TPTP Problem File: SLH0163^1.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_00293_011313__11954096_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1394 ( 633 unt; 120 typ; 0 def)
% Number of atoms : 3175 (1698 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 10621 ( 394 ~; 78 |; 243 &;8559 @)
% ( 0 <=>;1347 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 6 avg)
% Number of types : 11 ( 10 usr)
% Number of type conns : 471 ( 471 >; 0 *; 0 +; 0 <<)
% Number of symbols : 113 ( 110 usr; 14 con; 0-3 aty)
% Number of variables : 3564 ( 124 ^;3218 !; 222 ?;3564 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:14:06.133
%------------------------------------------------------------------------------
% Could-be-implicit typings (10)
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__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__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 (110)
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_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__List__Olist_It__Nat__Onat_J_J,type,
if_list_list_nat: $o > list_list_nat > list_list_nat > list_list_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_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
insert_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > list_nat > list_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_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__Num__Onum,type,
size_size_num: num > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Num_Oinc,type,
inc: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
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__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_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_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_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 (1262)
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_d_I2_J,axiom,
equiva3371634703666331078on_rgf @ ys ).
% d(2)
thf(fact_3_rgf__snoc,axiom,
! [Xs: list_nat,X: nat] :
( ( equiva3371634703666331078on_rgf @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= ( ( equiva3371634703666331078on_rgf @ Xs )
& ( ord_less_nat @ X @ ( plus_plus_nat @ ( equiva5889994315859557365_limit @ Xs ) @ one_one_nat ) ) ) ) ).
% rgf_snoc
thf(fact_4_Cons_Oprems_I2_J,axiom,
equiva3371634703666331078on_rgf @ ( append_nat @ ys @ ( cons_nat @ ya @ nil_nat ) ) ).
% Cons.prems(2)
thf(fact_5_b,axiom,
xs = ys ).
% b
thf(fact_6_Cons_Ohyps_I2_J,axiom,
( ( equiva3371634703666331078on_rgf @ xs )
=> ( ( equiva3371634703666331078on_rgf @ ys )
=> ( ( ( equiva2048684438135499664of_nat @ xs )
= ( equiva2048684438135499664of_nat @ ys ) )
=> ( xs = ys ) ) ) ) ).
% Cons.hyps(2)
thf(fact_7_Cons_Ohyps_I1_J,axiom,
( ( size_size_list_nat @ xs )
= ( size_size_list_nat @ ys ) ) ).
% Cons.hyps(1)
thf(fact_8_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_9_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_10_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_11_x__bound,axiom,
ord_less_nat @ xa @ ( plus_plus_nat @ ( equiva5889994315859557365_limit @ xs ) @ one_one_nat ) ).
% x_bound
thf(fact_12_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_13_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_14_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_15_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_16_a,axiom,
( ( equiva2048684438135499664of_nat @ xs )
= ( equiva2048684438135499664of_nat @ ys ) ) ).
% a
thf(fact_17_d_I1_J,axiom,
equiva3371634703666331078on_rgf @ xs ).
% d(1)
thf(fact_18_assms_I1_J,axiom,
( ( size_size_list_nat @ x )
= ( size_size_list_nat @ y ) ) ).
% assms(1)
thf(fact_19_assms_I4_J,axiom,
( ( equiva2048684438135499664of_nat @ x )
= ( equiva2048684438135499664of_nat @ y ) ) ).
% assms(4)
thf(fact_20_Cons_Oprems_I1_J,axiom,
equiva3371634703666331078on_rgf @ ( append_nat @ xs @ ( cons_nat @ xa @ nil_nat ) ) ).
% Cons.prems(1)
thf(fact_21_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_22_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_23_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_24_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_25_size__neq__size__imp__neq,axiom,
! [X: num,Y: num] :
( ( ( size_size_num @ X )
!= ( size_size_num @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_26_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_27_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_28_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_29_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_30_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,Xs2: list_nat,Y2: list_nat,Ys: list_list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( P @ Xs2 @ Ys )
=> ( P @ ( append_nat @ Xs2 @ ( 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_31_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,Xs2: list_list_nat,Y2: nat,Ys: list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ Xs2 @ Ys )
=> ( P @ ( append_list_nat @ Xs2 @ ( cons_list_nat @ X2 @ nil_list_nat ) ) @ ( append_nat @ Ys @ ( cons_nat @ Y2 @ nil_nat ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% list_induct_2_rev
thf(fact_32_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,Xs2: list_list_nat,Y2: list_nat,Ys: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( P @ Xs2 @ Ys )
=> ( P @ ( append_list_nat @ Xs2 @ ( 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_33_list__induct__2__rev,axiom,
! [X: list_nat,Y: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ X )
= ( size_size_list_nat @ Y ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat,Y2: nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ Xs2 @ Ys )
=> ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X2 @ nil_nat ) ) @ ( append_nat @ Ys @ ( cons_nat @ Y2 @ nil_nat ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% list_induct_2_rev
thf(fact_34_rgf__limit_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ~ ! [X2: nat,Xs2: list_nat] :
( X
!= ( cons_nat @ X2 @ Xs2 ) ) ) ).
% rgf_limit.cases
thf(fact_35_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_36_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_37_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_38_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_39_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_40_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B: nat,A: nat] :
( ( B3
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_41_group__cancel_Oadd1,axiom,
! [A3: nat,K: nat,A: nat,B: nat] :
( ( A3
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A3 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_42_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_43_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_44_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_45_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_46_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_47_Collect__mem__eq,axiom,
! [A3: set_list_nat] :
( ( collect_list_nat
@ ^ [X3: list_nat] : ( member_list_nat @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_48_Collect__mem__eq,axiom,
! [A3: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_49_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_50_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_51_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_52_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_53_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_54_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_55_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_56_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_57_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_58_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_59_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_60_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_61_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_62_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_63_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_64_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_65_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_66_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_67_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_68_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_69_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_70_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_71_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_72_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_73_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_74_length__append,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat] :
( ( size_s3023201423986296836st_nat @ ( append_list_nat @ Xs @ Ys2 ) )
= ( plus_plus_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ ( size_s3023201423986296836st_nat @ Ys2 ) ) ) ).
% length_append
thf(fact_75_length__append,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( size_size_list_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys2 ) ) ) ).
% length_append
thf(fact_76_append1__eq__conv,axiom,
! [Xs: list_list_nat,X: list_nat,Ys2: list_list_nat,Y: list_nat] :
( ( ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) )
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ Y @ nil_list_nat ) ) )
= ( ( Xs = Ys2 )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_77_append1__eq__conv,axiom,
! [Xs: list_nat,X: nat,Ys2: list_nat,Y: nat] :
( ( ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) )
= ( append_nat @ Ys2 @ ( cons_nat @ Y @ nil_nat ) ) )
= ( ( Xs = Ys2 )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_78__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_79_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_80_append__eq__append__conv,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat,Us: list_list_nat,Vs: list_list_nat] :
( ( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
| ( ( size_s3023201423986296836st_nat @ Us )
= ( size_s3023201423986296836st_nat @ Vs ) ) )
=> ( ( ( append_list_nat @ Xs @ Us )
= ( append_list_nat @ Ys2 @ Vs ) )
= ( ( Xs = Ys2 )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_81_append__eq__append__conv,axiom,
! [Xs: list_nat,Ys2: list_nat,Us: list_nat,Vs: list_nat] :
( ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
| ( ( size_size_list_nat @ Us )
= ( size_size_list_nat @ Vs ) ) )
=> ( ( ( append_nat @ Xs @ Us )
= ( append_nat @ Ys2 @ Vs ) )
= ( ( Xs = Ys2 )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_82_append_Oright__neutral,axiom,
! [A: list_list_nat] :
( ( append_list_nat @ A @ nil_list_nat )
= A ) ).
% append.right_neutral
thf(fact_83_append_Oright__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ A @ nil_nat )
= A ) ).
% append.right_neutral
thf(fact_84_append__Nil2,axiom,
! [Xs: list_list_nat] :
( ( append_list_nat @ Xs @ nil_list_nat )
= Xs ) ).
% append_Nil2
thf(fact_85_append__Nil2,axiom,
! [Xs: list_nat] :
( ( append_nat @ Xs @ nil_nat )
= Xs ) ).
% append_Nil2
thf(fact_86_append__self__conv,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat] :
( ( ( append_list_nat @ Xs @ Ys2 )
= Xs )
= ( Ys2 = nil_list_nat ) ) ).
% append_self_conv
thf(fact_87_append__self__conv,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( append_nat @ Xs @ Ys2 )
= Xs )
= ( Ys2 = nil_nat ) ) ).
% append_self_conv
thf(fact_88_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_89_self__append__conv,axiom,
! [Y: list_nat,Ys2: list_nat] :
( ( Y
= ( append_nat @ Y @ Ys2 ) )
= ( Ys2 = nil_nat ) ) ).
% self_append_conv
thf(fact_90_append__self__conv2,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat] :
( ( ( append_list_nat @ Xs @ Ys2 )
= Ys2 )
= ( Xs = nil_list_nat ) ) ).
% append_self_conv2
thf(fact_91_append__self__conv2,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( append_nat @ Xs @ Ys2 )
= Ys2 )
= ( Xs = nil_nat ) ) ).
% append_self_conv2
thf(fact_92_self__append__conv2,axiom,
! [Y: list_list_nat,Xs: list_list_nat] :
( ( Y
= ( append_list_nat @ Xs @ Y ) )
= ( Xs = nil_list_nat ) ) ).
% self_append_conv2
thf(fact_93_self__append__conv2,axiom,
! [Y: list_nat,Xs: list_nat] :
( ( Y
= ( append_nat @ Xs @ Y ) )
= ( Xs = nil_nat ) ) ).
% self_append_conv2
thf(fact_94_Nil__is__append__conv,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat] :
( ( nil_list_nat
= ( append_list_nat @ Xs @ Ys2 ) )
= ( ( Xs = nil_list_nat )
& ( Ys2 = nil_list_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_95_Nil__is__append__conv,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( nil_nat
= ( append_nat @ Xs @ Ys2 ) )
= ( ( Xs = nil_nat )
& ( Ys2 = nil_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_96_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_97_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_98_same__append__eq,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat,Zs: list_list_nat] :
( ( ( append_list_nat @ Xs @ Ys2 )
= ( append_list_nat @ Xs @ Zs ) )
= ( Ys2 = Zs ) ) ).
% same_append_eq
thf(fact_99_same__append__eq,axiom,
! [Xs: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( ( append_nat @ Xs @ Ys2 )
= ( append_nat @ Xs @ Zs ) )
= ( Ys2 = Zs ) ) ).
% same_append_eq
thf(fact_100_append__same__eq,axiom,
! [Ys2: list_list_nat,Xs: list_list_nat,Zs: list_list_nat] :
( ( ( append_list_nat @ Ys2 @ Xs )
= ( append_list_nat @ Zs @ Xs ) )
= ( Ys2 = Zs ) ) ).
% append_same_eq
thf(fact_101_append__same__eq,axiom,
! [Ys2: list_nat,Xs: list_nat,Zs: list_nat] :
( ( ( append_nat @ Ys2 @ Xs )
= ( append_nat @ Zs @ Xs ) )
= ( Ys2 = Zs ) ) ).
% append_same_eq
thf(fact_102_append__assoc,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat,Zs: list_list_nat] :
( ( append_list_nat @ ( append_list_nat @ Xs @ Ys2 ) @ Zs )
= ( append_list_nat @ Xs @ ( append_list_nat @ Ys2 @ Zs ) ) ) ).
% append_assoc
thf(fact_103_append__assoc,axiom,
! [Xs: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( append_nat @ ( append_nat @ Xs @ Ys2 ) @ Zs )
= ( append_nat @ Xs @ ( append_nat @ Ys2 @ Zs ) ) ) ).
% append_assoc
thf(fact_104_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_105_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_106_append__is__Nil__conv,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat] :
( ( ( append_list_nat @ Xs @ Ys2 )
= nil_list_nat )
= ( ( Xs = nil_list_nat )
& ( Ys2 = nil_list_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_107_append__is__Nil__conv,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( append_nat @ Xs @ Ys2 )
= nil_nat )
= ( ( Xs = nil_nat )
& ( Ys2 = nil_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_108_nth__append__length,axiom,
! [Xs: list_list_nat,X: list_nat,Ys2: list_list_nat] :
( ( nth_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ X @ Ys2 ) ) @ ( size_s3023201423986296836st_nat @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_109_nth__append__length,axiom,
! [Xs: list_nat,X: nat,Ys2: list_nat] :
( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ Ys2 ) ) @ ( size_size_list_nat @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_110_nth__append__length__plus,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat,N: nat] :
( ( nth_list_nat @ ( append_list_nat @ Xs @ Ys2 ) @ ( plus_plus_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ N ) )
= ( nth_list_nat @ Ys2 @ N ) ) ).
% nth_append_length_plus
thf(fact_111_nth__append__length__plus,axiom,
! [Xs: list_nat,Ys2: list_nat,N: nat] :
( ( nth_nat @ ( append_nat @ Xs @ Ys2 ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ N ) )
= ( nth_nat @ Ys2 @ N ) ) ).
% nth_append_length_plus
thf(fact_112_set__ConsD,axiom,
! [Y: list_nat,X: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ Y @ ( set_list_nat2 @ ( cons_list_nat @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_list_nat @ Y @ ( set_list_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_113_set__ConsD,axiom,
! [Y: nat,X: nat,Xs: list_nat] :
( ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member_nat @ Y @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_114_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_115_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_116_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_117_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_118_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_119_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_120_all__set__conv__all__nth,axiom,
! [Xs: list_list_nat,P: list_nat > $o] :
( ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( P @ X3 ) ) )
= ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( P @ ( nth_list_nat @ Xs @ I2 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_121_all__set__conv__all__nth,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( P @ X3 ) ) )
= ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I2 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_122_all__nth__imp__all__set,axiom,
! [Xs: list_list_nat,P: list_nat > $o,X: list_nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( P @ ( nth_list_nat @ Xs @ I3 ) ) )
=> ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_123_all__nth__imp__all__set,axiom,
! [Xs: list_nat,P: nat > $o,X: nat] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I3 ) ) )
=> ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_124_in__set__conv__nth,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
= ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
& ( ( nth_list_nat @ Xs @ I2 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_125_in__set__conv__nth,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
= ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I2 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_126_list__ball__nth,axiom,
! [N: nat,Xs: list_list_nat,P: list_nat > $o] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( P @ X2 ) )
=> ( P @ ( nth_list_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_127_list__ball__nth,axiom,
! [N: nat,Xs: list_nat,P: nat > $o] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( P @ X2 ) )
=> ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_128_nth__mem,axiom,
! [N: nat,Xs: list_list_nat] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( member_list_nat @ ( nth_list_nat @ Xs @ N ) @ ( set_list_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_129_nth__mem,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_130_list__eq__iff__nth__eq,axiom,
( ( ^ [Y3: list_list_nat,Z: list_list_nat] : ( Y3 = Z ) )
= ( ^ [Xs3: list_list_nat,Ys3: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs3 )
= ( size_s3023201423986296836st_nat @ Ys3 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs3 ) )
=> ( ( nth_list_nat @ Xs3 @ I2 )
= ( nth_list_nat @ Ys3 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_131_list__eq__iff__nth__eq,axiom,
( ( ^ [Y3: list_nat,Z: list_nat] : ( Y3 = Z ) )
= ( ^ [Xs3: list_nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys3 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs3 ) )
=> ( ( nth_nat @ Xs3 @ I2 )
= ( nth_nat @ Ys3 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_132_Skolem__list__nth,axiom,
! [K: nat,P: nat > list_nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X4: list_nat] : ( P @ I2 @ X4 ) ) )
= ( ? [Xs3: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs3 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_list_nat @ Xs3 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_133_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X4: nat] : ( P @ I2 @ X4 ) ) )
= ( ? [Xs3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_nat @ Xs3 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_134_nth__equalityI,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_list_nat @ Xs @ I3 )
= ( nth_list_nat @ Ys2 @ I3 ) ) )
=> ( Xs = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_135_nth__equalityI,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I3 )
= ( nth_nat @ Ys2 @ I3 ) ) )
=> ( Xs = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_136_split__list__first__prop__iff,axiom,
! [Xs: list_list_nat,P: list_nat > $o] :
( ( ? [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_list_nat,X3: list_nat] :
( ? [Zs2: list_list_nat] :
( Xs
= ( append_list_nat @ Ys3 @ ( cons_list_nat @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Y4: list_nat] :
( ( member_list_nat @ Y4 @ ( set_list_nat2 @ Ys3 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_137_split__list__first__prop__iff,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_nat,X3: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Y4: nat] :
( ( member_nat @ Y4 @ ( set_nat2 @ Ys3 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_138_split__list__last__prop__iff,axiom,
! [Xs: list_list_nat,P: list_nat > $o] :
( ( ? [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_list_nat,X3: list_nat,Zs2: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys3 @ ( cons_list_nat @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Y4: list_nat] :
( ( member_list_nat @ Y4 @ ( set_list_nat2 @ Zs2 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_139_split__list__last__prop__iff,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys3: list_nat,X3: nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Y4: nat] :
( ( member_nat @ Y4 @ ( set_nat2 @ Zs2 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_140_in__set__conv__decomp__first,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
= ( ? [Ys3: list_list_nat,Zs2: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys3 @ ( cons_list_nat @ X @ Zs2 ) ) )
& ~ ( member_list_nat @ X @ ( set_list_nat2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_141_in__set__conv__decomp__first,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys3: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs2 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_142_in__set__conv__decomp__last,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
= ( ? [Ys3: list_list_nat,Zs2: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys3 @ ( cons_list_nat @ X @ Zs2 ) ) )
& ~ ( member_list_nat @ X @ ( set_list_nat2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_143_in__set__conv__decomp__last,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys3: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs2 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_144_split__list__first__propE,axiom,
! [Xs: list_list_nat,P: list_nat > $o] :
( ? [X5: list_nat] :
( ( member_list_nat @ X5 @ ( set_list_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys: list_list_nat,X2: list_nat] :
( ? [Zs3: list_list_nat] :
( Xs
= ( append_list_nat @ Ys @ ( cons_list_nat @ X2 @ Zs3 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ ( set_list_nat2 @ Ys ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_145_split__list__first__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys: list_nat,X2: nat] :
( ? [Zs3: list_nat] :
( Xs
= ( append_nat @ Ys @ ( cons_nat @ X2 @ Zs3 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat2 @ Ys ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_146_split__list__last__propE,axiom,
! [Xs: list_list_nat,P: list_nat > $o] :
( ? [X5: list_nat] :
( ( member_list_nat @ X5 @ ( set_list_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys: list_list_nat,X2: list_nat,Zs3: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys @ ( cons_list_nat @ X2 @ Zs3 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ ( set_list_nat2 @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_147_split__list__last__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys: list_nat,X2: nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys @ ( cons_nat @ X2 @ Zs3 ) ) )
=> ( ( P @ X2 )
=> ~ ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat2 @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_148_split__list__first__prop,axiom,
! [Xs: list_list_nat,P: list_nat > $o] :
( ? [X5: list_nat] :
( ( member_list_nat @ X5 @ ( set_list_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys: list_list_nat,X2: list_nat] :
( ? [Zs3: list_list_nat] :
( Xs
= ( append_list_nat @ Ys @ ( cons_list_nat @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ ( set_list_nat2 @ Ys ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_149_split__list__first__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys: list_nat,X2: nat] :
( ? [Zs3: list_nat] :
( Xs
= ( append_nat @ Ys @ ( cons_nat @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat2 @ Ys ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_150_split__list__last__prop,axiom,
! [Xs: list_list_nat,P: list_nat > $o] :
( ? [X5: list_nat] :
( ( member_list_nat @ X5 @ ( set_list_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys: list_list_nat,X2: list_nat,Zs3: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys @ ( cons_list_nat @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Xa: list_nat] :
( ( member_list_nat @ Xa @ ( set_list_nat2 @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_151_split__list__last__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys: list_nat,X2: nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys @ ( cons_nat @ X2 @ Zs3 ) ) )
& ( P @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_nat2 @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_152_in__set__conv__decomp,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
= ( ? [Ys3: list_list_nat,Zs2: list_list_nat] :
( Xs
= ( append_list_nat @ Ys3 @ ( cons_list_nat @ X @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_153_in__set__conv__decomp,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
= ( ? [Ys3: list_nat,Zs2: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_154_append__Cons__eq__iff,axiom,
! [X: list_nat,Xs: list_list_nat,Ys2: list_list_nat,Xs4: list_list_nat,Ys4: list_list_nat] :
( ~ ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ( ~ ( member_list_nat @ X @ ( set_list_nat2 @ Ys2 ) )
=> ( ( ( append_list_nat @ Xs @ ( cons_list_nat @ X @ Ys2 ) )
= ( append_list_nat @ Xs4 @ ( cons_list_nat @ X @ Ys4 ) ) )
= ( ( Xs = Xs4 )
& ( Ys2 = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_155_append__Cons__eq__iff,axiom,
! [X: nat,Xs: list_nat,Ys2: list_nat,Xs4: list_nat,Ys4: list_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ~ ( member_nat @ X @ ( set_nat2 @ Ys2 ) )
=> ( ( ( append_nat @ Xs @ ( cons_nat @ X @ Ys2 ) )
= ( append_nat @ Xs4 @ ( cons_nat @ X @ Ys4 ) ) )
= ( ( Xs = Xs4 )
& ( Ys2 = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_156_split__list__propE,axiom,
! [Xs: list_list_nat,P: list_nat > $o] :
( ? [X5: list_nat] :
( ( member_list_nat @ X5 @ ( set_list_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys: list_list_nat,X2: list_nat] :
( ? [Zs3: list_list_nat] :
( Xs
= ( append_list_nat @ Ys @ ( cons_list_nat @ X2 @ Zs3 ) ) )
=> ~ ( P @ X2 ) ) ) ).
% split_list_propE
thf(fact_157_split__list__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys: list_nat,X2: nat] :
( ? [Zs3: list_nat] :
( Xs
= ( append_nat @ Ys @ ( cons_nat @ X2 @ Zs3 ) ) )
=> ~ ( P @ X2 ) ) ) ).
% split_list_propE
thf(fact_158_split__list__first,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ? [Ys: list_list_nat,Zs3: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys @ ( cons_list_nat @ X @ Zs3 ) ) )
& ~ ( member_list_nat @ X @ ( set_list_nat2 @ Ys ) ) ) ) ).
% split_list_first
thf(fact_159_split__list__first,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys @ ( cons_nat @ X @ Zs3 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Ys ) ) ) ) ).
% split_list_first
thf(fact_160_split__list__prop,axiom,
! [Xs: list_list_nat,P: list_nat > $o] :
( ? [X5: list_nat] :
( ( member_list_nat @ X5 @ ( set_list_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys: list_list_nat,X2: list_nat] :
( ? [Zs3: list_list_nat] :
( Xs
= ( append_list_nat @ Ys @ ( cons_list_nat @ X2 @ Zs3 ) ) )
& ( P @ X2 ) ) ) ).
% split_list_prop
thf(fact_161_split__list__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys: list_nat,X2: nat] :
( ? [Zs3: list_nat] :
( Xs
= ( append_nat @ Ys @ ( cons_nat @ X2 @ Zs3 ) ) )
& ( P @ X2 ) ) ) ).
% split_list_prop
thf(fact_162_split__list__last,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ? [Ys: list_list_nat,Zs3: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys @ ( cons_list_nat @ X @ Zs3 ) ) )
& ~ ( member_list_nat @ X @ ( set_list_nat2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_163_split__list__last,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys @ ( cons_nat @ X @ Zs3 ) ) )
& ~ ( member_nat @ X @ ( set_nat2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_164_split__list,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ? [Ys: list_list_nat,Zs3: list_list_nat] :
( Xs
= ( append_list_nat @ Ys @ ( cons_list_nat @ X @ Zs3 ) ) ) ) ).
% split_list
thf(fact_165_split__list,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ? [Ys: list_nat,Zs3: list_nat] :
( Xs
= ( append_nat @ Ys @ ( cons_nat @ X @ Zs3 ) ) ) ) ).
% split_list
thf(fact_166_rgf__limit__ge,axiom,
! [Y: nat,Xs: list_nat] :
( ( member_nat @ Y @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ Y @ ( equiva5889994315859557365_limit @ Xs ) ) ) ).
% rgf_limit_ge
thf(fact_167_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 ) )
& ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_s3023201423986296836st_nat @ X ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ( nth_list_nat @ X @ I2 )
= ( nth_list_nat @ X @ J2 ) )
= ( ( nth_list_nat @ Y @ I2 )
= ( nth_list_nat @ Y @ J2 ) ) ) ) ) ) ) ).
% kernel_of_eq
thf(fact_168_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 ) )
& ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_s3023201423986296836st_nat @ X ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ( nth_list_nat @ X @ I2 )
= ( nth_list_nat @ X @ J2 ) )
= ( ( nth_nat @ Y @ I2 )
= ( nth_nat @ Y @ J2 ) ) ) ) ) ) ) ).
% kernel_of_eq
thf(fact_169_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 ) )
& ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_size_list_nat @ X ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ( nth_nat @ X @ I2 )
= ( nth_nat @ X @ J2 ) )
= ( ( nth_list_nat @ Y @ I2 )
= ( nth_list_nat @ Y @ J2 ) ) ) ) ) ) ) ).
% kernel_of_eq
thf(fact_170_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 ) )
& ! [J2: nat] :
( ( ord_less_nat @ J2 @ ( size_size_list_nat @ X ) )
=> ! [I2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ( nth_nat @ X @ I2 )
= ( nth_nat @ X @ J2 ) )
= ( ( nth_nat @ Y @ I2 )
= ( nth_nat @ Y @ J2 ) ) ) ) ) ) ) ).
% kernel_of_eq
thf(fact_171_not__Cons__self2,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( cons_list_nat @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_172_not__Cons__self2,axiom,
! [X: nat,Xs: list_nat] :
( ( cons_nat @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_173_neq__if__length__neq,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
!= ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( Xs != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_174_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys2 ) )
=> ( Xs != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_175_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_list_nat] :
( ( size_s3023201423986296836st_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_176_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_nat] :
( ( size_size_list_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_177_append__eq__append__conv2,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat,Zs: list_list_nat,Ts: list_list_nat] :
( ( ( append_list_nat @ Xs @ Ys2 )
= ( append_list_nat @ Zs @ Ts ) )
= ( ? [Us2: list_list_nat] :
( ( ( Xs
= ( append_list_nat @ Zs @ Us2 ) )
& ( ( append_list_nat @ Us2 @ Ys2 )
= Ts ) )
| ( ( ( append_list_nat @ Xs @ Us2 )
= Zs )
& ( Ys2
= ( append_list_nat @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_178_append__eq__append__conv2,axiom,
! [Xs: list_nat,Ys2: list_nat,Zs: list_nat,Ts: list_nat] :
( ( ( append_nat @ Xs @ Ys2 )
= ( append_nat @ Zs @ Ts ) )
= ( ? [Us2: list_nat] :
( ( ( Xs
= ( append_nat @ Zs @ Us2 ) )
& ( ( append_nat @ Us2 @ Ys2 )
= Ts ) )
| ( ( ( append_nat @ Xs @ Us2 )
= Zs )
& ( Ys2
= ( append_nat @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_179_append__eq__appendI,axiom,
! [Xs: list_list_nat,Xs1: list_list_nat,Zs: list_list_nat,Ys2: list_list_nat,Us: list_list_nat] :
( ( ( append_list_nat @ Xs @ Xs1 )
= Zs )
=> ( ( Ys2
= ( append_list_nat @ Xs1 @ Us ) )
=> ( ( append_list_nat @ Xs @ Ys2 )
= ( append_list_nat @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_180_append__eq__appendI,axiom,
! [Xs: list_nat,Xs1: list_nat,Zs: list_nat,Ys2: list_nat,Us: list_nat] :
( ( ( append_nat @ Xs @ Xs1 )
= Zs )
=> ( ( Ys2
= ( append_nat @ Xs1 @ Us ) )
=> ( ( append_nat @ Xs @ Ys2 )
= ( append_nat @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_181_list__nonempty__induct,axiom,
! [Xs: list_list_nat,P: list_list_nat > $o] :
( ( Xs != nil_list_nat )
=> ( ! [X2: list_nat] : ( P @ ( cons_list_nat @ X2 @ nil_list_nat ) )
=> ( ! [X2: list_nat,Xs2: list_list_nat] :
( ( Xs2 != nil_list_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_list_nat @ X2 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_182_list__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
=> ( ! [X2: nat,Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_183_list__induct2_H,axiom,
! [P: list_nat > list_list_nat > $o,Xs: list_nat,Ys2: list_list_nat] :
( ( P @ nil_nat @ nil_list_nat )
=> ( ! [X2: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X2 @ Xs2 ) @ nil_list_nat )
=> ( ! [Y2: list_nat,Ys: list_list_nat] : ( P @ nil_nat @ ( cons_list_nat @ Y2 @ Ys ) )
=> ( ! [X2: nat,Xs2: list_nat,Y2: list_nat,Ys: list_list_nat] :
( ( P @ Xs2 @ Ys )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_list_nat @ Y2 @ Ys ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_184_list__induct2_H,axiom,
! [P: list_list_nat > list_nat > $o,Xs: list_list_nat,Ys2: list_nat] :
( ( P @ nil_list_nat @ nil_nat )
=> ( ! [X2: list_nat,Xs2: list_list_nat] : ( P @ ( cons_list_nat @ X2 @ Xs2 ) @ nil_nat )
=> ( ! [Y2: nat,Ys: list_nat] : ( P @ nil_list_nat @ ( cons_nat @ Y2 @ Ys ) )
=> ( ! [X2: list_nat,Xs2: list_list_nat,Y2: nat,Ys: list_nat] :
( ( P @ Xs2 @ Ys )
=> ( P @ ( cons_list_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_185_list__induct2_H,axiom,
! [P: list_list_nat > list_list_nat > $o,Xs: list_list_nat,Ys2: list_list_nat] :
( ( P @ nil_list_nat @ nil_list_nat )
=> ( ! [X2: list_nat,Xs2: list_list_nat] : ( P @ ( cons_list_nat @ X2 @ Xs2 ) @ nil_list_nat )
=> ( ! [Y2: list_nat,Ys: list_list_nat] : ( P @ nil_list_nat @ ( cons_list_nat @ Y2 @ Ys ) )
=> ( ! [X2: list_nat,Xs2: list_list_nat,Y2: list_nat,Ys: list_list_nat] :
( ( P @ Xs2 @ Ys )
=> ( P @ ( cons_list_nat @ X2 @ Xs2 ) @ ( cons_list_nat @ Y2 @ Ys ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_186_list__induct2_H,axiom,
! [P: list_nat > list_nat > $o,Xs: list_nat,Ys2: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X2 @ Xs2 ) @ nil_nat )
=> ( ! [Y2: nat,Ys: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y2 @ Ys ) )
=> ( ! [X2: nat,Xs2: list_nat,Y2: nat,Ys: list_nat] :
( ( P @ Xs2 @ Ys )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_187_neq__Nil__conv,axiom,
! [Xs: list_list_nat] :
( ( Xs != nil_list_nat )
= ( ? [Y4: list_nat,Ys3: list_list_nat] :
( Xs
= ( cons_list_nat @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_188_neq__Nil__conv,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
= ( ? [Y4: nat,Ys3: list_nat] :
( Xs
= ( cons_nat @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_189_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,Xs2: list_list_nat] :
( X
!= ( cons_list_nat @ X2 @ ( cons_list_nat @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_190_remdups__adj_Ocases,axiom,
! [X: list_nat] :
( ( X != nil_nat )
=> ( ! [X2: nat] :
( X
!= ( cons_nat @ X2 @ nil_nat ) )
=> ~ ! [X2: nat,Y2: nat,Xs2: list_nat] :
( X
!= ( cons_nat @ X2 @ ( cons_nat @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_191_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,Xs2: list_list_nat,Xss: list_list_list_nat] :
( X
!= ( cons_list_list_nat @ ( cons_list_nat @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_192_transpose_Ocases,axiom,
! [X: list_list_nat] :
( ( X != nil_list_nat )
=> ( ! [Xss: list_list_nat] :
( X
!= ( cons_list_nat @ nil_nat @ Xss ) )
=> ~ ! [X2: nat,Xs2: list_nat,Xss: list_list_nat] :
( X
!= ( cons_list_nat @ ( cons_nat @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_193_min__list_Ocases,axiom,
! [X: list_nat] :
( ! [X2: nat,Xs2: list_nat] :
( X
!= ( cons_nat @ X2 @ Xs2 ) )
=> ( X = nil_nat ) ) ).
% min_list.cases
thf(fact_194_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_195_list_Oexhaust,axiom,
! [Y: list_nat] :
( ( Y != nil_nat )
=> ~ ! [X212: nat,X222: list_nat] :
( Y
!= ( cons_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_196_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_197_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X22: list_nat] :
( ( List
= ( cons_nat @ X21 @ X22 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_198_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_199_list_Odistinct_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_200_Cons__eq__appendI,axiom,
! [X: list_nat,Xs1: list_list_nat,Ys2: list_list_nat,Xs: list_list_nat,Zs: list_list_nat] :
( ( ( cons_list_nat @ X @ Xs1 )
= Ys2 )
=> ( ( Xs
= ( append_list_nat @ Xs1 @ Zs ) )
=> ( ( cons_list_nat @ X @ Xs )
= ( append_list_nat @ Ys2 @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_201_Cons__eq__appendI,axiom,
! [X: nat,Xs1: list_nat,Ys2: list_nat,Xs: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X @ Xs1 )
= Ys2 )
=> ( ( Xs
= ( append_nat @ Xs1 @ Zs ) )
=> ( ( cons_nat @ X @ Xs )
= ( append_nat @ Ys2 @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_202_append__Cons,axiom,
! [X: list_nat,Xs: list_list_nat,Ys2: list_list_nat] :
( ( append_list_nat @ ( cons_list_nat @ X @ Xs ) @ Ys2 )
= ( cons_list_nat @ X @ ( append_list_nat @ Xs @ Ys2 ) ) ) ).
% append_Cons
thf(fact_203_append__Cons,axiom,
! [X: nat,Xs: list_nat,Ys2: list_nat] :
( ( append_nat @ ( cons_nat @ X @ Xs ) @ Ys2 )
= ( cons_nat @ X @ ( append_nat @ Xs @ Ys2 ) ) ) ).
% append_Cons
thf(fact_204_eq__Nil__appendI,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat] :
( ( Xs = Ys2 )
=> ( Xs
= ( append_list_nat @ nil_list_nat @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_205_eq__Nil__appendI,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( Xs = Ys2 )
=> ( Xs
= ( append_nat @ nil_nat @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_206_append_Oleft__neutral,axiom,
! [A: list_list_nat] :
( ( append_list_nat @ nil_list_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_207_append_Oleft__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ nil_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_208_append__Nil,axiom,
! [Ys2: list_list_nat] :
( ( append_list_nat @ nil_list_nat @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_209_append__Nil,axiom,
! [Ys2: list_nat] :
( ( append_nat @ nil_nat @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_210_length__induct,axiom,
! [P: list_list_nat > $o,Xs: list_list_nat] :
( ! [Xs2: list_list_nat] :
( ! [Ys5: list_list_nat] :
( ( ord_less_nat @ ( size_s3023201423986296836st_nat @ Ys5 ) @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( P @ Ys5 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_211_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs2: list_nat] :
( ! [Ys5: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys5 ) @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ Ys5 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_212_list__induct4,axiom,
! [Xs: 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 @ Xs )
= ( 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,Xs2: list_nat,Y2: nat,Ys: list_nat,Z3: nat,Zs3: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs3 @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys ) @ ( cons_nat @ Z3 @ Zs3 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_213_list__induct4,axiom,
! [Xs: 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 @ Xs )
= ( 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,Xs2: list_nat,Y2: nat,Ys: list_nat,Z3: nat,Zs3: list_nat,W: list_nat,Ws2: list_list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_s3023201423986296836st_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs3 @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys ) @ ( cons_nat @ Z3 @ Zs3 ) @ ( cons_list_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_214_list__induct4,axiom,
! [Xs: 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 @ Xs )
= ( 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,Xs2: list_nat,Y2: nat,Ys: list_nat,Z3: list_nat,Zs3: list_list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_s3023201423986296836st_nat @ Zs3 ) )
=> ( ( ( size_s3023201423986296836st_nat @ Zs3 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs3 @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys ) @ ( cons_list_nat @ Z3 @ Zs3 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_215_list__induct4,axiom,
! [Xs: 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 @ Xs )
= ( 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,Xs2: list_nat,Y2: list_nat,Ys: list_list_nat,Z3: nat,Zs3: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs3 @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_list_nat @ Y2 @ Ys ) @ ( cons_nat @ Z3 @ Zs3 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_216_list__induct4,axiom,
! [Xs: 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 @ Xs )
= ( 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,Xs2: list_list_nat,Y2: nat,Ys: list_nat,Z3: nat,Zs3: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs3 @ Ws2 )
=> ( P @ ( cons_list_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys ) @ ( cons_nat @ Z3 @ Zs3 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_217_list__induct4,axiom,
! [Xs: 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 @ Xs )
= ( 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,Xs2: list_nat,Y2: nat,Ys: list_nat,Z3: list_nat,Zs3: list_list_nat,W: list_nat,Ws2: list_list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_s3023201423986296836st_nat @ Zs3 ) )
=> ( ( ( size_s3023201423986296836st_nat @ Zs3 )
= ( size_s3023201423986296836st_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs3 @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys ) @ ( cons_list_nat @ Z3 @ Zs3 ) @ ( cons_list_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_218_list__induct4,axiom,
! [Xs: 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 @ Xs )
= ( 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,Xs2: list_nat,Y2: list_nat,Ys: list_list_nat,Z3: nat,Zs3: list_nat,W: list_nat,Ws2: list_list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_s3023201423986296836st_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs3 @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_list_nat @ Y2 @ Ys ) @ ( cons_nat @ Z3 @ Zs3 ) @ ( cons_list_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_219_list__induct4,axiom,
! [Xs: 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 @ Xs )
= ( 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,Xs2: list_nat,Y2: list_nat,Ys: list_list_nat,Z3: list_nat,Zs3: list_list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys )
= ( size_s3023201423986296836st_nat @ Zs3 ) )
=> ( ( ( size_s3023201423986296836st_nat @ Zs3 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs3 @ Ws2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_list_nat @ Y2 @ Ys ) @ ( cons_list_nat @ Z3 @ Zs3 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_220_list__induct4,axiom,
! [Xs: 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 @ Xs )
= ( 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,Xs2: list_list_nat,Y2: nat,Ys: list_nat,Z3: nat,Zs3: list_nat,W: list_nat,Ws2: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( ( size_size_list_nat @ Zs3 )
= ( size_s3023201423986296836st_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs3 @ Ws2 )
=> ( P @ ( cons_list_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys ) @ ( cons_nat @ Z3 @ Zs3 ) @ ( cons_list_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_221_list__induct4,axiom,
! [Xs: 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 @ Xs )
= ( 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,Xs2: list_list_nat,Y2: nat,Ys: list_nat,Z3: list_nat,Zs3: list_list_nat,W: nat,Ws2: list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_s3023201423986296836st_nat @ Zs3 ) )
=> ( ( ( size_s3023201423986296836st_nat @ Zs3 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys @ Zs3 @ Ws2 )
=> ( P @ ( cons_list_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys ) @ ( cons_list_nat @ Z3 @ Zs3 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_222_list__induct3,axiom,
! [Xs: list_nat,Ys2: list_nat,Zs: list_list_nat,P: list_nat > list_nat > list_list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_s3023201423986296836st_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_list_nat )
=> ( ! [X2: nat,Xs2: list_nat,Y2: nat,Ys: list_nat,Z3: list_nat,Zs3: list_list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_s3023201423986296836st_nat @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys @ Zs3 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys ) @ ( cons_list_nat @ Z3 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_223_list__induct3,axiom,
! [Xs: list_nat,Ys2: list_list_nat,Zs: list_nat,P: list_nat > list_list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_list_nat @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat,Y2: list_nat,Ys: list_list_nat,Z3: nat,Zs3: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys @ Zs3 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_list_nat @ Y2 @ Ys ) @ ( cons_nat @ Z3 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_224_list__induct3,axiom,
! [Xs: list_nat,Ys2: list_list_nat,Zs: list_list_nat,P: list_nat > list_list_nat > list_list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys2 )
= ( size_s3023201423986296836st_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_list_nat @ nil_list_nat )
=> ( ! [X2: nat,Xs2: list_nat,Y2: list_nat,Ys: list_list_nat,Z3: list_nat,Zs3: list_list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys )
= ( size_s3023201423986296836st_nat @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys @ Zs3 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_list_nat @ Y2 @ Ys ) @ ( cons_list_nat @ Z3 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_225_list__induct3,axiom,
! [Xs: list_list_nat,Ys2: list_nat,Zs: list_nat,P: list_list_nat > list_nat > list_nat > $o] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( 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,Xs2: list_list_nat,Y2: nat,Ys: list_nat,Z3: nat,Zs3: list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys @ Zs3 )
=> ( P @ ( cons_list_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys ) @ ( cons_nat @ Z3 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_226_list__induct3,axiom,
! [Xs: list_list_nat,Ys2: list_nat,Zs: list_list_nat,P: list_list_nat > list_nat > list_list_nat > $o] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( 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,Xs2: list_list_nat,Y2: nat,Ys: list_nat,Z3: list_nat,Zs3: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_s3023201423986296836st_nat @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys @ Zs3 )
=> ( P @ ( cons_list_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys ) @ ( cons_list_nat @ Z3 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_227_list__induct3,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat,Zs: list_nat,P: list_list_nat > list_list_nat > list_nat > $o] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( 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,Xs2: list_list_nat,Y2: list_nat,Ys: list_list_nat,Z3: nat,Zs3: list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys @ Zs3 )
=> ( P @ ( cons_list_nat @ X2 @ Xs2 ) @ ( cons_list_nat @ Y2 @ Ys ) @ ( cons_nat @ Z3 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_228_list__induct3,axiom,
! [Xs: 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 @ Xs )
= ( 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,Xs2: list_list_nat,Y2: list_nat,Ys: list_list_nat,Z3: list_nat,Zs3: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( ( size_s3023201423986296836st_nat @ Ys )
= ( size_s3023201423986296836st_nat @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys @ Zs3 )
=> ( P @ ( cons_list_nat @ X2 @ Xs2 ) @ ( cons_list_nat @ Y2 @ Ys ) @ ( cons_list_nat @ Z3 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_229_list__induct3,axiom,
! [Xs: list_nat,Ys2: list_nat,Zs: list_nat,P: list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( ( size_size_list_nat @ Ys2 )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat,Y2: nat,Ys: list_nat,Z3: nat,Zs3: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs3 ) )
=> ( ( P @ Xs2 @ Ys @ Zs3 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys ) @ ( cons_nat @ Z3 @ Zs3 ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_230_list__induct2,axiom,
! [Xs: list_nat,Ys2: list_list_nat,P: list_nat > list_list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( ( P @ nil_nat @ nil_list_nat )
=> ( ! [X2: nat,Xs2: list_nat,Y2: list_nat,Ys: list_list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( P @ Xs2 @ Ys )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_list_nat @ Y2 @ Ys ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_231_list__induct2,axiom,
! [Xs: list_list_nat,Ys2: list_nat,P: list_list_nat > list_nat > $o] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ nil_list_nat @ nil_nat )
=> ( ! [X2: list_nat,Xs2: list_list_nat,Y2: nat,Ys: list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ Xs2 @ Ys )
=> ( P @ ( cons_list_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_232_list__induct2,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat,P: list_list_nat > list_list_nat > $o] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( ( P @ nil_list_nat @ nil_list_nat )
=> ( ! [X2: list_nat,Xs2: list_list_nat,Y2: list_nat,Ys: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( ( P @ Xs2 @ Ys )
=> ( P @ ( cons_list_nat @ X2 @ Xs2 ) @ ( cons_list_nat @ Y2 @ Ys ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_233_list__induct2,axiom,
! [Xs: list_nat,Ys2: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat,Y2: nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ Xs2 @ Ys )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_234_rev__nonempty__induct,axiom,
! [Xs: list_list_nat,P: list_list_nat > $o] :
( ( Xs != nil_list_nat )
=> ( ! [X2: list_nat] : ( P @ ( cons_list_nat @ X2 @ nil_list_nat ) )
=> ( ! [X2: list_nat,Xs2: list_list_nat] :
( ( Xs2 != nil_list_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( append_list_nat @ Xs2 @ ( cons_list_nat @ X2 @ nil_list_nat ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_235_rev__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
=> ( ! [X2: nat,Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X2 @ nil_nat ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_236_append__eq__Cons__conv,axiom,
! [Ys2: list_list_nat,Zs: list_list_nat,X: list_nat,Xs: list_list_nat] :
( ( ( append_list_nat @ Ys2 @ Zs )
= ( cons_list_nat @ X @ Xs ) )
= ( ( ( Ys2 = nil_list_nat )
& ( Zs
= ( cons_list_nat @ X @ Xs ) ) )
| ? [Ys6: list_list_nat] :
( ( Ys2
= ( cons_list_nat @ X @ Ys6 ) )
& ( ( append_list_nat @ Ys6 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_237_append__eq__Cons__conv,axiom,
! [Ys2: list_nat,Zs: list_nat,X: nat,Xs: list_nat] :
( ( ( append_nat @ Ys2 @ Zs )
= ( cons_nat @ X @ Xs ) )
= ( ( ( Ys2 = nil_nat )
& ( Zs
= ( cons_nat @ X @ Xs ) ) )
| ? [Ys6: list_nat] :
( ( Ys2
= ( cons_nat @ X @ Ys6 ) )
& ( ( append_nat @ Ys6 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_238_Cons__eq__append__conv,axiom,
! [X: list_nat,Xs: list_list_nat,Ys2: list_list_nat,Zs: list_list_nat] :
( ( ( cons_list_nat @ X @ Xs )
= ( append_list_nat @ Ys2 @ Zs ) )
= ( ( ( Ys2 = nil_list_nat )
& ( ( cons_list_nat @ X @ Xs )
= Zs ) )
| ? [Ys6: list_list_nat] :
( ( ( cons_list_nat @ X @ Ys6 )
= Ys2 )
& ( Xs
= ( append_list_nat @ Ys6 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_239_Cons__eq__append__conv,axiom,
! [X: nat,Xs: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( append_nat @ Ys2 @ Zs ) )
= ( ( ( Ys2 = nil_nat )
& ( ( cons_nat @ X @ Xs )
= Zs ) )
| ? [Ys6: list_nat] :
( ( ( cons_nat @ X @ Ys6 )
= Ys2 )
& ( Xs
= ( append_nat @ Ys6 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_240_rev__exhaust,axiom,
! [Xs: list_list_nat] :
( ( Xs != nil_list_nat )
=> ~ ! [Ys: list_list_nat,Y2: list_nat] :
( Xs
!= ( append_list_nat @ Ys @ ( cons_list_nat @ Y2 @ nil_list_nat ) ) ) ) ).
% rev_exhaust
thf(fact_241_rev__exhaust,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ~ ! [Ys: list_nat,Y2: nat] :
( Xs
!= ( append_nat @ Ys @ ( cons_nat @ Y2 @ nil_nat ) ) ) ) ).
% rev_exhaust
thf(fact_242_rev__induct,axiom,
! [P: list_list_nat > $o,Xs: list_list_nat] :
( ( P @ nil_list_nat )
=> ( ! [X2: list_nat,Xs2: list_list_nat] :
( ( P @ Xs2 )
=> ( P @ ( append_list_nat @ Xs2 @ ( cons_list_nat @ X2 @ nil_list_nat ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_243_rev__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ( P @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat] :
( ( P @ Xs2 )
=> ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X2 @ nil_nat ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_244_same__length__different,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat] :
( ( Xs != Ys2 )
=> ( ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ? [Pre: list_list_nat,X2: list_nat,Xs5: list_list_nat,Y2: list_nat,Ys7: list_list_nat] :
( ( X2 != Y2 )
& ( Xs
= ( 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_245_same__length__different,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( Xs != Ys2 )
=> ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
=> ? [Pre: list_nat,X2: nat,Xs5: list_nat,Y2: nat,Ys7: list_nat] :
( ( X2 != Y2 )
& ( Xs
= ( 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_246_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_247_the__elem__set,axiom,
! [X: nat] :
( ( the_elem_nat @ ( set_nat2 @ ( cons_nat @ X @ nil_nat ) ) )
= X ) ).
% the_elem_set
thf(fact_248_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_249_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_250_bind__simps_I2_J,axiom,
! [X: nat,Xs: list_nat,F: nat > list_list_nat] :
( ( bind_nat_list_nat @ ( cons_nat @ X @ Xs ) @ F )
= ( append_list_nat @ ( F @ X ) @ ( bind_nat_list_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_251_bind__simps_I2_J,axiom,
! [X: list_nat,Xs: list_list_nat,F: list_nat > list_nat] :
( ( bind_list_nat_nat @ ( cons_list_nat @ X @ Xs ) @ F )
= ( append_nat @ ( F @ X ) @ ( bind_list_nat_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_252_bind__simps_I2_J,axiom,
! [X: list_nat,Xs: list_list_nat,F: list_nat > list_list_nat] :
( ( bind_l7796378977173581257st_nat @ ( cons_list_nat @ X @ Xs ) @ F )
= ( append_list_nat @ ( F @ X ) @ ( bind_l7796378977173581257st_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_253_bind__simps_I2_J,axiom,
! [X: nat,Xs: list_nat,F: nat > list_nat] :
( ( bind_nat_nat @ ( cons_nat @ X @ Xs ) @ F )
= ( append_nat @ ( F @ X ) @ ( bind_nat_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_254_gen__length__def,axiom,
( gen_length_list_nat
= ( ^ [N3: nat,Xs3: list_list_nat] : ( plus_plus_nat @ N3 @ ( size_s3023201423986296836st_nat @ Xs3 ) ) ) ) ).
% gen_length_def
thf(fact_255_gen__length__def,axiom,
( gen_length_nat
= ( ^ [N3: nat,Xs3: list_nat] : ( plus_plus_nat @ N3 @ ( size_size_list_nat @ Xs3 ) ) ) ) ).
% gen_length_def
thf(fact_256_set__swap,axiom,
! [I: nat,Xs: list_list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( set_list_nat2 @ ( list_update_list_nat @ ( list_update_list_nat @ Xs @ I @ ( nth_list_nat @ Xs @ J ) ) @ J @ ( nth_list_nat @ Xs @ I ) ) )
= ( set_list_nat2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_257_set__swap,axiom,
! [I: nat,Xs: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I ) ) )
= ( set_nat2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_258_maps__simps_I1_J,axiom,
! [F: nat > list_list_nat,X: nat,Xs: list_nat] :
( ( maps_nat_list_nat @ F @ ( cons_nat @ X @ Xs ) )
= ( append_list_nat @ ( F @ X ) @ ( maps_nat_list_nat @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_259_maps__simps_I1_J,axiom,
! [F: list_nat > list_nat,X: list_nat,Xs: list_list_nat] :
( ( maps_list_nat_nat @ F @ ( cons_list_nat @ X @ Xs ) )
= ( append_nat @ ( F @ X ) @ ( maps_list_nat_nat @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_260_maps__simps_I1_J,axiom,
! [F: list_nat > list_list_nat,X: list_nat,Xs: list_list_nat] :
( ( maps_l5785965478274863235st_nat @ F @ ( cons_list_nat @ X @ Xs ) )
= ( append_list_nat @ ( F @ X ) @ ( maps_l5785965478274863235st_nat @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_261_maps__simps_I1_J,axiom,
! [F: nat > list_nat,X: nat,Xs: list_nat] :
( ( maps_nat_nat @ F @ ( cons_nat @ X @ Xs ) )
= ( append_nat @ ( F @ X ) @ ( maps_nat_nat @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_262_not__in__set__insert,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ~ ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ( ( insert_list_nat @ X @ Xs )
= ( cons_list_nat @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_263_not__in__set__insert,axiom,
! [X: nat,Xs: list_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X @ Xs )
= ( cons_nat @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_264_insert__Nil,axiom,
! [X: list_nat] :
( ( insert_list_nat @ X @ nil_list_nat )
= ( cons_list_nat @ X @ nil_list_nat ) ) ).
% insert_Nil
thf(fact_265_insert__Nil,axiom,
! [X: nat] :
( ( insert_nat @ X @ nil_nat )
= ( cons_nat @ X @ nil_nat ) ) ).
% insert_Nil
thf(fact_266_list__update__overwrite,axiom,
! [Xs: list_nat,I: nat,X: nat,Y: nat] :
( ( list_update_nat @ ( list_update_nat @ Xs @ I @ X ) @ I @ Y )
= ( list_update_nat @ Xs @ I @ Y ) ) ).
% list_update_overwrite
thf(fact_267_list__update__nonempty,axiom,
! [Xs: list_list_nat,K: nat,X: list_nat] :
( ( ( list_update_list_nat @ Xs @ K @ X )
= nil_list_nat )
= ( Xs = nil_list_nat ) ) ).
% list_update_nonempty
thf(fact_268_list__update__nonempty,axiom,
! [Xs: list_nat,K: nat,X: nat] :
( ( ( list_update_nat @ Xs @ K @ X )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% list_update_nonempty
thf(fact_269_length__list__update,axiom,
! [Xs: list_list_nat,I: nat,X: list_nat] :
( ( size_s3023201423986296836st_nat @ ( list_update_list_nat @ Xs @ I @ X ) )
= ( size_s3023201423986296836st_nat @ Xs ) ) ).
% length_list_update
thf(fact_270_length__list__update,axiom,
! [Xs: list_nat,I: nat,X: nat] :
( ( size_size_list_nat @ ( list_update_nat @ Xs @ I @ X ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_list_update
thf(fact_271_list__update__id,axiom,
! [Xs: list_nat,I: nat] :
( ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ I ) )
= Xs ) ).
% list_update_id
thf(fact_272_nth__list__update__neq,axiom,
! [I: nat,J: nat,Xs: list_nat,X: nat] :
( ( I != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
= ( nth_nat @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_273_in__set__insert,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ( ( insert_list_nat @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_274_in__set__insert,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_275_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_276_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_277_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_278_bind__simps_I1_J,axiom,
! [F: nat > list_nat] :
( ( bind_nat_nat @ nil_nat @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_279_list__update__length,axiom,
! [Xs: list_list_nat,X: list_nat,Ys2: list_list_nat,Y: list_nat] :
( ( list_update_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ X @ Ys2 ) ) @ ( size_s3023201423986296836st_nat @ Xs ) @ Y )
= ( append_list_nat @ Xs @ ( cons_list_nat @ Y @ Ys2 ) ) ) ).
% list_update_length
thf(fact_280_list__update__length,axiom,
! [Xs: list_nat,X: nat,Ys2: list_nat,Y: nat] :
( ( list_update_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ Ys2 ) ) @ ( size_size_list_nat @ Xs ) @ Y )
= ( append_nat @ Xs @ ( cons_nat @ Y @ Ys2 ) ) ) ).
% list_update_length
thf(fact_281_nth__list__update__eq,axiom,
! [I: nat,Xs: list_list_nat,X: list_nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_list_nat @ ( list_update_list_nat @ Xs @ I @ X ) @ I )
= X ) ) ).
% nth_list_update_eq
thf(fact_282_nth__list__update__eq,axiom,
! [I: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ I )
= X ) ) ).
% nth_list_update_eq
thf(fact_283_list__update__swap,axiom,
! [I: nat,I4: nat,Xs: list_nat,X: nat,X6: nat] :
( ( I != I4 )
=> ( ( list_update_nat @ ( list_update_nat @ Xs @ I @ X ) @ I4 @ X6 )
= ( list_update_nat @ ( list_update_nat @ Xs @ I4 @ X6 ) @ I @ X ) ) ) ).
% list_update_swap
thf(fact_284_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_285_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_286_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_287_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_288_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_289_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_290_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_291_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_292_list__bind__cong,axiom,
! [Xs: list_nat,Ys2: list_nat,F: nat > list_nat,G: nat > list_nat] :
( ( Xs = Ys2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( bind_nat_nat @ Xs @ F )
= ( bind_nat_nat @ Ys2 @ G ) ) ) ) ).
% list_bind_cong
thf(fact_293_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_294_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_295_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_296_maps__simps_I2_J,axiom,
! [F: nat > list_nat] :
( ( maps_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% maps_simps(2)
thf(fact_297_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_list_nat @ N @ nil_list_nat )
= N ) ).
% gen_length_code(1)
thf(fact_298_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_nat @ N @ nil_nat )
= N ) ).
% gen_length_code(1)
thf(fact_299_set__update__memI,axiom,
! [N: nat,Xs: list_list_nat,X: list_nat] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( member_list_nat @ X @ ( set_list_nat2 @ ( list_update_list_nat @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_300_set__update__memI,axiom,
! [N: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_301_list__update__append1,axiom,
! [I: nat,Xs: list_list_nat,Ys2: list_list_nat,X: list_nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( list_update_list_nat @ ( append_list_nat @ Xs @ Ys2 ) @ I @ X )
= ( append_list_nat @ ( list_update_list_nat @ Xs @ I @ X ) @ Ys2 ) ) ) ).
% list_update_append1
thf(fact_302_list__update__append1,axiom,
! [I: nat,Xs: list_nat,Ys2: list_nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys2 ) @ I @ X )
= ( append_nat @ ( list_update_nat @ Xs @ I @ X ) @ Ys2 ) ) ) ).
% list_update_append1
thf(fact_303_nth__list__update,axiom,
! [I: nat,Xs: list_list_nat,J: nat,X: list_nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth_list_nat @ ( list_update_list_nat @ Xs @ I @ X ) @ J )
= X ) )
& ( ( I != J )
=> ( ( nth_list_nat @ ( list_update_list_nat @ Xs @ I @ X ) @ J )
= ( nth_list_nat @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_304_nth__list__update,axiom,
! [I: nat,Xs: list_nat,J: nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
= X ) )
& ( ( I != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
= ( nth_nat @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_305_list__update__same__conv,axiom,
! [I: nat,Xs: list_list_nat,X: list_nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( ( list_update_list_nat @ Xs @ I @ X )
= Xs )
= ( ( nth_list_nat @ Xs @ I )
= X ) ) ) ).
% list_update_same_conv
thf(fact_306_list__update__same__conv,axiom,
! [I: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ( list_update_nat @ Xs @ I @ X )
= Xs )
= ( ( nth_nat @ Xs @ I )
= X ) ) ) ).
% list_update_same_conv
thf(fact_307_List_Oinsert__def,axiom,
( insert_list_nat
= ( ^ [X3: list_nat,Xs3: list_list_nat] : ( if_list_list_nat @ ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs3 ) ) @ Xs3 @ ( cons_list_nat @ X3 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_308_List_Oinsert__def,axiom,
( insert_nat
= ( ^ [X3: nat,Xs3: list_nat] : ( if_list_nat @ ( member_nat @ X3 @ ( set_nat2 @ Xs3 ) ) @ Xs3 @ ( cons_nat @ X3 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_309_rgf__limit_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( equiva5889994315859557365_limit @ ( cons_nat @ X @ Xs ) )
= ( ord_max_nat @ ( plus_plus_nat @ X @ one_one_nat ) @ ( equiva5889994315859557365_limit @ Xs ) ) ) ).
% rgf_limit.simps(2)
thf(fact_310_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_311_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_312_max_Oabsorb4,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_max_num @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_313_max_Oabsorb4,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_max_nat @ A @ B )
= B ) ) ).
% max.absorb4
thf(fact_314_max_Oabsorb3,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_max_num @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_315_max_Oabsorb3,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_max_nat @ A @ B )
= A ) ) ).
% max.absorb3
thf(fact_316_max_Oidem,axiom,
! [A: num] :
( ( ord_max_num @ A @ A )
= A ) ).
% max.idem
thf(fact_317_max_Oidem,axiom,
! [A: nat] :
( ( ord_max_nat @ A @ A )
= A ) ).
% max.idem
thf(fact_318_max_Oleft__idem,axiom,
! [A: num,B: num] :
( ( ord_max_num @ A @ ( ord_max_num @ A @ B ) )
= ( ord_max_num @ A @ B ) ) ).
% max.left_idem
thf(fact_319_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_320_max_Oright__idem,axiom,
! [A: num,B: num] :
( ( ord_max_num @ ( ord_max_num @ A @ B ) @ B )
= ( ord_max_num @ A @ B ) ) ).
% max.right_idem
thf(fact_321_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_322_prefixes__snoc,axiom,
! [Xs: list_list_nat,X: list_nat] :
( ( prefixes_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) ) )
= ( append_list_list_nat @ ( prefixes_list_nat @ Xs ) @ ( cons_list_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) ) @ nil_list_list_nat ) ) ) ).
% prefixes_snoc
thf(fact_323_prefixes__snoc,axiom,
! [Xs: list_nat,X: nat] :
( ( prefixes_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= ( append_list_nat @ ( prefixes_nat @ Xs ) @ ( cons_list_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) @ nil_list_nat ) ) ) ).
% prefixes_snoc
thf(fact_324_rgf__limit_Oelims,axiom,
! [X: list_nat,Y: nat] :
( ( ( equiva5889994315859557365_limit @ X )
= Y )
=> ( ( ( X = nil_nat )
=> ( Y != zero_zero_nat ) )
=> ~ ! [X2: nat,Xs2: list_nat] :
( ( X
= ( cons_nat @ X2 @ Xs2 ) )
=> ( Y
!= ( ord_max_nat @ ( plus_plus_nat @ X2 @ one_one_nat ) @ ( equiva5889994315859557365_limit @ Xs2 ) ) ) ) ) ) ).
% rgf_limit.elims
thf(fact_325_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_326_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_327_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_328_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_329_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_330_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_331_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_332_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_333_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_334_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_335_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_336_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_337_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_338_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_339_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_340_max__nat_Oleft__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ zero_zero_nat @ A )
= A ) ).
% max_nat.left_neutral
thf(fact_341_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_342_max__nat_Oright__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ A @ zero_zero_nat )
= A ) ).
% max_nat.right_neutral
thf(fact_343_max__0L,axiom,
! [N: nat] :
( ( ord_max_nat @ zero_zero_nat @ N )
= N ) ).
% max_0L
thf(fact_344_max__0R,axiom,
! [N: nat] :
( ( ord_max_nat @ N @ zero_zero_nat )
= N ) ).
% max_0R
thf(fact_345_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_346_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_347_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_348_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_349_length__0__conv,axiom,
! [Xs: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_list_nat ) ) ).
% length_0_conv
thf(fact_350_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_351_max__0__1_I2_J,axiom,
( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
= one_one_nat ) ).
% max_0_1(2)
thf(fact_352_max__0__1_I1_J,axiom,
( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
= one_one_nat ) ).
% max_0_1(1)
thf(fact_353_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_354_nth__Cons__0,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( nth_list_nat @ ( cons_list_nat @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_355_nth__Cons__0,axiom,
! [X: nat,Xs: list_nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_356_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_357_length__greater__0__conv,axiom,
! [Xs: list_list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s3023201423986296836st_nat @ Xs ) )
= ( Xs != nil_list_nat ) ) ).
% length_greater_0_conv
thf(fact_358_length__greater__0__conv,axiom,
! [Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) )
= ( Xs != nil_nat ) ) ).
% length_greater_0_conv
thf(fact_359_length__prefixes,axiom,
! [Xs: list_list_nat] :
( ( size_s6248950052170075156st_nat @ ( prefixes_list_nat @ Xs ) )
= ( plus_plus_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ one_one_nat ) ) ).
% length_prefixes
thf(fact_360_length__prefixes,axiom,
! [Xs: list_nat] :
( ( size_s3023201423986296836st_nat @ ( prefixes_nat @ Xs ) )
= ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ).
% length_prefixes
thf(fact_361_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_362_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_363_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_364_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_365_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_366_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_367_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_368_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_369_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_370_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_371_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_372_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_373_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_374_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_375_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_376_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_377_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_378_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_379_prefixes__not__Nil,axiom,
! [Xs: list_nat] :
( ( prefixes_nat @ Xs )
!= nil_list_nat ) ).
% prefixes_not_Nil
thf(fact_380_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_381_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_382_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_383_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_384_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_385_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_386_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_387_list_Osize_I3_J,axiom,
( ( size_s3023201423986296836st_nat @ nil_list_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_388_list_Osize_I3_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_389_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_390_list__update__code_I2_J,axiom,
! [X: list_nat,Xs: list_list_nat,Y: list_nat] :
( ( list_update_list_nat @ ( cons_list_nat @ X @ Xs ) @ zero_zero_nat @ Y )
= ( cons_list_nat @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_391_list__update__code_I2_J,axiom,
! [X: nat,Xs: list_nat,Y: nat] :
( ( list_update_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat @ Y )
= ( cons_nat @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_392_rgf__limit_Osimps_I1_J,axiom,
( ( equiva5889994315859557365_limit @ nil_nat )
= zero_zero_nat ) ).
% rgf_limit.simps(1)
thf(fact_393_length__code,axiom,
( size_s3023201423986296836st_nat
= ( gen_length_list_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_394_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_395_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_396_length__pos__if__in__set,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_397_length__pos__if__in__set,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_398_max_Oleft__commute,axiom,
! [B: num,A: num,C: num] :
( ( ord_max_num @ B @ ( ord_max_num @ A @ C ) )
= ( ord_max_num @ A @ ( ord_max_num @ B @ C ) ) ) ).
% max.left_commute
thf(fact_399_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_400_max_Ocommute,axiom,
( ord_max_num
= ( ^ [A2: num,B2: num] : ( ord_max_num @ B2 @ A2 ) ) ) ).
% max.commute
thf(fact_401_max_Ocommute,axiom,
( ord_max_nat
= ( ^ [A2: nat,B2: nat] : ( ord_max_nat @ B2 @ A2 ) ) ) ).
% max.commute
thf(fact_402_max_Oassoc,axiom,
! [A: num,B: num,C: num] :
( ( ord_max_num @ ( ord_max_num @ A @ B ) @ C )
= ( ord_max_num @ A @ ( ord_max_num @ B @ C ) ) ) ).
% max.assoc
thf(fact_403_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_404_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_405_prefixes_Osimps_I1_J,axiom,
( ( prefixes_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% prefixes.simps(1)
thf(fact_406_prefixes__eq__snoc,axiom,
! [Ys2: list_list_nat,Xs: list_list_list_nat,X: list_list_nat] :
( ( ( prefixes_list_nat @ Ys2 )
= ( append_list_list_nat @ Xs @ ( cons_list_list_nat @ X @ nil_list_list_nat ) ) )
= ( ( ( ( Ys2 = nil_list_nat )
& ( Xs = nil_list_list_nat ) )
| ? [Z5: list_nat,Zs2: list_list_nat] :
( ( Ys2
= ( append_list_nat @ Zs2 @ ( cons_list_nat @ Z5 @ nil_list_nat ) ) )
& ( Xs
= ( prefixes_list_nat @ Zs2 ) ) ) )
& ( X = Ys2 ) ) ) ).
% prefixes_eq_snoc
thf(fact_407_prefixes__eq__snoc,axiom,
! [Ys2: list_nat,Xs: list_list_nat,X: list_nat] :
( ( ( prefixes_nat @ Ys2 )
= ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) ) )
= ( ( ( ( Ys2 = nil_nat )
& ( Xs = nil_list_nat ) )
| ? [Z5: nat,Zs2: list_nat] :
( ( Ys2
= ( append_nat @ Zs2 @ ( cons_nat @ Z5 @ nil_nat ) ) )
& ( Xs
= ( prefixes_nat @ Zs2 ) ) ) )
& ( X = Ys2 ) ) ) ).
% prefixes_eq_snoc
thf(fact_408_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_409_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_410_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_411_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_412_max_Ostrict__order__iff,axiom,
( ord_less_num
= ( ^ [B2: num,A2: num] :
( ( A2
= ( ord_max_num @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ) ).
% max.strict_order_iff
thf(fact_413_max_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( A2
= ( ord_max_nat @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ) ).
% max.strict_order_iff
thf(fact_414_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_415_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_416_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_417_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_418_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_419_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_420_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_421_n__lists_Osimps_I1_J,axiom,
! [Xs: list_list_nat] :
( ( n_lists_list_nat @ zero_zero_nat @ Xs )
= ( cons_list_list_nat @ nil_list_nat @ nil_list_list_nat ) ) ).
% n_lists.simps(1)
thf(fact_422_n__lists_Osimps_I1_J,axiom,
! [Xs: list_nat] :
( ( n_lists_nat @ zero_zero_nat @ Xs )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% n_lists.simps(1)
thf(fact_423_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_424_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_425_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_426_suffixes__eq__snoc,axiom,
! [Ys2: list_list_nat,Xs: list_list_list_nat,X: list_list_nat] :
( ( ( suffixes_list_nat @ Ys2 )
= ( append_list_list_nat @ Xs @ ( cons_list_list_nat @ X @ nil_list_list_nat ) ) )
= ( ( ( ( Ys2 = nil_list_nat )
& ( Xs = nil_list_list_nat ) )
| ? [Z5: list_nat,Zs2: list_list_nat] :
( ( Ys2
= ( cons_list_nat @ Z5 @ Zs2 ) )
& ( Xs
= ( suffixes_list_nat @ Zs2 ) ) ) )
& ( X = Ys2 ) ) ) ).
% suffixes_eq_snoc
thf(fact_427_suffixes__eq__snoc,axiom,
! [Ys2: list_nat,Xs: list_list_nat,X: list_nat] :
( ( ( suffixes_nat @ Ys2 )
= ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) ) )
= ( ( ( ( Ys2 = nil_nat )
& ( Xs = nil_list_nat ) )
| ? [Z5: nat,Zs2: list_nat] :
( ( Ys2
= ( cons_nat @ Z5 @ Zs2 ) )
& ( Xs
= ( suffixes_nat @ Zs2 ) ) ) )
& ( X = Ys2 ) ) ) ).
% suffixes_eq_snoc
thf(fact_428_suffixes__not__Nil,axiom,
! [Xs: list_nat] :
( ( suffixes_nat @ Xs )
!= nil_list_nat ) ).
% suffixes_not_Nil
thf(fact_429_length__n__lists__elem,axiom,
! [Ys2: list_list_nat,N: nat,Xs: list_list_nat] :
( ( member_list_list_nat @ Ys2 @ ( set_list_list_nat2 @ ( n_lists_list_nat @ N @ Xs ) ) )
=> ( ( size_s3023201423986296836st_nat @ Ys2 )
= N ) ) ).
% length_n_lists_elem
thf(fact_430_length__n__lists__elem,axiom,
! [Ys2: list_nat,N: nat,Xs: list_nat] :
( ( member_list_nat @ Ys2 @ ( set_list_nat2 @ ( n_lists_nat @ N @ Xs ) ) )
=> ( ( size_size_list_nat @ Ys2 )
= N ) ) ).
% length_n_lists_elem
thf(fact_431_verit__comp__simplify1_I1_J,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_432_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_433_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_434_suffixes_Osimps_I1_J,axiom,
( ( suffixes_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% suffixes.simps(1)
thf(fact_435_suffixes_Osimps_I2_J,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( suffixes_list_nat @ ( cons_list_nat @ X @ Xs ) )
= ( append_list_list_nat @ ( suffixes_list_nat @ Xs ) @ ( cons_list_list_nat @ ( cons_list_nat @ X @ Xs ) @ nil_list_list_nat ) ) ) ).
% suffixes.simps(2)
thf(fact_436_suffixes_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( suffixes_nat @ ( cons_nat @ X @ Xs ) )
= ( append_list_nat @ ( suffixes_nat @ Xs ) @ ( cons_list_nat @ ( cons_nat @ X @ Xs ) @ nil_list_nat ) ) ) ).
% suffixes.simps(2)
thf(fact_437_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_438_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_439_sublists_Osimps_I1_J,axiom,
( ( sublists_nat @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% sublists.simps(1)
thf(fact_440_rgf__limit_Opelims,axiom,
! [X: list_nat,Y: nat] :
( ( ( equiva5889994315859557365_limit @ X )
= Y )
=> ( ( accp_list_nat @ equiva5575797544161152836it_rel @ X )
=> ( ( ( X = nil_nat )
=> ( ( Y = zero_zero_nat )
=> ~ ( accp_list_nat @ equiva5575797544161152836it_rel @ nil_nat ) ) )
=> ~ ! [X2: nat,Xs2: list_nat] :
( ( X
= ( cons_nat @ X2 @ Xs2 ) )
=> ( ( Y
= ( ord_max_nat @ ( plus_plus_nat @ X2 @ one_one_nat ) @ ( equiva5889994315859557365_limit @ Xs2 ) ) )
=> ~ ( accp_list_nat @ equiva5575797544161152836it_rel @ ( cons_nat @ X2 @ Xs2 ) ) ) ) ) ) ) ).
% rgf_limit.pelims
thf(fact_441_nth__Cons__pos,axiom,
! [N: nat,X: list_nat,Xs: list_list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_list_nat @ ( cons_list_nat @ X @ Xs ) @ N )
= ( nth_list_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_442_nth__Cons__pos,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_443_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_444_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_445_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_446_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_447_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_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_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_453_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_454_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_455_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_456_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_457_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_458_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_459_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_460_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_461_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_462_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_463_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_464_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_465_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_466_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_467_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_468_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_469_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_470_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_471_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_472_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_473_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_474_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_475_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_476_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_477_nth__Cons_H,axiom,
! [N: nat,X: list_nat,Xs: list_list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( nth_list_nat @ ( cons_list_nat @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_list_nat @ ( cons_list_nat @ X @ Xs ) @ N )
= ( nth_list_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_478_nth__Cons_H,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_479_nth__append,axiom,
! [N: nat,Xs: list_list_nat,Ys2: list_list_nat] :
( ( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_list_nat @ ( append_list_nat @ Xs @ Ys2 ) @ N )
= ( nth_list_nat @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( nth_list_nat @ ( append_list_nat @ Xs @ Ys2 ) @ N )
= ( nth_list_nat @ Ys2 @ ( minus_minus_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_480_nth__append,axiom,
! [N: nat,Xs: list_nat,Ys2: list_nat] :
( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ Ys2 ) @ N )
= ( nth_nat @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ Ys2 ) @ N )
= ( nth_nat @ Ys2 @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_481_list__update__append,axiom,
! [N: nat,Xs: list_list_nat,Ys2: list_list_nat,X: list_nat] :
( ( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( list_update_list_nat @ ( append_list_nat @ Xs @ Ys2 ) @ N @ X )
= ( append_list_nat @ ( list_update_list_nat @ Xs @ N @ X ) @ Ys2 ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( list_update_list_nat @ ( append_list_nat @ Xs @ Ys2 ) @ N @ X )
= ( append_list_nat @ Xs @ ( list_update_list_nat @ Ys2 @ ( minus_minus_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) ) @ X ) ) ) ) ) ).
% list_update_append
thf(fact_482_list__update__append,axiom,
! [N: nat,Xs: list_nat,Ys2: list_nat,X: nat] :
( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys2 ) @ N @ X )
= ( append_nat @ ( list_update_nat @ Xs @ N @ X ) @ Ys2 ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys2 ) @ N @ X )
= ( append_nat @ Xs @ ( list_update_nat @ Ys2 @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ X ) ) ) ) ) ).
% list_update_append
thf(fact_483_in__set__product__lists__length,axiom,
! [Xs: list_list_nat,Xss2: list_list_list_nat] :
( ( member_list_list_nat @ Xs @ ( set_list_list_nat2 @ ( produc6783906451316923569st_nat @ Xss2 ) ) )
=> ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s6248950052170075156st_nat @ Xss2 ) ) ) ).
% in_set_product_lists_length
thf(fact_484_in__set__product__lists__length,axiom,
! [Xs: list_nat,Xss2: list_list_nat] :
( ( member_list_nat @ Xs @ ( set_list_nat2 @ ( product_lists_nat @ Xss2 ) ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Xss2 ) ) ) ).
% in_set_product_lists_length
thf(fact_485_nth__non__equal__first__eq,axiom,
! [X: list_nat,Y: list_nat,Xs: list_list_nat,N: nat] :
( ( X != Y )
=> ( ( ( nth_list_nat @ ( cons_list_nat @ X @ Xs ) @ N )
= Y )
= ( ( ( nth_list_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_486_nth__non__equal__first__eq,axiom,
! [X: nat,Y: nat,Xs: list_nat,N: nat] :
( ( X != Y )
=> ( ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= Y )
= ( ( ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_487_append__one__prefix,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat] :
( ( prefix_list_nat @ Xs @ Ys2 )
=> ( ( ord_less_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( prefix_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ ( nth_list_nat @ Ys2 @ ( size_s3023201423986296836st_nat @ Xs ) ) @ nil_list_nat ) ) @ Ys2 ) ) ) ).
% append_one_prefix
thf(fact_488_append__one__prefix,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( prefix_nat @ Xs @ Ys2 )
=> ( ( ord_less_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys2 ) )
=> ( prefix_nat @ ( append_nat @ Xs @ ( cons_nat @ ( nth_nat @ Ys2 @ ( size_size_list_nat @ Xs ) ) @ nil_nat ) ) @ Ys2 ) ) ) ).
% append_one_prefix
thf(fact_489_last__list__update,axiom,
! [Xs: list_list_nat,K: nat,X: list_nat] :
( ( Xs != nil_list_nat )
=> ( ( ( K
= ( minus_minus_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ one_one_nat ) )
=> ( ( last_list_nat @ ( list_update_list_nat @ Xs @ K @ X ) )
= X ) )
& ( ( K
!= ( minus_minus_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ one_one_nat ) )
=> ( ( last_list_nat @ ( list_update_list_nat @ Xs @ K @ X ) )
= ( last_list_nat @ Xs ) ) ) ) ) ).
% last_list_update
thf(fact_490_last__list__update,axiom,
! [Xs: list_nat,K: nat,X: nat] :
( ( Xs != nil_nat )
=> ( ( ( K
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
=> ( ( last_nat @ ( list_update_nat @ Xs @ K @ X ) )
= X ) )
& ( ( K
!= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
=> ( ( last_nat @ ( list_update_nat @ Xs @ K @ X ) )
= ( last_nat @ Xs ) ) ) ) ) ).
% last_list_update
thf(fact_491_last__conv__nth,axiom,
! [Xs: list_list_nat] :
( ( Xs != nil_list_nat )
=> ( ( last_list_nat @ Xs )
= ( nth_list_nat @ Xs @ ( minus_minus_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_492_last__conv__nth,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( last_nat @ Xs )
= ( nth_nat @ Xs @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_493_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,Xs3: 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 @ Xs3 @ Xs6 ) @ Xss22 ) ) )
& ( Ys2
= ( append_list_nat @ ( concat_list_nat @ Xss1 ) @ Xs3 ) )
& ( Zs
= ( append_list_nat @ Xs6 @ ( concat_list_nat @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_494_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,Xs3: list_nat,Xs6: list_nat,Xss22: list_list_nat] :
( ( Xss2
= ( append_list_nat @ Xss1 @ ( cons_list_nat @ ( append_nat @ Xs3 @ Xs6 ) @ Xss22 ) ) )
& ( Ys2
= ( append_nat @ ( concat_nat @ Xss1 ) @ Xs3 ) )
& ( Zs
= ( append_nat @ Xs6 @ ( concat_nat @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_495_nth__equal__first__eq,axiom,
! [X: list_nat,Xs: list_list_nat,N: nat] :
( ~ ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_s3023201423986296836st_nat @ Xs ) )
=> ( ( ( nth_list_nat @ ( cons_list_nat @ X @ Xs ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_496_nth__equal__first__eq,axiom,
! [X: nat,Xs: list_nat,N: nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
= X )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_497_prefix__order_Odual__order_Orefl,axiom,
! [A: list_nat] : ( prefix_nat @ A @ A ) ).
% prefix_order.dual_order.refl
thf(fact_498_prefix__order_Oorder__refl,axiom,
! [X: list_nat] : ( prefix_nat @ X @ X ) ).
% prefix_order.order_refl
thf(fact_499_last__prefixes,axiom,
! [Xs: list_nat] :
( ( last_list_nat @ ( prefixes_nat @ Xs ) )
= Xs ) ).
% last_prefixes
thf(fact_500_last__suffixes,axiom,
! [Xs: list_nat] :
( ( last_list_nat @ ( suffixes_nat @ Xs ) )
= Xs ) ).
% last_suffixes
thf(fact_501_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_502_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_503_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_504_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_505_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_506_Cons__prefix__Cons,axiom,
! [X: list_nat,Xs: list_list_nat,Y: list_nat,Ys2: list_list_nat] :
( ( prefix_list_nat @ ( cons_list_nat @ X @ Xs ) @ ( cons_list_nat @ Y @ Ys2 ) )
= ( ( X = Y )
& ( prefix_list_nat @ Xs @ Ys2 ) ) ) ).
% Cons_prefix_Cons
thf(fact_507_Cons__prefix__Cons,axiom,
! [X: nat,Xs: list_nat,Y: nat,Ys2: list_nat] :
( ( prefix_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys2 ) )
= ( ( X = Y )
& ( prefix_nat @ Xs @ Ys2 ) ) ) ).
% Cons_prefix_Cons
thf(fact_508_max_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_max_nat @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_509_max_Oabsorb1,axiom,
! [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_max_num @ A @ B )
= A ) ) ).
% max.absorb1
thf(fact_510_max_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_max_nat @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_511_max_Oabsorb2,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_max_num @ A @ B )
= B ) ) ).
% max.absorb2
thf(fact_512_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_513_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_514_prefix__code_I1_J,axiom,
! [Xs: list_list_nat] : ( prefix_list_nat @ nil_list_nat @ Xs ) ).
% prefix_code(1)
thf(fact_515_prefix__code_I1_J,axiom,
! [Xs: list_nat] : ( prefix_nat @ nil_nat @ Xs ) ).
% prefix_code(1)
thf(fact_516_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_517_prefix__bot_Oextremum__unique,axiom,
! [A: list_nat] :
( ( prefix_nat @ A @ nil_nat )
= ( A = nil_nat ) ) ).
% prefix_bot.extremum_unique
thf(fact_518_prefix__Nil,axiom,
! [Xs: list_list_nat] :
( ( prefix_list_nat @ Xs @ nil_list_nat )
= ( Xs = nil_list_nat ) ) ).
% prefix_Nil
thf(fact_519_prefix__Nil,axiom,
! [Xs: list_nat] :
( ( prefix_nat @ Xs @ nil_nat )
= ( Xs = nil_nat ) ) ).
% prefix_Nil
thf(fact_520_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_521_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_522_same__prefix__prefix,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat,Zs: list_list_nat] :
( ( prefix_list_nat @ ( append_list_nat @ Xs @ Ys2 ) @ ( append_list_nat @ Xs @ Zs ) )
= ( prefix_list_nat @ Ys2 @ Zs ) ) ).
% same_prefix_prefix
thf(fact_523_same__prefix__prefix,axiom,
! [Xs: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( prefix_nat @ ( append_nat @ Xs @ Ys2 ) @ ( append_nat @ Xs @ Zs ) )
= ( prefix_nat @ Ys2 @ Zs ) ) ).
% same_prefix_prefix
thf(fact_524_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_525_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_526_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_527_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_528_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_529_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_530_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_531_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_532_same__prefix__nil,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat] :
( ( prefix_list_nat @ ( append_list_nat @ Xs @ Ys2 ) @ Xs )
= ( Ys2 = nil_list_nat ) ) ).
% same_prefix_nil
thf(fact_533_same__prefix__nil,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( prefix_nat @ ( append_nat @ Xs @ Ys2 ) @ Xs )
= ( Ys2 = nil_nat ) ) ).
% same_prefix_nil
thf(fact_534_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_535_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_536_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_537_list__update__beyond,axiom,
! [Xs: list_list_nat,I: nat,X: list_nat] :
( ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ I )
=> ( ( list_update_list_nat @ Xs @ I @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_538_list__update__beyond,axiom,
! [Xs: list_nat,I: nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I )
=> ( ( list_update_nat @ Xs @ I @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_539_last__appendL,axiom,
! [Ys2: list_list_nat,Xs: list_list_nat] :
( ( Ys2 = nil_list_nat )
=> ( ( last_list_nat @ ( append_list_nat @ Xs @ Ys2 ) )
= ( last_list_nat @ Xs ) ) ) ).
% last_appendL
thf(fact_540_last__appendL,axiom,
! [Ys2: list_nat,Xs: list_nat] :
( ( Ys2 = nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( last_nat @ Xs ) ) ) ).
% last_appendL
thf(fact_541_last__appendR,axiom,
! [Ys2: list_list_nat,Xs: list_list_nat] :
( ( Ys2 != nil_list_nat )
=> ( ( last_list_nat @ ( append_list_nat @ Xs @ Ys2 ) )
= ( last_list_nat @ Ys2 ) ) ) ).
% last_appendR
thf(fact_542_last__appendR,axiom,
! [Ys2: list_nat,Xs: list_nat] :
( ( Ys2 != nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( last_nat @ Ys2 ) ) ) ).
% last_appendR
thf(fact_543_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_544_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_545_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_546_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_547_concat__append,axiom,
! [Xs: list_list_list_nat,Ys2: list_list_list_nat] :
( ( concat_list_nat @ ( append_list_list_nat @ Xs @ Ys2 ) )
= ( append_list_nat @ ( concat_list_nat @ Xs ) @ ( concat_list_nat @ Ys2 ) ) ) ).
% concat_append
thf(fact_548_concat__append,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat] :
( ( concat_nat @ ( append_list_nat @ Xs @ Ys2 ) )
= ( append_nat @ ( concat_nat @ Xs ) @ ( concat_nat @ Ys2 ) ) ) ).
% concat_append
thf(fact_549_in__set__prefixes,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( member_list_nat @ Xs @ ( set_list_nat2 @ ( prefixes_nat @ Ys2 ) ) )
= ( prefix_nat @ Xs @ Ys2 ) ) ).
% in_set_prefixes
thf(fact_550_prefix__snoc,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat,Y: list_nat] :
( ( prefix_list_nat @ Xs @ ( append_list_nat @ Ys2 @ ( cons_list_nat @ Y @ nil_list_nat ) ) )
= ( ( Xs
= ( append_list_nat @ Ys2 @ ( cons_list_nat @ Y @ nil_list_nat ) ) )
| ( prefix_list_nat @ Xs @ Ys2 ) ) ) ).
% prefix_snoc
thf(fact_551_prefix__snoc,axiom,
! [Xs: list_nat,Ys2: list_nat,Y: nat] :
( ( prefix_nat @ Xs @ ( append_nat @ Ys2 @ ( cons_nat @ Y @ nil_nat ) ) )
= ( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ Y @ nil_nat ) ) )
| ( prefix_nat @ Xs @ Ys2 ) ) ) ).
% prefix_snoc
thf(fact_552_last__snoc,axiom,
! [Xs: list_list_nat,X: list_nat] :
( ( last_list_nat @ ( append_list_nat @ Xs @ ( cons_list_nat @ X @ nil_list_nat ) ) )
= X ) ).
% last_snoc
thf(fact_553_last__snoc,axiom,
! [Xs: list_nat,X: nat] :
( ( last_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= X ) ).
% last_snoc
thf(fact_554_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 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_555_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_556_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_557_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_558_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_559_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_560_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_561_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_562_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_563_prefix__order_Odual__order_Oeq__iff,axiom,
( ( ^ [Y3: list_nat,Z: list_nat] : ( Y3 = Z ) )
= ( ^ [A2: list_nat,B2: list_nat] :
( ( prefix_nat @ B2 @ A2 )
& ( prefix_nat @ A2 @ B2 ) ) ) ) ).
% prefix_order.dual_order.eq_iff
thf(fact_564_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_565_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_566_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_567_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_568_prefix__order_Oorder__eq__iff,axiom,
( ( ^ [Y3: list_nat,Z: list_nat] : ( Y3 = Z ) )
= ( ^ [X3: list_nat,Y4: list_nat] :
( ( prefix_nat @ X3 @ Y4 )
& ( prefix_nat @ Y4 @ X3 ) ) ) ) ).
% prefix_order.order_eq_iff
thf(fact_569_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_570_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_571_prefix__order_Oeq__refl,axiom,
! [X: list_nat,Y: list_nat] :
( ( X = Y )
=> ( prefix_nat @ X @ Y ) ) ).
% prefix_order.eq_refl
thf(fact_572_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_573_prefix__order_Oeq__iff,axiom,
( ( ^ [Y3: list_nat,Z: list_nat] : ( Y3 = Z ) )
= ( ^ [A2: list_nat,B2: list_nat] :
( ( prefix_nat @ A2 @ B2 )
& ( prefix_nat @ B2 @ A2 ) ) ) ) ).
% prefix_order.eq_iff
thf(fact_574_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_575_prefix__length__le,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat] :
( ( prefix_list_nat @ Xs @ Ys2 )
=> ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ ( size_s3023201423986296836st_nat @ Ys2 ) ) ) ).
% prefix_length_le
thf(fact_576_prefix__length__le,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( prefix_nat @ Xs @ Ys2 )
=> ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys2 ) ) ) ).
% prefix_length_le
thf(fact_577_prefix__length__prefix,axiom,
! [Ps: list_list_nat,Xs: list_list_nat,Qs: list_list_nat] :
( ( prefix_list_nat @ Ps @ Xs )
=> ( ( prefix_list_nat @ Qs @ Xs )
=> ( ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Ps ) @ ( size_s3023201423986296836st_nat @ Qs ) )
=> ( prefix_list_nat @ Ps @ Qs ) ) ) ) ).
% prefix_length_prefix
thf(fact_578_prefix__length__prefix,axiom,
! [Ps: list_nat,Xs: list_nat,Qs: list_nat] :
( ( prefix_nat @ Ps @ Xs )
=> ( ( prefix_nat @ Qs @ Xs )
=> ( ( ord_less_eq_nat @ ( size_size_list_nat @ Ps ) @ ( size_size_list_nat @ Qs ) )
=> ( prefix_nat @ Ps @ Qs ) ) ) ) ).
% prefix_length_prefix
thf(fact_579_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_580_verit__la__disequality,axiom,
! [A: num,B: num] :
( ( A = B )
| ~ ( ord_less_eq_num @ A @ B )
| ~ ( ord_less_eq_num @ B @ A ) ) ).
% verit_la_disequality
thf(fact_581_verit__comp__simplify1_I2_J,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_582_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_583_verit__comp__simplify1_I2_J,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_584_prefix__bot_Obot__least,axiom,
! [A: list_list_nat] : ( prefix_list_nat @ nil_list_nat @ A ) ).
% prefix_bot.bot_least
thf(fact_585_prefix__bot_Obot__least,axiom,
! [A: list_nat] : ( prefix_nat @ nil_nat @ A ) ).
% prefix_bot.bot_least
thf(fact_586_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_587_prefix__bot_Oextremum__uniqueI,axiom,
! [A: list_nat] :
( ( prefix_nat @ A @ nil_nat )
=> ( A = nil_nat ) ) ).
% prefix_bot.extremum_uniqueI
thf(fact_588_Nil__prefix,axiom,
! [Xs: list_list_nat] : ( prefix_list_nat @ nil_list_nat @ Xs ) ).
% Nil_prefix
thf(fact_589_Nil__prefix,axiom,
! [Xs: list_nat] : ( prefix_nat @ nil_nat @ Xs ) ).
% Nil_prefix
thf(fact_590_prefixE,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat] :
( ( prefix_list_nat @ Xs @ Ys2 )
=> ~ ! [Zs3: list_list_nat] :
( Ys2
!= ( append_list_nat @ Xs @ Zs3 ) ) ) ).
% prefixE
thf(fact_591_prefixE,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( prefix_nat @ Xs @ Ys2 )
=> ~ ! [Zs3: list_nat] :
( Ys2
!= ( append_nat @ Xs @ Zs3 ) ) ) ).
% prefixE
thf(fact_592_prefixI,axiom,
! [Ys2: list_list_nat,Xs: list_list_nat,Zs: list_list_nat] :
( ( Ys2
= ( append_list_nat @ Xs @ Zs ) )
=> ( prefix_list_nat @ Xs @ Ys2 ) ) ).
% prefixI
thf(fact_593_prefixI,axiom,
! [Ys2: list_nat,Xs: list_nat,Zs: list_nat] :
( ( Ys2
= ( append_nat @ Xs @ Zs ) )
=> ( prefix_nat @ Xs @ Ys2 ) ) ).
% prefixI
thf(fact_594_prefix__def,axiom,
( prefix_list_nat
= ( ^ [Xs3: list_list_nat,Ys3: list_list_nat] :
? [Zs2: list_list_nat] :
( Ys3
= ( append_list_nat @ Xs3 @ Zs2 ) ) ) ) ).
% prefix_def
thf(fact_595_prefix__def,axiom,
( prefix_nat
= ( ^ [Xs3: list_nat,Ys3: list_nat] :
? [Zs2: list_nat] :
( Ys3
= ( append_nat @ Xs3 @ Zs2 ) ) ) ) ).
% prefix_def
thf(fact_596_prefix__append,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat,Zs: list_list_nat] :
( ( prefix_list_nat @ Xs @ ( append_list_nat @ Ys2 @ Zs ) )
= ( ( prefix_list_nat @ Xs @ Ys2 )
| ? [Us2: list_list_nat] :
( ( Xs
= ( append_list_nat @ Ys2 @ Us2 ) )
& ( prefix_list_nat @ Us2 @ Zs ) ) ) ) ).
% prefix_append
thf(fact_597_prefix__append,axiom,
! [Xs: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( prefix_nat @ Xs @ ( append_nat @ Ys2 @ Zs ) )
= ( ( prefix_nat @ Xs @ Ys2 )
| ? [Us2: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ Us2 ) )
& ( prefix_nat @ Us2 @ Zs ) ) ) ) ).
% prefix_append
thf(fact_598_prefix__prefix,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat,Zs: list_list_nat] :
( ( prefix_list_nat @ Xs @ Ys2 )
=> ( prefix_list_nat @ Xs @ ( append_list_nat @ Ys2 @ Zs ) ) ) ).
% prefix_prefix
thf(fact_599_prefix__prefix,axiom,
! [Xs: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( prefix_nat @ Xs @ Ys2 )
=> ( prefix_nat @ Xs @ ( append_nat @ Ys2 @ Zs ) ) ) ).
% prefix_prefix
thf(fact_600_append__prefixD,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat,Zs: list_list_nat] :
( ( prefix_list_nat @ ( append_list_nat @ Xs @ Ys2 ) @ Zs )
=> ( prefix_list_nat @ Xs @ Zs ) ) ).
% append_prefixD
thf(fact_601_append__prefixD,axiom,
! [Xs: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( prefix_nat @ ( append_nat @ Xs @ Ys2 ) @ Zs )
=> ( prefix_nat @ Xs @ Zs ) ) ).
% append_prefixD
thf(fact_602_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_603_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_604_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_605_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_606_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_607_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_608_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
? [C3: nat] :
( B2
= ( plus_plus_nat @ A2 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_609_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_610_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_611_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_612_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_613_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_614_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_615_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_616_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_617_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_618_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_619_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_620_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_621_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_622_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_623_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_624_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_625_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_626_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
& ( M5 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_627_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_628_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_629_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_630_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_631_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_632_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_633_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_634_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_635_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_636_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_637_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_638_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_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__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_641_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_642_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_643_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_644_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_645_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_646_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_647_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_648_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_649_max_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_max_nat @ A2 @ B2 )
= B2 ) ) ) ).
% max.absorb_iff2
thf(fact_650_max_Oabsorb__iff2,axiom,
( ord_less_eq_num
= ( ^ [A2: num,B2: num] :
( ( ord_max_num @ A2 @ B2 )
= B2 ) ) ) ).
% max.absorb_iff2
thf(fact_651_max_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_max_nat @ A2 @ B2 )
= A2 ) ) ) ).
% max.absorb_iff1
thf(fact_652_max_Oabsorb__iff1,axiom,
( ord_less_eq_num
= ( ^ [B2: num,A2: num] :
( ( ord_max_num @ A2 @ B2 )
= A2 ) ) ) ).
% max.absorb_iff1
thf(fact_653_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_654_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_655_max_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).
% max.cobounded2
thf(fact_656_max_Ocobounded2,axiom,
! [B: num,A: num] : ( ord_less_eq_num @ B @ ( ord_max_num @ A @ B ) ) ).
% max.cobounded2
thf(fact_657_max_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).
% max.cobounded1
thf(fact_658_max_Ocobounded1,axiom,
! [A: num,B: num] : ( ord_less_eq_num @ A @ ( ord_max_num @ A @ B ) ) ).
% max.cobounded1
thf(fact_659_max_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( A2
= ( ord_max_nat @ A2 @ B2 ) ) ) ) ).
% max.order_iff
thf(fact_660_max_Oorder__iff,axiom,
( ord_less_eq_num
= ( ^ [B2: num,A2: num] :
( A2
= ( ord_max_num @ A2 @ B2 ) ) ) ) ).
% max.order_iff
thf(fact_661_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_662_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_663_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_664_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_665_max_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( ord_max_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% max.orderI
thf(fact_666_max_OorderI,axiom,
! [A: num,B: num] :
( ( A
= ( ord_max_num @ A @ B ) )
=> ( ord_less_eq_num @ B @ A ) ) ).
% max.orderI
thf(fact_667_max_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( ord_max_nat @ A @ B ) ) ) ).
% max.orderE
thf(fact_668_max_OorderE,axiom,
! [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
=> ( A
= ( ord_max_num @ A @ B ) ) ) ).
% max.orderE
thf(fact_669_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_670_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_671_prefix__code_I2_J,axiom,
! [X: list_nat,Xs: list_list_nat] :
~ ( prefix_list_nat @ ( cons_list_nat @ X @ Xs ) @ nil_list_nat ) ).
% prefix_code(2)
thf(fact_672_prefix__code_I2_J,axiom,
! [X: nat,Xs: list_nat] :
~ ( prefix_nat @ ( cons_nat @ X @ Xs ) @ nil_nat ) ).
% prefix_code(2)
thf(fact_673_prefix__Cons,axiom,
! [Xs: list_list_nat,Y: list_nat,Ys2: list_list_nat] :
( ( prefix_list_nat @ Xs @ ( cons_list_nat @ Y @ Ys2 ) )
= ( ( Xs = nil_list_nat )
| ? [Zs2: list_list_nat] :
( ( Xs
= ( cons_list_nat @ Y @ Zs2 ) )
& ( prefix_list_nat @ Zs2 @ Ys2 ) ) ) ) ).
% prefix_Cons
thf(fact_674_prefix__Cons,axiom,
! [Xs: list_nat,Y: nat,Ys2: list_nat] :
( ( prefix_nat @ Xs @ ( cons_nat @ Y @ Ys2 ) )
= ( ( Xs = nil_nat )
| ? [Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Y @ Zs2 ) )
& ( prefix_nat @ Zs2 @ Ys2 ) ) ) ) ).
% prefix_Cons
thf(fact_675_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,Xs2: list_list_nat] :
( ( Ls
= ( cons_list_nat @ X2 @ Xs2 ) )
=> ( ( X2 = A4 )
=> ( prefix_list_nat @ As @ Xs2 ) ) ) )
=> ~ ! [A4: list_nat] :
( ? [As: list_list_nat] :
( Ps
= ( cons_list_nat @ A4 @ As ) )
=> ! [X2: list_nat] :
( ? [Xs2: list_list_nat] :
( Ls
= ( cons_list_nat @ X2 @ Xs2 ) )
=> ( X2 = A4 ) ) ) ) ) ) ).
% not_prefix_cases
thf(fact_676_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,Xs2: list_nat] :
( ( Ls
= ( cons_nat @ X2 @ Xs2 ) )
=> ( ( X2 = A4 )
=> ( prefix_nat @ As @ Xs2 ) ) ) )
=> ~ ! [A4: nat] :
( ? [As: list_nat] :
( Ps
= ( cons_nat @ A4 @ As ) )
=> ! [X2: nat] :
( ? [Xs2: list_nat] :
( Ls
= ( cons_nat @ X2 @ Xs2 ) )
=> ( X2 = A4 ) ) ) ) ) ) ).
% not_prefix_cases
thf(fact_677_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,Xs2: list_list_nat] : ( P @ ( cons_list_nat @ X2 @ Xs2 ) @ nil_list_nat )
=> ( ! [X2: list_nat,Xs2: list_list_nat,Y2: list_nat,Ys: list_list_nat] :
( ( X2 != Y2 )
=> ( P @ ( cons_list_nat @ X2 @ Xs2 ) @ ( cons_list_nat @ Y2 @ Ys ) ) )
=> ( ! [X2: list_nat,Xs2: list_list_nat,Y2: list_nat,Ys: list_list_nat] :
( ( X2 = Y2 )
=> ( ~ ( prefix_list_nat @ Xs2 @ Ys )
=> ( ( P @ Xs2 @ Ys )
=> ( P @ ( cons_list_nat @ X2 @ Xs2 ) @ ( cons_list_nat @ Y2 @ Ys ) ) ) ) )
=> ( P @ Ps @ Ls ) ) ) ) ) ).
% not_prefix_induct
thf(fact_678_not__prefix__induct,axiom,
! [Ps: list_nat,Ls: list_nat,P: list_nat > list_nat > $o] :
( ~ ( prefix_nat @ Ps @ Ls )
=> ( ! [X2: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X2 @ Xs2 ) @ nil_nat )
=> ( ! [X2: nat,Xs2: list_nat,Y2: nat,Ys: list_nat] :
( ( X2 != Y2 )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys ) ) )
=> ( ! [X2: nat,Xs2: list_nat,Y2: nat,Ys: list_nat] :
( ( X2 = Y2 )
=> ( ~ ( prefix_nat @ Xs2 @ Ys )
=> ( ( P @ Xs2 @ Ys )
=> ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys ) ) ) ) )
=> ( P @ Ps @ Ls ) ) ) ) ) ).
% not_prefix_induct
thf(fact_679_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_680_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_681_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_682_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_683_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_684_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_685_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_686_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_687_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_688_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_689_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_690_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_691_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_692_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_693_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_694_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_695_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_696_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_697_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_698_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_699_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_700_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_701_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_702_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_703_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_704_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_705_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_706_impossible__Cons,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat,X: list_nat] :
( ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ ( size_s3023201423986296836st_nat @ Ys2 ) )
=> ( Xs
!= ( cons_list_nat @ X @ Ys2 ) ) ) ).
% impossible_Cons
thf(fact_707_impossible__Cons,axiom,
! [Xs: list_nat,Ys2: list_nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys2 ) )
=> ( Xs
!= ( cons_nat @ X @ Ys2 ) ) ) ).
% impossible_Cons
thf(fact_708_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_709_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_710_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_711_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_712_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_713_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_714_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_715_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_716_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_717_concat_Osimps_I2_J,axiom,
! [X: list_list_nat,Xs: list_list_list_nat] :
( ( concat_list_nat @ ( cons_list_list_nat @ X @ Xs ) )
= ( append_list_nat @ X @ ( concat_list_nat @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_718_concat_Osimps_I2_J,axiom,
! [X: list_nat,Xs: list_list_nat] :
( ( concat_nat @ ( cons_list_nat @ X @ Xs ) )
= ( append_nat @ X @ ( concat_nat @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_719_last_Osimps,axiom,
! [Xs: list_list_nat,X: list_nat] :
( ( ( Xs = nil_list_nat )
=> ( ( last_list_nat @ ( cons_list_nat @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_list_nat )
=> ( ( last_list_nat @ ( cons_list_nat @ X @ Xs ) )
= ( last_list_nat @ Xs ) ) ) ) ).
% last.simps
thf(fact_720_last_Osimps,axiom,
! [Xs: list_nat,X: nat] :
( ( ( Xs = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs ) )
= ( last_nat @ Xs ) ) ) ) ).
% last.simps
thf(fact_721_last__ConsL,axiom,
! [Xs: list_list_nat,X: list_nat] :
( ( Xs = nil_list_nat )
=> ( ( last_list_nat @ ( cons_list_nat @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_722_last__ConsL,axiom,
! [Xs: list_nat,X: nat] :
( ( Xs = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_723_last__ConsR,axiom,
! [Xs: list_list_nat,X: list_nat] :
( ( Xs != nil_list_nat )
=> ( ( last_list_nat @ ( cons_list_nat @ X @ Xs ) )
= ( last_list_nat @ Xs ) ) ) ).
% last_ConsR
thf(fact_724_last__ConsR,axiom,
! [Xs: list_nat,X: nat] :
( ( Xs != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X @ Xs ) )
= ( last_nat @ Xs ) ) ) ).
% last_ConsR
thf(fact_725_concat_Osimps_I1_J,axiom,
( ( concat_list_nat @ nil_list_list_nat )
= nil_list_nat ) ).
% concat.simps(1)
thf(fact_726_concat_Osimps_I1_J,axiom,
( ( concat_nat @ nil_list_nat )
= nil_nat ) ).
% concat.simps(1)
thf(fact_727_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_728_last__in__set,axiom,
! [As2: list_nat] :
( ( As2 != nil_nat )
=> ( member_nat @ ( last_nat @ As2 ) @ ( set_nat2 @ As2 ) ) ) ).
% last_in_set
thf(fact_729_last__append,axiom,
! [Ys2: list_list_nat,Xs: list_list_nat] :
( ( ( Ys2 = nil_list_nat )
=> ( ( last_list_nat @ ( append_list_nat @ Xs @ Ys2 ) )
= ( last_list_nat @ Xs ) ) )
& ( ( Ys2 != nil_list_nat )
=> ( ( last_list_nat @ ( append_list_nat @ Xs @ Ys2 ) )
= ( last_list_nat @ Ys2 ) ) ) ) ).
% last_append
thf(fact_730_last__append,axiom,
! [Ys2: list_nat,Xs: list_nat] :
( ( ( Ys2 = nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( last_nat @ Xs ) ) )
& ( ( Ys2 != nil_nat )
=> ( ( last_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( last_nat @ Ys2 ) ) ) ) ).
% last_append
thf(fact_731_longest__common__suffix,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat] :
? [Ss: list_list_nat,Xs5: list_list_nat,Ys7: list_list_nat] :
( ( Xs
= ( 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_732_longest__common__suffix,axiom,
! [Xs: list_nat,Ys2: list_nat] :
? [Ss: list_nat,Xs5: list_nat,Ys7: list_nat] :
( ( Xs
= ( 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_733_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_734_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_735_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_736_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_737_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_738_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_739_rgf__def,axiom,
( equiva3371634703666331078on_rgf
= ( ^ [X3: list_nat] :
! [Ys3: list_nat,Y4: nat] :
( ( prefix_nat @ ( append_nat @ Ys3 @ ( cons_nat @ Y4 @ nil_nat ) ) @ X3 )
=> ( ord_less_eq_nat @ Y4 @ ( equiva5889994315859557365_limit @ Ys3 ) ) ) ) ) ).
% rgf_def
thf(fact_740_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_741_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,Xs2: 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 @ Xs2 @ Xs5 ) @ Xss23 ) ) )
& ( Ys2
= ( append_list_nat @ ( concat_list_nat @ Xss12 ) @ Xs2 ) )
& ( Zs
= ( append_list_nat @ Xs5 @ ( concat_list_nat @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_742_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,Xs2: list_nat,Xs5: list_nat,Xss23: list_list_nat] :
( ( Xss2
= ( append_list_nat @ Xss12 @ ( cons_list_nat @ ( append_nat @ Xs2 @ Xs5 ) @ Xss23 ) ) )
& ( Ys2
= ( append_nat @ ( concat_nat @ Xss12 ) @ Xs2 ) )
& ( Zs
= ( append_nat @ Xs5 @ ( concat_nat @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_743_max__def,axiom,
( ord_max_set_nat
= ( ^ [A2: set_nat,B2: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).
% max_def
thf(fact_744_max__def,axiom,
( ord_max_nat
= ( ^ [A2: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).
% max_def
thf(fact_745_max__def,axiom,
( ord_max_num
= ( ^ [A2: num,B2: num] : ( if_num @ ( ord_less_eq_num @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).
% max_def
thf(fact_746_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_747_max__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_max_nat @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_748_max__absorb1,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ( ( ord_max_num @ X @ Y )
= X ) ) ).
% max_absorb1
thf(fact_749_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_750_max__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_max_nat @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_751_max__absorb2,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_max_num @ X @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_752_subset__code_I1_J,axiom,
! [Xs: list_list_nat,B3: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ B3 )
= ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( member_list_nat @ X3 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_753_subset__code_I1_J,axiom,
! [Xs: list_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B3 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X3 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_754_set__subset__Cons,axiom,
! [Xs: list_list_nat,X: list_nat] : ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ ( set_list_nat2 @ ( cons_list_nat @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_755_set__subset__Cons,axiom,
! [Xs: list_nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_756_set__mono__prefix,axiom,
! [Xs: list_list_nat,Ys2: list_list_nat] :
( ( prefix_list_nat @ Xs @ Ys2 )
=> ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ ( set_list_nat2 @ Ys2 ) ) ) ).
% set_mono_prefix
thf(fact_757_set__mono__prefix,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( prefix_nat @ Xs @ Ys2 )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys2 ) ) ) ).
% set_mono_prefix
thf(fact_758_set__update__subsetI,axiom,
! [Xs: list_list_nat,A3: set_list_nat,X: list_nat,I: nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ A3 )
=> ( ( member_list_nat @ X @ A3 )
=> ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ ( list_update_list_nat @ Xs @ I @ X ) ) @ A3 ) ) ) ).
% set_update_subsetI
thf(fact_759_set__update__subsetI,axiom,
! [Xs: list_nat,A3: set_nat,X: nat,I: nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A3 )
=> ( ( member_nat @ X @ A3 )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X ) ) @ A3 ) ) ) ).
% set_update_subsetI
thf(fact_760_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_761_less__imp__neq,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_762_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_763_order_Oasym,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ~ ( ord_less_num @ B @ A ) ) ).
% order.asym
thf(fact_764_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_765_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_766_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_767_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_768_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_769_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_770_antisym__conv3,axiom,
! [Y: num,X: num] :
( ~ ( ord_less_num @ Y @ X )
=> ( ( ~ ( ord_less_num @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_771_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_772_linorder__cases,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_num @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_773_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_774_dual__order_Oasym,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ~ ( ord_less_num @ A @ B ) ) ).
% dual_order.asym
thf(fact_775_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_776_dual__order_Oirrefl,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% dual_order.irrefl
thf(fact_777_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_778_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_779_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_780_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_781_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_782_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_783_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_784_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_785_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_786_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_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: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_790_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_791_linorder__neqE,axiom,
! [X: num,Y: num] :
( ( X != Y )
=> ( ~ ( ord_less_num @ X @ Y )
=> ( ord_less_num @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_792_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_793_order__less__asym,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ~ ( ord_less_num @ Y @ X ) ) ).
% order_less_asym
thf(fact_794_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_795_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_796_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_797_order__less__asym_H,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ~ ( ord_less_num @ B @ A ) ) ).
% order_less_asym'
thf(fact_798_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_799_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_800_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_801_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_802_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_803_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_804_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_805_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_806_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_807_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_808_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_809_order__less__irrefl,axiom,
! [X: num] :
~ ( ord_less_num @ X @ X ) ).
% order_less_irrefl
thf(fact_810_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_811_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_812_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_813_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_814_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_815_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_816_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_817_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_818_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_819_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_820_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_821_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_822_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_823_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_824_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_825_order__less__imp__not__eq,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_826_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_827_order__less__imp__not__eq2,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_828_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_829_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_830_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_831_order__le__imp__less__or__eq,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_832_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_833_order__le__imp__less__or__eq,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_num @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_834_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_835_linorder__le__less__linear,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
| ( ord_less_num @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_836_order__less__le__subst2,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_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_le_subst2
thf(fact_837_order__less__le__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_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_le_subst2
thf(fact_838_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_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_le_subst2
thf(fact_839_order__less__le__subst2,axiom,
! [A: num,B: num,F: num > set_nat,C: set_nat] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_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_le_subst2
thf(fact_840_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_841_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_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_le_subst2
thf(fact_842_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_843_order__less__le__subst1,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_844_order__less__le__subst1,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_845_order__less__le__subst1,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_846_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_847_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_848_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_849_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_850_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_851_order__le__less__subst1,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_eq_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_le_less_subst1
thf(fact_852_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_853_order__less__le__trans,axiom,
! [X: num,Y: num,Z4: num] :
( ( ord_less_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ Z4 )
=> ( ord_less_num @ X @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_854_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_855_order__le__less__trans,axiom,
! [X: num,Y: num,Z4: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_num @ Y @ Z4 )
=> ( ord_less_num @ X @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_856_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_857_order__neq__le__trans,axiom,
! [A: num,B: num] :
( ( A != B )
=> ( ( ord_less_eq_num @ A @ B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_858_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_859_order__le__neq__trans,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( A != B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_860_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_861_order__less__imp__le,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_eq_num @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_862_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_863_linorder__not__less,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_num @ X @ Y ) )
= ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_not_less
thf(fact_864_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_865_linorder__not__le,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_eq_num @ X @ Y ) )
= ( ord_less_num @ Y @ X ) ) ).
% linorder_not_le
thf(fact_866_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_867_order__less__le,axiom,
( ord_less_num
= ( ^ [X3: num,Y4: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_868_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_869_order__le__less,axiom,
( ord_less_eq_num
= ( ^ [X3: num,Y4: num] :
( ( ord_less_num @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_870_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_871_dual__order_Ostrict__implies__order,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( ord_less_eq_num @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_872_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_873_order_Ostrict__implies__order,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( ord_less_eq_num @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_874_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ~ ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_875_dual__order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [B2: num,A2: num] :
( ( ord_less_eq_num @ B2 @ A2 )
& ~ ( ord_less_eq_num @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_876_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_877_dual__order_Ostrict__trans2,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_878_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_879_dual__order_Ostrict__trans1,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_880_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_881_dual__order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [B2: num,A2: num] :
( ( ord_less_eq_num @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_882_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_883_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [B2: num,A2: num] :
( ( ord_less_num @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_884_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_885_order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [A2: num,B2: num] :
( ( ord_less_eq_num @ A2 @ B2 )
& ~ ( ord_less_eq_num @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_886_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_887_order_Ostrict__trans2,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_888_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_889_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_890_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_891_order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [A2: num,B2: num] :
( ( ord_less_eq_num @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_892_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_893_order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [A2: num,B2: num] :
( ( ord_less_num @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_894_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_895_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_896_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_897_less__le__not__le,axiom,
( ord_less_num
= ( ^ [X3: num,Y4: num] :
( ( ord_less_eq_num @ X3 @ Y4 )
& ~ ( ord_less_eq_num @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_898_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_899_antisym__conv2,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ~ ( ord_less_num @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_900_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_901_antisym__conv1,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ( ord_less_eq_num @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_902_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_903_nless__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_num @ A @ B ) )
= ( ~ ( ord_less_eq_num @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_904_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_905_leI,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ord_less_eq_num @ Y @ X ) ) ).
% leI
thf(fact_906_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_907_leD,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ~ ( ord_less_num @ X @ Y ) ) ).
% leD
thf(fact_908_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_909_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 )
& ! [X5: nat] :
( ( ( ord_less_eq_nat @ A @ X5 )
& ( ord_less_nat @ X5 @ C2 ) )
=> ( P @ X5 ) )
& ! [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_910_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% pinf(6)
thf(fact_911_pinf_I6_J,axiom,
! [T: num] :
? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ Z3 @ X5 )
=> ~ ( ord_less_eq_num @ X5 @ T ) ) ).
% pinf(6)
thf(fact_912_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 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q3 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_913_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 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q3 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_914_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_915_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_916_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ~ ( ord_less_nat @ X5 @ T ) ) ).
% pinf(5)
thf(fact_917_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ord_less_nat @ T @ X5 ) ) ).
% pinf(7)
thf(fact_918_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 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q3 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_919_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 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q3 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_920_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_921_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_922_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ord_less_nat @ X5 @ T ) ) ).
% minf(5)
thf(fact_923_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ~ ( ord_less_nat @ T @ X5 ) ) ).
% minf(7)
thf(fact_924_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% minf(8)
thf(fact_925_minf_I8_J,axiom,
! [T: num] :
? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z3 )
=> ~ ( ord_less_eq_num @ T @ X5 ) ) ).
% minf(8)
thf(fact_926_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ord_less_eq_nat @ X5 @ T ) ) ).
% minf(6)
thf(fact_927_minf_I6_J,axiom,
! [T: num] :
? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ X5 @ Z3 )
=> ( ord_less_eq_num @ X5 @ T ) ) ).
% minf(6)
thf(fact_928_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ord_less_eq_nat @ T @ X5 ) ) ).
% pinf(8)
thf(fact_929_pinf_I8_J,axiom,
! [T: num] :
? [Z3: num] :
! [X5: num] :
( ( ord_less_num @ Z3 @ X5 )
=> ( ord_less_eq_num @ T @ X5 ) ) ).
% pinf(8)
thf(fact_930_append__butlast__last__id,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( append_nat @ ( butlast_nat @ Xs ) @ ( cons_nat @ ( last_nat @ Xs ) @ nil_nat ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_931_rotate1__length01,axiom,
! [Xs: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( ( rotate1_nat @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_932_stirling__row__code_I1_J,axiom,
( ( stirling_row @ zero_zero_nat )
= ( cons_nat @ one_one_nat @ nil_nat ) ) ).
% stirling_row_code(1)
thf(fact_933_rotate1__is__Nil__conv,axiom,
! [Xs: list_nat] :
( ( ( rotate1_nat @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% rotate1_is_Nil_conv
thf(fact_934_set__rotate1,axiom,
! [Xs: list_nat] :
( ( set_nat2 @ ( rotate1_nat @ Xs ) )
= ( set_nat2 @ Xs ) ) ).
% set_rotate1
thf(fact_935_length__rotate1,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rotate1_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rotate1
thf(fact_936_butlast__snoc,axiom,
! [Xs: list_nat,X: nat] :
( ( butlast_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_937_length__butlast,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( butlast_nat @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ).
% length_butlast
thf(fact_938_rotate1_Osimps_I1_J,axiom,
( ( rotate1_nat @ nil_nat )
= nil_nat ) ).
% rotate1.simps(1)
thf(fact_939_prefixeq__butlast,axiom,
! [Xs: list_nat] : ( prefix_nat @ ( butlast_nat @ Xs ) @ Xs ) ).
% prefixeq_butlast
thf(fact_940_in__set__butlastD,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ Xs ) ) )
=> ( member_nat @ X @ ( set_nat2 @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_941_butlast_Osimps_I1_J,axiom,
( ( butlast_nat @ nil_nat )
= nil_nat ) ).
% butlast.simps(1)
thf(fact_942_stirling__row__nonempty,axiom,
! [N: nat] :
( ( stirling_row @ N )
!= nil_nat ) ).
% stirling_row_nonempty
thf(fact_943_butlast_Osimps_I2_J,axiom,
! [Xs: list_nat,X: nat] :
( ( ( Xs = nil_nat )
=> ( ( butlast_nat @ ( cons_nat @ X @ Xs ) )
= nil_nat ) )
& ( ( Xs != nil_nat )
=> ( ( butlast_nat @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ ( butlast_nat @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_944_butlast__append,axiom,
! [Ys2: list_nat,Xs: list_nat] :
( ( ( Ys2 = nil_nat )
=> ( ( butlast_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( butlast_nat @ Xs ) ) )
& ( ( Ys2 != nil_nat )
=> ( ( butlast_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( append_nat @ Xs @ ( butlast_nat @ Ys2 ) ) ) ) ) ).
% butlast_append
thf(fact_945_in__set__butlast__appendI,axiom,
! [X: nat,Xs: list_nat,Ys2: list_nat] :
( ( ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ Xs ) ) )
| ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ Ys2 ) ) ) )
=> ( member_nat @ X @ ( set_nat2 @ ( butlast_nat @ ( append_nat @ Xs @ Ys2 ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_946_nth__butlast,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ ( butlast_nat @ Xs ) ) )
=> ( ( nth_nat @ ( butlast_nat @ Xs ) @ N )
= ( nth_nat @ Xs @ N ) ) ) ).
% nth_butlast
thf(fact_947_rotate1_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( rotate1_nat @ ( cons_nat @ X @ Xs ) )
= ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) ) ).
% rotate1.simps(2)
thf(fact_948_snoc__eq__iff__butlast,axiom,
! [Xs: list_nat,X: nat,Ys2: list_nat] :
( ( ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) )
= Ys2 )
= ( ( Ys2 != nil_nat )
& ( ( butlast_nat @ Ys2 )
= Xs )
& ( ( last_nat @ Ys2 )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_949_butlast__list__update,axiom,
! [K: nat,Xs: list_nat,X: nat] :
( ( ( K
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
=> ( ( butlast_nat @ ( list_update_nat @ Xs @ K @ X ) )
= ( butlast_nat @ Xs ) ) )
& ( ( K
!= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) )
=> ( ( butlast_nat @ ( list_update_nat @ Xs @ K @ X ) )
= ( list_update_nat @ ( butlast_nat @ Xs ) @ K @ X ) ) ) ) ).
% butlast_list_update
thf(fact_950_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_951_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_952_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_953_stirling__same,axiom,
! [N: nat] :
( ( stirling @ N @ N )
= one_one_nat ) ).
% stirling_same
thf(fact_954_stirling__less,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( stirling @ N @ K )
= zero_zero_nat ) ) ).
% stirling_less
thf(fact_955_stirling__0,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( stirling @ N @ zero_zero_nat )
= zero_zero_nat ) ) ).
% stirling_0
thf(fact_956_stirling_Osimps_I1_J,axiom,
( ( stirling @ zero_zero_nat @ zero_zero_nat )
= one_one_nat ) ).
% stirling.simps(1)
thf(fact_957_nth__Cons__numeral,axiom,
! [X: nat,Xs: list_nat,V: num] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ ( numeral_numeral_nat @ V ) )
= ( nth_nat @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) ) ) ).
% nth_Cons_numeral
thf(fact_958_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_959_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_960_nat_Oinject,axiom,
! [X23: nat,Y23: nat] :
( ( ( suc @ X23 )
= ( suc @ Y23 ) )
= ( X23 = Y23 ) ) ).
% nat.inject
thf(fact_961_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_962_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_963_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_964_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_965_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_966_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_967_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_968_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_969_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_970_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_971_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_972_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_973_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_974_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_975_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_976_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_977_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_978_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_979_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_980_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_981_nth__Cons__Suc,axiom,
! [X: nat,Xs: list_nat,N: nat] :
( ( nth_nat @ ( cons_nat @ X @ Xs ) @ ( suc @ N ) )
= ( nth_nat @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_982_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_983_gbinomial__0_I2_J,axiom,
! [K: nat] :
( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K ) )
= zero_zero_nat ) ).
% gbinomial_0(2)
thf(fact_984_gbinomial__0_I1_J,axiom,
! [A: nat] :
( ( gbinomial_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% gbinomial_0(1)
thf(fact_985_length__stirling__row,axiom,
! [N: nat] :
( ( size_size_list_nat @ ( stirling_row @ N ) )
= ( suc @ N ) ) ).
% length_stirling_row
thf(fact_986_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_987_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_988_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_989_length__suffixes,axiom,
! [Xs: list_nat] :
( ( size_s3023201423986296836st_nat @ ( suffixes_nat @ Xs ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_suffixes
thf(fact_990_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_991_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_le_numeral
thf(fact_992_zero__less__numeral,axiom,
! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_less_numeral
thf(fact_993_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_994_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_995_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_996_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_997_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_998_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_999_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1000_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_1001_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_1002_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_1003_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,Z3: nat] :
( ( R @ X2 @ Y2 )
=> ( ( R @ Y2 @ Z3 )
=> ( R @ X2 @ Z3 ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1004_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_1005_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1006_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_1007_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_1008_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1009_nat__arith_Osuc1,axiom,
! [A3: nat,K: nat,A: nat] :
( ( A3
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A3 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1010_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_1011_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_1012_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [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 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1013_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1014_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_1015_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_1016_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_1017_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_1018_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1019_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_1020_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_1021_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1022_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1023_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_1024_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_1025_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_1026_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_1027_enum__rgfs_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N2: nat] :
( X
!= ( suc @ N2 ) ) ) ).
% enum_rgfs.cases
thf(fact_1028_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_1029_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1030_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1031_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1032_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_1033_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_1034_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_1035_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1036_nat_OdiscI,axiom,
! [Nat: nat,X23: nat] :
( ( Nat
= ( suc @ X23 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1037_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1038_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1039_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( zero_zero_nat
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_1040_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_1041_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% one_le_numeral
thf(fact_1042_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_1043_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_1044_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_1045_lift__Suc__mono__le,axiom,
! [F: nat > num,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_num @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1046_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_1047_lift__Suc__antimono__le,axiom,
! [F: nat > num,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_num @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1048_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_1049_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_1050_length__Suc__conv,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y4: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y4 @ Ys3 ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_1051_Suc__length__conv,axiom,
! [N: nat,Xs: list_nat] :
( ( ( suc @ N )
= ( size_size_list_nat @ Xs ) )
= ( ? [Y4: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y4 @ 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 )
| ? [J2: nat] :
( ( M
= ( suc @ J2 ) )
& ( ord_less_nat @ J2 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1057_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_1058_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_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,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1062_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_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__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q4: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q4 ) ) ) ) ).
% less_natE
thf(fact_1069_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
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__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_1072_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_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__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_1078_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_1079_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
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__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
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,Xs: list_nat,I: nat,Y: nat] :
( ( list_update_nat @ ( cons_nat @ X @ Xs ) @ ( suc @ I ) @ Y )
= ( cons_nat @ X @ ( list_update_nat @ Xs @ 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,Xs: list_nat] :
( ( gen_length_nat @ N @ ( cons_nat @ X @ Xs ) )
= ( gen_length_nat @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_1088_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
= ( ? [X3: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ X3 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_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,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y4: nat,Ys3: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ Y4 @ nil_nat ) ) )
& ( ( size_size_list_nat @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_1094_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_1095_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_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,
! [Xs: list_nat,X: nat] :
( ( size_size_list_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_append_singleton
thf(fact_1098_length__Cons,axiom,
! [X: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X @ Xs ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_Cons
thf(fact_1099_SuccD,axiom,
! [K: nat,Kl: set_list_nat,Kl2: list_nat] :
( ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl @ Kl2 ) )
=> ( member_list_nat @ ( append_nat @ Kl2 @ ( cons_nat @ K @ nil_nat ) ) @ Kl ) ) ).
% SuccD
thf(fact_1100_SuccI,axiom,
! [Kl2: list_nat,K: nat,Kl: set_list_nat] :
( ( member_list_nat @ ( append_nat @ Kl2 @ ( cons_nat @ K @ nil_nat ) ) @ Kl )
=> ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl @ Kl2 ) ) ) ).
% SuccI
thf(fact_1101_empty__Shift,axiom,
! [Kl: set_list_nat,K: nat] :
( ( member_list_nat @ nil_nat @ Kl )
=> ( ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl @ nil_nat ) )
=> ( member_list_nat @ nil_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl @ K ) ) ) ) ).
% empty_Shift
thf(fact_1102_Succ__Shift,axiom,
! [Kl: set_list_nat,K: nat,Kl2: list_nat] :
( ( bNF_Gr6352880689984616693cc_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl @ K ) @ Kl2 )
= ( bNF_Gr6352880689984616693cc_nat @ Kl @ ( cons_nat @ K @ Kl2 ) ) ) ).
% Succ_Shift
thf(fact_1103_ShiftD,axiom,
! [Kl2: list_nat,Kl: set_list_nat,K: nat] :
( ( member_list_nat @ Kl2 @ ( bNF_Gr1872714664788909425ft_nat @ Kl @ K ) )
=> ( member_list_nat @ ( cons_nat @ K @ Kl2 ) @ Kl ) ) ).
% ShiftD
thf(fact_1104_take__Suc__conv__app__nth,axiom,
! [I: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( take_nat @ ( suc @ I ) @ Xs )
= ( append_nat @ ( take_nat @ I @ Xs ) @ ( cons_nat @ ( nth_nat @ Xs @ I ) @ nil_nat ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_1105_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_1106_take__Suc__Cons,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( take_nat @ ( suc @ N ) @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ ( take_nat @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_1107_take0,axiom,
( ( take_nat @ zero_zero_nat )
= ( ^ [Xs3: list_nat] : nil_nat ) ) ).
% take0
thf(fact_1108_take__eq__Nil,axiom,
! [N: nat,Xs: list_nat] :
( ( ( take_nat @ N @ Xs )
= nil_nat )
= ( ( N = zero_zero_nat )
| ( Xs = nil_nat ) ) ) ).
% take_eq_Nil
thf(fact_1109_take__eq__Nil2,axiom,
! [N: nat,Xs: list_nat] :
( ( nil_nat
= ( take_nat @ N @ Xs ) )
= ( ( N = zero_zero_nat )
| ( Xs = nil_nat ) ) ) ).
% take_eq_Nil2
thf(fact_1110_take__all__iff,axiom,
! [N: nat,Xs: list_nat] :
( ( ( take_nat @ N @ Xs )
= Xs )
= ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% take_all_iff
thf(fact_1111_take__all,axiom,
! [Xs: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
=> ( ( take_nat @ N @ Xs )
= Xs ) ) ).
% take_all
thf(fact_1112_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_1113_nth__take,axiom,
! [I: nat,N: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_nat @ ( take_nat @ N @ Xs ) @ I )
= ( nth_nat @ Xs @ I ) ) ) ).
% nth_take
thf(fact_1114_take__update__cancel,axiom,
! [N: nat,M: nat,Xs: list_nat,Y: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( take_nat @ N @ ( list_update_nat @ Xs @ M @ Y ) )
= ( take_nat @ N @ Xs ) ) ) ).
% take_update_cancel
thf(fact_1115_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( upt @ I @ J )
= nil_nat ) ) ).
% upt_conv_Nil
thf(fact_1116_length__upt,axiom,
! [I: nat,J: nat] :
( ( size_size_list_nat @ ( upt @ I @ J ) )
= ( minus_minus_nat @ J @ I ) ) ).
% length_upt
thf(fact_1117_take__append,axiom,
! [N: nat,Xs: list_nat,Ys2: list_nat] :
( ( take_nat @ N @ ( append_nat @ Xs @ Ys2 ) )
= ( append_nat @ ( take_nat @ N @ Xs ) @ ( take_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys2 ) ) ) ).
% take_append
thf(fact_1118_last__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( last_nat @ ( upt @ I @ J ) )
= ( minus_minus_nat @ J @ one_one_nat ) ) ) ).
% last_upt
thf(fact_1119_upt__eq__Nil__conv,axiom,
! [I: nat,J: nat] :
( ( ( upt @ I @ J )
= nil_nat )
= ( ( J = zero_zero_nat )
| ( ord_less_eq_nat @ J @ I ) ) ) ).
% upt_eq_Nil_conv
thf(fact_1120_nth__upt,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
=> ( ( nth_nat @ ( upt @ I @ J ) @ K )
= ( plus_plus_nat @ I @ K ) ) ) ).
% nth_upt
thf(fact_1121_take__Cons__numeral,axiom,
! [V: num,X: nat,Xs: list_nat] :
( ( take_nat @ ( numeral_numeral_nat @ V ) @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ ( take_nat @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) @ Xs ) ) ) ).
% take_Cons_numeral
thf(fact_1122_set__take__subset,axiom,
! [N: nat,Xs: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( take_nat @ N @ Xs ) ) @ ( set_nat2 @ Xs ) ) ).
% set_take_subset
thf(fact_1123_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ zero_zero_nat )
= nil_nat ) ).
% upt_0
thf(fact_1124_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_1125_take__0,axiom,
! [Xs: list_nat] :
( ( take_nat @ zero_zero_nat @ Xs )
= nil_nat ) ).
% take_0
thf(fact_1126_take__Nil,axiom,
! [N: nat] :
( ( take_nat @ N @ nil_nat )
= nil_nat ) ).
% take_Nil
thf(fact_1127_in__set__takeD,axiom,
! [X: nat,N: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ ( take_nat @ N @ Xs ) ) )
=> ( member_nat @ X @ ( set_nat2 @ Xs ) ) ) ).
% in_set_takeD
thf(fact_1128_take__equalityI,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ! [I3: nat] :
( ( take_nat @ I3 @ Xs )
= ( take_nat @ I3 @ Ys2 ) )
=> ( Xs = Ys2 ) ) ).
% take_equalityI
thf(fact_1129_take__update__swap,axiom,
! [M: nat,Xs: list_nat,N: nat,X: nat] :
( ( take_nat @ M @ ( list_update_nat @ Xs @ N @ X ) )
= ( list_update_nat @ ( take_nat @ M @ Xs ) @ N @ X ) ) ).
% take_update_swap
thf(fact_1130_take__is__prefix,axiom,
! [N: nat,Xs: list_nat] : ( prefix_nat @ ( take_nat @ N @ Xs ) @ Xs ) ).
% take_is_prefix
thf(fact_1131_set__take__subset__set__take,axiom,
! [M: nat,N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( take_nat @ M @ Xs ) ) @ ( set_nat2 @ ( take_nat @ N @ Xs ) ) ) ) ).
% set_take_subset_set_take
thf(fact_1132_take__butlast,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( take_nat @ N @ ( butlast_nat @ Xs ) )
= ( take_nat @ N @ Xs ) ) ) ).
% take_butlast
thf(fact_1133_upt__conv__Cons,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( upt @ I @ J )
= ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
% upt_conv_Cons
thf(fact_1134_upt__add__eq__append,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( plus_plus_nat @ J @ K ) )
= ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% upt_add_eq_append
thf(fact_1135_nth__take__lemma,axiom,
! [K: nat,Xs: list_nat,Ys2: list_nat] :
( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Ys2 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( ( nth_nat @ Xs @ I3 )
= ( nth_nat @ Ys2 @ I3 ) ) )
=> ( ( take_nat @ K @ Xs )
= ( take_nat @ K @ Ys2 ) ) ) ) ) ).
% nth_take_lemma
thf(fact_1136_butlast__conv__take,axiom,
( butlast_nat
= ( ^ [Xs3: list_nat] : ( take_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs3 ) @ one_one_nat ) @ Xs3 ) ) ) ).
% butlast_conv_take
thf(fact_1137_upt__eq__Cons__conv,axiom,
! [I: nat,J: nat,X: nat,Xs: list_nat] :
( ( ( upt @ I @ J )
= ( cons_nat @ X @ Xs ) )
= ( ( ord_less_nat @ I @ J )
& ( I = X )
& ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
= Xs ) ) ) ).
% upt_eq_Cons_conv
thf(fact_1138_upt__rec,axiom,
( upt
= ( ^ [I2: nat,J2: nat] : ( if_list_nat @ ( ord_less_nat @ I2 @ J2 ) @ ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J2 ) ) @ nil_nat ) ) ) ).
% upt_rec
thf(fact_1139_take__Cons_H,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( take_nat @ N @ ( cons_nat @ X @ Xs ) )
= nil_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( take_nat @ N @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ ( take_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ) ).
% take_Cons'
thf(fact_1140_butlast__take,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( butlast_nat @ ( take_nat @ N @ Xs ) )
= ( take_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ).
% butlast_take
thf(fact_1141_upt__Suc__append,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).
% upt_Suc_append
thf(fact_1142_upt__Suc,axiom,
! [I: nat,J: nat] :
( ( ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
& ( ~ ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= nil_nat ) ) ) ).
% upt_Suc
thf(fact_1143_upd__conv__take__nth__drop,axiom,
! [I: nat,Xs: list_nat,A: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ Xs @ I @ A )
= ( append_nat @ ( take_nat @ I @ Xs ) @ ( cons_nat @ A @ ( drop_nat @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_1144_id__take__nth__drop,axiom,
! [I: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( Xs
= ( append_nat @ ( take_nat @ I @ Xs ) @ ( cons_nat @ ( nth_nat @ Xs @ I ) @ ( drop_nat @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_1145_drop0,axiom,
( ( drop_nat @ zero_zero_nat )
= ( ^ [X3: list_nat] : X3 ) ) ).
% drop0
thf(fact_1146_drop__drop,axiom,
! [N: nat,M: nat,Xs: list_nat] :
( ( drop_nat @ N @ ( drop_nat @ M @ Xs ) )
= ( drop_nat @ ( plus_plus_nat @ N @ M ) @ Xs ) ) ).
% drop_drop
thf(fact_1147_drop__upt,axiom,
! [M: nat,I: nat,J: nat] :
( ( drop_nat @ M @ ( upt @ I @ J ) )
= ( upt @ ( plus_plus_nat @ I @ M ) @ J ) ) ).
% drop_upt
thf(fact_1148_drop__Suc__Cons,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( drop_nat @ ( suc @ N ) @ ( cons_nat @ X @ Xs ) )
= ( drop_nat @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_1149_length__drop,axiom,
! [N: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( drop_nat @ N @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% length_drop
thf(fact_1150_append__take__drop__id,axiom,
! [N: nat,Xs: list_nat] :
( ( append_nat @ ( take_nat @ N @ Xs ) @ ( drop_nat @ N @ Xs ) )
= Xs ) ).
% append_take_drop_id
thf(fact_1151_drop__update__cancel,axiom,
! [N: nat,M: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( drop_nat @ M @ ( list_update_nat @ Xs @ N @ X ) )
= ( drop_nat @ M @ Xs ) ) ) ).
% drop_update_cancel
thf(fact_1152_drop__all,axiom,
! [Xs: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
=> ( ( drop_nat @ N @ Xs )
= nil_nat ) ) ).
% drop_all
thf(fact_1153_drop__eq__Nil,axiom,
! [N: nat,Xs: list_nat] :
( ( ( drop_nat @ N @ Xs )
= nil_nat )
= ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_1154_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_nat] :
( ( nil_nat
= ( drop_nat @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_1155_drop__append,axiom,
! [N: nat,Xs: list_nat,Ys2: list_nat] :
( ( drop_nat @ N @ ( append_nat @ Xs @ Ys2 ) )
= ( append_nat @ ( drop_nat @ N @ Xs ) @ ( drop_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys2 ) ) ) ).
% drop_append
thf(fact_1156_last__drop,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( last_nat @ ( drop_nat @ N @ Xs ) )
= ( last_nat @ Xs ) ) ) ).
% last_drop
thf(fact_1157_drop__Cons__numeral,axiom,
! [V: num,X: nat,Xs: list_nat] :
( ( drop_nat @ ( numeral_numeral_nat @ V ) @ ( cons_nat @ X @ Xs ) )
= ( drop_nat @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) @ Xs ) ) ).
% drop_Cons_numeral
thf(fact_1158_nth__drop,axiom,
! [N: nat,Xs: list_nat,I: nat] :
( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( drop_nat @ N @ Xs ) @ I )
= ( nth_nat @ Xs @ ( plus_plus_nat @ N @ I ) ) ) ) ).
% nth_drop
thf(fact_1159_take__drop,axiom,
! [N: nat,M: nat,Xs: list_nat] :
( ( take_nat @ N @ ( drop_nat @ M @ Xs ) )
= ( drop_nat @ M @ ( take_nat @ ( plus_plus_nat @ N @ M ) @ Xs ) ) ) ).
% take_drop
thf(fact_1160_drop__take,axiom,
! [N: nat,M: nat,Xs: list_nat] :
( ( drop_nat @ N @ ( take_nat @ M @ Xs ) )
= ( take_nat @ ( minus_minus_nat @ M @ N ) @ ( drop_nat @ N @ Xs ) ) ) ).
% drop_take
thf(fact_1161_drop__butlast,axiom,
! [N: nat,Xs: list_nat] :
( ( drop_nat @ N @ ( butlast_nat @ Xs ) )
= ( butlast_nat @ ( drop_nat @ N @ Xs ) ) ) ).
% drop_butlast
thf(fact_1162_in__set__dropD,axiom,
! [X: nat,N: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ ( drop_nat @ N @ Xs ) ) )
=> ( member_nat @ X @ ( set_nat2 @ Xs ) ) ) ).
% in_set_dropD
thf(fact_1163_drop__0,axiom,
! [Xs: list_nat] :
( ( drop_nat @ zero_zero_nat @ Xs )
= Xs ) ).
% drop_0
thf(fact_1164_drop__Nil,axiom,
! [N: nat] :
( ( drop_nat @ N @ nil_nat )
= nil_nat ) ).
% drop_Nil
thf(fact_1165_nth__via__drop,axiom,
! [N: nat,Xs: list_nat,Y: nat,Ys2: list_nat] :
( ( ( drop_nat @ N @ Xs )
= ( cons_nat @ Y @ Ys2 ) )
=> ( ( nth_nat @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_1166_set__drop__subset,axiom,
! [N: nat,Xs: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( drop_nat @ N @ Xs ) ) @ ( set_nat2 @ Xs ) ) ).
% set_drop_subset
thf(fact_1167_set__drop__subset__set__drop,axiom,
! [N: nat,M: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( drop_nat @ M @ Xs ) ) @ ( set_nat2 @ ( drop_nat @ N @ Xs ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_1168_append__eq__conv__conj,axiom,
! [Xs: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( ( append_nat @ Xs @ Ys2 )
= Zs )
= ( ( Xs
= ( take_nat @ ( size_size_list_nat @ Xs ) @ Zs ) )
& ( Ys2
= ( drop_nat @ ( size_size_list_nat @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_1169_take__add,axiom,
! [I: nat,J: nat,Xs: list_nat] :
( ( take_nat @ ( plus_plus_nat @ I @ J ) @ Xs )
= ( append_nat @ ( take_nat @ I @ Xs ) @ ( take_nat @ J @ ( drop_nat @ I @ Xs ) ) ) ) ).
% take_add
thf(fact_1170_drop__update__swap,axiom,
! [M: nat,N: nat,Xs: list_nat,X: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( drop_nat @ M @ ( list_update_nat @ Xs @ N @ X ) )
= ( list_update_nat @ ( drop_nat @ M @ Xs ) @ ( minus_minus_nat @ N @ M ) @ X ) ) ) ).
% drop_update_swap
thf(fact_1171_drop__Cons_H,axiom,
! [N: nat,X: nat,Xs: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( drop_nat @ N @ ( cons_nat @ X @ Xs ) )
= ( cons_nat @ X @ Xs ) ) )
& ( ( N != zero_zero_nat )
=> ( ( drop_nat @ N @ ( cons_nat @ X @ Xs ) )
= ( drop_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ).
% drop_Cons'
thf(fact_1172_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_1173_Cons__nth__drop__Suc,axiom,
! [I: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( cons_nat @ ( nth_nat @ Xs @ I ) @ ( drop_nat @ ( suc @ I ) @ Xs ) )
= ( drop_nat @ I @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_1174_take__hd__drop,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( append_nat @ ( take_nat @ N @ Xs ) @ ( cons_nat @ ( hd_nat @ ( drop_nat @ N @ Xs ) ) @ nil_nat ) )
= ( take_nat @ ( suc @ N ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_1175_map__upt__eqI,axiom,
! [Xs: list_nat,N: nat,M: nat,F: nat > nat] :
( ( ( size_size_list_nat @ Xs )
= ( minus_minus_nat @ N @ M ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I3 )
= ( F @ ( plus_plus_nat @ M @ I3 ) ) ) )
=> ( ( map_nat_nat @ F @ ( upt @ M @ N ) )
= Xs ) ) ) ).
% map_upt_eqI
thf(fact_1176_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_1177_Nil__is__map__conv,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( nil_nat
= ( map_nat_nat @ F @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_1178_map__is__Nil__conv,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_1179_map__eq__conv,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Xs ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) ) ) ) ).
% map_eq_conv
thf(fact_1180_length__map,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( size_size_list_nat @ ( map_nat_nat @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_1181_map__append,axiom,
! [F: nat > nat,Xs: list_nat,Ys2: list_nat] :
( ( map_nat_nat @ F @ ( append_nat @ Xs @ Ys2 ) )
= ( append_nat @ ( map_nat_nat @ F @ Xs ) @ ( map_nat_nat @ F @ Ys2 ) ) ) ).
% map_append
thf(fact_1182_hd__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( hd_nat @ ( upt @ I @ J ) )
= I ) ) ).
% hd_upt
thf(fact_1183_hd__prefixes,axiom,
! [Xs: list_nat] :
( ( hd_list_nat @ ( prefixes_nat @ Xs ) )
= nil_nat ) ).
% hd_prefixes
thf(fact_1184_hd__suffixes,axiom,
! [Xs: list_nat] :
( ( hd_list_nat @ ( suffixes_nat @ Xs ) )
= nil_nat ) ).
% hd_suffixes
thf(fact_1185_hd__append2,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( Xs != nil_nat )
=> ( ( hd_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( hd_nat @ Xs ) ) ) ).
% hd_append2
thf(fact_1186_nth__map,axiom,
! [N: nat,Xs: list_nat,F: nat > nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( map_nat_nat @ F @ Xs ) @ N )
= ( F @ ( nth_nat @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_1187_hd__take,axiom,
! [J: nat,Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ J )
=> ( ( hd_nat @ ( take_nat @ J @ Xs ) )
= ( hd_nat @ Xs ) ) ) ).
% hd_take
thf(fact_1188_take__map,axiom,
! [N: nat,F: nat > nat,Xs: list_nat] :
( ( take_nat @ N @ ( map_nat_nat @ F @ Xs ) )
= ( map_nat_nat @ F @ ( take_nat @ N @ Xs ) ) ) ).
% take_map
thf(fact_1189_drop__map,axiom,
! [N: nat,F: nat > nat,Xs: list_nat] :
( ( drop_nat @ N @ ( map_nat_nat @ F @ Xs ) )
= ( map_nat_nat @ F @ ( drop_nat @ N @ Xs ) ) ) ).
% drop_map
thf(fact_1190_last__map,axiom,
! [Xs: list_nat,F: nat > nat] :
( ( Xs != nil_nat )
=> ( ( last_nat @ ( map_nat_nat @ F @ Xs ) )
= ( F @ ( last_nat @ Xs ) ) ) ) ).
% last_map
thf(fact_1191_hd__Nil__eq__last,axiom,
( ( hd_nat @ nil_nat )
= ( last_nat @ nil_nat ) ) ).
% hd_Nil_eq_last
thf(fact_1192_map__equality__iff,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys2: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys2 ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys2 ) )
=> ( ( F @ ( nth_nat @ Xs @ I2 ) )
= ( G @ ( nth_nat @ Ys2 @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_1193_hd__in__set,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( member_nat @ ( hd_nat @ Xs ) @ ( set_nat2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_1194_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_1195_hd__append,axiom,
! [Xs: list_nat,Ys2: list_nat] :
( ( ( Xs = nil_nat )
=> ( ( hd_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( hd_nat @ Ys2 ) ) )
& ( ( Xs != nil_nat )
=> ( ( hd_nat @ ( append_nat @ Xs @ Ys2 ) )
= ( hd_nat @ Xs ) ) ) ) ).
% hd_append
thf(fact_1196_longest__common__prefix,axiom,
! [Xs: list_nat,Ys2: list_nat] :
? [Ps2: list_nat,Xs5: list_nat,Ys7: list_nat] :
( ( Xs
= ( 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_1197_map__concat,axiom,
! [F: nat > nat,Xs: list_list_nat] :
( ( map_nat_nat @ F @ ( concat_nat @ Xs ) )
= ( concat_nat @ ( map_li7225945977422193158st_nat @ ( map_nat_nat @ F ) @ Xs ) ) ) ).
% map_concat
thf(fact_1198_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_1199_Cons__eq__map__D,axiom,
! [X: nat,Xs: list_nat,F: nat > nat,Ys2: list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( map_nat_nat @ F @ Ys2 ) )
=> ? [Z3: nat,Zs3: list_nat] :
( ( Ys2
= ( cons_nat @ Z3 @ Zs3 ) )
& ( X
= ( F @ Z3 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1200_map__eq__Cons__D,axiom,
! [F: nat > nat,Xs: list_nat,Y: nat,Ys2: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys2 ) )
=> ? [Z3: nat,Zs3: list_nat] :
( ( Xs
= ( cons_nat @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_nat_nat @ F @ Zs3 )
= Ys2 ) ) ) ).
% map_eq_Cons_D
thf(fact_1201_Cons__eq__map__conv,axiom,
! [X: nat,Xs: list_nat,F: nat > nat,Ys2: list_nat] :
( ( ( cons_nat @ X @ Xs )
= ( map_nat_nat @ F @ Ys2 ) )
= ( ? [Z5: nat,Zs2: list_nat] :
( ( Ys2
= ( cons_nat @ Z5 @ Zs2 ) )
& ( X
= ( F @ Z5 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1202_map__eq__Cons__conv,axiom,
! [F: nat > nat,Xs: list_nat,Y: nat,Ys2: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys2 ) )
= ( ? [Z5: nat,Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Z5 @ Zs2 ) )
& ( ( F @ Z5 )
= Y )
& ( ( map_nat_nat @ F @ Zs2 )
= Ys2 ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1203_list_Osimps_I8_J,axiom,
! [F: nat > nat] :
( ( map_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_1204_list_Omap__cong,axiom,
! [X: list_nat,Ya: list_nat,F: nat > nat,G: nat > nat] :
( ( X = Ya )
=> ( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ Ya ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_nat_nat @ F @ X )
= ( map_nat_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_1205_list_Omap__cong0,axiom,
! [X: list_nat,F: nat > nat,G: nat > nat] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ X ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_nat_nat @ F @ X )
= ( map_nat_nat @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_1206_list_Oinj__map__strong,axiom,
! [X: list_nat,Xa2: list_nat,F: nat > nat,Fa: nat > nat] :
( ! [Z3: nat,Za: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ X ) )
=> ( ( member_nat @ Za @ ( set_nat2 @ Xa2 ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( map_nat_nat @ F @ X )
= ( map_nat_nat @ Fa @ Xa2 ) )
=> ( X = Xa2 ) ) ) ).
% list.inj_map_strong
thf(fact_1207_list_Omap__ident__strong,axiom,
! [T: list_nat,F: nat > nat] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ T ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_nat_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_1208_map__ext,axiom,
! [Xs: list_nat,F: nat > nat,G: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_1209_map__idI,axiom,
! [Xs: list_nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_nat_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_1210_map__cong,axiom,
! [Xs: list_nat,Ys2: list_nat,F: nat > nat,G: nat > nat] :
( ( Xs = Ys2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Ys2 ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys2 ) ) ) ) ).
% map_cong
thf(fact_1211_ex__map__conv,axiom,
! [Ys2: list_nat,F: nat > nat] :
( ( ? [Xs3: list_nat] :
( Ys2
= ( map_nat_nat @ F @ Xs3 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Ys2 ) )
=> ? [Y4: nat] :
( X3
= ( F @ Y4 ) ) ) ) ) ).
% ex_map_conv
thf(fact_1212_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys2: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys2 ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1213_append__eq__map__conv,axiom,
! [Ys2: list_nat,Zs: list_nat,F: nat > nat,Xs: list_nat] :
( ( ( append_nat @ Ys2 @ Zs )
= ( map_nat_nat @ F @ Xs ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys2
= ( map_nat_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_nat @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_1214_map__eq__append__conv,axiom,
! [F: nat > nat,Xs: list_nat,Ys2: list_nat,Zs: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( append_nat @ Ys2 @ Zs ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys2
= ( map_nat_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_nat @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_1215_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_1216_hd__map,axiom,
! [Xs: list_nat,F: nat > nat] :
( ( Xs != nil_nat )
=> ( ( hd_nat @ ( map_nat_nat @ F @ Xs ) )
= ( F @ ( hd_nat @ Xs ) ) ) ) ).
% hd_map
thf(fact_1217_hd__concat,axiom,
! [Xs: list_list_nat] :
( ( Xs != nil_list_nat )
=> ( ( ( hd_list_nat @ Xs )
!= nil_nat )
=> ( ( hd_nat @ ( concat_nat @ Xs ) )
= ( hd_nat @ ( hd_list_nat @ Xs ) ) ) ) ) ).
% hd_concat
thf(fact_1218_map__update,axiom,
! [F: nat > nat,Xs: list_nat,K: nat,Y: nat] :
( ( map_nat_nat @ F @ ( list_update_nat @ Xs @ K @ Y ) )
= ( list_update_nat @ ( map_nat_nat @ F @ Xs ) @ 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_map__mono__prefix,axiom,
! [Xs: list_nat,Ys2: list_nat,F: nat > nat] :
( ( prefix_nat @ Xs @ Ys2 )
=> ( prefix_nat @ ( map_nat_nat @ F @ Xs ) @ ( map_nat_nat @ F @ Ys2 ) ) ) ).
% map_mono_prefix
thf(fact_1221_prefix__map__rightE,axiom,
! [Xs: list_nat,F: nat > nat,Ys2: list_nat] :
( ( prefix_nat @ Xs @ ( map_nat_nat @ F @ Ys2 ) )
=> ? [Xs5: list_nat] :
( ( prefix_nat @ Xs5 @ Ys2 )
& ( Xs
= ( map_nat_nat @ F @ Xs5 ) ) ) ) ).
% prefix_map_rightE
thf(fact_1222_rotate1__map,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( rotate1_nat @ ( map_nat_nat @ F @ Xs ) )
= ( map_nat_nat @ F @ ( rotate1_nat @ Xs ) ) ) ).
% rotate1_map
thf(fact_1223_map__butlast,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( map_nat_nat @ F @ ( butlast_nat @ Xs ) )
= ( butlast_nat @ ( map_nat_nat @ F @ Xs ) ) ) ).
% map_butlast
thf(fact_1224_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_1225_hd__conv__nth,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ( hd_nat @ Xs )
= ( nth_nat @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_1226_hd__drop__conv__nth,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( hd_nat @ ( drop_nat @ N @ Xs ) )
= ( nth_nat @ Xs @ N ) ) ) ).
% hd_drop_conv_nth
thf(fact_1227_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_1228_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_1229_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_1230_binomial__n__n,axiom,
! [N: nat] :
( ( binomial @ N @ N )
= one_one_nat ) ).
% binomial_n_n
thf(fact_1231_num__of__nat__numeral__eq,axiom,
! [Q2: num] :
( ( num_of_nat @ ( numeral_numeral_nat @ Q2 ) )
= Q2 ) ).
% num_of_nat_numeral_eq
thf(fact_1232_binomial__0__Suc,axiom,
! [K: nat] :
( ( binomial @ zero_zero_nat @ ( suc @ K ) )
= zero_zero_nat ) ).
% binomial_0_Suc
thf(fact_1233_binomial__1,axiom,
! [N: nat] :
( ( binomial @ N @ ( suc @ zero_zero_nat ) )
= N ) ).
% binomial_1
thf(fact_1234_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_1235_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_1236_binomial__n__0,axiom,
! [N: nat] :
( ( binomial @ N @ zero_zero_nat )
= one_one_nat ) ).
% binomial_n_0
thf(fact_1237_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_1238_choose__one,axiom,
! [N: nat] :
( ( binomial @ N @ one_one_nat )
= N ) ).
% choose_one
thf(fact_1239_binomial__eq__0,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( binomial @ N @ K )
= zero_zero_nat ) ) ).
% binomial_eq_0
thf(fact_1240_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_1241_prefixes_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( prefixes_nat @ ( cons_nat @ X @ Xs ) )
= ( cons_list_nat @ nil_nat @ ( map_li7225945977422193158st_nat @ ( cons_nat @ X ) @ ( prefixes_nat @ Xs ) ) ) ) ).
% prefixes.simps(2)
thf(fact_1242_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_1243_sublists_Osimps_I2_J,axiom,
! [X: nat,Xs: list_nat] :
( ( sublists_nat @ ( cons_nat @ X @ Xs ) )
= ( append_list_nat @ ( sublists_nat @ Xs ) @ ( map_li7225945977422193158st_nat @ ( cons_nat @ X ) @ ( prefixes_nat @ Xs ) ) ) ) ).
% sublists.simps(2)
thf(fact_1244_choose__reduce__nat,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( binomial @ N @ K )
= ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ) ).
% choose_reduce_nat
thf(fact_1245_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_1246_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_1247_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_1248_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_1249_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% Suc_numeral
thf(fact_1250_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_1251_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_1252_num__of__nat_Osimps_I1_J,axiom,
( ( num_of_nat @ zero_zero_nat )
= one ) ).
% num_of_nat.simps(1)
thf(fact_1253_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral_nat @ one )
= ( suc @ zero_zero_nat ) ) ).
% numeral_1_eq_Suc_0
thf(fact_1254_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_1255_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq_num @ X @ one )
= ( X = one ) ) ).
% le_num_One_iff
thf(fact_1256_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_1257_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_1258_num_Osize_I4_J,axiom,
( ( size_size_num @ one )
= zero_zero_nat ) ).
% num.size(4)
thf(fact_1259_num__of__nat__One,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ one_one_nat )
=> ( ( num_of_nat @ N )
= one ) ) ).
% num_of_nat_One
thf(fact_1260_Suc__nat__number__of__add,axiom,
! [V: num,N: nat] :
( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N ) ) ).
% Suc_nat_number_of_add
thf(fact_1261_num__of__nat_Osimps_I2_J,axiom,
! [N: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( num_of_nat @ ( suc @ N ) )
= ( inc @ ( num_of_nat @ N ) ) ) )
& ( ~ ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( num_of_nat @ ( suc @ N ) )
= one ) ) ) ).
% num_of_nat.simps(2)
% Helper facts (11)
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_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 ) ).
thf(help_If_3_1_If_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_T,axiom,
! [X: list_list_nat,Y: list_list_nat] :
( ( if_list_list_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J_T,axiom,
! [X: list_list_nat,Y: list_list_nat] :
( ( if_list_list_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_nat @ ya @ ( plus_plus_nat @ ( equiva5889994315859557365_limit @ ys ) @ one_one_nat ) ).
%------------------------------------------------------------------------------