TPTP Problem File: SLH0563^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Equivalence_Relation_Enumeration/0007_Equivalence_Relation_Enumeration/prob_00295_011397__11958572_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1401 ( 626 unt; 128 typ; 0 def)
% Number of atoms : 3213 (1702 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 10729 ( 396 ~; 73 |; 245 &;8639 @)
% ( 0 <=>;1376 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 6 avg)
% Number of types : 13 ( 12 usr)
% Number of type conns : 487 ( 487 >; 0 *; 0 +; 0 <<)
% Number of symbols : 119 ( 116 usr; 13 con; 0-3 aty)
% Number of variables : 3611 ( 118 ^;3274 !; 219 ?;3611 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:14:06.877
%------------------------------------------------------------------------------
% Could-be-implicit typings (12)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
list_list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
set_list_list_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__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
set_num: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (116)
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_OSucc_001t__Nat__Onat,type,
bNF_Gr6352880689984616693cc_nat: set_list_nat > list_nat > set_nat ).
thf(sy_c_Binomial_Obinomial,type,
binomial: nat > nat > nat ).
thf(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
gbinomial_nat: nat > nat > nat ).
thf(sy_c_Equivalence__Relation__Enumeration_Okernel__of_001t__List__Olist_It__Nat__Onat_J,type,
equiva6490762433048536736st_nat: list_list_nat > set_Pr1261947904930325089at_nat ).
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_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_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
plus_plus_num: num > num > num ).
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__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Num__Onum,type,
if_num: $o > num > num > num ).
thf(sy_c_If_001t__Set__Oset_It__Nat__Onat_J,type,
if_set_nat: $o > set_nat > set_nat > set_nat ).
thf(sy_c_List_Oappend_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
append_list_list_nat: list_list_list_nat > list_list_list_nat > list_list_list_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__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Obind_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
bind_l7796378977173581257st_nat: list_list_nat > ( list_nat > list_list_nat ) > list_list_nat ).
thf(sy_c_List_Obind_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
bind_list_nat_nat: list_list_nat > ( list_nat > list_nat ) > list_nat ).
thf(sy_c_List_Obind_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
bind_nat_list_nat: list_nat > ( nat > list_list_nat ) > list_list_nat ).
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_Obutlast_001t__Nat__Onat,type,
butlast_nat: list_nat > list_nat ).
thf(sy_c_List_Oconcat_001t__List__Olist_It__Nat__Onat_J,type,
concat_list_nat: list_list_list_nat > list_list_nat ).
thf(sy_c_List_Oconcat_001t__Nat__Onat,type,
concat_nat: list_list_nat > list_nat ).
thf(sy_c_List_Odrop_001t__Nat__Onat,type,
drop_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Ogen__length_001t__List__Olist_It__Nat__Onat_J,type,
gen_length_list_nat: nat > list_list_nat > nat ).
thf(sy_c_List_Ogen__length_001t__Nat__Onat,type,
gen_length_nat: nat > list_nat > nat ).
thf(sy_c_List_Olast_001t__List__Olist_It__Nat__Onat_J,type,
last_list_nat: list_list_nat > list_nat ).
thf(sy_c_List_Olast_001t__Nat__Onat,type,
last_nat: list_nat > nat ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
linord2614967742042102400et_nat: set_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
cons_list_list_nat: list_list_nat > list_list_list_nat > list_list_list_nat ).
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__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
nil_list_list_nat: list_list_list_nat ).
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__Nat__Onat,type,
nil_nat: list_nat ).
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__Nat__Onat,type,
hd_nat: list_nat > nat ).
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__Nat__Onat_001t__Nat__Onat,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_list_nat2: list_list_list_nat > set_list_list_nat ).
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__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist__update_001t__List__Olist_It__Nat__Onat_J,type,
list_update_list_nat: list_list_nat > nat > list_nat > list_list_nat ).
thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
list_update_nat: list_nat > nat > nat > list_nat ).
thf(sy_c_List_Omaps_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
maps_l5785965478274863235st_nat: ( list_nat > list_list_nat ) > list_list_nat > list_list_nat ).
thf(sy_c_List_Omaps_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
maps_list_nat_nat: ( list_nat > list_nat ) > list_list_nat > list_nat ).
thf(sy_c_List_Omaps_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
maps_nat_list_nat: ( nat > list_list_nat ) > list_nat > list_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_On__lists_001t__List__Olist_It__Nat__Onat_J,type,
n_lists_list_nat: nat > list_list_nat > list_list_list_nat ).
thf(sy_c_List_On__lists_001t__Nat__Onat,type,
n_lists_nat: nat > list_nat > list_list_nat ).
thf(sy_c_List_Onth_001t__List__Olist_It__Nat__Onat_J,type,
nth_list_nat: list_list_nat > nat > list_nat ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Oproduct__lists_001t__List__Olist_It__Nat__Onat_J,type,
produc6783906451316923569st_nat: list_list_list_nat > list_list_list_nat ).
thf(sy_c_List_Oproduct__lists_001t__Nat__Onat,type,
product_lists_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Orotate1_001t__Nat__Onat,type,
rotate1_nat: list_nat > list_nat ).
thf(sy_c_List_Otake_001t__Nat__Onat,type,
take_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > list_nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
size_s6248950052170075156st_nat: list_list_list_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__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Num_Onum__of__nat,type,
num_of_nat: nat > num ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
ord_less_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Num__Onum_J,type,
ord_less_set_num: set_num > set_num > $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__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__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__Num__Onum_J,type,
ord_less_eq_set_num: set_num > set_num > $o ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
ord_max_nat: nat > nat > nat ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Num__Onum,type,
ord_max_num: num > num > num ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Nat__Onat_J,type,
ord_max_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_Othe__elem_001t__List__Olist_It__Nat__Onat_J,type,
the_elem_list_nat: set_list_nat > list_nat ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
set_ord_atMost_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
set_or6659071591806873216st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
set_or5834768355832116004an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Num__Onum,type,
set_ord_lessThan_num: num > set_num ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_or890127255671739683et_nat: set_nat > set_set_nat ).
thf(sy_c_Stirling_Ostirling,type,
stirling: nat > nat > 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_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_Sublist_Oprefix_001t__List__Olist_It__Nat__Onat_J,type,
prefix_list_nat: list_list_nat > list_list_nat > $o ).
thf(sy_c_Sublist_Oprefix_001t__Nat__Onat,type,
prefix_nat: list_nat > list_nat > $o ).
thf(sy_c_Sublist_Oprefixes_001t__List__Olist_It__Nat__Onat_J,type,
prefixes_list_nat: list_list_nat > list_list_list_nat ).
thf(sy_c_Sublist_Oprefixes_001t__Nat__Onat,type,
prefixes_nat: list_nat > list_list_nat ).
thf(sy_c_Sublist_Osublists_001t__List__Olist_It__Nat__Onat_J,type,
sublists_list_nat: list_list_nat > list_list_list_nat ).
thf(sy_c_Sublist_Osublists_001t__Nat__Onat,type,
sublists_nat: list_nat > list_list_nat ).
thf(sy_c_Sublist_Osuffixes_001t__List__Olist_It__Nat__Onat_J,type,
suffixes_list_nat: list_list_nat > list_list_list_nat ).
thf(sy_c_Sublist_Osuffixes_001t__Nat__Onat,type,
suffixes_nat: list_nat > list_list_nat ).
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__List__Olist_It__Nat__Onat_J_J,type,
member_list_list_nat: list_list_nat > set_list_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__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Num__Onum,type,
member_num: num > set_num > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_v_x,type,
x: list_nat ).
thf(sy_v_xa____,type,
xa: nat ).
thf(sy_v_xs____,type,
xs: list_nat ).
thf(sy_v_y,type,
y: list_nat ).
thf(sy_v_ya____,type,
ya: nat ).
thf(sy_v_ys____,type,
ys: list_nat ).
% Relevant facts (1263)
thf(fact_0_assms_I3_J,axiom,
equiva3371634703666331078on_rgf @ y ).
% assms(3)
thf(fact_1_assms_I2_J,axiom,
equiva3371634703666331078on_rgf @ x ).
% assms(2)
thf(fact_2_x__bound,axiom,
ord_less_nat @ xa @ ( plus_plus_nat @ ( equiva5889994315859557365_limit @ xs ) @ one_one_nat ) ).
% x_bound
thf(fact_3_y__bound,axiom,
ord_less_nat @ ya @ ( plus_plus_nat @ ( equiva5889994315859557365_limit @ ys ) @ one_one_nat ) ).
% y_bound
thf(fact_4_Cons_Oprems_I1_J,axiom,
equiva3371634703666331078on_rgf @ ( append_nat @ xs @ ( cons_nat @ xa @ nil_nat ) ) ).
% Cons.prems(1)
thf(fact_5_Cons_Oprems_I2_J,axiom,
equiva3371634703666331078on_rgf @ ( append_nat @ ys @ ( cons_nat @ ya @ nil_nat ) ) ).
% Cons.prems(2)
thf(fact_6_c,axiom,
( ( ( member_nat @ xa @ ( set_nat2 @ xs ) )
=> ( xa = ya ) )
& ( ~ ( member_nat @ xa @ ( set_nat2 @ xs ) )
=> ~ ( member_nat @ ya @ ( set_nat2 @ ys ) ) ) ) ).
% c
thf(fact_7_assms_I4_J,axiom,
( ( equiva2048684438135499664of_nat @ x )
= ( equiva2048684438135499664of_nat @ y ) ) ).
% assms(4)
thf(fact_8_Cons_Oprems_I3_J,axiom,
( ( equiva2048684438135499664of_nat @ ( append_nat @ xs @ ( cons_nat @ xa @ nil_nat ) ) )
= ( equiva2048684438135499664of_nat @ ( append_nat @ ys @ ( cons_nat @ ya @ nil_nat ) ) ) ) ).
% Cons.prems(3)
thf(fact_9_a,axiom,
( ( equiva2048684438135499664of_nat @ xs )
= ( equiva2048684438135499664of_nat @ ys ) ) ).
% a
thf(fact_10_assms_I1_J,axiom,
( ( size_size_list_nat @ x )
= ( size_size_list_nat @ y ) ) ).
% assms(1)
thf(fact_11__092_060open_062_092_060And_062i_O_Ai_A_060_Alength_Axs_A_092_060Longrightarrow_062_A_Ixs_A_B_Ai_A_061_Ax_J_A_061_A_Iys_A_B_Ai_A_061_Ay_J_092_060close_062,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ xs ) )
=> ( ( ( nth_nat @ xs @ I )
= xa )
= ( ( nth_nat @ ys @ I )
= ya ) ) ) ).
% \<open>\<And>i. i < length xs \<Longrightarrow> (xs ! i = x) = (ys ! i = y)\<close>
thf(fact_12_kernel__of__eq__len,axiom,
! [X: list_list_nat,Y: list_list_nat] :
( ( ( equiva6490762433048536736st_nat @ X )
= ( equiva6490762433048536736st_nat @ Y ) )
=> ( ( size_s3023201423986296836st_nat @ X )
= ( size_s3023201423986296836st_nat @ Y ) ) ) ).
% kernel_of_eq_len
thf(fact_13_kernel__of__eq__len,axiom,
! [X: list_list_nat,Y: list_nat] :
( ( ( equiva6490762433048536736st_nat @ X )
= ( equiva2048684438135499664of_nat @ Y ) )
=> ( ( size_s3023201423986296836st_nat @ X )
= ( size_size_list_nat @ Y ) ) ) ).
% kernel_of_eq_len
thf(fact_14_kernel__of__eq__len,axiom,
! [X: list_nat,Y: list_list_nat] :
( ( ( equiva2048684438135499664of_nat @ X )
= ( equiva6490762433048536736st_nat @ Y ) )
=> ( ( size_size_list_nat @ X )
= ( size_s3023201423986296836st_nat @ Y ) ) ) ).
% kernel_of_eq_len
thf(fact_15_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_16_rgf__limit_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ~ ! [X2: nat,Xs: list_nat] :
( X
!= ( cons_nat @ X2 @ Xs ) ) ) ).
% rgf_limit.cases
thf(fact_17_rgf__limit__ge,axiom,
! [Y: nat,Xs2: list_nat] :
( ( member_nat @ Y @ ( set_nat2 @ Xs2 ) )
=> ( ord_less_nat @ Y @ ( equiva5889994315859557365_limit @ Xs2 ) ) ) ).
% rgf_limit_ge
thf(fact_18_d_I1_J,axiom,
equiva3371634703666331078on_rgf @ xs ).
% d(1)
thf(fact_19_d_I2_J,axiom,
equiva3371634703666331078on_rgf @ ys ).
% d(2)
thf(fact_20_b,axiom,
xs = ys ).
% b
thf(fact_21_rgf__imp__initial__segment,axiom,
! [Xs2: list_nat] :
( ( equiva3371634703666331078on_rgf @ Xs2 )
=> ( ( set_nat2 @ Xs2 )
= ( set_ord_lessThan_nat @ ( equiva5889994315859557365_limit @ Xs2 ) ) ) ) ).
% rgf_imp_initial_segment
thf(fact_22_Cons_Ohyps_I1_J,axiom,
( ( size_size_list_nat @ xs )
= ( size_size_list_nat @ ys ) ) ).
% Cons.hyps(1)
thf(fact_23_Cons_Ohyps_I2_J,axiom,
( ( equiva3371634703666331078on_rgf @ xs )
=> ( ( equiva3371634703666331078on_rgf @ ys )
=> ( ( ( equiva2048684438135499664of_nat @ xs )
= ( equiva2048684438135499664of_nat @ ys ) )
=> ( xs = ys ) ) ) ) ).
% Cons.hyps(2)
thf(fact_24_rgf__snoc,axiom,
! [Xs2: list_nat,X: nat] :
( ( equiva3371634703666331078on_rgf @ ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) ) )
= ( ( equiva3371634703666331078on_rgf @ Xs2 )
& ( ord_less_nat @ X @ ( plus_plus_nat @ ( equiva5889994315859557365_limit @ Xs2 ) @ one_one_nat ) ) ) ) ).
% rgf_snoc
thf(fact_25_kernel__of__eq,axiom,
! [X: list_list_nat,Y: list_list_nat] :
( ( ( equiva6490762433048536736st_nat @ X )
= ( equiva6490762433048536736st_nat @ Y ) )
= ( ( ( size_s3023201423986296836st_nat @ X )
= ( size_s3023201423986296836st_nat @ Y ) )
& ! [J: nat] :
( ( ord_less_nat @ J @ ( size_s3023201423986296836st_nat @ X ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ( nth_list_nat @ X @ I2 )
= ( nth_list_nat @ X @ J ) )
= ( ( nth_list_nat @ Y @ I2 )
= ( nth_list_nat @ Y @ J ) ) ) ) ) ) ) ).
% kernel_of_eq
thf(fact_26_kernel__of__eq,axiom,
! [X: list_list_nat,Y: list_nat] :
( ( ( equiva6490762433048536736st_nat @ X )
= ( equiva2048684438135499664of_nat @ Y ) )
= ( ( ( size_s3023201423986296836st_nat @ X )
= ( size_size_list_nat @ Y ) )
& ! [J: nat] :
( ( ord_less_nat @ J @ ( size_s3023201423986296836st_nat @ X ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ( nth_list_nat @ X @ I2 )
= ( nth_list_nat @ X @ J ) )
= ( ( nth_nat @ Y @ I2 )
= ( nth_nat @ Y @ J ) ) ) ) ) ) ) ).
% kernel_of_eq
thf(fact_27_kernel__of__eq,axiom,
! [X: list_nat,Y: list_list_nat] :
( ( ( equiva2048684438135499664of_nat @ X )
= ( equiva6490762433048536736st_nat @ Y ) )
= ( ( ( size_size_list_nat @ X )
= ( size_s3023201423986296836st_nat @ Y ) )
& ! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_nat @ X ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ( nth_nat @ X @ I2 )
= ( nth_nat @ X @ J ) )
= ( ( nth_list_nat @ Y @ I2 )
= ( nth_list_nat @ Y @ J ) ) ) ) ) ) ) ).
% kernel_of_eq
thf(fact_28_kernel__of__eq,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( equiva2048684438135499664of_nat @ X )
= ( equiva2048684438135499664of_nat @ Y ) )
= ( ( ( size_size_list_nat @ X )
= ( size_size_list_nat @ Y ) )
& ! [J: nat] :
( ( ord_less_nat @ J @ ( size_size_list_nat @ X ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ( nth_nat @ X @ I2 )
= ( nth_nat @ X @ J ) )
= ( ( nth_nat @ Y @ I2 )
= ( nth_nat @ Y @ J ) ) ) ) ) ) ) ).
% kernel_of_eq
thf(fact_29_list__induct__2__rev,axiom,
! [X: list_nat,Y: list_list_nat,P: list_nat > list_list_nat > $o] :
( ( ( size_size_list_nat @ X )
= ( size_s3023201423986296836st_nat @ Y ) )
=> ( ( P @ nil_nat @ nil_list_nat )
=> ( ! [X2: nat,Xs: list_nat,Y2: list_nat,Ys: list_list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) @ ( append_list_nat @ Ys @ ( cons_list_nat @ Y2 @ nil_list_nat ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% list_induct_2_rev
thf(fact_30_list__induct__2__rev,axiom,
! [X: list_list_nat,Y: list_nat,P: list_list_nat > list_nat > $o] :
( ( ( size_s3023201423986296836st_nat @ X )
= ( size_size_list_nat @ Y ) )
=> ( ( P @ nil_list_nat @ nil_nat )
=> ( ! [X2: list_nat,Xs: list_list_nat,Y2: nat,Ys: list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) ) @ ( append_nat @ Ys @ ( cons_nat @ Y2 @ nil_nat ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% list_induct_2_rev
thf(fact_31_list__induct__2__rev,axiom,
! [X: list_list_nat,Y: list_list_nat,P: list_list_nat > list_list_nat > $o] :
( ( ( size_s3023201423986296836st_nat @ X )
= ( size_s3023201423986296836st_nat @ Y ) )
=> ( ( P @ nil_list_nat @ nil_list_nat )
=> ( ! [X2: list_nat,Xs: list_list_nat,Y2: list_nat,Ys: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) ) @ ( append_list_nat @ Ys @ ( cons_list_nat @ Y2 @ nil_list_nat ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% list_induct_2_rev
thf(fact_32_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,Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) @ ( append_nat @ Ys @ ( cons_nat @ Y2 @ nil_nat ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% list_induct_2_rev
thf(fact_33_nth__append__length__plus,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat,N: nat] :
( ( nth_list_nat @ ( append_list_nat @ Xs2 @ Ys2 ) @ ( plus_plus_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ N ) )
= ( nth_list_nat @ Ys2 @ N ) ) ).
% nth_append_length_plus
thf(fact_34_nth__append__length__plus,axiom,
! [Xs2: list_nat,Ys2: list_nat,N: nat] :
( ( nth_nat @ ( append_nat @ Xs2 @ Ys2 ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs2 ) @ N ) )
= ( nth_nat @ Ys2 @ N ) ) ).
% nth_append_length_plus
thf(fact_35_nth__append__length,axiom,
! [Xs2: list_list_nat,X: list_nat,Ys2: list_list_nat] :
( ( nth_list_nat @ ( append_list_nat @ Xs2 @ ( cons_list_nat @ X @ Ys2 ) ) @ ( size_s3023201423986296836st_nat @ Xs2 ) )
= X ) ).
% nth_append_length
thf(fact_36_nth__append__length,axiom,
! [Xs2: list_nat,X: nat,Ys2: list_nat] :
( ( nth_nat @ ( append_nat @ Xs2 @ ( cons_nat @ X @ Ys2 ) ) @ ( size_size_list_nat @ Xs2 ) )
= X ) ).
% nth_append_length
thf(fact_37_length__append,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat] :
( ( size_s3023201423986296836st_nat @ ( append_list_nat @ Xs2 @ Ys2 ) )
= ( plus_plus_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ ( size_s3023201423986296836st_nat @ Ys2 ) ) ) ).
% length_append
thf(fact_38_length__append,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( size_size_list_nat @ ( append_nat @ Xs2 @ Ys2 ) )
= ( plus_plus_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ Ys2 ) ) ) ).
% length_append
thf(fact_39_append1__eq__conv,axiom,
! [Xs2: list_list_nat,X: list_nat,Ys2: list_list_nat,Y: list_nat] :
( ( ( append_list_nat @ Xs2 @ ( cons_list_nat @ X @ nil_list_nat ) )
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ Y @ nil_list_nat ) ) )
= ( ( Xs2 = Ys2 )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_40_append1__eq__conv,axiom,
! [Xs2: list_nat,X: nat,Ys2: list_nat,Y: nat] :
( ( ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) )
= ( append_nat @ Ys2 @ ( cons_nat @ Y @ nil_nat ) ) )
= ( ( Xs2 = Ys2 )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_41_append__eq__append__conv,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat,Us: list_list_nat,Vs: list_list_nat] :
( ( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
| ( ( size_s3023201423986296836st_nat @ Us )
= ( size_s3023201423986296836st_nat @ Vs ) ) )
=> ( ( ( append_list_nat @ Xs2 @ Us )
= ( append_list_nat @ Ys2 @ Vs ) )
= ( ( Xs2 = Ys2 )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_42_append__eq__append__conv,axiom,
! [Xs2: list_nat,Ys2: list_nat,Us: list_nat,Vs: list_nat] :
( ( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
| ( ( size_size_list_nat @ Us )
= ( size_size_list_nat @ Vs ) ) )
=> ( ( ( append_nat @ Xs2 @ Us )
= ( append_nat @ Ys2 @ Vs ) )
= ( ( Xs2 = Ys2 )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_43_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_44_append_Oright__neutral,axiom,
! [A: list_list_nat] :
( ( append_list_nat @ A @ nil_list_nat )
= A ) ).
% append.right_neutral
thf(fact_45_append_Oright__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ A @ nil_nat )
= A ) ).
% append.right_neutral
thf(fact_46_append__Nil2,axiom,
! [Xs2: list_list_nat] :
( ( append_list_nat @ Xs2 @ nil_list_nat )
= Xs2 ) ).
% append_Nil2
thf(fact_47_append__Nil2,axiom,
! [Xs2: list_nat] :
( ( append_nat @ Xs2 @ nil_nat )
= Xs2 ) ).
% append_Nil2
thf(fact_48_append__self__conv,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat] :
( ( ( append_list_nat @ Xs2 @ Ys2 )
= Xs2 )
= ( Ys2 = nil_list_nat ) ) ).
% append_self_conv
thf(fact_49_append__self__conv,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( ( append_nat @ Xs2 @ Ys2 )
= Xs2 )
= ( Ys2 = nil_nat ) ) ).
% append_self_conv
thf(fact_50_self__append__conv,axiom,
! [Y: list_list_nat,Ys2: list_list_nat] :
( ( Y
= ( append_list_nat @ Y @ Ys2 ) )
= ( Ys2 = nil_list_nat ) ) ).
% self_append_conv
thf(fact_51_self__append__conv,axiom,
! [Y: list_nat,Ys2: list_nat] :
( ( Y
= ( append_nat @ Y @ Ys2 ) )
= ( Ys2 = nil_nat ) ) ).
% self_append_conv
thf(fact_52_append__self__conv2,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat] :
( ( ( append_list_nat @ Xs2 @ Ys2 )
= Ys2 )
= ( Xs2 = nil_list_nat ) ) ).
% append_self_conv2
thf(fact_53_append__self__conv2,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( ( append_nat @ Xs2 @ Ys2 )
= Ys2 )
= ( Xs2 = nil_nat ) ) ).
% append_self_conv2
thf(fact_54_self__append__conv2,axiom,
! [Y: list_list_nat,Xs2: list_list_nat] :
( ( Y
= ( append_list_nat @ Xs2 @ Y ) )
= ( Xs2 = nil_list_nat ) ) ).
% self_append_conv2
thf(fact_55_self__append__conv2,axiom,
! [Y: list_nat,Xs2: list_nat] :
( ( Y
= ( append_nat @ Xs2 @ Y ) )
= ( Xs2 = nil_nat ) ) ).
% self_append_conv2
thf(fact_56_list_Oinject,axiom,
! [X21: list_nat,X22: list_list_nat,Y21: list_nat,Y22: list_list_nat] :
( ( ( cons_list_nat @ X21 @ X22 )
= ( cons_list_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_57_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_58_same__append__eq,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat,Zs: list_list_nat] :
( ( ( append_list_nat @ Xs2 @ Ys2 )
= ( append_list_nat @ Xs2 @ Zs ) )
= ( Ys2 = Zs ) ) ).
% same_append_eq
thf(fact_59_same__append__eq,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( ( append_nat @ Xs2 @ Ys2 )
= ( append_nat @ Xs2 @ Zs ) )
= ( Ys2 = Zs ) ) ).
% same_append_eq
thf(fact_60_append__same__eq,axiom,
! [Ys2: list_list_nat,Xs2: list_list_nat,Zs: list_list_nat] :
( ( ( append_list_nat @ Ys2 @ Xs2 )
= ( append_list_nat @ Zs @ Xs2 ) )
= ( Ys2 = Zs ) ) ).
% append_same_eq
thf(fact_61_append__same__eq,axiom,
! [Ys2: list_nat,Xs2: list_nat,Zs: list_nat] :
( ( ( append_nat @ Ys2 @ Xs2 )
= ( append_nat @ Zs @ Xs2 ) )
= ( Ys2 = Zs ) ) ).
% append_same_eq
thf(fact_62_append__assoc,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat,Zs: list_list_nat] :
( ( append_list_nat @ ( append_list_nat @ Xs2 @ Ys2 ) @ Zs )
= ( append_list_nat @ Xs2 @ ( append_list_nat @ Ys2 @ Zs ) ) ) ).
% append_assoc
thf(fact_63_append__assoc,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( append_nat @ ( append_nat @ Xs2 @ Ys2 ) @ Zs )
= ( append_nat @ Xs2 @ ( append_nat @ Ys2 @ Zs ) ) ) ).
% append_assoc
thf(fact_64_append_Oassoc,axiom,
! [A: list_list_nat,B: list_list_nat,C: list_list_nat] :
( ( append_list_nat @ ( append_list_nat @ A @ B ) @ C )
= ( append_list_nat @ A @ ( append_list_nat @ B @ C ) ) ) ).
% append.assoc
thf(fact_65_append_Oassoc,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( append_nat @ ( append_nat @ A @ B ) @ C )
= ( append_nat @ A @ ( append_nat @ B @ C ) ) ) ).
% append.assoc
thf(fact_66_append__is__Nil__conv,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat] :
( ( ( append_list_nat @ Xs2 @ Ys2 )
= nil_list_nat )
= ( ( Xs2 = nil_list_nat )
& ( Ys2 = nil_list_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_67_append__is__Nil__conv,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( ( append_nat @ Xs2 @ Ys2 )
= nil_nat )
= ( ( Xs2 = nil_nat )
& ( Ys2 = nil_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_68_Nil__is__append__conv,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat] :
( ( nil_list_nat
= ( append_list_nat @ Xs2 @ Ys2 ) )
= ( ( Xs2 = nil_list_nat )
& ( Ys2 = nil_list_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_69_Nil__is__append__conv,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( nil_nat
= ( append_nat @ Xs2 @ Ys2 ) )
= ( ( Xs2 = nil_nat )
& ( Ys2 = nil_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_70_mem__Collect__eq,axiom,
! [A: list_nat,P: list_nat > $o] :
( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_71_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_72_Collect__mem__eq,axiom,
! [A2: set_list_nat] :
( ( collect_list_nat
@ ^ [X3: list_nat] : ( member_list_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_73_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_74_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_75_not__Cons__self2,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( cons_list_nat @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_76_not__Cons__self2,axiom,
! [X: nat,Xs2: list_nat] :
( ( cons_nat @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_77_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_78_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_79_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_80_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_81_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_82_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_83_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_84_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_85_size__neq__size__imp__neq,axiom,
! [X: list_list_nat,Y: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ X )
!= ( size_s3023201423986296836st_nat @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_86_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_87_size__neq__size__imp__neq,axiom,
! [X: char,Y: char] :
( ( ( size_size_char @ X )
!= ( size_size_char @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_88_neq__if__length__neq,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
!= ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( Xs2 != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_89_neq__if__length__neq,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
!= ( size_size_list_nat @ Ys2 ) )
=> ( Xs2 != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_90_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs: list_list_nat] :
( ( size_s3023201423986296836st_nat @ Xs )
= N ) ).
% Ex_list_of_length
thf(fact_91_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs: list_nat] :
( ( size_size_list_nat @ Xs )
= N ) ).
% Ex_list_of_length
thf(fact_92_append__eq__append__conv2,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat,Zs: list_list_nat,Ts: list_list_nat] :
( ( ( append_list_nat @ Xs2 @ Ys2 )
= ( append_list_nat @ Zs @ Ts ) )
= ( ? [Us2: list_list_nat] :
( ( ( Xs2
= ( append_list_nat @ Zs @ Us2 ) )
& ( ( append_list_nat @ Us2 @ Ys2 )
= Ts ) )
| ( ( ( append_list_nat @ Xs2 @ Us2 )
= Zs )
& ( Ys2
= ( append_list_nat @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_93_append__eq__append__conv2,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs: list_nat,Ts: list_nat] :
( ( ( append_nat @ Xs2 @ Ys2 )
= ( append_nat @ Zs @ Ts ) )
= ( ? [Us2: list_nat] :
( ( ( Xs2
= ( append_nat @ Zs @ Us2 ) )
& ( ( append_nat @ Us2 @ Ys2 )
= Ts ) )
| ( ( ( append_nat @ Xs2 @ Us2 )
= Zs )
& ( Ys2
= ( append_nat @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_94_append__eq__appendI,axiom,
! [Xs2: list_list_nat,Xs1: list_list_nat,Zs: list_list_nat,Ys2: list_list_nat,Us: list_list_nat] :
( ( ( append_list_nat @ Xs2 @ Xs1 )
= Zs )
=> ( ( Ys2
= ( append_list_nat @ Xs1 @ Us ) )
=> ( ( append_list_nat @ Xs2 @ Ys2 )
= ( append_list_nat @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_95_append__eq__appendI,axiom,
! [Xs2: list_nat,Xs1: list_nat,Zs: list_nat,Ys2: list_nat,Us: list_nat] :
( ( ( append_nat @ Xs2 @ Xs1 )
= Zs )
=> ( ( Ys2
= ( append_nat @ Xs1 @ Us ) )
=> ( ( append_nat @ Xs2 @ Ys2 )
= ( append_nat @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_96_list__nonempty__induct,axiom,
! [Xs2: list_list_nat,P: list_list_nat > $o] :
( ( Xs2 != nil_list_nat )
=> ( ! [X2: list_nat] : ( P @ ( cons_list_nat @ X2 @ nil_list_nat ) )
=> ( ! [X2: list_nat,Xs: list_list_nat] :
( ( Xs != nil_list_nat )
=> ( ( P @ Xs )
=> ( P @ ( cons_list_nat @ X2 @ Xs ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_97_list__nonempty__induct,axiom,
! [Xs2: list_nat,P: list_nat > $o] :
( ( Xs2 != nil_nat )
=> ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
=> ( ! [X2: nat,Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( P @ Xs )
=> ( P @ ( cons_nat @ X2 @ Xs ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_98_list__induct2_H,axiom,
! [P: list_nat > list_list_nat > $o,Xs2: list_nat,Ys2: list_list_nat] :
( ( P @ nil_nat @ nil_list_nat )
=> ( ! [X2: nat,Xs: list_nat] : ( P @ ( cons_nat @ X2 @ Xs ) @ nil_list_nat )
=> ( ! [Y2: list_nat,Ys: list_list_nat] : ( P @ nil_nat @ ( cons_list_nat @ Y2 @ Ys ) )
=> ( ! [X2: nat,Xs: list_nat,Y2: list_nat,Ys: list_list_nat] :
( ( P @ Xs @ Ys )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y2 @ Ys ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_99_list__induct2_H,axiom,
! [P: list_list_nat > list_nat > $o,Xs2: list_list_nat,Ys2: list_nat] :
( ( P @ nil_list_nat @ nil_nat )
=> ( ! [X2: list_nat,Xs: list_list_nat] : ( P @ ( cons_list_nat @ X2 @ Xs ) @ nil_nat )
=> ( ! [Y2: nat,Ys: list_nat] : ( P @ nil_list_nat @ ( cons_nat @ Y2 @ Ys ) )
=> ( ! [X2: list_nat,Xs: list_list_nat,Y2: nat,Ys: list_nat] :
( ( P @ Xs @ Ys )
=> ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_100_list__induct2_H,axiom,
! [P: list_list_nat > list_list_nat > $o,Xs2: list_list_nat,Ys2: list_list_nat] :
( ( P @ nil_list_nat @ nil_list_nat )
=> ( ! [X2: list_nat,Xs: list_list_nat] : ( P @ ( cons_list_nat @ X2 @ Xs ) @ nil_list_nat )
=> ( ! [Y2: list_nat,Ys: list_list_nat] : ( P @ nil_list_nat @ ( cons_list_nat @ Y2 @ Ys ) )
=> ( ! [X2: list_nat,Xs: list_list_nat,Y2: list_nat,Ys: list_list_nat] :
( ( P @ Xs @ Ys )
=> ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y2 @ Ys ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_101_list__induct2_H,axiom,
! [P: list_nat > list_nat > $o,Xs2: list_nat,Ys2: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat] : ( P @ ( cons_nat @ X2 @ Xs ) @ nil_nat )
=> ( ! [Y2: nat,Ys: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y2 @ Ys ) )
=> ( ! [X2: nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( P @ Xs @ Ys )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_102_neq__Nil__conv,axiom,
! [Xs2: list_list_nat] :
( ( Xs2 != nil_list_nat )
= ( ? [Y3: list_nat,Ys3: list_list_nat] :
( Xs2
= ( cons_list_nat @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_103_neq__Nil__conv,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
= ( ? [Y3: nat,Ys3: list_nat] :
( Xs2
= ( cons_nat @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_104_remdups__adj_Ocases,axiom,
! [X: list_list_nat] :
( ( X != nil_list_nat )
=> ( ! [X2: list_nat] :
( X
!= ( cons_list_nat @ X2 @ nil_list_nat ) )
=> ~ ! [X2: list_nat,Y2: list_nat,Xs: list_list_nat] :
( X
!= ( cons_list_nat @ X2 @ ( cons_list_nat @ Y2 @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_105_remdups__adj_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ( ! [X2: nat] :
( X
!= ( cons_nat @ X2 @ nil_nat ) )
=> ~ ! [X2: nat,Y2: nat,Xs: list_nat] :
( X
!= ( cons_nat @ X2 @ ( cons_nat @ Y2 @ Xs ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_106_transpose_Ocases,axiom,
! [X: list_list_list_nat] :
( ( X != nil_list_list_nat )
=> ( ! [Xss: list_list_list_nat] :
( X
!= ( cons_list_list_nat @ nil_list_nat @ Xss ) )
=> ~ ! [X2: list_nat,Xs: list_list_nat,Xss: list_list_list_nat] :
( X
!= ( cons_list_list_nat @ ( cons_list_nat @ X2 @ Xs ) @ 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,Xs: list_nat,Xss: list_list_nat] :
( X
!= ( cons_list_nat @ ( cons_nat @ X2 @ Xs ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_108_min__list_Ocases,axiom,
! [X: list_nat] :
( ! [X2: nat,Xs: list_nat] :
( X
!= ( cons_nat @ X2 @ Xs ) )
=> ( X = nil_nat ) ) ).
% min_list.cases
thf(fact_109_list_Oexhaust,axiom,
! [Y: list_list_nat] :
( ( Y != nil_list_nat )
=> ~ ! [X212: list_nat,X222: list_list_nat] :
( Y
!= ( cons_list_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_110_list_Oexhaust,axiom,
! [Y: list_nat] :
( ( Y != nil_nat )
=> ~ ! [X212: nat,X222: list_nat] :
( Y
!= ( cons_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_111_list_OdiscI,axiom,
! [List: list_list_nat,X21: list_nat,X22: list_list_nat] :
( ( List
= ( cons_list_nat @ X21 @ X22 ) )
=> ( List != nil_list_nat ) ) ).
% list.discI
thf(fact_112_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X22: list_nat] :
( ( List
= ( cons_nat @ X21 @ X22 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_113_list_Odistinct_I1_J,axiom,
! [X21: list_nat,X22: list_list_nat] :
( nil_list_nat
!= ( cons_list_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_114_list_Odistinct_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_115_set__ConsD,axiom,
! [Y: list_nat,X: list_nat,Xs2: list_list_nat] :
( ( member_list_nat @ Y @ ( set_list_nat2 @ ( cons_list_nat @ X @ Xs2 ) ) )
=> ( ( Y = X )
| ( member_list_nat @ Y @ ( set_list_nat2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_116_set__ConsD,axiom,
! [Y: nat,X: nat,Xs2: list_nat] :
( ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X @ Xs2 ) ) )
=> ( ( Y = X )
| ( member_nat @ Y @ ( set_nat2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_117_list_Oset__cases,axiom,
! [E: list_nat,A: list_list_nat] :
( ( member_list_nat @ E @ ( set_list_nat2 @ A ) )
=> ( ! [Z2: list_list_nat] :
( A
!= ( cons_list_nat @ E @ Z2 ) )
=> ~ ! [Z1: list_nat,Z2: list_list_nat] :
( ( A
= ( cons_list_nat @ Z1 @ Z2 ) )
=> ~ ( member_list_nat @ E @ ( set_list_nat2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_118_list_Oset__cases,axiom,
! [E: nat,A: list_nat] :
( ( member_nat @ E @ ( set_nat2 @ A ) )
=> ( ! [Z2: list_nat] :
( A
!= ( cons_nat @ E @ Z2 ) )
=> ~ ! [Z1: nat,Z2: list_nat] :
( ( A
= ( cons_nat @ Z1 @ Z2 ) )
=> ~ ( member_nat @ E @ ( set_nat2 @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_119_list_Oset__intros_I1_J,axiom,
! [X21: list_nat,X22: list_list_nat] : ( member_list_nat @ X21 @ ( set_list_nat2 @ ( cons_list_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_120_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_121_list_Oset__intros_I2_J,axiom,
! [Y: list_nat,X22: list_list_nat,X21: list_nat] :
( ( member_list_nat @ Y @ ( set_list_nat2 @ X22 ) )
=> ( member_list_nat @ Y @ ( set_list_nat2 @ ( cons_list_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_122_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_123_Cons__eq__appendI,axiom,
! [X: list_nat,Xs1: list_list_nat,Ys2: list_list_nat,Xs2: list_list_nat,Zs: list_list_nat] :
( ( ( cons_list_nat @ X @ Xs1 )
= Ys2 )
=> ( ( Xs2
= ( append_list_nat @ Xs1 @ Zs ) )
=> ( ( cons_list_nat @ X @ Xs2 )
= ( append_list_nat @ Ys2 @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_124_Cons__eq__appendI,axiom,
! [X: nat,Xs1: list_nat,Ys2: list_nat,Xs2: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X @ Xs1 )
= Ys2 )
=> ( ( Xs2
= ( append_nat @ Xs1 @ Zs ) )
=> ( ( cons_nat @ X @ Xs2 )
= ( append_nat @ Ys2 @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_125_append__Cons,axiom,
! [X: list_nat,Xs2: list_list_nat,Ys2: list_list_nat] :
( ( append_list_nat @ ( cons_list_nat @ X @ Xs2 ) @ Ys2 )
= ( cons_list_nat @ X @ ( append_list_nat @ Xs2 @ Ys2 ) ) ) ).
% append_Cons
thf(fact_126_append__Cons,axiom,
! [X: nat,Xs2: list_nat,Ys2: list_nat] :
( ( append_nat @ ( cons_nat @ X @ Xs2 ) @ Ys2 )
= ( cons_nat @ X @ ( append_nat @ Xs2 @ Ys2 ) ) ) ).
% append_Cons
thf(fact_127_eq__Nil__appendI,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat] :
( ( Xs2 = Ys2 )
=> ( Xs2
= ( append_list_nat @ nil_list_nat @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_128_eq__Nil__appendI,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( Xs2 = Ys2 )
=> ( Xs2
= ( append_nat @ nil_nat @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_129_append_Oleft__neutral,axiom,
! [A: list_list_nat] :
( ( append_list_nat @ nil_list_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_130_append_Oleft__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ nil_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_131_append__Nil,axiom,
! [Ys2: list_list_nat] :
( ( append_list_nat @ nil_list_nat @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_132_append__Nil,axiom,
! [Ys2: list_nat] :
( ( append_nat @ nil_nat @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_133_length__induct,axiom,
! [P: list_list_nat > $o,Xs2: list_list_nat] :
( ! [Xs: list_list_nat] :
( ! [Ys4: list_list_nat] :
( ( ord_less_nat @ ( size_s3023201423986296836st_nat @ Ys4 ) @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( P @ Ys4 ) )
=> ( P @ Xs ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_134_length__induct,axiom,
! [P: list_nat > $o,Xs2: list_nat] :
( ! [Xs: list_nat] :
( ! [Ys4: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys4 ) @ ( size_size_list_nat @ Xs ) )
=> ( P @ Ys4 ) )
=> ( P @ Xs ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_135_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_136_trans__less__add2,axiom,
! [I: nat,J2: nat,M: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_less_add2
thf(fact_137_trans__less__add1,axiom,
! [I: nat,J2: nat,M: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_less_add1
thf(fact_138_add__less__mono1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_less_mono1
thf(fact_139_not__add__less2,axiom,
! [J2: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I ) @ I ) ).
% not_add_less2
thf(fact_140_not__add__less1,axiom,
! [I: nat,J2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J2 ) @ I ) ).
% not_add_less1
thf(fact_141_add__less__mono,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_less_mono
thf(fact_142_add__lessD1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J2 ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_143_list__induct4,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs: list_nat,Ws: list_nat,P: list_nat > list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat,Y2: nat,Ys: list_nat,Z: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs @ Ys @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_144_list__induct4,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs: list_nat,Ws: list_list_nat,P: list_nat > list_nat > list_nat > list_list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_s3023201423986296836st_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_list_nat )
=> ( ! [X2: nat,Xs: list_nat,Y2: nat,Ys: list_nat,Z: nat,Zs2: list_nat,W: list_nat,Ws2: list_list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_s3023201423986296836st_nat @ Ws2 ) )
=> ( ( P @ Xs @ Ys @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_list_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_145_list__induct4,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs: list_list_nat,Ws: list_nat,P: list_nat > list_nat > list_list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_s3023201423986296836st_nat @ Zs ) )
=> ( ( ( size_s3023201423986296836st_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_list_nat @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat,Y2: nat,Ys: list_nat,Z: list_nat,Zs2: list_list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_s3023201423986296836st_nat @ Zs2 ) )
=> ( ( ( size_s3023201423986296836st_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs @ Ys @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) @ ( cons_list_nat @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_146_list__induct4,axiom,
! [Xs2: list_nat,Ys2: list_list_nat,Zs: list_nat,Ws: list_nat,P: list_nat > list_list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_list_nat @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat,Y2: list_nat,Ys: list_list_nat,Z: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs @ Ys @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y2 @ Ys ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_147_list__induct4,axiom,
! [Xs2: list_list_nat,Ys2: list_nat,Zs: list_nat,Ws: list_nat,P: list_list_nat > list_nat > list_nat > list_nat > $o] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_list_nat @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X2: list_nat,Xs: list_list_nat,Y2: nat,Ys: list_nat,Z: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs @ Ys @ Zs2 @ Ws2 )
=> ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_148_list__induct4,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs: list_list_nat,Ws: list_list_nat,P: list_nat > list_nat > list_list_nat > list_list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_s3023201423986296836st_nat @ Zs ) )
=> ( ( ( size_s3023201423986296836st_nat @ Zs )
= ( size_s3023201423986296836st_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_list_nat @ nil_list_nat )
=> ( ! [X2: nat,Xs: list_nat,Y2: nat,Ys: list_nat,Z: list_nat,Zs2: list_list_nat,W: list_nat,Ws2: list_list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_s3023201423986296836st_nat @ Zs2 ) )
=> ( ( ( size_s3023201423986296836st_nat @ Zs2 )
= ( size_s3023201423986296836st_nat @ Ws2 ) )
=> ( ( P @ Xs @ Ys @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) @ ( cons_list_nat @ Z @ Zs2 ) @ ( cons_list_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_149_list__induct4,axiom,
! [Xs2: list_nat,Ys2: list_list_nat,Zs: list_nat,Ws: list_list_nat,P: list_nat > list_list_nat > list_nat > list_list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_s3023201423986296836st_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_list_nat @ nil_nat @ nil_list_nat )
=> ( ! [X2: nat,Xs: list_nat,Y2: list_nat,Ys: list_list_nat,Z: nat,Zs2: list_nat,W: list_nat,Ws2: list_list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_s3023201423986296836st_nat @ Ws2 ) )
=> ( ( P @ Xs @ Ys @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y2 @ Ys ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_list_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_150_list__induct4,axiom,
! [Xs2: list_nat,Ys2: list_list_nat,Zs: list_list_nat,Ws: list_nat,P: list_nat > list_list_nat > list_list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys2 )
= ( size_s3023201423986296836st_nat @ Zs ) )
=> ( ( ( size_s3023201423986296836st_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_list_nat @ nil_list_nat @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat,Y2: list_nat,Ys: list_list_nat,Z: list_nat,Zs2: list_list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys )
= ( size_s3023201423986296836st_nat @ Zs2 ) )
=> ( ( ( size_s3023201423986296836st_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs @ Ys @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y2 @ Ys ) @ ( cons_list_nat @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_151_list__induct4,axiom,
! [Xs2: list_list_nat,Ys2: list_nat,Zs: list_nat,Ws: list_list_nat,P: list_list_nat > list_nat > list_nat > list_list_nat > $o] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_s3023201423986296836st_nat @ Ws ) )
=> ( ( P @ nil_list_nat @ nil_nat @ nil_nat @ nil_list_nat )
=> ( ! [X2: list_nat,Xs: list_list_nat,Y2: nat,Ys: list_nat,Z: nat,Zs2: list_nat,W: list_nat,Ws2: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_s3023201423986296836st_nat @ Ws2 ) )
=> ( ( P @ Xs @ Ys @ Zs2 @ Ws2 )
=> ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_list_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_152_list__induct4,axiom,
! [Xs2: list_list_nat,Ys2: list_nat,Zs: list_list_nat,Ws: list_nat,P: list_list_nat > list_nat > list_list_nat > list_nat > $o] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_s3023201423986296836st_nat @ Zs ) )
=> ( ( ( size_s3023201423986296836st_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_list_nat @ nil_nat @ nil_list_nat @ nil_nat )
=> ( ! [X2: list_nat,Xs: list_list_nat,Y2: nat,Ys: list_nat,Z: list_nat,Zs2: list_list_nat,W: nat,Ws2: list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_s3023201423986296836st_nat @ Zs2 ) )
=> ( ( ( size_s3023201423986296836st_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs @ Ys @ Zs2 @ Ws2 )
=> ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) @ ( cons_list_nat @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_153_list__induct3,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs: list_list_nat,P: list_nat > list_nat > list_list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_s3023201423986296836st_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_list_nat )
=> ( ! [X2: nat,Xs: list_nat,Y2: nat,Ys: list_nat,Z: list_nat,Zs2: list_list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_s3023201423986296836st_nat @ Zs2 ) )
=> ( ( P @ Xs @ Ys @ Zs2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) @ ( cons_list_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_154_list__induct3,axiom,
! [Xs2: list_nat,Ys2: list_list_nat,Zs: list_nat,P: list_nat > list_list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_list_nat @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat,Y2: list_nat,Ys: list_list_nat,Z: nat,Zs2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs @ Ys @ Zs2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y2 @ Ys ) @ ( cons_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_155_list__induct3,axiom,
! [Xs2: list_nat,Ys2: list_list_nat,Zs: list_list_nat,P: list_nat > list_list_nat > list_list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys2 )
= ( size_s3023201423986296836st_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_list_nat @ nil_list_nat )
=> ( ! [X2: nat,Xs: list_nat,Y2: list_nat,Ys: list_list_nat,Z: list_nat,Zs2: list_list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys )
= ( size_s3023201423986296836st_nat @ Zs2 ) )
=> ( ( P @ Xs @ Ys @ Zs2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y2 @ Ys ) @ ( cons_list_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_156_list__induct3,axiom,
! [Xs2: list_list_nat,Ys2: list_nat,Zs: list_nat,P: list_list_nat > list_nat > list_nat > $o] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_list_nat @ nil_nat @ nil_nat )
=> ( ! [X2: list_nat,Xs: list_list_nat,Y2: nat,Ys: list_nat,Z: nat,Zs2: list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs @ Ys @ Zs2 )
=> ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) @ ( cons_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_157_list__induct3,axiom,
! [Xs2: list_list_nat,Ys2: list_nat,Zs: list_list_nat,P: list_list_nat > list_nat > list_list_nat > $o] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_s3023201423986296836st_nat @ Zs ) )
=> ( ( P @ nil_list_nat @ nil_nat @ nil_list_nat )
=> ( ! [X2: list_nat,Xs: list_list_nat,Y2: nat,Ys: list_nat,Z: list_nat,Zs2: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_s3023201423986296836st_nat @ Zs2 ) )
=> ( ( P @ Xs @ Ys @ Zs2 )
=> ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) @ ( cons_list_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_158_list__induct3,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat,Zs: list_nat,P: list_list_nat > list_list_nat > list_nat > $o] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_list_nat @ nil_list_nat @ nil_nat )
=> ( ! [X2: list_nat,Xs: list_list_nat,Y2: list_nat,Ys: list_list_nat,Z: nat,Zs2: list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs @ Ys @ Zs2 )
=> ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y2 @ Ys ) @ ( cons_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_159_list__induct3,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat,Zs: list_list_nat,P: list_list_nat > list_list_nat > list_list_nat > $o] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys2 )
= ( size_s3023201423986296836st_nat @ Zs ) )
=> ( ( P @ nil_list_nat @ nil_list_nat @ nil_list_nat )
=> ( ! [X2: list_nat,Xs: list_list_nat,Y2: list_nat,Ys: list_list_nat,Z: list_nat,Zs2: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys )
= ( size_s3023201423986296836st_nat @ Zs2 ) )
=> ( ( P @ Xs @ Ys @ Zs2 )
=> ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y2 @ Ys ) @ ( cons_list_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_160_list__induct3,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs: list_nat,P: list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat,Y2: nat,Ys: list_nat,Z: nat,Zs2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs @ Ys @ Zs2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) @ ( cons_nat @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_161_list__induct2,axiom,
! [Xs2: list_nat,Ys2: list_list_nat,P: list_nat > list_list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( ( P @ nil_nat @ nil_list_nat )
=> ( ! [X2: nat,Xs: list_nat,Y2: list_nat,Ys: list_list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y2 @ Ys ) ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_162_list__induct2,axiom,
! [Xs2: list_list_nat,Ys2: list_nat,P: list_list_nat > list_nat > $o] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ nil_list_nat @ nil_nat )
=> ( ! [X2: list_nat,Xs: list_list_nat,Y2: nat,Ys: list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_163_list__induct2,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat,P: list_list_nat > list_list_nat > $o] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( ( P @ nil_list_nat @ nil_list_nat )
=> ( ! [X2: list_nat,Xs: list_list_nat,Y2: list_nat,Ys: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y2 @ Ys ) ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_164_list__induct2,axiom,
! [Xs2: list_nat,Ys2: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_165_rev__nonempty__induct,axiom,
! [Xs2: list_list_nat,P: list_list_nat > $o] :
( ( Xs2 != nil_list_nat )
=> ( ! [X2: list_nat] : ( P @ ( cons_list_nat @ X2 @ nil_list_nat ) )
=> ( ! [X2: list_nat,Xs: list_list_nat] :
( ( Xs != nil_list_nat )
=> ( ( P @ Xs )
=> ( P @ ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% rev_nonempty_induct
thf(fact_166_rev__nonempty__induct,axiom,
! [Xs2: list_nat,P: list_nat > $o] :
( ( Xs2 != nil_nat )
=> ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
=> ( ! [X2: nat,Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( P @ Xs )
=> ( P @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% rev_nonempty_induct
thf(fact_167_append__eq__Cons__conv,axiom,
! [Ys2: list_list_nat,Zs: list_list_nat,X: list_nat,Xs2: list_list_nat] :
( ( ( append_list_nat @ Ys2 @ Zs )
= ( cons_list_nat @ X @ Xs2 ) )
= ( ( ( Ys2 = nil_list_nat )
& ( Zs
= ( cons_list_nat @ X @ Xs2 ) ) )
| ? [Ys5: list_list_nat] :
( ( Ys2
= ( cons_list_nat @ X @ Ys5 ) )
& ( ( append_list_nat @ Ys5 @ Zs )
= Xs2 ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_168_append__eq__Cons__conv,axiom,
! [Ys2: list_nat,Zs: list_nat,X: nat,Xs2: list_nat] :
( ( ( append_nat @ Ys2 @ Zs )
= ( cons_nat @ X @ Xs2 ) )
= ( ( ( Ys2 = nil_nat )
& ( Zs
= ( cons_nat @ X @ Xs2 ) ) )
| ? [Ys5: list_nat] :
( ( Ys2
= ( cons_nat @ X @ Ys5 ) )
& ( ( append_nat @ Ys5 @ Zs )
= Xs2 ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_169_Cons__eq__append__conv,axiom,
! [X: list_nat,Xs2: list_list_nat,Ys2: list_list_nat,Zs: list_list_nat] :
( ( ( cons_list_nat @ X @ Xs2 )
= ( append_list_nat @ Ys2 @ Zs ) )
= ( ( ( Ys2 = nil_list_nat )
& ( ( cons_list_nat @ X @ Xs2 )
= Zs ) )
| ? [Ys5: list_list_nat] :
( ( ( cons_list_nat @ X @ Ys5 )
= Ys2 )
& ( Xs2
= ( append_list_nat @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_170_Cons__eq__append__conv,axiom,
! [X: nat,Xs2: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X @ Xs2 )
= ( append_nat @ Ys2 @ Zs ) )
= ( ( ( Ys2 = nil_nat )
& ( ( cons_nat @ X @ Xs2 )
= Zs ) )
| ? [Ys5: list_nat] :
( ( ( cons_nat @ X @ Ys5 )
= Ys2 )
& ( Xs2
= ( append_nat @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_171_rev__exhaust,axiom,
! [Xs2: list_list_nat] :
( ( Xs2 != nil_list_nat )
=> ~ ! [Ys: list_list_nat,Y2: list_nat] :
( Xs2
!= ( append_list_nat @ Ys @ ( cons_list_nat @ Y2 @ nil_list_nat ) ) ) ) ).
% rev_exhaust
thf(fact_172_rev__exhaust,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ~ ! [Ys: list_nat,Y2: nat] :
( Xs2
!= ( append_nat @ Ys @ ( cons_nat @ Y2 @ nil_nat ) ) ) ) ).
% rev_exhaust
thf(fact_173_rev__induct,axiom,
! [P: list_list_nat > $o,Xs2: list_list_nat] :
( ( P @ nil_list_nat )
=> ( ! [X2: list_nat,Xs: list_list_nat] :
( ( P @ Xs )
=> ( P @ ( append_list_nat @ Xs @ ( cons_list_nat @ X2 @ nil_list_nat ) ) ) )
=> ( P @ Xs2 ) ) ) ).
% rev_induct
thf(fact_174_rev__induct,axiom,
! [P: list_nat > $o,Xs2: list_nat] :
( ( P @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat] :
( ( P @ Xs )
=> ( P @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) ) )
=> ( P @ Xs2 ) ) ) ).
% rev_induct
thf(fact_175_split__list__first__prop__iff,axiom,
! [Xs2: list_list_nat,P: list_nat > $o] :
( ( ? [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs2 ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_list_nat,X3: list_nat] :
( ? [Zs3: list_list_nat] :
( Xs2
= ( append_list_nat @ Ys3 @ ( cons_list_nat @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y3: list_nat] :
( ( member_list_nat @ Y3 @ ( set_list_nat2 @ Ys3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_176_split__list__first__prop__iff,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_nat,X3: nat] :
( ? [Zs3: list_nat] :
( Xs2
= ( 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_177_split__list__last__prop__iff,axiom,
! [Xs2: list_list_nat,P: list_nat > $o] :
( ( ? [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs2 ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_list_nat,X3: list_nat,Zs3: list_list_nat] :
( ( Xs2
= ( append_list_nat @ Ys3 @ ( cons_list_nat @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Y3: list_nat] :
( ( member_list_nat @ Y3 @ ( set_list_nat2 @ Zs3 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_178_split__list__last__prop__iff,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_nat,X3: nat,Zs3: list_nat] :
( ( Xs2
= ( 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_179_in__set__conv__decomp__first,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
= ( ? [Ys3: list_list_nat,Zs3: list_list_nat] :
( ( Xs2
= ( append_list_nat @ Ys3 @ ( cons_list_nat @ X @ Zs3 ) ) )
& ~ ( member_list_nat @ X @ ( set_list_nat2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_180_in__set__conv__decomp__first,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
= ( ? [Ys3: list_nat,Zs3: list_nat] :
( ( Xs2
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs3 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_181_in__set__conv__decomp__last,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
= ( ? [Ys3: list_list_nat,Zs3: list_list_nat] :
( ( Xs2
= ( append_list_nat @ Ys3 @ ( cons_list_nat @ X @ Zs3 ) ) )
& ~ ( member_list_nat @ X @ ( set_list_nat2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_182_in__set__conv__decomp__last,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
= ( ? [Ys3: list_nat,Zs3: list_nat] :
( ( Xs2
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs3 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_183_split__list__first__propE,axiom,
! [Xs2: list_list_nat,P: list_nat > $o] :
( ? [X4: list_nat] :
( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xs2 ) )
& ( P @ X4 ) )
=> ~ ! [Ys: list_list_nat,X2: list_nat] :
( ? [Zs2: list_list_nat] :
( Xs2
= ( append_list_nat @ Ys @ ( cons_list_nat @ X2 @ Zs2 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ ( set_list_nat2 @ Ys ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_184_split__list__first__propE,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
& ( P @ X4 ) )
=> ~ ! [Ys: list_nat,X2: nat] :
( ? [Zs2: list_nat] :
( Xs2
= ( append_nat @ Ys @ ( cons_nat @ X2 @ Zs2 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat2 @ Ys ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_185_split__list__last__propE,axiom,
! [Xs2: list_list_nat,P: list_nat > $o] :
( ? [X4: list_nat] :
( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xs2 ) )
& ( P @ X4 ) )
=> ~ ! [Ys: list_list_nat,X2: list_nat,Zs2: list_list_nat] :
( ( Xs2
= ( append_list_nat @ Ys @ ( cons_list_nat @ X2 @ Zs2 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ ( set_list_nat2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_186_split__list__last__propE,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
& ( P @ X4 ) )
=> ~ ! [Ys: list_nat,X2: nat,Zs2: list_nat] :
( ( Xs2
= ( append_nat @ Ys @ ( cons_nat @ X2 @ Zs2 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_187_split__list__first__prop,axiom,
! [Xs2: list_list_nat,P: list_nat > $o] :
( ? [X4: list_nat] :
( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xs2 ) )
& ( P @ X4 ) )
=> ? [Ys: list_list_nat,X2: list_nat] :
( ? [Zs2: list_list_nat] :
( Xs2
= ( append_list_nat @ Ys @ ( cons_list_nat @ X2 @ Zs2 ) ) )
& ( P @ X2 )
& ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ ( set_list_nat2 @ Ys ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_188_split__list__first__prop,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
& ( P @ X4 ) )
=> ? [Ys: list_nat,X2: nat] :
( ? [Zs2: list_nat] :
( Xs2
= ( append_nat @ Ys @ ( cons_nat @ X2 @ Zs2 ) ) )
& ( P @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat2 @ Ys ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_189_split__list__last__prop,axiom,
! [Xs2: list_list_nat,P: list_nat > $o] :
( ? [X4: list_nat] :
( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xs2 ) )
& ( P @ X4 ) )
=> ? [Ys: list_list_nat,X2: list_nat,Zs2: list_list_nat] :
( ( Xs2
= ( append_list_nat @ Ys @ ( cons_list_nat @ X2 @ Zs2 ) ) )
& ( P @ X2 )
& ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ ( set_list_nat2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_190_split__list__last__prop,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
& ( P @ X4 ) )
=> ? [Ys: list_nat,X2: nat,Zs2: list_nat] :
( ( Xs2
= ( append_nat @ Ys @ ( cons_nat @ X2 @ Zs2 ) ) )
& ( P @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat2 @ Zs2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_191_in__set__conv__decomp,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
= ( ? [Ys3: list_list_nat,Zs3: list_list_nat] :
( Xs2
= ( append_list_nat @ Ys3 @ ( cons_list_nat @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_192_in__set__conv__decomp,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
= ( ? [Ys3: list_nat,Zs3: list_nat] :
( Xs2
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_193_append__Cons__eq__iff,axiom,
! [X: list_nat,Xs2: list_list_nat,Ys2: list_list_nat,Xs3: list_list_nat,Ys6: list_list_nat] :
( ~ ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
=> ( ~ ( member_list_nat @ X @ ( set_list_nat2 @ Ys2 ) )
=> ( ( ( append_list_nat @ Xs2 @ ( cons_list_nat @ X @ Ys2 ) )
= ( append_list_nat @ Xs3 @ ( cons_list_nat @ X @ Ys6 ) ) )
= ( ( Xs2 = Xs3 )
& ( Ys2 = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_194_append__Cons__eq__iff,axiom,
! [X: nat,Xs2: list_nat,Ys2: list_nat,Xs3: list_nat,Ys6: list_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( ~ ( member_nat @ X @ ( set_nat2 @ Ys2 ) )
=> ( ( ( append_nat @ Xs2 @ ( cons_nat @ X @ Ys2 ) )
= ( append_nat @ Xs3 @ ( cons_nat @ X @ Ys6 ) ) )
= ( ( Xs2 = Xs3 )
& ( Ys2 = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_195_split__list__propE,axiom,
! [Xs2: list_list_nat,P: list_nat > $o] :
( ? [X4: list_nat] :
( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xs2 ) )
& ( P @ X4 ) )
=> ~ ! [Ys: list_list_nat,X2: list_nat] :
( ? [Zs2: list_list_nat] :
( Xs2
= ( append_list_nat @ Ys @ ( cons_list_nat @ X2 @ Zs2 ) ) )
=> ~ ( P @ X2 ) ) ) ).
% split_list_propE
thf(fact_196_split__list__propE,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
& ( P @ X4 ) )
=> ~ ! [Ys: list_nat,X2: nat] :
( ? [Zs2: list_nat] :
( Xs2
= ( append_nat @ Ys @ ( cons_nat @ X2 @ Zs2 ) ) )
=> ~ ( P @ X2 ) ) ) ).
% split_list_propE
thf(fact_197_split__list__first,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
=> ? [Ys: list_list_nat,Zs2: list_list_nat] :
( ( Xs2
= ( append_list_nat @ Ys @ ( cons_list_nat @ X @ Zs2 ) ) )
& ~ ( member_list_nat @ X @ ( set_list_nat2 @ Ys ) ) ) ) ).
% split_list_first
thf(fact_198_split__list__first,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ? [Ys: list_nat,Zs2: list_nat] :
( ( Xs2
= ( append_nat @ Ys @ ( cons_nat @ X @ Zs2 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Ys ) ) ) ) ).
% split_list_first
thf(fact_199_split__list__prop,axiom,
! [Xs2: list_list_nat,P: list_nat > $o] :
( ? [X4: list_nat] :
( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xs2 ) )
& ( P @ X4 ) )
=> ? [Ys: list_list_nat,X2: list_nat] :
( ? [Zs2: list_list_nat] :
( Xs2
= ( append_list_nat @ Ys @ ( cons_list_nat @ X2 @ Zs2 ) ) )
& ( P @ X2 ) ) ) ).
% split_list_prop
thf(fact_200_split__list__prop,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
& ( P @ X4 ) )
=> ? [Ys: list_nat,X2: nat] :
( ? [Zs2: list_nat] :
( Xs2
= ( append_nat @ Ys @ ( cons_nat @ X2 @ Zs2 ) ) )
& ( P @ X2 ) ) ) ).
% split_list_prop
thf(fact_201_split__list__last,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
=> ? [Ys: list_list_nat,Zs2: list_list_nat] :
( ( Xs2
= ( append_list_nat @ Ys @ ( cons_list_nat @ X @ Zs2 ) ) )
& ~ ( member_list_nat @ X @ ( set_list_nat2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_202_split__list__last,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ? [Ys: list_nat,Zs2: list_nat] :
( ( Xs2
= ( append_nat @ Ys @ ( cons_nat @ X @ Zs2 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_203_split__list,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
=> ? [Ys: list_list_nat,Zs2: list_list_nat] :
( Xs2
= ( append_list_nat @ Ys @ ( cons_list_nat @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_204_split__list,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ? [Ys: list_nat,Zs2: list_nat] :
( Xs2
= ( append_nat @ Ys @ ( cons_nat @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_205_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_list_nat,Z3: list_list_nat] : ( Y4 = Z3 ) )
= ( ^ [Xs4: list_list_nat,Ys3: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs4 )
= ( size_s3023201423986296836st_nat @ Ys3 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs4 ) )
=> ( ( nth_list_nat @ Xs4 @ I2 )
= ( nth_list_nat @ Ys3 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_206_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_nat,Z3: list_nat] : ( Y4 = Z3 ) )
= ( ^ [Xs4: list_nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs4 )
= ( size_size_list_nat @ Ys3 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs4 ) )
=> ( ( nth_nat @ Xs4 @ I2 )
= ( nth_nat @ Ys3 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_207_Skolem__list__nth,axiom,
! [K: nat,P: nat > list_nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X5: list_nat] : ( P @ I2 @ X5 ) ) )
= ( ? [Xs4: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs4 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_list_nat @ Xs4 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_208_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X5: nat] : ( P @ I2 @ X5 ) ) )
= ( ? [Xs4: list_nat] :
( ( ( size_size_list_nat @ Xs4 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_nat @ Xs4 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_209_nth__equalityI,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ( nth_list_nat @ Xs2 @ I3 )
= ( nth_list_nat @ Ys2 @ I3 ) ) )
=> ( Xs2 = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_210_nth__equalityI,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ Xs2 @ I3 )
= ( nth_nat @ Ys2 @ I3 ) ) )
=> ( Xs2 = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_211_same__length__different,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat] :
( ( Xs2 != Ys2 )
=> ( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ? [Pre: list_list_nat,X2: list_nat,Xs5: list_list_nat,Y2: list_nat,Ys7: list_list_nat] :
( ( X2 != Y2 )
& ( Xs2
= ( append_list_nat @ Pre @ ( append_list_nat @ ( cons_list_nat @ X2 @ nil_list_nat ) @ Xs5 ) ) )
& ( Ys2
= ( append_list_nat @ Pre @ ( append_list_nat @ ( cons_list_nat @ Y2 @ nil_list_nat ) @ Ys7 ) ) ) ) ) ) ).
% same_length_different
thf(fact_212_same__length__different,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( Xs2 != Ys2 )
=> ( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
=> ? [Pre: list_nat,X2: nat,Xs5: list_nat,Y2: nat,Ys7: list_nat] :
( ( X2 != Y2 )
& ( Xs2
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X2 @ nil_nat ) @ Xs5 ) ) )
& ( Ys2
= ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y2 @ nil_nat ) @ Ys7 ) ) ) ) ) ) ).
% same_length_different
thf(fact_213_all__set__conv__all__nth,axiom,
! [Xs2: list_list_nat,P: list_nat > $o] :
( ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs2 ) )
=> ( P @ X3 ) ) )
= ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( P @ ( nth_list_nat @ Xs2 @ I2 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_214_all__set__conv__all__nth,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
=> ( P @ X3 ) ) )
= ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ ( nth_nat @ Xs2 @ I2 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_215_all__nth__imp__all__set,axiom,
! [Xs2: list_list_nat,P: list_nat > $o,X: list_nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( P @ ( nth_list_nat @ Xs2 @ I3 ) ) )
=> ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_216_all__nth__imp__all__set,axiom,
! [Xs2: list_nat,P: nat > $o,X: nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ ( nth_nat @ Xs2 @ I3 ) ) )
=> ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_217_in__set__conv__nth,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
= ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
& ( ( nth_list_nat @ Xs2 @ I2 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_218_in__set__conv__nth,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
= ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
& ( ( nth_nat @ Xs2 @ I2 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_219_list__ball__nth,axiom,
! [N: nat,Xs2: list_list_nat,P: list_nat > $o] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs2 ) )
=> ( P @ X2 ) )
=> ( P @ ( nth_list_nat @ Xs2 @ N ) ) ) ) ).
% list_ball_nth
thf(fact_220_list__ball__nth,axiom,
! [N: nat,Xs2: list_nat,P: nat > $o] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( P @ X2 ) )
=> ( P @ ( nth_nat @ Xs2 @ N ) ) ) ) ).
% list_ball_nth
thf(fact_221_nth__mem,axiom,
! [N: nat,Xs2: list_list_nat] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( member_list_nat @ ( nth_list_nat @ Xs2 @ N ) @ ( set_list_nat2 @ Xs2 ) ) ) ).
% nth_mem
thf(fact_222_nth__mem,axiom,
! [N: nat,Xs2: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( member_nat @ ( nth_nat @ Xs2 @ N ) @ ( set_nat2 @ Xs2 ) ) ) ).
% nth_mem
thf(fact_223_lessThan__iff,axiom,
! [I: set_nat,K: set_nat] :
( ( member_set_nat @ I @ ( set_or890127255671739683et_nat @ K ) )
= ( ord_less_set_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_224_lessThan__iff,axiom,
! [I: num,K: num] :
( ( member_num @ I @ ( set_ord_lessThan_num @ K ) )
= ( ord_less_num @ I @ K ) ) ).
% lessThan_iff
thf(fact_225_lessThan__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
= ( ord_less_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_226_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_227_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_228_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_229_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_230_lessThan__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_lessThan_nat @ X )
= ( set_ord_lessThan_nat @ Y ) )
= ( X = Y ) ) ).
% lessThan_eq_iff
thf(fact_231_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_232_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_233_the__elem__set,axiom,
! [X: list_nat] :
( ( the_elem_list_nat @ ( set_list_nat2 @ ( cons_list_nat @ X @ nil_list_nat ) ) )
= X ) ).
% the_elem_set
thf(fact_234_the__elem__set,axiom,
! [X: nat] :
( ( the_elem_nat @ ( set_nat2 @ ( cons_nat @ X @ nil_nat ) ) )
= X ) ).
% the_elem_set
thf(fact_235_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_236_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( I = J2 )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_237_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_238_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A: nat] :
( ( B2
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_239_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_240_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_241_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_242_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_243_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_244_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_245_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_246_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( I = J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_247_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J2 )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_248_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_249_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_250_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_251_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_252_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_253_lessThan__strict__subset__iff,axiom,
! [M: num,N: num] :
( ( ord_less_set_num @ ( set_ord_lessThan_num @ M ) @ ( set_ord_lessThan_num @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_254_lessThan__strict__subset__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_255_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_256_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_257_product__lists_Osimps_I1_J,axiom,
( ( produc6783906451316923569st_nat @ nil_list_list_nat )
= ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) ).
% product_lists.simps(1)
thf(fact_258_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_259_bind__simps_I2_J,axiom,
! [X: nat,Xs2: list_nat,F: nat > list_list_nat] :
( ( bind_nat_list_nat @ ( cons_nat @ X @ Xs2 ) @ F )
= ( append_list_nat @ ( F @ X ) @ ( bind_nat_list_nat @ Xs2 @ F ) ) ) ).
% bind_simps(2)
thf(fact_260_bind__simps_I2_J,axiom,
! [X: list_nat,Xs2: list_list_nat,F: list_nat > list_nat] :
( ( bind_list_nat_nat @ ( cons_list_nat @ X @ Xs2 ) @ F )
= ( append_nat @ ( F @ X ) @ ( bind_list_nat_nat @ Xs2 @ F ) ) ) ).
% bind_simps(2)
thf(fact_261_bind__simps_I2_J,axiom,
! [X: list_nat,Xs2: list_list_nat,F: list_nat > list_list_nat] :
( ( bind_l7796378977173581257st_nat @ ( cons_list_nat @ X @ Xs2 ) @ F )
= ( append_list_nat @ ( F @ X ) @ ( bind_l7796378977173581257st_nat @ Xs2 @ F ) ) ) ).
% bind_simps(2)
thf(fact_262_bind__simps_I2_J,axiom,
! [X: nat,Xs2: list_nat,F: nat > list_nat] :
( ( bind_nat_nat @ ( cons_nat @ X @ Xs2 ) @ F )
= ( append_nat @ ( F @ X ) @ ( bind_nat_nat @ Xs2 @ F ) ) ) ).
% bind_simps(2)
thf(fact_263_gen__length__def,axiom,
( gen_length_list_nat
= ( ^ [N3: nat,Xs4: list_list_nat] : ( plus_plus_nat @ N3 @ ( size_s3023201423986296836st_nat @ Xs4 ) ) ) ) ).
% gen_length_def
thf(fact_264_gen__length__def,axiom,
( gen_length_nat
= ( ^ [N3: nat,Xs4: list_nat] : ( plus_plus_nat @ N3 @ ( size_size_list_nat @ Xs4 ) ) ) ) ).
% gen_length_def
thf(fact_265_prefixes__snoc,axiom,
! [Xs2: list_list_nat,X: list_nat] :
( ( prefixes_list_nat @ ( append_list_nat @ Xs2 @ ( cons_list_nat @ X @ nil_list_nat ) ) )
= ( append_list_list_nat @ ( prefixes_list_nat @ Xs2 ) @ ( cons_list_list_nat @ ( append_list_nat @ Xs2 @ ( cons_list_nat @ X @ nil_list_nat ) ) @ nil_list_list_nat ) ) ) ).
% prefixes_snoc
thf(fact_266_prefixes__snoc,axiom,
! [Xs2: list_nat,X: nat] :
( ( prefixes_nat @ ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) ) )
= ( append_list_nat @ ( prefixes_nat @ Xs2 ) @ ( cons_list_nat @ ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) ) @ nil_list_nat ) ) ) ).
% prefixes_snoc
thf(fact_267_set__swap,axiom,
! [I: nat,Xs2: list_list_nat,J2: nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ( ord_less_nat @ J2 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ( set_list_nat2 @ ( list_update_list_nat @ ( list_update_list_nat @ Xs2 @ I @ ( nth_list_nat @ Xs2 @ J2 ) ) @ J2 @ ( nth_list_nat @ Xs2 @ I ) ) )
= ( set_list_nat2 @ Xs2 ) ) ) ) ).
% set_swap
thf(fact_268_set__swap,axiom,
! [I: nat,Xs2: list_nat,J2: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs2 @ I @ ( nth_nat @ Xs2 @ J2 ) ) @ J2 @ ( nth_nat @ Xs2 @ I ) ) )
= ( set_nat2 @ Xs2 ) ) ) ) ).
% set_swap
thf(fact_269_maps__simps_I1_J,axiom,
! [F: nat > list_list_nat,X: nat,Xs2: list_nat] :
( ( maps_nat_list_nat @ F @ ( cons_nat @ X @ Xs2 ) )
= ( append_list_nat @ ( F @ X ) @ ( maps_nat_list_nat @ F @ Xs2 ) ) ) ).
% maps_simps(1)
thf(fact_270_maps__simps_I1_J,axiom,
! [F: list_nat > list_nat,X: list_nat,Xs2: list_list_nat] :
( ( maps_list_nat_nat @ F @ ( cons_list_nat @ X @ Xs2 ) )
= ( append_nat @ ( F @ X ) @ ( maps_list_nat_nat @ F @ Xs2 ) ) ) ).
% maps_simps(1)
thf(fact_271_maps__simps_I1_J,axiom,
! [F: list_nat > list_list_nat,X: list_nat,Xs2: list_list_nat] :
( ( maps_l5785965478274863235st_nat @ F @ ( cons_list_nat @ X @ Xs2 ) )
= ( append_list_nat @ ( F @ X ) @ ( maps_l5785965478274863235st_nat @ F @ Xs2 ) ) ) ).
% maps_simps(1)
thf(fact_272_maps__simps_I1_J,axiom,
! [F: nat > list_nat,X: nat,Xs2: list_nat] :
( ( maps_nat_nat @ F @ ( cons_nat @ X @ Xs2 ) )
= ( append_nat @ ( F @ X ) @ ( maps_nat_nat @ F @ Xs2 ) ) ) ).
% maps_simps(1)
thf(fact_273_list__update__overwrite,axiom,
! [Xs2: list_nat,I: nat,X: nat,Y: nat] :
( ( list_update_nat @ ( list_update_nat @ Xs2 @ I @ X ) @ I @ Y )
= ( list_update_nat @ Xs2 @ I @ Y ) ) ).
% list_update_overwrite
thf(fact_274_list__update__nonempty,axiom,
! [Xs2: list_list_nat,K: nat,X: list_nat] :
( ( ( list_update_list_nat @ Xs2 @ K @ X )
= nil_list_nat )
= ( Xs2 = nil_list_nat ) ) ).
% list_update_nonempty
thf(fact_275_list__update__nonempty,axiom,
! [Xs2: list_nat,K: nat,X: nat] :
( ( ( list_update_nat @ Xs2 @ K @ X )
= nil_nat )
= ( Xs2 = nil_nat ) ) ).
% list_update_nonempty
thf(fact_276_length__list__update,axiom,
! [Xs2: list_list_nat,I: nat,X: list_nat] :
( ( size_s3023201423986296836st_nat @ ( list_update_list_nat @ Xs2 @ I @ X ) )
= ( size_s3023201423986296836st_nat @ Xs2 ) ) ).
% length_list_update
thf(fact_277_length__list__update,axiom,
! [Xs2: list_nat,I: nat,X: nat] :
( ( size_size_list_nat @ ( list_update_nat @ Xs2 @ I @ X ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_list_update
thf(fact_278_nth__list__update__neq,axiom,
! [I: nat,J2: nat,Xs2: list_nat,X: nat] :
( ( I != J2 )
=> ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X ) @ J2 )
= ( nth_nat @ Xs2 @ J2 ) ) ) ).
% nth_list_update_neq
thf(fact_279_list__update__id,axiom,
! [Xs2: list_nat,I: nat] :
( ( list_update_nat @ Xs2 @ I @ ( nth_nat @ Xs2 @ I ) )
= Xs2 ) ).
% list_update_id
thf(fact_280_bind__simps_I1_J,axiom,
! [F: nat > list_list_nat] :
( ( bind_nat_list_nat @ nil_nat @ F )
= nil_list_nat ) ).
% bind_simps(1)
thf(fact_281_bind__simps_I1_J,axiom,
! [F: list_nat > list_nat] :
( ( bind_list_nat_nat @ nil_list_nat @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_282_bind__simps_I1_J,axiom,
! [F: list_nat > list_list_nat] :
( ( bind_l7796378977173581257st_nat @ nil_list_nat @ F )
= nil_list_nat ) ).
% bind_simps(1)
thf(fact_283_bind__simps_I1_J,axiom,
! [F: nat > list_nat] :
( ( bind_nat_nat @ nil_nat @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_284_list__update__length,axiom,
! [Xs2: list_list_nat,X: list_nat,Ys2: list_list_nat,Y: list_nat] :
( ( list_update_list_nat @ ( append_list_nat @ Xs2 @ ( cons_list_nat @ X @ Ys2 ) ) @ ( size_s3023201423986296836st_nat @ Xs2 ) @ Y )
= ( append_list_nat @ Xs2 @ ( cons_list_nat @ Y @ Ys2 ) ) ) ).
% list_update_length
thf(fact_285_list__update__length,axiom,
! [Xs2: list_nat,X: nat,Ys2: list_nat,Y: nat] :
( ( list_update_nat @ ( append_nat @ Xs2 @ ( cons_nat @ X @ Ys2 ) ) @ ( size_size_list_nat @ Xs2 ) @ Y )
= ( append_nat @ Xs2 @ ( cons_nat @ Y @ Ys2 ) ) ) ).
% list_update_length
thf(fact_286_nth__list__update__eq,axiom,
! [I: nat,Xs2: list_list_nat,X: list_nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ( nth_list_nat @ ( list_update_list_nat @ Xs2 @ I @ X ) @ I )
= X ) ) ).
% nth_list_update_eq
thf(fact_287_nth__list__update__eq,axiom,
! [I: nat,Xs2: list_nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X ) @ I )
= X ) ) ).
% nth_list_update_eq
thf(fact_288_length__prefixes,axiom,
! [Xs2: list_list_nat] :
( ( size_s6248950052170075156st_nat @ ( prefixes_list_nat @ Xs2 ) )
= ( plus_plus_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ one_one_nat ) ) ).
% length_prefixes
thf(fact_289_length__prefixes,axiom,
! [Xs2: list_nat] :
( ( size_s3023201423986296836st_nat @ ( prefixes_nat @ Xs2 ) )
= ( plus_plus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) ) ).
% length_prefixes
thf(fact_290_list__update__swap,axiom,
! [I: nat,I4: nat,Xs2: list_nat,X: nat,X6: nat] :
( ( I != I4 )
=> ( ( list_update_nat @ ( list_update_nat @ Xs2 @ I @ X ) @ I4 @ X6 )
= ( list_update_nat @ ( list_update_nat @ Xs2 @ I4 @ X6 ) @ I @ X ) ) ) ).
% list_update_swap
thf(fact_291_prefixes__not__Nil,axiom,
! [Xs2: list_nat] :
( ( prefixes_nat @ Xs2 )
!= nil_list_nat ) ).
% prefixes_not_Nil
thf(fact_292_max__add__distrib__right,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( plus_plus_nat @ X @ ( ord_max_nat @ Y @ Z4 ) )
= ( ord_max_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z4 ) ) ) ).
% max_add_distrib_right
thf(fact_293_max__add__distrib__left,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( plus_plus_nat @ ( ord_max_nat @ X @ Y ) @ Z4 )
= ( ord_max_nat @ ( plus_plus_nat @ X @ Z4 ) @ ( plus_plus_nat @ Y @ Z4 ) ) ) ).
% max_add_distrib_left
thf(fact_294_nat__add__max__right,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q2 ) )
= ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q2 ) ) ) ).
% nat_add_max_right
thf(fact_295_nat__add__max__left,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q2 )
= ( ord_max_nat @ ( plus_plus_nat @ M @ Q2 ) @ ( plus_plus_nat @ N @ Q2 ) ) ) ).
% nat_add_max_left
thf(fact_296_list__update_Osimps_I1_J,axiom,
! [I: nat,V: list_nat] :
( ( list_update_list_nat @ nil_list_nat @ I @ V )
= nil_list_nat ) ).
% list_update.simps(1)
thf(fact_297_list__update_Osimps_I1_J,axiom,
! [I: nat,V: nat] :
( ( list_update_nat @ nil_nat @ I @ V )
= nil_nat ) ).
% list_update.simps(1)
thf(fact_298_list__update__code_I1_J,axiom,
! [I: nat,Y: list_nat] :
( ( list_update_list_nat @ nil_list_nat @ I @ Y )
= nil_list_nat ) ).
% list_update_code(1)
thf(fact_299_list__update__code_I1_J,axiom,
! [I: nat,Y: nat] :
( ( list_update_nat @ nil_nat @ I @ Y )
= nil_nat ) ).
% list_update_code(1)
thf(fact_300_list__bind__cong,axiom,
! [Xs2: list_nat,Ys2: list_nat,F: nat > list_nat,G: nat > list_nat] :
( ( Xs2 = Ys2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( bind_nat_nat @ Xs2 @ F )
= ( bind_nat_nat @ Ys2 @ G ) ) ) ) ).
% list_bind_cong
thf(fact_301_in__set__product__lists__length,axiom,
! [Xs2: list_list_nat,Xss2: list_list_list_nat] :
( ( member_list_list_nat @ Xs2 @ ( set_list_list_nat2 @ ( produc6783906451316923569st_nat @ Xss2 ) ) )
=> ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_s6248950052170075156st_nat @ Xss2 ) ) ) ).
% in_set_product_lists_length
thf(fact_302_in__set__product__lists__length,axiom,
! [Xs2: list_nat,Xss2: list_list_nat] :
( ( member_list_nat @ Xs2 @ ( set_list_nat2 @ ( product_lists_nat @ Xss2 ) ) )
=> ( ( size_size_list_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Xss2 ) ) ) ).
% in_set_product_lists_length
thf(fact_303_prefixes_Osimps_I1_J,axiom,
( ( prefixes_list_nat @ nil_list_nat )
= ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) ).
% prefixes.simps(1)
thf(fact_304_prefixes_Osimps_I1_J,axiom,
( ( prefixes_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% prefixes.simps(1)
thf(fact_305_maps__simps_I2_J,axiom,
! [F: nat > list_list_nat] :
( ( maps_nat_list_nat @ F @ nil_nat )
= nil_list_nat ) ).
% maps_simps(2)
thf(fact_306_maps__simps_I2_J,axiom,
! [F: list_nat > list_nat] :
( ( maps_list_nat_nat @ F @ nil_list_nat )
= nil_nat ) ).
% maps_simps(2)
thf(fact_307_maps__simps_I2_J,axiom,
! [F: list_nat > list_list_nat] :
( ( maps_l5785965478274863235st_nat @ F @ nil_list_nat )
= nil_list_nat ) ).
% maps_simps(2)
thf(fact_308_maps__simps_I2_J,axiom,
! [F: nat > list_nat] :
( ( maps_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% maps_simps(2)
thf(fact_309_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_list_nat @ N @ nil_list_nat )
= N ) ).
% gen_length_code(1)
thf(fact_310_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_nat @ N @ nil_nat )
= N ) ).
% gen_length_code(1)
thf(fact_311_set__update__memI,axiom,
! [N: nat,Xs2: list_list_nat,X: list_nat] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( member_list_nat @ X @ ( set_list_nat2 @ ( list_update_list_nat @ Xs2 @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_312_set__update__memI,axiom,
! [N: nat,Xs2: list_nat,X: nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs2 @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_313_list__update__append1,axiom,
! [I: nat,Xs2: list_list_nat,Ys2: list_list_nat,X: list_nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ( list_update_list_nat @ ( append_list_nat @ Xs2 @ Ys2 ) @ I @ X )
= ( append_list_nat @ ( list_update_list_nat @ Xs2 @ I @ X ) @ Ys2 ) ) ) ).
% list_update_append1
thf(fact_314_list__update__append1,axiom,
! [I: nat,Xs2: list_nat,Ys2: list_nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( list_update_nat @ ( append_nat @ Xs2 @ Ys2 ) @ I @ X )
= ( append_nat @ ( list_update_nat @ Xs2 @ I @ X ) @ Ys2 ) ) ) ).
% list_update_append1
thf(fact_315_list__update__same__conv,axiom,
! [I: nat,Xs2: list_list_nat,X: list_nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ( ( list_update_list_nat @ Xs2 @ I @ X )
= Xs2 )
= ( ( nth_list_nat @ Xs2 @ I )
= X ) ) ) ).
% list_update_same_conv
thf(fact_316_list__update__same__conv,axiom,
! [I: nat,Xs2: list_nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ( list_update_nat @ Xs2 @ I @ X )
= Xs2 )
= ( ( nth_nat @ Xs2 @ I )
= X ) ) ) ).
% list_update_same_conv
thf(fact_317_nth__list__update,axiom,
! [I: nat,Xs2: list_list_nat,J2: nat,X: list_nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ( ( I = J2 )
=> ( ( nth_list_nat @ ( list_update_list_nat @ Xs2 @ I @ X ) @ J2 )
= X ) )
& ( ( I != J2 )
=> ( ( nth_list_nat @ ( list_update_list_nat @ Xs2 @ I @ X ) @ J2 )
= ( nth_list_nat @ Xs2 @ J2 ) ) ) ) ) ).
% nth_list_update
thf(fact_318_nth__list__update,axiom,
! [I: nat,Xs2: list_nat,J2: nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ( I = J2 )
=> ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X ) @ J2 )
= X ) )
& ( ( I != J2 )
=> ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X ) @ J2 )
= ( nth_nat @ Xs2 @ J2 ) ) ) ) ) ).
% nth_list_update
thf(fact_319_rgf__limit_Osimps_I2_J,axiom,
! [X: nat,Xs2: list_nat] :
( ( equiva5889994315859557365_limit @ ( cons_nat @ X @ Xs2 ) )
= ( ord_max_nat @ ( plus_plus_nat @ X @ one_one_nat ) @ ( equiva5889994315859557365_limit @ Xs2 ) ) ) ).
% rgf_limit.simps(2)
thf(fact_320_prefixes__eq__snoc,axiom,
! [Ys2: list_list_nat,Xs2: list_list_list_nat,X: list_list_nat] :
( ( ( prefixes_list_nat @ Ys2 )
= ( append_list_list_nat @ Xs2 @ ( cons_list_list_nat @ X @ nil_list_list_nat ) ) )
= ( ( ( ( Ys2 = nil_list_nat )
& ( Xs2 = nil_list_list_nat ) )
| ? [Z5: list_nat,Zs3: list_list_nat] :
( ( Ys2
= ( append_list_nat @ Zs3 @ ( cons_list_nat @ Z5 @ nil_list_nat ) ) )
& ( Xs2
= ( prefixes_list_nat @ Zs3 ) ) ) )
& ( X = Ys2 ) ) ) ).
% prefixes_eq_snoc
thf(fact_321_prefixes__eq__snoc,axiom,
! [Ys2: list_nat,Xs2: list_list_nat,X: list_nat] :
( ( ( prefixes_nat @ Ys2 )
= ( append_list_nat @ Xs2 @ ( cons_list_nat @ X @ nil_list_nat ) ) )
= ( ( ( ( Ys2 = nil_nat )
& ( Xs2 = nil_list_nat ) )
| ? [Z5: nat,Zs3: list_nat] :
( ( Ys2
= ( append_nat @ Zs3 @ ( cons_nat @ Z5 @ nil_nat ) ) )
& ( Xs2
= ( prefixes_nat @ Zs3 ) ) ) )
& ( X = Ys2 ) ) ) ).
% prefixes_eq_snoc
thf(fact_322_max_Oabsorb3,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_max_num @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_323_max_Oabsorb3,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_max_nat @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_324_max_Oabsorb4,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_max_num @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_325_max_Oabsorb4,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_max_nat @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_326_max__less__iff__conj,axiom,
! [X: num,Y: num,Z4: num] :
( ( ord_less_num @ ( ord_max_num @ X @ Y ) @ Z4 )
= ( ( ord_less_num @ X @ Z4 )
& ( ord_less_num @ Y @ Z4 ) ) ) ).
% max_less_iff_conj
thf(fact_327_max__less__iff__conj,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_nat @ ( ord_max_nat @ X @ Y ) @ Z4 )
= ( ( ord_less_nat @ X @ Z4 )
& ( ord_less_nat @ Y @ Z4 ) ) ) ).
% max_less_iff_conj
thf(fact_328_max_Oright__idem,axiom,
! [A: nat,B: nat] :
( ( ord_max_nat @ ( ord_max_nat @ A @ B ) @ B )
= ( ord_max_nat @ A @ B ) ) ).
% max.right_idem
thf(fact_329_max_Oleft__idem,axiom,
! [A: nat,B: nat] :
( ( ord_max_nat @ A @ ( ord_max_nat @ A @ B ) )
= ( ord_max_nat @ A @ B ) ) ).
% max.left_idem
thf(fact_330_max_Oidem,axiom,
! [A: nat] :
( ( ord_max_nat @ A @ A )
= A ) ).
% max.idem
thf(fact_331_rgf__limit_Oelims,axiom,
! [X: list_nat,Y: nat] :
( ( ( equiva5889994315859557365_limit @ X )
= Y )
=> ( ( ( X = nil_nat )
=> ( Y != zero_zero_nat ) )
=> ~ ! [X2: nat,Xs: list_nat] :
( ( X
= ( cons_nat @ X2 @ Xs ) )
=> ( Y
!= ( ord_max_nat @ ( plus_plus_nat @ X2 @ one_one_nat ) @ ( equiva5889994315859557365_limit @ Xs ) ) ) ) ) ) ).
% rgf_limit.elims
thf(fact_332_less__max__iff__disj,axiom,
! [Z4: num,X: num,Y: num] :
( ( ord_less_num @ Z4 @ ( ord_max_num @ X @ Y ) )
= ( ( ord_less_num @ Z4 @ X )
| ( ord_less_num @ Z4 @ Y ) ) ) ).
% less_max_iff_disj
thf(fact_333_less__max__iff__disj,axiom,
! [Z4: nat,X: nat,Y: nat] :
( ( ord_less_nat @ Z4 @ ( ord_max_nat @ X @ Y ) )
= ( ( ord_less_nat @ Z4 @ X )
| ( ord_less_nat @ Z4 @ Y ) ) ) ).
% less_max_iff_disj
thf(fact_334_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_335_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_336_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_337_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_338_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_339_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_340_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_341_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_342_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_343_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_344_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_345_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_346_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_347_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_348_max__0R,axiom,
! [N: nat] :
( ( ord_max_nat @ N @ zero_zero_nat )
= N ) ).
% max_0R
thf(fact_349_max__0L,axiom,
! [N: nat] :
( ( ord_max_nat @ zero_zero_nat @ N )
= N ) ).
% max_0L
thf(fact_350_max__nat_Oright__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ A @ zero_zero_nat )
= A ) ).
% max_nat.right_neutral
thf(fact_351_max__nat_Oneutr__eq__iff,axiom,
! [A: nat,B: nat] :
( ( zero_zero_nat
= ( ord_max_nat @ A @ B ) )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.neutr_eq_iff
thf(fact_352_max__nat_Oleft__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ zero_zero_nat @ A )
= A ) ).
% max_nat.left_neutral
thf(fact_353_max__nat_Oeq__neutr__iff,axiom,
! [A: nat,B: nat] :
( ( ( ord_max_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.eq_neutr_iff
thf(fact_354_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_355_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_356_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_357_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_358_length__0__conv,axiom,
! [Xs2: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= zero_zero_nat )
= ( Xs2 = nil_list_nat ) ) ).
% length_0_conv
thf(fact_359_length__0__conv,axiom,
! [Xs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= zero_zero_nat )
= ( Xs2 = nil_nat ) ) ).
% length_0_conv
thf(fact_360_max__0__1_I2_J,axiom,
( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
= one_one_nat ) ).
% max_0_1(2)
thf(fact_361_max__0__1_I1_J,axiom,
( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
= one_one_nat ) ).
% max_0_1(1)
thf(fact_362_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_363_nth__Cons__0,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( nth_list_nat @ ( cons_list_nat @ X @ Xs2 ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_364_nth__Cons__0,axiom,
! [X: nat,Xs2: list_nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_365_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_366_length__greater__0__conv,axiom,
! [Xs2: list_list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) )
= ( Xs2 != nil_list_nat ) ) ).
% length_greater_0_conv
thf(fact_367_length__greater__0__conv,axiom,
! [Xs2: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) )
= ( Xs2 != nil_nat ) ) ).
% length_greater_0_conv
thf(fact_368_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_369_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_370_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_371_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_372_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_373_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_374_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_375_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_376_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_377_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_378_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_379_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_380_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_381_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_382_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_383_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_384_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_385_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_386_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_387_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_388_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_389_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_390_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_391_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_392_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_393_list_Osize_I3_J,axiom,
( ( size_s3023201423986296836st_nat @ nil_list_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_394_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_395_less__imp__add__positive,axiom,
! [I: nat,J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J2 ) ) ) ).
% less_imp_add_positive
thf(fact_396_list__update__code_I2_J,axiom,
! [X: list_nat,Xs2: list_list_nat,Y: list_nat] :
( ( list_update_list_nat @ ( cons_list_nat @ X @ Xs2 ) @ zero_zero_nat @ Y )
= ( cons_list_nat @ Y @ Xs2 ) ) ).
% list_update_code(2)
thf(fact_397_list__update__code_I2_J,axiom,
! [X: nat,Xs2: list_nat,Y: nat] :
( ( list_update_nat @ ( cons_nat @ X @ Xs2 ) @ zero_zero_nat @ Y )
= ( cons_nat @ Y @ Xs2 ) ) ).
% list_update_code(2)
thf(fact_398_rgf__limit_Osimps_I1_J,axiom,
( ( equiva5889994315859557365_limit @ nil_nat )
= zero_zero_nat ) ).
% rgf_limit.simps(1)
thf(fact_399_length__code,axiom,
( size_s3023201423986296836st_nat
= ( gen_length_list_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_400_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_401_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_402_length__pos__if__in__set,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_403_length__pos__if__in__set,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_404_max_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_max_nat @ ( ord_max_nat @ A @ B ) @ C )
= ( ord_max_nat @ A @ ( ord_max_nat @ B @ C ) ) ) ).
% max.assoc
thf(fact_405_max_Ocommute,axiom,
( ord_max_nat
= ( ^ [A3: nat,B3: nat] : ( ord_max_nat @ B3 @ A3 ) ) ) ).
% max.commute
thf(fact_406_max_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_max_nat @ B @ ( ord_max_nat @ A @ C ) )
= ( ord_max_nat @ A @ ( ord_max_nat @ B @ C ) ) ) ).
% max.left_commute
thf(fact_407_max_Ostrict__coboundedI2,axiom,
! [C: num,B: num,A: num] :
( ( ord_less_num @ C @ B )
=> ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% max.strict_coboundedI2
thf(fact_408_max_Ostrict__coboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.strict_coboundedI2
thf(fact_409_max_Ostrict__coboundedI1,axiom,
! [C: num,A: num,B: num] :
( ( ord_less_num @ C @ A )
=> ( ord_less_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% max.strict_coboundedI1
thf(fact_410_max_Ostrict__coboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ C @ A )
=> ( ord_less_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.strict_coboundedI1
thf(fact_411_max_Ostrict__order__iff,axiom,
( ord_less_num
= ( ^ [B3: num,A3: num] :
( ( A3
= ( ord_max_num @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% max.strict_order_iff
thf(fact_412_max_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( A3
= ( ord_max_nat @ A3 @ B3 ) )
& ( A3 != B3 ) ) ) ) ).
% max.strict_order_iff
thf(fact_413_max_Ostrict__boundedE,axiom,
! [B: num,C: num,A: num] :
( ( ord_less_num @ ( ord_max_num @ B @ C ) @ A )
=> ~ ( ( ord_less_num @ B @ A )
=> ~ ( ord_less_num @ C @ A ) ) ) ).
% max.strict_boundedE
thf(fact_414_max_Ostrict__boundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ ( ord_max_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ C @ A ) ) ) ).
% max.strict_boundedE
thf(fact_415_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_list_nat @ N @ nil_list_nat )
= ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_list_nat @ N @ nil_list_nat )
= nil_list_list_nat ) ) ) ).
% n_lists_Nil
thf(fact_416_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_417_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_418_n__lists_Osimps_I1_J,axiom,
! [Xs2: list_list_nat] :
( ( n_lists_list_nat @ zero_zero_nat @ Xs2 )
= ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) ).
% n_lists.simps(1)
thf(fact_419_n__lists_Osimps_I1_J,axiom,
! [Xs2: list_nat] :
( ( n_lists_nat @ zero_zero_nat @ Xs2 )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% n_lists.simps(1)
thf(fact_420_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
= ( P @ B4 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
=> ( P @ A4 @ ( plus_plus_nat @ A4 @ B4 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_421_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_422_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_423_suffixes__eq__snoc,axiom,
! [Ys2: list_list_nat,Xs2: list_list_list_nat,X: list_list_nat] :
( ( ( suffixes_list_nat @ Ys2 )
= ( append_list_list_nat @ Xs2 @ ( cons_list_list_nat @ X @ nil_list_list_nat ) ) )
= ( ( ( ( Ys2 = nil_list_nat )
& ( Xs2 = nil_list_list_nat ) )
| ? [Z5: list_nat,Zs3: list_list_nat] :
( ( Ys2
= ( cons_list_nat @ Z5 @ Zs3 ) )
& ( Xs2
= ( suffixes_list_nat @ Zs3 ) ) ) )
& ( X = Ys2 ) ) ) ).
% suffixes_eq_snoc
thf(fact_424_suffixes__eq__snoc,axiom,
! [Ys2: list_nat,Xs2: list_list_nat,X: list_nat] :
( ( ( suffixes_nat @ Ys2 )
= ( append_list_nat @ Xs2 @ ( cons_list_nat @ X @ nil_list_nat ) ) )
= ( ( ( ( Ys2 = nil_nat )
& ( Xs2 = nil_list_nat ) )
| ? [Z5: nat,Zs3: list_nat] :
( ( Ys2
= ( cons_nat @ Z5 @ Zs3 ) )
& ( Xs2
= ( suffixes_nat @ Zs3 ) ) ) )
& ( X = Ys2 ) ) ) ).
% suffixes_eq_snoc
thf(fact_425_suffixes__not__Nil,axiom,
! [Xs2: list_nat] :
( ( suffixes_nat @ Xs2 )
!= nil_list_nat ) ).
% suffixes_not_Nil
thf(fact_426_verit__comp__simplify1_I1_J,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_427_verit__comp__simplify1_I1_J,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_428_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_429_suffixes_Osimps_I1_J,axiom,
( ( suffixes_list_nat @ nil_list_nat )
= ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) ).
% suffixes.simps(1)
thf(fact_430_suffixes_Osimps_I1_J,axiom,
( ( suffixes_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% suffixes.simps(1)
thf(fact_431_length__n__lists__elem,axiom,
! [Ys2: list_list_nat,N: nat,Xs2: list_list_nat] :
( ( member_list_list_nat @ Ys2 @ ( set_list_list_nat2 @ ( n_lists_list_nat @ N @ Xs2 ) ) )
=> ( ( size_s3023201423986296836st_nat @ Ys2 )
= N ) ) ).
% length_n_lists_elem
thf(fact_432_length__n__lists__elem,axiom,
! [Ys2: list_nat,N: nat,Xs2: list_nat] :
( ( member_list_nat @ Ys2 @ ( set_list_nat2 @ ( n_lists_nat @ N @ Xs2 ) ) )
=> ( ( size_size_list_nat @ Ys2 )
= N ) ) ).
% length_n_lists_elem
thf(fact_433_suffixes_Osimps_I2_J,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( suffixes_list_nat @ ( cons_list_nat @ X @ Xs2 ) )
= ( append_list_list_nat @ ( suffixes_list_nat @ Xs2 ) @ ( cons_list_list_nat @ ( cons_list_nat @ X @ Xs2 ) @ nil_list_list_nat ) ) ) ).
% suffixes.simps(2)
thf(fact_434_suffixes_Osimps_I2_J,axiom,
! [X: nat,Xs2: list_nat] :
( ( suffixes_nat @ ( cons_nat @ X @ Xs2 ) )
= ( append_list_nat @ ( suffixes_nat @ Xs2 ) @ ( cons_list_nat @ ( cons_nat @ X @ Xs2 ) @ nil_list_nat ) ) ) ).
% suffixes.simps(2)
thf(fact_435_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_436_sublists_Osimps_I1_J,axiom,
( ( sublists_list_nat @ nil_list_nat )
= ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) ).
% sublists.simps(1)
thf(fact_437_sublists_Osimps_I1_J,axiom,
( ( sublists_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% sublists.simps(1)
thf(fact_438_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,Xs: list_nat] :
( ( X
= ( cons_nat @ X2 @ Xs ) )
=> ( ( Y
= ( ord_max_nat @ ( plus_plus_nat @ X2 @ one_one_nat ) @ ( equiva5889994315859557365_limit @ Xs ) ) )
=> ~ ( accp_list_nat @ equiva5575797544161152836it_rel @ ( cons_nat @ X2 @ Xs ) ) ) ) ) ) ) ).
% rgf_limit.pelims
thf(fact_439_nth__Cons__pos,axiom,
! [N: nat,X: list_nat,Xs2: list_list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_list_nat @ ( cons_list_nat @ X @ Xs2 ) @ N )
= ( nth_list_nat @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_440_nth__Cons__pos,axiom,
! [N: nat,X: nat,Xs2: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ N )
= ( nth_nat @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_441_append__one__prefix,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat] :
( ( prefix_list_nat @ Xs2 @ Ys2 )
=> ( ( ord_less_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( prefix_list_nat @ ( append_list_nat @ Xs2 @ ( cons_list_nat @ ( nth_list_nat @ Ys2 @ ( size_s3023201423986296836st_nat @ Xs2 ) ) @ nil_list_nat ) ) @ Ys2 ) ) ) ).
% append_one_prefix
thf(fact_442_append__one__prefix,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( prefix_nat @ Xs2 @ Ys2 )
=> ( ( ord_less_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ Ys2 ) )
=> ( prefix_nat @ ( append_nat @ Xs2 @ ( cons_nat @ ( nth_nat @ Ys2 @ ( size_size_list_nat @ Xs2 ) ) @ nil_nat ) ) @ Ys2 ) ) ) ).
% append_one_prefix
thf(fact_443_prefix__order_Odual__order_Orefl,axiom,
! [A: list_nat] : ( prefix_nat @ A @ A ) ).
% prefix_order.dual_order.refl
thf(fact_444_prefix__order_Oorder__refl,axiom,
! [X: list_nat] : ( prefix_nat @ X @ X ) ).
% prefix_order.order_refl
thf(fact_445_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_446_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_447_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_448_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_449_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_450_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_451_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_452_Cons__prefix__Cons,axiom,
! [X: list_nat,Xs2: list_list_nat,Y: list_nat,Ys2: list_list_nat] :
( ( prefix_list_nat @ ( cons_list_nat @ X @ Xs2 ) @ ( cons_list_nat @ Y @ Ys2 ) )
= ( ( X = Y )
& ( prefix_list_nat @ Xs2 @ Ys2 ) ) ) ).
% Cons_prefix_Cons
thf(fact_453_Cons__prefix__Cons,axiom,
! [X: nat,Xs2: list_nat,Y: nat,Ys2: list_nat] :
( ( prefix_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) )
= ( ( X = Y )
& ( prefix_nat @ Xs2 @ Ys2 ) ) ) ).
% Cons_prefix_Cons
thf(fact_454_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_455_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_456_prefix__code_I1_J,axiom,
! [Xs2: list_list_nat] : ( prefix_list_nat @ nil_list_nat @ Xs2 ) ).
% prefix_code(1)
thf(fact_457_prefix__code_I1_J,axiom,
! [Xs2: list_nat] : ( prefix_nat @ nil_nat @ Xs2 ) ).
% prefix_code(1)
thf(fact_458_prefix__bot_Oextremum__unique,axiom,
! [A: list_list_nat] :
( ( prefix_list_nat @ A @ nil_list_nat )
= ( A = nil_list_nat ) ) ).
% prefix_bot.extremum_unique
thf(fact_459_prefix__bot_Oextremum__unique,axiom,
! [A: list_nat] :
( ( prefix_nat @ A @ nil_nat )
= ( A = nil_nat ) ) ).
% prefix_bot.extremum_unique
thf(fact_460_prefix__Nil,axiom,
! [Xs2: list_list_nat] :
( ( prefix_list_nat @ Xs2 @ nil_list_nat )
= ( Xs2 = nil_list_nat ) ) ).
% prefix_Nil
thf(fact_461_prefix__Nil,axiom,
! [Xs2: list_nat] :
( ( prefix_nat @ Xs2 @ nil_nat )
= ( Xs2 = nil_nat ) ) ).
% prefix_Nil
thf(fact_462_diff__diff__left,axiom,
! [I: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J2 ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% diff_diff_left
thf(fact_463_same__prefix__prefix,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat,Zs: list_list_nat] :
( ( prefix_list_nat @ ( append_list_nat @ Xs2 @ Ys2 ) @ ( append_list_nat @ Xs2 @ Zs ) )
= ( prefix_list_nat @ Ys2 @ Zs ) ) ).
% same_prefix_prefix
thf(fact_464_same__prefix__prefix,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( prefix_nat @ ( append_nat @ Xs2 @ Ys2 ) @ ( append_nat @ Xs2 @ Zs ) )
= ( prefix_nat @ Ys2 @ Zs ) ) ).
% same_prefix_prefix
thf(fact_465_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_466_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_467_same__prefix__nil,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat] :
( ( prefix_list_nat @ ( append_list_nat @ Xs2 @ Ys2 ) @ Xs2 )
= ( Ys2 = nil_list_nat ) ) ).
% same_prefix_nil
thf(fact_468_same__prefix__nil,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( prefix_nat @ ( append_nat @ Xs2 @ Ys2 ) @ Xs2 )
= ( Ys2 = nil_nat ) ) ).
% same_prefix_nil
thf(fact_469_in__set__prefixes,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( member_list_nat @ Xs2 @ ( set_list_nat2 @ ( prefixes_nat @ Ys2 ) ) )
= ( prefix_nat @ Xs2 @ Ys2 ) ) ).
% in_set_prefixes
thf(fact_470_prefix__snoc,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat,Y: list_nat] :
( ( prefix_list_nat @ Xs2 @ ( append_list_nat @ Ys2 @ ( cons_list_nat @ Y @ nil_list_nat ) ) )
= ( ( Xs2
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ Y @ nil_list_nat ) ) )
| ( prefix_list_nat @ Xs2 @ Ys2 ) ) ) ).
% prefix_snoc
thf(fact_471_prefix__snoc,axiom,
! [Xs2: list_nat,Ys2: list_nat,Y: nat] :
( ( prefix_nat @ Xs2 @ ( append_nat @ Ys2 @ ( cons_nat @ Y @ nil_nat ) ) )
= ( ( Xs2
= ( append_nat @ Ys2 @ ( cons_nat @ Y @ nil_nat ) ) )
| ( prefix_nat @ Xs2 @ Ys2 ) ) ) ).
% prefix_snoc
thf(fact_472_prefix__same__cases,axiom,
! [Xs_1: list_nat,Ys2: list_nat,Xs_2: list_nat] :
( ( prefix_nat @ Xs_1 @ Ys2 )
=> ( ( prefix_nat @ Xs_2 @ Ys2 )
=> ( ( prefix_nat @ Xs_1 @ Xs_2 )
| ( prefix_nat @ Xs_2 @ Xs_1 ) ) ) ) ).
% prefix_same_cases
thf(fact_473_prefix__order_Odual__order_Oantisym,axiom,
! [B: list_nat,A: list_nat] :
( ( prefix_nat @ B @ A )
=> ( ( prefix_nat @ A @ B )
=> ( A = B ) ) ) ).
% prefix_order.dual_order.antisym
thf(fact_474_prefix__order_Odual__order_Oeq__iff,axiom,
( ( ^ [Y4: list_nat,Z3: list_nat] : ( Y4 = Z3 ) )
= ( ^ [A3: list_nat,B3: list_nat] :
( ( prefix_nat @ B3 @ A3 )
& ( prefix_nat @ A3 @ B3 ) ) ) ) ).
% prefix_order.dual_order.eq_iff
thf(fact_475_prefix__order_Odual__order_Otrans,axiom,
! [B: list_nat,A: list_nat,C: list_nat] :
( ( prefix_nat @ B @ A )
=> ( ( prefix_nat @ C @ B )
=> ( prefix_nat @ C @ A ) ) ) ).
% prefix_order.dual_order.trans
thf(fact_476_prefix__order_Oord__le__eq__trans,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( prefix_nat @ A @ B )
=> ( ( B = C )
=> ( prefix_nat @ A @ C ) ) ) ).
% prefix_order.ord_le_eq_trans
thf(fact_477_prefix__order_Oord__eq__le__trans,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( A = B )
=> ( ( prefix_nat @ B @ C )
=> ( prefix_nat @ A @ C ) ) ) ).
% prefix_order.ord_eq_le_trans
thf(fact_478_prefix__order_Oorder__antisym,axiom,
! [X: list_nat,Y: list_nat] :
( ( prefix_nat @ X @ Y )
=> ( ( prefix_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% prefix_order.order_antisym
thf(fact_479_prefix__order_Oorder__eq__iff,axiom,
( ( ^ [Y4: list_nat,Z3: list_nat] : ( Y4 = Z3 ) )
= ( ^ [X3: list_nat,Y3: list_nat] :
( ( prefix_nat @ X3 @ Y3 )
& ( prefix_nat @ Y3 @ X3 ) ) ) ) ).
% prefix_order.order_eq_iff
thf(fact_480_prefix__order_Oantisym__conv,axiom,
! [Y: list_nat,X: list_nat] :
( ( prefix_nat @ Y @ X )
=> ( ( prefix_nat @ X @ Y )
= ( X = Y ) ) ) ).
% prefix_order.antisym_conv
thf(fact_481_prefix__order_Oorder__trans,axiom,
! [X: list_nat,Y: list_nat,Z4: list_nat] :
( ( prefix_nat @ X @ Y )
=> ( ( prefix_nat @ Y @ Z4 )
=> ( prefix_nat @ X @ Z4 ) ) ) ).
% prefix_order.order_trans
thf(fact_482_prefix__order_Oeq__refl,axiom,
! [X: list_nat,Y: list_nat] :
( ( X = Y )
=> ( prefix_nat @ X @ Y ) ) ).
% prefix_order.eq_refl
thf(fact_483_prefix__order_Oantisym,axiom,
! [A: list_nat,B: list_nat] :
( ( prefix_nat @ A @ B )
=> ( ( prefix_nat @ B @ A )
=> ( A = B ) ) ) ).
% prefix_order.antisym
thf(fact_484_prefix__order_Oeq__iff,axiom,
( ( ^ [Y4: list_nat,Z3: list_nat] : ( Y4 = Z3 ) )
= ( ^ [A3: list_nat,B3: list_nat] :
( ( prefix_nat @ A3 @ B3 )
& ( prefix_nat @ B3 @ A3 ) ) ) ) ).
% prefix_order.eq_iff
thf(fact_485_prefix__order_Otrans,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( prefix_nat @ A @ B )
=> ( ( prefix_nat @ B @ C )
=> ( prefix_nat @ A @ C ) ) ) ).
% prefix_order.trans
thf(fact_486_diff__commute,axiom,
! [I: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J2 ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J2 ) ) ).
% diff_commute
thf(fact_487_diff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_488_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_489_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_490_Nil__prefix,axiom,
! [Xs2: list_list_nat] : ( prefix_list_nat @ nil_list_nat @ Xs2 ) ).
% Nil_prefix
thf(fact_491_Nil__prefix,axiom,
! [Xs2: list_nat] : ( prefix_nat @ nil_nat @ Xs2 ) ).
% Nil_prefix
thf(fact_492_prefix__bot_Oextremum__uniqueI,axiom,
! [A: list_list_nat] :
( ( prefix_list_nat @ A @ nil_list_nat )
=> ( A = nil_list_nat ) ) ).
% prefix_bot.extremum_uniqueI
thf(fact_493_prefix__bot_Oextremum__uniqueI,axiom,
! [A: list_nat] :
( ( prefix_nat @ A @ nil_nat )
=> ( A = nil_nat ) ) ).
% prefix_bot.extremum_uniqueI
thf(fact_494_prefix__bot_Obot__least,axiom,
! [A: list_list_nat] : ( prefix_list_nat @ nil_list_nat @ A ) ).
% prefix_bot.bot_least
thf(fact_495_prefix__bot_Obot__least,axiom,
! [A: list_nat] : ( prefix_nat @ nil_nat @ A ) ).
% prefix_bot.bot_least
thf(fact_496_append__prefixD,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat,Zs: list_list_nat] :
( ( prefix_list_nat @ ( append_list_nat @ Xs2 @ Ys2 ) @ Zs )
=> ( prefix_list_nat @ Xs2 @ Zs ) ) ).
% append_prefixD
thf(fact_497_append__prefixD,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( prefix_nat @ ( append_nat @ Xs2 @ Ys2 ) @ Zs )
=> ( prefix_nat @ Xs2 @ Zs ) ) ).
% append_prefixD
thf(fact_498_prefix__prefix,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat,Zs: list_list_nat] :
( ( prefix_list_nat @ Xs2 @ Ys2 )
=> ( prefix_list_nat @ Xs2 @ ( append_list_nat @ Ys2 @ Zs ) ) ) ).
% prefix_prefix
thf(fact_499_prefix__prefix,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( prefix_nat @ Xs2 @ Ys2 )
=> ( prefix_nat @ Xs2 @ ( append_nat @ Ys2 @ Zs ) ) ) ).
% prefix_prefix
thf(fact_500_prefix__append,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat,Zs: list_list_nat] :
( ( prefix_list_nat @ Xs2 @ ( append_list_nat @ Ys2 @ Zs ) )
= ( ( prefix_list_nat @ Xs2 @ Ys2 )
| ? [Us2: list_list_nat] :
( ( Xs2
= ( append_list_nat @ Ys2 @ Us2 ) )
& ( prefix_list_nat @ Us2 @ Zs ) ) ) ) ).
% prefix_append
thf(fact_501_prefix__append,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( prefix_nat @ Xs2 @ ( append_nat @ Ys2 @ Zs ) )
= ( ( prefix_nat @ Xs2 @ Ys2 )
| ? [Us2: list_nat] :
( ( Xs2
= ( append_nat @ Ys2 @ Us2 ) )
& ( prefix_nat @ Us2 @ Zs ) ) ) ) ).
% prefix_append
thf(fact_502_prefix__def,axiom,
( prefix_list_nat
= ( ^ [Xs4: list_list_nat,Ys3: list_list_nat] :
? [Zs3: list_list_nat] :
( Ys3
= ( append_list_nat @ Xs4 @ Zs3 ) ) ) ) ).
% prefix_def
thf(fact_503_prefix__def,axiom,
( prefix_nat
= ( ^ [Xs4: list_nat,Ys3: list_nat] :
? [Zs3: list_nat] :
( Ys3
= ( append_nat @ Xs4 @ Zs3 ) ) ) ) ).
% prefix_def
thf(fact_504_prefixI,axiom,
! [Ys2: list_list_nat,Xs2: list_list_nat,Zs: list_list_nat] :
( ( Ys2
= ( append_list_nat @ Xs2 @ Zs ) )
=> ( prefix_list_nat @ Xs2 @ Ys2 ) ) ).
% prefixI
thf(fact_505_prefixI,axiom,
! [Ys2: list_nat,Xs2: list_nat,Zs: list_nat] :
( ( Ys2
= ( append_nat @ Xs2 @ Zs ) )
=> ( prefix_nat @ Xs2 @ Ys2 ) ) ).
% prefixI
thf(fact_506_prefixE,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat] :
( ( prefix_list_nat @ Xs2 @ Ys2 )
=> ~ ! [Zs2: list_list_nat] :
( Ys2
!= ( append_list_nat @ Xs2 @ Zs2 ) ) ) ).
% prefixE
thf(fact_507_prefixE,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( prefix_nat @ Xs2 @ Ys2 )
=> ~ ! [Zs2: list_nat] :
( Ys2
!= ( append_nat @ Xs2 @ Zs2 ) ) ) ).
% prefixE
thf(fact_508_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_509_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_510_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_511_less__imp__diff__less,axiom,
! [J2: nat,K: nat,N: nat] :
( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J2 @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_512_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_513_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_514_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_515_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_516_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_517_prefix__code_I2_J,axiom,
! [X: list_nat,Xs2: list_list_nat] :
~ ( prefix_list_nat @ ( cons_list_nat @ X @ Xs2 ) @ nil_list_nat ) ).
% prefix_code(2)
thf(fact_518_prefix__code_I2_J,axiom,
! [X: nat,Xs2: list_nat] :
~ ( prefix_nat @ ( cons_nat @ X @ Xs2 ) @ nil_nat ) ).
% prefix_code(2)
thf(fact_519_prefix__Cons,axiom,
! [Xs2: list_list_nat,Y: list_nat,Ys2: list_list_nat] :
( ( prefix_list_nat @ Xs2 @ ( cons_list_nat @ Y @ Ys2 ) )
= ( ( Xs2 = nil_list_nat )
| ? [Zs3: list_list_nat] :
( ( Xs2
= ( cons_list_nat @ Y @ Zs3 ) )
& ( prefix_list_nat @ Zs3 @ Ys2 ) ) ) ) ).
% prefix_Cons
thf(fact_520_prefix__Cons,axiom,
! [Xs2: list_nat,Y: nat,Ys2: list_nat] :
( ( prefix_nat @ Xs2 @ ( cons_nat @ Y @ Ys2 ) )
= ( ( Xs2 = nil_nat )
| ? [Zs3: list_nat] :
( ( Xs2
= ( cons_nat @ Y @ Zs3 ) )
& ( prefix_nat @ Zs3 @ Ys2 ) ) ) ) ).
% prefix_Cons
thf(fact_521_not__prefix__cases,axiom,
! [Ps: list_list_nat,Ls: list_list_nat] :
( ~ ( prefix_list_nat @ Ps @ Ls )
=> ( ( ( Ps != nil_list_nat )
=> ( Ls != nil_list_nat ) )
=> ( ! [A4: list_nat,As: list_list_nat] :
( ( Ps
= ( cons_list_nat @ A4 @ As ) )
=> ! [X2: list_nat,Xs: list_list_nat] :
( ( Ls
= ( cons_list_nat @ X2 @ Xs ) )
=> ( ( X2 = A4 )
=> ( prefix_list_nat @ As @ Xs ) ) ) )
=> ~ ! [A4: list_nat] :
( ? [As: list_list_nat] :
( Ps
= ( cons_list_nat @ A4 @ As ) )
=> ! [X2: list_nat] :
( ? [Xs: list_list_nat] :
( Ls
= ( cons_list_nat @ X2 @ Xs ) )
=> ( X2 = A4 ) ) ) ) ) ) ).
% not_prefix_cases
thf(fact_522_not__prefix__cases,axiom,
! [Ps: list_nat,Ls: list_nat] :
( ~ ( prefix_nat @ Ps @ Ls )
=> ( ( ( Ps != nil_nat )
=> ( Ls != nil_nat ) )
=> ( ! [A4: nat,As: list_nat] :
( ( Ps
= ( cons_nat @ A4 @ As ) )
=> ! [X2: nat,Xs: list_nat] :
( ( Ls
= ( cons_nat @ X2 @ Xs ) )
=> ( ( X2 = A4 )
=> ( prefix_nat @ As @ Xs ) ) ) )
=> ~ ! [A4: nat] :
( ? [As: list_nat] :
( Ps
= ( cons_nat @ A4 @ As ) )
=> ! [X2: nat] :
( ? [Xs: list_nat] :
( Ls
= ( cons_nat @ X2 @ Xs ) )
=> ( X2 = A4 ) ) ) ) ) ) ).
% not_prefix_cases
thf(fact_523_not__prefix__induct,axiom,
! [Ps: list_list_nat,Ls: list_list_nat,P: list_list_nat > list_list_nat > $o] :
( ~ ( prefix_list_nat @ Ps @ Ls )
=> ( ! [X2: list_nat,Xs: list_list_nat] : ( P @ ( cons_list_nat @ X2 @ Xs ) @ nil_list_nat )
=> ( ! [X2: list_nat,Xs: list_list_nat,Y2: list_nat,Ys: list_list_nat] :
( ( X2 != Y2 )
=> ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y2 @ Ys ) ) )
=> ( ! [X2: list_nat,Xs: list_list_nat,Y2: list_nat,Ys: list_list_nat] :
( ( X2 = Y2 )
=> ( ~ ( prefix_list_nat @ Xs @ Ys )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( cons_list_nat @ X2 @ Xs ) @ ( cons_list_nat @ Y2 @ Ys ) ) ) ) )
=> ( P @ Ps @ Ls ) ) ) ) ) ).
% not_prefix_induct
thf(fact_524_not__prefix__induct,axiom,
! [Ps: list_nat,Ls: list_nat,P: list_nat > list_nat > $o] :
( ~ ( prefix_nat @ Ps @ Ls )
=> ( ! [X2: nat,Xs: list_nat] : ( P @ ( cons_nat @ X2 @ Xs ) @ nil_nat )
=> ( ! [X2: nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( X2 != Y2 )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) ) )
=> ( ! [X2: nat,Xs: list_nat,Y2: nat,Ys: list_nat] :
( ( X2 = Y2 )
=> ( ~ ( prefix_nat @ Xs @ Ys )
=> ( ( P @ Xs @ Ys )
=> ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y2 @ Ys ) ) ) ) )
=> ( P @ Ps @ Ls ) ) ) ) ) ).
% not_prefix_induct
thf(fact_525_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_526_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_527_less__diff__conv,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J2 ) ) ).
% less_diff_conv
thf(fact_528_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_529_nat__minus__add__max,axiom,
! [N: nat,M: nat] :
( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
= ( ord_max_nat @ N @ M ) ) ).
% nat_minus_add_max
thf(fact_530_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
=> ( P @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_531_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
& ~ ( P @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_532_nth__Cons_H,axiom,
! [N: nat,X: list_nat,Xs2: list_list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( nth_list_nat @ ( cons_list_nat @ X @ Xs2 ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_list_nat @ ( cons_list_nat @ X @ Xs2 ) @ N )
= ( nth_list_nat @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_533_nth__Cons_H,axiom,
! [N: nat,X: nat,Xs2: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ N )
= ( nth_nat @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_534_nth__append,axiom,
! [N: nat,Xs2: list_list_nat,Ys2: list_list_nat] :
( ( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ( nth_list_nat @ ( append_list_nat @ Xs2 @ Ys2 ) @ N )
= ( nth_list_nat @ Xs2 @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ( nth_list_nat @ ( append_list_nat @ Xs2 @ Ys2 ) @ N )
= ( nth_list_nat @ Ys2 @ ( minus_minus_nat @ N @ ( size_s3023201423986296836st_nat @ Xs2 ) ) ) ) ) ) ).
% nth_append
thf(fact_535_nth__append,axiom,
! [N: nat,Xs2: list_nat,Ys2: list_nat] :
( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ ( append_nat @ Xs2 @ Ys2 ) @ N )
= ( nth_nat @ Xs2 @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ ( append_nat @ Xs2 @ Ys2 ) @ N )
= ( nth_nat @ Ys2 @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs2 ) ) ) ) ) ) ).
% nth_append
thf(fact_536_list__update__append,axiom,
! [N: nat,Xs2: list_list_nat,Ys2: list_list_nat,X: list_nat] :
( ( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ( list_update_list_nat @ ( append_list_nat @ Xs2 @ Ys2 ) @ N @ X )
= ( append_list_nat @ ( list_update_list_nat @ Xs2 @ N @ X ) @ Ys2 ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ( list_update_list_nat @ ( append_list_nat @ Xs2 @ Ys2 ) @ N @ X )
= ( append_list_nat @ Xs2 @ ( list_update_list_nat @ Ys2 @ ( minus_minus_nat @ N @ ( size_s3023201423986296836st_nat @ Xs2 ) ) @ X ) ) ) ) ) ).
% list_update_append
thf(fact_537_list__update__append,axiom,
! [N: nat,Xs2: list_nat,Ys2: list_nat,X: nat] :
( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( list_update_nat @ ( append_nat @ Xs2 @ Ys2 ) @ N @ X )
= ( append_nat @ ( list_update_nat @ Xs2 @ N @ X ) @ Ys2 ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( list_update_nat @ ( append_nat @ Xs2 @ Ys2 ) @ N @ X )
= ( append_nat @ Xs2 @ ( list_update_nat @ Ys2 @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs2 ) ) @ X ) ) ) ) ) ).
% list_update_append
thf(fact_538_nth__non__equal__first__eq,axiom,
! [X: list_nat,Y: list_nat,Xs2: list_list_nat,N: nat] :
( ( X != Y )
=> ( ( ( nth_list_nat @ ( cons_list_nat @ X @ Xs2 ) @ N )
= Y )
= ( ( ( nth_list_nat @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_539_nth__non__equal__first__eq,axiom,
! [X: nat,Y: nat,Xs2: list_nat,N: nat] :
( ( X != Y )
=> ( ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ N )
= Y )
= ( ( ( nth_nat @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_540_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_541_last__list__update,axiom,
! [Xs2: list_list_nat,K: nat,X: list_nat] :
( ( Xs2 != nil_list_nat )
=> ( ( ( K
= ( minus_minus_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ one_one_nat ) )
=> ( ( last_list_nat @ ( list_update_list_nat @ Xs2 @ K @ X ) )
= X ) )
& ( ( K
!= ( minus_minus_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ one_one_nat ) )
=> ( ( last_list_nat @ ( list_update_list_nat @ Xs2 @ K @ X ) )
= ( last_list_nat @ Xs2 ) ) ) ) ) ).
% last_list_update
thf(fact_542_last__list__update,axiom,
! [Xs2: list_nat,K: nat,X: nat] :
( ( Xs2 != nil_nat )
=> ( ( ( K
= ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) )
=> ( ( last_nat @ ( list_update_nat @ Xs2 @ K @ X ) )
= X ) )
& ( ( K
!= ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) )
=> ( ( last_nat @ ( list_update_nat @ Xs2 @ K @ X ) )
= ( last_nat @ Xs2 ) ) ) ) ) ).
% last_list_update
thf(fact_543_last__conv__nth,axiom,
! [Xs2: list_list_nat] :
( ( Xs2 != nil_list_nat )
=> ( ( last_list_nat @ Xs2 )
= ( nth_list_nat @ Xs2 @ ( minus_minus_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_544_last__conv__nth,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( last_nat @ Xs2 )
= ( nth_nat @ Xs2 @ ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_545_concat__eq__append__conv,axiom,
! [Xss2: list_list_list_nat,Ys2: list_list_nat,Zs: list_list_nat] :
( ( ( concat_list_nat @ Xss2 )
= ( append_list_nat @ Ys2 @ Zs ) )
= ( ( ( Xss2 = nil_list_list_nat )
=> ( ( Ys2 = nil_list_nat )
& ( Zs = nil_list_nat ) ) )
& ( ( Xss2 != nil_list_list_nat )
=> ? [Xss1: list_list_list_nat,Xs4: list_list_nat,Xs6: list_list_nat,Xss22: list_list_list_nat] :
( ( Xss2
= ( append_list_list_nat @ Xss1 @ ( cons_list_list_nat @ ( append_list_nat @ Xs4 @ Xs6 ) @ Xss22 ) ) )
& ( Ys2
= ( append_list_nat @ ( concat_list_nat @ Xss1 ) @ Xs4 ) )
& ( Zs
= ( append_list_nat @ Xs6 @ ( concat_list_nat @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_546_concat__eq__append__conv,axiom,
! [Xss2: list_list_nat,Ys2: list_nat,Zs: list_nat] :
( ( ( concat_nat @ Xss2 )
= ( append_nat @ Ys2 @ Zs ) )
= ( ( ( Xss2 = nil_list_nat )
=> ( ( Ys2 = 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 ) ) )
& ( Ys2
= ( append_nat @ ( concat_nat @ Xss1 ) @ Xs4 ) )
& ( Zs
= ( append_nat @ Xs6 @ ( concat_nat @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_547_nth__equal__first__eq,axiom,
! [X: list_nat,Xs2: list_list_nat,N: nat] :
( ~ ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
=> ( ( ord_less_eq_nat @ N @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ( ( nth_list_nat @ ( cons_list_nat @ X @ Xs2 ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_548_nth__equal__first__eq,axiom,
! [X: nat,Xs2: list_nat,N: nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_549_last__prefixes,axiom,
! [Xs2: list_nat] :
( ( last_list_nat @ ( prefixes_nat @ Xs2 ) )
= Xs2 ) ).
% last_prefixes
thf(fact_550_last__suffixes,axiom,
! [Xs2: list_nat] :
( ( last_list_nat @ ( suffixes_nat @ Xs2 ) )
= Xs2 ) ).
% last_suffixes
thf(fact_551_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_552_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_553_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_554_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_555_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_556_max_Obounded__iff,axiom,
! [B: num,C: num,A: num] :
( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
= ( ( ord_less_eq_num @ B @ A )
& ( ord_less_eq_num @ C @ A ) ) ) ).
% max.bounded_iff
thf(fact_557_max_Obounded__iff,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
= ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% max.bounded_iff
thf(fact_558_max_Oabsorb2,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_max_num @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_559_max_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_max_nat @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_560_max_Oabsorb1,axiom,
! [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_max_num @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_561_max_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_max_nat @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_562_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_563_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_564_lessThan__subset__iff,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X ) @ ( set_ord_lessThan_num @ Y ) )
= ( ord_less_eq_num @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_565_lessThan__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_566_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_567_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_568_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_569_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_570_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_571_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_572_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_573_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_574_Nat_Odiff__diff__right,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J2 ) ) ) ).
% Nat.diff_diff_right
thf(fact_575_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_576_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J2 ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_577_list__update__beyond,axiom,
! [Xs2: list_list_nat,I: nat,X: list_nat] :
( ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ I )
=> ( ( list_update_list_nat @ Xs2 @ I @ X )
= Xs2 ) ) ).
% list_update_beyond
thf(fact_578_list__update__beyond,axiom,
! [Xs2: list_nat,I: nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ I )
=> ( ( list_update_nat @ Xs2 @ I @ X )
= Xs2 ) ) ).
% list_update_beyond
thf(fact_579_last__appendR,axiom,
! [Ys2: list_list_nat,Xs2: list_list_nat] :
( ( Ys2 != nil_list_nat )
=> ( ( last_list_nat @ ( append_list_nat @ Xs2 @ Ys2 ) )
= ( last_list_nat @ Ys2 ) ) ) ).
% last_appendR
thf(fact_580_last__appendR,axiom,
! [Ys2: list_nat,Xs2: list_nat] :
( ( Ys2 != nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs2 @ Ys2 ) )
= ( last_nat @ Ys2 ) ) ) ).
% last_appendR
thf(fact_581_last__appendL,axiom,
! [Ys2: list_list_nat,Xs2: list_list_nat] :
( ( Ys2 = nil_list_nat )
=> ( ( last_list_nat @ ( append_list_nat @ Xs2 @ Ys2 ) )
= ( last_list_nat @ Xs2 ) ) ) ).
% last_appendL
thf(fact_582_last__appendL,axiom,
! [Ys2: list_nat,Xs2: list_nat] :
( ( Ys2 = nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs2 @ Ys2 ) )
= ( last_nat @ Xs2 ) ) ) ).
% last_appendL
thf(fact_583_Nil__eq__concat__conv,axiom,
! [Xss2: list_list_list_nat] :
( ( nil_list_nat
= ( concat_list_nat @ Xss2 ) )
= ( ! [X3: list_list_nat] :
( ( member_list_list_nat @ X3 @ ( set_list_list_nat2 @ Xss2 ) )
=> ( X3 = nil_list_nat ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_584_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_585_concat__eq__Nil__conv,axiom,
! [Xss2: list_list_list_nat] :
( ( ( concat_list_nat @ Xss2 )
= nil_list_nat )
= ( ! [X3: list_list_nat] :
( ( member_list_list_nat @ X3 @ ( set_list_list_nat2 @ Xss2 ) )
=> ( X3 = nil_list_nat ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_586_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_587_concat__append,axiom,
! [Xs2: list_list_list_nat,Ys2: list_list_list_nat] :
( ( concat_list_nat @ ( append_list_list_nat @ Xs2 @ Ys2 ) )
= ( append_list_nat @ ( concat_list_nat @ Xs2 ) @ ( concat_list_nat @ Ys2 ) ) ) ).
% concat_append
thf(fact_588_concat__append,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat] :
( ( concat_nat @ ( append_list_nat @ Xs2 @ Ys2 ) )
= ( append_nat @ ( concat_nat @ Xs2 ) @ ( concat_nat @ Ys2 ) ) ) ).
% concat_append
thf(fact_589_last__snoc,axiom,
! [Xs2: list_list_nat,X: list_nat] :
( ( last_list_nat @ ( append_list_nat @ Xs2 @ ( cons_list_nat @ X @ nil_list_nat ) ) )
= X ) ).
% last_snoc
thf(fact_590_last__snoc,axiom,
! [Xs2: list_nat,X: nat] :
( ( last_nat @ ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) ) )
= X ) ).
% last_snoc
thf(fact_591_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M3: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M3 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_592_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_593_le__trans,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_594_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_595_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_596_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_597_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 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_598_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_599_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_600_verit__comp__simplify1_I3_J,axiom,
! [B5: num,A5: num] :
( ( ~ ( ord_less_eq_num @ B5 @ A5 ) )
= ( ord_less_num @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_601_verit__comp__simplify1_I3_J,axiom,
! [B5: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B5 @ A5 ) )
= ( ord_less_nat @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_602_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_603_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_604_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
? [C3: nat] :
( B3
= ( plus_plus_nat @ A3 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_605_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_606_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_607_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_608_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_609_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_610_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( I = J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_611_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J2 )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_612_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_613_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_614_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_615_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_616_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_617_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J2: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_618_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_619_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_620_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
| ( M5 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_621_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_622_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
& ( M5 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_623_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N3: nat] :
? [K3: nat] :
( N3
= ( plus_plus_nat @ M5 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_624_trans__le__add2,axiom,
! [I: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_le_add2
thf(fact_625_trans__le__add1,axiom,
! [I: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_le_add1
thf(fact_626_add__le__mono1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% add_le_mono1
thf(fact_627_add__le__mono,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_le_mono
thf(fact_628_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_629_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_630_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_631_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_632_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_633_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_634_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_635_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_636_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_637_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_638_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_639_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_640_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_641_max_Omono,axiom,
! [C: num,A: num,D: num,B: num] :
( ( ord_less_eq_num @ C @ A )
=> ( ( ord_less_eq_num @ D @ B )
=> ( ord_less_eq_num @ ( ord_max_num @ C @ D ) @ ( ord_max_num @ A @ B ) ) ) ) ).
% max.mono
thf(fact_642_max_Omono,axiom,
! [C: nat,A: nat,D: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ( ord_less_eq_nat @ D @ B )
=> ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A @ B ) ) ) ) ).
% max.mono
thf(fact_643_max_OorderE,axiom,
! [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
=> ( A
= ( ord_max_num @ A @ B ) ) ) ).
% max.orderE
thf(fact_644_max_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( ord_max_nat @ A @ B ) ) ) ).
% max.orderE
thf(fact_645_max_OorderI,axiom,
! [A: num,B: num] :
( ( A
= ( ord_max_num @ A @ B ) )
=> ( ord_less_eq_num @ B @ A ) ) ).
% max.orderI
thf(fact_646_max_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( ord_max_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% max.orderI
thf(fact_647_max_OboundedE,axiom,
! [B: num,C: num,A: num] :
( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_num @ B @ A )
=> ~ ( ord_less_eq_num @ C @ A ) ) ) ).
% max.boundedE
thf(fact_648_max_OboundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_nat @ B @ A )
=> ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% max.boundedE
thf(fact_649_max_OboundedI,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ C @ A )
=> ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A ) ) ) ).
% max.boundedI
thf(fact_650_max_OboundedI,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A ) ) ) ).
% max.boundedI
thf(fact_651_max_Oorder__iff,axiom,
( ord_less_eq_num
= ( ^ [B3: num,A3: num] :
( A3
= ( ord_max_num @ A3 @ B3 ) ) ) ) ).
% max.order_iff
thf(fact_652_max_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( A3
= ( ord_max_nat @ A3 @ B3 ) ) ) ) ).
% max.order_iff
thf(fact_653_max_Ocobounded1,axiom,
! [A: num,B: num] : ( ord_less_eq_num @ A @ ( ord_max_num @ A @ B ) ) ).
% max.cobounded1
thf(fact_654_max_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).
% max.cobounded1
thf(fact_655_max_Ocobounded2,axiom,
! [B: num,A: num] : ( ord_less_eq_num @ B @ ( ord_max_num @ A @ B ) ) ).
% max.cobounded2
thf(fact_656_max_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).
% max.cobounded2
thf(fact_657_le__max__iff__disj,axiom,
! [Z4: num,X: num,Y: num] :
( ( ord_less_eq_num @ Z4 @ ( ord_max_num @ X @ Y ) )
= ( ( ord_less_eq_num @ Z4 @ X )
| ( ord_less_eq_num @ Z4 @ Y ) ) ) ).
% le_max_iff_disj
thf(fact_658_le__max__iff__disj,axiom,
! [Z4: nat,X: nat,Y: nat] :
( ( ord_less_eq_nat @ Z4 @ ( ord_max_nat @ X @ Y ) )
= ( ( ord_less_eq_nat @ Z4 @ X )
| ( ord_less_eq_nat @ Z4 @ Y ) ) ) ).
% le_max_iff_disj
thf(fact_659_max_Oabsorb__iff1,axiom,
( ord_less_eq_num
= ( ^ [B3: num,A3: num] :
( ( ord_max_num @ A3 @ B3 )
= A3 ) ) ) ).
% max.absorb_iff1
thf(fact_660_max_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_max_nat @ A3 @ B3 )
= A3 ) ) ) ).
% max.absorb_iff1
thf(fact_661_max_Oabsorb__iff2,axiom,
( ord_less_eq_num
= ( ^ [A3: num,B3: num] :
( ( ord_max_num @ A3 @ B3 )
= B3 ) ) ) ).
% max.absorb_iff2
thf(fact_662_max_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_max_nat @ A3 @ B3 )
= B3 ) ) ) ).
% max.absorb_iff2
thf(fact_663_max_OcoboundedI1,axiom,
! [C: num,A: num,B: num] :
( ( ord_less_eq_num @ C @ A )
=> ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_664_max_OcoboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.coboundedI1
thf(fact_665_max_OcoboundedI2,axiom,
! [C: num,B: num,A: num] :
( ( ord_less_eq_num @ C @ B )
=> ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_666_max_OcoboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).
% max.coboundedI2
thf(fact_667_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_668_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_669_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_670_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_671_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_672_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_673_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_674_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_675_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_676_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_677_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_678_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_679_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_680_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J2 )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_681_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J2 )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_682_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_683_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_684_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_685_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_686_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_687_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_688_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_689_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_690_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_691_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J2 @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J2 @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J2 ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_692_diff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% diff_add
thf(fact_693_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_694_impossible__Cons,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat,X: list_nat] :
( ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( Xs2
!= ( cons_list_nat @ X @ Ys2 ) ) ) ).
% impossible_Cons
thf(fact_695_impossible__Cons,axiom,
! [Xs2: list_nat,Ys2: list_nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ Ys2 ) )
=> ( Xs2
!= ( cons_nat @ X @ Ys2 ) ) ) ).
% impossible_Cons
thf(fact_696_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K2 )
=> ~ ( P @ I5 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_697_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_698_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_699_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_700_prefix__length__le,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat] :
( ( prefix_list_nat @ Xs2 @ Ys2 )
=> ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ ( size_s3023201423986296836st_nat @ Ys2 ) ) ) ).
% prefix_length_le
thf(fact_701_prefix__length__le,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( prefix_nat @ Xs2 @ Ys2 )
=> ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ Ys2 ) ) ) ).
% prefix_length_le
thf(fact_702_prefix__length__prefix,axiom,
! [Ps: list_list_nat,Xs2: list_list_nat,Qs: list_list_nat] :
( ( prefix_list_nat @ Ps @ Xs2 )
=> ( ( prefix_list_nat @ Qs @ Xs2 )
=> ( ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Ps ) @ ( size_s3023201423986296836st_nat @ Qs ) )
=> ( prefix_list_nat @ Ps @ Qs ) ) ) ) ).
% prefix_length_prefix
thf(fact_703_prefix__length__prefix,axiom,
! [Ps: list_nat,Xs2: list_nat,Qs: list_nat] :
( ( prefix_nat @ Ps @ Xs2 )
=> ( ( prefix_nat @ Qs @ Xs2 )
=> ( ( ord_less_eq_nat @ ( size_size_list_nat @ Ps ) @ ( size_size_list_nat @ Qs ) )
=> ( prefix_nat @ Ps @ Qs ) ) ) ) ).
% prefix_length_prefix
thf(fact_704_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ( minus_minus_nat @ J2 @ I )
= K )
= ( J2
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_705_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_706_Nat_Odiff__add__assoc,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J2 ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J2 @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_707_Nat_Ole__diff__conv2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J2 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_708_le__diff__conv,axiom,
! [J2: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J2 @ K ) @ I )
= ( ord_less_eq_nat @ J2 @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_709_concat_Osimps_I2_J,axiom,
! [X: list_list_nat,Xs2: list_list_list_nat] :
( ( concat_list_nat @ ( cons_list_list_nat @ X @ Xs2 ) )
= ( append_list_nat @ X @ ( concat_list_nat @ Xs2 ) ) ) ).
% concat.simps(2)
thf(fact_710_concat_Osimps_I2_J,axiom,
! [X: list_nat,Xs2: list_list_nat] :
( ( concat_nat @ ( cons_list_nat @ X @ Xs2 ) )
= ( append_nat @ X @ ( concat_nat @ Xs2 ) ) ) ).
% concat.simps(2)
thf(fact_711_last__ConsR,axiom,
! [Xs2: list_list_nat,X: list_nat] :
( ( Xs2 != nil_list_nat )
=> ( ( last_list_nat @ ( cons_list_nat @ X @ Xs2 ) )
= ( last_list_nat @ Xs2 ) ) ) ).
% last_ConsR
thf(fact_712_last__ConsR,axiom,
! [Xs2: list_nat,X: nat] :
( ( Xs2 != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs2 ) )
= ( last_nat @ Xs2 ) ) ) ).
% last_ConsR
thf(fact_713_last__ConsL,axiom,
! [Xs2: list_list_nat,X: list_nat] :
( ( Xs2 = nil_list_nat )
=> ( ( last_list_nat @ ( cons_list_nat @ X @ Xs2 ) )
= X ) ) ).
% last_ConsL
thf(fact_714_last__ConsL,axiom,
! [Xs2: list_nat,X: nat] :
( ( Xs2 = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs2 ) )
= X ) ) ).
% last_ConsL
thf(fact_715_last_Osimps,axiom,
! [Xs2: list_list_nat,X: list_nat] :
( ( ( Xs2 = nil_list_nat )
=> ( ( last_list_nat @ ( cons_list_nat @ X @ Xs2 ) )
= X ) )
& ( ( Xs2 != nil_list_nat )
=> ( ( last_list_nat @ ( cons_list_nat @ X @ Xs2 ) )
= ( last_list_nat @ Xs2 ) ) ) ) ).
% last.simps
thf(fact_716_last_Osimps,axiom,
! [Xs2: list_nat,X: nat] :
( ( ( Xs2 = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs2 ) )
= X ) )
& ( ( Xs2 != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs2 ) )
= ( last_nat @ Xs2 ) ) ) ) ).
% last.simps
thf(fact_717_concat_Osimps_I1_J,axiom,
( ( concat_list_nat @ nil_list_list_nat )
= nil_list_nat ) ).
% concat.simps(1)
thf(fact_718_concat_Osimps_I1_J,axiom,
( ( concat_nat @ nil_list_nat )
= nil_nat ) ).
% concat.simps(1)
thf(fact_719_last__in__set,axiom,
! [As2: list_list_nat] :
( ( As2 != nil_list_nat )
=> ( member_list_nat @ ( last_list_nat @ As2 ) @ ( set_list_nat2 @ As2 ) ) ) ).
% last_in_set
thf(fact_720_last__in__set,axiom,
! [As2: list_nat] :
( ( As2 != nil_nat )
=> ( member_nat @ ( last_nat @ As2 ) @ ( set_nat2 @ As2 ) ) ) ).
% last_in_set
thf(fact_721_longest__common__suffix,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat] :
? [Ss: list_list_nat,Xs5: list_list_nat,Ys7: list_list_nat] :
( ( Xs2
= ( append_list_nat @ Xs5 @ Ss ) )
& ( Ys2
= ( append_list_nat @ Ys7 @ Ss ) )
& ( ( Xs5 = nil_list_nat )
| ( Ys7 = nil_list_nat )
| ( ( last_list_nat @ Xs5 )
!= ( last_list_nat @ Ys7 ) ) ) ) ).
% longest_common_suffix
thf(fact_722_longest__common__suffix,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
? [Ss: list_nat,Xs5: list_nat,Ys7: list_nat] :
( ( Xs2
= ( append_nat @ Xs5 @ Ss ) )
& ( Ys2
= ( append_nat @ Ys7 @ Ss ) )
& ( ( Xs5 = nil_nat )
| ( Ys7 = nil_nat )
| ( ( last_nat @ Xs5 )
!= ( last_nat @ Ys7 ) ) ) ) ).
% longest_common_suffix
thf(fact_723_last__append,axiom,
! [Ys2: list_list_nat,Xs2: list_list_nat] :
( ( ( Ys2 = nil_list_nat )
=> ( ( last_list_nat @ ( append_list_nat @ Xs2 @ Ys2 ) )
= ( last_list_nat @ Xs2 ) ) )
& ( ( Ys2 != nil_list_nat )
=> ( ( last_list_nat @ ( append_list_nat @ Xs2 @ Ys2 ) )
= ( last_list_nat @ Ys2 ) ) ) ) ).
% last_append
thf(fact_724_last__append,axiom,
! [Ys2: list_nat,Xs2: list_nat] :
( ( ( Ys2 = nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs2 @ Ys2 ) )
= ( last_nat @ Xs2 ) ) )
& ( ( Ys2 != nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs2 @ Ys2 ) )
= ( last_nat @ Ys2 ) ) ) ) ).
% last_append
thf(fact_725_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_726_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_727_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_728_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_729_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_730_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_731_less__diff__conv2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J2 @ K ) @ I )
= ( ord_less_nat @ J2 @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_732_concat__eq__appendD,axiom,
! [Xss2: list_list_list_nat,Ys2: list_list_nat,Zs: list_list_nat] :
( ( ( concat_list_nat @ Xss2 )
= ( append_list_nat @ Ys2 @ Zs ) )
=> ( ( Xss2 != nil_list_list_nat )
=> ? [Xss12: list_list_list_nat,Xs: list_list_nat,Xs5: list_list_nat,Xss23: list_list_list_nat] :
( ( Xss2
= ( append_list_list_nat @ Xss12 @ ( cons_list_list_nat @ ( append_list_nat @ Xs @ Xs5 ) @ Xss23 ) ) )
& ( Ys2
= ( append_list_nat @ ( concat_list_nat @ Xss12 ) @ Xs ) )
& ( Zs
= ( append_list_nat @ Xs5 @ ( concat_list_nat @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_733_concat__eq__appendD,axiom,
! [Xss2: list_list_nat,Ys2: list_nat,Zs: list_nat] :
( ( ( concat_nat @ Xss2 )
= ( append_nat @ Ys2 @ Zs ) )
=> ( ( Xss2 != nil_list_nat )
=> ? [Xss12: list_list_nat,Xs: list_nat,Xs5: list_nat,Xss23: list_list_nat] :
( ( Xss2
= ( append_list_nat @ Xss12 @ ( cons_list_nat @ ( append_nat @ Xs @ Xs5 ) @ Xss23 ) ) )
& ( Ys2
= ( append_nat @ ( concat_nat @ Xss12 ) @ Xs ) )
& ( Zs
= ( append_nat @ Xs5 @ ( concat_nat @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_734_max__absorb2,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_max_set_nat @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_735_max__absorb2,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_max_num @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_736_max__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_max_nat @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_737_max__absorb1,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_max_set_nat @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_738_max__absorb1,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ( ( ord_max_num @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_739_max__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_max_nat @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_740_max__def,axiom,
( ord_max_set_nat
= ( ^ [A3: set_nat,B3: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).
% max_def
thf(fact_741_max__def,axiom,
( ord_max_num
= ( ^ [A3: num,B3: num] : ( if_num @ ( ord_less_eq_num @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).
% max_def
thf(fact_742_max__def,axiom,
( ord_max_nat
= ( ^ [A3: nat,B3: nat] : ( if_nat @ ( ord_less_eq_nat @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).
% max_def
thf(fact_743_subset__code_I1_J,axiom,
! [Xs2: list_list_nat,B2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs2 ) @ B2 )
= ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs2 ) )
=> ( member_list_nat @ X3 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_744_subset__code_I1_J,axiom,
! [Xs2: list_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ B2 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
=> ( member_nat @ X3 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_745_set__subset__Cons,axiom,
! [Xs2: list_list_nat,X: list_nat] : ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs2 ) @ ( set_list_nat2 @ ( cons_list_nat @ X @ Xs2 ) ) ) ).
% set_subset_Cons
thf(fact_746_set__subset__Cons,axiom,
! [Xs2: list_nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ ( set_nat2 @ ( cons_nat @ X @ Xs2 ) ) ) ).
% set_subset_Cons
thf(fact_747_set__mono__prefix,axiom,
! [Xs2: list_list_nat,Ys2: list_list_nat] :
( ( prefix_list_nat @ Xs2 @ Ys2 )
=> ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs2 ) @ ( set_list_nat2 @ Ys2 ) ) ) ).
% set_mono_prefix
thf(fact_748_set__mono__prefix,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( prefix_nat @ Xs2 @ Ys2 )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ ( set_nat2 @ Ys2 ) ) ) ).
% set_mono_prefix
thf(fact_749_set__update__subsetI,axiom,
! [Xs2: list_list_nat,A2: set_list_nat,X: list_nat,I: nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs2 ) @ A2 )
=> ( ( member_list_nat @ X @ A2 )
=> ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ ( list_update_list_nat @ Xs2 @ I @ X ) ) @ A2 ) ) ) ).
% set_update_subsetI
thf(fact_750_set__update__subsetI,axiom,
! [Xs2: list_nat,A2: set_nat,X: nat,I: nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs2 @ I @ X ) ) @ A2 ) ) ) ).
% set_update_subsetI
thf(fact_751_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_752_less__imp__neq,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_753_less__imp__neq,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_754_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_755_order_Oasym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ~ ( ord_less_set_nat @ B @ A ) ) ).
% order.asym
thf(fact_756_order_Oasym,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ~ ( ord_less_num @ B @ A ) ) ).
% order.asym
thf(fact_757_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_758_ord__eq__less__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( A = B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_759_ord__eq__less__trans,axiom,
! [A: num,B: num,C: num] :
( ( A = B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_760_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_761_ord__less__eq__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_762_ord__less__eq__trans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( B = C )
=> ( ord_less_num @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_763_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_764_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X2: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X2 )
=> ( P @ Y5 ) )
=> ( P @ X2 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_765_antisym__conv3,axiom,
! [Y: num,X: num] :
( ~ ( ord_less_num @ Y @ X )
=> ( ( ~ ( ord_less_num @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_766_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_767_linorder__cases,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_num @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_768_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_769_dual__order_Oasym,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ~ ( ord_less_set_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_770_dual__order_Oasym,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ~ ( ord_less_num @ A @ B ) ) ).
% dual_order.asym
thf(fact_771_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_772_dual__order_Oirrefl,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_773_dual__order_Oirrefl,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% dual_order.irrefl
thf(fact_774_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_775_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X7: nat] : ( P2 @ X7 ) )
= ( ^ [P3: nat > $o] :
? [N3: nat] :
( ( P3 @ N3 )
& ! [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ~ ( P3 @ M5 ) ) ) ) ) ).
% exists_least_iff
thf(fact_776_linorder__less__wlog,axiom,
! [P: num > num > $o,A: num,B: num] :
( ! [A4: num,B4: num] :
( ( ord_less_num @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: num] : ( P @ A4 @ A4 )
=> ( ! [A4: num,B4: num] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_777_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_778_order_Ostrict__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_779_order_Ostrict__trans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_780_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_781_not__less__iff__gr__or__eq,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_num @ X @ Y ) )
= ( ( ord_less_num @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_782_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_783_dual__order_Ostrict__trans,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_784_dual__order_Ostrict__trans,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_less_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_785_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_786_order_Ostrict__implies__not__eq,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_787_order_Ostrict__implies__not__eq,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_788_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_789_dual__order_Ostrict__implies__not__eq,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_790_dual__order_Ostrict__implies__not__eq,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_791_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_792_linorder__neqE,axiom,
! [X: num,Y: num] :
( ( X != Y )
=> ( ~ ( ord_less_num @ X @ Y )
=> ( ord_less_num @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_793_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_794_order__less__asym,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ~ ( ord_less_set_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_795_order__less__asym,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ~ ( ord_less_num @ Y @ X ) ) ).
% order_less_asym
thf(fact_796_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_797_linorder__neq__iff,axiom,
! [X: num,Y: num] :
( ( X != Y )
= ( ( ord_less_num @ X @ Y )
| ( ord_less_num @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_798_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_799_order__less__asym_H,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ~ ( ord_less_set_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_800_order__less__asym_H,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ~ ( ord_less_num @ B @ A ) ) ).
% order_less_asym'
thf(fact_801_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_802_order__less__trans,axiom,
! [X: set_nat,Y: set_nat,Z4: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ Y @ Z4 )
=> ( ord_less_set_nat @ X @ Z4 ) ) ) ).
% order_less_trans
thf(fact_803_order__less__trans,axiom,
! [X: num,Y: num,Z4: num] :
( ( ord_less_num @ X @ Y )
=> ( ( ord_less_num @ Y @ Z4 )
=> ( ord_less_num @ X @ Z4 ) ) ) ).
% order_less_trans
thf(fact_804_order__less__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z4 )
=> ( ord_less_nat @ X @ Z4 ) ) ) ).
% order_less_trans
thf(fact_805_ord__eq__less__subst,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_806_ord__eq__less__subst,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_807_ord__eq__less__subst,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_808_ord__eq__less__subst,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X2 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_809_ord__eq__less__subst,axiom,
! [A: num,F: set_nat > num,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X2 @ Y2 )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_810_ord__eq__less__subst,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_811_ord__eq__less__subst,axiom,
! [A: set_nat,F: num > set_nat,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_812_ord__eq__less__subst,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_813_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_814_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_815_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_816_ord__less__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_817_ord__less__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X2 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_818_ord__less__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > num,C: num] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X2 @ Y2 )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_819_ord__less__eq__subst,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_820_ord__less__eq__subst,axiom,
! [A: num,B: num,F: num > set_nat,C: set_nat] :
( ( ord_less_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_821_ord__less__eq__subst,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_822_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_823_order__less__irrefl,axiom,
! [X: set_nat] :
~ ( ord_less_set_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_824_order__less__irrefl,axiom,
! [X: num] :
~ ( ord_less_num @ X @ X ) ).
% order_less_irrefl
thf(fact_825_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_826_order__less__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_827_order__less__subst1,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_828_order__less__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_829_order__less__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X2 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_830_order__less__subst1,axiom,
! [A: set_nat,F: num > set_nat,B: num,C: num] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_831_order__less__subst1,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_832_order__less__subst1,axiom,
! [A: num,F: set_nat > num,B: set_nat,C: set_nat] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X2 @ Y2 )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_833_order__less__subst1,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_834_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_835_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_836_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_837_order__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_838_order__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X2 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_839_order__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > num,C: num] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_set_nat @ X2 @ Y2 )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_840_order__less__subst2,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_841_order__less__subst2,axiom,
! [A: num,B: num,F: num > set_nat,C: set_nat] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_842_order__less__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_num @ X2 @ Y2 )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_843_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_844_order__less__not__sym,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ~ ( ord_less_set_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_845_order__less__not__sym,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ~ ( ord_less_num @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_846_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_847_order__less__imp__triv,axiom,
! [X: set_nat,Y: set_nat,P: $o] :
( ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_848_order__less__imp__triv,axiom,
! [X: num,Y: num,P: $o] :
( ( ord_less_num @ X @ Y )
=> ( ( ord_less_num @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_849_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_850_linorder__less__linear,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
| ( X = Y )
| ( ord_less_num @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_851_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_852_order__less__imp__not__eq,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_853_order__less__imp__not__eq,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_854_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_855_order__less__imp__not__eq2,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_856_order__less__imp__not__eq2,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_857_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_858_order__less__imp__not__less,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ~ ( ord_less_set_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_859_order__less__imp__not__less,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ~ ( ord_less_num @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_860_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_861_leD,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ~ ( ord_less_set_nat @ X @ Y ) ) ).
% leD
thf(fact_862_leD,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ~ ( ord_less_num @ X @ Y ) ) ).
% leD
thf(fact_863_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_864_leI,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ord_less_eq_num @ Y @ X ) ) ).
% leI
thf(fact_865_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_866_nless__le,axiom,
! [A: set_nat,B: set_nat] :
( ( ~ ( ord_less_set_nat @ A @ B ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_867_nless__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_num @ A @ B ) )
= ( ~ ( ord_less_eq_num @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_868_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_869_antisym__conv1,axiom,
! [X: set_nat,Y: set_nat] :
( ~ ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_870_antisym__conv1,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ( ord_less_eq_num @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_871_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_872_antisym__conv2,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ~ ( ord_less_set_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_873_antisym__conv2,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ~ ( ord_less_num @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_874_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_875_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
& ~ ( ord_less_eq_set_nat @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_876_less__le__not__le,axiom,
( ord_less_num
= ( ^ [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
& ~ ( ord_less_eq_num @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_877_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
& ~ ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_878_not__le__imp__less,axiom,
! [Y: num,X: num] :
( ~ ( ord_less_eq_num @ Y @ X )
=> ( ord_less_num @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_879_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_880_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_set_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_881_order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [A3: num,B3: num] :
( ( ord_less_num @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_882_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_883_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_884_order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [A3: num,B3: num] :
( ( ord_less_eq_num @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_885_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_886_order_Ostrict__trans1,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_887_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_888_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_889_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_890_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_891_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_892_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_893_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_894_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_895_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_896_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_897_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_898_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_899_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_900_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_901_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_902_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_903_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_904_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z4 )
=> ( ord_less_nat @ X @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_905_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z4 )
=> ( ord_less_nat @ X @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_906_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_907_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_908_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_909_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_910_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_911_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_912_psubset__imp__ex__mem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ? [B4: nat] : ( member_nat @ B4 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_913_psubsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_914_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I5: nat] :
( ( ord_less_nat @ K2 @ I5 )
=> ( P @ I5 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_915_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ C2 ) )
=> ( P @ X4 ) )
& ! [D3: nat] :
( ! [X2: nat] :
( ( ( ord_less_eq_nat @ A @ X2 )
& ( ord_less_nat @ X2 @ D3 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_nat @ D3 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_916_pinf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ T ) ) ).
% pinf(6)
thf(fact_917_minf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z )
=> ~ ( ord_less_nat @ T @ X4 ) ) ).
% minf(7)
thf(fact_918_minf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z )
=> ( ord_less_nat @ X4 @ T ) ) ).
% minf(5)
thf(fact_919_minf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z )
=> ( X4 != T ) ) ).
% minf(4)
thf(fact_920_minf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z )
=> ( X4 != T ) ) ).
% minf(3)
thf(fact_921_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q3: nat > $o] :
( ? [Z6: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z6 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z6: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z6 )
=> ( ( Q @ X2 )
= ( Q3 @ X2 ) ) )
=> ? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q3 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_922_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q3: nat > $o] :
( ? [Z6: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z6 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z6: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z6 )
=> ( ( Q @ X2 )
= ( Q3 @ X2 ) ) )
=> ? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q3 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_923_pinf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z @ X4 )
=> ( ord_less_nat @ T @ X4 ) ) ).
% pinf(7)
thf(fact_924_pinf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z @ X4 )
=> ~ ( ord_less_nat @ X4 @ T ) ) ).
% pinf(5)
thf(fact_925_pinf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z @ X4 )
=> ( X4 != T ) ) ).
% pinf(4)
thf(fact_926_pinf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z @ X4 )
=> ( X4 != T ) ) ).
% pinf(3)
thf(fact_927_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q3: nat > $o] :
( ? [Z6: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z6 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z6: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z6 @ X2 )
=> ( ( Q @ X2 )
= ( Q3 @ X2 ) ) )
=> ? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q3 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_928_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q3: nat > $o] :
( ? [Z6: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z6 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z6: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z6 @ X2 )
=> ( ( Q @ X2 )
= ( Q3 @ X2 ) ) )
=> ? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q3 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_929_minf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z )
=> ~ ( ord_less_eq_nat @ T @ X4 ) ) ).
% minf(8)
thf(fact_930_minf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z )
=> ( ord_less_eq_nat @ X4 @ T ) ) ).
% minf(6)
thf(fact_931_pinf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z @ X4 )
=> ( ord_less_eq_nat @ T @ X4 ) ) ).
% pinf(8)
thf(fact_932_append__butlast__last__id,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( append_nat @ ( butlast_nat @ Xs2 ) @ ( cons_nat @ ( last_nat @ Xs2 ) @ nil_nat ) )
= Xs2 ) ) ).
% append_butlast_last_id
thf(fact_933_rotate1__length01,axiom,
! [Xs2: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat )
=> ( ( rotate1_nat @ Xs2 )
= Xs2 ) ) ).
% rotate1_length01
thf(fact_934_stirling__row__code_I1_J,axiom,
( ( stirling_row @ zero_zero_nat )
= ( cons_nat @ one_one_nat @ nil_nat ) ) ).
% stirling_row_code(1)
thf(fact_935_rotate1__is__Nil__conv,axiom,
! [Xs2: list_nat] :
( ( ( rotate1_nat @ Xs2 )
= nil_nat )
= ( Xs2 = nil_nat ) ) ).
% rotate1_is_Nil_conv
thf(fact_936_set__rotate1,axiom,
! [Xs2: list_nat] :
( ( set_nat2 @ ( rotate1_nat @ Xs2 ) )
= ( set_nat2 @ Xs2 ) ) ).
% set_rotate1
thf(fact_937_length__rotate1,axiom,
! [Xs2: list_nat] :
( ( size_size_list_nat @ ( rotate1_nat @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_rotate1
thf(fact_938_butlast__snoc,axiom,
! [Xs2: list_nat,X: nat] :
( ( butlast_nat @ ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) ) )
= Xs2 ) ).
% butlast_snoc
thf(fact_939_length__butlast,axiom,
! [Xs2: list_nat] :
( ( size_size_list_nat @ ( butlast_nat @ Xs2 ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) ) ).
% length_butlast
thf(fact_940_stirling__row__nonempty,axiom,
! [N: nat] :
( ( stirling_row @ N )
!= nil_nat ) ).
% stirling_row_nonempty
thf(fact_941_butlast_Osimps_I1_J,axiom,
( ( butlast_nat @ nil_nat )
= nil_nat ) ).
% butlast.simps(1)
thf(fact_942_in__set__butlastD,axiom,
! [X: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ Xs2 ) ) )
=> ( member_nat @ X @ ( set_nat2 @ Xs2 ) ) ) ).
% in_set_butlastD
thf(fact_943_prefixeq__butlast,axiom,
! [Xs2: list_nat] : ( prefix_nat @ ( butlast_nat @ Xs2 ) @ Xs2 ) ).
% prefixeq_butlast
thf(fact_944_rotate1_Osimps_I1_J,axiom,
( ( rotate1_nat @ nil_nat )
= nil_nat ) ).
% rotate1.simps(1)
thf(fact_945_butlast_Osimps_I2_J,axiom,
! [Xs2: list_nat,X: nat] :
( ( ( Xs2 = nil_nat )
=> ( ( butlast_nat @ ( cons_nat @ X @ Xs2 ) )
= nil_nat ) )
& ( ( Xs2 != nil_nat )
=> ( ( butlast_nat @ ( cons_nat @ X @ Xs2 ) )
= ( cons_nat @ X @ ( butlast_nat @ Xs2 ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_946_butlast__append,axiom,
! [Ys2: list_nat,Xs2: list_nat] :
( ( ( Ys2 = nil_nat )
=> ( ( butlast_nat @ ( append_nat @ Xs2 @ Ys2 ) )
= ( butlast_nat @ Xs2 ) ) )
& ( ( Ys2 != nil_nat )
=> ( ( butlast_nat @ ( append_nat @ Xs2 @ Ys2 ) )
= ( append_nat @ Xs2 @ ( butlast_nat @ Ys2 ) ) ) ) ) ).
% butlast_append
thf(fact_947_in__set__butlast__appendI,axiom,
! [X: nat,Xs2: list_nat,Ys2: list_nat] :
( ( ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ Xs2 ) ) )
| ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ Ys2 ) ) ) )
=> ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ ( append_nat @ Xs2 @ Ys2 ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_948_nth__butlast,axiom,
! [N: nat,Xs2: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ ( butlast_nat @ Xs2 ) ) )
=> ( ( nth_nat @ ( butlast_nat @ Xs2 ) @ N )
= ( nth_nat @ Xs2 @ N ) ) ) ).
% nth_butlast
thf(fact_949_rotate1_Osimps_I2_J,axiom,
! [X: nat,Xs2: list_nat] :
( ( rotate1_nat @ ( cons_nat @ X @ Xs2 ) )
= ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) ) ) ).
% rotate1.simps(2)
thf(fact_950_snoc__eq__iff__butlast,axiom,
! [Xs2: list_nat,X: nat,Ys2: list_nat] :
( ( ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) )
= Ys2 )
= ( ( Ys2 != nil_nat )
& ( ( butlast_nat @ Ys2 )
= Xs2 )
& ( ( last_nat @ Ys2 )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_951_butlast__list__update,axiom,
! [K: nat,Xs2: list_nat,X: nat] :
( ( ( K
= ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) )
=> ( ( butlast_nat @ ( list_update_nat @ Xs2 @ K @ X ) )
= ( butlast_nat @ Xs2 ) ) )
& ( ( K
!= ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) )
=> ( ( butlast_nat @ ( list_update_nat @ Xs2 @ K @ X ) )
= ( list_update_nat @ ( butlast_nat @ Xs2 ) @ K @ X ) ) ) ) ).
% butlast_list_update
thf(fact_952_stirling__row__aux_Osimps_I1_J,axiom,
! [N: nat,Y: nat] :
( ( stirling_row_aux_nat @ N @ Y @ nil_nat )
= ( cons_nat @ one_one_nat @ nil_nat ) ) ).
% stirling_row_aux.simps(1)
thf(fact_953_stirling__code,axiom,
( stirling
= ( ^ [N3: nat,K3: nat] : ( if_nat @ ( K3 = zero_zero_nat ) @ ( if_nat @ ( N3 = zero_zero_nat ) @ one_one_nat @ zero_zero_nat ) @ ( if_nat @ ( ord_less_nat @ N3 @ K3 ) @ zero_zero_nat @ ( if_nat @ ( K3 = N3 ) @ one_one_nat @ ( nth_nat @ ( stirling_row @ N3 ) @ K3 ) ) ) ) ) ) ).
% stirling_code
thf(fact_954_nth__stirling__row,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( nth_nat @ ( stirling_row @ N ) @ K )
= ( stirling @ N @ K ) ) ) ).
% nth_stirling_row
thf(fact_955_stirling__same,axiom,
! [N: nat] :
( ( stirling @ N @ N )
= one_one_nat ) ).
% stirling_same
thf(fact_956_stirling__less,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( stirling @ N @ K )
= zero_zero_nat ) ) ).
% stirling_less
thf(fact_957_stirling__0,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( stirling @ N @ zero_zero_nat )
= zero_zero_nat ) ) ).
% stirling_0
thf(fact_958_stirling_Osimps_I1_J,axiom,
( ( stirling @ zero_zero_nat @ zero_zero_nat )
= one_one_nat ) ).
% stirling.simps(1)
thf(fact_959_nth__Cons__numeral,axiom,
! [X: nat,Xs2: list_nat,V: num] :
( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ ( numeral_numeral_nat @ V ) )
= ( nth_nat @ Xs2 @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) ) ) ).
% nth_Cons_numeral
thf(fact_960_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_961_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_962_nat_Oinject,axiom,
! [X23: nat,Y23: nat] :
( ( ( suc @ X23 )
= ( suc @ Y23 ) )
= ( X23 = Y23 ) ) ).
% nat.inject
thf(fact_963_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_964_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_965_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_966_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_967_add__numeral__left,axiom,
! [V: num,W2: num,Z4: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W2 ) @ Z4 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W2 ) ) @ Z4 ) ) ).
% add_numeral_left
thf(fact_968_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_969_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_970_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_971_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_972_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_973_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_974_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_975_max__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( suc @ ( ord_max_nat @ M @ N ) ) ) ).
% max_Suc_Suc
thf(fact_976_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_977_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_978_max__number__of_I1_J,axiom,
! [U: num,V: num] :
( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
=> ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
= ( numeral_numeral_nat @ V ) ) )
& ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
=> ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
= ( numeral_numeral_nat @ U ) ) ) ) ).
% max_number_of(1)
thf(fact_979_max__0__1_I4_J,axiom,
! [X: num] :
( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ zero_zero_nat )
= ( numeral_numeral_nat @ X ) ) ).
% max_0_1(4)
thf(fact_980_max__0__1_I3_J,axiom,
! [X: num] :
( ( ord_max_nat @ zero_zero_nat @ ( numeral_numeral_nat @ X ) )
= ( numeral_numeral_nat @ X ) ) ).
% max_0_1(3)
thf(fact_981_max__0__1_I5_J,axiom,
! [X: num] :
( ( ord_max_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
= ( numeral_numeral_nat @ X ) ) ).
% max_0_1(5)
thf(fact_982_max__0__1_I6_J,axiom,
! [X: num] :
( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat )
= ( numeral_numeral_nat @ X ) ) ).
% max_0_1(6)
thf(fact_983_nth__Cons__Suc,axiom,
! [X: nat,Xs2: list_nat,N: nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ ( suc @ N ) )
= ( nth_nat @ Xs2 @ N ) ) ).
% nth_Cons_Suc
thf(fact_984_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_985_gbinomial__0_I2_J,axiom,
! [K: nat] :
( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K ) )
= zero_zero_nat ) ).
% gbinomial_0(2)
thf(fact_986_gbinomial__0_I1_J,axiom,
! [A: nat] :
( ( gbinomial_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% gbinomial_0(1)
thf(fact_987_length__stirling__row,axiom,
! [N: nat] :
( ( size_size_list_nat @ ( stirling_row @ N ) )
= ( suc @ N ) ) ).
% length_stirling_row
thf(fact_988_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_989_diff__Suc__diff__eq1,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J2 ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_990_diff__Suc__diff__eq2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J2 ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_991_length__suffixes,axiom,
! [Xs2: list_nat] :
( ( size_s3023201423986296836st_nat @ ( suffixes_nat @ Xs2 ) )
= ( suc @ ( size_size_list_nat @ Xs2 ) ) ) ).
% length_suffixes
thf(fact_992_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_993_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_le_numeral
thf(fact_994_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_995_zero__less__numeral,axiom,
! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_less_numeral
thf(fact_996_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% one_le_numeral
thf(fact_997_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_998_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% one_plus_numeral_commute
thf(fact_999_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1000_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1001_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_1002_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1003_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M4: nat] :
( M6
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_1004_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_1005_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1006_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_1007_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1008_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1009_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X2: nat] : ( R @ X2 @ X2 )
=> ( ! [X2: nat,Y2: nat,Z: nat] :
( ( R @ X2 @ Y2 )
=> ( ( R @ Y2 @ Z )
=> ( R @ X2 @ Z ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1010_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_1011_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1012_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1013_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1014_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1015_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1016_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_1017_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_1018_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_1019_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1020_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1021_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ N )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1022_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_1023_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1024_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_1025_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M7: nat] :
( ( M
= ( suc @ M7 ) )
& ( ord_less_nat @ N @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1026_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_1027_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_1028_less__trans__Suc,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1029_less__Suc__induct,axiom,
! [I: nat,J2: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J2 )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J3: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ( ord_less_nat @ J3 @ K2 )
=> ( ( P @ I3 @ J3 )
=> ( ( P @ J3 @ K2 )
=> ( P @ I3 @ K2 ) ) ) ) )
=> ( P @ I @ J2 ) ) ) ) ).
% less_Suc_induct
thf(fact_1030_strict__inc__induct,axiom,
! [I: nat,J2: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J2 )
=> ( ! [I3: nat] :
( ( J2
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_1031_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_1032_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_1033_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1034_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1035_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1036_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1037_diff__induct,axiom,
! [P: nat > nat > $o,M: 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 @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1038_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_1039_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1040_nat_OdiscI,axiom,
! [Nat: nat,X23: nat] :
( ( Nat
= ( suc @ X23 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1041_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1042_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1043_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( zero_zero_nat
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_1044_enum__rgfs_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N2: nat] :
( X
!= ( suc @ N2 ) ) ) ).
% enum_rgfs.cases
thf(fact_1045_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_1046_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1047_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1048_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_1049_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1050_length__Suc__conv,axiom,
! [Xs2: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( suc @ N ) )
= ( ? [Y3: nat,Ys3: list_nat] :
( ( Xs2
= ( cons_nat @ Y3 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_1051_Suc__length__conv,axiom,
! [N: nat,Xs2: list_nat] :
( ( ( suc @ N )
= ( size_size_list_nat @ Xs2 ) )
= ( ? [Y3: nat,Ys3: list_nat] :
( ( Xs2
= ( cons_nat @ Y3 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_1052_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ ( suc @ I2 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1053_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1054_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1055_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_1056_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J: nat] :
( ( M
= ( suc @ J ) )
& ( ord_less_nat @ J @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1057_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1058_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1059_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1060_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_1061_dec__induct,axiom,
! [I: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J2 ) ) ) ) ).
% dec_induct
thf(fact_1062_inc__induct,axiom,
! [I: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( P @ J2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J2 )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1063_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_1064_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_1065_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_1066_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1067_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_1068_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1069_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M5: nat,N3: nat] :
? [K3: nat] :
( N3
= ( suc @ ( plus_plus_nat @ M5 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1070_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1071_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1072_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q4: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q4 ) ) ) ) ).
% less_natE
thf(fact_1073_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1074_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1075_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1076_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1077_prefix__order_Olift__Suc__antimono__le,axiom,
! [F: nat > list_nat,N: nat,N4: nat] :
( ! [N2: nat] : ( prefix_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( prefix_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% prefix_order.lift_Suc_antimono_le
thf(fact_1078_prefix__order_Olift__Suc__mono__le,axiom,
! [F: nat > list_nat,N: nat,N4: nat] :
( ! [N2: nat] : ( prefix_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( prefix_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% prefix_order.lift_Suc_mono_le
thf(fact_1079_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1080_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1081_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1082_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1083_list__update__code_I3_J,axiom,
! [X: nat,Xs2: list_nat,I: nat,Y: nat] :
( ( list_update_nat @ ( cons_nat @ X @ Xs2 ) @ ( suc @ I ) @ Y )
= ( cons_nat @ X @ ( list_update_nat @ Xs2 @ I @ Y ) ) ) ).
% list_update_code(3)
thf(fact_1084_stirling_Osimps_I3_J,axiom,
! [N: nat] :
( ( stirling @ ( suc @ N ) @ zero_zero_nat )
= zero_zero_nat ) ).
% stirling.simps(3)
thf(fact_1085_stirling_Osimps_I2_J,axiom,
! [K: nat] :
( ( stirling @ zero_zero_nat @ ( suc @ K ) )
= zero_zero_nat ) ).
% stirling.simps(2)
thf(fact_1086_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_1087_gen__length__code_I2_J,axiom,
! [N: nat,X: nat,Xs2: list_nat] :
( ( gen_length_nat @ N @ ( cons_nat @ X @ Xs2 ) )
= ( gen_length_nat @ ( suc @ N ) @ Xs2 ) ) ).
% gen_length_code(2)
thf(fact_1088_Suc__le__length__iff,axiom,
! [N: nat,Xs2: list_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs2 ) )
= ( ? [X3: nat,Ys3: list_nat] :
( ( Xs2
= ( cons_nat @ X3 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_1089_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I5: nat] :
( ( ord_less_eq_nat @ I5 @ K2 )
=> ~ ( P @ I5 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1090_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_1091_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1092_list_Osize_I4_J,axiom,
! [X21: nat,X22: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_size_list_nat @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_1093_length__Suc__conv__rev,axiom,
! [Xs2: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( suc @ N ) )
= ( ? [Y3: nat,Ys3: list_nat] :
( ( Xs2
= ( append_nat @ Ys3 @ ( cons_nat @ Y3 @ nil_nat ) ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_1094_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_1095_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1096_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M5: nat,N3: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% add_eq_if
thf(fact_1097_length__append__singleton,axiom,
! [Xs2: list_nat,X: nat] :
( ( size_size_list_nat @ ( append_nat @ Xs2 @ ( cons_nat @ X @ nil_nat ) ) )
= ( suc @ ( size_size_list_nat @ Xs2 ) ) ) ).
% length_append_singleton
thf(fact_1098_length__Cons,axiom,
! [X: nat,Xs2: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X @ Xs2 ) )
= ( suc @ ( size_size_list_nat @ Xs2 ) ) ) ).
% length_Cons
thf(fact_1099_sorted__list__of__set__lessThan__Suc,axiom,
! [K: nat] :
( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K ) ) )
= ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K ) ) @ ( cons_nat @ K @ nil_nat ) ) ) ).
% sorted_list_of_set_lessThan_Suc
thf(fact_1100_nth__sorted__list__of__set__greaterThanLessThan,axiom,
! [N: nat,J2: nat,I: nat] :
( ( ord_less_nat @ N @ ( minus_minus_nat @ J2 @ ( suc @ I ) ) )
=> ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J2 ) ) @ N )
= ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_1101_greaterThanLessThan__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or5834768355832116004an_nat @ L @ U ) )
= ( ( ord_less_nat @ L @ I )
& ( ord_less_nat @ I @ U ) ) ) ).
% greaterThanLessThan_iff
thf(fact_1102_sorted__list__of__set__greaterThanLessThan,axiom,
! [I: nat,J2: nat] :
( ( ord_less_nat @ ( suc @ I ) @ J2 )
=> ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J2 ) )
= ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I ) @ J2 ) ) ) ) ) ).
% sorted_list_of_set_greaterThanLessThan
thf(fact_1103_nth__sorted__list__of__set__greaterThanAtMost,axiom,
! [N: nat,J2: nat,I: nat] :
( ( ord_less_nat @ N @ ( minus_minus_nat @ J2 @ I ) )
=> ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J2 ) ) @ N )
= ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_1104_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_1105_greaterThanAtMost__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or6659071591806873216st_nat @ L @ U ) )
= ( ( ord_less_nat @ L @ I )
& ( ord_less_eq_nat @ I @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_1106_Ioc__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or6659071591806873216st_nat @ A @ B ) @ ( set_or6659071591806873216st_nat @ C @ D ) )
= ( ( ord_less_eq_nat @ B @ A )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% Ioc_subset_iff
thf(fact_1107_Ioc__inj,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( set_or6659071591806873216st_nat @ A @ B )
= ( set_or6659071591806873216st_nat @ C @ D ) )
= ( ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ D @ C ) )
| ( ( A = C )
& ( B = D ) ) ) ) ).
% Ioc_inj
thf(fact_1108_sorted__list__of__set__greaterThanAtMost,axiom,
! [I: nat,J2: nat] :
( ( ord_less_eq_nat @ ( suc @ I ) @ J2 )
=> ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J2 ) )
= ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I ) @ J2 ) ) ) ) ) ).
% sorted_list_of_set_greaterThanAtMost
thf(fact_1109_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_1110_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_1111_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_1112_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_1113_sorted__list__of__set__atMost__Suc,axiom,
! [K: nat] :
( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K ) ) )
= ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K ) ) @ ( cons_nat @ ( suc @ K ) @ nil_nat ) ) ) ).
% sorted_list_of_set_atMost_Suc
thf(fact_1114_atMost__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_atMost_nat @ X )
= ( set_ord_atMost_nat @ Y ) )
= ( X = Y ) ) ).
% atMost_eq_iff
thf(fact_1115_atMost__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_atMost_nat @ K ) )
= ( ord_less_eq_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_1116_atMost__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X ) @ ( set_ord_atMost_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% atMost_subset_iff
thf(fact_1117_lessThan__Suc__atMost,axiom,
! [K: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K ) )
= ( set_ord_atMost_nat @ K ) ) ).
% lessThan_Suc_atMost
thf(fact_1118_Iic__subset__Iio__iff,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ A ) @ ( set_ord_lessThan_nat @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% Iic_subset_Iio_iff
thf(fact_1119_take__Suc__conv__app__nth,axiom,
! [I: nat,Xs2: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( take_nat @ ( suc @ I ) @ Xs2 )
= ( append_nat @ ( take_nat @ I @ Xs2 ) @ ( cons_nat @ ( nth_nat @ Xs2 @ I ) @ nil_nat ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_1120_upt__rec__numeral,axiom,
! [M: num,N: num] :
( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
=> ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
& ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
=> ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= nil_nat ) ) ) ).
% upt_rec_numeral
thf(fact_1121_take__Suc__Cons,axiom,
! [N: nat,X: nat,Xs2: list_nat] :
( ( take_nat @ ( suc @ N ) @ ( cons_nat @ X @ Xs2 ) )
= ( cons_nat @ X @ ( take_nat @ N @ Xs2 ) ) ) ).
% take_Suc_Cons
thf(fact_1122_take0,axiom,
( ( take_nat @ zero_zero_nat )
= ( ^ [Xs4: list_nat] : nil_nat ) ) ).
% take0
thf(fact_1123_take__eq__Nil,axiom,
! [N: nat,Xs2: list_nat] :
( ( ( take_nat @ N @ Xs2 )
= nil_nat )
= ( ( N = zero_zero_nat )
| ( Xs2 = nil_nat ) ) ) ).
% take_eq_Nil
thf(fact_1124_take__eq__Nil2,axiom,
! [N: nat,Xs2: list_nat] :
( ( nil_nat
= ( take_nat @ N @ Xs2 ) )
= ( ( N = zero_zero_nat )
| ( Xs2 = nil_nat ) ) ) ).
% take_eq_Nil2
thf(fact_1125_take__all,axiom,
! [Xs2: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N )
=> ( ( take_nat @ N @ Xs2 )
= Xs2 ) ) ).
% take_all
thf(fact_1126_take__all__iff,axiom,
! [N: nat,Xs2: list_nat] :
( ( ( take_nat @ N @ Xs2 )
= Xs2 )
= ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N ) ) ).
% take_all_iff
thf(fact_1127_take__upt,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M ) @ N )
=> ( ( take_nat @ M @ ( upt @ I @ N ) )
= ( upt @ I @ ( plus_plus_nat @ I @ M ) ) ) ) ).
% take_upt
thf(fact_1128_nth__take,axiom,
! [I: nat,N: nat,Xs2: list_nat] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_nat @ ( take_nat @ N @ Xs2 ) @ I )
= ( nth_nat @ Xs2 @ I ) ) ) ).
% nth_take
thf(fact_1129_take__update__cancel,axiom,
! [N: nat,M: nat,Xs2: list_nat,Y: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( take_nat @ N @ ( list_update_nat @ Xs2 @ M @ Y ) )
= ( take_nat @ N @ Xs2 ) ) ) ).
% take_update_cancel
thf(fact_1130_upt__conv__Nil,axiom,
! [J2: nat,I: nat] :
( ( ord_less_eq_nat @ J2 @ I )
=> ( ( upt @ I @ J2 )
= nil_nat ) ) ).
% upt_conv_Nil
thf(fact_1131_length__upt,axiom,
! [I: nat,J2: nat] :
( ( size_size_list_nat @ ( upt @ I @ J2 ) )
= ( minus_minus_nat @ J2 @ I ) ) ).
% length_upt
thf(fact_1132_take__append,axiom,
! [N: nat,Xs2: list_nat,Ys2: list_nat] :
( ( take_nat @ N @ ( append_nat @ Xs2 @ Ys2 ) )
= ( append_nat @ ( take_nat @ N @ Xs2 ) @ ( take_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs2 ) ) @ Ys2 ) ) ) ).
% take_append
thf(fact_1133_last__upt,axiom,
! [I: nat,J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( last_nat @ ( upt @ I @ J2 ) )
= ( minus_minus_nat @ J2 @ one_one_nat ) ) ) ).
% last_upt
thf(fact_1134_upt__eq__Nil__conv,axiom,
! [I: nat,J2: nat] :
( ( ( upt @ I @ J2 )
= nil_nat )
= ( ( J2 = zero_zero_nat )
| ( ord_less_eq_nat @ J2 @ I ) ) ) ).
% upt_eq_Nil_conv
thf(fact_1135_nth__upt,axiom,
! [I: nat,K: nat,J2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J2 )
=> ( ( nth_nat @ ( upt @ I @ J2 ) @ K )
= ( plus_plus_nat @ I @ K ) ) ) ).
% nth_upt
thf(fact_1136_take__Cons__numeral,axiom,
! [V: num,X: nat,Xs2: list_nat] :
( ( take_nat @ ( numeral_numeral_nat @ V ) @ ( cons_nat @ X @ Xs2 ) )
= ( cons_nat @ X @ ( take_nat @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) @ Xs2 ) ) ) ).
% take_Cons_numeral
thf(fact_1137_set__take__subset,axiom,
! [N: nat,Xs2: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( take_nat @ N @ Xs2 ) ) @ ( set_nat2 @ Xs2 ) ) ).
% set_take_subset
thf(fact_1138_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ zero_zero_nat )
= nil_nat ) ).
% upt_0
thf(fact_1139_upt__conv__Cons__Cons,axiom,
! [M: nat,N: nat,Ns: list_nat,Q2: nat] :
( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
= ( upt @ M @ Q2 ) )
= ( ( cons_nat @ N @ Ns )
= ( upt @ ( suc @ M ) @ Q2 ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_1140_take__equalityI,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ! [I3: nat] :
( ( take_nat @ I3 @ Xs2 )
= ( take_nat @ I3 @ Ys2 ) )
=> ( Xs2 = Ys2 ) ) ).
% take_equalityI
thf(fact_1141_take__update__swap,axiom,
! [M: nat,Xs2: list_nat,N: nat,X: nat] :
( ( take_nat @ M @ ( list_update_nat @ Xs2 @ N @ X ) )
= ( list_update_nat @ ( take_nat @ M @ Xs2 ) @ N @ X ) ) ).
% take_update_swap
thf(fact_1142_take__is__prefix,axiom,
! [N: nat,Xs2: list_nat] : ( prefix_nat @ ( take_nat @ N @ Xs2 ) @ Xs2 ) ).
% take_is_prefix
thf(fact_1143_in__set__takeD,axiom,
! [X: nat,N: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ ( take_nat @ N @ Xs2 ) ) )
=> ( member_nat @ X @ ( set_nat2 @ Xs2 ) ) ) ).
% in_set_takeD
thf(fact_1144_take__Nil,axiom,
! [N: nat] :
( ( take_nat @ N @ nil_nat )
= nil_nat ) ).
% take_Nil
thf(fact_1145_take__0,axiom,
! [Xs2: list_nat] :
( ( take_nat @ zero_zero_nat @ Xs2 )
= nil_nat ) ).
% take_0
thf(fact_1146_set__take__subset__set__take,axiom,
! [M: nat,N: nat,Xs2: list_nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( take_nat @ M @ Xs2 ) ) @ ( set_nat2 @ ( take_nat @ N @ Xs2 ) ) ) ) ).
% set_take_subset_set_take
thf(fact_1147_take__butlast,axiom,
! [N: nat,Xs2: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( take_nat @ N @ ( butlast_nat @ Xs2 ) )
= ( take_nat @ N @ Xs2 ) ) ) ).
% take_butlast
thf(fact_1148_upt__conv__Cons,axiom,
! [I: nat,J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( upt @ I @ J2 )
= ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J2 ) ) ) ) ).
% upt_conv_Cons
thf(fact_1149_upt__add__eq__append,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( upt @ I @ ( plus_plus_nat @ J2 @ K ) )
= ( append_nat @ ( upt @ I @ J2 ) @ ( upt @ J2 @ ( plus_plus_nat @ J2 @ K ) ) ) ) ) ).
% upt_add_eq_append
thf(fact_1150_atLeast__upt,axiom,
( set_ord_lessThan_nat
= ( ^ [N3: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N3 ) ) ) ) ).
% atLeast_upt
thf(fact_1151_greaterThanAtMost__upt,axiom,
( set_or6659071591806873216st_nat
= ( ^ [N3: nat,M5: nat] : ( set_nat2 @ ( upt @ ( suc @ N3 ) @ ( suc @ M5 ) ) ) ) ) ).
% greaterThanAtMost_upt
thf(fact_1152_greaterThanLessThan__upt,axiom,
( set_or5834768355832116004an_nat
= ( ^ [N3: nat,M5: nat] : ( set_nat2 @ ( upt @ ( suc @ N3 ) @ M5 ) ) ) ) ).
% greaterThanLessThan_upt
thf(fact_1153_nth__take__lemma,axiom,
! [K: nat,Xs2: list_nat,Ys2: list_nat] :
( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Ys2 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( ( nth_nat @ Xs2 @ I3 )
= ( nth_nat @ Ys2 @ I3 ) ) )
=> ( ( take_nat @ K @ Xs2 )
= ( take_nat @ K @ Ys2 ) ) ) ) ) ).
% nth_take_lemma
thf(fact_1154_butlast__conv__take,axiom,
( butlast_nat
= ( ^ [Xs4: list_nat] : ( take_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs4 ) @ one_one_nat ) @ Xs4 ) ) ) ).
% butlast_conv_take
thf(fact_1155_upt__eq__Cons__conv,axiom,
! [I: nat,J2: nat,X: nat,Xs2: list_nat] :
( ( ( upt @ I @ J2 )
= ( cons_nat @ X @ Xs2 ) )
= ( ( ord_less_nat @ I @ J2 )
& ( I = X )
& ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J2 )
= Xs2 ) ) ) ).
% upt_eq_Cons_conv
thf(fact_1156_atMost__upto,axiom,
( set_ord_atMost_nat
= ( ^ [N3: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N3 ) ) ) ) ) ).
% atMost_upto
thf(fact_1157_upt__rec,axiom,
( upt
= ( ^ [I2: nat,J: nat] : ( if_list_nat @ ( ord_less_nat @ I2 @ J ) @ ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J ) ) @ nil_nat ) ) ) ).
% upt_rec
thf(fact_1158_take__Cons_H,axiom,
! [N: nat,X: nat,Xs2: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( take_nat @ N @ ( cons_nat @ X @ Xs2 ) )
= nil_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( take_nat @ N @ ( cons_nat @ X @ Xs2 ) )
= ( cons_nat @ X @ ( take_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs2 ) ) ) ) ) ).
% take_Cons'
thf(fact_1159_butlast__take,axiom,
! [N: nat,Xs2: list_nat] :
( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( butlast_nat @ ( take_nat @ N @ Xs2 ) )
= ( take_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs2 ) ) ) ).
% butlast_take
thf(fact_1160_upt__Suc__append,axiom,
! [I: nat,J2: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( upt @ I @ ( suc @ J2 ) )
= ( append_nat @ ( upt @ I @ J2 ) @ ( cons_nat @ J2 @ nil_nat ) ) ) ) ).
% upt_Suc_append
thf(fact_1161_upt__Suc,axiom,
! [I: nat,J2: nat] :
( ( ( ord_less_eq_nat @ I @ J2 )
=> ( ( upt @ I @ ( suc @ J2 ) )
= ( append_nat @ ( upt @ I @ J2 ) @ ( cons_nat @ J2 @ nil_nat ) ) ) )
& ( ~ ( ord_less_eq_nat @ I @ J2 )
=> ( ( upt @ I @ ( suc @ J2 ) )
= nil_nat ) ) ) ).
% upt_Suc
thf(fact_1162_upd__conv__take__nth__drop,axiom,
! [I: nat,Xs2: list_nat,A: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( list_update_nat @ Xs2 @ I @ A )
= ( append_nat @ ( take_nat @ I @ Xs2 ) @ ( cons_nat @ A @ ( drop_nat @ ( suc @ I ) @ Xs2 ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_1163_id__take__nth__drop,axiom,
! [I: nat,Xs2: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( Xs2
= ( append_nat @ ( take_nat @ I @ Xs2 ) @ ( cons_nat @ ( nth_nat @ Xs2 @ I ) @ ( drop_nat @ ( suc @ I ) @ Xs2 ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_1164_drop0,axiom,
( ( drop_nat @ zero_zero_nat )
= ( ^ [X3: list_nat] : X3 ) ) ).
% drop0
thf(fact_1165_drop__drop,axiom,
! [N: nat,M: nat,Xs2: list_nat] :
( ( drop_nat @ N @ ( drop_nat @ M @ Xs2 ) )
= ( drop_nat @ ( plus_plus_nat @ N @ M ) @ Xs2 ) ) ).
% drop_drop
thf(fact_1166_drop__upt,axiom,
! [M: nat,I: nat,J2: nat] :
( ( drop_nat @ M @ ( upt @ I @ J2 ) )
= ( upt @ ( plus_plus_nat @ I @ M ) @ J2 ) ) ).
% drop_upt
thf(fact_1167_drop__Suc__Cons,axiom,
! [N: nat,X: nat,Xs2: list_nat] :
( ( drop_nat @ ( suc @ N ) @ ( cons_nat @ X @ Xs2 ) )
= ( drop_nat @ N @ Xs2 ) ) ).
% drop_Suc_Cons
thf(fact_1168_length__drop,axiom,
! [N: nat,Xs2: list_nat] :
( ( size_size_list_nat @ ( drop_nat @ N @ Xs2 ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ N ) ) ).
% length_drop
thf(fact_1169_append__take__drop__id,axiom,
! [N: nat,Xs2: list_nat] :
( ( append_nat @ ( take_nat @ N @ Xs2 ) @ ( drop_nat @ N @ Xs2 ) )
= Xs2 ) ).
% append_take_drop_id
thf(fact_1170_drop__update__cancel,axiom,
! [N: nat,M: nat,Xs2: list_nat,X: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( drop_nat @ M @ ( list_update_nat @ Xs2 @ N @ X ) )
= ( drop_nat @ M @ Xs2 ) ) ) ).
% drop_update_cancel
thf(fact_1171_drop__all,axiom,
! [Xs2: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N )
=> ( ( drop_nat @ N @ Xs2 )
= nil_nat ) ) ).
% drop_all
thf(fact_1172_drop__eq__Nil,axiom,
! [N: nat,Xs2: list_nat] :
( ( ( drop_nat @ N @ Xs2 )
= nil_nat )
= ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N ) ) ).
% drop_eq_Nil
thf(fact_1173_drop__eq__Nil2,axiom,
! [N: nat,Xs2: list_nat] :
( ( nil_nat
= ( drop_nat @ N @ Xs2 ) )
= ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_1174_drop__append,axiom,
! [N: nat,Xs2: list_nat,Ys2: list_nat] :
( ( drop_nat @ N @ ( append_nat @ Xs2 @ Ys2 ) )
= ( append_nat @ ( drop_nat @ N @ Xs2 ) @ ( drop_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs2 ) ) @ Ys2 ) ) ) ).
% drop_append
thf(fact_1175_last__drop,axiom,
! [N: nat,Xs2: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( last_nat @ ( drop_nat @ N @ Xs2 ) )
= ( last_nat @ Xs2 ) ) ) ).
% last_drop
thf(fact_1176_drop__Cons__numeral,axiom,
! [V: num,X: nat,Xs2: list_nat] :
( ( drop_nat @ ( numeral_numeral_nat @ V ) @ ( cons_nat @ X @ Xs2 ) )
= ( drop_nat @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) @ Xs2 ) ) ).
% drop_Cons_numeral
thf(fact_1177_nth__drop,axiom,
! [N: nat,Xs2: list_nat,I: nat] :
( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ ( drop_nat @ N @ Xs2 ) @ I )
= ( nth_nat @ Xs2 @ ( plus_plus_nat @ N @ I ) ) ) ) ).
% nth_drop
thf(fact_1178_take__drop,axiom,
! [N: nat,M: nat,Xs2: list_nat] :
( ( take_nat @ N @ ( drop_nat @ M @ Xs2 ) )
= ( drop_nat @ M @ ( take_nat @ ( plus_plus_nat @ N @ M ) @ Xs2 ) ) ) ).
% take_drop
thf(fact_1179_drop__take,axiom,
! [N: nat,M: nat,Xs2: list_nat] :
( ( drop_nat @ N @ ( take_nat @ M @ Xs2 ) )
= ( take_nat @ ( minus_minus_nat @ M @ N ) @ ( drop_nat @ N @ Xs2 ) ) ) ).
% drop_take
thf(fact_1180_drop__butlast,axiom,
! [N: nat,Xs2: list_nat] :
( ( drop_nat @ N @ ( butlast_nat @ Xs2 ) )
= ( butlast_nat @ ( drop_nat @ N @ Xs2 ) ) ) ).
% drop_butlast
thf(fact_1181_drop__Nil,axiom,
! [N: nat] :
( ( drop_nat @ N @ nil_nat )
= nil_nat ) ).
% drop_Nil
thf(fact_1182_drop__0,axiom,
! [Xs2: list_nat] :
( ( drop_nat @ zero_zero_nat @ Xs2 )
= Xs2 ) ).
% drop_0
thf(fact_1183_in__set__dropD,axiom,
! [X: nat,N: nat,Xs2: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ ( drop_nat @ N @ Xs2 ) ) )
=> ( member_nat @ X @ ( set_nat2 @ Xs2 ) ) ) ).
% in_set_dropD
thf(fact_1184_nth__via__drop,axiom,
! [N: nat,Xs2: list_nat,Y: nat,Ys2: list_nat] :
( ( ( drop_nat @ N @ Xs2 )
= ( cons_nat @ Y @ Ys2 ) )
=> ( ( nth_nat @ Xs2 @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_1185_set__drop__subset,axiom,
! [N: nat,Xs2: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( drop_nat @ N @ Xs2 ) ) @ ( set_nat2 @ Xs2 ) ) ).
% set_drop_subset
thf(fact_1186_set__drop__subset__set__drop,axiom,
! [N: nat,M: nat,Xs2: list_nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( drop_nat @ M @ Xs2 ) ) @ ( set_nat2 @ ( drop_nat @ N @ Xs2 ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_1187_append__eq__conv__conj,axiom,
! [Xs2: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( ( append_nat @ Xs2 @ Ys2 )
= Zs )
= ( ( Xs2
= ( take_nat @ ( size_size_list_nat @ Xs2 ) @ Zs ) )
& ( Ys2
= ( drop_nat @ ( size_size_list_nat @ Xs2 ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_1188_take__add,axiom,
! [I: nat,J2: nat,Xs2: list_nat] :
( ( take_nat @ ( plus_plus_nat @ I @ J2 ) @ Xs2 )
= ( append_nat @ ( take_nat @ I @ Xs2 ) @ ( take_nat @ J2 @ ( drop_nat @ I @ Xs2 ) ) ) ) ).
% take_add
thf(fact_1189_drop__update__swap,axiom,
! [M: nat,N: nat,Xs2: list_nat,X: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( drop_nat @ M @ ( list_update_nat @ Xs2 @ N @ X ) )
= ( list_update_nat @ ( drop_nat @ M @ Xs2 ) @ ( minus_minus_nat @ N @ M ) @ X ) ) ) ).
% drop_update_swap
thf(fact_1190_drop__Cons_H,axiom,
! [N: nat,X: nat,Xs2: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( drop_nat @ N @ ( cons_nat @ X @ Xs2 ) )
= ( cons_nat @ X @ Xs2 ) ) )
& ( ( N != zero_zero_nat )
=> ( ( drop_nat @ N @ ( cons_nat @ X @ Xs2 ) )
= ( drop_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs2 ) ) ) ) ).
% drop_Cons'
thf(fact_1191_append__eq__append__conv__if,axiom,
! [Xs_1: list_nat,Xs_2: list_nat,Ys_1: list_nat,Ys_2: list_nat] :
( ( ( append_nat @ Xs_1 @ Xs_2 )
= ( append_nat @ Ys_1 @ Ys_2 ) )
= ( ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs_1 ) @ ( size_size_list_nat @ Ys_1 ) )
=> ( ( Xs_1
= ( take_nat @ ( size_size_list_nat @ Xs_1 ) @ Ys_1 ) )
& ( Xs_2
= ( append_nat @ ( drop_nat @ ( size_size_list_nat @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ ( size_size_list_nat @ Xs_1 ) @ ( size_size_list_nat @ Ys_1 ) )
=> ( ( ( take_nat @ ( size_size_list_nat @ Ys_1 ) @ Xs_1 )
= Ys_1 )
& ( ( append_nat @ ( drop_nat @ ( size_size_list_nat @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
= Ys_2 ) ) ) ) ) ).
% append_eq_append_conv_if
thf(fact_1192_Cons__nth__drop__Suc,axiom,
! [I: nat,Xs2: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( cons_nat @ ( nth_nat @ Xs2 @ I ) @ ( drop_nat @ ( suc @ I ) @ Xs2 ) )
= ( drop_nat @ I @ Xs2 ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_1193_take__hd__drop,axiom,
! [N: nat,Xs2: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( append_nat @ ( take_nat @ N @ Xs2 ) @ ( cons_nat @ ( hd_nat @ ( drop_nat @ N @ Xs2 ) ) @ nil_nat ) )
= ( take_nat @ ( suc @ N ) @ Xs2 ) ) ) ).
% take_hd_drop
thf(fact_1194_map__upt__eqI,axiom,
! [Xs2: list_nat,N: nat,M: nat,F: nat > nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( minus_minus_nat @ N @ M ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ Xs2 @ I3 )
= ( F @ ( plus_plus_nat @ M @ I3 ) ) ) )
=> ( ( map_nat_nat @ F @ ( upt @ M @ N ) )
= Xs2 ) ) ) ).
% map_upt_eqI
thf(fact_1195_list_Omap__disc__iff,axiom,
! [F: nat > nat,A: list_nat] :
( ( ( map_nat_nat @ F @ A )
= nil_nat )
= ( A = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_1196_Nil__is__map__conv,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( nil_nat
= ( map_nat_nat @ F @ Xs2 ) )
= ( Xs2 = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_1197_map__is__Nil__conv,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= nil_nat )
= ( Xs2 = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_1198_map__eq__conv,axiom,
! [F: nat > nat,Xs2: list_nat,G: nat > nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Xs2 ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) ) ) ) ).
% map_eq_conv
thf(fact_1199_length__map,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( size_size_list_nat @ ( map_nat_nat @ F @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_map
thf(fact_1200_map__append,axiom,
! [F: nat > nat,Xs2: list_nat,Ys2: list_nat] :
( ( map_nat_nat @ F @ ( append_nat @ Xs2 @ Ys2 ) )
= ( append_nat @ ( map_nat_nat @ F @ Xs2 ) @ ( map_nat_nat @ F @ Ys2 ) ) ) ).
% map_append
thf(fact_1201_hd__upt,axiom,
! [I: nat,J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( hd_nat @ ( upt @ I @ J2 ) )
= I ) ) ).
% hd_upt
thf(fact_1202_hd__prefixes,axiom,
! [Xs2: list_nat] :
( ( hd_list_nat @ ( prefixes_nat @ Xs2 ) )
= nil_nat ) ).
% hd_prefixes
thf(fact_1203_hd__suffixes,axiom,
! [Xs2: list_nat] :
( ( hd_list_nat @ ( suffixes_nat @ Xs2 ) )
= nil_nat ) ).
% hd_suffixes
thf(fact_1204_hd__append2,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( hd_nat @ ( append_nat @ Xs2 @ Ys2 ) )
= ( hd_nat @ Xs2 ) ) ) ).
% hd_append2
thf(fact_1205_nth__map,axiom,
! [N: nat,Xs2: list_nat,F: nat > nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ ( map_nat_nat @ F @ Xs2 ) @ N )
= ( F @ ( nth_nat @ Xs2 @ N ) ) ) ) ).
% nth_map
thf(fact_1206_hd__take,axiom,
! [J2: nat,Xs2: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ J2 )
=> ( ( hd_nat @ ( take_nat @ J2 @ Xs2 ) )
= ( hd_nat @ Xs2 ) ) ) ).
% hd_take
thf(fact_1207_take__map,axiom,
! [N: nat,F: nat > nat,Xs2: list_nat] :
( ( take_nat @ N @ ( map_nat_nat @ F @ Xs2 ) )
= ( map_nat_nat @ F @ ( take_nat @ N @ Xs2 ) ) ) ).
% take_map
thf(fact_1208_drop__map,axiom,
! [N: nat,F: nat > nat,Xs2: list_nat] :
( ( drop_nat @ N @ ( map_nat_nat @ F @ Xs2 ) )
= ( map_nat_nat @ F @ ( drop_nat @ N @ Xs2 ) ) ) ).
% drop_map
thf(fact_1209_last__map,axiom,
! [Xs2: list_nat,F: nat > nat] :
( ( Xs2 != nil_nat )
=> ( ( last_nat @ ( map_nat_nat @ F @ Xs2 ) )
= ( F @ ( last_nat @ Xs2 ) ) ) ) ).
% last_map
thf(fact_1210_hd__Nil__eq__last,axiom,
( ( hd_nat @ nil_nat )
= ( last_nat @ nil_nat ) ) ).
% hd_Nil_eq_last
thf(fact_1211_list_Oset__sel_I1_J,axiom,
! [A: list_nat] :
( ( A != nil_nat )
=> ( member_nat @ ( hd_nat @ A ) @ ( set_nat2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_1212_hd__in__set,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( member_nat @ ( hd_nat @ Xs2 ) @ ( set_nat2 @ Xs2 ) ) ) ).
% hd_in_set
thf(fact_1213_hd__append,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
( ( ( Xs2 = nil_nat )
=> ( ( hd_nat @ ( append_nat @ Xs2 @ Ys2 ) )
= ( hd_nat @ Ys2 ) ) )
& ( ( Xs2 != nil_nat )
=> ( ( hd_nat @ ( append_nat @ Xs2 @ Ys2 ) )
= ( hd_nat @ Xs2 ) ) ) ) ).
% hd_append
thf(fact_1214_longest__common__prefix,axiom,
! [Xs2: list_nat,Ys2: list_nat] :
? [Ps2: list_nat,Xs5: list_nat,Ys7: list_nat] :
( ( Xs2
= ( append_nat @ Ps2 @ Xs5 ) )
& ( Ys2
= ( append_nat @ Ps2 @ Ys7 ) )
& ( ( Xs5 = nil_nat )
| ( Ys7 = nil_nat )
| ( ( hd_nat @ Xs5 )
!= ( hd_nat @ Ys7 ) ) ) ) ).
% longest_common_prefix
thf(fact_1215_hd__concat,axiom,
! [Xs2: list_list_nat] :
( ( Xs2 != nil_list_nat )
=> ( ( ( hd_list_nat @ Xs2 )
!= nil_nat )
=> ( ( hd_nat @ ( concat_nat @ Xs2 ) )
= ( hd_nat @ ( hd_list_nat @ Xs2 ) ) ) ) ) ).
% hd_concat
thf(fact_1216_hd__map,axiom,
! [Xs2: list_nat,F: nat > nat] :
( ( Xs2 != nil_nat )
=> ( ( hd_nat @ ( map_nat_nat @ F @ Xs2 ) )
= ( F @ ( hd_nat @ Xs2 ) ) ) ) ).
% hd_map
thf(fact_1217_list_Omap__sel_I1_J,axiom,
! [A: list_nat,F: nat > nat] :
( ( A != nil_nat )
=> ( ( hd_nat @ ( map_nat_nat @ F @ A ) )
= ( F @ ( hd_nat @ A ) ) ) ) ).
% list.map_sel(1)
thf(fact_1218_map__update,axiom,
! [F: nat > nat,Xs2: list_nat,K: nat,Y: nat] :
( ( map_nat_nat @ F @ ( list_update_nat @ Xs2 @ K @ Y ) )
= ( list_update_nat @ ( map_nat_nat @ F @ Xs2 ) @ K @ ( F @ Y ) ) ) ).
% map_update
thf(fact_1219_list_Osel_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( ( hd_nat @ ( cons_nat @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_1220_prefix__map__rightE,axiom,
! [Xs2: list_nat,F: nat > nat,Ys2: list_nat] :
( ( prefix_nat @ Xs2 @ ( map_nat_nat @ F @ Ys2 ) )
=> ? [Xs5: list_nat] :
( ( prefix_nat @ Xs5 @ Ys2 )
& ( Xs2
= ( map_nat_nat @ F @ Xs5 ) ) ) ) ).
% prefix_map_rightE
thf(fact_1221_map__mono__prefix,axiom,
! [Xs2: list_nat,Ys2: list_nat,F: nat > nat] :
( ( prefix_nat @ Xs2 @ Ys2 )
=> ( prefix_nat @ ( map_nat_nat @ F @ Xs2 ) @ ( map_nat_nat @ F @ Ys2 ) ) ) ).
% map_mono_prefix
thf(fact_1222_append__eq__map__conv,axiom,
! [Ys2: list_nat,Zs: list_nat,F: nat > nat,Xs2: list_nat] :
( ( ( append_nat @ Ys2 @ Zs )
= ( map_nat_nat @ F @ Xs2 ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs2
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys2
= ( map_nat_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_nat @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_1223_map__eq__append__conv,axiom,
! [F: nat > nat,Xs2: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( append_nat @ Ys2 @ Zs ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs2
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys2
= ( map_nat_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_nat @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_1224_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs2: list_nat,G: nat > nat,Ys2: list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Ys2 ) )
=> ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1225_list_Omap__cong,axiom,
! [X: list_nat,Ya: list_nat,F: nat > nat,G: nat > nat] :
( ( X = Ya )
=> ( ! [Z: nat] :
( ( member_nat @ Z @ ( set_nat2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_nat_nat @ F @ X )
= ( map_nat_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_1226_list_Omap__cong0,axiom,
! [X: list_nat,F: nat > nat,G: nat > nat] :
( ! [Z: nat] :
( ( member_nat @ Z @ ( set_nat2 @ X ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_nat_nat @ F @ X )
= ( map_nat_nat @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_1227_list_Oinj__map__strong,axiom,
! [X: list_nat,Xa2: list_nat,F: nat > nat,Fa: nat > nat] :
( ! [Z: nat,Za: nat] :
( ( member_nat @ Z @ ( set_nat2 @ X ) )
=> ( ( member_nat @ Za @ ( set_nat2 @ Xa2 ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_nat_nat @ F @ X )
= ( map_nat_nat @ Fa @ Xa2 ) )
=> ( X = Xa2 ) ) ) ).
% list.inj_map_strong
thf(fact_1228_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_1229_map__ext,axiom,
! [Xs2: list_nat,F: nat > nat,G: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_nat_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Xs2 ) ) ) ).
% map_ext
thf(fact_1230_map__idI,axiom,
! [Xs2: list_nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_nat_nat @ F @ Xs2 )
= Xs2 ) ) ).
% map_idI
thf(fact_1231_map__cong,axiom,
! [Xs2: list_nat,Ys2: list_nat,F: nat > nat,G: nat > nat] :
( ( Xs2 = Ys2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Ys2 ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_nat_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Ys2 ) ) ) ) ).
% map_cong
thf(fact_1232_ex__map__conv,axiom,
! [Ys2: list_nat,F: nat > nat] :
( ( ? [Xs4: list_nat] :
( Ys2
= ( map_nat_nat @ F @ Xs4 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Ys2 ) )
=> ? [Y3: nat] :
( X3
= ( F @ Y3 ) ) ) ) ) ).
% ex_map_conv
thf(fact_1233_list_Osimps_I8_J,axiom,
! [F: nat > nat] :
( ( map_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_1234_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_1235_Cons__eq__map__D,axiom,
! [X: nat,Xs2: list_nat,F: nat > nat,Ys2: list_nat] :
( ( ( cons_nat @ X @ Xs2 )
= ( map_nat_nat @ F @ Ys2 ) )
=> ? [Z: nat,Zs2: list_nat] :
( ( Ys2
= ( cons_nat @ Z @ Zs2 ) )
& ( X
= ( F @ Z ) )
& ( Xs2
= ( map_nat_nat @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1236_map__eq__Cons__D,axiom,
! [F: nat > nat,Xs2: list_nat,Y: nat,Ys2: list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( cons_nat @ Y @ Ys2 ) )
=> ? [Z: nat,Zs2: list_nat] :
( ( Xs2
= ( cons_nat @ Z @ Zs2 ) )
& ( ( F @ Z )
= Y )
& ( ( map_nat_nat @ F @ Zs2 )
= Ys2 ) ) ) ).
% map_eq_Cons_D
thf(fact_1237_Cons__eq__map__conv,axiom,
! [X: nat,Xs2: list_nat,F: nat > nat,Ys2: list_nat] :
( ( ( cons_nat @ X @ Xs2 )
= ( map_nat_nat @ F @ Ys2 ) )
= ( ? [Z5: nat,Zs3: list_nat] :
( ( Ys2
= ( cons_nat @ Z5 @ Zs3 ) )
& ( X
= ( F @ Z5 ) )
& ( Xs2
= ( map_nat_nat @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1238_map__eq__Cons__conv,axiom,
! [F: nat > nat,Xs2: list_nat,Y: nat,Ys2: list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( cons_nat @ Y @ Ys2 ) )
= ( ? [Z5: nat,Zs3: list_nat] :
( ( Xs2
= ( cons_nat @ Z5 @ Zs3 ) )
& ( ( F @ Z5 )
= Y )
& ( ( map_nat_nat @ F @ Zs3 )
= Ys2 ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1239_map__concat,axiom,
! [F: nat > nat,Xs2: list_list_nat] :
( ( map_nat_nat @ F @ ( concat_nat @ Xs2 ) )
= ( concat_nat @ ( map_li7225945977422193158st_nat @ ( map_nat_nat @ F ) @ Xs2 ) ) ) ).
% map_concat
thf(fact_1240_rotate1__map,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( rotate1_nat @ ( map_nat_nat @ F @ Xs2 ) )
= ( map_nat_nat @ F @ ( rotate1_nat @ Xs2 ) ) ) ).
% rotate1_map
thf(fact_1241_map__butlast,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( map_nat_nat @ F @ ( butlast_nat @ Xs2 ) )
= ( butlast_nat @ ( map_nat_nat @ F @ Xs2 ) ) ) ).
% map_butlast
thf(fact_1242_map__equality__iff,axiom,
! [F: nat > nat,Xs2: list_nat,G: nat > nat,Ys2: list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Ys2 ) )
= ( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys2 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys2 ) )
=> ( ( F @ ( nth_nat @ Xs2 @ I2 ) )
= ( G @ ( nth_nat @ Ys2 @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_1243_map__Suc__upt,axiom,
! [M: nat,N: nat] :
( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
= ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% map_Suc_upt
thf(fact_1244_hd__conv__nth,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( hd_nat @ Xs2 )
= ( nth_nat @ Xs2 @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_1245_hd__drop__conv__nth,axiom,
! [N: nat,Xs2: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( hd_nat @ ( drop_nat @ N @ Xs2 ) )
= ( nth_nat @ Xs2 @ N ) ) ) ).
% hd_drop_conv_nth
thf(fact_1246_nth__map__upt,axiom,
! [I: nat,N: nat,M: nat,F: nat > nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ N @ M ) )
=> ( ( nth_nat @ ( map_nat_nat @ F @ ( upt @ M @ N ) ) @ I )
= ( F @ ( plus_plus_nat @ M @ I ) ) ) ) ).
% nth_map_upt
thf(fact_1247_binomial__addition__formula,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( binomial @ N @ ( suc @ K ) )
= ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ).
% binomial_addition_formula
thf(fact_1248_num__of__nat__plus__distrib,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( num_of_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N ) ) ) ) ) ).
% num_of_nat_plus_distrib
thf(fact_1249_binomial__n__n,axiom,
! [N: nat] :
( ( binomial @ N @ N )
= one_one_nat ) ).
% binomial_n_n
thf(fact_1250_num__of__nat__numeral__eq,axiom,
! [Q2: num] :
( ( num_of_nat @ ( numeral_numeral_nat @ Q2 ) )
= Q2 ) ).
% num_of_nat_numeral_eq
thf(fact_1251_binomial__0__Suc,axiom,
! [K: nat] :
( ( binomial @ zero_zero_nat @ ( suc @ K ) )
= zero_zero_nat ) ).
% binomial_0_Suc
thf(fact_1252_binomial__1,axiom,
! [N: nat] :
( ( binomial @ N @ ( suc @ zero_zero_nat ) )
= N ) ).
% binomial_1
thf(fact_1253_binomial__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( ( binomial @ N @ K )
= zero_zero_nat )
= ( ord_less_nat @ N @ K ) ) ).
% binomial_eq_0_iff
thf(fact_1254_binomial__Suc__Suc,axiom,
! [N: nat,K: nat] :
( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
= ( plus_plus_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% binomial_Suc_Suc
thf(fact_1255_binomial__n__0,axiom,
! [N: nat] :
( ( binomial @ N @ zero_zero_nat )
= one_one_nat ) ).
% binomial_n_0
thf(fact_1256_zero__less__binomial__iff,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) )
= ( ord_less_eq_nat @ K @ N ) ) ).
% zero_less_binomial_iff
thf(fact_1257_zero__less__binomial,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) ) ) ).
% zero_less_binomial
thf(fact_1258_choose__one,axiom,
! [N: nat] :
( ( binomial @ N @ one_one_nat )
= N ) ).
% choose_one
thf(fact_1259_binomial__eq__0,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( binomial @ N @ K )
= zero_zero_nat ) ) ).
% binomial_eq_0
thf(fact_1260_prefixes_Osimps_I2_J,axiom,
! [X: nat,Xs2: list_nat] :
( ( prefixes_nat @ ( cons_nat @ X @ Xs2 ) )
= ( cons_list_nat @ nil_nat @ ( map_li7225945977422193158st_nat @ ( cons_nat @ X ) @ ( prefixes_nat @ Xs2 ) ) ) ) ).
% prefixes.simps(2)
thf(fact_1261_numeral__num__of__nat,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
= N ) ) ).
% numeral_num_of_nat
thf(fact_1262_sublists_Osimps_I2_J,axiom,
! [X: nat,Xs2: list_nat] :
( ( sublists_nat @ ( cons_nat @ X @ Xs2 ) )
= ( append_list_nat @ ( sublists_nat @ Xs2 ) @ ( map_li7225945977422193158st_nat @ ( cons_nat @ X ) @ ( prefixes_nat @ Xs2 ) ) ) ) ).
% sublists.simps(2)
% Helper facts (9)
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__Num__Onum_T,axiom,
! [X: num,Y: num] :
( ( if_num @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
! [X: num,Y: num] :
( ( if_num @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X: set_nat,Y: set_nat] :
( ( if_set_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
! [X: set_nat,Y: set_nat] :
( ( if_set_nat @ $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,
xa = ya ).
%------------------------------------------------------------------------------