removeAll_transition @ X3 @ Xs2 ) ) ) ).
% remove_code(1)
thf(fact_802_cong_Onfa__cong__axioms,axiom,
nfa_cong @ q0a @ ( suc @ q0a ) @ qfa @ qfa @ transsa @ ts_l ).
% cong.nfa_cong_axioms
thf(fact_803_cong_Otranss__eq,axiom,
! [Q: nat] :
( ( member_nat @ Q @ ( sq @ ( suc @ q0a ) @ ts_l ) )
=> ( ( nth_transition @ transsa @ ( minus_minus_nat @ Q @ q0a ) )
= ( nth_transition @ ts_l @ ( minus_minus_nat @ Q @ ( suc @ q0a ) ) ) ) ) ).
% cong.transs_eq
thf(fact_804_lift__Suc__antimono__le,axiom,
! [F: nat > set_nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_805_lift__Suc__mono__le,axiom,
! [F: nat > set_nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_set_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_set_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_806_nth__Cons__Suc,axiom,
! [X3: transition,Xs2: list_transition,N: nat] :
( ( nth_transition @ ( cons_transition @ X3 @ Xs2 ) @ ( suc @ N ) )
= ( nth_transition @ Xs2 @ N ) ) ).
% nth_Cons_Suc
thf(fact_807_nth__Cons__Suc,axiom,
! [X3: list_o,Xs2: list_list_o,N: nat] :
( ( nth_list_o @ ( cons_list_o @ X3 @ Xs2 ) @ ( suc @ N ) )
= ( nth_list_o @ Xs2 @ N ) ) ).
% nth_Cons_Suc
thf(fact_808_nfa__cong_Otranss__eq,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition,Q: nat] :
( ( nfa_cong @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( ( member_nat @ Q @ ( sq @ Q05 @ Transs3 ) )
=> ( ( nth_transition @ Transs @ ( minus_minus_nat @ Q @ Q0 ) )
= ( nth_transition @ Transs3 @ ( minus_minus_nat @ Q @ Q05 ) ) ) ) ) ).
% nfa_cong.transs_eq
thf(fact_809_nfa__cong_Oqf__eq,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition] :
( ( nfa_cong @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( Qf = Qf3 ) ) ).
% nfa_cong.qf_eq
thf(fact_810_nfa__cong_Oaxioms_I2_J,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition] :
( ( nfa_cong @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( nfa @ Q05 @ Qf3 @ Transs3 ) ) ).
% nfa_cong.axioms(2)
thf(fact_811_nfa__cong_Oaxioms_I1_J,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition] :
( ( nfa_cong @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( nfa @ Q0 @ Qf @ Transs ) ) ).
% nfa_cong.axioms(1)
thf(fact_812_nfa__cong__Plus_Oaxioms_I1_J,axiom,
! [Q0: nat,Q05: nat,Q06: nat,Qf: nat,Qf3: nat,Qf4: nat,Transs: list_transition,Transs3: list_transition,Transs4: list_transition] :
( ( nfa_cong_Plus @ Q0 @ Q05 @ Q06 @ Qf @ Qf3 @ Qf4 @ Transs @ Transs3 @ Transs4 )
=> ( nfa_cong @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 ) ) ).
% nfa_cong_Plus.axioms(1)
thf(fact_813_nfa__cong__Plus_Oaxioms_I2_J,axiom,
! [Q0: nat,Q05: nat,Q06: nat,Qf: nat,Qf3: nat,Qf4: nat,Transs: list_transition,Transs3: list_transition,Transs4: list_transition] :
( ( nfa_cong_Plus @ Q0 @ Q05 @ Q06 @ Qf @ Qf3 @ Qf4 @ Transs @ Transs3 @ Transs4 )
=> ( nfa_cong @ Q0 @ Q06 @ Qf @ Qf4 @ Transs @ Transs4 ) ) ).
% nfa_cong_Plus.axioms(2)
thf(fact_814_nfa__cong_H_Otranss__eq,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition,Q: nat] :
( ( nfa_cong2 @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( ( member_nat @ Q @ ( sq @ Q05 @ Transs3 ) )
=> ( ( nth_transition @ Transs @ ( minus_minus_nat @ Q @ Q0 ) )
= ( nth_transition @ Transs3 @ ( minus_minus_nat @ Q @ Q05 ) ) ) ) ) ).
% nfa_cong'.transs_eq
thf(fact_815_transs__q__in__set,axiom,
! [Q: nat,Q0: nat,Transs: list_transition] :
( ( member_nat @ Q @ ( sq @ Q0 @ Transs ) )
=> ( member_transition @ ( nth_transition @ Transs @ ( minus_minus_nat @ Q @ Q0 ) ) @ ( set_transition2 @ Transs ) ) ) ).
% transs_q_in_set
thf(fact_816_nfa__cong_Oq__Q__SQ__nfa_H__SQ,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition,Q: nat] :
( ( nfa_cong @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( ( member_nat @ Q @ ( q @ Q05 @ Qf3 @ Transs3 ) )
=> ( ( member_nat @ Q @ ( sq @ Q0 @ Transs ) )
= ( member_nat @ Q @ ( sq @ Q05 @ Transs3 ) ) ) ) ) ).
% nfa_cong.q_Q_SQ_nfa'_SQ
thf(fact_817_nfa__cong_OSQ__sub,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition] :
( ( nfa_cong @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( ord_less_eq_set_nat @ ( sq @ Q05 @ Transs3 ) @ ( sq @ Q0 @ Transs ) ) ) ).
% nfa_cong.SQ_sub
thf(fact_818_nfa__cong_Oeps__nfa_H__step__eps__closure,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition,Bs: list_o,Q: nat,Q2: nat] :
( ( nfa_cong @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( ( step_eps_closure @ Q0 @ Transs @ Bs @ Q @ Q2 )
=> ( ( member_nat @ Q @ ( q @ Q05 @ Qf3 @ Transs3 ) )
=> ( ( member_nat @ Q2 @ ( q @ Q05 @ Qf3 @ Transs3 ) )
& ( step_eps_closure @ Q05 @ Transs3 @ Bs @ Q @ Q2 ) ) ) ) ) ).
% nfa_cong.eps_nfa'_step_eps_closure
thf(fact_819_nfa__cong_Onfa_H__eps__step__eps__closure,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition,Bs: list_o,Q: nat,Q2: nat] :
( ( nfa_cong @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( ( step_eps_closure @ Q05 @ Transs3 @ Bs @ Q @ Q2 )
=> ( ( member_nat @ Q @ ( q @ Q05 @ Qf3 @ Transs3 ) )
=> ( ( member_nat @ Q2 @ ( q @ Q05 @ Qf3 @ Transs3 ) )
& ( step_eps_closure @ Q0 @ Transs @ Bs @ Q @ Q2 ) ) ) ) ) ).
% nfa_cong.nfa'_eps_step_eps_closure
thf(fact_820_nfa__cong_Ostep__eps__cong,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition,Q: nat,Bs: list_o,Q2: nat] :
( ( nfa_cong @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( ( member_nat @ Q @ ( q @ Q05 @ Qf3 @ Transs3 ) )
=> ( ( step_eps @ Q0 @ Transs @ Bs @ Q @ Q2 )
= ( step_eps @ Q05 @ Transs3 @ Bs @ Q @ Q2 ) ) ) ) ).
% nfa_cong.step_eps_cong
thf(fact_821_nfa__cong_Ostep__symb__cong,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition,Q: nat,Q2: nat] :
( ( nfa_cong @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( ( member_nat @ Q @ ( q @ Q05 @ Qf3 @ Transs3 ) )
=> ( ( step_symb @ Q0 @ Transs @ Q @ Q2 )
= ( step_symb @ Q05 @ Transs3 @ Q @ Q2 ) ) ) ) ).
% nfa_cong.step_symb_cong
thf(fact_822_nfa__cong_H__axioms__def,axiom,
( nfa_cong_axioms
= ( ^ [Q02: nat,Q03: nat,Qf5: nat,Transs2: list_transition,Transs5: list_transition] :
( ( ord_less_eq_set_nat @ ( sq @ Q03 @ Transs5 ) @ ( sq @ Q02 @ Transs2 ) )
& ( member_nat @ Qf5 @ ( sq @ Q02 @ Transs2 ) )
& ! [Q5: nat] :
( ( member_nat @ Q5 @ ( sq @ Q03 @ Transs5 ) )
=> ( ( nth_transition @ Transs2 @ ( minus_minus_nat @ Q5 @ Q02 ) )
= ( nth_transition @ Transs5 @ ( minus_minus_nat @ Q5 @ Q03 ) ) ) ) ) ) ) ).
% nfa_cong'_axioms_def
thf(fact_823_nfa__cong_H__axioms_Ointro,axiom,
! [Q05: nat,Transs3: list_transition,Q0: nat,Transs: list_transition,Qf3: nat] :
( ( ord_less_eq_set_nat @ ( sq @ Q05 @ Transs3 ) @ ( sq @ Q0 @ Transs ) )
=> ( ( member_nat @ Qf3 @ ( sq @ Q0 @ Transs ) )
=> ( ! [Q6: nat] :
( ( member_nat @ Q6 @ ( sq @ Q05 @ Transs3 ) )
=> ( ( nth_transition @ Transs @ ( minus_minus_nat @ Q6 @ Q0 ) )
= ( nth_transition @ Transs3 @ ( minus_minus_nat @ Q6 @ Q05 ) ) ) )
=> ( nfa_cong_axioms @ Q0 @ Q05 @ Qf3 @ Transs @ Transs3 ) ) ) ) ).
% nfa_cong'_axioms.intro
thf(fact_824_nfa__cong__Plus__def,axiom,
( nfa_cong_Plus
= ( ^ [Q02: nat,Q03: nat,Q04: nat,Qf2: nat,Qf5: nat,Qf6: nat,Transs2: list_transition,Transs5: list_transition,Transs6: list_transition] :
( ( nfa_cong @ Q02 @ Q03 @ Qf2 @ Qf5 @ Transs2 @ Transs5 )
& ( nfa_cong @ Q02 @ Q04 @ Qf2 @ Qf6 @ Transs2 @ Transs6 )
& ( nfa_cong_Plus_axioms @ Q02 @ Q03 @ Q04 @ Transs2 ) ) ) ) ).
% nfa_cong_Plus_def
thf(fact_825_nfa__cong__Plus_Ointro,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition,Q06: nat,Qf4: nat,Transs4: list_transition] :
( ( nfa_cong @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( ( nfa_cong @ Q0 @ Q06 @ Qf @ Qf4 @ Transs @ Transs4 )
=> ( ( nfa_cong_Plus_axioms @ Q0 @ Q05 @ Q06 @ Transs )
=> ( nfa_cong_Plus @ Q0 @ Q05 @ Q06 @ Qf @ Qf3 @ Qf4 @ Transs @ Transs3 @ Transs4 ) ) ) ) ).
% nfa_cong_Plus.intro
thf(fact_826_nfa__cong_Orun__accept__eps__cong,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition,R: set_nat,Bss: list_list_o,Bs: list_o] :
( ( nfa_cong @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( ( ord_less_eq_set_nat @ R @ ( q @ Q05 @ Qf3 @ Transs3 ) )
=> ( ( run_accept_eps @ Q0 @ Qf @ Transs @ R @ Bss @ Bs )
= ( run_accept_eps @ Q05 @ Qf3 @ Transs3 @ R @ Bss @ Bs ) ) ) ) ).
% nfa_cong.run_accept_eps_cong
thf(fact_827_nfa__cong_Ostep__eps__closure__set__cong,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition,R: set_nat,Bs: list_o] :
( ( nfa_cong @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( ( ord_less_eq_set_nat @ R @ ( q @ Q05 @ Qf3 @ Transs3 ) )
=> ( ( step_eps_closure_set @ Q0 @ Transs @ R @ Bs )
= ( step_eps_closure_set @ Q05 @ Transs3 @ R @ Bs ) ) ) ) ).
% nfa_cong.step_eps_closure_set_cong
thf(fact_828_nfa__cong_Odelta__cong,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition,R: set_nat,Bs: list_o] :
( ( nfa_cong @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( ( ord_less_eq_set_nat @ R @ ( q @ Q05 @ Qf3 @ Transs3 ) )
=> ( ( delta @ Q0 @ Transs @ R @ Bs )
= ( delta @ Q05 @ Transs3 @ R @ Bs ) ) ) ) ).
% nfa_cong.delta_cong
thf(fact_829_nfa__cong_Orun__cong,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition,R: set_nat,Bss: list_list_o] :
( ( nfa_cong @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( ( ord_less_eq_set_nat @ R @ ( q @ Q05 @ Qf3 @ Transs3 ) )
=> ( ( run @ Q0 @ Transs @ R @ Bss )
= ( run @ Q05 @ Transs3 @ R @ Bss ) ) ) ) ).
% nfa_cong.run_cong
thf(fact_830_nfa__cong_Ostep__symb__set__cong,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition,R: set_nat] :
( ( nfa_cong @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( ( ord_less_eq_set_nat @ R @ ( q @ Q05 @ Qf3 @ Transs3 ) )
=> ( ( step_symb_set @ Q0 @ Transs @ R )
= ( step_symb_set @ Q05 @ Transs3 @ R ) ) ) ) ).
% nfa_cong.step_symb_set_cong
thf(fact_831_nfa__cong_Oaccept__eps__cong,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition,R: set_nat,Bs: list_o] :
( ( nfa_cong @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( ( ord_less_eq_set_nat @ R @ ( q @ Q05 @ Qf3 @ Transs3 ) )
=> ( ( accept_eps @ Q0 @ Qf @ Transs @ R @ Bs )
= ( accept_eps @ Q05 @ Qf3 @ Transs3 @ R @ Bs ) ) ) ) ).
% nfa_cong.accept_eps_cong
thf(fact_832_nfa__cong__axioms__def,axiom,
( nfa_cong_axioms2
= ( ^ [Q02: nat,Q03: nat,Qf2: nat,Qf5: nat,Transs2: list_transition,Transs5: list_transition] :
( ( ord_less_eq_set_nat @ ( sq @ Q03 @ Transs5 ) @ ( sq @ Q02 @ Transs2 ) )
& ( Qf2 = Qf5 )
& ! [Q5: nat] :
( ( member_nat @ Q5 @ ( sq @ Q03 @ Transs5 ) )
=> ( ( nth_transition @ Transs2 @ ( minus_minus_nat @ Q5 @ Q02 ) )
= ( nth_transition @ Transs5 @ ( minus_minus_nat @ Q5 @ Q03 ) ) ) ) ) ) ) ).
% nfa_cong_axioms_def
thf(fact_833_nfa__cong__axioms_Ointro,axiom,
! [Q05: nat,Transs3: list_transition,Q0: nat,Transs: list_transition,Qf: nat,Qf3: nat] :
( ( ord_less_eq_set_nat @ ( sq @ Q05 @ Transs3 ) @ ( sq @ Q0 @ Transs ) )
=> ( ( Qf = Qf3 )
=> ( ! [Q6: nat] :
( ( member_nat @ Q6 @ ( sq @ Q05 @ Transs3 ) )
=> ( ( nth_transition @ Transs @ ( minus_minus_nat @ Q6 @ Q0 ) )
= ( nth_transition @ Transs3 @ ( minus_minus_nat @ Q6 @ Q05 ) ) ) )
=> ( nfa_cong_axioms2 @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 ) ) ) ) ).
% nfa_cong_axioms.intro
thf(fact_834_nfa__cong__Times_Ointro,axiom,
! [Q0: nat,Qf: nat,Q05: nat,Transs: list_transition,Transs3: list_transition,Transs4: list_transition] :
( ( nfa_cong2 @ Q0 @ Q0 @ Qf @ Q05 @ Transs @ Transs3 )
=> ( ( nfa_cong @ Q0 @ Q05 @ Qf @ Qf @ Transs @ Transs4 )
=> ( nfa_cong_Times @ Q0 @ Q05 @ Qf @ Transs @ Transs3 @ Transs4 ) ) ) ).
% nfa_cong_Times.intro
thf(fact_835_nfa__cong__Times_Oaxioms_I1_J,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Transs: list_transition,Transs3: list_transition,Transs4: list_transition] :
( ( nfa_cong_Times @ Q0 @ Q05 @ Qf @ Transs @ Transs3 @ Transs4 )
=> ( nfa_cong2 @ Q0 @ Q0 @ Qf @ Q05 @ Transs @ Transs3 ) ) ).
% nfa_cong_Times.axioms(1)
thf(fact_836_nfa__cong_Oaxioms_I3_J,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Qf3: nat,Transs: list_transition,Transs3: list_transition] :
( ( nfa_cong @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( nfa_cong_axioms2 @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 ) ) ).
% nfa_cong.axioms(3)
thf(fact_837_nfa__cong__Times_Oaxioms_I2_J,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Transs: list_transition,Transs3: list_transition,Transs4: list_transition] :
( ( nfa_cong_Times @ Q0 @ Q05 @ Qf @ Transs @ Transs3 @ Transs4 )
=> ( nfa_cong @ Q0 @ Q05 @ Qf @ Qf @ Transs @ Transs4 ) ) ).
% nfa_cong_Times.axioms(2)
thf(fact_838_nfa__cong_Ointro,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,Q05: nat,Qf3: nat,Transs3: list_transition] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ( ( nfa @ Q05 @ Qf3 @ Transs3 )
=> ( ( nfa_cong_axioms2 @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 )
=> ( nfa_cong @ Q0 @ Q05 @ Qf @ Qf3 @ Transs @ Transs3 ) ) ) ) ).
% nfa_cong.intro
thf(fact_839_nfa__cong__def,axiom,
( nfa_cong
= ( ^ [Q02: nat,Q03: nat,Qf2: nat,Qf5: nat,Transs2: list_transition,Transs5: list_transition] :
( ( nfa @ Q02 @ Qf2 @ Transs2 )
& ( nfa @ Q03 @ Qf5 @ Transs5 )
& ( nfa_cong_axioms2 @ Q02 @ Q03 @ Qf2 @ Qf5 @ Transs2 @ Transs5 ) ) ) ) ).
% nfa_cong_def
thf(fact_840_nfa__cong__Times__def,axiom,
( nfa_cong_Times
= ( ^ [Q02: nat,Q03: nat,Qf2: nat,Transs2: list_transition,Transs5: list_transition,Transs6: list_transition] :
( ( nfa_cong2 @ Q02 @ Q02 @ Qf2 @ Q03 @ Transs2 @ Transs5 )
& ( nfa_cong @ Q02 @ Q03 @ Qf2 @ Qf2 @ Transs2 @ Transs6 ) ) ) ) ).
% nfa_cong_Times_def
thf(fact_841_gen__length__code_I2_J,axiom,
! [N: nat,X3: list_o,Xs2: list_list_o] :
( ( gen_length_list_o @ N @ ( cons_list_o @ X3 @ Xs2 ) )
= ( gen_length_list_o @ ( suc @ N ) @ Xs2 ) ) ).
% gen_length_code(2)
thf(fact_842_nfa__cong__Star_Ortranclp__step__eps__q0__q0_H,axiom,
! [Q0: nat,Q05: nat,Qf: nat,Transs: list_transition,Transs3: list_transition,Bs: list_o,Q: nat,Q2: nat] :
( ( nfa_cong_Star @ Q0 @ Q05 @ Qf @ Transs @ Transs3 )
=> ( ( transi5422400438309235013lp_nat @ ( step_eps @ Q0 @ Transs @ Bs ) @ Q @ Q2 )
=> ( ( Q = Q0 )
=> ( ( member_nat @ Q2 @ ( insert_nat2 @ Q0 @ ( insert_nat2 @ Qf @ bot_bot_set_nat ) ) )
| ( ( member_nat @ Q2 @ ( sq @ Q05 @ Transs3 ) )
& ( transi5422400438309235013lp_nat @ ( step_eps @ Q05 @ Transs3 @ Bs ) @ Q05 @ Q2 ) ) ) ) ) ) ).
% nfa_cong_Star.rtranclp_step_eps_q0_q0'
thf(fact_843_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_844_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_o @ ( coset_o @ nil_o ) @ ( set_o2 @ nil_o ) ) ).
% subset_code(3)
thf(fact_845_subset__code_I3_J,axiom,
~ ( ord_le8419162016481440574sition @ ( coset_transition @ nil_transition ) @ ( set_transition2 @ nil_transition ) ) ).
% subset_code(3)
thf(fact_846_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_nat @ ( coset_nat @ nil_nat ) @ ( set_nat2 @ nil_nat ) ) ).
% subset_code(3)
thf(fact_847_rtranclp__closed__sub_H,axiom,
! [R: nat > nat > $o,Q: nat,Q2: nat] :
( ( transi5422400438309235013lp_nat @ R @ Q @ Q2 )
=> ( ( Q2 = Q )
| ? [Q7: nat] :
( ( R @ Q @ Q7 )
& ( transi5422400438309235013lp_nat @ R @ Q7 @ Q2 ) ) ) ) ).
% rtranclp_closed_sub'
thf(fact_848_rtranclp__unfold,axiom,
! [R: nat > nat > $o,X3: nat,Z2: nat] :
( ( transi5422400438309235013lp_nat @ R @ X3 @ Z2 )
=> ( ( X3 = Z2 )
| ? [Y: nat] :
( ( R @ X3 @ Y )
& ( transi5422400438309235013lp_nat @ R @ Y @ Z2 ) ) ) ) ).
% rtranclp_unfold
thf(fact_849_rtranclp__step,axiom,
! [R: transition > transition > $o,Q: transition,Q8: transition,X4: set_transition] :
( ( transi7192379879768945417sition @ R @ Q @ Q8 )
=> ( ! [Q3: transition] :
( ( R @ Q @ Q3 )
= ( member_transition @ Q3 @ X4 ) )
=> ( ( Q = Q8 )
| ? [X2: transition] :
( ( member_transition @ X2 @ X4 )
& ( R @ Q @ X2 )
& ( transi7192379879768945417sition @ R @ X2 @ Q8 ) ) ) ) ) ).
% rtranclp_step
thf(fact_850_rtranclp__step,axiom,
! [R: nat > nat > $o,Q: nat,Q8: nat,X4: set_nat] :
( ( transi5422400438309235013lp_nat @ R @ Q @ Q8 )
=> ( ! [Q3: nat] :
( ( R @ Q @ Q3 )
= ( member_nat @ Q3 @ X4 ) )
=> ( ( Q = Q8 )
| ? [X2: nat] :
( ( member_nat @ X2 @ X4 )
& ( R @ Q @ X2 )
& ( transi5422400438309235013lp_nat @ R @ X2 @ Q8 ) ) ) ) ) ).
% rtranclp_step
thf(fact_851_NFA_Ostep__eps__closure__def,axiom,
( step_eps_closure
= ( ^ [Q02: nat,Transs2: list_transition,Bs2: list_o] : ( transi5422400438309235013lp_nat @ ( step_eps @ Q02 @ Transs2 @ Bs2 ) ) ) ) ).
% NFA.step_eps_closure_def
thf(fact_852_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_le2281773382894182225sition @ N @ nil_transition )
= N ) ).
% gen_length_code(1)
thf(fact_853_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_list_o @ N @ nil_list_o )
= N ) ).
% gen_length_code(1)
thf(fact_854_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_o @ N @ nil_o )
= N ) ).
% gen_length_code(1)
thf(fact_855_subset__code_I2_J,axiom,
! [A2: set_transition,Ys: list_transition] :
( ( ord_le8419162016481440574sition @ A2 @ ( coset_transition @ Ys ) )
= ( ! [X: transition] :
( ( member_transition @ X @ ( set_transition2 @ Ys ) )
=> ~ ( member_transition @ X @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_856_subset__code_I2_J,axiom,
! [A2: set_nat,Ys: list_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( coset_nat @ Ys ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Ys ) )
=> ~ ( member_nat @ X @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_857_insert__code_I2_J,axiom,
! [X3: nat,Xs2: list_nat] :
( ( insert_nat2 @ X3 @ ( coset_nat @ Xs2 ) )
= ( coset_nat @ ( removeAll_nat @ X3 @ Xs2 ) ) ) ).
% insert_code(2)
thf(fact_858_rtranclp__reflclp,axiom,
! [R: nat > nat > $o] :
( ( transi5422400438309235013lp_nat
@ ( sup_sup_nat_nat_o @ R
@ ^ [Y2: nat,Z: nat] : ( Y2 = Z ) ) )
= ( transi5422400438309235013lp_nat @ R ) ) ).
% rtranclp_reflclp
thf(fact_859_rtranclp__reflclp__absorb,axiom,
! [R: nat > nat > $o] :
( ( sup_sup_nat_nat_o @ ( transi5422400438309235013lp_nat @ R )
@ ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( transi5422400438309235013lp_nat @ R ) ) ).
% rtranclp_reflclp_absorb
thf(fact_860_List_Oset__insert,axiom,
! [X3: nat,Xs2: list_nat] :
( ( set_nat2 @ ( insert_nat @ X3 @ Xs2 ) )
= ( insert_nat2 @ X3 @ ( set_nat2 @ Xs2 ) ) ) ).
% List.set_insert
thf(fact_861_List_Oset__insert,axiom,
! [X3: transition,Xs2: list_transition] :
( ( set_transition2 @ ( insert_transition @ X3 @ Xs2 ) )
= ( insert_transition2 @ X3 @ ( set_transition2 @ Xs2 ) ) ) ).
% List.set_insert
thf(fact_862_in__set__insert,axiom,
! [X3: nat,Xs2: list_nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
=> ( ( insert_nat @ X3 @ Xs2 )
= Xs2 ) ) ).
% in_set_insert
thf(fact_863_in__set__insert,axiom,
! [X3: transition,Xs2: list_transition] :
( ( member_transition @ X3 @ ( set_transition2 @ Xs2 ) )
=> ( ( insert_transition @ X3 @ Xs2 )
= Xs2 ) ) ).
% in_set_insert
thf(fact_864_insert__Nil,axiom,
! [X3: transition] :
( ( insert_transition @ X3 @ nil_transition )
= ( cons_transition @ X3 @ nil_transition ) ) ).
% insert_Nil
thf(fact_865_insert__Nil,axiom,
! [X3: $o] :
( ( insert_o @ X3 @ nil_o )
= ( cons_o @ X3 @ nil_o ) ) ).
% insert_Nil
thf(fact_866_insert__Nil,axiom,
! [X3: list_o] :
( ( insert_list_o @ X3 @ nil_list_o )
= ( cons_list_o @ X3 @ nil_list_o ) ) ).
% insert_Nil
thf(fact_867_not__in__set__insert,axiom,
! [X3: nat,Xs2: list_nat] :
( ~ ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
=> ( ( insert_nat @ X3 @ Xs2 )
= ( cons_nat @ X3 @ Xs2 ) ) ) ).
% not_in_set_insert
thf(fact_868_not__in__set__insert,axiom,
! [X3: transition,Xs2: list_transition] :
( ~ ( member_transition @ X3 @ ( set_transition2 @ Xs2 ) )
=> ( ( insert_transition @ X3 @ Xs2 )
= ( cons_transition @ X3 @ Xs2 ) ) ) ).
% not_in_set_insert
thf(fact_869_not__in__set__insert,axiom,
! [X3: list_o,Xs2: list_list_o] :
( ~ ( member_list_o @ X3 @ ( set_list_o2 @ Xs2 ) )
=> ( ( insert_list_o @ X3 @ Xs2 )
= ( cons_list_o @ X3 @ Xs2 ) ) ) ).
% not_in_set_insert
thf(fact_870_List_Oinsert__def,axiom,
( insert_nat
= ( ^ [X: nat,Xs3: list_nat] : ( if_list_nat @ ( member_nat @ X @ ( set_nat2 @ Xs3 ) ) @ Xs3 @ ( cons_nat @ X @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_871_List_Oinsert__def,axiom,
( insert_transition
= ( ^ [X: transition,Xs3: list_transition] : ( if_list_transition @ ( member_transition @ X @ ( set_transition2 @ Xs3 ) ) @ Xs3 @ ( cons_transition @ X @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_872_List_Oinsert__def,axiom,
( insert_list_o
= ( ^ [X: list_o,Xs3: list_list_o] : ( if_list_list_o @ ( member_list_o @ X @ ( set_list_o2 @ Xs3 ) ) @ Xs3 @ ( cons_list_o @ X @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_873_rtranclp__sup__rtranclp,axiom,
! [R: nat > nat > $o,S: nat > nat > $o] :
( ( transi5422400438309235013lp_nat @ ( sup_sup_nat_nat_o @ ( transi5422400438309235013lp_nat @ R ) @ ( transi5422400438309235013lp_nat @ S ) ) )
= ( transi5422400438309235013lp_nat @ ( sup_sup_nat_nat_o @ R @ S ) ) ) ).
% rtranclp_sup_rtranclp
thf(fact_874_Set_Ois__empty__def,axiom,
( is_empty_nat
= ( ^ [A3: set_nat] : ( A3 = bot_bot_set_nat ) ) ) ).
% Set.is_empty_def
thf(fact_875_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_876_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_877_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_878_transition_Oinject_I2_J,axiom,
! [X23: nat,Y23: nat] :
( ( ( symb_trans @ X23 )
= ( symb_trans @ Y23 ) )
= ( X23 = Y23 ) ) ).
% transition.inject(2)
thf(fact_879_transition_Odistinct_I5_J,axiom,
! [X23: nat,X31: nat,X32: nat] :
( ( symb_trans @ X23 )
!= ( split_trans @ X31 @ X32 ) ) ).
% transition.distinct(5)
thf(fact_880_is__empty__set,axiom,
! [Xs2: list_transition] :
( ( is_empty_transition @ ( set_transition2 @ Xs2 ) )
= ( null_transition @ Xs2 ) ) ).
% is_empty_set
thf(fact_881_state__set_Oelims,axiom,
! [X3: transition,Y3: set_nat] :
( ( ( state_set @ X3 )
= Y3 )
=> ( ! [S4: nat] :
( ? [Uu: nat] :
( X3
= ( eps_trans @ S4 @ Uu ) )
=> ( Y3
!= ( insert_nat2 @ S4 @ bot_bot_set_nat ) ) )
=> ( ! [S4: nat] :
( ( X3
= ( symb_trans @ S4 ) )
=> ( Y3
!= ( insert_nat2 @ S4 @ bot_bot_set_nat ) ) )
=> ~ ! [S4: nat,S5: nat] :
( ( X3
= ( split_trans @ S4 @ S5 ) )
=> ( Y3
!= ( insert_nat2 @ S4 @ ( insert_nat2 @ S5 @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% state_set.elims
thf(fact_882_fmla__set_Osimps_I2_J,axiom,
! [V: nat] :
( ( fmla_set @ ( symb_trans @ V ) )
= bot_bot_set_nat ) ).
% fmla_set.simps(2)
thf(fact_883_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_884_transition_Odistinct_I1_J,axiom,
! [X11: nat,X12: nat,X23: nat] :
( ( eps_trans @ X11 @ X12 )
!= ( symb_trans @ X23 ) ) ).
% transition.distinct(1)
thf(fact_885_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_886_fmla__set_Osimps_I1_J,axiom,
! [Uu2: nat,N: nat] :
( ( fmla_set @ ( eps_trans @ Uu2 @ N ) )
= ( insert_nat2 @ N @ bot_bot_set_nat ) ) ).
% fmla_set.simps(1)
thf(fact_887_null__rec_I1_J,axiom,
! [X3: list_o,Xs2: list_list_o] :
~ ( null_list_o @ ( cons_list_o @ X3 @ Xs2 ) ) ).
% null_rec(1)
thf(fact_888_eq__Nil__null,axiom,
! [Xs2: list_transition] :
( ( Xs2 = nil_transition )
= ( null_transition @ Xs2 ) ) ).
% eq_Nil_null
thf(fact_889_eq__Nil__null,axiom,
! [Xs2: list_list_o] :
( ( Xs2 = nil_list_o )
= ( null_list_o @ Xs2 ) ) ).
% eq_Nil_null
thf(fact_890_eq__Nil__null,axiom,
! [Xs2: list_o] :
( ( Xs2 = nil_o )
= ( null_o @ Xs2 ) ) ).
% eq_Nil_null
thf(fact_891_null__rec_I2_J,axiom,
null_transition @ nil_transition ).
% null_rec(2)
thf(fact_892_null__rec_I2_J,axiom,
null_list_o @ nil_list_o ).
% null_rec(2)
thf(fact_893_null__rec_I2_J,axiom,
null_o @ nil_o ).
% null_rec(2)
thf(fact_894_state__set_Ocases,axiom,
! [X3: transition] :
( ! [S4: nat,Uu: nat] :
( X3
!= ( eps_trans @ S4 @ Uu ) )
=> ( ! [S4: nat] :
( X3
!= ( symb_trans @ S4 ) )
=> ~ ! [S4: nat,S5: nat] :
( X3
!= ( split_trans @ S4 @ S5 ) ) ) ) ).
% state_set.cases
thf(fact_895_transition_Oexhaust,axiom,
! [Y3: transition] :
( ! [X112: nat,X122: nat] :
( Y3
!= ( eps_trans @ X112 @ X122 ) )
=> ( ! [X24: nat] :
( Y3
!= ( symb_trans @ X24 ) )
=> ~ ! [X312: nat,X322: nat] :
( Y3
!= ( split_trans @ X312 @ X322 ) ) ) ) ).
% transition.exhaust
thf(fact_896_fmla__set_Oelims,axiom,
! [X3: transition,Y3: set_nat] :
( ( ( fmla_set @ X3 )
= Y3 )
=> ( ! [Uu: nat,N3: nat] :
( ( X3
= ( eps_trans @ Uu @ N3 ) )
=> ( Y3
!= ( insert_nat2 @ N3 @ bot_bot_set_nat ) ) )
=> ( ( ? [V2: nat] :
( X3
= ( symb_trans @ V2 ) )
=> ( Y3 != bot_bot_set_nat ) )
=> ~ ( ? [V2: nat,Va: nat] :
( X3
= ( split_trans @ V2 @ Va ) )
=> ( Y3 != bot_bot_set_nat ) ) ) ) ) ).
% fmla_set.elims
thf(fact_897_state__set_Osimps_I1_J,axiom,
! [S2: nat,Uu2: nat] :
( ( state_set @ ( eps_trans @ S2 @ Uu2 ) )
= ( insert_nat2 @ S2 @ bot_bot_set_nat ) ) ).
% state_set.simps(1)
thf(fact_898_fmla__set_Osimps_I3_J,axiom,
! [V: nat,Va2: nat] :
( ( fmla_set @ ( split_trans @ V @ Va2 ) )
= bot_bot_set_nat ) ).
% fmla_set.simps(3)
thf(fact_899_fmla__set_Opelims,axiom,
! [X3: transition,Y3: set_nat] :
( ( ( fmla_set @ X3 )
= Y3 )
=> ( ( accp_transition @ fmla_set_rel @ X3 )
=> ( ! [Uu: nat,N3: nat] :
( ( X3
= ( eps_trans @ Uu @ N3 ) )
=> ( ( Y3
= ( insert_nat2 @ N3 @ bot_bot_set_nat ) )
=> ~ ( accp_transition @ fmla_set_rel @ ( eps_trans @ Uu @ N3 ) ) ) )
=> ( ! [V2: nat] :
( ( X3
= ( symb_trans @ V2 ) )
=> ( ( Y3 = bot_bot_set_nat )
=> ~ ( accp_transition @ fmla_set_rel @ ( symb_trans @ V2 ) ) ) )
=> ~ ! [V2: nat,Va: nat] :
( ( X3
= ( split_trans @ V2 @ Va ) )
=> ( ( Y3 = bot_bot_set_nat )
=> ~ ( accp_transition @ fmla_set_rel @ ( split_trans @ V2 @ Va ) ) ) ) ) ) ) ) ).
% fmla_set.pelims
thf(fact_900_state__set_Opelims,axiom,
! [X3: transition,Y3: set_nat] :
( ( ( state_set @ X3 )
= Y3 )
=> ( ( accp_transition @ state_set_rel @ X3 )
=> ( ! [S4: nat,Uu: nat] :
( ( X3
= ( eps_trans @ S4 @ Uu ) )
=> ( ( Y3
= ( insert_nat2 @ S4 @ bot_bot_set_nat ) )
=> ~ ( accp_transition @ state_set_rel @ ( eps_trans @ S4 @ Uu ) ) ) )
=> ( ! [S4: nat] :
( ( X3
= ( symb_trans @ S4 ) )
=> ( ( Y3
= ( insert_nat2 @ S4 @ bot_bot_set_nat ) )
=> ~ ( accp_transition @ state_set_rel @ ( symb_trans @ S4 ) ) ) )
=> ~ ! [S4: nat,S5: nat] :
( ( X3
= ( split_trans @ S4 @ S5 ) )
=> ( ( Y3
= ( insert_nat2 @ S4 @ ( insert_nat2 @ S5 @ bot_bot_set_nat ) ) )
=> ~ ( accp_transition @ state_set_rel @ ( split_trans @ S4 @ S5 ) ) ) ) ) ) ) ) ).
% state_set.pelims
thf(fact_901_list__ex1__simps_I1_J,axiom,
! [P: transition > $o] :
~ ( list_ex1_transition @ P @ nil_transition ) ).
% list_ex1_simps(1)
thf(fact_902_list__ex1__simps_I1_J,axiom,
! [P: list_o > $o] :
~ ( list_ex1_list_o @ P @ nil_list_o ) ).
% list_ex1_simps(1)
thf(fact_903_list__ex1__simps_I1_J,axiom,
! [P: $o > $o] :
~ ( list_ex1_o @ P @ nil_o ) ).
% list_ex1_simps(1)
thf(fact_904_list__ex1__iff,axiom,
( list_ex1_nat
= ( ^ [P2: nat > $o,Xs3: list_nat] :
? [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs3 ) )
& ( P2 @ X )
& ! [Y4: nat] :
( ( ( member_nat @ Y4 @ ( set_nat2 @ Xs3 ) )
& ( P2 @ Y4 ) )
=> ( Y4 = X ) ) ) ) ) ).
% list_ex1_iff
thf(fact_905_list__ex1__iff,axiom,
( list_ex1_transition
= ( ^ [P2: transition > $o,Xs3: list_transition] :
? [X: transition] :
( ( member_transition @ X @ ( set_transition2 @ Xs3 ) )
& ( P2 @ X )
& ! [Y4: transition] :
( ( ( member_transition @ Y4 @ ( set_transition2 @ Xs3 ) )
& ( P2 @ Y4 ) )
=> ( Y4 = X ) ) ) ) ) ).
% list_ex1_iff
thf(fact_906_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_907_subset__subseqs,axiom,
! [X4: set_transition,Xs2: list_transition] :
( ( ord_le8419162016481440574sition @ X4 @ ( set_transition2 @ Xs2 ) )
=> ( member7318969637299765063sition @ X4 @ ( image_4748612756971788127sition @ set_transition2 @ ( set_list_transition2 @ ( subseqs_transition @ Xs2 ) ) ) ) ) ).
% subset_subseqs
thf(fact_908_subset__subseqs,axiom,
! [X4: set_nat,Xs2: list_nat] :
( ( ord_less_eq_set_nat @ X4 @ ( set_nat2 @ Xs2 ) )
=> ( member_set_nat @ X4 @ ( image_1775855109352712557et_nat @ set_nat2 @ ( set_list_nat2 @ ( subseqs_nat @ Xs2 ) ) ) ) ) ).
% subset_subseqs
thf(fact_909_transpose__empty,axiom,
! [Xs2: list_list_transition] :
( ( ( transpose_transition @ Xs2 )
= nil_list_transition )
= ( ! [X: list_transition] :
( ( member1473516902542837997sition @ X @ ( set_list_transition2 @ Xs2 ) )
=> ( X = nil_transition ) ) ) ) ).
% transpose_empty
thf(fact_910_transpose__empty,axiom,
! [Xs2: list_list_list_o] :
( ( ( transpose_list_o @ Xs2 )
= nil_list_list_o )
= ( ! [X: list_list_o] :
( ( member_list_list_o @ X @ ( set_list_list_o2 @ Xs2 ) )
=> ( X = nil_list_o ) ) ) ) ).
% transpose_empty
thf(fact_911_transpose__empty,axiom,
! [Xs2: list_list_o] :
( ( ( transpose_o @ Xs2 )
= nil_list_o )
= ( ! [X: list_o] :
( ( member_list_o @ X @ ( set_list_o2 @ Xs2 ) )
=> ( X = nil_o ) ) ) ) ).
% transpose_empty
thf(fact_912_image__eqI,axiom,
! [B: nat,F: nat > nat,X3: nat,A2: set_nat] :
( ( B
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A2 )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_913_image__eqI,axiom,
! [B: transition,F: nat > transition,X3: nat,A2: set_nat] :
( ( B
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A2 )
=> ( member_transition @ B @ ( image_nat_transition @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_914_image__eqI,axiom,
! [B: nat,F: transition > nat,X3: transition,A2: set_transition] :
( ( B
= ( F @ X3 ) )
=> ( ( member_transition @ X3 @ A2 )
=> ( member_nat @ B @ ( image_transition_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_915_image__eqI,axiom,
! [B: transition,F: transition > transition,X3: transition,A2: set_transition] :
( ( B
= ( F @ X3 ) )
=> ( ( member_transition @ X3 @ A2 )
=> ( member_transition @ B @ ( image_5857460390510121477sition @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_916_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_917_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_918_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_919_image__insert,axiom,
! [F: nat > nat,A: nat,B2: set_nat] :
( ( image_nat_nat @ F @ ( insert_nat2 @ A @ B2 ) )
= ( insert_nat2 @ ( F @ A ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_920_insert__image,axiom,
! [X3: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( insert_nat2 @ ( F @ X3 ) @ ( image_nat_nat @ F @ A2 ) )
= ( image_nat_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_921_insert__image,axiom,
! [X3: transition,A2: set_transition,F: transition > nat] :
( ( member_transition @ X3 @ A2 )
=> ( ( insert_nat2 @ ( F @ X3 ) @ ( image_transition_nat @ F @ A2 ) )
= ( image_transition_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_922_subset__image__iff,axiom,
! [B2: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B2
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_923_subset__imageE,axiom,
! [B2: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A2 )
=> ( B2
!= ( image_nat_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_924_image__subsetI,axiom,
! [A2: set_nat,F: nat > transition,B2: set_transition] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_transition @ ( F @ X2 ) @ B2 ) )
=> ( ord_le8419162016481440574sition @ ( image_nat_transition @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_925_image__subsetI,axiom,
! [A2: set_transition,F: transition > transition,B2: set_transition] :
( ! [X2: transition] :
( ( member_transition @ X2 @ A2 )
=> ( member_transition @ ( F @ X2 ) @ B2 ) )
=> ( ord_le8419162016481440574sition @ ( image_5857460390510121477sition @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_926_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat,B2: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_927_image__subsetI,axiom,
! [A2: set_transition,F: transition > nat,B2: set_nat] :
( ! [X2: transition] :
( ( member_transition @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_transition_nat @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_928_image__mono,axiom,
! [A2: set_nat,B2: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_929_imageI,axiom,
! [X3: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ ( image_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_930_imageI,axiom,
! [X3: nat,A2: set_nat,F: nat > transition] :
( ( member_nat @ X3 @ A2 )
=> ( member_transition @ ( F @ X3 ) @ ( image_nat_transition @ F @ A2 ) ) ) ).
% imageI
thf(fact_931_imageI,axiom,
! [X3: transition,A2: set_transition,F: transition > nat] :
( ( member_transition @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ ( image_transition_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_932_imageI,axiom,
! [X3: transition,A2: set_transition,F: transition > transition] :
( ( member_transition @ X3 @ A2 )
=> ( member_transition @ ( F @ X3 ) @ ( image_5857460390510121477sition @ F @ A2 ) ) ) ).
% imageI
thf(fact_933_rev__image__eqI,axiom,
! [X3: nat,A2: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X3 @ A2 )
=> ( ( B
= ( F @ X3 ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_934_rev__image__eqI,axiom,
! [X3: nat,A2: set_nat,B: transition,F: nat > transition] :
( ( member_nat @ X3 @ A2 )
=> ( ( B
= ( F @ X3 ) )
=> ( member_transition @ B @ ( image_nat_transition @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_935_rev__image__eqI,axiom,
! [X3: transition,A2: set_transition,B: nat,F: transition > nat] :
( ( member_transition @ X3 @ A2 )
=> ( ( B
= ( F @ X3 ) )
=> ( member_nat @ B @ ( image_transition_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_936_rev__image__eqI,axiom,
! [X3: transition,A2: set_transition,B: transition,F: transition > transition] :
( ( member_transition @ X3 @ A2 )
=> ( ( B
= ( F @ X3 ) )
=> ( member_transition @ B @ ( image_5857460390510121477sition @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_937_can__select__def,axiom,
( can_select_nat
= ( ^ [P2: nat > $o,A3: set_nat] :
? [X: nat] :
( ( member_nat @ X @ A3 )
& ( P2 @ X )
& ! [Y4: nat] :
( ( ( member_nat @ Y4 @ A3 )
& ( P2 @ Y4 ) )
=> ( Y4 = X ) ) ) ) ) ).
% can_select_def
thf(fact_938_can__select__def,axiom,
( can_se3600352496914471099sition
= ( ^ [P2: transition > $o,A3: set_transition] :
? [X: transition] :
( ( member_transition @ X @ A3 )
& ( P2 @ X )
& ! [Y4: transition] :
( ( ( member_transition @ Y4 @ A3 )
& ( P2 @ Y4 ) )
=> ( Y4 = X ) ) ) ) ) ).
% can_select_def
thf(fact_939_transpose_Osimps_I1_J,axiom,
( ( transpose_o @ nil_list_o )
= nil_list_o ) ).
% transpose.simps(1)
thf(fact_940_image__Un,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( image_nat_nat @ F @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_Un
thf(fact_941_image__Int__subset,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( inf_inf_set_nat @ A2 @ B2 ) ) @ ( inf_inf_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_Int_subset
thf(fact_942_image__diff__subset,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B2 ) ) @ ( image_nat_nat @ F @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_943_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_944_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_945_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_946_all__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A2 )
=> ( P @ ( image_nat_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_947_subseqs__powset,axiom,
! [Xs2: list_transition] :
( ( image_4748612756971788127sition @ set_transition2 @ ( set_list_transition2 @ ( subseqs_transition @ Xs2 ) ) )
= ( pow_transition @ ( set_transition2 @ Xs2 ) ) ) ).
% subseqs_powset
thf(fact_948_pairwise__alt,axiom,
( pairwise_nat
= ( ^ [R2: nat > nat > $o,S6: set_nat] :
! [X: nat] :
( ( member_nat @ X @ S6 )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ ( minus_minus_set_nat @ S6 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) )
=> ( R2 @ X @ Y4 ) ) ) ) ) ).
% pairwise_alt
thf(fact_949_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_950_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_951_PowI,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( member_set_nat @ A2 @ ( pow_nat @ B2 ) ) ) ).
% PowI
thf(fact_952_Pow__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( member_set_nat @ A2 @ ( pow_nat @ B2 ) )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% Pow_iff
thf(fact_953_Pow__Int__eq,axiom,
! [A2: set_nat,B2: set_nat] :
( ( pow_nat @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( inf_inf_set_set_nat @ ( pow_nat @ A2 ) @ ( pow_nat @ B2 ) ) ) ).
% Pow_Int_eq
thf(fact_954_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_955_Un__Pow__subset,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_le6893508408891458716et_nat @ ( sup_sup_set_set_nat @ ( pow_nat @ A2 ) @ ( pow_nat @ B2 ) ) @ ( pow_nat @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% Un_Pow_subset
thf(fact_956_pairwise__empty,axiom,
! [P: nat > nat > $o] : ( pairwise_nat @ P @ bot_bot_set_nat ) ).
% pairwise_empty
thf(fact_957_Pow__bottom,axiom,
! [B2: set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( pow_nat @ B2 ) ) ).
% Pow_bottom
thf(fact_958_pairwiseI,axiom,
! [S: set_nat,R: nat > nat > $o] :
( ! [X2: nat,Y: nat] :
( ( member_nat @ X2 @ S )
=> ( ( member_nat @ Y @ S )
=> ( ( X2 != Y )
=> ( R @ X2 @ Y ) ) ) )
=> ( pairwise_nat @ R @ S ) ) ).
% pairwiseI
thf(fact_959_pairwiseI,axiom,
! [S: set_transition,R: transition > transition > $o] :
( ! [X2: transition,Y: transition] :
( ( member_transition @ X2 @ S )
=> ( ( member_transition @ Y @ S )
=> ( ( X2 != Y )
=> ( R @ X2 @ Y ) ) ) )
=> ( pairwise_transition @ R @ S ) ) ).
% pairwiseI
thf(fact_960_pairwiseD,axiom,
! [R: nat > nat > $o,S: set_nat,X3: nat,Y3: nat] :
( ( pairwise_nat @ R @ S )
=> ( ( member_nat @ X3 @ S )
=> ( ( member_nat @ Y3 @ S )
=> ( ( X3 != Y3 )
=> ( R @ X3 @ Y3 ) ) ) ) ) ).
% pairwiseD
thf(fact_961_pairwiseD,axiom,
! [R: transition > transition > $o,S: set_transition,X3: transition,Y3: transition] :
( ( pairwise_transition @ R @ S )
=> ( ( member_transition @ X3 @ S )
=> ( ( member_transition @ Y3 @ S )
=> ( ( X3 != Y3 )
=> ( R @ X3 @ Y3 ) ) ) ) ) ).
% pairwiseD
thf(fact_962_pairwise__insert,axiom,
! [R3: nat > nat > $o,X3: nat,S2: set_nat] :
( ( pairwise_nat @ R3 @ ( insert_nat2 @ X3 @ S2 ) )
= ( ! [Y4: nat] :
( ( ( member_nat @ Y4 @ S2 )
& ( Y4 != X3 ) )
=> ( ( R3 @ X3 @ Y4 )
& ( R3 @ Y4 @ X3 ) ) )
& ( pairwise_nat @ R3 @ S2 ) ) ) ).
% pairwise_insert
thf(fact_963_pairwise__insert,axiom,
! [R3: transition > transition > $o,X3: transition,S2: set_transition] :
( ( pairwise_transition @ R3 @ ( insert_transition2 @ X3 @ S2 ) )
= ( ! [Y4: transition] :
( ( ( member_transition @ Y4 @ S2 )
& ( Y4 != X3 ) )
=> ( ( R3 @ X3 @ Y4 )
& ( R3 @ Y4 @ X3 ) ) )
& ( pairwise_transition @ R3 @ S2 ) ) ) ).
% pairwise_insert
thf(fact_964_PowD,axiom,
! [A2: set_nat,B2: set_nat] :
( ( member_set_nat @ A2 @ ( pow_nat @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% PowD
thf(fact_965_Pow__mono,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( pow_nat @ A2 ) @ ( pow_nat @ B2 ) ) ) ).
% Pow_mono
thf(fact_966_pairwise__mono,axiom,
! [P: nat > nat > $o,A2: set_nat,Q4: nat > nat > $o,B2: set_nat] :
( ( pairwise_nat @ P @ A2 )
=> ( ! [X2: nat,Y: nat] :
( ( P @ X2 @ Y )
=> ( Q4 @ X2 @ Y ) )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( pairwise_nat @ Q4 @ B2 ) ) ) ) ).
% pairwise_mono
thf(fact_967_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_968_pairwise__singleton,axiom,
! [P: nat > nat > $o,A2: nat] : ( pairwise_nat @ P @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) ).
% pairwise_singleton
thf(fact_969_in__image__insert__iff,axiom,
! [B2: set_set_transition,X3: transition,A2: set_transition] :
( ! [C4: set_transition] :
( ( member7318969637299765063sition @ C4 @ B2 )
=> ~ ( member_transition @ X3 @ C4 ) )
=> ( ( member7318969637299765063sition @ A2 @ ( image_698392052263970309sition @ ( insert_transition2 @ X3 ) @ B2 ) )
= ( ( member_transition @ X3 @ A2 )
& ( member7318969637299765063sition @ ( minus_8944320859760356485sition @ A2 @ ( insert_transition2 @ X3 @ bot_bo301567166201926666sition ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_970_in__image__insert__iff,axiom,
! [B2: set_set_nat,X3: nat,A2: set_nat] :
( ! [C4: set_nat] :
( ( member_set_nat @ C4 @ B2 )
=> ~ ( member_nat @ X3 @ C4 ) )
=> ( ( member_set_nat @ A2 @ ( image_7916887816326733075et_nat @ ( insert_nat2 @ X3 ) @ B2 ) )
= ( ( member_nat @ X3 @ A2 )
& ( member_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_971_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_972_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_973_Pow__set_I1_J,axiom,
( ( pow_transition @ ( set_transition2 @ nil_transition ) )
= ( insert8494249028948967790sition @ bot_bo301567166201926666sition @ bot_bo1233527522848825322sition ) ) ).
% Pow_set(1)
thf(fact_974_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_975_psubset__insert__iff,axiom,
! [A2: set_transition,X3: transition,B2: set_transition] :
( ( ord_le5184432651266358346sition @ A2 @ ( insert_transition2 @ X3 @ B2 ) )
= ( ( ( member_transition @ X3 @ B2 )
=> ( ord_le5184432651266358346sition @ A2 @ B2 ) )
& ( ~ ( member_transition @ X3 @ B2 )
=> ( ( ( member_transition @ X3 @ A2 )
=> ( ord_le5184432651266358346sition @ ( minus_8944320859760356485sition @ A2 @ ( insert_transition2 @ X3 @ bot_bo301567166201926666sition ) ) @ B2 ) )
& ( ~ ( member_transition @ X3 @ A2 )
=> ( ord_le8419162016481440574sition @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_976_psubset__insert__iff,axiom,
! [A2: set_nat,X3: nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ ( insert_nat2 @ X3 @ B2 ) )
= ( ( ( member_nat @ X3 @ B2 )
=> ( ord_less_set_nat @ A2 @ B2 ) )
& ( ~ ( member_nat @ X3 @ B2 )
=> ( ( ( member_nat @ X3 @ A2 )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) @ B2 ) )
& ( ~ ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_977_remove__induct,axiom,
! [P: set_transition > $o,B2: set_transition] :
( ( P @ bot_bo301567166201926666sition )
=> ( ( ~ ( finite8165534619950747239sition @ B2 )
=> ( P @ B2 ) )
=> ( ! [A6: set_transition] :
( ( finite8165534619950747239sition @ A6 )
=> ( ( A6 != bot_bo301567166201926666sition )
=> ( ( ord_le8419162016481440574sition @ A6 @ B2 )
=> ( ! [X5: transition] :
( ( member_transition @ X5 @ A6 )
=> ( P @ ( minus_8944320859760356485sition @ A6 @ ( insert_transition2 @ X5 @ bot_bo301567166201926666sition ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_978_remove__induct,axiom,
! [P: set_nat > $o,B2: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B2 )
=> ( P @ B2 ) )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( A6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A6 @ B2 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat2 @ X5 @ bot_bot_set_nat ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_979_finite__remove__induct,axiom,
! [B2: set_transition,P: set_transition > $o] :
( ( finite8165534619950747239sition @ B2 )
=> ( ( P @ bot_bo301567166201926666sition )
=> ( ! [A6: set_transition] :
( ( finite8165534619950747239sition @ A6 )
=> ( ( A6 != bot_bo301567166201926666sition )
=> ( ( ord_le8419162016481440574sition @ A6 @ B2 )
=> ( ! [X5: transition] :
( ( member_transition @ X5 @ A6 )
=> ( P @ ( minus_8944320859760356485sition @ A6 @ ( insert_transition2 @ X5 @ bot_bo301567166201926666sition ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_980_finite__remove__induct,axiom,
! [B2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( A6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A6 @ B2 )
=> ( ! [X5: nat] :
( ( member_nat @ X5 @ A6 )
=> ( P @ ( minus_minus_set_nat @ A6 @ ( insert_nat2 @ X5 @ bot_bot_set_nat ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_981_finite__insert,axiom,
! [A: nat,A2: set_nat] :
( ( finite_finite_nat @ ( insert_nat2 @ A @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ).
% finite_insert
thf(fact_982_List_Ofinite__set,axiom,
! [Xs2: list_transition] : ( finite8165534619950747239sition @ ( set_transition2 @ Xs2 ) ) ).
% List.finite_set
thf(fact_983_List_Ofinite__set,axiom,
! [Xs2: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs2 ) ) ).
% List.finite_set
thf(fact_984_finite__Int,axiom,
! [F2: set_nat,G: set_nat] :
( ( ( finite_finite_nat @ F2 )
| ( finite_finite_nat @ G ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_985_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_986_finite__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_987_finite__Diff2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_988_psubsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_nat @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_989_finite__Diff__insert,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ B2 ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_990_finite__surj,axiom,
! [A2: set_nat,B2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
=> ( finite_finite_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_991_finite__subset__image,axiom,
! [B2: set_nat,F: nat > nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
=> ? [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A2 )
& ( finite_finite_nat @ C4 )
& ( B2
= ( image_nat_nat @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_992_ex__finite__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A2 )
& ( P @ ( image_nat_nat @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_993_all__finite__subset__image,axiom,
! [F: nat > nat,A2: set_nat,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A2 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A2 ) )
=> ( P @ ( image_nat_nat @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_994_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_995_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_996_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_997_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_998_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_999_finite_OinsertI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( insert_nat2 @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_1000_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_1001_rev__finite__subset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_1002_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_1003_finite__subset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_1004_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_eq_nat @ X2 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1005_finite__has__minimal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ( ord_less_eq_set_nat @ X2 @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1006_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_eq_nat @ A @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1007_finite__has__maximal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ( ord_less_eq_set_nat @ A @ X2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1008_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1009_finite__has__maximal,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1010_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1011_finite__has__minimal,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1012_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_1013_order__le__imp__less__or__eq,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ( ord_less_set_nat @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1014_order__less__le__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1015_order__le__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_1016_order__less__le__trans,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( ord_less_set_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_set_nat @ Y3 @ Z2 )
=> ( ord_less_set_nat @ X3 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_1017_order__le__less__trans,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ( ord_less_set_nat @ Y3 @ Z2 )
=> ( ord_less_set_nat @ X3 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_1018_order__neq__le__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( A != B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_1019_order__le__neq__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_1020_order__less__imp__le,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ X3 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_1021_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
& ( X != Y4 ) ) ) ) ).
% order_less_le
thf(fact_1022_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X: set_nat,Y4: set_nat] :
( ( ord_less_set_nat @ X @ Y4 )
| ( X = Y4 ) ) ) ) ).
% order_le_less
thf(fact_1023_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_1024_order_Ostrict__implies__order,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_1025_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B5: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
& ~ ( ord_less_eq_set_nat @ A4 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1026_dual__order_Ostrict__trans2,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_1027_dual__order_Ostrict__trans1,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_1028_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B5: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
& ( A4 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1029_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B5: set_nat,A4: set_nat] :
( ( ord_less_set_nat @ B5 @ A4 )
| ( A4 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1030_order_Ostrict__iff__not,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 ) ) ) ) ).
% order.strict_iff_not
thf(fact_1031_order_Ostrict__trans2,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1032_order_Ostrict__trans1,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1033_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1034_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
( ( ord_less_set_nat @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1035_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
& ~ ( ord_less_eq_set_nat @ Y4 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_1036_antisym__conv2,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ( ~ ( ord_less_set_nat @ X3 @ Y3 ) )
= ( X3 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_1037_antisym__conv1,axiom,
! [X3: set_nat,Y3: set_nat] :
( ~ ( ord_less_set_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_set_nat @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_1038_nless__le,axiom,
! [A: set_nat,B: set_nat] :
( ( ~ ( ord_less_set_nat @ A @ B ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1039_leD,axiom,
! [Y3: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X3 )
=> ~ ( ord_less_set_nat @ X3 @ Y3 ) ) ).
% leD
thf(fact_1040_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_1041_subset__psubset__trans,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_set_nat @ B2 @ C2 )
=> ( ord_less_set_nat @ A2 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_1042_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ~ ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_1043_psubset__subset__trans,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_set_nat @ A2 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_1044_psubset__imp__subset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_1045_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% psubset_eq
thf(fact_1046_psubsetE,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_1047_finite__list,axiom,
! [A2: set_transition] :
( ( finite8165534619950747239sition @ A2 )
=> ? [Xs: list_transition] :
( ( set_transition2 @ Xs )
= A2 ) ) ).
% finite_list
thf(fact_1048_finite__list,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ? [Xs: list_nat] :
( ( set_nat2 @ Xs )
= A2 ) ) ).
% finite_list
thf(fact_1049_psubset__imp__ex__mem,axiom,
! [A2: set_transition,B2: set_transition] :
( ( ord_le5184432651266358346sition @ A2 @ B2 )
=> ? [B7: transition] : ( member_transition @ B7 @ ( minus_8944320859760356485sition @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1050_psubset__imp__ex__mem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ? [B7: nat] : ( member_nat @ B7 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1051_psubsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_1052_psubsetD,axiom,
! [A2: set_transition,B2: set_transition,C: transition] :
( ( ord_le5184432651266358346sition @ A2 @ B2 )
=> ( ( member_transition @ C @ A2 )
=> ( member_transition @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_1053_finite__SQ,axiom,
! [Q0: nat,Transs: list_transition] : ( finite_finite_nat @ ( sq @ Q0 @ Transs ) ) ).
% finite_SQ
thf(fact_1054_finite__Q,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition] : ( finite_finite_nat @ ( q @ Q0 @ Qf @ Transs ) ) ).
% finite_Q
thf(fact_1055_inf_Ostrict__coboundedI2,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_1056_inf_Ostrict__coboundedI1,axiom,
! [A: set_nat,C: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ C )
=> ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_1057_inf_Ostrict__order__iff,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
( ( A4
= ( inf_inf_set_nat @ A4 @ B5 ) )
& ( A4 != B5 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_1058_inf_Ostrict__boundedE,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) )
=> ~ ( ( ord_less_set_nat @ A @ B )
=> ~ ( ord_less_set_nat @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_1059_inf_Oabsorb4,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( inf_inf_set_nat @ A @ B )
= B ) ) ).
% inf.absorb4
thf(fact_1060_inf_Oabsorb3,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( inf_inf_set_nat @ A @ B )
= A ) ) ).
% inf.absorb3
thf(fact_1061_less__infI2,axiom,
! [B: set_nat,X3: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ X3 )
=> ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X3 ) ) ).
% less_infI2
thf(fact_1062_less__infI1,axiom,
! [A: set_nat,X3: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ X3 )
=> ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X3 ) ) ).
% less_infI1
thf(fact_1063_not__psubset__empty,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_1064_bot_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_1065_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_1066_sup_Ostrict__coboundedI2,axiom,
! [C: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_1067_sup_Ostrict__coboundedI1,axiom,
! [C: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ C @ A )
=> ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_1068_sup_Ostrict__order__iff,axiom,
( ord_less_set_nat
= ( ^ [B5: set_nat,A4: set_nat] :
( ( A4
= ( sup_sup_set_nat @ A4 @ B5 ) )
& ( A4 != B5 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_1069_sup_Ostrict__boundedE,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_set_nat @ B @ A )
=> ~ ( ord_less_set_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_1070_sup_Oabsorb4,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_1071_sup_Oabsorb3,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( sup_sup_set_nat @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_1072_less__supI2,axiom,
! [X3: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ X3 @ B )
=> ( ord_less_set_nat @ X3 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% less_supI2
thf(fact_1073_less__supI1,axiom,
! [X3: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ X3 @ A )
=> ( ord_less_set_nat @ X3 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% less_supI1
thf(fact_1074_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_1075_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A4: set_nat] :
( ( A4 = bot_bot_set_nat )
| ? [A3: set_nat,B5: nat] :
( ( A4
= ( insert_nat2 @ B5 @ A3 ) )
& ( finite_finite_nat @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_1076_finite__induct,axiom,
! [F2: set_transition,P: set_transition > $o] :
( ( finite8165534619950747239sition @ F2 )
=> ( ( P @ bot_bo301567166201926666sition )
=> ( ! [X2: transition,F3: set_transition] :
( ( finite8165534619950747239sition @ F3 )
=> ( ~ ( member_transition @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_transition2 @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1077_finite__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat2 @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1078_finite__ne__induct,axiom,
! [F2: set_transition,P: set_transition > $o] :
( ( finite8165534619950747239sition @ F2 )
=> ( ( F2 != bot_bo301567166201926666sition )
=> ( ! [X2: transition] : ( P @ ( insert_transition2 @ X2 @ bot_bo301567166201926666sition ) )
=> ( ! [X2: transition,F3: set_transition] :
( ( finite8165534619950747239sition @ F3 )
=> ( ( F3 != bot_bo301567166201926666sition )
=> ( ~ ( member_transition @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_transition2 @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1079_finite__ne__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( F2 != bot_bot_set_nat )
=> ( ! [X2: nat] : ( P @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat2 @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1080_infinite__finite__induct,axiom,
! [P: set_transition > $o,A2: set_transition] :
( ! [A6: set_transition] :
( ~ ( finite8165534619950747239sition @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ bot_bo301567166201926666sition )
=> ( ! [X2: transition,F3: set_transition] :
( ( finite8165534619950747239sition @ F3 )
=> ( ~ ( member_transition @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_transition2 @ X2 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_1081_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 )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat2 @ X2 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_1082_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_1083_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_1084_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_1085_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_1086_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_1087_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_1088_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_1089_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_1090_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_1091_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_1092_infinite__growing,axiom,
! [X4: set_nat] :
( ( X4 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X4 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X4 )
& ( ord_less_nat @ X2 @ Xa ) ) )
=> ~ ( finite_finite_nat @ X4 ) ) ) ).
% infinite_growing
thf(fact_1093_ex__min__if__finite,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ S )
& ~ ? [Xa: nat] :
( ( member_nat @ Xa @ S )
& ( ord_less_nat @ Xa @ X2 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_1094_empty__in__Fpow,axiom,
! [A2: set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( finite_Fpow_nat @ A2 ) ) ).
% empty_in_Fpow
thf(fact_1095_Fpow__mono,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( finite_Fpow_nat @ A2 ) @ ( finite_Fpow_nat @ B2 ) ) ) ).
% Fpow_mono
thf(fact_1096_finite__transitivity__chain,axiom,
! [A2: set_transition,R: transition > transition > $o] :
( ( finite8165534619950747239sition @ A2 )
=> ( ! [X2: transition] :
~ ( R @ X2 @ X2 )
=> ( ! [X2: transition,Y: transition,Z3: transition] :
( ( R @ X2 @ Y )
=> ( ( R @ Y @ Z3 )
=> ( R @ X2 @ Z3 ) ) )
=> ( ! [X2: transition] :
( ( member_transition @ X2 @ A2 )
=> ? [Y6: transition] :
( ( member_transition @ Y6 @ A2 )
& ( R @ X2 @ Y6 ) ) )
=> ( A2 = bot_bo301567166201926666sition ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_1097_finite__transitivity__chain,axiom,
! [A2: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [X2: nat] :
~ ( R @ X2 @ X2 )
=> ( ! [X2: nat,Y: nat,Z3: nat] :
( ( R @ X2 @ Y )
=> ( ( R @ Y @ Z3 )
=> ( R @ X2 @ Z3 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ? [Y6: nat] :
( ( member_nat @ Y6 @ A2 )
& ( R @ X2 @ Y6 ) ) )
=> ( A2 = bot_bot_set_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_1098_verit__comp__simplify1_I2_J,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1099_Inf__fin_Oinsert__remove,axiom,
! [A2: set_set_nat,X3: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) )
= bot_bot_set_set_nat )
=> ( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X3 @ A2 ) )
= X3 ) )
& ( ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) )
!= bot_bot_set_set_nat )
=> ( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X3 @ A2 ) )
= ( inf_inf_set_nat @ X3 @ ( lattic3014633134055518761et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_1100_Inf__fin_Oinsert__remove,axiom,
! [A2: set_nat,X3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat2 @ X3 @ A2 ) )
= X3 ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat2 @ X3 @ A2 ) )
= ( inf_inf_nat @ X3 @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_1101_Inf__fin_Oremove,axiom,
! [A2: set_set_nat,X3: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ X3 @ A2 )
=> ( ( ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) )
= bot_bot_set_set_nat )
=> ( ( lattic3014633134055518761et_nat @ A2 )
= X3 ) )
& ( ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) )
!= bot_bot_set_set_nat )
=> ( ( lattic3014633134055518761et_nat @ A2 )
= ( inf_inf_set_nat @ X3 @ ( lattic3014633134055518761et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_1102_Inf__fin_Oremove,axiom,
! [A2: set_nat,X3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X3 @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A2 )
= X3 ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A2 )
= ( inf_inf_nat @ X3 @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_1103_Inf__fin_Osingleton,axiom,
! [X3: nat] :
( ( lattic5238388535129920115in_nat @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) )
= X3 ) ).
% Inf_fin.singleton
thf(fact_1104_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_1105_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_1106_Inf__fin_Oinsert,axiom,
! [A2: set_set_nat,X3: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X3 @ A2 ) )
= ( inf_inf_set_nat @ X3 @ ( lattic3014633134055518761et_nat @ A2 ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_1107_Inf__fin_Oinsert,axiom,
! [A2: set_nat,X3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat2 @ X3 @ A2 ) )
= ( inf_inf_nat @ X3 @ ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_1108_Inf__fin_Oin__idem,axiom,
! [A2: set_nat,X3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X3 @ A2 )
=> ( ( inf_inf_nat @ X3 @ ( lattic5238388535129920115in_nat @ A2 ) )
= ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_1109_Inf__fin_Oin__idem,axiom,
! [A2: set_set_nat,X3: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ X3 @ A2 )
=> ( ( inf_inf_set_nat @ X3 @ ( lattic3014633134055518761et_nat @ A2 ) )
= ( lattic3014633134055518761et_nat @ A2 ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_1110_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_1111_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_1112_Inf__fin_Obounded__iff,axiom,
! [A2: set_nat,X3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X3 @ ( lattic5238388535129920115in_nat @ A2 ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_less_eq_nat @ X3 @ X ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_1113_Inf__fin_Obounded__iff,axiom,
! [A2: set_set_nat,X3: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ( ord_less_eq_set_nat @ X3 @ ( lattic3014633134055518761et_nat @ A2 ) )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ X3 @ X ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_1114_Inf__fin_OboundedI,axiom,
! [A2: set_nat,X3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [A7: nat] :
( ( member_nat @ A7 @ A2 )
=> ( ord_less_eq_nat @ X3 @ A7 ) )
=> ( ord_less_eq_nat @ X3 @ ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1115_Inf__fin_OboundedI,axiom,
! [A2: set_set_nat,X3: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ! [A7: set_nat] :
( ( member_set_nat @ A7 @ A2 )
=> ( ord_less_eq_set_nat @ X3 @ A7 ) )
=> ( ord_less_eq_set_nat @ X3 @ ( lattic3014633134055518761et_nat @ A2 ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1116_Inf__fin_OboundedE,axiom,
! [A2: set_nat,X3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X3 @ ( lattic5238388535129920115in_nat @ A2 ) )
=> ! [A8: nat] :
( ( member_nat @ A8 @ A2 )
=> ( ord_less_eq_nat @ X3 @ A8 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1117_Inf__fin_OboundedE,axiom,
! [A2: set_set_nat,X3: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ( ord_less_eq_set_nat @ X3 @ ( lattic3014633134055518761et_nat @ A2 ) )
=> ! [A8: set_nat] :
( ( member_set_nat @ A8 @ A2 )
=> ( ord_less_eq_set_nat @ X3 @ A8 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1118_Inf__fin_Osubset__imp,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ord_less_eq_set_nat @ ( lattic3014633134055518761et_nat @ B2 ) @ ( lattic3014633134055518761et_nat @ A2 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_1119_Inf__fin_Osubset__imp,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ B2 ) @ ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_1120_Inf__fin_Ohom__commute,axiom,
! [H: set_nat > set_nat,N4: set_set_nat] :
( ! [X2: set_nat,Y: set_nat] :
( ( H @ ( inf_inf_set_nat @ X2 @ Y ) )
= ( inf_inf_set_nat @ ( H @ X2 ) @ ( H @ Y ) ) )
=> ( ( finite1152437895449049373et_nat @ N4 )
=> ( ( N4 != bot_bot_set_set_nat )
=> ( ( H @ ( lattic3014633134055518761et_nat @ N4 ) )
= ( lattic3014633134055518761et_nat @ ( image_7916887816326733075et_nat @ H @ N4 ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_1121_Inf__fin_Ohom__commute,axiom,
! [H: nat > nat,N4: set_nat] :
( ! [X2: nat,Y: nat] :
( ( H @ ( inf_inf_nat @ X2 @ Y ) )
= ( inf_inf_nat @ ( H @ X2 ) @ ( H @ Y ) ) )
=> ( ( finite_finite_nat @ N4 )
=> ( ( N4 != bot_bot_set_nat )
=> ( ( H @ ( lattic5238388535129920115in_nat @ N4 ) )
= ( lattic5238388535129920115in_nat @ ( image_nat_nat @ H @ N4 ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_1122_Inf__fin_Osubset,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( B2 != bot_bot_set_set_nat )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( ( inf_inf_set_nat @ ( lattic3014633134055518761et_nat @ B2 ) @ ( lattic3014633134055518761et_nat @ A2 ) )
= ( lattic3014633134055518761et_nat @ A2 ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_1123_Inf__fin_Osubset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( B2 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ B2 ) @ ( lattic5238388535129920115in_nat @ A2 ) )
= ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_1124_Inf__fin_Oclosed,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ! [X2: set_nat,Y: set_nat] : ( member_set_nat @ ( inf_inf_set_nat @ X2 @ Y ) @ ( insert_set_nat @ X2 @ ( insert_set_nat @ Y @ bot_bot_set_set_nat ) ) )
=> ( member_set_nat @ ( lattic3014633134055518761et_nat @ A2 ) @ A2 ) ) ) ) ).
% Inf_fin.closed
thf(fact_1125_Inf__fin_Oclosed,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [X2: nat,Y: nat] : ( member_nat @ ( inf_inf_nat @ X2 @ Y ) @ ( insert_nat2 @ X2 @ ( insert_nat2 @ Y @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic5238388535129920115in_nat @ A2 ) @ A2 ) ) ) ) ).
% Inf_fin.closed
thf(fact_1126_Inf__fin_Oinsert__not__elem,axiom,
! [A2: set_set_nat,X3: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ~ ( member_set_nat @ X3 @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X3 @ A2 ) )
= ( inf_inf_set_nat @ X3 @ ( lattic3014633134055518761et_nat @ A2 ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_1127_Inf__fin_Oinsert__not__elem,axiom,
! [A2: set_nat,X3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat @ X3 @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat2 @ X3 @ A2 ) )
= ( inf_inf_nat @ X3 @ ( lattic5238388535129920115in_nat @ A2 ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_1128_Inf__fin_Ounion,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( B2 != bot_bot_set_set_nat )
=> ( ( lattic3014633134055518761et_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) )
= ( inf_inf_set_nat @ ( lattic3014633134055518761et_nat @ A2 ) @ ( lattic3014633134055518761et_nat @ B2 ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_1129_Inf__fin_Ounion,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B2 )
=> ( ( B2 != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ A2 ) @ ( lattic5238388535129920115in_nat @ B2 ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_1130_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 )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ C )
=> ( ord_less_eq_set_nat @ X2 @ 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_1131_Sup__fin_Oinsert__remove,axiom,
! [A2: set_set_nat,X3: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) )
= bot_bot_set_set_nat )
=> ( ( lattic3835124923745554447et_nat @ ( insert_set_nat @ X3 @ A2 ) )
= X3 ) )
& ( ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) )
!= bot_bot_set_set_nat )
=> ( ( lattic3835124923745554447et_nat @ ( insert_set_nat @ X3 @ A2 ) )
= ( sup_sup_set_nat @ X3 @ ( lattic3835124923745554447et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_1132_Sup__fin_Oinsert__remove,axiom,
! [A2: set_nat,X3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat2 @ X3 @ A2 ) )
= X3 ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat2 @ X3 @ A2 ) )
= ( sup_sup_nat @ X3 @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_1133_Sup__fin_Osingleton,axiom,
! [X3: nat] :
( ( lattic1093996805478795353in_nat @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) )
= X3 ) ).
% Sup_fin.singleton
thf(fact_1134_inf__Sup__absorb,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ( ( inf_inf_nat @ A @ ( lattic1093996805478795353in_nat @ A2 ) )
= A ) ) ) ).
% inf_Sup_absorb
thf(fact_1135_inf__Sup__absorb,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ( ( inf_inf_set_nat @ A @ ( lattic3835124923745554447et_nat @ A2 ) )
= A ) ) ) ).
% inf_Sup_absorb
thf(fact_1136_Sup__fin_Oinsert,axiom,
! [A2: set_set_nat,X3: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ( lattic3835124923745554447et_nat @ ( insert_set_nat @ X3 @ A2 ) )
= ( sup_sup_set_nat @ X3 @ ( lattic3835124923745554447et_nat @ A2 ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_1137_Sup__fin_Oinsert,axiom,
! [A2: set_nat,X3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat2 @ X3 @ A2 ) )
= ( sup_sup_nat @ X3 @ ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_1138_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_1139_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_1140_Sup__fin_Oin__idem,axiom,
! [A2: set_nat,X3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X3 @ A2 )
=> ( ( sup_sup_nat @ X3 @ ( lattic1093996805478795353in_nat @ A2 ) )
= ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_1141_Sup__fin_Oin__idem,axiom,
! [A2: set_set_nat,X3: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ X3 @ A2 )
=> ( ( sup_sup_set_nat @ X3 @ ( lattic3835124923745554447et_nat @ A2 ) )
= ( lattic3835124923745554447et_nat @ A2 ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_1142_chainsD,axiom,
! [C: set_set_nat,S: set_set_nat,X3: set_nat,Y3: set_nat] :
( ( member_set_set_nat @ C @ ( chains_nat @ S ) )
=> ( ( member_set_nat @ X3 @ C )
=> ( ( member_set_nat @ Y3 @ C )
=> ( ( ord_less_eq_set_nat @ X3 @ Y3 )
| ( ord_less_eq_set_nat @ Y3 @ X3 ) ) ) ) ) ).
% chainsD
thf(fact_1143_Zorn__Lemma2,axiom,
! [A2: set_set_nat] :
( ! [X2: set_set_nat] :
( ( member_set_set_nat @ X2 @ ( chains_nat @ A2 ) )
=> ? [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
& ! [Xb: set_nat] :
( ( member_set_nat @ Xb @ X2 )
=> ( ord_less_eq_set_nat @ Xb @ Xa ) ) ) )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Xa )
=> ( Xa = X2 ) ) ) ) ) ).
% Zorn_Lemma2
thf(fact_1144_Sup__fin_OboundedE,axiom,
! [A2: set_nat,X3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ X3 )
=> ! [A8: nat] :
( ( member_nat @ A8 @ A2 )
=> ( ord_less_eq_nat @ A8 @ X3 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_1145_Sup__fin_OboundedE,axiom,
! [A2: set_set_nat,X3: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A2 ) @ X3 )
=> ! [A8: set_nat] :
( ( member_set_nat @ A8 @ A2 )
=> ( ord_less_eq_set_nat @ A8 @ X3 ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_1146_Sup__fin_OboundedI,axiom,
! [A2: set_nat,X3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [A7: nat] :
( ( member_nat @ A7 @ A2 )
=> ( ord_less_eq_nat @ A7 @ X3 ) )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ X3 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_1147_Sup__fin_OboundedI,axiom,
! [A2: set_set_nat,X3: 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 @ X3 ) )
=> ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A2 ) @ X3 ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_1148_Sup__fin_Obounded__iff,axiom,
! [A2: set_nat,X3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ X3 )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_less_eq_nat @ X @ X3 ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_1149_Sup__fin_Obounded__iff,axiom,
! [A2: set_set_nat,X3: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A2 ) @ X3 )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ X @ X3 ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_1150_Sup__fin_Osubset__imp,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A2 ) @ ( lattic3835124923745554447et_nat @ B2 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_1151_Sup__fin_Osubset__imp,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ ( lattic1093996805478795353in_nat @ B2 ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_1152_Sup__fin_Ohom__commute,axiom,
! [H: set_nat > set_nat,N4: set_set_nat] :
( ! [X2: set_nat,Y: set_nat] :
( ( H @ ( sup_sup_set_nat @ X2 @ Y ) )
= ( sup_sup_set_nat @ ( H @ X2 ) @ ( H @ Y ) ) )
=> ( ( finite1152437895449049373et_nat @ N4 )
=> ( ( N4 != bot_bot_set_set_nat )
=> ( ( H @ ( lattic3835124923745554447et_nat @ N4 ) )
= ( lattic3835124923745554447et_nat @ ( image_7916887816326733075et_nat @ H @ N4 ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_1153_Sup__fin_Ohom__commute,axiom,
! [H: nat > nat,N4: set_nat] :
( ! [X2: nat,Y: nat] :
( ( H @ ( sup_sup_nat @ X2 @ Y ) )
= ( sup_sup_nat @ ( H @ X2 ) @ ( H @ Y ) ) )
=> ( ( finite_finite_nat @ N4 )
=> ( ( N4 != bot_bot_set_nat )
=> ( ( H @ ( lattic1093996805478795353in_nat @ N4 ) )
= ( lattic1093996805478795353in_nat @ ( image_nat_nat @ H @ N4 ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_1154_Sup__fin_Osubset,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( B2 != bot_bot_set_set_nat )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ A2 )
=> ( ( sup_sup_set_nat @ ( lattic3835124923745554447et_nat @ B2 ) @ ( lattic3835124923745554447et_nat @ A2 ) )
= ( lattic3835124923745554447et_nat @ A2 ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_1155_Sup__fin_Osubset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( B2 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ B2 ) @ ( lattic1093996805478795353in_nat @ A2 ) )
= ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_1156_Sup__fin_Oinsert__not__elem,axiom,
! [A2: set_set_nat,X3: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ~ ( member_set_nat @ X3 @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ( lattic3835124923745554447et_nat @ ( insert_set_nat @ X3 @ A2 ) )
= ( sup_sup_set_nat @ X3 @ ( lattic3835124923745554447et_nat @ A2 ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_1157_Sup__fin_Oinsert__not__elem,axiom,
! [A2: set_nat,X3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ~ ( member_nat @ X3 @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat2 @ X3 @ A2 ) )
= ( sup_sup_nat @ X3 @ ( lattic1093996805478795353in_nat @ A2 ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_1158_Sup__fin_Oclosed,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ! [X2: set_nat,Y: set_nat] : ( member_set_nat @ ( sup_sup_set_nat @ X2 @ Y ) @ ( insert_set_nat @ X2 @ ( insert_set_nat @ Y @ bot_bot_set_set_nat ) ) )
=> ( member_set_nat @ ( lattic3835124923745554447et_nat @ A2 ) @ A2 ) ) ) ) ).
% Sup_fin.closed
thf(fact_1159_Sup__fin_Oclosed,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [X2: nat,Y: nat] : ( member_nat @ ( sup_sup_nat @ X2 @ Y ) @ ( insert_nat2 @ X2 @ ( insert_nat2 @ Y @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ A2 ) ) ) ) ).
% Sup_fin.closed
thf(fact_1160_Sup__fin_Ounion,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( B2 != bot_bot_set_set_nat )
=> ( ( lattic3835124923745554447et_nat @ ( sup_sup_set_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ ( lattic3835124923745554447et_nat @ A2 ) @ ( lattic3835124923745554447et_nat @ B2 ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_1161_Sup__fin_Ounion,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B2 )
=> ( ( B2 != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ ( lattic1093996805478795353in_nat @ B2 ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_1162_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_1163_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_1164_Sup__fin_Oremove,axiom,
! [A2: set_set_nat,X3: set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( member_set_nat @ X3 @ A2 )
=> ( ( ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) )
= bot_bot_set_set_nat )
=> ( ( lattic3835124923745554447et_nat @ A2 )
= X3 ) )
& ( ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) )
!= bot_bot_set_set_nat )
=> ( ( lattic3835124923745554447et_nat @ A2 )
= ( sup_sup_set_nat @ X3 @ ( lattic3835124923745554447et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_1165_Sup__fin_Oremove,axiom,
! [A2: set_nat,X3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X3 @ A2 )
=> ( ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A2 )
= X3 ) )
& ( ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A2 )
= ( sup_sup_nat @ X3 @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_1166_sup__fin__closed,axiom,
! [A2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A2 )
=> ( ( A2 != bot_bot_set_set_nat )
=> ( ! [X2: set_nat,Y: set_nat] :
( ( member_set_nat @ X2 @ A2 )
=> ( ( member_set_nat @ Y @ A2 )
=> ( member_set_nat @ ( sup_sup_set_nat @ X2 @ Y ) @ ( insert_set_nat @ X2 @ ( insert_set_nat @ Y @ bot_bot_set_set_nat ) ) ) ) )
=> ( member_set_nat @ ( lattic3835124923745554447et_nat @ A2 ) @ A2 ) ) ) ) ).
% sup_fin_closed
thf(fact_1167_sup__fin__closed,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [X2: nat,Y: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( member_nat @ ( sup_sup_nat @ X2 @ Y ) @ ( insert_nat2 @ X2 @ ( insert_nat2 @ Y @ bot_bot_set_nat ) ) ) ) )
=> ( member_nat @ ( lattic1093996805478795353in_nat @ A2 ) @ A2 ) ) ) ) ).
% sup_fin_closed
thf(fact_1168_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_1169_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_1170_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_1171_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_1172_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_1173_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_1174_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_1175_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( linord2614967742042102400et_nat @ A2 )
= ( linord2614967742042102400et_nat @ B2 ) )
=> ( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_1176_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_1177_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
! [A2: set_nat,X3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( linord2614967742042102400et_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) )
= ( remove1_nat @ X3 @ ( linord2614967742042102400et_nat @ A2 ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_1178_in__set__remove1,axiom,
! [A: nat,B: nat,Xs2: list_nat] :
( ( A != B )
=> ( ( member_nat @ A @ ( set_nat2 @ ( remove1_nat @ B @ Xs2 ) ) )
= ( member_nat @ A @ ( set_nat2 @ Xs2 ) ) ) ) ).
% in_set_remove1
thf(fact_1179_in__set__remove1,axiom,
! [A: transition,B: transition,Xs2: list_transition] :
( ( A != B )
=> ( ( member_transition @ A @ ( set_transition2 @ ( remove1_transition @ B @ Xs2 ) ) )
= ( member_transition @ A @ ( set_transition2 @ Xs2 ) ) ) ) ).
% in_set_remove1
thf(fact_1180_Min__singleton,axiom,
! [X3: nat] :
( ( lattic8721135487736765967in_nat @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) )
= X3 ) ).
% Min_singleton
thf(fact_1181_Min_Obounded__iff,axiom,
! [A2: set_nat,X3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X3 @ ( lattic8721135487736765967in_nat @ A2 ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_less_eq_nat @ X3 @ X ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_1182_Min__gr__iff,axiom,
! [A2: set_nat,X3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_nat @ X3 @ ( lattic8721135487736765967in_nat @ A2 ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_less_nat @ X3 @ X ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_1183_Min_OboundedI,axiom,
! [A2: set_nat,X3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ! [A7: nat] :
( ( member_nat @ A7 @ A2 )
=> ( ord_less_eq_nat @ X3 @ A7 ) )
=> ( ord_less_eq_nat @ X3 @ ( lattic8721135487736765967in_nat @ A2 ) ) ) ) ) ).
% Min.boundedI
thf(fact_1184_Min_OboundedE,axiom,
! [A2: set_nat,X3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X3 @ ( lattic8721135487736765967in_nat @ A2 ) )
=> ! [A8: nat] :
( ( member_nat @ A8 @ A2 )
=> ( ord_less_eq_nat @ X3 @ A8 ) ) ) ) ) ).
% Min.boundedE
thf(fact_1185_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 )
& ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_less_eq_nat @ M @ X ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_1186_Min__le__iff,axiom,
! [A2: set_nat,X3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ X3 )
= ( ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( ord_less_eq_nat @ X @ X3 ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_1187_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 )
& ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( ord_less_eq_nat @ M @ X ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_1188_Min__less__iff,axiom,
! [A2: set_nat,X3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( ord_less_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ X3 )
= ( ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( ord_less_nat @ X @ X3 ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_1189_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_1190_Min__le,axiom,
! [A2: set_nat,X3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A2 ) @ X3 ) ) ) ).
% Min_le
thf(fact_1191_Min__eqI,axiom,
! [A2: set_nat,X3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ! [Y: nat] :
( ( member_nat @ Y @ A2 )
=> ( ord_less_eq_nat @ X3 @ Y ) )
=> ( ( member_nat @ X3 @ A2 )
=> ( ( lattic8721135487736765967in_nat @ A2 )
= X3 ) ) ) ) ).
% Min_eqI
thf(fact_1192_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_1193_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_1194_set__remove1__subset,axiom,
! [X3: transition,Xs2: list_transition] : ( ord_le8419162016481440574sition @ ( set_transition2 @ ( remove1_transition @ X3 @ Xs2 ) ) @ ( set_transition2 @ Xs2 ) ) ).
% set_remove1_subset
thf(fact_1195_set__remove1__subset,axiom,
! [X3: nat,Xs2: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( remove1_nat @ X3 @ Xs2 ) ) @ ( set_nat2 @ Xs2 ) ) ).
% set_remove1_subset
thf(fact_1196_notin__set__remove1,axiom,
! [X3: nat,Xs2: list_nat,Y3: nat] :
( ~ ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
=> ~ ( member_nat @ X3 @ ( set_nat2 @ ( remove1_nat @ Y3 @ Xs2 ) ) ) ) ).
% notin_set_remove1
thf(fact_1197_notin__set__remove1,axiom,
! [X3: transition,Xs2: list_transition,Y3: transition] :
( ~ ( member_transition @ X3 @ ( set_transition2 @ Xs2 ) )
=> ~ ( member_transition @ X3 @ ( set_transition2 @ ( remove1_transition @ Y3 @ Xs2 ) ) ) ) ).
% notin_set_remove1
thf(fact_1198_remove1__idem,axiom,
! [X3: nat,Xs2: list_nat] :
( ~ ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
=> ( ( remove1_nat @ X3 @ Xs2 )
= Xs2 ) ) ).
% remove1_idem
thf(fact_1199_remove1__idem,axiom,
! [X3: transition,Xs2: list_transition] :
( ~ ( member_transition @ X3 @ ( set_transition2 @ Xs2 ) )
=> ( ( remove1_transition @ X3 @ Xs2 )
= Xs2 ) ) ).
% remove1_idem
thf(fact_1200_remove1_Osimps_I1_J,axiom,
! [X3: transition] :
( ( remove1_transition @ X3 @ nil_transition )
= nil_transition ) ).
% remove1.simps(1)
thf(fact_1201_remove1_Osimps_I1_J,axiom,
! [X3: list_o] :
( ( remove1_list_o @ X3 @ nil_list_o )
= nil_list_o ) ).
% remove1.simps(1)
thf(fact_1202_remove1_Osimps_I1_J,axiom,
! [X3: $o] :
( ( remove1_o @ X3 @ nil_o )
= nil_o ) ).
% remove1.simps(1)
thf(fact_1203_remove1_Osimps_I2_J,axiom,
! [X3: list_o,Y3: list_o,Xs2: list_list_o] :
( ( ( X3 = Y3 )
=> ( ( remove1_list_o @ X3 @ ( cons_list_o @ Y3 @ Xs2 ) )
= Xs2 ) )
& ( ( X3 != Y3 )
=> ( ( remove1_list_o @ X3 @ ( cons_list_o @ Y3 @ Xs2 ) )
= ( cons_list_o @ Y3 @ ( remove1_list_o @ X3 @ Xs2 ) ) ) ) ) ).
% remove1.simps(2)
thf(fact_1204_Min_Osubset__imp,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ B2 ) @ ( lattic8721135487736765967in_nat @ A2 ) ) ) ) ) ).
% Min.subset_imp
thf(fact_1205_Min__antimono,axiom,
! [M2: set_nat,N4: set_nat] :
( ( ord_less_eq_set_nat @ M2 @ N4 )
=> ( ( M2 != bot_bot_set_nat )
=> ( ( finite_finite_nat @ N4 )
=> ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ N4 ) @ ( lattic8721135487736765967in_nat @ M2 ) ) ) ) ) ).
% Min_antimono
thf(fact_1206_min__list__Min,axiom,
! [Xs2: list_o] :
( ( Xs2 != nil_o )
=> ( ( min_list_o @ Xs2 )
= ( lattic1973801136483472281_Min_o @ ( set_o2 @ Xs2 ) ) ) ) ).
% min_list_Min
thf(fact_1207_set__remove1__eq,axiom,
! [Xs2: list_transition,X3: transition] :
( ( distinct_transition @ Xs2 )
=> ( ( set_transition2 @ ( remove1_transition @ X3 @ Xs2 ) )
= ( minus_8944320859760356485sition @ ( set_transition2 @ Xs2 ) @ ( insert_transition2 @ X3 @ bot_bo301567166201926666sition ) ) ) ) ).
% set_remove1_eq
thf(fact_1208_set__remove1__eq,axiom,
! [Xs2: list_nat,X3: nat] :
( ( distinct_nat @ Xs2 )
=> ( ( set_nat2 @ ( remove1_nat @ X3 @ Xs2 ) )
= ( minus_minus_set_nat @ ( set_nat2 @ Xs2 ) @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) ) ).
% set_remove1_eq
thf(fact_1209_pred__on_Ochain__extend,axiom,
! [A2: set_transition,P: transition > transition > $o,C2: set_transition,Z2: transition] :
( ( pred_c3769398113492517609sition @ A2 @ P @ C2 )
=> ( ( member_transition @ Z2 @ A2 )
=> ( ! [X2: transition] :
( ( member_transition @ X2 @ C2 )
=> ( sup_su3408646655489644610tion_o @ P
@ ^ [Y2: transition,Z: transition] : ( Y2 = Z )
@ X2
@ Z2 ) )
=> ( pred_c3769398113492517609sition @ A2 @ P @ ( sup_su812053455038985074sition @ ( insert_transition2 @ Z2 @ bot_bo301567166201926666sition ) @ C2 ) ) ) ) ) ).
% pred_on.chain_extend
thf(fact_1210_pred__on_Ochain__extend,axiom,
! [A2: set_nat,P: nat > nat > $o,C2: set_nat,Z2: nat] :
( ( pred_chain_nat @ A2 @ P @ C2 )
=> ( ( member_nat @ Z2 @ A2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ C2 )
=> ( sup_sup_nat_nat_o @ P
@ ^ [Y2: nat,Z: nat] : ( Y2 = Z )
@ X2
@ Z2 ) )
=> ( pred_chain_nat @ A2 @ P @ ( sup_sup_set_nat @ ( insert_nat2 @ Z2 @ bot_bot_set_nat ) @ C2 ) ) ) ) ) ).
% pred_on.chain_extend
thf(fact_1211_subset__chain__insert,axiom,
! [A9: set_set_nat,B2: set_nat,B8: set_set_nat] :
( ( pred_chain_set_nat @ A9 @ ord_less_set_nat @ ( insert_set_nat @ B2 @ B8 ) )
= ( ( member_set_nat @ B2 @ A9 )
& ! [X: set_nat] :
( ( member_set_nat @ X @ B8 )
=> ( ( ord_less_eq_set_nat @ X @ B2 )
| ( ord_less_eq_set_nat @ B2 @ X ) ) )
& ( pred_chain_set_nat @ A9 @ ord_less_set_nat @ B8 ) ) ) ).
% subset_chain_insert
thf(fact_1212_finite__distinct__list,axiom,
! [A2: set_transition] :
( ( finite8165534619950747239sition @ A2 )
=> ? [Xs: list_transition] :
( ( ( set_transition2 @ Xs )
= A2 )
& ( distinct_transition @ Xs ) ) ) ).
% finite_distinct_list
thf(fact_1213_finite__distinct__list,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ? [Xs: list_nat] :
( ( ( set_nat2 @ Xs )
= A2 )
& ( distinct_nat @ Xs ) ) ) ).
% finite_distinct_list
thf(fact_1214_distinct__singleton,axiom,
! [X3: transition] : ( distinct_transition @ ( cons_transition @ X3 @ nil_transition ) ) ).
% distinct_singleton
thf(fact_1215_distinct__singleton,axiom,
! [X3: $o] : ( distinct_o @ ( cons_o @ X3 @ nil_o ) ) ).
% distinct_singleton
thf(fact_1216_distinct__singleton,axiom,
! [X3: list_o] : ( distinct_list_o @ ( cons_list_o @ X3 @ nil_list_o ) ) ).
% distinct_singleton
thf(fact_1217_distinct_Osimps_I2_J,axiom,
! [X3: nat,Xs2: list_nat] :
( ( distinct_nat @ ( cons_nat @ X3 @ Xs2 ) )
= ( ~ ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
& ( distinct_nat @ Xs2 ) ) ) ).
% distinct.simps(2)
thf(fact_1218_distinct_Osimps_I2_J,axiom,
! [X3: transition,Xs2: list_transition] :
( ( distinct_transition @ ( cons_transition @ X3 @ Xs2 ) )
= ( ~ ( member_transition @ X3 @ ( set_transition2 @ Xs2 ) )
& ( distinct_transition @ Xs2 ) ) ) ).
% distinct.simps(2)
thf(fact_1219_distinct_Osimps_I2_J,axiom,
! [X3: list_o,Xs2: list_list_o] :
( ( distinct_list_o @ ( cons_list_o @ X3 @ Xs2 ) )
= ( ~ ( member_list_o @ X3 @ ( set_list_o2 @ Xs2 ) )
& ( distinct_list_o @ Xs2 ) ) ) ).
% distinct.simps(2)
thf(fact_1220_distinct__length__2__or__more,axiom,
! [A: list_o,B: list_o,Xs2: list_list_o] :
( ( distinct_list_o @ ( cons_list_o @ A @ ( cons_list_o @ B @ Xs2 ) ) )
= ( ( A != B )
& ( distinct_list_o @ ( cons_list_o @ A @ Xs2 ) )
& ( distinct_list_o @ ( cons_list_o @ B @ Xs2 ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_1221_distinct_Osimps_I1_J,axiom,
distinct_transition @ nil_transition ).
% distinct.simps(1)
thf(fact_1222_distinct_Osimps_I1_J,axiom,
distinct_list_o @ nil_list_o ).
% distinct.simps(1)
thf(fact_1223_distinct_Osimps_I1_J,axiom,
distinct_o @ nil_o ).
% distinct.simps(1)
thf(fact_1224_subset__Zorn,axiom,
! [A2: set_set_nat] :
( ! [C4: set_set_nat] :
( ( pred_chain_set_nat @ A2 @ ord_less_set_nat @ C4 )
=> ? [X5: set_nat] :
( ( member_set_nat @ X5 @ A2 )
& ! [Xa2: set_nat] :
( ( member_set_nat @ Xa2 @ C4 )
=> ( ord_less_eq_set_nat @ Xa2 @ X5 ) ) ) )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Xa )
=> ( Xa = X2 ) ) ) ) ) ).
% subset_Zorn
thf(fact_1225_pred__on_Ochain__empty,axiom,
! [A2: set_nat,P: nat > nat > $o] : ( pred_chain_nat @ A2 @ P @ bot_bot_set_nat ) ).
% pred_on.chain_empty
thf(fact_1226_pred__on_Ochain__total,axiom,
! [A2: set_nat,P: nat > nat > $o,C2: set_nat,X3: nat,Y3: nat] :
( ( pred_chain_nat @ A2 @ P @ C2 )
=> ( ( member_nat @ X3 @ C2 )
=> ( ( member_nat @ Y3 @ C2 )
=> ( ( sup_sup_nat_nat_o @ P
@ ^ [Y2: nat,Z: nat] : ( Y2 = Z )
@ X3
@ Y3 )
| ( sup_sup_nat_nat_o @ P
@ ^ [Y2: nat,Z: nat] : ( Y2 = Z )
@ Y3
@ X3 ) ) ) ) ) ).
% pred_on.chain_total
thf(fact_1227_pred__on_Ochain__total,axiom,
! [A2: set_transition,P: transition > transition > $o,C2: set_transition,X3: transition,Y3: transition] :
( ( pred_c3769398113492517609sition @ A2 @ P @ C2 )
=> ( ( member_transition @ X3 @ C2 )
=> ( ( member_transition @ Y3 @ C2 )
=> ( ( sup_su3408646655489644610tion_o @ P
@ ^ [Y2: transition,Z: transition] : ( Y2 = Z )
@ X3
@ Y3 )
| ( sup_su3408646655489644610tion_o @ P
@ ^ [Y2: transition,Z: transition] : ( Y2 = Z )
@ Y3
@ X3 ) ) ) ) ) ).
% pred_on.chain_total
thf(fact_1228_pred__on_OchainI,axiom,
! [C2: set_transition,A2: set_transition,P: transition > transition > $o] :
( ( ord_le8419162016481440574sition @ C2 @ A2 )
=> ( ! [X2: transition,Y: transition] :
( ( member_transition @ X2 @ C2 )
=> ( ( member_transition @ Y @ C2 )
=> ( ( sup_su3408646655489644610tion_o @ P
@ ^ [Y2: transition,Z: transition] : ( Y2 = Z )
@ X2
@ Y )
| ( sup_su3408646655489644610tion_o @ P
@ ^ [Y2: transition,Z: transition] : ( Y2 = Z )
@ Y
@ X2 ) ) ) )
=> ( pred_c3769398113492517609sition @ A2 @ P @ C2 ) ) ) ).
% pred_on.chainI
thf(fact_1229_pred__on_OchainI,axiom,
! [C2: set_nat,A2: set_nat,P: nat > nat > $o] :
( ( ord_less_eq_set_nat @ C2 @ A2 )
=> ( ! [X2: nat,Y: nat] :
( ( member_nat @ X2 @ C2 )
=> ( ( member_nat @ Y @ C2 )
=> ( ( sup_sup_nat_nat_o @ P
@ ^ [Y2: nat,Z: nat] : ( Y2 = Z )
@ X2
@ Y )
| ( sup_sup_nat_nat_o @ P
@ ^ [Y2: nat,Z: nat] : ( Y2 = Z )
@ Y
@ X2 ) ) ) )
=> ( pred_chain_nat @ A2 @ P @ C2 ) ) ) ).
% pred_on.chainI
thf(fact_1230_pred__on_Ochain__def,axiom,
( pred_chain_nat
= ( ^ [A3: set_nat,P2: nat > nat > $o,C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A3 )
& ! [X: nat] :
( ( member_nat @ X @ C3 )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ C3 )
=> ( ( sup_sup_nat_nat_o @ P2
@ ^ [Y2: nat,Z: nat] : ( Y2 = Z )
@ X
@ Y4 )
| ( sup_sup_nat_nat_o @ P2
@ ^ [Y2: nat,Z: nat] : ( Y2 = Z )
@ Y4
@ X ) ) ) ) ) ) ) ).
% pred_on.chain_def
thf(fact_1231_subset__chain__def,axiom,
! [A9: set_set_nat,C5: set_set_nat] :
( ( pred_chain_set_nat @ A9 @ ord_less_set_nat @ C5 )
= ( ( ord_le6893508408891458716et_nat @ C5 @ A9 )
& ! [X: set_nat] :
( ( member_set_nat @ X @ C5 )
=> ! [Y4: set_nat] :
( ( member_set_nat @ Y4 @ C5 )
=> ( ( ord_less_eq_set_nat @ X @ Y4 )
| ( ord_less_eq_set_nat @ Y4 @ X ) ) ) ) ) ) ).
% subset_chain_def
thf(fact_1232_distinct__concat__iff,axiom,
! [Xs2: list_list_list_o] :
( ( distinct_list_o @ ( concat_list_o @ Xs2 ) )
= ( ( distinct_list_list_o @ ( remove3821550480258065712list_o @ nil_list_o @ Xs2 ) )
& ! [Ys3: list_list_o] :
( ( member_list_list_o @ Ys3 @ ( set_list_list_o2 @ Xs2 ) )
=> ( distinct_list_o @ Ys3 ) )
& ! [Ys3: list_list_o,Zs: list_list_o] :
( ( ( member_list_list_o @ Ys3 @ ( set_list_list_o2 @ Xs2 ) )
& ( member_list_list_o @ Zs @ ( set_list_list_o2 @ Xs2 ) )
& ( Ys3 != Zs ) )
=> ( ( inf_inf_set_list_o @ ( set_list_o2 @ Ys3 ) @ ( set_list_o2 @ Zs ) )
= bot_bot_set_list_o ) ) ) ) ).
% distinct_concat_iff
thf(fact_1233_distinct__concat__iff,axiom,
! [Xs2: list_list_o] :
( ( distinct_o @ ( concat_o @ Xs2 ) )
= ( ( distinct_list_o @ ( removeAll_list_o @ nil_o @ Xs2 ) )
& ! [Ys3: list_o] :
( ( member_list_o @ Ys3 @ ( set_list_o2 @ Xs2 ) )
=> ( distinct_o @ Ys3 ) )
& ! [Ys3: list_o,Zs: list_o] :
( ( ( member_list_o @ Ys3 @ ( set_list_o2 @ Xs2 ) )
& ( member_list_o @ Zs @ ( set_list_o2 @ Xs2 ) )
& ( Ys3 != Zs ) )
=> ( ( inf_inf_set_o @ ( set_o2 @ Ys3 ) @ ( set_o2 @ Zs ) )
= bot_bot_set_o ) ) ) ) ).
% distinct_concat_iff
thf(fact_1234_distinct__concat__iff,axiom,
! [Xs2: list_list_transition] :
( ( distinct_transition @ ( concat_transition @ Xs2 ) )
= ( ( distin4894176225816993341sition @ ( remove2429998804908088272sition @ nil_transition @ Xs2 ) )
& ! [Ys3: list_transition] :
( ( member1473516902542837997sition @ Ys3 @ ( set_list_transition2 @ Xs2 ) )
=> ( distinct_transition @ Ys3 ) )
& ! [Ys3: list_transition,Zs: list_transition] :
( ( ( member1473516902542837997sition @ Ys3 @ ( set_list_transition2 @ Xs2 ) )
& ( member1473516902542837997sition @ Zs @ ( set_list_transition2 @ Xs2 ) )
& ( Ys3 != Zs ) )
=> ( ( inf_in8814773338690644108sition @ ( set_transition2 @ Ys3 ) @ ( set_transition2 @ Zs ) )
= bot_bo301567166201926666sition ) ) ) ) ).
% distinct_concat_iff
thf(fact_1235_distinct__concat__iff,axiom,
! [Xs2: list_list_nat] :
( ( distinct_nat @ ( concat_nat @ Xs2 ) )
= ( ( distinct_list_nat @ ( removeAll_list_nat @ nil_nat @ Xs2 ) )
& ! [Ys3: list_nat] :
( ( member_list_nat @ Ys3 @ ( set_list_nat2 @ Xs2 ) )
=> ( distinct_nat @ Ys3 ) )
& ! [Ys3: list_nat,Zs: list_nat] :
( ( ( member_list_nat @ Ys3 @ ( set_list_nat2 @ Xs2 ) )
& ( member_list_nat @ Zs @ ( set_list_nat2 @ Xs2 ) )
& ( Ys3 != Zs ) )
=> ( ( inf_inf_set_nat @ ( set_nat2 @ Ys3 ) @ ( set_nat2 @ Zs ) )
= bot_bot_set_nat ) ) ) ) ).
% distinct_concat_iff
thf(fact_1236_distinct__list__update,axiom,
! [Xs2: list_transition,A: transition,I: nat] :
( ( distinct_transition @ Xs2 )
=> ( ~ ( member_transition @ A @ ( minus_8944320859760356485sition @ ( set_transition2 @ Xs2 ) @ ( insert_transition2 @ ( nth_transition @ Xs2 @ I ) @ bot_bo301567166201926666sition ) ) )
=> ( distinct_transition @ ( list_u2676915870505589036sition @ Xs2 @ I @ A ) ) ) ) ).
% distinct_list_update
thf(fact_1237_distinct__list__update,axiom,
! [Xs2: list_nat,A: nat,I: nat] :
( ( distinct_nat @ Xs2 )
=> ( ~ ( member_nat @ A @ ( minus_minus_set_nat @ ( set_nat2 @ Xs2 ) @ ( insert_nat2 @ ( nth_nat @ Xs2 @ I ) @ bot_bot_set_nat ) ) )
=> ( distinct_nat @ ( list_update_nat @ Xs2 @ I @ A ) ) ) ) ).
% distinct_list_update
thf(fact_1238_list__update__nonempty,axiom,
! [Xs2: list_transition,K: nat,X3: transition] :
( ( ( list_u2676915870505589036sition @ Xs2 @ K @ X3 )
= nil_transition )
= ( Xs2 = nil_transition ) ) ).
% list_update_nonempty
thf(fact_1239_list__update__nonempty,axiom,
! [Xs2: list_list_o,K: nat,X3: list_o] :
( ( ( list_update_list_o @ Xs2 @ K @ X3 )
= nil_list_o )
= ( Xs2 = nil_list_o ) ) ).
% list_update_nonempty
thf(fact_1240_list__update__nonempty,axiom,
! [Xs2: list_o,K: nat,X3: $o] :
( ( ( list_update_o @ Xs2 @ K @ X3 )
= nil_o )
= ( Xs2 = nil_o ) ) ).
% list_update_nonempty
thf(fact_1241_nth__list__update__neq,axiom,
! [I: nat,J: nat,Xs2: list_transition,X3: transition] :
( ( I != J )
=> ( ( nth_transition @ ( list_u2676915870505589036sition @ Xs2 @ I @ X3 ) @ J )
= ( nth_transition @ Xs2 @ J ) ) ) ).
% nth_list_update_neq
thf(fact_1242_list__update__id,axiom,
! [Xs2: list_transition,I: nat] :
( ( list_u2676915870505589036sition @ Xs2 @ I @ ( nth_transition @ Xs2 @ I ) )
= Xs2 ) ).
% list_update_id
thf(fact_1243_Nil__eq__concat__conv,axiom,
! [Xss2: list_list_transition] :
( ( nil_transition
= ( concat_transition @ Xss2 ) )
= ( ! [X: list_transition] :
( ( member1473516902542837997sition @ X @ ( set_list_transition2 @ Xss2 ) )
=> ( X = nil_transition ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_1244_Nil__eq__concat__conv,axiom,
! [Xss2: list_list_list_o] :
( ( nil_list_o
= ( concat_list_o @ Xss2 ) )
= ( ! [X: list_list_o] :
( ( member_list_list_o @ X @ ( set_list_list_o2 @ Xss2 ) )
=> ( X = nil_list_o ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_1245_Nil__eq__concat__conv,axiom,
! [Xss2: list_list_o] :
( ( nil_o
= ( concat_o @ Xss2 ) )
= ( ! [X: list_o] :
( ( member_list_o @ X @ ( set_list_o2 @ Xss2 ) )
=> ( X = nil_o ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_1246_concat__eq__Nil__conv,axiom,
! [Xss2: list_list_transition] :
( ( ( concat_transition @ Xss2 )
= nil_transition )
= ( ! [X: list_transition] :
( ( member1473516902542837997sition @ X @ ( set_list_transition2 @ Xss2 ) )
=> ( X = nil_transition ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_1247_concat__eq__Nil__conv,axiom,
! [Xss2: list_list_list_o] :
( ( ( concat_list_o @ Xss2 )
= nil_list_o )
= ( ! [X: list_list_o] :
( ( member_list_list_o @ X @ ( set_list_list_o2 @ Xss2 ) )
=> ( X = nil_list_o ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_1248_concat__eq__Nil__conv,axiom,
! [Xss2: list_list_o] :
( ( ( concat_o @ Xss2 )
= nil_o )
= ( ! [X: list_o] :
( ( member_list_o @ X @ ( set_list_o2 @ Xss2 ) )
=> ( X = nil_o ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_1249_set__update__subsetI,axiom,
! [Xs2: list_transition,A2: set_transition,X3: transition,I: nat] :
( ( ord_le8419162016481440574sition @ ( set_transition2 @ Xs2 ) @ A2 )
=> ( ( member_transition @ X3 @ A2 )
=> ( ord_le8419162016481440574sition @ ( set_transition2 @ ( list_u2676915870505589036sition @ Xs2 @ I @ X3 ) ) @ A2 ) ) ) ).
% set_update_subsetI
thf(fact_1250_set__update__subsetI,axiom,
! [Xs2: list_nat,A2: set_nat,X3: nat,I: nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
=> ( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs2 @ I @ X3 ) ) @ A2 ) ) ) ).
% set_update_subsetI
thf(fact_1251_list__update__code_I1_J,axiom,
! [I: nat,Y3: transition] :
( ( list_u2676915870505589036sition @ nil_transition @ I @ Y3 )
= nil_transition ) ).
% list_update_code(1)
thf(fact_1252_list__update__code_I1_J,axiom,
! [I: nat,Y3: list_o] :
( ( list_update_list_o @ nil_list_o @ I @ Y3 )
= nil_list_o ) ).
% list_update_code(1)
thf(fact_1253_list__update__code_I1_J,axiom,
! [I: nat,Y3: $o] :
( ( list_update_o @ nil_o @ I @ Y3 )
= nil_o ) ).
% list_update_code(1)
thf(fact_1254_list__update_Osimps_I1_J,axiom,
! [I: nat,V: transition] :
( ( list_u2676915870505589036sition @ nil_transition @ I @ V )
= nil_transition ) ).
% list_update.simps(1)
thf(fact_1255_list__update_Osimps_I1_J,axiom,
! [I: nat,V: list_o] :
( ( list_update_list_o @ nil_list_o @ I @ V )
= nil_list_o ) ).
% list_update.simps(1)
thf(fact_1256_list__update_Osimps_I1_J,axiom,
! [I: nat,V: $o] :
( ( list_update_o @ nil_o @ I @ V )
= nil_o ) ).
% list_update.simps(1)
thf(fact_1257_list__update__code_I3_J,axiom,
! [X3: list_o,Xs2: list_list_o,I: nat,Y3: list_o] :
( ( list_update_list_o @ ( cons_list_o @ X3 @ Xs2 ) @ ( suc @ I ) @ Y3 )
= ( cons_list_o @ X3 @ ( list_update_list_o @ Xs2 @ I @ Y3 ) ) ) ).
% list_update_code(3)
thf(fact_1258_concat_Osimps_I1_J,axiom,
( ( concat_transition @ nil_list_transition )
= nil_transition ) ).
% concat.simps(1)
thf(fact_1259_concat_Osimps_I1_J,axiom,
( ( concat_list_o @ nil_list_list_o )
= nil_list_o ) ).
% concat.simps(1)
thf(fact_1260_concat_Osimps_I1_J,axiom,
( ( concat_o @ nil_list_o )
= nil_o ) ).
% concat.simps(1)
thf(fact_1261_set__update__subset__insert,axiom,
! [Xs2: list_transition,I: nat,X3: transition] : ( ord_le8419162016481440574sition @ ( set_transition2 @ ( list_u2676915870505589036sition @ Xs2 @ I @ X3 ) ) @ ( insert_transition2 @ X3 @ ( set_transition2 @ Xs2 ) ) ) ).
% set_update_subset_insert
thf(fact_1262_set__update__subset__insert,axiom,
! [Xs2: list_nat,I: nat,X3: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs2 @ I @ X3 ) ) @ ( insert_nat2 @ X3 @ ( set_nat2 @ Xs2 ) ) ) ).
% set_update_subset_insert
thf(fact_1263_distinct__concat,axiom,
! [Xs2: list_list_transition] :
( ( distin4894176225816993341sition @ Xs2 )
=> ( ! [Ys2: list_transition] :
( ( member1473516902542837997sition @ Ys2 @ ( set_list_transition2 @ Xs2 ) )
=> ( distinct_transition @ Ys2 ) )
=> ( ! [Ys2: list_transition,Zs2: list_transition] :
( ( member1473516902542837997sition @ Ys2 @ ( set_list_transition2 @ Xs2 ) )
=> ( ( member1473516902542837997sition @ Zs2 @ ( set_list_transition2 @ Xs2 ) )
=> ( ( Ys2 != Zs2 )
=> ( ( inf_in8814773338690644108sition @ ( set_transition2 @ Ys2 ) @ ( set_transition2 @ Zs2 ) )
= bot_bo301567166201926666sition ) ) ) )
=> ( distinct_transition @ ( concat_transition @ Xs2 ) ) ) ) ) ).
% distinct_concat
thf(fact_1264_distinct__concat,axiom,
! [Xs2: list_list_nat] :
( ( distinct_list_nat @ Xs2 )
=> ( ! [Ys2: list_nat] :
( ( member_list_nat @ Ys2 @ ( set_list_nat2 @ Xs2 ) )
=> ( distinct_nat @ Ys2 ) )
=> ( ! [Ys2: list_nat,Zs2: list_nat] :
( ( member_list_nat @ Ys2 @ ( set_list_nat2 @ Xs2 ) )
=> ( ( member_list_nat @ Zs2 @ ( set_list_nat2 @ Xs2 ) )
=> ( ( Ys2 != Zs2 )
=> ( ( inf_inf_set_nat @ ( set_nat2 @ Ys2 ) @ ( set_nat2 @ Zs2 ) )
= bot_bot_set_nat ) ) ) )
=> ( distinct_nat @ ( concat_nat @ Xs2 ) ) ) ) ) ).
% distinct_concat
thf(fact_1265_set__update__distinct,axiom,
! [Xs2: list_transition,N: nat,X3: transition] :
( ( distinct_transition @ Xs2 )
=> ( ( ord_less_nat @ N @ ( size_s3613142680436377136sition @ Xs2 ) )
=> ( ( set_transition2 @ ( list_u2676915870505589036sition @ Xs2 @ N @ X3 ) )
= ( insert_transition2 @ X3 @ ( minus_8944320859760356485sition @ ( set_transition2 @ Xs2 ) @ ( insert_transition2 @ ( nth_transition @ Xs2 @ N ) @ bot_bo301567166201926666sition ) ) ) ) ) ) ).
% set_update_distinct
thf(fact_1266_set__update__distinct,axiom,
! [Xs2: list_nat,N: nat,X3: nat] :
( ( distinct_nat @ Xs2 )
=> ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( set_nat2 @ ( list_update_nat @ Xs2 @ N @ X3 ) )
= ( insert_nat2 @ X3 @ ( minus_minus_set_nat @ ( set_nat2 @ Xs2 ) @ ( insert_nat2 @ ( nth_nat @ Xs2 @ N ) @ bot_bot_set_nat ) ) ) ) ) ) ).
% set_update_distinct
thf(fact_1267_finite__subset__Union__chain,axiom,
! [A2: set_nat,B8: set_set_nat,A9: set_set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( comple7399068483239264473et_nat @ B8 ) )
=> ( ( B8 != bot_bot_set_set_nat )
=> ( ( pred_chain_set_nat @ A9 @ ord_less_set_nat @ B8 )
=> ~ ! [B4: set_nat] :
( ( member_set_nat @ B4 @ B8 )
=> ~ ( ord_less_eq_set_nat @ A2 @ B4 ) ) ) ) ) ) ).
% finite_subset_Union_chain
thf(fact_1268_cSup__singleton,axiom,
! [X3: nat] :
( ( complete_Sup_Sup_nat @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) )
= X3 ) ).
% cSup_singleton
thf(fact_1269_nth__list__update__eq,axiom,
! [I: nat,Xs2: list_transition,X3: transition] :
( ( ord_less_nat @ I @ ( size_s3613142680436377136sition @ Xs2 ) )
=> ( ( nth_transition @ ( list_u2676915870505589036sition @ Xs2 @ I @ X3 ) @ I )
= X3 ) ) ).
% nth_list_update_eq
thf(fact_1270_set__swap,axiom,
! [I: nat,Xs2: list_transition,J: nat] :
( ( ord_less_nat @ I @ ( size_s3613142680436377136sition @ Xs2 ) )
=> ( ( ord_less_nat @ J @ ( size_s3613142680436377136sition @ Xs2 ) )
=> ( ( set_transition2 @ ( list_u2676915870505589036sition @ ( list_u2676915870505589036sition @ Xs2 @ I @ ( nth_transition @ Xs2 @ J ) ) @ J @ ( nth_transition @ Xs2 @ I ) ) )
= ( set_transition2 @ Xs2 ) ) ) ) ).
% set_swap
thf(fact_1271_distinct__swap,axiom,
! [I: nat,Xs2: list_transition,J: nat] :
( ( ord_less_nat @ I @ ( size_s3613142680436377136sition @ Xs2 ) )
=> ( ( ord_less_nat @ J @ ( size_s3613142680436377136sition @ Xs2 ) )
=> ( ( distinct_transition @ ( list_u2676915870505589036sition @ ( list_u2676915870505589036sition @ Xs2 @ I @ ( nth_transition @ Xs2 @ J ) ) @ J @ ( nth_transition @ Xs2 @ I ) ) )
= ( distinct_transition @ Xs2 ) ) ) ) ).
% distinct_swap
thf(fact_1272_nth__eq__iff__index__eq,axiom,
! [Xs2: list_transition,I: nat,J: nat] :
( ( distinct_transition @ Xs2 )
=> ( ( ord_less_nat @ I @ ( size_s3613142680436377136sition @ Xs2 ) )
=> ( ( ord_less_nat @ J @ ( size_s3613142680436377136sition @ Xs2 ) )
=> ( ( ( nth_transition @ Xs2 @ I )
= ( nth_transition @ Xs2 @ J ) )
= ( I = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_1273_distinct__conv__nth,axiom,
( distinct_transition
= ( ^ [Xs3: list_transition] :
! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3613142680436377136sition @ Xs3 ) )
=> ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_s3613142680436377136sition @ Xs3 ) )
=> ( ( I2 != J2 )
=> ( ( nth_transition @ Xs3 @ I2 )
!= ( nth_transition @ Xs3 @ J2 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_1274_distinct__Ex1,axiom,
! [Xs2: list_nat,X3: nat] :
( ( distinct_nat @ Xs2 )
=> ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
=> ? [X2: nat] :
( ( ord_less_nat @ X2 @ ( size_size_list_nat @ Xs2 ) )
& ( ( nth_nat @ Xs2 @ X2 )
= X3 )
& ! [Y6: nat] :
( ( ( ord_less_nat @ Y6 @ ( size_size_list_nat @ Xs2 ) )
& ( ( nth_nat @ Xs2 @ Y6 )
= X3 ) )
=> ( Y6 = X2 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_1275_distinct__Ex1,axiom,
! [Xs2: list_transition,X3: transition] :
( ( distinct_transition @ Xs2 )
=> ( ( member_transition @ X3 @ ( set_transition2 @ Xs2 ) )
=> ? [X2: nat] :
( ( ord_less_nat @ X2 @ ( size_s3613142680436377136sition @ Xs2 ) )
& ( ( nth_transition @ Xs2 @ X2 )
= X3 )
& ! [Y6: nat] :
( ( ( ord_less_nat @ Y6 @ ( size_s3613142680436377136sition @ Xs2 ) )
& ( ( nth_transition @ Xs2 @ Y6 )
= X3 ) )
=> ( Y6 = X2 ) ) ) ) ) ).
% distinct_Ex1
% Helper facts (7)
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X3: list_nat,Y3: list_nat] :
( ( if_list_nat @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X3: list_nat,Y3: list_nat] :
( ( if_list_nat @ $true @ X3 @ Y3 )
= X3 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__NFA__Otransition_J_T,axiom,
! [X3: list_transition,Y3: list_transition] :
( ( if_list_transition @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__NFA__Otransition_J_T,axiom,
! [X3: list_transition,Y3: list_transition] :
( ( if_list_transition @ $true @ X3 @ Y3 )
= X3 ) ).
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,
! [X3: list_list_o,Y3: list_list_o] :
( ( if_list_list_o @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_I_Eo_J_J_T,axiom,
! [X3: list_list_o,Y3: list_list_o] :
( ( if_list_list_o @ $true @ X3 @ Y3 )
= X3 ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
! [Q9: nat] :
( ( step_eps @ q0a @ transsa @ bs @ qa @ Q9 )
=> ( ( step_eps_closure @ q0a @ transsa @ bs @ Q9 @ qfa )
=> thesis ) ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------