TPTP Problem File: SLH0388^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Eval_FO/0005_Ailamazyan/prob_05172_223613__16371778_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1410 ( 661 unt; 137 typ; 0 def)
% Number of atoms : 3234 (1470 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 9566 ( 372 ~; 58 |; 255 &;7735 @)
% ( 0 <=>;1146 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 14 ( 13 usr)
% Number of type conns : 495 ( 495 >; 0 *; 0 +; 0 <<)
% Number of symbols : 127 ( 124 usr; 17 con; 0-5 aty)
% Number of variables : 3354 ( 161 ^;3051 !; 142 ?;3354 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:13:24.082
%------------------------------------------------------------------------------
% Could-be-implicit typings (13)
thf(ty_n_t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
set_li6526943997496501093_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
set_se4904748513628223167_a_nat: $tType ).
thf(ty_n_t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
list_Sum_sum_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
set_Sum_sum_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
sum_sum_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (124)
thf(sy_c_Ailamazyan_Oad__agr__list_001tf__a_001t__Nat__Onat,type,
ad_agr_list_a_nat: set_a > list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o ).
thf(sy_c_Ailamazyan_Oad__agr__sets_001tf__a_001t__Nat__Onat,type,
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thf(sy_c_Ailamazyan_Oad__equiv__list_001tf__a_001t__Nat__Onat,type,
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thf(sy_c_Ailamazyan_Oadd__nth_001t__Nat__Onat,type,
add_nth_nat: nat > nat > list_nat > list_nat ).
thf(sy_c_Ailamazyan_Oadd__nth_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
add_nt4212672348507122516_a_nat: nat > sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_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__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
all_tu407047557562860027_a_nat: set_Sum_sum_a_nat > nat > set_li6526943997496501093_a_nat ).
thf(sy_c_Ailamazyan_Onats,type,
nats: list_nat > $o ).
thf(sy_c_Ailamazyan_Orem__nth_001t__Nat__Onat,type,
rem_nth_nat: nat > list_nat > list_nat ).
thf(sy_c_Ailamazyan_Orem__nth_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
rem_nt658808235856662061_a_nat: nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_Ailamazyan_Osp__equiv_001tf__a_001t__Nat__Onat,type,
sp_equiv_a_nat: ( nat > sum_sum_a_nat ) > ( nat > sum_sum_a_nat ) > set_nat > $o ).
thf(sy_c_Ailamazyan_Osp__equiv__list_001tf__a_001t__Nat__Onat,type,
sp_equiv_list_a_nat: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o ).
thf(sy_c_GCD_OGcd__class_OGcd_001t__Nat__Onat,type,
gcd_Gcd_nat: set_nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
minus_7954133019191499631st_nat: set_list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
minus_7395159227704179404_a_nat: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
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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_If_001t__List__Olist_It__Nat__Onat_J,type,
if_list_nat: $o > list_nat > list_nat > list_nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_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__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_List_Oappend_001t__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Oappend_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Obind_001t__Nat__Onat_001t__Nat__Onat,type,
bind_nat_nat: list_nat > ( nat > list_nat ) > list_nat ).
thf(sy_c_List_Ocoset_001t__Nat__Onat,type,
coset_nat: list_nat > set_nat ).
thf(sy_c_List_Ogen__length_001t__Nat__Onat,type,
gen_length_nat: nat > list_nat > nat ).
thf(sy_c_List_Ogen__length_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
gen_le1340941697924381074_a_nat: nat > list_Sum_sum_a_nat > nat ).
thf(sy_c_List_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
cons_Sum_sum_a_nat: sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
nil_Sum_sum_a_nat: list_Sum_sum_a_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
map_na823391071729141993_a_nat: ( nat > sum_sum_a_nat ) > list_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_Olist_Omap_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Nat__Onat,type,
map_Su5227373005390213643at_nat: ( sum_sum_a_nat > nat ) > list_Sum_sum_a_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
map_Su2790769393171190532_a_nat: ( sum_sum_a_nat > sum_sum_a_nat ) > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
set_Sum_sum_a_nat2: list_Sum_sum_a_nat > set_Sum_sum_a_nat ).
thf(sy_c_List_Omap__tailrec_001t__Nat__Onat_001t__Nat__Onat,type,
map_tailrec_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Omap__tailrec_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
map_ta1136998156224711455_a_nat: ( nat > sum_sum_a_nat ) > list_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_Omaps_001t__Nat__Onat_001t__Nat__Onat,type,
maps_nat_nat: ( nat > list_nat ) > list_nat > list_nat ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
nth_Sum_sum_a_nat: list_Sum_sum_a_nat > nat > sum_sum_a_nat ).
thf(sy_c_List_OremoveAll_001t__Nat__Onat,type,
removeAll_nat: nat > list_nat > list_nat ).
thf(sy_c_List_OremoveAll_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
remove3909449470355376139_a_nat: sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
replicate_nat: nat > nat > list_nat ).
thf(sy_c_List_Oreplicate_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
replic8955434655033810879_a_nat: nat > sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_Ounion_001t__Nat__Onat,type,
union_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > list_nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
size_s5283204784079214577_a_nat: list_Sum_sum_a_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
bot_bot_list_nat_o: list_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_M_Eo_J,type,
bot_bo9042073657639083596_nat_o: list_Sum_sum_a_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_M_Eo_J,type,
bot_bo7797463397293707474_nat_o: sum_sum_a_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__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
bot_bo1033123847703346641_a_nat: set_li6526943997496501093_a_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
bot_bo3438331934148233675_a_nat: set_Sum_sum_a_nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__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 ).
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Set_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Set_Opairwise_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Set_Opairwise_001t__Nat__Onat,type,
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thf(sy_c_Set_Oremove_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
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thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
the_elem_nat: set_nat > nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
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thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
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thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
set_or4236626031148496127et_nat: set_nat > set_set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
set_or677214730896469414_a_nat: set_Sum_sum_a_nat > set_se4904748513628223167_a_nat ).
thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_Itf__a_J,type,
set_ord_atMost_set_a: set_a > set_set_a ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
member408289922725080238_a_nat: list_Sum_sum_a_nat > set_li6526943997496501093_a_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_c_member_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
member8098812455498974984_a_nat: set_Sum_sum_a_nat > set_se4904748513628223167_a_nat > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
member_Sum_sum_a_nat: sum_sum_a_nat > set_Sum_sum_a_nat > $o ).
thf(sy_v_AD,type,
ad: set_a ).
thf(sy_v__092_060sigma_062____,type,
sigma: nat > sum_sum_a_nat ).
thf(sy_v__092_060tau_062____,type,
tau: nat > sum_sum_a_nat ).
thf(sy_v_i,type,
i: nat ).
thf(sy_v_n____,type,
n: nat ).
thf(sy_v_xs,type,
xs: list_Sum_sum_a_nat ).
thf(sy_v_ys,type,
ys: list_Sum_sum_a_nat ).
% Relevant facts (1267)
thf(fact_0_ad__agr__sets__comm,axiom,
! [FV: set_nat,S: set_nat,X: set_a,Sigma: nat > sum_sum_a_nat,Tau: nat > sum_sum_a_nat] :
( ( ad_agr_sets_a_nat @ FV @ S @ X @ Sigma @ Tau )
=> ( ad_agr_sets_a_nat @ FV @ S @ X @ Tau @ Sigma ) ) ).
% ad_agr_sets_comm
thf(fact_1_insert__Diff__single,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat2 @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) )
= ( insert_nat2 @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_2_Diff__insert0,axiom,
! [X2: list_nat,A2: set_list_nat,B: set_list_nat] :
( ~ ( member_list_nat @ X2 @ A2 )
=> ( ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ X2 @ B ) )
= ( minus_7954133019191499631st_nat @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_3_Diff__insert0,axiom,
! [X2: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ X2 @ A2 )
=> ( ( minus_7395159227704179404_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ X2 @ B ) )
= ( minus_7395159227704179404_a_nat @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_4_Diff__insert0,axiom,
! [X2: sum_sum_a_nat,A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ~ ( member_Sum_sum_a_nat @ X2 @ A2 )
=> ( ( minus_1134630996077396038_a_nat @ A2 @ ( insert_Sum_sum_a_nat @ X2 @ B ) )
= ( minus_1134630996077396038_a_nat @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_5_Diff__insert0,axiom,
! [X2: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ X2 @ A2 )
=> ( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X2 @ B ) )
= ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_6_insert__Diff1,axiom,
! [X2: list_nat,B: set_list_nat,A2: set_list_nat] :
( ( member_list_nat @ X2 @ B )
=> ( ( minus_7954133019191499631st_nat @ ( insert_list_nat @ X2 @ A2 ) @ B )
= ( minus_7954133019191499631st_nat @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_7_insert__Diff1,axiom,
! [X2: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ B )
=> ( ( minus_7395159227704179404_a_nat @ ( insert2950094090816004437_a_nat @ X2 @ A2 ) @ B )
= ( minus_7395159227704179404_a_nat @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_8_insert__Diff1,axiom,
! [X2: sum_sum_a_nat,B: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ B )
=> ( ( minus_1134630996077396038_a_nat @ ( insert_Sum_sum_a_nat @ X2 @ A2 ) @ B )
= ( minus_1134630996077396038_a_nat @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_9_insert__Diff1,axiom,
! [X2: nat,B: set_nat,A2: set_nat] :
( ( member_nat @ X2 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X2 @ A2 ) @ B )
= ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_10_Diff__empty,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Diff_empty
thf(fact_11_empty__Diff,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_12_Diff__cancel,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ A2 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_13_singletonI,axiom,
! [A: list_nat] : ( member_list_nat @ A @ ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ).
% singletonI
thf(fact_14_singletonI,axiom,
! [A: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ A @ ( insert2950094090816004437_a_nat @ A @ bot_bo1033123847703346641_a_nat ) ) ).
% singletonI
thf(fact_15_singletonI,axiom,
! [A: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ A @ ( insert_Sum_sum_a_nat @ A @ bot_bo3438331934148233675_a_nat ) ) ).
% singletonI
thf(fact_16_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_17_Diff__insert,axiom,
! [A2: set_nat,A: nat,B: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ B ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_18_insert__Diff,axiom,
! [A: list_nat,A2: set_list_nat] :
( ( member_list_nat @ A @ A2 )
=> ( ( insert_list_nat @ A @ ( minus_7954133019191499631st_nat @ A2 @ ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_19_insert__Diff,axiom,
! [A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A @ A2 )
=> ( ( insert2950094090816004437_a_nat @ A @ ( minus_7395159227704179404_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ A @ bot_bo1033123847703346641_a_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_20_insert__Diff,axiom,
! [A: sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A @ A2 )
=> ( ( insert_Sum_sum_a_nat @ A @ ( minus_1134630996077396038_a_nat @ A2 @ ( insert_Sum_sum_a_nat @ A @ bot_bo3438331934148233675_a_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_21_insert__Diff,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat2 @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_22_Diff__insert2,axiom,
! [A2: set_nat,A: nat,B: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ B ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) @ B ) ) ).
% Diff_insert2
thf(fact_23_Diff__insert__absorb,axiom,
! [X2: list_nat,A2: set_list_nat] :
( ~ ( member_list_nat @ X2 @ A2 )
=> ( ( minus_7954133019191499631st_nat @ ( insert_list_nat @ X2 @ A2 ) @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_24_Diff__insert__absorb,axiom,
! [X2: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ X2 @ A2 )
=> ( ( minus_7395159227704179404_a_nat @ ( insert2950094090816004437_a_nat @ X2 @ A2 ) @ ( insert2950094090816004437_a_nat @ X2 @ bot_bo1033123847703346641_a_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_25_Diff__insert__absorb,axiom,
! [X2: sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ~ ( member_Sum_sum_a_nat @ X2 @ A2 )
=> ( ( minus_1134630996077396038_a_nat @ ( insert_Sum_sum_a_nat @ X2 @ A2 ) @ ( insert_Sum_sum_a_nat @ X2 @ bot_bo3438331934148233675_a_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_26_Diff__insert__absorb,axiom,
! [X2: nat,A2: set_nat] :
( ~ ( member_nat @ X2 @ A2 )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X2 @ A2 ) @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_27_atMost__eq__iff,axiom,
! [X2: nat,Y: nat] :
( ( ( set_ord_atMost_nat @ X2 )
= ( set_ord_atMost_nat @ Y ) )
= ( X2 = Y ) ) ).
% atMost_eq_iff
thf(fact_28_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_29_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_30_all__not__in__conv,axiom,
! [A2: set_list_nat] :
( ( ! [X3: list_nat] :
~ ( member_list_nat @ X3 @ A2 ) )
= ( A2 = bot_bot_set_list_nat ) ) ).
% all_not_in_conv
thf(fact_31_all__not__in__conv,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ( ! [X3: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ X3 @ A2 ) )
= ( A2 = bot_bo1033123847703346641_a_nat ) ) ).
% all_not_in_conv
thf(fact_32_all__not__in__conv,axiom,
! [A2: set_Sum_sum_a_nat] :
( ( ! [X3: sum_sum_a_nat] :
~ ( member_Sum_sum_a_nat @ X3 @ A2 ) )
= ( A2 = bot_bo3438331934148233675_a_nat ) ) ).
% all_not_in_conv
thf(fact_33_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_34_empty__iff,axiom,
! [C: list_nat] :
~ ( member_list_nat @ C @ bot_bot_set_list_nat ) ).
% empty_iff
thf(fact_35_empty__iff,axiom,
! [C: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ C @ bot_bo1033123847703346641_a_nat ) ).
% empty_iff
thf(fact_36_empty__iff,axiom,
! [C: sum_sum_a_nat] :
~ ( member_Sum_sum_a_nat @ C @ bot_bo3438331934148233675_a_nat ) ).
% empty_iff
thf(fact_37_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_38_insert__absorb2,axiom,
! [X2: nat,A2: set_nat] :
( ( insert_nat2 @ X2 @ ( insert_nat2 @ X2 @ A2 ) )
= ( insert_nat2 @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_39_insert__iff,axiom,
! [A: list_nat,B2: list_nat,A2: set_list_nat] :
( ( member_list_nat @ A @ ( insert_list_nat @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_list_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_40_insert__iff,axiom,
! [A: list_Sum_sum_a_nat,B2: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A @ ( insert2950094090816004437_a_nat @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member408289922725080238_a_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_41_insert__iff,axiom,
! [A: sum_sum_a_nat,B2: sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A @ ( insert_Sum_sum_a_nat @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_Sum_sum_a_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_42_insert__iff,axiom,
! [A: nat,B2: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat2 @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_43_insertCI,axiom,
! [A: list_nat,B: set_list_nat,B2: list_nat] :
( ( ~ ( member_list_nat @ A @ B )
=> ( A = B2 ) )
=> ( member_list_nat @ A @ ( insert_list_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_44_insertCI,axiom,
! [A: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat,B2: list_Sum_sum_a_nat] :
( ( ~ ( member408289922725080238_a_nat @ A @ B )
=> ( A = B2 ) )
=> ( member408289922725080238_a_nat @ A @ ( insert2950094090816004437_a_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_45_insertCI,axiom,
! [A: sum_sum_a_nat,B: set_Sum_sum_a_nat,B2: sum_sum_a_nat] :
( ( ~ ( member_Sum_sum_a_nat @ A @ B )
=> ( A = B2 ) )
=> ( member_Sum_sum_a_nat @ A @ ( insert_Sum_sum_a_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_46_insertCI,axiom,
! [A: nat,B: set_nat,B2: nat] :
( ( ~ ( member_nat @ A @ B )
=> ( A = B2 ) )
=> ( member_nat @ A @ ( insert_nat2 @ B2 @ B ) ) ) ).
% insertCI
thf(fact_47_Diff__idemp,axiom,
! [A2: set_nat,B: set_nat] :
( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ B )
= ( minus_minus_set_nat @ A2 @ B ) ) ).
% Diff_idemp
thf(fact_48_Diff__iff,axiom,
! [C: list_nat,A2: set_list_nat,B: set_list_nat] :
( ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A2 @ B ) )
= ( ( member_list_nat @ C @ A2 )
& ~ ( member_list_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_49_Diff__iff,axiom,
! [C: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( minus_7395159227704179404_a_nat @ A2 @ B ) )
= ( ( member408289922725080238_a_nat @ C @ A2 )
& ~ ( member408289922725080238_a_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_50_Diff__iff,axiom,
! [C: sum_sum_a_nat,A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ ( minus_1134630996077396038_a_nat @ A2 @ B ) )
= ( ( member_Sum_sum_a_nat @ C @ A2 )
& ~ ( member_Sum_sum_a_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_51_Diff__iff,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B ) )
= ( ( member_nat @ C @ A2 )
& ~ ( member_nat @ C @ B ) ) ) ).
% Diff_iff
thf(fact_52_DiffI,axiom,
! [C: list_nat,A2: set_list_nat,B: set_list_nat] :
( ( member_list_nat @ C @ A2 )
=> ( ~ ( member_list_nat @ C @ B )
=> ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_53_DiffI,axiom,
! [C: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ A2 )
=> ( ~ ( member408289922725080238_a_nat @ C @ B )
=> ( member408289922725080238_a_nat @ C @ ( minus_7395159227704179404_a_nat @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_54_DiffI,axiom,
! [C: sum_sum_a_nat,A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ A2 )
=> ( ~ ( member_Sum_sum_a_nat @ C @ B )
=> ( member_Sum_sum_a_nat @ C @ ( minus_1134630996077396038_a_nat @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_55_DiffI,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_56_ex__in__conv,axiom,
! [A2: set_list_nat] :
( ( ? [X3: list_nat] : ( member_list_nat @ X3 @ A2 ) )
= ( A2 != bot_bot_set_list_nat ) ) ).
% ex_in_conv
thf(fact_57_ex__in__conv,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ( ? [X3: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X3 @ A2 ) )
= ( A2 != bot_bo1033123847703346641_a_nat ) ) ).
% ex_in_conv
thf(fact_58_ex__in__conv,axiom,
! [A2: set_Sum_sum_a_nat] :
( ( ? [X3: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X3 @ A2 ) )
= ( A2 != bot_bo3438331934148233675_a_nat ) ) ).
% ex_in_conv
thf(fact_59_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_60_equals0I,axiom,
! [A2: set_list_nat] :
( ! [Y2: list_nat] :
~ ( member_list_nat @ Y2 @ A2 )
=> ( A2 = bot_bot_set_list_nat ) ) ).
% equals0I
thf(fact_61_equals0I,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ! [Y2: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ Y2 @ A2 )
=> ( A2 = bot_bo1033123847703346641_a_nat ) ) ).
% equals0I
thf(fact_62_equals0I,axiom,
! [A2: set_Sum_sum_a_nat] :
( ! [Y2: sum_sum_a_nat] :
~ ( member_Sum_sum_a_nat @ Y2 @ A2 )
=> ( A2 = bot_bo3438331934148233675_a_nat ) ) ).
% equals0I
thf(fact_63_equals0I,axiom,
! [A2: set_nat] :
( ! [Y2: nat] :
~ ( member_nat @ Y2 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_64_equals0D,axiom,
! [A2: set_list_nat,A: list_nat] :
( ( A2 = bot_bot_set_list_nat )
=> ~ ( member_list_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_65_equals0D,axiom,
! [A2: set_li6526943997496501093_a_nat,A: list_Sum_sum_a_nat] :
( ( A2 = bot_bo1033123847703346641_a_nat )
=> ~ ( member408289922725080238_a_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_66_equals0D,axiom,
! [A2: set_Sum_sum_a_nat,A: sum_sum_a_nat] :
( ( A2 = bot_bo3438331934148233675_a_nat )
=> ~ ( member_Sum_sum_a_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_67_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_68_emptyE,axiom,
! [A: list_nat] :
~ ( member_list_nat @ A @ bot_bot_set_list_nat ) ).
% emptyE
thf(fact_69_emptyE,axiom,
! [A: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ A @ bot_bo1033123847703346641_a_nat ) ).
% emptyE
thf(fact_70_emptyE,axiom,
! [A: sum_sum_a_nat] :
~ ( member_Sum_sum_a_nat @ A @ bot_bo3438331934148233675_a_nat ) ).
% emptyE
thf(fact_71_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_72_mk__disjoint__insert,axiom,
! [A: list_nat,A2: set_list_nat] :
( ( member_list_nat @ A @ A2 )
=> ? [B3: set_list_nat] :
( ( A2
= ( insert_list_nat @ A @ B3 ) )
& ~ ( member_list_nat @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_73_mk__disjoint__insert,axiom,
! [A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A @ A2 )
=> ? [B3: set_li6526943997496501093_a_nat] :
( ( A2
= ( insert2950094090816004437_a_nat @ A @ B3 ) )
& ~ ( member408289922725080238_a_nat @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_74_mk__disjoint__insert,axiom,
! [A: sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A @ A2 )
=> ? [B3: set_Sum_sum_a_nat] :
( ( A2
= ( insert_Sum_sum_a_nat @ A @ B3 ) )
& ~ ( member_Sum_sum_a_nat @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_75_mk__disjoint__insert,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ? [B3: set_nat] :
( ( A2
= ( insert_nat2 @ A @ B3 ) )
& ~ ( member_nat @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_76_insert__commute,axiom,
! [X2: nat,Y: nat,A2: set_nat] :
( ( insert_nat2 @ X2 @ ( insert_nat2 @ Y @ A2 ) )
= ( insert_nat2 @ Y @ ( insert_nat2 @ X2 @ A2 ) ) ) ).
% insert_commute
thf(fact_77_insert__eq__iff,axiom,
! [A: list_nat,A2: set_list_nat,B2: list_nat,B: set_list_nat] :
( ~ ( member_list_nat @ A @ A2 )
=> ( ~ ( member_list_nat @ B2 @ B )
=> ( ( ( insert_list_nat @ A @ A2 )
= ( insert_list_nat @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C2: set_list_nat] :
( ( A2
= ( insert_list_nat @ B2 @ C2 ) )
& ~ ( member_list_nat @ B2 @ C2 )
& ( B
= ( insert_list_nat @ A @ C2 ) )
& ~ ( member_list_nat @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_78_insert__eq__iff,axiom,
! [A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ A @ A2 )
=> ( ~ ( member408289922725080238_a_nat @ B2 @ B )
=> ( ( ( insert2950094090816004437_a_nat @ A @ A2 )
= ( insert2950094090816004437_a_nat @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C2: set_li6526943997496501093_a_nat] :
( ( A2
= ( insert2950094090816004437_a_nat @ B2 @ C2 ) )
& ~ ( member408289922725080238_a_nat @ B2 @ C2 )
& ( B
= ( insert2950094090816004437_a_nat @ A @ C2 ) )
& ~ ( member408289922725080238_a_nat @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_79_insert__eq__iff,axiom,
! [A: sum_sum_a_nat,A2: set_Sum_sum_a_nat,B2: sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ~ ( member_Sum_sum_a_nat @ A @ A2 )
=> ( ~ ( member_Sum_sum_a_nat @ B2 @ B )
=> ( ( ( insert_Sum_sum_a_nat @ A @ A2 )
= ( insert_Sum_sum_a_nat @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C2: set_Sum_sum_a_nat] :
( ( A2
= ( insert_Sum_sum_a_nat @ B2 @ C2 ) )
& ~ ( member_Sum_sum_a_nat @ B2 @ C2 )
& ( B
= ( insert_Sum_sum_a_nat @ A @ C2 ) )
& ~ ( member_Sum_sum_a_nat @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_80_insert__eq__iff,axiom,
! [A: nat,A2: set_nat,B2: nat,B: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ B2 @ B )
=> ( ( ( insert_nat2 @ A @ A2 )
= ( insert_nat2 @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C2: set_nat] :
( ( A2
= ( insert_nat2 @ B2 @ C2 ) )
& ~ ( member_nat @ B2 @ C2 )
& ( B
= ( insert_nat2 @ A @ C2 ) )
& ~ ( member_nat @ A @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_81_insert__absorb,axiom,
! [A: list_nat,A2: set_list_nat] :
( ( member_list_nat @ A @ A2 )
=> ( ( insert_list_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_82_insert__absorb,axiom,
! [A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A @ A2 )
=> ( ( insert2950094090816004437_a_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_83_insert__absorb,axiom,
! [A: sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A @ A2 )
=> ( ( insert_Sum_sum_a_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_84_insert__absorb,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat2 @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_85_insert__ident,axiom,
! [X2: list_nat,A2: set_list_nat,B: set_list_nat] :
( ~ ( member_list_nat @ X2 @ A2 )
=> ( ~ ( member_list_nat @ X2 @ B )
=> ( ( ( insert_list_nat @ X2 @ A2 )
= ( insert_list_nat @ X2 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_86_insert__ident,axiom,
! [X2: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ X2 @ A2 )
=> ( ~ ( member408289922725080238_a_nat @ X2 @ B )
=> ( ( ( insert2950094090816004437_a_nat @ X2 @ A2 )
= ( insert2950094090816004437_a_nat @ X2 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_87_insert__ident,axiom,
! [X2: sum_sum_a_nat,A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ~ ( member_Sum_sum_a_nat @ X2 @ A2 )
=> ( ~ ( member_Sum_sum_a_nat @ X2 @ B )
=> ( ( ( insert_Sum_sum_a_nat @ X2 @ A2 )
= ( insert_Sum_sum_a_nat @ X2 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_88_insert__ident,axiom,
! [X2: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ X2 @ A2 )
=> ( ~ ( member_nat @ X2 @ B )
=> ( ( ( insert_nat2 @ X2 @ A2 )
= ( insert_nat2 @ X2 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_89_Set_Oset__insert,axiom,
! [X2: list_nat,A2: set_list_nat] :
( ( member_list_nat @ X2 @ A2 )
=> ~ ! [B3: set_list_nat] :
( ( A2
= ( insert_list_nat @ X2 @ B3 ) )
=> ( member_list_nat @ X2 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_90_Set_Oset__insert,axiom,
! [X2: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A2 )
=> ~ ! [B3: set_li6526943997496501093_a_nat] :
( ( A2
= ( insert2950094090816004437_a_nat @ X2 @ B3 ) )
=> ( member408289922725080238_a_nat @ X2 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_91_Set_Oset__insert,axiom,
! [X2: sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ A2 )
=> ~ ! [B3: set_Sum_sum_a_nat] :
( ( A2
= ( insert_Sum_sum_a_nat @ X2 @ B3 ) )
=> ( member_Sum_sum_a_nat @ X2 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_92_Set_Oset__insert,axiom,
! [X2: nat,A2: set_nat] :
( ( member_nat @ X2 @ A2 )
=> ~ ! [B3: set_nat] :
( ( A2
= ( insert_nat2 @ X2 @ B3 ) )
=> ( member_nat @ X2 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_93_insertI2,axiom,
! [A: list_nat,B: set_list_nat,B2: list_nat] :
( ( member_list_nat @ A @ B )
=> ( member_list_nat @ A @ ( insert_list_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_94_insertI2,axiom,
! [A: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat,B2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ A @ B )
=> ( member408289922725080238_a_nat @ A @ ( insert2950094090816004437_a_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_95_insertI2,axiom,
! [A: sum_sum_a_nat,B: set_Sum_sum_a_nat,B2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A @ B )
=> ( member_Sum_sum_a_nat @ A @ ( insert_Sum_sum_a_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_96_insertI2,axiom,
! [A: nat,B: set_nat,B2: nat] :
( ( member_nat @ A @ B )
=> ( member_nat @ A @ ( insert_nat2 @ B2 @ B ) ) ) ).
% insertI2
thf(fact_97_insertI1,axiom,
! [A: list_nat,B: set_list_nat] : ( member_list_nat @ A @ ( insert_list_nat @ A @ B ) ) ).
% insertI1
thf(fact_98_insertI1,axiom,
! [A: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] : ( member408289922725080238_a_nat @ A @ ( insert2950094090816004437_a_nat @ A @ B ) ) ).
% insertI1
thf(fact_99_insertI1,axiom,
! [A: sum_sum_a_nat,B: set_Sum_sum_a_nat] : ( member_Sum_sum_a_nat @ A @ ( insert_Sum_sum_a_nat @ A @ B ) ) ).
% insertI1
thf(fact_100_insertI1,axiom,
! [A: nat,B: set_nat] : ( member_nat @ A @ ( insert_nat2 @ A @ B ) ) ).
% insertI1
thf(fact_101_insertE,axiom,
! [A: list_nat,B2: list_nat,A2: set_list_nat] :
( ( member_list_nat @ A @ ( insert_list_nat @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_list_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_102_insertE,axiom,
! [A: list_Sum_sum_a_nat,B2: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A @ ( insert2950094090816004437_a_nat @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member408289922725080238_a_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_103_insertE,axiom,
! [A: sum_sum_a_nat,B2: sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A @ ( insert_Sum_sum_a_nat @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_Sum_sum_a_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_104_insertE,axiom,
! [A: nat,B2: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat2 @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_105_DiffD2,axiom,
! [C: list_nat,A2: set_list_nat,B: set_list_nat] :
( ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A2 @ B ) )
=> ~ ( member_list_nat @ C @ B ) ) ).
% DiffD2
thf(fact_106_DiffD2,axiom,
! [C: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( minus_7395159227704179404_a_nat @ A2 @ B ) )
=> ~ ( member408289922725080238_a_nat @ C @ B ) ) ).
% DiffD2
thf(fact_107_DiffD2,axiom,
! [C: sum_sum_a_nat,A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ ( minus_1134630996077396038_a_nat @ A2 @ B ) )
=> ~ ( member_Sum_sum_a_nat @ C @ B ) ) ).
% DiffD2
thf(fact_108_DiffD2,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B ) )
=> ~ ( member_nat @ C @ B ) ) ).
% DiffD2
thf(fact_109_DiffD1,axiom,
! [C: list_nat,A2: set_list_nat,B: set_list_nat] :
( ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A2 @ B ) )
=> ( member_list_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_110_DiffD1,axiom,
! [C: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( minus_7395159227704179404_a_nat @ A2 @ B ) )
=> ( member408289922725080238_a_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_111_DiffD1,axiom,
! [C: sum_sum_a_nat,A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ ( minus_1134630996077396038_a_nat @ A2 @ B ) )
=> ( member_Sum_sum_a_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_112_DiffD1,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B ) )
=> ( member_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_113_DiffE,axiom,
! [C: list_nat,A2: set_list_nat,B: set_list_nat] :
( ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A2 @ B ) )
=> ~ ( ( member_list_nat @ C @ A2 )
=> ( member_list_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_114_DiffE,axiom,
! [C: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( minus_7395159227704179404_a_nat @ A2 @ B ) )
=> ~ ( ( member408289922725080238_a_nat @ C @ A2 )
=> ( member408289922725080238_a_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_115_DiffE,axiom,
! [C: sum_sum_a_nat,A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ ( minus_1134630996077396038_a_nat @ A2 @ B ) )
=> ~ ( ( member_Sum_sum_a_nat @ C @ A2 )
=> ( member_Sum_sum_a_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_116_DiffE,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B ) ) ) ).
% DiffE
thf(fact_117_singleton__inject,axiom,
! [A: nat,B2: nat] :
( ( ( insert_nat2 @ A @ bot_bot_set_nat )
= ( insert_nat2 @ B2 @ bot_bot_set_nat ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_118_insert__not__empty,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat2 @ A @ A2 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_119_doubleton__eq__iff,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ( insert_nat2 @ A @ ( insert_nat2 @ B2 @ bot_bot_set_nat ) )
= ( insert_nat2 @ C @ ( insert_nat2 @ D @ bot_bot_set_nat ) ) )
= ( ( ( A = C )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_120_singleton__iff,axiom,
! [B2: list_nat,A: list_nat] :
( ( member_list_nat @ B2 @ ( insert_list_nat @ A @ bot_bot_set_list_nat ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_121_singleton__iff,axiom,
! [B2: list_Sum_sum_a_nat,A: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ B2 @ ( insert2950094090816004437_a_nat @ A @ bot_bo1033123847703346641_a_nat ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_122_singleton__iff,axiom,
! [B2: sum_sum_a_nat,A: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ B2 @ ( insert_Sum_sum_a_nat @ A @ bot_bo3438331934148233675_a_nat ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_123_singleton__iff,axiom,
! [B2: nat,A: nat] :
( ( member_nat @ B2 @ ( insert_nat2 @ A @ bot_bot_set_nat ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_124_singletonD,axiom,
! [B2: list_nat,A: list_nat] :
( ( member_list_nat @ B2 @ ( insert_list_nat @ A @ bot_bot_set_list_nat ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_125_singletonD,axiom,
! [B2: list_Sum_sum_a_nat,A: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ B2 @ ( insert2950094090816004437_a_nat @ A @ bot_bo1033123847703346641_a_nat ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_126_singletonD,axiom,
! [B2: sum_sum_a_nat,A: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ B2 @ ( insert_Sum_sum_a_nat @ A @ bot_bo3438331934148233675_a_nat ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_127_singletonD,axiom,
! [B2: nat,A: nat] :
( ( member_nat @ B2 @ ( insert_nat2 @ A @ bot_bot_set_nat ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_128_insert__Diff__if,axiom,
! [X2: list_nat,B: set_list_nat,A2: set_list_nat] :
( ( ( member_list_nat @ X2 @ B )
=> ( ( minus_7954133019191499631st_nat @ ( insert_list_nat @ X2 @ A2 ) @ B )
= ( minus_7954133019191499631st_nat @ A2 @ B ) ) )
& ( ~ ( member_list_nat @ X2 @ B )
=> ( ( minus_7954133019191499631st_nat @ ( insert_list_nat @ X2 @ A2 ) @ B )
= ( insert_list_nat @ X2 @ ( minus_7954133019191499631st_nat @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_129_insert__Diff__if,axiom,
! [X2: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ( member408289922725080238_a_nat @ X2 @ B )
=> ( ( minus_7395159227704179404_a_nat @ ( insert2950094090816004437_a_nat @ X2 @ A2 ) @ B )
= ( minus_7395159227704179404_a_nat @ A2 @ B ) ) )
& ( ~ ( member408289922725080238_a_nat @ X2 @ B )
=> ( ( minus_7395159227704179404_a_nat @ ( insert2950094090816004437_a_nat @ X2 @ A2 ) @ B )
= ( insert2950094090816004437_a_nat @ X2 @ ( minus_7395159227704179404_a_nat @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_130_insert__Diff__if,axiom,
! [X2: sum_sum_a_nat,B: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( ( member_Sum_sum_a_nat @ X2 @ B )
=> ( ( minus_1134630996077396038_a_nat @ ( insert_Sum_sum_a_nat @ X2 @ A2 ) @ B )
= ( minus_1134630996077396038_a_nat @ A2 @ B ) ) )
& ( ~ ( member_Sum_sum_a_nat @ X2 @ B )
=> ( ( minus_1134630996077396038_a_nat @ ( insert_Sum_sum_a_nat @ X2 @ A2 ) @ B )
= ( insert_Sum_sum_a_nat @ X2 @ ( minus_1134630996077396038_a_nat @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_131_insert__Diff__if,axiom,
! [X2: nat,B: set_nat,A2: set_nat] :
( ( ( member_nat @ X2 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X2 @ A2 ) @ B )
= ( minus_minus_set_nat @ A2 @ B ) ) )
& ( ~ ( member_nat @ X2 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat2 @ X2 @ A2 ) @ B )
= ( insert_nat2 @ X2 @ ( minus_minus_set_nat @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_132_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_133_mem__Collect__eq,axiom,
! [A: list_Sum_sum_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( member408289922725080238_a_nat @ A @ ( collec7555443234367654128_a_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_134_mem__Collect__eq,axiom,
! [A: sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( member_Sum_sum_a_nat @ A @ ( collec7073057861543223018_a_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_135_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_136_Collect__mem__eq,axiom,
! [A2: set_list_nat] :
( ( collect_list_nat
@ ^ [X3: list_nat] : ( member_list_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_137_Collect__mem__eq,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ( collec7555443234367654128_a_nat
@ ^ [X3: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_138_Collect__mem__eq,axiom,
! [A2: set_Sum_sum_a_nat] :
( ( collec7073057861543223018_a_nat
@ ^ [X3: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_139_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_140_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X4: nat] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_141_not__empty__eq__Iic__eq__empty,axiom,
! [H: nat] :
( bot_bot_set_nat
!= ( set_ord_atMost_nat @ H ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_142_bot__apply,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : bot_bot_o ) ) ).
% bot_apply
thf(fact_143_is__singletonI,axiom,
! [X2: nat] : ( is_singleton_nat @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_144_assms_I1_J,axiom,
ad_agr_list_a_nat @ ad @ xs @ ys ).
% assms(1)
thf(fact_145_is__singleton__def,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
? [X3: nat] :
( A3
= ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_def
thf(fact_146_is__singletonE,axiom,
! [A2: set_nat] :
( ( is_singleton_nat @ A2 )
=> ~ ! [X4: nat] :
( A2
!= ( insert_nat2 @ X4 @ bot_bot_set_nat ) ) ) ).
% is_singletonE
thf(fact_147_atMost__0,axiom,
( ( set_ord_atMost_nat @ zero_zero_nat )
= ( insert_nat2 @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% atMost_0
thf(fact_148_Set_Oremove__def,axiom,
( remove_nat
= ( ^ [X3: nat,A3: set_nat] : ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) ) ).
% Set.remove_def
thf(fact_149_the__elem__eq,axiom,
! [X2: nat] :
( ( the_elem_nat @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
= X2 ) ).
% the_elem_eq
thf(fact_150_pairwise__alt,axiom,
( pairwise_nat
= ( ^ [R: nat > nat > $o,S2: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ S2 )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ ( minus_minus_set_nat @ S2 @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) )
=> ( R @ X3 @ Y3 ) ) ) ) ) ).
% pairwise_alt
thf(fact_151_single__Diff__lessThan,axiom,
! [K: nat] :
( ( minus_minus_set_nat @ ( insert_nat2 @ K @ bot_bot_set_nat ) @ ( set_ord_lessThan_nat @ K ) )
= ( insert_nat2 @ K @ bot_bot_set_nat ) ) ).
% single_Diff_lessThan
thf(fact_152_bot__fun__def,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_153_lessThan__eq__iff,axiom,
! [X2: nat,Y: nat] :
( ( ( set_ord_lessThan_nat @ X2 )
= ( set_ord_lessThan_nat @ Y ) )
= ( X2 = Y ) ) ).
% lessThan_eq_iff
thf(fact_154_Set_Omember__remove,axiom,
! [X2: list_nat,Y: list_nat,A2: set_list_nat] :
( ( member_list_nat @ X2 @ ( remove_list_nat @ Y @ A2 ) )
= ( ( member_list_nat @ X2 @ A2 )
& ( X2 != Y ) ) ) ).
% Set.member_remove
thf(fact_155_Set_Omember__remove,axiom,
! [X2: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ ( remove5086202153292001386_a_nat @ Y @ A2 ) )
= ( ( member408289922725080238_a_nat @ X2 @ A2 )
& ( X2 != Y ) ) ) ).
% Set.member_remove
thf(fact_156_Set_Omember__remove,axiom,
! [X2: sum_sum_a_nat,Y: sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ ( remove_Sum_sum_a_nat @ Y @ A2 ) )
= ( ( member_Sum_sum_a_nat @ X2 @ A2 )
& ( X2 != Y ) ) ) ).
% Set.member_remove
thf(fact_157_Set_Omember__remove,axiom,
! [X2: nat,Y: nat,A2: set_nat] :
( ( member_nat @ X2 @ ( remove_nat @ Y @ A2 ) )
= ( ( member_nat @ X2 @ A2 )
& ( X2 != Y ) ) ) ).
% Set.member_remove
thf(fact_158_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_159_pairwiseD,axiom,
! [R2: list_nat > list_nat > $o,S: set_list_nat,X2: list_nat,Y: list_nat] :
( ( pairwise_list_nat @ R2 @ S )
=> ( ( member_list_nat @ X2 @ S )
=> ( ( member_list_nat @ Y @ S )
=> ( ( X2 != Y )
=> ( R2 @ X2 @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_160_pairwiseD,axiom,
! [R2: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o,S: set_li6526943997496501093_a_nat,X2: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat] :
( ( pairwi4897900009783174640_a_nat @ R2 @ S )
=> ( ( member408289922725080238_a_nat @ X2 @ S )
=> ( ( member408289922725080238_a_nat @ Y @ S )
=> ( ( X2 != Y )
=> ( R2 @ X2 @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_161_pairwiseD,axiom,
! [R2: sum_sum_a_nat > sum_sum_a_nat > $o,S: set_Sum_sum_a_nat,X2: sum_sum_a_nat,Y: sum_sum_a_nat] :
( ( pairwi7370142813935713258_a_nat @ R2 @ S )
=> ( ( member_Sum_sum_a_nat @ X2 @ S )
=> ( ( member_Sum_sum_a_nat @ Y @ S )
=> ( ( X2 != Y )
=> ( R2 @ X2 @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_162_pairwiseD,axiom,
! [R2: nat > nat > $o,S: set_nat,X2: nat,Y: nat] :
( ( pairwise_nat @ R2 @ S )
=> ( ( member_nat @ X2 @ S )
=> ( ( member_nat @ Y @ S )
=> ( ( X2 != Y )
=> ( R2 @ X2 @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_163_pairwiseI,axiom,
! [S: set_list_nat,R2: list_nat > list_nat > $o] :
( ! [X4: list_nat,Y2: list_nat] :
( ( member_list_nat @ X4 @ S )
=> ( ( member_list_nat @ Y2 @ S )
=> ( ( X4 != Y2 )
=> ( R2 @ X4 @ Y2 ) ) ) )
=> ( pairwise_list_nat @ R2 @ S ) ) ).
% pairwiseI
thf(fact_164_pairwiseI,axiom,
! [S: set_li6526943997496501093_a_nat,R2: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
( ! [X4: list_Sum_sum_a_nat,Y2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ S )
=> ( ( member408289922725080238_a_nat @ Y2 @ S )
=> ( ( X4 != Y2 )
=> ( R2 @ X4 @ Y2 ) ) ) )
=> ( pairwi4897900009783174640_a_nat @ R2 @ S ) ) ).
% pairwiseI
thf(fact_165_pairwiseI,axiom,
! [S: set_Sum_sum_a_nat,R2: sum_sum_a_nat > sum_sum_a_nat > $o] :
( ! [X4: sum_sum_a_nat,Y2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ S )
=> ( ( member_Sum_sum_a_nat @ Y2 @ S )
=> ( ( X4 != Y2 )
=> ( R2 @ X4 @ Y2 ) ) ) )
=> ( pairwi7370142813935713258_a_nat @ R2 @ S ) ) ).
% pairwiseI
thf(fact_166_pairwiseI,axiom,
! [S: set_nat,R2: nat > nat > $o] :
( ! [X4: nat,Y2: nat] :
( ( member_nat @ X4 @ S )
=> ( ( member_nat @ Y2 @ S )
=> ( ( X4 != Y2 )
=> ( R2 @ X4 @ Y2 ) ) ) )
=> ( pairwise_nat @ R2 @ S ) ) ).
% pairwiseI
thf(fact_167_pairwise__def,axiom,
( pairwise_nat
= ( ^ [R: nat > nat > $o,S2: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ S2 )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ S2 )
=> ( ( X3 != Y3 )
=> ( R @ X3 @ Y3 ) ) ) ) ) ) ).
% pairwise_def
thf(fact_168_ad__agr__list__trans,axiom,
! [X: set_a,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
( ( ad_agr_list_a_nat @ X @ Xs @ Ys )
=> ( ( ad_agr_list_a_nat @ X @ Ys @ Zs )
=> ( ad_agr_list_a_nat @ X @ Xs @ Zs ) ) ) ).
% ad_agr_list_trans
thf(fact_169_ad__agr__list__refl,axiom,
! [X: set_a,Xs: list_Sum_sum_a_nat] : ( ad_agr_list_a_nat @ X @ Xs @ Xs ) ).
% ad_agr_list_refl
thf(fact_170_ad__agr__list__comm,axiom,
! [X: set_a,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ad_agr_list_a_nat @ X @ Xs @ Ys )
=> ( ad_agr_list_a_nat @ X @ Ys @ Xs ) ) ).
% ad_agr_list_comm
thf(fact_171_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_172_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_173_pairwise__empty,axiom,
! [P: nat > nat > $o] : ( pairwise_nat @ P @ bot_bot_set_nat ) ).
% pairwise_empty
thf(fact_174_pairwise__insert,axiom,
! [R3: list_nat > list_nat > $o,X2: list_nat,S3: set_list_nat] :
( ( pairwise_list_nat @ R3 @ ( insert_list_nat @ X2 @ S3 ) )
= ( ! [Y3: list_nat] :
( ( ( member_list_nat @ Y3 @ S3 )
& ( Y3 != X2 ) )
=> ( ( R3 @ X2 @ Y3 )
& ( R3 @ Y3 @ X2 ) ) )
& ( pairwise_list_nat @ R3 @ S3 ) ) ) ).
% pairwise_insert
thf(fact_175_pairwise__insert,axiom,
! [R3: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o,X2: list_Sum_sum_a_nat,S3: set_li6526943997496501093_a_nat] :
( ( pairwi4897900009783174640_a_nat @ R3 @ ( insert2950094090816004437_a_nat @ X2 @ S3 ) )
= ( ! [Y3: list_Sum_sum_a_nat] :
( ( ( member408289922725080238_a_nat @ Y3 @ S3 )
& ( Y3 != X2 ) )
=> ( ( R3 @ X2 @ Y3 )
& ( R3 @ Y3 @ X2 ) ) )
& ( pairwi4897900009783174640_a_nat @ R3 @ S3 ) ) ) ).
% pairwise_insert
thf(fact_176_pairwise__insert,axiom,
! [R3: sum_sum_a_nat > sum_sum_a_nat > $o,X2: sum_sum_a_nat,S3: set_Sum_sum_a_nat] :
( ( pairwi7370142813935713258_a_nat @ R3 @ ( insert_Sum_sum_a_nat @ X2 @ S3 ) )
= ( ! [Y3: sum_sum_a_nat] :
( ( ( member_Sum_sum_a_nat @ Y3 @ S3 )
& ( Y3 != X2 ) )
=> ( ( R3 @ X2 @ Y3 )
& ( R3 @ Y3 @ X2 ) ) )
& ( pairwi7370142813935713258_a_nat @ R3 @ S3 ) ) ) ).
% pairwise_insert
thf(fact_177_pairwise__insert,axiom,
! [R3: nat > nat > $o,X2: nat,S3: set_nat] :
( ( pairwise_nat @ R3 @ ( insert_nat2 @ X2 @ S3 ) )
= ( ! [Y3: nat] :
( ( ( member_nat @ Y3 @ S3 )
& ( Y3 != X2 ) )
=> ( ( R3 @ X2 @ Y3 )
& ( R3 @ Y3 @ X2 ) ) )
& ( pairwise_nat @ R3 @ S3 ) ) ) ).
% pairwise_insert
thf(fact_178_is__singleton__the__elem,axiom,
( is_singleton_nat
= ( ^ [A3: set_nat] :
( A3
= ( insert_nat2 @ ( the_elem_nat @ A3 ) @ bot_bot_set_nat ) ) ) ) ).
% is_singleton_the_elem
thf(fact_179_Iio__eq__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan_nat @ N )
= bot_bot_set_nat )
= ( N = bot_bot_nat ) ) ).
% Iio_eq_empty_iff
thf(fact_180_is__singletonI_H,axiom,
! [A2: set_list_nat] :
( ( A2 != bot_bot_set_list_nat )
=> ( ! [X4: list_nat,Y2: list_nat] :
( ( member_list_nat @ X4 @ A2 )
=> ( ( member_list_nat @ Y2 @ A2 )
=> ( X4 = Y2 ) ) )
=> ( is_sin2641923865335537900st_nat @ A2 ) ) ) ).
% is_singletonI'
thf(fact_181_is__singletonI_H,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ( A2 != bot_bo1033123847703346641_a_nat )
=> ( ! [X4: list_Sum_sum_a_nat,Y2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A2 )
=> ( ( member408289922725080238_a_nat @ Y2 @ A2 )
=> ( X4 = Y2 ) ) )
=> ( is_sin2231188923920309881_a_nat @ A2 ) ) ) ).
% is_singletonI'
thf(fact_182_is__singletonI_H,axiom,
! [A2: set_Sum_sum_a_nat] :
( ( A2 != bot_bo3438331934148233675_a_nat )
=> ( ! [X4: sum_sum_a_nat,Y2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ A2 )
=> ( ( member_Sum_sum_a_nat @ Y2 @ A2 )
=> ( X4 = Y2 ) ) )
=> ( is_sin5176708635568246003_a_nat @ A2 ) ) ) ).
% is_singletonI'
thf(fact_183_is__singletonI_H,axiom,
! [A2: set_nat] :
( ( A2 != bot_bot_set_nat )
=> ( ! [X4: nat,Y2: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( member_nat @ Y2 @ A2 )
=> ( X4 = Y2 ) ) )
=> ( is_singleton_nat @ A2 ) ) ) ).
% is_singletonI'
thf(fact_184_pairwise__singleton,axiom,
! [P: nat > nat > $o,A2: nat] : ( pairwise_nat @ P @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) ).
% pairwise_singleton
thf(fact_185_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_186_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_187_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_188_len__ys,axiom,
( n
= ( size_s5283204784079214577_a_nat @ ys ) ) ).
% len_ys
thf(fact_189_n__def,axiom,
( n
= ( size_s5283204784079214577_a_nat @ xs ) ) ).
% n_def
thf(fact_190__092_060tau_062__def,axiom,
( ys
= ( map_na823391071729141993_a_nat @ tau @ ( upt @ zero_zero_nat @ n ) ) ) ).
% \<tau>_def
thf(fact_191__092_060sigma_062__def,axiom,
( xs
= ( map_na823391071729141993_a_nat @ sigma @ ( upt @ zero_zero_nat @ n ) ) ) ).
% \<sigma>_def
thf(fact_192_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_193_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_194__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062_092_060sigma_062_O_Axs_A_061_Amap_A_092_060sigma_062_A_0910_O_O_060n_093_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Sigma2: nat > sum_sum_a_nat] :
( xs
!= ( map_na823391071729141993_a_nat @ Sigma2 @ ( upt @ zero_zero_nat @ n ) ) ) ).
% \<open>\<And>thesis. (\<And>\<sigma>. xs = map \<sigma> [0..<n] \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_195__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062_092_060tau_062_O_Ays_A_061_Amap_A_092_060tau_062_A_0910_O_O_060n_093_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Tau2: nat > sum_sum_a_nat] :
( ys
!= ( map_na823391071729141993_a_nat @ Tau2 @ ( upt @ zero_zero_nat @ n ) ) ) ).
% \<open>\<And>thesis. (\<And>\<tau>. ys = map \<tau> [0..<n] \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_196_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_197_size__neq__size__imp__neq,axiom,
! [X2: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ X2 )
!= ( size_s5283204784079214577_a_nat @ Y ) )
=> ( X2 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_198_size__neq__size__imp__neq,axiom,
! [X2: list_nat,Y: list_nat] :
( ( ( size_size_list_nat @ X2 )
!= ( size_size_list_nat @ Y ) )
=> ( X2 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_199_size__neq__size__imp__neq,axiom,
! [X2: char,Y: char] :
( ( ( size_size_char @ X2 )
!= ( size_size_char @ Y ) )
=> ( X2 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_200_ad__agr__list__length,axiom,
! [X: set_a,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ad_agr_list_a_nat @ X @ Xs @ Ys )
=> ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).
% ad_agr_list_length
thf(fact_201_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_202_diff__right__commute,axiom,
! [A: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B2 )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B2 ) @ C ) ) ).
% diff_right_commute
thf(fact_203_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_204_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_205_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_206_set__n_I1_J,axiom,
( ( set_nat2 @ ( upt @ zero_zero_nat @ n ) )
= ( minus_minus_set_nat @ ( set_ord_atMost_nat @ n ) @ ( insert_nat2 @ n @ bot_bot_set_nat ) ) ) ).
% set_n(1)
thf(fact_207_length__map,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( map_Su2790769393171190532_a_nat @ F @ Xs ) )
= ( size_s5283204784079214577_a_nat @ Xs ) ) ).
% length_map
thf(fact_208_length__map,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat] :
( ( size_s5283204784079214577_a_nat @ ( map_na823391071729141993_a_nat @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_209_length__map,axiom,
! [F: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat] :
( ( size_size_list_nat @ ( map_Su5227373005390213643at_nat @ F @ Xs ) )
= ( size_s5283204784079214577_a_nat @ Xs ) ) ).
% length_map
thf(fact_210_length__map,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( size_size_list_nat @ ( map_nat_nat @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_211_assms_I2_J,axiom,
ord_less_eq_nat @ i @ ( size_s5283204784079214577_a_nat @ xs ) ).
% assms(2)
thf(fact_212_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_213_bot__empty__eq,axiom,
( bot_bot_list_nat_o
= ( ^ [X3: list_nat] : ( member_list_nat @ X3 @ bot_bot_set_list_nat ) ) ) ).
% bot_empty_eq
thf(fact_214_bot__empty__eq,axiom,
( bot_bo9042073657639083596_nat_o
= ( ^ [X3: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X3 @ bot_bo1033123847703346641_a_nat ) ) ) ).
% bot_empty_eq
thf(fact_215_bot__empty__eq,axiom,
( bot_bo7797463397293707474_nat_o
= ( ^ [X3: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X3 @ bot_bo3438331934148233675_a_nat ) ) ) ).
% bot_empty_eq
thf(fact_216_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : ( member_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_217_map__eq__imp__length__eq,axiom,
! [F: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat,G: sum_sum_a_nat > nat,Ys: list_Sum_sum_a_nat] :
( ( ( map_Su5227373005390213643at_nat @ F @ Xs )
= ( map_Su5227373005390213643at_nat @ G @ Ys ) )
=> ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_218_map__eq__imp__length__eq,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat,G: sum_sum_a_nat > sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( map_Su2790769393171190532_a_nat @ F @ Xs )
= ( map_Su2790769393171190532_a_nat @ G @ Ys ) )
=> ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_219_map__eq__imp__length__eq,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat,G: nat > sum_sum_a_nat,Ys: list_nat] :
( ( ( map_Su2790769393171190532_a_nat @ F @ Xs )
= ( map_na823391071729141993_a_nat @ G @ Ys ) )
=> ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_220_map__eq__imp__length__eq,axiom,
! [F: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_Su5227373005390213643at_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) )
=> ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_221_map__eq__imp__length__eq,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat,G: sum_sum_a_nat > sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( map_na823391071729141993_a_nat @ F @ Xs )
= ( map_Su2790769393171190532_a_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_222_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs: list_nat,G: sum_sum_a_nat > nat,Ys: list_Sum_sum_a_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_Su5227373005390213643at_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_223_map__eq__imp__length__eq,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat,G: nat > sum_sum_a_nat,Ys: list_nat] :
( ( ( map_na823391071729141993_a_nat @ F @ Xs )
= ( map_na823391071729141993_a_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_224_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_225_ivl__disj__un__singleton_I2_J,axiom,
! [U: nat] :
( ( sup_sup_set_nat @ ( set_ord_lessThan_nat @ U ) @ ( insert_nat2 @ U @ bot_bot_set_nat ) )
= ( set_ord_atMost_nat @ U ) ) ).
% ivl_disj_un_singleton(2)
thf(fact_226_ad__agr__list__link,axiom,
! [Ns: list_nat,AD: set_a,Sigma: nat > sum_sum_a_nat,Tau: nat > sum_sum_a_nat] :
( ( ad_agr_sets_a_nat @ ( set_nat2 @ Ns ) @ ( set_nat2 @ Ns ) @ AD @ Sigma @ Tau )
= ( ad_agr_list_a_nat @ AD @ ( map_na823391071729141993_a_nat @ Sigma @ Ns ) @ ( map_na823391071729141993_a_nat @ Tau @ Ns ) ) ) ).
% ad_agr_list_link
thf(fact_227_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_Sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_228_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_nat] :
( ( size_size_list_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_229_neq__if__length__neq,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
!= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_230_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_231_order__refl,axiom,
! [X2: set_nat] : ( ord_less_eq_set_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_232_order__refl,axiom,
! [X2: set_a] : ( ord_less_eq_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_233_order__refl,axiom,
! [X2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_234_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_235_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_236_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_237_dual__order_Orefl,axiom,
! [A: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ A @ A ) ).
% dual_order.refl
thf(fact_238_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_239_sup_Oidem,axiom,
! [A: nat] :
( ( sup_sup_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_240_sup_Oidem,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_241_sup__idem,axiom,
! [X2: nat] :
( ( sup_sup_nat @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_242_sup__idem,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_243_sup_Oleft__idem,axiom,
! [A: nat,B2: nat] :
( ( sup_sup_nat @ A @ ( sup_sup_nat @ A @ B2 ) )
= ( sup_sup_nat @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_244_sup_Oleft__idem,axiom,
! [A: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B2 ) )
= ( sup_sup_set_nat @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_245_sup__left__idem,axiom,
! [X2: nat,Y: nat] :
( ( sup_sup_nat @ X2 @ ( sup_sup_nat @ X2 @ Y ) )
= ( sup_sup_nat @ X2 @ Y ) ) ).
% sup_left_idem
thf(fact_246_sup__left__idem,axiom,
! [X2: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y ) )
= ( sup_sup_set_nat @ X2 @ Y ) ) ).
% sup_left_idem
thf(fact_247_sup_Oright__idem,axiom,
! [A: nat,B2: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ A @ B2 ) @ B2 )
= ( sup_sup_nat @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_248_sup_Oright__idem,axiom,
! [A: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ B2 )
= ( sup_sup_set_nat @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_249_UnCI,axiom,
! [C: list_nat,B: set_list_nat,A2: set_list_nat] :
( ( ~ ( member_list_nat @ C @ B )
=> ( member_list_nat @ C @ A2 ) )
=> ( member_list_nat @ C @ ( sup_sup_set_list_nat @ A2 @ B ) ) ) ).
% UnCI
thf(fact_250_UnCI,axiom,
! [C: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ~ ( member408289922725080238_a_nat @ C @ B )
=> ( member408289922725080238_a_nat @ C @ A2 ) )
=> ( member408289922725080238_a_nat @ C @ ( sup_su4083067149120280889_a_nat @ A2 @ B ) ) ) ).
% UnCI
thf(fact_251_UnCI,axiom,
! [C: sum_sum_a_nat,B: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( ~ ( member_Sum_sum_a_nat @ C @ B )
=> ( member_Sum_sum_a_nat @ C @ A2 ) )
=> ( member_Sum_sum_a_nat @ C @ ( sup_su6804446743777130803_a_nat @ A2 @ B ) ) ) ).
% UnCI
thf(fact_252_UnCI,axiom,
! [C: nat,B: set_nat,A2: set_nat] :
( ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ A2 ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnCI
thf(fact_253_Un__iff,axiom,
! [C: list_nat,A2: set_list_nat,B: set_list_nat] :
( ( member_list_nat @ C @ ( sup_sup_set_list_nat @ A2 @ B ) )
= ( ( member_list_nat @ C @ A2 )
| ( member_list_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_254_Un__iff,axiom,
! [C: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( sup_su4083067149120280889_a_nat @ A2 @ B ) )
= ( ( member408289922725080238_a_nat @ C @ A2 )
| ( member408289922725080238_a_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_255_Un__iff,axiom,
! [C: sum_sum_a_nat,A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ ( sup_su6804446743777130803_a_nat @ A2 @ B ) )
= ( ( member_Sum_sum_a_nat @ C @ A2 )
| ( member_Sum_sum_a_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_256_Un__iff,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) )
= ( ( member_nat @ C @ A2 )
| ( member_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_257_i__le__n,axiom,
ord_less_eq_nat @ i @ n ).
% i_le_n
thf(fact_258_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_259_le__sup__iff,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X2 @ Y ) @ Z )
= ( ( ord_less_eq_set_a @ X2 @ Z )
& ( ord_less_eq_set_a @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_260_le__sup__iff,axiom,
! [X2: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ X2 @ Y ) @ Z )
= ( ( ord_le1325389633284124927_a_nat @ X2 @ Z )
& ( ord_le1325389633284124927_a_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_261_le__sup__iff,axiom,
! [X2: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X2 @ Y ) @ Z )
= ( ( ord_less_eq_set_nat @ X2 @ Z )
& ( ord_less_eq_set_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_262_le__sup__iff,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X2 @ Z )
& ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_263_sup_Obounded__iff,axiom,
! [B2: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A )
= ( ( ord_less_eq_set_a @ B2 @ A )
& ( ord_less_eq_set_a @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_264_sup_Obounded__iff,axiom,
! [B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ B2 @ C ) @ A )
= ( ( ord_le1325389633284124927_a_nat @ B2 @ A )
& ( ord_le1325389633284124927_a_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_265_sup_Obounded__iff,axiom,
! [B2: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A )
= ( ( ord_less_eq_set_nat @ B2 @ A )
& ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_266_sup_Obounded__iff,axiom,
! [B2: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A )
= ( ( ord_less_eq_nat @ B2 @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_267_sup__bot__left,axiom,
! [X2: nat > $o] :
( ( sup_sup_nat_o @ bot_bot_nat_o @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_268_sup__bot__left,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_269_sup__bot__right,axiom,
! [X2: nat > $o] :
( ( sup_sup_nat_o @ X2 @ bot_bot_nat_o )
= X2 ) ).
% sup_bot_right
thf(fact_270_sup__bot__right,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ bot_bot_set_nat )
= X2 ) ).
% sup_bot_right
thf(fact_271_bot__eq__sup__iff,axiom,
! [X2: nat > $o,Y: nat > $o] :
( ( bot_bot_nat_o
= ( sup_sup_nat_o @ X2 @ Y ) )
= ( ( X2 = bot_bot_nat_o )
& ( Y = bot_bot_nat_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_272_bot__eq__sup__iff,axiom,
! [X2: set_nat,Y: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X2 @ Y ) )
= ( ( X2 = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_273_sup__eq__bot__iff,axiom,
! [X2: nat > $o,Y: nat > $o] :
( ( ( sup_sup_nat_o @ X2 @ Y )
= bot_bot_nat_o )
= ( ( X2 = bot_bot_nat_o )
& ( Y = bot_bot_nat_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_274_sup__eq__bot__iff,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ( sup_sup_set_nat @ X2 @ Y )
= bot_bot_set_nat )
= ( ( X2 = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_275_sup__bot_Oeq__neutr__iff,axiom,
! [A: nat > $o,B2: nat > $o] :
( ( ( sup_sup_nat_o @ A @ B2 )
= bot_bot_nat_o )
= ( ( A = bot_bot_nat_o )
& ( B2 = bot_bot_nat_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_276_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_nat,B2: set_nat] :
( ( ( sup_sup_set_nat @ A @ B2 )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_277_sup__bot_Oleft__neutral,axiom,
! [A: nat > $o] :
( ( sup_sup_nat_o @ bot_bot_nat_o @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_278_sup__bot_Oleft__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_279_sup__bot_Oneutr__eq__iff,axiom,
! [A: nat > $o,B2: nat > $o] :
( ( bot_bot_nat_o
= ( sup_sup_nat_o @ A @ B2 ) )
= ( ( A = bot_bot_nat_o )
& ( B2 = bot_bot_nat_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_280_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A @ B2 ) )
= ( ( A = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_281_sup__bot_Oright__neutral,axiom,
! [A: nat > $o] :
( ( sup_sup_nat_o @ A @ bot_bot_nat_o )
= A ) ).
% sup_bot.right_neutral
thf(fact_282_sup__bot_Oright__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_283_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_284_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_285_map__eq__conv,axiom,
! [F: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat,G: sum_sum_a_nat > nat] :
( ( ( map_Su5227373005390213643at_nat @ F @ Xs )
= ( map_Su5227373005390213643at_nat @ G @ Xs ) )
= ( ! [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) ) ) ) ).
% map_eq_conv
thf(fact_286_map__eq__conv,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat,G: sum_sum_a_nat > sum_sum_a_nat] :
( ( ( map_Su2790769393171190532_a_nat @ F @ Xs )
= ( map_Su2790769393171190532_a_nat @ G @ Xs ) )
= ( ! [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) ) ) ) ).
% map_eq_conv
thf(fact_287_map__eq__conv,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat,G: nat > sum_sum_a_nat] :
( ( ( map_na823391071729141993_a_nat @ F @ Xs )
= ( map_na823391071729141993_a_nat @ G @ Xs ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) ) ) ) ).
% map_eq_conv
thf(fact_288_map__eq__conv,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Xs ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) ) ) ) ).
% map_eq_conv
thf(fact_289_lessThan__subset__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X2 ) @ ( set_ord_lessThan_nat @ Y ) )
= ( ord_less_eq_nat @ X2 @ Y ) ) ).
% lessThan_subset_iff
thf(fact_290_Un__empty,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_291_Un__insert__left,axiom,
! [A: nat,B: set_nat,C3: set_nat] :
( ( sup_sup_set_nat @ ( insert_nat2 @ A @ B ) @ C3 )
= ( insert_nat2 @ A @ ( sup_sup_set_nat @ B @ C3 ) ) ) ).
% Un_insert_left
thf(fact_292_Un__insert__right,axiom,
! [A2: set_nat,A: nat,B: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( insert_nat2 @ A @ B ) )
= ( insert_nat2 @ A @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_293_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_294_atMost__subset__iff,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_or4236626031148496127et_nat @ X2 ) @ ( set_or4236626031148496127et_nat @ Y ) )
= ( ord_less_eq_set_nat @ X2 @ Y ) ) ).
% atMost_subset_iff
thf(fact_295_atMost__subset__iff,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_ord_atMost_set_a @ X2 ) @ ( set_ord_atMost_set_a @ Y ) )
= ( ord_less_eq_set_a @ X2 @ Y ) ) ).
% atMost_subset_iff
thf(fact_296_atMost__subset__iff,axiom,
! [X2: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( ord_le7974500612278410847_a_nat @ ( set_or677214730896469414_a_nat @ X2 ) @ ( set_or677214730896469414_a_nat @ Y ) )
= ( ord_le1325389633284124927_a_nat @ X2 @ Y ) ) ).
% atMost_subset_iff
thf(fact_297_atMost__subset__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X2 ) @ ( set_ord_atMost_nat @ Y ) )
= ( ord_less_eq_nat @ X2 @ Y ) ) ).
% atMost_subset_iff
thf(fact_298_atMost__iff,axiom,
! [I: set_nat,K: set_nat] :
( ( member_set_nat @ I @ ( set_or4236626031148496127et_nat @ K ) )
= ( ord_less_eq_set_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_299_atMost__iff,axiom,
! [I: set_a,K: set_a] :
( ( member_set_a @ I @ ( set_ord_atMost_set_a @ K ) )
= ( ord_less_eq_set_a @ I @ K ) ) ).
% atMost_iff
thf(fact_300_atMost__iff,axiom,
! [I: set_Sum_sum_a_nat,K: set_Sum_sum_a_nat] :
( ( member8098812455498974984_a_nat @ I @ ( set_or677214730896469414_a_nat @ K ) )
= ( ord_le1325389633284124927_a_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_301_atMost__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_atMost_nat @ K ) )
= ( ord_less_eq_nat @ I @ K ) ) ).
% atMost_iff
thf(fact_302_Un__Diff__cancel,axiom,
! [A2: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( minus_minus_set_nat @ B @ A2 ) )
= ( sup_sup_set_nat @ A2 @ B ) ) ).
% Un_Diff_cancel
thf(fact_303_Un__Diff__cancel2,axiom,
! [B: set_nat,A2: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B @ A2 ) @ A2 )
= ( sup_sup_set_nat @ B @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_304_length__upt,axiom,
! [I: nat,J: nat] :
( ( size_size_list_nat @ ( upt @ I @ J ) )
= ( minus_minus_nat @ J @ I ) ) ).
% length_upt
thf(fact_305_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_306_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_307_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_308_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_309_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_310_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_311_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_312_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B2 ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_313_UnE,axiom,
! [C: list_nat,A2: set_list_nat,B: set_list_nat] :
( ( member_list_nat @ C @ ( sup_sup_set_list_nat @ A2 @ B ) )
=> ( ~ ( member_list_nat @ C @ A2 )
=> ( member_list_nat @ C @ B ) ) ) ).
% UnE
thf(fact_314_UnE,axiom,
! [C: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ ( sup_su4083067149120280889_a_nat @ A2 @ B ) )
=> ( ~ ( member408289922725080238_a_nat @ C @ A2 )
=> ( member408289922725080238_a_nat @ C @ B ) ) ) ).
% UnE
thf(fact_315_UnE,axiom,
! [C: sum_sum_a_nat,A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ ( sup_su6804446743777130803_a_nat @ A2 @ B ) )
=> ( ~ ( member_Sum_sum_a_nat @ C @ A2 )
=> ( member_Sum_sum_a_nat @ C @ B ) ) ) ).
% UnE
thf(fact_316_UnE,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) )
=> ( ~ ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B ) ) ) ).
% UnE
thf(fact_317_UnI1,axiom,
! [C: list_nat,A2: set_list_nat,B: set_list_nat] :
( ( member_list_nat @ C @ A2 )
=> ( member_list_nat @ C @ ( sup_sup_set_list_nat @ A2 @ B ) ) ) ).
% UnI1
thf(fact_318_UnI1,axiom,
! [C: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ A2 )
=> ( member408289922725080238_a_nat @ C @ ( sup_su4083067149120280889_a_nat @ A2 @ B ) ) ) ).
% UnI1
thf(fact_319_UnI1,axiom,
! [C: sum_sum_a_nat,A2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ A2 )
=> ( member_Sum_sum_a_nat @ C @ ( sup_su6804446743777130803_a_nat @ A2 @ B ) ) ) ).
% UnI1
thf(fact_320_UnI1,axiom,
! [C: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnI1
thf(fact_321_UnI2,axiom,
! [C: list_nat,B: set_list_nat,A2: set_list_nat] :
( ( member_list_nat @ C @ B )
=> ( member_list_nat @ C @ ( sup_sup_set_list_nat @ A2 @ B ) ) ) ).
% UnI2
thf(fact_322_UnI2,axiom,
! [C: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C @ B )
=> ( member408289922725080238_a_nat @ C @ ( sup_su4083067149120280889_a_nat @ A2 @ B ) ) ) ).
% UnI2
thf(fact_323_UnI2,axiom,
! [C: sum_sum_a_nat,B: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ B )
=> ( member_Sum_sum_a_nat @ C @ ( sup_su6804446743777130803_a_nat @ A2 @ B ) ) ) ).
% UnI2
thf(fact_324_UnI2,axiom,
! [C: nat,B: set_nat,A2: set_nat] :
( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnI2
thf(fact_325_bex__Un,axiom,
! [A2: set_nat,B: set_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( sup_sup_set_nat @ A2 @ B ) )
& ( P @ X3 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ X3 ) )
| ? [X3: nat] :
( ( member_nat @ X3 @ B )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_326_ball__Un,axiom,
! [A2: set_nat,B: set_nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( sup_sup_set_nat @ A2 @ B ) )
=> ( P @ X3 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( P @ X3 ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ B )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_327_Un__assoc,axiom,
! [A2: set_nat,B: set_nat,C3: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C3 )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C3 ) ) ) ).
% Un_assoc
thf(fact_328_Un__absorb,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_329_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B4: set_nat] : ( sup_sup_set_nat @ B4 @ A3 ) ) ) ).
% Un_commute
thf(fact_330_Un__left__absorb,axiom,
! [A2: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B ) )
= ( sup_sup_set_nat @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_331_Un__left__commute,axiom,
! [A2: set_nat,B: set_nat,C3: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C3 ) )
= ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A2 @ C3 ) ) ) ).
% Un_left_commute
thf(fact_332_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M2: nat] :
( ( P @ X2 )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M2 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_333_inf__sup__aci_I8_J,axiom,
! [X2: nat,Y: nat] :
( ( sup_sup_nat @ X2 @ ( sup_sup_nat @ X2 @ Y ) )
= ( sup_sup_nat @ X2 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_334_inf__sup__aci_I8_J,axiom,
! [X2: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y ) )
= ( sup_sup_set_nat @ X2 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_335_inf__sup__aci_I7_J,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( sup_sup_nat @ X2 @ ( sup_sup_nat @ Y @ Z ) )
= ( sup_sup_nat @ Y @ ( sup_sup_nat @ X2 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_336_inf__sup__aci_I7_J,axiom,
! [X2: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ Y @ Z ) )
= ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X2 @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_337_inf__sup__aci_I6_J,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ X2 @ Y ) @ Z )
= ( sup_sup_nat @ X2 @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_338_inf__sup__aci_I6_J,axiom,
! [X2: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X2 @ Y ) @ Z )
= ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_339_inf__sup__aci_I5_J,axiom,
( sup_sup_nat
= ( ^ [X3: nat,Y3: nat] : ( sup_sup_nat @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_340_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat
= ( ^ [X3: set_nat,Y3: set_nat] : ( sup_sup_set_nat @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_341_inf__sup__ord_I4_J,axiom,
! [Y: set_a,X2: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_342_inf__sup__ord_I4_J,axiom,
! [Y: set_Sum_sum_a_nat,X2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ Y @ ( sup_su6804446743777130803_a_nat @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_343_inf__sup__ord_I4_J,axiom,
! [Y: set_nat,X2: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_344_inf__sup__ord_I4_J,axiom,
! [Y: nat,X2: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_345_inf__sup__ord_I3_J,axiom,
! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_346_inf__sup__ord_I3_J,axiom,
! [X2: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ X2 @ ( sup_su6804446743777130803_a_nat @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_347_inf__sup__ord_I3_J,axiom,
! [X2: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_348_inf__sup__ord_I3_J,axiom,
! [X2: nat,Y: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_349_nle__le,axiom,
! [A: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_350_le__cases3,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_351_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat,Z2: set_nat] : ( Y5 = Z2 ) )
= ( ^ [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
& ( ord_less_eq_set_nat @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_352_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
& ( ord_less_eq_set_a @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_353_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_Sum_sum_a_nat,Z2: set_Sum_sum_a_nat] : ( Y5 = Z2 ) )
= ( ^ [X3: set_Sum_sum_a_nat,Y3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X3 @ Y3 )
& ( ord_le1325389633284124927_a_nat @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_354_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_355_ord__eq__le__trans,axiom,
! [A: set_nat,B2: set_nat,C: set_nat] :
( ( A = B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_356_ord__eq__le__trans,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( A = B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_357_ord__eq__le__trans,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( A = B2 )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ C )
=> ( ord_le1325389633284124927_a_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_358_ord__eq__le__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( A = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_359_ord__le__eq__trans,axiom,
! [A: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_360_ord__le__eq__trans,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_361_ord__le__eq__trans,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_le1325389633284124927_a_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_362_ord__le__eq__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_363_order__antisym,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_364_order__antisym,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_365_order__antisym,axiom,
! [X2: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X2 @ Y )
=> ( ( ord_le1325389633284124927_a_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_366_order__antisym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_367_order_Otrans,axiom,
! [A: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_368_order_Otrans,axiom,
! [A: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_369_order_Otrans,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B2 )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ C )
=> ( ord_le1325389633284124927_a_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_370_order_Otrans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_371_order__trans,axiom,
! [X2: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( ord_less_eq_set_nat @ Y @ Z )
=> ( ord_less_eq_set_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_372_order__trans,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z )
=> ( ord_less_eq_set_a @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_373_order__trans,axiom,
! [X2: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X2 @ Y )
=> ( ( ord_le1325389633284124927_a_nat @ Y @ Z )
=> ( ord_le1325389633284124927_a_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_374_order__trans,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_375_le__supE,axiom,
! [A: set_a,B2: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B2 ) @ X2 )
=> ~ ( ( ord_less_eq_set_a @ A @ X2 )
=> ~ ( ord_less_eq_set_a @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_376_le__supE,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,X2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) @ X2 )
=> ~ ( ( ord_le1325389633284124927_a_nat @ A @ X2 )
=> ~ ( ord_le1325389633284124927_a_nat @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_377_le__supE,axiom,
! [A: set_nat,B2: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ X2 )
=> ~ ( ( ord_less_eq_set_nat @ A @ X2 )
=> ~ ( ord_less_eq_set_nat @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_378_le__supE,axiom,
! [A: nat,B2: nat,X2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B2 ) @ X2 )
=> ~ ( ( ord_less_eq_nat @ A @ X2 )
=> ~ ( ord_less_eq_nat @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_379_le__supI,axiom,
! [A: set_a,X2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ X2 )
=> ( ( ord_less_eq_set_a @ B2 @ X2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_380_le__supI,axiom,
! [A: set_Sum_sum_a_nat,X2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ X2 )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ X2 )
=> ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_381_le__supI,axiom,
! [A: set_nat,X2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ X2 )
=> ( ( ord_less_eq_set_nat @ B2 @ X2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_382_le__supI,axiom,
! [A: nat,X2: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ X2 )
=> ( ( ord_less_eq_nat @ B2 @ X2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_383_sup__ge1,axiom,
! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ X2 @ Y ) ) ).
% sup_ge1
thf(fact_384_sup__ge1,axiom,
! [X2: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ X2 @ ( sup_su6804446743777130803_a_nat @ X2 @ Y ) ) ).
% sup_ge1
thf(fact_385_sup__ge1,axiom,
! [X2: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y ) ) ).
% sup_ge1
thf(fact_386_sup__ge1,axiom,
! [X2: nat,Y: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y ) ) ).
% sup_ge1
thf(fact_387_sup__ge2,axiom,
! [Y: set_a,X2: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X2 @ Y ) ) ).
% sup_ge2
thf(fact_388_sup__ge2,axiom,
! [Y: set_Sum_sum_a_nat,X2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ Y @ ( sup_su6804446743777130803_a_nat @ X2 @ Y ) ) ).
% sup_ge2
thf(fact_389_sup__ge2,axiom,
! [Y: set_nat,X2: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X2 @ Y ) ) ).
% sup_ge2
thf(fact_390_sup__ge2,axiom,
! [Y: nat,X2: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X2 @ Y ) ) ).
% sup_ge2
thf(fact_391_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B2: nat] :
( ! [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ A4 @ B5 )
=> ( P @ A4 @ B5 ) )
=> ( ! [A4: nat,B5: nat] :
( ( P @ B5 @ A4 )
=> ( P @ A4 @ B5 ) )
=> ( P @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_392_le__supI1,axiom,
! [X2: set_a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X2 @ A )
=> ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_393_le__supI1,axiom,
! [X2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X2 @ A )
=> ( ord_le1325389633284124927_a_nat @ X2 @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_394_le__supI1,axiom,
! [X2: set_nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ A )
=> ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_395_le__supI1,axiom,
! [X2: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ X2 @ A )
=> ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_396_le__supI2,axiom,
! [X2: set_a,B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ X2 @ B2 )
=> ( ord_less_eq_set_a @ X2 @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_397_le__supI2,axiom,
! [X2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X2 @ B2 )
=> ( ord_le1325389633284124927_a_nat @ X2 @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_398_le__supI2,axiom,
! [X2: set_nat,B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ B2 )
=> ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_399_le__supI2,axiom,
! [X2: nat,B2: nat,A: nat] :
( ( ord_less_eq_nat @ X2 @ B2 )
=> ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_400_sup_Omono,axiom,
! [C: set_a,A: set_a,D: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C @ A )
=> ( ( ord_less_eq_set_a @ D @ B2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D ) @ ( sup_sup_set_a @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_401_sup_Omono,axiom,
! [C: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat,D: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C @ A )
=> ( ( ord_le1325389633284124927_a_nat @ D @ B2 )
=> ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ C @ D ) @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_402_sup_Omono,axiom,
! [C: set_nat,A: set_nat,D: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C @ A )
=> ( ( ord_less_eq_set_nat @ D @ B2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C @ D ) @ ( sup_sup_set_nat @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_403_sup_Omono,axiom,
! [C: nat,A: nat,D: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ( ord_less_eq_nat @ D @ B2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_404_sup__mono,axiom,
! [A: set_a,C: set_a,B2: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B2 @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B2 ) @ ( sup_sup_set_a @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_405_sup__mono,axiom,
! [A: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,D: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ C )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ D )
=> ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ A @ B2 ) @ ( sup_su6804446743777130803_a_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_406_sup__mono,axiom,
! [A: set_nat,C: set_nat,B2: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ D )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ ( sup_sup_set_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_407_sup__mono,axiom,
! [A: nat,C: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B2 ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_408_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_nat,Z2: set_nat] : ( Y5 = Z2 ) )
= ( ^ [A5: set_nat,B6: set_nat] :
( ( ord_less_eq_set_nat @ B6 @ A5 )
& ( ord_less_eq_set_nat @ A5 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_409_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [A5: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ A5 )
& ( ord_less_eq_set_a @ A5 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_410_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_Sum_sum_a_nat,Z2: set_Sum_sum_a_nat] : ( Y5 = Z2 ) )
= ( ^ [A5: set_Sum_sum_a_nat,B6: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B6 @ A5 )
& ( ord_le1325389633284124927_a_nat @ A5 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_411_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A5: nat,B6: nat] :
( ( ord_less_eq_nat @ B6 @ A5 )
& ( ord_less_eq_nat @ A5 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_412_sup_Oassoc,axiom,
! [A: nat,B2: nat,C: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ A @ B2 ) @ C )
= ( sup_sup_nat @ A @ ( sup_sup_nat @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_413_sup_Oassoc,axiom,
! [A: set_nat,B2: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ C )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_414_sup__assoc,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ X2 @ Y ) @ Z )
= ( sup_sup_nat @ X2 @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_415_sup__assoc,axiom,
! [X2: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X2 @ Y ) @ Z )
= ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_416_sup__least,axiom,
! [Y: set_a,X2: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ Y @ X2 )
=> ( ( ord_less_eq_set_a @ Z @ X2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X2 ) ) ) ).
% sup_least
thf(fact_417_sup__least,axiom,
! [Y: set_Sum_sum_a_nat,X2: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ Y @ X2 )
=> ( ( ord_le1325389633284124927_a_nat @ Z @ X2 )
=> ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ Y @ Z ) @ X2 ) ) ) ).
% sup_least
thf(fact_418_sup__least,axiom,
! [Y: set_nat,X2: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X2 )
=> ( ( ord_less_eq_set_nat @ Z @ X2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y @ Z ) @ X2 ) ) ) ).
% sup_least
thf(fact_419_sup__least,axiom,
! [Y: nat,X2: nat,Z: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ Z @ X2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X2 ) ) ) ).
% sup_least
thf(fact_420_dual__order_Oantisym,axiom,
! [B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_421_dual__order_Oantisym,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_422_dual__order_Oantisym,axiom,
! [B2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B2 @ A )
=> ( ( ord_le1325389633284124927_a_nat @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_423_dual__order_Oantisym,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_424_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y3: set_a] :
( ( sup_sup_set_a @ X3 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_425_le__iff__sup,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [X3: set_Sum_sum_a_nat,Y3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X3 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_426_le__iff__sup,axiom,
( ord_less_eq_set_nat
= ( ^ [X3: set_nat,Y3: set_nat] :
( ( sup_sup_set_nat @ X3 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_427_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y3: nat] :
( ( sup_sup_nat @ X3 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_428_sup_OorderE,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( A
= ( sup_sup_set_a @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_429_sup_OorderE,axiom,
! [B2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B2 @ A )
=> ( A
= ( sup_su6804446743777130803_a_nat @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_430_sup_OorderE,axiom,
! [B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( A
= ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_431_sup_OorderE,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( A
= ( sup_sup_nat @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_432_sup_OorderI,axiom,
! [A: set_a,B2: set_a] :
( ( A
= ( sup_sup_set_a @ A @ B2 ) )
=> ( ord_less_eq_set_a @ B2 @ A ) ) ).
% sup.orderI
thf(fact_433_sup_OorderI,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( A
= ( sup_su6804446743777130803_a_nat @ A @ B2 ) )
=> ( ord_le1325389633284124927_a_nat @ B2 @ A ) ) ).
% sup.orderI
thf(fact_434_sup_OorderI,axiom,
! [A: set_nat,B2: set_nat] :
( ( A
= ( sup_sup_set_nat @ A @ B2 ) )
=> ( ord_less_eq_set_nat @ B2 @ A ) ) ).
% sup.orderI
thf(fact_435_sup_OorderI,axiom,
! [A: nat,B2: nat] :
( ( A
= ( sup_sup_nat @ A @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A ) ) ).
% sup.orderI
thf(fact_436_sup__unique,axiom,
! [F: set_a > set_a > set_a,X2: set_a,Y: set_a] :
( ! [X4: set_a,Y2: set_a] : ( ord_less_eq_set_a @ X4 @ ( F @ X4 @ Y2 ) )
=> ( ! [X4: set_a,Y2: set_a] : ( ord_less_eq_set_a @ Y2 @ ( F @ X4 @ Y2 ) )
=> ( ! [X4: set_a,Y2: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X4 )
=> ( ( ord_less_eq_set_a @ Z3 @ X4 )
=> ( ord_less_eq_set_a @ ( F @ Y2 @ Z3 ) @ X4 ) ) )
=> ( ( sup_sup_set_a @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_437_sup__unique,axiom,
! [F: set_Sum_sum_a_nat > set_Sum_sum_a_nat > set_Sum_sum_a_nat,X2: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ! [X4: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ X4 @ ( F @ X4 @ Y2 ) )
=> ( ! [X4: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ Y2 @ ( F @ X4 @ Y2 ) )
=> ( ! [X4: set_Sum_sum_a_nat,Y2: set_Sum_sum_a_nat,Z3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ Y2 @ X4 )
=> ( ( ord_le1325389633284124927_a_nat @ Z3 @ X4 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ Y2 @ Z3 ) @ X4 ) ) )
=> ( ( sup_su6804446743777130803_a_nat @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_438_sup__unique,axiom,
! [F: set_nat > set_nat > set_nat,X2: set_nat,Y: set_nat] :
( ! [X4: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ X4 @ ( F @ X4 @ Y2 ) )
=> ( ! [X4: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ Y2 @ ( F @ X4 @ Y2 ) )
=> ( ! [X4: set_nat,Y2: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X4 )
=> ( ( ord_less_eq_set_nat @ Z3 @ X4 )
=> ( ord_less_eq_set_nat @ ( F @ Y2 @ Z3 ) @ X4 ) ) )
=> ( ( sup_sup_set_nat @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_439_sup__unique,axiom,
! [F: nat > nat > nat,X2: nat,Y: nat] :
( ! [X4: nat,Y2: nat] : ( ord_less_eq_nat @ X4 @ ( F @ X4 @ Y2 ) )
=> ( ! [X4: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ ( F @ X4 @ Y2 ) )
=> ( ! [X4: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y2 @ X4 )
=> ( ( ord_less_eq_nat @ Z3 @ X4 )
=> ( ord_less_eq_nat @ ( F @ Y2 @ Z3 ) @ X4 ) ) )
=> ( ( sup_sup_nat @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_440_dual__order_Otrans,axiom,
! [B2: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( ord_less_eq_set_nat @ C @ B2 )
=> ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_441_dual__order_Otrans,axiom,
! [B2: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_442_dual__order_Otrans,axiom,
! [B2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B2 @ A )
=> ( ( ord_le1325389633284124927_a_nat @ C @ B2 )
=> ( ord_le1325389633284124927_a_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_443_dual__order_Otrans,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_444_sup_Oabsorb1,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( sup_sup_set_a @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_445_sup_Oabsorb1,axiom,
! [B2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B2 @ A )
=> ( ( sup_su6804446743777130803_a_nat @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_446_sup_Oabsorb1,axiom,
! [B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( sup_sup_set_nat @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_447_sup_Oabsorb1,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( sup_sup_nat @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_448_sup_Oabsorb2,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( sup_sup_set_a @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_449_sup_Oabsorb2,axiom,
! [A: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B2 )
=> ( ( sup_su6804446743777130803_a_nat @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_450_sup_Oabsorb2,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( sup_sup_set_nat @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_451_sup_Oabsorb2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( sup_sup_nat @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_452_sup_Ocommute,axiom,
( sup_sup_nat
= ( ^ [A5: nat,B6: nat] : ( sup_sup_nat @ B6 @ A5 ) ) ) ).
% sup.commute
thf(fact_453_sup_Ocommute,axiom,
( sup_sup_set_nat
= ( ^ [A5: set_nat,B6: set_nat] : ( sup_sup_set_nat @ B6 @ A5 ) ) ) ).
% sup.commute
thf(fact_454_sup__absorb1,axiom,
! [Y: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X2 )
=> ( ( sup_sup_set_nat @ X2 @ Y )
= X2 ) ) ).
% sup_absorb1
thf(fact_455_sup__absorb1,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( sup_sup_nat @ X2 @ Y )
= X2 ) ) ).
% sup_absorb1
thf(fact_456_sup__absorb2,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
=> ( ( sup_sup_set_nat @ X2 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_457_sup__absorb2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( sup_sup_nat @ X2 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_458_sup__commute,axiom,
( sup_sup_set_nat
= ( ^ [X3: set_nat,Y3: set_nat] : ( sup_sup_set_nat @ Y3 @ X3 ) ) ) ).
% sup_commute
thf(fact_459_sup_OboundedE,axiom,
! [B2: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A )
=> ~ ( ( ord_less_eq_set_nat @ B2 @ A )
=> ~ ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_460_sup_OboundedE,axiom,
! [B2: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A )
=> ~ ( ( ord_less_eq_nat @ B2 @ A )
=> ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_461_sup_OboundedI,axiom,
! [B2: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( ord_less_eq_set_nat @ C @ A )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_462_sup_OboundedI,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_463_sup_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [B6: set_nat,A5: set_nat] :
( A5
= ( sup_sup_set_nat @ A5 @ B6 ) ) ) ) ).
% sup.order_iff
thf(fact_464_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B6: nat,A5: nat] :
( A5
= ( sup_sup_nat @ A5 @ B6 ) ) ) ) ).
% sup.order_iff
thf(fact_465_sup_Ocobounded1,axiom,
! [A: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_466_sup_Ocobounded1,axiom,
! [A: nat,B2: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_467_sup_Ocobounded2,axiom,
! [B2: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B2 @ ( sup_sup_set_nat @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_468_sup_Ocobounded2,axiom,
! [B2: nat,A: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_469_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [B6: set_nat,A5: set_nat] :
( ( sup_sup_set_nat @ A5 @ B6 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_470_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B6: nat,A5: nat] :
( ( sup_sup_nat @ A5 @ B6 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_471_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B6: set_nat] :
( ( sup_sup_set_nat @ A5 @ B6 )
= B6 ) ) ) ).
% sup.absorb_iff2
thf(fact_472_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B6: nat] :
( ( sup_sup_nat @ A5 @ B6 )
= B6 ) ) ) ).
% sup.absorb_iff2
thf(fact_473_sup_OcoboundedI1,axiom,
! [C: set_nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C @ A )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_474_sup_OcoboundedI1,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_475_sup_OcoboundedI2,axiom,
! [C: set_nat,B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ C @ B2 )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_476_sup_OcoboundedI2,axiom,
! [C: nat,B2: nat,A: nat] :
( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_477_sup_Oleft__commute,axiom,
! [B2: set_nat,A: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A @ C ) )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_478_sup__left__commute,axiom,
! [X2: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( sup_sup_set_nat @ Y @ Z ) )
= ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X2 @ Z ) ) ) ).
% sup_left_commute
thf(fact_479_antisym,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_480_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A5: nat,B6: nat] :
( ( ord_less_eq_nat @ A5 @ B6 )
& ( ord_less_eq_nat @ B6 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_481_order__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_482_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_483_order__eq__refl,axiom,
! [X2: nat,Y: nat] :
( ( X2 = Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_484_linorder__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_485_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_486_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_487_linorder__le__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_488_order__antisym__conv,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_489_ex__map__conv,axiom,
! [Ys: list_Sum_sum_a_nat,F: nat > sum_sum_a_nat] :
( ( ? [Xs3: list_nat] :
( Ys
= ( map_na823391071729141993_a_nat @ F @ Xs3 ) ) )
= ( ! [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ ( set_Sum_sum_a_nat2 @ Ys ) )
=> ? [Y3: nat] :
( X3
= ( F @ Y3 ) ) ) ) ) ).
% ex_map_conv
thf(fact_490_ex__map__conv,axiom,
! [Ys: list_nat,F: nat > nat] :
( ( ? [Xs3: list_nat] :
( Ys
= ( map_nat_nat @ F @ Xs3 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
=> ? [Y3: nat] :
( X3
= ( F @ Y3 ) ) ) ) ) ).
% ex_map_conv
thf(fact_491_map__cong,axiom,
! [Xs: list_nat,Ys: list_nat,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ( Xs = Ys )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Ys ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_na823391071729141993_a_nat @ F @ Xs )
= ( map_na823391071729141993_a_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_492_map__cong,axiom,
! [Xs: list_nat,Ys: list_nat,F: nat > nat,G: nat > nat] :
( ( Xs = Ys )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Ys ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_493_map__idI,axiom,
! [Xs: list_nat,F: nat > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X4 )
= X4 ) )
=> ( ( map_nat_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_494_map__ext,axiom,
! [Xs: list_nat,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_na823391071729141993_a_nat @ F @ Xs )
= ( map_na823391071729141993_a_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_495_map__ext,axiom,
! [Xs: list_nat,F: nat > nat,G: nat > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_496_list_Omap__ident__strong,axiom,
! [T: list_nat,F: nat > nat] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ T ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_nat_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_497_list_Oinj__map__strong,axiom,
! [X2: list_nat,Xa: list_nat,F: nat > sum_sum_a_nat,Fa: nat > sum_sum_a_nat] :
( ! [Z3: nat,Za: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ X2 ) )
=> ( ( member_nat @ Za @ ( set_nat2 @ Xa ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( map_na823391071729141993_a_nat @ F @ X2 )
= ( map_na823391071729141993_a_nat @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_498_list_Oinj__map__strong,axiom,
! [X2: list_nat,Xa: list_nat,F: nat > nat,Fa: nat > nat] :
( ! [Z3: nat,Za: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ X2 ) )
=> ( ( member_nat @ Za @ ( set_nat2 @ Xa ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( map_nat_nat @ F @ X2 )
= ( map_nat_nat @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_499_list_Omap__cong0,axiom,
! [X2: list_nat,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ X2 ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_na823391071729141993_a_nat @ F @ X2 )
= ( map_na823391071729141993_a_nat @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_500_list_Omap__cong0,axiom,
! [X2: list_nat,F: nat > nat,G: nat > nat] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ X2 ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_nat_nat @ F @ X2 )
= ( map_nat_nat @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_501_list_Omap__cong,axiom,
! [X2: list_nat,Ya: list_nat,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ( X2 = Ya )
=> ( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ Ya ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_na823391071729141993_a_nat @ F @ X2 )
= ( map_na823391071729141993_a_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_502_list_Omap__cong,axiom,
! [X2: list_nat,Ya: list_nat,F: nat > nat,G: nat > nat] :
( ( X2 = Ya )
=> ( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ Ya ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_nat_nat @ F @ X2 )
= ( map_nat_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_503_Un__empty__left,axiom,
! [B: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_504_Un__empty__right,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Un_empty_right
thf(fact_505_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_506_Un__Diff,axiom,
! [A2: set_nat,B: set_nat,C3: set_nat] :
( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C3 )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A2 @ C3 ) @ ( minus_minus_set_nat @ B @ C3 ) ) ) ).
% Un_Diff
thf(fact_507_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_508_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_509_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_510_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_511_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_512_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_513_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_514_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_515_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_516_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_517_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_518_le__diff__iff_H,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_519_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_520_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_521_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_522_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_523_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_524_insert__is__Un,axiom,
( insert_nat2
= ( ^ [A5: nat] : ( sup_sup_set_nat @ ( insert_nat2 @ A5 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_525_Un__singleton__iff,axiom,
! [A2: set_nat,B: set_nat,X2: nat] :
( ( ( sup_sup_set_nat @ A2 @ B )
= ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
= ( ( ( A2 = bot_bot_set_nat )
& ( B
= ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) )
| ( ( A2
= ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
& ( B = bot_bot_set_nat ) )
| ( ( A2
= ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
& ( B
= ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_526_singleton__Un__iff,axiom,
! [X2: nat,A2: set_nat,B: set_nat] :
( ( ( insert_nat2 @ X2 @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A2 @ B ) )
= ( ( ( A2 = bot_bot_set_nat )
& ( B
= ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) )
| ( ( A2
= ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
& ( B = bot_bot_set_nat ) )
| ( ( A2
= ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
& ( B
= ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_527_atLeast__upt,axiom,
( set_ord_lessThan_nat
= ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N2 ) ) ) ) ).
% atLeast_upt
thf(fact_528_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_529_diff__shunt__var,axiom,
! [X2: set_nat,Y: set_nat] :
( ( ( minus_minus_set_nat @ X2 @ Y )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X2 @ Y ) ) ).
% diff_shunt_var
thf(fact_530_nats__def,axiom,
( nats
= ( ^ [Ns2: list_nat] :
( Ns2
= ( upt @ zero_zero_nat @ ( size_size_list_nat @ Ns2 ) ) ) ) ) ).
% nats_def
thf(fact_531_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C4: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_532_set__removeAll,axiom,
! [X2: nat,Xs: list_nat] :
( ( set_nat2 @ ( removeAll_nat @ X2 @ Xs ) )
= ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ).
% set_removeAll
thf(fact_533_boolean__algebra_Odisj__zero__right,axiom,
! [X2: set_nat] :
( ( sup_sup_set_nat @ X2 @ bot_bot_set_nat )
= X2 ) ).
% boolean_algebra.disj_zero_right
thf(fact_534_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_535_set__n_I2_J,axiom,
( ( set_nat2 @ ( append_nat @ ( upt @ zero_zero_nat @ i ) @ ( cons_nat @ n @ ( upt @ i @ n ) ) ) )
= ( set_ord_atMost_nat @ n ) ) ).
% set_n(2)
thf(fact_536_length__code,axiom,
( size_s5283204784079214577_a_nat
= ( gen_le1340941697924381074_a_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_537_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_538_list_Oinject,axiom,
! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
( ( ( cons_nat @ X21 @ X22 )
= ( cons_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_539_subsetI,axiom,
! [A2: set_nat,B: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_nat @ X4 @ B ) )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_540_same__append__eq,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_541_append__same__eq,axiom,
! [Ys: list_nat,Xs: list_nat,Zs: list_nat] :
( ( ( append_nat @ Ys @ Xs )
= ( append_nat @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_542_append__assoc,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( append_nat @ ( append_nat @ Xs @ Ys ) @ Zs )
= ( append_nat @ Xs @ ( append_nat @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_543_append_Oassoc,axiom,
! [A: list_nat,B2: list_nat,C: list_nat] :
( ( append_nat @ ( append_nat @ A @ B2 ) @ C )
= ( append_nat @ A @ ( append_nat @ B2 @ C ) ) ) ).
% append.assoc
thf(fact_544_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_545_subset__empty,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_546_insert__subset,axiom,
! [X2: nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat2 @ X2 @ A2 ) @ B )
= ( ( member_nat @ X2 @ B )
& ( ord_less_eq_set_nat @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_547_Un__subset__iff,axiom,
! [A2: set_nat,B: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C3 )
= ( ( ord_less_eq_set_nat @ A2 @ C3 )
& ( ord_less_eq_set_nat @ B @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_548_append__eq__append__conv,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Us: list_Sum_sum_a_nat,Vs: list_Sum_sum_a_nat] :
( ( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
| ( ( size_s5283204784079214577_a_nat @ Us )
= ( size_s5283204784079214577_a_nat @ Vs ) ) )
=> ( ( ( append_Sum_sum_a_nat @ Xs @ Us )
= ( append_Sum_sum_a_nat @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_549_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_550_map__append,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat,Ys: list_nat] :
( ( map_na823391071729141993_a_nat @ F @ ( append_nat @ Xs @ Ys ) )
= ( append_Sum_sum_a_nat @ ( map_na823391071729141993_a_nat @ F @ Xs ) @ ( map_na823391071729141993_a_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_551_map__append,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
( ( map_nat_nat @ F @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( map_nat_nat @ F @ Xs ) @ ( map_nat_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_552_removeAll__id,axiom,
! [X2: nat,Xs: list_nat] :
( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( removeAll_nat @ X2 @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_553_removeAll__append,axiom,
! [X2: nat,Xs: list_nat,Ys: list_nat] :
( ( removeAll_nat @ X2 @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( removeAll_nat @ X2 @ Xs ) @ ( removeAll_nat @ X2 @ Ys ) ) ) ).
% removeAll_append
thf(fact_554_list_Osimps_I15_J,axiom,
! [X21: nat,X22: list_nat] :
( ( set_nat2 @ ( cons_nat @ X21 @ X22 ) )
= ( insert_nat2 @ X21 @ ( set_nat2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_555_singleton__insert__inj__eq_H,axiom,
! [A: nat,A2: set_nat,B2: nat] :
( ( ( insert_nat2 @ A @ A2 )
= ( insert_nat2 @ B2 @ bot_bot_set_nat ) )
= ( ( A = B2 )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ B2 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_556_singleton__insert__inj__eq,axiom,
! [B2: nat,A: nat,A2: set_nat] :
( ( ( insert_nat2 @ B2 @ bot_bot_set_nat )
= ( insert_nat2 @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ B2 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_557_Diff__eq__empty__iff,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( minus_minus_set_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_558_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_559_in__mono,axiom,
! [A2: set_nat,B: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_nat @ X2 @ A2 )
=> ( member_nat @ X2 @ B ) ) ) ).
% in_mono
thf(fact_560_subsetD,axiom,
! [A2: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_561_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( member_nat @ X3 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_562_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A3 )
=> ( member_nat @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_563_split__list__first__prop__iff,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys2: list_nat,X3: nat] :
( ? [Zs2: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Y3: nat] :
( ( member_nat @ Y3 @ ( set_nat2 @ Ys2 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_564_split__list__last__prop__iff,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
& ( P @ X3 ) ) )
= ( ? [Ys2: list_nat,X3: nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs2 ) ) )
& ( P @ X3 )
& ! [Y3: nat] :
( ( member_nat @ Y3 @ ( set_nat2 @ Zs2 ) )
=> ~ ( P @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_565_in__set__conv__decomp__first,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [Ys2: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs2 ) ) )
& ~ ( member_nat @ X2 @ ( set_nat2 @ Ys2 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_566_in__set__conv__decomp__last,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [Ys2: list_nat,Zs2: list_nat] :
( ( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs2 ) ) )
& ~ ( member_nat @ X2 @ ( set_nat2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_567_split__list__first__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys3: list_nat,X4: nat] :
( ? [Zs3: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X4 @ Zs3 ) ) )
=> ( ( P @ X4 )
=> ~ ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( set_nat2 @ Ys3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_568_append__eq__append__conv2,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ts: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Zs @ Ts ) )
= ( ? [Us2: list_nat] :
( ( ( Xs
= ( append_nat @ Zs @ Us2 ) )
& ( ( append_nat @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_nat @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_nat @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_569_split__list__last__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys3: list_nat,X4: nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X4 @ Zs3 ) ) )
=> ( ( P @ X4 )
=> ~ ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( set_nat2 @ Zs3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_570_split__list__first__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys3: list_nat,X4: nat] :
( ? [Zs3: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X4 @ Zs3 ) ) )
& ( P @ X4 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( set_nat2 @ Ys3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_571_split__list__last__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys3: list_nat,X4: nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X4 @ Zs3 ) ) )
& ( P @ X4 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ ( set_nat2 @ Zs3 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_572_in__set__conv__decomp,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [Ys2: list_nat,Zs2: list_nat] :
( Xs
= ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_573_append__Cons__eq__iff,axiom,
! [X2: nat,Xs: list_nat,Ys: list_nat,Xs4: list_nat,Ys4: list_nat] :
( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ~ ( member_nat @ X2 @ ( set_nat2 @ Ys ) )
=> ( ( ( append_nat @ Xs @ ( cons_nat @ X2 @ Ys ) )
= ( append_nat @ Xs4 @ ( cons_nat @ X2 @ Ys4 ) ) )
= ( ( Xs = Xs4 )
& ( Ys = Ys4 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_574_append__eq__appendI,axiom,
! [Xs: list_nat,Xs1: list_nat,Zs: list_nat,Ys: list_nat,Us: list_nat] :
( ( ( append_nat @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_nat @ Xs1 @ Us ) )
=> ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_575_split__list__propE,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ~ ! [Ys3: list_nat,X4: nat] :
( ? [Zs3: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X4 @ Zs3 ) ) )
=> ~ ( P @ X4 ) ) ) ).
% split_list_propE
thf(fact_576_split__list__first,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ? [Ys3: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X2 @ Zs3 ) ) )
& ~ ( member_nat @ X2 @ ( set_nat2 @ Ys3 ) ) ) ) ).
% split_list_first
thf(fact_577_split__list__prop,axiom,
! [Xs: list_nat,P: nat > $o] :
( ? [X5: nat] :
( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
& ( P @ X5 ) )
=> ? [Ys3: list_nat,X4: nat] :
( ? [Zs3: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X4 @ Zs3 ) ) )
& ( P @ X4 ) ) ) ).
% split_list_prop
thf(fact_578_split__list__last,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ? [Ys3: list_nat,Zs3: list_nat] :
( ( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X2 @ Zs3 ) ) )
& ~ ( member_nat @ X2 @ ( set_nat2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_579_set__subset__Cons,axiom,
! [Xs: list_nat,X2: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_580_Cons__eq__appendI,axiom,
! [X2: nat,Xs1: list_nat,Ys: list_nat,Xs: list_nat,Zs: list_nat] :
( ( ( cons_nat @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_nat @ Xs1 @ Zs ) )
=> ( ( cons_nat @ X2 @ Xs )
= ( append_nat @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_581_not__Cons__self2,axiom,
! [X2: nat,Xs: list_nat] :
( ( cons_nat @ X2 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_582_split__list,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ? [Ys3: list_nat,Zs3: list_nat] :
( Xs
= ( append_nat @ Ys3 @ ( cons_nat @ X2 @ Zs3 ) ) ) ) ).
% split_list
thf(fact_583_append__Cons,axiom,
! [X2: nat,Xs: list_nat,Ys: list_nat] :
( ( append_nat @ ( cons_nat @ X2 @ Xs ) @ Ys )
= ( cons_nat @ X2 @ ( append_nat @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_584_removeAll_Osimps_I2_J,axiom,
! [X2: nat,Y: nat,Xs: list_nat] :
( ( ( X2 = Y )
=> ( ( removeAll_nat @ X2 @ ( cons_nat @ Y @ Xs ) )
= ( removeAll_nat @ X2 @ Xs ) ) )
& ( ( X2 != Y )
=> ( ( removeAll_nat @ X2 @ ( cons_nat @ Y @ Xs ) )
= ( cons_nat @ Y @ ( removeAll_nat @ X2 @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_585_append__eq__map__conv,axiom,
! [Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat,F: nat > sum_sum_a_nat,Xs: list_nat] :
( ( ( append_Sum_sum_a_nat @ Ys @ Zs )
= ( map_na823391071729141993_a_nat @ F @ Xs ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys
= ( map_na823391071729141993_a_nat @ F @ Us2 ) )
& ( Zs
= ( map_na823391071729141993_a_nat @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_586_append__eq__map__conv,axiom,
! [Ys: list_nat,Zs: list_nat,F: nat > nat,Xs: list_nat] :
( ( ( append_nat @ Ys @ Zs )
= ( map_nat_nat @ F @ Xs ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys
= ( map_nat_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_nat @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_587_map__eq__append__conv,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat,Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
( ( ( map_na823391071729141993_a_nat @ F @ Xs )
= ( append_Sum_sum_a_nat @ Ys @ Zs ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys
= ( map_na823391071729141993_a_nat @ F @ Us2 ) )
& ( Zs
= ( map_na823391071729141993_a_nat @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_588_map__eq__append__conv,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( append_nat @ Ys @ Zs ) )
= ( ? [Us2: list_nat,Vs2: list_nat] :
( ( Xs
= ( append_nat @ Us2 @ Vs2 ) )
& ( Ys
= ( map_nat_nat @ F @ Us2 ) )
& ( Zs
= ( map_nat_nat @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_589_list_Oset__intros_I2_J,axiom,
! [Y: nat,X22: list_nat,X21: nat] :
( ( member_nat @ Y @ ( set_nat2 @ X22 ) )
=> ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_590_list_Oset__intros_I1_J,axiom,
! [X21: nat,X22: list_nat] : ( member_nat @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_591_list_Oset__cases,axiom,
! [E: nat,A: list_nat] :
( ( member_nat @ E @ ( set_nat2 @ A ) )
=> ( ! [Z22: list_nat] :
( A
!= ( cons_nat @ E @ Z22 ) )
=> ~ ! [Z1: nat,Z22: list_nat] :
( ( A
= ( cons_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat @ E @ ( set_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_592_set__ConsD,axiom,
! [Y: nat,X2: nat,Xs: list_nat] :
( ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X2 @ Xs ) ) )
=> ( ( Y = X2 )
| ( member_nat @ Y @ ( set_nat2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_593_list_Osimps_I9_J,axiom,
! [F: nat > sum_sum_a_nat,X21: nat,X22: list_nat] :
( ( map_na823391071729141993_a_nat @ F @ ( cons_nat @ X21 @ X22 ) )
= ( cons_Sum_sum_a_nat @ ( F @ X21 ) @ ( map_na823391071729141993_a_nat @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_594_list_Osimps_I9_J,axiom,
! [F: nat > nat,X21: nat,X22: list_nat] :
( ( map_nat_nat @ F @ ( cons_nat @ X21 @ X22 ) )
= ( cons_nat @ ( F @ X21 ) @ ( map_nat_nat @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_595_Cons__eq__map__D,axiom,
! [X2: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,F: nat > sum_sum_a_nat,Ys: list_nat] :
( ( ( cons_Sum_sum_a_nat @ X2 @ Xs )
= ( map_na823391071729141993_a_nat @ F @ Ys ) )
=> ? [Z3: nat,Zs3: list_nat] :
( ( Ys
= ( cons_nat @ Z3 @ Zs3 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_na823391071729141993_a_nat @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_596_Cons__eq__map__D,axiom,
! [X2: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_nat_nat @ F @ Ys ) )
=> ? [Z3: nat,Zs3: list_nat] :
( ( Ys
= ( cons_nat @ Z3 @ Zs3 ) )
& ( X2
= ( F @ Z3 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_597_map__eq__Cons__D,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat,Y: sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( map_na823391071729141993_a_nat @ F @ Xs )
= ( cons_Sum_sum_a_nat @ Y @ Ys ) )
=> ? [Z3: nat,Zs3: list_nat] :
( ( Xs
= ( cons_nat @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_na823391071729141993_a_nat @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_598_map__eq__Cons__D,axiom,
! [F: nat > nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
=> ? [Z3: nat,Zs3: list_nat] :
( ( Xs
= ( cons_nat @ Z3 @ Zs3 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_nat_nat @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_599_Cons__eq__map__conv,axiom,
! [X2: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,F: nat > sum_sum_a_nat,Ys: list_nat] :
( ( ( cons_Sum_sum_a_nat @ X2 @ Xs )
= ( map_na823391071729141993_a_nat @ F @ Ys ) )
= ( ? [Z4: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z4 @ Zs2 ) )
& ( X2
= ( F @ Z4 ) )
& ( Xs
= ( map_na823391071729141993_a_nat @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_600_Cons__eq__map__conv,axiom,
! [X2: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( map_nat_nat @ F @ Ys ) )
= ( ? [Z4: nat,Zs2: list_nat] :
( ( Ys
= ( cons_nat @ Z4 @ Zs2 ) )
& ( X2
= ( F @ Z4 ) )
& ( Xs
= ( map_nat_nat @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_601_map__eq__Cons__conv,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat,Y: sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( map_na823391071729141993_a_nat @ F @ Xs )
= ( cons_Sum_sum_a_nat @ Y @ Ys ) )
= ( ? [Z4: nat,Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Z4 @ Zs2 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_na823391071729141993_a_nat @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_602_map__eq__Cons__conv,axiom,
! [F: nat > nat,Xs: list_nat,Y: nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( cons_nat @ Y @ Ys ) )
= ( ? [Z4: nat,Zs2: list_nat] :
( ( Xs
= ( cons_nat @ Z4 @ Zs2 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_nat_nat @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_603_subset__code_I1_J,axiom,
! [Xs: list_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_604_subset__insertI2,axiom,
! [A2: set_nat,B: set_nat,B2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_605_subset__insertI,axiom,
! [B: set_nat,A: nat] : ( ord_less_eq_set_nat @ B @ ( insert_nat2 @ A @ B ) ) ).
% subset_insertI
thf(fact_606_subset__insert,axiom,
! [X2: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ X2 @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ X2 @ B ) )
= ( ord_less_eq_set_nat @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_607_insert__mono,axiom,
! [C3: set_nat,D2: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ C3 @ D2 )
=> ( ord_less_eq_set_nat @ ( insert_nat2 @ A @ C3 ) @ ( insert_nat2 @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_608_subset__Un__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B4: set_nat] :
( ( sup_sup_set_nat @ A3 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_609_subset__UnE,axiom,
! [C3: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ ( sup_sup_set_nat @ A2 @ B ) )
=> ~ ! [A6: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ A2 )
=> ! [B7: set_nat] :
( ( ord_less_eq_set_nat @ B7 @ B )
=> ( C3
!= ( sup_sup_set_nat @ A6 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_610_Un__absorb2,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( sup_sup_set_nat @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_611_Un__absorb1,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( sup_sup_set_nat @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_612_Un__upper2,axiom,
! [B: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A2 @ B ) ) ).
% Un_upper2
thf(fact_613_Un__upper1,axiom,
! [A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B ) ) ).
% Un_upper1
thf(fact_614_Un__least,axiom,
! [A2: set_nat,C3: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C3 )
=> ( ( ord_less_eq_set_nat @ B @ C3 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C3 ) ) ) ).
% Un_least
thf(fact_615_Un__mono,axiom,
! [A2: set_nat,C3: set_nat,B: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C3 )
=> ( ( ord_less_eq_set_nat @ B @ D2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ ( sup_sup_set_nat @ C3 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_616_double__diff,axiom,
! [A2: set_nat,B: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C3 )
=> ( ( minus_minus_set_nat @ B @ ( minus_minus_set_nat @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_617_Diff__subset,axiom,
! [A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_618_Diff__mono,axiom,
! [A2: set_nat,C3: set_nat,D2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C3 )
=> ( ( ord_less_eq_set_nat @ D2 @ B )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ ( minus_minus_set_nat @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_619_ad__agr__list__mono,axiom,
! [X: set_a,Y6: set_a,Ys: list_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( ord_less_eq_set_a @ X @ Y6 )
=> ( ( ad_agr_list_a_nat @ Y6 @ Ys @ Xs )
=> ( ad_agr_list_a_nat @ X @ Ys @ Xs ) ) ) ).
% ad_agr_list_mono
thf(fact_620_ad__agr__sets__mono,axiom,
! [X: set_a,Y6: set_a,FV: set_nat,S: set_nat,Sigma: nat > sum_sum_a_nat,Tau: nat > sum_sum_a_nat] :
( ( ord_less_eq_set_a @ X @ Y6 )
=> ( ( ad_agr_sets_a_nat @ FV @ S @ Y6 @ Sigma @ Tau )
=> ( ad_agr_sets_a_nat @ FV @ S @ X @ Sigma @ Tau ) ) ) ).
% ad_agr_sets_mono
thf(fact_621_ad__agr__sets__mono_H,axiom,
! [S: set_nat,S4: set_nat,FV: set_nat,X: set_a,Sigma: nat > sum_sum_a_nat,Tau: nat > sum_sum_a_nat] :
( ( ord_less_eq_set_nat @ S @ S4 )
=> ( ( ad_agr_sets_a_nat @ FV @ S4 @ X @ Sigma @ Tau )
=> ( ad_agr_sets_a_nat @ FV @ S @ X @ Sigma @ Tau ) ) ) ).
% ad_agr_sets_mono'
thf(fact_622_impossible__Cons,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,X2: sum_sum_a_nat] :
( ( ord_less_eq_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( Xs
!= ( cons_Sum_sum_a_nat @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_623_impossible__Cons,axiom,
! [Xs: list_nat,Ys: list_nat,X2: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) )
=> ( Xs
!= ( cons_nat @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_624_subset__singleton__iff,axiom,
! [X: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X @ ( insert_nat2 @ A @ bot_bot_set_nat ) )
= ( ( X = bot_bot_set_nat )
| ( X
= ( insert_nat2 @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_625_subset__singletonD,axiom,
! [A2: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
=> ( ( A2 = bot_bot_set_nat )
| ( A2
= ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_626_subset__Diff__insert,axiom,
! [A2: set_nat,B: set_nat,X2: nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B @ ( insert_nat2 @ X2 @ C3 ) ) )
= ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B @ C3 ) )
& ~ ( member_nat @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_627_Diff__subset__conv,axiom,
! [A2: set_nat,B: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ C3 )
= ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C3 ) ) ) ).
% Diff_subset_conv
thf(fact_628_Diff__partition,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( sup_sup_set_nat @ A2 @ ( minus_minus_set_nat @ B @ A2 ) )
= B ) ) ).
% Diff_partition
thf(fact_629_length__removeAll__less__eq,axiom,
! [X2: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] : ( ord_less_eq_nat @ ( size_s5283204784079214577_a_nat @ ( remove3909449470355376139_a_nat @ X2 @ Xs ) ) @ ( size_s5283204784079214577_a_nat @ Xs ) ) ).
% length_removeAll_less_eq
thf(fact_630_length__removeAll__less__eq,axiom,
! [X2: nat,Xs: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( removeAll_nat @ X2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% length_removeAll_less_eq
thf(fact_631_remove__code_I1_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( remove_nat @ X2 @ ( set_nat2 @ Xs ) )
= ( set_nat2 @ ( removeAll_nat @ X2 @ Xs ) ) ) ).
% remove_code(1)
thf(fact_632_Diff__single__insert,axiom,
! [A2: set_nat,X2: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) @ B )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ X2 @ B ) ) ) ).
% Diff_single_insert
thf(fact_633_subset__insert__iff,axiom,
! [A2: set_nat,X2: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ X2 @ B ) )
= ( ( ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) @ B ) )
& ( ~ ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_634_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_nat,K: set_nat,A: set_nat,B2: set_nat] :
( ( A2
= ( sup_sup_set_nat @ K @ A ) )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_635_boolean__algebra__cancel_Osup2,axiom,
! [B: set_nat,K: set_nat,B2: set_nat,A: set_nat] :
( ( B
= ( sup_sup_set_nat @ K @ B2 ) )
=> ( ( sup_sup_set_nat @ A @ B )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_636_ad__agr__list__subset,axiom,
! [Ms: list_nat,Ns: list_nat,X: set_a,Sigma: nat > sum_sum_a_nat,Sigma3: nat > sum_sum_a_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Ms ) @ ( set_nat2 @ Ns ) )
=> ( ( ad_agr_list_a_nat @ X @ ( map_na823391071729141993_a_nat @ Sigma @ Ns ) @ ( map_na823391071729141993_a_nat @ Sigma3 @ Ns ) )
=> ( ad_agr_list_a_nat @ X @ ( map_na823391071729141993_a_nat @ Sigma @ Ms ) @ ( map_na823391071729141993_a_nat @ Sigma3 @ Ms ) ) ) ) ).
% ad_agr_list_subset
thf(fact_637_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C4: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_638_insert__subsetI,axiom,
! [X2: nat,A2: set_nat,X: set_nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( ord_less_eq_set_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( insert_nat2 @ X2 @ X ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_639_subset__emptyI,axiom,
! [A2: set_nat] :
( ! [X4: nat] :
~ ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_640_in__set__simps_I1_J,axiom,
! [X2: nat,Y: nat,Z: nat,Ys: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ ( cons_nat @ Y @ ( cons_nat @ Z @ Ys ) ) ) )
= ( ( X2 = Y )
| ( member_nat @ X2 @ ( set_nat2 @ ( cons_nat @ Z @ Ys ) ) ) ) ) ).
% in_set_simps(1)
thf(fact_641_bind__simps_I2_J,axiom,
! [X2: nat,Xs: list_nat,F: nat > list_nat] :
( ( bind_nat_nat @ ( cons_nat @ X2 @ Xs ) @ F )
= ( append_nat @ ( F @ X2 ) @ ( bind_nat_nat @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_642_all__tuplesI,axiom,
! [Vs: list_Sum_sum_a_nat,N: nat,Xs: set_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Vs )
= N )
=> ( ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Vs ) @ Xs )
=> ( member408289922725080238_a_nat @ Vs @ ( all_tu407047557562860027_a_nat @ Xs @ N ) ) ) ) ).
% all_tuplesI
thf(fact_643_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_644_all__tuplesD,axiom,
! [Vs: list_Sum_sum_a_nat,Xs: set_Sum_sum_a_nat,N: nat] :
( ( member408289922725080238_a_nat @ Vs @ ( all_tu407047557562860027_a_nat @ Xs @ N ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Vs )
= N )
& ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Vs ) @ Xs ) ) ) ).
% all_tuplesD
thf(fact_645_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_646_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_647_sp__equiv__list__subset,axiom,
! [Ms: list_nat,Ns: list_nat,Sigma: nat > sum_sum_a_nat,Sigma3: nat > sum_sum_a_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Ms ) @ ( set_nat2 @ Ns ) )
=> ( ( sp_equiv_list_a_nat @ ( map_na823391071729141993_a_nat @ Sigma @ Ns ) @ ( map_na823391071729141993_a_nat @ Sigma3 @ Ns ) )
=> ( sp_equiv_list_a_nat @ ( map_na823391071729141993_a_nat @ Sigma @ Ms ) @ ( map_na823391071729141993_a_nat @ Sigma3 @ Ms ) ) ) ) ).
% sp_equiv_list_subset
thf(fact_648_maps__simps_I1_J,axiom,
! [F: nat > list_nat,X2: nat,Xs: list_nat] :
( ( maps_nat_nat @ F @ ( cons_nat @ X2 @ Xs ) )
= ( append_nat @ ( F @ X2 ) @ ( maps_nat_nat @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_649_set__replicate,axiom,
! [N: nat,X2: nat] :
( ( N != zero_zero_nat )
=> ( ( set_nat2 @ ( replicate_nat @ N @ X2 ) )
= ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ).
% set_replicate
thf(fact_650_map__eq__map__tailrec,axiom,
map_na823391071729141993_a_nat = map_ta1136998156224711455_a_nat ).
% map_eq_map_tailrec
thf(fact_651_map__eq__map__tailrec,axiom,
map_nat_nat = map_tailrec_nat_nat ).
% map_eq_map_tailrec
thf(fact_652_nth__equal__first__eq,axiom,
! [X2: sum_sum_a_nat,Xs: list_Sum_sum_a_nat,N: nat] :
( ~ ( member_Sum_sum_a_nat @ X2 @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( ( nth_Sum_sum_a_nat @ ( cons_Sum_sum_a_nat @ X2 @ Xs ) @ N )
= X2 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_653_nth__equal__first__eq,axiom,
! [X2: nat,Xs: list_nat,N: nat] :
( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
= X2 )
= ( N = zero_zero_nat ) ) ) ) ).
% nth_equal_first_eq
thf(fact_654_length__replicate,axiom,
! [N: nat,X2: sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( replic8955434655033810879_a_nat @ N @ X2 ) )
= N ) ).
% length_replicate
thf(fact_655_length__replicate,axiom,
! [N: nat,X2: nat] :
( ( size_size_list_nat @ ( replicate_nat @ N @ X2 ) )
= N ) ).
% length_replicate
thf(fact_656_map__replicate,axiom,
! [F: nat > sum_sum_a_nat,N: nat,X2: nat] :
( ( map_na823391071729141993_a_nat @ F @ ( replicate_nat @ N @ X2 ) )
= ( replic8955434655033810879_a_nat @ N @ ( F @ X2 ) ) ) ).
% map_replicate
thf(fact_657_map__replicate,axiom,
! [F: nat > nat,N: nat,X2: nat] :
( ( map_nat_nat @ F @ ( replicate_nat @ N @ X2 ) )
= ( replicate_nat @ N @ ( F @ X2 ) ) ) ).
% map_replicate
thf(fact_658_nth__Cons__0,axiom,
! [X2: nat,Xs: list_nat] :
( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_659_in__set__replicate,axiom,
! [X2: nat,N: nat,Y: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ ( replicate_nat @ N @ Y ) ) )
= ( ( X2 = Y )
& ( N != zero_zero_nat ) ) ) ).
% in_set_replicate
thf(fact_660_Bex__set__replicate,axiom,
! [N: nat,A: nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
& ( P @ X3 ) ) )
= ( ( P @ A )
& ( N != zero_zero_nat ) ) ) ).
% Bex_set_replicate
thf(fact_661_Ball__set__replicate,axiom,
! [N: nat,A: nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
=> ( P @ X3 ) ) )
= ( ( P @ A )
| ( N = zero_zero_nat ) ) ) ).
% Ball_set_replicate
thf(fact_662_nth__append__length,axiom,
! [Xs: list_Sum_sum_a_nat,X2: sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( nth_Sum_sum_a_nat @ ( append_Sum_sum_a_nat @ Xs @ ( cons_Sum_sum_a_nat @ X2 @ Ys ) ) @ ( size_s5283204784079214577_a_nat @ Xs ) )
= X2 ) ).
% nth_append_length
thf(fact_663_nth__append__length,axiom,
! [Xs: list_nat,X2: nat,Ys: list_nat] :
( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ Ys ) ) @ ( size_size_list_nat @ Xs ) )
= X2 ) ).
% nth_append_length
thf(fact_664_append__replicate__commute,axiom,
! [N: nat,X2: nat,K: nat] :
( ( append_nat @ ( replicate_nat @ N @ X2 ) @ ( replicate_nat @ K @ X2 ) )
= ( append_nat @ ( replicate_nat @ K @ X2 ) @ ( replicate_nat @ N @ X2 ) ) ) ).
% append_replicate_commute
thf(fact_665_replicate__app__Cons__same,axiom,
! [N: nat,X2: nat,Xs: list_nat] :
( ( append_nat @ ( replicate_nat @ N @ X2 ) @ ( cons_nat @ X2 @ Xs ) )
= ( cons_nat @ X2 @ ( append_nat @ ( replicate_nat @ N @ X2 ) @ Xs ) ) ) ).
% replicate_app_Cons_same
thf(fact_666_replicate__length__same,axiom,
! [Xs: list_Sum_sum_a_nat,X2: sum_sum_a_nat] :
( ! [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( X4 = X2 ) )
=> ( ( replic8955434655033810879_a_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ X2 )
= Xs ) ) ).
% replicate_length_same
thf(fact_667_replicate__length__same,axiom,
! [Xs: list_nat,X2: nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ( X4 = X2 ) )
=> ( ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X2 )
= Xs ) ) ).
% replicate_length_same
thf(fact_668_replicate__eqI,axiom,
! [Xs: list_Sum_sum_a_nat,N: nat,X2: sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= N )
=> ( ! [Y2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Y2 @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( Y2 = X2 ) )
=> ( Xs
= ( replic8955434655033810879_a_nat @ N @ X2 ) ) ) ) ).
% replicate_eqI
thf(fact_669_replicate__eqI,axiom,
! [Xs: list_nat,N: nat,X2: nat] :
( ( ( size_size_list_nat @ Xs )
= N )
=> ( ! [Y2: nat] :
( ( member_nat @ Y2 @ ( set_nat2 @ Xs ) )
=> ( Y2 = X2 ) )
=> ( Xs
= ( replicate_nat @ N @ X2 ) ) ) ) ).
% replicate_eqI
thf(fact_670_set__replicate__conv__if,axiom,
! [N: nat,X2: nat] :
( ( ( N = zero_zero_nat )
=> ( ( set_nat2 @ ( replicate_nat @ N @ X2 ) )
= bot_bot_set_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( set_nat2 @ ( replicate_nat @ N @ X2 ) )
= ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ) ).
% set_replicate_conv_if
thf(fact_671_length__nth__simps_I3_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ zero_zero_nat )
= X2 ) ).
% length_nth_simps(3)
thf(fact_672_sp__equiv__list__link,axiom,
! [Sigma: nat > sum_sum_a_nat,Ns: list_nat,Tau: nat > sum_sum_a_nat] :
( ( sp_equiv_list_a_nat @ ( map_na823391071729141993_a_nat @ Sigma @ Ns ) @ ( map_na823391071729141993_a_nat @ Tau @ Ns ) )
= ( sp_equiv_a_nat @ Sigma @ Tau @ ( set_nat2 @ Ns ) ) ) ).
% sp_equiv_list_link
thf(fact_673_ad__agr__list__def,axiom,
( ad_agr_list_a_nat
= ( ^ [X6: set_a,Xs3: list_Sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs3 )
= ( size_s5283204784079214577_a_nat @ Ys2 ) )
& ( ad_equiv_list_a_nat @ X6 @ Xs3 @ Ys2 )
& ( sp_equiv_list_a_nat @ Xs3 @ Ys2 ) ) ) ) ).
% ad_agr_list_def
thf(fact_674_List_Oset__insert,axiom,
! [X2: nat,Xs: list_nat] :
( ( set_nat2 @ ( insert_nat @ X2 @ Xs ) )
= ( insert_nat2 @ X2 @ ( set_nat2 @ Xs ) ) ) ).
% List.set_insert
thf(fact_675_not__in__set__insert,axiom,
! [X2: nat,Xs: list_nat] :
( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X2 @ Xs )
= ( cons_nat @ X2 @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_676_in__set__insert,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( insert_nat @ X2 @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_677_List_Oinsert__def,axiom,
( insert_nat
= ( ^ [X3: nat,Xs3: list_nat] : ( if_list_nat @ ( member_nat @ X3 @ ( set_nat2 @ Xs3 ) ) @ Xs3 @ ( cons_nat @ X3 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_678_nth__Cons_H,axiom,
! [N: nat,X2: nat,Xs: list_nat] :
( ( ( N = zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
= X2 ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_679_atMost__upto,axiom,
( set_ord_atMost_nat
= ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N2 ) ) ) ) ) ).
% atMost_upto
thf(fact_680_nth__append,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( ord_less_nat @ N @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( nth_Sum_sum_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) @ N )
= ( nth_Sum_sum_a_nat @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( nth_Sum_sum_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) @ N )
= ( nth_Sum_sum_a_nat @ Ys @ ( minus_minus_nat @ N @ ( size_s5283204784079214577_a_nat @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_681_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_682_set__replicate__Suc,axiom,
! [N: nat,X2: nat] :
( ( set_nat2 @ ( replicate_nat @ ( suc @ N ) @ X2 ) )
= ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ).
% set_replicate_Suc
thf(fact_683_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_684_nat_Oinject,axiom,
! [X23: nat,Y23: nat] :
( ( ( suc @ X23 )
= ( suc @ Y23 ) )
= ( X23 = Y23 ) ) ).
% nat.inject
thf(fact_685_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_686_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_687_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_688_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_689_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_690_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_691_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_692_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_693_lessThan__iff,axiom,
! [I: nat,K: nat] :
( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
= ( ord_less_nat @ I @ K ) ) ).
% lessThan_iff
thf(fact_694_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_695_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_696_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_697_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_698_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_699_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_700_nth__Cons__Suc,axiom,
! [X2: nat,Xs: list_nat,N: nat] :
( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ ( suc @ N ) )
= ( nth_nat @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_701_length__nth__simps_I4_J,axiom,
! [X2: nat,Xs: list_nat,N: nat] :
( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ ( suc @ N ) )
= ( nth_nat @ Xs @ N ) ) ).
% length_nth_simps(4)
thf(fact_702_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_703_nth__replicate,axiom,
! [I: nat,N: nat,X2: nat] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_nat @ ( replicate_nat @ N @ X2 ) @ I )
= X2 ) ) ).
% nth_replicate
thf(fact_704_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_705_nth__map,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat,F: sum_sum_a_nat > nat] :
( ( ord_less_nat @ N @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( nth_nat @ ( map_Su5227373005390213643at_nat @ F @ Xs ) @ N )
= ( F @ ( nth_Sum_sum_a_nat @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_706_nth__map,axiom,
! [N: nat,Xs: list_nat,F: nat > sum_sum_a_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_Sum_sum_a_nat @ ( map_na823391071729141993_a_nat @ F @ Xs ) @ N )
= ( F @ ( nth_nat @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_707_nth__map,axiom,
! [N: nat,Xs: list_nat,F: nat > nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( map_nat_nat @ F @ Xs ) @ N )
= ( F @ ( nth_nat @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_708_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_709_nth__Cons__pos,axiom,
! [N: nat,X2: nat,Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
= ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_710_Comparator__Generator_OAll__less__Suc,axiom,
! [X2: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ X2 ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ X2 )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% Comparator_Generator.All_less_Suc
thf(fact_711_forall__finite_I2_J,axiom,
! [P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ zero_zero_nat ) )
=> ( P @ I2 ) ) )
= ( P @ zero_zero_nat ) ) ).
% forall_finite(2)
thf(fact_712_forall__finite_I3_J,axiom,
! [X2: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ ( suc @ X2 ) ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ X2 ) )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% forall_finite(3)
thf(fact_713_leD,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y ) ) ).
% leD
thf(fact_714_leI,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% leI
thf(fact_715_nless__le,axiom,
! [A: nat,B2: nat] :
( ( ~ ( ord_less_nat @ A @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A @ B2 )
| ( A = B2 ) ) ) ).
% nless_le
thf(fact_716_antisym__conv1,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_717_antisym__conv2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_718_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
& ~ ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_719_not__le__imp__less,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y @ X2 )
=> ( ord_less_nat @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_720_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B6: nat] :
( ( ord_less_nat @ A5 @ B6 )
| ( A5 = B6 ) ) ) ) ).
% order.order_iff_strict
thf(fact_721_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B6: nat] :
( ( ord_less_eq_nat @ A5 @ B6 )
& ( A5 != B6 ) ) ) ) ).
% order.strict_iff_order
thf(fact_722_order_Ostrict__trans1,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_723_order_Ostrict__trans2,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_724_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B6: nat] :
( ( ord_less_eq_nat @ A5 @ B6 )
& ~ ( ord_less_eq_nat @ B6 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_725_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B6: nat,A5: nat] :
( ( ord_less_nat @ B6 @ A5 )
| ( A5 = B6 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_726_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B6: nat,A5: nat] :
( ( ord_less_eq_nat @ B6 @ A5 )
& ( A5 != B6 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_727_dual__order_Ostrict__trans1,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_728_dual__order_Ostrict__trans2,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_729_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B6: nat,A5: nat] :
( ( ord_less_eq_nat @ B6 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B6 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_730_order_Ostrict__implies__order,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% order.strict_implies_order
thf(fact_731_dual__order_Ostrict__implies__order,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( ord_less_eq_nat @ B2 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_732_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_733_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_734_linorder__not__le,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y ) )
= ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_735_linorder__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_736_order__less__imp__le,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_737_order__le__neq__trans,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_nat @ A @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_738_order__neq__le__trans,axiom,
! [A: nat,B2: nat] :
( ( A != B2 )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_nat @ A @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_739_order__le__less__trans,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_740_order__less__le__trans,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_741_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_742_order__le__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_743_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_eq_nat @ X4 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_744_order__less__le__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_745_linorder__le__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_746_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_747_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_748_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_749_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_750_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_751_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_752_upt__conv__Cons,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( upt @ I @ J )
= ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
% upt_conv_Cons
thf(fact_753_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_754_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_755_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_756_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_757_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_758_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X4: nat,Y2: nat] :
( ( P @ X4 @ Y2 )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y2 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_759_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_760_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_761_nat_OdiscI,axiom,
! [Nat: nat,X23: nat] :
( ( Nat
= ( suc @ X23 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_762_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_763_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_764_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( zero_zero_nat
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_765_bot_Oextremum__strict,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% bot.extremum_strict
thf(fact_766_bot_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_767_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_768_bot_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_769_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_770_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_771_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_772_Suc__le__D,axiom,
! [N: nat,M4: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M4 )
=> ? [M3: nat] :
( M4
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_773_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_774_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_775_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_776_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N3 )
=> ( P @ M5 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_777_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_778_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X4: nat] : ( R2 @ X4 @ X4 )
=> ( ! [X4: nat,Y2: nat,Z3: nat] :
( ( R2 @ X4 @ Y2 )
=> ( ( R2 @ Y2 @ Z3 )
=> ( R2 @ X4 @ Z3 ) ) )
=> ( ! [N3: nat] : ( R2 @ N3 @ ( suc @ N3 ) )
=> ( R2 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_779_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_780_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_781_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_782_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_783_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_784_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_785_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_786_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_787_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_788_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_789_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_790_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_791_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_792_lessThan__strict__subset__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% lessThan_strict_subset_iff
thf(fact_793_order__less__imp__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_794_order__less__imp__not__eq2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_795_order__less__imp__not__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_796_linorder__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_797_order__less__imp__triv,axiom,
! [X2: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_798_order__less__not__sym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_799_order__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_800_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_801_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_802_ord__less__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_803_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X4: nat,Y2: nat] :
( ( ord_less_nat @ X4 @ Y2 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_804_order__less__trans,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_805_order__less__asym_H,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ~ ( ord_less_nat @ B2 @ A ) ) ).
% order_less_asym'
thf(fact_806_linorder__neq__iff,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
= ( ( ord_less_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_807_order__less__asym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_808_linorder__neqE,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_809_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( A != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_810_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( A != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_811_dual__order_Ostrict__trans,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_812_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ( ord_less_nat @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_813_order_Ostrict__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_814_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B2: nat] :
( ! [A4: nat,B5: nat] :
( ( ord_less_nat @ A4 @ B5 )
=> ( P @ A4 @ B5 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B5: nat] :
( ( P @ B5 @ A4 )
=> ( P @ A4 @ B5 ) )
=> ( P @ A @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_815_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X7: nat] : ( P2 @ X7 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M6: nat] :
( ( ord_less_nat @ M6 @ N2 )
=> ~ ( P3 @ M6 ) ) ) ) ) ).
% exists_least_iff
thf(fact_816_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_817_dual__order_Oasym,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ B2 @ A )
=> ~ ( ord_less_nat @ A @ B2 ) ) ).
% dual_order.asym
thf(fact_818_linorder__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_819_antisym__conv3,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_nat @ Y @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_820_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X4: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X4 )
=> ( P @ Y4 ) )
=> ( P @ X4 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_821_ord__less__eq__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_822_ord__eq__less__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( A = B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_823_order_Oasym,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ~ ( ord_less_nat @ B2 @ A ) ) ).
% order.asym
thf(fact_824_less__imp__neq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_825_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_826_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_827_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I4: nat] :
( ( J
= ( suc @ I4 ) )
=> ( P @ I4 ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ J )
=> ( ( P @ ( suc @ I4 ) )
=> ( P @ I4 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_828_linorder__neqE__nat,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_829_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
& ~ ( P @ M5 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_830_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( P @ M5 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_831_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_832_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I4: nat] : ( P @ I4 @ ( suc @ I4 ) )
=> ( ! [I4: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I4 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I4 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_833_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_834_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_835_less__not__refl3,axiom,
! [S3: nat,T: nat] :
( ( ord_less_nat @ S3 @ T )
=> ( S3 != T ) ) ).
% less_not_refl3
thf(fact_836_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_837_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_838_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_839_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_840_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M7: nat] :
( ( M
= ( suc @ M7 ) )
& ( ord_less_nat @ N @ M7 ) ) ) ) ).
% Suc_less_eq2
thf(fact_841_Nat_OAll__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ I2 ) ) ) ) ).
% Nat.All_less_Suc
thf(fact_842_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_843_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_844_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_845_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_846_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ N )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_847_Suc__inject,axiom,
! [X2: nat,Y: nat] :
( ( ( suc @ X2 )
= ( suc @ Y ) )
=> ( X2 = Y ) ) ).
% Suc_inject
thf(fact_848_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_849_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_850_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_851_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_852_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_853_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_854_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_855_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_856_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_857_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ ( suc @ I2 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_858_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_859_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_860_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_861_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_862_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_863_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_864_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_865_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_866_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_867_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_868_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_869_sup_Ostrict__coboundedI2,axiom,
! [C: set_nat,B2: set_nat,A: set_nat] :
( ( ord_less_set_nat @ C @ B2 )
=> ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_870_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B2: nat,A: nat] :
( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_871_sup_Ostrict__coboundedI1,axiom,
! [C: set_nat,A: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ C @ A )
=> ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_872_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_nat @ C @ A )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_873_sup_Ostrict__order__iff,axiom,
( ord_less_set_nat
= ( ^ [B6: set_nat,A5: set_nat] :
( ( A5
= ( sup_sup_set_nat @ A5 @ B6 ) )
& ( A5 != B6 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_874_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B6: nat,A5: nat] :
( ( A5
= ( sup_sup_nat @ A5 @ B6 ) )
& ( A5 != B6 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_875_sup_Ostrict__boundedE,axiom,
! [B2: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A )
=> ~ ( ( ord_less_set_nat @ B2 @ A )
=> ~ ( ord_less_set_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_876_sup_Ostrict__boundedE,axiom,
! [B2: nat,C: nat,A: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B2 @ C ) @ A )
=> ~ ( ( ord_less_nat @ B2 @ A )
=> ~ ( ord_less_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_877_sup_Oabsorb4,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ( ( sup_sup_set_nat @ A @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_878_sup_Oabsorb4,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( sup_sup_nat @ A @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_879_sup_Oabsorb3,axiom,
! [B2: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B2 @ A )
=> ( ( sup_sup_set_nat @ A @ B2 )
= A ) ) ).
% sup.absorb3
thf(fact_880_sup_Oabsorb3,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( ( sup_sup_nat @ A @ B2 )
= A ) ) ).
% sup.absorb3
thf(fact_881_less__supI2,axiom,
! [X2: set_nat,B2: set_nat,A: set_nat] :
( ( ord_less_set_nat @ X2 @ B2 )
=> ( ord_less_set_nat @ X2 @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% less_supI2
thf(fact_882_less__supI2,axiom,
! [X2: nat,B2: nat,A: nat] :
( ( ord_less_nat @ X2 @ B2 )
=> ( ord_less_nat @ X2 @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% less_supI2
thf(fact_883_less__supI1,axiom,
! [X2: set_nat,A: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ X2 @ A )
=> ( ord_less_set_nat @ X2 @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% less_supI1
thf(fact_884_less__supI1,axiom,
! [X2: nat,A: nat,B2: nat] :
( ( ord_less_nat @ X2 @ A )
=> ( ord_less_nat @ X2 @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% less_supI1
thf(fact_885_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_886_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_887_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_888_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_889_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_890_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_891_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_892_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N3 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_893_forall__finite_I1_J,axiom,
! [P: nat > $o,I3: nat] :
( ( ord_less_nat @ I3 @ zero_zero_nat )
=> ( P @ I3 ) ) ).
% forall_finite(1)
thf(fact_894_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I4: nat,J2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ord_less_nat @ ( F @ I4 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_895_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_896_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_897_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N2: nat] :
( ( ord_less_nat @ M6 @ N2 )
| ( M6 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_898_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_899_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M6: nat,N2: nat] :
( ( ord_less_eq_nat @ M6 @ N2 )
& ( M6 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_900_length__induct,axiom,
! [P: list_Sum_sum_a_nat > $o,Xs: list_Sum_sum_a_nat] :
( ! [Xs2: list_Sum_sum_a_nat] :
( ! [Ys5: list_Sum_sum_a_nat] :
( ( ord_less_nat @ ( size_s5283204784079214577_a_nat @ Ys5 ) @ ( size_s5283204784079214577_a_nat @ Xs2 ) )
=> ( P @ Ys5 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_901_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs2: list_nat] :
( ! [Ys5: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys5 ) @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ Ys5 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_902_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_903_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_904_map__Suc__upt,axiom,
! [M: nat,N: nat] :
( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
= ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% map_Suc_upt
thf(fact_905_nth__non__equal__first__eq,axiom,
! [X2: nat,Y: nat,Xs: list_nat,N: nat] :
( ( X2 != Y )
=> ( ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
= Y )
= ( ( ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_906_Cons__replicate__eq,axiom,
! [X2: nat,Xs: list_nat,N: nat,Y: nat] :
( ( ( cons_nat @ X2 @ Xs )
= ( replicate_nat @ N @ Y ) )
= ( ( X2 = Y )
& ( ord_less_nat @ zero_zero_nat @ N )
& ( Xs
= ( replicate_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ X2 ) ) ) ) ).
% Cons_replicate_eq
thf(fact_907_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_908_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_909_Suc__length__conv,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat] :
( ( ( suc @ N )
= ( size_s5283204784079214577_a_nat @ Xs ) )
= ( ? [Y3: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
( ( Xs
= ( cons_Sum_sum_a_nat @ Y3 @ Ys2 ) )
& ( ( size_s5283204784079214577_a_nat @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_910_Suc__length__conv,axiom,
! [N: nat,Xs: list_nat] :
( ( ( suc @ N )
= ( size_size_list_nat @ Xs ) )
= ( ? [Y3: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ Y3 @ Ys2 ) )
& ( ( size_size_list_nat @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_911_length__Suc__conv,axiom,
! [Xs: list_Sum_sum_a_nat,N: nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y3: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
( ( Xs
= ( cons_Sum_sum_a_nat @ Y3 @ Ys2 ) )
& ( ( size_s5283204784079214577_a_nat @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_912_length__Suc__conv,axiom,
! [Xs: list_nat,N: nat] :
( ( ( size_size_list_nat @ Xs )
= ( suc @ N ) )
= ( ? [Y3: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ Y3 @ Ys2 ) )
& ( ( size_size_list_nat @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_913_length__nth__simps_I2_J,axiom,
! [X2: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( cons_Sum_sum_a_nat @ X2 @ Xs ) )
= ( suc @ ( size_s5283204784079214577_a_nat @ Xs ) ) ) ).
% length_nth_simps(2)
thf(fact_914_length__nth__simps_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X2 @ Xs ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_nth_simps(2)
thf(fact_915_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_916_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_917_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_918_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_919_diff__less__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_920_replicate__Suc,axiom,
! [N: nat,X2: nat] :
( ( replicate_nat @ ( suc @ N ) @ X2 )
= ( cons_nat @ X2 @ ( replicate_nat @ N @ X2 ) ) ) ).
% replicate_Suc
thf(fact_921_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_Sum_sum_a_nat,Z2: list_Sum_sum_a_nat] : ( Y5 = Z2 ) )
= ( ^ [Xs3: list_Sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs3 )
= ( size_s5283204784079214577_a_nat @ Ys2 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s5283204784079214577_a_nat @ Xs3 ) )
=> ( ( nth_Sum_sum_a_nat @ Xs3 @ I2 )
= ( nth_Sum_sum_a_nat @ Ys2 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_922_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_nat,Z2: list_nat] : ( Y5 = Z2 ) )
= ( ^ [Xs3: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys2 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs3 ) )
=> ( ( nth_nat @ Xs3 @ I2 )
= ( nth_nat @ Ys2 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_923_Skolem__list__nth,axiom,
! [K: nat,P: nat > sum_sum_a_nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X6: sum_sum_a_nat] : ( P @ I2 @ X6 ) ) )
= ( ? [Xs3: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs3 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_Sum_sum_a_nat @ Xs3 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_924_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X6: nat] : ( P @ I2 @ X6 ) ) )
= ( ? [Xs3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_nat @ Xs3 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_925_nth__equalityI,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( nth_Sum_sum_a_nat @ Xs @ I4 )
= ( nth_Sum_sum_a_nat @ Ys @ I4 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_926_nth__equalityI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I4 )
= ( nth_nat @ Ys @ I4 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_927_lessThan__Suc__atMost,axiom,
! [K: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K ) )
= ( set_ord_atMost_nat @ K ) ) ).
% lessThan_Suc_atMost
thf(fact_928_lessThan__Suc,axiom,
! [K: nat] :
( ( set_ord_lessThan_nat @ ( suc @ K ) )
= ( insert_nat2 @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).
% lessThan_Suc
thf(fact_929_atMost__Suc,axiom,
! [K: nat] :
( ( set_ord_atMost_nat @ ( suc @ K ) )
= ( insert_nat2 @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).
% atMost_Suc
thf(fact_930_upt__conv__Cons__Cons,axiom,
! [M: nat,N: nat,Ns: list_nat,Q2: nat] :
( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
= ( upt @ M @ Q2 ) )
= ( ( cons_nat @ N @ Ns )
= ( upt @ ( suc @ M ) @ Q2 ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_931_gen__length__code_I2_J,axiom,
! [N: nat,X2: nat,Xs: list_nat] :
( ( gen_length_nat @ N @ ( cons_nat @ X2 @ Xs ) )
= ( gen_length_nat @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_932_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s5283204784079214577_a_nat @ Xs ) )
= ( ? [X3: sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
( ( Xs
= ( cons_Sum_sum_a_nat @ X3 @ Ys2 ) )
& ( ord_less_eq_nat @ N @ ( size_s5283204784079214577_a_nat @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_933_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
= ( ? [X3: nat,Ys2: list_nat] :
( ( Xs
= ( cons_nat @ X3 @ Ys2 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_934_length__pos__if__in__set,axiom,
! [X2: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s5283204784079214577_a_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_935_length__pos__if__in__set,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_936_nth__mem,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat] :
( ( ord_less_nat @ N @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( member_Sum_sum_a_nat @ ( nth_Sum_sum_a_nat @ Xs @ N ) @ ( set_Sum_sum_a_nat2 @ Xs ) ) ) ).
% nth_mem
thf(fact_937_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_938_list__ball__nth,axiom,
! [N: nat,Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( ord_less_nat @ N @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ! [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( P @ X4 ) )
=> ( P @ ( nth_Sum_sum_a_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_939_list__ball__nth,axiom,
! [N: nat,Xs: list_nat,P: nat > $o] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ( P @ X4 ) )
=> ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_940_in__set__conv__nth,axiom,
! [X2: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ ( set_Sum_sum_a_nat2 @ Xs ) )
= ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
& ( ( nth_Sum_sum_a_nat @ Xs @ I2 )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_941_in__set__conv__nth,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
= ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
& ( ( nth_nat @ Xs @ I2 )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_942_all__nth__imp__all__set,axiom,
! [Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o,X2: sum_sum_a_nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( P @ ( nth_Sum_sum_a_nat @ Xs @ I4 ) ) )
=> ( ( member_Sum_sum_a_nat @ X2 @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_943_all__nth__imp__all__set,axiom,
! [Xs: list_nat,P: nat > $o,X2: nat] :
( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I4 ) ) )
=> ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_944_all__set__conv__all__nth,axiom,
! [Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( ! [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( P @ X3 ) ) )
= ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( P @ ( nth_Sum_sum_a_nat @ Xs @ I2 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_945_all__set__conv__all__nth,axiom,
! [Xs: list_nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( P @ X3 ) ) )
= ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( P @ ( nth_nat @ Xs @ I2 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_946_map__equality__iff,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat,G: nat > sum_sum_a_nat,Ys: list_nat] :
( ( ( map_Su2790769393171190532_a_nat @ F @ Xs )
= ( map_na823391071729141993_a_nat @ G @ Ys ) )
= ( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( ( F @ ( nth_Sum_sum_a_nat @ Xs @ I2 ) )
= ( G @ ( nth_nat @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_947_map__equality__iff,axiom,
! [F: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_Su5227373005390213643at_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) )
= ( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( ( F @ ( nth_Sum_sum_a_nat @ Xs @ I2 ) )
= ( G @ ( nth_nat @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_948_map__equality__iff,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat,G: sum_sum_a_nat > sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( map_na823391071729141993_a_nat @ F @ Xs )
= ( map_Su2790769393171190532_a_nat @ G @ Ys ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( F @ ( nth_nat @ Xs @ I2 ) )
= ( G @ ( nth_Sum_sum_a_nat @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_949_map__equality__iff,axiom,
! [F: nat > nat,Xs: list_nat,G: sum_sum_a_nat > nat,Ys: list_Sum_sum_a_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_Su5227373005390213643at_nat @ G @ Ys ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( F @ ( nth_nat @ Xs @ I2 ) )
= ( G @ ( nth_Sum_sum_a_nat @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_950_map__equality__iff,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat,G: nat > sum_sum_a_nat,Ys: list_nat] :
( ( ( map_na823391071729141993_a_nat @ F @ Xs )
= ( map_na823391071729141993_a_nat @ G @ Ys ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( ( F @ ( nth_nat @ Xs @ I2 ) )
= ( G @ ( nth_nat @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_951_map__equality__iff,axiom,
! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ G @ Ys ) )
= ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( ( F @ ( nth_nat @ Xs @ I2 ) )
= ( G @ ( nth_nat @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_952_Iic__subset__Iio__iff,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ A ) @ ( set_ord_lessThan_nat @ B2 ) )
= ( ord_less_nat @ A @ B2 ) ) ).
% Iic_subset_Iio_iff
thf(fact_953_subset__code_I2_J,axiom,
! [A2: set_nat,Ys: list_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( coset_nat @ Ys ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_954_length__removeAll__less,axiom,
! [X2: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( ord_less_nat @ ( size_s5283204784079214577_a_nat @ ( remove3909449470355376139_a_nat @ X2 @ Xs ) ) @ ( size_s5283204784079214577_a_nat @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_955_length__removeAll__less,axiom,
! [X2: nat,Xs: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ ( size_size_list_nat @ ( removeAll_nat @ X2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_956_insert__code_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( insert_nat2 @ X2 @ ( coset_nat @ Xs ) )
= ( coset_nat @ ( removeAll_nat @ X2 @ Xs ) ) ) ).
% insert_code(2)
thf(fact_957_length__Cons,axiom,
! [X2: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( cons_Sum_sum_a_nat @ X2 @ Xs ) )
= ( suc @ ( size_s5283204784079214577_a_nat @ Xs ) ) ) ).
% length_Cons
thf(fact_958_length__Cons,axiom,
! [X2: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X2 @ Xs ) )
= ( suc @ ( size_size_list_nat @ Xs ) ) ) ).
% length_Cons
thf(fact_959_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_960_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_961_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_962_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_963_not__psubset__empty,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_964_psubsetD,axiom,
! [A2: set_nat,B: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B ) ) ) ).
% psubsetD
thf(fact_965_psubset__imp__ex__mem,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ B )
=> ? [B5: nat] : ( member_nat @ B5 @ ( minus_minus_set_nat @ B @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_966_psubset__insert__iff,axiom,
! [A2: set_nat,X2: nat,B: set_nat] :
( ( ord_less_set_nat @ A2 @ ( insert_nat2 @ X2 @ B ) )
= ( ( ( member_nat @ X2 @ B )
=> ( ord_less_set_nat @ A2 @ B ) )
& ( ~ ( member_nat @ X2 @ B )
=> ( ( ( member_nat @ X2 @ A2 )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) @ B ) )
& ( ~ ( member_nat @ X2 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_967_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_968_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_969_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K2 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_970_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N3: nat] :
( ~ ( P @ N3 )
& ( P @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_971_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% minf(8)
thf(fact_972_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ord_less_eq_nat @ X5 @ T ) ) ).
% minf(6)
thf(fact_973_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ord_less_eq_nat @ T @ X5 ) ) ).
% pinf(8)
thf(fact_974_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% pinf(6)
thf(fact_975_complete__interval,axiom,
! [A: nat,B2: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B2 )
=> ( ( P @ A )
=> ( ~ ( P @ B2 )
=> ? [C5: nat] :
( ( ord_less_eq_nat @ A @ C5 )
& ( ord_less_eq_nat @ C5 @ B2 )
& ! [X5: nat] :
( ( ( ord_less_eq_nat @ A @ X5 )
& ( ord_less_nat @ X5 @ C5 ) )
=> ( P @ X5 ) )
& ! [D3: nat] :
( ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ D3 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_nat @ D3 @ C5 ) ) ) ) ) ) ).
% complete_interval
thf(fact_976_verit__comp__simplify1_I3_J,axiom,
! [B8: nat,A7: nat] :
( ( ~ ( ord_less_eq_nat @ B8 @ A7 ) )
= ( ord_less_nat @ A7 @ B8 ) ) ).
% verit_comp_simplify1(3)
thf(fact_977_map__upt__eqI,axiom,
! [Xs: list_Sum_sum_a_nat,N: nat,M: nat,F: nat > sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( minus_minus_nat @ N @ M ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( nth_Sum_sum_a_nat @ Xs @ I4 )
= ( F @ ( plus_plus_nat @ M @ I4 ) ) ) )
=> ( ( map_na823391071729141993_a_nat @ F @ ( upt @ M @ N ) )
= Xs ) ) ) ).
% map_upt_eqI
thf(fact_978_map__upt__eqI,axiom,
! [Xs: list_nat,N: nat,M: nat,F: nat > nat] :
( ( ( size_size_list_nat @ Xs )
= ( minus_minus_nat @ N @ M ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I4 )
= ( F @ ( plus_plus_nat @ M @ I4 ) ) ) )
=> ( ( map_nat_nat @ F @ ( upt @ M @ N ) )
= Xs ) ) ) ).
% map_upt_eqI
thf(fact_979_add__left__cancel,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ A @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_980_add__right__cancel,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_981_add__le__cancel__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_cancel_right
thf(fact_982_add__le__cancel__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_cancel_left
thf(fact_983_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_984_add__cancel__left__left,axiom,
! [B2: nat,A: nat] :
( ( ( plus_plus_nat @ B2 @ A )
= A )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_985_add__cancel__left__right,axiom,
! [A: nat,B2: nat] :
( ( ( plus_plus_nat @ A @ B2 )
= A )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_986_add__cancel__right__left,axiom,
! [A: nat,B2: nat] :
( ( A
= ( plus_plus_nat @ B2 @ A ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_987_add__cancel__right__right,axiom,
! [A: nat,B2: nat] :
( ( A
= ( plus_plus_nat @ A @ B2 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_988_add__eq__0__iff__both__eq__0,axiom,
! [X2: nat,Y: nat] :
( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_989_zero__eq__add__iff__both__eq__0,axiom,
! [X2: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X2 @ Y ) )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_990_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_991_add__less__cancel__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_nat @ A @ B2 ) ) ).
% add_less_cancel_left
thf(fact_992_add__less__cancel__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_nat @ A @ B2 ) ) ).
% add_less_cancel_right
thf(fact_993_add__diff__cancel__right_H,axiom,
! [A: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
= A ) ).
% add_diff_cancel_right'
thf(fact_994_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( minus_minus_nat @ A @ B2 ) ) ).
% add_diff_cancel_right
thf(fact_995_add__diff__cancel__left_H,axiom,
! [A: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B2 ) @ A )
= B2 ) ).
% add_diff_cancel_left'
thf(fact_996_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
= ( minus_minus_nat @ A @ B2 ) ) ).
% add_diff_cancel_left
thf(fact_997_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_998_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_999_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_1000_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_1001_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_1002_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_1003_add__le__same__cancel1,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_1004_add__le__same__cancel2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_1005_le__add__same__cancel1,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_1006_le__add__same__cancel2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_1007_less__add__same__cancel2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_1008_less__add__same__cancel1,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_1009_add__less__same__cancel2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_1010_add__less__same__cancel1,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_1011_le__add__diff__inverse,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A @ B2 ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_1012_le__add__diff__inverse2,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B2 ) @ B2 )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_1013_diff__add__zero,axiom,
! [A: nat,B2: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_1014_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1015_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_1016_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_1017_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_1018_length__append,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).
% length_append
thf(fact_1019_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_1020_nth__upt,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
=> ( ( nth_nat @ ( upt @ I @ J ) @ K )
= ( plus_plus_nat @ I @ K ) ) ) ).
% nth_upt
thf(fact_1021_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_1022_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_1023_nth__append__length__plus,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,N: nat] :
( ( nth_Sum_sum_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ N ) )
= ( nth_Sum_sum_a_nat @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_1024_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_1025_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_1026_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_1027_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_1028_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_1029_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1030_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1031_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_1032_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_1033_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_nat @ A @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_1034_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_nat @ A @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_1035_add__strict__right__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_1036_add__strict__left__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_1037_add__strict__mono,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1038_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_1039_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_1040_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_1041_add__decreasing,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_1042_add__increasing,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_1043_add__decreasing2,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_1044_add__increasing2,axiom,
! [C: nat,B2: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B2 @ A )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_1045_add__nonneg__nonneg,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1046_add__nonpos__nonpos,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_1047_add__nonneg__eq__0__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1048_add__nonpos__eq__0__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1049_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_1050_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_1051_add__le__less__mono,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1052_add__less__le__mono,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1053_add__neg__neg,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_1054_add__pos__pos,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% add_pos_pos
thf(fact_1055_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ~ ! [C5: nat] :
( ( B2
= ( plus_plus_nat @ A @ C5 ) )
=> ( C5 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_1056_pos__add__strict,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_1057_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( minus_minus_nat @ B2 @ A )
= C )
= ( B2
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1058_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B2 @ A ) )
= B2 ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1059_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B2 @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1060_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1061_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1062_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1063_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1064_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B2 @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1065_le__add__diff,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_1066_diff__add,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A ) @ A )
= B2 ) ) ).
% diff_add
thf(fact_1067_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_1068_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_1069_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B2: nat] :
( ~ ( ord_less_nat @ A @ B2 )
=> ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A @ B2 ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1070_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_1071_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1072_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1073_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M6: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus_nat @ M6 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1074_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_1075_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_1076_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_1077_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_1078_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1079_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_1080_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_1081_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1082_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1083_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_1084_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_1085_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_1086_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1087_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1088_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_1089_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_1090_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B6: nat] :
? [C4: nat] :
( B6
= ( plus_plus_nat @ A5 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_1091_add__right__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_1092_less__eqE,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ~ ! [C5: nat] :
( B2
!= ( plus_plus_nat @ A @ C5 ) ) ) ).
% less_eqE
thf(fact_1093_add__left__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_1094_add__mono,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_mono
thf(fact_1095_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_1096_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_1097_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_1098_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_1099_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_1100_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_1101_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_1102_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B2: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_1103_group__cancel_Oadd2,axiom,
! [B: nat,K: nat,B2: nat,A: nat] :
( ( B
= ( plus_plus_nat @ K @ B2 ) )
=> ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_1104_add_Oassoc,axiom,
! [A: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_1105_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A5: nat,B6: nat] : ( plus_plus_nat @ B6 @ A5 ) ) ) ).
% add.commute
thf(fact_1106_add_Oleft__commute,axiom,
! [B2: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_1107_add__left__imp__eq,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ A @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_1108_add__right__imp__eq,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_1109_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_1110_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1111_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1112_diff__diff__eq,axiom,
! [A: nat,B2: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B2 ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% diff_diff_eq
thf(fact_1113_add__implies__diff,axiom,
! [C: nat,B2: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B2 )
= A )
=> ( C
= ( minus_minus_nat @ A @ B2 ) ) ) ).
% add_implies_diff
thf(fact_1114_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1115_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1116_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1117_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_1118_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1119_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1120_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M6: nat,N2: nat] :
? [K3: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M6 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1121_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1122_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1123_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1124_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1125_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1126_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1127_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_1128_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_1129_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_1130_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_1131_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_1132_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_1133_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_1134_replicate__add,axiom,
! [N: nat,M: nat,X2: nat] :
( ( replicate_nat @ ( plus_plus_nat @ N @ M ) @ X2 )
= ( append_nat @ ( replicate_nat @ N @ X2 ) @ ( replicate_nat @ M @ X2 ) ) ) ).
% replicate_add
thf(fact_1135_gen__length__def,axiom,
( gen_le1340941697924381074_a_nat
= ( ^ [N2: nat,Xs3: list_Sum_sum_a_nat] : ( plus_plus_nat @ N2 @ ( size_s5283204784079214577_a_nat @ Xs3 ) ) ) ) ).
% gen_length_def
thf(fact_1136_gen__length__def,axiom,
( gen_length_nat
= ( ^ [N2: nat,Xs3: list_nat] : ( plus_plus_nat @ N2 @ ( size_size_list_nat @ Xs3 ) ) ) ) ).
% gen_length_def
thf(fact_1137_add__neg__nonpos,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_1138_add__nonneg__pos,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% add_nonneg_pos
thf(fact_1139_add__nonpos__neg,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_1140_add__pos__nonneg,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% add_pos_nonneg
thf(fact_1141_add__strict__increasing,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1142_add__strict__increasing2,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1143_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_1144_verit__la__disequality,axiom,
! [A: nat,B2: nat] :
( ( A = B2 )
| ~ ( ord_less_eq_nat @ A @ B2 )
| ~ ( ord_less_eq_nat @ B2 @ A ) ) ).
% verit_la_disequality
thf(fact_1145_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1146_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A @ B2 ) )
= ( ( ( ord_less_nat @ A @ B2 )
=> ( P @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A
= ( plus_plus_nat @ B2 @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_1147_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A @ B2 ) )
= ( ~ ( ( ( ord_less_nat @ A @ B2 )
& ~ ( P @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A
= ( plus_plus_nat @ B2 @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1148_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_1149_upt__add__eq__append,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( plus_plus_nat @ J @ K ) )
= ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% upt_add_eq_append
thf(fact_1150_list_Osize_I4_J,axiom,
! [X21: sum_sum_a_nat,X22: list_Sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( cons_Sum_sum_a_nat @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_s5283204784079214577_a_nat @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_1151_list_Osize_I4_J,axiom,
! [X21: nat,X22: list_nat] :
( ( size_size_list_nat @ ( cons_nat @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_size_list_nat @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_1152_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M6: nat,N2: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% add_eq_if
thf(fact_1153_upt__eq__Cons__conv,axiom,
! [I: nat,J: nat,X2: nat,Xs: list_nat] :
( ( ( upt @ I @ J )
= ( cons_nat @ X2 @ Xs ) )
= ( ( ord_less_nat @ I @ J )
& ( I = X2 )
& ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
= Xs ) ) ) ).
% upt_eq_Cons_conv
thf(fact_1154_nth__map__upt,axiom,
! [I: nat,N: nat,M: nat,F: nat > sum_sum_a_nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ N @ M ) )
=> ( ( nth_Sum_sum_a_nat @ ( map_na823391071729141993_a_nat @ F @ ( upt @ M @ N ) ) @ I )
= ( F @ ( plus_plus_nat @ M @ I ) ) ) ) ).
% nth_map_upt
thf(fact_1155_nth__map__upt,axiom,
! [I: nat,N: nat,M: nat,F: nat > nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ N @ M ) )
=> ( ( nth_nat @ ( map_nat_nat @ F @ ( upt @ M @ N ) ) @ I )
= ( F @ ( plus_plus_nat @ M @ I ) ) ) ) ).
% nth_map_upt
thf(fact_1156_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B2: nat] :
( ! [A4: nat,B5: nat] :
( ( P @ A4 @ B5 )
= ( P @ B5 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B5: nat] :
( ( P @ A4 @ B5 )
=> ( P @ A4 @ ( plus_plus_nat @ A4 @ B5 ) ) )
=> ( P @ A @ B2 ) ) ) ) ).
% Euclid_induct
thf(fact_1157_add__0__iff,axiom,
! [B2: nat,A: nat] :
( ( B2
= ( plus_plus_nat @ B2 @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_1158_rem__nth__length,axiom,
! [I: nat,Xs: list_Sum_sum_a_nat] :
( ( ord_less_nat @ I @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( size_s5283204784079214577_a_nat @ ( rem_nt658808235856662061_a_nat @ I @ Xs ) )
= ( minus_minus_nat @ ( size_s5283204784079214577_a_nat @ Xs ) @ one_one_nat ) ) ) ).
% rem_nth_length
thf(fact_1159_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_1160_rem__nth_Osimps_I2_J,axiom,
! [X2: nat,Xs: list_nat] :
( ( rem_nth_nat @ zero_zero_nat @ ( cons_nat @ X2 @ Xs ) )
= Xs ) ).
% rem_nth.simps(2)
thf(fact_1161_rem__nth_Osimps_I3_J,axiom,
! [N: nat,X2: nat,Xs: list_nat] :
( ( rem_nth_nat @ ( suc @ N ) @ ( cons_nat @ X2 @ Xs ) )
= ( cons_nat @ X2 @ ( rem_nth_nat @ N @ Xs ) ) ) ).
% rem_nth.simps(3)
thf(fact_1162_add__nth__rem__nth__self,axiom,
! [I: nat,Xs: list_Sum_sum_a_nat] :
( ( ord_less_nat @ I @ ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( add_nt4212672348507122516_a_nat @ I @ ( nth_Sum_sum_a_nat @ Xs @ I ) @ ( rem_nt658808235856662061_a_nat @ I @ Xs ) )
= Xs ) ) ).
% add_nth_rem_nth_self
thf(fact_1163_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_1164_atLeast1__atMost__eq__remove0,axiom,
! [N: nat] :
( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N )
= ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N ) @ ( insert_nat2 @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% atLeast1_atMost_eq_remove0
thf(fact_1165_Icc__eq__Icc,axiom,
! [L: nat,H: nat,L2: nat,H2: nat] :
( ( ( set_or1269000886237332187st_nat @ L @ H )
= ( set_or1269000886237332187st_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_eq_nat @ L @ H )
& ~ ( ord_less_eq_nat @ L2 @ H2 ) ) ) ) ).
% Icc_eq_Icc
thf(fact_1166_atLeastAtMost__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I )
& ( ord_less_eq_nat @ I @ U ) ) ) ).
% atLeastAtMost_iff
thf(fact_1167_atLeastatMost__empty__iff2,axiom,
! [A: nat,B2: nat] :
( ( bot_bot_set_nat
= ( set_or1269000886237332187st_nat @ A @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A @ B2 ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_1168_atLeastatMost__empty__iff,axiom,
! [A: nat,B2: nat] :
( ( ( set_or1269000886237332187st_nat @ A @ B2 )
= bot_bot_set_nat )
= ( ~ ( ord_less_eq_nat @ A @ B2 ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_1169_atLeastatMost__subset__iff,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B2 ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ~ ( ord_less_eq_nat @ A @ B2 )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B2 @ D ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_1170_atLeastatMost__empty,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( ( set_or1269000886237332187st_nat @ A @ B2 )
= bot_bot_set_nat ) ) ).
% atLeastatMost_empty
thf(fact_1171_atLeastAtMost__singleton,axiom,
! [A: nat] :
( ( set_or1269000886237332187st_nat @ A @ A )
= ( insert_nat2 @ A @ bot_bot_set_nat ) ) ).
% atLeastAtMost_singleton
thf(fact_1172_atLeastAtMost__singleton__iff,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ( set_or1269000886237332187st_nat @ A @ B2 )
= ( insert_nat2 @ C @ bot_bot_set_nat ) )
= ( ( A = B2 )
& ( B2 = C ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_1173_Icc__subset__Iic__iff,axiom,
! [L: nat,H: nat,H2: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L @ H ) @ ( set_ord_atMost_nat @ H2 ) )
= ( ~ ( ord_less_eq_nat @ L @ H )
| ( ord_less_eq_nat @ H @ H2 ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_1174_ivl__disj__un__two__touch_I4_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_or1269000886237332187st_nat @ L @ M ) @ ( set_or1269000886237332187st_nat @ M @ U ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_1175_atLeastAtMost__singleton_H,axiom,
! [A: nat,B2: nat] :
( ( A = B2 )
=> ( ( set_or1269000886237332187st_nat @ A @ B2 )
= ( insert_nat2 @ A @ bot_bot_set_nat ) ) ) ).
% atLeastAtMost_singleton'
thf(fact_1176_all__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M6: nat] :
( ( ord_less_eq_nat @ M6 @ N )
=> ( P @ M6 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
=> ( P @ X3 ) ) ) ) ).
% all_nat_less
thf(fact_1177_ex__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M6: nat] :
( ( ord_less_eq_nat @ M6 @ N )
& ( P @ M6 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
& ( P @ X3 ) ) ) ) ).
% ex_nat_less
thf(fact_1178_atMost__atLeast0,axiom,
( set_ord_atMost_nat
= ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).
% atMost_atLeast0
thf(fact_1179_atLeastatMost__psubset__iff,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B2 ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
= ( ( ~ ( ord_less_eq_nat @ A @ B2 )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B2 @ D )
& ( ( ord_less_nat @ C @ A )
| ( ord_less_nat @ B2 @ D ) ) ) )
& ( ord_less_eq_nat @ C @ D ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_1180_atLeast0__atMost__Suc,axiom,
! [N: nat] :
( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
= ( insert_nat2 @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% atLeast0_atMost_Suc
thf(fact_1181_Icc__eq__insert__lb__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( set_or1269000886237332187st_nat @ M @ N )
= ( insert_nat2 @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ).
% Icc_eq_insert_lb_nat
thf(fact_1182_atLeastAtMostSuc__conv,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) )
= ( insert_nat2 @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ).
% atLeastAtMostSuc_conv
thf(fact_1183_atLeastAtMost__insertL,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( insert_nat2 @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
= ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% atLeastAtMost_insertL
thf(fact_1184_atLeastAtMost__upt,axiom,
( set_or1269000886237332187st_nat
= ( ^ [N2: nat,M6: nat] : ( set_nat2 @ ( upt @ N2 @ ( suc @ M6 ) ) ) ) ) ).
% atLeastAtMost_upt
thf(fact_1185_add__nth_Osimps_I1_J,axiom,
! [A: nat,Xs: list_nat] :
( ( add_nth_nat @ zero_zero_nat @ A @ Xs )
= ( cons_nat @ A @ Xs ) ) ).
% add_nth.simps(1)
thf(fact_1186_ivl__disj__un__one_I4_J,axiom,
! [L: nat,U: nat] :
( ( ord_less_eq_nat @ L @ U )
=> ( ( sup_sup_set_nat @ ( set_ord_lessThan_nat @ L ) @ ( set_or1269000886237332187st_nat @ L @ U ) )
= ( set_ord_atMost_nat @ U ) ) ) ).
% ivl_disj_un_one(4)
thf(fact_1187_add__nth__length,axiom,
! [I: nat,Zs: list_Sum_sum_a_nat,Z: sum_sum_a_nat] :
( ( ord_less_eq_nat @ I @ ( size_s5283204784079214577_a_nat @ Zs ) )
=> ( ( size_s5283204784079214577_a_nat @ ( add_nt4212672348507122516_a_nat @ I @ Z @ Zs ) )
= ( suc @ ( size_s5283204784079214577_a_nat @ Zs ) ) ) ) ).
% add_nth_length
thf(fact_1188_add__nth__length,axiom,
! [I: nat,Zs: list_nat,Z: nat] :
( ( ord_less_eq_nat @ I @ ( size_size_list_nat @ Zs ) )
=> ( ( size_size_list_nat @ ( add_nth_nat @ I @ Z @ Zs ) )
= ( suc @ ( size_size_list_nat @ Zs ) ) ) ) ).
% add_nth_length
thf(fact_1189_rem__nth__add__nth,axiom,
! [I: nat,Zs: list_Sum_sum_a_nat,Z: sum_sum_a_nat] :
( ( ord_less_eq_nat @ I @ ( size_s5283204784079214577_a_nat @ Zs ) )
=> ( ( rem_nt658808235856662061_a_nat @ I @ ( add_nt4212672348507122516_a_nat @ I @ Z @ Zs ) )
= Zs ) ) ).
% rem_nth_add_nth
thf(fact_1190_rem__nth__add__nth,axiom,
! [I: nat,Zs: list_nat,Z: nat] :
( ( ord_less_eq_nat @ I @ ( size_size_list_nat @ Zs ) )
=> ( ( rem_nth_nat @ I @ ( add_nth_nat @ I @ Z @ Zs ) )
= Zs ) ) ).
% rem_nth_add_nth
thf(fact_1191_atLeast1__lessThan__eq__remove0,axiom,
! [N: nat] :
( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N )
= ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( insert_nat2 @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% atLeast1_lessThan_eq_remove0
thf(fact_1192_Gcd__0__iff,axiom,
! [A2: set_nat] :
( ( ( gcd_Gcd_nat @ A2 )
= zero_zero_nat )
= ( ord_less_eq_set_nat @ A2 @ ( insert_nat2 @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% Gcd_0_iff
thf(fact_1193_atLeastLessThan__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I )
& ( ord_less_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_1194_atLeastLessThan__empty,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( set_or4665077453230672383an_nat @ A @ B2 )
= bot_bot_set_nat ) ) ).
% atLeastLessThan_empty
thf(fact_1195_ivl__subset,axiom,
! [I: nat,J: nat,M: nat,N: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ J @ I )
| ( ( ord_less_eq_nat @ M @ I )
& ( ord_less_eq_nat @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_1196_atLeastLessThan__empty__iff2,axiom,
! [A: nat,B2: nat] :
( ( bot_bot_set_nat
= ( set_or4665077453230672383an_nat @ A @ B2 ) )
= ( ~ ( ord_less_nat @ A @ B2 ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_1197_atLeastLessThan__empty__iff,axiom,
! [A: nat,B2: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B2 )
= bot_bot_set_nat )
= ( ~ ( ord_less_nat @ A @ B2 ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_1198_ivl__diff,axiom,
! [I: nat,N: nat,M: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ I @ M ) @ ( set_or4665077453230672383an_nat @ I @ N ) )
= ( set_or4665077453230672383an_nat @ N @ M ) ) ) ).
% ivl_diff
thf(fact_1199_Gcd__empty,axiom,
( ( gcd_Gcd_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Gcd_empty
thf(fact_1200_lessThan__minus__lessThan,axiom,
! [N: nat,M: nat] :
( ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( set_ord_lessThan_nat @ M ) )
= ( set_or4665077453230672383an_nat @ M @ N ) ) ).
% lessThan_minus_lessThan
thf(fact_1201_atLeastLessThan__singleton,axiom,
! [M: nat] :
( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
= ( insert_nat2 @ M @ bot_bot_set_nat ) ) ).
% atLeastLessThan_singleton
thf(fact_1202_ivl__disj__un__two_I7_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_or4665077453230672383an_nat @ L @ M ) @ ( set_or1269000886237332187st_nat @ M @ U ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(7)
thf(fact_1203_atLeastLessThanSuc__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% atLeastLessThanSuc_atLeastAtMost
thf(fact_1204_atLeastLessThan__add__Un,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K ) )
= ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% atLeastLessThan_add_Un
thf(fact_1205_atLeastLessThan__upt,axiom,
( set_or4665077453230672383an_nat
= ( ^ [I2: nat,J3: nat] : ( set_nat2 @ ( upt @ I2 @ J3 ) ) ) ) ).
% atLeastLessThan_upt
thf(fact_1206_ivl__disj__un__one_I2_J,axiom,
! [L: nat,U: nat] :
( ( ord_less_eq_nat @ L @ U )
=> ( ( sup_sup_set_nat @ ( set_ord_lessThan_nat @ L ) @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( set_ord_lessThan_nat @ U ) ) ) ).
% ivl_disj_un_one(2)
thf(fact_1207_atLeastLessThan0,axiom,
! [M: nat] :
( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
= bot_bot_set_nat ) ).
% atLeastLessThan0
thf(fact_1208_atLeastLessThan__eq__iff,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ( set_or4665077453230672383an_nat @ A @ B2 )
= ( set_or4665077453230672383an_nat @ C @ D ) )
= ( ( A = C )
& ( B2 = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_1209_Ico__eq__Ico,axiom,
! [L: nat,H: nat,L2: nat,H2: nat] :
( ( ( set_or4665077453230672383an_nat @ L @ H )
= ( set_or4665077453230672383an_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_nat @ L @ H )
& ~ ( ord_less_nat @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_1210_atLeastLessThan__inj_I1_J,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B2 )
= ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_1211_atLeastLessThan__inj_I2_J,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B2 )
= ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( B2 = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_1212_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M6: nat] :
( ( ord_less_nat @ M6 @ N )
& ( P @ M6 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P @ X3 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_1213_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M6: nat] :
( ( ord_less_nat @ M6 @ N )
=> ( P @ M6 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P @ X3 ) ) ) ) ).
% all_nat_less_eq
thf(fact_1214_lessThan__atLeast0,axiom,
( set_ord_lessThan_nat
= ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% lessThan_atLeast0
thf(fact_1215_ivl__disj__un__two_I3_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_or4665077453230672383an_nat @ L @ M ) @ ( set_or4665077453230672383an_nat @ M @ U ) )
= ( set_or4665077453230672383an_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two(3)
thf(fact_1216_atLeastLessThan__subset__iff,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A @ B2 ) @ ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_eq_nat @ B2 @ A )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B2 @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_1217_atLeast0__lessThan__Suc,axiom,
! [N: nat] :
( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
= ( insert_nat2 @ N @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% atLeast0_lessThan_Suc
thf(fact_1218_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B2 ) @ ( set_or4665077453230672383an_nat @ C @ D ) )
= ( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_nat @ B2 @ D ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_1219_ivl__disj__un__two__touch_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_or1269000886237332187st_nat @ L @ M ) @ ( set_or4665077453230672383an_nat @ M @ U ) )
= ( set_or4665077453230672383an_nat @ L @ U ) ) ) ) ).
% ivl_disj_un_two_touch(2)
thf(fact_1220_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
( set_or4665077453230672383an_nat
= ( ^ [A5: nat,B6: nat] : ( minus_minus_set_nat @ ( set_or1269000886237332187st_nat @ A5 @ B6 ) @ ( insert_nat2 @ B6 @ bot_bot_set_nat ) ) ) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_1221_atLeastLessThanSuc,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_eq_nat @ M @ N )
=> ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
= ( insert_nat2 @ N @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ M @ N )
=> ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
= bot_bot_set_nat ) ) ) ).
% atLeastLessThanSuc
thf(fact_1222_ivl__disj__un__singleton_I6_J,axiom,
! [L: nat,U: nat] :
( ( ord_less_eq_nat @ L @ U )
=> ( ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ L @ U ) @ ( insert_nat2 @ U @ bot_bot_set_nat ) )
= ( set_or1269000886237332187st_nat @ L @ U ) ) ) ).
% ivl_disj_un_singleton(6)
thf(fact_1223_upt__Suc__append,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).
% upt_Suc_append
thf(fact_1224_list_Omap__disc__iff,axiom,
! [F: nat > sum_sum_a_nat,A: list_nat] :
( ( ( map_na823391071729141993_a_nat @ F @ A )
= nil_Sum_sum_a_nat )
= ( A = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_1225_list_Omap__disc__iff,axiom,
! [F: nat > nat,A: list_nat] :
( ( ( map_nat_nat @ F @ A )
= nil_nat )
= ( A = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_1226_Nil__is__map__conv,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat] :
( ( nil_Sum_sum_a_nat
= ( map_na823391071729141993_a_nat @ F @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_1227_Nil__is__map__conv,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( nil_nat
= ( map_nat_nat @ F @ Xs ) )
= ( Xs = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_1228_map__is__Nil__conv,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat] :
( ( ( map_na823391071729141993_a_nat @ F @ Xs )
= nil_Sum_sum_a_nat )
= ( Xs = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_1229_map__is__Nil__conv,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= nil_nat )
= ( Xs = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_1230_append_Oright__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ A @ nil_nat )
= A ) ).
% append.right_neutral
thf(fact_1231_append__Nil2,axiom,
! [Xs: list_nat] :
( ( append_nat @ Xs @ nil_nat )
= Xs ) ).
% append_Nil2
thf(fact_1232_append__self__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Xs )
= ( Ys = nil_nat ) ) ).
% append_self_conv
thf(fact_1233_self__append__conv,axiom,
! [Y: list_nat,Ys: list_nat] :
( ( Y
= ( append_nat @ Y @ Ys ) )
= ( Ys = nil_nat ) ) ).
% self_append_conv
thf(fact_1234_append__self__conv2,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= Ys )
= ( Xs = nil_nat ) ) ).
% append_self_conv2
thf(fact_1235_self__append__conv2,axiom,
! [Y: list_nat,Xs: list_nat] :
( ( Y
= ( append_nat @ Xs @ Y ) )
= ( Xs = nil_nat ) ) ).
% self_append_conv2
thf(fact_1236_Nil__is__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( nil_nat
= ( append_nat @ Xs @ Ys ) )
= ( ( Xs = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_1237_append__is__Nil__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= nil_nat )
= ( ( Xs = nil_nat )
& ( Ys = nil_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_1238_bind__simps_I1_J,axiom,
! [F: nat > list_nat] :
( ( bind_nat_nat @ nil_nat @ F )
= nil_nat ) ).
% bind_simps(1)
thf(fact_1239_List_Oset__empty,axiom,
! [Xs: list_nat] :
( ( ( set_nat2 @ Xs )
= bot_bot_set_nat )
= ( Xs = nil_nat ) ) ).
% List.set_empty
thf(fact_1240_set__empty2,axiom,
! [Xs: list_nat] :
( ( bot_bot_set_nat
= ( set_nat2 @ Xs ) )
= ( Xs = nil_nat ) ) ).
% set_empty2
thf(fact_1241_length__0__conv,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_Sum_sum_a_nat ) ) ).
% length_0_conv
thf(fact_1242_length__0__conv,axiom,
! [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= zero_zero_nat )
= ( Xs = nil_nat ) ) ).
% length_0_conv
thf(fact_1243_append1__eq__conv,axiom,
! [Xs: list_nat,X2: nat,Ys: list_nat,Y: nat] :
( ( ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) )
= ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) )
= ( ( Xs = Ys )
& ( X2 = Y ) ) ) ).
% append1_eq_conv
thf(fact_1244_empty__replicate,axiom,
! [N: nat,X2: nat] :
( ( nil_nat
= ( replicate_nat @ N @ X2 ) )
= ( N = zero_zero_nat ) ) ).
% empty_replicate
thf(fact_1245_replicate__empty,axiom,
! [N: nat,X2: nat] :
( ( ( replicate_nat @ N @ X2 )
= nil_nat )
= ( N = zero_zero_nat ) ) ).
% replicate_empty
thf(fact_1246_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( upt @ I @ J )
= nil_nat ) ) ).
% upt_conv_Nil
thf(fact_1247_insert__Nil,axiom,
! [X2: nat] :
( ( insert_nat @ X2 @ nil_nat )
= ( cons_nat @ X2 @ nil_nat ) ) ).
% insert_Nil
thf(fact_1248_length__greater__0__conv,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s5283204784079214577_a_nat @ Xs ) )
= ( Xs != nil_Sum_sum_a_nat ) ) ).
% length_greater_0_conv
thf(fact_1249_length__greater__0__conv,axiom,
! [Xs: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) )
= ( Xs != nil_nat ) ) ).
% length_greater_0_conv
thf(fact_1250_upt__eq__Nil__conv,axiom,
! [I: nat,J: nat] :
( ( ( upt @ I @ J )
= nil_nat )
= ( ( J = zero_zero_nat )
| ( ord_less_eq_nat @ J @ I ) ) ) ).
% upt_eq_Nil_conv
thf(fact_1251_rem__nth_Osimps_I1_J,axiom,
! [Uu: nat] :
( ( rem_nth_nat @ Uu @ nil_nat )
= nil_nat ) ).
% rem_nth.simps(1)
thf(fact_1252_list_Osimps_I8_J,axiom,
! [F: nat > sum_sum_a_nat] :
( ( map_na823391071729141993_a_nat @ F @ nil_nat )
= nil_Sum_sum_a_nat ) ).
% list.simps(8)
thf(fact_1253_list_Osimps_I8_J,axiom,
! [F: nat > nat] :
( ( map_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_1254_list_Odistinct_I1_J,axiom,
! [X21: nat,X22: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_1255_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X22: list_nat] :
( ( List
= ( cons_nat @ X21 @ X22 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_1256_list_Oexhaust,axiom,
! [Y: list_nat] :
( ( Y != nil_nat )
=> ~ ! [X212: nat,X222: list_nat] :
( Y
!= ( cons_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_1257_min__list_Ocases,axiom,
! [X2: list_nat] :
( ! [X4: nat,Xs2: list_nat] :
( X2
!= ( cons_nat @ X4 @ Xs2 ) )
=> ( X2 = nil_nat ) ) ).
% min_list.cases
thf(fact_1258_neq__Nil__conv,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
= ( ? [Y3: nat,Ys2: list_nat] :
( Xs
= ( cons_nat @ Y3 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_1259_list__induct2_H,axiom,
! [P: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [X4: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X4 @ Xs2 ) @ nil_nat )
=> ( ! [Y2: nat,Ys3: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y2 @ Ys3 ) )
=> ( ! [X4: nat,Xs2: list_nat,Y2: nat,Ys3: list_nat] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_1260_list__nonempty__induct,axiom,
! [Xs: list_nat,P: list_nat > $o] :
( ( Xs != nil_nat )
=> ( ! [X4: nat] : ( P @ ( cons_nat @ X4 @ nil_nat ) )
=> ( ! [X4: nat,Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_nat @ X4 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_1261_in__set__simps_I3_J,axiom,
! [X2: nat] :
~ ( member_nat @ X2 @ ( set_nat2 @ nil_nat ) ) ).
% in_set_simps(3)
thf(fact_1262_append__Nil,axiom,
! [Ys: list_nat] :
( ( append_nat @ nil_nat @ Ys )
= Ys ) ).
% append_Nil
thf(fact_1263_append_Oleft__neutral,axiom,
! [A: list_nat] :
( ( append_nat @ nil_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_1264_eq__Nil__appendI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs = Ys )
=> ( Xs
= ( append_nat @ nil_nat @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_1265_removeAll_Osimps_I1_J,axiom,
! [X2: nat] :
( ( removeAll_nat @ X2 @ nil_nat )
= nil_nat ) ).
% removeAll.simps(1)
thf(fact_1266_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_nat @ N @ nil_nat )
= N ) ).
% gen_length_code(1)
% Helper facts (5)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X2: list_nat,Y: list_nat] :
( ( if_list_nat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
! [X2: list_nat,Y: list_nat] :
( ( if_list_nat @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ad_agr_sets_a_nat @ ( minus_minus_set_nat @ ( set_ord_atMost_nat @ n ) @ ( insert_nat2 @ n @ bot_bot_set_nat ) ) @ ( minus_minus_set_nat @ ( set_ord_atMost_nat @ n ) @ ( insert_nat2 @ n @ bot_bot_set_nat ) ) @ ad @ sigma @ tau ).
%------------------------------------------------------------------------------