TPTP Problem File: SLH0689^1.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_00623_029203__16658066_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 852 ( 345 unt; 157 typ; 0 def)
% Number of atoms : 1765 ( 909 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 8365 ( 205 ~; 39 |; 113 &;7237 @)
% ( 0 <=>; 771 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 8 avg)
% Number of types : 25 ( 24 usr)
% Number of type conns : 404 ( 404 >; 0 *; 0 +; 0 <<)
% Number of symbols : 136 ( 133 usr; 35 con; 0-9 aty)
% Number of variables : 2577 ( 136 ^;2391 !; 50 ?;2577 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:53:53.523
%------------------------------------------------------------------------------
% Could-be-implicit typings (24)
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__MDL__Oformula_Itf__a_Mtf__t_J_J_J_J,type,
produc238307629227160457la_a_t: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__MDL__Oformula_Itf__a_Mtf__t_J_J_J,type,
produc8388488633478513124la_a_t: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_I_Eo_J_Mt__List__Olist_It__List__Olist_I_Eo_J_J_J,type,
produc7334440844593340333list_o: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__NFA__Otransition_Mt__List__Olist_It__NFA__Otransition_J_J,type,
produc7282413182419550381sition: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_I_Eo_J_J_J,type,
list_list_list_o: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__List__Olist_I_Eo_J_J,type,
produc6454500794699219245list_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__List__Olist_It__MDL__Oformula_Itf__a_Mtf__t_J_J,type,
list_formula_a_t: $tType ).
thf(ty_n_t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__t_J_J,type,
set_formula_a_t: $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__NFA__Otransition_J,type,
set_transition: $tType ).
thf(ty_n_t__Trace__Otrace_Itf__a_Mtf__t_J,type,
trace_a_t: $tType ).
thf(ty_n_t__MDL__Oformula_Itf__a_Mtf__t_J,type,
formula_a_t: $tType ).
thf(ty_n_t__MDL__Oregex_Itf__a_Mtf__t_J,type,
regex_a_t: $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 (133)
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__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_If_001t__Set__Oset_It__Nat__Onat_J,type,
if_set_nat: $o > set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__List__Olist_I_Eo_J_J,type,
sup_sup_set_list_o: set_list_o > set_list_o > set_list_o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__t_J_J,type,
sup_su887667473539889925la_a_t: set_formula_a_t > set_formula_a_t > set_formula_a_t ).
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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
sup_su6327502436637775413at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_List_Oappend_001_Eo,type,
append_o: list_o > list_o > list_o ).
thf(sy_c_List_Oappend_001t__List__Olist_I_Eo_J,type,
append_list_o: list_list_o > list_list_o > list_list_o ).
thf(sy_c_List_Oappend_001t__MDL__Oformula_Itf__a_Mtf__t_J,type,
append_formula_a_t: list_formula_a_t > list_formula_a_t > list_formula_a_t ).
thf(sy_c_List_Oappend_001t__NFA__Otransition,type,
append_transition: list_transition > list_transition > list_transition ).
thf(sy_c_List_Oappend_001t__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_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__MDL__Oformula_Itf__a_Mtf__t_J,type,
cons_formula_a_t: formula_a_t > list_formula_a_t > list_formula_a_t ).
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__MDL__Oformula_Itf__a_Mtf__t_J,type,
nil_formula_a_t: list_formula_a_t ).
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__MDL__Oformula_Itf__a_Mtf__t_J,type,
set_formula_a_t2: list_formula_a_t > set_formula_a_t ).
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_Onth_001_Eo,type,
nth_o: list_o > nat > $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__MDL__Oformula_Itf__a_Mtf__t_J,type,
nth_formula_a_t: list_formula_a_t > nat > formula_a_t ).
thf(sy_c_List_Onth_001t__NFA__Otransition,type,
nth_transition: list_transition > nat > transition ).
thf(sy_c_MDL_OMDL_Omatch_001tf__a_001tf__t,type,
match_a_t: trace_a_t > regex_a_t > set_Pr1261947904930325089at_nat ).
thf(sy_c_MDL_OMDL_Osat_001tf__a_001tf__t,type,
sat_a_t: trace_a_t > formula_a_t > nat > $o ).
thf(sy_c_MDL_Oeps_001tf__a_001tf__t,type,
eps_a_t: regex_a_t > $o ).
thf(sy_c_MDL_Oregex_OStar_001tf__a_001tf__t,type,
star_a_t: regex_a_t > regex_a_t ).
thf(sy_c_MDL_Oregex_OTimes_001tf__a_001tf__t,type,
times_a_t: regex_a_t > regex_a_t > regex_a_t ).
thf(sy_c_MDL_Owf__regex_001tf__a_001tf__t,type,
wf_regex_a_t: regex_a_t > $o ).
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_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_Ostep__eps,type,
step_eps: nat > list_transition > list_o > nat > nat > $o ).
thf(sy_c_NFA_Ostep__eps__closure,type,
step_eps_closure: nat > list_transition > list_o > nat > nat > $o ).
thf(sy_c_NFA_Ostep__eps__closure__set,type,
step_eps_closure_set: nat > list_transition > set_nat > list_o > set_nat ).
thf(sy_c_NFA_Ostep__eps__set,type,
step_eps_set: nat > list_transition > list_o > set_nat > set_nat ).
thf(sy_c_NFA_Ostep__eps__sucs,type,
step_eps_sucs: nat > list_transition > list_o > nat > set_nat ).
thf(sy_c_NFA_Ostep__symb,type,
step_symb: nat > list_transition > nat > nat > $o ).
thf(sy_c_NFA_Ostep__symb__set,type,
step_symb_set: nat > list_transition > set_nat > set_nat ).
thf(sy_c_NFA_Ostep__symb__sucs,type,
step_symb_sucs: nat > list_transition > nat > set_nat ).
thf(sy_c_NFA_Otransition_Ocase__transition_001_Eo,type,
case_transition_o: ( nat > nat > $o ) > ( nat > $o ) > ( nat > nat > $o ) > transition > $o ).
thf(sy_c_NFA_Otransition_Ocase__transition_001t__Set__Oset_It__Nat__Onat_J,type,
case_t7109905494440202798et_nat: ( nat > nat > set_nat ) > ( nat > set_nat ) > ( nat > nat > set_nat ) > transition > set_nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
size_size_list_o: list_o > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_I_Eo_J_J,type,
size_s2710708370519433104list_o: list_list_o > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__t_J_J,type,
size_s8846756101701226951la_a_t: list_formula_a_t > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__NFA__Otransition_J,type,
size_s3613142680436377136sition: list_transition > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_Eo_J,type,
bot_bot_set_o: set_o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_I_Eo_J_J,type,
bot_bot_set_list_o: set_list_o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__NFA__Otransition_J,type,
bot_bo301567166201926666sition: set_transition ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
bot_bo2099793752762293965at_nat: set_Pr1261947904930325089at_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__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > 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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le3146513528884898305at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Product__Type_OPair_001_Eo_001t__List__Olist_I_Eo_J,type,
produc7263596898809104029list_o: $o > list_o > produc6454500794699219245list_o ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_I_Eo_J_001t__List__Olist_It__List__Olist_I_Eo_J_J,type,
produc1974676357355975453list_o: list_o > list_list_o > produc7334440844593340333list_o ).
thf(sy_c_Product__Type_OPair_001t__NFA__Otransition_001t__List__Olist_It__NFA__Otransition_J,type,
produc9190507990355890461sition: transition > list_transition > produc7282413182419550381sition ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__t_J_J,type,
produc9017461973804568604la_a_t: nat > list_formula_a_t > produc8388488633478513124la_a_t ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__MDL__Oformula_Itf__a_Mtf__t_J_J_J,type,
produc8654416511292156347la_a_t: nat > produc8388488633478513124la_a_t > produc238307629227160457la_a_t ).
thf(sy_c_Relation_Orelcomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
relcomp_nat_nat_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
collec3392354462482085612at_nat: ( product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat ).
thf(sy_c_Set_Oinsert_001t__List__Olist_I_Eo_J,type,
insert_list_o: list_o > set_list_o > set_list_o ).
thf(sy_c_Set_Oinsert_001t__NFA__Otransition,type,
insert_transition: transition > set_transition > set_transition ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
insert8211810215607154385at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Temporal_OMDL_OIH_001tf__a_001tf__t,type,
iH_a_t: trace_a_t > regex_a_t > nat > nat > list_formula_a_t > list_transition > list_list_o > list_o > nat > $o ).
thf(sy_c_Temporal_Obuild__nfa__impl_001tf__a_001tf__t,type,
build_nfa_impl_a_t: regex_a_t > produc238307629227160457la_a_t > list_transition ).
thf(sy_c_Temporal_Ocollect__subfmlas_001tf__a_001tf__t,type,
collect_subfmlas_a_t: regex_a_t > list_formula_a_t > list_formula_a_t ).
thf(sy_c_Temporal_Ostate__cnt_001tf__a_001tf__t,type,
state_cnt_a_t: regex_a_t > nat ).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Nat__Onat,type,
transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Wellfounded_Opred__nat,type,
pred_nat: set_Pr1261947904930325089at_nat ).
thf(sy_c_member_001t__List__Olist_I_Eo_J,type,
member_list_o: list_o > set_list_o > $o ).
thf(sy_c_member_001t__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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_v__092_060sigma_062,type,
sigma: trace_a_t ).
thf(sy_v_bs,type,
bs: list_o ).
thf(sy_v_bss,type,
bss: list_list_o ).
thf(sy_v_i,type,
i: nat ).
thf(sy_v_phis,type,
phis: list_formula_a_t ).
thf(sy_v_phis_H____,type,
phis2: list_formula_a_t ).
thf(sy_v_phisa____,type,
phisa: list_formula_a_t ).
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_r,type,
r: regex_a_t ).
thf(sy_v_ra____,type,
ra: regex_a_t ).
thf(sy_v_s____,type,
s: regex_a_t ).
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 ).
thf(sy_v_ts__r____,type,
ts_r: list_transition ).
% Relevant facts (691)
thf(fact_0_assms_I1_J,axiom,
wf_regex_a_t @ r ).
% assms(1)
thf(fact_1_cong_Oright_Oqf__eq,axiom,
qfa = qfa ).
% cong.right.qf_eq
thf(fact_2_Times_Oprems_I1_J,axiom,
wf_regex_a_t @ ( times_a_t @ ra @ s ) ).
% Times.prems(1)
thf(fact_3_Times_Oprems_I3_J,axiom,
step_eps_closure @ q0a @ transsa @ bs @ qa @ qfa ).
% Times.prems(3)
thf(fact_4_right_Oqf__not__in__SQ,axiom,
~ ( member_nat @ qfa @ ( sq @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_r ) ) ).
% right.qf_not_in_SQ
thf(fact_5_Times_OIH_I1_J,axiom,
! [Q0: nat,Qf: nat,Phis: list_formula_a_t,Transs: list_transition,Q: nat] :
( ( wf_regex_a_t @ ra )
=> ( ( iH_a_t @ sigma @ ra @ Q0 @ Qf @ Phis @ Transs @ bss @ bs @ i )
=> ( ( step_eps_closure @ Q0 @ Transs @ bs @ Q @ Qf )
=> ( step_eps_closure @ Q0 @ Transs @ nil_o @ Q @ Qf ) ) ) ) ).
% Times.IH(1)
thf(fact_6_Times_OIH_I2_J,axiom,
! [Q0: nat,Qf: nat,Phis: list_formula_a_t,Transs: list_transition,Q: nat] :
( ( wf_regex_a_t @ s )
=> ( ( iH_a_t @ sigma @ s @ Q0 @ Qf @ Phis @ Transs @ bss @ bs @ i )
=> ( ( step_eps_closure @ Q0 @ Transs @ bs @ Q @ Qf )
=> ( step_eps_closure @ Q0 @ Transs @ nil_o @ Q @ Qf ) ) ) ) ).
% Times.IH(2)
thf(fact_7_Times_Oprems_I2_J,axiom,
iH_a_t @ sigma @ ( times_a_t @ ra @ s ) @ q0a @ qfa @ phisa @ transsa @ bss @ bs @ i ).
% Times.prems(2)
thf(fact_8_right_Ostep__eps__closure__qf,axiom,
! [Bs: list_o,Q: nat,Q2: nat] :
( ( step_eps_closure @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_r @ Bs @ Q @ Q2 )
=> ( ( Q = qfa )
=> ( Q = Q2 ) ) ) ).
% right.step_eps_closure_qf
thf(fact_9_MDL_OIH_Ocong,axiom,
iH_a_t = iH_a_t ).
% MDL.IH.cong
thf(fact_10_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_11_ts__r__def,axiom,
( ts_r
= ( build_nfa_impl_a_t @ s @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ( produc9017461973804568604la_a_t @ qfa @ ( collect_subfmlas_a_t @ ra @ phisa ) ) ) ) ) ).
% ts_r_def
thf(fact_12_cong_Oqf_H__in__SQ,axiom,
member_nat @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ( sq @ q0a @ transsa ) ).
% cong.qf'_in_SQ
thf(fact_13_base_Oqf__not__in__SQ,axiom,
~ ( member_nat @ qfa @ ( sq @ q0a @ transsa ) ) ).
% base.qf_not_in_SQ
thf(fact_14_right_Ostep__symb__qf,axiom,
! [Q: nat] :
~ ( step_symb @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_r @ qfa @ Q ) ).
% right.step_symb_qf
thf(fact_15_right_Ostep__eps__qf,axiom,
! [Bs: list_o,Q: nat] :
~ ( step_eps @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_r @ Bs @ qfa @ Q ) ).
% right.step_eps_qf
thf(fact_16_right_Onfa__axioms,axiom,
nfa @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ qfa @ ts_r ).
% right.nfa_axioms
thf(fact_17_collect,axiom,
( ( collect_subfmlas_a_t @ ( times_a_t @ ra @ s ) @ phisa )
= ( append_formula_a_t @ ( collect_subfmlas_a_t @ ra @ phisa ) @ phis2 ) ) ).
% collect
thf(fact_18_left__IH,axiom,
iH_a_t @ sigma @ ra @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ phisa @ ts_l @ bss @ bs @ i ).
% left_IH
thf(fact_19_assms_I2_J,axiom,
iH_a_t @ sigma @ r @ q0 @ qf @ phis @ transs @ bss @ bs @ i ).
% assms(2)
thf(fact_20_cong_Oright_Onfa__cong__axioms,axiom,
nfa_cong @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ qfa @ qfa @ transsa @ ts_r ).
% cong.right.nfa_cong_axioms
thf(fact_21_qf__not__in__SQ,axiom,
~ ( member_nat @ qfa @ ( sq @ q0a @ ( build_nfa_impl_a_t @ ( times_a_t @ ra @ s ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) ) ) ).
% qf_not_in_SQ
thf(fact_22_cong_Oq__SQ__SQ__nfa_H__SQ,axiom,
! [Q: nat] :
( ( member_nat @ Q @ ( sq @ q0a @ ts_l ) )
=> ( ( member_nat @ Q @ ( sq @ q0a @ transsa ) )
= ( member_nat @ Q @ ( sq @ q0a @ ts_l ) ) ) ) ).
% cong.q_SQ_SQ_nfa'_SQ
thf(fact_23_base_Onfa__axioms,axiom,
nfa @ q0a @ qfa @ transsa ).
% base.nfa_axioms
thf(fact_24_base_Ostep__eps__qf,axiom,
! [Bs: list_o,Q: nat] :
~ ( step_eps @ q0a @ transsa @ Bs @ qfa @ Q ) ).
% base.step_eps_qf
thf(fact_25_cong_Ostep__symb__cong__Q,axiom,
! [Q: nat,Q2: nat] :
( ( step_symb @ q0a @ ts_l @ Q @ Q2 )
=> ( step_symb @ q0a @ transsa @ Q @ Q2 ) ) ).
% cong.step_symb_cong_Q
thf(fact_26_base_Ostep__symb__qf,axiom,
! [Q: nat] :
~ ( step_symb @ q0a @ transsa @ qfa @ Q ) ).
% base.step_symb_qf
thf(fact_27_cong_Ostep__eps__cong__SQ,axiom,
! [Q: nat,Bs: list_o,Q2: nat] :
( ( member_nat @ Q @ ( sq @ q0a @ ts_l ) )
=> ( ( step_eps @ q0a @ transsa @ Bs @ Q @ Q2 )
= ( step_eps @ q0a @ ts_l @ Bs @ Q @ Q2 ) ) ) ).
% cong.step_eps_cong_SQ
thf(fact_28_cong_Ostep__symb__cong__SQ,axiom,
! [Q: nat,Q2: nat] :
( ( member_nat @ Q @ ( sq @ q0a @ ts_l ) )
=> ( ( step_symb @ q0a @ transsa @ Q @ Q2 )
= ( step_symb @ q0a @ ts_l @ Q @ Q2 ) ) ) ).
% cong.step_symb_cong_SQ
thf(fact_29_left_Oqf__not__in__SQ,axiom,
~ ( member_nat @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ( sq @ q0a @ ts_l ) ) ).
% left.qf_not_in_SQ
thf(fact_30_left_Ostep__eps__closure__qf,axiom,
! [Bs: list_o,Q: nat,Q2: nat] :
( ( step_eps_closure @ q0a @ ts_l @ Bs @ Q @ Q2 )
=> ( ( Q
= ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) )
=> ( Q = Q2 ) ) ) ).
% left.step_eps_closure_qf
thf(fact_31_left_Onfa__axioms,axiom,
nfa @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_l ).
% left.nfa_axioms
thf(fact_32_left_Ostep__eps__qf,axiom,
! [Bs: list_o,Q: nat] :
~ ( step_eps @ q0a @ ts_l @ Bs @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ Q ) ).
% left.step_eps_qf
thf(fact_33_left_Ostep__symb__qf,axiom,
! [Q: nat] :
~ ( step_symb @ q0a @ ts_l @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ Q ) ).
% left.step_symb_qf
thf(fact_34_assms_I3_J,axiom,
step_eps_closure @ q0 @ transs @ bs @ q2 @ qf ).
% assms(3)
thf(fact_35__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062phis_H_O_Acollect__subfmlas_A_ITimes_Ar_As_J_Aphis_A_061_Acollect__subfmlas_Ar_Aphis_A_064_Aphis_H_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Phis2: list_formula_a_t] :
( ( collect_subfmlas_a_t @ ( times_a_t @ ra @ s ) @ phisa )
!= ( append_formula_a_t @ ( collect_subfmlas_a_t @ ra @ phisa ) @ Phis2 ) ) ).
% \<open>\<And>thesis. (\<And>phis'. collect_subfmlas (Times r s) phis = collect_subfmlas r phis @ phis' \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_36_ts__l__def,axiom,
( ts_l
= ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ phisa ) ) ) ) ).
% ts_l_def
thf(fact_37_transs__def,axiom,
( transsa
= ( build_nfa_impl_a_t @ ( times_a_t @ ra @ s ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) ) ).
% transs_def
thf(fact_38_cong_Onfa__cong__Times__axioms,axiom,
nfa_cong_Times @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ qfa @ transsa @ ts_l @ ts_r ).
% cong.nfa_cong_Times_axioms
thf(fact_39_collect__subfmlas__app,axiom,
! [R: regex_a_t,Phis: list_formula_a_t] :
? [Phis2: list_formula_a_t] :
( ( collect_subfmlas_a_t @ R @ Phis )
= ( append_formula_a_t @ Phis @ Phis2 ) ) ).
% collect_subfmlas_app
thf(fact_40_collect__subfmlas_Osimps_I4_J,axiom,
! [R: regex_a_t,S: regex_a_t,Phis: list_formula_a_t] :
( ( collect_subfmlas_a_t @ ( times_a_t @ R @ S ) @ Phis )
= ( collect_subfmlas_a_t @ S @ ( collect_subfmlas_a_t @ R @ Phis ) ) ) ).
% collect_subfmlas.simps(4)
thf(fact_41_state__cnt_Osimps_I4_J,axiom,
! [R: regex_a_t,S: regex_a_t] :
( ( state_cnt_a_t @ ( times_a_t @ R @ S ) )
= ( plus_plus_nat @ ( state_cnt_a_t @ R ) @ ( state_cnt_a_t @ S ) ) ) ).
% state_cnt.simps(4)
thf(fact_42_cong_Onfa__cong_H__axioms,axiom,
nfa_cong2 @ q0a @ q0a @ qfa @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ transsa @ ts_l ).
% cong.nfa_cong'_axioms
thf(fact_43_case__left,axiom,
! [Q: nat] :
( ( step_eps_closure @ q0a @ ts_l @ bs @ Q @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) )
=> ( ( member_nat @ Q @ ( q @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_l ) )
=> ( ( wf_regex_a_t @ ra )
=> ( step_eps_closure @ q0a @ transsa @ nil_o @ Q @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) ) ) ) ) ).
% case_left
thf(fact_44_cong_Oright_OSQ__sub,axiom,
ord_less_eq_set_nat @ ( sq @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_r ) @ ( sq @ q0a @ transsa ) ).
% cong.right.SQ_sub
thf(fact_45_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_46_mem__Collect__eq,axiom,
! [A: transition,P: transition > $o] :
( ( member_transition @ A @ ( collect_transition @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_47_mem__Collect__eq,axiom,
! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_48_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_49_Collect__mem__eq,axiom,
! [A2: set_transition] :
( ( collect_transition
@ ^ [X: transition] : ( member_transition @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_50_Collect__mem__eq,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ( collec3392354462482085612at_nat
@ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_51_cong_Oright_Ostep__symb__cong,axiom,
! [Q: nat,Q2: nat] :
( ( member_nat @ Q @ ( q @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ qfa @ ts_r ) )
=> ( ( step_symb @ q0a @ transsa @ Q @ Q2 )
= ( step_symb @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_r @ Q @ Q2 ) ) ) ).
% cong.right.step_symb_cong
thf(fact_52_cong_Oright_Ostep__eps__cong,axiom,
! [Q: nat,Bs: list_o,Q2: nat] :
( ( member_nat @ Q @ ( q @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ qfa @ ts_r ) )
=> ( ( step_eps @ q0a @ transsa @ Bs @ Q @ Q2 )
= ( step_eps @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_r @ Bs @ Q @ Q2 ) ) ) ).
% cong.right.step_eps_cong
thf(fact_53_cong_Oright_Onfa_H__eps__step__eps__closure,axiom,
! [Bs: list_o,Q: nat,Q2: nat] :
( ( step_eps_closure @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_r @ Bs @ Q @ Q2 )
=> ( ( member_nat @ Q @ ( q @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ qfa @ ts_r ) )
=> ( ( member_nat @ Q2 @ ( q @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ qfa @ ts_r ) )
& ( step_eps_closure @ q0a @ transsa @ Bs @ Q @ Q2 ) ) ) ) ).
% cong.right.nfa'_eps_step_eps_closure
thf(fact_54_cong_Oright_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 @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ qfa @ ts_r ) )
=> ( ( member_nat @ Q2 @ ( q @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ qfa @ ts_r ) )
& ( step_eps_closure @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_r @ Bs @ Q @ Q2 ) ) ) ) ).
% cong.right.eps_nfa'_step_eps_closure
thf(fact_55_cong_Oright_Oq__Q__SQ__nfa_H__SQ,axiom,
! [Q: nat] :
( ( member_nat @ Q @ ( q @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ qfa @ ts_r ) )
=> ( ( member_nat @ Q @ ( sq @ q0a @ transsa ) )
= ( member_nat @ Q @ ( sq @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_r ) ) ) ) ).
% cong.right.q_Q_SQ_nfa'_SQ
thf(fact_56_right_Ostep__symb__closed,axiom,
! [Q: nat,Q2: nat] :
( ( step_symb @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_r @ Q @ Q2 )
=> ( member_nat @ Q2 @ ( q @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ qfa @ ts_r ) ) ) ).
% right.step_symb_closed
thf(fact_57_right_Ostep__eps__closed,axiom,
! [Bs: list_o,Q: nat,Q2: nat] :
( ( step_eps @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_r @ Bs @ Q @ Q2 )
=> ( member_nat @ Q2 @ ( q @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ qfa @ ts_r ) ) ) ).
% right.step_eps_closed
thf(fact_58_right_Ostep__eps__closure__closed,axiom,
! [Bs: list_o,Q: nat,Q2: nat] :
( ( step_eps_closure @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_r @ Bs @ Q @ Q2 )
=> ( ( Q != Q2 )
=> ( member_nat @ Q2 @ ( q @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ qfa @ ts_r ) ) ) ) ).
% right.step_eps_closure_closed
thf(fact_59_cong_OSQ__sub,axiom,
ord_less_eq_set_nat @ ( sq @ q0a @ ts_l ) @ ( sq @ q0a @ transsa ) ).
% cong.SQ_sub
thf(fact_60_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_61_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_62_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_63_left_Ostep__eps__closure__closed,axiom,
! [Bs: list_o,Q: nat,Q2: nat] :
( ( step_eps_closure @ q0a @ ts_l @ Bs @ Q @ Q2 )
=> ( ( Q != Q2 )
=> ( member_nat @ Q2 @ ( q @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_l ) ) ) ) ).
% left.step_eps_closure_closed
thf(fact_64_left_Ostep__eps__closed,axiom,
! [Bs: list_o,Q: nat,Q2: nat] :
( ( step_eps @ q0a @ ts_l @ Bs @ Q @ Q2 )
=> ( member_nat @ Q2 @ ( q @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_l ) ) ) ).
% left.step_eps_closed
thf(fact_65_left_Ostep__symb__closed,axiom,
! [Q: nat,Q2: nat] :
( ( step_symb @ q0a @ ts_l @ Q @ Q2 )
=> ( member_nat @ Q2 @ ( q @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_l ) ) ) ).
% left.step_symb_closed
thf(fact_66_qf__left__Q,axiom,
~ ( member_nat @ qfa @ ( q @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_l ) ) ).
% qf_left_Q
thf(fact_67_cong_Oeps__nfa_H__step__eps__closure__cong,axiom,
! [Bs: list_o,Q: nat,Q2: nat] :
( ( step_eps_closure @ q0a @ transsa @ Bs @ Q @ Q2 )
=> ( ( member_nat @ Q @ ( q @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_l ) )
=> ( ( ( member_nat @ Q2 @ ( q @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_l ) )
& ( step_eps_closure @ q0a @ ts_l @ Bs @ Q @ Q2 ) )
| ( ( step_eps_closure @ q0a @ ts_l @ Bs @ Q @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) )
& ( step_eps_closure @ q0a @ transsa @ Bs @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ Q2 ) ) ) ) ) ).
% cong.eps_nfa'_step_eps_closure_cong
thf(fact_68_cong_Onfa_H__eps__step__eps__closure__cong,axiom,
! [Bs: list_o,Q: nat,Q2: nat] :
( ( step_eps_closure @ q0a @ ts_l @ Bs @ Q @ Q2 )
=> ( ( member_nat @ Q @ ( q @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_l ) )
=> ( ( member_nat @ Q2 @ ( q @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_l ) )
& ( step_eps_closure @ q0a @ transsa @ Bs @ Q @ Q2 ) ) ) ) ).
% cong.nfa'_eps_step_eps_closure_cong
thf(fact_69_cong_Onfa_H__step__eps__closure__cong,axiom,
! [Bs: list_o,Q: nat,Q2: nat] :
( ( step_eps_closure @ q0a @ ts_l @ Bs @ Q @ Q2 )
=> ( ( member_nat @ Q @ ( q @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_l ) )
=> ( step_eps_closure @ q0a @ transsa @ Bs @ Q @ Q2 ) ) ) ).
% cong.nfa'_step_eps_closure_cong
thf(fact_70_cong_Ostep__eps__cong__Q,axiom,
! [Q: nat,Bs: list_o,Q2: nat] :
( ( member_nat @ Q @ ( q @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_l ) )
=> ( ( step_eps @ q0a @ ts_l @ Bs @ Q @ Q2 )
=> ( step_eps @ q0a @ transsa @ Bs @ Q @ Q2 ) ) ) ).
% cong.step_eps_cong_Q
thf(fact_71_cong_Onfa_H__Q__sub__Q,axiom,
ord_less_eq_set_nat @ ( q @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_l ) @ ( q @ q0a @ qfa @ transsa ) ).
% cong.nfa'_Q_sub_Q
thf(fact_72_base_Ostep__eps__set__closed,axiom,
! [Bs: list_o,R2: set_nat] : ( ord_less_eq_set_nat @ ( step_eps_set @ q0a @ transsa @ Bs @ R2 ) @ ( q @ q0a @ qfa @ transsa ) ) ).
% base.step_eps_set_closed
thf(fact_73_left_Ostep__eps__set__closed,axiom,
! [Bs: list_o,R2: set_nat] : ( ord_less_eq_set_nat @ ( step_eps_set @ q0a @ ts_l @ Bs @ R2 ) @ ( q @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_l ) ) ).
% left.step_eps_set_closed
thf(fact_74_right_Ostep__eps__set__closed,axiom,
! [Bs: list_o,R2: set_nat] : ( ord_less_eq_set_nat @ ( step_eps_set @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_r @ Bs @ R2 ) @ ( q @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ qfa @ ts_r ) ) ).
% right.step_eps_set_closed
thf(fact_75_cong_Oright_Oaccept__eps__cong,axiom,
! [R2: set_nat,Bs: list_o] :
( ( ord_less_eq_set_nat @ R2 @ ( q @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ qfa @ ts_r ) )
=> ( ( accept_eps @ q0a @ qfa @ transsa @ R2 @ Bs )
= ( accept_eps @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ qfa @ ts_r @ R2 @ Bs ) ) ) ).
% cong.right.accept_eps_cong
thf(fact_76_cong_Oright_Ostep__symb__set__cong,axiom,
! [R2: set_nat] :
( ( ord_less_eq_set_nat @ R2 @ ( q @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ qfa @ ts_r ) )
=> ( ( step_symb_set @ q0a @ transsa @ R2 )
= ( step_symb_set @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_r @ R2 ) ) ) ).
% cong.right.step_symb_set_cong
thf(fact_77_right_Ostep__symb__set__closed,axiom,
! [R2: set_nat] : ( ord_less_eq_set_nat @ ( step_symb_set @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_r @ R2 ) @ ( q @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ qfa @ ts_r ) ) ).
% right.step_symb_set_closed
thf(fact_78_cong_Oright_Orun__cong,axiom,
! [R2: set_nat,Bss: list_list_o] :
( ( ord_less_eq_set_nat @ R2 @ ( q @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ qfa @ ts_r ) )
=> ( ( run @ q0a @ transsa @ R2 @ Bss )
= ( run @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_r @ R2 @ Bss ) ) ) ).
% cong.right.run_cong
thf(fact_79_left_Ostep__symb__set__closed,axiom,
! [R2: set_nat] : ( ord_less_eq_set_nat @ ( step_symb_set @ q0a @ ts_l @ R2 ) @ ( q @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_l ) ) ).
% left.step_symb_set_closed
thf(fact_80_right_Orun__closed,axiom,
! [R2: set_nat,Bss: list_list_o] :
( ( ord_less_eq_set_nat @ R2 @ ( q @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ qfa @ ts_r ) )
=> ( ord_less_eq_set_nat @ ( run @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_r @ R2 @ Bss ) @ ( q @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ qfa @ ts_r ) ) ) ).
% right.run_closed
thf(fact_81_cong_Oright_Odelta__cong,axiom,
! [R2: set_nat,Bs: list_o] :
( ( ord_less_eq_set_nat @ R2 @ ( q @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ qfa @ ts_r ) )
=> ( ( delta @ q0a @ transsa @ R2 @ Bs )
= ( delta @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_r @ R2 @ Bs ) ) ) ).
% cong.right.delta_cong
thf(fact_82_cong_Oright_Ostep__eps__closure__set__cong,axiom,
! [R2: set_nat,Bs: list_o] :
( ( ord_less_eq_set_nat @ R2 @ ( q @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ qfa @ ts_r ) )
=> ( ( step_eps_closure_set @ q0a @ transsa @ R2 @ Bs )
= ( step_eps_closure_set @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_r @ R2 @ Bs ) ) ) ).
% cong.right.step_eps_closure_set_cong
thf(fact_83_right_Odelta__closed,axiom,
! [R2: set_nat,Bs: list_o] : ( ord_less_eq_set_nat @ ( delta @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_r @ R2 @ Bs ) @ ( q @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ qfa @ ts_r ) ) ).
% right.delta_closed
thf(fact_84_cong_Onfa_H__delta__sub__delta,axiom,
! [R2: set_nat,Bs: list_o] :
( ( ord_less_eq_set_nat @ R2 @ ( q @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_l ) )
=> ( ord_less_eq_set_nat @ ( delta @ q0a @ ts_l @ R2 @ Bs ) @ ( delta @ q0a @ transsa @ R2 @ Bs ) ) ) ).
% cong.nfa'_delta_sub_delta
thf(fact_85_left_Orun__closed,axiom,
! [R2: set_nat,Bss: list_list_o] :
( ( ord_less_eq_set_nat @ R2 @ ( q @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_l ) )
=> ( ord_less_eq_set_nat @ ( run @ q0a @ ts_l @ R2 @ Bss ) @ ( q @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_l ) ) ) ).
% left.run_closed
thf(fact_86_cong_Ostep__symb__set__cong__Q,axiom,
! [R2: set_nat] : ( ord_less_eq_set_nat @ ( step_symb_set @ q0a @ ts_l @ R2 ) @ ( step_symb_set @ q0a @ transsa @ R2 ) ) ).
% cong.step_symb_set_cong_Q
thf(fact_87_base_Ostep__eps__closure__set__closed,axiom,
! [R2: set_nat,Bs: list_o] :
( ( ord_less_eq_set_nat @ R2 @ ( q @ q0a @ qfa @ transsa ) )
=> ( ord_less_eq_set_nat @ ( step_eps_closure_set @ q0a @ transsa @ R2 @ Bs ) @ ( q @ q0a @ qfa @ transsa ) ) ) ).
% base.step_eps_closure_set_closed
thf(fact_88_base_Odelta__closed,axiom,
! [R2: set_nat,Bs: list_o] : ( ord_less_eq_set_nat @ ( delta @ q0a @ transsa @ R2 @ Bs ) @ ( q @ q0a @ qfa @ transsa ) ) ).
% base.delta_closed
thf(fact_89_base_Orun__closed,axiom,
! [R2: set_nat,Bss: list_list_o] :
( ( ord_less_eq_set_nat @ R2 @ ( q @ q0a @ qfa @ transsa ) )
=> ( ord_less_eq_set_nat @ ( run @ q0a @ transsa @ R2 @ Bss ) @ ( q @ q0a @ qfa @ transsa ) ) ) ).
% base.run_closed
thf(fact_90_base_Ostep__symb__set__closed,axiom,
! [R2: set_nat] : ( ord_less_eq_set_nat @ ( step_symb_set @ q0a @ transsa @ R2 ) @ ( q @ q0a @ qfa @ transsa ) ) ).
% base.step_symb_set_closed
thf(fact_91_cong_Ostep__symb__set__cong__SQ,axiom,
! [R2: set_nat] :
( ( ord_less_eq_set_nat @ R2 @ ( sq @ q0a @ ts_l ) )
=> ( ( step_symb_set @ q0a @ transsa @ R2 )
= ( step_symb_set @ q0a @ ts_l @ R2 ) ) ) ).
% cong.step_symb_set_cong_SQ
thf(fact_92_left_Ostep__eps__closure__set__closed,axiom,
! [R2: set_nat,Bs: list_o] :
( ( ord_less_eq_set_nat @ R2 @ ( q @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_l ) )
=> ( ord_less_eq_set_nat @ ( step_eps_closure_set @ q0a @ ts_l @ R2 @ Bs ) @ ( q @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_l ) ) ) ).
% left.step_eps_closure_set_closed
thf(fact_93_left_Odelta__closed,axiom,
! [R2: set_nat,Bs: list_o] : ( ord_less_eq_set_nat @ ( delta @ q0a @ ts_l @ R2 @ Bs ) @ ( q @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_l ) ) ).
% left.delta_closed
thf(fact_94_cong_Ostep__eps__closure__set__cong__unreach,axiom,
! [R2: set_nat,Bs: list_o] :
( ( ord_less_eq_set_nat @ R2 @ ( q @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_l ) )
=> ( ~ ( member_nat @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ( step_eps_closure_set @ q0a @ ts_l @ R2 @ Bs ) )
=> ( ( step_eps_closure_set @ q0a @ transsa @ R2 @ Bs )
= ( step_eps_closure_set @ q0a @ ts_l @ R2 @ Bs ) ) ) ) ).
% cong.step_eps_closure_set_cong_unreach
thf(fact_95_cong_Onfa_H__step__eps__closure__set__sub,axiom,
! [R2: set_nat,Bs: list_o] :
( ( ord_less_eq_set_nat @ R2 @ ( q @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_l ) )
=> ( ord_less_eq_set_nat @ ( step_eps_closure_set @ q0a @ ts_l @ R2 @ Bs ) @ ( step_eps_closure_set @ q0a @ transsa @ R2 @ Bs ) ) ) ).
% cong.nfa'_step_eps_closure_set_sub
thf(fact_96_right_Ostep__eps__closure__set__closed,axiom,
! [R2: set_nat,Bs: list_o] :
( ( ord_less_eq_set_nat @ R2 @ ( q @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ qfa @ ts_r ) )
=> ( ord_less_eq_set_nat @ ( step_eps_closure_set @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_r @ R2 @ Bs ) @ ( q @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ qfa @ ts_r ) ) ) ).
% right.step_eps_closure_set_closed
thf(fact_97_cong_Odelta__cong__unreach,axiom,
! [R2: set_nat,Bs: list_o] :
( ( ord_less_eq_set_nat @ R2 @ ( q @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_l ) )
=> ( ~ ( accept_eps @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_l @ R2 @ Bs )
=> ( ( delta @ q0a @ transsa @ R2 @ Bs )
= ( delta @ q0a @ ts_l @ R2 @ Bs ) ) ) ) ).
% cong.delta_cong_unreach
thf(fact_98_right_Orun__closed__Cons,axiom,
! [R2: set_nat,Bs: list_o,Bss: list_list_o] : ( ord_less_eq_set_nat @ ( run @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_r @ R2 @ ( cons_list_o @ Bs @ Bss ) ) @ ( q @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ qfa @ ts_r ) ) ).
% right.run_closed_Cons
thf(fact_99_right_Ostep__eps__closure__set__closed__union,axiom,
! [R2: set_nat,Bs: list_o] : ( ord_less_eq_set_nat @ ( step_eps_closure_set @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_r @ R2 @ Bs ) @ ( sup_sup_set_nat @ R2 @ ( q @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ qfa @ ts_r ) ) ) ).
% right.step_eps_closure_set_closed_union
thf(fact_100_left_Orun__closed__Cons,axiom,
! [R2: set_nat,Bs: list_o,Bss: list_list_o] : ( ord_less_eq_set_nat @ ( run @ q0a @ ts_l @ R2 @ ( cons_list_o @ Bs @ Bss ) ) @ ( q @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_l ) ) ).
% left.run_closed_Cons
thf(fact_101_cong_Oright_Orun__accept__eps__cong,axiom,
! [R2: set_nat,Bss: list_list_o,Bs: list_o] :
( ( ord_less_eq_set_nat @ R2 @ ( q @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ qfa @ ts_r ) )
=> ( ( run_accept_eps @ q0a @ qfa @ transsa @ R2 @ Bss @ Bs )
= ( run_accept_eps @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ qfa @ ts_r @ R2 @ Bss @ Bs ) ) ) ).
% cong.right.run_accept_eps_cong
thf(fact_102_left_Ostep__eps__closure__set__closed__union,axiom,
! [R2: set_nat,Bs: list_o] : ( ord_less_eq_set_nat @ ( step_eps_closure_set @ q0a @ ts_l @ R2 @ Bs ) @ ( sup_sup_set_nat @ R2 @ ( q @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_l ) ) ) ).
% left.step_eps_closure_set_closed_union
thf(fact_103_base_Orun__closed__Cons,axiom,
! [R2: set_nat,Bs: list_o,Bss: list_list_o] : ( ord_less_eq_set_nat @ ( run @ q0a @ transsa @ R2 @ ( cons_list_o @ Bs @ Bss ) ) @ ( q @ q0a @ qfa @ transsa ) ) ).
% base.run_closed_Cons
thf(fact_104_base_Ostep__eps__closure__set__closed__union,axiom,
! [R2: set_nat,Bs: list_o] : ( ord_less_eq_set_nat @ ( step_eps_closure_set @ q0a @ transsa @ R2 @ Bs ) @ ( sup_sup_set_nat @ R2 @ ( q @ q0a @ qfa @ transsa ) ) ) ).
% base.step_eps_closure_set_closed_union
thf(fact_105_nfa__cong_H_Odelta__cong__unreach,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition,R2: set_nat,Bs: list_o] :
( ( nfa_cong2 @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( ( ord_less_eq_set_nat @ R2 @ ( q @ Q02 @ Qf2 @ Transs2 ) )
=> ( ~ ( accept_eps @ Q02 @ Qf2 @ Transs2 @ R2 @ Bs )
=> ( ( delta @ Q0 @ Transs @ R2 @ Bs )
= ( delta @ Q02 @ Transs2 @ R2 @ Bs ) ) ) ) ) ).
% nfa_cong'.delta_cong_unreach
thf(fact_106_len__ts_I3_J,axiom,
( ( size_s3613142680436377136sition @ transsa )
= ( plus_plus_nat @ ( state_cnt_a_t @ ra ) @ ( state_cnt_a_t @ s ) ) ) ).
% len_ts(3)
thf(fact_107_append_Oright__neutral,axiom,
! [A: list_formula_a_t] :
( ( append_formula_a_t @ A @ nil_formula_a_t )
= A ) ).
% append.right_neutral
thf(fact_108_append_Oright__neutral,axiom,
! [A: list_o] :
( ( append_o @ A @ nil_o )
= A ) ).
% append.right_neutral
thf(fact_109_append_Oright__neutral,axiom,
! [A: list_transition] :
( ( append_transition @ A @ nil_transition )
= A ) ).
% append.right_neutral
thf(fact_110_append_Oright__neutral,axiom,
! [A: list_list_o] :
( ( append_list_o @ A @ nil_list_o )
= A ) ).
% append.right_neutral
thf(fact_111_left_Otranss__not__Nil,axiom,
ts_l != nil_transition ).
% left.transs_not_Nil
thf(fact_112_ts__nonempty_I1_J,axiom,
ts_l != nil_transition ).
% ts_nonempty(1)
thf(fact_113_base_Otranss__not__Nil,axiom,
transsa != nil_transition ).
% base.transs_not_Nil
thf(fact_114_ts__nonempty_I2_J,axiom,
ts_r != nil_transition ).
% ts_nonempty(2)
thf(fact_115_right_Otranss__not__Nil,axiom,
ts_r != nil_transition ).
% right.transs_not_Nil
thf(fact_116_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_117_same__append__eq,axiom,
! [Xs: list_formula_a_t,Ys: list_formula_a_t,Zs: list_formula_a_t] :
( ( ( append_formula_a_t @ Xs @ Ys )
= ( append_formula_a_t @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_118_same__append__eq,axiom,
! [Xs: list_transition,Ys: list_transition,Zs: list_transition] :
( ( ( append_transition @ Xs @ Ys )
= ( append_transition @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_119_same__append__eq,axiom,
! [Xs: list_list_o,Ys: list_list_o,Zs: list_list_o] :
( ( ( append_list_o @ Xs @ Ys )
= ( append_list_o @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_120_append__same__eq,axiom,
! [Ys: list_formula_a_t,Xs: list_formula_a_t,Zs: list_formula_a_t] :
( ( ( append_formula_a_t @ Ys @ Xs )
= ( append_formula_a_t @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_121_append__same__eq,axiom,
! [Ys: list_transition,Xs: list_transition,Zs: list_transition] :
( ( ( append_transition @ Ys @ Xs )
= ( append_transition @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_122_append__same__eq,axiom,
! [Ys: list_list_o,Xs: list_list_o,Zs: list_list_o] :
( ( ( append_list_o @ Ys @ Xs )
= ( append_list_o @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_123_append__assoc,axiom,
! [Xs: list_formula_a_t,Ys: list_formula_a_t,Zs: list_formula_a_t] :
( ( append_formula_a_t @ ( append_formula_a_t @ Xs @ Ys ) @ Zs )
= ( append_formula_a_t @ Xs @ ( append_formula_a_t @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_124_append__assoc,axiom,
! [Xs: list_transition,Ys: list_transition,Zs: list_transition] :
( ( append_transition @ ( append_transition @ Xs @ Ys ) @ Zs )
= ( append_transition @ Xs @ ( append_transition @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_125_append__assoc,axiom,
! [Xs: list_list_o,Ys: list_list_o,Zs: list_list_o] :
( ( append_list_o @ ( append_list_o @ Xs @ Ys ) @ Zs )
= ( append_list_o @ Xs @ ( append_list_o @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_126_append_Oassoc,axiom,
! [A: list_formula_a_t,B: list_formula_a_t,C: list_formula_a_t] :
( ( append_formula_a_t @ ( append_formula_a_t @ A @ B ) @ C )
= ( append_formula_a_t @ A @ ( append_formula_a_t @ B @ C ) ) ) ).
% append.assoc
thf(fact_127_append_Oassoc,axiom,
! [A: list_transition,B: list_transition,C: list_transition] :
( ( append_transition @ ( append_transition @ A @ B ) @ C )
= ( append_transition @ A @ ( append_transition @ B @ C ) ) ) ).
% append.assoc
thf(fact_128_append_Oassoc,axiom,
! [A: list_list_o,B: list_list_o,C: list_list_o] :
( ( append_list_o @ ( append_list_o @ A @ B ) @ C )
= ( append_list_o @ A @ ( append_list_o @ B @ C ) ) ) ).
% append.assoc
thf(fact_129_transs__eq,axiom,
( transsa
= ( append_transition @ ts_l @ ts_r ) ) ).
% transs_eq
thf(fact_130_len__ts_I1_J,axiom,
( ( size_s3613142680436377136sition @ ts_l )
= ( state_cnt_a_t @ ra ) ) ).
% len_ts(1)
thf(fact_131_len__ts_I2_J,axiom,
( ( size_s3613142680436377136sition @ ts_r )
= ( state_cnt_a_t @ s ) ) ).
% len_ts(2)
thf(fact_132_append__is__Nil__conv,axiom,
! [Xs: list_formula_a_t,Ys: list_formula_a_t] :
( ( ( append_formula_a_t @ Xs @ Ys )
= nil_formula_a_t )
= ( ( Xs = nil_formula_a_t )
& ( Ys = nil_formula_a_t ) ) ) ).
% append_is_Nil_conv
thf(fact_133_append__is__Nil__conv,axiom,
! [Xs: list_o,Ys: list_o] :
( ( ( append_o @ Xs @ Ys )
= nil_o )
= ( ( Xs = nil_o )
& ( Ys = nil_o ) ) ) ).
% append_is_Nil_conv
thf(fact_134_append__is__Nil__conv,axiom,
! [Xs: list_transition,Ys: list_transition] :
( ( ( append_transition @ Xs @ Ys )
= nil_transition )
= ( ( Xs = nil_transition )
& ( Ys = nil_transition ) ) ) ).
% append_is_Nil_conv
thf(fact_135_append__is__Nil__conv,axiom,
! [Xs: list_list_o,Ys: list_list_o] :
( ( ( append_list_o @ Xs @ Ys )
= nil_list_o )
= ( ( Xs = nil_list_o )
& ( Ys = nil_list_o ) ) ) ).
% append_is_Nil_conv
thf(fact_136_Nil__is__append__conv,axiom,
! [Xs: list_formula_a_t,Ys: list_formula_a_t] :
( ( nil_formula_a_t
= ( append_formula_a_t @ Xs @ Ys ) )
= ( ( Xs = nil_formula_a_t )
& ( Ys = nil_formula_a_t ) ) ) ).
% Nil_is_append_conv
thf(fact_137_Nil__is__append__conv,axiom,
! [Xs: list_o,Ys: list_o] :
( ( nil_o
= ( append_o @ Xs @ Ys ) )
= ( ( Xs = nil_o )
& ( Ys = nil_o ) ) ) ).
% Nil_is_append_conv
thf(fact_138_Nil__is__append__conv,axiom,
! [Xs: list_transition,Ys: list_transition] :
( ( nil_transition
= ( append_transition @ Xs @ Ys ) )
= ( ( Xs = nil_transition )
& ( Ys = nil_transition ) ) ) ).
% Nil_is_append_conv
thf(fact_139_Nil__is__append__conv,axiom,
! [Xs: list_list_o,Ys: list_list_o] :
( ( nil_list_o
= ( append_list_o @ Xs @ Ys ) )
= ( ( Xs = nil_list_o )
& ( Ys = nil_list_o ) ) ) ).
% Nil_is_append_conv
thf(fact_140_self__append__conv2,axiom,
! [Y: list_formula_a_t,Xs: list_formula_a_t] :
( ( Y
= ( append_formula_a_t @ Xs @ Y ) )
= ( Xs = nil_formula_a_t ) ) ).
% self_append_conv2
thf(fact_141_self__append__conv2,axiom,
! [Y: list_o,Xs: list_o] :
( ( Y
= ( append_o @ Xs @ Y ) )
= ( Xs = nil_o ) ) ).
% self_append_conv2
thf(fact_142_self__append__conv2,axiom,
! [Y: list_transition,Xs: list_transition] :
( ( Y
= ( append_transition @ Xs @ Y ) )
= ( Xs = nil_transition ) ) ).
% self_append_conv2
thf(fact_143_self__append__conv2,axiom,
! [Y: list_list_o,Xs: list_list_o] :
( ( Y
= ( append_list_o @ Xs @ Y ) )
= ( Xs = nil_list_o ) ) ).
% self_append_conv2
thf(fact_144_append__self__conv2,axiom,
! [Xs: list_formula_a_t,Ys: list_formula_a_t] :
( ( ( append_formula_a_t @ Xs @ Ys )
= Ys )
= ( Xs = nil_formula_a_t ) ) ).
% append_self_conv2
thf(fact_145_append__self__conv2,axiom,
! [Xs: list_o,Ys: list_o] :
( ( ( append_o @ Xs @ Ys )
= Ys )
= ( Xs = nil_o ) ) ).
% append_self_conv2
thf(fact_146_append__self__conv2,axiom,
! [Xs: list_transition,Ys: list_transition] :
( ( ( append_transition @ Xs @ Ys )
= Ys )
= ( Xs = nil_transition ) ) ).
% append_self_conv2
thf(fact_147_append__self__conv2,axiom,
! [Xs: list_list_o,Ys: list_list_o] :
( ( ( append_list_o @ Xs @ Ys )
= Ys )
= ( Xs = nil_list_o ) ) ).
% append_self_conv2
thf(fact_148_self__append__conv,axiom,
! [Y: list_formula_a_t,Ys: list_formula_a_t] :
( ( Y
= ( append_formula_a_t @ Y @ Ys ) )
= ( Ys = nil_formula_a_t ) ) ).
% self_append_conv
thf(fact_149_self__append__conv,axiom,
! [Y: list_o,Ys: list_o] :
( ( Y
= ( append_o @ Y @ Ys ) )
= ( Ys = nil_o ) ) ).
% self_append_conv
thf(fact_150_self__append__conv,axiom,
! [Y: list_transition,Ys: list_transition] :
( ( Y
= ( append_transition @ Y @ Ys ) )
= ( Ys = nil_transition ) ) ).
% self_append_conv
thf(fact_151_self__append__conv,axiom,
! [Y: list_list_o,Ys: list_list_o] :
( ( Y
= ( append_list_o @ Y @ Ys ) )
= ( Ys = nil_list_o ) ) ).
% self_append_conv
thf(fact_152_append__self__conv,axiom,
! [Xs: list_formula_a_t,Ys: list_formula_a_t] :
( ( ( append_formula_a_t @ Xs @ Ys )
= Xs )
= ( Ys = nil_formula_a_t ) ) ).
% append_self_conv
thf(fact_153_append__self__conv,axiom,
! [Xs: list_o,Ys: list_o] :
( ( ( append_o @ Xs @ Ys )
= Xs )
= ( Ys = nil_o ) ) ).
% append_self_conv
thf(fact_154_append__self__conv,axiom,
! [Xs: list_transition,Ys: list_transition] :
( ( ( append_transition @ Xs @ Ys )
= Xs )
= ( Ys = nil_transition ) ) ).
% append_self_conv
thf(fact_155_append__self__conv,axiom,
! [Xs: list_list_o,Ys: list_list_o] :
( ( ( append_list_o @ Xs @ Ys )
= Xs )
= ( Ys = nil_list_o ) ) ).
% append_self_conv
thf(fact_156_append__Nil2,axiom,
! [Xs: list_formula_a_t] :
( ( append_formula_a_t @ Xs @ nil_formula_a_t )
= Xs ) ).
% append_Nil2
thf(fact_157_append__Nil2,axiom,
! [Xs: list_o] :
( ( append_o @ Xs @ nil_o )
= Xs ) ).
% append_Nil2
thf(fact_158_append__Nil2,axiom,
! [Xs: list_transition] :
( ( append_transition @ Xs @ nil_transition )
= Xs ) ).
% append_Nil2
thf(fact_159_append__Nil2,axiom,
! [Xs: list_list_o] :
( ( append_list_o @ Xs @ nil_list_o )
= Xs ) ).
% append_Nil2
thf(fact_160_append__eq__append__conv,axiom,
! [Xs: list_transition,Ys: list_transition,Us: list_transition,Vs: list_transition] :
( ( ( ( size_s3613142680436377136sition @ Xs )
= ( size_s3613142680436377136sition @ Ys ) )
| ( ( size_s3613142680436377136sition @ Us )
= ( size_s3613142680436377136sition @ Vs ) ) )
=> ( ( ( append_transition @ Xs @ Us )
= ( append_transition @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_161_append__eq__append__conv,axiom,
! [Xs: list_list_o,Ys: list_list_o,Us: list_list_o,Vs: list_list_o] :
( ( ( ( size_s2710708370519433104list_o @ Xs )
= ( size_s2710708370519433104list_o @ Ys ) )
| ( ( size_s2710708370519433104list_o @ Us )
= ( size_s2710708370519433104list_o @ Vs ) ) )
=> ( ( ( append_list_o @ Xs @ Us )
= ( append_list_o @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_162_append__eq__append__conv,axiom,
! [Xs: list_formula_a_t,Ys: list_formula_a_t,Us: list_formula_a_t,Vs: list_formula_a_t] :
( ( ( ( size_s8846756101701226951la_a_t @ Xs )
= ( size_s8846756101701226951la_a_t @ Ys ) )
| ( ( size_s8846756101701226951la_a_t @ Us )
= ( size_s8846756101701226951la_a_t @ Vs ) ) )
=> ( ( ( append_formula_a_t @ Xs @ Us )
= ( append_formula_a_t @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_163_append__eq__append__conv,axiom,
! [Xs: list_o,Ys: list_o,Us: list_o,Vs: list_o] :
( ( ( ( size_size_list_o @ Xs )
= ( size_size_list_o @ Ys ) )
| ( ( size_size_list_o @ Us )
= ( size_size_list_o @ Vs ) ) )
=> ( ( ( append_o @ Xs @ Us )
= ( append_o @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_164_append1__eq__conv,axiom,
! [Xs: list_formula_a_t,X2: formula_a_t,Ys: list_formula_a_t,Y: formula_a_t] :
( ( ( append_formula_a_t @ Xs @ ( cons_formula_a_t @ X2 @ nil_formula_a_t ) )
= ( append_formula_a_t @ Ys @ ( cons_formula_a_t @ Y @ nil_formula_a_t ) ) )
= ( ( Xs = Ys )
& ( X2 = Y ) ) ) ).
% append1_eq_conv
thf(fact_165_append1__eq__conv,axiom,
! [Xs: list_o,X2: $o,Ys: list_o,Y: $o] :
( ( ( append_o @ Xs @ ( cons_o @ X2 @ nil_o ) )
= ( append_o @ Ys @ ( cons_o @ Y @ nil_o ) ) )
= ( ( Xs = Ys )
& ( X2 = Y ) ) ) ).
% append1_eq_conv
thf(fact_166_append1__eq__conv,axiom,
! [Xs: list_transition,X2: transition,Ys: list_transition,Y: transition] :
( ( ( append_transition @ Xs @ ( cons_transition @ X2 @ nil_transition ) )
= ( append_transition @ Ys @ ( cons_transition @ Y @ nil_transition ) ) )
= ( ( Xs = Ys )
& ( X2 = Y ) ) ) ).
% append1_eq_conv
thf(fact_167_append1__eq__conv,axiom,
! [Xs: list_list_o,X2: list_o,Ys: list_list_o,Y: list_o] :
( ( ( append_list_o @ Xs @ ( cons_list_o @ X2 @ nil_list_o ) )
= ( append_list_o @ Ys @ ( cons_list_o @ Y @ nil_list_o ) ) )
= ( ( Xs = Ys )
& ( X2 = Y ) ) ) ).
% append1_eq_conv
thf(fact_168_length__append,axiom,
! [Xs: list_transition,Ys: list_transition] :
( ( size_s3613142680436377136sition @ ( append_transition @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_s3613142680436377136sition @ Xs ) @ ( size_s3613142680436377136sition @ Ys ) ) ) ).
% length_append
thf(fact_169_length__append,axiom,
! [Xs: list_list_o,Ys: list_list_o] :
( ( size_s2710708370519433104list_o @ ( append_list_o @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_s2710708370519433104list_o @ Xs ) @ ( size_s2710708370519433104list_o @ Ys ) ) ) ).
% length_append
thf(fact_170_length__append,axiom,
! [Xs: list_formula_a_t,Ys: list_formula_a_t] :
( ( size_s8846756101701226951la_a_t @ ( append_formula_a_t @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_s8846756101701226951la_a_t @ Xs ) @ ( size_s8846756101701226951la_a_t @ Ys ) ) ) ).
% length_append
thf(fact_171_length__append,axiom,
! [Xs: list_o,Ys: list_o] :
( ( size_size_list_o @ ( append_o @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).
% length_append
thf(fact_172_run__accept__eps__Cons,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,R2: set_nat,Bs: list_o,Bss: list_list_o,Cs: list_o] :
( ( run_accept_eps @ Q0 @ Qf @ Transs @ R2 @ ( cons_list_o @ Bs @ Bss ) @ Cs )
= ( run_accept_eps @ Q0 @ Qf @ Transs @ ( delta @ Q0 @ Transs @ R2 @ Bs ) @ Bss @ Cs ) ) ).
% run_accept_eps_Cons
thf(fact_173_run__accept__eps__Cons__eps,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,R2: set_nat,Cs: list_o,Css: list_list_o,Bs: list_o] :
( ( run_accept_eps @ Q0 @ Qf @ Transs @ ( step_eps_closure_set @ Q0 @ Transs @ R2 @ Cs ) @ ( cons_list_o @ Cs @ Css ) @ Bs )
= ( run_accept_eps @ Q0 @ Qf @ Transs @ R2 @ ( cons_list_o @ Cs @ Css ) @ Bs ) ) ).
% run_accept_eps_Cons_eps
thf(fact_174_run__accept__eps__Cons__delta__cong,axiom,
! [Q0: nat,Transs: list_transition,R2: set_nat,Bs: list_o,S2: set_nat,Qf: nat,Bss: list_list_o,Cs: list_o] :
( ( ( delta @ Q0 @ Transs @ R2 @ Bs )
= ( delta @ Q0 @ Transs @ S2 @ Bs ) )
=> ( ( run_accept_eps @ Q0 @ Qf @ Transs @ R2 @ ( cons_list_o @ Bs @ Bss ) @ Cs )
= ( run_accept_eps @ Q0 @ Qf @ Transs @ S2 @ ( cons_list_o @ Bs @ Bss ) @ Cs ) ) ) ).
% run_accept_eps_Cons_delta_cong
thf(fact_175_run__accept__eps__Nil__eps,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,R2: set_nat,Bs: list_o] :
( ( run_accept_eps @ Q0 @ Qf @ Transs @ ( step_eps_closure_set @ Q0 @ Transs @ R2 @ Bs ) @ nil_list_o @ Bs )
= ( run_accept_eps @ Q0 @ Qf @ Transs @ R2 @ nil_list_o @ Bs ) ) ).
% run_accept_eps_Nil_eps
thf(fact_176_run__accept__eps__Nil,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,R2: set_nat,Cs: list_o] :
( ( run_accept_eps @ Q0 @ Qf @ Transs @ R2 @ nil_list_o @ Cs )
= ( accept_eps @ Q0 @ Qf @ Transs @ R2 @ Cs ) ) ).
% run_accept_eps_Nil
thf(fact_177_transpose_Ocases,axiom,
! [X2: list_list_transition] :
( ( X2 != nil_list_transition )
=> ( ! [Xss: list_list_transition] :
( X2
!= ( cons_list_transition @ nil_transition @ Xss ) )
=> ~ ! [X3: transition,Xs2: list_transition,Xss: list_list_transition] :
( X2
!= ( cons_list_transition @ ( cons_transition @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_178_transpose_Ocases,axiom,
! [X2: list_list_list_o] :
( ( X2 != nil_list_list_o )
=> ( ! [Xss: list_list_list_o] :
( X2
!= ( cons_list_list_o @ nil_list_o @ Xss ) )
=> ~ ! [X3: list_o,Xs2: list_list_o,Xss: list_list_list_o] :
( X2
!= ( cons_list_list_o @ ( cons_list_o @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_179_transpose_Ocases,axiom,
! [X2: list_list_o] :
( ( X2 != nil_list_o )
=> ( ! [Xss: list_list_o] :
( X2
!= ( cons_list_o @ nil_o @ Xss ) )
=> ~ ! [X3: $o,Xs2: list_o,Xss: list_list_o] :
( X2
!= ( cons_list_o @ ( cons_o @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_180_run__accept__eps__split,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,R2: set_nat,S2: set_nat,Bss: list_list_o,Bs: list_o] :
( ( run_accept_eps @ Q0 @ Qf @ Transs @ ( sup_sup_set_nat @ R2 @ S2 ) @ Bss @ Bs )
= ( ( run_accept_eps @ Q0 @ Qf @ Transs @ R2 @ Bss @ Bs )
| ( run_accept_eps @ Q0 @ Qf @ Transs @ S2 @ Bss @ Bs ) ) ) ).
% run_accept_eps_split
thf(fact_181_neq__if__length__neq,axiom,
! [Xs: list_transition,Ys: list_transition] :
( ( ( size_s3613142680436377136sition @ Xs )
!= ( size_s3613142680436377136sition @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_182_neq__if__length__neq,axiom,
! [Xs: list_list_o,Ys: list_list_o] :
( ( ( size_s2710708370519433104list_o @ Xs )
!= ( size_s2710708370519433104list_o @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_183_neq__if__length__neq,axiom,
! [Xs: list_formula_a_t,Ys: list_formula_a_t] :
( ( ( size_s8846756101701226951la_a_t @ Xs )
!= ( size_s8846756101701226951la_a_t @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_184_neq__if__length__neq,axiom,
! [Xs: list_o,Ys: list_o] :
( ( ( size_size_list_o @ Xs )
!= ( size_size_list_o @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_185_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_transition] :
( ( size_s3613142680436377136sition @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_186_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_list_o] :
( ( size_s2710708370519433104list_o @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_187_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_formula_a_t] :
( ( size_s8846756101701226951la_a_t @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_188_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_o] :
( ( size_size_list_o @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_189_not__Cons__self2,axiom,
! [X2: list_o,Xs: list_list_o] :
( ( cons_list_o @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_190_list__induct2,axiom,
! [Xs: list_transition,Ys: list_transition,P: list_transition > list_transition > $o] :
( ( ( size_s3613142680436377136sition @ Xs )
= ( size_s3613142680436377136sition @ Ys ) )
=> ( ( P @ nil_transition @ nil_transition )
=> ( ! [X3: transition,Xs2: list_transition,Y2: transition,Ys2: list_transition] :
( ( ( size_s3613142680436377136sition @ Xs2 )
= ( size_s3613142680436377136sition @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_transition @ X3 @ Xs2 ) @ ( cons_transition @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_191_list__induct2,axiom,
! [Xs: list_transition,Ys: list_o,P: list_transition > list_o > $o] :
( ( ( size_s3613142680436377136sition @ Xs )
= ( size_size_list_o @ Ys ) )
=> ( ( P @ nil_transition @ nil_o )
=> ( ! [X3: transition,Xs2: list_transition,Y2: $o,Ys2: list_o] :
( ( ( size_s3613142680436377136sition @ Xs2 )
= ( size_size_list_o @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_transition @ X3 @ Xs2 ) @ ( cons_o @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_192_list__induct2,axiom,
! [Xs: list_o,Ys: list_transition,P: list_o > list_transition > $o] :
( ( ( size_size_list_o @ Xs )
= ( size_s3613142680436377136sition @ Ys ) )
=> ( ( P @ nil_o @ nil_transition )
=> ( ! [X3: $o,Xs2: list_o,Y2: transition,Ys2: list_transition] :
( ( ( size_size_list_o @ Xs2 )
= ( size_s3613142680436377136sition @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_o @ X3 @ Xs2 ) @ ( cons_transition @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_193_list__induct2,axiom,
! [Xs: list_o,Ys: list_o,P: list_o > list_o > $o] :
( ( ( size_size_list_o @ Xs )
= ( size_size_list_o @ Ys ) )
=> ( ( P @ nil_o @ nil_o )
=> ( ! [X3: $o,Xs2: list_o,Y2: $o,Ys2: list_o] :
( ( ( size_size_list_o @ Xs2 )
= ( size_size_list_o @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_o @ X3 @ Xs2 ) @ ( cons_o @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_194_list__induct2,axiom,
! [Xs: list_transition,Ys: list_list_o,P: list_transition > list_list_o > $o] :
( ( ( size_s3613142680436377136sition @ Xs )
= ( size_s2710708370519433104list_o @ Ys ) )
=> ( ( P @ nil_transition @ nil_list_o )
=> ( ! [X3: transition,Xs2: list_transition,Y2: list_o,Ys2: list_list_o] :
( ( ( size_s3613142680436377136sition @ Xs2 )
= ( size_s2710708370519433104list_o @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_transition @ X3 @ Xs2 ) @ ( cons_list_o @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_195_list__induct2,axiom,
! [Xs: list_list_o,Ys: list_transition,P: list_list_o > list_transition > $o] :
( ( ( size_s2710708370519433104list_o @ Xs )
= ( size_s3613142680436377136sition @ Ys ) )
=> ( ( P @ nil_list_o @ nil_transition )
=> ( ! [X3: list_o,Xs2: list_list_o,Y2: transition,Ys2: list_transition] :
( ( ( size_s2710708370519433104list_o @ Xs2 )
= ( size_s3613142680436377136sition @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_list_o @ X3 @ Xs2 ) @ ( cons_transition @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_196_list__induct2,axiom,
! [Xs: list_list_o,Ys: list_o,P: list_list_o > list_o > $o] :
( ( ( size_s2710708370519433104list_o @ Xs )
= ( size_size_list_o @ Ys ) )
=> ( ( P @ nil_list_o @ nil_o )
=> ( ! [X3: list_o,Xs2: list_list_o,Y2: $o,Ys2: list_o] :
( ( ( size_s2710708370519433104list_o @ Xs2 )
= ( size_size_list_o @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_list_o @ X3 @ Xs2 ) @ ( cons_o @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_197_list__induct2,axiom,
! [Xs: list_o,Ys: list_list_o,P: list_o > list_list_o > $o] :
( ( ( size_size_list_o @ Xs )
= ( size_s2710708370519433104list_o @ Ys ) )
=> ( ( P @ nil_o @ nil_list_o )
=> ( ! [X3: $o,Xs2: list_o,Y2: list_o,Ys2: list_list_o] :
( ( ( size_size_list_o @ Xs2 )
= ( size_s2710708370519433104list_o @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_o @ X3 @ Xs2 ) @ ( cons_list_o @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_198_list__induct2,axiom,
! [Xs: list_transition,Ys: list_formula_a_t,P: list_transition > list_formula_a_t > $o] :
( ( ( size_s3613142680436377136sition @ Xs )
= ( size_s8846756101701226951la_a_t @ Ys ) )
=> ( ( P @ nil_transition @ nil_formula_a_t )
=> ( ! [X3: transition,Xs2: list_transition,Y2: formula_a_t,Ys2: list_formula_a_t] :
( ( ( size_s3613142680436377136sition @ Xs2 )
= ( size_s8846756101701226951la_a_t @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_transition @ X3 @ Xs2 ) @ ( cons_formula_a_t @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_199_list__induct2,axiom,
! [Xs: list_list_o,Ys: list_list_o,P: list_list_o > list_list_o > $o] :
( ( ( size_s2710708370519433104list_o @ Xs )
= ( size_s2710708370519433104list_o @ Ys ) )
=> ( ( P @ nil_list_o @ nil_list_o )
=> ( ! [X3: list_o,Xs2: list_list_o,Y2: list_o,Ys2: list_list_o] :
( ( ( size_s2710708370519433104list_o @ Xs2 )
= ( size_s2710708370519433104list_o @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_list_o @ X3 @ Xs2 ) @ ( cons_list_o @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_200_list__induct3,axiom,
! [Xs: list_transition,Ys: list_transition,Zs: list_transition,P: list_transition > list_transition > list_transition > $o] :
( ( ( size_s3613142680436377136sition @ Xs )
= ( size_s3613142680436377136sition @ Ys ) )
=> ( ( ( size_s3613142680436377136sition @ Ys )
= ( size_s3613142680436377136sition @ Zs ) )
=> ( ( P @ nil_transition @ nil_transition @ nil_transition )
=> ( ! [X3: transition,Xs2: list_transition,Y2: transition,Ys2: list_transition,Z: transition,Zs2: list_transition] :
( ( ( size_s3613142680436377136sition @ Xs2 )
= ( size_s3613142680436377136sition @ Ys2 ) )
=> ( ( ( size_s3613142680436377136sition @ Ys2 )
= ( size_s3613142680436377136sition @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_transition @ X3 @ Xs2 ) @ ( cons_transition @ Y2 @ Ys2 ) @ ( cons_transition @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_201_list__induct3,axiom,
! [Xs: list_transition,Ys: list_transition,Zs: list_o,P: list_transition > list_transition > list_o > $o] :
( ( ( size_s3613142680436377136sition @ Xs )
= ( size_s3613142680436377136sition @ Ys ) )
=> ( ( ( size_s3613142680436377136sition @ Ys )
= ( size_size_list_o @ Zs ) )
=> ( ( P @ nil_transition @ nil_transition @ nil_o )
=> ( ! [X3: transition,Xs2: list_transition,Y2: transition,Ys2: list_transition,Z: $o,Zs2: list_o] :
( ( ( size_s3613142680436377136sition @ Xs2 )
= ( size_s3613142680436377136sition @ Ys2 ) )
=> ( ( ( size_s3613142680436377136sition @ Ys2 )
= ( size_size_list_o @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_transition @ X3 @ Xs2 ) @ ( cons_transition @ Y2 @ Ys2 ) @ ( cons_o @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_202_list__induct3,axiom,
! [Xs: list_transition,Ys: list_o,Zs: list_transition,P: list_transition > list_o > list_transition > $o] :
( ( ( size_s3613142680436377136sition @ Xs )
= ( size_size_list_o @ Ys ) )
=> ( ( ( size_size_list_o @ Ys )
= ( size_s3613142680436377136sition @ Zs ) )
=> ( ( P @ nil_transition @ nil_o @ nil_transition )
=> ( ! [X3: transition,Xs2: list_transition,Y2: $o,Ys2: list_o,Z: transition,Zs2: list_transition] :
( ( ( size_s3613142680436377136sition @ Xs2 )
= ( size_size_list_o @ Ys2 ) )
=> ( ( ( size_size_list_o @ Ys2 )
= ( size_s3613142680436377136sition @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_transition @ X3 @ Xs2 ) @ ( cons_o @ Y2 @ Ys2 ) @ ( cons_transition @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_203_list__induct3,axiom,
! [Xs: list_transition,Ys: list_o,Zs: list_o,P: list_transition > list_o > list_o > $o] :
( ( ( size_s3613142680436377136sition @ Xs )
= ( size_size_list_o @ Ys ) )
=> ( ( ( size_size_list_o @ Ys )
= ( size_size_list_o @ Zs ) )
=> ( ( P @ nil_transition @ nil_o @ nil_o )
=> ( ! [X3: transition,Xs2: list_transition,Y2: $o,Ys2: list_o,Z: $o,Zs2: list_o] :
( ( ( size_s3613142680436377136sition @ Xs2 )
= ( size_size_list_o @ Ys2 ) )
=> ( ( ( size_size_list_o @ Ys2 )
= ( size_size_list_o @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_transition @ X3 @ Xs2 ) @ ( cons_o @ Y2 @ Ys2 ) @ ( cons_o @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_204_list__induct3,axiom,
! [Xs: list_o,Ys: list_transition,Zs: list_transition,P: list_o > list_transition > list_transition > $o] :
( ( ( size_size_list_o @ Xs )
= ( size_s3613142680436377136sition @ Ys ) )
=> ( ( ( size_s3613142680436377136sition @ Ys )
= ( size_s3613142680436377136sition @ Zs ) )
=> ( ( P @ nil_o @ nil_transition @ nil_transition )
=> ( ! [X3: $o,Xs2: list_o,Y2: transition,Ys2: list_transition,Z: transition,Zs2: list_transition] :
( ( ( size_size_list_o @ Xs2 )
= ( size_s3613142680436377136sition @ Ys2 ) )
=> ( ( ( size_s3613142680436377136sition @ Ys2 )
= ( size_s3613142680436377136sition @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_o @ X3 @ Xs2 ) @ ( cons_transition @ Y2 @ Ys2 ) @ ( cons_transition @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_205_list__induct3,axiom,
! [Xs: list_o,Ys: list_transition,Zs: list_o,P: list_o > list_transition > list_o > $o] :
( ( ( size_size_list_o @ Xs )
= ( size_s3613142680436377136sition @ Ys ) )
=> ( ( ( size_s3613142680436377136sition @ Ys )
= ( size_size_list_o @ Zs ) )
=> ( ( P @ nil_o @ nil_transition @ nil_o )
=> ( ! [X3: $o,Xs2: list_o,Y2: transition,Ys2: list_transition,Z: $o,Zs2: list_o] :
( ( ( size_size_list_o @ Xs2 )
= ( size_s3613142680436377136sition @ Ys2 ) )
=> ( ( ( size_s3613142680436377136sition @ Ys2 )
= ( size_size_list_o @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_o @ X3 @ Xs2 ) @ ( cons_transition @ Y2 @ Ys2 ) @ ( cons_o @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_206_list__induct3,axiom,
! [Xs: list_o,Ys: list_o,Zs: list_transition,P: list_o > list_o > list_transition > $o] :
( ( ( size_size_list_o @ Xs )
= ( size_size_list_o @ Ys ) )
=> ( ( ( size_size_list_o @ Ys )
= ( size_s3613142680436377136sition @ Zs ) )
=> ( ( P @ nil_o @ nil_o @ nil_transition )
=> ( ! [X3: $o,Xs2: list_o,Y2: $o,Ys2: list_o,Z: transition,Zs2: list_transition] :
( ( ( size_size_list_o @ Xs2 )
= ( size_size_list_o @ Ys2 ) )
=> ( ( ( size_size_list_o @ Ys2 )
= ( size_s3613142680436377136sition @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_o @ X3 @ Xs2 ) @ ( cons_o @ Y2 @ Ys2 ) @ ( cons_transition @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_207_list__induct3,axiom,
! [Xs: list_o,Ys: list_o,Zs: list_o,P: list_o > list_o > list_o > $o] :
( ( ( size_size_list_o @ Xs )
= ( size_size_list_o @ Ys ) )
=> ( ( ( size_size_list_o @ Ys )
= ( size_size_list_o @ Zs ) )
=> ( ( P @ nil_o @ nil_o @ nil_o )
=> ( ! [X3: $o,Xs2: list_o,Y2: $o,Ys2: list_o,Z: $o,Zs2: list_o] :
( ( ( size_size_list_o @ Xs2 )
= ( size_size_list_o @ Ys2 ) )
=> ( ( ( size_size_list_o @ Ys2 )
= ( size_size_list_o @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_o @ X3 @ Xs2 ) @ ( cons_o @ Y2 @ Ys2 ) @ ( cons_o @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_208_list__induct3,axiom,
! [Xs: list_transition,Ys: list_transition,Zs: list_list_o,P: list_transition > list_transition > list_list_o > $o] :
( ( ( size_s3613142680436377136sition @ Xs )
= ( size_s3613142680436377136sition @ Ys ) )
=> ( ( ( size_s3613142680436377136sition @ Ys )
= ( size_s2710708370519433104list_o @ Zs ) )
=> ( ( P @ nil_transition @ nil_transition @ nil_list_o )
=> ( ! [X3: transition,Xs2: list_transition,Y2: transition,Ys2: list_transition,Z: list_o,Zs2: list_list_o] :
( ( ( size_s3613142680436377136sition @ Xs2 )
= ( size_s3613142680436377136sition @ Ys2 ) )
=> ( ( ( size_s3613142680436377136sition @ Ys2 )
= ( size_s2710708370519433104list_o @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_transition @ X3 @ Xs2 ) @ ( cons_transition @ Y2 @ Ys2 ) @ ( cons_list_o @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_209_list__induct3,axiom,
! [Xs: list_transition,Ys: list_list_o,Zs: list_transition,P: list_transition > list_list_o > list_transition > $o] :
( ( ( size_s3613142680436377136sition @ Xs )
= ( size_s2710708370519433104list_o @ Ys ) )
=> ( ( ( size_s2710708370519433104list_o @ Ys )
= ( size_s3613142680436377136sition @ Zs ) )
=> ( ( P @ nil_transition @ nil_list_o @ nil_transition )
=> ( ! [X3: transition,Xs2: list_transition,Y2: list_o,Ys2: list_list_o,Z: transition,Zs2: list_transition] :
( ( ( size_s3613142680436377136sition @ Xs2 )
= ( size_s2710708370519433104list_o @ Ys2 ) )
=> ( ( ( size_s2710708370519433104list_o @ Ys2 )
= ( size_s3613142680436377136sition @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_transition @ X3 @ Xs2 ) @ ( cons_list_o @ Y2 @ Ys2 ) @ ( cons_transition @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_210_list__induct4,axiom,
! [Xs: list_transition,Ys: list_transition,Zs: list_transition,Ws: list_transition,P: list_transition > list_transition > list_transition > list_transition > $o] :
( ( ( size_s3613142680436377136sition @ Xs )
= ( size_s3613142680436377136sition @ Ys ) )
=> ( ( ( size_s3613142680436377136sition @ Ys )
= ( size_s3613142680436377136sition @ Zs ) )
=> ( ( ( size_s3613142680436377136sition @ Zs )
= ( size_s3613142680436377136sition @ Ws ) )
=> ( ( P @ nil_transition @ nil_transition @ nil_transition @ nil_transition )
=> ( ! [X3: transition,Xs2: list_transition,Y2: transition,Ys2: list_transition,Z: transition,Zs2: list_transition,W: transition,Ws2: list_transition] :
( ( ( size_s3613142680436377136sition @ Xs2 )
= ( size_s3613142680436377136sition @ Ys2 ) )
=> ( ( ( size_s3613142680436377136sition @ Ys2 )
= ( size_s3613142680436377136sition @ Zs2 ) )
=> ( ( ( size_s3613142680436377136sition @ Zs2 )
= ( size_s3613142680436377136sition @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_transition @ X3 @ Xs2 ) @ ( cons_transition @ Y2 @ Ys2 ) @ ( cons_transition @ Z @ Zs2 ) @ ( cons_transition @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_211_list__induct4,axiom,
! [Xs: list_transition,Ys: list_transition,Zs: list_transition,Ws: list_o,P: list_transition > list_transition > list_transition > list_o > $o] :
( ( ( size_s3613142680436377136sition @ Xs )
= ( size_s3613142680436377136sition @ Ys ) )
=> ( ( ( size_s3613142680436377136sition @ Ys )
= ( size_s3613142680436377136sition @ Zs ) )
=> ( ( ( size_s3613142680436377136sition @ Zs )
= ( size_size_list_o @ Ws ) )
=> ( ( P @ nil_transition @ nil_transition @ nil_transition @ nil_o )
=> ( ! [X3: transition,Xs2: list_transition,Y2: transition,Ys2: list_transition,Z: transition,Zs2: list_transition,W: $o,Ws2: list_o] :
( ( ( size_s3613142680436377136sition @ Xs2 )
= ( size_s3613142680436377136sition @ Ys2 ) )
=> ( ( ( size_s3613142680436377136sition @ Ys2 )
= ( size_s3613142680436377136sition @ Zs2 ) )
=> ( ( ( size_s3613142680436377136sition @ Zs2 )
= ( size_size_list_o @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_transition @ X3 @ Xs2 ) @ ( cons_transition @ Y2 @ Ys2 ) @ ( cons_transition @ Z @ Zs2 ) @ ( cons_o @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_212_list__induct4,axiom,
! [Xs: list_transition,Ys: list_transition,Zs: list_o,Ws: list_transition,P: list_transition > list_transition > list_o > list_transition > $o] :
( ( ( size_s3613142680436377136sition @ Xs )
= ( size_s3613142680436377136sition @ Ys ) )
=> ( ( ( size_s3613142680436377136sition @ Ys )
= ( size_size_list_o @ Zs ) )
=> ( ( ( size_size_list_o @ Zs )
= ( size_s3613142680436377136sition @ Ws ) )
=> ( ( P @ nil_transition @ nil_transition @ nil_o @ nil_transition )
=> ( ! [X3: transition,Xs2: list_transition,Y2: transition,Ys2: list_transition,Z: $o,Zs2: list_o,W: transition,Ws2: list_transition] :
( ( ( size_s3613142680436377136sition @ Xs2 )
= ( size_s3613142680436377136sition @ Ys2 ) )
=> ( ( ( size_s3613142680436377136sition @ Ys2 )
= ( size_size_list_o @ Zs2 ) )
=> ( ( ( size_size_list_o @ Zs2 )
= ( size_s3613142680436377136sition @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_transition @ X3 @ Xs2 ) @ ( cons_transition @ Y2 @ Ys2 ) @ ( cons_o @ Z @ Zs2 ) @ ( cons_transition @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_213_list__induct4,axiom,
! [Xs: list_transition,Ys: list_transition,Zs: list_o,Ws: list_o,P: list_transition > list_transition > list_o > list_o > $o] :
( ( ( size_s3613142680436377136sition @ Xs )
= ( size_s3613142680436377136sition @ Ys ) )
=> ( ( ( size_s3613142680436377136sition @ Ys )
= ( size_size_list_o @ Zs ) )
=> ( ( ( size_size_list_o @ Zs )
= ( size_size_list_o @ Ws ) )
=> ( ( P @ nil_transition @ nil_transition @ nil_o @ nil_o )
=> ( ! [X3: transition,Xs2: list_transition,Y2: transition,Ys2: list_transition,Z: $o,Zs2: list_o,W: $o,Ws2: list_o] :
( ( ( size_s3613142680436377136sition @ Xs2 )
= ( size_s3613142680436377136sition @ Ys2 ) )
=> ( ( ( size_s3613142680436377136sition @ Ys2 )
= ( size_size_list_o @ Zs2 ) )
=> ( ( ( size_size_list_o @ Zs2 )
= ( size_size_list_o @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_transition @ X3 @ Xs2 ) @ ( cons_transition @ Y2 @ Ys2 ) @ ( cons_o @ Z @ Zs2 ) @ ( cons_o @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_214_list__induct4,axiom,
! [Xs: list_transition,Ys: list_o,Zs: list_transition,Ws: list_transition,P: list_transition > list_o > list_transition > list_transition > $o] :
( ( ( size_s3613142680436377136sition @ Xs )
= ( size_size_list_o @ Ys ) )
=> ( ( ( size_size_list_o @ Ys )
= ( size_s3613142680436377136sition @ Zs ) )
=> ( ( ( size_s3613142680436377136sition @ Zs )
= ( size_s3613142680436377136sition @ Ws ) )
=> ( ( P @ nil_transition @ nil_o @ nil_transition @ nil_transition )
=> ( ! [X3: transition,Xs2: list_transition,Y2: $o,Ys2: list_o,Z: transition,Zs2: list_transition,W: transition,Ws2: list_transition] :
( ( ( size_s3613142680436377136sition @ Xs2 )
= ( size_size_list_o @ Ys2 ) )
=> ( ( ( size_size_list_o @ Ys2 )
= ( size_s3613142680436377136sition @ Zs2 ) )
=> ( ( ( size_s3613142680436377136sition @ Zs2 )
= ( size_s3613142680436377136sition @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_transition @ X3 @ Xs2 ) @ ( cons_o @ Y2 @ Ys2 ) @ ( cons_transition @ Z @ Zs2 ) @ ( cons_transition @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_215_list__induct4,axiom,
! [Xs: list_transition,Ys: list_o,Zs: list_transition,Ws: list_o,P: list_transition > list_o > list_transition > list_o > $o] :
( ( ( size_s3613142680436377136sition @ Xs )
= ( size_size_list_o @ Ys ) )
=> ( ( ( size_size_list_o @ Ys )
= ( size_s3613142680436377136sition @ Zs ) )
=> ( ( ( size_s3613142680436377136sition @ Zs )
= ( size_size_list_o @ Ws ) )
=> ( ( P @ nil_transition @ nil_o @ nil_transition @ nil_o )
=> ( ! [X3: transition,Xs2: list_transition,Y2: $o,Ys2: list_o,Z: transition,Zs2: list_transition,W: $o,Ws2: list_o] :
( ( ( size_s3613142680436377136sition @ Xs2 )
= ( size_size_list_o @ Ys2 ) )
=> ( ( ( size_size_list_o @ Ys2 )
= ( size_s3613142680436377136sition @ Zs2 ) )
=> ( ( ( size_s3613142680436377136sition @ Zs2 )
= ( size_size_list_o @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_transition @ X3 @ Xs2 ) @ ( cons_o @ Y2 @ Ys2 ) @ ( cons_transition @ Z @ Zs2 ) @ ( cons_o @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_216_list__induct4,axiom,
! [Xs: list_transition,Ys: list_o,Zs: list_o,Ws: list_transition,P: list_transition > list_o > list_o > list_transition > $o] :
( ( ( size_s3613142680436377136sition @ Xs )
= ( size_size_list_o @ Ys ) )
=> ( ( ( size_size_list_o @ Ys )
= ( size_size_list_o @ Zs ) )
=> ( ( ( size_size_list_o @ Zs )
= ( size_s3613142680436377136sition @ Ws ) )
=> ( ( P @ nil_transition @ nil_o @ nil_o @ nil_transition )
=> ( ! [X3: transition,Xs2: list_transition,Y2: $o,Ys2: list_o,Z: $o,Zs2: list_o,W: transition,Ws2: list_transition] :
( ( ( size_s3613142680436377136sition @ Xs2 )
= ( size_size_list_o @ Ys2 ) )
=> ( ( ( size_size_list_o @ Ys2 )
= ( size_size_list_o @ Zs2 ) )
=> ( ( ( size_size_list_o @ Zs2 )
= ( size_s3613142680436377136sition @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_transition @ X3 @ Xs2 ) @ ( cons_o @ Y2 @ Ys2 ) @ ( cons_o @ Z @ Zs2 ) @ ( cons_transition @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_217_list__induct4,axiom,
! [Xs: list_transition,Ys: list_o,Zs: list_o,Ws: list_o,P: list_transition > list_o > list_o > list_o > $o] :
( ( ( size_s3613142680436377136sition @ Xs )
= ( size_size_list_o @ Ys ) )
=> ( ( ( size_size_list_o @ Ys )
= ( size_size_list_o @ Zs ) )
=> ( ( ( size_size_list_o @ Zs )
= ( size_size_list_o @ Ws ) )
=> ( ( P @ nil_transition @ nil_o @ nil_o @ nil_o )
=> ( ! [X3: transition,Xs2: list_transition,Y2: $o,Ys2: list_o,Z: $o,Zs2: list_o,W: $o,Ws2: list_o] :
( ( ( size_s3613142680436377136sition @ Xs2 )
= ( size_size_list_o @ Ys2 ) )
=> ( ( ( size_size_list_o @ Ys2 )
= ( size_size_list_o @ Zs2 ) )
=> ( ( ( size_size_list_o @ Zs2 )
= ( size_size_list_o @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_transition @ X3 @ Xs2 ) @ ( cons_o @ Y2 @ Ys2 ) @ ( cons_o @ Z @ Zs2 ) @ ( cons_o @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_218_list__induct4,axiom,
! [Xs: list_o,Ys: list_transition,Zs: list_transition,Ws: list_transition,P: list_o > list_transition > list_transition > list_transition > $o] :
( ( ( size_size_list_o @ Xs )
= ( size_s3613142680436377136sition @ Ys ) )
=> ( ( ( size_s3613142680436377136sition @ Ys )
= ( size_s3613142680436377136sition @ Zs ) )
=> ( ( ( size_s3613142680436377136sition @ Zs )
= ( size_s3613142680436377136sition @ Ws ) )
=> ( ( P @ nil_o @ nil_transition @ nil_transition @ nil_transition )
=> ( ! [X3: $o,Xs2: list_o,Y2: transition,Ys2: list_transition,Z: transition,Zs2: list_transition,W: transition,Ws2: list_transition] :
( ( ( size_size_list_o @ Xs2 )
= ( size_s3613142680436377136sition @ Ys2 ) )
=> ( ( ( size_s3613142680436377136sition @ Ys2 )
= ( size_s3613142680436377136sition @ Zs2 ) )
=> ( ( ( size_s3613142680436377136sition @ Zs2 )
= ( size_s3613142680436377136sition @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_o @ X3 @ Xs2 ) @ ( cons_transition @ Y2 @ Ys2 ) @ ( cons_transition @ Z @ Zs2 ) @ ( cons_transition @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_219_list__induct4,axiom,
! [Xs: list_o,Ys: list_transition,Zs: list_transition,Ws: list_o,P: list_o > list_transition > list_transition > list_o > $o] :
( ( ( size_size_list_o @ Xs )
= ( size_s3613142680436377136sition @ Ys ) )
=> ( ( ( size_s3613142680436377136sition @ Ys )
= ( size_s3613142680436377136sition @ Zs ) )
=> ( ( ( size_s3613142680436377136sition @ Zs )
= ( size_size_list_o @ Ws ) )
=> ( ( P @ nil_o @ nil_transition @ nil_transition @ nil_o )
=> ( ! [X3: $o,Xs2: list_o,Y2: transition,Ys2: list_transition,Z: transition,Zs2: list_transition,W: $o,Ws2: list_o] :
( ( ( size_size_list_o @ Xs2 )
= ( size_s3613142680436377136sition @ Ys2 ) )
=> ( ( ( size_s3613142680436377136sition @ Ys2 )
= ( size_s3613142680436377136sition @ Zs2 ) )
=> ( ( ( size_s3613142680436377136sition @ Zs2 )
= ( size_size_list_o @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_o @ X3 @ Xs2 ) @ ( cons_transition @ Y2 @ Ys2 ) @ ( cons_transition @ Z @ Zs2 ) @ ( cons_o @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_220_list__nonempty__induct,axiom,
! [Xs: list_o,P: list_o > $o] :
( ( Xs != nil_o )
=> ( ! [X3: $o] : ( P @ ( cons_o @ X3 @ nil_o ) )
=> ( ! [X3: $o,Xs2: list_o] :
( ( Xs2 != nil_o )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_o @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_221_list__nonempty__induct,axiom,
! [Xs: list_transition,P: list_transition > $o] :
( ( Xs != nil_transition )
=> ( ! [X3: transition] : ( P @ ( cons_transition @ X3 @ nil_transition ) )
=> ( ! [X3: transition,Xs2: list_transition] :
( ( Xs2 != nil_transition )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_transition @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_222_list__nonempty__induct,axiom,
! [Xs: list_list_o,P: list_list_o > $o] :
( ( Xs != nil_list_o )
=> ( ! [X3: list_o] : ( P @ ( cons_list_o @ X3 @ nil_list_o ) )
=> ( ! [X3: list_o,Xs2: list_list_o] :
( ( Xs2 != nil_list_o )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_list_o @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_223_list__induct2_H,axiom,
! [P: list_o > list_o > $o,Xs: list_o,Ys: list_o] :
( ( P @ nil_o @ nil_o )
=> ( ! [X3: $o,Xs2: list_o] : ( P @ ( cons_o @ X3 @ Xs2 ) @ nil_o )
=> ( ! [Y2: $o,Ys2: list_o] : ( P @ nil_o @ ( cons_o @ Y2 @ Ys2 ) )
=> ( ! [X3: $o,Xs2: list_o,Y2: $o,Ys2: list_o] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_o @ X3 @ Xs2 ) @ ( cons_o @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_224_list__induct2_H,axiom,
! [P: list_o > list_transition > $o,Xs: list_o,Ys: list_transition] :
( ( P @ nil_o @ nil_transition )
=> ( ! [X3: $o,Xs2: list_o] : ( P @ ( cons_o @ X3 @ Xs2 ) @ nil_transition )
=> ( ! [Y2: transition,Ys2: list_transition] : ( P @ nil_o @ ( cons_transition @ Y2 @ Ys2 ) )
=> ( ! [X3: $o,Xs2: list_o,Y2: transition,Ys2: list_transition] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_o @ X3 @ Xs2 ) @ ( cons_transition @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_225_list__induct2_H,axiom,
! [P: list_transition > list_o > $o,Xs: list_transition,Ys: list_o] :
( ( P @ nil_transition @ nil_o )
=> ( ! [X3: transition,Xs2: list_transition] : ( P @ ( cons_transition @ X3 @ Xs2 ) @ nil_o )
=> ( ! [Y2: $o,Ys2: list_o] : ( P @ nil_transition @ ( cons_o @ Y2 @ Ys2 ) )
=> ( ! [X3: transition,Xs2: list_transition,Y2: $o,Ys2: list_o] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_transition @ X3 @ Xs2 ) @ ( cons_o @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_226_list__induct2_H,axiom,
! [P: list_transition > list_transition > $o,Xs: list_transition,Ys: list_transition] :
( ( P @ nil_transition @ nil_transition )
=> ( ! [X3: transition,Xs2: list_transition] : ( P @ ( cons_transition @ X3 @ Xs2 ) @ nil_transition )
=> ( ! [Y2: transition,Ys2: list_transition] : ( P @ nil_transition @ ( cons_transition @ Y2 @ Ys2 ) )
=> ( ! [X3: transition,Xs2: list_transition,Y2: transition,Ys2: list_transition] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_transition @ X3 @ Xs2 ) @ ( cons_transition @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_227_list__induct2_H,axiom,
! [P: list_o > list_list_o > $o,Xs: list_o,Ys: list_list_o] :
( ( P @ nil_o @ nil_list_o )
=> ( ! [X3: $o,Xs2: list_o] : ( P @ ( cons_o @ X3 @ Xs2 ) @ nil_list_o )
=> ( ! [Y2: list_o,Ys2: list_list_o] : ( P @ nil_o @ ( cons_list_o @ Y2 @ Ys2 ) )
=> ( ! [X3: $o,Xs2: list_o,Y2: list_o,Ys2: list_list_o] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_o @ X3 @ Xs2 ) @ ( cons_list_o @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_228_list__induct2_H,axiom,
! [P: list_transition > list_list_o > $o,Xs: list_transition,Ys: list_list_o] :
( ( P @ nil_transition @ nil_list_o )
=> ( ! [X3: transition,Xs2: list_transition] : ( P @ ( cons_transition @ X3 @ Xs2 ) @ nil_list_o )
=> ( ! [Y2: list_o,Ys2: list_list_o] : ( P @ nil_transition @ ( cons_list_o @ Y2 @ Ys2 ) )
=> ( ! [X3: transition,Xs2: list_transition,Y2: list_o,Ys2: list_list_o] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_transition @ X3 @ Xs2 ) @ ( cons_list_o @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_229_list__induct2_H,axiom,
! [P: list_list_o > list_o > $o,Xs: list_list_o,Ys: list_o] :
( ( P @ nil_list_o @ nil_o )
=> ( ! [X3: list_o,Xs2: list_list_o] : ( P @ ( cons_list_o @ X3 @ Xs2 ) @ nil_o )
=> ( ! [Y2: $o,Ys2: list_o] : ( P @ nil_list_o @ ( cons_o @ Y2 @ Ys2 ) )
=> ( ! [X3: list_o,Xs2: list_list_o,Y2: $o,Ys2: list_o] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_list_o @ X3 @ Xs2 ) @ ( cons_o @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_230_list__induct2_H,axiom,
! [P: list_list_o > list_transition > $o,Xs: list_list_o,Ys: list_transition] :
( ( P @ nil_list_o @ nil_transition )
=> ( ! [X3: list_o,Xs2: list_list_o] : ( P @ ( cons_list_o @ X3 @ Xs2 ) @ nil_transition )
=> ( ! [Y2: transition,Ys2: list_transition] : ( P @ nil_list_o @ ( cons_transition @ Y2 @ Ys2 ) )
=> ( ! [X3: list_o,Xs2: list_list_o,Y2: transition,Ys2: list_transition] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_list_o @ X3 @ Xs2 ) @ ( cons_transition @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_231_list__induct2_H,axiom,
! [P: list_list_o > list_list_o > $o,Xs: list_list_o,Ys: list_list_o] :
( ( P @ nil_list_o @ nil_list_o )
=> ( ! [X3: list_o,Xs2: list_list_o] : ( P @ ( cons_list_o @ X3 @ Xs2 ) @ nil_list_o )
=> ( ! [Y2: list_o,Ys2: list_list_o] : ( P @ nil_list_o @ ( cons_list_o @ Y2 @ Ys2 ) )
=> ( ! [X3: list_o,Xs2: list_list_o,Y2: list_o,Ys2: list_list_o] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_list_o @ X3 @ Xs2 ) @ ( cons_list_o @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_232_neq__Nil__conv,axiom,
! [Xs: list_o] :
( ( Xs != nil_o )
= ( ? [Y3: $o,Ys3: list_o] :
( Xs
= ( cons_o @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_233_neq__Nil__conv,axiom,
! [Xs: list_transition] :
( ( Xs != nil_transition )
= ( ? [Y3: transition,Ys3: list_transition] :
( Xs
= ( cons_transition @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_234_neq__Nil__conv,axiom,
! [Xs: list_list_o] :
( ( Xs != nil_list_o )
= ( ? [Y3: list_o,Ys3: list_list_o] :
( Xs
= ( cons_list_o @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_235_remdups__adj_Ocases,axiom,
! [X2: list_o] :
( ( X2 != nil_o )
=> ( ! [X3: $o] :
( X2
!= ( cons_o @ X3 @ nil_o ) )
=> ~ ! [X3: $o,Y2: $o,Xs2: list_o] :
( X2
!= ( cons_o @ X3 @ ( cons_o @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_236_remdups__adj_Ocases,axiom,
! [X2: list_transition] :
( ( X2 != nil_transition )
=> ( ! [X3: transition] :
( X2
!= ( cons_transition @ X3 @ nil_transition ) )
=> ~ ! [X3: transition,Y2: transition,Xs2: list_transition] :
( X2
!= ( cons_transition @ X3 @ ( cons_transition @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_237_remdups__adj_Ocases,axiom,
! [X2: list_list_o] :
( ( X2 != nil_list_o )
=> ( ! [X3: list_o] :
( X2
!= ( cons_list_o @ X3 @ nil_list_o ) )
=> ~ ! [X3: list_o,Y2: list_o,Xs2: list_list_o] :
( X2
!= ( cons_list_o @ X3 @ ( cons_list_o @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_238_list__split_Ocases,axiom,
! [X2: list_o] :
( ( X2 != nil_o )
=> ~ ! [X3: $o,Xs2: list_o] :
( X2
!= ( cons_o @ X3 @ Xs2 ) ) ) ).
% list_split.cases
thf(fact_239_list__split_Ocases,axiom,
! [X2: list_transition] :
( ( X2 != nil_transition )
=> ~ ! [X3: transition,Xs2: list_transition] :
( X2
!= ( cons_transition @ X3 @ Xs2 ) ) ) ).
% list_split.cases
thf(fact_240_list__split_Ocases,axiom,
! [X2: list_list_o] :
( ( X2 != nil_list_o )
=> ~ ! [X3: list_o,Xs2: list_list_o] :
( X2
!= ( cons_list_o @ X3 @ Xs2 ) ) ) ).
% list_split.cases
thf(fact_241_min__list_Ocases,axiom,
! [X2: list_o] :
( ! [X3: $o,Xs2: list_o] :
( X2
!= ( cons_o @ X3 @ Xs2 ) )
=> ( X2 = nil_o ) ) ).
% min_list.cases
thf(fact_242_list_Oexhaust,axiom,
! [Y: list_o] :
( ( Y != nil_o )
=> ~ ! [X212: $o,X222: list_o] :
( Y
!= ( cons_o @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_243_list_Oexhaust,axiom,
! [Y: list_transition] :
( ( Y != nil_transition )
=> ~ ! [X212: transition,X222: list_transition] :
( Y
!= ( cons_transition @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_244_list_Oexhaust,axiom,
! [Y: list_list_o] :
( ( Y != nil_list_o )
=> ~ ! [X212: list_o,X222: list_list_o] :
( Y
!= ( cons_list_o @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_245_list_OdiscI,axiom,
! [List: list_o,X21: $o,X22: list_o] :
( ( List
= ( cons_o @ X21 @ X22 ) )
=> ( List != nil_o ) ) ).
% list.discI
thf(fact_246_list_OdiscI,axiom,
! [List: list_transition,X21: transition,X22: list_transition] :
( ( List
= ( cons_transition @ X21 @ X22 ) )
=> ( List != nil_transition ) ) ).
% list.discI
thf(fact_247_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_248_list_Odistinct_I1_J,axiom,
! [X21: $o,X22: list_o] :
( nil_o
!= ( cons_o @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_249_list_Odistinct_I1_J,axiom,
! [X21: transition,X22: list_transition] :
( nil_transition
!= ( cons_transition @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_250_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_251_Cons__eq__appendI,axiom,
! [X2: formula_a_t,Xs1: list_formula_a_t,Ys: list_formula_a_t,Xs: list_formula_a_t,Zs: list_formula_a_t] :
( ( ( cons_formula_a_t @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_formula_a_t @ Xs1 @ Zs ) )
=> ( ( cons_formula_a_t @ X2 @ Xs )
= ( append_formula_a_t @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_252_Cons__eq__appendI,axiom,
! [X2: transition,Xs1: list_transition,Ys: list_transition,Xs: list_transition,Zs: list_transition] :
( ( ( cons_transition @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_transition @ Xs1 @ Zs ) )
=> ( ( cons_transition @ X2 @ Xs )
= ( append_transition @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_253_Cons__eq__appendI,axiom,
! [X2: list_o,Xs1: list_list_o,Ys: list_list_o,Xs: list_list_o,Zs: list_list_o] :
( ( ( cons_list_o @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_list_o @ Xs1 @ Zs ) )
=> ( ( cons_list_o @ X2 @ Xs )
= ( append_list_o @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_254_append__Cons,axiom,
! [X2: formula_a_t,Xs: list_formula_a_t,Ys: list_formula_a_t] :
( ( append_formula_a_t @ ( cons_formula_a_t @ X2 @ Xs ) @ Ys )
= ( cons_formula_a_t @ X2 @ ( append_formula_a_t @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_255_append__Cons,axiom,
! [X2: transition,Xs: list_transition,Ys: list_transition] :
( ( append_transition @ ( cons_transition @ X2 @ Xs ) @ Ys )
= ( cons_transition @ X2 @ ( append_transition @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_256_append__Cons,axiom,
! [X2: list_o,Xs: list_list_o,Ys: list_list_o] :
( ( append_list_o @ ( cons_list_o @ X2 @ Xs ) @ Ys )
= ( cons_list_o @ X2 @ ( append_list_o @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_257_step__eps__closure__set__split,axiom,
! [Q0: nat,Transs: list_transition,R2: set_nat,S2: set_nat,Bs: list_o] :
( ( step_eps_closure_set @ Q0 @ Transs @ ( sup_sup_set_nat @ R2 @ S2 ) @ Bs )
= ( sup_sup_set_nat @ ( step_eps_closure_set @ Q0 @ Transs @ R2 @ Bs ) @ ( step_eps_closure_set @ Q0 @ Transs @ S2 @ Bs ) ) ) ).
% step_eps_closure_set_split
thf(fact_258_delta__split,axiom,
! [Q0: nat,Transs: list_transition,R2: set_nat,S2: set_nat,Bs: list_o] :
( ( delta @ Q0 @ Transs @ ( sup_sup_set_nat @ R2 @ S2 ) @ Bs )
= ( sup_sup_set_nat @ ( delta @ Q0 @ Transs @ R2 @ Bs ) @ ( delta @ Q0 @ Transs @ S2 @ Bs ) ) ) ).
% delta_split
thf(fact_259_run__comp,axiom,
! [Q0: nat,Transs: list_transition,R2: set_nat,Bss: list_list_o,Css: list_list_o] :
( ( run @ Q0 @ Transs @ R2 @ ( append_list_o @ Bss @ Css ) )
= ( run @ Q0 @ Transs @ ( run @ Q0 @ Transs @ R2 @ Bss ) @ Css ) ) ).
% run_comp
thf(fact_260_run__split,axiom,
! [Q0: nat,Transs: list_transition,R2: set_nat,S2: set_nat,Bss: list_list_o] :
( ( run @ Q0 @ Transs @ ( sup_sup_set_nat @ R2 @ S2 ) @ Bss )
= ( sup_sup_set_nat @ ( run @ Q0 @ Transs @ R2 @ Bss ) @ ( run @ Q0 @ Transs @ S2 @ Bss ) ) ) ).
% run_split
thf(fact_261_step__symb__set__split,axiom,
! [Q0: nat,Transs: list_transition,R2: set_nat,S2: set_nat] :
( ( step_symb_set @ Q0 @ Transs @ ( sup_sup_set_nat @ R2 @ S2 ) )
= ( sup_sup_set_nat @ ( step_symb_set @ Q0 @ Transs @ R2 ) @ ( step_symb_set @ Q0 @ Transs @ S2 ) ) ) ).
% step_symb_set_split
thf(fact_262_run__Nil,axiom,
! [Q0: nat,Transs: list_transition,R2: set_nat] :
( ( run @ Q0 @ Transs @ R2 @ nil_list_o )
= R2 ) ).
% run_Nil
thf(fact_263_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_264_accept__eps__split,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,R2: set_nat,S2: set_nat,Bs: list_o] :
( ( accept_eps @ Q0 @ Qf @ Transs @ ( sup_sup_set_nat @ R2 @ S2 ) @ Bs )
= ( ( accept_eps @ Q0 @ Qf @ Transs @ R2 @ Bs )
| ( accept_eps @ Q0 @ Qf @ Transs @ S2 @ Bs ) ) ) ).
% accept_eps_split
thf(fact_265_same__length__different,axiom,
! [Xs: list_transition,Ys: list_transition] :
( ( Xs != Ys )
=> ( ( ( size_s3613142680436377136sition @ Xs )
= ( size_s3613142680436377136sition @ Ys ) )
=> ? [Pre: list_transition,X3: transition,Xs3: list_transition,Y2: transition,Ys4: list_transition] :
( ( X3 != Y2 )
& ( Xs
= ( append_transition @ Pre @ ( append_transition @ ( cons_transition @ X3 @ nil_transition ) @ Xs3 ) ) )
& ( Ys
= ( append_transition @ Pre @ ( append_transition @ ( cons_transition @ Y2 @ nil_transition ) @ Ys4 ) ) ) ) ) ) ).
% same_length_different
thf(fact_266_same__length__different,axiom,
! [Xs: list_list_o,Ys: list_list_o] :
( ( Xs != Ys )
=> ( ( ( size_s2710708370519433104list_o @ Xs )
= ( size_s2710708370519433104list_o @ Ys ) )
=> ? [Pre: list_list_o,X3: list_o,Xs3: list_list_o,Y2: list_o,Ys4: list_list_o] :
( ( X3 != Y2 )
& ( Xs
= ( append_list_o @ Pre @ ( append_list_o @ ( cons_list_o @ X3 @ nil_list_o ) @ Xs3 ) ) )
& ( Ys
= ( append_list_o @ Pre @ ( append_list_o @ ( cons_list_o @ Y2 @ nil_list_o ) @ Ys4 ) ) ) ) ) ) ).
% same_length_different
thf(fact_267_same__length__different,axiom,
! [Xs: list_formula_a_t,Ys: list_formula_a_t] :
( ( Xs != Ys )
=> ( ( ( size_s8846756101701226951la_a_t @ Xs )
= ( size_s8846756101701226951la_a_t @ Ys ) )
=> ? [Pre: list_formula_a_t,X3: formula_a_t,Xs3: list_formula_a_t,Y2: formula_a_t,Ys4: list_formula_a_t] :
( ( X3 != Y2 )
& ( Xs
= ( append_formula_a_t @ Pre @ ( append_formula_a_t @ ( cons_formula_a_t @ X3 @ nil_formula_a_t ) @ Xs3 ) ) )
& ( Ys
= ( append_formula_a_t @ Pre @ ( append_formula_a_t @ ( cons_formula_a_t @ Y2 @ nil_formula_a_t ) @ Ys4 ) ) ) ) ) ) ).
% same_length_different
thf(fact_268_same__length__different,axiom,
! [Xs: list_o,Ys: list_o] :
( ( Xs != Ys )
=> ( ( ( size_size_list_o @ Xs )
= ( size_size_list_o @ Ys ) )
=> ? [Pre: list_o,X3: $o,Xs3: list_o,Y2: $o,Ys4: list_o] :
( ( X3 != Y2 )
& ( Xs
= ( append_o @ Pre @ ( append_o @ ( cons_o @ X3 @ nil_o ) @ Xs3 ) ) )
& ( Ys
= ( append_o @ Pre @ ( append_o @ ( cons_o @ Y2 @ nil_o ) @ Ys4 ) ) ) ) ) ) ).
% same_length_different
thf(fact_269_rev__nonempty__induct,axiom,
! [Xs: list_formula_a_t,P: list_formula_a_t > $o] :
( ( Xs != nil_formula_a_t )
=> ( ! [X3: formula_a_t] : ( P @ ( cons_formula_a_t @ X3 @ nil_formula_a_t ) )
=> ( ! [X3: formula_a_t,Xs2: list_formula_a_t] :
( ( Xs2 != nil_formula_a_t )
=> ( ( P @ Xs2 )
=> ( P @ ( append_formula_a_t @ Xs2 @ ( cons_formula_a_t @ X3 @ nil_formula_a_t ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_270_rev__nonempty__induct,axiom,
! [Xs: list_o,P: list_o > $o] :
( ( Xs != nil_o )
=> ( ! [X3: $o] : ( P @ ( cons_o @ X3 @ nil_o ) )
=> ( ! [X3: $o,Xs2: list_o] :
( ( Xs2 != nil_o )
=> ( ( P @ Xs2 )
=> ( P @ ( append_o @ Xs2 @ ( cons_o @ X3 @ nil_o ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_271_rev__nonempty__induct,axiom,
! [Xs: list_transition,P: list_transition > $o] :
( ( Xs != nil_transition )
=> ( ! [X3: transition] : ( P @ ( cons_transition @ X3 @ nil_transition ) )
=> ( ! [X3: transition,Xs2: list_transition] :
( ( Xs2 != nil_transition )
=> ( ( P @ Xs2 )
=> ( P @ ( append_transition @ Xs2 @ ( cons_transition @ X3 @ nil_transition ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_272_rev__nonempty__induct,axiom,
! [Xs: list_list_o,P: list_list_o > $o] :
( ( Xs != nil_list_o )
=> ( ! [X3: list_o] : ( P @ ( cons_list_o @ X3 @ nil_list_o ) )
=> ( ! [X3: list_o,Xs2: list_list_o] :
( ( Xs2 != nil_list_o )
=> ( ( P @ Xs2 )
=> ( P @ ( append_list_o @ Xs2 @ ( cons_list_o @ X3 @ nil_list_o ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_273_append__eq__Cons__conv,axiom,
! [Ys: list_formula_a_t,Zs: list_formula_a_t,X2: formula_a_t,Xs: list_formula_a_t] :
( ( ( append_formula_a_t @ Ys @ Zs )
= ( cons_formula_a_t @ X2 @ Xs ) )
= ( ( ( Ys = nil_formula_a_t )
& ( Zs
= ( cons_formula_a_t @ X2 @ Xs ) ) )
| ? [Ys5: list_formula_a_t] :
( ( Ys
= ( cons_formula_a_t @ X2 @ Ys5 ) )
& ( ( append_formula_a_t @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_274_append__eq__Cons__conv,axiom,
! [Ys: list_o,Zs: list_o,X2: $o,Xs: list_o] :
( ( ( append_o @ Ys @ Zs )
= ( cons_o @ X2 @ Xs ) )
= ( ( ( Ys = nil_o )
& ( Zs
= ( cons_o @ X2 @ Xs ) ) )
| ? [Ys5: list_o] :
( ( Ys
= ( cons_o @ X2 @ Ys5 ) )
& ( ( append_o @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_275_append__eq__Cons__conv,axiom,
! [Ys: list_transition,Zs: list_transition,X2: transition,Xs: list_transition] :
( ( ( append_transition @ Ys @ Zs )
= ( cons_transition @ X2 @ Xs ) )
= ( ( ( Ys = nil_transition )
& ( Zs
= ( cons_transition @ X2 @ Xs ) ) )
| ? [Ys5: list_transition] :
( ( Ys
= ( cons_transition @ X2 @ Ys5 ) )
& ( ( append_transition @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_276_append__eq__Cons__conv,axiom,
! [Ys: list_list_o,Zs: list_list_o,X2: list_o,Xs: list_list_o] :
( ( ( append_list_o @ Ys @ Zs )
= ( cons_list_o @ X2 @ Xs ) )
= ( ( ( Ys = nil_list_o )
& ( Zs
= ( cons_list_o @ X2 @ Xs ) ) )
| ? [Ys5: list_list_o] :
( ( Ys
= ( cons_list_o @ X2 @ Ys5 ) )
& ( ( append_list_o @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_277_Cons__eq__append__conv,axiom,
! [X2: formula_a_t,Xs: list_formula_a_t,Ys: list_formula_a_t,Zs: list_formula_a_t] :
( ( ( cons_formula_a_t @ X2 @ Xs )
= ( append_formula_a_t @ Ys @ Zs ) )
= ( ( ( Ys = nil_formula_a_t )
& ( ( cons_formula_a_t @ X2 @ Xs )
= Zs ) )
| ? [Ys5: list_formula_a_t] :
( ( ( cons_formula_a_t @ X2 @ Ys5 )
= Ys )
& ( Xs
= ( append_formula_a_t @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_278_Cons__eq__append__conv,axiom,
! [X2: $o,Xs: list_o,Ys: list_o,Zs: list_o] :
( ( ( cons_o @ X2 @ Xs )
= ( append_o @ Ys @ Zs ) )
= ( ( ( Ys = nil_o )
& ( ( cons_o @ X2 @ Xs )
= Zs ) )
| ? [Ys5: list_o] :
( ( ( cons_o @ X2 @ Ys5 )
= Ys )
& ( Xs
= ( append_o @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_279_Cons__eq__append__conv,axiom,
! [X2: transition,Xs: list_transition,Ys: list_transition,Zs: list_transition] :
( ( ( cons_transition @ X2 @ Xs )
= ( append_transition @ Ys @ Zs ) )
= ( ( ( Ys = nil_transition )
& ( ( cons_transition @ X2 @ Xs )
= Zs ) )
| ? [Ys5: list_transition] :
( ( ( cons_transition @ X2 @ Ys5 )
= Ys )
& ( Xs
= ( append_transition @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_280_Cons__eq__append__conv,axiom,
! [X2: list_o,Xs: list_list_o,Ys: list_list_o,Zs: list_list_o] :
( ( ( cons_list_o @ X2 @ Xs )
= ( append_list_o @ Ys @ Zs ) )
= ( ( ( Ys = nil_list_o )
& ( ( cons_list_o @ X2 @ Xs )
= Zs ) )
| ? [Ys5: list_list_o] :
( ( ( cons_list_o @ X2 @ Ys5 )
= Ys )
& ( Xs
= ( append_list_o @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_281_rev__exhaust,axiom,
! [Xs: list_formula_a_t] :
( ( Xs != nil_formula_a_t )
=> ~ ! [Ys2: list_formula_a_t,Y2: formula_a_t] :
( Xs
!= ( append_formula_a_t @ Ys2 @ ( cons_formula_a_t @ Y2 @ nil_formula_a_t ) ) ) ) ).
% rev_exhaust
thf(fact_282_rev__exhaust,axiom,
! [Xs: list_o] :
( ( Xs != nil_o )
=> ~ ! [Ys2: list_o,Y2: $o] :
( Xs
!= ( append_o @ Ys2 @ ( cons_o @ Y2 @ nil_o ) ) ) ) ).
% rev_exhaust
thf(fact_283_rev__exhaust,axiom,
! [Xs: list_transition] :
( ( Xs != nil_transition )
=> ~ ! [Ys2: list_transition,Y2: transition] :
( Xs
!= ( append_transition @ Ys2 @ ( cons_transition @ Y2 @ nil_transition ) ) ) ) ).
% rev_exhaust
thf(fact_284_rev__exhaust,axiom,
! [Xs: list_list_o] :
( ( Xs != nil_list_o )
=> ~ ! [Ys2: list_list_o,Y2: list_o] :
( Xs
!= ( append_list_o @ Ys2 @ ( cons_list_o @ Y2 @ nil_list_o ) ) ) ) ).
% rev_exhaust
thf(fact_285_rev__induct,axiom,
! [P: list_formula_a_t > $o,Xs: list_formula_a_t] :
( ( P @ nil_formula_a_t )
=> ( ! [X3: formula_a_t,Xs2: list_formula_a_t] :
( ( P @ Xs2 )
=> ( P @ ( append_formula_a_t @ Xs2 @ ( cons_formula_a_t @ X3 @ nil_formula_a_t ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_286_rev__induct,axiom,
! [P: list_o > $o,Xs: list_o] :
( ( P @ nil_o )
=> ( ! [X3: $o,Xs2: list_o] :
( ( P @ Xs2 )
=> ( P @ ( append_o @ Xs2 @ ( cons_o @ X3 @ nil_o ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_287_rev__induct,axiom,
! [P: list_transition > $o,Xs: list_transition] :
( ( P @ nil_transition )
=> ( ! [X3: transition,Xs2: list_transition] :
( ( P @ Xs2 )
=> ( P @ ( append_transition @ Xs2 @ ( cons_transition @ X3 @ nil_transition ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_288_rev__induct,axiom,
! [P: list_list_o > $o,Xs: list_list_o] :
( ( P @ nil_list_o )
=> ( ! [X3: list_o,Xs2: list_list_o] :
( ( P @ Xs2 )
=> ( P @ ( append_list_o @ Xs2 @ ( cons_list_o @ X3 @ nil_list_o ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_289_step__eps__closure__set__flip,axiom,
! [Q0: nat,Transs: list_transition,R2: set_nat,Bs: list_o,S2: set_nat] :
( ( ( step_eps_closure_set @ Q0 @ Transs @ R2 @ Bs )
= ( sup_sup_set_nat @ R2 @ S2 ) )
=> ( ord_less_eq_set_nat @ ( step_eps_closure_set @ Q0 @ Transs @ S2 @ Bs ) @ ( sup_sup_set_nat @ R2 @ S2 ) ) ) ).
% step_eps_closure_set_flip
thf(fact_290_NFA_Orun__accept__eps__def,axiom,
( run_accept_eps
= ( ^ [Q03: nat,Qf3: nat,Transs3: list_transition,R3: set_nat,Bss2: list_list_o] : ( accept_eps @ Q03 @ Qf3 @ Transs3 @ ( run @ Q03 @ Transs3 @ R3 @ Bss2 ) ) ) ) ).
% NFA.run_accept_eps_def
thf(fact_291_run__Cons,axiom,
! [Q0: nat,Transs: list_transition,R2: set_nat,Bs: list_o,Bss: list_list_o] :
( ( run @ Q0 @ Transs @ R2 @ ( cons_list_o @ Bs @ Bss ) )
= ( run @ Q0 @ Transs @ ( delta @ Q0 @ Transs @ R2 @ Bs ) @ Bss ) ) ).
% run_Cons
thf(fact_292_delta__step__symb__set__absorb,axiom,
( delta
= ( ^ [Q03: nat,Transs3: list_transition,R3: set_nat,Bs2: list_o] : ( sup_sup_set_nat @ ( delta @ Q03 @ Transs3 @ R3 @ Bs2 ) @ ( step_symb_set @ Q03 @ Transs3 @ R3 ) ) ) ) ).
% delta_step_symb_set_absorb
thf(fact_293_pos_Ocases,axiom,
! [X2: produc6454500794699219245list_o] :
( ! [A3: $o] :
( X2
!= ( produc7263596898809104029list_o @ A3 @ nil_o ) )
=> ~ ! [A3: $o,X3: $o,Xs2: list_o] :
( X2
!= ( produc7263596898809104029list_o @ A3 @ ( cons_o @ X3 @ Xs2 ) ) ) ) ).
% pos.cases
thf(fact_294_pos_Ocases,axiom,
! [X2: produc7282413182419550381sition] :
( ! [A3: transition] :
( X2
!= ( produc9190507990355890461sition @ A3 @ nil_transition ) )
=> ~ ! [A3: transition,X3: transition,Xs2: list_transition] :
( X2
!= ( produc9190507990355890461sition @ A3 @ ( cons_transition @ X3 @ Xs2 ) ) ) ) ).
% pos.cases
thf(fact_295_pos_Ocases,axiom,
! [X2: produc7334440844593340333list_o] :
( ! [A3: list_o] :
( X2
!= ( produc1974676357355975453list_o @ A3 @ nil_list_o ) )
=> ~ ! [A3: list_o,X3: list_o,Xs2: list_list_o] :
( X2
!= ( produc1974676357355975453list_o @ A3 @ ( cons_list_o @ X3 @ Xs2 ) ) ) ) ).
% pos.cases
thf(fact_296_nfa__cong_Orun__accept__eps__cong,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition,R2: set_nat,Bss: list_list_o,Bs: list_o] :
( ( nfa_cong @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( ( ord_less_eq_set_nat @ R2 @ ( q @ Q02 @ Qf2 @ Transs2 ) )
=> ( ( run_accept_eps @ Q0 @ Qf @ Transs @ R2 @ Bss @ Bs )
= ( run_accept_eps @ Q02 @ Qf2 @ Transs2 @ R2 @ Bss @ Bs ) ) ) ) ).
% nfa_cong.run_accept_eps_cong
thf(fact_297_delta__eps__split,axiom,
! [Q0: nat,Transs: list_transition,R2: set_nat,Bs: list_o,S2: set_nat] :
( ( ( step_eps_closure_set @ Q0 @ Transs @ R2 @ Bs )
= ( sup_sup_set_nat @ R2 @ S2 ) )
=> ( ( delta @ Q0 @ Transs @ R2 @ Bs )
= ( sup_sup_set_nat @ ( step_symb_set @ Q0 @ Transs @ R2 ) @ ( delta @ Q0 @ Transs @ S2 @ Bs ) ) ) ) ).
% delta_eps_split
thf(fact_298_nfa_Ostep__eps__closure__set__closed__union,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,R2: set_nat,Bs: list_o] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ( ord_less_eq_set_nat @ ( step_eps_closure_set @ Q0 @ Transs @ R2 @ Bs ) @ ( sup_sup_set_nat @ R2 @ ( q @ Q0 @ Qf @ Transs ) ) ) ) ).
% nfa.step_eps_closure_set_closed_union
thf(fact_299_nfa_Orun__closed__Cons,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,R2: set_nat,Bs: list_o,Bss: list_list_o] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ( ord_less_eq_set_nat @ ( run @ Q0 @ Transs @ R2 @ ( cons_list_o @ Bs @ Bss ) ) @ ( q @ Q0 @ Qf @ Transs ) ) ) ).
% nfa.run_closed_Cons
thf(fact_300_append__eq__append__conv2,axiom,
! [Xs: list_formula_a_t,Ys: list_formula_a_t,Zs: list_formula_a_t,Ts: list_formula_a_t] :
( ( ( append_formula_a_t @ Xs @ Ys )
= ( append_formula_a_t @ Zs @ Ts ) )
= ( ? [Us2: list_formula_a_t] :
( ( ( Xs
= ( append_formula_a_t @ Zs @ Us2 ) )
& ( ( append_formula_a_t @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_formula_a_t @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_formula_a_t @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_301_append__eq__append__conv2,axiom,
! [Xs: list_transition,Ys: list_transition,Zs: list_transition,Ts: list_transition] :
( ( ( append_transition @ Xs @ Ys )
= ( append_transition @ Zs @ Ts ) )
= ( ? [Us2: list_transition] :
( ( ( Xs
= ( append_transition @ Zs @ Us2 ) )
& ( ( append_transition @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_transition @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_transition @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_302_append__eq__append__conv2,axiom,
! [Xs: list_list_o,Ys: list_list_o,Zs: list_list_o,Ts: list_list_o] :
( ( ( append_list_o @ Xs @ Ys )
= ( append_list_o @ Zs @ Ts ) )
= ( ? [Us2: list_list_o] :
( ( ( Xs
= ( append_list_o @ Zs @ Us2 ) )
& ( ( append_list_o @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_list_o @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_list_o @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_303_append__eq__appendI,axiom,
! [Xs: list_formula_a_t,Xs1: list_formula_a_t,Zs: list_formula_a_t,Ys: list_formula_a_t,Us: list_formula_a_t] :
( ( ( append_formula_a_t @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_formula_a_t @ Xs1 @ Us ) )
=> ( ( append_formula_a_t @ Xs @ Ys )
= ( append_formula_a_t @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_304_append__eq__appendI,axiom,
! [Xs: list_transition,Xs1: list_transition,Zs: list_transition,Ys: list_transition,Us: list_transition] :
( ( ( append_transition @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_transition @ Xs1 @ Us ) )
=> ( ( append_transition @ Xs @ Ys )
= ( append_transition @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_305_append__eq__appendI,axiom,
! [Xs: list_list_o,Xs1: list_list_o,Zs: list_list_o,Ys: list_list_o,Us: list_list_o] :
( ( ( append_list_o @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_list_o @ Xs1 @ Us ) )
=> ( ( append_list_o @ Xs @ Ys )
= ( append_list_o @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_306_build__nfa__impl__not__Nil,axiom,
! [R: regex_a_t,Q0: nat,Qf: nat,Phis: list_formula_a_t] :
( ( build_nfa_impl_a_t @ R @ ( produc8654416511292156347la_a_t @ Q0 @ ( produc9017461973804568604la_a_t @ Qf @ Phis ) ) )
!= nil_transition ) ).
% build_nfa_impl_not_Nil
thf(fact_307_step__eps__closure__set__idem,axiom,
! [Q0: nat,Transs: list_transition,R2: set_nat,Bs: list_o] :
( ( step_eps_closure_set @ Q0 @ Transs @ ( step_eps_closure_set @ Q0 @ Transs @ R2 @ Bs ) @ Bs )
= ( step_eps_closure_set @ Q0 @ Transs @ R2 @ Bs ) ) ).
% step_eps_closure_set_idem
thf(fact_308_nfa__cong_Oqf__eq,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition] :
( ( nfa_cong @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( Qf = Qf2 ) ) ).
% nfa_cong.qf_eq
thf(fact_309_build__nfa__impl__state__cnt,axiom,
! [R: regex_a_t,Q0: nat,Qf: nat,Phis: list_formula_a_t] :
( ( size_s3613142680436377136sition @ ( build_nfa_impl_a_t @ R @ ( produc8654416511292156347la_a_t @ Q0 @ ( produc9017461973804568604la_a_t @ Qf @ Phis ) ) ) )
= ( state_cnt_a_t @ R ) ) ).
% build_nfa_impl_state_cnt
thf(fact_310_eq__Nil__appendI,axiom,
! [Xs: list_formula_a_t,Ys: list_formula_a_t] :
( ( Xs = Ys )
=> ( Xs
= ( append_formula_a_t @ nil_formula_a_t @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_311_eq__Nil__appendI,axiom,
! [Xs: list_o,Ys: list_o] :
( ( Xs = Ys )
=> ( Xs
= ( append_o @ nil_o @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_312_eq__Nil__appendI,axiom,
! [Xs: list_transition,Ys: list_transition] :
( ( Xs = Ys )
=> ( Xs
= ( append_transition @ nil_transition @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_313_eq__Nil__appendI,axiom,
! [Xs: list_list_o,Ys: list_list_o] :
( ( Xs = Ys )
=> ( Xs
= ( append_list_o @ nil_list_o @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_314_append_Oleft__neutral,axiom,
! [A: list_formula_a_t] :
( ( append_formula_a_t @ nil_formula_a_t @ A )
= A ) ).
% append.left_neutral
thf(fact_315_append_Oleft__neutral,axiom,
! [A: list_o] :
( ( append_o @ nil_o @ A )
= A ) ).
% append.left_neutral
thf(fact_316_append_Oleft__neutral,axiom,
! [A: list_transition] :
( ( append_transition @ nil_transition @ A )
= A ) ).
% append.left_neutral
thf(fact_317_append_Oleft__neutral,axiom,
! [A: list_list_o] :
( ( append_list_o @ nil_list_o @ A )
= A ) ).
% append.left_neutral
thf(fact_318_append__Nil,axiom,
! [Ys: list_formula_a_t] :
( ( append_formula_a_t @ nil_formula_a_t @ Ys )
= Ys ) ).
% append_Nil
thf(fact_319_append__Nil,axiom,
! [Ys: list_o] :
( ( append_o @ nil_o @ Ys )
= Ys ) ).
% append_Nil
thf(fact_320_append__Nil,axiom,
! [Ys: list_transition] :
( ( append_transition @ nil_transition @ Ys )
= Ys ) ).
% append_Nil
thf(fact_321_append__Nil,axiom,
! [Ys: list_list_o] :
( ( append_list_o @ nil_list_o @ Ys )
= Ys ) ).
% append_Nil
thf(fact_322_step__eps__closure__set__refl,axiom,
! [R2: set_nat,Q0: nat,Transs: list_transition,Bs: list_o] : ( ord_less_eq_set_nat @ R2 @ ( step_eps_closure_set @ Q0 @ Transs @ R2 @ Bs ) ) ).
% step_eps_closure_set_refl
thf(fact_323_step__eps__closure__set__mono,axiom,
! [R2: set_nat,S2: set_nat,Q0: nat,Transs: list_transition,Bs: list_o] :
( ( ord_less_eq_set_nat @ R2 @ S2 )
=> ( ord_less_eq_set_nat @ ( step_eps_closure_set @ Q0 @ Transs @ R2 @ Bs ) @ ( step_eps_closure_set @ Q0 @ Transs @ S2 @ Bs ) ) ) ).
% step_eps_closure_set_mono
thf(fact_324_delta__eps,axiom,
! [Q0: nat,Transs: list_transition,R2: set_nat,Bs: list_o] :
( ( delta @ Q0 @ Transs @ ( step_eps_closure_set @ Q0 @ Transs @ R2 @ Bs ) @ Bs )
= ( delta @ Q0 @ Transs @ R2 @ Bs ) ) ).
% delta_eps
thf(fact_325_step__symb__set__mono,axiom,
! [R2: set_nat,S2: set_nat,Q0: nat,Transs: list_transition] :
( ( ord_less_eq_set_nat @ R2 @ S2 )
=> ( ord_less_eq_set_nat @ ( step_symb_set @ Q0 @ Transs @ R2 ) @ ( step_symb_set @ Q0 @ Transs @ S2 ) ) ) ).
% step_symb_set_mono
thf(fact_326_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_327_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_328_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_329_step__eps__closure__set__step__id,axiom,
! [R2: set_nat,Q0: nat,Transs: list_transition,Bs: list_o] :
( ! [Q3: nat,Q4: nat] :
( ( member_nat @ Q3 @ R2 )
=> ~ ( step_eps @ Q0 @ Transs @ Bs @ Q3 @ Q4 ) )
=> ( ( step_eps_closure_set @ Q0 @ Transs @ R2 @ Bs )
= R2 ) ) ).
% step_eps_closure_set_step_id
thf(fact_330_step__step__eps__closure,axiom,
! [Q0: nat,Transs: list_transition,Bs: list_o,Q: nat,Q2: nat,R2: set_nat] :
( ( step_eps @ Q0 @ Transs @ Bs @ Q @ Q2 )
=> ( ( member_nat @ Q @ R2 )
=> ( member_nat @ Q2 @ ( step_eps_closure_set @ Q0 @ Transs @ R2 @ Bs ) ) ) ) ).
% step_step_eps_closure
thf(fact_331_nfa__cong_H_Oq__SQ__SQ__nfa_H__SQ,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition,Q: nat] :
( ( nfa_cong2 @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( ( member_nat @ Q @ ( sq @ Q02 @ Transs2 ) )
=> ( ( member_nat @ Q @ ( sq @ Q0 @ Transs ) )
= ( member_nat @ Q @ ( sq @ Q02 @ Transs2 ) ) ) ) ) ).
% nfa_cong'.q_SQ_SQ_nfa'_SQ
thf(fact_332_nfa__cong_H_Oqf_H__in__SQ,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition] :
( ( nfa_cong2 @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( member_nat @ Qf2 @ ( sq @ Q0 @ Transs ) ) ) ).
% nfa_cong'.qf'_in_SQ
thf(fact_333_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_334_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_335_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 )
=> ( ! [Q4: nat] :
~ ( step_eps @ Q0 @ Transs @ Bs @ Q @ Q4 )
=> ( Q = Q2 ) ) ) ).
% step_eps_closure_empty
thf(fact_336_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_337_NFA_Oaccept__eps__def,axiom,
( accept_eps
= ( ^ [Q03: nat,Qf3: nat,Transs3: list_transition,R3: set_nat,Bs2: list_o] : ( member_nat @ Qf3 @ ( step_eps_closure_set @ Q03 @ Transs3 @ R3 @ Bs2 ) ) ) ) ).
% NFA.accept_eps_def
thf(fact_338_nfa__cong_H_Oaxioms_I1_J,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition] :
( ( nfa_cong2 @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( nfa @ Q0 @ Qf @ Transs ) ) ).
% nfa_cong'.axioms(1)
thf(fact_339_nfa__cong_H_Oaxioms_I2_J,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition] :
( ( nfa_cong2 @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( nfa @ Q02 @ Qf2 @ Transs2 ) ) ).
% nfa_cong'.axioms(2)
thf(fact_340_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_341_nfa__cong_Oaxioms_I1_J,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition] :
( ( nfa_cong @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( nfa @ Q0 @ Qf @ Transs ) ) ).
% nfa_cong.axioms(1)
thf(fact_342_nfa__cong_Oaxioms_I2_J,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition] :
( ( nfa_cong @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( nfa @ Q02 @ Qf2 @ Transs2 ) ) ).
% nfa_cong.axioms(2)
thf(fact_343_nfa__cong_H_Ostep__symb__cong__Q,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition,Q: nat,Q2: nat] :
( ( nfa_cong2 @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( ( step_symb @ Q02 @ Transs2 @ Q @ Q2 )
=> ( step_symb @ Q0 @ Transs @ Q @ Q2 ) ) ) ).
% nfa_cong'.step_symb_cong_Q
thf(fact_344_step__eps__accept__eps,axiom,
! [Q0: nat,Transs: list_transition,Bs: list_o,Q: nat,Qf: nat,R2: set_nat] :
( ( step_eps @ Q0 @ Transs @ Bs @ Q @ Qf )
=> ( ( member_nat @ Q @ R2 )
=> ( accept_eps @ Q0 @ Qf @ Transs @ R2 @ Bs ) ) ) ).
% step_eps_accept_eps
thf(fact_345_step__eps__set__mono,axiom,
! [R2: set_nat,S2: set_nat,Q0: nat,Transs: list_transition,Bs: list_o] :
( ( ord_less_eq_set_nat @ R2 @ S2 )
=> ( ord_less_eq_set_nat @ ( step_eps_set @ Q0 @ Transs @ Bs @ R2 ) @ ( step_eps_set @ Q0 @ Transs @ Bs @ S2 ) ) ) ).
% step_eps_set_mono
thf(fact_346_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_347_nfa__cong__Times_Oaxioms_I1_J,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Transs: list_transition,Transs2: list_transition,Transs4: list_transition] :
( ( nfa_cong_Times @ Q0 @ Q02 @ Qf @ Transs @ Transs2 @ Transs4 )
=> ( nfa_cong2 @ Q0 @ Q0 @ Qf @ Q02 @ Transs @ Transs2 ) ) ).
% nfa_cong_Times.axioms(1)
thf(fact_348_nfa__cong__Times_Oaxioms_I2_J,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Transs: list_transition,Transs2: list_transition,Transs4: list_transition] :
( ( nfa_cong_Times @ Q0 @ Q02 @ Qf @ Transs @ Transs2 @ Transs4 )
=> ( nfa_cong @ Q0 @ Q02 @ Qf @ Qf @ Transs @ Transs4 ) ) ).
% nfa_cong_Times.axioms(2)
thf(fact_349_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_350_delta__sub__eps__mono,axiom,
! [S2: set_nat,Q0: nat,Transs: list_transition,R2: set_nat,Bs: list_o] :
( ( ord_less_eq_set_nat @ S2 @ ( step_eps_closure_set @ Q0 @ Transs @ R2 @ Bs ) )
=> ( ord_less_eq_set_nat @ ( delta @ Q0 @ Transs @ S2 @ Bs ) @ ( delta @ Q0 @ Transs @ R2 @ Bs ) ) ) ).
% delta_sub_eps_mono
thf(fact_351_nfa__cong_H_Onfa_H__Q__sub__Q,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition] :
( ( nfa_cong2 @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( ord_less_eq_set_nat @ ( q @ Q02 @ Qf2 @ Transs2 ) @ ( q @ Q0 @ Qf @ Transs ) ) ) ).
% nfa_cong'.nfa'_Q_sub_Q
thf(fact_352_nfa__cong_H_OSQ__sub,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition] :
( ( nfa_cong2 @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( ord_less_eq_set_nat @ ( sq @ Q02 @ Transs2 ) @ ( sq @ Q0 @ Transs ) ) ) ).
% nfa_cong'.SQ_sub
thf(fact_353_nfa__cong_Oq__Q__SQ__nfa_H__SQ,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition,Q: nat] :
( ( nfa_cong @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( ( member_nat @ Q @ ( q @ Q02 @ Qf2 @ Transs2 ) )
=> ( ( member_nat @ Q @ ( sq @ Q0 @ Transs ) )
= ( member_nat @ Q @ ( sq @ Q02 @ Transs2 ) ) ) ) ) ).
% nfa_cong.q_Q_SQ_nfa'_SQ
thf(fact_354_nfa__cong_OSQ__sub,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition] :
( ( nfa_cong @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( ord_less_eq_set_nat @ ( sq @ Q02 @ Transs2 ) @ ( sq @ Q0 @ Transs ) ) ) ).
% nfa_cong.SQ_sub
thf(fact_355_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_356_NFA_Odelta__def,axiom,
( delta
= ( ^ [Q03: nat,Transs3: list_transition,R3: set_nat,Bs2: list_o] : ( step_symb_set @ Q03 @ Transs3 @ ( step_eps_closure_set @ Q03 @ Transs3 @ R3 @ Bs2 ) ) ) ) ).
% NFA.delta_def
thf(fact_357_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_358_nfa__cong_H_Onfa_H__step__eps__closure__cong,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition,Bs: list_o,Q: nat,Q2: nat] :
( ( nfa_cong2 @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( ( step_eps_closure @ Q02 @ Transs2 @ Bs @ Q @ Q2 )
=> ( ( member_nat @ Q @ ( q @ Q02 @ Qf2 @ Transs2 ) )
=> ( step_eps_closure @ Q0 @ Transs @ Bs @ Q @ Q2 ) ) ) ) ).
% nfa_cong'.nfa'_step_eps_closure_cong
thf(fact_359_nfa__cong_H_Oeps__nfa_H__step__eps__closure__cong,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition,Bs: list_o,Q: nat,Q2: nat] :
( ( nfa_cong2 @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( ( step_eps_closure @ Q0 @ Transs @ Bs @ Q @ Q2 )
=> ( ( member_nat @ Q @ ( q @ Q02 @ Qf2 @ Transs2 ) )
=> ( ( ( member_nat @ Q2 @ ( q @ Q02 @ Qf2 @ Transs2 ) )
& ( step_eps_closure @ Q02 @ Transs2 @ Bs @ Q @ Q2 ) )
| ( ( step_eps_closure @ Q02 @ Transs2 @ Bs @ Q @ Qf2 )
& ( step_eps_closure @ Q0 @ Transs @ Bs @ Qf2 @ Q2 ) ) ) ) ) ) ).
% nfa_cong'.eps_nfa'_step_eps_closure_cong
thf(fact_360_nfa__cong_H_Onfa_H__eps__step__eps__closure__cong,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition,Bs: list_o,Q: nat,Q2: nat] :
( ( nfa_cong2 @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( ( step_eps_closure @ Q02 @ Transs2 @ Bs @ Q @ Q2 )
=> ( ( member_nat @ Q @ ( q @ Q02 @ Qf2 @ Transs2 ) )
=> ( ( member_nat @ Q2 @ ( q @ Q02 @ Qf2 @ Transs2 ) )
& ( step_eps_closure @ Q0 @ Transs @ Bs @ Q @ Q2 ) ) ) ) ) ).
% nfa_cong'.nfa'_eps_step_eps_closure_cong
thf(fact_361_nfa__cong_H_Ostep__eps__cong__Q,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition,Q: nat,Bs: list_o,Q2: nat] :
( ( nfa_cong2 @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( ( member_nat @ Q @ ( q @ Q02 @ Qf2 @ Transs2 ) )
=> ( ( step_eps @ Q02 @ Transs2 @ Bs @ Q @ Q2 )
=> ( step_eps @ Q0 @ Transs @ Bs @ Q @ Q2 ) ) ) ) ).
% nfa_cong'.step_eps_cong_Q
thf(fact_362_nfa__cong_H_Ostep__symb__set__cong__Q,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition,R2: set_nat] :
( ( nfa_cong2 @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( ord_less_eq_set_nat @ ( step_symb_set @ Q02 @ Transs2 @ R2 ) @ ( step_symb_set @ Q0 @ Transs @ R2 ) ) ) ).
% nfa_cong'.step_symb_set_cong_Q
thf(fact_363_nfa__cong_Onfa_H__eps__step__eps__closure,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition,Bs: list_o,Q: nat,Q2: nat] :
( ( nfa_cong @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( ( step_eps_closure @ Q02 @ Transs2 @ Bs @ Q @ Q2 )
=> ( ( member_nat @ Q @ ( q @ Q02 @ Qf2 @ Transs2 ) )
=> ( ( member_nat @ Q2 @ ( q @ Q02 @ Qf2 @ Transs2 ) )
& ( step_eps_closure @ Q0 @ Transs @ Bs @ Q @ Q2 ) ) ) ) ) ).
% nfa_cong.nfa'_eps_step_eps_closure
thf(fact_364_nfa__cong_Oeps__nfa_H__step__eps__closure,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition,Bs: list_o,Q: nat,Q2: nat] :
( ( nfa_cong @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( ( step_eps_closure @ Q0 @ Transs @ Bs @ Q @ Q2 )
=> ( ( member_nat @ Q @ ( q @ Q02 @ Qf2 @ Transs2 ) )
=> ( ( member_nat @ Q2 @ ( q @ Q02 @ Qf2 @ Transs2 ) )
& ( step_eps_closure @ Q02 @ Transs2 @ Bs @ Q @ Q2 ) ) ) ) ) ).
% nfa_cong.eps_nfa'_step_eps_closure
thf(fact_365_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_366_nfa__cong_Ostep__eps__cong,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition,Q: nat,Bs: list_o,Q2: nat] :
( ( nfa_cong @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( ( member_nat @ Q @ ( q @ Q02 @ Qf2 @ Transs2 ) )
=> ( ( step_eps @ Q0 @ Transs @ Bs @ Q @ Q2 )
= ( step_eps @ Q02 @ Transs2 @ Bs @ Q @ Q2 ) ) ) ) ).
% nfa_cong.step_eps_cong
thf(fact_367_nfa__cong_H_Ostep__eps__cong__SQ,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition,Q: nat,Bs: list_o,Q2: nat] :
( ( nfa_cong2 @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( ( member_nat @ Q @ ( sq @ Q02 @ Transs2 ) )
=> ( ( step_eps @ Q0 @ Transs @ Bs @ Q @ Q2 )
= ( step_eps @ Q02 @ Transs2 @ Bs @ Q @ Q2 ) ) ) ) ).
% nfa_cong'.step_eps_cong_SQ
thf(fact_368_nfa__cong_Ostep__symb__cong,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition,Q: nat,Q2: nat] :
( ( nfa_cong @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( ( member_nat @ Q @ ( q @ Q02 @ Qf2 @ Transs2 ) )
=> ( ( step_symb @ Q0 @ Transs @ Q @ Q2 )
= ( step_symb @ Q02 @ Transs2 @ Q @ Q2 ) ) ) ) ).
% nfa_cong.step_symb_cong
thf(fact_369_nfa__cong_H_Ostep__symb__cong__SQ,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition,Q: nat,Q2: nat] :
( ( nfa_cong2 @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( ( member_nat @ Q @ ( sq @ Q02 @ Transs2 ) )
=> ( ( step_symb @ Q0 @ Transs @ Q @ Q2 )
= ( step_symb @ Q02 @ Transs2 @ Q @ Q2 ) ) ) ) ).
% nfa_cong'.step_symb_cong_SQ
thf(fact_370_step__eps__closure__set__mono_H,axiom,
! [Q0: nat,Transs: list_transition,R2: set_nat,Bs: list_o] : ( ord_less_eq_set_nat @ ( step_eps_closure_set @ Q0 @ Transs @ R2 @ nil_o ) @ ( step_eps_closure_set @ Q0 @ Transs @ R2 @ Bs ) ) ).
% step_eps_closure_set_mono'
thf(fact_371_nfa__cong__Times__def,axiom,
( nfa_cong_Times
= ( ^ [Q03: nat,Q04: nat,Qf3: nat,Transs3: list_transition,Transs5: list_transition,Transs6: list_transition] :
( ( nfa_cong2 @ Q03 @ Q03 @ Qf3 @ Q04 @ Transs3 @ Transs5 )
& ( nfa_cong @ Q03 @ Q04 @ Qf3 @ Qf3 @ Transs3 @ Transs6 ) ) ) ) ).
% nfa_cong_Times_def
thf(fact_372_nfa__cong__Times_Ointro,axiom,
! [Q0: nat,Qf: nat,Q02: nat,Transs: list_transition,Transs2: list_transition,Transs4: list_transition] :
( ( nfa_cong2 @ Q0 @ Q0 @ Qf @ Q02 @ Transs @ Transs2 )
=> ( ( nfa_cong @ Q0 @ Q02 @ Qf @ Qf @ Transs @ Transs4 )
=> ( nfa_cong_Times @ Q0 @ Q02 @ Qf @ Transs @ Transs2 @ Transs4 ) ) ) ).
% nfa_cong_Times.intro
thf(fact_373_nfa_Ostep__eps__closure__set__closed,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,R2: set_nat,Bs: list_o] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ( ( ord_less_eq_set_nat @ R2 @ ( q @ Q0 @ Qf @ Transs ) )
=> ( ord_less_eq_set_nat @ ( step_eps_closure_set @ Q0 @ Transs @ R2 @ Bs ) @ ( q @ Q0 @ Qf @ Transs ) ) ) ) ).
% nfa.step_eps_closure_set_closed
thf(fact_374_nfa_Odelta__closed,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,R2: set_nat,Bs: list_o] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ( ord_less_eq_set_nat @ ( delta @ Q0 @ Transs @ R2 @ Bs ) @ ( q @ Q0 @ Qf @ Transs ) ) ) ).
% nfa.delta_closed
thf(fact_375_nfa__cong_H_Ostep__eps__closure__set__cong__unreach,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition,R2: set_nat,Bs: list_o] :
( ( nfa_cong2 @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( ( ord_less_eq_set_nat @ R2 @ ( q @ Q02 @ Qf2 @ Transs2 ) )
=> ( ~ ( member_nat @ Qf2 @ ( step_eps_closure_set @ Q02 @ Transs2 @ R2 @ Bs ) )
=> ( ( step_eps_closure_set @ Q0 @ Transs @ R2 @ Bs )
= ( step_eps_closure_set @ Q02 @ Transs2 @ R2 @ Bs ) ) ) ) ) ).
% nfa_cong'.step_eps_closure_set_cong_unreach
thf(fact_376_nfa__cong_H_Onfa_H__step__eps__closure__set__sub,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition,R2: set_nat,Bs: list_o] :
( ( nfa_cong2 @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( ( ord_less_eq_set_nat @ R2 @ ( q @ Q02 @ Qf2 @ Transs2 ) )
=> ( ord_less_eq_set_nat @ ( step_eps_closure_set @ Q02 @ Transs2 @ R2 @ Bs ) @ ( step_eps_closure_set @ Q0 @ Transs @ R2 @ Bs ) ) ) ) ).
% nfa_cong'.nfa'_step_eps_closure_set_sub
thf(fact_377_nfa__cong_H_Onfa_H__delta__sub__delta,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition,R2: set_nat,Bs: list_o] :
( ( nfa_cong2 @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( ( ord_less_eq_set_nat @ R2 @ ( q @ Q02 @ Qf2 @ Transs2 ) )
=> ( ord_less_eq_set_nat @ ( delta @ Q02 @ Transs2 @ R2 @ Bs ) @ ( delta @ Q0 @ Transs @ R2 @ Bs ) ) ) ) ).
% nfa_cong'.nfa'_delta_sub_delta
thf(fact_378_nfa_Orun__closed,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,R2: set_nat,Bss: list_list_o] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ( ( ord_less_eq_set_nat @ R2 @ ( q @ Q0 @ Qf @ Transs ) )
=> ( ord_less_eq_set_nat @ ( run @ Q0 @ Transs @ R2 @ Bss ) @ ( q @ Q0 @ Qf @ Transs ) ) ) ) ).
% nfa.run_closed
thf(fact_379_nfa__cong_Ostep__eps__closure__set__cong,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition,R2: set_nat,Bs: list_o] :
( ( nfa_cong @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( ( ord_less_eq_set_nat @ R2 @ ( q @ Q02 @ Qf2 @ Transs2 ) )
=> ( ( step_eps_closure_set @ Q0 @ Transs @ R2 @ Bs )
= ( step_eps_closure_set @ Q02 @ Transs2 @ R2 @ Bs ) ) ) ) ).
% nfa_cong.step_eps_closure_set_cong
thf(fact_380_nfa__cong_Odelta__cong,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition,R2: set_nat,Bs: list_o] :
( ( nfa_cong @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( ( ord_less_eq_set_nat @ R2 @ ( q @ Q02 @ Qf2 @ Transs2 ) )
=> ( ( delta @ Q0 @ Transs @ R2 @ Bs )
= ( delta @ Q02 @ Transs2 @ R2 @ Bs ) ) ) ) ).
% nfa_cong.delta_cong
thf(fact_381_nfa_Ostep__symb__set__closed,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,R2: set_nat] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ( ord_less_eq_set_nat @ ( step_symb_set @ Q0 @ Transs @ R2 ) @ ( q @ Q0 @ Qf @ Transs ) ) ) ).
% nfa.step_symb_set_closed
thf(fact_382_nfa__cong_Orun__cong,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition,R2: set_nat,Bss: list_list_o] :
( ( nfa_cong @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( ( ord_less_eq_set_nat @ R2 @ ( q @ Q02 @ Qf2 @ Transs2 ) )
=> ( ( run @ Q0 @ Transs @ R2 @ Bss )
= ( run @ Q02 @ Transs2 @ R2 @ Bss ) ) ) ) ).
% nfa_cong.run_cong
thf(fact_383_nfa__cong_Ostep__symb__set__cong,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition,R2: set_nat] :
( ( nfa_cong @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( ( ord_less_eq_set_nat @ R2 @ ( q @ Q02 @ Qf2 @ Transs2 ) )
=> ( ( step_symb_set @ Q0 @ Transs @ R2 )
= ( step_symb_set @ Q02 @ Transs2 @ R2 ) ) ) ) ).
% nfa_cong.step_symb_set_cong
thf(fact_384_nfa__cong_H_Ostep__symb__set__cong__SQ,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition,R2: set_nat] :
( ( nfa_cong2 @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( ( ord_less_eq_set_nat @ R2 @ ( sq @ Q02 @ Transs2 ) )
=> ( ( step_symb_set @ Q0 @ Transs @ R2 )
= ( step_symb_set @ Q02 @ Transs2 @ R2 ) ) ) ) ).
% nfa_cong'.step_symb_set_cong_SQ
thf(fact_385_nfa__cong_Oaccept__eps__cong,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition,R2: set_nat,Bs: list_o] :
( ( nfa_cong @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( ( ord_less_eq_set_nat @ R2 @ ( q @ Q02 @ Qf2 @ Transs2 ) )
=> ( ( accept_eps @ Q0 @ Qf @ Transs @ R2 @ Bs )
= ( accept_eps @ Q02 @ Qf2 @ Transs2 @ R2 @ Bs ) ) ) ) ).
% nfa_cong.accept_eps_cong
thf(fact_386_nfa_Ostep__eps__set__closed,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,Bs: list_o,R2: set_nat] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ( ord_less_eq_set_nat @ ( step_eps_set @ Q0 @ Transs @ Bs @ R2 ) @ ( q @ Q0 @ Qf @ Transs ) ) ) ).
% nfa.step_eps_set_closed
thf(fact_387_nth__append__length__plus,axiom,
! [Xs: list_transition,Ys: list_transition,N: nat] :
( ( nth_transition @ ( append_transition @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_s3613142680436377136sition @ Xs ) @ N ) )
= ( nth_transition @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_388_nth__append__length__plus,axiom,
! [Xs: list_list_o,Ys: list_list_o,N: nat] :
( ( nth_list_o @ ( append_list_o @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_s2710708370519433104list_o @ Xs ) @ N ) )
= ( nth_list_o @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_389_nth__append__length__plus,axiom,
! [Xs: list_formula_a_t,Ys: list_formula_a_t,N: nat] :
( ( nth_formula_a_t @ ( append_formula_a_t @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_s8846756101701226951la_a_t @ Xs ) @ N ) )
= ( nth_formula_a_t @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_390_nth__append__length__plus,axiom,
! [Xs: list_o,Ys: list_o,N: nat] :
( ( nth_o @ ( append_o @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_o @ Xs ) @ N ) )
= ( nth_o @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_391_nth__append__length,axiom,
! [Xs: list_transition,X2: transition,Ys: list_transition] :
( ( nth_transition @ ( append_transition @ Xs @ ( cons_transition @ X2 @ Ys ) ) @ ( size_s3613142680436377136sition @ Xs ) )
= X2 ) ).
% nth_append_length
thf(fact_392_nth__append__length,axiom,
! [Xs: list_list_o,X2: list_o,Ys: list_list_o] :
( ( nth_list_o @ ( append_list_o @ Xs @ ( cons_list_o @ X2 @ Ys ) ) @ ( size_s2710708370519433104list_o @ Xs ) )
= X2 ) ).
% nth_append_length
thf(fact_393_nth__append__length,axiom,
! [Xs: list_formula_a_t,X2: formula_a_t,Ys: list_formula_a_t] :
( ( nth_formula_a_t @ ( append_formula_a_t @ Xs @ ( cons_formula_a_t @ X2 @ Ys ) ) @ ( size_s8846756101701226951la_a_t @ Xs ) )
= X2 ) ).
% nth_append_length
thf(fact_394_nth__append__length,axiom,
! [Xs: list_o,X2: $o,Ys: list_o] :
( ( nth_o @ ( append_o @ Xs @ ( cons_o @ X2 @ Ys ) ) @ ( size_size_list_o @ Xs ) )
= X2 ) ).
% nth_append_length
thf(fact_395_set__append,axiom,
! [Xs: list_formula_a_t,Ys: list_formula_a_t] :
( ( set_formula_a_t2 @ ( append_formula_a_t @ Xs @ Ys ) )
= ( sup_su887667473539889925la_a_t @ ( set_formula_a_t2 @ Xs ) @ ( set_formula_a_t2 @ Ys ) ) ) ).
% set_append
thf(fact_396_set__append,axiom,
! [Xs: list_transition,Ys: list_transition] :
( ( set_transition2 @ ( append_transition @ Xs @ Ys ) )
= ( sup_su812053455038985074sition @ ( set_transition2 @ Xs ) @ ( set_transition2 @ Ys ) ) ) ).
% set_append
thf(fact_397_set__append,axiom,
! [Xs: list_list_o,Ys: list_list_o] :
( ( set_list_o2 @ ( append_list_o @ Xs @ Ys ) )
= ( sup_sup_set_list_o @ ( set_list_o2 @ Xs ) @ ( set_list_o2 @ Ys ) ) ) ).
% set_append
thf(fact_398_set__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( set_nat2 @ ( append_nat @ Xs @ Ys ) )
= ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) ) ).
% set_append
thf(fact_399_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_400_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_401_sup_Obounded__iff,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
= ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_402_le__sup__iff,axiom,
! [X2: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X2 @ Y ) @ Z2 )
= ( ( ord_less_eq_set_nat @ X2 @ Z2 )
& ( ord_less_eq_set_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_403_le__sup__iff,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ Y ) @ Z2 )
= ( ( ord_less_eq_nat @ X2 @ Z2 )
& ( ord_less_eq_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_404_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_405_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_406_cong_Oaccept__eps__nfa_H__run,axiom,
! [R2: set_nat,Bss: list_list_o,Bs: list_o] :
( ( ord_less_eq_set_nat @ R2 @ ( q @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_l ) )
=> ( ( accept_eps @ q0a @ qfa @ transsa @ ( run @ q0a @ ts_l @ R2 @ Bss ) @ Bs )
= ( ( accept_eps @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_l @ ( run @ q0a @ ts_l @ R2 @ Bss ) @ Bs )
& ( accept_eps @ q0a @ qfa @ transsa @ ( run @ q0a @ transsa @ ( insert_nat @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ bot_bot_set_nat ) @ nil_list_o ) @ Bs ) ) ) ) ).
% cong.accept_eps_nfa'_run
thf(fact_407_cong_Odelta__cong__reach,axiom,
! [R2: set_nat,Bs: list_o] :
( ( ord_less_eq_set_nat @ R2 @ ( q @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_l ) )
=> ( ( accept_eps @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_l @ R2 @ Bs )
=> ( ( delta @ q0a @ transsa @ R2 @ Bs )
= ( sup_sup_set_nat @ ( delta @ q0a @ ts_l @ R2 @ Bs ) @ ( delta @ q0a @ transsa @ ( insert_nat @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ bot_bot_set_nat ) @ Bs ) ) ) ) ) ).
% cong.delta_cong_reach
thf(fact_408_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_409_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_410_empty__iff,axiom,
! [C: transition] :
~ ( member_transition @ C @ bot_bo301567166201926666sition ) ).
% empty_iff
thf(fact_411_empty__iff,axiom,
! [C: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ C @ bot_bo2099793752762293965at_nat ) ).
% empty_iff
thf(fact_412_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_413_all__not__in__conv,axiom,
! [A2: set_transition] :
( ( ! [X: transition] :
~ ( member_transition @ X @ A2 ) )
= ( A2 = bot_bo301567166201926666sition ) ) ).
% all_not_in_conv
thf(fact_414_all__not__in__conv,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ( ! [X: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ X @ A2 ) )
= ( A2 = bot_bo2099793752762293965at_nat ) ) ).
% all_not_in_conv
thf(fact_415_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_416_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_417_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_418_subsetI,axiom,
! [A2: set_transition,B2: set_transition] :
( ! [X3: transition] :
( ( member_transition @ X3 @ A2 )
=> ( member_transition @ X3 @ B2 ) )
=> ( ord_le8419162016481440574sition @ A2 @ B2 ) ) ).
% subsetI
thf(fact_419_subsetI,axiom,
! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ A2 )
=> ( member8440522571783428010at_nat @ X3 @ B2 ) )
=> ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_420_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ X3 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_421_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_422_insertCI,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( ~ ( member_nat @ A @ B2 )
=> ( A = B ) )
=> ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_423_insertCI,axiom,
! [A: transition,B2: set_transition,B: transition] :
( ( ~ ( member_transition @ A @ B2 )
=> ( A = B ) )
=> ( member_transition @ A @ ( insert_transition @ B @ B2 ) ) ) ).
% insertCI
thf(fact_424_insertCI,axiom,
! [A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
( ( ~ ( member8440522571783428010at_nat @ A @ B2 )
=> ( A = B ) )
=> ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_425_insert__iff,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
= ( ( A = B )
| ( member_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_426_insert__iff,axiom,
! [A: transition,B: transition,A2: set_transition] :
( ( member_transition @ A @ ( insert_transition @ B @ A2 ) )
= ( ( A = B )
| ( member_transition @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_427_insert__iff,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ A2 ) )
= ( ( A = B )
| ( member8440522571783428010at_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_428_insert__absorb2,axiom,
! [X2: nat,A2: set_nat] :
( ( insert_nat @ X2 @ ( insert_nat @ X2 @ A2 ) )
= ( insert_nat @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_429_sup_Oidem,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_430_sup__idem,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_431_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_432_sup__left__idem,axiom,
! [X2: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y ) )
= ( sup_sup_set_nat @ X2 @ Y ) ) ).
% sup_left_idem
thf(fact_433_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_434_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_435_UnCI,axiom,
! [C: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
( ( ~ ( member8440522571783428010at_nat @ C @ B2 )
=> ( member8440522571783428010at_nat @ C @ A2 ) )
=> ( member8440522571783428010at_nat @ C @ ( sup_su6327502436637775413at_nat @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_436_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_437_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_438_Un__iff,axiom,
! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ C @ ( sup_su6327502436637775413at_nat @ A2 @ B2 ) )
= ( ( member8440522571783428010at_nat @ C @ A2 )
| ( member8440522571783428010at_nat @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_439_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_440_left_Oq0__sub__SQ,axiom,
ord_less_eq_set_nat @ ( insert_nat @ q0a @ bot_bot_set_nat ) @ ( sq @ q0a @ ts_l ) ).
% left.q0_sub_SQ
thf(fact_441_base_Oq0__sub__SQ,axiom,
ord_less_eq_set_nat @ ( insert_nat @ q0a @ bot_bot_set_nat ) @ ( sq @ q0a @ transsa ) ).
% base.q0_sub_SQ
thf(fact_442_base_Ostep__eps__closure__set__qf,axiom,
! [Bs: list_o] :
( ( step_eps_closure_set @ q0a @ transsa @ ( insert_nat @ qfa @ bot_bot_set_nat ) @ Bs )
= ( insert_nat @ qfa @ bot_bot_set_nat ) ) ).
% base.step_eps_closure_set_qf
thf(fact_443_base_Odelta__qf,axiom,
! [Bs: list_o] :
( ( delta @ q0a @ transsa @ ( insert_nat @ qfa @ bot_bot_set_nat ) @ Bs )
= bot_bot_set_nat ) ).
% base.delta_qf
thf(fact_444_base_Ostep__symb__set__qf,axiom,
( ( step_symb_set @ q0a @ transsa @ ( insert_nat @ qfa @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ).
% base.step_symb_set_qf
thf(fact_445_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_446_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_447_singletonI,axiom,
! [A: transition] : ( member_transition @ A @ ( insert_transition @ A @ bot_bo301567166201926666sition ) ) ).
% singletonI
thf(fact_448_singletonI,axiom,
! [A: product_prod_nat_nat] : ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).
% singletonI
thf(fact_449_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_450_sup__bot__left,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_451_sup__bot__right,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ bot_bot_set_nat )
= X2 ) ).
% sup_bot_right
thf(fact_452_bot__eq__sup__iff,axiom,
! [X2: set_nat,Y: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X2 @ Y ) )
= ( ( X2 = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_453_sup__eq__bot__iff,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ( sup_sup_set_nat @ X2 @ Y )
= bot_bot_set_nat )
= ( ( X2 = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_454_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_455_sup__bot_Oleft__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_456_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_457_sup__bot_Oright__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_458_insert__subset,axiom,
! [X2: transition,A2: set_transition,B2: set_transition] :
( ( ord_le8419162016481440574sition @ ( insert_transition @ X2 @ A2 ) @ B2 )
= ( ( member_transition @ X2 @ B2 )
& ( ord_le8419162016481440574sition @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_459_insert__subset,axiom,
! [X2: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ X2 @ A2 ) @ B2 )
= ( ( member8440522571783428010at_nat @ X2 @ B2 )
& ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_460_insert__subset,axiom,
! [X2: nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ A2 ) @ B2 )
= ( ( member_nat @ X2 @ B2 )
& ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_461_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_462_Un__insert__left,axiom,
! [A: nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( insert_nat @ A @ B2 ) @ C2 )
= ( insert_nat @ A @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_463_Un__insert__right,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( insert_nat @ A @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_464_base_Oq0__sub__Q,axiom,
ord_less_eq_set_nat @ ( insert_nat @ q0a @ bot_bot_set_nat ) @ ( q @ q0a @ qfa @ transsa ) ).
% base.q0_sub_Q
thf(fact_465_base_Orun__accept__eps__qf__many,axiom,
! [Bs: list_o,Bss: list_list_o,Cs: list_o] :
~ ( run_accept_eps @ q0a @ qfa @ transsa @ ( insert_nat @ qfa @ bot_bot_set_nat ) @ ( cons_list_o @ Bs @ Bss ) @ Cs ) ).
% base.run_accept_eps_qf_many
thf(fact_466_base_Orun__qf__many,axiom,
! [Bs: list_o,Bss: list_list_o] :
( ( run @ q0a @ transsa @ ( insert_nat @ qfa @ bot_bot_set_nat ) @ ( cons_list_o @ Bs @ Bss ) )
= bot_bot_set_nat ) ).
% base.run_qf_many
thf(fact_467_base_Orun__accept__eps__qf__one,axiom,
! [Bs: list_o] : ( run_accept_eps @ q0a @ qfa @ transsa @ ( insert_nat @ qfa @ bot_bot_set_nat ) @ nil_list_o @ Bs ) ).
% base.run_accept_eps_qf_one
thf(fact_468_left_Ostep__eps__closure__set__qf,axiom,
! [Bs: list_o] :
( ( step_eps_closure_set @ q0a @ ts_l @ ( insert_nat @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ bot_bot_set_nat ) @ Bs )
= ( insert_nat @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ bot_bot_set_nat ) ) ).
% left.step_eps_closure_set_qf
thf(fact_469_left_Odelta__qf,axiom,
! [Bs: list_o] :
( ( delta @ q0a @ ts_l @ ( insert_nat @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ bot_bot_set_nat ) @ Bs )
= bot_bot_set_nat ) ).
% left.delta_qf
thf(fact_470_left_Ostep__symb__set__qf,axiom,
( ( step_symb_set @ q0a @ ts_l @ ( insert_nat @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ).
% left.step_symb_set_qf
thf(fact_471_left_Oq0__sub__Q,axiom,
ord_less_eq_set_nat @ ( insert_nat @ q0a @ bot_bot_set_nat ) @ ( q @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_l ) ).
% left.q0_sub_Q
thf(fact_472_right_Oq0__sub__SQ,axiom,
ord_less_eq_set_nat @ ( insert_nat @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ bot_bot_set_nat ) @ ( sq @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_r ) ).
% right.q0_sub_SQ
thf(fact_473_left_Orun__accept__eps__qf__many,axiom,
! [Bs: list_o,Bss: list_list_o,Cs: list_o] :
~ ( run_accept_eps @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_l @ ( insert_nat @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ bot_bot_set_nat ) @ ( cons_list_o @ Bs @ Bss ) @ Cs ) ).
% left.run_accept_eps_qf_many
thf(fact_474_left_Orun__qf__many,axiom,
! [Bs: list_o,Bss: list_list_o] :
( ( run @ q0a @ ts_l @ ( insert_nat @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ bot_bot_set_nat ) @ ( cons_list_o @ Bs @ Bss ) )
= bot_bot_set_nat ) ).
% left.run_qf_many
thf(fact_475_left_Orun__accept__eps__qf__one,axiom,
! [Bs: list_o] : ( run_accept_eps @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_l @ ( insert_nat @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ bot_bot_set_nat ) @ nil_list_o @ Bs ) ).
% left.run_accept_eps_qf_one
thf(fact_476_right_Ostep__eps__closure__set__qf,axiom,
! [Bs: list_o] :
( ( step_eps_closure_set @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_r @ ( insert_nat @ qfa @ bot_bot_set_nat ) @ Bs )
= ( insert_nat @ qfa @ bot_bot_set_nat ) ) ).
% right.step_eps_closure_set_qf
thf(fact_477_right_Odelta__qf,axiom,
! [Bs: list_o] :
( ( delta @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_r @ ( insert_nat @ qfa @ bot_bot_set_nat ) @ Bs )
= bot_bot_set_nat ) ).
% right.delta_qf
thf(fact_478_singleton__insert__inj__eq_H,axiom,
! [A: nat,A2: set_nat,B: nat] :
( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B @ bot_bot_set_nat ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_479_singleton__insert__inj__eq,axiom,
! [B: nat,A: nat,A2: set_nat] :
( ( ( insert_nat @ B @ bot_bot_set_nat )
= ( insert_nat @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_480_right_Ostep__symb__set__qf,axiom,
( ( step_symb_set @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_r @ ( insert_nat @ qfa @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ).
% right.step_symb_set_qf
thf(fact_481_set__empty,axiom,
! [Xs: list_o] :
( ( ( set_o2 @ Xs )
= bot_bot_set_o )
= ( Xs = nil_o ) ) ).
% set_empty
thf(fact_482_set__empty,axiom,
! [Xs: list_transition] :
( ( ( set_transition2 @ Xs )
= bot_bo301567166201926666sition )
= ( Xs = nil_transition ) ) ).
% set_empty
thf(fact_483_set__empty,axiom,
! [Xs: list_list_o] :
( ( ( set_list_o2 @ Xs )
= bot_bot_set_list_o )
= ( Xs = nil_list_o ) ) ).
% set_empty
thf(fact_484_set__empty,axiom,
! [Xs: list_nat] :
( ( ( set_nat2 @ Xs )
= bot_bot_set_nat )
= ( Xs = nil_nat ) ) ).
% set_empty
thf(fact_485_set__empty2,axiom,
! [Xs: list_o] :
( ( bot_bot_set_o
= ( set_o2 @ Xs ) )
= ( Xs = nil_o ) ) ).
% set_empty2
thf(fact_486_set__empty2,axiom,
! [Xs: list_transition] :
( ( bot_bo301567166201926666sition
= ( set_transition2 @ Xs ) )
= ( Xs = nil_transition ) ) ).
% set_empty2
thf(fact_487_set__empty2,axiom,
! [Xs: list_list_o] :
( ( bot_bot_set_list_o
= ( set_list_o2 @ Xs ) )
= ( Xs = nil_list_o ) ) ).
% set_empty2
thf(fact_488_set__empty2,axiom,
! [Xs: list_nat] :
( ( bot_bot_set_nat
= ( set_nat2 @ Xs ) )
= ( Xs = nil_nat ) ) ).
% set_empty2
thf(fact_489_list_Osimps_I15_J,axiom,
! [X21: nat,X22: list_nat] :
( ( set_nat2 @ ( cons_nat @ X21 @ X22 ) )
= ( insert_nat @ X21 @ ( set_nat2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_490_list_Osimps_I15_J,axiom,
! [X21: transition,X22: list_transition] :
( ( set_transition2 @ ( cons_transition @ X21 @ X22 ) )
= ( insert_transition @ X21 @ ( set_transition2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_491_list_Osimps_I15_J,axiom,
! [X21: list_o,X22: list_list_o] :
( ( set_list_o2 @ ( cons_list_o @ X21 @ X22 ) )
= ( insert_list_o @ X21 @ ( set_list_o2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_492_right_Oq0__sub__Q,axiom,
ord_less_eq_set_nat @ ( insert_nat @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ bot_bot_set_nat ) @ ( q @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ qfa @ ts_r ) ).
% right.q0_sub_Q
thf(fact_493_right_Orun__accept__eps__qf__many,axiom,
! [Bs: list_o,Bss: list_list_o,Cs: list_o] :
~ ( run_accept_eps @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ qfa @ ts_r @ ( insert_nat @ qfa @ bot_bot_set_nat ) @ ( cons_list_o @ Bs @ Bss ) @ Cs ) ).
% right.run_accept_eps_qf_many
thf(fact_494_right_Orun__qf__many,axiom,
! [Bs: list_o,Bss: list_list_o] :
( ( run @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_r @ ( insert_nat @ qfa @ bot_bot_set_nat ) @ ( cons_list_o @ Bs @ Bss ) )
= bot_bot_set_nat ) ).
% right.run_qf_many
thf(fact_495_right_Orun__accept__eps__qf__one,axiom,
! [Bs: list_o] : ( run_accept_eps @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ qfa @ ts_r @ ( insert_nat @ qfa @ bot_bot_set_nat ) @ nil_list_o @ Bs ) ).
% right.run_accept_eps_qf_one
thf(fact_496_cong_Ostep__eps__closure__set__cong__reach,axiom,
! [R2: set_nat,Bs: list_o] :
( ( ord_less_eq_set_nat @ R2 @ ( q @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_l ) )
=> ( ( member_nat @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ( step_eps_closure_set @ q0a @ ts_l @ R2 @ Bs ) )
=> ( ( step_eps_closure_set @ q0a @ transsa @ R2 @ Bs )
= ( sup_sup_set_nat @ ( step_eps_closure_set @ q0a @ ts_l @ R2 @ Bs ) @ ( step_eps_closure_set @ q0a @ transsa @ ( insert_nat @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ bot_bot_set_nat ) @ Bs ) ) ) ) ) ).
% cong.step_eps_closure_set_cong_reach
thf(fact_497_singleton__Un__iff,axiom,
! [X2: nat,A2: set_nat,B2: set_nat] :
( ( ( insert_nat @ X2 @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( ( A2 = bot_bot_set_nat )
& ( B2
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
| ( ( A2
= ( insert_nat @ X2 @ bot_bot_set_nat ) )
& ( B2 = bot_bot_set_nat ) )
| ( ( A2
= ( insert_nat @ X2 @ bot_bot_set_nat ) )
& ( B2
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_498_Un__singleton__iff,axiom,
! [A2: set_nat,B2: set_nat,X2: nat] :
( ( ( sup_sup_set_nat @ A2 @ B2 )
= ( insert_nat @ X2 @ bot_bot_set_nat ) )
= ( ( ( A2 = bot_bot_set_nat )
& ( B2
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
| ( ( A2
= ( insert_nat @ X2 @ bot_bot_set_nat ) )
& ( B2 = bot_bot_set_nat ) )
| ( ( A2
= ( insert_nat @ X2 @ bot_bot_set_nat ) )
& ( B2
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_499_insert__is__Un,axiom,
( insert_nat
= ( ^ [A4: nat] : ( sup_sup_set_nat @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_500_subset__singleton__iff,axiom,
! [X4: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X4 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( ( X4 = bot_bot_set_nat )
| ( X4
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_501_subset__singletonD,axiom,
! [A2: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
=> ( ( A2 = bot_bot_set_nat )
| ( A2
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_502_emptyE,axiom,
! [A: transition] :
~ ( member_transition @ A @ bot_bo301567166201926666sition ) ).
% emptyE
thf(fact_503_emptyE,axiom,
! [A: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ A @ bot_bo2099793752762293965at_nat ) ).
% emptyE
thf(fact_504_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_505_insertE,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
=> ( ( A != B )
=> ( member_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_506_insertE,axiom,
! [A: transition,B: transition,A2: set_transition] :
( ( member_transition @ A @ ( insert_transition @ B @ A2 ) )
=> ( ( A != B )
=> ( member_transition @ A @ A2 ) ) ) ).
% insertE
thf(fact_507_insertE,axiom,
! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ A2 ) )
=> ( ( A != B )
=> ( member8440522571783428010at_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_508_equals0D,axiom,
! [A2: set_transition,A: transition] :
( ( A2 = bot_bo301567166201926666sition )
=> ~ ( member_transition @ A @ A2 ) ) ).
% equals0D
thf(fact_509_equals0D,axiom,
! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
( ( A2 = bot_bo2099793752762293965at_nat )
=> ~ ( member8440522571783428010at_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_510_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_511_equals0I,axiom,
! [A2: set_transition] :
( ! [Y2: transition] :
~ ( member_transition @ Y2 @ A2 )
=> ( A2 = bot_bo301567166201926666sition ) ) ).
% equals0I
thf(fact_512_equals0I,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ! [Y2: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ Y2 @ A2 )
=> ( A2 = bot_bo2099793752762293965at_nat ) ) ).
% equals0I
thf(fact_513_equals0I,axiom,
! [A2: set_nat] :
( ! [Y2: nat] :
~ ( member_nat @ Y2 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_514_insertI1,axiom,
! [A: nat,B2: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B2 ) ) ).
% insertI1
thf(fact_515_insertI1,axiom,
! [A: transition,B2: set_transition] : ( member_transition @ A @ ( insert_transition @ A @ B2 ) ) ).
% insertI1
thf(fact_516_insertI1,axiom,
! [A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] : ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ A @ B2 ) ) ).
% insertI1
thf(fact_517_insertI2,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( member_nat @ A @ B2 )
=> ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_518_insertI2,axiom,
! [A: transition,B2: set_transition,B: transition] :
( ( member_transition @ A @ B2 )
=> ( member_transition @ A @ ( insert_transition @ B @ B2 ) ) ) ).
% insertI2
thf(fact_519_insertI2,axiom,
! [A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A @ B2 )
=> ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_520_ex__in__conv,axiom,
! [A2: set_transition] :
( ( ? [X: transition] : ( member_transition @ X @ A2 ) )
= ( A2 != bot_bo301567166201926666sition ) ) ).
% ex_in_conv
thf(fact_521_ex__in__conv,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ( ? [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A2 ) )
= ( A2 != bot_bo2099793752762293965at_nat ) ) ).
% ex_in_conv
thf(fact_522_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X: nat] : ( member_nat @ X @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_523_Set_Oset__insert,axiom,
! [X2: nat,A2: set_nat] :
( ( member_nat @ X2 @ A2 )
=> ~ ! [B3: set_nat] :
( ( A2
= ( insert_nat @ X2 @ B3 ) )
=> ( member_nat @ X2 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_524_Set_Oset__insert,axiom,
! [X2: transition,A2: set_transition] :
( ( member_transition @ X2 @ A2 )
=> ~ ! [B3: set_transition] :
( ( A2
= ( insert_transition @ X2 @ B3 ) )
=> ( member_transition @ X2 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_525_Set_Oset__insert,axiom,
! [X2: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ X2 @ A2 )
=> ~ ! [B3: set_Pr1261947904930325089at_nat] :
( ( A2
= ( insert8211810215607154385at_nat @ X2 @ B3 ) )
=> ( member8440522571783428010at_nat @ X2 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_526_singletonD,axiom,
! [B: transition,A: transition] :
( ( member_transition @ B @ ( insert_transition @ A @ bot_bo301567166201926666sition ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_527_singletonD,axiom,
! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_528_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_529_insert__ident,axiom,
! [X2: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X2 @ A2 )
=> ( ~ ( member_nat @ X2 @ B2 )
=> ( ( ( insert_nat @ X2 @ A2 )
= ( insert_nat @ X2 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_530_insert__ident,axiom,
! [X2: transition,A2: set_transition,B2: set_transition] :
( ~ ( member_transition @ X2 @ A2 )
=> ( ~ ( member_transition @ X2 @ B2 )
=> ( ( ( insert_transition @ X2 @ A2 )
= ( insert_transition @ X2 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_531_insert__ident,axiom,
! [X2: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
( ~ ( member8440522571783428010at_nat @ X2 @ A2 )
=> ( ~ ( member8440522571783428010at_nat @ X2 @ B2 )
=> ( ( ( insert8211810215607154385at_nat @ X2 @ A2 )
= ( insert8211810215607154385at_nat @ X2 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_532_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_nat @ Qf @ bot_bot_set_nat ) @ nil_list_o @ Bs ) ) ).
% nfa.run_accept_eps_qf_one
thf(fact_533_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_nat @ Qf @ bot_bot_set_nat ) @ Bs )
= ( insert_nat @ Qf @ bot_bot_set_nat ) ) ) ).
% nfa.step_eps_closure_set_qf
thf(fact_534_nfa_Odelta__qf,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,Bs: list_o] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ( ( delta @ Q0 @ Transs @ ( insert_nat @ Qf @ bot_bot_set_nat ) @ Bs )
= bot_bot_set_nat ) ) ).
% nfa.delta_qf
thf(fact_535_nfa_Ostep__symb__set__qf,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ( ( step_symb_set @ Q0 @ Transs @ ( insert_nat @ Qf @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ) ).
% nfa.step_symb_set_qf
thf(fact_536_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_537_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_538_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_539_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_540_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_541_nfa_Oq0__sub__Q,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ( ord_less_eq_set_nat @ ( insert_nat @ Q0 @ bot_bot_set_nat ) @ ( q @ Q0 @ Qf @ Transs ) ) ) ).
% nfa.q0_sub_Q
thf(fact_542_nfa_Oq0__sub__SQ,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ( ord_less_eq_set_nat @ ( insert_nat @ Q0 @ bot_bot_set_nat ) @ ( sq @ Q0 @ Transs ) ) ) ).
% nfa.q0_sub_SQ
thf(fact_543_NFA_OQ__def,axiom,
( q
= ( ^ [Q03: nat,Qf3: nat,Transs3: list_transition] : ( sup_sup_set_nat @ ( sq @ Q03 @ Transs3 ) @ ( insert_nat @ Qf3 @ bot_bot_set_nat ) ) ) ) ).
% NFA.Q_def
thf(fact_544_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_nat @ Qf @ bot_bot_set_nat ) @ ( cons_list_o @ Bs @ Bss ) @ Cs ) ) ).
% nfa.run_accept_eps_qf_many
thf(fact_545_step__eps__closure__set__unfold,axiom,
! [Q0: nat,Transs: list_transition,Bs: list_o,Q: nat,X4: set_nat] :
( ! [Q4: nat] :
( ( step_eps @ Q0 @ Transs @ Bs @ Q @ Q4 )
= ( member_nat @ Q4 @ X4 ) )
=> ( ( step_eps_closure_set @ Q0 @ Transs @ ( insert_nat @ Q @ bot_bot_set_nat ) @ Bs )
= ( sup_sup_set_nat @ ( insert_nat @ Q @ bot_bot_set_nat ) @ ( step_eps_closure_set @ Q0 @ Transs @ X4 @ Bs ) ) ) ) ).
% step_eps_closure_set_unfold
thf(fact_546_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_nat @ Qf @ bot_bot_set_nat ) @ ( cons_list_o @ Bs @ Bss ) )
= bot_bot_set_nat ) ) ).
% nfa.run_qf_many
thf(fact_547_nfa__cong_H_Ostep__eps__closure__set__cong__reach,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition,R2: set_nat,Bs: list_o] :
( ( nfa_cong2 @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( ( ord_less_eq_set_nat @ R2 @ ( q @ Q02 @ Qf2 @ Transs2 ) )
=> ( ( member_nat @ Qf2 @ ( step_eps_closure_set @ Q02 @ Transs2 @ R2 @ Bs ) )
=> ( ( step_eps_closure_set @ Q0 @ Transs @ R2 @ Bs )
= ( sup_sup_set_nat @ ( step_eps_closure_set @ Q02 @ Transs2 @ R2 @ Bs ) @ ( step_eps_closure_set @ Q0 @ Transs @ ( insert_nat @ Qf2 @ bot_bot_set_nat ) @ Bs ) ) ) ) ) ) ).
% nfa_cong'.step_eps_closure_set_cong_reach
thf(fact_548_nfa__cong_H_Oaccept__eps__nfa_H__run,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition,R2: set_nat,Bss: list_list_o,Bs: list_o] :
( ( nfa_cong2 @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( ( ord_less_eq_set_nat @ R2 @ ( q @ Q02 @ Qf2 @ Transs2 ) )
=> ( ( accept_eps @ Q0 @ Qf @ Transs @ ( run @ Q02 @ Transs2 @ R2 @ Bss ) @ Bs )
= ( ( accept_eps @ Q02 @ Qf2 @ Transs2 @ ( run @ Q02 @ Transs2 @ R2 @ Bss ) @ Bs )
& ( accept_eps @ Q0 @ Qf @ Transs @ ( run @ Q0 @ Transs @ ( insert_nat @ Qf2 @ bot_bot_set_nat ) @ nil_list_o ) @ Bs ) ) ) ) ) ).
% nfa_cong'.accept_eps_nfa'_run
thf(fact_549_nfa__cong_H_Odelta__cong__reach,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition,R2: set_nat,Bs: list_o] :
( ( nfa_cong2 @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( ( ord_less_eq_set_nat @ R2 @ ( q @ Q02 @ Qf2 @ Transs2 ) )
=> ( ( accept_eps @ Q02 @ Qf2 @ Transs2 @ R2 @ Bs )
=> ( ( delta @ Q0 @ Transs @ R2 @ Bs )
= ( sup_sup_set_nat @ ( delta @ Q02 @ Transs2 @ R2 @ Bs ) @ ( delta @ Q0 @ Transs @ ( insert_nat @ Qf2 @ bot_bot_set_nat ) @ Bs ) ) ) ) ) ) ).
% nfa_cong'.delta_cong_reach
thf(fact_550_run__accept__eps__Nil__eps__split,axiom,
! [Q0: nat,Transs: list_transition,R2: set_nat,Bs: list_o,S2: set_nat,Qf: nat] :
( ( ( step_eps_closure_set @ Q0 @ Transs @ R2 @ Bs )
= ( sup_sup_set_nat @ R2 @ S2 ) )
=> ( ( ( step_symb_set @ Q0 @ Transs @ R2 )
= bot_bot_set_nat )
=> ( ~ ( member_nat @ Qf @ R2 )
=> ( ( run_accept_eps @ Q0 @ Qf @ Transs @ R2 @ nil_list_o @ Bs )
= ( run_accept_eps @ Q0 @ Qf @ Transs @ S2 @ nil_list_o @ Bs ) ) ) ) ) ).
% run_accept_eps_Nil_eps_split
thf(fact_551_run__accept__eps__Cons__eps__split,axiom,
! [Q0: nat,Transs: list_transition,R2: set_nat,Cs: list_o,S2: set_nat,Qf: nat,Css: list_list_o,Bs: list_o] :
( ( ( step_eps_closure_set @ Q0 @ Transs @ R2 @ Cs )
= ( sup_sup_set_nat @ R2 @ S2 ) )
=> ( ( ( step_symb_set @ Q0 @ Transs @ R2 )
= bot_bot_set_nat )
=> ( ~ ( member_nat @ Qf @ R2 )
=> ( ( run_accept_eps @ Q0 @ Qf @ Transs @ R2 @ ( cons_list_o @ Cs @ Css ) @ Bs )
= ( run_accept_eps @ Q0 @ Qf @ Transs @ S2 @ ( cons_list_o @ Cs @ Css ) @ Bs ) ) ) ) ) ).
% run_accept_eps_Cons_eps_split
thf(fact_552_run__eps__split,axiom,
! [Q0: nat,Transs: list_transition,R2: set_nat,Bs: list_o,S2: set_nat,Bss: list_list_o] :
( ( ( step_eps_closure_set @ Q0 @ Transs @ R2 @ Bs )
= ( sup_sup_set_nat @ R2 @ S2 ) )
=> ( ( ( step_symb_set @ Q0 @ Transs @ R2 )
= bot_bot_set_nat )
=> ( ( run @ Q0 @ Transs @ R2 @ ( cons_list_o @ Bs @ Bss ) )
= ( run @ Q0 @ Transs @ S2 @ ( cons_list_o @ Bs @ Bss ) ) ) ) ) ).
% run_eps_split
thf(fact_553_right_Ostate__closed,axiom,
! [T: transition] :
( ( member_transition @ T @ ( set_transition2 @ ts_r ) )
=> ( ord_less_eq_set_nat @ ( state_set @ T ) @ ( q @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ qfa @ ts_r ) ) ) ).
% right.state_closed
thf(fact_554_left_Ostate__closed,axiom,
! [T: transition] :
( ( member_transition @ T @ ( set_transition2 @ ts_l ) )
=> ( ord_less_eq_set_nat @ ( state_set @ T ) @ ( q @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_l ) ) ) ).
% left.state_closed
thf(fact_555_base_Ostate__closed,axiom,
! [T: transition] :
( ( member_transition @ T @ ( set_transition2 @ transsa ) )
=> ( ord_less_eq_set_nat @ ( state_set @ T ) @ ( q @ q0a @ qfa @ transsa ) ) ) ).
% base.state_closed
thf(fact_556_right_OQ__diff__qf__SQ,axiom,
( ( minus_minus_set_nat @ ( q @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ qfa @ ts_r ) @ ( insert_nat @ qfa @ bot_bot_set_nat ) )
= ( sq @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_r ) ) ).
% right.Q_diff_qf_SQ
thf(fact_557_left_OQ__diff__qf__SQ,axiom,
( ( minus_minus_set_nat @ ( q @ q0a @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_l ) @ ( insert_nat @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ bot_bot_set_nat ) )
= ( sq @ q0a @ ts_l ) ) ).
% left.Q_diff_qf_SQ
thf(fact_558_cong_Oright_Otranss__eq,axiom,
! [Q: nat] :
( ( member_nat @ Q @ ( sq @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ ts_r ) )
=> ( ( nth_transition @ transsa @ ( minus_minus_nat @ Q @ q0a ) )
= ( nth_transition @ ts_r @ ( minus_minus_nat @ Q @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) ) ) ) ) ).
% cong.right.transs_eq
thf(fact_559_base_OQ__diff__qf__SQ,axiom,
( ( minus_minus_set_nat @ ( q @ q0a @ qfa @ transsa ) @ ( insert_nat @ qfa @ bot_bot_set_nat ) )
= ( sq @ q0a @ transsa ) ) ).
% base.Q_diff_qf_SQ
thf(fact_560_cong_Otranss__eq,axiom,
! [Q: nat] :
( ( member_nat @ Q @ ( sq @ q0a @ ts_l ) )
=> ( ( nth_transition @ transsa @ ( minus_minus_nat @ Q @ q0a ) )
= ( nth_transition @ ts_l @ ( minus_minus_nat @ Q @ q0a ) ) ) ) ).
% cong.transs_eq
thf(fact_561_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_562_nfa__cong_H_Otranss__eq,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition,Q: nat] :
( ( nfa_cong2 @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( ( member_nat @ Q @ ( sq @ Q02 @ Transs2 ) )
=> ( ( nth_transition @ Transs @ ( minus_minus_nat @ Q @ Q0 ) )
= ( nth_transition @ Transs2 @ ( minus_minus_nat @ Q @ Q02 ) ) ) ) ) ).
% nfa_cong'.transs_eq
thf(fact_563_nfa__cong_Otranss__eq,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition,Q: nat] :
( ( nfa_cong @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( ( member_nat @ Q @ ( sq @ Q02 @ Transs2 ) )
=> ( ( nth_transition @ Transs @ ( minus_minus_nat @ Q @ Q0 ) )
= ( nth_transition @ Transs2 @ ( minus_minus_nat @ Q @ Q02 ) ) ) ) ) ).
% nfa_cong.transs_eq
thf(fact_564_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_nat @ Qf @ bot_bot_set_nat ) )
= ( sq @ Q0 @ Transs ) ) ) ).
% nfa.Q_diff_qf_SQ
thf(fact_565_nfa_Ostate__closed,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,T: transition] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ( ( member_transition @ T @ ( set_transition2 @ Transs ) )
=> ( ord_less_eq_set_nat @ ( state_set @ T ) @ ( q @ Q0 @ Qf @ Transs ) ) ) ) ).
% nfa.state_closed
thf(fact_566_nfa_Ointro,axiom,
! [Transs: list_transition,Q0: nat,Qf: nat] :
( ! [T2: transition] :
( ( member_transition @ T2 @ ( set_transition2 @ Transs ) )
=> ( ord_less_eq_set_nat @ ( state_set @ T2 ) @ ( q @ Q0 @ Qf @ Transs ) ) )
=> ( ( Transs != nil_transition )
=> ( ~ ( member_nat @ Qf @ ( sq @ Q0 @ Transs ) )
=> ( nfa @ Q0 @ Qf @ Transs ) ) ) ) ).
% nfa.intro
thf(fact_567_nfa__def,axiom,
( nfa
= ( ^ [Q03: nat,Qf3: nat,Transs3: list_transition] :
( ! [T3: transition] :
( ( member_transition @ T3 @ ( set_transition2 @ Transs3 ) )
=> ( ord_less_eq_set_nat @ ( state_set @ T3 ) @ ( q @ Q03 @ Qf3 @ Transs3 ) ) )
& ( Transs3 != nil_transition )
& ~ ( member_nat @ Qf3 @ ( sq @ Q03 @ Transs3 ) ) ) ) ) ).
% nfa_def
thf(fact_568_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_569_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_570_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_571_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_572_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_573_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_574_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_575_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_576_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_577_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_578_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_579_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_580_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
? [K2: nat] :
( N2
= ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_581_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_582_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_583_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_584_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_585_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_586_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_587_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_588_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_589_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_590_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_591_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_592_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_593_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_594_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_595_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_596_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_597_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_598_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_599_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_600_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_601_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_602_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_603_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_604_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_605_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_606_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_607_q__in__SQ,axiom,
! [Q: nat,Q0: nat,Transs: list_transition] :
( ( member_nat @ Q @ ( sq @ Q0 @ Transs ) )
= ( ( ord_less_eq_nat @ Q0 @ Q )
& ( ord_less_nat @ Q @ ( plus_plus_nat @ Q0 @ ( size_s3613142680436377136sition @ Transs ) ) ) ) ) ).
% q_in_SQ
thf(fact_608_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_609_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_610_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_611_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_612_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_613_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_614_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_615_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_616_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_617_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_618_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_619_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_620_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_621_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_622_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_623_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_624_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_625_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_626_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_627_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_628_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_629_nfa__correct,axiom,
! [R: regex_a_t,Q0: nat,Qf: nat,Phis: list_formula_a_t,Transs: list_transition,Bss: list_list_o,Bs: list_o,I: nat] :
( ( iH_a_t @ sigma @ R @ Q0 @ Qf @ Phis @ Transs @ Bss @ Bs @ I )
=> ( ( run_accept_eps @ Q0 @ Qf @ Transs @ ( insert_nat @ Q0 @ bot_bot_set_nat ) @ Bss @ Bs )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ ( plus_plus_nat @ I @ ( size_s2710708370519433104list_o @ Bss ) ) ) @ ( match_a_t @ sigma @ R ) ) ) ) ).
% nfa_correct
thf(fact_630_match__le,axiom,
! [I: nat,J: nat,R: regex_a_t] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ ( match_a_t @ sigma @ R ) )
=> ( ord_less_eq_nat @ I @ J ) ) ).
% match_le
thf(fact_631_match__Times,axiom,
! [I: nat,N: nat,R: regex_a_t,S: regex_a_t] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ ( plus_plus_nat @ I @ N ) ) @ ( match_a_t @ sigma @ ( times_a_t @ R @ S ) ) )
= ( ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ ( plus_plus_nat @ I @ K2 ) ) @ ( match_a_t @ sigma @ R ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ I @ N ) ) @ ( match_a_t @ sigma @ S ) ) ) ) ) ).
% match_Times
thf(fact_632_wf__regex__eps__match,axiom,
! [R: regex_a_t,I: nat] :
( ( wf_regex_a_t @ R )
=> ( ( eps_a_t @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ I ) @ ( match_a_t @ sigma @ R ) ) ) ) ).
% wf_regex_eps_match
thf(fact_633_match__refl__eps,axiom,
! [I: nat,R: regex_a_t] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ I ) @ ( match_a_t @ sigma @ R ) )
=> ( eps_a_t @ R ) ) ).
% match_refl_eps
thf(fact_634_NFA_OSQ__def,axiom,
( sq
= ( ^ [Q03: nat,Transs3: list_transition] : ( set_or4665077453230672383an_nat @ Q03 @ ( plus_plus_nat @ Q03 @ ( size_s3613142680436377136sition @ Transs3 ) ) ) ) ) ).
% NFA.SQ_def
thf(fact_635_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M4: nat] :
( ( P @ X2 )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M4 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_636_atLeastLessThan__add__Un,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K ) )
= ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% atLeastLessThan_add_Un
thf(fact_637_IH__def,axiom,
! [R: regex_a_t,Q0: nat,Qf: nat,Phis: list_formula_a_t,Transs: list_transition,Bss: list_list_o,Bs: list_o,I: nat] :
( ( iH_a_t @ sigma @ R @ Q0 @ Qf @ Phis @ Transs @ Bss @ Bs @ I )
= ( ( Transs
= ( build_nfa_impl_a_t @ R @ ( produc8654416511292156347la_a_t @ Q0 @ ( produc9017461973804568604la_a_t @ Qf @ Phis ) ) ) )
& ! [X: list_o] :
( ( member_list_o @ X @ ( set_list_o2 @ Bss ) )
=> ( ord_less_eq_nat @ ( size_s8846756101701226951la_a_t @ ( collect_subfmlas_a_t @ R @ Phis ) ) @ ( size_size_list_o @ X ) ) )
& ( ord_less_eq_nat @ ( size_s8846756101701226951la_a_t @ ( collect_subfmlas_a_t @ R @ Phis ) ) @ ( size_size_list_o @ Bs ) )
& ~ ( member_nat @ Qf @ ( sq @ Q0 @ ( build_nfa_impl_a_t @ R @ ( produc8654416511292156347la_a_t @ Q0 @ ( produc9017461973804568604la_a_t @ Qf @ Phis ) ) ) ) )
& ! [K2: nat] :
( ( ord_less_nat @ K2 @ ( size_s8846756101701226951la_a_t @ ( collect_subfmlas_a_t @ R @ Phis ) ) )
=> ( ( nth_o @ Bs @ K2 )
= ( sat_a_t @ sigma @ ( nth_formula_a_t @ ( collect_subfmlas_a_t @ R @ Phis ) @ K2 ) @ ( plus_plus_nat @ I @ ( size_s2710708370519433104list_o @ Bss ) ) ) ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ ( size_s2710708370519433104list_o @ Bss ) )
=> ! [K2: nat] :
( ( ord_less_nat @ K2 @ ( size_s8846756101701226951la_a_t @ ( collect_subfmlas_a_t @ R @ Phis ) ) )
=> ( ( nth_o @ ( nth_list_o @ Bss @ J3 ) @ K2 )
= ( sat_a_t @ sigma @ ( nth_formula_a_t @ ( collect_subfmlas_a_t @ R @ Phis ) @ K2 ) @ ( plus_plus_nat @ I @ J3 ) ) ) ) ) ) ) ).
% IH_def
thf(fact_638_match__Star__unfold,axiom,
! [I: nat,J: nat,R: regex_a_t] :
( ( ord_less_nat @ I @ J )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ ( match_a_t @ sigma @ ( star_a_t @ R ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ I @ J ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ X3 ) @ ( match_a_t @ sigma @ ( star_a_t @ R ) ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ J ) @ ( match_a_t @ sigma @ R ) ) ) ) ) ).
% match_Star_unfold
thf(fact_639_step__eps__closure__set__code,axiom,
( step_eps_closure_set
= ( ^ [Q03: nat,Transs3: list_transition,R3: set_nat,Bs2: list_o] :
( if_set_nat
@ ( R3
= ( sup_sup_set_nat @ R3 @ ( step_eps_set @ Q03 @ Transs3 @ Bs2 @ R3 ) ) )
@ R3
@ ( step_eps_closure_set @ Q03 @ Transs3 @ ( sup_sup_set_nat @ R3 @ ( step_eps_set @ Q03 @ Transs3 @ Bs2 @ R3 ) ) @ Bs2 ) ) ) ) ).
% step_eps_closure_set_code
thf(fact_640_NFA_Ostep__eps__def,axiom,
( step_eps
= ( ^ [Q03: nat,Transs3: list_transition,Bs2: list_o,Q5: nat,Q6: nat] :
( ( member_nat @ Q5 @ ( sq @ Q03 @ Transs3 ) )
& ( case_transition_o
@ ^ [P2: nat,N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_o @ Bs2 ) )
& ( nth_o @ Bs2 @ N2 )
& ( P2 = Q6 ) )
@ ^ [Nat: nat] : $false
@ ^ [P2: nat,P3: nat] :
( ( P2 = Q6 )
| ( P3 = Q6 ) )
@ ( nth_transition @ Transs3 @ ( minus_minus_nat @ Q5 @ Q03 ) ) ) ) ) ) ).
% NFA.step_eps_def
thf(fact_641_NFA_Ostep__symb__def,axiom,
( step_symb
= ( ^ [Q03: nat,Transs3: list_transition,Q5: nat,Q6: nat] :
( ( member_nat @ Q5 @ ( sq @ Q03 @ Transs3 ) )
& ( case_transition_o
@ ^ [Nat1: nat,Nat2: nat] : $false
@ ^ [P2: nat] : ( P2 = Q6 )
@ ^ [Nat1: nat,Nat2: nat] : $false
@ ( nth_transition @ Transs3 @ ( minus_minus_nat @ Q5 @ Q03 ) ) ) ) ) ) ).
% NFA.step_symb_def
thf(fact_642_NFA_Ostep__eps__sucs__def,axiom,
( step_eps_sucs
= ( ^ [Q03: nat,Transs3: list_transition,Bs2: list_o,Q5: nat] :
( if_set_nat @ ( member_nat @ Q5 @ ( sq @ Q03 @ Transs3 ) )
@ ( case_t7109905494440202798et_nat
@ ^ [P2: nat,N2: nat] :
( if_set_nat
@ ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Bs2 ) )
& ( nth_o @ Bs2 @ N2 ) )
@ ( insert_nat @ P2 @ bot_bot_set_nat )
@ bot_bot_set_nat )
@ ^ [Nat: nat] : bot_bot_set_nat
@ ^ [P2: nat,P3: nat] : ( insert_nat @ P2 @ ( insert_nat @ P3 @ bot_bot_set_nat ) )
@ ( nth_transition @ Transs3 @ ( minus_minus_nat @ Q5 @ Q03 ) ) )
@ bot_bot_set_nat ) ) ) ).
% NFA.step_eps_sucs_def
thf(fact_643_NFA_Ostep__symb__sucs__def,axiom,
( step_symb_sucs
= ( ^ [Q03: nat,Transs3: list_transition,Q5: nat] :
( if_set_nat @ ( member_nat @ Q5 @ ( sq @ Q03 @ Transs3 ) )
@ ( case_t7109905494440202798et_nat
@ ^ [Nat1: nat,Nat2: nat] : bot_bot_set_nat
@ ^ [P2: nat] : ( insert_nat @ P2 @ bot_bot_set_nat )
@ ^ [Nat1: nat,Nat2: nat] : bot_bot_set_nat
@ ( nth_transition @ Transs3 @ ( minus_minus_nat @ Q5 @ Q03 ) ) )
@ bot_bot_set_nat ) ) ) ).
% NFA.step_symb_sucs_def
thf(fact_644_step__eps__sucs__sound,axiom,
! [Q2: nat,Q0: nat,Transs: list_transition,Bs: list_o,Q: nat] :
( ( member_nat @ Q2 @ ( step_eps_sucs @ Q0 @ Transs @ Bs @ Q ) )
= ( step_eps @ Q0 @ Transs @ Bs @ Q @ Q2 ) ) ).
% step_eps_sucs_sound
thf(fact_645_step__symb__sucs__sound,axiom,
! [Q2: nat,Q0: nat,Transs: list_transition,Q: nat] :
( ( member_nat @ Q2 @ ( step_symb_sucs @ Q0 @ Transs @ Q ) )
= ( step_symb @ Q0 @ Transs @ Q @ Q2 ) ) ).
% step_symb_sucs_sound
thf(fact_646_nfa__cong__Star__axioms__def,axiom,
( nfa_cong_Star_axioms
= ( ^ [Q03: nat,Q04: nat,Qf3: nat,Transs3: list_transition] :
( ! [Bs2: list_o,Q5: nat] :
( ( step_eps @ Q03 @ Transs3 @ Bs2 @ Q03 @ Q5 )
= ( member_nat @ Q5 @ ( insert_nat @ Q04 @ ( insert_nat @ Qf3 @ bot_bot_set_nat ) ) ) )
& ! [Q5: nat] :
~ ( step_symb @ Q03 @ Transs3 @ Q03 @ Q5 ) ) ) ) ).
% nfa_cong_Star_axioms_def
thf(fact_647_nfa__cong__Plus__axioms__def,axiom,
( nfa_cong_Plus_axioms
= ( ^ [Q03: nat,Q04: nat,Q05: nat,Transs3: list_transition] :
( ! [Bs2: list_o,Q5: nat] :
( ( step_eps @ Q03 @ Transs3 @ Bs2 @ Q03 @ Q5 )
= ( member_nat @ Q5 @ ( insert_nat @ Q04 @ ( insert_nat @ Q05 @ bot_bot_set_nat ) ) ) )
& ! [Q5: nat] :
~ ( step_symb @ Q03 @ Transs3 @ Q03 @ Q5 ) ) ) ) ).
% nfa_cong_Plus_axioms_def
thf(fact_648_nfa__cong__Plus__axioms_Ointro,axiom,
! [Q0: nat,Transs: list_transition,Q02: nat,Q06: nat] :
( ! [Bs3: list_o,Q3: nat] :
( ( step_eps @ Q0 @ Transs @ Bs3 @ Q0 @ Q3 )
= ( member_nat @ Q3 @ ( insert_nat @ Q02 @ ( insert_nat @ Q06 @ bot_bot_set_nat ) ) ) )
=> ( ! [Q3: nat] :
~ ( step_symb @ Q0 @ Transs @ Q0 @ Q3 )
=> ( nfa_cong_Plus_axioms @ Q0 @ Q02 @ Q06 @ Transs ) ) ) ).
% nfa_cong_Plus_axioms.intro
thf(fact_649_nfa__cong__Star__axioms_Ointro,axiom,
! [Q0: nat,Transs: list_transition,Q02: nat,Qf: nat] :
( ! [Bs3: list_o,Q3: nat] :
( ( step_eps @ Q0 @ Transs @ Bs3 @ Q0 @ Q3 )
= ( member_nat @ Q3 @ ( insert_nat @ Q02 @ ( insert_nat @ Qf @ bot_bot_set_nat ) ) ) )
=> ( ! [Q3: nat] :
~ ( step_symb @ Q0 @ Transs @ Q0 @ Q3 )
=> ( nfa_cong_Star_axioms @ Q0 @ Q02 @ Qf @ Transs ) ) ) ).
% nfa_cong_Star_axioms.intro
thf(fact_650_nfa__cong_H__axioms__def,axiom,
( nfa_cong_axioms
= ( ^ [Q03: nat,Q04: nat,Qf4: nat,Transs3: list_transition,Transs5: list_transition] :
( ( ord_less_eq_set_nat @ ( sq @ Q04 @ Transs5 ) @ ( sq @ Q03 @ Transs3 ) )
& ( member_nat @ Qf4 @ ( sq @ Q03 @ Transs3 ) )
& ! [Q5: nat] :
( ( member_nat @ Q5 @ ( sq @ Q04 @ Transs5 ) )
=> ( ( nth_transition @ Transs3 @ ( minus_minus_nat @ Q5 @ Q03 ) )
= ( nth_transition @ Transs5 @ ( minus_minus_nat @ Q5 @ Q04 ) ) ) ) ) ) ) ).
% nfa_cong'_axioms_def
thf(fact_651_nfa__cong__axioms__def,axiom,
( nfa_cong_axioms2
= ( ^ [Q03: nat,Q04: nat,Qf3: nat,Qf4: nat,Transs3: list_transition,Transs5: list_transition] :
( ( ord_less_eq_set_nat @ ( sq @ Q04 @ Transs5 ) @ ( sq @ Q03 @ Transs3 ) )
& ( Qf3 = Qf4 )
& ! [Q5: nat] :
( ( member_nat @ Q5 @ ( sq @ Q04 @ Transs5 ) )
=> ( ( nth_transition @ Transs3 @ ( minus_minus_nat @ Q5 @ Q03 ) )
= ( nth_transition @ Transs5 @ ( minus_minus_nat @ Q5 @ Q04 ) ) ) ) ) ) ) ).
% nfa_cong_axioms_def
thf(fact_652_nfa__cong_H_Oaxioms_I3_J,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition] :
( ( nfa_cong2 @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( nfa_cong_axioms @ Q0 @ Q02 @ Qf2 @ Transs @ Transs2 ) ) ).
% nfa_cong'.axioms(3)
thf(fact_653_nfa__cong_Oaxioms_I3_J,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition] :
( ( nfa_cong @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( nfa_cong_axioms2 @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 ) ) ).
% nfa_cong.axioms(3)
thf(fact_654_nfa__cong_H__def,axiom,
( nfa_cong2
= ( ^ [Q03: nat,Q04: nat,Qf3: nat,Qf4: nat,Transs3: list_transition,Transs5: list_transition] :
( ( nfa @ Q03 @ Qf3 @ Transs3 )
& ( nfa @ Q04 @ Qf4 @ Transs5 )
& ( nfa_cong_axioms @ Q03 @ Q04 @ Qf4 @ Transs3 @ Transs5 ) ) ) ) ).
% nfa_cong'_def
thf(fact_655_nfa__cong_H_Ointro,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,Q02: nat,Qf2: nat,Transs2: list_transition] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ( ( nfa @ Q02 @ Qf2 @ Transs2 )
=> ( ( nfa_cong_axioms @ Q0 @ Q02 @ Qf2 @ Transs @ Transs2 )
=> ( nfa_cong2 @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 ) ) ) ) ).
% nfa_cong'.intro
thf(fact_656_nfa__cong__def,axiom,
( nfa_cong
= ( ^ [Q03: nat,Q04: nat,Qf3: nat,Qf4: nat,Transs3: list_transition,Transs5: list_transition] :
( ( nfa @ Q03 @ Qf3 @ Transs3 )
& ( nfa @ Q04 @ Qf4 @ Transs5 )
& ( nfa_cong_axioms2 @ Q03 @ Q04 @ Qf3 @ Qf4 @ Transs3 @ Transs5 ) ) ) ) ).
% nfa_cong_def
thf(fact_657_nfa__cong_Ointro,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,Q02: nat,Qf2: nat,Transs2: list_transition] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ( ( nfa @ Q02 @ Qf2 @ Transs2 )
=> ( ( nfa_cong_axioms2 @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( nfa_cong @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 ) ) ) ) ).
% nfa_cong.intro
thf(fact_658_nfa__cong__axioms_Ointro,axiom,
! [Q02: nat,Transs2: list_transition,Q0: nat,Transs: list_transition,Qf: nat,Qf2: nat] :
( ( ord_less_eq_set_nat @ ( sq @ Q02 @ Transs2 ) @ ( sq @ Q0 @ Transs ) )
=> ( ( Qf = Qf2 )
=> ( ! [Q3: nat] :
( ( member_nat @ Q3 @ ( sq @ Q02 @ Transs2 ) )
=> ( ( nth_transition @ Transs @ ( minus_minus_nat @ Q3 @ Q0 ) )
= ( nth_transition @ Transs2 @ ( minus_minus_nat @ Q3 @ Q02 ) ) ) )
=> ( nfa_cong_axioms2 @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 ) ) ) ) ).
% nfa_cong_axioms.intro
thf(fact_659_nfa__cong_H__axioms_Ointro,axiom,
! [Q02: nat,Transs2: list_transition,Q0: nat,Transs: list_transition,Qf2: nat] :
( ( ord_less_eq_set_nat @ ( sq @ Q02 @ Transs2 ) @ ( sq @ Q0 @ Transs ) )
=> ( ( member_nat @ Qf2 @ ( sq @ Q0 @ Transs ) )
=> ( ! [Q3: nat] :
( ( member_nat @ Q3 @ ( sq @ Q02 @ Transs2 ) )
=> ( ( nth_transition @ Transs @ ( minus_minus_nat @ Q3 @ Q0 ) )
= ( nth_transition @ Transs2 @ ( minus_minus_nat @ Q3 @ Q02 ) ) ) )
=> ( nfa_cong_axioms @ Q0 @ Q02 @ Qf2 @ Transs @ Transs2 ) ) ) ) ).
% nfa_cong'_axioms.intro
thf(fact_660_match_Osimps_I5_J,axiom,
! [R: regex_a_t] :
( ( match_a_t @ sigma @ ( star_a_t @ R ) )
= ( transi2905341329935302413cl_nat @ ( match_a_t @ sigma @ R ) ) ) ).
% match.simps(5)
thf(fact_661_NFA_Oaccept__def,axiom,
( accept
= ( ^ [Q03: nat,Qf3: nat,Transs3: list_transition,R3: set_nat] : ( accept_eps @ Q03 @ Qf3 @ Transs3 @ R3 @ nil_o ) ) ) ).
% NFA.accept_def
thf(fact_662_match_Osimps_I4_J,axiom,
! [R: regex_a_t,S: regex_a_t] :
( ( match_a_t @ sigma @ ( times_a_t @ R @ S ) )
= ( relcomp_nat_nat_nat @ ( match_a_t @ sigma @ R ) @ ( match_a_t @ sigma @ S ) ) ) ).
% match.simps(4)
thf(fact_663_NFA_Orun__accept__def,axiom,
( run_accept
= ( ^ [Q03: nat,Qf3: nat,Transs3: list_transition,R3: set_nat,Bss2: list_list_o] : ( accept @ Q03 @ Qf3 @ Transs3 @ ( run @ Q03 @ Transs3 @ R3 @ Bss2 ) ) ) ) ).
% NFA.run_accept_def
thf(fact_664_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K3 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K3 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_665_pred__nat__trancl__eq__le,axiom,
! [M: nat,N: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% pred_nat_trancl_eq_le
thf(fact_666_nfa__cong__Plus_Orun__accept__eps__Nil__cong,axiom,
! [Q0: nat,Q02: nat,Q06: nat,Qf: nat,Qf2: nat,Qf5: nat,Transs: list_transition,Transs2: list_transition,Transs4: list_transition,Bs: list_o] :
( ( nfa_cong_Plus @ Q0 @ Q02 @ Q06 @ Qf @ Qf2 @ Qf5 @ Transs @ Transs2 @ Transs4 )
=> ( ( run_accept_eps @ Q0 @ Qf @ Transs @ ( insert_nat @ Q0 @ bot_bot_set_nat ) @ nil_list_o @ Bs )
= ( ( run_accept_eps @ Q02 @ Qf2 @ Transs2 @ ( insert_nat @ Q02 @ bot_bot_set_nat ) @ nil_list_o @ Bs )
| ( run_accept_eps @ Q06 @ Qf5 @ Transs4 @ ( insert_nat @ Q06 @ bot_bot_set_nat ) @ nil_list_o @ Bs ) ) ) ) ).
% nfa_cong_Plus.run_accept_eps_Nil_cong
thf(fact_667_nfa__cong__Plus_Ostep__eps__closure__set__q0,axiom,
! [Q0: nat,Q02: nat,Q06: nat,Qf: nat,Qf2: nat,Qf5: nat,Transs: list_transition,Transs2: list_transition,Transs4: list_transition,Bs: list_o] :
( ( nfa_cong_Plus @ Q0 @ Q02 @ Q06 @ Qf @ Qf2 @ Qf5 @ Transs @ Transs2 @ Transs4 )
=> ( ( step_eps_closure_set @ Q0 @ Transs @ ( insert_nat @ Q0 @ bot_bot_set_nat ) @ Bs )
= ( sup_sup_set_nat @ ( insert_nat @ Q0 @ bot_bot_set_nat ) @ ( sup_sup_set_nat @ ( step_eps_closure_set @ Q02 @ Transs2 @ ( insert_nat @ Q02 @ bot_bot_set_nat ) @ Bs ) @ ( step_eps_closure_set @ Q06 @ Transs4 @ ( insert_nat @ Q06 @ bot_bot_set_nat ) @ Bs ) ) ) ) ) ).
% nfa_cong_Plus.step_eps_closure_set_q0
thf(fact_668_nfa__cong__Plus_Oaxioms_I3_J,axiom,
! [Q0: nat,Q02: nat,Q06: nat,Qf: nat,Qf2: nat,Qf5: nat,Transs: list_transition,Transs2: list_transition,Transs4: list_transition] :
( ( nfa_cong_Plus @ Q0 @ Q02 @ Q06 @ Qf @ Qf2 @ Qf5 @ Transs @ Transs2 @ Transs4 )
=> ( nfa_cong_Plus_axioms @ Q0 @ Q02 @ Q06 @ Transs ) ) ).
% nfa_cong_Plus.axioms(3)
thf(fact_669_nfa__cong__Plus_Oaxioms_I1_J,axiom,
! [Q0: nat,Q02: nat,Q06: nat,Qf: nat,Qf2: nat,Qf5: nat,Transs: list_transition,Transs2: list_transition,Transs4: list_transition] :
( ( nfa_cong_Plus @ Q0 @ Q02 @ Q06 @ Qf @ Qf2 @ Qf5 @ Transs @ Transs2 @ Transs4 )
=> ( nfa_cong @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 ) ) ).
% nfa_cong_Plus.axioms(1)
thf(fact_670_nfa__cong__Plus_Oaxioms_I2_J,axiom,
! [Q0: nat,Q02: nat,Q06: nat,Qf: nat,Qf2: nat,Qf5: nat,Transs: list_transition,Transs2: list_transition,Transs4: list_transition] :
( ( nfa_cong_Plus @ Q0 @ Q02 @ Q06 @ Qf @ Qf2 @ Qf5 @ Transs @ Transs2 @ Transs4 )
=> ( nfa_cong @ Q0 @ Q06 @ Qf @ Qf5 @ Transs @ Transs4 ) ) ).
% nfa_cong_Plus.axioms(2)
thf(fact_671_nfa__cong__Plus_Ostep__symb__q0,axiom,
! [Q0: nat,Q02: nat,Q06: nat,Qf: nat,Qf2: nat,Qf5: nat,Transs: list_transition,Transs2: list_transition,Transs4: list_transition,Q: nat] :
( ( nfa_cong_Plus @ Q0 @ Q02 @ Q06 @ Qf @ Qf2 @ Qf5 @ Transs @ Transs2 @ Transs4 )
=> ~ ( step_symb @ Q0 @ Transs @ Q0 @ Q ) ) ).
% nfa_cong_Plus.step_symb_q0
thf(fact_672_nfa__cong__Plus_Oqf__not__q0,axiom,
! [Q0: nat,Q02: nat,Q06: nat,Qf: nat,Qf2: nat,Qf5: nat,Transs: list_transition,Transs2: list_transition,Transs4: list_transition] :
( ( nfa_cong_Plus @ Q0 @ Q02 @ Q06 @ Qf @ Qf2 @ Qf5 @ Transs @ Transs2 @ Transs4 )
=> ~ ( member_nat @ Qf @ ( insert_nat @ Q0 @ bot_bot_set_nat ) ) ) ).
% nfa_cong_Plus.qf_not_q0
thf(fact_673_nfa__cong__Plus__def,axiom,
( nfa_cong_Plus
= ( ^ [Q03: nat,Q04: nat,Q05: nat,Qf3: nat,Qf4: nat,Qf6: nat,Transs3: list_transition,Transs5: list_transition,Transs6: list_transition] :
( ( nfa_cong @ Q03 @ Q04 @ Qf3 @ Qf4 @ Transs3 @ Transs5 )
& ( nfa_cong @ Q03 @ Q05 @ Qf3 @ Qf6 @ Transs3 @ Transs6 )
& ( nfa_cong_Plus_axioms @ Q03 @ Q04 @ Q05 @ Transs3 ) ) ) ) ).
% nfa_cong_Plus_def
thf(fact_674_nfa__cong__Plus_Ointro,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Qf2: nat,Transs: list_transition,Transs2: list_transition,Q06: nat,Qf5: nat,Transs4: list_transition] :
( ( nfa_cong @ Q0 @ Q02 @ Qf @ Qf2 @ Transs @ Transs2 )
=> ( ( nfa_cong @ Q0 @ Q06 @ Qf @ Qf5 @ Transs @ Transs4 )
=> ( ( nfa_cong_Plus_axioms @ Q0 @ Q02 @ Q06 @ Transs )
=> ( nfa_cong_Plus @ Q0 @ Q02 @ Q06 @ Qf @ Qf2 @ Qf5 @ Transs @ Transs2 @ Transs4 ) ) ) ) ).
% nfa_cong_Plus.intro
thf(fact_675_nfa__cong__Plus_Orun__accept__eps__cong,axiom,
! [Q0: nat,Q02: nat,Q06: nat,Qf: nat,Qf2: nat,Qf5: nat,Transs: list_transition,Transs2: list_transition,Transs4: list_transition,Bss: list_list_o,Bs: list_o] :
( ( nfa_cong_Plus @ Q0 @ Q02 @ Q06 @ Qf @ Qf2 @ Qf5 @ Transs @ Transs2 @ Transs4 )
=> ( ( run_accept_eps @ Q0 @ Qf @ Transs @ ( insert_nat @ Q0 @ bot_bot_set_nat ) @ Bss @ Bs )
= ( ( run_accept_eps @ Q02 @ Qf2 @ Transs2 @ ( insert_nat @ Q02 @ bot_bot_set_nat ) @ Bss @ Bs )
| ( run_accept_eps @ Q06 @ Qf5 @ Transs4 @ ( insert_nat @ Q06 @ bot_bot_set_nat ) @ Bss @ Bs ) ) ) ) ).
% nfa_cong_Plus.run_accept_eps_cong
thf(fact_676_nfa__cong__Plus_Ostep__symb__set__q0,axiom,
! [Q0: nat,Q02: nat,Q06: nat,Qf: nat,Qf2: nat,Qf5: nat,Transs: list_transition,Transs2: list_transition,Transs4: list_transition] :
( ( nfa_cong_Plus @ Q0 @ Q02 @ Q06 @ Qf @ Qf2 @ Qf5 @ Transs @ Transs2 @ Transs4 )
=> ( ( step_symb_set @ Q0 @ Transs @ ( insert_nat @ Q0 @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ) ).
% nfa_cong_Plus.step_symb_set_q0
thf(fact_677_nfa__cong__Plus_Ostep__eps__q0,axiom,
! [Q0: nat,Q02: nat,Q06: nat,Qf: nat,Qf2: nat,Qf5: nat,Transs: list_transition,Transs2: list_transition,Transs4: list_transition,Bs: list_o,Q: nat] :
( ( nfa_cong_Plus @ Q0 @ Q02 @ Q06 @ Qf @ Qf2 @ Qf5 @ Transs @ Transs2 @ Transs4 )
=> ( ( step_eps @ Q0 @ Transs @ Bs @ Q0 @ Q )
= ( member_nat @ Q @ ( insert_nat @ Q02 @ ( insert_nat @ Q06 @ bot_bot_set_nat ) ) ) ) ) ).
% nfa_cong_Plus.step_eps_q0
thf(fact_678_nfa__cong__Plus_Orun__accept__eps__Cons__cong,axiom,
! [Q0: nat,Q02: nat,Q06: nat,Qf: nat,Qf2: nat,Qf5: nat,Transs: list_transition,Transs2: list_transition,Transs4: list_transition,Cs: list_o,Css: list_list_o,Bs: list_o] :
( ( nfa_cong_Plus @ Q0 @ Q02 @ Q06 @ Qf @ Qf2 @ Qf5 @ Transs @ Transs2 @ Transs4 )
=> ( ( run_accept_eps @ Q0 @ Qf @ Transs @ ( insert_nat @ Q0 @ bot_bot_set_nat ) @ ( cons_list_o @ Cs @ Css ) @ Bs )
= ( ( run_accept_eps @ Q02 @ Qf2 @ Transs2 @ ( insert_nat @ Q02 @ bot_bot_set_nat ) @ ( cons_list_o @ Cs @ Css ) @ Bs )
| ( run_accept_eps @ Q06 @ Qf5 @ Transs4 @ ( insert_nat @ Q06 @ bot_bot_set_nat ) @ ( cons_list_o @ Cs @ Css ) @ Bs ) ) ) ) ).
% nfa_cong_Plus.run_accept_eps_Cons_cong
thf(fact_679_nfa__cong__Star_Orun__accept__eps__Nil,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Transs: list_transition,Transs2: list_transition,Bs: list_o] :
( ( nfa_cong_Star @ Q0 @ Q02 @ Qf @ Transs @ Transs2 )
=> ( run_accept_eps @ Q0 @ Qf @ Transs @ ( insert_nat @ Q0 @ bot_bot_set_nat ) @ nil_list_o @ Bs ) ) ).
% nfa_cong_Star.run_accept_eps_Nil
thf(fact_680_nfa__cong__Star_Oaxioms_I2_J,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Transs: list_transition,Transs2: list_transition] :
( ( nfa_cong_Star @ Q0 @ Q02 @ Qf @ Transs @ Transs2 )
=> ( nfa_cong_Star_axioms @ Q0 @ Q02 @ Qf @ Transs ) ) ).
% nfa_cong_Star.axioms(2)
thf(fact_681_nfa__cong__Star_Oaxioms_I1_J,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Transs: list_transition,Transs2: list_transition] :
( ( nfa_cong_Star @ Q0 @ Q02 @ Qf @ Transs @ Transs2 )
=> ( nfa_cong2 @ Q0 @ Q02 @ Qf @ Q0 @ Transs @ Transs2 ) ) ).
% nfa_cong_Star.axioms(1)
thf(fact_682_nfa__cong__Star_Ostep__symb__q0,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Transs: list_transition,Transs2: list_transition,Q: nat] :
( ( nfa_cong_Star @ Q0 @ Q02 @ Qf @ Transs @ Transs2 )
=> ~ ( step_symb @ Q0 @ Transs @ Q0 @ Q ) ) ).
% nfa_cong_Star.step_symb_q0
thf(fact_683_nfa__cong__Star_Ointro,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Transs: list_transition,Transs2: list_transition] :
( ( nfa_cong2 @ Q0 @ Q02 @ Qf @ Q0 @ Transs @ Transs2 )
=> ( ( nfa_cong_Star_axioms @ Q0 @ Q02 @ Qf @ Transs )
=> ( nfa_cong_Star @ Q0 @ Q02 @ Qf @ Transs @ Transs2 ) ) ) ).
% nfa_cong_Star.intro
thf(fact_684_nfa__cong__Star__def,axiom,
( nfa_cong_Star
= ( ^ [Q03: nat,Q04: nat,Qf3: nat,Transs3: list_transition,Transs5: list_transition] :
( ( nfa_cong2 @ Q03 @ Q04 @ Qf3 @ Q03 @ Transs3 @ Transs5 )
& ( nfa_cong_Star_axioms @ Q03 @ Q04 @ Qf3 @ Transs3 ) ) ) ) ).
% nfa_cong_Star_def
thf(fact_685_nfa__cong__Star_Odelta__q0__q0_H,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Transs: list_transition,Transs2: list_transition,Bs: list_o] :
( ( nfa_cong_Star @ Q0 @ Q02 @ Qf @ Transs @ Transs2 )
=> ( ( delta @ Q0 @ Transs @ ( insert_nat @ Q0 @ bot_bot_set_nat ) @ Bs )
= ( delta @ Q0 @ Transs @ ( insert_nat @ Q02 @ bot_bot_set_nat ) @ Bs ) ) ) ).
% nfa_cong_Star.delta_q0_q0'
thf(fact_686_nfa__cong__Star_Ostep__eps__q0,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Transs: list_transition,Transs2: list_transition,Bs: list_o,Q: nat] :
( ( nfa_cong_Star @ Q0 @ Q02 @ Qf @ Transs @ Transs2 )
=> ( ( step_eps @ Q0 @ Transs @ Bs @ Q0 @ Q )
= ( member_nat @ Q @ ( insert_nat @ Q02 @ ( insert_nat @ Qf @ bot_bot_set_nat ) ) ) ) ) ).
% nfa_cong_Star.step_eps_q0
thf(fact_687_nfa__cong__Star_Ostep__symb__set__q0,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Transs: list_transition,Transs2: list_transition] :
( ( nfa_cong_Star @ Q0 @ Q02 @ Qf @ Transs @ Transs2 )
=> ( ( step_symb_set @ Q0 @ Transs @ ( insert_nat @ Q0 @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ) ).
% nfa_cong_Star.step_symb_set_q0
thf(fact_688_nfa__cong__Star_Odelta__sub__nfa_H__delta,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Transs: list_transition,Transs2: list_transition,Bs: list_o] :
( ( nfa_cong_Star @ Q0 @ Q02 @ Qf @ Transs @ Transs2 )
=> ( ord_less_eq_set_nat @ ( delta @ Q0 @ Transs @ ( insert_nat @ Q0 @ bot_bot_set_nat ) @ Bs ) @ ( delta @ Q02 @ Transs2 @ ( insert_nat @ Q02 @ bot_bot_set_nat ) @ Bs ) ) ) ).
% nfa_cong_Star.delta_sub_nfa'_delta
thf(fact_689_nfa__cong__Star_Ostep__eps__closure__set__q0__split,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Transs: list_transition,Transs2: list_transition,Bs: list_o] :
( ( nfa_cong_Star @ Q0 @ Q02 @ Qf @ Transs @ Transs2 )
=> ( ( step_eps_closure_set @ Q0 @ Transs @ ( insert_nat @ Q0 @ bot_bot_set_nat ) @ Bs )
= ( sup_sup_set_nat @ ( insert_nat @ Q0 @ ( insert_nat @ Qf @ bot_bot_set_nat ) ) @ ( step_eps_closure_set @ Q0 @ Transs @ ( insert_nat @ Q02 @ bot_bot_set_nat ) @ Bs ) ) ) ) ).
% nfa_cong_Star.step_eps_closure_set_q0_split
thf(fact_690_nfa__cong__Star_Ostep__eps__closure__set__q0,axiom,
! [Q0: nat,Q02: nat,Qf: nat,Transs: list_transition,Transs2: list_transition,Bs: list_o] :
( ( nfa_cong_Star @ Q0 @ Q02 @ Qf @ Transs @ Transs2 )
=> ( ord_less_eq_set_nat @ ( step_eps_closure_set @ Q0 @ Transs @ ( insert_nat @ Q0 @ bot_bot_set_nat ) @ Bs ) @ ( sup_sup_set_nat @ ( insert_nat @ Q0 @ ( insert_nat @ Qf @ bot_bot_set_nat ) ) @ ( inf_inf_set_nat @ ( step_eps_closure_set @ Q02 @ Transs2 @ ( insert_nat @ Q02 @ bot_bot_set_nat ) @ Bs ) @ ( sq @ Q02 @ Transs2 ) ) ) ) ) ).
% nfa_cong_Star.step_eps_closure_set_q0
% Helper facts (3)
thf(help_If_3_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X2: set_nat,Y: set_nat] :
( ( if_set_nat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X2: set_nat,Y: set_nat] :
( ( if_set_nat @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
iH_a_t @ sigma @ s @ ( plus_plus_nat @ q0a @ ( state_cnt_a_t @ ra ) ) @ qfa @ ( collect_subfmlas_a_t @ ra @ phisa ) @ ts_r @ bss @ bs @ i ).
%------------------------------------------------------------------------------