TPTP Problem File: SLH0922^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Equivalence_Relation_Enumeration/0007_Equivalence_Relation_Enumeration/prob_00367_014048__12072804_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1504 ( 614 unt; 226 typ; 0 def)
% Number of atoms : 3601 (2055 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 11250 ( 418 ~; 63 |; 395 &;8804 @)
% ( 0 <=>;1570 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 6 avg)
% Number of types : 22 ( 21 usr)
% Number of type conns : 954 ( 954 >; 0 *; 0 +; 0 <<)
% Number of symbols : 208 ( 205 usr; 29 con; 0-4 aty)
% Number of variables : 3656 ( 46 ^;3261 !; 349 ?;3656 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:14:34.093
%------------------------------------------------------------------------------
% Could-be-implicit typings (21)
thf(ty_n_t__Set__Oset_I_062_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J_J,type,
set_Pr1882883127215053275t_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Product____Type__Ounit_Mt__Nat__Onat_J_J,type,
set_Product_unit_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Product____Type__Ounit_J_J,type,
set_nat_Product_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Ounit_J_J,type,
set_set_Product_unit: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_nat_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Ounit_J,type,
list_Product_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Ounit_J,type,
set_Product_unit: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Product____Type__Ounit,type,
product_unit: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (205)
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_001tf__a,type,
bNF_Greatest_Shift_a: set_list_a > a > set_list_a ).
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_001tf__a,type,
bNF_Greatest_Succ_a: set_list_a > list_a > set_a ).
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_Itf__a_J,type,
comple2307003609928055243_set_a: set_set_a > set_a ).
thf(sy_c_Equivalence__Relation__Enumeration_Okernel__of_001t__Nat__Onat,type,
equiva2048684438135499664of_nat: list_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Equivalence__Relation__Enumeration_Okernel__of_001tf__a,type,
equiva2867628904822520638l_of_a: list_a > set_Pr1261947904930325089at_nat ).
thf(sy_c_Equivalence__Relation__Enumeration_Orgf,type,
equiva3371634703666331078on_rgf: list_nat > $o ).
thf(sy_c_Equivalence__Relation__Enumeration_Orgf__limit,type,
equiva5889994315859557365_limit: list_nat > nat ).
thf(sy_c_Equivalence__Relation__Enumeration_Orgf__limit__rel,type,
equiva5575797544161152836it_rel: list_nat > list_nat > $o ).
thf(sy_c_Finite__Set_OFpow_001t__Nat__Onat,type,
finite_Fpow_nat: set_nat > set_set_nat ).
thf(sy_c_Finite__Set_OFpow_001tf__a,type,
finite_Fpow_a: set_a > set_set_a ).
thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Nat__Onat_J,type,
finite_card_list_nat: set_list_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__List__Olist_Itf__a_J,type,
finite_card_list_a: set_list_a > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Ounit,type,
finite410649719033368117t_unit: set_Product_unit > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite2115694454571419734at_nat: set_nat_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_Mt__Product____Type__Ounit_J,type,
finite4257689694021357085t_unit: set_nat_Product_unit > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Product____Type__Ounit_Mt__Nat__Onat_J,type,
finite4332129999517832055it_nat: set_Product_unit_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J,type,
finite6665322292308856380t_unit: set_Pr1882883127215053275t_unit > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Ounit,type,
finite4290736615968046902t_unit: set_Product_unit > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
inj_on3049792774292151987st_nat: ( list_nat > list_nat ) > set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_Itf__a_J_001t__List__Olist_It__Nat__Onat_J,type,
inj_on6731145966573583411st_nat: ( list_a > list_nat ) > set_list_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
inj_on_list_a_list_a: ( list_a > list_a ) > set_list_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Product____Type__Ounit,type,
inj_on7061601236592826506t_unit: ( nat > product_unit ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001tf__a,type,
inj_on_nat_a: ( nat > a ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Product____Type__Ounit_001t__Nat__Onat,type,
inj_on8430439091780834860it_nat: ( product_unit > nat ) > set_Product_unit > $o ).
thf(sy_c_Fun_Oinj__on_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
inj_on8151373323710067377t_unit: ( product_unit > product_unit ) > set_Product_unit > $o ).
thf(sy_c_Fun_Oinj__on_001t__Product____Type__Ounit_001tf__a,type,
inj_on8151663806560157602unit_a: ( product_unit > a ) > set_Product_unit > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
inj_on4604407203859583615et_nat: ( set_nat > set_nat ) > set_set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Nat__Onat_J,type,
inj_on_set_a_set_nat: ( set_a > set_nat ) > set_set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001t__Nat__Onat,type,
inj_on_a_nat: ( a > nat ) > set_a > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001tf__a,type,
inj_on_a_a: ( a > a ) > set_a > $o ).
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_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
if_list_nat: $o > list_nat > list_nat > list_nat ).
thf(sy_c_If_001t__List__Olist_Itf__a_J,type,
if_list_a: $o > list_a > list_a > list_a ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
lattic8265883725875713057ax_nat: set_nat > nat ).
thf(sy_c_List_Oappend_001t__List__Olist_It__Nat__Onat_J,type,
append_list_nat: list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Oappend_001t__List__Olist_Itf__a_J,type,
append_list_a: list_list_a > list_list_a > list_list_a ).
thf(sy_c_List_Oappend_001t__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Oappend_001tf__a,type,
append_a: list_a > list_a > list_a ).
thf(sy_c_List_Obind_001t__Nat__Onat_001t__Nat__Onat,type,
bind_nat_nat: list_nat > ( nat > list_nat ) > list_nat ).
thf(sy_c_List_Obind_001t__Nat__Onat_001tf__a,type,
bind_nat_a: list_nat > ( nat > list_a ) > list_a ).
thf(sy_c_List_Obind_001tf__a_001t__Nat__Onat,type,
bind_a_nat: list_a > ( a > list_nat ) > list_nat ).
thf(sy_c_List_Obind_001tf__a_001tf__a,type,
bind_a_a: list_a > ( a > list_a ) > list_a ).
thf(sy_c_List_Obutlast_001t__Nat__Onat,type,
butlast_nat: list_nat > list_nat ).
thf(sy_c_List_Obutlast_001tf__a,type,
butlast_a: list_a > list_a ).
thf(sy_c_List_Ocan__select_001t__Nat__Onat,type,
can_select_nat: ( nat > $o ) > set_nat > $o ).
thf(sy_c_List_Ocan__select_001tf__a,type,
can_select_a: ( a > $o ) > set_a > $o ).
thf(sy_c_List_Oconcat_001t__Nat__Onat,type,
concat_nat: list_list_nat > list_nat ).
thf(sy_c_List_Oconcat_001tf__a,type,
concat_a: list_list_a > list_a ).
thf(sy_c_List_Ocoset_001t__Nat__Onat,type,
coset_nat: list_nat > set_nat ).
thf(sy_c_List_Ocoset_001t__Product____Type__Ounit,type,
coset_Product_unit: list_Product_unit > set_Product_unit ).
thf(sy_c_List_Ocoset_001tf__a,type,
coset_a: list_a > set_a ).
thf(sy_c_List_Odistinct__adj_001t__Nat__Onat,type,
distinct_adj_nat: list_nat > $o ).
thf(sy_c_List_Odistinct__adj_001tf__a,type,
distinct_adj_a: list_a > $o ).
thf(sy_c_List_Ogen__length_001t__Nat__Onat,type,
gen_length_nat: nat > list_nat > nat ).
thf(sy_c_List_Ogen__length_001tf__a,type,
gen_length_a: nat > list_a > nat ).
thf(sy_c_List_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oinsert_001tf__a,type,
insert_a: a > list_a > list_a ).
thf(sy_c_List_Olast_001t__Nat__Onat,type,
last_nat: list_nat > nat ).
thf(sy_c_List_Olast_001tf__a,type,
last_a: list_a > a ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Nat__Onat_J,type,
cons_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_Itf__a_J,type,
cons_list_a: list_a > list_list_a > list_list_a ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Nat__Onat_J,type,
nil_list_nat: list_list_nat ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_Itf__a_J,type,
nil_list_a: list_list_a ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Ounit,type,
nil_Product_unit: list_Product_unit ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist_Ohd_001t__List__Olist_It__Nat__Onat_J,type,
hd_list_nat: list_list_nat > list_nat ).
thf(sy_c_List_Olist_Ohd_001t__List__Olist_Itf__a_J,type,
hd_list_a: list_list_a > list_a ).
thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
hd_nat: list_nat > nat ).
thf(sy_c_List_Olist_Ohd_001tf__a,type,
hd_a: list_a > a ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
map_li7225945977422193158st_nat: ( list_nat > list_nat ) > list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_Itf__a_J_001t__List__Olist_It__Nat__Onat_J,type,
map_list_a_list_nat: ( list_a > list_nat ) > list_list_a > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
map_list_a_list_a: ( list_a > list_a ) > list_list_a > list_list_a ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001tf__a,type,
map_nat_a: ( nat > a ) > list_nat > list_a ).
thf(sy_c_List_Olist_Omap_001tf__a_001t__Nat__Onat,type,
map_a_nat: ( a > nat ) > list_a > list_nat ).
thf(sy_c_List_Olist_Omap_001tf__a_001tf__a,type,
map_a_a: ( a > a ) > list_a > list_a ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
set_list_nat2: list_list_nat > set_list_nat ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_Itf__a_J,type,
set_list_a2: list_list_a > set_list_a ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Ounit,type,
set_Product_unit2: list_Product_unit > set_Product_unit ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_List_Olist__ex1_001t__Nat__Onat,type,
list_ex1_nat: ( nat > $o ) > list_nat > $o ).
thf(sy_c_List_Olist__ex1_001tf__a,type,
list_ex1_a: ( a > $o ) > list_a > $o ).
thf(sy_c_List_Omap__tailrec_001tf__a_001t__Nat__Onat,type,
map_tailrec_a_nat: ( a > nat ) > list_a > list_nat ).
thf(sy_c_List_Omaps_001t__Nat__Onat_001t__Nat__Onat,type,
maps_nat_nat: ( nat > list_nat ) > list_nat > list_nat ).
thf(sy_c_List_Omaps_001t__Nat__Onat_001tf__a,type,
maps_nat_a: ( nat > list_a ) > list_nat > list_a ).
thf(sy_c_List_Omaps_001tf__a_001t__Nat__Onat,type,
maps_a_nat: ( a > list_nat ) > list_a > list_nat ).
thf(sy_c_List_Omaps_001tf__a_001tf__a,type,
maps_a_a: ( a > list_a ) > list_a > list_a ).
thf(sy_c_List_On__lists_001t__Nat__Onat,type,
n_lists_nat: nat > list_nat > list_list_nat ).
thf(sy_c_List_On__lists_001tf__a,type,
n_lists_a: nat > list_a > list_list_a ).
thf(sy_c_List_Oproduct__lists_001t__Nat__Onat,type,
product_lists_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Oproduct__lists_001tf__a,type,
product_lists_a: list_list_a > list_list_a ).
thf(sy_c_List_Oremdups__adj_001t__Nat__Onat,type,
remdups_adj_nat: list_nat > list_nat ).
thf(sy_c_List_Oremdups__adj_001tf__a,type,
remdups_adj_a: list_a > list_a ).
thf(sy_c_List_OremoveAll_001t__Nat__Onat,type,
removeAll_nat: nat > list_nat > list_nat ).
thf(sy_c_List_OremoveAll_001tf__a,type,
removeAll_a: a > list_a > list_a ).
thf(sy_c_List_Oreplicate_001t__List__Olist_It__Nat__Onat_J,type,
replicate_list_nat: nat > list_nat > list_list_nat ).
thf(sy_c_List_Oreplicate_001t__List__Olist_Itf__a_J,type,
replicate_list_a: nat > list_a > list_list_a ).
thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
replicate_nat: nat > nat > list_nat ).
thf(sy_c_List_Oreplicate_001t__Product____Type__Ounit,type,
replic7505510843043721677t_unit: nat > product_unit > list_Product_unit ).
thf(sy_c_List_Oreplicate_001tf__a,type,
replicate_a: nat > a > list_a ).
thf(sy_c_List_Osubseqs_001t__Nat__Onat,type,
subseqs_nat: list_nat > list_list_nat ).
thf(sy_c_List_Osubseqs_001tf__a,type,
subseqs_a: list_a > list_list_a ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
size_s3023201423986296836st_nat: list_list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
size_s349497388124573686list_a: list_list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Ounit_J,type,
size_s245203480648594047t_unit: list_Product_unit > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
size_size_list_a: list_a > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Ounit_J,type,
bot_bo3957492148770167129t_unit: set_Product_unit ).
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_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Ounit_J,type,
ord_le3507040750410214029t_unit: set_Product_unit > set_Product_unit > $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_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
ord_max_nat: nat > nat > nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
top_top_set_nat_nat: set_nat_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Product____Type__Ounit_J_J,type,
top_to8442108875268333988t_unit: set_nat_Product_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Product____Type__Ounit_Mt__Nat__Onat_J_J,type,
top_to5871476398150932990it_nat: set_Product_unit_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J_J,type,
top_to658657236369668235t_unit: set_Pr1882883127215053275t_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
top_top_set_list_nat: set_list_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
top_top_set_list_a: set_list_a ).
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__Product____Type__Ounit_J,type,
top_to1996260823553986621t_unit: set_Product_unit ).
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__Product____Type__Ounit_J_J,type,
top_to1767297665138865437t_unit: set_set_Product_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
top_top_set_a: set_a ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OPow_001t__Nat__Onat,type,
pow_nat: set_nat > set_set_nat ).
thf(sy_c_Set_OPow_001t__Product____Type__Ounit,type,
pow_Product_unit: set_Product_unit > set_set_Product_unit ).
thf(sy_c_Set_OPow_001tf__a,type,
pow_a: set_a > set_set_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_1775855109352712557et_nat: ( list_nat > set_nat ) > set_list_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
image_list_a_set_a: ( list_a > set_a ) > set_list_a > set_set_a ).
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__Product____Type__Ounit,type,
image_8730104196221521654t_unit: ( nat > product_unit ) > set_nat > set_Product_unit ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__a,type,
image_nat_a: ( nat > a ) > set_nat > set_a ).
thf(sy_c_Set_Oimage_001t__Product____Type__Ounit_001t__Nat__Onat,type,
image_875570014554754200it_nat: ( product_unit > nat ) > set_Product_unit > set_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
image_405062704495631173t_unit: ( product_unit > product_unit ) > set_Product_unit > set_Product_unit ).
thf(sy_c_Set_Oimage_001t__Product____Type__Ounit_001tf__a,type,
image_Product_unit_a: ( product_unit > a ) > set_Product_unit > set_a ).
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_Itf__a_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_set_a_set_nat: ( set_a > set_nat ) > set_set_a > set_set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
image_a_nat: ( a > nat ) > set_a > set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat2: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Ounit,type,
insert_Product_unit: product_unit > set_Product_unit > set_Product_unit ).
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_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a2: a > set_a > set_a ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $o ).
thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
remove_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oremove_001tf__a,type,
remove_a: a > set_a > set_a ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set_Othe__elem_001tf__a,type,
the_elem_a: set_a > a ).
thf(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
set_ord_atLeast_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
set_or1210151606488870762an_nat: nat > set_nat ).
thf(sy_c_Stirling_Ostirling__row,type,
stirling_row: nat > list_nat ).
thf(sy_c_Stirling_Ostirling__row__aux_001t__Nat__Onat,type,
stirling_row_aux_nat: nat > nat > list_nat > list_nat ).
thf(sy_c_Sublist_Oprefix_001t__Nat__Onat,type,
prefix_nat: list_nat > list_nat > $o ).
thf(sy_c_Sublist_Oprefixes_001t__Nat__Onat,type,
prefixes_nat: list_nat > list_list_nat ).
thf(sy_c_Sublist_Oprefixes_001tf__a,type,
prefixes_a: list_a > list_list_a ).
thf(sy_c_Sublist_Osublists_001t__Nat__Onat,type,
sublists_nat: list_nat > list_list_nat ).
thf(sy_c_Sublist_Osublists_001tf__a,type,
sublists_a: list_a > list_list_a ).
thf(sy_c_Sublist_Osuffixes_001t__Nat__Onat,type,
suffixes_nat: list_nat > list_list_nat ).
thf(sy_c_Sublist_Osuffixes_001tf__a,type,
suffixes_a: list_a > list_list_a ).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
accp_list_nat: ( list_nat > list_nat > $o ) > list_nat > $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_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Ounit,type,
member_Product_unit: product_unit > set_Product_unit > $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_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_f_H____,type,
f: a > nat ).
thf(sy_v_f____,type,
f2: a > nat ).
thf(sy_v_n____,type,
n: nat ).
thf(sy_v_x1____,type,
x1: list_a ).
thf(sy_v_x2____,type,
x2: a ).
thf(sy_v_xa____,type,
xa: list_a ).
% Relevant facts (1270)
thf(fact_0_False,axiom,
~ ( member_a @ x2 @ ( set_a2 @ x1 ) ) ).
% False
thf(fact_1__092_060open_062inj__on_Af_H_A_Iinsert_Ax2_A_Iset_Ax1_J_J_092_060close_062,axiom,
inj_on_a_nat @ f @ ( insert_a2 @ x2 @ ( set_a2 @ x1 ) ) ).
% \<open>inj_on f' (insert x2 (set x1))\<close>
thf(fact_2_x__def,axiom,
( xa
= ( append_a @ x1 @ ( cons_a @ x2 @ nil_a ) ) ) ).
% x_def
thf(fact_3_inj__f,axiom,
inj_on_a_nat @ f2 @ ( set_a2 @ x1 ) ).
% inj_f
thf(fact_4__092_060open_062inj__on_Af_H_A_Iset_Ax1_J_092_060close_062,axiom,
inj_on_a_nat @ f @ ( set_a2 @ x1 ) ).
% \<open>inj_on f' (set x1)\<close>
thf(fact_5__092_060open_062f_H_Ax2_A_092_060notin_062_Af_H_A_096_Aset_Ax1_092_060close_062,axiom,
~ ( member_nat @ ( f @ x2 ) @ ( image_a_nat @ f @ ( set_a2 @ x1 ) ) ) ).
% \<open>f' x2 \<notin> f' ` set x1\<close>
thf(fact_6_inj__onD,axiom,
! [F: a > nat,A: set_a,X: a,Y: a] :
( ( inj_on_a_nat @ F @ A )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_a @ X @ A )
=> ( ( member_a @ Y @ A )
=> ( X = Y ) ) ) ) ) ).
% inj_onD
thf(fact_7_inj__onD,axiom,
! [F: nat > nat,A: set_nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( X = Y ) ) ) ) ) ).
% inj_onD
thf(fact_8_inj__onI,axiom,
! [A: set_a,F: a > nat] :
( ! [X2: a,Y2: a] :
( ( member_a @ X2 @ A )
=> ( ( member_a @ Y2 @ A )
=> ( ( ( F @ X2 )
= ( F @ Y2 ) )
=> ( X2 = Y2 ) ) ) )
=> ( inj_on_a_nat @ F @ A ) ) ).
% inj_onI
thf(fact_9_inj__onI,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X2: nat,Y2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y2 @ A )
=> ( ( ( F @ X2 )
= ( F @ Y2 ) )
=> ( X2 = Y2 ) ) ) )
=> ( inj_on_nat_nat @ F @ A ) ) ).
% inj_onI
thf(fact_10_inj__on__def,axiom,
( inj_on_a_nat
= ( ^ [F2: a > nat,A2: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ! [Y3: a] :
( ( member_a @ Y3 @ A2 )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y3 ) )
=> ( X3 = Y3 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_11_inj__on__def,axiom,
( inj_on_nat_nat
= ( ^ [F2: nat > nat,A2: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ A2 )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y3 ) )
=> ( X3 = Y3 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_12_inj__on__cong,axiom,
! [A: set_a,F: a > nat,G: a > nat] :
( ! [A3: a] :
( ( member_a @ A3 @ A )
=> ( ( F @ A3 )
= ( G @ A3 ) ) )
=> ( ( inj_on_a_nat @ F @ A )
= ( inj_on_a_nat @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_13_inj__on__cong,axiom,
! [A: set_nat,F: nat > nat,G: nat > nat] :
( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ( ( F @ A3 )
= ( G @ A3 ) ) )
=> ( ( inj_on_nat_nat @ F @ A )
= ( inj_on_nat_nat @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_14_inj__on__eq__iff,axiom,
! [F: a > nat,A: set_a,X: a,Y: a] :
( ( inj_on_a_nat @ F @ A )
=> ( ( member_a @ X @ A )
=> ( ( member_a @ Y @ A )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_15_inj__on__eq__iff,axiom,
! [F: nat > nat,A: set_nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_16_inj__on__contraD,axiom,
! [F: a > nat,A: set_a,X: a,Y: a] :
( ( inj_on_a_nat @ F @ A )
=> ( ( X != Y )
=> ( ( member_a @ X @ A )
=> ( ( member_a @ Y @ A )
=> ( ( F @ X )
!= ( F @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_17_inj__on__contraD,axiom,
! [F: nat > nat,A: set_nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( X != Y )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( F @ X )
!= ( F @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_18_inj__on__inverseI,axiom,
! [A: set_a,G: nat > a,F: a > nat] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( G @ ( F @ X2 ) )
= X2 ) )
=> ( inj_on_a_nat @ F @ A ) ) ).
% inj_on_inverseI
thf(fact_19_inj__on__inverseI,axiom,
! [A: set_nat,G: nat > nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( G @ ( F @ X2 ) )
= X2 ) )
=> ( inj_on_nat_nat @ F @ A ) ) ).
% inj_on_inverseI
thf(fact_20__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062f_O_A_092_060lbrakk_062inj__on_Af_A_Iset_Ax1_J_059_Argf_A_Imap_Af_Ax1_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [F3: a > nat] :
( ( inj_on_a_nat @ F3 @ ( set_a2 @ x1 ) )
=> ~ ( equiva3371634703666331078on_rgf @ ( map_a_nat @ F3 @ x1 ) ) ) ).
% \<open>\<And>thesis. (\<And>f. \<lbrakk>inj_on f (set x1); rgf (map f x1)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_21_Suc_Ohyps_I2_J,axiom,
( ( suc @ n )
= ( size_size_list_a @ xa ) ) ).
% Suc.hyps(2)
thf(fact_22_f_H__def,axiom,
( f
= ( ^ [Y3: a] : ( if_nat @ ( member_a @ Y3 @ ( set_a2 @ x1 ) ) @ ( f2 @ Y3 ) @ ( equiva5889994315859557365_limit @ ( map_a_nat @ f2 @ x1 ) ) ) ) ) ).
% f'_def
thf(fact_23_list_Oset__intros_I2_J,axiom,
! [Y: a,X22: list_a,X21: a] :
( ( member_a @ Y @ ( set_a2 @ X22 ) )
=> ( member_a @ Y @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_24_list_Oset__intros_I2_J,axiom,
! [Y: nat,X22: list_nat,X21: nat] :
( ( member_nat @ Y @ ( set_nat2 @ X22 ) )
=> ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_25_l__x1,axiom,
( ( size_size_list_a @ x1 )
= n ) ).
% l_x1
thf(fact_26_pc__f,axiom,
equiva3371634703666331078on_rgf @ ( map_a_nat @ f2 @ x1 ) ).
% pc_f
thf(fact_27_list_Oinject,axiom,
! [X21: a,X22: list_a,Y21: a,Y22: list_a] :
( ( ( cons_a @ X21 @ X22 )
= ( cons_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_28_list_Oinject,axiom,
! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
( ( ( cons_nat @ X21 @ X22 )
= ( cons_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_29_append_Oassoc,axiom,
! [A4: list_a,B: list_a,C: list_a] :
( ( append_a @ ( append_a @ A4 @ B ) @ C )
= ( append_a @ A4 @ ( append_a @ B @ C ) ) ) ).
% append.assoc
thf(fact_30_append_Oassoc,axiom,
! [A4: list_nat,B: list_nat,C: list_nat] :
( ( append_nat @ ( append_nat @ A4 @ B ) @ C )
= ( append_nat @ A4 @ ( append_nat @ B @ C ) ) ) ).
% append.assoc
thf(fact_31_append__assoc,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( append_a @ ( append_a @ Xs @ Ys ) @ Zs )
= ( append_a @ Xs @ ( append_a @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_32_append__assoc,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( append_nat @ ( append_nat @ Xs @ Ys ) @ Zs )
= ( append_nat @ Xs @ ( append_nat @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_33_append__same__eq,axiom,
! [Ys: list_a,Xs: list_a,Zs: list_a] :
( ( ( append_a @ Ys @ Xs )
= ( append_a @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_34_append__same__eq,axiom,
! [Ys: list_nat,Xs: list_nat,Zs: list_nat] :
( ( ( append_nat @ Ys @ Xs )
= ( append_nat @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_35_same__append__eq,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( append_a @ Xs @ Ys )
= ( append_a @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_36_same__append__eq,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_37__092_060open_062rgf__limit_A_Imap_Af_Ax1_J_A_092_060notin_062_Aset_A_Imap_Af_Ax1_J_092_060close_062,axiom,
~ ( member_nat @ ( equiva5889994315859557365_limit @ ( map_a_nat @ f2 @ x1 ) ) @ ( set_nat2 @ ( map_a_nat @ f2 @ x1 ) ) ) ).
% \<open>rgf_limit (map f x1) \<notin> set (map f x1)\<close>
thf(fact_38_append_Oright__neutral,axiom,
! [A4: list_a] :
( ( append_a @ A4 @ nil_a )
= A4 ) ).
% append.right_neutral
thf(fact_39_append_Oright__neutral,axiom,
! [A4: list_nat] :
( ( append_nat @ A4 @ nil_nat )
= A4 ) ).
% append.right_neutral
thf(fact_40_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_41_append__Nil2,axiom,
! [Xs: list_nat] :
( ( append_nat @ Xs @ nil_nat )
= Xs ) ).
% append_Nil2
thf(fact_42_append__self__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_43_append__self__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_nat ) ) ).
% append_self_conv
thf(fact_44_self__append__conv,axiom,
! [Y: list_a,Ys: list_a] :
( ( Y
= ( append_a @ Y @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_45_self__append__conv,axiom,
! [Y: list_nat,Ys: list_nat] :
( ( Y
= ( append_nat @ Y @ Ys ) )
= ( Ys = nil_nat ) ) ).
% self_append_conv
thf(fact_46_append__self__conv2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_47_append__self__conv2,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_nat ) ) ).
% append_self_conv2
thf(fact_48_self__append__conv2,axiom,
! [Y: list_a,Xs: list_a] :
( ( Y
= ( append_a @ Xs @ Y ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_49_self__append__conv2,axiom,
! [Y: list_nat,Xs: list_nat] :
( ( Y
= ( append_nat @ Xs @ Y ) )
= ( Xs = nil_nat ) ) ).
% self_append_conv2
thf(fact_50_Nil__is__append__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( nil_a
= ( append_a @ Xs @ Ys ) )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% Nil_is_append_conv
thf(fact_51_Nil__is__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( nil_nat
= ( append_nat @ Xs @ Ys ) )
= ( ( Xs = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_52_append__is__Nil__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= nil_a )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% append_is_Nil_conv
thf(fact_53_append__is__Nil__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= nil_nat )
= ( ( Xs = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_54_list_Omap__disc__iff,axiom,
! [F: a > a,A4: list_a] :
( ( ( map_a_a @ F @ A4 )
= nil_a )
= ( A4 = nil_a ) ) ).
% list.map_disc_iff
thf(fact_55_list_Omap__disc__iff,axiom,
! [F: nat > a,A4: list_nat] :
( ( ( map_nat_a @ F @ A4 )
= nil_a )
= ( A4 = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_56_list_Omap__disc__iff,axiom,
! [F: nat > nat,A4: list_nat] :
( ( ( map_nat_nat @ F @ A4 )
= nil_nat )
= ( A4 = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_57_list_Omap__disc__iff,axiom,
! [F: a > nat,A4: list_a] :
( ( ( map_a_nat @ F @ A4 )
= nil_nat )
= ( A4 = nil_a ) ) ).
% list.map_disc_iff
thf(fact_58_Nil__is__map__conv,axiom,
! [F: a > a,Xs: list_a] :
( ( nil_a
= ( map_a_a @ F @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_map_conv
thf(fact_59_Nil__is__map__conv,axiom,
! [F: nat > a,Xs: list_nat] :
( ( nil_a
= ( map_nat_a @ F @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_60_Nil__is__map__conv,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( nil_nat
= ( map_nat_nat @ F @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_61_Nil__is__map__conv,axiom,
! [F: a > nat,Xs: list_a] :
( ( nil_nat
= ( map_a_nat @ F @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_map_conv
thf(fact_62_map__is__Nil__conv,axiom,
! [F: a > a,Xs: list_a] :
( ( ( map_a_a @ F @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% map_is_Nil_conv
thf(fact_63_map__is__Nil__conv,axiom,
! [F: nat > a,Xs: list_nat] :
( ( ( map_nat_a @ F @ Xs )
= nil_a )
= ( Xs = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_64_map__is__Nil__conv,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_65_map__is__Nil__conv,axiom,
! [F: a > nat,Xs: list_a] :
( ( ( map_a_nat @ F @ Xs )
= nil_nat )
= ( Xs = nil_a ) ) ).
% map_is_Nil_conv
thf(fact_66_append__eq__append__conv,axiom,
! [Xs: list_a,Ys: list_a,Us: list_a,Vs: list_a] :
( ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
| ( ( size_size_list_a @ Us )
= ( size_size_list_a @ Vs ) ) )
=> ( ( ( append_a @ Xs @ Us )
= ( append_a @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_67_append__eq__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat,Us: list_nat,Vs: list_nat] :
( ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
| ( ( size_size_list_nat @ Us )
= ( size_size_list_nat @ Vs ) ) )
=> ( ( ( append_nat @ Xs @ Us )
= ( append_nat @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_68_map__eq__conv,axiom,
! [F: a > nat,Xs: list_a,G: a > nat] :
( ( ( map_a_nat @ F @ Xs )
= ( map_a_nat @ G @ Xs ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) ) ) ) ).
% map_eq_conv
thf(fact_69_length__map,axiom,
! [F: a > a,Xs: list_a] :
( ( size_size_list_a @ ( map_a_a @ F @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_map
thf(fact_70_length__map,axiom,
! [F: nat > a,Xs: list_nat] :
( ( size_size_list_a @ ( map_nat_a @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_71_length__map,axiom,
! [F: a > nat,Xs: list_a] :
( ( size_size_list_nat @ ( map_a_nat @ F @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% length_map
thf(fact_72_length__map,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( size_size_list_nat @ ( map_nat_nat @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_73_map__append,axiom,
! [F: a > a,Xs: list_a,Ys: list_a] :
( ( map_a_a @ F @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( map_a_a @ F @ Xs ) @ ( map_a_a @ F @ Ys ) ) ) ).
% map_append
thf(fact_74_map__append,axiom,
! [F: nat > a,Xs: list_nat,Ys: list_nat] :
( ( map_nat_a @ F @ ( append_nat @ Xs @ Ys ) )
= ( append_a @ ( map_nat_a @ F @ Xs ) @ ( map_nat_a @ F @ Ys ) ) ) ).
% map_append
thf(fact_75_map__append,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
( ( map_nat_nat @ F @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( map_nat_nat @ F @ Xs ) @ ( map_nat_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_76_map__append,axiom,
! [F: a > nat,Xs: list_a,Ys: list_a] :
( ( map_a_nat @ F @ ( append_a @ Xs @ Ys ) )
= ( append_nat @ ( map_a_nat @ F @ Xs ) @ ( map_a_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_77_Suc_Ohyps_I1_J,axiom,
! [X: list_a] :
( ( n
= ( size_size_list_a @ X ) )
=> ? [F3: a > nat] :
( ( inj_on_a_nat @ F3 @ ( set_a2 @ X ) )
& ( equiva3371634703666331078on_rgf @ ( map_a_nat @ F3 @ X ) ) ) ) ).
% Suc.hyps(1)
thf(fact_78_list_Osimps_I15_J,axiom,
! [X21: a,X22: list_a] :
( ( set_a2 @ ( cons_a @ X21 @ X22 ) )
= ( insert_a2 @ X21 @ ( set_a2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_79_list_Osimps_I15_J,axiom,
! [X21: nat,X22: list_nat] :
( ( set_nat2 @ ( cons_nat @ X21 @ X22 ) )
= ( insert_nat2 @ X21 @ ( set_nat2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_80_append1__eq__conv,axiom,
! [Xs: list_a,X: a,Ys: list_a,Y: a] :
( ( ( append_a @ Xs @ ( cons_a @ X @ nil_a ) )
= ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_81_append1__eq__conv,axiom,
! [Xs: list_nat,X: nat,Ys: list_nat,Y: nat] :
( ( ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) )
= ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_82_list_Oset__map,axiom,
! [F: a > a,V: list_a] :
( ( set_a2 @ ( map_a_a @ F @ V ) )
= ( image_a_a @ F @ ( set_a2 @ V ) ) ) ).
% list.set_map
thf(fact_83_list_Oset__map,axiom,
! [F: nat > a,V: list_nat] :
( ( set_a2 @ ( map_nat_a @ F @ V ) )
= ( image_nat_a @ F @ ( set_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_84_list_Oset__map,axiom,
! [F: a > nat,V: list_a] :
( ( set_nat2 @ ( map_a_nat @ F @ V ) )
= ( image_a_nat @ F @ ( set_a2 @ V ) ) ) ).
% list.set_map
thf(fact_85_list_Oset__map,axiom,
! [F: nat > nat,V: list_nat] :
( ( set_nat2 @ ( map_nat_nat @ F @ V ) )
= ( image_nat_nat @ F @ ( set_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_86__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062x1_Ax2_O_A_092_060lbrakk_062x_A_061_Ax1_A_064_A_091x2_093_059_Alength_Ax1_A_061_An_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [X1: list_a] :
( ? [X23: a] :
( xa
= ( append_a @ X1 @ ( cons_a @ X23 @ nil_a ) ) )
=> ( ( size_size_list_a @ X1 )
!= n ) ) ).
% \<open>\<And>thesis. (\<And>x1 x2. \<lbrakk>x = x1 @ [x2]; length x1 = n\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_87_mem__Collect__eq,axiom,
! [A4: a,P: a > $o] :
( ( member_a @ A4 @ ( collect_a @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_88_mem__Collect__eq,axiom,
! [A4: nat,P: nat > $o] :
( ( member_nat @ A4 @ ( collect_nat @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_89_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_90_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_91_list_Osimps_I9_J,axiom,
! [F: a > a,X21: a,X22: list_a] :
( ( map_a_a @ F @ ( cons_a @ X21 @ X22 ) )
= ( cons_a @ ( F @ X21 ) @ ( map_a_a @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_92_list_Osimps_I9_J,axiom,
! [F: a > nat,X21: a,X22: list_a] :
( ( map_a_nat @ F @ ( cons_a @ X21 @ X22 ) )
= ( cons_nat @ ( F @ X21 ) @ ( map_a_nat @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_93_list_Osimps_I9_J,axiom,
! [F: nat > a,X21: nat,X22: list_nat] :
( ( map_nat_a @ F @ ( cons_nat @ X21 @ X22 ) )
= ( cons_a @ ( F @ X21 ) @ ( map_nat_a @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_94_list_Osimps_I9_J,axiom,
! [F: nat > nat,X21: nat,X22: list_nat] :
( ( map_nat_nat @ F @ ( cons_nat @ X21 @ X22 ) )
= ( cons_nat @ ( F @ X21 ) @ ( map_nat_nat @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_95_list_Osimps_I8_J,axiom,
! [F: a > a] :
( ( map_a_a @ F @ nil_a )
= nil_a ) ).
% list.simps(8)
thf(fact_96_list_Osimps_I8_J,axiom,
! [F: nat > a] :
( ( map_nat_a @ F @ nil_nat )
= nil_a ) ).
% list.simps(8)
thf(fact_97_list_Osimps_I8_J,axiom,
! [F: nat > nat] :
( ( map_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_98_list_Osimps_I8_J,axiom,
! [F: a > nat] :
( ( map_a_nat @ F @ nil_a )
= nil_nat ) ).
% list.simps(8)
thf(fact_99_list_Odistinct_I1_J,axiom,
! [X21: a,X22: list_a] :
( nil_a
!= ( cons_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_100_list_Odistinct_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_101_list_OdiscI,axiom,
! [List: list_a,X21: a,X22: list_a] :
( ( List
= ( cons_a @ X21 @ X22 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_102_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X22: list_nat] :
( ( List
= ( cons_nat @ X21 @ X22 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_103_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X212: a,X222: list_a] :
( Y
!= ( cons_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_104_list_Oexhaust,axiom,
! [Y: list_nat] :
( ( Y != nil_nat )
=> ~ ! [X212: nat,X222: list_nat] :
( Y
!= ( cons_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_105_min__list_Ocases,axiom,
! [X: list_nat] :
( ! [X2: nat,Xs2: list_nat] :
( X
!= ( cons_nat @ X2 @ Xs2 ) )
=> ( X = nil_nat ) ) ).
% min_list.cases
thf(fact_106_transpose_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X2: a,Xs2: list_a,Xss: list_list_a] :
( X
!= ( cons_list_a @ ( cons_a @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_107_transpose_Ocases,axiom,
! [X: list_list_nat] :
( ( X != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X2: nat,Xs2: list_nat,Xss: list_list_nat] :
( X
!= ( cons_list_nat @ ( cons_nat @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_108_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_109_append__Nil,axiom,
! [Ys: list_nat] :
( ( append_nat @ nil_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_110_remdups__adj_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ( ! [X2: a] :
( X
!= ( cons_a @ X2 @ nil_a ) )
=> ~ ! [X2: a,Y2: a,Xs2: list_a] :
( X
!= ( cons_a @ X2 @ ( cons_a @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_111_remdups__adj_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ( ! [X2: nat] :
( X
!= ( cons_nat @ X2 @ nil_nat ) )
=> ~ ! [X2: nat,Y2: nat,Xs2: list_nat] :
( X
!= ( cons_nat @ X2 @ ( cons_nat @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_112_append__Cons,axiom,
! [X: a,Xs: list_a,Ys: list_a] :
( ( append_a @ ( cons_a @ X @ Xs ) @ Ys )
= ( cons_a @ X @ ( append_a @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_113_append__Cons,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat] :
( ( append_nat @ ( cons_nat @ X @ Xs ) @ Ys )
= ( cons_nat @ X @ ( append_nat @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_114_append_Oleft__neutral,axiom,
! [A4: list_a] :
( ( append_a @ nil_a @ A4 )
= A4 ) ).
% append.left_neutral
thf(fact_115_append_Oleft__neutral,axiom,
! [A4: list_nat] :
( ( append_nat @ nil_nat @ A4 )
= A4 ) ).
% append.left_neutral
thf(fact_116_image__set,axiom,
! [F: a > a,Xs: list_a] :
( ( image_a_a @ F @ ( set_a2 @ Xs ) )
= ( set_a2 @ ( map_a_a @ F @ Xs ) ) ) ).
% image_set
thf(fact_117_image__set,axiom,
! [F: a > nat,Xs: list_a] :
( ( image_a_nat @ F @ ( set_a2 @ Xs ) )
= ( set_nat2 @ ( map_a_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_118_image__set,axiom,
! [F: nat > a,Xs: list_nat] :
( ( image_nat_a @ F @ ( set_nat2 @ Xs ) )
= ( set_a2 @ ( map_nat_a @ F @ Xs ) ) ) ).
% image_set
thf(fact_119_image__set,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( image_nat_nat @ F @ ( set_nat2 @ Xs ) )
= ( set_nat2 @ ( map_nat_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_120_rev__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ( P @ nil_a )
=> ( ! [X2: a,Xs2: list_a] :
( ( P @ Xs2 )
=> ( P @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_121_rev__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ( P @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat] :
( ( P @ Xs2 )
=> ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X2 @ nil_nat ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_122_rev__exhaust,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ~ ! [Ys2: list_a,Y2: a] :
( Xs
!= ( append_a @ Ys2 @ ( cons_a @ Y2 @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_123_rev__exhaust,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ~ ! [Ys2: list_nat,Y2: nat] :
( Xs
!= ( append_nat @ Ys2 @ ( cons_nat @ Y2 @ nil_nat ) ) ) ) ).
% rev_exhaust
thf(fact_124_inj__on__Cons1,axiom,
! [X: a,A: set_list_a] : ( inj_on_list_a_list_a @ ( cons_a @ X ) @ A ) ).
% inj_on_Cons1
thf(fact_125_inj__on__Cons1,axiom,
! [X: nat,A: set_list_nat] : ( inj_on3049792774292151987st_nat @ ( cons_nat @ X ) @ A ) ).
% inj_on_Cons1
thf(fact_126_list__induct2,axiom,
! [Xs: list_a,Ys: list_a,P: list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y2: a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_127_list__induct2,axiom,
! [Xs: list_a,Ys: list_nat,P: list_a > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_a @ nil_nat )
=> ( ! [X2: a,Xs2: list_a,Y2: nat,Ys2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_128_list__induct2,axiom,
! [Xs: list_nat,Ys: list_a,P: list_nat > list_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_nat @ nil_a )
=> ( ! [X2: nat,Xs2: list_nat,Y2: a,Ys2: list_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_129_list__induct2,axiom,
! [Xs: list_nat,Ys: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat,Y2: nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_130_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,P: list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_a @ nil_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y2: a,Ys2: list_a,Z: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_131_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_nat,P: list_a > list_a > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_a @ nil_a @ nil_nat )
=> ( ! [X2: a,Xs2: list_a,Y2: a,Ys2: list_a,Z: nat,Zs2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) @ ( cons_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_132_list__induct3,axiom,
! [Xs: list_a,Ys: list_nat,Zs: list_a,P: list_a > list_nat > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_a @ nil_nat @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y2: nat,Ys2: list_nat,Z: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_133_list__induct3,axiom,
! [Xs: list_a,Ys: list_nat,Zs: list_nat,P: list_a > list_nat > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_a @ nil_nat @ nil_nat )
=> ( ! [X2: a,Xs2: list_a,Y2: nat,Ys2: list_nat,Z: nat,Zs2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) @ ( cons_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_134_list__induct3,axiom,
! [Xs: list_nat,Ys: list_a,Zs: list_a,P: list_nat > list_a > list_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_nat @ nil_a @ nil_a )
=> ( ! [X2: nat,Xs2: list_nat,Y2: a,Ys2: list_a,Z: a,Zs2: list_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_135_list__induct3,axiom,
! [Xs: list_nat,Ys: list_a,Zs: list_nat,P: list_nat > list_a > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_a @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat,Y2: a,Ys2: list_a,Z: nat,Zs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) @ ( cons_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_136_list__induct3,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_a,P: list_nat > list_nat > list_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_a )
=> ( ! [X2: nat,Xs2: list_nat,Y2: nat,Ys2: list_nat,Z: a,Zs2: list_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_137_list__induct3,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat,P: list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat,Y2: nat,Ys2: list_nat,Z: nat,Zs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) @ ( cons_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_138_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ws: list_a,P: list_a > list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y2: a,Ys2: list_a,Z: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_139_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ws: list_nat,P: list_a > list_a > list_a > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_a @ nil_a @ nil_a @ nil_nat )
=> ( ! [X2: a,Xs2: list_a,Y2: a,Ys2: list_a,Z: a,Zs2: list_a,W: nat,Ws2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_140_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_nat,Ws: list_a,P: list_a > list_a > list_nat > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_a @ nil_nat @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y2: a,Ys2: list_a,Z: nat,Zs2: list_nat,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_141_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_nat,Ws: list_nat,P: list_a > list_a > list_nat > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_a @ nil_a @ nil_nat @ nil_nat )
=> ( ! [X2: a,Xs2: list_a,Y2: a,Ys2: list_a,Z: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_142_list__induct4,axiom,
! [Xs: list_a,Ys: list_nat,Zs: list_a,Ws: list_a,P: list_a > list_nat > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_nat @ nil_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y2: nat,Ys2: list_nat,Z: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_143_list__induct4,axiom,
! [Xs: list_a,Ys: list_nat,Zs: list_a,Ws: list_nat,P: list_a > list_nat > list_a > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_a @ nil_nat @ nil_a @ nil_nat )
=> ( ! [X2: a,Xs2: list_a,Y2: nat,Ys2: list_nat,Z: a,Zs2: list_a,W: nat,Ws2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_144_list__induct4,axiom,
! [Xs: list_a,Ys: list_nat,Zs: list_nat,Ws: list_a,P: list_a > list_nat > list_nat > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_nat @ nil_nat @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y2: nat,Ys2: list_nat,Z: nat,Zs2: list_nat,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_145_list__induct4,axiom,
! [Xs: list_a,Ys: list_nat,Zs: list_nat,Ws: list_nat,P: list_a > list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_a @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X2: a,Xs2: list_a,Y2: nat,Ys2: list_nat,Z: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_146_list__induct4,axiom,
! [Xs: list_nat,Ys: list_a,Zs: list_a,Ws: list_a,P: list_nat > list_a > list_a > list_a > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_nat @ nil_a @ nil_a @ nil_a )
=> ( ! [X2: nat,Xs2: list_nat,Y2: a,Ys2: list_a,Z: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_147_list__induct4,axiom,
! [Xs: list_nat,Ys: list_a,Zs: list_a,Ws: list_nat,P: list_nat > list_a > list_a > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_a @ nil_a @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat,Y2: a,Ys2: list_a,Z: a,Zs2: list_a,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_148_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y3: a,Ys3: list_a] :
( Xs
= ( cons_a @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_149_neq__Nil__conv,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
= ( ? [Y3: nat,Ys3: list_nat] :
( Xs
= ( cons_nat @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_150_Cons__eq__map__D,axiom,
! [X: a,Xs: list_a,F: a > a,Ys: list_a] :
( ( ( cons_a @ X @ Xs )
= ( map_a_a @ F @ Ys ) )
=> ? [Z: a,Zs2: list_a] :
( ( Ys
= ( cons_a @ Z @ Zs2 ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_a_a @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_151_Cons__eq__map__D,axiom,
! [X: a,Xs: list_a,F: nat > a,Ys: list_nat] :
( ( ( cons_a @ X @ Xs )
= ( map_nat_a @ F @ Ys ) )
=> ? [Z: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z @ Zs2 ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_nat_a @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_152_Cons__eq__map__D,axiom,
! [X: nat,Xs: list_nat,F: a > nat,Ys: list_a] :
( ( ( cons_nat @ X @ Xs )
= ( map_a_nat @ F @ Ys ) )
=> ? [Z: a,Zs2: list_a] :
( ( Ys
= ( cons_a @ Z @ Zs2 ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_a_nat @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_153_Cons__eq__map__D,axiom,
! [X: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( map_nat_nat @ F @ Ys ) )
=> ? [Z: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z @ Zs2 ) )
& ( X
= ( F @ Z ) )
& ( Xs
= ( map_nat_nat @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_154_list__induct2_H,axiom,
! [P: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
( ( P @ nil_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a] : ( P @ ( cons_a @ X2 @ Xs2 ) @ nil_a )
=> ( ! [Y2: a,Ys2: list_a] : ( P @ nil_a @ ( cons_a @ Y2 @ Ys2 ) )
=> ( ! [X2: a,Xs2: list_a,Y2: a,Ys2: list_a] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_155_list__induct2_H,axiom,
! [P: list_a > list_nat > $o,Xs: list_a,Ys: list_nat] :
( ( P @ nil_a @ nil_nat )
=> ( ! [X2: a,Xs2: list_a] : ( P @ ( cons_a @ X2 @ Xs2 ) @ nil_nat )
=> ( ! [Y2: nat,Ys2: list_nat] : ( P @ nil_a @ ( cons_nat @ Y2 @ Ys2 ) )
=> ( ! [X2: a,Xs2: list_a,Y2: nat,Ys2: list_nat] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_156_list__induct2_H,axiom,
! [P: list_nat > list_a > $o,Xs: list_nat,Ys: list_a] :
( ( P @ nil_nat @ nil_a )
=> ( ! [X2: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X2 @ Xs2 ) @ nil_a )
=> ( ! [Y2: a,Ys2: list_a] : ( P @ nil_nat @ ( cons_a @ Y2 @ Ys2 ) )
=> ( ! [X2: nat,Xs2: list_nat,Y2: a,Ys2: list_a] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_157_list__induct2_H,axiom,
! [P: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X2 @ Xs2 ) @ nil_nat )
=> ( ! [Y2: nat,Ys2: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y2 @ Ys2 ) )
=> ( ! [X2: nat,Xs2: list_nat,Y2: nat,Ys2: list_nat] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_158_map__eq__Cons__D,axiom,
! [F: a > a,Xs: list_a,Y: a,Ys: list_a] :
( ( ( map_a_a @ F @ Xs )
= ( cons_a @ Y @ Ys ) )
=> ? [Z: a,Zs2: list_a] :
( ( Xs
= ( cons_a @ Z @ Zs2 ) )
& ( ( F @ Z )
= Y )
& ( ( map_a_a @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_159_map__eq__Cons__D,axiom,
! [F: nat > a,Xs: list_nat,Y: a,Ys: list_a] :
( ( ( map_nat_a @ F @ Xs )
= ( cons_a @ Y @ Ys ) )
=> ? [Z: nat,Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Z @ Zs2 ) )
& ( ( F @ Z )
= Y )
& ( ( map_nat_a @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_160_map__eq__Cons__D,axiom,
! [F: a > nat,Xs: list_a,Y: nat,Ys: list_nat] :
( ( ( map_a_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
=> ? [Z: a,Zs2: list_a] :
( ( Xs
= ( cons_a @ Z @ Zs2 ) )
& ( ( F @ Z )
= Y )
& ( ( map_a_nat @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_161_map__eq__Cons__D,axiom,
! [F: nat > nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
=> ? [Z: nat,Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Z @ Zs2 ) )
& ( ( F @ Z )
= Y )
& ( ( map_nat_nat @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_162_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_163_eq__Nil__appendI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append_nat @ nil_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_164_not__Cons__self2,axiom,
! [X: a,Xs: list_a] :
( ( cons_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_165_not__Cons__self2,axiom,
! [X: nat,Xs: list_nat] :
( ( cons_nat @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_166_Cons__eq__appendI,axiom,
! [X: a,Xs1: list_a,Ys: list_a,Xs: list_a,Zs: list_a] :
( ( ( cons_a @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_a @ Xs1 @ Zs ) )
=> ( ( cons_a @ X @ Xs )
= ( append_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_167_Cons__eq__appendI,axiom,
! [X: nat,Xs1: list_nat,Ys: list_nat,Xs: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_nat @ Xs1 @ Zs ) )
=> ( ( cons_nat @ X @ Xs )
= ( append_nat @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_168_Suc__length__conv,axiom,
! [N: nat,Xs: list_a] :
( ( ( suc @ N )
= ( size_size_list_a @ Xs ) )
= ( ? [Y3: a,Ys3: list_a] :
( ( Xs
= ( cons_a @ Y3 @ Ys3 ) )
& ( ( size_size_list_a @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_169_Suc__length__conv,axiom,
! [N: nat,Xs: list_nat] :
( ( ( suc @ N )
= ( size_size_list_nat @ Xs ) )
= ( ? [Y3: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y3 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_170_length__Suc__conv,axiom,
! [Xs: list_a,N: nat] :
( ( ( size_size_list_a @ Xs )
= ( suc @ N ) )
= ( ? [Y3: a,Ys3: list_a] :
( ( Xs
= ( cons_a @ Y3 @ Ys3 ) )
& ( ( size_size_list_a @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_171_length__Suc__conv,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y3: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y3 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_172_Cons__eq__map__conv,axiom,
! [X: a,Xs: list_a,F: a > a,Ys: list_a] :
( ( ( cons_a @ X @ Xs )
= ( map_a_a @ F @ Ys ) )
= ( ? [Z2: a,Zs3: list_a] :
( ( Ys
= ( cons_a @ Z2 @ Zs3 ) )
& ( X
= ( F @ Z2 ) )
& ( Xs
= ( map_a_a @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_173_Cons__eq__map__conv,axiom,
! [X: a,Xs: list_a,F: nat > a,Ys: list_nat] :
( ( ( cons_a @ X @ Xs )
= ( map_nat_a @ F @ Ys ) )
= ( ? [Z2: nat,Zs3: list_nat] :
( ( Ys
= ( cons_nat @ Z2 @ Zs3 ) )
& ( X
= ( F @ Z2 ) )
& ( Xs
= ( map_nat_a @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_174_Cons__eq__map__conv,axiom,
! [X: nat,Xs: list_nat,F: a > nat,Ys: list_a] :
( ( ( cons_nat @ X @ Xs )
= ( map_a_nat @ F @ Ys ) )
= ( ? [Z2: a,Zs3: list_a] :
( ( Ys
= ( cons_a @ Z2 @ Zs3 ) )
& ( X
= ( F @ Z2 ) )
& ( Xs
= ( map_a_nat @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_175_Cons__eq__map__conv,axiom,
! [X: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( map_nat_nat @ F @ Ys ) )
= ( ? [Z2: nat,Zs3: list_nat] :
( ( Ys
= ( cons_nat @ Z2 @ Zs3 ) )
& ( X
= ( F @ Z2 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_176_map__eq__Cons__conv,axiom,
! [F: a > a,Xs: list_a,Y: a,Ys: list_a] :
( ( ( map_a_a @ F @ Xs )
= ( cons_a @ Y @ Ys ) )
= ( ? [Z2: a,Zs3: list_a] :
( ( Xs
= ( cons_a @ Z2 @ Zs3 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_a_a @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_177_map__eq__Cons__conv,axiom,
! [F: nat > a,Xs: list_nat,Y: a,Ys: list_a] :
( ( ( map_nat_a @ F @ Xs )
= ( cons_a @ Y @ Ys ) )
= ( ? [Z2: nat,Zs3: list_nat] :
( ( Xs
= ( cons_nat @ Z2 @ Zs3 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_nat_a @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_178_map__eq__Cons__conv,axiom,
! [F: a > nat,Xs: list_a,Y: nat,Ys: list_nat] :
( ( ( map_a_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
= ( ? [Z2: a,Zs3: list_a] :
( ( Xs
= ( cons_a @ Z2 @ Zs3 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_a_nat @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_179_map__eq__Cons__conv,axiom,
! [F: nat > nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
= ( ? [Z2: nat,Zs3: list_nat] :
( ( Xs
= ( cons_nat @ Z2 @ Zs3 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_nat_nat @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_180_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_a] :
( ( size_size_list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_181_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_nat] :
( ( size_size_list_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_182_append__eq__appendI,axiom,
! [Xs: list_a,Xs1: list_a,Zs: list_a,Ys: list_a,Us: list_a] :
( ( ( append_a @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_a @ Xs1 @ Us ) )
=> ( ( append_a @ Xs @ Ys )
= ( append_a @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_183_append__eq__appendI,axiom,
! [Xs: list_nat,Xs1: list_nat,Zs: list_nat,Ys: list_nat,Us: list_nat] :
( ( ( append_nat @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_nat @ Xs1 @ Us ) )
=> ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_184_neq__if__length__neq,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
!= ( size_size_list_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_185_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_186_append__eq__map__conv,axiom,
! [Ys: list_a,Zs: list_a,F: a > a,Xs: list_a] :
( ( ( append_a @ Ys @ Zs )
= ( map_a_a @ F @ Xs ) )
= ( ? [Us2: list_a,Vs2: list_a] :
( ( Xs
= ( append_a @ Us2 @ Vs2 ) )
& ( Ys
= ( map_a_a @ F @ Us2 ) )
& ( Zs
= ( map_a_a @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_187_append__eq__map__conv,axiom,
! [Ys: list_a,Zs: list_a,F: nat > a,Xs: list_nat] :
( ( ( append_a @ Ys @ Zs )
= ( map_nat_a @ F @ Xs ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys
= ( map_nat_a @ F @ Us2 ) )
& ( Zs
= ( map_nat_a @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_188_append__eq__map__conv,axiom,
! [Ys: list_nat,Zs: list_nat,F: nat > nat,Xs: list_nat] :
( ( ( append_nat @ Ys @ Zs )
= ( map_nat_nat @ F @ Xs ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys
= ( map_nat_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_nat @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_189_append__eq__map__conv,axiom,
! [Ys: list_nat,Zs: list_nat,F: a > nat,Xs: list_a] :
( ( ( append_nat @ Ys @ Zs )
= ( map_a_nat @ F @ Xs ) )
= ( ? [Us2: list_a,Vs2: list_a] :
( ( Xs
= ( append_a @ Us2 @ Vs2 ) )
& ( Ys
= ( map_a_nat @ F @ Us2 ) )
& ( Zs
= ( map_a_nat @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_190_map__eq__append__conv,axiom,
! [F: a > a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( map_a_a @ F @ Xs )
= ( append_a @ Ys @ Zs ) )
= ( ? [Us2: list_a,Vs2: list_a] :
( ( Xs
= ( append_a @ Us2 @ Vs2 ) )
& ( Ys
= ( map_a_a @ F @ Us2 ) )
& ( Zs
= ( map_a_a @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_191_map__eq__append__conv,axiom,
! [F: nat > a,Xs: list_nat,Ys: list_a,Zs: list_a] :
( ( ( map_nat_a @ F @ Xs )
= ( append_a @ Ys @ Zs ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys
= ( map_nat_a @ F @ Us2 ) )
& ( Zs
= ( map_nat_a @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_192_map__eq__append__conv,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( append_nat @ Ys @ Zs ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys
= ( map_nat_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_nat @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_193_map__eq__append__conv,axiom,
! [F: a > nat,Xs: list_a,Ys: list_nat,Zs: list_nat] :
( ( ( map_a_nat @ F @ Xs )
= ( append_nat @ Ys @ Zs ) )
= ( ? [Us2: list_a,Vs2: list_a] :
( ( Xs
= ( append_a @ Us2 @ Vs2 ) )
& ( Ys
= ( map_a_nat @ F @ Us2 ) )
& ( Zs
= ( map_a_nat @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_194_Cons__eq__append__conv,axiom,
! [X: a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( cons_a @ X @ Xs )
= ( append_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_a )
& ( ( cons_a @ X @ Xs )
= Zs ) )
| ? [Ys4: list_a] :
( ( ( cons_a @ X @ Ys4 )
= Ys )
& ( Xs
= ( append_a @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_195_Cons__eq__append__conv,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( append_nat @ Ys @ Zs ) )
= ( ( ( Ys = nil_nat )
& ( ( cons_nat @ X @ Xs )
= Zs ) )
| ? [Ys4: list_nat] :
( ( ( cons_nat @ X @ Ys4 )
= Ys )
& ( Xs
= ( append_nat @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_196_append__eq__Cons__conv,axiom,
! [Ys: list_a,Zs: list_a,X: a,Xs: list_a] :
( ( ( append_a @ Ys @ Zs )
= ( cons_a @ X @ Xs ) )
= ( ( ( Ys = nil_a )
& ( Zs
= ( cons_a @ X @ Xs ) ) )
| ? [Ys4: list_a] :
( ( Ys
= ( cons_a @ X @ Ys4 ) )
& ( ( append_a @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_197_append__eq__Cons__conv,axiom,
! [Ys: list_nat,Zs: list_nat,X: nat,Xs: list_nat] :
( ( ( append_nat @ Ys @ Zs )
= ( cons_nat @ X @ Xs ) )
= ( ( ( Ys = nil_nat )
& ( Zs
= ( cons_nat @ X @ Xs ) ) )
| ? [Ys4: list_nat] :
( ( Ys
= ( cons_nat @ X @ Ys4 ) )
& ( ( append_nat @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_198_length__Suc__conv__rev,axiom,
! [Xs: list_a,N: nat] :
( ( ( size_size_list_a @ Xs )
= ( suc @ N ) )
= ( ? [Y3: a,Ys3: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ Y3 @ nil_a ) ) )
& ( ( size_size_list_a @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_199_length__Suc__conv__rev,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y3: nat,Ys3: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ Y3 @ nil_nat ) ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_200_rev__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X2: a] : ( P @ ( cons_a @ X2 @ nil_a ) )
=> ( ! [X2: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P @ Xs2 )
=> ( P @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_201_rev__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
=> ( ! [X2: nat,Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X2 @ nil_nat ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_202_list__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X2: a] : ( P @ ( cons_a @ X2 @ nil_a ) )
=> ( ! [X2: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_a @ X2 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_203_list__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
=> ( ! [X2: nat,Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_204_map__eq__imp__length__eq,axiom,
! [F: a > nat,Xs: list_a,G: a > nat,Ys: list_a] :
( ( ( map_a_nat @ F @ Xs )
= ( map_a_nat @ G @ Ys ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_205_map__eq__imp__length__eq,axiom,
! [F: a > nat,Xs: list_a,G: nat > nat,Ys: list_nat] :
( ( ( map_a_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_206_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs: list_nat,G: a > nat,Ys: list_a] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_a_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_207_same__length__different,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != Ys )
=> ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ? [Pre: list_a,X2: a,Xs3: list_a,Y2: a,Ys5: list_a] :
( ( X2 != Y2 )
& ( Xs
= ( append_a @ Pre @ ( append_a @ ( cons_a @ X2 @ nil_a ) @ Xs3 ) ) )
& ( Ys
= ( append_a @ Pre @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_208_same__length__different,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs != Ys )
=> ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ? [Pre: list_nat,X2: nat,Xs3: list_nat,Y2: nat,Ys5: list_nat] :
( ( X2 != Y2 )
& ( Xs
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X2 @ nil_nat ) @ Xs3 ) ) )
& ( Ys
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y2 @ nil_nat ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_209_append__eq__append__conv2,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ts: list_a] :
( ( ( append_a @ Xs @ Ys )
= ( append_a @ Zs @ Ts ) )
= ( ? [Us2: list_a] :
( ( ( Xs
= ( append_a @ Zs @ Us2 ) )
& ( ( append_a @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_a @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_a @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_210_append__eq__append__conv2,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ts: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Zs @ Ts ) )
= ( ? [Us2: list_nat] :
( ( ( Xs
= ( append_nat @ Zs @ Us2 ) )
& ( ( append_nat @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_nat @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_nat @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_211_list__induct__2__rev,axiom,
! [X: list_a,Y: list_a,P: list_a > list_a > $o] :
( ( ( size_size_list_a @ X )
= ( size_size_list_a @ Y ) )
=> ( ( P @ nil_a @ nil_a )
=> ( ! [X2: a,Xs2: list_a,Y2: a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) @ ( append_a @ Ys2 @ ( cons_a @ Y2 @ nil_a ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% list_induct_2_rev
thf(fact_212_list__induct__2__rev,axiom,
! [X: list_a,Y: list_nat,P: list_a > list_nat > $o] :
( ( ( size_size_list_a @ X )
= ( size_size_list_nat @ Y ) )
=> ( ( P @ nil_a @ nil_nat )
=> ( ! [X2: a,Xs2: list_a,Y2: nat,Ys2: list_nat] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) @ ( append_nat @ Ys2 @ ( cons_nat @ Y2 @ nil_nat ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% list_induct_2_rev
thf(fact_213_list__induct__2__rev,axiom,
! [X: list_nat,Y: list_a,P: list_nat > list_a > $o] :
( ( ( size_size_list_nat @ X )
= ( size_size_list_a @ Y ) )
=> ( ( P @ nil_nat @ nil_a )
=> ( ! [X2: nat,Xs2: list_nat,Y2: a,Ys2: list_a] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X2 @ nil_nat ) ) @ ( append_a @ Ys2 @ ( cons_a @ Y2 @ nil_a ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% list_induct_2_rev
thf(fact_214_list__induct__2__rev,axiom,
! [X: list_nat,Y: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ X )
= ( size_size_list_nat @ Y ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat,Y2: nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X2 @ nil_nat ) ) @ ( append_nat @ Ys2 @ ( cons_nat @ Y2 @ nil_nat ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% list_induct_2_rev
thf(fact_215_inj__img__insertE,axiom,
! [F: a > a,A: set_a,X: a,B2: set_a] :
( ( inj_on_a_a @ F @ A )
=> ( ~ ( member_a @ X @ B2 )
=> ( ( ( insert_a2 @ X @ B2 )
= ( image_a_a @ F @ A ) )
=> ~ ! [X4: a,A5: set_a] :
( ~ ( member_a @ X4 @ A5 )
=> ( ( A
= ( insert_a2 @ X4 @ A5 ) )
=> ( ( X
= ( F @ X4 ) )
=> ( B2
!= ( image_a_a @ F @ A5 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_216_inj__img__insertE,axiom,
! [F: nat > a,A: set_nat,X: a,B2: set_a] :
( ( inj_on_nat_a @ F @ A )
=> ( ~ ( member_a @ X @ B2 )
=> ( ( ( insert_a2 @ X @ B2 )
= ( image_nat_a @ F @ A ) )
=> ~ ! [X4: nat,A5: set_nat] :
( ~ ( member_nat @ X4 @ A5 )
=> ( ( A
= ( insert_nat2 @ X4 @ A5 ) )
=> ( ( X
= ( F @ X4 ) )
=> ( B2
!= ( image_nat_a @ F @ A5 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_217_inj__img__insertE,axiom,
! [F: a > nat,A: set_a,X: nat,B2: set_nat] :
( ( inj_on_a_nat @ F @ A )
=> ( ~ ( member_nat @ X @ B2 )
=> ( ( ( insert_nat2 @ X @ B2 )
= ( image_a_nat @ F @ A ) )
=> ~ ! [X4: a,A5: set_a] :
( ~ ( member_a @ X4 @ A5 )
=> ( ( A
= ( insert_a2 @ X4 @ A5 ) )
=> ( ( X
= ( F @ X4 ) )
=> ( B2
!= ( image_a_nat @ F @ A5 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_218_inj__img__insertE,axiom,
! [F: nat > nat,A: set_nat,X: nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ~ ( member_nat @ X @ B2 )
=> ( ( ( insert_nat2 @ X @ B2 )
= ( image_nat_nat @ F @ A ) )
=> ~ ! [X4: nat,A5: set_nat] :
( ~ ( member_nat @ X4 @ A5 )
=> ( ( A
= ( insert_nat2 @ X4 @ A5 ) )
=> ( ( X
= ( F @ X4 ) )
=> ( B2
!= ( image_nat_nat @ F @ A5 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_219_ex__map__conv,axiom,
! [Ys: list_nat,F: a > nat] :
( ( ? [Xs4: list_a] :
( Ys
= ( map_a_nat @ F @ Xs4 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
=> ? [Y3: a] :
( X3
= ( F @ Y3 ) ) ) ) ) ).
% ex_map_conv
thf(fact_220_map__cong,axiom,
! [Xs: list_a,Ys: list_a,F: a > nat,G: a > nat] :
( ( Xs = Ys )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Ys ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_a_nat @ F @ Xs )
= ( map_a_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_221_map__idI,axiom,
! [Xs: list_a,F: a > a] :
( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_a_a @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_222_map__idI,axiom,
! [Xs: list_nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_nat_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_223_map__ext,axiom,
! [Xs: list_a,F: a > nat,G: a > nat] :
( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_a_nat @ F @ Xs )
= ( map_a_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_224_list_Omap__ident__strong,axiom,
! [T: list_a,F: a > a] :
( ! [Z: a] :
( ( member_a @ Z @ ( set_a2 @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_a_a @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_225_list_Omap__ident__strong,axiom,
! [T: list_nat,F: nat > nat] :
( ! [Z: nat] :
( ( member_nat @ Z @ ( set_nat2 @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_nat_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_226_list_Oinj__map__strong,axiom,
! [X: list_a,Xa: list_a,F: a > nat,Fa: a > nat] :
( ! [Z: a,Za: a] :
( ( member_a @ Z @ ( set_a2 @ X ) )
=> ( ( member_a @ Za @ ( set_a2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_a_nat @ F @ X )
= ( map_a_nat @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_227_list_Omap__cong0,axiom,
! [X: list_a,F: a > nat,G: a > nat] :
( ! [Z: a] :
( ( member_a @ Z @ ( set_a2 @ X ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_a_nat @ F @ X )
= ( map_a_nat @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_228_list_Omap__cong,axiom,
! [X: list_a,Ya: list_a,F: a > nat,G: a > nat] :
( ( X = Ya )
=> ( ! [Z: a] :
( ( member_a @ Z @ ( set_a2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_a_nat @ F @ X )
= ( map_a_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_229_inj__on__image__iff,axiom,
! [A: set_a,G: a > nat,F: a > a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ! [Xa2: a] :
( ( member_a @ Xa2 @ A )
=> ( ( ( G @ ( F @ X2 ) )
= ( G @ ( F @ Xa2 ) ) )
= ( ( G @ X2 )
= ( G @ Xa2 ) ) ) ) )
=> ( ( inj_on_a_a @ F @ A )
=> ( ( inj_on_a_nat @ G @ ( image_a_a @ F @ A ) )
= ( inj_on_a_nat @ G @ A ) ) ) ) ).
% inj_on_image_iff
thf(fact_230_inj__on__image__iff,axiom,
! [A: set_nat,G: nat > nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A )
=> ( ( ( G @ ( F @ X2 ) )
= ( G @ ( F @ Xa2 ) ) )
= ( ( G @ X2 )
= ( G @ Xa2 ) ) ) ) )
=> ( ( inj_on_nat_nat @ F @ A )
=> ( ( inj_on_nat_nat @ G @ ( image_nat_nat @ F @ A ) )
= ( inj_on_nat_nat @ G @ A ) ) ) ) ).
% inj_on_image_iff
thf(fact_231_split__list__first__prop__iff,axiom,
! [Xs: list_a,P: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_a,X3: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y3: a] :
( ( member_a @ Y3 @ ( set_a2 @ Ys3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_232_split__list__first__prop__iff,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_nat,X3: nat] :
( ? [Zs3: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y3: nat] :
( ( member_nat @ Y3 @ ( set_nat2 @ Ys3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_233_split__list__last__prop__iff,axiom,
! [Xs: list_a,P: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_a,X3: a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y3: a] :
( ( member_a @ Y3 @ ( set_a2 @ Zs3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_234_split__list__last__prop__iff,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_nat,X3: nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y3: nat] :
( ( member_nat @ Y3 @ ( set_nat2 @ Zs3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_235_in__set__conv__decomp__first,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [Ys3: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs3 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_236_in__set__conv__decomp__first,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys3: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs3 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_237_in__set__conv__decomp__last,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [Ys3: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs3 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_238_in__set__conv__decomp__last,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys3: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs3 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_239_split__list__first__propE,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_a,X2: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa3: a] :
( ( member_a @ Xa3 @ ( set_a2 @ Ys2 ) )
=> ~ ( P @ Xa3 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_240_split__list__first__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_nat,X2: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs2 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa3: nat] :
( ( member_nat @ Xa3 @ ( set_nat2 @ Ys2 ) )
=> ~ ( P @ Xa3 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_241_split__list__last__propE,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_a,X2: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa3: a] :
( ( member_a @ Xa3 @ ( set_a2 @ Zs2 ) )
=> ~ ( P @ Xa3 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_242_split__list__last__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_nat,X2: nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs2 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa3: nat] :
( ( member_nat @ Xa3 @ ( set_nat2 @ Zs2 ) )
=> ~ ( P @ Xa3 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_243_split__list__first__prop,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys2: list_a,X2: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) )
& ( P @ X2 )
& ! [Xa3: a] :
( ( member_a @ Xa3 @ ( set_a2 @ Ys2 ) )
=> ~ ( P @ Xa3 ) ) ) ) ).
% split_list_first_prop
thf(fact_244_split__list__first__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys2: list_nat,X2: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs2 ) ) )
& ( P @ X2 )
& ! [Xa3: nat] :
( ( member_nat @ Xa3 @ ( set_nat2 @ Ys2 ) )
=> ~ ( P @ Xa3 ) ) ) ) ).
% split_list_first_prop
thf(fact_245_split__list__last__prop,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys2: list_a,X2: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) )
& ( P @ X2 )
& ! [Xa3: a] :
( ( member_a @ Xa3 @ ( set_a2 @ Zs2 ) )
=> ~ ( P @ Xa3 ) ) ) ) ).
% split_list_last_prop
thf(fact_246_split__list__last__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys2: list_nat,X2: nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs2 ) ) )
& ( P @ X2 )
& ! [Xa3: nat] :
( ( member_nat @ Xa3 @ ( set_nat2 @ Zs2 ) )
=> ~ ( P @ Xa3 ) ) ) ) ).
% split_list_last_prop
thf(fact_247_in__set__conv__decomp,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
= ( ? [Ys3: list_a,Zs3: list_a] :
( Xs
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_248_in__set__conv__decomp,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys3: list_nat,Zs3: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_249_append__Cons__eq__iff,axiom,
! [X: a,Xs: list_a,Ys: list_a,Xs5: list_a,Ys6: list_a] :
( ~ ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ~ ( member_a @ X @ ( set_a2 @ Ys ) )
=> ( ( ( append_a @ Xs @ ( cons_a @ X @ Ys ) )
= ( append_a @ Xs5 @ ( cons_a @ X @ Ys6 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_250_append__Cons__eq__iff,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat,Xs5: list_nat,Ys6: list_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ~ ( member_nat @ X @ ( set_nat2 @ Ys ) )
=> ( ( ( append_nat @ Xs @ ( cons_nat @ X @ Ys ) )
= ( append_nat @ Xs5 @ ( cons_nat @ X @ Ys6 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_251_split__list__propE,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_a,X2: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) )
=> ~ ( P @ X2 ) ) ) ).
% split_list_propE
thf(fact_252_split__list__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys2: list_nat,X2: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs2 ) ) )
=> ~ ( P @ X2 ) ) ) ).
% split_list_propE
thf(fact_253_split__list__first,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ? [Ys2: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X @ Zs2 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_254_split__list__first,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys2: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs2 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_255_split__list__prop,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys2: list_a,X2: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) )
& ( P @ X2 ) ) ) ).
% split_list_prop
thf(fact_256_split__list__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys2: list_nat,X2: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs2 ) ) )
& ( P @ X2 ) ) ) ).
% split_list_prop
thf(fact_257_split__list__last,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ? [Ys2: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X @ Zs2 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_258_split__list__last,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys2: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs2 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_259_split__list,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ? [Ys2: list_a,Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_260_split__list,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys2: list_nat,Zs2: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_261_set__ConsD,axiom,
! [Y: a,X: a,Xs: list_a] :
( ( member_a @ Y @ ( set_a2 @ ( cons_a @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_a @ Y @ ( set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_262_set__ConsD,axiom,
! [Y: nat,X: nat,Xs: list_nat] :
( ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_nat @ Y @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_263_list_Oset__cases,axiom,
! [E: a,A4: list_a] :
( ( member_a @ E @ ( set_a2 @ A4 ) )
=> ( ! [Z22: list_a] :
( A4
!= ( cons_a @ E @ Z22 ) )
=> ~ ! [Z1: a,Z22: list_a] :
( ( A4
= ( cons_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( set_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_264_list_Oset__cases,axiom,
! [E: nat,A4: list_nat] :
( ( member_nat @ E @ ( set_nat2 @ A4 ) )
=> ( ! [Z22: list_nat] :
( A4
!= ( cons_nat @ E @ Z22 ) )
=> ~ ! [Z1: nat,Z22: list_nat] :
( ( A4
= ( cons_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat @ E @ ( set_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_265_list_Oset__intros_I1_J,axiom,
! [X21: a,X22: list_a] : ( member_a @ X21 @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_266_list_Oset__intros_I1_J,axiom,
! [X21: nat,X22: list_nat] : ( member_nat @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_267_length__append__singleton,axiom,
! [Xs: list_a,X: a] :
( ( size_size_list_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ).
% length_append_singleton
thf(fact_268_length__append__singleton,axiom,
! [Xs: list_nat,X: nat] :
( ( size_size_list_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_append_singleton
thf(fact_269_image__insert,axiom,
! [F: a > a,A4: a,B2: set_a] :
( ( image_a_a @ F @ ( insert_a2 @ A4 @ B2 ) )
= ( insert_a2 @ ( F @ A4 ) @ ( image_a_a @ F @ B2 ) ) ) ).
% image_insert
thf(fact_270_image__insert,axiom,
! [F: a > nat,A4: a,B2: set_a] :
( ( image_a_nat @ F @ ( insert_a2 @ A4 @ B2 ) )
= ( insert_nat2 @ ( F @ A4 ) @ ( image_a_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_271_image__insert,axiom,
! [F: nat > a,A4: nat,B2: set_nat] :
( ( image_nat_a @ F @ ( insert_nat2 @ A4 @ B2 ) )
= ( insert_a2 @ ( F @ A4 ) @ ( image_nat_a @ F @ B2 ) ) ) ).
% image_insert
thf(fact_272_image__insert,axiom,
! [F: nat > nat,A4: nat,B2: set_nat] :
( ( image_nat_nat @ F @ ( insert_nat2 @ A4 @ B2 ) )
= ( insert_nat2 @ ( F @ A4 ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_273_insert__image,axiom,
! [X: a,A: set_a,F: a > a] :
( ( member_a @ X @ A )
=> ( ( insert_a2 @ ( F @ X ) @ ( image_a_a @ F @ A ) )
= ( image_a_a @ F @ A ) ) ) ).
% insert_image
thf(fact_274_insert__image,axiom,
! [X: a,A: set_a,F: a > nat] :
( ( member_a @ X @ A )
=> ( ( insert_nat2 @ ( F @ X ) @ ( image_a_nat @ F @ A ) )
= ( image_a_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_275_insert__image,axiom,
! [X: nat,A: set_nat,F: nat > a] :
( ( member_nat @ X @ A )
=> ( ( insert_a2 @ ( F @ X ) @ ( image_nat_a @ F @ A ) )
= ( image_nat_a @ F @ A ) ) ) ).
% insert_image
thf(fact_276_insert__image,axiom,
! [X: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( ( insert_nat2 @ ( F @ X ) @ ( image_nat_nat @ F @ A ) )
= ( image_nat_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_277_length__Cons,axiom,
! [X: a,Xs: list_a] :
( ( size_size_list_a @ ( cons_a @ X @ Xs ) )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ).
% length_Cons
thf(fact_278_length__Cons,axiom,
! [X: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X @ Xs ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_Cons
thf(fact_279_insertCI,axiom,
! [A4: a,B2: set_a,B: a] :
( ( ~ ( member_a @ A4 @ B2 )
=> ( A4 = B ) )
=> ( member_a @ A4 @ ( insert_a2 @ B @ B2 ) ) ) ).
% insertCI
thf(fact_280_insertCI,axiom,
! [A4: nat,B2: set_nat,B: nat] :
( ( ~ ( member_nat @ A4 @ B2 )
=> ( A4 = B ) )
=> ( member_nat @ A4 @ ( insert_nat2 @ B @ B2 ) ) ) ).
% insertCI
thf(fact_281_insert__iff,axiom,
! [A4: a,B: a,A: set_a] :
( ( member_a @ A4 @ ( insert_a2 @ B @ A ) )
= ( ( A4 = B )
| ( member_a @ A4 @ A ) ) ) ).
% insert_iff
thf(fact_282_insert__iff,axiom,
! [A4: nat,B: nat,A: set_nat] :
( ( member_nat @ A4 @ ( insert_nat2 @ B @ A ) )
= ( ( A4 = B )
| ( member_nat @ A4 @ A ) ) ) ).
% insert_iff
thf(fact_283_insert__absorb2,axiom,
! [X: a,A: set_a] :
( ( insert_a2 @ X @ ( insert_a2 @ X @ A ) )
= ( insert_a2 @ X @ A ) ) ).
% insert_absorb2
thf(fact_284_insert__absorb2,axiom,
! [X: nat,A: set_nat] :
( ( insert_nat2 @ X @ ( insert_nat2 @ X @ A ) )
= ( insert_nat2 @ X @ A ) ) ).
% insert_absorb2
thf(fact_285_nat_Oinject,axiom,
! [X24: nat,Y23: nat] :
( ( ( suc @ X24 )
= ( suc @ Y23 ) )
= ( X24 = Y23 ) ) ).
% nat.inject
thf(fact_286_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_287_image__eqI,axiom,
! [B: a,F: a > a,X: a,A: set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_a @ X @ A )
=> ( member_a @ B @ ( image_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_288_image__eqI,axiom,
! [B: nat,F: a > nat,X: a,A: set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_a @ X @ A )
=> ( member_nat @ B @ ( image_a_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_289_image__eqI,axiom,
! [B: a,F: nat > a,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_a @ B @ ( image_nat_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_290_image__eqI,axiom,
! [B: nat,F: nat > nat,X: nat,A: set_nat] :
( ( B
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_291_the__elem__set,axiom,
! [X: a] :
( ( the_elem_a @ ( set_a2 @ ( cons_a @ X @ nil_a ) ) )
= X ) ).
% the_elem_set
thf(fact_292_the__elem__set,axiom,
! [X: nat] :
( ( the_elem_nat @ ( set_nat2 @ ( cons_nat @ X @ nil_nat ) ) )
= X ) ).
% the_elem_set
thf(fact_293_rev__image__eqI,axiom,
! [X: a,A: set_a,B: a,F: a > a] :
( ( member_a @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_a_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_294_rev__image__eqI,axiom,
! [X: a,A: set_a,B: nat,F: a > nat] :
( ( member_a @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_a_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_295_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: a,F: nat > a] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_nat_a @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_296_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B: nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_297_ball__imageD,axiom,
! [F: a > nat,A: set_a,P: nat > $o] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( image_a_nat @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X5: a] :
( ( member_a @ X5 @ A )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_298_ball__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X5: nat] :
( ( member_nat @ X5 @ A )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_299_image__cong,axiom,
! [M: set_a,N2: set_a,F: a > nat,G: a > nat] :
( ( M = N2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_a_nat @ F @ M )
= ( image_a_nat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_300_image__cong,axiom,
! [M: set_nat,N2: set_nat,F: nat > nat,G: nat > nat] :
( ( M = N2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N2 )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_nat_nat @ F @ M )
= ( image_nat_nat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_301_bex__imageD,axiom,
! [F: a > nat,A: set_a,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( image_a_nat @ F @ A ) )
& ( P @ X5 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_302_bex__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( image_nat_nat @ F @ A ) )
& ( P @ X5 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_303_image__iff,axiom,
! [Z3: nat,F: a > nat,A: set_a] :
( ( member_nat @ Z3 @ ( image_a_nat @ F @ A ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A )
& ( Z3
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_304_image__iff,axiom,
! [Z3: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ Z3 @ ( image_nat_nat @ F @ A ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( Z3
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_305_imageI,axiom,
! [X: a,A: set_a,F: a > a] :
( ( member_a @ X @ A )
=> ( member_a @ ( F @ X ) @ ( image_a_a @ F @ A ) ) ) ).
% imageI
thf(fact_306_imageI,axiom,
! [X: a,A: set_a,F: a > nat] :
( ( member_a @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_a_nat @ F @ A ) ) ) ).
% imageI
thf(fact_307_imageI,axiom,
! [X: nat,A: set_nat,F: nat > a] :
( ( member_nat @ X @ A )
=> ( member_a @ ( F @ X ) @ ( image_nat_a @ F @ A ) ) ) ).
% imageI
thf(fact_308_imageI,axiom,
! [X: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_309_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_310_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_311_inj__Suc,axiom,
! [N2: set_nat] : ( inj_on_nat_nat @ suc @ N2 ) ).
% inj_Suc
thf(fact_312_mk__disjoint__insert,axiom,
! [A4: a,A: set_a] :
( ( member_a @ A4 @ A )
=> ? [B3: set_a] :
( ( A
= ( insert_a2 @ A4 @ B3 ) )
& ~ ( member_a @ A4 @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_313_mk__disjoint__insert,axiom,
! [A4: nat,A: set_nat] :
( ( member_nat @ A4 @ A )
=> ? [B3: set_nat] :
( ( A
= ( insert_nat2 @ A4 @ B3 ) )
& ~ ( member_nat @ A4 @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_314_insert__commute,axiom,
! [X: a,Y: a,A: set_a] :
( ( insert_a2 @ X @ ( insert_a2 @ Y @ A ) )
= ( insert_a2 @ Y @ ( insert_a2 @ X @ A ) ) ) ).
% insert_commute
thf(fact_315_insert__commute,axiom,
! [X: nat,Y: nat,A: set_nat] :
( ( insert_nat2 @ X @ ( insert_nat2 @ Y @ A ) )
= ( insert_nat2 @ Y @ ( insert_nat2 @ X @ A ) ) ) ).
% insert_commute
thf(fact_316_insert__eq__iff,axiom,
! [A4: a,A: set_a,B: a,B2: set_a] :
( ~ ( member_a @ A4 @ A )
=> ( ~ ( member_a @ B @ B2 )
=> ( ( ( insert_a2 @ A4 @ A )
= ( insert_a2 @ B @ B2 ) )
= ( ( ( A4 = B )
=> ( A = B2 ) )
& ( ( A4 != B )
=> ? [C2: set_a] :
( ( A
= ( insert_a2 @ B @ C2 ) )
& ~ ( member_a @ B @ C2 )
& ( B2
= ( insert_a2 @ A4 @ C2 ) )
& ~ ( member_a @ A4 @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_317_insert__eq__iff,axiom,
! [A4: nat,A: set_nat,B: nat,B2: set_nat] :
( ~ ( member_nat @ A4 @ A )
=> ( ~ ( member_nat @ B @ B2 )
=> ( ( ( insert_nat2 @ A4 @ A )
= ( insert_nat2 @ B @ B2 ) )
= ( ( ( A4 = B )
=> ( A = B2 ) )
& ( ( A4 != B )
=> ? [C2: set_nat] :
( ( A
= ( insert_nat2 @ B @ C2 ) )
& ~ ( member_nat @ B @ C2 )
& ( B2
= ( insert_nat2 @ A4 @ C2 ) )
& ~ ( member_nat @ A4 @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_318_insert__absorb,axiom,
! [A4: a,A: set_a] :
( ( member_a @ A4 @ A )
=> ( ( insert_a2 @ A4 @ A )
= A ) ) ).
% insert_absorb
thf(fact_319_insert__absorb,axiom,
! [A4: nat,A: set_nat] :
( ( member_nat @ A4 @ A )
=> ( ( insert_nat2 @ A4 @ A )
= A ) ) ).
% insert_absorb
thf(fact_320_insert__ident,axiom,
! [X: a,A: set_a,B2: set_a] :
( ~ ( member_a @ X @ A )
=> ( ~ ( member_a @ X @ B2 )
=> ( ( ( insert_a2 @ X @ A )
= ( insert_a2 @ X @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_321_insert__ident,axiom,
! [X: nat,A: set_nat,B2: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ~ ( member_nat @ X @ B2 )
=> ( ( ( insert_nat2 @ X @ A )
= ( insert_nat2 @ X @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_322_Set_Oset__insert,axiom,
! [X: a,A: set_a] :
( ( member_a @ X @ A )
=> ~ ! [B3: set_a] :
( ( A
= ( insert_a2 @ X @ B3 ) )
=> ( member_a @ X @ B3 ) ) ) ).
% Set.set_insert
thf(fact_323_Set_Oset__insert,axiom,
! [X: nat,A: set_nat] :
( ( member_nat @ X @ A )
=> ~ ! [B3: set_nat] :
( ( A
= ( insert_nat2 @ X @ B3 ) )
=> ( member_nat @ X @ B3 ) ) ) ).
% Set.set_insert
thf(fact_324_insertI2,axiom,
! [A4: a,B2: set_a,B: a] :
( ( member_a @ A4 @ B2 )
=> ( member_a @ A4 @ ( insert_a2 @ B @ B2 ) ) ) ).
% insertI2
thf(fact_325_insertI2,axiom,
! [A4: nat,B2: set_nat,B: nat] :
( ( member_nat @ A4 @ B2 )
=> ( member_nat @ A4 @ ( insert_nat2 @ B @ B2 ) ) ) ).
% insertI2
thf(fact_326_insertI1,axiom,
! [A4: a,B2: set_a] : ( member_a @ A4 @ ( insert_a2 @ A4 @ B2 ) ) ).
% insertI1
thf(fact_327_insertI1,axiom,
! [A4: nat,B2: set_nat] : ( member_nat @ A4 @ ( insert_nat2 @ A4 @ B2 ) ) ).
% insertI1
thf(fact_328_insertE,axiom,
! [A4: a,B: a,A: set_a] :
( ( member_a @ A4 @ ( insert_a2 @ B @ A ) )
=> ( ( A4 != B )
=> ( member_a @ A4 @ A ) ) ) ).
% insertE
thf(fact_329_insertE,axiom,
! [A4: nat,B: nat,A: set_nat] :
( ( member_nat @ A4 @ ( insert_nat2 @ B @ A ) )
=> ( ( A4 != B )
=> ( member_nat @ A4 @ A ) ) ) ).
% insertE
thf(fact_330_size__neq__size__imp__neq,axiom,
! [X: list_a,Y: list_a] :
( ( ( size_size_list_a @ X )
!= ( size_size_list_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_331_size__neq__size__imp__neq,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( size_size_list_nat @ X )
!= ( size_size_list_nat @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_332_SuccI,axiom,
! [Kl: list_a,K: a,Kl2: set_list_a] :
( ( member_list_a @ ( append_a @ Kl @ ( cons_a @ K @ nil_a ) ) @ Kl2 )
=> ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_333_SuccI,axiom,
! [Kl: list_nat,K: nat,Kl2: set_list_nat] :
( ( member_list_nat @ ( append_nat @ Kl @ ( cons_nat @ K @ nil_nat ) ) @ Kl2 )
=> ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ Kl ) ) ) ).
% SuccI
thf(fact_334_SuccD,axiom,
! [K: a,Kl2: set_list_a,Kl: list_a] :
( ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl2 @ Kl ) )
=> ( member_list_a @ ( append_a @ Kl @ ( cons_a @ K @ nil_a ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_335_SuccD,axiom,
! [K: nat,Kl2: set_list_nat,Kl: list_nat] :
( ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ Kl ) )
=> ( member_list_nat @ ( append_nat @ Kl @ ( cons_nat @ K @ nil_nat ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_336_product__lists_Osimps_I1_J,axiom,
( ( product_lists_a @ nil_list_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% product_lists.simps(1)
thf(fact_337_product__lists_Osimps_I1_J,axiom,
( ( product_lists_nat @ nil_list_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% product_lists.simps(1)
thf(fact_338_bind__simps_I2_J,axiom,
! [X: a,Xs: list_a,F: a > list_a] :
( ( bind_a_a @ ( cons_a @ X @ Xs ) @ F )
= ( append_a @ ( F @ X ) @ ( bind_a_a @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_339_bind__simps_I2_J,axiom,
! [X: a,Xs: list_a,F: a > list_nat] :
( ( bind_a_nat @ ( cons_a @ X @ Xs ) @ F )
= ( append_nat @ ( F @ X ) @ ( bind_a_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_340_bind__simps_I2_J,axiom,
! [X: nat,Xs: list_nat,F: nat > list_a] :
( ( bind_nat_a @ ( cons_nat @ X @ Xs ) @ F )
= ( append_a @ ( F @ X ) @ ( bind_nat_a @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_341_bind__simps_I2_J,axiom,
! [X: nat,Xs: list_nat,F: nat > list_nat] :
( ( bind_nat_nat @ ( cons_nat @ X @ Xs ) @ F )
= ( append_nat @ ( F @ X ) @ ( bind_nat_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_342_kernel__of__under__inj__map,axiom,
! [F: a > nat,X: list_a] :
( ( inj_on_a_nat @ F @ ( set_a2 @ X ) )
=> ( ( equiva2867628904822520638l_of_a @ X )
= ( equiva2048684438135499664of_nat @ ( map_a_nat @ F @ X ) ) ) ) ).
% kernel_of_under_inj_map
thf(fact_343_kernel__of__under__inj__map,axiom,
! [F: nat > nat,X: list_nat] :
( ( inj_on_nat_nat @ F @ ( set_nat2 @ X ) )
=> ( ( equiva2048684438135499664of_nat @ X )
= ( equiva2048684438135499664of_nat @ ( map_nat_nat @ F @ X ) ) ) ) ).
% kernel_of_under_inj_map
thf(fact_344_map__removeAll__inj__on,axiom,
! [F: a > nat,X: a,Xs: list_a] :
( ( inj_on_a_nat @ F @ ( insert_a2 @ X @ ( set_a2 @ Xs ) ) )
=> ( ( map_a_nat @ F @ ( removeAll_a @ X @ Xs ) )
= ( removeAll_nat @ ( F @ X ) @ ( map_a_nat @ F @ Xs ) ) ) ) ).
% map_removeAll_inj_on
thf(fact_345_map__removeAll__inj__on,axiom,
! [F: nat > nat,X: nat,Xs: list_nat] :
( ( inj_on_nat_nat @ F @ ( insert_nat2 @ X @ ( set_nat2 @ Xs ) ) )
=> ( ( map_nat_nat @ F @ ( removeAll_nat @ X @ Xs ) )
= ( removeAll_nat @ ( F @ X ) @ ( map_nat_nat @ F @ Xs ) ) ) ) ).
% map_removeAll_inj_on
thf(fact_346_prefixes__snoc,axiom,
! [Xs: list_a,X: a] :
( ( prefixes_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= ( append_list_a @ ( prefixes_a @ Xs ) @ ( cons_list_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) @ nil_list_a ) ) ) ).
% prefixes_snoc
thf(fact_347_prefixes__snoc,axiom,
! [Xs: list_nat,X: nat] :
( ( prefixes_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= ( append_list_nat @ ( prefixes_nat @ Xs ) @ ( cons_list_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) @ nil_list_nat ) ) ) ).
% prefixes_snoc
thf(fact_348_removeAll__id,axiom,
! [X: a,Xs: list_a] :
( ~ ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ( removeAll_a @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_349_removeAll__id,axiom,
! [X: nat,Xs: list_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( removeAll_nat @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_350_removeAll__append,axiom,
! [X: a,Xs: list_a,Ys: list_a] :
( ( removeAll_a @ X @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( removeAll_a @ X @ Xs ) @ ( removeAll_a @ X @ Ys ) ) ) ).
% removeAll_append
thf(fact_351_removeAll__append,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat] :
( ( removeAll_nat @ X @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( removeAll_nat @ X @ Xs ) @ ( removeAll_nat @ X @ Ys ) ) ) ).
% removeAll_append
thf(fact_352_bind__simps_I1_J,axiom,
! [F: a > list_a] :
( ( bind_a_a @ nil_a @ F )
= nil_a ) ).
% bind_simps(1)
thf(fact_353_bind__simps_I1_J,axiom,
! [F: a > list_nat] :
( ( bind_a_nat @ nil_a @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_354_bind__simps_I1_J,axiom,
! [F: nat > list_a] :
( ( bind_nat_a @ nil_nat @ F )
= nil_a ) ).
% bind_simps(1)
thf(fact_355_bind__simps_I1_J,axiom,
! [F: nat > list_nat] :
( ( bind_nat_nat @ nil_nat @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_356_removeAll_Osimps_I2_J,axiom,
! [X: a,Y: a,Xs: list_a] :
( ( ( X = Y )
=> ( ( removeAll_a @ X @ ( cons_a @ Y @ Xs ) )
= ( removeAll_a @ X @ Xs ) ) )
& ( ( X != Y )
=> ( ( removeAll_a @ X @ ( cons_a @ Y @ Xs ) )
= ( cons_a @ Y @ ( removeAll_a @ X @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_357_removeAll_Osimps_I2_J,axiom,
! [X: nat,Y: nat,Xs: list_nat] :
( ( ( X = Y )
=> ( ( removeAll_nat @ X @ ( cons_nat @ Y @ Xs ) )
= ( removeAll_nat @ X @ Xs ) ) )
& ( ( X != Y )
=> ( ( removeAll_nat @ X @ ( cons_nat @ Y @ Xs ) )
= ( cons_nat @ Y @ ( removeAll_nat @ X @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_358_removeAll_Osimps_I1_J,axiom,
! [X: a] :
( ( removeAll_a @ X @ nil_a )
= nil_a ) ).
% removeAll.simps(1)
thf(fact_359_removeAll_Osimps_I1_J,axiom,
! [X: nat] :
( ( removeAll_nat @ X @ nil_nat )
= nil_nat ) ).
% removeAll.simps(1)
thf(fact_360_kernel__of__eq__len,axiom,
! [X: list_a,Y: list_a] :
( ( ( equiva2867628904822520638l_of_a @ X )
= ( equiva2867628904822520638l_of_a @ Y ) )
=> ( ( size_size_list_a @ X )
= ( size_size_list_a @ Y ) ) ) ).
% kernel_of_eq_len
thf(fact_361_kernel__of__eq__len,axiom,
! [X: list_a,Y: list_nat] :
( ( ( equiva2867628904822520638l_of_a @ X )
= ( equiva2048684438135499664of_nat @ Y ) )
=> ( ( size_size_list_a @ X )
= ( size_size_list_nat @ Y ) ) ) ).
% kernel_of_eq_len
thf(fact_362_kernel__of__eq__len,axiom,
! [X: list_nat,Y: list_a] :
( ( ( equiva2048684438135499664of_nat @ X )
= ( equiva2867628904822520638l_of_a @ Y ) )
=> ( ( size_size_list_nat @ X )
= ( size_size_list_a @ Y ) ) ) ).
% kernel_of_eq_len
thf(fact_363_kernel__of__eq__len,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( equiva2048684438135499664of_nat @ X )
= ( equiva2048684438135499664of_nat @ Y ) )
=> ( ( size_size_list_nat @ X )
= ( size_size_list_nat @ Y ) ) ) ).
% kernel_of_eq_len
thf(fact_364_in__set__product__lists__length,axiom,
! [Xs: list_a,Xss2: list_list_a] :
( ( member_list_a @ Xs @ ( set_list_a2 @ ( product_lists_a @ Xss2 ) ) )
=> ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Xss2 ) ) ) ).
% in_set_product_lists_length
thf(fact_365_in__set__product__lists__length,axiom,
! [Xs: list_nat,Xss2: list_list_nat] :
( ( member_list_nat @ Xs @ ( set_list_nat2 @ ( product_lists_nat @ Xss2 ) ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Xss2 ) ) ) ).
% in_set_product_lists_length
thf(fact_366_kernel__of__inj__on__rgfs__aux,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( size_size_list_nat @ X )
= ( size_size_list_nat @ Y ) )
=> ( ( equiva3371634703666331078on_rgf @ X )
=> ( ( equiva3371634703666331078on_rgf @ Y )
=> ( ( ( equiva2048684438135499664of_nat @ X )
= ( equiva2048684438135499664of_nat @ Y ) )
=> ( X = Y ) ) ) ) ) ).
% kernel_of_inj_on_rgfs_aux
thf(fact_367_prefixes_Osimps_I2_J,axiom,
! [X: a,Xs: list_a] :
( ( prefixes_a @ ( cons_a @ X @ Xs ) )
= ( cons_list_a @ nil_a @ ( map_list_a_list_a @ ( cons_a @ X ) @ ( prefixes_a @ Xs ) ) ) ) ).
% prefixes.simps(2)
thf(fact_368_prefixes_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( prefixes_nat @ ( cons_nat @ X @ Xs ) )
= ( cons_list_nat @ nil_nat @ ( map_li7225945977422193158st_nat @ ( cons_nat @ X ) @ ( prefixes_nat @ Xs ) ) ) ) ).
% prefixes.simps(2)
thf(fact_369_prefixes_Osimps_I1_J,axiom,
( ( prefixes_a @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% prefixes.simps(1)
thf(fact_370_prefixes_Osimps_I1_J,axiom,
( ( prefixes_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% prefixes.simps(1)
thf(fact_371_prefixes__eq__snoc,axiom,
! [Ys: list_a,Xs: list_list_a,X: list_a] :
( ( ( prefixes_a @ Ys )
= ( append_list_a @ Xs @ ( cons_list_a @ X @ nil_list_a ) ) )
= ( ( ( ( Ys = nil_a )
& ( Xs = nil_list_a ) )
| ? [Z2: a,Zs3: list_a] :
( ( Ys
= ( append_a @ Zs3 @ ( cons_a @ Z2 @ nil_a ) ) )
& ( Xs
= ( prefixes_a @ Zs3 ) ) ) )
& ( X = Ys ) ) ) ).
% prefixes_eq_snoc
thf(fact_372_prefixes__eq__snoc,axiom,
! [Ys: list_nat,Xs: list_list_nat,X: list_nat] :
( ( ( prefixes_nat @ Ys )
= ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) ) )
= ( ( ( ( Ys = nil_nat )
& ( Xs = nil_list_nat ) )
| ? [Z2: nat,Zs3: list_nat] :
( ( Ys
= ( append_nat @ Zs3 @ ( cons_nat @ Z2 @ nil_nat ) ) )
& ( Xs
= ( prefixes_nat @ Zs3 ) ) ) )
& ( X = Ys ) ) ) ).
% prefixes_eq_snoc
thf(fact_373_empty__Shift,axiom,
! [Kl2: set_list_a,K: a] :
( ( member_list_a @ nil_a @ Kl2 )
=> ( ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl2 @ nil_a ) )
=> ( member_list_a @ nil_a @ ( bNF_Greatest_Shift_a @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_374_empty__Shift,axiom,
! [Kl2: set_list_nat,K: nat] :
( ( member_list_nat @ nil_nat @ Kl2 )
=> ( ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ nil_nat ) )
=> ( member_list_nat @ nil_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl2 @ K ) ) ) ) ).
% empty_Shift
thf(fact_375_Succ__Shift,axiom,
! [Kl2: set_list_a,K: a,Kl: list_a] :
( ( bNF_Greatest_Succ_a @ ( bNF_Greatest_Shift_a @ Kl2 @ K ) @ Kl )
= ( bNF_Greatest_Succ_a @ Kl2 @ ( cons_a @ K @ Kl ) ) ) ).
% Succ_Shift
thf(fact_376_Succ__Shift,axiom,
! [Kl2: set_list_nat,K: nat,Kl: list_nat] :
( ( bNF_Gr6352880689984616693cc_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl2 @ K ) @ Kl )
= ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ ( cons_nat @ K @ Kl ) ) ) ).
% Succ_Shift
thf(fact_377_sublists_Osimps_I2_J,axiom,
! [X: a,Xs: list_a] :
( ( sublists_a @ ( cons_a @ X @ Xs ) )
= ( append_list_a @ ( sublists_a @ Xs ) @ ( map_list_a_list_a @ ( cons_a @ X ) @ ( prefixes_a @ Xs ) ) ) ) ).
% sublists.simps(2)
thf(fact_378_sublists_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( sublists_nat @ ( cons_nat @ X @ Xs ) )
= ( append_list_nat @ ( sublists_nat @ Xs ) @ ( map_li7225945977422193158st_nat @ ( cons_nat @ X ) @ ( prefixes_nat @ Xs ) ) ) ) ).
% sublists.simps(2)
thf(fact_379_suffixes__eq__snoc,axiom,
! [Ys: list_a,Xs: list_list_a,X: list_a] :
( ( ( suffixes_a @ Ys )
= ( append_list_a @ Xs @ ( cons_list_a @ X @ nil_list_a ) ) )
= ( ( ( ( Ys = nil_a )
& ( Xs = nil_list_a ) )
| ? [Z2: a,Zs3: list_a] :
( ( Ys
= ( cons_a @ Z2 @ Zs3 ) )
& ( Xs
= ( suffixes_a @ Zs3 ) ) ) )
& ( X = Ys ) ) ) ).
% suffixes_eq_snoc
thf(fact_380_suffixes__eq__snoc,axiom,
! [Ys: list_nat,Xs: list_list_nat,X: list_nat] :
( ( ( suffixes_nat @ Ys )
= ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) ) )
= ( ( ( ( Ys = nil_nat )
& ( Xs = nil_list_nat ) )
| ? [Z2: nat,Zs3: list_nat] :
( ( Ys
= ( cons_nat @ Z2 @ Zs3 ) )
& ( Xs
= ( suffixes_nat @ Zs3 ) ) ) )
& ( X = Ys ) ) ) ).
% suffixes_eq_snoc
thf(fact_381_sublists_Osimps_I1_J,axiom,
( ( sublists_a @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% sublists.simps(1)
thf(fact_382_sublists_Osimps_I1_J,axiom,
( ( sublists_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% sublists.simps(1)
thf(fact_383_concat__eq__append__conv,axiom,
! [Xss2: list_list_a,Ys: list_a,Zs: list_a] :
( ( ( concat_a @ Xss2 )
= ( append_a @ Ys @ Zs ) )
= ( ( ( Xss2 = nil_list_a )
=> ( ( Ys = nil_a )
& ( Zs = nil_a ) ) )
& ( ( Xss2 != nil_list_a )
=> ? [Xss1: list_list_a,Xs4: list_a,Xs6: list_a,Xss22: list_list_a] :
( ( Xss2
= ( append_list_a @ Xss1 @ ( cons_list_a @ ( append_a @ Xs4 @ Xs6 ) @ Xss22 ) ) )
& ( Ys
= ( append_a @ ( concat_a @ Xss1 ) @ Xs4 ) )
& ( Zs
= ( append_a @ Xs6 @ ( concat_a @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_384_concat__eq__append__conv,axiom,
! [Xss2: list_list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( concat_nat @ Xss2 )
= ( append_nat @ Ys @ Zs ) )
= ( ( ( Xss2 = nil_list_nat )
=> ( ( Ys = nil_nat )
& ( Zs = nil_nat ) ) )
& ( ( Xss2 != nil_list_nat )
=> ? [Xss1: list_list_nat,Xs4: list_nat,Xs6: list_nat,Xss22: list_list_nat] :
( ( Xss2
= ( append_list_nat @ Xss1 @ ( cons_list_nat @ ( append_nat @ Xs4 @ Xs6 ) @ Xss22 ) ) )
& ( Ys
= ( append_nat @ ( concat_nat @ Xss1 ) @ Xs4 ) )
& ( Zs
= ( append_nat @ Xs6 @ ( concat_nat @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_385_suffixes_Osimps_I2_J,axiom,
! [X: a,Xs: list_a] :
( ( suffixes_a @ ( cons_a @ X @ Xs ) )
= ( append_list_a @ ( suffixes_a @ Xs ) @ ( cons_list_a @ ( cons_a @ X @ Xs ) @ nil_list_a ) ) ) ).
% suffixes.simps(2)
thf(fact_386_suffixes_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( suffixes_nat @ ( cons_nat @ X @ Xs ) )
= ( append_list_nat @ ( suffixes_nat @ Xs ) @ ( cons_list_nat @ ( cons_nat @ X @ Xs ) @ nil_list_nat ) ) ) ).
% suffixes.simps(2)
thf(fact_387_gen__length__code_I2_J,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( gen_length_a @ N @ ( cons_a @ X @ Xs ) )
= ( gen_length_a @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_388_gen__length__code_I2_J,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( gen_length_nat @ N @ ( cons_nat @ X @ Xs ) )
= ( gen_length_nat @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_389_concat__eq__Nil__conv,axiom,
! [Xss2: list_list_a] :
( ( ( concat_a @ Xss2 )
= nil_a )
= ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xss2 ) )
=> ( X3 = nil_a ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_390_concat__eq__Nil__conv,axiom,
! [Xss2: list_list_nat] :
( ( ( concat_nat @ Xss2 )
= nil_nat )
= ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xss2 ) )
=> ( X3 = nil_nat ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_391_Nil__eq__concat__conv,axiom,
! [Xss2: list_list_a] :
( ( nil_a
= ( concat_a @ Xss2 ) )
= ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xss2 ) )
=> ( X3 = nil_a ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_392_Nil__eq__concat__conv,axiom,
! [Xss2: list_list_nat] :
( ( nil_nat
= ( concat_nat @ Xss2 ) )
= ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xss2 ) )
=> ( X3 = nil_nat ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_393_concat__append,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( concat_a @ ( append_list_a @ Xs @ Ys ) )
= ( append_a @ ( concat_a @ Xs ) @ ( concat_a @ Ys ) ) ) ).
% concat_append
thf(fact_394_concat__append,axiom,
! [Xs: list_list_nat,Ys: list_list_nat] :
( ( concat_nat @ ( append_list_nat @ Xs @ Ys ) )
= ( append_nat @ ( concat_nat @ Xs ) @ ( concat_nat @ Ys ) ) ) ).
% concat_append
thf(fact_395_length__suffixes,axiom,
! [Xs: list_a] :
( ( size_s349497388124573686list_a @ ( suffixes_a @ Xs ) )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ).
% length_suffixes
thf(fact_396_length__suffixes,axiom,
! [Xs: list_nat] :
( ( size_s3023201423986296836st_nat @ ( suffixes_nat @ Xs ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_suffixes
thf(fact_397_map__concat,axiom,
! [F: a > nat,Xs: list_list_a] :
( ( map_a_nat @ F @ ( concat_a @ Xs ) )
= ( concat_nat @ ( map_list_a_list_nat @ ( map_a_nat @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_398_concat_Osimps_I2_J,axiom,
! [X: list_a,Xs: list_list_a] :
( ( concat_a @ ( cons_list_a @ X @ Xs ) )
= ( append_a @ X @ ( concat_a @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_399_concat_Osimps_I2_J,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( concat_nat @ ( cons_list_nat @ X @ Xs ) )
= ( append_nat @ X @ ( concat_nat @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_400_concat_Osimps_I1_J,axiom,
( ( concat_a @ nil_list_a )
= nil_a ) ).
% concat.simps(1)
thf(fact_401_concat_Osimps_I1_J,axiom,
( ( concat_nat @ nil_list_nat )
= nil_nat ) ).
% concat.simps(1)
thf(fact_402_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_a @ N @ nil_a )
= N ) ).
% gen_length_code(1)
thf(fact_403_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_nat @ N @ nil_nat )
= N ) ).
% gen_length_code(1)
thf(fact_404_ShiftD,axiom,
! [Kl: list_a,Kl2: set_list_a,K: a] :
( ( member_list_a @ Kl @ ( bNF_Greatest_Shift_a @ Kl2 @ K ) )
=> ( member_list_a @ ( cons_a @ K @ Kl ) @ Kl2 ) ) ).
% ShiftD
thf(fact_405_ShiftD,axiom,
! [Kl: list_nat,Kl2: set_list_nat,K: nat] :
( ( member_list_nat @ Kl @ ( bNF_Gr1872714664788909425ft_nat @ Kl2 @ K ) )
=> ( member_list_nat @ ( cons_nat @ K @ Kl ) @ Kl2 ) ) ).
% ShiftD
thf(fact_406_suffixes_Osimps_I1_J,axiom,
( ( suffixes_a @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% suffixes.simps(1)
thf(fact_407_suffixes_Osimps_I1_J,axiom,
( ( suffixes_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% suffixes.simps(1)
thf(fact_408_concat__eq__appendD,axiom,
! [Xss2: list_list_a,Ys: list_a,Zs: list_a] :
( ( ( concat_a @ Xss2 )
= ( append_a @ Ys @ Zs ) )
=> ( ( Xss2 != nil_list_a )
=> ? [Xss12: list_list_a,Xs2: list_a,Xs3: list_a,Xss23: list_list_a] :
( ( Xss2
= ( append_list_a @ Xss12 @ ( cons_list_a @ ( append_a @ Xs2 @ Xs3 ) @ Xss23 ) ) )
& ( Ys
= ( append_a @ ( concat_a @ Xss12 ) @ Xs2 ) )
& ( Zs
= ( append_a @ Xs3 @ ( concat_a @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_409_concat__eq__appendD,axiom,
! [Xss2: list_list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( concat_nat @ Xss2 )
= ( append_nat @ Ys @ Zs ) )
=> ( ( Xss2 != nil_list_nat )
=> ? [Xss12: list_list_nat,Xs2: list_nat,Xs3: list_nat,Xss23: list_list_nat] :
( ( Xss2
= ( append_list_nat @ Xss12 @ ( cons_list_nat @ ( append_nat @ Xs2 @ Xs3 ) @ Xss23 ) ) )
& ( Ys
= ( append_nat @ ( concat_nat @ Xss12 ) @ Xs2 ) )
& ( Zs
= ( append_nat @ Xs3 @ ( concat_nat @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_410_card__set__suffixes,axiom,
! [Xs: list_a] :
( ( finite_card_list_a @ ( set_list_a2 @ ( suffixes_a @ Xs ) ) )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ).
% card_set_suffixes
thf(fact_411_card__set__suffixes,axiom,
! [Xs: list_nat] :
( ( finite_card_list_nat @ ( set_list_nat2 @ ( suffixes_nat @ Xs ) ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% card_set_suffixes
thf(fact_412_card__set__prefixes,axiom,
! [Xs: list_a] :
( ( finite_card_list_a @ ( set_list_a2 @ ( prefixes_a @ Xs ) ) )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ).
% card_set_prefixes
thf(fact_413_card__set__prefixes,axiom,
! [Xs: list_nat] :
( ( finite_card_list_nat @ ( set_list_nat2 @ ( prefixes_nat @ Xs ) ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% card_set_prefixes
thf(fact_414_List_Oset__insert,axiom,
! [X: a,Xs: list_a] :
( ( set_a2 @ ( insert_a @ X @ Xs ) )
= ( insert_a2 @ X @ ( set_a2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_415_List_Oset__insert,axiom,
! [X: nat,Xs: list_nat] :
( ( set_nat2 @ ( insert_nat @ X @ Xs ) )
= ( insert_nat2 @ X @ ( set_nat2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_416_maps__simps_I1_J,axiom,
! [F: a > list_a,X: a,Xs: list_a] :
( ( maps_a_a @ F @ ( cons_a @ X @ Xs ) )
= ( append_a @ ( F @ X ) @ ( maps_a_a @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_417_maps__simps_I1_J,axiom,
! [F: a > list_nat,X: a,Xs: list_a] :
( ( maps_a_nat @ F @ ( cons_a @ X @ Xs ) )
= ( append_nat @ ( F @ X ) @ ( maps_a_nat @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_418_maps__simps_I1_J,axiom,
! [F: nat > list_a,X: nat,Xs: list_nat] :
( ( maps_nat_a @ F @ ( cons_nat @ X @ Xs ) )
= ( append_a @ ( F @ X ) @ ( maps_nat_a @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_419_maps__simps_I1_J,axiom,
! [F: nat > list_nat,X: nat,Xs: list_nat] :
( ( maps_nat_nat @ F @ ( cons_nat @ X @ Xs ) )
= ( append_nat @ ( F @ X ) @ ( maps_nat_nat @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_420_not__in__set__insert,axiom,
! [X: a,Xs: list_a] :
( ~ ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ( insert_a @ X @ Xs )
= ( cons_a @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_421_not__in__set__insert,axiom,
! [X: nat,Xs: list_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X @ Xs )
= ( cons_nat @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_422_subseqs_Osimps_I1_J,axiom,
( ( subseqs_a @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% subseqs.simps(1)
thf(fact_423_subseqs_Osimps_I1_J,axiom,
( ( subseqs_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% subseqs.simps(1)
thf(fact_424_in__set__insert,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ( insert_a @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_425_in__set__insert,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_426_insert__Nil,axiom,
! [X: a] :
( ( insert_a @ X @ nil_a )
= ( cons_a @ X @ nil_a ) ) ).
% insert_Nil
thf(fact_427_insert__Nil,axiom,
! [X: nat] :
( ( insert_nat @ X @ nil_nat )
= ( cons_nat @ X @ nil_nat ) ) ).
% insert_Nil
thf(fact_428_maps__simps_I2_J,axiom,
! [F: a > list_a] :
( ( maps_a_a @ F @ nil_a )
= nil_a ) ).
% maps_simps(2)
thf(fact_429_maps__simps_I2_J,axiom,
! [F: a > list_nat] :
( ( maps_a_nat @ F @ nil_a )
= nil_nat ) ).
% maps_simps(2)
thf(fact_430_maps__simps_I2_J,axiom,
! [F: nat > list_a] :
( ( maps_nat_a @ F @ nil_nat )
= nil_a ) ).
% maps_simps(2)
thf(fact_431_maps__simps_I2_J,axiom,
! [F: nat > list_nat] :
( ( maps_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% maps_simps(2)
thf(fact_432_Cons__in__subseqsD,axiom,
! [Y: a,Ys: list_a,Xs: list_a] :
( ( member_list_a @ ( cons_a @ Y @ Ys ) @ ( set_list_a2 @ ( subseqs_a @ Xs ) ) )
=> ( member_list_a @ Ys @ ( set_list_a2 @ ( subseqs_a @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_433_Cons__in__subseqsD,axiom,
! [Y: nat,Ys: list_nat,Xs: list_nat] :
( ( member_list_nat @ ( cons_nat @ Y @ Ys ) @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) )
=> ( member_list_nat @ Ys @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_434_List_Oinsert__def,axiom,
( insert_a
= ( ^ [X3: a,Xs4: list_a] : ( if_list_a @ ( member_a @ X3 @ ( set_a2 @ Xs4 ) ) @ Xs4 @ ( cons_a @ X3 @ Xs4 ) ) ) ) ).
% List.insert_def
thf(fact_435_List_Oinsert__def,axiom,
( insert_nat
= ( ^ [X3: nat,Xs4: list_nat] : ( if_list_nat @ ( member_nat @ X3 @ ( set_nat2 @ Xs4 ) ) @ Xs4 @ ( cons_nat @ X3 @ Xs4 ) ) ) ) ).
% List.insert_def
thf(fact_436_card__image,axiom,
! [F: product_unit > product_unit,A: set_Product_unit] :
( ( inj_on8151373323710067377t_unit @ F @ A )
=> ( ( finite410649719033368117t_unit @ ( image_405062704495631173t_unit @ F @ A ) )
= ( finite410649719033368117t_unit @ A ) ) ) ).
% card_image
thf(fact_437_card__image,axiom,
! [F: nat > product_unit,A: set_nat] :
( ( inj_on7061601236592826506t_unit @ F @ A )
=> ( ( finite410649719033368117t_unit @ ( image_8730104196221521654t_unit @ F @ A ) )
= ( finite_card_nat @ A ) ) ) ).
% card_image
thf(fact_438_card__image,axiom,
! [F: a > nat,A: set_a] :
( ( inj_on_a_nat @ F @ A )
=> ( ( finite_card_nat @ ( image_a_nat @ F @ A ) )
= ( finite_card_a @ A ) ) ) ).
% card_image
thf(fact_439_card__image,axiom,
! [F: product_unit > nat,A: set_Product_unit] :
( ( inj_on8430439091780834860it_nat @ F @ A )
=> ( ( finite_card_nat @ ( image_875570014554754200it_nat @ F @ A ) )
= ( finite410649719033368117t_unit @ A ) ) ) ).
% card_image
thf(fact_440_card__image,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( finite_card_nat @ ( image_nat_nat @ F @ A ) )
= ( finite_card_nat @ A ) ) ) ).
% card_image
thf(fact_441_map__eq__map__tailrec,axiom,
map_a_nat = map_tailrec_a_nat ).
% map_eq_map_tailrec
thf(fact_442_remove__code_I1_J,axiom,
! [X: a,Xs: list_a] :
( ( remove_a @ X @ ( set_a2 @ Xs ) )
= ( set_a2 @ ( removeAll_a @ X @ Xs ) ) ) ).
% remove_code(1)
thf(fact_443_remove__code_I1_J,axiom,
! [X: nat,Xs: list_nat] :
( ( remove_nat @ X @ ( set_nat2 @ Xs ) )
= ( set_nat2 @ ( removeAll_nat @ X @ Xs ) ) ) ).
% remove_code(1)
thf(fact_444_inj__on__mapI,axiom,
! [F: a > nat,A: set_list_a] :
( ( inj_on_a_nat @ F @ ( comple2307003609928055243_set_a @ ( image_list_a_set_a @ set_a2 @ A ) ) )
=> ( inj_on6731145966573583411st_nat @ ( map_a_nat @ F ) @ A ) ) ).
% inj_on_mapI
thf(fact_445_inj__on__mapI,axiom,
! [F: nat > nat,A: set_list_nat] :
( ( inj_on_nat_nat @ F @ ( comple7399068483239264473et_nat @ ( image_1775855109352712557et_nat @ set_nat2 @ A ) ) )
=> ( inj_on3049792774292151987st_nat @ ( map_nat_nat @ F ) @ A ) ) ).
% inj_on_mapI
thf(fact_446_length__n__lists__elem,axiom,
! [Ys: list_a,N: nat,Xs: list_a] :
( ( member_list_a @ Ys @ ( set_list_a2 @ ( n_lists_a @ N @ Xs ) ) )
=> ( ( size_size_list_a @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_447_length__n__lists__elem,axiom,
! [Ys: list_nat,N: nat,Xs: list_nat] :
( ( member_list_nat @ Ys @ ( set_list_nat2 @ ( n_lists_nat @ N @ Xs ) ) )
=> ( ( size_size_list_nat @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_448_list__ex1__simps_I1_J,axiom,
! [P: a > $o] :
~ ( list_ex1_a @ P @ nil_a ) ).
% list_ex1_simps(1)
thf(fact_449_list__ex1__simps_I1_J,axiom,
! [P: nat > $o] :
~ ( list_ex1_nat @ P @ nil_nat ) ).
% list_ex1_simps(1)
thf(fact_450_butlast__snoc,axiom,
! [Xs: list_a,X: a] :
( ( butlast_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_451_butlast__snoc,axiom,
! [Xs: list_nat,X: nat] :
( ( butlast_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_452_last__snoc,axiom,
! [Xs: list_a,X: a] :
( ( last_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= X ) ).
% last_snoc
thf(fact_453_last__snoc,axiom,
! [Xs: list_nat,X: nat] :
( ( last_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= X ) ).
% last_snoc
thf(fact_454_member__remove,axiom,
! [X: a,Y: a,A: set_a] :
( ( member_a @ X @ ( remove_a @ Y @ A ) )
= ( ( member_a @ X @ A )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_455_member__remove,axiom,
! [X: nat,Y: nat,A: set_nat] :
( ( member_nat @ X @ ( remove_nat @ Y @ A ) )
= ( ( member_nat @ X @ A )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_456_last__appendR,axiom,
! [Ys: list_a,Xs: list_a] :
( ( Ys != nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Ys ) ) ) ).
% last_appendR
thf(fact_457_last__appendR,axiom,
! [Ys: list_nat,Xs: list_nat] :
( ( Ys != nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
= ( last_nat @ Ys ) ) ) ).
% last_appendR
thf(fact_458_last__appendL,axiom,
! [Ys: list_a,Xs: list_a] :
( ( Ys = nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Xs ) ) ) ).
% last_appendL
thf(fact_459_last__appendL,axiom,
! [Ys: list_nat,Xs: list_nat] :
( ( Ys = nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
= ( last_nat @ Xs ) ) ) ).
% last_appendL
thf(fact_460_append__butlast__last__id,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( append_a @ ( butlast_a @ Xs ) @ ( cons_a @ ( last_a @ Xs ) @ nil_a ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_461_append__butlast__last__id,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( append_nat @ ( butlast_nat @ Xs ) @ ( cons_nat @ ( last_nat @ Xs ) @ nil_nat ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_462_butlast_Osimps_I1_J,axiom,
( ( butlast_a @ nil_a )
= nil_a ) ).
% butlast.simps(1)
thf(fact_463_butlast_Osimps_I1_J,axiom,
( ( butlast_nat @ nil_nat )
= nil_nat ) ).
% butlast.simps(1)
thf(fact_464_in__set__butlastD,axiom,
! [X: a,Xs: list_a] :
( ( member_a @ X @ ( set_a2 @ ( butlast_a @ Xs ) ) )
=> ( member_a @ X @ ( set_a2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_465_in__set__butlastD,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ Xs ) ) )
=> ( member_nat @ X @ ( set_nat2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_466_map__butlast,axiom,
! [F: a > nat,Xs: list_a] :
( ( map_a_nat @ F @ ( butlast_a @ Xs ) )
= ( butlast_nat @ ( map_a_nat @ F @ Xs ) ) ) ).
% map_butlast
thf(fact_467_snoc__eq__iff__butlast,axiom,
! [Xs: list_a,X: a,Ys: list_a] :
( ( ( append_a @ Xs @ ( cons_a @ X @ nil_a ) )
= Ys )
= ( ( Ys != nil_a )
& ( ( butlast_a @ Ys )
= Xs )
& ( ( last_a @ Ys )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_468_snoc__eq__iff__butlast,axiom,
! [Xs: list_nat,X: nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) )
= Ys )
= ( ( Ys != nil_nat )
& ( ( butlast_nat @ Ys )
= Xs )
& ( ( last_nat @ Ys )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_469_last__ConsR,axiom,
! [Xs: list_a,X: a] :
( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= ( last_a @ Xs ) ) ) ).
% last_ConsR
thf(fact_470_last__ConsR,axiom,
! [Xs: list_nat,X: nat] :
( ( Xs != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs ) )
= ( last_nat @ Xs ) ) ) ).
% last_ConsR
thf(fact_471_last__ConsL,axiom,
! [Xs: list_a,X: a] :
( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_472_last__ConsL,axiom,
! [Xs: list_nat,X: nat] :
( ( Xs = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_473_last_Osimps,axiom,
! [Xs: list_a,X: a] :
( ( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= ( last_a @ Xs ) ) ) ) ).
% last.simps
thf(fact_474_last_Osimps,axiom,
! [Xs: list_nat,X: nat] :
( ( ( Xs = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs ) )
= ( last_nat @ Xs ) ) ) ) ).
% last.simps
thf(fact_475_last__in__set,axiom,
! [As: list_a] :
( ( As != nil_a )
=> ( member_a @ ( last_a @ As ) @ ( set_a2 @ As ) ) ) ).
% last_in_set
thf(fact_476_last__in__set,axiom,
! [As: list_nat] :
( ( As != nil_nat )
=> ( member_nat @ ( last_nat @ As ) @ ( set_nat2 @ As ) ) ) ).
% last_in_set
thf(fact_477_longest__common__suffix,axiom,
! [Xs: list_a,Ys: list_a] :
? [Ss: list_a,Xs3: list_a,Ys5: list_a] :
( ( Xs
= ( append_a @ Xs3 @ Ss ) )
& ( Ys
= ( append_a @ Ys5 @ Ss ) )
& ( ( Xs3 = nil_a )
| ( Ys5 = nil_a )
| ( ( last_a @ Xs3 )
!= ( last_a @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_478_longest__common__suffix,axiom,
! [Xs: list_nat,Ys: list_nat] :
? [Ss: list_nat,Xs3: list_nat,Ys5: list_nat] :
( ( Xs
= ( append_nat @ Xs3 @ Ss ) )
& ( Ys
= ( append_nat @ Ys5 @ Ss ) )
& ( ( Xs3 = nil_nat )
| ( Ys5 = nil_nat )
| ( ( last_nat @ Xs3 )
!= ( last_nat @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_479_last__append,axiom,
! [Ys: list_a,Xs: list_a] :
( ( ( Ys = nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Xs ) ) )
& ( ( Ys != nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Ys ) ) ) ) ).
% last_append
thf(fact_480_last__append,axiom,
! [Ys: list_nat,Xs: list_nat] :
( ( ( Ys = nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
= ( last_nat @ Xs ) ) )
& ( ( Ys != nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
= ( last_nat @ Ys ) ) ) ) ).
% last_append
thf(fact_481_last__map,axiom,
! [Xs: list_a,F: a > nat] :
( ( Xs != nil_a )
=> ( ( last_nat @ ( map_a_nat @ F @ Xs ) )
= ( F @ ( last_a @ Xs ) ) ) ) ).
% last_map
thf(fact_482_butlast_Osimps_I2_J,axiom,
! [Xs: list_a,X: a] :
( ( ( Xs = nil_a )
=> ( ( butlast_a @ ( cons_a @ X @ Xs ) )
= nil_a ) )
& ( ( Xs != nil_a )
=> ( ( butlast_a @ ( cons_a @ X @ Xs ) )
= ( cons_a @ X @ ( butlast_a @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_483_butlast_Osimps_I2_J,axiom,
! [Xs: list_nat,X: nat] :
( ( ( Xs = nil_nat )
=> ( ( butlast_nat @ ( cons_nat @ X @ Xs ) )
= nil_nat ) )
& ( ( Xs != nil_nat )
=> ( ( butlast_nat @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ ( butlast_nat @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_484_butlast__append,axiom,
! [Ys: list_a,Xs: list_a] :
( ( ( Ys = nil_a )
=> ( ( butlast_a @ ( append_a @ Xs @ Ys ) )
= ( butlast_a @ Xs ) ) )
& ( ( Ys != nil_a )
=> ( ( butlast_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ Xs @ ( butlast_a @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_485_butlast__append,axiom,
! [Ys: list_nat,Xs: list_nat] :
( ( ( Ys = nil_nat )
=> ( ( butlast_nat @ ( append_nat @ Xs @ Ys ) )
= ( butlast_nat @ Xs ) ) )
& ( ( Ys != nil_nat )
=> ( ( butlast_nat @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ Xs @ ( butlast_nat @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_486_in__set__butlast__appendI,axiom,
! [X: a,Xs: list_a,Ys: list_a] :
( ( ( member_a @ X @ ( set_a2 @ ( butlast_a @ Xs ) ) )
| ( member_a @ X @ ( set_a2 @ ( butlast_a @ Ys ) ) ) )
=> ( member_a @ X @ ( set_a2 @ ( butlast_a @ ( append_a @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_487_in__set__butlast__appendI,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat] :
( ( ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ Xs ) ) )
| ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ Ys ) ) ) )
=> ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ ( append_nat @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_488_list__ex1__iff,axiom,
( list_ex1_a
= ( ^ [P2: a > $o,Xs4: list_a] :
? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs4 ) )
& ( P2 @ X3 )
& ! [Y3: a] :
( ( ( member_a @ Y3 @ ( set_a2 @ Xs4 ) )
& ( P2 @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_489_list__ex1__iff,axiom,
( list_ex1_nat
= ( ^ [P2: nat > $o,Xs4: list_nat] :
? [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs4 ) )
& ( P2 @ X3 )
& ! [Y3: nat] :
( ( ( member_nat @ Y3 @ ( set_nat2 @ Xs4 ) )
& ( P2 @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_490_inj__on__image,axiom,
! [F: a > nat,A: set_set_a] :
( ( inj_on_a_nat @ F @ ( comple2307003609928055243_set_a @ A ) )
=> ( inj_on_set_a_set_nat @ ( image_a_nat @ F ) @ A ) ) ).
% inj_on_image
thf(fact_491_inj__on__image,axiom,
! [F: nat > nat,A: set_set_nat] :
( ( inj_on_nat_nat @ F @ ( comple7399068483239264473et_nat @ A ) )
=> ( inj_on4604407203859583615et_nat @ ( image_nat_nat @ F ) @ A ) ) ).
% inj_on_image
thf(fact_492_can__select__set__list__ex1,axiom,
! [P: a > $o,A: list_a] :
( ( can_select_a @ P @ ( set_a2 @ A ) )
= ( list_ex1_a @ P @ A ) ) ).
% can_select_set_list_ex1
thf(fact_493_can__select__set__list__ex1,axiom,
! [P: nat > $o,A: list_nat] :
( ( can_select_nat @ P @ ( set_nat2 @ A ) )
= ( list_ex1_nat @ P @ A ) ) ).
% can_select_set_list_ex1
thf(fact_494_SUP__cong,axiom,
! [A: set_a,B2: set_a,C3: a > nat,D: a > nat] :
( ( A = B2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B2 )
=> ( ( C3 @ X2 )
= ( D @ X2 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_a_nat @ C3 @ A ) )
= ( complete_Sup_Sup_nat @ ( image_a_nat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_495_SUP__cong,axiom,
! [A: set_nat,B2: set_nat,C3: nat > nat,D: nat > nat] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C3 @ X2 )
= ( D @ X2 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ C3 @ A ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).
% SUP_cong
thf(fact_496_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_a @ N @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_a @ N @ nil_a )
= nil_list_a ) ) ) ).
% n_lists_Nil
thf(fact_497_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_nat @ N @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_nat @ N @ nil_nat )
= nil_list_nat ) ) ) ).
% n_lists_Nil
thf(fact_498_subseqs__powset,axiom,
! [Xs: list_a] :
( ( image_list_a_set_a @ set_a2 @ ( set_list_a2 @ ( subseqs_a @ Xs ) ) )
= ( pow_a @ ( set_a2 @ Xs ) ) ) ).
% subseqs_powset
thf(fact_499_subseqs__powset,axiom,
! [Xs: list_nat] :
( ( image_1775855109352712557et_nat @ set_nat2 @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) )
= ( pow_nat @ ( set_nat2 @ Xs ) ) ) ).
% subseqs_powset
thf(fact_500_subset__subseqs,axiom,
! [X6: set_a,Xs: list_a] :
( ( ord_less_eq_set_a @ X6 @ ( set_a2 @ Xs ) )
=> ( member_set_a @ X6 @ ( image_list_a_set_a @ set_a2 @ ( set_list_a2 @ ( subseqs_a @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_501_subset__subseqs,axiom,
! [X6: set_nat,Xs: list_nat] :
( ( ord_less_eq_set_nat @ X6 @ ( set_nat2 @ Xs ) )
=> ( member_set_nat @ X6 @ ( image_1775855109352712557et_nat @ set_nat2 @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_502_n__lists_Osimps_I1_J,axiom,
! [Xs: list_a] :
( ( n_lists_a @ zero_zero_nat @ Xs )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% n_lists.simps(1)
thf(fact_503_n__lists_Osimps_I1_J,axiom,
! [Xs: list_nat] :
( ( n_lists_nat @ zero_zero_nat @ Xs )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% n_lists.simps(1)
thf(fact_504_subsetI,axiom,
! [A: set_a,B2: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ X2 @ B2 ) )
=> ( ord_less_eq_set_a @ A @ B2 ) ) ).
% subsetI
thf(fact_505_subsetI,axiom,
! [A: set_nat,B2: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B2 ) )
=> ( ord_less_eq_set_nat @ A @ B2 ) ) ).
% subsetI
thf(fact_506_insert__subset,axiom,
! [X: a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( insert_a2 @ X @ A ) @ B2 )
= ( ( member_a @ X @ B2 )
& ( ord_less_eq_set_a @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_507_insert__subset,axiom,
! [X: nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat2 @ X @ A ) @ B2 )
= ( ( member_nat @ X @ B2 )
& ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_508_length__0__conv,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_a ) ) ).
% length_0_conv
thf(fact_509_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_510_image__Pow__mono,axiom,
! [F: a > nat,A: set_a,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_set_a_set_nat @ ( image_a_nat @ F ) @ ( pow_a @ A ) ) @ ( pow_nat @ B2 ) ) ) ).
% image_Pow_mono
thf(fact_511_image__Pow__mono,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( pow_nat @ A ) ) @ ( pow_nat @ B2 ) ) ) ).
% image_Pow_mono
thf(fact_512_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_513_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_514_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B4: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A2 )
=> ( member_a @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_515_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B4: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A2 )
=> ( member_nat @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_516_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B4: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_517_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B4: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_518_subsetD,axiom,
! [A: set_a,B2: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_519_subsetD,axiom,
! [A: set_nat,B2: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_520_in__mono,axiom,
! [A: set_a,B2: set_a,X: a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( member_a @ X @ A )
=> ( member_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_521_in__mono,axiom,
! [A: set_nat,B2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_522_can__select__def,axiom,
( can_select_a
= ( ^ [P2: a > $o,A2: set_a] :
? [X3: a] :
( ( member_a @ X3 @ A2 )
& ( P2 @ X3 )
& ! [Y3: a] :
( ( ( member_a @ Y3 @ A2 )
& ( P2 @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% can_select_def
thf(fact_523_can__select__def,axiom,
( can_select_nat
= ( ^ [P2: nat > $o,A2: set_nat] :
? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P2 @ X3 )
& ! [Y3: nat] :
( ( ( member_nat @ Y3 @ A2 )
& ( P2 @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ).
% can_select_def
thf(fact_524_image__mono,axiom,
! [A: set_a,B2: set_a,F: a > nat] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ ( image_a_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_525_image__mono,axiom,
! [A: set_nat,B2: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_mono
thf(fact_526_image__subsetI,axiom,
! [A: set_a,F: a > a,B2: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_527_image__subsetI,axiom,
! [A: set_a,F: a > nat,B2: set_nat] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_528_image__subsetI,axiom,
! [A: set_nat,F: nat > a,B2: set_a] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_a @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_529_image__subsetI,axiom,
! [A: set_nat,F: nat > nat,B2: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B2 ) ) ).
% image_subsetI
thf(fact_530_subset__imageE,axiom,
! [B2: set_nat,F: a > nat,A: set_a] :
( ( ord_less_eq_set_nat @ B2 @ ( image_a_nat @ F @ A ) )
=> ~ ! [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A )
=> ( B2
!= ( image_a_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_531_subset__imageE,axiom,
! [B2: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
=> ~ ! [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
=> ( B2
!= ( image_nat_nat @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_532_image__subset__iff,axiom,
! [F: a > nat,A: set_a,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ B2 )
= ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_533_image__subset__iff,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B2 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_534_subset__image__iff,axiom,
! [B2: set_nat,F: a > nat,A: set_a] :
( ( ord_less_eq_set_nat @ B2 @ ( image_a_nat @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B2
= ( image_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_535_subset__image__iff,axiom,
! [B2: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B2
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_536_all__subset__image,axiom,
! [F: a > nat,A: set_a,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_a_nat @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A )
=> ( P @ ( image_a_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_537_all__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A )
=> ( P @ ( image_nat_nat @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_538_insert__mono,axiom,
! [C3: set_a,D: set_a,A4: a] :
( ( ord_less_eq_set_a @ C3 @ D )
=> ( ord_less_eq_set_a @ ( insert_a2 @ A4 @ C3 ) @ ( insert_a2 @ A4 @ D ) ) ) ).
% insert_mono
thf(fact_539_insert__mono,axiom,
! [C3: set_nat,D: set_nat,A4: nat] :
( ( ord_less_eq_set_nat @ C3 @ D )
=> ( ord_less_eq_set_nat @ ( insert_nat2 @ A4 @ C3 ) @ ( insert_nat2 @ A4 @ D ) ) ) ).
% insert_mono
thf(fact_540_subset__insert,axiom,
! [X: a,A: set_a,B2: set_a] :
( ~ ( member_a @ X @ A )
=> ( ( ord_less_eq_set_a @ A @ ( insert_a2 @ X @ B2 ) )
= ( ord_less_eq_set_a @ A @ B2 ) ) ) ).
% subset_insert
thf(fact_541_subset__insert,axiom,
! [X: nat,A: set_nat,B2: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ X @ B2 ) )
= ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% subset_insert
thf(fact_542_subset__insertI,axiom,
! [B2: set_a,A4: a] : ( ord_less_eq_set_a @ B2 @ ( insert_a2 @ A4 @ B2 ) ) ).
% subset_insertI
thf(fact_543_subset__insertI,axiom,
! [B2: set_nat,A4: nat] : ( ord_less_eq_set_nat @ B2 @ ( insert_nat2 @ A4 @ B2 ) ) ).
% subset_insertI
thf(fact_544_subset__insertI2,axiom,
! [A: set_a,B2: set_a,B: a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ord_less_eq_set_a @ A @ ( insert_a2 @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_545_subset__insertI2,axiom,
! [A: set_nat,B2: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_546_image__Pow__surj,axiom,
! [F: a > nat,A: set_a,B2: set_nat] :
( ( ( image_a_nat @ F @ A )
= B2 )
=> ( ( image_set_a_set_nat @ ( image_a_nat @ F ) @ ( pow_a @ A ) )
= ( pow_nat @ B2 ) ) ) ).
% image_Pow_surj
thf(fact_547_image__Pow__surj,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( ( image_nat_nat @ F @ A )
= B2 )
=> ( ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( pow_nat @ A ) )
= ( pow_nat @ B2 ) ) ) ).
% image_Pow_surj
thf(fact_548_subset__code_I1_J,axiom,
! [Xs: list_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B2 )
= ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( member_a @ X3 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_549_subset__code_I1_J,axiom,
! [Xs: list_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B2 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X3 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_550_subset__inj__on,axiom,
! [F: a > nat,B2: set_a,A: set_a] :
( ( inj_on_a_nat @ F @ B2 )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( inj_on_a_nat @ F @ A ) ) ) ).
% subset_inj_on
thf(fact_551_subset__inj__on,axiom,
! [F: nat > nat,B2: set_nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( inj_on_nat_nat @ F @ A ) ) ) ).
% subset_inj_on
thf(fact_552_inj__on__subset,axiom,
! [F: a > nat,A: set_a,B2: set_a] :
( ( inj_on_a_nat @ F @ A )
=> ( ( ord_less_eq_set_a @ B2 @ A )
=> ( inj_on_a_nat @ F @ B2 ) ) ) ).
% inj_on_subset
thf(fact_553_inj__on__subset,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( inj_on_nat_nat @ F @ B2 ) ) ) ).
% inj_on_subset
thf(fact_554_enum__rgfs_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N4: nat] :
( X
!= ( suc @ N4 ) ) ) ).
% enum_rgfs.cases
thf(fact_555_nat_Odistinct_I1_J,axiom,
! [X24: nat] :
( zero_zero_nat
!= ( suc @ X24 ) ) ).
% nat.distinct(1)
thf(fact_556_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_557_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_558_nat_OdiscI,axiom,
! [Nat: nat,X24: nat] :
( ( Nat
= ( suc @ X24 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_559_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_560_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_561_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N: nat] :
( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X2: nat,Y2: nat] :
( ( P @ X2 @ Y2 )
=> ( P @ ( suc @ X2 ) @ ( suc @ Y2 ) ) )
=> ( P @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_562_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_563_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_564_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_565_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_566_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_567_rgf__limit_Osimps_I1_J,axiom,
( ( equiva5889994315859557365_limit @ nil_nat )
= zero_zero_nat ) ).
% rgf_limit.simps(1)
thf(fact_568_set__subset__Cons,axiom,
! [Xs: list_a,X: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ ( cons_a @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_569_set__subset__Cons,axiom,
! [Xs: list_nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_570_inj__on__image__eq__iff,axiom,
! [F: a > nat,C3: set_a,A: set_a,B2: set_a] :
( ( inj_on_a_nat @ F @ C3 )
=> ( ( ord_less_eq_set_a @ A @ C3 )
=> ( ( ord_less_eq_set_a @ B2 @ C3 )
=> ( ( ( image_a_nat @ F @ A )
= ( image_a_nat @ F @ B2 ) )
= ( A = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_571_inj__on__image__eq__iff,axiom,
! [F: nat > nat,C3: set_nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ C3 )
=> ( ( ord_less_eq_set_nat @ A @ C3 )
=> ( ( ord_less_eq_set_nat @ B2 @ C3 )
=> ( ( ( image_nat_nat @ F @ A )
= ( image_nat_nat @ F @ B2 ) )
= ( A = B2 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_572_inj__on__image__mem__iff,axiom,
! [F: a > a,B2: set_a,A4: a,A: set_a] :
( ( inj_on_a_a @ F @ B2 )
=> ( ( member_a @ A4 @ B2 )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( member_a @ ( F @ A4 ) @ ( image_a_a @ F @ A ) )
= ( member_a @ A4 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_573_inj__on__image__mem__iff,axiom,
! [F: a > nat,B2: set_a,A4: a,A: set_a] :
( ( inj_on_a_nat @ F @ B2 )
=> ( ( member_a @ A4 @ B2 )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( member_nat @ ( F @ A4 ) @ ( image_a_nat @ F @ A ) )
= ( member_a @ A4 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_574_inj__on__image__mem__iff,axiom,
! [F: nat > a,B2: set_nat,A4: nat,A: set_nat] :
( ( inj_on_nat_a @ F @ B2 )
=> ( ( member_nat @ A4 @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_a @ ( F @ A4 ) @ ( image_nat_a @ F @ A ) )
= ( member_nat @ A4 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_575_inj__on__image__mem__iff,axiom,
! [F: nat > nat,B2: set_nat,A4: nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ B2 )
=> ( ( member_nat @ A4 @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat @ ( F @ A4 ) @ ( image_nat_nat @ F @ A ) )
= ( member_nat @ A4 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_576_inj__on__image__Pow,axiom,
! [F: a > nat,A: set_a] :
( ( inj_on_a_nat @ F @ A )
=> ( inj_on_set_a_set_nat @ ( image_a_nat @ F ) @ ( pow_a @ A ) ) ) ).
% inj_on_image_Pow
thf(fact_577_inj__on__image__Pow,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( inj_on4604407203859583615et_nat @ ( image_nat_nat @ F ) @ ( pow_nat @ A ) ) ) ).
% inj_on_image_Pow
thf(fact_578_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_579_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_580_length__code,axiom,
( size_size_list_a
= ( gen_length_a @ zero_zero_nat ) ) ).
% length_code
thf(fact_581_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_582_Inf_OINF__cong,axiom,
! [A: set_a,B2: set_a,C3: a > nat,D: a > nat,Inf: set_nat > nat] :
( ( A = B2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B2 )
=> ( ( C3 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_a_nat @ C3 @ A ) )
= ( Inf @ ( image_a_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_583_Inf_OINF__cong,axiom,
! [A: set_nat,B2: set_nat,C3: nat > nat,D: nat > nat,Inf: set_nat > nat] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C3 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Inf @ ( image_nat_nat @ C3 @ A ) )
= ( Inf @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_584_Sup_OSUP__cong,axiom,
! [A: set_a,B2: set_a,C3: a > nat,D: a > nat,Sup: set_nat > nat] :
( ( A = B2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B2 )
=> ( ( C3 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_a_nat @ C3 @ A ) )
= ( Sup @ ( image_a_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_585_Sup_OSUP__cong,axiom,
! [A: set_nat,B2: set_nat,C3: nat > nat,D: nat > nat,Sup: set_nat > nat] :
( ( A = B2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( C3 @ X2 )
= ( D @ X2 ) ) )
=> ( ( Sup @ ( image_nat_nat @ C3 @ A ) )
= ( Sup @ ( image_nat_nat @ D @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_586_inj__on__impl__inj__on__image,axiom,
! [F: a > nat,A: set_a,X6: set_set_a] :
( ( inj_on_a_nat @ F @ A )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ X6 )
=> ( ord_less_eq_set_a @ X2 @ A ) )
=> ( inj_on_set_a_set_nat @ ( image_a_nat @ F ) @ X6 ) ) ) ).
% inj_on_impl_inj_on_image
thf(fact_587_inj__on__impl__inj__on__image,axiom,
! [F: nat > nat,A: set_nat,X6: set_set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ X6 )
=> ( ord_less_eq_set_nat @ X2 @ A ) )
=> ( inj_on4604407203859583615et_nat @ ( image_nat_nat @ F ) @ X6 ) ) ) ).
% inj_on_impl_inj_on_image
thf(fact_588_subset__image__inj,axiom,
! [S: set_nat,F: a > nat,T3: set_a] :
( ( ord_less_eq_set_nat @ S @ ( image_a_nat @ F @ T3 ) )
= ( ? [U: set_a] :
( ( ord_less_eq_set_a @ U @ T3 )
& ( inj_on_a_nat @ F @ U )
& ( S
= ( image_a_nat @ F @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_589_subset__image__inj,axiom,
! [S: set_nat,F: nat > nat,T3: set_nat] :
( ( ord_less_eq_set_nat @ S @ ( image_nat_nat @ F @ T3 ) )
= ( ? [U: set_nat] :
( ( ord_less_eq_set_nat @ U @ T3 )
& ( inj_on_nat_nat @ F @ U )
& ( S
= ( image_nat_nat @ F @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_590_insert__subsetI,axiom,
! [X: a,A: set_a,X6: set_a] :
( ( member_a @ X @ A )
=> ( ( ord_less_eq_set_a @ X6 @ A )
=> ( ord_less_eq_set_a @ ( insert_a2 @ X @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_591_insert__subsetI,axiom,
! [X: nat,A: set_nat,X6: set_nat] :
( ( member_nat @ X @ A )
=> ( ( ord_less_eq_set_nat @ X6 @ A )
=> ( ord_less_eq_set_nat @ ( insert_nat2 @ X @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_592_zero__notin__Suc__image,axiom,
! [A: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A ) ) ).
% zero_notin_Suc_image
thf(fact_593_inj__on__image__Fpow,axiom,
! [F: a > nat,A: set_a] :
( ( inj_on_a_nat @ F @ A )
=> ( inj_on_set_a_set_nat @ ( image_a_nat @ F ) @ ( finite_Fpow_a @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_594_inj__on__image__Fpow,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( inj_on4604407203859583615et_nat @ ( image_nat_nat @ F ) @ ( finite_Fpow_nat @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_595_Suc__le__mono,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ).
% Suc_le_mono
thf(fact_596_bot__nat__0_Oextremum,axiom,
! [A4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A4 ) ).
% bot_nat_0.extremum
thf(fact_597_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_598_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_599_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_600_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_601_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_602_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_603_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_604_transitive__stepwise__le,axiom,
! [M2: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ! [X2: nat] : ( R @ X2 @ X2 )
=> ( ! [X2: nat,Y2: nat,Z: nat] :
( ( R @ X2 @ Y2 )
=> ( ( R @ Y2 @ Z )
=> ( R @ X2 @ Z ) ) )
=> ( ! [N4: nat] : ( R @ N4 @ ( suc @ N4 ) )
=> ( R @ M2 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_605_nat__induct__at__least,axiom,
! [M2: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( P @ M2 )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_606_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ! [M4: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N4 )
=> ( P @ M4 ) )
=> ( P @ N4 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_607_not__less__eq__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_608_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_609_le__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M2 @ N )
| ( M2
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_610_Suc__le__D,axiom,
! [N: nat,M5: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M5 )
=> ? [M3: nat] :
( M5
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_611_le__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_612_le__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N )
=> ( M2
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_613_Suc__leD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_leD
thf(fact_614_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_615_bot__nat__0_Oextremum__unique,axiom,
! [A4: nat] :
( ( ord_less_eq_nat @ A4 @ zero_zero_nat )
= ( A4 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_616_bot__nat__0_Oextremum__uniqueI,axiom,
! [A4: nat] :
( ( ord_less_eq_nat @ A4 @ zero_zero_nat )
=> ( A4 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_617_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_618_impossible__Cons,axiom,
! [Xs: list_a,Ys: list_a,X: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) )
=> ( Xs
!= ( cons_a @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_619_impossible__Cons,axiom,
! [Xs: list_nat,Ys: list_nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) )
=> ( Xs
!= ( cons_nat @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_620_card__insert__le,axiom,
! [A: set_a,X: a] : ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ ( insert_a2 @ X @ A ) ) ) ).
% card_insert_le
thf(fact_621_card__insert__le,axiom,
! [A: set_Product_unit,X: product_unit] : ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ ( insert_Product_unit @ X @ A ) ) ) ).
% card_insert_le
thf(fact_622_card__insert__le,axiom,
! [A: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ ( insert_nat2 @ X @ A ) ) ) ).
% card_insert_le
thf(fact_623_length__removeAll__less__eq,axiom,
! [X: a,Xs: list_a] : ( ord_less_eq_nat @ ( size_size_list_a @ ( removeAll_a @ X @ Xs ) ) @ ( size_size_list_a @ Xs ) ) ).
% length_removeAll_less_eq
thf(fact_624_length__removeAll__less__eq,axiom,
! [X: nat,Xs: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( removeAll_nat @ X @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% length_removeAll_less_eq
thf(fact_625_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_a @ Xs ) )
= ( ? [X3: a,Ys3: list_a] :
( ( Xs
= ( cons_a @ X3 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_a @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_626_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
= ( ? [X3: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ X3 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_627_card__length,axiom,
! [Xs: list_Product_unit] : ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( set_Product_unit2 @ Xs ) ) @ ( size_s245203480648594047t_unit @ Xs ) ) ).
% card_length
thf(fact_628_card__length,axiom,
! [Xs: list_a] : ( ord_less_eq_nat @ ( finite_card_a @ ( set_a2 @ Xs ) ) @ ( size_size_list_a @ Xs ) ) ).
% card_length
thf(fact_629_card__length,axiom,
! [Xs: list_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( set_nat2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% card_length
thf(fact_630_injectivity__image,axiom,
! [F: a > nat,A: set_a,G: a > nat,Invert: nat > a] :
( ( ( image_a_nat @ F @ A )
= ( image_a_nat @ G @ A ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ( ( Invert @ ( F @ X2 ) )
= X2 )
& ( ( Invert @ ( G @ X2 ) )
= X2 ) ) )
=> ! [X5: a] :
( ( member_a @ X5 @ A )
=> ( ( F @ X5 )
= ( G @ X5 ) ) ) ) ) ).
% injectivity_image
thf(fact_631_injectivity__image,axiom,
! [F: nat > nat,A: set_nat,G: nat > nat,Invert: nat > nat] :
( ( ( image_nat_nat @ F @ A )
= ( image_nat_nat @ G @ A ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( ( Invert @ ( F @ X2 ) )
= X2 )
& ( ( Invert @ ( G @ X2 ) )
= X2 ) ) )
=> ! [X5: nat] :
( ( member_nat @ X5 @ A )
=> ( ( F @ X5 )
= ( G @ X5 ) ) ) ) ) ).
% injectivity_image
thf(fact_632_image__Fpow__mono,axiom,
! [F: a > nat,A: set_a,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_set_a_set_nat @ ( image_a_nat @ F ) @ ( finite_Fpow_a @ A ) ) @ ( finite_Fpow_nat @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_633_image__Fpow__mono,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( finite_Fpow_nat @ A ) ) @ ( finite_Fpow_nat @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_634_length__stirling__row,axiom,
! [N: nat] :
( ( size_size_list_nat @ ( stirling_row @ N ) )
= ( suc @ N ) ) ).
% length_stirling_row
thf(fact_635_card__set__1__iff__replicate,axiom,
! [Xs: list_Product_unit] :
( ( ( finite410649719033368117t_unit @ ( set_Product_unit2 @ Xs ) )
= ( suc @ zero_zero_nat ) )
= ( ( Xs != nil_Product_unit )
& ? [X3: product_unit] :
( Xs
= ( replic7505510843043721677t_unit @ ( size_s245203480648594047t_unit @ Xs ) @ X3 ) ) ) ) ).
% card_set_1_iff_replicate
thf(fact_636_card__set__1__iff__replicate,axiom,
! [Xs: list_a] :
( ( ( finite_card_a @ ( set_a2 @ Xs ) )
= ( suc @ zero_zero_nat ) )
= ( ( Xs != nil_a )
& ? [X3: a] :
( Xs
= ( replicate_a @ ( size_size_list_a @ Xs ) @ X3 ) ) ) ) ).
% card_set_1_iff_replicate
thf(fact_637_card__set__1__iff__replicate,axiom,
! [Xs: list_nat] :
( ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
= ( suc @ zero_zero_nat ) )
= ( ( Xs != nil_nat )
& ? [X3: nat] :
( Xs
= ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X3 ) ) ) ) ).
% card_set_1_iff_replicate
thf(fact_638_card__le__inj,axiom,
! [A: set_Product_unit,B2: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( finite4290736615968046902t_unit @ B2 )
=> ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ B2 ) )
=> ? [F3: product_unit > product_unit] :
( ( ord_le3507040750410214029t_unit @ ( image_405062704495631173t_unit @ F3 @ A ) @ B2 )
& ( inj_on8151373323710067377t_unit @ F3 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_639_card__le__inj,axiom,
! [A: set_a,B2: set_nat] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_nat @ B2 ) )
=> ? [F3: a > nat] :
( ( ord_less_eq_set_nat @ ( image_a_nat @ F3 @ A ) @ B2 )
& ( inj_on_a_nat @ F3 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_640_card__le__inj,axiom,
! [A: set_Product_unit,B2: set_nat] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite_card_nat @ B2 ) )
=> ? [F3: product_unit > nat] :
( ( ord_less_eq_set_nat @ ( image_875570014554754200it_nat @ F3 @ A ) @ B2 )
& ( inj_on8430439091780834860it_nat @ F3 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_641_card__le__inj,axiom,
! [A: set_nat,B2: set_Product_unit] :
( ( finite_finite_nat @ A )
=> ( ( finite4290736615968046902t_unit @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite410649719033368117t_unit @ B2 ) )
=> ? [F3: nat > product_unit] :
( ( ord_le3507040750410214029t_unit @ ( image_8730104196221521654t_unit @ F3 @ A ) @ B2 )
& ( inj_on7061601236592826506t_unit @ F3 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_642_card__le__inj,axiom,
! [A: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B2 ) )
=> ? [F3: nat > nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ A ) @ B2 )
& ( inj_on_nat_nat @ F3 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_643_card__inj__on__le,axiom,
! [F: product_unit > product_unit,A: set_Product_unit,B2: set_Product_unit] :
( ( inj_on8151373323710067377t_unit @ F @ A )
=> ( ( ord_le3507040750410214029t_unit @ ( image_405062704495631173t_unit @ F @ A ) @ B2 )
=> ( ( finite4290736615968046902t_unit @ B2 )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ B2 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_644_card__inj__on__le,axiom,
! [F: nat > product_unit,A: set_nat,B2: set_Product_unit] :
( ( inj_on7061601236592826506t_unit @ F @ A )
=> ( ( ord_le3507040750410214029t_unit @ ( image_8730104196221521654t_unit @ F @ A ) @ B2 )
=> ( ( finite4290736615968046902t_unit @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite410649719033368117t_unit @ B2 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_645_card__inj__on__le,axiom,
! [F: a > nat,A: set_a,B2: set_nat] :
( ( inj_on_a_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_646_card__inj__on__le,axiom,
! [F: product_unit > nat,A: set_Product_unit,B2: set_nat] :
( ( inj_on8430439091780834860it_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ ( image_875570014554754200it_nat @ F @ A ) @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_647_card__inj__on__le,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_648_inj__on__iff__card__le,axiom,
! [A: set_Product_unit,B2: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( finite4290736615968046902t_unit @ B2 )
=> ( ( ? [F2: product_unit > product_unit] :
( ( inj_on8151373323710067377t_unit @ F2 @ A )
& ( ord_le3507040750410214029t_unit @ ( image_405062704495631173t_unit @ F2 @ A ) @ B2 ) ) )
= ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ B2 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_649_inj__on__iff__card__le,axiom,
! [A: set_a,B2: set_nat] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ? [F2: a > nat] :
( ( inj_on_a_nat @ F2 @ A )
& ( ord_less_eq_set_nat @ ( image_a_nat @ F2 @ A ) @ B2 ) ) )
= ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_650_inj__on__iff__card__le,axiom,
! [A: set_Product_unit,B2: set_nat] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ? [F2: product_unit > nat] :
( ( inj_on8430439091780834860it_nat @ F2 @ A )
& ( ord_less_eq_set_nat @ ( image_875570014554754200it_nat @ F2 @ A ) @ B2 ) ) )
= ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_651_inj__on__iff__card__le,axiom,
! [A: set_nat,B2: set_Product_unit] :
( ( finite_finite_nat @ A )
=> ( ( finite4290736615968046902t_unit @ B2 )
=> ( ( ? [F2: nat > product_unit] :
( ( inj_on7061601236592826506t_unit @ F2 @ A )
& ( ord_le3507040750410214029t_unit @ ( image_8730104196221521654t_unit @ F2 @ A ) @ B2 ) ) )
= ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite410649719033368117t_unit @ B2 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_652_inj__on__iff__card__le,axiom,
! [A: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ? [F2: nat > nat] :
( ( inj_on_nat_nat @ F2 @ A )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A ) @ B2 ) ) )
= ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_653_remdups__adj__length__ge1,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_size_list_a @ ( remdups_adj_a @ Xs ) ) ) ) ).
% remdups_adj_length_ge1
thf(fact_654_remdups__adj__length__ge1,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_size_list_nat @ ( remdups_adj_nat @ Xs ) ) ) ) ).
% remdups_adj_length_ge1
thf(fact_655_finite__imageI,axiom,
! [F4: set_a,H: a > nat] :
( ( finite_finite_a @ F4 )
=> ( finite_finite_nat @ ( image_a_nat @ H @ F4 ) ) ) ).
% finite_imageI
thf(fact_656_finite__imageI,axiom,
! [F4: set_nat,H: nat > nat] :
( ( finite_finite_nat @ F4 )
=> ( finite_finite_nat @ ( image_nat_nat @ H @ F4 ) ) ) ).
% finite_imageI
thf(fact_657_finite__insert,axiom,
! [A4: a,A: set_a] :
( ( finite_finite_a @ ( insert_a2 @ A4 @ A ) )
= ( finite_finite_a @ A ) ) ).
% finite_insert
thf(fact_658_finite__insert,axiom,
! [A4: nat,A: set_nat] :
( ( finite_finite_nat @ ( insert_nat2 @ A4 @ A ) )
= ( finite_finite_nat @ A ) ) ).
% finite_insert
thf(fact_659_List_Ofinite__set,axiom,
! [Xs: list_a] : ( finite_finite_a @ ( set_a2 @ Xs ) ) ).
% List.finite_set
thf(fact_660_List_Ofinite__set,axiom,
! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_661_concat__replicate__trivial,axiom,
! [I: nat] :
( ( concat_a @ ( replicate_list_a @ I @ nil_a ) )
= nil_a ) ).
% concat_replicate_trivial
thf(fact_662_concat__replicate__trivial,axiom,
! [I: nat] :
( ( concat_nat @ ( replicate_list_nat @ I @ nil_nat ) )
= nil_nat ) ).
% concat_replicate_trivial
thf(fact_663_remdups__adj__Nil__iff,axiom,
! [Xs: list_a] :
( ( ( remdups_adj_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% remdups_adj_Nil_iff
thf(fact_664_remdups__adj__Nil__iff,axiom,
! [Xs: list_nat] :
( ( ( remdups_adj_nat @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% remdups_adj_Nil_iff
thf(fact_665_length__replicate,axiom,
! [N: nat,X: a] :
( ( size_size_list_a @ ( replicate_a @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_666_length__replicate,axiom,
! [N: nat,X: nat] :
( ( size_size_list_nat @ ( replicate_nat @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_667_remdups__adj__set,axiom,
! [Xs: list_a] :
( ( set_a2 @ ( remdups_adj_a @ Xs ) )
= ( set_a2 @ Xs ) ) ).
% remdups_adj_set
thf(fact_668_remdups__adj__set,axiom,
! [Xs: list_nat] :
( ( set_nat2 @ ( remdups_adj_nat @ Xs ) )
= ( set_nat2 @ Xs ) ) ).
% remdups_adj_set
thf(fact_669_map__replicate,axiom,
! [F: a > nat,N: nat,X: a] :
( ( map_a_nat @ F @ ( replicate_a @ N @ X ) )
= ( replicate_nat @ N @ ( F @ X ) ) ) ).
% map_replicate
thf(fact_670_card_Oinfinite,axiom,
! [A: set_Product_unit] :
( ~ ( finite4290736615968046902t_unit @ A )
=> ( ( finite410649719033368117t_unit @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_671_card_Oinfinite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_card_nat @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_672_replicate__empty,axiom,
! [N: nat,X: a] :
( ( ( replicate_a @ N @ X )
= nil_a )
= ( N = zero_zero_nat ) ) ).
% replicate_empty
thf(fact_673_replicate__empty,axiom,
! [N: nat,X: nat] :
( ( ( replicate_nat @ N @ X )
= nil_nat )
= ( N = zero_zero_nat ) ) ).
% replicate_empty
thf(fact_674_empty__replicate,axiom,
! [N: nat,X: a] :
( ( nil_a
= ( replicate_a @ N @ X ) )
= ( N = zero_zero_nat ) ) ).
% empty_replicate
thf(fact_675_empty__replicate,axiom,
! [N: nat,X: nat] :
( ( nil_nat
= ( replicate_nat @ N @ X ) )
= ( N = zero_zero_nat ) ) ).
% empty_replicate
thf(fact_676_in__set__replicate,axiom,
! [X: a,N: nat,Y: a] :
( ( member_a @ X @ ( set_a2 @ ( replicate_a @ N @ Y ) ) )
= ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_677_in__set__replicate,axiom,
! [X: nat,N: nat,Y: nat] :
( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N @ Y ) ) )
= ( ( X = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_678_Bex__set__replicate,axiom,
! [N: nat,A4: a,P: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ ( replicate_a @ N @ A4 ) ) )
& ( P @ X3 ) ) )
= ( ( P @ A4 )
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_679_Bex__set__replicate,axiom,
! [N: nat,A4: nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ ( replicate_nat @ N @ A4 ) ) )
& ( P @ X3 ) ) )
= ( ( P @ A4 )
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_680_Ball__set__replicate,axiom,
! [N: nat,A4: a,P: a > $o] :
( ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ ( replicate_a @ N @ A4 ) ) )
=> ( P @ X3 ) ) )
= ( ( P @ A4 )
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_681_Ball__set__replicate,axiom,
! [N: nat,A4: nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ ( replicate_nat @ N @ A4 ) ) )
=> ( P @ X3 ) ) )
= ( ( P @ A4 )
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_682_card__insert__disjoint,axiom,
! [A: set_a,X: a] :
( ( finite_finite_a @ A )
=> ( ~ ( member_a @ X @ A )
=> ( ( finite_card_a @ ( insert_a2 @ X @ A ) )
= ( suc @ ( finite_card_a @ A ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_683_card__insert__disjoint,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ~ ( member_Product_unit @ X @ A )
=> ( ( finite410649719033368117t_unit @ ( insert_Product_unit @ X @ A ) )
= ( suc @ ( finite410649719033368117t_unit @ A ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_684_card__insert__disjoint,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat @ X @ A )
=> ( ( finite_card_nat @ ( insert_nat2 @ X @ A ) )
= ( suc @ ( finite_card_nat @ A ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_685_remdups__adj_Osimps_I3_J,axiom,
! [X: a,Y: a,Xs: list_a] :
( ( ( X = Y )
=> ( ( remdups_adj_a @ ( cons_a @ X @ ( cons_a @ Y @ Xs ) ) )
= ( remdups_adj_a @ ( cons_a @ X @ Xs ) ) ) )
& ( ( X != Y )
=> ( ( remdups_adj_a @ ( cons_a @ X @ ( cons_a @ Y @ Xs ) ) )
= ( cons_a @ X @ ( remdups_adj_a @ ( cons_a @ Y @ Xs ) ) ) ) ) ) ).
% remdups_adj.simps(3)
thf(fact_686_remdups__adj_Osimps_I3_J,axiom,
! [X: nat,Y: nat,Xs: list_nat] :
( ( ( X = Y )
=> ( ( remdups_adj_nat @ ( cons_nat @ X @ ( cons_nat @ Y @ Xs ) ) )
= ( remdups_adj_nat @ ( cons_nat @ X @ Xs ) ) ) )
& ( ( X != Y )
=> ( ( remdups_adj_nat @ ( cons_nat @ X @ ( cons_nat @ Y @ Xs ) ) )
= ( cons_nat @ X @ ( remdups_adj_nat @ ( cons_nat @ Y @ Xs ) ) ) ) ) ) ).
% remdups_adj.simps(3)
thf(fact_687_append__replicate__commute,axiom,
! [N: nat,X: a,K: nat] :
( ( append_a @ ( replicate_a @ N @ X ) @ ( replicate_a @ K @ X ) )
= ( append_a @ ( replicate_a @ K @ X ) @ ( replicate_a @ N @ X ) ) ) ).
% append_replicate_commute
thf(fact_688_append__replicate__commute,axiom,
! [N: nat,X: nat,K: nat] :
( ( append_nat @ ( replicate_nat @ N @ X ) @ ( replicate_nat @ K @ X ) )
= ( append_nat @ ( replicate_nat @ K @ X ) @ ( replicate_nat @ N @ X ) ) ) ).
% append_replicate_commute
thf(fact_689_remdups__adj_Osimps_I1_J,axiom,
( ( remdups_adj_a @ nil_a )
= nil_a ) ).
% remdups_adj.simps(1)
thf(fact_690_remdups__adj_Osimps_I1_J,axiom,
( ( remdups_adj_nat @ nil_nat )
= nil_nat ) ).
% remdups_adj.simps(1)
thf(fact_691_finite_OinsertI,axiom,
! [A: set_a,A4: a] :
( ( finite_finite_a @ A )
=> ( finite_finite_a @ ( insert_a2 @ A4 @ A ) ) ) ).
% finite.insertI
thf(fact_692_finite_OinsertI,axiom,
! [A: set_nat,A4: nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( insert_nat2 @ A4 @ A ) ) ) ).
% finite.insertI
thf(fact_693_finite__list,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ? [Xs2: list_a] :
( ( set_a2 @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_694_finite__list,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ? [Xs2: list_nat] :
( ( set_nat2 @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_695_remdups__adj__replicate,axiom,
! [N: nat,X: a] :
( ( ( N = zero_zero_nat )
=> ( ( remdups_adj_a @ ( replicate_a @ N @ X ) )
= nil_a ) )
& ( ( N != zero_zero_nat )
=> ( ( remdups_adj_a @ ( replicate_a @ N @ X ) )
= ( cons_a @ X @ nil_a ) ) ) ) ).
% remdups_adj_replicate
thf(fact_696_remdups__adj__replicate,axiom,
! [N: nat,X: nat] :
( ( ( N = zero_zero_nat )
=> ( ( remdups_adj_nat @ ( replicate_nat @ N @ X ) )
= nil_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( remdups_adj_nat @ ( replicate_nat @ N @ X ) )
= ( cons_nat @ X @ nil_nat ) ) ) ) ).
% remdups_adj_replicate
thf(fact_697_remdups__adj__singleton,axiom,
! [Xs: list_a,X: a] :
( ( ( remdups_adj_a @ Xs )
= ( cons_a @ X @ nil_a ) )
=> ( Xs
= ( replicate_a @ ( size_size_list_a @ Xs ) @ X ) ) ) ).
% remdups_adj_singleton
thf(fact_698_remdups__adj__singleton,axiom,
! [Xs: list_nat,X: nat] :
( ( ( remdups_adj_nat @ Xs )
= ( cons_nat @ X @ nil_nat ) )
=> ( Xs
= ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X ) ) ) ).
% remdups_adj_singleton
thf(fact_699_replicate__Suc,axiom,
! [N: nat,X: a] :
( ( replicate_a @ ( suc @ N ) @ X )
= ( cons_a @ X @ ( replicate_a @ N @ X ) ) ) ).
% replicate_Suc
thf(fact_700_replicate__Suc,axiom,
! [N: nat,X: nat] :
( ( replicate_nat @ ( suc @ N ) @ X )
= ( cons_nat @ X @ ( replicate_nat @ N @ X ) ) ) ).
% replicate_Suc
thf(fact_701_replicate__0,axiom,
! [X: a] :
( ( replicate_a @ zero_zero_nat @ X )
= nil_a ) ).
% replicate_0
thf(fact_702_replicate__0,axiom,
! [X: nat] :
( ( replicate_nat @ zero_zero_nat @ X )
= nil_nat ) ).
% replicate_0
thf(fact_703_replicate__app__Cons__same,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( append_a @ ( replicate_a @ N @ X ) @ ( cons_a @ X @ Xs ) )
= ( cons_a @ X @ ( append_a @ ( replicate_a @ N @ X ) @ Xs ) ) ) ).
% replicate_app_Cons_same
thf(fact_704_replicate__app__Cons__same,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( append_nat @ ( replicate_nat @ N @ X ) @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ ( append_nat @ ( replicate_nat @ N @ X ) @ Xs ) ) ) ).
% replicate_app_Cons_same
thf(fact_705_replicate__eqI,axiom,
! [Xs: list_a,N: nat,X: a] :
( ( ( size_size_list_a @ Xs )
= N )
=> ( ! [Y2: a] :
( ( member_a @ Y2 @ ( set_a2 @ Xs ) )
=> ( Y2 = X ) )
=> ( Xs
= ( replicate_a @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_706_replicate__eqI,axiom,
! [Xs: list_nat,N: nat,X: nat] :
( ( ( size_size_list_nat @ Xs )
= N )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ ( set_nat2 @ Xs ) )
=> ( Y2 = X ) )
=> ( Xs
= ( replicate_nat @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_707_replicate__length__same,axiom,
! [Xs: list_a,X: a] :
( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( X2 = X ) )
=> ( ( replicate_a @ ( size_size_list_a @ Xs ) @ X )
= Xs ) ) ).
% replicate_length_same
thf(fact_708_replicate__length__same,axiom,
! [Xs: list_nat,X: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( X2 = X ) )
=> ( ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X )
= Xs ) ) ).
% replicate_length_same
thf(fact_709_remdups__adj_Osimps_I2_J,axiom,
! [X: a] :
( ( remdups_adj_a @ ( cons_a @ X @ nil_a ) )
= ( cons_a @ X @ nil_a ) ) ).
% remdups_adj.simps(2)
thf(fact_710_remdups__adj_Osimps_I2_J,axiom,
! [X: nat] :
( ( remdups_adj_nat @ ( cons_nat @ X @ nil_nat ) )
= ( cons_nat @ X @ nil_nat ) ) ).
% remdups_adj.simps(2)
thf(fact_711_remdups__adj_Oelims,axiom,
! [X: list_a,Y: list_a] :
( ( ( remdups_adj_a @ X )
= Y )
=> ( ( ( X = nil_a )
=> ( Y != nil_a ) )
=> ( ! [X2: a] :
( ( X
= ( cons_a @ X2 @ nil_a ) )
=> ( Y
!= ( cons_a @ X2 @ nil_a ) ) )
=> ~ ! [X2: a,Y2: a,Xs2: list_a] :
( ( X
= ( cons_a @ X2 @ ( cons_a @ Y2 @ Xs2 ) ) )
=> ~ ( ( ( X2 = Y2 )
=> ( Y
= ( remdups_adj_a @ ( cons_a @ X2 @ Xs2 ) ) ) )
& ( ( X2 != Y2 )
=> ( Y
= ( cons_a @ X2 @ ( remdups_adj_a @ ( cons_a @ Y2 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% remdups_adj.elims
thf(fact_712_remdups__adj_Oelims,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( remdups_adj_nat @ X )
= Y )
=> ( ( ( X = nil_nat )
=> ( Y != nil_nat ) )
=> ( ! [X2: nat] :
( ( X
= ( cons_nat @ X2 @ nil_nat ) )
=> ( Y
!= ( cons_nat @ X2 @ nil_nat ) ) )
=> ~ ! [X2: nat,Y2: nat,Xs2: list_nat] :
( ( X
= ( cons_nat @ X2 @ ( cons_nat @ Y2 @ Xs2 ) ) )
=> ~ ( ( ( X2 = Y2 )
=> ( Y
= ( remdups_adj_nat @ ( cons_nat @ X2 @ Xs2 ) ) ) )
& ( ( X2 != Y2 )
=> ( Y
= ( cons_nat @ X2 @ ( remdups_adj_nat @ ( cons_nat @ Y2 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% remdups_adj.elims
thf(fact_713_all__finite__subset__image,axiom,
! [F: a > nat,A: set_a,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_a_nat @ F @ A ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A ) )
=> ( P @ ( image_a_nat @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_714_all__finite__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_nat] :
( ( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A ) )
=> ( P @ ( image_nat_nat @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_715_ex__finite__subset__image,axiom,
! [F: a > nat,A: set_a,P: set_nat > $o] :
( ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_a_nat @ F @ A ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A )
& ( P @ ( image_a_nat @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_716_ex__finite__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ ( image_nat_nat @ F @ A ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_nat] :
( ( finite_finite_nat @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A )
& ( P @ ( image_nat_nat @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_717_finite__subset__image,axiom,
! [B2: set_nat,F: a > nat,A: set_a] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_a_nat @ F @ A ) )
=> ? [C4: set_a] :
( ( ord_less_eq_set_a @ C4 @ A )
& ( finite_finite_a @ C4 )
& ( B2
= ( image_a_nat @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_718_finite__subset__image,axiom,
! [B2: set_nat,F: nat > nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
=> ? [C4: set_nat] :
( ( ord_less_eq_set_nat @ C4 @ A )
& ( finite_finite_nat @ C4 )
& ( B2
= ( image_nat_nat @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_719_finite__surj,axiom,
! [A: set_a,B2: set_nat,F: a > nat] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_a_nat @ F @ A ) )
=> ( finite_finite_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_720_finite__surj,axiom,
! [A: set_nat,B2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
=> ( finite_finite_nat @ B2 ) ) ) ).
% finite_surj
thf(fact_721_remdups__adj__length,axiom,
! [Xs: list_a] : ( ord_less_eq_nat @ ( size_size_list_a @ ( remdups_adj_a @ Xs ) ) @ ( size_size_list_a @ Xs ) ) ).
% remdups_adj_length
thf(fact_722_remdups__adj__length,axiom,
! [Xs: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( remdups_adj_nat @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% remdups_adj_length
thf(fact_723_finite__imageD,axiom,
! [F: a > nat,A: set_a] :
( ( finite_finite_nat @ ( image_a_nat @ F @ A ) )
=> ( ( inj_on_a_nat @ F @ A )
=> ( finite_finite_a @ A ) ) ) ).
% finite_imageD
thf(fact_724_finite__imageD,axiom,
! [F: nat > nat,A: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F @ A ) )
=> ( ( inj_on_nat_nat @ F @ A )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_imageD
thf(fact_725_finite__image__iff,axiom,
! [F: a > nat,A: set_a] :
( ( inj_on_a_nat @ F @ A )
=> ( ( finite_finite_nat @ ( image_a_nat @ F @ A ) )
= ( finite_finite_a @ A ) ) ) ).
% finite_image_iff
thf(fact_726_finite__image__iff,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F @ A ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_image_iff
thf(fact_727_card__subset__eq,axiom,
! [B2: set_Product_unit,A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ B2 )
=> ( ( ord_le3507040750410214029t_unit @ A @ B2 )
=> ( ( ( finite410649719033368117t_unit @ A )
= ( finite410649719033368117t_unit @ B2 ) )
=> ( A = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_728_card__subset__eq,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ( finite_card_nat @ A )
= ( finite_card_nat @ B2 ) )
=> ( A = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_729_infinite__arbitrarily__large,axiom,
! [A: set_Product_unit,N: nat] :
( ~ ( finite4290736615968046902t_unit @ A )
=> ? [B3: set_Product_unit] :
( ( finite4290736615968046902t_unit @ B3 )
& ( ( finite410649719033368117t_unit @ B3 )
= N )
& ( ord_le3507040750410214029t_unit @ B3 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_730_infinite__arbitrarily__large,axiom,
! [A: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A )
=> ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ( finite_card_nat @ B3 )
= N )
& ( ord_less_eq_set_nat @ B3 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_731_card__le__if__inj__on__rel,axiom,
! [B2: set_a,A: set_a,R2: a > a > $o] :
( ( finite_finite_a @ B2 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A )
=> ? [B5: a] :
( ( member_a @ B5 @ B2 )
& ( R2 @ A3 @ B5 ) ) )
=> ( ! [A1: a,A22: a,B6: a] :
( ( member_a @ A1 @ A )
=> ( ( member_a @ A22 @ A )
=> ( ( member_a @ B6 @ B2 )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_732_card__le__if__inj__on__rel,axiom,
! [B2: set_Product_unit,A: set_a,R2: a > product_unit > $o] :
( ( finite4290736615968046902t_unit @ B2 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A )
=> ? [B5: product_unit] :
( ( member_Product_unit @ B5 @ B2 )
& ( R2 @ A3 @ B5 ) ) )
=> ( ! [A1: a,A22: a,B6: product_unit] :
( ( member_a @ A1 @ A )
=> ( ( member_a @ A22 @ A )
=> ( ( member_Product_unit @ B6 @ B2 )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite410649719033368117t_unit @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_733_card__le__if__inj__on__rel,axiom,
! [B2: set_a,A: set_Product_unit,R2: product_unit > a > $o] :
( ( finite_finite_a @ B2 )
=> ( ! [A3: product_unit] :
( ( member_Product_unit @ A3 @ A )
=> ? [B5: a] :
( ( member_a @ B5 @ B2 )
& ( R2 @ A3 @ B5 ) ) )
=> ( ! [A1: product_unit,A22: product_unit,B6: a] :
( ( member_Product_unit @ A1 @ A )
=> ( ( member_Product_unit @ A22 @ A )
=> ( ( member_a @ B6 @ B2 )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_734_card__le__if__inj__on__rel,axiom,
! [B2: set_Product_unit,A: set_Product_unit,R2: product_unit > product_unit > $o] :
( ( finite4290736615968046902t_unit @ B2 )
=> ( ! [A3: product_unit] :
( ( member_Product_unit @ A3 @ A )
=> ? [B5: product_unit] :
( ( member_Product_unit @ B5 @ B2 )
& ( R2 @ A3 @ B5 ) ) )
=> ( ! [A1: product_unit,A22: product_unit,B6: product_unit] :
( ( member_Product_unit @ A1 @ A )
=> ( ( member_Product_unit @ A22 @ A )
=> ( ( member_Product_unit @ B6 @ B2 )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_735_card__le__if__inj__on__rel,axiom,
! [B2: set_a,A: set_nat,R2: nat > a > $o] :
( ( finite_finite_a @ B2 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ? [B5: a] :
( ( member_a @ B5 @ B2 )
& ( R2 @ A3 @ B5 ) ) )
=> ( ! [A1: nat,A22: nat,B6: a] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_a @ B6 @ B2 )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_736_card__le__if__inj__on__rel,axiom,
! [B2: set_Product_unit,A: set_nat,R2: nat > product_unit > $o] :
( ( finite4290736615968046902t_unit @ B2 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ? [B5: product_unit] :
( ( member_Product_unit @ B5 @ B2 )
& ( R2 @ A3 @ B5 ) ) )
=> ( ! [A1: nat,A22: nat,B6: product_unit] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_Product_unit @ B6 @ B2 )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite410649719033368117t_unit @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_737_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A: set_a,R2: a > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A3: a] :
( ( member_a @ A3 @ A )
=> ? [B5: nat] :
( ( member_nat @ B5 @ B2 )
& ( R2 @ A3 @ B5 ) ) )
=> ( ! [A1: a,A22: a,B6: nat] :
( ( member_a @ A1 @ A )
=> ( ( member_a @ A22 @ A )
=> ( ( member_nat @ B6 @ B2 )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_738_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A: set_Product_unit,R2: product_unit > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A3: product_unit] :
( ( member_Product_unit @ A3 @ A )
=> ? [B5: nat] :
( ( member_nat @ B5 @ B2 )
& ( R2 @ A3 @ B5 ) ) )
=> ( ! [A1: product_unit,A22: product_unit,B6: nat] :
( ( member_Product_unit @ A1 @ A )
=> ( ( member_Product_unit @ A22 @ A )
=> ( ( member_nat @ B6 @ B2 )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_739_card__le__if__inj__on__rel,axiom,
! [B2: set_nat,A: set_nat,R2: nat > nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A )
=> ? [B5: nat] :
( ( member_nat @ B5 @ B2 )
& ( R2 @ A3 @ B5 ) ) )
=> ( ! [A1: nat,A22: nat,B6: nat] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_nat @ B6 @ B2 )
=> ( ( R2 @ A1 @ B6 )
=> ( ( R2 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_740_comm__append__are__replicate,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= ( append_a @ Ys @ Xs ) )
=> ? [M3: nat,N4: nat,Zs2: list_a] :
( ( ( concat_a @ ( replicate_list_a @ M3 @ Zs2 ) )
= Xs )
& ( ( concat_a @ ( replicate_list_a @ N4 @ Zs2 ) )
= Ys ) ) ) ).
% comm_append_are_replicate
thf(fact_741_comm__append__are__replicate,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Ys @ Xs ) )
=> ? [M3: nat,N4: nat,Zs2: list_nat] :
( ( ( concat_nat @ ( replicate_list_nat @ M3 @ Zs2 ) )
= Xs )
& ( ( concat_nat @ ( replicate_list_nat @ N4 @ Zs2 ) )
= Ys ) ) ) ).
% comm_append_are_replicate
thf(fact_742_replicate__append__same,axiom,
! [I: nat,X: a] :
( ( append_a @ ( replicate_a @ I @ X ) @ ( cons_a @ X @ nil_a ) )
= ( cons_a @ X @ ( replicate_a @ I @ X ) ) ) ).
% replicate_append_same
thf(fact_743_replicate__append__same,axiom,
! [I: nat,X: nat] :
( ( append_nat @ ( replicate_nat @ I @ X ) @ ( cons_nat @ X @ nil_nat ) )
= ( cons_nat @ X @ ( replicate_nat @ I @ X ) ) ) ).
% replicate_append_same
thf(fact_744_remdups__adj__append__two,axiom,
! [Xs: list_a,X: a,Y: a] :
( ( remdups_adj_a @ ( append_a @ Xs @ ( cons_a @ X @ ( cons_a @ Y @ nil_a ) ) ) )
= ( append_a @ ( remdups_adj_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) ) @ ( if_list_a @ ( X = Y ) @ nil_a @ ( cons_a @ Y @ nil_a ) ) ) ) ).
% remdups_adj_append_two
thf(fact_745_remdups__adj__append__two,axiom,
! [Xs: list_nat,X: nat,Y: nat] :
( ( remdups_adj_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ ( cons_nat @ Y @ nil_nat ) ) ) )
= ( append_nat @ ( remdups_adj_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) ) @ ( if_list_nat @ ( X = Y ) @ nil_nat @ ( cons_nat @ Y @ nil_nat ) ) ) ) ).
% remdups_adj_append_two
thf(fact_746_finite__surj__inj,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( image_nat_nat @ F @ A ) )
=> ( inj_on_nat_nat @ F @ A ) ) ) ).
% finite_surj_inj
thf(fact_747_inj__on__finite,axiom,
! [F: a > nat,A: set_a,B2: set_nat] :
( ( inj_on_a_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_a @ A ) ) ) ) ).
% inj_on_finite
thf(fact_748_inj__on__finite,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A ) ) ) ) ).
% inj_on_finite
thf(fact_749_endo__inj__surj,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ A )
=> ( ( inj_on_nat_nat @ F @ A )
=> ( ( image_nat_nat @ F @ A )
= A ) ) ) ) ).
% endo_inj_surj
thf(fact_750_card__Suc__eq__finite,axiom,
! [A: set_a,K: nat] :
( ( ( finite_card_a @ A )
= ( suc @ K ) )
= ( ? [B7: a,B4: set_a] :
( ( A
= ( insert_a2 @ B7 @ B4 ) )
& ~ ( member_a @ B7 @ B4 )
& ( ( finite_card_a @ B4 )
= K )
& ( finite_finite_a @ B4 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_751_card__Suc__eq__finite,axiom,
! [A: set_Product_unit,K: nat] :
( ( ( finite410649719033368117t_unit @ A )
= ( suc @ K ) )
= ( ? [B7: product_unit,B4: set_Product_unit] :
( ( A
= ( insert_Product_unit @ B7 @ B4 ) )
& ~ ( member_Product_unit @ B7 @ B4 )
& ( ( finite410649719033368117t_unit @ B4 )
= K )
& ( finite4290736615968046902t_unit @ B4 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_752_card__Suc__eq__finite,axiom,
! [A: set_nat,K: nat] :
( ( ( finite_card_nat @ A )
= ( suc @ K ) )
= ( ? [B7: nat,B4: set_nat] :
( ( A
= ( insert_nat2 @ B7 @ B4 ) )
& ~ ( member_nat @ B7 @ B4 )
& ( ( finite_card_nat @ B4 )
= K )
& ( finite_finite_nat @ B4 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_753_card__insert__if,axiom,
! [A: set_a,X: a] :
( ( finite_finite_a @ A )
=> ( ( ( member_a @ X @ A )
=> ( ( finite_card_a @ ( insert_a2 @ X @ A ) )
= ( finite_card_a @ A ) ) )
& ( ~ ( member_a @ X @ A )
=> ( ( finite_card_a @ ( insert_a2 @ X @ A ) )
= ( suc @ ( finite_card_a @ A ) ) ) ) ) ) ).
% card_insert_if
thf(fact_754_card__insert__if,axiom,
! [A: set_Product_unit,X: product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( ( member_Product_unit @ X @ A )
=> ( ( finite410649719033368117t_unit @ ( insert_Product_unit @ X @ A ) )
= ( finite410649719033368117t_unit @ A ) ) )
& ( ~ ( member_Product_unit @ X @ A )
=> ( ( finite410649719033368117t_unit @ ( insert_Product_unit @ X @ A ) )
= ( suc @ ( finite410649719033368117t_unit @ A ) ) ) ) ) ) ).
% card_insert_if
thf(fact_755_card__insert__if,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( ( member_nat @ X @ A )
=> ( ( finite_card_nat @ ( insert_nat2 @ X @ A ) )
= ( finite_card_nat @ A ) ) )
& ( ~ ( member_nat @ X @ A )
=> ( ( finite_card_nat @ ( insert_nat2 @ X @ A ) )
= ( suc @ ( finite_card_nat @ A ) ) ) ) ) ) ).
% card_insert_if
thf(fact_756_card__image__le,axiom,
! [A: set_Product_unit,F: product_unit > product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) @ ( finite410649719033368117t_unit @ A ) ) ) ).
% card_image_le
thf(fact_757_card__image__le,axiom,
! [A: set_a,F: a > nat] :
( ( finite_finite_a @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_a_nat @ F @ A ) ) @ ( finite_card_a @ A ) ) ) ).
% card_image_le
thf(fact_758_card__image__le,axiom,
! [A: set_Product_unit,F: product_unit > nat] :
( ( finite4290736615968046902t_unit @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_875570014554754200it_nat @ F @ A ) ) @ ( finite410649719033368117t_unit @ A ) ) ) ).
% card_image_le
thf(fact_759_card__image__le,axiom,
! [A: set_nat,F: nat > product_unit] :
( ( finite_finite_nat @ A )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( image_8730104196221521654t_unit @ F @ A ) ) @ ( finite_card_nat @ A ) ) ) ).
% card_image_le
thf(fact_760_card__image__le,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_nat_nat @ F @ A ) ) @ ( finite_card_nat @ A ) ) ) ).
% card_image_le
thf(fact_761_card__mono,axiom,
! [B2: set_Product_unit,A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ B2 )
=> ( ( ord_le3507040750410214029t_unit @ A @ B2 )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ B2 ) ) ) ) ).
% card_mono
thf(fact_762_card__mono,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% card_mono
thf(fact_763_card__seteq,axiom,
! [B2: set_Product_unit,A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ B2 )
=> ( ( ord_le3507040750410214029t_unit @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ B2 ) @ ( finite410649719033368117t_unit @ A ) )
=> ( A = B2 ) ) ) ) ).
% card_seteq
thf(fact_764_card__seteq,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat @ A ) )
=> ( A = B2 ) ) ) ) ).
% card_seteq
thf(fact_765_exists__subset__between,axiom,
! [A: set_Product_unit,N: nat,C3: set_Product_unit] :
( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite410649719033368117t_unit @ C3 ) )
=> ( ( ord_le3507040750410214029t_unit @ A @ C3 )
=> ( ( finite4290736615968046902t_unit @ C3 )
=> ? [B3: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ B3 )
& ( ord_le3507040750410214029t_unit @ B3 @ C3 )
& ( ( finite410649719033368117t_unit @ B3 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_766_exists__subset__between,axiom,
! [A: set_nat,N: nat,C3: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C3 ) )
=> ( ( ord_less_eq_set_nat @ A @ C3 )
=> ( ( finite_finite_nat @ C3 )
=> ? [B3: set_nat] :
( ( ord_less_eq_set_nat @ A @ B3 )
& ( ord_less_eq_set_nat @ B3 @ C3 )
& ( ( finite_card_nat @ B3 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_767_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_Product_unit] :
( ( ord_less_eq_nat @ N @ ( finite410649719033368117t_unit @ S ) )
=> ~ ! [T4: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ T4 @ S )
=> ( ( ( finite410649719033368117t_unit @ T4 )
= N )
=> ~ ( finite4290736615968046902t_unit @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_768_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S ) )
=> ~ ! [T4: set_nat] :
( ( ord_less_eq_set_nat @ T4 @ S )
=> ( ( ( finite_card_nat @ T4 )
= N )
=> ~ ( finite_finite_nat @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_769_finite__if__finite__subsets__card__bdd,axiom,
! [F4: set_Product_unit,C3: nat] :
( ! [G2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ G2 @ F4 )
=> ( ( finite4290736615968046902t_unit @ G2 )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ G2 ) @ C3 ) ) )
=> ( ( finite4290736615968046902t_unit @ F4 )
& ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ F4 ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_770_finite__if__finite__subsets__card__bdd,axiom,
! [F4: set_nat,C3: nat] :
( ! [G2: set_nat] :
( ( ord_less_eq_set_nat @ G2 @ F4 )
=> ( ( finite_finite_nat @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G2 ) @ C3 ) ) )
=> ( ( finite_finite_nat @ F4 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F4 ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_771_inj__on__iff__eq__card,axiom,
! [A: set_Product_unit,F: product_unit > product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( inj_on8151373323710067377t_unit @ F @ A )
= ( ( finite410649719033368117t_unit @ ( image_405062704495631173t_unit @ F @ A ) )
= ( finite410649719033368117t_unit @ A ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_772_inj__on__iff__eq__card,axiom,
! [A: set_a,F: a > nat] :
( ( finite_finite_a @ A )
=> ( ( inj_on_a_nat @ F @ A )
= ( ( finite_card_nat @ ( image_a_nat @ F @ A ) )
= ( finite_card_a @ A ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_773_inj__on__iff__eq__card,axiom,
! [A: set_Product_unit,F: product_unit > nat] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( inj_on8430439091780834860it_nat @ F @ A )
= ( ( finite_card_nat @ ( image_875570014554754200it_nat @ F @ A ) )
= ( finite410649719033368117t_unit @ A ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_774_inj__on__iff__eq__card,axiom,
! [A: set_nat,F: nat > product_unit] :
( ( finite_finite_nat @ A )
=> ( ( inj_on7061601236592826506t_unit @ F @ A )
= ( ( finite410649719033368117t_unit @ ( image_8730104196221521654t_unit @ F @ A ) )
= ( finite_card_nat @ A ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_775_inj__on__iff__eq__card,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( inj_on_nat_nat @ F @ A )
= ( ( finite_card_nat @ ( image_nat_nat @ F @ A ) )
= ( finite_card_nat @ A ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_776_eq__card__imp__inj__on,axiom,
! [A: set_Product_unit,F: product_unit > product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( ( finite410649719033368117t_unit @ ( image_405062704495631173t_unit @ F @ A ) )
= ( finite410649719033368117t_unit @ A ) )
=> ( inj_on8151373323710067377t_unit @ F @ A ) ) ) ).
% eq_card_imp_inj_on
thf(fact_777_eq__card__imp__inj__on,axiom,
! [A: set_a,F: a > nat] :
( ( finite_finite_a @ A )
=> ( ( ( finite_card_nat @ ( image_a_nat @ F @ A ) )
= ( finite_card_a @ A ) )
=> ( inj_on_a_nat @ F @ A ) ) ) ).
% eq_card_imp_inj_on
thf(fact_778_eq__card__imp__inj__on,axiom,
! [A: set_Product_unit,F: product_unit > nat] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( ( finite_card_nat @ ( image_875570014554754200it_nat @ F @ A ) )
= ( finite410649719033368117t_unit @ A ) )
=> ( inj_on8430439091780834860it_nat @ F @ A ) ) ) ).
% eq_card_imp_inj_on
thf(fact_779_eq__card__imp__inj__on,axiom,
! [A: set_nat,F: nat > product_unit] :
( ( finite_finite_nat @ A )
=> ( ( ( finite410649719033368117t_unit @ ( image_8730104196221521654t_unit @ F @ A ) )
= ( finite_card_nat @ A ) )
=> ( inj_on7061601236592826506t_unit @ F @ A ) ) ) ).
% eq_card_imp_inj_on
thf(fact_780_eq__card__imp__inj__on,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ( finite_card_nat @ ( image_nat_nat @ F @ A ) )
= ( finite_card_nat @ A ) )
=> ( inj_on_nat_nat @ F @ A ) ) ) ).
% eq_card_imp_inj_on
thf(fact_781_surj__card__le,axiom,
! [A: set_Product_unit,B2: set_Product_unit,F: product_unit > product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( ord_le3507040750410214029t_unit @ B2 @ ( image_405062704495631173t_unit @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ B2 ) @ ( finite410649719033368117t_unit @ A ) ) ) ) ).
% surj_card_le
thf(fact_782_surj__card__le,axiom,
! [A: set_a,B2: set_nat,F: a > nat] :
( ( finite_finite_a @ A )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_a_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_a @ A ) ) ) ) ).
% surj_card_le
thf(fact_783_surj__card__le,axiom,
! [A: set_Product_unit,B2: set_nat,F: product_unit > nat] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_875570014554754200it_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite410649719033368117t_unit @ A ) ) ) ) ).
% surj_card_le
thf(fact_784_surj__card__le,axiom,
! [A: set_nat,B2: set_Product_unit,F: nat > product_unit] :
( ( finite_finite_nat @ A )
=> ( ( ord_le3507040750410214029t_unit @ B2 @ ( image_8730104196221521654t_unit @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ B2 ) @ ( finite_card_nat @ A ) ) ) ) ).
% surj_card_le
thf(fact_785_surj__card__le,axiom,
! [A: set_nat,B2: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat @ A ) ) ) ) ).
% surj_card_le
thf(fact_786_card__le__Suc0__iff__eq,axiom,
! [A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( suc @ zero_zero_nat ) )
= ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A )
=> ! [Y3: product_unit] :
( ( member_Product_unit @ Y3 @ A )
=> ( X3 = Y3 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_787_card__le__Suc0__iff__eq,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( suc @ zero_zero_nat ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ A )
=> ( X3 = Y3 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_788_card__le__Suc__iff,axiom,
! [N: nat,A: set_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_a @ A ) )
= ( ? [A6: a,B4: set_a] :
( ( A
= ( insert_a2 @ A6 @ B4 ) )
& ~ ( member_a @ A6 @ B4 )
& ( ord_less_eq_nat @ N @ ( finite_card_a @ B4 ) )
& ( finite_finite_a @ B4 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_789_card__le__Suc__iff,axiom,
! [N: nat,A: set_Product_unit] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite410649719033368117t_unit @ A ) )
= ( ? [A6: product_unit,B4: set_Product_unit] :
( ( A
= ( insert_Product_unit @ A6 @ B4 ) )
& ~ ( member_Product_unit @ A6 @ B4 )
& ( ord_less_eq_nat @ N @ ( finite410649719033368117t_unit @ B4 ) )
& ( finite4290736615968046902t_unit @ B4 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_790_card__le__Suc__iff,axiom,
! [N: nat,A: set_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_nat @ A ) )
= ( ? [A6: nat,B4: set_nat] :
( ( A
= ( insert_nat2 @ A6 @ B4 ) )
& ~ ( member_nat @ A6 @ B4 )
& ( ord_less_eq_nat @ N @ ( finite_card_nat @ B4 ) )
& ( finite_finite_nat @ B4 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_791_card__bij__eq,axiom,
! [F: product_unit > product_unit,A: set_Product_unit,B2: set_Product_unit,G: product_unit > product_unit] :
( ( inj_on8151373323710067377t_unit @ F @ A )
=> ( ( ord_le3507040750410214029t_unit @ ( image_405062704495631173t_unit @ F @ A ) @ B2 )
=> ( ( inj_on8151373323710067377t_unit @ G @ B2 )
=> ( ( ord_le3507040750410214029t_unit @ ( image_405062704495631173t_unit @ G @ B2 ) @ A )
=> ( ( finite4290736615968046902t_unit @ A )
=> ( ( finite4290736615968046902t_unit @ B2 )
=> ( ( finite410649719033368117t_unit @ A )
= ( finite410649719033368117t_unit @ B2 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_792_card__bij__eq,axiom,
! [F: a > nat,A: set_a,B2: set_nat,G: nat > a] :
( ( inj_on_a_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ B2 )
=> ( ( inj_on_nat_a @ G @ B2 )
=> ( ( ord_less_eq_set_a @ ( image_nat_a @ G @ B2 ) @ A )
=> ( ( finite_finite_a @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ( finite_card_a @ A )
= ( finite_card_nat @ B2 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_793_card__bij__eq,axiom,
! [F: product_unit > nat,A: set_Product_unit,B2: set_nat,G: nat > product_unit] :
( ( inj_on8430439091780834860it_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ ( image_875570014554754200it_nat @ F @ A ) @ B2 )
=> ( ( inj_on7061601236592826506t_unit @ G @ B2 )
=> ( ( ord_le3507040750410214029t_unit @ ( image_8730104196221521654t_unit @ G @ B2 ) @ A )
=> ( ( finite4290736615968046902t_unit @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ( finite410649719033368117t_unit @ A )
= ( finite_card_nat @ B2 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_794_card__bij__eq,axiom,
! [F: nat > a,A: set_nat,B2: set_a,G: a > nat] :
( ( inj_on_nat_a @ F @ A )
=> ( ( ord_less_eq_set_a @ ( image_nat_a @ F @ A ) @ B2 )
=> ( ( inj_on_a_nat @ G @ B2 )
=> ( ( ord_less_eq_set_nat @ ( image_a_nat @ G @ B2 ) @ A )
=> ( ( finite_finite_nat @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ( finite_card_nat @ A )
= ( finite_card_a @ B2 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_795_card__bij__eq,axiom,
! [F: nat > product_unit,A: set_nat,B2: set_Product_unit,G: product_unit > nat] :
( ( inj_on7061601236592826506t_unit @ F @ A )
=> ( ( ord_le3507040750410214029t_unit @ ( image_8730104196221521654t_unit @ F @ A ) @ B2 )
=> ( ( inj_on8430439091780834860it_nat @ G @ B2 )
=> ( ( ord_less_eq_set_nat @ ( image_875570014554754200it_nat @ G @ B2 ) @ A )
=> ( ( finite_finite_nat @ A )
=> ( ( finite4290736615968046902t_unit @ B2 )
=> ( ( finite_card_nat @ A )
= ( finite410649719033368117t_unit @ B2 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_796_card__bij__eq,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat,G: nat > nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B2 )
=> ( ( inj_on_nat_nat @ G @ B2 )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ G @ B2 ) @ A )
=> ( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ( finite_card_nat @ A )
= ( finite_card_nat @ B2 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_797_surjective__iff__injective__gen,axiom,
! [S: set_Product_unit,T3: set_Product_unit,F: product_unit > product_unit] :
( ( finite4290736615968046902t_unit @ S )
=> ( ( finite4290736615968046902t_unit @ T3 )
=> ( ( ( finite410649719033368117t_unit @ S )
= ( finite410649719033368117t_unit @ T3 ) )
=> ( ( ord_le3507040750410214029t_unit @ ( image_405062704495631173t_unit @ F @ S ) @ T3 )
=> ( ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ T3 )
=> ? [Y3: product_unit] :
( ( member_Product_unit @ Y3 @ S )
& ( ( F @ Y3 )
= X3 ) ) ) )
= ( inj_on8151373323710067377t_unit @ F @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_798_surjective__iff__injective__gen,axiom,
! [S: set_a,T3: set_nat,F: a > nat] :
( ( finite_finite_a @ S )
=> ( ( finite_finite_nat @ T3 )
=> ( ( ( finite_card_a @ S )
= ( finite_card_nat @ T3 ) )
=> ( ( ord_less_eq_set_nat @ ( image_a_nat @ F @ S ) @ T3 )
=> ( ( ! [X3: nat] :
( ( member_nat @ X3 @ T3 )
=> ? [Y3: a] :
( ( member_a @ Y3 @ S )
& ( ( F @ Y3 )
= X3 ) ) ) )
= ( inj_on_a_nat @ F @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_799_surjective__iff__injective__gen,axiom,
! [S: set_Product_unit,T3: set_nat,F: product_unit > nat] :
( ( finite4290736615968046902t_unit @ S )
=> ( ( finite_finite_nat @ T3 )
=> ( ( ( finite410649719033368117t_unit @ S )
= ( finite_card_nat @ T3 ) )
=> ( ( ord_less_eq_set_nat @ ( image_875570014554754200it_nat @ F @ S ) @ T3 )
=> ( ( ! [X3: nat] :
( ( member_nat @ X3 @ T3 )
=> ? [Y3: product_unit] :
( ( member_Product_unit @ Y3 @ S )
& ( ( F @ Y3 )
= X3 ) ) ) )
= ( inj_on8430439091780834860it_nat @ F @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_800_surjective__iff__injective__gen,axiom,
! [S: set_nat,T3: set_Product_unit,F: nat > product_unit] :
( ( finite_finite_nat @ S )
=> ( ( finite4290736615968046902t_unit @ T3 )
=> ( ( ( finite_card_nat @ S )
= ( finite410649719033368117t_unit @ T3 ) )
=> ( ( ord_le3507040750410214029t_unit @ ( image_8730104196221521654t_unit @ F @ S ) @ T3 )
=> ( ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ T3 )
=> ? [Y3: nat] :
( ( member_nat @ Y3 @ S )
& ( ( F @ Y3 )
= X3 ) ) ) )
= ( inj_on7061601236592826506t_unit @ F @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_801_surjective__iff__injective__gen,axiom,
! [S: set_nat,T3: set_nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( finite_finite_nat @ T3 )
=> ( ( ( finite_card_nat @ S )
= ( finite_card_nat @ T3 ) )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ S ) @ T3 )
=> ( ( ! [X3: nat] :
( ( member_nat @ X3 @ T3 )
=> ? [Y3: nat] :
( ( member_nat @ Y3 @ S )
& ( ( F @ Y3 )
= X3 ) ) ) )
= ( inj_on_nat_nat @ F @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_802_stirling__row__code_I2_J,axiom,
! [N: nat] :
( ( stirling_row @ ( suc @ N ) )
= ( stirling_row_aux_nat @ N @ zero_zero_nat @ ( stirling_row @ N ) ) ) ).
% stirling_row_code(2)
thf(fact_803_remdups__adj__singleton__iff,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ ( remdups_adj_a @ Xs ) )
= ( suc @ zero_zero_nat ) )
= ( ( Xs != nil_a )
& ( Xs
= ( replicate_a @ ( size_size_list_a @ Xs ) @ ( hd_a @ Xs ) ) ) ) ) ).
% remdups_adj_singleton_iff
thf(fact_804_remdups__adj__singleton__iff,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ ( remdups_adj_nat @ Xs ) )
= ( suc @ zero_zero_nat ) )
= ( ( Xs != nil_nat )
& ( Xs
= ( replicate_nat @ ( size_size_list_nat @ Xs ) @ ( hd_nat @ Xs ) ) ) ) ) ).
% remdups_adj_singleton_iff
thf(fact_805_set__replicate,axiom,
! [N: nat,X: a] :
( ( N != zero_zero_nat )
=> ( ( set_a2 @ ( replicate_a @ N @ X ) )
= ( insert_a2 @ X @ bot_bot_set_a ) ) ) ).
% set_replicate
thf(fact_806_set__replicate,axiom,
! [N: nat,X: nat] :
( ( N != zero_zero_nat )
=> ( ( set_nat2 @ ( replicate_nat @ N @ X ) )
= ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ).
% set_replicate
thf(fact_807_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_a @ ( coset_a @ nil_a ) @ ( set_a2 @ nil_a ) ) ).
% subset_code(3)
thf(fact_808_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_nat @ ( coset_nat @ nil_nat ) @ ( set_nat2 @ nil_nat ) ) ).
% subset_code(3)
thf(fact_809_infinite__countable__subset,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ? [F3: nat > nat] :
( ( inj_on_nat_nat @ F3 @ top_top_set_nat )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ top_top_set_nat ) @ S ) ) ) ).
% infinite_countable_subset
thf(fact_810_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_811_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_812_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_813_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_814_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_815_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_816_UNIV__I,axiom,
! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% UNIV_I
thf(fact_817_UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_818_UNIV__I,axiom,
! [X: product_unit] : ( member_Product_unit @ X @ top_to1996260823553986621t_unit ) ).
% UNIV_I
thf(fact_819_image__is__empty,axiom,
! [F: a > nat,A: set_a] :
( ( ( image_a_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_820_image__is__empty,axiom,
! [F: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_821_empty__is__image,axiom,
! [F: a > nat,A: set_a] :
( ( bot_bot_set_nat
= ( image_a_nat @ F @ A ) )
= ( A = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_822_empty__is__image,axiom,
! [F: nat > nat,A: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_823_image__empty,axiom,
! [F: a > nat] :
( ( image_a_nat @ F @ bot_bot_set_a )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_824_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_825_subset__empty,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_826_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_827_singletonI,axiom,
! [A4: a] : ( member_a @ A4 @ ( insert_a2 @ A4 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_828_singletonI,axiom,
! [A4: nat] : ( member_nat @ A4 @ ( insert_nat2 @ A4 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_829_inj__on__empty,axiom,
! [F: a > nat] : ( inj_on_a_nat @ F @ bot_bot_set_a ) ).
% inj_on_empty
thf(fact_830_inj__on__empty,axiom,
! [F: nat > nat] : ( inj_on_nat_nat @ F @ bot_bot_set_nat ) ).
% inj_on_empty
thf(fact_831_Pow__empty,axiom,
( ( pow_nat @ bot_bot_set_nat )
= ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat ) ) ).
% Pow_empty
thf(fact_832_Pow__singleton__iff,axiom,
! [X6: set_nat,Y5: set_nat] :
( ( ( pow_nat @ X6 )
= ( insert_set_nat @ Y5 @ bot_bot_set_set_nat ) )
= ( ( X6 = bot_bot_set_nat )
& ( Y5 = bot_bot_set_nat ) ) ) ).
% Pow_singleton_iff
thf(fact_833_Pow__UNIV,axiom,
( ( pow_nat @ top_top_set_nat )
= top_top_set_set_nat ) ).
% Pow_UNIV
thf(fact_834_Pow__UNIV,axiom,
( ( pow_Product_unit @ top_to1996260823553986621t_unit )
= top_to1767297665138865437t_unit ) ).
% Pow_UNIV
thf(fact_835_hd__prefixes,axiom,
! [Xs: list_a] :
( ( hd_list_a @ ( prefixes_a @ Xs ) )
= nil_a ) ).
% hd_prefixes
thf(fact_836_hd__prefixes,axiom,
! [Xs: list_nat] :
( ( hd_list_nat @ ( prefixes_nat @ Xs ) )
= nil_nat ) ).
% hd_prefixes
thf(fact_837_hd__suffixes,axiom,
! [Xs: list_a] :
( ( hd_list_a @ ( suffixes_a @ Xs ) )
= nil_a ) ).
% hd_suffixes
thf(fact_838_hd__suffixes,axiom,
! [Xs: list_nat] :
( ( hd_list_nat @ ( suffixes_nat @ Xs ) )
= nil_nat ) ).
% hd_suffixes
thf(fact_839_singleton__insert__inj__eq_H,axiom,
! [A4: a,A: set_a,B: a] :
( ( ( insert_a2 @ A4 @ A )
= ( insert_a2 @ B @ bot_bot_set_a ) )
= ( ( A4 = B )
& ( ord_less_eq_set_a @ A @ ( insert_a2 @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_840_singleton__insert__inj__eq_H,axiom,
! [A4: nat,A: set_nat,B: nat] :
( ( ( insert_nat2 @ A4 @ A )
= ( insert_nat2 @ B @ bot_bot_set_nat ) )
= ( ( A4 = B )
& ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_841_singleton__insert__inj__eq,axiom,
! [B: a,A4: a,A: set_a] :
( ( ( insert_a2 @ B @ bot_bot_set_a )
= ( insert_a2 @ A4 @ A ) )
= ( ( A4 = B )
& ( ord_less_eq_set_a @ A @ ( insert_a2 @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_842_singleton__insert__inj__eq,axiom,
! [B: nat,A4: nat,A: set_nat] :
( ( ( insert_nat2 @ B @ bot_bot_set_nat )
= ( insert_nat2 @ A4 @ A ) )
= ( ( A4 = B )
& ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_843_set__empty,axiom,
! [Xs: list_a] :
( ( ( set_a2 @ Xs )
= bot_bot_set_a )
= ( Xs = nil_a ) ) ).
% set_empty
thf(fact_844_set__empty,axiom,
! [Xs: list_nat] :
( ( ( set_nat2 @ Xs )
= bot_bot_set_nat )
= ( Xs = nil_nat ) ) ).
% set_empty
thf(fact_845_set__empty2,axiom,
! [Xs: list_a] :
( ( bot_bot_set_a
= ( set_a2 @ Xs ) )
= ( Xs = nil_a ) ) ).
% set_empty2
thf(fact_846_set__empty2,axiom,
! [Xs: list_nat] :
( ( bot_bot_set_nat
= ( set_nat2 @ Xs ) )
= ( Xs = nil_nat ) ) ).
% set_empty2
thf(fact_847_card_Oempty,axiom,
( ( finite410649719033368117t_unit @ bot_bo3957492148770167129t_unit )
= zero_zero_nat ) ).
% card.empty
thf(fact_848_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_849_inj__map__eq__map,axiom,
! [F: a > nat,Xs: list_a,Ys: list_a] :
( ( inj_on_a_nat @ F @ top_top_set_a )
=> ( ( ( map_a_nat @ F @ Xs )
= ( map_a_nat @ F @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_map_eq_map
thf(fact_850_inj__map__eq__map,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ F @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_map_eq_map
thf(fact_851_hd__append2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Xs ) ) ) ).
% hd_append2
thf(fact_852_hd__append2,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs != nil_nat )
=> ( ( hd_nat @ ( append_nat @ Xs @ Ys ) )
= ( hd_nat @ Xs ) ) ) ).
% hd_append2
thf(fact_853_the__elem__eq,axiom,
! [X: a] :
( ( the_elem_a @ ( insert_a2 @ X @ bot_bot_set_a ) )
= X ) ).
% the_elem_eq
thf(fact_854_the__elem__eq,axiom,
! [X: nat] :
( ( the_elem_nat @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= X ) ).
% the_elem_eq
thf(fact_855_card__0__eq,axiom,
! [A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( ( finite410649719033368117t_unit @ A )
= zero_zero_nat )
= ( A = bot_bo3957492148770167129t_unit ) ) ) ).
% card_0_eq
thf(fact_856_card__0__eq,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( ( finite_card_nat @ A )
= zero_zero_nat )
= ( A = bot_bot_set_nat ) ) ) ).
% card_0_eq
thf(fact_857_inj__mapI,axiom,
! [F: a > nat] :
( ( inj_on_a_nat @ F @ top_top_set_a )
=> ( inj_on6731145966573583411st_nat @ ( map_a_nat @ F ) @ top_top_set_list_a ) ) ).
% inj_mapI
thf(fact_858_inj__mapI,axiom,
! [F: nat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( inj_on3049792774292151987st_nat @ ( map_nat_nat @ F ) @ top_top_set_list_nat ) ) ).
% inj_mapI
thf(fact_859_inj__map,axiom,
! [F: a > nat] :
( ( inj_on6731145966573583411st_nat @ ( map_a_nat @ F ) @ top_top_set_list_a )
= ( inj_on_a_nat @ F @ top_top_set_a ) ) ).
% inj_map
thf(fact_860_inj__map,axiom,
! [F: nat > nat] :
( ( inj_on3049792774292151987st_nat @ ( map_nat_nat @ F ) @ top_top_set_list_nat )
= ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ).
% inj_map
thf(fact_861_range__eq__singletonD,axiom,
! [F: a > nat,A4: nat,X: a] :
( ( ( image_a_nat @ F @ top_top_set_a )
= ( insert_nat2 @ A4 @ bot_bot_set_nat ) )
=> ( ( F @ X )
= A4 ) ) ).
% range_eq_singletonD
thf(fact_862_range__eq__singletonD,axiom,
! [F: nat > a,A4: a,X: nat] :
( ( ( image_nat_a @ F @ top_top_set_nat )
= ( insert_a2 @ A4 @ bot_bot_set_a ) )
=> ( ( F @ X )
= A4 ) ) ).
% range_eq_singletonD
thf(fact_863_range__eq__singletonD,axiom,
! [F: nat > nat,A4: nat,X: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= ( insert_nat2 @ A4 @ bot_bot_set_nat ) )
=> ( ( F @ X )
= A4 ) ) ).
% range_eq_singletonD
thf(fact_864_range__eq__singletonD,axiom,
! [F: product_unit > a,A4: a,X: product_unit] :
( ( ( image_Product_unit_a @ F @ top_to1996260823553986621t_unit )
= ( insert_a2 @ A4 @ bot_bot_set_a ) )
=> ( ( F @ X )
= A4 ) ) ).
% range_eq_singletonD
thf(fact_865_range__eq__singletonD,axiom,
! [F: product_unit > nat,A4: nat,X: product_unit] :
( ( ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit )
= ( insert_nat2 @ A4 @ bot_bot_set_nat ) )
=> ( ( F @ X )
= A4 ) ) ).
% range_eq_singletonD
thf(fact_866_emptyE,axiom,
! [A4: a] :
~ ( member_a @ A4 @ bot_bot_set_a ) ).
% emptyE
thf(fact_867_emptyE,axiom,
! [A4: nat] :
~ ( member_nat @ A4 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_868_equals0D,axiom,
! [A: set_a,A4: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A4 @ A ) ) ).
% equals0D
thf(fact_869_equals0D,axiom,
! [A: set_nat,A4: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A4 @ A ) ) ).
% equals0D
thf(fact_870_equals0I,axiom,
! [A: set_a] :
( ! [Y2: a] :
~ ( member_a @ Y2 @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_871_equals0I,axiom,
! [A: set_nat] :
( ! [Y2: nat] :
~ ( member_nat @ Y2 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_872_UNIV__eq__I,axiom,
! [A: set_a] :
( ! [X2: a] : ( member_a @ X2 @ A )
=> ( top_top_set_a = A ) ) ).
% UNIV_eq_I
thf(fact_873_UNIV__eq__I,axiom,
! [A: set_nat] :
( ! [X2: nat] : ( member_nat @ X2 @ A )
=> ( top_top_set_nat = A ) ) ).
% UNIV_eq_I
thf(fact_874_UNIV__eq__I,axiom,
! [A: set_Product_unit] :
( ! [X2: product_unit] : ( member_Product_unit @ X2 @ A )
=> ( top_to1996260823553986621t_unit = A ) ) ).
% UNIV_eq_I
thf(fact_875_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_876_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_877_UNIV__witness,axiom,
? [X2: a] : ( member_a @ X2 @ top_top_set_a ) ).
% UNIV_witness
thf(fact_878_UNIV__witness,axiom,
? [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_879_UNIV__witness,axiom,
? [X2: product_unit] : ( member_Product_unit @ X2 @ top_to1996260823553986621t_unit ) ).
% UNIV_witness
thf(fact_880_empty__not__UNIV,axiom,
bot_bot_set_nat != top_top_set_nat ).
% empty_not_UNIV
thf(fact_881_empty__not__UNIV,axiom,
bot_bo3957492148770167129t_unit != top_to1996260823553986621t_unit ).
% empty_not_UNIV
thf(fact_882_UNIV__coset,axiom,
( top_top_set_a
= ( coset_a @ nil_a ) ) ).
% UNIV_coset
thf(fact_883_UNIV__coset,axiom,
( top_top_set_nat
= ( coset_nat @ nil_nat ) ) ).
% UNIV_coset
thf(fact_884_UNIV__coset,axiom,
( top_to1996260823553986621t_unit
= ( coset_Product_unit @ nil_Product_unit ) ) ).
% UNIV_coset
thf(fact_885_list_Osel_I1_J,axiom,
! [X21: a,X22: list_a] :
( ( hd_a @ ( cons_a @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_886_list_Osel_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( ( hd_nat @ ( cons_nat @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_887_range__eqI,axiom,
! [B: nat,F: a > nat,X: a] :
( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_a_nat @ F @ top_top_set_a ) ) ) ).
% range_eqI
thf(fact_888_range__eqI,axiom,
! [B: a,F: nat > a,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_nat_a @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_889_range__eqI,axiom,
! [B: nat,F: nat > nat,X: nat] :
( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_890_range__eqI,axiom,
! [B: a,F: product_unit > a,X: product_unit] :
( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_Product_unit_a @ F @ top_to1996260823553986621t_unit ) ) ) ).
% range_eqI
thf(fact_891_range__eqI,axiom,
! [B: nat,F: product_unit > nat,X: product_unit] :
( ( B
= ( F @ X ) )
=> ( member_nat @ B @ ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit ) ) ) ).
% range_eqI
thf(fact_892_rangeI,axiom,
! [F: a > nat,X: a] : ( member_nat @ ( F @ X ) @ ( image_a_nat @ F @ top_top_set_a ) ) ).
% rangeI
thf(fact_893_rangeI,axiom,
! [F: nat > a,X: nat] : ( member_a @ ( F @ X ) @ ( image_nat_a @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_894_rangeI,axiom,
! [F: nat > nat,X: nat] : ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_895_rangeI,axiom,
! [F: product_unit > a,X: product_unit] : ( member_a @ ( F @ X ) @ ( image_Product_unit_a @ F @ top_to1996260823553986621t_unit ) ) ).
% rangeI
thf(fact_896_rangeI,axiom,
! [F: product_unit > nat,X: product_unit] : ( member_nat @ ( F @ X ) @ ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit ) ) ).
% rangeI
thf(fact_897_surj__def,axiom,
! [F: a > nat] :
( ( ( image_a_nat @ F @ top_top_set_a )
= top_top_set_nat )
= ( ! [Y3: nat] :
? [X3: a] :
( Y3
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_898_surj__def,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ! [Y3: nat] :
? [X3: nat] :
( Y3
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_899_surj__def,axiom,
! [F: nat > product_unit] :
( ( ( image_8730104196221521654t_unit @ F @ top_top_set_nat )
= top_to1996260823553986621t_unit )
= ( ! [Y3: product_unit] :
? [X3: nat] :
( Y3
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_900_surj__def,axiom,
! [F: product_unit > nat] :
( ( ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit )
= top_top_set_nat )
= ( ! [Y3: nat] :
? [X3: product_unit] :
( Y3
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_901_surj__def,axiom,
! [F: product_unit > product_unit] :
( ( ( image_405062704495631173t_unit @ F @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit )
= ( ! [Y3: product_unit] :
? [X3: product_unit] :
( Y3
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_902_surjI,axiom,
! [G: a > nat,F: nat > a] :
( ! [X2: nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_a_nat @ G @ top_top_set_a )
= top_top_set_nat ) ) ).
% surjI
thf(fact_903_surjI,axiom,
! [G: nat > nat,F: nat > nat] :
( ! [X2: nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_904_surjI,axiom,
! [G: nat > product_unit,F: product_unit > nat] :
( ! [X2: product_unit] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_8730104196221521654t_unit @ G @ top_top_set_nat )
= top_to1996260823553986621t_unit ) ) ).
% surjI
thf(fact_905_surjI,axiom,
! [G: product_unit > nat,F: nat > product_unit] :
( ! [X2: nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_875570014554754200it_nat @ G @ top_to1996260823553986621t_unit )
= top_top_set_nat ) ) ).
% surjI
thf(fact_906_surjI,axiom,
! [G: product_unit > product_unit,F: product_unit > product_unit] :
( ! [X2: product_unit] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_405062704495631173t_unit @ G @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit ) ) ).
% surjI
thf(fact_907_surjE,axiom,
! [F: a > nat,Y: nat] :
( ( ( image_a_nat @ F @ top_top_set_a )
= top_top_set_nat )
=> ~ ! [X2: a] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_908_surjE,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ~ ! [X2: nat] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_909_surjE,axiom,
! [F: nat > product_unit,Y: product_unit] :
( ( ( image_8730104196221521654t_unit @ F @ top_top_set_nat )
= top_to1996260823553986621t_unit )
=> ~ ! [X2: nat] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_910_surjE,axiom,
! [F: product_unit > nat,Y: nat] :
( ( ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit )
= top_top_set_nat )
=> ~ ! [X2: product_unit] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_911_surjE,axiom,
! [F: product_unit > product_unit,Y: product_unit] :
( ( ( image_405062704495631173t_unit @ F @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit )
=> ~ ! [X2: product_unit] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_912_surjD,axiom,
! [F: a > nat,Y: nat] :
( ( ( image_a_nat @ F @ top_top_set_a )
= top_top_set_nat )
=> ? [X2: a] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_913_surjD,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ? [X2: nat] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_914_surjD,axiom,
! [F: nat > product_unit,Y: product_unit] :
( ( ( image_8730104196221521654t_unit @ F @ top_top_set_nat )
= top_to1996260823553986621t_unit )
=> ? [X2: nat] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_915_surjD,axiom,
! [F: product_unit > nat,Y: nat] :
( ( ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit )
= top_top_set_nat )
=> ? [X2: product_unit] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_916_surjD,axiom,
! [F: product_unit > product_unit,Y: product_unit] :
( ( ( image_405062704495631173t_unit @ F @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit )
=> ? [X2: product_unit] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_917_subset__UNIV,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_918_subset__UNIV,axiom,
! [A: set_Product_unit] : ( ord_le3507040750410214029t_unit @ A @ top_to1996260823553986621t_unit ) ).
% subset_UNIV
thf(fact_919_insert__UNIV,axiom,
! [X: a] :
( ( insert_a2 @ X @ top_top_set_a )
= top_top_set_a ) ).
% insert_UNIV
thf(fact_920_insert__UNIV,axiom,
! [X: nat] :
( ( insert_nat2 @ X @ top_top_set_nat )
= top_top_set_nat ) ).
% insert_UNIV
thf(fact_921_insert__UNIV,axiom,
! [X: product_unit] :
( ( insert_Product_unit @ X @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit ) ).
% insert_UNIV
thf(fact_922_singleton__inject,axiom,
! [A4: a,B: a] :
( ( ( insert_a2 @ A4 @ bot_bot_set_a )
= ( insert_a2 @ B @ bot_bot_set_a ) )
=> ( A4 = B ) ) ).
% singleton_inject
thf(fact_923_singleton__inject,axiom,
! [A4: nat,B: nat] :
( ( ( insert_nat2 @ A4 @ bot_bot_set_nat )
= ( insert_nat2 @ B @ bot_bot_set_nat ) )
=> ( A4 = B ) ) ).
% singleton_inject
thf(fact_924_insert__not__empty,axiom,
! [A4: a,A: set_a] :
( ( insert_a2 @ A4 @ A )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_925_insert__not__empty,axiom,
! [A4: nat,A: set_nat] :
( ( insert_nat2 @ A4 @ A )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_926_doubleton__eq__iff,axiom,
! [A4: a,B: a,C: a,D2: a] :
( ( ( insert_a2 @ A4 @ ( insert_a2 @ B @ bot_bot_set_a ) )
= ( insert_a2 @ C @ ( insert_a2 @ D2 @ bot_bot_set_a ) ) )
= ( ( ( A4 = C )
& ( B = D2 ) )
| ( ( A4 = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_927_doubleton__eq__iff,axiom,
! [A4: nat,B: nat,C: nat,D2: nat] :
( ( ( insert_nat2 @ A4 @ ( insert_nat2 @ B @ bot_bot_set_nat ) )
= ( insert_nat2 @ C @ ( insert_nat2 @ D2 @ bot_bot_set_nat ) ) )
= ( ( ( A4 = C )
& ( B = D2 ) )
| ( ( A4 = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_928_singleton__iff,axiom,
! [B: a,A4: a] :
( ( member_a @ B @ ( insert_a2 @ A4 @ bot_bot_set_a ) )
= ( B = A4 ) ) ).
% singleton_iff
thf(fact_929_singleton__iff,axiom,
! [B: nat,A4: nat] :
( ( member_nat @ B @ ( insert_nat2 @ A4 @ bot_bot_set_nat ) )
= ( B = A4 ) ) ).
% singleton_iff
thf(fact_930_singletonD,axiom,
! [B: a,A4: a] :
( ( member_a @ B @ ( insert_a2 @ A4 @ bot_bot_set_a ) )
=> ( B = A4 ) ) ).
% singletonD
thf(fact_931_singletonD,axiom,
! [B: nat,A4: nat] :
( ( member_nat @ B @ ( insert_nat2 @ A4 @ bot_bot_set_nat ) )
=> ( B = A4 ) ) ).
% singletonD
thf(fact_932_injD,axiom,
! [F: a > nat,X: a,Y: a] :
( ( inj_on_a_nat @ F @ top_top_set_a )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( X = Y ) ) ) ).
% injD
thf(fact_933_injD,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( X = Y ) ) ) ).
% injD
thf(fact_934_injI,axiom,
! [F: a > nat] :
( ! [X2: a,Y2: a] :
( ( ( F @ X2 )
= ( F @ Y2 ) )
=> ( X2 = Y2 ) )
=> ( inj_on_a_nat @ F @ top_top_set_a ) ) ).
% injI
thf(fact_935_injI,axiom,
! [F: nat > nat] :
( ! [X2: nat,Y2: nat] :
( ( ( F @ X2 )
= ( F @ Y2 ) )
=> ( X2 = Y2 ) )
=> ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ).
% injI
thf(fact_936_inj__eq,axiom,
! [F: a > nat,X: a,Y: a] :
( ( inj_on_a_nat @ F @ top_top_set_a )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% inj_eq
thf(fact_937_inj__eq,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% inj_eq
thf(fact_938_inj__def,axiom,
! [F: a > nat] :
( ( inj_on_a_nat @ F @ top_top_set_a )
= ( ! [X3: a,Y3: a] :
( ( ( F @ X3 )
= ( F @ Y3 ) )
=> ( X3 = Y3 ) ) ) ) ).
% inj_def
thf(fact_939_inj__def,axiom,
! [F: nat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
= ( ! [X3: nat,Y3: nat] :
( ( ( F @ X3 )
= ( F @ Y3 ) )
=> ( X3 = Y3 ) ) ) ) ).
% inj_def
thf(fact_940_Pow__bottom,axiom,
! [B2: set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( pow_nat @ B2 ) ) ).
% Pow_bottom
thf(fact_941_hd__concat,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
=> ( ( ( hd_list_a @ Xs )
!= nil_a )
=> ( ( hd_a @ ( concat_a @ Xs ) )
= ( hd_a @ ( hd_list_a @ Xs ) ) ) ) ) ).
% hd_concat
thf(fact_942_hd__concat,axiom,
! [Xs: list_list_nat] :
( ( Xs != nil_list_nat )
=> ( ( ( hd_list_nat @ Xs )
!= nil_nat )
=> ( ( hd_nat @ ( concat_nat @ Xs ) )
= ( hd_nat @ ( hd_list_nat @ Xs ) ) ) ) ) ).
% hd_concat
thf(fact_943_Pow__set_I1_J,axiom,
( ( pow_a @ ( set_a2 @ nil_a ) )
= ( insert_set_a @ bot_bot_set_a @ bot_bot_set_set_a ) ) ).
% Pow_set(1)
thf(fact_944_Pow__set_I1_J,axiom,
( ( pow_nat @ ( set_nat2 @ nil_nat ) )
= ( insert_set_nat @ bot_bot_set_nat @ bot_bot_set_set_nat ) ) ).
% Pow_set(1)
thf(fact_945_inj__mapD,axiom,
! [F: a > nat] :
( ( inj_on6731145966573583411st_nat @ ( map_a_nat @ F ) @ top_top_set_list_a )
=> ( inj_on_a_nat @ F @ top_top_set_a ) ) ).
% inj_mapD
thf(fact_946_inj__mapD,axiom,
! [F: nat > nat] :
( ( inj_on3049792774292151987st_nat @ ( map_nat_nat @ F ) @ top_top_set_list_nat )
=> ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ).
% inj_mapD
thf(fact_947_finite__fun__UNIVD1,axiom,
( ( finite2115694454571419734at_nat @ top_top_set_nat_nat )
=> ( ( ( finite_card_nat @ top_top_set_nat )
!= ( suc @ zero_zero_nat ) )
=> ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_fun_UNIVD1
thf(fact_948_finite__fun__UNIVD1,axiom,
( ( finite4332129999517832055it_nat @ top_to5871476398150932990it_nat )
=> ( ( ( finite_card_nat @ top_top_set_nat )
!= ( suc @ zero_zero_nat ) )
=> ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_fun_UNIVD1
thf(fact_949_finite__fun__UNIVD1,axiom,
( ( finite4257689694021357085t_unit @ top_to8442108875268333988t_unit )
=> ( ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
!= ( suc @ zero_zero_nat ) )
=> ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_fun_UNIVD1
thf(fact_950_finite__fun__UNIVD1,axiom,
( ( finite6665322292308856380t_unit @ top_to658657236369668235t_unit )
=> ( ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
!= ( suc @ zero_zero_nat ) )
=> ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_fun_UNIVD1
thf(fact_951_list_Oset__sel_I1_J,axiom,
! [A4: list_a] :
( ( A4 != nil_a )
=> ( member_a @ ( hd_a @ A4 ) @ ( set_a2 @ A4 ) ) ) ).
% list.set_sel(1)
thf(fact_952_list_Oset__sel_I1_J,axiom,
! [A4: list_nat] :
( ( A4 != nil_nat )
=> ( member_nat @ ( hd_nat @ A4 ) @ ( set_nat2 @ A4 ) ) ) ).
% list.set_sel(1)
thf(fact_953_hd__in__set,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( member_a @ ( hd_a @ Xs ) @ ( set_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_954_hd__in__set,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( member_nat @ ( hd_nat @ Xs ) @ ( set_nat2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_955_longest__common__prefix,axiom,
! [Xs: list_a,Ys: list_a] :
? [Ps: list_a,Xs3: list_a,Ys5: list_a] :
( ( Xs
= ( append_a @ Ps @ Xs3 ) )
& ( Ys
= ( append_a @ Ps @ Ys5 ) )
& ( ( Xs3 = nil_a )
| ( Ys5 = nil_a )
| ( ( hd_a @ Xs3 )
!= ( hd_a @ Ys5 ) ) ) ) ).
% longest_common_prefix
thf(fact_956_longest__common__prefix,axiom,
! [Xs: list_nat,Ys: list_nat] :
? [Ps: list_nat,Xs3: list_nat,Ys5: list_nat] :
( ( Xs
= ( append_nat @ Ps @ Xs3 ) )
& ( Ys
= ( append_nat @ Ps @ Ys5 ) )
& ( ( Xs3 = nil_nat )
| ( Ys5 = nil_nat )
| ( ( hd_nat @ Xs3 )
!= ( hd_nat @ Ys5 ) ) ) ) ).
% longest_common_prefix
thf(fact_957_hd__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( Xs = nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Ys ) ) )
& ( ( Xs != nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Xs ) ) ) ) ).
% hd_append
thf(fact_958_hd__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( Xs = nil_nat )
=> ( ( hd_nat @ ( append_nat @ Xs @ Ys ) )
= ( hd_nat @ Ys ) ) )
& ( ( Xs != nil_nat )
=> ( ( hd_nat @ ( append_nat @ Xs @ Ys ) )
= ( hd_nat @ Xs ) ) ) ) ).
% hd_append
thf(fact_959_hd__map,axiom,
! [Xs: list_a,F: a > nat] :
( ( Xs != nil_a )
=> ( ( hd_nat @ ( map_a_nat @ F @ Xs ) )
= ( F @ ( hd_a @ Xs ) ) ) ) ).
% hd_map
thf(fact_960_list_Omap__sel_I1_J,axiom,
! [A4: list_a,F: a > nat] :
( ( A4 != nil_a )
=> ( ( hd_nat @ ( map_a_nat @ F @ A4 ) )
= ( F @ ( hd_a @ A4 ) ) ) ) ).
% list.map_sel(1)
thf(fact_961_range__subsetD,axiom,
! [F: a > nat,B2: set_nat,I: a] :
( ( ord_less_eq_set_nat @ ( image_a_nat @ F @ top_top_set_a ) @ B2 )
=> ( member_nat @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_962_range__subsetD,axiom,
! [F: nat > a,B2: set_a,I: nat] :
( ( ord_less_eq_set_a @ ( image_nat_a @ F @ top_top_set_nat ) @ B2 )
=> ( member_a @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_963_range__subsetD,axiom,
! [F: nat > nat,B2: set_nat,I: nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ top_top_set_nat ) @ B2 )
=> ( member_nat @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_964_range__subsetD,axiom,
! [F: product_unit > a,B2: set_a,I: product_unit] :
( ( ord_less_eq_set_a @ ( image_Product_unit_a @ F @ top_to1996260823553986621t_unit ) @ B2 )
=> ( member_a @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_965_range__subsetD,axiom,
! [F: product_unit > nat,B2: set_nat,I: product_unit] :
( ( ord_less_eq_set_nat @ ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit ) @ B2 )
=> ( member_nat @ ( F @ I ) @ B2 ) ) ).
% range_subsetD
thf(fact_966_finite_Ocases,axiom,
! [A4: set_a] :
( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ~ ! [A7: set_a] :
( ? [A3: a] :
( A4
= ( insert_a2 @ A3 @ A7 ) )
=> ~ ( finite_finite_a @ A7 ) ) ) ) ).
% finite.cases
thf(fact_967_finite_Ocases,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ~ ! [A7: set_nat] :
( ? [A3: nat] :
( A4
= ( insert_nat2 @ A3 @ A7 ) )
=> ~ ( finite_finite_nat @ A7 ) ) ) ) ).
% finite.cases
thf(fact_968_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A6: set_a] :
( ( A6 = bot_bot_set_a )
| ? [A2: set_a,B7: a] :
( ( A6
= ( insert_a2 @ B7 @ A2 ) )
& ( finite_finite_a @ A2 ) ) ) ) ) ).
% finite.simps
thf(fact_969_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A6: set_nat] :
( ( A6 = bot_bot_set_nat )
| ? [A2: set_nat,B7: nat] :
( ( A6
= ( insert_nat2 @ B7 @ A2 ) )
& ( finite_finite_nat @ A2 ) ) ) ) ) ).
% finite.simps
thf(fact_970_finite__induct,axiom,
! [F4: set_a,P: set_a > $o] :
( ( finite_finite_a @ F4 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,F5: set_a] :
( ( finite_finite_a @ F5 )
=> ( ~ ( member_a @ X2 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_a2 @ X2 @ F5 ) ) ) ) )
=> ( P @ F4 ) ) ) ) ).
% finite_induct
thf(fact_971_finite__induct,axiom,
! [F4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,F5: set_nat] :
( ( finite_finite_nat @ F5 )
=> ( ~ ( member_nat @ X2 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_nat2 @ X2 @ F5 ) ) ) ) )
=> ( P @ F4 ) ) ) ) ).
% finite_induct
thf(fact_972_finite__ne__induct,axiom,
! [F4: set_a,P: set_a > $o] :
( ( finite_finite_a @ F4 )
=> ( ( F4 != bot_bot_set_a )
=> ( ! [X2: a] : ( P @ ( insert_a2 @ X2 @ bot_bot_set_a ) )
=> ( ! [X2: a,F5: set_a] :
( ( finite_finite_a @ F5 )
=> ( ( F5 != bot_bot_set_a )
=> ( ~ ( member_a @ X2 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_a2 @ X2 @ F5 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_973_finite__ne__induct,axiom,
! [F4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( F4 != bot_bot_set_nat )
=> ( ! [X2: nat] : ( P @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
=> ( ! [X2: nat,F5: set_nat] :
( ( finite_finite_nat @ F5 )
=> ( ( F5 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X2 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_nat2 @ X2 @ F5 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_974_infinite__finite__induct,axiom,
! [P: set_a > $o,A: set_a] :
( ! [A7: set_a] :
( ~ ( finite_finite_a @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,F5: set_a] :
( ( finite_finite_a @ F5 )
=> ( ~ ( member_a @ X2 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_a2 @ X2 @ F5 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_975_infinite__finite__induct,axiom,
! [P: set_nat > $o,A: set_nat] :
( ! [A7: set_nat] :
( ~ ( finite_finite_nat @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,F5: set_nat] :
( ( finite_finite_nat @ F5 )
=> ( ~ ( member_nat @ X2 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_nat2 @ X2 @ F5 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_976_subset__singleton__iff,axiom,
! [X6: set_a,A4: a] :
( ( ord_less_eq_set_a @ X6 @ ( insert_a2 @ A4 @ bot_bot_set_a ) )
= ( ( X6 = bot_bot_set_a )
| ( X6
= ( insert_a2 @ A4 @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_977_subset__singleton__iff,axiom,
! [X6: set_nat,A4: nat] :
( ( ord_less_eq_set_nat @ X6 @ ( insert_nat2 @ A4 @ bot_bot_set_nat ) )
= ( ( X6 = bot_bot_set_nat )
| ( X6
= ( insert_nat2 @ A4 @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_978_subset__singletonD,axiom,
! [A: set_a,X: a] :
( ( ord_less_eq_set_a @ A @ ( insert_a2 @ X @ bot_bot_set_a ) )
=> ( ( A = bot_bot_set_a )
| ( A
= ( insert_a2 @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_979_subset__singletonD,axiom,
! [A: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
=> ( ( A = bot_bot_set_nat )
| ( A
= ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_980_empty__set,axiom,
( bot_bot_set_a
= ( set_a2 @ nil_a ) ) ).
% empty_set
thf(fact_981_empty__set,axiom,
( bot_bot_set_nat
= ( set_nat2 @ nil_nat ) ) ).
% empty_set
thf(fact_982_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( ( finite_card_nat @ A )
= ( finite_card_nat @ top_top_set_nat ) )
=> ( A = top_top_set_nat ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_983_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( ( finite410649719033368117t_unit @ A )
= ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit ) )
=> ( A = top_to1996260823553986621t_unit ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_984_inj__image__mem__iff,axiom,
! [F: a > a,A4: a,A: set_a] :
( ( inj_on_a_a @ F @ top_top_set_a )
=> ( ( member_a @ ( F @ A4 ) @ ( image_a_a @ F @ A ) )
= ( member_a @ A4 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_985_inj__image__mem__iff,axiom,
! [F: a > nat,A4: a,A: set_a] :
( ( inj_on_a_nat @ F @ top_top_set_a )
=> ( ( member_nat @ ( F @ A4 ) @ ( image_a_nat @ F @ A ) )
= ( member_a @ A4 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_986_inj__image__mem__iff,axiom,
! [F: nat > a,A4: nat,A: set_nat] :
( ( inj_on_nat_a @ F @ top_top_set_nat )
=> ( ( member_a @ ( F @ A4 ) @ ( image_nat_a @ F @ A ) )
= ( member_nat @ A4 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_987_inj__image__mem__iff,axiom,
! [F: nat > nat,A4: nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( member_nat @ ( F @ A4 ) @ ( image_nat_nat @ F @ A ) )
= ( member_nat @ A4 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_988_inj__image__mem__iff,axiom,
! [F: product_unit > a,A4: product_unit,A: set_Product_unit] :
( ( inj_on8151663806560157602unit_a @ F @ top_to1996260823553986621t_unit )
=> ( ( member_a @ ( F @ A4 ) @ ( image_Product_unit_a @ F @ A ) )
= ( member_Product_unit @ A4 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_989_inj__image__mem__iff,axiom,
! [F: product_unit > nat,A4: product_unit,A: set_Product_unit] :
( ( inj_on8430439091780834860it_nat @ F @ top_to1996260823553986621t_unit )
=> ( ( member_nat @ ( F @ A4 ) @ ( image_875570014554754200it_nat @ F @ A ) )
= ( member_Product_unit @ A4 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_990_inj__image__eq__iff,axiom,
! [F: a > nat,A: set_a,B2: set_a] :
( ( inj_on_a_nat @ F @ top_top_set_a )
=> ( ( ( image_a_nat @ F @ A )
= ( image_a_nat @ F @ B2 ) )
= ( A = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_991_inj__image__eq__iff,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ( image_nat_nat @ F @ A )
= ( image_nat_nat @ F @ B2 ) )
= ( A = B2 ) ) ) ).
% inj_image_eq_iff
thf(fact_992_range__ex1__eq,axiom,
! [F: a > nat,B: nat] :
( ( inj_on_a_nat @ F @ top_top_set_a )
=> ( ( member_nat @ B @ ( image_a_nat @ F @ top_top_set_a ) )
= ( ? [X3: a] :
( ( B
= ( F @ X3 ) )
& ! [Y3: a] :
( ( B
= ( F @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_993_range__ex1__eq,axiom,
! [F: nat > a,B: a] :
( ( inj_on_nat_a @ F @ top_top_set_nat )
=> ( ( member_a @ B @ ( image_nat_a @ F @ top_top_set_nat ) )
= ( ? [X3: nat] :
( ( B
= ( F @ X3 ) )
& ! [Y3: nat] :
( ( B
= ( F @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_994_range__ex1__eq,axiom,
! [F: nat > nat,B: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( member_nat @ B @ ( image_nat_nat @ F @ top_top_set_nat ) )
= ( ? [X3: nat] :
( ( B
= ( F @ X3 ) )
& ! [Y3: nat] :
( ( B
= ( F @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_995_range__ex1__eq,axiom,
! [F: product_unit > a,B: a] :
( ( inj_on8151663806560157602unit_a @ F @ top_to1996260823553986621t_unit )
=> ( ( member_a @ B @ ( image_Product_unit_a @ F @ top_to1996260823553986621t_unit ) )
= ( ? [X3: product_unit] :
( ( B
= ( F @ X3 ) )
& ! [Y3: product_unit] :
( ( B
= ( F @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_996_range__ex1__eq,axiom,
! [F: product_unit > nat,B: nat] :
( ( inj_on8430439091780834860it_nat @ F @ top_to1996260823553986621t_unit )
=> ( ( member_nat @ B @ ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit ) )
= ( ? [X3: product_unit] :
( ( B
= ( F @ X3 ) )
& ! [Y3: product_unit] :
( ( B
= ( F @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_997_hd__Nil__eq__last,axiom,
( ( hd_a @ nil_a )
= ( last_a @ nil_a ) ) ).
% hd_Nil_eq_last
thf(fact_998_hd__Nil__eq__last,axiom,
( ( hd_nat @ nil_nat )
= ( last_nat @ nil_nat ) ) ).
% hd_Nil_eq_last
thf(fact_999_map__injective,axiom,
! [F: a > nat,Xs: list_a,Ys: list_a] :
( ( ( map_a_nat @ F @ Xs )
= ( map_a_nat @ F @ Ys ) )
=> ( ( inj_on_a_nat @ F @ top_top_set_a )
=> ( Xs = Ys ) ) ) ).
% map_injective
thf(fact_1000_map__injective,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ F @ Ys ) )
=> ( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( Xs = Ys ) ) ) ).
% map_injective
thf(fact_1001_the__elem__image__unique,axiom,
! [A: set_a,F: a > nat,X: a] :
( ( A != bot_bot_set_a )
=> ( ! [Y2: a] :
( ( member_a @ Y2 @ A )
=> ( ( F @ Y2 )
= ( F @ X ) ) )
=> ( ( the_elem_nat @ ( image_a_nat @ F @ A ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_1002_the__elem__image__unique,axiom,
! [A: set_nat,F: nat > nat,X: nat] :
( ( A != bot_bot_set_nat )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ A )
=> ( ( F @ Y2 )
= ( F @ X ) ) )
=> ( ( the_elem_nat @ ( image_nat_nat @ F @ A ) )
= ( F @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_1003_finite__subset__induct,axiom,
! [F4: set_a,A: set_a,P: set_a > $o] :
( ( finite_finite_a @ F4 )
=> ( ( ord_less_eq_set_a @ F4 @ A )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F5: set_a] :
( ( finite_finite_a @ F5 )
=> ( ( member_a @ A3 @ A )
=> ( ~ ( member_a @ A3 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_a2 @ A3 @ F5 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1004_finite__subset__induct,axiom,
! [F4: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( ord_less_eq_set_nat @ F4 @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A3: nat,F5: set_nat] :
( ( finite_finite_nat @ F5 )
=> ( ( member_nat @ A3 @ A )
=> ( ~ ( member_nat @ A3 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_nat2 @ A3 @ F5 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1005_finite__subset__induct_H,axiom,
! [F4: set_a,A: set_a,P: set_a > $o] :
( ( finite_finite_a @ F4 )
=> ( ( ord_less_eq_set_a @ F4 @ A )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A3: a,F5: set_a] :
( ( finite_finite_a @ F5 )
=> ( ( member_a @ A3 @ A )
=> ( ( ord_less_eq_set_a @ F5 @ A )
=> ( ~ ( member_a @ A3 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_a2 @ A3 @ F5 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1006_finite__subset__induct_H,axiom,
! [F4: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F4 )
=> ( ( ord_less_eq_set_nat @ F4 @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A3: nat,F5: set_nat] :
( ( finite_finite_nat @ F5 )
=> ( ( member_nat @ A3 @ A )
=> ( ( ord_less_eq_set_nat @ F5 @ A )
=> ( ~ ( member_nat @ A3 @ F5 )
=> ( ( P @ F5 )
=> ( P @ ( insert_nat2 @ A3 @ F5 ) ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1007_finite__UNIV__inj__surj,axiom,
! [F: nat > nat] :
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_1008_finite__UNIV__inj__surj,axiom,
! [F: product_unit > product_unit] :
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( inj_on8151373323710067377t_unit @ F @ top_to1996260823553986621t_unit )
=> ( ( image_405062704495631173t_unit @ F @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_1009_finite__UNIV__surj__inj,axiom,
! [F: nat > nat] :
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_1010_finite__UNIV__surj__inj,axiom,
! [F: product_unit > product_unit] :
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( ( image_405062704495631173t_unit @ F @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit )
=> ( inj_on8151373323710067377t_unit @ F @ top_to1996260823553986621t_unit ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_1011_inj__image__subset__iff,axiom,
! [F: a > nat,A: set_a,B2: set_a] :
( ( inj_on_a_nat @ F @ top_top_set_a )
=> ( ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A ) @ ( image_a_nat @ F @ B2 ) )
= ( ord_less_eq_set_a @ A @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_1012_inj__image__subset__iff,axiom,
! [F: nat > nat,A: set_nat,B2: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B2 ) )
= ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% inj_image_subset_iff
thf(fact_1013_card__eq__0__iff,axiom,
! [A: set_Product_unit] :
( ( ( finite410649719033368117t_unit @ A )
= zero_zero_nat )
= ( ( A = bot_bo3957492148770167129t_unit )
| ~ ( finite4290736615968046902t_unit @ A ) ) ) ).
% card_eq_0_iff
thf(fact_1014_card__eq__0__iff,axiom,
! [A: set_nat] :
( ( ( finite_card_nat @ A )
= zero_zero_nat )
= ( ( A = bot_bot_set_nat )
| ~ ( finite_finite_nat @ A ) ) ) ).
% card_eq_0_iff
thf(fact_1015_inj__on__iff__surj,axiom,
! [A: set_a,A8: set_nat] :
( ( A != bot_bot_set_a )
=> ( ( ? [F2: a > nat] :
( ( inj_on_a_nat @ F2 @ A )
& ( ord_less_eq_set_nat @ ( image_a_nat @ F2 @ A ) @ A8 ) ) )
= ( ? [G3: nat > a] :
( ( image_nat_a @ G3 @ A8 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_1016_inj__on__iff__surj,axiom,
! [A: set_nat,A8: set_a] :
( ( A != bot_bot_set_nat )
=> ( ( ? [F2: nat > a] :
( ( inj_on_nat_a @ F2 @ A )
& ( ord_less_eq_set_a @ ( image_nat_a @ F2 @ A ) @ A8 ) ) )
= ( ? [G3: a > nat] :
( ( image_a_nat @ G3 @ A8 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_1017_inj__on__iff__surj,axiom,
! [A: set_nat,A8: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ( ? [F2: nat > nat] :
( ( inj_on_nat_nat @ F2 @ A )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A ) @ A8 ) ) )
= ( ? [G3: nat > nat] :
( ( image_nat_nat @ G3 @ A8 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_1018_subset__code_I2_J,axiom,
! [A: set_a,Ys: list_a] :
( ( ord_less_eq_set_a @ A @ ( coset_a @ Ys ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Ys ) )
=> ~ ( member_a @ X3 @ A ) ) ) ) ).
% subset_code(2)
thf(fact_1019_subset__code_I2_J,axiom,
! [A: set_nat,Ys: list_nat] :
( ( ord_less_eq_set_nat @ A @ ( coset_nat @ Ys ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
=> ~ ( member_nat @ X3 @ A ) ) ) ) ).
% subset_code(2)
thf(fact_1020_remdups__adj__map__injective,axiom,
! [F: a > nat,Xs: list_a] :
( ( inj_on_a_nat @ F @ top_top_set_a )
=> ( ( remdups_adj_nat @ ( map_a_nat @ F @ Xs ) )
= ( map_a_nat @ F @ ( remdups_adj_a @ Xs ) ) ) ) ).
% remdups_adj_map_injective
thf(fact_1021_remdups__adj__map__injective,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( remdups_adj_nat @ ( map_nat_nat @ F @ Xs ) )
= ( map_nat_nat @ F @ ( remdups_adj_nat @ Xs ) ) ) ) ).
% remdups_adj_map_injective
thf(fact_1022_map__removeAll__inj,axiom,
! [F: a > nat,X: a,Xs: list_a] :
( ( inj_on_a_nat @ F @ top_top_set_a )
=> ( ( map_a_nat @ F @ ( removeAll_a @ X @ Xs ) )
= ( removeAll_nat @ ( F @ X ) @ ( map_a_nat @ F @ Xs ) ) ) ) ).
% map_removeAll_inj
thf(fact_1023_map__removeAll__inj,axiom,
! [F: nat > nat,X: nat,Xs: list_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( map_nat_nat @ F @ ( removeAll_nat @ X @ Xs ) )
= ( removeAll_nat @ ( F @ X ) @ ( map_nat_nat @ F @ Xs ) ) ) ) ).
% map_removeAll_inj
thf(fact_1024_insert__code_I2_J,axiom,
! [X: a,Xs: list_a] :
( ( insert_a2 @ X @ ( coset_a @ Xs ) )
= ( coset_a @ ( removeAll_a @ X @ Xs ) ) ) ).
% insert_code(2)
thf(fact_1025_insert__code_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( insert_nat2 @ X @ ( coset_nat @ Xs ) )
= ( coset_nat @ ( removeAll_nat @ X @ Xs ) ) ) ).
% insert_code(2)
thf(fact_1026_card__Suc__eq,axiom,
! [A: set_a,K: nat] :
( ( ( finite_card_a @ A )
= ( suc @ K ) )
= ( ? [B7: a,B4: set_a] :
( ( A
= ( insert_a2 @ B7 @ B4 ) )
& ~ ( member_a @ B7 @ B4 )
& ( ( finite_card_a @ B4 )
= K )
& ( ( K = zero_zero_nat )
=> ( B4 = bot_bot_set_a ) ) ) ) ) ).
% card_Suc_eq
thf(fact_1027_card__Suc__eq,axiom,
! [A: set_Product_unit,K: nat] :
( ( ( finite410649719033368117t_unit @ A )
= ( suc @ K ) )
= ( ? [B7: product_unit,B4: set_Product_unit] :
( ( A
= ( insert_Product_unit @ B7 @ B4 ) )
& ~ ( member_Product_unit @ B7 @ B4 )
& ( ( finite410649719033368117t_unit @ B4 )
= K )
& ( ( K = zero_zero_nat )
=> ( B4 = bot_bo3957492148770167129t_unit ) ) ) ) ) ).
% card_Suc_eq
thf(fact_1028_card__Suc__eq,axiom,
! [A: set_nat,K: nat] :
( ( ( finite_card_nat @ A )
= ( suc @ K ) )
= ( ? [B7: nat,B4: set_nat] :
( ( A
= ( insert_nat2 @ B7 @ B4 ) )
& ~ ( member_nat @ B7 @ B4 )
& ( ( finite_card_nat @ B4 )
= K )
& ( ( K = zero_zero_nat )
=> ( B4 = bot_bot_set_nat ) ) ) ) ) ).
% card_Suc_eq
thf(fact_1029_card__eq__SucD,axiom,
! [A: set_a,K: nat] :
( ( ( finite_card_a @ A )
= ( suc @ K ) )
=> ? [B6: a,B3: set_a] :
( ( A
= ( insert_a2 @ B6 @ B3 ) )
& ~ ( member_a @ B6 @ B3 )
& ( ( finite_card_a @ B3 )
= K )
& ( ( K = zero_zero_nat )
=> ( B3 = bot_bot_set_a ) ) ) ) ).
% card_eq_SucD
thf(fact_1030_card__eq__SucD,axiom,
! [A: set_Product_unit,K: nat] :
( ( ( finite410649719033368117t_unit @ A )
= ( suc @ K ) )
=> ? [B6: product_unit,B3: set_Product_unit] :
( ( A
= ( insert_Product_unit @ B6 @ B3 ) )
& ~ ( member_Product_unit @ B6 @ B3 )
& ( ( finite410649719033368117t_unit @ B3 )
= K )
& ( ( K = zero_zero_nat )
=> ( B3 = bot_bo3957492148770167129t_unit ) ) ) ) ).
% card_eq_SucD
thf(fact_1031_card__eq__SucD,axiom,
! [A: set_nat,K: nat] :
( ( ( finite_card_nat @ A )
= ( suc @ K ) )
=> ? [B6: nat,B3: set_nat] :
( ( A
= ( insert_nat2 @ B6 @ B3 ) )
& ~ ( member_nat @ B6 @ B3 )
& ( ( finite_card_nat @ B3 )
= K )
& ( ( K = zero_zero_nat )
=> ( B3 = bot_bot_set_nat ) ) ) ) ).
% card_eq_SucD
thf(fact_1032_card__1__singleton__iff,axiom,
! [A: set_a] :
( ( ( finite_card_a @ A )
= ( suc @ zero_zero_nat ) )
= ( ? [X3: a] :
( A
= ( insert_a2 @ X3 @ bot_bot_set_a ) ) ) ) ).
% card_1_singleton_iff
thf(fact_1033_card__1__singleton__iff,axiom,
! [A: set_Product_unit] :
( ( ( finite410649719033368117t_unit @ A )
= ( suc @ zero_zero_nat ) )
= ( ? [X3: product_unit] :
( A
= ( insert_Product_unit @ X3 @ bot_bo3957492148770167129t_unit ) ) ) ) ).
% card_1_singleton_iff
thf(fact_1034_card__1__singleton__iff,axiom,
! [A: set_nat] :
( ( ( finite_card_nat @ A )
= ( suc @ zero_zero_nat ) )
= ( ? [X3: nat] :
( A
= ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) ) ).
% card_1_singleton_iff
thf(fact_1035_remdups__adj__append_H,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( Xs = nil_a )
| ( Ys = nil_a )
| ( ( last_a @ Xs )
!= ( hd_a @ Ys ) ) )
=> ( ( remdups_adj_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( remdups_adj_a @ Xs ) @ ( remdups_adj_a @ Ys ) ) ) ) ).
% remdups_adj_append'
thf(fact_1036_remdups__adj__append_H,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( Xs = nil_nat )
| ( Ys = nil_nat )
| ( ( last_nat @ Xs )
!= ( hd_nat @ Ys ) ) )
=> ( ( remdups_adj_nat @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( remdups_adj_nat @ Xs ) @ ( remdups_adj_nat @ Ys ) ) ) ) ).
% remdups_adj_append'
thf(fact_1037_set__replicate__Suc,axiom,
! [N: nat,X: a] :
( ( set_a2 @ ( replicate_a @ ( suc @ N ) @ X ) )
= ( insert_a2 @ X @ bot_bot_set_a ) ) ).
% set_replicate_Suc
thf(fact_1038_set__replicate__Suc,axiom,
! [N: nat,X: nat] :
( ( set_nat2 @ ( replicate_nat @ ( suc @ N ) @ X ) )
= ( insert_nat2 @ X @ bot_bot_set_nat ) ) ).
% set_replicate_Suc
thf(fact_1039_set__replicate__conv__if,axiom,
! [N: nat,X: a] :
( ( ( N = zero_zero_nat )
=> ( ( set_a2 @ ( replicate_a @ N @ X ) )
= bot_bot_set_a ) )
& ( ( N != zero_zero_nat )
=> ( ( set_a2 @ ( replicate_a @ N @ X ) )
= ( insert_a2 @ X @ bot_bot_set_a ) ) ) ) ).
% set_replicate_conv_if
thf(fact_1040_set__replicate__conv__if,axiom,
! [N: nat,X: nat] :
( ( ( N = zero_zero_nat )
=> ( ( set_nat2 @ ( replicate_nat @ N @ X ) )
= bot_bot_set_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( set_nat2 @ ( replicate_nat @ N @ X ) )
= ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ).
% set_replicate_conv_if
thf(fact_1041_infinite__iff__countable__subset,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ? [F2: nat > nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ top_top_set_nat ) @ S ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_1042_cSup__singleton,axiom,
! [X: nat] :
( ( complete_Sup_Sup_nat @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
= X ) ).
% cSup_singleton
thf(fact_1043_finite__ranking__induct,axiom,
! [S: set_a,P: set_a > $o,F: a > nat] :
( ( finite_finite_a @ S )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,S2: set_a] :
( ( finite_finite_a @ S2 )
=> ( ! [Y4: a] :
( ( member_a @ Y4 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y4 ) @ ( F @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_a2 @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1044_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,S2: set_nat] :
( ( finite_finite_nat @ S2 )
=> ( ! [Y4: nat] :
( ( member_nat @ Y4 @ S2 )
=> ( ord_less_eq_nat @ ( F @ Y4 ) @ ( F @ X2 ) ) )
=> ( ( P @ S2 )
=> ( P @ ( insert_nat2 @ X2 @ S2 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1045_cSUP__least,axiom,
! [A: set_a,F: a > nat,M: nat] :
( ( A != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ M ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_a_nat @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1046_cSUP__least,axiom,
! [A: set_nat,F: nat > nat,M: nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ M ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_1047_is__singleton__the__elem,axiom,
( is_singleton_a
= ( ^ [A2: set_a] :
( A2
= ( insert_a2 @ ( the_elem_a @ A2 ) @ bot_bot_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_1048_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A2: set_nat] :
( A2
= ( insert_nat2 @ ( the_elem_nat @ A2 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_1049_is__singletonI,axiom,
! [X: a] : ( is_singleton_a @ ( insert_a2 @ X @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_1050_is__singletonI,axiom,
! [X: nat] : ( is_singleton_nat @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_1051_is__singletonI_H,axiom,
! [A: set_a] :
( ( A != bot_bot_set_a )
=> ( ! [X2: a,Y2: a] :
( ( member_a @ X2 @ A )
=> ( ( member_a @ Y2 @ A )
=> ( X2 = Y2 ) ) )
=> ( is_singleton_a @ A ) ) ) ).
% is_singletonI'
thf(fact_1052_is__singletonI_H,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X2: nat,Y2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y2 @ A )
=> ( X2 = Y2 ) ) )
=> ( is_singleton_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_1053_is__singletonE,axiom,
! [A: set_a] :
( ( is_singleton_a @ A )
=> ~ ! [X2: a] :
( A
!= ( insert_a2 @ X2 @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_1054_is__singletonE,axiom,
! [A: set_nat] :
( ( is_singleton_nat @ A )
=> ~ ! [X2: nat] :
( A
!= ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_1055_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A2: set_a] :
? [X3: a] :
( A2
= ( insert_a2 @ X3 @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_1056_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A2: set_nat] :
? [X3: nat] :
( A2
= ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_1057_elim__singleton,axiom,
! [X: a,S3: a,X7: a,T: a] :
( ( ( member_a @ X @ ( insert_a2 @ S3 @ bot_bot_set_a ) )
& ( member_a @ X7 @ ( insert_a2 @ T @ bot_bot_set_a ) ) )
=> ( ( X = S3 )
& ( X7 = T ) ) ) ).
% elim_singleton
thf(fact_1058_elim__singleton,axiom,
! [X: a,S3: a,X7: nat,T: nat] :
( ( ( member_a @ X @ ( insert_a2 @ S3 @ bot_bot_set_a ) )
& ( member_nat @ X7 @ ( insert_nat2 @ T @ bot_bot_set_nat ) ) )
=> ( ( X = S3 )
& ( X7 = T ) ) ) ).
% elim_singleton
thf(fact_1059_elim__singleton,axiom,
! [X: nat,S3: nat,X7: a,T: a] :
( ( ( member_nat @ X @ ( insert_nat2 @ S3 @ bot_bot_set_nat ) )
& ( member_a @ X7 @ ( insert_a2 @ T @ bot_bot_set_a ) ) )
=> ( ( X = S3 )
& ( X7 = T ) ) ) ).
% elim_singleton
thf(fact_1060_elim__singleton,axiom,
! [X: nat,S3: nat,X7: nat,T: nat] :
( ( ( member_nat @ X @ ( insert_nat2 @ S3 @ bot_bot_set_nat ) )
& ( member_nat @ X7 @ ( insert_nat2 @ T @ bot_bot_set_nat ) ) )
=> ( ( X = S3 )
& ( X7 = T ) ) ) ).
% elim_singleton
thf(fact_1061_distinct__adj__append__iff,axiom,
! [Xs: list_a,Ys: list_a] :
( ( distinct_adj_a @ ( append_a @ Xs @ Ys ) )
= ( ( distinct_adj_a @ Xs )
& ( distinct_adj_a @ Ys )
& ( ( Xs = nil_a )
| ( Ys = nil_a )
| ( ( last_a @ Xs )
!= ( hd_a @ Ys ) ) ) ) ) ).
% distinct_adj_append_iff
thf(fact_1062_distinct__adj__append__iff,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( distinct_adj_nat @ ( append_nat @ Xs @ Ys ) )
= ( ( distinct_adj_nat @ Xs )
& ( distinct_adj_nat @ Ys )
& ( ( Xs = nil_nat )
| ( Ys = nil_nat )
| ( ( last_nat @ Xs )
!= ( hd_nat @ Ys ) ) ) ) ) ).
% distinct_adj_append_iff
thf(fact_1063_greaterThan__0,axiom,
( ( set_or1210151606488870762an_nat @ zero_zero_nat )
= ( image_nat_nat @ suc @ top_top_set_nat ) ) ).
% greaterThan_0
thf(fact_1064_set__removeAll,axiom,
! [X: a,Xs: list_a] :
( ( set_a2 @ ( removeAll_a @ X @ Xs ) )
= ( minus_minus_set_a @ ( set_a2 @ Xs ) @ ( insert_a2 @ X @ bot_bot_set_a ) ) ) ).
% set_removeAll
thf(fact_1065_set__removeAll,axiom,
! [X: nat,Xs: list_nat] :
( ( set_nat2 @ ( removeAll_nat @ X @ Xs ) )
= ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ).
% set_removeAll
thf(fact_1066_DiffI,axiom,
! [C: a,A: set_a,B2: set_a] :
( ( member_a @ C @ A )
=> ( ~ ( member_a @ C @ B2 )
=> ( member_a @ C @ ( minus_minus_set_a @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_1067_DiffI,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ A )
=> ( ~ ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_1068_Diff__iff,axiom,
! [C: a,A: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B2 ) )
= ( ( member_a @ C @ A )
& ~ ( member_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_1069_Diff__iff,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B2 ) )
= ( ( member_nat @ C @ A )
& ~ ( member_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_1070_Diff__idemp,axiom,
! [A: set_nat,B2: set_nat] :
( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B2 ) @ B2 )
= ( minus_minus_set_nat @ A @ B2 ) ) ).
% Diff_idemp
thf(fact_1071_Diff__cancel,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ A )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_1072_empty__Diff,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_1073_greaterThan__Suc,axiom,
! [K: nat] :
( ( set_or1210151606488870762an_nat @ ( suc @ K ) )
= ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K ) @ ( insert_nat2 @ ( suc @ K ) @ bot_bot_set_nat ) ) ) ).
% greaterThan_Suc
thf(fact_1074_diff__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_Suc_Suc
thf(fact_1075_Suc__diff__diff,axiom,
! [M2: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1076_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1077_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1078_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1079_Suc__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_eq
thf(fact_1080_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1081_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1082_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_1083_bot__nat__0_Onot__eq__extremum,axiom,
! [A4: nat] :
( ( A4 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A4 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1084_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1085_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1086_diff__is__0__eq,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% diff_is_0_eq
thf(fact_1087_diff__is__0__eq_H,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1088_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_1089_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1090_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1091_diff__less,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).
% diff_less
thf(fact_1092_diff__less__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_1093_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1094_diff__less__Suc,axiom,
! [M2: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_1095_Suc__diff__Suc,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N ) ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1096_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1097_less__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1098_diff__less__mono,axiom,
! [A4: nat,B: nat,C: nat] :
( ( ord_less_nat @ A4 @ B )
=> ( ( ord_less_eq_nat @ C @ A4 )
=> ( ord_less_nat @ ( minus_minus_nat @ A4 @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1099_eq__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1100_le__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1101_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1102_diff__le__mono,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1103_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_1104_le__diff__iff_H,axiom,
! [A4: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A4 @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A4 ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A4 ) ) ) ) ).
% le_diff_iff'
thf(fact_1105_diff__le__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_1106_diffs0__imp__equal,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M2 )
= zero_zero_nat )
=> ( M2 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1107_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_1108_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1109_not__less__less__Suc__eq,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1110_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_1111_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1112_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1113_Suc__less__SucD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_SucD
thf(fact_1114_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_1115_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M2 )
= ( ? [M6: nat] :
( ( M2
= ( suc @ M6 ) )
& ( ord_less_nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1116_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_1117_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_nat @ M2 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_1118_less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) ) ) ).
% less_Suc_eq
thf(fact_1119_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ N )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1120_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1121_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_1122_Suc__lessI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ( suc @ M2 )
!= N )
=> ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_1123_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1124_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_1125_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1126_bot__nat__0_Oextremum__strict,axiom,
! [A4: nat] :
~ ( ord_less_nat @ A4 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1127_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1128_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1129_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1130_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1131_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1132_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ~ ( P @ N4 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_1133_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1134_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1135_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1136_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M7: nat,N5: nat] :
( ( ord_less_nat @ M7 @ N5 )
| ( M7 = N5 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1137_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_1138_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M7: nat,N5: nat] :
( ( ord_less_eq_nat @ M7 @ N5 )
& ( M7 != N5 ) ) ) ) ).
% nat_less_le
thf(fact_1139_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_1140_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ~ ( P @ N4 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
& ~ ( P @ M4 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_1141_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
=> ( P @ M4 ) )
=> ( P @ N4 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_1142_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_1143_less__not__refl3,axiom,
! [S3: nat,T: nat] :
( ( ord_less_nat @ S3 @ T )
=> ( S3 != T ) ) ).
% less_not_refl3
thf(fact_1144_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_1145_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_1146_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_1147_Suc__diff__le,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1148_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1149_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M7: nat] :
( N
= ( suc @ M7 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1150_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1151_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_1152_less__Suc__eq__0__disj,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( M2 = zero_zero_nat )
| ? [J3: nat] :
( ( M2
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1153_le__imp__less__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1154_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N5: nat] : ( ord_less_eq_nat @ ( suc @ N5 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1155_less__Suc__eq__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_Suc_eq_le
thf(fact_1156_le__less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_1157_Suc__le__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_lessD
thf(fact_1158_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I @ N4 )
=> ( ( ord_less_nat @ N4 @ J )
=> ( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1159_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I @ N4 )
=> ( ( ord_less_nat @ N4 @ J )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1160_Suc__le__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_eq
thf(fact_1161_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_1162_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1163_rgf__limit__ge,axiom,
! [Y: nat,Xs: list_nat] :
( ( member_nat @ Y @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ Y @ ( equiva5889994315859557365_limit @ Xs ) ) ) ).
% rgf_limit_ge
thf(fact_1164_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1165_atLeast__Suc,axiom,
! [K: nat] :
( ( set_ord_atLeast_nat @ ( suc @ K ) )
= ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K ) @ ( insert_nat2 @ K @ bot_bot_set_nat ) ) ) ).
% atLeast_Suc
thf(fact_1166_card__UNIV__unit,axiom,
( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
= one_one_nat ) ).
% card_UNIV_unit
thf(fact_1167_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1168_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1169_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1170_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_1171_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1172_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1173_mono__Suc,axiom,
monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ suc ).
% mono_Suc
thf(fact_1174_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_1175_atLeast__Suc__greaterThan,axiom,
! [K: nat] :
( ( set_ord_atLeast_nat @ ( suc @ K ) )
= ( set_or1210151606488870762an_nat @ K ) ) ).
% atLeast_Suc_greaterThan
thf(fact_1176_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1177_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1178_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1179_rgf__snoc,axiom,
! [Xs: list_nat,X: nat] :
( ( equiva3371634703666331078on_rgf @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= ( ( equiva3371634703666331078on_rgf @ Xs )
& ( ord_less_nat @ X @ ( plus_plus_nat @ ( equiva5889994315859557365_limit @ Xs ) @ one_one_nat ) ) ) ) ).
% rgf_snoc
thf(fact_1180_add__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc_right
thf(fact_1181_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1182_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_1183_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1184_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1185_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1186_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1187_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1188_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1189_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1190_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1191_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1192_diff__add__0,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1193_Nat_Odiff__cancel,axiom,
! [K: nat,M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% Nat.diff_cancel
thf(fact_1194_diff__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_cancel2
thf(fact_1195_diff__add__inverse,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M2 ) @ N )
= M2 ) ).
% diff_add_inverse
thf(fact_1196_diff__add__inverse2,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ N )
= M2 ) ).
% diff_add_inverse2
thf(fact_1197_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% less_add_eq_less
thf(fact_1198_trans__less__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_1199_trans__less__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_1200_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1201_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1202_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1203_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1204_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1205_card__le__Suc__Max,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( ord_less_eq_nat @ ( finite_card_nat @ S ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S ) ) ) ) ).
% card_le_Suc_Max
thf(fact_1206_one__is__add,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M2 @ N ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1207_add__is__1,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1208_less__imp__Suc__add,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1209_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M7: nat,N5: nat] :
? [K3: nat] :
( N5
= ( suc @ ( plus_plus_nat @ M7 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1210_less__add__Suc2,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M2 @ I ) ) ) ).
% less_add_Suc2
thf(fact_1211_less__add__Suc1,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_1212_less__natE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ~ ! [Q: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M2 @ Q ) ) ) ) ).
% less_natE
thf(fact_1213_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1214_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M3: nat,N4: nat] :
( ( ord_less_nat @ M3 @ N4 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N4 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1215_add__Suc__shift,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( plus_plus_nat @ M2 @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1216_add__Suc,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc
thf(fact_1217_nat__arith_Osuc1,axiom,
! [A: nat,K: nat,A4: nat] :
( ( A
= ( plus_plus_nat @ K @ A4 ) )
=> ( ( suc @ A )
= ( plus_plus_nat @ K @ ( suc @ A4 ) ) ) ) ).
% nat_arith.suc1
thf(fact_1218_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1219_add__eq__self__zero,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= M2 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1220_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M7: nat,N5: nat] :
? [K3: nat] :
( N5
= ( plus_plus_nat @ M7 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1221_trans__le__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_1222_trans__le__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_1223_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1224_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1225_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N4: nat] :
( L
= ( plus_plus_nat @ K @ N4 ) ) ) ).
% le_Suc_ex
thf(fact_1226_add__leD2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1227_add__leD1,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% add_leD1
thf(fact_1228_le__add2,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).
% le_add2
thf(fact_1229_le__add1,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).
% le_add1
thf(fact_1230_add__leE,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M2 @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1231_add__diff__inverse__nat,axiom,
! [M2: nat,N: nat] :
( ~ ( ord_less_nat @ M2 @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M2 @ N ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_1232_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1233_Suc__eq__plus1,axiom,
( suc
= ( ^ [N5: nat] : ( plus_plus_nat @ N5 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1234_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1235_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1236_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1237_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1238_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1239_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1240_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1241_nat__diff__split__asm,axiom,
! [P: nat > $o,A4: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A4 @ B ) )
= ( ~ ( ( ( ord_less_nat @ A4 @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A4
= ( plus_plus_nat @ B @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1242_nat__diff__split,axiom,
! [P: nat > $o,A4: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A4 @ B ) )
= ( ( ( ord_less_nat @ A4 @ B )
=> ( P @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A4
= ( plus_plus_nat @ B @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_1243_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1244_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M7: nat,N5: nat] : ( if_nat @ ( M7 = zero_zero_nat ) @ N5 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M7 @ one_one_nat ) @ N5 ) ) ) ) ) ).
% add_eq_if
thf(fact_1245_rgf__def,axiom,
( equiva3371634703666331078on_rgf
= ( ^ [X3: list_nat] :
! [Ys3: list_nat,Y3: nat] :
( ( prefix_nat @ ( append_nat @ Ys3 @ ( cons_nat @ Y3 @ nil_nat ) ) @ X3 )
=> ( ord_less_eq_nat @ Y3 @ ( equiva5889994315859557365_limit @ Ys3 ) ) ) ) ) ).
% rgf_def
thf(fact_1246_rgf__limit__snoc,axiom,
! [X: list_nat,Y: nat] :
( ( equiva5889994315859557365_limit @ ( append_nat @ X @ ( cons_nat @ Y @ nil_nat ) ) )
= ( ord_max_nat @ ( plus_plus_nat @ Y @ one_one_nat ) @ ( equiva5889994315859557365_limit @ X ) ) ) ).
% rgf_limit_snoc
thf(fact_1247_max__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_max_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( suc @ ( ord_max_nat @ M2 @ N ) ) ) ).
% max_Suc_Suc
thf(fact_1248_max__0R,axiom,
! [N: nat] :
( ( ord_max_nat @ N @ zero_zero_nat )
= N ) ).
% max_0R
thf(fact_1249_max__0L,axiom,
! [N: nat] :
( ( ord_max_nat @ zero_zero_nat @ N )
= N ) ).
% max_0L
thf(fact_1250_max__nat_Oright__neutral,axiom,
! [A4: nat] :
( ( ord_max_nat @ A4 @ zero_zero_nat )
= A4 ) ).
% max_nat.right_neutral
thf(fact_1251_max__nat_Oneutr__eq__iff,axiom,
! [A4: nat,B: nat] :
( ( zero_zero_nat
= ( ord_max_nat @ A4 @ B ) )
= ( ( A4 = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.neutr_eq_iff
thf(fact_1252_max__nat_Oleft__neutral,axiom,
! [A4: nat] :
( ( ord_max_nat @ zero_zero_nat @ A4 )
= A4 ) ).
% max_nat.left_neutral
thf(fact_1253_max__nat_Oeq__neutr__iff,axiom,
! [A4: nat,B: nat] :
( ( ( ord_max_nat @ A4 @ B )
= zero_zero_nat )
= ( ( A4 = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.eq_neutr_iff
thf(fact_1254_nat__minus__add__max,axiom,
! [N: nat,M2: nat] :
( ( plus_plus_nat @ ( minus_minus_nat @ N @ M2 ) @ M2 )
= ( ord_max_nat @ N @ M2 ) ) ).
% nat_minus_add_max
thf(fact_1255_nat__add__max__right,axiom,
! [M2: nat,N: nat,Q2: nat] :
( ( plus_plus_nat @ M2 @ ( ord_max_nat @ N @ Q2 ) )
= ( ord_max_nat @ ( plus_plus_nat @ M2 @ N ) @ ( plus_plus_nat @ M2 @ Q2 ) ) ) ).
% nat_add_max_right
thf(fact_1256_nat__add__max__left,axiom,
! [M2: nat,N: nat,Q2: nat] :
( ( plus_plus_nat @ ( ord_max_nat @ M2 @ N ) @ Q2 )
= ( ord_max_nat @ ( plus_plus_nat @ M2 @ Q2 ) @ ( plus_plus_nat @ N @ Q2 ) ) ) ).
% nat_add_max_left
thf(fact_1257_rgf__limit_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( equiva5889994315859557365_limit @ ( cons_nat @ X @ Xs ) )
= ( ord_max_nat @ ( plus_plus_nat @ X @ one_one_nat ) @ ( equiva5889994315859557365_limit @ Xs ) ) ) ).
% rgf_limit.simps(2)
thf(fact_1258_rgf__limit_Oelims,axiom,
! [X: list_nat,Y: nat] :
( ( ( equiva5889994315859557365_limit @ X )
= Y )
=> ( ( ( X = nil_nat )
=> ( Y != zero_zero_nat ) )
=> ~ ! [X2: nat,Xs2: list_nat] :
( ( X
= ( cons_nat @ X2 @ Xs2 ) )
=> ( Y
!= ( ord_max_nat @ ( plus_plus_nat @ X2 @ one_one_nat ) @ ( equiva5889994315859557365_limit @ Xs2 ) ) ) ) ) ) ).
% rgf_limit.elims
thf(fact_1259_rgf__limit_Opelims,axiom,
! [X: list_nat,Y: nat] :
( ( ( equiva5889994315859557365_limit @ X )
= Y )
=> ( ( accp_list_nat @ equiva5575797544161152836it_rel @ X )
=> ( ( ( X = nil_nat )
=> ( ( Y = zero_zero_nat )
=> ~ ( accp_list_nat @ equiva5575797544161152836it_rel @ nil_nat ) ) )
=> ~ ! [X2: nat,Xs2: list_nat] :
( ( X
= ( cons_nat @ X2 @ Xs2 ) )
=> ( ( Y
= ( ord_max_nat @ ( plus_plus_nat @ X2 @ one_one_nat ) @ ( equiva5889994315859557365_limit @ Xs2 ) ) )
=> ~ ( accp_list_nat @ equiva5575797544161152836it_rel @ ( cons_nat @ X2 @ Xs2 ) ) ) ) ) ) ) ).
% rgf_limit.pelims
thf(fact_1260_mono__times__nat,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ ( times_times_nat @ N ) ) ) ).
% mono_times_nat
thf(fact_1261_mult__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M2 @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M2 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1262_mult__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N ) )
= ( ( M2 = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1263_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1264_mult__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1265_nat__mult__eq__1__iff,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= one_one_nat )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1266_nat__1__eq__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M2 @ N ) )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1267_one__eq__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M2 @ N ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1268_mult__eq__1__iff,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1269_nat__0__less__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
% Helper facts (7)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y: list_a] :
( ( if_list_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y: list_a] :
( ( if_list_a @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( if_list_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
inj_on_a_nat @ f @ ( set_a2 @ xa ) ).
%------------------------------------------------------------------------------