TPTP Problem File: SLH0021^1.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 : Eval_FO/0005_Ailamazyan/prob_04784_205781__16280046_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1371 ( 591 unt; 101 typ; 0 def)
% Number of atoms : 3325 (1133 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 10044 ( 327 ~; 52 |; 202 &;8091 @)
% ( 0 <=>;1372 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 6 avg)
% Number of types : 10 ( 9 usr)
% Number of type conns : 327 ( 327 >; 0 *; 0 +; 0 <<)
% Number of symbols : 93 ( 92 usr; 13 con; 0-6 aty)
% Number of variables : 3276 ( 193 ^;3004 !; 79 ?;3276 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:13:08.791
%------------------------------------------------------------------------------
% Could-be-implicit typings (9)
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
set_list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_list_set_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
list_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (92)
thf(sy_c_Ailamazyan_Oadd__nth_001t__Nat__Onat,type,
add_nth_nat: nat > nat > list_nat > list_nat ).
thf(sy_c_Ailamazyan_Oall__tuples_001t__List__Olist_It__Nat__Onat_J,type,
all_tuples_list_nat: set_list_nat > nat > set_list_list_nat ).
thf(sy_c_Ailamazyan_Oall__tuples_001t__Nat__Onat,type,
all_tuples_nat: set_nat > nat > set_list_nat ).
thf(sy_c_Ailamazyan_Oall__tuples_001t__Set__Oset_It__Nat__Onat_J,type,
all_tuples_set_nat: set_set_nat > nat > set_list_set_nat ).
thf(sy_c_Ailamazyan_Orem__nth_001t__Nat__Onat,type,
rem_nth_nat: nat > list_nat > list_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_Finite__Set_Ofinite_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
finite8170528100393595399st_nat: set_list_list_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Nat__Onat_J,type,
finite8100373058378681591st_nat: set_list_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
finite1091814263879798189et_nat: set_list_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
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_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_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_Eo_J,type,
sup_sup_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
sup_sup_set_list_nat: set_list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
sup_sup_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Nat__Onat,type,
lattic7446932960582359483at_nat: ( nat > nat ) > set_nat > nat ).
thf(sy_c_List_Oappend_001t__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Odistinct_001t__List__Olist_It__Nat__Onat_J,type,
distinct_list_nat: list_list_nat > $o ).
thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
distinct_nat: list_nat > $o ).
thf(sy_c_List_Odistinct_001t__Set__Oset_It__Nat__Onat_J,type,
distinct_set_nat: list_set_nat > $o ).
thf(sy_c_List_Ogen__length_001t__Nat__Onat,type,
gen_length_nat: nat > list_nat > nat ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
linord2614967742042102400et_nat: set_nat > list_nat ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
set_list_nat2: list_list_nat > set_list_nat ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
set_set_nat2: list_set_nat > set_set_nat ).
thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
list_update_nat: list_nat > nat > nat > list_nat ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Oremove1_001t__Nat__Onat,type,
remove1_nat: nat > list_nat > list_nat ).
thf(sy_c_List_OremoveAll_001t__Nat__Onat,type,
removeAll_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Orev_001t__Nat__Onat,type,
rev_nat: list_nat > list_nat ).
thf(sy_c_List_Orotate1_001t__Nat__Onat,type,
rotate1_nat: list_nat > list_nat ).
thf(sy_c_List_Orotate_001t__Nat__Onat,type,
rotate_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Osorted__wrt_001t__List__Olist_It__Nat__Onat_J,type,
sorted_wrt_list_nat: ( list_nat > list_nat > $o ) > list_list_nat > $o ).
thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
thf(sy_c_List_Osorted__wrt_001t__Set__Oset_It__Nat__Onat_J,type,
sorted_wrt_set_nat: ( set_nat > set_nat > $o ) > list_set_nat > $o ).
thf(sy_c_List_Osplice_001t__Nat__Onat,type,
splice_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Ounion_001t__Nat__Onat,type,
union_nat: list_nat > list_nat > list_nat ).
thf(sy_c_Nat_OSuc,type,
suc: 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_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_001_Eo,type,
bot_bot_o: $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_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__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > 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_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Nat__Onat_J,type,
order_5724808138429204845et_nat: ( set_nat > $o ) > set_nat ).
thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
modulo_modulo_nat: nat > nat > 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_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set__Impl_Oord__class_Oquicksort_001t__Nat__Onat,type,
set_or9089632773640736191rt_nat: list_nat > list_nat ).
thf(sy_c_Set__Impl_Oord__class_Oquicksort__acc_001t__Nat__Onat,type,
set_or5558937660843164036cc_nat: list_nat > list_nat > list_nat ).
thf(sy_c_Set__Impl_Oord__class_Oquicksort__part_001t__Nat__Onat,type,
set_or1804217446461887602rt_nat: list_nat > nat > list_nat > list_nat > list_nat > list_nat > list_nat ).
thf(sy_c_Set__Impl_Oord__class_Oremdups__sorted_001t__Nat__Onat,type,
set_or6599480164596245535ed_nat: list_nat > list_nat ).
thf(sy_c_Set__Impl_Oord__class_Osorted__list__subset_001t__Nat__Onat,type,
set_or6742139631805365739et_nat: ( nat > nat > $o ) > list_nat > list_nat > $o ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
set_or6659071591806873216st_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or7074010630789208630et_nat: set_nat > set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
set_or5834768355832116004an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Set__Oset_It__Nat__Onat_J,type,
set_or8625682525731655386et_nat: set_nat > set_nat > set_set_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_both____,type,
both: list_nat ).
thf(sy_v_ns_092_060phi_062_H____,type,
ns_phi: list_nat ).
thf(sy_v_ns_092_060phi_062____,type,
ns_phi2: list_nat ).
thf(sy_v_ns_092_060psi_062_H____,type,
ns_psi: list_nat ).
thf(sy_v_ns_092_060psi_062____,type,
ns_psi2: list_nat ).
thf(sy_v_ns____,type,
ns: list_nat ).
% Relevant facts (1269)
thf(fact_0_ns__sd_I1_J,axiom,
( ( sorted_wrt_nat @ ord_less_eq_nat @ ns )
& ( distinct_nat @ ns ) ) ).
% ns_sd(1)
thf(fact_1_ns__sd_H,axiom,
( ( sorted_wrt_nat @ ord_less_eq_nat @ ns_psi )
& ( distinct_nat @ ns_psi ) ) ).
% ns_sd'
thf(fact_2_aux_I3_J,axiom,
( ( sorted_wrt_nat @ ord_less_eq_nat @ both )
& ( distinct_nat @ both ) ) ).
% aux(3)
thf(fact_3_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_4_order__refl,axiom,
! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).
% order_refl
thf(fact_5_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_6_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_7_aux_I2_J,axiom,
( ( sorted_wrt_nat @ ord_less_eq_nat @ ns_phi )
& ( distinct_nat @ ns_phi ) ) ).
% aux(2)
thf(fact_8_ns__sd_I3_J,axiom,
( ( sorted_wrt_nat @ ord_less_eq_nat @ ns_psi2 )
& ( distinct_nat @ ns_psi2 ) ) ).
% ns_sd(3)
thf(fact_9_ns__sd_I2_J,axiom,
( ( sorted_wrt_nat @ ord_less_eq_nat @ ns_phi2 )
& ( distinct_nat @ ns_phi2 ) ) ).
% ns_sd(2)
thf(fact_10_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ Xs )
=> ( distinct_nat @ Xs ) ) ).
% sorted_list_of_set.distinct_if_distinct_map
thf(fact_11_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_12_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_13_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_14_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_15_order__antisym__conv,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_16_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_17_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_18_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_19_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_20_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_21_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_22_ord__eq__le__subst,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_23_ord__eq__le__subst,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_24_ord__eq__le__subst,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_25_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_26_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_27_order__eq__refl,axiom,
! [X: set_nat,Y: set_nat] :
( ( X = Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_28_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_29_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_30_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_31_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_32_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_33_order__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_34_order__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_35_order__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_36_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_37_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_38_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_39_antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_40_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_41_dual__order_Otrans,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_42_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_43_dual__order_Oantisym,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_44_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_45_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
& ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_46_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_47_order__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_48_order__trans,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z2 )
=> ( ord_less_eq_set_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_49_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_50_order_Otrans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_51_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_52_order__antisym,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_53_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_54_ord__le__eq__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_55_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_56_ord__eq__le__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( A = B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_57_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_58_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
& ( ord_less_eq_set_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_59_le__cases3,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_60_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_61_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_62_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_63_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_64_ns__sd_I8_J,axiom,
ord_less_eq_set_nat @ ( set_nat2 @ ns_psi2 ) @ ( set_nat2 @ both ) ).
% ns_sd(8)
thf(fact_65_ns__sd_I6_J,axiom,
ord_less_eq_set_nat @ ( set_nat2 @ ns ) @ ( set_nat2 @ both ) ).
% ns_sd(6)
thf(fact_66_ns__sd_I7_J,axiom,
ord_less_eq_set_nat @ ( set_nat2 @ ns_phi ) @ ( set_nat2 @ ns_psi2 ) ).
% ns_sd(7)
thf(fact_67_mem__Collect__eq,axiom,
! [A: list_nat,P: list_nat > $o] :
( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_68_mem__Collect__eq,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_69_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_70_Collect__mem__eq,axiom,
! [A4: set_list_nat] :
( ( collect_list_nat
@ ^ [X3: list_nat] : ( member_list_nat @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_71_Collect__mem__eq,axiom,
! [A4: set_set_nat] :
( ( collect_set_nat
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_72_Collect__mem__eq,axiom,
! [A4: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_73_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_74_ns__sd_I5_J,axiom,
ord_less_eq_set_nat @ ( set_nat2 @ ns ) @ ( set_nat2 @ ns_psi2 ) ).
% ns_sd(5)
thf(fact_75_ns__sd_I4_J,axiom,
ord_less_eq_set_nat @ ( set_nat2 @ ns ) @ ( set_nat2 @ ns_phi2 ) ).
% ns_sd(4)
thf(fact_76_distinct__union,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( distinct_nat @ ( union_nat @ Xs @ Ys ) )
= ( distinct_nat @ Ys ) ) ).
% distinct_union
thf(fact_77_aux6_I2_J,axiom,
( ( sup_sup_set_nat @ ( set_nat2 @ ns_psi2 ) @ ( set_nat2 @ ns_psi ) )
= ( set_nat2 @ both ) ) ).
% aux6(2)
thf(fact_78_len__ns_092_060psi_062,axiom,
( ( plus_plus_nat @ ( size_size_list_nat @ ns ) @ ( size_size_list_nat @ ns_phi ) )
= ( size_size_list_nat @ ns_psi2 ) ) ).
% len_ns\<psi>
thf(fact_79_aux_I5_J,axiom,
( ( sup_sup_set_nat @ ( set_nat2 @ ns_phi2 ) @ ( set_nat2 @ ns_phi ) )
= ( set_nat2 @ both ) ) ).
% aux(5)
thf(fact_80_aux3_I2_J,axiom,
( ( sup_sup_set_nat @ ( set_nat2 @ ns_phi ) @ ( set_nat2 @ ns ) )
= ( set_nat2 @ ns_psi2 ) ) ).
% aux3(2)
thf(fact_81_aux4_I2_J,axiom,
( ( sup_sup_set_nat @ ( set_nat2 @ ns ) @ ( set_nat2 @ ns_phi ) )
= ( set_nat2 @ ns_psi2 ) ) ).
% aux4(2)
thf(fact_82_Greatest__equality,axiom,
! [P: set_nat > $o,X: set_nat] :
( ( P @ X )
=> ( ! [Y2: set_nat] :
( ( P @ Y2 )
=> ( ord_less_eq_set_nat @ Y2 @ X ) )
=> ( ( order_5724808138429204845et_nat @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_83_Greatest__equality,axiom,
! [P: nat > $o,X: nat] :
( ( P @ X )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ( order_Greatest_nat @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_84_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_85_set__union,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( set_nat2 @ ( union_nat @ Xs @ Ys ) )
= ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) ) ).
% set_union
thf(fact_86_subset__code_I1_J,axiom,
! [Xs: list_list_nat,B4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ B4 )
= ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
=> ( member_list_nat @ X3 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_87_subset__code_I1_J,axiom,
! [Xs: list_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ B4 )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs ) )
=> ( member_set_nat @ X3 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_88_subset__code_I1_J,axiom,
! [Xs: list_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B4 )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X3 @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_89_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_nat] :
( ( size_size_list_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_90_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_91_size__neq__size__imp__neq,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( size_size_list_nat @ X )
!= ( size_size_list_nat @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_92_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_93_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_94_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_95_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_96_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_97_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_98_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_99_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_100_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_101_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_102_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N3: nat] :
? [K2: nat] :
( N3
= ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_103_sorted__wrt__mono__rel,axiom,
! [Xs: list_list_nat,P: list_nat > list_nat > $o,Q: list_nat > list_nat > $o] :
( ! [X2: list_nat,Y2: list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs ) )
=> ( ( member_list_nat @ Y2 @ ( set_list_nat2 @ Xs ) )
=> ( ( P @ X2 @ Y2 )
=> ( Q @ X2 @ Y2 ) ) ) )
=> ( ( sorted_wrt_list_nat @ P @ Xs )
=> ( sorted_wrt_list_nat @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_104_sorted__wrt__mono__rel,axiom,
! [Xs: list_set_nat,P: set_nat > set_nat > $o,Q: set_nat > set_nat > $o] :
( ! [X2: set_nat,Y2: set_nat] :
( ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs ) )
=> ( ( member_set_nat @ Y2 @ ( set_set_nat2 @ Xs ) )
=> ( ( P @ X2 @ Y2 )
=> ( Q @ X2 @ Y2 ) ) ) )
=> ( ( sorted_wrt_set_nat @ P @ Xs )
=> ( sorted_wrt_set_nat @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_105_sorted__wrt__mono__rel,axiom,
! [Xs: list_nat,P: nat > nat > $o,Q: nat > nat > $o] :
( ! [X2: nat,Y2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( member_nat @ Y2 @ ( set_nat2 @ Xs ) )
=> ( ( P @ X2 @ Y2 )
=> ( Q @ X2 @ Y2 ) ) ) )
=> ( ( sorted_wrt_nat @ P @ Xs )
=> ( sorted_wrt_nat @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_106_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_107_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_108_GreatestI__ex__nat,axiom,
! [P: nat > $o,B: nat] :
( ? [X_1: nat] : ( P @ X_1 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_109_sorted__distinct__set__unique,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( distinct_nat @ Xs )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ys )
=> ( ( distinct_nat @ Ys )
=> ( ( ( set_nat2 @ Xs )
= ( set_nat2 @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ).
% sorted_distinct_set_unique
thf(fact_110_GreatestI2__order,axiom,
! [P: set_nat > $o,X: set_nat,Q: set_nat > $o] :
( ( P @ X )
=> ( ! [Y2: set_nat] :
( ( P @ Y2 )
=> ( ord_less_eq_set_nat @ Y2 @ X ) )
=> ( ! [X2: set_nat] :
( ( P @ X2 )
=> ( ! [Y5: set_nat] :
( ( P @ Y5 )
=> ( ord_less_eq_set_nat @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_5724808138429204845et_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_111_GreatestI2__order,axiom,
! [P: nat > $o,X: nat,Q: nat > $o] :
( ( P @ X )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X2 ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_112_Un__subset__iff,axiom,
! [A4: set_nat,B4: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A4 @ B4 ) @ C2 )
= ( ( ord_less_eq_set_nat @ A4 @ C2 )
& ( ord_less_eq_set_nat @ B4 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_113_le__sup__iff,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_nat @ X @ Z2 )
& ( ord_less_eq_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_114_le__sup__iff,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_set_nat @ X @ Z2 )
& ( ord_less_eq_set_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_115_sup_Obounded__iff,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
= ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_116_sup_Obounded__iff,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
= ( ( ord_less_eq_set_nat @ B @ A )
& ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_117_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_118_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_119_Un__iff,axiom,
! [C: list_nat,A4: set_list_nat,B4: set_list_nat] :
( ( member_list_nat @ C @ ( sup_sup_set_list_nat @ A4 @ B4 ) )
= ( ( member_list_nat @ C @ A4 )
| ( member_list_nat @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_120_Un__iff,axiom,
! [C: set_nat,A4: set_set_nat,B4: set_set_nat] :
( ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A4 @ B4 ) )
= ( ( member_set_nat @ C @ A4 )
| ( member_set_nat @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_121_Un__iff,axiom,
! [C: nat,A4: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A4 @ B4 ) )
= ( ( member_nat @ C @ A4 )
| ( member_nat @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_122_UnCI,axiom,
! [C: list_nat,B4: set_list_nat,A4: set_list_nat] :
( ( ~ ( member_list_nat @ C @ B4 )
=> ( member_list_nat @ C @ A4 ) )
=> ( member_list_nat @ C @ ( sup_sup_set_list_nat @ A4 @ B4 ) ) ) ).
% UnCI
thf(fact_123_UnCI,axiom,
! [C: set_nat,B4: set_set_nat,A4: set_set_nat] :
( ( ~ ( member_set_nat @ C @ B4 )
=> ( member_set_nat @ C @ A4 ) )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A4 @ B4 ) ) ) ).
% UnCI
thf(fact_124_UnCI,axiom,
! [C: nat,B4: set_nat,A4: set_nat] :
( ( ~ ( member_nat @ C @ B4 )
=> ( member_nat @ C @ A4 ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A4 @ B4 ) ) ) ).
% UnCI
thf(fact_125_sup_Oright__idem,axiom,
! [A: nat,B: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ A @ B ) @ B )
= ( sup_sup_nat @ A @ B ) ) ).
% sup.right_idem
thf(fact_126_sup_Oright__idem,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ B )
= ( sup_sup_set_nat @ A @ B ) ) ).
% sup.right_idem
thf(fact_127_sup__left__idem,axiom,
! [X: nat,Y: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ X @ Y ) )
= ( sup_sup_nat @ X @ Y ) ) ).
% sup_left_idem
thf(fact_128_sup__left__idem,axiom,
! [X: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
= ( sup_sup_set_nat @ X @ Y ) ) ).
% sup_left_idem
thf(fact_129_sup_Oleft__idem,axiom,
! [A: nat,B: nat] :
( ( sup_sup_nat @ A @ ( sup_sup_nat @ A @ B ) )
= ( sup_sup_nat @ A @ B ) ) ).
% sup.left_idem
thf(fact_130_sup_Oleft__idem,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ A @ B ) ) ).
% sup.left_idem
thf(fact_131_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_132_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_133_subset__antisym,axiom,
! [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% subset_antisym
thf(fact_134_subsetI,axiom,
! [A4: set_list_nat,B4: set_list_nat] :
( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ A4 )
=> ( member_list_nat @ X2 @ B4 ) )
=> ( ord_le6045566169113846134st_nat @ A4 @ B4 ) ) ).
% subsetI
thf(fact_135_subsetI,axiom,
! [A4: set_set_nat,B4: set_set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A4 )
=> ( member_set_nat @ X2 @ B4 ) )
=> ( ord_le6893508408891458716et_nat @ A4 @ B4 ) ) ).
% subsetI
thf(fact_136_subsetI,axiom,
! [A4: set_nat,B4: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A4 )
=> ( member_nat @ X2 @ B4 ) )
=> ( ord_less_eq_set_nat @ A4 @ B4 ) ) ).
% subsetI
thf(fact_137_sup_Oidem,axiom,
! [A: nat] :
( ( sup_sup_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_138_sup_Oidem,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_139_sup__idem,axiom,
! [X: nat] :
( ( sup_sup_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_140_sup__idem,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_141_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_142_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_143_group__cancel_Oadd1,axiom,
! [A4: nat,K: nat,A: nat,B: nat] :
( ( A4
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A4 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_144_group__cancel_Oadd2,axiom,
! [B4: nat,K: nat,B: nat,A: nat] :
( ( B4
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B4 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_145_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_146_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_147_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_148_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_149_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_150_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_151_set__eq__subset,axiom,
( ( ^ [Y3: set_nat,Z: set_nat] : ( Y3 = Z ) )
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_152_subset__trans,axiom,
! [A4: set_nat,B4: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ C2 )
=> ( ord_less_eq_set_nat @ A4 @ C2 ) ) ) ).
% subset_trans
thf(fact_153_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_154_subset__refl,axiom,
! [A4: set_nat] : ( ord_less_eq_set_nat @ A4 @ A4 ) ).
% subset_refl
thf(fact_155_subset__iff,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A5: set_list_nat,B5: set_list_nat] :
! [T: list_nat] :
( ( member_list_nat @ T @ A5 )
=> ( member_list_nat @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_156_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A5: set_set_nat,B5: set_set_nat] :
! [T: set_nat] :
( ( member_set_nat @ T @ A5 )
=> ( member_set_nat @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_157_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A5 )
=> ( member_nat @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_158_Set_OequalityD2,axiom,
! [A4: set_nat,B4: set_nat] :
( ( A4 = B4 )
=> ( ord_less_eq_set_nat @ B4 @ A4 ) ) ).
% Set.equalityD2
thf(fact_159_equalityD1,axiom,
! [A4: set_nat,B4: set_nat] :
( ( A4 = B4 )
=> ( ord_less_eq_set_nat @ A4 @ B4 ) ) ).
% equalityD1
thf(fact_160_subset__eq,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A5: set_list_nat,B5: set_list_nat] :
! [X3: list_nat] :
( ( member_list_nat @ X3 @ A5 )
=> ( member_list_nat @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_161_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A5: set_set_nat,B5: set_set_nat] :
! [X3: set_nat] :
( ( member_set_nat @ X3 @ A5 )
=> ( member_set_nat @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_162_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A5 )
=> ( member_nat @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_163_equalityE,axiom,
! [A4: set_nat,B4: set_nat] :
( ( A4 = B4 )
=> ~ ( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ~ ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ).
% equalityE
thf(fact_164_subsetD,axiom,
! [A4: set_list_nat,B4: set_list_nat,C: list_nat] :
( ( ord_le6045566169113846134st_nat @ A4 @ B4 )
=> ( ( member_list_nat @ C @ A4 )
=> ( member_list_nat @ C @ B4 ) ) ) ).
% subsetD
thf(fact_165_subsetD,axiom,
! [A4: set_set_nat,B4: set_set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
=> ( ( member_set_nat @ C @ A4 )
=> ( member_set_nat @ C @ B4 ) ) ) ).
% subsetD
thf(fact_166_subsetD,axiom,
! [A4: set_nat,B4: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( ( member_nat @ C @ A4 )
=> ( member_nat @ C @ B4 ) ) ) ).
% subsetD
thf(fact_167_in__mono,axiom,
! [A4: set_list_nat,B4: set_list_nat,X: list_nat] :
( ( ord_le6045566169113846134st_nat @ A4 @ B4 )
=> ( ( member_list_nat @ X @ A4 )
=> ( member_list_nat @ X @ B4 ) ) ) ).
% in_mono
thf(fact_168_in__mono,axiom,
! [A4: set_set_nat,B4: set_set_nat,X: set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
=> ( ( member_set_nat @ X @ A4 )
=> ( member_set_nat @ X @ B4 ) ) ) ).
% in_mono
thf(fact_169_in__mono,axiom,
! [A4: set_nat,B4: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( ( member_nat @ X @ A4 )
=> ( member_nat @ X @ B4 ) ) ) ).
% in_mono
thf(fact_170_inf__sup__aci_I8_J,axiom,
! [X: nat,Y: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ X @ Y ) )
= ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_171_inf__sup__aci_I8_J,axiom,
! [X: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
= ( sup_sup_set_nat @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_172_inf__sup__aci_I7_J,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z2 ) )
= ( sup_sup_nat @ Y @ ( sup_sup_nat @ X @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_173_inf__sup__aci_I7_J,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z2 ) )
= ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_174_inf__sup__aci_I6_J,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ X @ Y ) @ Z2 )
= ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_175_inf__sup__aci_I6_J,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z2 )
= ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_176_inf__sup__aci_I5_J,axiom,
( sup_sup_nat
= ( ^ [X3: nat,Y4: nat] : ( sup_sup_nat @ Y4 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_177_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] : ( sup_sup_set_nat @ Y4 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_178_sup_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ A @ B ) @ C )
= ( sup_sup_nat @ A @ ( sup_sup_nat @ B @ C ) ) ) ).
% sup.assoc
thf(fact_179_sup_Oassoc,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C ) ) ) ).
% sup.assoc
thf(fact_180_sup__assoc,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ X @ Y ) @ Z2 )
= ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_181_sup__assoc,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z2 )
= ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_182_sup_Ocommute,axiom,
( sup_sup_nat
= ( ^ [A2: nat,B2: nat] : ( sup_sup_nat @ B2 @ A2 ) ) ) ).
% sup.commute
thf(fact_183_sup_Ocommute,axiom,
( sup_sup_set_nat
= ( ^ [A2: set_nat,B2: set_nat] : ( sup_sup_set_nat @ B2 @ A2 ) ) ) ).
% sup.commute
thf(fact_184_sup__commute,axiom,
( sup_sup_nat
= ( ^ [X3: nat,Y4: nat] : ( sup_sup_nat @ Y4 @ X3 ) ) ) ).
% sup_commute
thf(fact_185_sup__commute,axiom,
( sup_sup_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] : ( sup_sup_set_nat @ Y4 @ X3 ) ) ) ).
% sup_commute
thf(fact_186_sup_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( sup_sup_nat @ B @ ( sup_sup_nat @ A @ C ) )
= ( sup_sup_nat @ A @ ( sup_sup_nat @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_187_sup_Oleft__commute,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A @ C ) )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_188_sup__left__commute,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z2 ) )
= ( sup_sup_nat @ Y @ ( sup_sup_nat @ X @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_189_sup__left__commute,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z2 ) )
= ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_190_UnE,axiom,
! [C: list_nat,A4: set_list_nat,B4: set_list_nat] :
( ( member_list_nat @ C @ ( sup_sup_set_list_nat @ A4 @ B4 ) )
=> ( ~ ( member_list_nat @ C @ A4 )
=> ( member_list_nat @ C @ B4 ) ) ) ).
% UnE
thf(fact_191_UnE,axiom,
! [C: set_nat,A4: set_set_nat,B4: set_set_nat] :
( ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A4 @ B4 ) )
=> ( ~ ( member_set_nat @ C @ A4 )
=> ( member_set_nat @ C @ B4 ) ) ) ).
% UnE
thf(fact_192_UnE,axiom,
! [C: nat,A4: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A4 @ B4 ) )
=> ( ~ ( member_nat @ C @ A4 )
=> ( member_nat @ C @ B4 ) ) ) ).
% UnE
thf(fact_193_UnI1,axiom,
! [C: list_nat,A4: set_list_nat,B4: set_list_nat] :
( ( member_list_nat @ C @ A4 )
=> ( member_list_nat @ C @ ( sup_sup_set_list_nat @ A4 @ B4 ) ) ) ).
% UnI1
thf(fact_194_UnI1,axiom,
! [C: set_nat,A4: set_set_nat,B4: set_set_nat] :
( ( member_set_nat @ C @ A4 )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A4 @ B4 ) ) ) ).
% UnI1
thf(fact_195_UnI1,axiom,
! [C: nat,A4: set_nat,B4: set_nat] :
( ( member_nat @ C @ A4 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A4 @ B4 ) ) ) ).
% UnI1
thf(fact_196_UnI2,axiom,
! [C: list_nat,B4: set_list_nat,A4: set_list_nat] :
( ( member_list_nat @ C @ B4 )
=> ( member_list_nat @ C @ ( sup_sup_set_list_nat @ A4 @ B4 ) ) ) ).
% UnI2
thf(fact_197_UnI2,axiom,
! [C: set_nat,B4: set_set_nat,A4: set_set_nat] :
( ( member_set_nat @ C @ B4 )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A4 @ B4 ) ) ) ).
% UnI2
thf(fact_198_UnI2,axiom,
! [C: nat,B4: set_nat,A4: set_nat] :
( ( member_nat @ C @ B4 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A4 @ B4 ) ) ) ).
% UnI2
thf(fact_199_bex__Un,axiom,
! [A4: set_nat,B4: set_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( sup_sup_set_nat @ A4 @ B4 ) )
& ( P @ X3 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A4 )
& ( P @ X3 ) )
| ? [X3: nat] :
( ( member_nat @ X3 @ B4 )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_200_ball__Un,axiom,
! [A4: set_nat,B4: set_nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( sup_sup_set_nat @ A4 @ B4 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( P @ X3 ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ B4 )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_201_Un__assoc,axiom,
! [A4: set_nat,B4: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A4 @ B4 ) @ C2 )
= ( sup_sup_set_nat @ A4 @ ( sup_sup_set_nat @ B4 @ C2 ) ) ) ).
% Un_assoc
thf(fact_202_Un__absorb,axiom,
! [A4: set_nat] :
( ( sup_sup_set_nat @ A4 @ A4 )
= A4 ) ).
% Un_absorb
thf(fact_203_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A5: set_nat,B5: set_nat] : ( sup_sup_set_nat @ B5 @ A5 ) ) ) ).
% Un_commute
thf(fact_204_Un__left__absorb,axiom,
! [A4: set_nat,B4: set_nat] :
( ( sup_sup_set_nat @ A4 @ ( sup_sup_set_nat @ A4 @ B4 ) )
= ( sup_sup_set_nat @ A4 @ B4 ) ) ).
% Un_left_absorb
thf(fact_205_Un__left__commute,axiom,
! [A4: set_nat,B4: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ A4 @ ( sup_sup_set_nat @ B4 @ C2 ) )
= ( sup_sup_set_nat @ B4 @ ( sup_sup_set_nat @ A4 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_206_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_207_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_208_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
? [C3: nat] :
( B2
= ( plus_plus_nat @ A2 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_209_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_210_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C4: nat] :
( B
!= ( plus_plus_nat @ A @ C4 ) ) ) ).
% less_eqE
thf(fact_211_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_212_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_213_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_214_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_215_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_216_sup_OcoboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_217_sup_OcoboundedI2,axiom,
! [C: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_218_sup_OcoboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_219_sup_OcoboundedI1,axiom,
! [C: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ A )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_220_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ) ).
% sup.absorb_iff2
thf(fact_221_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ B2 )
= B2 ) ) ) ).
% sup.absorb_iff2
thf(fact_222_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ) ).
% sup.absorb_iff1
thf(fact_223_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [B2: set_nat,A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ B2 )
= A2 ) ) ) ).
% sup.absorb_iff1
thf(fact_224_sup_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_225_sup_Ocobounded2,axiom,
! [B: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_226_sup_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_227_sup_Ocobounded1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_228_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( A2
= ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).
% sup.order_iff
thf(fact_229_sup_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [B2: set_nat,A2: set_nat] :
( A2
= ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).
% sup.order_iff
thf(fact_230_sup_OboundedI,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_231_sup_OboundedI,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ A )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_232_sup_OboundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_nat @ B @ A )
=> ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_233_sup_OboundedE,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_set_nat @ B @ A )
=> ~ ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_234_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_235_sup__absorb2,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( sup_sup_set_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_236_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_237_sup__absorb1,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( sup_sup_set_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_238_sup_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_239_sup_Oabsorb2,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_240_sup_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_241_sup_Oabsorb1,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( sup_sup_set_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_242_sup__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ X2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat @ Z3 @ X2 )
=> ( ord_less_eq_nat @ ( F @ Y2 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_243_sup__unique,axiom,
! [F: set_nat > set_nat > set_nat,X: set_nat,Y: set_nat] :
( ! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ Y2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: set_nat,Y2: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_set_nat @ Z3 @ X2 )
=> ( ord_less_eq_set_nat @ ( F @ Y2 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup_set_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_244_sup_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( sup_sup_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_245_sup_OorderI,axiom,
! [A: set_nat,B: set_nat] :
( ( A
= ( sup_sup_set_nat @ A @ B ) )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_246_sup_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( sup_sup_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_247_sup_OorderE,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( A
= ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_248_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( sup_sup_nat @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_249_le__iff__sup,axiom,
( ord_less_eq_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( sup_sup_set_nat @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_250_sup__least,axiom,
! [Y: nat,X: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z2 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_251_sup__least,axiom,
! [Y: set_nat,X: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ Z2 @ X )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_252_sup__mono,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_253_sup__mono,axiom,
! [A: set_nat,C: set_nat,B: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B @ D )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_254_sup_Omono,axiom,
! [C: nat,A: nat,D: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ( ord_less_eq_nat @ D @ B )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_255_sup_Omono,axiom,
! [C: set_nat,A: set_nat,D: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ A )
=> ( ( ord_less_eq_set_nat @ D @ B )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C @ D ) @ ( sup_sup_set_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_256_le__supI2,axiom,
! [X: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_257_le__supI2,axiom,
! [X: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ X @ B )
=> ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_258_le__supI1,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_259_le__supI1,axiom,
! [X: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ X @ A )
=> ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_260_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_261_sup__ge2,axiom,
! [Y: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_262_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_263_sup__ge1,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_264_le__supI,axiom,
! [A: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_265_le__supI,axiom,
! [A: set_nat,X: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ X )
=> ( ( ord_less_eq_set_nat @ B @ X )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_266_le__supE,axiom,
! [A: nat,B: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_nat @ A @ X )
=> ~ ( ord_less_eq_nat @ B @ X ) ) ) ).
% le_supE
thf(fact_267_le__supE,axiom,
! [A: set_nat,B: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_set_nat @ A @ X )
=> ~ ( ord_less_eq_set_nat @ B @ X ) ) ) ).
% le_supE
thf(fact_268_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_269_inf__sup__ord_I3_J,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_270_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_271_inf__sup__ord_I4_J,axiom,
! [Y: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_272_subset__Un__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( sup_sup_set_nat @ A5 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_273_subset__UnE,axiom,
! [C2: set_nat,A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A4 @ B4 ) )
=> ~ ! [A6: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ A4 )
=> ! [B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ B4 )
=> ( C2
!= ( sup_sup_set_nat @ A6 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_274_Un__absorb2,axiom,
! [B4: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
=> ( ( sup_sup_set_nat @ A4 @ B4 )
= A4 ) ) ).
% Un_absorb2
thf(fact_275_Un__absorb1,axiom,
! [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( ( sup_sup_set_nat @ A4 @ B4 )
= B4 ) ) ).
% Un_absorb1
thf(fact_276_Un__upper2,axiom,
! [B4: set_nat,A4: set_nat] : ( ord_less_eq_set_nat @ B4 @ ( sup_sup_set_nat @ A4 @ B4 ) ) ).
% Un_upper2
thf(fact_277_Un__upper1,axiom,
! [A4: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ A4 @ ( sup_sup_set_nat @ A4 @ B4 ) ) ).
% Un_upper1
thf(fact_278_Un__least,axiom,
! [A4: set_nat,C2: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ C2 )
=> ( ( ord_less_eq_set_nat @ B4 @ C2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A4 @ B4 ) @ C2 ) ) ) ).
% Un_least
thf(fact_279_Un__mono,axiom,
! [A4: set_nat,C2: set_nat,B4: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ C2 )
=> ( ( ord_less_eq_set_nat @ B4 @ D2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A4 @ B4 ) @ ( sup_sup_set_nat @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_280_gen__length__def,axiom,
( gen_length_nat
= ( ^ [N3: nat,Xs3: list_nat] : ( plus_plus_nat @ N3 @ ( size_size_list_nat @ Xs3 ) ) ) ) ).
% gen_length_def
thf(fact_281_all__tuplesI,axiom,
! [Vs: list_nat,N: nat,Xs: set_nat] :
( ( ( size_size_list_nat @ Vs )
= N )
=> ( ( ord_less_eq_set_nat @ ( set_nat2 @ Vs ) @ Xs )
=> ( member_list_nat @ Vs @ ( all_tuples_nat @ Xs @ N ) ) ) ) ).
% all_tuplesI
thf(fact_282_all__tuplesD,axiom,
! [Vs: list_nat,Xs: set_nat,N: nat] :
( ( member_list_nat @ Vs @ ( all_tuples_nat @ Xs @ N ) )
=> ( ( ( size_size_list_nat @ Vs )
= N )
& ( ord_less_eq_set_nat @ ( set_nat2 @ Vs ) @ Xs ) ) ) ).
% all_tuplesD
thf(fact_283_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( distinct_nat @ Xs )
=> ( ( linord2614967742042102400et_nat @ ( set_nat2 @ Xs ) )
= Xs ) ) ) ).
% sorted_list_of_set.idem_if_sorted_distinct
thf(fact_284_sorted01,axiom,
! [Xs: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ Xs ) ) ).
% sorted01
thf(fact_285_sorted__list__subset__correct,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( distinct_nat @ Xs )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ys )
=> ( ( distinct_nat @ Ys )
=> ( ( set_or6742139631805365739et_nat
@ ^ [Y3: nat,Z: nat] : ( Y3 = Z )
@ Xs
@ Ys )
= ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) ) ) ) ) ) ).
% sorted_list_subset_correct
thf(fact_286_length__splice,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( size_size_list_nat @ ( splice_nat @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_splice
thf(fact_287_finite__sorted__distinct__unique,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ? [X2: list_nat] :
( ( ( set_nat2 @ X2 )
= A4 )
& ( sorted_wrt_nat @ ord_less_eq_nat @ X2 )
& ( distinct_nat @ X2 )
& ! [Y5: list_nat] :
( ( ( ( set_nat2 @ Y5 )
= A4 )
& ( sorted_wrt_nat @ ord_less_eq_nat @ Y5 )
& ( distinct_nat @ Y5 ) )
=> ( Y5 = X2 ) ) ) ) ).
% finite_sorted_distinct_unique
thf(fact_288_aux6_I1_J,axiom,
( ( inf_inf_set_nat @ ( set_nat2 @ ns_psi2 ) @ ( set_nat2 @ ns_psi ) )
= bot_bot_set_nat ) ).
% aux6(1)
thf(fact_289_set__remdups__sorted,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( set_nat2 @ ( set_or6599480164596245535ed_nat @ Xs ) )
= ( set_nat2 @ Xs ) ) ) ).
% set_remdups_sorted
thf(fact_290_bot__apply,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : bot_bot_o ) ) ).
% bot_apply
thf(fact_291_empty__iff,axiom,
! [C: list_nat] :
~ ( member_list_nat @ C @ bot_bot_set_list_nat ) ).
% empty_iff
thf(fact_292_empty__iff,axiom,
! [C: set_nat] :
~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).
% empty_iff
thf(fact_293_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_294_all__not__in__conv,axiom,
! [A4: set_list_nat] :
( ( ! [X3: list_nat] :
~ ( member_list_nat @ X3 @ A4 ) )
= ( A4 = bot_bot_set_list_nat ) ) ).
% all_not_in_conv
thf(fact_295_all__not__in__conv,axiom,
! [A4: set_set_nat] :
( ( ! [X3: set_nat] :
~ ( member_set_nat @ X3 @ A4 ) )
= ( A4 = bot_bot_set_set_nat ) ) ).
% all_not_in_conv
thf(fact_296_all__not__in__conv,axiom,
! [A4: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A4 ) )
= ( A4 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_297_Collect__empty__eq,axiom,
! [P: set_nat > $o] :
( ( ( collect_set_nat @ P )
= bot_bot_set_set_nat )
= ( ! [X3: set_nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_298_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_299_empty__Collect__eq,axiom,
! [P: set_nat > $o] :
( ( bot_bot_set_set_nat
= ( collect_set_nat @ P ) )
= ( ! [X3: set_nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_300_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_301_inf_Oidem,axiom,
! [A: nat] :
( ( inf_inf_nat @ A @ A )
= A ) ).
% inf.idem
thf(fact_302_inf_Oidem,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ A )
= A ) ).
% inf.idem
thf(fact_303_inf__idem,axiom,
! [X: nat] :
( ( inf_inf_nat @ X @ X )
= X ) ).
% inf_idem
thf(fact_304_inf__idem,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ X )
= X ) ).
% inf_idem
thf(fact_305_inf_Oleft__idem,axiom,
! [A: nat,B: nat] :
( ( inf_inf_nat @ A @ ( inf_inf_nat @ A @ B ) )
= ( inf_inf_nat @ A @ B ) ) ).
% inf.left_idem
thf(fact_306_inf_Oleft__idem,axiom,
! [A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ A @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ).
% inf.left_idem
thf(fact_307_inf__left__idem,axiom,
! [X: nat,Y: nat] :
( ( inf_inf_nat @ X @ ( inf_inf_nat @ X @ Y ) )
= ( inf_inf_nat @ X @ Y ) ) ).
% inf_left_idem
thf(fact_308_inf__left__idem,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ X @ Y ) )
= ( inf_inf_set_nat @ X @ Y ) ) ).
% inf_left_idem
thf(fact_309_inf_Oright__idem,axiom,
! [A: nat,B: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ A @ B ) @ B )
= ( inf_inf_nat @ A @ B ) ) ).
% inf.right_idem
thf(fact_310_inf_Oright__idem,axiom,
! [A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B ) @ B )
= ( inf_inf_set_nat @ A @ B ) ) ).
% inf.right_idem
thf(fact_311_inf__right__idem,axiom,
! [X: nat,Y: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Y )
= ( inf_inf_nat @ X @ Y ) ) ).
% inf_right_idem
thf(fact_312_inf__right__idem,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y )
= ( inf_inf_set_nat @ X @ Y ) ) ).
% inf_right_idem
thf(fact_313_IntI,axiom,
! [C: list_nat,A4: set_list_nat,B4: set_list_nat] :
( ( member_list_nat @ C @ A4 )
=> ( ( member_list_nat @ C @ B4 )
=> ( member_list_nat @ C @ ( inf_inf_set_list_nat @ A4 @ B4 ) ) ) ) ).
% IntI
thf(fact_314_IntI,axiom,
! [C: set_nat,A4: set_set_nat,B4: set_set_nat] :
( ( member_set_nat @ C @ A4 )
=> ( ( member_set_nat @ C @ B4 )
=> ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A4 @ B4 ) ) ) ) ).
% IntI
thf(fact_315_IntI,axiom,
! [C: nat,A4: set_nat,B4: set_nat] :
( ( member_nat @ C @ A4 )
=> ( ( member_nat @ C @ B4 )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A4 @ B4 ) ) ) ) ).
% IntI
thf(fact_316_Int__iff,axiom,
! [C: list_nat,A4: set_list_nat,B4: set_list_nat] :
( ( member_list_nat @ C @ ( inf_inf_set_list_nat @ A4 @ B4 ) )
= ( ( member_list_nat @ C @ A4 )
& ( member_list_nat @ C @ B4 ) ) ) ).
% Int_iff
thf(fact_317_Int__iff,axiom,
! [C: set_nat,A4: set_set_nat,B4: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A4 @ B4 ) )
= ( ( member_set_nat @ C @ A4 )
& ( member_set_nat @ C @ B4 ) ) ) ).
% Int_iff
thf(fact_318_Int__iff,axiom,
! [C: nat,A4: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A4 @ B4 ) )
= ( ( member_nat @ C @ A4 )
& ( member_nat @ C @ B4 ) ) ) ).
% Int_iff
thf(fact_319_aux_I4_J,axiom,
( ( inf_inf_set_nat @ ( set_nat2 @ ns_phi2 ) @ ( set_nat2 @ ns_phi ) )
= bot_bot_set_nat ) ).
% aux(4)
thf(fact_320_aux3_I1_J,axiom,
( ( inf_inf_set_nat @ ( set_nat2 @ ns_phi ) @ ( set_nat2 @ ns ) )
= bot_bot_set_nat ) ).
% aux3(1)
thf(fact_321_aux4_I1_J,axiom,
( ( inf_inf_set_nat @ ( set_nat2 @ ns ) @ ( set_nat2 @ ns_phi ) )
= bot_bot_set_nat ) ).
% aux4(1)
thf(fact_322_le__inf__iff,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_323_le__inf__iff,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z2 ) )
= ( ( ord_less_eq_set_nat @ X @ Y )
& ( ord_less_eq_set_nat @ X @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_324_inf_Obounded__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_325_inf_Obounded__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) )
= ( ( ord_less_eq_set_nat @ A @ B )
& ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_326_subset__empty,axiom,
! [A4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ bot_bot_set_set_nat )
= ( A4 = bot_bot_set_set_nat ) ) ).
% subset_empty
thf(fact_327_subset__empty,axiom,
! [A4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ bot_bot_set_nat )
= ( A4 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_328_empty__subsetI,axiom,
! [A4: set_set_nat] : ( ord_le6893508408891458716et_nat @ bot_bot_set_set_nat @ A4 ) ).
% empty_subsetI
thf(fact_329_empty__subsetI,axiom,
! [A4: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A4 ) ).
% empty_subsetI
thf(fact_330_List_Ofinite__set,axiom,
! [Xs: list_list_nat] : ( finite8100373058378681591st_nat @ ( set_list_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_331_List_Ofinite__set,axiom,
! [Xs: list_set_nat] : ( finite1152437895449049373et_nat @ ( set_set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_332_List_Ofinite__set,axiom,
! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_333_sup__bot_Oright__neutral,axiom,
! [A: set_set_nat] :
( ( sup_sup_set_set_nat @ A @ bot_bot_set_set_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_334_sup__bot_Oright__neutral,axiom,
! [A: nat > $o] :
( ( sup_sup_nat_o @ A @ bot_bot_nat_o )
= A ) ).
% sup_bot.right_neutral
thf(fact_335_sup__bot_Oright__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_336_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( bot_bot_set_set_nat
= ( sup_sup_set_set_nat @ A @ B ) )
= ( ( A = bot_bot_set_set_nat )
& ( B = bot_bot_set_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_337_sup__bot_Oneutr__eq__iff,axiom,
! [A: nat > $o,B: nat > $o] :
( ( bot_bot_nat_o
= ( sup_sup_nat_o @ A @ B ) )
= ( ( A = bot_bot_nat_o )
& ( B = bot_bot_nat_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_338_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A @ B ) )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_339_sup__bot_Oleft__neutral,axiom,
! [A: set_set_nat] :
( ( sup_sup_set_set_nat @ bot_bot_set_set_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_340_sup__bot_Oleft__neutral,axiom,
! [A: nat > $o] :
( ( sup_sup_nat_o @ bot_bot_nat_o @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_341_sup__bot_Oleft__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_342_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ( sup_sup_set_set_nat @ A @ B )
= bot_bot_set_set_nat )
= ( ( A = bot_bot_set_set_nat )
& ( B = bot_bot_set_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_343_sup__bot_Oeq__neutr__iff,axiom,
! [A: nat > $o,B: nat > $o] :
( ( ( sup_sup_nat_o @ A @ B )
= bot_bot_nat_o )
= ( ( A = bot_bot_nat_o )
& ( B = bot_bot_nat_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_344_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_345_sup__eq__bot__iff,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( ( sup_sup_set_set_nat @ X @ Y )
= bot_bot_set_set_nat )
= ( ( X = bot_bot_set_set_nat )
& ( Y = bot_bot_set_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_346_sup__eq__bot__iff,axiom,
! [X: nat > $o,Y: nat > $o] :
( ( ( sup_sup_nat_o @ X @ Y )
= bot_bot_nat_o )
= ( ( X = bot_bot_nat_o )
& ( Y = bot_bot_nat_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_347_sup__eq__bot__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( sup_sup_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ( X = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_348_bot__eq__sup__iff,axiom,
! [X: set_set_nat,Y: set_set_nat] :
( ( bot_bot_set_set_nat
= ( sup_sup_set_set_nat @ X @ Y ) )
= ( ( X = bot_bot_set_set_nat )
& ( Y = bot_bot_set_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_349_bot__eq__sup__iff,axiom,
! [X: nat > $o,Y: nat > $o] :
( ( bot_bot_nat_o
= ( sup_sup_nat_o @ X @ Y ) )
= ( ( X = bot_bot_nat_o )
& ( Y = bot_bot_nat_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_350_bot__eq__sup__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X @ Y ) )
= ( ( X = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_351_sup__bot__right,axiom,
! [X: set_set_nat] :
( ( sup_sup_set_set_nat @ X @ bot_bot_set_set_nat )
= X ) ).
% sup_bot_right
thf(fact_352_sup__bot__right,axiom,
! [X: nat > $o] :
( ( sup_sup_nat_o @ X @ bot_bot_nat_o )
= X ) ).
% sup_bot_right
thf(fact_353_sup__bot__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% sup_bot_right
thf(fact_354_sup__bot__left,axiom,
! [X: set_set_nat] :
( ( sup_sup_set_set_nat @ bot_bot_set_set_nat @ X )
= X ) ).
% sup_bot_left
thf(fact_355_sup__bot__left,axiom,
! [X: nat > $o] :
( ( sup_sup_nat_o @ bot_bot_nat_o @ X )
= X ) ).
% sup_bot_left
thf(fact_356_sup__bot__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X )
= X ) ).
% sup_bot_left
thf(fact_357_inf__bot__left,axiom,
! [X: set_set_nat] :
( ( inf_inf_set_set_nat @ bot_bot_set_set_nat @ X )
= bot_bot_set_set_nat ) ).
% inf_bot_left
thf(fact_358_inf__bot__left,axiom,
! [X: nat > $o] :
( ( inf_inf_nat_o @ bot_bot_nat_o @ X )
= bot_bot_nat_o ) ).
% inf_bot_left
thf(fact_359_inf__bot__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_360_inf__bot__right,axiom,
! [X: set_set_nat] :
( ( inf_inf_set_set_nat @ X @ bot_bot_set_set_nat )
= bot_bot_set_set_nat ) ).
% inf_bot_right
thf(fact_361_inf__bot__right,axiom,
! [X: nat > $o] :
( ( inf_inf_nat_o @ X @ bot_bot_nat_o )
= bot_bot_nat_o ) ).
% inf_bot_right
thf(fact_362_inf__bot__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_363_Un__empty,axiom,
! [A4: set_set_nat,B4: set_set_nat] :
( ( ( sup_sup_set_set_nat @ A4 @ B4 )
= bot_bot_set_set_nat )
= ( ( A4 = bot_bot_set_set_nat )
& ( B4 = bot_bot_set_set_nat ) ) ) ).
% Un_empty
thf(fact_364_Un__empty,axiom,
! [A4: set_nat,B4: set_nat] :
( ( ( sup_sup_set_nat @ A4 @ B4 )
= bot_bot_set_nat )
= ( ( A4 = bot_bot_set_nat )
& ( B4 = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_365_sup__inf__absorb,axiom,
! [X: nat,Y: nat] :
( ( sup_sup_nat @ X @ ( inf_inf_nat @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_366_sup__inf__absorb,axiom,
! [X: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_367_inf__sup__absorb,axiom,
! [X: nat,Y: nat] :
( ( inf_inf_nat @ X @ ( sup_sup_nat @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_368_inf__sup__absorb,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_369_Int__subset__iff,axiom,
! [C2: set_nat,A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A4 @ B4 ) )
= ( ( ord_less_eq_set_nat @ C2 @ A4 )
& ( ord_less_eq_set_nat @ C2 @ B4 ) ) ) ).
% Int_subset_iff
thf(fact_370_Un__Int__eq_I1_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_371_Un__Int__eq_I2_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_372_Un__Int__eq_I3_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( inf_inf_set_nat @ S @ ( sup_sup_set_nat @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_373_Un__Int__eq_I4_J,axiom,
! [T2: set_nat,S: set_nat] :
( ( inf_inf_set_nat @ T2 @ ( sup_sup_set_nat @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_374_Int__Un__eq_I1_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_375_Int__Un__eq_I2_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_376_Int__Un__eq_I3_J,axiom,
! [S: set_nat,T2: set_nat] :
( ( sup_sup_set_nat @ S @ ( inf_inf_set_nat @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_377_Int__Un__eq_I4_J,axiom,
! [T2: set_nat,S: set_nat] :
( ( sup_sup_set_nat @ T2 @ ( inf_inf_set_nat @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_378_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( set_nat2 @ ( linord2614967742042102400et_nat @ A4 ) )
= A4 ) ) ).
% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_379_inf__sup__aci_I4_J,axiom,
! [X: nat,Y: nat] :
( ( inf_inf_nat @ X @ ( inf_inf_nat @ X @ Y ) )
= ( inf_inf_nat @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_380_inf__sup__aci_I4_J,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ X @ Y ) )
= ( inf_inf_set_nat @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_381_inf__sup__aci_I3_J,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) )
= ( inf_inf_nat @ Y @ ( inf_inf_nat @ X @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_382_inf__sup__aci_I3_J,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z2 ) )
= ( inf_inf_set_nat @ Y @ ( inf_inf_set_nat @ X @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_383_inf__sup__aci_I2_J,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Z2 )
= ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_384_inf__sup__aci_I2_J,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Z2 )
= ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_385_inf__sup__aci_I1_J,axiom,
( inf_inf_nat
= ( ^ [X3: nat,Y4: nat] : ( inf_inf_nat @ Y4 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_386_inf__sup__aci_I1_J,axiom,
( inf_inf_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] : ( inf_inf_set_nat @ Y4 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_387_inf_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ A @ B ) @ C )
= ( inf_inf_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ).
% inf.assoc
thf(fact_388_inf_Oassoc,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C )
= ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) ) ) ).
% inf.assoc
thf(fact_389_inf__assoc,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Z2 )
= ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) ) ) ).
% inf_assoc
thf(fact_390_inf__assoc,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Z2 )
= ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z2 ) ) ) ).
% inf_assoc
thf(fact_391_inf_Ocommute,axiom,
( inf_inf_nat
= ( ^ [A2: nat,B2: nat] : ( inf_inf_nat @ B2 @ A2 ) ) ) ).
% inf.commute
thf(fact_392_inf_Ocommute,axiom,
( inf_inf_set_nat
= ( ^ [A2: set_nat,B2: set_nat] : ( inf_inf_set_nat @ B2 @ A2 ) ) ) ).
% inf.commute
thf(fact_393_inf__commute,axiom,
( inf_inf_nat
= ( ^ [X3: nat,Y4: nat] : ( inf_inf_nat @ Y4 @ X3 ) ) ) ).
% inf_commute
thf(fact_394_inf__commute,axiom,
( inf_inf_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] : ( inf_inf_set_nat @ Y4 @ X3 ) ) ) ).
% inf_commute
thf(fact_395_inf_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( inf_inf_nat @ B @ ( inf_inf_nat @ A @ C ) )
= ( inf_inf_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_396_inf_Oleft__commute,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( inf_inf_set_nat @ B @ ( inf_inf_set_nat @ A @ C ) )
= ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_397_inf__left__commute,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) )
= ( inf_inf_nat @ Y @ ( inf_inf_nat @ X @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_398_inf__left__commute,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z2 ) )
= ( inf_inf_set_nat @ Y @ ( inf_inf_set_nat @ X @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_399_IntE,axiom,
! [C: list_nat,A4: set_list_nat,B4: set_list_nat] :
( ( member_list_nat @ C @ ( inf_inf_set_list_nat @ A4 @ B4 ) )
=> ~ ( ( member_list_nat @ C @ A4 )
=> ~ ( member_list_nat @ C @ B4 ) ) ) ).
% IntE
thf(fact_400_IntE,axiom,
! [C: set_nat,A4: set_set_nat,B4: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A4 @ B4 ) )
=> ~ ( ( member_set_nat @ C @ A4 )
=> ~ ( member_set_nat @ C @ B4 ) ) ) ).
% IntE
thf(fact_401_IntE,axiom,
! [C: nat,A4: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A4 @ B4 ) )
=> ~ ( ( member_nat @ C @ A4 )
=> ~ ( member_nat @ C @ B4 ) ) ) ).
% IntE
thf(fact_402_IntD1,axiom,
! [C: list_nat,A4: set_list_nat,B4: set_list_nat] :
( ( member_list_nat @ C @ ( inf_inf_set_list_nat @ A4 @ B4 ) )
=> ( member_list_nat @ C @ A4 ) ) ).
% IntD1
thf(fact_403_IntD1,axiom,
! [C: set_nat,A4: set_set_nat,B4: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A4 @ B4 ) )
=> ( member_set_nat @ C @ A4 ) ) ).
% IntD1
thf(fact_404_IntD1,axiom,
! [C: nat,A4: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A4 @ B4 ) )
=> ( member_nat @ C @ A4 ) ) ).
% IntD1
thf(fact_405_IntD2,axiom,
! [C: list_nat,A4: set_list_nat,B4: set_list_nat] :
( ( member_list_nat @ C @ ( inf_inf_set_list_nat @ A4 @ B4 ) )
=> ( member_list_nat @ C @ B4 ) ) ).
% IntD2
thf(fact_406_IntD2,axiom,
! [C: set_nat,A4: set_set_nat,B4: set_set_nat] :
( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A4 @ B4 ) )
=> ( member_set_nat @ C @ B4 ) ) ).
% IntD2
thf(fact_407_IntD2,axiom,
! [C: nat,A4: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A4 @ B4 ) )
=> ( member_nat @ C @ B4 ) ) ).
% IntD2
thf(fact_408_emptyE,axiom,
! [A: list_nat] :
~ ( member_list_nat @ A @ bot_bot_set_list_nat ) ).
% emptyE
thf(fact_409_emptyE,axiom,
! [A: set_nat] :
~ ( member_set_nat @ A @ bot_bot_set_set_nat ) ).
% emptyE
thf(fact_410_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_411_equals0D,axiom,
! [A4: set_list_nat,A: list_nat] :
( ( A4 = bot_bot_set_list_nat )
=> ~ ( member_list_nat @ A @ A4 ) ) ).
% equals0D
thf(fact_412_equals0D,axiom,
! [A4: set_set_nat,A: set_nat] :
( ( A4 = bot_bot_set_set_nat )
=> ~ ( member_set_nat @ A @ A4 ) ) ).
% equals0D
thf(fact_413_equals0D,axiom,
! [A4: set_nat,A: nat] :
( ( A4 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A4 ) ) ).
% equals0D
thf(fact_414_equals0I,axiom,
! [A4: set_list_nat] :
( ! [Y2: list_nat] :
~ ( member_list_nat @ Y2 @ A4 )
=> ( A4 = bot_bot_set_list_nat ) ) ).
% equals0I
thf(fact_415_equals0I,axiom,
! [A4: set_set_nat] :
( ! [Y2: set_nat] :
~ ( member_set_nat @ Y2 @ A4 )
=> ( A4 = bot_bot_set_set_nat ) ) ).
% equals0I
thf(fact_416_equals0I,axiom,
! [A4: set_nat] :
( ! [Y2: nat] :
~ ( member_nat @ Y2 @ A4 )
=> ( A4 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_417_Int__assoc,axiom,
! [A4: set_nat,B4: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A4 @ B4 ) @ C2 )
= ( inf_inf_set_nat @ A4 @ ( inf_inf_set_nat @ B4 @ C2 ) ) ) ).
% Int_assoc
thf(fact_418_Int__absorb,axiom,
! [A4: set_nat] :
( ( inf_inf_set_nat @ A4 @ A4 )
= A4 ) ).
% Int_absorb
thf(fact_419_Int__emptyI,axiom,
! [A4: set_list_nat,B4: set_list_nat] :
( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ A4 )
=> ~ ( member_list_nat @ X2 @ B4 ) )
=> ( ( inf_inf_set_list_nat @ A4 @ B4 )
= bot_bot_set_list_nat ) ) ).
% Int_emptyI
thf(fact_420_Int__emptyI,axiom,
! [A4: set_set_nat,B4: set_set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A4 )
=> ~ ( member_set_nat @ X2 @ B4 ) )
=> ( ( inf_inf_set_set_nat @ A4 @ B4 )
= bot_bot_set_set_nat ) ) ).
% Int_emptyI
thf(fact_421_Int__emptyI,axiom,
! [A4: set_nat,B4: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A4 )
=> ~ ( member_nat @ X2 @ B4 ) )
=> ( ( inf_inf_set_nat @ A4 @ B4 )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_422_ex__in__conv,axiom,
! [A4: set_list_nat] :
( ( ? [X3: list_nat] : ( member_list_nat @ X3 @ A4 ) )
= ( A4 != bot_bot_set_list_nat ) ) ).
% ex_in_conv
thf(fact_423_ex__in__conv,axiom,
! [A4: set_set_nat] :
( ( ? [X3: set_nat] : ( member_set_nat @ X3 @ A4 ) )
= ( A4 != bot_bot_set_set_nat ) ) ).
% ex_in_conv
thf(fact_424_ex__in__conv,axiom,
! [A4: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A4 ) )
= ( A4 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_425_Int__commute,axiom,
( inf_inf_set_nat
= ( ^ [A5: set_nat,B5: set_nat] : ( inf_inf_set_nat @ B5 @ A5 ) ) ) ).
% Int_commute
thf(fact_426_disjoint__iff,axiom,
! [A4: set_list_nat,B4: set_list_nat] :
( ( ( inf_inf_set_list_nat @ A4 @ B4 )
= bot_bot_set_list_nat )
= ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A4 )
=> ~ ( member_list_nat @ X3 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_427_disjoint__iff,axiom,
! [A4: set_set_nat,B4: set_set_nat] :
( ( ( inf_inf_set_set_nat @ A4 @ B4 )
= bot_bot_set_set_nat )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A4 )
=> ~ ( member_set_nat @ X3 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_428_disjoint__iff,axiom,
! [A4: set_nat,B4: set_nat] :
( ( ( inf_inf_set_nat @ A4 @ B4 )
= bot_bot_set_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ~ ( member_nat @ X3 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_429_Int__empty__left,axiom,
! [B4: set_set_nat] :
( ( inf_inf_set_set_nat @ bot_bot_set_set_nat @ B4 )
= bot_bot_set_set_nat ) ).
% Int_empty_left
thf(fact_430_Int__empty__left,axiom,
! [B4: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B4 )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_431_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_432_Int__empty__right,axiom,
! [A4: set_set_nat] :
( ( inf_inf_set_set_nat @ A4 @ bot_bot_set_set_nat )
= bot_bot_set_set_nat ) ).
% Int_empty_right
thf(fact_433_Int__empty__right,axiom,
! [A4: set_nat] :
( ( inf_inf_set_nat @ A4 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_434_Int__left__absorb,axiom,
! [A4: set_nat,B4: set_nat] :
( ( inf_inf_set_nat @ A4 @ ( inf_inf_set_nat @ A4 @ B4 ) )
= ( inf_inf_set_nat @ A4 @ B4 ) ) ).
% Int_left_absorb
thf(fact_435_Int__left__commute,axiom,
! [A4: set_nat,B4: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ A4 @ ( inf_inf_set_nat @ B4 @ C2 ) )
= ( inf_inf_set_nat @ B4 @ ( inf_inf_set_nat @ A4 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_436_disjoint__iff__not__equal,axiom,
! [A4: set_set_nat,B4: set_set_nat] :
( ( ( inf_inf_set_set_nat @ A4 @ B4 )
= bot_bot_set_set_nat )
= ( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ A4 )
=> ! [Y4: set_nat] :
( ( member_set_nat @ Y4 @ B4 )
=> ( X3 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_437_disjoint__iff__not__equal,axiom,
! [A4: set_nat,B4: set_nat] :
( ( ( inf_inf_set_nat @ A4 @ B4 )
= bot_bot_set_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ B4 )
=> ( X3 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_438_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
! [A4: set_nat,B4: set_nat] :
( ( ( linord2614967742042102400et_nat @ A4 )
= ( linord2614967742042102400et_nat @ B4 ) )
=> ( ( finite_finite_nat @ A4 )
=> ( ( finite_finite_nat @ B4 )
=> ( A4 = B4 ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_439_bot__fun__def,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_440_all__tuples__finite,axiom,
! [Xs: set_list_nat,N: nat] :
( ( finite8100373058378681591st_nat @ Xs )
=> ( finite8170528100393595399st_nat @ ( all_tuples_list_nat @ Xs @ N ) ) ) ).
% all_tuples_finite
thf(fact_441_all__tuples__finite,axiom,
! [Xs: set_set_nat,N: nat] :
( ( finite1152437895449049373et_nat @ Xs )
=> ( finite1091814263879798189et_nat @ ( all_tuples_set_nat @ Xs @ N ) ) ) ).
% all_tuples_finite
thf(fact_442_all__tuples__finite,axiom,
! [Xs: set_nat,N: nat] :
( ( finite_finite_nat @ Xs )
=> ( finite8100373058378681591st_nat @ ( all_tuples_nat @ Xs @ N ) ) ) ).
% all_tuples_finite
thf(fact_443_bot_Oextremum,axiom,
! [A: set_set_nat] : ( ord_le6893508408891458716et_nat @ bot_bot_set_set_nat @ A ) ).
% bot.extremum
thf(fact_444_bot_Oextremum,axiom,
! [A: nat > $o] : ( ord_less_eq_nat_o @ bot_bot_nat_o @ A ) ).
% bot.extremum
thf(fact_445_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_446_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_447_bot_Oextremum__unique,axiom,
! [A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ bot_bot_set_set_nat )
= ( A = bot_bot_set_set_nat ) ) ).
% bot.extremum_unique
thf(fact_448_bot_Oextremum__unique,axiom,
! [A: nat > $o] :
( ( ord_less_eq_nat_o @ A @ bot_bot_nat_o )
= ( A = bot_bot_nat_o ) ) ).
% bot.extremum_unique
thf(fact_449_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_450_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_451_bot_Oextremum__uniqueI,axiom,
! [A: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ bot_bot_set_set_nat )
=> ( A = bot_bot_set_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_452_bot_Oextremum__uniqueI,axiom,
! [A: nat > $o] :
( ( ord_less_eq_nat_o @ A @ bot_bot_nat_o )
=> ( A = bot_bot_nat_o ) ) ).
% bot.extremum_uniqueI
thf(fact_453_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_454_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_455_inf__sup__ord_I2_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_456_inf__sup__ord_I2_J,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_457_inf__sup__ord_I1_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_458_inf__sup__ord_I1_J,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_459_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_460_inf__le1,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_461_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_462_inf__le2,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_463_le__infE,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) )
=> ~ ( ( ord_less_eq_nat @ X @ A )
=> ~ ( ord_less_eq_nat @ X @ B ) ) ) ).
% le_infE
thf(fact_464_le__infE,axiom,
! [X: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ A @ B ) )
=> ~ ( ( ord_less_eq_set_nat @ X @ A )
=> ~ ( ord_less_eq_set_nat @ X @ B ) ) ) ).
% le_infE
thf(fact_465_le__infI,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% le_infI
thf(fact_466_le__infI,axiom,
! [X: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ X @ A )
=> ( ( ord_less_eq_set_nat @ X @ B )
=> ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% le_infI
thf(fact_467_inf__mono,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_468_inf__mono,axiom,
! [A: set_nat,C: set_nat,B: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B @ D )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_469_le__infI1,axiom,
! [A: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_470_le__infI1,axiom,
! [A: set_nat,X: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ X )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_471_le__infI2,axiom,
! [B: nat,X: nat,A: nat] :
( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_472_le__infI2,axiom,
! [B: set_nat,X: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ X )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_473_inf_OorderE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( A
= ( inf_inf_nat @ A @ B ) ) ) ).
% inf.orderE
thf(fact_474_inf_OorderE,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( A
= ( inf_inf_set_nat @ A @ B ) ) ) ).
% inf.orderE
thf(fact_475_inf_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( inf_inf_nat @ A @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% inf.orderI
thf(fact_476_inf_OorderI,axiom,
! [A: set_nat,B: set_nat] :
( ( A
= ( inf_inf_set_nat @ A @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% inf.orderI
thf(fact_477_inf__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ X2 @ Z3 )
=> ( ord_less_eq_nat @ X2 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_478_inf__unique,axiom,
! [F: set_nat > set_nat > set_nat,X: set_nat,Y: set_nat] :
( ! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( F @ X2 @ Y2 ) @ X2 )
=> ( ! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( F @ X2 @ Y2 ) @ Y2 )
=> ( ! [X2: set_nat,Y2: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Z3 )
=> ( ord_less_eq_set_nat @ X2 @ ( F @ Y2 @ Z3 ) ) ) )
=> ( ( inf_inf_set_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_479_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( inf_inf_nat @ X3 @ Y4 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_480_le__iff__inf,axiom,
( ord_less_eq_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( inf_inf_set_nat @ X3 @ Y4 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_481_inf_Oabsorb1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( inf_inf_nat @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_482_inf_Oabsorb1,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( inf_inf_set_nat @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_483_inf_Oabsorb2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( inf_inf_nat @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_484_inf_Oabsorb2,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( inf_inf_set_nat @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_485_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_486_inf__absorb1,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( inf_inf_set_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_487_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_488_inf__absorb2,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( inf_inf_set_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_489_inf_OboundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_nat @ A @ B )
=> ~ ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_490_inf_OboundedE,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ~ ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_491_inf_OboundedI,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_492_inf_OboundedI,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ A @ C )
=> ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_493_inf__greatest,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z2 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_494_inf__greatest,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ X @ Z2 )
=> ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_495_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( A2
= ( inf_inf_nat @ A2 @ B2 ) ) ) ) ).
% inf.order_iff
thf(fact_496_inf_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
( A2
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% inf.order_iff
thf(fact_497_inf_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_498_inf_Ocobounded1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_499_inf_Ocobounded2,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_500_inf_Ocobounded2,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_501_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( inf_inf_nat @ A2 @ B2 )
= A2 ) ) ) ).
% inf.absorb_iff1
thf(fact_502_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ A2 @ B2 )
= A2 ) ) ) ).
% inf.absorb_iff1
thf(fact_503_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( inf_inf_nat @ A2 @ B2 )
= B2 ) ) ) ).
% inf.absorb_iff2
thf(fact_504_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [B2: set_nat,A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ B2 )
= B2 ) ) ) ).
% inf.absorb_iff2
thf(fact_505_inf_OcoboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_506_inf_OcoboundedI1,axiom,
! [A: set_nat,C: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_507_inf_OcoboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_508_inf_OcoboundedI2,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_509_sup__inf__distrib2,axiom,
! [Y: nat,Z2: nat,X: nat] :
( ( sup_sup_nat @ ( inf_inf_nat @ Y @ Z2 ) @ X )
= ( inf_inf_nat @ ( sup_sup_nat @ Y @ X ) @ ( sup_sup_nat @ Z2 @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_510_sup__inf__distrib2,axiom,
! [Y: set_nat,Z2: set_nat,X: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ Z2 ) @ X )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ X ) @ ( sup_sup_set_nat @ Z2 @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_511_sup__inf__distrib1,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) )
= ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_512_sup__inf__distrib1,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_513_inf__sup__distrib2,axiom,
! [Y: nat,Z2: nat,X: nat] :
( ( inf_inf_nat @ ( sup_sup_nat @ Y @ Z2 ) @ X )
= ( sup_sup_nat @ ( inf_inf_nat @ Y @ X ) @ ( inf_inf_nat @ Z2 @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_514_inf__sup__distrib2,axiom,
! [Y: set_nat,Z2: set_nat,X: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ Z2 ) @ X )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ X ) @ ( inf_inf_set_nat @ Z2 @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_515_inf__sup__distrib1,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z2 ) )
= ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_516_inf__sup__distrib1,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_517_distrib__imp2,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ! [X2: nat,Y2: nat,Z3: nat] :
( ( sup_sup_nat @ X2 @ ( inf_inf_nat @ Y2 @ Z3 ) )
= ( inf_inf_nat @ ( sup_sup_nat @ X2 @ Y2 ) @ ( sup_sup_nat @ X2 @ Z3 ) ) )
=> ( ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z2 ) )
= ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_518_distrib__imp2,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ! [X2: set_nat,Y2: set_nat,Z3: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( inf_inf_set_nat @ Y2 @ Z3 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X2 @ Y2 ) @ ( sup_sup_set_nat @ X2 @ Z3 ) ) )
=> ( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_519_distrib__imp1,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ! [X2: nat,Y2: nat,Z3: nat] :
( ( inf_inf_nat @ X2 @ ( sup_sup_nat @ Y2 @ Z3 ) )
= ( sup_sup_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ ( inf_inf_nat @ X2 @ Z3 ) ) )
=> ( ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) )
= ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_520_distrib__imp1,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ! [X2: set_nat,Y2: set_nat,Z3: set_nat] :
( ( inf_inf_set_nat @ X2 @ ( sup_sup_set_nat @ Y2 @ Z3 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X2 @ Y2 ) @ ( inf_inf_set_nat @ X2 @ Z3 ) ) )
=> ( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_521_Int__mono,axiom,
! [A4: set_nat,C2: set_nat,B4: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ C2 )
=> ( ( ord_less_eq_set_nat @ B4 @ D2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A4 @ B4 ) @ ( inf_inf_set_nat @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_522_Int__lower1,axiom,
! [A4: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A4 @ B4 ) @ A4 ) ).
% Int_lower1
thf(fact_523_Int__lower2,axiom,
! [A4: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A4 @ B4 ) @ B4 ) ).
% Int_lower2
thf(fact_524_Int__absorb1,axiom,
! [B4: set_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
=> ( ( inf_inf_set_nat @ A4 @ B4 )
= B4 ) ) ).
% Int_absorb1
thf(fact_525_Int__absorb2,axiom,
! [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( ( inf_inf_set_nat @ A4 @ B4 )
= A4 ) ) ).
% Int_absorb2
thf(fact_526_Int__greatest,axiom,
! [C2: set_nat,A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A4 )
=> ( ( ord_less_eq_set_nat @ C2 @ B4 )
=> ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A4 @ B4 ) ) ) ) ).
% Int_greatest
thf(fact_527_Int__Collect__mono,axiom,
! [A4: set_list_nat,B4: set_list_nat,P: list_nat > $o,Q: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ A4 @ B4 )
=> ( ! [X2: list_nat] :
( ( member_list_nat @ X2 @ A4 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le6045566169113846134st_nat @ ( inf_inf_set_list_nat @ A4 @ ( collect_list_nat @ P ) ) @ ( inf_inf_set_list_nat @ B4 @ ( collect_list_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_528_Int__Collect__mono,axiom,
! [A4: set_set_nat,B4: set_set_nat,P: set_nat > $o,Q: set_nat > $o] :
( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
=> ( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A4 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A4 @ ( collect_set_nat @ P ) ) @ ( inf_inf_set_set_nat @ B4 @ ( collect_set_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_529_Int__Collect__mono,axiom,
! [A4: set_nat,B4: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A4 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A4 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B4 @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_530_finite__list,axiom,
! [A4: set_list_nat] :
( ( finite8100373058378681591st_nat @ A4 )
=> ? [Xs2: list_list_nat] :
( ( set_list_nat2 @ Xs2 )
= A4 ) ) ).
% finite_list
thf(fact_531_finite__list,axiom,
! [A4: set_set_nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ? [Xs2: list_set_nat] :
( ( set_set_nat2 @ Xs2 )
= A4 ) ) ).
% finite_list
thf(fact_532_finite__list,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ? [Xs2: list_nat] :
( ( set_nat2 @ Xs2 )
= A4 ) ) ).
% finite_list
thf(fact_533_Un__Int__distrib2,axiom,
! [B4: set_nat,C2: set_nat,A4: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ B4 @ C2 ) @ A4 )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ B4 @ A4 ) @ ( sup_sup_set_nat @ C2 @ A4 ) ) ) ).
% Un_Int_distrib2
thf(fact_534_Int__Un__distrib2,axiom,
! [B4: set_nat,C2: set_nat,A4: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ B4 @ C2 ) @ A4 )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ B4 @ A4 ) @ ( inf_inf_set_nat @ C2 @ A4 ) ) ) ).
% Int_Un_distrib2
thf(fact_535_Un__Int__distrib,axiom,
! [A4: set_nat,B4: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ A4 @ ( inf_inf_set_nat @ B4 @ C2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ A4 @ B4 ) @ ( sup_sup_set_nat @ A4 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_536_Int__Un__distrib,axiom,
! [A4: set_nat,B4: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ A4 @ ( sup_sup_set_nat @ B4 @ C2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ A4 @ B4 ) @ ( inf_inf_set_nat @ A4 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_537_Un__Int__crazy,axiom,
! [A4: set_nat,B4: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ A4 @ B4 ) @ ( inf_inf_set_nat @ B4 @ C2 ) ) @ ( inf_inf_set_nat @ C2 @ A4 ) )
= ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ A4 @ B4 ) @ ( sup_sup_set_nat @ B4 @ C2 ) ) @ ( sup_sup_set_nat @ C2 @ A4 ) ) ) ).
% Un_Int_crazy
thf(fact_538_Un__empty__right,axiom,
! [A4: set_set_nat] :
( ( sup_sup_set_set_nat @ A4 @ bot_bot_set_set_nat )
= A4 ) ).
% Un_empty_right
thf(fact_539_Un__empty__right,axiom,
! [A4: set_nat] :
( ( sup_sup_set_nat @ A4 @ bot_bot_set_nat )
= A4 ) ).
% Un_empty_right
thf(fact_540_Un__empty__left,axiom,
! [B4: set_set_nat] :
( ( sup_sup_set_set_nat @ bot_bot_set_set_nat @ B4 )
= B4 ) ).
% Un_empty_left
thf(fact_541_Un__empty__left,axiom,
! [B4: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B4 )
= B4 ) ).
% Un_empty_left
thf(fact_542_sorted__list__of__set_Odistinct__sorted__key__list__of__set,axiom,
! [A4: set_nat] : ( distinct_nat @ ( linord2614967742042102400et_nat @ A4 ) ) ).
% sorted_list_of_set.distinct_sorted_key_list_of_set
thf(fact_543_sorted__remdups__sorted,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( set_or6599480164596245535ed_nat @ Xs ) ) ) ).
% sorted_remdups_sorted
thf(fact_544_distrib__inf__le,axiom,
! [X: nat,Y: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z2 ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_545_distrib__inf__le,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z2 ) ) @ ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_546_distrib__sup__le,axiom,
! [X: nat,Y: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z2 ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_547_distrib__sup__le,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z2 ) ) @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_548_Un__Int__assoc__eq,axiom,
! [A4: set_nat,B4: set_nat,C2: set_nat] :
( ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A4 @ B4 ) @ C2 )
= ( inf_inf_set_nat @ A4 @ ( sup_sup_set_nat @ B4 @ C2 ) ) )
= ( ord_less_eq_set_nat @ C2 @ A4 ) ) ).
% Un_Int_assoc_eq
thf(fact_549_finite__distinct__list,axiom,
! [A4: set_list_nat] :
( ( finite8100373058378681591st_nat @ A4 )
=> ? [Xs2: list_list_nat] :
( ( ( set_list_nat2 @ Xs2 )
= A4 )
& ( distinct_list_nat @ Xs2 ) ) ) ).
% finite_distinct_list
thf(fact_550_finite__distinct__list,axiom,
! [A4: set_set_nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ? [Xs2: list_set_nat] :
( ( ( set_set_nat2 @ Xs2 )
= A4 )
& ( distinct_set_nat @ Xs2 ) ) ) ).
% finite_distinct_list
thf(fact_551_finite__distinct__list,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ? [Xs2: list_nat] :
( ( ( set_nat2 @ Xs2 )
= A4 )
& ( distinct_nat @ Xs2 ) ) ) ).
% finite_distinct_list
thf(fact_552_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
! [A4: set_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( linord2614967742042102400et_nat @ A4 ) ) ).
% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_553_distinct__remdups__sorted,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( distinct_nat @ ( set_or6599480164596245535ed_nat @ Xs ) ) ) ).
% distinct_remdups_sorted
thf(fact_554_sorted__wrt01,axiom,
! [Xs: list_nat,P: nat > nat > $o] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( sorted_wrt_nat @ P @ Xs ) ) ).
% sorted_wrt01
thf(fact_555_all__tuples__setD,axiom,
! [Vs: list_nat,Xs: set_nat,N: nat] :
( ( member_list_nat @ Vs @ ( all_tuples_nat @ Xs @ N ) )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ Vs ) @ Xs ) ) ).
% all_tuples_setD
thf(fact_556_finite__Int,axiom,
! [F2: set_list_nat,G: set_list_nat] :
( ( ( finite8100373058378681591st_nat @ F2 )
| ( finite8100373058378681591st_nat @ G ) )
=> ( finite8100373058378681591st_nat @ ( inf_inf_set_list_nat @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_557_finite__Int,axiom,
! [F2: set_set_nat,G: set_set_nat] :
( ( ( finite1152437895449049373et_nat @ F2 )
| ( finite1152437895449049373et_nat @ G ) )
=> ( finite1152437895449049373et_nat @ ( inf_inf_set_set_nat @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_558_finite__Int,axiom,
! [F2: set_nat,G: set_nat] :
( ( ( finite_finite_nat @ F2 )
| ( finite_finite_nat @ G ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_559_finite__Un,axiom,
! [F2: set_list_nat,G: set_list_nat] :
( ( finite8100373058378681591st_nat @ ( sup_sup_set_list_nat @ F2 @ G ) )
= ( ( finite8100373058378681591st_nat @ F2 )
& ( finite8100373058378681591st_nat @ G ) ) ) ).
% finite_Un
thf(fact_560_finite__Un,axiom,
! [F2: set_set_nat,G: set_set_nat] :
( ( finite1152437895449049373et_nat @ ( sup_sup_set_set_nat @ F2 @ G ) )
= ( ( finite1152437895449049373et_nat @ F2 )
& ( finite1152437895449049373et_nat @ G ) ) ) ).
% finite_Un
thf(fact_561_finite__Un,axiom,
! [F2: set_nat,G: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G ) )
= ( ( finite_finite_nat @ F2 )
& ( finite_finite_nat @ G ) ) ) ).
% finite_Un
thf(fact_562_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_set_nat] :
( ( inf_inf_set_set_nat @ X @ bot_bot_set_set_nat )
= bot_bot_set_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_563_boolean__algebra_Oconj__zero__right,axiom,
! [X: nat > $o] :
( ( inf_inf_nat_o @ X @ bot_bot_nat_o )
= bot_bot_nat_o ) ).
% boolean_algebra.conj_zero_right
thf(fact_564_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_565_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_set_nat] :
( ( inf_inf_set_set_nat @ bot_bot_set_set_nat @ X )
= bot_bot_set_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_566_boolean__algebra_Oconj__zero__left,axiom,
! [X: nat > $o] :
( ( inf_inf_nat_o @ bot_bot_nat_o @ X )
= bot_bot_nat_o ) ).
% boolean_algebra.conj_zero_left
thf(fact_567_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_568_finite__has__maximal,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_569_finite__has__maximal,axiom,
! [A4: set_set_nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ( A4 != bot_bot_set_set_nat )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A4 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A4 )
=> ( ( ord_less_eq_set_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_570_finite__has__minimal,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A4 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_571_finite__has__minimal,axiom,
! [A4: set_set_nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ( A4 != bot_bot_set_set_nat )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A4 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A4 )
=> ( ( ord_less_eq_set_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_572_infinite__Un,axiom,
! [S: set_list_nat,T2: set_list_nat] :
( ( ~ ( finite8100373058378681591st_nat @ ( sup_sup_set_list_nat @ S @ T2 ) ) )
= ( ~ ( finite8100373058378681591st_nat @ S )
| ~ ( finite8100373058378681591st_nat @ T2 ) ) ) ).
% infinite_Un
thf(fact_573_infinite__Un,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( ~ ( finite1152437895449049373et_nat @ ( sup_sup_set_set_nat @ S @ T2 ) ) )
= ( ~ ( finite1152437895449049373et_nat @ S )
| ~ ( finite1152437895449049373et_nat @ T2 ) ) ) ).
% infinite_Un
thf(fact_574_infinite__Un,axiom,
! [S: set_nat,T2: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) )
= ( ~ ( finite_finite_nat @ S )
| ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_Un
thf(fact_575_bot__set__def,axiom,
( bot_bot_set_set_nat
= ( collect_set_nat @ bot_bot_set_nat_o ) ) ).
% bot_set_def
thf(fact_576_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_577_boolean__algebra__cancel_Osup1,axiom,
! [A4: nat,K: nat,A: nat,B: nat] :
( ( A4
= ( sup_sup_nat @ K @ A ) )
=> ( ( sup_sup_nat @ A4 @ B )
= ( sup_sup_nat @ K @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_578_boolean__algebra__cancel_Osup1,axiom,
! [A4: set_nat,K: set_nat,A: set_nat,B: set_nat] :
( ( A4
= ( sup_sup_set_nat @ K @ A ) )
=> ( ( sup_sup_set_nat @ A4 @ B )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_579_boolean__algebra__cancel_Osup2,axiom,
! [B4: nat,K: nat,B: nat,A: nat] :
( ( B4
= ( sup_sup_nat @ K @ B ) )
=> ( ( sup_sup_nat @ A @ B4 )
= ( sup_sup_nat @ K @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_580_boolean__algebra__cancel_Osup2,axiom,
! [B4: set_nat,K: set_nat,B: set_nat,A: set_nat] :
( ( B4
= ( sup_sup_set_nat @ K @ B ) )
=> ( ( sup_sup_set_nat @ A @ B4 )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_581_boolean__algebra__cancel_Oinf1,axiom,
! [A4: nat,K: nat,A: nat,B: nat] :
( ( A4
= ( inf_inf_nat @ K @ A ) )
=> ( ( inf_inf_nat @ A4 @ B )
= ( inf_inf_nat @ K @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_582_boolean__algebra__cancel_Oinf1,axiom,
! [A4: set_nat,K: set_nat,A: set_nat,B: set_nat] :
( ( A4
= ( inf_inf_set_nat @ K @ A ) )
=> ( ( inf_inf_set_nat @ A4 @ B )
= ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_583_boolean__algebra__cancel_Oinf2,axiom,
! [B4: nat,K: nat,B: nat,A: nat] :
( ( B4
= ( inf_inf_nat @ K @ B ) )
=> ( ( inf_inf_nat @ A @ B4 )
= ( inf_inf_nat @ K @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_584_boolean__algebra__cancel_Oinf2,axiom,
! [B4: set_nat,K: set_nat,B: set_nat,A: set_nat] :
( ( B4
= ( inf_inf_set_nat @ K @ B ) )
=> ( ( inf_inf_set_nat @ A @ B4 )
= ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_585_finite__has__minimal2,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A4 )
& ( ord_less_eq_nat @ X2 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_586_finite__has__minimal2,axiom,
! [A4: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ( member_set_nat @ A @ A4 )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A4 )
& ( ord_less_eq_set_nat @ X2 @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A4 )
=> ( ( ord_less_eq_set_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_587_finite__has__maximal2,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A @ A4 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A4 )
& ( ord_less_eq_nat @ A @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A4 )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_588_finite__has__maximal2,axiom,
! [A4: set_set_nat,A: set_nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ( member_set_nat @ A @ A4 )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A4 )
& ( ord_less_eq_set_nat @ A @ X2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A4 )
=> ( ( ord_less_eq_set_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_589_finite_OemptyI,axiom,
finite8100373058378681591st_nat @ bot_bot_set_list_nat ).
% finite.emptyI
thf(fact_590_finite_OemptyI,axiom,
finite1152437895449049373et_nat @ bot_bot_set_set_nat ).
% finite.emptyI
thf(fact_591_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_592_infinite__imp__nonempty,axiom,
! [S: set_list_nat] :
( ~ ( finite8100373058378681591st_nat @ S )
=> ( S != bot_bot_set_list_nat ) ) ).
% infinite_imp_nonempty
thf(fact_593_infinite__imp__nonempty,axiom,
! [S: set_set_nat] :
( ~ ( finite1152437895449049373et_nat @ S )
=> ( S != bot_bot_set_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_594_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_595_rev__finite__subset,axiom,
! [B4: set_list_nat,A4: set_list_nat] :
( ( finite8100373058378681591st_nat @ B4 )
=> ( ( ord_le6045566169113846134st_nat @ A4 @ B4 )
=> ( finite8100373058378681591st_nat @ A4 ) ) ) ).
% rev_finite_subset
thf(fact_596_rev__finite__subset,axiom,
! [B4: set_set_nat,A4: set_set_nat] :
( ( finite1152437895449049373et_nat @ B4 )
=> ( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
=> ( finite1152437895449049373et_nat @ A4 ) ) ) ).
% rev_finite_subset
thf(fact_597_rev__finite__subset,axiom,
! [B4: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( finite_finite_nat @ A4 ) ) ) ).
% rev_finite_subset
thf(fact_598_infinite__super,axiom,
! [S: set_list_nat,T2: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ S @ T2 )
=> ( ~ ( finite8100373058378681591st_nat @ S )
=> ~ ( finite8100373058378681591st_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_599_infinite__super,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ S @ T2 )
=> ( ~ ( finite1152437895449049373et_nat @ S )
=> ~ ( finite1152437895449049373et_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_600_infinite__super,axiom,
! [S: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_601_finite__subset,axiom,
! [A4: set_list_nat,B4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A4 @ B4 )
=> ( ( finite8100373058378681591st_nat @ B4 )
=> ( finite8100373058378681591st_nat @ A4 ) ) ) ).
% finite_subset
thf(fact_602_finite__subset,axiom,
! [A4: set_set_nat,B4: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
=> ( ( finite1152437895449049373et_nat @ B4 )
=> ( finite1152437895449049373et_nat @ A4 ) ) ) ).
% finite_subset
thf(fact_603_finite__subset,axiom,
! [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( ( finite_finite_nat @ B4 )
=> ( finite_finite_nat @ A4 ) ) ) ).
% finite_subset
thf(fact_604_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_set_nat] :
( ( sup_sup_set_set_nat @ X @ bot_bot_set_set_nat )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_605_boolean__algebra_Odisj__zero__right,axiom,
! [X: nat > $o] :
( ( sup_sup_nat_o @ X @ bot_bot_nat_o )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_606_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_607_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z2 ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_608_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z2 ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_609_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_nat,Z2: set_nat,X: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ Z2 ) @ X )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ X ) @ ( inf_inf_set_nat @ Z2 @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_610_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_nat,Z2: set_nat,X: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ Z2 ) @ X )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ X ) @ ( sup_sup_set_nat @ Z2 @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_611_finite__UnI,axiom,
! [F2: set_list_nat,G: set_list_nat] :
( ( finite8100373058378681591st_nat @ F2 )
=> ( ( finite8100373058378681591st_nat @ G )
=> ( finite8100373058378681591st_nat @ ( sup_sup_set_list_nat @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_612_finite__UnI,axiom,
! [F2: set_set_nat,G: set_set_nat] :
( ( finite1152437895449049373et_nat @ F2 )
=> ( ( finite1152437895449049373et_nat @ G )
=> ( finite1152437895449049373et_nat @ ( sup_sup_set_set_nat @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_613_finite__UnI,axiom,
! [F2: set_nat,G: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( finite_finite_nat @ G )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G ) ) ) ) ).
% finite_UnI
thf(fact_614_Un__infinite,axiom,
! [S: set_list_nat,T2: set_list_nat] :
( ~ ( finite8100373058378681591st_nat @ S )
=> ~ ( finite8100373058378681591st_nat @ ( sup_sup_set_list_nat @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_615_Un__infinite,axiom,
! [S: set_set_nat,T2: set_set_nat] :
( ~ ( finite1152437895449049373et_nat @ S )
=> ~ ( finite1152437895449049373et_nat @ ( sup_sup_set_set_nat @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_616_Un__infinite,axiom,
! [S: set_nat,T2: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) ) ).
% Un_infinite
thf(fact_617_subset__emptyI,axiom,
! [A4: set_list_nat] :
( ! [X2: list_nat] :
~ ( member_list_nat @ X2 @ A4 )
=> ( ord_le6045566169113846134st_nat @ A4 @ bot_bot_set_list_nat ) ) ).
% subset_emptyI
thf(fact_618_subset__emptyI,axiom,
! [A4: set_set_nat] :
( ! [X2: set_nat] :
~ ( member_set_nat @ X2 @ A4 )
=> ( ord_le6893508408891458716et_nat @ A4 @ bot_bot_set_set_nat ) ) ).
% subset_emptyI
thf(fact_619_subset__emptyI,axiom,
! [A4: set_nat] :
( ! [X2: nat] :
~ ( member_nat @ X2 @ A4 )
=> ( ord_less_eq_set_nat @ A4 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_620_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_621_rotate1__length01,axiom,
! [Xs: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( ( rotate1_nat @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_622_sorted__quicksort,axiom,
! [Xs: list_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( set_or9089632773640736191rt_nat @ Xs ) ) ).
% sorted_quicksort
thf(fact_623_distinct__quicksort,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ ( set_or9089632773640736191rt_nat @ Xs ) )
= ( distinct_nat @ Xs ) ) ).
% distinct_quicksort
thf(fact_624_card__Un__disjoint,axiom,
! [A4: set_list_nat,B4: set_list_nat] :
( ( finite8100373058378681591st_nat @ A4 )
=> ( ( finite8100373058378681591st_nat @ B4 )
=> ( ( ( inf_inf_set_list_nat @ A4 @ B4 )
= bot_bot_set_list_nat )
=> ( ( finite_card_list_nat @ ( sup_sup_set_list_nat @ A4 @ B4 ) )
= ( plus_plus_nat @ ( finite_card_list_nat @ A4 ) @ ( finite_card_list_nat @ B4 ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_625_card__Un__disjoint,axiom,
! [A4: set_set_nat,B4: set_set_nat] :
( ( finite1152437895449049373et_nat @ A4 )
=> ( ( finite1152437895449049373et_nat @ B4 )
=> ( ( ( inf_inf_set_set_nat @ A4 @ B4 )
= bot_bot_set_set_nat )
=> ( ( finite_card_set_nat @ ( sup_sup_set_set_nat @ A4 @ B4 ) )
= ( plus_plus_nat @ ( finite_card_set_nat @ A4 ) @ ( finite_card_set_nat @ B4 ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_626_card__Un__disjoint,axiom,
! [A4: set_nat,B4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( finite_finite_nat @ B4 )
=> ( ( ( inf_inf_set_nat @ A4 @ B4 )
= bot_bot_set_nat )
=> ( ( finite_card_nat @ ( sup_sup_set_nat @ A4 @ B4 ) )
= ( plus_plus_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_627_set__rotate1,axiom,
! [Xs: list_nat] :
( ( set_nat2 @ ( rotate1_nat @ Xs ) )
= ( set_nat2 @ Xs ) ) ).
% set_rotate1
thf(fact_628_length__rotate1,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rotate1_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rotate1
thf(fact_629_distinct1__rotate,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ ( rotate1_nat @ Xs ) )
= ( distinct_nat @ Xs ) ) ).
% distinct1_rotate
thf(fact_630_set__quicksort,axiom,
! [Xs: list_nat] :
( ( set_nat2 @ ( set_or9089632773640736191rt_nat @ Xs ) )
= ( set_nat2 @ Xs ) ) ).
% set_quicksort
thf(fact_631_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
! [A4: set_nat] :
( ( size_size_list_nat @ ( linord2614967742042102400et_nat @ A4 ) )
= ( finite_card_nat @ A4 ) ) ).
% sorted_list_of_set.length_sorted_key_list_of_set
thf(fact_632_card__subset__eq,axiom,
! [B4: set_list_nat,A4: set_list_nat] :
( ( finite8100373058378681591st_nat @ B4 )
=> ( ( ord_le6045566169113846134st_nat @ A4 @ B4 )
=> ( ( ( finite_card_list_nat @ A4 )
= ( finite_card_list_nat @ B4 ) )
=> ( A4 = B4 ) ) ) ) ).
% card_subset_eq
thf(fact_633_card__subset__eq,axiom,
! [B4: set_set_nat,A4: set_set_nat] :
( ( finite1152437895449049373et_nat @ B4 )
=> ( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
=> ( ( ( finite_card_set_nat @ A4 )
= ( finite_card_set_nat @ B4 ) )
=> ( A4 = B4 ) ) ) ) ).
% card_subset_eq
thf(fact_634_card__subset__eq,axiom,
! [B4: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( ( ( finite_card_nat @ A4 )
= ( finite_card_nat @ B4 ) )
=> ( A4 = B4 ) ) ) ) ).
% card_subset_eq
thf(fact_635_infinite__arbitrarily__large,axiom,
! [A4: set_list_nat,N: nat] :
( ~ ( finite8100373058378681591st_nat @ A4 )
=> ? [B7: set_list_nat] :
( ( finite8100373058378681591st_nat @ B7 )
& ( ( finite_card_list_nat @ B7 )
= N )
& ( ord_le6045566169113846134st_nat @ B7 @ A4 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_636_infinite__arbitrarily__large,axiom,
! [A4: set_set_nat,N: nat] :
( ~ ( finite1152437895449049373et_nat @ A4 )
=> ? [B7: set_set_nat] :
( ( finite1152437895449049373et_nat @ B7 )
& ( ( finite_card_set_nat @ B7 )
= N )
& ( ord_le6893508408891458716et_nat @ B7 @ A4 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_637_infinite__arbitrarily__large,axiom,
! [A4: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A4 )
=> ? [B7: set_nat] :
( ( finite_finite_nat @ B7 )
& ( ( finite_card_nat @ B7 )
= N )
& ( ord_less_eq_set_nat @ B7 @ A4 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_638_card__mono,axiom,
! [B4: set_list_nat,A4: set_list_nat] :
( ( finite8100373058378681591st_nat @ B4 )
=> ( ( ord_le6045566169113846134st_nat @ A4 @ B4 )
=> ( ord_less_eq_nat @ ( finite_card_list_nat @ A4 ) @ ( finite_card_list_nat @ B4 ) ) ) ) ).
% card_mono
thf(fact_639_card__mono,axiom,
! [B4: set_set_nat,A4: set_set_nat] :
( ( finite1152437895449049373et_nat @ B4 )
=> ( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ A4 ) @ ( finite_card_set_nat @ B4 ) ) ) ) ).
% card_mono
thf(fact_640_card__mono,axiom,
! [B4: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) ) ) ) ).
% card_mono
thf(fact_641_card__seteq,axiom,
! [B4: set_list_nat,A4: set_list_nat] :
( ( finite8100373058378681591st_nat @ B4 )
=> ( ( ord_le6045566169113846134st_nat @ A4 @ B4 )
=> ( ( ord_less_eq_nat @ ( finite_card_list_nat @ B4 ) @ ( finite_card_list_nat @ A4 ) )
=> ( A4 = B4 ) ) ) ) ).
% card_seteq
thf(fact_642_card__seteq,axiom,
! [B4: set_set_nat,A4: set_set_nat] :
( ( finite1152437895449049373et_nat @ B4 )
=> ( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_nat @ B4 ) @ ( finite_card_set_nat @ A4 ) )
=> ( A4 = B4 ) ) ) ) ).
% card_seteq
thf(fact_643_card__seteq,axiom,
! [B4: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B4 ) @ ( finite_card_nat @ A4 ) )
=> ( A4 = B4 ) ) ) ) ).
% card_seteq
thf(fact_644_exists__subset__between,axiom,
! [A4: set_list_nat,N: nat,C2: set_list_nat] :
( ( ord_less_eq_nat @ ( finite_card_list_nat @ A4 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_list_nat @ C2 ) )
=> ( ( ord_le6045566169113846134st_nat @ A4 @ C2 )
=> ( ( finite8100373058378681591st_nat @ C2 )
=> ? [B7: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ A4 @ B7 )
& ( ord_le6045566169113846134st_nat @ B7 @ C2 )
& ( ( finite_card_list_nat @ B7 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_645_exists__subset__between,axiom,
! [A4: set_set_nat,N: nat,C2: set_set_nat] :
( ( ord_less_eq_nat @ ( finite_card_set_nat @ A4 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_set_nat @ C2 ) )
=> ( ( ord_le6893508408891458716et_nat @ A4 @ C2 )
=> ( ( finite1152437895449049373et_nat @ C2 )
=> ? [B7: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A4 @ B7 )
& ( ord_le6893508408891458716et_nat @ B7 @ C2 )
& ( ( finite_card_set_nat @ B7 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_646_exists__subset__between,axiom,
! [A4: set_nat,N: nat,C2: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C2 ) )
=> ( ( ord_less_eq_set_nat @ A4 @ C2 )
=> ( ( finite_finite_nat @ C2 )
=> ? [B7: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B7 )
& ( ord_less_eq_set_nat @ B7 @ C2 )
& ( ( finite_card_nat @ B7 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_647_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S ) )
=> ~ ! [T3: set_nat] :
( ( ord_less_eq_set_nat @ T3 @ S )
=> ( ( ( finite_card_nat @ T3 )
= N )
=> ~ ( finite_finite_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_648_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_nat,C2: nat] :
( ! [G2: set_nat] :
( ( ord_less_eq_set_nat @ G2 @ F2 )
=> ( ( finite_finite_nat @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G2 ) @ C2 ) ) )
=> ( ( finite_finite_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_649_card__length,axiom,
! [Xs: list_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( set_nat2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% card_length
thf(fact_650_card__Un__le,axiom,
! [A4: set_nat,B4: set_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( sup_sup_set_nat @ A4 @ B4 ) ) @ ( plus_plus_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) ) ) ).
% card_Un_le
thf(fact_651_card__distinct,axiom,
! [Xs: list_nat] :
( ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
= ( size_size_list_nat @ Xs ) )
=> ( distinct_nat @ Xs ) ) ).
% card_distinct
thf(fact_652_distinct__card,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ Xs )
=> ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ) ).
% distinct_card
thf(fact_653_card__Un__Int,axiom,
! [A4: set_nat,B4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( finite_finite_nat @ B4 )
=> ( ( plus_plus_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) )
= ( plus_plus_nat @ ( finite_card_nat @ ( sup_sup_set_nat @ A4 @ B4 ) ) @ ( finite_card_nat @ ( inf_inf_set_nat @ A4 @ B4 ) ) ) ) ) ) ).
% card_Un_Int
thf(fact_654_arb__finite__subset,axiom,
! [Y6: set_nat,N: nat] :
( ( finite_finite_nat @ Y6 )
=> ? [X4: set_nat] :
( ( ( inf_inf_set_nat @ Y6 @ X4 )
= bot_bot_set_nat )
& ( finite_finite_nat @ X4 )
& ( ord_less_eq_nat @ N @ ( finite_card_nat @ X4 ) ) ) ) ).
% arb_finite_subset
thf(fact_655_card__le__if__inj__on__rel,axiom,
! [B4: set_nat,A4: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ B4 )
=> ( ! [A3: nat] :
( ( member_nat @ A3 @ A4 )
=> ? [B8: nat] :
( ( member_nat @ B8 @ B4 )
& ( R @ A3 @ B8 ) ) )
=> ( ! [A1: nat,A22: nat,B3: nat] :
( ( member_nat @ A1 @ A4 )
=> ( ( member_nat @ A22 @ A4 )
=> ( ( member_nat @ B3 @ B4 )
=> ( ( R @ A1 @ B3 )
=> ( ( R @ A22 @ B3 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_656_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : ( member_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_657_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_658_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
! [A4: set_nat,L: list_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ( sorted_wrt_nat @ ord_less_nat @ L )
& ( ( set_nat2 @ L )
= A4 )
& ( ( size_size_list_nat @ L )
= ( finite_card_nat @ A4 ) ) )
= ( ( linord2614967742042102400et_nat @ A4 )
= L ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_unique
thf(fact_659_psubsetI,axiom,
! [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( ( A4 != B4 )
=> ( ord_less_set_nat @ A4 @ B4 ) ) ) ).
% psubsetI
thf(fact_660_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_661_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_662_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_663_psubset__card__mono,axiom,
! [B4: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_set_nat @ A4 @ B4 )
=> ( ord_less_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) ) ) ) ).
% psubset_card_mono
thf(fact_664_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_665_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_666_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_667_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_668_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_669_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_670_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_671_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_672_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_673_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_674_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_675_order__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_676_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_677_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_678_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_679_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_680_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_681_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_682_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_683_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_684_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_685_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_686_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N3: nat] :
( ( P3 @ N3 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ~ ( P3 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_687_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_688_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_689_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_690_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_691_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X2: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X2 )
=> ( P @ Y5 ) )
=> ( P @ X2 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_692_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_693_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_694_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_695_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_696_gt__ex,axiom,
! [X: nat] :
? [X_12: nat] : ( ord_less_nat @ X @ X_12 ) ).
% gt_ex
thf(fact_697_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_698_order__le__imp__less__or__eq,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_699_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_700_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_701_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_702_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_703_order__less__le__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_704_order__less__le__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_705_order__less__le__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_706_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_707_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_708_order__le__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_709_order__le__less__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ ( F @ B ) @ C )
=> ( ! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_710_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_711_order__le__less__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_712_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_713_order__less__le__trans,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z2 )
=> ( ord_less_set_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_714_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_715_order__le__less__trans,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_set_nat @ Y @ Z2 )
=> ( ord_less_set_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_716_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_717_order__neq__le__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( A != B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_718_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_719_order__le__neq__trans,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_set_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_720_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_721_order__less__imp__le,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_set_nat @ X @ Y )
=> ( ord_less_eq_set_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_722_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_723_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_724_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_725_order__less__le,axiom,
( ord_less_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_726_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_727_order__le__less,axiom,
( ord_less_eq_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( ord_less_set_nat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_728_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_729_dual__order_Ostrict__implies__order,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_730_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_731_order_Ostrict__implies__order,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_732_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ~ ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_733_dual__order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
& ~ ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_734_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_735_dual__order_Ostrict__trans2,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_736_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_737_dual__order_Ostrict__trans1,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_738_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_739_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_740_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_741_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [B2: set_nat,A2: set_nat] :
( ( ord_less_set_nat @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_742_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_743_order_Ostrict__iff__not,axiom,
( ord_less_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
& ~ ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_744_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_745_order_Ostrict__trans2,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_746_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_747_order_Ostrict__trans1,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_748_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_749_order_Ostrict__iff__order,axiom,
( ord_less_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_750_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_751_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_752_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_753_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_754_less__le__not__le,axiom,
( ord_less_set_nat
= ( ^ [X3: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y4 )
& ~ ( ord_less_eq_set_nat @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_755_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_756_antisym__conv2,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ~ ( ord_less_set_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_757_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_758_antisym__conv1,axiom,
! [X: set_nat,Y: set_nat] :
( ~ ( ord_less_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_759_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_760_nless__le,axiom,
! [A: set_nat,B: set_nat] :
( ( ~ ( ord_less_set_nat @ A @ B ) )
= ( ~ ( ord_less_eq_set_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_761_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_762_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_763_leD,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ~ ( ord_less_set_nat @ X @ Y ) ) ).
% leD
thf(fact_764_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_765_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_766_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_767_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_768_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_769_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_770_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_771_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_772_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_773_bot_Onot__eq__extremum,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
= ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_774_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_775_bot_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_776_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_777_not__psubset__empty,axiom,
! [A4: set_nat] :
~ ( ord_less_set_nat @ A4 @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_778_finite__psubset__induct,axiom,
! [A4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ! [A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ! [B9: set_nat] :
( ( ord_less_set_nat @ B9 @ A7 )
=> ( P @ B9 ) )
=> ( P @ A7 ) ) )
=> ( P @ A4 ) ) ) ).
% finite_psubset_induct
thf(fact_779_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_set_nat @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_780_subset__psubset__trans,axiom,
! [A4: set_nat,B4: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( ( ord_less_set_nat @ B4 @ C2 )
=> ( ord_less_set_nat @ A4 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_781_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ~ ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_782_psubset__subset__trans,axiom,
! [A4: set_nat,B4: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A4 @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ C2 )
=> ( ord_less_set_nat @ A4 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_783_psubset__imp__subset,axiom,
! [A4: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ A4 @ B4 )
=> ( ord_less_eq_set_nat @ A4 @ B4 ) ) ).
% psubset_imp_subset
thf(fact_784_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_785_psubsetE,axiom,
! [A4: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ A4 @ B4 )
=> ~ ( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ).
% psubsetE
thf(fact_786_less__supI1,axiom,
! [X: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ X @ A )
=> ( ord_less_set_nat @ X @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% less_supI1
thf(fact_787_less__supI1,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_nat @ X @ A )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI1
thf(fact_788_less__supI2,axiom,
! [X: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ X @ B )
=> ( ord_less_set_nat @ X @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% less_supI2
thf(fact_789_less__supI2,axiom,
! [X: nat,B: nat,A: nat] :
( ( ord_less_nat @ X @ B )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI2
thf(fact_790_sup_Oabsorb3,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( sup_sup_set_nat @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_791_sup_Oabsorb3,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_792_sup_Oabsorb4,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_793_sup_Oabsorb4,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_794_sup_Ostrict__boundedE,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_set_nat @ B @ A )
=> ~ ( ord_less_set_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_795_sup_Ostrict__boundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_796_sup_Ostrict__order__iff,axiom,
( ord_less_set_nat
= ( ^ [B2: set_nat,A2: set_nat] :
( ( A2
= ( sup_sup_set_nat @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_797_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_798_sup_Ostrict__coboundedI1,axiom,
! [C: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ C @ A )
=> ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_799_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ C @ A )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_800_sup_Ostrict__coboundedI2,axiom,
! [C: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_801_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_802_inf_Ostrict__coboundedI2,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ C )
=> ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_803_inf_Ostrict__coboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_804_inf_Ostrict__coboundedI1,axiom,
! [A: set_nat,C: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ C )
=> ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_805_inf_Ostrict__coboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ A @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_806_inf_Ostrict__order__iff,axiom,
( ord_less_set_nat
= ( ^ [A2: set_nat,B2: set_nat] :
( ( A2
= ( inf_inf_set_nat @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_807_inf_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_808_inf_Ostrict__boundedE,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) )
=> ~ ( ( ord_less_set_nat @ A @ B )
=> ~ ( ord_less_set_nat @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_809_inf_Ostrict__boundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_810_inf_Oabsorb4,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( inf_inf_set_nat @ A @ B )
= B ) ) ).
% inf.absorb4
thf(fact_811_inf_Oabsorb4,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( inf_inf_nat @ A @ B )
= B ) ) ).
% inf.absorb4
thf(fact_812_inf_Oabsorb3,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( inf_inf_set_nat @ A @ B )
= A ) ) ).
% inf.absorb3
thf(fact_813_inf_Oabsorb3,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( inf_inf_nat @ A @ B )
= A ) ) ).
% inf.absorb3
thf(fact_814_less__infI2,axiom,
! [B: set_nat,X: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ X )
=> ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X ) ) ).
% less_infI2
thf(fact_815_less__infI2,axiom,
! [B: nat,X: nat,A: nat] :
( ( ord_less_nat @ B @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% less_infI2
thf(fact_816_less__infI1,axiom,
! [A: set_nat,X: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ X )
=> ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X ) ) ).
% less_infI1
thf(fact_817_less__infI1,axiom,
! [A: nat,X: nat,B: nat] :
( ( ord_less_nat @ A @ X )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).
% less_infI1
thf(fact_818_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_819_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_820_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_821_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
| ( M2 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_822_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_823_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
& ( M2 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_824_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs2: list_nat] :
( ! [Ys2: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_825_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_826_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_827_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_828_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_829_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_830_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_831_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_832_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_833_card__psubset,axiom,
! [B4: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) )
=> ( ord_less_set_nat @ A4 @ B4 ) ) ) ) ).
% card_psubset
thf(fact_834_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_835_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_836_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_837_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_838_strict__sorted__imp__sorted,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ Xs ) ) ).
% strict_sorted_imp_sorted
thf(fact_839_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_840_strict__sorted__equal,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs )
=> ( ( sorted_wrt_nat @ ord_less_nat @ Ys )
=> ( ( ( set_nat2 @ Ys )
= ( set_nat2 @ Xs ) )
=> ( Ys = Xs ) ) ) ) ).
% strict_sorted_equal
thf(fact_841_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
! [A4: set_nat] : ( sorted_wrt_nat @ ord_less_nat @ ( linord2614967742042102400et_nat @ A4 ) ) ).
% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_842_finite__maxlen,axiom,
! [M4: set_list_nat] :
( ( finite8100373058378681591st_nat @ M4 )
=> ? [N2: nat] :
! [X6: list_nat] :
( ( member_list_nat @ X6 @ M4 )
=> ( ord_less_nat @ ( size_size_list_nat @ X6 ) @ N2 ) ) ) ).
% finite_maxlen
thf(fact_843_strict__sorted__iff,axiom,
! [L: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ L )
= ( ( sorted_wrt_nat @ ord_less_eq_nat @ L )
& ( distinct_nat @ L ) ) ) ).
% strict_sorted_iff
thf(fact_844_arb__element,axiom,
! [Y6: set_nat] :
( ( finite_finite_nat @ Y6 )
=> ? [X2: nat] :
~ ( member_nat @ X2 @ Y6 ) ) ).
% arb_element
thf(fact_845_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M4: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M4 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_846_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M2: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N4 )
=> ( ord_less_eq_nat @ X3 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_847_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ~ ! [L2: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ L2 )
=> ( ( ( set_nat2 @ L2 )
= A4 )
=> ( ( size_size_list_nat @ L2 )
!= ( finite_card_nat @ A4 ) ) ) ) ) ).
% sorted_list_of_set.finite_set_strict_sorted
thf(fact_848_infinite__growing,axiom,
! [X7: set_nat] :
( ( X7 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X7 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ X7 )
& ( ord_less_nat @ X2 @ Xa ) ) )
=> ~ ( finite_finite_nat @ X7 ) ) ) ).
% infinite_growing
thf(fact_849_ex__min__if__finite,axiom,
! [S: set_nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ S )
& ~ ? [Xa: nat] :
( ( member_nat @ Xa @ S )
& ( ord_less_nat @ Xa @ X2 ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_850_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_851_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_852_card__gt__1D,axiom,
! [A4: set_nat] :
( ( ord_less_nat @ one_one_nat @ ( finite_card_nat @ A4 ) )
=> ? [X2: nat,Y2: nat] :
( ( member_nat @ X2 @ A4 )
& ( member_nat @ Y2 @ A4 )
& ( X2 != Y2 ) ) ) ).
% card_gt_1D
thf(fact_853_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_854_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N2 )
& ~ ( P @ M5 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_855_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N2 )
=> ( P @ M5 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_856_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_857_less__not__refl3,axiom,
! [S2: nat,T4: nat] :
( ( ord_less_nat @ S2 @ T4 )
=> ( S2 != T4 ) ) ).
% less_not_refl3
thf(fact_858_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_859_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_860_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_861_psubsetD,axiom,
! [A4: set_nat,B4: set_nat,C: nat] :
( ( ord_less_set_nat @ A4 @ B4 )
=> ( ( member_nat @ C @ A4 )
=> ( member_nat @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_862_bounded__nat__set__is__finite,axiom,
! [N5: set_nat,N: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ N5 )
=> ( ord_less_nat @ X2 @ N ) )
=> ( finite_finite_nat @ N5 ) ) ).
% bounded_nat_set_is_finite
thf(fact_863_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N4: set_nat] :
? [M2: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N4 )
=> ( ord_less_nat @ X3 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_864_arg__min__if__finite_I2_J,axiom,
! [S: set_nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ~ ? [X6: nat] :
( ( member_nat @ X6 @ S )
& ( ord_less_nat @ ( F @ X6 ) @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_865_arg__min__least,axiom,
! [S: set_nat,Y: nat,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( S != bot_bot_set_nat )
=> ( ( member_nat @ Y @ S )
=> ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).
% arg_min_least
thf(fact_866_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K3 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K3 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_867_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C4: nat] :
( ( ord_less_eq_nat @ A @ C4 )
& ( ord_less_eq_nat @ C4 @ B )
& ! [X6: nat] :
( ( ( ord_less_eq_nat @ A @ X6 )
& ( ord_less_nat @ X6 @ C4 ) )
=> ( P @ X6 ) )
& ! [D3: nat] :
( ! [X2: nat] :
( ( ( ord_less_eq_nat @ A @ X2 )
& ( ord_less_nat @ X2 @ D3 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_nat @ D3 @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_868_minf_I8_J,axiom,
! [T4: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ~ ( ord_less_eq_nat @ T4 @ X6 ) ) ).
% minf(8)
thf(fact_869_minf_I6_J,axiom,
! [T4: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ord_less_eq_nat @ X6 @ T4 ) ) ).
% minf(6)
thf(fact_870_pinf_I8_J,axiom,
! [T4: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ord_less_eq_nat @ T4 @ X6 ) ) ).
% pinf(8)
thf(fact_871_pinf_I6_J,axiom,
! [T4: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ~ ( ord_less_eq_nat @ X6 @ T4 ) ) ).
% pinf(6)
thf(fact_872_verit__comp__simplify1_I3_J,axiom,
! [B10: nat,A8: nat] :
( ( ~ ( ord_less_eq_nat @ B10 @ A8 ) )
= ( ord_less_nat @ A8 @ B10 ) ) ).
% verit_comp_simplify1(3)
thf(fact_873_sorted__quicksort__acc,axiom,
! [Ac: list_nat,Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Ac )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( set_nat2 @ Ac ) )
=> ( ord_less_nat @ X2 @ Xa2 ) ) )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( set_or5558937660843164036cc_nat @ Ac @ Xs ) ) ) ) ).
% sorted_quicksort_acc
thf(fact_874_is__singleton__altdef,axiom,
( is_singleton_nat
= ( ^ [A5: set_nat] :
( ( finite_card_nat @ A5 )
= one_one_nat ) ) ) ).
% is_singleton_altdef
thf(fact_875_rotate__length01,axiom,
! [Xs: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
=> ( ( rotate_nat @ N @ Xs )
= Xs ) ) ).
% rotate_length01
thf(fact_876_set__rotate,axiom,
! [N: nat,Xs: list_nat] :
( ( set_nat2 @ ( rotate_nat @ N @ Xs ) )
= ( set_nat2 @ Xs ) ) ).
% set_rotate
thf(fact_877_length__rotate,axiom,
! [N: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( rotate_nat @ N @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rotate
thf(fact_878_distinct__rotate,axiom,
! [N: nat,Xs: list_nat] :
( ( distinct_nat @ ( rotate_nat @ N @ Xs ) )
= ( distinct_nat @ Xs ) ) ).
% distinct_rotate
thf(fact_879_set__quicksort__acc,axiom,
! [Ac: list_nat,Xs: list_nat] :
( ( set_nat2 @ ( set_or5558937660843164036cc_nat @ Ac @ Xs ) )
= ( sup_sup_set_nat @ ( set_nat2 @ Ac ) @ ( set_nat2 @ Xs ) ) ) ).
% set_quicksort_acc
thf(fact_880_is__singletonI_H,axiom,
! [A4: set_nat] :
( ( A4 != bot_bot_set_nat )
=> ( ! [X2: nat,Y2: nat] :
( ( member_nat @ X2 @ A4 )
=> ( ( member_nat @ Y2 @ A4 )
=> ( X2 = Y2 ) ) )
=> ( is_singleton_nat @ A4 ) ) ) ).
% is_singletonI'
thf(fact_881_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_882_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_883_verit__comp__simplify1_I2_J,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_884_distinct__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( distinct_nat @ ( append_nat @ Xs @ Ys ) )
= ( ( distinct_nat @ Xs )
& ( distinct_nat @ Ys )
& ( ( inf_inf_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) )
= bot_bot_set_nat ) ) ) ).
% distinct_append
thf(fact_885_sorted__iff__nth__mono,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I4: nat,J3: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I4 ) @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_886_append__eq__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat,Us: list_nat,Vs: list_nat] :
( ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
| ( ( size_size_list_nat @ Us )
= ( size_size_list_nat @ Vs ) ) )
=> ( ( ( append_nat @ Xs @ Us )
= ( append_nat @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_887_set__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( set_nat2 @ ( append_nat @ Xs @ Ys ) )
= ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) ) ).
% set_append
thf(fact_888_length__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( size_size_list_nat @ ( append_nat @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% length_append
thf(fact_889_nth__append__length__plus,axiom,
! [Xs: list_nat,Ys: list_nat,N: nat] :
( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ N ) )
= ( nth_nat @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_890_nth__equalityI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I2 )
= ( nth_nat @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_891_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X8: nat] : ( P @ I4 @ X8 ) ) )
= ( ? [Xs3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P @ I4 @ ( nth_nat @ Xs3 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_892_list__eq__iff__nth__eq,axiom,
( ( ^ [Y3: list_nat,Z: list_nat] : ( Y3 = Z ) )
= ( ^ [Xs3: list_nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs3 ) )
=> ( ( nth_nat @ Xs3 @ I4 )
= ( nth_nat @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_893_sorted__wrt__append,axiom,
! [P: nat > nat > $o,Xs: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ P @ ( append_nat @ Xs @ Ys ) )
= ( ( sorted_wrt_nat @ P @ Xs )
& ( sorted_wrt_nat @ P @ Ys )
& ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ ( set_nat2 @ Ys ) )
=> ( P @ X3 @ Y4 ) ) ) ) ) ).
% sorted_wrt_append
thf(fact_894_rotate__append,axiom,
! [L: list_nat,Q2: list_nat] :
( ( rotate_nat @ ( size_size_list_nat @ L ) @ ( append_nat @ L @ Q2 ) )
= ( append_nat @ Q2 @ L ) ) ).
% rotate_append
thf(fact_895_distinct__quicksort__acc,axiom,
! [Ac: list_nat,Xs: list_nat] :
( ( distinct_nat @ ( set_or5558937660843164036cc_nat @ Ac @ Xs ) )
= ( distinct_nat @ ( append_nat @ Ac @ Xs ) ) ) ).
% distinct_quicksort_acc
thf(fact_896_all__set__conv__all__nth,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( P @ X3 ) ) )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I4 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_897_all__nth__imp__all__set,axiom,
! [Xs: list_nat,P: nat > $o,X: nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I2 ) ) )
=> ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_898_in__set__conv__nth,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_899_list__ball__nth,axiom,
! [N: nat,Xs: list_nat,P: nat > $o] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( P @ X2 ) )
=> ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_900_nth__mem,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_901_nth__eq__iff__index__eq,axiom,
! [Xs: list_nat,I: nat,J: nat] :
( ( distinct_nat @ Xs )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( ( nth_nat @ Xs @ I )
= ( nth_nat @ Xs @ J ) )
= ( I = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_902_distinct__conv__nth,axiom,
( distinct_nat
= ( ^ [Xs3: list_nat] :
! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs3 ) )
=> ! [J3: nat] :
( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs3 ) )
=> ( ( I4 != J3 )
=> ( ( nth_nat @ Xs3 @ I4 )
!= ( nth_nat @ Xs3 @ J3 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_903_sorted__wrt__iff__nth__less,axiom,
( sorted_wrt_nat
= ( ^ [P3: nat > nat > $o,Xs3: list_nat] :
! [I4: nat,J3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs3 ) )
=> ( P3 @ ( nth_nat @ Xs3 @ I4 ) @ ( nth_nat @ Xs3 @ J3 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_904_sorted__wrt__nth__less,axiom,
! [P: nat > nat > $o,Xs: list_nat,I: nat,J: nat] :
( ( sorted_wrt_nat @ P @ Xs )
=> ( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_905_sorted__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( append_nat @ Xs @ Ys ) )
= ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
& ( sorted_wrt_nat @ ord_less_eq_nat @ Ys )
& ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ ( set_nat2 @ Ys ) )
=> ( ord_less_eq_nat @ X3 @ Y4 ) ) ) ) ) ).
% sorted_append
thf(fact_906_sorted__iff__nth__mono__less,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I4: nat,J3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I4 ) @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_907_distinct__Ex1,axiom,
! [Xs: list_nat,X: nat] :
( ( distinct_nat @ Xs )
=> ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ? [X2: nat] :
( ( ord_less_nat @ X2 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ X2 )
= X )
& ! [Y5: nat] :
( ( ( ord_less_nat @ Y5 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ Y5 )
= X ) )
=> ( Y5 = X2 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_908_sorted__wrt__less__idx,axiom,
! [Ns: list_nat,I: nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Ns )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
=> ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).
% sorted_wrt_less_idx
thf(fact_909_sorted__nth__mono,axiom,
! [Xs: list_nat,I: nat,J: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_910_sorted__rev__nth__mono,axiom,
! [Xs: list_nat,I: nat,J: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ J ) @ ( nth_nat @ Xs @ I ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_911_sorted__rev__iff__nth__mono,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
= ( ! [I4: nat,J3: nat] :
( ( ord_less_eq_nat @ I4 @ J3 )
=> ( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ J3 ) @ ( nth_nat @ Xs @ I4 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_912_nth__rotate,axiom,
! [N: nat,Xs: list_nat,M: nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( rotate_nat @ M @ Xs ) @ N )
= ( nth_nat @ Xs @ ( modulo_modulo_nat @ ( plus_plus_nat @ M @ N ) @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% nth_rotate
thf(fact_913_set__rev,axiom,
! [Xs: list_nat] :
( ( set_nat2 @ ( rev_nat @ Xs ) )
= ( set_nat2 @ Xs ) ) ).
% set_rev
thf(fact_914_length__rev,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rev_nat @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_rev
thf(fact_915_distinct__rev,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ ( rev_nat @ Xs ) )
= ( distinct_nat @ Xs ) ) ).
% distinct_rev
thf(fact_916_rotate__conv__mod,axiom,
( rotate_nat
= ( ^ [N3: nat,Xs3: list_nat] : ( rotate_nat @ ( modulo_modulo_nat @ N3 @ ( size_size_list_nat @ Xs3 ) ) @ Xs3 ) ) ) ).
% rotate_conv_mod
thf(fact_917_mod__add__self2,axiom,
! [A: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_add_self2
thf(fact_918_mod__add__self1,axiom,
! [B: nat,A: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( modulo_modulo_nat @ A @ B ) ) ).
% mod_add_self1
thf(fact_919_mod__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ M ) ).
% mod_less_eq_dividend
thf(fact_920_mod__add__eq,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% mod_add_eq
thf(fact_921_mod__add__cong,axiom,
! [A: nat,C: nat,A8: nat,B: nat,B10: nat] :
( ( ( modulo_modulo_nat @ A @ C )
= ( modulo_modulo_nat @ A8 @ C ) )
=> ( ( ( modulo_modulo_nat @ B @ C )
= ( modulo_modulo_nat @ B10 @ C ) )
=> ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A8 @ B10 ) @ C ) ) ) ) ).
% mod_add_cong
thf(fact_922_mod__add__left__eq,axiom,
! [A: nat,C: nat,B: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% mod_add_left_eq
thf(fact_923_mod__add__right__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
= ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% mod_add_right_eq
thf(fact_924_sorted__rev__iff__nth__Suc,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
= ( ! [I4: nat] :
( ( ord_less_nat @ ( suc @ I4 ) @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ ( suc @ I4 ) ) @ ( nth_nat @ Xs @ I4 ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_925_nth__rotate1,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( rotate1_nat @ Xs ) @ N )
= ( nth_nat @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% nth_rotate1
thf(fact_926_sorted__iff__nth__Suc,axiom,
! [Xs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
= ( ! [I4: nat] :
( ( ord_less_nat @ ( suc @ I4 ) @ ( size_size_list_nat @ Xs ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I4 ) @ ( nth_nat @ Xs @ ( suc @ I4 ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_927_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_928_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_929_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_930_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_931_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_932_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_933_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_934_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_935_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_936_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K3: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K3 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K3 )
=> ( P @ I2 @ K3 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_937_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_938_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_939_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_940_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_941_Nat_OAll__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ I4 ) ) ) ) ).
% Nat.All_less_Suc
thf(fact_942_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_943_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_944_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ N )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_945_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_946_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_947_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_948_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_949_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_950_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_951_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_952_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_953_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_954_Suc__le__D,axiom,
! [N: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
=> ? [M3: nat] :
( M7
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_955_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_956_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_957_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_958_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N2 )
=> ( P @ M5 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_959_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_960_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X2: nat] : ( R2 @ X2 @ X2 )
=> ( ! [X2: nat,Y2: nat,Z3: nat] :
( ( R2 @ X2 @ Y2 )
=> ( ( R2 @ Y2 @ Z3 )
=> ( R2 @ X2 @ Z3 ) ) )
=> ( ! [N2: nat] : ( R2 @ N2 @ ( suc @ N2 ) )
=> ( R2 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_961_nat__arith_Osuc1,axiom,
! [A4: nat,K: nat,A: nat] :
( ( A4
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A4 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_962_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_963_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_964_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_965_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_966_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N6: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N6 )
=> ( ord_less_eq_nat @ ( F @ N6 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_967_lift__Suc__antimono__le,axiom,
! [F: nat > set_nat,N: nat,N6: nat] :
( ! [N2: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N6 )
=> ( ord_less_eq_set_nat @ ( F @ N6 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_968_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N6: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N6 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_969_lift__Suc__mono__le,axiom,
! [F: nat > set_nat,N: nat,N6: nat] :
( ! [N2: nat] : ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N6 )
=> ( ord_less_eq_set_nat @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_970_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N6: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N6 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N6 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_971_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_972_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_973_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_974_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_975_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_976_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_977_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_978_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_979_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_980_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_981_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_982_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_983_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_984_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M2: nat,N3: nat] :
? [K2: nat] :
( N3
= ( suc @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_985_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K3: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_986_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_987_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_988_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_989_mod__Suc__le__divisor,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N ) ) @ N ) ).
% mod_Suc_le_divisor
thf(fact_990_ex__out,axiom,
! [X7: set_nat] :
( ( finite_finite_nat @ X7 )
=> ? [K3: nat] :
( ~ ( member_nat @ K3 @ X7 )
& ( ord_less_nat @ K3 @ ( suc @ ( finite_card_nat @ X7 ) ) ) ) ) ).
% ex_out
thf(fact_991_rev__nth,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( rev_nat @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ ( suc @ N ) ) ) ) ) ).
% rev_nth
thf(fact_992_add__nth__length,axiom,
! [I: nat,Zs: list_nat,Z2: nat] :
( ( ord_less_eq_nat @ I @ ( size_size_list_nat @ Zs ) )
=> ( ( size_size_list_nat @ ( add_nth_nat @ I @ Z2 @ Zs ) )
= ( suc @ ( size_size_list_nat @ Zs ) ) ) ) ).
% add_nth_length
thf(fact_993_distinct__swap,axiom,
! [I: nat,Xs: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( distinct_nat @ ( list_update_nat @ ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I ) ) )
= ( distinct_nat @ Xs ) ) ) ) ).
% distinct_swap
thf(fact_994_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_995_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_996_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_997_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_998_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_999_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_1000_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1001_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1002_length__list__update,axiom,
! [Xs: list_nat,I: nat,X: nat] :
( ( size_size_list_nat @ ( list_update_nat @ Xs @ I @ X ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_list_update
thf(fact_1003_list__update__id,axiom,
! [Xs: list_nat,I: nat] :
( ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ I ) )
= Xs ) ).
% list_update_id
thf(fact_1004_nth__list__update__neq,axiom,
! [I: nat,J: nat,Xs: list_nat,X: nat] :
( ( I != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
= ( nth_nat @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_1005_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_1006_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_1007_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1008_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1009_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1010_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1011_list__update__beyond,axiom,
! [Xs: list_nat,I: nat,X: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I )
=> ( ( list_update_nat @ Xs @ I @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_1012_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1013_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1014_nth__list__update__eq,axiom,
! [I: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ I )
= X ) ) ).
% nth_list_update_eq
thf(fact_1015_set__swap,axiom,
! [I: nat,Xs: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
=> ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I ) ) )
= ( set_nat2 @ Xs ) ) ) ) ).
% set_swap
thf(fact_1016_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1017_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_1018_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_1019_diff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_1020_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_1021_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1022_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1023_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_1024_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_1025_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_1026_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1027_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1028_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_1029_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_1030_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_1031_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_1032_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_1033_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_1034_list__update__append,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat,X: nat] :
( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ N @ X )
= ( append_nat @ ( list_update_nat @ Xs @ N @ X ) @ Ys ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ N @ X )
= ( append_nat @ Xs @ ( list_update_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ X ) ) ) ) ) ).
% list_update_append
thf(fact_1035_rev__update,axiom,
! [K: nat,Xs: list_nat,Y: nat] :
( ( ord_less_nat @ K @ ( size_size_list_nat @ Xs ) )
=> ( ( rev_nat @ ( list_update_nat @ Xs @ K @ Y ) )
= ( list_update_nat @ ( rev_nat @ Xs ) @ ( minus_minus_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ K ) @ one_one_nat ) @ Y ) ) ) ).
% rev_update
thf(fact_1036_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1037_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1038_diff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% diff_add
thf(fact_1039_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_1040_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1041_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1042_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1043_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1044_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1045_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1046_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1047_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1048_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1049_diff__shunt__var,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( minus_minus_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X @ Y ) ) ).
% diff_shunt_var
thf(fact_1050_set__update__subsetI,axiom,
! [Xs: list_nat,A4: set_nat,X: nat,I: nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A4 )
=> ( ( member_nat @ X @ A4 )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X ) ) @ A4 ) ) ) ).
% set_update_subsetI
thf(fact_1051_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_1052_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1053_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_1054_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_1055_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1056_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_1057_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1058_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1059_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1060_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1061_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1062_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1063_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_1064_le__mod__geq,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( modulo_modulo_nat @ M @ N )
= ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% le_mod_geq
thf(fact_1065_set__update__memI,axiom,
! [N: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_1066_list__update__append1,axiom,
! [I: nat,Xs: list_nat,Ys: list_nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ I @ X )
= ( append_nat @ ( list_update_nat @ Xs @ I @ X ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_1067_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1068_nth__list__update,axiom,
! [I: nat,Xs: list_nat,J: nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
= X ) )
& ( ( I != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
= ( nth_nat @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_1069_list__update__same__conv,axiom,
! [I: nat,Xs: list_nat,X: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( ( list_update_nat @ Xs @ I @ X )
= Xs )
= ( ( nth_nat @ Xs @ I )
= X ) ) ) ).
% list_update_same_conv
thf(fact_1070_nth__append,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat] :
( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
= ( nth_nat @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
= ( nth_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_1071_rotate__rev,axiom,
! [N: nat,Xs: list_nat] :
( ( rotate_nat @ N @ ( rev_nat @ Xs ) )
= ( rev_nat @ ( rotate_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Xs ) ) ) @ Xs ) ) ) ).
% rotate_rev
thf(fact_1072_add__nth__rem__nth__self,axiom,
! [I: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( add_nth_nat @ I @ ( nth_nat @ Xs @ I ) @ ( rem_nth_nat @ I @ Xs ) )
= Xs ) ) ).
% add_nth_rem_nth_self
thf(fact_1073_rem__nth__length,axiom,
! [I: nat,Xs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
=> ( ( size_size_list_nat @ ( rem_nth_nat @ I @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ) ).
% rem_nth_length
thf(fact_1074_Diff__cancel,axiom,
! [A4: set_nat] :
( ( minus_minus_set_nat @ A4 @ A4 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_1075_empty__Diff,axiom,
! [A4: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A4 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_1076_Diff__empty,axiom,
! [A4: set_nat] :
( ( minus_minus_set_nat @ A4 @ bot_bot_set_nat )
= A4 ) ).
% Diff_empty
thf(fact_1077_finite__Diff2,axiom,
! [B4: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ B4 ) )
= ( finite_finite_nat @ A4 ) ) ) ).
% finite_Diff2
thf(fact_1078_finite__Diff,axiom,
! [A4: set_nat,B4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ B4 ) ) ) ).
% finite_Diff
thf(fact_1079_Un__Diff__cancel,axiom,
! [A4: set_nat,B4: set_nat] :
( ( sup_sup_set_nat @ A4 @ ( minus_minus_set_nat @ B4 @ A4 ) )
= ( sup_sup_set_nat @ A4 @ B4 ) ) ).
% Un_Diff_cancel
thf(fact_1080_Un__Diff__cancel2,axiom,
! [B4: set_nat,A4: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B4 @ A4 ) @ A4 )
= ( sup_sup_set_nat @ B4 @ A4 ) ) ).
% Un_Diff_cancel2
thf(fact_1081_Diff__eq__empty__iff,axiom,
! [A4: set_nat,B4: set_nat] :
( ( ( minus_minus_set_nat @ A4 @ B4 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A4 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_1082_Diff__disjoint,axiom,
! [A4: set_nat,B4: set_nat] :
( ( inf_inf_set_nat @ A4 @ ( minus_minus_set_nat @ B4 @ A4 ) )
= bot_bot_set_nat ) ).
% Diff_disjoint
thf(fact_1083_Diff__Int__distrib2,axiom,
! [A4: set_nat,B4: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ ( minus_minus_set_nat @ A4 @ B4 ) @ C2 )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ A4 @ C2 ) @ ( inf_inf_set_nat @ B4 @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_1084_Diff__Int__distrib,axiom,
! [C2: set_nat,A4: set_nat,B4: set_nat] :
( ( inf_inf_set_nat @ C2 @ ( minus_minus_set_nat @ A4 @ B4 ) )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ C2 @ A4 ) @ ( inf_inf_set_nat @ C2 @ B4 ) ) ) ).
% Diff_Int_distrib
thf(fact_1085_Diff__Diff__Int,axiom,
! [A4: set_nat,B4: set_nat] :
( ( minus_minus_set_nat @ A4 @ ( minus_minus_set_nat @ A4 @ B4 ) )
= ( inf_inf_set_nat @ A4 @ B4 ) ) ).
% Diff_Diff_Int
thf(fact_1086_Diff__Int2,axiom,
! [A4: set_nat,C2: set_nat,B4: set_nat] :
( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A4 @ C2 ) @ ( inf_inf_set_nat @ B4 @ C2 ) )
= ( minus_minus_set_nat @ ( inf_inf_set_nat @ A4 @ C2 ) @ B4 ) ) ).
% Diff_Int2
thf(fact_1087_Int__Diff,axiom,
! [A4: set_nat,B4: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A4 @ B4 ) @ C2 )
= ( inf_inf_set_nat @ A4 @ ( minus_minus_set_nat @ B4 @ C2 ) ) ) ).
% Int_Diff
thf(fact_1088_Un__Diff,axiom,
! [A4: set_nat,B4: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A4 @ B4 ) @ C2 )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A4 @ C2 ) @ ( minus_minus_set_nat @ B4 @ C2 ) ) ) ).
% Un_Diff
thf(fact_1089_double__diff,axiom,
! [A4: set_nat,B4: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ C2 )
=> ( ( minus_minus_set_nat @ B4 @ ( minus_minus_set_nat @ C2 @ A4 ) )
= A4 ) ) ) ).
% double_diff
thf(fact_1090_Diff__subset,axiom,
! [A4: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ B4 ) @ A4 ) ).
% Diff_subset
thf(fact_1091_Diff__mono,axiom,
! [A4: set_nat,C2: set_nat,D2: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ C2 )
=> ( ( ord_less_eq_set_nat @ D2 @ B4 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ B4 ) @ ( minus_minus_set_nat @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_1092_Diff__infinite__finite,axiom,
! [T2: set_nat,S: set_nat] :
( ( finite_finite_nat @ T2 )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1093_psubset__imp__ex__mem,axiom,
! [A4: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ A4 @ B4 )
=> ? [B3: nat] : ( member_nat @ B3 @ ( minus_minus_set_nat @ B4 @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1094_Int__Diff__disjoint,axiom,
! [A4: set_nat,B4: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A4 @ B4 ) @ ( minus_minus_set_nat @ A4 @ B4 ) )
= bot_bot_set_nat ) ).
% Int_Diff_disjoint
thf(fact_1095_Diff__triv,axiom,
! [A4: set_nat,B4: set_nat] :
( ( ( inf_inf_set_nat @ A4 @ B4 )
= bot_bot_set_nat )
=> ( ( minus_minus_set_nat @ A4 @ B4 )
= A4 ) ) ).
% Diff_triv
thf(fact_1096_Diff__subset__conv,axiom,
! [A4: set_nat,B4: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ B4 ) @ C2 )
= ( ord_less_eq_set_nat @ A4 @ ( sup_sup_set_nat @ B4 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_1097_Diff__partition,axiom,
! [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( ( sup_sup_set_nat @ A4 @ ( minus_minus_set_nat @ B4 @ A4 ) )
= B4 ) ) ).
% Diff_partition
thf(fact_1098_Diff__Un,axiom,
! [A4: set_nat,B4: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ A4 @ ( sup_sup_set_nat @ B4 @ C2 ) )
= ( inf_inf_set_nat @ ( minus_minus_set_nat @ A4 @ B4 ) @ ( minus_minus_set_nat @ A4 @ C2 ) ) ) ).
% Diff_Un
thf(fact_1099_Diff__Int,axiom,
! [A4: set_nat,B4: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ A4 @ ( inf_inf_set_nat @ B4 @ C2 ) )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A4 @ B4 ) @ ( minus_minus_set_nat @ A4 @ C2 ) ) ) ).
% Diff_Int
thf(fact_1100_Int__Diff__Un,axiom,
! [A4: set_nat,B4: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A4 @ B4 ) @ ( minus_minus_set_nat @ A4 @ B4 ) )
= A4 ) ).
% Int_Diff_Un
thf(fact_1101_Un__Diff__Int,axiom,
! [A4: set_nat,B4: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ A4 @ B4 ) @ ( inf_inf_set_nat @ A4 @ B4 ) )
= A4 ) ).
% Un_Diff_Int
thf(fact_1102_card__less__sym__Diff,axiom,
! [A4: set_nat,B4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( finite_finite_nat @ B4 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ B4 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B4 @ A4 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_1103_card__le__sym__Diff,axiom,
! [A4: set_nat,B4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ B4 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B4 @ A4 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_1104_card__set__minus,axiom,
! [Xs: list_nat,X7: set_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ X7 ) ) @ ( size_size_list_nat @ Xs ) ) ).
% card_set_minus
thf(fact_1105_card__Diff__subset,axiom,
! [B4: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ A4 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ B4 ) )
= ( minus_minus_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_1106_diff__card__le__card__Diff,axiom,
! [B4: set_nat,A4: set_nat] :
( ( finite_finite_nat @ B4 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B4 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ B4 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1107_card__Diff__subset__Int,axiom,
! [A4: set_nat,B4: set_nat] :
( ( finite_finite_nat @ ( inf_inf_set_nat @ A4 @ B4 ) )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ B4 ) )
= ( minus_minus_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ ( inf_inf_set_nat @ A4 @ B4 ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_1108_rem__nth__add__nth,axiom,
! [I: nat,Zs: list_nat,Z2: nat] :
( ( ord_less_eq_nat @ I @ ( size_size_list_nat @ Zs ) )
=> ( ( rem_nth_nat @ I @ ( add_nth_nat @ I @ Z2 @ Zs ) )
= Zs ) ) ).
% rem_nth_add_nth
thf(fact_1109_set__update__distinct,axiom,
! [Xs: list_nat,N: nat,X: nat] :
( ( distinct_nat @ Xs )
=> ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( set_nat2 @ ( list_update_nat @ Xs @ N @ X ) )
= ( insert_nat @ X @ ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat @ ( nth_nat @ Xs @ N ) @ bot_bot_set_nat ) ) ) ) ) ) ).
% set_update_distinct
thf(fact_1110_nth__sorted__list__of__set__greaterThanLessThan,axiom,
! [N: nat,J: nat,I: nat] :
( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
=> ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N )
= ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_1111_insertCI,axiom,
! [A: nat,B4: set_nat,B: nat] :
( ( ~ ( member_nat @ A @ B4 )
=> ( A = B ) )
=> ( member_nat @ A @ ( insert_nat @ B @ B4 ) ) ) ).
% insertCI
thf(fact_1112_insert__iff,axiom,
! [A: nat,B: nat,A4: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A4 ) )
= ( ( A = B )
| ( member_nat @ A @ A4 ) ) ) ).
% insert_iff
thf(fact_1113_DiffI,axiom,
! [C: nat,A4: set_nat,B4: set_nat] :
( ( member_nat @ C @ A4 )
=> ( ~ ( member_nat @ C @ B4 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B4 ) ) ) ) ).
% DiffI
thf(fact_1114_Diff__iff,axiom,
! [C: nat,A4: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B4 ) )
= ( ( member_nat @ C @ A4 )
& ~ ( member_nat @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_1115_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_1116_finite__insert,axiom,
! [A: nat,A4: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A @ A4 ) )
= ( finite_finite_nat @ A4 ) ) ).
% finite_insert
thf(fact_1117_insert__subset,axiom,
! [X: nat,A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A4 ) @ B4 )
= ( ( member_nat @ X @ B4 )
& ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ).
% insert_subset
thf(fact_1118_Un__insert__left,axiom,
! [A: nat,B4: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( insert_nat @ A @ B4 ) @ C2 )
= ( insert_nat @ A @ ( sup_sup_set_nat @ B4 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_1119_Un__insert__right,axiom,
! [A4: set_nat,A: nat,B4: set_nat] :
( ( sup_sup_set_nat @ A4 @ ( insert_nat @ A @ B4 ) )
= ( insert_nat @ A @ ( sup_sup_set_nat @ A4 @ B4 ) ) ) ).
% Un_insert_right
thf(fact_1120_Int__insert__right__if1,axiom,
! [A: nat,A4: set_nat,B4: set_nat] :
( ( member_nat @ A @ A4 )
=> ( ( inf_inf_set_nat @ A4 @ ( insert_nat @ A @ B4 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A4 @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1121_Int__insert__right__if0,axiom,
! [A: nat,A4: set_nat,B4: set_nat] :
( ~ ( member_nat @ A @ A4 )
=> ( ( inf_inf_set_nat @ A4 @ ( insert_nat @ A @ B4 ) )
= ( inf_inf_set_nat @ A4 @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_1122_insert__inter__insert,axiom,
! [A: nat,A4: set_nat,B4: set_nat] :
( ( inf_inf_set_nat @ ( insert_nat @ A @ A4 ) @ ( insert_nat @ A @ B4 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A4 @ B4 ) ) ) ).
% insert_inter_insert
thf(fact_1123_Int__insert__left__if1,axiom,
! [A: nat,C2: set_nat,B4: set_nat] :
( ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B4 ) @ C2 )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B4 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1124_Int__insert__left__if0,axiom,
! [A: nat,C2: set_nat,B4: set_nat] :
( ~ ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B4 ) @ C2 )
= ( inf_inf_set_nat @ B4 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1125_Diff__insert0,axiom,
! [X: nat,A4: set_nat,B4: set_nat] :
( ~ ( member_nat @ X @ A4 )
=> ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ B4 ) )
= ( minus_minus_set_nat @ A4 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_1126_insert__Diff1,axiom,
! [X: nat,B4: set_nat,A4: set_nat] :
( ( member_nat @ X @ B4 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A4 ) @ B4 )
= ( minus_minus_set_nat @ A4 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_1127_finite__greaterThanLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or5834768355832116004an_nat @ L @ U ) ) ).
% finite_greaterThanLessThan
thf(fact_1128_singleton__insert__inj__eq_H,axiom,
! [A: nat,A4: set_nat,B: nat] :
( ( ( insert_nat @ A @ A4 )
= ( insert_nat @ B @ bot_bot_set_nat ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_1129_singleton__insert__inj__eq,axiom,
! [B: nat,A: nat,A4: set_nat] :
( ( ( insert_nat @ B @ bot_bot_set_nat )
= ( insert_nat @ A @ A4 ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_1130_disjoint__insert_I2_J,axiom,
! [A4: set_nat,B: nat,B4: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A4 @ ( insert_nat @ B @ B4 ) ) )
= ( ~ ( member_nat @ B @ A4 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A4 @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1131_disjoint__insert_I1_J,axiom,
! [B4: set_nat,A: nat,A4: set_nat] :
( ( ( inf_inf_set_nat @ B4 @ ( insert_nat @ A @ A4 ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B4 )
& ( ( inf_inf_set_nat @ B4 @ A4 )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_1132_insert__disjoint_I2_J,axiom,
! [A: nat,A4: set_nat,B4: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A @ A4 ) @ B4 ) )
= ( ~ ( member_nat @ A @ B4 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A4 @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1133_insert__disjoint_I1_J,axiom,
! [A: nat,A4: set_nat,B4: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A @ A4 ) @ B4 )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A @ B4 )
& ( ( inf_inf_set_nat @ A4 @ B4 )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_1134_insert__Diff__single,axiom,
! [A: nat,A4: set_nat] :
( ( insert_nat @ A @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( insert_nat @ A @ A4 ) ) ).
% insert_Diff_single
thf(fact_1135_finite__Diff__insert,axiom,
! [A4: set_nat,A: nat,B4: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ B4 ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ B4 ) ) ) ).
% finite_Diff_insert
thf(fact_1136_greaterThanLessThan__empty,axiom,
! [L: set_nat,K: set_nat] :
( ( ord_less_eq_set_nat @ L @ K )
=> ( ( set_or8625682525731655386et_nat @ K @ L )
= bot_bot_set_set_nat ) ) ).
% greaterThanLessThan_empty
thf(fact_1137_greaterThanLessThan__empty,axiom,
! [L: nat,K: nat] :
( ( ord_less_eq_nat @ L @ K )
=> ( ( set_or5834768355832116004an_nat @ K @ L )
= bot_bot_set_nat ) ) ).
% greaterThanLessThan_empty
thf(fact_1138_is__singletonI,axiom,
! [X: nat] : ( is_singleton_nat @ ( insert_nat @ X @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_1139_card__insert__disjoint,axiom,
! [A4: set_nat,X: nat] :
( ( finite_finite_nat @ A4 )
=> ( ~ ( member_nat @ X @ A4 )
=> ( ( finite_card_nat @ ( insert_nat @ X @ A4 ) )
= ( suc @ ( finite_card_nat @ A4 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_1140_card__Diff__insert,axiom,
! [A: nat,A4: set_nat,B4: set_nat] :
( ( member_nat @ A @ A4 )
=> ( ~ ( member_nat @ A @ B4 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ B4 ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ B4 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1141_subset__Diff__insert,axiom,
! [A4: set_nat,B4: set_nat,X: nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ ( minus_minus_set_nat @ B4 @ ( insert_nat @ X @ C2 ) ) )
= ( ( ord_less_eq_set_nat @ A4 @ ( minus_minus_set_nat @ B4 @ C2 ) )
& ~ ( member_nat @ X @ A4 ) ) ) ).
% subset_Diff_insert
thf(fact_1142_DiffE,axiom,
! [C: nat,A4: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B4 ) )
=> ~ ( ( member_nat @ C @ A4 )
=> ( member_nat @ C @ B4 ) ) ) ).
% DiffE
thf(fact_1143_DiffD1,axiom,
! [C: nat,A4: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B4 ) )
=> ( member_nat @ C @ A4 ) ) ).
% DiffD1
thf(fact_1144_DiffD2,axiom,
! [C: nat,A4: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B4 ) )
=> ~ ( member_nat @ C @ B4 ) ) ).
% DiffD2
thf(fact_1145_insert__Diff__if,axiom,
! [X: nat,B4: set_nat,A4: set_nat] :
( ( ( member_nat @ X @ B4 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A4 ) @ B4 )
= ( minus_minus_set_nat @ A4 @ B4 ) ) )
& ( ~ ( member_nat @ X @ B4 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A4 ) @ B4 )
= ( insert_nat @ X @ ( minus_minus_set_nat @ A4 @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_1146_subset__insert__iff,axiom,
! [A4: set_nat,X: nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X @ B4 ) )
= ( ( ( member_nat @ X @ A4 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B4 ) )
& ( ~ ( member_nat @ X @ A4 )
=> ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_1147_Diff__single__insert,axiom,
! [A4: set_nat,X: nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B4 )
=> ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X @ B4 ) ) ) ).
% Diff_single_insert
thf(fact_1148_Diff__insert__absorb,axiom,
! [X: nat,A4: set_nat] :
( ~ ( member_nat @ X @ A4 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A4 ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_1149_Diff__insert2,axiom,
! [A4: set_nat,A: nat,B4: set_nat] :
( ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ B4 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B4 ) ) ).
% Diff_insert2
thf(fact_1150_insert__Diff,axiom,
! [A: nat,A4: set_nat] :
( ( member_nat @ A @ A4 )
=> ( ( insert_nat @ A @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_1151_Diff__insert,axiom,
! [A4: set_nat,A: nat,B4: set_nat] :
( ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ B4 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A4 @ B4 ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_1152_finite__empty__induct,axiom,
! [A4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( P @ A4 )
=> ( ! [A3: nat,A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ( member_nat @ A3 @ A7 )
=> ( ( P @ A7 )
=> ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_1153_infinite__coinduct,axiom,
! [X7: set_nat > $o,A4: set_nat] :
( ( X7 @ A4 )
=> ( ! [A7: set_nat] :
( ( X7 @ A7 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ A7 )
& ( ( X7 @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A4 ) ) ) ).
% infinite_coinduct
thf(fact_1154_infinite__remove,axiom,
! [S: set_nat,A: nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_1155_subset__singleton__iff,axiom,
! [X7: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X7 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( ( X7 = bot_bot_set_nat )
| ( X7
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_1156_subset__singletonD,axiom,
! [A4: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ( A4 = bot_bot_set_nat )
| ( A4
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_1157_singleton__Un__iff,axiom,
! [X: nat,A4: set_nat,B4: set_nat] :
( ( ( insert_nat @ X @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A4 @ B4 ) )
= ( ( ( A4 = bot_bot_set_nat )
& ( B4
= ( insert_nat @ X @ bot_bot_set_nat ) ) )
| ( ( A4
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B4 = bot_bot_set_nat ) )
| ( ( A4
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B4
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1158_Un__singleton__iff,axiom,
! [A4: set_nat,B4: set_nat,X: nat] :
( ( ( sup_sup_set_nat @ A4 @ B4 )
= ( insert_nat @ X @ bot_bot_set_nat ) )
= ( ( ( A4 = bot_bot_set_nat )
& ( B4
= ( insert_nat @ X @ bot_bot_set_nat ) ) )
| ( ( A4
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B4 = bot_bot_set_nat ) )
| ( ( A4
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B4
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1159_insert__is__Un,axiom,
( insert_nat
= ( ^ [A2: nat] : ( sup_sup_set_nat @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_1160_card__insert__le,axiom,
! [A4: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ ( insert_nat @ X @ A4 ) ) ) ).
% card_insert_le
thf(fact_1161_singleton__inject,axiom,
! [A: nat,B: nat] :
( ( ( insert_nat @ A @ bot_bot_set_nat )
= ( insert_nat @ B @ bot_bot_set_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_1162_insert__not__empty,axiom,
! [A: nat,A4: set_nat] :
( ( insert_nat @ A @ A4 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_1163_doubleton__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_1164_singleton__iff,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_1165_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_1166_insertE,axiom,
! [A: nat,B: nat,A4: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A4 ) )
=> ( ( A != B )
=> ( member_nat @ A @ A4 ) ) ) ).
% insertE
thf(fact_1167_insertI1,axiom,
! [A: nat,B4: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B4 ) ) ).
% insertI1
thf(fact_1168_insertI2,axiom,
! [A: nat,B4: set_nat,B: nat] :
( ( member_nat @ A @ B4 )
=> ( member_nat @ A @ ( insert_nat @ B @ B4 ) ) ) ).
% insertI2
thf(fact_1169_Set_Oset__insert,axiom,
! [X: nat,A4: set_nat] :
( ( member_nat @ X @ A4 )
=> ~ ! [B7: set_nat] :
( ( A4
= ( insert_nat @ X @ B7 ) )
=> ( member_nat @ X @ B7 ) ) ) ).
% Set.set_insert
thf(fact_1170_insert__ident,axiom,
! [X: nat,A4: set_nat,B4: set_nat] :
( ~ ( member_nat @ X @ A4 )
=> ( ~ ( member_nat @ X @ B4 )
=> ( ( ( insert_nat @ X @ A4 )
= ( insert_nat @ X @ B4 ) )
= ( A4 = B4 ) ) ) ) ).
% insert_ident
thf(fact_1171_insert__absorb,axiom,
! [A: nat,A4: set_nat] :
( ( member_nat @ A @ A4 )
=> ( ( insert_nat @ A @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_1172_insert__eq__iff,axiom,
! [A: nat,A4: set_nat,B: nat,B4: set_nat] :
( ~ ( member_nat @ A @ A4 )
=> ( ~ ( member_nat @ B @ B4 )
=> ( ( ( insert_nat @ A @ A4 )
= ( insert_nat @ B @ B4 ) )
= ( ( ( A = B )
=> ( A4 = B4 ) )
& ( ( A != B )
=> ? [C5: set_nat] :
( ( A4
= ( insert_nat @ B @ C5 ) )
& ~ ( member_nat @ B @ C5 )
& ( B4
= ( insert_nat @ A @ C5 ) )
& ~ ( member_nat @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1173_mk__disjoint__insert,axiom,
! [A: nat,A4: set_nat] :
( ( member_nat @ A @ A4 )
=> ? [B7: set_nat] :
( ( A4
= ( insert_nat @ A @ B7 ) )
& ~ ( member_nat @ A @ B7 ) ) ) ).
% mk_disjoint_insert
thf(fact_1174_Int__insert__right,axiom,
! [A: nat,A4: set_nat,B4: set_nat] :
( ( ( member_nat @ A @ A4 )
=> ( ( inf_inf_set_nat @ A4 @ ( insert_nat @ A @ B4 ) )
= ( insert_nat @ A @ ( inf_inf_set_nat @ A4 @ B4 ) ) ) )
& ( ~ ( member_nat @ A @ A4 )
=> ( ( inf_inf_set_nat @ A4 @ ( insert_nat @ A @ B4 ) )
= ( inf_inf_set_nat @ A4 @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_1175_Int__insert__left,axiom,
! [A: nat,C2: set_nat,B4: set_nat] :
( ( ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B4 ) @ C2 )
= ( insert_nat @ A @ ( inf_inf_set_nat @ B4 @ C2 ) ) ) )
& ( ~ ( member_nat @ A @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A @ B4 ) @ C2 )
= ( inf_inf_set_nat @ B4 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1176_subset__insertI2,axiom,
! [A4: set_nat,B4: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ B @ B4 ) ) ) ).
% subset_insertI2
thf(fact_1177_subset__insertI,axiom,
! [B4: set_nat,A: nat] : ( ord_less_eq_set_nat @ B4 @ ( insert_nat @ A @ B4 ) ) ).
% subset_insertI
thf(fact_1178_subset__insert,axiom,
! [X: nat,A4: set_nat,B4: set_nat] :
( ~ ( member_nat @ X @ A4 )
=> ( ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X @ B4 ) )
= ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ).
% subset_insert
thf(fact_1179_insert__mono,axiom,
! [C2: set_nat,D2: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ C2 @ D2 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A @ C2 ) @ ( insert_nat @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_1180_insert__subsetI,axiom,
! [X: nat,A4: set_nat,X7: set_nat] :
( ( member_nat @ X @ A4 )
=> ( ( ord_less_eq_set_nat @ X7 @ A4 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ X @ X7 ) @ A4 ) ) ) ).
% insert_subsetI
thf(fact_1181_finite_OinsertI,axiom,
! [A4: set_nat,A: nat] :
( ( finite_finite_nat @ A4 )
=> ( finite_finite_nat @ ( insert_nat @ A @ A4 ) ) ) ).
% finite.insertI
thf(fact_1182_finite_Ocases,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ~ ! [A7: set_nat] :
( ? [A3: nat] :
( A
= ( insert_nat @ A3 @ A7 ) )
=> ~ ( finite_finite_nat @ A7 ) ) ) ) ).
% finite.cases
thf(fact_1183_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A2: set_nat] :
( ( A2 = bot_bot_set_nat )
| ? [A5: set_nat,B2: nat] :
( ( A2
= ( insert_nat @ B2 @ A5 ) )
& ( finite_finite_nat @ A5 ) ) ) ) ) ).
% finite.simps
thf(fact_1184_finite__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1185_finite__ne__induct,axiom,
! [F2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( F2 != bot_bot_set_nat )
=> ( ! [X2: nat] : ( P @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1186_infinite__finite__induct,axiom,
! [P: set_nat > $o,A4: set_nat] :
( ! [A7: set_nat] :
( ~ ( finite_finite_nat @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ X2 @ F3 ) ) ) ) )
=> ( P @ A4 ) ) ) ) ).
% infinite_finite_induct
thf(fact_1187_is__singletonE,axiom,
! [A4: set_nat] :
( ( is_singleton_nat @ A4 )
=> ~ ! [X2: nat] :
( A4
!= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_1188_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A5: set_nat] :
? [X3: nat] :
( A5
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_1189_finite__ranking__induct,axiom,
! [S: set_nat,P: set_nat > $o,F: nat > nat] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,S3: set_nat] :
( ( finite_finite_nat @ S3 )
=> ( ! [Y5: nat] :
( ( member_nat @ Y5 @ S3 )
=> ( ord_less_eq_nat @ ( F @ Y5 ) @ ( F @ X2 ) ) )
=> ( ( P @ S3 )
=> ( P @ ( insert_nat @ X2 @ S3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_ranking_induct
thf(fact_1190_finite__linorder__max__induct,axiom,
! [A4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B3: nat,A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A7 )
=> ( ord_less_nat @ X6 @ B3 ) )
=> ( ( P @ A7 )
=> ( P @ ( insert_nat @ B3 @ A7 ) ) ) ) )
=> ( P @ A4 ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_1191_finite__linorder__min__induct,axiom,
! [A4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A4 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [B3: nat,A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A7 )
=> ( ord_less_nat @ B3 @ X6 ) )
=> ( ( P @ A7 )
=> ( P @ ( insert_nat @ B3 @ A7 ) ) ) ) )
=> ( P @ A4 ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_1192_finite__subset__induct,axiom,
! [F2: set_nat,A4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A4 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A3 @ A4 )
=> ( ~ ( member_nat @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A3 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1193_finite__subset__induct_H,axiom,
! [F2: set_nat,A4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A4 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A3 @ A4 )
=> ( ( ord_less_eq_set_nat @ F3 @ A4 )
=> ( ~ ( member_nat @ A3 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat @ A3 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1194_remove__induct,axiom,
! [P: set_nat > $o,B4: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B4 )
=> ( P @ B4 ) )
=> ( ! [A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ( A7 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A7 @ B4 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A7 )
=> ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% remove_induct
thf(fact_1195_finite__remove__induct,axiom,
! [B4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B4 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ( A7 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A7 @ B4 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A7 )
=> ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% finite_remove_induct
thf(fact_1196_card__Suc__eq__finite,axiom,
! [A4: set_nat,K: nat] :
( ( ( finite_card_nat @ A4 )
= ( suc @ K ) )
= ( ? [B2: nat,B5: set_nat] :
( ( A4
= ( insert_nat @ B2 @ B5 ) )
& ~ ( member_nat @ B2 @ B5 )
& ( ( finite_card_nat @ B5 )
= K )
& ( finite_finite_nat @ B5 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_1197_card__insert__if,axiom,
! [A4: set_nat,X: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( ( member_nat @ X @ A4 )
=> ( ( finite_card_nat @ ( insert_nat @ X @ A4 ) )
= ( finite_card_nat @ A4 ) ) )
& ( ~ ( member_nat @ X @ A4 )
=> ( ( finite_card_nat @ ( insert_nat @ X @ A4 ) )
= ( suc @ ( finite_card_nat @ A4 ) ) ) ) ) ) ).
% card_insert_if
thf(fact_1198_card__Diff1__le,axiom,
! [A4: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A4 ) ) ).
% card_Diff1_le
thf(fact_1199_card__eq__1__iff,axiom,
! [A4: set_nat] :
( ( ( finite_card_nat @ A4 )
= one_one_nat )
= ( ? [X3: nat] :
( A4
= ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).
% card_eq_1_iff
thf(fact_1200_card__1__singletonE,axiom,
! [A4: set_nat] :
( ( ( finite_card_nat @ A4 )
= one_one_nat )
=> ~ ! [X2: nat] :
( A4
!= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ).
% card_1_singletonE
thf(fact_1201_finite__induct__select,axiom,
! [S: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ S )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [T3: set_nat] :
( ( ord_less_set_nat @ T3 @ S )
=> ( ( P @ T3 )
=> ? [X6: nat] :
( ( member_nat @ X6 @ ( minus_minus_set_nat @ S @ T3 ) )
& ( P @ ( insert_nat @ X6 @ T3 ) ) ) ) )
=> ( P @ S ) ) ) ) ).
% finite_induct_select
thf(fact_1202_psubset__insert__iff,axiom,
! [A4: set_nat,X: nat,B4: set_nat] :
( ( ord_less_set_nat @ A4 @ ( insert_nat @ X @ B4 ) )
= ( ( ( member_nat @ X @ B4 )
=> ( ord_less_set_nat @ A4 @ B4 ) )
& ( ~ ( member_nat @ X @ B4 )
=> ( ( ( member_nat @ X @ A4 )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B4 ) )
& ( ~ ( member_nat @ X @ A4 )
=> ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1203_set__update__subset__insert,axiom,
! [Xs: list_nat,I: nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X ) ) @ ( insert_nat @ X @ ( set_nat2 @ Xs ) ) ) ).
% set_update_subset_insert
thf(fact_1204_card_Oremove,axiom,
! [A4: set_nat,X: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ X @ A4 )
=> ( ( finite_card_nat @ A4 )
= ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% card.remove
thf(fact_1205_card_Oinsert__remove,axiom,
! [A4: set_nat,X: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( finite_card_nat @ ( insert_nat @ X @ A4 ) )
= ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_1206_card__Suc__Diff1,axiom,
! [A4: set_nat,X: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ X @ A4 )
=> ( ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) )
= ( finite_card_nat @ A4 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_1207_card__Diff1__less__iff,axiom,
! [A4: set_nat,X: nat] :
( ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A4 ) )
= ( ( finite_finite_nat @ A4 )
& ( member_nat @ X @ A4 ) ) ) ).
% card_Diff1_less_iff
thf(fact_1208_card__Diff2__less,axiom,
! [A4: set_nat,X: nat,Y: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ X @ A4 )
=> ( ( member_nat @ Y @ A4 )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( insert_nat @ Y @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A4 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_1209_card__Diff1__less,axiom,
! [A4: set_nat,X: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ X @ A4 )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A4 ) ) ) ) ).
% card_Diff1_less
thf(fact_1210_card__le__Suc__iff,axiom,
! [N: nat,A4: set_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_nat @ A4 ) )
= ( ? [A2: nat,B5: set_nat] :
( ( A4
= ( insert_nat @ A2 @ B5 ) )
& ~ ( member_nat @ A2 @ B5 )
& ( ord_less_eq_nat @ N @ ( finite_card_nat @ B5 ) )
& ( finite_finite_nat @ B5 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_1211_card__Diff__singleton__if,axiom,
! [X: nat,A4: set_nat] :
( ( ( member_nat @ X @ A4 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A4 ) @ one_one_nat ) ) )
& ( ~ ( member_nat @ X @ A4 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( finite_card_nat @ A4 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1212_card__Diff__singleton,axiom,
! [X: nat,A4: set_nat] :
( ( member_nat @ X @ A4 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A4 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1213_distinct__list__update,axiom,
! [Xs: list_nat,A: nat,I: nat] :
( ( distinct_nat @ Xs )
=> ( ~ ( member_nat @ A @ ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat @ ( nth_nat @ Xs @ I ) @ bot_bot_set_nat ) ) )
=> ( distinct_nat @ ( list_update_nat @ Xs @ I @ A ) ) ) ) ).
% distinct_list_update
thf(fact_1214_sorted__quicksort__part,axiom,
! [Ac: list_nat,Lts: list_nat,X: nat,Eqs: list_nat,Gts: list_nat,Zs: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Ac )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( set_nat2 @ Lts ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( set_nat2 @ Eqs ) ) @ ( set_nat2 @ Gts ) ) @ ( set_nat2 @ Zs ) ) )
=> ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( set_nat2 @ Ac ) )
=> ( ord_less_nat @ X2 @ Xa2 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Lts ) )
=> ( ord_less_nat @ X2 @ X ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Eqs ) )
=> ( X2 = X ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Gts ) )
=> ( ord_less_nat @ X @ X2 ) )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( set_or1804217446461887602rt_nat @ Ac @ X @ Lts @ Eqs @ Gts @ Zs ) ) ) ) ) ) ) ).
% sorted_quicksort_part
thf(fact_1215_set__remove1__eq,axiom,
! [Xs: list_nat,X: nat] :
( ( distinct_nat @ Xs )
=> ( ( set_nat2 @ ( remove1_nat @ X @ Xs ) )
= ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% set_remove1_eq
thf(fact_1216_in__set__remove1,axiom,
! [A: nat,B: nat,Xs: list_nat] :
( ( A != B )
=> ( ( member_nat @ A @ ( set_nat2 @ ( remove1_nat @ B @ Xs ) ) )
= ( member_nat @ A @ ( set_nat2 @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_1217_set__quicksort__part,axiom,
! [Ac: list_nat,X: nat,Lts: list_nat,Eqs: list_nat,Gts: list_nat,Zs: list_nat] :
( ( set_nat2 @ ( set_or1804217446461887602rt_nat @ Ac @ X @ Lts @ Eqs @ Gts @ Zs ) )
= ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( set_nat2 @ Ac ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( set_nat2 @ Lts ) ) @ ( set_nat2 @ Eqs ) ) @ ( set_nat2 @ Gts ) ) @ ( set_nat2 @ Zs ) ) ) ).
% set_quicksort_part
thf(fact_1218_remove1__idem,axiom,
! [X: nat,Xs: list_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( remove1_nat @ X @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_1219_notin__set__remove1,axiom,
! [X: nat,Xs: list_nat,Y: nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ~ ( member_nat @ X @ ( set_nat2 @ ( remove1_nat @ Y @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_1220_distinct__remove1,axiom,
! [Xs: list_nat,X: nat] :
( ( distinct_nat @ Xs )
=> ( distinct_nat @ ( remove1_nat @ X @ Xs ) ) ) ).
% distinct_remove1
thf(fact_1221_remove1__append,axiom,
! [X: nat,Xs: list_nat,Ys: list_nat] :
( ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( remove1_nat @ X @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( remove1_nat @ X @ Xs ) @ Ys ) ) )
& ( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( remove1_nat @ X @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ Xs @ ( remove1_nat @ X @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_1222_sorted__remove1,axiom,
! [Xs: list_nat,A: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( remove1_nat @ A @ Xs ) ) ) ).
% sorted_remove1
thf(fact_1223_set__remove1__subset,axiom,
! [X: nat,Xs: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( remove1_nat @ X @ Xs ) ) @ ( set_nat2 @ Xs ) ) ).
% set_remove1_subset
thf(fact_1224_length__remove1,axiom,
! [X: nat,Xs: list_nat] :
( ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( size_size_list_nat @ ( remove1_nat @ X @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) )
& ( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( size_size_list_nat @ ( remove1_nat @ X @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ) ) ).
% length_remove1
thf(fact_1225_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
! [A4: set_nat,X: nat] :
( ( finite_finite_nat @ A4 )
=> ( ( linord2614967742042102400et_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( remove1_nat @ X @ ( linord2614967742042102400et_nat @ A4 ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_1226_nth__sorted__list__of__set__greaterThanAtMost,axiom,
! [N: nat,J: nat,I: nat] :
( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I ) )
=> ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N )
= ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_1227_set__removeAll,axiom,
! [X: nat,Xs: list_nat] :
( ( set_nat2 @ ( removeAll_nat @ X @ Xs ) )
= ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).
% set_removeAll
thf(fact_1228_removeAll__id,axiom,
! [X: nat,Xs: list_nat] :
( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( removeAll_nat @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_1229_finite__greaterThanAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or6659071591806873216st_nat @ L @ U ) ) ).
% finite_greaterThanAtMost
thf(fact_1230_greaterThanAtMost__iff,axiom,
! [I: set_nat,L: set_nat,U: set_nat] :
( ( member_set_nat @ I @ ( set_or7074010630789208630et_nat @ L @ U ) )
= ( ( ord_less_set_nat @ L @ I )
& ( ord_less_eq_set_nat @ I @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_1231_greaterThanAtMost__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or6659071591806873216st_nat @ L @ U ) )
= ( ( ord_less_nat @ L @ I )
& ( ord_less_eq_nat @ I @ U ) ) ) ).
% greaterThanAtMost_iff
thf(fact_1232_greaterThanAtMost__empty,axiom,
! [L: set_nat,K: set_nat] :
( ( ord_less_eq_set_nat @ L @ K )
=> ( ( set_or7074010630789208630et_nat @ K @ L )
= bot_bot_set_set_nat ) ) ).
% greaterThanAtMost_empty
thf(fact_1233_greaterThanAtMost__empty,axiom,
! [L: nat,K: nat] :
( ( ord_less_eq_nat @ L @ K )
=> ( ( set_or6659071591806873216st_nat @ K @ L )
= bot_bot_set_nat ) ) ).
% greaterThanAtMost_empty
thf(fact_1234_greaterThanAtMost__empty__iff2,axiom,
! [K: nat,L: nat] :
( ( bot_bot_set_nat
= ( set_or6659071591806873216st_nat @ K @ L ) )
= ( ~ ( ord_less_nat @ K @ L ) ) ) ).
% greaterThanAtMost_empty_iff2
thf(fact_1235_greaterThanAtMost__empty__iff,axiom,
! [K: nat,L: nat] :
( ( ( set_or6659071591806873216st_nat @ K @ L )
= bot_bot_set_nat )
= ( ~ ( ord_less_nat @ K @ L ) ) ) ).
% greaterThanAtMost_empty_iff
thf(fact_1236_distinct__remove1__removeAll,axiom,
! [Xs: list_nat,X: nat] :
( ( distinct_nat @ Xs )
=> ( ( remove1_nat @ X @ Xs )
= ( removeAll_nat @ X @ Xs ) ) ) ).
% distinct_remove1_removeAll
thf(fact_1237_length__removeAll__less__eq,axiom,
! [X: nat,Xs: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( removeAll_nat @ X @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% length_removeAll_less_eq
thf(fact_1238_Ioc__disjoint,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( inf_inf_set_nat @ ( set_or6659071591806873216st_nat @ A @ B ) @ ( set_or6659071591806873216st_nat @ C @ D ) )
= bot_bot_set_nat )
= ( ( ord_less_eq_nat @ B @ A )
| ( ord_less_eq_nat @ D @ C )
| ( ord_less_eq_nat @ B @ C )
| ( ord_less_eq_nat @ D @ A ) ) ) ).
% Ioc_disjoint
thf(fact_1239_Ioc__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or6659071591806873216st_nat @ A @ B ) @ ( set_or6659071591806873216st_nat @ C @ D ) )
= ( ( ord_less_eq_nat @ B @ A )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% Ioc_subset_iff
thf(fact_1240_ivl__disj__un__two_I6_J,axiom,
! [L: nat,M: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M )
=> ( ( ord_less_eq_nat @ M @ U )
=> ( ( sup_sup_set_nat @ ( set_or6659071591806873216st_nat @ L @ M ) @ ( set_or6659071591806873216st_nat @ M @ U ) )
= ( set_or6659071591806873216st_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(6)
thf(fact_1241_ivl__disj__int__two_I6_J,axiom,
! [L: nat,M: nat,U: nat] :
( ( inf_inf_set_nat @ ( set_or6659071591806873216st_nat @ L @ M ) @ ( set_or6659071591806873216st_nat @ M @ U ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_two(6)
thf(fact_1242_distinct__removeAll,axiom,
! [Xs: list_nat,X: nat] :
( ( distinct_nat @ Xs )
=> ( distinct_nat @ ( removeAll_nat @ X @ Xs ) ) ) ).
% distinct_removeAll
thf(fact_1243_Ioc__inj,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( set_or6659071591806873216st_nat @ A @ B )
= ( set_or6659071591806873216st_nat @ C @ D ) )
= ( ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ D @ C ) )
| ( ( A = C )
& ( B = D ) ) ) ) ).
% Ioc_inj
thf(fact_1244_ivl__disj__int__two_I2_J,axiom,
! [L: nat,M: nat,U: nat] :
( ( inf_inf_set_nat @ ( set_or6659071591806873216st_nat @ L @ M ) @ ( set_or5834768355832116004an_nat @ M @ U ) )
= bot_bot_set_nat ) ).
% ivl_disj_int_two(2)
thf(fact_1245_length__removeAll__less,axiom,
! [X: nat,Xs: list_nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ ( size_size_list_nat @ ( removeAll_nat @ X @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_1246_ivl__disj__un__two_I2_J,axiom,
! [L: nat,M: nat,U: nat] :
( ( ord_less_eq_nat @ L @ M )
=> ( ( ord_less_nat @ M @ U )
=> ( ( sup_sup_set_nat @ ( set_or6659071591806873216st_nat @ L @ M ) @ ( set_or5834768355832116004an_nat @ M @ U ) )
= ( set_or5834768355832116004an_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_1247_ivl__disj__un__singleton_I4_J,axiom,
! [L: nat,U: nat] :
( ( ord_less_nat @ L @ U )
=> ( ( sup_sup_set_nat @ ( set_or5834768355832116004an_nat @ L @ U ) @ ( insert_nat @ U @ bot_bot_set_nat ) )
= ( set_or6659071591806873216st_nat @ L @ U ) ) ) ).
% ivl_disj_un_singleton(4)
thf(fact_1248_card__insert__le__m1,axiom,
! [N: nat,Y: set_nat,X: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( insert_nat @ X @ Y ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_1249_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_1250_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_1251_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_1252_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_1253_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_1254_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_1255_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_1256_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_1257_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_1258_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_1259_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_1260_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_1261_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1262_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1263_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1264_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1265_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1266_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1267_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_1268_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
% Conjectures (1)
thf(conj_0,conjecture,
( ( sorted_wrt_nat @ ord_less_eq_nat @ ns )
& ( distinct_nat @ ns ) ) ).
%------------------------------------------------------------------------------