TPTP Problem File: SLH0255^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : FOL_Seq_Calc2/0017_Countermodel/prob_00092_003195__12897760_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1589 ( 711 unt; 308 typ; 0 def)
% Number of atoms : 3188 (1773 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 9279 ( 340 ~; 60 |; 266 &;7542 @)
% ( 0 <=>;1071 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 5 avg)
% Number of types : 25 ( 24 usr)
% Number of type conns : 1302 (1302 >; 0 *; 0 +; 0 <<)
% Number of symbols : 287 ( 284 usr; 32 con; 0-5 aty)
% Number of variables : 3594 ( 401 ^;3081 !; 112 ?;3594 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 08:44:45.642
%------------------------------------------------------------------------------
% Could-be-implicit typings (24)
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_J,type,
list_list_list_fm: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_J,type,
set_list_list_fm: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J_J,type,
list_set_list_fm: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__SeCaV__Otm_J_J,type,
list_list_tm: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
list_list_fm: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__SeCaV__Otm_J_J,type,
set_list_tm: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
set_list_fm: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__SeCaV__Otm_J_J,type,
list_set_tm: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__SeCaV__Ofm_J_J,type,
list_set_fm: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
list_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__SeCaV__Otm_J_J,type,
set_set_tm: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__List__Olist_It__SeCaV__Otm_J,type,
list_tm: $tType ).
thf(ty_n_t__List__Olist_It__SeCaV__Ofm_J,type,
list_fm: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__SeCaV__Otm_J,type,
set_tm: $tType ).
thf(ty_n_t__Set__Oset_It__SeCaV__Ofm_J,type,
set_fm: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Prover__Orule,type,
rule: $tType ).
thf(ty_n_t__SeCaV__Otm,type,
tm: $tType ).
thf(ty_n_t__SeCaV__Ofm,type,
fm: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (284)
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__List__Olist_It__SeCaV__Ofm_J,type,
bNF_Gr4365904581682047384ist_fm: set_list_list_fm > list_fm > set_list_list_fm ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Nat__Onat,type,
bNF_Gr1872714664788909425ft_nat: set_list_nat > nat > set_list_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__SeCaV__Ofm,type,
bNF_Gr6607445516916672786ift_fm: set_list_fm > fm > set_list_fm ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__SeCaV__Otm,type,
bNF_Gr6607445516917591172ift_tm: set_list_tm > tm > set_list_tm ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__List__Olist_It__SeCaV__Ofm_J,type,
bNF_Gr8387611704671093012ist_fm: set_list_list_fm > list_list_fm > set_list_fm ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Nat__Onat,type,
bNF_Gr6352880689984616693cc_nat: set_list_nat > list_nat > set_nat ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__SeCaV__Ofm,type,
bNF_Greatest_Succ_fm: set_list_fm > list_fm > set_fm ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__SeCaV__Otm,type,
bNF_Greatest_Succ_tm: set_list_tm > list_tm > set_tm ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
comple7399068483239264473et_nat: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__SeCaV__Otm_J,type,
comple2138885804642794802set_tm: set_set_tm > set_tm ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Fun_Ofun__upd_001t__Nat__Onat_001_062_It__List__Olist_It__SeCaV__Otm_J_Mt__SeCaV__Otm_J,type,
fun_up3737067029978700398_tm_tm: ( nat > list_tm > tm ) > nat > ( list_tm > tm ) > nat > list_tm > tm ).
thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Nat__Onat,type,
monotone_on_nat_nat: set_nat > ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat ) > $o ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_HOL_OUniq_001_Eo,type,
uniq_o: ( $o > $o ) > $o ).
thf(sy_c_HOL_OUniq_001t__List__Olist_It__SeCaV__Ofm_J,type,
uniq_list_fm: ( list_fm > $o ) > $o ).
thf(sy_c_HOL_OUniq_001t__Nat__Onat,type,
uniq_nat: ( nat > $o ) > $o ).
thf(sy_c_HOL_OUniq_001t__SeCaV__Otm,type,
uniq_tm: ( tm > $o ) > $o ).
thf(sy_c_Hilbert__Choice_Oinv__into_001_Eo_001_Eo,type,
hilbert_inv_into_o_o: set_o > ( $o > $o ) > $o > $o ).
thf(sy_c_Hilbert__Choice_Oinv__into_001_Eo_001t__Nat__Onat,type,
hilber770666707827806236_o_nat: set_o > ( $o > nat ) > nat > $o ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__List__Olist_It__SeCaV__Ofm_J_001t__Nat__Onat,type,
hilber1704621864209052157fm_nat: set_list_fm > ( list_fm > nat ) > nat > list_fm ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__List__Olist_It__SeCaV__Ofm_J_001t__SeCaV__Otm,type,
hilber1817404320317758200_fm_tm: set_list_fm > ( list_fm > tm ) > tm > list_fm ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Nat__Onat_001_Eo,type,
hilber3873338068935991546_nat_o: set_nat > ( nat > $o ) > $o > nat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Nat__Onat_001t__Nat__Onat,type,
hilber3633877196798814958at_nat: set_nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Nat__Onat_001t__SeCaV__Otm,type,
hilber5150249191788879431nat_tm: set_nat > ( nat > tm ) > tm > nat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__SeCaV__Ofm_001t__Set__Oset_It__Nat__Onat_J,type,
hilber5456991776183730989et_nat: set_fm > ( fm > set_nat ) > set_nat > fm ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__SeCaV__Ofm_001t__Set__Oset_It__SeCaV__Otm_J,type,
hilber6286924743720013790set_tm: set_fm > ( fm > set_tm ) > set_tm > fm ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__SeCaV__Otm_001t__Nat__Onat,type,
hilber3344278766186360553tm_nat: set_tm > ( tm > nat ) > nat > tm ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__SeCaV__Otm_001t__SeCaV__Otm,type,
hilber7929195366230530316_tm_tm: set_tm > ( tm > tm ) > tm > tm ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__SeCaV__Otm_001t__Set__Oset_It__Nat__Onat_J,type,
hilber8933797207411865247et_nat: set_tm > ( tm > set_nat ) > set_nat > tm ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__SeCaV__Otm_001t__Set__Oset_It__SeCaV__Otm_J,type,
hilber5702680533203024108set_tm: set_tm > ( tm > set_tm ) > set_tm > tm ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Set__Oset_It__Nat__Onat_J_001t__SeCaV__Ofm,type,
hilber6297111070489569439nat_fm: set_set_nat > ( set_nat > fm ) > fm > set_nat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Set__Oset_It__Nat__Onat_J_001t__SeCaV__Otm,type,
hilber6297111070490487825nat_tm: set_set_nat > ( set_nat > tm ) > tm > set_nat ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Set__Oset_It__SeCaV__Otm_J_001t__SeCaV__Ofm,type,
hilber6730722110952398266_tm_fm: set_set_tm > ( set_tm > fm ) > fm > set_tm ).
thf(sy_c_Hilbert__Choice_Oinv__into_001t__Set__Oset_It__SeCaV__Otm_J_001t__SeCaV__Otm,type,
hilber6730722110953316652_tm_tm: set_set_tm > ( set_tm > tm ) > tm > set_tm ).
thf(sy_c_Hintikka_Oterms,type,
terms: set_fm > set_tm ).
thf(sy_c_If_001t__SeCaV__Otm,type,
if_tm: $o > tm > tm > tm ).
thf(sy_c_If_001t__Set__Oset_It__SeCaV__Otm_J,type,
if_set_tm: $o > set_tm > set_tm > set_tm ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__List__Olist_It__SeCaV__Ofm_J_M_Eo_J,type,
sup_sup_list_fm_o: ( list_fm > $o ) > ( list_fm > $o ) > list_fm > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_Eo_J,type,
sup_sup_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__SeCaV__Otm_M_Eo_J,type,
sup_sup_tm_o: ( tm > $o ) > ( tm > $o ) > tm > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_Eo_J,type,
sup_sup_set_o: set_o > set_o > set_o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
sup_sup_set_list_fm: set_list_fm > set_list_fm > set_list_fm ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__SeCaV__Ofm_J,type,
sup_sup_set_fm: set_fm > set_fm > set_fm ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__SeCaV__Otm_J,type,
sup_sup_set_tm: set_tm > set_tm > set_tm ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
sup_sup_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__SeCaV__Otm_J_J,type,
sup_sup_set_set_tm: set_set_tm > set_set_tm > set_set_tm ).
thf(sy_c_List_Oappend_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
append_list_list_fm: list_list_list_fm > list_list_list_fm > list_list_list_fm ).
thf(sy_c_List_Oappend_001t__List__Olist_It__SeCaV__Ofm_J,type,
append_list_fm: list_list_fm > list_list_fm > list_list_fm ).
thf(sy_c_List_Oappend_001t__List__Olist_It__SeCaV__Otm_J,type,
append_list_tm: list_list_tm > list_list_tm > list_list_tm ).
thf(sy_c_List_Oappend_001t__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Oappend_001t__SeCaV__Ofm,type,
append_fm: list_fm > list_fm > list_fm ).
thf(sy_c_List_Oappend_001t__SeCaV__Otm,type,
append_tm: list_tm > list_tm > list_tm ).
thf(sy_c_List_Obind_001t__List__Olist_It__SeCaV__Ofm_J_001t__List__Olist_It__SeCaV__Ofm_J,type,
bind_list_fm_list_fm: list_list_fm > ( list_fm > list_list_fm ) > list_list_fm ).
thf(sy_c_List_Obind_001t__List__Olist_It__SeCaV__Ofm_J_001t__SeCaV__Ofm,type,
bind_list_fm_fm: list_list_fm > ( list_fm > list_fm ) > list_fm ).
thf(sy_c_List_Obind_001t__List__Olist_It__SeCaV__Ofm_J_001t__SeCaV__Otm,type,
bind_list_fm_tm: list_list_fm > ( list_fm > list_tm ) > list_tm ).
thf(sy_c_List_Obind_001t__SeCaV__Ofm_001t__List__Olist_It__SeCaV__Ofm_J,type,
bind_fm_list_fm: list_fm > ( fm > list_list_fm ) > list_list_fm ).
thf(sy_c_List_Obind_001t__SeCaV__Ofm_001t__SeCaV__Ofm,type,
bind_fm_fm: list_fm > ( fm > list_fm ) > list_fm ).
thf(sy_c_List_Obind_001t__SeCaV__Ofm_001t__SeCaV__Otm,type,
bind_fm_tm: list_fm > ( fm > list_tm ) > list_tm ).
thf(sy_c_List_Obind_001t__SeCaV__Otm_001t__List__Olist_It__SeCaV__Ofm_J,type,
bind_tm_list_fm: list_tm > ( tm > list_list_fm ) > list_list_fm ).
thf(sy_c_List_Obind_001t__SeCaV__Otm_001t__SeCaV__Ofm,type,
bind_tm_fm: list_tm > ( tm > list_fm ) > list_fm ).
thf(sy_c_List_Obind_001t__SeCaV__Otm_001t__SeCaV__Otm,type,
bind_tm_tm: list_tm > ( tm > list_tm ) > list_tm ).
thf(sy_c_List_Oconcat_001t__SeCaV__Ofm,type,
concat_fm: list_list_fm > list_fm ).
thf(sy_c_List_Oconcat_001t__SeCaV__Otm,type,
concat_tm: list_list_tm > list_tm ).
thf(sy_c_List_Oinsert_001t__List__Olist_It__SeCaV__Ofm_J,type,
insert_list_fm: list_fm > list_list_fm > list_list_fm ).
thf(sy_c_List_Oinsert_001t__SeCaV__Ofm,type,
insert_fm: fm > list_fm > list_fm ).
thf(sy_c_List_Oinsert_001t__SeCaV__Otm,type,
insert_tm: tm > list_tm > list_tm ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
cons_list_list_fm: list_list_fm > list_list_list_fm > list_list_list_fm ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__SeCaV__Ofm_J,type,
cons_list_fm: list_fm > list_list_fm > list_list_fm ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__SeCaV__Otm_J,type,
cons_list_tm: list_tm > list_list_tm > list_list_tm ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__SeCaV__Ofm,type,
cons_fm: fm > list_fm > list_fm ).
thf(sy_c_List_Olist_OCons_001t__SeCaV__Otm,type,
cons_tm: tm > list_tm > list_tm ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
nil_list_list_fm: list_list_list_fm ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__SeCaV__Ofm_J,type,
nil_list_fm: list_list_fm ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__SeCaV__Otm_J,type,
nil_list_tm: list_list_tm ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001t__SeCaV__Ofm,type,
nil_fm: list_fm ).
thf(sy_c_List_Olist_ONil_001t__SeCaV__Otm,type,
nil_tm: list_tm ).
thf(sy_c_List_Olist_ONil_001t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
nil_set_list_fm: list_set_list_fm ).
thf(sy_c_List_Olist_ONil_001t__Set__Oset_It__Nat__Onat_J,type,
nil_set_nat: list_set_nat ).
thf(sy_c_List_Olist_ONil_001t__Set__Oset_It__SeCaV__Ofm_J,type,
nil_set_fm: list_set_fm ).
thf(sy_c_List_Olist_ONil_001t__Set__Oset_It__SeCaV__Otm_J,type,
nil_set_tm: list_set_tm ).
thf(sy_c_List_Olist_Ocase__list_001t__List__Olist_It__SeCaV__Otm_J_001t__SeCaV__Otm,type,
case_list_list_tm_tm: list_tm > ( tm > list_tm > list_tm ) > list_tm > list_tm ).
thf(sy_c_List_Olist_Olist__all_001t__List__Olist_It__SeCaV__Ofm_J,type,
list_all_list_fm: ( list_fm > $o ) > list_list_fm > $o ).
thf(sy_c_List_Olist_Olist__all_001t__SeCaV__Ofm,type,
list_all_fm: ( fm > $o ) > list_fm > $o ).
thf(sy_c_List_Olist_Olist__all_001t__SeCaV__Otm,type,
list_all_tm: ( tm > $o ) > list_tm > $o ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
map_li4351931137408529412ist_fm: ( list_list_fm > list_list_fm ) > list_list_list_fm > list_list_list_fm ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__SeCaV__Ofm_J_001t__List__Olist_It__SeCaV__Ofm_J,type,
map_list_fm_list_fm: ( list_fm > list_fm ) > list_list_fm > list_list_fm ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__SeCaV__Ofm_J_001t__SeCaV__Ofm,type,
map_list_fm_fm: ( list_fm > fm ) > list_list_fm > list_fm ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__SeCaV__Ofm_J_001t__SeCaV__Otm,type,
map_list_fm_tm: ( list_fm > tm ) > list_list_fm > list_tm ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__SeCaV__Otm_J_001t__List__Olist_It__SeCaV__Otm_J,type,
map_list_tm_list_tm: ( list_tm > list_tm ) > list_list_tm > list_list_tm ).
thf(sy_c_List_Olist_Omap_001t__SeCaV__Ofm_001t__List__Olist_It__SeCaV__Ofm_J,type,
map_fm_list_fm: ( fm > list_fm ) > list_fm > list_list_fm ).
thf(sy_c_List_Olist_Omap_001t__SeCaV__Ofm_001t__List__Olist_It__SeCaV__Otm_J,type,
map_fm_list_tm: ( fm > list_tm ) > list_fm > list_list_tm ).
thf(sy_c_List_Olist_Omap_001t__SeCaV__Ofm_001t__SeCaV__Ofm,type,
map_fm_fm: ( fm > fm ) > list_fm > list_fm ).
thf(sy_c_List_Olist_Omap_001t__SeCaV__Ofm_001t__SeCaV__Otm,type,
map_fm_tm: ( fm > tm ) > list_fm > list_tm ).
thf(sy_c_List_Olist_Omap_001t__SeCaV__Otm_001t__List__Olist_It__SeCaV__Ofm_J,type,
map_tm_list_fm: ( tm > list_fm ) > list_tm > list_list_fm ).
thf(sy_c_List_Olist_Omap_001t__SeCaV__Otm_001t__List__Olist_It__SeCaV__Otm_J,type,
map_tm_list_tm: ( tm > list_tm ) > list_tm > list_list_tm ).
thf(sy_c_List_Olist_Omap_001t__SeCaV__Otm_001t__SeCaV__Ofm,type,
map_tm_fm: ( tm > fm ) > list_tm > list_fm ).
thf(sy_c_List_Olist_Omap_001t__SeCaV__Otm_001t__SeCaV__Otm,type,
map_tm_tm: ( tm > tm ) > list_tm > list_tm ).
thf(sy_c_List_Olist_Omap_001t__SeCaV__Otm_001t__Set__Oset_It__Nat__Onat_J,type,
map_tm_set_nat: ( tm > set_nat ) > list_tm > list_set_nat ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__SeCaV__Ofm_J,type,
set_list_fm2: list_list_fm > set_list_fm ).
thf(sy_c_List_Olist_Oset_001t__SeCaV__Ofm,type,
set_fm2: list_fm > set_fm ).
thf(sy_c_List_Olist_Oset_001t__SeCaV__Otm,type,
set_tm2: list_tm > set_tm ).
thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
set_set_nat2: list_set_nat > set_set_nat ).
thf(sy_c_List_Olist__ex1_001t__List__Olist_It__SeCaV__Ofm_J,type,
list_ex1_list_fm: ( list_fm > $o ) > list_list_fm > $o ).
thf(sy_c_List_Olist__ex1_001t__SeCaV__Ofm,type,
list_ex1_fm: ( fm > $o ) > list_fm > $o ).
thf(sy_c_List_Olist__ex1_001t__SeCaV__Otm,type,
list_ex1_tm: ( tm > $o ) > list_tm > $o ).
thf(sy_c_List_Olistset_001t__List__Olist_It__SeCaV__Ofm_J,type,
listset_list_fm: list_set_list_fm > set_list_list_fm ).
thf(sy_c_List_Olistset_001t__SeCaV__Ofm,type,
listset_fm: list_set_fm > set_list_fm ).
thf(sy_c_List_Olistset_001t__SeCaV__Otm,type,
listset_tm: list_set_tm > set_list_tm ).
thf(sy_c_List_Oproduct__lists_001t__List__Olist_It__SeCaV__Ofm_J,type,
produc373462945560358120ist_fm: list_list_list_fm > list_list_list_fm ).
thf(sy_c_List_Oproduct__lists_001t__SeCaV__Ofm,type,
product_lists_fm: list_list_fm > list_list_fm ).
thf(sy_c_List_Oproduct__lists_001t__SeCaV__Otm,type,
product_lists_tm: list_list_tm > list_list_tm ).
thf(sy_c_List_Oremdups_001t__SeCaV__Otm,type,
remdups_tm: list_tm > list_tm ).
thf(sy_c_List_Oset__Cons_001t__List__Olist_It__SeCaV__Ofm_J,type,
set_Cons_list_fm: set_list_fm > set_list_list_fm > set_list_list_fm ).
thf(sy_c_List_Oset__Cons_001t__SeCaV__Ofm,type,
set_Cons_fm: set_fm > set_list_fm > set_list_fm ).
thf(sy_c_List_Oset__Cons_001t__SeCaV__Otm,type,
set_Cons_tm: set_tm > set_list_tm > set_list_tm ).
thf(sy_c_List_Osubseqs_001t__List__Olist_It__SeCaV__Ofm_J,type,
subseqs_list_fm: list_list_fm > list_list_list_fm ).
thf(sy_c_List_Osubseqs_001t__SeCaV__Ofm,type,
subseqs_fm: list_fm > list_list_fm ).
thf(sy_c_List_Osubseqs_001t__SeCaV__Otm,type,
subseqs_tm: list_tm > list_list_tm ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__SeCaV__Otm,type,
size_size_tm: tm > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_Eo_M_Eo_J,type,
bot_bot_o_o: $o > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__List__Olist_It__SeCaV__Ofm_J_M_Eo_J,type,
bot_bot_list_fm_o: list_fm > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__SeCaV__Otm_M_Eo_J,type,
bot_bot_tm_o: tm > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: 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_It__List__Olist_It__SeCaV__Ofm_J_J_J,type,
bot_bo6959746583267231307ist_fm: set_list_list_fm ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
bot_bot_set_list_fm: set_list_fm ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__SeCaV__Otm_J_J,type,
bot_bot_set_list_tm: set_list_tm ).
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__SeCaV__Ofm_J,type,
bot_bot_set_fm: set_fm ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__SeCaV__Otm_J,type,
bot_bot_set_tm: set_tm ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__SeCaV__Otm_J_J,type,
bot_bot_set_set_tm: set_set_tm ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_It__SeCaV__Ofm_J_M_Eo_J,type,
ord_le6518561683347902116t_fm_o: ( list_fm > $o ) > ( list_fm > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_It__SeCaV__Otm_J_M_Eo_J,type,
ord_le2468657205176945586t_tm_o: ( list_tm > $o ) > ( list_tm > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__SeCaV__Ofm_M_Eo_J,type,
ord_less_eq_fm_o: ( fm > $o ) > ( fm > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__SeCaV__Otm_M_Eo_J,type,
ord_less_eq_tm_o: ( tm > $o ) > ( tm > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
ord_less_eq_set_o: set_o > set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
ord_le7838213414353715577ist_fm: set_list_fm > set_list_fm > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__SeCaV__Ofm_J,type,
ord_less_eq_set_fm: set_fm > set_fm > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__SeCaV__Otm_J,type,
ord_less_eq_set_tm: set_tm > set_tm > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__SeCaV__Otm_J_J,type,
ord_le5601931644483074373set_tm: set_set_tm > set_set_tm > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_I_Eo_M_Eo_J,type,
top_top_o_o: $o > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__List__Olist_It__SeCaV__Ofm_J_M_Eo_J,type,
top_top_list_fm_o: list_fm > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
top_top_nat_o: nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__SeCaV__Otm_M_Eo_J,type,
top_top_tm_o: tm > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
top_top_set_o: set_o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__List__Olist_It__SeCaV__Ofm_J_J,type,
top_top_set_list_fm: set_list_fm ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__SeCaV__Ofm_J,type,
top_top_set_fm: set_fm ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__SeCaV__Otm_J,type,
top_top_set_tm: set_tm ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
top_top_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__SeCaV__Otm_J_J,type,
top_top_set_set_tm: set_set_tm ).
thf(sy_c_Prover_Ochildren,type,
children: list_tm > rule > list_fm > list_list_fm ).
thf(sy_c_Prover_Olist__prod_001t__SeCaV__Ofm,type,
list_prod_fm: list_list_fm > list_list_fm > list_list_fm ).
thf(sy_c_Prover_Oparts,type,
parts: list_tm > rule > fm > list_list_fm ).
thf(sy_c_Prover_OsubtermFm,type,
subtermFm: fm > list_tm ).
thf(sy_c_Prover_OsubtermTm,type,
subtermTm: tm > list_tm ).
thf(sy_c_Prover_Osubterms,type,
subterms: list_fm > list_tm ).
thf(sy_c_SeCaV_Oinc__list,type,
inc_list: list_tm > list_tm ).
thf(sy_c_SeCaV_Oinc__term,type,
inc_term: tm > tm ).
thf(sy_c_SeCaV_Oliftt,type,
liftt: tm > tm ).
thf(sy_c_SeCaV_Oliftts,type,
liftts: list_tm > list_tm ).
thf(sy_c_SeCaV_Onew__list,type,
new_list: nat > list_tm > $o ).
thf(sy_c_SeCaV_Onew__term,type,
new_term: nat > tm > $o ).
thf(sy_c_SeCaV_Onews,type,
news: nat > list_fm > $o ).
thf(sy_c_SeCaV_Oparams,type,
params: fm > set_nat ).
thf(sy_c_SeCaV_Oparamst,type,
paramst: tm > set_nat ).
thf(sy_c_SeCaV_Oparamst_H,type,
paramst2: tm > set_nat ).
thf(sy_c_SeCaV_Oparamst_H_H,type,
paramst3: tm > set_nat ).
thf(sy_c_SeCaV_Oparamst_H_H__rel,type,
paramst_rel: tm > tm > $o ).
thf(sy_c_SeCaV_Oparamsts,type,
paramsts: list_tm > set_nat ).
thf(sy_c_SeCaV_Osemantics__list_001t__List__Olist_It__SeCaV__Ofm_J,type,
semant3092164258205821607ist_fm: ( nat > list_fm ) > ( nat > list_list_fm > list_fm ) > list_tm > list_list_fm ).
thf(sy_c_SeCaV_Osemantics__list_001t__SeCaV__Ofm,type,
semantics_list_fm: ( nat > fm ) > ( nat > list_fm > fm ) > list_tm > list_fm ).
thf(sy_c_SeCaV_Osemantics__list_001t__SeCaV__Otm,type,
semantics_list_tm: ( nat > tm ) > ( nat > list_tm > tm ) > list_tm > list_tm ).
thf(sy_c_SeCaV_Osemantics__term_001t__List__Olist_It__SeCaV__Ofm_J,type,
semant91140444983244153ist_fm: ( nat > list_fm ) > ( nat > list_list_fm > list_fm ) > tm > list_fm ).
thf(sy_c_SeCaV_Osemantics__term_001t__SeCaV__Ofm,type,
semantics_term_fm: ( nat > fm ) > ( nat > list_fm > fm ) > tm > fm ).
thf(sy_c_SeCaV_Osemantics__term_001t__SeCaV__Otm,type,
semantics_term_tm: ( nat > tm ) > ( nat > list_tm > tm ) > tm > tm ).
thf(sy_c_SeCaV_Oshift_001t__Nat__Onat_001t__SeCaV__Otm,type,
shift_nat_tm: ( nat > tm ) > nat > tm > nat > tm ).
thf(sy_c_SeCaV_Osub,type,
sub: nat > tm > fm > fm ).
thf(sy_c_SeCaV_Osub__list,type,
sub_list: nat > tm > list_tm > list_tm ).
thf(sy_c_SeCaV_Osub__term,type,
sub_term: nat > tm > tm > tm ).
thf(sy_c_SeCaV_Osubstt,type,
substt: tm > tm > nat > tm ).
thf(sy_c_SeCaV_Osubstts,type,
substts: list_tm > tm > nat > list_tm ).
thf(sy_c_SeCaV_Otm_OFun,type,
fun: nat > list_tm > tm ).
thf(sy_c_SeCaV_Otm_OVar,type,
var: nat > tm ).
thf(sy_c_SeCaV_Otm_Osize__tm,type,
size_tm: tm > nat ).
thf(sy_c_Set_OCollect_001_Eo,type,
collect_o: ( $o > $o ) > set_o ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
collect_list_list_fm: ( list_list_fm > $o ) > set_list_list_fm ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__SeCaV__Ofm_J,type,
collect_list_fm: ( list_fm > $o ) > set_list_fm ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__SeCaV__Otm_J,type,
collect_list_tm: ( list_tm > $o ) > set_list_tm ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__SeCaV__Ofm,type,
collect_fm: ( fm > $o ) > set_fm ).
thf(sy_c_Set_OCollect_001t__SeCaV__Otm,type,
collect_tm: ( tm > $o ) > set_tm ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__SeCaV__Otm_J,type,
collect_set_tm: ( set_tm > $o ) > set_set_tm ).
thf(sy_c_Set_Oimage_001_Eo_001_Eo,type,
image_o_o: ( $o > $o ) > set_o > set_o ).
thf(sy_c_Set_Oimage_001_Eo_001t__List__Olist_It__SeCaV__Ofm_J,type,
image_o_list_fm: ( $o > list_fm ) > set_o > set_list_fm ).
thf(sy_c_Set_Oimage_001_Eo_001t__Nat__Onat,type,
image_o_nat: ( $o > nat ) > set_o > set_nat ).
thf(sy_c_Set_Oimage_001_Eo_001t__SeCaV__Otm,type,
image_o_tm: ( $o > tm ) > set_o > set_tm ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__SeCaV__Ofm_J_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
image_4162782105507059915ist_fm: ( list_fm > list_list_fm ) > set_list_fm > set_list_list_fm ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__SeCaV__Ofm_J_001t__List__Olist_It__SeCaV__Ofm_J,type,
image_9148017957442633541ist_fm: ( list_fm > list_fm ) > set_list_fm > set_list_fm ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__SeCaV__Ofm_J_001t__Nat__Onat,type,
image_list_fm_nat: ( list_fm > nat ) > set_list_fm > set_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__SeCaV__Ofm_J_001t__SeCaV__Otm,type,
image_list_fm_tm: ( list_fm > tm ) > set_list_fm > set_tm ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_Eo,type,
image_nat_o: ( nat > $o ) > set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__List__Olist_It__SeCaV__Ofm_J,type,
image_nat_list_fm: ( nat > list_fm ) > set_nat > set_list_fm ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__SeCaV__Otm,type,
image_nat_tm: ( nat > tm ) > set_nat > set_tm ).
thf(sy_c_Set_Oimage_001t__SeCaV__Ofm_001t__List__Olist_It__SeCaV__Ofm_J,type,
image_fm_list_fm: ( fm > list_fm ) > set_fm > set_list_fm ).
thf(sy_c_Set_Oimage_001t__SeCaV__Ofm_001t__SeCaV__Ofm,type,
image_fm_fm: ( fm > fm ) > set_fm > set_fm ).
thf(sy_c_Set_Oimage_001t__SeCaV__Ofm_001t__SeCaV__Otm,type,
image_fm_tm: ( fm > tm ) > set_fm > set_tm ).
thf(sy_c_Set_Oimage_001t__SeCaV__Ofm_001t__Set__Oset_It__Nat__Onat_J,type,
image_fm_set_nat: ( fm > set_nat ) > set_fm > set_set_nat ).
thf(sy_c_Set_Oimage_001t__SeCaV__Ofm_001t__Set__Oset_It__SeCaV__Otm_J,type,
image_fm_set_tm: ( fm > set_tm ) > set_fm > set_set_tm ).
thf(sy_c_Set_Oimage_001t__SeCaV__Otm_001_Eo,type,
image_tm_o: ( tm > $o ) > set_tm > set_o ).
thf(sy_c_Set_Oimage_001t__SeCaV__Otm_001t__List__Olist_It__SeCaV__Ofm_J,type,
image_tm_list_fm: ( tm > list_fm ) > set_tm > set_list_fm ).
thf(sy_c_Set_Oimage_001t__SeCaV__Otm_001t__List__Olist_It__SeCaV__Otm_J,type,
image_tm_list_tm: ( tm > list_tm ) > set_tm > set_list_tm ).
thf(sy_c_Set_Oimage_001t__SeCaV__Otm_001t__Nat__Onat,type,
image_tm_nat: ( tm > nat ) > set_tm > set_nat ).
thf(sy_c_Set_Oimage_001t__SeCaV__Otm_001t__SeCaV__Ofm,type,
image_tm_fm: ( tm > fm ) > set_tm > set_fm ).
thf(sy_c_Set_Oimage_001t__SeCaV__Otm_001t__SeCaV__Otm,type,
image_tm_tm: ( tm > tm ) > set_tm > set_tm ).
thf(sy_c_Set_Oimage_001t__SeCaV__Otm_001t__Set__Oset_It__Nat__Onat_J,type,
image_tm_set_nat: ( tm > set_nat ) > set_tm > set_set_nat ).
thf(sy_c_Set_Oimage_001t__SeCaV__Otm_001t__Set__Oset_It__SeCaV__Otm_J,type,
image_tm_set_tm: ( tm > set_tm ) > set_tm > set_set_tm ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__SeCaV__Ofm,type,
image_set_nat_fm: ( set_nat > fm ) > set_set_nat > set_fm ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__SeCaV__Otm,type,
image_set_nat_tm: ( set_nat > tm ) > set_set_nat > set_tm ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__SeCaV__Otm_J_001t__SeCaV__Ofm,type,
image_set_tm_fm: ( set_tm > fm ) > set_set_tm > set_fm ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__SeCaV__Otm_J_001t__SeCaV__Otm,type,
image_set_tm_tm: ( set_tm > tm ) > set_set_tm > set_tm ).
thf(sy_c_Set_Oinsert_001_Eo,type,
insert_o: $o > set_o > set_o ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
insert_list_list_fm: list_list_fm > set_list_list_fm > set_list_list_fm ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__SeCaV__Ofm_J,type,
insert_list_fm2: list_fm > set_list_fm > set_list_fm ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__SeCaV__Otm_J,type,
insert_list_tm: list_tm > set_list_tm > set_list_tm ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__SeCaV__Ofm,type,
insert_fm2: fm > set_fm > set_fm ).
thf(sy_c_Set_Oinsert_001t__SeCaV__Otm,type,
insert_tm2: tm > set_tm > set_tm ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
insert_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__SeCaV__Otm_J,type,
insert_set_tm: set_tm > set_set_tm > set_set_tm ).
thf(sy_c_Set_Ois__empty_001_Eo,type,
is_empty_o: set_o > $o ).
thf(sy_c_Set_Ois__empty_001t__Nat__Onat,type,
is_empty_nat: set_nat > $o ).
thf(sy_c_Set_Ois__empty_001t__SeCaV__Otm,type,
is_empty_tm: set_tm > $o ).
thf(sy_c_Set_Ois__singleton_001_Eo,type,
is_singleton_o: set_o > $o ).
thf(sy_c_Set_Ois__singleton_001t__List__Olist_It__SeCaV__Ofm_J,type,
is_singleton_list_fm: set_list_fm > $o ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__SeCaV__Otm,type,
is_singleton_tm: set_tm > $o ).
thf(sy_c_Set_Othe__elem_001_Eo,type,
the_elem_o: set_o > $o ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set_Othe__elem_001t__SeCaV__Otm,type,
the_elem_tm: set_tm > tm ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_It__Nat__Onat_J,type,
the_elem_set_nat: set_set_nat > set_nat ).
thf(sy_c_Set_Othe__elem_001t__Set__Oset_It__SeCaV__Otm_J,type,
the_elem_set_tm: set_set_tm > set_tm ).
thf(sy_c_Sublist_Oprefixes_001t__List__Olist_It__SeCaV__Ofm_J,type,
prefixes_list_fm: list_list_fm > list_list_list_fm ).
thf(sy_c_Sublist_Oprefixes_001t__SeCaV__Ofm,type,
prefixes_fm: list_fm > list_list_fm ).
thf(sy_c_Sublist_Oprefixes_001t__SeCaV__Otm,type,
prefixes_tm: list_tm > list_list_tm ).
thf(sy_c_Sublist_Osublists_001t__List__Olist_It__SeCaV__Ofm_J,type,
sublists_list_fm: list_list_fm > list_list_list_fm ).
thf(sy_c_Sublist_Osublists_001t__SeCaV__Ofm,type,
sublists_fm: list_fm > list_list_fm ).
thf(sy_c_Sublist_Osublists_001t__SeCaV__Otm,type,
sublists_tm: list_tm > list_list_tm ).
thf(sy_c_Sublist_Osuffixes_001t__List__Olist_It__SeCaV__Ofm_J,type,
suffixes_list_fm: list_list_fm > list_list_list_fm ).
thf(sy_c_Sublist_Osuffixes_001t__SeCaV__Ofm,type,
suffixes_fm: list_fm > list_list_fm ).
thf(sy_c_Sublist_Osuffixes_001t__SeCaV__Otm,type,
suffixes_tm: list_tm > list_list_tm ).
thf(sy_c_Wellfounded_Oaccp_001t__SeCaV__Otm,type,
accp_tm: ( tm > tm > $o ) > tm > $o ).
thf(sy_c_fChoice_001_Eo,type,
fChoice_o: ( $o > $o ) > $o ).
thf(sy_c_fChoice_001t__List__Olist_It__SeCaV__Ofm_J,type,
fChoice_list_fm: ( list_fm > $o ) > list_fm ).
thf(sy_c_fChoice_001t__Nat__Onat,type,
fChoice_nat: ( nat > $o ) > nat ).
thf(sy_c_fChoice_001t__SeCaV__Otm,type,
fChoice_tm: ( tm > $o ) > tm ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__SeCaV__Ofm_J_J,type,
member_list_list_fm: list_list_fm > set_list_list_fm > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__SeCaV__Ofm_J,type,
member_list_fm: list_fm > set_list_fm > $o ).
thf(sy_c_member_001t__List__Olist_It__SeCaV__Otm_J,type,
member_list_tm: list_tm > set_list_tm > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__SeCaV__Ofm,type,
member_fm: fm > set_fm > $o ).
thf(sy_c_member_001t__SeCaV__Otm,type,
member_tm: tm > set_tm > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__SeCaV__Otm_J,type,
member_set_tm: set_tm > set_set_tm > $o ).
thf(sy_v_S,type,
s: set_fm ).
thf(sy_v_n____,type,
n: nat ).
% Relevant facts (1271)
thf(fact_0_some__equality,axiom,
! [P: tm > $o,A: tm] :
( ( P @ A )
=> ( ! [X: tm] :
( ( P @ X )
=> ( X = A ) )
=> ( ( fChoice_tm @ P )
= A ) ) ) ).
% some_equality
thf(fact_1_some__eq__trivial,axiom,
! [X2: tm] :
( ( fChoice_tm
@ ^ [Y: tm] : ( Y = X2 ) )
= X2 ) ).
% some_eq_trivial
thf(fact_2_some__sym__eq__trivial,axiom,
! [X2: tm] :
( ( fChoice_tm
@ ( ^ [Y2: tm,Z: tm] : ( Y2 = Z )
@ X2 ) )
= X2 ) ).
% some_sym_eq_trivial
thf(fact_3_tm_Oinject_I2_J,axiom,
! [X22: nat,Y22: nat] :
( ( ( var @ X22 )
= ( var @ Y22 ) )
= ( X22 = Y22 ) ) ).
% tm.inject(2)
thf(fact_4_verit__sko__ex_H,axiom,
! [P: tm > $o,A2: $o] :
( ( ( P @ ( fChoice_tm @ P ) )
= A2 )
=> ( ( ? [X3: tm] : ( P @ X3 ) )
= A2 ) ) ).
% verit_sko_ex'
thf(fact_5_verit__sko__forall,axiom,
( ( ^ [P2: tm > $o] :
! [X4: tm] : ( P2 @ X4 ) )
= ( ^ [P3: tm > $o] :
( P3
@ ( fChoice_tm
@ ^ [X5: tm] :
~ ( P3 @ X5 ) ) ) ) ) ).
% verit_sko_forall
thf(fact_6_someI2,axiom,
! [P: tm > $o,A: tm,Q: tm > $o] :
( ( P @ A )
=> ( ! [X: tm] :
( ( P @ X )
=> ( Q @ X ) )
=> ( Q @ ( fChoice_tm @ P ) ) ) ) ).
% someI2
thf(fact_7_verit__sko__forall_H,axiom,
! [P: tm > $o,A2: $o] :
( ( ( P
@ ( fChoice_tm
@ ^ [X5: tm] :
~ ( P @ X5 ) ) )
= A2 )
=> ( ( ! [X3: tm] : ( P @ X3 ) )
= A2 ) ) ).
% verit_sko_forall'
thf(fact_8_verit__sko__forall_H_H,axiom,
! [B: tm,A2: tm,P: tm > $o] :
( ( B = A2 )
=> ( ( ( fChoice_tm @ P )
= A2 )
= ( ( fChoice_tm @ P )
= B ) ) ) ).
% verit_sko_forall''
thf(fact_9_someI__ex,axiom,
! [P: tm > $o] :
( ? [X_1: tm] : ( P @ X_1 )
=> ( P @ ( fChoice_tm @ P ) ) ) ).
% someI_ex
thf(fact_10_someI2__ex,axiom,
! [P: tm > $o,Q: tm > $o] :
( ? [X_1: tm] : ( P @ X_1 )
=> ( ! [X: tm] :
( ( P @ X )
=> ( Q @ X ) )
=> ( Q @ ( fChoice_tm @ P ) ) ) ) ).
% someI2_ex
thf(fact_11_someI2__bex,axiom,
! [A2: set_nat,P: nat > $o,Q: nat > $o] :
( ? [X6: nat] :
( ( member_nat @ X6 @ A2 )
& ( P @ X6 ) )
=> ( ! [X: nat] :
( ( ( member_nat @ X @ A2 )
& ( P @ X ) )
=> ( Q @ X ) )
=> ( Q
@ ( fChoice_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( P @ X5 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_12_someI2__bex,axiom,
! [A2: set_list_fm,P: list_fm > $o,Q: list_fm > $o] :
( ? [X6: list_fm] :
( ( member_list_fm @ X6 @ A2 )
& ( P @ X6 ) )
=> ( ! [X: list_fm] :
( ( ( member_list_fm @ X @ A2 )
& ( P @ X ) )
=> ( Q @ X ) )
=> ( Q
@ ( fChoice_list_fm
@ ^ [X5: list_fm] :
( ( member_list_fm @ X5 @ A2 )
& ( P @ X5 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_13_someI2__bex,axiom,
! [A2: set_tm,P: tm > $o,Q: tm > $o] :
( ? [X6: tm] :
( ( member_tm @ X6 @ A2 )
& ( P @ X6 ) )
=> ( ! [X: tm] :
( ( ( member_tm @ X @ A2 )
& ( P @ X ) )
=> ( Q @ X ) )
=> ( Q
@ ( fChoice_tm
@ ^ [X5: tm] :
( ( member_tm @ X5 @ A2 )
& ( P @ X5 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_14_some__eq__imp,axiom,
! [P: tm > $o,A: tm,B2: tm] :
( ( ( fChoice_tm @ P )
= A )
=> ( ( P @ B2 )
=> ( P @ A ) ) ) ).
% some_eq_imp
thf(fact_15_tfl__some,axiom,
! [P4: tm > $o,X6: tm] :
( ( P4 @ X6 )
=> ( P4 @ ( fChoice_tm @ P4 ) ) ) ).
% tfl_some
thf(fact_16_Eps__cong,axiom,
! [P: tm > $o,Q: tm > $o] :
( ! [X: tm] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( fChoice_tm @ P )
= ( fChoice_tm @ Q ) ) ) ).
% Eps_cong
thf(fact_17_someI,axiom,
! [P: tm > $o,X2: tm] :
( ( P @ X2 )
=> ( P @ ( fChoice_tm @ P ) ) ) ).
% someI
thf(fact_18_verit__sko__forall__indirect2,axiom,
! [X2: tm,P: tm > $o,P5: tm > $o] :
( ( X2
= ( fChoice_tm
@ ^ [X5: tm] :
~ ( P @ X5 ) ) )
=> ( ! [X: tm] :
( ( P @ X )
= ( P5 @ X ) )
=> ( ( ! [X3: tm] : ( P5 @ X3 ) )
= ( P @ X2 ) ) ) ) ).
% verit_sko_forall_indirect2
thf(fact_19_verit__sko__forall__indirect,axiom,
! [X2: tm,P: tm > $o] :
( ( X2
= ( fChoice_tm
@ ^ [X5: tm] :
~ ( P @ X5 ) ) )
=> ( ( ! [X3: tm] : ( P @ X3 ) )
= ( P @ X2 ) ) ) ).
% verit_sko_forall_indirect
thf(fact_20_some1__equality,axiom,
! [P: tm > $o,A: tm] :
( ? [X6: tm] :
( ( P @ X6 )
& ! [Y3: tm] :
( ( P @ Y3 )
=> ( Y3 = X6 ) ) )
=> ( ( P @ A )
=> ( ( fChoice_tm @ P )
= A ) ) ) ).
% some1_equality
thf(fact_21_verit__sko__ex__indirect2,axiom,
! [X2: tm,P: tm > $o,P5: tm > $o] :
( ( X2
= ( fChoice_tm @ P ) )
=> ( ! [X: tm] :
( ( P @ X )
= ( P5 @ X ) )
=> ( ( ? [X3: tm] : ( P5 @ X3 ) )
= ( P @ X2 ) ) ) ) ).
% verit_sko_ex_indirect2
thf(fact_22_verit__sko__ex__indirect,axiom,
! [X2: tm,P: tm > $o] :
( ( X2
= ( fChoice_tm @ P ) )
=> ( ( ? [X3: tm] : ( P @ X3 ) )
= ( P @ X2 ) ) ) ).
% verit_sko_ex_indirect
thf(fact_23_some__eq__ex,axiom,
! [P: tm > $o] :
( ( P @ ( fChoice_tm @ P ) )
= ( ? [X3: tm] : ( P @ X3 ) ) ) ).
% some_eq_ex
thf(fact_24_new__term_Osimps_I1_J,axiom,
! [C: nat,N: nat] : ( new_term @ C @ ( var @ N ) ) ).
% new_term.simps(1)
thf(fact_25_Nitpick_OEps__psimp,axiom,
! [P: tm > $o,X2: tm,Y4: tm] :
( ( P @ X2 )
=> ( ~ ( P @ Y4 )
=> ( ( ( fChoice_tm @ P )
= Y4 )
=> ( ( fChoice_tm @ P )
= X2 ) ) ) ) ).
% Nitpick.Eps_psimp
thf(fact_26_usemantics__E_I1_J,axiom,
! [T: tm,S: set_fm] :
( ( member_tm @ T @ ( terms @ S ) )
=> ( ( semantics_term_tm
@ ^ [N2: nat] :
( if_tm @ ( member_tm @ ( var @ N2 ) @ ( terms @ S ) ) @ ( var @ N2 )
@ ( fChoice_tm
@ ^ [T2: tm] : ( member_tm @ T2 @ ( terms @ S ) ) ) )
@ ^ [I: nat,L: list_tm] :
( if_tm @ ( member_tm @ ( fun @ I @ L ) @ ( terms @ S ) ) @ ( fun @ I @ L )
@ ( fChoice_tm
@ ^ [T2: tm] : ( member_tm @ T2 @ ( terms @ S ) ) ) )
@ T )
= T ) ) ).
% usemantics_E(1)
thf(fact_27_semantics__term_Osimps_I1_J,axiom,
! [E: nat > tm,F: nat > list_tm > tm,N: nat] :
( ( semantics_term_tm @ E @ F @ ( var @ N ) )
= ( E @ N ) ) ).
% semantics_term.simps(1)
thf(fact_28_some__in__eq,axiom,
! [A2: set_list_fm] :
( ( member_list_fm
@ ( fChoice_list_fm
@ ^ [X5: list_fm] : ( member_list_fm @ X5 @ A2 ) )
@ A2 )
= ( A2 != bot_bot_set_list_fm ) ) ).
% some_in_eq
thf(fact_29_some__in__eq,axiom,
! [A2: set_tm] :
( ( member_tm
@ ( fChoice_tm
@ ^ [X5: tm] : ( member_tm @ X5 @ A2 ) )
@ A2 )
= ( A2 != bot_bot_set_tm ) ) ).
% some_in_eq
thf(fact_30_some__in__eq,axiom,
! [A2: set_nat] :
( ( member_nat
@ ( fChoice_nat
@ ^ [X5: nat] : ( member_nat @ X5 @ A2 ) )
@ A2 )
= ( A2 != bot_bot_set_nat ) ) ).
% some_in_eq
thf(fact_31_some__in__eq,axiom,
! [A2: set_o] :
( ( member_o
@ ( fChoice_o
@ ^ [X5: $o] : ( member_o @ X5 @ A2 ) )
@ A2 )
= ( A2 != bot_bot_set_o ) ) ).
% some_in_eq
thf(fact_32_usemantics__E_I2_J,axiom,
! [S: set_fm,Ts: list_tm] :
( ( list_all_tm
@ ^ [T2: tm] : ( member_tm @ T2 @ ( terms @ S ) )
@ Ts )
=> ( ( semantics_list_tm
@ ^ [N2: nat] :
( if_tm @ ( member_tm @ ( var @ N2 ) @ ( terms @ S ) ) @ ( var @ N2 )
@ ( fChoice_tm
@ ^ [T2: tm] : ( member_tm @ T2 @ ( terms @ S ) ) ) )
@ ^ [I: nat,L: list_tm] :
( if_tm @ ( member_tm @ ( fun @ I @ L ) @ ( terms @ S ) ) @ ( fun @ I @ L )
@ ( fChoice_tm
@ ^ [T2: tm] : ( member_tm @ T2 @ ( terms @ S ) ) ) )
@ Ts )
= Ts ) ) ).
% usemantics_E(2)
thf(fact_33_terms__ne,axiom,
! [S: set_fm] :
( ( terms @ S )
!= bot_bot_set_tm ) ).
% terms_ne
thf(fact_34_tm_Oinject_I1_J,axiom,
! [X11: nat,X12: list_tm,Y11: nat,Y12: list_tm] :
( ( ( fun @ X11 @ X12 )
= ( fun @ Y11 @ Y12 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 ) ) ) ).
% tm.inject(1)
thf(fact_35_semantics__term_Osimps_I2_J,axiom,
! [E: nat > tm,F: nat > list_tm > tm,I2: nat,L2: list_tm] :
( ( semantics_term_tm @ E @ F @ ( fun @ I2 @ L2 ) )
= ( F @ I2 @ ( semantics_list_tm @ E @ F @ L2 ) ) ) ).
% semantics_term.simps(2)
thf(fact_36_tm_Odistinct_I1_J,axiom,
! [X11: nat,X12: list_tm,X22: nat] :
( ( fun @ X11 @ X12 )
!= ( var @ X22 ) ) ).
% tm.distinct(1)
thf(fact_37_tm_Oexhaust,axiom,
! [Y4: tm] :
( ! [X112: nat,X122: list_tm] :
( Y4
!= ( fun @ X112 @ X122 ) )
=> ~ ! [X23: nat] :
( Y4
!= ( var @ X23 ) ) ) ).
% tm.exhaust
thf(fact_38_paramst_H_H_Ocases,axiom,
! [X2: tm] :
( ! [N3: nat] :
( X2
!= ( var @ N3 ) )
=> ~ ! [A3: nat,Ts2: list_tm] :
( X2
!= ( fun @ A3 @ Ts2 ) ) ) ).
% paramst''.cases
thf(fact_39_empty__Collect__eq,axiom,
! [P: list_fm > $o] :
( ( bot_bot_set_list_fm
= ( collect_list_fm @ P ) )
= ( ! [X5: list_fm] :
~ ( P @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_40_empty__Collect__eq,axiom,
! [P: tm > $o] :
( ( bot_bot_set_tm
= ( collect_tm @ P ) )
= ( ! [X5: tm] :
~ ( P @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_41_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X5: nat] :
~ ( P @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_42_empty__Collect__eq,axiom,
! [P: $o > $o] :
( ( bot_bot_set_o
= ( collect_o @ P ) )
= ( ! [X5: $o] :
~ ( P @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_43_Collect__empty__eq,axiom,
! [P: list_fm > $o] :
( ( ( collect_list_fm @ P )
= bot_bot_set_list_fm )
= ( ! [X5: list_fm] :
~ ( P @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_44_Collect__empty__eq,axiom,
! [P: tm > $o] :
( ( ( collect_tm @ P )
= bot_bot_set_tm )
= ( ! [X5: tm] :
~ ( P @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_45_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X5: nat] :
~ ( P @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_46_Collect__empty__eq,axiom,
! [P: $o > $o] :
( ( ( collect_o @ P )
= bot_bot_set_o )
= ( ! [X5: $o] :
~ ( P @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_47_all__not__in__conv,axiom,
! [A2: set_list_fm] :
( ( ! [X5: list_fm] :
~ ( member_list_fm @ X5 @ A2 ) )
= ( A2 = bot_bot_set_list_fm ) ) ).
% all_not_in_conv
thf(fact_48_all__not__in__conv,axiom,
! [A2: set_tm] :
( ( ! [X5: tm] :
~ ( member_tm @ X5 @ A2 ) )
= ( A2 = bot_bot_set_tm ) ) ).
% all_not_in_conv
thf(fact_49_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X5: nat] :
~ ( member_nat @ X5 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_50_all__not__in__conv,axiom,
! [A2: set_o] :
( ( ! [X5: $o] :
~ ( member_o @ X5 @ A2 ) )
= ( A2 = bot_bot_set_o ) ) ).
% all_not_in_conv
thf(fact_51_mem__Collect__eq,axiom,
! [A: tm,P: tm > $o] :
( ( member_tm @ A @ ( collect_tm @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_52_mem__Collect__eq,axiom,
! [A: list_fm,P: list_fm > $o] :
( ( member_list_fm @ A @ ( collect_list_fm @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_53_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_54_Collect__mem__eq,axiom,
! [A2: set_tm] :
( ( collect_tm
@ ^ [X5: tm] : ( member_tm @ X5 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_55_Collect__mem__eq,axiom,
! [A2: set_list_fm] :
( ( collect_list_fm
@ ^ [X5: list_fm] : ( member_list_fm @ X5 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_56_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X5: nat] : ( member_nat @ X5 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_57_Collect__cong,axiom,
! [P: list_fm > $o,Q: list_fm > $o] :
( ! [X: list_fm] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_list_fm @ P )
= ( collect_list_fm @ Q ) ) ) ).
% Collect_cong
thf(fact_58_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_59_empty__iff,axiom,
! [C: list_fm] :
~ ( member_list_fm @ C @ bot_bot_set_list_fm ) ).
% empty_iff
thf(fact_60_empty__iff,axiom,
! [C: tm] :
~ ( member_tm @ C @ bot_bot_set_tm ) ).
% empty_iff
thf(fact_61_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_62_empty__iff,axiom,
! [C: $o] :
~ ( member_o @ C @ bot_bot_set_o ) ).
% empty_iff
thf(fact_63_new__term_Osimps_I2_J,axiom,
! [C: nat,I2: nat,L2: list_tm] :
( ( new_term @ C @ ( fun @ I2 @ L2 ) )
= ( ( I2 != C )
& ( ( I2 != C )
=> ( new_list @ C @ L2 ) ) ) ) ).
% new_term.simps(2)
thf(fact_64_list_Opred__True,axiom,
( ( list_all_tm
@ ^ [Uu: tm] : $true )
= ( ^ [Uu: list_tm] : $true ) ) ).
% list.pred_True
thf(fact_65_list_Opred__True,axiom,
( ( list_all_fm
@ ^ [Uu: fm] : $true )
= ( ^ [Uu: list_fm] : $true ) ) ).
% list.pred_True
thf(fact_66_empty__def,axiom,
( bot_bot_set_list_fm
= ( collect_list_fm
@ ^ [X5: list_fm] : $false ) ) ).
% empty_def
thf(fact_67_empty__def,axiom,
( bot_bot_set_tm
= ( collect_tm
@ ^ [X5: tm] : $false ) ) ).
% empty_def
thf(fact_68_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X5: nat] : $false ) ) ).
% empty_def
thf(fact_69_empty__def,axiom,
( bot_bot_set_o
= ( collect_o
@ ^ [X5: $o] : $false ) ) ).
% empty_def
thf(fact_70_s1_I1_J,axiom,
( new_term
= ( ^ [C2: nat,T2: tm] :
~ ( member_nat @ C2 @ ( paramst @ T2 ) ) ) ) ).
% s1(1)
thf(fact_71_paramst_H_Osimps_I1_J,axiom,
! [N: nat] :
( ( paramst2 @ ( var @ N ) )
= bot_bot_set_nat ) ).
% paramst'.simps(1)
thf(fact_72_ex__in__conv,axiom,
! [A2: set_list_fm] :
( ( ? [X5: list_fm] : ( member_list_fm @ X5 @ A2 ) )
= ( A2 != bot_bot_set_list_fm ) ) ).
% ex_in_conv
thf(fact_73_ex__in__conv,axiom,
! [A2: set_tm] :
( ( ? [X5: tm] : ( member_tm @ X5 @ A2 ) )
= ( A2 != bot_bot_set_tm ) ) ).
% ex_in_conv
thf(fact_74_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X5: nat] : ( member_nat @ X5 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_75_ex__in__conv,axiom,
! [A2: set_o] :
( ( ? [X5: $o] : ( member_o @ X5 @ A2 ) )
= ( A2 != bot_bot_set_o ) ) ).
% ex_in_conv
thf(fact_76_p1,axiom,
paramst2 = paramst ).
% p1
thf(fact_77_bot__set__def,axiom,
( bot_bot_set_list_fm
= ( collect_list_fm @ bot_bot_list_fm_o ) ) ).
% bot_set_def
thf(fact_78_bot__set__def,axiom,
( bot_bot_set_tm
= ( collect_tm @ bot_bot_tm_o ) ) ).
% bot_set_def
thf(fact_79_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_80_bot__set__def,axiom,
( bot_bot_set_o
= ( collect_o @ bot_bot_o_o ) ) ).
% bot_set_def
thf(fact_81_paramst_Osimps_I1_J,axiom,
! [N: nat] :
( ( paramst @ ( var @ N ) )
= bot_bot_set_nat ) ).
% paramst.simps(1)
thf(fact_82_emptyE,axiom,
! [A: list_fm] :
~ ( member_list_fm @ A @ bot_bot_set_list_fm ) ).
% emptyE
thf(fact_83_emptyE,axiom,
! [A: tm] :
~ ( member_tm @ A @ bot_bot_set_tm ) ).
% emptyE
thf(fact_84_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_85_emptyE,axiom,
! [A: $o] :
~ ( member_o @ A @ bot_bot_set_o ) ).
% emptyE
thf(fact_86_equals0D,axiom,
! [A2: set_list_fm,A: list_fm] :
( ( A2 = bot_bot_set_list_fm )
=> ~ ( member_list_fm @ A @ A2 ) ) ).
% equals0D
thf(fact_87_equals0D,axiom,
! [A2: set_tm,A: tm] :
( ( A2 = bot_bot_set_tm )
=> ~ ( member_tm @ A @ A2 ) ) ).
% equals0D
thf(fact_88_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_89_equals0D,axiom,
! [A2: set_o,A: $o] :
( ( A2 = bot_bot_set_o )
=> ~ ( member_o @ A @ A2 ) ) ).
% equals0D
thf(fact_90_equals0I,axiom,
! [A2: set_list_fm] :
( ! [Y3: list_fm] :
~ ( member_list_fm @ Y3 @ A2 )
=> ( A2 = bot_bot_set_list_fm ) ) ).
% equals0I
thf(fact_91_equals0I,axiom,
! [A2: set_tm] :
( ! [Y3: tm] :
~ ( member_tm @ Y3 @ A2 )
=> ( A2 = bot_bot_set_tm ) ) ).
% equals0I
thf(fact_92_equals0I,axiom,
! [A2: set_nat] :
( ! [Y3: nat] :
~ ( member_nat @ Y3 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_93_equals0I,axiom,
! [A2: set_o] :
( ! [Y3: $o] :
~ ( member_o @ Y3 @ A2 )
=> ( A2 = bot_bot_set_o ) ) ).
% equals0I
thf(fact_94_Set_Ois__empty__def,axiom,
( is_empty_tm
= ( ^ [A4: set_tm] : ( A4 = bot_bot_set_tm ) ) ) ).
% Set.is_empty_def
thf(fact_95_Set_Ois__empty__def,axiom,
( is_empty_nat
= ( ^ [A4: set_nat] : ( A4 = bot_bot_set_nat ) ) ) ).
% Set.is_empty_def
thf(fact_96_Set_Ois__empty__def,axiom,
( is_empty_o
= ( ^ [A4: set_o] : ( A4 = bot_bot_set_o ) ) ) ).
% Set.is_empty_def
thf(fact_97_paramst_H_H_Osimps_I1_J,axiom,
! [N: nat] :
( ( paramst3 @ ( var @ N ) )
= bot_bot_set_nat ) ).
% paramst''.simps(1)
thf(fact_98_upd__lemma_H_I1_J,axiom,
! [N: nat,T: tm,E: nat > tm,F: nat > list_tm > tm,Z2: list_tm > tm] :
( ~ ( member_nat @ N @ ( paramst @ T ) )
=> ( ( semantics_term_tm @ E @ ( fun_up3737067029978700398_tm_tm @ F @ N @ Z2 ) @ T )
= ( semantics_term_tm @ E @ F @ T ) ) ) ).
% upd_lemma'(1)
thf(fact_99_p1_H,axiom,
paramst3 = paramst ).
% p1'
thf(fact_100_s1_I2_J,axiom,
( new_list
= ( ^ [C2: nat,L: list_tm] :
~ ( member_nat @ C2 @ ( paramsts @ L ) ) ) ) ).
% s1(2)
thf(fact_101_new__list_Osimps_I2_J,axiom,
! [C: nat,T: tm,L2: list_tm] :
( ( new_list @ C @ ( cons_tm @ T @ L2 ) )
= ( ( ( new_term @ C @ T )
=> ( new_list @ C @ L2 ) )
& ( new_term @ C @ T ) ) ) ).
% new_list.simps(2)
thf(fact_102_bot__empty__eq,axiom,
( bot_bot_list_fm_o
= ( ^ [X5: list_fm] : ( member_list_fm @ X5 @ bot_bot_set_list_fm ) ) ) ).
% bot_empty_eq
thf(fact_103_bot__empty__eq,axiom,
( bot_bot_tm_o
= ( ^ [X5: tm] : ( member_tm @ X5 @ bot_bot_set_tm ) ) ) ).
% bot_empty_eq
thf(fact_104_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X5: nat] : ( member_nat @ X5 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_105_bot__empty__eq,axiom,
( bot_bot_o_o
= ( ^ [X5: $o] : ( member_o @ X5 @ bot_bot_set_o ) ) ) ).
% bot_empty_eq
thf(fact_106_Collect__empty__eq__bot,axiom,
! [P: list_fm > $o] :
( ( ( collect_list_fm @ P )
= bot_bot_set_list_fm )
= ( P = bot_bot_list_fm_o ) ) ).
% Collect_empty_eq_bot
thf(fact_107_Collect__empty__eq__bot,axiom,
! [P: tm > $o] :
( ( ( collect_tm @ P )
= bot_bot_set_tm )
= ( P = bot_bot_tm_o ) ) ).
% Collect_empty_eq_bot
thf(fact_108_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_109_Collect__empty__eq__bot,axiom,
! [P: $o > $o] :
( ( ( collect_o @ P )
= bot_bot_set_o )
= ( P = bot_bot_o_o ) ) ).
% Collect_empty_eq_bot
thf(fact_110_is__singletonI_H,axiom,
! [A2: set_list_fm] :
( ( A2 != bot_bot_set_list_fm )
=> ( ! [X: list_fm,Y3: list_fm] :
( ( member_list_fm @ X @ A2 )
=> ( ( member_list_fm @ Y3 @ A2 )
=> ( X = Y3 ) ) )
=> ( is_singleton_list_fm @ A2 ) ) ) ).
% is_singletonI'
thf(fact_111_is__singletonI_H,axiom,
! [A2: set_tm] :
( ( A2 != bot_bot_set_tm )
=> ( ! [X: tm,Y3: tm] :
( ( member_tm @ X @ A2 )
=> ( ( member_tm @ Y3 @ A2 )
=> ( X = Y3 ) ) )
=> ( is_singleton_tm @ A2 ) ) ) ).
% is_singletonI'
thf(fact_112_is__singletonI_H,axiom,
! [A2: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X: nat,Y3: nat] :
( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y3 @ A2 )
=> ( X = Y3 ) ) )
=> ( is_singleton_nat @ A2 ) ) ) ).
% is_singletonI'
thf(fact_113_is__singletonI_H,axiom,
! [A2: set_o] :
( ( A2 != bot_bot_set_o )
=> ( ! [X: $o,Y3: $o] :
( ( member_o @ X @ A2 )
=> ( ( member_o @ Y3 @ A2 )
=> ( X = Y3 ) ) )
=> ( is_singleton_o @ A2 ) ) ) ).
% is_singletonI'
thf(fact_114_subst__lemma_H_I2_J,axiom,
! [E: nat > tm,F: nat > list_tm > tm,Ts: list_tm,U: tm,I2: nat] :
( ( semantics_list_tm @ E @ F @ ( substts @ Ts @ U @ I2 ) )
= ( semantics_list_tm @ ( shift_nat_tm @ E @ I2 @ ( semantics_term_tm @ E @ F @ U ) ) @ F @ Ts ) ) ).
% subst_lemma'(2)
thf(fact_115_UNIV__I,axiom,
! [X2: tm] : ( member_tm @ X2 @ top_top_set_tm ) ).
% UNIV_I
thf(fact_116_UNIV__I,axiom,
! [X2: list_fm] : ( member_list_fm @ X2 @ top_top_set_list_fm ) ).
% UNIV_I
thf(fact_117_UNIV__I,axiom,
! [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% UNIV_I
thf(fact_118_UNIV__I,axiom,
! [X2: $o] : ( member_o @ X2 @ top_top_set_o ) ).
% UNIV_I
thf(fact_119_list_Oinject,axiom,
! [X21: tm,X222: list_tm,Y21: tm,Y222: list_tm] :
( ( ( cons_tm @ X21 @ X222 )
= ( cons_tm @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_120_list_Oinject,axiom,
! [X21: fm,X222: list_fm,Y21: fm,Y222: list_fm] :
( ( ( cons_fm @ X21 @ X222 )
= ( cons_fm @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_121_list_Oinject,axiom,
! [X21: list_fm,X222: list_list_fm,Y21: list_fm,Y222: list_list_fm] :
( ( ( cons_list_fm @ X21 @ X222 )
= ( cons_list_fm @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_122_list_Opred__inject_I2_J,axiom,
! [P: list_fm > $o,A: list_fm,Aa: list_list_fm] :
( ( list_all_list_fm @ P @ ( cons_list_fm @ A @ Aa ) )
= ( ( P @ A )
& ( list_all_list_fm @ P @ Aa ) ) ) ).
% list.pred_inject(2)
thf(fact_123_list_Opred__inject_I2_J,axiom,
! [P: tm > $o,A: tm,Aa: list_tm] :
( ( list_all_tm @ P @ ( cons_tm @ A @ Aa ) )
= ( ( P @ A )
& ( list_all_tm @ P @ Aa ) ) ) ).
% list.pred_inject(2)
thf(fact_124_list_Opred__inject_I2_J,axiom,
! [P: fm > $o,A: fm,Aa: list_fm] :
( ( list_all_fm @ P @ ( cons_fm @ A @ Aa ) )
= ( ( P @ A )
& ( list_all_fm @ P @ Aa ) ) ) ).
% list.pred_inject(2)
thf(fact_125_list__all__simps_I1_J,axiom,
! [P: list_fm > $o,X2: list_fm,Xs: list_list_fm] :
( ( list_all_list_fm @ P @ ( cons_list_fm @ X2 @ Xs ) )
= ( ( P @ X2 )
& ( list_all_list_fm @ P @ Xs ) ) ) ).
% list_all_simps(1)
thf(fact_126_list__all__simps_I1_J,axiom,
! [P: tm > $o,X2: tm,Xs: list_tm] :
( ( list_all_tm @ P @ ( cons_tm @ X2 @ Xs ) )
= ( ( P @ X2 )
& ( list_all_tm @ P @ Xs ) ) ) ).
% list_all_simps(1)
thf(fact_127_list__all__simps_I1_J,axiom,
! [P: fm > $o,X2: fm,Xs: list_fm] :
( ( list_all_fm @ P @ ( cons_fm @ X2 @ Xs ) )
= ( ( P @ X2 )
& ( list_all_fm @ P @ Xs ) ) ) ).
% list_all_simps(1)
thf(fact_128_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_list_fm
@ ^ [S2: list_fm] : P )
= top_top_set_list_fm ) )
& ( ~ P
=> ( ( collect_list_fm
@ ^ [S2: list_fm] : P )
= bot_bot_set_list_fm ) ) ) ).
% Collect_const
thf(fact_129_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_tm
@ ^ [S2: tm] : P )
= top_top_set_tm ) )
& ( ~ P
=> ( ( collect_tm
@ ^ [S2: tm] : P )
= bot_bot_set_tm ) ) ) ).
% Collect_const
thf(fact_130_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_nat
@ ^ [S2: nat] : P )
= top_top_set_nat ) )
& ( ~ P
=> ( ( collect_nat
@ ^ [S2: nat] : P )
= bot_bot_set_nat ) ) ) ).
% Collect_const
thf(fact_131_Collect__const,axiom,
! [P: $o] :
( ( P
=> ( ( collect_o
@ ^ [S2: $o] : P )
= top_top_set_o ) )
& ( ~ P
=> ( ( collect_o
@ ^ [S2: $o] : P )
= bot_bot_set_o ) ) ) ).
% Collect_const
thf(fact_132_inv__identity,axiom,
( ( hilber3633877196798814958at_nat @ top_top_set_nat
@ ^ [A5: nat] : A5 )
= ( ^ [A5: nat] : A5 ) ) ).
% inv_identity
thf(fact_133_inv__identity,axiom,
( ( hilbert_inv_into_o_o @ top_top_set_o
@ ^ [A5: $o] : A5 )
= ( ^ [A5: $o] : A5 ) ) ).
% inv_identity
thf(fact_134_upd__lemma_H_I2_J,axiom,
! [N: nat,Ts: list_tm,E: nat > tm,F: nat > list_tm > tm,Z2: list_tm > tm] :
( ~ ( member_nat @ N @ ( paramsts @ Ts ) )
=> ( ( semantics_list_tm @ E @ ( fun_up3737067029978700398_tm_tm @ F @ N @ Z2 ) @ Ts )
= ( semantics_list_tm @ E @ F @ Ts ) ) ) ).
% upd_lemma'(2)
thf(fact_135_UNIV__eq__I,axiom,
! [A2: set_tm] :
( ! [X: tm] : ( member_tm @ X @ A2 )
=> ( top_top_set_tm = A2 ) ) ).
% UNIV_eq_I
thf(fact_136_UNIV__eq__I,axiom,
! [A2: set_list_fm] :
( ! [X: list_fm] : ( member_list_fm @ X @ A2 )
=> ( top_top_set_list_fm = A2 ) ) ).
% UNIV_eq_I
thf(fact_137_UNIV__eq__I,axiom,
! [A2: set_nat] :
( ! [X: nat] : ( member_nat @ X @ A2 )
=> ( top_top_set_nat = A2 ) ) ).
% UNIV_eq_I
thf(fact_138_UNIV__eq__I,axiom,
! [A2: set_o] :
( ! [X: $o] : ( member_o @ X @ A2 )
=> ( top_top_set_o = A2 ) ) ).
% UNIV_eq_I
thf(fact_139_UNIV__witness,axiom,
? [X: tm] : ( member_tm @ X @ top_top_set_tm ) ).
% UNIV_witness
thf(fact_140_UNIV__witness,axiom,
? [X: list_fm] : ( member_list_fm @ X @ top_top_set_list_fm ) ).
% UNIV_witness
thf(fact_141_UNIV__witness,axiom,
? [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_142_UNIV__witness,axiom,
? [X: $o] : ( member_o @ X @ top_top_set_o ) ).
% UNIV_witness
thf(fact_143_not__Cons__self2,axiom,
! [X2: tm,Xs: list_tm] :
( ( cons_tm @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_144_not__Cons__self2,axiom,
! [X2: fm,Xs: list_fm] :
( ( cons_fm @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_145_not__Cons__self2,axiom,
! [X2: list_fm,Xs: list_list_fm] :
( ( cons_list_fm @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_146_UNIV__def,axiom,
( top_top_set_list_fm
= ( collect_list_fm
@ ^ [X5: list_fm] : $true ) ) ).
% UNIV_def
thf(fact_147_UNIV__def,axiom,
( top_top_set_nat
= ( collect_nat
@ ^ [X5: nat] : $true ) ) ).
% UNIV_def
thf(fact_148_UNIV__def,axiom,
( top_top_set_o
= ( collect_o
@ ^ [X5: $o] : $true ) ) ).
% UNIV_def
thf(fact_149_empty__not__UNIV,axiom,
bot_bot_set_tm != top_top_set_tm ).
% empty_not_UNIV
thf(fact_150_empty__not__UNIV,axiom,
bot_bot_set_nat != top_top_set_nat ).
% empty_not_UNIV
thf(fact_151_empty__not__UNIV,axiom,
bot_bot_set_o != top_top_set_o ).
% empty_not_UNIV
thf(fact_152_semantics__list_Osimps_I2_J,axiom,
! [E: nat > fm,F: nat > list_fm > fm,T: tm,L2: list_tm] :
( ( semantics_list_fm @ E @ F @ ( cons_tm @ T @ L2 ) )
= ( cons_fm @ ( semantics_term_fm @ E @ F @ T ) @ ( semantics_list_fm @ E @ F @ L2 ) ) ) ).
% semantics_list.simps(2)
thf(fact_153_semantics__list_Osimps_I2_J,axiom,
! [E: nat > list_fm,F: nat > list_list_fm > list_fm,T: tm,L2: list_tm] :
( ( semant3092164258205821607ist_fm @ E @ F @ ( cons_tm @ T @ L2 ) )
= ( cons_list_fm @ ( semant91140444983244153ist_fm @ E @ F @ T ) @ ( semant3092164258205821607ist_fm @ E @ F @ L2 ) ) ) ).
% semantics_list.simps(2)
thf(fact_154_semantics__list_Osimps_I2_J,axiom,
! [E: nat > tm,F: nat > list_tm > tm,T: tm,L2: list_tm] :
( ( semantics_list_tm @ E @ F @ ( cons_tm @ T @ L2 ) )
= ( cons_tm @ ( semantics_term_tm @ E @ F @ T ) @ ( semantics_list_tm @ E @ F @ L2 ) ) ) ).
% semantics_list.simps(2)
thf(fact_155_subst__lemma_H_I1_J,axiom,
! [E: nat > tm,F: nat > list_tm > tm,T: tm,U: tm,I2: nat] :
( ( semantics_term_tm @ E @ F @ ( substt @ T @ U @ I2 ) )
= ( semantics_term_tm @ ( shift_nat_tm @ E @ I2 @ ( semantics_term_tm @ E @ F @ U ) ) @ F @ T ) ) ).
% subst_lemma'(1)
thf(fact_156_list__ex1__simps_I2_J,axiom,
! [P: list_fm > $o,X2: list_fm,Xs: list_list_fm] :
( ( list_ex1_list_fm @ P @ ( cons_list_fm @ X2 @ Xs ) )
= ( ( ( P @ X2 )
=> ( list_all_list_fm
@ ^ [Y: list_fm] :
( ~ ( P @ Y )
| ( X2 = Y ) )
@ Xs ) )
& ( ~ ( P @ X2 )
=> ( list_ex1_list_fm @ P @ Xs ) ) ) ) ).
% list_ex1_simps(2)
thf(fact_157_list__ex1__simps_I2_J,axiom,
! [P: tm > $o,X2: tm,Xs: list_tm] :
( ( list_ex1_tm @ P @ ( cons_tm @ X2 @ Xs ) )
= ( ( ( P @ X2 )
=> ( list_all_tm
@ ^ [Y: tm] :
( ~ ( P @ Y )
| ( X2 = Y ) )
@ Xs ) )
& ( ~ ( P @ X2 )
=> ( list_ex1_tm @ P @ Xs ) ) ) ) ).
% list_ex1_simps(2)
thf(fact_158_list__ex1__simps_I2_J,axiom,
! [P: fm > $o,X2: fm,Xs: list_fm] :
( ( list_ex1_fm @ P @ ( cons_fm @ X2 @ Xs ) )
= ( ( ( P @ X2 )
=> ( list_all_fm
@ ^ [Y: fm] :
( ~ ( P @ Y )
| ( X2 = Y ) )
@ Xs ) )
& ( ~ ( P @ X2 )
=> ( list_ex1_fm @ P @ Xs ) ) ) ) ).
% list_ex1_simps(2)
thf(fact_159_iso__tuple__UNIV__I,axiom,
! [X2: tm] : ( member_tm @ X2 @ top_top_set_tm ) ).
% iso_tuple_UNIV_I
thf(fact_160_iso__tuple__UNIV__I,axiom,
! [X2: list_fm] : ( member_list_fm @ X2 @ top_top_set_list_fm ) ).
% iso_tuple_UNIV_I
thf(fact_161_iso__tuple__UNIV__I,axiom,
! [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_162_iso__tuple__UNIV__I,axiom,
! [X2: $o] : ( member_o @ X2 @ top_top_set_o ) ).
% iso_tuple_UNIV_I
thf(fact_163_substts_Osimps_I2_J,axiom,
! [T: tm,Ts: list_tm,S3: tm,K: nat] :
( ( substts @ ( cons_tm @ T @ Ts ) @ S3 @ K )
= ( cons_tm @ ( substt @ T @ S3 @ K ) @ ( substts @ Ts @ S3 @ K ) ) ) ).
% substts.simps(2)
thf(fact_164_substt_Osimps_I2_J,axiom,
! [A: nat,Ts: list_tm,S3: tm,K: nat] :
( ( substt @ ( fun @ A @ Ts ) @ S3 @ K )
= ( fun @ A @ ( substts @ Ts @ S3 @ K ) ) ) ).
% substt.simps(2)
thf(fact_165_s5_I2_J,axiom,
( sub_list
= ( ^ [V: nat,S2: tm,L: list_tm] : ( substts @ L @ S2 @ V ) ) ) ).
% s5(2)
thf(fact_166_top__empty__eq,axiom,
( top_top_tm_o
= ( ^ [X5: tm] : ( member_tm @ X5 @ top_top_set_tm ) ) ) ).
% top_empty_eq
thf(fact_167_top__empty__eq,axiom,
( top_top_list_fm_o
= ( ^ [X5: list_fm] : ( member_list_fm @ X5 @ top_top_set_list_fm ) ) ) ).
% top_empty_eq
thf(fact_168_top__empty__eq,axiom,
( top_top_nat_o
= ( ^ [X5: nat] : ( member_nat @ X5 @ top_top_set_nat ) ) ) ).
% top_empty_eq
thf(fact_169_top__empty__eq,axiom,
( top_top_o_o
= ( ^ [X5: $o] : ( member_o @ X5 @ top_top_set_o ) ) ) ).
% top_empty_eq
thf(fact_170_top__set__def,axiom,
( top_top_set_list_fm
= ( collect_list_fm @ top_top_list_fm_o ) ) ).
% top_set_def
thf(fact_171_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_172_top__set__def,axiom,
( top_top_set_o
= ( collect_o @ top_top_o_o ) ) ).
% top_set_def
thf(fact_173_sub__list_Osimps_I2_J,axiom,
! [V2: nat,S3: tm,T: tm,L2: list_tm] :
( ( sub_list @ V2 @ S3 @ ( cons_tm @ T @ L2 ) )
= ( cons_tm @ ( sub_term @ V2 @ S3 @ T ) @ ( sub_list @ V2 @ S3 @ L2 ) ) ) ).
% sub_list.simps(2)
thf(fact_174_s5_I1_J,axiom,
( sub_term
= ( ^ [V: nat,S2: tm,T2: tm] : ( substt @ T2 @ S2 @ V ) ) ) ).
% s5(1)
thf(fact_175_sub__term_Osimps_I2_J,axiom,
! [V2: nat,S3: tm,I2: nat,L2: list_tm] :
( ( sub_term @ V2 @ S3 @ ( fun @ I2 @ L2 ) )
= ( fun @ I2 @ ( sub_list @ V2 @ S3 @ L2 ) ) ) ).
% sub_term.simps(2)
thf(fact_176_Shift__def,axiom,
( bNF_Gr6607445516917591172ift_tm
= ( ^ [Kl: set_list_tm,K2: tm] :
( collect_list_tm
@ ^ [Kl2: list_tm] : ( member_list_tm @ ( cons_tm @ K2 @ Kl2 ) @ Kl ) ) ) ) ).
% Shift_def
thf(fact_177_Shift__def,axiom,
( bNF_Gr6607445516916672786ift_fm
= ( ^ [Kl: set_list_fm,K2: fm] :
( collect_list_fm
@ ^ [Kl2: list_fm] : ( member_list_fm @ ( cons_fm @ K2 @ Kl2 ) @ Kl ) ) ) ) ).
% Shift_def
thf(fact_178_Shift__def,axiom,
( bNF_Gr4365904581682047384ist_fm
= ( ^ [Kl: set_list_list_fm,K2: list_fm] :
( collect_list_list_fm
@ ^ [Kl2: list_list_fm] : ( member_list_list_fm @ ( cons_list_fm @ K2 @ Kl2 ) @ Kl ) ) ) ) ).
% Shift_def
thf(fact_179_paramsts_Osimps_I2_J,axiom,
! [T: tm,Ts: list_tm] :
( ( paramsts @ ( cons_tm @ T @ Ts ) )
= ( sup_sup_set_nat @ ( paramst @ T ) @ ( paramsts @ Ts ) ) ) ).
% paramsts.simps(2)
thf(fact_180_ShiftD,axiom,
! [Kl3: list_tm,Kl4: set_list_tm,K: tm] :
( ( member_list_tm @ Kl3 @ ( bNF_Gr6607445516917591172ift_tm @ Kl4 @ K ) )
=> ( member_list_tm @ ( cons_tm @ K @ Kl3 ) @ Kl4 ) ) ).
% ShiftD
thf(fact_181_ShiftD,axiom,
! [Kl3: list_fm,Kl4: set_list_fm,K: fm] :
( ( member_list_fm @ Kl3 @ ( bNF_Gr6607445516916672786ift_fm @ Kl4 @ K ) )
=> ( member_list_fm @ ( cons_fm @ K @ Kl3 ) @ Kl4 ) ) ).
% ShiftD
thf(fact_182_ShiftD,axiom,
! [Kl3: list_list_fm,Kl4: set_list_list_fm,K: list_fm] :
( ( member_list_list_fm @ Kl3 @ ( bNF_Gr4365904581682047384ist_fm @ Kl4 @ K ) )
=> ( member_list_list_fm @ ( cons_list_fm @ K @ Kl3 ) @ Kl4 ) ) ).
% ShiftD
thf(fact_183_UnCI,axiom,
! [C: tm,B: set_tm,A2: set_tm] :
( ( ~ ( member_tm @ C @ B )
=> ( member_tm @ C @ A2 ) )
=> ( member_tm @ C @ ( sup_sup_set_tm @ A2 @ B ) ) ) ).
% UnCI
thf(fact_184_UnCI,axiom,
! [C: list_fm,B: set_list_fm,A2: set_list_fm] :
( ( ~ ( member_list_fm @ C @ B )
=> ( member_list_fm @ C @ A2 ) )
=> ( member_list_fm @ C @ ( sup_sup_set_list_fm @ A2 @ B ) ) ) ).
% UnCI
thf(fact_185_UnCI,axiom,
! [C: nat,B: set_nat,A2: set_nat] :
( ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ A2 ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnCI
thf(fact_186_Un__iff,axiom,
! [C: tm,A2: set_tm,B: set_tm] :
( ( member_tm @ C @ ( sup_sup_set_tm @ A2 @ B ) )
= ( ( member_tm @ C @ A2 )
| ( member_tm @ C @ B ) ) ) ).
% Un_iff
thf(fact_187_Un__iff,axiom,
! [C: list_fm,A2: set_list_fm,B: set_list_fm] :
( ( member_list_fm @ C @ ( sup_sup_set_list_fm @ A2 @ B ) )
= ( ( member_list_fm @ C @ A2 )
| ( member_list_fm @ C @ B ) ) ) ).
% Un_iff
thf(fact_188_Un__iff,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) )
= ( ( member_nat @ C @ A2 )
| ( member_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_189_Un__empty,axiom,
! [A2: set_tm,B: set_tm] :
( ( ( sup_sup_set_tm @ A2 @ B )
= bot_bot_set_tm )
= ( ( A2 = bot_bot_set_tm )
& ( B = bot_bot_set_tm ) ) ) ).
% Un_empty
thf(fact_190_Un__empty,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_191_Un__empty,axiom,
! [A2: set_o,B: set_o] :
( ( ( sup_sup_set_o @ A2 @ B )
= bot_bot_set_o )
= ( ( A2 = bot_bot_set_o )
& ( B = bot_bot_set_o ) ) ) ).
% Un_empty
thf(fact_192_list__all__simps_I2_J,axiom,
! [P: list_fm > $o] : ( list_all_list_fm @ P @ nil_list_fm ) ).
% list_all_simps(2)
thf(fact_193_list__all__simps_I2_J,axiom,
! [P: tm > $o] : ( list_all_tm @ P @ nil_tm ) ).
% list_all_simps(2)
thf(fact_194_list__all__simps_I2_J,axiom,
! [P: fm > $o] : ( list_all_fm @ P @ nil_fm ) ).
% list_all_simps(2)
thf(fact_195_list__ex1__simps_I1_J,axiom,
! [P: tm > $o] :
~ ( list_ex1_tm @ P @ nil_tm ) ).
% list_ex1_simps(1)
thf(fact_196_list__ex1__simps_I1_J,axiom,
! [P: fm > $o] :
~ ( list_ex1_fm @ P @ nil_fm ) ).
% list_ex1_simps(1)
thf(fact_197_list__ex1__simps_I1_J,axiom,
! [P: list_fm > $o] :
~ ( list_ex1_list_fm @ P @ nil_list_fm ) ).
% list_ex1_simps(1)
thf(fact_198_transpose_Ocases,axiom,
! [X2: list_list_tm] :
( ( X2 != nil_list_tm )
=> ( ! [Xss: list_list_tm] :
( X2
!= ( cons_list_tm @ nil_tm @ Xss ) )
=> ~ ! [X: tm,Xs2: list_tm,Xss: list_list_tm] :
( X2
!= ( cons_list_tm @ ( cons_tm @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_199_transpose_Ocases,axiom,
! [X2: list_list_list_fm] :
( ( X2 != nil_list_list_fm )
=> ( ! [Xss: list_list_list_fm] :
( X2
!= ( cons_list_list_fm @ nil_list_fm @ Xss ) )
=> ~ ! [X: list_fm,Xs2: list_list_fm,Xss: list_list_list_fm] :
( X2
!= ( cons_list_list_fm @ ( cons_list_fm @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_200_transpose_Ocases,axiom,
! [X2: list_list_fm] :
( ( X2 != nil_list_fm )
=> ( ! [Xss: list_list_fm] :
( X2
!= ( cons_list_fm @ nil_fm @ Xss ) )
=> ~ ! [X: fm,Xs2: list_fm,Xss: list_list_fm] :
( X2
!= ( cons_list_fm @ ( cons_fm @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_201_UnE,axiom,
! [C: tm,A2: set_tm,B: set_tm] :
( ( member_tm @ C @ ( sup_sup_set_tm @ A2 @ B ) )
=> ( ~ ( member_tm @ C @ A2 )
=> ( member_tm @ C @ B ) ) ) ).
% UnE
thf(fact_202_UnE,axiom,
! [C: list_fm,A2: set_list_fm,B: set_list_fm] :
( ( member_list_fm @ C @ ( sup_sup_set_list_fm @ A2 @ B ) )
=> ( ~ ( member_list_fm @ C @ A2 )
=> ( member_list_fm @ C @ B ) ) ) ).
% UnE
thf(fact_203_UnE,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) )
=> ( ~ ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B ) ) ) ).
% UnE
thf(fact_204_UnI1,axiom,
! [C: tm,A2: set_tm,B: set_tm] :
( ( member_tm @ C @ A2 )
=> ( member_tm @ C @ ( sup_sup_set_tm @ A2 @ B ) ) ) ).
% UnI1
thf(fact_205_UnI1,axiom,
! [C: list_fm,A2: set_list_fm,B: set_list_fm] :
( ( member_list_fm @ C @ A2 )
=> ( member_list_fm @ C @ ( sup_sup_set_list_fm @ A2 @ B ) ) ) ).
% UnI1
thf(fact_206_UnI1,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnI1
thf(fact_207_UnI2,axiom,
! [C: tm,B: set_tm,A2: set_tm] :
( ( member_tm @ C @ B )
=> ( member_tm @ C @ ( sup_sup_set_tm @ A2 @ B ) ) ) ).
% UnI2
thf(fact_208_UnI2,axiom,
! [C: list_fm,B: set_list_fm,A2: set_list_fm] :
( ( member_list_fm @ C @ B )
=> ( member_list_fm @ C @ ( sup_sup_set_list_fm @ A2 @ B ) ) ) ).
% UnI2
thf(fact_209_UnI2,axiom,
! [C: nat,B: set_nat,A2: set_nat] :
( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnI2
thf(fact_210_bex__Un,axiom,
! [A2: set_nat,B: set_nat,P: nat > $o] :
( ( ? [X5: nat] :
( ( member_nat @ X5 @ ( sup_sup_set_nat @ A2 @ B ) )
& ( P @ X5 ) ) )
= ( ? [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( P @ X5 ) )
| ? [X5: nat] :
( ( member_nat @ X5 @ B )
& ( P @ X5 ) ) ) ) ).
% bex_Un
thf(fact_211_ball__Un,axiom,
! [A2: set_nat,B: set_nat,P: nat > $o] :
( ( ! [X5: nat] :
( ( member_nat @ X5 @ ( sup_sup_set_nat @ A2 @ B ) )
=> ( P @ X5 ) ) )
= ( ! [X5: nat] :
( ( member_nat @ X5 @ A2 )
=> ( P @ X5 ) )
& ! [X5: nat] :
( ( member_nat @ X5 @ B )
=> ( P @ X5 ) ) ) ) ).
% ball_Un
thf(fact_212_Un__assoc,axiom,
! [A2: set_nat,B: set_nat,C3: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C3 )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C3 ) ) ) ).
% Un_assoc
thf(fact_213_Un__absorb,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_214_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B3: set_nat] : ( sup_sup_set_nat @ B3 @ A4 ) ) ) ).
% Un_commute
thf(fact_215_Un__left__absorb,axiom,
! [A2: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B ) )
= ( sup_sup_set_nat @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_216_Un__left__commute,axiom,
! [A2: set_nat,B: set_nat,C3: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C3 ) )
= ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A2 @ C3 ) ) ) ).
% Un_left_commute
thf(fact_217_Collect__disj__eq,axiom,
! [P: list_fm > $o,Q: list_fm > $o] :
( ( collect_list_fm
@ ^ [X5: list_fm] :
( ( P @ X5 )
| ( Q @ X5 ) ) )
= ( sup_sup_set_list_fm @ ( collect_list_fm @ P ) @ ( collect_list_fm @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_218_Collect__disj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X5: nat] :
( ( P @ X5 )
| ( Q @ X5 ) ) )
= ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_219_Un__def,axiom,
( sup_sup_set_tm
= ( ^ [A4: set_tm,B3: set_tm] :
( collect_tm
@ ^ [X5: tm] :
( ( member_tm @ X5 @ A4 )
| ( member_tm @ X5 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_220_Un__def,axiom,
( sup_sup_set_list_fm
= ( ^ [A4: set_list_fm,B3: set_list_fm] :
( collect_list_fm
@ ^ [X5: list_fm] :
( ( member_list_fm @ X5 @ A4 )
| ( member_list_fm @ X5 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_221_Un__def,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A4 )
| ( member_nat @ X5 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_222_semantics__list_Osimps_I1_J,axiom,
! [E: nat > fm,F: nat > list_fm > fm] :
( ( semantics_list_fm @ E @ F @ nil_tm )
= nil_fm ) ).
% semantics_list.simps(1)
thf(fact_223_semantics__list_Osimps_I1_J,axiom,
! [E: nat > list_fm,F: nat > list_list_fm > list_fm] :
( ( semant3092164258205821607ist_fm @ E @ F @ nil_tm )
= nil_list_fm ) ).
% semantics_list.simps(1)
thf(fact_224_semantics__list_Osimps_I1_J,axiom,
! [E: nat > tm,F: nat > list_tm > tm] :
( ( semantics_list_tm @ E @ F @ nil_tm )
= nil_tm ) ).
% semantics_list.simps(1)
thf(fact_225_Un__UNIV__left,axiom,
! [B: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ B )
= top_top_set_nat ) ).
% Un_UNIV_left
thf(fact_226_Un__UNIV__left,axiom,
! [B: set_o] :
( ( sup_sup_set_o @ top_top_set_o @ B )
= top_top_set_o ) ).
% Un_UNIV_left
thf(fact_227_Un__UNIV__right,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ top_top_set_nat )
= top_top_set_nat ) ).
% Un_UNIV_right
thf(fact_228_Un__UNIV__right,axiom,
! [A2: set_o] :
( ( sup_sup_set_o @ A2 @ top_top_set_o )
= top_top_set_o ) ).
% Un_UNIV_right
thf(fact_229_Un__empty__right,axiom,
! [A2: set_tm] :
( ( sup_sup_set_tm @ A2 @ bot_bot_set_tm )
= A2 ) ).
% Un_empty_right
thf(fact_230_Un__empty__right,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Un_empty_right
thf(fact_231_Un__empty__right,axiom,
! [A2: set_o] :
( ( sup_sup_set_o @ A2 @ bot_bot_set_o )
= A2 ) ).
% Un_empty_right
thf(fact_232_Un__empty__left,axiom,
! [B: set_tm] :
( ( sup_sup_set_tm @ bot_bot_set_tm @ B )
= B ) ).
% Un_empty_left
thf(fact_233_Un__empty__left,axiom,
! [B: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_234_Un__empty__left,axiom,
! [B: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ B )
= B ) ).
% Un_empty_left
thf(fact_235_list_Odistinct_I1_J,axiom,
! [X21: tm,X222: list_tm] :
( nil_tm
!= ( cons_tm @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_236_list_Odistinct_I1_J,axiom,
! [X21: fm,X222: list_fm] :
( nil_fm
!= ( cons_fm @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_237_list_Odistinct_I1_J,axiom,
! [X21: list_fm,X222: list_list_fm] :
( nil_list_fm
!= ( cons_list_fm @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_238_list_OdiscI,axiom,
! [List: list_tm,X21: tm,X222: list_tm] :
( ( List
= ( cons_tm @ X21 @ X222 ) )
=> ( List != nil_tm ) ) ).
% list.discI
thf(fact_239_list_OdiscI,axiom,
! [List: list_fm,X21: fm,X222: list_fm] :
( ( List
= ( cons_fm @ X21 @ X222 ) )
=> ( List != nil_fm ) ) ).
% list.discI
thf(fact_240_list_OdiscI,axiom,
! [List: list_list_fm,X21: list_fm,X222: list_list_fm] :
( ( List
= ( cons_list_fm @ X21 @ X222 ) )
=> ( List != nil_list_fm ) ) ).
% list.discI
thf(fact_241_list_Oexhaust,axiom,
! [Y4: list_tm] :
( ( Y4 != nil_tm )
=> ~ ! [X212: tm,X223: list_tm] :
( Y4
!= ( cons_tm @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_242_list_Oexhaust,axiom,
! [Y4: list_fm] :
( ( Y4 != nil_fm )
=> ~ ! [X212: fm,X223: list_fm] :
( Y4
!= ( cons_fm @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_243_list_Oexhaust,axiom,
! [Y4: list_list_fm] :
( ( Y4 != nil_list_fm )
=> ~ ! [X212: list_fm,X223: list_list_fm] :
( Y4
!= ( cons_list_fm @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_244_remdups__adj_Ocases,axiom,
! [X2: list_tm] :
( ( X2 != nil_tm )
=> ( ! [X: tm] :
( X2
!= ( cons_tm @ X @ nil_tm ) )
=> ~ ! [X: tm,Y3: tm,Xs2: list_tm] :
( X2
!= ( cons_tm @ X @ ( cons_tm @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_245_remdups__adj_Ocases,axiom,
! [X2: list_fm] :
( ( X2 != nil_fm )
=> ( ! [X: fm] :
( X2
!= ( cons_fm @ X @ nil_fm ) )
=> ~ ! [X: fm,Y3: fm,Xs2: list_fm] :
( X2
!= ( cons_fm @ X @ ( cons_fm @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_246_remdups__adj_Ocases,axiom,
! [X2: list_list_fm] :
( ( X2 != nil_list_fm )
=> ( ! [X: list_fm] :
( X2
!= ( cons_list_fm @ X @ nil_list_fm ) )
=> ~ ! [X: list_fm,Y3: list_fm,Xs2: list_list_fm] :
( X2
!= ( cons_list_fm @ X @ ( cons_list_fm @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_247_neq__Nil__conv,axiom,
! [Xs: list_tm] :
( ( Xs != nil_tm )
= ( ? [Y: tm,Ys: list_tm] :
( Xs
= ( cons_tm @ Y @ Ys ) ) ) ) ).
% neq_Nil_conv
thf(fact_248_neq__Nil__conv,axiom,
! [Xs: list_fm] :
( ( Xs != nil_fm )
= ( ? [Y: fm,Ys: list_fm] :
( Xs
= ( cons_fm @ Y @ Ys ) ) ) ) ).
% neq_Nil_conv
thf(fact_249_neq__Nil__conv,axiom,
! [Xs: list_list_fm] :
( ( Xs != nil_list_fm )
= ( ? [Y: list_fm,Ys: list_list_fm] :
( Xs
= ( cons_list_fm @ Y @ Ys ) ) ) ) ).
% neq_Nil_conv
thf(fact_250_list__induct2_H,axiom,
! [P: list_tm > list_tm > $o,Xs: list_tm,Ys2: list_tm] :
( ( P @ nil_tm @ nil_tm )
=> ( ! [X: tm,Xs2: list_tm] : ( P @ ( cons_tm @ X @ Xs2 ) @ nil_tm )
=> ( ! [Y3: tm,Ys3: list_tm] : ( P @ nil_tm @ ( cons_tm @ Y3 @ Ys3 ) )
=> ( ! [X: tm,Xs2: list_tm,Y3: tm,Ys3: list_tm] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_tm @ X @ Xs2 ) @ ( cons_tm @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_251_list__induct2_H,axiom,
! [P: list_tm > list_fm > $o,Xs: list_tm,Ys2: list_fm] :
( ( P @ nil_tm @ nil_fm )
=> ( ! [X: tm,Xs2: list_tm] : ( P @ ( cons_tm @ X @ Xs2 ) @ nil_fm )
=> ( ! [Y3: fm,Ys3: list_fm] : ( P @ nil_tm @ ( cons_fm @ Y3 @ Ys3 ) )
=> ( ! [X: tm,Xs2: list_tm,Y3: fm,Ys3: list_fm] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_tm @ X @ Xs2 ) @ ( cons_fm @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_252_list__induct2_H,axiom,
! [P: list_tm > list_list_fm > $o,Xs: list_tm,Ys2: list_list_fm] :
( ( P @ nil_tm @ nil_list_fm )
=> ( ! [X: tm,Xs2: list_tm] : ( P @ ( cons_tm @ X @ Xs2 ) @ nil_list_fm )
=> ( ! [Y3: list_fm,Ys3: list_list_fm] : ( P @ nil_tm @ ( cons_list_fm @ Y3 @ Ys3 ) )
=> ( ! [X: tm,Xs2: list_tm,Y3: list_fm,Ys3: list_list_fm] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_tm @ X @ Xs2 ) @ ( cons_list_fm @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_253_list__induct2_H,axiom,
! [P: list_fm > list_tm > $o,Xs: list_fm,Ys2: list_tm] :
( ( P @ nil_fm @ nil_tm )
=> ( ! [X: fm,Xs2: list_fm] : ( P @ ( cons_fm @ X @ Xs2 ) @ nil_tm )
=> ( ! [Y3: tm,Ys3: list_tm] : ( P @ nil_fm @ ( cons_tm @ Y3 @ Ys3 ) )
=> ( ! [X: fm,Xs2: list_fm,Y3: tm,Ys3: list_tm] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_fm @ X @ Xs2 ) @ ( cons_tm @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_254_list__induct2_H,axiom,
! [P: list_fm > list_fm > $o,Xs: list_fm,Ys2: list_fm] :
( ( P @ nil_fm @ nil_fm )
=> ( ! [X: fm,Xs2: list_fm] : ( P @ ( cons_fm @ X @ Xs2 ) @ nil_fm )
=> ( ! [Y3: fm,Ys3: list_fm] : ( P @ nil_fm @ ( cons_fm @ Y3 @ Ys3 ) )
=> ( ! [X: fm,Xs2: list_fm,Y3: fm,Ys3: list_fm] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_fm @ X @ Xs2 ) @ ( cons_fm @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_255_list__induct2_H,axiom,
! [P: list_fm > list_list_fm > $o,Xs: list_fm,Ys2: list_list_fm] :
( ( P @ nil_fm @ nil_list_fm )
=> ( ! [X: fm,Xs2: list_fm] : ( P @ ( cons_fm @ X @ Xs2 ) @ nil_list_fm )
=> ( ! [Y3: list_fm,Ys3: list_list_fm] : ( P @ nil_fm @ ( cons_list_fm @ Y3 @ Ys3 ) )
=> ( ! [X: fm,Xs2: list_fm,Y3: list_fm,Ys3: list_list_fm] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_fm @ X @ Xs2 ) @ ( cons_list_fm @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_256_list__induct2_H,axiom,
! [P: list_list_fm > list_tm > $o,Xs: list_list_fm,Ys2: list_tm] :
( ( P @ nil_list_fm @ nil_tm )
=> ( ! [X: list_fm,Xs2: list_list_fm] : ( P @ ( cons_list_fm @ X @ Xs2 ) @ nil_tm )
=> ( ! [Y3: tm,Ys3: list_tm] : ( P @ nil_list_fm @ ( cons_tm @ Y3 @ Ys3 ) )
=> ( ! [X: list_fm,Xs2: list_list_fm,Y3: tm,Ys3: list_tm] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_list_fm @ X @ Xs2 ) @ ( cons_tm @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_257_list__induct2_H,axiom,
! [P: list_list_fm > list_fm > $o,Xs: list_list_fm,Ys2: list_fm] :
( ( P @ nil_list_fm @ nil_fm )
=> ( ! [X: list_fm,Xs2: list_list_fm] : ( P @ ( cons_list_fm @ X @ Xs2 ) @ nil_fm )
=> ( ! [Y3: fm,Ys3: list_fm] : ( P @ nil_list_fm @ ( cons_fm @ Y3 @ Ys3 ) )
=> ( ! [X: list_fm,Xs2: list_list_fm,Y3: fm,Ys3: list_fm] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_list_fm @ X @ Xs2 ) @ ( cons_fm @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_258_list__induct2_H,axiom,
! [P: list_list_fm > list_list_fm > $o,Xs: list_list_fm,Ys2: list_list_fm] :
( ( P @ nil_list_fm @ nil_list_fm )
=> ( ! [X: list_fm,Xs2: list_list_fm] : ( P @ ( cons_list_fm @ X @ Xs2 ) @ nil_list_fm )
=> ( ! [Y3: list_fm,Ys3: list_list_fm] : ( P @ nil_list_fm @ ( cons_list_fm @ Y3 @ Ys3 ) )
=> ( ! [X: list_fm,Xs2: list_list_fm,Y3: list_fm,Ys3: list_list_fm] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_list_fm @ X @ Xs2 ) @ ( cons_list_fm @ Y3 @ Ys3 ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_259_list__nonempty__induct,axiom,
! [Xs: list_tm,P: list_tm > $o] :
( ( Xs != nil_tm )
=> ( ! [X: tm] : ( P @ ( cons_tm @ X @ nil_tm ) )
=> ( ! [X: tm,Xs2: list_tm] :
( ( Xs2 != nil_tm )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_tm @ X @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_260_list__nonempty__induct,axiom,
! [Xs: list_fm,P: list_fm > $o] :
( ( Xs != nil_fm )
=> ( ! [X: fm] : ( P @ ( cons_fm @ X @ nil_fm ) )
=> ( ! [X: fm,Xs2: list_fm] :
( ( Xs2 != nil_fm )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_fm @ X @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_261_list__nonempty__induct,axiom,
! [Xs: list_list_fm,P: list_list_fm > $o] :
( ( Xs != nil_list_fm )
=> ( ! [X: list_fm] : ( P @ ( cons_list_fm @ X @ nil_list_fm ) )
=> ( ! [X: list_fm,Xs2: list_list_fm] :
( ( Xs2 != nil_list_fm )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_list_fm @ X @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_262_list_Opred__inject_I1_J,axiom,
! [P: list_fm > $o] : ( list_all_list_fm @ P @ nil_list_fm ) ).
% list.pred_inject(1)
thf(fact_263_list_Opred__inject_I1_J,axiom,
! [P: tm > $o] : ( list_all_tm @ P @ nil_tm ) ).
% list.pred_inject(1)
thf(fact_264_list_Opred__inject_I1_J,axiom,
! [P: fm > $o] : ( list_all_fm @ P @ nil_fm ) ).
% list.pred_inject(1)
thf(fact_265_sub__list_Osimps_I1_J,axiom,
! [V2: nat,S3: tm] :
( ( sub_list @ V2 @ S3 @ nil_tm )
= nil_tm ) ).
% sub_list.simps(1)
thf(fact_266_substts_Osimps_I1_J,axiom,
! [S3: tm,K: nat] :
( ( substts @ nil_tm @ S3 @ K )
= nil_tm ) ).
% substts.simps(1)
thf(fact_267_new__list_Osimps_I1_J,axiom,
! [C: nat] : ( new_list @ C @ nil_tm ) ).
% new_list.simps(1)
thf(fact_268_paramsts_Osimps_I1_J,axiom,
( ( paramsts @ nil_tm )
= bot_bot_set_nat ) ).
% paramsts.simps(1)
thf(fact_269_sup__bot_Oright__neutral,axiom,
! [A: set_tm] :
( ( sup_sup_set_tm @ A @ bot_bot_set_tm )
= A ) ).
% sup_bot.right_neutral
thf(fact_270_sup__bot_Oright__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_271_sup__bot_Oright__neutral,axiom,
! [A: set_o] :
( ( sup_sup_set_o @ A @ bot_bot_set_o )
= A ) ).
% sup_bot.right_neutral
thf(fact_272_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_tm,B2: set_tm] :
( ( bot_bot_set_tm
= ( sup_sup_set_tm @ A @ B2 ) )
= ( ( A = bot_bot_set_tm )
& ( B2 = bot_bot_set_tm ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_273_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A @ B2 ) )
= ( ( A = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_274_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_o,B2: set_o] :
( ( bot_bot_set_o
= ( sup_sup_set_o @ A @ B2 ) )
= ( ( A = bot_bot_set_o )
& ( B2 = bot_bot_set_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_275_sup__bot_Oleft__neutral,axiom,
! [A: set_tm] :
( ( sup_sup_set_tm @ bot_bot_set_tm @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_276_sup__bot_Oleft__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_277_sup__bot_Oleft__neutral,axiom,
! [A: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_278_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_tm,B2: set_tm] :
( ( ( sup_sup_set_tm @ A @ B2 )
= bot_bot_set_tm )
= ( ( A = bot_bot_set_tm )
& ( B2 = bot_bot_set_tm ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_279_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_nat,B2: set_nat] :
( ( ( sup_sup_set_nat @ A @ B2 )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_280_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_o,B2: set_o] :
( ( ( sup_sup_set_o @ A @ B2 )
= bot_bot_set_o )
= ( ( A = bot_bot_set_o )
& ( B2 = bot_bot_set_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_281_sup__eq__bot__iff,axiom,
! [X2: set_tm,Y4: set_tm] :
( ( ( sup_sup_set_tm @ X2 @ Y4 )
= bot_bot_set_tm )
= ( ( X2 = bot_bot_set_tm )
& ( Y4 = bot_bot_set_tm ) ) ) ).
% sup_eq_bot_iff
thf(fact_282_sup__eq__bot__iff,axiom,
! [X2: set_nat,Y4: set_nat] :
( ( ( sup_sup_set_nat @ X2 @ Y4 )
= bot_bot_set_nat )
= ( ( X2 = bot_bot_set_nat )
& ( Y4 = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_283_sup__eq__bot__iff,axiom,
! [X2: set_o,Y4: set_o] :
( ( ( sup_sup_set_o @ X2 @ Y4 )
= bot_bot_set_o )
= ( ( X2 = bot_bot_set_o )
& ( Y4 = bot_bot_set_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_284_bot__eq__sup__iff,axiom,
! [X2: set_tm,Y4: set_tm] :
( ( bot_bot_set_tm
= ( sup_sup_set_tm @ X2 @ Y4 ) )
= ( ( X2 = bot_bot_set_tm )
& ( Y4 = bot_bot_set_tm ) ) ) ).
% bot_eq_sup_iff
thf(fact_285_bot__eq__sup__iff,axiom,
! [X2: set_nat,Y4: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X2 @ Y4 ) )
= ( ( X2 = bot_bot_set_nat )
& ( Y4 = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_286_bot__eq__sup__iff,axiom,
! [X2: set_o,Y4: set_o] :
( ( bot_bot_set_o
= ( sup_sup_set_o @ X2 @ Y4 ) )
= ( ( X2 = bot_bot_set_o )
& ( Y4 = bot_bot_set_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_287_sup__bot__right,axiom,
! [X2: set_tm] :
( ( sup_sup_set_tm @ X2 @ bot_bot_set_tm )
= X2 ) ).
% sup_bot_right
thf(fact_288_sup__bot__right,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ bot_bot_set_nat )
= X2 ) ).
% sup_bot_right
thf(fact_289_sup__bot__right,axiom,
! [X2: set_o] :
( ( sup_sup_set_o @ X2 @ bot_bot_set_o )
= X2 ) ).
% sup_bot_right
thf(fact_290_sup__bot__left,axiom,
! [X2: set_tm] :
( ( sup_sup_set_tm @ bot_bot_set_tm @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_291_sup__bot__left,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_292_sup__bot__left,axiom,
! [X2: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_293_boolean__algebra_Odisj__one__right,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ top_top_set_nat )
= top_top_set_nat ) ).
% boolean_algebra.disj_one_right
thf(fact_294_boolean__algebra_Odisj__one__right,axiom,
! [X2: set_o] :
( ( sup_sup_set_o @ X2 @ top_top_set_o )
= top_top_set_o ) ).
% boolean_algebra.disj_one_right
thf(fact_295_sup__top__left,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ X2 )
= top_top_set_nat ) ).
% sup_top_left
thf(fact_296_sup__top__left,axiom,
! [X2: set_o] :
( ( sup_sup_set_o @ top_top_set_o @ X2 )
= top_top_set_o ) ).
% sup_top_left
thf(fact_297_sup__top__right,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ top_top_set_nat )
= top_top_set_nat ) ).
% sup_top_right
thf(fact_298_sup__top__right,axiom,
! [X2: set_o] :
( ( sup_sup_set_o @ X2 @ top_top_set_o )
= top_top_set_o ) ).
% sup_top_right
thf(fact_299_boolean__algebra_Odisj__one__left,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ X2 )
= top_top_set_nat ) ).
% boolean_algebra.disj_one_left
thf(fact_300_boolean__algebra_Odisj__one__left,axiom,
! [X2: set_o] :
( ( sup_sup_set_o @ top_top_set_o @ X2 )
= top_top_set_o ) ).
% boolean_algebra.disj_one_left
thf(fact_301_sup__Un__eq,axiom,
! [R: set_tm,S: set_tm] :
( ( sup_sup_tm_o
@ ^ [X5: tm] : ( member_tm @ X5 @ R )
@ ^ [X5: tm] : ( member_tm @ X5 @ S ) )
= ( ^ [X5: tm] : ( member_tm @ X5 @ ( sup_sup_set_tm @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_302_sup__Un__eq,axiom,
! [R: set_list_fm,S: set_list_fm] :
( ( sup_sup_list_fm_o
@ ^ [X5: list_fm] : ( member_list_fm @ X5 @ R )
@ ^ [X5: list_fm] : ( member_list_fm @ X5 @ S ) )
= ( ^ [X5: list_fm] : ( member_list_fm @ X5 @ ( sup_sup_set_list_fm @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_303_sup__Un__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( sup_sup_nat_o
@ ^ [X5: nat] : ( member_nat @ X5 @ R )
@ ^ [X5: nat] : ( member_nat @ X5 @ S ) )
= ( ^ [X5: nat] : ( member_nat @ X5 @ ( sup_sup_set_nat @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_304_sup__set__def,axiom,
( sup_sup_set_tm
= ( ^ [A4: set_tm,B3: set_tm] :
( collect_tm
@ ( sup_sup_tm_o
@ ^ [X5: tm] : ( member_tm @ X5 @ A4 )
@ ^ [X5: tm] : ( member_tm @ X5 @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_305_sup__set__def,axiom,
( sup_sup_set_list_fm
= ( ^ [A4: set_list_fm,B3: set_list_fm] :
( collect_list_fm
@ ( sup_sup_list_fm_o
@ ^ [X5: list_fm] : ( member_list_fm @ X5 @ A4 )
@ ^ [X5: list_fm] : ( member_list_fm @ X5 @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_306_sup__set__def,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( collect_nat
@ ( sup_sup_nat_o
@ ^ [X5: nat] : ( member_nat @ X5 @ A4 )
@ ^ [X5: nat] : ( member_nat @ X5 @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_307_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_tm] :
( ( sup_sup_set_tm @ X2 @ bot_bot_set_tm )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_308_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ bot_bot_set_nat )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_309_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_o] :
( ( sup_sup_set_o @ X2 @ bot_bot_set_o )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_310_empty__Shift,axiom,
! [Kl4: set_list_nat,K: nat] :
( ( member_list_nat @ nil_nat @ Kl4 )
=> ( ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl4 @ nil_nat ) )
=> ( member_list_nat @ nil_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl4 @ K ) ) ) ) ).
% empty_Shift
thf(fact_311_empty__Shift,axiom,
! [Kl4: set_list_tm,K: tm] :
( ( member_list_tm @ nil_tm @ Kl4 )
=> ( ( member_tm @ K @ ( bNF_Greatest_Succ_tm @ Kl4 @ nil_tm ) )
=> ( member_list_tm @ nil_tm @ ( bNF_Gr6607445516917591172ift_tm @ Kl4 @ K ) ) ) ) ).
% empty_Shift
thf(fact_312_empty__Shift,axiom,
! [Kl4: set_list_fm,K: fm] :
( ( member_list_fm @ nil_fm @ Kl4 )
=> ( ( member_fm @ K @ ( bNF_Greatest_Succ_fm @ Kl4 @ nil_fm ) )
=> ( member_list_fm @ nil_fm @ ( bNF_Gr6607445516916672786ift_fm @ Kl4 @ K ) ) ) ) ).
% empty_Shift
thf(fact_313_empty__Shift,axiom,
! [Kl4: set_list_list_fm,K: list_fm] :
( ( member_list_list_fm @ nil_list_fm @ Kl4 )
=> ( ( member_list_fm @ K @ ( bNF_Gr8387611704671093012ist_fm @ Kl4 @ nil_list_fm ) )
=> ( member_list_list_fm @ nil_list_fm @ ( bNF_Gr4365904581682047384ist_fm @ Kl4 @ K ) ) ) ) ).
% empty_Shift
thf(fact_314_Succ__Shift,axiom,
! [Kl4: set_list_tm,K: tm,Kl3: list_tm] :
( ( bNF_Greatest_Succ_tm @ ( bNF_Gr6607445516917591172ift_tm @ Kl4 @ K ) @ Kl3 )
= ( bNF_Greatest_Succ_tm @ Kl4 @ ( cons_tm @ K @ Kl3 ) ) ) ).
% Succ_Shift
thf(fact_315_Succ__Shift,axiom,
! [Kl4: set_list_fm,K: fm,Kl3: list_fm] :
( ( bNF_Greatest_Succ_fm @ ( bNF_Gr6607445516916672786ift_fm @ Kl4 @ K ) @ Kl3 )
= ( bNF_Greatest_Succ_fm @ Kl4 @ ( cons_fm @ K @ Kl3 ) ) ) ).
% Succ_Shift
thf(fact_316_Succ__Shift,axiom,
! [Kl4: set_list_list_fm,K: list_fm,Kl3: list_list_fm] :
( ( bNF_Gr8387611704671093012ist_fm @ ( bNF_Gr4365904581682047384ist_fm @ Kl4 @ K ) @ Kl3 )
= ( bNF_Gr8387611704671093012ist_fm @ Kl4 @ ( cons_list_fm @ K @ Kl3 ) ) ) ).
% Succ_Shift
thf(fact_317_paramst_Osimps_I2_J,axiom,
! [A: nat,Ts: list_tm] :
( ( paramst @ ( fun @ A @ Ts ) )
= ( sup_sup_set_nat @ ( insert_nat @ A @ bot_bot_set_nat ) @ ( paramsts @ Ts ) ) ) ).
% paramst.simps(2)
thf(fact_318_product__lists_Osimps_I1_J,axiom,
( ( product_lists_tm @ nil_list_tm )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) ).
% product_lists.simps(1)
thf(fact_319_product__lists_Osimps_I1_J,axiom,
( ( produc373462945560358120ist_fm @ nil_list_list_fm )
= ( cons_list_list_fm @ nil_list_fm @ nil_list_list_fm ) ) ).
% product_lists.simps(1)
thf(fact_320_product__lists_Osimps_I1_J,axiom,
( ( product_lists_fm @ nil_list_fm )
= ( cons_list_fm @ nil_fm @ nil_list_fm ) ) ).
% product_lists.simps(1)
thf(fact_321_insert__Nil,axiom,
! [X2: tm] :
( ( insert_tm @ X2 @ nil_tm )
= ( cons_tm @ X2 @ nil_tm ) ) ).
% insert_Nil
thf(fact_322_insert__Nil,axiom,
! [X2: fm] :
( ( insert_fm @ X2 @ nil_fm )
= ( cons_fm @ X2 @ nil_fm ) ) ).
% insert_Nil
thf(fact_323_insert__Nil,axiom,
! [X2: list_fm] :
( ( insert_list_fm @ X2 @ nil_list_fm )
= ( cons_list_fm @ X2 @ nil_list_fm ) ) ).
% insert_Nil
thf(fact_324_insertCI,axiom,
! [A: $o,B: set_o,B2: $o] :
( ( ~ ( member_o @ A @ B )
=> ( A = B2 ) )
=> ( member_o @ A @ ( insert_o @ B2 @ B ) ) ) ).
% insertCI
thf(fact_325_insertCI,axiom,
! [A: tm,B: set_tm,B2: tm] :
( ( ~ ( member_tm @ A @ B )
=> ( A = B2 ) )
=> ( member_tm @ A @ ( insert_tm2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_326_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_327_insertCI,axiom,
! [A: list_fm,B: set_list_fm,B2: list_fm] :
( ( ~ ( member_list_fm @ A @ B )
=> ( A = B2 ) )
=> ( member_list_fm @ A @ ( insert_list_fm2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_328_insert__iff,axiom,
! [A: $o,B2: $o,A2: set_o] :
( ( member_o @ A @ ( insert_o @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_o @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_329_insert__iff,axiom,
! [A: tm,B2: tm,A2: set_tm] :
( ( member_tm @ A @ ( insert_tm2 @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_tm @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_330_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_331_insert__iff,axiom,
! [A: list_fm,B2: list_fm,A2: set_list_fm] :
( ( member_list_fm @ A @ ( insert_list_fm2 @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_list_fm @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_332_insert__absorb2,axiom,
! [X2: nat,A2: set_nat] :
( ( insert_nat @ X2 @ ( insert_nat @ X2 @ A2 ) )
= ( insert_nat @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_333_insert__absorb2,axiom,
! [X2: $o,A2: set_o] :
( ( insert_o @ X2 @ ( insert_o @ X2 @ A2 ) )
= ( insert_o @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_334_insert__absorb2,axiom,
! [X2: tm,A2: set_tm] :
( ( insert_tm2 @ X2 @ ( insert_tm2 @ X2 @ A2 ) )
= ( insert_tm2 @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_335_singletonI,axiom,
! [A: list_fm] : ( member_list_fm @ A @ ( insert_list_fm2 @ A @ bot_bot_set_list_fm ) ) ).
% singletonI
thf(fact_336_singletonI,axiom,
! [A: tm] : ( member_tm @ A @ ( insert_tm2 @ A @ bot_bot_set_tm ) ) ).
% singletonI
thf(fact_337_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_338_singletonI,axiom,
! [A: $o] : ( member_o @ A @ ( insert_o @ A @ bot_bot_set_o ) ) ).
% singletonI
thf(fact_339_Un__insert__left,axiom,
! [A: $o,B: set_o,C3: set_o] :
( ( sup_sup_set_o @ ( insert_o @ A @ B ) @ C3 )
= ( insert_o @ A @ ( sup_sup_set_o @ B @ C3 ) ) ) ).
% Un_insert_left
thf(fact_340_Un__insert__left,axiom,
! [A: tm,B: set_tm,C3: set_tm] :
( ( sup_sup_set_tm @ ( insert_tm2 @ A @ B ) @ C3 )
= ( insert_tm2 @ A @ ( sup_sup_set_tm @ B @ C3 ) ) ) ).
% Un_insert_left
thf(fact_341_Un__insert__left,axiom,
! [A: nat,B: set_nat,C3: set_nat] :
( ( sup_sup_set_nat @ ( insert_nat @ A @ B ) @ C3 )
= ( insert_nat @ A @ ( sup_sup_set_nat @ B @ C3 ) ) ) ).
% Un_insert_left
thf(fact_342_Un__insert__right,axiom,
! [A2: set_o,A: $o,B: set_o] :
( ( sup_sup_set_o @ A2 @ ( insert_o @ A @ B ) )
= ( insert_o @ A @ ( sup_sup_set_o @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_343_Un__insert__right,axiom,
! [A2: set_tm,A: tm,B: set_tm] :
( ( sup_sup_set_tm @ A2 @ ( insert_tm2 @ A @ B ) )
= ( insert_tm2 @ A @ ( sup_sup_set_tm @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_344_Un__insert__right,axiom,
! [A2: set_nat,A: nat,B: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( insert_nat @ A @ B ) )
= ( insert_nat @ A @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_345_singleton__conv2,axiom,
! [A: list_fm] :
( ( collect_list_fm
@ ( ^ [Y2: list_fm,Z: list_fm] : ( Y2 = Z )
@ A ) )
= ( insert_list_fm2 @ A @ bot_bot_set_list_fm ) ) ).
% singleton_conv2
thf(fact_346_singleton__conv2,axiom,
! [A: tm] :
( ( collect_tm
@ ( ^ [Y2: tm,Z: tm] : ( Y2 = Z )
@ A ) )
= ( insert_tm2 @ A @ bot_bot_set_tm ) ) ).
% singleton_conv2
thf(fact_347_singleton__conv2,axiom,
! [A: nat] :
( ( collect_nat
@ ( ^ [Y2: nat,Z: nat] : ( Y2 = Z )
@ A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_348_singleton__conv2,axiom,
! [A: $o] :
( ( collect_o
@ ( ^ [Y2: $o,Z: $o] : ( Y2 = Z )
@ A ) )
= ( insert_o @ A @ bot_bot_set_o ) ) ).
% singleton_conv2
thf(fact_349_singleton__conv,axiom,
! [A: list_fm] :
( ( collect_list_fm
@ ^ [X5: list_fm] : ( X5 = A ) )
= ( insert_list_fm2 @ A @ bot_bot_set_list_fm ) ) ).
% singleton_conv
thf(fact_350_singleton__conv,axiom,
! [A: tm] :
( ( collect_tm
@ ^ [X5: tm] : ( X5 = A ) )
= ( insert_tm2 @ A @ bot_bot_set_tm ) ) ).
% singleton_conv
thf(fact_351_singleton__conv,axiom,
! [A: nat] :
( ( collect_nat
@ ^ [X5: nat] : ( X5 = A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_352_singleton__conv,axiom,
! [A: $o] :
( ( collect_o
@ ^ [X5: $o] : ( X5 = A ) )
= ( insert_o @ A @ bot_bot_set_o ) ) ).
% singleton_conv
thf(fact_353_is__singletonI,axiom,
! [X2: tm] : ( is_singleton_tm @ ( insert_tm2 @ X2 @ bot_bot_set_tm ) ) ).
% is_singletonI
thf(fact_354_is__singletonI,axiom,
! [X2: nat] : ( is_singleton_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_355_is__singletonI,axiom,
! [X2: $o] : ( is_singleton_o @ ( insert_o @ X2 @ bot_bot_set_o ) ) ).
% is_singletonI
thf(fact_356_insertE,axiom,
! [A: $o,B2: $o,A2: set_o] :
( ( member_o @ A @ ( insert_o @ B2 @ A2 ) )
=> ( ( A = ~ B2 )
=> ( member_o @ A @ A2 ) ) ) ).
% insertE
thf(fact_357_insertE,axiom,
! [A: tm,B2: tm,A2: set_tm] :
( ( member_tm @ A @ ( insert_tm2 @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_tm @ A @ A2 ) ) ) ).
% insertE
thf(fact_358_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_359_insertE,axiom,
! [A: list_fm,B2: list_fm,A2: set_list_fm] :
( ( member_list_fm @ A @ ( insert_list_fm2 @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_list_fm @ A @ A2 ) ) ) ).
% insertE
thf(fact_360_insertI1,axiom,
! [A: $o,B: set_o] : ( member_o @ A @ ( insert_o @ A @ B ) ) ).
% insertI1
thf(fact_361_insertI1,axiom,
! [A: tm,B: set_tm] : ( member_tm @ A @ ( insert_tm2 @ A @ B ) ) ).
% insertI1
thf(fact_362_insertI1,axiom,
! [A: nat,B: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B ) ) ).
% insertI1
thf(fact_363_insertI1,axiom,
! [A: list_fm,B: set_list_fm] : ( member_list_fm @ A @ ( insert_list_fm2 @ A @ B ) ) ).
% insertI1
thf(fact_364_insertI2,axiom,
! [A: $o,B: set_o,B2: $o] :
( ( member_o @ A @ B )
=> ( member_o @ A @ ( insert_o @ B2 @ B ) ) ) ).
% insertI2
thf(fact_365_insertI2,axiom,
! [A: tm,B: set_tm,B2: tm] :
( ( member_tm @ A @ B )
=> ( member_tm @ A @ ( insert_tm2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_366_insertI2,axiom,
! [A: nat,B: set_nat,B2: nat] :
( ( member_nat @ A @ B )
=> ( member_nat @ A @ ( insert_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_367_insertI2,axiom,
! [A: list_fm,B: set_list_fm,B2: list_fm] :
( ( member_list_fm @ A @ B )
=> ( member_list_fm @ A @ ( insert_list_fm2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_368_Set_Oset__insert,axiom,
! [X2: $o,A2: set_o] :
( ( member_o @ X2 @ A2 )
=> ~ ! [B4: set_o] :
( ( A2
= ( insert_o @ X2 @ B4 ) )
=> ( member_o @ X2 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_369_Set_Oset__insert,axiom,
! [X2: tm,A2: set_tm] :
( ( member_tm @ X2 @ A2 )
=> ~ ! [B4: set_tm] :
( ( A2
= ( insert_tm2 @ X2 @ B4 ) )
=> ( member_tm @ X2 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_370_Set_Oset__insert,axiom,
! [X2: nat,A2: set_nat] :
( ( member_nat @ X2 @ A2 )
=> ~ ! [B4: set_nat] :
( ( A2
= ( insert_nat @ X2 @ B4 ) )
=> ( member_nat @ X2 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_371_Set_Oset__insert,axiom,
! [X2: list_fm,A2: set_list_fm] :
( ( member_list_fm @ X2 @ A2 )
=> ~ ! [B4: set_list_fm] :
( ( A2
= ( insert_list_fm2 @ X2 @ B4 ) )
=> ( member_list_fm @ X2 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_372_insert__ident,axiom,
! [X2: $o,A2: set_o,B: set_o] :
( ~ ( member_o @ X2 @ A2 )
=> ( ~ ( member_o @ X2 @ B )
=> ( ( ( insert_o @ X2 @ A2 )
= ( insert_o @ X2 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_373_insert__ident,axiom,
! [X2: tm,A2: set_tm,B: set_tm] :
( ~ ( member_tm @ X2 @ A2 )
=> ( ~ ( member_tm @ X2 @ B )
=> ( ( ( insert_tm2 @ X2 @ A2 )
= ( insert_tm2 @ X2 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_374_insert__ident,axiom,
! [X2: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ X2 @ A2 )
=> ( ~ ( member_nat @ X2 @ B )
=> ( ( ( insert_nat @ X2 @ A2 )
= ( insert_nat @ X2 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_375_insert__ident,axiom,
! [X2: list_fm,A2: set_list_fm,B: set_list_fm] :
( ~ ( member_list_fm @ X2 @ A2 )
=> ( ~ ( member_list_fm @ X2 @ B )
=> ( ( ( insert_list_fm2 @ X2 @ A2 )
= ( insert_list_fm2 @ X2 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_376_insert__absorb,axiom,
! [A: $o,A2: set_o] :
( ( member_o @ A @ A2 )
=> ( ( insert_o @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_377_insert__absorb,axiom,
! [A: tm,A2: set_tm] :
( ( member_tm @ A @ A2 )
=> ( ( insert_tm2 @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_378_insert__absorb,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_379_insert__absorb,axiom,
! [A: list_fm,A2: set_list_fm] :
( ( member_list_fm @ A @ A2 )
=> ( ( insert_list_fm2 @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_380_insert__eq__iff,axiom,
! [A: $o,A2: set_o,B2: $o,B: set_o] :
( ~ ( member_o @ A @ A2 )
=> ( ~ ( member_o @ B2 @ B )
=> ( ( ( insert_o @ A @ A2 )
= ( insert_o @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A = ~ B2 )
=> ? [C4: set_o] :
( ( A2
= ( insert_o @ B2 @ C4 ) )
& ~ ( member_o @ B2 @ C4 )
& ( B
= ( insert_o @ A @ C4 ) )
& ~ ( member_o @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_381_insert__eq__iff,axiom,
! [A: tm,A2: set_tm,B2: tm,B: set_tm] :
( ~ ( member_tm @ A @ A2 )
=> ( ~ ( member_tm @ B2 @ B )
=> ( ( ( insert_tm2 @ A @ A2 )
= ( insert_tm2 @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C4: set_tm] :
( ( A2
= ( insert_tm2 @ B2 @ C4 ) )
& ~ ( member_tm @ B2 @ C4 )
& ( B
= ( insert_tm2 @ A @ C4 ) )
& ~ ( member_tm @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_382_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_383_insert__eq__iff,axiom,
! [A: list_fm,A2: set_list_fm,B2: list_fm,B: set_list_fm] :
( ~ ( member_list_fm @ A @ A2 )
=> ( ~ ( member_list_fm @ B2 @ B )
=> ( ( ( insert_list_fm2 @ A @ A2 )
= ( insert_list_fm2 @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C4: set_list_fm] :
( ( A2
= ( insert_list_fm2 @ B2 @ C4 ) )
& ~ ( member_list_fm @ B2 @ C4 )
& ( B
= ( insert_list_fm2 @ A @ C4 ) )
& ~ ( member_list_fm @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_384_insert__commute,axiom,
! [X2: nat,Y4: nat,A2: set_nat] :
( ( insert_nat @ X2 @ ( insert_nat @ Y4 @ A2 ) )
= ( insert_nat @ Y4 @ ( insert_nat @ X2 @ A2 ) ) ) ).
% insert_commute
thf(fact_385_insert__commute,axiom,
! [X2: $o,Y4: $o,A2: set_o] :
( ( insert_o @ X2 @ ( insert_o @ Y4 @ A2 ) )
= ( insert_o @ Y4 @ ( insert_o @ X2 @ A2 ) ) ) ).
% insert_commute
thf(fact_386_insert__commute,axiom,
! [X2: tm,Y4: tm,A2: set_tm] :
( ( insert_tm2 @ X2 @ ( insert_tm2 @ Y4 @ A2 ) )
= ( insert_tm2 @ Y4 @ ( insert_tm2 @ X2 @ A2 ) ) ) ).
% insert_commute
thf(fact_387_mk__disjoint__insert,axiom,
! [A: $o,A2: set_o] :
( ( member_o @ A @ A2 )
=> ? [B4: set_o] :
( ( A2
= ( insert_o @ A @ B4 ) )
& ~ ( member_o @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_388_mk__disjoint__insert,axiom,
! [A: tm,A2: set_tm] :
( ( member_tm @ A @ A2 )
=> ? [B4: set_tm] :
( ( A2
= ( insert_tm2 @ A @ B4 ) )
& ~ ( member_tm @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_389_mk__disjoint__insert,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ? [B4: set_nat] :
( ( A2
= ( insert_nat @ A @ B4 ) )
& ~ ( member_nat @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_390_mk__disjoint__insert,axiom,
! [A: list_fm,A2: set_list_fm] :
( ( member_list_fm @ A @ A2 )
=> ? [B4: set_list_fm] :
( ( A2
= ( insert_list_fm2 @ A @ B4 ) )
& ~ ( member_list_fm @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_391_insert__compr,axiom,
( insert_o
= ( ^ [A5: $o,B3: set_o] :
( collect_o
@ ^ [X5: $o] :
( ( X5 = A5 )
| ( member_o @ X5 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_392_insert__compr,axiom,
( insert_tm2
= ( ^ [A5: tm,B3: set_tm] :
( collect_tm
@ ^ [X5: tm] :
( ( X5 = A5 )
| ( member_tm @ X5 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_393_insert__compr,axiom,
( insert_list_fm2
= ( ^ [A5: list_fm,B3: set_list_fm] :
( collect_list_fm
@ ^ [X5: list_fm] :
( ( X5 = A5 )
| ( member_list_fm @ X5 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_394_insert__compr,axiom,
( insert_nat
= ( ^ [A5: nat,B3: set_nat] :
( collect_nat
@ ^ [X5: nat] :
( ( X5 = A5 )
| ( member_nat @ X5 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_395_insert__Collect,axiom,
! [A: $o,P: $o > $o] :
( ( insert_o @ A @ ( collect_o @ P ) )
= ( collect_o
@ ^ [U2: $o] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_396_insert__Collect,axiom,
! [A: tm,P: tm > $o] :
( ( insert_tm2 @ A @ ( collect_tm @ P ) )
= ( collect_tm
@ ^ [U2: tm] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_397_insert__Collect,axiom,
! [A: list_fm,P: list_fm > $o] :
( ( insert_list_fm2 @ A @ ( collect_list_fm @ P ) )
= ( collect_list_fm
@ ^ [U2: list_fm] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_398_insert__Collect,axiom,
! [A: nat,P: nat > $o] :
( ( insert_nat @ A @ ( collect_nat @ P ) )
= ( collect_nat
@ ^ [U2: nat] :
( ( U2 != A )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_399_insert__UNIV,axiom,
! [X2: tm] :
( ( insert_tm2 @ X2 @ top_top_set_tm )
= top_top_set_tm ) ).
% insert_UNIV
thf(fact_400_insert__UNIV,axiom,
! [X2: nat] :
( ( insert_nat @ X2 @ top_top_set_nat )
= top_top_set_nat ) ).
% insert_UNIV
thf(fact_401_insert__UNIV,axiom,
! [X2: $o] :
( ( insert_o @ X2 @ top_top_set_o )
= top_top_set_o ) ).
% insert_UNIV
thf(fact_402_singleton__inject,axiom,
! [A: tm,B2: tm] :
( ( ( insert_tm2 @ A @ bot_bot_set_tm )
= ( insert_tm2 @ B2 @ bot_bot_set_tm ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_403_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_404_singleton__inject,axiom,
! [A: $o,B2: $o] :
( ( ( insert_o @ A @ bot_bot_set_o )
= ( insert_o @ B2 @ bot_bot_set_o ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_405_insert__not__empty,axiom,
! [A: tm,A2: set_tm] :
( ( insert_tm2 @ A @ A2 )
!= bot_bot_set_tm ) ).
% insert_not_empty
thf(fact_406_insert__not__empty,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ A2 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_407_insert__not__empty,axiom,
! [A: $o,A2: set_o] :
( ( insert_o @ A @ A2 )
!= bot_bot_set_o ) ).
% insert_not_empty
thf(fact_408_doubleton__eq__iff,axiom,
! [A: tm,B2: tm,C: tm,D: tm] :
( ( ( insert_tm2 @ A @ ( insert_tm2 @ B2 @ bot_bot_set_tm ) )
= ( insert_tm2 @ C @ ( insert_tm2 @ D @ bot_bot_set_tm ) ) )
= ( ( ( A = C )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_409_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_410_doubleton__eq__iff,axiom,
! [A: $o,B2: $o,C: $o,D: $o] :
( ( ( insert_o @ A @ ( insert_o @ B2 @ bot_bot_set_o ) )
= ( insert_o @ C @ ( insert_o @ D @ bot_bot_set_o ) ) )
= ( ( ( A = C )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_411_singleton__iff,axiom,
! [B2: list_fm,A: list_fm] :
( ( member_list_fm @ B2 @ ( insert_list_fm2 @ A @ bot_bot_set_list_fm ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_412_singleton__iff,axiom,
! [B2: tm,A: tm] :
( ( member_tm @ B2 @ ( insert_tm2 @ A @ bot_bot_set_tm ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_413_singleton__iff,axiom,
! [B2: nat,A: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_414_singleton__iff,axiom,
! [B2: $o,A: $o] :
( ( member_o @ B2 @ ( insert_o @ A @ bot_bot_set_o ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_415_singletonD,axiom,
! [B2: list_fm,A: list_fm] :
( ( member_list_fm @ B2 @ ( insert_list_fm2 @ A @ bot_bot_set_list_fm ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_416_singletonD,axiom,
! [B2: tm,A: tm] :
( ( member_tm @ B2 @ ( insert_tm2 @ A @ bot_bot_set_tm ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_417_singletonD,axiom,
! [B2: nat,A: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_418_singletonD,axiom,
! [B2: $o,A: $o] :
( ( member_o @ B2 @ ( insert_o @ A @ bot_bot_set_o ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_419_Collect__conv__if2,axiom,
! [P: list_fm > $o,A: list_fm] :
( ( ( P @ A )
=> ( ( collect_list_fm
@ ^ [X5: list_fm] :
( ( A = X5 )
& ( P @ X5 ) ) )
= ( insert_list_fm2 @ A @ bot_bot_set_list_fm ) ) )
& ( ~ ( P @ A )
=> ( ( collect_list_fm
@ ^ [X5: list_fm] :
( ( A = X5 )
& ( P @ X5 ) ) )
= bot_bot_set_list_fm ) ) ) ).
% Collect_conv_if2
thf(fact_420_Collect__conv__if2,axiom,
! [P: tm > $o,A: tm] :
( ( ( P @ A )
=> ( ( collect_tm
@ ^ [X5: tm] :
( ( A = X5 )
& ( P @ X5 ) ) )
= ( insert_tm2 @ A @ bot_bot_set_tm ) ) )
& ( ~ ( P @ A )
=> ( ( collect_tm
@ ^ [X5: tm] :
( ( A = X5 )
& ( P @ X5 ) ) )
= bot_bot_set_tm ) ) ) ).
% Collect_conv_if2
thf(fact_421_Collect__conv__if2,axiom,
! [P: nat > $o,A: nat] :
( ( ( P @ A )
=> ( ( collect_nat
@ ^ [X5: nat] :
( ( A = X5 )
& ( P @ X5 ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_nat
@ ^ [X5: nat] :
( ( A = X5 )
& ( P @ X5 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_422_Collect__conv__if2,axiom,
! [P: $o > $o,A: $o] :
( ( ( P @ A )
=> ( ( collect_o
@ ^ [X5: $o] :
( ( A = X5 )
& ( P @ X5 ) ) )
= ( insert_o @ A @ bot_bot_set_o ) ) )
& ( ~ ( P @ A )
=> ( ( collect_o
@ ^ [X5: $o] :
( ( A = X5 )
& ( P @ X5 ) ) )
= bot_bot_set_o ) ) ) ).
% Collect_conv_if2
thf(fact_423_Collect__conv__if,axiom,
! [P: list_fm > $o,A: list_fm] :
( ( ( P @ A )
=> ( ( collect_list_fm
@ ^ [X5: list_fm] :
( ( X5 = A )
& ( P @ X5 ) ) )
= ( insert_list_fm2 @ A @ bot_bot_set_list_fm ) ) )
& ( ~ ( P @ A )
=> ( ( collect_list_fm
@ ^ [X5: list_fm] :
( ( X5 = A )
& ( P @ X5 ) ) )
= bot_bot_set_list_fm ) ) ) ).
% Collect_conv_if
thf(fact_424_Collect__conv__if,axiom,
! [P: tm > $o,A: tm] :
( ( ( P @ A )
=> ( ( collect_tm
@ ^ [X5: tm] :
( ( X5 = A )
& ( P @ X5 ) ) )
= ( insert_tm2 @ A @ bot_bot_set_tm ) ) )
& ( ~ ( P @ A )
=> ( ( collect_tm
@ ^ [X5: tm] :
( ( X5 = A )
& ( P @ X5 ) ) )
= bot_bot_set_tm ) ) ) ).
% Collect_conv_if
thf(fact_425_Collect__conv__if,axiom,
! [P: nat > $o,A: nat] :
( ( ( P @ A )
=> ( ( collect_nat
@ ^ [X5: nat] :
( ( X5 = A )
& ( P @ X5 ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_nat
@ ^ [X5: nat] :
( ( X5 = A )
& ( P @ X5 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_426_Collect__conv__if,axiom,
! [P: $o > $o,A: $o] :
( ( ( P @ A )
=> ( ( collect_o
@ ^ [X5: $o] :
( ( X5 = A )
& ( P @ X5 ) ) )
= ( insert_o @ A @ bot_bot_set_o ) ) )
& ( ~ ( P @ A )
=> ( ( collect_o
@ ^ [X5: $o] :
( ( X5 = A )
& ( P @ X5 ) ) )
= bot_bot_set_o ) ) ) ).
% Collect_conv_if
thf(fact_427_insert__def,axiom,
( insert_o
= ( ^ [A5: $o] :
( sup_sup_set_o
@ ( collect_o
@ ^ [X5: $o] : ( X5 = A5 ) ) ) ) ) ).
% insert_def
thf(fact_428_insert__def,axiom,
( insert_tm2
= ( ^ [A5: tm] :
( sup_sup_set_tm
@ ( collect_tm
@ ^ [X5: tm] : ( X5 = A5 ) ) ) ) ) ).
% insert_def
thf(fact_429_insert__def,axiom,
( insert_list_fm2
= ( ^ [A5: list_fm] :
( sup_sup_set_list_fm
@ ( collect_list_fm
@ ^ [X5: list_fm] : ( X5 = A5 ) ) ) ) ) ).
% insert_def
thf(fact_430_insert__def,axiom,
( insert_nat
= ( ^ [A5: nat] :
( sup_sup_set_nat
@ ( collect_nat
@ ^ [X5: nat] : ( X5 = A5 ) ) ) ) ) ).
% insert_def
thf(fact_431_singleton__Un__iff,axiom,
! [X2: tm,A2: set_tm,B: set_tm] :
( ( ( insert_tm2 @ X2 @ bot_bot_set_tm )
= ( sup_sup_set_tm @ A2 @ B ) )
= ( ( ( A2 = bot_bot_set_tm )
& ( B
= ( insert_tm2 @ X2 @ bot_bot_set_tm ) ) )
| ( ( A2
= ( insert_tm2 @ X2 @ bot_bot_set_tm ) )
& ( B = bot_bot_set_tm ) )
| ( ( A2
= ( insert_tm2 @ X2 @ bot_bot_set_tm ) )
& ( B
= ( insert_tm2 @ X2 @ bot_bot_set_tm ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_432_singleton__Un__iff,axiom,
! [X2: nat,A2: set_nat,B: set_nat] :
( ( ( insert_nat @ X2 @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A2 @ B ) )
= ( ( ( A2 = bot_bot_set_nat )
& ( B
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
| ( ( A2
= ( insert_nat @ X2 @ bot_bot_set_nat ) )
& ( B = bot_bot_set_nat ) )
| ( ( A2
= ( insert_nat @ X2 @ bot_bot_set_nat ) )
& ( B
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_433_singleton__Un__iff,axiom,
! [X2: $o,A2: set_o,B: set_o] :
( ( ( insert_o @ X2 @ bot_bot_set_o )
= ( sup_sup_set_o @ A2 @ B ) )
= ( ( ( A2 = bot_bot_set_o )
& ( B
= ( insert_o @ X2 @ bot_bot_set_o ) ) )
| ( ( A2
= ( insert_o @ X2 @ bot_bot_set_o ) )
& ( B = bot_bot_set_o ) )
| ( ( A2
= ( insert_o @ X2 @ bot_bot_set_o ) )
& ( B
= ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_434_Un__singleton__iff,axiom,
! [A2: set_tm,B: set_tm,X2: tm] :
( ( ( sup_sup_set_tm @ A2 @ B )
= ( insert_tm2 @ X2 @ bot_bot_set_tm ) )
= ( ( ( A2 = bot_bot_set_tm )
& ( B
= ( insert_tm2 @ X2 @ bot_bot_set_tm ) ) )
| ( ( A2
= ( insert_tm2 @ X2 @ bot_bot_set_tm ) )
& ( B = bot_bot_set_tm ) )
| ( ( A2
= ( insert_tm2 @ X2 @ bot_bot_set_tm ) )
& ( B
= ( insert_tm2 @ X2 @ bot_bot_set_tm ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_435_Un__singleton__iff,axiom,
! [A2: set_nat,B: set_nat,X2: nat] :
( ( ( sup_sup_set_nat @ A2 @ B )
= ( insert_nat @ X2 @ bot_bot_set_nat ) )
= ( ( ( A2 = bot_bot_set_nat )
& ( B
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
| ( ( A2
= ( insert_nat @ X2 @ bot_bot_set_nat ) )
& ( B = bot_bot_set_nat ) )
| ( ( A2
= ( insert_nat @ X2 @ bot_bot_set_nat ) )
& ( B
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_436_Un__singleton__iff,axiom,
! [A2: set_o,B: set_o,X2: $o] :
( ( ( sup_sup_set_o @ A2 @ B )
= ( insert_o @ X2 @ bot_bot_set_o ) )
= ( ( ( A2 = bot_bot_set_o )
& ( B
= ( insert_o @ X2 @ bot_bot_set_o ) ) )
| ( ( A2
= ( insert_o @ X2 @ bot_bot_set_o ) )
& ( B = bot_bot_set_o ) )
| ( ( A2
= ( insert_o @ X2 @ bot_bot_set_o ) )
& ( B
= ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_437_insert__is__Un,axiom,
( insert_tm2
= ( ^ [A5: tm] : ( sup_sup_set_tm @ ( insert_tm2 @ A5 @ bot_bot_set_tm ) ) ) ) ).
% insert_is_Un
thf(fact_438_insert__is__Un,axiom,
( insert_nat
= ( ^ [A5: nat] : ( sup_sup_set_nat @ ( insert_nat @ A5 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_439_insert__is__Un,axiom,
( insert_o
= ( ^ [A5: $o] : ( sup_sup_set_o @ ( insert_o @ A5 @ bot_bot_set_o ) ) ) ) ).
% insert_is_Un
thf(fact_440_is__singletonE,axiom,
! [A2: set_tm] :
( ( is_singleton_tm @ A2 )
=> ~ ! [X: tm] :
( A2
!= ( insert_tm2 @ X @ bot_bot_set_tm ) ) ) ).
% is_singletonE
thf(fact_441_is__singletonE,axiom,
! [A2: set_nat] :
( ( is_singleton_nat @ A2 )
=> ~ ! [X: nat] :
( A2
!= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_442_is__singletonE,axiom,
! [A2: set_o] :
( ( is_singleton_o @ A2 )
=> ~ ! [X: $o] :
( A2
!= ( insert_o @ X @ bot_bot_set_o ) ) ) ).
% is_singletonE
thf(fact_443_is__singleton__def,axiom,
( is_singleton_tm
= ( ^ [A4: set_tm] :
? [X5: tm] :
( A4
= ( insert_tm2 @ X5 @ bot_bot_set_tm ) ) ) ) ).
% is_singleton_def
thf(fact_444_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A4: set_nat] :
? [X5: nat] :
( A4
= ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_445_is__singleton__def,axiom,
( is_singleton_o
= ( ^ [A4: set_o] :
? [X5: $o] :
( A4
= ( insert_o @ X5 @ bot_bot_set_o ) ) ) ) ).
% is_singleton_def
thf(fact_446_is__singleton__the__elem,axiom,
( is_singleton_tm
= ( ^ [A4: set_tm] :
( A4
= ( insert_tm2 @ ( the_elem_tm @ A4 ) @ bot_bot_set_tm ) ) ) ) ).
% is_singleton_the_elem
thf(fact_447_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A4: set_nat] :
( A4
= ( insert_nat @ ( the_elem_nat @ A4 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_448_is__singleton__the__elem,axiom,
( is_singleton_o
= ( ^ [A4: set_o] :
( A4
= ( insert_o @ ( the_elem_o @ A4 ) @ bot_bot_set_o ) ) ) ) ).
% is_singleton_the_elem
thf(fact_449_sublists_Osimps_I1_J,axiom,
( ( sublists_tm @ nil_tm )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) ).
% sublists.simps(1)
thf(fact_450_sublists_Osimps_I1_J,axiom,
( ( sublists_list_fm @ nil_list_fm )
= ( cons_list_list_fm @ nil_list_fm @ nil_list_list_fm ) ) ).
% sublists.simps(1)
thf(fact_451_sublists_Osimps_I1_J,axiom,
( ( sublists_fm @ nil_fm )
= ( cons_list_fm @ nil_fm @ nil_list_fm ) ) ).
% sublists.simps(1)
thf(fact_452_the__elem__eq,axiom,
! [X2: tm] :
( ( the_elem_tm @ ( insert_tm2 @ X2 @ bot_bot_set_tm ) )
= X2 ) ).
% the_elem_eq
thf(fact_453_the__elem__eq,axiom,
! [X2: nat] :
( ( the_elem_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= X2 ) ).
% the_elem_eq
thf(fact_454_the__elem__eq,axiom,
! [X2: $o] :
( ( the_elem_o @ ( insert_o @ X2 @ bot_bot_set_o ) )
= X2 ) ).
% the_elem_eq
thf(fact_455_subseqs_Osimps_I1_J,axiom,
( ( subseqs_tm @ nil_tm )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) ).
% subseqs.simps(1)
thf(fact_456_subseqs_Osimps_I1_J,axiom,
( ( subseqs_list_fm @ nil_list_fm )
= ( cons_list_list_fm @ nil_list_fm @ nil_list_list_fm ) ) ).
% subseqs.simps(1)
thf(fact_457_subseqs_Osimps_I1_J,axiom,
( ( subseqs_fm @ nil_fm )
= ( cons_list_fm @ nil_fm @ nil_list_fm ) ) ).
% subseqs.simps(1)
thf(fact_458_suffixes_Osimps_I1_J,axiom,
( ( suffixes_tm @ nil_tm )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) ).
% suffixes.simps(1)
thf(fact_459_suffixes_Osimps_I1_J,axiom,
( ( suffixes_list_fm @ nil_list_fm )
= ( cons_list_list_fm @ nil_list_fm @ nil_list_list_fm ) ) ).
% suffixes.simps(1)
thf(fact_460_suffixes_Osimps_I1_J,axiom,
( ( suffixes_fm @ nil_fm )
= ( cons_list_fm @ nil_fm @ nil_list_fm ) ) ).
% suffixes.simps(1)
thf(fact_461_paramst__sub__term_I2_J,axiom,
! [M: nat,S3: tm,L2: list_tm] : ( ord_less_eq_set_nat @ ( paramsts @ ( sub_list @ M @ S3 @ L2 ) ) @ ( sup_sup_set_nat @ ( paramst @ S3 ) @ ( paramsts @ L2 ) ) ) ).
% paramst_sub_term(2)
thf(fact_462_range__constant,axiom,
! [X2: set_nat] :
( ( image_tm_set_nat
@ ^ [Uu: tm] : X2
@ top_top_set_tm )
= ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ).
% range_constant
thf(fact_463_range__constant,axiom,
! [X2: set_tm] :
( ( image_tm_set_tm
@ ^ [Uu: tm] : X2
@ top_top_set_tm )
= ( insert_set_tm @ X2 @ bot_bot_set_set_tm ) ) ).
% range_constant
thf(fact_464_range__constant,axiom,
! [X2: set_tm] :
( ( image_fm_set_tm
@ ^ [Uu: fm] : X2
@ top_top_set_fm )
= ( insert_set_tm @ X2 @ bot_bot_set_set_tm ) ) ).
% range_constant
thf(fact_465_range__constant,axiom,
! [X2: set_nat] :
( ( image_fm_set_nat
@ ^ [Uu: fm] : X2
@ top_top_set_fm )
= ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ).
% range_constant
thf(fact_466_range__constant,axiom,
! [X2: tm] :
( ( image_nat_tm
@ ^ [Uu: nat] : X2
@ top_top_set_nat )
= ( insert_tm2 @ X2 @ bot_bot_set_tm ) ) ).
% range_constant
thf(fact_467_range__constant,axiom,
! [X2: nat] :
( ( image_nat_nat
@ ^ [Uu: nat] : X2
@ top_top_set_nat )
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ).
% range_constant
thf(fact_468_range__constant,axiom,
! [X2: $o] :
( ( image_nat_o
@ ^ [Uu: nat] : X2
@ top_top_set_nat )
= ( insert_o @ X2 @ bot_bot_set_o ) ) ).
% range_constant
thf(fact_469_range__constant,axiom,
! [X2: tm] :
( ( image_o_tm
@ ^ [Uu: $o] : X2
@ top_top_set_o )
= ( insert_tm2 @ X2 @ bot_bot_set_tm ) ) ).
% range_constant
thf(fact_470_range__constant,axiom,
! [X2: nat] :
( ( image_o_nat
@ ^ [Uu: $o] : X2
@ top_top_set_o )
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ).
% range_constant
thf(fact_471_range__constant,axiom,
! [X2: $o] :
( ( image_o_o
@ ^ [Uu: $o] : X2
@ top_top_set_o )
= ( insert_o @ X2 @ bot_bot_set_o ) ) ).
% range_constant
thf(fact_472_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_473_dual__order_Orefl,axiom,
! [A: set_tm] : ( ord_less_eq_set_tm @ A @ A ) ).
% dual_order.refl
thf(fact_474_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_475_order__refl,axiom,
! [X2: set_nat] : ( ord_less_eq_set_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_476_order__refl,axiom,
! [X2: set_tm] : ( ord_less_eq_set_tm @ X2 @ X2 ) ).
% order_refl
thf(fact_477_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_478_image__eqI,axiom,
! [B2: tm,F: tm > tm,X2: tm,A2: set_tm] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_tm @ X2 @ A2 )
=> ( member_tm @ B2 @ ( image_tm_tm @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_479_image__eqI,axiom,
! [B2: nat,F: tm > nat,X2: tm,A2: set_tm] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_tm @ X2 @ A2 )
=> ( member_nat @ B2 @ ( image_tm_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_480_image__eqI,axiom,
! [B2: tm,F: nat > tm,X2: nat,A2: set_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_tm @ B2 @ ( image_nat_tm @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_481_image__eqI,axiom,
! [B2: nat,F: nat > nat,X2: nat,A2: set_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_482_image__eqI,axiom,
! [B2: set_tm,F: fm > set_tm,X2: fm,A2: set_fm] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_fm @ X2 @ A2 )
=> ( member_set_tm @ B2 @ ( image_fm_set_tm @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_483_image__eqI,axiom,
! [B2: set_nat,F: fm > set_nat,X2: fm,A2: set_fm] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_fm @ X2 @ A2 )
=> ( member_set_nat @ B2 @ ( image_fm_set_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_484_image__eqI,axiom,
! [B2: set_nat,F: tm > set_nat,X2: tm,A2: set_tm] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_tm @ X2 @ A2 )
=> ( member_set_nat @ B2 @ ( image_tm_set_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_485_image__eqI,axiom,
! [B2: set_tm,F: tm > set_tm,X2: tm,A2: set_tm] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_tm @ X2 @ A2 )
=> ( member_set_tm @ B2 @ ( image_tm_set_tm @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_486_image__eqI,axiom,
! [B2: list_fm,F: tm > list_fm,X2: tm,A2: set_tm] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_tm @ X2 @ A2 )
=> ( member_list_fm @ B2 @ ( image_tm_list_fm @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_487_image__eqI,axiom,
! [B2: list_fm,F: nat > list_fm,X2: nat,A2: set_nat] :
( ( B2
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_list_fm @ B2 @ ( image_nat_list_fm @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_488_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_489_subset__antisym,axiom,
! [A2: set_tm,B: set_tm] :
( ( ord_less_eq_set_tm @ A2 @ B )
=> ( ( ord_less_eq_set_tm @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_490_subsetI,axiom,
! [A2: set_list_fm,B: set_list_fm] :
( ! [X: list_fm] :
( ( member_list_fm @ X @ A2 )
=> ( member_list_fm @ X @ B ) )
=> ( ord_le7838213414353715577ist_fm @ A2 @ B ) ) ).
% subsetI
thf(fact_491_subsetI,axiom,
! [A2: set_nat,B: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B ) )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_492_subsetI,axiom,
! [A2: set_tm,B: set_tm] :
( ! [X: tm] :
( ( member_tm @ X @ A2 )
=> ( member_tm @ X @ B ) )
=> ( ord_less_eq_set_tm @ A2 @ B ) ) ).
% subsetI
thf(fact_493_image__ident,axiom,
! [Y5: set_nat] :
( ( image_nat_nat
@ ^ [X5: nat] : X5
@ Y5 )
= Y5 ) ).
% image_ident
thf(fact_494_image__empty,axiom,
! [F: tm > tm] :
( ( image_tm_tm @ F @ bot_bot_set_tm )
= bot_bot_set_tm ) ).
% image_empty
thf(fact_495_image__empty,axiom,
! [F: tm > nat] :
( ( image_tm_nat @ F @ bot_bot_set_tm )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_496_image__empty,axiom,
! [F: tm > $o] :
( ( image_tm_o @ F @ bot_bot_set_tm )
= bot_bot_set_o ) ).
% image_empty
thf(fact_497_image__empty,axiom,
! [F: nat > tm] :
( ( image_nat_tm @ F @ bot_bot_set_nat )
= bot_bot_set_tm ) ).
% image_empty
thf(fact_498_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_499_image__empty,axiom,
! [F: nat > $o] :
( ( image_nat_o @ F @ bot_bot_set_nat )
= bot_bot_set_o ) ).
% image_empty
thf(fact_500_image__empty,axiom,
! [F: $o > tm] :
( ( image_o_tm @ F @ bot_bot_set_o )
= bot_bot_set_tm ) ).
% image_empty
thf(fact_501_image__empty,axiom,
! [F: $o > nat] :
( ( image_o_nat @ F @ bot_bot_set_o )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_502_image__empty,axiom,
! [F: $o > $o] :
( ( image_o_o @ F @ bot_bot_set_o )
= bot_bot_set_o ) ).
% image_empty
thf(fact_503_image__empty,axiom,
! [F: fm > set_tm] :
( ( image_fm_set_tm @ F @ bot_bot_set_fm )
= bot_bot_set_set_tm ) ).
% image_empty
thf(fact_504_empty__is__image,axiom,
! [F: tm > tm,A2: set_tm] :
( ( bot_bot_set_tm
= ( image_tm_tm @ F @ A2 ) )
= ( A2 = bot_bot_set_tm ) ) ).
% empty_is_image
thf(fact_505_empty__is__image,axiom,
! [F: nat > tm,A2: set_nat] :
( ( bot_bot_set_tm
= ( image_nat_tm @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_506_empty__is__image,axiom,
! [F: $o > tm,A2: set_o] :
( ( bot_bot_set_tm
= ( image_o_tm @ F @ A2 ) )
= ( A2 = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_507_empty__is__image,axiom,
! [F: tm > nat,A2: set_tm] :
( ( bot_bot_set_nat
= ( image_tm_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_tm ) ) ).
% empty_is_image
thf(fact_508_empty__is__image,axiom,
! [F: nat > nat,A2: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_509_empty__is__image,axiom,
! [F: $o > nat,A2: set_o] :
( ( bot_bot_set_nat
= ( image_o_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_510_empty__is__image,axiom,
! [F: tm > $o,A2: set_tm] :
( ( bot_bot_set_o
= ( image_tm_o @ F @ A2 ) )
= ( A2 = bot_bot_set_tm ) ) ).
% empty_is_image
thf(fact_511_empty__is__image,axiom,
! [F: nat > $o,A2: set_nat] :
( ( bot_bot_set_o
= ( image_nat_o @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_512_empty__is__image,axiom,
! [F: $o > $o,A2: set_o] :
( ( bot_bot_set_o
= ( image_o_o @ F @ A2 ) )
= ( A2 = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_513_empty__is__image,axiom,
! [F: fm > set_tm,A2: set_fm] :
( ( bot_bot_set_set_tm
= ( image_fm_set_tm @ F @ A2 ) )
= ( A2 = bot_bot_set_fm ) ) ).
% empty_is_image
thf(fact_514_image__is__empty,axiom,
! [F: tm > tm,A2: set_tm] :
( ( ( image_tm_tm @ F @ A2 )
= bot_bot_set_tm )
= ( A2 = bot_bot_set_tm ) ) ).
% image_is_empty
thf(fact_515_image__is__empty,axiom,
! [F: nat > tm,A2: set_nat] :
( ( ( image_nat_tm @ F @ A2 )
= bot_bot_set_tm )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_516_image__is__empty,axiom,
! [F: $o > tm,A2: set_o] :
( ( ( image_o_tm @ F @ A2 )
= bot_bot_set_tm )
= ( A2 = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_517_image__is__empty,axiom,
! [F: tm > nat,A2: set_tm] :
( ( ( image_tm_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_tm ) ) ).
% image_is_empty
thf(fact_518_image__is__empty,axiom,
! [F: nat > nat,A2: set_nat] :
( ( ( image_nat_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_519_image__is__empty,axiom,
! [F: $o > nat,A2: set_o] :
( ( ( image_o_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_520_image__is__empty,axiom,
! [F: tm > $o,A2: set_tm] :
( ( ( image_tm_o @ F @ A2 )
= bot_bot_set_o )
= ( A2 = bot_bot_set_tm ) ) ).
% image_is_empty
thf(fact_521_image__is__empty,axiom,
! [F: nat > $o,A2: set_nat] :
( ( ( image_nat_o @ F @ A2 )
= bot_bot_set_o )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_522_image__is__empty,axiom,
! [F: $o > $o,A2: set_o] :
( ( ( image_o_o @ F @ A2 )
= bot_bot_set_o )
= ( A2 = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_523_image__is__empty,axiom,
! [F: fm > set_tm,A2: set_fm] :
( ( ( image_fm_set_tm @ F @ A2 )
= bot_bot_set_set_tm )
= ( A2 = bot_bot_set_fm ) ) ).
% image_is_empty
thf(fact_524_subset__empty,axiom,
! [A2: set_o] :
( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
= ( A2 = bot_bot_set_o ) ) ).
% subset_empty
thf(fact_525_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_526_subset__empty,axiom,
! [A2: set_tm] :
( ( ord_less_eq_set_tm @ A2 @ bot_bot_set_tm )
= ( A2 = bot_bot_set_tm ) ) ).
% subset_empty
thf(fact_527_empty__subsetI,axiom,
! [A2: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A2 ) ).
% empty_subsetI
thf(fact_528_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_529_empty__subsetI,axiom,
! [A2: set_tm] : ( ord_less_eq_set_tm @ bot_bot_set_tm @ A2 ) ).
% empty_subsetI
thf(fact_530_insert__image,axiom,
! [X2: tm,A2: set_tm,F: tm > nat] :
( ( member_tm @ X2 @ A2 )
=> ( ( insert_nat @ ( F @ X2 ) @ ( image_tm_nat @ F @ A2 ) )
= ( image_tm_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_531_insert__image,axiom,
! [X2: tm,A2: set_tm,F: tm > $o] :
( ( member_tm @ X2 @ A2 )
=> ( ( insert_o @ ( F @ X2 ) @ ( image_tm_o @ F @ A2 ) )
= ( image_tm_o @ F @ A2 ) ) ) ).
% insert_image
thf(fact_532_insert__image,axiom,
! [X2: tm,A2: set_tm,F: tm > tm] :
( ( member_tm @ X2 @ A2 )
=> ( ( insert_tm2 @ ( F @ X2 ) @ ( image_tm_tm @ F @ A2 ) )
= ( image_tm_tm @ F @ A2 ) ) ) ).
% insert_image
thf(fact_533_insert__image,axiom,
! [X2: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( insert_nat @ ( F @ X2 ) @ ( image_nat_nat @ F @ A2 ) )
= ( image_nat_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_534_insert__image,axiom,
! [X2: nat,A2: set_nat,F: nat > $o] :
( ( member_nat @ X2 @ A2 )
=> ( ( insert_o @ ( F @ X2 ) @ ( image_nat_o @ F @ A2 ) )
= ( image_nat_o @ F @ A2 ) ) ) ).
% insert_image
thf(fact_535_insert__image,axiom,
! [X2: nat,A2: set_nat,F: nat > tm] :
( ( member_nat @ X2 @ A2 )
=> ( ( insert_tm2 @ ( F @ X2 ) @ ( image_nat_tm @ F @ A2 ) )
= ( image_nat_tm @ F @ A2 ) ) ) ).
% insert_image
thf(fact_536_insert__image,axiom,
! [X2: fm,A2: set_fm,F: fm > set_tm] :
( ( member_fm @ X2 @ A2 )
=> ( ( insert_set_tm @ ( F @ X2 ) @ ( image_fm_set_tm @ F @ A2 ) )
= ( image_fm_set_tm @ F @ A2 ) ) ) ).
% insert_image
thf(fact_537_insert__image,axiom,
! [X2: fm,A2: set_fm,F: fm > set_nat] :
( ( member_fm @ X2 @ A2 )
=> ( ( insert_set_nat @ ( F @ X2 ) @ ( image_fm_set_nat @ F @ A2 ) )
= ( image_fm_set_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_538_insert__image,axiom,
! [X2: tm,A2: set_tm,F: tm > set_nat] :
( ( member_tm @ X2 @ A2 )
=> ( ( insert_set_nat @ ( F @ X2 ) @ ( image_tm_set_nat @ F @ A2 ) )
= ( image_tm_set_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_539_insert__image,axiom,
! [X2: tm,A2: set_tm,F: tm > set_tm] :
( ( member_tm @ X2 @ A2 )
=> ( ( insert_set_tm @ ( F @ X2 ) @ ( image_tm_set_tm @ F @ A2 ) )
= ( image_tm_set_tm @ F @ A2 ) ) ) ).
% insert_image
thf(fact_540_image__insert,axiom,
! [F: nat > nat,A: nat,B: set_nat] :
( ( image_nat_nat @ F @ ( insert_nat @ A @ B ) )
= ( insert_nat @ ( F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_541_image__insert,axiom,
! [F: nat > $o,A: nat,B: set_nat] :
( ( image_nat_o @ F @ ( insert_nat @ A @ B ) )
= ( insert_o @ ( F @ A ) @ ( image_nat_o @ F @ B ) ) ) ).
% image_insert
thf(fact_542_image__insert,axiom,
! [F: nat > tm,A: nat,B: set_nat] :
( ( image_nat_tm @ F @ ( insert_nat @ A @ B ) )
= ( insert_tm2 @ ( F @ A ) @ ( image_nat_tm @ F @ B ) ) ) ).
% image_insert
thf(fact_543_image__insert,axiom,
! [F: $o > nat,A: $o,B: set_o] :
( ( image_o_nat @ F @ ( insert_o @ A @ B ) )
= ( insert_nat @ ( F @ A ) @ ( image_o_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_544_image__insert,axiom,
! [F: $o > $o,A: $o,B: set_o] :
( ( image_o_o @ F @ ( insert_o @ A @ B ) )
= ( insert_o @ ( F @ A ) @ ( image_o_o @ F @ B ) ) ) ).
% image_insert
thf(fact_545_image__insert,axiom,
! [F: $o > tm,A: $o,B: set_o] :
( ( image_o_tm @ F @ ( insert_o @ A @ B ) )
= ( insert_tm2 @ ( F @ A ) @ ( image_o_tm @ F @ B ) ) ) ).
% image_insert
thf(fact_546_image__insert,axiom,
! [F: tm > nat,A: tm,B: set_tm] :
( ( image_tm_nat @ F @ ( insert_tm2 @ A @ B ) )
= ( insert_nat @ ( F @ A ) @ ( image_tm_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_547_image__insert,axiom,
! [F: tm > $o,A: tm,B: set_tm] :
( ( image_tm_o @ F @ ( insert_tm2 @ A @ B ) )
= ( insert_o @ ( F @ A ) @ ( image_tm_o @ F @ B ) ) ) ).
% image_insert
thf(fact_548_image__insert,axiom,
! [F: tm > tm,A: tm,B: set_tm] :
( ( image_tm_tm @ F @ ( insert_tm2 @ A @ B ) )
= ( insert_tm2 @ ( F @ A ) @ ( image_tm_tm @ F @ B ) ) ) ).
% image_insert
thf(fact_549_image__insert,axiom,
! [F: fm > set_tm,A: fm,B: set_fm] :
( ( image_fm_set_tm @ F @ ( insert_fm2 @ A @ B ) )
= ( insert_set_tm @ ( F @ A ) @ ( image_fm_set_tm @ F @ B ) ) ) ).
% image_insert
thf(fact_550_insert__subset,axiom,
! [X2: $o,A2: set_o,B: set_o] :
( ( ord_less_eq_set_o @ ( insert_o @ X2 @ A2 ) @ B )
= ( ( member_o @ X2 @ B )
& ( ord_less_eq_set_o @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_551_insert__subset,axiom,
! [X2: list_fm,A2: set_list_fm,B: set_list_fm] :
( ( ord_le7838213414353715577ist_fm @ ( insert_list_fm2 @ X2 @ A2 ) @ B )
= ( ( member_list_fm @ X2 @ B )
& ( ord_le7838213414353715577ist_fm @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_552_insert__subset,axiom,
! [X2: nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ A2 ) @ B )
= ( ( member_nat @ X2 @ B )
& ( ord_less_eq_set_nat @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_553_insert__subset,axiom,
! [X2: tm,A2: set_tm,B: set_tm] :
( ( ord_less_eq_set_tm @ ( insert_tm2 @ X2 @ A2 ) @ B )
= ( ( member_tm @ X2 @ B )
& ( ord_less_eq_set_tm @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_554_Un__subset__iff,axiom,
! [A2: set_nat,B: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C3 )
= ( ( ord_less_eq_set_nat @ A2 @ C3 )
& ( ord_less_eq_set_nat @ B @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_555_Un__subset__iff,axiom,
! [A2: set_tm,B: set_tm,C3: set_tm] :
( ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ A2 @ B ) @ C3 )
= ( ( ord_less_eq_set_tm @ A2 @ C3 )
& ( ord_less_eq_set_tm @ B @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_556_singleton__insert__inj__eq,axiom,
! [B2: $o,A: $o,A2: set_o] :
( ( ( insert_o @ B2 @ bot_bot_set_o )
= ( insert_o @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_less_eq_set_o @ A2 @ ( insert_o @ B2 @ bot_bot_set_o ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_557_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_558_singleton__insert__inj__eq,axiom,
! [B2: tm,A: tm,A2: set_tm] :
( ( ( insert_tm2 @ B2 @ bot_bot_set_tm )
= ( insert_tm2 @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_less_eq_set_tm @ A2 @ ( insert_tm2 @ B2 @ bot_bot_set_tm ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_559_singleton__insert__inj__eq_H,axiom,
! [A: $o,A2: set_o,B2: $o] :
( ( ( insert_o @ A @ A2 )
= ( insert_o @ B2 @ bot_bot_set_o ) )
= ( ( A = B2 )
& ( ord_less_eq_set_o @ A2 @ ( insert_o @ B2 @ bot_bot_set_o ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_560_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_561_singleton__insert__inj__eq_H,axiom,
! [A: tm,A2: set_tm,B2: tm] :
( ( ( insert_tm2 @ A @ A2 )
= ( insert_tm2 @ B2 @ bot_bot_set_tm ) )
= ( ( A = B2 )
& ( ord_less_eq_set_tm @ A2 @ ( insert_tm2 @ B2 @ bot_bot_set_tm ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_562_range__subsetD,axiom,
! [F: tm > set_nat,B: set_set_nat,I2: tm] :
( ( ord_le6893508408891458716et_nat @ ( image_tm_set_nat @ F @ top_top_set_tm ) @ B )
=> ( member_set_nat @ ( F @ I2 ) @ B ) ) ).
% range_subsetD
thf(fact_563_range__subsetD,axiom,
! [F: tm > set_tm,B: set_set_tm,I2: tm] :
( ( ord_le5601931644483074373set_tm @ ( image_tm_set_tm @ F @ top_top_set_tm ) @ B )
=> ( member_set_tm @ ( F @ I2 ) @ B ) ) ).
% range_subsetD
thf(fact_564_range__subsetD,axiom,
! [F: fm > set_tm,B: set_set_tm,I2: fm] :
( ( ord_le5601931644483074373set_tm @ ( image_fm_set_tm @ F @ top_top_set_fm ) @ B )
=> ( member_set_tm @ ( F @ I2 ) @ B ) ) ).
% range_subsetD
thf(fact_565_range__subsetD,axiom,
! [F: fm > set_nat,B: set_set_nat,I2: fm] :
( ( ord_le6893508408891458716et_nat @ ( image_fm_set_nat @ F @ top_top_set_fm ) @ B )
=> ( member_set_nat @ ( F @ I2 ) @ B ) ) ).
% range_subsetD
thf(fact_566_range__subsetD,axiom,
! [F: nat > list_fm,B: set_list_fm,I2: nat] :
( ( ord_le7838213414353715577ist_fm @ ( image_nat_list_fm @ F @ top_top_set_nat ) @ B )
=> ( member_list_fm @ ( F @ I2 ) @ B ) ) ).
% range_subsetD
thf(fact_567_range__subsetD,axiom,
! [F: $o > list_fm,B: set_list_fm,I2: $o] :
( ( ord_le7838213414353715577ist_fm @ ( image_o_list_fm @ F @ top_top_set_o ) @ B )
=> ( member_list_fm @ ( F @ I2 ) @ B ) ) ).
% range_subsetD
thf(fact_568_range__subsetD,axiom,
! [F: nat > nat,B: set_nat,I2: nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ top_top_set_nat ) @ B )
=> ( member_nat @ ( F @ I2 ) @ B ) ) ).
% range_subsetD
thf(fact_569_range__subsetD,axiom,
! [F: $o > nat,B: set_nat,I2: $o] :
( ( ord_less_eq_set_nat @ ( image_o_nat @ F @ top_top_set_o ) @ B )
=> ( member_nat @ ( F @ I2 ) @ B ) ) ).
% range_subsetD
thf(fact_570_range__subsetD,axiom,
! [F: nat > tm,B: set_tm,I2: nat] :
( ( ord_less_eq_set_tm @ ( image_nat_tm @ F @ top_top_set_nat ) @ B )
=> ( member_tm @ ( F @ I2 ) @ B ) ) ).
% range_subsetD
thf(fact_571_range__subsetD,axiom,
! [F: $o > tm,B: set_tm,I2: $o] :
( ( ord_less_eq_set_tm @ ( image_o_tm @ F @ top_top_set_o ) @ B )
=> ( member_tm @ ( F @ I2 ) @ B ) ) ).
% range_subsetD
thf(fact_572_Compr__image__eq,axiom,
! [F: tm > tm,A2: set_tm,P: tm > $o] :
( ( collect_tm
@ ^ [X5: tm] :
( ( member_tm @ X5 @ ( image_tm_tm @ F @ A2 ) )
& ( P @ X5 ) ) )
= ( image_tm_tm @ F
@ ( collect_tm
@ ^ [X5: tm] :
( ( member_tm @ X5 @ A2 )
& ( P @ ( F @ X5 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_573_Compr__image__eq,axiom,
! [F: nat > tm,A2: set_nat,P: tm > $o] :
( ( collect_tm
@ ^ [X5: tm] :
( ( member_tm @ X5 @ ( image_nat_tm @ F @ A2 ) )
& ( P @ X5 ) ) )
= ( image_nat_tm @ F
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( P @ ( F @ X5 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_574_Compr__image__eq,axiom,
! [F: tm > nat,A2: set_tm,P: nat > $o] :
( ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ ( image_tm_nat @ F @ A2 ) )
& ( P @ X5 ) ) )
= ( image_tm_nat @ F
@ ( collect_tm
@ ^ [X5: tm] :
( ( member_tm @ X5 @ A2 )
& ( P @ ( F @ X5 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_575_Compr__image__eq,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X5 ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( P @ ( F @ X5 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_576_Compr__image__eq,axiom,
! [F: fm > set_tm,A2: set_fm,P: set_tm > $o] :
( ( collect_set_tm
@ ^ [X5: set_tm] :
( ( member_set_tm @ X5 @ ( image_fm_set_tm @ F @ A2 ) )
& ( P @ X5 ) ) )
= ( image_fm_set_tm @ F
@ ( collect_fm
@ ^ [X5: fm] :
( ( member_fm @ X5 @ A2 )
& ( P @ ( F @ X5 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_577_Compr__image__eq,axiom,
! [F: fm > set_nat,A2: set_fm,P: set_nat > $o] :
( ( collect_set_nat
@ ^ [X5: set_nat] :
( ( member_set_nat @ X5 @ ( image_fm_set_nat @ F @ A2 ) )
& ( P @ X5 ) ) )
= ( image_fm_set_nat @ F
@ ( collect_fm
@ ^ [X5: fm] :
( ( member_fm @ X5 @ A2 )
& ( P @ ( F @ X5 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_578_Compr__image__eq,axiom,
! [F: tm > set_nat,A2: set_tm,P: set_nat > $o] :
( ( collect_set_nat
@ ^ [X5: set_nat] :
( ( member_set_nat @ X5 @ ( image_tm_set_nat @ F @ A2 ) )
& ( P @ X5 ) ) )
= ( image_tm_set_nat @ F
@ ( collect_tm
@ ^ [X5: tm] :
( ( member_tm @ X5 @ A2 )
& ( P @ ( F @ X5 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_579_Compr__image__eq,axiom,
! [F: tm > set_tm,A2: set_tm,P: set_tm > $o] :
( ( collect_set_tm
@ ^ [X5: set_tm] :
( ( member_set_tm @ X5 @ ( image_tm_set_tm @ F @ A2 ) )
& ( P @ X5 ) ) )
= ( image_tm_set_tm @ F
@ ( collect_tm
@ ^ [X5: tm] :
( ( member_tm @ X5 @ A2 )
& ( P @ ( F @ X5 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_580_Compr__image__eq,axiom,
! [F: list_fm > tm,A2: set_list_fm,P: tm > $o] :
( ( collect_tm
@ ^ [X5: tm] :
( ( member_tm @ X5 @ ( image_list_fm_tm @ F @ A2 ) )
& ( P @ X5 ) ) )
= ( image_list_fm_tm @ F
@ ( collect_list_fm
@ ^ [X5: list_fm] :
( ( member_list_fm @ X5 @ A2 )
& ( P @ ( F @ X5 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_581_Compr__image__eq,axiom,
! [F: tm > list_fm,A2: set_tm,P: list_fm > $o] :
( ( collect_list_fm
@ ^ [X5: list_fm] :
( ( member_list_fm @ X5 @ ( image_tm_list_fm @ F @ A2 ) )
& ( P @ X5 ) ) )
= ( image_tm_list_fm @ F
@ ( collect_tm
@ ^ [X5: tm] :
( ( member_tm @ X5 @ A2 )
& ( P @ ( F @ X5 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_582_image__image,axiom,
! [F: nat > nat,G: nat > nat,A2: set_nat] :
( ( image_nat_nat @ F @ ( image_nat_nat @ G @ A2 ) )
= ( image_nat_nat
@ ^ [X5: nat] : ( F @ ( G @ X5 ) )
@ A2 ) ) ).
% image_image
thf(fact_583_image__image,axiom,
! [F: tm > set_nat,G: tm > tm,A2: set_tm] :
( ( image_tm_set_nat @ F @ ( image_tm_tm @ G @ A2 ) )
= ( image_tm_set_nat
@ ^ [X5: tm] : ( F @ ( G @ X5 ) )
@ A2 ) ) ).
% image_image
thf(fact_584_image__image,axiom,
! [F: tm > set_nat,G: fm > tm,A2: set_fm] :
( ( image_tm_set_nat @ F @ ( image_fm_tm @ G @ A2 ) )
= ( image_fm_set_nat
@ ^ [X5: fm] : ( F @ ( G @ X5 ) )
@ A2 ) ) ).
% image_image
thf(fact_585_image__image,axiom,
! [F: tm > set_tm,G: tm > tm,A2: set_tm] :
( ( image_tm_set_tm @ F @ ( image_tm_tm @ G @ A2 ) )
= ( image_tm_set_tm
@ ^ [X5: tm] : ( F @ ( G @ X5 ) )
@ A2 ) ) ).
% image_image
thf(fact_586_image__image,axiom,
! [F: tm > set_tm,G: fm > tm,A2: set_fm] :
( ( image_tm_set_tm @ F @ ( image_fm_tm @ G @ A2 ) )
= ( image_fm_set_tm
@ ^ [X5: fm] : ( F @ ( G @ X5 ) )
@ A2 ) ) ).
% image_image
thf(fact_587_image__image,axiom,
! [F: fm > set_tm,G: tm > fm,A2: set_tm] :
( ( image_fm_set_tm @ F @ ( image_tm_fm @ G @ A2 ) )
= ( image_tm_set_tm
@ ^ [X5: tm] : ( F @ ( G @ X5 ) )
@ A2 ) ) ).
% image_image
thf(fact_588_image__image,axiom,
! [F: fm > set_tm,G: fm > fm,A2: set_fm] :
( ( image_fm_set_tm @ F @ ( image_fm_fm @ G @ A2 ) )
= ( image_fm_set_tm
@ ^ [X5: fm] : ( F @ ( G @ X5 ) )
@ A2 ) ) ).
% image_image
thf(fact_589_image__image,axiom,
! [F: fm > set_nat,G: tm > fm,A2: set_tm] :
( ( image_fm_set_nat @ F @ ( image_tm_fm @ G @ A2 ) )
= ( image_tm_set_nat
@ ^ [X5: tm] : ( F @ ( G @ X5 ) )
@ A2 ) ) ).
% image_image
thf(fact_590_image__image,axiom,
! [F: fm > set_nat,G: fm > fm,A2: set_fm] :
( ( image_fm_set_nat @ F @ ( image_fm_fm @ G @ A2 ) )
= ( image_fm_set_nat
@ ^ [X5: fm] : ( F @ ( G @ X5 ) )
@ A2 ) ) ).
% image_image
thf(fact_591_image__image,axiom,
! [F: set_nat > set_nat,G: tm > set_nat,A2: set_tm] :
( ( image_7916887816326733075et_nat @ F @ ( image_tm_set_nat @ G @ A2 ) )
= ( image_tm_set_nat
@ ^ [X5: tm] : ( F @ ( G @ X5 ) )
@ A2 ) ) ).
% image_image
thf(fact_592_imageE,axiom,
! [B2: tm,F: tm > tm,A2: set_tm] :
( ( member_tm @ B2 @ ( image_tm_tm @ F @ A2 ) )
=> ~ ! [X: tm] :
( ( B2
= ( F @ X ) )
=> ~ ( member_tm @ X @ A2 ) ) ) ).
% imageE
thf(fact_593_imageE,axiom,
! [B2: tm,F: nat > tm,A2: set_nat] :
( ( member_tm @ B2 @ ( image_nat_tm @ F @ A2 ) )
=> ~ ! [X: nat] :
( ( B2
= ( F @ X ) )
=> ~ ( member_nat @ X @ A2 ) ) ) ).
% imageE
thf(fact_594_imageE,axiom,
! [B2: nat,F: tm > nat,A2: set_tm] :
( ( member_nat @ B2 @ ( image_tm_nat @ F @ A2 ) )
=> ~ ! [X: tm] :
( ( B2
= ( F @ X ) )
=> ~ ( member_tm @ X @ A2 ) ) ) ).
% imageE
thf(fact_595_imageE,axiom,
! [B2: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [X: nat] :
( ( B2
= ( F @ X ) )
=> ~ ( member_nat @ X @ A2 ) ) ) ).
% imageE
thf(fact_596_imageE,axiom,
! [B2: set_tm,F: fm > set_tm,A2: set_fm] :
( ( member_set_tm @ B2 @ ( image_fm_set_tm @ F @ A2 ) )
=> ~ ! [X: fm] :
( ( B2
= ( F @ X ) )
=> ~ ( member_fm @ X @ A2 ) ) ) ).
% imageE
thf(fact_597_imageE,axiom,
! [B2: set_nat,F: fm > set_nat,A2: set_fm] :
( ( member_set_nat @ B2 @ ( image_fm_set_nat @ F @ A2 ) )
=> ~ ! [X: fm] :
( ( B2
= ( F @ X ) )
=> ~ ( member_fm @ X @ A2 ) ) ) ).
% imageE
thf(fact_598_imageE,axiom,
! [B2: set_nat,F: tm > set_nat,A2: set_tm] :
( ( member_set_nat @ B2 @ ( image_tm_set_nat @ F @ A2 ) )
=> ~ ! [X: tm] :
( ( B2
= ( F @ X ) )
=> ~ ( member_tm @ X @ A2 ) ) ) ).
% imageE
thf(fact_599_imageE,axiom,
! [B2: set_tm,F: tm > set_tm,A2: set_tm] :
( ( member_set_tm @ B2 @ ( image_tm_set_tm @ F @ A2 ) )
=> ~ ! [X: tm] :
( ( B2
= ( F @ X ) )
=> ~ ( member_tm @ X @ A2 ) ) ) ).
% imageE
thf(fact_600_imageE,axiom,
! [B2: tm,F: list_fm > tm,A2: set_list_fm] :
( ( member_tm @ B2 @ ( image_list_fm_tm @ F @ A2 ) )
=> ~ ! [X: list_fm] :
( ( B2
= ( F @ X ) )
=> ~ ( member_list_fm @ X @ A2 ) ) ) ).
% imageE
thf(fact_601_imageE,axiom,
! [B2: nat,F: list_fm > nat,A2: set_list_fm] :
( ( member_nat @ B2 @ ( image_list_fm_nat @ F @ A2 ) )
=> ~ ! [X: list_fm] :
( ( B2
= ( F @ X ) )
=> ~ ( member_list_fm @ X @ A2 ) ) ) ).
% imageE
thf(fact_602_image__inv__into__cancel,axiom,
! [F: set_tm > fm,A2: set_set_tm,A6: set_fm,B5: set_fm] :
( ( ( image_set_tm_fm @ F @ A2 )
= A6 )
=> ( ( ord_less_eq_set_fm @ B5 @ A6 )
=> ( ( image_set_tm_fm @ F @ ( image_fm_set_tm @ ( hilber6730722110952398266_tm_fm @ A2 @ F ) @ B5 ) )
= B5 ) ) ) ).
% image_inv_into_cancel
thf(fact_603_image__inv__into__cancel,axiom,
! [F: set_nat > fm,A2: set_set_nat,A6: set_fm,B5: set_fm] :
( ( ( image_set_nat_fm @ F @ A2 )
= A6 )
=> ( ( ord_less_eq_set_fm @ B5 @ A6 )
=> ( ( image_set_nat_fm @ F @ ( image_fm_set_nat @ ( hilber6297111070489569439nat_fm @ A2 @ F ) @ B5 ) )
= B5 ) ) ) ).
% image_inv_into_cancel
thf(fact_604_image__inv__into__cancel,axiom,
! [F: tm > set_nat,A2: set_tm,A6: set_set_nat,B5: set_set_nat] :
( ( ( image_tm_set_nat @ F @ A2 )
= A6 )
=> ( ( ord_le6893508408891458716et_nat @ B5 @ A6 )
=> ( ( image_tm_set_nat @ F @ ( image_set_nat_tm @ ( hilber8933797207411865247et_nat @ A2 @ F ) @ B5 ) )
= B5 ) ) ) ).
% image_inv_into_cancel
thf(fact_605_image__inv__into__cancel,axiom,
! [F: tm > set_tm,A2: set_tm,A6: set_set_tm,B5: set_set_tm] :
( ( ( image_tm_set_tm @ F @ A2 )
= A6 )
=> ( ( ord_le5601931644483074373set_tm @ B5 @ A6 )
=> ( ( image_tm_set_tm @ F @ ( image_set_tm_tm @ ( hilber5702680533203024108set_tm @ A2 @ F ) @ B5 ) )
= B5 ) ) ) ).
% image_inv_into_cancel
thf(fact_606_image__inv__into__cancel,axiom,
! [F: fm > set_tm,A2: set_fm,A6: set_set_tm,B5: set_set_tm] :
( ( ( image_fm_set_tm @ F @ A2 )
= A6 )
=> ( ( ord_le5601931644483074373set_tm @ B5 @ A6 )
=> ( ( image_fm_set_tm @ F @ ( image_set_tm_fm @ ( hilber6286924743720013790set_tm @ A2 @ F ) @ B5 ) )
= B5 ) ) ) ).
% image_inv_into_cancel
thf(fact_607_image__inv__into__cancel,axiom,
! [F: fm > set_nat,A2: set_fm,A6: set_set_nat,B5: set_set_nat] :
( ( ( image_fm_set_nat @ F @ A2 )
= A6 )
=> ( ( ord_le6893508408891458716et_nat @ B5 @ A6 )
=> ( ( image_fm_set_nat @ F @ ( image_set_nat_fm @ ( hilber5456991776183730989et_nat @ A2 @ F ) @ B5 ) )
= B5 ) ) ) ).
% image_inv_into_cancel
thf(fact_608_image__inv__into__cancel,axiom,
! [F: nat > nat,A2: set_nat,A6: set_nat,B5: set_nat] :
( ( ( image_nat_nat @ F @ A2 )
= A6 )
=> ( ( ord_less_eq_set_nat @ B5 @ A6 )
=> ( ( image_nat_nat @ F @ ( image_nat_nat @ ( hilber3633877196798814958at_nat @ A2 @ F ) @ B5 ) )
= B5 ) ) ) ).
% image_inv_into_cancel
thf(fact_609_image__inv__into__cancel,axiom,
! [F: set_nat > tm,A2: set_set_nat,A6: set_tm,B5: set_tm] :
( ( ( image_set_nat_tm @ F @ A2 )
= A6 )
=> ( ( ord_less_eq_set_tm @ B5 @ A6 )
=> ( ( image_set_nat_tm @ F @ ( image_tm_set_nat @ ( hilber6297111070490487825nat_tm @ A2 @ F ) @ B5 ) )
= B5 ) ) ) ).
% image_inv_into_cancel
thf(fact_610_image__inv__into__cancel,axiom,
! [F: set_tm > tm,A2: set_set_tm,A6: set_tm,B5: set_tm] :
( ( ( image_set_tm_tm @ F @ A2 )
= A6 )
=> ( ( ord_less_eq_set_tm @ B5 @ A6 )
=> ( ( image_set_tm_tm @ F @ ( image_tm_set_tm @ ( hilber6730722110953316652_tm_tm @ A2 @ F ) @ B5 ) )
= B5 ) ) ) ).
% image_inv_into_cancel
thf(fact_611_verit__la__disequality,axiom,
! [A: nat,B2: nat] :
( ( A = B2 )
| ~ ( ord_less_eq_nat @ A @ B2 )
| ~ ( ord_less_eq_nat @ B2 @ A ) ) ).
% verit_la_disequality
thf(fact_612_verit__comp__simplify1_I2_J,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_613_verit__comp__simplify1_I2_J,axiom,
! [A: set_tm] : ( ord_less_eq_set_tm @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_614_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_615_order__antisym__conv,axiom,
! [Y4: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y4 @ X2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Y4 )
= ( X2 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_616_order__antisym__conv,axiom,
! [Y4: set_tm,X2: set_tm] :
( ( ord_less_eq_set_tm @ Y4 @ X2 )
=> ( ( ord_less_eq_set_tm @ X2 @ Y4 )
= ( X2 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_617_order__antisym__conv,axiom,
! [Y4: nat,X2: nat] :
( ( ord_less_eq_nat @ Y4 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y4 )
= ( X2 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_618_linorder__le__cases,axiom,
! [X2: nat,Y4: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X2 ) ) ).
% linorder_le_cases
thf(fact_619_ord__le__eq__subst,axiom,
! [A: set_nat,B2: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_620_ord__le__eq__subst,axiom,
! [A: set_nat,B2: set_nat,F: set_nat > set_tm,C: set_tm] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_tm @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_621_ord__le__eq__subst,axiom,
! [A: set_nat,B2: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_622_ord__le__eq__subst,axiom,
! [A: set_tm,B2: set_tm,F: set_tm > set_nat,C: set_nat] :
( ( ord_less_eq_set_tm @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_tm,Y3: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_623_ord__le__eq__subst,axiom,
! [A: set_tm,B2: set_tm,F: set_tm > set_tm,C: set_tm] :
( ( ord_less_eq_set_tm @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_tm,Y3: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y3 )
=> ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_tm @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_624_ord__le__eq__subst,axiom,
! [A: set_tm,B2: set_tm,F: set_tm > nat,C: nat] :
( ( ord_less_eq_set_tm @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_tm,Y3: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_625_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_626_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > set_tm,C: set_tm] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_tm @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_627_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_628_ord__eq__le__subst,axiom,
! [A: set_nat,F: set_nat > set_nat,B2: set_nat,C: set_nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_629_ord__eq__le__subst,axiom,
! [A: set_tm,F: set_nat > set_tm,B2: set_nat,C: set_nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_tm @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_630_ord__eq__le__subst,axiom,
! [A: nat,F: set_nat > nat,B2: set_nat,C: set_nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_631_ord__eq__le__subst,axiom,
! [A: set_nat,F: set_tm > set_nat,B2: set_tm,C: set_tm] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_tm @ B2 @ C )
=> ( ! [X: set_tm,Y3: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_632_ord__eq__le__subst,axiom,
! [A: set_tm,F: set_tm > set_tm,B2: set_tm,C: set_tm] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_tm @ B2 @ C )
=> ( ! [X: set_tm,Y3: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y3 )
=> ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_tm @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_633_ord__eq__le__subst,axiom,
! [A: nat,F: set_tm > nat,B2: set_tm,C: set_tm] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_tm @ B2 @ C )
=> ( ! [X: set_tm,Y3: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_634_ord__eq__le__subst,axiom,
! [A: set_nat,F: nat > set_nat,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_635_ord__eq__le__subst,axiom,
! [A: set_tm,F: nat > set_tm,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_tm @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_636_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_637_linorder__linear,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
| ( ord_less_eq_nat @ Y4 @ X2 ) ) ).
% linorder_linear
thf(fact_638_order__eq__refl,axiom,
! [X2: set_nat,Y4: set_nat] :
( ( X2 = Y4 )
=> ( ord_less_eq_set_nat @ X2 @ Y4 ) ) ).
% order_eq_refl
thf(fact_639_order__eq__refl,axiom,
! [X2: set_tm,Y4: set_tm] :
( ( X2 = Y4 )
=> ( ord_less_eq_set_tm @ X2 @ Y4 ) ) ).
% order_eq_refl
thf(fact_640_order__eq__refl,axiom,
! [X2: nat,Y4: nat] :
( ( X2 = Y4 )
=> ( ord_less_eq_nat @ X2 @ Y4 ) ) ).
% order_eq_refl
thf(fact_641_order__subst2,axiom,
! [A: set_nat,B2: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_642_order__subst2,axiom,
! [A: set_nat,B2: set_nat,F: set_nat > set_tm,C: set_tm] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_tm @ ( F @ B2 ) @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_tm @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_643_order__subst2,axiom,
! [A: set_nat,B2: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_644_order__subst2,axiom,
! [A: set_tm,B2: set_tm,F: set_tm > set_nat,C: set_nat] :
( ( ord_less_eq_set_tm @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X: set_tm,Y3: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_645_order__subst2,axiom,
! [A: set_tm,B2: set_tm,F: set_tm > set_tm,C: set_tm] :
( ( ord_less_eq_set_tm @ A @ B2 )
=> ( ( ord_less_eq_set_tm @ ( F @ B2 ) @ C )
=> ( ! [X: set_tm,Y3: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y3 )
=> ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_tm @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_646_order__subst2,axiom,
! [A: set_tm,B2: set_tm,F: set_tm > nat,C: nat] :
( ( ord_less_eq_set_tm @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X: set_tm,Y3: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_647_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_648_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > set_tm,C: set_tm] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_set_tm @ ( F @ B2 ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_tm @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_649_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_650_order__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_651_order__subst1,axiom,
! [A: set_nat,F: set_tm > set_nat,B2: set_tm,C: set_tm] :
( ( ord_less_eq_set_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_tm @ B2 @ C )
=> ( ! [X: set_tm,Y3: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_652_order__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B2: nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_653_order__subst1,axiom,
! [A: set_tm,F: set_nat > set_tm,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_tm @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_tm @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_654_order__subst1,axiom,
! [A: set_tm,F: set_tm > set_tm,B2: set_tm,C: set_tm] :
( ( ord_less_eq_set_tm @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_tm @ B2 @ C )
=> ( ! [X: set_tm,Y3: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y3 )
=> ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_tm @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_655_order__subst1,axiom,
! [A: set_tm,F: nat > set_tm,B2: nat,C: nat] :
( ( ord_less_eq_set_tm @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_set_tm @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_tm @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_656_order__subst1,axiom,
! [A: nat,F: set_nat > nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_657_order__subst1,axiom,
! [A: nat,F: set_tm > nat,B2: set_tm,C: set_tm] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_tm @ B2 @ C )
=> ( ! [X: set_tm,Y3: set_tm] :
( ( ord_less_eq_set_tm @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_658_order__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_659_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [A5: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B6 )
& ( ord_less_eq_set_nat @ B6 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_660_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_tm,Z: set_tm] : ( Y2 = Z ) )
= ( ^ [A5: set_tm,B6: set_tm] :
( ( ord_less_eq_set_tm @ A5 @ B6 )
& ( ord_less_eq_set_tm @ B6 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_661_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A5: nat,B6: nat] :
( ( ord_less_eq_nat @ A5 @ B6 )
& ( ord_less_eq_nat @ B6 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_662_subset__image__iff,axiom,
! [B: set_set_tm,F: fm > set_tm,A2: set_fm] :
( ( ord_le5601931644483074373set_tm @ B @ ( image_fm_set_tm @ F @ A2 ) )
= ( ? [AA: set_fm] :
( ( ord_less_eq_set_fm @ AA @ A2 )
& ( B
= ( image_fm_set_tm @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_663_subset__image__iff,axiom,
! [B: set_set_nat,F: fm > set_nat,A2: set_fm] :
( ( ord_le6893508408891458716et_nat @ B @ ( image_fm_set_nat @ F @ A2 ) )
= ( ? [AA: set_fm] :
( ( ord_less_eq_set_fm @ AA @ A2 )
& ( B
= ( image_fm_set_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_664_subset__image__iff,axiom,
! [B: set_set_nat,F: tm > set_nat,A2: set_tm] :
( ( ord_le6893508408891458716et_nat @ B @ ( image_tm_set_nat @ F @ A2 ) )
= ( ? [AA: set_tm] :
( ( ord_less_eq_set_tm @ AA @ A2 )
& ( B
= ( image_tm_set_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_665_subset__image__iff,axiom,
! [B: set_set_tm,F: tm > set_tm,A2: set_tm] :
( ( ord_le5601931644483074373set_tm @ B @ ( image_tm_set_tm @ F @ A2 ) )
= ( ? [AA: set_tm] :
( ( ord_less_eq_set_tm @ AA @ A2 )
& ( B
= ( image_tm_set_tm @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_666_subset__image__iff,axiom,
! [B: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_667_subset__image__iff,axiom,
! [B: set_nat,F: tm > nat,A2: set_tm] :
( ( ord_less_eq_set_nat @ B @ ( image_tm_nat @ F @ A2 ) )
= ( ? [AA: set_tm] :
( ( ord_less_eq_set_tm @ AA @ A2 )
& ( B
= ( image_tm_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_668_subset__image__iff,axiom,
! [B: set_tm,F: nat > tm,A2: set_nat] :
( ( ord_less_eq_set_tm @ B @ ( image_nat_tm @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B
= ( image_nat_tm @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_669_subset__image__iff,axiom,
! [B: set_tm,F: tm > tm,A2: set_tm] :
( ( ord_less_eq_set_tm @ B @ ( image_tm_tm @ F @ A2 ) )
= ( ? [AA: set_tm] :
( ( ord_less_eq_set_tm @ AA @ A2 )
& ( B
= ( image_tm_tm @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_670_image__subset__iff,axiom,
! [F: tm > set_nat,A2: set_tm,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_tm_set_nat @ F @ A2 ) @ B )
= ( ! [X5: tm] :
( ( member_tm @ X5 @ A2 )
=> ( member_set_nat @ ( F @ X5 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_671_image__subset__iff,axiom,
! [F: tm > set_tm,A2: set_tm,B: set_set_tm] :
( ( ord_le5601931644483074373set_tm @ ( image_tm_set_tm @ F @ A2 ) @ B )
= ( ! [X5: tm] :
( ( member_tm @ X5 @ A2 )
=> ( member_set_tm @ ( F @ X5 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_672_image__subset__iff,axiom,
! [F: fm > set_tm,A2: set_fm,B: set_set_tm] :
( ( ord_le5601931644483074373set_tm @ ( image_fm_set_tm @ F @ A2 ) @ B )
= ( ! [X5: fm] :
( ( member_fm @ X5 @ A2 )
=> ( member_set_tm @ ( F @ X5 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_673_image__subset__iff,axiom,
! [F: fm > set_nat,A2: set_fm,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( image_fm_set_nat @ F @ A2 ) @ B )
= ( ! [X5: fm] :
( ( member_fm @ X5 @ A2 )
=> ( member_set_nat @ ( F @ X5 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_674_image__subset__iff,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B )
= ( ! [X5: nat] :
( ( member_nat @ X5 @ A2 )
=> ( member_nat @ ( F @ X5 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_675_Collect__mono__iff,axiom,
! [P: list_fm > $o,Q: list_fm > $o] :
( ( ord_le7838213414353715577ist_fm @ ( collect_list_fm @ P ) @ ( collect_list_fm @ Q ) )
= ( ! [X5: list_fm] :
( ( P @ X5 )
=> ( Q @ X5 ) ) ) ) ).
% Collect_mono_iff
thf(fact_676_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X5: nat] :
( ( P @ X5 )
=> ( Q @ X5 ) ) ) ) ).
% Collect_mono_iff
thf(fact_677_Collect__mono__iff,axiom,
! [P: tm > $o,Q: tm > $o] :
( ( ord_less_eq_set_tm @ ( collect_tm @ P ) @ ( collect_tm @ Q ) )
= ( ! [X5: tm] :
( ( P @ X5 )
=> ( Q @ X5 ) ) ) ) ).
% Collect_mono_iff
thf(fact_678_subset__imageE,axiom,
! [B: set_set_tm,F: fm > set_tm,A2: set_fm] :
( ( ord_le5601931644483074373set_tm @ B @ ( image_fm_set_tm @ F @ A2 ) )
=> ~ ! [C5: set_fm] :
( ( ord_less_eq_set_fm @ C5 @ A2 )
=> ( B
!= ( image_fm_set_tm @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_679_subset__imageE,axiom,
! [B: set_set_nat,F: fm > set_nat,A2: set_fm] :
( ( ord_le6893508408891458716et_nat @ B @ ( image_fm_set_nat @ F @ A2 ) )
=> ~ ! [C5: set_fm] :
( ( ord_less_eq_set_fm @ C5 @ A2 )
=> ( B
!= ( image_fm_set_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_680_subset__imageE,axiom,
! [B: set_set_nat,F: tm > set_nat,A2: set_tm] :
( ( ord_le6893508408891458716et_nat @ B @ ( image_tm_set_nat @ F @ A2 ) )
=> ~ ! [C5: set_tm] :
( ( ord_less_eq_set_tm @ C5 @ A2 )
=> ( B
!= ( image_tm_set_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_681_subset__imageE,axiom,
! [B: set_set_tm,F: tm > set_tm,A2: set_tm] :
( ( ord_le5601931644483074373set_tm @ B @ ( image_tm_set_tm @ F @ A2 ) )
=> ~ ! [C5: set_tm] :
( ( ord_less_eq_set_tm @ C5 @ A2 )
=> ( B
!= ( image_tm_set_tm @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_682_subset__imageE,axiom,
! [B: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A2 )
=> ( B
!= ( image_nat_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_683_subset__imageE,axiom,
! [B: set_nat,F: tm > nat,A2: set_tm] :
( ( ord_less_eq_set_nat @ B @ ( image_tm_nat @ F @ A2 ) )
=> ~ ! [C5: set_tm] :
( ( ord_less_eq_set_tm @ C5 @ A2 )
=> ( B
!= ( image_tm_nat @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_684_subset__imageE,axiom,
! [B: set_tm,F: nat > tm,A2: set_nat] :
( ( ord_less_eq_set_tm @ B @ ( image_nat_tm @ F @ A2 ) )
=> ~ ! [C5: set_nat] :
( ( ord_less_eq_set_nat @ C5 @ A2 )
=> ( B
!= ( image_nat_tm @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_685_subset__imageE,axiom,
! [B: set_tm,F: tm > tm,A2: set_tm] :
( ( ord_less_eq_set_tm @ B @ ( image_tm_tm @ F @ A2 ) )
=> ~ ! [C5: set_tm] :
( ( ord_less_eq_set_tm @ C5 @ A2 )
=> ( B
!= ( image_tm_tm @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_686_set__eq__subset,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [A4: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_687_set__eq__subset,axiom,
( ( ^ [Y2: set_tm,Z: set_tm] : ( Y2 = Z ) )
= ( ^ [A4: set_tm,B3: set_tm] :
( ( ord_less_eq_set_tm @ A4 @ B3 )
& ( ord_less_eq_set_tm @ B3 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_688_rev__image__eqI,axiom,
! [X2: tm,A2: set_tm,B2: tm,F: tm > tm] :
( ( member_tm @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_tm @ B2 @ ( image_tm_tm @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_689_rev__image__eqI,axiom,
! [X2: tm,A2: set_tm,B2: nat,F: tm > nat] :
( ( member_tm @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_nat @ B2 @ ( image_tm_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_690_rev__image__eqI,axiom,
! [X2: nat,A2: set_nat,B2: tm,F: nat > tm] :
( ( member_nat @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_tm @ B2 @ ( image_nat_tm @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_691_rev__image__eqI,axiom,
! [X2: nat,A2: set_nat,B2: nat,F: nat > nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_692_rev__image__eqI,axiom,
! [X2: fm,A2: set_fm,B2: set_tm,F: fm > set_tm] :
( ( member_fm @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_set_tm @ B2 @ ( image_fm_set_tm @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_693_rev__image__eqI,axiom,
! [X2: fm,A2: set_fm,B2: set_nat,F: fm > set_nat] :
( ( member_fm @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_set_nat @ B2 @ ( image_fm_set_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_694_rev__image__eqI,axiom,
! [X2: tm,A2: set_tm,B2: set_nat,F: tm > set_nat] :
( ( member_tm @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_set_nat @ B2 @ ( image_tm_set_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_695_rev__image__eqI,axiom,
! [X2: tm,A2: set_tm,B2: set_tm,F: tm > set_tm] :
( ( member_tm @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_set_tm @ B2 @ ( image_tm_set_tm @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_696_rev__image__eqI,axiom,
! [X2: tm,A2: set_tm,B2: list_fm,F: tm > list_fm] :
( ( member_tm @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_list_fm @ B2 @ ( image_tm_list_fm @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_697_rev__image__eqI,axiom,
! [X2: nat,A2: set_nat,B2: list_fm,F: nat > list_fm] :
( ( member_nat @ X2 @ A2 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member_list_fm @ B2 @ ( image_nat_list_fm @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_698_image__subsetI,axiom,
! [A2: set_tm,F: tm > nat,B: set_nat] :
( ! [X: tm] :
( ( member_tm @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_tm_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_699_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat,B: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_700_image__subsetI,axiom,
! [A2: set_tm,F: tm > tm,B: set_tm] :
( ! [X: tm] :
( ( member_tm @ X @ A2 )
=> ( member_tm @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_tm @ ( image_tm_tm @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_701_image__subsetI,axiom,
! [A2: set_nat,F: nat > tm,B: set_tm] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_tm @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_tm @ ( image_nat_tm @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_702_image__subsetI,axiom,
! [A2: set_fm,F: fm > set_tm,B: set_set_tm] :
( ! [X: fm] :
( ( member_fm @ X @ A2 )
=> ( member_set_tm @ ( F @ X ) @ B ) )
=> ( ord_le5601931644483074373set_tm @ ( image_fm_set_tm @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_703_image__subsetI,axiom,
! [A2: set_fm,F: fm > set_nat,B: set_set_nat] :
( ! [X: fm] :
( ( member_fm @ X @ A2 )
=> ( member_set_nat @ ( F @ X ) @ B ) )
=> ( ord_le6893508408891458716et_nat @ ( image_fm_set_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_704_image__subsetI,axiom,
! [A2: set_tm,F: tm > set_nat,B: set_set_nat] :
( ! [X: tm] :
( ( member_tm @ X @ A2 )
=> ( member_set_nat @ ( F @ X ) @ B ) )
=> ( ord_le6893508408891458716et_nat @ ( image_tm_set_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_705_image__subsetI,axiom,
! [A2: set_tm,F: tm > set_tm,B: set_set_tm] :
( ! [X: tm] :
( ( member_tm @ X @ A2 )
=> ( member_set_tm @ ( F @ X ) @ B ) )
=> ( ord_le5601931644483074373set_tm @ ( image_tm_set_tm @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_706_image__subsetI,axiom,
! [A2: set_tm,F: tm > list_fm,B: set_list_fm] :
( ! [X: tm] :
( ( member_tm @ X @ A2 )
=> ( member_list_fm @ ( F @ X ) @ B ) )
=> ( ord_le7838213414353715577ist_fm @ ( image_tm_list_fm @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_707_image__subsetI,axiom,
! [A2: set_nat,F: nat > list_fm,B: set_list_fm] :
( ! [X: nat] :
( ( member_nat @ X @ A2 )
=> ( member_list_fm @ ( F @ X ) @ B ) )
=> ( ord_le7838213414353715577ist_fm @ ( image_nat_list_fm @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_708_antisym,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_709_antisym,axiom,
! [A: set_tm,B2: set_tm] :
( ( ord_less_eq_set_tm @ A @ B2 )
=> ( ( ord_less_eq_set_tm @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_710_antisym,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_711_subset__trans,axiom,
! [A2: set_nat,B: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C3 )
=> ( ord_less_eq_set_nat @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_712_subset__trans,axiom,
! [A2: set_tm,B: set_tm,C3: set_tm] :
( ( ord_less_eq_set_tm @ A2 @ B )
=> ( ( ord_less_eq_set_tm @ B @ C3 )
=> ( ord_less_eq_set_tm @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_713_Collect__mono,axiom,
! [P: list_fm > $o,Q: list_fm > $o] :
( ! [X: list_fm] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_le7838213414353715577ist_fm @ ( collect_list_fm @ P ) @ ( collect_list_fm @ Q ) ) ) ).
% Collect_mono
thf(fact_714_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_715_Collect__mono,axiom,
! [P: tm > $o,Q: tm > $o] :
( ! [X: tm] :
( ( P @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_tm @ ( collect_tm @ P ) @ ( collect_tm @ Q ) ) ) ).
% Collect_mono
thf(fact_716_subset__refl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_717_subset__refl,axiom,
! [A2: set_tm] : ( ord_less_eq_set_tm @ A2 @ A2 ) ).
% subset_refl
thf(fact_718_ball__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ! [X: nat] :
( ( member_nat @ X @ ( image_nat_nat @ F @ A2 ) )
=> ( P @ X ) )
=> ! [X6: nat] :
( ( member_nat @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_719_ball__imageD,axiom,
! [F: tm > set_nat,A2: set_tm,P: set_nat > $o] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ ( image_tm_set_nat @ F @ A2 ) )
=> ( P @ X ) )
=> ! [X6: tm] :
( ( member_tm @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_720_ball__imageD,axiom,
! [F: tm > set_tm,A2: set_tm,P: set_tm > $o] :
( ! [X: set_tm] :
( ( member_set_tm @ X @ ( image_tm_set_tm @ F @ A2 ) )
=> ( P @ X ) )
=> ! [X6: tm] :
( ( member_tm @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_721_ball__imageD,axiom,
! [F: fm > set_tm,A2: set_fm,P: set_tm > $o] :
( ! [X: set_tm] :
( ( member_set_tm @ X @ ( image_fm_set_tm @ F @ A2 ) )
=> ( P @ X ) )
=> ! [X6: fm] :
( ( member_fm @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_722_ball__imageD,axiom,
! [F: fm > set_nat,A2: set_fm,P: set_nat > $o] :
( ! [X: set_nat] :
( ( member_set_nat @ X @ ( image_fm_set_nat @ F @ A2 ) )
=> ( P @ X ) )
=> ! [X6: fm] :
( ( member_fm @ X6 @ A2 )
=> ( P @ ( F @ X6 ) ) ) ) ).
% ball_imageD
thf(fact_723_subset__iff,axiom,
( ord_le7838213414353715577ist_fm
= ( ^ [A4: set_list_fm,B3: set_list_fm] :
! [T2: list_fm] :
( ( member_list_fm @ T2 @ A4 )
=> ( member_list_fm @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_724_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A4 )
=> ( member_nat @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_725_subset__iff,axiom,
( ord_less_eq_set_tm
= ( ^ [A4: set_tm,B3: set_tm] :
! [T2: tm] :
( ( member_tm @ T2 @ A4 )
=> ( member_tm @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_726_image__mono,axiom,
! [A2: set_fm,B: set_fm,F: fm > set_tm] :
( ( ord_less_eq_set_fm @ A2 @ B )
=> ( ord_le5601931644483074373set_tm @ ( image_fm_set_tm @ F @ A2 ) @ ( image_fm_set_tm @ F @ B ) ) ) ).
% image_mono
thf(fact_727_image__mono,axiom,
! [A2: set_fm,B: set_fm,F: fm > set_nat] :
( ( ord_less_eq_set_fm @ A2 @ B )
=> ( ord_le6893508408891458716et_nat @ ( image_fm_set_nat @ F @ A2 ) @ ( image_fm_set_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_728_image__mono,axiom,
! [A2: set_nat,B: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_729_image__mono,axiom,
! [A2: set_nat,B: set_nat,F: nat > tm] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_eq_set_tm @ ( image_nat_tm @ F @ A2 ) @ ( image_nat_tm @ F @ B ) ) ) ).
% image_mono
thf(fact_730_image__mono,axiom,
! [A2: set_tm,B: set_tm,F: tm > set_nat] :
( ( ord_less_eq_set_tm @ A2 @ B )
=> ( ord_le6893508408891458716et_nat @ ( image_tm_set_nat @ F @ A2 ) @ ( image_tm_set_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_731_image__mono,axiom,
! [A2: set_tm,B: set_tm,F: tm > set_tm] :
( ( ord_less_eq_set_tm @ A2 @ B )
=> ( ord_le5601931644483074373set_tm @ ( image_tm_set_tm @ F @ A2 ) @ ( image_tm_set_tm @ F @ B ) ) ) ).
% image_mono
thf(fact_732_image__mono,axiom,
! [A2: set_tm,B: set_tm,F: tm > nat] :
( ( ord_less_eq_set_tm @ A2 @ B )
=> ( ord_less_eq_set_nat @ ( image_tm_nat @ F @ A2 ) @ ( image_tm_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_733_image__mono,axiom,
! [A2: set_tm,B: set_tm,F: tm > tm] :
( ( ord_less_eq_set_tm @ A2 @ B )
=> ( ord_less_eq_set_tm @ ( image_tm_tm @ F @ A2 ) @ ( image_tm_tm @ F @ B ) ) ) ).
% image_mono
thf(fact_734_image__cong,axiom,
! [M2: set_fm,N4: set_fm,F: fm > set_tm,G: fm > set_tm] :
( ( M2 = N4 )
=> ( ! [X: fm] :
( ( member_fm @ X @ N4 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_fm_set_tm @ F @ M2 )
= ( image_fm_set_tm @ G @ N4 ) ) ) ) ).
% image_cong
thf(fact_735_image__cong,axiom,
! [M2: set_fm,N4: set_fm,F: fm > set_nat,G: fm > set_nat] :
( ( M2 = N4 )
=> ( ! [X: fm] :
( ( member_fm @ X @ N4 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_fm_set_nat @ F @ M2 )
= ( image_fm_set_nat @ G @ N4 ) ) ) ) ).
% image_cong
thf(fact_736_image__cong,axiom,
! [M2: set_tm,N4: set_tm,F: tm > set_nat,G: tm > set_nat] :
( ( M2 = N4 )
=> ( ! [X: tm] :
( ( member_tm @ X @ N4 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_tm_set_nat @ F @ M2 )
= ( image_tm_set_nat @ G @ N4 ) ) ) ) ).
% image_cong
thf(fact_737_image__cong,axiom,
! [M2: set_tm,N4: set_tm,F: tm > set_tm,G: tm > set_tm] :
( ( M2 = N4 )
=> ( ! [X: tm] :
( ( member_tm @ X @ N4 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_tm_set_tm @ F @ M2 )
= ( image_tm_set_tm @ G @ N4 ) ) ) ) ).
% image_cong
thf(fact_738_image__cong,axiom,
! [M2: set_nat,N4: set_nat,F: nat > nat,G: nat > nat] :
( ( M2 = N4 )
=> ( ! [X: nat] :
( ( member_nat @ X @ N4 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image_nat_nat @ F @ M2 )
= ( image_nat_nat @ G @ N4 ) ) ) ) ).
% image_cong
thf(fact_739_equalityD2,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 = B )
=> ( ord_less_eq_set_nat @ B @ A2 ) ) ).
% equalityD2
thf(fact_740_equalityD2,axiom,
! [A2: set_tm,B: set_tm] :
( ( A2 = B )
=> ( ord_less_eq_set_tm @ B @ A2 ) ) ).
% equalityD2
thf(fact_741_equalityD1,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 = B )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% equalityD1
thf(fact_742_equalityD1,axiom,
! [A2: set_tm,B: set_tm] :
( ( A2 = B )
=> ( ord_less_eq_set_tm @ A2 @ B ) ) ).
% equalityD1
thf(fact_743_bex__imageD,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ? [X6: nat] :
( ( member_nat @ X6 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X: nat] :
( ( member_nat @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_744_bex__imageD,axiom,
! [F: tm > set_nat,A2: set_tm,P: set_nat > $o] :
( ? [X6: set_nat] :
( ( member_set_nat @ X6 @ ( image_tm_set_nat @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X: tm] :
( ( member_tm @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_745_bex__imageD,axiom,
! [F: tm > set_tm,A2: set_tm,P: set_tm > $o] :
( ? [X6: set_tm] :
( ( member_set_tm @ X6 @ ( image_tm_set_tm @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X: tm] :
( ( member_tm @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_746_bex__imageD,axiom,
! [F: fm > set_tm,A2: set_fm,P: set_tm > $o] :
( ? [X6: set_tm] :
( ( member_set_tm @ X6 @ ( image_fm_set_tm @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X: fm] :
( ( member_fm @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_747_bex__imageD,axiom,
! [F: fm > set_nat,A2: set_fm,P: set_nat > $o] :
( ? [X6: set_nat] :
( ( member_set_nat @ X6 @ ( image_fm_set_nat @ F @ A2 ) )
& ( P @ X6 ) )
=> ? [X: fm] :
( ( member_fm @ X @ A2 )
& ( P @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_748_subset__eq,axiom,
( ord_le7838213414353715577ist_fm
= ( ^ [A4: set_list_fm,B3: set_list_fm] :
! [X5: list_fm] :
( ( member_list_fm @ X5 @ A4 )
=> ( member_list_fm @ X5 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_749_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
! [X5: nat] :
( ( member_nat @ X5 @ A4 )
=> ( member_nat @ X5 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_750_subset__eq,axiom,
( ord_less_eq_set_tm
= ( ^ [A4: set_tm,B3: set_tm] :
! [X5: tm] :
( ( member_tm @ X5 @ A4 )
=> ( member_tm @ X5 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_751_image__iff,axiom,
! [Z2: set_nat,F: tm > set_nat,A2: set_tm] :
( ( member_set_nat @ Z2 @ ( image_tm_set_nat @ F @ A2 ) )
= ( ? [X5: tm] :
( ( member_tm @ X5 @ A2 )
& ( Z2
= ( F @ X5 ) ) ) ) ) ).
% image_iff
thf(fact_752_image__iff,axiom,
! [Z2: set_tm,F: tm > set_tm,A2: set_tm] :
( ( member_set_tm @ Z2 @ ( image_tm_set_tm @ F @ A2 ) )
= ( ? [X5: tm] :
( ( member_tm @ X5 @ A2 )
& ( Z2
= ( F @ X5 ) ) ) ) ) ).
% image_iff
thf(fact_753_image__iff,axiom,
! [Z2: set_tm,F: fm > set_tm,A2: set_fm] :
( ( member_set_tm @ Z2 @ ( image_fm_set_tm @ F @ A2 ) )
= ( ? [X5: fm] :
( ( member_fm @ X5 @ A2 )
& ( Z2
= ( F @ X5 ) ) ) ) ) ).
% image_iff
thf(fact_754_image__iff,axiom,
! [Z2: set_nat,F: fm > set_nat,A2: set_fm] :
( ( member_set_nat @ Z2 @ ( image_fm_set_nat @ F @ A2 ) )
= ( ? [X5: fm] :
( ( member_fm @ X5 @ A2 )
& ( Z2
= ( F @ X5 ) ) ) ) ) ).
% image_iff
thf(fact_755_image__iff,axiom,
! [Z2: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ Z2 @ ( image_nat_nat @ F @ A2 ) )
= ( ? [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( Z2
= ( F @ X5 ) ) ) ) ) ).
% image_iff
thf(fact_756_equalityE,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 = B )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A2 ) ) ) ).
% equalityE
thf(fact_757_equalityE,axiom,
! [A2: set_tm,B: set_tm] :
( ( A2 = B )
=> ~ ( ( ord_less_eq_set_tm @ A2 @ B )
=> ~ ( ord_less_eq_set_tm @ B @ A2 ) ) ) ).
% equalityE
thf(fact_758_subsetD,axiom,
! [A2: set_list_fm,B: set_list_fm,C: list_fm] :
( ( ord_le7838213414353715577ist_fm @ A2 @ B )
=> ( ( member_list_fm @ C @ A2 )
=> ( member_list_fm @ C @ B ) ) ) ).
% subsetD
thf(fact_759_subsetD,axiom,
! [A2: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_760_subsetD,axiom,
! [A2: set_tm,B: set_tm,C: tm] :
( ( ord_less_eq_set_tm @ A2 @ B )
=> ( ( member_tm @ C @ A2 )
=> ( member_tm @ C @ B ) ) ) ).
% subsetD
thf(fact_761_in__mono,axiom,
! [A2: set_list_fm,B: set_list_fm,X2: list_fm] :
( ( ord_le7838213414353715577ist_fm @ A2 @ B )
=> ( ( member_list_fm @ X2 @ A2 )
=> ( member_list_fm @ X2 @ B ) ) ) ).
% in_mono
thf(fact_762_in__mono,axiom,
! [A2: set_nat,B: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_nat @ X2 @ B ) ) ) ).
% in_mono
thf(fact_763_in__mono,axiom,
! [A2: set_tm,B: set_tm,X2: tm] :
( ( ord_less_eq_set_tm @ A2 @ B )
=> ( ( member_tm @ X2 @ A2 )
=> ( member_tm @ X2 @ B ) ) ) ).
% in_mono
thf(fact_764_imageI,axiom,
! [X2: tm,A2: set_tm,F: tm > tm] :
( ( member_tm @ X2 @ A2 )
=> ( member_tm @ ( F @ X2 ) @ ( image_tm_tm @ F @ A2 ) ) ) ).
% imageI
thf(fact_765_imageI,axiom,
! [X2: tm,A2: set_tm,F: tm > nat] :
( ( member_tm @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( image_tm_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_766_imageI,axiom,
! [X2: nat,A2: set_nat,F: nat > tm] :
( ( member_nat @ X2 @ A2 )
=> ( member_tm @ ( F @ X2 ) @ ( image_nat_tm @ F @ A2 ) ) ) ).
% imageI
thf(fact_767_imageI,axiom,
! [X2: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ ( F @ X2 ) @ ( image_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_768_imageI,axiom,
! [X2: fm,A2: set_fm,F: fm > set_tm] :
( ( member_fm @ X2 @ A2 )
=> ( member_set_tm @ ( F @ X2 ) @ ( image_fm_set_tm @ F @ A2 ) ) ) ).
% imageI
thf(fact_769_imageI,axiom,
! [X2: fm,A2: set_fm,F: fm > set_nat] :
( ( member_fm @ X2 @ A2 )
=> ( member_set_nat @ ( F @ X2 ) @ ( image_fm_set_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_770_imageI,axiom,
! [X2: tm,A2: set_tm,F: tm > set_nat] :
( ( member_tm @ X2 @ A2 )
=> ( member_set_nat @ ( F @ X2 ) @ ( image_tm_set_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_771_imageI,axiom,
! [X2: tm,A2: set_tm,F: tm > set_tm] :
( ( member_tm @ X2 @ A2 )
=> ( member_set_tm @ ( F @ X2 ) @ ( image_tm_set_tm @ F @ A2 ) ) ) ).
% imageI
thf(fact_772_imageI,axiom,
! [X2: tm,A2: set_tm,F: tm > list_fm] :
( ( member_tm @ X2 @ A2 )
=> ( member_list_fm @ ( F @ X2 ) @ ( image_tm_list_fm @ F @ A2 ) ) ) ).
% imageI
thf(fact_773_imageI,axiom,
! [X2: nat,A2: set_nat,F: nat > list_fm] :
( ( member_nat @ X2 @ A2 )
=> ( member_list_fm @ ( F @ X2 ) @ ( image_nat_list_fm @ F @ A2 ) ) ) ).
% imageI
thf(fact_774_dual__order_Otrans,axiom,
! [B2: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( ord_less_eq_set_nat @ C @ B2 )
=> ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_775_dual__order_Otrans,axiom,
! [B2: set_tm,A: set_tm,C: set_tm] :
( ( ord_less_eq_set_tm @ B2 @ A )
=> ( ( ord_less_eq_set_tm @ C @ B2 )
=> ( ord_less_eq_set_tm @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_776_dual__order_Otrans,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_777_dual__order_Oantisym,axiom,
! [B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_778_dual__order_Oantisym,axiom,
! [B2: set_tm,A: set_tm] :
( ( ord_less_eq_set_tm @ B2 @ A )
=> ( ( ord_less_eq_set_tm @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_779_dual__order_Oantisym,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_780_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [A5: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A5 )
& ( ord_less_eq_set_nat @ A5 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_781_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: set_tm,Z: set_tm] : ( Y2 = Z ) )
= ( ^ [A5: set_tm,B6: set_tm] :
( ( ord_less_eq_set_tm @ B6 @ A5 )
& ( ord_less_eq_set_tm @ A5 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_782_dual__order_Oeq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A5: nat,B6: nat] :
( ( ord_less_eq_nat @ B6 @ A5 )
& ( ord_less_eq_nat @ A5 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_783_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B2: nat] :
( ! [A3: nat,B7: nat] :
( ( ord_less_eq_nat @ A3 @ B7 )
=> ( P @ A3 @ B7 ) )
=> ( ! [A3: nat,B7: nat] :
( ( P @ B7 @ A3 )
=> ( P @ A3 @ B7 ) )
=> ( P @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_784_order__trans,axiom,
! [X2: set_nat,Y4: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y4 )
=> ( ( ord_less_eq_set_nat @ Y4 @ Z2 )
=> ( ord_less_eq_set_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_785_order__trans,axiom,
! [X2: set_tm,Y4: set_tm,Z2: set_tm] :
( ( ord_less_eq_set_tm @ X2 @ Y4 )
=> ( ( ord_less_eq_set_tm @ Y4 @ Z2 )
=> ( ord_less_eq_set_tm @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_786_order__trans,axiom,
! [X2: nat,Y4: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ Z2 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_787_order_Otrans,axiom,
! [A: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_788_order_Otrans,axiom,
! [A: set_tm,B2: set_tm,C: set_tm] :
( ( ord_less_eq_set_tm @ A @ B2 )
=> ( ( ord_less_eq_set_tm @ B2 @ C )
=> ( ord_less_eq_set_tm @ A @ C ) ) ) ).
% order.trans
thf(fact_789_order_Otrans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_790_order__antisym,axiom,
! [X2: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y4 )
=> ( ( ord_less_eq_set_nat @ Y4 @ X2 )
=> ( X2 = Y4 ) ) ) ).
% order_antisym
thf(fact_791_order__antisym,axiom,
! [X2: set_tm,Y4: set_tm] :
( ( ord_less_eq_set_tm @ X2 @ Y4 )
=> ( ( ord_less_eq_set_tm @ Y4 @ X2 )
=> ( X2 = Y4 ) ) ) ).
% order_antisym
thf(fact_792_order__antisym,axiom,
! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ X2 )
=> ( X2 = Y4 ) ) ) ).
% order_antisym
thf(fact_793_ord__le__eq__trans,axiom,
! [A: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_794_ord__le__eq__trans,axiom,
! [A: set_tm,B2: set_tm,C: set_tm] :
( ( ord_less_eq_set_tm @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_tm @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_795_ord__le__eq__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_796_ord__eq__le__trans,axiom,
! [A: set_nat,B2: set_nat,C: set_nat] :
( ( A = B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_797_ord__eq__le__trans,axiom,
! [A: set_tm,B2: set_tm,C: set_tm] :
( ( A = B2 )
=> ( ( ord_less_eq_set_tm @ B2 @ C )
=> ( ord_less_eq_set_tm @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_798_ord__eq__le__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( A = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_799_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [X5: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X5 @ Y )
& ( ord_less_eq_set_nat @ Y @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_800_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: set_tm,Z: set_tm] : ( Y2 = Z ) )
= ( ^ [X5: set_tm,Y: set_tm] :
( ( ord_less_eq_set_tm @ X5 @ Y )
& ( ord_less_eq_set_tm @ Y @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_801_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [X5: nat,Y: nat] :
( ( ord_less_eq_nat @ X5 @ Y )
& ( ord_less_eq_nat @ Y @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_802_le__cases3,axiom,
! [X2: nat,Y4: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y4 )
=> ~ ( ord_less_eq_nat @ Y4 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y4 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y4 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y4 )
=> ~ ( ord_less_eq_nat @ Y4 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y4 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_803_nle__le,axiom,
! [A: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_804_the__elem__image__unique,axiom,
! [A2: set_fm,F: fm > set_tm,X2: fm] :
( ( A2 != bot_bot_set_fm )
=> ( ! [Y3: fm] :
( ( member_fm @ Y3 @ A2 )
=> ( ( F @ Y3 )
= ( F @ X2 ) ) )
=> ( ( the_elem_set_tm @ ( image_fm_set_tm @ F @ A2 ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_805_the__elem__image__unique,axiom,
! [A2: set_fm,F: fm > set_nat,X2: fm] :
( ( A2 != bot_bot_set_fm )
=> ( ! [Y3: fm] :
( ( member_fm @ Y3 @ A2 )
=> ( ( F @ Y3 )
= ( F @ X2 ) ) )
=> ( ( the_elem_set_nat @ ( image_fm_set_nat @ F @ A2 ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_806_the__elem__image__unique,axiom,
! [A2: set_tm,F: tm > set_nat,X2: tm] :
( ( A2 != bot_bot_set_tm )
=> ( ! [Y3: tm] :
( ( member_tm @ Y3 @ A2 )
=> ( ( F @ Y3 )
= ( F @ X2 ) ) )
=> ( ( the_elem_set_nat @ ( image_tm_set_nat @ F @ A2 ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_807_the__elem__image__unique,axiom,
! [A2: set_tm,F: tm > set_tm,X2: tm] :
( ( A2 != bot_bot_set_tm )
=> ( ! [Y3: tm] :
( ( member_tm @ Y3 @ A2 )
=> ( ( F @ Y3 )
= ( F @ X2 ) ) )
=> ( ( the_elem_set_tm @ ( image_tm_set_tm @ F @ A2 ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_808_the__elem__image__unique,axiom,
! [A2: set_nat,F: nat > nat,X2: nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ A2 )
=> ( ( F @ Y3 )
= ( F @ X2 ) ) )
=> ( ( the_elem_nat @ ( image_nat_nat @ F @ A2 ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_809_subset__Collect__iff,axiom,
! [B: set_list_fm,A2: set_list_fm,P: list_fm > $o] :
( ( ord_le7838213414353715577ist_fm @ B @ A2 )
=> ( ( ord_le7838213414353715577ist_fm @ B
@ ( collect_list_fm
@ ^ [X5: list_fm] :
( ( member_list_fm @ X5 @ A2 )
& ( P @ X5 ) ) ) )
= ( ! [X5: list_fm] :
( ( member_list_fm @ X5 @ B )
=> ( P @ X5 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_810_subset__Collect__iff,axiom,
! [B: set_nat,A2: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_less_eq_set_nat @ B
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( P @ X5 ) ) ) )
= ( ! [X5: nat] :
( ( member_nat @ X5 @ B )
=> ( P @ X5 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_811_subset__Collect__iff,axiom,
! [B: set_tm,A2: set_tm,P: tm > $o] :
( ( ord_less_eq_set_tm @ B @ A2 )
=> ( ( ord_less_eq_set_tm @ B
@ ( collect_tm
@ ^ [X5: tm] :
( ( member_tm @ X5 @ A2 )
& ( P @ X5 ) ) ) )
= ( ! [X5: tm] :
( ( member_tm @ X5 @ B )
=> ( P @ X5 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_812_subset__CollectI,axiom,
! [B: set_list_fm,A2: set_list_fm,Q: list_fm > $o,P: list_fm > $o] :
( ( ord_le7838213414353715577ist_fm @ B @ A2 )
=> ( ! [X: list_fm] :
( ( member_list_fm @ X @ B )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_le7838213414353715577ist_fm
@ ( collect_list_fm
@ ^ [X5: list_fm] :
( ( member_list_fm @ X5 @ B )
& ( Q @ X5 ) ) )
@ ( collect_list_fm
@ ^ [X5: list_fm] :
( ( member_list_fm @ X5 @ A2 )
& ( P @ X5 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_813_subset__CollectI,axiom,
! [B: set_nat,A2: set_nat,Q: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ B )
& ( Q @ X5 ) ) )
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( P @ X5 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_814_subset__CollectI,axiom,
! [B: set_tm,A2: set_tm,Q: tm > $o,P: tm > $o] :
( ( ord_less_eq_set_tm @ B @ A2 )
=> ( ! [X: tm] :
( ( member_tm @ X @ B )
=> ( ( Q @ X )
=> ( P @ X ) ) )
=> ( ord_less_eq_set_tm
@ ( collect_tm
@ ^ [X5: tm] :
( ( member_tm @ X5 @ B )
& ( Q @ X5 ) ) )
@ ( collect_tm
@ ^ [X5: tm] :
( ( member_tm @ X5 @ A2 )
& ( P @ X5 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_815_Collect__subset,axiom,
! [A2: set_list_fm,P: list_fm > $o] :
( ord_le7838213414353715577ist_fm
@ ( collect_list_fm
@ ^ [X5: list_fm] :
( ( member_list_fm @ X5 @ A2 )
& ( P @ X5 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_816_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ A2 )
& ( P @ X5 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_817_Collect__subset,axiom,
! [A2: set_tm,P: tm > $o] :
( ord_less_eq_set_tm
@ ( collect_tm
@ ^ [X5: tm] :
( ( member_tm @ X5 @ A2 )
& ( P @ X5 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_818_surjD,axiom,
! [F: tm > set_nat,Y4: set_nat] :
( ( ( image_tm_set_nat @ F @ top_top_set_tm )
= top_top_set_set_nat )
=> ? [X: tm] :
( Y4
= ( F @ X ) ) ) ).
% surjD
thf(fact_819_surjD,axiom,
! [F: tm > set_tm,Y4: set_tm] :
( ( ( image_tm_set_tm @ F @ top_top_set_tm )
= top_top_set_set_tm )
=> ? [X: tm] :
( Y4
= ( F @ X ) ) ) ).
% surjD
thf(fact_820_surjD,axiom,
! [F: fm > set_tm,Y4: set_tm] :
( ( ( image_fm_set_tm @ F @ top_top_set_fm )
= top_top_set_set_tm )
=> ? [X: fm] :
( Y4
= ( F @ X ) ) ) ).
% surjD
thf(fact_821_surjD,axiom,
! [F: fm > set_nat,Y4: set_nat] :
( ( ( image_fm_set_nat @ F @ top_top_set_fm )
= top_top_set_set_nat )
=> ? [X: fm] :
( Y4
= ( F @ X ) ) ) ).
% surjD
thf(fact_822_surjD,axiom,
! [F: nat > nat,Y4: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ? [X: nat] :
( Y4
= ( F @ X ) ) ) ).
% surjD
thf(fact_823_surjD,axiom,
! [F: nat > $o,Y4: $o] :
( ( ( image_nat_o @ F @ top_top_set_nat )
= top_top_set_o )
=> ? [X: nat] :
( Y4
= ( F @ X ) ) ) ).
% surjD
thf(fact_824_surjD,axiom,
! [F: $o > nat,Y4: nat] :
( ( ( image_o_nat @ F @ top_top_set_o )
= top_top_set_nat )
=> ? [X: $o] :
( Y4
= ( F @ X ) ) ) ).
% surjD
thf(fact_825_surjD,axiom,
! [F: $o > $o,Y4: $o] :
( ( ( image_o_o @ F @ top_top_set_o )
= top_top_set_o )
=> ? [X: $o] :
( Y4
= ( F @ X ) ) ) ).
% surjD
thf(fact_826_surjE,axiom,
! [F: tm > set_nat,Y4: set_nat] :
( ( ( image_tm_set_nat @ F @ top_top_set_tm )
= top_top_set_set_nat )
=> ~ ! [X: tm] :
( Y4
!= ( F @ X ) ) ) ).
% surjE
thf(fact_827_surjE,axiom,
! [F: tm > set_tm,Y4: set_tm] :
( ( ( image_tm_set_tm @ F @ top_top_set_tm )
= top_top_set_set_tm )
=> ~ ! [X: tm] :
( Y4
!= ( F @ X ) ) ) ).
% surjE
thf(fact_828_surjE,axiom,
! [F: fm > set_tm,Y4: set_tm] :
( ( ( image_fm_set_tm @ F @ top_top_set_fm )
= top_top_set_set_tm )
=> ~ ! [X: fm] :
( Y4
!= ( F @ X ) ) ) ).
% surjE
thf(fact_829_surjE,axiom,
! [F: fm > set_nat,Y4: set_nat] :
( ( ( image_fm_set_nat @ F @ top_top_set_fm )
= top_top_set_set_nat )
=> ~ ! [X: fm] :
( Y4
!= ( F @ X ) ) ) ).
% surjE
thf(fact_830_surjE,axiom,
! [F: nat > nat,Y4: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ~ ! [X: nat] :
( Y4
!= ( F @ X ) ) ) ).
% surjE
thf(fact_831_surjE,axiom,
! [F: nat > $o,Y4: $o] :
( ( ( image_nat_o @ F @ top_top_set_nat )
= top_top_set_o )
=> ~ ! [X: nat] :
( Y4
= ( ~ ( F @ X ) ) ) ) ).
% surjE
thf(fact_832_surjE,axiom,
! [F: $o > nat,Y4: nat] :
( ( ( image_o_nat @ F @ top_top_set_o )
= top_top_set_nat )
=> ~ ! [X: $o] :
( Y4
!= ( F @ X ) ) ) ).
% surjE
thf(fact_833_surjE,axiom,
! [F: $o > $o,Y4: $o] :
( ( ( image_o_o @ F @ top_top_set_o )
= top_top_set_o )
=> ~ ! [X: $o] :
( Y4
= ( ~ ( F @ X ) ) ) ) ).
% surjE
thf(fact_834_surjI,axiom,
! [G: tm > set_nat,F: set_nat > tm] :
( ! [X: set_nat] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_tm_set_nat @ G @ top_top_set_tm )
= top_top_set_set_nat ) ) ).
% surjI
thf(fact_835_surjI,axiom,
! [G: tm > set_tm,F: set_tm > tm] :
( ! [X: set_tm] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_tm_set_tm @ G @ top_top_set_tm )
= top_top_set_set_tm ) ) ).
% surjI
thf(fact_836_surjI,axiom,
! [G: fm > set_tm,F: set_tm > fm] :
( ! [X: set_tm] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_fm_set_tm @ G @ top_top_set_fm )
= top_top_set_set_tm ) ) ).
% surjI
thf(fact_837_surjI,axiom,
! [G: fm > set_nat,F: set_nat > fm] :
( ! [X: set_nat] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_fm_set_nat @ G @ top_top_set_fm )
= top_top_set_set_nat ) ) ).
% surjI
thf(fact_838_surjI,axiom,
! [G: nat > nat,F: nat > nat] :
( ! [X: nat] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_839_surjI,axiom,
! [G: nat > $o,F: $o > nat] :
( ! [X: $o] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_nat_o @ G @ top_top_set_nat )
= top_top_set_o ) ) ).
% surjI
thf(fact_840_surjI,axiom,
! [G: $o > nat,F: nat > $o] :
( ! [X: nat] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_o_nat @ G @ top_top_set_o )
= top_top_set_nat ) ) ).
% surjI
thf(fact_841_surjI,axiom,
! [G: $o > $o,F: $o > $o] :
( ! [X: $o] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image_o_o @ G @ top_top_set_o )
= top_top_set_o ) ) ).
% surjI
thf(fact_842_surj__def,axiom,
! [F: tm > set_nat] :
( ( ( image_tm_set_nat @ F @ top_top_set_tm )
= top_top_set_set_nat )
= ( ! [Y: set_nat] :
? [X5: tm] :
( Y
= ( F @ X5 ) ) ) ) ).
% surj_def
thf(fact_843_surj__def,axiom,
! [F: tm > set_tm] :
( ( ( image_tm_set_tm @ F @ top_top_set_tm )
= top_top_set_set_tm )
= ( ! [Y: set_tm] :
? [X5: tm] :
( Y
= ( F @ X5 ) ) ) ) ).
% surj_def
thf(fact_844_surj__def,axiom,
! [F: fm > set_tm] :
( ( ( image_fm_set_tm @ F @ top_top_set_fm )
= top_top_set_set_tm )
= ( ! [Y: set_tm] :
? [X5: fm] :
( Y
= ( F @ X5 ) ) ) ) ).
% surj_def
thf(fact_845_surj__def,axiom,
! [F: fm > set_nat] :
( ( ( image_fm_set_nat @ F @ top_top_set_fm )
= top_top_set_set_nat )
= ( ! [Y: set_nat] :
? [X5: fm] :
( Y
= ( F @ X5 ) ) ) ) ).
% surj_def
thf(fact_846_surj__def,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ! [Y: nat] :
? [X5: nat] :
( Y
= ( F @ X5 ) ) ) ) ).
% surj_def
thf(fact_847_surj__def,axiom,
! [F: nat > $o] :
( ( ( image_nat_o @ F @ top_top_set_nat )
= top_top_set_o )
= ( ! [Y: $o] :
? [X5: nat] :
( Y
= ( F @ X5 ) ) ) ) ).
% surj_def
thf(fact_848_surj__def,axiom,
! [F: $o > nat] :
( ( ( image_o_nat @ F @ top_top_set_o )
= top_top_set_nat )
= ( ! [Y: nat] :
? [X5: $o] :
( Y
= ( F @ X5 ) ) ) ) ).
% surj_def
thf(fact_849_surj__def,axiom,
! [F: $o > $o] :
( ( ( image_o_o @ F @ top_top_set_o )
= top_top_set_o )
= ( ! [Y: $o] :
? [X5: $o] :
( Y
= ( F @ X5 ) ) ) ) ).
% surj_def
thf(fact_850_range__eqI,axiom,
! [B2: set_nat,F: tm > set_nat,X2: tm] :
( ( B2
= ( F @ X2 ) )
=> ( member_set_nat @ B2 @ ( image_tm_set_nat @ F @ top_top_set_tm ) ) ) ).
% range_eqI
thf(fact_851_range__eqI,axiom,
! [B2: set_tm,F: tm > set_tm,X2: tm] :
( ( B2
= ( F @ X2 ) )
=> ( member_set_tm @ B2 @ ( image_tm_set_tm @ F @ top_top_set_tm ) ) ) ).
% range_eqI
thf(fact_852_range__eqI,axiom,
! [B2: set_tm,F: fm > set_tm,X2: fm] :
( ( B2
= ( F @ X2 ) )
=> ( member_set_tm @ B2 @ ( image_fm_set_tm @ F @ top_top_set_fm ) ) ) ).
% range_eqI
thf(fact_853_range__eqI,axiom,
! [B2: set_nat,F: fm > set_nat,X2: fm] :
( ( B2
= ( F @ X2 ) )
=> ( member_set_nat @ B2 @ ( image_fm_set_nat @ F @ top_top_set_fm ) ) ) ).
% range_eqI
thf(fact_854_range__eqI,axiom,
! [B2: tm,F: nat > tm,X2: nat] :
( ( B2
= ( F @ X2 ) )
=> ( member_tm @ B2 @ ( image_nat_tm @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_855_range__eqI,axiom,
! [B2: nat,F: nat > nat,X2: nat] :
( ( B2
= ( F @ X2 ) )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_856_range__eqI,axiom,
! [B2: list_fm,F: nat > list_fm,X2: nat] :
( ( B2
= ( F @ X2 ) )
=> ( member_list_fm @ B2 @ ( image_nat_list_fm @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_857_range__eqI,axiom,
! [B2: tm,F: $o > tm,X2: $o] :
( ( B2
= ( F @ X2 ) )
=> ( member_tm @ B2 @ ( image_o_tm @ F @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_858_range__eqI,axiom,
! [B2: nat,F: $o > nat,X2: $o] :
( ( B2
= ( F @ X2 ) )
=> ( member_nat @ B2 @ ( image_o_nat @ F @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_859_range__eqI,axiom,
! [B2: list_fm,F: $o > list_fm,X2: $o] :
( ( B2
= ( F @ X2 ) )
=> ( member_list_fm @ B2 @ ( image_o_list_fm @ F @ top_top_set_o ) ) ) ).
% range_eqI
thf(fact_860_rangeI,axiom,
! [F: tm > set_nat,X2: tm] : ( member_set_nat @ ( F @ X2 ) @ ( image_tm_set_nat @ F @ top_top_set_tm ) ) ).
% rangeI
thf(fact_861_rangeI,axiom,
! [F: tm > set_tm,X2: tm] : ( member_set_tm @ ( F @ X2 ) @ ( image_tm_set_tm @ F @ top_top_set_tm ) ) ).
% rangeI
thf(fact_862_rangeI,axiom,
! [F: fm > set_tm,X2: fm] : ( member_set_tm @ ( F @ X2 ) @ ( image_fm_set_tm @ F @ top_top_set_fm ) ) ).
% rangeI
thf(fact_863_rangeI,axiom,
! [F: fm > set_nat,X2: fm] : ( member_set_nat @ ( F @ X2 ) @ ( image_fm_set_nat @ F @ top_top_set_fm ) ) ).
% rangeI
thf(fact_864_rangeI,axiom,
! [F: nat > tm,X2: nat] : ( member_tm @ ( F @ X2 ) @ ( image_nat_tm @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_865_rangeI,axiom,
! [F: nat > nat,X2: nat] : ( member_nat @ ( F @ X2 ) @ ( image_nat_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_866_rangeI,axiom,
! [F: nat > list_fm,X2: nat] : ( member_list_fm @ ( F @ X2 ) @ ( image_nat_list_fm @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_867_rangeI,axiom,
! [F: $o > tm,X2: $o] : ( member_tm @ ( F @ X2 ) @ ( image_o_tm @ F @ top_top_set_o ) ) ).
% rangeI
thf(fact_868_rangeI,axiom,
! [F: $o > nat,X2: $o] : ( member_nat @ ( F @ X2 ) @ ( image_o_nat @ F @ top_top_set_o ) ) ).
% rangeI
thf(fact_869_rangeI,axiom,
! [F: $o > list_fm,X2: $o] : ( member_list_fm @ ( F @ X2 ) @ ( image_o_list_fm @ F @ top_top_set_o ) ) ).
% rangeI
thf(fact_870_image__Un,axiom,
! [F: tm > set_nat,A2: set_tm,B: set_tm] :
( ( image_tm_set_nat @ F @ ( sup_sup_set_tm @ A2 @ B ) )
= ( sup_sup_set_set_nat @ ( image_tm_set_nat @ F @ A2 ) @ ( image_tm_set_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_871_image__Un,axiom,
! [F: tm > set_tm,A2: set_tm,B: set_tm] :
( ( image_tm_set_tm @ F @ ( sup_sup_set_tm @ A2 @ B ) )
= ( sup_sup_set_set_tm @ ( image_tm_set_tm @ F @ A2 ) @ ( image_tm_set_tm @ F @ B ) ) ) ).
% image_Un
thf(fact_872_image__Un,axiom,
! [F: fm > set_tm,A2: set_fm,B: set_fm] :
( ( image_fm_set_tm @ F @ ( sup_sup_set_fm @ A2 @ B ) )
= ( sup_sup_set_set_tm @ ( image_fm_set_tm @ F @ A2 ) @ ( image_fm_set_tm @ F @ B ) ) ) ).
% image_Un
thf(fact_873_image__Un,axiom,
! [F: fm > set_nat,A2: set_fm,B: set_fm] :
( ( image_fm_set_nat @ F @ ( sup_sup_set_fm @ A2 @ B ) )
= ( sup_sup_set_set_nat @ ( image_fm_set_nat @ F @ A2 ) @ ( image_fm_set_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_874_image__Un,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat] :
( ( image_nat_nat @ F @ ( sup_sup_set_nat @ A2 @ B ) )
= ( sup_sup_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_875_top_Oextremum__uniqueI,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ top_top_set_o @ A )
=> ( A = top_top_set_o ) ) ).
% top.extremum_uniqueI
thf(fact_876_top_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
=> ( A = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_877_top_Oextremum__uniqueI,axiom,
! [A: set_tm] :
( ( ord_less_eq_set_tm @ top_top_set_tm @ A )
=> ( A = top_top_set_tm ) ) ).
% top.extremum_uniqueI
thf(fact_878_top_Oextremum__unique,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ top_top_set_o @ A )
= ( A = top_top_set_o ) ) ).
% top.extremum_unique
thf(fact_879_top_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
= ( A = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_880_top_Oextremum__unique,axiom,
! [A: set_tm] :
( ( ord_less_eq_set_tm @ top_top_set_tm @ A )
= ( A = top_top_set_tm ) ) ).
% top.extremum_unique
thf(fact_881_top__greatest,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ A @ top_top_set_o ) ).
% top_greatest
thf(fact_882_top__greatest,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% top_greatest
thf(fact_883_top__greatest,axiom,
! [A: set_tm] : ( ord_less_eq_set_tm @ A @ top_top_set_tm ) ).
% top_greatest
thf(fact_884_bot_Oextremum,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).
% bot.extremum
thf(fact_885_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_886_bot_Oextremum,axiom,
! [A: set_tm] : ( ord_less_eq_set_tm @ bot_bot_set_tm @ A ) ).
% bot.extremum
thf(fact_887_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_888_bot_Oextremum__unique,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
= ( A = bot_bot_set_o ) ) ).
% bot.extremum_unique
thf(fact_889_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_890_bot_Oextremum__unique,axiom,
! [A: set_tm] :
( ( ord_less_eq_set_tm @ A @ bot_bot_set_tm )
= ( A = bot_bot_set_tm ) ) ).
% bot.extremum_unique
thf(fact_891_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_892_bot_Oextremum__uniqueI,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
=> ( A = bot_bot_set_o ) ) ).
% bot.extremum_uniqueI
thf(fact_893_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_894_bot_Oextremum__uniqueI,axiom,
! [A: set_tm] :
( ( ord_less_eq_set_tm @ A @ bot_bot_set_tm )
=> ( A = bot_bot_set_tm ) ) ).
% bot.extremum_uniqueI
thf(fact_895_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_896_subset__UNIV,axiom,
! [A2: set_o] : ( ord_less_eq_set_o @ A2 @ top_top_set_o ) ).
% subset_UNIV
thf(fact_897_subset__UNIV,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_898_subset__UNIV,axiom,
! [A2: set_tm] : ( ord_less_eq_set_tm @ A2 @ top_top_set_tm ) ).
% subset_UNIV
thf(fact_899_subset__insertI2,axiom,
! [A2: set_o,B: set_o,B2: $o] :
( ( ord_less_eq_set_o @ A2 @ B )
=> ( ord_less_eq_set_o @ A2 @ ( insert_o @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_900_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_901_subset__insertI2,axiom,
! [A2: set_tm,B: set_tm,B2: tm] :
( ( ord_less_eq_set_tm @ A2 @ B )
=> ( ord_less_eq_set_tm @ A2 @ ( insert_tm2 @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_902_subset__insertI,axiom,
! [B: set_o,A: $o] : ( ord_less_eq_set_o @ B @ ( insert_o @ A @ B ) ) ).
% subset_insertI
thf(fact_903_subset__insertI,axiom,
! [B: set_nat,A: nat] : ( ord_less_eq_set_nat @ B @ ( insert_nat @ A @ B ) ) ).
% subset_insertI
thf(fact_904_subset__insertI,axiom,
! [B: set_tm,A: tm] : ( ord_less_eq_set_tm @ B @ ( insert_tm2 @ A @ B ) ) ).
% subset_insertI
thf(fact_905_subset__insert,axiom,
! [X2: $o,A2: set_o,B: set_o] :
( ~ ( member_o @ X2 @ A2 )
=> ( ( ord_less_eq_set_o @ A2 @ ( insert_o @ X2 @ B ) )
= ( ord_less_eq_set_o @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_906_subset__insert,axiom,
! [X2: list_fm,A2: set_list_fm,B: set_list_fm] :
( ~ ( member_list_fm @ X2 @ A2 )
=> ( ( ord_le7838213414353715577ist_fm @ A2 @ ( insert_list_fm2 @ X2 @ B ) )
= ( ord_le7838213414353715577ist_fm @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_907_subset__insert,axiom,
! [X2: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ X2 @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X2 @ B ) )
= ( ord_less_eq_set_nat @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_908_subset__insert,axiom,
! [X2: tm,A2: set_tm,B: set_tm] :
( ~ ( member_tm @ X2 @ A2 )
=> ( ( ord_less_eq_set_tm @ A2 @ ( insert_tm2 @ X2 @ B ) )
= ( ord_less_eq_set_tm @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_909_insert__mono,axiom,
! [C3: set_o,D2: set_o,A: $o] :
( ( ord_less_eq_set_o @ C3 @ D2 )
=> ( ord_less_eq_set_o @ ( insert_o @ A @ C3 ) @ ( insert_o @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_910_insert__mono,axiom,
! [C3: set_nat,D2: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ C3 @ D2 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A @ C3 ) @ ( insert_nat @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_911_insert__mono,axiom,
! [C3: set_tm,D2: set_tm,A: tm] :
( ( ord_less_eq_set_tm @ C3 @ D2 )
=> ( ord_less_eq_set_tm @ ( insert_tm2 @ A @ C3 ) @ ( insert_tm2 @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_912_inv__into__injective,axiom,
! [A2: set_tm,F: tm > set_nat,X2: set_nat,Y4: set_nat] :
( ( ( hilber8933797207411865247et_nat @ A2 @ F @ X2 )
= ( hilber8933797207411865247et_nat @ A2 @ F @ Y4 ) )
=> ( ( member_set_nat @ X2 @ ( image_tm_set_nat @ F @ A2 ) )
=> ( ( member_set_nat @ Y4 @ ( image_tm_set_nat @ F @ A2 ) )
=> ( X2 = Y4 ) ) ) ) ).
% inv_into_injective
thf(fact_913_inv__into__injective,axiom,
! [A2: set_tm,F: tm > set_tm,X2: set_tm,Y4: set_tm] :
( ( ( hilber5702680533203024108set_tm @ A2 @ F @ X2 )
= ( hilber5702680533203024108set_tm @ A2 @ F @ Y4 ) )
=> ( ( member_set_tm @ X2 @ ( image_tm_set_tm @ F @ A2 ) )
=> ( ( member_set_tm @ Y4 @ ( image_tm_set_tm @ F @ A2 ) )
=> ( X2 = Y4 ) ) ) ) ).
% inv_into_injective
thf(fact_914_inv__into__injective,axiom,
! [A2: set_fm,F: fm > set_tm,X2: set_tm,Y4: set_tm] :
( ( ( hilber6286924743720013790set_tm @ A2 @ F @ X2 )
= ( hilber6286924743720013790set_tm @ A2 @ F @ Y4 ) )
=> ( ( member_set_tm @ X2 @ ( image_fm_set_tm @ F @ A2 ) )
=> ( ( member_set_tm @ Y4 @ ( image_fm_set_tm @ F @ A2 ) )
=> ( X2 = Y4 ) ) ) ) ).
% inv_into_injective
thf(fact_915_inv__into__injective,axiom,
! [A2: set_fm,F: fm > set_nat,X2: set_nat,Y4: set_nat] :
( ( ( hilber5456991776183730989et_nat @ A2 @ F @ X2 )
= ( hilber5456991776183730989et_nat @ A2 @ F @ Y4 ) )
=> ( ( member_set_nat @ X2 @ ( image_fm_set_nat @ F @ A2 ) )
=> ( ( member_set_nat @ Y4 @ ( image_fm_set_nat @ F @ A2 ) )
=> ( X2 = Y4 ) ) ) ) ).
% inv_into_injective
thf(fact_916_inv__into__injective,axiom,
! [A2: set_nat,F: nat > nat,X2: nat,Y4: nat] :
( ( ( hilber3633877196798814958at_nat @ A2 @ F @ X2 )
= ( hilber3633877196798814958at_nat @ A2 @ F @ Y4 ) )
=> ( ( member_nat @ X2 @ ( image_nat_nat @ F @ A2 ) )
=> ( ( member_nat @ Y4 @ ( image_nat_nat @ F @ A2 ) )
=> ( X2 = Y4 ) ) ) ) ).
% inv_into_injective
thf(fact_917_inv__into__into,axiom,
! [X2: tm,F: tm > tm,A2: set_tm] :
( ( member_tm @ X2 @ ( image_tm_tm @ F @ A2 ) )
=> ( member_tm @ ( hilber7929195366230530316_tm_tm @ A2 @ F @ X2 ) @ A2 ) ) ).
% inv_into_into
thf(fact_918_inv__into__into,axiom,
! [X2: tm,F: nat > tm,A2: set_nat] :
( ( member_tm @ X2 @ ( image_nat_tm @ F @ A2 ) )
=> ( member_nat @ ( hilber5150249191788879431nat_tm @ A2 @ F @ X2 ) @ A2 ) ) ).
% inv_into_into
thf(fact_919_inv__into__into,axiom,
! [X2: nat,F: tm > nat,A2: set_tm] :
( ( member_nat @ X2 @ ( image_tm_nat @ F @ A2 ) )
=> ( member_tm @ ( hilber3344278766186360553tm_nat @ A2 @ F @ X2 ) @ A2 ) ) ).
% inv_into_into
thf(fact_920_inv__into__into,axiom,
! [X2: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F @ A2 ) )
=> ( member_nat @ ( hilber3633877196798814958at_nat @ A2 @ F @ X2 ) @ A2 ) ) ).
% inv_into_into
thf(fact_921_inv__into__into,axiom,
! [X2: set_tm,F: fm > set_tm,A2: set_fm] :
( ( member_set_tm @ X2 @ ( image_fm_set_tm @ F @ A2 ) )
=> ( member_fm @ ( hilber6286924743720013790set_tm @ A2 @ F @ X2 ) @ A2 ) ) ).
% inv_into_into
thf(fact_922_inv__into__into,axiom,
! [X2: set_nat,F: fm > set_nat,A2: set_fm] :
( ( member_set_nat @ X2 @ ( image_fm_set_nat @ F @ A2 ) )
=> ( member_fm @ ( hilber5456991776183730989et_nat @ A2 @ F @ X2 ) @ A2 ) ) ).
% inv_into_into
thf(fact_923_inv__into__into,axiom,
! [X2: set_nat,F: tm > set_nat,A2: set_tm] :
( ( member_set_nat @ X2 @ ( image_tm_set_nat @ F @ A2 ) )
=> ( member_tm @ ( hilber8933797207411865247et_nat @ A2 @ F @ X2 ) @ A2 ) ) ).
% inv_into_into
thf(fact_924_inv__into__into,axiom,
! [X2: set_tm,F: tm > set_tm,A2: set_tm] :
( ( member_set_tm @ X2 @ ( image_tm_set_tm @ F @ A2 ) )
=> ( member_tm @ ( hilber5702680533203024108set_tm @ A2 @ F @ X2 ) @ A2 ) ) ).
% inv_into_into
thf(fact_925_inv__into__into,axiom,
! [X2: tm,F: list_fm > tm,A2: set_list_fm] :
( ( member_tm @ X2 @ ( image_list_fm_tm @ F @ A2 ) )
=> ( member_list_fm @ ( hilber1817404320317758200_fm_tm @ A2 @ F @ X2 ) @ A2 ) ) ).
% inv_into_into
thf(fact_926_inv__into__into,axiom,
! [X2: nat,F: list_fm > nat,A2: set_list_fm] :
( ( member_nat @ X2 @ ( image_list_fm_nat @ F @ A2 ) )
=> ( member_list_fm @ ( hilber1704621864209052157fm_nat @ A2 @ F @ X2 ) @ A2 ) ) ).
% inv_into_into
thf(fact_927_f__inv__into__f,axiom,
! [Y4: set_nat,F: tm > set_nat,A2: set_tm] :
( ( member_set_nat @ Y4 @ ( image_tm_set_nat @ F @ A2 ) )
=> ( ( F @ ( hilber8933797207411865247et_nat @ A2 @ F @ Y4 ) )
= Y4 ) ) ).
% f_inv_into_f
thf(fact_928_f__inv__into__f,axiom,
! [Y4: set_tm,F: tm > set_tm,A2: set_tm] :
( ( member_set_tm @ Y4 @ ( image_tm_set_tm @ F @ A2 ) )
=> ( ( F @ ( hilber5702680533203024108set_tm @ A2 @ F @ Y4 ) )
= Y4 ) ) ).
% f_inv_into_f
thf(fact_929_f__inv__into__f,axiom,
! [Y4: set_tm,F: fm > set_tm,A2: set_fm] :
( ( member_set_tm @ Y4 @ ( image_fm_set_tm @ F @ A2 ) )
=> ( ( F @ ( hilber6286924743720013790set_tm @ A2 @ F @ Y4 ) )
= Y4 ) ) ).
% f_inv_into_f
thf(fact_930_f__inv__into__f,axiom,
! [Y4: set_nat,F: fm > set_nat,A2: set_fm] :
( ( member_set_nat @ Y4 @ ( image_fm_set_nat @ F @ A2 ) )
=> ( ( F @ ( hilber5456991776183730989et_nat @ A2 @ F @ Y4 ) )
= Y4 ) ) ).
% f_inv_into_f
thf(fact_931_f__inv__into__f,axiom,
! [Y4: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ Y4 @ ( image_nat_nat @ F @ A2 ) )
=> ( ( F @ ( hilber3633877196798814958at_nat @ A2 @ F @ Y4 ) )
= Y4 ) ) ).
% f_inv_into_f
thf(fact_932_subset__Un__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( ( sup_sup_set_nat @ A4 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_933_subset__Un__eq,axiom,
( ord_less_eq_set_tm
= ( ^ [A4: set_tm,B3: set_tm] :
( ( sup_sup_set_tm @ A4 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_934_subset__UnE,axiom,
! [C3: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ ( sup_sup_set_nat @ A2 @ B ) )
=> ~ ! [A7: set_nat] :
( ( ord_less_eq_set_nat @ A7 @ A2 )
=> ! [B8: set_nat] :
( ( ord_less_eq_set_nat @ B8 @ B )
=> ( C3
!= ( sup_sup_set_nat @ A7 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_935_subset__UnE,axiom,
! [C3: set_tm,A2: set_tm,B: set_tm] :
( ( ord_less_eq_set_tm @ C3 @ ( sup_sup_set_tm @ A2 @ B ) )
=> ~ ! [A7: set_tm] :
( ( ord_less_eq_set_tm @ A7 @ A2 )
=> ! [B8: set_tm] :
( ( ord_less_eq_set_tm @ B8 @ B )
=> ( C3
!= ( sup_sup_set_tm @ A7 @ B8 ) ) ) ) ) ).
% subset_UnE
thf(fact_936_Un__absorb2,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( sup_sup_set_nat @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_937_Un__absorb2,axiom,
! [B: set_tm,A2: set_tm] :
( ( ord_less_eq_set_tm @ B @ A2 )
=> ( ( sup_sup_set_tm @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_938_Un__absorb1,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( sup_sup_set_nat @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_939_Un__absorb1,axiom,
! [A2: set_tm,B: set_tm] :
( ( ord_less_eq_set_tm @ A2 @ B )
=> ( ( sup_sup_set_tm @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_940_Un__upper2,axiom,
! [B: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A2 @ B ) ) ).
% Un_upper2
thf(fact_941_Un__upper2,axiom,
! [B: set_tm,A2: set_tm] : ( ord_less_eq_set_tm @ B @ ( sup_sup_set_tm @ A2 @ B ) ) ).
% Un_upper2
thf(fact_942_Un__upper1,axiom,
! [A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B ) ) ).
% Un_upper1
thf(fact_943_Un__upper1,axiom,
! [A2: set_tm,B: set_tm] : ( ord_less_eq_set_tm @ A2 @ ( sup_sup_set_tm @ A2 @ B ) ) ).
% Un_upper1
thf(fact_944_Un__least,axiom,
! [A2: set_nat,C3: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C3 )
=> ( ( ord_less_eq_set_nat @ B @ C3 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C3 ) ) ) ).
% Un_least
thf(fact_945_Un__least,axiom,
! [A2: set_tm,C3: set_tm,B: set_tm] :
( ( ord_less_eq_set_tm @ A2 @ C3 )
=> ( ( ord_less_eq_set_tm @ B @ C3 )
=> ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ A2 @ B ) @ C3 ) ) ) ).
% Un_least
thf(fact_946_Un__mono,axiom,
! [A2: set_nat,C3: set_nat,B: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C3 )
=> ( ( ord_less_eq_set_nat @ B @ D2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ ( sup_sup_set_nat @ C3 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_947_Un__mono,axiom,
! [A2: set_tm,C3: set_tm,B: set_tm,D2: set_tm] :
( ( ord_less_eq_set_tm @ A2 @ C3 )
=> ( ( ord_less_eq_set_tm @ B @ D2 )
=> ( ord_less_eq_set_tm @ ( sup_sup_set_tm @ A2 @ B ) @ ( sup_sup_set_tm @ C3 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_948_range__composition,axiom,
! [F: nat > nat,G: nat > nat] :
( ( image_nat_nat
@ ^ [X5: nat] : ( F @ ( G @ X5 ) )
@ top_top_set_nat )
= ( image_nat_nat @ F @ ( image_nat_nat @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_949_range__composition,axiom,
! [F: nat > nat,G: $o > nat] :
( ( image_o_nat
@ ^ [X5: $o] : ( F @ ( G @ X5 ) )
@ top_top_set_o )
= ( image_nat_nat @ F @ ( image_o_nat @ G @ top_top_set_o ) ) ) ).
% range_composition
thf(fact_950_range__composition,axiom,
! [F: tm > set_nat,G: tm > tm] :
( ( image_tm_set_nat
@ ^ [X5: tm] : ( F @ ( G @ X5 ) )
@ top_top_set_tm )
= ( image_tm_set_nat @ F @ ( image_tm_tm @ G @ top_top_set_tm ) ) ) ).
% range_composition
thf(fact_951_range__composition,axiom,
! [F: fm > set_nat,G: tm > fm] :
( ( image_tm_set_nat
@ ^ [X5: tm] : ( F @ ( G @ X5 ) )
@ top_top_set_tm )
= ( image_fm_set_nat @ F @ ( image_tm_fm @ G @ top_top_set_tm ) ) ) ).
% range_composition
thf(fact_952_range__composition,axiom,
! [F: tm > set_tm,G: tm > tm] :
( ( image_tm_set_tm
@ ^ [X5: tm] : ( F @ ( G @ X5 ) )
@ top_top_set_tm )
= ( image_tm_set_tm @ F @ ( image_tm_tm @ G @ top_top_set_tm ) ) ) ).
% range_composition
thf(fact_953_range__composition,axiom,
! [F: fm > set_tm,G: tm > fm] :
( ( image_tm_set_tm
@ ^ [X5: tm] : ( F @ ( G @ X5 ) )
@ top_top_set_tm )
= ( image_fm_set_tm @ F @ ( image_tm_fm @ G @ top_top_set_tm ) ) ) ).
% range_composition
thf(fact_954_range__composition,axiom,
! [F: tm > set_tm,G: fm > tm] :
( ( image_fm_set_tm
@ ^ [X5: fm] : ( F @ ( G @ X5 ) )
@ top_top_set_fm )
= ( image_tm_set_tm @ F @ ( image_fm_tm @ G @ top_top_set_fm ) ) ) ).
% range_composition
thf(fact_955_range__composition,axiom,
! [F: fm > set_tm,G: fm > fm] :
( ( image_fm_set_tm
@ ^ [X5: fm] : ( F @ ( G @ X5 ) )
@ top_top_set_fm )
= ( image_fm_set_tm @ F @ ( image_fm_fm @ G @ top_top_set_fm ) ) ) ).
% range_composition
thf(fact_956_range__composition,axiom,
! [F: tm > set_nat,G: fm > tm] :
( ( image_fm_set_nat
@ ^ [X5: fm] : ( F @ ( G @ X5 ) )
@ top_top_set_fm )
= ( image_tm_set_nat @ F @ ( image_fm_tm @ G @ top_top_set_fm ) ) ) ).
% range_composition
thf(fact_957_range__composition,axiom,
! [F: fm > set_nat,G: fm > fm] :
( ( image_fm_set_nat
@ ^ [X5: fm] : ( F @ ( G @ X5 ) )
@ top_top_set_fm )
= ( image_fm_set_nat @ F @ ( image_fm_fm @ G @ top_top_set_fm ) ) ) ).
% range_composition
thf(fact_958_rangeE,axiom,
! [B2: set_nat,F: tm > set_nat] :
( ( member_set_nat @ B2 @ ( image_tm_set_nat @ F @ top_top_set_tm ) )
=> ~ ! [X: tm] :
( B2
!= ( F @ X ) ) ) ).
% rangeE
thf(fact_959_rangeE,axiom,
! [B2: set_tm,F: tm > set_tm] :
( ( member_set_tm @ B2 @ ( image_tm_set_tm @ F @ top_top_set_tm ) )
=> ~ ! [X: tm] :
( B2
!= ( F @ X ) ) ) ).
% rangeE
thf(fact_960_rangeE,axiom,
! [B2: set_tm,F: fm > set_tm] :
( ( member_set_tm @ B2 @ ( image_fm_set_tm @ F @ top_top_set_fm ) )
=> ~ ! [X: fm] :
( B2
!= ( F @ X ) ) ) ).
% rangeE
thf(fact_961_rangeE,axiom,
! [B2: set_nat,F: fm > set_nat] :
( ( member_set_nat @ B2 @ ( image_fm_set_nat @ F @ top_top_set_fm ) )
=> ~ ! [X: fm] :
( B2
!= ( F @ X ) ) ) ).
% rangeE
thf(fact_962_rangeE,axiom,
! [B2: tm,F: nat > tm] :
( ( member_tm @ B2 @ ( image_nat_tm @ F @ top_top_set_nat ) )
=> ~ ! [X: nat] :
( B2
!= ( F @ X ) ) ) ).
% rangeE
thf(fact_963_rangeE,axiom,
! [B2: nat,F: nat > nat] :
( ( member_nat @ B2 @ ( image_nat_nat @ F @ top_top_set_nat ) )
=> ~ ! [X: nat] :
( B2
!= ( F @ X ) ) ) ).
% rangeE
thf(fact_964_rangeE,axiom,
! [B2: list_fm,F: nat > list_fm] :
( ( member_list_fm @ B2 @ ( image_nat_list_fm @ F @ top_top_set_nat ) )
=> ~ ! [X: nat] :
( B2
!= ( F @ X ) ) ) ).
% rangeE
thf(fact_965_rangeE,axiom,
! [B2: tm,F: $o > tm] :
( ( member_tm @ B2 @ ( image_o_tm @ F @ top_top_set_o ) )
=> ~ ! [X: $o] :
( B2
!= ( F @ X ) ) ) ).
% rangeE
thf(fact_966_rangeE,axiom,
! [B2: nat,F: $o > nat] :
( ( member_nat @ B2 @ ( image_o_nat @ F @ top_top_set_o ) )
=> ~ ! [X: $o] :
( B2
!= ( F @ X ) ) ) ).
% rangeE
thf(fact_967_rangeE,axiom,
! [B2: list_fm,F: $o > list_fm] :
( ( member_list_fm @ B2 @ ( image_o_list_fm @ F @ top_top_set_o ) )
=> ~ ! [X: $o] :
( B2
!= ( F @ X ) ) ) ).
% rangeE
thf(fact_968_surj__imp__inv__eq,axiom,
! [F: tm > set_nat,G: set_nat > tm] :
( ( ( image_tm_set_nat @ F @ top_top_set_tm )
= top_top_set_set_nat )
=> ( ! [X: tm] :
( ( G @ ( F @ X ) )
= X )
=> ( ( hilber8933797207411865247et_nat @ top_top_set_tm @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_969_surj__imp__inv__eq,axiom,
! [F: tm > set_tm,G: set_tm > tm] :
( ( ( image_tm_set_tm @ F @ top_top_set_tm )
= top_top_set_set_tm )
=> ( ! [X: tm] :
( ( G @ ( F @ X ) )
= X )
=> ( ( hilber5702680533203024108set_tm @ top_top_set_tm @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_970_surj__imp__inv__eq,axiom,
! [F: fm > set_tm,G: set_tm > fm] :
( ( ( image_fm_set_tm @ F @ top_top_set_fm )
= top_top_set_set_tm )
=> ( ! [X: fm] :
( ( G @ ( F @ X ) )
= X )
=> ( ( hilber6286924743720013790set_tm @ top_top_set_fm @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_971_surj__imp__inv__eq,axiom,
! [F: fm > set_nat,G: set_nat > fm] :
( ( ( image_fm_set_nat @ F @ top_top_set_fm )
= top_top_set_set_nat )
=> ( ! [X: fm] :
( ( G @ ( F @ X ) )
= X )
=> ( ( hilber5456991776183730989et_nat @ top_top_set_fm @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_972_surj__imp__inv__eq,axiom,
! [F: nat > nat,G: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ! [X: nat] :
( ( G @ ( F @ X ) )
= X )
=> ( ( hilber3633877196798814958at_nat @ top_top_set_nat @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_973_surj__imp__inv__eq,axiom,
! [F: nat > $o,G: $o > nat] :
( ( ( image_nat_o @ F @ top_top_set_nat )
= top_top_set_o )
=> ( ! [X: nat] :
( ( G @ ( F @ X ) )
= X )
=> ( ( hilber3873338068935991546_nat_o @ top_top_set_nat @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_974_surj__imp__inv__eq,axiom,
! [F: $o > nat,G: nat > $o] :
( ( ( image_o_nat @ F @ top_top_set_o )
= top_top_set_nat )
=> ( ! [X: $o] :
( ( G @ ( F @ X ) )
= X )
=> ( ( hilber770666707827806236_o_nat @ top_top_set_o @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_975_surj__imp__inv__eq,axiom,
! [F: $o > $o,G: $o > $o] :
( ( ( image_o_o @ F @ top_top_set_o )
= top_top_set_o )
=> ( ! [X: $o] :
( ( G @ ( F @ X ) )
= X )
=> ( ( hilbert_inv_into_o_o @ top_top_set_o @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_976_image__f__inv__f,axiom,
! [F: nat > nat,A2: set_nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ( image_nat_nat @ F @ ( image_nat_nat @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F ) @ A2 ) )
= A2 ) ) ).
% image_f_inv_f
thf(fact_977_image__f__inv__f,axiom,
! [F: nat > $o,A2: set_o] :
( ( ( image_nat_o @ F @ top_top_set_nat )
= top_top_set_o )
=> ( ( image_nat_o @ F @ ( image_o_nat @ ( hilber3873338068935991546_nat_o @ top_top_set_nat @ F ) @ A2 ) )
= A2 ) ) ).
% image_f_inv_f
thf(fact_978_image__f__inv__f,axiom,
! [F: $o > nat,A2: set_nat] :
( ( ( image_o_nat @ F @ top_top_set_o )
= top_top_set_nat )
=> ( ( image_o_nat @ F @ ( image_nat_o @ ( hilber770666707827806236_o_nat @ top_top_set_o @ F ) @ A2 ) )
= A2 ) ) ).
% image_f_inv_f
thf(fact_979_image__f__inv__f,axiom,
! [F: $o > $o,A2: set_o] :
( ( ( image_o_o @ F @ top_top_set_o )
= top_top_set_o )
=> ( ( image_o_o @ F @ ( image_o_o @ ( hilbert_inv_into_o_o @ top_top_set_o @ F ) @ A2 ) )
= A2 ) ) ).
% image_f_inv_f
thf(fact_980_image__f__inv__f,axiom,
! [F: set_nat > tm,A2: set_tm] :
( ( ( image_set_nat_tm @ F @ top_top_set_set_nat )
= top_top_set_tm )
=> ( ( image_set_nat_tm @ F @ ( image_tm_set_nat @ ( hilber6297111070490487825nat_tm @ top_top_set_set_nat @ F ) @ A2 ) )
= A2 ) ) ).
% image_f_inv_f
thf(fact_981_image__f__inv__f,axiom,
! [F: set_tm > tm,A2: set_tm] :
( ( ( image_set_tm_tm @ F @ top_top_set_set_tm )
= top_top_set_tm )
=> ( ( image_set_tm_tm @ F @ ( image_tm_set_tm @ ( hilber6730722110953316652_tm_tm @ top_top_set_set_tm @ F ) @ A2 ) )
= A2 ) ) ).
% image_f_inv_f
thf(fact_982_image__f__inv__f,axiom,
! [F: set_tm > fm,A2: set_fm] :
( ( ( image_set_tm_fm @ F @ top_top_set_set_tm )
= top_top_set_fm )
=> ( ( image_set_tm_fm @ F @ ( image_fm_set_tm @ ( hilber6730722110952398266_tm_fm @ top_top_set_set_tm @ F ) @ A2 ) )
= A2 ) ) ).
% image_f_inv_f
thf(fact_983_image__f__inv__f,axiom,
! [F: set_nat > fm,A2: set_fm] :
( ( ( image_set_nat_fm @ F @ top_top_set_set_nat )
= top_top_set_fm )
=> ( ( image_set_nat_fm @ F @ ( image_fm_set_nat @ ( hilber6297111070489569439nat_fm @ top_top_set_set_nat @ F ) @ A2 ) )
= A2 ) ) ).
% image_f_inv_f
thf(fact_984_image__f__inv__f,axiom,
! [F: tm > set_nat,A2: set_set_nat] :
( ( ( image_tm_set_nat @ F @ top_top_set_tm )
= top_top_set_set_nat )
=> ( ( image_tm_set_nat @ F @ ( image_set_nat_tm @ ( hilber8933797207411865247et_nat @ top_top_set_tm @ F ) @ A2 ) )
= A2 ) ) ).
% image_f_inv_f
thf(fact_985_image__f__inv__f,axiom,
! [F: tm > set_tm,A2: set_set_tm] :
( ( ( image_tm_set_tm @ F @ top_top_set_tm )
= top_top_set_set_tm )
=> ( ( image_tm_set_tm @ F @ ( image_set_tm_tm @ ( hilber5702680533203024108set_tm @ top_top_set_tm @ F ) @ A2 ) )
= A2 ) ) ).
% image_f_inv_f
thf(fact_986_surj__iff__all,axiom,
! [F: tm > set_nat] :
( ( ( image_tm_set_nat @ F @ top_top_set_tm )
= top_top_set_set_nat )
= ( ! [X5: set_nat] :
( ( F @ ( hilber8933797207411865247et_nat @ top_top_set_tm @ F @ X5 ) )
= X5 ) ) ) ).
% surj_iff_all
thf(fact_987_surj__iff__all,axiom,
! [F: tm > set_tm] :
( ( ( image_tm_set_tm @ F @ top_top_set_tm )
= top_top_set_set_tm )
= ( ! [X5: set_tm] :
( ( F @ ( hilber5702680533203024108set_tm @ top_top_set_tm @ F @ X5 ) )
= X5 ) ) ) ).
% surj_iff_all
thf(fact_988_surj__iff__all,axiom,
! [F: fm > set_tm] :
( ( ( image_fm_set_tm @ F @ top_top_set_fm )
= top_top_set_set_tm )
= ( ! [X5: set_tm] :
( ( F @ ( hilber6286924743720013790set_tm @ top_top_set_fm @ F @ X5 ) )
= X5 ) ) ) ).
% surj_iff_all
thf(fact_989_surj__iff__all,axiom,
! [F: fm > set_nat] :
( ( ( image_fm_set_nat @ F @ top_top_set_fm )
= top_top_set_set_nat )
= ( ! [X5: set_nat] :
( ( F @ ( hilber5456991776183730989et_nat @ top_top_set_fm @ F @ X5 ) )
= X5 ) ) ) ).
% surj_iff_all
thf(fact_990_surj__iff__all,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ! [X5: nat] :
( ( F @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ X5 ) )
= X5 ) ) ) ).
% surj_iff_all
thf(fact_991_surj__iff__all,axiom,
! [F: nat > $o] :
( ( ( image_nat_o @ F @ top_top_set_nat )
= top_top_set_o )
= ( ! [X5: $o] :
( ( F @ ( hilber3873338068935991546_nat_o @ top_top_set_nat @ F @ X5 ) )
= X5 ) ) ) ).
% surj_iff_all
thf(fact_992_surj__iff__all,axiom,
! [F: $o > nat] :
( ( ( image_o_nat @ F @ top_top_set_o )
= top_top_set_nat )
= ( ! [X5: nat] :
( ( F @ ( hilber770666707827806236_o_nat @ top_top_set_o @ F @ X5 ) )
= X5 ) ) ) ).
% surj_iff_all
thf(fact_993_surj__iff__all,axiom,
! [F: $o > $o] :
( ( ( image_o_o @ F @ top_top_set_o )
= top_top_set_o )
= ( ! [X5: $o] :
( ( F @ ( hilbert_inv_into_o_o @ top_top_set_o @ F @ X5 ) )
= X5 ) ) ) ).
% surj_iff_all
thf(fact_994_surj__f__inv__f,axiom,
! [F: tm > set_nat,Y4: set_nat] :
( ( ( image_tm_set_nat @ F @ top_top_set_tm )
= top_top_set_set_nat )
=> ( ( F @ ( hilber8933797207411865247et_nat @ top_top_set_tm @ F @ Y4 ) )
= Y4 ) ) ).
% surj_f_inv_f
thf(fact_995_surj__f__inv__f,axiom,
! [F: tm > set_tm,Y4: set_tm] :
( ( ( image_tm_set_tm @ F @ top_top_set_tm )
= top_top_set_set_tm )
=> ( ( F @ ( hilber5702680533203024108set_tm @ top_top_set_tm @ F @ Y4 ) )
= Y4 ) ) ).
% surj_f_inv_f
thf(fact_996_surj__f__inv__f,axiom,
! [F: fm > set_tm,Y4: set_tm] :
( ( ( image_fm_set_tm @ F @ top_top_set_fm )
= top_top_set_set_tm )
=> ( ( F @ ( hilber6286924743720013790set_tm @ top_top_set_fm @ F @ Y4 ) )
= Y4 ) ) ).
% surj_f_inv_f
thf(fact_997_surj__f__inv__f,axiom,
! [F: fm > set_nat,Y4: set_nat] :
( ( ( image_fm_set_nat @ F @ top_top_set_fm )
= top_top_set_set_nat )
=> ( ( F @ ( hilber5456991776183730989et_nat @ top_top_set_fm @ F @ Y4 ) )
= Y4 ) ) ).
% surj_f_inv_f
thf(fact_998_surj__f__inv__f,axiom,
! [F: nat > nat,Y4: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ( F @ ( hilber3633877196798814958at_nat @ top_top_set_nat @ F @ Y4 ) )
= Y4 ) ) ).
% surj_f_inv_f
thf(fact_999_surj__f__inv__f,axiom,
! [F: nat > $o,Y4: $o] :
( ( ( image_nat_o @ F @ top_top_set_nat )
= top_top_set_o )
=> ( ( F @ ( hilber3873338068935991546_nat_o @ top_top_set_nat @ F @ Y4 ) )
= Y4 ) ) ).
% surj_f_inv_f
thf(fact_1000_surj__f__inv__f,axiom,
! [F: $o > nat,Y4: nat] :
( ( ( image_o_nat @ F @ top_top_set_o )
= top_top_set_nat )
=> ( ( F @ ( hilber770666707827806236_o_nat @ top_top_set_o @ F @ Y4 ) )
= Y4 ) ) ).
% surj_f_inv_f
thf(fact_1001_surj__f__inv__f,axiom,
! [F: $o > $o,Y4: $o] :
( ( ( image_o_o @ F @ top_top_set_o )
= top_top_set_o )
=> ( ( F @ ( hilbert_inv_into_o_o @ top_top_set_o @ F @ Y4 ) )
= Y4 ) ) ).
% surj_f_inv_f
thf(fact_1002_subset__singletonD,axiom,
! [A2: set_o,X2: $o] :
( ( ord_less_eq_set_o @ A2 @ ( insert_o @ X2 @ bot_bot_set_o ) )
=> ( ( A2 = bot_bot_set_o )
| ( A2
= ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ).
% subset_singletonD
thf(fact_1003_subset__singletonD,axiom,
! [A2: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
=> ( ( A2 = bot_bot_set_nat )
| ( A2
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_1004_subset__singletonD,axiom,
! [A2: set_tm,X2: tm] :
( ( ord_less_eq_set_tm @ A2 @ ( insert_tm2 @ X2 @ bot_bot_set_tm ) )
=> ( ( A2 = bot_bot_set_tm )
| ( A2
= ( insert_tm2 @ X2 @ bot_bot_set_tm ) ) ) ) ).
% subset_singletonD
thf(fact_1005_subset__singleton__iff,axiom,
! [X7: set_o,A: $o] :
( ( ord_less_eq_set_o @ X7 @ ( insert_o @ A @ bot_bot_set_o ) )
= ( ( X7 = bot_bot_set_o )
| ( X7
= ( insert_o @ A @ bot_bot_set_o ) ) ) ) ).
% subset_singleton_iff
thf(fact_1006_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_1007_subset__singleton__iff,axiom,
! [X7: set_tm,A: tm] :
( ( ord_less_eq_set_tm @ X7 @ ( insert_tm2 @ A @ bot_bot_set_tm ) )
= ( ( X7 = bot_bot_set_tm )
| ( X7
= ( insert_tm2 @ A @ bot_bot_set_tm ) ) ) ) ).
% subset_singleton_iff
thf(fact_1008_image__constant,axiom,
! [X2: tm,A2: set_tm,C: tm] :
( ( member_tm @ X2 @ A2 )
=> ( ( image_tm_tm
@ ^ [X5: tm] : C
@ A2 )
= ( insert_tm2 @ C @ bot_bot_set_tm ) ) ) ).
% image_constant
thf(fact_1009_image__constant,axiom,
! [X2: nat,A2: set_nat,C: tm] :
( ( member_nat @ X2 @ A2 )
=> ( ( image_nat_tm
@ ^ [X5: nat] : C
@ A2 )
= ( insert_tm2 @ C @ bot_bot_set_tm ) ) ) ).
% image_constant
thf(fact_1010_image__constant,axiom,
! [X2: tm,A2: set_tm,C: nat] :
( ( member_tm @ X2 @ A2 )
=> ( ( image_tm_nat
@ ^ [X5: tm] : C
@ A2 )
= ( insert_nat @ C @ bot_bot_set_nat ) ) ) ).
% image_constant
thf(fact_1011_image__constant,axiom,
! [X2: nat,A2: set_nat,C: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( image_nat_nat
@ ^ [X5: nat] : C
@ A2 )
= ( insert_nat @ C @ bot_bot_set_nat ) ) ) ).
% image_constant
thf(fact_1012_image__constant,axiom,
! [X2: tm,A2: set_tm,C: $o] :
( ( member_tm @ X2 @ A2 )
=> ( ( image_tm_o
@ ^ [X5: tm] : C
@ A2 )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ).
% image_constant
thf(fact_1013_image__constant,axiom,
! [X2: nat,A2: set_nat,C: $o] :
( ( member_nat @ X2 @ A2 )
=> ( ( image_nat_o
@ ^ [X5: nat] : C
@ A2 )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ).
% image_constant
thf(fact_1014_image__constant,axiom,
! [X2: fm,A2: set_fm,C: set_tm] :
( ( member_fm @ X2 @ A2 )
=> ( ( image_fm_set_tm
@ ^ [X5: fm] : C
@ A2 )
= ( insert_set_tm @ C @ bot_bot_set_set_tm ) ) ) ).
% image_constant
thf(fact_1015_image__constant,axiom,
! [X2: fm,A2: set_fm,C: set_nat] :
( ( member_fm @ X2 @ A2 )
=> ( ( image_fm_set_nat
@ ^ [X5: fm] : C
@ A2 )
= ( insert_set_nat @ C @ bot_bot_set_set_nat ) ) ) ).
% image_constant
thf(fact_1016_image__constant,axiom,
! [X2: tm,A2: set_tm,C: set_nat] :
( ( member_tm @ X2 @ A2 )
=> ( ( image_tm_set_nat
@ ^ [X5: tm] : C
@ A2 )
= ( insert_set_nat @ C @ bot_bot_set_set_nat ) ) ) ).
% image_constant
thf(fact_1017_image__constant,axiom,
! [X2: tm,A2: set_tm,C: set_tm] :
( ( member_tm @ X2 @ A2 )
=> ( ( image_tm_set_tm
@ ^ [X5: tm] : C
@ A2 )
= ( insert_set_tm @ C @ bot_bot_set_set_tm ) ) ) ).
% image_constant
thf(fact_1018_image__constant__conv,axiom,
! [A2: set_tm,C: tm] :
( ( ( A2 = bot_bot_set_tm )
=> ( ( image_tm_tm
@ ^ [X5: tm] : C
@ A2 )
= bot_bot_set_tm ) )
& ( ( A2 != bot_bot_set_tm )
=> ( ( image_tm_tm
@ ^ [X5: tm] : C
@ A2 )
= ( insert_tm2 @ C @ bot_bot_set_tm ) ) ) ) ).
% image_constant_conv
thf(fact_1019_image__constant__conv,axiom,
! [A2: set_tm,C: nat] :
( ( ( A2 = bot_bot_set_tm )
=> ( ( image_tm_nat
@ ^ [X5: tm] : C
@ A2 )
= bot_bot_set_nat ) )
& ( ( A2 != bot_bot_set_tm )
=> ( ( image_tm_nat
@ ^ [X5: tm] : C
@ A2 )
= ( insert_nat @ C @ bot_bot_set_nat ) ) ) ) ).
% image_constant_conv
thf(fact_1020_image__constant__conv,axiom,
! [A2: set_tm,C: $o] :
( ( ( A2 = bot_bot_set_tm )
=> ( ( image_tm_o
@ ^ [X5: tm] : C
@ A2 )
= bot_bot_set_o ) )
& ( ( A2 != bot_bot_set_tm )
=> ( ( image_tm_o
@ ^ [X5: tm] : C
@ A2 )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ) ).
% image_constant_conv
thf(fact_1021_image__constant__conv,axiom,
! [A2: set_nat,C: tm] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( image_nat_tm
@ ^ [X5: nat] : C
@ A2 )
= bot_bot_set_tm ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( image_nat_tm
@ ^ [X5: nat] : C
@ A2 )
= ( insert_tm2 @ C @ bot_bot_set_tm ) ) ) ) ).
% image_constant_conv
thf(fact_1022_image__constant__conv,axiom,
! [A2: set_nat,C: nat] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( image_nat_nat
@ ^ [X5: nat] : C
@ A2 )
= bot_bot_set_nat ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( image_nat_nat
@ ^ [X5: nat] : C
@ A2 )
= ( insert_nat @ C @ bot_bot_set_nat ) ) ) ) ).
% image_constant_conv
thf(fact_1023_image__constant__conv,axiom,
! [A2: set_nat,C: $o] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( image_nat_o
@ ^ [X5: nat] : C
@ A2 )
= bot_bot_set_o ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( image_nat_o
@ ^ [X5: nat] : C
@ A2 )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ) ).
% image_constant_conv
thf(fact_1024_image__constant__conv,axiom,
! [A2: set_o,C: tm] :
( ( ( A2 = bot_bot_set_o )
=> ( ( image_o_tm
@ ^ [X5: $o] : C
@ A2 )
= bot_bot_set_tm ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( image_o_tm
@ ^ [X5: $o] : C
@ A2 )
= ( insert_tm2 @ C @ bot_bot_set_tm ) ) ) ) ).
% image_constant_conv
thf(fact_1025_image__constant__conv,axiom,
! [A2: set_o,C: nat] :
( ( ( A2 = bot_bot_set_o )
=> ( ( image_o_nat
@ ^ [X5: $o] : C
@ A2 )
= bot_bot_set_nat ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( image_o_nat
@ ^ [X5: $o] : C
@ A2 )
= ( insert_nat @ C @ bot_bot_set_nat ) ) ) ) ).
% image_constant_conv
thf(fact_1026_image__constant__conv,axiom,
! [A2: set_o,C: $o] :
( ( ( A2 = bot_bot_set_o )
=> ( ( image_o_o
@ ^ [X5: $o] : C
@ A2 )
= bot_bot_set_o ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( image_o_o
@ ^ [X5: $o] : C
@ A2 )
= ( insert_o @ C @ bot_bot_set_o ) ) ) ) ).
% image_constant_conv
thf(fact_1027_image__constant__conv,axiom,
! [A2: set_fm,C: set_tm] :
( ( ( A2 = bot_bot_set_fm )
=> ( ( image_fm_set_tm
@ ^ [X5: fm] : C
@ A2 )
= bot_bot_set_set_tm ) )
& ( ( A2 != bot_bot_set_fm )
=> ( ( image_fm_set_tm
@ ^ [X5: fm] : C
@ A2 )
= ( insert_set_tm @ C @ bot_bot_set_set_tm ) ) ) ) ).
% image_constant_conv
thf(fact_1028_range__eq__singletonD,axiom,
! [F: tm > set_nat,A: set_nat,X2: tm] :
( ( ( image_tm_set_nat @ F @ top_top_set_tm )
= ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
=> ( ( F @ X2 )
= A ) ) ).
% range_eq_singletonD
thf(fact_1029_range__eq__singletonD,axiom,
! [F: tm > set_tm,A: set_tm,X2: tm] :
( ( ( image_tm_set_tm @ F @ top_top_set_tm )
= ( insert_set_tm @ A @ bot_bot_set_set_tm ) )
=> ( ( F @ X2 )
= A ) ) ).
% range_eq_singletonD
thf(fact_1030_range__eq__singletonD,axiom,
! [F: fm > set_tm,A: set_tm,X2: fm] :
( ( ( image_fm_set_tm @ F @ top_top_set_fm )
= ( insert_set_tm @ A @ bot_bot_set_set_tm ) )
=> ( ( F @ X2 )
= A ) ) ).
% range_eq_singletonD
thf(fact_1031_range__eq__singletonD,axiom,
! [F: fm > set_nat,A: set_nat,X2: fm] :
( ( ( image_fm_set_nat @ F @ top_top_set_fm )
= ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
=> ( ( F @ X2 )
= A ) ) ).
% range_eq_singletonD
thf(fact_1032_range__eq__singletonD,axiom,
! [F: nat > tm,A: tm,X2: nat] :
( ( ( image_nat_tm @ F @ top_top_set_nat )
= ( insert_tm2 @ A @ bot_bot_set_tm ) )
=> ( ( F @ X2 )
= A ) ) ).
% range_eq_singletonD
thf(fact_1033_range__eq__singletonD,axiom,
! [F: nat > nat,A: nat,X2: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( ( F @ X2 )
= A ) ) ).
% range_eq_singletonD
thf(fact_1034_range__eq__singletonD,axiom,
! [F: nat > $o,A: $o,X2: nat] :
( ( ( image_nat_o @ F @ top_top_set_nat )
= ( insert_o @ A @ bot_bot_set_o ) )
=> ( ( F @ X2 )
= A ) ) ).
% range_eq_singletonD
thf(fact_1035_range__eq__singletonD,axiom,
! [F: $o > tm,A: tm,X2: $o] :
( ( ( image_o_tm @ F @ top_top_set_o )
= ( insert_tm2 @ A @ bot_bot_set_tm ) )
=> ( ( F @ X2 )
= A ) ) ).
% range_eq_singletonD
thf(fact_1036_range__eq__singletonD,axiom,
! [F: $o > nat,A: nat,X2: $o] :
( ( ( image_o_nat @ F @ top_top_set_o )
= ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( ( F @ X2 )
= A ) ) ).
% range_eq_singletonD
thf(fact_1037_range__eq__singletonD,axiom,
! [F: $o > $o,A: $o,X2: $o] :
( ( ( image_o_o @ F @ top_top_set_o )
= ( insert_o @ A @ bot_bot_set_o ) )
=> ( ( F @ X2 )
= A ) ) ).
% range_eq_singletonD
thf(fact_1038_paramst__sub__term_I1_J,axiom,
! [M: nat,S3: tm,T: tm] : ( ord_less_eq_set_nat @ ( paramst @ ( sub_term @ M @ S3 @ T ) ) @ ( sup_sup_set_nat @ ( paramst @ S3 ) @ ( paramst @ T ) ) ) ).
% paramst_sub_term(1)
thf(fact_1039_set__Cons__sing__Nil,axiom,
! [A2: set_tm] :
( ( set_Cons_tm @ A2 @ ( insert_list_tm @ nil_tm @ bot_bot_set_list_tm ) )
= ( image_tm_list_tm
@ ^ [X5: tm] : ( cons_tm @ X5 @ nil_tm )
@ A2 ) ) ).
% set_Cons_sing_Nil
thf(fact_1040_set__Cons__sing__Nil,axiom,
! [A2: set_fm] :
( ( set_Cons_fm @ A2 @ ( insert_list_fm2 @ nil_fm @ bot_bot_set_list_fm ) )
= ( image_fm_list_fm
@ ^ [X5: fm] : ( cons_fm @ X5 @ nil_fm )
@ A2 ) ) ).
% set_Cons_sing_Nil
thf(fact_1041_set__Cons__sing__Nil,axiom,
! [A2: set_list_fm] :
( ( set_Cons_list_fm @ A2 @ ( insert_list_list_fm @ nil_list_fm @ bot_bo6959746583267231307ist_fm ) )
= ( image_4162782105507059915ist_fm
@ ^ [X5: list_fm] : ( cons_list_fm @ X5 @ nil_list_fm )
@ A2 ) ) ).
% set_Cons_sing_Nil
thf(fact_1042_image__Collect__subsetI,axiom,
! [P: tm > $o,F: tm > set_nat,B: set_set_nat] :
( ! [X: tm] :
( ( P @ X )
=> ( member_set_nat @ ( F @ X ) @ B ) )
=> ( ord_le6893508408891458716et_nat @ ( image_tm_set_nat @ F @ ( collect_tm @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1043_image__Collect__subsetI,axiom,
! [P: tm > $o,F: tm > set_tm,B: set_set_tm] :
( ! [X: tm] :
( ( P @ X )
=> ( member_set_tm @ ( F @ X ) @ B ) )
=> ( ord_le5601931644483074373set_tm @ ( image_tm_set_tm @ F @ ( collect_tm @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1044_image__Collect__subsetI,axiom,
! [P: fm > $o,F: fm > set_tm,B: set_set_tm] :
( ! [X: fm] :
( ( P @ X )
=> ( member_set_tm @ ( F @ X ) @ B ) )
=> ( ord_le5601931644483074373set_tm @ ( image_fm_set_tm @ F @ ( collect_fm @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1045_image__Collect__subsetI,axiom,
! [P: fm > $o,F: fm > set_nat,B: set_set_nat] :
( ! [X: fm] :
( ( P @ X )
=> ( member_set_nat @ ( F @ X ) @ B ) )
=> ( ord_le6893508408891458716et_nat @ ( image_fm_set_nat @ F @ ( collect_fm @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1046_image__Collect__subsetI,axiom,
! [P: list_fm > $o,F: list_fm > list_fm,B: set_list_fm] :
( ! [X: list_fm] :
( ( P @ X )
=> ( member_list_fm @ ( F @ X ) @ B ) )
=> ( ord_le7838213414353715577ist_fm @ ( image_9148017957442633541ist_fm @ F @ ( collect_list_fm @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1047_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > list_fm,B: set_list_fm] :
( ! [X: nat] :
( ( P @ X )
=> ( member_list_fm @ ( F @ X ) @ B ) )
=> ( ord_le7838213414353715577ist_fm @ ( image_nat_list_fm @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1048_image__Collect__subsetI,axiom,
! [P: list_fm > $o,F: list_fm > nat,B: set_nat] :
( ! [X: list_fm] :
( ( P @ X )
=> ( member_nat @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_list_fm_nat @ F @ ( collect_list_fm @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1049_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > nat,B: set_nat] :
( ! [X: nat] :
( ( P @ X )
=> ( member_nat @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1050_image__Collect__subsetI,axiom,
! [P: list_fm > $o,F: list_fm > tm,B: set_tm] :
( ! [X: list_fm] :
( ( P @ X )
=> ( member_tm @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_tm @ ( image_list_fm_tm @ F @ ( collect_list_fm @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1051_image__Collect__subsetI,axiom,
! [P: nat > $o,F: nat > tm,B: set_tm] :
( ! [X: nat] :
( ( P @ X )
=> ( member_tm @ ( F @ X ) @ B ) )
=> ( ord_less_eq_set_tm @ ( image_nat_tm @ F @ ( collect_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1052_subset__emptyI,axiom,
! [A2: set_list_fm] :
( ! [X: list_fm] :
~ ( member_list_fm @ X @ A2 )
=> ( ord_le7838213414353715577ist_fm @ A2 @ bot_bot_set_list_fm ) ) ).
% subset_emptyI
thf(fact_1053_subset__emptyI,axiom,
! [A2: set_o] :
( ! [X: $o] :
~ ( member_o @ X @ A2 )
=> ( ord_less_eq_set_o @ A2 @ bot_bot_set_o ) ) ).
% subset_emptyI
thf(fact_1054_subset__emptyI,axiom,
! [A2: set_nat] :
( ! [X: nat] :
~ ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_1055_subset__emptyI,axiom,
! [A2: set_tm] :
( ! [X: tm] :
~ ( member_tm @ X @ A2 )
=> ( ord_less_eq_set_tm @ A2 @ bot_bot_set_tm ) ) ).
% subset_emptyI
thf(fact_1056_suffixes__eq__snoc,axiom,
! [Ys2: list_tm,Xs: list_list_tm,X2: list_tm] :
( ( ( suffixes_tm @ Ys2 )
= ( append_list_tm @ Xs @ ( cons_list_tm @ X2 @ nil_list_tm ) ) )
= ( ( ( ( Ys2 = nil_tm )
& ( Xs = nil_list_tm ) )
| ? [Z3: tm,Zs: list_tm] :
( ( Ys2
= ( cons_tm @ Z3 @ Zs ) )
& ( Xs
= ( suffixes_tm @ Zs ) ) ) )
& ( X2 = Ys2 ) ) ) ).
% suffixes_eq_snoc
thf(fact_1057_suffixes__eq__snoc,axiom,
! [Ys2: list_list_fm,Xs: list_list_list_fm,X2: list_list_fm] :
( ( ( suffixes_list_fm @ Ys2 )
= ( append_list_list_fm @ Xs @ ( cons_list_list_fm @ X2 @ nil_list_list_fm ) ) )
= ( ( ( ( Ys2 = nil_list_fm )
& ( Xs = nil_list_list_fm ) )
| ? [Z3: list_fm,Zs: list_list_fm] :
( ( Ys2
= ( cons_list_fm @ Z3 @ Zs ) )
& ( Xs
= ( suffixes_list_fm @ Zs ) ) ) )
& ( X2 = Ys2 ) ) ) ).
% suffixes_eq_snoc
thf(fact_1058_suffixes__eq__snoc,axiom,
! [Ys2: list_fm,Xs: list_list_fm,X2: list_fm] :
( ( ( suffixes_fm @ Ys2 )
= ( append_list_fm @ Xs @ ( cons_list_fm @ X2 @ nil_list_fm ) ) )
= ( ( ( ( Ys2 = nil_fm )
& ( Xs = nil_list_fm ) )
| ? [Z3: fm,Zs: list_fm] :
( ( Ys2
= ( cons_fm @ Z3 @ Zs ) )
& ( Xs
= ( suffixes_fm @ Zs ) ) ) )
& ( X2 = Ys2 ) ) ) ).
% suffixes_eq_snoc
thf(fact_1059_paramst__liftt_I2_J,axiom,
! [Ts: list_tm] :
( ( paramsts @ ( liftts @ Ts ) )
= ( paramsts @ Ts ) ) ).
% paramst_liftt(2)
thf(fact_1060_subset__singleton__iff__Uniq,axiom,
! [A2: set_list_fm] :
( ( ? [A5: list_fm] : ( ord_le7838213414353715577ist_fm @ A2 @ ( insert_list_fm2 @ A5 @ bot_bot_set_list_fm ) ) )
= ( uniq_list_fm
@ ^ [X5: list_fm] : ( member_list_fm @ X5 @ A2 ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_1061_subset__singleton__iff__Uniq,axiom,
! [A2: set_o] :
( ( ? [A5: $o] : ( ord_less_eq_set_o @ A2 @ ( insert_o @ A5 @ bot_bot_set_o ) ) )
= ( uniq_o
@ ^ [X5: $o] : ( member_o @ X5 @ A2 ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_1062_subset__singleton__iff__Uniq,axiom,
! [A2: set_nat] :
( ( ? [A5: nat] : ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ A5 @ bot_bot_set_nat ) ) )
= ( uniq_nat
@ ^ [X5: nat] : ( member_nat @ X5 @ A2 ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_1063_subset__singleton__iff__Uniq,axiom,
! [A2: set_tm] :
( ( ? [A5: tm] : ( ord_less_eq_set_tm @ A2 @ ( insert_tm2 @ A5 @ bot_bot_set_tm ) ) )
= ( uniq_tm
@ ^ [X5: tm] : ( member_tm @ X5 @ A2 ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_1064_conj__subset__def,axiom,
! [A2: set_list_fm,P: list_fm > $o,Q: list_fm > $o] :
( ( ord_le7838213414353715577ist_fm @ A2
@ ( collect_list_fm
@ ^ [X5: list_fm] :
( ( P @ X5 )
& ( Q @ X5 ) ) ) )
= ( ( ord_le7838213414353715577ist_fm @ A2 @ ( collect_list_fm @ P ) )
& ( ord_le7838213414353715577ist_fm @ A2 @ ( collect_list_fm @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_1065_conj__subset__def,axiom,
! [A2: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A2
@ ( collect_nat
@ ^ [X5: nat] :
( ( P @ X5 )
& ( Q @ X5 ) ) ) )
= ( ( ord_less_eq_set_nat @ A2 @ ( collect_nat @ P ) )
& ( ord_less_eq_set_nat @ A2 @ ( collect_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_1066_conj__subset__def,axiom,
! [A2: set_tm,P: tm > $o,Q: tm > $o] :
( ( ord_less_eq_set_tm @ A2
@ ( collect_tm
@ ^ [X5: tm] :
( ( P @ X5 )
& ( Q @ X5 ) ) ) )
= ( ( ord_less_eq_set_tm @ A2 @ ( collect_tm @ P ) )
& ( ord_less_eq_set_tm @ A2 @ ( collect_tm @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_1067_append_Oassoc,axiom,
! [A: list_tm,B2: list_tm,C: list_tm] :
( ( append_tm @ ( append_tm @ A @ B2 ) @ C )
= ( append_tm @ A @ ( append_tm @ B2 @ C ) ) ) ).
% append.assoc
thf(fact_1068_append_Oassoc,axiom,
! [A: list_fm,B2: list_fm,C: list_fm] :
( ( append_fm @ ( append_fm @ A @ B2 ) @ C )
= ( append_fm @ A @ ( append_fm @ B2 @ C ) ) ) ).
% append.assoc
thf(fact_1069_append__assoc,axiom,
! [Xs: list_tm,Ys2: list_tm,Zs2: list_tm] :
( ( append_tm @ ( append_tm @ Xs @ Ys2 ) @ Zs2 )
= ( append_tm @ Xs @ ( append_tm @ Ys2 @ Zs2 ) ) ) ).
% append_assoc
thf(fact_1070_append__assoc,axiom,
! [Xs: list_fm,Ys2: list_fm,Zs2: list_fm] :
( ( append_fm @ ( append_fm @ Xs @ Ys2 ) @ Zs2 )
= ( append_fm @ Xs @ ( append_fm @ Ys2 @ Zs2 ) ) ) ).
% append_assoc
thf(fact_1071_append__same__eq,axiom,
! [Ys2: list_tm,Xs: list_tm,Zs2: list_tm] :
( ( ( append_tm @ Ys2 @ Xs )
= ( append_tm @ Zs2 @ Xs ) )
= ( Ys2 = Zs2 ) ) ).
% append_same_eq
thf(fact_1072_append__same__eq,axiom,
! [Ys2: list_fm,Xs: list_fm,Zs2: list_fm] :
( ( ( append_fm @ Ys2 @ Xs )
= ( append_fm @ Zs2 @ Xs ) )
= ( Ys2 = Zs2 ) ) ).
% append_same_eq
thf(fact_1073_same__append__eq,axiom,
! [Xs: list_tm,Ys2: list_tm,Zs2: list_tm] :
( ( ( append_tm @ Xs @ Ys2 )
= ( append_tm @ Xs @ Zs2 ) )
= ( Ys2 = Zs2 ) ) ).
% same_append_eq
thf(fact_1074_same__append__eq,axiom,
! [Xs: list_fm,Ys2: list_fm,Zs2: list_fm] :
( ( ( append_fm @ Xs @ Ys2 )
= ( append_fm @ Xs @ Zs2 ) )
= ( Ys2 = Zs2 ) ) ).
% same_append_eq
thf(fact_1075_append_Oright__neutral,axiom,
! [A: list_tm] :
( ( append_tm @ A @ nil_tm )
= A ) ).
% append.right_neutral
thf(fact_1076_append_Oright__neutral,axiom,
! [A: list_fm] :
( ( append_fm @ A @ nil_fm )
= A ) ).
% append.right_neutral
thf(fact_1077_append_Oright__neutral,axiom,
! [A: list_list_fm] :
( ( append_list_fm @ A @ nil_list_fm )
= A ) ).
% append.right_neutral
thf(fact_1078_append__Nil2,axiom,
! [Xs: list_tm] :
( ( append_tm @ Xs @ nil_tm )
= Xs ) ).
% append_Nil2
thf(fact_1079_append__Nil2,axiom,
! [Xs: list_fm] :
( ( append_fm @ Xs @ nil_fm )
= Xs ) ).
% append_Nil2
thf(fact_1080_append__Nil2,axiom,
! [Xs: list_list_fm] :
( ( append_list_fm @ Xs @ nil_list_fm )
= Xs ) ).
% append_Nil2
thf(fact_1081_append__self__conv,axiom,
! [Xs: list_tm,Ys2: list_tm] :
( ( ( append_tm @ Xs @ Ys2 )
= Xs )
= ( Ys2 = nil_tm ) ) ).
% append_self_conv
thf(fact_1082_append__self__conv,axiom,
! [Xs: list_fm,Ys2: list_fm] :
( ( ( append_fm @ Xs @ Ys2 )
= Xs )
= ( Ys2 = nil_fm ) ) ).
% append_self_conv
thf(fact_1083_append__self__conv,axiom,
! [Xs: list_list_fm,Ys2: list_list_fm] :
( ( ( append_list_fm @ Xs @ Ys2 )
= Xs )
= ( Ys2 = nil_list_fm ) ) ).
% append_self_conv
thf(fact_1084_self__append__conv,axiom,
! [Y4: list_tm,Ys2: list_tm] :
( ( Y4
= ( append_tm @ Y4 @ Ys2 ) )
= ( Ys2 = nil_tm ) ) ).
% self_append_conv
thf(fact_1085_self__append__conv,axiom,
! [Y4: list_fm,Ys2: list_fm] :
( ( Y4
= ( append_fm @ Y4 @ Ys2 ) )
= ( Ys2 = nil_fm ) ) ).
% self_append_conv
thf(fact_1086_self__append__conv,axiom,
! [Y4: list_list_fm,Ys2: list_list_fm] :
( ( Y4
= ( append_list_fm @ Y4 @ Ys2 ) )
= ( Ys2 = nil_list_fm ) ) ).
% self_append_conv
thf(fact_1087_append__self__conv2,axiom,
! [Xs: list_tm,Ys2: list_tm] :
( ( ( append_tm @ Xs @ Ys2 )
= Ys2 )
= ( Xs = nil_tm ) ) ).
% append_self_conv2
thf(fact_1088_append__self__conv2,axiom,
! [Xs: list_fm,Ys2: list_fm] :
( ( ( append_fm @ Xs @ Ys2 )
= Ys2 )
= ( Xs = nil_fm ) ) ).
% append_self_conv2
thf(fact_1089_append__self__conv2,axiom,
! [Xs: list_list_fm,Ys2: list_list_fm] :
( ( ( append_list_fm @ Xs @ Ys2 )
= Ys2 )
= ( Xs = nil_list_fm ) ) ).
% append_self_conv2
thf(fact_1090_self__append__conv2,axiom,
! [Y4: list_tm,Xs: list_tm] :
( ( Y4
= ( append_tm @ Xs @ Y4 ) )
= ( Xs = nil_tm ) ) ).
% self_append_conv2
thf(fact_1091_self__append__conv2,axiom,
! [Y4: list_fm,Xs: list_fm] :
( ( Y4
= ( append_fm @ Xs @ Y4 ) )
= ( Xs = nil_fm ) ) ).
% self_append_conv2
thf(fact_1092_self__append__conv2,axiom,
! [Y4: list_list_fm,Xs: list_list_fm] :
( ( Y4
= ( append_list_fm @ Xs @ Y4 ) )
= ( Xs = nil_list_fm ) ) ).
% self_append_conv2
thf(fact_1093_Nil__is__append__conv,axiom,
! [Xs: list_tm,Ys2: list_tm] :
( ( nil_tm
= ( append_tm @ Xs @ Ys2 ) )
= ( ( Xs = nil_tm )
& ( Ys2 = nil_tm ) ) ) ).
% Nil_is_append_conv
thf(fact_1094_Nil__is__append__conv,axiom,
! [Xs: list_fm,Ys2: list_fm] :
( ( nil_fm
= ( append_fm @ Xs @ Ys2 ) )
= ( ( Xs = nil_fm )
& ( Ys2 = nil_fm ) ) ) ).
% Nil_is_append_conv
thf(fact_1095_Nil__is__append__conv,axiom,
! [Xs: list_list_fm,Ys2: list_list_fm] :
( ( nil_list_fm
= ( append_list_fm @ Xs @ Ys2 ) )
= ( ( Xs = nil_list_fm )
& ( Ys2 = nil_list_fm ) ) ) ).
% Nil_is_append_conv
thf(fact_1096_append__is__Nil__conv,axiom,
! [Xs: list_tm,Ys2: list_tm] :
( ( ( append_tm @ Xs @ Ys2 )
= nil_tm )
= ( ( Xs = nil_tm )
& ( Ys2 = nil_tm ) ) ) ).
% append_is_Nil_conv
thf(fact_1097_append__is__Nil__conv,axiom,
! [Xs: list_fm,Ys2: list_fm] :
( ( ( append_fm @ Xs @ Ys2 )
= nil_fm )
= ( ( Xs = nil_fm )
& ( Ys2 = nil_fm ) ) ) ).
% append_is_Nil_conv
thf(fact_1098_append__is__Nil__conv,axiom,
! [Xs: list_list_fm,Ys2: list_list_fm] :
( ( ( append_list_fm @ Xs @ Ys2 )
= nil_list_fm )
= ( ( Xs = nil_list_fm )
& ( Ys2 = nil_list_fm ) ) ) ).
% append_is_Nil_conv
thf(fact_1099_list__all__append,axiom,
! [P: tm > $o,Xs: list_tm,Ys2: list_tm] :
( ( list_all_tm @ P @ ( append_tm @ Xs @ Ys2 ) )
= ( ( list_all_tm @ P @ Xs )
& ( list_all_tm @ P @ Ys2 ) ) ) ).
% list_all_append
thf(fact_1100_list__all__append,axiom,
! [P: fm > $o,Xs: list_fm,Ys2: list_fm] :
( ( list_all_fm @ P @ ( append_fm @ Xs @ Ys2 ) )
= ( ( list_all_fm @ P @ Xs )
& ( list_all_fm @ P @ Ys2 ) ) ) ).
% list_all_append
thf(fact_1101_append1__eq__conv,axiom,
! [Xs: list_tm,X2: tm,Ys2: list_tm,Y4: tm] :
( ( ( append_tm @ Xs @ ( cons_tm @ X2 @ nil_tm ) )
= ( append_tm @ Ys2 @ ( cons_tm @ Y4 @ nil_tm ) ) )
= ( ( Xs = Ys2 )
& ( X2 = Y4 ) ) ) ).
% append1_eq_conv
thf(fact_1102_append1__eq__conv,axiom,
! [Xs: list_fm,X2: fm,Ys2: list_fm,Y4: fm] :
( ( ( append_fm @ Xs @ ( cons_fm @ X2 @ nil_fm ) )
= ( append_fm @ Ys2 @ ( cons_fm @ Y4 @ nil_fm ) ) )
= ( ( Xs = Ys2 )
& ( X2 = Y4 ) ) ) ).
% append1_eq_conv
thf(fact_1103_append1__eq__conv,axiom,
! [Xs: list_list_fm,X2: list_fm,Ys2: list_list_fm,Y4: list_fm] :
( ( ( append_list_fm @ Xs @ ( cons_list_fm @ X2 @ nil_list_fm ) )
= ( append_list_fm @ Ys2 @ ( cons_list_fm @ Y4 @ nil_list_fm ) ) )
= ( ( Xs = Ys2 )
& ( X2 = Y4 ) ) ) ).
% append1_eq_conv
thf(fact_1104_list_Opred__mono,axiom,
! [P: tm > $o,Pa: tm > $o] :
( ( ord_less_eq_tm_o @ P @ Pa )
=> ( ord_le2468657205176945586t_tm_o @ ( list_all_tm @ P ) @ ( list_all_tm @ Pa ) ) ) ).
% list.pred_mono
thf(fact_1105_list_Opred__mono,axiom,
! [P: fm > $o,Pa: fm > $o] :
( ( ord_less_eq_fm_o @ P @ Pa )
=> ( ord_le6518561683347902116t_fm_o @ ( list_all_fm @ P ) @ ( list_all_fm @ Pa ) ) ) ).
% list.pred_mono
thf(fact_1106_append__eq__appendI,axiom,
! [Xs: list_tm,Xs1: list_tm,Zs2: list_tm,Ys2: list_tm,Us: list_tm] :
( ( ( append_tm @ Xs @ Xs1 )
= Zs2 )
=> ( ( Ys2
= ( append_tm @ Xs1 @ Us ) )
=> ( ( append_tm @ Xs @ Ys2 )
= ( append_tm @ Zs2 @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_1107_append__eq__appendI,axiom,
! [Xs: list_fm,Xs1: list_fm,Zs2: list_fm,Ys2: list_fm,Us: list_fm] :
( ( ( append_fm @ Xs @ Xs1 )
= Zs2 )
=> ( ( Ys2
= ( append_fm @ Xs1 @ Us ) )
=> ( ( append_fm @ Xs @ Ys2 )
= ( append_fm @ Zs2 @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_1108_append__eq__append__conv2,axiom,
! [Xs: list_tm,Ys2: list_tm,Zs2: list_tm,Ts: list_tm] :
( ( ( append_tm @ Xs @ Ys2 )
= ( append_tm @ Zs2 @ Ts ) )
= ( ? [Us2: list_tm] :
( ( ( Xs
= ( append_tm @ Zs2 @ Us2 ) )
& ( ( append_tm @ Us2 @ Ys2 )
= Ts ) )
| ( ( ( append_tm @ Xs @ Us2 )
= Zs2 )
& ( Ys2
= ( append_tm @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_1109_append__eq__append__conv2,axiom,
! [Xs: list_fm,Ys2: list_fm,Zs2: list_fm,Ts: list_fm] :
( ( ( append_fm @ Xs @ Ys2 )
= ( append_fm @ Zs2 @ Ts ) )
= ( ? [Us2: list_fm] :
( ( ( Xs
= ( append_fm @ Zs2 @ Us2 ) )
& ( ( append_fm @ Us2 @ Ys2 )
= Ts ) )
| ( ( ( append_fm @ Xs @ Us2 )
= Zs2 )
& ( Ys2
= ( append_fm @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_1110_append__Cons,axiom,
! [X2: tm,Xs: list_tm,Ys2: list_tm] :
( ( append_tm @ ( cons_tm @ X2 @ Xs ) @ Ys2 )
= ( cons_tm @ X2 @ ( append_tm @ Xs @ Ys2 ) ) ) ).
% append_Cons
thf(fact_1111_append__Cons,axiom,
! [X2: fm,Xs: list_fm,Ys2: list_fm] :
( ( append_fm @ ( cons_fm @ X2 @ Xs ) @ Ys2 )
= ( cons_fm @ X2 @ ( append_fm @ Xs @ Ys2 ) ) ) ).
% append_Cons
thf(fact_1112_append__Cons,axiom,
! [X2: list_fm,Xs: list_list_fm,Ys2: list_list_fm] :
( ( append_list_fm @ ( cons_list_fm @ X2 @ Xs ) @ Ys2 )
= ( cons_list_fm @ X2 @ ( append_list_fm @ Xs @ Ys2 ) ) ) ).
% append_Cons
thf(fact_1113_Cons__eq__appendI,axiom,
! [X2: tm,Xs1: list_tm,Ys2: list_tm,Xs: list_tm,Zs2: list_tm] :
( ( ( cons_tm @ X2 @ Xs1 )
= Ys2 )
=> ( ( Xs
= ( append_tm @ Xs1 @ Zs2 ) )
=> ( ( cons_tm @ X2 @ Xs )
= ( append_tm @ Ys2 @ Zs2 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_1114_Cons__eq__appendI,axiom,
! [X2: fm,Xs1: list_fm,Ys2: list_fm,Xs: list_fm,Zs2: list_fm] :
( ( ( cons_fm @ X2 @ Xs1 )
= Ys2 )
=> ( ( Xs
= ( append_fm @ Xs1 @ Zs2 ) )
=> ( ( cons_fm @ X2 @ Xs )
= ( append_fm @ Ys2 @ Zs2 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_1115_Cons__eq__appendI,axiom,
! [X2: list_fm,Xs1: list_list_fm,Ys2: list_list_fm,Xs: list_list_fm,Zs2: list_list_fm] :
( ( ( cons_list_fm @ X2 @ Xs1 )
= Ys2 )
=> ( ( Xs
= ( append_list_fm @ Xs1 @ Zs2 ) )
=> ( ( cons_list_fm @ X2 @ Xs )
= ( append_list_fm @ Ys2 @ Zs2 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_1116_append__Nil,axiom,
! [Ys2: list_tm] :
( ( append_tm @ nil_tm @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_1117_append__Nil,axiom,
! [Ys2: list_fm] :
( ( append_fm @ nil_fm @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_1118_append__Nil,axiom,
! [Ys2: list_list_fm] :
( ( append_list_fm @ nil_list_fm @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_1119_append_Oleft__neutral,axiom,
! [A: list_tm] :
( ( append_tm @ nil_tm @ A )
= A ) ).
% append.left_neutral
thf(fact_1120_append_Oleft__neutral,axiom,
! [A: list_fm] :
( ( append_fm @ nil_fm @ A )
= A ) ).
% append.left_neutral
thf(fact_1121_append_Oleft__neutral,axiom,
! [A: list_list_fm] :
( ( append_list_fm @ nil_list_fm @ A )
= A ) ).
% append.left_neutral
thf(fact_1122_eq__Nil__appendI,axiom,
! [Xs: list_tm,Ys2: list_tm] :
( ( Xs = Ys2 )
=> ( Xs
= ( append_tm @ nil_tm @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_1123_eq__Nil__appendI,axiom,
! [Xs: list_fm,Ys2: list_fm] :
( ( Xs = Ys2 )
=> ( Xs
= ( append_fm @ nil_fm @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_1124_eq__Nil__appendI,axiom,
! [Xs: list_list_fm,Ys2: list_list_fm] :
( ( Xs = Ys2 )
=> ( Xs
= ( append_list_fm @ nil_list_fm @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_1125_less__eq__set__def,axiom,
( ord_le7838213414353715577ist_fm
= ( ^ [A4: set_list_fm,B3: set_list_fm] :
( ord_le6518561683347902116t_fm_o
@ ^ [X5: list_fm] : ( member_list_fm @ X5 @ A4 )
@ ^ [X5: list_fm] : ( member_list_fm @ X5 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_1126_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( ord_less_eq_nat_o
@ ^ [X5: nat] : ( member_nat @ X5 @ A4 )
@ ^ [X5: nat] : ( member_nat @ X5 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_1127_less__eq__set__def,axiom,
( ord_less_eq_set_tm
= ( ^ [A4: set_tm,B3: set_tm] :
( ord_less_eq_tm_o
@ ^ [X5: tm] : ( member_tm @ X5 @ A4 )
@ ^ [X5: tm] : ( member_tm @ X5 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_1128_pred__subset__eq,axiom,
! [R: set_list_fm,S: set_list_fm] :
( ( ord_le6518561683347902116t_fm_o
@ ^ [X5: list_fm] : ( member_list_fm @ X5 @ R )
@ ^ [X5: list_fm] : ( member_list_fm @ X5 @ S ) )
= ( ord_le7838213414353715577ist_fm @ R @ S ) ) ).
% pred_subset_eq
thf(fact_1129_pred__subset__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X5: nat] : ( member_nat @ X5 @ R )
@ ^ [X5: nat] : ( member_nat @ X5 @ S ) )
= ( ord_less_eq_set_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_1130_pred__subset__eq,axiom,
! [R: set_tm,S: set_tm] :
( ( ord_less_eq_tm_o
@ ^ [X5: tm] : ( member_tm @ X5 @ R )
@ ^ [X5: tm] : ( member_tm @ X5 @ S ) )
= ( ord_less_eq_set_tm @ R @ S ) ) ).
% pred_subset_eq
thf(fact_1131_rev__induct,axiom,
! [P: list_tm > $o,Xs: list_tm] :
( ( P @ nil_tm )
=> ( ! [X: tm,Xs2: list_tm] :
( ( P @ Xs2 )
=> ( P @ ( append_tm @ Xs2 @ ( cons_tm @ X @ nil_tm ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_1132_rev__induct,axiom,
! [P: list_fm > $o,Xs: list_fm] :
( ( P @ nil_fm )
=> ( ! [X: fm,Xs2: list_fm] :
( ( P @ Xs2 )
=> ( P @ ( append_fm @ Xs2 @ ( cons_fm @ X @ nil_fm ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_1133_rev__induct,axiom,
! [P: list_list_fm > $o,Xs: list_list_fm] :
( ( P @ nil_list_fm )
=> ( ! [X: list_fm,Xs2: list_list_fm] :
( ( P @ Xs2 )
=> ( P @ ( append_list_fm @ Xs2 @ ( cons_list_fm @ X @ nil_list_fm ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_1134_rev__exhaust,axiom,
! [Xs: list_tm] :
( ( Xs != nil_tm )
=> ~ ! [Ys3: list_tm,Y3: tm] :
( Xs
!= ( append_tm @ Ys3 @ ( cons_tm @ Y3 @ nil_tm ) ) ) ) ).
% rev_exhaust
thf(fact_1135_rev__exhaust,axiom,
! [Xs: list_fm] :
( ( Xs != nil_fm )
=> ~ ! [Ys3: list_fm,Y3: fm] :
( Xs
!= ( append_fm @ Ys3 @ ( cons_fm @ Y3 @ nil_fm ) ) ) ) ).
% rev_exhaust
thf(fact_1136_rev__exhaust,axiom,
! [Xs: list_list_fm] :
( ( Xs != nil_list_fm )
=> ~ ! [Ys3: list_list_fm,Y3: list_fm] :
( Xs
!= ( append_list_fm @ Ys3 @ ( cons_list_fm @ Y3 @ nil_list_fm ) ) ) ) ).
% rev_exhaust
thf(fact_1137_Cons__eq__append__conv,axiom,
! [X2: tm,Xs: list_tm,Ys2: list_tm,Zs2: list_tm] :
( ( ( cons_tm @ X2 @ Xs )
= ( append_tm @ Ys2 @ Zs2 ) )
= ( ( ( Ys2 = nil_tm )
& ( ( cons_tm @ X2 @ Xs )
= Zs2 ) )
| ? [Ys4: list_tm] :
( ( ( cons_tm @ X2 @ Ys4 )
= Ys2 )
& ( Xs
= ( append_tm @ Ys4 @ Zs2 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_1138_Cons__eq__append__conv,axiom,
! [X2: fm,Xs: list_fm,Ys2: list_fm,Zs2: list_fm] :
( ( ( cons_fm @ X2 @ Xs )
= ( append_fm @ Ys2 @ Zs2 ) )
= ( ( ( Ys2 = nil_fm )
& ( ( cons_fm @ X2 @ Xs )
= Zs2 ) )
| ? [Ys4: list_fm] :
( ( ( cons_fm @ X2 @ Ys4 )
= Ys2 )
& ( Xs
= ( append_fm @ Ys4 @ Zs2 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_1139_Cons__eq__append__conv,axiom,
! [X2: list_fm,Xs: list_list_fm,Ys2: list_list_fm,Zs2: list_list_fm] :
( ( ( cons_list_fm @ X2 @ Xs )
= ( append_list_fm @ Ys2 @ Zs2 ) )
= ( ( ( Ys2 = nil_list_fm )
& ( ( cons_list_fm @ X2 @ Xs )
= Zs2 ) )
| ? [Ys4: list_list_fm] :
( ( ( cons_list_fm @ X2 @ Ys4 )
= Ys2 )
& ( Xs
= ( append_list_fm @ Ys4 @ Zs2 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_1140_append__eq__Cons__conv,axiom,
! [Ys2: list_tm,Zs2: list_tm,X2: tm,Xs: list_tm] :
( ( ( append_tm @ Ys2 @ Zs2 )
= ( cons_tm @ X2 @ Xs ) )
= ( ( ( Ys2 = nil_tm )
& ( Zs2
= ( cons_tm @ X2 @ Xs ) ) )
| ? [Ys4: list_tm] :
( ( Ys2
= ( cons_tm @ X2 @ Ys4 ) )
& ( ( append_tm @ Ys4 @ Zs2 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_1141_append__eq__Cons__conv,axiom,
! [Ys2: list_fm,Zs2: list_fm,X2: fm,Xs: list_fm] :
( ( ( append_fm @ Ys2 @ Zs2 )
= ( cons_fm @ X2 @ Xs ) )
= ( ( ( Ys2 = nil_fm )
& ( Zs2
= ( cons_fm @ X2 @ Xs ) ) )
| ? [Ys4: list_fm] :
( ( Ys2
= ( cons_fm @ X2 @ Ys4 ) )
& ( ( append_fm @ Ys4 @ Zs2 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_1142_append__eq__Cons__conv,axiom,
! [Ys2: list_list_fm,Zs2: list_list_fm,X2: list_fm,Xs: list_list_fm] :
( ( ( append_list_fm @ Ys2 @ Zs2 )
= ( cons_list_fm @ X2 @ Xs ) )
= ( ( ( Ys2 = nil_list_fm )
& ( Zs2
= ( cons_list_fm @ X2 @ Xs ) ) )
| ? [Ys4: list_list_fm] :
( ( Ys2
= ( cons_list_fm @ X2 @ Ys4 ) )
& ( ( append_list_fm @ Ys4 @ Zs2 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_1143_rev__nonempty__induct,axiom,
! [Xs: list_tm,P: list_tm > $o] :
( ( Xs != nil_tm )
=> ( ! [X: tm] : ( P @ ( cons_tm @ X @ nil_tm ) )
=> ( ! [X: tm,Xs2: list_tm] :
( ( Xs2 != nil_tm )
=> ( ( P @ Xs2 )
=> ( P @ ( append_tm @ Xs2 @ ( cons_tm @ X @ nil_tm ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_1144_rev__nonempty__induct,axiom,
! [Xs: list_fm,P: list_fm > $o] :
( ( Xs != nil_fm )
=> ( ! [X: fm] : ( P @ ( cons_fm @ X @ nil_fm ) )
=> ( ! [X: fm,Xs2: list_fm] :
( ( Xs2 != nil_fm )
=> ( ( P @ Xs2 )
=> ( P @ ( append_fm @ Xs2 @ ( cons_fm @ X @ nil_fm ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_1145_rev__nonempty__induct,axiom,
! [Xs: list_list_fm,P: list_list_fm > $o] :
( ( Xs != nil_list_fm )
=> ( ! [X: list_fm] : ( P @ ( cons_list_fm @ X @ nil_list_fm ) )
=> ( ! [X: list_fm,Xs2: list_list_fm] :
( ( Xs2 != nil_list_fm )
=> ( ( P @ Xs2 )
=> ( P @ ( append_list_fm @ Xs2 @ ( cons_list_fm @ X @ nil_list_fm ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_1146_liftts_Osimps_I1_J,axiom,
( ( liftts @ nil_tm )
= nil_tm ) ).
% liftts.simps(1)
thf(fact_1147_SuccD,axiom,
! [K: nat,Kl4: set_list_nat,Kl3: list_nat] :
( ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl4 @ Kl3 ) )
=> ( member_list_nat @ ( append_nat @ Kl3 @ ( cons_nat @ K @ nil_nat ) ) @ Kl4 ) ) ).
% SuccD
thf(fact_1148_SuccD,axiom,
! [K: tm,Kl4: set_list_tm,Kl3: list_tm] :
( ( member_tm @ K @ ( bNF_Greatest_Succ_tm @ Kl4 @ Kl3 ) )
=> ( member_list_tm @ ( append_tm @ Kl3 @ ( cons_tm @ K @ nil_tm ) ) @ Kl4 ) ) ).
% SuccD
thf(fact_1149_SuccD,axiom,
! [K: fm,Kl4: set_list_fm,Kl3: list_fm] :
( ( member_fm @ K @ ( bNF_Greatest_Succ_fm @ Kl4 @ Kl3 ) )
=> ( member_list_fm @ ( append_fm @ Kl3 @ ( cons_fm @ K @ nil_fm ) ) @ Kl4 ) ) ).
% SuccD
thf(fact_1150_SuccD,axiom,
! [K: list_fm,Kl4: set_list_list_fm,Kl3: list_list_fm] :
( ( member_list_fm @ K @ ( bNF_Gr8387611704671093012ist_fm @ Kl4 @ Kl3 ) )
=> ( member_list_list_fm @ ( append_list_fm @ Kl3 @ ( cons_list_fm @ K @ nil_list_fm ) ) @ Kl4 ) ) ).
% SuccD
thf(fact_1151_SuccI,axiom,
! [Kl3: list_nat,K: nat,Kl4: set_list_nat] :
( ( member_list_nat @ ( append_nat @ Kl3 @ ( cons_nat @ K @ nil_nat ) ) @ Kl4 )
=> ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl4 @ Kl3 ) ) ) ).
% SuccI
thf(fact_1152_SuccI,axiom,
! [Kl3: list_tm,K: tm,Kl4: set_list_tm] :
( ( member_list_tm @ ( append_tm @ Kl3 @ ( cons_tm @ K @ nil_tm ) ) @ Kl4 )
=> ( member_tm @ K @ ( bNF_Greatest_Succ_tm @ Kl4 @ Kl3 ) ) ) ).
% SuccI
thf(fact_1153_SuccI,axiom,
! [Kl3: list_fm,K: fm,Kl4: set_list_fm] :
( ( member_list_fm @ ( append_fm @ Kl3 @ ( cons_fm @ K @ nil_fm ) ) @ Kl4 )
=> ( member_fm @ K @ ( bNF_Greatest_Succ_fm @ Kl4 @ Kl3 ) ) ) ).
% SuccI
thf(fact_1154_SuccI,axiom,
! [Kl3: list_list_fm,K: list_fm,Kl4: set_list_list_fm] :
( ( member_list_list_fm @ ( append_list_fm @ Kl3 @ ( cons_list_fm @ K @ nil_list_fm ) ) @ Kl4 )
=> ( member_list_fm @ K @ ( bNF_Gr8387611704671093012ist_fm @ Kl4 @ Kl3 ) ) ) ).
% SuccI
thf(fact_1155_Succ__def,axiom,
( bNF_Gr6352880689984616693cc_nat
= ( ^ [Kl: set_list_nat,Kl2: list_nat] :
( collect_nat
@ ^ [K2: nat] : ( member_list_nat @ ( append_nat @ Kl2 @ ( cons_nat @ K2 @ nil_nat ) ) @ Kl ) ) ) ) ).
% Succ_def
thf(fact_1156_Succ__def,axiom,
( bNF_Greatest_Succ_tm
= ( ^ [Kl: set_list_tm,Kl2: list_tm] :
( collect_tm
@ ^ [K2: tm] : ( member_list_tm @ ( append_tm @ Kl2 @ ( cons_tm @ K2 @ nil_tm ) ) @ Kl ) ) ) ) ).
% Succ_def
thf(fact_1157_Succ__def,axiom,
( bNF_Greatest_Succ_fm
= ( ^ [Kl: set_list_fm,Kl2: list_fm] :
( collect_fm
@ ^ [K2: fm] : ( member_list_fm @ ( append_fm @ Kl2 @ ( cons_fm @ K2 @ nil_fm ) ) @ Kl ) ) ) ) ).
% Succ_def
thf(fact_1158_Succ__def,axiom,
( bNF_Gr8387611704671093012ist_fm
= ( ^ [Kl: set_list_list_fm,Kl2: list_list_fm] :
( collect_list_fm
@ ^ [K2: list_fm] : ( member_list_list_fm @ ( append_list_fm @ Kl2 @ ( cons_list_fm @ K2 @ nil_list_fm ) ) @ Kl ) ) ) ) ).
% Succ_def
thf(fact_1159_suffixes_Osimps_I2_J,axiom,
! [X2: tm,Xs: list_tm] :
( ( suffixes_tm @ ( cons_tm @ X2 @ Xs ) )
= ( append_list_tm @ ( suffixes_tm @ Xs ) @ ( cons_list_tm @ ( cons_tm @ X2 @ Xs ) @ nil_list_tm ) ) ) ).
% suffixes.simps(2)
thf(fact_1160_suffixes_Osimps_I2_J,axiom,
! [X2: fm,Xs: list_fm] :
( ( suffixes_fm @ ( cons_fm @ X2 @ Xs ) )
= ( append_list_fm @ ( suffixes_fm @ Xs ) @ ( cons_list_fm @ ( cons_fm @ X2 @ Xs ) @ nil_list_fm ) ) ) ).
% suffixes.simps(2)
thf(fact_1161_suffixes_Osimps_I2_J,axiom,
! [X2: list_fm,Xs: list_list_fm] :
( ( suffixes_list_fm @ ( cons_list_fm @ X2 @ Xs ) )
= ( append_list_list_fm @ ( suffixes_list_fm @ Xs ) @ ( cons_list_list_fm @ ( cons_list_fm @ X2 @ Xs ) @ nil_list_list_fm ) ) ) ).
% suffixes.simps(2)
thf(fact_1162_Collect__restrict,axiom,
! [X7: set_list_fm,P: list_fm > $o] :
( ord_le7838213414353715577ist_fm
@ ( collect_list_fm
@ ^ [X5: list_fm] :
( ( member_list_fm @ X5 @ X7 )
& ( P @ X5 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_1163_Collect__restrict,axiom,
! [X7: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ X7 )
& ( P @ X5 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_1164_Collect__restrict,axiom,
! [X7: set_tm,P: tm > $o] :
( ord_less_eq_set_tm
@ ( collect_tm
@ ^ [X5: tm] :
( ( member_tm @ X5 @ X7 )
& ( P @ X5 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_1165_prop__restrict,axiom,
! [X2: list_fm,Z4: set_list_fm,X7: set_list_fm,P: list_fm > $o] :
( ( member_list_fm @ X2 @ Z4 )
=> ( ( ord_le7838213414353715577ist_fm @ Z4
@ ( collect_list_fm
@ ^ [X5: list_fm] :
( ( member_list_fm @ X5 @ X7 )
& ( P @ X5 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_1166_prop__restrict,axiom,
! [X2: nat,Z4: set_nat,X7: set_nat,P: nat > $o] :
( ( member_nat @ X2 @ Z4 )
=> ( ( ord_less_eq_set_nat @ Z4
@ ( collect_nat
@ ^ [X5: nat] :
( ( member_nat @ X5 @ X7 )
& ( P @ X5 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_1167_prop__restrict,axiom,
! [X2: tm,Z4: set_tm,X7: set_tm,P: tm > $o] :
( ( member_tm @ X2 @ Z4 )
=> ( ( ord_less_eq_set_tm @ Z4
@ ( collect_tm
@ ^ [X5: tm] :
( ( member_tm @ X5 @ X7 )
& ( P @ X5 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_1168_prefixes__snoc,axiom,
! [Xs: list_tm,X2: tm] :
( ( prefixes_tm @ ( append_tm @ Xs @ ( cons_tm @ X2 @ nil_tm ) ) )
= ( append_list_tm @ ( prefixes_tm @ Xs ) @ ( cons_list_tm @ ( append_tm @ Xs @ ( cons_tm @ X2 @ nil_tm ) ) @ nil_list_tm ) ) ) ).
% prefixes_snoc
thf(fact_1169_prefixes__snoc,axiom,
! [Xs: list_fm,X2: fm] :
( ( prefixes_fm @ ( append_fm @ Xs @ ( cons_fm @ X2 @ nil_fm ) ) )
= ( append_list_fm @ ( prefixes_fm @ Xs ) @ ( cons_list_fm @ ( append_fm @ Xs @ ( cons_fm @ X2 @ nil_fm ) ) @ nil_list_fm ) ) ) ).
% prefixes_snoc
thf(fact_1170_prefixes__snoc,axiom,
! [Xs: list_list_fm,X2: list_fm] :
( ( prefixes_list_fm @ ( append_list_fm @ Xs @ ( cons_list_fm @ X2 @ nil_list_fm ) ) )
= ( append_list_list_fm @ ( prefixes_list_fm @ Xs ) @ ( cons_list_list_fm @ ( append_list_fm @ Xs @ ( cons_list_fm @ X2 @ nil_list_fm ) ) @ nil_list_list_fm ) ) ) ).
% prefixes_snoc
thf(fact_1171_listset_Osimps_I1_J,axiom,
( ( listset_tm @ nil_set_tm )
= ( insert_list_tm @ nil_tm @ bot_bot_set_list_tm ) ) ).
% listset.simps(1)
thf(fact_1172_listset_Osimps_I1_J,axiom,
( ( listset_fm @ nil_set_fm )
= ( insert_list_fm2 @ nil_fm @ bot_bot_set_list_fm ) ) ).
% listset.simps(1)
thf(fact_1173_listset_Osimps_I1_J,axiom,
( ( listset_list_fm @ nil_set_list_fm )
= ( insert_list_list_fm @ nil_list_fm @ bot_bo6959746583267231307ist_fm ) ) ).
% listset.simps(1)
thf(fact_1174_prefixes__eq__snoc,axiom,
! [Ys2: list_tm,Xs: list_list_tm,X2: list_tm] :
( ( ( prefixes_tm @ Ys2 )
= ( append_list_tm @ Xs @ ( cons_list_tm @ X2 @ nil_list_tm ) ) )
= ( ( ( ( Ys2 = nil_tm )
& ( Xs = nil_list_tm ) )
| ? [Z3: tm,Zs: list_tm] :
( ( Ys2
= ( append_tm @ Zs @ ( cons_tm @ Z3 @ nil_tm ) ) )
& ( Xs
= ( prefixes_tm @ Zs ) ) ) )
& ( X2 = Ys2 ) ) ) ).
% prefixes_eq_snoc
thf(fact_1175_prefixes__eq__snoc,axiom,
! [Ys2: list_list_fm,Xs: list_list_list_fm,X2: list_list_fm] :
( ( ( prefixes_list_fm @ Ys2 )
= ( append_list_list_fm @ Xs @ ( cons_list_list_fm @ X2 @ nil_list_list_fm ) ) )
= ( ( ( ( Ys2 = nil_list_fm )
& ( Xs = nil_list_list_fm ) )
| ? [Z3: list_fm,Zs: list_list_fm] :
( ( Ys2
= ( append_list_fm @ Zs @ ( cons_list_fm @ Z3 @ nil_list_fm ) ) )
& ( Xs
= ( prefixes_list_fm @ Zs ) ) ) )
& ( X2 = Ys2 ) ) ) ).
% prefixes_eq_snoc
thf(fact_1176_prefixes__eq__snoc,axiom,
! [Ys2: list_fm,Xs: list_list_fm,X2: list_fm] :
( ( ( prefixes_fm @ Ys2 )
= ( append_list_fm @ Xs @ ( cons_list_fm @ X2 @ nil_list_fm ) ) )
= ( ( ( ( Ys2 = nil_fm )
& ( Xs = nil_list_fm ) )
| ? [Z3: fm,Zs: list_fm] :
( ( Ys2
= ( append_fm @ Zs @ ( cons_fm @ Z3 @ nil_fm ) ) )
& ( Xs
= ( prefixes_fm @ Zs ) ) ) )
& ( X2 = Ys2 ) ) ) ).
% prefixes_eq_snoc
thf(fact_1177_s4_I2_J,axiom,
inc_list = liftts ).
% s4(2)
thf(fact_1178_bind__simps_I2_J,axiom,
! [X2: tm,Xs: list_tm,F: tm > list_tm] :
( ( bind_tm_tm @ ( cons_tm @ X2 @ Xs ) @ F )
= ( append_tm @ ( F @ X2 ) @ ( bind_tm_tm @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_1179_bind__simps_I2_J,axiom,
! [X2: tm,Xs: list_tm,F: tm > list_fm] :
( ( bind_tm_fm @ ( cons_tm @ X2 @ Xs ) @ F )
= ( append_fm @ ( F @ X2 ) @ ( bind_tm_fm @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_1180_bind__simps_I2_J,axiom,
! [X2: fm,Xs: list_fm,F: fm > list_tm] :
( ( bind_fm_tm @ ( cons_fm @ X2 @ Xs ) @ F )
= ( append_tm @ ( F @ X2 ) @ ( bind_fm_tm @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_1181_bind__simps_I2_J,axiom,
! [X2: fm,Xs: list_fm,F: fm > list_fm] :
( ( bind_fm_fm @ ( cons_fm @ X2 @ Xs ) @ F )
= ( append_fm @ ( F @ X2 ) @ ( bind_fm_fm @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_1182_bind__simps_I2_J,axiom,
! [X2: list_fm,Xs: list_list_fm,F: list_fm > list_tm] :
( ( bind_list_fm_tm @ ( cons_list_fm @ X2 @ Xs ) @ F )
= ( append_tm @ ( F @ X2 ) @ ( bind_list_fm_tm @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_1183_bind__simps_I2_J,axiom,
! [X2: list_fm,Xs: list_list_fm,F: list_fm > list_fm] :
( ( bind_list_fm_fm @ ( cons_list_fm @ X2 @ Xs ) @ F )
= ( append_fm @ ( F @ X2 ) @ ( bind_list_fm_fm @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_1184_bind__simps_I1_J,axiom,
! [F: tm > list_tm] :
( ( bind_tm_tm @ nil_tm @ F )
= nil_tm ) ).
% bind_simps(1)
thf(fact_1185_bind__simps_I1_J,axiom,
! [F: tm > list_fm] :
( ( bind_tm_fm @ nil_tm @ F )
= nil_fm ) ).
% bind_simps(1)
thf(fact_1186_bind__simps_I1_J,axiom,
! [F: tm > list_list_fm] :
( ( bind_tm_list_fm @ nil_tm @ F )
= nil_list_fm ) ).
% bind_simps(1)
thf(fact_1187_bind__simps_I1_J,axiom,
! [F: fm > list_tm] :
( ( bind_fm_tm @ nil_fm @ F )
= nil_tm ) ).
% bind_simps(1)
thf(fact_1188_bind__simps_I1_J,axiom,
! [F: fm > list_fm] :
( ( bind_fm_fm @ nil_fm @ F )
= nil_fm ) ).
% bind_simps(1)
thf(fact_1189_bind__simps_I1_J,axiom,
! [F: fm > list_list_fm] :
( ( bind_fm_list_fm @ nil_fm @ F )
= nil_list_fm ) ).
% bind_simps(1)
thf(fact_1190_bind__simps_I1_J,axiom,
! [F: list_fm > list_tm] :
( ( bind_list_fm_tm @ nil_list_fm @ F )
= nil_tm ) ).
% bind_simps(1)
thf(fact_1191_bind__simps_I1_J,axiom,
! [F: list_fm > list_fm] :
( ( bind_list_fm_fm @ nil_list_fm @ F )
= nil_fm ) ).
% bind_simps(1)
thf(fact_1192_bind__simps_I1_J,axiom,
! [F: list_fm > list_list_fm] :
( ( bind_list_fm_list_fm @ nil_list_fm @ F )
= nil_list_fm ) ).
% bind_simps(1)
thf(fact_1193_inc__list_Osimps_I1_J,axiom,
( ( inc_list @ nil_tm )
= nil_tm ) ).
% inc_list.simps(1)
thf(fact_1194_prefixes_Osimps_I1_J,axiom,
( ( prefixes_tm @ nil_tm )
= ( cons_list_tm @ nil_tm @ nil_list_tm ) ) ).
% prefixes.simps(1)
thf(fact_1195_prefixes_Osimps_I1_J,axiom,
( ( prefixes_list_fm @ nil_list_fm )
= ( cons_list_list_fm @ nil_list_fm @ nil_list_list_fm ) ) ).
% prefixes.simps(1)
thf(fact_1196_prefixes_Osimps_I1_J,axiom,
( ( prefixes_fm @ nil_fm )
= ( cons_list_fm @ nil_fm @ nil_list_fm ) ) ).
% prefixes.simps(1)
thf(fact_1197_inc__list_Osimps_I2_J,axiom,
! [T: tm,L2: list_tm] :
( ( inc_list @ ( cons_tm @ T @ L2 ) )
= ( cons_tm @ ( inc_term @ T ) @ ( inc_list @ L2 ) ) ) ).
% inc_list.simps(2)
thf(fact_1198_inc__term_Osimps_I2_J,axiom,
! [I2: nat,L2: list_tm] :
( ( inc_term @ ( fun @ I2 @ L2 ) )
= ( fun @ I2 @ ( inc_list @ L2 ) ) ) ).
% inc_term.simps(2)
thf(fact_1199_suffixes__snoc,axiom,
! [Xs: list_tm,X2: tm] :
( ( suffixes_tm @ ( append_tm @ Xs @ ( cons_tm @ X2 @ nil_tm ) ) )
= ( cons_list_tm @ nil_tm
@ ( map_list_tm_list_tm
@ ^ [Ys: list_tm] : ( append_tm @ Ys @ ( cons_tm @ X2 @ nil_tm ) )
@ ( suffixes_tm @ Xs ) ) ) ) ).
% suffixes_snoc
thf(fact_1200_suffixes__snoc,axiom,
! [Xs: list_fm,X2: fm] :
( ( suffixes_fm @ ( append_fm @ Xs @ ( cons_fm @ X2 @ nil_fm ) ) )
= ( cons_list_fm @ nil_fm
@ ( map_list_fm_list_fm
@ ^ [Ys: list_fm] : ( append_fm @ Ys @ ( cons_fm @ X2 @ nil_fm ) )
@ ( suffixes_fm @ Xs ) ) ) ) ).
% suffixes_snoc
thf(fact_1201_suffixes__snoc,axiom,
! [Xs: list_list_fm,X2: list_fm] :
( ( suffixes_list_fm @ ( append_list_fm @ Xs @ ( cons_list_fm @ X2 @ nil_list_fm ) ) )
= ( cons_list_list_fm @ nil_list_fm
@ ( map_li4351931137408529412ist_fm
@ ^ [Ys: list_list_fm] : ( append_list_fm @ Ys @ ( cons_list_fm @ X2 @ nil_list_fm ) )
@ ( suffixes_list_fm @ Xs ) ) ) ) ).
% suffixes_snoc
thf(fact_1202_lift__lemma_I2_J,axiom,
! [E: nat > tm,X2: tm,F: nat > list_tm > tm,Ts: list_tm] :
( ( semantics_list_tm @ ( shift_nat_tm @ E @ zero_zero_nat @ X2 ) @ F @ ( liftts @ Ts ) )
= ( semantics_list_tm @ E @ F @ Ts ) ) ).
% lift_lemma(2)
thf(fact_1203_liftts_Osimps_I2_J,axiom,
! [T: tm,Ts: list_tm] :
( ( liftts @ ( cons_tm @ T @ Ts ) )
= ( cons_tm @ ( liftt @ T ) @ ( liftts @ Ts ) ) ) ).
% liftts.simps(2)
thf(fact_1204_list_Omap__disc__iff,axiom,
! [F: tm > tm,A: list_tm] :
( ( ( map_tm_tm @ F @ A )
= nil_tm )
= ( A = nil_tm ) ) ).
% list.map_disc_iff
thf(fact_1205_list_Omap__disc__iff,axiom,
! [F: fm > tm,A: list_fm] :
( ( ( map_fm_tm @ F @ A )
= nil_tm )
= ( A = nil_fm ) ) ).
% list.map_disc_iff
thf(fact_1206_list_Omap__disc__iff,axiom,
! [F: tm > fm,A: list_tm] :
( ( ( map_tm_fm @ F @ A )
= nil_fm )
= ( A = nil_tm ) ) ).
% list.map_disc_iff
thf(fact_1207_list_Omap__disc__iff,axiom,
! [F: fm > fm,A: list_fm] :
( ( ( map_fm_fm @ F @ A )
= nil_fm )
= ( A = nil_fm ) ) ).
% list.map_disc_iff
thf(fact_1208_list_Omap__disc__iff,axiom,
! [F: list_fm > tm,A: list_list_fm] :
( ( ( map_list_fm_tm @ F @ A )
= nil_tm )
= ( A = nil_list_fm ) ) ).
% list.map_disc_iff
thf(fact_1209_list_Omap__disc__iff,axiom,
! [F: list_fm > fm,A: list_list_fm] :
( ( ( map_list_fm_fm @ F @ A )
= nil_fm )
= ( A = nil_list_fm ) ) ).
% list.map_disc_iff
thf(fact_1210_list_Omap__disc__iff,axiom,
! [F: tm > list_fm,A: list_tm] :
( ( ( map_tm_list_fm @ F @ A )
= nil_list_fm )
= ( A = nil_tm ) ) ).
% list.map_disc_iff
thf(fact_1211_list_Omap__disc__iff,axiom,
! [F: fm > list_fm,A: list_fm] :
( ( ( map_fm_list_fm @ F @ A )
= nil_list_fm )
= ( A = nil_fm ) ) ).
% list.map_disc_iff
thf(fact_1212_list_Omap__disc__iff,axiom,
! [F: tm > set_nat,A: list_tm] :
( ( ( map_tm_set_nat @ F @ A )
= nil_set_nat )
= ( A = nil_tm ) ) ).
% list.map_disc_iff
thf(fact_1213_list_Omap__disc__iff,axiom,
! [F: fm > list_tm,A: list_fm] :
( ( ( map_fm_list_tm @ F @ A )
= nil_list_tm )
= ( A = nil_fm ) ) ).
% list.map_disc_iff
thf(fact_1214_Nil__is__map__conv,axiom,
! [F: tm > tm,Xs: list_tm] :
( ( nil_tm
= ( map_tm_tm @ F @ Xs ) )
= ( Xs = nil_tm ) ) ).
% Nil_is_map_conv
thf(fact_1215_Nil__is__map__conv,axiom,
! [F: fm > tm,Xs: list_fm] :
( ( nil_tm
= ( map_fm_tm @ F @ Xs ) )
= ( Xs = nil_fm ) ) ).
% Nil_is_map_conv
thf(fact_1216_Nil__is__map__conv,axiom,
! [F: tm > fm,Xs: list_tm] :
( ( nil_fm
= ( map_tm_fm @ F @ Xs ) )
= ( Xs = nil_tm ) ) ).
% Nil_is_map_conv
thf(fact_1217_Nil__is__map__conv,axiom,
! [F: fm > fm,Xs: list_fm] :
( ( nil_fm
= ( map_fm_fm @ F @ Xs ) )
= ( Xs = nil_fm ) ) ).
% Nil_is_map_conv
thf(fact_1218_Nil__is__map__conv,axiom,
! [F: list_fm > tm,Xs: list_list_fm] :
( ( nil_tm
= ( map_list_fm_tm @ F @ Xs ) )
= ( Xs = nil_list_fm ) ) ).
% Nil_is_map_conv
thf(fact_1219_Nil__is__map__conv,axiom,
! [F: list_fm > fm,Xs: list_list_fm] :
( ( nil_fm
= ( map_list_fm_fm @ F @ Xs ) )
= ( Xs = nil_list_fm ) ) ).
% Nil_is_map_conv
thf(fact_1220_Nil__is__map__conv,axiom,
! [F: tm > list_fm,Xs: list_tm] :
( ( nil_list_fm
= ( map_tm_list_fm @ F @ Xs ) )
= ( Xs = nil_tm ) ) ).
% Nil_is_map_conv
thf(fact_1221_Nil__is__map__conv,axiom,
! [F: fm > list_fm,Xs: list_fm] :
( ( nil_list_fm
= ( map_fm_list_fm @ F @ Xs ) )
= ( Xs = nil_fm ) ) ).
% Nil_is_map_conv
thf(fact_1222_Nil__is__map__conv,axiom,
! [F: tm > set_nat,Xs: list_tm] :
( ( nil_set_nat
= ( map_tm_set_nat @ F @ Xs ) )
= ( Xs = nil_tm ) ) ).
% Nil_is_map_conv
thf(fact_1223_Nil__is__map__conv,axiom,
! [F: fm > list_tm,Xs: list_fm] :
( ( nil_list_tm
= ( map_fm_list_tm @ F @ Xs ) )
= ( Xs = nil_fm ) ) ).
% Nil_is_map_conv
thf(fact_1224_map__is__Nil__conv,axiom,
! [F: fm > list_fm,Xs: list_fm] :
( ( ( map_fm_list_fm @ F @ Xs )
= nil_list_fm )
= ( Xs = nil_fm ) ) ).
% map_is_Nil_conv
thf(fact_1225_map__is__Nil__conv,axiom,
! [F: list_fm > list_fm,Xs: list_list_fm] :
( ( ( map_list_fm_list_fm @ F @ Xs )
= nil_list_fm )
= ( Xs = nil_list_fm ) ) ).
% map_is_Nil_conv
thf(fact_1226_map__is__Nil__conv,axiom,
! [F: tm > set_nat,Xs: list_tm] :
( ( ( map_tm_set_nat @ F @ Xs )
= nil_set_nat )
= ( Xs = nil_tm ) ) ).
% map_is_Nil_conv
thf(fact_1227_map__is__Nil__conv,axiom,
! [F: fm > list_tm,Xs: list_fm] :
( ( ( map_fm_list_tm @ F @ Xs )
= nil_list_tm )
= ( Xs = nil_fm ) ) ).
% map_is_Nil_conv
thf(fact_1228_map__is__Nil__conv,axiom,
! [F: tm > list_tm,Xs: list_tm] :
( ( ( map_tm_list_tm @ F @ Xs )
= nil_list_tm )
= ( Xs = nil_tm ) ) ).
% map_is_Nil_conv
thf(fact_1229_paramst__liftt_I1_J,axiom,
! [T: tm] :
( ( paramst @ ( liftt @ T ) )
= ( paramst @ T ) ) ).
% paramst_liftt(1)
thf(fact_1230_s4_I1_J,axiom,
inc_term = liftt ).
% s4(1)
thf(fact_1231_liftt_Osimps_I2_J,axiom,
! [A: nat,Ts: list_tm] :
( ( liftt @ ( fun @ A @ Ts ) )
= ( fun @ A @ ( liftts @ Ts ) ) ) ).
% liftt.simps(2)
thf(fact_1232_tm_Osize__gen_I2_J,axiom,
! [X22: nat] :
( ( size_tm @ ( var @ X22 ) )
= zero_zero_nat ) ).
% tm.size_gen(2)
thf(fact_1233_subtermTm_Osimps_I2_J,axiom,
! [N: nat] :
( ( subtermTm @ ( var @ N ) )
= ( cons_tm @ ( var @ N ) @ nil_tm ) ) ).
% subtermTm.simps(2)
thf(fact_1234_liftt_Osimps_I1_J,axiom,
! [I2: nat] :
( ( liftt @ ( var @ I2 ) )
= ( var @ ( suc @ I2 ) ) ) ).
% liftt.simps(1)
thf(fact_1235_UNIV__nat__eq,axiom,
( top_top_set_nat
= ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ top_top_set_nat ) ) ) ).
% UNIV_nat_eq
thf(fact_1236_tm_Osize_I4_J,axiom,
! [X22: nat] :
( ( size_size_tm @ ( var @ X22 ) )
= zero_zero_nat ) ).
% tm.size(4)
thf(fact_1237_sub__term__const__transfer_I1_J,axiom,
! [M: nat,A: nat,T: tm,S3: tm] :
( ( ( sub_term @ M @ ( fun @ A @ nil_tm ) @ T )
!= ( sub_term @ M @ S3 @ T ) )
=> ( member_tm @ ( fun @ A @ nil_tm ) @ ( set_tm2 @ ( subtermTm @ ( sub_term @ M @ ( fun @ A @ nil_tm ) @ T ) ) ) ) ) ).
% sub_term_const_transfer(1)
thf(fact_1238_subterm__Fun__refl,axiom,
! [Ts: list_tm,N: nat] : ( ord_less_eq_set_tm @ ( set_tm2 @ Ts ) @ ( set_tm2 @ ( subtermTm @ ( fun @ N @ Ts ) ) ) ) ).
% subterm_Fun_refl
thf(fact_1239_terms__downwards__closed,axiom,
! [T: tm,S: set_fm] :
( ( member_tm @ T @ ( terms @ S ) )
=> ( ord_less_eq_set_tm @ ( set_tm2 @ ( subtermTm @ T ) ) @ ( terms @ S ) ) ) ).
% terms_downwards_closed
thf(fact_1240_paramsts__subset,axiom,
! [A2: list_tm,B: list_tm] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ A2 ) @ ( set_tm2 @ B ) )
=> ( ord_less_eq_set_nat @ ( paramsts @ A2 ) @ ( paramsts @ B ) ) ) ).
% paramsts_subset
thf(fact_1241_paramst__subtermTm_I1_J,axiom,
! [T: tm,X6: nat] :
( ( member_nat @ X6 @ ( paramst @ T ) )
=> ? [L3: list_tm] : ( member_tm @ ( fun @ X6 @ L3 ) @ ( set_tm2 @ ( subtermTm @ T ) ) ) ) ).
% paramst_subtermTm(1)
thf(fact_1242_paramst_H_H_Oelims,axiom,
! [X2: tm,Y4: set_nat] :
( ( ( paramst3 @ X2 )
= Y4 )
=> ( ( ? [N3: nat] :
( X2
= ( var @ N3 ) )
=> ( Y4 != bot_bot_set_nat ) )
=> ~ ! [A3: nat,Ts2: list_tm] :
( ( X2
= ( fun @ A3 @ Ts2 ) )
=> ( Y4
!= ( sup_sup_set_nat @ ( insert_nat @ A3 @ bot_bot_set_nat ) @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ paramst3 @ ( set_tm2 @ Ts2 ) ) ) ) ) ) ) ) ).
% paramst''.elims
thf(fact_1243_paramst_H_H_Osimps_I2_J,axiom,
! [A: nat,Ts: list_tm] :
( ( paramst3 @ ( fun @ A @ Ts ) )
= ( sup_sup_set_nat @ ( insert_nat @ A @ bot_bot_set_nat ) @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ paramst3 @ ( set_tm2 @ Ts ) ) ) ) ) ).
% paramst''.simps(2)
thf(fact_1244_p0,axiom,
( paramsts
= ( ^ [Ts3: list_tm] : ( comple7399068483239264473et_nat @ ( set_set_nat2 @ ( map_tm_set_nat @ paramst @ Ts3 ) ) ) ) ) ).
% p0
thf(fact_1245_paramst_H_Osimps_I2_J,axiom,
! [A: nat,Ts: list_tm] :
( ( paramst2 @ ( fun @ A @ Ts ) )
= ( sup_sup_set_nat @ ( insert_nat @ A @ bot_bot_set_nat ) @ ( comple7399068483239264473et_nat @ ( set_set_nat2 @ ( map_tm_set_nat @ paramst2 @ Ts ) ) ) ) ) ).
% paramst'.simps(2)
thf(fact_1246_paramst__subtermTm_I2_J,axiom,
! [Ts: list_tm,X6: nat] :
( ( member_nat @ X6 @ ( paramsts @ Ts ) )
=> ? [L3: list_tm] :
( member_tm @ ( fun @ X6 @ L3 )
@ ( comple2138885804642794802set_tm
@ ( image_tm_set_tm
@ ^ [T2: tm] : ( set_tm2 @ ( subtermTm @ T2 ) )
@ ( set_tm2 @ Ts ) ) ) ) ) ).
% paramst_subtermTm(2)
thf(fact_1247_sub__term__const__transfer_I2_J,axiom,
! [M: nat,A: nat,Ts: list_tm,S3: tm] :
( ( ( sub_list @ M @ ( fun @ A @ nil_tm ) @ Ts )
!= ( sub_list @ M @ S3 @ Ts ) )
=> ( member_tm @ ( fun @ A @ nil_tm )
@ ( comple2138885804642794802set_tm
@ ( image_tm_set_tm
@ ^ [T2: tm] : ( set_tm2 @ ( subtermTm @ T2 ) )
@ ( set_tm2 @ ( sub_list @ M @ ( fun @ A @ nil_tm ) @ Ts ) ) ) ) ) ) ).
% sub_term_const_transfer(2)
thf(fact_1248_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_1249_UNIV__bool,axiom,
( top_top_set_o
= ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).
% UNIV_bool
thf(fact_1250_paramst_H_H_Opelims,axiom,
! [X2: tm,Y4: set_nat] :
( ( ( paramst3 @ X2 )
= Y4 )
=> ( ( accp_tm @ paramst_rel @ X2 )
=> ( ! [N3: nat] :
( ( X2
= ( var @ N3 ) )
=> ( ( Y4 = bot_bot_set_nat )
=> ~ ( accp_tm @ paramst_rel @ ( var @ N3 ) ) ) )
=> ~ ! [A3: nat,Ts2: list_tm] :
( ( X2
= ( fun @ A3 @ Ts2 ) )
=> ( ( Y4
= ( sup_sup_set_nat @ ( insert_nat @ A3 @ bot_bot_set_nat ) @ ( comple7399068483239264473et_nat @ ( image_tm_set_nat @ paramst3 @ ( set_tm2 @ Ts2 ) ) ) ) )
=> ~ ( accp_tm @ paramst_rel @ ( fun @ A3 @ Ts2 ) ) ) ) ) ) ) ).
% paramst''.pelims
thf(fact_1251_terms__def,axiom,
( terms
= ( ^ [H: set_fm] :
( if_set_tm
@ ( ( comple2138885804642794802set_tm
@ ( image_fm_set_tm
@ ^ [P6: fm] : ( set_tm2 @ ( subtermFm @ P6 ) )
@ H ) )
= bot_bot_set_tm )
@ ( insert_tm2 @ ( fun @ zero_zero_nat @ nil_tm ) @ bot_bot_set_tm )
@ ( comple2138885804642794802set_tm
@ ( image_fm_set_tm
@ ^ [P6: fm] : ( set_tm2 @ ( subtermFm @ P6 ) )
@ H ) ) ) ) ) ).
% terms_def
thf(fact_1252_fun__arguments__subterm,axiom,
! [N: nat,Ts: list_tm,P7: fm] :
( ( member_tm @ ( fun @ N @ Ts ) @ ( set_tm2 @ ( subtermFm @ P7 ) ) )
=> ( ord_less_eq_set_tm @ ( set_tm2 @ Ts ) @ ( set_tm2 @ ( subtermFm @ P7 ) ) ) ) ).
% fun_arguments_subterm
thf(fact_1253_terms__cases,axiom,
! [T: tm,S: set_fm] :
( ( member_tm @ T @ ( terms @ S ) )
=> ( ( T
= ( fun @ zero_zero_nat @ nil_tm ) )
| ? [X: fm] :
( ( member_fm @ X @ S )
& ( member_tm @ T @ ( set_tm2 @ ( subtermFm @ X ) ) ) ) ) ) ).
% terms_cases
thf(fact_1254_set__subterms,axiom,
! [Z2: list_fm] :
( ( ( ( comple2138885804642794802set_tm
@ ( image_fm_set_tm
@ ^ [P6: fm] : ( set_tm2 @ ( subtermFm @ P6 ) )
@ ( set_fm2 @ Z2 ) ) )
= bot_bot_set_tm )
=> ( ( set_tm2 @ ( subterms @ Z2 ) )
= ( insert_tm2 @ ( fun @ zero_zero_nat @ nil_tm ) @ bot_bot_set_tm ) ) )
& ( ( ( comple2138885804642794802set_tm
@ ( image_fm_set_tm
@ ^ [P6: fm] : ( set_tm2 @ ( subtermFm @ P6 ) )
@ ( set_fm2 @ Z2 ) ) )
!= bot_bot_set_tm )
=> ( ( set_tm2 @ ( subterms @ Z2 ) )
= ( comple2138885804642794802set_tm
@ ( image_fm_set_tm
@ ^ [P6: fm] : ( set_tm2 @ ( subtermFm @ P6 ) )
@ ( set_fm2 @ Z2 ) ) ) ) ) ) ).
% set_subterms
thf(fact_1255_subtermFm__subset__params,axiom,
! [P7: fm,A2: list_tm] :
( ( ord_less_eq_set_tm @ ( set_tm2 @ ( subtermFm @ P7 ) ) @ ( set_tm2 @ A2 ) )
=> ( ord_less_eq_set_nat @ ( params @ P7 ) @ ( paramsts @ A2 ) ) ) ).
% subtermFm_subset_params
thf(fact_1256_params__subtermFm,axiom,
! [P7: fm,X6: nat] :
( ( member_nat @ X6 @ ( params @ P7 ) )
=> ? [L3: list_tm] : ( member_tm @ ( fun @ X6 @ L3 ) @ ( set_tm2 @ ( subtermFm @ P7 ) ) ) ) ).
% params_subtermFm
thf(fact_1257_subterms__def,axiom,
( subterms
= ( ^ [Z3: list_fm] : ( case_list_list_tm_tm @ ( cons_tm @ ( fun @ zero_zero_nat @ nil_tm ) @ nil_tm ) @ cons_tm @ ( remdups_tm @ ( concat_tm @ ( map_fm_list_tm @ subtermFm @ Z3 ) ) ) ) ) ) ).
% subterms_def
thf(fact_1258_news__paramss,axiom,
( news
= ( ^ [I: nat,Z3: list_fm] :
~ ( member_nat @ I @ ( comple7399068483239264473et_nat @ ( image_fm_set_nat @ params @ ( set_fm2 @ Z3 ) ) ) ) ) ) ).
% news_paramss
thf(fact_1259_s3,axiom,
( news
= ( ^ [C2: nat] :
( list_all_fm
@ ^ [P6: fm] :
~ ( member_nat @ C2 @ ( params @ P6 ) ) ) ) ) ).
% s3
thf(fact_1260_news_Osimps_I1_J,axiom,
! [C: nat] : ( news @ C @ nil_fm ) ).
% news.simps(1)
thf(fact_1261_subtermTm_Osimps_I1_J,axiom,
! [N: nat,Ts: list_tm] :
( ( subtermTm @ ( fun @ N @ Ts ) )
= ( cons_tm @ ( fun @ N @ Ts ) @ ( remdups_tm @ ( concat_tm @ ( map_tm_list_tm @ subtermTm @ Ts ) ) ) ) ) ).
% subtermTm.simps(1)
thf(fact_1262_children_Osimps_I2_J,axiom,
! [A2: list_tm,R2: rule,P7: fm,Z2: list_fm] :
( ( children @ A2 @ R2 @ ( cons_fm @ P7 @ Z2 ) )
= ( list_prod_fm @ ( parts @ A2 @ R2 @ P7 ) @ ( children @ ( remdups_tm @ ( append_tm @ A2 @ ( concat_tm @ ( map_fm_list_tm @ subtermFm @ ( concat_fm @ ( parts @ A2 @ R2 @ P7 ) ) ) ) ) ) @ R2 @ Z2 ) ) ) ).
% children.simps(2)
thf(fact_1263_children_Osimps_I1_J,axiom,
! [Uu2: list_tm,Uv: rule] :
( ( children @ Uu2 @ Uv @ nil_fm )
= ( cons_list_fm @ nil_fm @ nil_list_fm ) ) ).
% children.simps(1)
thf(fact_1264_set__children__Cons,axiom,
! [A2: list_tm,R2: rule,P7: fm,Z2: list_fm] :
( ( set_list_fm2 @ ( children @ A2 @ R2 @ ( cons_fm @ P7 @ Z2 ) ) )
= ( collect_list_fm
@ ^ [Uu: list_fm] :
? [Hs: list_fm,Ts3: list_fm] :
( ( Uu
= ( append_fm @ Hs @ Ts3 ) )
& ( member_list_fm @ Hs @ ( set_list_fm2 @ ( parts @ A2 @ R2 @ P7 ) ) )
& ( member_list_fm @ Ts3 @ ( set_list_fm2 @ ( children @ ( remdups_tm @ ( append_tm @ A2 @ ( concat_tm @ ( map_fm_list_tm @ subtermFm @ ( concat_fm @ ( parts @ A2 @ R2 @ P7 ) ) ) ) ) ) @ R2 @ Z2 ) ) ) ) ) ) ).
% set_children_Cons
thf(fact_1265_params__sub,axiom,
! [M: nat,T: tm,P7: fm] : ( ord_less_eq_set_nat @ ( params @ ( sub @ M @ T @ P7 ) ) @ ( sup_sup_set_nat @ ( paramst @ T ) @ ( params @ P7 ) ) ) ).
% params_sub
thf(fact_1266_sub__const__transfer,axiom,
! [M: nat,A: nat,P7: fm,T: tm] :
( ( ( sub @ M @ ( fun @ A @ nil_tm ) @ P7 )
!= ( sub @ M @ T @ P7 ) )
=> ( member_tm @ ( fun @ A @ nil_tm ) @ ( set_tm2 @ ( subtermFm @ ( sub @ M @ ( fun @ A @ nil_tm ) @ P7 ) ) ) ) ) ).
% sub_const_transfer
thf(fact_1267_mono__Suc,axiom,
monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ suc ).
% mono_Suc
thf(fact_1268_strict__mono__imp__increasing,axiom,
! [F: nat > nat,N: nat] :
( ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_nat @ ord_less_nat @ F )
=> ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).
% strict_mono_imp_increasing
thf(fact_1269_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_1270_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% finite_Collect_le_nat
% Helper facts (9)
thf(help_fChoice_1_1_fChoice_001_Eo_T,axiom,
! [P: $o > $o] :
( ( P @ ( fChoice_o @ P ) )
= ( ? [X3: $o] : ( P @ X3 ) ) ) ).
thf(help_If_2_1_If_001t__SeCaV__Otm_T,axiom,
! [X2: tm,Y4: tm] :
( ( if_tm @ $false @ X2 @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__SeCaV__Otm_T,axiom,
! [X2: tm,Y4: tm] :
( ( if_tm @ $true @ X2 @ Y4 )
= X2 ) ).
thf(help_fChoice_1_1_fChoice_001t__Nat__Onat_T,axiom,
! [P: nat > $o] :
( ( P @ ( fChoice_nat @ P ) )
= ( ? [X3: nat] : ( P @ X3 ) ) ) ).
thf(help_fChoice_1_1_fChoice_001t__SeCaV__Otm_T,axiom,
! [P: tm > $o] :
( ( P @ ( fChoice_tm @ P ) )
= ( ? [X3: tm] : ( P @ X3 ) ) ) ).
thf(help_If_3_1_If_001t__Set__Oset_It__SeCaV__Otm_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__SeCaV__Otm_J_T,axiom,
! [X2: set_tm,Y4: set_tm] :
( ( if_set_tm @ $false @ X2 @ Y4 )
= Y4 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__SeCaV__Otm_J_T,axiom,
! [X2: set_tm,Y4: set_tm] :
( ( if_set_tm @ $true @ X2 @ Y4 )
= X2 ) ).
thf(help_fChoice_1_1_fChoice_001t__List__Olist_It__SeCaV__Ofm_J_T,axiom,
! [P: list_fm > $o] :
( ( P @ ( fChoice_list_fm @ P ) )
= ( ? [X3: list_fm] : ( P @ X3 ) ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ~ ( member_tm @ ( var @ n ) @ ( terms @ s ) )
=> ( member_tm
@ ( fChoice_tm
@ ^ [T2: tm] : ( member_tm @ T2 @ ( terms @ s ) ) )
@ ( terms @ s ) ) ) ).
%------------------------------------------------------------------------------