TPTP Problem File: SCT169_3.p
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%------------------------------------------------------------------------------
% File : SCT169_3 : TPTP v8.2.0. Released v5.3.0.
% Domain : Social Choice Theory
% Problem : Arrow's Impossibility Theorem line 29, 1000 axioms selected
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla11]
% Names : arrow_1000_tff_l29 [Bla11]
% Status : Theorem
% Rating : 0.89 v8.2.0, 1.00 v7.1.0, 0.75 v7.0.0, 1.00 v5.5.0, 0.75 v5.4.0, 1.00 v5.3.0
% Syntax : Number of formulae : 1636 ( 446 unt; 369 typ; 0 def)
% Number of atoms : 2819 (1211 equ)
% Maximal formula atoms : 8 ( 1 avg)
% Number of connectives : 1904 ( 352 ~; 76 |; 236 &)
% ( 305 <=>; 935 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 5 avg)
% Maximal term depth : 9 ( 2 avg)
% Number of types : 89 ( 88 usr)
% Number of type conns : 359 ( 200 >; 159 *; 0 +; 0 <<)
% Number of predicates : 2 ( 1 usr; 0 prp; 1-2 aty)
% Number of functors : 280 ( 280 usr; 81 con; 0-3 aty)
% Number of variables : 3275 (3066 !; 209 ?;3275 :)
% SPC : TF0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 17:16:26
%------------------------------------------------------------------------------
%----Should-be-implicit typings (88)
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tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
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tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__List__Olist_Itc__HOL__Obool_J_J,type,
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tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__List__Olist_Itc__Nat__Onat_J_J,type,
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tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J,type,
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tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__Option__Ooption_Itc__HOL__Obool_J_J,type,
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tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__Option__Ooption_Itc__Nat__Onat_J_J,type,
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tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_J,type,
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tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__fun_Itc__Nat__Onat_Mtc__Nat__Onat_J_J,type,
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tff(ty_ty_tc__fun_Itc__Nat__Onat_Mtc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_,type,
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fun_fu1537843567t_bool: $tType ).
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fun_fu1085384691l_bool: $tType ).
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tff(ty_ty_tc__fun_Itc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J,type,
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tff(ty_ty_tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J,type,
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%----Explicit typings (281)
tff(sy_c_Big__Operators_Olattice_OSup__fin_000tc__Nat__Onat,type,
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tff(sy_c_Big__Operators_Olinorder__class_OMax_000tc__Nat__Onat,type,
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tff(sy_c_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Obool_J_000tc__024,type,
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tff(sy_c_COMBB_000tc__List__Olist_Itc__Arrow____Order____Mirabelle____effngqiqbs__Oa,type,
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tff(sy_c_COMBB_000tc__List__Olist_Itc__Arrow____Order____Mirabelle____effngqiqbs__Oa_025,type,
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tff(sy_c_COMBB_000tc__List__Olist_Itc__HOL__Obool_J_000tc__List__Olist_Itc__HOL__Obo,type,
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tff(sy_c_COMBB_000tc__List__Olist_Itc__Nat__Onat_J_000tc__List__Olist_Itc__Nat__Onat,type,
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tff(sy_c_COMBC_000tc__HOL__Obool_000tc__HOL__Obool_000tc__HOL__Obool,type,
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tff(sy_c_COMBC_000tc__HOL__Obool_000tc__List__Olist_Itc__HOL__Obool_J_000tc__List__O,type,
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tff(sy_c_COMBC_000tc__List__Olist_Itc__HOL__Obool_J_000tc__List__Olist_Itc__HOL__Obo,type,
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tff(sy_c_COMBC_000tc__List__Olist_Itc__Nat__Onat_J_000tc__List__Olist_Itc__Nat__Onat,type,
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tff(sy_c_COMBC_000tc__Nat__Onat_000tc__List__Olist_Itc__HOL__Obool_J_000tc__List__Ol,type,
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tff(sy_c_COMBC_000tc__Nat__Onat_000tc__Nat__Onat_000tc__HOL__Obool,type,
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tff(sy_c_COMBC_000tc__Nat__Onat_000tc__Nat__Onat_000tc__Nat__Onat,type,
cOMBC_nat_nat_nat: fun_nat_fun_nat_nat > fun_nat_fun_nat_nat ).
tff(sy_c_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__H,type,
cOMBC_226598744l_bool: fun_na1436237685l_bool > fun_fu821463397t_bool ).
tff(sy_c_COMBI_000tc__Nat__Onat,type,
cOMBI_nat: fun_nat_nat ).
tff(sy_c_COMBK_000tc__fun_Itc__List__Olist_Itc__Arrow____Order____Mirabelle____effng,type,
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tff(sy_c_COMBK_000tc__fun_Itc__List__Olist_Itc__Nat__Onat_J_Mtc__List__Olist_Itc__Na,type,
cOMBK_1717478773at_nat: fun_li1155097699st_nat > fun_na941842582st_nat ).
tff(sy_c_COMBS_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt_000tc__HOL__Ob,type,
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tff(sy_c_COMBS_000tc__HOL__Obool_000tc__HOL__Obool_000tc__HOL__Obool,type,
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tff(sy_c_COMBS_000tc__Nat__Onat_000tc__HOL__Obool_000tc__HOL__Obool,type,
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tff(sy_c_Groups_Ominus__class_Ominus_000tc__Nat__Onat,type,
minus_minus_nat: fun_nat_fun_nat_nat ).
tff(sy_c_Groups_Ominus__class_Ominus_000tc__fun_Itc__Arrow____Order____Mirabelle____,type,
minus_1875147572t_bool: fun_fu2100440475t_bool ).
tff(sy_c_Groups_Ominus__class_Ominus_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J,type,
minus_2067140911t_bool: fun_fu915923926t_bool ).
tff(sy_c_Groups_Oplus__class_Oplus_000tc__Nat__Onat,type,
plus_plus_nat: fun_nat_fun_nat_nat ).
tff(sy_c_Groups_Ozero__class_Ozero_000tc__Nat__Onat,type,
zero_zero_nat: nat ).
tff(sy_c_HOL_Obool_Obool__size,type,
bool_size: fun_bool_nat ).
tff(sy_c_HOL_Oequal__class_Oequal_000tc__List__Olist_Itc__Arrow____Order____Mirabell,type,
equal_631514850le_alt: fun_li495175091t_bool ).
tff(sy_c_HOL_Oequal__class_Oequal_000tc__List__Olist_Itc__Nat__Onat_J,type,
equal_equal_list_nat: fun_li981111539t_bool ).
tff(sy_c_If_000tc__List__Olist_Itc__Arrow____Order____Mirabelle____effngqiqbs__Oalt_,type,
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tff(sy_c_If_000tc__List__Olist_Itc__HOL__Obool_J,type,
if_list_bool: ( bool * list_bool * list_bool ) > list_bool ).
tff(sy_c_If_000tc__List__Olist_Itc__Nat__Onat_J,type,
if_list_nat: ( bool * list_nat * list_nat ) > list_nat ).
tff(sy_c_List_OListMem_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt,type,
listMe1496348587le_alt: arrow_1893660431le_alt > fun_li1109619974t_bool ).
tff(sy_c_List_OListMem_000tc__HOL__Obool,type,
listMem_bool: bool > fun_list_bool_bool ).
tff(sy_c_List_OListMem_000tc__Nat__Onat,type,
listMem_nat: nat > fun_list_nat_bool ).
tff(sy_c_List_Oappend_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt,type,
append1597383892le_alt: fun_li264309018le_alt ).
tff(sy_c_List_Oappend_000tc__HOL__Obool,type,
append_bool: fun_li662746748t_bool ).
tff(sy_c_List_Oappend_000tc__Nat__Onat,type,
append_nat: fun_li511653589st_nat ).
tff(sy_c_List_Obutlast_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt,type,
butlas1693249291le_alt: list_A114658244le_alt > list_A114658244le_alt ).
tff(sy_c_List_Obutlast_000tc__HOL__Obool,type,
butlast_bool: list_bool > list_bool ).
tff(sy_c_List_Obutlast_000tc__Nat__Onat,type,
butlast_nat: list_nat > list_nat ).
tff(sy_c_List_Odistinct_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt,type,
distin1654625714le_alt: fun_li1109619974t_bool ).
tff(sy_c_List_Odistinct_000tc__HOL__Obool,type,
distinct_bool: fun_list_bool_bool ).
tff(sy_c_List_Odistinct_000tc__Nat__Onat,type,
distinct_nat: fun_list_nat_bool ).
tff(sy_c_List_OdropWhile_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt,type,
dropWh587599712le_alt: ( fun_Ar17477381t_bool * list_A114658244le_alt ) > list_A114658244le_alt ).
tff(sy_c_List_OdropWhile_000tc__HOL__Obool,type,
dropWhile_bool: ( fun_bool_bool * list_bool ) > list_bool ).
tff(sy_c_List_OdropWhile_000tc__Nat__Onat,type,
dropWhile_nat: ( fun_nat_bool * list_nat ) > list_nat ).
tff(sy_c_List_Odrop_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt,type,
drop_A617527551le_alt: fun_na960272160le_alt ).
tff(sy_c_List_Odrop_000tc__HOL__Obool,type,
drop_bool: fun_na1279735280t_bool ).
tff(sy_c_List_Odrop_000tc__Nat__Onat,type,
drop_nat: fun_na941842582st_nat ).
tff(sy_c_List_Ofoldl_000tc__List__Olist_Itc__Arrow____Order____Mirabelle____effngqiq,type,
foldl_1844370748le_alt: ( fun_li164935701le_alt * list_A114658244le_alt * list_A114658244le_alt ) > list_A114658244le_alt ).
tff(sy_c_List_Ofoldl_000tc__List__Olist_Itc__HOL__Obool_J_000tc__HOL__Obool,type,
foldl_list_bool_bool: ( fun_li85477609t_bool * list_bool * list_bool ) > list_bool ).
tff(sy_c_List_Ofoldl_000tc__List__Olist_Itc__Nat__Onat_J_000tc__Nat__Onat,type,
foldl_list_nat_nat: ( fun_li1816930438st_nat * list_nat * list_nat ) > list_nat ).
tff(sy_c_List_Ofoldl_000tc__Nat__Onat_000tc__Nat__Onat,type,
foldl_nat_nat: ( fun_nat_fun_nat_nat * nat * list_nat ) > nat ).
tff(sy_c_List_Ofoldr_000tc__Nat__Onat_000tc__Nat__Onat,type,
foldr_nat_nat: ( fun_nat_fun_nat_nat * list_nat ) > fun_nat_nat ).
tff(sy_c_List_Ohd_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt,type,
hd_Arr1236501138le_alt: fun_li1641155132le_alt ).
tff(sy_c_List_Ohd_000tc__HOL__Obool,type,
hd_bool: fun_list_bool_bool ).
tff(sy_c_List_Ohd_000tc__Nat__Onat,type,
hd_nat: fun_list_nat_nat ).
tff(sy_c_List_Oinsert_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt,type,
insert1391384533le_alt: fun_Ar1448189019le_alt ).
tff(sy_c_List_Oinsert_000tc__HOL__Obool,type,
insert_bool: fun_bo1175651857t_bool ).
tff(sy_c_List_Oinsert_000tc__Nat__Onat,type,
insert_nat: fun_na941842582st_nat ).
tff(sy_c_List_Olast_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt,type,
last_A488133080le_alt: fun_li1641155132le_alt ).
tff(sy_c_List_Olast_000tc__HOL__Obool,type,
last_bool: fun_list_bool_bool ).
tff(sy_c_List_Olast_000tc__Nat__Onat,type,
last_nat: fun_list_nat_nat ).
tff(sy_c_List_Olinorder__class_Oinsort__insert__key_000tc__Nat__Onat_000tc__Nat__Ona,type,
linord297828738at_nat: fun_nat_nat > fun_na941842582st_nat ).
tff(sy_c_List_Olinorder__class_Oinsort__key_000tc__Nat__Onat_000tc__Nat__Onat,type,
linord1850739451at_nat: fun_nat_nat > fun_na941842582st_nat ).
tff(sy_c_List_Olinorder__class_Osort__key_000tc__Nat__Onat_000tc__Nat__Onat,type,
linord852754176at_nat: ( fun_nat_nat * list_nat ) > list_nat ).
tff(sy_c_List_Olinorder__class_Osorted_000tc__Nat__Onat,type,
linorder_sorted_nat: fun_list_nat_bool ).
tff(sy_c_List_Olist_OCons_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt,type,
cons_A1647044463le_alt: fun_Ar1448189019le_alt ).
tff(sy_c_List_Olist_OCons_000tc__HOL__Obool,type,
cons_bool: fun_bo1175651857t_bool ).
tff(sy_c_List_Olist_OCons_000tc__List__Olist_Itc__Arrow____Order____Mirabelle____eff,type,
cons_l782000996le_alt: fun_li638184490le_alt ).
tff(sy_c_List_Olist_OCons_000tc__List__Olist_Itc__Nat__Onat_J,type,
cons_list_nat: fun_li1325124399st_nat ).
tff(sy_c_List_Olist_OCons_000tc__Nat__Onat,type,
cons_nat: fun_na941842582st_nat ).
tff(sy_c_List_Olist_ONil_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt,type,
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tff(sy_c_List_Olist_ONil_000tc__HOL__Obool,type,
nil_bool: list_bool ).
tff(sy_c_List_Olist_ONil_000tc__Nat__Onat,type,
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tff(sy_c_List_Olist_Olist__case_000tc__List__Olist_Itc__Arrow____Order____Mirabelle_,type,
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tff(sy_c_List_Olist_Olist__case_000tc__List__Olist_Itc__HOL__Obool_J_000tc__HOL__Obo,type,
list_c1750880501l_bool: ( list_bool * fun_bo1175651857t_bool * list_bool ) > list_bool ).
tff(sy_c_List_Olist_Olist__case_000tc__List__Olist_Itc__Nat__Onat_J_000tc__Nat__Onat,type,
list_c554461815at_nat: ( list_nat * fun_na941842582st_nat * list_nat ) > list_nat ).
tff(sy_c_List_Olist_Olist__size_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oa,type,
list_s1668597313le_alt: ( fun_Ar1091137528lt_nat * list_A114658244le_alt ) > nat ).
tff(sy_c_List_Olist_Olist__size_000tc__HOL__Obool,type,
list_size_bool: ( fun_bool_nat * list_bool ) > nat ).
tff(sy_c_List_Olist_Olist__size_000tc__Nat__Onat,type,
list_size_nat: ( fun_nat_nat * list_nat ) > nat ).
tff(sy_c_List_Olist__all_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt,type,
list_a1988415615le_alt: fun_Ar17477381t_bool > fun_li1109619974t_bool ).
tff(sy_c_List_Olist__all_000tc__HOL__Obool,type,
list_all_bool: fun_bool_bool > fun_list_bool_bool ).
tff(sy_c_List_Olist__all_000tc__Nat__Onat,type,
list_all_nat: fun_nat_bool > fun_list_nat_bool ).
tff(sy_c_List_Olist__ex1_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt,type,
list_e1657546370le_alt: fun_Ar17477381t_bool > fun_li1109619974t_bool ).
tff(sy_c_List_Olist__ex1_000tc__HOL__Obool,type,
list_ex1_bool: fun_bool_bool > fun_list_bool_bool ).
tff(sy_c_List_Olist__ex1_000tc__Nat__Onat,type,
list_ex1_nat: fun_nat_bool > fun_list_nat_bool ).
tff(sy_c_List_Olistrelp_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt,type,
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tff(sy_c_List_Olistrelp_000tc__HOL__Obool,type,
listrelp_bool: ( fun_bo1549164019l_bool * list_bool ) > fun_list_bool_bool ).
tff(sy_c_List_Olistrelp_000tc__Nat__Onat,type,
listrelp_nat: ( fun_nat_fun_nat_bool * list_nat ) > fun_list_nat_bool ).
tff(sy_c_List_Omap_000tc__Nat__Onat_000tc__Nat__Onat,type,
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tff(sy_c_List_Omap__filter_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt_00,type,
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tff(sy_c_List_Omap__filter_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt_00_026,type,
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tff(sy_c_List_Omap__filter_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt_00_027,type,
map_fi1320516869lt_nat: ( fun_Ar63580199on_nat * list_A114658244le_alt ) > list_nat ).
tff(sy_c_List_Omap__filter_000tc__HOL__Obool_000tc__Arrow____Order____Mirabelle____e,type,
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tff(sy_c_List_Omap__filter_000tc__HOL__Obool_000tc__Nat__Onat,type,
map_filter_bool_nat: ( fun_bool_option_nat * list_bool ) > list_nat ).
tff(sy_c_List_Omap__filter_000tc__Nat__Onat_000tc__Arrow____Order____Mirabelle____ef,type,
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tff(sy_c_List_Omap__filter_000tc__Nat__Onat_000tc__HOL__Obool,type,
map_filter_nat_bool: ( fun_nat_option_bool * list_nat ) > list_bool ).
tff(sy_c_List_Omap__filter_000tc__Nat__Onat_000tc__Nat__Onat,type,
map_filter_nat_nat: ( fun_nat_option_nat * list_nat ) > list_nat ).
tff(sy_c_List_Omaps_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt_000tc__Ar,type,
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tff(sy_c_List_Omaps_000tc__HOL__Obool_000tc__Nat__Onat,type,
maps_bool_nat: ( fun_bool_list_nat * list_bool ) > list_nat ).
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maps_n398774214le_alt: ( fun_na86871399le_alt * list_nat ) > list_A114658244le_alt ).
tff(sy_c_List_Omember_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt,type,
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tff(sy_c_List_Omember_000tc__HOL__Obool,type,
member_bool_1: fun_li229471966l_bool ).
tff(sy_c_List_Omember_000tc__Nat__Onat,type,
member_nat_1: fun_li123523364t_bool ).
tff(sy_c_List_Onth_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt,type,
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tff(sy_c_List_Onth_000tc__HOL__Obool,type,
nth_bool: list_bool > fun_nat_bool ).
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tff(sy_c_List_Onth_000tc__List__Olist_Itc__Nat__Onat_J,type,
nth_list_nat: list_list_nat > fun_nat_list_nat ).
tff(sy_c_List_Onth_000tc__Nat__Onat,type,
nth_nat: list_nat > fun_nat_nat ).
tff(sy_c_List_Onull_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt,type,
null_A791782855le_alt: fun_li1109619974t_bool ).
tff(sy_c_List_Onull_000tc__HOL__Obool,type,
null_bool: fun_list_bool_bool ).
tff(sy_c_List_Onull_000tc__Nat__Onat,type,
null_nat: fun_list_nat_bool ).
tff(sy_c_List_Oremove1_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt,type,
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tff(sy_c_List_Oremove1_000tc__HOL__Obool,type,
remove1_bool: fun_bo1175651857t_bool ).
tff(sy_c_List_Oremove1_000tc__Nat__Onat,type,
remove1_nat: fun_na941842582st_nat ).
tff(sy_c_List_Oreplicate_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt,type,
replic782356601le_alt: nat > fun_Ar849108972le_alt ).
tff(sy_c_List_Oreplicate_000tc__HOL__Obool,type,
replicate_bool: nat > fun_bool_list_bool ).
tff(sy_c_List_Oreplicate_000tc__Nat__Onat,type,
replicate_nat: nat > fun_nat_list_nat ).
tff(sy_c_List_Orev_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt,type,
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tff(sy_c_List_Orev_000tc__HOL__Obool,type,
rev_bool: list_bool > list_bool ).
tff(sy_c_List_Orev_000tc__Nat__Onat,type,
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tff(sy_c_List_Orotate1_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt,type,
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tff(sy_c_List_Orotate1_000tc__HOL__Obool,type,
rotate1_bool: list_bool > list_bool ).
tff(sy_c_List_Orotate1_000tc__Nat__Onat,type,
rotate1_nat: list_nat > list_nat ).
tff(sy_c_List_Oset_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt,type,
set_Ar1995755744le_alt: fun_li297692846t_bool ).
tff(sy_c_List_Oset_000tc__HOL__Obool,type,
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tff(sy_c_List_Oset_000tc__Nat__Onat,type,
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tff(sy_c_List_Osplice_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt,type,
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tff(sy_c_List_Osplice_000tc__HOL__Obool,type,
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tff(sy_c_List_Osplice_000tc__Nat__Onat,type,
splice_nat: ( list_nat * list_nat ) > list_nat ).
tff(sy_c_List_Osublist_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt,type,
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tff(sy_c_List_Osublist_000tc__HOL__Obool,type,
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tff(sy_c_List_Osublist_000tc__Nat__Onat,type,
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tff(sy_c_List_OtakeWhile_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt,type,
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tff(sy_c_List_OtakeWhile_000tc__HOL__Obool,type,
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tff(sy_c_List_OtakeWhile_000tc__Nat__Onat,type,
takeWhile_nat: ( fun_nat_bool * list_nat ) > list_nat ).
tff(sy_c_List_Otake_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt,type,
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tff(sy_c_List_Otl_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt,type,
tl_Arr1883752598le_alt: list_A114658244le_alt > list_A114658244le_alt ).
tff(sy_c_List_Otl_000tc__HOL__Obool,type,
tl_bool: list_bool > list_bool ).
tff(sy_c_List_Otl_000tc__Nat__Onat,type,
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tff(sy_c_List_Oupt,type,
upt: nat > fun_nat_list_nat ).
tff(sy_c_Nat_OSuc,type,
suc: fun_nat_nat ).
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tff(sy_c_Nat_Onat_Onat__case_000tc__HOL__Obool,type,
nat_case_bool: ( bool * fun_nat_bool ) > fun_nat_bool ).
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tff(sy_c_Nat_Onat_Onat__case_000tc__List__Olist_Itc__HOL__Obool_J,type,
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tff(sy_c_Nat_Onat_Onat__case_000tc__List__Olist_Itc__Nat__Onat_J,type,
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hAPP_f1730770594t_bool: ( fun_fu915923926t_bool * fun_nat_bool ) > fun_fu821463397t_bool ).
tff(sy_c_hBOOL,type,
hBOOL: bool > $o ).
tff(sy_c_member_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt,type,
member1502664802le_alt: fun_Ar569529141l_bool ).
tff(sy_c_member_000tc__HOL__Obool,type,
member_bool: fun_bo492988759l_bool ).
tff(sy_c_member_000tc__Nat__Onat,type,
member_nat: fun_na1436237685l_bool ).
tff(sy_v_a,type,
a: arrow_1893660431le_alt ).
tff(sy_v_b,type,
b: arrow_1893660431le_alt ).
%----Relevant facts (1200)
tff(fact_0_alt3,axiom,
? [A_74: arrow_1893660431le_alt,B_1: arrow_1893660431le_alt,C: arrow_1893660431le_alt] : hBOOL(hAPP_l499746980t_bool(distin1654625714le_alt,hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,A_74),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,B_1),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,C),nil_Ar557011903le_alt))))) ).
tff(fact_1_distinct_Osimps_I1_J,axiom,
hBOOL(hAPP_l499746980t_bool(distin1654625714le_alt,nil_Ar557011903le_alt)) ).
tff(fact_2_distinct_Osimps_I1_J,axiom,
hBOOL(hAPP_list_nat_bool(distinct_nat,nil_nat)) ).
tff(fact_3_list_Osimps_I2_J,axiom,
! [A_84: bool,List_2: list_bool] : nil_bool != hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,A_84),List_2) ).
tff(fact_4_list_Osimps_I2_J,axiom,
! [A_85: arrow_1893660431le_alt,List_4: list_A114658244le_alt] : nil_Ar557011903le_alt != hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,A_85),List_4) ).
tff(fact_5_list_Osimps_I2_J,axiom,
! [A_85: nat,List_4: list_nat] : nil_nat != hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,A_85),List_4) ).
tff(fact_6_list_Osimps_I3_J,axiom,
! [A_84: bool,List_2: list_bool] : hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,A_84),List_2) != nil_bool ).
tff(fact_7_list_Osimps_I3_J,axiom,
! [A_84: arrow_1893660431le_alt,List_2: list_A114658244le_alt] : hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,A_84),List_2) != nil_Ar557011903le_alt ).
tff(fact_8_list_Osimps_I3_J,axiom,
! [A_84: nat,List_2: list_nat] : hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,A_84),List_2) != nil_nat ).
tff(fact_9_neq__Nil__conv,axiom,
! [Xs: list_bool] :
( ( Xs != nil_bool )
<=> ? [Y_25: bool,Ys_6: list_bool] : Xs = hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,Y_25),Ys_6) ) ).
tff(fact_10_neq__Nil__conv,axiom,
! [Xs: list_A114658244le_alt] :
( ( Xs != nil_Ar557011903le_alt )
<=> ? [Y_25: arrow_1893660431le_alt,Ys_6: list_A114658244le_alt] : Xs = hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,Y_25),Ys_6) ) ).
tff(fact_11_neq__Nil__conv,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
<=> ? [Y_25: nat,Ys_6: list_nat] : Xs = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,Y_25),Ys_6) ) ).
tff(fact_12_list_Oexhaust,axiom,
! [Y_2: list_bool] :
( ( Y_2 != nil_bool )
=> ~ ! [A_74: bool,List_3: list_bool] : Y_2 != hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,A_74),List_3) ) ).
tff(fact_13_list_Oexhaust,axiom,
! [Y_68: list_A114658244le_alt] :
( ( Y_68 != nil_Ar557011903le_alt )
=> ~ ! [A_74: arrow_1893660431le_alt,List_3: list_A114658244le_alt] : Y_68 != hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,A_74),List_3) ) ).
tff(fact_14_list_Oexhaust,axiom,
! [Y_68: list_nat] :
( ( Y_68 != nil_nat )
=> ~ ! [A_74: nat,List_3: list_nat] : Y_68 != hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,A_74),List_3) ) ).
tff(fact_15_not__Cons__self,axiom,
! [Xs: list_bool,X_1: bool] : Xs != hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs) ).
tff(fact_16_not__Cons__self,axiom,
! [Xs_97: list_A114658244le_alt,X_126: arrow_1893660431le_alt] : Xs_97 != hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_126),Xs_97) ).
tff(fact_17_not__Cons__self,axiom,
! [Xs_97: list_nat,X_126: nat] : Xs_97 != hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_126),Xs_97) ).
tff(fact_18_not__Cons__self2,axiom,
! [X_1: bool,Xs: list_bool] : hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs) != Xs ).
tff(fact_19_not__Cons__self2,axiom,
! [X_125: arrow_1893660431le_alt,Xs_96: list_A114658244le_alt] : hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_125),Xs_96) != Xs_96 ).
tff(fact_20_not__Cons__self2,axiom,
! [X_125: nat,Xs_96: list_nat] : hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_125),Xs_96) != Xs_96 ).
tff(fact_21_list_Oinject,axiom,
! [Aa: bool,List_1: list_bool,A_84: bool,List_2: list_bool] :
( ( hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,Aa),List_1) = hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,A_84),List_2) )
<=> ( ( hBOOL(Aa)
<=> hBOOL(A_84) )
& ( List_1 = List_2 ) ) ) ).
tff(fact_22_list_Oinject,axiom,
! [Aa: arrow_1893660431le_alt,List_1: list_A114658244le_alt,A_84: arrow_1893660431le_alt,List_2: list_A114658244le_alt] :
( ( hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,Aa),List_1) = hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,A_84),List_2) )
<=> ( ( Aa = A_84 )
& ( List_1 = List_2 ) ) ) ).
tff(fact_23_list_Oinject,axiom,
! [Aa: nat,List_1: list_nat,A_84: nat,List_2: list_nat] :
( ( hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,Aa),List_1) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,A_84),List_2) )
<=> ( ( Aa = A_84 )
& ( List_1 = List_2 ) ) ) ).
tff(fact_24_splice_Osimps_I2_J,axiom,
! [V_1: bool,Va_1: list_bool] : splice_bool(hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,V_1),Va_1),nil_bool) = hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,V_1),Va_1) ).
tff(fact_25_splice_Osimps_I2_J,axiom,
! [V: arrow_1893660431le_alt,Va: list_A114658244le_alt] : splice791716242le_alt(hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,V),Va),nil_Ar557011903le_alt) = hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,V),Va) ).
tff(fact_26_splice_Osimps_I2_J,axiom,
! [V: nat,Va: list_nat] : splice_nat(hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,V),Va),nil_nat) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,V),Va) ).
tff(fact_27_insert__Nil,axiom,
! [X_1: bool] : hAPP_l1189022293t_bool(hAPP_b994696797t_bool(insert_bool,X_1),nil_bool) = hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),nil_bool) ).
tff(fact_28_insert__Nil,axiom,
! [X_124: arrow_1893660431le_alt] : hAPP_l84111627le_alt(hAPP_A436723315le_alt(insert1391384533le_alt,X_124),nil_Ar557011903le_alt) = hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_124),nil_Ar557011903le_alt) ).
tff(fact_29_insert__Nil,axiom,
! [X_124: nat] : hAPP_l248265089st_nat(hAPP_n280362926st_nat(insert_nat,X_124),nil_nat) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_124),nil_nat) ).
tff(fact_30_list__nonempty__induct,axiom,
! [P_1: fun_list_bool_bool,Xs: list_bool] :
( ( Xs != nil_bool )
=> ( ! [X_4: bool] : hBOOL(hAPP_list_bool_bool(P_1,hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_4),nil_bool)))
=> ( ! [X_4: bool,Xs_19: list_bool] :
( ( Xs_19 != nil_bool )
=> ( hBOOL(hAPP_list_bool_bool(P_1,Xs_19))
=> hBOOL(hAPP_list_bool_bool(P_1,hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_4),Xs_19))) ) )
=> hBOOL(hAPP_list_bool_bool(P_1,Xs)) ) ) ) ).
tff(fact_31_list__nonempty__induct,axiom,
! [P_1: fun_li1109619974t_bool,Xs: list_A114658244le_alt] :
( ( Xs != nil_Ar557011903le_alt )
=> ( ! [X_4: arrow_1893660431le_alt] : hBOOL(hAPP_l499746980t_bool(P_1,hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_4),nil_Ar557011903le_alt)))
=> ( ! [X_4: arrow_1893660431le_alt,Xs_19: list_A114658244le_alt] :
( ( Xs_19 != nil_Ar557011903le_alt )
=> ( hBOOL(hAPP_l499746980t_bool(P_1,Xs_19))
=> hBOOL(hAPP_l499746980t_bool(P_1,hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_4),Xs_19))) ) )
=> hBOOL(hAPP_l499746980t_bool(P_1,Xs)) ) ) ) ).
tff(fact_32_list__nonempty__induct,axiom,
! [P_1: fun_list_nat_bool,Xs: list_nat] :
( ( Xs != nil_nat )
=> ( ! [X_4: nat] : hBOOL(hAPP_list_nat_bool(P_1,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_4),nil_nat)))
=> ( ! [X_4: nat,Xs_19: list_nat] :
( ( Xs_19 != nil_nat )
=> ( hBOOL(hAPP_list_nat_bool(P_1,Xs_19))
=> hBOOL(hAPP_list_nat_bool(P_1,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_4),Xs_19))) ) )
=> hBOOL(hAPP_list_nat_bool(P_1,Xs)) ) ) ) ).
tff(fact_33_distinct__butlast,axiom,
! [Xs_95: list_A114658244le_alt] :
( ( Xs_95 != nil_Ar557011903le_alt )
=> ( hBOOL(hAPP_l499746980t_bool(distin1654625714le_alt,Xs_95))
=> hBOOL(hAPP_l499746980t_bool(distin1654625714le_alt,butlas1693249291le_alt(Xs_95))) ) ) ).
tff(fact_34_distinct__butlast,axiom,
! [Xs_95: list_nat] :
( ( Xs_95 != nil_nat )
=> ( hBOOL(hAPP_list_nat_bool(distinct_nat,Xs_95))
=> hBOOL(hAPP_list_nat_bool(distinct_nat,butlast_nat(Xs_95))) ) ) ).
tff(fact_35_butlast_Osimps_I1_J,axiom,
butlas1693249291le_alt(nil_Ar557011903le_alt) = nil_Ar557011903le_alt ).
tff(fact_36_butlast_Osimps_I1_J,axiom,
butlast_nat(nil_nat) = nil_nat ).
tff(fact_37_distinct__insert,axiom,
! [X_123: arrow_1893660431le_alt,Xs_94: list_A114658244le_alt] :
( hBOOL(hAPP_l499746980t_bool(distin1654625714le_alt,Xs_94))
=> hBOOL(hAPP_l499746980t_bool(distin1654625714le_alt,hAPP_l84111627le_alt(hAPP_A436723315le_alt(insert1391384533le_alt,X_123),Xs_94))) ) ).
tff(fact_38_distinct__insert,axiom,
! [X_123: nat,Xs_94: list_nat] :
( hBOOL(hAPP_list_nat_bool(distinct_nat,Xs_94))
=> hBOOL(hAPP_list_nat_bool(distinct_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(insert_nat,X_123),Xs_94))) ) ).
tff(fact_39_splice_Osimps_I3_J,axiom,
! [X_1: bool,Xs: list_bool,Y_2: bool,Ys: list_bool] : splice_bool(hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,Y_2),Ys)) = hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,Y_2),splice_bool(Xs,Ys))) ).
tff(fact_40_splice_Osimps_I3_J,axiom,
! [X_122: arrow_1893660431le_alt,Xs_93: list_A114658244le_alt,Y_67: arrow_1893660431le_alt,Ys_31: list_A114658244le_alt] : splice791716242le_alt(hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_122),Xs_93),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,Y_67),Ys_31)) = hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_122),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,Y_67),splice791716242le_alt(Xs_93,Ys_31))) ).
tff(fact_41_splice_Osimps_I3_J,axiom,
! [X_122: nat,Xs_93: list_nat,Y_67: nat,Ys_31: list_nat] : splice_nat(hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_122),Xs_93),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,Y_67),Ys_31)) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_122),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,Y_67),splice_nat(Xs_93,Ys_31))) ).
tff(fact_42_splice__Nil2,axiom,
! [Xs_92: list_A114658244le_alt] : splice791716242le_alt(Xs_92,nil_Ar557011903le_alt) = Xs_92 ).
tff(fact_43_splice__Nil2,axiom,
! [Xs_92: list_nat] : splice_nat(Xs_92,nil_nat) = Xs_92 ).
tff(fact_44_splice_Osimps_I1_J,axiom,
! [Ys_30: list_A114658244le_alt] : splice791716242le_alt(nil_Ar557011903le_alt,Ys_30) = Ys_30 ).
tff(fact_45_splice_Osimps_I1_J,axiom,
! [Ys_30: list_nat] : splice_nat(nil_nat,Ys_30) = Ys_30 ).
tff(fact_46_butlast_Osimps_I2_J,axiom,
! [X_1: bool,Xs: list_bool] :
( ( ( Xs = nil_bool )
=> ( butlast_bool(hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs)) = nil_bool ) )
& ( ( Xs != nil_bool )
=> ( butlast_bool(hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs)) = hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),butlast_bool(Xs)) ) ) ) ).
tff(fact_47_butlast_Osimps_I2_J,axiom,
! [X_121: arrow_1893660431le_alt,Xs_91: list_A114658244le_alt] :
( ( ( Xs_91 = nil_Ar557011903le_alt )
=> ( butlas1693249291le_alt(hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_121),Xs_91)) = nil_Ar557011903le_alt ) )
& ( ( Xs_91 != nil_Ar557011903le_alt )
=> ( butlas1693249291le_alt(hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_121),Xs_91)) = hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_121),butlas1693249291le_alt(Xs_91)) ) ) ) ).
tff(fact_48_butlast_Osimps_I2_J,axiom,
! [X_121: nat,Xs_91: list_nat] :
( ( ( Xs_91 = nil_nat )
=> ( butlast_nat(hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_121),Xs_91)) = nil_nat ) )
& ( ( Xs_91 != nil_nat )
=> ( butlast_nat(hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_121),Xs_91)) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_121),butlast_nat(Xs_91)) ) ) ) ).
tff(fact_49_list_Osimps_I4_J,axiom,
! [F1: list_A114658244le_alt,F2: fun_Ar1448189019le_alt] : list_c1024366775le_alt(F1,F2,nil_Ar557011903le_alt) = F1 ).
tff(fact_50_list_Osimps_I4_J,axiom,
! [F1: list_nat,F2: fun_na941842582st_nat] : list_c554461815at_nat(F1,F2,nil_nat) = F1 ).
tff(fact_51_list_Osimps_I5_J,axiom,
! [F1: list_A114658244le_alt,F2: fun_Ar1448189019le_alt,Aa: arrow_1893660431le_alt,List_1: list_A114658244le_alt] : list_c1024366775le_alt(F1,F2,hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,Aa),List_1)) = hAPP_l84111627le_alt(hAPP_A436723315le_alt(F2,Aa),List_1) ).
tff(fact_52_list_Osimps_I5_J,axiom,
! [F1: list_nat,F2: fun_na941842582st_nat,Aa: nat,List_1: list_nat] : list_c554461815at_nat(F1,F2,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,Aa),List_1)) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(F2,Aa),List_1) ).
tff(fact_53_eq__Nil__null,axiom,
! [Xs: list_A114658244le_alt] :
( ( Xs = nil_Ar557011903le_alt )
<=> hBOOL(hAPP_l499746980t_bool(null_A791782855le_alt,Xs)) ) ).
tff(fact_54_eq__Nil__null,axiom,
! [Xs: list_nat] :
( ( Xs = nil_nat )
<=> hBOOL(hAPP_list_nat_bool(null_nat,Xs)) ) ).
tff(fact_55_List_Onull__def,axiom,
! [Xs: list_A114658244le_alt] :
( hBOOL(hAPP_l499746980t_bool(null_A791782855le_alt,Xs))
<=> ( Xs = nil_Ar557011903le_alt ) ) ).
tff(fact_56_List_Onull__def,axiom,
! [Xs: list_nat] :
( hBOOL(hAPP_list_nat_bool(null_nat,Xs))
<=> ( Xs = nil_nat ) ) ).
tff(fact_57_null__rec_I2_J,axiom,
hBOOL(hAPP_l499746980t_bool(null_A791782855le_alt,nil_Ar557011903le_alt)) ).
tff(fact_58_null__rec_I2_J,axiom,
hBOOL(hAPP_list_nat_bool(null_nat,nil_nat)) ).
tff(fact_59_null__rec_I1_J,axiom,
! [X_1: bool,Xs: list_bool] : ~ hBOOL(hAPP_list_bool_bool(null_bool,hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs))) ).
tff(fact_60_null__rec_I1_J,axiom,
! [X_120: arrow_1893660431le_alt,Xs_90: list_A114658244le_alt] : ~ hBOOL(hAPP_l499746980t_bool(null_A791782855le_alt,hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_120),Xs_90))) ).
tff(fact_61_null__rec_I1_J,axiom,
! [X_120: nat,Xs_90: list_nat] : ~ hBOOL(hAPP_list_nat_bool(null_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_120),Xs_90))) ).
tff(fact_62_butlast__snoc,axiom,
! [Xs: list_bool,X_1: bool] : butlast_bool(hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,Xs),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),nil_bool))) = Xs ).
tff(fact_63_butlast__snoc,axiom,
! [Xs_89: list_A114658244le_alt,X_119: arrow_1893660431le_alt] : butlas1693249291le_alt(hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs_89),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_119),nil_Ar557011903le_alt))) = Xs_89 ).
tff(fact_64_butlast__snoc,axiom,
! [Xs_89: list_nat,X_119: nat] : butlast_nat(hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs_89),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_119),nil_nat))) = Xs_89 ).
tff(fact_65_sorted__single,axiom,
! [X_118: nat] : hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_118),nil_nat))) ).
tff(fact_66_last__ConsL,axiom,
! [X_1: bool,Xs: list_bool] :
( ( Xs = nil_bool )
=> ( hBOOL(hAPP_list_bool_bool(last_bool,hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs)))
<=> hBOOL(X_1) ) ) ).
tff(fact_67_last__ConsL,axiom,
! [X_117: arrow_1893660431le_alt,Xs_88: list_A114658244le_alt] :
( ( Xs_88 = nil_Ar557011903le_alt )
=> ( hAPP_l1749541000le_alt(last_A488133080le_alt,hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_117),Xs_88)) = X_117 ) ) ).
tff(fact_68_last__ConsL,axiom,
! [X_117: nat,Xs_88: list_nat] :
( ( Xs_88 = nil_nat )
=> ( hAPP_list_nat_nat(last_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_117),Xs_88)) = X_117 ) ) ).
tff(fact_69_sorted_ONil,axiom,
hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,nil_nat)) ).
tff(fact_70_append__assoc,axiom,
! [Xs_87: list_A114658244le_alt,Ys_29: list_A114658244le_alt,Zs_6: list_A114658244le_alt] : hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs_87),Ys_29)),Zs_6) = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs_87),hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Ys_29),Zs_6)) ).
tff(fact_71_append__assoc,axiom,
! [Xs_87: list_nat,Ys_29: list_nat,Zs_6: list_nat] : hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs_87),Ys_29)),Zs_6) = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs_87),hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Ys_29),Zs_6)) ).
tff(fact_72_append__eq__append__conv2,axiom,
! [Xs: list_A114658244le_alt,Ys: list_A114658244le_alt,Zs: list_A114658244le_alt,Ts: list_A114658244le_alt] :
( ( hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs),Ys) = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Zs),Ts) )
<=> ? [Us_1: list_A114658244le_alt] :
( ( ( Xs = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Zs),Us_1) )
& ( hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Us_1),Ys) = Ts ) )
| ( ( hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs),Us_1) = Zs )
& ( Ys = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Us_1),Ts) ) ) ) ) ).
tff(fact_73_append__eq__append__conv2,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ts: list_nat] :
( ( hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs),Ys) = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Zs),Ts) )
<=> ? [Us_1: list_nat] :
( ( ( Xs = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Zs),Us_1) )
& ( hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Us_1),Ys) = Ts ) )
| ( ( hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs),Us_1) = Zs )
& ( Ys = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Us_1),Ts) ) ) ) ) ).
tff(fact_74_same__append__eq,axiom,
! [Xs: list_A114658244le_alt,Ys: list_A114658244le_alt,Zs: list_A114658244le_alt] :
( ( hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs),Ys) = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs),Zs) )
<=> ( Ys = Zs ) ) ).
tff(fact_75_same__append__eq,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs),Ys) = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs),Zs) )
<=> ( Ys = Zs ) ) ).
tff(fact_76_append__same__eq,axiom,
! [Ys: list_A114658244le_alt,Xs: list_A114658244le_alt,Zs: list_A114658244le_alt] :
( ( hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Ys),Xs) = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Zs),Xs) )
<=> ( Ys = Zs ) ) ).
tff(fact_77_append__same__eq,axiom,
! [Ys: list_nat,Xs: list_nat,Zs: list_nat] :
( ( hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Ys),Xs) = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Zs),Xs) )
<=> ( Ys = Zs ) ) ).
tff(fact_78_append__eq__appendI,axiom,
! [Ys_28: list_A114658244le_alt,Us: list_A114658244le_alt,Xs_86: list_A114658244le_alt,Xs1_2: list_A114658244le_alt,Zs_5: list_A114658244le_alt] :
( ( hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs_86),Xs1_2) = Zs_5 )
=> ( ( Ys_28 = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs1_2),Us) )
=> ( hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs_86),Ys_28) = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Zs_5),Us) ) ) ) ).
tff(fact_79_append__eq__appendI,axiom,
! [Ys_28: list_nat,Us: list_nat,Xs_86: list_nat,Xs1_2: list_nat,Zs_5: list_nat] :
( ( hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs_86),Xs1_2) = Zs_5 )
=> ( ( Ys_28 = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs1_2),Us) )
=> ( hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs_86),Ys_28) = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Zs_5),Us) ) ) ) ).
tff(fact_80_last__appendL,axiom,
! [Xs_85: list_A114658244le_alt,Ys_27: list_A114658244le_alt] :
( ( Ys_27 = nil_Ar557011903le_alt )
=> ( hAPP_l1749541000le_alt(last_A488133080le_alt,hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs_85),Ys_27)) = hAPP_l1749541000le_alt(last_A488133080le_alt,Xs_85) ) ) ).
tff(fact_81_last__appendL,axiom,
! [Xs_85: list_nat,Ys_27: list_nat] :
( ( Ys_27 = nil_nat )
=> ( hAPP_list_nat_nat(last_nat,hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs_85),Ys_27)) = hAPP_list_nat_nat(last_nat,Xs_85) ) ) ).
tff(fact_82_last__appendR,axiom,
! [Xs_84: list_A114658244le_alt,Ys_26: list_A114658244le_alt] :
( ( Ys_26 != nil_Ar557011903le_alt )
=> ( hAPP_l1749541000le_alt(last_A488133080le_alt,hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs_84),Ys_26)) = hAPP_l1749541000le_alt(last_A488133080le_alt,Ys_26) ) ) ).
tff(fact_83_last__appendR,axiom,
! [Xs_84: list_nat,Ys_26: list_nat] :
( ( Ys_26 != nil_nat )
=> ( hAPP_list_nat_nat(last_nat,hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs_84),Ys_26)) = hAPP_list_nat_nat(last_nat,Ys_26) ) ) ).
tff(fact_84_last__append,axiom,
! [Xs_83: list_A114658244le_alt,Ys_25: list_A114658244le_alt] :
( ( ( Ys_25 = nil_Ar557011903le_alt )
=> ( hAPP_l1749541000le_alt(last_A488133080le_alt,hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs_83),Ys_25)) = hAPP_l1749541000le_alt(last_A488133080le_alt,Xs_83) ) )
& ( ( Ys_25 != nil_Ar557011903le_alt )
=> ( hAPP_l1749541000le_alt(last_A488133080le_alt,hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs_83),Ys_25)) = hAPP_l1749541000le_alt(last_A488133080le_alt,Ys_25) ) ) ) ).
tff(fact_85_last__append,axiom,
! [Xs_83: list_nat,Ys_25: list_nat] :
( ( ( Ys_25 = nil_nat )
=> ( hAPP_list_nat_nat(last_nat,hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs_83),Ys_25)) = hAPP_list_nat_nat(last_nat,Xs_83) ) )
& ( ( Ys_25 != nil_nat )
=> ( hAPP_list_nat_nat(last_nat,hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs_83),Ys_25)) = hAPP_list_nat_nat(last_nat,Ys_25) ) ) ) ).
tff(fact_86_maps__simps_I1_J,axiom,
! [F: fun_Ar849108972le_alt,X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt] : maps_A652163713le_alt(F,hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),Xs)) = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,hAPP_A1382181054le_alt(F,X_1)),maps_A652163713le_alt(F,Xs)) ).
tff(fact_87_maps__simps_I1_J,axiom,
! [F: fun_na86871399le_alt,X_1: nat,Xs: list_nat] : maps_n398774214le_alt(F,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Xs)) = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,hAPP_n2007559481le_alt(F,X_1)),maps_n398774214le_alt(F,Xs)) ).
tff(fact_88_maps__simps_I1_J,axiom,
! [F: fun_bool_list_nat,X_1: bool,Xs: list_bool] : maps_bool_nat(F,hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs)) = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,hAPP_bool_list_nat(F,X_1)),maps_bool_nat(F,Xs)) ).
tff(fact_89_last__snoc,axiom,
! [Xs: list_bool,X_1: bool] :
( hBOOL(hAPP_list_bool_bool(last_bool,hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,Xs),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),nil_bool))))
<=> hBOOL(X_1) ) ).
tff(fact_90_last__snoc,axiom,
! [Xs_82: list_A114658244le_alt,X_116: arrow_1893660431le_alt] : hAPP_l1749541000le_alt(last_A488133080le_alt,hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs_82),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_116),nil_Ar557011903le_alt))) = X_116 ).
tff(fact_91_last__snoc,axiom,
! [Xs_82: list_nat,X_116: nat] : hAPP_list_nat_nat(last_nat,hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs_82),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_116),nil_nat))) = X_116 ).
tff(fact_92_append__Cons,axiom,
! [X_1: bool,Xs: list_bool,Ys: list_bool] : hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs)),Ys) = hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,Xs),Ys)) ).
tff(fact_93_append__Cons,axiom,
! [X_115: arrow_1893660431le_alt,Xs_81: list_A114658244le_alt,Ys_24: list_A114658244le_alt] : hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_115),Xs_81)),Ys_24) = hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_115),hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs_81),Ys_24)) ).
tff(fact_94_append__Cons,axiom,
! [X_115: nat,Xs_81: list_nat,Ys_24: list_nat] : hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_115),Xs_81)),Ys_24) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_115),hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs_81),Ys_24)) ).
tff(fact_95_Cons__eq__appendI,axiom,
! [Xs: list_bool,Zs: list_bool,X_1: bool,Xs1_1: list_bool,Ys: list_bool] :
( ( hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs1_1) = Ys )
=> ( ( Xs = hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,Xs1_1),Zs) )
=> ( hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs) = hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,Ys),Zs) ) ) ) ).
tff(fact_96_Cons__eq__appendI,axiom,
! [Xs_80: list_A114658244le_alt,Zs_4: list_A114658244le_alt,X_114: arrow_1893660431le_alt,Xs1: list_A114658244le_alt,Ys_23: list_A114658244le_alt] :
( ( hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_114),Xs1) = Ys_23 )
=> ( ( Xs_80 = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs1),Zs_4) )
=> ( hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_114),Xs_80) = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Ys_23),Zs_4) ) ) ) ).
tff(fact_97_Cons__eq__appendI,axiom,
! [Xs_80: list_nat,Zs_4: list_nat,X_114: nat,Xs1: list_nat,Ys_23: list_nat] :
( ( hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_114),Xs1) = Ys_23 )
=> ( ( Xs_80 = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs1),Zs_4) )
=> ( hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_114),Xs_80) = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Ys_23),Zs_4) ) ) ) ).
tff(fact_98_append__Nil,axiom,
! [Ys_22: list_A114658244le_alt] : hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,nil_Ar557011903le_alt),Ys_22) = Ys_22 ).
tff(fact_99_append__Nil,axiom,
! [Ys_22: list_nat] : hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,nil_nat),Ys_22) = Ys_22 ).
tff(fact_100_Nil__is__append__conv,axiom,
! [Xs: list_A114658244le_alt,Ys: list_A114658244le_alt] :
( ( nil_Ar557011903le_alt = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs),Ys) )
<=> ( ( Xs = nil_Ar557011903le_alt )
& ( Ys = nil_Ar557011903le_alt ) ) ) ).
tff(fact_101_Nil__is__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( nil_nat = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs),Ys) )
<=> ( ( Xs = nil_nat )
& ( Ys = nil_nat ) ) ) ).
tff(fact_102_append__Nil2,axiom,
! [Xs_79: list_A114658244le_alt] : hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs_79),nil_Ar557011903le_alt) = Xs_79 ).
tff(fact_103_append__Nil2,axiom,
! [Xs_79: list_nat] : hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs_79),nil_nat) = Xs_79 ).
tff(fact_104_self__append__conv,axiom,
! [Xs: list_A114658244le_alt,Ys: list_A114658244le_alt] :
( ( Xs = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs),Ys) )
<=> ( Ys = nil_Ar557011903le_alt ) ) ).
tff(fact_105_self__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs),Ys) )
<=> ( Ys = nil_nat ) ) ).
tff(fact_106_self__append__conv2,axiom,
! [Ys: list_A114658244le_alt,Xs: list_A114658244le_alt] :
( ( Ys = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs),Ys) )
<=> ( Xs = nil_Ar557011903le_alt ) ) ).
tff(fact_107_self__append__conv2,axiom,
! [Ys: list_nat,Xs: list_nat] :
( ( Ys = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs),Ys) )
<=> ( Xs = nil_nat ) ) ).
tff(fact_108_append__is__Nil__conv,axiom,
! [Xs: list_A114658244le_alt,Ys: list_A114658244le_alt] :
( ( hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs),Ys) = nil_Ar557011903le_alt )
<=> ( ( Xs = nil_Ar557011903le_alt )
& ( Ys = nil_Ar557011903le_alt ) ) ) ).
tff(fact_109_append__is__Nil__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs),Ys) = nil_nat )
<=> ( ( Xs = nil_nat )
& ( Ys = nil_nat ) ) ) ).
tff(fact_110_append__self__conv,axiom,
! [Xs: list_A114658244le_alt,Ys: list_A114658244le_alt] :
( ( hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs),Ys) = Xs )
<=> ( Ys = nil_Ar557011903le_alt ) ) ).
tff(fact_111_append__self__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs),Ys) = Xs )
<=> ( Ys = nil_nat ) ) ).
tff(fact_112_append__self__conv2,axiom,
! [Xs: list_A114658244le_alt,Ys: list_A114658244le_alt] :
( ( hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs),Ys) = Ys )
<=> ( Xs = nil_Ar557011903le_alt ) ) ).
tff(fact_113_append__self__conv2,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs),Ys) = Ys )
<=> ( Xs = nil_nat ) ) ).
tff(fact_114_eq__Nil__appendI,axiom,
! [Xs_78: list_A114658244le_alt,Ys_21: list_A114658244le_alt] :
( ( Xs_78 = Ys_21 )
=> ( Xs_78 = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,nil_Ar557011903le_alt),Ys_21) ) ) ).
tff(fact_115_eq__Nil__appendI,axiom,
! [Xs_78: list_nat,Ys_21: list_nat] :
( ( Xs_78 = Ys_21 )
=> ( Xs_78 = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,nil_nat),Ys_21) ) ) ).
tff(fact_116_snoc__eq__iff__butlast,axiom,
! [Xs: list_bool,X_1: bool,Ys: list_bool] :
( ( hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,Xs),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),nil_bool)) = Ys )
<=> ( ( Ys != nil_bool )
& ( butlast_bool(Ys) = Xs )
& ( hBOOL(hAPP_list_bool_bool(last_bool,Ys))
<=> hBOOL(X_1) ) ) ) ).
tff(fact_117_snoc__eq__iff__butlast,axiom,
! [Xs: list_A114658244le_alt,X_1: arrow_1893660431le_alt,Ys: list_A114658244le_alt] :
( ( hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),nil_Ar557011903le_alt)) = Ys )
<=> ( ( Ys != nil_Ar557011903le_alt )
& ( butlas1693249291le_alt(Ys) = Xs )
& ( hAPP_l1749541000le_alt(last_A488133080le_alt,Ys) = X_1 ) ) ) ).
tff(fact_118_snoc__eq__iff__butlast,axiom,
! [Xs: list_nat,X_1: nat,Ys: list_nat] :
( ( hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),nil_nat)) = Ys )
<=> ( ( Ys != nil_nat )
& ( butlast_nat(Ys) = Xs )
& ( hAPP_list_nat_nat(last_nat,Ys) = X_1 ) ) ) ).
tff(fact_119_append__butlast__last__id,axiom,
! [Xs: list_bool] :
( ( Xs != nil_bool )
=> ( hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,butlast_bool(Xs)),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,hAPP_list_bool_bool(last_bool,Xs)),nil_bool)) = Xs ) ) ).
tff(fact_120_append__butlast__last__id,axiom,
! [Xs_77: list_A114658244le_alt] :
( ( Xs_77 != nil_Ar557011903le_alt )
=> ( hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,butlas1693249291le_alt(Xs_77)),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,hAPP_l1749541000le_alt(last_A488133080le_alt,Xs_77)),nil_Ar557011903le_alt)) = Xs_77 ) ) ).
tff(fact_121_append__butlast__last__id,axiom,
! [Xs_77: list_nat] :
( ( Xs_77 != nil_nat )
=> ( hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,butlast_nat(Xs_77)),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,hAPP_list_nat_nat(last_nat,Xs_77)),nil_nat)) = Xs_77 ) ) ).
tff(fact_122_append__eq__Cons__conv,axiom,
! [Ys: list_bool,Zs: list_bool,X_1: bool,Xs: list_bool] :
( ( hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,Ys),Zs) = hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs) )
<=> ( ( ( Ys = nil_bool )
& ( Zs = hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs) ) )
| ? [Ys_20: list_bool] :
( ( Ys = hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Ys_20) )
& ( hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,Ys_20),Zs) = Xs ) ) ) ) ).
tff(fact_123_append__eq__Cons__conv,axiom,
! [Ys: list_A114658244le_alt,Zs: list_A114658244le_alt,X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt] :
( ( hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Ys),Zs) = hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),Xs) )
<=> ( ( ( Ys = nil_Ar557011903le_alt )
& ( Zs = hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),Xs) ) )
| ? [Ys_20: list_A114658244le_alt] :
( ( Ys = hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),Ys_20) )
& ( hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Ys_20),Zs) = Xs ) ) ) ) ).
tff(fact_124_append__eq__Cons__conv,axiom,
! [Ys: list_nat,Zs: list_nat,X_1: nat,Xs: list_nat] :
( ( hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Ys),Zs) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Xs) )
<=> ( ( ( Ys = nil_nat )
& ( Zs = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Xs) ) )
| ? [Ys_20: list_nat] :
( ( Ys = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Ys_20) )
& ( hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Ys_20),Zs) = Xs ) ) ) ) ).
tff(fact_125_Cons__eq__append__conv,axiom,
! [X_1: bool,Xs: list_bool,Ys: list_bool,Zs: list_bool] :
( ( hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs) = hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,Ys),Zs) )
<=> ( ( ( Ys = nil_bool )
& ( hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs) = Zs ) )
| ? [Ys_20: list_bool] :
( ( hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Ys_20) = Ys )
& ( Xs = hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,Ys_20),Zs) ) ) ) ) ).
tff(fact_126_Cons__eq__append__conv,axiom,
! [X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt,Ys: list_A114658244le_alt,Zs: list_A114658244le_alt] :
( ( hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),Xs) = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Ys),Zs) )
<=> ( ( ( Ys = nil_Ar557011903le_alt )
& ( hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),Xs) = Zs ) )
| ? [Ys_20: list_A114658244le_alt] :
( ( hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),Ys_20) = Ys )
& ( Xs = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Ys_20),Zs) ) ) ) ) ).
tff(fact_127_Cons__eq__append__conv,axiom,
! [X_1: nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Xs) = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Ys),Zs) )
<=> ( ( ( Ys = nil_nat )
& ( hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Xs) = Zs ) )
| ? [Ys_20: list_nat] :
( ( hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Ys_20) = Ys )
& ( Xs = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Ys_20),Zs) ) ) ) ) ).
tff(fact_128_append1__eq__conv,axiom,
! [Xs: list_bool,X_1: bool,Ys: list_bool,Y_2: bool] :
( ( hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,Xs),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),nil_bool)) = hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,Ys),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,Y_2),nil_bool)) )
<=> ( ( Xs = Ys )
& ( hBOOL(X_1)
<=> hBOOL(Y_2) ) ) ) ).
tff(fact_129_append1__eq__conv,axiom,
! [Xs: list_A114658244le_alt,X_1: arrow_1893660431le_alt,Ys: list_A114658244le_alt,Y_2: arrow_1893660431le_alt] :
( ( hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),nil_Ar557011903le_alt)) = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Ys),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,Y_2),nil_Ar557011903le_alt)) )
<=> ( ( Xs = Ys )
& ( X_1 = Y_2 ) ) ) ).
tff(fact_130_append1__eq__conv,axiom,
! [Xs: list_nat,X_1: nat,Ys: list_nat,Y_2: nat] :
( ( hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),nil_nat)) = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Ys),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,Y_2),nil_nat)) )
<=> ( ( Xs = Ys )
& ( X_1 = Y_2 ) ) ) ).
tff(fact_131_butlast__append,axiom,
! [Xs_76: list_A114658244le_alt,Ys_19: list_A114658244le_alt] :
( ( ( Ys_19 = nil_Ar557011903le_alt )
=> ( butlas1693249291le_alt(hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs_76),Ys_19)) = butlas1693249291le_alt(Xs_76) ) )
& ( ( Ys_19 != nil_Ar557011903le_alt )
=> ( butlas1693249291le_alt(hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs_76),Ys_19)) = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs_76),butlas1693249291le_alt(Ys_19)) ) ) ) ).
tff(fact_132_butlast__append,axiom,
! [Xs_76: list_nat,Ys_19: list_nat] :
( ( ( Ys_19 = nil_nat )
=> ( butlast_nat(hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs_76),Ys_19)) = butlast_nat(Xs_76) ) )
& ( ( Ys_19 != nil_nat )
=> ( butlast_nat(hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs_76),Ys_19)) = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs_76),butlast_nat(Ys_19)) ) ) ) ).
tff(fact_133_sorted__butlast,axiom,
! [Xs_75: list_nat] :
( ( Xs_75 != nil_nat )
=> ( hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,Xs_75))
=> hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,butlast_nat(Xs_75))) ) ) ).
tff(fact_134_last_Osimps,axiom,
! [X_1: bool,Xs: list_bool] :
( hBOOL(hAPP_list_bool_bool(last_bool,hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs)))
<=> ( ( ( Xs = nil_bool )
=> hBOOL(X_1) )
& ( ( Xs != nil_bool )
=> hBOOL(hAPP_list_bool_bool(last_bool,Xs)) ) ) ) ).
tff(fact_135_last_Osimps,axiom,
! [X_113: arrow_1893660431le_alt,Xs_74: list_A114658244le_alt] :
( ( ( Xs_74 = nil_Ar557011903le_alt )
=> ( hAPP_l1749541000le_alt(last_A488133080le_alt,hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_113),Xs_74)) = X_113 ) )
& ( ( Xs_74 != nil_Ar557011903le_alt )
=> ( hAPP_l1749541000le_alt(last_A488133080le_alt,hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_113),Xs_74)) = hAPP_l1749541000le_alt(last_A488133080le_alt,Xs_74) ) ) ) ).
tff(fact_136_last_Osimps,axiom,
! [X_113: nat,Xs_74: list_nat] :
( ( ( Xs_74 = nil_nat )
=> ( hAPP_list_nat_nat(last_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_113),Xs_74)) = X_113 ) )
& ( ( Xs_74 != nil_nat )
=> ( hAPP_list_nat_nat(last_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_113),Xs_74)) = hAPP_list_nat_nat(last_nat,Xs_74) ) ) ) ).
tff(fact_137_last__ConsR,axiom,
! [X_1: bool,Xs: list_bool] :
( ( Xs != nil_bool )
=> ( hBOOL(hAPP_list_bool_bool(last_bool,hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs)))
<=> hBOOL(hAPP_list_bool_bool(last_bool,Xs)) ) ) ).
tff(fact_138_last__ConsR,axiom,
! [X_112: arrow_1893660431le_alt,Xs_73: list_A114658244le_alt] :
( ( Xs_73 != nil_Ar557011903le_alt )
=> ( hAPP_l1749541000le_alt(last_A488133080le_alt,hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_112),Xs_73)) = hAPP_l1749541000le_alt(last_A488133080le_alt,Xs_73) ) ) ).
tff(fact_139_last__ConsR,axiom,
! [X_112: nat,Xs_73: list_nat] :
( ( Xs_73 != nil_nat )
=> ( hAPP_list_nat_nat(last_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_112),Xs_73)) = hAPP_list_nat_nat(last_nat,Xs_73) ) ) ).
tff(fact_140_rev__induct,axiom,
! [Xs: list_bool,P_1: fun_list_bool_bool] :
( hBOOL(hAPP_list_bool_bool(P_1,nil_bool))
=> ( ! [X_4: bool,Xs_19: list_bool] :
( hBOOL(hAPP_list_bool_bool(P_1,Xs_19))
=> hBOOL(hAPP_list_bool_bool(P_1,hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,Xs_19),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_4),nil_bool)))) )
=> hBOOL(hAPP_list_bool_bool(P_1,Xs)) ) ) ).
tff(fact_141_rev__induct,axiom,
! [Xs: list_A114658244le_alt,P_1: fun_li1109619974t_bool] :
( hBOOL(hAPP_l499746980t_bool(P_1,nil_Ar557011903le_alt))
=> ( ! [X_4: arrow_1893660431le_alt,Xs_19: list_A114658244le_alt] :
( hBOOL(hAPP_l499746980t_bool(P_1,Xs_19))
=> hBOOL(hAPP_l499746980t_bool(P_1,hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs_19),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_4),nil_Ar557011903le_alt)))) )
=> hBOOL(hAPP_l499746980t_bool(P_1,Xs)) ) ) ).
tff(fact_142_rev__induct,axiom,
! [Xs: list_nat,P_1: fun_list_nat_bool] :
( hBOOL(hAPP_list_nat_bool(P_1,nil_nat))
=> ( ! [X_4: nat,Xs_19: list_nat] :
( hBOOL(hAPP_list_nat_bool(P_1,Xs_19))
=> hBOOL(hAPP_list_nat_bool(P_1,hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs_19),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_4),nil_nat)))) )
=> hBOOL(hAPP_list_nat_bool(P_1,Xs)) ) ) ).
tff(fact_143_rev__cases,axiom,
! [Xs: list_bool] :
( ( Xs != nil_bool )
=> ~ ! [Ys_6: list_bool,Y_25: bool] : Xs != hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,Ys_6),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,Y_25),nil_bool)) ) ).
tff(fact_144_rev__cases,axiom,
! [Xs_72: list_A114658244le_alt] :
( ( Xs_72 != nil_Ar557011903le_alt )
=> ~ ! [Ys_6: list_A114658244le_alt,Y_25: arrow_1893660431le_alt] : Xs_72 != hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Ys_6),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,Y_25),nil_Ar557011903le_alt)) ) ).
tff(fact_145_rev__cases,axiom,
! [Xs_72: list_nat] :
( ( Xs_72 != nil_nat )
=> ~ ! [Ys_6: list_nat,Y_25: nat] : Xs_72 != hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Ys_6),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,Y_25),nil_nat)) ) ).
tff(fact_146_equal__Nil__null,axiom,
! [Xs: list_A114658244le_alt] :
( hBOOL(hAPP_l499746980t_bool(hAPP_l82091775t_bool(equal_631514850le_alt,Xs),nil_Ar557011903le_alt))
<=> hBOOL(hAPP_l499746980t_bool(null_A791782855le_alt,Xs)) ) ).
tff(fact_147_equal__Nil__null,axiom,
! [Xs: list_nat] :
( hBOOL(hAPP_list_nat_bool(hAPP_l120351551t_bool(equal_equal_list_nat,Xs),nil_nat))
<=> hBOOL(hAPP_list_nat_bool(null_nat,Xs)) ) ).
tff(fact_148_rotate__simps,axiom,
! [X_1: bool,Xs: list_bool] :
( ( rotate1753650700le_alt(nil_Ar557011903le_alt) = nil_Ar557011903le_alt )
& ( rotate1_bool(hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs)) = hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,Xs),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),nil_bool)) ) ) ).
tff(fact_149_rotate__simps,axiom,
! [X_1: bool,Xs: list_bool] :
( ( rotate1_nat(nil_nat) = nil_nat )
& ( rotate1_bool(hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs)) = hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,Xs),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),nil_bool)) ) ) ).
tff(fact_150_rotate__simps,axiom,
! [X_111: arrow_1893660431le_alt,Xs_71: list_A114658244le_alt] :
( ( rotate1_nat(nil_nat) = nil_nat )
& ( rotate1753650700le_alt(hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_111),Xs_71)) = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs_71),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_111),nil_Ar557011903le_alt)) ) ) ).
tff(fact_151_rotate__simps,axiom,
! [X_111: arrow_1893660431le_alt,Xs_71: list_A114658244le_alt] :
( ( rotate1753650700le_alt(nil_Ar557011903le_alt) = nil_Ar557011903le_alt )
& ( rotate1753650700le_alt(hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_111),Xs_71)) = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs_71),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_111),nil_Ar557011903le_alt)) ) ) ).
tff(fact_152_rotate__simps,axiom,
! [X_111: nat,Xs_71: list_nat] :
( ( rotate1_nat(nil_nat) = nil_nat )
& ( rotate1_nat(hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_111),Xs_71)) = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs_71),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_111),nil_nat)) ) ) ).
tff(fact_153_rotate__simps,axiom,
! [X_111: nat,Xs_71: list_nat] :
( ( rotate1753650700le_alt(nil_Ar557011903le_alt) = nil_Ar557011903le_alt )
& ( rotate1_nat(hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_111),Xs_71)) = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs_71),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_111),nil_nat)) ) ) ).
tff(fact_154_not__distinct__decomp,axiom,
! [Ws_1: list_bool] :
( ~ hBOOL(hAPP_list_bool_bool(distinct_bool,Ws_1))
=> ? [Xs_19: list_bool,Ys_6: list_bool,Zs_2: list_bool,Y_25: bool] : Ws_1 = hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,Xs_19),hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,Y_25),nil_bool)),hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,Ys_6),hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,Y_25),nil_bool)),Zs_2)))) ) ).
tff(fact_155_not__distinct__decomp,axiom,
! [Ws: list_A114658244le_alt] :
( ~ hBOOL(hAPP_l499746980t_bool(distin1654625714le_alt,Ws))
=> ? [Xs_19: list_A114658244le_alt,Ys_6: list_A114658244le_alt,Zs_2: list_A114658244le_alt,Y_25: arrow_1893660431le_alt] : Ws = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs_19),hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,Y_25),nil_Ar557011903le_alt)),hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Ys_6),hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,Y_25),nil_Ar557011903le_alt)),Zs_2)))) ) ).
tff(fact_156_not__distinct__decomp,axiom,
! [Ws: list_nat] :
( ~ hBOOL(hAPP_list_nat_bool(distinct_nat,Ws))
=> ? [Xs_19: list_nat,Ys_6: list_nat,Zs_2: list_nat,Y_25: nat] : Ws = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs_19),hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,Y_25),nil_nat)),hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Ys_6),hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,Y_25),nil_nat)),Zs_2)))) ) ).
tff(fact_157_replicate__append__same,axiom,
! [I_1: nat,X_1: bool] : hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,hAPP_bool_list_bool(replicate_bool(I_1),X_1)),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),nil_bool)) = hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),hAPP_bool_list_bool(replicate_bool(I_1),X_1)) ).
tff(fact_158_replicate__append__same,axiom,
! [I_13: nat,X_110: arrow_1893660431le_alt] : hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,hAPP_A1382181054le_alt(replic782356601le_alt(I_13),X_110)),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_110),nil_Ar557011903le_alt)) = hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_110),hAPP_A1382181054le_alt(replic782356601le_alt(I_13),X_110)) ).
tff(fact_159_replicate__append__same,axiom,
! [I_13: nat,X_110: nat] : hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,hAPP_nat_list_nat(replicate_nat(I_13),X_110)),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_110),nil_nat)) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_110),hAPP_nat_list_nat(replicate_nat(I_13),X_110)) ).
tff(fact_160_rev_Osimps_I2_J,axiom,
! [X_1: bool,Xs: list_bool] : rev_bool(hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs)) = hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,rev_bool(Xs)),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),nil_bool)) ).
tff(fact_161_rev_Osimps_I2_J,axiom,
! [X_109: arrow_1893660431le_alt,Xs_70: list_A114658244le_alt] : rev_Ar377224735le_alt(hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_109),Xs_70)) = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,rev_Ar377224735le_alt(Xs_70)),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_109),nil_Ar557011903le_alt)) ).
tff(fact_162_rev_Osimps_I2_J,axiom,
! [X_109: nat,Xs_70: list_nat] : rev_nat(hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_109),Xs_70)) = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,rev_nat(Xs_70)),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_109),nil_nat)) ).
tff(fact_163_rev__eq__Cons__iff,axiom,
! [Xs: list_bool,Y_2: bool,Ys: list_bool] :
( ( rev_bool(Xs) = hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,Y_2),Ys) )
<=> ( Xs = hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,rev_bool(Ys)),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,Y_2),nil_bool)) ) ) ).
tff(fact_164_rev__eq__Cons__iff,axiom,
! [Xs: list_A114658244le_alt,Y_2: arrow_1893660431le_alt,Ys: list_A114658244le_alt] :
( ( rev_Ar377224735le_alt(Xs) = hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,Y_2),Ys) )
<=> ( Xs = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,rev_Ar377224735le_alt(Ys)),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,Y_2),nil_Ar557011903le_alt)) ) ) ).
tff(fact_165_rev__eq__Cons__iff,axiom,
! [Xs: list_nat,Y_2: nat,Ys: list_nat] :
( ( rev_nat(Xs) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,Y_2),Ys) )
<=> ( Xs = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,rev_nat(Ys)),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,Y_2),nil_nat)) ) ) ).
tff(fact_166_hd__append2,axiom,
! [Ys_18: list_A114658244le_alt,Xs_69: list_A114658244le_alt] :
( ( Xs_69 != nil_Ar557011903le_alt )
=> ( hAPP_l1749541000le_alt(hd_Arr1236501138le_alt,hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs_69),Ys_18)) = hAPP_l1749541000le_alt(hd_Arr1236501138le_alt,Xs_69) ) ) ).
tff(fact_167_hd__append2,axiom,
! [Ys_18: list_nat,Xs_69: list_nat] :
( ( Xs_69 != nil_nat )
=> ( hAPP_list_nat_nat(hd_nat,hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs_69),Ys_18)) = hAPP_list_nat_nat(hd_nat,Xs_69) ) ) ).
tff(fact_168_hd__append,axiom,
! [Ys_17: list_A114658244le_alt,Xs_68: list_A114658244le_alt] :
( ( ( Xs_68 = nil_Ar557011903le_alt )
=> ( hAPP_l1749541000le_alt(hd_Arr1236501138le_alt,hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs_68),Ys_17)) = hAPP_l1749541000le_alt(hd_Arr1236501138le_alt,Ys_17) ) )
& ( ( Xs_68 != nil_Ar557011903le_alt )
=> ( hAPP_l1749541000le_alt(hd_Arr1236501138le_alt,hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs_68),Ys_17)) = hAPP_l1749541000le_alt(hd_Arr1236501138le_alt,Xs_68) ) ) ) ).
tff(fact_169_hd__append,axiom,
! [Ys_17: list_nat,Xs_68: list_nat] :
( ( ( Xs_68 = nil_nat )
=> ( hAPP_list_nat_nat(hd_nat,hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs_68),Ys_17)) = hAPP_list_nat_nat(hd_nat,Ys_17) ) )
& ( ( Xs_68 != nil_nat )
=> ( hAPP_list_nat_nat(hd_nat,hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs_68),Ys_17)) = hAPP_list_nat_nat(hd_nat,Xs_68) ) ) ) ).
tff(fact_170_tl__append2,axiom,
! [Ys_16: list_A114658244le_alt,Xs_67: list_A114658244le_alt] :
( ( Xs_67 != nil_Ar557011903le_alt )
=> ( tl_Arr1883752598le_alt(hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs_67),Ys_16)) = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,tl_Arr1883752598le_alt(Xs_67)),Ys_16) ) ) ).
tff(fact_171_tl__append2,axiom,
! [Ys_16: list_nat,Xs_67: list_nat] :
( ( Xs_67 != nil_nat )
=> ( tl_nat(hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs_67),Ys_16)) = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,tl_nat(Xs_67)),Ys_16) ) ) ).
tff(fact_172_rev__is__rev__conv,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( rev_nat(Xs) = rev_nat(Ys) )
<=> ( Xs = Ys ) ) ).
tff(fact_173_rev__is__rev__conv,axiom,
! [Xs: list_A114658244le_alt,Ys: list_A114658244le_alt] :
( ( rev_Ar377224735le_alt(Xs) = rev_Ar377224735le_alt(Ys) )
<=> ( Xs = Ys ) ) ).
tff(fact_174_rev__swap,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( rev_nat(Xs) = Ys )
<=> ( Xs = rev_nat(Ys) ) ) ).
tff(fact_175_rev__swap,axiom,
! [Xs: list_A114658244le_alt,Ys: list_A114658244le_alt] :
( ( rev_Ar377224735le_alt(Xs) = Ys )
<=> ( Xs = rev_Ar377224735le_alt(Ys) ) ) ).
tff(fact_176_mem__def,axiom,
! [X_1: arrow_1893660431le_alt,A: fun_Ar17477381t_bool] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),A))
<=> hBOOL(hAPP_A1748732503t_bool(A,X_1)) ) ).
tff(fact_177_mem__def,axiom,
! [X_1: nat,A: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
<=> hBOOL(hAPP_nat_bool(A,X_1)) ) ).
tff(fact_178_Collect__def,axiom,
! [P_1: fun_nat_bool] : collect_nat(P_1) = P_1 ).
tff(fact_179_rev__replicate,axiom,
! [N_44: nat,X_108: nat] : rev_nat(hAPP_nat_list_nat(replicate_nat(N_44),X_108)) = hAPP_nat_list_nat(replicate_nat(N_44),X_108) ).
tff(fact_180_rev__replicate,axiom,
! [N_44: nat,X_108: arrow_1893660431le_alt] : rev_Ar377224735le_alt(hAPP_A1382181054le_alt(replic782356601le_alt(N_44),X_108)) = hAPP_A1382181054le_alt(replic782356601le_alt(N_44),X_108) ).
tff(fact_181_rev__rev__ident,axiom,
! [Xs_66: list_nat] : rev_nat(rev_nat(Xs_66)) = Xs_66 ).
tff(fact_182_rev__rev__ident,axiom,
! [Xs_66: list_A114658244le_alt] : rev_Ar377224735le_alt(rev_Ar377224735le_alt(Xs_66)) = Xs_66 ).
tff(fact_183_equal__list__def,axiom,
! [X_1: list_nat,Y_2: list_nat] :
( hBOOL(hAPP_list_nat_bool(hAPP_l120351551t_bool(equal_equal_list_nat,X_1),Y_2))
<=> ( X_1 = Y_2 ) ) ).
tff(fact_184_equal__list__def,axiom,
! [X_1: list_A114658244le_alt,Y_2: list_A114658244le_alt] :
( hBOOL(hAPP_l499746980t_bool(hAPP_l82091775t_bool(equal_631514850le_alt,X_1),Y_2))
<=> ( X_1 = Y_2 ) ) ).
tff(fact_185_rev__is__Nil__conv,axiom,
! [Xs: list_A114658244le_alt] :
( ( rev_Ar377224735le_alt(Xs) = nil_Ar557011903le_alt )
<=> ( Xs = nil_Ar557011903le_alt ) ) ).
tff(fact_186_rev__is__Nil__conv,axiom,
! [Xs: list_nat] :
( ( rev_nat(Xs) = nil_nat )
<=> ( Xs = nil_nat ) ) ).
tff(fact_187_Nil__is__rev__conv,axiom,
! [Xs: list_A114658244le_alt] :
( ( nil_Ar557011903le_alt = rev_Ar377224735le_alt(Xs) )
<=> ( Xs = nil_Ar557011903le_alt ) ) ).
tff(fact_188_Nil__is__rev__conv,axiom,
! [Xs: list_nat] :
( ( nil_nat = rev_nat(Xs) )
<=> ( Xs = nil_nat ) ) ).
tff(fact_189_rev_Osimps_I1_J,axiom,
rev_Ar377224735le_alt(nil_Ar557011903le_alt) = nil_Ar557011903le_alt ).
tff(fact_190_rev_Osimps_I1_J,axiom,
rev_nat(nil_nat) = nil_nat ).
tff(fact_191_rev__append,axiom,
! [Xs_65: list_A114658244le_alt,Ys_15: list_A114658244le_alt] : rev_Ar377224735le_alt(hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs_65),Ys_15)) = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,rev_Ar377224735le_alt(Ys_15)),rev_Ar377224735le_alt(Xs_65)) ).
tff(fact_192_rev__append,axiom,
! [Xs_65: list_nat,Ys_15: list_nat] : rev_nat(hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs_65),Ys_15)) = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,rev_nat(Ys_15)),rev_nat(Xs_65)) ).
tff(fact_193_distinct__rev,axiom,
! [Xs: list_A114658244le_alt] :
( hBOOL(hAPP_l499746980t_bool(distin1654625714le_alt,rev_Ar377224735le_alt(Xs)))
<=> hBOOL(hAPP_l499746980t_bool(distin1654625714le_alt,Xs)) ) ).
tff(fact_194_distinct__rev,axiom,
! [Xs: list_nat] :
( hBOOL(hAPP_list_nat_bool(distinct_nat,rev_nat(Xs)))
<=> hBOOL(hAPP_list_nat_bool(distinct_nat,Xs)) ) ).
tff(fact_195_last__rev,axiom,
! [Xs_64: list_A114658244le_alt] :
( ( Xs_64 != nil_Ar557011903le_alt )
=> ( hAPP_l1749541000le_alt(last_A488133080le_alt,rev_Ar377224735le_alt(Xs_64)) = hAPP_l1749541000le_alt(hd_Arr1236501138le_alt,Xs_64) ) ) ).
tff(fact_196_last__rev,axiom,
! [Xs_64: list_nat] :
( ( Xs_64 != nil_nat )
=> ( hAPP_list_nat_nat(last_nat,rev_nat(Xs_64)) = hAPP_list_nat_nat(hd_nat,Xs_64) ) ) ).
tff(fact_197_hd__rev,axiom,
! [Xs_63: list_A114658244le_alt] :
( ( Xs_63 != nil_Ar557011903le_alt )
=> ( hAPP_l1749541000le_alt(hd_Arr1236501138le_alt,rev_Ar377224735le_alt(Xs_63)) = hAPP_l1749541000le_alt(last_A488133080le_alt,Xs_63) ) ) ).
tff(fact_198_hd__rev,axiom,
! [Xs_63: list_nat] :
( ( Xs_63 != nil_nat )
=> ( hAPP_list_nat_nat(hd_nat,rev_nat(Xs_63)) = hAPP_list_nat_nat(last_nat,Xs_63) ) ) ).
tff(fact_199_append__replicate__commute,axiom,
! [N_43: nat,X_107: arrow_1893660431le_alt,K_6: nat] : hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,hAPP_A1382181054le_alt(replic782356601le_alt(N_43),X_107)),hAPP_A1382181054le_alt(replic782356601le_alt(K_6),X_107)) = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,hAPP_A1382181054le_alt(replic782356601le_alt(K_6),X_107)),hAPP_A1382181054le_alt(replic782356601le_alt(N_43),X_107)) ).
tff(fact_200_append__replicate__commute,axiom,
! [N_43: nat,X_107: nat,K_6: nat] : hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,hAPP_nat_list_nat(replicate_nat(N_43),X_107)),hAPP_nat_list_nat(replicate_nat(K_6),X_107)) = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,hAPP_nat_list_nat(replicate_nat(K_6),X_107)),hAPP_nat_list_nat(replicate_nat(N_43),X_107)) ).
tff(fact_201_rotate1__hd__tl,axiom,
! [Xs: list_bool] :
( ( Xs != nil_bool )
=> ( rotate1_bool(Xs) = hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,tl_bool(Xs)),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,hAPP_list_bool_bool(hd_bool,Xs)),nil_bool)) ) ) ).
tff(fact_202_rotate1__hd__tl,axiom,
! [Xs_62: list_A114658244le_alt] :
( ( Xs_62 != nil_Ar557011903le_alt )
=> ( rotate1753650700le_alt(Xs_62) = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,tl_Arr1883752598le_alt(Xs_62)),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,hAPP_l1749541000le_alt(hd_Arr1236501138le_alt,Xs_62)),nil_Ar557011903le_alt)) ) ) ).
tff(fact_203_rotate1__hd__tl,axiom,
! [Xs_62: list_nat] :
( ( Xs_62 != nil_nat )
=> ( rotate1_nat(Xs_62) = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,tl_nat(Xs_62)),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,hAPP_list_nat_nat(hd_nat,Xs_62)),nil_nat)) ) ) ).
tff(fact_204_tl_Osimps_I2_J,axiom,
! [X_1: bool,Xs: list_bool] : tl_bool(hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs)) = Xs ).
tff(fact_205_tl_Osimps_I2_J,axiom,
! [X_106: arrow_1893660431le_alt,Xs_61: list_A114658244le_alt] : tl_Arr1883752598le_alt(hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_106),Xs_61)) = Xs_61 ).
tff(fact_206_tl_Osimps_I2_J,axiom,
! [X_106: nat,Xs_61: list_nat] : tl_nat(hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_106),Xs_61)) = Xs_61 ).
tff(fact_207_tl_Osimps_I1_J,axiom,
tl_Arr1883752598le_alt(nil_Ar557011903le_alt) = nil_Ar557011903le_alt ).
tff(fact_208_tl_Osimps_I1_J,axiom,
tl_nat(nil_nat) = nil_nat ).
tff(fact_209_hd_Osimps,axiom,
! [X_1: bool,Xs: list_bool] :
( hBOOL(hAPP_list_bool_bool(hd_bool,hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs)))
<=> hBOOL(X_1) ) ).
tff(fact_210_hd_Osimps,axiom,
! [X_105: arrow_1893660431le_alt,Xs_60: list_A114658244le_alt] : hAPP_l1749541000le_alt(hd_Arr1236501138le_alt,hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_105),Xs_60)) = X_105 ).
tff(fact_211_hd_Osimps,axiom,
! [X_105: nat,Xs_60: list_nat] : hAPP_list_nat_nat(hd_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_105),Xs_60)) = X_105 ).
tff(fact_212_distinct__tl,axiom,
! [Xs_59: list_A114658244le_alt] :
( hBOOL(hAPP_l499746980t_bool(distin1654625714le_alt,Xs_59))
=> hBOOL(hAPP_l499746980t_bool(distin1654625714le_alt,tl_Arr1883752598le_alt(Xs_59))) ) ).
tff(fact_213_distinct__tl,axiom,
! [Xs_59: list_nat] :
( hBOOL(hAPP_list_nat_bool(distinct_nat,Xs_59))
=> hBOOL(hAPP_list_nat_bool(distinct_nat,tl_nat(Xs_59))) ) ).
tff(fact_214_sorted__tl,axiom,
! [Xs_58: list_nat] :
( hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,Xs_58))
=> hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,tl_nat(Xs_58))) ) ).
tff(fact_215_rotate1__is__Nil__conv,axiom,
! [Xs: list_A114658244le_alt] :
( ( rotate1753650700le_alt(Xs) = nil_Ar557011903le_alt )
<=> ( Xs = nil_Ar557011903le_alt ) ) ).
tff(fact_216_rotate1__is__Nil__conv,axiom,
! [Xs: list_nat] :
( ( rotate1_nat(Xs) = nil_nat )
<=> ( Xs = nil_nat ) ) ).
tff(fact_217_distinct1__rotate,axiom,
! [Xs: list_A114658244le_alt] :
( hBOOL(hAPP_l499746980t_bool(distin1654625714le_alt,rotate1753650700le_alt(Xs)))
<=> hBOOL(hAPP_l499746980t_bool(distin1654625714le_alt,Xs)) ) ).
tff(fact_218_distinct1__rotate,axiom,
! [Xs: list_nat] :
( hBOOL(hAPP_list_nat_bool(distinct_nat,rotate1_nat(Xs)))
<=> hBOOL(hAPP_list_nat_bool(distinct_nat,Xs)) ) ).
tff(fact_219_singleton__rev__conv,axiom,
! [X_1: bool,Xs: list_bool] :
( ( hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),nil_bool) = rev_bool(Xs) )
<=> ( Xs = hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),nil_bool) ) ) ).
tff(fact_220_singleton__rev__conv,axiom,
! [X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt] :
( ( hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),nil_Ar557011903le_alt) = rev_Ar377224735le_alt(Xs) )
<=> ( Xs = hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),nil_Ar557011903le_alt) ) ) ).
tff(fact_221_singleton__rev__conv,axiom,
! [X_1: nat,Xs: list_nat] :
( ( hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),nil_nat) = rev_nat(Xs) )
<=> ( Xs = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),nil_nat) ) ) ).
tff(fact_222_rev__singleton__conv,axiom,
! [Xs: list_bool,X_1: bool] :
( ( rev_bool(Xs) = hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),nil_bool) )
<=> ( Xs = hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),nil_bool) ) ) ).
tff(fact_223_rev__singleton__conv,axiom,
! [Xs: list_A114658244le_alt,X_1: arrow_1893660431le_alt] :
( ( rev_Ar377224735le_alt(Xs) = hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),nil_Ar557011903le_alt) )
<=> ( Xs = hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),nil_Ar557011903le_alt) ) ) ).
tff(fact_224_rev__singleton__conv,axiom,
! [Xs: list_nat,X_1: nat] :
( ( rev_nat(Xs) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),nil_nat) )
<=> ( Xs = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),nil_nat) ) ) ).
tff(fact_225_replicate__app__Cons__same,axiom,
! [N_1: nat,X_1: bool,Xs: list_bool] : hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,hAPP_bool_list_bool(replicate_bool(N_1),X_1)),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs)) = hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,hAPP_bool_list_bool(replicate_bool(N_1),X_1)),Xs)) ).
tff(fact_226_replicate__app__Cons__same,axiom,
! [N_42: nat,X_104: arrow_1893660431le_alt,Xs_57: list_A114658244le_alt] : hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,hAPP_A1382181054le_alt(replic782356601le_alt(N_42),X_104)),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_104),Xs_57)) = hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_104),hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,hAPP_A1382181054le_alt(replic782356601le_alt(N_42),X_104)),Xs_57)) ).
tff(fact_227_replicate__app__Cons__same,axiom,
! [N_42: nat,X_104: nat,Xs_57: list_nat] : hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,hAPP_nat_list_nat(replicate_nat(N_42),X_104)),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_104),Xs_57)) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_104),hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,hAPP_nat_list_nat(replicate_nat(N_42),X_104)),Xs_57)) ).
tff(fact_228_equal,axiom,
equal_equal_list_nat = fequal_list_nat ).
tff(fact_229_equal,axiom,
equal_631514850le_alt = fequal928191109le_alt ).
tff(fact_230_equal__refl,axiom,
! [X_103: list_nat] : hBOOL(hAPP_list_nat_bool(hAPP_l120351551t_bool(equal_equal_list_nat,X_103),X_103)) ).
tff(fact_231_equal__refl,axiom,
! [X_103: list_A114658244le_alt] : hBOOL(hAPP_l499746980t_bool(hAPP_l82091775t_bool(equal_631514850le_alt,X_103),X_103)) ).
tff(fact_232_equal__eq,axiom,
! [X_1: list_nat,Y_2: list_nat] :
( hBOOL(hAPP_list_nat_bool(hAPP_l120351551t_bool(equal_equal_list_nat,X_1),Y_2))
<=> ( X_1 = Y_2 ) ) ).
tff(fact_233_equal__eq,axiom,
! [X_1: list_A114658244le_alt,Y_2: list_A114658244le_alt] :
( hBOOL(hAPP_l499746980t_bool(hAPP_l82091775t_bool(equal_631514850le_alt,X_1),Y_2))
<=> ( X_1 = Y_2 ) ) ).
tff(fact_234_eq__equal,axiom,
fequal_list_nat = equal_equal_list_nat ).
tff(fact_235_eq__equal,axiom,
fequal928191109le_alt = equal_631514850le_alt ).
tff(fact_236_hd__dropWhile,axiom,
! [P_1: fun_Ar17477381t_bool,Xs: list_A114658244le_alt] :
( ( dropWh587599712le_alt(P_1,Xs) != nil_Ar557011903le_alt )
=> ~ hBOOL(hAPP_A1748732503t_bool(P_1,hAPP_l1749541000le_alt(hd_Arr1236501138le_alt,dropWh587599712le_alt(P_1,Xs)))) ) ).
tff(fact_237_hd__dropWhile,axiom,
! [P_1: fun_nat_bool,Xs: list_nat] :
( ( dropWhile_nat(P_1,Xs) != nil_nat )
=> ~ hBOOL(hAPP_nat_bool(P_1,hAPP_list_nat_nat(hd_nat,dropWhile_nat(P_1,Xs)))) ) ).
tff(fact_238_list__ex1__simps_I1_J,axiom,
! [P_1: fun_Ar17477381t_bool] : ~ hBOOL(hAPP_l499746980t_bool(list_e1657546370le_alt(P_1),nil_Ar557011903le_alt)) ).
tff(fact_239_list__ex1__simps_I1_J,axiom,
! [P_1: fun_nat_bool] : ~ hBOOL(hAPP_list_nat_bool(list_ex1_nat(P_1),nil_nat)) ).
tff(fact_240_member__rec_I2_J,axiom,
! [Y_66: arrow_1893660431le_alt] : ~ hBOOL(hAPP_A1748732503t_bool(hAPP_l2018657868t_bool(member775785940le_alt,nil_Ar557011903le_alt),Y_66)) ).
tff(fact_241_member__rec_I2_J,axiom,
! [Y_66: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_l1821533378t_bool(member_nat_1,nil_nat),Y_66)) ).
tff(fact_242_insert,axiom,
! [Y_2: bool,X_1: bool,Xs: list_bool] :
( hBOOL(hAPP_list_bool_bool(listMem_bool(X_1),Xs))
=> hBOOL(hAPP_list_bool_bool(listMem_bool(X_1),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,Y_2),Xs))) ) ).
tff(fact_243_insert,axiom,
! [Y_65: arrow_1893660431le_alt,X_102: arrow_1893660431le_alt,Xs_56: list_A114658244le_alt] :
( hBOOL(hAPP_l499746980t_bool(listMe1496348587le_alt(X_102),Xs_56))
=> hBOOL(hAPP_l499746980t_bool(listMe1496348587le_alt(X_102),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,Y_65),Xs_56))) ) ).
tff(fact_244_insert,axiom,
! [Y_65: nat,X_102: nat,Xs_56: list_nat] :
( hBOOL(hAPP_list_nat_bool(listMem_nat(X_102),Xs_56))
=> hBOOL(hAPP_list_nat_bool(listMem_nat(X_102),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,Y_65),Xs_56))) ) ).
tff(fact_245_member__rec_I1_J,axiom,
! [X_1: bool,Xs: list_bool,Y_2: bool] :
( hBOOL(hAPP_bool_bool(hAPP_l96088310l_bool(member_bool_1,hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs)),Y_2))
<=> ( ( hBOOL(X_1)
<=> hBOOL(Y_2) )
| hBOOL(hAPP_bool_bool(hAPP_l96088310l_bool(member_bool_1,Xs),Y_2)) ) ) ).
tff(fact_246_member__rec_I1_J,axiom,
! [X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt,Y_2: arrow_1893660431le_alt] :
( hBOOL(hAPP_A1748732503t_bool(hAPP_l2018657868t_bool(member775785940le_alt,hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),Xs)),Y_2))
<=> ( ( X_1 = Y_2 )
| hBOOL(hAPP_A1748732503t_bool(hAPP_l2018657868t_bool(member775785940le_alt,Xs),Y_2)) ) ) ).
tff(fact_247_member__rec_I1_J,axiom,
! [X_1: nat,Xs: list_nat,Y_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_l1821533378t_bool(member_nat_1,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Xs)),Y_2))
<=> ( ( X_1 = Y_2 )
| hBOOL(hAPP_nat_bool(hAPP_l1821533378t_bool(member_nat_1,Xs),Y_2)) ) ) ).
tff(fact_248_elem,axiom,
! [X_1: bool,Xs: list_bool] : hBOOL(hAPP_list_bool_bool(listMem_bool(X_1),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs))) ).
tff(fact_249_elem,axiom,
! [X_101: arrow_1893660431le_alt,Xs_55: list_A114658244le_alt] : hBOOL(hAPP_l499746980t_bool(listMe1496348587le_alt(X_101),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_101),Xs_55))) ).
tff(fact_250_elem,axiom,
! [X_101: nat,Xs_55: list_nat] : hBOOL(hAPP_list_nat_bool(listMem_nat(X_101),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_101),Xs_55))) ).
tff(fact_251_dropWhile_Osimps_I2_J,axiom,
! [Xs: list_bool,P_1: fun_bool_bool,X_1: bool] :
( ( hBOOL(hAPP_bool_bool(P_1,X_1))
=> ( dropWhile_bool(P_1,hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs)) = dropWhile_bool(P_1,Xs) ) )
& ( ~ hBOOL(hAPP_bool_bool(P_1,X_1))
=> ( dropWhile_bool(P_1,hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs)) = hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs) ) ) ) ).
tff(fact_252_dropWhile_Osimps_I2_J,axiom,
! [Xs: list_A114658244le_alt,P_1: fun_Ar17477381t_bool,X_1: arrow_1893660431le_alt] :
( ( hBOOL(hAPP_A1748732503t_bool(P_1,X_1))
=> ( dropWh587599712le_alt(P_1,hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),Xs)) = dropWh587599712le_alt(P_1,Xs) ) )
& ( ~ hBOOL(hAPP_A1748732503t_bool(P_1,X_1))
=> ( dropWh587599712le_alt(P_1,hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),Xs)) = hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),Xs) ) ) ) ).
tff(fact_253_dropWhile_Osimps_I2_J,axiom,
! [Xs: list_nat,P_1: fun_nat_bool,X_1: nat] :
( ( hBOOL(hAPP_nat_bool(P_1,X_1))
=> ( dropWhile_nat(P_1,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Xs)) = dropWhile_nat(P_1,Xs) ) )
& ( ~ hBOOL(hAPP_nat_bool(P_1,X_1))
=> ( dropWhile_nat(P_1,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Xs)) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Xs) ) ) ) ).
tff(fact_254_dropWhile_Osimps_I1_J,axiom,
! [P_1: fun_Ar17477381t_bool] : dropWh587599712le_alt(P_1,nil_Ar557011903le_alt) = nil_Ar557011903le_alt ).
tff(fact_255_dropWhile_Osimps_I1_J,axiom,
! [P_1: fun_nat_bool] : dropWhile_nat(P_1,nil_nat) = nil_nat ).
tff(fact_256_distinct__dropWhile,axiom,
! [P_1: fun_Ar17477381t_bool,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_l499746980t_bool(distin1654625714le_alt,Xs))
=> hBOOL(hAPP_l499746980t_bool(distin1654625714le_alt,dropWh587599712le_alt(P_1,Xs))) ) ).
tff(fact_257_distinct__dropWhile,axiom,
! [P_1: fun_nat_bool,Xs: list_nat] :
( hBOOL(hAPP_list_nat_bool(distinct_nat,Xs))
=> hBOOL(hAPP_list_nat_bool(distinct_nat,dropWhile_nat(P_1,Xs))) ) ).
tff(fact_258_sorted__dropWhile,axiom,
! [P_1: fun_nat_bool,Xs: list_nat] :
( hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,Xs))
=> hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,dropWhile_nat(P_1,Xs))) ) ).
tff(fact_259_dropWhile__eq__Cons__conv,axiom,
! [P_1: fun_bool_bool,Xs: list_bool,Y_2: bool,Ys: list_bool] :
( ( dropWhile_bool(P_1,Xs) = hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,Y_2),Ys) )
<=> ( ( Xs = hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,takeWhile_bool(P_1,Xs)),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,Y_2),Ys)) )
& ~ hBOOL(hAPP_bool_bool(P_1,Y_2)) ) ) ).
tff(fact_260_dropWhile__eq__Cons__conv,axiom,
! [P_1: fun_Ar17477381t_bool,Xs: list_A114658244le_alt,Y_2: arrow_1893660431le_alt,Ys: list_A114658244le_alt] :
( ( dropWh587599712le_alt(P_1,Xs) = hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,Y_2),Ys) )
<=> ( ( Xs = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,takeWh967109304le_alt(P_1,Xs)),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,Y_2),Ys)) )
& ~ hBOOL(hAPP_A1748732503t_bool(P_1,Y_2)) ) ) ).
tff(fact_261_dropWhile__eq__Cons__conv,axiom,
! [P_1: fun_nat_bool,Xs: list_nat,Y_2: nat,Ys: list_nat] :
( ( dropWhile_nat(P_1,Xs) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,Y_2),Ys) )
<=> ( ( Xs = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,takeWhile_nat(P_1,Xs)),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,Y_2),Ys)) )
& ~ hBOOL(hAPP_nat_bool(P_1,Y_2)) ) ) ).
tff(fact_262_ListMem_Osimps,axiom,
! [A1: bool,A2: list_bool] :
( hBOOL(hAPP_list_bool_bool(listMem_bool(A1),A2))
<=> ( ? [X_4: bool,Xs_19: list_bool] :
( ( hBOOL(A1)
<=> hBOOL(X_4) )
& ( A2 = hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_4),Xs_19) ) )
| ? [X_4: bool,Xs_19: list_bool,Y_25: bool] :
( ( hBOOL(A1)
<=> hBOOL(X_4) )
& ( A2 = hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,Y_25),Xs_19) )
& hBOOL(hAPP_list_bool_bool(listMem_bool(X_4),Xs_19)) ) ) ) ).
tff(fact_263_ListMem_Osimps,axiom,
! [A1: arrow_1893660431le_alt,A2: list_A114658244le_alt] :
( hBOOL(hAPP_l499746980t_bool(listMe1496348587le_alt(A1),A2))
<=> ( ? [X_4: arrow_1893660431le_alt,Xs_19: list_A114658244le_alt] :
( ( A1 = X_4 )
& ( A2 = hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_4),Xs_19) ) )
| ? [X_4: arrow_1893660431le_alt,Xs_19: list_A114658244le_alt,Y_25: arrow_1893660431le_alt] :
( ( A1 = X_4 )
& ( A2 = hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,Y_25),Xs_19) )
& hBOOL(hAPP_l499746980t_bool(listMe1496348587le_alt(X_4),Xs_19)) ) ) ) ).
tff(fact_264_ListMem_Osimps,axiom,
! [A1: nat,A2: list_nat] :
( hBOOL(hAPP_list_nat_bool(listMem_nat(A1),A2))
<=> ( ? [X_4: nat,Xs_19: list_nat] :
( ( A1 = X_4 )
& ( A2 = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_4),Xs_19) ) )
| ? [X_4: nat,Xs_19: list_nat,Y_25: nat] :
( ( A1 = X_4 )
& ( A2 = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,Y_25),Xs_19) )
& hBOOL(hAPP_list_nat_bool(listMem_nat(X_4),Xs_19)) ) ) ) ).
tff(fact_265_takeWhile__dropWhile__id,axiom,
! [P_1: fun_Ar17477381t_bool,Xs: list_A114658244le_alt] : hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,takeWh967109304le_alt(P_1,Xs)),dropWh587599712le_alt(P_1,Xs)) = Xs ).
tff(fact_266_takeWhile__dropWhile__id,axiom,
! [P_1: fun_nat_bool,Xs: list_nat] : hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,takeWhile_nat(P_1,Xs)),dropWhile_nat(P_1,Xs)) = Xs ).
tff(fact_267_hd__replicate,axiom,
! [X_100: arrow_1893660431le_alt,N_41: nat] :
( ( N_41 != zero_zero_nat )
=> ( hAPP_l1749541000le_alt(hd_Arr1236501138le_alt,hAPP_A1382181054le_alt(replic782356601le_alt(N_41),X_100)) = X_100 ) ) ).
tff(fact_268_hd__replicate,axiom,
! [X_100: nat,N_41: nat] :
( ( N_41 != zero_zero_nat )
=> ( hAPP_list_nat_nat(hd_nat,hAPP_nat_list_nat(replicate_nat(N_41),X_100)) = X_100 ) ) ).
tff(fact_269_last__replicate,axiom,
! [X_99: nat,N_40: nat] :
( ( N_40 != zero_zero_nat )
=> ( hAPP_list_nat_nat(last_nat,hAPP_nat_list_nat(replicate_nat(N_40),X_99)) = X_99 ) ) ).
tff(fact_270_last__replicate,axiom,
! [X_99: arrow_1893660431le_alt,N_40: nat] :
( ( N_40 != zero_zero_nat )
=> ( hAPP_l1749541000le_alt(last_A488133080le_alt,hAPP_A1382181054le_alt(replic782356601le_alt(N_40),X_99)) = X_99 ) ) ).
tff(fact_271_list__induct2_H,axiom,
! [Xs: list_A114658244le_alt,Ys: list_bool,P_1: fun_li1957118493l_bool] :
( hBOOL(hAPP_list_bool_bool(hAPP_l2076432827l_bool(P_1,nil_Ar557011903le_alt),nil_bool))
=> ( ! [X_4: arrow_1893660431le_alt,Xs_19: list_A114658244le_alt] : hBOOL(hAPP_list_bool_bool(hAPP_l2076432827l_bool(P_1,hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_4),Xs_19)),nil_bool))
=> ( ! [Y_25: bool,Ys_6: list_bool] : hBOOL(hAPP_list_bool_bool(hAPP_l2076432827l_bool(P_1,nil_Ar557011903le_alt),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,Y_25),Ys_6)))
=> ( ! [X_4: arrow_1893660431le_alt,Xs_19: list_A114658244le_alt,Y_25: bool,Ys_6: list_bool] :
( hBOOL(hAPP_list_bool_bool(hAPP_l2076432827l_bool(P_1,Xs_19),Ys_6))
=> hBOOL(hAPP_list_bool_bool(hAPP_l2076432827l_bool(P_1,hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_4),Xs_19)),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,Y_25),Ys_6))) )
=> hBOOL(hAPP_list_bool_bool(hAPP_l2076432827l_bool(P_1,Xs),Ys)) ) ) ) ) ).
tff(fact_272_list__induct2_H,axiom,
! [Xs: list_nat,Ys: list_bool,P_1: fun_li972600984l_bool] :
( hBOOL(hAPP_list_bool_bool(hAPP_l1101248310l_bool(P_1,nil_nat),nil_bool))
=> ( ! [X_4: nat,Xs_19: list_nat] : hBOOL(hAPP_list_bool_bool(hAPP_l1101248310l_bool(P_1,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_4),Xs_19)),nil_bool))
=> ( ! [Y_25: bool,Ys_6: list_bool] : hBOOL(hAPP_list_bool_bool(hAPP_l1101248310l_bool(P_1,nil_nat),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,Y_25),Ys_6)))
=> ( ! [X_4: nat,Xs_19: list_nat,Y_25: bool,Ys_6: list_bool] :
( hBOOL(hAPP_list_bool_bool(hAPP_l1101248310l_bool(P_1,Xs_19),Ys_6))
=> hBOOL(hAPP_list_bool_bool(hAPP_l1101248310l_bool(P_1,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_4),Xs_19)),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,Y_25),Ys_6))) )
=> hBOOL(hAPP_list_bool_bool(hAPP_l1101248310l_bool(P_1,Xs),Ys)) ) ) ) ) ).
tff(fact_273_list__induct2_H,axiom,
! [Xs: list_bool,Ys: list_A114658244le_alt,P_1: fun_li1854140997t_bool] :
( hBOOL(hAPP_l499746980t_bool(hAPP_l799327709t_bool(P_1,nil_bool),nil_Ar557011903le_alt))
=> ( ! [X_4: bool,Xs_19: list_bool] : hBOOL(hAPP_l499746980t_bool(hAPP_l799327709t_bool(P_1,hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_4),Xs_19)),nil_Ar557011903le_alt))
=> ( ! [Y_25: arrow_1893660431le_alt,Ys_6: list_A114658244le_alt] : hBOOL(hAPP_l499746980t_bool(hAPP_l799327709t_bool(P_1,nil_bool),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,Y_25),Ys_6)))
=> ( ! [X_4: bool,Xs_19: list_bool,Y_25: arrow_1893660431le_alt,Ys_6: list_A114658244le_alt] :
( hBOOL(hAPP_l499746980t_bool(hAPP_l799327709t_bool(P_1,Xs_19),Ys_6))
=> hBOOL(hAPP_l499746980t_bool(hAPP_l799327709t_bool(P_1,hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_4),Xs_19)),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,Y_25),Ys_6))) )
=> hBOOL(hAPP_l499746980t_bool(hAPP_l799327709t_bool(P_1,Xs),Ys)) ) ) ) ) ).
tff(fact_274_list__induct2_H,axiom,
! [Xs: list_bool,Ys: list_nat,P_1: fun_li513084864t_bool] :
( hBOOL(hAPP_list_nat_bool(hAPP_l909187928t_bool(P_1,nil_bool),nil_nat))
=> ( ! [X_4: bool,Xs_19: list_bool] : hBOOL(hAPP_list_nat_bool(hAPP_l909187928t_bool(P_1,hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_4),Xs_19)),nil_nat))
=> ( ! [Y_25: nat,Ys_6: list_nat] : hBOOL(hAPP_list_nat_bool(hAPP_l909187928t_bool(P_1,nil_bool),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,Y_25),Ys_6)))
=> ( ! [X_4: bool,Xs_19: list_bool,Y_25: nat,Ys_6: list_nat] :
( hBOOL(hAPP_list_nat_bool(hAPP_l909187928t_bool(P_1,Xs_19),Ys_6))
=> hBOOL(hAPP_list_nat_bool(hAPP_l909187928t_bool(P_1,hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_4),Xs_19)),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,Y_25),Ys_6))) )
=> hBOOL(hAPP_list_nat_bool(hAPP_l909187928t_bool(P_1,Xs),Ys)) ) ) ) ) ).
tff(fact_275_listrelp_ONil,axiom,
! [R_2: fun_Ar331856045t_bool] : hBOOL(hAPP_l499746980t_bool(listre483979801le_alt(R_2,nil_Ar557011903le_alt),nil_Ar557011903le_alt)) ).
tff(fact_276_listrelp_ONil,axiom,
! [R_2: fun_nat_fun_nat_bool] : hBOOL(hAPP_list_nat_bool(listrelp_nat(R_2,nil_nat),nil_nat)) ).
tff(fact_277_listrelp_OCons,axiom,
! [Xs: list_bool,Ys: list_bool,R_2: fun_bo1549164019l_bool,X_1: bool,Y_2: bool] :
( hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(R_2,X_1),Y_2))
=> ( hBOOL(hAPP_list_bool_bool(listrelp_bool(R_2,Xs),Ys))
=> hBOOL(hAPP_list_bool_bool(listrelp_bool(R_2,hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs)),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,Y_2),Ys))) ) ) ).
tff(fact_278_listrelp_OCons,axiom,
! [Xs: list_A114658244le_alt,Ys: list_A114658244le_alt,R_2: fun_Ar331856045t_bool,X_1: arrow_1893660431le_alt,Y_2: arrow_1893660431le_alt] :
( hBOOL(hAPP_A1748732503t_bool(hAPP_A2063800319t_bool(R_2,X_1),Y_2))
=> ( hBOOL(hAPP_l499746980t_bool(listre483979801le_alt(R_2,Xs),Ys))
=> hBOOL(hAPP_l499746980t_bool(listre483979801le_alt(R_2,hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),Xs)),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,Y_2),Ys))) ) ) ).
tff(fact_279_listrelp_OCons,axiom,
! [Xs: list_nat,Ys: list_nat,R_2: fun_nat_fun_nat_bool,X_1: nat,Y_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(R_2,X_1),Y_2))
=> ( hBOOL(hAPP_list_nat_bool(listrelp_nat(R_2,Xs),Ys))
=> hBOOL(hAPP_list_nat_bool(listrelp_nat(R_2,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Xs)),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,Y_2),Ys))) ) ) ).
tff(fact_280_takeWhile_Osimps_I1_J,axiom,
! [P_1: fun_Ar17477381t_bool] : takeWh967109304le_alt(P_1,nil_Ar557011903le_alt) = nil_Ar557011903le_alt ).
tff(fact_281_takeWhile_Osimps_I1_J,axiom,
! [P_1: fun_nat_bool] : takeWhile_nat(P_1,nil_nat) = nil_nat ).
tff(fact_282_replicate__eq__replicate,axiom,
! [M_1: nat,X_1: nat,N_1: nat,Y_2: nat] :
( ( hAPP_nat_list_nat(replicate_nat(M_1),X_1) = hAPP_nat_list_nat(replicate_nat(N_1),Y_2) )
<=> ( ( M_1 = N_1 )
& ( ( M_1 != zero_zero_nat )
=> ( X_1 = Y_2 ) ) ) ) ).
tff(fact_283_replicate__eq__replicate,axiom,
! [M_1: nat,X_1: arrow_1893660431le_alt,N_1: nat,Y_2: arrow_1893660431le_alt] :
( ( hAPP_A1382181054le_alt(replic782356601le_alt(M_1),X_1) = hAPP_A1382181054le_alt(replic782356601le_alt(N_1),Y_2) )
<=> ( ( M_1 = N_1 )
& ( ( M_1 != zero_zero_nat )
=> ( X_1 = Y_2 ) ) ) ) ).
tff(fact_284_distinct__takeWhile,axiom,
! [P_1: fun_Ar17477381t_bool,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_l499746980t_bool(distin1654625714le_alt,Xs))
=> hBOOL(hAPP_l499746980t_bool(distin1654625714le_alt,takeWh967109304le_alt(P_1,Xs))) ) ).
tff(fact_285_distinct__takeWhile,axiom,
! [P_1: fun_nat_bool,Xs: list_nat] :
( hBOOL(hAPP_list_nat_bool(distinct_nat,Xs))
=> hBOOL(hAPP_list_nat_bool(distinct_nat,takeWhile_nat(P_1,Xs))) ) ).
tff(fact_286_sorted__takeWhile,axiom,
! [P_1: fun_nat_bool,Xs: list_nat] :
( hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,Xs))
=> hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,takeWhile_nat(P_1,Xs))) ) ).
tff(fact_287_replicate__0,axiom,
! [X_98: arrow_1893660431le_alt] : hAPP_A1382181054le_alt(replic782356601le_alt(zero_zero_nat),X_98) = nil_Ar557011903le_alt ).
tff(fact_288_replicate__0,axiom,
! [X_98: nat] : hAPP_nat_list_nat(replicate_nat(zero_zero_nat),X_98) = nil_nat ).
tff(fact_289_empty__replicate,axiom,
! [N_1: nat,X_1: arrow_1893660431le_alt] :
( ( nil_Ar557011903le_alt = hAPP_A1382181054le_alt(replic782356601le_alt(N_1),X_1) )
<=> ( N_1 = zero_zero_nat ) ) ).
tff(fact_290_empty__replicate,axiom,
! [N_1: nat,X_1: nat] :
( ( nil_nat = hAPP_nat_list_nat(replicate_nat(N_1),X_1) )
<=> ( N_1 = zero_zero_nat ) ) ).
tff(fact_291_replicate__empty,axiom,
! [N_1: nat,X_1: arrow_1893660431le_alt] :
( ( hAPP_A1382181054le_alt(replic782356601le_alt(N_1),X_1) = nil_Ar557011903le_alt )
<=> ( N_1 = zero_zero_nat ) ) ).
tff(fact_292_replicate__empty,axiom,
! [N_1: nat,X_1: nat] :
( ( hAPP_nat_list_nat(replicate_nat(N_1),X_1) = nil_nat )
<=> ( N_1 = zero_zero_nat ) ) ).
tff(fact_293_takeWhile_Osimps_I2_J,axiom,
! [Xs: list_bool,P_1: fun_bool_bool,X_1: bool] :
( ( hBOOL(hAPP_bool_bool(P_1,X_1))
=> ( takeWhile_bool(P_1,hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs)) = hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),takeWhile_bool(P_1,Xs)) ) )
& ( ~ hBOOL(hAPP_bool_bool(P_1,X_1))
=> ( takeWhile_bool(P_1,hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs)) = nil_bool ) ) ) ).
tff(fact_294_takeWhile_Osimps_I2_J,axiom,
! [Xs: list_A114658244le_alt,P_1: fun_Ar17477381t_bool,X_1: arrow_1893660431le_alt] :
( ( hBOOL(hAPP_A1748732503t_bool(P_1,X_1))
=> ( takeWh967109304le_alt(P_1,hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),Xs)) = hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),takeWh967109304le_alt(P_1,Xs)) ) )
& ( ~ hBOOL(hAPP_A1748732503t_bool(P_1,X_1))
=> ( takeWh967109304le_alt(P_1,hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),Xs)) = nil_Ar557011903le_alt ) ) ) ).
tff(fact_295_takeWhile_Osimps_I2_J,axiom,
! [Xs: list_nat,P_1: fun_nat_bool,X_1: nat] :
( ( hBOOL(hAPP_nat_bool(P_1,X_1))
=> ( takeWhile_nat(P_1,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Xs)) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),takeWhile_nat(P_1,Xs)) ) )
& ( ~ hBOOL(hAPP_nat_bool(P_1,X_1))
=> ( takeWhile_nat(P_1,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Xs)) = nil_nat ) ) ) ).
tff(fact_296_takeWhile__tail,axiom,
! [Xs: list_bool,L_2: list_bool,P_1: fun_bool_bool,X_1: bool] :
( ~ hBOOL(hAPP_bool_bool(P_1,X_1))
=> ( takeWhile_bool(P_1,hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,Xs),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),L_2))) = takeWhile_bool(P_1,Xs) ) ) ).
tff(fact_297_takeWhile__tail,axiom,
! [Xs: list_A114658244le_alt,L_2: list_A114658244le_alt,P_1: fun_Ar17477381t_bool,X_1: arrow_1893660431le_alt] :
( ~ hBOOL(hAPP_A1748732503t_bool(P_1,X_1))
=> ( takeWh967109304le_alt(P_1,hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),L_2))) = takeWh967109304le_alt(P_1,Xs) ) ) ).
tff(fact_298_takeWhile__tail,axiom,
! [Xs: list_nat,L_2: list_nat,P_1: fun_nat_bool,X_1: nat] :
( ~ hBOOL(hAPP_nat_bool(P_1,X_1))
=> ( takeWhile_nat(P_1,hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),L_2))) = takeWhile_nat(P_1,Xs) ) ) ).
tff(fact_299_sublist__singleton,axiom,
! [X_1: bool,A: fun_nat_bool] :
( ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,zero_zero_nat),A))
=> ( sublist_bool(hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),nil_bool),A) = hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),nil_bool) ) )
& ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,zero_zero_nat),A))
=> ( sublist_bool(hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),nil_bool),A) = nil_bool ) ) ) ).
tff(fact_300_sublist__singleton,axiom,
! [X_1: arrow_1893660431le_alt,A: fun_nat_bool] :
( ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,zero_zero_nat),A))
=> ( sublis1318771396le_alt(hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),nil_Ar557011903le_alt),A) = hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),nil_Ar557011903le_alt) ) )
& ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,zero_zero_nat),A))
=> ( sublis1318771396le_alt(hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),nil_Ar557011903le_alt),A) = nil_Ar557011903le_alt ) ) ) ).
tff(fact_301_sublist__singleton,axiom,
! [X_1: nat,A: fun_nat_bool] :
( ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,zero_zero_nat),A))
=> ( sublist_nat(hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),nil_nat),A) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),nil_nat) ) )
& ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,zero_zero_nat),A))
=> ( sublist_nat(hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),nil_nat),A) = nil_nat ) ) ) ).
tff(fact_302_listrelp_Osimps,axiom,
! [R_2: fun_bo1549164019l_bool,A1: list_bool,A2: list_bool] :
( hBOOL(hAPP_list_bool_bool(listrelp_bool(R_2,A1),A2))
<=> ( ( ( A1 = nil_bool )
& ( A2 = nil_bool ) )
| ? [X_4: bool,Y_25: bool,Xs_19: list_bool,Ys_6: list_bool] :
( ( A1 = hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_4),Xs_19) )
& ( A2 = hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,Y_25),Ys_6) )
& hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(R_2,X_4),Y_25))
& hBOOL(hAPP_list_bool_bool(listrelp_bool(R_2,Xs_19),Ys_6)) ) ) ) ).
tff(fact_303_listrelp_Osimps,axiom,
! [R_2: fun_Ar331856045t_bool,A1: list_A114658244le_alt,A2: list_A114658244le_alt] :
( hBOOL(hAPP_l499746980t_bool(listre483979801le_alt(R_2,A1),A2))
<=> ( ( ( A1 = nil_Ar557011903le_alt )
& ( A2 = nil_Ar557011903le_alt ) )
| ? [X_4: arrow_1893660431le_alt,Y_25: arrow_1893660431le_alt,Xs_19: list_A114658244le_alt,Ys_6: list_A114658244le_alt] :
( ( A1 = hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_4),Xs_19) )
& ( A2 = hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,Y_25),Ys_6) )
& hBOOL(hAPP_A1748732503t_bool(hAPP_A2063800319t_bool(R_2,X_4),Y_25))
& hBOOL(hAPP_l499746980t_bool(listre483979801le_alt(R_2,Xs_19),Ys_6)) ) ) ) ).
tff(fact_304_listrelp_Osimps,axiom,
! [R_2: fun_nat_fun_nat_bool,A1: list_nat,A2: list_nat] :
( hBOOL(hAPP_list_nat_bool(listrelp_nat(R_2,A1),A2))
<=> ( ( ( A1 = nil_nat )
& ( A2 = nil_nat ) )
| ? [X_4: nat,Y_25: nat,Xs_19: list_nat,Ys_6: list_nat] :
( ( A1 = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_4),Xs_19) )
& ( A2 = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,Y_25),Ys_6) )
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(R_2,X_4),Y_25))
& hBOOL(hAPP_list_nat_bool(listrelp_nat(R_2,Xs_19),Ys_6)) ) ) ) ).
tff(fact_305_list_Osize_I1_J,axiom,
! [Fa: fun_Ar1091137528lt_nat] : list_s1668597313le_alt(Fa,nil_Ar557011903le_alt) = zero_zero_nat ).
tff(fact_306_list_Osize_I1_J,axiom,
! [Fa: fun_nat_nat] : list_size_nat(Fa,nil_nat) = zero_zero_nat ).
tff(fact_307_zero__reorient,axiom,
! [X_1: nat] :
( ( zero_zero_nat = X_1 )
<=> ( X_1 = zero_zero_nat ) ) ).
tff(fact_308_hd__conv__nth,axiom,
! [Xs: list_bool] :
( ( Xs != nil_bool )
=> ( hBOOL(hAPP_list_bool_bool(hd_bool,Xs))
<=> hBOOL(hAPP_nat_bool(nth_bool(Xs),zero_zero_nat)) ) ) ).
tff(fact_309_hd__conv__nth,axiom,
! [Xs_54: list_A114658244le_alt] :
( ( Xs_54 != nil_Ar557011903le_alt )
=> ( hAPP_l1749541000le_alt(hd_Arr1236501138le_alt,Xs_54) = hAPP_n832063962le_alt(nth_Ar1544180416le_alt(Xs_54),zero_zero_nat) ) ) ).
tff(fact_310_hd__conv__nth,axiom,
! [Xs_54: list_nat] :
( ( Xs_54 != nil_nat )
=> ( hAPP_list_nat_nat(hd_nat,Xs_54) = hAPP_nat_nat(nth_nat(Xs_54),zero_zero_nat) ) ) ).
tff(fact_311_insort__key_Osimps_I1_J,axiom,
! [F: fun_nat_nat,X_1: nat] : hAPP_l248265089st_nat(hAPP_n280362926st_nat(linord1850739451at_nat(F),X_1),nil_nat) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),nil_nat) ).
tff(fact_312_takeWhile__not__last,axiom,
! [Xs: list_A114658244le_alt] :
( ( Xs != nil_Ar557011903le_alt )
=> ( hBOOL(hAPP_l499746980t_bool(distin1654625714le_alt,Xs))
=> ( takeWh967109304le_alt(cOMBB_388242475le_alt(fNot,hAPP_A2063800319t_bool(cOMBC_1826973210t_bool(fequal1590852494le_alt),hAPP_l1749541000le_alt(last_A488133080le_alt,Xs))),Xs) = butlas1693249291le_alt(Xs) ) ) ) ).
tff(fact_313_takeWhile__not__last,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> ( hBOOL(hAPP_list_nat_bool(distinct_nat,Xs))
=> ( takeWhile_nat(cOMBB_bool_bool_nat(fNot,hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(fequal_nat),hAPP_list_nat_nat(last_nat,Xs))),Xs) = butlast_nat(Xs) ) ) ) ).
tff(fact_314_List_Oinsert__def,axiom,
! [X_1: bool,Xs: list_bool] :
( ( hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,X_1),hAPP_l96088310l_bool(set_bool,Xs)))
=> ( hAPP_l1189022293t_bool(hAPP_b994696797t_bool(insert_bool,X_1),Xs) = Xs ) )
& ( ~ hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,X_1),hAPP_l96088310l_bool(set_bool,Xs)))
=> ( hAPP_l1189022293t_bool(hAPP_b994696797t_bool(insert_bool,X_1),Xs) = hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs) ) ) ) ).
tff(fact_315_List_Oinsert__def,axiom,
! [X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt] :
( ( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
=> ( hAPP_l84111627le_alt(hAPP_A436723315le_alt(insert1391384533le_alt,X_1),Xs) = Xs ) )
& ( ~ hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
=> ( hAPP_l84111627le_alt(hAPP_A436723315le_alt(insert1391384533le_alt,X_1),Xs) = hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),Xs) ) ) ) ).
tff(fact_316_List_Oinsert__def,axiom,
! [X_1: nat,Xs: list_nat] :
( ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Xs)))
=> ( hAPP_l248265089st_nat(hAPP_n280362926st_nat(insert_nat,X_1),Xs) = Xs ) )
& ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Xs)))
=> ( hAPP_l248265089st_nat(hAPP_n280362926st_nat(insert_nat,X_1),Xs) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Xs) ) ) ) ).
tff(fact_317_distinct__insort,axiom,
! [F: fun_nat_nat,X_1: nat,Xs: list_nat] :
( hBOOL(hAPP_list_nat_bool(distinct_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(linord1850739451at_nat(F),X_1),Xs)))
<=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Xs)))
& hBOOL(hAPP_list_nat_bool(distinct_nat,Xs)) ) ) ).
tff(fact_318_insort__left__comm,axiom,
! [X_1: nat,Y_2: nat,Xs: list_nat] : hAPP_l248265089st_nat(hAPP_n280362926st_nat(linord1850739451at_nat(cOMBI_nat),X_1),hAPP_l248265089st_nat(hAPP_n280362926st_nat(linord1850739451at_nat(cOMBI_nat),Y_2),Xs)) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(linord1850739451at_nat(cOMBI_nat),Y_2),hAPP_l248265089st_nat(hAPP_n280362926st_nat(linord1850739451at_nat(cOMBI_nat),X_1),Xs)) ).
tff(fact_319_notin__set__sublistI,axiom,
! [I_11: fun_nat_bool,X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt] :
( ~ hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
=> ~ hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,sublis1318771396le_alt(Xs,I_11)))) ) ).
tff(fact_320_notin__set__sublistI,axiom,
! [I_11: fun_nat_bool,X_1: nat,Xs: list_nat] :
( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Xs)))
=> ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,sublist_nat(Xs,I_11)))) ) ).
tff(fact_321_insort__key__left__comm,axiom,
! [Xs: list_nat,F: fun_nat_nat,X_1: nat,Y_2: nat] :
( ( hAPP_nat_nat(F,X_1) != hAPP_nat_nat(F,Y_2) )
=> ( hAPP_l248265089st_nat(hAPP_n280362926st_nat(linord1850739451at_nat(F),Y_2),hAPP_l248265089st_nat(hAPP_n280362926st_nat(linord1850739451at_nat(F),X_1),Xs)) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(linord1850739451at_nat(F),X_1),hAPP_l248265089st_nat(hAPP_n280362926st_nat(linord1850739451at_nat(F),Y_2),Xs)) ) ) ).
tff(fact_322_in__set__sublistD,axiom,
! [X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt,I_11: fun_nat_bool] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,sublis1318771396le_alt(Xs,I_11))))
=> hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs))) ) ).
tff(fact_323_in__set__sublistD,axiom,
! [X_1: nat,Xs: list_nat,I_11: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,sublist_nat(Xs,I_11))))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Xs))) ) ).
tff(fact_324_sorted__insort,axiom,
! [X_1: nat,Xs: list_nat] :
( hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(linord1850739451at_nat(cOMBI_nat),X_1),Xs)))
<=> hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,Xs)) ) ).
tff(fact_325_tl__append,axiom,
! [Xs: list_A114658244le_alt,Ys: list_A114658244le_alt] : tl_Arr1883752598le_alt(hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs),Ys)) = list_c1024366775le_alt(tl_Arr1883752598le_alt(Ys),cOMBK_930998768le_alt(hAPP_l86320102le_alt(cOMBC_1833749623le_alt(append1597383892le_alt),Ys)),Xs) ).
tff(fact_326_tl__append,axiom,
! [Xs: list_nat,Ys: list_nat] : tl_nat(hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs),Ys)) = list_c554461815at_nat(tl_nat(Ys),cOMBK_1717478773at_nat(hAPP_l710375585st_nat(cOMBC_950362600st_nat(append_nat),Ys)),Xs) ).
tff(fact_327_set__ConsD,axiom,
! [Y_2: bool,X_1: bool,Xs: list_bool] :
( hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,Y_2),hAPP_l96088310l_bool(set_bool,hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs))))
=> ( ( hBOOL(Y_2)
<=> hBOOL(X_1) )
| hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,Y_2),hAPP_l96088310l_bool(set_bool,Xs))) ) ) ).
tff(fact_328_set__ConsD,axiom,
! [Y_2: arrow_1893660431le_alt,X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,Y_2),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),Xs))))
=> ( ( Y_2 = X_1 )
| hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,Y_2),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs))) ) ) ).
tff(fact_329_set__ConsD,axiom,
! [Y_2: nat,X_1: nat,Xs: list_nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Y_2),hAPP_l1821533378t_bool(set_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Xs))))
=> ( ( Y_2 = X_1 )
| hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Y_2),hAPP_l1821533378t_bool(set_nat,Xs))) ) ) ).
tff(fact_330_set__rev,axiom,
! [Xs: list_A114658244le_alt] : hAPP_l2018657868t_bool(set_Ar1995755744le_alt,rev_Ar377224735le_alt(Xs)) = hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs) ).
tff(fact_331_set__rev,axiom,
! [Xs: list_nat] : hAPP_l1821533378t_bool(set_nat,rev_nat(Xs)) = hAPP_l1821533378t_bool(set_nat,Xs) ).
tff(fact_332_takeWhile__eq__all__conv,axiom,
! [P_1: fun_Ar17477381t_bool,Xs: list_A114658244le_alt] :
( ( takeWh967109304le_alt(P_1,Xs) = Xs )
<=> ! [X_4: arrow_1893660431le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_4),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
=> hBOOL(hAPP_A1748732503t_bool(P_1,X_4)) ) ) ).
tff(fact_333_takeWhile__eq__all__conv,axiom,
! [P_1: fun_nat_bool,Xs: list_nat] :
( ( takeWhile_nat(P_1,Xs) = Xs )
<=> ! [X_4: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_4),hAPP_l1821533378t_bool(set_nat,Xs)))
=> hBOOL(hAPP_nat_bool(P_1,X_4)) ) ) ).
tff(fact_334_set__takeWhileD,axiom,
! [X_1: arrow_1893660431le_alt,P_1: fun_Ar17477381t_bool,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,takeWh967109304le_alt(P_1,Xs))))
=> ( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
& hBOOL(hAPP_A1748732503t_bool(P_1,X_1)) ) ) ).
tff(fact_335_set__takeWhileD,axiom,
! [X_1: nat,P_1: fun_nat_bool,Xs: list_nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,takeWhile_nat(P_1,Xs))))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Xs)))
& hBOOL(hAPP_nat_bool(P_1,X_1)) ) ) ).
tff(fact_336_in__set__butlastD,axiom,
! [X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,butlas1693249291le_alt(Xs))))
=> hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs))) ) ).
tff(fact_337_in__set__butlastD,axiom,
! [X_1: nat,Xs: list_nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,butlast_nat(Xs))))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Xs))) ) ).
tff(fact_338_insort__not__Nil,axiom,
! [F: fun_nat_nat,Aa: nat,Xs: list_nat] : hAPP_l248265089st_nat(hAPP_n280362926st_nat(linord1850739451at_nat(F),Aa),Xs) != nil_nat ).
tff(fact_339_set__rotate1,axiom,
! [Xs: list_A114658244le_alt] : hAPP_l2018657868t_bool(set_Ar1995755744le_alt,rotate1753650700le_alt(Xs)) = hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs) ).
tff(fact_340_set__rotate1,axiom,
! [Xs: list_nat] : hAPP_l1821533378t_bool(set_nat,rotate1_nat(Xs)) = hAPP_l1821533378t_bool(set_nat,Xs) ).
tff(fact_341_in__set__insert,axiom,
! [X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
=> ( hAPP_l84111627le_alt(hAPP_A436723315le_alt(insert1391384533le_alt,X_1),Xs) = Xs ) ) ).
tff(fact_342_in__set__insert,axiom,
! [X_1: nat,Xs: list_nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Xs)))
=> ( hAPP_l248265089st_nat(hAPP_n280362926st_nat(insert_nat,X_1),Xs) = Xs ) ) ).
tff(fact_343_member__set,axiom,
member775785940le_alt = set_Ar1995755744le_alt ).
tff(fact_344_member__set,axiom,
member_nat_1 = set_nat ).
tff(fact_345_list__ex1__iff,axiom,
! [P_1: fun_Ar17477381t_bool,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_l499746980t_bool(list_e1657546370le_alt(P_1),Xs))
<=> ? [X_4: arrow_1893660431le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_4),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
& hBOOL(hAPP_A1748732503t_bool(P_1,X_4))
& ! [Y_25: arrow_1893660431le_alt] :
( ( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,Y_25),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
& hBOOL(hAPP_A1748732503t_bool(P_1,Y_25)) )
=> ( Y_25 = X_4 ) ) ) ) ).
tff(fact_346_list__ex1__iff,axiom,
! [P_1: fun_nat_bool,Xs: list_nat] :
( hBOOL(hAPP_list_nat_bool(list_ex1_nat(P_1),Xs))
<=> ? [X_4: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_4),hAPP_l1821533378t_bool(set_nat,Xs)))
& hBOOL(hAPP_nat_bool(P_1,X_4))
& ! [Y_25: nat] :
( ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Y_25),hAPP_l1821533378t_bool(set_nat,Xs)))
& hBOOL(hAPP_nat_bool(P_1,Y_25)) )
=> ( Y_25 = X_4 ) ) ) ) ).
tff(fact_347_List_Omember__def,axiom,
! [Xs: list_A114658244le_alt,X_1: arrow_1893660431le_alt] :
( hBOOL(hAPP_A1748732503t_bool(hAPP_l2018657868t_bool(member775785940le_alt,Xs),X_1))
<=> hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs))) ) ).
tff(fact_348_List_Omember__def,axiom,
! [Xs: list_nat,X_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_l1821533378t_bool(member_nat_1,Xs),X_1))
<=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Xs))) ) ).
tff(fact_349_ListMem__iff,axiom,
! [X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_l499746980t_bool(listMe1496348587le_alt(X_1),Xs))
<=> hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs))) ) ).
tff(fact_350_ListMem__iff,axiom,
! [X_1: nat,Xs: list_nat] :
( hBOOL(hAPP_list_nat_bool(listMem_nat(X_1),Xs))
<=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Xs))) ) ).
tff(fact_351_in__set__member,axiom,
! [X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
<=> hBOOL(hAPP_A1748732503t_bool(hAPP_l2018657868t_bool(member775785940le_alt,Xs),X_1)) ) ).
tff(fact_352_in__set__member,axiom,
! [X_1: nat,Xs: list_nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Xs)))
<=> hBOOL(hAPP_nat_bool(hAPP_l1821533378t_bool(member_nat_1,Xs),X_1)) ) ).
tff(fact_353_dropWhile__neq__rev,axiom,
! [X_1: bool,Xs: list_bool] :
( hBOOL(hAPP_list_bool_bool(distinct_bool,Xs))
=> ( hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,X_1),hAPP_l96088310l_bool(set_bool,Xs)))
=> ( dropWhile_bool(cOMBB_bool_bool_bool(fNot,hAPP_b589554111l_bool(cOMBC_bool_bool_bool(fequal_bool),X_1)),rev_bool(Xs)) = hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),rev_bool(takeWhile_bool(cOMBB_bool_bool_bool(fNot,hAPP_b589554111l_bool(cOMBC_bool_bool_bool(fequal_bool),X_1)),Xs))) ) ) ) ).
tff(fact_354_dropWhile__neq__rev,axiom,
! [X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_l499746980t_bool(distin1654625714le_alt,Xs))
=> ( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
=> ( dropWh587599712le_alt(cOMBB_388242475le_alt(fNot,hAPP_A2063800319t_bool(cOMBC_1826973210t_bool(fequal1590852494le_alt),X_1)),rev_Ar377224735le_alt(Xs)) = hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),rev_Ar377224735le_alt(takeWh967109304le_alt(cOMBB_388242475le_alt(fNot,hAPP_A2063800319t_bool(cOMBC_1826973210t_bool(fequal1590852494le_alt),X_1)),Xs))) ) ) ) ).
tff(fact_355_dropWhile__neq__rev,axiom,
! [X_1: nat,Xs: list_nat] :
( hBOOL(hAPP_list_nat_bool(distinct_nat,Xs))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Xs)))
=> ( dropWhile_nat(cOMBB_bool_bool_nat(fNot,hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(fequal_nat),X_1)),rev_nat(Xs)) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),rev_nat(takeWhile_nat(cOMBB_bool_bool_nat(fNot,hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(fequal_nat),X_1)),Xs))) ) ) ) ).
tff(fact_356_takeWhile__neq__rev,axiom,
! [X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_l499746980t_bool(distin1654625714le_alt,Xs))
=> ( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
=> ( takeWh967109304le_alt(cOMBB_388242475le_alt(fNot,hAPP_A2063800319t_bool(cOMBC_1826973210t_bool(fequal1590852494le_alt),X_1)),rev_Ar377224735le_alt(Xs)) = rev_Ar377224735le_alt(tl_Arr1883752598le_alt(dropWh587599712le_alt(cOMBB_388242475le_alt(fNot,hAPP_A2063800319t_bool(cOMBC_1826973210t_bool(fequal1590852494le_alt),X_1)),Xs))) ) ) ) ).
tff(fact_357_takeWhile__neq__rev,axiom,
! [X_1: nat,Xs: list_nat] :
( hBOOL(hAPP_list_nat_bool(distinct_nat,Xs))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Xs)))
=> ( takeWhile_nat(cOMBB_bool_bool_nat(fNot,hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(fequal_nat),X_1)),rev_nat(Xs)) = rev_nat(tl_nat(dropWhile_nat(cOMBB_bool_bool_nat(fNot,hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(fequal_nat),X_1)),Xs))) ) ) ) ).
tff(fact_358_sublist__nil,axiom,
! [A: fun_nat_bool] : sublis1318771396le_alt(nil_Ar557011903le_alt,A) = nil_Ar557011903le_alt ).
tff(fact_359_sublist__nil,axiom,
! [A: fun_nat_bool] : sublist_nat(nil_nat,A) = nil_nat ).
tff(fact_360_distinct__sublistI,axiom,
! [I_11: fun_nat_bool,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_l499746980t_bool(distin1654625714le_alt,Xs))
=> hBOOL(hAPP_l499746980t_bool(distin1654625714le_alt,sublis1318771396le_alt(Xs,I_11))) ) ).
tff(fact_361_distinct__sublistI,axiom,
! [I_11: fun_nat_bool,Xs: list_nat] :
( hBOOL(hAPP_list_nat_bool(distinct_nat,Xs))
=> hBOOL(hAPP_list_nat_bool(distinct_nat,sublist_nat(Xs,I_11))) ) ).
tff(fact_362_rotate1__def,axiom,
! [Xs: list_bool] : rotate1_bool(Xs) = list_c1750880501l_bool(nil_bool,cOMBB_1663860504l_bool(cOMBC_1789016513t_bool(append_bool),hAPP_l1045755451t_bool(cOMBC_80330970t_bool(cons_bool),nil_bool)),Xs) ).
tff(fact_363_rotate1__def,axiom,
! [Xs: list_A114658244le_alt] : rotate1753650700le_alt(Xs) = list_c1024366775le_alt(nil_Ar557011903le_alt,cOMBB_1076781858le_alt(cOMBC_1833749623le_alt(append1597383892le_alt),hAPP_l1822938419le_alt(cOMBC_673369480le_alt(cons_A1647044463le_alt),nil_Ar557011903le_alt)),Xs) ).
tff(fact_364_rotate1__def,axiom,
! [Xs: list_nat] : rotate1_nat(Xs) = list_c554461815at_nat(nil_nat,cOMBB_472649634at_nat(cOMBC_950362600st_nat(append_nat),hAPP_l383341732st_nat(cOMBC_1204473209st_nat(cons_nat),nil_nat)),Xs) ).
tff(fact_365_nth__Cons__0,axiom,
! [X_1: bool,Xs: list_bool] :
( hBOOL(hAPP_nat_bool(nth_bool(hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs)),zero_zero_nat))
<=> hBOOL(X_1) ) ).
tff(fact_366_nth__Cons__0,axiom,
! [X_97: arrow_1893660431le_alt,Xs_53: list_A114658244le_alt] : hAPP_n832063962le_alt(nth_Ar1544180416le_alt(hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_97),Xs_53)),zero_zero_nat) = X_97 ).
tff(fact_367_nth__Cons__0,axiom,
! [X_97: nat,Xs_53: list_nat] : hAPP_nat_nat(nth_nat(hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_97),Xs_53)),zero_zero_nat) = X_97 ).
tff(fact_368_distinct_Osimps_I2_J,axiom,
! [X_1: bool,Xs: list_bool] :
( hBOOL(hAPP_list_bool_bool(distinct_bool,hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs)))
<=> ( ~ hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,X_1),hAPP_l96088310l_bool(set_bool,Xs)))
& hBOOL(hAPP_list_bool_bool(distinct_bool,Xs)) ) ) ).
tff(fact_369_distinct_Osimps_I2_J,axiom,
! [X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_l499746980t_bool(distin1654625714le_alt,hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),Xs)))
<=> ( ~ hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
& hBOOL(hAPP_l499746980t_bool(distin1654625714le_alt,Xs)) ) ) ).
tff(fact_370_distinct_Osimps_I2_J,axiom,
! [X_1: nat,Xs: list_nat] :
( hBOOL(hAPP_list_nat_bool(distinct_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Xs)))
<=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Xs)))
& hBOOL(hAPP_list_nat_bool(distinct_nat,Xs)) ) ) ).
tff(fact_371_in__set__replicate,axiom,
! [X_1: arrow_1893660431le_alt,N_1: nat,Y_2: arrow_1893660431le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,hAPP_A1382181054le_alt(replic782356601le_alt(N_1),Y_2))))
<=> ( ( X_1 = Y_2 )
& ( N_1 != zero_zero_nat ) ) ) ).
tff(fact_372_in__set__replicate,axiom,
! [X_1: nat,N_1: nat,Y_2: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,hAPP_nat_list_nat(replicate_nat(N_1),Y_2))))
<=> ( ( X_1 = Y_2 )
& ( N_1 != zero_zero_nat ) ) ) ).
tff(fact_373_Bex__set__replicate,axiom,
! [P_1: fun_Ar17477381t_bool,N_1: nat,Aa: arrow_1893660431le_alt] :
( ? [X_4: arrow_1893660431le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_4),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,hAPP_A1382181054le_alt(replic782356601le_alt(N_1),Aa))))
& hBOOL(hAPP_A1748732503t_bool(P_1,X_4)) )
<=> ( hBOOL(hAPP_A1748732503t_bool(P_1,Aa))
& ( N_1 != zero_zero_nat ) ) ) ).
tff(fact_374_Bex__set__replicate,axiom,
! [P_1: fun_nat_bool,N_1: nat,Aa: nat] :
( ? [X_4: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_4),hAPP_l1821533378t_bool(set_nat,hAPP_nat_list_nat(replicate_nat(N_1),Aa))))
& hBOOL(hAPP_nat_bool(P_1,X_4)) )
<=> ( hBOOL(hAPP_nat_bool(P_1,Aa))
& ( N_1 != zero_zero_nat ) ) ) ).
tff(fact_375_Ball__set__replicate,axiom,
! [P_1: fun_Ar17477381t_bool,N_1: nat,Aa: arrow_1893660431le_alt] :
( ! [X_4: arrow_1893660431le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_4),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,hAPP_A1382181054le_alt(replic782356601le_alt(N_1),Aa))))
=> hBOOL(hAPP_A1748732503t_bool(P_1,X_4)) )
<=> ( hBOOL(hAPP_A1748732503t_bool(P_1,Aa))
| ( N_1 = zero_zero_nat ) ) ) ).
tff(fact_376_Ball__set__replicate,axiom,
! [P_1: fun_nat_bool,N_1: nat,Aa: nat] :
( ! [X_4: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_4),hAPP_l1821533378t_bool(set_nat,hAPP_nat_list_nat(replicate_nat(N_1),Aa))))
=> hBOOL(hAPP_nat_bool(P_1,X_4)) )
<=> ( hBOOL(hAPP_nat_bool(P_1,Aa))
| ( N_1 = zero_zero_nat ) ) ) ).
tff(fact_377_sorted__distinct__set__unique,axiom,
! [Ys: list_nat,Xs: list_nat] :
( hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,Xs))
=> ( hBOOL(hAPP_list_nat_bool(distinct_nat,Xs))
=> ( hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,Ys))
=> ( hBOOL(hAPP_list_nat_bool(distinct_nat,Ys))
=> ( ( hAPP_l1821533378t_bool(set_nat,Xs) = hAPP_l1821533378t_bool(set_nat,Ys) )
=> ( Xs = Ys ) ) ) ) ) ) ).
tff(fact_378_takeWhile__append1,axiom,
! [Ys: list_A114658244le_alt,P_1: fun_Ar17477381t_bool,X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
=> ( ~ hBOOL(hAPP_A1748732503t_bool(P_1,X_1))
=> ( takeWh967109304le_alt(P_1,hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs),Ys)) = takeWh967109304le_alt(P_1,Xs) ) ) ) ).
tff(fact_379_takeWhile__append1,axiom,
! [Ys: list_nat,P_1: fun_nat_bool,X_1: nat,Xs: list_nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Xs)))
=> ( ~ hBOOL(hAPP_nat_bool(P_1,X_1))
=> ( takeWhile_nat(P_1,hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs),Ys)) = takeWhile_nat(P_1,Xs) ) ) ) ).
tff(fact_380_last__in__set,axiom,
! [As: list_A114658244le_alt] :
( ( As != nil_Ar557011903le_alt )
=> hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,hAPP_l1749541000le_alt(last_A488133080le_alt,As)),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,As))) ) ).
tff(fact_381_last__in__set,axiom,
! [As: list_nat] :
( ( As != nil_nat )
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_list_nat_nat(last_nat,As)),hAPP_l1821533378t_bool(set_nat,As))) ) ).
tff(fact_382_dropWhile__eq__Nil__conv,axiom,
! [P_1: fun_Ar17477381t_bool,Xs: list_A114658244le_alt] :
( ( dropWh587599712le_alt(P_1,Xs) = nil_Ar557011903le_alt )
<=> ! [X_4: arrow_1893660431le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_4),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
=> hBOOL(hAPP_A1748732503t_bool(P_1,X_4)) ) ) ).
tff(fact_383_dropWhile__eq__Nil__conv,axiom,
! [P_1: fun_nat_bool,Xs: list_nat] :
( ( dropWhile_nat(P_1,Xs) = nil_nat )
<=> ! [X_4: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_4),hAPP_l1821533378t_bool(set_nat,Xs)))
=> hBOOL(hAPP_nat_bool(P_1,X_4)) ) ) ).
tff(fact_384_in__set__butlast__appendI,axiom,
! [Ys: list_A114658244le_alt,X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt] :
( ( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,butlas1693249291le_alt(Xs))))
| hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,butlas1693249291le_alt(Ys)))) )
=> hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,butlas1693249291le_alt(hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs),Ys))))) ) ).
tff(fact_385_in__set__butlast__appendI,axiom,
! [Ys: list_nat,X_1: nat,Xs: list_nat] :
( ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,butlast_nat(Xs))))
| hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,butlast_nat(Ys)))) )
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,butlast_nat(hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs),Ys))))) ) ).
tff(fact_386_hd__in__set,axiom,
! [Xs: list_A114658244le_alt] :
( ( Xs != nil_Ar557011903le_alt )
=> hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,hAPP_l1749541000le_alt(hd_Arr1236501138le_alt,Xs)),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs))) ) ).
tff(fact_387_hd__in__set,axiom,
! [Xs: list_nat] :
( ( Xs != nil_nat )
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_list_nat_nat(hd_nat,Xs)),hAPP_l1821533378t_bool(set_nat,Xs))) ) ).
tff(fact_388_dropWhile__append1,axiom,
! [Ys: list_A114658244le_alt,P_1: fun_Ar17477381t_bool,X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
=> ( ~ hBOOL(hAPP_A1748732503t_bool(P_1,X_1))
=> ( dropWh587599712le_alt(P_1,hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs),Ys)) = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,dropWh587599712le_alt(P_1,Xs)),Ys) ) ) ) ).
tff(fact_389_dropWhile__append1,axiom,
! [Ys: list_nat,P_1: fun_nat_bool,X_1: nat,Xs: list_nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Xs)))
=> ( ~ hBOOL(hAPP_nat_bool(P_1,X_1))
=> ( dropWhile_nat(P_1,hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs),Ys)) = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,dropWhile_nat(P_1,Xs)),Ys) ) ) ) ).
tff(fact_390_not__in__set__insert,axiom,
! [X_1: bool,Xs: list_bool] :
( ~ hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,X_1),hAPP_l96088310l_bool(set_bool,Xs)))
=> ( hAPP_l1189022293t_bool(hAPP_b994696797t_bool(insert_bool,X_1),Xs) = hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs) ) ) ).
tff(fact_391_not__in__set__insert,axiom,
! [X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt] :
( ~ hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
=> ( hAPP_l84111627le_alt(hAPP_A436723315le_alt(insert1391384533le_alt,X_1),Xs) = hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),Xs) ) ) ).
tff(fact_392_not__in__set__insert,axiom,
! [X_1: nat,Xs: list_nat] :
( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Xs)))
=> ( hAPP_l248265089st_nat(hAPP_n280362926st_nat(insert_nat,X_1),Xs) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Xs) ) ) ).
tff(fact_393_dropWhile__append2,axiom,
! [Ys: list_A114658244le_alt,P_1: fun_Ar17477381t_bool,Xs: list_A114658244le_alt] :
( ! [X_4: arrow_1893660431le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_4),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
=> hBOOL(hAPP_A1748732503t_bool(P_1,X_4)) )
=> ( dropWh587599712le_alt(P_1,hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs),Ys)) = dropWh587599712le_alt(P_1,Ys) ) ) ).
tff(fact_394_dropWhile__append2,axiom,
! [Ys: list_nat,P_1: fun_nat_bool,Xs: list_nat] :
( ! [X_4: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_4),hAPP_l1821533378t_bool(set_nat,Xs)))
=> hBOOL(hAPP_nat_bool(P_1,X_4)) )
=> ( dropWhile_nat(P_1,hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs),Ys)) = dropWhile_nat(P_1,Ys) ) ) ).
tff(fact_395_takeWhile__append2,axiom,
! [Ys: list_A114658244le_alt,P_1: fun_Ar17477381t_bool,Xs: list_A114658244le_alt] :
( ! [X_4: arrow_1893660431le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_4),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
=> hBOOL(hAPP_A1748732503t_bool(P_1,X_4)) )
=> ( takeWh967109304le_alt(P_1,hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs),Ys)) = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs),takeWh967109304le_alt(P_1,Ys)) ) ) ).
tff(fact_396_takeWhile__append2,axiom,
! [Ys: list_nat,P_1: fun_nat_bool,Xs: list_nat] :
( ! [X_4: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_4),hAPP_l1821533378t_bool(set_nat,Xs)))
=> hBOOL(hAPP_nat_bool(P_1,X_4)) )
=> ( takeWhile_nat(P_1,hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs),Ys)) = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs),takeWhile_nat(P_1,Ys)) ) ) ).
tff(fact_397_in__set__conv__decomp,axiom,
! [X_1: bool,Xs: list_bool] :
( hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,X_1),hAPP_l96088310l_bool(set_bool,Xs)))
<=> ? [Ys_6: list_bool,Zs_2: list_bool] : Xs = hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,Ys_6),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Zs_2)) ) ).
tff(fact_398_in__set__conv__decomp,axiom,
! [X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
<=> ? [Ys_6: list_A114658244le_alt,Zs_2: list_A114658244le_alt] : Xs = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Ys_6),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),Zs_2)) ) ).
tff(fact_399_in__set__conv__decomp,axiom,
! [X_1: nat,Xs: list_nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Xs)))
<=> ? [Ys_6: list_nat,Zs_2: list_nat] : Xs = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Ys_6),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Zs_2)) ) ).
tff(fact_400_in__set__conv__decomp__last,axiom,
! [X_1: bool,Xs: list_bool] :
( hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,X_1),hAPP_l96088310l_bool(set_bool,Xs)))
<=> ? [Ys_6: list_bool,Zs_2: list_bool] :
( ( Xs = hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,Ys_6),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Zs_2)) )
& ~ hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,X_1),hAPP_l96088310l_bool(set_bool,Zs_2))) ) ) ).
tff(fact_401_in__set__conv__decomp__last,axiom,
! [X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
<=> ? [Ys_6: list_A114658244le_alt,Zs_2: list_A114658244le_alt] :
( ( Xs = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Ys_6),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),Zs_2)) )
& ~ hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Zs_2))) ) ) ).
tff(fact_402_in__set__conv__decomp__last,axiom,
! [X_1: nat,Xs: list_nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Xs)))
<=> ? [Ys_6: list_nat,Zs_2: list_nat] :
( ( Xs = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Ys_6),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Zs_2)) )
& ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Zs_2))) ) ) ).
tff(fact_403_in__set__conv__decomp__first,axiom,
! [X_1: bool,Xs: list_bool] :
( hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,X_1),hAPP_l96088310l_bool(set_bool,Xs)))
<=> ? [Ys_6: list_bool,Zs_2: list_bool] :
( ( Xs = hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,Ys_6),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Zs_2)) )
& ~ hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,X_1),hAPP_l96088310l_bool(set_bool,Ys_6))) ) ) ).
tff(fact_404_in__set__conv__decomp__first,axiom,
! [X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
<=> ? [Ys_6: list_A114658244le_alt,Zs_2: list_A114658244le_alt] :
( ( Xs = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Ys_6),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),Zs_2)) )
& ~ hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Ys_6))) ) ) ).
tff(fact_405_in__set__conv__decomp__first,axiom,
! [X_1: nat,Xs: list_nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Xs)))
<=> ? [Ys_6: list_nat,Zs_2: list_nat] :
( ( Xs = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Ys_6),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Zs_2)) )
& ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Ys_6))) ) ) ).
tff(fact_406_split__list__propE,axiom,
! [P_1: fun_bool_bool,Xs: list_bool] :
( ? [X_4: bool] :
( hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,X_4),hAPP_l96088310l_bool(set_bool,Xs)))
& hBOOL(hAPP_bool_bool(P_1,X_4)) )
=> ~ ! [Ys_6: list_bool,X_4: bool] :
( ? [Zs_2: list_bool] : Xs = hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,Ys_6),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_4),Zs_2))
=> ~ hBOOL(hAPP_bool_bool(P_1,X_4)) ) ) ).
tff(fact_407_split__list__propE,axiom,
! [P_1: fun_Ar17477381t_bool,Xs: list_A114658244le_alt] :
( ? [X_4: arrow_1893660431le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_4),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
& hBOOL(hAPP_A1748732503t_bool(P_1,X_4)) )
=> ~ ! [Ys_6: list_A114658244le_alt,X_4: arrow_1893660431le_alt] :
( ? [Zs_2: list_A114658244le_alt] : Xs = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Ys_6),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_4),Zs_2))
=> ~ hBOOL(hAPP_A1748732503t_bool(P_1,X_4)) ) ) ).
tff(fact_408_split__list__propE,axiom,
! [P_1: fun_nat_bool,Xs: list_nat] :
( ? [X_4: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_4),hAPP_l1821533378t_bool(set_nat,Xs)))
& hBOOL(hAPP_nat_bool(P_1,X_4)) )
=> ~ ! [Ys_6: list_nat,X_4: nat] :
( ? [Zs_2: list_nat] : Xs = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Ys_6),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_4),Zs_2))
=> ~ hBOOL(hAPP_nat_bool(P_1,X_4)) ) ) ).
tff(fact_409_dropWhile__cong,axiom,
! [Q_1: fun_Ar17477381t_bool,P_1: fun_Ar17477381t_bool,L_2: list_A114658244le_alt,K_3: list_A114658244le_alt] :
( ( L_2 = K_3 )
=> ( ! [X_4: arrow_1893660431le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_4),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,L_2)))
=> ( hBOOL(hAPP_A1748732503t_bool(P_1,X_4))
<=> hBOOL(hAPP_A1748732503t_bool(Q_1,X_4)) ) )
=> ( dropWh587599712le_alt(P_1,L_2) = dropWh587599712le_alt(Q_1,K_3) ) ) ) ).
tff(fact_410_dropWhile__cong,axiom,
! [Q_1: fun_nat_bool,P_1: fun_nat_bool,L_2: list_nat,K_3: list_nat] :
( ( L_2 = K_3 )
=> ( ! [X_4: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_4),hAPP_l1821533378t_bool(set_nat,L_2)))
=> ( hBOOL(hAPP_nat_bool(P_1,X_4))
<=> hBOOL(hAPP_nat_bool(Q_1,X_4)) ) )
=> ( dropWhile_nat(P_1,L_2) = dropWhile_nat(Q_1,K_3) ) ) ) ).
tff(fact_411_takeWhile__cong,axiom,
! [Q_1: fun_Ar17477381t_bool,P_1: fun_Ar17477381t_bool,L_2: list_A114658244le_alt,K_3: list_A114658244le_alt] :
( ( L_2 = K_3 )
=> ( ! [X_4: arrow_1893660431le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_4),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,L_2)))
=> ( hBOOL(hAPP_A1748732503t_bool(P_1,X_4))
<=> hBOOL(hAPP_A1748732503t_bool(Q_1,X_4)) ) )
=> ( takeWh967109304le_alt(P_1,L_2) = takeWh967109304le_alt(Q_1,K_3) ) ) ) ).
tff(fact_412_takeWhile__cong,axiom,
! [Q_1: fun_nat_bool,P_1: fun_nat_bool,L_2: list_nat,K_3: list_nat] :
( ( L_2 = K_3 )
=> ( ! [X_4: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_4),hAPP_l1821533378t_bool(set_nat,L_2)))
=> ( hBOOL(hAPP_nat_bool(P_1,X_4))
<=> hBOOL(hAPP_nat_bool(Q_1,X_4)) ) )
=> ( takeWhile_nat(P_1,L_2) = takeWhile_nat(Q_1,K_3) ) ) ) ).
tff(fact_413_map__filter__simps_I1_J,axiom,
! [F: fun_Ar1450809516n_bool,X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt] : map_fi966861858t_bool(F,hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),Xs)) = option718651152l_bool(map_fi966861858t_bool(F,Xs),hAPP_l1045755451t_bool(cOMBC_80330970t_bool(cons_bool),map_fi966861858t_bool(F,Xs)),hAPP_A1193192004n_bool(F,X_1)) ).
tff(fact_414_map__filter__simps_I1_J,axiom,
! [F: fun_Ar63580199on_nat,X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt] : map_fi1320516869lt_nat(F,hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),Xs)) = option306705938at_nat(map_fi1320516869lt_nat(F,Xs),hAPP_l383341732st_nat(cOMBC_1204473209st_nat(cons_nat),map_fi1320516869lt_nat(F,Xs)),hAPP_A1602729209on_nat(F,X_1)) ).
tff(fact_415_map__filter__simps_I1_J,axiom,
! [F: fun_Ar294405804le_alt,X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt] : map_fi886559370le_alt(F,hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),Xs)) = option1112736594le_alt(map_fi886559370le_alt(F,Xs),hAPP_l1822938419le_alt(cOMBC_673369480le_alt(cons_A1647044463le_alt),map_fi886559370le_alt(F,Xs)),hAPP_A1567920766le_alt(F,X_1)) ).
tff(fact_416_map__filter__simps_I1_J,axiom,
! [F: fun_nat_option_bool,X_1: nat,Xs: list_nat] : map_filter_nat_bool(F,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Xs)) = option718651152l_bool(map_filter_nat_bool(F,Xs),hAPP_l1045755451t_bool(cOMBC_80330970t_bool(cons_bool),map_filter_nat_bool(F,Xs)),hAPP_nat_option_bool(F,X_1)) ).
tff(fact_417_map__filter__simps_I1_J,axiom,
! [F: fun_nat_option_nat,X_1: nat,Xs: list_nat] : map_filter_nat_nat(F,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Xs)) = option306705938at_nat(map_filter_nat_nat(F,Xs),hAPP_l383341732st_nat(cOMBC_1204473209st_nat(cons_nat),map_filter_nat_nat(F,Xs)),hAPP_nat_option_nat(F,X_1)) ).
tff(fact_418_map__filter__simps_I1_J,axiom,
! [F: fun_na1650596135le_alt,X_1: nat,Xs: list_nat] : map_fi733515087le_alt(F,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Xs)) = option1112736594le_alt(map_fi733515087le_alt(F,Xs),hAPP_l1822938419le_alt(cOMBC_673369480le_alt(cons_A1647044463le_alt),map_fi733515087le_alt(F,Xs)),hAPP_n4766713le_alt(F,X_1)) ).
tff(fact_419_map__filter__simps_I1_J,axiom,
! [F: fun_bo646066118le_alt,X_1: bool,Xs: list_bool] : map_fi280134778le_alt(F,hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs)) = option1112736594le_alt(map_fi280134778le_alt(F,Xs),hAPP_l1822938419le_alt(cOMBC_673369480le_alt(cons_A1647044463le_alt),map_fi280134778le_alt(F,Xs)),hAPP_b935842788le_alt(F,X_1)) ).
tff(fact_420_map__filter__simps_I1_J,axiom,
! [F: fun_bool_option_nat,X_1: bool,Xs: list_bool] : map_filter_bool_nat(F,hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs)) = option306705938at_nat(map_filter_bool_nat(F,Xs),hAPP_l383341732st_nat(cOMBC_1204473209st_nat(cons_nat),map_filter_bool_nat(F,Xs)),hAPP_bool_option_nat(F,X_1)) ).
tff(fact_421_split__list__first__propE,axiom,
! [P_1: fun_bool_bool,Xs: list_bool] :
( ? [X_4: bool] :
( hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,X_4),hAPP_l96088310l_bool(set_bool,Xs)))
& hBOOL(hAPP_bool_bool(P_1,X_4)) )
=> ~ ! [Ys_6: list_bool,X_4: bool] :
( ? [Zs_2: list_bool] : Xs = hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,Ys_6),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_4),Zs_2))
=> ( hBOOL(hAPP_bool_bool(P_1,X_4))
=> ~ ! [Xa: bool] :
( hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,Xa),hAPP_l96088310l_bool(set_bool,Ys_6)))
=> ~ hBOOL(hAPP_bool_bool(P_1,Xa)) ) ) ) ) ).
tff(fact_422_split__list__first__propE,axiom,
! [P_1: fun_Ar17477381t_bool,Xs: list_A114658244le_alt] :
( ? [X_4: arrow_1893660431le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_4),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
& hBOOL(hAPP_A1748732503t_bool(P_1,X_4)) )
=> ~ ! [Ys_6: list_A114658244le_alt,X_4: arrow_1893660431le_alt] :
( ? [Zs_2: list_A114658244le_alt] : Xs = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Ys_6),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_4),Zs_2))
=> ( hBOOL(hAPP_A1748732503t_bool(P_1,X_4))
=> ~ ! [Xa: arrow_1893660431le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,Xa),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Ys_6)))
=> ~ hBOOL(hAPP_A1748732503t_bool(P_1,Xa)) ) ) ) ) ).
tff(fact_423_split__list__first__propE,axiom,
! [P_1: fun_nat_bool,Xs: list_nat] :
( ? [X_4: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_4),hAPP_l1821533378t_bool(set_nat,Xs)))
& hBOOL(hAPP_nat_bool(P_1,X_4)) )
=> ~ ! [Ys_6: list_nat,X_4: nat] :
( ? [Zs_2: list_nat] : Xs = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Ys_6),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_4),Zs_2))
=> ( hBOOL(hAPP_nat_bool(P_1,X_4))
=> ~ ! [Xa: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Xa),hAPP_l1821533378t_bool(set_nat,Ys_6)))
=> ~ hBOOL(hAPP_nat_bool(P_1,Xa)) ) ) ) ) ).
tff(fact_424_split__list__last__propE,axiom,
! [P_1: fun_bool_bool,Xs: list_bool] :
( ? [X_4: bool] :
( hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,X_4),hAPP_l96088310l_bool(set_bool,Xs)))
& hBOOL(hAPP_bool_bool(P_1,X_4)) )
=> ~ ! [Ys_6: list_bool,X_4: bool,Zs_2: list_bool] :
( ( Xs = hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,Ys_6),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_4),Zs_2)) )
=> ( hBOOL(hAPP_bool_bool(P_1,X_4))
=> ~ ! [Xa: bool] :
( hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,Xa),hAPP_l96088310l_bool(set_bool,Zs_2)))
=> ~ hBOOL(hAPP_bool_bool(P_1,Xa)) ) ) ) ) ).
tff(fact_425_split__list__last__propE,axiom,
! [P_1: fun_Ar17477381t_bool,Xs: list_A114658244le_alt] :
( ? [X_4: arrow_1893660431le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_4),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
& hBOOL(hAPP_A1748732503t_bool(P_1,X_4)) )
=> ~ ! [Ys_6: list_A114658244le_alt,X_4: arrow_1893660431le_alt,Zs_2: list_A114658244le_alt] :
( ( Xs = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Ys_6),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_4),Zs_2)) )
=> ( hBOOL(hAPP_A1748732503t_bool(P_1,X_4))
=> ~ ! [Xa: arrow_1893660431le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,Xa),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Zs_2)))
=> ~ hBOOL(hAPP_A1748732503t_bool(P_1,Xa)) ) ) ) ) ).
tff(fact_426_split__list__last__propE,axiom,
! [P_1: fun_nat_bool,Xs: list_nat] :
( ? [X_4: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_4),hAPP_l1821533378t_bool(set_nat,Xs)))
& hBOOL(hAPP_nat_bool(P_1,X_4)) )
=> ~ ! [Ys_6: list_nat,X_4: nat,Zs_2: list_nat] :
( ( Xs = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Ys_6),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_4),Zs_2)) )
=> ( hBOOL(hAPP_nat_bool(P_1,X_4))
=> ~ ! [Xa: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Xa),hAPP_l1821533378t_bool(set_nat,Zs_2)))
=> ~ hBOOL(hAPP_nat_bool(P_1,Xa)) ) ) ) ) ).
tff(fact_427_split__list__first,axiom,
! [X_1: bool,Xs: list_bool] :
( hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,X_1),hAPP_l96088310l_bool(set_bool,Xs)))
=> ? [Ys_6: list_bool,Zs_2: list_bool] :
( ( Xs = hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,Ys_6),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Zs_2)) )
& ~ hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,X_1),hAPP_l96088310l_bool(set_bool,Ys_6))) ) ) ).
tff(fact_428_split__list__first,axiom,
! [X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
=> ? [Ys_6: list_A114658244le_alt,Zs_2: list_A114658244le_alt] :
( ( Xs = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Ys_6),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),Zs_2)) )
& ~ hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Ys_6))) ) ) ).
tff(fact_429_split__list__first,axiom,
! [X_1: nat,Xs: list_nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Xs)))
=> ? [Ys_6: list_nat,Zs_2: list_nat] :
( ( Xs = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Ys_6),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Zs_2)) )
& ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Ys_6))) ) ) ).
tff(fact_430_split__list__last,axiom,
! [X_1: bool,Xs: list_bool] :
( hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,X_1),hAPP_l96088310l_bool(set_bool,Xs)))
=> ? [Ys_6: list_bool,Zs_2: list_bool] :
( ( Xs = hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,Ys_6),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Zs_2)) )
& ~ hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,X_1),hAPP_l96088310l_bool(set_bool,Zs_2))) ) ) ).
tff(fact_431_split__list__last,axiom,
! [X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
=> ? [Ys_6: list_A114658244le_alt,Zs_2: list_A114658244le_alt] :
( ( Xs = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Ys_6),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),Zs_2)) )
& ~ hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Zs_2))) ) ) ).
tff(fact_432_split__list__last,axiom,
! [X_1: nat,Xs: list_nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Xs)))
=> ? [Ys_6: list_nat,Zs_2: list_nat] :
( ( Xs = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Ys_6),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Zs_2)) )
& ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Zs_2))) ) ) ).
tff(fact_433_split__list,axiom,
! [X_1: bool,Xs: list_bool] :
( hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,X_1),hAPP_l96088310l_bool(set_bool,Xs)))
=> ? [Ys_6: list_bool,Zs_2: list_bool] : Xs = hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,Ys_6),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Zs_2)) ) ).
tff(fact_434_split__list,axiom,
! [X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
=> ? [Ys_6: list_A114658244le_alt,Zs_2: list_A114658244le_alt] : Xs = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Ys_6),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),Zs_2)) ) ).
tff(fact_435_split__list,axiom,
! [X_1: nat,Xs: list_nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Xs)))
=> ? [Ys_6: list_nat,Zs_2: list_nat] : Xs = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Ys_6),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Zs_2)) ) ).
tff(fact_436_split__list__first__prop__iff,axiom,
! [P_1: fun_bool_bool,Xs: list_bool] :
( ? [X_4: bool] :
( hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,X_4),hAPP_l96088310l_bool(set_bool,Xs)))
& hBOOL(hAPP_bool_bool(P_1,X_4)) )
<=> ? [Ys_6: list_bool,X_4: bool] :
( ? [Zs_2: list_bool] : Xs = hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,Ys_6),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_4),Zs_2))
& hBOOL(hAPP_bool_bool(P_1,X_4))
& ! [Xa: bool] :
( hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,Xa),hAPP_l96088310l_bool(set_bool,Ys_6)))
=> ~ hBOOL(hAPP_bool_bool(P_1,Xa)) ) ) ) ).
tff(fact_437_split__list__first__prop__iff,axiom,
! [P_1: fun_Ar17477381t_bool,Xs: list_A114658244le_alt] :
( ? [X_4: arrow_1893660431le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_4),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
& hBOOL(hAPP_A1748732503t_bool(P_1,X_4)) )
<=> ? [Ys_6: list_A114658244le_alt,X_4: arrow_1893660431le_alt] :
( ? [Zs_2: list_A114658244le_alt] : Xs = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Ys_6),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_4),Zs_2))
& hBOOL(hAPP_A1748732503t_bool(P_1,X_4))
& ! [Xa: arrow_1893660431le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,Xa),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Ys_6)))
=> ~ hBOOL(hAPP_A1748732503t_bool(P_1,Xa)) ) ) ) ).
tff(fact_438_split__list__first__prop__iff,axiom,
! [P_1: fun_nat_bool,Xs: list_nat] :
( ? [X_4: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_4),hAPP_l1821533378t_bool(set_nat,Xs)))
& hBOOL(hAPP_nat_bool(P_1,X_4)) )
<=> ? [Ys_6: list_nat,X_4: nat] :
( ? [Zs_2: list_nat] : Xs = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Ys_6),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_4),Zs_2))
& hBOOL(hAPP_nat_bool(P_1,X_4))
& ! [Xa: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Xa),hAPP_l1821533378t_bool(set_nat,Ys_6)))
=> ~ hBOOL(hAPP_nat_bool(P_1,Xa)) ) ) ) ).
tff(fact_439_split__list__last__prop__iff,axiom,
! [P_1: fun_bool_bool,Xs: list_bool] :
( ? [X_4: bool] :
( hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,X_4),hAPP_l96088310l_bool(set_bool,Xs)))
& hBOOL(hAPP_bool_bool(P_1,X_4)) )
<=> ? [Ys_6: list_bool,X_4: bool,Zs_2: list_bool] :
( ( Xs = hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,Ys_6),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_4),Zs_2)) )
& hBOOL(hAPP_bool_bool(P_1,X_4))
& ! [Xa: bool] :
( hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,Xa),hAPP_l96088310l_bool(set_bool,Zs_2)))
=> ~ hBOOL(hAPP_bool_bool(P_1,Xa)) ) ) ) ).
tff(fact_440_split__list__last__prop__iff,axiom,
! [P_1: fun_Ar17477381t_bool,Xs: list_A114658244le_alt] :
( ? [X_4: arrow_1893660431le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_4),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
& hBOOL(hAPP_A1748732503t_bool(P_1,X_4)) )
<=> ? [Ys_6: list_A114658244le_alt,X_4: arrow_1893660431le_alt,Zs_2: list_A114658244le_alt] :
( ( Xs = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Ys_6),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_4),Zs_2)) )
& hBOOL(hAPP_A1748732503t_bool(P_1,X_4))
& ! [Xa: arrow_1893660431le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,Xa),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Zs_2)))
=> ~ hBOOL(hAPP_A1748732503t_bool(P_1,Xa)) ) ) ) ).
tff(fact_441_split__list__last__prop__iff,axiom,
! [P_1: fun_nat_bool,Xs: list_nat] :
( ? [X_4: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_4),hAPP_l1821533378t_bool(set_nat,Xs)))
& hBOOL(hAPP_nat_bool(P_1,X_4)) )
<=> ? [Ys_6: list_nat,X_4: nat,Zs_2: list_nat] :
( ( Xs = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Ys_6),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_4),Zs_2)) )
& hBOOL(hAPP_nat_bool(P_1,X_4))
& ! [Xa: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Xa),hAPP_l1821533378t_bool(set_nat,Zs_2)))
=> ~ hBOOL(hAPP_nat_bool(P_1,Xa)) ) ) ) ).
tff(fact_442_sublist__Cons,axiom,
! [X_1: bool,L_2: list_bool,A: fun_nat_bool] : sublist_bool(hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),L_2),A) = hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,if_list_bool(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,zero_zero_nat),A),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),nil_bool),nil_bool)),sublist_bool(L_2,collect_nat(hAPP_f800510211t_bool(cOMBC_226598744l_bool(cOMBB_1146692694ol_nat(member_nat,suc)),A)))) ).
tff(fact_443_sublist__Cons,axiom,
! [X_1: arrow_1893660431le_alt,L_2: list_A114658244le_alt,A: fun_nat_bool] : sublis1318771396le_alt(hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),L_2),A) = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,if_lis1472843348le_alt(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,zero_zero_nat),A),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),nil_Ar557011903le_alt),nil_Ar557011903le_alt)),sublis1318771396le_alt(L_2,collect_nat(hAPP_f800510211t_bool(cOMBC_226598744l_bool(cOMBB_1146692694ol_nat(member_nat,suc)),A)))) ).
tff(fact_444_sublist__Cons,axiom,
! [X_1: nat,L_2: list_nat,A: fun_nat_bool] : sublist_nat(hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),L_2),A) = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,if_list_nat(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,zero_zero_nat),A),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),nil_nat),nil_nat)),sublist_nat(L_2,collect_nat(hAPP_f800510211t_bool(cOMBC_226598744l_bool(cOMBB_1146692694ol_nat(member_nat,suc)),A)))) ).
tff(fact_445_nth__Cons__Suc,axiom,
! [X_1: bool,Xs: list_bool,N_1: nat] :
( hBOOL(hAPP_nat_bool(nth_bool(hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs)),hAPP_nat_nat(suc,N_1)))
<=> hBOOL(hAPP_nat_bool(nth_bool(Xs),N_1)) ) ).
tff(fact_446_nth__Cons__Suc,axiom,
! [X_96: arrow_1893660431le_alt,Xs_52: list_A114658244le_alt,N_39: nat] : hAPP_n832063962le_alt(nth_Ar1544180416le_alt(hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_96),Xs_52)),hAPP_nat_nat(suc,N_39)) = hAPP_n832063962le_alt(nth_Ar1544180416le_alt(Xs_52),N_39) ).
tff(fact_447_nth__Cons__Suc,axiom,
! [X_96: nat,Xs_52: list_nat,N_39: nat] : hAPP_nat_nat(nth_nat(hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_96),Xs_52)),hAPP_nat_nat(suc,N_39)) = hAPP_nat_nat(nth_nat(Xs_52),N_39) ).
tff(fact_448_replicate__Suc,axiom,
! [N_1: nat,X_1: bool] : hAPP_bool_list_bool(replicate_bool(hAPP_nat_nat(suc,N_1)),X_1) = hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),hAPP_bool_list_bool(replicate_bool(N_1),X_1)) ).
tff(fact_449_replicate__Suc,axiom,
! [N_38: nat,X_95: arrow_1893660431le_alt] : hAPP_A1382181054le_alt(replic782356601le_alt(hAPP_nat_nat(suc,N_38)),X_95) = hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_95),hAPP_A1382181054le_alt(replic782356601le_alt(N_38),X_95)) ).
tff(fact_450_replicate__Suc,axiom,
! [N_38: nat,X_95: nat] : hAPP_nat_list_nat(replicate_nat(hAPP_nat_nat(suc,N_38)),X_95) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_95),hAPP_nat_list_nat(replicate_nat(N_38),X_95)) ).
tff(fact_451_Suc__neq__Zero,axiom,
! [M: nat] : hAPP_nat_nat(suc,M) != zero_zero_nat ).
tff(fact_452_Zero__neq__Suc,axiom,
! [M: nat] : zero_zero_nat != hAPP_nat_nat(suc,M) ).
tff(fact_453_nat_Osimps_I3_J,axiom,
! [Nat_5: nat] : hAPP_nat_nat(suc,Nat_5) != zero_zero_nat ).
tff(fact_454_Suc__not__Zero,axiom,
! [M: nat] : hAPP_nat_nat(suc,M) != zero_zero_nat ).
tff(fact_455_nat_Osimps_I2_J,axiom,
! [Nat_4: nat] : zero_zero_nat != hAPP_nat_nat(suc,Nat_4) ).
tff(fact_456_Zero__not__Suc,axiom,
! [M: nat] : zero_zero_nat != hAPP_nat_nat(suc,M) ).
tff(fact_457_nth_Osimps,axiom,
! [X_1: list_nat,Xs: list_list_nat,N_1: nat] : hAPP_nat_list_nat(nth_list_nat(hAPP_l1734754389st_nat(hAPP_l1872445307st_nat(cons_list_nat,X_1),Xs)),N_1) = hAPP_nat_list_nat(nat_case_list_nat(X_1,nth_list_nat(Xs)),N_1) ).
tff(fact_458_nth_Osimps,axiom,
! [X_1: list_A114658244le_alt,Xs: list_l2028444773le_alt,N_1: nat] : hAPP_n2007559481le_alt(nth_li103876819le_alt(hAPP_l1913469717le_alt(hAPP_l1201992822le_alt(cons_l782000996le_alt,X_1),Xs)),N_1) = hAPP_n2007559481le_alt(nat_ca146784400le_alt(X_1,nth_li103876819le_alt(Xs)),N_1) ).
tff(fact_459_nth_Osimps,axiom,
! [X_1: bool,Xs: list_bool,N_1: nat] :
( hBOOL(hAPP_nat_bool(nth_bool(hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs)),N_1))
<=> hBOOL(hAPP_nat_bool(nat_case_bool(X_1,nth_bool(Xs)),N_1)) ) ).
tff(fact_460_nth_Osimps,axiom,
! [X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt,N_1: nat] : hAPP_n832063962le_alt(nth_Ar1544180416le_alt(hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),Xs)),N_1) = hAPP_n832063962le_alt(nat_ca822049091le_alt(X_1,nth_Ar1544180416le_alt(Xs)),N_1) ).
tff(fact_461_nth_Osimps,axiom,
! [X_1: nat,Xs: list_nat,N_1: nat] : hAPP_nat_nat(nth_nat(hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Xs)),N_1) = hAPP_nat_nat(nat_case_nat(X_1,nth_nat(Xs)),N_1) ).
tff(fact_462_nat__case__0,axiom,
! [F1: list_nat,F2: fun_nat_list_nat] : hAPP_nat_list_nat(nat_case_list_nat(F1,F2),zero_zero_nat) = F1 ).
tff(fact_463_nat__case__0,axiom,
! [F1: list_A114658244le_alt,F2: fun_na86871399le_alt] : hAPP_n2007559481le_alt(nat_ca146784400le_alt(F1,F2),zero_zero_nat) = F1 ).
tff(fact_464_nat__case__0,axiom,
! [F1: arrow_1893660431le_alt,F2: fun_na665391426le_alt] : hAPP_n832063962le_alt(nat_ca822049091le_alt(F1,F2),zero_zero_nat) = F1 ).
tff(fact_465_nat__case__0,axiom,
! [F1: nat,F2: fun_nat_nat] : hAPP_nat_nat(nat_case_nat(F1,F2),zero_zero_nat) = F1 ).
tff(fact_466_nat__case__0,axiom,
! [F1: bool,F2: fun_nat_bool] :
( hBOOL(hAPP_nat_bool(nat_case_bool(F1,F2),zero_zero_nat))
<=> hBOOL(F1) ) ).
tff(fact_467_nat__case__Suc,axiom,
! [F1: list_nat,F2: fun_nat_list_nat,Nat_3: nat] : hAPP_nat_list_nat(nat_case_list_nat(F1,F2),hAPP_nat_nat(suc,Nat_3)) = hAPP_nat_list_nat(F2,Nat_3) ).
tff(fact_468_nat__case__Suc,axiom,
! [F1: list_A114658244le_alt,F2: fun_na86871399le_alt,Nat_3: nat] : hAPP_n2007559481le_alt(nat_ca146784400le_alt(F1,F2),hAPP_nat_nat(suc,Nat_3)) = hAPP_n2007559481le_alt(F2,Nat_3) ).
tff(fact_469_nat__case__Suc,axiom,
! [F1: arrow_1893660431le_alt,F2: fun_na665391426le_alt,Nat_3: nat] : hAPP_n832063962le_alt(nat_ca822049091le_alt(F1,F2),hAPP_nat_nat(suc,Nat_3)) = hAPP_n832063962le_alt(F2,Nat_3) ).
tff(fact_470_nat__case__Suc,axiom,
! [F1: nat,F2: fun_nat_nat,Nat_3: nat] : hAPP_nat_nat(nat_case_nat(F1,F2),hAPP_nat_nat(suc,Nat_3)) = hAPP_nat_nat(F2,Nat_3) ).
tff(fact_471_nat__case__Suc,axiom,
! [F1: bool,F2: fun_nat_bool,Nat_3: nat] :
( hBOOL(hAPP_nat_bool(nat_case_bool(F1,F2),hAPP_nat_nat(suc,Nat_3)))
<=> hBOOL(hAPP_nat_bool(F2,Nat_3)) ) ).
tff(fact_472_n__not__Suc__n,axiom,
! [N: nat] : N != hAPP_nat_nat(suc,N) ).
tff(fact_473_Suc__n__not__n,axiom,
! [N: nat] : hAPP_nat_nat(suc,N) != N ).
tff(fact_474_nat_Oinject,axiom,
! [Nat_3: nat,Nat_2: nat] :
( ( hAPP_nat_nat(suc,Nat_3) = hAPP_nat_nat(suc,Nat_2) )
<=> ( Nat_3 = Nat_2 ) ) ).
tff(fact_475_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( hAPP_nat_nat(suc,X) = hAPP_nat_nat(suc,Y) )
=> ( X = Y ) ) ).
tff(fact_476_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M_5: nat] : N = hAPP_nat_nat(suc,M_5) ) ).
tff(fact_477_nat__induct,axiom,
! [N_1: nat,P_1: fun_nat_bool] :
( hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))
=> ( ! [N_19: nat] :
( hBOOL(hAPP_nat_bool(P_1,N_19))
=> hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,N_19))) )
=> hBOOL(hAPP_nat_bool(P_1,N_1)) ) ) ).
tff(fact_478_zero__induct,axiom,
! [P_1: fun_nat_bool,K_3: nat] :
( hBOOL(hAPP_nat_bool(P_1,K_3))
=> ( ! [N_19: nat] :
( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,N_19)))
=> hBOOL(hAPP_nat_bool(P_1,N_19)) )
=> hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) ) ) ).
tff(fact_479_nat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat_1: nat] : Y != hAPP_nat_nat(suc,Nat_1) ) ).
tff(fact_480_take__Suc,axiom,
! [N_1: nat,Xs: list_bool] :
( ( Xs != nil_bool )
=> ( hAPP_l1189022293t_bool(hAPP_n479129736t_bool(take_bool,hAPP_nat_nat(suc,N_1)),Xs) = hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,hAPP_list_bool_bool(hd_bool,Xs)),hAPP_l1189022293t_bool(hAPP_n479129736t_bool(take_bool,N_1),tl_bool(Xs))) ) ) ).
tff(fact_481_take__Suc,axiom,
! [N_37: nat,Xs_51: list_A114658244le_alt] :
( ( Xs_51 != nil_Ar557011903le_alt )
=> ( hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,hAPP_nat_nat(suc,N_37)),Xs_51) = hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,hAPP_l1749541000le_alt(hd_Arr1236501138le_alt,Xs_51)),hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,N_37),tl_Arr1883752598le_alt(Xs_51))) ) ) ).
tff(fact_482_take__Suc,axiom,
! [N_37: nat,Xs_51: list_nat] :
( ( Xs_51 != nil_nat )
=> ( hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,hAPP_nat_nat(suc,N_37)),Xs_51) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,hAPP_list_nat_nat(hd_nat,Xs_51)),hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,N_37),tl_nat(Xs_51))) ) ) ).
tff(fact_483_insort__insert__insort,axiom,
! [X_1: nat,Xs: list_nat] :
( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Xs)))
=> ( hAPP_l248265089st_nat(hAPP_n280362926st_nat(linord297828738at_nat(cOMBI_nat),X_1),Xs) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(linord1850739451at_nat(cOMBI_nat),X_1),Xs) ) ) ).
tff(fact_484_list__ex1__simps_I2_J,axiom,
! [P_1: fun_bool_bool,X_1: bool,Xs: list_bool] :
( hBOOL(hAPP_list_bool_bool(list_ex1_bool(P_1),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs)))
<=> ( ( hBOOL(hAPP_bool_bool(P_1,X_1))
=> hBOOL(hAPP_list_bool_bool(list_all_bool(cOMBS_bool_bool_bool(cOMBB_1914122979l_bool(fdisj,cOMBB_bool_bool_bool(fNot,P_1)),hAPP_b589554111l_bool(fequal_bool,X_1))),Xs)) )
& ( ~ hBOOL(hAPP_bool_bool(P_1,X_1))
=> hBOOL(hAPP_list_bool_bool(list_ex1_bool(P_1),Xs)) ) ) ) ).
tff(fact_485_list__ex1__simps_I2_J,axiom,
! [P_1: fun_Ar17477381t_bool,X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_l499746980t_bool(list_e1657546370le_alt(P_1),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),Xs)))
<=> ( ( hBOOL(hAPP_A1748732503t_bool(P_1,X_1))
=> hBOOL(hAPP_l499746980t_bool(list_a1988415615le_alt(cOMBS_119208490l_bool(cOMBB_1234541833le_alt(fdisj,cOMBB_388242475le_alt(fNot,P_1)),hAPP_A2063800319t_bool(fequal1590852494le_alt,X_1))),Xs)) )
& ( ~ hBOOL(hAPP_A1748732503t_bool(P_1,X_1))
=> hBOOL(hAPP_l499746980t_bool(list_e1657546370le_alt(P_1),Xs)) ) ) ) ).
tff(fact_486_list__ex1__simps_I2_J,axiom,
! [P_1: fun_nat_bool,X_1: nat,Xs: list_nat] :
( hBOOL(hAPP_list_nat_bool(list_ex1_nat(P_1),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Xs)))
<=> ( ( hBOOL(hAPP_nat_bool(P_1,X_1))
=> hBOOL(hAPP_list_nat_bool(list_all_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fdisj,cOMBB_bool_bool_nat(fNot,P_1)),hAPP_n1699378549t_bool(fequal_nat,X_1))),Xs)) )
& ( ~ hBOOL(hAPP_nat_bool(P_1,X_1))
=> hBOOL(hAPP_list_nat_bool(list_ex1_nat(P_1),Xs)) ) ) ) ).
tff(fact_487_insort__remove1,axiom,
! [Aa: nat,Xs: list_nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Aa),hAPP_l1821533378t_bool(set_nat,Xs)))
=> ( hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,Xs))
=> ( hAPP_l248265089st_nat(hAPP_n280362926st_nat(linord1850739451at_nat(cOMBI_nat),Aa),hAPP_l248265089st_nat(hAPP_n280362926st_nat(remove1_nat,Aa),Xs)) = Xs ) ) ) ).
tff(fact_488_remove1__commute,axiom,
! [X_94: nat,Y_64: nat,Zs_3: list_nat] : hAPP_l248265089st_nat(hAPP_n280362926st_nat(remove1_nat,X_94),hAPP_l248265089st_nat(hAPP_n280362926st_nat(remove1_nat,Y_64),Zs_3)) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(remove1_nat,Y_64),hAPP_l248265089st_nat(hAPP_n280362926st_nat(remove1_nat,X_94),Zs_3)) ).
tff(fact_489_remove1__commute,axiom,
! [X_94: arrow_1893660431le_alt,Y_64: arrow_1893660431le_alt,Zs_3: list_A114658244le_alt] : hAPP_l84111627le_alt(hAPP_A436723315le_alt(remove861161045le_alt,X_94),hAPP_l84111627le_alt(hAPP_A436723315le_alt(remove861161045le_alt,Y_64),Zs_3)) = hAPP_l84111627le_alt(hAPP_A436723315le_alt(remove861161045le_alt,Y_64),hAPP_l84111627le_alt(hAPP_A436723315le_alt(remove861161045le_alt,X_94),Zs_3)) ).
tff(fact_490_take__Nil,axiom,
! [N_36: nat] : hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,N_36),nil_Ar557011903le_alt) = nil_Ar557011903le_alt ).
tff(fact_491_take__Nil,axiom,
! [N_36: nat] : hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,N_36),nil_nat) = nil_nat ).
tff(fact_492_in__set__takeD,axiom,
! [X_1: arrow_1893660431le_alt,N_1: nat,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,N_1),Xs))))
=> hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs))) ) ).
tff(fact_493_in__set__takeD,axiom,
! [X_1: nat,N_1: nat,Xs: list_nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,N_1),Xs))))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Xs))) ) ).
tff(fact_494_distinct__take,axiom,
! [I_12: nat,Xs_50: list_A114658244le_alt] :
( hBOOL(hAPP_l499746980t_bool(distin1654625714le_alt,Xs_50))
=> hBOOL(hAPP_l499746980t_bool(distin1654625714le_alt,hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,I_12),Xs_50))) ) ).
tff(fact_495_distinct__take,axiom,
! [I_12: nat,Xs_50: list_nat] :
( hBOOL(hAPP_list_nat_bool(distinct_nat,Xs_50))
=> hBOOL(hAPP_list_nat_bool(distinct_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,I_12),Xs_50))) ) ).
tff(fact_496_sorted__take,axiom,
! [N_35: nat,Xs_49: list_nat] :
( hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,Xs_49))
=> hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,N_35),Xs_49))) ) ).
tff(fact_497_remove1_Osimps_I2_J,axiom,
! [Xs: list_bool,Y_2: bool,X_1: bool] :
( ( ( hBOOL(X_1)
<=> hBOOL(Y_2) )
=> ( hAPP_l1189022293t_bool(hAPP_b994696797t_bool(remove1_bool,X_1),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,Y_2),Xs)) = Xs ) )
& ( ( hBOOL(X_1)
<=> ~ hBOOL(Y_2) )
=> ( hAPP_l1189022293t_bool(hAPP_b994696797t_bool(remove1_bool,X_1),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,Y_2),Xs)) = hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,Y_2),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(remove1_bool,X_1),Xs)) ) ) ) ).
tff(fact_498_remove1_Osimps_I2_J,axiom,
! [Xs_48: list_A114658244le_alt,X_93: arrow_1893660431le_alt,Y_63: arrow_1893660431le_alt] :
( ( ( X_93 = Y_63 )
=> ( hAPP_l84111627le_alt(hAPP_A436723315le_alt(remove861161045le_alt,X_93),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,Y_63),Xs_48)) = Xs_48 ) )
& ( ( X_93 != Y_63 )
=> ( hAPP_l84111627le_alt(hAPP_A436723315le_alt(remove861161045le_alt,X_93),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,Y_63),Xs_48)) = hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,Y_63),hAPP_l84111627le_alt(hAPP_A436723315le_alt(remove861161045le_alt,X_93),Xs_48)) ) ) ) ).
tff(fact_499_remove1_Osimps_I2_J,axiom,
! [Xs_48: list_nat,X_93: nat,Y_63: nat] :
( ( ( X_93 = Y_63 )
=> ( hAPP_l248265089st_nat(hAPP_n280362926st_nat(remove1_nat,X_93),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,Y_63),Xs_48)) = Xs_48 ) )
& ( ( X_93 != Y_63 )
=> ( hAPP_l248265089st_nat(hAPP_n280362926st_nat(remove1_nat,X_93),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,Y_63),Xs_48)) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,Y_63),hAPP_l248265089st_nat(hAPP_n280362926st_nat(remove1_nat,X_93),Xs_48)) ) ) ) ).
tff(fact_500_remove1_Osimps_I1_J,axiom,
! [X_92: arrow_1893660431le_alt] : hAPP_l84111627le_alt(hAPP_A436723315le_alt(remove861161045le_alt,X_92),nil_Ar557011903le_alt) = nil_Ar557011903le_alt ).
tff(fact_501_remove1_Osimps_I1_J,axiom,
! [X_92: nat] : hAPP_l248265089st_nat(hAPP_n280362926st_nat(remove1_nat,X_92),nil_nat) = nil_nat ).
tff(fact_502_in__set__remove1,axiom,
! [Xs: list_A114658244le_alt,Aa: arrow_1893660431le_alt,Ba: arrow_1893660431le_alt] :
( ( Aa != Ba )
=> ( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,Aa),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,hAPP_l84111627le_alt(hAPP_A436723315le_alt(remove861161045le_alt,Ba),Xs))))
<=> hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,Aa),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs))) ) ) ).
tff(fact_503_in__set__remove1,axiom,
! [Xs: list_nat,Aa: nat,Ba: nat] :
( ( Aa != Ba )
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Aa),hAPP_l1821533378t_bool(set_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(remove1_nat,Ba),Xs))))
<=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Aa),hAPP_l1821533378t_bool(set_nat,Xs))) ) ) ).
tff(fact_504_notin__set__remove1,axiom,
! [Y_2: arrow_1893660431le_alt,X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt] :
( ~ hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
=> ~ hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,hAPP_l84111627le_alt(hAPP_A436723315le_alt(remove861161045le_alt,Y_2),Xs)))) ) ).
tff(fact_505_notin__set__remove1,axiom,
! [Y_2: nat,X_1: nat,Xs: list_nat] :
( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Xs)))
=> ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(remove1_nat,Y_2),Xs)))) ) ).
tff(fact_506_remove1__idem,axiom,
! [X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt] :
( ~ hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
=> ( hAPP_l84111627le_alt(hAPP_A436723315le_alt(remove861161045le_alt,X_1),Xs) = Xs ) ) ).
tff(fact_507_remove1__idem,axiom,
! [X_1: nat,Xs: list_nat] :
( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Xs)))
=> ( hAPP_l248265089st_nat(hAPP_n280362926st_nat(remove1_nat,X_1),Xs) = Xs ) ) ).
tff(fact_508_distinct__remove1,axiom,
! [X_91: arrow_1893660431le_alt,Xs_47: list_A114658244le_alt] :
( hBOOL(hAPP_l499746980t_bool(distin1654625714le_alt,Xs_47))
=> hBOOL(hAPP_l499746980t_bool(distin1654625714le_alt,hAPP_l84111627le_alt(hAPP_A436723315le_alt(remove861161045le_alt,X_91),Xs_47))) ) ).
tff(fact_509_distinct__remove1,axiom,
! [X_91: nat,Xs_47: list_nat] :
( hBOOL(hAPP_list_nat_bool(distinct_nat,Xs_47))
=> hBOOL(hAPP_list_nat_bool(distinct_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(remove1_nat,X_91),Xs_47))) ) ).
tff(fact_510_sorted__remove1,axiom,
! [A_83: nat,Xs_46: list_nat] :
( hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,Xs_46))
=> hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(remove1_nat,A_83),Xs_46))) ) ).
tff(fact_511_list__all__simps_I1_J,axiom,
! [P_1: fun_bool_bool,X_1: bool,Xs: list_bool] :
( hBOOL(hAPP_list_bool_bool(list_all_bool(P_1),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs)))
<=> ( hBOOL(hAPP_bool_bool(P_1,X_1))
& hBOOL(hAPP_list_bool_bool(list_all_bool(P_1),Xs)) ) ) ).
tff(fact_512_list__all__simps_I1_J,axiom,
! [P_1: fun_Ar17477381t_bool,X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_l499746980t_bool(list_a1988415615le_alt(P_1),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),Xs)))
<=> ( hBOOL(hAPP_A1748732503t_bool(P_1,X_1))
& hBOOL(hAPP_l499746980t_bool(list_a1988415615le_alt(P_1),Xs)) ) ) ).
tff(fact_513_list__all__simps_I1_J,axiom,
! [P_1: fun_nat_bool,X_1: nat,Xs: list_nat] :
( hBOOL(hAPP_list_nat_bool(list_all_nat(P_1),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Xs)))
<=> ( hBOOL(hAPP_nat_bool(P_1,X_1))
& hBOOL(hAPP_list_nat_bool(list_all_nat(P_1),Xs)) ) ) ).
tff(fact_514_list__all__simps_I2_J,axiom,
! [P_1: fun_Ar17477381t_bool] : hBOOL(hAPP_l499746980t_bool(list_a1988415615le_alt(P_1),nil_Ar557011903le_alt)) ).
tff(fact_515_list__all__simps_I2_J,axiom,
! [P_1: fun_nat_bool] : hBOOL(hAPP_list_nat_bool(list_all_nat(P_1),nil_nat)) ).
tff(fact_516_list__all__iff,axiom,
! [P_1: fun_Ar17477381t_bool,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_l499746980t_bool(list_a1988415615le_alt(P_1),Xs))
<=> ! [X_4: arrow_1893660431le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_4),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
=> hBOOL(hAPP_A1748732503t_bool(P_1,X_4)) ) ) ).
tff(fact_517_list__all__iff,axiom,
! [P_1: fun_nat_bool,Xs: list_nat] :
( hBOOL(hAPP_list_nat_bool(list_all_nat(P_1),Xs))
<=> ! [X_4: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_4),hAPP_l1821533378t_bool(set_nat,Xs)))
=> hBOOL(hAPP_nat_bool(P_1,X_4)) ) ) ).
tff(fact_518_list__all__append,axiom,
! [P_1: fun_Ar17477381t_bool,Xs: list_A114658244le_alt,Ys: list_A114658244le_alt] :
( hBOOL(hAPP_l499746980t_bool(list_a1988415615le_alt(P_1),hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs),Ys)))
<=> ( hBOOL(hAPP_l499746980t_bool(list_a1988415615le_alt(P_1),Xs))
& hBOOL(hAPP_l499746980t_bool(list_a1988415615le_alt(P_1),Ys)) ) ) ).
tff(fact_519_list__all__append,axiom,
! [P_1: fun_nat_bool,Xs: list_nat,Ys: list_nat] :
( hBOOL(hAPP_list_nat_bool(list_all_nat(P_1),hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs),Ys)))
<=> ( hBOOL(hAPP_list_nat_bool(list_all_nat(P_1),Xs))
& hBOOL(hAPP_list_nat_bool(list_all_nat(P_1),Ys)) ) ) ).
tff(fact_520_list__all__rev,axiom,
! [P_1: fun_nat_bool,Xs: list_nat] :
( hBOOL(hAPP_list_nat_bool(list_all_nat(P_1),rev_nat(Xs)))
<=> hBOOL(hAPP_list_nat_bool(list_all_nat(P_1),Xs)) ) ).
tff(fact_521_list__all__rev,axiom,
! [P_1: fun_Ar17477381t_bool,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_l499746980t_bool(list_a1988415615le_alt(P_1),rev_Ar377224735le_alt(Xs)))
<=> hBOOL(hAPP_l499746980t_bool(list_a1988415615le_alt(P_1),Xs)) ) ).
tff(fact_522_take__0,axiom,
! [Xs_45: list_A114658244le_alt] : hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,zero_zero_nat),Xs_45) = nil_Ar557011903le_alt ).
tff(fact_523_take__0,axiom,
! [Xs_45: list_nat] : hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,zero_zero_nat),Xs_45) = nil_nat ).
tff(fact_524_take__eq__Nil,axiom,
! [N_1: nat,Xs: list_A114658244le_alt] :
( ( hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,N_1),Xs) = nil_Ar557011903le_alt )
<=> ( ( N_1 = zero_zero_nat )
| ( Xs = nil_Ar557011903le_alt ) ) ) ).
tff(fact_525_take__eq__Nil,axiom,
! [N_1: nat,Xs: list_nat] :
( ( hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,N_1),Xs) = nil_nat )
<=> ( ( N_1 = zero_zero_nat )
| ( Xs = nil_nat ) ) ) ).
tff(fact_526_take__Suc__Cons,axiom,
! [N_1: nat,X_1: bool,Xs: list_bool] : hAPP_l1189022293t_bool(hAPP_n479129736t_bool(take_bool,hAPP_nat_nat(suc,N_1)),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs)) = hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),hAPP_l1189022293t_bool(hAPP_n479129736t_bool(take_bool,N_1),Xs)) ).
tff(fact_527_take__Suc__Cons,axiom,
! [N_34: nat,X_90: arrow_1893660431le_alt,Xs_44: list_A114658244le_alt] : hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,hAPP_nat_nat(suc,N_34)),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_90),Xs_44)) = hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_90),hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,N_34),Xs_44)) ).
tff(fact_528_take__Suc__Cons,axiom,
! [N_34: nat,X_90: nat,Xs_44: list_nat] : hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,hAPP_nat_nat(suc,N_34)),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_90),Xs_44)) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_90),hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,N_34),Xs_44)) ).
tff(fact_529_remove1__insort,axiom,
! [X_1: nat,Xs: list_nat] : hAPP_l248265089st_nat(hAPP_n280362926st_nat(remove1_nat,X_1),hAPP_l248265089st_nat(hAPP_n280362926st_nat(linord1850739451at_nat(cOMBI_nat),X_1),Xs)) = Xs ).
tff(fact_530_take__tl,axiom,
! [N_33: nat,Xs_43: list_A114658244le_alt] : hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,N_33),tl_Arr1883752598le_alt(Xs_43)) = tl_Arr1883752598le_alt(hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,hAPP_nat_nat(suc,N_33)),Xs_43)) ).
tff(fact_531_take__tl,axiom,
! [N_33: nat,Xs_43: list_nat] : hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,N_33),tl_nat(Xs_43)) = tl_nat(hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,hAPP_nat_nat(suc,N_33)),Xs_43)) ).
tff(fact_532_take__Cons,axiom,
! [N_1: nat,X_1: bool,Xs: list_bool] : hAPP_l1189022293t_bool(hAPP_n479129736t_bool(take_bool,N_1),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs)) = hAPP_nat_list_bool(nat_case_list_bool(nil_bool,cOMBB_1891801486ol_nat(hAPP_b994696797t_bool(cons_bool,X_1),hAPP_l1656175496t_bool(cOMBC_1038295629t_bool(take_bool),Xs))),N_1) ).
tff(fact_533_take__Cons,axiom,
! [N_1: nat,X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt] : hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,N_1),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),Xs)) = hAPP_n2007559481le_alt(nat_ca146784400le_alt(nil_Ar557011903le_alt,cOMBB_138440792lt_nat(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),hAPP_l1210406190le_alt(cOMBC_869637379le_alt(take_A2032349095le_alt),Xs))),N_1) ).
tff(fact_534_take__Cons,axiom,
! [N_1: nat,X_1: nat,Xs: list_nat] : hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,N_1),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Xs)) = hAPP_nat_list_nat(nat_case_list_nat(nil_nat,cOMBB_850496866at_nat(hAPP_n280362926st_nat(cons_nat,X_1),hAPP_l383341732st_nat(cOMBC_1204473209st_nat(take_nat),Xs))),N_1) ).
tff(fact_535_distinct__insort__insert,axiom,
! [F: fun_nat_nat,X_1: nat,Xs: list_nat] :
( hBOOL(hAPP_list_nat_bool(distinct_nat,Xs))
=> hBOOL(hAPP_list_nat_bool(distinct_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(linord297828738at_nat(F),X_1),Xs))) ) ).
tff(fact_536_remove1__append,axiom,
! [Ys: list_A114658244le_alt,X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt] :
( ( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
=> ( hAPP_l84111627le_alt(hAPP_A436723315le_alt(remove861161045le_alt,X_1),hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs),Ys)) = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,hAPP_l84111627le_alt(hAPP_A436723315le_alt(remove861161045le_alt,X_1),Xs)),Ys) ) )
& ( ~ hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
=> ( hAPP_l84111627le_alt(hAPP_A436723315le_alt(remove861161045le_alt,X_1),hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs),Ys)) = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs),hAPP_l84111627le_alt(hAPP_A436723315le_alt(remove861161045le_alt,X_1),Ys)) ) ) ) ).
tff(fact_537_remove1__append,axiom,
! [Ys: list_nat,X_1: nat,Xs: list_nat] :
( ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Xs)))
=> ( hAPP_l248265089st_nat(hAPP_n280362926st_nat(remove1_nat,X_1),hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs),Ys)) = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(remove1_nat,X_1),Xs)),Ys) ) )
& ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Xs)))
=> ( hAPP_l248265089st_nat(hAPP_n280362926st_nat(remove1_nat,X_1),hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs),Ys)) = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs),hAPP_l248265089st_nat(hAPP_n280362926st_nat(remove1_nat,X_1),Ys)) ) ) ) ).
tff(fact_538_insort__insert__triv,axiom,
! [X_1: nat,Xs: list_nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Xs)))
=> ( hAPP_l248265089st_nat(hAPP_n280362926st_nat(linord297828738at_nat(cOMBI_nat),X_1),Xs) = Xs ) ) ).
tff(fact_539_sorted__insort__insert,axiom,
! [X_1: nat,Xs: list_nat] :
( hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,Xs))
=> hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(linord297828738at_nat(cOMBI_nat),X_1),Xs))) ) ).
tff(fact_540_list__all__cong,axiom,
! [G: fun_Ar17477381t_bool,F: fun_Ar17477381t_bool,Xs: list_A114658244le_alt,Ys: list_A114658244le_alt] :
( ( Xs = Ys )
=> ( ! [X_4: arrow_1893660431le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_4),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Ys)))
=> ( hBOOL(hAPP_A1748732503t_bool(F,X_4))
<=> hBOOL(hAPP_A1748732503t_bool(G,X_4)) ) )
=> ( hBOOL(hAPP_l499746980t_bool(list_a1988415615le_alt(F),Xs))
<=> hBOOL(hAPP_l499746980t_bool(list_a1988415615le_alt(G),Ys)) ) ) ) ).
tff(fact_541_list__all__cong,axiom,
! [G: fun_nat_bool,F: fun_nat_bool,Xs: list_nat,Ys: list_nat] :
( ( Xs = Ys )
=> ( ! [X_4: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_4),hAPP_l1821533378t_bool(set_nat,Ys)))
=> ( hBOOL(hAPP_nat_bool(F,X_4))
<=> hBOOL(hAPP_nat_bool(G,X_4)) ) )
=> ( hBOOL(hAPP_list_nat_bool(list_all_nat(F),Xs))
<=> hBOOL(hAPP_list_nat_bool(list_all_nat(G),Ys)) ) ) ) ).
tff(fact_542_take__equalityI,axiom,
! [Xs_42: list_A114658244le_alt,Ys_14: list_A114658244le_alt] :
( ! [I_4: nat] : hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,I_4),Xs_42) = hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,I_4),Ys_14)
=> ( Xs_42 = Ys_14 ) ) ).
tff(fact_543_take__equalityI,axiom,
! [Xs_42: list_nat,Ys_14: list_nat] :
( ! [I_4: nat] : hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,I_4),Xs_42) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,I_4),Ys_14)
=> ( Xs_42 = Ys_14 ) ) ).
tff(fact_544_list_Osize_I2_J,axiom,
! [Fa: fun_bool_nat,Aa: bool,List_1: list_bool] : list_size_bool(Fa,hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,Aa),List_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_bool_nat(Fa,Aa)),list_size_bool(Fa,List_1))),hAPP_nat_nat(suc,zero_zero_nat)) ).
tff(fact_545_list_Osize_I2_J,axiom,
! [Fa: fun_Ar1091137528lt_nat,Aa: arrow_1893660431le_alt,List_1: list_A114658244le_alt] : list_s1668597313le_alt(Fa,hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,Aa),List_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_A1419065744lt_nat(Fa,Aa)),list_s1668597313le_alt(Fa,List_1))),hAPP_nat_nat(suc,zero_zero_nat)) ).
tff(fact_546_list_Osize_I2_J,axiom,
! [Fa: fun_nat_nat,Aa: nat,List_1: list_nat] : list_size_nat(Fa,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,Aa),List_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(Fa,Aa)),list_size_nat(Fa,List_1))),hAPP_nat_nat(suc,zero_zero_nat)) ).
tff(fact_547_split__list__prop,axiom,
! [P_1: fun_Ar17477381t_bool,Xs: list_A114658244le_alt] :
( ? [X_4: arrow_1893660431le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_4),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
& hBOOL(hAPP_A1748732503t_bool(P_1,X_4)) )
=> ? [Ys_6: list_A114658244le_alt,X_4: arrow_1893660431le_alt] :
( ? [Zs_2: list_A114658244le_alt] : Xs = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Ys_6),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_4),Zs_2))
& hBOOL(hAPP_A1748732503t_bool(P_1,X_4)) ) ) ).
tff(fact_548_split__list__prop,axiom,
! [P_1: fun_nat_bool,Xs: list_nat] :
( ? [X_4: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_4),hAPP_l1821533378t_bool(set_nat,Xs)))
& hBOOL(hAPP_nat_bool(P_1,X_4)) )
=> ? [Ys_6: list_nat,X_4: nat] :
( ? [Zs_2: list_nat] : Xs = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Ys_6),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_4),Zs_2))
& hBOOL(hAPP_nat_bool(P_1,X_4)) ) ) ).
tff(fact_549_split__list__prop,axiom,
! [P_1: fun_bool_bool,Xs: list_bool] :
( ? [X_4: bool] :
( hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,X_4),hAPP_l96088310l_bool(set_bool,Xs)))
& hBOOL(hAPP_bool_bool(P_1,X_4)) )
=> ? [Ys_6: list_bool,X_4: bool] :
( ? [Zs_2: list_bool] : Xs = hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,Ys_6),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_4),Zs_2))
& hBOOL(hAPP_bool_bool(P_1,X_4)) ) ) ).
tff(fact_550_split__list__last__prop,axiom,
! [P_1: fun_Ar17477381t_bool,Xs: list_A114658244le_alt] :
( ? [X_4: arrow_1893660431le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_4),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
& hBOOL(hAPP_A1748732503t_bool(P_1,X_4)) )
=> ? [Ys_6: list_A114658244le_alt,X_4: arrow_1893660431le_alt,Zs_2: list_A114658244le_alt] :
( ( Xs = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Ys_6),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_4),Zs_2)) )
& hBOOL(hAPP_A1748732503t_bool(P_1,X_4))
& ! [Xa: arrow_1893660431le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,Xa),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Zs_2)))
=> ~ hBOOL(hAPP_A1748732503t_bool(P_1,Xa)) ) ) ) ).
tff(fact_551_split__list__last__prop,axiom,
! [P_1: fun_nat_bool,Xs: list_nat] :
( ? [X_4: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_4),hAPP_l1821533378t_bool(set_nat,Xs)))
& hBOOL(hAPP_nat_bool(P_1,X_4)) )
=> ? [Ys_6: list_nat,X_4: nat,Zs_2: list_nat] :
( ( Xs = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Ys_6),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_4),Zs_2)) )
& hBOOL(hAPP_nat_bool(P_1,X_4))
& ! [Xa: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Xa),hAPP_l1821533378t_bool(set_nat,Zs_2)))
=> ~ hBOOL(hAPP_nat_bool(P_1,Xa)) ) ) ) ).
tff(fact_552_split__list__last__prop,axiom,
! [P_1: fun_bool_bool,Xs: list_bool] :
( ? [X_4: bool] :
( hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,X_4),hAPP_l96088310l_bool(set_bool,Xs)))
& hBOOL(hAPP_bool_bool(P_1,X_4)) )
=> ? [Ys_6: list_bool,X_4: bool,Zs_2: list_bool] :
( ( Xs = hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,Ys_6),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_4),Zs_2)) )
& hBOOL(hAPP_bool_bool(P_1,X_4))
& ! [Xa: bool] :
( hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,Xa),hAPP_l96088310l_bool(set_bool,Zs_2)))
=> ~ hBOOL(hAPP_bool_bool(P_1,Xa)) ) ) ) ).
tff(fact_553_split__list__first__prop,axiom,
! [P_1: fun_Ar17477381t_bool,Xs: list_A114658244le_alt] :
( ? [X_4: arrow_1893660431le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_4),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
& hBOOL(hAPP_A1748732503t_bool(P_1,X_4)) )
=> ? [Ys_6: list_A114658244le_alt,X_4: arrow_1893660431le_alt] :
( ? [Zs_2: list_A114658244le_alt] : Xs = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Ys_6),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_4),Zs_2))
& hBOOL(hAPP_A1748732503t_bool(P_1,X_4))
& ! [Xa: arrow_1893660431le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,Xa),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Ys_6)))
=> ~ hBOOL(hAPP_A1748732503t_bool(P_1,Xa)) ) ) ) ).
tff(fact_554_split__list__first__prop,axiom,
! [P_1: fun_nat_bool,Xs: list_nat] :
( ? [X_4: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_4),hAPP_l1821533378t_bool(set_nat,Xs)))
& hBOOL(hAPP_nat_bool(P_1,X_4)) )
=> ? [Ys_6: list_nat,X_4: nat] :
( ? [Zs_2: list_nat] : Xs = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Ys_6),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_4),Zs_2))
& hBOOL(hAPP_nat_bool(P_1,X_4))
& ! [Xa: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Xa),hAPP_l1821533378t_bool(set_nat,Ys_6)))
=> ~ hBOOL(hAPP_nat_bool(P_1,Xa)) ) ) ) ).
tff(fact_555_split__list__first__prop,axiom,
! [P_1: fun_bool_bool,Xs: list_bool] :
( ? [X_4: bool] :
( hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,X_4),hAPP_l96088310l_bool(set_bool,Xs)))
& hBOOL(hAPP_bool_bool(P_1,X_4)) )
=> ? [Ys_6: list_bool,X_4: bool] :
( ? [Zs_2: list_bool] : Xs = hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,Ys_6),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_4),Zs_2))
& hBOOL(hAPP_bool_bool(P_1,X_4))
& ! [Xa: bool] :
( hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,Xa),hAPP_l96088310l_bool(set_bool,Ys_6)))
=> ~ hBOOL(hAPP_bool_bool(P_1,Xa)) ) ) ) ).
tff(fact_556_rev__foldl__cons,axiom,
! [Xs: list_nat] : rev_nat(Xs) = foldl_list_nat_nat(cOMBC_1204473209st_nat(cons_nat),nil_nat,Xs) ).
tff(fact_557_rev__foldl__cons,axiom,
! [Xs: list_A114658244le_alt] : rev_Ar377224735le_alt(Xs) = foldl_1844370748le_alt(cOMBC_673369480le_alt(cons_A1647044463le_alt),nil_Ar557011903le_alt,Xs) ).
tff(fact_558_rev__foldl__cons,axiom,
! [Xs: list_bool] : rev_bool(Xs) = foldl_list_bool_bool(cOMBC_80330970t_bool(cons_bool),nil_bool,Xs) ).
tff(fact_559_foldl__absorb0,axiom,
! [X_1: nat,Zs: list_nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_1),foldl_nat_nat(plus_plus_nat,zero_zero_nat,Zs)) = foldl_nat_nat(plus_plus_nat,X_1,Zs) ).
tff(fact_560_add__right__imp__eq,axiom,
! [B_70: nat,A_82: nat,C_43: nat] :
( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_70),A_82) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_43),A_82) )
=> ( B_70 = C_43 ) ) ).
tff(fact_561_add__imp__eq,axiom,
! [A_81: nat,B_69: nat,C_42: nat] :
( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_81),B_69) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_81),C_42) )
=> ( B_69 = C_42 ) ) ).
tff(fact_562_add__left__imp__eq,axiom,
! [A_80: nat,B_68: nat,C_41: nat] :
( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_80),B_68) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_80),C_41) )
=> ( B_68 = C_41 ) ) ).
tff(fact_563_add__right__cancel,axiom,
! [Ba: nat,Aa: nat,C_2: nat] :
( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ba),Aa) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_2),Aa) )
<=> ( Ba = C_2 ) ) ).
tff(fact_564_add__left__cancel,axiom,
! [Aa: nat,Ba: nat,C_2: nat] :
( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Aa),Ba) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Aa),C_2) )
<=> ( Ba = C_2 ) ) ).
tff(fact_565_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_79: nat,B_67: nat,C_40: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_79),B_67)),C_40) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_79),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_67),C_40)) ).
tff(fact_566_foldl__assoc,axiom,
! [X_1: nat,Y_2: nat,Zs: list_nat] : foldl_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_1),Y_2),Zs) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_1),foldl_nat_nat(plus_plus_nat,Y_2,Zs)) ).
tff(fact_567_nat__add__right__cancel,axiom,
! [M_1: nat,K_3: nat,N_1: nat] :
( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),K_3) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_1),K_3) )
<=> ( M_1 = N_1 ) ) ).
tff(fact_568_nat__add__left__cancel,axiom,
! [K_3: nat,M_1: nat,N_1: nat] :
( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_3),M_1) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_3),N_1) )
<=> ( M_1 = N_1 ) ) ).
tff(fact_569_nat__add__assoc,axiom,
! [M: nat,N: nat,K: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)),K) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),K)) ).
tff(fact_570_nat__add__left__commute,axiom,
! [X: nat,Y: nat,Z_3: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z_3)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X),Z_3)) ).
tff(fact_571_nat__add__commute,axiom,
! [M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M) ).
tff(fact_572_add_Ocomm__neutral,axiom,
! [A_78: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_78),zero_zero_nat) = A_78 ).
tff(fact_573_add__0__right,axiom,
! [A_77: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_77),zero_zero_nat) = A_77 ).
tff(fact_574_add__0,axiom,
! [A_76: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A_76) = A_76 ).
tff(fact_575_add__0__left,axiom,
! [A_75: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A_75) = A_75 ).
tff(fact_576_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N) = M )
=> ( N = zero_zero_nat ) ) ).
tff(fact_577_add__is__0,axiom,
! [M_1: nat,N_1: nat] :
( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1) = zero_zero_nat )
<=> ( ( M_1 = zero_zero_nat )
& ( N_1 = zero_zero_nat ) ) ) ).
tff(fact_578_Nat_Oadd__0__right,axiom,
! [M: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),zero_zero_nat) = M ).
tff(fact_579_plus__nat_Oadd__0,axiom,
! [N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),N) = N ).
tff(fact_580_add__Suc__shift,axiom,
! [M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(suc,M)),N) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),hAPP_nat_nat(suc,N)) ).
tff(fact_581_add__Suc,axiom,
! [M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(suc,M)),N) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)) ).
tff(fact_582_add__Suc__right,axiom,
! [M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)) ).
tff(fact_583_foldl__Cons,axiom,
! [F: fun_nat_fun_nat_nat,Aa: nat,X_1: nat,Xs: list_nat] : foldl_nat_nat(F,Aa,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Xs)) = foldl_nat_nat(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,Aa),X_1),Xs) ).
tff(fact_584_foldl__Nil,axiom,
! [F: fun_nat_fun_nat_nat,Aa: nat] : foldl_nat_nat(F,Aa,nil_nat) = Aa ).
tff(fact_585_foldl__append,axiom,
! [F: fun_nat_fun_nat_nat,Aa: nat,Xs: list_nat,Ys: list_nat] : foldl_nat_nat(F,Aa,hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs),Ys)) = foldl_nat_nat(F,foldl_nat_nat(F,Aa,Xs),Ys) ).
tff(fact_586_one__is__add,axiom,
! [M_1: nat,N_1: nat] :
( ( hAPP_nat_nat(suc,zero_zero_nat) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1) )
<=> ( ( ( M_1 = hAPP_nat_nat(suc,zero_zero_nat) )
& ( N_1 = zero_zero_nat ) )
| ( ( M_1 = zero_zero_nat )
& ( N_1 = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ) ).
tff(fact_587_add__is__1,axiom,
! [M_1: nat,N_1: nat] :
( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1) = hAPP_nat_nat(suc,zero_zero_nat) )
<=> ( ( ( M_1 = hAPP_nat_nat(suc,zero_zero_nat) )
& ( N_1 = zero_zero_nat ) )
| ( ( M_1 = zero_zero_nat )
& ( N_1 = hAPP_nat_nat(suc,zero_zero_nat) ) ) ) ) ).
tff(fact_588_replicate__add,axiom,
! [N_32: nat,M_8: nat,X_89: arrow_1893660431le_alt] : hAPP_A1382181054le_alt(replic782356601le_alt(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_32),M_8)),X_89) = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,hAPP_A1382181054le_alt(replic782356601le_alt(N_32),X_89)),hAPP_A1382181054le_alt(replic782356601le_alt(M_8),X_89)) ).
tff(fact_589_replicate__add,axiom,
! [N_32: nat,M_8: nat,X_89: nat] : hAPP_nat_list_nat(replicate_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_32),M_8)),X_89) = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,hAPP_nat_list_nat(replicate_nat(N_32),X_89)),hAPP_nat_list_nat(replicate_nat(M_8),X_89)) ).
tff(fact_590_foldl__rev,axiom,
! [S_2: nat,Xs: list_nat,F: fun_nat_fun_nat_nat] :
( ! [X_4: nat,Y_25: nat,S_3: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,S_3),X_4)),Y_25) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,S_3),Y_25)),X_4)
=> ( foldl_nat_nat(F,S_2,rev_nat(Xs)) = foldl_nat_nat(F,S_2,Xs) ) ) ).
tff(fact_591_foldl__cong,axiom,
! [F: fun_nat_fun_nat_nat,G: fun_nat_fun_nat_nat,L_2: list_nat,K_3: list_nat,Aa: nat,Ba: nat] :
( ( Aa = Ba )
=> ( ( L_2 = K_3 )
=> ( ! [A_74: nat,X_4: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_4),hAPP_l1821533378t_bool(set_nat,L_2)))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,A_74),X_4) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(G,A_74),X_4) ) )
=> ( foldl_nat_nat(F,Aa,L_2) = foldl_nat_nat(G,Ba,K_3) ) ) ) ) ).
tff(fact_592_foldl__weak__invariant,axiom,
! [F: fun_nat_fun_nat_nat,Xs: list_nat,P_1: fun_nat_bool,S_2: nat] :
( hBOOL(hAPP_nat_bool(P_1,S_2))
=> ( ! [S_3: nat,X_4: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_4),hAPP_l1821533378t_bool(set_nat,Xs)))
=> ( hBOOL(hAPP_nat_bool(P_1,S_3))
=> hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,S_3),X_4))) ) )
=> hBOOL(hAPP_nat_bool(P_1,foldl_nat_nat(F,S_2,Xs))) ) ) ).
tff(fact_593_sort__foldl__insort,axiom,
! [Xs: list_nat] : linord852754176at_nat(cOMBI_nat,Xs) = foldl_list_nat_nat(cOMBC_1204473209st_nat(linord1850739451at_nat(cOMBI_nat)),nil_nat,Xs) ).
tff(fact_594_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A_73: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A_73) = A_73 ).
tff(fact_595_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A_72: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_72),zero_zero_nat) = A_72 ).
tff(fact_596_add__0__iff,axiom,
! [Ba: nat,Aa: nat] :
( ( Ba = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ba),Aa) )
<=> ( Aa = zero_zero_nat ) ) ).
tff(fact_597_sum__eq__0__conv,axiom,
! [M_1: nat,Ns: list_nat] :
( ( foldl_nat_nat(plus_plus_nat,M_1,Ns) = zero_zero_nat )
<=> ( ( M_1 = zero_zero_nat )
& ! [X_4: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_4),hAPP_l1821533378t_bool(set_nat,Ns)))
=> ( X_4 = zero_zero_nat ) ) ) ) ).
tff(fact_598_sorted__sort,axiom,
! [Xs: list_nat] : hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,linord852754176at_nat(cOMBI_nat,Xs))) ).
tff(fact_599_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_71: nat,B_66: nat,C_39: nat,D_9: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_71),B_66)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_39),D_9)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_71),C_39)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_66),D_9)) ).
tff(fact_600_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_70: nat,B_65: nat,C_38: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_70),B_65)),C_38) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_70),C_38)),B_65) ).
tff(fact_601_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_69: nat,B_64: nat,C_37: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_69),B_64)),C_37) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_69),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_64),C_37)) ).
tff(fact_602_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_68: nat,C_36: nat,D_8: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_68),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_36),D_8)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_68),C_36)),D_8) ).
tff(fact_603_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_67: nat,C_35: nat,D_7: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_67),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_35),D_7)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_35),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_67),D_7)) ).
tff(fact_604_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_66: nat,C_34: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_66),C_34) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_34),A_66) ).
tff(fact_605_sort__key__simps_I2_J,axiom,
! [F: fun_nat_nat,X_1: nat,Xs: list_nat] : linord852754176at_nat(F,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Xs)) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(linord1850739451at_nat(F),X_1),linord852754176at_nat(F,Xs)) ).
tff(fact_606_foldl__fun__comm,axiom,
! [S_2: nat,Xs: list_nat,X_1: nat,F: fun_nat_fun_nat_nat] :
( ! [X_4: nat,Y_25: nat,S_3: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,S_3),X_4)),Y_25) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,S_3),Y_25)),X_4)
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,foldl_nat_nat(F,S_2,Xs)),X_1) = foldl_nat_nat(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,S_2),X_1),Xs) ) ) ).
tff(fact_607_nat_Osize_I2_J,axiom,
! [Nat: nat] : hAPP_nat_nat(nat_size,hAPP_nat_nat(suc,Nat)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(nat_size,Nat)),hAPP_nat_nat(suc,zero_zero_nat)) ).
tff(fact_608_foldl__foldr1,axiom,
! [Xs: list_nat] : foldl_nat_nat(plus_plus_nat,zero_zero_nat,Xs) = hAPP_nat_nat(foldr_nat_nat(plus_plus_nat,Xs),zero_zero_nat) ).
tff(fact_609_foldl__foldr1__lemma,axiom,
! [Aa: nat,Xs: list_nat] : foldl_nat_nat(plus_plus_nat,Aa,Xs) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Aa),hAPP_nat_nat(foldr_nat_nat(plus_plus_nat,Xs),zero_zero_nat)) ).
tff(fact_610_foldl__invariant,axiom,
! [F: fun_nat_fun_nat_nat,P_1: fun_nat_bool,Xs: list_nat,Q_1: fun_nat_bool,X_1: nat] :
( hBOOL(hAPP_nat_bool(Q_1,X_1))
=> ( ! [X_4: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_4),hAPP_l1821533378t_bool(set_nat,Xs)))
=> hBOOL(hAPP_nat_bool(P_1,X_4)) )
=> ( ! [X_4: nat,Y_25: nat] :
( ( hBOOL(hAPP_nat_bool(P_1,X_4))
& hBOOL(hAPP_nat_bool(Q_1,Y_25)) )
=> hBOOL(hAPP_nat_bool(Q_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(F,Y_25),X_4))) )
=> hBOOL(hAPP_nat_bool(Q_1,foldl_nat_nat(F,X_1,Xs))) ) ) ) ).
tff(fact_611_nat_Osize_I1_J,axiom,
hAPP_nat_nat(nat_size,zero_zero_nat) = zero_zero_nat ).
tff(fact_612_foldr__conv__foldl,axiom,
! [Xs: list_nat,Aa: nat] : hAPP_nat_nat(foldr_nat_nat(plus_plus_nat,Xs),Aa) = foldl_nat_nat(plus_plus_nat,Aa,Xs) ).
tff(fact_613_foldr__foldl,axiom,
! [F: fun_nat_fun_nat_nat,Xs: list_nat,Aa: nat] : hAPP_nat_nat(foldr_nat_nat(F,Xs),Aa) = foldl_nat_nat(cOMBC_nat_nat_nat(F),Aa,rev_nat(Xs)) ).
tff(fact_614_foldl__foldr,axiom,
! [F: fun_nat_fun_nat_nat,Aa: nat,Xs: list_nat] : foldl_nat_nat(F,Aa,Xs) = hAPP_nat_nat(foldr_nat_nat(cOMBC_nat_nat_nat(F),rev_nat(Xs)),Aa) ).
tff(fact_615_nat_Osize_I4_J,axiom,
! [Nat: nat] : hAPP_nat_nat(size_size_nat,hAPP_nat_nat(suc,Nat)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(size_size_nat,Nat)),hAPP_nat_nat(suc,zero_zero_nat)) ).
tff(fact_616_foldr__max__sorted,axiom,
! [Y_2: nat,Xs: list_nat] :
( hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,rev_nat(Xs)))
=> ( ( ( Xs = nil_nat )
=> ( hAPP_nat_nat(foldr_nat_nat(ord_max_nat,Xs),Y_2) = Y_2 ) )
& ( ( Xs != nil_nat )
=> ( hAPP_nat_nat(foldr_nat_nat(ord_max_nat,Xs),Y_2) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_nat_nat(nth_nat(Xs),zero_zero_nat)),Y_2) ) ) ) ) ).
tff(fact_617_elem__le__sum,axiom,
! [N_1: nat,Ns: list_nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,N_1),hAPP_l1821533378t_bool(set_nat,Ns)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),foldl_nat_nat(plus_plus_nat,zero_zero_nat,Ns))) ) ).
tff(fact_618_bool_Osize_I1_J,axiom,
hAPP_bool_nat(bool_size,fTrue) = zero_zero_nat ).
tff(fact_619_bool_Osize_I2_J,axiom,
hAPP_bool_nat(bool_size,fFalse) = zero_zero_nat ).
tff(fact_620_le0,axiom,
! [N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N)) ).
tff(fact_621_nat__size,axiom,
! [N: nat] : hAPP_nat_nat(size_size_nat,N) = N ).
tff(fact_622_le__refl,axiom,
! [N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),N)) ).
tff(fact_623_nat__le__linear,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M)) ) ).
tff(fact_624_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).
tff(fact_625_le__trans,axiom,
! [K: nat,I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),K)) ) ) ).
tff(fact_626_le__antisym,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
=> ( M = N ) ) ) ).
tff(fact_627_max__add__distrib__left,axiom,
! [X_88: nat,Y_62: nat,Z_26: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_88),Y_62)),Z_26) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_88),Z_26)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y_62),Z_26)) ).
tff(fact_628_max__0R,axiom,
! [N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,N),zero_zero_nat) = N ).
tff(fact_629_max__0L,axiom,
! [N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,zero_zero_nat),N) = N ).
tff(fact_630_max__Suc__Suc,axiom,
! [M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,M),N)) ).
tff(fact_631_add__le__cancel__right,axiom,
! [Aa: nat,C_2: nat,Ba: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Aa),C_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ba),C_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Aa),Ba)) ) ).
tff(fact_632_add__le__cancel__left,axiom,
! [C_2: nat,Aa: nat,Ba: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_2),Aa)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_2),Ba)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Aa),Ba)) ) ).
tff(fact_633_add__right__mono,axiom,
! [C_33: nat,A_65: nat,B_63: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_65),B_63))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_65),C_33)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_63),C_33))) ) ).
tff(fact_634_add__left__mono,axiom,
! [C_32: nat,A_64: nat,B_62: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_64),B_62))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_32),A_64)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_32),B_62))) ) ).
tff(fact_635_add__mono,axiom,
! [C_31: nat,D_6: nat,A_63: nat,B_61: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_63),B_61))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_31),D_6))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_63),C_31)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_61),D_6))) ) ) ).
tff(fact_636_add__le__imp__le__right,axiom,
! [A_62: nat,C_30: nat,B_60: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_62),C_30)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_60),C_30)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_62),B_60)) ) ).
tff(fact_637_add__le__imp__le__left,axiom,
! [C_29: nat,A_61: nat,B_59: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_29),A_61)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_29),B_59)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_61),B_59)) ) ).
tff(fact_638_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),N)) ).
tff(fact_639_le__0__eq,axiom,
! [N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),zero_zero_nat))
<=> ( N_1 = zero_zero_nat ) ) ).
tff(fact_640_Suc__n__not__le__n,axiom,
! [N: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N)),N)) ).
tff(fact_641_not__less__eq__eq,axiom,
! [M_1: nat,N_1: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N_1)),M_1)) ) ).
tff(fact_642_le__Suc__eq,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),hAPP_nat_nat(suc,N_1)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
| ( M_1 = hAPP_nat_nat(suc,N_1) ) ) ) ).
tff(fact_643_Suc__le__mono,axiom,
! [N_1: nat,M_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N_1)),hAPP_nat_nat(suc,M_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_1)) ) ).
tff(fact_644_le__SucI,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(suc,N))) ) ).
tff(fact_645_le__SucE,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(suc,N)))
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> ( M = hAPP_nat_nat(suc,N) ) ) ) ).
tff(fact_646_Suc__leD,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M)),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).
tff(fact_647_le__add2,axiom,
! [N: nat,M: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N))) ).
tff(fact_648_le__add1,axiom,
! [N: nat,M: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M))) ).
tff(fact_649_le__iff__add,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
<=> ? [K_2: nat] : N_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),K_2) ) ).
tff(fact_650_nat__add__left__cancel__le,axiom,
! [K_3: nat,M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_3),M_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_3),N_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1)) ) ).
tff(fact_651_trans__le__add1,axiom,
! [M: nat,I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),M))) ) ).
tff(fact_652_trans__le__add2,axiom,
! [M: nat,I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),J))) ) ).
tff(fact_653_add__le__mono1,axiom,
! [K: nat,I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ).
tff(fact_654_add__le__mono,axiom,
! [K: nat,L: nat,I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),L))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ) ).
tff(fact_655_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K)),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N)) ) ).
tff(fact_656_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K)),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).
tff(fact_657_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K)),N))
=> ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N)) ) ) ).
tff(fact_658_max__Suc1,axiom,
! [N_1: nat,M_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_nat_nat(suc,N_1)),M_1) = hAPP_nat_nat(nat_case_nat(hAPP_nat_nat(suc,N_1),cOMBB_nat_nat_nat(suc,hAPP_nat_fun_nat_nat(ord_max_nat,N_1))),M_1) ).
tff(fact_659_max__Suc2,axiom,
! [M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,M_1),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(nat_case_nat(hAPP_nat_nat(suc,N_1),cOMBB_nat_nat_nat(suc,hAPP_nat_fun_nat_nat(cOMBC_nat_nat_nat(ord_max_nat),N_1))),M_1) ).
tff(fact_660_less__eq__nat_Osimps_I2_J,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M_1)),N_1))
<=> hBOOL(hAPP_nat_bool(nat_case_bool(fFalse,hAPP_n1699378549t_bool(ord_less_eq_nat,M_1)),N_1)) ) ).
tff(fact_661_add__nonpos__nonpos,axiom,
! [B_58: nat,A_60: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_60),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_58),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_60),B_58)),zero_zero_nat)) ) ) ).
tff(fact_662_add__increasing2,axiom,
! [B_57: nat,A_59: nat,C_28: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),C_28))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_57),A_59))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_57),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_59),C_28))) ) ) ).
tff(fact_663_add__increasing,axiom,
! [B_56: nat,C_27: nat,A_58: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_58))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_56),C_27))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_56),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_58),C_27))) ) ) ).
tff(fact_664_add__nonneg__eq__0__iff,axiom,
! [Y_2: nat,X_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),X_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),Y_2))
=> ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_1),Y_2) = zero_zero_nat )
<=> ( ( X_1 = zero_zero_nat )
& ( Y_2 = zero_zero_nat ) ) ) ) ) ).
tff(fact_665_add__nonneg__nonneg,axiom,
! [B_55: nat,A_57: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_57))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_55))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_57),B_55))) ) ) ).
tff(fact_666_sorted__many,axiom,
! [Zs_1: list_nat,X_87: nat,Y_61: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_87),Y_61))
=> ( hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,Y_61),Zs_1)))
=> hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_87),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,Y_61),Zs_1)))) ) ) ).
tff(fact_667_sorted__many__eq,axiom,
! [X_1: nat,Y_2: nat,Zs: list_nat] :
( hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,Y_2),Zs))))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2))
& hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,Y_2),Zs))) ) ) ).
tff(fact_668_insort__key_Osimps_I2_J,axiom,
! [Ys: list_nat,F: fun_nat_nat,X_1: nat,Y_2: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,X_1)),hAPP_nat_nat(F,Y_2)))
=> ( hAPP_l248265089st_nat(hAPP_n280362926st_nat(linord1850739451at_nat(F),X_1),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,Y_2),Ys)) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,Y_2),Ys)) ) )
& ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,X_1)),hAPP_nat_nat(F,Y_2)))
=> ( hAPP_l248265089st_nat(hAPP_n280362926st_nat(linord1850739451at_nat(F),X_1),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,Y_2),Ys)) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,Y_2),hAPP_l248265089st_nat(hAPP_n280362926st_nat(linord1850739451at_nat(F),X_1),Ys)) ) ) ) ).
tff(fact_669_list__size__estimation_H,axiom,
! [Y_2: nat,F: fun_Ar1091137528lt_nat,X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_2),hAPP_A1419065744lt_nat(F,X_1)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_2),list_s1668597313le_alt(F,Xs))) ) ) ).
tff(fact_670_list__size__estimation_H,axiom,
! [Y_2: nat,F: fun_nat_nat,X_1: nat,Xs: list_nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Xs)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_2),hAPP_nat_nat(F,X_1)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_2),list_size_nat(F,Xs))) ) ) ).
tff(fact_671_nat_Osize_I3_J,axiom,
hAPP_nat_nat(size_size_nat,zero_zero_nat) = zero_zero_nat ).
tff(fact_672_start__le__sum,axiom,
! [Ns: list_nat,M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),foldl_nat_nat(plus_plus_nat,N_1,Ns))) ) ).
tff(fact_673_sorted__Cons,axiom,
! [X_1: nat,Xs: list_nat] :
( hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Xs)))
<=> ( hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,Xs))
& ! [X_4: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_4),hAPP_l1821533378t_bool(set_nat,Xs)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),X_4)) ) ) ) ).
tff(fact_674_sorted__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs),Ys)))
<=> ( hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,Xs))
& hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,Ys))
& ! [X_4: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_4),hAPP_l1821533378t_bool(set_nat,Xs)))
=> ! [Xa: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Xa),hAPP_l1821533378t_bool(set_nat,Ys)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_4),Xa)) ) ) ) ) ).
tff(fact_675_sorted_Osimps,axiom,
! [Aa: list_nat] :
( hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,Aa))
<=> ( ( Aa = nil_nat )
| ? [Xs_19: list_nat,X_4: nat] :
( ( Aa = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_4),Xs_19) )
& ! [Xa: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,Xa),hAPP_l1821533378t_bool(set_nat,Xs_19)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_4),Xa)) )
& hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,Xs_19)) ) ) ) ).
tff(fact_676_insort__is__Cons,axiom,
! [F: fun_nat_nat,Aa: nat,Xs: list_nat] :
( ! [X_4: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_4),hAPP_l1821533378t_bool(set_nat,Xs)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,Aa)),hAPP_nat_nat(F,X_4))) )
=> ( hAPP_l248265089st_nat(hAPP_n280362926st_nat(linord1850739451at_nat(F),Aa),Xs) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,Aa),Xs) ) ) ).
tff(fact_677_sorted_OCons,axiom,
! [X_1: nat,Xs: list_nat] :
( ! [X_4: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_4),hAPP_l1821533378t_bool(set_nat,Xs)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),X_4)) )
=> ( hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,Xs))
=> hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Xs))) ) ) ).
tff(fact_678_lift__Suc__mono__le,axiom,
! [N_1: nat,N_20: nat,F: fun_nat_nat] :
( ! [N_19: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,N_19)),hAPP_nat_nat(F,hAPP_nat_nat(suc,N_19))))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),N_20))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,N_1)),hAPP_nat_nat(F,N_20))) ) ) ).
tff(fact_679_Suc__le__D,axiom,
! [N: nat,M_7: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N)),M_7))
=> ? [M_5: nat] : M_7 = hAPP_nat_nat(suc,M_5) ) ).
tff(fact_680_order__refl,axiom,
! [X_86: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_86),X_86)) ).
tff(fact_681_set__subset__Cons,axiom,
! [Xs: list_A114658244le_alt,X_1: arrow_1893660431le_alt] : hBOOL(hAPP_f739344229l_bool(hAPP_f1549168575l_bool(ord_le1779845435t_bool,hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_1),Xs)))) ).
tff(fact_682_set__subset__Cons,axiom,
! [Xs: list_nat,X_1: nat] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_l1821533378t_bool(set_nat,Xs)),hAPP_l1821533378t_bool(set_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Xs)))) ).
tff(fact_683_set__subset__Cons,axiom,
! [Xs: list_bool,X_1: bool] : hBOOL(hAPP_f387058535l_bool(hAPP_f612708895l_bool(ord_le742797417l_bool,hAPP_l96088310l_bool(set_bool,Xs)),hAPP_l96088310l_bool(set_bool,hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs)))) ).
tff(fact_684_set__take__subset,axiom,
! [N_1: nat,Xs: list_A114658244le_alt] : hBOOL(hAPP_f739344229l_bool(hAPP_f1549168575l_bool(ord_le1779845435t_bool,hAPP_l2018657868t_bool(set_Ar1995755744le_alt,hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,N_1),Xs))),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs))) ).
tff(fact_685_set__take__subset,axiom,
! [N_1: nat,Xs: list_nat] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_l1821533378t_bool(set_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,N_1),Xs))),hAPP_l1821533378t_bool(set_nat,Xs))) ).
tff(fact_686_set__remove1__subset,axiom,
! [X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt] : hBOOL(hAPP_f739344229l_bool(hAPP_f1549168575l_bool(ord_le1779845435t_bool,hAPP_l2018657868t_bool(set_Ar1995755744le_alt,hAPP_l84111627le_alt(hAPP_A436723315le_alt(remove861161045le_alt,X_1),Xs))),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs))) ).
tff(fact_687_set__remove1__subset,axiom,
! [X_1: nat,Xs: list_nat] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_l1821533378t_bool(set_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(remove1_nat,X_1),Xs))),hAPP_l1821533378t_bool(set_nat,Xs))) ).
tff(fact_688_set__sublist__subset,axiom,
! [Xs: list_A114658244le_alt,I_11: fun_nat_bool] : hBOOL(hAPP_f739344229l_bool(hAPP_f1549168575l_bool(ord_le1779845435t_bool,hAPP_l2018657868t_bool(set_Ar1995755744le_alt,sublis1318771396le_alt(Xs,I_11))),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs))) ).
tff(fact_689_set__sublist__subset,axiom,
! [Xs: list_nat,I_11: fun_nat_bool] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_l1821533378t_bool(set_nat,sublist_nat(Xs,I_11))),hAPP_l1821533378t_bool(set_nat,Xs))) ).
tff(fact_690_linorder__linear,axiom,
! [X_85: nat,Y_60: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_85),Y_60))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_60),X_85)) ) ).
tff(fact_691_order__eq__iff,axiom,
! [X_1: nat,Y_2: nat] :
( ( X_1 = Y_2 )
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_2),X_1)) ) ) ).
tff(fact_692_order__eq__refl,axiom,
! [X_84: nat,Y_59: nat] :
( ( X_84 = Y_59 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_84),Y_59)) ) ).
tff(fact_693_order__antisym__conv,axiom,
! [Y_2: nat,X_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_2),X_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2))
<=> ( X_1 = Y_2 ) ) ) ).
tff(fact_694_ord__eq__le__trans,axiom,
! [C_26: nat,A_56: nat,B_54: nat] :
( ( A_56 = B_54 )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_54),C_26))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_56),C_26)) ) ) ).
tff(fact_695_xt1_I3_J,axiom,
! [C_25: nat,A_55: nat,B_53: nat] :
( ( A_55 = B_53 )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_25),B_53))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_25),A_55)) ) ) ).
tff(fact_696_ord__le__eq__trans,axiom,
! [C_24: nat,A_54: nat,B_52: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_54),B_52))
=> ( ( B_52 = C_24 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_54),C_24)) ) ) ).
tff(fact_697_xt1_I4_J,axiom,
! [C_23: nat,B_51: nat,A_53: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_51),A_53))
=> ( ( B_51 = C_23 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_23),A_53)) ) ) ).
tff(fact_698_order__antisym,axiom,
! [X_83: nat,Y_58: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_83),Y_58))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_58),X_83))
=> ( X_83 = Y_58 ) ) ) ).
tff(fact_699_order__trans,axiom,
! [Z_25: nat,X_82: nat,Y_57: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_82),Y_57))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_57),Z_25))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_82),Z_25)) ) ) ).
tff(fact_700_xt1_I5_J,axiom,
! [Y_56: nat,X_81: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_56),X_81))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_81),Y_56))
=> ( X_81 = Y_56 ) ) ) ).
tff(fact_701_xt1_I6_J,axiom,
! [Z_24: nat,Y_55: nat,X_80: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_55),X_80))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_24),Y_55))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_24),X_80)) ) ) ).
tff(fact_702_linorder__le__cases,axiom,
! [X_79: nat,Y_54: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_79),Y_54))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_54),X_79)) ) ).
tff(fact_703_set__take__subset__set__take,axiom,
! [Xs: list_A114658244le_alt,M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
=> hBOOL(hAPP_f739344229l_bool(hAPP_f1549168575l_bool(ord_le1779845435t_bool,hAPP_l2018657868t_bool(set_Ar1995755744le_alt,hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,M_1),Xs))),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,N_1),Xs)))) ) ).
tff(fact_704_set__take__subset__set__take,axiom,
! [Xs: list_nat,M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_l1821533378t_bool(set_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,M_1),Xs))),hAPP_l1821533378t_bool(set_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,N_1),Xs)))) ) ).
tff(fact_705_le__max__iff__disj,axiom,
! [Z: nat,X_1: nat,Y_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_1),Y_2)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z),X_1))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z),Y_2)) ) ) ).
tff(fact_706_max__leastR,axiom,
! [X_78: nat,Least_3: nat] :
( ! [X_4: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Least_3),X_4))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_78),Least_3) = X_78 ) ) ).
tff(fact_707_max__leastL,axiom,
! [X_77: nat,Least_2: nat] :
( ! [X_4: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Least_2),X_4))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,Least_2),X_77) = X_77 ) ) ).
tff(fact_708_max__ord__max,axiom,
ord_max_nat = max_nat(ord_less_eq_nat) ).
tff(fact_709_termination__basic__simps_I3_J,axiom,
! [Z_3: nat,X: nat,Y: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z_3))) ) ).
tff(fact_710_termination__basic__simps_I4_J,axiom,
! [Y: nat,X: nat,Z_3: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Z_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z_3))) ) ).
tff(fact_711_subsetD,axiom,
! [C_2: arrow_1893660431le_alt,A: fun_Ar17477381t_bool,B: fun_Ar17477381t_bool] :
( hBOOL(hAPP_f739344229l_bool(hAPP_f1549168575l_bool(ord_le1779845435t_bool,A),B))
=> ( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,C_2),A))
=> hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,C_2),B)) ) ) ).
tff(fact_712_subsetD,axiom,
! [C_2: nat,A: fun_nat_bool,B: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),A))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),B)) ) ) ).
tff(fact_713_min__max_Ole__supE,axiom,
! [A_52: nat,B_50: nat,X_76: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,A_52),B_50)),X_76))
=> ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_52),X_76))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_50),X_76)) ) ) ).
tff(fact_714_min__max_Osup__mono,axiom,
! [B_49: nat,D_5: nat,A_51: nat,C_22: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_51),C_22))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_49),D_5))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,A_51),B_49)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,C_22),D_5))) ) ) ).
tff(fact_715_in__mono,axiom,
! [X_1: arrow_1893660431le_alt,A: fun_Ar17477381t_bool,B: fun_Ar17477381t_bool] :
( hBOOL(hAPP_f739344229l_bool(hAPP_f1549168575l_bool(ord_le1779845435t_bool,A),B))
=> ( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),A))
=> hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),B)) ) ) ).
tff(fact_716_in__mono,axiom,
! [X_1: nat,A: fun_nat_bool,B: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B)) ) ) ).
tff(fact_717_set__rev__mp,axiom,
! [B: fun_Ar17477381t_bool,X_1: arrow_1893660431le_alt,A: fun_Ar17477381t_bool] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),A))
=> ( hBOOL(hAPP_f739344229l_bool(hAPP_f1549168575l_bool(ord_le1779845435t_bool,A),B))
=> hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),B)) ) ) ).
tff(fact_718_set__rev__mp,axiom,
! [B: fun_nat_bool,X_1: nat,A: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B)) ) ) ).
tff(fact_719_set__mp,axiom,
! [X_1: arrow_1893660431le_alt,A: fun_Ar17477381t_bool,B: fun_Ar17477381t_bool] :
( hBOOL(hAPP_f739344229l_bool(hAPP_f1549168575l_bool(ord_le1779845435t_bool,A),B))
=> ( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),A))
=> hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),B)) ) ) ).
tff(fact_720_set__mp,axiom,
! [X_1: nat,A: fun_nat_bool,B: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B)) ) ) ).
tff(fact_721_min__max_Osup_Oidem,axiom,
! [A_50: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,A_50),A_50) = A_50 ).
tff(fact_722_min__max_Osup_Ocommute,axiom,
! [A_49: nat,B_48: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,A_49),B_48) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,B_48),A_49) ).
tff(fact_723_min__max_Osup__commute,axiom,
! [X_75: nat,Y_53: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_75),Y_53) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,Y_53),X_75) ).
tff(fact_724_min__max_Osup_Oleft__idem,axiom,
! [A_48: nat,B_47: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,A_48),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,A_48),B_47)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,A_48),B_47) ).
tff(fact_725_min__max_Osup__left__idem,axiom,
! [X_74: nat,Y_52: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_74),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_74),Y_52)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_74),Y_52) ).
tff(fact_726_min__max_Osup_Oleft__commute,axiom,
! [B_46: nat,A_47: nat,C_21: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,B_46),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,A_47),C_21)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,A_47),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,B_46),C_21)) ).
tff(fact_727_min__max_Osup__left__commute,axiom,
! [X_73: nat,Y_51: nat,Z_23: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_73),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,Y_51),Z_23)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,Y_51),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_73),Z_23)) ).
tff(fact_728_min__max_Osup_Oassoc,axiom,
! [A_46: nat,B_45: nat,C_20: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,A_46),B_45)),C_20) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,A_46),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,B_45),C_20)) ).
tff(fact_729_min__max_Osup__assoc,axiom,
! [X_72: nat,Y_50: nat,Z_22: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_72),Y_50)),Z_22) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_72),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,Y_50),Z_22)) ).
tff(fact_730_le__maxI1,axiom,
! [X_71: nat,Y_49: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_71),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_71),Y_49))) ).
tff(fact_731_le__maxI2,axiom,
! [Y_48: nat,X_70: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_48),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_70),Y_48))) ).
tff(fact_732_min__max_Ole__iff__sup,axiom,
! [X_1: nat,Y_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2))
<=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_1),Y_2) = Y_2 ) ) ).
tff(fact_733_min__max_Ole__sup__iff,axiom,
! [X_1: nat,Y_2: nat,Z: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_1),Y_2)),Z))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Z))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_2),Z)) ) ) ).
tff(fact_734_min__max_Ole__supI1,axiom,
! [B_44: nat,X_69: nat,A_45: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_69),A_45))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_69),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,A_45),B_44))) ) ).
tff(fact_735_min__max_Ole__supI2,axiom,
! [A_44: nat,X_68: nat,B_43: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_68),B_43))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_68),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,A_44),B_43))) ) ).
tff(fact_736_min__max_Osup__absorb2,axiom,
! [X_67: nat,Y_47: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_67),Y_47))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_67),Y_47) = Y_47 ) ) ).
tff(fact_737_min__max_Osup__absorb1,axiom,
! [Y_46: nat,X_66: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_46),X_66))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_66),Y_46) = X_66 ) ) ).
tff(fact_738_min__max_Ole__supI,axiom,
! [B_42: nat,A_43: nat,X_65: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_43),X_65))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_42),X_65))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,A_43),B_42)),X_65)) ) ) ).
tff(fact_739_min__max_Osup__least,axiom,
! [Z_21: nat,Y_45: nat,X_64: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_45),X_64))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_21),X_64))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,Y_45),Z_21)),X_64)) ) ) ).
tff(fact_740_subsetI,axiom,
! [B: fun_Ar17477381t_bool,A: fun_Ar17477381t_bool] :
( ! [X_4: arrow_1893660431le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_4),A))
=> hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_4),B)) )
=> hBOOL(hAPP_f739344229l_bool(hAPP_f1549168575l_bool(ord_le1779845435t_bool,A),B)) ) ).
tff(fact_741_subsetI,axiom,
! [B: fun_nat_bool,A: fun_nat_bool] :
( ! [X_4: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_4),A))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_4),B)) )
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) ) ).
tff(fact_742_weak__decr__stable,axiom,
! [F: fun_nat_nat] :
( ! [I_4: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,hAPP_nat_nat(suc,I_4))),hAPP_nat_nat(F,I_4)))
=> ? [I_4: nat] :
! [K_2: nat] : hAPP_nat_nat(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_4),K_2)) = hAPP_nat_nat(F,I_4) ) ).
tff(fact_743_Max__fin__set__fold,axiom,
! [X_1: nat,Xs: list_nat] : big_linorder_Max_nat(hAPP_l1821533378t_bool(set_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Xs))) = foldl_nat_nat(ord_max_nat,X_1,Xs) ).
tff(fact_744_Collect__mono,axiom,
! [Q_1: fun_nat_bool,P_1: fun_nat_bool] :
( ! [X_4: nat] :
( hBOOL(hAPP_nat_bool(P_1,X_4))
=> hBOOL(hAPP_nat_bool(Q_1,X_4)) )
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,collect_nat(P_1)),collect_nat(Q_1))) ) ).
tff(fact_745_pred__subset__eq,axiom,
! [R_1: fun_Ar17477381t_bool,S_1: fun_Ar17477381t_bool] :
( hBOOL(hAPP_f739344229l_bool(hAPP_f1549168575l_bool(ord_le1779845435t_bool,hAPP_f1592536333t_bool(cOMBC_1310341090l_bool(member1502664802le_alt),R_1)),hAPP_f1592536333t_bool(cOMBC_1310341090l_bool(member1502664802le_alt),S_1)))
<=> hBOOL(hAPP_f739344229l_bool(hAPP_f1549168575l_bool(ord_le1779845435t_bool,R_1),S_1)) ) ).
tff(fact_746_pred__subset__eq,axiom,
! [R_1: fun_nat_bool,S_1: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),R_1)),hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),S_1)))
<=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,R_1),S_1)) ) ).
tff(fact_747_min__max_OSup__fin__set__fold,axiom,
! [X_1: nat,Xs: list_nat] : big_Sup_fin_nat(ord_max_nat,hAPP_l1821533378t_bool(set_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_1),Xs))) = foldl_nat_nat(ord_max_nat,X_1,Xs) ).
tff(fact_748_xt1_I15_J,axiom,
! [C_2: nat,Aa: nat,F: fun_nat_nat,Ba: nat] :
( ( Aa = hAPP_nat_nat(F,Ba) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_2),Ba))
=> ( ! [X_4: nat,Y_25: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_25),X_4))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,Y_25)),hAPP_nat_nat(F,X_4))) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,C_2)),Aa)) ) ) ) ).
tff(fact_749_xt1_I16_J,axiom,
! [F: fun_nat_nat,C_2: nat,Ba: nat,Aa: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Ba),Aa))
=> ( ( hAPP_nat_nat(F,Ba) = C_2 )
=> ( ! [X_4: nat,Y_25: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_25),X_4))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,Y_25)),hAPP_nat_nat(F,X_4))) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_2),hAPP_nat_nat(F,Aa))) ) ) ) ).
tff(fact_750_full__nat__induct,axiom,
! [N_1: nat,P_1: fun_nat_bool] :
( ! [N_19: nat] :
( ! [M_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M_5)),N_19))
=> hBOOL(hAPP_nat_bool(P_1,M_5)) )
=> hBOOL(hAPP_nat_bool(P_1,N_19)) )
=> hBOOL(hAPP_nat_bool(P_1,N_1)) ) ).
tff(fact_751_dual__min,axiom,
min_nat(cOMBC_nat_nat_bool(ord_less_eq_nat)) = ord_max_nat ).
tff(fact_752_list_Osize_I4_J,axiom,
! [Aa: bool,List_1: list_bool] : hAPP_list_bool_nat(size_size_list_bool,hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,Aa),List_1)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_list_bool_nat(size_size_list_bool,List_1)),hAPP_nat_nat(suc,zero_zero_nat)) ).
tff(fact_753_list_Osize_I4_J,axiom,
! [A_42: nat,List: list_nat] : hAPP_list_nat_nat(size_size_list_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,A_42),List)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_list_nat_nat(size_size_list_nat,List)),hAPP_nat_nat(suc,zero_zero_nat)) ).
tff(fact_754_list_Osize_I4_J,axiom,
! [A_42: arrow_1893660431le_alt,List: list_A114658244le_alt] : hAPP_l1084348419lt_nat(size_s2005684270le_alt,hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,A_42),List)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_l1084348419lt_nat(size_s2005684270le_alt,List)),hAPP_nat_nat(suc,zero_zero_nat)) ).
tff(fact_755_diff__Suc,axiom,
! [M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(nat_case_nat(zero_zero_nat,cOMBI_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1)) ).
tff(fact_756_min__Suc2,axiom,
! [M_1: nat,N_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,M_1),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(nat_case_nat(zero_zero_nat,cOMBB_nat_nat_nat(suc,hAPP_nat_fun_nat_nat(cOMBC_nat_nat_nat(ord_min_nat),N_1))),M_1) ).
tff(fact_757_min__max_Ole__infE,axiom,
! [X_63: nat,A_41: nat,B_41: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_63),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_41),B_41)))
=> ~ ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_63),A_41))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_63),B_41)) ) ) ).
tff(fact_758_min__max_Oinf__mono,axiom,
! [B_40: nat,D_4: nat,A_40: nat,C_19: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_40),C_19))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_40),D_4))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_40),B_40)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,C_19),D_4))) ) ) ).
tff(fact_759_min__max_Oinf__greatest,axiom,
! [Z_20: nat,X_62: nat,Y_44: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_62),Y_44))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_62),Z_20))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_62),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,Y_44),Z_20))) ) ) ).
tff(fact_760_min__max_Ole__infI,axiom,
! [B_39: nat,X_61: nat,A_39: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_61),A_39))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_61),B_39))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_61),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_39),B_39))) ) ) ).
tff(fact_761_min__max_Oinf__absorb2,axiom,
! [Y_43: nat,X_60: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_43),X_60))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_60),Y_43) = Y_43 ) ) ).
tff(fact_762_min__max_Oinf__absorb1,axiom,
! [X_59: nat,Y_42: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_59),Y_42))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_59),Y_42) = X_59 ) ) ).
tff(fact_763_min__max_Ole__infI2,axiom,
! [A_38: nat,B_38: nat,X_58: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_38),X_58))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_38),B_38)),X_58)) ) ).
tff(fact_764_min__max_Ole__infI1,axiom,
! [B_37: nat,A_37: nat,X_57: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_37),X_57))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_37),B_37)),X_57)) ) ).
tff(fact_765_min__max_Ole__inf__iff,axiom,
! [X_1: nat,Y_2: nat,Z: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,Y_2),Z)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Z)) ) ) ).
tff(fact_766_min__max_Ole__iff__inf,axiom,
! [X_1: nat,Y_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2))
<=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_1),Y_2) = X_1 ) ) ).
tff(fact_767_min__max_Oinf__le2,axiom,
! [X_56: nat,Y_41: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_56),Y_41)),Y_41)) ).
tff(fact_768_min__max_Oinf__le1,axiom,
! [X_55: nat,Y_40: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_55),Y_40)),X_55)) ).
tff(fact_769_min__max_Oinf_Oidem,axiom,
! [A_36: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_36),A_36) = A_36 ).
tff(fact_770_min__max_Oinf_Ocommute,axiom,
! [A_35: nat,B_36: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_35),B_36) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,B_36),A_35) ).
tff(fact_771_min__max_Oinf__commute,axiom,
! [X_54: nat,Y_39: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_54),Y_39) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,Y_39),X_54) ).
tff(fact_772_min__max_Oinf_Oleft__idem,axiom,
! [A_34: nat,B_35: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_34),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_34),B_35)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_34),B_35) ).
tff(fact_773_min__max_Oinf__left__idem,axiom,
! [X_53: nat,Y_38: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_53),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_53),Y_38)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_53),Y_38) ).
tff(fact_774_min__max_Oinf_Oleft__commute,axiom,
! [B_34: nat,A_33: nat,C_18: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,B_34),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_33),C_18)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_33),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,B_34),C_18)) ).
tff(fact_775_min__max_Oinf__left__commute,axiom,
! [X_52: nat,Y_37: nat,Z_19: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_52),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,Y_37),Z_19)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,Y_37),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_52),Z_19)) ).
tff(fact_776_min__max_Oinf_Oassoc,axiom,
! [A_32: nat,B_33: nat,C_17: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_32),B_33)),C_17) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_32),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,B_33),C_17)) ).
tff(fact_777_min__max_Oinf__assoc,axiom,
! [X_51: nat,Y_36: nat,Z_18: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_51),Y_36)),Z_18) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_51),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,Y_36),Z_18)) ).
tff(fact_778_min__max_Osup__inf__absorb,axiom,
! [X_50: nat,Y_35: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_50),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_50),Y_35)) = X_50 ).
tff(fact_779_min__max_Oinf__sup__absorb,axiom,
! [X_49: nat,Y_34: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_49),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_49),Y_34)) = X_49 ).
tff(fact_780_min__max_Osup__inf__distrib1,axiom,
! [X_48: nat,Y_33: nat,Z_17: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_48),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,Y_33),Z_17)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_48),Y_33)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_48),Z_17)) ).
tff(fact_781_min__max_Oinf__sup__distrib1,axiom,
! [X_47: nat,Y_32: nat,Z_16: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_47),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,Y_32),Z_16)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_47),Y_32)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_47),Z_16)) ).
tff(fact_782_min__max_Osup__inf__distrib2,axiom,
! [Y_31: nat,Z_15: nat,X_46: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,Y_31),Z_15)),X_46) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,Y_31),X_46)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,Z_15),X_46)) ).
tff(fact_783_min__max_Oinf__sup__distrib2,axiom,
! [Y_30: nat,Z_14: nat,X_45: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,Y_30),Z_14)),X_45) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,Y_30),X_45)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,Z_14),X_45)) ).
tff(fact_784_length__rev,axiom,
! [Xs_41: list_nat] : hAPP_list_nat_nat(size_size_list_nat,rev_nat(Xs_41)) = hAPP_list_nat_nat(size_size_list_nat,Xs_41) ).
tff(fact_785_length__rev,axiom,
! [Xs_41: list_A114658244le_alt] : hAPP_l1084348419lt_nat(size_s2005684270le_alt,rev_Ar377224735le_alt(Xs_41)) = hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs_41) ).
tff(fact_786_take__append,axiom,
! [N_31: nat,Xs_40: list_A114658244le_alt,Ys_13: list_A114658244le_alt] : hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,N_31),hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs_40),Ys_13)) = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,N_31),Xs_40)),hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_31),hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs_40))),Ys_13)) ).
tff(fact_787_take__append,axiom,
! [N_31: nat,Xs_40: list_nat,Ys_13: list_nat] : hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,N_31),hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs_40),Ys_13)) = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,N_31),Xs_40)),hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_31),hAPP_list_nat_nat(size_size_list_nat,Xs_40))),Ys_13)) ).
tff(fact_788_min__add__distrib__left,axiom,
! [X_44: nat,Y_29: nat,Z_13: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_44),Y_29)),Z_13) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,X_44),Z_13)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y_29),Z_13)) ).
tff(fact_789_min__0L,axiom,
! [N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,zero_zero_nat),N) = zero_zero_nat ).
tff(fact_790_min__0R,axiom,
! [N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,N),zero_zero_nat) = zero_zero_nat ).
tff(fact_791_min__Suc__Suc,axiom,
! [M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,M),N)) ).
tff(fact_792_diff__0__eq__0,axiom,
! [N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),N) = zero_zero_nat ).
tff(fact_793_minus__nat_Odiff__0,axiom,
! [M: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),zero_zero_nat) = M ).
tff(fact_794_diff__self__eq__0,axiom,
! [M: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),M) = zero_zero_nat ).
tff(fact_795_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat )
=> ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),M) = zero_zero_nat )
=> ( M = N ) ) ) ).
tff(fact_796_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),N)),hAPP_nat_nat(suc,K)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),K) ).
tff(fact_797_diff__Suc__Suc,axiom,
! [M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N) ).
tff(fact_798_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),K)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N) ).
tff(fact_799_diff__cancel,axiom,
! [K: nat,M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),M)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),N)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N) ).
tff(fact_800_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),J)),K) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),K)) ).
tff(fact_801_diff__add__inverse,axiom,
! [N: nat,M: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M)),N) = M ).
tff(fact_802_diff__add__inverse2,axiom,
! [M: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),N)),N) = M ).
tff(fact_803_diff__commute,axiom,
! [I: nat,J: nat,K: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),J)),K) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),K)),J) ).
tff(fact_804_min__diff,axiom,
! [M: nat,I: nat,N: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),I)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),I)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,M),N)),I) ).
tff(fact_805_Min_Oidem,axiom,
! [X_43: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_43),X_43) = X_43 ).
tff(fact_806_length__take,axiom,
! [N_30: nat,Xs_39: list_A114658244le_alt] : hAPP_l1084348419lt_nat(size_s2005684270le_alt,hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,N_30),Xs_39)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs_39)),N_30) ).
tff(fact_807_length__take,axiom,
! [N_30: nat,Xs_39: list_nat] : hAPP_list_nat_nat(size_size_list_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,N_30),Xs_39)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_list_nat_nat(size_size_list_nat,Xs_39)),N_30) ).
tff(fact_808_neq__if__length__neq,axiom,
! [Xs_38: list_nat,Ys_12: list_nat] :
( ( hAPP_list_nat_nat(size_size_list_nat,Xs_38) != hAPP_list_nat_nat(size_size_list_nat,Ys_12) )
=> ( Xs_38 != Ys_12 ) ) ).
tff(fact_809_take__take,axiom,
! [N_29: nat,M_6: nat,Xs_37: list_A114658244le_alt] : hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,N_29),hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,M_6),Xs_37)) = hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,N_29),M_6)),Xs_37) ).
tff(fact_810_take__take,axiom,
! [N_29: nat,M_6: nat,Xs_37: list_nat] : hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,N_29),hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,M_6),Xs_37)) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,N_29),M_6)),Xs_37) ).
tff(fact_811_diff__le__self,axiom,
! [M: nat,N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),M)) ).
tff(fact_812_diff__le__mono2,axiom,
! [L: nat,M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L),M))) ) ).
tff(fact_813_diff__le__mono,axiom,
! [L: nat,M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),L)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),L))) ) ).
tff(fact_814_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),N))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),I)) = I ) ) ).
tff(fact_815_eq__diff__iff,axiom,
! [N_1: nat,K_3: nat,M_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_3),M_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_3),N_1))
=> ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),K_3) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),K_3) )
<=> ( M_1 = N_1 ) ) ) ) ).
tff(fact_816_Nat_Odiff__diff__eq,axiom,
! [N: nat,K: nat,M: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),M))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),K)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N) ) ) ) ).
tff(fact_817_le__diff__iff,axiom,
! [N_1: nat,K_3: nat,M_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_3),M_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_3),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),K_3)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),K_3)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1)) ) ) ) ).
tff(fact_818_min__ord__min,axiom,
ord_min_nat = min_nat(ord_less_eq_nat) ).
tff(fact_819_min__le__iff__disj,axiom,
! [X_1: nat,Y_2: nat,Z: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_1),Y_2)),Z))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Z))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_2),Z)) ) ) ).
tff(fact_820_length__replicate,axiom,
! [N_28: nat,X_42: nat] : hAPP_list_nat_nat(size_size_list_nat,hAPP_nat_list_nat(replicate_nat(N_28),X_42)) = N_28 ).
tff(fact_821_length__replicate,axiom,
! [N_28: nat,X_42: arrow_1893660431le_alt] : hAPP_l1084348419lt_nat(size_s2005684270le_alt,hAPP_A1382181054le_alt(replic782356601le_alt(N_28),X_42)) = N_28 ).
tff(fact_822_length__rotate1,axiom,
! [Xs_36: list_nat] : hAPP_list_nat_nat(size_size_list_nat,rotate1_nat(Xs_36)) = hAPP_list_nat_nat(size_size_list_nat,Xs_36) ).
tff(fact_823_length__rotate1,axiom,
! [Xs_36: list_A114658244le_alt] : hAPP_l1084348419lt_nat(size_s2005684270le_alt,rotate1753650700le_alt(Xs_36)) = hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs_36) ).
tff(fact_824_min__max_Odistrib__inf__le,axiom,
! [X_41: nat,Y_28: nat,Z_12: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_41),Y_28)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_41),Z_12))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_41),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,Y_28),Z_12)))) ).
tff(fact_825_min__max_Odistrib__sup__le,axiom,
! [X_40: nat,Y_27: nat,Z_11: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_40),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,Y_27),Z_11))),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_40),Y_27)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_40),Z_11)))) ).
tff(fact_826_diff__add__0,axiom,
! [N: nat,M: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M)) = zero_zero_nat ).
tff(fact_827_list_Osize_I3_J,axiom,
hAPP_list_nat_nat(size_size_list_nat,nil_nat) = zero_zero_nat ).
tff(fact_828_list_Osize_I3_J,axiom,
hAPP_l1084348419lt_nat(size_s2005684270le_alt,nil_Ar557011903le_alt) = zero_zero_nat ).
tff(fact_829_length__0__conv,axiom,
! [Xs: list_nat] :
( ( hAPP_list_nat_nat(size_size_list_nat,Xs) = zero_zero_nat )
<=> ( Xs = nil_nat ) ) ).
tff(fact_830_length__0__conv,axiom,
! [Xs: list_A114658244le_alt] :
( ( hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs) = zero_zero_nat )
<=> ( Xs = nil_Ar557011903le_alt ) ) ).
tff(fact_831_diff__is__0__eq,axiom,
! [M_1: nat,N_1: nat] :
( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),N_1) = zero_zero_nat )
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1)) ) ).
tff(fact_832_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N) = zero_zero_nat ) ) ).
tff(fact_833_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,M)),N) = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)) ) ) ).
tff(fact_834_impossible__Cons,axiom,
! [X_1: bool,Xs: list_bool,Ys: list_bool] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_list_bool_nat(size_size_list_bool,Xs)),hAPP_list_bool_nat(size_size_list_bool,Ys)))
=> ( Xs != hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Ys) ) ) ).
tff(fact_835_impossible__Cons,axiom,
! [X_39: nat,Xs_35: list_nat,Ys_11: list_nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_list_nat_nat(size_size_list_nat,Xs_35)),hAPP_list_nat_nat(size_size_list_nat,Ys_11)))
=> ( Xs_35 != hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_39),Ys_11) ) ) ).
tff(fact_836_impossible__Cons,axiom,
! [X_39: arrow_1893660431le_alt,Xs_35: list_A114658244le_alt,Ys_11: list_A114658244le_alt] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs_35)),hAPP_l1084348419lt_nat(size_s2005684270le_alt,Ys_11)))
=> ( Xs_35 != hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_39),Ys_11) ) ) ).
tff(fact_837_diff__diff__right,axiom,
! [I: nat,K: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ) ).
tff(fact_838_le__diff__conv,axiom,
! [J_1: nat,K_3: nat,I_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),K_3)),I_1))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_3))) ) ).
tff(fact_839_le__add__diff,axiom,
! [M: nat,K: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),M)),K))) ) ).
tff(fact_840_le__add__diff__inverse,axiom,
! [N: nat,M: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)) = M ) ) ).
tff(fact_841_add__diff__assoc,axiom,
! [I: nat,K: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J)),K) ) ) ).
tff(fact_842_le__diff__conv2,axiom,
! [I_1: nat,K_3: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_3),J_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),K_3)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_3)),J_1)) ) ) ).
tff(fact_843_le__add__diff__inverse2,axiom,
! [N: nat,M: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N),M))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),N) = M ) ) ).
tff(fact_844_le__imp__diff__is__add,axiom,
! [K_3: nat,I_1: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_1))
=> ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),I_1) = K_3 )
<=> ( J_1 = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_3),I_1) ) ) ) ).
tff(fact_845_diff__add__assoc,axiom,
! [I: nat,K: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J)),K) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K)) ) ) ).
tff(fact_846_add__diff__assoc2,axiom,
! [I: nat,K: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K)),I) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),I)),K) ) ) ).
tff(fact_847_diff__add__assoc2,axiom,
! [I: nat,K: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),I)),K) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K)),I) ) ) ).
tff(fact_848_length__append,axiom,
! [Xs_34: list_A114658244le_alt,Ys_10: list_A114658244le_alt] : hAPP_l1084348419lt_nat(size_s2005684270le_alt,hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs_34),Ys_10)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs_34)),hAPP_l1084348419lt_nat(size_s2005684270le_alt,Ys_10)) ).
tff(fact_849_length__append,axiom,
! [Xs_34: list_nat,Ys_10: list_nat] : hAPP_list_nat_nat(size_size_list_nat,hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs_34),Ys_10)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_list_nat_nat(size_size_list_nat,Xs_34)),hAPP_list_nat_nat(size_size_list_nat,Ys_10)) ).
tff(fact_850_take__all,axiom,
! [Xs_33: list_A114658244le_alt,N_27: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs_33)),N_27))
=> ( hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,N_27),Xs_33) = Xs_33 ) ) ).
tff(fact_851_take__all,axiom,
! [Xs_33: list_nat,N_27: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_list_nat_nat(size_size_list_nat,Xs_33)),N_27))
=> ( hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,N_27),Xs_33) = Xs_33 ) ) ).
tff(fact_852_take__replicate,axiom,
! [I_10: nat,K_5: nat,X_38: arrow_1893660431le_alt] : hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,I_10),hAPP_A1382181054le_alt(replic782356601le_alt(K_5),X_38)) = hAPP_A1382181054le_alt(replic782356601le_alt(hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,I_10),K_5)),X_38) ).
tff(fact_853_take__replicate,axiom,
! [I_10: nat,K_5: nat,X_38: nat] : hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,I_10),hAPP_nat_list_nat(replicate_nat(K_5),X_38)) = hAPP_nat_list_nat(replicate_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,I_10),K_5)),X_38) ).
tff(fact_854_length__takeWhile__le,axiom,
! [P_1: fun_nat_bool,Xs: list_nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_list_nat_nat(size_size_list_nat,takeWhile_nat(P_1,Xs))),hAPP_list_nat_nat(size_size_list_nat,Xs))) ).
tff(fact_855_length__takeWhile__le,axiom,
! [P_1: fun_Ar17477381t_bool,Xs: list_A114658244le_alt] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_l1084348419lt_nat(size_s2005684270le_alt,takeWh967109304le_alt(P_1,Xs))),hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs))) ).
tff(fact_856_length__insort,axiom,
! [F: fun_nat_nat,X_1: nat,Xs: list_nat] : hAPP_list_nat_nat(size_size_list_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(linord1850739451at_nat(F),X_1),Xs)) = hAPP_nat_nat(suc,hAPP_list_nat_nat(size_size_list_nat,Xs)) ).
tff(fact_857_length__dropWhile__le,axiom,
! [P_1: fun_nat_bool,Xs: list_nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_list_nat_nat(size_size_list_nat,dropWhile_nat(P_1,Xs))),hAPP_list_nat_nat(size_size_list_nat,Xs))) ).
tff(fact_858_length__dropWhile__le,axiom,
! [P_1: fun_Ar17477381t_bool,Xs: list_A114658244le_alt] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_l1084348419lt_nat(size_s2005684270le_alt,dropWh587599712le_alt(P_1,Xs))),hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs))) ).
tff(fact_859_nat__minus__add__max,axiom,
! [N: nat,M: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),M)),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,N),M) ).
tff(fact_860_takeWhile__eq__take,axiom,
! [P_1: fun_Ar17477381t_bool,Xs: list_A114658244le_alt] : takeWh967109304le_alt(P_1,Xs) = hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,hAPP_l1084348419lt_nat(size_s2005684270le_alt,takeWh967109304le_alt(P_1,Xs))),Xs) ).
tff(fact_861_takeWhile__eq__take,axiom,
! [P_1: fun_nat_bool,Xs: list_nat] : takeWhile_nat(P_1,Xs) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,hAPP_list_nat_nat(size_size_list_nat,takeWhile_nat(P_1,Xs))),Xs) ).
tff(fact_862_dual__max,axiom,
max_nat(cOMBC_nat_nat_bool(ord_less_eq_nat)) = ord_min_nat ).
tff(fact_863_length__splice,axiom,
! [Xs_32: list_nat,Ys_9: list_nat] : hAPP_list_nat_nat(size_size_list_nat,splice_nat(Xs_32,Ys_9)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_list_nat_nat(size_size_list_nat,Xs_32)),hAPP_list_nat_nat(size_size_list_nat,Ys_9)) ).
tff(fact_864_length__splice,axiom,
! [Xs_32: list_A114658244le_alt,Ys_9: list_A114658244le_alt] : hAPP_l1084348419lt_nat(size_s2005684270le_alt,splice791716242le_alt(Xs_32,Ys_9)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs_32)),hAPP_l1084348419lt_nat(size_s2005684270le_alt,Ys_9)) ).
tff(fact_865_diff__Suc__diff__eq2,axiom,
! [M: nat,K: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K))),M) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(suc,J)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),M)) ) ) ).
tff(fact_866_diff__Suc__diff__eq1,axiom,
! [M: nat,K: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),J))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),K))) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K)),hAPP_nat_nat(suc,J)) ) ) ).
tff(fact_867_nth__append__length,axiom,
! [Xs: list_bool,X_1: bool,Ys: list_bool] :
( hBOOL(hAPP_nat_bool(nth_bool(hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,Xs),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Ys))),hAPP_list_bool_nat(size_size_list_bool,Xs)))
<=> hBOOL(X_1) ) ).
tff(fact_868_nth__append__length,axiom,
! [Xs_31: list_A114658244le_alt,X_37: arrow_1893660431le_alt,Ys_8: list_A114658244le_alt] : hAPP_n832063962le_alt(nth_Ar1544180416le_alt(hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs_31),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_37),Ys_8))),hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs_31)) = X_37 ).
tff(fact_869_nth__append__length,axiom,
! [Xs_31: list_nat,X_37: nat,Ys_8: list_nat] : hAPP_nat_nat(nth_nat(hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs_31),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_37),Ys_8))),hAPP_list_nat_nat(size_size_list_nat,Xs_31)) = X_37 ).
tff(fact_870_nth__append__length__plus,axiom,
! [Xs: list_bool,Ys: list_bool,N_1: nat] :
( hBOOL(hAPP_nat_bool(nth_bool(hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,Xs),Ys)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_list_bool_nat(size_size_list_bool,Xs)),N_1)))
<=> hBOOL(hAPP_nat_bool(nth_bool(Ys),N_1)) ) ).
tff(fact_871_nth__append__length__plus,axiom,
! [Xs_30: list_A114658244le_alt,Ys_7: list_A114658244le_alt,N_26: nat] : hAPP_n832063962le_alt(nth_Ar1544180416le_alt(hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs_30),Ys_7)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs_30)),N_26)) = hAPP_n832063962le_alt(nth_Ar1544180416le_alt(Ys_7),N_26) ).
tff(fact_872_nth__append__length__plus,axiom,
! [Xs_30: list_nat,Ys_7: list_nat,N_26: nat] : hAPP_nat_nat(nth_nat(hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs_30),Ys_7)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_list_nat_nat(size_size_list_nat,Xs_30)),N_26)) = hAPP_nat_nat(nth_nat(Ys_7),N_26) ).
tff(fact_873_Max_Oidem,axiom,
! [X_36: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_36),X_36) = X_36 ).
tff(fact_874_sublist__append,axiom,
! [L_2: list_A114658244le_alt,L_1: list_A114658244le_alt,A: fun_nat_bool] : sublis1318771396le_alt(hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,L_2),L_1),A) = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,sublis1318771396le_alt(L_2,A)),sublis1318771396le_alt(L_1,collect_nat(hAPP_f800510211t_bool(cOMBC_226598744l_bool(cOMBB_1146692694ol_nat(member_nat,hAPP_nat_fun_nat_nat(cOMBC_nat_nat_nat(plus_plus_nat),hAPP_l1084348419lt_nat(size_s2005684270le_alt,L_2)))),A)))) ).
tff(fact_875_sublist__append,axiom,
! [L_2: list_nat,L_1: list_nat,A: fun_nat_bool] : sublist_nat(hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,L_2),L_1),A) = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,sublist_nat(L_2,A)),sublist_nat(L_1,collect_nat(hAPP_f800510211t_bool(cOMBC_226598744l_bool(cOMBB_1146692694ol_nat(member_nat,hAPP_nat_fun_nat_nat(cOMBC_nat_nat_nat(plus_plus_nat),hAPP_list_nat_nat(size_size_list_nat,L_2)))),A)))) ).
tff(fact_876_min__Suc1,axiom,
! [N_1: nat,M_1: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(suc,N_1)),M_1) = hAPP_nat_nat(nat_case_nat(zero_zero_nat,cOMBB_nat_nat_nat(suc,hAPP_nat_fun_nat_nat(ord_min_nat,N_1))),M_1) ).
tff(fact_877_Suc__length__conv,axiom,
! [N_1: nat,Xs: list_bool] :
( ( hAPP_nat_nat(suc,N_1) = hAPP_list_bool_nat(size_size_list_bool,Xs) )
<=> ? [Y_25: bool,Ys_6: list_bool] :
( ( Xs = hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,Y_25),Ys_6) )
& ( hAPP_list_bool_nat(size_size_list_bool,Ys_6) = N_1 ) ) ) ).
tff(fact_878_Suc__length__conv,axiom,
! [N_1: nat,Xs: list_nat] :
( ( hAPP_nat_nat(suc,N_1) = hAPP_list_nat_nat(size_size_list_nat,Xs) )
<=> ? [Y_25: nat,Ys_6: list_nat] :
( ( Xs = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,Y_25),Ys_6) )
& ( hAPP_list_nat_nat(size_size_list_nat,Ys_6) = N_1 ) ) ) ).
tff(fact_879_Suc__length__conv,axiom,
! [N_1: nat,Xs: list_A114658244le_alt] :
( ( hAPP_nat_nat(suc,N_1) = hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs) )
<=> ? [Y_25: arrow_1893660431le_alt,Ys_6: list_A114658244le_alt] :
( ( Xs = hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,Y_25),Ys_6) )
& ( hAPP_l1084348419lt_nat(size_s2005684270le_alt,Ys_6) = N_1 ) ) ) ).
tff(fact_880_length__Suc__conv,axiom,
! [Xs: list_bool,N_1: nat] :
( ( hAPP_list_bool_nat(size_size_list_bool,Xs) = hAPP_nat_nat(suc,N_1) )
<=> ? [Y_25: bool,Ys_6: list_bool] :
( ( Xs = hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,Y_25),Ys_6) )
& ( hAPP_list_bool_nat(size_size_list_bool,Ys_6) = N_1 ) ) ) ).
tff(fact_881_length__Suc__conv,axiom,
! [Xs: list_nat,N_1: nat] :
( ( hAPP_list_nat_nat(size_size_list_nat,Xs) = hAPP_nat_nat(suc,N_1) )
<=> ? [Y_25: nat,Ys_6: list_nat] :
( ( Xs = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,Y_25),Ys_6) )
& ( hAPP_list_nat_nat(size_size_list_nat,Ys_6) = N_1 ) ) ) ).
tff(fact_882_length__Suc__conv,axiom,
! [Xs: list_A114658244le_alt,N_1: nat] :
( ( hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs) = hAPP_nat_nat(suc,N_1) )
<=> ? [Y_25: arrow_1893660431le_alt,Ys_6: list_A114658244le_alt] :
( ( Xs = hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,Y_25),Ys_6) )
& ( hAPP_l1084348419lt_nat(size_s2005684270le_alt,Ys_6) = N_1 ) ) ) ).
tff(fact_883_min__max_Odistrib__imp2,axiom,
! [X_35: nat,Y_26: nat,Z_10: nat] :
( ! [X_4: nat,Y_25: nat,Z_9: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_4),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,Y_25),Z_9)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_4),Y_25)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_4),Z_9))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_35),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,Y_26),Z_10)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_35),Y_26)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_35),Z_10)) ) ) ).
tff(fact_884_min__max_Odistrib__imp1,axiom,
! [X_34: nat,Y_24: nat,Z_8: nat] :
( ! [X_4: nat,Y_25: nat,Z_9: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_4),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,Y_25),Z_9)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_4),Y_25)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_4),Z_9))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_34),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,Y_24),Z_8)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_34),Y_24)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_34),Z_8)) ) ) ).
tff(fact_885_zero__induct__lemma,axiom,
! [I_1: nat,P_1: fun_nat_bool,K_3: nat] :
( hBOOL(hAPP_nat_bool(P_1,K_3))
=> ( ! [N_19: nat] :
( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,N_19)))
=> hBOOL(hAPP_nat_bool(P_1,N_19)) )
=> hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,K_3),I_1))) ) ) ).
tff(fact_886_min__leastL,axiom,
! [X_33: nat,Least_1: nat] :
( ! [X_4: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Least_1),X_4))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,Least_1),X_33) = Least_1 ) ) ).
tff(fact_887_DiffI,axiom,
! [B: fun_Ar17477381t_bool,C_2: arrow_1893660431le_alt,A: fun_Ar17477381t_bool] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,C_2),A))
=> ( ~ hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,C_2),B))
=> hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,C_2),hAPP_f1592536333t_bool(hAPP_f571726183t_bool(minus_1875147572t_bool,A),B))) ) ) ).
tff(fact_888_DiffI,axiom,
! [B: fun_nat_bool,C_2: nat,A: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),A))
=> ( ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),B))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B))) ) ) ).
tff(fact_889_DiffE,axiom,
! [C_2: arrow_1893660431le_alt,A: fun_Ar17477381t_bool,B: fun_Ar17477381t_bool] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,C_2),hAPP_f1592536333t_bool(hAPP_f571726183t_bool(minus_1875147572t_bool,A),B)))
=> ~ ( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,C_2),A))
=> hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,C_2),B)) ) ) ).
tff(fact_890_DiffE,axiom,
! [C_2: nat,A: fun_nat_bool,B: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))
=> ~ ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),A))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),B)) ) ) ).
tff(fact_891_set__diff__eq,axiom,
! [A: fun_Ar17477381t_bool,B: fun_Ar17477381t_bool] : hAPP_f1592536333t_bool(hAPP_f571726183t_bool(minus_1875147572t_bool,A),B) = collec12892580le_alt(cOMBS_119208490l_bool(cOMBB_1234541833le_alt(fconj,hAPP_f1592536333t_bool(cOMBC_1310341090l_bool(member1502664802le_alt),A)),cOMBB_388242475le_alt(fNot,hAPP_f1592536333t_bool(cOMBC_1310341090l_bool(member1502664802le_alt),B)))) ).
tff(fact_892_set__diff__eq,axiom,
! [A: fun_nat_bool,B: fun_nat_bool] : hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B) = collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),A)),cOMBB_bool_bool_nat(fNot,hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),B)))) ).
tff(fact_893_Diff__iff,axiom,
! [C_2: arrow_1893660431le_alt,A: fun_Ar17477381t_bool,B: fun_Ar17477381t_bool] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,C_2),hAPP_f1592536333t_bool(hAPP_f571726183t_bool(minus_1875147572t_bool,A),B)))
<=> ( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,C_2),A))
& ~ hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,C_2),B)) ) ) ).
tff(fact_894_Diff__iff,axiom,
! [C_2: nat,A: fun_nat_bool,B: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))
<=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),A))
& ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),B)) ) ) ).
tff(fact_895_DiffD1,axiom,
! [C_2: arrow_1893660431le_alt,A: fun_Ar17477381t_bool,B: fun_Ar17477381t_bool] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,C_2),hAPP_f1592536333t_bool(hAPP_f571726183t_bool(minus_1875147572t_bool,A),B)))
=> hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,C_2),A)) ) ).
tff(fact_896_DiffD1,axiom,
! [C_2: nat,A: fun_nat_bool,B: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),A)) ) ).
tff(fact_897_DiffD2,axiom,
! [C_2: arrow_1893660431le_alt,A: fun_Ar17477381t_bool,B: fun_Ar17477381t_bool] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,C_2),hAPP_f1592536333t_bool(hAPP_f571726183t_bool(minus_1875147572t_bool,A),B)))
=> ~ hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,C_2),B)) ) ).
tff(fact_898_DiffD2,axiom,
! [C_2: nat,A: fun_nat_bool,B: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,A),B)))
=> ~ hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),B)) ) ).
tff(fact_899_min__leastR,axiom,
! [X_32: nat,Least: nat] :
( ! [X_4: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Least),X_4))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_32),Least) = Least ) ) ).
tff(fact_900_take__Suc__conv__app__nth,axiom,
! [I_1: nat,Xs: list_bool] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),hAPP_list_bool_nat(size_size_list_bool,Xs)))
=> ( hAPP_l1189022293t_bool(hAPP_n479129736t_bool(take_bool,hAPP_nat_nat(suc,I_1)),Xs) = hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,hAPP_l1189022293t_bool(hAPP_n479129736t_bool(take_bool,I_1),Xs)),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,hAPP_nat_bool(nth_bool(Xs),I_1)),nil_bool)) ) ) ).
tff(fact_901_take__Suc__conv__app__nth,axiom,
! [I_9: nat,Xs_29: list_A114658244le_alt] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_9),hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs_29)))
=> ( hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,hAPP_nat_nat(suc,I_9)),Xs_29) = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,I_9),Xs_29)),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,hAPP_n832063962le_alt(nth_Ar1544180416le_alt(Xs_29),I_9)),nil_Ar557011903le_alt)) ) ) ).
tff(fact_902_take__Suc__conv__app__nth,axiom,
! [I_9: nat,Xs_29: list_nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_9),hAPP_list_nat_nat(size_size_list_nat,Xs_29)))
=> ( hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,hAPP_nat_nat(suc,I_9)),Xs_29) = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,I_9),Xs_29)),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,hAPP_nat_nat(nth_nat(Xs_29),I_9)),nil_nat)) ) ) ).
tff(fact_903_sorted__rev__nth__mono,axiom,
! [I_8: nat,J_4: nat,Xs_28: list_nat] :
( hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,rev_nat(Xs_28)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_8),J_4))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_4),hAPP_list_nat_nat(size_size_list_nat,Xs_28)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(nth_nat(Xs_28),J_4)),hAPP_nat_nat(nth_nat(Xs_28),I_8))) ) ) ) ).
tff(fact_904_less__zeroE,axiom,
! [N: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat)) ).
tff(fact_905_lessI,axiom,
! [N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(suc,N))) ).
tff(fact_906_Suc__mono,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N))) ) ).
tff(fact_907_zero__less__Suc,axiom,
! [N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(suc,N))) ).
tff(fact_908_less__imp__diff__less,axiom,
! [N: nat,J: nat,K: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),N)),K)) ) ).
tff(fact_909_diff__less__mono2,axiom,
! [L: nat,M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),L))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L),N)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,L),M))) ) ) ).
tff(fact_910_xt1_I8_J,axiom,
! [Z_7: nat,Y_23: nat,X_31: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_23),X_31))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_7),Y_23))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_7),X_31)) ) ) ).
tff(fact_911_order__le__less__trans,axiom,
! [Z_6: nat,X_30: nat,Y_22: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_30),Y_22))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_22),Z_6))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_30),Z_6)) ) ) ).
tff(fact_912_xt1_I7_J,axiom,
! [Z_5: nat,Y_21: nat,X_29: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_21),X_29))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Z_5),Y_21))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_5),X_29)) ) ) ).
tff(fact_913_order__less__le__trans,axiom,
! [Z_4: nat,X_28: nat,Y_20: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_28),Y_20))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_20),Z_4))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_28),Z_4)) ) ) ).
tff(fact_914_xt1_I11_J,axiom,
! [B_32: nat,A_31: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_32),A_31))
=> ( ( A_31 != B_32 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_32),A_31)) ) ) ).
tff(fact_915_order__le__neq__trans,axiom,
! [A_30: nat,B_31: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_30),B_31))
=> ( ( A_30 != B_31 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_30),B_31)) ) ) ).
tff(fact_916_order__le__imp__less__or__eq,axiom,
! [X_27: nat,Y_19: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_27),Y_19))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_27),Y_19))
| ( X_27 = Y_19 ) ) ) ).
tff(fact_917_linorder__antisym__conv2,axiom,
! [X_1: nat,Y_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2))
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2))
<=> ( X_1 = Y_2 ) ) ) ).
tff(fact_918_termination__basic__simps_I5_J,axiom,
! [X: nat,Y: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X),Y)) ) ).
tff(fact_919_order__less__imp__le,axiom,
! [X_26: nat,Y_18: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_26),Y_18))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_26),Y_18)) ) ).
tff(fact_920_leD,axiom,
! [Y_17: nat,X_25: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_17),X_25))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_25),Y_17)) ) ).
tff(fact_921_xt1_I12_J,axiom,
! [A_29: nat,B_30: nat] :
( ( A_29 != B_30 )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_30),A_29))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_30),A_29)) ) ) ).
tff(fact_922_order__neq__le__trans,axiom,
! [A_28: nat,B_29: nat] :
( ( A_28 != B_29 )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_28),B_29))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_28),B_29)) ) ) ).
tff(fact_923_linorder__antisym__conv1,axiom,
! [X_1: nat,Y_2: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2))
<=> ( X_1 = Y_2 ) ) ) ).
tff(fact_924_not__leE,axiom,
! [Y_16: nat,X_24: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_16),X_24))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_24),Y_16)) ) ).
tff(fact_925_leI,axiom,
! [X_23: nat,Y_15: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_23),Y_15))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_15),X_23)) ) ).
tff(fact_926_order__le__less,axiom,
! [X_1: nat,Y_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2))
| ( X_1 = Y_2 ) ) ) ).
tff(fact_927_less__le__not__le,axiom,
! [X_1: nat,Y_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2))
& ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_2),X_1)) ) ) ).
tff(fact_928_order__less__le,axiom,
! [X_1: nat,Y_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2))
& ( X_1 != Y_2 ) ) ) ).
tff(fact_929_linorder__le__less__linear,axiom,
! [X_22: nat,Y_14: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_22),Y_14))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_14),X_22)) ) ).
tff(fact_930_linorder__not__le,axiom,
! [X_1: nat,Y_2: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),Y_2))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_2),X_1)) ) ).
tff(fact_931_linorder__not__less,axiom,
! [X_1: nat,Y_2: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Y_2),X_1)) ) ).
tff(fact_932_termination__basic__simps_I2_J,axiom,
! [Y: nat,X: nat,Z_3: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Z_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z_3))) ) ).
tff(fact_933_termination__basic__simps_I1_J,axiom,
! [Z_3: nat,X: nat,Y: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Y),Z_3))) ) ).
tff(fact_934_not__less0,axiom,
! [N: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat)) ).
tff(fact_935_neq0__conv,axiom,
! [N_1: nat] :
( ( N_1 != zero_zero_nat )
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) ) ).
tff(fact_936_less__nat__zero__code,axiom,
! [N: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),zero_zero_nat)) ).
tff(fact_937_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> ( N != zero_zero_nat ) ) ).
tff(fact_938_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N)) ) ).
tff(fact_939_not__less__eq,axiom,
! [M_1: nat,N_1: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(suc,M_1))) ) ).
tff(fact_940_less__Suc__eq,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),hAPP_nat_nat(suc,N_1)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
| ( M_1 = N_1 ) ) ) ).
tff(fact_941_Suc__less__eq,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M_1)),hAPP_nat_nat(suc,N_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) ) ).
tff(fact_942_not__less__less__Suc__eq,axiom,
! [N_1: nat,M_1: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),M_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(suc,M_1)))
<=> ( N_1 = M_1 ) ) ) ).
tff(fact_943_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),M))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),hAPP_nat_nat(suc,M)))
=> ( M = N ) ) ) ).
tff(fact_944_less__SucI,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,N))) ) ).
tff(fact_945_Suc__lessI,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> ( ( hAPP_nat_nat(suc,M) != N )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),N)) ) ) ).
tff(fact_946_less__trans__Suc,axiom,
! [K: nat,I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,I)),K)) ) ) ).
tff(fact_947_less__SucE,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,N)))
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> ( M = N ) ) ) ).
tff(fact_948_Suc__lessD,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ).
tff(fact_949_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,M)),hAPP_nat_nat(suc,N)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ).
tff(fact_950_not__add__less1,axiom,
! [I: nat,J: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J)),I)) ).
tff(fact_951_not__add__less2,axiom,
! [J: nat,I: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),I)),I)) ).
tff(fact_952_nat__add__left__cancel__less,axiom,
! [K_3: nat,M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_3),M_1)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K_3),N_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) ) ).
tff(fact_953_trans__less__add1,axiom,
! [M: nat,I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),M))) ) ).
tff(fact_954_trans__less__add2,axiom,
! [M: nat,I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),J))) ) ).
tff(fact_955_add__less__mono1,axiom,
! [K: nat,I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ).
tff(fact_956_add__less__mono,axiom,
! [K: nat,L: nat,I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K),L))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),L))) ) ) ).
tff(fact_957_less__add__eq__less,axiom,
! [M: nat,N: nat,K: nat,L: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,K),L))
=> ( ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),L) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,K),N) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ) ).
tff(fact_958_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),J)),K))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),K)) ) ).
tff(fact_959_less__max__iff__disj,axiom,
! [Z: nat,X_1: nat,Y_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_1),Y_2)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z),X_1))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z),Y_2)) ) ) ).
tff(fact_960_max__less__iff__conj,axiom,
! [X_1: nat,Y_2: nat,Z: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,X_1),Y_2)),Z))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Z))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_2),Z)) ) ) ).
tff(fact_961_nat__less__le,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
& ( M_1 != N_1 ) ) ) ).
tff(fact_962_le__eq__less__or__eq,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
| ( M_1 = N_1 ) ) ) ).
tff(fact_963_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).
tff(fact_964_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> ( ( M != N )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ) ).
tff(fact_965_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
| ( M = N ) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N)) ) ).
tff(fact_966_nat__less__cases,axiom,
! [P_1: fun_nat_fun_nat_bool,M_1: nat,N_1: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N_1),M_1)) )
=> ( ( ( M_1 = N_1 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N_1),M_1)) )
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),M_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N_1),M_1)) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,N_1),M_1)) ) ) ) ).
tff(fact_967_less__not__refl3,axiom,
! [S: nat,T: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,S),T))
=> ( S != T ) ) ).
tff(fact_968_less__not__refl2,axiom,
! [N: nat,M: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),M))
=> ( M != N ) ) ).
tff(fact_969_less__irrefl__nat,axiom,
! [N: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N)) ).
tff(fact_970_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X),Y))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y),X)) ) ) ).
tff(fact_971_nat__neq__iff,axiom,
! [M_1: nat,N_1: nat] :
( ( M_1 != N_1 )
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),M_1)) ) ) ).
tff(fact_972_less__not__refl,axiom,
! [N: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N),N)) ).
tff(fact_973_add__less__cancel__right,axiom,
! [Aa: nat,C_2: nat,Ba: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Aa),C_2)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ba),C_2)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Aa),Ba)) ) ).
tff(fact_974_add__less__cancel__left,axiom,
! [C_2: nat,Aa: nat,Ba: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_2),Aa)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_2),Ba)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Aa),Ba)) ) ).
tff(fact_975_add__strict__right__mono,axiom,
! [C_16: nat,A_27: nat,B_28: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_27),B_28))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_27),C_16)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_28),C_16))) ) ).
tff(fact_976_add__strict__left__mono,axiom,
! [C_15: nat,A_26: nat,B_27: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_26),B_27))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_15),A_26)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_15),B_27))) ) ).
tff(fact_977_add__strict__mono,axiom,
! [C_14: nat,D_3: nat,A_25: nat,B_26: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_25),B_26))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_14),D_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_25),C_14)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_26),D_3))) ) ) ).
tff(fact_978_add__less__imp__less__right,axiom,
! [A_24: nat,C_13: nat,B_25: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_24),C_13)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_25),C_13)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_24),B_25)) ) ).
tff(fact_979_add__less__imp__less__left,axiom,
! [C_12: nat,A_23: nat,B_24: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_12),A_23)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,C_12),B_24)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_23),B_24)) ) ).
tff(fact_980_linorder__cases,axiom,
! [X_21: nat,Y_13: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_21),Y_13))
=> ( ( X_21 != Y_13 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_13),X_21)) ) ) ).
tff(fact_981_order__less__asym,axiom,
! [X_20: nat,Y_12: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_20),Y_12))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_12),X_20)) ) ).
tff(fact_982_xt1_I10_J,axiom,
! [Z_2: nat,Y_11: nat,X_19: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_11),X_19))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_2),Y_11))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z_2),X_19)) ) ) ).
tff(fact_983_order__less__trans,axiom,
! [Z_1: nat,X_18: nat,Y_10: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_18),Y_10))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_10),Z_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_18),Z_1)) ) ) ).
tff(fact_984_xt1_I2_J,axiom,
! [C_11: nat,B_23: nat,A_22: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_23),A_22))
=> ( ( B_23 = C_11 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_11),A_22)) ) ) ).
tff(fact_985_ord__less__eq__trans,axiom,
! [C_10: nat,A_21: nat,B_22: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_21),B_22))
=> ( ( B_22 = C_10 )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_21),C_10)) ) ) ).
tff(fact_986_xt1_I1_J,axiom,
! [C_9: nat,A_20: nat,B_21: nat] :
( ( A_20 = B_21 )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_9),B_21))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_9),A_20)) ) ) ).
tff(fact_987_ord__eq__less__trans,axiom,
! [C_8: nat,A_19: nat,B_20: nat] :
( ( A_19 = B_20 )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_20),C_8))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_19),C_8)) ) ) ).
tff(fact_988_xt1_I9_J,axiom,
! [B_19: nat,A_18: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_19),A_18))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_18),B_19)) ) ).
tff(fact_989_order__less__asym_H,axiom,
! [A_17: nat,B_18: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_17),B_18))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_18),A_17)) ) ).
tff(fact_990_order__less__imp__triv,axiom,
! [P_1: bool,X_1: nat,Y_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_2),X_1))
=> hBOOL(P_1) ) ) ).
tff(fact_991_order__less__imp__not__eq2,axiom,
! [X_17: nat,Y_9: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_17),Y_9))
=> ( Y_9 != X_17 ) ) ).
tff(fact_992_order__less__imp__not__eq,axiom,
! [X_16: nat,Y_8: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_16),Y_8))
=> ( X_16 != Y_8 ) ) ).
tff(fact_993_order__less__imp__not__less,axiom,
! [X_15: nat,Y_7: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_15),Y_7))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_7),X_15)) ) ).
tff(fact_994_order__less__not__sym,axiom,
! [X_14: nat,Y_6: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_14),Y_6))
=> ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_6),X_14)) ) ).
tff(fact_995_less__imp__neq,axiom,
! [X_13: nat,Y_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_13),Y_5))
=> ( X_13 != Y_5 ) ) ).
tff(fact_996_linorder__neqE,axiom,
! [X_12: nat,Y_4: nat] :
( ( X_12 != Y_4 )
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_12),Y_4))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_4),X_12)) ) ) ).
tff(fact_997_linorder__antisym__conv3,axiom,
! [Y_2: nat,X_1: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_2),X_1))
=> ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2))
<=> ( X_1 = Y_2 ) ) ) ).
tff(fact_998_linorder__less__linear,axiom,
! [X_11: nat,Y_3: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_11),Y_3))
| ( X_11 = Y_3 )
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_3),X_11)) ) ).
tff(fact_999_not__less__iff__gr__or__eq,axiom,
! [X_1: nat,Y_2: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_2),X_1))
| ( X_1 = Y_2 ) ) ) ).
tff(fact_1000_linorder__neq__iff,axiom,
! [X_1: nat,Y_2: nat] :
( ( X_1 != Y_2 )
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Y_2))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_2),X_1)) ) ) ).
tff(fact_1001_order__less__irrefl,axiom,
! [X_10: nat] : ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_10),X_10)) ).
tff(fact_1002_min__max_Oless__supI2,axiom,
! [A_16: nat,X_9: nat,B_17: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_9),B_17))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_9),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,A_16),B_17))) ) ).
tff(fact_1003_min__max_Oless__supI1,axiom,
! [B_16: nat,X_8: nat,A_15: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_8),A_15))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_8),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_max_nat,A_15),B_16))) ) ).
tff(fact_1004_min__less__iff__disj,axiom,
! [X_1: nat,Y_2: nat,Z: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_1),Y_2)),Z))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,X_1),Z))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_2),Z)) ) ) ).
tff(fact_1005_min__less__iff__conj,axiom,
! [Z: nat,X_1: nat,Y_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z),hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,X_1),Y_2)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z),X_1))
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Z),Y_2)) ) ) ).
tff(fact_1006_min__max_Oless__infI2,axiom,
! [A_14: nat,B_15: nat,X_7: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_15),X_7))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_14),B_15)),X_7)) ) ).
tff(fact_1007_min__max_Oless__infI1,axiom,
! [B_14: nat,A_13: nat,X_6: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_13),X_6))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(ord_min_nat,A_13),B_14)),X_6)) ) ).
tff(fact_1008_add__le__less__mono,axiom,
! [C_7: nat,D_2: nat,A_12: nat,B_13: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_12),B_13))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,C_7),D_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_12),C_7)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_13),D_2))) ) ) ).
tff(fact_1009_add__less__le__mono,axiom,
! [C_6: nat,D_1: nat,A_11: nat,B_12: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_11),B_12))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_6),D_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_11),C_6)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,B_12),D_1))) ) ) ).
tff(fact_1010_add__pos__pos,axiom,
! [B_11: nat,A_10: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_10))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_11))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_10),B_11))) ) ) ).
tff(fact_1011_add__neg__neg,axiom,
! [B_10: nat,A_9: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_9),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_10),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_9),B_10)),zero_zero_nat)) ) ) ).
tff(fact_1012_gr0__conv__Suc,axiom,
! [N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1))
<=> ? [M_5: nat] : N_1 = hAPP_nat_nat(suc,M_5) ) ).
tff(fact_1013_less__Suc0,axiom,
! [N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(suc,zero_zero_nat)))
<=> ( N_1 = zero_zero_nat ) ) ).
tff(fact_1014_less__Suc__eq__0__disj,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),hAPP_nat_nat(suc,N_1)))
<=> ( ( M_1 = zero_zero_nat )
| ? [J_2: nat] :
( ( M_1 = hAPP_nat_nat(suc,J_2) )
& hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_2),N_1)) ) ) ) ).
tff(fact_1015_add__gr__0,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),N_1)))
<=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M_1))
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N_1)) ) ) ).
tff(fact_1016_less__add__Suc1,axiom,
! [I: nat,M: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),M)))) ).
tff(fact_1017_less__add__Suc2,axiom,
! [I: nat,M: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),I)))) ).
tff(fact_1018_less__iff__Suc__add,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1))
<=> ? [K_2: nat] : N_1 = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),K_2)) ) ).
tff(fact_1019_less__eq__Suc__le,axiom,
! [N_1: nat,M_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),M_1))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N_1)),M_1)) ) ).
tff(fact_1020_less__Suc__eq__le,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),hAPP_nat_nat(suc,N_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1)) ) ).
tff(fact_1021_Suc__le__eq,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M_1)),N_1))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) ) ).
tff(fact_1022_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),hAPP_nat_nat(suc,N))) ) ).
tff(fact_1023_Suc__leI,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M)),N)) ) ).
tff(fact_1024_le__less__Suc__eq,axiom,
! [M_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_1),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_nat_nat(suc,M_1)))
<=> ( N_1 = M_1 ) ) ) ).
tff(fact_1025_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M)),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N)) ) ).
tff(fact_1026_diff__less,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),M))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),M)) ) ) ).
tff(fact_1027_zero__less__diff,axiom,
! [N_1: nat,M_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),M_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) ) ).
tff(fact_1028_diff__less__Suc,axiom,
! [M: nat,N: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)),hAPP_nat_nat(suc,M))) ).
tff(fact_1029_add__diff__inverse,axiom,
! [M: nat,N: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M),N)) = M ) ) ).
tff(fact_1030_less__diff__conv,axiom,
! [I_1: nat,J_1: nat,K_3: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J_1),K_3)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I_1),K_3)),J_1)) ) ).
tff(fact_1031_diff__less__mono,axiom,
! [C_5: nat,A_8: nat,B_9: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_8),B_9))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,C_5),A_8))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,A_8),C_5)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,B_9),C_5))) ) ) ).
tff(fact_1032_less__diff__iff,axiom,
! [N_1: nat,K_3: nat,M_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_3),M_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_3),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_1),K_3)),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),K_3)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_1),N_1)) ) ) ) ).
tff(fact_1033_list__eq__iff__nth__eq,axiom,
! [Xs: list_bool,Ys: list_bool] :
( ( Xs = Ys )
<=> ( ( hAPP_list_bool_nat(size_size_list_bool,Xs) = hAPP_list_bool_nat(size_size_list_bool,Ys) )
& ! [I_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),hAPP_list_bool_nat(size_size_list_bool,Xs)))
=> ( hBOOL(hAPP_nat_bool(nth_bool(Xs),I_4))
<=> hBOOL(hAPP_nat_bool(nth_bool(Ys),I_4)) ) ) ) ) ).
tff(fact_1034_list__eq__iff__nth__eq,axiom,
! [Xs: list_A114658244le_alt,Ys: list_A114658244le_alt] :
( ( Xs = Ys )
<=> ( ( hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs) = hAPP_l1084348419lt_nat(size_s2005684270le_alt,Ys) )
& ! [I_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs)))
=> ( hAPP_n832063962le_alt(nth_Ar1544180416le_alt(Xs),I_4) = hAPP_n832063962le_alt(nth_Ar1544180416le_alt(Ys),I_4) ) ) ) ) ).
tff(fact_1035_list__eq__iff__nth__eq,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( Xs = Ys )
<=> ( ( hAPP_list_nat_nat(size_size_list_nat,Xs) = hAPP_list_nat_nat(size_size_list_nat,Ys) )
& ! [I_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),hAPP_list_nat_nat(size_size_list_nat,Xs)))
=> ( hAPP_nat_nat(nth_nat(Xs),I_4) = hAPP_nat_nat(nth_nat(Ys),I_4) ) ) ) ) ).
tff(fact_1036_nth__take,axiom,
! [Xs: list_bool,I_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),N_1))
=> ( hBOOL(hAPP_nat_bool(nth_bool(hAPP_l1189022293t_bool(hAPP_n479129736t_bool(take_bool,N_1),Xs)),I_1))
<=> hBOOL(hAPP_nat_bool(nth_bool(Xs),I_1)) ) ) ).
tff(fact_1037_nth__take,axiom,
! [Xs_27: list_A114658244le_alt,I_7: nat,N_25: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_7),N_25))
=> ( hAPP_n832063962le_alt(nth_Ar1544180416le_alt(hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,N_25),Xs_27)),I_7) = hAPP_n832063962le_alt(nth_Ar1544180416le_alt(Xs_27),I_7) ) ) ).
tff(fact_1038_nth__take,axiom,
! [Xs_27: list_nat,I_7: nat,N_25: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_7),N_25))
=> ( hAPP_nat_nat(nth_nat(hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,N_25),Xs_27)),I_7) = hAPP_nat_nat(nth_nat(Xs_27),I_7) ) ) ).
tff(fact_1039_nth__replicate,axiom,
! [X_1: bool,I_1: nat,N_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),N_1))
=> ( hBOOL(hAPP_nat_bool(nth_bool(hAPP_bool_list_bool(replicate_bool(N_1),X_1)),I_1))
<=> hBOOL(X_1) ) ) ).
tff(fact_1040_nth__replicate,axiom,
! [X_5: arrow_1893660431le_alt,I_6: nat,N_24: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_6),N_24))
=> ( hAPP_n832063962le_alt(nth_Ar1544180416le_alt(hAPP_A1382181054le_alt(replic782356601le_alt(N_24),X_5)),I_6) = X_5 ) ) ).
tff(fact_1041_nth__replicate,axiom,
! [X_5: nat,I_6: nat,N_24: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_6),N_24))
=> ( hAPP_nat_nat(nth_nat(hAPP_nat_list_nat(replicate_nat(N_24),X_5)),I_6) = X_5 ) ) ).
tff(fact_1042_list__size__estimation,axiom,
! [Y_2: nat,F: fun_Ar1091137528lt_nat,X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_2),hAPP_A1419065744lt_nat(F,X_1)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_2),list_s1668597313le_alt(F,Xs))) ) ) ).
tff(fact_1043_list__size__estimation,axiom,
! [Y_2: nat,F: fun_nat_nat,X_1: nat,Xs: list_nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Xs)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_2),hAPP_nat_nat(F,X_1)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Y_2),list_size_nat(F,Xs))) ) ) ).
tff(fact_1044_less__eq__Suc__le__raw,axiom,
! [X_4: nat] : hAPP_n1699378549t_bool(ord_less_nat,X_4) = hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,X_4)) ).
tff(fact_1045_add__nonpos__neg,axiom,
! [B_8: nat,A_7: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,A_7),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_8),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_7),B_8)),zero_zero_nat)) ) ) ).
tff(fact_1046_add__neg__nonpos,axiom,
! [B_7: nat,A_6: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,A_6),zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_7),zero_zero_nat))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_6),B_7)),zero_zero_nat)) ) ) ).
tff(fact_1047_add__strict__increasing2,axiom,
! [B_6: nat,C_4: nat,A_5: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_5))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_6),C_4))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_6),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_5),C_4))) ) ) ).
tff(fact_1048_add__strict__increasing,axiom,
! [B_5: nat,C_3: nat,A_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_4))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,B_5),C_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_5),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_4),C_3))) ) ) ).
tff(fact_1049_add__nonneg__pos,axiom,
! [B_4: nat,A_3: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),A_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),B_4))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_3),B_4))) ) ) ).
tff(fact_1050_add__pos__nonneg,axiom,
! [B_3: nat,A_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,zero_zero_nat),B_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_2),B_3))) ) ) ).
tff(fact_1051_length__greater__0__conv,axiom,
! [Xs: list_nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_list_nat_nat(size_size_list_nat,Xs)))
<=> ( Xs != nil_nat ) ) ).
tff(fact_1052_length__greater__0__conv,axiom,
! [Xs: list_A114658244le_alt] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs)))
<=> ( Xs != nil_Ar557011903le_alt ) ) ).
tff(fact_1053_length__pos__if__in__set,axiom,
! [X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs))) ) ).
tff(fact_1054_length__pos__if__in__set,axiom,
! [X_1: nat,Xs: list_nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Xs)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),hAPP_list_nat_nat(size_size_list_nat,Xs))) ) ).
tff(fact_1055_Suc__pred,axiom,
! [N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ( hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,zero_zero_nat))) = N ) ) ).
tff(fact_1056_diff__Suc__less,axiom,
! [I: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N),hAPP_nat_nat(suc,I))),N)) ) ).
tff(fact_1057_nat__diff__split__asm,axiom,
! [P_1: fun_nat_bool,Aa: nat,Ba: nat] :
( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Aa),Ba)))
<=> ~ ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Aa),Ba))
& ~ hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) )
| ? [D: nat] :
( ( Aa = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ba),D) )
& ~ hBOOL(hAPP_nat_bool(P_1,D)) ) ) ) ).
tff(fact_1058_nat__diff__split,axiom,
! [P_1: fun_nat_bool,Aa: nat,Ba: nat] :
( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,Aa),Ba)))
<=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,Aa),Ba))
=> hBOOL(hAPP_nat_bool(P_1,zero_zero_nat)) )
& ! [D: nat] :
( ( Aa = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,Ba),D) )
=> hBOOL(hAPP_nat_bool(P_1,D)) ) ) ) ).
tff(fact_1059_in__set__conv__nth,axiom,
! [X_1: bool,Xs: list_bool] :
( hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,X_1),hAPP_l96088310l_bool(set_bool,Xs)))
<=> ? [I_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),hAPP_list_bool_nat(size_size_list_bool,Xs)))
& ( hBOOL(hAPP_nat_bool(nth_bool(Xs),I_4))
<=> hBOOL(X_1) ) ) ) ).
tff(fact_1060_in__set__conv__nth,axiom,
! [X_1: arrow_1893660431le_alt,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
<=> ? [I_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs)))
& ( hAPP_n832063962le_alt(nth_Ar1544180416le_alt(Xs),I_4) = X_1 ) ) ) ).
tff(fact_1061_in__set__conv__nth,axiom,
! [X_1: nat,Xs: list_nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Xs)))
<=> ? [I_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),hAPP_list_nat_nat(size_size_list_nat,Xs)))
& ( hAPP_nat_nat(nth_nat(Xs),I_4) = X_1 ) ) ) ).
tff(fact_1062_all__set__conv__all__nth,axiom,
! [P_1: fun_bool_bool,Xs: list_bool] :
( ! [X_4: bool] :
( hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,X_4),hAPP_l96088310l_bool(set_bool,Xs)))
=> hBOOL(hAPP_bool_bool(P_1,X_4)) )
<=> ! [I_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),hAPP_list_bool_nat(size_size_list_bool,Xs)))
=> hBOOL(hAPP_bool_bool(P_1,hAPP_nat_bool(nth_bool(Xs),I_4))) ) ) ).
tff(fact_1063_all__set__conv__all__nth,axiom,
! [P_1: fun_Ar17477381t_bool,Xs: list_A114658244le_alt] :
( ! [X_4: arrow_1893660431le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_4),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
=> hBOOL(hAPP_A1748732503t_bool(P_1,X_4)) )
<=> ! [I_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs)))
=> hBOOL(hAPP_A1748732503t_bool(P_1,hAPP_n832063962le_alt(nth_Ar1544180416le_alt(Xs),I_4))) ) ) ).
tff(fact_1064_all__set__conv__all__nth,axiom,
! [P_1: fun_nat_bool,Xs: list_nat] :
( ! [X_4: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_4),hAPP_l1821533378t_bool(set_nat,Xs)))
=> hBOOL(hAPP_nat_bool(P_1,X_4)) )
<=> ! [I_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),hAPP_list_nat_nat(size_size_list_nat,Xs)))
=> hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(nth_nat(Xs),I_4))) ) ) ).
tff(fact_1065_nth__mem,axiom,
! [N_1: nat,Xs: list_bool] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_list_bool_nat(size_size_list_bool,Xs)))
=> hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,hAPP_nat_bool(nth_bool(Xs),N_1)),hAPP_l96088310l_bool(set_bool,Xs))) ) ).
tff(fact_1066_nth__mem,axiom,
! [N_1: nat,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs)))
=> hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,hAPP_n832063962le_alt(nth_Ar1544180416le_alt(Xs),N_1)),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs))) ) ).
tff(fact_1067_nth__mem,axiom,
! [N_1: nat,Xs: list_nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_list_nat_nat(size_size_list_nat,Xs)))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,hAPP_nat_nat(nth_nat(Xs),N_1)),hAPP_l1821533378t_bool(set_nat,Xs))) ) ).
tff(fact_1068_distinct__conv__nth,axiom,
! [Xs: list_bool] :
( hBOOL(hAPP_list_bool_bool(distinct_bool,Xs))
<=> ! [I_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),hAPP_list_bool_nat(size_size_list_bool,Xs)))
=> ! [J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_2),hAPP_list_bool_nat(size_size_list_bool,Xs)))
=> ( ( I_4 != J_2 )
=> ~ ( hBOOL(hAPP_nat_bool(nth_bool(Xs),I_4))
<=> hBOOL(hAPP_nat_bool(nth_bool(Xs),J_2)) ) ) ) ) ) ).
tff(fact_1069_distinct__conv__nth,axiom,
! [Xs: list_A114658244le_alt] :
( hBOOL(hAPP_l499746980t_bool(distin1654625714le_alt,Xs))
<=> ! [I_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs)))
=> ! [J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_2),hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs)))
=> ( ( I_4 != J_2 )
=> ( hAPP_n832063962le_alt(nth_Ar1544180416le_alt(Xs),I_4) != hAPP_n832063962le_alt(nth_Ar1544180416le_alt(Xs),J_2) ) ) ) ) ) ).
tff(fact_1070_distinct__conv__nth,axiom,
! [Xs: list_nat] :
( hBOOL(hAPP_list_nat_bool(distinct_nat,Xs))
<=> ! [I_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),hAPP_list_nat_nat(size_size_list_nat,Xs)))
=> ! [J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_2),hAPP_list_nat_nat(size_size_list_nat,Xs)))
=> ( ( I_4 != J_2 )
=> ( hAPP_nat_nat(nth_nat(Xs),I_4) != hAPP_nat_nat(nth_nat(Xs),J_2) ) ) ) ) ) ).
tff(fact_1071_nth__eq__iff__index__eq,axiom,
! [J_1: nat,I_1: nat,Xs: list_bool] :
( hBOOL(hAPP_list_bool_bool(distinct_bool,Xs))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),hAPP_list_bool_nat(size_size_list_bool,Xs)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),hAPP_list_bool_nat(size_size_list_bool,Xs)))
=> ( ( hBOOL(hAPP_nat_bool(nth_bool(Xs),I_1))
<=> hBOOL(hAPP_nat_bool(nth_bool(Xs),J_1)) )
<=> ( I_1 = J_1 ) ) ) ) ) ).
tff(fact_1072_nth__eq__iff__index__eq,axiom,
! [J_1: nat,I_1: nat,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_l499746980t_bool(distin1654625714le_alt,Xs))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs)))
=> ( ( hAPP_n832063962le_alt(nth_Ar1544180416le_alt(Xs),I_1) = hAPP_n832063962le_alt(nth_Ar1544180416le_alt(Xs),J_1) )
<=> ( I_1 = J_1 ) ) ) ) ) ).
tff(fact_1073_nth__eq__iff__index__eq,axiom,
! [J_1: nat,I_1: nat,Xs: list_nat] :
( hBOOL(hAPP_list_nat_bool(distinct_nat,Xs))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_1),hAPP_list_nat_nat(size_size_list_nat,Xs)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),hAPP_list_nat_nat(size_size_list_nat,Xs)))
=> ( ( hAPP_nat_nat(nth_nat(Xs),I_1) = hAPP_nat_nat(nth_nat(Xs),J_1) )
<=> ( I_1 = J_1 ) ) ) ) ) ).
tff(fact_1074_nth__length__takeWhile,axiom,
! [P_1: fun_bool_bool,Xs: list_bool] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_list_bool_nat(size_size_list_bool,takeWhile_bool(P_1,Xs))),hAPP_list_bool_nat(size_size_list_bool,Xs)))
=> ~ hBOOL(hAPP_bool_bool(P_1,hAPP_nat_bool(nth_bool(Xs),hAPP_list_bool_nat(size_size_list_bool,takeWhile_bool(P_1,Xs))))) ) ).
tff(fact_1075_nth__length__takeWhile,axiom,
! [P_1: fun_Ar17477381t_bool,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_l1084348419lt_nat(size_s2005684270le_alt,takeWh967109304le_alt(P_1,Xs))),hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs)))
=> ~ hBOOL(hAPP_A1748732503t_bool(P_1,hAPP_n832063962le_alt(nth_Ar1544180416le_alt(Xs),hAPP_l1084348419lt_nat(size_s2005684270le_alt,takeWh967109304le_alt(P_1,Xs))))) ) ).
tff(fact_1076_nth__length__takeWhile,axiom,
! [P_1: fun_nat_bool,Xs: list_nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_list_nat_nat(size_size_list_nat,takeWhile_nat(P_1,Xs))),hAPP_list_nat_nat(size_size_list_nat,Xs)))
=> ~ hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(nth_nat(Xs),hAPP_list_nat_nat(size_size_list_nat,takeWhile_nat(P_1,Xs))))) ) ).
tff(fact_1077_takeWhile__nth,axiom,
! [J_1: nat,P_1: fun_bool_bool,Xs: list_bool] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),hAPP_list_bool_nat(size_size_list_bool,takeWhile_bool(P_1,Xs))))
=> ( hBOOL(hAPP_nat_bool(nth_bool(takeWhile_bool(P_1,Xs)),J_1))
<=> hBOOL(hAPP_nat_bool(nth_bool(Xs),J_1)) ) ) ).
tff(fact_1078_takeWhile__nth,axiom,
! [J_1: nat,P_1: fun_Ar17477381t_bool,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),hAPP_l1084348419lt_nat(size_s2005684270le_alt,takeWh967109304le_alt(P_1,Xs))))
=> ( hAPP_n832063962le_alt(nth_Ar1544180416le_alt(takeWh967109304le_alt(P_1,Xs)),J_1) = hAPP_n832063962le_alt(nth_Ar1544180416le_alt(Xs),J_1) ) ) ).
tff(fact_1079_takeWhile__nth,axiom,
! [J_1: nat,P_1: fun_nat_bool,Xs: list_nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),hAPP_list_nat_nat(size_size_list_nat,takeWhile_nat(P_1,Xs))))
=> ( hAPP_nat_nat(nth_nat(takeWhile_nat(P_1,Xs)),J_1) = hAPP_nat_nat(nth_nat(Xs),J_1) ) ) ).
tff(fact_1080_take__butlast,axiom,
! [N_23: nat,Xs_26: list_A114658244le_alt] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_23),hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs_26)))
=> ( hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,N_23),butlas1693249291le_alt(Xs_26)) = hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,N_23),Xs_26) ) ) ).
tff(fact_1081_take__butlast,axiom,
! [N_23: nat,Xs_26: list_nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_23),hAPP_list_nat_nat(size_size_list_nat,Xs_26)))
=> ( hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,N_23),butlast_nat(Xs_26)) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,N_23),Xs_26) ) ) ).
tff(fact_1082_list__all__length,axiom,
! [P_1: fun_bool_bool,Xs: list_bool] :
( hBOOL(hAPP_list_bool_bool(list_all_bool(P_1),Xs))
<=> ! [N_19: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_19),hAPP_list_bool_nat(size_size_list_bool,Xs)))
=> hBOOL(hAPP_bool_bool(P_1,hAPP_nat_bool(nth_bool(Xs),N_19))) ) ) ).
tff(fact_1083_list__all__length,axiom,
! [P_1: fun_Ar17477381t_bool,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_l499746980t_bool(list_a1988415615le_alt(P_1),Xs))
<=> ! [N_19: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_19),hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs)))
=> hBOOL(hAPP_A1748732503t_bool(P_1,hAPP_n832063962le_alt(nth_Ar1544180416le_alt(Xs),N_19))) ) ) ).
tff(fact_1084_list__all__length,axiom,
! [P_1: fun_nat_bool,Xs: list_nat] :
( hBOOL(hAPP_list_nat_bool(list_all_nat(P_1),Xs))
<=> ! [N_19: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_19),hAPP_list_nat_nat(size_size_list_nat,Xs)))
=> hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(nth_nat(Xs),N_19))) ) ) ).
tff(fact_1085_nth__append,axiom,
! [Xs: list_bool,Ys: list_bool,N_1: nat] :
( hBOOL(hAPP_nat_bool(nth_bool(hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,Xs),Ys)),N_1))
<=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_list_bool_nat(size_size_list_bool,Xs)))
=> hBOOL(hAPP_nat_bool(nth_bool(Xs),N_1)) )
& ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_list_bool_nat(size_size_list_bool,Xs)))
=> hBOOL(hAPP_nat_bool(nth_bool(Ys),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_1),hAPP_list_bool_nat(size_size_list_bool,Xs)))) ) ) ) ).
tff(fact_1086_nth__append,axiom,
! [Ys_5: list_A114658244le_alt,N_22: nat,Xs_25: list_A114658244le_alt] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_22),hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs_25)))
=> ( hAPP_n832063962le_alt(nth_Ar1544180416le_alt(hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs_25),Ys_5)),N_22) = hAPP_n832063962le_alt(nth_Ar1544180416le_alt(Xs_25),N_22) ) )
& ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_22),hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs_25)))
=> ( hAPP_n832063962le_alt(nth_Ar1544180416le_alt(hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs_25),Ys_5)),N_22) = hAPP_n832063962le_alt(nth_Ar1544180416le_alt(Ys_5),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_22),hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs_25))) ) ) ) ).
tff(fact_1087_nth__append,axiom,
! [Ys_5: list_nat,N_22: nat,Xs_25: list_nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_22),hAPP_list_nat_nat(size_size_list_nat,Xs_25)))
=> ( hAPP_nat_nat(nth_nat(hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs_25),Ys_5)),N_22) = hAPP_nat_nat(nth_nat(Xs_25),N_22) ) )
& ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_22),hAPP_list_nat_nat(size_size_list_nat,Xs_25)))
=> ( hAPP_nat_nat(nth_nat(hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs_25),Ys_5)),N_22) = hAPP_nat_nat(nth_nat(Ys_5),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_22),hAPP_list_nat_nat(size_size_list_nat,Xs_25))) ) ) ) ).
tff(fact_1088_sorted__nth__mono,axiom,
! [I_5: nat,J_3: nat,Xs_24: list_nat] :
( hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,Xs_24))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_5),J_3))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_3),hAPP_list_nat_nat(size_size_list_nat,Xs_24)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(nth_nat(Xs_24),I_5)),hAPP_nat_nat(nth_nat(Xs_24),J_3))) ) ) ) ).
tff(fact_1089_sorted__equals__nth__mono,axiom,
! [Xs: list_nat] :
( hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,Xs))
<=> ! [J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_2),hAPP_list_nat_nat(size_size_list_nat,Xs)))
=> ! [I_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_4),J_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(nth_nat(Xs),I_4)),hAPP_nat_nat(nth_nat(Xs),J_2))) ) ) ) ).
tff(fact_1090_rev__nth,axiom,
! [N_1: nat,Xs: list_bool] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_list_bool_nat(size_size_list_bool,Xs)))
=> ( hBOOL(hAPP_nat_bool(nth_bool(rev_bool(Xs)),N_1))
<=> hBOOL(hAPP_nat_bool(nth_bool(Xs),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_list_bool_nat(size_size_list_bool,Xs)),hAPP_nat_nat(suc,N_1)))) ) ) ).
tff(fact_1091_rev__nth,axiom,
! [N_21: nat,Xs_23: list_A114658244le_alt] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_21),hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs_23)))
=> ( hAPP_n832063962le_alt(nth_Ar1544180416le_alt(rev_Ar377224735le_alt(Xs_23)),N_21) = hAPP_n832063962le_alt(nth_Ar1544180416le_alt(Xs_23),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs_23)),hAPP_nat_nat(suc,N_21))) ) ) ).
tff(fact_1092_rev__nth,axiom,
! [N_21: nat,Xs_23: list_nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_21),hAPP_list_nat_nat(size_size_list_nat,Xs_23)))
=> ( hAPP_nat_nat(nth_nat(rev_nat(Xs_23)),N_21) = hAPP_nat_nat(nth_nat(Xs_23),hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_list_nat_nat(size_size_list_nat,Xs_23)),hAPP_nat_nat(suc,N_21))) ) ) ).
tff(fact_1093_dropWhile__nth,axiom,
! [J_1: nat,P_1: fun_bool_bool,Xs: list_bool] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),hAPP_list_bool_nat(size_size_list_bool,dropWhile_bool(P_1,Xs))))
=> ( hBOOL(hAPP_nat_bool(nth_bool(dropWhile_bool(P_1,Xs)),J_1))
<=> hBOOL(hAPP_nat_bool(nth_bool(Xs),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_1),hAPP_list_bool_nat(size_size_list_bool,takeWhile_bool(P_1,Xs))))) ) ) ).
tff(fact_1094_dropWhile__nth,axiom,
! [J_1: nat,P_1: fun_Ar17477381t_bool,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),hAPP_l1084348419lt_nat(size_s2005684270le_alt,dropWh587599712le_alt(P_1,Xs))))
=> ( hAPP_n832063962le_alt(nth_Ar1544180416le_alt(dropWh587599712le_alt(P_1,Xs)),J_1) = hAPP_n832063962le_alt(nth_Ar1544180416le_alt(Xs),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_1),hAPP_l1084348419lt_nat(size_s2005684270le_alt,takeWh967109304le_alt(P_1,Xs)))) ) ) ).
tff(fact_1095_dropWhile__nth,axiom,
! [J_1: nat,P_1: fun_nat_bool,Xs: list_nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_1),hAPP_list_nat_nat(size_size_list_nat,dropWhile_nat(P_1,Xs))))
=> ( hAPP_nat_nat(nth_nat(dropWhile_nat(P_1,Xs)),J_1) = hAPP_nat_nat(nth_nat(Xs),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J_1),hAPP_list_nat_nat(size_size_list_nat,takeWhile_nat(P_1,Xs)))) ) ) ).
tff(fact_1096_sorted__nth__monoI,axiom,
! [Xs_22: list_nat] :
( ! [I_4: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_4),J_2))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,J_2),hAPP_list_nat_nat(size_size_list_nat,Xs_22)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(nth_nat(Xs_22),I_4)),hAPP_nat_nat(nth_nat(Xs_22),J_2))) ) )
=> hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,Xs_22)) ) ).
tff(fact_1097_takeWhile__eq__take__P__nth,axiom,
! [P_1: fun_bool_bool,Xs: list_bool,N_1: nat] :
( ! [I_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),hAPP_list_bool_nat(size_size_list_bool,Xs)))
=> hBOOL(hAPP_bool_bool(P_1,hAPP_nat_bool(nth_bool(Xs),I_4))) ) )
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_list_bool_nat(size_size_list_bool,Xs)))
=> ~ hBOOL(hAPP_bool_bool(P_1,hAPP_nat_bool(nth_bool(Xs),N_1))) )
=> ( takeWhile_bool(P_1,Xs) = hAPP_l1189022293t_bool(hAPP_n479129736t_bool(take_bool,N_1),Xs) ) ) ) ).
tff(fact_1098_takeWhile__eq__take__P__nth,axiom,
! [P_1: fun_Ar17477381t_bool,Xs: list_A114658244le_alt,N_1: nat] :
( ! [I_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs)))
=> hBOOL(hAPP_A1748732503t_bool(P_1,hAPP_n832063962le_alt(nth_Ar1544180416le_alt(Xs),I_4))) ) )
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs)))
=> ~ hBOOL(hAPP_A1748732503t_bool(P_1,hAPP_n832063962le_alt(nth_Ar1544180416le_alt(Xs),N_1))) )
=> ( takeWh967109304le_alt(P_1,Xs) = hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,N_1),Xs) ) ) ) ).
tff(fact_1099_takeWhile__eq__take__P__nth,axiom,
! [P_1: fun_nat_bool,Xs: list_nat,N_1: nat] :
( ! [I_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),N_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),hAPP_list_nat_nat(size_size_list_nat,Xs)))
=> hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(nth_nat(Xs),I_4))) ) )
=> ( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_list_nat_nat(size_size_list_nat,Xs)))
=> ~ hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(nth_nat(Xs),N_1))) )
=> ( takeWhile_nat(P_1,Xs) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,N_1),Xs) ) ) ) ).
tff(fact_1100_length__takeWhile__less__P__nth,axiom,
! [P_1: fun_bool_bool,Xs: list_bool,J_1: nat] :
( ! [I_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),J_1))
=> hBOOL(hAPP_bool_bool(P_1,hAPP_nat_bool(nth_bool(Xs),I_4))) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_1),hAPP_list_bool_nat(size_size_list_bool,Xs)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_1),hAPP_list_bool_nat(size_size_list_bool,takeWhile_bool(P_1,Xs)))) ) ) ).
tff(fact_1101_length__takeWhile__less__P__nth,axiom,
! [P_1: fun_Ar17477381t_bool,Xs: list_A114658244le_alt,J_1: nat] :
( ! [I_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),J_1))
=> hBOOL(hAPP_A1748732503t_bool(P_1,hAPP_n832063962le_alt(nth_Ar1544180416le_alt(Xs),I_4))) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_1),hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_1),hAPP_l1084348419lt_nat(size_s2005684270le_alt,takeWh967109304le_alt(P_1,Xs)))) ) ) ).
tff(fact_1102_length__takeWhile__less__P__nth,axiom,
! [P_1: fun_nat_bool,Xs: list_nat,J_1: nat] :
( ! [I_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),J_1))
=> hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(nth_nat(Xs),I_4))) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_1),hAPP_list_nat_nat(size_size_list_nat,Xs)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_1),hAPP_list_nat_nat(size_size_list_nat,takeWhile_nat(P_1,Xs)))) ) ) ).
tff(fact_1103_nth__take__lemma,axiom,
! [Ys: list_bool,K_3: nat,Xs: list_bool] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_3),hAPP_list_bool_nat(size_size_list_bool,Xs)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_3),hAPP_list_bool_nat(size_size_list_bool,Ys)))
=> ( ! [I_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),K_3))
=> ( hBOOL(hAPP_nat_bool(nth_bool(Xs),I_4))
<=> hBOOL(hAPP_nat_bool(nth_bool(Ys),I_4)) ) )
=> ( hAPP_l1189022293t_bool(hAPP_n479129736t_bool(take_bool,K_3),Xs) = hAPP_l1189022293t_bool(hAPP_n479129736t_bool(take_bool,K_3),Ys) ) ) ) ) ).
tff(fact_1104_nth__take__lemma,axiom,
! [Ys_4: list_A114658244le_alt,K_4: nat,Xs_21: list_A114658244le_alt] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_4),hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs_21)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_4),hAPP_l1084348419lt_nat(size_s2005684270le_alt,Ys_4)))
=> ( ! [I_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),K_4))
=> ( hAPP_n832063962le_alt(nth_Ar1544180416le_alt(Xs_21),I_4) = hAPP_n832063962le_alt(nth_Ar1544180416le_alt(Ys_4),I_4) ) )
=> ( hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,K_4),Xs_21) = hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,K_4),Ys_4) ) ) ) ) ).
tff(fact_1105_nth__take__lemma,axiom,
! [Ys_4: list_nat,K_4: nat,Xs_21: list_nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_4),hAPP_list_nat_nat(size_size_list_nat,Xs_21)))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_4),hAPP_list_nat_nat(size_size_list_nat,Ys_4)))
=> ( ! [I_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),K_4))
=> ( hAPP_nat_nat(nth_nat(Xs_21),I_4) = hAPP_nat_nat(nth_nat(Ys_4),I_4) ) )
=> ( hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,K_4),Xs_21) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,K_4),Ys_4) ) ) ) ) ).
tff(fact_1106_list__size__pointwise,axiom,
! [F: fun_Ar1091137528lt_nat,G: fun_Ar1091137528lt_nat,Xs: list_A114658244le_alt] :
( ! [X_4: arrow_1893660431le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_4),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_A1419065744lt_nat(F,X_4)),hAPP_A1419065744lt_nat(G,X_4))) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,list_s1668597313le_alt(F,Xs)),list_s1668597313le_alt(G,Xs))) ) ).
tff(fact_1107_list__size__pointwise,axiom,
! [F: fun_nat_nat,G: fun_nat_nat,Xs: list_nat] :
( ! [X_4: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_4),hAPP_l1821533378t_bool(set_nat,Xs)))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,X_4)),hAPP_nat_nat(G,X_4))) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,list_size_nat(F,Xs)),list_size_nat(G,Xs))) ) ).
tff(fact_1108_psubsetD,axiom,
! [C_2: arrow_1893660431le_alt,A: fun_Ar17477381t_bool,B: fun_Ar17477381t_bool] :
( hBOOL(hAPP_f739344229l_bool(hAPP_f1549168575l_bool(ord_le123846959t_bool,A),B))
=> ( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,C_2),A))
=> hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,C_2),B)) ) ) ).
tff(fact_1109_psubsetD,axiom,
! [C_2: nat,A: fun_nat_bool,B: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,A),B))
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),A))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_2),B)) ) ) ).
tff(fact_1110_all__nth__imp__all__set,axiom,
! [X_1: bool,P_1: fun_bool_bool,Xs: list_bool] :
( ! [I_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),hAPP_list_bool_nat(size_size_list_bool,Xs)))
=> hBOOL(hAPP_bool_bool(P_1,hAPP_nat_bool(nth_bool(Xs),I_4))) )
=> ( hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,X_1),hAPP_l96088310l_bool(set_bool,Xs)))
=> hBOOL(hAPP_bool_bool(P_1,X_1)) ) ) ).
tff(fact_1111_all__nth__imp__all__set,axiom,
! [X_1: arrow_1893660431le_alt,P_1: fun_Ar17477381t_bool,Xs: list_A114658244le_alt] :
( ! [I_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs)))
=> hBOOL(hAPP_A1748732503t_bool(P_1,hAPP_n832063962le_alt(nth_Ar1544180416le_alt(Xs),I_4))) )
=> ( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
=> hBOOL(hAPP_A1748732503t_bool(P_1,X_1)) ) ) ).
tff(fact_1112_all__nth__imp__all__set,axiom,
! [X_1: nat,P_1: fun_nat_bool,Xs: list_nat] :
( ! [I_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),hAPP_list_nat_nat(size_size_list_nat,Xs)))
=> hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(nth_nat(Xs),I_4))) )
=> ( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Xs)))
=> hBOOL(hAPP_nat_bool(P_1,X_1)) ) ) ).
tff(fact_1113_list__ball__nth,axiom,
! [P_1: fun_bool_bool,N_1: nat,Xs: list_bool] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_list_bool_nat(size_size_list_bool,Xs)))
=> ( ! [X_4: bool] :
( hBOOL(hAPP_f387058535l_bool(hAPP_b1787118453l_bool(member_bool,X_4),hAPP_l96088310l_bool(set_bool,Xs)))
=> hBOOL(hAPP_bool_bool(P_1,X_4)) )
=> hBOOL(hAPP_bool_bool(P_1,hAPP_nat_bool(nth_bool(Xs),N_1))) ) ) ).
tff(fact_1114_list__ball__nth,axiom,
! [P_1: fun_Ar17477381t_bool,N_1: nat,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs)))
=> ( ! [X_4: arrow_1893660431le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_4),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs)))
=> hBOOL(hAPP_A1748732503t_bool(P_1,X_4)) )
=> hBOOL(hAPP_A1748732503t_bool(P_1,hAPP_n832063962le_alt(nth_Ar1544180416le_alt(Xs),N_1))) ) ) ).
tff(fact_1115_list__ball__nth,axiom,
! [P_1: fun_nat_bool,N_1: nat,Xs: list_nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_list_nat_nat(size_size_list_nat,Xs)))
=> ( ! [X_4: nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_4),hAPP_l1821533378t_bool(set_nat,Xs)))
=> hBOOL(hAPP_nat_bool(P_1,X_4)) )
=> hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(nth_nat(Xs),N_1))) ) ) ).
tff(fact_1116_nth__equalityI,axiom,
! [Xs: list_bool,Ys: list_bool] :
( ( hAPP_list_bool_nat(size_size_list_bool,Xs) = hAPP_list_bool_nat(size_size_list_bool,Ys) )
=> ( ! [I_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),hAPP_list_bool_nat(size_size_list_bool,Xs)))
=> ( hBOOL(hAPP_nat_bool(nth_bool(Xs),I_4))
<=> hBOOL(hAPP_nat_bool(nth_bool(Ys),I_4)) ) )
=> ( Xs = Ys ) ) ) ).
tff(fact_1117_nth__equalityI,axiom,
! [Xs_20: list_A114658244le_alt,Ys_3: list_A114658244le_alt] :
( ( hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs_20) = hAPP_l1084348419lt_nat(size_s2005684270le_alt,Ys_3) )
=> ( ! [I_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs_20)))
=> ( hAPP_n832063962le_alt(nth_Ar1544180416le_alt(Xs_20),I_4) = hAPP_n832063962le_alt(nth_Ar1544180416le_alt(Ys_3),I_4) ) )
=> ( Xs_20 = Ys_3 ) ) ) ).
tff(fact_1118_nth__equalityI,axiom,
! [Xs_20: list_nat,Ys_3: list_nat] :
( ( hAPP_list_nat_nat(size_size_list_nat,Xs_20) = hAPP_list_nat_nat(size_size_list_nat,Ys_3) )
=> ( ! [I_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),hAPP_list_nat_nat(size_size_list_nat,Xs_20)))
=> ( hAPP_nat_nat(nth_nat(Xs_20),I_4) = hAPP_nat_nat(nth_nat(Ys_3),I_4) ) )
=> ( Xs_20 = Ys_3 ) ) ) ).
tff(fact_1119_Skolem__list__nth,axiom,
! [P_1: fun_na1469252690l_bool,K_3: nat] :
( ! [I_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),K_3))
=> ? [X1: bool] : hBOOL(hAPP_bool_bool(hAPP_n1006566506l_bool(P_1,I_4),X1)) )
<=> ? [Xs_19: list_bool] :
( ( hAPP_list_bool_nat(size_size_list_bool,Xs_19) = K_3 )
& ! [I_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),K_3))
=> hBOOL(hAPP_bool_bool(hAPP_n1006566506l_bool(P_1,I_4),hAPP_nat_bool(nth_bool(Xs_19),I_4))) ) ) ) ).
tff(fact_1120_Skolem__list__nth,axiom,
! [P_1: fun_na2015178792t_bool,K_3: nat] :
( ! [I_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),K_3))
=> ? [X1: arrow_1893660431le_alt] : hBOOL(hAPP_A1748732503t_bool(hAPP_n1201661562t_bool(P_1,I_4),X1)) )
<=> ? [Xs_19: list_A114658244le_alt] :
( ( hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs_19) = K_3 )
& ! [I_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),K_3))
=> hBOOL(hAPP_A1748732503t_bool(hAPP_n1201661562t_bool(P_1,I_4),hAPP_n832063962le_alt(nth_Ar1544180416le_alt(Xs_19),I_4))) ) ) ) ).
tff(fact_1121_Skolem__list__nth,axiom,
! [P_1: fun_nat_fun_nat_bool,K_3: nat] :
( ! [I_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),K_3))
=> ? [X1: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,I_4),X1)) )
<=> ? [Xs_19: list_nat] :
( ( hAPP_list_nat_nat(size_size_list_nat,Xs_19) = K_3 )
& ! [I_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),K_3))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(P_1,I_4),hAPP_nat_nat(nth_nat(Xs_19),I_4))) ) ) ) ).
tff(fact_1122_mono__nat__linear__lb,axiom,
! [M_1: nat,K_3: nat,F: fun_nat_nat] :
( ! [M_5: nat,N_19: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M_5),N_19))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,M_5)),hAPP_nat_nat(F,N_19))) )
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,hAPP_nat_nat(F,M_1)),K_3)),hAPP_nat_nat(F,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M_1),K_3)))) ) ).
tff(fact_1123_inc__induct,axiom,
! [P_1: fun_nat_bool,I_1: nat,J_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_1))
=> ( hBOOL(hAPP_nat_bool(P_1,J_1))
=> ( ! [I_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),J_1))
=> ( hBOOL(hAPP_nat_bool(P_1,hAPP_nat_nat(suc,I_4)))
=> hBOOL(hAPP_nat_bool(P_1,I_4)) ) )
=> hBOOL(hAPP_nat_bool(P_1,I_1)) ) ) ) ).
tff(fact_1124_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,M),N))
=> ? [K_2: nat] : N = hAPP_nat_nat(suc,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,M),K_2)) ) ).
tff(fact_1125_ex__least__nat__le,axiom,
! [N_1: nat,P_1: fun_nat_bool] :
( ~ hBOOL(hAPP_nat_bool(P_1,zero_zero_nat))
=> ( hBOOL(hAPP_nat_bool(P_1,N_1))
=> ? [K_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_2),N_1))
& ! [I_4: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),K_2))
=> ~ hBOOL(hAPP_nat_bool(P_1,I_4)) )
& hBOOL(hAPP_nat_bool(P_1,K_2)) ) ) ) ).
tff(fact_1126_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> ? [K_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),K_2))
& ( hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K_2) = J ) ) ) ).
tff(fact_1127_gr0__implies__Suc,axiom,
! [N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),N))
=> ? [M_5: nat] : N = hAPP_nat_nat(suc,M_5) ) ).
tff(fact_1128_lift__Suc__mono__less,axiom,
! [N_1: nat,N_20: nat,F: fun_nat_nat] :
( ! [N_19: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,N_19)),hAPP_nat_nat(F,hAPP_nat_nat(suc,N_19))))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),N_20))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,N_1)),hAPP_nat_nat(F,N_20))) ) ) ).
tff(fact_1129_lift__Suc__mono__less__iff,axiom,
! [N_1: nat,M_1: nat,F: fun_nat_nat] :
( ! [N_19: nat] : hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,N_19)),hAPP_nat_nat(F,hAPP_nat_nat(suc,N_19))))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,N_1)),hAPP_nat_nat(F,M_1)))
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),M_1)) ) ) ).
tff(fact_1130_pos__add__strict,axiom,
! [B_2: nat,C_1: nat,A_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,zero_zero_nat),A_1))
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_2),C_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,B_2),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,A_1),C_1))) ) ) ).
tff(fact_1131_psubset__imp__ex__mem,axiom,
! [A: fun_Ar17477381t_bool,B: fun_Ar17477381t_bool] :
( hBOOL(hAPP_f739344229l_bool(hAPP_f1549168575l_bool(ord_le123846959t_bool,A),B))
=> ? [B_1: arrow_1893660431le_alt] : hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,B_1),hAPP_f1592536333t_bool(hAPP_f571726183t_bool(minus_1875147572t_bool,B),A))) ) ).
tff(fact_1132_psubset__imp__ex__mem,axiom,
! [A: fun_nat_bool,B: fun_nat_bool] :
( hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le382113706t_bool,A),B))
=> ? [B_1: nat] : hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,B_1),hAPP_f800510211t_bool(hAPP_f1730770594t_bool(minus_2067140911t_bool,B),A))) ) ).
tff(fact_1133_less__mono__imp__le__mono,axiom,
! [I_1: nat,J_1: nat,F: fun_nat_nat] :
( ! [I_4: nat,J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I_4),J_2))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(F,I_4)),hAPP_nat_nat(F,J_2))) )
=> ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J_1))
=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,I_1)),hAPP_nat_nat(F,J_1))) ) ) ).
tff(fact_1134_take__hd__drop,axiom,
! [N_18: nat,Xs_18: list_A114658244le_alt] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_18),hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs_18)))
=> ( hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,N_18),Xs_18)),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,hAPP_l1749541000le_alt(hd_Arr1236501138le_alt,hAPP_l84111627le_alt(hAPP_n372949560le_alt(drop_A617527551le_alt,N_18),Xs_18))),nil_Ar557011903le_alt)) = hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,hAPP_nat_nat(suc,N_18)),Xs_18) ) ) ).
tff(fact_1135_take__hd__drop,axiom,
! [N_18: nat,Xs_18: list_nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_18),hAPP_list_nat_nat(size_size_list_nat,Xs_18)))
=> ( hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,N_18),Xs_18)),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,hAPP_list_nat_nat(hd_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,N_18),Xs_18))),nil_nat)) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,hAPP_nat_nat(suc,N_18)),Xs_18) ) ) ).
tff(fact_1136_take__hd__drop,axiom,
! [N_1: nat,Xs: list_bool] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,N_1),hAPP_list_bool_nat(size_size_list_bool,Xs)))
=> ( hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(append_bool,hAPP_l1189022293t_bool(hAPP_n479129736t_bool(take_bool,N_1),Xs)),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,hAPP_list_bool_bool(hd_bool,hAPP_l1189022293t_bool(hAPP_n479129736t_bool(drop_bool,N_1),Xs))),nil_bool)) = hAPP_l1189022293t_bool(hAPP_n479129736t_bool(take_bool,hAPP_nat_nat(suc,N_1)),Xs) ) ) ).
tff(fact_1137_drop__Suc__Cons,axiom,
! [N_17: nat,X_3: nat,Xs_17: list_nat] : hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,hAPP_nat_nat(suc,N_17)),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,X_3),Xs_17)) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,N_17),Xs_17) ).
tff(fact_1138_drop__Suc__Cons,axiom,
! [N_1: nat,X_1: bool,Xs: list_bool] : hAPP_l1189022293t_bool(hAPP_n479129736t_bool(drop_bool,hAPP_nat_nat(suc,N_1)),hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,X_1),Xs)) = hAPP_l1189022293t_bool(hAPP_n479129736t_bool(drop_bool,N_1),Xs) ).
tff(fact_1139_drop__Suc__Cons,axiom,
! [N_17: nat,X_3: arrow_1893660431le_alt,Xs_17: list_A114658244le_alt] : hAPP_l84111627le_alt(hAPP_n372949560le_alt(drop_A617527551le_alt,hAPP_nat_nat(suc,N_17)),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,X_3),Xs_17)) = hAPP_l84111627le_alt(hAPP_n372949560le_alt(drop_A617527551le_alt,N_17),Xs_17) ).
tff(fact_1140_set__drop__subset,axiom,
! [N_1: nat,Xs: list_A114658244le_alt] : hBOOL(hAPP_f739344229l_bool(hAPP_f1549168575l_bool(ord_le1779845435t_bool,hAPP_l2018657868t_bool(set_Ar1995755744le_alt,hAPP_l84111627le_alt(hAPP_n372949560le_alt(drop_A617527551le_alt,N_1),Xs))),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs))) ).
tff(fact_1141_set__drop__subset,axiom,
! [N_1: nat,Xs: list_nat] : hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_l1821533378t_bool(set_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,N_1),Xs))),hAPP_l1821533378t_bool(set_nat,Xs))) ).
tff(fact_1142_length__drop,axiom,
! [N_16: nat,Xs_16: list_nat] : hAPP_list_nat_nat(size_size_list_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,N_16),Xs_16)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,hAPP_list_nat_nat(size_size_list_nat,Xs_16)),N_16) ).
tff(fact_1143_nth__via__drop,axiom,
! [N_1: nat,Xs: list_bool,Y_2: bool,Ys: list_bool] :
( ( hAPP_l1189022293t_bool(hAPP_n479129736t_bool(drop_bool,N_1),Xs) = hAPP_l1189022293t_bool(hAPP_b994696797t_bool(cons_bool,Y_2),Ys) )
=> ( hBOOL(hAPP_nat_bool(nth_bool(Xs),N_1))
<=> hBOOL(Y_2) ) ) ).
tff(fact_1144_nth__via__drop,axiom,
! [N_15: nat,Xs_15: list_A114658244le_alt,Y_1: arrow_1893660431le_alt,Ys_2: list_A114658244le_alt] :
( ( hAPP_l84111627le_alt(hAPP_n372949560le_alt(drop_A617527551le_alt,N_15),Xs_15) = hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,Y_1),Ys_2) )
=> ( hAPP_n832063962le_alt(nth_Ar1544180416le_alt(Xs_15),N_15) = Y_1 ) ) ).
tff(fact_1145_nth__via__drop,axiom,
! [N_15: nat,Xs_15: list_nat,Y_1: nat,Ys_2: list_nat] :
( ( hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,N_15),Xs_15) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,Y_1),Ys_2) )
=> ( hAPP_nat_nat(nth_nat(Xs_15),N_15) = Y_1 ) ) ).
tff(fact_1146_take__drop,axiom,
! [N_14: nat,M_4: nat,Xs_14: list_nat] : hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,N_14),hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,M_4),Xs_14)) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,M_4),hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_14),M_4)),Xs_14)) ).
tff(fact_1147_take__drop,axiom,
! [N_14: nat,M_4: nat,Xs_14: list_A114658244le_alt] : hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,N_14),hAPP_l84111627le_alt(hAPP_n372949560le_alt(drop_A617527551le_alt,M_4),Xs_14)) = hAPP_l84111627le_alt(hAPP_n372949560le_alt(drop_A617527551le_alt,M_4),hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_14),M_4)),Xs_14)) ).
tff(fact_1148_drop__take,axiom,
! [N_13: nat,M_3: nat,Xs_13: list_nat] : hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,N_13),hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,M_3),Xs_13)) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_3),N_13)),hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,N_13),Xs_13)) ).
tff(fact_1149_drop__take,axiom,
! [N_13: nat,M_3: nat,Xs_13: list_A114658244le_alt] : hAPP_l84111627le_alt(hAPP_n372949560le_alt(drop_A617527551le_alt,N_13),hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,M_3),Xs_13)) = hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,M_3),N_13)),hAPP_l84111627le_alt(hAPP_n372949560le_alt(drop_A617527551le_alt,N_13),Xs_13)) ).
tff(fact_1150_append__take__drop__id,axiom,
! [N_12: nat,Xs_12: list_nat] : hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,N_12),Xs_12)),hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,N_12),Xs_12)) = Xs_12 ).
tff(fact_1151_append__take__drop__id,axiom,
! [N_12: nat,Xs_12: list_A114658244le_alt] : hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,N_12),Xs_12)),hAPP_l84111627le_alt(hAPP_n372949560le_alt(drop_A617527551le_alt,N_12),Xs_12)) = Xs_12 ).
tff(fact_1152_drop__replicate,axiom,
! [I_3: nat,K_1: nat,X_2: nat] : hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,I_3),hAPP_nat_list_nat(replicate_nat(K_1),X_2)) = hAPP_nat_list_nat(replicate_nat(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,K_1),I_3)),X_2) ).
tff(fact_1153_drop__replicate,axiom,
! [I_3: nat,K_1: nat,X_2: arrow_1893660431le_alt] : hAPP_l84111627le_alt(hAPP_n372949560le_alt(drop_A617527551le_alt,I_3),hAPP_A1382181054le_alt(replic782356601le_alt(K_1),X_2)) = hAPP_A1382181054le_alt(replic782356601le_alt(hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,K_1),I_3)),X_2) ).
tff(fact_1154_drop__Suc,axiom,
! [N_11: nat,Xs_11: list_nat] : hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,hAPP_nat_nat(suc,N_11)),Xs_11) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,N_11),tl_nat(Xs_11)) ).
tff(fact_1155_drop__Suc,axiom,
! [N_11: nat,Xs_11: list_A114658244le_alt] : hAPP_l84111627le_alt(hAPP_n372949560le_alt(drop_A617527551le_alt,hAPP_nat_nat(suc,N_11)),Xs_11) = hAPP_l84111627le_alt(hAPP_n372949560le_alt(drop_A617527551le_alt,N_11),tl_Arr1883752598le_alt(Xs_11)) ).
tff(fact_1156_drop__butlast,axiom,
! [N_10: nat,Xs_10: list_nat] : hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,N_10),butlast_nat(Xs_10)) = butlast_nat(hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,N_10),Xs_10)) ).
tff(fact_1157_drop__butlast,axiom,
! [N_10: nat,Xs_10: list_A114658244le_alt] : hAPP_l84111627le_alt(hAPP_n372949560le_alt(drop_A617527551le_alt,N_10),butlas1693249291le_alt(Xs_10)) = butlas1693249291le_alt(hAPP_l84111627le_alt(hAPP_n372949560le_alt(drop_A617527551le_alt,N_10),Xs_10)) ).
tff(fact_1158_butlast__drop,axiom,
! [N_9: nat,Xs_9: list_nat] : butlast_nat(hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,N_9),Xs_9)) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,N_9),butlast_nat(Xs_9)) ).
tff(fact_1159_butlast__drop,axiom,
! [N_9: nat,Xs_9: list_A114658244le_alt] : butlas1693249291le_alt(hAPP_l84111627le_alt(hAPP_n372949560le_alt(drop_A617527551le_alt,N_9),Xs_9)) = hAPP_l84111627le_alt(hAPP_n372949560le_alt(drop_A617527551le_alt,N_9),butlas1693249291le_alt(Xs_9)) ).
tff(fact_1160_distinct__drop,axiom,
! [I_2: nat,Xs_8: list_nat] :
( hBOOL(hAPP_list_nat_bool(distinct_nat,Xs_8))
=> hBOOL(hAPP_list_nat_bool(distinct_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,I_2),Xs_8))) ) ).
tff(fact_1161_distinct__drop,axiom,
! [I_2: nat,Xs_8: list_A114658244le_alt] :
( hBOOL(hAPP_l499746980t_bool(distin1654625714le_alt,Xs_8))
=> hBOOL(hAPP_l499746980t_bool(distin1654625714le_alt,hAPP_l84111627le_alt(hAPP_n372949560le_alt(drop_A617527551le_alt,I_2),Xs_8))) ) ).
tff(fact_1162_sorted__drop,axiom,
! [N_8: nat,Xs_7: list_nat] :
( hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,Xs_7))
=> hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,N_8),Xs_7))) ) ).
tff(fact_1163_drop__Nil,axiom,
! [N_7: nat] : hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,N_7),nil_nat) = nil_nat ).
tff(fact_1164_drop__Nil,axiom,
! [N_7: nat] : hAPP_l84111627le_alt(hAPP_n372949560le_alt(drop_A617527551le_alt,N_7),nil_Ar557011903le_alt) = nil_Ar557011903le_alt ).
tff(fact_1165_in__set__dropD,axiom,
! [X_1: arrow_1893660431le_alt,N_1: nat,Xs: list_A114658244le_alt] :
( hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,hAPP_l84111627le_alt(hAPP_n372949560le_alt(drop_A617527551le_alt,N_1),Xs))))
=> hBOOL(hAPP_f739344229l_bool(hAPP_A1501726285l_bool(member1502664802le_alt,X_1),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,Xs))) ) ).
tff(fact_1166_in__set__dropD,axiom,
! [X_1: nat,N_1: nat,Xs: list_nat] :
( hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,N_1),Xs))))
=> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),hAPP_l1821533378t_bool(set_nat,Xs))) ) ).
tff(fact_1167_drop__drop,axiom,
! [N_6: nat,M_2: nat,Xs_6: list_nat] : hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,N_6),hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,M_2),Xs_6)) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,N_6),M_2)),Xs_6) ).
tff(fact_1168_drop__0,axiom,
! [Xs_5: list_nat] : hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,zero_zero_nat),Xs_5) = Xs_5 ).
tff(fact_1169_tl__drop,axiom,
! [N_5: nat,Xs_4: list_nat] : tl_nat(hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,N_5),Xs_4)) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,N_5),tl_nat(Xs_4)) ).
tff(fact_1170_tl__drop,axiom,
! [N_5: nat,Xs_4: list_A114658244le_alt] : tl_Arr1883752598le_alt(hAPP_l84111627le_alt(hAPP_n372949560le_alt(drop_A617527551le_alt,N_5),Xs_4)) = hAPP_l84111627le_alt(hAPP_n372949560le_alt(drop_A617527551le_alt,N_5),tl_Arr1883752598le_alt(Xs_4)) ).
tff(fact_1171_drop__tl,axiom,
! [N_4: nat,Xs_3: list_nat] : hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,N_4),tl_nat(Xs_3)) = tl_nat(hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,N_4),Xs_3)) ).
tff(fact_1172_drop__tl,axiom,
! [N_4: nat,Xs_3: list_A114658244le_alt] : hAPP_l84111627le_alt(hAPP_n372949560le_alt(drop_A617527551le_alt,N_4),tl_Arr1883752598le_alt(Xs_3)) = tl_Arr1883752598le_alt(hAPP_l84111627le_alt(hAPP_n372949560le_alt(drop_A617527551le_alt,N_4),Xs_3)) ).
tff(fact_1173_set__drop__subset__set__drop,axiom,
! [Xs: list_A114658244le_alt,N_1: nat,M_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_1))
=> hBOOL(hAPP_f739344229l_bool(hAPP_f1549168575l_bool(ord_le1779845435t_bool,hAPP_l2018657868t_bool(set_Ar1995755744le_alt,hAPP_l84111627le_alt(hAPP_n372949560le_alt(drop_A617527551le_alt,M_1),Xs))),hAPP_l2018657868t_bool(set_Ar1995755744le_alt,hAPP_l84111627le_alt(hAPP_n372949560le_alt(drop_A617527551le_alt,N_1),Xs)))) ) ).
tff(fact_1174_set__drop__subset__set__drop,axiom,
! [Xs: list_nat,N_1: nat,M_1: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_1))
=> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_l1821533378t_bool(set_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,M_1),Xs))),hAPP_l1821533378t_bool(set_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,N_1),Xs)))) ) ).
tff(fact_1175_drop__eq__Nil,axiom,
! [N_1: nat,Xs: list_nat] :
( ( hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,N_1),Xs) = nil_nat )
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_list_nat_nat(size_size_list_nat,Xs)),N_1)) ) ).
tff(fact_1176_drop__eq__Nil,axiom,
! [N_1: nat,Xs: list_A114658244le_alt] :
( ( hAPP_l84111627le_alt(hAPP_n372949560le_alt(drop_A617527551le_alt,N_1),Xs) = nil_Ar557011903le_alt )
<=> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs)),N_1)) ) ).
tff(fact_1177_drop__all,axiom,
! [Xs_2: list_nat,N_3: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_list_nat_nat(size_size_list_nat,Xs_2)),N_3))
=> ( hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,N_3),Xs_2) = nil_nat ) ) ).
tff(fact_1178_drop__all,axiom,
! [Xs_2: list_A114658244le_alt,N_3: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs_2)),N_3))
=> ( hAPP_l84111627le_alt(hAPP_n372949560le_alt(drop_A617527551le_alt,N_3),Xs_2) = nil_Ar557011903le_alt ) ) ).
tff(fact_1179_drop__append,axiom,
! [N_2: nat,Xs_1: list_nat,Ys_1: list_nat] : hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,N_2),hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,Xs_1),Ys_1)) = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,N_2),Xs_1)),hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_2),hAPP_list_nat_nat(size_size_list_nat,Xs_1))),Ys_1)) ).
tff(fact_1180_drop__append,axiom,
! [N_2: nat,Xs_1: list_A114658244le_alt,Ys_1: list_A114658244le_alt] : hAPP_l84111627le_alt(hAPP_n372949560le_alt(drop_A617527551le_alt,N_2),hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs_1),Ys_1)) = hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,hAPP_l84111627le_alt(hAPP_n372949560le_alt(drop_A617527551le_alt,N_2),Xs_1)),hAPP_l84111627le_alt(hAPP_n372949560le_alt(drop_A617527551le_alt,hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,N_2),hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs_1))),Ys_1)) ).
tff(fact_1181_append__eq__conv__conj,axiom,
! [Xs: list_A114658244le_alt,Ys: list_A114658244le_alt,Zs: list_A114658244le_alt] :
( ( hAPP_l84111627le_alt(hAPP_l86320102le_alt(append1597383892le_alt,Xs),Ys) = Zs )
<=> ( ( Xs = hAPP_l84111627le_alt(hAPP_n372949560le_alt(take_A2032349095le_alt,hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs)),Zs) )
& ( Ys = hAPP_l84111627le_alt(hAPP_n372949560le_alt(drop_A617527551le_alt,hAPP_l1084348419lt_nat(size_s2005684270le_alt,Xs)),Zs) ) ) ) ).
tff(fact_1182_lessE,axiom,
! [I: nat,K: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),K))
=> ( ( K != hAPP_nat_nat(suc,I) )
=> ~ ! [J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2))
=> ( K != hAPP_nat_nat(suc,J_2) ) ) ) ) ).
tff(fact_1183_Suc__lessE,axiom,
! [I: nat,K: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(suc,I)),K))
=> ~ ! [J_2: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J_2))
=> ( K != hAPP_nat_nat(suc,J_2) ) ) ) ).
tff(fact_1184_map__Suc__upt,axiom,
! [M_1: nat,N_1: nat] : map_nat_nat(suc,hAPP_nat_list_nat(upt(M_1),N_1)) = hAPP_nat_list_nat(upt(hAPP_nat_nat(suc,M_1)),hAPP_nat_nat(suc,N_1)) ).
tff(fact_1185_sorted__upt,axiom,
! [I: nat,J: nat] : hBOOL(hAPP_list_nat_bool(linorder_sorted_nat,hAPP_nat_list_nat(upt(I),J))) ).
tff(fact_1186_nth__upt,axiom,
! [I: nat,K: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K)),J))
=> ( hAPP_nat_nat(nth_nat(hAPP_nat_list_nat(upt(I),J)),K) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),K) ) ) ).
tff(fact_1187_upt__0,axiom,
! [I: nat] : hAPP_nat_list_nat(upt(I),zero_zero_nat) = nil_nat ).
tff(fact_1188_upt__rec,axiom,
! [I: nat,J: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> ( hAPP_nat_list_nat(upt(I),J) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,I),hAPP_nat_list_nat(upt(hAPP_nat_nat(suc,I)),J)) ) )
& ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> ( hAPP_nat_list_nat(upt(I),J) = nil_nat ) ) ) ).
tff(fact_1189_upt__conv__Cons,axiom,
! [I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> ( hAPP_nat_list_nat(upt(I),J) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,I),hAPP_nat_list_nat(upt(hAPP_nat_nat(suc,I)),J)) ) ) ).
tff(fact_1190_drop__upt,axiom,
! [M: nat,I: nat,J: nat] : hAPP_l248265089st_nat(hAPP_n280362926st_nat(drop_nat,M),hAPP_nat_list_nat(upt(I),J)) = hAPP_nat_list_nat(upt(hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),M)),J) ).
tff(fact_1191_hd__upt,axiom,
! [I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_nat,I),J))
=> ( hAPP_list_nat_nat(hd_nat,hAPP_nat_list_nat(upt(I),J)) = I ) ) ).
tff(fact_1192_distinct__upt,axiom,
! [I: nat,J: nat] : hBOOL(hAPP_list_nat_bool(distinct_nat,hAPP_nat_list_nat(upt(I),J))) ).
tff(fact_1193_length__upt,axiom,
! [I: nat,J: nat] : hAPP_list_nat_nat(size_size_list_nat,hAPP_nat_list_nat(upt(I),J)) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(minus_minus_nat,J),I) ).
tff(fact_1194_take__upt,axiom,
! [I: nat,M: nat,N: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),M)),N))
=> ( hAPP_l248265089st_nat(hAPP_n280362926st_nat(take_nat,M),hAPP_nat_list_nat(upt(I),N)) = hAPP_nat_list_nat(upt(I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,I),M)) ) ) ).
tff(fact_1195_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),I))
=> ( hAPP_nat_list_nat(upt(I),J) = nil_nat ) ) ).
tff(fact_1196_upt__eq__Nil__conv,axiom,
! [I_1: nat,J_1: nat] :
( ( hAPP_nat_list_nat(upt(I_1),J_1) = nil_nat )
<=> ( ( J_1 = zero_zero_nat )
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J_1),I_1)) ) ) ).
tff(fact_1197_upt__Suc,axiom,
! [I: nat,J: nat] :
( ( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
=> ( hAPP_nat_list_nat(upt(I),hAPP_nat_nat(suc,J)) = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,hAPP_nat_list_nat(upt(I),J)),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,J),nil_nat)) ) )
& ( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
=> ( hAPP_nat_list_nat(upt(I),hAPP_nat_nat(suc,J)) = nil_nat ) ) ) ).
tff(fact_1198_upt__Suc__append,axiom,
! [I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
=> ( hAPP_nat_list_nat(upt(I),hAPP_nat_nat(suc,J)) = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,hAPP_nat_list_nat(upt(I),J)),hAPP_l248265089st_nat(hAPP_n280362926st_nat(cons_nat,J),nil_nat)) ) ) ).
tff(fact_1199_upt__add__eq__append,axiom,
! [K: nat,I: nat,J: nat] :
( hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I),J))
=> ( hAPP_nat_list_nat(upt(I),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),K)) = hAPP_l248265089st_nat(hAPP_l710375585st_nat(append_nat,hAPP_nat_list_nat(upt(I),J)),hAPP_nat_list_nat(upt(J),hAPP_nat_nat(hAPP_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).
%----Helper facts (65)
tff(help_fNot_1_1_U,axiom,
! [P: bool] :
( ~ hBOOL(hAPP_bool_bool(fNot,P))
| ~ hBOOL(P) ) ).
tff(help_fNot_2_1_U,axiom,
! [P: bool] :
( hBOOL(P)
| hBOOL(hAPP_bool_bool(fNot,P)) ) ).
tff(help_fTrue_1_1_U,axiom,
hBOOL(fTrue) ).
tff(help_fTrue_1_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
tff(help_fconj_1_1_U,axiom,
! [Q: bool,P: bool] :
( ~ hBOOL(P)
| ~ hBOOL(Q)
| hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q)) ) ).
tff(help_fconj_2_1_U,axiom,
! [P: bool,Q: bool] :
( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q))
| hBOOL(P) ) ).
tff(help_fconj_3_1_U,axiom,
! [P: bool,Q: bool] :
( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q))
| hBOOL(Q) ) ).
tff(help_fdisj_1_1_U,axiom,
! [Q: bool,P: bool] :
( ~ hBOOL(P)
| hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fdisj,P),Q)) ) ).
tff(help_fdisj_2_1_U,axiom,
! [P: bool,Q: bool] :
( ~ hBOOL(Q)
| hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fdisj,P),Q)) ) ).
tff(help_fdisj_3_1_U,axiom,
! [P: bool,Q: bool] :
( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fdisj,P),Q))
| hBOOL(P)
| hBOOL(Q) ) ).
tff(help_fFalse_1_1_U,axiom,
~ hBOOL(fFalse) ).
tff(help_fFalse_1_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
tff(help_COMBI_1_1_COMBI_000tc__Nat__Onat_U,axiom,
! [P: nat] : hAPP_nat_nat(cOMBI_nat,P) = P ).
tff(help_fequal_1_1_fequal_000tc__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ~ hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(fequal_nat,X),Y))
| ( X = Y ) ) ).
tff(help_fequal_2_1_fequal_000tc__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
| hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(fequal_nat,X),Y)) ) ).
tff(help_fequal_1_1_fequal_000tc__HOL__Obool_T,axiom,
! [X: bool,Y: bool] :
( ~ hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fequal_bool,X),Y))
| ( X = Y ) ) ).
tff(help_fequal_2_1_fequal_000tc__HOL__Obool_T,axiom,
! [X: bool,Y: bool] :
( ( X != Y )
| hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fequal_bool,X),Y)) ) ).
tff(help_If_1_1_If_000tc__List__Olist_Itc__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] : if_list_nat(fTrue,X,Y) = X ).
tff(help_If_2_1_If_000tc__List__Olist_Itc__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] : if_list_nat(fFalse,X,Y) = Y ).
tff(help_If_3_1_If_000tc__List__Olist_Itc__Nat__Onat_J_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
tff(help_If_1_1_If_000tc__List__Olist_Itc__HOL__Obool_J_T,axiom,
! [X: list_bool,Y: list_bool] : if_list_bool(fTrue,X,Y) = X ).
tff(help_If_2_1_If_000tc__List__Olist_Itc__HOL__Obool_J_T,axiom,
! [X: list_bool,Y: list_bool] : if_list_bool(fFalse,X,Y) = Y ).
tff(help_If_3_1_If_000tc__List__Olist_Itc__HOL__Obool_J_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
tff(help_fequal_1_1_fequal_000tc__List__Olist_Itc__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ~ hBOOL(hAPP_list_nat_bool(hAPP_l120351551t_bool(fequal_list_nat,X),Y))
| ( X = Y ) ) ).
tff(help_fequal_2_1_fequal_000tc__List__Olist_Itc__Nat__Onat_J_T,axiom,
! [X: list_nat,Y: list_nat] :
( ( X != Y )
| hBOOL(hAPP_list_nat_bool(hAPP_l120351551t_bool(fequal_list_nat,X),Y)) ) ).
tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__Nat__Onat_000tc__Nat__Onat_U,axiom,
! [P: fun_nat_nat,Q: fun_nat_nat,R: nat] : hAPP_nat_nat(cOMBB_nat_nat_nat(P,Q),R) = hAPP_nat_nat(P,hAPP_nat_nat(Q,R)) ).
tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Nat__Onat_000tc__Nat__Onat_U,axiom,
! [P: fun_nat_fun_nat_nat,Q: nat,R: nat] : hAPP_nat_nat(hAPP_nat_fun_nat_nat(cOMBC_nat_nat_nat(P),Q),R) = hAPP_nat_nat(hAPP_nat_fun_nat_nat(P,R),Q) ).
tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Nat__Onat_000tc__HOL__Obool_U,axiom,
! [P: fun_nat_fun_nat_bool,Q: nat,R: nat] : hAPP_nat_bool(hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(P),Q),R) = hAPP_nat_bool(hAPP_n1699378549t_bool(P,R),Q) ).
tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Nat__Onat_U,axiom,
! [P: fun_bool_bool,Q: fun_nat_bool,R: nat] : hAPP_nat_bool(cOMBB_bool_bool_nat(P,Q),R) = hAPP_bool_bool(P,hAPP_nat_bool(Q,R)) ).
tff(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
! [P: fun_na1469252690l_bool,Q: fun_nat_bool,R: nat] : hAPP_nat_bool(cOMBS_nat_bool_bool(P,Q),R) = hAPP_bool_bool(hAPP_n1006566506l_bool(P,R),hAPP_nat_bool(Q,R)) ).
tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
! [P: fun_bool_bool,Q: fun_bool_bool,R: bool] : hAPP_bool_bool(cOMBB_bool_bool_bool(P,Q),R) = hAPP_bool_bool(P,hAPP_bool_bool(Q,R)) ).
tff(help_COMBC_1_1_COMBC_000tc__HOL__Obool_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
! [P: fun_bo1549164019l_bool,Q: bool,R: bool] : hAPP_bool_bool(hAPP_b589554111l_bool(cOMBC_bool_bool_bool(P),Q),R) = hAPP_bool_bool(hAPP_b589554111l_bool(P,R),Q) ).
tff(help_COMBS_1_1_COMBS_000tc__HOL__Obool_000tc__HOL__Obool_000tc__HOL__Obool_U,axiom,
! [P: fun_bo1549164019l_bool,Q: fun_bool_bool,R: bool] : hAPP_bool_bool(cOMBS_bool_bool_bool(P,Q),R) = hAPP_bool_bool(hAPP_b589554111l_bool(P,R),hAPP_bool_bool(Q,R)) ).
tff(help_fequal_1_1_fequal_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt_T,axiom,
! [X: arrow_1893660431le_alt,Y: arrow_1893660431le_alt] :
( ~ hBOOL(hAPP_A1748732503t_bool(hAPP_A2063800319t_bool(fequal1590852494le_alt,X),Y))
| ( X = Y ) ) ).
tff(help_fequal_2_1_fequal_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt_T,axiom,
! [X: arrow_1893660431le_alt,Y: arrow_1893660431le_alt] :
( ( X != Y )
| hBOOL(hAPP_A1748732503t_bool(hAPP_A2063800319t_bool(fequal1590852494le_alt,X),Y)) ) ).
tff(help_If_1_1_If_000tc__List__Olist_Itc__Arrow____Order____Mirabelle____effngqiqbs,axiom,
! [X: list_A114658244le_alt,Y: list_A114658244le_alt] : if_lis1472843348le_alt(fTrue,X,Y) = X ).
tff(help_If_2_1_If_000tc__List__Olist_Itc__Arrow____Order____Mirabelle____effngqiqbs,axiom,
! [X: list_A114658244le_alt,Y: list_A114658244le_alt] : if_lis1472843348le_alt(fFalse,X,Y) = Y ).
tff(help_If_3_1_If_000tc__List__Olist_Itc__Arrow____Order____Mirabelle____effngqiqbs,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
tff(help_fequal_1_1_fequal_000tc__List__Olist_Itc__Arrow____Order____Mirabelle____ef,axiom,
! [X: list_A114658244le_alt,Y: list_A114658244le_alt] :
( ~ hBOOL(hAPP_l499746980t_bool(hAPP_l82091775t_bool(fequal928191109le_alt,X),Y))
| ( X = Y ) ) ).
tff(help_fequal_2_1_fequal_000tc__List__Olist_Itc__Arrow____Order____Mirabelle____ef,axiom,
! [X: list_A114658244le_alt,Y: list_A114658244le_alt] :
( ( X != Y )
| hBOOL(hAPP_l499746980t_bool(hAPP_l82091775t_bool(fequal928191109le_alt,X),Y)) ) ).
tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_,axiom,
! [P: fun_na1436237685l_bool,Q: fun_nat_bool,R: nat] : hAPP_nat_bool(hAPP_f800510211t_bool(cOMBC_226598744l_bool(P),Q),R) = hAPP_f54304608l_bool(hAPP_n215258509l_bool(P,R),Q) ).
tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo,axiom,
! [P: fun_bo1549164019l_bool,Q: fun_nat_bool,R: nat] : hAPP_n1006566506l_bool(cOMBB_1015721476ol_nat(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_nat_bool(Q,R)) ).
tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_050,axiom,
! [P: fun_bo1549164019l_bool,Q: fun_bool_bool,R: bool] : hAPP_b589554111l_bool(cOMBB_1914122979l_bool(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_bool_bool(Q,R)) ).
tff(help_COMBB_1_1_COMBB_000tc__List__Olist_Itc__Nat__Onat_J_000tc__List__Olist_Itc_,axiom,
! [P: fun_li1155097699st_nat,Q: fun_nat_list_nat,R: nat] : hAPP_nat_list_nat(cOMBB_850496866at_nat(P,Q),R) = hAPP_l248265089st_nat(P,hAPP_nat_list_nat(Q,R)) ).
tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__List__Olist_Itc__Nat__Onat_J_000tc_,axiom,
! [P: fun_na941842582st_nat,Q: list_nat,R: nat] : hAPP_nat_list_nat(hAPP_l383341732st_nat(cOMBC_1204473209st_nat(P),Q),R) = hAPP_l248265089st_nat(hAPP_n280362926st_nat(P,R),Q) ).
tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Arrow____Order__,axiom,
! [P: fun_bool_bool,Q: fun_Ar17477381t_bool,R: arrow_1893660431le_alt] : hAPP_A1748732503t_bool(cOMBB_388242475le_alt(P,Q),R) = hAPP_bool_bool(P,hAPP_A1748732503t_bool(Q,R)) ).
tff(help_COMBB_1_1_COMBB_000tc__List__Olist_Itc__HOL__Obool_J_000tc__List__Olist_Itc,axiom,
! [P: fun_li107039421t_bool,Q: fun_nat_list_bool,R: nat] : hAPP_nat_list_bool(cOMBB_1891801486ol_nat(P,Q),R) = hAPP_l1189022293t_bool(P,hAPP_nat_list_bool(Q,R)) ).
tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__List__Olist_Itc__HOL__Obool_J_000tc,axiom,
! [P: fun_na1279735280t_bool,Q: list_bool,R: nat] : hAPP_nat_list_bool(hAPP_l1656175496t_bool(cOMBC_1038295629t_bool(P),Q),R) = hAPP_l1189022293t_bool(hAPP_n479129736t_bool(P,R),Q) ).
tff(help_COMBS_1_1_COMBS_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt_000t,axiom,
! [P: fun_Ar1433468301l_bool,Q: fun_Ar17477381t_bool,R: arrow_1893660431le_alt] : hAPP_A1748732503t_bool(cOMBS_119208490l_bool(P,Q),R) = hAPP_bool_bool(hAPP_A463015077l_bool(P,R),hAPP_A1748732503t_bool(Q,R)) ).
tff(help_COMBC_1_1_COMBC_000tc__HOL__Obool_000tc__List__Olist_Itc__HOL__Obool_J_000t,axiom,
! [P: fun_bo1175651857t_bool,Q: list_bool,R: bool] : hAPP_bool_list_bool(hAPP_l1045755451t_bool(cOMBC_80330970t_bool(P),Q),R) = hAPP_l1189022293t_bool(hAPP_b994696797t_bool(P,R),Q) ).
tff(help_COMBK_1_1_COMBK_000tc__fun_Itc__List__Olist_Itc__Nat__Onat_J_Mtc__List__Oli,axiom,
! [P: fun_li1155097699st_nat,Q: nat] : hAPP_n280362926st_nat(cOMBK_1717478773at_nat(P),Q) = P ).
tff(help_COMBB_1_1_COMBB_000tc__Nat__Onat_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HO,axiom,
! [P: fun_na1436237685l_bool,Q: fun_nat_nat,R: nat] : hAPP_n215258509l_bool(cOMBB_1146692694ol_nat(P,Q),R) = hAPP_n215258509l_bool(P,hAPP_nat_nat(Q,R)) ).
tff(help_COMBC_1_1_COMBC_000tc__List__Olist_Itc__Nat__Onat_J_000tc__List__Olist_Itc_,axiom,
! [P: fun_li511653589st_nat,Q: list_nat,R: list_nat] : hAPP_l248265089st_nat(hAPP_l710375585st_nat(cOMBC_950362600st_nat(P),Q),R) = hAPP_l248265089st_nat(hAPP_l710375585st_nat(P,R),Q) ).
tff(help_COMBC_1_1_COMBC_000tc__List__Olist_Itc__HOL__Obool_J_000tc__List__Olist_Itc,axiom,
! [P: fun_li662746748t_bool,Q: list_bool,R: list_bool] : hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(cOMBC_1789016513t_bool(P),Q),R) = hAPP_l1189022293t_bool(hAPP_l1947611924t_bool(P,R),Q) ).
tff(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_051,axiom,
! [P: fun_bo1549164019l_bool,Q: fun_Ar17477381t_bool,R: arrow_1893660431le_alt] : hAPP_A463015077l_bool(cOMBB_1234541833le_alt(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_A1748732503t_bool(Q,R)) ).
tff(help_COMBC_1_1_COMBC_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt_000t,axiom,
! [P: fun_Ar331856045t_bool,Q: arrow_1893660431le_alt,R: arrow_1893660431le_alt] : hAPP_A1748732503t_bool(hAPP_A2063800319t_bool(cOMBC_1826973210t_bool(P),Q),R) = hAPP_A1748732503t_bool(hAPP_A2063800319t_bool(P,R),Q) ).
tff(help_COMBB_1_1_COMBB_000tc__List__Olist_Itc__Nat__Onat_J_000tc__fun_Itc__List__O,axiom,
! [P: fun_li511653589st_nat,Q: fun_nat_list_nat,R: nat] : hAPP_n280362926st_nat(cOMBB_472649634at_nat(P,Q),R) = hAPP_l710375585st_nat(P,hAPP_nat_list_nat(Q,R)) ).
tff(help_COMBB_1_1_COMBB_000tc__List__Olist_Itc__HOL__Obool_J_000tc__fun_Itc__List__,axiom,
! [P: fun_li662746748t_bool,Q: fun_bool_list_bool,R: bool] : hAPP_b994696797t_bool(cOMBB_1663860504l_bool(P,Q),R) = hAPP_l1947611924t_bool(P,hAPP_bool_list_bool(Q,R)) ).
tff(help_COMBC_1_1_COMBC_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt_000t_052,axiom,
! [P: fun_Ar569529141l_bool,Q: fun_Ar17477381t_bool,R: arrow_1893660431le_alt] : hAPP_A1748732503t_bool(hAPP_f1592536333t_bool(cOMBC_1310341090l_bool(P),Q),R) = hAPP_f739344229l_bool(hAPP_A1501726285l_bool(P,R),Q) ).
tff(help_COMBB_1_1_COMBB_000tc__List__Olist_Itc__Arrow____Order____Mirabelle____effn,axiom,
! [P: fun_li90789869le_alt,Q: fun_na86871399le_alt,R: nat] : hAPP_n2007559481le_alt(cOMBB_138440792lt_nat(P,Q),R) = hAPP_l84111627le_alt(P,hAPP_n2007559481le_alt(Q,R)) ).
tff(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__List__Olist_Itc__Arrow____Order____,axiom,
! [P: fun_na960272160le_alt,Q: list_A114658244le_alt,R: nat] : hAPP_n2007559481le_alt(hAPP_l1210406190le_alt(cOMBC_869637379le_alt(P),Q),R) = hAPP_l84111627le_alt(hAPP_n372949560le_alt(P,R),Q) ).
tff(help_COMBC_1_1_COMBC_000tc__Arrow____Order____Mirabelle____effngqiqbs__Oalt_000t_053,axiom,
! [P: fun_Ar1448189019le_alt,Q: list_A114658244le_alt,R: arrow_1893660431le_alt] : hAPP_A1382181054le_alt(hAPP_l1822938419le_alt(cOMBC_673369480le_alt(P),Q),R) = hAPP_l84111627le_alt(hAPP_A436723315le_alt(P,R),Q) ).
tff(help_COMBK_1_1_COMBK_000tc__fun_Itc__List__Olist_Itc__Arrow____Order____Mirabell,axiom,
! [P: fun_li90789869le_alt,Q: arrow_1893660431le_alt] : hAPP_A436723315le_alt(cOMBK_930998768le_alt(P),Q) = P ).
tff(help_COMBC_1_1_COMBC_000tc__List__Olist_Itc__Arrow____Order____Mirabelle____effn,axiom,
! [P: fun_li264309018le_alt,Q: list_A114658244le_alt,R: list_A114658244le_alt] : hAPP_l84111627le_alt(hAPP_l86320102le_alt(cOMBC_1833749623le_alt(P),Q),R) = hAPP_l84111627le_alt(hAPP_l86320102le_alt(P,R),Q) ).
tff(help_COMBB_1_1_COMBB_000tc__List__Olist_Itc__Arrow____Order____Mirabelle____effn_054,axiom,
! [P: fun_li264309018le_alt,Q: fun_Ar849108972le_alt,R: arrow_1893660431le_alt] : hAPP_A436723315le_alt(cOMBB_1076781858le_alt(P,Q),R) = hAPP_l86320102le_alt(P,hAPP_A1382181054le_alt(Q,R)) ).
%----Conjectures (2)
tff(conj_0,hypothesis,
a != b ).
tff(conj_1,conjecture,
? [C: arrow_1893660431le_alt] : hBOOL(hAPP_l499746980t_bool(distin1654625714le_alt,hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,a),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,b),hAPP_l84111627le_alt(hAPP_A436723315le_alt(cons_A1647044463le_alt,C),nil_Ar557011903le_alt))))) ).
%------------------------------------------------------------------------------