TPTP Problem File: SLH0925^1.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_00193_007375__11869196_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1374 ( 682 unt; 107 typ; 0 def)
% Number of atoms : 3247 (1389 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 9168 ( 366 ~; 63 |; 214 &;7416 @)
% ( 0 <=>;1109 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 12 ( 11 usr)
% Number of type conns : 462 ( 462 >; 0 *; 0 +; 0 <<)
% Number of symbols : 99 ( 96 usr; 13 con; 0-3 aty)
% Number of variables : 3203 ( 293 ^;2832 !; 78 ?;3203 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:13:46.976
%------------------------------------------------------------------------------
% Could-be-implicit typings (11)
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
list_P6011104703257516679at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
set_set_list_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
set_nat_o: $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__Nat__Onat,type,
nat: $tType ).
% Explicit typings (96)
thf(sy_c_Equivalence__Relation__Enumeration_Oenum__rgfs,type,
equiva7426478223624825838m_rgfs: nat > list_list_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_Finite__Set_Ocard_001t__List__Olist_It__Nat__Onat_J,type,
finite_card_list_nat: set_list_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Nat__Onat_J,type,
finite_card_set_nat: set_set_nat > nat ).
thf(sy_c_GCD_OGcd__class_OGcd_001t__Nat__Onat,type,
gcd_Gcd_nat: set_nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
minus_7954133019191499631st_nat: set_list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list_001t__Nat__Onat,type,
groups4561878855575611511st_nat: list_nat > nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
inf_inf_list_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > list_nat > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Nat__Onat_M_Eo_J,type,
inf_inf_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
inf_inf_set_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > set_nat > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_Eo,type,
inf_inf_o: $o > $o > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
inf_inf_set_list_nat: set_list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
inf_inf_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_List_Ocount__list_001t__List__Olist_It__Nat__Onat_J,type,
count_list_list_nat: list_list_nat > list_nat > nat ).
thf(sy_c_List_Ocount__list_001t__Nat__Onat,type,
count_list_nat: list_nat > nat > nat ).
thf(sy_c_List_Ogen__length_001t__Nat__Onat,type,
gen_length_nat: 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__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
map_list_nat_nat: ( list_nat > nat ) > list_list_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
map_li9084805350295849819at_nat: ( list_nat > product_prod_nat_nat ) > list_list_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
map_nat_list_nat: ( nat > list_nat ) > 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_Omap_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
map_na7298421622053143531at_nat: ( nat > product_prod_nat_nat ) > list_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
map_Pr7251875764006242973st_nat: ( product_prod_nat_nat > list_nat ) > list_P6011104703257516679at_nat > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
map_Pr3938374229010428429at_nat: ( product_prod_nat_nat > nat ) > list_P6011104703257516679at_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
map_Pr8058819605623181956at_nat: ( product_prod_nat_nat > product_prod_nat_nat ) > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Olist_Osize__list_001t__List__Olist_It__Nat__Onat_J,type,
size_list_list_nat: ( list_nat > nat ) > list_list_nat > nat ).
thf(sy_c_List_Olist_Osize__list_001t__Nat__Onat,type,
size_list_nat: ( nat > nat ) > list_nat > nat ).
thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Nat__Onat,type,
product_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
size_s3023201423986296836st_nat: list_list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
bot_bot_list_nat_o: list_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
bot_bot_set_nat_o: set_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
bot_bot_set_nat_o2: set_nat_o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
bot_bot_set_list_nat: set_list_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Nat__Onat_J_J_J,type,
bot_bo3886227569956363488st_nat: set_set_list_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
ord_less_list_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
ord_less_set_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).
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__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
ord_le1190675801316882794st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $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_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
zero_n2687167440665602831ol_nat: $o > 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_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
image_7976474329151083847st_nat: ( list_nat > list_nat ) > set_list_nat > set_list_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
image_list_nat_nat: ( list_nat > nat ) > set_list_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
image_nat_list_nat: ( nat > list_nat ) > set_nat > set_list_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
insert_list_nat: list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
insert_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Set_Ois__empty_001t__List__Olist_It__Nat__Onat_J,type,
is_empty_list_nat: set_list_nat > $o ).
thf(sy_c_Set_Ois__empty_001t__Nat__Onat,type,
is_empty_nat: set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__List__Olist_It__Nat__Onat_J,type,
is_sin2641923865335537900st_nat: set_list_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Othe__elem_001t__List__Olist_It__Nat__Onat_J,type,
the_elem_list_nat: set_list_nat > list_nat ).
thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001_062_It__Nat__Onat_M_Eo_J,type,
set_or8293666589767078672_nat_o: ( nat > $o ) > ( nat > $o ) > set_nat_o ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_or5210331525979695045st_nat: set_list_nat > set_list_nat > set_set_list_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_or3540276404033026485et_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Stirling_OStirling,type,
stirling: nat > nat > nat ).
thf(sy_c_Stirling_Ostirling,type,
stirling2: 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_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_v_x,type,
x: list_nat ).
thf(sy_v_xa____,type,
xa: nat ).
thf(sy_v_xs____,type,
xs: list_nat ).
% Relevant facts (1263)
thf(fact_0_assms,axiom,
equiva3371634703666331078on_rgf @ x ).
% assms
thf(fact_1_snoc_OIH,axiom,
( ( equiva3371634703666331078on_rgf @ xs )
=> ( ( count_list_list_nat @ ( equiva7426478223624825838m_rgfs @ ( size_size_list_nat @ xs ) ) @ xs )
= one_one_nat ) ) ).
% snoc.IH
thf(fact_2_calculation,axiom,
( one_one_nat
= ( count_list_list_nat @ ( equiva7426478223624825838m_rgfs @ ( size_size_list_nat @ xs ) ) @ xs ) ) ).
% calculation
thf(fact_3_a,axiom,
( ( finite_card_nat @ ( inf_inf_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( plus_plus_nat @ ( equiva5889994315859557365_limit @ xs ) @ one_one_nat ) ) @ ( insert_nat @ xa @ bot_bot_set_nat ) ) )
= one_one_nat ) ).
% a
thf(fact_4_b,axiom,
equiva3371634703666331078on_rgf @ xs ).
% b
thf(fact_5_sum__list__0,axiom,
! [Xs: list_nat] :
( ( groups4561878855575611511st_nat
@ ( map_nat_nat
@ ^ [X: nat] : zero_zero_nat
@ Xs ) )
= zero_zero_nat ) ).
% sum_list_0
thf(fact_6_sum__list__0,axiom,
! [Xs: list_list_nat] :
( ( groups4561878855575611511st_nat
@ ( map_list_nat_nat
@ ^ [X: list_nat] : zero_zero_nat
@ Xs ) )
= zero_zero_nat ) ).
% sum_list_0
thf(fact_7_card_Oempty,axiom,
( ( finite_card_set_nat @ bot_bot_set_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_8_card_Oempty,axiom,
( ( finite_card_list_nat @ bot_bot_set_list_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_9_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_10_disjoint__insert_I2_J,axiom,
! [A: set_set_nat,B: set_nat,B2: set_set_nat] :
( ( bot_bot_set_set_nat
= ( inf_inf_set_set_nat @ A @ ( insert_set_nat @ B @ B2 ) ) )
= ( ~ ( member_set_nat @ B @ A )
& ( bot_bot_set_set_nat
= ( inf_inf_set_set_nat @ A @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_11_disjoint__insert_I2_J,axiom,
! [A: set_list_nat,B: list_nat,B2: set_list_nat] :
( ( bot_bot_set_list_nat
= ( inf_inf_set_list_nat @ A @ ( insert_list_nat @ B @ B2 ) ) )
= ( ~ ( member_list_nat @ B @ A )
& ( bot_bot_set_list_nat
= ( inf_inf_set_list_nat @ A @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_12_disjoint__insert_I2_J,axiom,
! [A: set_nat,B: nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ ( insert_nat @ B @ B2 ) ) )
= ( ~ ( member_nat @ B @ A )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_13_disjoint__insert_I1_J,axiom,
! [B2: set_set_nat,A2: set_nat,A: set_set_nat] :
( ( ( inf_inf_set_set_nat @ B2 @ ( insert_set_nat @ A2 @ A ) )
= bot_bot_set_set_nat )
= ( ~ ( member_set_nat @ A2 @ B2 )
& ( ( inf_inf_set_set_nat @ B2 @ A )
= bot_bot_set_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_14_disjoint__insert_I1_J,axiom,
! [B2: set_list_nat,A2: list_nat,A: set_list_nat] :
( ( ( inf_inf_set_list_nat @ B2 @ ( insert_list_nat @ A2 @ A ) )
= bot_bot_set_list_nat )
= ( ~ ( member_list_nat @ A2 @ B2 )
& ( ( inf_inf_set_list_nat @ B2 @ A )
= bot_bot_set_list_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_15_disjoint__insert_I1_J,axiom,
! [B2: set_nat,A2: nat,A: set_nat] :
( ( ( inf_inf_set_nat @ B2 @ ( insert_nat @ A2 @ A ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B2 )
& ( ( inf_inf_set_nat @ B2 @ A )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_16_insert__disjoint_I2_J,axiom,
! [A2: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( bot_bot_set_set_nat
= ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ A ) @ B2 ) )
= ( ~ ( member_set_nat @ A2 @ B2 )
& ( bot_bot_set_set_nat
= ( inf_inf_set_set_nat @ A @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_17_insert__disjoint_I2_J,axiom,
! [A2: list_nat,A: set_list_nat,B2: set_list_nat] :
( ( bot_bot_set_list_nat
= ( inf_inf_set_list_nat @ ( insert_list_nat @ A2 @ A ) @ B2 ) )
= ( ~ ( member_list_nat @ A2 @ B2 )
& ( bot_bot_set_list_nat
= ( inf_inf_set_list_nat @ A @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_18_insert__disjoint_I2_J,axiom,
! [A2: nat,A: set_nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B2 ) )
= ( ~ ( member_nat @ A2 @ B2 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_19_insert__disjoint_I1_J,axiom,
! [A2: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ A ) @ B2 )
= bot_bot_set_set_nat )
= ( ~ ( member_set_nat @ A2 @ B2 )
& ( ( inf_inf_set_set_nat @ A @ B2 )
= bot_bot_set_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_20_insert__disjoint_I1_J,axiom,
! [A2: nat,A: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B2 )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B2 )
& ( ( inf_inf_set_nat @ A @ B2 )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_21_insert__disjoint_I1_J,axiom,
! [A2: list_nat,A: set_list_nat,B2: set_list_nat] :
( ( ( inf_inf_set_list_nat @ ( insert_list_nat @ A2 @ A ) @ B2 )
= bot_bot_set_list_nat )
= ( ~ ( member_list_nat @ A2 @ B2 )
& ( ( inf_inf_set_list_nat @ A @ B2 )
= bot_bot_set_list_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_22__092_060open_062x_A_060_Argf__limit_Axs_A_L_A1_092_060close_062,axiom,
ord_less_nat @ xa @ ( plus_plus_nat @ ( equiva5889994315859557365_limit @ xs ) @ one_one_nat ) ).
% \<open>x < rgf_limit xs + 1\<close>
thf(fact_23_singleton__conv,axiom,
! [A2: set_nat] :
( ( collect_set_nat
@ ^ [X: set_nat] : ( X = A2 ) )
= ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) ).
% singleton_conv
thf(fact_24_singleton__conv,axiom,
! [A2: nat] :
( ( collect_nat
@ ^ [X: nat] : ( X = A2 ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_25_singleton__conv,axiom,
! [A2: list_nat] :
( ( collect_list_nat
@ ^ [X: list_nat] : ( X = A2 ) )
= ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) ) ).
% singleton_conv
thf(fact_26_empty__Collect__eq,axiom,
! [P: set_nat > $o] :
( ( bot_bot_set_set_nat
= ( collect_set_nat @ P ) )
= ( ! [X: set_nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_27_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_28_empty__Collect__eq,axiom,
! [P: list_nat > $o] :
( ( bot_bot_set_list_nat
= ( collect_list_nat @ P ) )
= ( ! [X: list_nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_29_Collect__empty__eq,axiom,
! [P: set_nat > $o] :
( ( ( collect_set_nat @ P )
= bot_bot_set_set_nat )
= ( ! [X: set_nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_30_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_31_Collect__empty__eq,axiom,
! [P: list_nat > $o] :
( ( ( collect_list_nat @ P )
= bot_bot_set_list_nat )
= ( ! [X: list_nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_32_all__not__in__conv,axiom,
! [A: set_set_nat] :
( ( ! [X: set_nat] :
~ ( member_set_nat @ X @ A ) )
= ( A = bot_bot_set_set_nat ) ) ).
% all_not_in_conv
thf(fact_33_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X: nat] :
~ ( member_nat @ X @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_34_all__not__in__conv,axiom,
! [A: set_list_nat] :
( ( ! [X: list_nat] :
~ ( member_list_nat @ X @ A ) )
= ( A = bot_bot_set_list_nat ) ) ).
% all_not_in_conv
thf(fact_35_empty__iff,axiom,
! [C: set_nat] :
~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).
% empty_iff
thf(fact_36_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_37_empty__iff,axiom,
! [C: list_nat] :
~ ( member_list_nat @ C @ bot_bot_set_list_nat ) ).
% empty_iff
thf(fact_38_insert__absorb2,axiom,
! [X2: list_nat,A: set_list_nat] :
( ( insert_list_nat @ X2 @ ( insert_list_nat @ X2 @ A ) )
= ( insert_list_nat @ X2 @ A ) ) ).
% insert_absorb2
thf(fact_39_insert__absorb2,axiom,
! [X2: nat,A: set_nat] :
( ( insert_nat @ X2 @ ( insert_nat @ X2 @ A ) )
= ( insert_nat @ X2 @ A ) ) ).
% insert_absorb2
thf(fact_40_insert__iff,axiom,
! [A2: list_nat,B: list_nat,A: set_list_nat] :
( ( member_list_nat @ A2 @ ( insert_list_nat @ B @ A ) )
= ( ( A2 = B )
| ( member_list_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_41_insert__iff,axiom,
! [A2: set_nat,B: set_nat,A: set_set_nat] :
( ( member_set_nat @ A2 @ ( insert_set_nat @ B @ A ) )
= ( ( A2 = B )
| ( member_set_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_42_insert__iff,axiom,
! [A2: nat,B: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B @ A ) )
= ( ( A2 = B )
| ( member_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_43_insertCI,axiom,
! [A2: list_nat,B2: set_list_nat,B: list_nat] :
( ( ~ ( member_list_nat @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member_list_nat @ A2 @ ( insert_list_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_44_insertCI,axiom,
! [A2: set_nat,B2: set_set_nat,B: set_nat] :
( ( ~ ( member_set_nat @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member_set_nat @ A2 @ ( insert_set_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_45_insertCI,axiom,
! [A2: nat,B2: set_nat,B: nat] :
( ( ~ ( member_nat @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member_nat @ A2 @ ( insert_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_46_Int__iff,axiom,
! [C: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A @ B2 ) )
= ( ( member_set_nat @ C @ A )
& ( member_set_nat @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_47_Int__iff,axiom,
! [C: list_nat,A: set_list_nat,B2: set_list_nat] :
( ( member_list_nat @ C @ ( inf_inf_set_list_nat @ A @ B2 ) )
= ( ( member_list_nat @ C @ A )
& ( member_list_nat @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_48_Int__iff,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B2 ) )
= ( ( member_nat @ C @ A )
& ( member_nat @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_49_IntI,axiom,
! [C: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ A )
=> ( ( member_set_nat @ C @ B2 )
=> ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_50_IntI,axiom,
! [C: list_nat,A: set_list_nat,B2: set_list_nat] :
( ( member_list_nat @ C @ A )
=> ( ( member_list_nat @ C @ B2 )
=> ( member_list_nat @ C @ ( inf_inf_set_list_nat @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_51_IntI,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ A )
=> ( ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A @ B2 ) ) ) ) ).
% IntI
thf(fact_52_mult__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A2 @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A2 = B ) ) ) ).
% mult_cancel_right
thf(fact_53_mult__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ( times_times_nat @ C @ A2 )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A2 = B ) ) ) ).
% mult_cancel_left
thf(fact_54_mult__eq__0__iff,axiom,
! [A2: nat,B: nat] :
( ( ( times_times_nat @ A2 @ B )
= zero_zero_nat )
= ( ( A2 = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_55_mult__zero__right,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_56_mult__zero__left,axiom,
! [A2: nat] :
( ( times_times_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_57_singletonI,axiom,
! [A2: set_nat] : ( member_set_nat @ A2 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) ).
% singletonI
thf(fact_58_singletonI,axiom,
! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_59_singletonI,axiom,
! [A2: list_nat] : ( member_list_nat @ A2 @ ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) ) ).
% singletonI
thf(fact_60_Int__insert__right__if1,axiom,
! [A2: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ A2 @ A )
=> ( ( inf_inf_set_set_nat @ A @ ( insert_set_nat @ A2 @ B2 ) )
= ( insert_set_nat @ A2 @ ( inf_inf_set_set_nat @ A @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_61_Int__insert__right__if1,axiom,
! [A2: list_nat,A: set_list_nat,B2: set_list_nat] :
( ( member_list_nat @ A2 @ A )
=> ( ( inf_inf_set_list_nat @ A @ ( insert_list_nat @ A2 @ B2 ) )
= ( insert_list_nat @ A2 @ ( inf_inf_set_list_nat @ A @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_62_Int__insert__right__if1,axiom,
! [A2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B2 ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_63_Int__insert__right__if0,axiom,
! [A2: set_nat,A: set_set_nat,B2: set_set_nat] :
( ~ ( member_set_nat @ A2 @ A )
=> ( ( inf_inf_set_set_nat @ A @ ( insert_set_nat @ A2 @ B2 ) )
= ( inf_inf_set_set_nat @ A @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_64_Int__insert__right__if0,axiom,
! [A2: list_nat,A: set_list_nat,B2: set_list_nat] :
( ~ ( member_list_nat @ A2 @ A )
=> ( ( inf_inf_set_list_nat @ A @ ( insert_list_nat @ A2 @ B2 ) )
= ( inf_inf_set_list_nat @ A @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_65_Int__insert__right__if0,axiom,
! [A2: nat,A: set_nat,B2: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B2 ) )
= ( inf_inf_set_nat @ A @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_66_insert__inter__insert,axiom,
! [A2: list_nat,A: set_list_nat,B2: set_list_nat] :
( ( inf_inf_set_list_nat @ ( insert_list_nat @ A2 @ A ) @ ( insert_list_nat @ A2 @ B2 ) )
= ( insert_list_nat @ A2 @ ( inf_inf_set_list_nat @ A @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_67_insert__inter__insert,axiom,
! [A2: nat,A: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ ( insert_nat @ A2 @ B2 ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_68_Int__insert__left__if1,axiom,
! [A2: set_nat,C2: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ A2 @ C2 )
=> ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ B2 ) @ C2 )
= ( insert_set_nat @ A2 @ ( inf_inf_set_set_nat @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_69_Int__insert__left__if1,axiom,
! [A2: list_nat,C2: set_list_nat,B2: set_list_nat] :
( ( member_list_nat @ A2 @ C2 )
=> ( ( inf_inf_set_list_nat @ ( insert_list_nat @ A2 @ B2 ) @ C2 )
= ( insert_list_nat @ A2 @ ( inf_inf_set_list_nat @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_70_Int__insert__left__if1,axiom,
! [A2: nat,C2: set_nat,B2: set_nat] :
( ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B2 ) @ C2 )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_71_Int__insert__left__if0,axiom,
! [A2: set_nat,C2: set_set_nat,B2: set_set_nat] :
( ~ ( member_set_nat @ A2 @ C2 )
=> ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_set_nat @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_72_Int__insert__left__if0,axiom,
! [A2: list_nat,C2: set_list_nat,B2: set_list_nat] :
( ~ ( member_list_nat @ A2 @ C2 )
=> ( ( inf_inf_set_list_nat @ ( insert_list_nat @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_list_nat @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_73_Int__insert__left__if0,axiom,
! [A2: nat,C2: set_nat,B2: set_nat] :
( ~ ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_nat @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_74_of__bool__eq__0__iff,axiom,
! [P: $o] :
( ( ( zero_n2687167440665602831ol_nat @ P )
= zero_zero_nat )
= ~ P ) ).
% of_bool_eq_0_iff
thf(fact_75_of__bool__eq_I1_J,axiom,
( ( zero_n2687167440665602831ol_nat @ $false )
= zero_zero_nat ) ).
% of_bool_eq(1)
thf(fact_76_of__bool__less__iff,axiom,
! [P: $o,Q: $o] :
( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
= ( ~ P
& Q ) ) ).
% of_bool_less_iff
thf(fact_77_of__bool__eq__1__iff,axiom,
! [P: $o] :
( ( ( zero_n2687167440665602831ol_nat @ P )
= one_one_nat )
= P ) ).
% of_bool_eq_1_iff
thf(fact_78_of__bool__eq_I2_J,axiom,
( ( zero_n2687167440665602831ol_nat @ $true )
= one_one_nat ) ).
% of_bool_eq(2)
thf(fact_79_singleton__conv2,axiom,
! [A2: set_nat] :
( ( collect_set_nat
@ ( ^ [Y: set_nat,Z: set_nat] : ( Y = Z )
@ A2 ) )
= ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) ).
% singleton_conv2
thf(fact_80_singleton__conv2,axiom,
! [A2: nat] :
( ( collect_nat
@ ( ^ [Y: nat,Z: nat] : ( Y = Z )
@ A2 ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_81_singleton__conv2,axiom,
! [A2: list_nat] :
( ( collect_list_nat
@ ( ^ [Y: list_nat,Z: list_nat] : ( Y = Z )
@ A2 ) )
= ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) ) ).
% singleton_conv2
thf(fact_82_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_83_zero__less__of__bool__iff,axiom,
! [P: $o] :
( ( ord_less_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
= P ) ).
% zero_less_of_bool_iff
thf(fact_84_mem__Collect__eq,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A2 @ ( collect_set_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_85_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_86_mem__Collect__eq,axiom,
! [A2: list_nat,P: list_nat > $o] :
( ( member_list_nat @ A2 @ ( collect_list_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_87_Collect__mem__eq,axiom,
! [A: set_set_nat] :
( ( collect_set_nat
@ ^ [X: set_nat] : ( member_set_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_88_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_89_Collect__mem__eq,axiom,
! [A: set_list_nat] :
( ( collect_list_nat
@ ^ [X: list_nat] : ( member_list_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_90_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_91_Collect__cong,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ! [X3: list_nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_list_nat @ P )
= ( collect_list_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_92_of__bool__less__one__iff,axiom,
! [P: $o] :
( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat )
= ~ P ) ).
% of_bool_less_one_iff
thf(fact_93_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_94_mult__strict__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_95_mult__strict__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_96_zero__less__mult__pos2,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_97_zero__less__mult__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_98_mult__pos__neg2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A2 ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_99_mult__pos__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_100_mult__pos__neg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_101_mult__neg__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_102_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_103_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_104_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_105_add__mono1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_106_less__add__one,axiom,
! [A2: nat] : ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ).
% less_add_one
thf(fact_107_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_108_ex__in__conv,axiom,
! [A: set_set_nat] :
( ( ? [X: set_nat] : ( member_set_nat @ X @ A ) )
= ( A != bot_bot_set_set_nat ) ) ).
% ex_in_conv
thf(fact_109_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X: nat] : ( member_nat @ X @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_110_ex__in__conv,axiom,
! [A: set_list_nat] :
( ( ? [X: list_nat] : ( member_list_nat @ X @ A ) )
= ( A != bot_bot_set_list_nat ) ) ).
% ex_in_conv
thf(fact_111_equals0I,axiom,
! [A: set_set_nat] :
( ! [Y2: set_nat] :
~ ( member_set_nat @ Y2 @ A )
=> ( A = bot_bot_set_set_nat ) ) ).
% equals0I
thf(fact_112_equals0I,axiom,
! [A: set_nat] :
( ! [Y2: nat] :
~ ( member_nat @ Y2 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_113_equals0I,axiom,
! [A: set_list_nat] :
( ! [Y2: list_nat] :
~ ( member_list_nat @ Y2 @ A )
=> ( A = bot_bot_set_list_nat ) ) ).
% equals0I
thf(fact_114_equals0D,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( A = bot_bot_set_set_nat )
=> ~ ( member_set_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_115_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_116_equals0D,axiom,
! [A: set_list_nat,A2: list_nat] :
( ( A = bot_bot_set_list_nat )
=> ~ ( member_list_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_117_emptyE,axiom,
! [A2: set_nat] :
~ ( member_set_nat @ A2 @ bot_bot_set_set_nat ) ).
% emptyE
thf(fact_118_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_119_emptyE,axiom,
! [A2: list_nat] :
~ ( member_list_nat @ A2 @ bot_bot_set_list_nat ) ).
% emptyE
thf(fact_120_mk__disjoint__insert,axiom,
! [A2: list_nat,A: set_list_nat] :
( ( member_list_nat @ A2 @ A )
=> ? [B3: set_list_nat] :
( ( A
= ( insert_list_nat @ A2 @ B3 ) )
& ~ ( member_list_nat @ A2 @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_121_mk__disjoint__insert,axiom,
! [A2: set_nat,A: set_set_nat] :
( ( member_set_nat @ A2 @ A )
=> ? [B3: set_set_nat] :
( ( A
= ( insert_set_nat @ A2 @ B3 ) )
& ~ ( member_set_nat @ A2 @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_122_mk__disjoint__insert,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ? [B3: set_nat] :
( ( A
= ( insert_nat @ A2 @ B3 ) )
& ~ ( member_nat @ A2 @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_123_insert__commute,axiom,
! [X2: list_nat,Y3: list_nat,A: set_list_nat] :
( ( insert_list_nat @ X2 @ ( insert_list_nat @ Y3 @ A ) )
= ( insert_list_nat @ Y3 @ ( insert_list_nat @ X2 @ A ) ) ) ).
% insert_commute
thf(fact_124_insert__commute,axiom,
! [X2: nat,Y3: nat,A: set_nat] :
( ( insert_nat @ X2 @ ( insert_nat @ Y3 @ A ) )
= ( insert_nat @ Y3 @ ( insert_nat @ X2 @ A ) ) ) ).
% insert_commute
thf(fact_125_insert__eq__iff,axiom,
! [A2: list_nat,A: set_list_nat,B: list_nat,B2: set_list_nat] :
( ~ ( member_list_nat @ A2 @ A )
=> ( ~ ( member_list_nat @ B @ B2 )
=> ( ( ( insert_list_nat @ A2 @ A )
= ( insert_list_nat @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C3: set_list_nat] :
( ( A
= ( insert_list_nat @ B @ C3 ) )
& ~ ( member_list_nat @ B @ C3 )
& ( B2
= ( insert_list_nat @ A2 @ C3 ) )
& ~ ( member_list_nat @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_126_insert__eq__iff,axiom,
! [A2: set_nat,A: set_set_nat,B: set_nat,B2: set_set_nat] :
( ~ ( member_set_nat @ A2 @ A )
=> ( ~ ( member_set_nat @ B @ B2 )
=> ( ( ( insert_set_nat @ A2 @ A )
= ( insert_set_nat @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C3: set_set_nat] :
( ( A
= ( insert_set_nat @ B @ C3 ) )
& ~ ( member_set_nat @ B @ C3 )
& ( B2
= ( insert_set_nat @ A2 @ C3 ) )
& ~ ( member_set_nat @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_127_insert__eq__iff,axiom,
! [A2: nat,A: set_nat,B: nat,B2: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ~ ( member_nat @ B @ B2 )
=> ( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C3: set_nat] :
( ( A
= ( insert_nat @ B @ C3 ) )
& ~ ( member_nat @ B @ C3 )
& ( B2
= ( insert_nat @ A2 @ C3 ) )
& ~ ( member_nat @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_128_insert__absorb,axiom,
! [A2: list_nat,A: set_list_nat] :
( ( member_list_nat @ A2 @ A )
=> ( ( insert_list_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_129_insert__absorb,axiom,
! [A2: set_nat,A: set_set_nat] :
( ( member_set_nat @ A2 @ A )
=> ( ( insert_set_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_130_insert__absorb,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_131_insert__ident,axiom,
! [X2: list_nat,A: set_list_nat,B2: set_list_nat] :
( ~ ( member_list_nat @ X2 @ A )
=> ( ~ ( member_list_nat @ X2 @ B2 )
=> ( ( ( insert_list_nat @ X2 @ A )
= ( insert_list_nat @ X2 @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_132_insert__ident,axiom,
! [X2: set_nat,A: set_set_nat,B2: set_set_nat] :
( ~ ( member_set_nat @ X2 @ A )
=> ( ~ ( member_set_nat @ X2 @ B2 )
=> ( ( ( insert_set_nat @ X2 @ A )
= ( insert_set_nat @ X2 @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_133_insert__ident,axiom,
! [X2: nat,A: set_nat,B2: set_nat] :
( ~ ( member_nat @ X2 @ A )
=> ( ~ ( member_nat @ X2 @ B2 )
=> ( ( ( insert_nat @ X2 @ A )
= ( insert_nat @ X2 @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_134_Set_Oset__insert,axiom,
! [X2: list_nat,A: set_list_nat] :
( ( member_list_nat @ X2 @ A )
=> ~ ! [B3: set_list_nat] :
( ( A
= ( insert_list_nat @ X2 @ B3 ) )
=> ( member_list_nat @ X2 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_135_Set_Oset__insert,axiom,
! [X2: set_nat,A: set_set_nat] :
( ( member_set_nat @ X2 @ A )
=> ~ ! [B3: set_set_nat] :
( ( A
= ( insert_set_nat @ X2 @ B3 ) )
=> ( member_set_nat @ X2 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_136_Set_Oset__insert,axiom,
! [X2: nat,A: set_nat] :
( ( member_nat @ X2 @ A )
=> ~ ! [B3: set_nat] :
( ( A
= ( insert_nat @ X2 @ B3 ) )
=> ( member_nat @ X2 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_137_insertI2,axiom,
! [A2: list_nat,B2: set_list_nat,B: list_nat] :
( ( member_list_nat @ A2 @ B2 )
=> ( member_list_nat @ A2 @ ( insert_list_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_138_insertI2,axiom,
! [A2: set_nat,B2: set_set_nat,B: set_nat] :
( ( member_set_nat @ A2 @ B2 )
=> ( member_set_nat @ A2 @ ( insert_set_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_139_insertI2,axiom,
! [A2: nat,B2: set_nat,B: nat] :
( ( member_nat @ A2 @ B2 )
=> ( member_nat @ A2 @ ( insert_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_140_insertI1,axiom,
! [A2: list_nat,B2: set_list_nat] : ( member_list_nat @ A2 @ ( insert_list_nat @ A2 @ B2 ) ) ).
% insertI1
thf(fact_141_insertI1,axiom,
! [A2: set_nat,B2: set_set_nat] : ( member_set_nat @ A2 @ ( insert_set_nat @ A2 @ B2 ) ) ).
% insertI1
thf(fact_142_insertI1,axiom,
! [A2: nat,B2: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B2 ) ) ).
% insertI1
thf(fact_143_insertE,axiom,
! [A2: list_nat,B: list_nat,A: set_list_nat] :
( ( member_list_nat @ A2 @ ( insert_list_nat @ B @ A ) )
=> ( ( A2 != B )
=> ( member_list_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_144_insertE,axiom,
! [A2: set_nat,B: set_nat,A: set_set_nat] :
( ( member_set_nat @ A2 @ ( insert_set_nat @ B @ A ) )
=> ( ( A2 != B )
=> ( member_set_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_145_insertE,axiom,
! [A2: nat,B: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B @ A ) )
=> ( ( A2 != B )
=> ( member_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_146_Int__left__commute,axiom,
! [A: set_list_nat,B2: set_list_nat,C2: set_list_nat] :
( ( inf_inf_set_list_nat @ A @ ( inf_inf_set_list_nat @ B2 @ C2 ) )
= ( inf_inf_set_list_nat @ B2 @ ( inf_inf_set_list_nat @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_147_Int__left__commute,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B2 @ C2 ) )
= ( inf_inf_set_nat @ B2 @ ( inf_inf_set_nat @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_148_Int__left__absorb,axiom,
! [A: set_list_nat,B2: set_list_nat] :
( ( inf_inf_set_list_nat @ A @ ( inf_inf_set_list_nat @ A @ B2 ) )
= ( inf_inf_set_list_nat @ A @ B2 ) ) ).
% Int_left_absorb
thf(fact_149_Int__left__absorb,axiom,
! [A: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ A @ B2 ) )
= ( inf_inf_set_nat @ A @ B2 ) ) ).
% Int_left_absorb
thf(fact_150_Int__commute,axiom,
( inf_inf_set_list_nat
= ( ^ [A3: set_list_nat,B4: set_list_nat] : ( inf_inf_set_list_nat @ B4 @ A3 ) ) ) ).
% Int_commute
thf(fact_151_Int__commute,axiom,
( inf_inf_set_nat
= ( ^ [A3: set_nat,B4: set_nat] : ( inf_inf_set_nat @ B4 @ A3 ) ) ) ).
% Int_commute
thf(fact_152_Int__absorb,axiom,
! [A: set_list_nat] :
( ( inf_inf_set_list_nat @ A @ A )
= A ) ).
% Int_absorb
thf(fact_153_Int__absorb,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ A )
= A ) ).
% Int_absorb
thf(fact_154_Int__assoc,axiom,
! [A: set_list_nat,B2: set_list_nat,C2: set_list_nat] :
( ( inf_inf_set_list_nat @ ( inf_inf_set_list_nat @ A @ B2 ) @ C2 )
= ( inf_inf_set_list_nat @ A @ ( inf_inf_set_list_nat @ B2 @ C2 ) ) ) ).
% Int_assoc
thf(fact_155_Int__assoc,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ C2 )
= ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) ).
% Int_assoc
thf(fact_156_IntD2,axiom,
! [C: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A @ B2 ) )
=> ( member_set_nat @ C @ B2 ) ) ).
% IntD2
thf(fact_157_IntD2,axiom,
! [C: list_nat,A: set_list_nat,B2: set_list_nat] :
( ( member_list_nat @ C @ ( inf_inf_set_list_nat @ A @ B2 ) )
=> ( member_list_nat @ C @ B2 ) ) ).
% IntD2
thf(fact_158_IntD2,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B2 ) )
=> ( member_nat @ C @ B2 ) ) ).
% IntD2
thf(fact_159_IntD1,axiom,
! [C: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A @ B2 ) )
=> ( member_set_nat @ C @ A ) ) ).
% IntD1
thf(fact_160_IntD1,axiom,
! [C: list_nat,A: set_list_nat,B2: set_list_nat] :
( ( member_list_nat @ C @ ( inf_inf_set_list_nat @ A @ B2 ) )
=> ( member_list_nat @ C @ A ) ) ).
% IntD1
thf(fact_161_IntD1,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B2 ) )
=> ( member_nat @ C @ A ) ) ).
% IntD1
thf(fact_162_IntE,axiom,
! [C: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A @ B2 ) )
=> ~ ( ( member_set_nat @ C @ A )
=> ~ ( member_set_nat @ C @ B2 ) ) ) ).
% IntE
thf(fact_163_IntE,axiom,
! [C: list_nat,A: set_list_nat,B2: set_list_nat] :
( ( member_list_nat @ C @ ( inf_inf_set_list_nat @ A @ B2 ) )
=> ~ ( ( member_list_nat @ C @ A )
=> ~ ( member_list_nat @ C @ B2 ) ) ) ).
% IntE
thf(fact_164_IntE,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B2 ) )
=> ~ ( ( member_nat @ C @ A )
=> ~ ( member_nat @ C @ B2 ) ) ) ).
% IntE
thf(fact_165_of__bool__eq__iff,axiom,
! [P2: $o,Q2: $o] :
( ( ( zero_n2687167440665602831ol_nat @ P2 )
= ( zero_n2687167440665602831ol_nat @ Q2 ) )
= ( P2 = Q2 ) ) ).
% of_bool_eq_iff
thf(fact_166_empty__def,axiom,
( bot_bot_set_set_nat
= ( collect_set_nat
@ ^ [X: set_nat] : $false ) ) ).
% empty_def
thf(fact_167_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X: nat] : $false ) ) ).
% empty_def
thf(fact_168_empty__def,axiom,
( bot_bot_set_list_nat
= ( collect_list_nat
@ ^ [X: list_nat] : $false ) ) ).
% empty_def
thf(fact_169_insert__Collect,axiom,
! [A2: nat,P: nat > $o] :
( ( insert_nat @ A2 @ ( collect_nat @ P ) )
= ( collect_nat
@ ^ [U: nat] :
( ( U != A2 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_170_insert__Collect,axiom,
! [A2: list_nat,P: list_nat > $o] :
( ( insert_list_nat @ A2 @ ( collect_list_nat @ P ) )
= ( collect_list_nat
@ ^ [U: list_nat] :
( ( U != A2 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_171_insert__compr,axiom,
( insert_set_nat
= ( ^ [A4: set_nat,B4: set_set_nat] :
( collect_set_nat
@ ^ [X: set_nat] :
( ( X = A4 )
| ( member_set_nat @ X @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_172_insert__compr,axiom,
( insert_nat
= ( ^ [A4: nat,B4: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( X = A4 )
| ( member_nat @ X @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_173_insert__compr,axiom,
( insert_list_nat
= ( ^ [A4: list_nat,B4: set_list_nat] :
( collect_list_nat
@ ^ [X: list_nat] :
( ( X = A4 )
| ( member_list_nat @ X @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_174_Collect__conj__eq,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ( collect_list_nat
@ ^ [X: list_nat] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_inf_set_list_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_175_Collect__conj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_176_Int__Collect,axiom,
! [X2: set_nat,A: set_set_nat,P: set_nat > $o] :
( ( member_set_nat @ X2 @ ( inf_inf_set_set_nat @ A @ ( collect_set_nat @ P ) ) )
= ( ( member_set_nat @ X2 @ A )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_177_Int__Collect,axiom,
! [X2: list_nat,A: set_list_nat,P: list_nat > $o] :
( ( member_list_nat @ X2 @ ( inf_inf_set_list_nat @ A @ ( collect_list_nat @ P ) ) )
= ( ( member_list_nat @ X2 @ A )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_178_Int__Collect,axiom,
! [X2: nat,A: set_nat,P: nat > $o] :
( ( member_nat @ X2 @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) )
= ( ( member_nat @ X2 @ A )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_179_Int__def,axiom,
( inf_inf_set_set_nat
= ( ^ [A3: set_set_nat,B4: set_set_nat] :
( collect_set_nat
@ ^ [X: set_nat] :
( ( member_set_nat @ X @ A3 )
& ( member_set_nat @ X @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_180_Int__def,axiom,
( inf_inf_set_list_nat
= ( ^ [A3: set_list_nat,B4: set_list_nat] :
( collect_list_nat
@ ^ [X: list_nat] :
( ( member_list_nat @ X @ A3 )
& ( member_list_nat @ X @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_181_Int__def,axiom,
( inf_inf_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A3 )
& ( member_nat @ X @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_182_mult__right__cancel,axiom,
! [C: nat,A2: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A2 @ C )
= ( times_times_nat @ B @ C ) )
= ( A2 = B ) ) ) ).
% mult_right_cancel
thf(fact_183_mult__left__cancel,axiom,
! [C: nat,A2: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A2 )
= ( times_times_nat @ C @ B ) )
= ( A2 = B ) ) ) ).
% mult_left_cancel
thf(fact_184_no__zero__divisors,axiom,
! [A2: nat,B: nat] :
( ( A2 != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A2 @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_185_divisors__zero,axiom,
! [A2: nat,B: nat] :
( ( ( times_times_nat @ A2 @ B )
= zero_zero_nat )
=> ( ( A2 = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_186_mult__not__zero,axiom,
! [A2: nat,B: nat] :
( ( ( times_times_nat @ A2 @ B )
!= zero_zero_nat )
=> ( ( A2 != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_187_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_188_combine__common__factor,axiom,
! [A2: nat,E: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A2 @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_189_distrib__right,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_190_distrib__left,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ A2 @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ B ) @ ( times_times_nat @ A2 @ C ) ) ) ).
% distrib_left
thf(fact_191_comm__semiring__class_Odistrib,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_192_singleton__inject,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( insert_set_nat @ A2 @ bot_bot_set_set_nat )
= ( insert_set_nat @ B @ bot_bot_set_set_nat ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_193_singleton__inject,axiom,
! [A2: nat,B: nat] :
( ( ( insert_nat @ A2 @ bot_bot_set_nat )
= ( insert_nat @ B @ bot_bot_set_nat ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_194_singleton__inject,axiom,
! [A2: list_nat,B: list_nat] :
( ( ( insert_list_nat @ A2 @ bot_bot_set_list_nat )
= ( insert_list_nat @ B @ bot_bot_set_list_nat ) )
=> ( A2 = B ) ) ).
% singleton_inject
thf(fact_195_insert__not__empty,axiom,
! [A2: set_nat,A: set_set_nat] :
( ( insert_set_nat @ A2 @ A )
!= bot_bot_set_set_nat ) ).
% insert_not_empty
thf(fact_196_insert__not__empty,axiom,
! [A2: nat,A: set_nat] :
( ( insert_nat @ A2 @ A )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_197_insert__not__empty,axiom,
! [A2: list_nat,A: set_list_nat] :
( ( insert_list_nat @ A2 @ A )
!= bot_bot_set_list_nat ) ).
% insert_not_empty
thf(fact_198_doubleton__eq__iff,axiom,
! [A2: set_nat,B: set_nat,C: set_nat,D: set_nat] :
( ( ( insert_set_nat @ A2 @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) )
= ( insert_set_nat @ C @ ( insert_set_nat @ D @ bot_bot_set_set_nat ) ) )
= ( ( ( A2 = C )
& ( B = D ) )
| ( ( A2 = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_199_doubleton__eq__iff,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ( insert_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
= ( ( ( A2 = C )
& ( B = D ) )
| ( ( A2 = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_200_doubleton__eq__iff,axiom,
! [A2: list_nat,B: list_nat,C: list_nat,D: list_nat] :
( ( ( insert_list_nat @ A2 @ ( insert_list_nat @ B @ bot_bot_set_list_nat ) )
= ( insert_list_nat @ C @ ( insert_list_nat @ D @ bot_bot_set_list_nat ) ) )
= ( ( ( A2 = C )
& ( B = D ) )
| ( ( A2 = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_201_singleton__iff,axiom,
! [B: set_nat,A2: set_nat] :
( ( member_set_nat @ B @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_202_singleton__iff,axiom,
! [B: nat,A2: nat] :
( ( member_nat @ B @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_203_singleton__iff,axiom,
! [B: list_nat,A2: list_nat] :
( ( member_list_nat @ B @ ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_204_singletonD,axiom,
! [B: set_nat,A2: set_nat] :
( ( member_set_nat @ B @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_205_singletonD,axiom,
! [B: nat,A2: nat] :
( ( member_nat @ B @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_206_singletonD,axiom,
! [B: list_nat,A2: list_nat] :
( ( member_list_nat @ B @ ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_207_disjoint__iff__not__equal,axiom,
! [A: set_set_nat,B2: set_set_nat] :
( ( ( inf_inf_set_set_nat @ A @ B2 )
= bot_bot_set_set_nat )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ! [Y4: set_nat] :
( ( member_set_nat @ Y4 @ B2 )
=> ( X != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_208_disjoint__iff__not__equal,axiom,
! [A: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A @ B2 )
= bot_bot_set_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ B2 )
=> ( X != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_209_disjoint__iff__not__equal,axiom,
! [A: set_list_nat,B2: set_list_nat] :
( ( ( inf_inf_set_list_nat @ A @ B2 )
= bot_bot_set_list_nat )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ A )
=> ! [Y4: list_nat] :
( ( member_list_nat @ Y4 @ B2 )
=> ( X != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_210_Int__empty__right,axiom,
! [A: set_set_nat] :
( ( inf_inf_set_set_nat @ A @ bot_bot_set_set_nat )
= bot_bot_set_set_nat ) ).
% Int_empty_right
thf(fact_211_Int__empty__right,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_212_Int__empty__right,axiom,
! [A: set_list_nat] :
( ( inf_inf_set_list_nat @ A @ bot_bot_set_list_nat )
= bot_bot_set_list_nat ) ).
% Int_empty_right
thf(fact_213_Int__empty__left,axiom,
! [B2: set_set_nat] :
( ( inf_inf_set_set_nat @ bot_bot_set_set_nat @ B2 )
= bot_bot_set_set_nat ) ).
% Int_empty_left
thf(fact_214_Int__empty__left,axiom,
! [B2: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B2 )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_215_Int__empty__left,axiom,
! [B2: set_list_nat] :
( ( inf_inf_set_list_nat @ bot_bot_set_list_nat @ B2 )
= bot_bot_set_list_nat ) ).
% Int_empty_left
thf(fact_216_disjoint__iff,axiom,
! [A: set_set_nat,B2: set_set_nat] :
( ( ( inf_inf_set_set_nat @ A @ B2 )
= bot_bot_set_set_nat )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ~ ( member_set_nat @ X @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_217_disjoint__iff,axiom,
! [A: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A @ B2 )
= bot_bot_set_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ~ ( member_nat @ X @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_218_disjoint__iff,axiom,
! [A: set_list_nat,B2: set_list_nat] :
( ( ( inf_inf_set_list_nat @ A @ B2 )
= bot_bot_set_list_nat )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ A )
=> ~ ( member_list_nat @ X @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_219_Int__emptyI,axiom,
! [A: set_set_nat,B2: set_set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A )
=> ~ ( member_set_nat @ X3 @ B2 ) )
=> ( ( inf_inf_set_set_nat @ A @ B2 )
= bot_bot_set_set_nat ) ) ).
% Int_emptyI
thf(fact_220_Int__emptyI,axiom,
! [A: set_nat,B2: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ~ ( member_nat @ X3 @ B2 ) )
=> ( ( inf_inf_set_nat @ A @ B2 )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_221_Int__emptyI,axiom,
! [A: set_list_nat,B2: set_list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
=> ~ ( member_list_nat @ X3 @ B2 ) )
=> ( ( inf_inf_set_list_nat @ A @ B2 )
= bot_bot_set_list_nat ) ) ).
% Int_emptyI
thf(fact_222_Int__insert__right,axiom,
! [A2: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( ( member_set_nat @ A2 @ A )
=> ( ( inf_inf_set_set_nat @ A @ ( insert_set_nat @ A2 @ B2 ) )
= ( insert_set_nat @ A2 @ ( inf_inf_set_set_nat @ A @ B2 ) ) ) )
& ( ~ ( member_set_nat @ A2 @ A )
=> ( ( inf_inf_set_set_nat @ A @ ( insert_set_nat @ A2 @ B2 ) )
= ( inf_inf_set_set_nat @ A @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_223_Int__insert__right,axiom,
! [A2: list_nat,A: set_list_nat,B2: set_list_nat] :
( ( ( member_list_nat @ A2 @ A )
=> ( ( inf_inf_set_list_nat @ A @ ( insert_list_nat @ A2 @ B2 ) )
= ( insert_list_nat @ A2 @ ( inf_inf_set_list_nat @ A @ B2 ) ) ) )
& ( ~ ( member_list_nat @ A2 @ A )
=> ( ( inf_inf_set_list_nat @ A @ ( insert_list_nat @ A2 @ B2 ) )
= ( inf_inf_set_list_nat @ A @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_224_Int__insert__right,axiom,
! [A2: nat,A: set_nat,B2: set_nat] :
( ( ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B2 ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B2 ) ) ) )
& ( ~ ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B2 ) )
= ( inf_inf_set_nat @ A @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_225_Int__insert__left,axiom,
! [A2: set_nat,C2: set_set_nat,B2: set_set_nat] :
( ( ( member_set_nat @ A2 @ C2 )
=> ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ B2 ) @ C2 )
= ( insert_set_nat @ A2 @ ( inf_inf_set_set_nat @ B2 @ C2 ) ) ) )
& ( ~ ( member_set_nat @ A2 @ C2 )
=> ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_set_nat @ B2 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_226_Int__insert__left,axiom,
! [A2: list_nat,C2: set_list_nat,B2: set_list_nat] :
( ( ( member_list_nat @ A2 @ C2 )
=> ( ( inf_inf_set_list_nat @ ( insert_list_nat @ A2 @ B2 ) @ C2 )
= ( insert_list_nat @ A2 @ ( inf_inf_set_list_nat @ B2 @ C2 ) ) ) )
& ( ~ ( member_list_nat @ A2 @ C2 )
=> ( ( inf_inf_set_list_nat @ ( insert_list_nat @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_list_nat @ B2 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_227_Int__insert__left,axiom,
! [A2: nat,C2: set_nat,B2: set_nat] :
( ( ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B2 ) @ C2 )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) )
& ( ~ ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_nat @ B2 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_228_of__bool__conj,axiom,
! [P: $o,Q: $o] :
( ( zero_n2687167440665602831ol_nat
@ ( P
& Q ) )
= ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).
% of_bool_conj
thf(fact_229_lambda__zero,axiom,
( ( ^ [H: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_230_lambda__one,axiom,
( ( ^ [X: nat] : X )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_231_Collect__conv__if2,axiom,
! [P: set_nat > $o,A2: set_nat] :
( ( ( P @ A2 )
=> ( ( collect_set_nat
@ ^ [X: set_nat] :
( ( A2 = X )
& ( P @ X ) ) )
= ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_set_nat
@ ^ [X: set_nat] :
( ( A2 = X )
& ( P @ X ) ) )
= bot_bot_set_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_232_Collect__conv__if2,axiom,
! [P: nat > $o,A2: nat] :
( ( ( P @ A2 )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( A2 = X )
& ( P @ X ) ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( A2 = X )
& ( P @ X ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_233_Collect__conv__if2,axiom,
! [P: list_nat > $o,A2: list_nat] :
( ( ( P @ A2 )
=> ( ( collect_list_nat
@ ^ [X: list_nat] :
( ( A2 = X )
& ( P @ X ) ) )
= ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_list_nat
@ ^ [X: list_nat] :
( ( A2 = X )
& ( P @ X ) ) )
= bot_bot_set_list_nat ) ) ) ).
% Collect_conv_if2
thf(fact_234_Collect__conv__if,axiom,
! [P: set_nat > $o,A2: set_nat] :
( ( ( P @ A2 )
=> ( ( collect_set_nat
@ ^ [X: set_nat] :
( ( X = A2 )
& ( P @ X ) ) )
= ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_set_nat
@ ^ [X: set_nat] :
( ( X = A2 )
& ( P @ X ) ) )
= bot_bot_set_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_235_Collect__conv__if,axiom,
! [P: nat > $o,A2: nat] :
( ( ( P @ A2 )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( X = A2 )
& ( P @ X ) ) )
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_nat
@ ^ [X: nat] :
( ( X = A2 )
& ( P @ X ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_236_Collect__conv__if,axiom,
! [P: list_nat > $o,A2: list_nat] :
( ( ( P @ A2 )
=> ( ( collect_list_nat
@ ^ [X: list_nat] :
( ( X = A2 )
& ( P @ X ) ) )
= ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_list_nat
@ ^ [X: list_nat] :
( ( X = A2 )
& ( P @ X ) ) )
= bot_bot_set_list_nat ) ) ) ).
% Collect_conv_if
thf(fact_237_split__of__bool__asm,axiom,
! [P: nat > $o,P2: $o] :
( ( P @ ( zero_n2687167440665602831ol_nat @ P2 ) )
= ( ~ ( ( P2
& ~ ( P @ one_one_nat ) )
| ( ~ P2
& ~ ( P @ zero_zero_nat ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_238_split__of__bool,axiom,
! [P: nat > $o,P2: $o] :
( ( P @ ( zero_n2687167440665602831ol_nat @ P2 ) )
= ( ( P2
=> ( P @ one_one_nat ) )
& ( ~ P2
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_of_bool
thf(fact_239_of__bool__def,axiom,
( zero_n2687167440665602831ol_nat
= ( ^ [P3: $o] : ( if_nat @ P3 @ one_one_nat @ zero_zero_nat ) ) ) ).
% of_bool_def
thf(fact_240_sum__list__mult__const,axiom,
! [F: nat > nat,C: nat,Xs: list_nat] :
( ( groups4561878855575611511st_nat
@ ( map_nat_nat
@ ^ [X: nat] : ( times_times_nat @ ( F @ X ) @ C )
@ Xs ) )
= ( times_times_nat @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ F @ Xs ) ) @ C ) ) ).
% sum_list_mult_const
thf(fact_241_sum__list__mult__const,axiom,
! [F: list_nat > nat,C: nat,Xs: list_list_nat] :
( ( groups4561878855575611511st_nat
@ ( map_list_nat_nat
@ ^ [X: list_nat] : ( times_times_nat @ ( F @ X ) @ C )
@ Xs ) )
= ( times_times_nat @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ F @ Xs ) ) @ C ) ) ).
% sum_list_mult_const
thf(fact_242_sum__list__const__mult,axiom,
! [C: nat,F: nat > nat,Xs: list_nat] :
( ( groups4561878855575611511st_nat
@ ( map_nat_nat
@ ^ [X: nat] : ( times_times_nat @ C @ ( F @ X ) )
@ Xs ) )
= ( times_times_nat @ C @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ F @ Xs ) ) ) ) ).
% sum_list_const_mult
thf(fact_243_sum__list__const__mult,axiom,
! [C: nat,F: list_nat > nat,Xs: list_list_nat] :
( ( groups4561878855575611511st_nat
@ ( map_list_nat_nat
@ ^ [X: list_nat] : ( times_times_nat @ C @ ( F @ X ) )
@ Xs ) )
= ( times_times_nat @ C @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ F @ Xs ) ) ) ) ).
% sum_list_const_mult
thf(fact_244_sum__list__addf,axiom,
! [F: nat > nat,G: nat > nat,Xs: list_nat] :
( ( groups4561878855575611511st_nat
@ ( map_nat_nat
@ ^ [X: nat] : ( plus_plus_nat @ ( F @ X ) @ ( G @ X ) )
@ Xs ) )
= ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ F @ Xs ) ) @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ G @ Xs ) ) ) ) ).
% sum_list_addf
thf(fact_245_sum__list__addf,axiom,
! [F: list_nat > nat,G: list_nat > nat,Xs: list_list_nat] :
( ( groups4561878855575611511st_nat
@ ( map_list_nat_nat
@ ^ [X: list_nat] : ( plus_plus_nat @ ( F @ X ) @ ( G @ X ) )
@ Xs ) )
= ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ F @ Xs ) ) @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ G @ Xs ) ) ) ) ).
% sum_list_addf
thf(fact_246_card__1__singletonE,axiom,
! [A: set_set_nat] :
( ( ( finite_card_set_nat @ A )
= one_one_nat )
=> ~ ! [X3: set_nat] :
( A
!= ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) ) ) ).
% card_1_singletonE
thf(fact_247_card__1__singletonE,axiom,
! [A: set_nat] :
( ( ( finite_card_nat @ A )
= one_one_nat )
=> ~ ! [X3: nat] :
( A
!= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% card_1_singletonE
thf(fact_248_card__1__singletonE,axiom,
! [A: set_list_nat] :
( ( ( finite_card_list_nat @ A )
= one_one_nat )
=> ~ ! [X3: list_nat] :
( A
!= ( insert_list_nat @ X3 @ bot_bot_set_list_nat ) ) ) ).
% card_1_singletonE
thf(fact_249_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_250_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_251_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_252_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_253_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_254_atLeastLessThan__empty__iff2,axiom,
! [A2: set_list_nat,B: set_list_nat] :
( ( bot_bo3886227569956363488st_nat
= ( set_or5210331525979695045st_nat @ A2 @ B ) )
= ( ~ ( ord_le1190675801316882794st_nat @ A2 @ B ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_255_atLeastLessThan__empty__iff2,axiom,
! [A2: nat > $o,B: nat > $o] :
( ( bot_bot_set_nat_o2
= ( set_or8293666589767078672_nat_o @ A2 @ B ) )
= ( ~ ( ord_less_nat_o @ A2 @ B ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_256_atLeastLessThan__empty__iff2,axiom,
! [A2: set_nat,B: set_nat] :
( ( bot_bot_set_set_nat
= ( set_or3540276404033026485et_nat @ A2 @ B ) )
= ( ~ ( ord_less_set_nat @ A2 @ B ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_257_atLeastLessThan__empty__iff2,axiom,
! [A2: nat,B: nat] :
( ( bot_bot_set_nat
= ( set_or4665077453230672383an_nat @ A2 @ B ) )
= ( ~ ( ord_less_nat @ A2 @ B ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_258_atLeastLessThan__empty__iff,axiom,
! [A2: set_list_nat,B: set_list_nat] :
( ( ( set_or5210331525979695045st_nat @ A2 @ B )
= bot_bo3886227569956363488st_nat )
= ( ~ ( ord_le1190675801316882794st_nat @ A2 @ B ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_259_atLeastLessThan__empty__iff,axiom,
! [A2: nat > $o,B: nat > $o] :
( ( ( set_or8293666589767078672_nat_o @ A2 @ B )
= bot_bot_set_nat_o2 )
= ( ~ ( ord_less_nat_o @ A2 @ B ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_260_atLeastLessThan__empty__iff,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( set_or3540276404033026485et_nat @ A2 @ B )
= bot_bot_set_set_nat )
= ( ~ ( ord_less_set_nat @ A2 @ B ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_261_atLeastLessThan__empty__iff,axiom,
! [A2: nat,B: nat] :
( ( ( set_or4665077453230672383an_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ~ ( ord_less_nat @ A2 @ B ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_262_less__add__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_263_less__add__same__cancel1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_264_add__less__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_265_add__left__cancel,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_266_add__right__cancel,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_267_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_268_add_Oright__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.right_neutral
thf(fact_269_add__cancel__left__left,axiom,
! [B: nat,A2: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_270_add__cancel__left__right,axiom,
! [A2: nat,B: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= A2 )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_271_add__cancel__right__left,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ B @ A2 ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_272_add__cancel__right__right,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ A2 @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_273_add__eq__0__iff__both__eq__0,axiom,
! [X2: nat,Y3: nat] :
( ( ( plus_plus_nat @ X2 @ Y3 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_274_zero__eq__add__iff__both__eq__0,axiom,
! [X2: nat,Y3: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X2 @ Y3 ) )
= ( ( X2 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_275_add__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% add_0
thf(fact_276_add__less__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_left
thf(fact_277_add__less__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% add_less_cancel_right
thf(fact_278_mult_Oright__neutral,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ one_one_nat )
= A2 ) ).
% mult.right_neutral
thf(fact_279_mult__1,axiom,
! [A2: nat] :
( ( times_times_nat @ one_one_nat @ A2 )
= A2 ) ).
% mult_1
thf(fact_280_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_281_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_282_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_283_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_284_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_285_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_286_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_287_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_288_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_289_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_290_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_291_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_292_add__less__same__cancel1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_293_not__psubset__empty,axiom,
! [A: set_set_nat] :
~ ( ord_less_set_set_nat @ A @ bot_bot_set_set_nat ) ).
% not_psubset_empty
thf(fact_294_not__psubset__empty,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_295_not__psubset__empty,axiom,
! [A: set_list_nat] :
~ ( ord_le1190675801316882794st_nat @ A @ bot_bot_set_list_nat ) ).
% not_psubset_empty
thf(fact_296_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_297_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A2 @ B ) @ C )
= ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_298_mult_Oassoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A2 @ B ) @ C )
= ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_299_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A4: nat,B5: nat] : ( times_times_nat @ B5 @ A4 ) ) ) ).
% mult.commute
thf(fact_300_mult_Oleft__commute,axiom,
! [B: nat,A2: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A2 @ C ) )
= ( times_times_nat @ A2 @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_301_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_302_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( I2 = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I2 @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_303_group__cancel_Oadd1,axiom,
! [A: nat,K: nat,A2: nat,B: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_304_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A2: nat] :
( ( B2
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_305_add_Oassoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_306_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B5: nat] : ( plus_plus_nat @ B5 @ A4 ) ) ) ).
% add.commute
thf(fact_307_add_Oleft__commute,axiom,
! [B: nat,A2: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A2 @ C ) )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_308_add__left__imp__eq,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ A2 @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_309_add__right__imp__eq,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_310_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_311_linorder__neqE__nat,axiom,
! [X2: nat,Y3: nat] :
( ( X2 != Y3 )
=> ( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_312_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_313_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_314_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_315_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_316_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_317_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_318_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_319_size__neq__size__imp__neq,axiom,
! [X2: list_list_nat,Y3: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ X2 )
!= ( size_s3023201423986296836st_nat @ Y3 ) )
=> ( X2 != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_320_size__neq__size__imp__neq,axiom,
! [X2: list_P6011104703257516679at_nat,Y3: list_P6011104703257516679at_nat] :
( ( ( size_s5460976970255530739at_nat @ X2 )
!= ( size_s5460976970255530739at_nat @ Y3 ) )
=> ( X2 != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_321_size__neq__size__imp__neq,axiom,
! [X2: list_nat,Y3: list_nat] :
( ( ( size_size_list_nat @ X2 )
!= ( size_size_list_nat @ Y3 ) )
=> ( X2 != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_322_size__neq__size__imp__neq,axiom,
! [X2: char,Y3: char] :
( ( ( size_size_char @ X2 )
!= ( size_size_char @ Y3 ) )
=> ( X2 != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_323_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_324_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_325_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_326_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_327_comm__monoid__add__class_Oadd__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_328_add_Ocomm__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.comm_neutral
thf(fact_329_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_330_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( I2 = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_331_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_332_add__strict__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_333_add__strict__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_334_add__strict__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_335_add__less__imp__less__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_left
thf(fact_336_add__less__imp__less__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A2 @ B ) ) ).
% add_less_imp_less_right
thf(fact_337_comm__monoid__mult__class_Omult__1,axiom,
! [A2: nat] :
( ( times_times_nat @ one_one_nat @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_338_mult_Ocomm__neutral,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ one_one_nat )
= A2 ) ).
% mult.comm_neutral
thf(fact_339_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_340_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_341_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_342_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_343_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_344_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_345_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_346_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_347_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_348_atLeastLessThan__inj_I2_J,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A2 @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_349_atLeastLessThan__inj_I1_J,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A2 @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( A2 = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_350_Ico__eq__Ico,axiom,
! [L: nat,H2: nat,L2: nat,H3: nat] :
( ( ( set_or4665077453230672383an_nat @ L @ H2 )
= ( set_or4665077453230672383an_nat @ L2 @ H3 ) )
= ( ( ( L = L2 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_nat @ L @ H2 )
& ~ ( ord_less_nat @ L2 @ H3 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_351_atLeastLessThan__eq__iff,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ( set_or4665077453230672383an_nat @ A2 @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
= ( ( A2 = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_352_add__lessD1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
=> ( ord_less_nat @ I2 @ K ) ) ).
% add_lessD1
thf(fact_353_add__less__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_354_not__add__less1,axiom,
! [I2: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).
% not_add_less1
thf(fact_355_not__add__less2,axiom,
! [J: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_356_add__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_357_trans__less__add1,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_358_trans__less__add2,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_359_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_360_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_361_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_362_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_363_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_364_left__add__mult__distrib,axiom,
! [I2: nat,U2: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I2 @ U2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I2 @ J ) @ U2 ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_365_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_366_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_367_add__neg__neg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_368_add__pos__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_pos_pos
thf(fact_369_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ! [C4: nat] :
( ( B
= ( plus_plus_nat @ A2 @ C4 ) )
=> ( C4 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_370_pos__add__strict,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_371_less__imp__add__positive,axiom,
! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I2 @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_372_mult__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_373_mult__less__mono2,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_374_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_375_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_376_ivl__disj__int__two_I3_J,axiom,
! [L: set_nat,M: set_nat,U2: set_nat] :
( ( inf_inf_set_set_nat @ ( set_or3540276404033026485et_nat @ L @ M ) @ ( set_or3540276404033026485et_nat @ M @ U2 ) )
= bot_bot_set_set_nat ) ).
% ivl_disj_int_two(3)
thf(fact_377_ivl__disj__int__two_I3_J,axiom,
! [L: nat,M: nat,U2: nat] :
( ( inf_inf_set_nat @ ( set_or4665077453230672383an_nat @ L @ M ) @ ( set_or4665077453230672383an_nat @ M @ U2 ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_two(3)
thf(fact_378_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_379_atLeastLessThan0,axiom,
! [M: nat] :
( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
= bot_bot_set_nat ) ).
% atLeastLessThan0
thf(fact_380_length__map,axiom,
! [F: product_prod_nat_nat > nat,Xs: list_P6011104703257516679at_nat] :
( ( size_size_list_nat @ ( map_Pr3938374229010428429at_nat @ F @ Xs ) )
= ( size_s5460976970255530739at_nat @ Xs ) ) ).
% length_map
thf(fact_381_length__map,axiom,
! [F: nat > list_nat,Xs: list_nat] :
( ( size_s3023201423986296836st_nat @ ( map_nat_list_nat @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_382_length__map,axiom,
! [F: list_nat > list_nat,Xs: list_list_nat] :
( ( size_s3023201423986296836st_nat @ ( map_li7225945977422193158st_nat @ F @ Xs ) )
= ( size_s3023201423986296836st_nat @ Xs ) ) ).
% length_map
thf(fact_383_length__map,axiom,
! [F: product_prod_nat_nat > list_nat,Xs: list_P6011104703257516679at_nat] :
( ( size_s3023201423986296836st_nat @ ( map_Pr7251875764006242973st_nat @ F @ Xs ) )
= ( size_s5460976970255530739at_nat @ Xs ) ) ).
% length_map
thf(fact_384_length__map,axiom,
! [F: nat > product_prod_nat_nat,Xs: list_nat] :
( ( size_s5460976970255530739at_nat @ ( map_na7298421622053143531at_nat @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_385_length__map,axiom,
! [F: list_nat > product_prod_nat_nat,Xs: list_list_nat] :
( ( size_s5460976970255530739at_nat @ ( map_li9084805350295849819at_nat @ F @ Xs ) )
= ( size_s3023201423986296836st_nat @ Xs ) ) ).
% length_map
thf(fact_386_length__map,axiom,
! [F: product_prod_nat_nat > product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
( ( size_s5460976970255530739at_nat @ ( map_Pr8058819605623181956at_nat @ F @ Xs ) )
= ( size_s5460976970255530739at_nat @ Xs ) ) ).
% length_map
thf(fact_387_length__map,axiom,
! [F: list_nat > nat,Xs: list_list_nat] :
( ( size_size_list_nat @ ( map_list_nat_nat @ F @ Xs ) )
= ( size_s3023201423986296836st_nat @ Xs ) ) ).
% length_map
thf(fact_388_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_389_inf__bot__left,axiom,
! [X2: list_nat > $o] :
( ( inf_inf_list_nat_o @ bot_bot_list_nat_o @ X2 )
= bot_bot_list_nat_o ) ).
% inf_bot_left
thf(fact_390_inf__bot__left,axiom,
! [X2: nat > $o] :
( ( inf_inf_nat_o @ bot_bot_nat_o @ X2 )
= bot_bot_nat_o ) ).
% inf_bot_left
thf(fact_391_inf__bot__left,axiom,
! [X2: set_set_nat] :
( ( inf_inf_set_set_nat @ bot_bot_set_set_nat @ X2 )
= bot_bot_set_set_nat ) ).
% inf_bot_left
thf(fact_392_inf__bot__left,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X2 )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_393_inf__bot__left,axiom,
! [X2: set_list_nat] :
( ( inf_inf_set_list_nat @ bot_bot_set_list_nat @ X2 )
= bot_bot_set_list_nat ) ).
% inf_bot_left
thf(fact_394_inf__bot__right,axiom,
! [X2: list_nat > $o] :
( ( inf_inf_list_nat_o @ X2 @ bot_bot_list_nat_o )
= bot_bot_list_nat_o ) ).
% inf_bot_right
thf(fact_395_inf__bot__right,axiom,
! [X2: nat > $o] :
( ( inf_inf_nat_o @ X2 @ bot_bot_nat_o )
= bot_bot_nat_o ) ).
% inf_bot_right
thf(fact_396_inf__bot__right,axiom,
! [X2: set_set_nat] :
( ( inf_inf_set_set_nat @ X2 @ bot_bot_set_set_nat )
= bot_bot_set_set_nat ) ).
% inf_bot_right
thf(fact_397_inf__bot__right,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ X2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_398_inf__bot__right,axiom,
! [X2: set_list_nat] :
( ( inf_inf_set_list_nat @ X2 @ bot_bot_set_list_nat )
= bot_bot_set_list_nat ) ).
% inf_bot_right
thf(fact_399_boolean__algebra_Oconj__zero__left,axiom,
! [X2: list_nat > $o] :
( ( inf_inf_list_nat_o @ bot_bot_list_nat_o @ X2 )
= bot_bot_list_nat_o ) ).
% boolean_algebra.conj_zero_left
thf(fact_400_boolean__algebra_Oconj__zero__left,axiom,
! [X2: nat > $o] :
( ( inf_inf_nat_o @ bot_bot_nat_o @ X2 )
= bot_bot_nat_o ) ).
% boolean_algebra.conj_zero_left
thf(fact_401_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_set_nat] :
( ( inf_inf_set_set_nat @ bot_bot_set_set_nat @ X2 )
= bot_bot_set_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_402_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X2 )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_403_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_list_nat] :
( ( inf_inf_set_list_nat @ bot_bot_set_list_nat @ X2 )
= bot_bot_set_list_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_404_boolean__algebra_Oconj__zero__right,axiom,
! [X2: list_nat > $o] :
( ( inf_inf_list_nat_o @ X2 @ bot_bot_list_nat_o )
= bot_bot_list_nat_o ) ).
% boolean_algebra.conj_zero_right
thf(fact_405_boolean__algebra_Oconj__zero__right,axiom,
! [X2: nat > $o] :
( ( inf_inf_nat_o @ X2 @ bot_bot_nat_o )
= bot_bot_nat_o ) ).
% boolean_algebra.conj_zero_right
thf(fact_406_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_set_nat] :
( ( inf_inf_set_set_nat @ X2 @ bot_bot_set_set_nat )
= bot_bot_set_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_407_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ X2 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_408_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_list_nat] :
( ( inf_inf_set_list_nat @ X2 @ bot_bot_set_list_nat )
= bot_bot_set_list_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_409_map__ident,axiom,
( ( map_nat_nat
@ ^ [X: nat] : X )
= ( ^ [Xs2: list_nat] : Xs2 ) ) ).
% map_ident
thf(fact_410_map__ident,axiom,
( ( map_li7225945977422193158st_nat
@ ^ [X: list_nat] : X )
= ( ^ [Xs2: list_list_nat] : Xs2 ) ) ).
% map_ident
thf(fact_411_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M3: nat] :
( ( ord_less_nat @ M3 @ N )
& ( P @ M3 ) ) )
= ( ? [X: nat] :
( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P @ X ) ) ) ) ).
% ex_nat_less_eq
thf(fact_412_inf__apply,axiom,
( inf_inf_nat_o
= ( ^ [F2: nat > $o,G2: nat > $o,X: nat] : ( inf_inf_o @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ).
% inf_apply
thf(fact_413_inf__apply,axiom,
( inf_inf_list_nat_o
= ( ^ [F2: list_nat > $o,G2: list_nat > $o,X: list_nat] : ( inf_inf_o @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ).
% inf_apply
thf(fact_414_inf__right__idem,axiom,
! [X2: set_list_nat,Y3: set_list_nat] :
( ( inf_inf_set_list_nat @ ( inf_inf_set_list_nat @ X2 @ Y3 ) @ Y3 )
= ( inf_inf_set_list_nat @ X2 @ Y3 ) ) ).
% inf_right_idem
thf(fact_415_inf__right__idem,axiom,
! [X2: nat > $o,Y3: nat > $o] :
( ( inf_inf_nat_o @ ( inf_inf_nat_o @ X2 @ Y3 ) @ Y3 )
= ( inf_inf_nat_o @ X2 @ Y3 ) ) ).
% inf_right_idem
thf(fact_416_inf__right__idem,axiom,
! [X2: list_nat > $o,Y3: list_nat > $o] :
( ( inf_inf_list_nat_o @ ( inf_inf_list_nat_o @ X2 @ Y3 ) @ Y3 )
= ( inf_inf_list_nat_o @ X2 @ Y3 ) ) ).
% inf_right_idem
thf(fact_417_inf__right__idem,axiom,
! [X2: nat,Y3: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ X2 @ Y3 ) @ Y3 )
= ( inf_inf_nat @ X2 @ Y3 ) ) ).
% inf_right_idem
thf(fact_418_inf__right__idem,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ Y3 )
= ( inf_inf_set_nat @ X2 @ Y3 ) ) ).
% inf_right_idem
thf(fact_419_inf_Oright__idem,axiom,
! [A2: set_list_nat,B: set_list_nat] :
( ( inf_inf_set_list_nat @ ( inf_inf_set_list_nat @ A2 @ B ) @ B )
= ( inf_inf_set_list_nat @ A2 @ B ) ) ).
% inf.right_idem
thf(fact_420_inf_Oright__idem,axiom,
! [A2: nat > $o,B: nat > $o] :
( ( inf_inf_nat_o @ ( inf_inf_nat_o @ A2 @ B ) @ B )
= ( inf_inf_nat_o @ A2 @ B ) ) ).
% inf.right_idem
thf(fact_421_inf_Oright__idem,axiom,
! [A2: list_nat > $o,B: list_nat > $o] :
( ( inf_inf_list_nat_o @ ( inf_inf_list_nat_o @ A2 @ B ) @ B )
= ( inf_inf_list_nat_o @ A2 @ B ) ) ).
% inf.right_idem
thf(fact_422_inf_Oright__idem,axiom,
! [A2: nat,B: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ A2 @ B ) @ B )
= ( inf_inf_nat @ A2 @ B ) ) ).
% inf.right_idem
thf(fact_423_inf_Oright__idem,axiom,
! [A2: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ B )
= ( inf_inf_set_nat @ A2 @ B ) ) ).
% inf.right_idem
thf(fact_424_inf__left__idem,axiom,
! [X2: set_list_nat,Y3: set_list_nat] :
( ( inf_inf_set_list_nat @ X2 @ ( inf_inf_set_list_nat @ X2 @ Y3 ) )
= ( inf_inf_set_list_nat @ X2 @ Y3 ) ) ).
% inf_left_idem
thf(fact_425_inf__left__idem,axiom,
! [X2: nat > $o,Y3: nat > $o] :
( ( inf_inf_nat_o @ X2 @ ( inf_inf_nat_o @ X2 @ Y3 ) )
= ( inf_inf_nat_o @ X2 @ Y3 ) ) ).
% inf_left_idem
thf(fact_426_inf__left__idem,axiom,
! [X2: list_nat > $o,Y3: list_nat > $o] :
( ( inf_inf_list_nat_o @ X2 @ ( inf_inf_list_nat_o @ X2 @ Y3 ) )
= ( inf_inf_list_nat_o @ X2 @ Y3 ) ) ).
% inf_left_idem
thf(fact_427_inf__left__idem,axiom,
! [X2: nat,Y3: nat] :
( ( inf_inf_nat @ X2 @ ( inf_inf_nat @ X2 @ Y3 ) )
= ( inf_inf_nat @ X2 @ Y3 ) ) ).
% inf_left_idem
thf(fact_428_inf__left__idem,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( inf_inf_set_nat @ X2 @ ( inf_inf_set_nat @ X2 @ Y3 ) )
= ( inf_inf_set_nat @ X2 @ Y3 ) ) ).
% inf_left_idem
thf(fact_429_inf_Oleft__idem,axiom,
! [A2: set_list_nat,B: set_list_nat] :
( ( inf_inf_set_list_nat @ A2 @ ( inf_inf_set_list_nat @ A2 @ B ) )
= ( inf_inf_set_list_nat @ A2 @ B ) ) ).
% inf.left_idem
thf(fact_430_inf_Oleft__idem,axiom,
! [A2: nat > $o,B: nat > $o] :
( ( inf_inf_nat_o @ A2 @ ( inf_inf_nat_o @ A2 @ B ) )
= ( inf_inf_nat_o @ A2 @ B ) ) ).
% inf.left_idem
thf(fact_431_inf_Oleft__idem,axiom,
! [A2: list_nat > $o,B: list_nat > $o] :
( ( inf_inf_list_nat_o @ A2 @ ( inf_inf_list_nat_o @ A2 @ B ) )
= ( inf_inf_list_nat_o @ A2 @ B ) ) ).
% inf.left_idem
thf(fact_432_inf_Oleft__idem,axiom,
! [A2: nat,B: nat] :
( ( inf_inf_nat @ A2 @ ( inf_inf_nat @ A2 @ B ) )
= ( inf_inf_nat @ A2 @ B ) ) ).
% inf.left_idem
thf(fact_433_inf_Oleft__idem,axiom,
! [A2: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ A2 @ B ) )
= ( inf_inf_set_nat @ A2 @ B ) ) ).
% inf.left_idem
thf(fact_434_inf__idem,axiom,
! [X2: set_list_nat] :
( ( inf_inf_set_list_nat @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_435_inf__idem,axiom,
! [X2: nat > $o] :
( ( inf_inf_nat_o @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_436_inf__idem,axiom,
! [X2: list_nat > $o] :
( ( inf_inf_list_nat_o @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_437_inf__idem,axiom,
! [X2: nat] :
( ( inf_inf_nat @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_438_inf__idem,axiom,
! [X2: set_nat] :
( ( inf_inf_set_nat @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_439_inf_Oidem,axiom,
! [A2: set_list_nat] :
( ( inf_inf_set_list_nat @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_440_inf_Oidem,axiom,
! [A2: nat > $o] :
( ( inf_inf_nat_o @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_441_inf_Oidem,axiom,
! [A2: list_nat > $o] :
( ( inf_inf_list_nat_o @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_442_inf_Oidem,axiom,
! [A2: nat] :
( ( inf_inf_nat @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_443_inf_Oidem,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_444_psubsetD,axiom,
! [A: set_set_nat,B2: set_set_nat,C: set_nat] :
( ( ord_less_set_set_nat @ A @ B2 )
=> ( ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_445_psubsetD,axiom,
! [A: set_list_nat,B2: set_list_nat,C: list_nat] :
( ( ord_le1190675801316882794st_nat @ A @ B2 )
=> ( ( member_list_nat @ C @ A )
=> ( member_list_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_446_psubsetD,axiom,
! [A: set_nat,B2: set_nat,C: nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_447_less__set__def,axiom,
( ord_less_set_set_nat
= ( ^ [A3: set_set_nat,B4: set_set_nat] :
( ord_less_set_nat_o
@ ^ [X: set_nat] : ( member_set_nat @ X @ A3 )
@ ^ [X: set_nat] : ( member_set_nat @ X @ B4 ) ) ) ) ).
% less_set_def
thf(fact_448_less__set__def,axiom,
( ord_le1190675801316882794st_nat
= ( ^ [A3: set_list_nat,B4: set_list_nat] :
( ord_less_list_nat_o
@ ^ [X: list_nat] : ( member_list_nat @ X @ A3 )
@ ^ [X: list_nat] : ( member_list_nat @ X @ B4 ) ) ) ) ).
% less_set_def
thf(fact_449_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
( ord_less_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A3 )
@ ^ [X: nat] : ( member_nat @ X @ B4 ) ) ) ) ).
% less_set_def
thf(fact_450_psubset__trans,axiom,
! [A: set_list_nat,B2: set_list_nat,C2: set_list_nat] :
( ( ord_le1190675801316882794st_nat @ A @ B2 )
=> ( ( ord_le1190675801316882794st_nat @ B2 @ C2 )
=> ( ord_le1190675801316882794st_nat @ A @ C2 ) ) ) ).
% psubset_trans
thf(fact_451_psubset__trans,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ( ( ord_less_set_nat @ B2 @ C2 )
=> ( ord_less_set_nat @ A @ C2 ) ) ) ).
% psubset_trans
thf(fact_452_bot__set__def,axiom,
( bot_bot_set_set_nat
= ( collect_set_nat @ bot_bot_set_nat_o ) ) ).
% bot_set_def
thf(fact_453_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_454_bot__set__def,axiom,
( bot_bot_set_list_nat
= ( collect_list_nat @ bot_bot_list_nat_o ) ) ).
% bot_set_def
thf(fact_455_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_456_inf__set__def,axiom,
( inf_inf_set_set_nat
= ( ^ [A3: set_set_nat,B4: set_set_nat] :
( collect_set_nat
@ ( inf_inf_set_nat_o
@ ^ [X: set_nat] : ( member_set_nat @ X @ A3 )
@ ^ [X: set_nat] : ( member_set_nat @ X @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_457_inf__set__def,axiom,
( inf_inf_set_list_nat
= ( ^ [A3: set_list_nat,B4: set_list_nat] :
( collect_list_nat
@ ( inf_inf_list_nat_o
@ ^ [X: list_nat] : ( member_list_nat @ X @ A3 )
@ ^ [X: list_nat] : ( member_list_nat @ X @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_458_inf__set__def,axiom,
( inf_inf_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
( collect_nat
@ ( inf_inf_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A3 )
@ ^ [X: nat] : ( member_nat @ X @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_459_inf__fun__def,axiom,
( inf_inf_nat_o
= ( ^ [F2: nat > $o,G2: nat > $o,X: nat] : ( inf_inf_o @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ).
% inf_fun_def
thf(fact_460_inf__fun__def,axiom,
( inf_inf_list_nat_o
= ( ^ [F2: list_nat > $o,G2: list_nat > $o,X: list_nat] : ( inf_inf_o @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ).
% inf_fun_def
thf(fact_461_inf__left__commute,axiom,
! [X2: set_list_nat,Y3: set_list_nat,Z2: set_list_nat] :
( ( inf_inf_set_list_nat @ X2 @ ( inf_inf_set_list_nat @ Y3 @ Z2 ) )
= ( inf_inf_set_list_nat @ Y3 @ ( inf_inf_set_list_nat @ X2 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_462_inf__left__commute,axiom,
! [X2: nat > $o,Y3: nat > $o,Z2: nat > $o] :
( ( inf_inf_nat_o @ X2 @ ( inf_inf_nat_o @ Y3 @ Z2 ) )
= ( inf_inf_nat_o @ Y3 @ ( inf_inf_nat_o @ X2 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_463_inf__left__commute,axiom,
! [X2: list_nat > $o,Y3: list_nat > $o,Z2: list_nat > $o] :
( ( inf_inf_list_nat_o @ X2 @ ( inf_inf_list_nat_o @ Y3 @ Z2 ) )
= ( inf_inf_list_nat_o @ Y3 @ ( inf_inf_list_nat_o @ X2 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_464_inf__left__commute,axiom,
! [X2: nat,Y3: nat,Z2: nat] :
( ( inf_inf_nat @ X2 @ ( inf_inf_nat @ Y3 @ Z2 ) )
= ( inf_inf_nat @ Y3 @ ( inf_inf_nat @ X2 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_465_inf__left__commute,axiom,
! [X2: set_nat,Y3: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ X2 @ ( inf_inf_set_nat @ Y3 @ Z2 ) )
= ( inf_inf_set_nat @ Y3 @ ( inf_inf_set_nat @ X2 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_466_inf_Oleft__commute,axiom,
! [B: set_list_nat,A2: set_list_nat,C: set_list_nat] :
( ( inf_inf_set_list_nat @ B @ ( inf_inf_set_list_nat @ A2 @ C ) )
= ( inf_inf_set_list_nat @ A2 @ ( inf_inf_set_list_nat @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_467_inf_Oleft__commute,axiom,
! [B: nat > $o,A2: nat > $o,C: nat > $o] :
( ( inf_inf_nat_o @ B @ ( inf_inf_nat_o @ A2 @ C ) )
= ( inf_inf_nat_o @ A2 @ ( inf_inf_nat_o @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_468_inf_Oleft__commute,axiom,
! [B: list_nat > $o,A2: list_nat > $o,C: list_nat > $o] :
( ( inf_inf_list_nat_o @ B @ ( inf_inf_list_nat_o @ A2 @ C ) )
= ( inf_inf_list_nat_o @ A2 @ ( inf_inf_list_nat_o @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_469_inf_Oleft__commute,axiom,
! [B: nat,A2: nat,C: nat] :
( ( inf_inf_nat @ B @ ( inf_inf_nat @ A2 @ C ) )
= ( inf_inf_nat @ A2 @ ( inf_inf_nat @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_470_inf_Oleft__commute,axiom,
! [B: set_nat,A2: set_nat,C: set_nat] :
( ( inf_inf_set_nat @ B @ ( inf_inf_set_nat @ A2 @ C ) )
= ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_471_boolean__algebra__cancel_Oinf2,axiom,
! [B2: set_list_nat,K: set_list_nat,B: set_list_nat,A2: set_list_nat] :
( ( B2
= ( inf_inf_set_list_nat @ K @ B ) )
=> ( ( inf_inf_set_list_nat @ A2 @ B2 )
= ( inf_inf_set_list_nat @ K @ ( inf_inf_set_list_nat @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_472_boolean__algebra__cancel_Oinf2,axiom,
! [B2: nat > $o,K: nat > $o,B: nat > $o,A2: nat > $o] :
( ( B2
= ( inf_inf_nat_o @ K @ B ) )
=> ( ( inf_inf_nat_o @ A2 @ B2 )
= ( inf_inf_nat_o @ K @ ( inf_inf_nat_o @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_473_boolean__algebra__cancel_Oinf2,axiom,
! [B2: list_nat > $o,K: list_nat > $o,B: list_nat > $o,A2: list_nat > $o] :
( ( B2
= ( inf_inf_list_nat_o @ K @ B ) )
=> ( ( inf_inf_list_nat_o @ A2 @ B2 )
= ( inf_inf_list_nat_o @ K @ ( inf_inf_list_nat_o @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_474_boolean__algebra__cancel_Oinf2,axiom,
! [B2: nat,K: nat,B: nat,A2: nat] :
( ( B2
= ( inf_inf_nat @ K @ B ) )
=> ( ( inf_inf_nat @ A2 @ B2 )
= ( inf_inf_nat @ K @ ( inf_inf_nat @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_475_boolean__algebra__cancel_Oinf2,axiom,
! [B2: set_nat,K: set_nat,B: set_nat,A2: set_nat] :
( ( B2
= ( inf_inf_set_nat @ K @ B ) )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_476_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_list_nat,K: set_list_nat,A2: set_list_nat,B: set_list_nat] :
( ( A
= ( inf_inf_set_list_nat @ K @ A2 ) )
=> ( ( inf_inf_set_list_nat @ A @ B )
= ( inf_inf_set_list_nat @ K @ ( inf_inf_set_list_nat @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_477_boolean__algebra__cancel_Oinf1,axiom,
! [A: nat > $o,K: nat > $o,A2: nat > $o,B: nat > $o] :
( ( A
= ( inf_inf_nat_o @ K @ A2 ) )
=> ( ( inf_inf_nat_o @ A @ B )
= ( inf_inf_nat_o @ K @ ( inf_inf_nat_o @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_478_boolean__algebra__cancel_Oinf1,axiom,
! [A: list_nat > $o,K: list_nat > $o,A2: list_nat > $o,B: list_nat > $o] :
( ( A
= ( inf_inf_list_nat_o @ K @ A2 ) )
=> ( ( inf_inf_list_nat_o @ A @ B )
= ( inf_inf_list_nat_o @ K @ ( inf_inf_list_nat_o @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_479_boolean__algebra__cancel_Oinf1,axiom,
! [A: nat,K: nat,A2: nat,B: nat] :
( ( A
= ( inf_inf_nat @ K @ A2 ) )
=> ( ( inf_inf_nat @ A @ B )
= ( inf_inf_nat @ K @ ( inf_inf_nat @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_480_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_nat,K: set_nat,A2: set_nat,B: set_nat] :
( ( A
= ( inf_inf_set_nat @ K @ A2 ) )
=> ( ( inf_inf_set_nat @ A @ B )
= ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_481_inf__commute,axiom,
( inf_inf_set_list_nat
= ( ^ [X: set_list_nat,Y4: set_list_nat] : ( inf_inf_set_list_nat @ Y4 @ X ) ) ) ).
% inf_commute
thf(fact_482_inf__commute,axiom,
( inf_inf_nat_o
= ( ^ [X: nat > $o,Y4: nat > $o] : ( inf_inf_nat_o @ Y4 @ X ) ) ) ).
% inf_commute
thf(fact_483_inf__commute,axiom,
( inf_inf_list_nat_o
= ( ^ [X: list_nat > $o,Y4: list_nat > $o] : ( inf_inf_list_nat_o @ Y4 @ X ) ) ) ).
% inf_commute
thf(fact_484_inf__commute,axiom,
( inf_inf_nat
= ( ^ [X: nat,Y4: nat] : ( inf_inf_nat @ Y4 @ X ) ) ) ).
% inf_commute
thf(fact_485_inf__commute,axiom,
( inf_inf_set_nat
= ( ^ [X: set_nat,Y4: set_nat] : ( inf_inf_set_nat @ Y4 @ X ) ) ) ).
% inf_commute
thf(fact_486_inf_Ocommute,axiom,
( inf_inf_set_list_nat
= ( ^ [A4: set_list_nat,B5: set_list_nat] : ( inf_inf_set_list_nat @ B5 @ A4 ) ) ) ).
% inf.commute
thf(fact_487_inf_Ocommute,axiom,
( inf_inf_nat_o
= ( ^ [A4: nat > $o,B5: nat > $o] : ( inf_inf_nat_o @ B5 @ A4 ) ) ) ).
% inf.commute
thf(fact_488_inf_Ocommute,axiom,
( inf_inf_list_nat_o
= ( ^ [A4: list_nat > $o,B5: list_nat > $o] : ( inf_inf_list_nat_o @ B5 @ A4 ) ) ) ).
% inf.commute
thf(fact_489_inf_Ocommute,axiom,
( inf_inf_nat
= ( ^ [A4: nat,B5: nat] : ( inf_inf_nat @ B5 @ A4 ) ) ) ).
% inf.commute
thf(fact_490_inf_Ocommute,axiom,
( inf_inf_set_nat
= ( ^ [A4: set_nat,B5: set_nat] : ( inf_inf_set_nat @ B5 @ A4 ) ) ) ).
% inf.commute
thf(fact_491_inf__assoc,axiom,
! [X2: set_list_nat,Y3: set_list_nat,Z2: set_list_nat] :
( ( inf_inf_set_list_nat @ ( inf_inf_set_list_nat @ X2 @ Y3 ) @ Z2 )
= ( inf_inf_set_list_nat @ X2 @ ( inf_inf_set_list_nat @ Y3 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_492_inf__assoc,axiom,
! [X2: nat > $o,Y3: nat > $o,Z2: nat > $o] :
( ( inf_inf_nat_o @ ( inf_inf_nat_o @ X2 @ Y3 ) @ Z2 )
= ( inf_inf_nat_o @ X2 @ ( inf_inf_nat_o @ Y3 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_493_inf__assoc,axiom,
! [X2: list_nat > $o,Y3: list_nat > $o,Z2: list_nat > $o] :
( ( inf_inf_list_nat_o @ ( inf_inf_list_nat_o @ X2 @ Y3 ) @ Z2 )
= ( inf_inf_list_nat_o @ X2 @ ( inf_inf_list_nat_o @ Y3 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_494_inf__assoc,axiom,
! [X2: nat,Y3: nat,Z2: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ X2 @ Y3 ) @ Z2 )
= ( inf_inf_nat @ X2 @ ( inf_inf_nat @ Y3 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_495_inf__assoc,axiom,
! [X2: set_nat,Y3: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ Z2 )
= ( inf_inf_set_nat @ X2 @ ( inf_inf_set_nat @ Y3 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_496_inf_Oassoc,axiom,
! [A2: list_nat > $o,B: list_nat > $o,C: list_nat > $o] :
( ( inf_inf_list_nat_o @ ( inf_inf_list_nat_o @ A2 @ B ) @ C )
= ( inf_inf_list_nat_o @ A2 @ ( inf_inf_list_nat_o @ B @ C ) ) ) ).
% inf.assoc
thf(fact_497_inf_Oassoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ A2 @ B ) @ C )
= ( inf_inf_nat @ A2 @ ( inf_inf_nat @ B @ C ) ) ) ).
% inf.assoc
thf(fact_498_inf_Oassoc,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ C )
= ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ B @ C ) ) ) ).
% inf.assoc
thf(fact_499_inf__sup__aci_I1_J,axiom,
( inf_inf_set_nat
= ( ^ [X: set_nat,Y4: set_nat] : ( inf_inf_set_nat @ Y4 @ X ) ) ) ).
% inf_sup_aci(1)
thf(fact_500_inf__sup__aci_I2_J,axiom,
! [X2: set_nat,Y3: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ Z2 )
= ( inf_inf_set_nat @ X2 @ ( inf_inf_set_nat @ Y3 @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_501_inf__sup__aci_I3_J,axiom,
! [X2: set_nat,Y3: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ X2 @ ( inf_inf_set_nat @ Y3 @ Z2 ) )
= ( inf_inf_set_nat @ Y3 @ ( inf_inf_set_nat @ X2 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_502_inf__sup__aci_I4_J,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( inf_inf_set_nat @ X2 @ ( inf_inf_set_nat @ X2 @ Y3 ) )
= ( inf_inf_set_nat @ X2 @ Y3 ) ) ).
% inf_sup_aci(4)
thf(fact_503_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_504_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_nat] :
( ( size_size_list_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_505_inf_Ostrict__coboundedI2,axiom,
! [B: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_506_inf_Ostrict__coboundedI2,axiom,
! [B: nat,C: nat,A2: nat] :
( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_507_inf_Ostrict__coboundedI1,axiom,
! [A2: set_nat,C: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ C )
=> ( ord_less_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_508_inf_Ostrict__coboundedI1,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_nat @ A2 @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_509_inf_Ostrict__order__iff,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
( ( A4
= ( inf_inf_set_nat @ A4 @ B5 ) )
& ( A4 != B5 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_510_inf_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A4: nat,B5: nat] :
( ( A4
= ( inf_inf_nat @ A4 @ B5 ) )
& ( A4 != B5 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_511_inf_Ostrict__boundedE,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ ( inf_inf_set_nat @ B @ C ) )
=> ~ ( ( ord_less_set_nat @ A2 @ B )
=> ~ ( ord_less_set_nat @ A2 @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_512_inf_Ostrict__boundedE,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ A2 @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_513_inf_Oabsorb4,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ B )
= B ) ) ).
% inf.absorb4
thf(fact_514_inf_Oabsorb4,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( inf_inf_nat @ A2 @ B )
= B ) ) ).
% inf.absorb4
thf(fact_515_inf_Oabsorb3,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( inf_inf_set_nat @ A2 @ B )
= A2 ) ) ).
% inf.absorb3
thf(fact_516_inf_Oabsorb3,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( inf_inf_nat @ A2 @ B )
= A2 ) ) ).
% inf.absorb3
thf(fact_517_less__infI2,axiom,
! [B: set_nat,X2: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B @ X2 )
=> ( ord_less_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ X2 ) ) ).
% less_infI2
thf(fact_518_less__infI2,axiom,
! [B: nat,X2: nat,A2: nat] :
( ( ord_less_nat @ B @ X2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B ) @ X2 ) ) ).
% less_infI2
thf(fact_519_less__infI1,axiom,
! [A2: set_nat,X2: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ X2 )
=> ( ord_less_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ X2 ) ) ).
% less_infI1
thf(fact_520_less__infI1,axiom,
! [A2: nat,X2: nat,B: nat] :
( ( ord_less_nat @ A2 @ X2 )
=> ( ord_less_nat @ ( inf_inf_nat @ A2 @ B ) @ X2 ) ) ).
% less_infI1
thf(fact_521_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs3: list_nat] :
( ! [Ys2: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs3 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_522_map__eq__imp__length__eq,axiom,
! [F: list_nat > nat,Xs: list_list_nat,G: list_nat > nat,Ys: list_list_nat] :
( ( ( map_list_nat_nat @ F @ Xs )
= ( map_list_nat_nat @ G @ Ys ) )
=> ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_523_map__eq__imp__length__eq,axiom,
! [F: list_nat > nat,Xs: list_list_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_list_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) )
=> ( ( size_s3023201423986296836st_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_524_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs: list_nat,G: list_nat > nat,Ys: list_list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_list_nat_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s3023201423986296836st_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_525_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( P @ M3 ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P @ X ) ) ) ) ).
% all_nat_less_eq
thf(fact_526_add__scale__eq__noteq,axiom,
! [R: nat,A2: nat,B: nat,C: nat,D: nat] :
( ( R != zero_zero_nat )
=> ( ( ( A2 = B )
& ( C != D ) )
=> ( ( plus_plus_nat @ A2 @ ( times_times_nat @ R @ C ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_527_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_528_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C5: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_529_the__elem__eq,axiom,
! [X2: nat] :
( ( the_elem_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
= X2 ) ).
% the_elem_eq
thf(fact_530_the__elem__eq,axiom,
! [X2: list_nat] :
( ( the_elem_list_nat @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) )
= X2 ) ).
% the_elem_eq
thf(fact_531_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_532_add__0__iff,axiom,
! [B: nat,A2: nat] :
( ( B
= ( plus_plus_nat @ B @ A2 ) )
= ( A2 = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_533_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_534_crossproduct__eq,axiom,
! [W: nat,Y3: nat,X2: nat,Z2: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y3 ) @ ( times_times_nat @ X2 @ Z2 ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z2 ) @ ( times_times_nat @ X2 @ Y3 ) ) )
= ( ( W = X2 )
| ( Y3 = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_535_crossproduct__noteq,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ( A2 != B )
& ( C != D ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D ) )
!= ( plus_plus_nat @ ( times_times_nat @ A2 @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_536_elim__singleton,axiom,
! [X2: nat,S: nat,X4: nat,T: nat] :
( ( ( member_nat @ X2 @ ( insert_nat @ S @ bot_bot_set_nat ) )
& ( member_nat @ X4 @ ( insert_nat @ T @ bot_bot_set_nat ) ) )
=> ( ( X2 = S )
& ( X4 = T ) ) ) ).
% elim_singleton
thf(fact_537_elim__singleton,axiom,
! [X2: nat,S: nat,X4: list_nat,T: list_nat] :
( ( ( member_nat @ X2 @ ( insert_nat @ S @ bot_bot_set_nat ) )
& ( member_list_nat @ X4 @ ( insert_list_nat @ T @ bot_bot_set_list_nat ) ) )
=> ( ( X2 = S )
& ( X4 = T ) ) ) ).
% elim_singleton
thf(fact_538_elim__singleton,axiom,
! [X2: list_nat,S: list_nat,X4: nat,T: nat] :
( ( ( member_list_nat @ X2 @ ( insert_list_nat @ S @ bot_bot_set_list_nat ) )
& ( member_nat @ X4 @ ( insert_nat @ T @ bot_bot_set_nat ) ) )
=> ( ( X2 = S )
& ( X4 = T ) ) ) ).
% elim_singleton
thf(fact_539_elim__singleton,axiom,
! [X2: list_nat,S: list_nat,X4: list_nat,T: list_nat] :
( ( ( member_list_nat @ X2 @ ( insert_list_nat @ S @ bot_bot_set_list_nat ) )
& ( member_list_nat @ X4 @ ( insert_list_nat @ T @ bot_bot_set_list_nat ) ) )
=> ( ( X2 = S )
& ( X4 = T ) ) ) ).
% elim_singleton
thf(fact_540_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C5: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_541_inf__Int__eq,axiom,
! [R2: set_nat,S2: set_nat] :
( ( inf_inf_nat_o
@ ^ [X: nat] : ( member_nat @ X @ R2 )
@ ^ [X: nat] : ( member_nat @ X @ S2 ) )
= ( ^ [X: nat] : ( member_nat @ X @ ( inf_inf_set_nat @ R2 @ S2 ) ) ) ) ).
% inf_Int_eq
thf(fact_542_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_543_Collect__empty__eq__bot,axiom,
! [P: list_nat > $o] :
( ( ( collect_list_nat @ P )
= bot_bot_set_list_nat )
= ( P = bot_bot_list_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_544_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X: nat] : ( member_nat @ X @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_545_bot__empty__eq,axiom,
( bot_bot_list_nat_o
= ( ^ [X: list_nat] : ( member_list_nat @ X @ bot_bot_set_list_nat ) ) ) ).
% bot_empty_eq
thf(fact_546_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
( A3
= ( insert_nat @ ( the_elem_nat @ A3 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_547_is__singleton__the__elem,axiom,
( is_sin2641923865335537900st_nat
= ( ^ [A3: set_list_nat] :
( A3
= ( insert_list_nat @ ( the_elem_list_nat @ A3 ) @ bot_bot_set_list_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_548_is__singletonI,axiom,
! [X2: nat] : ( is_singleton_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_549_is__singletonI,axiom,
! [X2: list_nat] : ( is_sin2641923865335537900st_nat @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) ).
% is_singletonI
thf(fact_550_is__singletonI_H,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X3: nat,Y2: nat] :
( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y2 @ A )
=> ( X3 = Y2 ) ) )
=> ( is_singleton_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_551_is__singletonI_H,axiom,
! [A: set_list_nat] :
( ( A != bot_bot_set_list_nat )
=> ( ! [X3: list_nat,Y2: list_nat] :
( ( member_list_nat @ X3 @ A )
=> ( ( member_list_nat @ Y2 @ A )
=> ( X3 = Y2 ) ) )
=> ( is_sin2641923865335537900st_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_552_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_553_less__imp__neq,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% less_imp_neq
thf(fact_554_order_Oasym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order.asym
thf(fact_555_ord__eq__less__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_556_ord__less__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_557_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_558_antisym__conv3,axiom,
! [Y3: nat,X2: nat] :
( ~ ( ord_less_nat @ Y3 @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv3
thf(fact_559_linorder__cases,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ( X2 != Y3 )
=> ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_cases
thf(fact_560_dual__order_Oasym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ~ ( ord_less_nat @ A2 @ B ) ) ).
% dual_order.asym
thf(fact_561_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_562_exists__least__iff,axiom,
( ( ^ [P4: nat > $o] :
? [X5: nat] : ( P4 @ X5 ) )
= ( ^ [P5: nat > $o] :
? [N3: nat] :
( ( P5 @ N3 )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ~ ( P5 @ M3 ) ) ) ) ) ).
% exists_least_iff
thf(fact_563_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A5: nat,B6: nat] :
( ( ord_less_nat @ A5 @ B6 )
=> ( P @ A5 @ B6 ) )
=> ( ! [A5: nat] : ( P @ A5 @ A5 )
=> ( ! [A5: nat,B6: nat] :
( ( P @ B6 @ A5 )
=> ( P @ A5 @ B6 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_564_order_Ostrict__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_565_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( ( ord_less_nat @ Y3 @ X2 )
| ( X2 = Y3 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_566_dual__order_Ostrict__trans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_567_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( A2 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_568_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( A2 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_569_linorder__neqE,axiom,
! [X2: nat,Y3: nat] :
( ( X2 != Y3 )
=> ( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_570_order__less__asym,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% order_less_asym
thf(fact_571_linorder__neq__iff,axiom,
! [X2: nat,Y3: nat] :
( ( X2 != Y3 )
= ( ( ord_less_nat @ X2 @ Y3 )
| ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_572_order__less__asym_H,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ~ ( ord_less_nat @ B @ A2 ) ) ).
% order_less_asym'
thf(fact_573_order__less__trans,axiom,
! [X2: nat,Y3: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_574_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_575_ord__less__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_576_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_577_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_578_order__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_579_order__less__not__sym,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% order_less_not_sym
thf(fact_580_order__less__imp__triv,axiom,
! [X2: nat,Y3: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_581_linorder__less__linear,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
| ( X2 = Y3 )
| ( ord_less_nat @ Y3 @ X2 ) ) ).
% linorder_less_linear
thf(fact_582_order__less__imp__not__eq,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( X2 != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_583_order__less__imp__not__eq2,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( Y3 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_584_order__less__imp__not__less,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_585_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
? [X: nat] :
( A3
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_586_is__singleton__def,axiom,
( is_sin2641923865335537900st_nat
= ( ^ [A3: set_list_nat] :
? [X: list_nat] :
( A3
= ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) ) ) ).
% is_singleton_def
thf(fact_587_is__singletonE,axiom,
! [A: set_nat] :
( ( is_singleton_nat @ A )
=> ~ ! [X3: nat] :
( A
!= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_588_is__singletonE,axiom,
! [A: set_list_nat] :
( ( is_sin2641923865335537900st_nat @ A )
=> ~ ! [X3: list_nat] :
( A
!= ( insert_list_nat @ X3 @ bot_bot_set_list_nat ) ) ) ).
% is_singletonE
thf(fact_589_is__singleton__altdef,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
( ( finite_card_nat @ A3 )
= one_one_nat ) ) ) ).
% is_singleton_altdef
thf(fact_590_is__singleton__altdef,axiom,
( is_sin2641923865335537900st_nat
= ( ^ [A3: set_list_nat] :
( ( finite_card_list_nat @ A3 )
= one_one_nat ) ) ) ).
% is_singleton_altdef
thf(fact_591_bot_Onot__eq__extremum,axiom,
! [A2: set_nat] :
( ( A2 != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_592_bot_Onot__eq__extremum,axiom,
! [A2: set_list_nat] :
( ( A2 != bot_bot_set_list_nat )
= ( ord_le1190675801316882794st_nat @ bot_bot_set_list_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_593_bot_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_594_bot_Oextremum__strict,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_595_bot_Oextremum__strict,axiom,
! [A2: set_list_nat] :
~ ( ord_le1190675801316882794st_nat @ A2 @ bot_bot_set_list_nat ) ).
% bot.extremum_strict
thf(fact_596_bot_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_597_Euclid__induct,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A5: nat,B6: nat] :
( ( P @ A5 @ B6 )
= ( P @ B6 @ A5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ zero_zero_nat )
=> ( ! [A5: nat,B6: nat] :
( ( P @ A5 @ B6 )
=> ( P @ A5 @ ( plus_plus_nat @ A5 @ B6 ) ) )
=> ( P @ A2 @ B ) ) ) ) ).
% Euclid_induct
thf(fact_598_verit__sum__simplify,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% verit_sum_simplify
thf(fact_599_size__list__conv__sum__list,axiom,
( size_list_list_nat
= ( ^ [F2: list_nat > nat,Xs2: list_list_nat] : ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ F2 @ Xs2 ) ) @ ( size_s3023201423986296836st_nat @ Xs2 ) ) ) ) ).
% size_list_conv_sum_list
thf(fact_600_size__list__conv__sum__list,axiom,
( size_list_nat
= ( ^ [F2: nat > nat,Xs2: list_nat] : ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ F2 @ Xs2 ) ) @ ( size_size_list_nat @ Xs2 ) ) ) ) ).
% size_list_conv_sum_list
thf(fact_601_length__product,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( size_s5460976970255530739at_nat @ ( product_nat_nat @ Xs @ Ys ) )
= ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_product
thf(fact_602_gen__length__def,axiom,
( gen_length_nat
= ( ^ [N3: nat,Xs2: list_nat] : ( plus_plus_nat @ N3 @ ( size_size_list_nat @ Xs2 ) ) ) ) ).
% gen_length_def
thf(fact_603_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_604_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_605_sum__list__Suc,axiom,
! [F: list_nat > nat,Xs: list_list_nat] :
( ( groups4561878855575611511st_nat
@ ( map_list_nat_nat
@ ^ [X: list_nat] : ( suc @ ( F @ X ) )
@ Xs ) )
= ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_list_nat_nat @ F @ Xs ) ) @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ).
% sum_list_Suc
thf(fact_606_sum__list__Suc,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( groups4561878855575611511st_nat
@ ( map_nat_nat
@ ^ [X: nat] : ( suc @ ( F @ X ) )
@ Xs ) )
= ( plus_plus_nat @ ( groups4561878855575611511st_nat @ ( map_nat_nat @ F @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ) ).
% sum_list_Suc
thf(fact_607_Set_Ois__empty__def,axiom,
( is_empty_nat
= ( ^ [A3: set_nat] : ( A3 = bot_bot_set_nat ) ) ) ).
% Set.is_empty_def
thf(fact_608_Set_Ois__empty__def,axiom,
( is_empty_list_nat
= ( ^ [A3: set_list_nat] : ( A3 = bot_bot_set_list_nat ) ) ) ).
% Set.is_empty_def
thf(fact_609_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_610_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_611_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_612_order__refl,axiom,
! [X2: set_nat] : ( ord_less_eq_set_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_613_empty__subsetI,axiom,
! [A: set_list_nat] : ( ord_le6045566169113846134st_nat @ bot_bot_set_list_nat @ A ) ).
% empty_subsetI
thf(fact_614_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_615_subset__empty,axiom,
! [A: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ bot_bot_set_list_nat )
= ( A = bot_bot_set_list_nat ) ) ).
% subset_empty
thf(fact_616_subset__empty,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_617_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_618_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_619_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_620_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_621_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_622_insert__subset,axiom,
! [X2: nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ A ) @ B2 )
= ( ( member_nat @ X2 @ B2 )
& ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_623_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_624_Int__subset__iff,axiom,
! [C2: set_nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A @ B2 ) )
= ( ( ord_less_eq_set_nat @ C2 @ A )
& ( ord_less_eq_set_nat @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_625_psubsetI,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_set_nat @ A @ B2 ) ) ) ).
% psubsetI
thf(fact_626_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_627_add__le__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_right
thf(fact_628_add__le__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_cancel_left
thf(fact_629_inf_Obounded__iff,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A2 @ B )
& ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_630_inf_Obounded__iff,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B @ C ) )
= ( ( ord_less_eq_set_nat @ A2 @ B )
& ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_631_le__inf__iff,axiom,
! [X2: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y3 @ Z2 ) )
= ( ( ord_less_eq_nat @ X2 @ Y3 )
& ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_632_le__inf__iff,axiom,
! [X2: set_nat,Y3: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ Y3 @ Z2 ) )
= ( ( ord_less_eq_set_nat @ X2 @ Y3 )
& ( ord_less_eq_set_nat @ X2 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_633_singleton__insert__inj__eq_H,axiom,
! [A2: list_nat,A: set_list_nat,B: list_nat] :
( ( ( insert_list_nat @ A2 @ A )
= ( insert_list_nat @ B @ bot_bot_set_list_nat ) )
= ( ( A2 = B )
& ( ord_le6045566169113846134st_nat @ A @ ( insert_list_nat @ B @ bot_bot_set_list_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_634_singleton__insert__inj__eq_H,axiom,
! [A2: nat,A: set_nat,B: nat] :
( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B @ bot_bot_set_nat ) )
= ( ( A2 = B )
& ( ord_less_eq_set_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_635_singleton__insert__inj__eq,axiom,
! [B: list_nat,A2: list_nat,A: set_list_nat] :
( ( ( insert_list_nat @ B @ bot_bot_set_list_nat )
= ( insert_list_nat @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_le6045566169113846134st_nat @ A @ ( insert_list_nat @ B @ bot_bot_set_list_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_636_singleton__insert__inj__eq,axiom,
! [B: nat,A2: nat,A: set_nat] :
( ( ( insert_nat @ B @ bot_bot_set_nat )
= ( insert_nat @ A2 @ A ) )
= ( ( A2 = B )
& ( ord_less_eq_set_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_637_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_638_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_639_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_640_ivl__subset,axiom,
! [I2: nat,J: nat,M: nat,N: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I2 @ J ) @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ J @ I2 )
| ( ( ord_less_eq_nat @ M @ I2 )
& ( ord_less_eq_nat @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_641_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_642_of__bool__less__eq__iff,axiom,
! [P: $o,Q: $o] :
( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
= ( P
=> Q ) ) ).
% of_bool_less_eq_iff
thf(fact_643_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_eq_nat @ I @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_644_le__add__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_645_le__add__same__cancel1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_646_add__le__same__cancel2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_647_add__le__same__cancel1,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_648_atLeastLessThan__iff,axiom,
! [I2: set_nat,L: set_nat,U2: set_nat] :
( ( member_set_nat @ I2 @ ( set_or3540276404033026485et_nat @ L @ U2 ) )
= ( ( ord_less_eq_set_nat @ L @ I2 )
& ( ord_less_set_nat @ I2 @ U2 ) ) ) ).
% atLeastLessThan_iff
thf(fact_649_atLeastLessThan__iff,axiom,
! [I2: nat,L: nat,U2: nat] :
( ( member_nat @ I2 @ ( set_or4665077453230672383an_nat @ L @ U2 ) )
= ( ( ord_less_eq_nat @ L @ I2 )
& ( ord_less_nat @ I2 @ U2 ) ) ) ).
% atLeastLessThan_iff
thf(fact_650_atLeastLessThan__empty,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( set_or3540276404033026485et_nat @ A2 @ B )
= bot_bot_set_set_nat ) ) ).
% atLeastLessThan_empty
thf(fact_651_atLeastLessThan__empty,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( set_or4665077453230672383an_nat @ A2 @ B )
= bot_bot_set_nat ) ) ).
% atLeastLessThan_empty
thf(fact_652_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_653_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_654_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_655_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_656_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_657_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_658_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_659_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_660_atLeastLessThan__singleton,axiom,
! [M: nat] :
( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
= ( insert_nat @ M @ bot_bot_set_nat ) ) ).
% atLeastLessThan_singleton
thf(fact_661_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_662_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_663_dec__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P @ I2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_664_inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% inc_induct
thf(fact_665_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_666_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_667_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_668_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_669_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_670_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_671_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R2 @ X3 @ X3 )
=> ( ! [X3: nat,Y2: nat,Z3: nat] :
( ( R2 @ X3 @ Y2 )
=> ( ( R2 @ Y2 @ Z3 )
=> ( R2 @ X3 @ Z3 ) ) )
=> ( ! [N2: nat] : ( R2 @ N2 @ ( suc @ N2 ) )
=> ( R2 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_672_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_673_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_674_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_675_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_676_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_677_Suc__inject,axiom,
! [X2: nat,Y3: nat] :
( ( ( suc @ X2 )
= ( suc @ Y3 ) )
=> ( X2 = Y3 ) ) ).
% Suc_inject
thf(fact_678_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_679_Suc__le__D,axiom,
! [N: nat,M4: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M4 )
=> ? [M5: nat] :
( M4
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_680_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_681_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_682_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_683_atLeastLessThan__subset__iff,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A2 @ B ) @ ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_eq_nat @ B @ A2 )
| ( ( ord_less_eq_nat @ C @ A2 )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_684_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_685_lift__Suc__antimono__le,axiom,
! [F: nat > set_nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_set_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_686_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_687_lift__Suc__mono__le,axiom,
! [F: nat > set_nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_set_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_688_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_689_order__antisym__conv,axiom,
! [Y3: nat,X2: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_690_order__antisym__conv,axiom,
! [Y3: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_691_linorder__le__cases,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% linorder_le_cases
thf(fact_692_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_693_ord__le__eq__subst,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_694_ord__le__eq__subst,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_695_ord__le__eq__subst,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_696_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_697_ord__eq__le__subst,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_698_ord__eq__le__subst,axiom,
! [A2: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_699_ord__eq__le__subst,axiom,
! [A2: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_700_linorder__linear,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% linorder_linear
thf(fact_701_order__eq__refl,axiom,
! [X2: nat,Y3: nat] :
( ( X2 = Y3 )
=> ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_702_order__eq__refl,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( X2 = Y3 )
=> ( ord_less_eq_set_nat @ X2 @ Y3 ) ) ).
% order_eq_refl
thf(fact_703_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_704_order__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_705_order__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_706_order__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_707_order__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_708_order__subst1,axiom,
! [A2: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_709_order__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_710_order__subst1,axiom,
! [A2: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_711_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ A4 @ B5 )
& ( ord_less_eq_nat @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_712_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y: set_nat,Z: set_nat] : ( Y = Z ) )
= ( ^ [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_713_antisym,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_714_antisym,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% antisym
thf(fact_715_dual__order_Otrans,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_716_dual__order_Otrans,axiom,
! [B: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_717_dual__order_Oantisym,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_718_dual__order_Oantisym,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% dual_order.antisym
thf(fact_719_dual__order_Oeq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A4 )
& ( ord_less_eq_nat @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_720_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_nat,Z: set_nat] : ( Y = Z ) )
= ( ^ [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
& ( ord_less_eq_set_nat @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_721_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B: nat] :
( ! [A5: nat,B6: nat] :
( ( ord_less_eq_nat @ A5 @ B6 )
=> ( P @ A5 @ B6 ) )
=> ( ! [A5: nat,B6: nat] :
( ( P @ B6 @ A5 )
=> ( P @ A5 @ B6 ) )
=> ( P @ A2 @ B ) ) ) ).
% linorder_wlog
thf(fact_722_order__trans,axiom,
! [X2: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z2 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_723_order__trans,axiom,
! [X2: set_nat,Y3: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_set_nat @ Y3 @ Z2 )
=> ( ord_less_eq_set_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_724_order_Otrans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_725_order_Otrans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_726_order__antisym,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_727_order__antisym,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_set_nat @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% order_antisym
thf(fact_728_ord__le__eq__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_729_ord__le__eq__trans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_730_ord__eq__le__trans,axiom,
! [A2: nat,B: nat,C: nat] :
( ( A2 = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_731_ord__eq__le__trans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( A2 = B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_732_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : ( Y = Z ) )
= ( ^ [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_733_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y: set_nat,Z: set_nat] : ( Y = Z ) )
= ( ^ [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
& ( ord_less_eq_set_nat @ Y4 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_734_le__cases3,axiom,
! [X2: nat,Y3: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y3 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_735_nle__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B ) )
= ( ( ord_less_eq_nat @ B @ A2 )
& ( B != A2 ) ) ) ).
% nle_le
thf(fact_736_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_737_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_738_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_739_order__le__imp__less__or__eq,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ( ord_less_set_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_740_linorder__le__less__linear,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
| ( ord_less_nat @ Y3 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_741_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_742_order__less__le__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_743_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_744_order__less__le__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_745_order__less__le__subst1,axiom,
! [A2: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_746_order__less__le__subst1,axiom,
! [A2: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_747_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_748_order__le__less__subst2,axiom,
! [A2: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_749_order__le__less__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_750_order__le__less__subst2,axiom,
! [A2: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_751_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_752_order__le__less__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_753_order__less__le__trans,axiom,
! [X2: nat,Y3: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_754_order__less__le__trans,axiom,
! [X2: set_nat,Y3: set_nat,Z2: set_nat] :
( ( ord_less_set_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_set_nat @ Y3 @ Z2 )
=> ( ord_less_set_nat @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_755_order__le__less__trans,axiom,
! [X2: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_756_order__le__less__trans,axiom,
! [X2: set_nat,Y3: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ( ord_less_set_nat @ Y3 @ Z2 )
=> ( ord_less_set_nat @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_757_order__neq__le__trans,axiom,
! [A2: nat,B: nat] :
( ( A2 != B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_758_order__neq__le__trans,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 != B )
=> ( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_set_nat @ A2 @ B ) ) ) ).
% order_neq_le_trans
thf(fact_759_order__le__neq__trans,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_760_order__le__neq__trans,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less_set_nat @ A2 @ B ) ) ) ).
% order_le_neq_trans
thf(fact_761_order__less__imp__le,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_762_order__less__imp__le,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( ord_less_set_nat @ X2 @ Y3 )
=> ( ord_less_eq_set_nat @ X2 @ Y3 ) ) ).
% order_less_imp_le
thf(fact_763_linorder__not__less,axiom,
! [X2: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% linorder_not_less
thf(fact_764_linorder__not__le,axiom,
! [X2: nat,Y3: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y3 ) )
= ( ord_less_nat @ Y3 @ X2 ) ) ).
% linorder_not_le
thf(fact_765_order__less__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
& ( X != Y4 ) ) ) ) ).
% order_less_le
thf(fact_766_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
& ( X != Y4 ) ) ) ) ).
% order_less_le
thf(fact_767_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
| ( X = Y4 ) ) ) ) ).
% order_le_less
thf(fact_768_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X: set_nat,Y4: set_nat] :
( ( ord_less_set_nat @ X @ Y4 )
| ( X = Y4 ) ) ) ) ).
% order_le_less
thf(fact_769_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A2: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_770_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B @ A2 )
=> ( ord_less_eq_set_nat @ B @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_771_order_Ostrict__implies__order,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_772_order_Ostrict__implies__order,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% order.strict_implies_order
thf(fact_773_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B5: nat,A4: nat] :
( ( ord_less_eq_nat @ B5 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_774_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B5: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
& ~ ( ord_less_eq_set_nat @ A4 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_775_dual__order_Ostrict__trans2,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B @ A2 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_776_dual__order_Ostrict__trans2,axiom,
! [B: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_777_dual__order_Ostrict__trans1,axiom,
! [B: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_778_dual__order_Ostrict__trans1,axiom,
! [B: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_779_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B5: nat,A4: nat] :
( ( ord_less_eq_nat @ B5 @ A4 )
& ( A4 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_780_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B5: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A4 )
& ( A4 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_781_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A4: nat] :
( ( ord_less_nat @ B5 @ A4 )
| ( A4 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_782_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B5: set_nat,A4: set_nat] :
( ( ord_less_set_nat @ B5 @ A4 )
| ( A4 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_783_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ A4 @ B5 )
& ~ ( ord_less_eq_nat @ B5 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_784_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
& ~ ( ord_less_eq_set_nat @ B5 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_785_order_Ostrict__trans2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_786_order_Ostrict__trans2,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_787_order_Ostrict__trans1,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_788_order_Ostrict__trans1,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_789_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_790_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_791_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B5: nat] :
( ( ord_less_nat @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_792_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
( ( ord_less_set_nat @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_793_not__le__imp__less,axiom,
! [Y3: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ord_less_nat @ X2 @ Y3 ) ) ).
% not_le_imp_less
thf(fact_794_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_795_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
& ~ ( ord_less_eq_set_nat @ Y4 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_796_antisym__conv2,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_797_antisym__conv2,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ( ~ ( ord_less_set_nat @ X2 @ Y3 ) )
= ( X2 = Y3 ) ) ) ).
% antisym_conv2
thf(fact_798_antisym__conv1,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_799_antisym__conv1,axiom,
! [X2: set_nat,Y3: set_nat] :
( ~ ( ord_less_set_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_set_nat @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ).
% antisym_conv1
thf(fact_800_nless__le,axiom,
! [A2: nat,B: nat] :
( ( ~ ( ord_less_nat @ A2 @ B ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_801_nless__le,axiom,
! [A2: set_nat,B: set_nat] :
( ( ~ ( ord_less_set_nat @ A2 @ B ) )
= ( ~ ( ord_less_eq_set_nat @ A2 @ B )
| ( A2 = B ) ) ) ).
% nless_le
thf(fact_802_leI,axiom,
! [X2: nat,Y3: nat] :
( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% leI
thf(fact_803_leD,axiom,
! [Y3: nat,X2: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y3 ) ) ).
% leD
thf(fact_804_leD,axiom,
! [Y3: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X2 )
=> ~ ( ord_less_set_nat @ X2 @ Y3 ) ) ).
% leD
thf(fact_805_verit__comp__simplify1_I3_J,axiom,
! [B7: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B7 @ A6 ) )
= ( ord_less_nat @ A6 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_806_enum__rgfs_Ocases,axiom,
! [X2: nat] :
( ( X2 != zero_zero_nat )
=> ~ ! [N2: nat] :
( X2
!= ( suc @ N2 ) ) ) ).
% enum_rgfs.cases
thf(fact_807_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_808_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_809_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_810_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_811_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_812_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X3: nat,Y2: nat] :
( ( P @ X3 @ Y2 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y2 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_813_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_814_old_Onat_Oexhaust,axiom,
! [Y3: nat] :
( ( Y3 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y3
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_815_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_816_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_817_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_818_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_819_add__le__imp__le__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_right
thf(fact_820_add__le__imp__le__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% add_le_imp_le_left
thf(fact_821_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B5: nat] :
? [C5: nat] :
( B5
= ( plus_plus_nat @ A4 @ C5 ) ) ) ) ).
% le_iff_add
thf(fact_822_add__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_823_less__eqE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ~ ! [C4: nat] :
( B
!= ( plus_plus_nat @ A2 @ C4 ) ) ) ).
% less_eqE
thf(fact_824_add__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_825_add__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_826_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_827_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( I2 = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_828_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_829_bot_Oextremum,axiom,
! [A2: set_list_nat] : ( ord_le6045566169113846134st_nat @ bot_bot_set_list_nat @ A2 ) ).
% bot.extremum
thf(fact_830_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_831_bot_Oextremum,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% bot.extremum
thf(fact_832_bot_Oextremum__unique,axiom,
! [A2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ bot_bot_set_list_nat )
= ( A2 = bot_bot_set_list_nat ) ) ).
% bot.extremum_unique
thf(fact_833_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_834_bot_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_835_bot_Oextremum__uniqueI,axiom,
! [A2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A2 @ bot_bot_set_list_nat )
=> ( A2 = bot_bot_set_list_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_836_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_837_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
=> ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_838_Nat_OlessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( K
!= ( suc @ I2 ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_839_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_840_Suc__lessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_841_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_842_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_843_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_844_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
& ( P @ I ) ) )
= ( ( P @ N )
| ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( P @ I ) ) ) ) ).
% Ex_less_Suc
thf(fact_845_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_846_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_847_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
=> ( P @ I ) ) )
= ( ( P @ N )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P @ I ) ) ) ) ).
% All_less_Suc
thf(fact_848_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M6: nat] :
( ( M
= ( suc @ M6 ) )
& ( ord_less_nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_849_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_850_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_851_less__trans__Suc,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_852_less__Suc__induct,axiom,
! [I2: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I4: nat] : ( P @ I4 @ ( suc @ I4 ) )
=> ( ! [I4: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I4 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I4 @ K2 ) ) ) ) )
=> ( P @ I2 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_853_strict__inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I4: nat] :
( ( J
= ( suc @ I4 ) )
=> ( P @ I4 ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ( P @ ( suc @ I4 ) )
=> ( P @ I4 ) ) )
=> ( P @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_854_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_855_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_856_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_857_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_858_nat__arith_Osuc1,axiom,
! [A: nat,K: nat,A2: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( suc @ A )
= ( plus_plus_nat @ K @ ( suc @ A2 ) ) ) ) ).
% nat_arith.suc1
thf(fact_859_inf__sup__ord_I2_J,axiom,
! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_860_inf__sup__ord_I2_J,axiom,
! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_861_inf__sup__ord_I1_J,axiom,
! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y3 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_862_inf__sup__ord_I1_J,axiom,
! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_863_inf__le1,axiom,
! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y3 ) @ X2 ) ).
% inf_le1
thf(fact_864_inf__le1,axiom,
! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ X2 ) ).
% inf_le1
thf(fact_865_inf__le2,axiom,
! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_866_inf__le2,axiom,
! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_867_le__infE,axiom,
! [X2: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A2 @ B ) )
=> ~ ( ( ord_less_eq_nat @ X2 @ A2 )
=> ~ ( ord_less_eq_nat @ X2 @ B ) ) ) ).
% le_infE
thf(fact_868_le__infE,axiom,
! [X2: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ A2 @ B ) )
=> ~ ( ( ord_less_eq_set_nat @ X2 @ A2 )
=> ~ ( ord_less_eq_set_nat @ X2 @ B ) ) ) ).
% le_infE
thf(fact_869_le__infI,axiom,
! [X2: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ X2 @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ B )
=> ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A2 @ B ) ) ) ) ).
% le_infI
thf(fact_870_le__infI,axiom,
! [X2: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ A2 )
=> ( ( ord_less_eq_set_nat @ X2 @ B )
=> ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% le_infI
thf(fact_871_inf__mono,axiom,
! [A2: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_872_inf__mono,axiom,
! [A2: set_nat,C: set_nat,B: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B @ D )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ ( inf_inf_set_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_873_le__infI1,axiom,
! [A2: nat,X2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ X2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ X2 ) ) ).
% le_infI1
thf(fact_874_le__infI1,axiom,
! [A2: set_nat,X2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ X2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ X2 ) ) ).
% le_infI1
thf(fact_875_le__infI2,axiom,
! [B: nat,X2: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ X2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ X2 ) ) ).
% le_infI2
thf(fact_876_le__infI2,axiom,
! [B: set_nat,X2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ X2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ X2 ) ) ).
% le_infI2
thf(fact_877_inf_OorderE,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( A2
= ( inf_inf_nat @ A2 @ B ) ) ) ).
% inf.orderE
thf(fact_878_inf_OorderE,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( A2
= ( inf_inf_set_nat @ A2 @ B ) ) ) ).
% inf.orderE
thf(fact_879_inf_OorderI,axiom,
! [A2: nat,B: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B ) )
=> ( ord_less_eq_nat @ A2 @ B ) ) ).
% inf.orderI
thf(fact_880_inf_OorderI,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2
= ( inf_inf_set_nat @ A2 @ B ) )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% inf.orderI
thf(fact_881_inf__unique,axiom,
! [F: nat > nat > nat,X2: nat,Y3: nat] :
( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_nat @ X3 @ Z3 )
=> ( ord_less_eq_nat @ X3 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X2 @ Y3 )
= ( F @ X2 @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_882_inf__unique,axiom,
! [F: set_nat > set_nat > set_nat,X2: set_nat,Y3: set_nat] :
( ! [X3: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( F @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( F @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: set_nat,Y2: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y2 )
=> ( ( ord_less_eq_set_nat @ X3 @ Z3 )
=> ( ord_less_eq_set_nat @ X3 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_set_nat @ X2 @ Y3 )
= ( F @ X2 @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_883_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y4: nat] :
( ( inf_inf_nat @ X @ Y4 )
= X ) ) ) ).
% le_iff_inf
thf(fact_884_le__iff__inf,axiom,
( ord_less_eq_set_nat
= ( ^ [X: set_nat,Y4: set_nat] :
( ( inf_inf_set_nat @ X @ Y4 )
= X ) ) ) ).
% le_iff_inf
thf(fact_885_inf_Oabsorb1,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( inf_inf_nat @ A2 @ B )
= A2 ) ) ).
% inf.absorb1
thf(fact_886_inf_Oabsorb1,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( inf_inf_set_nat @ A2 @ B )
= A2 ) ) ).
% inf.absorb1
thf(fact_887_inf_Oabsorb2,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( inf_inf_nat @ A2 @ B )
= B ) ) ).
% inf.absorb2
thf(fact_888_inf_Oabsorb2,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ B )
= B ) ) ).
% inf.absorb2
thf(fact_889_inf__absorb1,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( inf_inf_nat @ X2 @ Y3 )
= X2 ) ) ).
% inf_absorb1
thf(fact_890_inf__absorb1,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ( inf_inf_set_nat @ X2 @ Y3 )
= X2 ) ) ).
% inf_absorb1
thf(fact_891_inf__absorb2,axiom,
! [Y3: nat,X2: nat] :
( ( ord_less_eq_nat @ Y3 @ X2 )
=> ( ( inf_inf_nat @ X2 @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_892_inf__absorb2,axiom,
! [Y3: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X2 )
=> ( ( inf_inf_set_nat @ X2 @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_893_inf_OboundedE,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_nat @ A2 @ B )
=> ~ ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_894_inf_OboundedE,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B )
=> ~ ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_895_inf_OboundedI,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_896_inf_OboundedI,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_897_inf__greatest,axiom,
! [X2: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y3 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_898_inf__greatest,axiom,
! [X2: set_nat,Y3: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y3 )
=> ( ( ord_less_eq_set_nat @ X2 @ Z2 )
=> ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ Y3 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_899_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B5: nat] :
( A4
= ( inf_inf_nat @ A4 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_900_inf_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
( A4
= ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_901_inf_Ocobounded1,axiom,
! [A2: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ A2 ) ).
% inf.cobounded1
thf(fact_902_inf_Ocobounded1,axiom,
! [A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ A2 ) ).
% inf.cobounded1
thf(fact_903_inf_Ocobounded2,axiom,
! [A2: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ B ) ).
% inf.cobounded2
thf(fact_904_inf_Ocobounded2,axiom,
! [A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ B ) ).
% inf.cobounded2
thf(fact_905_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B5: nat] :
( ( inf_inf_nat @ A4 @ B5 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_906_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B5: set_nat] :
( ( inf_inf_set_nat @ A4 @ B5 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_907_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A4: nat] :
( ( inf_inf_nat @ A4 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_908_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [B5: set_nat,A4: set_nat] :
( ( inf_inf_set_nat @ A4 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_909_inf_OcoboundedI1,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_910_inf_OcoboundedI1,axiom,
! [A2: set_nat,C: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_911_inf_OcoboundedI2,axiom,
! [B: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_912_inf_OcoboundedI2,axiom,
! [B: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_913_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_914_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_915_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_916_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_917_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_918_subset__insertI2,axiom,
! [A: set_nat,B2: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ord_less_eq_set_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_919_subset__insertI,axiom,
! [B2: set_nat,A2: nat] : ( ord_less_eq_set_nat @ B2 @ ( insert_nat @ A2 @ B2 ) ) ).
% subset_insertI
thf(fact_920_subset__insert,axiom,
! [X2: nat,A: set_nat,B2: set_nat] :
( ~ ( member_nat @ X2 @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X2 @ B2 ) )
= ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% subset_insert
thf(fact_921_insert__mono,axiom,
! [C2: set_nat,D2: set_nat,A2: nat] :
( ( ord_less_eq_set_nat @ C2 @ D2 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A2 @ C2 ) @ ( insert_nat @ A2 @ D2 ) ) ) ).
% insert_mono
thf(fact_922_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_923_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_924_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_925_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_926_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_927_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J: nat] :
( ! [I4: nat,J2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ord_less_nat @ ( F @ I4 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_928_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
? [K3: nat] :
( N3
= ( plus_plus_nat @ M3 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_929_trans__le__add2,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_930_trans__le__add1,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_931_add__le__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_932_add__le__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_933_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_934_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_935_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_936_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_937_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_938_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_939_Int__Collect__mono,axiom,
! [A: set_list_nat,B2: set_list_nat,P: list_nat > $o,Q: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ A @ B2 )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le6045566169113846134st_nat @ ( inf_inf_set_list_nat @ A @ ( collect_list_nat @ P ) ) @ ( inf_inf_set_list_nat @ B2 @ ( collect_list_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_940_Int__Collect__mono,axiom,
! [A: set_nat,B2: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B2 @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_941_Int__greatest,axiom,
! [C2: set_nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A )
=> ( ( ord_less_eq_set_nat @ C2 @ B2 )
=> ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_942_Int__absorb2,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( inf_inf_set_nat @ A @ B2 )
= A ) ) ).
% Int_absorb2
thf(fact_943_Int__absorb1,axiom,
! [B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( inf_inf_set_nat @ A @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_944_Int__lower2,axiom,
! [A: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_945_Int__lower1,axiom,
! [A: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ A ) ).
% Int_lower1
thf(fact_946_Int__mono,axiom,
! [A: set_nat,C2: set_nat,B2: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ D2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ ( inf_inf_set_nat @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_947_mult__le__mono2,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_948_mult__le__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_949_mult__le__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_950_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_951_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_952_psubsetE,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ~ ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ord_less_eq_set_nat @ B2 @ A ) ) ) ).
% psubsetE
thf(fact_953_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
& ( A3 != B4 ) ) ) ) ).
% psubset_eq
thf(fact_954_psubset__imp__subset,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ( ord_less_eq_set_nat @ A @ B2 ) ) ).
% psubset_imp_subset
thf(fact_955_psubset__subset__trans,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_set_nat @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_956_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
& ~ ( ord_less_eq_set_nat @ B4 @ A3 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_957_subset__psubset__trans,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_set_nat @ B2 @ C2 )
=> ( ord_less_set_nat @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_958_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ A3 @ B4 )
| ( A3 = B4 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_959_subset__eq__atLeast0__lessThan__card,axiom,
! [N5: set_nat,N: nat] :
( ( ord_less_eq_set_nat @ N5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ N5 ) @ N ) ) ).
% subset_eq_atLeast0_lessThan_card
thf(fact_960_atLeastLessThanSuc,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_eq_nat @ M @ N )
=> ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
= ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ M @ N )
=> ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
= bot_bot_set_nat ) ) ) ).
% atLeastLessThanSuc
thf(fact_961_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_962_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_963_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_964_mult__nonneg__nonpos2,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A2 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_965_mult__nonpos__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_966_mult__nonneg__nonpos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_967_mult__nonneg__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_968_split__mult__neg__le,axiom,
! [A2: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_969_mult__right__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_970_mult__left__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_971_mult__mono_H,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_972_mult__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_973_add__nonpos__eq__0__iff,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y3 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X2 @ Y3 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_974_add__nonneg__eq__0__iff,axiom,
! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
=> ( ( ( plus_plus_nat @ X2 @ Y3 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_975_add__nonpos__nonpos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_976_add__nonneg__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_977_add__increasing2,axiom,
! [C: nat,B: nat,A2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A2 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_978_add__decreasing2,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_979_add__increasing,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_980_add__decreasing,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_981_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_982_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_983_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_984_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_985_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_986_add__le__less__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_987_add__less__le__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_988_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
& ( P @ I ) ) )
= ( ( P @ zero_zero_nat )
| ? [I: nat] :
( ( ord_less_nat @ I @ N )
& ( P @ ( suc @ I ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_989_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_990_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I: nat] :
( ( ord_less_nat @ I @ ( suc @ N ) )
=> ( P @ I ) ) )
= ( ( P @ zero_zero_nat )
& ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( P @ ( suc @ I ) ) ) ) ) ).
% All_less_Suc2
thf(fact_991_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_992_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_993_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_994_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_995_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).
% less_natE
thf(fact_996_less__add__Suc1,axiom,
! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M ) ) ) ).
% less_add_Suc1
thf(fact_997_less__add__Suc2,axiom,
! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M @ I2 ) ) ) ).
% less_add_Suc2
thf(fact_998_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
? [K3: nat] :
( N3
= ( suc @ ( plus_plus_nat @ M3 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_999_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_1000_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1001_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1002_subset__singleton__iff,axiom,
! [X6: set_list_nat,A2: list_nat] :
( ( ord_le6045566169113846134st_nat @ X6 @ ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) )
= ( ( X6 = bot_bot_set_list_nat )
| ( X6
= ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_1003_subset__singleton__iff,axiom,
! [X6: set_nat,A2: nat] :
( ( ord_less_eq_set_nat @ X6 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( ( X6 = bot_bot_set_nat )
| ( X6
= ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_1004_subset__singletonD,axiom,
! [A: set_list_nat,X2: list_nat] :
( ( ord_le6045566169113846134st_nat @ A @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) )
=> ( ( A = bot_bot_set_list_nat )
| ( A
= ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) ) ) ).
% subset_singletonD
thf(fact_1005_subset__singletonD,axiom,
! [A: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
=> ( ( A = bot_bot_set_nat )
| ( A
= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_1006_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_1007_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1008_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1009_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1010_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1011_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M5: nat,N2: nat] :
( ( ord_less_nat @ M5 @ N2 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1012_zero__less__eq__of__bool,axiom,
! [P: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) ) ).
% zero_less_eq_of_bool
thf(fact_1013_card__insert__le,axiom,
! [A: set_nat,X2: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ ( insert_nat @ X2 @ A ) ) ) ).
% card_insert_le
thf(fact_1014_card__insert__le,axiom,
! [A: set_list_nat,X2: list_nat] : ( ord_less_eq_nat @ ( finite_card_list_nat @ A ) @ ( finite_card_list_nat @ ( insert_list_nat @ X2 @ A ) ) ) ).
% card_insert_le
thf(fact_1015_of__bool__less__eq__one,axiom,
! [P: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat ) ).
% of_bool_less_eq_one
thf(fact_1016_count__le__length,axiom,
! [Xs: list_list_nat,X2: list_nat] : ( ord_less_eq_nat @ ( count_list_list_nat @ Xs @ X2 ) @ ( size_s3023201423986296836st_nat @ Xs ) ) ).
% count_le_length
thf(fact_1017_count__le__length,axiom,
! [Xs: list_nat,X2: nat] : ( ord_less_eq_nat @ ( count_list_nat @ Xs @ X2 ) @ ( size_size_list_nat @ Xs ) ) ).
% count_le_length
thf(fact_1018_count__list__map__ge,axiom,
! [Xs: list_list_nat,X2: list_nat,F: list_nat > nat] : ( ord_less_eq_nat @ ( count_list_list_nat @ Xs @ X2 ) @ ( count_list_nat @ ( map_list_nat_nat @ F @ Xs ) @ ( F @ X2 ) ) ) ).
% count_list_map_ge
thf(fact_1019_count__list__map__ge,axiom,
! [Xs: list_list_nat,X2: list_nat,F: list_nat > list_nat] : ( ord_less_eq_nat @ ( count_list_list_nat @ Xs @ X2 ) @ ( count_list_list_nat @ ( map_li7225945977422193158st_nat @ F @ Xs ) @ ( F @ X2 ) ) ) ).
% count_list_map_ge
thf(fact_1020_mult__less__le__imp__less,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1021_mult__le__less__imp__less,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1022_mult__right__le__imp__le,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A2 @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_1023_mult__left__le__imp__le,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A2 @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_1024_mult__strict__mono_H,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1025_mult__right__less__imp__less,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_1026_mult__strict__mono,axiom,
! [A2: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1027_mult__left__less__imp__less,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A2 @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_1028_add__strict__increasing2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1029_add__strict__increasing,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1030_add__pos__nonneg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1031_add__nonpos__neg,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_1032_add__nonneg__pos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1033_add__neg__nonpos,axiom,
! [A2: nat,B: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_1034_mult__left__le,axiom,
! [C: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ A2 ) ) ) ).
% mult_left_le
thf(fact_1035_mult__le__one,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_1036_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1037_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1038_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_1039_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_1040_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1041_atLeast0__lessThan__Suc,axiom,
! [N: nat] :
( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
= ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% atLeast0_lessThan_Suc
thf(fact_1042_subset__card__intvl__is__intvl,axiom,
! [A: set_nat,K: nat] :
( ( ord_less_eq_set_nat @ A @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A ) ) ) )
=> ( A
= ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A ) ) ) ) ) ).
% subset_card_intvl_is_intvl
thf(fact_1043_card__less__Suc2,axiom,
! [M7: set_nat,I2: nat] :
( ~ ( member_nat @ zero_zero_nat @ M7 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ ( suc @ K3 ) @ M7 )
& ( ord_less_nat @ K3 @ I2 ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ K3 @ M7 )
& ( ord_less_nat @ K3 @ ( suc @ I2 ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_1044_card__less__Suc,axiom,
! [M7: set_nat,I2: nat] :
( ( member_nat @ zero_zero_nat @ M7 )
=> ( ( suc
@ ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ ( suc @ K3 ) @ M7 )
& ( ord_less_nat @ K3 @ I2 ) ) ) ) )
= ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ K3 @ M7 )
& ( ord_less_nat @ K3 @ ( suc @ I2 ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_1045_card__less,axiom,
! [M7: set_nat,I2: nat] :
( ( member_nat @ zero_zero_nat @ M7 )
=> ( ( finite_card_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( member_nat @ K3 @ M7 )
& ( ord_less_nat @ K3 @ ( suc @ I2 ) ) ) ) )
!= zero_zero_nat ) ) ).
% card_less
thf(fact_1046_card__1__singleton__iff,axiom,
! [A: set_nat] :
( ( ( finite_card_nat @ A )
= ( suc @ zero_zero_nat ) )
= ( ? [X: nat] :
( A
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% card_1_singleton_iff
thf(fact_1047_card__1__singleton__iff,axiom,
! [A: set_list_nat] :
( ( ( finite_card_list_nat @ A )
= ( suc @ zero_zero_nat ) )
= ( ? [X: list_nat] :
( A
= ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) ) ) ).
% card_1_singleton_iff
thf(fact_1048_card__eq__SucD,axiom,
! [A: set_nat,K: nat] :
( ( ( finite_card_nat @ A )
= ( suc @ K ) )
=> ? [B6: nat,B3: set_nat] :
( ( A
= ( insert_nat @ B6 @ B3 ) )
& ~ ( member_nat @ B6 @ B3 )
& ( ( finite_card_nat @ B3 )
= K )
& ( ( K = zero_zero_nat )
=> ( B3 = bot_bot_set_nat ) ) ) ) ).
% card_eq_SucD
thf(fact_1049_card__eq__SucD,axiom,
! [A: set_list_nat,K: nat] :
( ( ( finite_card_list_nat @ A )
= ( suc @ K ) )
=> ? [B6: list_nat,B3: set_list_nat] :
( ( A
= ( insert_list_nat @ B6 @ B3 ) )
& ~ ( member_list_nat @ B6 @ B3 )
& ( ( finite_card_list_nat @ B3 )
= K )
& ( ( K = zero_zero_nat )
=> ( B3 = bot_bot_set_list_nat ) ) ) ) ).
% card_eq_SucD
thf(fact_1050_card__Suc__eq,axiom,
! [A: set_nat,K: nat] :
( ( ( finite_card_nat @ A )
= ( suc @ K ) )
= ( ? [B5: nat,B4: set_nat] :
( ( A
= ( insert_nat @ B5 @ B4 ) )
& ~ ( member_nat @ B5 @ B4 )
& ( ( finite_card_nat @ B4 )
= K )
& ( ( K = zero_zero_nat )
=> ( B4 = bot_bot_set_nat ) ) ) ) ) ).
% card_Suc_eq
thf(fact_1051_card__Suc__eq,axiom,
! [A: set_list_nat,K: nat] :
( ( ( finite_card_list_nat @ A )
= ( suc @ K ) )
= ( ? [B5: list_nat,B4: set_list_nat] :
( ( A
= ( insert_list_nat @ B5 @ B4 ) )
& ~ ( member_list_nat @ B5 @ B4 )
& ( ( finite_card_list_nat @ B4 )
= K )
& ( ( K = zero_zero_nat )
=> ( B4 = bot_bot_set_list_nat ) ) ) ) ) ).
% card_Suc_eq
thf(fact_1052_subset__antisym,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_1053_subsetI,axiom,
! [A: set_nat,B2: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B2 ) )
=> ( ord_less_eq_set_nat @ A @ B2 ) ) ).
% subsetI
thf(fact_1054_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M7: nat] :
( ( P @ X2 )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M7 ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X7: nat] :
( ( P @ X7 )
=> ( ord_less_eq_nat @ X7 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1055_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 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1056_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_1057_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_1058_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_1059_le__trans,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_1060_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1061_Collect__subset,axiom,
! [A: set_list_nat,P: list_nat > $o] :
( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X: list_nat] :
( ( member_list_nat @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_1062_Collect__subset,axiom,
! [A: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_1063_Collect__mono__iff,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) )
= ( ! [X: list_nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_1064_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_1065_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A3 )
@ ^ [X: nat] : ( member_nat @ X @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_1066_set__eq__subset,axiom,
( ( ^ [Y: set_nat,Z: set_nat] : ( Y = Z ) )
= ( ^ [A3: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_1067_subset__trans,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_1068_Collect__mono,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ! [X3: list_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_1069_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_1070_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_1071_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A3 )
=> ( member_nat @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_1072_equalityD2,axiom,
! [A: set_nat,B2: set_nat] :
( ( A = B2 )
=> ( ord_less_eq_set_nat @ B2 @ A ) ) ).
% equalityD2
thf(fact_1073_equalityD1,axiom,
! [A: set_nat,B2: set_nat] :
( ( A = B2 )
=> ( ord_less_eq_set_nat @ A @ B2 ) ) ).
% equalityD1
thf(fact_1074_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A3 )
=> ( member_nat @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_1075_equalityE,axiom,
! [A: set_nat,B2: set_nat] :
( ( A = B2 )
=> ~ ( ( ord_less_eq_set_nat @ A @ B2 )
=> ~ ( ord_less_eq_set_nat @ B2 @ A ) ) ) ).
% equalityE
thf(fact_1076_subsetD,axiom,
! [A: set_nat,B2: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_1077_in__mono,axiom,
! [A: set_nat,B2: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_1078_pred__subset__eq,axiom,
! [R2: set_nat,S2: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ R2 )
@ ^ [X: nat] : ( member_nat @ X @ S2 ) )
= ( ord_less_eq_set_nat @ R2 @ S2 ) ) ).
% pred_subset_eq
thf(fact_1079_length__stirling__row,axiom,
! [N: nat] :
( ( size_size_list_nat @ ( stirling_row @ N ) )
= ( suc @ N ) ) ).
% length_stirling_row
thf(fact_1080_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 )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K2 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_1081_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_1082_Stirling_Oelims,axiom,
! [X2: nat,Xa: nat,Y3: nat] :
( ( ( stirling @ X2 @ Xa )
= Y3 )
=> ( ( ( X2 = zero_zero_nat )
=> ( ( Xa = zero_zero_nat )
=> ( Y3 != one_one_nat ) ) )
=> ( ( ( X2 = zero_zero_nat )
=> ( ? [K2: nat] :
( Xa
= ( suc @ K2 ) )
=> ( Y3 != zero_zero_nat ) ) )
=> ( ( ? [N2: nat] :
( X2
= ( suc @ N2 ) )
=> ( ( Xa = zero_zero_nat )
=> ( Y3 != zero_zero_nat ) ) )
=> ~ ! [N2: nat] :
( ( X2
= ( suc @ N2 ) )
=> ! [K2: nat] :
( ( Xa
= ( suc @ K2 ) )
=> ( Y3
!= ( plus_plus_nat @ ( times_times_nat @ ( suc @ K2 ) @ ( stirling @ N2 @ ( suc @ K2 ) ) ) @ ( stirling @ N2 @ K2 ) ) ) ) ) ) ) ) ) ).
% Stirling.elims
thf(fact_1083_Stirling__same,axiom,
! [N: nat] :
( ( stirling @ N @ N )
= one_one_nat ) ).
% Stirling_same
thf(fact_1084_Stirling__less,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( stirling @ N @ K )
= zero_zero_nat ) ) ).
% Stirling_less
thf(fact_1085_Stirling__1,axiom,
! [N: nat] :
( ( stirling @ ( suc @ N ) @ ( suc @ zero_zero_nat ) )
= one_one_nat ) ).
% Stirling_1
thf(fact_1086_Stirling_Osimps_I3_J,axiom,
! [N: nat] :
( ( stirling @ ( suc @ N ) @ zero_zero_nat )
= zero_zero_nat ) ).
% Stirling.simps(3)
thf(fact_1087_Stirling_Osimps_I2_J,axiom,
! [K: nat] :
( ( stirling @ zero_zero_nat @ ( suc @ K ) )
= zero_zero_nat ) ).
% Stirling.simps(2)
thf(fact_1088_Stirling_Osimps_I1_J,axiom,
( ( stirling @ zero_zero_nat @ zero_zero_nat )
= one_one_nat ) ).
% Stirling.simps(1)
thf(fact_1089_Stirling_Osimps_I4_J,axiom,
! [N: nat,K: nat] :
( ( stirling @ ( suc @ N ) @ ( suc @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( suc @ K ) @ ( stirling @ N @ ( suc @ K ) ) ) @ ( stirling @ N @ K ) ) ) ).
% Stirling.simps(4)
thf(fact_1090_insert__subsetI,axiom,
! [X2: nat,A: set_nat,X6: set_nat] :
( ( member_nat @ X2 @ A )
=> ( ( ord_less_eq_set_nat @ X6 @ A )
=> ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ X6 ) @ A ) ) ) ).
% insert_subsetI
thf(fact_1091_subset__emptyI,axiom,
! [A: set_list_nat] :
( ! [X3: list_nat] :
~ ( member_list_nat @ X3 @ A )
=> ( ord_le6045566169113846134st_nat @ A @ bot_bot_set_list_nat ) ) ).
% subset_emptyI
thf(fact_1092_subset__emptyI,axiom,
! [A: set_nat] :
( ! [X3: nat] :
~ ( member_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ A @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_1093_Collect__restrict,axiom,
! [X6: set_list_nat,P: list_nat > $o] :
( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X: list_nat] :
( ( member_list_nat @ X @ X6 )
& ( P @ X ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_1094_Collect__restrict,axiom,
! [X6: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ X6 )
& ( P @ X ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_1095_prop__restrict,axiom,
! [X2: list_nat,Z4: set_list_nat,X6: set_list_nat,P: list_nat > $o] :
( ( member_list_nat @ X2 @ Z4 )
=> ( ( ord_le6045566169113846134st_nat @ Z4
@ ( collect_list_nat
@ ^ [X: list_nat] :
( ( member_list_nat @ X @ X6 )
& ( P @ X ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_1096_prop__restrict,axiom,
! [X2: nat,Z4: set_nat,X6: set_nat,P: nat > $o] :
( ( member_nat @ X2 @ Z4 )
=> ( ( ord_less_eq_set_nat @ Z4
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ X6 )
& ( P @ X ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_1097_stirling_Oelims,axiom,
! [X2: nat,Xa: nat,Y3: nat] :
( ( ( stirling2 @ X2 @ Xa )
= Y3 )
=> ( ( ( X2 = zero_zero_nat )
=> ( ( Xa = zero_zero_nat )
=> ( Y3 != one_one_nat ) ) )
=> ( ( ( X2 = zero_zero_nat )
=> ( ? [K2: nat] :
( Xa
= ( suc @ K2 ) )
=> ( Y3 != zero_zero_nat ) ) )
=> ( ( ? [N2: nat] :
( X2
= ( suc @ N2 ) )
=> ( ( Xa = zero_zero_nat )
=> ( Y3 != zero_zero_nat ) ) )
=> ~ ! [N2: nat] :
( ( X2
= ( suc @ N2 ) )
=> ! [K2: nat] :
( ( Xa
= ( suc @ K2 ) )
=> ( Y3
!= ( plus_plus_nat @ ( times_times_nat @ N2 @ ( stirling2 @ N2 @ ( suc @ K2 ) ) ) @ ( stirling2 @ N2 @ K2 ) ) ) ) ) ) ) ) ) ).
% stirling.elims
thf(fact_1098_complete__interval,axiom,
! [A2: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A2 @ B )
=> ( ( P @ A2 )
=> ( ~ ( P @ B )
=> ? [C4: nat] :
( ( ord_less_eq_nat @ A2 @ C4 )
& ( ord_less_eq_nat @ C4 @ B )
& ! [X7: nat] :
( ( ( ord_less_eq_nat @ A2 @ X7 )
& ( ord_less_nat @ X7 @ C4 ) )
=> ( P @ X7 ) )
& ! [D3: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A2 @ X3 )
& ( ord_less_nat @ X3 @ D3 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D3 @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1099_stirling__same,axiom,
! [N: nat] :
( ( stirling2 @ N @ N )
= one_one_nat ) ).
% stirling_same
thf(fact_1100_stirling__less,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( stirling2 @ N @ K )
= zero_zero_nat ) ) ).
% stirling_less
thf(fact_1101_stirling__0,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( stirling2 @ N @ zero_zero_nat )
= zero_zero_nat ) ) ).
% stirling_0
thf(fact_1102_stirling_Osimps_I3_J,axiom,
! [N: nat] :
( ( stirling2 @ ( suc @ N ) @ zero_zero_nat )
= zero_zero_nat ) ).
% stirling.simps(3)
thf(fact_1103_stirling_Osimps_I2_J,axiom,
! [K: nat] :
( ( stirling2 @ zero_zero_nat @ ( suc @ K ) )
= zero_zero_nat ) ).
% stirling.simps(2)
thf(fact_1104_stirling_Osimps_I1_J,axiom,
( ( stirling2 @ zero_zero_nat @ zero_zero_nat )
= one_one_nat ) ).
% stirling.simps(1)
thf(fact_1105_stirling_Osimps_I4_J,axiom,
! [N: nat,K: nat] :
( ( stirling2 @ ( suc @ N ) @ ( suc @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ N @ ( stirling2 @ N @ ( suc @ K ) ) ) @ ( stirling2 @ N @ K ) ) ) ).
% stirling.simps(4)
thf(fact_1106_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X7 ) ) ).
% minf(8)
thf(fact_1107_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ( ord_less_eq_nat @ X7 @ T ) ) ).
% minf(6)
thf(fact_1108_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ~ ( ord_less_nat @ T @ X7 ) ) ).
% minf(7)
thf(fact_1109_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ( ord_less_nat @ X7 @ T ) ) ).
% minf(5)
thf(fact_1110_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ( X7 != T ) ) ).
% minf(4)
thf(fact_1111_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ( X7 != T ) ) ).
% minf(3)
thf(fact_1112_minf_I2_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q4: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P6 @ X7 )
| ( Q4 @ X7 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1113_minf_I1_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q4: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P6 @ X7 )
& ( Q4 @ X7 ) ) ) ) ) ) ).
% minf(1)
thf(fact_1114_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ( ord_less_nat @ T @ X7 ) ) ).
% pinf(7)
thf(fact_1115_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ~ ( ord_less_nat @ X7 @ T ) ) ).
% pinf(5)
thf(fact_1116_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ( X7 != T ) ) ).
% pinf(4)
thf(fact_1117_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ( X7 != T ) ) ).
% pinf(3)
thf(fact_1118_pinf_I2_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q4: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P6 @ X7 )
| ( Q4 @ X7 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1119_pinf_I1_J,axiom,
! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q4: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q4 @ X3 ) ) )
=> ? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P6 @ X7 )
& ( Q4 @ X7 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1120_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ~ ( ord_less_eq_nat @ X7 @ T ) ) ).
% pinf(6)
thf(fact_1121_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ( ord_less_eq_nat @ T @ X7 ) ) ).
% pinf(8)
thf(fact_1122_Gcd__0__iff,axiom,
! [A: set_nat] :
( ( ( gcd_Gcd_nat @ A )
= zero_zero_nat )
= ( ord_less_eq_set_nat @ A @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% Gcd_0_iff
thf(fact_1123_card__insert__le__m1,axiom,
! [N: nat,Y3: set_nat,X2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ Y3 ) @ ( minus_minus_nat @ N @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( insert_nat @ X2 @ Y3 ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_1124_card__insert__le__m1,axiom,
! [N: nat,Y3: set_list_nat,X2: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( finite_card_list_nat @ Y3 ) @ ( minus_minus_nat @ N @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite_card_list_nat @ ( insert_list_nat @ X2 @ Y3 ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_1125_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ A2 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1126_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_1127_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_1128_add__diff__cancel__right_H,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_1129_add__diff__cancel__right,axiom,
! [A2: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A2 @ B ) ) ).
% add_diff_cancel_right
thf(fact_1130_add__diff__cancel__left_H,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ A2 )
= B ) ).
% add_diff_cancel_left'
thf(fact_1131_add__diff__cancel__left,axiom,
! [C: nat,A2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A2 @ B ) ) ).
% add_diff_cancel_left
thf(fact_1132_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_1133_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_1134_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1135_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1136_diff__diff__cancel,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_1137_diff__diff__left,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1138_le__add__diff__inverse2,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A2 @ B ) @ B )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_1139_le__add__diff__inverse,axiom,
! [B: nat,A2: nat] :
( ( ord_less_eq_nat @ B @ A2 )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A2 @ B ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_1140_diff__add__zero,axiom,
! [A2: nat,B: nat] :
( ( minus_minus_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_1141_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_1142_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_1143_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_1144_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1145_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1146_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1147_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1148_Gcd__empty,axiom,
( ( gcd_Gcd_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Gcd_empty
thf(fact_1149_card__atLeastLessThan,axiom,
! [L: nat,U2: nat] :
( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L @ U2 ) )
= ( minus_minus_nat @ U2 @ L ) ) ).
% card_atLeastLessThan
thf(fact_1150_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_1151_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I2 @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1152_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I2 )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1153_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_1154_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_1155_diff__less__mono,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1156_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_1157_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1158_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1159_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_1160_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_1161_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_1162_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_1163_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_1164_le__diff__conv,axiom,
! [J: nat,K: nat,I2: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ).
% le_diff_conv
thf(fact_1165_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1166_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
= ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1167_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1168_Nat_Ole__imp__diff__is__add,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ( minus_minus_nat @ J @ I2 )
= K )
= ( J
= ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1169_add__le__imp__le__diff,axiom,
! [I2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1170_add__le__add__imp__diff__le,axiom,
! [I2: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ 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_1171_diff__add,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ A2 )
= B ) ) ).
% diff_add
thf(fact_1172_le__add__diff,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 ) ) ) ).
% le_add_diff
thf(fact_1173_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1174_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1175_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A2 )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1176_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1177_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1178_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1179_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A2: nat,B: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( plus_plus_nat @ A2 @ ( minus_minus_nat @ B @ A2 ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1180_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A2: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ord_less_eq_nat @ A2 @ B )
=> ( ( ( minus_minus_nat @ B @ A2 )
= C )
= ( B
= ( plus_plus_nat @ C @ A2 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1181_diff__shunt__var,axiom,
! [X2: set_list_nat,Y3: set_list_nat] :
( ( ( minus_7954133019191499631st_nat @ X2 @ Y3 )
= bot_bot_set_list_nat )
= ( ord_le6045566169113846134st_nat @ X2 @ Y3 ) ) ).
% diff_shunt_var
thf(fact_1182_diff__shunt__var,axiom,
! [X2: set_nat,Y3: set_nat] :
( ( ( minus_minus_set_nat @ X2 @ Y3 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X2 @ Y3 ) ) ).
% diff_shunt_var
thf(fact_1183_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I2: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I2 ) ) ) ) ).
% zero_induct_lemma
thf(fact_1184_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_1185_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1186_diff__right__commute,axiom,
! [A2: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_1187_diff__commute,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J ) ) ).
% diff_commute
thf(fact_1188_diff__diff__eq,axiom,
! [A2: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C )
= ( minus_minus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_1189_add__implies__diff,axiom,
! [C: nat,B: nat,A2: nat] :
( ( ( plus_plus_nat @ C @ B )
= A2 )
=> ( C
= ( minus_minus_nat @ A2 @ B ) ) ) ).
% add_implies_diff
thf(fact_1190_right__diff__distrib_H,axiom,
! [A2: nat,B: nat,C: nat] :
( ( times_times_nat @ A2 @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A2 @ B ) @ ( times_times_nat @ A2 @ C ) ) ) ).
% right_diff_distrib'
thf(fact_1191_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A2: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A2 )
= ( minus_minus_nat @ ( times_times_nat @ B @ A2 ) @ ( times_times_nat @ C @ A2 ) ) ) ).
% left_diff_distrib'
thf(fact_1192_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_1193_less__diff__conv,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1194_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_1195_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_1196_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_1197_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_1198_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1199_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A2: nat,B: nat] :
( ~ ( ord_less_nat @ A2 @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A2 @ B ) )
= A2 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1200_Gcd__1,axiom,
! [A: set_nat] :
( ( member_nat @ one_one_nat @ A )
=> ( ( gcd_Gcd_nat @ A )
= one_one_nat ) ) ).
% Gcd_1
thf(fact_1201_Gcd__nat__eq__one,axiom,
! [N5: set_nat] :
( ( member_nat @ one_one_nat @ N5 )
=> ( ( gcd_Gcd_nat @ N5 )
= one_one_nat ) ) ).
% Gcd_nat_eq_one
thf(fact_1202_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1203_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1204_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1205_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1206_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_1207_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_1208_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_1209_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_1210_diff__Suc__less,axiom,
! [N: nat,I2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1211_nat__diff__split,axiom,
! [P: nat > $o,A2: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B ) )
= ( ( ( ord_less_nat @ A2 @ B )
=> ( P @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A2
= ( plus_plus_nat @ B @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_1212_nat__diff__split__asm,axiom,
! [P: nat > $o,A2: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B ) )
= ( ~ ( ( ( ord_less_nat @ A2 @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A2
= ( plus_plus_nat @ B @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1213_less__diff__conv2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1214_nat__diff__add__eq2,axiom,
! [I2: nat,J: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U2 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U2 ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1215_nat__diff__add__eq1,axiom,
! [J: nat,I2: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U2 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U2 ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1216_nat__le__add__iff2,axiom,
! [I2: nat,J: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U2 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U2 ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1217_nat__le__add__iff1,axiom,
! [J: nat,I2: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U2 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U2 ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1218_nat__eq__add__iff2,axiom,
! [I2: nat,J: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U2 ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U2 ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1219_nat__eq__add__iff1,axiom,
! [J: nat,I2: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U2 ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U2 ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_1220_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_1221_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_1222_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% add_eq_if
thf(fact_1223_nat__less__add__iff1,axiom,
! [J: nat,I2: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U2 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U2 ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1224_nat__less__add__iff2,axiom,
! [I2: nat,J: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U2 ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U2 ) @ N ) )
= ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U2 ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1225_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1226_card__length__sum__list__rec,axiom,
! [M: nat,N5: nat] :
( ( ord_less_eq_nat @ one_one_nat @ M )
=> ( ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L3: list_nat] :
( ( ( size_size_list_nat @ L3 )
= M )
& ( ( groups4561878855575611511st_nat @ L3 )
= N5 ) ) ) )
= ( plus_plus_nat
@ ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L3: list_nat] :
( ( ( size_size_list_nat @ L3 )
= ( minus_minus_nat @ M @ one_one_nat ) )
& ( ( groups4561878855575611511st_nat @ L3 )
= N5 ) ) ) )
@ ( finite_card_list_nat
@ ( collect_list_nat
@ ^ [L3: list_nat] :
( ( ( size_size_list_nat @ L3 )
= M )
& ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L3 ) @ one_one_nat )
= N5 ) ) ) ) ) ) ) ).
% card_length_sum_list_rec
thf(fact_1227_image__minus__const__atLeastLessThan__nat,axiom,
! [C: nat,Y3: nat,X2: nat] :
( ( ( ord_less_nat @ C @ Y3 )
=> ( ( image_nat_nat
@ ^ [I: nat] : ( minus_minus_nat @ I @ C )
@ ( set_or4665077453230672383an_nat @ X2 @ Y3 ) )
= ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X2 @ C ) @ ( minus_minus_nat @ Y3 @ C ) ) ) )
& ( ~ ( ord_less_nat @ C @ Y3 )
=> ( ( ( ord_less_nat @ X2 @ Y3 )
=> ( ( image_nat_nat
@ ^ [I: nat] : ( minus_minus_nat @ I @ C )
@ ( set_or4665077453230672383an_nat @ X2 @ Y3 ) )
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
& ( ~ ( ord_less_nat @ X2 @ Y3 )
=> ( ( image_nat_nat
@ ^ [I: nat] : ( minus_minus_nat @ I @ C )
@ ( set_or4665077453230672383an_nat @ X2 @ Y3 ) )
= bot_bot_set_nat ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_1228_image__eqI,axiom,
! [B: nat,F: nat > nat,X2: nat,A: set_nat] :
( ( B
= ( F @ X2 ) )
=> ( ( member_nat @ X2 @ A )
=> ( member_nat @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_1229_Diff__cancel,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ A )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_1230_Diff__cancel,axiom,
! [A: set_list_nat] :
( ( minus_7954133019191499631st_nat @ A @ A )
= bot_bot_set_list_nat ) ).
% Diff_cancel
thf(fact_1231_empty__Diff,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_1232_empty__Diff,axiom,
! [A: set_list_nat] :
( ( minus_7954133019191499631st_nat @ bot_bot_set_list_nat @ A )
= bot_bot_set_list_nat ) ).
% empty_Diff
thf(fact_1233_Diff__empty,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Diff_empty
thf(fact_1234_Diff__empty,axiom,
! [A: set_list_nat] :
( ( minus_7954133019191499631st_nat @ A @ bot_bot_set_list_nat )
= A ) ).
% Diff_empty
thf(fact_1235_insert__Diff1,axiom,
! [X2: nat,B2: set_nat,A: set_nat] :
( ( member_nat @ X2 @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A ) @ B2 )
= ( minus_minus_set_nat @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_1236_Diff__insert0,axiom,
! [X2: nat,A: set_nat,B2: set_nat] :
( ~ ( member_nat @ X2 @ A )
=> ( ( minus_minus_set_nat @ A @ ( insert_nat @ X2 @ B2 ) )
= ( minus_minus_set_nat @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1237_image__ident,axiom,
! [Y6: set_nat] :
( ( image_nat_nat
@ ^ [X: nat] : X
@ Y6 )
= Y6 ) ).
% image_ident
thf(fact_1238_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_1239_image__empty,axiom,
! [F: nat > list_nat] :
( ( image_nat_list_nat @ F @ bot_bot_set_nat )
= bot_bot_set_list_nat ) ).
% image_empty
thf(fact_1240_image__empty,axiom,
! [F: list_nat > nat] :
( ( image_list_nat_nat @ F @ bot_bot_set_list_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_1241_image__empty,axiom,
! [F: list_nat > list_nat] :
( ( image_7976474329151083847st_nat @ F @ bot_bot_set_list_nat )
= bot_bot_set_list_nat ) ).
% image_empty
thf(fact_1242_empty__is__image,axiom,
! [F: nat > nat,A: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_1243_empty__is__image,axiom,
! [F: list_nat > nat,A: set_list_nat] :
( ( bot_bot_set_nat
= ( image_list_nat_nat @ F @ A ) )
= ( A = bot_bot_set_list_nat ) ) ).
% empty_is_image
thf(fact_1244_empty__is__image,axiom,
! [F: nat > list_nat,A: set_nat] :
( ( bot_bot_set_list_nat
= ( image_nat_list_nat @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_1245_empty__is__image,axiom,
! [F: list_nat > list_nat,A: set_list_nat] :
( ( bot_bot_set_list_nat
= ( image_7976474329151083847st_nat @ F @ A ) )
= ( A = bot_bot_set_list_nat ) ) ).
% empty_is_image
thf(fact_1246_image__is__empty,axiom,
! [F: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_1247_image__is__empty,axiom,
! [F: list_nat > nat,A: set_list_nat] :
( ( ( image_list_nat_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_list_nat ) ) ).
% image_is_empty
thf(fact_1248_image__is__empty,axiom,
! [F: nat > list_nat,A: set_nat] :
( ( ( image_nat_list_nat @ F @ A )
= bot_bot_set_list_nat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_1249_image__is__empty,axiom,
! [F: list_nat > list_nat,A: set_list_nat] :
( ( ( image_7976474329151083847st_nat @ F @ A )
= bot_bot_set_list_nat )
= ( A = bot_bot_set_list_nat ) ) ).
% image_is_empty
thf(fact_1250_Diff__eq__empty__iff,axiom,
! [A: set_list_nat,B2: set_list_nat] :
( ( ( minus_7954133019191499631st_nat @ A @ B2 )
= bot_bot_set_list_nat )
= ( ord_le6045566169113846134st_nat @ A @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_1251_Diff__eq__empty__iff,axiom,
! [A: set_nat,B2: set_nat] :
( ( ( minus_minus_set_nat @ A @ B2 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_1252_image__insert,axiom,
! [F: nat > nat,A2: nat,B2: set_nat] :
( ( image_nat_nat @ F @ ( insert_nat @ A2 @ B2 ) )
= ( insert_nat @ ( F @ A2 ) @ ( image_nat_nat @ F @ B2 ) ) ) ).
% image_insert
thf(fact_1253_insert__image,axiom,
! [X2: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X2 @ A )
=> ( ( insert_nat @ ( F @ X2 ) @ ( image_nat_nat @ F @ A ) )
= ( image_nat_nat @ F @ A ) ) ) ).
% insert_image
thf(fact_1254_insert__Diff__single,axiom,
! [A2: nat,A: set_nat] :
( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= ( insert_nat @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_1255_insert__Diff__single,axiom,
! [A2: list_nat,A: set_list_nat] :
( ( insert_list_nat @ A2 @ ( minus_7954133019191499631st_nat @ A @ ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) ) )
= ( insert_list_nat @ A2 @ A ) ) ).
% insert_Diff_single
thf(fact_1256_ivl__diff,axiom,
! [I2: nat,N: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ N )
=> ( ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ I2 @ M ) @ ( set_or4665077453230672383an_nat @ I2 @ N ) )
= ( set_or4665077453230672383an_nat @ N @ M ) ) ) ).
% ivl_diff
thf(fact_1257_Diff__disjoint,axiom,
! [A: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ A @ ( minus_minus_set_nat @ B2 @ A ) )
= bot_bot_set_nat ) ).
% Diff_disjoint
thf(fact_1258_Diff__disjoint,axiom,
! [A: set_list_nat,B2: set_list_nat] :
( ( inf_inf_set_list_nat @ A @ ( minus_7954133019191499631st_nat @ B2 @ A ) )
= bot_bot_set_list_nat ) ).
% Diff_disjoint
thf(fact_1259_image__add__0,axiom,
! [S2: set_nat] :
( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S2 )
= S2 ) ).
% image_add_0
thf(fact_1260_image__add__atLeastLessThan,axiom,
! [K: nat,I2: nat,J: nat] :
( ( image_nat_nat @ ( plus_plus_nat @ K ) @ ( set_or4665077453230672383an_nat @ I2 @ J ) )
= ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% image_add_atLeastLessThan
thf(fact_1261_image__Suc__atLeastLessThan,axiom,
! [I2: nat,J: nat] :
( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I2 @ J ) )
= ( set_or4665077453230672383an_nat @ ( suc @ I2 ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastLessThan
thf(fact_1262_image__add__atLeastLessThan_H,axiom,
! [K: nat,I2: nat,J: nat] :
( ( image_nat_nat
@ ^ [N3: nat] : ( plus_plus_nat @ N3 @ K )
@ ( set_or4665077453230672383an_nat @ I2 @ J ) )
= ( set_or4665077453230672383an_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% image_add_atLeastLessThan'
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y3: nat] :
( ( if_nat @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y3: nat] :
( ( if_nat @ $true @ X2 @ Y3 )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( count_list_list_nat @ ( equiva7426478223624825838m_rgfs @ ( size_size_list_nat @ xs ) ) @ xs )
= ( groups4561878855575611511st_nat
@ ( map_list_nat_nat
@ ^ [R1: list_nat] : ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( xs = R1 ) ) @ ( finite_card_nat @ ( inf_inf_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( plus_plus_nat @ ( equiva5889994315859557365_limit @ xs ) @ one_one_nat ) ) @ ( insert_nat @ xa @ bot_bot_set_nat ) ) ) )
@ ( equiva7426478223624825838m_rgfs @ ( size_size_list_nat @ xs ) ) ) ) ) ).
%------------------------------------------------------------------------------