TPTP Problem File: SLH0440^1.p
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%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : FO_Theory_Rewriting/0082_FOR_Check/prob_00786_036949__18962980_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1418 ( 557 unt; 161 typ; 0 def)
% Number of atoms : 3202 (1338 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10227 ( 320 ~; 42 |; 136 &;8299 @)
% ( 0 <=>;1430 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 6 avg)
% Number of types : 12 ( 11 usr)
% Number of type conns : 796 ( 796 >; 0 *; 0 +; 0 <<)
% Number of symbols : 153 ( 150 usr; 8 con; 0-3 aty)
% Number of variables : 3428 ( 252 ^;3107 !; 69 ?;3428 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:03:28.350
%------------------------------------------------------------------------------
% Could-be-implicit typings (11)
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
list_list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
set_list_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_J,type,
list_list_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_list_set_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
list_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
list_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (150)
thf(sy_c_Basic__Utils_Ofilter__rev__nth_001t__Nat__Onat,type,
basic_4353017870094810967th_nat: ( nat > $o ) > list_nat > nat > nat ).
thf(sy_c_FOR__Check_Oupd__bruijn,type,
fOR_upd_bruijn: list_nat > list_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
minus_1139252259498527702_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > list_nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
minus_7954133019191499631st_nat: set_list_nat > set_list_nat > set_list_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_List_Obutlast_001t__List__Olist_It__Nat__Onat_J,type,
butlast_list_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Obutlast_001t__Nat__Onat,type,
butlast_nat: list_nat > list_nat ).
thf(sy_c_List_Obutlast_001t__Set__Oset_It__Nat__Onat_J,type,
butlast_set_nat: list_set_nat > list_set_nat ).
thf(sy_c_List_Ocoset_001t__List__Olist_It__Nat__Onat_J,type,
coset_list_nat: list_list_nat > set_list_nat ).
thf(sy_c_List_Ocoset_001t__Nat__Onat,type,
coset_nat: list_nat > set_nat ).
thf(sy_c_List_Ofilter_001t__List__Olist_It__Nat__Onat_J,type,
filter_list_nat: ( list_nat > $o ) > list_list_nat > list_list_nat ).
thf(sy_c_List_Ofilter_001t__Nat__Onat,type,
filter_nat: ( nat > $o ) > list_nat > list_nat ).
thf(sy_c_List_Ogen__length_001t__List__Olist_It__Nat__Onat_J,type,
gen_length_list_nat: nat > list_list_nat > nat ).
thf(sy_c_List_Ogen__length_001t__Nat__Onat,type,
gen_length_nat: nat > list_nat > nat ).
thf(sy_c_List_Ogen__length_001t__Set__Oset_It__Nat__Onat_J,type,
gen_length_set_nat: nat > list_set_nat > nat ).
thf(sy_c_List_Olast_001t__List__Olist_It__Nat__Onat_J,type,
last_list_nat: list_list_nat > list_nat ).
thf(sy_c_List_Olast_001t__Nat__Onat,type,
last_nat: list_nat > nat ).
thf(sy_c_List_Olast_001t__Set__Oset_It__Nat__Onat_J,type,
last_set_nat: list_set_nat > set_nat ).
thf(sy_c_List_Olinorder__class_Oinsort__insert__key_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
linord8148156736319587652st_nat: ( list_nat > list_nat ) > list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Olinorder__class_Oinsort__insert__key_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
linord3253225449353161780at_nat: ( list_nat > nat ) > list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Olinorder__class_Oinsort__insert__key_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
linord1014268402190876596st_nat: ( nat > list_nat ) > nat > list_nat > list_nat ).
thf(sy_c_List_Olinorder__class_Oinsort__insert__key_001t__Nat__Onat_001t__Nat__Onat,type,
linord1921536354676448932at_nat: ( nat > nat ) > nat > list_nat > list_nat ).
thf(sy_c_List_Olinorder__class_Oinsort__key_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
linord3838877917356005949st_nat: ( list_nat > list_nat ) > list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Olinorder__class_Oinsort__key_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
linord5978504541935096237at_nat: ( list_nat > nat ) > list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Olinorder__class_Oinsort__key_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
linord3739547494772811053st_nat: ( nat > list_nat ) > nat > list_nat > list_nat ).
thf(sy_c_List_Olinorder__class_Oinsort__key_001t__Nat__Onat_001t__Nat__Onat,type,
linord8961336180081300637at_nat: ( nat > nat ) > nat > list_nat > list_nat ).
thf(sy_c_List_Olinorder__class_Osort__key_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
linord1364705781165306818st_nat: ( list_nat > list_nat ) > list_list_nat > list_list_nat ).
thf(sy_c_List_Olinorder__class_Osort__key_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
linord770824052286894514at_nat: ( list_nat > nat ) > list_list_nat > list_list_nat ).
thf(sy_c_List_Olinorder__class_Osort__key_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
linord7755239041979385138st_nat: ( nat > list_nat ) > list_nat > list_nat ).
thf(sy_c_List_Olinorder__class_Osort__key_001t__Nat__Onat_001t__Nat__Onat,type,
linord738340561235409698at_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olinorder__class_Ostable__sort__key_001t__Nat__Onat_001t__Nat__Onat,type,
linord227665693835759911at_nat: ( ( nat > nat ) > list_nat > list_nat ) > $o ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Nat__Onat_J,type,
cons_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Nat__Onat_J,type,
nil_list_nat: list_list_nat ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001t__Set__Oset_It__Nat__Onat_J,type,
nil_set_nat: list_set_nat ).
thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
hd_nat: list_nat > nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
map_li7225945977422193158st_nat: ( list_nat > list_nat ) > list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
map_list_nat_nat: ( list_nat > nat ) > list_list_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
map_list_nat_set_nat: ( list_nat > set_nat ) > list_list_nat > list_set_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
map_nat_list_nat: ( nat > list_nat ) > list_nat > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
map_nat_set_nat: ( nat > set_nat ) > list_nat > list_set_nat ).
thf(sy_c_List_Olist_Omap_001t__Set__Oset_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
map_set_nat_list_nat: ( set_nat > list_nat ) > list_set_nat > list_list_nat ).
thf(sy_c_List_Olist_Omap_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
map_set_nat_nat: ( set_nat > nat ) > list_set_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
map_set_nat_set_nat: ( set_nat > set_nat ) > list_set_nat > list_set_nat ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_list_nat2: list_list_list_nat > set_list_list_nat ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
set_list_nat2: list_list_nat > set_list_nat ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
set_list_set_nat2: list_list_set_nat > set_list_set_nat ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
set_set_nat2: list_set_nat > set_set_nat ).
thf(sy_c_List_Olist_Osize__list_001t__List__Olist_It__Nat__Onat_J,type,
size_list_list_nat: ( list_nat > nat ) > list_list_nat > nat ).
thf(sy_c_List_Olist_Osize__list_001t__Nat__Onat,type,
size_list_nat: ( nat > nat ) > list_nat > nat ).
thf(sy_c_List_Olist_Otl_001t__List__Olist_It__Nat__Onat_J,type,
tl_list_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
tl_nat: list_nat > list_nat ).
thf(sy_c_List_Olist_Otl_001t__Set__Oset_It__Nat__Onat_J,type,
tl_set_nat: list_set_nat > list_set_nat ).
thf(sy_c_List_Olist__update_001t__List__Olist_It__Nat__Onat_J,type,
list_update_list_nat: list_list_nat > nat > list_nat > list_list_nat ).
thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
list_update_nat: list_nat > nat > nat > list_nat ).
thf(sy_c_List_Olist__update_001t__Set__Oset_It__Nat__Onat_J,type,
list_update_set_nat: list_set_nat > nat > set_nat > list_set_nat ).
thf(sy_c_List_Olistset_001t__Nat__Onat,type,
listset_nat: list_set_nat > set_list_nat ).
thf(sy_c_List_Omap__tailrec_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
map_ta9145449198693458768st_nat: ( list_nat > list_nat ) > list_list_nat > list_list_nat ).
thf(sy_c_List_Omap__tailrec_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
map_ta354985537753071680at_nat: ( list_nat > nat ) > list_list_nat > list_nat ).
thf(sy_c_List_Omap__tailrec_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
map_ta7339400527445562304st_nat: ( nat > list_nat ) > list_nat > list_list_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_On__lists_001t__List__Olist_It__Nat__Onat_J,type,
n_lists_list_nat: nat > list_list_nat > list_list_list_nat ).
thf(sy_c_List_On__lists_001t__Nat__Onat,type,
n_lists_nat: nat > list_nat > list_list_nat ).
thf(sy_c_List_On__lists_001t__Set__Oset_It__Nat__Onat_J,type,
n_lists_set_nat: nat > list_set_nat > list_list_set_nat ).
thf(sy_c_List_Onth_001t__List__Olist_It__Nat__Onat_J,type,
nth_list_nat: list_list_nat > nat > list_nat ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001t__Set__Oset_It__Nat__Onat_J,type,
nth_set_nat: list_set_nat > nat > set_nat ).
thf(sy_c_List_Oproduct__lists_001t__Nat__Onat,type,
product_lists_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Oremove1_001t__List__Olist_It__Nat__Onat_J,type,
remove1_list_nat: list_nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Oremove1_001t__Nat__Onat,type,
remove1_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Oremove1_001t__Set__Oset_It__Nat__Onat_J,type,
remove1_set_nat: set_nat > list_set_nat > list_set_nat ).
thf(sy_c_List_Oreplicate_001t__List__Olist_It__Nat__Onat_J,type,
replicate_list_nat: nat > list_nat > list_list_nat ).
thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
replicate_nat: nat > nat > list_nat ).
thf(sy_c_List_Oreplicate_001t__Set__Oset_It__Nat__Onat_J,type,
replicate_set_nat: nat > set_nat > list_set_nat ).
thf(sy_c_List_Orev_001t__Nat__Onat,type,
rev_nat: list_nat > list_nat ).
thf(sy_c_List_Orotate1_001t__List__Olist_It__Nat__Onat_J,type,
rotate1_list_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Orotate1_001t__Nat__Onat,type,
rotate1_nat: list_nat > list_nat ).
thf(sy_c_List_Orotate1_001t__Set__Oset_It__Nat__Onat_J,type,
rotate1_set_nat: list_set_nat > list_set_nat ).
thf(sy_c_List_Orotate_001t__List__Olist_It__Nat__Onat_J,type,
rotate_list_nat: nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Orotate_001t__Nat__Onat,type,
rotate_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Orotate_001t__Set__Oset_It__Nat__Onat_J,type,
rotate_set_nat: nat > list_set_nat > list_set_nat ).
thf(sy_c_List_Osorted__wrt_001t__List__Olist_It__Nat__Onat_J,type,
sorted_wrt_list_nat: ( list_nat > list_nat > $o ) > list_list_nat > $o ).
thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
thf(sy_c_List_Osorted__wrt_001t__Set__Oset_It__Nat__Onat_J,type,
sorted_wrt_set_nat: ( set_nat > set_nat > $o ) > list_set_nat > $o ).
thf(sy_c_List_OtakeWhile_001t__List__Olist_It__Nat__Onat_J,type,
takeWhile_list_nat: ( list_nat > $o ) > list_list_nat > list_list_nat ).
thf(sy_c_List_OtakeWhile_001t__Nat__Onat,type,
takeWhile_nat: ( nat > $o ) > list_nat > list_nat ).
thf(sy_c_List_Otake_001t__List__Olist_It__Nat__Onat_J,type,
take_list_nat: nat > list_list_nat > list_list_nat ).
thf(sy_c_List_Otake_001t__Nat__Onat,type,
take_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Otake_001t__Set__Oset_It__Nat__Onat_J,type,
take_set_nat: nat > list_set_nat > list_set_nat ).
thf(sy_c_List_Otranspose_001t__Nat__Onat,type,
transpose_nat: list_list_nat > list_list_nat ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > list_nat ).
thf(sy_c_Missing__List_Oadjust__idx,type,
missing_adjust_idx: nat > nat > nat ).
thf(sy_c_Missing__List_Oadjust__idx__rev,type,
missin3815256168798769645dx_rev: nat > nat > nat ).
thf(sy_c_Missing__List_Oconcat__lists_001t__Nat__Onat,type,
missin4567272213201432058ts_nat: list_list_nat > list_list_nat ).
thf(sy_c_Missing__List_Ogenerate__lists_001t__List__Olist_It__Nat__Onat_J,type,
missin2779806319917622601st_nat: nat > list_list_nat > list_list_list_nat ).
thf(sy_c_Missing__List_Ogenerate__lists_001t__Nat__Onat,type,
missin2047014633743487673ts_nat: nat > list_nat > list_list_nat ).
thf(sy_c_Missing__List_Ogenerate__lists_001t__Set__Oset_It__Nat__Onat_J,type,
missin5440160964871002735et_nat: nat > list_set_nat > list_list_set_nat ).
thf(sy_c_Missing__List_Olist__diff_001t__List__Olist_It__Nat__Onat_J,type,
missin6169663638126051804st_nat: list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_Missing__List_Olist__diff_001t__Nat__Onat,type,
missin818507234016924876ff_nat: list_nat > list_nat > list_nat ).
thf(sy_c_Missing__List_Omin__list_001t__Nat__Onat,type,
missing_min_list_nat: list_nat > nat ).
thf(sy_c_Missing__List_Opermut_001t__Nat__Onat,type,
missing_permut_nat: list_nat > ( nat > nat ) > list_nat ).
thf(sy_c_Missing__List_Opermut__aux_001t__Nat__Onat,type,
missin1888654203714970382ux_nat: list_nat > ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_Missing__List_Oremdups__sort_001t__List__Olist_It__Nat__Onat_J,type,
missin2640887136080315381st_nat: list_list_nat > list_list_nat ).
thf(sy_c_Missing__List_Oremdups__sort_001t__Nat__Onat,type,
missin6101193410121742181rt_nat: list_nat > list_nat ).
thf(sy_c_Missing__List_Oremove__nth_001t__Nat__Onat,type,
missin7175274867594579095th_nat: nat > list_nat > list_nat ).
thf(sy_c_Missing__List_Osubtract__list__sorted_001t__List__Olist_It__Nat__Onat_J,type,
missin8418467895465290280st_nat: list_list_nat > list_list_nat > list_list_nat ).
thf(sy_c_Missing__List_Osubtract__list__sorted_001t__Nat__Onat,type,
missin6424796737333596952ed_nat: list_nat > list_nat > list_nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
size_s3023201423986296836st_nat: list_list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
size_s3254054031482475050et_nat: list_set_nat > nat ).
thf(sy_c_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat,type,
ord_Least_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__List__Olist_It__Nat__Onat_J,type,
ord_less_list_nat: list_nat > list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
ord_le1520216061033275535_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
ord_le6806709344281226192st_nat: list_list_nat > list_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__List__Olist_It__Nat__Onat_J,type,
ord_less_eq_list_nat: list_nat > list_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__List__Olist_It__Nat__Onat_J_J,type,
ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__List__Olist_It__Nat__Onat_J,type,
order_691136778212554055st_nat: ( list_nat > $o ) > list_nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Set__Oset_It__Nat__Onat_J,type,
order_5724808138429204845et_nat: ( set_nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
collec5989764272469232197st_nat: ( list_list_nat > $o ) > set_list_list_nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
collect_list_set_nat: ( list_set_nat > $o ) > set_list_set_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set__Impl_Oord__class_Oquicksort_001t__List__Olist_It__Nat__Onat_J,type,
set_or6919174964970218831st_nat: list_list_nat > list_list_nat ).
thf(sy_c_Set__Impl_Oord__class_Oquicksort_001t__Nat__Onat,type,
set_or9089632773640736191rt_nat: list_nat > list_nat ).
thf(sy_c_Set__Impl_Oord__class_Oremdups__sorted_001t__List__Olist_It__Nat__Onat_J,type,
set_or4609908739687369647st_nat: list_list_nat > list_list_nat ).
thf(sy_c_Set__Impl_Oord__class_Oremdups__sorted_001t__Nat__Onat,type,
set_or6599480164596245535ed_nat: list_nat > list_nat ).
thf(sy_c_Utils_Otrancl__listp_001t__Nat__Onat,type,
trancl_listp_nat: ( nat > nat > $o ) > list_nat > list_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
member_list_list_nat: list_list_nat > set_list_list_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
member_list_set_nat: list_set_nat > set_list_set_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_v_xs,type,
xs: list_nat ).
% Relevant facts (1252)
thf(fact_0_upd__bruijn__def,axiom,
( fOR_upd_bruijn
= ( ^ [Vs: list_nat] :
( tl_nat
@ ( map_nat_nat
@ ^ [X: nat] : ( minus_minus_nat @ X @ one_one_nat )
@ Vs ) ) ) ) ).
% upd_bruijn_def
thf(fact_1_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_2_map__ident,axiom,
( ( map_li7225945977422193158st_nat
@ ^ [X: list_nat] : X )
= ( ^ [Xs: list_list_nat] : Xs ) ) ).
% map_ident
thf(fact_3_map__ident,axiom,
( ( map_nat_nat
@ ^ [X: nat] : X )
= ( ^ [Xs: list_nat] : Xs ) ) ).
% map_ident
thf(fact_4_sorted__map,axiom,
! [F: list_nat > nat,Xs2: list_list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_list_nat_nat @ F @ Xs2 ) )
= ( sorted_wrt_list_nat
@ ^ [X: list_nat,Y: list_nat] : ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
@ Xs2 ) ) ).
% sorted_map
thf(fact_5_sorted__map,axiom,
! [F: list_nat > list_nat,Xs2: list_list_nat] :
( ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ ( map_li7225945977422193158st_nat @ F @ Xs2 ) )
= ( sorted_wrt_list_nat
@ ^ [X: list_nat,Y: list_nat] : ( ord_less_eq_list_nat @ ( F @ X ) @ ( F @ Y ) )
@ Xs2 ) ) ).
% sorted_map
thf(fact_6_sorted__map,axiom,
! [F: nat > list_nat,Xs2: list_nat] :
( ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ ( map_nat_list_nat @ F @ Xs2 ) )
= ( sorted_wrt_nat
@ ^ [X: nat,Y: nat] : ( ord_less_eq_list_nat @ ( F @ X ) @ ( F @ Y ) )
@ Xs2 ) ) ).
% sorted_map
thf(fact_7_sorted__map,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ Xs2 ) )
= ( sorted_wrt_nat
@ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
@ Xs2 ) ) ).
% sorted_map
thf(fact_8_sorted__tl,axiom,
! [Xs2: list_list_nat] :
( ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ Xs2 )
=> ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ ( tl_list_nat @ Xs2 ) ) ) ).
% sorted_tl
thf(fact_9_sorted__tl,axiom,
! [Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( tl_nat @ Xs2 ) ) ) ).
% sorted_tl
thf(fact_10_sorted__wrt__map,axiom,
! [R: list_nat > list_nat > $o,F: list_nat > list_nat,Xs2: list_list_nat] :
( ( sorted_wrt_list_nat @ R @ ( map_li7225945977422193158st_nat @ F @ Xs2 ) )
= ( sorted_wrt_list_nat
@ ^ [X: list_nat,Y: list_nat] : ( R @ ( F @ X ) @ ( F @ Y ) )
@ Xs2 ) ) ).
% sorted_wrt_map
thf(fact_11_sorted__wrt__map,axiom,
! [R: list_nat > list_nat > $o,F: nat > list_nat,Xs2: list_nat] :
( ( sorted_wrt_list_nat @ R @ ( map_nat_list_nat @ F @ Xs2 ) )
= ( sorted_wrt_nat
@ ^ [X: nat,Y: nat] : ( R @ ( F @ X ) @ ( F @ Y ) )
@ Xs2 ) ) ).
% sorted_wrt_map
thf(fact_12_sorted__wrt__map,axiom,
! [R: nat > nat > $o,F: list_nat > nat,Xs2: list_list_nat] :
( ( sorted_wrt_nat @ R @ ( map_list_nat_nat @ F @ Xs2 ) )
= ( sorted_wrt_list_nat
@ ^ [X: list_nat,Y: list_nat] : ( R @ ( F @ X ) @ ( F @ Y ) )
@ Xs2 ) ) ).
% sorted_wrt_map
thf(fact_13_sorted__wrt__map,axiom,
! [R: nat > nat > $o,F: nat > nat,Xs2: list_nat] :
( ( sorted_wrt_nat @ R @ ( map_nat_nat @ F @ Xs2 ) )
= ( sorted_wrt_nat
@ ^ [X: nat,Y: nat] : ( R @ ( F @ X ) @ ( F @ Y ) )
@ Xs2 ) ) ).
% sorted_wrt_map
thf(fact_14_order__refl,axiom,
! [X2: set_nat] : ( ord_less_eq_set_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_15_order__refl,axiom,
! [X2: list_nat] : ( ord_less_eq_list_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_16_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_17_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_18_dual__order_Orefl,axiom,
! [A: list_nat] : ( ord_less_eq_list_nat @ A @ A ) ).
% dual_order.refl
thf(fact_19_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_20_map__tl,axiom,
! [F: nat > list_nat,Xs2: list_nat] :
( ( map_nat_list_nat @ F @ ( tl_nat @ Xs2 ) )
= ( tl_list_nat @ ( map_nat_list_nat @ F @ Xs2 ) ) ) ).
% map_tl
thf(fact_21_map__tl,axiom,
! [F: list_nat > list_nat,Xs2: list_list_nat] :
( ( map_li7225945977422193158st_nat @ F @ ( tl_list_nat @ Xs2 ) )
= ( tl_list_nat @ ( map_li7225945977422193158st_nat @ F @ Xs2 ) ) ) ).
% map_tl
thf(fact_22_map__tl,axiom,
! [F: list_nat > nat,Xs2: list_list_nat] :
( ( map_list_nat_nat @ F @ ( tl_list_nat @ Xs2 ) )
= ( tl_nat @ ( map_list_nat_nat @ F @ Xs2 ) ) ) ).
% map_tl
thf(fact_23_map__tl,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( map_nat_nat @ F @ ( tl_nat @ Xs2 ) )
= ( tl_nat @ ( map_nat_nat @ F @ Xs2 ) ) ) ).
% map_tl
thf(fact_24_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_25_order__antisym__conv,axiom,
! [Y2: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_26_order__antisym__conv,axiom,
! [Y2: list_nat,X2: list_nat] :
( ( ord_less_eq_list_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_list_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_27_order__antisym__conv,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_28_linorder__le__cases,axiom,
! [X2: list_nat,Y2: list_nat] :
( ~ ( ord_less_eq_list_nat @ X2 @ Y2 )
=> ( ord_less_eq_list_nat @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_29_linorder__le__cases,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_30_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_31_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > list_nat,C: list_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_list_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_list_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_32_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_33_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_34_ord__le__eq__subst,axiom,
! [A: set_nat,B: set_nat,F: set_nat > list_nat,C: list_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_less_eq_list_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_list_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_35_ord__le__eq__subst,axiom,
! [A: list_nat,B: list_nat,F: list_nat > nat,C: nat] :
( ( ord_less_eq_list_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: list_nat,Y3: list_nat] :
( ( ord_less_eq_list_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_36_ord__le__eq__subst,axiom,
! [A: list_nat,B: list_nat,F: list_nat > set_nat,C: set_nat] :
( ( ord_less_eq_list_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: list_nat,Y3: list_nat] :
( ( ord_less_eq_list_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_37_ord__le__eq__subst,axiom,
! [A: list_nat,B: list_nat,F: list_nat > list_nat,C: list_nat] :
( ( ord_less_eq_list_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: list_nat,Y3: list_nat] :
( ( ord_less_eq_list_nat @ X3 @ Y3 )
=> ( ord_less_eq_list_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_list_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_38_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_39_ord__eq__le__subst,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_40_ord__eq__le__subst,axiom,
! [A: list_nat,F: nat > list_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_list_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_list_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_41_ord__eq__le__subst,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_42_ord__eq__le__subst,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_43_ord__eq__le__subst,axiom,
! [A: list_nat,F: set_nat > list_nat,B: set_nat,C: set_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_less_eq_list_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_list_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_44_ord__eq__le__subst,axiom,
! [A: nat,F: list_nat > nat,B: list_nat,C: list_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_list_nat @ B @ C )
=> ( ! [X3: list_nat,Y3: list_nat] :
( ( ord_less_eq_list_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_45_ord__eq__le__subst,axiom,
! [A: set_nat,F: list_nat > set_nat,B: list_nat,C: list_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_list_nat @ B @ C )
=> ( ! [X3: list_nat,Y3: list_nat] :
( ( ord_less_eq_list_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_46_ord__eq__le__subst,axiom,
! [A: list_nat,F: list_nat > list_nat,B: list_nat,C: list_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_list_nat @ B @ C )
=> ( ! [X3: list_nat,Y3: list_nat] :
( ( ord_less_eq_list_nat @ X3 @ Y3 )
=> ( ord_less_eq_list_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_list_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_47_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_48_linorder__linear,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( ord_less_eq_list_nat @ X2 @ Y2 )
| ( ord_less_eq_list_nat @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_49_linorder__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_50_order__eq__refl,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( X2 = Y2 )
=> ( ord_less_eq_set_nat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_51_order__eq__refl,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( X2 = Y2 )
=> ( ord_less_eq_list_nat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_52_order__eq__refl,axiom,
! [X2: nat,Y2: nat] :
( ( X2 = Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_53_order__subst2,axiom,
! [A: nat,B: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_54_order__subst2,axiom,
! [A: nat,B: nat,F: nat > list_nat,C: list_nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_list_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_list_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_list_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_55_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_56_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_57_order__subst2,axiom,
! [A: set_nat,B: set_nat,F: set_nat > list_nat,C: list_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_list_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_less_eq_list_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_list_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_58_order__subst2,axiom,
! [A: list_nat,B: list_nat,F: list_nat > nat,C: nat] :
( ( ord_less_eq_list_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: list_nat,Y3: list_nat] :
( ( ord_less_eq_list_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_59_order__subst2,axiom,
! [A: list_nat,B: list_nat,F: list_nat > set_nat,C: set_nat] :
( ( ord_less_eq_list_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ ( F @ B ) @ C )
=> ( ! [X3: list_nat,Y3: list_nat] :
( ( ord_less_eq_list_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_60_order__subst2,axiom,
! [A: list_nat,B: list_nat,F: list_nat > list_nat,C: list_nat] :
( ( ord_less_eq_list_nat @ A @ B )
=> ( ( ord_less_eq_list_nat @ ( F @ B ) @ C )
=> ( ! [X3: list_nat,Y3: list_nat] :
( ( ord_less_eq_list_nat @ X3 @ Y3 )
=> ( ord_less_eq_list_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_list_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_61_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_62_order__subst1,axiom,
! [A: nat,F: set_nat > nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_63_order__subst1,axiom,
! [A: nat,F: list_nat > nat,B: list_nat,C: list_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_list_nat @ B @ C )
=> ( ! [X3: list_nat,Y3: list_nat] :
( ( ord_less_eq_list_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_64_order__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B: nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_65_order__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_66_order__subst1,axiom,
! [A: set_nat,F: list_nat > set_nat,B: list_nat,C: list_nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_list_nat @ B @ C )
=> ( ! [X3: list_nat,Y3: list_nat] :
( ( ord_less_eq_list_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_67_order__subst1,axiom,
! [A: list_nat,F: nat > list_nat,B: nat,C: nat] :
( ( ord_less_eq_list_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_list_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_list_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_68_order__subst1,axiom,
! [A: list_nat,F: set_nat > list_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_list_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ord_less_eq_list_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_list_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_69_order__subst1,axiom,
! [A: list_nat,F: list_nat > list_nat,B: list_nat,C: list_nat] :
( ( ord_less_eq_list_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_list_nat @ B @ C )
=> ( ! [X3: list_nat,Y3: list_nat] :
( ( ord_less_eq_list_nat @ X3 @ Y3 )
=> ( ord_less_eq_list_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_list_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_70_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_71_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
& ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_72_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: list_nat,Z: list_nat] : ( Y4 = Z ) )
= ( ^ [A2: list_nat,B2: list_nat] :
( ( ord_less_eq_list_nat @ A2 @ B2 )
& ( ord_less_eq_list_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_73_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_74_antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_75_antisym,axiom,
! [A: list_nat,B: list_nat] :
( ( ord_less_eq_list_nat @ A @ B )
=> ( ( ord_less_eq_list_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_76_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_77_dual__order_Otrans,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_78_dual__order_Otrans,axiom,
! [B: list_nat,A: list_nat,C: list_nat] :
( ( ord_less_eq_list_nat @ B @ A )
=> ( ( ord_less_eq_list_nat @ C @ B )
=> ( ord_less_eq_list_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_79_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_80_dual__order_Oantisym,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_81_dual__order_Oantisym,axiom,
! [B: list_nat,A: list_nat] :
( ( ord_less_eq_list_nat @ B @ A )
=> ( ( ord_less_eq_list_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_82_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_83_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
& ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_84_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: list_nat,Z: list_nat] : ( Y4 = Z ) )
= ( ^ [A2: list_nat,B2: list_nat] :
( ( ord_less_eq_list_nat @ B2 @ A2 )
& ( ord_less_eq_list_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_85_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_86_linorder__wlog,axiom,
! [P: list_nat > list_nat > $o,A: list_nat,B: list_nat] :
( ! [A3: list_nat,B3: list_nat] :
( ( ord_less_eq_list_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: list_nat,B3: list_nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_87_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_88_order__trans,axiom,
! [X2: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ Z2 )
=> ( ord_less_eq_set_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_89_order__trans,axiom,
! [X2: list_nat,Y2: list_nat,Z2: list_nat] :
( ( ord_less_eq_list_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_list_nat @ Y2 @ Z2 )
=> ( ord_less_eq_list_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_90_order__trans,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_91_order_Otrans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_92_order_Otrans,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( ord_less_eq_list_nat @ A @ B )
=> ( ( ord_less_eq_list_nat @ B @ C )
=> ( ord_less_eq_list_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_93_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_94_order__antisym,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_95_order__antisym,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( ord_less_eq_list_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_list_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_96_order__antisym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_97_ord__le__eq__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_98_ord__le__eq__trans,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( ord_less_eq_list_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_list_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_99_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_100_ord__eq__le__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( A = B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_101_ord__eq__le__trans,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( A = B )
=> ( ( ord_less_eq_list_nat @ B @ C )
=> ( ord_less_eq_list_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_102_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_103_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
& ( ord_less_eq_set_nat @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_104_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: list_nat,Z: list_nat] : ( Y4 = Z ) )
= ( ^ [X: list_nat,Y: list_nat] :
( ( ord_less_eq_list_nat @ X @ Y )
& ( ord_less_eq_list_nat @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_105_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_106_le__cases3,axiom,
! [X2: list_nat,Y2: list_nat,Z2: list_nat] :
( ( ( ord_less_eq_list_nat @ X2 @ Y2 )
=> ~ ( ord_less_eq_list_nat @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_list_nat @ Y2 @ X2 )
=> ~ ( ord_less_eq_list_nat @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_list_nat @ X2 @ Z2 )
=> ~ ( ord_less_eq_list_nat @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_list_nat @ Z2 @ Y2 )
=> ~ ( ord_less_eq_list_nat @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_list_nat @ Y2 @ Z2 )
=> ~ ( ord_less_eq_list_nat @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_list_nat @ Z2 @ X2 )
=> ~ ( ord_less_eq_list_nat @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_107_le__cases3,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_108_nle__le,axiom,
! [A: list_nat,B: list_nat] :
( ( ~ ( ord_less_eq_list_nat @ A @ B ) )
= ( ( ord_less_eq_list_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_109_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_110_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_111_diff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_112_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_113_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_114_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_115_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_116_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_117_Collect__mem__eq,axiom,
! [A4: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_118_Collect__mem__eq,axiom,
! [A4: set_list_nat] :
( ( collect_list_nat
@ ^ [X: list_nat] : ( member_list_nat @ X @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_119_Collect__cong,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ! [X3: list_nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_list_nat @ P )
= ( collect_list_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_120_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_121_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_122_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_123_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_124_list_Omap__ident,axiom,
! [T: list_list_nat] :
( ( map_li7225945977422193158st_nat
@ ^ [X: list_nat] : X
@ T )
= T ) ).
% list.map_ident
thf(fact_125_list_Omap__ident,axiom,
! [T: list_nat] :
( ( map_nat_nat
@ ^ [X: nat] : X
@ T )
= T ) ).
% list.map_ident
thf(fact_126_sorted__wrt__true,axiom,
! [Xs2: list_nat] :
( sorted_wrt_nat
@ ^ [Uu: nat,Uv: nat] : $true
@ Xs2 ) ).
% sorted_wrt_true
thf(fact_127_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_128_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_129_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_130_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_131_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_132_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_133_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_134_sorted__insort__insert__key,axiom,
! [F: list_nat > nat,Xs2: list_list_nat,X2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_list_nat_nat @ F @ Xs2 ) )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_list_nat_nat @ F @ ( linord3253225449353161780at_nat @ F @ X2 @ Xs2 ) ) ) ) ).
% sorted_insort_insert_key
thf(fact_135_sorted__insort__insert__key,axiom,
! [F: nat > list_nat,Xs2: list_nat,X2: nat] :
( ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ ( map_nat_list_nat @ F @ Xs2 ) )
=> ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ ( map_nat_list_nat @ F @ ( linord1014268402190876596st_nat @ F @ X2 @ Xs2 ) ) ) ) ).
% sorted_insort_insert_key
thf(fact_136_sorted__insort__insert__key,axiom,
! [F: list_nat > list_nat,Xs2: list_list_nat,X2: list_nat] :
( ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ ( map_li7225945977422193158st_nat @ F @ Xs2 ) )
=> ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ ( map_li7225945977422193158st_nat @ F @ ( linord8148156736319587652st_nat @ F @ X2 @ Xs2 ) ) ) ) ).
% sorted_insort_insert_key
thf(fact_137_sorted__insort__insert__key,axiom,
! [F: nat > nat,Xs2: list_nat,X2: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ Xs2 ) )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ ( linord1921536354676448932at_nat @ F @ X2 @ Xs2 ) ) ) ) ).
% sorted_insort_insert_key
thf(fact_138_upd__bruijn__length,axiom,
! [Vs2: list_nat] :
( ( size_size_list_nat @ ( fOR_upd_bruijn @ Vs2 ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Vs2 ) @ one_one_nat ) ) ).
% upd_bruijn_length
thf(fact_139_map__eq__map__tailrec,axiom,
map_nat_list_nat = map_ta7339400527445562304st_nat ).
% map_eq_map_tailrec
thf(fact_140_map__eq__map__tailrec,axiom,
map_li7225945977422193158st_nat = map_ta9145449198693458768st_nat ).
% map_eq_map_tailrec
thf(fact_141_map__eq__map__tailrec,axiom,
map_list_nat_nat = map_ta354985537753071680at_nat ).
% map_eq_map_tailrec
thf(fact_142_map__eq__map__tailrec,axiom,
map_nat_nat = map_tailrec_nat_nat ).
% map_eq_map_tailrec
thf(fact_143_sorted__insort__insert,axiom,
! [Xs2: list_list_nat,X2: list_nat] :
( ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ Xs2 )
=> ( sorted_wrt_list_nat @ ord_less_eq_list_nat
@ ( linord8148156736319587652st_nat
@ ^ [X: list_nat] : X
@ X2
@ Xs2 ) ) ) ).
% sorted_insort_insert
thf(fact_144_sorted__insort__insert,axiom,
! [Xs2: list_nat,X2: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
=> ( sorted_wrt_nat @ ord_less_eq_nat
@ ( linord1921536354676448932at_nat
@ ^ [X: nat] : X
@ X2
@ Xs2 ) ) ) ).
% sorted_insort_insert
thf(fact_145_sorted01,axiom,
! [Xs2: list_list_nat] :
( ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ one_one_nat )
=> ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ Xs2 ) ) ).
% sorted01
thf(fact_146_sorted01,axiom,
! [Xs2: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 ) ) ).
% sorted01
thf(fact_147_length__tl,axiom,
! [Xs2: list_list_nat] :
( ( size_s3023201423986296836st_nat @ ( tl_list_nat @ Xs2 ) )
= ( minus_minus_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ one_one_nat ) ) ).
% length_tl
thf(fact_148_length__tl,axiom,
! [Xs2: list_set_nat] :
( ( size_s3254054031482475050et_nat @ ( tl_set_nat @ Xs2 ) )
= ( minus_minus_nat @ ( size_s3254054031482475050et_nat @ Xs2 ) @ one_one_nat ) ) ).
% length_tl
thf(fact_149_length__tl,axiom,
! [Xs2: list_nat] :
( ( size_size_list_nat @ ( tl_nat @ Xs2 ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) ) ).
% length_tl
thf(fact_150_tl__replicate,axiom,
! [N: nat,X2: nat] :
( ( tl_nat @ ( replicate_nat @ N @ X2 ) )
= ( replicate_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ X2 ) ) ).
% tl_replicate
thf(fact_151_sorted__map__remove1,axiom,
! [F: list_nat > nat,Xs2: list_list_nat,X2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_list_nat_nat @ F @ Xs2 ) )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_list_nat_nat @ F @ ( remove1_list_nat @ X2 @ Xs2 ) ) ) ) ).
% sorted_map_remove1
thf(fact_152_sorted__map__remove1,axiom,
! [F: nat > list_nat,Xs2: list_nat,X2: nat] :
( ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ ( map_nat_list_nat @ F @ Xs2 ) )
=> ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ ( map_nat_list_nat @ F @ ( remove1_nat @ X2 @ Xs2 ) ) ) ) ).
% sorted_map_remove1
thf(fact_153_sorted__map__remove1,axiom,
! [F: list_nat > list_nat,Xs2: list_list_nat,X2: list_nat] :
( ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ ( map_li7225945977422193158st_nat @ F @ Xs2 ) )
=> ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ ( map_li7225945977422193158st_nat @ F @ ( remove1_list_nat @ X2 @ Xs2 ) ) ) ) ).
% sorted_map_remove1
thf(fact_154_sorted__map__remove1,axiom,
! [F: nat > nat,Xs2: list_nat,X2: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ Xs2 ) )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ ( remove1_nat @ X2 @ Xs2 ) ) ) ) ).
% sorted_map_remove1
thf(fact_155_Greatest__equality,axiom,
! [P: set_nat > $o,X2: set_nat] :
( ( P @ X2 )
=> ( ! [Y3: set_nat] :
( ( P @ Y3 )
=> ( ord_less_eq_set_nat @ Y3 @ X2 ) )
=> ( ( order_5724808138429204845et_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_156_Greatest__equality,axiom,
! [P: list_nat > $o,X2: list_nat] :
( ( P @ X2 )
=> ( ! [Y3: list_nat] :
( ( P @ Y3 )
=> ( ord_less_eq_list_nat @ Y3 @ X2 ) )
=> ( ( order_691136778212554055st_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_157_Greatest__equality,axiom,
! [P: nat > $o,X2: nat] :
( ( P @ X2 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) )
=> ( ( order_Greatest_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_158_GreatestI2__order,axiom,
! [P: set_nat > $o,X2: set_nat,Q: set_nat > $o] :
( ( P @ X2 )
=> ( ! [Y3: set_nat] :
( ( P @ Y3 )
=> ( ord_less_eq_set_nat @ Y3 @ X2 ) )
=> ( ! [X3: set_nat] :
( ( P @ X3 )
=> ( ! [Y5: set_nat] :
( ( P @ Y5 )
=> ( ord_less_eq_set_nat @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_5724808138429204845et_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_159_GreatestI2__order,axiom,
! [P: list_nat > $o,X2: list_nat,Q: list_nat > $o] :
( ( P @ X2 )
=> ( ! [Y3: list_nat] :
( ( P @ Y3 )
=> ( ord_less_eq_list_nat @ Y3 @ X2 ) )
=> ( ! [X3: list_nat] :
( ( P @ X3 )
=> ( ! [Y5: list_nat] :
( ( P @ Y5 )
=> ( ord_less_eq_list_nat @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_691136778212554055st_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_160_GreatestI2__order,axiom,
! [P: nat > $o,X2: nat,Q: nat > $o] :
( ( P @ X2 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_161_one__natural_Orsp,axiom,
one_one_nat = one_one_nat ).
% one_natural.rsp
thf(fact_162_length__map,axiom,
! [F: list_nat > nat,Xs2: list_list_nat] :
( ( size_size_list_nat @ ( map_list_nat_nat @ F @ Xs2 ) )
= ( size_s3023201423986296836st_nat @ Xs2 ) ) ).
% length_map
thf(fact_163_length__map,axiom,
! [F: set_nat > nat,Xs2: list_set_nat] :
( ( size_size_list_nat @ ( map_set_nat_nat @ F @ Xs2 ) )
= ( size_s3254054031482475050et_nat @ Xs2 ) ) ).
% length_map
thf(fact_164_length__map,axiom,
! [F: nat > list_nat,Xs2: list_nat] :
( ( size_s3023201423986296836st_nat @ ( map_nat_list_nat @ F @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_map
thf(fact_165_length__map,axiom,
! [F: list_nat > list_nat,Xs2: list_list_nat] :
( ( size_s3023201423986296836st_nat @ ( map_li7225945977422193158st_nat @ F @ Xs2 ) )
= ( size_s3023201423986296836st_nat @ Xs2 ) ) ).
% length_map
thf(fact_166_length__map,axiom,
! [F: set_nat > list_nat,Xs2: list_set_nat] :
( ( size_s3023201423986296836st_nat @ ( map_set_nat_list_nat @ F @ Xs2 ) )
= ( size_s3254054031482475050et_nat @ Xs2 ) ) ).
% length_map
thf(fact_167_length__map,axiom,
! [F: nat > set_nat,Xs2: list_nat] :
( ( size_s3254054031482475050et_nat @ ( map_nat_set_nat @ F @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_map
thf(fact_168_length__map,axiom,
! [F: list_nat > set_nat,Xs2: list_list_nat] :
( ( size_s3254054031482475050et_nat @ ( map_list_nat_set_nat @ F @ Xs2 ) )
= ( size_s3023201423986296836st_nat @ Xs2 ) ) ).
% length_map
thf(fact_169_length__map,axiom,
! [F: set_nat > set_nat,Xs2: list_set_nat] :
( ( size_s3254054031482475050et_nat @ ( map_set_nat_set_nat @ F @ Xs2 ) )
= ( size_s3254054031482475050et_nat @ Xs2 ) ) ).
% length_map
thf(fact_170_length__map,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( size_size_list_nat @ ( map_nat_nat @ F @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_map
thf(fact_171_length__replicate,axiom,
! [N: nat,X2: list_nat] :
( ( size_s3023201423986296836st_nat @ ( replicate_list_nat @ N @ X2 ) )
= N ) ).
% length_replicate
thf(fact_172_length__replicate,axiom,
! [N: nat,X2: set_nat] :
( ( size_s3254054031482475050et_nat @ ( replicate_set_nat @ N @ X2 ) )
= N ) ).
% length_replicate
thf(fact_173_length__replicate,axiom,
! [N: nat,X2: nat] :
( ( size_size_list_nat @ ( replicate_nat @ N @ X2 ) )
= N ) ).
% length_replicate
thf(fact_174_map__replicate,axiom,
! [F: list_nat > list_nat,N: nat,X2: list_nat] :
( ( map_li7225945977422193158st_nat @ F @ ( replicate_list_nat @ N @ X2 ) )
= ( replicate_list_nat @ N @ ( F @ X2 ) ) ) ).
% map_replicate
thf(fact_175_map__replicate,axiom,
! [F: list_nat > nat,N: nat,X2: list_nat] :
( ( map_list_nat_nat @ F @ ( replicate_list_nat @ N @ X2 ) )
= ( replicate_nat @ N @ ( F @ X2 ) ) ) ).
% map_replicate
thf(fact_176_map__replicate,axiom,
! [F: nat > list_nat,N: nat,X2: nat] :
( ( map_nat_list_nat @ F @ ( replicate_nat @ N @ X2 ) )
= ( replicate_list_nat @ N @ ( F @ X2 ) ) ) ).
% map_replicate
thf(fact_177_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_178_remove1__commute,axiom,
! [X2: nat,Y2: nat,Zs: list_nat] :
( ( remove1_nat @ X2 @ ( remove1_nat @ Y2 @ Zs ) )
= ( remove1_nat @ Y2 @ ( remove1_nat @ X2 @ Zs ) ) ) ).
% remove1_commute
thf(fact_179_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_list_nat] :
( ( size_s3023201423986296836st_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_180_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_set_nat] :
( ( size_s3254054031482475050et_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_181_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_nat] :
( ( size_size_list_nat @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_182_neq__if__length__neq,axiom,
! [Xs2: list_list_nat,Ys: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
!= ( size_s3023201423986296836st_nat @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_183_neq__if__length__neq,axiom,
! [Xs2: list_set_nat,Ys: list_set_nat] :
( ( ( size_s3254054031482475050et_nat @ Xs2 )
!= ( size_s3254054031482475050et_nat @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_184_neq__if__length__neq,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_185_size__neq__size__imp__neq,axiom,
! [X2: list_list_nat,Y2: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ X2 )
!= ( size_s3023201423986296836st_nat @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_186_size__neq__size__imp__neq,axiom,
! [X2: list_set_nat,Y2: list_set_nat] :
( ( ( size_s3254054031482475050et_nat @ X2 )
!= ( size_s3254054031482475050et_nat @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_187_size__neq__size__imp__neq,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( ( size_size_list_nat @ X2 )
!= ( size_size_list_nat @ Y2 ) )
=> ( X2 != Y2 ) ) ).
% size_neq_size_imp_neq
thf(fact_188_map__replicate__const,axiom,
! [K: list_nat,Lst: list_nat] :
( ( map_nat_list_nat
@ ^ [X: nat] : K
@ Lst )
= ( replicate_list_nat @ ( size_size_list_nat @ Lst ) @ K ) ) ).
% map_replicate_const
thf(fact_189_map__replicate__const,axiom,
! [K: list_nat,Lst: list_list_nat] :
( ( map_li7225945977422193158st_nat
@ ^ [X: list_nat] : K
@ Lst )
= ( replicate_list_nat @ ( size_s3023201423986296836st_nat @ Lst ) @ K ) ) ).
% map_replicate_const
thf(fact_190_map__replicate__const,axiom,
! [K: nat,Lst: list_list_nat] :
( ( map_list_nat_nat
@ ^ [X: list_nat] : K
@ Lst )
= ( replicate_nat @ ( size_s3023201423986296836st_nat @ Lst ) @ K ) ) ).
% map_replicate_const
thf(fact_191_map__replicate__const,axiom,
! [K: nat,Lst: list_set_nat] :
( ( map_set_nat_nat
@ ^ [X: set_nat] : K
@ Lst )
= ( replicate_nat @ ( size_s3254054031482475050et_nat @ Lst ) @ K ) ) ).
% map_replicate_const
thf(fact_192_map__replicate__const,axiom,
! [K: nat,Lst: list_nat] :
( ( map_nat_nat
@ ^ [X: nat] : K
@ Lst )
= ( replicate_nat @ ( size_size_list_nat @ Lst ) @ K ) ) ).
% map_replicate_const
thf(fact_193_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs2: list_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_194_map__eq__imp__length__eq,axiom,
! [F: nat > list_nat,Xs2: list_nat,G: nat > list_nat,Ys: list_nat] :
( ( ( map_nat_list_nat @ F @ Xs2 )
= ( map_nat_list_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_195_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs2: list_nat,G: list_nat > nat,Ys: list_list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( map_list_nat_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_196_map__eq__imp__length__eq,axiom,
! [F: nat > nat,Xs2: list_nat,G: set_nat > nat,Ys: list_set_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( map_set_nat_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs2 )
= ( size_s3254054031482475050et_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_197_map__eq__imp__length__eq,axiom,
! [F: list_nat > nat,Xs2: list_list_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_list_nat_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Ys ) )
=> ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_198_map__eq__imp__length__eq,axiom,
! [F: set_nat > nat,Xs2: list_set_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_set_nat_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Ys ) )
=> ( ( size_s3254054031482475050et_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_199_map__eq__imp__length__eq,axiom,
! [F: nat > list_nat,Xs2: list_nat,G: list_nat > list_nat,Ys: list_list_nat] :
( ( ( map_nat_list_nat @ F @ Xs2 )
= ( map_li7225945977422193158st_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_200_map__eq__imp__length__eq,axiom,
! [F: nat > list_nat,Xs2: list_nat,G: set_nat > list_nat,Ys: list_set_nat] :
( ( ( map_nat_list_nat @ F @ Xs2 )
= ( map_set_nat_list_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs2 )
= ( size_s3254054031482475050et_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_201_map__eq__imp__length__eq,axiom,
! [F: list_nat > list_nat,Xs2: list_list_nat,G: nat > list_nat,Ys: list_nat] :
( ( ( map_li7225945977422193158st_nat @ F @ Xs2 )
= ( map_nat_list_nat @ G @ Ys ) )
=> ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_202_map__eq__imp__length__eq,axiom,
! [F: list_nat > nat,Xs2: list_list_nat,G: list_nat > nat,Ys: list_list_nat] :
( ( ( map_list_nat_nat @ F @ Xs2 )
= ( map_list_nat_nat @ G @ Ys ) )
=> ( ( size_s3023201423986296836st_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_203_sorted__replicate,axiom,
! [N: nat,X2: list_nat] : ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ ( replicate_list_nat @ N @ X2 ) ) ).
% sorted_replicate
thf(fact_204_sorted__replicate,axiom,
! [N: nat,X2: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( replicate_nat @ N @ X2 ) ) ).
% sorted_replicate
thf(fact_205_sorted__remove1,axiom,
! [Xs2: list_list_nat,A: list_nat] :
( ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ Xs2 )
=> ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ ( remove1_list_nat @ A @ Xs2 ) ) ) ).
% sorted_remove1
thf(fact_206_sorted__remove1,axiom,
! [Xs2: list_nat,A: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( remove1_nat @ A @ Xs2 ) ) ) ).
% sorted_remove1
thf(fact_207_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_208_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_209_GreatestI__ex__nat,axiom,
! [P: nat > $o,B: nat] :
( ? [X_1: nat] : ( P @ X_1 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_210_sorted__wrt01,axiom,
! [Xs2: list_list_nat,P: list_nat > list_nat > $o] :
( ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ one_one_nat )
=> ( sorted_wrt_list_nat @ P @ Xs2 ) ) ).
% sorted_wrt01
thf(fact_211_sorted__wrt01,axiom,
! [Xs2: list_set_nat,P: set_nat > set_nat > $o] :
( ( ord_less_eq_nat @ ( size_s3254054031482475050et_nat @ Xs2 ) @ one_one_nat )
=> ( sorted_wrt_set_nat @ P @ Xs2 ) ) ).
% sorted_wrt01
thf(fact_212_sorted__wrt01,axiom,
! [Xs2: list_nat,P: nat > nat > $o] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat )
=> ( sorted_wrt_nat @ P @ Xs2 ) ) ).
% sorted_wrt01
thf(fact_213_rotate1__length01,axiom,
! [Xs2: list_list_nat] :
( ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ one_one_nat )
=> ( ( rotate1_list_nat @ Xs2 )
= Xs2 ) ) ).
% rotate1_length01
thf(fact_214_rotate1__length01,axiom,
! [Xs2: list_set_nat] :
( ( ord_less_eq_nat @ ( size_s3254054031482475050et_nat @ Xs2 ) @ one_one_nat )
=> ( ( rotate1_set_nat @ Xs2 )
= Xs2 ) ) ).
% rotate1_length01
thf(fact_215_rotate1__length01,axiom,
! [Xs2: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat )
=> ( ( rotate1_nat @ Xs2 )
= Xs2 ) ) ).
% rotate1_length01
thf(fact_216_rotate__length01,axiom,
! [Xs2: list_list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ one_one_nat )
=> ( ( rotate_list_nat @ N @ Xs2 )
= Xs2 ) ) ).
% rotate_length01
thf(fact_217_rotate__length01,axiom,
! [Xs2: list_set_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_s3254054031482475050et_nat @ Xs2 ) @ one_one_nat )
=> ( ( rotate_set_nat @ N @ Xs2 )
= Xs2 ) ) ).
% rotate_length01
thf(fact_218_rotate__length01,axiom,
! [Xs2: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat )
=> ( ( rotate_nat @ N @ Xs2 )
= Xs2 ) ) ).
% rotate_length01
thf(fact_219_length__butlast,axiom,
! [Xs2: list_list_nat] :
( ( size_s3023201423986296836st_nat @ ( butlast_list_nat @ Xs2 ) )
= ( minus_minus_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ one_one_nat ) ) ).
% length_butlast
thf(fact_220_length__butlast,axiom,
! [Xs2: list_set_nat] :
( ( size_s3254054031482475050et_nat @ ( butlast_set_nat @ Xs2 ) )
= ( minus_minus_nat @ ( size_s3254054031482475050et_nat @ Xs2 ) @ one_one_nat ) ) ).
% length_butlast
thf(fact_221_length__butlast,axiom,
! [Xs2: list_nat] :
( ( size_size_list_nat @ ( butlast_nat @ Xs2 ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) ) ).
% length_butlast
thf(fact_222_length__remove1,axiom,
! [X2: list_nat,Xs2: list_list_nat] :
( ( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs2 ) )
=> ( ( size_s3023201423986296836st_nat @ ( remove1_list_nat @ X2 @ Xs2 ) )
= ( minus_minus_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ one_one_nat ) ) )
& ( ~ ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs2 ) )
=> ( ( size_s3023201423986296836st_nat @ ( remove1_list_nat @ X2 @ Xs2 ) )
= ( size_s3023201423986296836st_nat @ Xs2 ) ) ) ) ).
% length_remove1
thf(fact_223_length__remove1,axiom,
! [X2: set_nat,Xs2: list_set_nat] :
( ( ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs2 ) )
=> ( ( size_s3254054031482475050et_nat @ ( remove1_set_nat @ X2 @ Xs2 ) )
= ( minus_minus_nat @ ( size_s3254054031482475050et_nat @ Xs2 ) @ one_one_nat ) ) )
& ( ~ ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs2 ) )
=> ( ( size_s3254054031482475050et_nat @ ( remove1_set_nat @ X2 @ Xs2 ) )
= ( size_s3254054031482475050et_nat @ Xs2 ) ) ) ) ).
% length_remove1
thf(fact_224_length__remove1,axiom,
! [X2: nat,Xs2: list_nat] :
( ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ( size_size_list_nat @ ( remove1_nat @ X2 @ Xs2 ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) ) )
& ( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ( size_size_list_nat @ ( remove1_nat @ X2 @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ) ) ).
% length_remove1
thf(fact_225_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M2: nat] :
( ( P @ X2 )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M2 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_226_sorted__sort__key,axiom,
! [F: list_nat > nat,Xs2: list_list_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_list_nat_nat @ F @ ( linord770824052286894514at_nat @ F @ Xs2 ) ) ) ).
% sorted_sort_key
thf(fact_227_sorted__sort__key,axiom,
! [F: nat > list_nat,Xs2: list_nat] : ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ ( map_nat_list_nat @ F @ ( linord7755239041979385138st_nat @ F @ Xs2 ) ) ) ).
% sorted_sort_key
thf(fact_228_sorted__sort__key,axiom,
! [F: list_nat > list_nat,Xs2: list_list_nat] : ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ ( map_li7225945977422193158st_nat @ F @ ( linord1364705781165306818st_nat @ F @ Xs2 ) ) ) ).
% sorted_sort_key
thf(fact_229_sorted__sort__key,axiom,
! [F: nat > nat,Xs2: list_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ ( linord738340561235409698at_nat @ F @ Xs2 ) ) ) ).
% sorted_sort_key
thf(fact_230_sort__key__id__if__sorted,axiom,
! [F: list_nat > nat,Xs2: list_list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_list_nat_nat @ F @ Xs2 ) )
=> ( ( linord770824052286894514at_nat @ F @ Xs2 )
= Xs2 ) ) ).
% sort_key_id_if_sorted
thf(fact_231_sort__key__id__if__sorted,axiom,
! [F: nat > list_nat,Xs2: list_nat] :
( ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ ( map_nat_list_nat @ F @ Xs2 ) )
=> ( ( linord7755239041979385138st_nat @ F @ Xs2 )
= Xs2 ) ) ).
% sort_key_id_if_sorted
thf(fact_232_sort__key__id__if__sorted,axiom,
! [F: list_nat > list_nat,Xs2: list_list_nat] :
( ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ ( map_li7225945977422193158st_nat @ F @ Xs2 ) )
=> ( ( linord1364705781165306818st_nat @ F @ Xs2 )
= Xs2 ) ) ).
% sort_key_id_if_sorted
thf(fact_233_sort__key__id__if__sorted,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ Xs2 ) )
=> ( ( linord738340561235409698at_nat @ F @ Xs2 )
= Xs2 ) ) ).
% sort_key_id_if_sorted
thf(fact_234_map__eq__conv,axiom,
! [F: nat > list_nat,Xs2: list_nat,G: nat > list_nat] :
( ( ( map_nat_list_nat @ F @ Xs2 )
= ( map_nat_list_nat @ G @ Xs2 ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_235_map__eq__conv,axiom,
! [F: list_nat > list_nat,Xs2: list_list_nat,G: list_nat > list_nat] :
( ( ( map_li7225945977422193158st_nat @ F @ Xs2 )
= ( map_li7225945977422193158st_nat @ G @ Xs2 ) )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_236_map__eq__conv,axiom,
! [F: list_nat > nat,Xs2: list_list_nat,G: list_nat > nat] :
( ( ( map_list_nat_nat @ F @ Xs2 )
= ( map_list_nat_nat @ G @ Xs2 ) )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_237_map__eq__conv,axiom,
! [F: nat > nat,Xs2: list_nat,G: nat > nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Xs2 ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_238_length__sort,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( size_size_list_nat @ ( linord738340561235409698at_nat @ F @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_sort
thf(fact_239_set__sort,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( set_nat2 @ ( linord738340561235409698at_nat @ F @ Xs2 ) )
= ( set_nat2 @ Xs2 ) ) ).
% set_sort
thf(fact_240_in__set__remove1,axiom,
! [A: nat,B: nat,Xs2: list_nat] :
( ( A != B )
=> ( ( member_nat @ A @ ( set_nat2 @ ( remove1_nat @ B @ Xs2 ) ) )
= ( member_nat @ A @ ( set_nat2 @ Xs2 ) ) ) ) ).
% in_set_remove1
thf(fact_241_in__set__remove1,axiom,
! [A: list_nat,B: list_nat,Xs2: list_list_nat] :
( ( A != B )
=> ( ( member_list_nat @ A @ ( set_list_nat2 @ ( remove1_list_nat @ B @ Xs2 ) ) )
= ( member_list_nat @ A @ ( set_list_nat2 @ Xs2 ) ) ) ) ).
% in_set_remove1
thf(fact_242_length__rotate,axiom,
! [N: nat,Xs2: list_list_nat] :
( ( size_s3023201423986296836st_nat @ ( rotate_list_nat @ N @ Xs2 ) )
= ( size_s3023201423986296836st_nat @ Xs2 ) ) ).
% length_rotate
thf(fact_243_length__rotate,axiom,
! [N: nat,Xs2: list_set_nat] :
( ( size_s3254054031482475050et_nat @ ( rotate_set_nat @ N @ Xs2 ) )
= ( size_s3254054031482475050et_nat @ Xs2 ) ) ).
% length_rotate
thf(fact_244_length__rotate,axiom,
! [N: nat,Xs2: list_nat] :
( ( size_size_list_nat @ ( rotate_nat @ N @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_rotate
thf(fact_245_set__rotate,axiom,
! [N: nat,Xs2: list_nat] :
( ( set_nat2 @ ( rotate_nat @ N @ Xs2 ) )
= ( set_nat2 @ Xs2 ) ) ).
% set_rotate
thf(fact_246_set__rotate,axiom,
! [N: nat,Xs2: list_list_nat] :
( ( set_list_nat2 @ ( rotate_list_nat @ N @ Xs2 ) )
= ( set_list_nat2 @ Xs2 ) ) ).
% set_rotate
thf(fact_247_length__rotate1,axiom,
! [Xs2: list_list_nat] :
( ( size_s3023201423986296836st_nat @ ( rotate1_list_nat @ Xs2 ) )
= ( size_s3023201423986296836st_nat @ Xs2 ) ) ).
% length_rotate1
thf(fact_248_length__rotate1,axiom,
! [Xs2: list_set_nat] :
( ( size_s3254054031482475050et_nat @ ( rotate1_set_nat @ Xs2 ) )
= ( size_s3254054031482475050et_nat @ Xs2 ) ) ).
% length_rotate1
thf(fact_249_length__rotate1,axiom,
! [Xs2: list_nat] :
( ( size_size_list_nat @ ( rotate1_nat @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_rotate1
thf(fact_250_set__rotate1,axiom,
! [Xs2: list_nat] :
( ( set_nat2 @ ( rotate1_nat @ Xs2 ) )
= ( set_nat2 @ Xs2 ) ) ).
% set_rotate1
thf(fact_251_set__rotate1,axiom,
! [Xs2: list_list_nat] :
( ( set_list_nat2 @ ( rotate1_list_nat @ Xs2 ) )
= ( set_list_nat2 @ Xs2 ) ) ).
% set_rotate1
thf(fact_252_rotate1__replicate,axiom,
! [N: nat,A: nat] :
( ( rotate1_nat @ ( replicate_nat @ N @ A ) )
= ( replicate_nat @ N @ A ) ) ).
% rotate1_replicate
thf(fact_253_subset__code_I1_J,axiom,
! [Xs2: list_list_nat,B4: set_list_nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs2 ) @ B4 )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
=> ( member_list_nat @ X @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_254_subset__code_I1_J,axiom,
! [Xs2: list_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ B4 )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( member_nat @ X @ B4 ) ) ) ) ).
% subset_code(1)
thf(fact_255_rotate1__rotate__swap,axiom,
! [N: nat,Xs2: list_nat] :
( ( rotate1_nat @ ( rotate_nat @ N @ Xs2 ) )
= ( rotate_nat @ N @ ( rotate1_nat @ Xs2 ) ) ) ).
% rotate1_rotate_swap
thf(fact_256_in__set__butlastD,axiom,
! [X2: nat,Xs2: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ ( butlast_nat @ Xs2 ) ) )
=> ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) ) ) ).
% in_set_butlastD
thf(fact_257_in__set__butlastD,axiom,
! [X2: list_nat,Xs2: list_list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ ( butlast_list_nat @ Xs2 ) ) )
=> ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs2 ) ) ) ).
% in_set_butlastD
thf(fact_258_sort__key__const,axiom,
! [C: nat,Xs2: list_nat] :
( ( linord738340561235409698at_nat
@ ^ [X: nat] : C
@ Xs2 )
= Xs2 ) ).
% sort_key_const
thf(fact_259_ex__map__conv,axiom,
! [Ys: list_nat,F: list_nat > nat] :
( ( ? [Xs: list_list_nat] :
( Ys
= ( map_list_nat_nat @ F @ Xs ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Ys ) )
=> ? [Y: list_nat] :
( X
= ( F @ Y ) ) ) ) ) ).
% ex_map_conv
thf(fact_260_ex__map__conv,axiom,
! [Ys: list_list_nat,F: nat > list_nat] :
( ( ? [Xs: list_nat] :
( Ys
= ( map_nat_list_nat @ F @ Xs ) ) )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Ys ) )
=> ? [Y: nat] :
( X
= ( F @ Y ) ) ) ) ) ).
% ex_map_conv
thf(fact_261_ex__map__conv,axiom,
! [Ys: list_list_nat,F: list_nat > list_nat] :
( ( ? [Xs: list_list_nat] :
( Ys
= ( map_li7225945977422193158st_nat @ F @ Xs ) ) )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Ys ) )
=> ? [Y: list_nat] :
( X
= ( F @ Y ) ) ) ) ) ).
% ex_map_conv
thf(fact_262_ex__map__conv,axiom,
! [Ys: list_nat,F: nat > nat] :
( ( ? [Xs: list_nat] :
( Ys
= ( map_nat_nat @ F @ Xs ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Ys ) )
=> ? [Y: nat] :
( X
= ( F @ Y ) ) ) ) ) ).
% ex_map_conv
thf(fact_263_map__cong,axiom,
! [Xs2: list_nat,Ys: list_nat,F: nat > list_nat,G: nat > list_nat] :
( ( Xs2 = Ys )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_nat_list_nat @ F @ Xs2 )
= ( map_nat_list_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_264_map__cong,axiom,
! [Xs2: list_list_nat,Ys: list_list_nat,F: list_nat > list_nat,G: list_nat > list_nat] :
( ( Xs2 = Ys )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_li7225945977422193158st_nat @ F @ Xs2 )
= ( map_li7225945977422193158st_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_265_map__cong,axiom,
! [Xs2: list_list_nat,Ys: list_list_nat,F: list_nat > nat,G: list_nat > nat] :
( ( Xs2 = Ys )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_list_nat_nat @ F @ Xs2 )
= ( map_list_nat_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_266_map__cong,axiom,
! [Xs2: list_nat,Ys: list_nat,F: nat > nat,G: nat > nat] :
( ( Xs2 = Ys )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_nat_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_267_map__idI,axiom,
! [Xs2: list_list_nat,F: list_nat > list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs2 ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_li7225945977422193158st_nat @ F @ Xs2 )
= Xs2 ) ) ).
% map_idI
thf(fact_268_map__idI,axiom,
! [Xs2: list_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_nat_nat @ F @ Xs2 )
= Xs2 ) ) ).
% map_idI
thf(fact_269_map__ext,axiom,
! [Xs2: list_nat,F: nat > list_nat,G: nat > list_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_nat_list_nat @ F @ Xs2 )
= ( map_nat_list_nat @ G @ Xs2 ) ) ) ).
% map_ext
thf(fact_270_map__ext,axiom,
! [Xs2: list_list_nat,F: list_nat > list_nat,G: list_nat > list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs2 ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_li7225945977422193158st_nat @ F @ Xs2 )
= ( map_li7225945977422193158st_nat @ G @ Xs2 ) ) ) ).
% map_ext
thf(fact_271_map__ext,axiom,
! [Xs2: list_list_nat,F: list_nat > nat,G: list_nat > nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs2 ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_list_nat_nat @ F @ Xs2 )
= ( map_list_nat_nat @ G @ Xs2 ) ) ) ).
% map_ext
thf(fact_272_map__ext,axiom,
! [Xs2: list_nat,F: nat > nat,G: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_nat_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Xs2 ) ) ) ).
% map_ext
thf(fact_273_list_Omap__ident__strong,axiom,
! [T: list_list_nat,F: list_nat > list_nat] :
( ! [Z3: list_nat] :
( ( member_list_nat @ Z3 @ ( set_list_nat2 @ T ) )
=> ( ( F @ Z3 )
= Z3 ) )
=> ( ( map_li7225945977422193158st_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_274_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_275_list_Oinj__map__strong,axiom,
! [X2: list_nat,Xa: list_nat,F: nat > list_nat,Fa: nat > list_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_list_nat @ F @ X2 )
= ( map_nat_list_nat @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_276_list_Oinj__map__strong,axiom,
! [X2: list_list_nat,Xa: list_list_nat,F: list_nat > list_nat,Fa: list_nat > list_nat] :
( ! [Z3: list_nat,Za: list_nat] :
( ( member_list_nat @ Z3 @ ( set_list_nat2 @ X2 ) )
=> ( ( member_list_nat @ Za @ ( set_list_nat2 @ Xa ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( map_li7225945977422193158st_nat @ F @ X2 )
= ( map_li7225945977422193158st_nat @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_277_list_Oinj__map__strong,axiom,
! [X2: list_list_nat,Xa: list_list_nat,F: list_nat > nat,Fa: list_nat > nat] :
( ! [Z3: list_nat,Za: list_nat] :
( ( member_list_nat @ Z3 @ ( set_list_nat2 @ X2 ) )
=> ( ( member_list_nat @ Za @ ( set_list_nat2 @ Xa ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( map_list_nat_nat @ F @ X2 )
= ( map_list_nat_nat @ Fa @ Xa ) )
=> ( X2 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_278_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_279_list_Omap__cong0,axiom,
! [X2: list_nat,F: nat > list_nat,G: nat > list_nat] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ X2 ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_nat_list_nat @ F @ X2 )
= ( map_nat_list_nat @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_280_list_Omap__cong0,axiom,
! [X2: list_list_nat,F: list_nat > list_nat,G: list_nat > list_nat] :
( ! [Z3: list_nat] :
( ( member_list_nat @ Z3 @ ( set_list_nat2 @ X2 ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_li7225945977422193158st_nat @ F @ X2 )
= ( map_li7225945977422193158st_nat @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_281_list_Omap__cong0,axiom,
! [X2: list_list_nat,F: list_nat > nat,G: list_nat > nat] :
( ! [Z3: list_nat] :
( ( member_list_nat @ Z3 @ ( set_list_nat2 @ X2 ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_list_nat_nat @ F @ X2 )
= ( map_list_nat_nat @ G @ X2 ) ) ) ).
% list.map_cong0
thf(fact_282_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_283_list_Omap__cong,axiom,
! [X2: list_nat,Ya: list_nat,F: nat > list_nat,G: nat > list_nat] :
( ( X2 = Ya )
=> ( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( set_nat2 @ Ya ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_nat_list_nat @ F @ X2 )
= ( map_nat_list_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_284_list_Omap__cong,axiom,
! [X2: list_list_nat,Ya: list_list_nat,F: list_nat > list_nat,G: list_nat > list_nat] :
( ( X2 = Ya )
=> ( ! [Z3: list_nat] :
( ( member_list_nat @ Z3 @ ( set_list_nat2 @ Ya ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_li7225945977422193158st_nat @ F @ X2 )
= ( map_li7225945977422193158st_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_285_list_Omap__cong,axiom,
! [X2: list_list_nat,Ya: list_list_nat,F: list_nat > nat,G: list_nat > nat] :
( ( X2 = Ya )
=> ( ! [Z3: list_nat] :
( ( member_list_nat @ Z3 @ ( set_list_nat2 @ Ya ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_list_nat_nat @ F @ X2 )
= ( map_list_nat_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_286_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_287_sorted__wrt__mono__rel,axiom,
! [Xs2: list_list_nat,P: list_nat > list_nat > $o,Q: list_nat > list_nat > $o] :
( ! [X3: list_nat,Y3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs2 ) )
=> ( ( member_list_nat @ Y3 @ ( set_list_nat2 @ Xs2 ) )
=> ( ( P @ X3 @ Y3 )
=> ( Q @ X3 @ Y3 ) ) ) )
=> ( ( sorted_wrt_list_nat @ P @ Xs2 )
=> ( sorted_wrt_list_nat @ Q @ Xs2 ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_288_sorted__wrt__mono__rel,axiom,
! [Xs2: list_nat,P: nat > nat > $o,Q: nat > nat > $o] :
( ! [X3: nat,Y3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
=> ( ( member_nat @ Y3 @ ( set_nat2 @ Xs2 ) )
=> ( ( P @ X3 @ Y3 )
=> ( Q @ X3 @ Y3 ) ) ) )
=> ( ( sorted_wrt_nat @ P @ Xs2 )
=> ( sorted_wrt_nat @ Q @ Xs2 ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_289_set__remove1__subset,axiom,
! [X2: list_nat,Xs2: list_list_nat] : ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ ( remove1_list_nat @ X2 @ Xs2 ) ) @ ( set_list_nat2 @ Xs2 ) ) ).
% set_remove1_subset
thf(fact_290_set__remove1__subset,axiom,
! [X2: nat,Xs2: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( remove1_nat @ X2 @ Xs2 ) ) @ ( set_nat2 @ Xs2 ) ) ).
% set_remove1_subset
thf(fact_291_notin__set__remove1,axiom,
! [X2: nat,Xs2: list_nat,Y2: nat] :
( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ~ ( member_nat @ X2 @ ( set_nat2 @ ( remove1_nat @ Y2 @ Xs2 ) ) ) ) ).
% notin_set_remove1
thf(fact_292_notin__set__remove1,axiom,
! [X2: list_nat,Xs2: list_list_nat,Y2: list_nat] :
( ~ ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs2 ) )
=> ~ ( member_list_nat @ X2 @ ( set_list_nat2 @ ( remove1_list_nat @ Y2 @ Xs2 ) ) ) ) ).
% notin_set_remove1
thf(fact_293_remove1__idem,axiom,
! [X2: nat,Xs2: list_nat] :
( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ( remove1_nat @ X2 @ Xs2 )
= Xs2 ) ) ).
% remove1_idem
thf(fact_294_remove1__idem,axiom,
! [X2: list_nat,Xs2: list_list_nat] :
( ~ ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs2 ) )
=> ( ( remove1_list_nat @ X2 @ Xs2 )
= Xs2 ) ) ).
% remove1_idem
thf(fact_295_map__butlast,axiom,
! [F: nat > list_nat,Xs2: list_nat] :
( ( map_nat_list_nat @ F @ ( butlast_nat @ Xs2 ) )
= ( butlast_list_nat @ ( map_nat_list_nat @ F @ Xs2 ) ) ) ).
% map_butlast
thf(fact_296_map__butlast,axiom,
! [F: list_nat > list_nat,Xs2: list_list_nat] :
( ( map_li7225945977422193158st_nat @ F @ ( butlast_list_nat @ Xs2 ) )
= ( butlast_list_nat @ ( map_li7225945977422193158st_nat @ F @ Xs2 ) ) ) ).
% map_butlast
thf(fact_297_map__butlast,axiom,
! [F: list_nat > nat,Xs2: list_list_nat] :
( ( map_list_nat_nat @ F @ ( butlast_list_nat @ Xs2 ) )
= ( butlast_nat @ ( map_list_nat_nat @ F @ Xs2 ) ) ) ).
% map_butlast
thf(fact_298_map__butlast,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( map_nat_nat @ F @ ( butlast_nat @ Xs2 ) )
= ( butlast_nat @ ( map_nat_nat @ F @ Xs2 ) ) ) ).
% map_butlast
thf(fact_299_rotate__map,axiom,
! [N: nat,F: nat > list_nat,Xs2: list_nat] :
( ( rotate_list_nat @ N @ ( map_nat_list_nat @ F @ Xs2 ) )
= ( map_nat_list_nat @ F @ ( rotate_nat @ N @ Xs2 ) ) ) ).
% rotate_map
thf(fact_300_rotate__map,axiom,
! [N: nat,F: list_nat > list_nat,Xs2: list_list_nat] :
( ( rotate_list_nat @ N @ ( map_li7225945977422193158st_nat @ F @ Xs2 ) )
= ( map_li7225945977422193158st_nat @ F @ ( rotate_list_nat @ N @ Xs2 ) ) ) ).
% rotate_map
thf(fact_301_rotate__map,axiom,
! [N: nat,F: list_nat > nat,Xs2: list_list_nat] :
( ( rotate_nat @ N @ ( map_list_nat_nat @ F @ Xs2 ) )
= ( map_list_nat_nat @ F @ ( rotate_list_nat @ N @ Xs2 ) ) ) ).
% rotate_map
thf(fact_302_rotate__map,axiom,
! [N: nat,F: nat > nat,Xs2: list_nat] :
( ( rotate_nat @ N @ ( map_nat_nat @ F @ Xs2 ) )
= ( map_nat_nat @ F @ ( rotate_nat @ N @ Xs2 ) ) ) ).
% rotate_map
thf(fact_303_butlast__tl,axiom,
! [Xs2: list_nat] :
( ( butlast_nat @ ( tl_nat @ Xs2 ) )
= ( tl_nat @ ( butlast_nat @ Xs2 ) ) ) ).
% butlast_tl
thf(fact_304_rotate1__map,axiom,
! [F: nat > list_nat,Xs2: list_nat] :
( ( rotate1_list_nat @ ( map_nat_list_nat @ F @ Xs2 ) )
= ( map_nat_list_nat @ F @ ( rotate1_nat @ Xs2 ) ) ) ).
% rotate1_map
thf(fact_305_rotate1__map,axiom,
! [F: list_nat > list_nat,Xs2: list_list_nat] :
( ( rotate1_list_nat @ ( map_li7225945977422193158st_nat @ F @ Xs2 ) )
= ( map_li7225945977422193158st_nat @ F @ ( rotate1_list_nat @ Xs2 ) ) ) ).
% rotate1_map
thf(fact_306_rotate1__map,axiom,
! [F: list_nat > nat,Xs2: list_list_nat] :
( ( rotate1_nat @ ( map_list_nat_nat @ F @ Xs2 ) )
= ( map_list_nat_nat @ F @ ( rotate1_list_nat @ Xs2 ) ) ) ).
% rotate1_map
thf(fact_307_rotate1__map,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( rotate1_nat @ ( map_nat_nat @ F @ Xs2 ) )
= ( map_nat_nat @ F @ ( rotate1_nat @ Xs2 ) ) ) ).
% rotate1_map
thf(fact_308_insort__insert__triv,axiom,
! [X2: list_nat,Xs2: list_list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs2 ) )
=> ( ( linord8148156736319587652st_nat
@ ^ [X: list_nat] : X
@ X2
@ Xs2 )
= Xs2 ) ) ).
% insort_insert_triv
thf(fact_309_insort__insert__triv,axiom,
! [X2: nat,Xs2: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ( linord1921536354676448932at_nat
@ ^ [X: nat] : X
@ X2
@ Xs2 )
= Xs2 ) ) ).
% insort_insert_triv
thf(fact_310_replicate__length__same,axiom,
! [Xs2: list_list_nat,X2: list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs2 ) )
=> ( X3 = X2 ) )
=> ( ( replicate_list_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ X2 )
= Xs2 ) ) ).
% replicate_length_same
thf(fact_311_replicate__length__same,axiom,
! [Xs2: list_set_nat,X2: set_nat] :
( ! [X3: set_nat] :
( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs2 ) )
=> ( X3 = X2 ) )
=> ( ( replicate_set_nat @ ( size_s3254054031482475050et_nat @ Xs2 ) @ X2 )
= Xs2 ) ) ).
% replicate_length_same
thf(fact_312_replicate__length__same,axiom,
! [Xs2: list_nat,X2: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
=> ( X3 = X2 ) )
=> ( ( replicate_nat @ ( size_size_list_nat @ Xs2 ) @ X2 )
= Xs2 ) ) ).
% replicate_length_same
thf(fact_313_replicate__eqI,axiom,
! [Xs2: list_list_nat,N: nat,X2: list_nat] :
( ( ( size_s3023201423986296836st_nat @ Xs2 )
= N )
=> ( ! [Y3: list_nat] :
( ( member_list_nat @ Y3 @ ( set_list_nat2 @ Xs2 ) )
=> ( Y3 = X2 ) )
=> ( Xs2
= ( replicate_list_nat @ N @ X2 ) ) ) ) ).
% replicate_eqI
thf(fact_314_replicate__eqI,axiom,
! [Xs2: list_set_nat,N: nat,X2: set_nat] :
( ( ( size_s3254054031482475050et_nat @ Xs2 )
= N )
=> ( ! [Y3: set_nat] :
( ( member_set_nat @ Y3 @ ( set_set_nat2 @ Xs2 ) )
=> ( Y3 = X2 ) )
=> ( Xs2
= ( replicate_set_nat @ N @ X2 ) ) ) ) ).
% replicate_eqI
thf(fact_315_replicate__eqI,axiom,
! [Xs2: list_nat,N: nat,X2: nat] :
( ( ( size_size_list_nat @ Xs2 )
= N )
=> ( ! [Y3: nat] :
( ( member_nat @ Y3 @ ( set_nat2 @ Xs2 ) )
=> ( Y3 = X2 ) )
=> ( Xs2
= ( replicate_nat @ N @ X2 ) ) ) ) ).
% replicate_eqI
thf(fact_316_sorted__sort__id,axiom,
! [Xs2: list_list_nat] :
( ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ Xs2 )
=> ( ( linord1364705781165306818st_nat
@ ^ [X: list_nat] : X
@ Xs2 )
= Xs2 ) ) ).
% sorted_sort_id
thf(fact_317_sorted__sort__id,axiom,
! [Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
=> ( ( linord738340561235409698at_nat
@ ^ [X: nat] : X
@ Xs2 )
= Xs2 ) ) ).
% sorted_sort_id
thf(fact_318_sorted__sort,axiom,
! [Xs2: list_list_nat] :
( sorted_wrt_list_nat @ ord_less_eq_list_nat
@ ( linord1364705781165306818st_nat
@ ^ [X: list_nat] : X
@ Xs2 ) ) ).
% sorted_sort
thf(fact_319_sorted__sort,axiom,
! [Xs2: list_nat] :
( sorted_wrt_nat @ ord_less_eq_nat
@ ( linord738340561235409698at_nat
@ ^ [X: nat] : X
@ Xs2 ) ) ).
% sorted_sort
thf(fact_320_set__generate__lists,axiom,
! [N: nat,Xs2: list_list_nat] :
( ( set_list_list_nat2 @ ( missin2779806319917622601st_nat @ N @ Xs2 ) )
= ( collec5989764272469232197st_nat
@ ^ [As: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ As )
= N )
& ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ As ) @ ( set_list_nat2 @ Xs2 ) ) ) ) ) ).
% set_generate_lists
thf(fact_321_set__generate__lists,axiom,
! [N: nat,Xs2: list_set_nat] :
( ( set_list_set_nat2 @ ( missin5440160964871002735et_nat @ N @ Xs2 ) )
= ( collect_list_set_nat
@ ^ [As: list_set_nat] :
( ( ( size_s3254054031482475050et_nat @ As )
= N )
& ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ As ) @ ( set_set_nat2 @ Xs2 ) ) ) ) ) ).
% set_generate_lists
thf(fact_322_set__generate__lists,axiom,
! [N: nat,Xs2: list_nat] :
( ( set_list_nat2 @ ( missin2047014633743487673ts_nat @ N @ Xs2 ) )
= ( collect_list_nat
@ ^ [As: list_nat] :
( ( ( size_size_list_nat @ As )
= N )
& ( ord_less_eq_set_nat @ ( set_nat2 @ As ) @ ( set_nat2 @ Xs2 ) ) ) ) ) ).
% set_generate_lists
thf(fact_323_stable__sort__key__sort__key,axiom,
linord227665693835759911at_nat @ linord738340561235409698at_nat ).
% stable_sort_key_sort_key
thf(fact_324_butlast__take,axiom,
! [N: nat,Xs2: list_list_nat] :
( ( ord_less_eq_nat @ N @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ( butlast_list_nat @ ( take_list_nat @ N @ Xs2 ) )
= ( take_list_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs2 ) ) ) ).
% butlast_take
thf(fact_325_butlast__take,axiom,
! [N: nat,Xs2: list_set_nat] :
( ( ord_less_eq_nat @ N @ ( size_s3254054031482475050et_nat @ Xs2 ) )
=> ( ( butlast_set_nat @ ( take_set_nat @ N @ Xs2 ) )
= ( take_set_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs2 ) ) ) ).
% butlast_take
thf(fact_326_butlast__take,axiom,
! [N: nat,Xs2: list_nat] :
( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( butlast_nat @ ( take_nat @ N @ Xs2 ) )
= ( take_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs2 ) ) ) ).
% butlast_take
thf(fact_327_insort__remove1,axiom,
! [A: list_nat,Xs2: list_list_nat] :
( ( member_list_nat @ A @ ( set_list_nat2 @ Xs2 ) )
=> ( ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ Xs2 )
=> ( ( linord3838877917356005949st_nat
@ ^ [X: list_nat] : X
@ A
@ ( remove1_list_nat @ A @ Xs2 ) )
= Xs2 ) ) ) ).
% insort_remove1
thf(fact_328_insort__remove1,axiom,
! [A: nat,Xs2: list_nat] :
( ( member_nat @ A @ ( set_nat2 @ Xs2 ) )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
=> ( ( linord8961336180081300637at_nat
@ ^ [X: nat] : X
@ A
@ ( remove1_nat @ A @ Xs2 ) )
= Xs2 ) ) ) ).
% insort_remove1
thf(fact_329_butlast__list__update,axiom,
! [K: nat,Xs2: list_list_nat,X2: list_nat] :
( ( ( K
= ( minus_minus_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ one_one_nat ) )
=> ( ( butlast_list_nat @ ( list_update_list_nat @ Xs2 @ K @ X2 ) )
= ( butlast_list_nat @ Xs2 ) ) )
& ( ( K
!= ( minus_minus_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ one_one_nat ) )
=> ( ( butlast_list_nat @ ( list_update_list_nat @ Xs2 @ K @ X2 ) )
= ( list_update_list_nat @ ( butlast_list_nat @ Xs2 ) @ K @ X2 ) ) ) ) ).
% butlast_list_update
thf(fact_330_butlast__list__update,axiom,
! [K: nat,Xs2: list_set_nat,X2: set_nat] :
( ( ( K
= ( minus_minus_nat @ ( size_s3254054031482475050et_nat @ Xs2 ) @ one_one_nat ) )
=> ( ( butlast_set_nat @ ( list_update_set_nat @ Xs2 @ K @ X2 ) )
= ( butlast_set_nat @ Xs2 ) ) )
& ( ( K
!= ( minus_minus_nat @ ( size_s3254054031482475050et_nat @ Xs2 ) @ one_one_nat ) )
=> ( ( butlast_set_nat @ ( list_update_set_nat @ Xs2 @ K @ X2 ) )
= ( list_update_set_nat @ ( butlast_set_nat @ Xs2 ) @ K @ X2 ) ) ) ) ).
% butlast_list_update
thf(fact_331_butlast__list__update,axiom,
! [K: nat,Xs2: list_nat,X2: nat] :
( ( ( K
= ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) )
=> ( ( butlast_nat @ ( list_update_nat @ Xs2 @ K @ X2 ) )
= ( butlast_nat @ Xs2 ) ) )
& ( ( K
!= ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) )
=> ( ( butlast_nat @ ( list_update_nat @ Xs2 @ K @ X2 ) )
= ( list_update_nat @ ( butlast_nat @ Xs2 ) @ K @ X2 ) ) ) ) ).
% butlast_list_update
thf(fact_332_butlast__conv__take,axiom,
( butlast_list_nat
= ( ^ [Xs: list_list_nat] : ( take_list_nat @ ( minus_minus_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ one_one_nat ) @ Xs ) ) ) ).
% butlast_conv_take
thf(fact_333_butlast__conv__take,axiom,
( butlast_set_nat
= ( ^ [Xs: list_set_nat] : ( take_set_nat @ ( minus_minus_nat @ ( size_s3254054031482475050et_nat @ Xs ) @ one_one_nat ) @ Xs ) ) ) ).
% butlast_conv_take
thf(fact_334_butlast__conv__take,axiom,
( butlast_nat
= ( ^ [Xs: list_nat] : ( take_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) @ Xs ) ) ) ).
% butlast_conv_take
thf(fact_335_set__n__lists,axiom,
! [N: nat,Xs2: list_list_nat] :
( ( set_list_list_nat2 @ ( n_lists_list_nat @ N @ Xs2 ) )
= ( collec5989764272469232197st_nat
@ ^ [Ys2: list_list_nat] :
( ( ( size_s3023201423986296836st_nat @ Ys2 )
= N )
& ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Ys2 ) @ ( set_list_nat2 @ Xs2 ) ) ) ) ) ).
% set_n_lists
thf(fact_336_set__n__lists,axiom,
! [N: nat,Xs2: list_set_nat] :
( ( set_list_set_nat2 @ ( n_lists_set_nat @ N @ Xs2 ) )
= ( collect_list_set_nat
@ ^ [Ys2: list_set_nat] :
( ( ( size_s3254054031482475050et_nat @ Ys2 )
= N )
& ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Ys2 ) @ ( set_set_nat2 @ Xs2 ) ) ) ) ) ).
% set_n_lists
thf(fact_337_set__n__lists,axiom,
! [N: nat,Xs2: list_nat] :
( ( set_list_nat2 @ ( n_lists_nat @ N @ Xs2 ) )
= ( collect_list_nat
@ ^ [Ys2: list_nat] :
( ( ( size_size_list_nat @ Ys2 )
= N )
& ( ord_less_eq_set_nat @ ( set_nat2 @ Ys2 ) @ ( set_nat2 @ Xs2 ) ) ) ) ) ).
% set_n_lists
thf(fact_338_sorted__butlast,axiom,
! [Xs2: list_list_nat] :
( ( Xs2 != nil_list_nat )
=> ( ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ Xs2 )
=> ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ ( butlast_list_nat @ Xs2 ) ) ) ) ).
% sorted_butlast
thf(fact_339_sorted__butlast,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( butlast_nat @ Xs2 ) ) ) ) ).
% sorted_butlast
thf(fact_340_set__remdups__sorted,axiom,
! [Xs2: list_list_nat] :
( ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ Xs2 )
=> ( ( set_list_nat2 @ ( set_or4609908739687369647st_nat @ Xs2 ) )
= ( set_list_nat2 @ Xs2 ) ) ) ).
% set_remdups_sorted
thf(fact_341_set__remdups__sorted,axiom,
! [Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
=> ( ( set_nat2 @ ( set_or6599480164596245535ed_nat @ Xs2 ) )
= ( set_nat2 @ Xs2 ) ) ) ).
% set_remdups_sorted
thf(fact_342_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_343_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_344_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_345_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_346_list__update__overwrite,axiom,
! [Xs2: list_nat,I: nat,X2: nat,Y2: nat] :
( ( list_update_nat @ ( list_update_nat @ Xs2 @ I @ X2 ) @ I @ Y2 )
= ( list_update_nat @ Xs2 @ I @ Y2 ) ) ).
% list_update_overwrite
thf(fact_347_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_348_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_349_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_350_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_351_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_352_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_353_list_Omap__disc__iff,axiom,
! [F: nat > list_nat,A: list_nat] :
( ( ( map_nat_list_nat @ F @ A )
= nil_list_nat )
= ( A = nil_nat ) ) ).
% list.map_disc_iff
thf(fact_354_list_Omap__disc__iff,axiom,
! [F: list_nat > list_nat,A: list_list_nat] :
( ( ( map_li7225945977422193158st_nat @ F @ A )
= nil_list_nat )
= ( A = nil_list_nat ) ) ).
% list.map_disc_iff
thf(fact_355_list_Omap__disc__iff,axiom,
! [F: list_nat > nat,A: list_list_nat] :
( ( ( map_list_nat_nat @ F @ A )
= nil_nat )
= ( A = nil_list_nat ) ) ).
% list.map_disc_iff
thf(fact_356_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_357_Nil__is__map__conv,axiom,
! [F: nat > list_nat,Xs2: list_nat] :
( ( nil_list_nat
= ( map_nat_list_nat @ F @ Xs2 ) )
= ( Xs2 = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_358_Nil__is__map__conv,axiom,
! [F: list_nat > list_nat,Xs2: list_list_nat] :
( ( nil_list_nat
= ( map_li7225945977422193158st_nat @ F @ Xs2 ) )
= ( Xs2 = nil_list_nat ) ) ).
% Nil_is_map_conv
thf(fact_359_Nil__is__map__conv,axiom,
! [F: list_nat > nat,Xs2: list_list_nat] :
( ( nil_nat
= ( map_list_nat_nat @ F @ Xs2 ) )
= ( Xs2 = nil_list_nat ) ) ).
% Nil_is_map_conv
thf(fact_360_Nil__is__map__conv,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( nil_nat
= ( map_nat_nat @ F @ Xs2 ) )
= ( Xs2 = nil_nat ) ) ).
% Nil_is_map_conv
thf(fact_361_map__is__Nil__conv,axiom,
! [F: nat > list_nat,Xs2: list_nat] :
( ( ( map_nat_list_nat @ F @ Xs2 )
= nil_list_nat )
= ( Xs2 = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_362_map__is__Nil__conv,axiom,
! [F: list_nat > list_nat,Xs2: list_list_nat] :
( ( ( map_li7225945977422193158st_nat @ F @ Xs2 )
= nil_list_nat )
= ( Xs2 = nil_list_nat ) ) ).
% map_is_Nil_conv
thf(fact_363_map__is__Nil__conv,axiom,
! [F: list_nat > nat,Xs2: list_list_nat] :
( ( ( map_list_nat_nat @ F @ Xs2 )
= nil_nat )
= ( Xs2 = nil_list_nat ) ) ).
% map_is_Nil_conv
thf(fact_364_map__is__Nil__conv,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= nil_nat )
= ( Xs2 = nil_nat ) ) ).
% map_is_Nil_conv
thf(fact_365_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_366_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_367_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_368_list__update__nonempty,axiom,
! [Xs2: list_list_nat,K: nat,X2: list_nat] :
( ( ( list_update_list_nat @ Xs2 @ K @ X2 )
= nil_list_nat )
= ( Xs2 = nil_list_nat ) ) ).
% list_update_nonempty
thf(fact_369_list__update__nonempty,axiom,
! [Xs2: list_nat,K: nat,X2: nat] :
( ( ( list_update_nat @ Xs2 @ K @ X2 )
= nil_nat )
= ( Xs2 = nil_nat ) ) ).
% list_update_nonempty
thf(fact_370_length__list__update,axiom,
! [Xs2: list_list_nat,I: nat,X2: list_nat] :
( ( size_s3023201423986296836st_nat @ ( list_update_list_nat @ Xs2 @ I @ X2 ) )
= ( size_s3023201423986296836st_nat @ Xs2 ) ) ).
% length_list_update
thf(fact_371_length__list__update,axiom,
! [Xs2: list_set_nat,I: nat,X2: set_nat] :
( ( size_s3254054031482475050et_nat @ ( list_update_set_nat @ Xs2 @ I @ X2 ) )
= ( size_s3254054031482475050et_nat @ Xs2 ) ) ).
% length_list_update
thf(fact_372_length__list__update,axiom,
! [Xs2: list_nat,I: nat,X2: nat] :
( ( size_size_list_nat @ ( list_update_nat @ Xs2 @ I @ X2 ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_list_update
thf(fact_373_sort__key__simps_I1_J,axiom,
! [F: nat > nat] :
( ( linord738340561235409698at_nat @ F @ nil_nat )
= nil_nat ) ).
% sort_key_simps(1)
thf(fact_374_rotate__is__Nil__conv,axiom,
! [N: nat,Xs2: list_list_nat] :
( ( ( rotate_list_nat @ N @ Xs2 )
= nil_list_nat )
= ( Xs2 = nil_list_nat ) ) ).
% rotate_is_Nil_conv
thf(fact_375_rotate__is__Nil__conv,axiom,
! [N: nat,Xs2: list_nat] :
( ( ( rotate_nat @ N @ Xs2 )
= nil_nat )
= ( Xs2 = nil_nat ) ) ).
% rotate_is_Nil_conv
thf(fact_376_rotate1__is__Nil__conv,axiom,
! [Xs2: list_list_nat] :
( ( ( rotate1_list_nat @ Xs2 )
= nil_list_nat )
= ( Xs2 = nil_list_nat ) ) ).
% rotate1_is_Nil_conv
thf(fact_377_rotate1__is__Nil__conv,axiom,
! [Xs2: list_nat] :
( ( ( rotate1_nat @ Xs2 )
= nil_nat )
= ( Xs2 = nil_nat ) ) ).
% rotate1_is_Nil_conv
thf(fact_378_remove1__insort__key,axiom,
! [X2: nat,F: nat > nat,Xs2: list_nat] :
( ( remove1_nat @ X2 @ ( linord8961336180081300637at_nat @ F @ X2 @ Xs2 ) )
= Xs2 ) ).
% remove1_insort_key
thf(fact_379_take__all__iff,axiom,
! [N: nat,Xs2: list_list_nat] :
( ( ( take_list_nat @ N @ Xs2 )
= Xs2 )
= ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ N ) ) ).
% take_all_iff
thf(fact_380_take__all__iff,axiom,
! [N: nat,Xs2: list_set_nat] :
( ( ( take_set_nat @ N @ Xs2 )
= Xs2 )
= ( ord_less_eq_nat @ ( size_s3254054031482475050et_nat @ Xs2 ) @ N ) ) ).
% take_all_iff
thf(fact_381_take__all__iff,axiom,
! [N: nat,Xs2: list_nat] :
( ( ( take_nat @ N @ Xs2 )
= Xs2 )
= ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N ) ) ).
% take_all_iff
thf(fact_382_take__all,axiom,
! [Xs2: list_list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ N )
=> ( ( take_list_nat @ N @ Xs2 )
= Xs2 ) ) ).
% take_all
thf(fact_383_take__all,axiom,
! [Xs2: list_set_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_s3254054031482475050et_nat @ Xs2 ) @ N )
=> ( ( take_set_nat @ N @ Xs2 )
= Xs2 ) ) ).
% take_all
thf(fact_384_take__all,axiom,
! [Xs2: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N )
=> ( ( take_nat @ N @ Xs2 )
= Xs2 ) ) ).
% take_all
thf(fact_385_list__update__beyond,axiom,
! [Xs2: list_list_nat,I: nat,X2: list_nat] :
( ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ I )
=> ( ( list_update_list_nat @ Xs2 @ I @ X2 )
= Xs2 ) ) ).
% list_update_beyond
thf(fact_386_list__update__beyond,axiom,
! [Xs2: list_set_nat,I: nat,X2: set_nat] :
( ( ord_less_eq_nat @ ( size_s3254054031482475050et_nat @ Xs2 ) @ I )
=> ( ( list_update_set_nat @ Xs2 @ I @ X2 )
= Xs2 ) ) ).
% list_update_beyond
thf(fact_387_list__update__beyond,axiom,
! [Xs2: list_nat,I: nat,X2: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ I )
=> ( ( list_update_nat @ Xs2 @ I @ X2 )
= Xs2 ) ) ).
% list_update_beyond
thf(fact_388_take__update__cancel,axiom,
! [N: nat,M: nat,Xs2: list_nat,Y2: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( take_nat @ N @ ( list_update_nat @ Xs2 @ M @ Y2 ) )
= ( take_nat @ N @ Xs2 ) ) ) ).
% take_update_cancel
thf(fact_389_list__update__code_I1_J,axiom,
! [I: nat,Y2: list_nat] :
( ( list_update_list_nat @ nil_list_nat @ I @ Y2 )
= nil_list_nat ) ).
% list_update_code(1)
thf(fact_390_list__update__code_I1_J,axiom,
! [I: nat,Y2: nat] :
( ( list_update_nat @ nil_nat @ I @ Y2 )
= nil_nat ) ).
% list_update_code(1)
thf(fact_391_list__update_Osimps_I1_J,axiom,
! [I: nat,V: list_nat] :
( ( list_update_list_nat @ nil_list_nat @ I @ V )
= nil_list_nat ) ).
% list_update.simps(1)
thf(fact_392_list__update_Osimps_I1_J,axiom,
! [I: nat,V: nat] :
( ( list_update_nat @ nil_nat @ I @ V )
= nil_nat ) ).
% list_update.simps(1)
thf(fact_393_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_394_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_395_take__Nil,axiom,
! [N: nat] :
( ( take_list_nat @ N @ nil_list_nat )
= nil_list_nat ) ).
% take_Nil
thf(fact_396_take__Nil,axiom,
! [N: nat] :
( ( take_nat @ N @ nil_nat )
= nil_nat ) ).
% take_Nil
thf(fact_397_group__cancel_Oadd1,axiom,
! [A4: nat,K: nat,A: nat,B: nat] :
( ( A4
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A4 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_398_group__cancel_Oadd2,axiom,
! [B4: nat,K: nat,B: nat,A: nat] :
( ( B4
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B4 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_399_insort__not__Nil,axiom,
! [F: nat > nat,A: nat,Xs2: list_nat] :
( ( linord8961336180081300637at_nat @ F @ A @ Xs2 )
!= nil_nat ) ).
% insort_not_Nil
thf(fact_400_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_401_insort__key__left__comm,axiom,
! [F: nat > nat,X2: nat,Y2: nat,Xs2: list_nat] :
( ( ( F @ X2 )
!= ( F @ Y2 ) )
=> ( ( linord8961336180081300637at_nat @ F @ Y2 @ ( linord8961336180081300637at_nat @ F @ X2 @ Xs2 ) )
= ( linord8961336180081300637at_nat @ F @ X2 @ ( linord8961336180081300637at_nat @ F @ Y2 @ Xs2 ) ) ) ) ).
% insort_key_left_comm
thf(fact_402_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_403_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_404_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_405_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_406_take__equalityI,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ! [I2: nat] :
( ( take_nat @ I2 @ Xs2 )
= ( take_nat @ I2 @ Ys ) )
=> ( Xs2 = Ys ) ) ).
% take_equalityI
thf(fact_407_list__update__swap,axiom,
! [I: nat,I3: nat,Xs2: list_nat,X2: nat,X5: nat] :
( ( I != I3 )
=> ( ( list_update_nat @ ( list_update_nat @ Xs2 @ I @ X2 ) @ I3 @ X5 )
= ( list_update_nat @ ( list_update_nat @ Xs2 @ I3 @ X5 ) @ I @ X2 ) ) ) ).
% list_update_swap
thf(fact_408_take__update__swap,axiom,
! [M: nat,Xs2: list_nat,N: nat,X2: nat] :
( ( take_nat @ M @ ( list_update_nat @ Xs2 @ N @ X2 ) )
= ( list_update_nat @ ( take_nat @ M @ Xs2 ) @ N @ X2 ) ) ).
% take_update_swap
thf(fact_409_remdups__sorted_Osimps_I1_J,axiom,
( ( set_or4609908739687369647st_nat @ nil_list_nat )
= nil_list_nat ) ).
% remdups_sorted.simps(1)
thf(fact_410_remdups__sorted_Osimps_I1_J,axiom,
( ( set_or6599480164596245535ed_nat @ nil_nat )
= nil_nat ) ).
% remdups_sorted.simps(1)
thf(fact_411_insort__left__comm,axiom,
! [X2: nat,Y2: nat,Xs2: list_nat] :
( ( linord8961336180081300637at_nat
@ ^ [X: nat] : X
@ X2
@ ( linord8961336180081300637at_nat
@ ^ [X: nat] : X
@ Y2
@ Xs2 ) )
= ( linord8961336180081300637at_nat
@ ^ [X: nat] : X
@ Y2
@ ( linord8961336180081300637at_nat
@ ^ [X: nat] : X
@ X2
@ Xs2 ) ) ) ).
% insort_left_comm
thf(fact_412_in__set__takeD,axiom,
! [X2: list_nat,N: nat,Xs2: list_list_nat] :
( ( member_list_nat @ X2 @ ( set_list_nat2 @ ( take_list_nat @ N @ Xs2 ) ) )
=> ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs2 ) ) ) ).
% in_set_takeD
thf(fact_413_in__set__takeD,axiom,
! [X2: nat,N: nat,Xs2: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ ( take_nat @ N @ Xs2 ) ) )
=> ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) ) ) ).
% in_set_takeD
thf(fact_414_set__take__subset,axiom,
! [N: nat,Xs2: list_list_nat] : ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ ( take_list_nat @ N @ Xs2 ) ) @ ( set_list_nat2 @ Xs2 ) ) ).
% set_take_subset
thf(fact_415_set__take__subset,axiom,
! [N: nat,Xs2: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( take_nat @ N @ Xs2 ) ) @ ( set_nat2 @ Xs2 ) ) ).
% set_take_subset
thf(fact_416_set__update__subsetI,axiom,
! [Xs2: list_list_nat,A4: set_list_nat,X2: list_nat,I: nat] :
( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs2 ) @ A4 )
=> ( ( member_list_nat @ X2 @ A4 )
=> ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ ( list_update_list_nat @ Xs2 @ I @ X2 ) ) @ A4 ) ) ) ).
% set_update_subsetI
thf(fact_417_set__update__subsetI,axiom,
! [Xs2: list_nat,A4: set_nat,X2: nat,I: nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A4 )
=> ( ( member_nat @ X2 @ A4 )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs2 @ I @ X2 ) ) @ A4 ) ) ) ).
% set_update_subsetI
thf(fact_418_take__map,axiom,
! [N: nat,F: nat > list_nat,Xs2: list_nat] :
( ( take_list_nat @ N @ ( map_nat_list_nat @ F @ Xs2 ) )
= ( map_nat_list_nat @ F @ ( take_nat @ N @ Xs2 ) ) ) ).
% take_map
thf(fact_419_take__map,axiom,
! [N: nat,F: list_nat > list_nat,Xs2: list_list_nat] :
( ( take_list_nat @ N @ ( map_li7225945977422193158st_nat @ F @ Xs2 ) )
= ( map_li7225945977422193158st_nat @ F @ ( take_list_nat @ N @ Xs2 ) ) ) ).
% take_map
thf(fact_420_take__map,axiom,
! [N: nat,F: list_nat > nat,Xs2: list_list_nat] :
( ( take_nat @ N @ ( map_list_nat_nat @ F @ Xs2 ) )
= ( map_list_nat_nat @ F @ ( take_list_nat @ N @ Xs2 ) ) ) ).
% take_map
thf(fact_421_take__map,axiom,
! [N: nat,F: nat > nat,Xs2: list_nat] :
( ( take_nat @ N @ ( map_nat_nat @ F @ Xs2 ) )
= ( map_nat_nat @ F @ ( take_nat @ N @ Xs2 ) ) ) ).
% take_map
thf(fact_422_map__update,axiom,
! [F: list_nat > list_nat,Xs2: list_list_nat,K: nat,Y2: list_nat] :
( ( map_li7225945977422193158st_nat @ F @ ( list_update_list_nat @ Xs2 @ K @ Y2 ) )
= ( list_update_list_nat @ ( map_li7225945977422193158st_nat @ F @ Xs2 ) @ K @ ( F @ Y2 ) ) ) ).
% map_update
thf(fact_423_map__update,axiom,
! [F: list_nat > nat,Xs2: list_list_nat,K: nat,Y2: list_nat] :
( ( map_list_nat_nat @ F @ ( list_update_list_nat @ Xs2 @ K @ Y2 ) )
= ( list_update_nat @ ( map_list_nat_nat @ F @ Xs2 ) @ K @ ( F @ Y2 ) ) ) ).
% map_update
thf(fact_424_map__update,axiom,
! [F: nat > list_nat,Xs2: list_nat,K: nat,Y2: nat] :
( ( map_nat_list_nat @ F @ ( list_update_nat @ Xs2 @ K @ Y2 ) )
= ( list_update_list_nat @ ( map_nat_list_nat @ F @ Xs2 ) @ K @ ( F @ Y2 ) ) ) ).
% map_update
thf(fact_425_map__update,axiom,
! [F: nat > nat,Xs2: list_nat,K: nat,Y2: nat] :
( ( map_nat_nat @ F @ ( list_update_nat @ Xs2 @ K @ Y2 ) )
= ( list_update_nat @ ( map_nat_nat @ F @ Xs2 ) @ K @ ( F @ Y2 ) ) ) ).
% map_update
thf(fact_426_sorted__wrt__take,axiom,
! [F: nat > nat > $o,Xs2: list_nat,N: nat] :
( ( sorted_wrt_nat @ F @ Xs2 )
=> ( sorted_wrt_nat @ F @ ( take_nat @ N @ Xs2 ) ) ) ).
% sorted_wrt_take
thf(fact_427_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_428_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_429_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_430_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_431_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_432_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_433_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_434_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
? [C3: nat] :
( B2
= ( plus_plus_nat @ A2 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_435_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_436_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_437_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_438_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_439_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_440_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_441_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_442_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_443_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_444_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_445_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_446_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_447_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_448_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_449_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N3: nat] :
? [K2: nat] :
( N3
= ( plus_plus_nat @ M4 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_450_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_451_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_452_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_453_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_454_list_Osimps_I8_J,axiom,
! [F: nat > list_nat] :
( ( map_nat_list_nat @ F @ nil_nat )
= nil_list_nat ) ).
% list.simps(8)
thf(fact_455_list_Osimps_I8_J,axiom,
! [F: list_nat > list_nat] :
( ( map_li7225945977422193158st_nat @ F @ nil_list_nat )
= nil_list_nat ) ).
% list.simps(8)
thf(fact_456_list_Osimps_I8_J,axiom,
! [F: list_nat > nat] :
( ( map_list_nat_nat @ F @ nil_list_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_457_list_Osimps_I8_J,axiom,
! [F: nat > nat] :
( ( map_nat_nat @ F @ nil_nat )
= nil_nat ) ).
% list.simps(8)
thf(fact_458_sorted__wrt_Osimps_I1_J,axiom,
! [P: list_nat > list_nat > $o] : ( sorted_wrt_list_nat @ P @ nil_list_nat ) ).
% sorted_wrt.simps(1)
thf(fact_459_sorted__wrt_Osimps_I1_J,axiom,
! [P: nat > nat > $o] : ( sorted_wrt_nat @ P @ nil_nat ) ).
% sorted_wrt.simps(1)
thf(fact_460_list_Osel_I2_J,axiom,
( ( tl_list_nat @ nil_list_nat )
= nil_list_nat ) ).
% list.sel(2)
thf(fact_461_list_Osel_I2_J,axiom,
( ( tl_nat @ nil_nat )
= nil_nat ) ).
% list.sel(2)
thf(fact_462_rotate__rotate,axiom,
! [M: nat,N: nat,Xs2: list_nat] :
( ( rotate_nat @ M @ ( rotate_nat @ N @ Xs2 ) )
= ( rotate_nat @ ( plus_plus_nat @ M @ N ) @ Xs2 ) ) ).
% rotate_rotate
thf(fact_463_remove1_Osimps_I1_J,axiom,
! [X2: list_nat] :
( ( remove1_list_nat @ X2 @ nil_list_nat )
= nil_list_nat ) ).
% remove1.simps(1)
thf(fact_464_remove1_Osimps_I1_J,axiom,
! [X2: nat] :
( ( remove1_nat @ X2 @ nil_nat )
= nil_nat ) ).
% remove1.simps(1)
thf(fact_465_butlast_Osimps_I1_J,axiom,
( ( butlast_list_nat @ nil_list_nat )
= nil_list_nat ) ).
% butlast.simps(1)
thf(fact_466_butlast_Osimps_I1_J,axiom,
( ( butlast_nat @ nil_nat )
= nil_nat ) ).
% butlast.simps(1)
thf(fact_467_rotate1_Osimps_I1_J,axiom,
( ( rotate1_list_nat @ nil_list_nat )
= nil_list_nat ) ).
% rotate1.simps(1)
thf(fact_468_rotate1_Osimps_I1_J,axiom,
( ( rotate1_nat @ nil_nat )
= nil_nat ) ).
% rotate1.simps(1)
thf(fact_469_length__n__lists__elem,axiom,
! [Ys: list_list_nat,N: nat,Xs2: list_list_nat] :
( ( member_list_list_nat @ Ys @ ( set_list_list_nat2 @ ( n_lists_list_nat @ N @ Xs2 ) ) )
=> ( ( size_s3023201423986296836st_nat @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_470_length__n__lists__elem,axiom,
! [Ys: list_set_nat,N: nat,Xs2: list_set_nat] :
( ( member_list_set_nat @ Ys @ ( set_list_set_nat2 @ ( n_lists_set_nat @ N @ Xs2 ) ) )
=> ( ( size_s3254054031482475050et_nat @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_471_length__n__lists__elem,axiom,
! [Ys: list_nat,N: nat,Xs2: list_nat] :
( ( member_list_nat @ Ys @ ( set_list_nat2 @ ( n_lists_nat @ N @ Xs2 ) ) )
=> ( ( size_size_list_nat @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_472_sorted__remdups__sorted,axiom,
! [Xs2: list_list_nat] :
( ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ Xs2 )
=> ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ ( set_or4609908739687369647st_nat @ Xs2 ) ) ) ).
% sorted_remdups_sorted
thf(fact_473_sorted__remdups__sorted,axiom,
! [Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( set_or6599480164596245535ed_nat @ Xs2 ) ) ) ).
% sorted_remdups_sorted
thf(fact_474_set__take__subset__set__take,axiom,
! [M: nat,N: nat,Xs2: list_list_nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ ( take_list_nat @ M @ Xs2 ) ) @ ( set_list_nat2 @ ( take_list_nat @ N @ Xs2 ) ) ) ) ).
% set_take_subset_set_take
thf(fact_475_set__take__subset__set__take,axiom,
! [M: nat,N: nat,Xs2: list_nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( take_nat @ M @ Xs2 ) ) @ ( set_nat2 @ ( take_nat @ N @ Xs2 ) ) ) ) ).
% set_take_subset_set_take
thf(fact_476_sorted__take,axiom,
! [Xs2: list_list_nat,N: nat] :
( ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ Xs2 )
=> ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ ( take_list_nat @ N @ Xs2 ) ) ) ).
% sorted_take
thf(fact_477_sorted__take,axiom,
! [Xs2: list_nat,N: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( take_nat @ N @ Xs2 ) ) ) ).
% sorted_take
thf(fact_478_diff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% diff_add
thf(fact_479_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_480_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_481_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_482_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_483_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_484_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_485_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_486_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_487_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_488_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_489_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_490_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_491_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_492_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_493_sorted0,axiom,
sorted_wrt_list_nat @ ord_less_eq_list_nat @ nil_list_nat ).
% sorted0
thf(fact_494_sorted0,axiom,
sorted_wrt_nat @ ord_less_eq_nat @ nil_nat ).
% sorted0
thf(fact_495_list_Oset__sel_I2_J,axiom,
! [A: list_list_nat,X2: list_nat] :
( ( A != nil_list_nat )
=> ( ( member_list_nat @ X2 @ ( set_list_nat2 @ ( tl_list_nat @ A ) ) )
=> ( member_list_nat @ X2 @ ( set_list_nat2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_496_list_Oset__sel_I2_J,axiom,
! [A: list_nat,X2: nat] :
( ( A != nil_nat )
=> ( ( member_nat @ X2 @ ( set_nat2 @ ( tl_nat @ A ) ) )
=> ( member_nat @ X2 @ ( set_nat2 @ A ) ) ) ) ).
% list.set_sel(2)
thf(fact_497_list_Omap__sel_I2_J,axiom,
! [A: list_nat,F: nat > list_nat] :
( ( A != nil_nat )
=> ( ( tl_list_nat @ ( map_nat_list_nat @ F @ A ) )
= ( map_nat_list_nat @ F @ ( tl_nat @ A ) ) ) ) ).
% list.map_sel(2)
thf(fact_498_list_Omap__sel_I2_J,axiom,
! [A: list_list_nat,F: list_nat > list_nat] :
( ( A != nil_list_nat )
=> ( ( tl_list_nat @ ( map_li7225945977422193158st_nat @ F @ A ) )
= ( map_li7225945977422193158st_nat @ F @ ( tl_list_nat @ A ) ) ) ) ).
% list.map_sel(2)
thf(fact_499_list_Omap__sel_I2_J,axiom,
! [A: list_list_nat,F: list_nat > nat] :
( ( A != nil_list_nat )
=> ( ( tl_nat @ ( map_list_nat_nat @ F @ A ) )
= ( map_list_nat_nat @ F @ ( tl_list_nat @ A ) ) ) ) ).
% list.map_sel(2)
thf(fact_500_list_Omap__sel_I2_J,axiom,
! [A: list_nat,F: nat > nat] :
( ( A != nil_nat )
=> ( ( tl_nat @ ( map_nat_nat @ F @ A ) )
= ( map_nat_nat @ F @ ( tl_nat @ A ) ) ) ) ).
% list.map_sel(2)
thf(fact_501_sorted__insort,axiom,
! [X2: list_nat,Xs2: list_list_nat] :
( ( sorted_wrt_list_nat @ ord_less_eq_list_nat
@ ( linord3838877917356005949st_nat
@ ^ [X: list_nat] : X
@ X2
@ Xs2 ) )
= ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ Xs2 ) ) ).
% sorted_insort
thf(fact_502_sorted__insort,axiom,
! [X2: nat,Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat
@ ( linord8961336180081300637at_nat
@ ^ [X: nat] : X
@ X2
@ Xs2 ) )
= ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 ) ) ).
% sorted_insort
thf(fact_503_insort__insert__insort,axiom,
! [X2: list_nat,Xs2: list_list_nat] :
( ~ ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs2 ) )
=> ( ( linord8148156736319587652st_nat
@ ^ [X: list_nat] : X
@ X2
@ Xs2 )
= ( linord3838877917356005949st_nat
@ ^ [X: list_nat] : X
@ X2
@ Xs2 ) ) ) ).
% insort_insert_insort
thf(fact_504_insort__insert__insort,axiom,
! [X2: nat,Xs2: list_nat] :
( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ( linord1921536354676448932at_nat
@ ^ [X: nat] : X
@ X2
@ Xs2 )
= ( linord8961336180081300637at_nat
@ ^ [X: nat] : X
@ X2
@ Xs2 ) ) ) ).
% insort_insert_insort
thf(fact_505_tl__take,axiom,
! [N: nat,Xs2: list_nat] :
( ( tl_nat @ ( take_nat @ N @ Xs2 ) )
= ( take_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( tl_nat @ Xs2 ) ) ) ).
% tl_take
thf(fact_506_sorted__insort__key,axiom,
! [F: list_nat > nat,X2: list_nat,Xs2: list_list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_list_nat_nat @ F @ ( linord5978504541935096237at_nat @ F @ X2 @ Xs2 ) ) )
= ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_list_nat_nat @ F @ Xs2 ) ) ) ).
% sorted_insort_key
thf(fact_507_sorted__insort__key,axiom,
! [F: nat > list_nat,X2: nat,Xs2: list_nat] :
( ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ ( map_nat_list_nat @ F @ ( linord3739547494772811053st_nat @ F @ X2 @ Xs2 ) ) )
= ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ ( map_nat_list_nat @ F @ Xs2 ) ) ) ).
% sorted_insort_key
thf(fact_508_sorted__insort__key,axiom,
! [F: list_nat > list_nat,X2: list_nat,Xs2: list_list_nat] :
( ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ ( map_li7225945977422193158st_nat @ F @ ( linord3838877917356005949st_nat @ F @ X2 @ Xs2 ) ) )
= ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ ( map_li7225945977422193158st_nat @ F @ Xs2 ) ) ) ).
% sorted_insort_key
thf(fact_509_sorted__insort__key,axiom,
! [F: nat > nat,X2: nat,Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ ( linord8961336180081300637at_nat @ F @ X2 @ Xs2 ) ) )
= ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ Xs2 ) ) ) ).
% sorted_insort_key
thf(fact_510_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_511_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_512_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_513_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_514_subset__antisym,axiom,
! [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% subset_antisym
thf(fact_515_subsetI,axiom,
! [A4: set_list_nat,B4: set_list_nat] :
( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ A4 )
=> ( member_list_nat @ X3 @ B4 ) )
=> ( ord_le6045566169113846134st_nat @ A4 @ B4 ) ) ).
% subsetI
thf(fact_516_subsetI,axiom,
! [A4: set_nat,B4: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A4 )
=> ( member_nat @ X3 @ B4 ) )
=> ( ord_less_eq_set_nat @ A4 @ B4 ) ) ).
% subsetI
thf(fact_517_le__Nil,axiom,
! [X2: list_list_nat] :
( ( ord_le6806709344281226192st_nat @ X2 @ nil_list_nat )
= ( X2 = nil_list_nat ) ) ).
% le_Nil
thf(fact_518_le__Nil,axiom,
! [X2: list_nat] :
( ( ord_less_eq_list_nat @ X2 @ nil_nat )
= ( X2 = nil_nat ) ) ).
% le_Nil
thf(fact_519_in__set__simps_I3_J,axiom,
! [X2: list_nat] :
~ ( member_list_nat @ X2 @ ( set_list_nat2 @ nil_list_nat ) ) ).
% in_set_simps(3)
thf(fact_520_in__set__simps_I3_J,axiom,
! [X2: nat] :
~ ( member_nat @ X2 @ ( set_nat2 @ nil_nat ) ) ).
% in_set_simps(3)
thf(fact_521_less__eq__list__code_I2_J,axiom,
! [Xs2: list_list_nat] : ( ord_le6806709344281226192st_nat @ nil_list_nat @ Xs2 ) ).
% less_eq_list_code(2)
thf(fact_522_less__eq__list__code_I2_J,axiom,
! [Xs2: list_nat] : ( ord_less_eq_list_nat @ nil_nat @ Xs2 ) ).
% less_eq_list_code(2)
thf(fact_523_Nil__le__Cons,axiom,
! [X2: list_list_nat] : ( ord_le6806709344281226192st_nat @ nil_list_nat @ X2 ) ).
% Nil_le_Cons
thf(fact_524_Nil__le__Cons,axiom,
! [X2: list_nat] : ( ord_less_eq_list_nat @ nil_nat @ X2 ) ).
% Nil_le_Cons
thf(fact_525_in__mono,axiom,
! [A4: set_list_nat,B4: set_list_nat,X2: list_nat] :
( ( ord_le6045566169113846134st_nat @ A4 @ B4 )
=> ( ( member_list_nat @ X2 @ A4 )
=> ( member_list_nat @ X2 @ B4 ) ) ) ).
% in_mono
thf(fact_526_in__mono,axiom,
! [A4: set_nat,B4: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( ( member_nat @ X2 @ A4 )
=> ( member_nat @ X2 @ B4 ) ) ) ).
% in_mono
thf(fact_527_subsetD,axiom,
! [A4: set_list_nat,B4: set_list_nat,C: list_nat] :
( ( ord_le6045566169113846134st_nat @ A4 @ B4 )
=> ( ( member_list_nat @ C @ A4 )
=> ( member_list_nat @ C @ B4 ) ) ) ).
% subsetD
thf(fact_528_subsetD,axiom,
! [A4: set_nat,B4: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( ( member_nat @ C @ A4 )
=> ( member_nat @ C @ B4 ) ) ) ).
% subsetD
thf(fact_529_Diff__mono,axiom,
! [A4: set_nat,C4: set_nat,D2: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ C4 )
=> ( ( ord_less_eq_set_nat @ D2 @ B4 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ B4 ) @ ( minus_minus_set_nat @ C4 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_530_equalityE,axiom,
! [A4: set_nat,B4: set_nat] :
( ( A4 = B4 )
=> ~ ( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ~ ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ).
% equalityE
thf(fact_531_subset__eq,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A5: set_list_nat,B5: set_list_nat] :
! [X: list_nat] :
( ( member_list_nat @ X @ A5 )
=> ( member_list_nat @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_532_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A5 )
=> ( member_nat @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_533_equalityD1,axiom,
! [A4: set_nat,B4: set_nat] :
( ( A4 = B4 )
=> ( ord_less_eq_set_nat @ A4 @ B4 ) ) ).
% equalityD1
thf(fact_534_Set_OequalityD2,axiom,
! [A4: set_nat,B4: set_nat] :
( ( A4 = B4 )
=> ( ord_less_eq_set_nat @ B4 @ A4 ) ) ).
% Set.equalityD2
thf(fact_535_subset__iff,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A5: set_list_nat,B5: set_list_nat] :
! [T2: list_nat] :
( ( member_list_nat @ T2 @ A5 )
=> ( member_list_nat @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_536_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A5 )
=> ( member_nat @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_537_Diff__subset,axiom,
! [A4: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ B4 ) @ A4 ) ).
% Diff_subset
thf(fact_538_double__diff,axiom,
! [A4: set_nat,B4: set_nat,C4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ C4 )
=> ( ( minus_minus_set_nat @ B4 @ ( minus_minus_set_nat @ C4 @ A4 ) )
= A4 ) ) ) ).
% double_diff
thf(fact_539_subset__refl,axiom,
! [A4: set_nat] : ( ord_less_eq_set_nat @ A4 @ A4 ) ).
% subset_refl
thf(fact_540_Collect__mono,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ! [X3: list_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_541_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_542_subset__trans,axiom,
! [A4: set_nat,B4: set_nat,C4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
=> ( ( ord_less_eq_set_nat @ B4 @ C4 )
=> ( ord_less_eq_set_nat @ A4 @ C4 ) ) ) ).
% subset_trans
thf(fact_543_set__eq__subset,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_544_Collect__mono__iff,axiom,
! [P: list_nat > $o,Q: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) )
= ( ! [X: list_nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_545_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_546_Collect__subset,axiom,
! [A4: set_list_nat,P: list_nat > $o] :
( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X: list_nat] :
( ( member_list_nat @ X @ A4 )
& ( P @ X ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_547_Collect__subset,axiom,
! [A4: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A4 )
& ( P @ X ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_548_less__eq__set__def,axiom,
( ord_le6045566169113846134st_nat
= ( ^ [A5: set_list_nat,B5: set_list_nat] :
( ord_le1520216061033275535_nat_o
@ ^ [X: list_nat] : ( member_list_nat @ X @ A5 )
@ ^ [X: list_nat] : ( member_list_nat @ X @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_549_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A5 )
@ ^ [X: nat] : ( member_nat @ X @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_550_set__subtract__list__sorted,axiom,
! [Xs2: list_list_nat,Ys: list_list_nat] :
( ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ Xs2 )
=> ( ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ Ys )
=> ( ( set_list_nat2 @ ( missin8418467895465290280st_nat @ Xs2 @ Ys ) )
= ( minus_7954133019191499631st_nat @ ( set_list_nat2 @ Xs2 ) @ ( set_list_nat2 @ Ys ) ) ) ) ) ).
% set_subtract_list_sorted
thf(fact_551_set__subtract__list__sorted,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ys )
=> ( ( set_nat2 @ ( missin6424796737333596952ed_nat @ Xs2 @ Ys ) )
= ( minus_minus_set_nat @ ( set_nat2 @ Xs2 ) @ ( set_nat2 @ Ys ) ) ) ) ) ).
% set_subtract_list_sorted
thf(fact_552_remdups__sort_I2_J,axiom,
! [Xs2: list_list_nat] :
( ( set_list_nat2 @ ( missin2640887136080315381st_nat @ Xs2 ) )
= ( set_list_nat2 @ Xs2 ) ) ).
% remdups_sort(2)
thf(fact_553_remdups__sort_I2_J,axiom,
! [Xs2: list_nat] :
( ( set_nat2 @ ( missin6101193410121742181rt_nat @ Xs2 ) )
= ( set_nat2 @ Xs2 ) ) ).
% remdups_sort(2)
thf(fact_554_set__list__diff,axiom,
! [Xs2: list_list_nat,Ys: list_list_nat] :
( ( set_list_nat2 @ ( missin6169663638126051804st_nat @ Xs2 @ Ys ) )
= ( minus_7954133019191499631st_nat @ ( set_list_nat2 @ Xs2 ) @ ( set_list_nat2 @ Ys ) ) ) ).
% set_list_diff
thf(fact_555_set__list__diff,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( set_nat2 @ ( missin818507234016924876ff_nat @ Xs2 @ Ys ) )
= ( minus_minus_set_nat @ ( set_nat2 @ Xs2 ) @ ( set_nat2 @ Ys ) ) ) ).
% set_list_diff
thf(fact_556_remdups__sort_I1_J,axiom,
! [Xs2: list_list_nat] : ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ ( missin2640887136080315381st_nat @ Xs2 ) ) ).
% remdups_sort(1)
thf(fact_557_remdups__sort_I1_J,axiom,
! [Xs2: list_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( missin6101193410121742181rt_nat @ Xs2 ) ) ).
% remdups_sort(1)
thf(fact_558_last__list__update,axiom,
! [Xs2: list_list_nat,K: nat,X2: list_nat] :
( ( Xs2 != nil_list_nat )
=> ( ( ( K
= ( minus_minus_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ one_one_nat ) )
=> ( ( last_list_nat @ ( list_update_list_nat @ Xs2 @ K @ X2 ) )
= X2 ) )
& ( ( K
!= ( minus_minus_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ one_one_nat ) )
=> ( ( last_list_nat @ ( list_update_list_nat @ Xs2 @ K @ X2 ) )
= ( last_list_nat @ Xs2 ) ) ) ) ) ).
% last_list_update
thf(fact_559_last__list__update,axiom,
! [Xs2: list_set_nat,K: nat,X2: set_nat] :
( ( Xs2 != nil_set_nat )
=> ( ( ( K
= ( minus_minus_nat @ ( size_s3254054031482475050et_nat @ Xs2 ) @ one_one_nat ) )
=> ( ( last_set_nat @ ( list_update_set_nat @ Xs2 @ K @ X2 ) )
= X2 ) )
& ( ( K
!= ( minus_minus_nat @ ( size_s3254054031482475050et_nat @ Xs2 ) @ one_one_nat ) )
=> ( ( last_set_nat @ ( list_update_set_nat @ Xs2 @ K @ X2 ) )
= ( last_set_nat @ Xs2 ) ) ) ) ) ).
% last_list_update
thf(fact_560_last__list__update,axiom,
! [Xs2: list_nat,K: nat,X2: nat] :
( ( Xs2 != nil_nat )
=> ( ( ( K
= ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) )
=> ( ( last_nat @ ( list_update_nat @ Xs2 @ K @ X2 ) )
= X2 ) )
& ( ( K
!= ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) )
=> ( ( last_nat @ ( list_update_nat @ Xs2 @ K @ X2 ) )
= ( last_nat @ Xs2 ) ) ) ) ) ).
% last_list_update
thf(fact_561_sorted__quicksort,axiom,
! [Xs2: list_list_nat] : ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ ( set_or6919174964970218831st_nat @ Xs2 ) ) ).
% sorted_quicksort
thf(fact_562_sorted__quicksort,axiom,
! [Xs2: list_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( set_or9089632773640736191rt_nat @ Xs2 ) ) ).
% sorted_quicksort
thf(fact_563_gen__length__def,axiom,
( gen_length_list_nat
= ( ^ [N3: nat,Xs: list_list_nat] : ( plus_plus_nat @ N3 @ ( size_s3023201423986296836st_nat @ Xs ) ) ) ) ).
% gen_length_def
thf(fact_564_gen__length__def,axiom,
( gen_length_set_nat
= ( ^ [N3: nat,Xs: list_set_nat] : ( plus_plus_nat @ N3 @ ( size_s3254054031482475050et_nat @ Xs ) ) ) ) ).
% gen_length_def
thf(fact_565_gen__length__def,axiom,
( gen_length_nat
= ( ^ [N3: nat,Xs: list_nat] : ( plus_plus_nat @ N3 @ ( size_size_list_nat @ Xs ) ) ) ) ).
% gen_length_def
thf(fact_566_Diff__idemp,axiom,
! [A4: set_nat,B4: set_nat] :
( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A4 @ B4 ) @ B4 )
= ( minus_minus_set_nat @ A4 @ B4 ) ) ).
% Diff_idemp
thf(fact_567_Diff__iff,axiom,
! [C: list_nat,A4: set_list_nat,B4: set_list_nat] :
( ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A4 @ B4 ) )
= ( ( member_list_nat @ C @ A4 )
& ~ ( member_list_nat @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_568_Diff__iff,axiom,
! [C: nat,A4: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B4 ) )
= ( ( member_nat @ C @ A4 )
& ~ ( member_nat @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_569_DiffI,axiom,
! [C: list_nat,A4: set_list_nat,B4: set_list_nat] :
( ( member_list_nat @ C @ A4 )
=> ( ~ ( member_list_nat @ C @ B4 )
=> ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A4 @ B4 ) ) ) ) ).
% DiffI
thf(fact_570_DiffI,axiom,
! [C: nat,A4: set_nat,B4: set_nat] :
( ( member_nat @ C @ A4 )
=> ( ~ ( member_nat @ C @ B4 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B4 ) ) ) ) ).
% DiffI
thf(fact_571_set__quicksort,axiom,
! [Xs2: list_list_nat] :
( ( set_list_nat2 @ ( set_or6919174964970218831st_nat @ Xs2 ) )
= ( set_list_nat2 @ Xs2 ) ) ).
% set_quicksort
thf(fact_572_set__quicksort,axiom,
! [Xs2: list_nat] :
( ( set_nat2 @ ( set_or9089632773640736191rt_nat @ Xs2 ) )
= ( set_nat2 @ Xs2 ) ) ).
% set_quicksort
thf(fact_573_minus__set__def,axiom,
( minus_7954133019191499631st_nat
= ( ^ [A5: set_list_nat,B5: set_list_nat] :
( collect_list_nat
@ ( minus_1139252259498527702_nat_o
@ ^ [X: list_nat] : ( member_list_nat @ X @ A5 )
@ ^ [X: list_nat] : ( member_list_nat @ X @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_574_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A5 )
@ ^ [X: nat] : ( member_nat @ X @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_575_set__diff__eq,axiom,
( minus_7954133019191499631st_nat
= ( ^ [A5: set_list_nat,B5: set_list_nat] :
( collect_list_nat
@ ^ [X: list_nat] :
( ( member_list_nat @ X @ A5 )
& ~ ( member_list_nat @ X @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_576_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A5 )
& ~ ( member_nat @ X @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_577_DiffD2,axiom,
! [C: list_nat,A4: set_list_nat,B4: set_list_nat] :
( ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A4 @ B4 ) )
=> ~ ( member_list_nat @ C @ B4 ) ) ).
% DiffD2
thf(fact_578_DiffD2,axiom,
! [C: nat,A4: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B4 ) )
=> ~ ( member_nat @ C @ B4 ) ) ).
% DiffD2
thf(fact_579_DiffD1,axiom,
! [C: list_nat,A4: set_list_nat,B4: set_list_nat] :
( ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A4 @ B4 ) )
=> ( member_list_nat @ C @ A4 ) ) ).
% DiffD1
thf(fact_580_DiffD1,axiom,
! [C: nat,A4: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B4 ) )
=> ( member_nat @ C @ A4 ) ) ).
% DiffD1
thf(fact_581_DiffE,axiom,
! [C: list_nat,A4: set_list_nat,B4: set_list_nat] :
( ( member_list_nat @ C @ ( minus_7954133019191499631st_nat @ A4 @ B4 ) )
=> ~ ( ( member_list_nat @ C @ A4 )
=> ( member_list_nat @ C @ B4 ) ) ) ).
% DiffE
thf(fact_582_DiffE,axiom,
! [C: nat,A4: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B4 ) )
=> ~ ( ( member_nat @ C @ A4 )
=> ( member_nat @ C @ B4 ) ) ) ).
% DiffE
thf(fact_583_subtract__list__sorted_Osimps_I2_J,axiom,
! [Ys: list_list_nat] :
( ( missin8418467895465290280st_nat @ nil_list_nat @ Ys )
= nil_list_nat ) ).
% subtract_list_sorted.simps(2)
thf(fact_584_subtract__list__sorted_Osimps_I2_J,axiom,
! [Ys: list_nat] :
( ( missin6424796737333596952ed_nat @ nil_nat @ Ys )
= nil_nat ) ).
% subtract_list_sorted.simps(2)
thf(fact_585_last__in__set,axiom,
! [As2: list_list_nat] :
( ( As2 != nil_list_nat )
=> ( member_list_nat @ ( last_list_nat @ As2 ) @ ( set_list_nat2 @ As2 ) ) ) ).
% last_in_set
thf(fact_586_last__in__set,axiom,
! [As2: list_nat] :
( ( As2 != nil_nat )
=> ( member_nat @ ( last_nat @ As2 ) @ ( set_nat2 @ As2 ) ) ) ).
% last_in_set
thf(fact_587_last__map,axiom,
! [Xs2: list_nat,F: nat > list_nat] :
( ( Xs2 != nil_nat )
=> ( ( last_list_nat @ ( map_nat_list_nat @ F @ Xs2 ) )
= ( F @ ( last_nat @ Xs2 ) ) ) ) ).
% last_map
thf(fact_588_last__map,axiom,
! [Xs2: list_list_nat,F: list_nat > list_nat] :
( ( Xs2 != nil_list_nat )
=> ( ( last_list_nat @ ( map_li7225945977422193158st_nat @ F @ Xs2 ) )
= ( F @ ( last_list_nat @ Xs2 ) ) ) ) ).
% last_map
thf(fact_589_last__map,axiom,
! [Xs2: list_list_nat,F: list_nat > nat] :
( ( Xs2 != nil_list_nat )
=> ( ( last_nat @ ( map_list_nat_nat @ F @ Xs2 ) )
= ( F @ ( last_list_nat @ Xs2 ) ) ) ) ).
% last_map
thf(fact_590_last__map,axiom,
! [Xs2: list_nat,F: nat > nat] :
( ( Xs2 != nil_nat )
=> ( ( last_nat @ ( map_nat_nat @ F @ Xs2 ) )
= ( F @ ( last_nat @ Xs2 ) ) ) ) ).
% last_map
thf(fact_591_last__tl,axiom,
! [Xs2: list_list_nat] :
( ( ( Xs2 = nil_list_nat )
| ( ( tl_list_nat @ Xs2 )
!= nil_list_nat ) )
=> ( ( last_list_nat @ ( tl_list_nat @ Xs2 ) )
= ( last_list_nat @ Xs2 ) ) ) ).
% last_tl
thf(fact_592_last__tl,axiom,
! [Xs2: list_nat] :
( ( ( Xs2 = nil_nat )
| ( ( tl_nat @ Xs2 )
!= nil_nat ) )
=> ( ( last_nat @ ( tl_nat @ Xs2 ) )
= ( last_nat @ Xs2 ) ) ) ).
% last_tl
thf(fact_593_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_list_nat @ N @ nil_list_nat )
= N ) ).
% gen_length_code(1)
thf(fact_594_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_nat @ N @ nil_nat )
= N ) ).
% gen_length_code(1)
thf(fact_595_subset__subtract__listed__sorted,axiom,
! [Xs2: list_list_nat,Ys: list_list_nat] : ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ ( missin8418467895465290280st_nat @ Xs2 @ Ys ) ) @ ( set_list_nat2 @ Xs2 ) ) ).
% subset_subtract_listed_sorted
thf(fact_596_subset__subtract__listed__sorted,axiom,
! [Xs2: list_nat,Ys: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( missin6424796737333596952ed_nat @ Xs2 @ Ys ) ) @ ( set_nat2 @ Xs2 ) ) ).
% subset_subtract_listed_sorted
thf(fact_597_list__diff_Osimps_I1_J,axiom,
! [Ys: list_list_nat] :
( ( missin6169663638126051804st_nat @ nil_list_nat @ Ys )
= nil_list_nat ) ).
% list_diff.simps(1)
thf(fact_598_list__diff_Osimps_I1_J,axiom,
! [Ys: list_nat] :
( ( missin818507234016924876ff_nat @ nil_nat @ Ys )
= nil_nat ) ).
% list_diff.simps(1)
thf(fact_599_quicksort__conv__sort,axiom,
( set_or9089632773640736191rt_nat
= ( linord738340561235409698at_nat
@ ^ [X: nat] : X ) ) ).
% quicksort_conv_sort
thf(fact_600_sorted__last,axiom,
! [Xs2: list_list_nat,X2: list_nat] :
( ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ Xs2 )
=> ( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xs2 ) )
=> ( ord_less_eq_list_nat @ X2 @ ( last_list_nat @ Xs2 ) ) ) ) ).
% sorted_last
thf(fact_601_sorted__last,axiom,
! [Xs2: list_nat,X2: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
=> ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ord_less_eq_nat @ X2 @ ( last_nat @ Xs2 ) ) ) ) ).
% sorted_last
thf(fact_602_pred__subset__eq,axiom,
! [R: set_list_nat,S: set_list_nat] :
( ( ord_le1520216061033275535_nat_o
@ ^ [X: list_nat] : ( member_list_nat @ X @ R )
@ ^ [X: list_nat] : ( member_list_nat @ X @ S ) )
= ( ord_le6045566169113846134st_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_603_pred__subset__eq,axiom,
! [R: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ R )
@ ^ [X: nat] : ( member_nat @ X @ S ) )
= ( ord_less_eq_set_nat @ R @ S ) ) ).
% pred_subset_eq
thf(fact_604_last__conv__nth,axiom,
! [Xs2: list_list_nat] :
( ( Xs2 != nil_list_nat )
=> ( ( last_list_nat @ Xs2 )
= ( nth_list_nat @ Xs2 @ ( minus_minus_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_605_last__conv__nth,axiom,
! [Xs2: list_set_nat] :
( ( Xs2 != nil_set_nat )
=> ( ( last_set_nat @ Xs2 )
= ( nth_set_nat @ Xs2 @ ( minus_minus_nat @ ( size_s3254054031482475050et_nat @ Xs2 ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_606_last__conv__nth,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( last_nat @ Xs2 )
= ( nth_nat @ Xs2 @ ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_607_subset__code_I3_J,axiom,
~ ( ord_le6045566169113846134st_nat @ ( coset_list_nat @ nil_list_nat ) @ ( set_list_nat2 @ nil_list_nat ) ) ).
% subset_code(3)
thf(fact_608_subset__code_I3_J,axiom,
~ ( ord_less_eq_set_nat @ ( coset_nat @ nil_nat ) @ ( set_nat2 @ nil_nat ) ) ).
% subset_code(3)
thf(fact_609_subset__Collect__iff,axiom,
! [B4: set_list_nat,A4: set_list_nat,P: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ B4 @ A4 )
=> ( ( ord_le6045566169113846134st_nat @ B4
@ ( collect_list_nat
@ ^ [X: list_nat] :
( ( member_list_nat @ X @ A4 )
& ( P @ X ) ) ) )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ B4 )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_610_subset__Collect__iff,axiom,
! [B4: set_nat,A4: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
=> ( ( ord_less_eq_set_nat @ B4
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A4 )
& ( P @ X ) ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ B4 )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_611_subset__CollectI,axiom,
! [B4: set_list_nat,A4: set_list_nat,Q: list_nat > $o,P: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ B4 @ A4 )
=> ( ! [X3: list_nat] :
( ( member_list_nat @ X3 @ B4 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X: list_nat] :
( ( member_list_nat @ X @ B4 )
& ( Q @ X ) ) )
@ ( collect_list_nat
@ ^ [X: list_nat] :
( ( member_list_nat @ X @ A4 )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_612_subset__CollectI,axiom,
! [B4: set_nat,A4: set_nat,Q: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B4 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ B4 )
& ( Q @ X ) ) )
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A4 )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_613_nth__list__update__neq,axiom,
! [I: nat,J: nat,Xs2: list_list_nat,X2: list_nat] :
( ( I != J )
=> ( ( nth_list_nat @ ( list_update_list_nat @ Xs2 @ I @ X2 ) @ J )
= ( nth_list_nat @ Xs2 @ J ) ) ) ).
% nth_list_update_neq
thf(fact_614_nth__list__update__neq,axiom,
! [I: nat,J: nat,Xs2: list_set_nat,X2: set_nat] :
( ( I != J )
=> ( ( nth_set_nat @ ( list_update_set_nat @ Xs2 @ I @ X2 ) @ J )
= ( nth_set_nat @ Xs2 @ J ) ) ) ).
% nth_list_update_neq
thf(fact_615_nth__list__update__neq,axiom,
! [I: nat,J: nat,Xs2: list_nat,X2: nat] :
( ( I != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X2 ) @ J )
= ( nth_nat @ Xs2 @ J ) ) ) ).
% nth_list_update_neq
thf(fact_616_list__update__id,axiom,
! [Xs2: list_list_nat,I: nat] :
( ( list_update_list_nat @ Xs2 @ I @ ( nth_list_nat @ Xs2 @ I ) )
= Xs2 ) ).
% list_update_id
thf(fact_617_list__update__id,axiom,
! [Xs2: list_set_nat,I: nat] :
( ( list_update_set_nat @ Xs2 @ I @ ( nth_set_nat @ Xs2 @ I ) )
= Xs2 ) ).
% list_update_id
thf(fact_618_list__update__id,axiom,
! [Xs2: list_nat,I: nat] :
( ( list_update_nat @ Xs2 @ I @ ( nth_nat @ Xs2 @ I ) )
= Xs2 ) ).
% list_update_id
thf(fact_619_subset__code_I2_J,axiom,
! [A4: set_list_nat,Ys: list_list_nat] :
( ( ord_le6045566169113846134st_nat @ A4 @ ( coset_list_nat @ Ys ) )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Ys ) )
=> ~ ( member_list_nat @ X @ A4 ) ) ) ) ).
% subset_code(2)
thf(fact_620_subset__code_I2_J,axiom,
! [A4: set_nat,Ys: list_nat] :
( ( ord_less_eq_set_nat @ A4 @ ( coset_nat @ Ys ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Ys ) )
=> ~ ( member_nat @ X @ A4 ) ) ) ) ).
% subset_code(2)
thf(fact_621_set__swap,axiom,
! [I: nat,Xs2: list_list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ( ord_less_nat @ J @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ( set_list_nat2 @ ( list_update_list_nat @ ( list_update_list_nat @ Xs2 @ I @ ( nth_list_nat @ Xs2 @ J ) ) @ J @ ( nth_list_nat @ Xs2 @ I ) ) )
= ( set_list_nat2 @ Xs2 ) ) ) ) ).
% set_swap
thf(fact_622_set__swap,axiom,
! [I: nat,Xs2: list_set_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_s3254054031482475050et_nat @ Xs2 ) )
=> ( ( ord_less_nat @ J @ ( size_s3254054031482475050et_nat @ Xs2 ) )
=> ( ( set_set_nat2 @ ( list_update_set_nat @ ( list_update_set_nat @ Xs2 @ I @ ( nth_set_nat @ Xs2 @ J ) ) @ J @ ( nth_set_nat @ Xs2 @ I ) ) )
= ( set_set_nat2 @ Xs2 ) ) ) ) ).
% set_swap
thf(fact_623_set__swap,axiom,
! [I: nat,Xs2: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs2 @ I @ ( nth_nat @ Xs2 @ J ) ) @ J @ ( nth_nat @ Xs2 @ I ) ) )
= ( set_nat2 @ Xs2 ) ) ) ) ).
% set_swap
thf(fact_624_sorted__iff__nth__mono,axiom,
! [Xs2: list_list_nat] :
( ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ Xs2 )
= ( ! [I4: nat,J2: nat] :
( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ord_less_eq_list_nat @ ( nth_list_nat @ Xs2 @ I4 ) @ ( nth_list_nat @ Xs2 @ J2 ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_625_sorted__iff__nth__mono,axiom,
! [Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
= ( ! [I4: nat,J2: nat] :
( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs2 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ I4 ) @ ( nth_nat @ Xs2 @ J2 ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_626_sorted__nth__mono,axiom,
! [Xs2: list_list_nat,I: nat,J: nat] :
( ( sorted_wrt_list_nat @ ord_less_eq_list_nat @ Xs2 )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ord_less_eq_list_nat @ ( nth_list_nat @ Xs2 @ I ) @ ( nth_list_nat @ Xs2 @ J ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_627_sorted__nth__mono,axiom,
! [Xs2: list_nat,I: nat,J: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ I ) @ ( nth_nat @ Xs2 @ J ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_628_conj__subset__def,axiom,
! [A4: set_list_nat,P: list_nat > $o,Q: list_nat > $o] :
( ( ord_le6045566169113846134st_nat @ A4
@ ( collect_list_nat
@ ^ [X: list_nat] :
( ( P @ X )
& ( Q @ X ) ) ) )
= ( ( ord_le6045566169113846134st_nat @ A4 @ ( collect_list_nat @ P ) )
& ( ord_le6045566169113846134st_nat @ A4 @ ( collect_list_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_629_conj__subset__def,axiom,
! [A4: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A4
@ ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
& ( Q @ X ) ) ) )
= ( ( ord_less_eq_set_nat @ A4 @ ( collect_nat @ P ) )
& ( ord_less_eq_set_nat @ A4 @ ( collect_nat @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_630_prop__restrict,axiom,
! [X2: list_nat,Z4: set_list_nat,X6: set_list_nat,P: list_nat > $o] :
( ( member_list_nat @ X2 @ Z4 )
=> ( ( ord_le6045566169113846134st_nat @ Z4
@ ( collect_list_nat
@ ^ [X: list_nat] :
( ( member_list_nat @ X @ X6 )
& ( P @ X ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_631_prop__restrict,axiom,
! [X2: nat,Z4: set_nat,X6: set_nat,P: nat > $o] :
( ( member_nat @ X2 @ Z4 )
=> ( ( ord_less_eq_set_nat @ Z4
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ X6 )
& ( P @ X ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_632_Collect__restrict,axiom,
! [X6: set_list_nat,P: list_nat > $o] :
( ord_le6045566169113846134st_nat
@ ( collect_list_nat
@ ^ [X: list_nat] :
( ( member_list_nat @ X @ X6 )
& ( P @ X ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_633_Collect__restrict,axiom,
! [X6: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ X6 )
& ( P @ X ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_634_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_635_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_636_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_637_nth__take,axiom,
! [I: nat,N: nat,Xs2: list_list_nat] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_list_nat @ ( take_list_nat @ N @ Xs2 ) @ I )
= ( nth_list_nat @ Xs2 @ I ) ) ) ).
% nth_take
thf(fact_638_nth__take,axiom,
! [I: nat,N: nat,Xs2: list_set_nat] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_set_nat @ ( take_set_nat @ N @ Xs2 ) @ I )
= ( nth_set_nat @ Xs2 @ I ) ) ) ).
% nth_take
thf(fact_639_nth__take,axiom,
! [I: nat,N: nat,Xs2: list_nat] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_nat @ ( take_nat @ N @ Xs2 ) @ I )
= ( nth_nat @ Xs2 @ I ) ) ) ).
% nth_take
thf(fact_640_nth__replicate,axiom,
! [I: nat,N: nat,X2: list_nat] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_list_nat @ ( replicate_list_nat @ N @ X2 ) @ I )
= X2 ) ) ).
% nth_replicate
thf(fact_641_nth__replicate,axiom,
! [I: nat,N: nat,X2: set_nat] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_set_nat @ ( replicate_set_nat @ N @ X2 ) @ I )
= X2 ) ) ).
% nth_replicate
thf(fact_642_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_643_nth__map,axiom,
! [N: nat,Xs2: list_nat,F: nat > set_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_set_nat @ ( map_nat_set_nat @ F @ Xs2 ) @ N )
= ( F @ ( nth_nat @ Xs2 @ N ) ) ) ) ).
% nth_map
thf(fact_644_nth__map,axiom,
! [N: nat,Xs2: list_nat,F: nat > list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_list_nat @ ( map_nat_list_nat @ F @ Xs2 ) @ N )
= ( F @ ( nth_nat @ Xs2 @ N ) ) ) ) ).
% nth_map
thf(fact_645_nth__map,axiom,
! [N: nat,Xs2: list_list_nat,F: list_nat > set_nat] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ( nth_set_nat @ ( map_list_nat_set_nat @ F @ Xs2 ) @ N )
= ( F @ ( nth_list_nat @ Xs2 @ N ) ) ) ) ).
% nth_map
thf(fact_646_nth__map,axiom,
! [N: nat,Xs2: list_list_nat,F: list_nat > list_nat] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ( nth_list_nat @ ( map_li7225945977422193158st_nat @ F @ Xs2 ) @ N )
= ( F @ ( nth_list_nat @ Xs2 @ N ) ) ) ) ).
% nth_map
thf(fact_647_nth__map,axiom,
! [N: nat,Xs2: list_list_nat,F: list_nat > nat] :
( ( ord_less_nat @ N @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ( nth_nat @ ( map_list_nat_nat @ F @ Xs2 ) @ N )
= ( F @ ( nth_list_nat @ Xs2 @ N ) ) ) ) ).
% nth_map
thf(fact_648_nth__map,axiom,
! [N: nat,Xs2: list_set_nat,F: set_nat > nat] :
( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs2 ) )
=> ( ( nth_nat @ ( map_set_nat_nat @ F @ Xs2 ) @ N )
= ( F @ ( nth_set_nat @ Xs2 @ N ) ) ) ) ).
% nth_map
thf(fact_649_nth__map,axiom,
! [N: nat,Xs2: list_set_nat,F: set_nat > list_nat] :
( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs2 ) )
=> ( ( nth_list_nat @ ( map_set_nat_list_nat @ F @ Xs2 ) @ N )
= ( F @ ( nth_set_nat @ Xs2 @ N ) ) ) ) ).
% nth_map
thf(fact_650_nth__map,axiom,
! [N: nat,Xs2: list_set_nat,F: set_nat > set_nat] :
( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs2 ) )
=> ( ( nth_set_nat @ ( map_set_nat_set_nat @ F @ Xs2 ) @ N )
= ( F @ ( nth_set_nat @ Xs2 @ N ) ) ) ) ).
% nth_map
thf(fact_651_nth__map,axiom,
! [N: nat,Xs2: list_nat,F: nat > nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ ( map_nat_nat @ F @ Xs2 ) @ N )
= ( F @ ( nth_nat @ Xs2 @ N ) ) ) ) ).
% nth_map
thf(fact_652_nth__list__update__eq,axiom,
! [I: nat,Xs2: list_list_nat,X2: list_nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ( nth_list_nat @ ( list_update_list_nat @ Xs2 @ I @ X2 ) @ I )
= X2 ) ) ).
% nth_list_update_eq
thf(fact_653_nth__list__update__eq,axiom,
! [I: nat,Xs2: list_set_nat,X2: set_nat] :
( ( ord_less_nat @ I @ ( size_s3254054031482475050et_nat @ Xs2 ) )
=> ( ( nth_set_nat @ ( list_update_set_nat @ Xs2 @ I @ X2 ) @ I )
= X2 ) ) ).
% nth_list_update_eq
thf(fact_654_nth__list__update__eq,axiom,
! [I: nat,Xs2: list_nat,X2: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X2 ) @ I )
= X2 ) ) ).
% nth_list_update_eq
thf(fact_655_gt__ex,axiom,
! [X2: nat] :
? [X_12: nat] : ( ord_less_nat @ X2 @ X_12 ) ).
% gt_ex
thf(fact_656_less__imp__neq,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( ord_less_list_nat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_657_less__imp__neq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% less_imp_neq
thf(fact_658_order_Oasym,axiom,
! [A: list_nat,B: list_nat] :
( ( ord_less_list_nat @ A @ B )
=> ~ ( ord_less_list_nat @ B @ A ) ) ).
% order.asym
thf(fact_659_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_660_ord__eq__less__trans,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( A = B )
=> ( ( ord_less_list_nat @ B @ C )
=> ( ord_less_list_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_661_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_662_ord__less__eq__trans,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( ord_less_list_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_list_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_663_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_664_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_665_antisym__conv3,axiom,
! [Y2: list_nat,X2: list_nat] :
( ~ ( ord_less_list_nat @ Y2 @ X2 )
=> ( ( ~ ( ord_less_list_nat @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_666_antisym__conv3,axiom,
! [Y2: nat,X2: nat] :
( ~ ( ord_less_nat @ Y2 @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_667_linorder__cases,axiom,
! [X2: list_nat,Y2: list_nat] :
( ~ ( ord_less_list_nat @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_list_nat @ Y2 @ X2 ) ) ) ).
% linorder_cases
thf(fact_668_linorder__cases,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_cases
thf(fact_669_dual__order_Oasym,axiom,
! [B: list_nat,A: list_nat] :
( ( ord_less_list_nat @ B @ A )
=> ~ ( ord_less_list_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_670_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_671_dual__order_Oirrefl,axiom,
! [A: list_nat] :
~ ( ord_less_list_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_672_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_673_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X7: nat] : ( P2 @ X7 ) )
= ( ^ [P3: nat > $o] :
? [N3: nat] :
( ( P3 @ N3 )
& ! [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ~ ( P3 @ M4 ) ) ) ) ) ).
% exists_least_iff
thf(fact_674_linorder__less__wlog,axiom,
! [P: list_nat > list_nat > $o,A: list_nat,B: list_nat] :
( ! [A3: list_nat,B3: list_nat] :
( ( ord_less_list_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: list_nat] : ( P @ A3 @ A3 )
=> ( ! [A3: list_nat,B3: list_nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_675_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_676_order_Ostrict__trans,axiom,
! [A: list_nat,B: list_nat,C: list_nat] :
( ( ord_less_list_nat @ A @ B )
=> ( ( ord_less_list_nat @ B @ C )
=> ( ord_less_list_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_677_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_678_not__less__iff__gr__or__eq,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( ~ ( ord_less_list_nat @ X2 @ Y2 ) )
= ( ( ord_less_list_nat @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_679_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( ( ord_less_nat @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_680_dual__order_Ostrict__trans,axiom,
! [B: list_nat,A: list_nat,C: list_nat] :
( ( ord_less_list_nat @ B @ A )
=> ( ( ord_less_list_nat @ C @ B )
=> ( ord_less_list_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_681_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_682_order_Ostrict__implies__not__eq,axiom,
! [A: list_nat,B: list_nat] :
( ( ord_less_list_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_683_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_684_dual__order_Ostrict__implies__not__eq,axiom,
! [B: list_nat,A: list_nat] :
( ( ord_less_list_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_685_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_686_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_687_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_688_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_689_less__not__refl3,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( S2 != T ) ) ).
% less_not_refl3
thf(fact_690_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_691_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N2 )
=> ( P @ M5 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_692_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N2 )
& ~ ( P @ M5 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_693_linorder__neqE__nat,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_694_linorder__neqE,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_list_nat @ X2 @ Y2 )
=> ( ord_less_list_nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_695_linorder__neqE,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE
thf(fact_696_order__less__asym,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( ord_less_list_nat @ X2 @ Y2 )
=> ~ ( ord_less_list_nat @ Y2 @ X2 ) ) ).
% order_less_asym
thf(fact_697_order__less__asym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_asym
thf(fact_698_linorder__neq__iff,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( X2 != Y2 )
= ( ( ord_less_list_nat @ X2 @ Y2 )
| ( ord_less_list_nat @ Y2 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_699_linorder__neq__iff,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
= ( ( ord_less_nat @ X2 @ Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_700_order__less__asym_H,axiom,
! [A: list_nat,B: list_nat] :
( ( ord_less_list_nat @ A @ B )
=> ~ ( ord_less_list_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_701_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_702_order__less__trans,axiom,
! [X2: list_nat,Y2: list_nat,Z2: list_nat] :
( ( ord_less_list_nat @ X2 @ Y2 )
=> ( ( ord_less_list_nat @ Y2 @ Z2 )
=> ( ord_less_list_nat @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_703_order__less__trans,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_704_ord__eq__less__subst,axiom,
! [A: list_nat,F: nat > list_nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_list_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_list_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_705_ord__eq__less__subst,axiom,
! [A: nat,F: list_nat > nat,B: list_nat,C: list_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_list_nat @ B @ C )
=> ( ! [X3: list_nat,Y3: list_nat] :
( ( ord_less_list_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_706_ord__eq__less__subst,axiom,
! [A: list_nat,F: list_nat > list_nat,B: list_nat,C: list_nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_list_nat @ B @ C )
=> ( ! [X3: list_nat,Y3: list_nat] :
( ( ord_less_list_nat @ X3 @ Y3 )
=> ( ord_less_list_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_list_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_707_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_708_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > list_nat,C: list_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_list_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_list_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_709_ord__less__eq__subst,axiom,
! [A: list_nat,B: list_nat,F: list_nat > nat,C: nat] :
( ( ord_less_list_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: list_nat,Y3: list_nat] :
( ( ord_less_list_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_710_ord__less__eq__subst,axiom,
! [A: list_nat,B: list_nat,F: list_nat > list_nat,C: list_nat] :
( ( ord_less_list_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: list_nat,Y3: list_nat] :
( ( ord_less_list_nat @ X3 @ Y3 )
=> ( ord_less_list_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_list_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_711_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_712_order__less__irrefl,axiom,
! [X2: list_nat] :
~ ( ord_less_list_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_713_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_714_order__less__subst1,axiom,
! [A: nat,F: list_nat > nat,B: list_nat,C: list_nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_list_nat @ B @ C )
=> ( ! [X3: list_nat,Y3: list_nat] :
( ( ord_less_list_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_715_order__less__subst1,axiom,
! [A: list_nat,F: nat > list_nat,B: nat,C: nat] :
( ( ord_less_list_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_list_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_list_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_716_order__less__subst1,axiom,
! [A: list_nat,F: list_nat > list_nat,B: list_nat,C: list_nat] :
( ( ord_less_list_nat @ A @ ( F @ B ) )
=> ( ( ord_less_list_nat @ B @ C )
=> ( ! [X3: list_nat,Y3: list_nat] :
( ( ord_less_list_nat @ X3 @ Y3 )
=> ( ord_less_list_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_list_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_717_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_718_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > list_nat,C: list_nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_list_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_list_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_list_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_719_order__less__subst2,axiom,
! [A: list_nat,B: list_nat,F: list_nat > nat,C: nat] :
( ( ord_less_list_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: list_nat,Y3: list_nat] :
( ( ord_less_list_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_720_order__less__subst2,axiom,
! [A: list_nat,B: list_nat,F: list_nat > list_nat,C: list_nat] :
( ( ord_less_list_nat @ A @ B )
=> ( ( ord_less_list_nat @ ( F @ B ) @ C )
=> ( ! [X3: list_nat,Y3: list_nat] :
( ( ord_less_list_nat @ X3 @ Y3 )
=> ( ord_less_list_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_list_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_721_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_722_order__less__not__sym,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( ord_less_list_nat @ X2 @ Y2 )
=> ~ ( ord_less_list_nat @ Y2 @ X2 ) ) ).
% order_less_not_sym
thf(fact_723_order__less__not__sym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_not_sym
thf(fact_724_order__less__imp__triv,axiom,
! [X2: list_nat,Y2: list_nat,P: $o] :
( ( ord_less_list_nat @ X2 @ Y2 )
=> ( ( ord_less_list_nat @ Y2 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_725_order__less__imp__triv,axiom,
! [X2: nat,Y2: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_726_linorder__less__linear,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( ord_less_list_nat @ X2 @ Y2 )
| ( X2 = Y2 )
| ( ord_less_list_nat @ Y2 @ X2 ) ) ).
% linorder_less_linear
thf(fact_727_linorder__less__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_less_linear
thf(fact_728_order__less__imp__not__eq,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( ord_less_list_nat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_729_order__less__imp__not__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ).
% order_less_imp_not_eq
thf(fact_730_order__less__imp__not__eq2,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( ord_less_list_nat @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_731_order__less__imp__not__eq2,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_732_order__less__imp__not__less,axiom,
! [X2: list_nat,Y2: list_nat] :
( ( ord_less_list_nat @ X2 @ Y2 )
=> ~ ( ord_less_list_nat @ Y2 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_733_order__less__imp__not__less,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_734_leD,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y2 ) ) ).
% leD
thf(fact_735_leI,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% leI
thf(fact_736_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_737_antisym__conv1,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_738_antisym__conv2,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ).
% antisym_conv2
thf(fact_739_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ~ ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% less_le_not_le
thf(fact_740_not__le__imp__less,axiom,
! [Y2: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ord_less_nat @ X2 @ Y2 ) ) ).
% not_le_imp_less
thf(fact_741_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_742_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_743_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_744_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_745_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_746_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_747_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_748_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_749_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_750_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ~ ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_751_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_752_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_753_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_less
thf(fact_754_order__less__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ( X != Y ) ) ) ) ).
% order_less_le
thf(fact_755_linorder__not__le,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y2 ) )
= ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_not_le
thf(fact_756_linorder__not__less,axiom,
! [X2: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_not_less
thf(fact_757_order__less__imp__le,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% order_less_imp_le
thf(fact_758_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_759_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_760_order__le__less__trans,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_761_order__less__le__trans,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_762_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_763_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_764_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_765_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_766_linorder__le__less__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_nat @ Y2 @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_767_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_768_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_769_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_770_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_771_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_772_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_773_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_774_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_775_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_776_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_777_length__induct,axiom,
! [P: list_nat > $o,Xs2: list_nat] :
( ! [Xs3: list_nat] :
( ! [Ys3: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys3 ) @ ( size_size_list_nat @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_778_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_779_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_780_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_781_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_782_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_783_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_784_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_785_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_786_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
& ( M4 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_787_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_788_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
| ( M4 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_789_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_790_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_791_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_792_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_793_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_794_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_795_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_796_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_797_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_798_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_799_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_800_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_801_strict__sorted__imp__sorted,axiom,
! [Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs2 )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 ) ) ).
% strict_sorted_imp_sorted
thf(fact_802_nth__equalityI,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ Xs2 @ I2 )
= ( nth_nat @ Ys @ I2 ) ) )
=> ( Xs2 = Ys ) ) ) ).
% nth_equalityI
thf(fact_803_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X8: nat] : ( P @ I4 @ X8 ) ) )
= ( ? [Xs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P @ I4 @ ( nth_nat @ Xs @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_804_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_nat,Z: list_nat] : ( Y4 = Z ) )
= ( ^ [Xs: list_nat,Ys2: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I4 )
= ( nth_nat @ Ys2 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_805_Ex__list__of__length__P,axiom,
! [N: nat,P: nat > nat > $o] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ? [X4: nat] : ( P @ X4 @ I2 ) )
=> ? [Xs3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ N )
=> ( P @ ( nth_nat @ Xs3 @ I5 ) @ I5 ) ) ) ) ).
% Ex_list_of_length_P
thf(fact_806_strict__sorted__simps_I1_J,axiom,
sorted_wrt_nat @ ord_less_nat @ nil_nat ).
% strict_sorted_simps(1)
thf(fact_807_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_808_strict__sorted__equal,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Xs2 )
=> ( ( sorted_wrt_nat @ ord_less_nat @ Ys )
=> ( ( ( set_nat2 @ Ys )
= ( set_nat2 @ Xs2 ) )
=> ( Ys = Xs2 ) ) ) ) ).
% strict_sorted_equal
thf(fact_809_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_810_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_811_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_812_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_813_ex__set__conv__ex__nth,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ( ? [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
& ( P @ X ) ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
& ( P @ ( nth_nat @ Xs2 @ I4 ) ) ) ) ) ).
% ex_set_conv_ex_nth
thf(fact_814_all__set__conv__all__nth,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ( P @ X ) ) )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ ( nth_nat @ Xs2 @ I4 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_815_all__nth__imp__all__set,axiom,
! [Xs2: list_nat,P: nat > $o,X2: nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ ( nth_nat @ Xs2 @ I2 ) ) )
=> ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_816_in__set__conv__nth,axiom,
! [X2: nat,Xs2: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
= ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
& ( ( nth_nat @ Xs2 @ I4 )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_817_list__ball__nth,axiom,
! [N: nat,Xs2: list_nat,P: nat > $o] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
=> ( P @ X3 ) )
=> ( P @ ( nth_nat @ Xs2 @ N ) ) ) ) ).
% list_ball_nth
thf(fact_818_nth__mem,axiom,
! [N: nat,Xs2: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( member_nat @ ( nth_nat @ Xs2 @ N ) @ ( set_nat2 @ Xs2 ) ) ) ).
% nth_mem
thf(fact_819_map__eq__conv_H,axiom,
! [F: nat > nat,Xs2: list_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Ys ) )
= ( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( F @ ( nth_nat @ Xs2 @ I4 ) )
= ( G @ ( nth_nat @ Ys @ I4 ) ) ) ) ) ) ).
% map_eq_conv'
thf(fact_820_map__nth__conv,axiom,
! [F: nat > nat,Ss: list_nat,G: nat > nat,Ts: list_nat] :
( ( ( map_nat_nat @ F @ Ss )
= ( map_nat_nat @ G @ Ts ) )
=> ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Ss ) )
=> ( ( F @ ( nth_nat @ Ss @ I5 ) )
= ( G @ ( nth_nat @ Ts @ I5 ) ) ) ) ) ).
% map_nth_conv
thf(fact_821_map__nth__eq__conv,axiom,
! [Xs2: list_nat,Ys: list_nat,F: nat > nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( map_nat_nat @ F @ Xs2 )
= Ys )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys ) )
=> ( ( F @ ( nth_nat @ Xs2 @ I4 ) )
= ( nth_nat @ Ys @ I4 ) ) ) ) ) ) ).
% map_nth_eq_conv
thf(fact_822_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_823_sorted__wrt__nth__less,axiom,
! [P: nat > nat > $o,Xs2: list_nat,I: nat,J: nat] :
( ( sorted_wrt_nat @ P @ Xs2 )
=> ( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ ( nth_nat @ Xs2 @ I ) @ ( nth_nat @ Xs2 @ J ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_824_sorted__wrt__iff__nth__less,axiom,
( sorted_wrt_nat
= ( ^ [P3: nat > nat > $o,Xs: list_nat] :
! [I4: nat,J2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
=> ( P3 @ ( nth_nat @ Xs @ I4 ) @ ( nth_nat @ Xs @ J2 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_825_set__update__memI,axiom,
! [N: nat,Xs2: list_nat,X2: nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( member_nat @ X2 @ ( set_nat2 @ ( list_update_nat @ Xs2 @ N @ X2 ) ) ) ) ).
% set_update_memI
thf(fact_826_nth__list__update,axiom,
! [I: nat,Xs2: list_nat,J: nat,X2: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ( I = J )
=> ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X2 ) @ J )
= X2 ) )
& ( ( I != J )
=> ( ( nth_nat @ ( list_update_nat @ Xs2 @ I @ X2 ) @ J )
= ( nth_nat @ Xs2 @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_827_list__update__same__conv,axiom,
! [I: nat,Xs2: list_nat,X2: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ( list_update_nat @ Xs2 @ I @ X2 )
= Xs2 )
= ( ( nth_nat @ Xs2 @ I )
= X2 ) ) ) ).
% list_update_same_conv
thf(fact_828_parallel__list__update,axiom,
! [N: nat,R2: nat > nat > $o,P4: list_nat > $o,Xs2: list_nat,Ys: list_nat] :
( ! [Xs3: list_nat,I2: nat,Y3: nat] :
( ( ( size_size_list_nat @ Xs3 )
= N )
=> ( ( ord_less_nat @ I2 @ N )
=> ( ( R2 @ ( nth_nat @ Xs3 @ I2 ) @ Y3 )
=> ( ( P4 @ Xs3 )
=> ( P4 @ ( list_update_nat @ Xs3 @ I2 @ Y3 ) ) ) ) ) )
=> ( ( ( size_size_list_nat @ Xs2 )
= N )
=> ( ( P4 @ Xs2 )
=> ( ( ( size_size_list_nat @ Ys )
= N )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( R2 @ ( nth_nat @ Xs2 @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) )
=> ( P4 @ Ys ) ) ) ) ) ) ).
% parallel_list_update
thf(fact_829_nth__butlast,axiom,
! [N: nat,Xs2: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ ( butlast_nat @ Xs2 ) ) )
=> ( ( nth_nat @ ( butlast_nat @ Xs2 ) @ N )
= ( nth_nat @ Xs2 @ N ) ) ) ).
% nth_butlast
thf(fact_830_take__butlast,axiom,
! [N: nat,Xs2: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( take_nat @ N @ ( butlast_nat @ Xs2 ) )
= ( take_nat @ N @ Xs2 ) ) ) ).
% take_butlast
thf(fact_831_sorted__wrt__less__idx,axiom,
! [Ns: list_nat,I: nat] :
( ( sorted_wrt_nat @ ord_less_nat @ Ns )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
=> ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).
% sorted_wrt_less_idx
thf(fact_832_sorted__iff__nth__mono__less,axiom,
! [Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
= ( ! [I4: nat,J2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs2 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ I4 ) @ ( nth_nat @ Xs2 @ J2 ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_833_nth__take__prefix,axiom,
! [Ys: list_nat,Xs2: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Ys ) @ ( size_size_list_nat @ Xs2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( ( nth_nat @ Xs2 @ I2 )
= ( nth_nat @ Ys @ I2 ) ) )
=> ( ( take_nat @ ( size_size_list_nat @ Ys ) @ Xs2 )
= Ys ) ) ) ).
% nth_take_prefix
thf(fact_834_nth__take__lemma,axiom,
! [K: nat,Xs2: list_nat,Ys: list_nat] :
( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ord_less_eq_nat @ K @ ( size_size_list_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( nth_nat @ Xs2 @ I2 )
= ( nth_nat @ Ys @ I2 ) ) )
=> ( ( take_nat @ K @ Xs2 )
= ( take_nat @ K @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_835_set__take__nth,axiom,
! [X2: nat,I: nat,Xs2: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ ( take_nat @ I @ Xs2 ) ) )
=> ? [J3: nat] :
( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs2 ) )
& ( ord_less_nat @ J3 @ I )
& ( ( nth_nat @ Xs2 @ J3 )
= X2 ) ) ) ).
% set_take_nth
thf(fact_836_set__list__subset__nth__conv,axiom,
! [Xs2: list_nat,A4: set_nat,I: nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A4 )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( member_nat @ ( nth_nat @ Xs2 @ I ) @ A4 ) ) ) ).
% set_list_subset_nth_conv
thf(fact_837_set__list__subset__eq__nth__conv,axiom,
! [Xs2: list_nat,A4: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A4 )
= ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
=> ( member_nat @ ( nth_nat @ Xs2 @ I4 ) @ A4 ) ) ) ) ).
% set_list_subset_eq_nth_conv
thf(fact_838_subseteq__set__conv__nth,axiom,
! [Ss: list_nat,T3: set_nat] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ss ) )
=> ( member_nat @ ( nth_nat @ Ss @ I4 ) @ T3 ) ) )
= ( ord_less_eq_set_nat @ ( set_nat2 @ Ss ) @ T3 ) ) ).
% subseteq_set_conv_nth
thf(fact_839_inv__to__set,axiom,
! [Ss: list_nat,S: set_nat] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ss ) )
=> ( member_nat @ ( nth_nat @ Ss @ I4 ) @ S ) ) )
= ( ord_less_eq_set_nat @ ( set_nat2 @ Ss ) @ S ) ) ).
% inv_to_set
thf(fact_840_map__eq__nth__conv,axiom,
! [F: nat > nat,Xs2: list_nat,G: nat > nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Ys ) )
= ( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys ) )
=> ( ( F @ ( nth_nat @ Xs2 @ I4 ) )
= ( G @ ( nth_nat @ Ys @ I4 ) ) ) ) ) ) ).
% map_eq_nth_conv
thf(fact_841_not__less__Nil,axiom,
! [X2: list_nat] :
~ ( ord_less_list_nat @ X2 @ nil_nat ) ).
% not_less_Nil
thf(fact_842_less__list__code_I1_J,axiom,
! [Xs2: list_nat] :
~ ( ord_less_list_nat @ Xs2 @ nil_nat ) ).
% less_list_code(1)
thf(fact_843_nth__equalityE,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( Xs2 = Ys )
=> ~ ( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ~ ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ Xs2 @ I5 )
= ( nth_nat @ Ys @ I5 ) ) ) ) ) ).
% nth_equalityE
thf(fact_844_in__set__idx,axiom,
! [X2: nat,Xs2: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
& ( ( nth_nat @ Xs2 @ I2 )
= X2 ) ) ) ).
% in_set_idx
thf(fact_845_nth__map__conv,axiom,
! [Xs2: list_nat,Ys: list_nat,F: nat > nat,G: nat > nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( F @ ( nth_nat @ Xs2 @ I2 ) )
= ( G @ ( nth_nat @ Ys @ I2 ) ) ) )
=> ( ( map_nat_nat @ F @ Xs2 )
= ( map_nat_nat @ G @ Ys ) ) ) ) ).
% nth_map_conv
thf(fact_846_permut__sound,axiom,
! [I: nat,As2: list_nat,F: nat > nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ As2 ) )
=> ( ( nth_nat @ ( missing_permut_nat @ As2 @ F ) @ I )
= ( nth_nat @ As2 @ ( F @ I ) ) ) ) ).
% permut_sound
thf(fact_847_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
=> ( ! [I5: nat] :
( ( ord_less_nat @ K3 @ I5 )
=> ( P @ I5 ) )
=> ( P @ K3 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_848_inf__pigeonhole__principle,axiom,
! [N: nat,F: nat > nat > $o] :
( ! [K3: nat] :
? [I5: nat] :
( ( ord_less_nat @ I5 @ N )
& ( F @ K3 @ I5 ) )
=> ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ! [K4: nat] :
? [K5: nat] :
( ( ord_less_eq_nat @ K4 @ K5 )
& ( F @ K5 @ I2 ) ) ) ) ).
% inf_pigeonhole_principle
thf(fact_849_Utils_Otrancl__listp_Ocases,axiom,
! [R3: nat > nat > $o,A1: list_nat,A22: list_nat] :
( ( trancl_listp_nat @ R3 @ A1 @ A22 )
=> ( ( ( ( size_size_list_nat @ A1 )
= ( size_size_list_nat @ A22 ) )
=> ~ ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_nat @ A22 ) )
=> ( R3 @ ( nth_nat @ A1 @ I5 ) @ ( nth_nat @ A22 @ I5 ) ) ) )
=> ~ ! [Ys4: list_nat,I2: nat,Z3: nat] :
( ( A22
= ( list_update_nat @ Ys4 @ I2 @ Z3 ) )
=> ( ( trancl_listp_nat @ R3 @ A1 @ Ys4 )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys4 ) )
=> ~ ( R3 @ ( nth_nat @ Ys4 @ I2 ) @ Z3 ) ) ) ) ) ) ).
% Utils.trancl_listp.cases
thf(fact_850_Utils_Otrancl__listp_Osimps,axiom,
( trancl_listp_nat
= ( ^ [R4: nat > nat > $o,A12: list_nat,A23: list_nat] :
( ? [Xs: list_nat,Ys2: list_nat] :
( ( A12 = Xs )
& ( A23 = Ys2 )
& ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys2 ) )
=> ( R4 @ ( nth_nat @ Xs @ I4 ) @ ( nth_nat @ Ys2 @ I4 ) ) ) )
| ? [Xs: list_nat,Ys2: list_nat,I4: nat,Z5: nat] :
( ( A12 = Xs )
& ( A23
= ( list_update_nat @ Ys2 @ I4 @ Z5 ) )
& ( trancl_listp_nat @ R4 @ Xs @ Ys2 )
& ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys2 ) )
& ( R4 @ ( nth_nat @ Ys2 @ I4 ) @ Z5 ) ) ) ) ) ).
% Utils.trancl_listp.simps
thf(fact_851_Utils_Otrancl__listp_Olist__trancl,axiom,
! [R3: nat > nat > $o,Xs2: list_nat,Ys: list_nat,I: nat,Z2: nat] :
( ( trancl_listp_nat @ R3 @ Xs2 @ Ys )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
=> ( ( R3 @ ( nth_nat @ Ys @ I ) @ Z2 )
=> ( trancl_listp_nat @ R3 @ Xs2 @ ( list_update_nat @ Ys @ I @ Z2 ) ) ) ) ) ).
% Utils.trancl_listp.list_trancl
thf(fact_852_Utils_Otrancl__listp_Obase,axiom,
! [Xs2: list_nat,Ys: list_nat,R3: nat > nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( R3 @ ( nth_nat @ Xs2 @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) )
=> ( trancl_listp_nat @ R3 @ Xs2 @ Ys ) ) ) ).
% Utils.trancl_listp.base
thf(fact_853_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X4 ) ) ).
% minf(8)
thf(fact_854_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z3 )
=> ( ord_less_eq_nat @ X4 @ T ) ) ).
% minf(6)
thf(fact_855_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ( ord_less_eq_nat @ T @ X4 ) ) ).
% pinf(8)
thf(fact_856_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z3 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ T ) ) ).
% pinf(6)
thf(fact_857_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ C2 ) )
=> ( P @ X4 ) )
& ! [D3: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D3 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D3 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_858_verit__comp__simplify1_I3_J,axiom,
! [B6: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
= ( ord_less_nat @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_859_permut__aux__sound,axiom,
! [I: nat,As2: list_nat,F: nat > nat,Bs: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ As2 ) )
=> ( ( nth_nat @ ( missin1888654203714970382ux_nat @ As2 @ F @ Bs ) @ I )
= ( nth_nat @ Bs @ ( F @ I ) ) ) ) ).
% permut_aux_sound
thf(fact_860_remove__nth__length,axiom,
! [N: nat,Xs2: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( size_size_list_nat @ ( missin7175274867594579095th_nat @ N @ Xs2 ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat ) ) ) ).
% remove_nth_length
thf(fact_861_permut__aux_Osimps_I1_J,axiom,
! [Uu2: nat > nat,Uv2: list_nat] :
( ( missin1888654203714970382ux_nat @ nil_nat @ Uu2 @ Uv2 )
= nil_nat ) ).
% permut_aux.simps(1)
thf(fact_862_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_863_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_864_remove__nth__id,axiom,
! [Xs2: list_nat,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N )
=> ( ( missin7175274867594579095th_nat @ N @ Xs2 )
= Xs2 ) ) ).
% remove_nth_id
thf(fact_865_remove__nth__sound__l,axiom,
! [P4: nat,N: nat,Xs2: list_nat] :
( ( ord_less_nat @ P4 @ N )
=> ( ( nth_nat @ ( missin7175274867594579095th_nat @ N @ Xs2 ) @ P4 )
= ( nth_nat @ Xs2 @ P4 ) ) ) ).
% remove_nth_sound_l
thf(fact_866_remove__nth__P__compat,axiom,
! [As2: list_nat,Bs: list_nat,P: nat > nat > $o,P4: nat] :
( ( ( size_size_list_nat @ As2 )
= ( size_size_list_nat @ Bs ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ As2 ) )
=> ( P @ ( nth_nat @ As2 @ I2 ) @ ( nth_nat @ Bs @ I2 ) ) )
=> ! [I5: nat] :
( ( ord_less_nat @ I5 @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ P4 @ As2 ) ) )
=> ( P @ ( nth_nat @ ( missin7175274867594579095th_nat @ P4 @ As2 ) @ I5 ) @ ( nth_nat @ ( missin7175274867594579095th_nat @ P4 @ Bs ) @ I5 ) ) ) ) ) ).
% remove_nth_P_compat
thf(fact_867_adjust__idx__rev__nth,axiom,
! [I: nat,Xs2: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( J != I )
=> ( ( nth_nat @ ( missin7175274867594579095th_nat @ I @ Xs2 ) @ ( missin3815256168798769645dx_rev @ I @ J ) )
= ( nth_nat @ Xs2 @ J ) ) ) ) ).
% adjust_idx_rev_nth
thf(fact_868_adjust__idx__nth,axiom,
! [I: nat,Xs2: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ ( missin7175274867594579095th_nat @ I @ Xs2 ) @ J )
= ( nth_nat @ Xs2 @ ( missing_adjust_idx @ I @ J ) ) ) ) ).
% adjust_idx_nth
thf(fact_869_adjust__idx__rev__length,axiom,
! [I: nat,Xs2: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ( ( J != I )
=> ( ord_less_nat @ ( missin3815256168798769645dx_rev @ I @ J ) @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ I @ Xs2 ) ) ) ) ) ) ).
% adjust_idx_rev_length
thf(fact_870_adjust__idx__length,axiom,
! [I: nat,Xs2: list_nat,J: nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ I @ Xs2 ) ) )
=> ( ord_less_nat @ ( missing_adjust_idx @ I @ J ) @ ( size_size_list_nat @ Xs2 ) ) ) ) ).
% adjust_idx_length
thf(fact_871_adjust__idx__i,axiom,
! [I: nat,J: nat] :
( ( missing_adjust_idx @ I @ J )
!= I ) ).
% adjust_idx_i
thf(fact_872_adjust__idx__rev1,axiom,
! [I: nat,J: nat] :
( ( missin3815256168798769645dx_rev @ I @ ( missing_adjust_idx @ I @ J ) )
= J ) ).
% adjust_idx_rev1
thf(fact_873_adjust__idx__rev2,axiom,
! [J: nat,I: nat] :
( ( J != I )
=> ( ( missing_adjust_idx @ I @ ( missin3815256168798769645dx_rev @ I @ J ) )
= J ) ) ).
% adjust_idx_rev2
thf(fact_874_filter__rev__nth__butlast,axiom,
! [P: nat > $o,Xs2: list_nat,I: nat] :
( ~ ( P @ ( last_nat @ Xs2 ) )
=> ( ( basic_4353017870094810967th_nat @ P @ Xs2 @ I )
= ( basic_4353017870094810967th_nat @ P @ ( butlast_nat @ Xs2 ) @ I ) ) ) ).
% filter_rev_nth_butlast
thf(fact_875_min__list__nth,axiom,
! [Xs2: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) )
=> ( ord_less_eq_nat @ ( missing_min_list_nat @ Xs2 ) @ ( missing_min_list_nat @ Ys ) ) ) ) ).
% min_list_nth
thf(fact_876_map__upt__eqI,axiom,
! [Xs2: list_nat,N: nat,M: nat,F: nat > nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( minus_minus_nat @ N @ M ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ Xs2 @ I2 )
= ( F @ ( plus_plus_nat @ M @ I2 ) ) ) )
=> ( ( map_nat_nat @ F @ ( upt @ M @ N ) )
= Xs2 ) ) ) ).
% map_upt_eqI
thf(fact_877_sorted__rev__nth__mono,axiom,
! [Xs2: list_nat,I: nat,J: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs2 ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ J ) @ ( nth_nat @ Xs2 @ I ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_878_sort__upt,axiom,
! [M: nat,N: nat] :
( ( linord738340561235409698at_nat
@ ^ [X: nat] : X
@ ( upt @ M @ N ) )
= ( upt @ M @ N ) ) ).
% sort_upt
thf(fact_879_Nil__is__rev__conv,axiom,
! [Xs2: list_nat] :
( ( nil_nat
= ( rev_nat @ Xs2 ) )
= ( Xs2 = nil_nat ) ) ).
% Nil_is_rev_conv
thf(fact_880_rev__is__Nil__conv,axiom,
! [Xs2: list_nat] :
( ( ( rev_nat @ Xs2 )
= nil_nat )
= ( Xs2 = nil_nat ) ) ).
% rev_is_Nil_conv
thf(fact_881_length__rev,axiom,
! [Xs2: list_nat] :
( ( size_size_list_nat @ ( rev_nat @ Xs2 ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% length_rev
thf(fact_882_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( upt @ I @ J )
= nil_nat ) ) ).
% upt_conv_Nil
thf(fact_883_take__upt,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M ) @ N )
=> ( ( take_nat @ M @ ( upt @ I @ N ) )
= ( upt @ I @ ( plus_plus_nat @ I @ M ) ) ) ) ).
% take_upt
thf(fact_884_length__upt,axiom,
! [I: nat,J: nat] :
( ( size_size_list_nat @ ( upt @ I @ J ) )
= ( minus_minus_nat @ J @ I ) ) ).
% length_upt
thf(fact_885_butlast__rev,axiom,
! [Xs2: list_nat] :
( ( butlast_nat @ ( rev_nat @ Xs2 ) )
= ( rev_nat @ ( tl_nat @ Xs2 ) ) ) ).
% butlast_rev
thf(fact_886_last__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( last_nat @ ( upt @ I @ J ) )
= ( minus_minus_nat @ J @ one_one_nat ) ) ) ).
% last_upt
thf(fact_887_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_888_sorted__wrt__rev,axiom,
! [P: nat > nat > $o,Xs2: list_nat] :
( ( sorted_wrt_nat @ P @ ( rev_nat @ Xs2 ) )
= ( sorted_wrt_nat
@ ^ [X: nat,Y: nat] : ( P @ Y @ X )
@ Xs2 ) ) ).
% sorted_wrt_rev
thf(fact_889_rev__map,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( rev_nat @ ( map_nat_nat @ F @ Xs2 ) )
= ( map_nat_nat @ F @ ( rev_nat @ Xs2 ) ) ) ).
% rev_map
thf(fact_890_rev_Osimps_I1_J,axiom,
( ( rev_nat @ nil_nat )
= nil_nat ) ).
% rev.simps(1)
thf(fact_891_sorted__wrt__upt,axiom,
! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N ) ) ).
% sorted_wrt_upt
thf(fact_892_sorted__upt,axiom,
! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N ) ) ).
% sorted_upt
thf(fact_893_min__list,axiom,
! [X2: nat,Xs2: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ord_less_eq_nat @ ( missing_min_list_nat @ Xs2 ) @ X2 ) ) ).
% min_list
thf(fact_894_min__list__ex,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( member_nat @ ( missing_min_list_nat @ Xs2 ) @ ( set_nat2 @ Xs2 ) ) ) ).
% min_list_ex
thf(fact_895_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_896_rev__update,axiom,
! [K: nat,Xs2: list_nat,Y2: nat] :
( ( ord_less_nat @ K @ ( size_size_list_nat @ Xs2 ) )
=> ( ( rev_nat @ ( list_update_nat @ Xs2 @ K @ Y2 ) )
= ( list_update_nat @ ( rev_nat @ Xs2 ) @ ( minus_minus_nat @ ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ K ) @ one_one_nat ) @ Y2 ) ) ) ).
% rev_update
thf(fact_897_sorted__rev__iff__nth__mono,axiom,
! [Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs2 ) )
= ( ! [I4: nat,J2: nat] :
( ( ord_less_eq_nat @ I4 @ J2 )
=> ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs2 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ J2 ) @ ( nth_nat @ Xs2 @ I4 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_898_take__upt__idx,axiom,
! [I: nat,Ls: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Ls ) )
=> ( ( take_nat @ I @ Ls )
= ( map_nat_nat @ ( nth_nat @ Ls ) @ ( upt @ zero_zero_nat @ I ) ) ) ) ).
% take_upt_idx
thf(fact_899_sorted__rev__iff__nth__Suc,axiom,
! [Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs2 ) )
= ( ! [I4: nat] :
( ( ord_less_nat @ ( suc @ I4 ) @ ( size_size_list_nat @ Xs2 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ ( suc @ I4 ) ) @ ( nth_nat @ Xs2 @ I4 ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_900_map__upt__len__conv,axiom,
! [F: nat > nat,Xs2: list_nat] :
( ( map_nat_nat
@ ^ [I4: nat] : ( F @ ( nth_nat @ Xs2 @ I4 ) )
@ ( upt @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) )
= ( map_nat_nat @ F @ Xs2 ) ) ).
% map_upt_len_conv
thf(fact_901_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_902_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_903_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_904_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_905_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_906_zero__eq__add__iff__both__eq__0,axiom,
! [X2: nat,Y2: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X2 @ Y2 ) )
= ( ( X2 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_907_add__eq__0__iff__both__eq__0,axiom,
! [X2: nat,Y2: nat] :
( ( ( plus_plus_nat @ X2 @ Y2 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_908_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_909_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_910_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_911_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_912_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_913_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_914_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_915_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_916_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_917_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_918_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_919_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_920_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_921_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_922_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_923_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_924_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_925_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_926_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_927_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_928_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_929_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_930_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_931_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_932_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_933_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_934_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_935_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_936_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_937_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_938_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_939_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_940_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_941_length__0__conv,axiom,
! [Xs2: list_nat] :
( ( ( size_size_list_nat @ Xs2 )
= zero_zero_nat )
= ( Xs2 = nil_nat ) ) ).
% length_0_conv
thf(fact_942_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_943_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_944_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_945_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_946_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_947_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_948_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_949_take0,axiom,
( ( take_nat @ zero_zero_nat )
= ( ^ [Xs: list_nat] : nil_nat ) ) ).
% take0
thf(fact_950_take__eq__Nil,axiom,
! [N: nat,Xs2: list_nat] :
( ( ( take_nat @ N @ Xs2 )
= nil_nat )
= ( ( N = zero_zero_nat )
| ( Xs2 = nil_nat ) ) ) ).
% take_eq_Nil
thf(fact_951_take__eq__Nil2,axiom,
! [N: nat,Xs2: list_nat] :
( ( nil_nat
= ( take_nat @ N @ Xs2 ) )
= ( ( N = zero_zero_nat )
| ( Xs2 = nil_nat ) ) ) ).
% take_eq_Nil2
thf(fact_952_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_953_empty__replicate,axiom,
! [N: nat,X2: nat] :
( ( nil_nat
= ( replicate_nat @ N @ X2 ) )
= ( N = zero_zero_nat ) ) ).
% empty_replicate
thf(fact_954_replicate__empty,axiom,
! [N: nat,X2: nat] :
( ( ( replicate_nat @ N @ X2 )
= nil_nat )
= ( N = zero_zero_nat ) ) ).
% replicate_empty
thf(fact_955_last__replicate,axiom,
! [N: nat,X2: nat] :
( ( N != zero_zero_nat )
=> ( ( last_nat @ ( replicate_nat @ N @ X2 ) )
= X2 ) ) ).
% last_replicate
thf(fact_956_length__greater__0__conv,axiom,
! [Xs2: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) )
= ( Xs2 != nil_nat ) ) ).
% length_greater_0_conv
thf(fact_957_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_958_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_959_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_960_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_961_tl__upt,axiom,
! [M: nat,N: nat] :
( ( tl_nat @ ( upt @ M @ N ) )
= ( upt @ ( suc @ M ) @ N ) ) ).
% tl_upt
thf(fact_962_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_963_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ zero_zero_nat )
= nil_nat ) ).
% upt_0
thf(fact_964_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_965_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M4: nat,N3: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% add_eq_if
thf(fact_966_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_967_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_968_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J2: nat] :
( ( M
= ( suc @ J2 ) )
& ( ord_less_nat @ J2 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_969_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ! [I5: nat] :
( ( ord_less_eq_nat @ I5 @ K3 )
=> ~ ( P @ I5 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_970_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_971_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_972_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_973_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_974_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J3: nat,K3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ( ord_less_nat @ J3 @ K3 )
=> ( ( P @ I2 @ J3 )
=> ( ( P @ J3 @ K3 )
=> ( P @ I2 @ K3 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_975_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_976_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_977_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_978_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M6: nat] :
( ( M
= ( suc @ M6 ) )
& ( ord_less_nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_979_Nat_OAll__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ I4 ) ) ) ) ).
% Nat.All_less_Suc
thf(fact_980_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_981_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_982_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ N )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_983_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_984_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_985_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_986_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_987_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_988_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_989_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_990_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_991_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_992_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_993_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_994_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N2 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_995_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_996_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_997_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_998_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_999_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1000_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1001_nat__arith_Osuc1,axiom,
! [A4: nat,K: nat,A: nat] :
( ( A4
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A4 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1002_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1003_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_1004_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z3: nat] :
( ( R @ X3 @ Y3 )
=> ( ( R @ Y3 @ Z3 )
=> ( R @ X3 @ Z3 ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1005_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1006_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N2 )
=> ( P @ M5 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1007_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_1008_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1009_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_1010_Suc__le__D,axiom,
! [N: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
=> ? [M3: nat] :
( M7
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_1011_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1012_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_1013_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1014_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1015_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_1016_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_1017_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_1018_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_1019_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_1020_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1021_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_1022_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1023_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_1024_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_1025_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1026_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_1027_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1028_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_1029_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_1030_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1031_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1032_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1033_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1034_old_Onat_Oexhaust,axiom,
! [Y2: nat] :
( ( Y2 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y2
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1035_Suc__inject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
=> ( X2 = Y2 ) ) ).
% Suc_inject
thf(fact_1036_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_1037_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1038_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1039_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1040_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1041_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1042_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1043_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_1044_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_1045_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_1046_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1047_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_1048_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_1049_adjust__idx__rev__def,axiom,
( missin3815256168798769645dx_rev
= ( ^ [I4: nat,J2: nat] : ( if_nat @ ( ord_less_nat @ J2 @ I4 ) @ J2 @ ( minus_minus_nat @ J2 @ ( suc @ zero_zero_nat ) ) ) ) ) ).
% adjust_idx_rev_def
thf(fact_1050_map__decr__upt,axiom,
! [M: nat,N: nat] :
( ( map_nat_nat
@ ^ [N3: nat] : ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) )
@ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
= ( upt @ M @ N ) ) ).
% map_decr_upt
thf(fact_1051_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1052_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_1053_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1054_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1055_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q2: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).
% less_natE
thf(fact_1056_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1057_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1058_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M4: nat,N3: nat] :
? [K2: nat] :
( N3
= ( suc @ ( plus_plus_nat @ M4 @ K2 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1059_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K3: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1060_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_1061_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1062_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_1063_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_1064_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_1065_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1066_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1067_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_1068_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1069_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_1070_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1071_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_1072_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1073_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1074_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1075_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_1076_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_1077_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_1078_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1079_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_1080_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1081_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_1082_add__nonneg__eq__0__iff,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y2 )
=> ( ( ( plus_plus_nat @ X2 @ Y2 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1083_add__nonpos__eq__0__iff,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y2 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X2 @ Y2 )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y2 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1084_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_1085_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_1086_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1087_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_1088_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_1089_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_1090_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_1091_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_1092_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_1093_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_1094_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1095_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K3 )
=> ~ ( P @ I5 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1096_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_1097_take__tl,axiom,
! [N: nat,Xs2: list_nat] :
( ( take_nat @ N @ ( tl_nat @ Xs2 ) )
= ( tl_nat @ ( take_nat @ ( suc @ N ) @ Xs2 ) ) ) ).
% take_tl
thf(fact_1098_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_1099_take__0,axiom,
! [Xs2: list_nat] :
( ( take_nat @ zero_zero_nat @ Xs2 )
= nil_nat ) ).
% take_0
thf(fact_1100_replicate__0,axiom,
! [X2: nat] :
( ( replicate_nat @ zero_zero_nat @ X2 )
= nil_nat ) ).
% replicate_0
thf(fact_1101_pres__sorted__dec,axiom,
! [Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
=> ( sorted_wrt_nat @ ord_less_eq_nat
@ ( map_nat_nat
@ ^ [X: nat] : ( minus_minus_nat @ X @ ( suc @ zero_zero_nat ) )
@ Xs2 ) ) ) ).
% pres_sorted_dec
thf(fact_1102_adjust__idx__def,axiom,
( missing_adjust_idx
= ( ^ [I4: nat,J2: nat] : ( if_nat @ ( ord_less_nat @ J2 @ I4 ) @ J2 @ ( suc @ J2 ) ) ) ) ).
% adjust_idx_def
thf(fact_1103_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_1104_length__code,axiom,
( size_size_list_nat
= ( gen_length_nat @ zero_zero_nat ) ) ).
% length_code
thf(fact_1105_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_1106_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1107_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_1108_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1109_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1110_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1111_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_1112_length__pos__if__in__set,axiom,
! [X2: nat,Xs2: list_nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_1113_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1114_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_1115_remove__nth__len,axiom,
! [I: nat,Xs2: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( size_size_list_nat @ Xs2 )
= ( suc @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ I @ Xs2 ) ) ) ) ) ).
% remove_nth_len
thf(fact_1116_map__upt__add_H,axiom,
! [F: nat > nat,A: nat,B: nat] :
( ( map_nat_nat @ F @ ( upt @ A @ ( plus_plus_nat @ A @ B ) ) )
= ( map_nat_nat
@ ^ [I4: nat] : ( F @ ( plus_plus_nat @ A @ I4 ) )
@ ( upt @ zero_zero_nat @ B ) ) ) ).
% map_upt_add'
thf(fact_1117_map__replicate__trivial,axiom,
! [X2: nat,I: nat] :
( ( map_nat_nat
@ ^ [I4: nat] : X2
@ ( upt @ zero_zero_nat @ I ) )
= ( replicate_nat @ I @ X2 ) ) ).
% map_replicate_trivial
thf(fact_1118_nth__tl,axiom,
! [N: nat,Xs2: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ ( tl_nat @ Xs2 ) ) )
=> ( ( nth_nat @ ( tl_nat @ Xs2 ) @ N )
= ( nth_nat @ Xs2 @ ( suc @ N ) ) ) ) ).
% nth_tl
thf(fact_1119_nth__remove__nth__conv,axiom,
! [I: nat,N: nat,Xs2: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ ( missin7175274867594579095th_nat @ N @ Xs2 ) ) )
=> ( ( nth_nat @ ( missin7175274867594579095th_nat @ N @ Xs2 ) @ I )
= ( nth_nat @ Xs2 @ ( if_nat @ ( ord_less_nat @ I @ N ) @ I @ ( suc @ I ) ) ) ) ) ).
% nth_remove_nth_conv
thf(fact_1120_map__add__upt,axiom,
! [N: nat,M: nat] :
( ( map_nat_nat
@ ^ [I4: nat] : ( plus_plus_nat @ I4 @ N )
@ ( upt @ zero_zero_nat @ M ) )
= ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).
% map_add_upt
thf(fact_1121_map__nth,axiom,
! [Xs2: list_nat] :
( ( map_nat_nat @ ( nth_nat @ Xs2 ) @ ( upt @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) )
= Xs2 ) ).
% map_nth
thf(fact_1122_sorted__iff__nth__Suc,axiom,
! [Xs2: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
= ( ! [I4: nat] :
( ( ord_less_nat @ ( suc @ I4 ) @ ( size_size_list_nat @ Xs2 ) )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs2 @ I4 ) @ ( nth_nat @ Xs2 @ ( suc @ I4 ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_1123_rev__nth,axiom,
! [N: nat,Xs2: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ ( rev_nat @ Xs2 ) @ N )
= ( nth_nat @ Xs2 @ ( minus_minus_nat @ ( size_size_list_nat @ Xs2 ) @ ( suc @ N ) ) ) ) ) ).
% rev_nth
thf(fact_1124_remove__nth__sound__r,axiom,
! [N: nat,P4: nat,Xs2: list_nat] :
( ( ord_less_eq_nat @ N @ P4 )
=> ( ( ord_less_nat @ P4 @ ( size_size_list_nat @ Xs2 ) )
=> ( ( nth_nat @ ( missin7175274867594579095th_nat @ N @ Xs2 ) @ P4 )
= ( nth_nat @ Xs2 @ ( suc @ P4 ) ) ) ) ) ).
% remove_nth_sound_r
thf(fact_1125_map__upt__len__same__len__conv,axiom,
! [Xs2: list_nat,Ys: list_nat,F: nat > nat] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( map_nat_nat
@ ^ [I4: nat] : ( F @ ( nth_nat @ Xs2 @ I4 ) )
@ ( upt @ zero_zero_nat @ ( size_size_list_nat @ Ys ) ) )
= ( map_nat_nat @ F @ Xs2 ) ) ) ).
% map_upt_len_same_len_conv
thf(fact_1126_RRn__Automata_Olast__nthI,axiom,
! [I: nat,Ts: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Ts ) )
=> ( ~ ( ord_less_nat @ I @ ( minus_minus_nat @ ( size_size_list_nat @ Ts ) @ ( suc @ zero_zero_nat ) ) )
=> ( ( nth_nat @ Ts @ I )
= ( last_nat @ Ts ) ) ) ) ).
% RRn_Automata.last_nthI
thf(fact_1127_Nitpick_Osize__list__simp_I2_J,axiom,
( size_size_list_nat
= ( ^ [Xs: list_nat] : ( if_nat @ ( Xs = nil_nat ) @ zero_zero_nat @ ( suc @ ( size_size_list_nat @ ( tl_nat @ Xs ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_1128_length__nth__simps_I1_J,axiom,
( ( size_size_list_nat @ nil_nat )
= zero_zero_nat ) ).
% length_nth_simps(1)
thf(fact_1129_transpose__rectangle,axiom,
! [Xs2: list_list_nat,N: nat] :
( ( ( Xs2 = nil_list_nat )
=> ( N = zero_zero_nat ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ( size_size_list_nat @ ( nth_list_nat @ Xs2 @ I2 ) )
= N ) )
=> ( ( transpose_nat @ Xs2 )
= ( map_nat_list_nat
@ ^ [I4: nat] :
( map_nat_nat
@ ^ [J2: nat] : ( nth_nat @ ( nth_list_nat @ Xs2 @ J2 ) @ I4 )
@ ( upt @ zero_zero_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) ) )
@ ( upt @ zero_zero_nat @ N ) ) ) ) ) ).
% transpose_rectangle
thf(fact_1130_transpose__map__map,axiom,
! [F: nat > nat,Xs2: list_list_nat] :
( ( transpose_nat @ ( map_li7225945977422193158st_nat @ ( map_nat_nat @ F ) @ Xs2 ) )
= ( map_li7225945977422193158st_nat @ ( map_nat_nat @ F ) @ ( transpose_nat @ Xs2 ) ) ) ).
% transpose_map_map
thf(fact_1131_transpose__empty,axiom,
! [Xs2: list_list_nat] :
( ( ( transpose_nat @ Xs2 )
= nil_list_nat )
= ( ! [X: list_nat] :
( ( member_list_nat @ X @ ( set_list_nat2 @ Xs2 ) )
=> ( X = nil_nat ) ) ) ) ).
% transpose_empty
thf(fact_1132_sorted__transpose,axiom,
! [Xs2: list_list_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ ( map_list_nat_nat @ size_size_list_nat @ ( transpose_nat @ Xs2 ) ) ) ) ).
% sorted_transpose
thf(fact_1133_length__transpose__sorted,axiom,
! [Xs2: list_list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ ( map_list_nat_nat @ size_size_list_nat @ Xs2 ) ) )
=> ( ( ( Xs2 = nil_list_nat )
=> ( ( size_s3023201423986296836st_nat @ ( transpose_nat @ Xs2 ) )
= zero_zero_nat ) )
& ( ( Xs2 != nil_list_nat )
=> ( ( size_s3023201423986296836st_nat @ ( transpose_nat @ Xs2 ) )
= ( size_size_list_nat @ ( nth_list_nat @ Xs2 @ zero_zero_nat ) ) ) ) ) ) ).
% length_transpose_sorted
thf(fact_1134_set__concat__lists,axiom,
! [Xs2: list_list_nat] :
( ( set_list_nat2 @ ( missin4567272213201432058ts_nat @ Xs2 ) )
= ( collect_list_nat
@ ^ [As: list_nat] :
( ( ( size_size_list_nat @ As )
= ( size_s3023201423986296836st_nat @ Xs2 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( member_nat @ ( nth_nat @ As @ I4 ) @ ( set_nat2 @ ( nth_list_nat @ Xs2 @ I4 ) ) ) ) ) ) ) ).
% set_concat_lists
thf(fact_1135_nth__nth__transpose__sorted,axiom,
! [Xs2: list_list_nat,I: nat,J: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ ( map_list_nat_nat @ size_size_list_nat @ Xs2 ) ) )
=> ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ ( transpose_nat @ Xs2 ) ) )
=> ( ( ord_less_nat @ J
@ ( size_s3023201423986296836st_nat
@ ( filter_list_nat
@ ^ [Ys2: list_nat] : ( ord_less_nat @ I @ ( size_size_list_nat @ Ys2 ) )
@ Xs2 ) ) )
=> ( ( nth_nat @ ( nth_list_nat @ ( transpose_nat @ Xs2 ) @ I ) @ J )
= ( nth_nat @ ( nth_list_nat @ Xs2 @ J ) @ I ) ) ) ) ) ).
% nth_nth_transpose_sorted
thf(fact_1136_filter__False,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
=> ~ ( P @ X3 ) )
=> ( ( filter_nat @ P @ Xs2 )
= nil_nat ) ) ).
% filter_False
thf(fact_1137_Containers__Auxiliary_Oempty__filter__conv,axiom,
! [P: nat > $o,Xs2: list_nat] :
( ( nil_nat
= ( filter_nat @ P @ Xs2 ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ~ ( P @ X ) ) ) ) ).
% Containers_Auxiliary.empty_filter_conv
thf(fact_1138_filter__empty__conv,axiom,
! [P: nat > $o,Xs2: list_nat] :
( ( ( filter_nat @ P @ Xs2 )
= nil_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
=> ~ ( P @ X ) ) ) ) ).
% filter_empty_conv
thf(fact_1139_length__filter__le,axiom,
! [P: nat > $o,Xs2: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( filter_nat @ P @ Xs2 ) ) @ ( size_size_list_nat @ Xs2 ) ) ).
% length_filter_le
thf(fact_1140_filter__replicate,axiom,
! [P: nat > $o,X2: nat,N: nat] :
( ( ( P @ X2 )
=> ( ( filter_nat @ P @ ( replicate_nat @ N @ X2 ) )
= ( replicate_nat @ N @ X2 ) ) )
& ( ~ ( P @ X2 )
=> ( ( filter_nat @ P @ ( replicate_nat @ N @ X2 ) )
= nil_nat ) ) ) ).
% filter_replicate
thf(fact_1141_sum__length__filter__compl,axiom,
! [P: nat > $o,Xs2: list_nat] :
( ( plus_plus_nat @ ( size_size_list_nat @ ( filter_nat @ P @ Xs2 ) )
@ ( size_size_list_nat
@ ( filter_nat
@ ^ [X: nat] :
~ ( P @ X )
@ Xs2 ) ) )
= ( size_size_list_nat @ Xs2 ) ) ).
% sum_length_filter_compl
thf(fact_1142_replicate__length__filter,axiom,
! [X2: nat,Xs2: list_nat] :
( ( replicate_nat
@ ( size_size_list_nat
@ ( filter_nat
@ ( ^ [Y4: nat,Z: nat] : ( Y4 = Z )
@ X2 )
@ Xs2 ) )
@ X2 )
= ( filter_nat
@ ( ^ [Y4: nat,Z: nat] : ( Y4 = Z )
@ X2 )
@ Xs2 ) ) ).
% replicate_length_filter
thf(fact_1143_length__filter__less,axiom,
! [X2: nat,Xs2: list_nat,P: nat > $o] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
=> ( ~ ( P @ X2 )
=> ( ord_less_nat @ ( size_size_list_nat @ ( filter_nat @ P @ Xs2 ) ) @ ( size_size_list_nat @ Xs2 ) ) ) ) ).
% length_filter_less
thf(fact_1144_filter_Osimps_I1_J,axiom,
! [P: nat > $o] :
( ( filter_nat @ P @ nil_nat )
= nil_nat ) ).
% filter.simps(1)
thf(fact_1145_sorted__filter,axiom,
! [F: nat > nat,Xs2: list_nat,P: nat > $o] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ Xs2 ) )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ ( filter_nat @ P @ Xs2 ) ) ) ) ).
% sorted_filter
thf(fact_1146_filter__sort,axiom,
! [P: nat > $o,F: nat > nat,Xs2: list_nat] :
( ( filter_nat @ P @ ( linord738340561235409698at_nat @ F @ Xs2 ) )
= ( linord738340561235409698at_nat @ F @ ( filter_nat @ P @ Xs2 ) ) ) ).
% filter_sort
thf(fact_1147_sort__key__stable,axiom,
! [F: nat > nat,K: nat,Xs2: list_nat] :
( ( filter_nat
@ ^ [Y: nat] :
( ( F @ Y )
= K )
@ ( linord738340561235409698at_nat @ F @ Xs2 ) )
= ( filter_nat
@ ^ [Y: nat] :
( ( F @ Y )
= K )
@ Xs2 ) ) ).
% sort_key_stable
thf(fact_1148_sorted__wrt__filter,axiom,
! [F: nat > nat > $o,Xs2: list_nat,P: nat > $o] :
( ( sorted_wrt_nat @ F @ Xs2 )
=> ( sorted_wrt_nat @ F @ ( filter_nat @ P @ Xs2 ) ) ) ).
% sorted_wrt_filter
thf(fact_1149_sorted__same,axiom,
! [G: list_nat > nat,Xs2: list_nat] :
( sorted_wrt_nat @ ord_less_eq_nat
@ ( filter_nat
@ ^ [X: nat] :
( X
= ( G @ Xs2 ) )
@ Xs2 ) ) ).
% sorted_same
thf(fact_1150_sorted__map__same,axiom,
! [F: nat > nat,G: list_nat > nat,Xs2: list_nat] :
( sorted_wrt_nat @ ord_less_eq_nat
@ ( map_nat_nat @ F
@ ( filter_nat
@ ^ [X: nat] :
( ( F @ X )
= ( G @ Xs2 ) )
@ Xs2 ) ) ) ).
% sorted_map_same
thf(fact_1151_filter__insort,axiom,
! [F: nat > nat,Xs2: list_nat,P: nat > $o,X2: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ Xs2 ) )
=> ( ( P @ X2 )
=> ( ( filter_nat @ P @ ( linord8961336180081300637at_nat @ F @ X2 @ Xs2 ) )
= ( linord8961336180081300637at_nat @ F @ X2 @ ( filter_nat @ P @ Xs2 ) ) ) ) ) ).
% filter_insort
thf(fact_1152_filter__rev__nth__idx,axiom,
! [I: nat,Xs2: list_nat,P: nat > $o,Ys: list_nat] :
( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
=> ( ( P @ ( nth_nat @ Xs2 @ I ) )
=> ( ( Ys
= ( filter_nat @ P @ Xs2 ) )
=> ( ( ( nth_nat @ Xs2 @ I )
= ( nth_nat @ Ys @ ( basic_4353017870094810967th_nat @ P @ Xs2 @ I ) ) )
& ( ord_less_nat @ ( basic_4353017870094810967th_nat @ P @ Xs2 @ I ) @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).
% filter_rev_nth_idx
thf(fact_1153_filter__rev__nth__def,axiom,
( basic_4353017870094810967th_nat
= ( ^ [P3: nat > $o,Xs: list_nat,I4: nat] : ( minus_minus_nat @ ( size_size_list_nat @ ( filter_nat @ P3 @ ( take_nat @ ( suc @ I4 ) @ Xs ) ) ) @ one_one_nat ) ) ) ).
% filter_rev_nth_def
thf(fact_1154_nth__transpose,axiom,
! [I: nat,Xs2: list_list_nat] :
( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ ( transpose_nat @ Xs2 ) ) )
=> ( ( nth_list_nat @ ( transpose_nat @ Xs2 ) @ I )
= ( map_list_nat_nat
@ ^ [Xs: list_nat] : ( nth_nat @ Xs @ I )
@ ( filter_list_nat
@ ^ [Ys2: list_nat] : ( ord_less_nat @ I @ ( size_size_list_nat @ Ys2 ) )
@ Xs2 ) ) ) ) ).
% nth_transpose
thf(fact_1155_transpose__column__length,axiom,
! [Xs2: list_list_nat,I: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ ( map_list_nat_nat @ size_size_list_nat @ Xs2 ) ) )
=> ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ( size_s3023201423986296836st_nat
@ ( filter_list_nat
@ ^ [Ys2: list_nat] : ( ord_less_nat @ I @ ( size_size_list_nat @ Ys2 ) )
@ ( transpose_nat @ Xs2 ) ) )
= ( size_size_list_nat @ ( nth_list_nat @ Xs2 @ I ) ) ) ) ) ).
% transpose_column_length
thf(fact_1156_transpose__column,axiom,
! [Xs2: list_list_nat,I: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ ( map_list_nat_nat @ size_size_list_nat @ Xs2 ) ) )
=> ( ( ord_less_nat @ I @ ( size_s3023201423986296836st_nat @ Xs2 ) )
=> ( ( map_list_nat_nat
@ ^ [Ys2: list_nat] : ( nth_nat @ Ys2 @ I )
@ ( filter_list_nat
@ ^ [Ys2: list_nat] : ( ord_less_nat @ I @ ( size_size_list_nat @ Ys2 ) )
@ ( transpose_nat @ Xs2 ) ) )
= ( nth_list_nat @ Xs2 @ I ) ) ) ) ).
% transpose_column
thf(fact_1157_listset,axiom,
( listset_nat
= ( ^ [Xs: list_set_nat] :
( collect_list_nat
@ ^ [Ys2: list_nat] :
( ( ( size_size_list_nat @ Ys2 )
= ( size_s3254054031482475050et_nat @ Xs ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s3254054031482475050et_nat @ Xs ) )
=> ( member_nat @ ( nth_nat @ Ys2 @ I4 ) @ ( nth_set_nat @ Xs @ I4 ) ) ) ) ) ) ) ).
% listset
thf(fact_1158_transpose__transpose,axiom,
! [Xs2: list_list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ ( map_list_nat_nat @ size_size_list_nat @ Xs2 ) ) )
=> ( ( transpose_nat @ ( transpose_nat @ Xs2 ) )
= ( takeWhile_list_nat
@ ^ [X: list_nat] : ( X != nil_nat )
@ Xs2 ) ) ) ).
% transpose_transpose
thf(fact_1159_takeWhile__replicate,axiom,
! [P: nat > $o,X2: nat,N: nat] :
( ( ( P @ X2 )
=> ( ( takeWhile_nat @ P @ ( replicate_nat @ N @ X2 ) )
= ( replicate_nat @ N @ X2 ) ) )
& ( ~ ( P @ X2 )
=> ( ( takeWhile_nat @ P @ ( replicate_nat @ N @ X2 ) )
= nil_nat ) ) ) ).
% takeWhile_replicate
thf(fact_1160_sorted__takeWhile,axiom,
! [Xs2: list_nat,P: nat > $o] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs2 )
=> ( sorted_wrt_nat @ ord_less_eq_nat @ ( takeWhile_nat @ P @ Xs2 ) ) ) ).
% sorted_takeWhile
thf(fact_1161_takeWhile_Osimps_I1_J,axiom,
! [P: nat > $o] :
( ( takeWhile_nat @ P @ nil_nat )
= nil_nat ) ).
% takeWhile.simps(1)
thf(fact_1162_takeWhile__eq__take,axiom,
( takeWhile_nat
= ( ^ [P3: nat > $o,Xs: list_nat] : ( take_nat @ ( size_size_list_nat @ ( takeWhile_nat @ P3 @ Xs ) ) @ Xs ) ) ) ).
% takeWhile_eq_take
thf(fact_1163_length__takeWhile__le,axiom,
! [P: nat > $o,Xs2: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( takeWhile_nat @ P @ Xs2 ) ) @ ( size_size_list_nat @ Xs2 ) ) ).
% length_takeWhile_le
thf(fact_1164_nth__length__takeWhile,axiom,
! [P: nat > $o,Xs2: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ ( takeWhile_nat @ P @ Xs2 ) ) @ ( size_size_list_nat @ Xs2 ) )
=> ~ ( P @ ( nth_nat @ Xs2 @ ( size_size_list_nat @ ( takeWhile_nat @ P @ Xs2 ) ) ) ) ) ).
% nth_length_takeWhile
thf(fact_1165_takeWhile__nth,axiom,
! [J: nat,P: nat > $o,Xs2: list_nat] :
( ( ord_less_nat @ J @ ( size_size_list_nat @ ( takeWhile_nat @ P @ Xs2 ) ) )
=> ( ( nth_nat @ ( takeWhile_nat @ P @ Xs2 ) @ J )
= ( nth_nat @ Xs2 @ J ) ) ) ).
% takeWhile_nth
thf(fact_1166_length__takeWhile__less__P__nth,axiom,
! [J: nat,P: nat > $o,Xs2: list_nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( P @ ( nth_nat @ Xs2 @ I2 ) ) )
=> ( ( ord_less_eq_nat @ J @ ( size_size_list_nat @ Xs2 ) )
=> ( ord_less_eq_nat @ J @ ( size_size_list_nat @ ( takeWhile_nat @ P @ Xs2 ) ) ) ) ) ).
% length_takeWhile_less_P_nth
thf(fact_1167_takeWhile__eq__take__P__nth,axiom,
! [N: nat,Xs2: list_nat,P: nat > $o] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ ( nth_nat @ Xs2 @ I2 ) ) ) )
=> ( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
=> ~ ( P @ ( nth_nat @ Xs2 @ N ) ) )
=> ( ( takeWhile_nat @ P @ Xs2 )
= ( take_nat @ N @ Xs2 ) ) ) ) ).
% takeWhile_eq_take_P_nth
thf(fact_1168_filter__equals__takeWhile__sorted__rev,axiom,
! [F: nat > nat,Xs2: list_nat,T: nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ ( map_nat_nat @ F @ Xs2 ) ) )
=> ( ( filter_nat
@ ^ [X: nat] : ( ord_less_nat @ T @ ( F @ X ) )
@ Xs2 )
= ( takeWhile_nat
@ ^ [X: nat] : ( ord_less_nat @ T @ ( F @ X ) )
@ Xs2 ) ) ) ).
% filter_equals_takeWhile_sorted_rev
thf(fact_1169_insort__key__remove1,axiom,
! [A: nat,Xs2: list_nat,F: nat > nat] :
( ( member_nat @ A @ ( set_nat2 @ Xs2 ) )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ Xs2 ) )
=> ( ( ( hd_nat
@ ( filter_nat
@ ^ [X: nat] :
( ( F @ A )
= ( F @ X ) )
@ Xs2 ) )
= A )
=> ( ( linord8961336180081300637at_nat @ F @ A @ ( remove1_nat @ A @ Xs2 ) )
= Xs2 ) ) ) ) ).
% insort_key_remove1
thf(fact_1170_hd__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( hd_nat @ ( upt @ I @ J ) )
= I ) ) ).
% hd_upt
thf(fact_1171_hd__replicate,axiom,
! [N: nat,X2: nat] :
( ( N != zero_zero_nat )
=> ( ( hd_nat @ ( replicate_nat @ N @ X2 ) )
= X2 ) ) ).
% hd_replicate
thf(fact_1172_hd__take,axiom,
! [J: nat,Xs2: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ J )
=> ( ( hd_nat @ ( take_nat @ J @ Xs2 ) )
= ( hd_nat @ Xs2 ) ) ) ).
% hd_take
thf(fact_1173_hd__map,axiom,
! [Xs2: list_nat,F: nat > nat] :
( ( Xs2 != nil_nat )
=> ( ( hd_nat @ ( map_nat_nat @ F @ Xs2 ) )
= ( F @ ( hd_nat @ Xs2 ) ) ) ) ).
% hd_map
thf(fact_1174_list_Omap__sel_I1_J,axiom,
! [A: list_nat,F: nat > nat] :
( ( A != nil_nat )
=> ( ( hd_nat @ ( map_nat_nat @ F @ A ) )
= ( F @ ( hd_nat @ A ) ) ) ) ).
% list.map_sel(1)
thf(fact_1175_takeWhile__eq__Nil__iff,axiom,
! [P: nat > $o,Xs2: list_nat] :
( ( ( takeWhile_nat @ P @ Xs2 )
= nil_nat )
= ( ( Xs2 = nil_nat )
| ~ ( P @ ( hd_nat @ Xs2 ) ) ) ) ).
% takeWhile_eq_Nil_iff
thf(fact_1176_hd__conv__nth,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( hd_nat @ Xs2 )
= ( nth_nat @ Xs2 @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_1177_list_Oset__sel_I1_J,axiom,
! [A: list_nat] :
( ( A != nil_nat )
=> ( member_nat @ ( hd_nat @ A ) @ ( set_nat2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_1178_hd__in__set,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( member_nat @ ( hd_nat @ Xs2 ) @ ( set_nat2 @ Xs2 ) ) ) ).
% hd_in_set
thf(fact_1179_list_Oexpand,axiom,
! [List: list_nat,List2: list_nat] :
( ( ( List = nil_nat )
= ( List2 = nil_nat ) )
=> ( ( ( List != nil_nat )
=> ( ( List2 != nil_nat )
=> ( ( ( hd_nat @ List )
= ( hd_nat @ List2 ) )
& ( ( tl_nat @ List )
= ( tl_nat @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_1180_hd__Nil__eq__last,axiom,
( ( hd_nat @ nil_nat )
= ( last_nat @ nil_nat ) ) ).
% hd_Nil_eq_last
thf(fact_1181_hd__rev,axiom,
! [Xs2: list_nat] :
( ( hd_nat @ ( rev_nat @ Xs2 ) )
= ( last_nat @ Xs2 ) ) ).
% hd_rev
thf(fact_1182_last__rev,axiom,
! [Xs2: list_nat] :
( ( last_nat @ ( rev_nat @ Xs2 ) )
= ( hd_nat @ Xs2 ) ) ).
% last_rev
thf(fact_1183_Least__eq__0,axiom,
! [P: nat > $o] :
( ( P @ zero_zero_nat )
=> ( ( ord_Least_nat @ P )
= zero_zero_nat ) ) ).
% Least_eq_0
thf(fact_1184_Least__le,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ord_less_eq_nat @ ( ord_Least_nat @ P ) @ K ) ) ).
% Least_le
thf(fact_1185_not__less__Least,axiom,
! [K: nat,P: nat > $o] :
( ( ord_less_nat @ K @ ( ord_Least_nat @ P ) )
=> ~ ( P @ K ) ) ).
% not_less_Least
thf(fact_1186_Least__Suc2,axiom,
! [P: nat > $o,N: nat,Q: nat > $o,M: nat] :
( ( P @ N )
=> ( ( Q @ M )
=> ( ~ ( P @ zero_zero_nat )
=> ( ! [K3: nat] :
( ( P @ ( suc @ K3 ) )
= ( Q @ K3 ) )
=> ( ( ord_Least_nat @ P )
= ( suc @ ( ord_Least_nat @ Q ) ) ) ) ) ) ) ).
% Least_Suc2
thf(fact_1187_Least__Suc,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ( ( ord_Least_nat @ P )
= ( suc
@ ( ord_Least_nat
@ ^ [M4: nat] : ( P @ ( suc @ M4 ) ) ) ) ) ) ) ).
% Least_Suc
thf(fact_1188_LeastI2__wellorder__ex,axiom,
! [P: nat > $o,Q: nat > $o] :
( ? [X_1: nat] : ( P @ X_1 )
=> ( ! [A3: nat] :
( ( P @ A3 )
=> ( ! [B7: nat] :
( ( P @ B7 )
=> ( ord_less_eq_nat @ A3 @ B7 ) )
=> ( Q @ A3 ) ) )
=> ( Q @ ( ord_Least_nat @ P ) ) ) ) ).
% LeastI2_wellorder_ex
thf(fact_1189_LeastI2__wellorder,axiom,
! [P: nat > $o,A: nat,Q: nat > $o] :
( ( P @ A )
=> ( ! [A3: nat] :
( ( P @ A3 )
=> ( ! [B7: nat] :
( ( P @ B7 )
=> ( ord_less_eq_nat @ A3 @ B7 ) )
=> ( Q @ A3 ) ) )
=> ( Q @ ( ord_Least_nat @ P ) ) ) ) ).
% LeastI2_wellorder
thf(fact_1190_Least__equality,axiom,
! [P: nat > $o,X2: nat] :
( ( P @ X2 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ X2 @ Y3 ) )
=> ( ( ord_Least_nat @ P )
= X2 ) ) ) ).
% Least_equality
thf(fact_1191_LeastI2__order,axiom,
! [P: nat > $o,X2: nat,Q: nat > $o] :
( ( P @ X2 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ X2 @ Y3 ) )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ X3 @ Y5 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( ord_Least_nat @ P ) ) ) ) ) ).
% LeastI2_order
thf(fact_1192_Least1__le,axiom,
! [P: nat > $o,Z2: nat] :
( ? [X4: nat] :
( ( P @ X4 )
& ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ X4 @ Y3 ) )
& ! [Y3: nat] :
( ( ( P @ Y3 )
& ! [Ya2: nat] :
( ( P @ Ya2 )
=> ( ord_less_eq_nat @ Y3 @ Ya2 ) ) )
=> ( Y3 = X4 ) ) )
=> ( ( P @ Z2 )
=> ( ord_less_eq_nat @ ( ord_Least_nat @ P ) @ Z2 ) ) ) ).
% Least1_le
thf(fact_1193_Least1I,axiom,
! [P: nat > $o] :
( ? [X4: nat] :
( ( P @ X4 )
& ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ X4 @ Y3 ) )
& ! [Y3: nat] :
( ( ( P @ Y3 )
& ! [Ya2: nat] :
( ( P @ Ya2 )
=> ( ord_less_eq_nat @ Y3 @ Ya2 ) ) )
=> ( Y3 = X4 ) ) )
=> ( P @ ( ord_Least_nat @ P ) ) ) ).
% Least1I
thf(fact_1194_LeastI,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( P @ ( ord_Least_nat @ P ) ) ) ).
% LeastI
thf(fact_1195_LeastI2,axiom,
! [P: nat > $o,A: nat,Q: nat > $o] :
( ( P @ A )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( ord_Least_nat @ P ) ) ) ) ).
% LeastI2
thf(fact_1196_LeastI__ex,axiom,
! [P: nat > $o] :
( ? [X_1: nat] : ( P @ X_1 )
=> ( P @ ( ord_Least_nat @ P ) ) ) ).
% LeastI_ex
thf(fact_1197_LeastI2__ex,axiom,
! [P: nat > $o,Q: nat > $o] :
( ? [X_1: nat] : ( P @ X_1 )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( ord_Least_nat @ P ) ) ) ) ).
% LeastI2_ex
thf(fact_1198_Nitpick_Osize__list__simp_I1_J,axiom,
( size_list_nat
= ( ^ [F2: nat > nat,Xs: list_nat] : ( if_nat @ ( Xs = nil_nat ) @ zero_zero_nat @ ( suc @ ( plus_plus_nat @ ( F2 @ ( hd_nat @ Xs ) ) @ ( size_list_nat @ F2 @ ( tl_nat @ Xs ) ) ) ) ) ) ) ).
% Nitpick.size_list_simp(1)
thf(fact_1199_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( P @ A3 @ B3 )
= ( P @ B3 @ A3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
=> ( ! [A3: nat,B3: nat] :
( ( P @ A3 @ B3 )
=> ( P @ A3 @ ( plus_plus_nat @ A3 @ B3 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1200_list_Osize__gen_I1_J,axiom,
! [X2: nat > nat] :
( ( size_list_nat @ X2 @ nil_nat )
= zero_zero_nat ) ).
% list.size_gen(1)
thf(fact_1201_size__simp2,axiom,
! [T: list_nat,Ts: list_list_nat,S2: list_nat] :
( ( member_list_nat @ T @ ( set_list_nat2 @ Ts ) )
=> ( ord_less_nat @ ( size_size_list_nat @ T ) @ ( suc @ ( suc @ ( plus_plus_nat @ ( size_size_list_nat @ S2 ) @ ( size_list_list_nat @ size_size_list_nat @ Ts ) ) ) ) ) ) ).
% size_simp2
thf(fact_1202_size__simp1,axiom,
! [T: list_nat,Ts: list_list_nat] :
( ( member_list_nat @ T @ ( set_list_nat2 @ Ts ) )
=> ( ord_less_nat @ ( size_size_list_nat @ T ) @ ( suc @ ( size_list_list_nat @ size_size_list_nat @ Ts ) ) ) ) ).
% size_simp1
thf(fact_1203_triangle__Suc,axiom,
! [N: nat] :
( ( nat_triangle @ ( suc @ N ) )
= ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% triangle_Suc
thf(fact_1204_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_1205_nth__Cons__pos,axiom,
! [N: nat,X2: nat,Xs2: list_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_nat @ ( cons_nat @ X2 @ Xs2 ) @ N )
= ( nth_nat @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_1206_in__set__product__lists__length,axiom,
! [Xs2: list_nat,Xss: list_list_nat] :
( ( member_list_nat @ Xs2 @ ( set_list_nat2 @ ( product_lists_nat @ Xss ) ) )
=> ( ( size_size_list_nat @ Xs2 )
= ( size_s3023201423986296836st_nat @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_1207_nth__Cons__0,axiom,
! [X2: nat,Xs2: list_nat] :
( ( nth_nat @ ( cons_nat @ X2 @ Xs2 ) @ zero_zero_nat )
= X2 ) ).
% nth_Cons_0
thf(fact_1208_nth__Cons__Suc,axiom,
! [X2: nat,Xs2: list_nat,N: nat] :
( ( nth_nat @ ( cons_nat @ X2 @ Xs2 ) @ ( suc @ N ) )
= ( nth_nat @ Xs2 @ N ) ) ).
% nth_Cons_Suc
thf(fact_1209_take__Suc__Cons,axiom,
! [N: nat,X2: nat,Xs2: list_nat] :
( ( take_nat @ ( suc @ N ) @ ( cons_nat @ X2 @ Xs2 ) )
= ( cons_nat @ X2 @ ( take_nat @ N @ Xs2 ) ) ) ).
% take_Suc_Cons
thf(fact_1210_singleton__rev__conv,axiom,
! [X2: nat,Xs2: list_nat] :
( ( ( cons_nat @ X2 @ nil_nat )
= ( rev_nat @ Xs2 ) )
= ( ( cons_nat @ X2 @ nil_nat )
= Xs2 ) ) ).
% singleton_rev_conv
thf(fact_1211_rev__singleton__conv,axiom,
! [Xs2: list_nat,X2: nat] :
( ( ( rev_nat @ Xs2 )
= ( cons_nat @ X2 @ nil_nat ) )
= ( Xs2
= ( cons_nat @ X2 @ nil_nat ) ) ) ).
% rev_singleton_conv
thf(fact_1212_Cons__less__Cons,axiom,
! [A: nat,X2: list_nat,B: nat,Y2: list_nat] :
( ( ord_less_list_nat @ ( cons_nat @ A @ X2 ) @ ( cons_nat @ B @ Y2 ) )
= ( ( ord_less_nat @ A @ B )
| ( ( A = B )
& ( ord_less_list_nat @ X2 @ Y2 ) ) ) ) ).
% Cons_less_Cons
thf(fact_1213_sort__key__simps_I2_J,axiom,
! [F: nat > nat,X2: nat,Xs2: list_nat] :
( ( linord738340561235409698at_nat @ F @ ( cons_nat @ X2 @ Xs2 ) )
= ( linord8961336180081300637at_nat @ F @ X2 @ ( linord738340561235409698at_nat @ F @ Xs2 ) ) ) ).
% sort_key_simps(2)
thf(fact_1214_Cons__le__Cons,axiom,
! [A: nat,X2: list_nat,B: nat,Y2: list_nat] :
( ( ord_less_eq_list_nat @ ( cons_nat @ A @ X2 ) @ ( cons_nat @ B @ Y2 ) )
= ( ( ord_less_nat @ A @ B )
| ( ( A = B )
& ( ord_less_eq_list_nat @ X2 @ Y2 ) ) ) ) ).
% Cons_le_Cons
thf(fact_1215_hd__Cons__tl,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ( cons_nat @ ( hd_nat @ Xs2 ) @ ( tl_nat @ Xs2 ) )
= Xs2 ) ) ).
% hd_Cons_tl
thf(fact_1216_list_Ocollapse,axiom,
! [List: list_nat] :
( ( List != nil_nat )
=> ( ( cons_nat @ ( hd_nat @ List ) @ ( tl_nat @ List ) )
= List ) ) ).
% list.collapse
thf(fact_1217_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_nat @ N @ nil_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_nat @ N @ nil_nat )
= nil_list_nat ) ) ) ).
% n_lists_Nil
thf(fact_1218_list__induct4,axiom,
! [Xs2: list_nat,Ys: list_nat,Zs: list_nat,Ws: list_nat,P: list_nat > list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Ws ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs3: list_nat,Y3: nat,Ys4: list_nat,Z3: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( ( size_size_list_nat @ Zs2 )
= ( size_size_list_nat @ Ws2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z3 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1219_list__induct3,axiom,
! [Xs2: list_nat,Ys: list_nat,Zs: list_nat,P: list_nat > list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Zs ) )
=> ( ( P @ nil_nat @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs3: list_nat,Y3: nat,Ys4: list_nat,Z3: nat,Zs2: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( ( size_size_list_nat @ Ys4 )
= ( size_size_list_nat @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys4 @ Zs2 )
=> ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) @ ( cons_nat @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_1220_list__induct2,axiom,
! [Xs2: list_nat,Ys: list_nat,P: list_nat > list_nat > $o] :
( ( ( size_size_list_nat @ Xs2 )
= ( size_size_list_nat @ Ys ) )
=> ( ( P @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs3: list_nat,Y3: nat,Ys4: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys4 ) )
=> ( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1221_in__set__simps_I2_J,axiom,
! [X2: nat,Y2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ ( cons_nat @ Y2 @ nil_nat ) ) )
= ( X2 = Y2 ) ) ).
% in_set_simps(2)
thf(fact_1222_sorted__wrt1,axiom,
! [P: nat > nat > $o,X2: nat] : ( sorted_wrt_nat @ P @ ( cons_nat @ X2 @ nil_nat ) ) ).
% sorted_wrt1
thf(fact_1223_Nil__tl,axiom,
! [Xs2: list_nat] :
( ( nil_nat
= ( tl_nat @ Xs2 ) )
= ( ( Xs2 = nil_nat )
| ? [X: nat] :
( Xs2
= ( cons_nat @ X @ nil_nat ) ) ) ) ).
% Nil_tl
thf(fact_1224_tl__Nil,axiom,
! [Xs2: list_nat] :
( ( ( tl_nat @ Xs2 )
= nil_nat )
= ( ( Xs2 = nil_nat )
| ? [X: nat] :
( Xs2
= ( cons_nat @ X @ nil_nat ) ) ) ) ).
% tl_Nil
thf(fact_1225_last_Osimps,axiom,
! [Xs2: list_nat,X2: nat] :
( ( ( Xs2 = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X2 @ Xs2 ) )
= X2 ) )
& ( ( Xs2 != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X2 @ Xs2 ) )
= ( last_nat @ Xs2 ) ) ) ) ).
% last.simps
thf(fact_1226_last__ConsL,axiom,
! [Xs2: list_nat,X2: nat] :
( ( Xs2 = nil_nat )
=> ( ( last_nat @ ( cons_nat @ X2 @ Xs2 ) )
= X2 ) ) ).
% last_ConsL
thf(fact_1227_last__ConsR,axiom,
! [Xs2: list_nat,X2: nat] :
( ( Xs2 != nil_nat )
=> ( ( last_nat @ ( cons_nat @ X2 @ Xs2 ) )
= ( last_nat @ Xs2 ) ) ) ).
% last_ConsR
thf(fact_1228_less__list__code_I2_J,axiom,
! [X2: nat,Xs2: list_nat] : ( ord_less_list_nat @ nil_nat @ ( cons_nat @ X2 @ Xs2 ) ) ).
% less_list_code(2)
thf(fact_1229_Nil__less__Cons,axiom,
! [A: nat,X2: list_nat] : ( ord_less_list_nat @ nil_nat @ ( cons_nat @ A @ X2 ) ) ).
% Nil_less_Cons
thf(fact_1230_butlast_Osimps_I2_J,axiom,
! [Xs2: list_nat,X2: nat] :
( ( ( Xs2 = nil_nat )
=> ( ( butlast_nat @ ( cons_nat @ X2 @ Xs2 ) )
= nil_nat ) )
& ( ( Xs2 != nil_nat )
=> ( ( butlast_nat @ ( cons_nat @ X2 @ Xs2 ) )
= ( cons_nat @ X2 @ ( butlast_nat @ Xs2 ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_1231_Missing__List_Omin__list_Osimps_I1_J,axiom,
! [X2: nat] :
( ( missing_min_list_nat @ ( cons_nat @ X2 @ nil_nat ) )
= X2 ) ).
% Missing_List.min_list.simps(1)
thf(fact_1232_subtract__list__sorted_Osimps_I3_J,axiom,
! [V: nat,Va: list_nat] :
( ( missin6424796737333596952ed_nat @ ( cons_nat @ V @ Va ) @ nil_nat )
= ( cons_nat @ V @ Va ) ) ).
% subtract_list_sorted.simps(3)
thf(fact_1233_remdups__sorted_Osimps_I2_J,axiom,
! [X2: nat] :
( ( set_or6599480164596245535ed_nat @ ( cons_nat @ X2 @ nil_nat ) )
= ( cons_nat @ X2 @ nil_nat ) ) ).
% remdups_sorted.simps(2)
thf(fact_1234_less__eq__list__code_I1_J,axiom,
! [X2: nat,Xs2: list_nat] :
~ ( ord_less_eq_list_nat @ ( cons_nat @ X2 @ Xs2 ) @ nil_nat ) ).
% less_eq_list_code(1)
thf(fact_1235_transpose_Osimps_I2_J,axiom,
! [Xss: list_list_nat] :
( ( transpose_nat @ ( cons_list_nat @ nil_nat @ Xss ) )
= ( transpose_nat @ Xss ) ) ).
% transpose.simps(2)
thf(fact_1236_transpose_Ocases,axiom,
! [X2: list_list_nat] :
( ( X2 != nil_list_nat )
=> ( ! [Xss2: list_list_nat] :
( X2
!= ( cons_list_nat @ nil_nat @ Xss2 ) )
=> ~ ! [X3: nat,Xs3: list_nat,Xss2: list_list_nat] :
( X2
!= ( cons_list_nat @ ( cons_nat @ X3 @ Xs3 ) @ Xss2 ) ) ) ) ).
% transpose.cases
thf(fact_1237_product__lists_Osimps_I1_J,axiom,
( ( product_lists_nat @ nil_list_nat )
= ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).
% product_lists.simps(1)
thf(fact_1238_list_Odistinct_I1_J,axiom,
! [X21: nat,X222: list_nat] :
( nil_nat
!= ( cons_nat @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_1239_list_OdiscI,axiom,
! [List: list_nat,X21: nat,X222: list_nat] :
( ( List
= ( cons_nat @ X21 @ X222 ) )
=> ( List != nil_nat ) ) ).
% list.discI
thf(fact_1240_list_Oexhaust,axiom,
! [Y2: list_nat] :
( ( Y2 != nil_nat )
=> ~ ! [X212: nat,X223: list_nat] :
( Y2
!= ( cons_nat @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_1241_List_Omin__list_Ocases,axiom,
! [X2: list_nat] :
( ! [X3: nat,Xs3: list_nat] :
( X2
!= ( cons_nat @ X3 @ Xs3 ) )
=> ( X2 = nil_nat ) ) ).
% List.min_list.cases
thf(fact_1242_neq__Nil__conv,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
= ( ? [Y: nat,Ys2: list_nat] :
( Xs2
= ( cons_nat @ Y @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_1243_list__induct2_H,axiom,
! [P: list_nat > list_nat > $o,Xs2: list_nat,Ys: list_nat] :
( ( P @ nil_nat @ nil_nat )
=> ( ! [X3: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X3 @ Xs3 ) @ nil_nat )
=> ( ! [Y3: nat,Ys4: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y3 @ Ys4 ) )
=> ( ! [X3: nat,Xs3: list_nat,Y3: nat,Ys4: list_nat] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) ) )
=> ( P @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_1244_list__nonempty__induct,axiom,
! [Xs2: list_nat,P: list_nat > $o] :
( ( Xs2 != nil_nat )
=> ( ! [X3: nat] : ( P @ ( cons_nat @ X3 @ nil_nat ) )
=> ( ! [X3: nat,Xs3: list_nat] :
( ( Xs3 != nil_nat )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_nat @ X3 @ Xs3 ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_1245_list__4__cases,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ! [X3: nat] :
( Xs2
!= ( cons_nat @ X3 @ nil_nat ) )
=> ( ! [X3: nat,Y3: nat] :
( Xs2
!= ( cons_nat @ X3 @ ( cons_nat @ Y3 @ nil_nat ) ) )
=> ~ ! [X3: nat,Y3: nat,Z3: nat,Zs2: list_nat] :
( Xs2
!= ( cons_nat @ X3 @ ( cons_nat @ Y3 @ ( cons_nat @ Z3 @ Zs2 ) ) ) ) ) ) ) ).
% list_4_cases
thf(fact_1246_list__3__cases,axiom,
! [Xs2: list_nat] :
( ( Xs2 != nil_nat )
=> ( ! [X3: nat] :
( Xs2
!= ( cons_nat @ X3 @ nil_nat ) )
=> ~ ! [X3: nat,Y3: nat,Ys4: list_nat] :
( Xs2
!= ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Ys4 ) ) ) ) ) ).
% list_3_cases
thf(fact_1247_remdups__sorted_Ocases,axiom,
! [X2: list_nat] :
( ( X2 != nil_nat )
=> ( ! [X3: nat] :
( X2
!= ( cons_nat @ X3 @ nil_nat ) )
=> ~ ! [X3: nat,Y3: nat,Xs3: list_nat] :
( X2
!= ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs3 ) ) ) ) ) ).
% remdups_sorted.cases
thf(fact_1248_ord_Oremdups__sorted_Ocases,axiom,
! [X2: list_nat] :
( ( X2 != nil_nat )
=> ( ! [X3: nat] :
( X2
!= ( cons_nat @ X3 @ nil_nat ) )
=> ~ ! [X3: nat,Y3: nat,Xs3: list_nat] :
( X2
!= ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs3 ) ) ) ) ) ).
% ord.remdups_sorted.cases
thf(fact_1249_Missing__List_Omin__list_Ocases,axiom,
! [X2: list_nat] :
( ! [X3: nat] :
( X2
!= ( cons_nat @ X3 @ nil_nat ) )
=> ( ! [X3: nat,V2: nat,Va2: list_nat] :
( X2
!= ( cons_nat @ X3 @ ( cons_nat @ V2 @ Va2 ) ) )
=> ( X2 = nil_nat ) ) ) ).
% Missing_List.min_list.cases
thf(fact_1250_takeWhile_Osimps_I2_J,axiom,
! [P: nat > $o,X2: nat,Xs2: list_nat] :
( ( ( P @ X2 )
=> ( ( takeWhile_nat @ P @ ( cons_nat @ X2 @ Xs2 ) )
= ( cons_nat @ X2 @ ( takeWhile_nat @ P @ Xs2 ) ) ) )
& ( ~ ( P @ X2 )
=> ( ( takeWhile_nat @ P @ ( cons_nat @ X2 @ Xs2 ) )
= nil_nat ) ) ) ).
% takeWhile.simps(2)
thf(fact_1251_Suc__le__length__iff,axiom,
! [N: nat,Xs2: list_nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs2 ) )
= ( ? [X: nat,Ys2: list_nat] :
( ( Xs2
= ( cons_nat @ X @ Ys2 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y2: nat] :
( ( if_nat @ $false @ X2 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y2: nat] :
( ( if_nat @ $true @ X2 @ Y2 )
= X2 ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
sorted_wrt_nat @ ord_less_eq_nat @ xs ).
thf(conj_1,conjecture,
( sorted_wrt_nat @ ord_less_eq_nat
@ ( tl_nat
@ ( map_nat_nat
@ ^ [X: nat] : ( minus_minus_nat @ X @ one_one_nat )
@ xs ) ) ) ).
%------------------------------------------------------------------------------