TPTP Problem File: SLH0690^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% 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_00406_018405__16620836_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 964 ( 317 unt; 204 typ; 0 def)
% Number of atoms : 1910 ( 817 equ; 0 cnn)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 9055 ( 288 ~; 32 |; 96 &;7683 @)
% ( 0 <=>; 956 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 8 avg)
% Number of types : 37 ( 36 usr)
% Number of type conns : 517 ( 517 >; 0 *; 0 +; 0 <<)
% Number of symbols : 169 ( 168 usr; 30 con; 0-9 aty)
% Number of variables : 2703 ( 101 ^;2552 !; 50 ?;2703 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 14:53:42.904
%------------------------------------------------------------------------------
% Could-be-implicit typings (36)
thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__MDL__Oformula_Itf__a_Mtf__t_J_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__MDL__Oformula_Itf__a_Mtf__t_J_J_J_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__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_J_J,type,
produc6783461406735195739la_a_t: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__List__Olist_It__MDL__Oformula_Itf__a_Mtf__t_J_J_Mt__List__Olist_It__MDL__Oformula_Itf__a_Mtf__t_J_J_J_J_Mt__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_J,type,
produc920861655927507678la_a_t: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
produc4471711990508489141at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__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_J,type,
produc3148207259879549498la_a_t: $tType ).
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_I_062_It__List__Olist_I_Eo_J_M_062_It__List__Olist_I_Eo_J_M_Eo_J_J_Mt__List__Olist_It__List__Olist_I_Eo_J_J_J,type,
produc867208691148388573list_o: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__NFA__Otransition_M_062_It__NFA__Otransition_M_Eo_J_J_Mt__List__Olist_It__NFA__Otransition_J_J,type,
produc5558917482089586557sition: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__List__Olist_I_Eo_J_J_Mt__List__Olist_It__List__Olist_I_Eo_J_J_J,type,
produc4690905340047322919list_o: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__NFA__Otransition_J_Mt__List__Olist_It__NFA__Otransition_J_J,type,
produc1202413579072790439sition: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc7248412053542808358at_nat: $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__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
produc7819656566062154093et_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_I_Eo_M_062_I_Eo_M_Eo_J_J_Mt__List__Olist_I_Eo_J_J,type,
produc8642409424279824599list_o: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_I_Eo_J_Mt__List__Olist_I_Eo_J_J,type,
produc7102631898165422375list_o: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
list_P6011104703257516679at_nat: $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__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
produc7491599851749785783at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
produc2400336064389900727et_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__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 (168)
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_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_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__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_List_Odrop_001_Eo,type,
drop_o: nat > list_o > list_o ).
thf(sy_c_List_Odrop_001t__List__Olist_I_Eo_J,type,
drop_list_o: nat > list_list_o > list_list_o ).
thf(sy_c_List_Odrop_001t__MDL__Oformula_Itf__a_Mtf__t_J,type,
drop_formula_a_t: nat > list_formula_a_t > list_formula_a_t ).
thf(sy_c_List_Odrop_001t__NFA__Otransition,type,
drop_transition: nat > list_transition > list_transition ).
thf(sy_c_List_Odrop_001t__Nat__Onat,type,
drop_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Odrop_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
drop_P8868858903918902087at_nat: nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_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_OCons_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
cons_P6512896166579812791at_nat: product_prod_nat_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_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_ONil_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
nil_Pr5478986624290739719at_nat: list_P6011104703257516679at_nat ).
thf(sy_c_List_Olist_Ohd_001_Eo,type,
hd_o: list_o > $o ).
thf(sy_c_List_Olist_Ohd_001t__List__Olist_I_Eo_J,type,
hd_list_o: list_list_o > list_o ).
thf(sy_c_List_Olist_Ohd_001t__MDL__Oformula_Itf__a_Mtf__t_J,type,
hd_formula_a_t: list_formula_a_t > formula_a_t ).
thf(sy_c_List_Olist_Ohd_001t__NFA__Otransition,type,
hd_transition: list_transition > transition ).
thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
hd_nat: list_nat > nat ).
thf(sy_c_List_Olist_Ohd_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
hd_Pro3460610213475200108at_nat: list_P6011104703257516679at_nat > product_prod_nat_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_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
set_Pr5648618587558075414at_nat: list_P6011104703257516679at_nat > set_Pr1261947904930325089at_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_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
nth_Pr7617993195940197384at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat ).
thf(sy_c_List_Otake_001_Eo,type,
take_o: nat > list_o > list_o ).
thf(sy_c_List_Otake_001t__List__Olist_I_Eo_J,type,
take_list_o: nat > list_list_o > list_list_o ).
thf(sy_c_List_Otake_001t__MDL__Oformula_Itf__a_Mtf__t_J,type,
take_formula_a_t: nat > list_formula_a_t > list_formula_a_t ).
thf(sy_c_List_Otake_001t__NFA__Otransition,type,
take_transition: nat > list_transition > list_transition ).
thf(sy_c_List_Otake_001t__Nat__Onat,type,
take_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Otake_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
take_P2173866234530122223at_nat: nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).
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_Orderive_001tf__a_001tf__t,type,
rderive_a_t: regex_a_t > regex_a_t ).
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_Ofmla__set,type,
fmla_set: transition > set_nat ).
thf(sy_c_NFA_Onfa,type,
nfa: nat > nat > list_transition > $o ).
thf(sy_c_NFA_Onfa__cong_H,type,
nfa_cong: 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__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__symb,type,
step_symb: nat > list_transition > nat > nat > $o ).
thf(sy_c_NFA_Ostep__symb__set,type,
step_symb_set: nat > list_transition > set_nat > set_nat ).
thf(sy_c_NFA_Otransition_Osplit__trans,type,
split_trans: nat > nat > transition ).
thf(sy_c_Nat_OSuc,type,
suc: nat > 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_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__MDL__Oformula_Itf__a_Mtf__t_J,type,
size_s4016968051272393527la_a_t: formula_a_t > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__MDL__Oregex_Itf__a_Mtf__t_J,type,
size_size_regex_a_t: regex_a_t > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
bot_bo2769642828321324397at_nat: product_prod_nat_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
bot_bo3047382831089536473et_nat: produc7819656566062154093et_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_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
ord_le1203424502768444845at_nat: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
ord_le4872869340735563107at_nat: produc7491599851749785783at_nat > produc7491599851749785783at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__NFA__Otransition_J,type,
ord_le5184432651266358346sition: set_transition > set_transition > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le7866589430770878221at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
ord_le8460144461188290721at_nat: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
ord_le4284901688344473943et_nat: produc2400336064389900727et_nat > produc2400336064389900727et_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
ord_le152793438849583191at_nat: produc7491599851749785783at_nat > produc7491599851749785783at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
ord_le2041963031926835469et_nat: produc7819656566062154093et_nat > produc7819656566062154093et_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
ord_less_eq_set_o: set_o > set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_I_Eo_J_J,type,
ord_le6901083488122529182list_o: set_list_o > set_list_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__MDL__Oformula_Itf__a_Mtf__t_J_J,type,
ord_le7457455060544393785la_a_t: set_formula_a_t > set_formula_a_t > $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_062_I_Eo_M_062_I_Eo_M_Eo_J_J_001t__List__Olist_I_Eo_J,type,
produc8744836578217649351list_o: ( $o > $o > $o ) > list_o > produc8642409424279824599list_o ).
thf(sy_c_Product__Type_OPair_001_062_It__List__Olist_I_Eo_J_M_062_It__List__Olist_I_Eo_J_M_Eo_J_J_001t__List__Olist_It__List__Olist_I_Eo_J_J,type,
produc5708246048495597389list_o: ( list_o > list_o > $o ) > list_list_o > produc867208691148388573list_o ).
thf(sy_c_Product__Type_OPair_001_062_It__NFA__Otransition_M_062_It__NFA__Otransition_M_Eo_J_J_001t__List__Olist_It__NFA__Otransition_J,type,
produc3648819927465945709sition: ( transition > transition > $o ) > list_transition > produc5558917482089586557sition ).
thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__List__Olist_It__MDL__Oformula_Itf__a_Mtf__t_J_J_Mt__List__Olist_It__MDL__Oformula_Itf__a_Mtf__t_J_J_J_J_001t__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,
produc4053737076228309398la_a_t: ( nat > list_formula_a_t > list_formula_a_t ) > produc238307629227160457la_a_t > produc920861655927507678la_a_t ).
thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc3209952032786966637at_nat: ( nat > nat > nat ) > produc7248412053542808358at_nat > produc4471711990508489141at_nat ).
thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__MDL__Oformula_Itf__a_Mtf__t_J_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__MDL__Oformula_Itf__a_Mtf__t_J_J_J_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__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_J,type,
produc8352854409798802573la_a_t: ( nat > produc8388488633478513124la_a_t > produc8388488633478513124la_a_t ) > produc3148207259879549498la_a_t > produc6783461406735195739la_a_t ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_I_Eo_J_001t__List__Olist_I_Eo_J,type,
produc8435520187683070743list_o: list_o > list_o > produc7102631898165422375list_o ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_I_Eo_J_J_001t__List__Olist_It__List__Olist_I_Eo_J_J,type,
produc2957346356364703511list_o: list_list_o > list_list_o > produc4690905340047322919list_o ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__NFA__Otransition_J_001t__List__Olist_It__NFA__Otransition_J,type,
produc6690642445140780183sition: list_transition > list_transition > produc1202413579072790439sition ).
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_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
produc487386426758144856at_nat: nat > product_prod_nat_nat > produc7248412053542808358at_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__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,
produc8590409963084727026la_a_t: nat > produc238307629227160457la_a_t > produc3148207259879549498la_a_t ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
produc4207506657711014383et_nat: nat > set_nat > produc2400336064389900727et_nat ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
produc641871753055645167at_nat: set_nat > nat > produc7491599851749785783at_nat ).
thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
produc4532415448927165861et_nat: set_nat > set_nat > produc7819656566062154093et_nat ).
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_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_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_Transitive__Closure_Ortrancl_001t__Nat__Onat,type,
transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__List__Olist_I_Eo_J,type,
member_list_o: list_o > set_list_o > $o ).
thf(sy_c_member_001t__MDL__Oformula_Itf__a_Mtf__t_J,type,
member_formula_a_t: formula_a_t > set_formula_a_t > $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_bsa____,type,
bsa: list_o ).
thf(sy_v_bsb____,type,
bsb: list_o ).
thf(sy_v_bssa____,type,
bssa: list_list_o ).
thf(sy_v_bssb____,type,
bssb: list_list_o ).
thf(sy_v_ia____,type,
ia: nat ).
thf(sy_v_ib____,type,
ib: nat ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_phisa____,type,
phisa: list_formula_a_t ).
thf(sy_v_q0a____,type,
q0a: nat ).
thf(sy_v_qfa____,type,
qfa: nat ).
thf(sy_v_ra____,type,
ra: regex_a_t ).
thf(sy_v_transsa____,type,
transsa: list_transition ).
% Relevant facts (758)
thf(fact_0_left_Oqf__not__in__SQ,axiom,
~ ( member_nat @ q0a @ ( sq @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) ) ) ).
% left.qf_not_in_SQ
thf(fact_1_length__collect__subfmlas,axiom,
! [Phis: list_formula_a_t,R: regex_a_t] : ( ord_less_eq_nat @ ( size_s8846756101701226951la_a_t @ Phis ) @ ( size_s8846756101701226951la_a_t @ ( collect_subfmlas_a_t @ R @ Phis ) ) ) ).
% length_collect_subfmlas
thf(fact_2_left_Ostep__symb__qf,axiom,
! [Q: nat] :
~ ( step_symb @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) @ q0a @ Q ) ).
% left.step_symb_qf
thf(fact_3_left_Ostep__eps__closure__qf,axiom,
! [Bs: list_o,Q: nat,Q2: nat] :
( ( step_eps_closure @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) @ Bs @ Q @ Q2 )
=> ( ( Q = q0a )
=> ( Q = Q2 ) ) ) ).
% left.step_eps_closure_qf
thf(fact_4_left__IH,axiom,
iH_a_t @ sigma @ ra @ ( plus_plus_nat @ q0a @ one_one_nat ) @ q0a @ phisa @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) @ bssb @ bsb @ ib ).
% left_IH
thf(fact_5_nth__drop,axiom,
! [N: nat,Xs: list_list_o,I: nat] :
( ( ord_less_eq_nat @ N @ ( size_s2710708370519433104list_o @ Xs ) )
=> ( ( nth_list_o @ ( drop_list_o @ N @ Xs ) @ I )
= ( nth_list_o @ Xs @ ( plus_plus_nat @ N @ I ) ) ) ) ).
% nth_drop
thf(fact_6_nth__drop,axiom,
! [N: nat,Xs: list_formula_a_t,I: nat] :
( ( ord_less_eq_nat @ N @ ( size_s8846756101701226951la_a_t @ Xs ) )
=> ( ( nth_formula_a_t @ ( drop_formula_a_t @ N @ Xs ) @ I )
= ( nth_formula_a_t @ Xs @ ( plus_plus_nat @ N @ I ) ) ) ) ).
% nth_drop
thf(fact_7_nth__drop,axiom,
! [N: nat,Xs: list_o,I: nat] :
( ( ord_less_eq_nat @ N @ ( size_size_list_o @ Xs ) )
=> ( ( nth_o @ ( drop_o @ N @ Xs ) @ I )
= ( nth_o @ Xs @ ( plus_plus_nat @ N @ I ) ) ) ) ).
% nth_drop
thf(fact_8_nth__drop,axiom,
! [N: nat,Xs: list_transition,I: nat] :
( ( ord_less_eq_nat @ N @ ( size_s3613142680436377136sition @ Xs ) )
=> ( ( nth_transition @ ( drop_transition @ N @ Xs ) @ I )
= ( nth_transition @ Xs @ ( plus_plus_nat @ N @ I ) ) ) ) ).
% nth_drop
thf(fact_9_left_Ostep__eps__qf,axiom,
! [Bs: list_o,Q: nat] :
~ ( step_eps @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) @ Bs @ q0a @ Q ) ).
% left.step_eps_qf
thf(fact_10_left_Onfa__axioms,axiom,
nfa @ ( plus_plus_nat @ q0a @ one_one_nat ) @ q0a @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) ).
% left.nfa_axioms
thf(fact_11_left_Otranss__not__Nil,axiom,
( ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) )
!= nil_transition ) ).
% left.transs_not_Nil
thf(fact_12_nth__take,axiom,
! [I: nat,N: nat,Xs: list_o] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_o @ ( take_o @ N @ Xs ) @ I )
= ( nth_o @ Xs @ I ) ) ) ).
% nth_take
thf(fact_13_nth__take,axiom,
! [I: nat,N: nat,Xs: list_formula_a_t] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_formula_a_t @ ( take_formula_a_t @ N @ Xs ) @ I )
= ( nth_formula_a_t @ Xs @ I ) ) ) ).
% nth_take
thf(fact_14_nth__take,axiom,
! [I: nat,N: nat,Xs: list_transition] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_transition @ ( take_transition @ N @ Xs ) @ I )
= ( nth_transition @ Xs @ I ) ) ) ).
% nth_take
thf(fact_15_nth__take,axiom,
! [I: nat,N: nat,Xs: list_list_o] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_list_o @ ( take_list_o @ N @ Xs ) @ I )
= ( nth_list_o @ Xs @ I ) ) ) ).
% nth_take
thf(fact_16_take__all,axiom,
! [Xs: list_list_o,N: nat] :
( ( ord_less_eq_nat @ ( size_s2710708370519433104list_o @ Xs ) @ N )
=> ( ( take_list_o @ N @ Xs )
= Xs ) ) ).
% take_all
thf(fact_17_take__all,axiom,
! [Xs: list_formula_a_t,N: nat] :
( ( ord_less_eq_nat @ ( size_s8846756101701226951la_a_t @ Xs ) @ N )
=> ( ( take_formula_a_t @ N @ Xs )
= Xs ) ) ).
% take_all
thf(fact_18_take__all,axiom,
! [Xs: list_o,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ N )
=> ( ( take_o @ N @ Xs )
= Xs ) ) ).
% take_all
thf(fact_19_take__all,axiom,
! [Xs: list_transition,N: nat] :
( ( ord_less_eq_nat @ ( size_s3613142680436377136sition @ Xs ) @ N )
=> ( ( take_transition @ N @ Xs )
= Xs ) ) ).
% take_all
thf(fact_20_take__all__iff,axiom,
! [N: nat,Xs: list_list_o] :
( ( ( take_list_o @ N @ Xs )
= Xs )
= ( ord_less_eq_nat @ ( size_s2710708370519433104list_o @ Xs ) @ N ) ) ).
% take_all_iff
thf(fact_21_take__all__iff,axiom,
! [N: nat,Xs: list_formula_a_t] :
( ( ( take_formula_a_t @ N @ Xs )
= Xs )
= ( ord_less_eq_nat @ ( size_s8846756101701226951la_a_t @ Xs ) @ N ) ) ).
% take_all_iff
thf(fact_22_take__all__iff,axiom,
! [N: nat,Xs: list_o] :
( ( ( take_o @ N @ Xs )
= Xs )
= ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ N ) ) ).
% take_all_iff
thf(fact_23_take__all__iff,axiom,
! [N: nat,Xs: list_transition] :
( ( ( take_transition @ N @ Xs )
= Xs )
= ( ord_less_eq_nat @ ( size_s3613142680436377136sition @ Xs ) @ N ) ) ).
% take_all_iff
thf(fact_24_less__prod__simp,axiom,
! [X1: nat,Y1: nat,X2: nat,Y2: nat] :
( ( ord_le1203424502768444845at_nat @ ( product_Pair_nat_nat @ X1 @ Y1 ) @ ( product_Pair_nat_nat @ X2 @ Y2 ) )
= ( ( ord_less_nat @ X1 @ X2 )
| ( ( ord_less_eq_nat @ X1 @ X2 )
& ( ord_less_nat @ Y1 @ Y2 ) ) ) ) ).
% less_prod_simp
thf(fact_25_less__prod__simp,axiom,
! [X1: set_nat,Y1: nat,X2: set_nat,Y2: nat] :
( ( ord_le4872869340735563107at_nat @ ( produc641871753055645167at_nat @ X1 @ Y1 ) @ ( produc641871753055645167at_nat @ X2 @ Y2 ) )
= ( ( ord_less_set_nat @ X1 @ X2 )
| ( ( ord_less_eq_set_nat @ X1 @ X2 )
& ( ord_less_nat @ Y1 @ Y2 ) ) ) ) ).
% less_prod_simp
thf(fact_26_less__eq__prod__simp,axiom,
! [X1: nat,Y1: nat,X2: nat,Y2: nat] :
( ( ord_le8460144461188290721at_nat @ ( product_Pair_nat_nat @ X1 @ Y1 ) @ ( product_Pair_nat_nat @ X2 @ Y2 ) )
= ( ( ord_less_nat @ X1 @ X2 )
| ( ( ord_less_eq_nat @ X1 @ X2 )
& ( ord_less_eq_nat @ Y1 @ Y2 ) ) ) ) ).
% less_eq_prod_simp
thf(fact_27_less__eq__prod__simp,axiom,
! [X1: nat,Y1: set_nat,X2: nat,Y2: set_nat] :
( ( ord_le4284901688344473943et_nat @ ( produc4207506657711014383et_nat @ X1 @ Y1 ) @ ( produc4207506657711014383et_nat @ X2 @ Y2 ) )
= ( ( ord_less_nat @ X1 @ X2 )
| ( ( ord_less_eq_nat @ X1 @ X2 )
& ( ord_less_eq_set_nat @ Y1 @ Y2 ) ) ) ) ).
% less_eq_prod_simp
thf(fact_28_less__eq__prod__simp,axiom,
! [X1: set_nat,Y1: nat,X2: set_nat,Y2: nat] :
( ( ord_le152793438849583191at_nat @ ( produc641871753055645167at_nat @ X1 @ Y1 ) @ ( produc641871753055645167at_nat @ X2 @ Y2 ) )
= ( ( ord_less_set_nat @ X1 @ X2 )
| ( ( ord_less_eq_set_nat @ X1 @ X2 )
& ( ord_less_eq_nat @ Y1 @ Y2 ) ) ) ) ).
% less_eq_prod_simp
thf(fact_29_less__eq__prod__simp,axiom,
! [X1: set_nat,Y1: set_nat,X2: set_nat,Y2: set_nat] :
( ( ord_le2041963031926835469et_nat @ ( produc4532415448927165861et_nat @ X1 @ Y1 ) @ ( produc4532415448927165861et_nat @ X2 @ Y2 ) )
= ( ( ord_less_set_nat @ X1 @ X2 )
| ( ( ord_less_eq_set_nat @ X1 @ X2 )
& ( ord_less_eq_set_nat @ Y1 @ Y2 ) ) ) ) ).
% less_eq_prod_simp
thf(fact_30_drop__drop,axiom,
! [N: nat,M: nat,Xs: list_list_o] :
( ( drop_list_o @ N @ ( drop_list_o @ M @ Xs ) )
= ( drop_list_o @ ( plus_plus_nat @ N @ M ) @ Xs ) ) ).
% drop_drop
thf(fact_31_drop__drop,axiom,
! [N: nat,M: nat,Xs: list_transition] :
( ( drop_transition @ N @ ( drop_transition @ M @ Xs ) )
= ( drop_transition @ ( plus_plus_nat @ N @ M ) @ Xs ) ) ).
% drop_drop
thf(fact_32_drop__drop,axiom,
! [N: nat,M: nat,Xs: list_o] :
( ( drop_o @ N @ ( drop_o @ M @ Xs ) )
= ( drop_o @ ( plus_plus_nat @ N @ M ) @ Xs ) ) ).
% drop_drop
thf(fact_33_drop__drop,axiom,
! [N: nat,M: nat,Xs: list_formula_a_t] :
( ( drop_formula_a_t @ N @ ( drop_formula_a_t @ M @ Xs ) )
= ( drop_formula_a_t @ ( plus_plus_nat @ N @ M ) @ Xs ) ) ).
% drop_drop
thf(fact_34_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_list_o] :
( ( nil_list_o
= ( drop_list_o @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_s2710708370519433104list_o @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_35_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_formula_a_t] :
( ( nil_formula_a_t
= ( drop_formula_a_t @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_s8846756101701226951la_a_t @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_36_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_o] :
( ( nil_o
= ( drop_o @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_37_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_transition] :
( ( nil_transition
= ( drop_transition @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_s3613142680436377136sition @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_38_drop__eq__Nil,axiom,
! [N: nat,Xs: list_list_o] :
( ( ( drop_list_o @ N @ Xs )
= nil_list_o )
= ( ord_less_eq_nat @ ( size_s2710708370519433104list_o @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_39_drop__eq__Nil,axiom,
! [N: nat,Xs: list_formula_a_t] :
( ( ( drop_formula_a_t @ N @ Xs )
= nil_formula_a_t )
= ( ord_less_eq_nat @ ( size_s8846756101701226951la_a_t @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_40_drop__eq__Nil,axiom,
! [N: nat,Xs: list_o] :
( ( ( drop_o @ N @ Xs )
= nil_o )
= ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_41_drop__eq__Nil,axiom,
! [N: nat,Xs: list_transition] :
( ( ( drop_transition @ N @ Xs )
= nil_transition )
= ( ord_less_eq_nat @ ( size_s3613142680436377136sition @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_42_drop__all,axiom,
! [Xs: list_list_o,N: nat] :
( ( ord_less_eq_nat @ ( size_s2710708370519433104list_o @ Xs ) @ N )
=> ( ( drop_list_o @ N @ Xs )
= nil_list_o ) ) ).
% drop_all
thf(fact_43_drop__all,axiom,
! [Xs: list_formula_a_t,N: nat] :
( ( ord_less_eq_nat @ ( size_s8846756101701226951la_a_t @ Xs ) @ N )
=> ( ( drop_formula_a_t @ N @ Xs )
= nil_formula_a_t ) ) ).
% drop_all
thf(fact_44_drop__all,axiom,
! [Xs: list_o,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ N )
=> ( ( drop_o @ N @ Xs )
= nil_o ) ) ).
% drop_all
thf(fact_45_drop__all,axiom,
! [Xs: list_transition,N: nat] :
( ( ord_less_eq_nat @ ( size_s3613142680436377136sition @ Xs ) @ N )
=> ( ( drop_transition @ N @ Xs )
= nil_transition ) ) ).
% drop_all
thf(fact_46_subset__code_I1_J,axiom,
! [Xs: list_P6011104703257516679at_nat,B: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ B )
= ( ! [X: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ( member8440522571783428010at_nat @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_47_subset__code_I1_J,axiom,
! [Xs: list_list_o,B: set_list_o] :
( ( ord_le6901083488122529182list_o @ ( set_list_o2 @ Xs ) @ B )
= ( ! [X: list_o] :
( ( member_list_o @ X @ ( set_list_o2 @ Xs ) )
=> ( member_list_o @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_48_subset__code_I1_J,axiom,
! [Xs: list_transition,B: set_transition] :
( ( ord_le8419162016481440574sition @ ( set_transition2 @ Xs ) @ B )
= ( ! [X: transition] :
( ( member_transition @ X @ ( set_transition2 @ Xs ) )
=> ( member_transition @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_49_subset__code_I1_J,axiom,
! [Xs: list_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_50_take__Nil,axiom,
! [N: nat] :
( ( take_formula_a_t @ N @ nil_formula_a_t )
= nil_formula_a_t ) ).
% take_Nil
thf(fact_51_take__Nil,axiom,
! [N: nat] :
( ( take_transition @ N @ nil_transition )
= nil_transition ) ).
% take_Nil
thf(fact_52_take__Nil,axiom,
! [N: nat] :
( ( take_o @ N @ nil_o )
= nil_o ) ).
% take_Nil
thf(fact_53_take__Nil,axiom,
! [N: nat] :
( ( take_list_o @ N @ nil_list_o )
= nil_list_o ) ).
% take_Nil
thf(fact_54_drop__Nil,axiom,
! [N: nat] :
( ( drop_formula_a_t @ N @ nil_formula_a_t )
= nil_formula_a_t ) ).
% drop_Nil
thf(fact_55_drop__Nil,axiom,
! [N: nat] :
( ( drop_transition @ N @ nil_transition )
= nil_transition ) ).
% drop_Nil
thf(fact_56_drop__Nil,axiom,
! [N: nat] :
( ( drop_o @ N @ nil_o )
= nil_o ) ).
% drop_Nil
thf(fact_57_drop__Nil,axiom,
! [N: nat] :
( ( drop_list_o @ N @ nil_list_o )
= nil_list_o ) ).
% drop_Nil
thf(fact_58_MDL_OIH_Ocong,axiom,
iH_a_t = iH_a_t ).
% MDL.IH.cong
thf(fact_59_set__take__subset,axiom,
! [N: nat,Xs: list_formula_a_t] : ( ord_le7457455060544393785la_a_t @ ( set_formula_a_t2 @ ( take_formula_a_t @ N @ Xs ) ) @ ( set_formula_a_t2 @ Xs ) ) ).
% set_take_subset
thf(fact_60_set__take__subset,axiom,
! [N: nat,Xs: list_o] : ( ord_less_eq_set_o @ ( set_o2 @ ( take_o @ N @ Xs ) ) @ ( set_o2 @ Xs ) ) ).
% set_take_subset
thf(fact_61_set__take__subset,axiom,
! [N: nat,Xs: list_list_o] : ( ord_le6901083488122529182list_o @ ( set_list_o2 @ ( take_list_o @ N @ Xs ) ) @ ( set_list_o2 @ Xs ) ) ).
% set_take_subset
thf(fact_62_set__take__subset,axiom,
! [N: nat,Xs: list_transition] : ( ord_le8419162016481440574sition @ ( set_transition2 @ ( take_transition @ N @ Xs ) ) @ ( set_transition2 @ Xs ) ) ).
% set_take_subset
thf(fact_63_set__take__subset,axiom,
! [N: nat,Xs: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( take_nat @ N @ Xs ) ) @ ( set_nat2 @ Xs ) ) ).
% set_take_subset
thf(fact_64_set__drop__subset,axiom,
! [N: nat,Xs: list_o] : ( ord_less_eq_set_o @ ( set_o2 @ ( drop_o @ N @ Xs ) ) @ ( set_o2 @ Xs ) ) ).
% set_drop_subset
thf(fact_65_set__drop__subset,axiom,
! [N: nat,Xs: list_formula_a_t] : ( ord_le7457455060544393785la_a_t @ ( set_formula_a_t2 @ ( drop_formula_a_t @ N @ Xs ) ) @ ( set_formula_a_t2 @ Xs ) ) ).
% set_drop_subset
thf(fact_66_set__drop__subset,axiom,
! [N: nat,Xs: list_list_o] : ( ord_le6901083488122529182list_o @ ( set_list_o2 @ ( drop_list_o @ N @ Xs ) ) @ ( set_list_o2 @ Xs ) ) ).
% set_drop_subset
thf(fact_67_set__drop__subset,axiom,
! [N: nat,Xs: list_transition] : ( ord_le8419162016481440574sition @ ( set_transition2 @ ( drop_transition @ N @ Xs ) ) @ ( set_transition2 @ Xs ) ) ).
% set_drop_subset
thf(fact_68_set__drop__subset,axiom,
! [N: nat,Xs: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( drop_nat @ N @ Xs ) ) @ ( set_nat2 @ Xs ) ) ).
% set_drop_subset
thf(fact_69_hd__in__set,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( member_nat @ ( hd_nat @ Xs ) @ ( set_nat2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_70_hd__in__set,axiom,
! [Xs: list_P6011104703257516679at_nat] :
( ( Xs != nil_Pr5478986624290739719at_nat )
=> ( member8440522571783428010at_nat @ ( hd_Pro3460610213475200108at_nat @ Xs ) @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).
% hd_in_set
thf(fact_71_hd__in__set,axiom,
! [Xs: list_o] :
( ( Xs != nil_o )
=> ( member_o @ ( hd_o @ Xs ) @ ( set_o2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_72_hd__in__set,axiom,
! [Xs: list_list_o] :
( ( Xs != nil_list_o )
=> ( member_list_o @ ( hd_list_o @ Xs ) @ ( set_list_o2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_73_hd__in__set,axiom,
! [Xs: list_transition] :
( ( Xs != nil_transition )
=> ( member_transition @ ( hd_transition @ Xs ) @ ( set_transition2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_74_list_Oset__sel_I1_J,axiom,
! [A: list_nat] :
( ( A != nil_nat )
=> ( member_nat @ ( hd_nat @ A ) @ ( set_nat2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_75_list_Oset__sel_I1_J,axiom,
! [A: list_P6011104703257516679at_nat] :
( ( A != nil_Pr5478986624290739719at_nat )
=> ( member8440522571783428010at_nat @ ( hd_Pro3460610213475200108at_nat @ A ) @ ( set_Pr5648618587558075414at_nat @ A ) ) ) ).
% list.set_sel(1)
thf(fact_76_list_Oset__sel_I1_J,axiom,
! [A: list_o] :
( ( A != nil_o )
=> ( member_o @ ( hd_o @ A ) @ ( set_o2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_77_list_Oset__sel_I1_J,axiom,
! [A: list_list_o] :
( ( A != nil_list_o )
=> ( member_list_o @ ( hd_list_o @ A ) @ ( set_list_o2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_78_list_Oset__sel_I1_J,axiom,
! [A: list_transition] :
( ( A != nil_transition )
=> ( member_transition @ ( hd_transition @ A ) @ ( set_transition2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_79_set__take__subset__set__take,axiom,
! [M: nat,N: nat,Xs: list_formula_a_t] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_le7457455060544393785la_a_t @ ( set_formula_a_t2 @ ( take_formula_a_t @ M @ Xs ) ) @ ( set_formula_a_t2 @ ( take_formula_a_t @ N @ Xs ) ) ) ) ).
% set_take_subset_set_take
thf(fact_80_set__take__subset__set__take,axiom,
! [M: nat,N: nat,Xs: list_o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_set_o @ ( set_o2 @ ( take_o @ M @ Xs ) ) @ ( set_o2 @ ( take_o @ N @ Xs ) ) ) ) ).
% set_take_subset_set_take
thf(fact_81_set__take__subset__set__take,axiom,
! [M: nat,N: nat,Xs: list_list_o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_le6901083488122529182list_o @ ( set_list_o2 @ ( take_list_o @ M @ Xs ) ) @ ( set_list_o2 @ ( take_list_o @ N @ Xs ) ) ) ) ).
% set_take_subset_set_take
thf(fact_82_set__take__subset__set__take,axiom,
! [M: nat,N: nat,Xs: list_transition] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_le8419162016481440574sition @ ( set_transition2 @ ( take_transition @ M @ Xs ) ) @ ( set_transition2 @ ( take_transition @ N @ Xs ) ) ) ) ).
% set_take_subset_set_take
thf(fact_83_set__take__subset__set__take,axiom,
! [M: nat,N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( take_nat @ M @ Xs ) ) @ ( set_nat2 @ ( take_nat @ N @ Xs ) ) ) ) ).
% set_take_subset_set_take
thf(fact_84_set__drop__subset__set__drop,axiom,
! [N: nat,M: nat,Xs: list_o] :
( ( ord_less_eq_nat @ N @ M )
=> ( ord_less_eq_set_o @ ( set_o2 @ ( drop_o @ M @ Xs ) ) @ ( set_o2 @ ( drop_o @ N @ Xs ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_85_set__drop__subset__set__drop,axiom,
! [N: nat,M: nat,Xs: list_formula_a_t] :
( ( ord_less_eq_nat @ N @ M )
=> ( ord_le7457455060544393785la_a_t @ ( set_formula_a_t2 @ ( drop_formula_a_t @ M @ Xs ) ) @ ( set_formula_a_t2 @ ( drop_formula_a_t @ N @ Xs ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_86_set__drop__subset__set__drop,axiom,
! [N: nat,M: nat,Xs: list_list_o] :
( ( ord_less_eq_nat @ N @ M )
=> ( ord_le6901083488122529182list_o @ ( set_list_o2 @ ( drop_list_o @ M @ Xs ) ) @ ( set_list_o2 @ ( drop_list_o @ N @ Xs ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_87_set__drop__subset__set__drop,axiom,
! [N: nat,M: nat,Xs: list_transition] :
( ( ord_less_eq_nat @ N @ M )
=> ( ord_le8419162016481440574sition @ ( set_transition2 @ ( drop_transition @ M @ Xs ) ) @ ( set_transition2 @ ( drop_transition @ N @ Xs ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_88_set__drop__subset__set__drop,axiom,
! [N: nat,M: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( drop_nat @ M @ Xs ) ) @ ( set_nat2 @ ( drop_nat @ N @ Xs ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_89_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_90_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_91_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_92_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_93_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_list_o] :
( ( size_s2710708370519433104list_o @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_94_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_formula_a_t] :
( ( size_s8846756101701226951la_a_t @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_95_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_o] :
( ( size_size_list_o @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_96_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_transition] :
( ( size_s3613142680436377136sition @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_97_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_98_take__equalityI,axiom,
! [Xs: list_list_o,Ys: list_list_o] :
( ! [I2: nat] :
( ( take_list_o @ I2 @ Xs )
= ( take_list_o @ I2 @ Ys ) )
=> ( Xs = Ys ) ) ).
% take_equalityI
thf(fact_99_take__equalityI,axiom,
! [Xs: list_transition,Ys: list_transition] :
( ! [I2: nat] :
( ( take_transition @ I2 @ Xs )
= ( take_transition @ I2 @ Ys ) )
=> ( Xs = Ys ) ) ).
% take_equalityI
thf(fact_100_take__equalityI,axiom,
! [Xs: list_formula_a_t,Ys: list_formula_a_t] :
( ! [I2: nat] :
( ( take_formula_a_t @ I2 @ Xs )
= ( take_formula_a_t @ I2 @ Ys ) )
=> ( Xs = Ys ) ) ).
% take_equalityI
thf(fact_101_take__equalityI,axiom,
! [Xs: list_o,Ys: list_o] :
( ! [I2: nat] :
( ( take_o @ I2 @ Xs )
= ( take_o @ I2 @ Ys ) )
=> ( Xs = Ys ) ) ).
% take_equalityI
thf(fact_102_length__induct,axiom,
! [P: list_list_o > $o,Xs: list_list_o] :
( ! [Xs2: list_list_o] :
( ! [Ys2: list_list_o] :
( ( ord_less_nat @ ( size_s2710708370519433104list_o @ Ys2 ) @ ( size_s2710708370519433104list_o @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_103_length__induct,axiom,
! [P: list_formula_a_t > $o,Xs: list_formula_a_t] :
( ! [Xs2: list_formula_a_t] :
( ! [Ys2: list_formula_a_t] :
( ( ord_less_nat @ ( size_s8846756101701226951la_a_t @ Ys2 ) @ ( size_s8846756101701226951la_a_t @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_104_length__induct,axiom,
! [P: list_o > $o,Xs: list_o] :
( ! [Xs2: list_o] :
( ! [Ys2: list_o] :
( ( ord_less_nat @ ( size_size_list_o @ Ys2 ) @ ( size_size_list_o @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_105_length__induct,axiom,
! [P: list_transition > $o,Xs: list_transition] :
( ! [Xs2: list_transition] :
( ! [Ys2: list_transition] :
( ( ord_less_nat @ ( size_s3613142680436377136sition @ Ys2 ) @ ( size_s3613142680436377136sition @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_106_in__set__takeD,axiom,
! [X3: formula_a_t,N: nat,Xs: list_formula_a_t] :
( ( member_formula_a_t @ X3 @ ( set_formula_a_t2 @ ( take_formula_a_t @ N @ Xs ) ) )
=> ( member_formula_a_t @ X3 @ ( set_formula_a_t2 @ Xs ) ) ) ).
% in_set_takeD
thf(fact_107_in__set__takeD,axiom,
! [X3: $o,N: nat,Xs: list_o] :
( ( member_o @ X3 @ ( set_o2 @ ( take_o @ N @ Xs ) ) )
=> ( member_o @ X3 @ ( set_o2 @ Xs ) ) ) ).
% in_set_takeD
thf(fact_108_in__set__takeD,axiom,
! [X3: nat,N: nat,Xs: list_nat] :
( ( member_nat @ X3 @ ( set_nat2 @ ( take_nat @ N @ Xs ) ) )
=> ( member_nat @ X3 @ ( set_nat2 @ Xs ) ) ) ).
% in_set_takeD
thf(fact_109_in__set__takeD,axiom,
! [X3: product_prod_nat_nat,N: nat,Xs: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ ( take_P2173866234530122223at_nat @ N @ Xs ) ) )
=> ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).
% in_set_takeD
thf(fact_110_in__set__takeD,axiom,
! [X3: list_o,N: nat,Xs: list_list_o] :
( ( member_list_o @ X3 @ ( set_list_o2 @ ( take_list_o @ N @ Xs ) ) )
=> ( member_list_o @ X3 @ ( set_list_o2 @ Xs ) ) ) ).
% in_set_takeD
thf(fact_111_in__set__takeD,axiom,
! [X3: transition,N: nat,Xs: list_transition] :
( ( member_transition @ X3 @ ( set_transition2 @ ( take_transition @ N @ Xs ) ) )
=> ( member_transition @ X3 @ ( set_transition2 @ Xs ) ) ) ).
% in_set_takeD
thf(fact_112_in__set__dropD,axiom,
! [X3: $o,N: nat,Xs: list_o] :
( ( member_o @ X3 @ ( set_o2 @ ( drop_o @ N @ Xs ) ) )
=> ( member_o @ X3 @ ( set_o2 @ Xs ) ) ) ).
% in_set_dropD
thf(fact_113_in__set__dropD,axiom,
! [X3: formula_a_t,N: nat,Xs: list_formula_a_t] :
( ( member_formula_a_t @ X3 @ ( set_formula_a_t2 @ ( drop_formula_a_t @ N @ Xs ) ) )
=> ( member_formula_a_t @ X3 @ ( set_formula_a_t2 @ Xs ) ) ) ).
% in_set_dropD
thf(fact_114_in__set__dropD,axiom,
! [X3: nat,N: nat,Xs: list_nat] :
( ( member_nat @ X3 @ ( set_nat2 @ ( drop_nat @ N @ Xs ) ) )
=> ( member_nat @ X3 @ ( set_nat2 @ Xs ) ) ) ).
% in_set_dropD
thf(fact_115_in__set__dropD,axiom,
! [X3: product_prod_nat_nat,N: nat,Xs: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ ( drop_P8868858903918902087at_nat @ N @ Xs ) ) )
=> ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).
% in_set_dropD
thf(fact_116_in__set__dropD,axiom,
! [X3: list_o,N: nat,Xs: list_list_o] :
( ( member_list_o @ X3 @ ( set_list_o2 @ ( drop_list_o @ N @ Xs ) ) )
=> ( member_list_o @ X3 @ ( set_list_o2 @ Xs ) ) ) ).
% in_set_dropD
thf(fact_117_in__set__dropD,axiom,
! [X3: transition,N: nat,Xs: list_transition] :
( ( member_transition @ X3 @ ( set_transition2 @ ( drop_transition @ N @ Xs ) ) )
=> ( member_transition @ X3 @ ( set_transition2 @ Xs ) ) ) ).
% in_set_dropD
thf(fact_118_list__eq__iff__nth__eq,axiom,
( ( ^ [Y: list_list_o,Z: list_list_o] : ( Y = Z ) )
= ( ^ [Xs3: list_list_o,Ys3: list_list_o] :
( ( ( size_s2710708370519433104list_o @ Xs3 )
= ( size_s2710708370519433104list_o @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s2710708370519433104list_o @ Xs3 ) )
=> ( ( nth_list_o @ Xs3 @ I3 )
= ( nth_list_o @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_119_list__eq__iff__nth__eq,axiom,
( ( ^ [Y: list_formula_a_t,Z: list_formula_a_t] : ( Y = Z ) )
= ( ^ [Xs3: list_formula_a_t,Ys3: list_formula_a_t] :
( ( ( size_s8846756101701226951la_a_t @ Xs3 )
= ( size_s8846756101701226951la_a_t @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s8846756101701226951la_a_t @ Xs3 ) )
=> ( ( nth_formula_a_t @ Xs3 @ I3 )
= ( nth_formula_a_t @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_120_list__eq__iff__nth__eq,axiom,
( ( ^ [Y: list_o,Z: list_o] : ( Y = Z ) )
= ( ^ [Xs3: list_o,Ys3: list_o] :
( ( ( size_size_list_o @ Xs3 )
= ( size_size_list_o @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs3 ) )
=> ( ( nth_o @ Xs3 @ I3 )
= ( nth_o @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_121_list__eq__iff__nth__eq,axiom,
( ( ^ [Y: list_transition,Z: list_transition] : ( Y = Z ) )
= ( ^ [Xs3: list_transition,Ys3: list_transition] :
( ( ( size_s3613142680436377136sition @ Xs3 )
= ( size_s3613142680436377136sition @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3613142680436377136sition @ Xs3 ) )
=> ( ( nth_transition @ Xs3 @ I3 )
= ( nth_transition @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_122_Skolem__list__nth,axiom,
! [K: nat,P: nat > list_o > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X4: list_o] : ( P @ I3 @ X4 ) ) )
= ( ? [Xs3: list_list_o] :
( ( ( size_s2710708370519433104list_o @ Xs3 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_list_o @ Xs3 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_123_Skolem__list__nth,axiom,
! [K: nat,P: nat > formula_a_t > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X4: formula_a_t] : ( P @ I3 @ X4 ) ) )
= ( ? [Xs3: list_formula_a_t] :
( ( ( size_s8846756101701226951la_a_t @ Xs3 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_formula_a_t @ Xs3 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_124_Skolem__list__nth,axiom,
! [K: nat,P: nat > $o > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X4: $o] : ( P @ I3 @ X4 ) ) )
= ( ? [Xs3: list_o] :
( ( ( size_size_list_o @ Xs3 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_o @ Xs3 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_125_Skolem__list__nth,axiom,
! [K: nat,P: nat > transition > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X4: transition] : ( P @ I3 @ X4 ) ) )
= ( ? [Xs3: list_transition] :
( ( ( size_s3613142680436377136sition @ Xs3 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_transition @ Xs3 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_126_nth__equalityI,axiom,
! [Xs: list_list_o,Ys: list_list_o] :
( ( ( size_s2710708370519433104list_o @ Xs )
= ( size_s2710708370519433104list_o @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s2710708370519433104list_o @ Xs ) )
=> ( ( nth_list_o @ Xs @ I2 )
= ( nth_list_o @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_127_nth__equalityI,axiom,
! [Xs: list_formula_a_t,Ys: list_formula_a_t] :
( ( ( size_s8846756101701226951la_a_t @ Xs )
= ( size_s8846756101701226951la_a_t @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s8846756101701226951la_a_t @ Xs ) )
=> ( ( nth_formula_a_t @ Xs @ I2 )
= ( nth_formula_a_t @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_128_nth__equalityI,axiom,
! [Xs: list_o,Ys: list_o] :
( ( ( size_size_list_o @ Xs )
= ( size_size_list_o @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
=> ( ( nth_o @ Xs @ I2 )
= ( nth_o @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_129_nth__equalityI,axiom,
! [Xs: list_transition,Ys: list_transition] :
( ( ( size_s3613142680436377136sition @ Xs )
= ( size_s3613142680436377136sition @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3613142680436377136sition @ Xs ) )
=> ( ( nth_transition @ Xs @ I2 )
= ( nth_transition @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_130_take__drop,axiom,
! [N: nat,M: nat,Xs: list_list_o] :
( ( take_list_o @ N @ ( drop_list_o @ M @ Xs ) )
= ( drop_list_o @ M @ ( take_list_o @ ( plus_plus_nat @ N @ M ) @ Xs ) ) ) ).
% take_drop
thf(fact_131_take__drop,axiom,
! [N: nat,M: nat,Xs: list_transition] :
( ( take_transition @ N @ ( drop_transition @ M @ Xs ) )
= ( drop_transition @ M @ ( take_transition @ ( plus_plus_nat @ N @ M ) @ Xs ) ) ) ).
% take_drop
thf(fact_132_take__drop,axiom,
! [N: nat,M: nat,Xs: list_formula_a_t] :
( ( take_formula_a_t @ N @ ( drop_formula_a_t @ M @ Xs ) )
= ( drop_formula_a_t @ M @ ( take_formula_a_t @ ( plus_plus_nat @ N @ M ) @ Xs ) ) ) ).
% take_drop
thf(fact_133_take__drop,axiom,
! [N: nat,M: nat,Xs: list_o] :
( ( take_o @ N @ ( drop_o @ M @ Xs ) )
= ( drop_o @ M @ ( take_o @ ( plus_plus_nat @ N @ M ) @ Xs ) ) ) ).
% take_drop
thf(fact_134_all__set__conv__all__nth,axiom,
! [Xs: list_list_o,P: list_o > $o] :
( ( ! [X: list_o] :
( ( member_list_o @ X @ ( set_list_o2 @ Xs ) )
=> ( P @ X ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s2710708370519433104list_o @ Xs ) )
=> ( P @ ( nth_list_o @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_135_all__set__conv__all__nth,axiom,
! [Xs: list_formula_a_t,P: formula_a_t > $o] :
( ( ! [X: formula_a_t] :
( ( member_formula_a_t @ X @ ( set_formula_a_t2 @ Xs ) )
=> ( P @ X ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s8846756101701226951la_a_t @ Xs ) )
=> ( P @ ( nth_formula_a_t @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_136_all__set__conv__all__nth,axiom,
! [Xs: list_o,P: $o > $o] :
( ( ! [X: $o] :
( ( member_o @ X @ ( set_o2 @ Xs ) )
=> ( P @ X ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
=> ( P @ ( nth_o @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_137_all__set__conv__all__nth,axiom,
! [Xs: list_transition,P: transition > $o] :
( ( ! [X: transition] :
( ( member_transition @ X @ ( set_transition2 @ Xs ) )
=> ( P @ X ) ) )
= ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3613142680436377136sition @ Xs ) )
=> ( P @ ( nth_transition @ Xs @ I3 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_138_all__nth__imp__all__set,axiom,
! [Xs: list_nat,P: nat > $o,X3: nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I2 ) ) )
=> ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( P @ X3 ) ) ) ).
% all_nth_imp_all_set
thf(fact_139_all__nth__imp__all__set,axiom,
! [Xs: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,X3: product_prod_nat_nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ I2 ) ) )
=> ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
=> ( P @ X3 ) ) ) ).
% all_nth_imp_all_set
thf(fact_140_all__nth__imp__all__set,axiom,
! [Xs: list_list_o,P: list_o > $o,X3: list_o] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s2710708370519433104list_o @ Xs ) )
=> ( P @ ( nth_list_o @ Xs @ I2 ) ) )
=> ( ( member_list_o @ X3 @ ( set_list_o2 @ Xs ) )
=> ( P @ X3 ) ) ) ).
% all_nth_imp_all_set
thf(fact_141_all__nth__imp__all__set,axiom,
! [Xs: list_formula_a_t,P: formula_a_t > $o,X3: formula_a_t] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s8846756101701226951la_a_t @ Xs ) )
=> ( P @ ( nth_formula_a_t @ Xs @ I2 ) ) )
=> ( ( member_formula_a_t @ X3 @ ( set_formula_a_t2 @ Xs ) )
=> ( P @ X3 ) ) ) ).
% all_nth_imp_all_set
thf(fact_142_all__nth__imp__all__set,axiom,
! [Xs: list_o,P: $o > $o,X3: $o] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
=> ( P @ ( nth_o @ Xs @ I2 ) ) )
=> ( ( member_o @ X3 @ ( set_o2 @ Xs ) )
=> ( P @ X3 ) ) ) ).
% all_nth_imp_all_set
thf(fact_143_all__nth__imp__all__set,axiom,
! [Xs: list_transition,P: transition > $o,X3: transition] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3613142680436377136sition @ Xs ) )
=> ( P @ ( nth_transition @ Xs @ I2 ) ) )
=> ( ( member_transition @ X3 @ ( set_transition2 @ Xs ) )
=> ( P @ X3 ) ) ) ).
% all_nth_imp_all_set
thf(fact_144_in__set__conv__nth,axiom,
! [X3: nat,Xs: list_nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I3 )
= X3 ) ) ) ) ).
% in_set_conv_nth
thf(fact_145_in__set__conv__nth,axiom,
! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs ) )
& ( ( nth_Pr7617993195940197384at_nat @ Xs @ I3 )
= X3 ) ) ) ) ).
% in_set_conv_nth
thf(fact_146_in__set__conv__nth,axiom,
! [X3: list_o,Xs: list_list_o] :
( ( member_list_o @ X3 @ ( set_list_o2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s2710708370519433104list_o @ Xs ) )
& ( ( nth_list_o @ Xs @ I3 )
= X3 ) ) ) ) ).
% in_set_conv_nth
thf(fact_147_in__set__conv__nth,axiom,
! [X3: formula_a_t,Xs: list_formula_a_t] :
( ( member_formula_a_t @ X3 @ ( set_formula_a_t2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s8846756101701226951la_a_t @ Xs ) )
& ( ( nth_formula_a_t @ Xs @ I3 )
= X3 ) ) ) ) ).
% in_set_conv_nth
thf(fact_148_in__set__conv__nth,axiom,
! [X3: $o,Xs: list_o] :
( ( member_o @ X3 @ ( set_o2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
& ( ( nth_o @ Xs @ I3 )
= X3 ) ) ) ) ).
% in_set_conv_nth
thf(fact_149_in__set__conv__nth,axiom,
! [X3: transition,Xs: list_transition] :
( ( member_transition @ X3 @ ( set_transition2 @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3613142680436377136sition @ Xs ) )
& ( ( nth_transition @ Xs @ I3 )
= X3 ) ) ) ) ).
% in_set_conv_nth
thf(fact_150_list__ball__nth,axiom,
! [N: nat,Xs: list_list_o,P: list_o > $o] :
( ( ord_less_nat @ N @ ( size_s2710708370519433104list_o @ Xs ) )
=> ( ! [X5: list_o] :
( ( member_list_o @ X5 @ ( set_list_o2 @ Xs ) )
=> ( P @ X5 ) )
=> ( P @ ( nth_list_o @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_151_list__ball__nth,axiom,
! [N: nat,Xs: list_formula_a_t,P: formula_a_t > $o] :
( ( ord_less_nat @ N @ ( size_s8846756101701226951la_a_t @ Xs ) )
=> ( ! [X5: formula_a_t] :
( ( member_formula_a_t @ X5 @ ( set_formula_a_t2 @ Xs ) )
=> ( P @ X5 ) )
=> ( P @ ( nth_formula_a_t @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_152_list__ball__nth,axiom,
! [N: nat,Xs: list_o,P: $o > $o] :
( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
=> ( ! [X5: $o] :
( ( member_o @ X5 @ ( set_o2 @ Xs ) )
=> ( P @ X5 ) )
=> ( P @ ( nth_o @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_153_list__ball__nth,axiom,
! [N: nat,Xs: list_transition,P: transition > $o] :
( ( ord_less_nat @ N @ ( size_s3613142680436377136sition @ Xs ) )
=> ( ! [X5: transition] :
( ( member_transition @ X5 @ ( set_transition2 @ Xs ) )
=> ( P @ X5 ) )
=> ( P @ ( nth_transition @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_154_nth__mem,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_155_nth__mem,axiom,
! [N: nat,Xs: list_P6011104703257516679at_nat] :
( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs ) )
=> ( member8440522571783428010at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ N ) @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).
% nth_mem
thf(fact_156_nth__mem,axiom,
! [N: nat,Xs: list_list_o] :
( ( ord_less_nat @ N @ ( size_s2710708370519433104list_o @ Xs ) )
=> ( member_list_o @ ( nth_list_o @ Xs @ N ) @ ( set_list_o2 @ Xs ) ) ) ).
% nth_mem
thf(fact_157_nth__mem,axiom,
! [N: nat,Xs: list_formula_a_t] :
( ( ord_less_nat @ N @ ( size_s8846756101701226951la_a_t @ Xs ) )
=> ( member_formula_a_t @ ( nth_formula_a_t @ Xs @ N ) @ ( set_formula_a_t2 @ Xs ) ) ) ).
% nth_mem
thf(fact_158_nth__mem,axiom,
! [N: nat,Xs: list_o] :
( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
=> ( member_o @ ( nth_o @ Xs @ N ) @ ( set_o2 @ Xs ) ) ) ).
% nth_mem
thf(fact_159_nth__mem,axiom,
! [N: nat,Xs: list_transition] :
( ( ord_less_nat @ N @ ( size_s3613142680436377136sition @ Xs ) )
=> ( member_transition @ ( nth_transition @ Xs @ N ) @ ( set_transition2 @ Xs ) ) ) ).
% nth_mem
thf(fact_160_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_161_mem__Collect__eq,axiom,
! [A: transition,P: transition > $o] :
( ( member_transition @ A @ ( collect_transition @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_162_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_163_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_164_Collect__mem__eq,axiom,
! [A2: set_transition] :
( ( collect_transition
@ ^ [X: transition] : ( member_transition @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_165_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_166_nth__take__lemma,axiom,
! [K: nat,Xs: list_list_o,Ys: list_list_o] :
( ( ord_less_eq_nat @ K @ ( size_s2710708370519433104list_o @ Xs ) )
=> ( ( ord_less_eq_nat @ K @ ( size_s2710708370519433104list_o @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( nth_list_o @ Xs @ I2 )
= ( nth_list_o @ Ys @ I2 ) ) )
=> ( ( take_list_o @ K @ Xs )
= ( take_list_o @ K @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_167_nth__take__lemma,axiom,
! [K: nat,Xs: list_formula_a_t,Ys: list_formula_a_t] :
( ( ord_less_eq_nat @ K @ ( size_s8846756101701226951la_a_t @ Xs ) )
=> ( ( ord_less_eq_nat @ K @ ( size_s8846756101701226951la_a_t @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( nth_formula_a_t @ Xs @ I2 )
= ( nth_formula_a_t @ Ys @ I2 ) ) )
=> ( ( take_formula_a_t @ K @ Xs )
= ( take_formula_a_t @ K @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_168_nth__take__lemma,axiom,
! [K: nat,Xs: list_o,Ys: list_o] :
( ( ord_less_eq_nat @ K @ ( size_size_list_o @ Xs ) )
=> ( ( ord_less_eq_nat @ K @ ( size_size_list_o @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( nth_o @ Xs @ I2 )
= ( nth_o @ Ys @ I2 ) ) )
=> ( ( take_o @ K @ Xs )
= ( take_o @ K @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_169_nth__take__lemma,axiom,
! [K: nat,Xs: list_transition,Ys: list_transition] :
( ( ord_less_eq_nat @ K @ ( size_s3613142680436377136sition @ Xs ) )
=> ( ( ord_less_eq_nat @ K @ ( size_s3613142680436377136sition @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( nth_transition @ Xs @ I2 )
= ( nth_transition @ Ys @ I2 ) ) )
=> ( ( take_transition @ K @ Xs )
= ( take_transition @ K @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_170_hd__drop__conv__nth,axiom,
! [N: nat,Xs: list_list_o] :
( ( ord_less_nat @ N @ ( size_s2710708370519433104list_o @ Xs ) )
=> ( ( hd_list_o @ ( drop_list_o @ N @ Xs ) )
= ( nth_list_o @ Xs @ N ) ) ) ).
% hd_drop_conv_nth
thf(fact_171_hd__drop__conv__nth,axiom,
! [N: nat,Xs: list_formula_a_t] :
( ( ord_less_nat @ N @ ( size_s8846756101701226951la_a_t @ Xs ) )
=> ( ( hd_formula_a_t @ ( drop_formula_a_t @ N @ Xs ) )
= ( nth_formula_a_t @ Xs @ N ) ) ) ).
% hd_drop_conv_nth
thf(fact_172_hd__drop__conv__nth,axiom,
! [N: nat,Xs: list_o] :
( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
=> ( ( hd_o @ ( drop_o @ N @ Xs ) )
= ( nth_o @ Xs @ N ) ) ) ).
% hd_drop_conv_nth
thf(fact_173_hd__drop__conv__nth,axiom,
! [N: nat,Xs: list_transition] :
( ( ord_less_nat @ N @ ( size_s3613142680436377136sition @ Xs ) )
=> ( ( hd_transition @ ( drop_transition @ N @ Xs ) )
= ( nth_transition @ Xs @ N ) ) ) ).
% hd_drop_conv_nth
thf(fact_174_left_Ostep__symb__closed,axiom,
! [Q: nat,Q2: nat] :
( ( step_symb @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) @ Q @ Q2 )
=> ( member_nat @ Q2 @ ( q @ ( plus_plus_nat @ q0a @ one_one_nat ) @ q0a @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) ) ) ) ).
% left.step_symb_closed
thf(fact_175_left_Ostep__eps__closure__closed,axiom,
! [Bs: list_o,Q: nat,Q2: nat] :
( ( step_eps_closure @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) @ Bs @ Q @ Q2 )
=> ( ( Q != Q2 )
=> ( member_nat @ Q2 @ ( q @ ( plus_plus_nat @ q0a @ one_one_nat ) @ q0a @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) ) ) ) ) ).
% left.step_eps_closure_closed
thf(fact_176_left_Ostep__eps__closed,axiom,
! [Bs: list_o,Q: nat,Q2: nat] :
( ( step_eps @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) @ Bs @ Q @ Q2 )
=> ( member_nat @ Q2 @ ( q @ ( plus_plus_nat @ q0a @ one_one_nat ) @ q0a @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) ) ) ) ).
% left.step_eps_closed
thf(fact_177_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_178_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_179_add__less__cancel__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_nat @ A @ B2 ) ) ).
% add_less_cancel_right
thf(fact_180_add__less__cancel__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_nat @ A @ B2 ) ) ).
% add_less_cancel_left
thf(fact_181_add__le__cancel__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_cancel_right
thf(fact_182_add__le__cancel__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_cancel_left
thf(fact_183_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_184_add__left__cancel,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ A @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_185_add__right__cancel,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_186_left_Ostep__eps__set__closed,axiom,
! [Bs: list_o,R2: set_nat] : ( ord_less_eq_set_nat @ ( step_eps_set @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) @ Bs @ R2 ) @ ( q @ ( plus_plus_nat @ q0a @ one_one_nat ) @ q0a @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) ) ) ).
% left.step_eps_set_closed
thf(fact_187_left_Ostate__closed,axiom,
! [T: transition] :
( ( member_transition @ T @ ( set_transition2 @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) ) )
=> ( ord_less_eq_set_nat @ ( state_set @ T ) @ ( q @ ( plus_plus_nat @ q0a @ one_one_nat ) @ q0a @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) ) ) ) ).
% left.state_closed
thf(fact_188_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_189_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_190_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_191_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_192_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_193_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_194_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_195_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B2: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_196_group__cancel_Oadd2,axiom,
! [B: nat,K: nat,B2: nat,A: nat] :
( ( B
= ( plus_plus_nat @ K @ B2 ) )
=> ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_197_add_Oassoc,axiom,
! [A: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_198_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_199_add_Oleft__commute,axiom,
! [B2: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_200_add__left__imp__eq,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ A @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_201_add__right__imp__eq,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_202_one__reorient,axiom,
! [X3: nat] :
( ( one_one_nat = X3 )
= ( X3 = one_one_nat ) ) ).
% one_reorient
thf(fact_203_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_204_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_205_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_206_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_207_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_208_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_209_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_210_linorder__neqE__nat,axiom,
! [X3: nat,Y3: nat] :
( ( X3 != Y3 )
=> ( ~ ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ Y3 @ X3 ) ) ) ).
% linorder_neqE_nat
thf(fact_211_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_212_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_213_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_214_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_215_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_216_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B2 ) )
=> ? [X5: nat] :
( ( P @ X5 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X5 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_217_bounded__Max__nat,axiom,
! [P: nat > $o,X3: nat,M3: nat] :
( ( P @ X3 )
=> ( ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M3 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_218_size__neq__size__imp__neq,axiom,
! [X3: list_list_o,Y3: list_list_o] :
( ( ( size_s2710708370519433104list_o @ X3 )
!= ( size_s2710708370519433104list_o @ Y3 ) )
=> ( X3 != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_219_size__neq__size__imp__neq,axiom,
! [X3: list_formula_a_t,Y3: list_formula_a_t] :
( ( ( size_s8846756101701226951la_a_t @ X3 )
!= ( size_s8846756101701226951la_a_t @ Y3 ) )
=> ( X3 != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_220_size__neq__size__imp__neq,axiom,
! [X3: list_o,Y3: list_o] :
( ( ( size_size_list_o @ X3 )
!= ( size_size_list_o @ Y3 ) )
=> ( X3 != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_221_size__neq__size__imp__neq,axiom,
! [X3: list_transition,Y3: list_transition] :
( ( ( size_s3613142680436377136sition @ X3 )
!= ( size_s3613142680436377136sition @ Y3 ) )
=> ( X3 != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_222_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_223_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_224_add__mono__thms__linordered__semiring_I1_J,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_mono_thms_linordered_semiring(1)
thf(fact_225_ordered__ab__semigroup__add__class_Oadd__mono,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% ordered_ab_semigroup_add_class.add_mono
thf(fact_226_add__left__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_227_less__eqE,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ~ ! [C2: nat] :
( B2
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_228_add__right__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_229_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
? [C3: nat] :
( B3
= ( plus_plus_nat @ A3 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_230_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_231_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_232_add__mono__thms__linordered__field_I5_J,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_mono_thms_linordered_field(5)
thf(fact_233_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_234_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_235_add__strict__mono,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_236_add__strict__left__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_237_add__strict__right__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_238_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_nat @ A @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_239_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_nat @ A @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_240_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
& ( M5 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_241_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_242_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
| ( M5 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_243_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_244_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_245_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_246_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_247_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_248_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_249_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_250_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_251_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_252_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_253_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_254_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_255_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_256_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_257_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_258_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_259_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_260_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_261_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_262_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_263_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_264_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N3: nat] :
? [K2: nat] :
( N3
= ( plus_plus_nat @ M5 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_265_fold__atLeastAtMost__nat_Ocases,axiom,
! [X3: produc6783461406735195739la_a_t] :
~ ! [F2: nat > produc8388488633478513124la_a_t > produc8388488633478513124la_a_t,A4: nat,B4: nat,Acc: produc8388488633478513124la_a_t] :
( X3
!= ( produc8352854409798802573la_a_t @ F2 @ ( produc8590409963084727026la_a_t @ A4 @ ( produc8654416511292156347la_a_t @ B4 @ Acc ) ) ) ) ).
% fold_atLeastAtMost_nat.cases
thf(fact_266_fold__atLeastAtMost__nat_Ocases,axiom,
! [X3: produc4471711990508489141at_nat] :
~ ! [F2: nat > nat > nat,A4: nat,B4: nat,Acc: nat] :
( X3
!= ( produc3209952032786966637at_nat @ F2 @ ( produc487386426758144856at_nat @ A4 @ ( product_Pair_nat_nat @ B4 @ Acc ) ) ) ) ).
% fold_atLeastAtMost_nat.cases
thf(fact_267_fold__atLeastAtMost__nat_Ocases,axiom,
! [X3: produc920861655927507678la_a_t] :
~ ! [F2: nat > list_formula_a_t > list_formula_a_t,A4: nat,B4: nat,Acc: list_formula_a_t] :
( X3
!= ( produc4053737076228309398la_a_t @ F2 @ ( produc8654416511292156347la_a_t @ A4 @ ( produc9017461973804568604la_a_t @ B4 @ Acc ) ) ) ) ).
% fold_atLeastAtMost_nat.cases
thf(fact_268_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_269_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_270_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_271_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_272_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_273_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_274_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_275_step__eps__closure__empty,axiom,
! [Q0: nat,Transs: list_transition,Bs: list_o,Q: nat,Q2: nat] :
( ( step_eps_closure @ Q0 @ Transs @ Bs @ Q @ Q2 )
=> ( ! [Q3: nat] :
~ ( step_eps @ Q0 @ Transs @ Bs @ Q @ Q3 )
=> ( Q = Q2 ) ) ) ).
% step_eps_closure_empty
thf(fact_276_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_277_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_278_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_279_add__le__less__mono,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_280_add__less__le__mono,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_281_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_282_left_Ostep__symb__set__closed,axiom,
! [R2: set_nat] : ( ord_less_eq_set_nat @ ( step_symb_set @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) @ R2 ) @ ( q @ ( plus_plus_nat @ q0a @ one_one_nat ) @ q0a @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) ) ) ).
% left.step_symb_set_closed
thf(fact_283_left_Orun__closed,axiom,
! [R2: set_nat,Bss: list_list_o] :
( ( ord_less_eq_set_nat @ R2 @ ( q @ ( plus_plus_nat @ q0a @ one_one_nat ) @ q0a @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) ) )
=> ( ord_less_eq_set_nat @ ( run @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) @ R2 @ Bss ) @ ( q @ ( plus_plus_nat @ q0a @ one_one_nat ) @ q0a @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) ) ) ) ).
% left.run_closed
thf(fact_284_left_Odelta__closed,axiom,
! [R2: set_nat,Bs: list_o] : ( ord_less_eq_set_nat @ ( delta @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) @ R2 @ Bs ) @ ( q @ ( plus_plus_nat @ q0a @ one_one_nat ) @ q0a @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) ) ) ).
% left.delta_closed
thf(fact_285_left_Ostep__eps__closure__set__closed,axiom,
! [R2: set_nat,Bs: list_o] :
( ( ord_less_eq_set_nat @ R2 @ ( q @ ( plus_plus_nat @ q0a @ one_one_nat ) @ q0a @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) ) )
=> ( ord_less_eq_set_nat @ ( step_eps_closure_set @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) @ R2 @ Bs ) @ ( q @ ( plus_plus_nat @ q0a @ one_one_nat ) @ q0a @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) ) ) ) ).
% left.step_eps_closure_set_closed
thf(fact_286_build__nfa__impl__fmla__set,axiom,
! [T: transition,R: regex_a_t,Q0: nat,Qf: nat,Phis: list_formula_a_t,N: nat] :
( ( member_transition @ T @ ( set_transition2 @ ( build_nfa_impl_a_t @ R @ ( produc8654416511292156347la_a_t @ Q0 @ ( produc9017461973804568604la_a_t @ Qf @ Phis ) ) ) ) )
=> ( ( member_nat @ N @ ( fmla_set @ T ) )
=> ( ord_less_nat @ N @ ( size_s8846756101701226951la_a_t @ ( collect_subfmlas_a_t @ R @ Phis ) ) ) ) ) ).
% build_nfa_impl_fmla_set
thf(fact_287_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_288_add__mono1,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B2 @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_289_left_Orun__closed__Cons,axiom,
! [R2: set_nat,Bs: list_o,Bss: list_list_o] : ( ord_less_eq_set_nat @ ( run @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) @ R2 @ ( cons_list_o @ Bs @ Bss ) ) @ ( q @ ( plus_plus_nat @ q0a @ one_one_nat ) @ q0a @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) ) ) ).
% left.run_closed_Cons
thf(fact_290_left_Oq0__sub__SQ,axiom,
ord_less_eq_set_nat @ ( insert_nat @ ( plus_plus_nat @ q0a @ one_one_nat ) @ bot_bot_set_nat ) @ ( sq @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) ) ).
% left.q0_sub_SQ
thf(fact_291_subsetI,axiom,
! [A2: set_transition,B: set_transition] :
( ! [X5: transition] :
( ( member_transition @ X5 @ A2 )
=> ( member_transition @ X5 @ B ) )
=> ( ord_le8419162016481440574sition @ A2 @ B ) ) ).
% subsetI
thf(fact_292_subsetI,axiom,
! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ! [X5: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X5 @ A2 )
=> ( member8440522571783428010at_nat @ X5 @ B ) )
=> ( ord_le3146513528884898305at_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_293_subsetI,axiom,
! [A2: set_nat,B: set_nat] :
( ! [X5: nat] :
( ( member_nat @ X5 @ A2 )
=> ( member_nat @ X5 @ B ) )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_294_psubsetI,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_set_nat @ A2 @ B ) ) ) ).
% psubsetI
thf(fact_295_empty__iff,axiom,
! [C: transition] :
~ ( member_transition @ C @ bot_bo301567166201926666sition ) ).
% empty_iff
thf(fact_296_empty__iff,axiom,
! [C: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ C @ bot_bo2099793752762293965at_nat ) ).
% empty_iff
thf(fact_297_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_298_all__not__in__conv,axiom,
! [A2: set_transition] :
( ( ! [X: transition] :
~ ( member_transition @ X @ A2 ) )
= ( A2 = bot_bo301567166201926666sition ) ) ).
% all_not_in_conv
thf(fact_299_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_300_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_301_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_302_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_303_subset__antisym,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_304_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_305_insertCI,axiom,
! [A: nat,B: set_nat,B2: nat] :
( ( ~ ( member_nat @ A @ B )
=> ( A = B2 ) )
=> ( member_nat @ A @ ( insert_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_306_insertCI,axiom,
! [A: transition,B: set_transition,B2: transition] :
( ( ~ ( member_transition @ A @ B )
=> ( A = B2 ) )
=> ( member_transition @ A @ ( insert_transition @ B2 @ B ) ) ) ).
% insertCI
thf(fact_307_insertCI,axiom,
! [A: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat,B2: product_prod_nat_nat] :
( ( ~ ( member8440522571783428010at_nat @ A @ B )
=> ( A = B2 ) )
=> ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_308_insert__iff,axiom,
! [A: nat,B2: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_309_insert__iff,axiom,
! [A: transition,B2: transition,A2: set_transition] :
( ( member_transition @ A @ ( insert_transition @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_transition @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_310_insert__iff,axiom,
! [A: product_prod_nat_nat,B2: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member8440522571783428010at_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_311_insert__absorb2,axiom,
! [X3: nat,A2: set_nat] :
( ( insert_nat @ X3 @ ( insert_nat @ X3 @ A2 ) )
= ( insert_nat @ X3 @ A2 ) ) ).
% insert_absorb2
thf(fact_312_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_313_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_314_singletonI,axiom,
! [A: transition] : ( member_transition @ A @ ( insert_transition @ A @ bot_bo301567166201926666sition ) ) ).
% singletonI
thf(fact_315_singletonI,axiom,
! [A: product_prod_nat_nat] : ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).
% singletonI
thf(fact_316_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_317_insert__subset,axiom,
! [X3: transition,A2: set_transition,B: set_transition] :
( ( ord_le8419162016481440574sition @ ( insert_transition @ X3 @ A2 ) @ B )
= ( ( member_transition @ X3 @ B )
& ( ord_le8419162016481440574sition @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_318_insert__subset,axiom,
! [X3: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ X3 @ A2 ) @ B )
= ( ( member8440522571783428010at_nat @ X3 @ B )
& ( ord_le3146513528884898305at_nat @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_319_insert__subset,axiom,
! [X3: nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X3 @ A2 ) @ B )
= ( ( member_nat @ X3 @ B )
& ( ord_less_eq_set_nat @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_320_singleton__insert__inj__eq,axiom,
! [B2: nat,A: nat,A2: set_nat] :
( ( ( insert_nat @ B2 @ bot_bot_set_nat )
= ( insert_nat @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_321_singleton__insert__inj__eq_H,axiom,
! [A: nat,A2: set_nat,B2: nat] :
( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B2 @ bot_bot_set_nat ) )
= ( ( A = B2 )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_322_set__empty,axiom,
! [Xs: list_o] :
( ( ( set_o2 @ Xs )
= bot_bot_set_o )
= ( Xs = nil_o ) ) ).
% set_empty
thf(fact_323_set__empty,axiom,
! [Xs: list_list_o] :
( ( ( set_list_o2 @ Xs )
= bot_bot_set_list_o )
= ( Xs = nil_list_o ) ) ).
% set_empty
thf(fact_324_set__empty,axiom,
! [Xs: list_transition] :
( ( ( set_transition2 @ Xs )
= bot_bo301567166201926666sition )
= ( Xs = nil_transition ) ) ).
% set_empty
thf(fact_325_set__empty,axiom,
! [Xs: list_nat] :
( ( ( set_nat2 @ Xs )
= bot_bot_set_nat )
= ( Xs = nil_nat ) ) ).
% set_empty
thf(fact_326_set__empty2,axiom,
! [Xs: list_o] :
( ( bot_bot_set_o
= ( set_o2 @ Xs ) )
= ( Xs = nil_o ) ) ).
% set_empty2
thf(fact_327_set__empty2,axiom,
! [Xs: list_list_o] :
( ( bot_bot_set_list_o
= ( set_list_o2 @ Xs ) )
= ( Xs = nil_list_o ) ) ).
% set_empty2
thf(fact_328_set__empty2,axiom,
! [Xs: list_transition] :
( ( bot_bo301567166201926666sition
= ( set_transition2 @ Xs ) )
= ( Xs = nil_transition ) ) ).
% set_empty2
thf(fact_329_set__empty2,axiom,
! [Xs: list_nat] :
( ( bot_bot_set_nat
= ( set_nat2 @ Xs ) )
= ( Xs = nil_nat ) ) ).
% set_empty2
thf(fact_330_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_331_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_332_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_333_left_Ostep__eps__closure__set__qf,axiom,
! [Bs: list_o] :
( ( step_eps_closure_set @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) @ ( insert_nat @ q0a @ bot_bot_set_nat ) @ Bs )
= ( insert_nat @ q0a @ bot_bot_set_nat ) ) ).
% left.step_eps_closure_set_qf
thf(fact_334_left_Odelta__qf,axiom,
! [Bs: list_o] :
( ( delta @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) @ ( insert_nat @ q0a @ bot_bot_set_nat ) @ Bs )
= bot_bot_set_nat ) ).
% left.delta_qf
thf(fact_335_left_Ostep__symb__set__qf,axiom,
( ( step_symb_set @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) @ ( insert_nat @ q0a @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ).
% left.step_symb_set_qf
thf(fact_336_left_Oq0__sub__Q,axiom,
ord_less_eq_set_nat @ ( insert_nat @ ( plus_plus_nat @ q0a @ one_one_nat ) @ bot_bot_set_nat ) @ ( q @ ( plus_plus_nat @ q0a @ one_one_nat ) @ q0a @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) ) ).
% left.q0_sub_Q
thf(fact_337_left_Orun__qf__many,axiom,
! [Bs: list_o,Bss: list_list_o] :
( ( run @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) @ ( insert_nat @ q0a @ bot_bot_set_nat ) @ ( cons_list_o @ Bs @ Bss ) )
= bot_bot_set_nat ) ).
% left.run_qf_many
thf(fact_338_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_339_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_340_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_341_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_342_run__Nil,axiom,
! [Q0: nat,Transs: list_transition,R2: set_nat] :
( ( run @ Q0 @ Transs @ R2 @ nil_list_o )
= R2 ) ).
% run_Nil
thf(fact_343_subset__singletonD,axiom,
! [A2: set_nat,X3: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
=> ( ( A2 = bot_bot_set_nat )
| ( A2
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_344_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_345_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_346_subset__singleton__iff,axiom,
! [X7: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X7 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( ( X7 = bot_bot_set_nat )
| ( X7
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_347_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_348_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_349_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_350_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_351_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_352_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_353_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_354_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_355_NFA_Odelta__def,axiom,
( delta
= ( ^ [Q02: nat,Transs2: list_transition,R3: set_nat,Bs2: list_o] : ( step_symb_set @ Q02 @ Transs2 @ ( step_eps_closure_set @ Q02 @ Transs2 @ R3 @ Bs2 ) ) ) ) ).
% NFA.delta_def
thf(fact_356_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_357_bot__prod__def,axiom,
( bot_bo2769642828321324397at_nat
= ( product_Pair_nat_nat @ bot_bot_nat @ bot_bot_nat ) ) ).
% bot_prod_def
thf(fact_358_bot__prod__def,axiom,
( bot_bo3047382831089536473et_nat
= ( produc4532415448927165861et_nat @ bot_bot_set_nat @ bot_bot_set_nat ) ) ).
% bot_prod_def
thf(fact_359_emptyE,axiom,
! [A: transition] :
~ ( member_transition @ A @ bot_bo301567166201926666sition ) ).
% emptyE
thf(fact_360_emptyE,axiom,
! [A: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ A @ bot_bo2099793752762293965at_nat ) ).
% emptyE
thf(fact_361_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_362_insertE,axiom,
! [A: nat,B2: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_363_insertE,axiom,
! [A: transition,B2: transition,A2: set_transition] :
( ( member_transition @ A @ ( insert_transition @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_transition @ A @ A2 ) ) ) ).
% insertE
thf(fact_364_insertE,axiom,
! [A: product_prod_nat_nat,B2: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member8440522571783428010at_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_365_equals0D,axiom,
! [A2: set_transition,A: transition] :
( ( A2 = bot_bo301567166201926666sition )
=> ~ ( member_transition @ A @ A2 ) ) ).
% equals0D
thf(fact_366_equals0D,axiom,
! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
( ( A2 = bot_bo2099793752762293965at_nat )
=> ~ ( member8440522571783428010at_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_367_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_368_equals0I,axiom,
! [A2: set_transition] :
( ! [Y4: transition] :
~ ( member_transition @ Y4 @ A2 )
=> ( A2 = bot_bo301567166201926666sition ) ) ).
% equals0I
thf(fact_369_equals0I,axiom,
! [A2: set_Pr1261947904930325089at_nat] :
( ! [Y4: product_prod_nat_nat] :
~ ( member8440522571783428010at_nat @ Y4 @ A2 )
=> ( A2 = bot_bo2099793752762293965at_nat ) ) ).
% equals0I
thf(fact_370_equals0I,axiom,
! [A2: set_nat] :
( ! [Y4: nat] :
~ ( member_nat @ Y4 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_371_insertI1,axiom,
! [A: nat,B: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B ) ) ).
% insertI1
thf(fact_372_insertI1,axiom,
! [A: transition,B: set_transition] : ( member_transition @ A @ ( insert_transition @ A @ B ) ) ).
% insertI1
thf(fact_373_insertI1,axiom,
! [A: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] : ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ A @ B ) ) ).
% insertI1
thf(fact_374_insertI2,axiom,
! [A: nat,B: set_nat,B2: nat] :
( ( member_nat @ A @ B )
=> ( member_nat @ A @ ( insert_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_375_insertI2,axiom,
! [A: transition,B: set_transition,B2: transition] :
( ( member_transition @ A @ B )
=> ( member_transition @ A @ ( insert_transition @ B2 @ B ) ) ) ).
% insertI2
thf(fact_376_insertI2,axiom,
! [A: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat,B2: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ A @ B )
=> ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_377_ex__in__conv,axiom,
! [A2: set_transition] :
( ( ? [X: transition] : ( member_transition @ X @ A2 ) )
= ( A2 != bot_bo301567166201926666sition ) ) ).
% ex_in_conv
thf(fact_378_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_379_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X: nat] : ( member_nat @ X @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_380_Set_Oset__insert,axiom,
! [X3: nat,A2: set_nat] :
( ( member_nat @ X3 @ A2 )
=> ~ ! [B5: set_nat] :
( ( A2
= ( insert_nat @ X3 @ B5 ) )
=> ( member_nat @ X3 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_381_Set_Oset__insert,axiom,
! [X3: transition,A2: set_transition] :
( ( member_transition @ X3 @ A2 )
=> ~ ! [B5: set_transition] :
( ( A2
= ( insert_transition @ X3 @ B5 ) )
=> ( member_transition @ X3 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_382_Set_Oset__insert,axiom,
! [X3: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ X3 @ A2 )
=> ~ ! [B5: set_Pr1261947904930325089at_nat] :
( ( A2
= ( insert8211810215607154385at_nat @ X3 @ B5 ) )
=> ( member8440522571783428010at_nat @ X3 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_383_singletonD,axiom,
! [B2: transition,A: transition] :
( ( member_transition @ B2 @ ( insert_transition @ A @ bot_bo301567166201926666sition ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_384_singletonD,axiom,
! [B2: product_prod_nat_nat,A: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_385_singletonD,axiom,
! [B2: nat,A: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_386_insert__ident,axiom,
! [X3: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ X3 @ A2 )
=> ( ~ ( member_nat @ X3 @ B )
=> ( ( ( insert_nat @ X3 @ A2 )
= ( insert_nat @ X3 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_387_insert__ident,axiom,
! [X3: transition,A2: set_transition,B: set_transition] :
( ~ ( member_transition @ X3 @ A2 )
=> ( ~ ( member_transition @ X3 @ B )
=> ( ( ( insert_transition @ X3 @ A2 )
= ( insert_transition @ X3 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_388_insert__ident,axiom,
! [X3: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ~ ( member8440522571783428010at_nat @ X3 @ A2 )
=> ( ~ ( member8440522571783428010at_nat @ X3 @ B )
=> ( ( ( insert8211810215607154385at_nat @ X3 @ A2 )
= ( insert8211810215607154385at_nat @ X3 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_389_insert__absorb,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_390_insert__absorb,axiom,
! [A: transition,A2: set_transition] :
( ( member_transition @ A @ A2 )
=> ( ( insert_transition @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_391_insert__absorb,axiom,
! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A @ A2 )
=> ( ( insert8211810215607154385at_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_392_insert__eq__iff,axiom,
! [A: nat,A2: set_nat,B2: nat,B: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ B2 @ B )
=> ( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C4: set_nat] :
( ( A2
= ( insert_nat @ B2 @ C4 ) )
& ~ ( member_nat @ B2 @ C4 )
& ( B
= ( insert_nat @ A @ C4 ) )
& ~ ( member_nat @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_393_insert__eq__iff,axiom,
! [A: transition,A2: set_transition,B2: transition,B: set_transition] :
( ~ ( member_transition @ A @ A2 )
=> ( ~ ( member_transition @ B2 @ B )
=> ( ( ( insert_transition @ A @ A2 )
= ( insert_transition @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C4: set_transition] :
( ( A2
= ( insert_transition @ B2 @ C4 ) )
& ~ ( member_transition @ B2 @ C4 )
& ( B
= ( insert_transition @ A @ C4 ) )
& ~ ( member_transition @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_394_insert__eq__iff,axiom,
! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: product_prod_nat_nat,B: set_Pr1261947904930325089at_nat] :
( ~ ( member8440522571783428010at_nat @ A @ A2 )
=> ( ~ ( member8440522571783428010at_nat @ B2 @ B )
=> ( ( ( insert8211810215607154385at_nat @ A @ A2 )
= ( insert8211810215607154385at_nat @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C4: set_Pr1261947904930325089at_nat] :
( ( A2
= ( insert8211810215607154385at_nat @ B2 @ C4 ) )
& ~ ( member8440522571783428010at_nat @ B2 @ C4 )
& ( B
= ( insert8211810215607154385at_nat @ A @ C4 ) )
& ~ ( member8440522571783428010at_nat @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_395_singleton__iff,axiom,
! [B2: transition,A: transition] :
( ( member_transition @ B2 @ ( insert_transition @ A @ bot_bo301567166201926666sition ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_396_singleton__iff,axiom,
! [B2: product_prod_nat_nat,A: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ B2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_397_singleton__iff,axiom,
! [B2: nat,A: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_398_insert__commute,axiom,
! [X3: nat,Y3: nat,A2: set_nat] :
( ( insert_nat @ X3 @ ( insert_nat @ Y3 @ A2 ) )
= ( insert_nat @ Y3 @ ( insert_nat @ X3 @ A2 ) ) ) ).
% insert_commute
thf(fact_399_doubleton__eq__iff,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ( insert_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
= ( ( ( A = C )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_400_insert__not__empty,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ A2 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_401_singleton__inject,axiom,
! [A: nat,B2: nat] :
( ( ( insert_nat @ A @ bot_bot_set_nat )
= ( insert_nat @ B2 @ bot_bot_set_nat ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_402_mk__disjoint__insert,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ? [B5: set_nat] :
( ( A2
= ( insert_nat @ A @ B5 ) )
& ~ ( member_nat @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_403_mk__disjoint__insert,axiom,
! [A: transition,A2: set_transition] :
( ( member_transition @ A @ A2 )
=> ? [B5: set_transition] :
( ( A2
= ( insert_transition @ A @ B5 ) )
& ~ ( member_transition @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_404_mk__disjoint__insert,axiom,
! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
( ( member8440522571783428010at_nat @ A @ A2 )
=> ? [B5: set_Pr1261947904930325089at_nat] :
( ( A2
= ( insert8211810215607154385at_nat @ A @ B5 ) )
& ~ ( member8440522571783428010at_nat @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_405_not__psubset__empty,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_406_not__Cons__self2,axiom,
! [X3: list_o,Xs: list_list_o] :
( ( cons_list_o @ X3 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_407_psubsetD,axiom,
! [A2: set_nat,B: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B ) ) ) ).
% psubsetD
thf(fact_408_psubsetD,axiom,
! [A2: set_transition,B: set_transition,C: transition] :
( ( ord_le5184432651266358346sition @ A2 @ B )
=> ( ( member_transition @ C @ A2 )
=> ( member_transition @ C @ B ) ) ) ).
% psubsetD
thf(fact_409_psubsetD,axiom,
! [A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
( ( ord_le7866589430770878221at_nat @ A2 @ B )
=> ( ( member8440522571783428010at_nat @ C @ A2 )
=> ( member8440522571783428010at_nat @ C @ B ) ) ) ).
% psubsetD
thf(fact_410_transpose_Ocases,axiom,
! [X3: list_list_transition] :
( ( X3 != nil_list_transition )
=> ( ! [Xss: list_list_transition] :
( X3
!= ( cons_list_transition @ nil_transition @ Xss ) )
=> ~ ! [X5: transition,Xs2: list_transition,Xss: list_list_transition] :
( X3
!= ( cons_list_transition @ ( cons_transition @ X5 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_411_transpose_Ocases,axiom,
! [X3: list_list_list_o] :
( ( X3 != nil_list_list_o )
=> ( ! [Xss: list_list_list_o] :
( X3
!= ( cons_list_list_o @ nil_list_o @ Xss ) )
=> ~ ! [X5: list_o,Xs2: list_list_o,Xss: list_list_list_o] :
( X3
!= ( cons_list_list_o @ ( cons_list_o @ X5 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_412_transpose_Ocases,axiom,
! [X3: list_list_o] :
( ( X3 != nil_list_o )
=> ( ! [Xss: list_list_o] :
( X3
!= ( cons_list_o @ nil_o @ Xss ) )
=> ~ ! [X5: $o,Xs2: list_o,Xss: list_list_o] :
( X3
!= ( cons_list_o @ ( cons_o @ X5 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_413_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_414_subset__insertI2,axiom,
! [A2: set_nat,B: set_nat,B2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_415_subset__insertI,axiom,
! [B: set_nat,A: nat] : ( ord_less_eq_set_nat @ B @ ( insert_nat @ A @ B ) ) ).
% subset_insertI
thf(fact_416_subset__insert,axiom,
! [X3: transition,A2: set_transition,B: set_transition] :
( ~ ( member_transition @ X3 @ A2 )
=> ( ( ord_le8419162016481440574sition @ A2 @ ( insert_transition @ X3 @ B ) )
= ( ord_le8419162016481440574sition @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_417_subset__insert,axiom,
! [X3: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
( ~ ( member8440522571783428010at_nat @ X3 @ A2 )
=> ( ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ X3 @ B ) )
= ( ord_le3146513528884898305at_nat @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_418_subset__insert,axiom,
! [X3: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ X3 @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X3 @ B ) )
= ( ord_less_eq_set_nat @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_419_insert__mono,axiom,
! [C5: set_nat,D2: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ C5 @ D2 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A @ C5 ) @ ( insert_nat @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_420_list_Odistinct_I1_J,axiom,
! [X21: transition,X22: list_transition] :
( nil_transition
!= ( cons_transition @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_421_list_Odistinct_I1_J,axiom,
! [X21: $o,X22: list_o] :
( nil_o
!= ( cons_o @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_422_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_423_list_OdiscI,axiom,
! [List: list_transition,X21: transition,X22: list_transition] :
( ( List
= ( cons_transition @ X21 @ X22 ) )
=> ( List != nil_transition ) ) ).
% list.discI
thf(fact_424_list_OdiscI,axiom,
! [List: list_o,X21: $o,X22: list_o] :
( ( List
= ( cons_o @ X21 @ X22 ) )
=> ( List != nil_o ) ) ).
% list.discI
thf(fact_425_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_426_list_Oexhaust,axiom,
! [Y3: list_transition] :
( ( Y3 != nil_transition )
=> ~ ! [X212: transition,X222: list_transition] :
( Y3
!= ( cons_transition @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_427_list_Oexhaust,axiom,
! [Y3: list_o] :
( ( Y3 != nil_o )
=> ~ ! [X212: $o,X222: list_o] :
( Y3
!= ( cons_o @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_428_list_Oexhaust,axiom,
! [Y3: list_list_o] :
( ( Y3 != nil_list_o )
=> ~ ! [X212: list_o,X222: list_list_o] :
( Y3
!= ( cons_list_o @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_429_splice_Ocases,axiom,
! [X3: produc1202413579072790439sition] :
( ! [Ys4: list_transition] :
( X3
!= ( produc6690642445140780183sition @ nil_transition @ Ys4 ) )
=> ~ ! [X5: transition,Xs2: list_transition,Ys4: list_transition] :
( X3
!= ( produc6690642445140780183sition @ ( cons_transition @ X5 @ Xs2 ) @ Ys4 ) ) ) ).
% splice.cases
thf(fact_430_splice_Ocases,axiom,
! [X3: produc7102631898165422375list_o] :
( ! [Ys4: list_o] :
( X3
!= ( produc8435520187683070743list_o @ nil_o @ Ys4 ) )
=> ~ ! [X5: $o,Xs2: list_o,Ys4: list_o] :
( X3
!= ( produc8435520187683070743list_o @ ( cons_o @ X5 @ Xs2 ) @ Ys4 ) ) ) ).
% splice.cases
thf(fact_431_splice_Ocases,axiom,
! [X3: produc4690905340047322919list_o] :
( ! [Ys4: list_list_o] :
( X3
!= ( produc2957346356364703511list_o @ nil_list_o @ Ys4 ) )
=> ~ ! [X5: list_o,Xs2: list_list_o,Ys4: list_list_o] :
( X3
!= ( produc2957346356364703511list_o @ ( cons_list_o @ X5 @ Xs2 ) @ Ys4 ) ) ) ).
% splice.cases
thf(fact_432_min__list_Ocases,axiom,
! [X3: list_o] :
( ! [X5: $o,Xs2: list_o] :
( X3
!= ( cons_o @ X5 @ Xs2 ) )
=> ( X3 = nil_o ) ) ).
% min_list.cases
thf(fact_433_shuffles_Ocases,axiom,
! [X3: produc1202413579072790439sition] :
( ! [Ys4: list_transition] :
( X3
!= ( produc6690642445140780183sition @ nil_transition @ Ys4 ) )
=> ( ! [Xs2: list_transition] :
( X3
!= ( produc6690642445140780183sition @ Xs2 @ nil_transition ) )
=> ~ ! [X5: transition,Xs2: list_transition,Y4: transition,Ys4: list_transition] :
( X3
!= ( produc6690642445140780183sition @ ( cons_transition @ X5 @ Xs2 ) @ ( cons_transition @ Y4 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_434_shuffles_Ocases,axiom,
! [X3: produc7102631898165422375list_o] :
( ! [Ys4: list_o] :
( X3
!= ( produc8435520187683070743list_o @ nil_o @ Ys4 ) )
=> ( ! [Xs2: list_o] :
( X3
!= ( produc8435520187683070743list_o @ Xs2 @ nil_o ) )
=> ~ ! [X5: $o,Xs2: list_o,Y4: $o,Ys4: list_o] :
( X3
!= ( produc8435520187683070743list_o @ ( cons_o @ X5 @ Xs2 ) @ ( cons_o @ Y4 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_435_shuffles_Ocases,axiom,
! [X3: produc4690905340047322919list_o] :
( ! [Ys4: list_list_o] :
( X3
!= ( produc2957346356364703511list_o @ nil_list_o @ Ys4 ) )
=> ( ! [Xs2: list_list_o] :
( X3
!= ( produc2957346356364703511list_o @ Xs2 @ nil_list_o ) )
=> ~ ! [X5: list_o,Xs2: list_list_o,Y4: list_o,Ys4: list_list_o] :
( X3
!= ( produc2957346356364703511list_o @ ( cons_list_o @ X5 @ Xs2 ) @ ( cons_list_o @ Y4 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_436_sorted__wrt_Ocases,axiom,
! [X3: produc5558917482089586557sition] :
( ! [P2: transition > transition > $o] :
( X3
!= ( produc3648819927465945709sition @ P2 @ nil_transition ) )
=> ~ ! [P2: transition > transition > $o,X5: transition,Ys4: list_transition] :
( X3
!= ( produc3648819927465945709sition @ P2 @ ( cons_transition @ X5 @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_437_sorted__wrt_Ocases,axiom,
! [X3: produc8642409424279824599list_o] :
( ! [P2: $o > $o > $o] :
( X3
!= ( produc8744836578217649351list_o @ P2 @ nil_o ) )
=> ~ ! [P2: $o > $o > $o,X5: $o,Ys4: list_o] :
( X3
!= ( produc8744836578217649351list_o @ P2 @ ( cons_o @ X5 @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_438_sorted__wrt_Ocases,axiom,
! [X3: produc867208691148388573list_o] :
( ! [P2: list_o > list_o > $o] :
( X3
!= ( produc5708246048495597389list_o @ P2 @ nil_list_o ) )
=> ~ ! [P2: list_o > list_o > $o,X5: list_o,Ys4: list_list_o] :
( X3
!= ( produc5708246048495597389list_o @ P2 @ ( cons_list_o @ X5 @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_439_remdups__adj_Ocases,axiom,
! [X3: list_transition] :
( ( X3 != nil_transition )
=> ( ! [X5: transition] :
( X3
!= ( cons_transition @ X5 @ nil_transition ) )
=> ~ ! [X5: transition,Y4: transition,Xs2: list_transition] :
( X3
!= ( cons_transition @ X5 @ ( cons_transition @ Y4 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_440_remdups__adj_Ocases,axiom,
! [X3: list_o] :
( ( X3 != nil_o )
=> ( ! [X5: $o] :
( X3
!= ( cons_o @ X5 @ nil_o ) )
=> ~ ! [X5: $o,Y4: $o,Xs2: list_o] :
( X3
!= ( cons_o @ X5 @ ( cons_o @ Y4 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_441_remdups__adj_Ocases,axiom,
! [X3: list_list_o] :
( ( X3 != nil_list_o )
=> ( ! [X5: list_o] :
( X3
!= ( cons_list_o @ X5 @ nil_list_o ) )
=> ~ ! [X5: list_o,Y4: list_o,Xs2: list_list_o] :
( X3
!= ( cons_list_o @ X5 @ ( cons_list_o @ Y4 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_442_successively_Ocases,axiom,
! [X3: produc5558917482089586557sition] :
( ! [P2: transition > transition > $o] :
( X3
!= ( produc3648819927465945709sition @ P2 @ nil_transition ) )
=> ( ! [P2: transition > transition > $o,X5: transition] :
( X3
!= ( produc3648819927465945709sition @ P2 @ ( cons_transition @ X5 @ nil_transition ) ) )
=> ~ ! [P2: transition > transition > $o,X5: transition,Y4: transition,Xs2: list_transition] :
( X3
!= ( produc3648819927465945709sition @ P2 @ ( cons_transition @ X5 @ ( cons_transition @ Y4 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_443_successively_Ocases,axiom,
! [X3: produc8642409424279824599list_o] :
( ! [P2: $o > $o > $o] :
( X3
!= ( produc8744836578217649351list_o @ P2 @ nil_o ) )
=> ( ! [P2: $o > $o > $o,X5: $o] :
( X3
!= ( produc8744836578217649351list_o @ P2 @ ( cons_o @ X5 @ nil_o ) ) )
=> ~ ! [P2: $o > $o > $o,X5: $o,Y4: $o,Xs2: list_o] :
( X3
!= ( produc8744836578217649351list_o @ P2 @ ( cons_o @ X5 @ ( cons_o @ Y4 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_444_successively_Ocases,axiom,
! [X3: produc867208691148388573list_o] :
( ! [P2: list_o > list_o > $o] :
( X3
!= ( produc5708246048495597389list_o @ P2 @ nil_list_o ) )
=> ( ! [P2: list_o > list_o > $o,X5: list_o] :
( X3
!= ( produc5708246048495597389list_o @ P2 @ ( cons_list_o @ X5 @ nil_list_o ) ) )
=> ~ ! [P2: list_o > list_o > $o,X5: list_o,Y4: list_o,Xs2: list_list_o] :
( X3
!= ( produc5708246048495597389list_o @ P2 @ ( cons_list_o @ X5 @ ( cons_list_o @ Y4 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_445_neq__Nil__conv,axiom,
! [Xs: list_transition] :
( ( Xs != nil_transition )
= ( ? [Y6: transition,Ys3: list_transition] :
( Xs
= ( cons_transition @ Y6 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_446_neq__Nil__conv,axiom,
! [Xs: list_o] :
( ( Xs != nil_o )
= ( ? [Y6: $o,Ys3: list_o] :
( Xs
= ( cons_o @ Y6 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_447_neq__Nil__conv,axiom,
! [Xs: list_list_o] :
( ( Xs != nil_list_o )
= ( ? [Y6: list_o,Ys3: list_list_o] :
( Xs
= ( cons_list_o @ Y6 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_448_list__induct2_H,axiom,
! [P: list_transition > list_transition > $o,Xs: list_transition,Ys: list_transition] :
( ( P @ nil_transition @ nil_transition )
=> ( ! [X5: transition,Xs2: list_transition] : ( P @ ( cons_transition @ X5 @ Xs2 ) @ nil_transition )
=> ( ! [Y4: transition,Ys4: list_transition] : ( P @ nil_transition @ ( cons_transition @ Y4 @ Ys4 ) )
=> ( ! [X5: transition,Xs2: list_transition,Y4: transition,Ys4: list_transition] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_transition @ X5 @ Xs2 ) @ ( cons_transition @ Y4 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_449_list__induct2_H,axiom,
! [P: list_transition > list_o > $o,Xs: list_transition,Ys: list_o] :
( ( P @ nil_transition @ nil_o )
=> ( ! [X5: transition,Xs2: list_transition] : ( P @ ( cons_transition @ X5 @ Xs2 ) @ nil_o )
=> ( ! [Y4: $o,Ys4: list_o] : ( P @ nil_transition @ ( cons_o @ Y4 @ Ys4 ) )
=> ( ! [X5: transition,Xs2: list_transition,Y4: $o,Ys4: list_o] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_transition @ X5 @ Xs2 ) @ ( cons_o @ Y4 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_450_list__induct2_H,axiom,
! [P: list_o > list_transition > $o,Xs: list_o,Ys: list_transition] :
( ( P @ nil_o @ nil_transition )
=> ( ! [X5: $o,Xs2: list_o] : ( P @ ( cons_o @ X5 @ Xs2 ) @ nil_transition )
=> ( ! [Y4: transition,Ys4: list_transition] : ( P @ nil_o @ ( cons_transition @ Y4 @ Ys4 ) )
=> ( ! [X5: $o,Xs2: list_o,Y4: transition,Ys4: list_transition] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_o @ X5 @ Xs2 ) @ ( cons_transition @ Y4 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_451_list__induct2_H,axiom,
! [P: list_o > list_o > $o,Xs: list_o,Ys: list_o] :
( ( P @ nil_o @ nil_o )
=> ( ! [X5: $o,Xs2: list_o] : ( P @ ( cons_o @ X5 @ Xs2 ) @ nil_o )
=> ( ! [Y4: $o,Ys4: list_o] : ( P @ nil_o @ ( cons_o @ Y4 @ Ys4 ) )
=> ( ! [X5: $o,Xs2: list_o,Y4: $o,Ys4: list_o] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_o @ X5 @ Xs2 ) @ ( cons_o @ Y4 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_452_list__induct2_H,axiom,
! [P: list_transition > list_list_o > $o,Xs: list_transition,Ys: list_list_o] :
( ( P @ nil_transition @ nil_list_o )
=> ( ! [X5: transition,Xs2: list_transition] : ( P @ ( cons_transition @ X5 @ Xs2 ) @ nil_list_o )
=> ( ! [Y4: list_o,Ys4: list_list_o] : ( P @ nil_transition @ ( cons_list_o @ Y4 @ Ys4 ) )
=> ( ! [X5: transition,Xs2: list_transition,Y4: list_o,Ys4: list_list_o] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_transition @ X5 @ Xs2 ) @ ( cons_list_o @ Y4 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_453_list__induct2_H,axiom,
! [P: list_o > list_list_o > $o,Xs: list_o,Ys: list_list_o] :
( ( P @ nil_o @ nil_list_o )
=> ( ! [X5: $o,Xs2: list_o] : ( P @ ( cons_o @ X5 @ Xs2 ) @ nil_list_o )
=> ( ! [Y4: list_o,Ys4: list_list_o] : ( P @ nil_o @ ( cons_list_o @ Y4 @ Ys4 ) )
=> ( ! [X5: $o,Xs2: list_o,Y4: list_o,Ys4: list_list_o] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_o @ X5 @ Xs2 ) @ ( cons_list_o @ Y4 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_454_list__induct2_H,axiom,
! [P: list_list_o > list_transition > $o,Xs: list_list_o,Ys: list_transition] :
( ( P @ nil_list_o @ nil_transition )
=> ( ! [X5: list_o,Xs2: list_list_o] : ( P @ ( cons_list_o @ X5 @ Xs2 ) @ nil_transition )
=> ( ! [Y4: transition,Ys4: list_transition] : ( P @ nil_list_o @ ( cons_transition @ Y4 @ Ys4 ) )
=> ( ! [X5: list_o,Xs2: list_list_o,Y4: transition,Ys4: list_transition] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_list_o @ X5 @ Xs2 ) @ ( cons_transition @ Y4 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_455_list__induct2_H,axiom,
! [P: list_list_o > list_o > $o,Xs: list_list_o,Ys: list_o] :
( ( P @ nil_list_o @ nil_o )
=> ( ! [X5: list_o,Xs2: list_list_o] : ( P @ ( cons_list_o @ X5 @ Xs2 ) @ nil_o )
=> ( ! [Y4: $o,Ys4: list_o] : ( P @ nil_list_o @ ( cons_o @ Y4 @ Ys4 ) )
=> ( ! [X5: list_o,Xs2: list_list_o,Y4: $o,Ys4: list_o] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_list_o @ X5 @ Xs2 ) @ ( cons_o @ Y4 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_456_list__induct2_H,axiom,
! [P: list_list_o > list_list_o > $o,Xs: list_list_o,Ys: list_list_o] :
( ( P @ nil_list_o @ nil_list_o )
=> ( ! [X5: list_o,Xs2: list_list_o] : ( P @ ( cons_list_o @ X5 @ Xs2 ) @ nil_list_o )
=> ( ! [Y4: list_o,Ys4: list_list_o] : ( P @ nil_list_o @ ( cons_list_o @ Y4 @ Ys4 ) )
=> ( ! [X5: list_o,Xs2: list_list_o,Y4: list_o,Ys4: list_list_o] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_list_o @ X5 @ Xs2 ) @ ( cons_list_o @ Y4 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_457_list__nonempty__induct,axiom,
! [Xs: list_transition,P: list_transition > $o] :
( ( Xs != nil_transition )
=> ( ! [X5: transition] : ( P @ ( cons_transition @ X5 @ nil_transition ) )
=> ( ! [X5: transition,Xs2: list_transition] :
( ( Xs2 != nil_transition )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_transition @ X5 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_458_list__nonempty__induct,axiom,
! [Xs: list_o,P: list_o > $o] :
( ( Xs != nil_o )
=> ( ! [X5: $o] : ( P @ ( cons_o @ X5 @ nil_o ) )
=> ( ! [X5: $o,Xs2: list_o] :
( ( Xs2 != nil_o )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_o @ X5 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_459_list__nonempty__induct,axiom,
! [Xs: list_list_o,P: list_list_o > $o] :
( ( Xs != nil_list_o )
=> ( ! [X5: list_o] : ( P @ ( cons_list_o @ X5 @ nil_list_o ) )
=> ( ! [X5: list_o,Xs2: list_list_o] :
( ( Xs2 != nil_list_o )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_list_o @ X5 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_460_list__split_Ocases,axiom,
! [X3: list_transition] :
( ( X3 != nil_transition )
=> ~ ! [X5: transition,Xs2: list_transition] :
( X3
!= ( cons_transition @ X5 @ Xs2 ) ) ) ).
% list_split.cases
thf(fact_461_list__split_Ocases,axiom,
! [X3: list_o] :
( ( X3 != nil_o )
=> ~ ! [X5: $o,Xs2: list_o] :
( X3
!= ( cons_o @ X5 @ Xs2 ) ) ) ).
% list_split.cases
thf(fact_462_list__split_Ocases,axiom,
! [X3: list_list_o] :
( ( X3 != nil_list_o )
=> ~ ! [X5: list_o,Xs2: list_list_o] :
( X3
!= ( cons_list_o @ X5 @ Xs2 ) ) ) ).
% list_split.cases
thf(fact_463_set__ConsD,axiom,
! [Y3: nat,X3: nat,Xs: list_nat] :
( ( member_nat @ Y3 @ ( set_nat2 @ ( cons_nat @ X3 @ Xs ) ) )
=> ( ( Y3 = X3 )
| ( member_nat @ Y3 @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_464_set__ConsD,axiom,
! [Y3: product_prod_nat_nat,X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ Y3 @ ( set_Pr5648618587558075414at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) ) )
=> ( ( Y3 = X3 )
| ( member8440522571783428010at_nat @ Y3 @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_465_set__ConsD,axiom,
! [Y3: transition,X3: transition,Xs: list_transition] :
( ( member_transition @ Y3 @ ( set_transition2 @ ( cons_transition @ X3 @ Xs ) ) )
=> ( ( Y3 = X3 )
| ( member_transition @ Y3 @ ( set_transition2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_466_set__ConsD,axiom,
! [Y3: list_o,X3: list_o,Xs: list_list_o] :
( ( member_list_o @ Y3 @ ( set_list_o2 @ ( cons_list_o @ X3 @ Xs ) ) )
=> ( ( Y3 = X3 )
| ( member_list_o @ Y3 @ ( set_list_o2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_467_list_Oset__cases,axiom,
! [E: nat,A: list_nat] :
( ( member_nat @ E @ ( set_nat2 @ A ) )
=> ( ! [Z2: list_nat] :
( A
!= ( cons_nat @ E @ Z2 ) )
=> ~ ! [Z1: nat,Z2: list_nat] :
( ( A
= ( cons_nat @ Z1 @ Z2 ) )
=> ~ ( member_nat @ E @ ( set_nat2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_468_list_Oset__cases,axiom,
! [E: product_prod_nat_nat,A: list_P6011104703257516679at_nat] :
( ( member8440522571783428010at_nat @ E @ ( set_Pr5648618587558075414at_nat @ A ) )
=> ( ! [Z2: list_P6011104703257516679at_nat] :
( A
!= ( cons_P6512896166579812791at_nat @ E @ Z2 ) )
=> ~ ! [Z1: product_prod_nat_nat,Z2: list_P6011104703257516679at_nat] :
( ( A
= ( cons_P6512896166579812791at_nat @ Z1 @ Z2 ) )
=> ~ ( member8440522571783428010at_nat @ E @ ( set_Pr5648618587558075414at_nat @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_469_list_Oset__cases,axiom,
! [E: transition,A: list_transition] :
( ( member_transition @ E @ ( set_transition2 @ A ) )
=> ( ! [Z2: list_transition] :
( A
!= ( cons_transition @ E @ Z2 ) )
=> ~ ! [Z1: transition,Z2: list_transition] :
( ( A
= ( cons_transition @ Z1 @ Z2 ) )
=> ~ ( member_transition @ E @ ( set_transition2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_470_list_Oset__cases,axiom,
! [E: list_o,A: list_list_o] :
( ( member_list_o @ E @ ( set_list_o2 @ A ) )
=> ( ! [Z2: list_list_o] :
( A
!= ( cons_list_o @ E @ Z2 ) )
=> ~ ! [Z1: list_o,Z2: list_list_o] :
( ( A
= ( cons_list_o @ Z1 @ Z2 ) )
=> ~ ( member_list_o @ E @ ( set_list_o2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_471_list_Oset__intros_I1_J,axiom,
! [X21: nat,X22: list_nat] : ( member_nat @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_472_list_Oset__intros_I1_J,axiom,
! [X21: product_prod_nat_nat,X22: list_P6011104703257516679at_nat] : ( member8440522571783428010at_nat @ X21 @ ( set_Pr5648618587558075414at_nat @ ( cons_P6512896166579812791at_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_473_list_Oset__intros_I1_J,axiom,
! [X21: transition,X22: list_transition] : ( member_transition @ X21 @ ( set_transition2 @ ( cons_transition @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_474_list_Oset__intros_I1_J,axiom,
! [X21: list_o,X22: list_list_o] : ( member_list_o @ X21 @ ( set_list_o2 @ ( cons_list_o @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_475_list_Oset__intros_I2_J,axiom,
! [Y3: nat,X22: list_nat,X21: nat] :
( ( member_nat @ Y3 @ ( set_nat2 @ X22 ) )
=> ( member_nat @ Y3 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_476_list_Oset__intros_I2_J,axiom,
! [Y3: product_prod_nat_nat,X22: list_P6011104703257516679at_nat,X21: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ Y3 @ ( set_Pr5648618587558075414at_nat @ X22 ) )
=> ( member8440522571783428010at_nat @ Y3 @ ( set_Pr5648618587558075414at_nat @ ( cons_P6512896166579812791at_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_477_list_Oset__intros_I2_J,axiom,
! [Y3: transition,X22: list_transition,X21: transition] :
( ( member_transition @ Y3 @ ( set_transition2 @ X22 ) )
=> ( member_transition @ Y3 @ ( set_transition2 @ ( cons_transition @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_478_list_Oset__intros_I2_J,axiom,
! [Y3: list_o,X22: list_list_o,X21: list_o] :
( ( member_list_o @ Y3 @ ( set_list_o2 @ X22 ) )
=> ( member_list_o @ Y3 @ ( set_list_o2 @ ( cons_list_o @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_479_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_480_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_481_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_482_list_Osel_I1_J,axiom,
! [X21: list_o,X22: list_list_o] :
( ( hd_list_o @ ( cons_list_o @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_483_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_484_step__eps__closure__set__step__id,axiom,
! [R2: set_nat,Q0: nat,Transs: list_transition,Bs: list_o] :
( ! [Q4: nat,Q3: nat] :
( ( member_nat @ Q4 @ R2 )
=> ~ ( step_eps @ Q0 @ Transs @ Bs @ Q4 @ Q3 ) )
=> ( ( step_eps_closure_set @ Q0 @ Transs @ R2 @ Bs )
= R2 ) ) ).
% step_eps_closure_set_step_id
thf(fact_485_collect__subfmlas__size,axiom,
! [X3: formula_a_t,R: regex_a_t] :
( ( member_formula_a_t @ X3 @ ( set_formula_a_t2 @ ( collect_subfmlas_a_t @ R @ nil_formula_a_t ) ) )
=> ( ord_less_nat @ ( size_s4016968051272393527la_a_t @ X3 ) @ ( size_size_regex_a_t @ R ) ) ) ).
% collect_subfmlas_size
thf(fact_486_set__subset__Cons,axiom,
! [Xs: list_transition,X3: transition] : ( ord_le8419162016481440574sition @ ( set_transition2 @ Xs ) @ ( set_transition2 @ ( cons_transition @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_487_set__subset__Cons,axiom,
! [Xs: list_list_o,X3: list_o] : ( ord_le6901083488122529182list_o @ ( set_list_o2 @ Xs ) @ ( set_list_o2 @ ( cons_list_o @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_488_set__subset__Cons,axiom,
! [Xs: list_nat,X3: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X3 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_489_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 )
=> ( ! [X5: $o,Xs2: list_o,Y4: $o,Ys4: list_o] :
( ( ( size_size_list_o @ Xs2 )
= ( size_size_list_o @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_o @ X5 @ Xs2 ) @ ( cons_o @ Y4 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_490_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 )
=> ( ! [X5: $o,Xs2: list_o,Y4: transition,Ys4: list_transition] :
( ( ( size_size_list_o @ Xs2 )
= ( size_s3613142680436377136sition @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_o @ X5 @ Xs2 ) @ ( cons_transition @ Y4 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_491_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 )
=> ( ! [X5: transition,Xs2: list_transition,Y4: $o,Ys4: list_o] :
( ( ( size_s3613142680436377136sition @ Xs2 )
= ( size_size_list_o @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_transition @ X5 @ Xs2 ) @ ( cons_o @ Y4 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_492_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 )
=> ( ! [X5: transition,Xs2: list_transition,Y4: transition,Ys4: list_transition] :
( ( ( size_s3613142680436377136sition @ Xs2 )
= ( size_s3613142680436377136sition @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_transition @ X5 @ Xs2 ) @ ( cons_transition @ Y4 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_493_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 )
=> ( ! [X5: list_o,Xs2: list_list_o,Y4: $o,Ys4: list_o] :
( ( ( size_s2710708370519433104list_o @ Xs2 )
= ( size_size_list_o @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_list_o @ X5 @ Xs2 ) @ ( cons_o @ Y4 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_494_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 )
=> ( ! [X5: list_o,Xs2: list_list_o,Y4: transition,Ys4: list_transition] :
( ( ( size_s2710708370519433104list_o @ Xs2 )
= ( size_s3613142680436377136sition @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_list_o @ X5 @ Xs2 ) @ ( cons_transition @ Y4 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_495_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 )
=> ( ! [X5: $o,Xs2: list_o,Y4: list_o,Ys4: list_list_o] :
( ( ( size_size_list_o @ Xs2 )
= ( size_s2710708370519433104list_o @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_o @ X5 @ Xs2 ) @ ( cons_list_o @ Y4 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_496_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 )
=> ( ! [X5: transition,Xs2: list_transition,Y4: list_o,Ys4: list_list_o] :
( ( ( size_s3613142680436377136sition @ Xs2 )
= ( size_s2710708370519433104list_o @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_transition @ X5 @ Xs2 ) @ ( cons_list_o @ Y4 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_497_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 )
=> ( ! [X5: list_o,Xs2: list_list_o,Y4: list_o,Ys4: list_list_o] :
( ( ( size_s2710708370519433104list_o @ Xs2 )
= ( size_s2710708370519433104list_o @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_list_o @ X5 @ Xs2 ) @ ( cons_list_o @ Y4 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_498_list__induct2,axiom,
! [Xs: list_formula_a_t,Ys: list_o,P: list_formula_a_t > list_o > $o] :
( ( ( size_s8846756101701226951la_a_t @ Xs )
= ( size_size_list_o @ Ys ) )
=> ( ( P @ nil_formula_a_t @ nil_o )
=> ( ! [X5: formula_a_t,Xs2: list_formula_a_t,Y4: $o,Ys4: list_o] :
( ( ( size_s8846756101701226951la_a_t @ Xs2 )
= ( size_size_list_o @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_formula_a_t @ X5 @ Xs2 ) @ ( cons_o @ Y4 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_499_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 )
=> ( ! [X5: $o,Xs2: list_o,Y4: $o,Ys4: list_o,Z3: $o,Zs2: list_o] :
( ( ( size_size_list_o @ Xs2 )
= ( size_size_list_o @ Ys4 ) )
=> ( ( ( size_size_list_o @ Ys4 )
= ( size_size_list_o @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_o @ X5 @ Xs2 ) @ ( cons_o @ Y4 @ Ys4 ) @ ( cons_o @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_500_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 )
=> ( ! [X5: $o,Xs2: list_o,Y4: $o,Ys4: list_o,Z3: transition,Zs2: list_transition] :
( ( ( size_size_list_o @ Xs2 )
= ( size_size_list_o @ Ys4 ) )
=> ( ( ( size_size_list_o @ Ys4 )
= ( size_s3613142680436377136sition @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_o @ X5 @ Xs2 ) @ ( cons_o @ Y4 @ Ys4 ) @ ( cons_transition @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_501_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 )
=> ( ! [X5: $o,Xs2: list_o,Y4: transition,Ys4: list_transition,Z3: $o,Zs2: list_o] :
( ( ( size_size_list_o @ Xs2 )
= ( size_s3613142680436377136sition @ Ys4 ) )
=> ( ( ( size_s3613142680436377136sition @ Ys4 )
= ( size_size_list_o @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_o @ X5 @ Xs2 ) @ ( cons_transition @ Y4 @ Ys4 ) @ ( cons_o @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_502_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 )
=> ( ! [X5: $o,Xs2: list_o,Y4: transition,Ys4: list_transition,Z3: transition,Zs2: list_transition] :
( ( ( size_size_list_o @ Xs2 )
= ( size_s3613142680436377136sition @ Ys4 ) )
=> ( ( ( size_s3613142680436377136sition @ Ys4 )
= ( size_s3613142680436377136sition @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_o @ X5 @ Xs2 ) @ ( cons_transition @ Y4 @ Ys4 ) @ ( cons_transition @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_503_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 )
=> ( ! [X5: transition,Xs2: list_transition,Y4: $o,Ys4: list_o,Z3: $o,Zs2: list_o] :
( ( ( size_s3613142680436377136sition @ Xs2 )
= ( size_size_list_o @ Ys4 ) )
=> ( ( ( size_size_list_o @ Ys4 )
= ( size_size_list_o @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_transition @ X5 @ Xs2 ) @ ( cons_o @ Y4 @ Ys4 ) @ ( cons_o @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_504_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 )
=> ( ! [X5: transition,Xs2: list_transition,Y4: $o,Ys4: list_o,Z3: transition,Zs2: list_transition] :
( ( ( size_s3613142680436377136sition @ Xs2 )
= ( size_size_list_o @ Ys4 ) )
=> ( ( ( size_size_list_o @ Ys4 )
= ( size_s3613142680436377136sition @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_transition @ X5 @ Xs2 ) @ ( cons_o @ Y4 @ Ys4 ) @ ( cons_transition @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_505_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 )
=> ( ! [X5: transition,Xs2: list_transition,Y4: transition,Ys4: list_transition,Z3: $o,Zs2: list_o] :
( ( ( size_s3613142680436377136sition @ Xs2 )
= ( size_s3613142680436377136sition @ Ys4 ) )
=> ( ( ( size_s3613142680436377136sition @ Ys4 )
= ( size_size_list_o @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_transition @ X5 @ Xs2 ) @ ( cons_transition @ Y4 @ Ys4 ) @ ( cons_o @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_506_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 )
=> ( ! [X5: transition,Xs2: list_transition,Y4: transition,Ys4: list_transition,Z3: transition,Zs2: list_transition] :
( ( ( size_s3613142680436377136sition @ Xs2 )
= ( size_s3613142680436377136sition @ Ys4 ) )
=> ( ( ( size_s3613142680436377136sition @ Ys4 )
= ( size_s3613142680436377136sition @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_transition @ X5 @ Xs2 ) @ ( cons_transition @ Y4 @ Ys4 ) @ ( cons_transition @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_507_list__induct3,axiom,
! [Xs: list_list_o,Ys: list_o,Zs: list_o,P: list_list_o > list_o > list_o > $o] :
( ( ( size_s2710708370519433104list_o @ Xs )
= ( size_size_list_o @ Ys ) )
=> ( ( ( size_size_list_o @ Ys )
= ( size_size_list_o @ Zs ) )
=> ( ( P @ nil_list_o @ nil_o @ nil_o )
=> ( ! [X5: list_o,Xs2: list_list_o,Y4: $o,Ys4: list_o,Z3: $o,Zs2: list_o] :
( ( ( size_s2710708370519433104list_o @ Xs2 )
= ( size_size_list_o @ Ys4 ) )
=> ( ( ( size_size_list_o @ Ys4 )
= ( size_size_list_o @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_list_o @ X5 @ Xs2 ) @ ( cons_o @ Y4 @ Ys4 ) @ ( cons_o @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_508_list__induct3,axiom,
! [Xs: list_list_o,Ys: list_o,Zs: list_transition,P: list_list_o > list_o > list_transition > $o] :
( ( ( size_s2710708370519433104list_o @ Xs )
= ( size_size_list_o @ Ys ) )
=> ( ( ( size_size_list_o @ Ys )
= ( size_s3613142680436377136sition @ Zs ) )
=> ( ( P @ nil_list_o @ nil_o @ nil_transition )
=> ( ! [X5: list_o,Xs2: list_list_o,Y4: $o,Ys4: list_o,Z3: transition,Zs2: list_transition] :
( ( ( size_s2710708370519433104list_o @ Xs2 )
= ( size_size_list_o @ Ys4 ) )
=> ( ( ( size_size_list_o @ Ys4 )
= ( size_s3613142680436377136sition @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_list_o @ X5 @ Xs2 ) @ ( cons_o @ Y4 @ Ys4 ) @ ( cons_transition @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_509_list__induct4,axiom,
! [Xs: list_o,Ys: list_o,Zs: list_o,Ws: list_o,P: list_o > 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 ) )
=> ( ( ( size_size_list_o @ Zs )
= ( size_size_list_o @ Ws ) )
=> ( ( P @ nil_o @ nil_o @ nil_o @ nil_o )
=> ( ! [X5: $o,Xs2: list_o,Y4: $o,Ys4: list_o,Z3: $o,Zs2: list_o,W: $o,Ws2: list_o] :
( ( ( size_size_list_o @ Xs2 )
= ( size_size_list_o @ Ys4 ) )
=> ( ( ( size_size_list_o @ Ys4 )
= ( size_size_list_o @ Zs2 ) )
=> ( ( ( size_size_list_o @ Zs2 )
= ( size_size_list_o @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_o @ X5 @ Xs2 ) @ ( cons_o @ Y4 @ Ys4 ) @ ( cons_o @ Z3 @ Zs2 ) @ ( cons_o @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_510_list__induct4,axiom,
! [Xs: list_o,Ys: list_o,Zs: list_o,Ws: list_transition,P: list_o > list_o > list_o > list_transition > $o] :
( ( ( size_size_list_o @ 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_o @ nil_o @ nil_o @ nil_transition )
=> ( ! [X5: $o,Xs2: list_o,Y4: $o,Ys4: list_o,Z3: $o,Zs2: list_o,W: transition,Ws2: list_transition] :
( ( ( size_size_list_o @ Xs2 )
= ( size_size_list_o @ Ys4 ) )
=> ( ( ( size_size_list_o @ Ys4 )
= ( size_size_list_o @ Zs2 ) )
=> ( ( ( size_size_list_o @ Zs2 )
= ( size_s3613142680436377136sition @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_o @ X5 @ Xs2 ) @ ( cons_o @ Y4 @ Ys4 ) @ ( cons_o @ Z3 @ Zs2 ) @ ( cons_transition @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_511_list__induct4,axiom,
! [Xs: list_o,Ys: list_o,Zs: list_transition,Ws: list_o,P: list_o > list_o > list_transition > list_o > $o] :
( ( ( size_size_list_o @ Xs )
= ( size_size_list_o @ Ys ) )
=> ( ( ( size_size_list_o @ Ys )
= ( size_s3613142680436377136sition @ Zs ) )
=> ( ( ( size_s3613142680436377136sition @ Zs )
= ( size_size_list_o @ Ws ) )
=> ( ( P @ nil_o @ nil_o @ nil_transition @ nil_o )
=> ( ! [X5: $o,Xs2: list_o,Y4: $o,Ys4: list_o,Z3: transition,Zs2: list_transition,W: $o,Ws2: list_o] :
( ( ( size_size_list_o @ Xs2 )
= ( size_size_list_o @ Ys4 ) )
=> ( ( ( size_size_list_o @ Ys4 )
= ( size_s3613142680436377136sition @ Zs2 ) )
=> ( ( ( size_s3613142680436377136sition @ Zs2 )
= ( size_size_list_o @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_o @ X5 @ Xs2 ) @ ( cons_o @ Y4 @ Ys4 ) @ ( cons_transition @ Z3 @ Zs2 ) @ ( cons_o @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_512_list__induct4,axiom,
! [Xs: list_o,Ys: list_o,Zs: list_transition,Ws: list_transition,P: list_o > list_o > list_transition > list_transition > $o] :
( ( ( size_size_list_o @ Xs )
= ( size_size_list_o @ Ys ) )
=> ( ( ( size_size_list_o @ Ys )
= ( size_s3613142680436377136sition @ Zs ) )
=> ( ( ( size_s3613142680436377136sition @ Zs )
= ( size_s3613142680436377136sition @ Ws ) )
=> ( ( P @ nil_o @ nil_o @ nil_transition @ nil_transition )
=> ( ! [X5: $o,Xs2: list_o,Y4: $o,Ys4: list_o,Z3: transition,Zs2: list_transition,W: transition,Ws2: list_transition] :
( ( ( size_size_list_o @ Xs2 )
= ( size_size_list_o @ Ys4 ) )
=> ( ( ( size_size_list_o @ Ys4 )
= ( size_s3613142680436377136sition @ Zs2 ) )
=> ( ( ( size_s3613142680436377136sition @ Zs2 )
= ( size_s3613142680436377136sition @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_o @ X5 @ Xs2 ) @ ( cons_o @ Y4 @ Ys4 ) @ ( cons_transition @ Z3 @ Zs2 ) @ ( cons_transition @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_513_list__induct4,axiom,
! [Xs: list_o,Ys: list_transition,Zs: list_o,Ws: list_o,P: list_o > list_transition > list_o > list_o > $o] :
( ( ( size_size_list_o @ Xs )
= ( size_s3613142680436377136sition @ Ys ) )
=> ( ( ( size_s3613142680436377136sition @ Ys )
= ( size_size_list_o @ Zs ) )
=> ( ( ( size_size_list_o @ Zs )
= ( size_size_list_o @ Ws ) )
=> ( ( P @ nil_o @ nil_transition @ nil_o @ nil_o )
=> ( ! [X5: $o,Xs2: list_o,Y4: transition,Ys4: list_transition,Z3: $o,Zs2: list_o,W: $o,Ws2: list_o] :
( ( ( size_size_list_o @ Xs2 )
= ( size_s3613142680436377136sition @ Ys4 ) )
=> ( ( ( size_s3613142680436377136sition @ Ys4 )
= ( size_size_list_o @ Zs2 ) )
=> ( ( ( size_size_list_o @ Zs2 )
= ( size_size_list_o @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_o @ X5 @ Xs2 ) @ ( cons_transition @ Y4 @ Ys4 ) @ ( cons_o @ Z3 @ Zs2 ) @ ( cons_o @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_514_list__induct4,axiom,
! [Xs: list_o,Ys: list_transition,Zs: list_o,Ws: list_transition,P: list_o > list_transition > list_o > list_transition > $o] :
( ( ( size_size_list_o @ Xs )
= ( size_s3613142680436377136sition @ Ys ) )
=> ( ( ( size_s3613142680436377136sition @ Ys )
= ( size_size_list_o @ Zs ) )
=> ( ( ( size_size_list_o @ Zs )
= ( size_s3613142680436377136sition @ Ws ) )
=> ( ( P @ nil_o @ nil_transition @ nil_o @ nil_transition )
=> ( ! [X5: $o,Xs2: list_o,Y4: transition,Ys4: list_transition,Z3: $o,Zs2: list_o,W: transition,Ws2: list_transition] :
( ( ( size_size_list_o @ Xs2 )
= ( size_s3613142680436377136sition @ Ys4 ) )
=> ( ( ( size_s3613142680436377136sition @ Ys4 )
= ( size_size_list_o @ Zs2 ) )
=> ( ( ( size_size_list_o @ Zs2 )
= ( size_s3613142680436377136sition @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_o @ X5 @ Xs2 ) @ ( cons_transition @ Y4 @ Ys4 ) @ ( cons_o @ Z3 @ Zs2 ) @ ( cons_transition @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_515_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 )
=> ( ! [X5: $o,Xs2: list_o,Y4: transition,Ys4: list_transition,Z3: transition,Zs2: list_transition,W: $o,Ws2: list_o] :
( ( ( size_size_list_o @ Xs2 )
= ( size_s3613142680436377136sition @ Ys4 ) )
=> ( ( ( size_s3613142680436377136sition @ Ys4 )
= ( size_s3613142680436377136sition @ Zs2 ) )
=> ( ( ( size_s3613142680436377136sition @ Zs2 )
= ( size_size_list_o @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_o @ X5 @ Xs2 ) @ ( cons_transition @ Y4 @ Ys4 ) @ ( cons_transition @ Z3 @ Zs2 ) @ ( cons_o @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_516_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 )
=> ( ! [X5: $o,Xs2: list_o,Y4: transition,Ys4: list_transition,Z3: transition,Zs2: list_transition,W: transition,Ws2: list_transition] :
( ( ( size_size_list_o @ Xs2 )
= ( size_s3613142680436377136sition @ Ys4 ) )
=> ( ( ( size_s3613142680436377136sition @ Ys4 )
= ( size_s3613142680436377136sition @ Zs2 ) )
=> ( ( ( size_s3613142680436377136sition @ Zs2 )
= ( size_s3613142680436377136sition @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_o @ X5 @ Xs2 ) @ ( cons_transition @ Y4 @ Ys4 ) @ ( cons_transition @ Z3 @ Zs2 ) @ ( cons_transition @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_517_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 )
=> ( ! [X5: transition,Xs2: list_transition,Y4: $o,Ys4: list_o,Z3: $o,Zs2: list_o,W: $o,Ws2: list_o] :
( ( ( size_s3613142680436377136sition @ Xs2 )
= ( size_size_list_o @ Ys4 ) )
=> ( ( ( size_size_list_o @ Ys4 )
= ( size_size_list_o @ Zs2 ) )
=> ( ( ( size_size_list_o @ Zs2 )
= ( size_size_list_o @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_transition @ X5 @ Xs2 ) @ ( cons_o @ Y4 @ Ys4 ) @ ( cons_o @ Z3 @ Zs2 ) @ ( cons_o @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_518_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 )
=> ( ! [X5: transition,Xs2: list_transition,Y4: $o,Ys4: list_o,Z3: $o,Zs2: list_o,W: transition,Ws2: list_transition] :
( ( ( size_s3613142680436377136sition @ Xs2 )
= ( size_size_list_o @ Ys4 ) )
=> ( ( ( size_size_list_o @ Ys4 )
= ( size_size_list_o @ Zs2 ) )
=> ( ( ( size_size_list_o @ Zs2 )
= ( size_s3613142680436377136sition @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_transition @ X5 @ Xs2 ) @ ( cons_o @ Y4 @ Ys4 ) @ ( cons_o @ Z3 @ Zs2 ) @ ( cons_transition @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_519_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_520_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_521_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_522_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_523_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_524_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_525_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_526_nfa__def,axiom,
( nfa
= ( ^ [Q02: nat,Qf2: nat,Transs2: list_transition] :
( ! [T3: transition] :
( ( member_transition @ T3 @ ( set_transition2 @ Transs2 ) )
=> ( ord_less_eq_set_nat @ ( state_set @ T3 ) @ ( q @ Q02 @ Qf2 @ Transs2 ) ) )
& ( Transs2 != nil_transition )
& ~ ( member_nat @ Qf2 @ ( sq @ Q02 @ Transs2 ) ) ) ) ) ).
% nfa_def
thf(fact_527_left_OQ__diff__qf__SQ,axiom,
( ( minus_minus_set_nat @ ( q @ ( plus_plus_nat @ q0a @ one_one_nat ) @ q0a @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) ) @ ( insert_nat @ q0a @ bot_bot_set_nat ) )
= ( sq @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) ) ) ).
% left.Q_diff_qf_SQ
thf(fact_528_left_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 @ one_one_nat ) @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) @ R2 @ Bs ) @ ( sup_sup_set_nat @ R2 @ ( q @ ( plus_plus_nat @ q0a @ one_one_nat ) @ q0a @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) ) ) ) ).
% left.step_eps_closure_set_closed_union
thf(fact_529_left_Orun__accept__eps__qf__many,axiom,
! [Bs: list_o,Bss: list_list_o,Cs: list_o] :
~ ( run_accept_eps @ ( plus_plus_nat @ q0a @ one_one_nat ) @ q0a @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) @ ( insert_nat @ q0a @ bot_bot_set_nat ) @ ( cons_list_o @ Bs @ Bss ) @ Cs ) ).
% left.run_accept_eps_qf_many
thf(fact_530_left_Orun__accept__eps__qf__one,axiom,
! [Bs: list_o] : ( run_accept_eps @ ( plus_plus_nat @ q0a @ one_one_nat ) @ q0a @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) @ ( insert_nat @ q0a @ bot_bot_set_nat ) @ nil_list_o @ Bs ) ).
% left.run_accept_eps_qf_one
thf(fact_531_nfa__cong__Plus__axioms_Ointro,axiom,
! [Q0: nat,Transs: list_transition,Q03: nat,Q04: nat] :
( ! [Bs3: list_o,Q4: nat] :
( ( step_eps @ Q0 @ Transs @ Bs3 @ Q0 @ Q4 )
= ( member_nat @ Q4 @ ( insert_nat @ Q03 @ ( insert_nat @ Q04 @ bot_bot_set_nat ) ) ) )
=> ( ! [Q4: nat] :
~ ( step_symb @ Q0 @ Transs @ Q0 @ Q4 )
=> ( nfa_cong_Plus_axioms @ Q0 @ Q03 @ Q04 @ Transs ) ) ) ).
% nfa_cong_Plus_axioms.intro
thf(fact_532_nfa__cong__Star__axioms_Ointro,axiom,
! [Q0: nat,Transs: list_transition,Q03: nat,Qf: nat] :
( ! [Bs3: list_o,Q4: nat] :
( ( step_eps @ Q0 @ Transs @ Bs3 @ Q0 @ Q4 )
= ( member_nat @ Q4 @ ( insert_nat @ Q03 @ ( insert_nat @ Qf @ bot_bot_set_nat ) ) ) )
=> ( ! [Q4: nat] :
~ ( step_symb @ Q0 @ Transs @ Q0 @ Q4 )
=> ( nfa_cong_Star_axioms @ Q0 @ Q03 @ Qf @ Transs ) ) ) ).
% nfa_cong_Star_axioms.intro
thf(fact_533_nfa__cong__Plus__axioms__def,axiom,
( nfa_cong_Plus_axioms
= ( ^ [Q02: nat,Q05: nat,Q06: nat,Transs2: list_transition] :
( ! [Bs2: list_o,Q5: nat] :
( ( step_eps @ Q02 @ Transs2 @ Bs2 @ Q02 @ Q5 )
= ( member_nat @ Q5 @ ( insert_nat @ Q05 @ ( insert_nat @ Q06 @ bot_bot_set_nat ) ) ) )
& ! [Q5: nat] :
~ ( step_symb @ Q02 @ Transs2 @ Q02 @ Q5 ) ) ) ) ).
% nfa_cong_Plus_axioms_def
thf(fact_534_nfa__cong__Star__axioms__def,axiom,
( nfa_cong_Star_axioms
= ( ^ [Q02: nat,Q05: nat,Qf2: nat,Transs2: list_transition] :
( ! [Bs2: list_o,Q5: nat] :
( ( step_eps @ Q02 @ Transs2 @ Bs2 @ Q02 @ Q5 )
= ( member_nat @ Q5 @ ( insert_nat @ Q05 @ ( insert_nat @ Qf2 @ bot_bot_set_nat ) ) ) )
& ! [Q5: nat] :
~ ( step_symb @ Q02 @ Transs2 @ Q02 @ Q5 ) ) ) ) ).
% nfa_cong_Star_axioms_def
thf(fact_535_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_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_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_538_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_539_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_540_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_541_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_542_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_543_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_544_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_545_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_546_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_547_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_548_delta__step__symb__set__absorb,axiom,
( delta
= ( ^ [Q02: nat,Transs2: list_transition,R3: set_nat,Bs2: list_o] : ( sup_sup_set_nat @ ( delta @ Q02 @ Transs2 @ R3 @ Bs2 ) @ ( step_symb_set @ Q02 @ Transs2 @ R3 ) ) ) ) ).
% delta_step_symb_set_absorb
thf(fact_549_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_550_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_551_NFA_OQ__def,axiom,
( q
= ( ^ [Q02: nat,Qf2: nat,Transs2: list_transition] : ( sup_sup_set_nat @ ( sq @ Q02 @ Transs2 ) @ ( insert_nat @ Qf2 @ bot_bot_set_nat ) ) ) ) ).
% NFA.Q_def
thf(fact_552_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_553_step__eps__closure__set__unfold,axiom,
! [Q0: nat,Transs: list_transition,Bs: list_o,Q: nat,X7: set_nat] :
( ! [Q3: nat] :
( ( step_eps @ Q0 @ Transs @ Bs @ Q @ Q3 )
= ( member_nat @ Q3 @ X7 ) )
=> ( ( 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 @ X7 @ Bs ) ) ) ) ).
% step_eps_closure_set_unfold
thf(fact_554_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_555_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_556_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_557_Star_Ohyps,axiom,
! [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 @ ra @ 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 @ ra ) ) ) ) ).
% Star.hyps
thf(fact_558_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_559_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_560_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_561_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_562_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_563_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_564_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_565_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_566_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_567_le__diff__iff_H,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_568_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_569_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_570_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_571_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_572_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_573_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_574_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_575_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_576_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_577_diff__less__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_578_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_579_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_580_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_581_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_582_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_583_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_584_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_585_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_586_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_587_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_588_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_589_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_590_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_591_NFA_OSQ__def,axiom,
( sq
= ( ^ [Q02: nat,Transs2: list_transition] : ( set_or4665077453230672383an_nat @ Q02 @ ( plus_plus_nat @ Q02 @ ( size_s3613142680436377136sition @ Transs2 ) ) ) ) ) ).
% NFA.SQ_def
thf(fact_592_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 ) ) )
=> ? [X5: nat] :
( ( member_nat @ X5 @ ( set_or4665077453230672383an_nat @ I @ J ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ X5 ) @ ( match_a_t @ sigma @ ( star_a_t @ R ) ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ J ) @ ( match_a_t @ sigma @ R ) ) ) ) ) ).
% match_Star_unfold
thf(fact_593_nfa__cong_H__axioms__def,axiom,
( nfa_cong_axioms
= ( ^ [Q02: nat,Q05: nat,Qf3: nat,Transs2: list_transition,Transs3: list_transition] :
( ( ord_less_eq_set_nat @ ( sq @ Q05 @ Transs3 ) @ ( sq @ Q02 @ Transs2 ) )
& ( member_nat @ Qf3 @ ( sq @ Q02 @ Transs2 ) )
& ! [Q5: nat] :
( ( member_nat @ Q5 @ ( sq @ Q05 @ Transs3 ) )
=> ( ( nth_transition @ Transs2 @ ( minus_minus_nat @ Q5 @ Q02 ) )
= ( nth_transition @ Transs3 @ ( minus_minus_nat @ Q5 @ Q05 ) ) ) ) ) ) ) ).
% nfa_cong'_axioms_def
thf(fact_594_nfa__cong__axioms__def,axiom,
( nfa_cong_axioms2
= ( ^ [Q02: nat,Q05: nat,Qf2: nat,Qf3: nat,Transs2: list_transition,Transs3: list_transition] :
( ( ord_less_eq_set_nat @ ( sq @ Q05 @ Transs3 ) @ ( sq @ Q02 @ Transs2 ) )
& ( Qf2 = Qf3 )
& ! [Q5: nat] :
( ( member_nat @ Q5 @ ( sq @ Q05 @ Transs3 ) )
=> ( ( nth_transition @ Transs2 @ ( minus_minus_nat @ Q5 @ Q02 ) )
= ( nth_transition @ Transs3 @ ( minus_minus_nat @ Q5 @ Q05 ) ) ) ) ) ) ) ).
% nfa_cong_axioms_def
thf(fact_595_nfa__cong__axioms_Ointro,axiom,
! [Q03: nat,Transs4: list_transition,Q0: nat,Transs: list_transition,Qf: nat,Qf4: nat] :
( ( ord_less_eq_set_nat @ ( sq @ Q03 @ Transs4 ) @ ( sq @ Q0 @ Transs ) )
=> ( ( Qf = Qf4 )
=> ( ! [Q4: nat] :
( ( member_nat @ Q4 @ ( sq @ Q03 @ Transs4 ) )
=> ( ( nth_transition @ Transs @ ( minus_minus_nat @ Q4 @ Q0 ) )
= ( nth_transition @ Transs4 @ ( minus_minus_nat @ Q4 @ Q03 ) ) ) )
=> ( nfa_cong_axioms2 @ Q0 @ Q03 @ Qf @ Qf4 @ Transs @ Transs4 ) ) ) ) ).
% nfa_cong_axioms.intro
thf(fact_596_nfa__cong_H__axioms_Ointro,axiom,
! [Q03: nat,Transs4: list_transition,Q0: nat,Transs: list_transition,Qf4: nat] :
( ( ord_less_eq_set_nat @ ( sq @ Q03 @ Transs4 ) @ ( sq @ Q0 @ Transs ) )
=> ( ( member_nat @ Qf4 @ ( sq @ Q0 @ Transs ) )
=> ( ! [Q4: nat] :
( ( member_nat @ Q4 @ ( sq @ Q03 @ Transs4 ) )
=> ( ( nth_transition @ Transs @ ( minus_minus_nat @ Q4 @ Q0 ) )
= ( nth_transition @ Transs4 @ ( minus_minus_nat @ Q4 @ Q03 ) ) ) )
=> ( nfa_cong_axioms @ Q0 @ Q03 @ Qf4 @ Transs @ Transs4 ) ) ) ) ).
% nfa_cong'_axioms.intro
thf(fact_597__C1_Ohyps_C,axiom,
! [M2: nat] :
( ( ord_less_nat @ M2 @ ( size_s2710708370519433104list_o @ bssb ) )
=> ! [X6: list_list_o] :
( ( M2
= ( size_s2710708370519433104list_o @ X6 ) )
=> ( ! [Xa: nat,Xb: nat,Xc: list_formula_a_t,Xd: list_transition,Xe: list_list_o,Xf: list_o,Xg: nat] :
( ( iH_a_t @ sigma @ ra @ Xa @ Xb @ Xc @ Xd @ Xe @ Xf @ Xg )
=> ( ( run_accept_eps @ Xa @ Xb @ Xd @ ( insert_nat @ Xa @ bot_bot_set_nat ) @ Xe @ Xf )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Xg @ ( plus_plus_nat @ Xg @ ( size_s2710708370519433104list_o @ Xe ) ) ) @ ( match_a_t @ sigma @ ra ) ) ) )
=> ! [Xa2: list_o,Xb2: nat] :
( ( iH_a_t @ sigma @ ( star_a_t @ ra ) @ q0a @ qfa @ phisa @ transsa @ X6 @ Xa2 @ Xb2 )
=> ( ( run_accept_eps @ q0a @ qfa @ transsa @ ( insert_nat @ q0a @ bot_bot_set_nat ) @ X6 @ Xa2 )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Xb2 @ ( plus_plus_nat @ Xb2 @ ( size_s2710708370519433104list_o @ X6 ) ) ) @ ( match_a_t @ sigma @ ( star_a_t @ ra ) ) ) ) ) ) ) ) ).
% "1.hyps"
thf(fact_598_match__Star,axiom,
! [I: nat,N: nat,R: regex_a_t] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ ( plus_plus_nat @ I @ ( suc @ N ) ) ) @ ( match_a_t @ sigma @ ( star_a_t @ R ) ) )
= ( ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ ( plus_plus_nat @ ( plus_plus_nat @ I @ one_one_nat ) @ K2 ) ) @ ( match_a_t @ sigma @ R ) )
& ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( plus_plus_nat @ I @ one_one_nat ) @ K2 ) @ ( plus_plus_nat @ I @ ( suc @ N ) ) ) @ ( match_a_t @ sigma @ ( star_a_t @ R ) ) ) ) ) ) ).
% match_Star
thf(fact_599__C1_Oprems_C_I2_J,axiom,
iH_a_t @ sigma @ ( star_a_t @ ra ) @ q0a @ qfa @ phisa @ transsa @ bssb @ bsb @ ib ).
% "1.prems"(2)
thf(fact_600_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_601_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_602_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_603_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_604_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_605_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_606_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_607_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_608_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_609_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_610_qf__not__in__SQ,axiom,
~ ( member_nat @ qfa @ ( sq @ q0a @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) ) ) ).
% qf_not_in_SQ
thf(fact_611_base_Onfa__axioms,axiom,
nfa @ q0a @ qfa @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) ).
% base.nfa_axioms
thf(fact_612_base_Ostep__eps__qf,axiom,
! [Bs: list_o,Q: nat] :
~ ( step_eps @ q0a @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) @ Bs @ qfa @ Q ) ).
% base.step_eps_qf
thf(fact_613_base_Ostep__eps__closure__qf,axiom,
! [Bs: list_o,Q: nat,Q2: nat] :
( ( step_eps_closure @ q0a @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) @ Bs @ Q @ Q2 )
=> ( ( Q = qfa )
=> ( Q = Q2 ) ) ) ).
% base.step_eps_closure_qf
thf(fact_614_base_Ostep__symb__qf,axiom,
! [Q: nat] :
~ ( step_symb @ q0a @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) @ qfa @ Q ) ).
% base.step_symb_qf
thf(fact_615_base_Otranss__not__Nil,axiom,
( ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) )
!= nil_transition ) ).
% base.transs_not_Nil
thf(fact_616_base_Ostep__eps__closed,axiom,
! [Bs: list_o,Q: nat,Q2: nat] :
( ( step_eps @ q0a @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) @ Bs @ Q @ Q2 )
=> ( member_nat @ Q2 @ ( q @ q0a @ qfa @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) ) ) ) ).
% base.step_eps_closed
thf(fact_617_base_Ostep__eps__closure__closed,axiom,
! [Bs: list_o,Q: nat,Q2: nat] :
( ( step_eps_closure @ q0a @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) @ Bs @ Q @ Q2 )
=> ( ( Q != Q2 )
=> ( member_nat @ Q2 @ ( q @ q0a @ qfa @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) ) ) ) ) ).
% base.step_eps_closure_closed
thf(fact_618_base_Ostep__symb__closed,axiom,
! [Q: nat,Q2: nat] :
( ( step_symb @ q0a @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) @ Q @ Q2 )
=> ( member_nat @ Q2 @ ( q @ q0a @ qfa @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) ) ) ) ).
% base.step_symb_closed
thf(fact_619_transs__def,axiom,
( transsa
= ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) ) ).
% transs_def
thf(fact_620_base_Ostep__eps__closure__set__qf,axiom,
! [Bs: list_o] :
( ( step_eps_closure_set @ q0a @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) @ ( insert_nat @ qfa @ bot_bot_set_nat ) @ Bs )
= ( insert_nat @ qfa @ bot_bot_set_nat ) ) ).
% base.step_eps_closure_set_qf
thf(fact_621_base_Odelta__qf,axiom,
! [Bs: list_o] :
( ( delta @ q0a @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) @ ( insert_nat @ qfa @ bot_bot_set_nat ) @ Bs )
= bot_bot_set_nat ) ).
% base.delta_qf
thf(fact_622_base_Ostep__symb__set__qf,axiom,
( ( step_symb_set @ q0a @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) @ ( insert_nat @ qfa @ bot_bot_set_nat ) )
= bot_bot_set_nat ) ).
% base.step_symb_set_qf
thf(fact_623_base_Ostep__eps__closure__set__closed,axiom,
! [R2: set_nat,Bs: list_o] :
( ( ord_less_eq_set_nat @ R2 @ ( q @ q0a @ qfa @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) ) )
=> ( ord_less_eq_set_nat @ ( step_eps_closure_set @ q0a @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) @ R2 @ Bs ) @ ( q @ q0a @ qfa @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) ) ) ) ).
% base.step_eps_closure_set_closed
thf(fact_624_base_Odelta__closed,axiom,
! [R2: set_nat,Bs: list_o] : ( ord_less_eq_set_nat @ ( delta @ q0a @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) @ R2 @ Bs ) @ ( q @ q0a @ qfa @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) ) ) ).
% base.delta_closed
thf(fact_625_base_Orun__closed,axiom,
! [R2: set_nat,Bss: list_list_o] :
( ( ord_less_eq_set_nat @ R2 @ ( q @ q0a @ qfa @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) ) )
=> ( ord_less_eq_set_nat @ ( run @ q0a @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) @ R2 @ Bss ) @ ( q @ q0a @ qfa @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) ) ) ) ).
% base.run_closed
thf(fact_626_base_Ostep__symb__set__closed,axiom,
! [R2: set_nat] : ( ord_less_eq_set_nat @ ( step_symb_set @ q0a @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) @ R2 ) @ ( q @ q0a @ qfa @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) ) ) ).
% base.step_symb_set_closed
thf(fact_627_base_Ostep__eps__set__closed,axiom,
! [Bs: list_o,R2: set_nat] : ( ord_less_eq_set_nat @ ( step_eps_set @ q0a @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) @ Bs @ R2 ) @ ( q @ q0a @ qfa @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) ) ) ).
% base.step_eps_set_closed
thf(fact_628_base_Oq0__sub__Q,axiom,
ord_less_eq_set_nat @ ( insert_nat @ q0a @ bot_bot_set_nat ) @ ( q @ q0a @ qfa @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) ) ).
% base.q0_sub_Q
thf(fact_629_base_Oq0__sub__SQ,axiom,
ord_less_eq_set_nat @ ( insert_nat @ q0a @ bot_bot_set_nat ) @ ( sq @ q0a @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) ) ).
% base.q0_sub_SQ
thf(fact_630_base_Orun__accept__eps__qf__many,axiom,
! [Bs: list_o,Bss: list_list_o,Cs: list_o] :
~ ( run_accept_eps @ q0a @ qfa @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) @ ( insert_nat @ qfa @ bot_bot_set_nat ) @ ( cons_list_o @ Bs @ Bss ) @ Cs ) ).
% base.run_accept_eps_qf_many
thf(fact_631_base_Orun__qf__many,axiom,
! [Bs: list_o,Bss: list_list_o] :
( ( run @ q0a @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) @ ( insert_nat @ qfa @ bot_bot_set_nat ) @ ( cons_list_o @ Bs @ Bss ) )
= bot_bot_set_nat ) ).
% base.run_qf_many
thf(fact_632_base_Orun__accept__eps__qf__one,axiom,
! [Bs: list_o] : ( run_accept_eps @ q0a @ qfa @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) @ ( insert_nat @ qfa @ bot_bot_set_nat ) @ nil_list_o @ Bs ) ).
% base.run_accept_eps_qf_one
thf(fact_633_base_Ostep__eps__closure__set__closed__union,axiom,
! [R2: set_nat,Bs: list_o] : ( ord_less_eq_set_nat @ ( step_eps_closure_set @ q0a @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) @ R2 @ Bs ) @ ( sup_sup_set_nat @ R2 @ ( q @ q0a @ qfa @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) ) ) ) ).
% base.step_eps_closure_set_closed_union
thf(fact_634_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_635_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_636_base_Orun__closed__Cons,axiom,
! [R2: set_nat,Bs: list_o,Bss: list_list_o] : ( ord_less_eq_set_nat @ ( run @ q0a @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) @ R2 @ ( cons_list_o @ Bs @ Bss ) ) @ ( q @ q0a @ qfa @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) ) ) ).
% base.run_closed_Cons
thf(fact_637_base_Ostate__closed,axiom,
! [T: transition] :
( ( member_transition @ T @ ( set_transition2 @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) ) )
=> ( ord_less_eq_set_nat @ ( state_set @ T ) @ ( q @ q0a @ qfa @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) ) ) ) ).
% base.state_closed
thf(fact_638_atLeastLessThan__singleton,axiom,
! [M: nat] :
( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
= ( insert_nat @ M @ bot_bot_set_nat ) ) ).
% atLeastLessThan_singleton
thf(fact_639_base_OQ__diff__qf__SQ,axiom,
( ( minus_minus_set_nat @ ( q @ q0a @ qfa @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) ) @ ( insert_nat @ qfa @ bot_bot_set_nat ) )
= ( sq @ q0a @ ( build_nfa_impl_a_t @ ( star_a_t @ ra ) @ ( produc8654416511292156347la_a_t @ q0a @ ( produc9017461973804568604la_a_t @ qfa @ phisa ) ) ) ) ) ).
% base.Q_diff_qf_SQ
thf(fact_640_Star_Oprems,axiom,
iH_a_t @ sigma @ ( star_a_t @ ra ) @ q0a @ qfa @ phisa @ transsa @ bssa @ bsa @ ia ).
% Star.prems
thf(fact_641_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_642_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_643_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_644_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M4: nat] :
( M6
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_645_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_646_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_647_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_648_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_649_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_650_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X5: nat] : ( R2 @ X5 @ X5 )
=> ( ! [X5: nat,Y4: nat,Z3: nat] :
( ( R2 @ X5 @ Y4 )
=> ( ( R2 @ Y4 @ Z3 )
=> ( R2 @ X5 @ Z3 ) ) )
=> ( ! [N2: nat] : ( R2 @ N2 @ ( suc @ N2 ) )
=> ( R2 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_651_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_652_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_653_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K3: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K3 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K3 )
=> ( P @ I2 @ K3 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_654_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_655_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_656_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_657_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M7: nat] :
( ( M
= ( suc @ M7 ) )
& ( ord_less_nat @ N @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_658_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_659_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_660_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_661_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ N )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_662_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_663_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_664_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_665_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_666_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_667_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_668_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_669_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_670_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_671_Suc__inject,axiom,
! [X3: nat,Y3: nat] :
( ( ( suc @ X3 )
= ( suc @ Y3 ) )
=> ( X3 = Y3 ) ) ).
% Suc_inject
thf(fact_672_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_673_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_674_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_675_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_676_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_677_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_678_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_679_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_680_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_681_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_682_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_683_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q4: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q4 ) ) ) ) ).
% less_natE
thf(fact_684_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_685_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_686_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M5: nat,N3: nat] :
? [K2: nat] :
( N3
= ( suc @ ( plus_plus_nat @ M5 @ K2 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_687_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K3: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_688_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_689_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_690_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_691_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_692_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_693_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_694_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_695_atLeastLessThanSuc,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_eq_nat @ M @ N )
=> ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
= ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ M @ N )
=> ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
= bot_bot_set_nat ) ) ) ).
% atLeastLessThanSuc
thf(fact_696_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_697_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_698_match__rderive,axiom,
! [R: regex_a_t,I: nat,J: nat] :
( ( wf_regex_a_t @ R )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ ( suc @ J ) ) @ ( match_a_t @ sigma @ R ) )
= ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ ( match_a_t @ sigma @ ( rderive_a_t @ R ) ) ) ) ) ) ).
% match_rderive
thf(fact_699_prod__decode__aux_Ocases,axiom,
! [X3: product_prod_nat_nat] :
~ ! [K3: nat,M4: nat] :
( X3
!= ( product_Pair_nat_nat @ K3 @ M4 ) ) ).
% prod_decode_aux.cases
thf(fact_700_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_701_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_702_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 )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K3 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K3 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_703_nfa__cong_H_Orun__accept__eps__cong__Cons__sub__simp,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Qf4: nat,Transs: list_transition,Transs4: list_transition,R2: set_nat,Bs: list_o,Bss: list_list_o,Cs: list_o] :
( ( nfa_cong @ Q0 @ Q03 @ Qf @ Qf4 @ Transs @ Transs4 )
=> ( ( ord_less_eq_set_nat @ R2 @ ( q @ Q03 @ Qf4 @ Transs4 ) )
=> ( ( ord_less_eq_set_nat @ ( delta @ Q0 @ Transs @ ( insert_nat @ Qf4 @ bot_bot_set_nat ) @ Bs ) @ ( delta @ Q03 @ Transs4 @ R2 @ Bs ) )
=> ( ( run_accept_eps @ Q0 @ Qf @ Transs @ R2 @ ( cons_list_o @ Bs @ Bss ) @ Cs )
= ( ( ( run_accept_eps @ Q03 @ Qf4 @ Transs4 @ R2 @ ( cons_list_o @ Bs @ Bss ) @ Cs )
& ( run_accept_eps @ Q0 @ Qf @ Transs @ ( insert_nat @ Qf4 @ bot_bot_set_nat ) @ nil_list_o @ Cs ) )
| ? [X: nat] :
( ( member_nat @ X @ ( set_ord_lessThan_nat @ ( size_s2710708370519433104list_o @ Bss ) ) )
& ( run_accept_eps @ Q03 @ Qf4 @ Transs4 @ R2 @ ( take_list_o @ ( suc @ X ) @ ( cons_list_o @ Bs @ Bss ) ) @ ( hd_list_o @ ( drop_list_o @ X @ Bss ) ) )
& ( run_accept_eps @ Q0 @ Qf @ Transs @ ( insert_nat @ Qf4 @ bot_bot_set_nat ) @ ( drop_list_o @ X @ Bss ) @ Cs ) ) ) ) ) ) ) ).
% nfa_cong'.run_accept_eps_cong_Cons_sub_simp
thf(fact_704_nfa__cong_H_Onfa_H__Q__sub__Q,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Qf4: nat,Transs: list_transition,Transs4: list_transition] :
( ( nfa_cong @ Q0 @ Q03 @ Qf @ Qf4 @ Transs @ Transs4 )
=> ( ord_less_eq_set_nat @ ( q @ Q03 @ Qf4 @ Transs4 ) @ ( q @ Q0 @ Qf @ Transs ) ) ) ).
% nfa_cong'.nfa'_Q_sub_Q
thf(fact_705_nfa__cong_H_OSQ__sub,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Qf4: nat,Transs: list_transition,Transs4: list_transition] :
( ( nfa_cong @ Q0 @ Q03 @ Qf @ Qf4 @ Transs @ Transs4 )
=> ( ord_less_eq_set_nat @ ( sq @ Q03 @ Transs4 ) @ ( sq @ Q0 @ Transs ) ) ) ).
% nfa_cong'.SQ_sub
thf(fact_706_nfa__cong_H_Oaxioms_I3_J,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Qf4: nat,Transs: list_transition,Transs4: list_transition] :
( ( nfa_cong @ Q0 @ Q03 @ Qf @ Qf4 @ Transs @ Transs4 )
=> ( nfa_cong_axioms @ Q0 @ Q03 @ Qf4 @ Transs @ Transs4 ) ) ).
% nfa_cong'.axioms(3)
thf(fact_707_lessThan__Suc,axiom,
! [K: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K ) )
= ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).
% lessThan_Suc
thf(fact_708_nfa__cong_H_Ostep__symb__cong__Q,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Qf4: nat,Transs: list_transition,Transs4: list_transition,Q: nat,Q2: nat] :
( ( nfa_cong @ Q0 @ Q03 @ Qf @ Qf4 @ Transs @ Transs4 )
=> ( ( step_symb @ Q03 @ Transs4 @ Q @ Q2 )
=> ( step_symb @ Q0 @ Transs @ Q @ Q2 ) ) ) ).
% nfa_cong'.step_symb_cong_Q
thf(fact_709_nfa__cong_H_Oaxioms_I2_J,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Qf4: nat,Transs: list_transition,Transs4: list_transition] :
( ( nfa_cong @ Q0 @ Q03 @ Qf @ Qf4 @ Transs @ Transs4 )
=> ( nfa @ Q03 @ Qf4 @ Transs4 ) ) ).
% nfa_cong'.axioms(2)
thf(fact_710_nfa__cong_H_Oaxioms_I1_J,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Qf4: nat,Transs: list_transition,Transs4: list_transition] :
( ( nfa_cong @ Q0 @ Q03 @ Qf @ Qf4 @ Transs @ Transs4 )
=> ( nfa @ Q0 @ Qf @ Transs ) ) ).
% nfa_cong'.axioms(1)
thf(fact_711_nfa__cong_H_Oq__SQ__SQ__nfa_H__SQ,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Qf4: nat,Transs: list_transition,Transs4: list_transition,Q: nat] :
( ( nfa_cong @ Q0 @ Q03 @ Qf @ Qf4 @ Transs @ Transs4 )
=> ( ( member_nat @ Q @ ( sq @ Q03 @ Transs4 ) )
=> ( ( member_nat @ Q @ ( sq @ Q0 @ Transs ) )
= ( member_nat @ Q @ ( sq @ Q03 @ Transs4 ) ) ) ) ) ).
% nfa_cong'.q_SQ_SQ_nfa'_SQ
thf(fact_712_nfa__cong_H_Oqf_H__in__SQ,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Qf4: nat,Transs: list_transition,Transs4: list_transition] :
( ( nfa_cong @ Q0 @ Q03 @ Qf @ Qf4 @ Transs @ Transs4 )
=> ( member_nat @ Qf4 @ ( sq @ Q0 @ Transs ) ) ) ).
% nfa_cong'.qf'_in_SQ
thf(fact_713_nfa__cong_H_Ostep__symb__set__cong__Q,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Qf4: nat,Transs: list_transition,Transs4: list_transition,R2: set_nat] :
( ( nfa_cong @ Q0 @ Q03 @ Qf @ Qf4 @ Transs @ Transs4 )
=> ( ord_less_eq_set_nat @ ( step_symb_set @ Q03 @ Transs4 @ R2 ) @ ( step_symb_set @ Q0 @ Transs @ R2 ) ) ) ).
% nfa_cong'.step_symb_set_cong_Q
thf(fact_714_nfa__cong_H_Ostep__eps__cong__Q,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Qf4: nat,Transs: list_transition,Transs4: list_transition,Q: nat,Bs: list_o,Q2: nat] :
( ( nfa_cong @ Q0 @ Q03 @ Qf @ Qf4 @ Transs @ Transs4 )
=> ( ( member_nat @ Q @ ( q @ Q03 @ Qf4 @ Transs4 ) )
=> ( ( step_eps @ Q03 @ Transs4 @ Bs @ Q @ Q2 )
=> ( step_eps @ Q0 @ Transs @ Bs @ Q @ Q2 ) ) ) ) ).
% nfa_cong'.step_eps_cong_Q
thf(fact_715_nfa__cong_H_Ostep__eps__cong__SQ,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Qf4: nat,Transs: list_transition,Transs4: list_transition,Q: nat,Bs: list_o,Q2: nat] :
( ( nfa_cong @ Q0 @ Q03 @ Qf @ Qf4 @ Transs @ Transs4 )
=> ( ( member_nat @ Q @ ( sq @ Q03 @ Transs4 ) )
=> ( ( step_eps @ Q0 @ Transs @ Bs @ Q @ Q2 )
= ( step_eps @ Q03 @ Transs4 @ Bs @ Q @ Q2 ) ) ) ) ).
% nfa_cong'.step_eps_cong_SQ
thf(fact_716_nfa__cong_H_Onfa_H__step__eps__closure__cong,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Qf4: nat,Transs: list_transition,Transs4: list_transition,Bs: list_o,Q: nat,Q2: nat] :
( ( nfa_cong @ Q0 @ Q03 @ Qf @ Qf4 @ Transs @ Transs4 )
=> ( ( step_eps_closure @ Q03 @ Transs4 @ Bs @ Q @ Q2 )
=> ( ( member_nat @ Q @ ( q @ Q03 @ Qf4 @ Transs4 ) )
=> ( step_eps_closure @ Q0 @ Transs @ Bs @ Q @ Q2 ) ) ) ) ).
% nfa_cong'.nfa'_step_eps_closure_cong
thf(fact_717_nfa__cong_H_Oeps__nfa_H__step__eps__closure__cong,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Qf4: nat,Transs: list_transition,Transs4: list_transition,Bs: list_o,Q: nat,Q2: nat] :
( ( nfa_cong @ Q0 @ Q03 @ Qf @ Qf4 @ Transs @ Transs4 )
=> ( ( step_eps_closure @ Q0 @ Transs @ Bs @ Q @ Q2 )
=> ( ( member_nat @ Q @ ( q @ Q03 @ Qf4 @ Transs4 ) )
=> ( ( ( member_nat @ Q2 @ ( q @ Q03 @ Qf4 @ Transs4 ) )
& ( step_eps_closure @ Q03 @ Transs4 @ Bs @ Q @ Q2 ) )
| ( ( step_eps_closure @ Q03 @ Transs4 @ Bs @ Q @ Qf4 )
& ( step_eps_closure @ Q0 @ Transs @ Bs @ Qf4 @ Q2 ) ) ) ) ) ) ).
% nfa_cong'.eps_nfa'_step_eps_closure_cong
thf(fact_718_nfa__cong_H_Onfa_H__eps__step__eps__closure__cong,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Qf4: nat,Transs: list_transition,Transs4: list_transition,Bs: list_o,Q: nat,Q2: nat] :
( ( nfa_cong @ Q0 @ Q03 @ Qf @ Qf4 @ Transs @ Transs4 )
=> ( ( step_eps_closure @ Q03 @ Transs4 @ Bs @ Q @ Q2 )
=> ( ( member_nat @ Q @ ( q @ Q03 @ Qf4 @ Transs4 ) )
=> ( ( member_nat @ Q2 @ ( q @ Q03 @ Qf4 @ Transs4 ) )
& ( step_eps_closure @ Q0 @ Transs @ Bs @ Q @ Q2 ) ) ) ) ) ).
% nfa_cong'.nfa'_eps_step_eps_closure_cong
thf(fact_719_nfa__cong_H_Ostep__symb__cong__SQ,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Qf4: nat,Transs: list_transition,Transs4: list_transition,Q: nat,Q2: nat] :
( ( nfa_cong @ Q0 @ Q03 @ Qf @ Qf4 @ Transs @ Transs4 )
=> ( ( member_nat @ Q @ ( sq @ Q03 @ Transs4 ) )
=> ( ( step_symb @ Q0 @ Transs @ Q @ Q2 )
= ( step_symb @ Q03 @ Transs4 @ Q @ Q2 ) ) ) ) ).
% nfa_cong'.step_symb_cong_SQ
thf(fact_720_nfa__cong_H_Ointro,axiom,
! [Q0: nat,Qf: nat,Transs: list_transition,Q03: nat,Qf4: nat,Transs4: list_transition] :
( ( nfa @ Q0 @ Qf @ Transs )
=> ( ( nfa @ Q03 @ Qf4 @ Transs4 )
=> ( ( nfa_cong_axioms @ Q0 @ Q03 @ Qf4 @ Transs @ Transs4 )
=> ( nfa_cong @ Q0 @ Q03 @ Qf @ Qf4 @ Transs @ Transs4 ) ) ) ) ).
% nfa_cong'.intro
thf(fact_721_nfa__cong_H__def,axiom,
( nfa_cong
= ( ^ [Q02: nat,Q05: nat,Qf2: nat,Qf3: nat,Transs2: list_transition,Transs3: list_transition] :
( ( nfa @ Q02 @ Qf2 @ Transs2 )
& ( nfa @ Q05 @ Qf3 @ Transs3 )
& ( nfa_cong_axioms @ Q02 @ Q05 @ Qf3 @ Transs2 @ Transs3 ) ) ) ) ).
% nfa_cong'_def
thf(fact_722_nfa__cong_H_Onfa_H__step__eps__closure__set__sub,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Qf4: nat,Transs: list_transition,Transs4: list_transition,R2: set_nat,Bs: list_o] :
( ( nfa_cong @ Q0 @ Q03 @ Qf @ Qf4 @ Transs @ Transs4 )
=> ( ( ord_less_eq_set_nat @ R2 @ ( q @ Q03 @ Qf4 @ Transs4 ) )
=> ( ord_less_eq_set_nat @ ( step_eps_closure_set @ Q03 @ Transs4 @ R2 @ Bs ) @ ( step_eps_closure_set @ Q0 @ Transs @ R2 @ Bs ) ) ) ) ).
% nfa_cong'.nfa'_step_eps_closure_set_sub
thf(fact_723_nfa__cong_H_Ostep__eps__closure__set__cong__unreach,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Qf4: nat,Transs: list_transition,Transs4: list_transition,R2: set_nat,Bs: list_o] :
( ( nfa_cong @ Q0 @ Q03 @ Qf @ Qf4 @ Transs @ Transs4 )
=> ( ( ord_less_eq_set_nat @ R2 @ ( q @ Q03 @ Qf4 @ Transs4 ) )
=> ( ~ ( member_nat @ Qf4 @ ( step_eps_closure_set @ Q03 @ Transs4 @ R2 @ Bs ) )
=> ( ( step_eps_closure_set @ Q0 @ Transs @ R2 @ Bs )
= ( step_eps_closure_set @ Q03 @ Transs4 @ R2 @ Bs ) ) ) ) ) ).
% nfa_cong'.step_eps_closure_set_cong_unreach
thf(fact_724_nfa__cong_H_Onfa_H__delta__sub__delta,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Qf4: nat,Transs: list_transition,Transs4: list_transition,R2: set_nat,Bs: list_o] :
( ( nfa_cong @ Q0 @ Q03 @ Qf @ Qf4 @ Transs @ Transs4 )
=> ( ( ord_less_eq_set_nat @ R2 @ ( q @ Q03 @ Qf4 @ Transs4 ) )
=> ( ord_less_eq_set_nat @ ( delta @ Q03 @ Transs4 @ R2 @ Bs ) @ ( delta @ Q0 @ Transs @ R2 @ Bs ) ) ) ) ).
% nfa_cong'.nfa'_delta_sub_delta
thf(fact_725_nfa__cong_H_Ostep__symb__set__cong__SQ,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Qf4: nat,Transs: list_transition,Transs4: list_transition,R2: set_nat] :
( ( nfa_cong @ Q0 @ Q03 @ Qf @ Qf4 @ Transs @ Transs4 )
=> ( ( ord_less_eq_set_nat @ R2 @ ( sq @ Q03 @ Transs4 ) )
=> ( ( step_symb_set @ Q0 @ Transs @ R2 )
= ( step_symb_set @ Q03 @ Transs4 @ R2 ) ) ) ) ).
% nfa_cong'.step_symb_set_cong_SQ
thf(fact_726_nfa__cong_H_Otranss__eq,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Qf4: nat,Transs: list_transition,Transs4: list_transition,Q: nat] :
( ( nfa_cong @ Q0 @ Q03 @ Qf @ Qf4 @ Transs @ Transs4 )
=> ( ( member_nat @ Q @ ( sq @ Q03 @ Transs4 ) )
=> ( ( nth_transition @ Transs @ ( minus_minus_nat @ Q @ Q0 ) )
= ( nth_transition @ Transs4 @ ( minus_minus_nat @ Q @ Q03 ) ) ) ) ) ).
% nfa_cong'.transs_eq
thf(fact_727_nfa__cong_H_Ostep__eps__closure__set__cong__reach,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Qf4: nat,Transs: list_transition,Transs4: list_transition,R2: set_nat,Bs: list_o] :
( ( nfa_cong @ Q0 @ Q03 @ Qf @ Qf4 @ Transs @ Transs4 )
=> ( ( ord_less_eq_set_nat @ R2 @ ( q @ Q03 @ Qf4 @ Transs4 ) )
=> ( ( member_nat @ Qf4 @ ( step_eps_closure_set @ Q03 @ Transs4 @ R2 @ Bs ) )
=> ( ( step_eps_closure_set @ Q0 @ Transs @ R2 @ Bs )
= ( sup_sup_set_nat @ ( step_eps_closure_set @ Q03 @ Transs4 @ R2 @ Bs ) @ ( step_eps_closure_set @ Q0 @ Transs @ ( insert_nat @ Qf4 @ bot_bot_set_nat ) @ Bs ) ) ) ) ) ) ).
% nfa_cong'.step_eps_closure_set_cong_reach
thf(fact_728_nfa__cong__Times_Orun__accept__eps__cong,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Transs: list_transition,Transs4: list_transition,Transs5: list_transition,Bss: list_list_o,Bs: list_o] :
( ( nfa_cong_Times @ Q0 @ Q03 @ Qf @ Transs @ Transs4 @ Transs5 )
=> ( ( run_accept_eps @ Q0 @ Qf @ Transs @ ( insert_nat @ Q0 @ bot_bot_set_nat ) @ Bss @ Bs )
= ( ( ( run_accept_eps @ Q0 @ Q03 @ Transs4 @ ( insert_nat @ Q0 @ bot_bot_set_nat ) @ Bss @ Bs )
& ( run_accept_eps @ Q03 @ Qf @ Transs5 @ ( insert_nat @ Q03 @ bot_bot_set_nat ) @ nil_list_o @ Bs ) )
| ? [X: nat] :
( ( member_nat @ X @ ( set_ord_lessThan_nat @ ( size_s2710708370519433104list_o @ Bss ) ) )
& ( run_accept_eps @ Q0 @ Q03 @ Transs4 @ ( insert_nat @ Q0 @ bot_bot_set_nat ) @ ( take_list_o @ X @ Bss ) @ ( hd_list_o @ ( drop_list_o @ X @ Bss ) ) )
& ( run_accept_eps @ Q03 @ Qf @ Transs5 @ ( insert_nat @ Q03 @ bot_bot_set_nat ) @ ( drop_list_o @ X @ Bss ) @ Bs ) ) ) ) ) ).
% nfa_cong_Times.run_accept_eps_cong
thf(fact_729_nfa__cong__Star_Orun__accept__eps__cong__Cons,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Transs: list_transition,Transs4: list_transition,Bs: list_o,Bss: list_list_o,Cs: list_o] :
( ( nfa_cong_Star @ Q0 @ Q03 @ Qf @ Transs @ Transs4 )
=> ( ( run_accept_eps @ Q0 @ Qf @ Transs @ ( insert_nat @ Q0 @ bot_bot_set_nat ) @ ( cons_list_o @ Bs @ Bss ) @ Cs )
= ( ( ( run_accept_eps @ Q03 @ Q0 @ Transs4 @ ( insert_nat @ Q03 @ bot_bot_set_nat ) @ ( cons_list_o @ Bs @ Bss ) @ Cs )
& ( run_accept_eps @ Q0 @ Qf @ Transs @ ( insert_nat @ Q0 @ bot_bot_set_nat ) @ nil_list_o @ Cs ) )
| ? [X: nat] :
( ( member_nat @ X @ ( set_ord_lessThan_nat @ ( size_s2710708370519433104list_o @ Bss ) ) )
& ( run_accept_eps @ Q03 @ Q0 @ Transs4 @ ( insert_nat @ Q03 @ bot_bot_set_nat ) @ ( take_list_o @ ( suc @ X ) @ ( cons_list_o @ Bs @ Bss ) ) @ ( hd_list_o @ ( drop_list_o @ X @ Bss ) ) )
& ( run_accept_eps @ Q0 @ Qf @ Transs @ ( insert_nat @ Q0 @ bot_bot_set_nat ) @ ( drop_list_o @ X @ Bss ) @ Cs ) ) ) ) ) ).
% nfa_cong_Star.run_accept_eps_cong_Cons
thf(fact_730_nfa__cong__Star_Oaxioms_I1_J,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Transs: list_transition,Transs4: list_transition] :
( ( nfa_cong_Star @ Q0 @ Q03 @ Qf @ Transs @ Transs4 )
=> ( nfa_cong @ Q0 @ Q03 @ Qf @ Q0 @ Transs @ Transs4 ) ) ).
% nfa_cong_Star.axioms(1)
thf(fact_731_nfa__cong__Star_Ostep__symb__q0,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Transs: list_transition,Transs4: list_transition,Q: nat] :
( ( nfa_cong_Star @ Q0 @ Q03 @ Qf @ Transs @ Transs4 )
=> ~ ( step_symb @ Q0 @ Transs @ Q0 @ Q ) ) ).
% nfa_cong_Star.step_symb_q0
thf(fact_732_nfa__cong__Star_Oaxioms_I2_J,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Transs: list_transition,Transs4: list_transition] :
( ( nfa_cong_Star @ Q0 @ Q03 @ Qf @ Transs @ Transs4 )
=> ( nfa_cong_Star_axioms @ Q0 @ Q03 @ Qf @ Transs ) ) ).
% nfa_cong_Star.axioms(2)
thf(fact_733_nfa__cong__Times_Oaxioms_I1_J,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Transs: list_transition,Transs4: list_transition,Transs5: list_transition] :
( ( nfa_cong_Times @ Q0 @ Q03 @ Qf @ Transs @ Transs4 @ Transs5 )
=> ( nfa_cong @ Q0 @ Q0 @ Qf @ Q03 @ Transs @ Transs4 ) ) ).
% nfa_cong_Times.axioms(1)
thf(fact_734_nfa__cong__Star_Ointro,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Transs: list_transition,Transs4: list_transition] :
( ( nfa_cong @ Q0 @ Q03 @ Qf @ Q0 @ Transs @ Transs4 )
=> ( ( nfa_cong_Star_axioms @ Q0 @ Q03 @ Qf @ Transs )
=> ( nfa_cong_Star @ Q0 @ Q03 @ Qf @ Transs @ Transs4 ) ) ) ).
% nfa_cong_Star.intro
thf(fact_735_nfa__cong__Star__def,axiom,
( nfa_cong_Star
= ( ^ [Q02: nat,Q05: nat,Qf2: nat,Transs2: list_transition,Transs3: list_transition] :
( ( nfa_cong @ Q02 @ Q05 @ Qf2 @ Q02 @ Transs2 @ Transs3 )
& ( nfa_cong_Star_axioms @ Q02 @ Q05 @ Qf2 @ Transs2 ) ) ) ) ).
% nfa_cong_Star_def
thf(fact_736_nfa__cong__Star_Odelta__q0__q0_H,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Transs: list_transition,Transs4: list_transition,Bs: list_o] :
( ( nfa_cong_Star @ Q0 @ Q03 @ Qf @ Transs @ Transs4 )
=> ( ( delta @ Q0 @ Transs @ ( insert_nat @ Q0 @ bot_bot_set_nat ) @ Bs )
= ( delta @ Q0 @ Transs @ ( insert_nat @ Q03 @ bot_bot_set_nat ) @ Bs ) ) ) ).
% nfa_cong_Star.delta_q0_q0'
thf(fact_737_nfa__cong__Star_Ostep__eps__q0,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Transs: list_transition,Transs4: list_transition,Bs: list_o,Q: nat] :
( ( nfa_cong_Star @ Q0 @ Q03 @ Qf @ Transs @ Transs4 )
=> ( ( step_eps @ Q0 @ Transs @ Bs @ Q0 @ Q )
= ( member_nat @ Q @ ( insert_nat @ Q03 @ ( insert_nat @ Qf @ bot_bot_set_nat ) ) ) ) ) ).
% nfa_cong_Star.step_eps_q0
thf(fact_738_nfa__cong__Star_Ostep__symb__set__q0,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Transs: list_transition,Transs4: list_transition] :
( ( nfa_cong_Star @ Q0 @ Q03 @ Qf @ Transs @ Transs4 )
=> ( ( 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_739_nfa__cong__Star_Odelta__sub__nfa_H__delta,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Transs: list_transition,Transs4: list_transition,Bs: list_o] :
( ( nfa_cong_Star @ Q0 @ Q03 @ Qf @ Transs @ Transs4 )
=> ( ord_less_eq_set_nat @ ( delta @ Q0 @ Transs @ ( insert_nat @ Q0 @ bot_bot_set_nat ) @ Bs ) @ ( delta @ Q03 @ Transs4 @ ( insert_nat @ Q03 @ bot_bot_set_nat ) @ Bs ) ) ) ).
% nfa_cong_Star.delta_sub_nfa'_delta
thf(fact_740_nfa__cong__Star_Ostep__eps__closure__set__q0__split,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Transs: list_transition,Transs4: list_transition,Bs: list_o] :
( ( nfa_cong_Star @ Q0 @ Q03 @ Qf @ Transs @ Transs4 )
=> ( ( 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 @ Q03 @ bot_bot_set_nat ) @ Bs ) ) ) ) ).
% nfa_cong_Star.step_eps_closure_set_q0_split
thf(fact_741_nfa__cong__Star_Orun__accept__eps__Nil,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Transs: list_transition,Transs4: list_transition,Bs: list_o] :
( ( nfa_cong_Star @ Q0 @ Q03 @ Qf @ Transs @ Transs4 )
=> ( 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_742_nfa__cong_H_Oaccept__eps__nfa_H__run,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Qf4: nat,Transs: list_transition,Transs4: list_transition,R2: set_nat,Bss: list_list_o,Bs: list_o] :
( ( nfa_cong @ Q0 @ Q03 @ Qf @ Qf4 @ Transs @ Transs4 )
=> ( ( ord_less_eq_set_nat @ R2 @ ( q @ Q03 @ Qf4 @ Transs4 ) )
=> ( ( accept_eps @ Q0 @ Qf @ Transs @ ( run @ Q03 @ Transs4 @ R2 @ Bss ) @ Bs )
= ( ( accept_eps @ Q03 @ Qf4 @ Transs4 @ ( run @ Q03 @ Transs4 @ R2 @ Bss ) @ Bs )
& ( accept_eps @ Q0 @ Qf @ Transs @ ( run @ Q0 @ Transs @ ( insert_nat @ Qf4 @ bot_bot_set_nat ) @ nil_list_o ) @ Bs ) ) ) ) ) ).
% nfa_cong'.accept_eps_nfa'_run
thf(fact_743_nfa__cong_H_Odelta__cong__reach,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Qf4: nat,Transs: list_transition,Transs4: list_transition,R2: set_nat,Bs: list_o] :
( ( nfa_cong @ Q0 @ Q03 @ Qf @ Qf4 @ Transs @ Transs4 )
=> ( ( ord_less_eq_set_nat @ R2 @ ( q @ Q03 @ Qf4 @ Transs4 ) )
=> ( ( accept_eps @ Q03 @ Qf4 @ Transs4 @ R2 @ Bs )
=> ( ( delta @ Q0 @ Transs @ R2 @ Bs )
= ( sup_sup_set_nat @ ( delta @ Q03 @ Transs4 @ R2 @ Bs ) @ ( delta @ Q0 @ Transs @ ( insert_nat @ Qf4 @ bot_bot_set_nat ) @ Bs ) ) ) ) ) ) ).
% nfa_cong'.delta_cong_reach
thf(fact_744_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_745_NFA_Oaccept__eps__def,axiom,
( accept_eps
= ( ^ [Q02: nat,Qf2: nat,Transs2: list_transition,R3: set_nat,Bs2: list_o] : ( member_nat @ Qf2 @ ( step_eps_closure_set @ Q02 @ Transs2 @ R3 @ Bs2 ) ) ) ) ).
% NFA.accept_eps_def
thf(fact_746_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_747_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_748_NFA_Orun__accept__eps__def,axiom,
( run_accept_eps
= ( ^ [Q02: nat,Qf2: nat,Transs2: list_transition,R3: set_nat,Bss2: list_list_o] : ( accept_eps @ Q02 @ Qf2 @ Transs2 @ ( run @ Q02 @ Transs2 @ R3 @ Bss2 ) ) ) ) ).
% NFA.run_accept_eps_def
thf(fact_749_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_750_nfa__cong_H_Odelta__cong__unreach,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Qf4: nat,Transs: list_transition,Transs4: list_transition,R2: set_nat,Bs: list_o] :
( ( nfa_cong @ Q0 @ Q03 @ Qf @ Qf4 @ Transs @ Transs4 )
=> ( ( ord_less_eq_set_nat @ R2 @ ( q @ Q03 @ Qf4 @ Transs4 ) )
=> ( ~ ( accept_eps @ Q03 @ Qf4 @ Transs4 @ R2 @ Bs )
=> ( ( delta @ Q0 @ Transs @ R2 @ Bs )
= ( delta @ Q03 @ Transs4 @ R2 @ Bs ) ) ) ) ) ).
% nfa_cong'.delta_cong_unreach
thf(fact_751_NFA_Oaccept__def,axiom,
( accept
= ( ^ [Q02: nat,Qf2: nat,Transs2: list_transition,R3: set_nat] : ( accept_eps @ Q02 @ Qf2 @ Transs2 @ R3 @ nil_o ) ) ) ).
% NFA.accept_def
thf(fact_752_NFA_Orun__accept__def,axiom,
( run_accept
= ( ^ [Q02: nat,Qf2: nat,Transs2: list_transition,R3: set_nat,Bss2: list_list_o] : ( accept @ Q02 @ Qf2 @ Transs2 @ ( run @ Q02 @ Transs2 @ R3 @ Bss2 ) ) ) ) ).
% NFA.run_accept_def
thf(fact_753_transition_Oinject_I3_J,axiom,
! [X31: nat,X32: nat,Y31: nat,Y32: nat] :
( ( ( split_trans @ X31 @ X32 )
= ( split_trans @ Y31 @ Y32 ) )
= ( ( X31 = Y31 )
& ( X32 = Y32 ) ) ) ).
% transition.inject(3)
thf(fact_754_fmla__set_Osimps_I3_J,axiom,
! [V: nat,Va: nat] :
( ( fmla_set @ ( split_trans @ V @ Va ) )
= bot_bot_set_nat ) ).
% fmla_set.simps(3)
thf(fact_755_state__set_Osimps_I3_J,axiom,
! [S: nat,S3: nat] :
( ( state_set @ ( split_trans @ S @ S3 ) )
= ( insert_nat @ S @ ( insert_nat @ S3 @ bot_bot_set_nat ) ) ) ).
% state_set.simps(3)
thf(fact_756_nfa__cong__Star_Ostep__eps__closure__set__q0,axiom,
! [Q0: nat,Q03: nat,Qf: nat,Transs: list_transition,Transs4: list_transition,Bs: list_o] :
( ( nfa_cong_Star @ Q0 @ Q03 @ Qf @ Transs @ Transs4 )
=> ( 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 @ Q03 @ Transs4 @ ( insert_nat @ Q03 @ bot_bot_set_nat ) @ Bs ) @ ( sq @ Q03 @ Transs4 ) ) ) ) ) ).
% nfa_cong_Star.step_eps_closure_set_q0
thf(fact_757_step__symb__set__proj,axiom,
( step_symb_set
= ( ^ [Q02: nat,Transs2: list_transition,R3: set_nat] : ( step_symb_set @ Q02 @ Transs2 @ ( inf_inf_set_nat @ R3 @ ( sq @ Q02 @ Transs2 ) ) ) ) ) ).
% step_symb_set_proj
% Conjectures (2)
thf(conj_0,hypothesis,
ord_less_nat @ n @ ( size_s2710708370519433104list_o @ bssb ) ).
thf(conj_1,conjecture,
( ! [X5: list_o] :
( ( member_list_o @ X5 @ ( set_list_o2 @ ( take_list_o @ n @ bssb ) ) )
=> ( ord_less_eq_nat @ ( size_s8846756101701226951la_a_t @ ( collect_subfmlas_a_t @ ra @ phisa ) ) @ ( size_size_list_o @ X5 ) ) )
& ( ord_less_eq_nat @ ( size_s8846756101701226951la_a_t @ ( collect_subfmlas_a_t @ ra @ phisa ) ) @ ( size_size_list_o @ ( hd_list_o @ ( drop_list_o @ n @ bssb ) ) ) )
& ~ ( member_nat @ q0a @ ( sq @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( build_nfa_impl_a_t @ ra @ ( produc8654416511292156347la_a_t @ ( plus_plus_nat @ q0a @ one_one_nat ) @ ( produc9017461973804568604la_a_t @ q0a @ phisa ) ) ) ) )
& ! [K3: nat] :
( ~ ( ord_less_nat @ K3 @ ( size_s8846756101701226951la_a_t @ ( collect_subfmlas_a_t @ ra @ phisa ) ) )
| ( ( nth_o @ ( hd_list_o @ ( drop_list_o @ n @ bssb ) ) @ K3 )
!= ( ~ ( sat_a_t @ sigma @ ( nth_formula_a_t @ ( collect_subfmlas_a_t @ ra @ phisa ) @ K3 ) @ ( plus_plus_nat @ ib @ ( size_s2710708370519433104list_o @ ( take_list_o @ n @ bssb ) ) ) ) ) ) )
& ! [J2: nat] :
( ~ ( ord_less_nat @ J2 @ ( size_s2710708370519433104list_o @ ( take_list_o @ n @ bssb ) ) )
| ! [K3: nat] :
( ~ ( ord_less_nat @ K3 @ ( size_s8846756101701226951la_a_t @ ( collect_subfmlas_a_t @ ra @ phisa ) ) )
| ( ( nth_o @ ( nth_list_o @ ( take_list_o @ n @ bssb ) @ J2 ) @ K3 )
!= ( ~ ( sat_a_t @ sigma @ ( nth_formula_a_t @ ( collect_subfmlas_a_t @ ra @ phisa ) @ K3 ) @ ( plus_plus_nat @ ib @ J2 ) ) ) ) ) ) ) ).
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